file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Order/Hom/CompleteLattice.lean
|
sSupHom.copy_eq
|
[] |
[
285,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
284,
1
] |
Mathlib/CategoryTheory/Over.lean
|
CategoryTheory.Under.UnderMorphism.ext
|
[
{
"state_after": "T : Type u₁\ninst✝ : Category T\nX✝ X : T\nU V : Under X\nf g : U ⟶ V\nleft✝ : U.left ⟶ V.left\nb : U.right ⟶ V.right\nw✝ : (Functor.fromPUnit X).map left✝ ≫ V.hom = U.hom ≫ (𝟭 T).map b\nh : (CommaMorphism.mk left✝ b).right = g.right\n⊢ CommaMorphism.mk left✝ b = g",
"state_before": "T : Type u₁\ninst✝ : Category T\nX✝ X : T\nU V : Under X\nf g : U ⟶ V\nh : f.right = g.right\n⊢ f = g",
"tactic": "let ⟨_,b,_⟩ := f"
},
{
"state_after": "T : Type u₁\ninst✝ : Category T\nX✝ X : T\nU V : Under X\nf g : U ⟶ V\nleft✝¹ : U.left ⟶ V.left\nb : U.right ⟶ V.right\nw✝¹ : (Functor.fromPUnit X).map left✝¹ ≫ V.hom = U.hom ≫ (𝟭 T).map b\nleft✝ : U.left ⟶ V.left\ne : U.right ⟶ V.right\nw✝ : (Functor.fromPUnit X).map left✝ ≫ V.hom = U.hom ≫ (𝟭 T).map e\nh : (CommaMorphism.mk left✝¹ b).right = (CommaMorphism.mk left✝ e).right\n⊢ CommaMorphism.mk left✝¹ b = CommaMorphism.mk left✝ e",
"state_before": "T : Type u₁\ninst✝ : Category T\nX✝ X : T\nU V : Under X\nf g : U ⟶ V\nleft✝ : U.left ⟶ V.left\nb : U.right ⟶ V.right\nw✝ : (Functor.fromPUnit X).map left✝ ≫ V.hom = U.hom ≫ (𝟭 T).map b\nh : (CommaMorphism.mk left✝ b).right = g.right\n⊢ CommaMorphism.mk left✝ b = g",
"tactic": "let ⟨_,e,_⟩ := g"
},
{
"state_after": "case e_left\nT : Type u₁\ninst✝ : Category T\nX✝ X : T\nU V : Under X\nf g : U ⟶ V\nleft✝¹ : U.left ⟶ V.left\nb : U.right ⟶ V.right\nw✝¹ : (Functor.fromPUnit X).map left✝¹ ≫ V.hom = U.hom ≫ (𝟭 T).map b\nleft✝ : U.left ⟶ V.left\ne : U.right ⟶ V.right\nw✝ : (Functor.fromPUnit X).map left✝ ≫ V.hom = U.hom ≫ (𝟭 T).map e\nh : (CommaMorphism.mk left✝¹ b).right = (CommaMorphism.mk left✝ e).right\n⊢ left✝¹ = left✝",
"state_before": "T : Type u₁\ninst✝ : Category T\nX✝ X : T\nU V : Under X\nf g : U ⟶ V\nleft✝¹ : U.left ⟶ V.left\nb : U.right ⟶ V.right\nw✝¹ : (Functor.fromPUnit X).map left✝¹ ≫ V.hom = U.hom ≫ (𝟭 T).map b\nleft✝ : U.left ⟶ V.left\ne : U.right ⟶ V.right\nw✝ : (Functor.fromPUnit X).map left✝ ≫ V.hom = U.hom ≫ (𝟭 T).map e\nh : (CommaMorphism.mk left✝¹ b).right = (CommaMorphism.mk left✝ e).right\n⊢ CommaMorphism.mk left✝¹ b = CommaMorphism.mk left✝ e",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_left\nT : Type u₁\ninst✝ : Category T\nX✝ X : T\nU V : Under X\nf g : U ⟶ V\nleft✝¹ : U.left ⟶ V.left\nb : U.right ⟶ V.right\nw✝¹ : (Functor.fromPUnit X).map left✝¹ ≫ V.hom = U.hom ≫ (𝟭 T).map b\nleft✝ : U.left ⟶ V.left\ne : U.right ⟶ V.right\nw✝ : (Functor.fromPUnit X).map left✝ ≫ V.hom = U.hom ≫ (𝟭 T).map e\nh : (CommaMorphism.mk left✝¹ b).right = (CommaMorphism.mk left✝ e).right\n⊢ left✝¹ = left✝",
"tactic": "simp only [eq_iff_true_of_subsingleton]"
}
] |
[
332,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
329,
1
] |
Mathlib/Data/Nat/Cast/Basic.lean
|
Sum.elim_natCast_natCast
|
[] |
[
335,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
333,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.sameCycle_one
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.31727\nα : Type u_1\nβ : Type ?u.31733\nf g : Perm α\np : α → Prop\nx y z : α\n⊢ SameCycle 1 x y ↔ x = y",
"tactic": "simp [SameCycle]"
}
] |
[
107,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.lt_div
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.232451\nβ : Type ?u.232454\nγ : Type ?u.232457\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b c : Ordinal\nh : c ≠ 0\n⊢ a < b / c ↔ c * succ a ≤ b",
"tactic": "rw [← not_le, div_le h, not_lt]"
}
] |
[
902,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
901,
1
] |
Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean
|
Finset.card_pow_div_pow_le'
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA B✝ C : Finset α\nhA : Finset.Nonempty A\nB : Finset α\nm n : ℕ\n⊢ ↑(card (B⁻¹ ^ m / B⁻¹ ^ n)) ≤ (↑(card (A * B⁻¹)) / ↑(card A)) ^ (m + n) * ↑(card A)",
"state_before": "α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA B✝ C : Finset α\nhA : Finset.Nonempty A\nB : Finset α\nm n : ℕ\n⊢ ↑(card (B ^ m / B ^ n)) ≤ (↑(card (A / B)) / ↑(card A)) ^ (m + n) * ↑(card A)",
"tactic": "rw [← card_inv, inv_div', ← inv_pow, ← inv_pow, div_eq_mul_inv A]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA B✝ C : Finset α\nhA : Finset.Nonempty A\nB : Finset α\nm n : ℕ\n⊢ ↑(card (B⁻¹ ^ m / B⁻¹ ^ n)) ≤ (↑(card (A * B⁻¹)) / ↑(card A)) ^ (m + n) * ↑(card A)",
"tactic": "exact card_pow_div_pow_le hA _ _ _"
}
] |
[
244,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
241,
1
] |
Std/Data/List/Basic.lean
|
List.eraseIdx_eq_eraseIdxTR
|
[
{
"state_after": "case h.h.h\nα : Type u_1\nl : List α\nn : Nat\n⊢ eraseIdx l n = eraseIdxTR l n",
"state_before": "⊢ @eraseIdx = @eraseIdxTR",
"tactic": "funext α l n"
},
{
"state_after": "case h.h.h\nα : Type u_1\nl : List α\nn : Nat\n⊢ eraseIdx l n = eraseIdxTR.go l l n #[]",
"state_before": "case h.h.h\nα : Type u_1\nl : List α\nn : Nat\n⊢ eraseIdx l n = eraseIdxTR l n",
"tactic": "simp [eraseIdxTR]"
},
{
"state_after": "case h.h.h\nα : Type u_1\nl : List α\nn : Nat\n⊢ ∀ (xs : List α) (acc : Array α), l = acc.data ++ xs → eraseIdxTR.go l xs n acc = acc.data ++ eraseIdx xs n",
"state_before": "case h.h.h\nα : Type u_1\nl : List α\nn : Nat\n⊢ eraseIdx l n = eraseIdxTR.go l l n #[]",
"tactic": "suffices ∀ xs acc, l = acc.data ++ xs → eraseIdxTR.go l xs n acc = acc.data ++ xs.eraseIdx n from\n (this l #[] (by simp)).symm"
},
{
"state_after": "case h.h.h\nα : Type u_1\nl : List α\nn : Nat\nxs : List α\n⊢ ∀ (acc : Array α), l = acc.data ++ xs → eraseIdxTR.go l xs n acc = acc.data ++ eraseIdx xs n",
"state_before": "case h.h.h\nα : Type u_1\nl : List α\nn : Nat\n⊢ ∀ (xs : List α) (acc : Array α), l = acc.data ++ xs → eraseIdxTR.go l xs n acc = acc.data ++ eraseIdx xs n",
"tactic": "intro xs"
},
{
"state_after": "no goals",
"state_before": "case h.h.h\nα : Type u_1\nl : List α\nn : Nat\nxs : List α\n⊢ ∀ (acc : Array α), l = acc.data ++ xs → eraseIdxTR.go l xs n acc = acc.data ++ eraseIdx xs n",
"tactic": "induction xs generalizing n with intro acc h\n| nil => simp [eraseIdx, eraseIdxTR.go, h]\n| cons x xs IH =>\nmatch n with\n| 0 => simp [eraseIdx, eraseIdxTR.go]\n| n+1 =>\nsimp [eraseIdx, eraseIdxTR.go]\nrw [IH]; simp; simp; exact h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl : List α\nn : Nat\nthis : ∀ (xs : List α) (acc : Array α), l = acc.data ++ xs → eraseIdxTR.go l xs n acc = acc.data ++ eraseIdx xs n\n⊢ l = #[].data ++ l",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case h.h.h.nil\nα : Type u_1\nl : List α\nn : Nat\nacc : Array α\nh : l = acc.data ++ []\n⊢ eraseIdxTR.go l [] n acc = acc.data ++ eraseIdx [] n",
"tactic": "simp [eraseIdx, eraseIdxTR.go, h]"
},
{
"state_after": "no goals",
"state_before": "case h.h.h.cons\nα : Type u_1\nl : List α\nx : α\nxs : List α\nIH : ∀ (n : Nat) (acc : Array α), l = acc.data ++ xs → eraseIdxTR.go l xs n acc = acc.data ++ eraseIdx xs n\nn : Nat\nacc : Array α\nh : l = acc.data ++ x :: xs\n⊢ eraseIdxTR.go l (x :: xs) n acc = acc.data ++ eraseIdx (x :: xs) n",
"tactic": "match n with\n| 0 => simp [eraseIdx, eraseIdxTR.go]\n| n+1 =>\n simp [eraseIdx, eraseIdxTR.go]\n rw [IH]; simp; simp; exact h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl : List α\nx : α\nxs : List α\nIH : ∀ (n : Nat) (acc : Array α), l = acc.data ++ xs → eraseIdxTR.go l xs n acc = acc.data ++ eraseIdx xs n\nn : Nat\nacc : Array α\nh : l = acc.data ++ x :: xs\n⊢ eraseIdxTR.go l (x :: xs) 0 acc = acc.data ++ eraseIdx (x :: xs) 0",
"tactic": "simp [eraseIdx, eraseIdxTR.go]"
},
{
"state_after": "α : Type u_1\nl : List α\nx : α\nxs : List α\nIH : ∀ (n : Nat) (acc : Array α), l = acc.data ++ xs → eraseIdxTR.go l xs n acc = acc.data ++ eraseIdx xs n\nn✝ : Nat\nacc : Array α\nh : l = acc.data ++ x :: xs\nn : Nat\n⊢ eraseIdxTR.go l xs n (Array.push acc x) = acc.data ++ x :: eraseIdx xs n",
"state_before": "α : Type u_1\nl : List α\nx : α\nxs : List α\nIH : ∀ (n : Nat) (acc : Array α), l = acc.data ++ xs → eraseIdxTR.go l xs n acc = acc.data ++ eraseIdx xs n\nn✝ : Nat\nacc : Array α\nh : l = acc.data ++ x :: xs\nn : Nat\n⊢ eraseIdxTR.go l (x :: xs) (n + 1) acc = acc.data ++ eraseIdx (x :: xs) (n + 1)",
"tactic": "simp [eraseIdx, eraseIdxTR.go]"
},
{
"state_after": "α : Type u_1\nl : List α\nx : α\nxs : List α\nIH : ∀ (n : Nat) (acc : Array α), l = acc.data ++ xs → eraseIdxTR.go l xs n acc = acc.data ++ eraseIdx xs n\nn✝ : Nat\nacc : Array α\nh : l = acc.data ++ x :: xs\nn : Nat\n⊢ (Array.push acc x).data ++ eraseIdx xs n = acc.data ++ x :: eraseIdx xs n\n\ncase a\nα : Type u_1\nl : List α\nx : α\nxs : List α\nIH : ∀ (n : Nat) (acc : Array α), l = acc.data ++ xs → eraseIdxTR.go l xs n acc = acc.data ++ eraseIdx xs n\nn✝ : Nat\nacc : Array α\nh : l = acc.data ++ x :: xs\nn : Nat\n⊢ l = (Array.push acc x).data ++ xs",
"state_before": "α : Type u_1\nl : List α\nx : α\nxs : List α\nIH : ∀ (n : Nat) (acc : Array α), l = acc.data ++ xs → eraseIdxTR.go l xs n acc = acc.data ++ eraseIdx xs n\nn✝ : Nat\nacc : Array α\nh : l = acc.data ++ x :: xs\nn : Nat\n⊢ eraseIdxTR.go l xs n (Array.push acc x) = acc.data ++ x :: eraseIdx xs n",
"tactic": "rw [IH]"
},
{
"state_after": "case a\nα : Type u_1\nl : List α\nx : α\nxs : List α\nIH : ∀ (n : Nat) (acc : Array α), l = acc.data ++ xs → eraseIdxTR.go l xs n acc = acc.data ++ eraseIdx xs n\nn✝ : Nat\nacc : Array α\nh : l = acc.data ++ x :: xs\nn : Nat\n⊢ l = (Array.push acc x).data ++ xs",
"state_before": "α : Type u_1\nl : List α\nx : α\nxs : List α\nIH : ∀ (n : Nat) (acc : Array α), l = acc.data ++ xs → eraseIdxTR.go l xs n acc = acc.data ++ eraseIdx xs n\nn✝ : Nat\nacc : Array α\nh : l = acc.data ++ x :: xs\nn : Nat\n⊢ (Array.push acc x).data ++ eraseIdx xs n = acc.data ++ x :: eraseIdx xs n\n\ncase a\nα : Type u_1\nl : List α\nx : α\nxs : List α\nIH : ∀ (n : Nat) (acc : Array α), l = acc.data ++ xs → eraseIdxTR.go l xs n acc = acc.data ++ eraseIdx xs n\nn✝ : Nat\nacc : Array α\nh : l = acc.data ++ x :: xs\nn : Nat\n⊢ l = (Array.push acc x).data ++ xs",
"tactic": "simp"
},
{
"state_after": "case a\nα : Type u_1\nl : List α\nx : α\nxs : List α\nIH : ∀ (n : Nat) (acc : Array α), l = acc.data ++ xs → eraseIdxTR.go l xs n acc = acc.data ++ eraseIdx xs n\nn✝ : Nat\nacc : Array α\nh : l = acc.data ++ x :: xs\nn : Nat\n⊢ l = acc.data ++ x :: xs",
"state_before": "case a\nα : Type u_1\nl : List α\nx : α\nxs : List α\nIH : ∀ (n : Nat) (acc : Array α), l = acc.data ++ xs → eraseIdxTR.go l xs n acc = acc.data ++ eraseIdx xs n\nn✝ : Nat\nacc : Array α\nh : l = acc.data ++ x :: xs\nn : Nat\n⊢ l = (Array.push acc x).data ++ xs",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nl : List α\nx : α\nxs : List α\nIH : ∀ (n : Nat) (acc : Array α), l = acc.data ++ xs → eraseIdxTR.go l xs n acc = acc.data ++ eraseIdx xs n\nn✝ : Nat\nacc : Array α\nh : l = acc.data ++ x :: xs\nn : Nat\n⊢ l = acc.data ++ x :: xs",
"tactic": "exact h"
}
] |
[
72,
35
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
61,
10
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.mk'_spec'_mk
|
[] |
[
279,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.hasPullback_of_preservesPullback
|
[] |
[
109,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieIdeal.incl_coe
|
[] |
[
1121,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1120,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.image_mem_of_mem_comap
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.259120\nι : Sort x\nf✝ f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm : α → β\nm' : β → γ\ns : Set α\nt : Set β\nf : Filter α\nc : β → α\nh : range c ∈ f\nW : Set β\nW_in : W ∈ comap c f\n⊢ c '' W ∈ map c (comap c f)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.259120\nι : Sort x\nf✝ f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm : α → β\nm' : β → γ\ns : Set α\nt : Set β\nf : Filter α\nc : β → α\nh : range c ∈ f\nW : Set β\nW_in : W ∈ comap c f\n⊢ c '' W ∈ f",
"tactic": "rw [← map_comap_of_mem h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.259120\nι : Sort x\nf✝ f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm : α → β\nm' : β → γ\ns : Set α\nt : Set β\nf : Filter α\nc : β → α\nh : range c ∈ f\nW : Set β\nW_in : W ∈ comap c f\n⊢ c '' W ∈ map c (comap c f)",
"tactic": "exact image_mem_map W_in"
}
] |
[
2289,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2286,
1
] |
Mathlib/AlgebraicGeometry/SheafedSpace.lean
|
AlgebraicGeometry.SheafedSpace.id_c_app
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX : SheafedSpace C\nU : (Opens ↑↑X.toPresheafedSpace)ᵒᵖ\n⊢ X.presheaf.obj U = ((𝟙 X).base _* X.presheaf).obj U",
"tactic": "aesop_cat"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX : SheafedSpace C\nU : (Opens ↑↑X.toPresheafedSpace)ᵒᵖ\n⊢ (𝟙 X).c.app U =\n eqToHom (_ : X.presheaf.obj U = X.presheaf.obj ((Functor.op (Opens.map (𝟙 ↑X.toPresheafedSpace))).obj U))",
"tactic": "aesop_cat"
}
] |
[
133,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.conj_bit0
|
[
{
"state_after": "no goals",
"state_before": "z : ℂ\n⊢ (↑(starRingEnd ℂ) (bit0 z)).re = (bit0 (↑(starRingEnd ℂ) z)).re ∧\n (↑(starRingEnd ℂ) (bit0 z)).im = (bit0 (↑(starRingEnd ℂ) z)).im",
"tactic": "simp [bit0]"
}
] |
[
530,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
529,
1
] |
Mathlib/Algebra/Hom/Units.lean
|
IsUnit.mul_eq_one_iff_inv_eq
|
[] |
[
362,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
11
] |
Mathlib/Topology/UniformSpace/Completion.lean
|
UniformSpace.Completion.ext
|
[] |
[
521,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
519,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
Wbtw.left_mem_affineSpan_of_right_ne
|
[] |
[
749,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
747,
1
] |
Mathlib/Data/Sym/Basic.lean
|
Sym.coe_replicate
|
[] |
[
285,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
284,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
ContinuousAt.norm'
|
[] |
[
1218,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1217,
1
] |
Mathlib/Data/Real/CauSeqCompletion.lean
|
CauSeq.lt_lim
|
[] |
[
458,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
457,
1
] |
Mathlib/Analysis/NormedSpace/FiniteDimension.lean
|
continuousOn_clm_apply
|
[
{
"state_after": "𝕜 : Type u\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : FiniteDimensional 𝕜 E\nf : X → E →L[𝕜] F\ns : Set X\nh : ∀ (y : E), ContinuousOn (fun x => ↑(f x) y) s\n⊢ ContinuousOn f s",
"state_before": "𝕜 : Type u\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : FiniteDimensional 𝕜 E\nf : X → E →L[𝕜] F\ns : Set X\n⊢ ContinuousOn f s ↔ ∀ (y : E), ContinuousOn (fun x => ↑(f x) y) s",
"tactic": "refine' ⟨fun h y => (ContinuousLinearMap.apply 𝕜 F y).continuous.comp_continuousOn h, fun h => _⟩"
},
{
"state_after": "𝕜 : Type u\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : FiniteDimensional 𝕜 E\nf : X → E →L[𝕜] F\ns : Set X\nh : ∀ (y : E), ContinuousOn (fun x => ↑(f x) y) s\nd : ℕ := finrank 𝕜 E\n⊢ ContinuousOn f s",
"state_before": "𝕜 : Type u\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : FiniteDimensional 𝕜 E\nf : X → E →L[𝕜] F\ns : Set X\nh : ∀ (y : E), ContinuousOn (fun x => ↑(f x) y) s\n⊢ ContinuousOn f s",
"tactic": "let d := finrank 𝕜 E"
},
{
"state_after": "𝕜 : Type u\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : FiniteDimensional 𝕜 E\nf : X → E →L[𝕜] F\ns : Set X\nh : ∀ (y : E), ContinuousOn (fun x => ↑(f x) y) s\nd : ℕ := finrank 𝕜 E\nhd : d = finrank 𝕜 (Fin d → 𝕜)\n⊢ ContinuousOn f s",
"state_before": "𝕜 : Type u\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : FiniteDimensional 𝕜 E\nf : X → E →L[𝕜] F\ns : Set X\nh : ∀ (y : E), ContinuousOn (fun x => ↑(f x) y) s\nd : ℕ := finrank 𝕜 E\n⊢ ContinuousOn f s",
"tactic": "have hd : d = finrank 𝕜 (Fin d → 𝕜) := (finrank_fin_fun 𝕜).symm"
},
{
"state_after": "𝕜 : Type u\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : FiniteDimensional 𝕜 E\nf : X → E →L[𝕜] F\ns : Set X\nh : ∀ (y : E), ContinuousOn (fun x => ↑(f x) y) s\nd : ℕ := finrank 𝕜 E\nhd : d = finrank 𝕜 (Fin d → 𝕜)\ne₁ : E ≃L[𝕜] Fin d → 𝕜 := ContinuousLinearEquiv.ofFinrankEq hd\n⊢ ContinuousOn f s",
"state_before": "𝕜 : Type u\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : FiniteDimensional 𝕜 E\nf : X → E →L[𝕜] F\ns : Set X\nh : ∀ (y : E), ContinuousOn (fun x => ↑(f x) y) s\nd : ℕ := finrank 𝕜 E\nhd : d = finrank 𝕜 (Fin d → 𝕜)\n⊢ ContinuousOn f s",
"tactic": "let e₁ : E ≃L[𝕜] Fin d → 𝕜 := ContinuousLinearEquiv.ofFinrankEq hd"
},
{
"state_after": "𝕜 : Type u\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : FiniteDimensional 𝕜 E\nf : X → E →L[𝕜] F\ns : Set X\nh : ∀ (y : E), ContinuousOn (fun x => ↑(f x) y) s\nd : ℕ := finrank 𝕜 E\nhd : d = finrank 𝕜 (Fin d → 𝕜)\ne₁ : E ≃L[𝕜] Fin d → 𝕜 := ContinuousLinearEquiv.ofFinrankEq hd\ne₂ : (E →L[𝕜] F) ≃L[𝕜] Fin d → F :=\n ContinuousLinearEquiv.trans (ContinuousLinearEquiv.arrowCongr e₁ 1) (ContinuousLinearEquiv.piRing (Fin d))\n⊢ ContinuousOn f s",
"state_before": "𝕜 : Type u\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : FiniteDimensional 𝕜 E\nf : X → E →L[𝕜] F\ns : Set X\nh : ∀ (y : E), ContinuousOn (fun x => ↑(f x) y) s\nd : ℕ := finrank 𝕜 E\nhd : d = finrank 𝕜 (Fin d → 𝕜)\ne₁ : E ≃L[𝕜] Fin d → 𝕜 := ContinuousLinearEquiv.ofFinrankEq hd\n⊢ ContinuousOn f s",
"tactic": "let e₂ : (E →L[𝕜] F) ≃L[𝕜] Fin d → F :=\n (e₁.arrowCongr (1 : F ≃L[𝕜] F)).trans (ContinuousLinearEquiv.piRing (Fin d))"
},
{
"state_after": "𝕜 : Type u\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : FiniteDimensional 𝕜 E\nf : X → E →L[𝕜] F\ns : Set X\nh : ∀ (y : E), ContinuousOn (fun x => ↑(f x) y) s\nd : ℕ := finrank 𝕜 E\nhd : d = finrank 𝕜 (Fin d → 𝕜)\ne₁ : E ≃L[𝕜] Fin d → 𝕜 := ContinuousLinearEquiv.ofFinrankEq hd\ne₂ : (E →L[𝕜] F) ≃L[𝕜] Fin d → F :=\n ContinuousLinearEquiv.trans (ContinuousLinearEquiv.arrowCongr e₁ 1) (ContinuousLinearEquiv.piRing (Fin d))\n⊢ ContinuousOn ((↑(ContinuousLinearEquiv.symm e₂) ∘ ↑e₂) ∘ f) s",
"state_before": "𝕜 : Type u\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : FiniteDimensional 𝕜 E\nf : X → E →L[𝕜] F\ns : Set X\nh : ∀ (y : E), ContinuousOn (fun x => ↑(f x) y) s\nd : ℕ := finrank 𝕜 E\nhd : d = finrank 𝕜 (Fin d → 𝕜)\ne₁ : E ≃L[𝕜] Fin d → 𝕜 := ContinuousLinearEquiv.ofFinrankEq hd\ne₂ : (E →L[𝕜] F) ≃L[𝕜] Fin d → F :=\n ContinuousLinearEquiv.trans (ContinuousLinearEquiv.arrowCongr e₁ 1) (ContinuousLinearEquiv.piRing (Fin d))\n⊢ ContinuousOn f s",
"tactic": "rw [← Function.comp.left_id f, ← e₂.symm_comp_self]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : FiniteDimensional 𝕜 E\nf : X → E →L[𝕜] F\ns : Set X\nh : ∀ (y : E), ContinuousOn (fun x => ↑(f x) y) s\nd : ℕ := finrank 𝕜 E\nhd : d = finrank 𝕜 (Fin d → 𝕜)\ne₁ : E ≃L[𝕜] Fin d → 𝕜 := ContinuousLinearEquiv.ofFinrankEq hd\ne₂ : (E →L[𝕜] F) ≃L[𝕜] Fin d → F :=\n ContinuousLinearEquiv.trans (ContinuousLinearEquiv.arrowCongr e₁ 1) (ContinuousLinearEquiv.piRing (Fin d))\n⊢ ContinuousOn ((↑(ContinuousLinearEquiv.symm e₂) ∘ ↑e₂) ∘ f) s",
"tactic": "exact e₂.symm.continuous.comp_continuousOn (continuousOn_pi.mpr fun i => h _)"
}
] |
[
585,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
576,
1
] |
Mathlib/GroupTheory/PresentedGroup.lean
|
PresentedGroup.toGroup.of
|
[] |
[
84,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Topology/Order/Basic.lean
|
Monotone.map_csInf_of_continuousAt
|
[] |
[
2813,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2811,
1
] |
Mathlib/Data/Multiset/Nodup.lean
|
Multiset.Nodup.not_mem_erase
|
[] |
[
187,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.IsSt.inv
|
[
{
"state_after": "x : ℝ*\nhi : ¬Infinitesimal x\nhr : IsSt x 0\n⊢ False",
"state_before": "x : ℝ*\nr : ℝ\nhi : ¬Infinitesimal x\nhr : IsSt x r\n⊢ r ≠ 0",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "x : ℝ*\nhi : ¬Infinitesimal x\nhr : IsSt x 0\n⊢ False",
"tactic": "exact hi hr"
}
] |
[
789,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
788,
1
] |
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
|
Matrix.isUnit_det_of_invertible
|
[] |
[
209,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/Algebra/RingQuot.lean
|
RingQuot.Rel.smul
|
[
{
"state_after": "no goals",
"state_before": "R : Type u₁\ninst✝³ : Semiring R\nS : Type u₂\ninst✝² : CommSemiring S\nA : Type u₃\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nr : A → A → Prop\nk : S\na b : A\nh : Rel r a b\n⊢ Rel r (k • a) (k • b)",
"tactic": "simp only [Algebra.smul_def, Rel.mul_right h]"
}
] |
[
85,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/GroupTheory/Sylow.lean
|
Sylow.normalizer_normalizer
|
[
{
"state_after": "G : Type u\nα : Type v\nβ : Type w\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nthis :\n Normal (normalizer ↑(Sylow.subtype P (_ : ↑P ≤ normalizer (normalizer ↑P)))) →\n Normal ↑(Sylow.subtype P (_ : ↑P ≤ normalizer (normalizer ↑P)))\n⊢ normalizer (normalizer ↑P) = normalizer ↑P",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\n⊢ normalizer (normalizer ↑P) = normalizer ↑P",
"tactic": "have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))"
},
{
"state_after": "G : Type u\nα : Type v\nβ : Type w\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nthis : True → subgroupOf (normalizer ↑P) (normalizer (normalizer ↑P)) = ⊤\n⊢ normalizer (normalizer ↑P) = normalizer ↑P",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nthis :\n Normal (normalizer ↑(Sylow.subtype P (_ : ↑P ≤ normalizer (normalizer ↑P)))) →\n Normal ↑(Sylow.subtype P (_ : ↑P ≤ normalizer (normalizer ↑P)))\n⊢ normalizer (normalizer ↑P) = normalizer ↑P",
"tactic": "simp_rw [← normalizer_eq_top, Sylow.coe_subtype, ← subgroupOf_normalizer_eq le_normalizer, ←\n subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this"
},
{
"state_after": "G : Type u\nα : Type v\nβ : Type w\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nthis : True → subgroupOf (normalizer ↑P) (normalizer (normalizer ↑P)) = ⊤\n⊢ Subgroup.map (Subgroup.subtype (normalizer (normalizer ↑P)))\n (subgroupOf (normalizer ↑P) (normalizer (normalizer ↑P))) =\n normalizer ↑P",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nthis : True → subgroupOf (normalizer ↑P) (normalizer (normalizer ↑P)) = ⊤\n⊢ normalizer (normalizer ↑P) = normalizer ↑P",
"tactic": "rw [← subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,\n ← this trivial]"
},
{
"state_after": "no goals",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nthis : True → subgroupOf (normalizer ↑P) (normalizer (normalizer ↑P)) = ⊤\n⊢ Subgroup.map (Subgroup.subtype (normalizer (normalizer ↑P)))\n (subgroupOf (normalizer ↑P) (normalizer (normalizer ↑P))) =\n normalizer ↑P",
"tactic": "exact map_comap_eq_self (le_normalizer.trans (ge_of_eq (subtype_range _)))"
}
] |
[
738,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
731,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Icc_self
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.30276\ninst✝ : PartialOrder α\na✝ b c a : α\n⊢ ∀ (x : α), x ∈ Icc a a ↔ x ∈ {a}",
"tactic": "simp [Icc, le_antisymm_iff, and_comm]"
}
] |
[
760,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
759,
1
] |
Mathlib/Data/Polynomial/AlgebraMap.lean
|
Polynomial.eval₂_int_castRingHom_X
|
[] |
[
148,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Topology/UniformSpace/Completion.lean
|
UniformSpace.Completion.denseRange_coe₂
|
[] |
[
485,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
483,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
IsGLB.ciInf_eq
|
[] |
[
559,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
558,
1
] |
Mathlib/Topology/UniformSpace/Cauchy.lean
|
SequentiallyComplete.setSeq_mem
|
[] |
[
657,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
656,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
ConcaveOn.translate_left
|
[] |
[
304,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
302,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.set_walk_length_toFinset_eq
|
[
{
"state_after": "case a\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nn : ℕ\nu v : V\np : Walk G u v\n⊢ p ∈ Set.toFinset {p | Walk.length p = n} ↔ p ∈ finsetWalkLength G n u v",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nn : ℕ\nu v : V\n⊢ Set.toFinset {p | Walk.length p = n} = finsetWalkLength G n u v",
"tactic": "ext p"
},
{
"state_after": "no goals",
"state_before": "case a\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝¹ : DecidableEq V\ninst✝ : LocallyFinite G\nn : ℕ\nu v : V\np : Walk G u v\n⊢ p ∈ Set.toFinset {p | Walk.length p = n} ↔ p ∈ finsetWalkLength G n u v",
"tactic": "simp [← coe_finsetWalkLength_eq]"
}
] |
[
2383,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2380,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometryEquiv.mul_def
|
[] |
[
891,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
890,
1
] |
Mathlib/Data/Pi/Algebra.lean
|
Function.extend_one
|
[
{
"state_after": "no goals",
"state_before": "I : Type u\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nf✝ : I → Type v₁\ng : I → Type v₂\nh : I → Type v₃\nx y : (i : I) → f✝ i\ni : I\ninst✝ : One γ\nf : α → β\nx✝ : β\n⊢ extend f 1 1 x✝ = OfNat.ofNat 1 x✝",
"tactic": "apply ite_self"
}
] |
[
363,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
362,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
PowerSeries.isUnit_constantCoeff
|
[] |
[
1642,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1641,
1
] |
Mathlib/Algebra/Order/Field/Power.lean
|
zpow_lt_iff_lt
|
[] |
[
82,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.two_nsmul_eq_iff
|
[
{
"state_after": "no goals",
"state_before": "ψ θ : Angle\n⊢ 2 • ψ = 2 • θ ↔ ψ = θ ∨ ψ = θ + ↑π",
"tactic": "simp_rw [← coe_nat_zsmul, Nat.cast_ofNat, two_zsmul_eq_iff]"
}
] |
[
195,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/RingTheory/Coprime/Basic.lean
|
isCoprime_group_smul_right
|
[] |
[
237,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.card_one
|
[] |
[
146,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/LinearAlgebra/Dual.lean
|
Submodule.dualAnnihilator_top
|
[
{
"state_after": "R : Type u\nM : Type v\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nW : Submodule R M\n⊢ dualAnnihilator ⊤ ≤ ⊥",
"state_before": "R : Type u\nM : Type v\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nW : Submodule R M\n⊢ dualAnnihilator ⊤ = ⊥",
"tactic": "rw [eq_bot_iff]"
},
{
"state_after": "R : Type u\nM : Type v\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nW : Submodule R M\nv : Module.Dual R M\n⊢ v ∈ dualAnnihilator ⊤ → v ∈ ⊥",
"state_before": "R : Type u\nM : Type v\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nW : Submodule R M\n⊢ dualAnnihilator ⊤ ≤ ⊥",
"tactic": "intro v"
},
{
"state_after": "R : Type u\nM : Type v\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nW : Submodule R M\nv : Module.Dual R M\n⊢ (∀ (w : M), ↑v w = 0) → v = 0",
"state_before": "R : Type u\nM : Type v\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nW : Submodule R M\nv : Module.Dual R M\n⊢ v ∈ dualAnnihilator ⊤ → v ∈ ⊥",
"tactic": "simp_rw [mem_dualAnnihilator, mem_bot, mem_top, forall_true_left]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nM : Type v\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nW : Submodule R M\nv : Module.Dual R M\n⊢ (∀ (w : M), ↑v w = 0) → v = 0",
"tactic": "exact fun h => LinearMap.ext h"
}
] |
[
861,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
857,
1
] |
Mathlib/Data/Fintype/Card.lean
|
Fintype.false
|
[] |
[
934,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
933,
11
] |
Mathlib/Data/Nat/Sqrt.lean
|
Nat.sqrt_isSqrt
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ Nat.IsSqrt 0 (sqrt 0)",
"tactic": "simp [IsSqrt]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ Nat.IsSqrt 1 (sqrt 1)",
"tactic": "simp [IsSqrt]"
},
{
"state_after": "n✝ n : ℕ\nh : ¬n + 2 ≤ 1\n⊢ Nat.IsSqrt (n + 2) (sqrt (n + 2))",
"state_before": "n✝ n : ℕ\n⊢ Nat.IsSqrt (n + 2) (sqrt (n + 2))",
"tactic": "have h : ¬ (n + 2) ≤ 1 := by simp"
},
{
"state_after": "n✝ n : ℕ\nh : ¬n + 2 ≤ 1\n⊢ sqrt.iter (n + 2) ((n + 2) / 2) * sqrt.iter (n + 2) ((n + 2) / 2) ≤ n + 2 ∧\n n + 2 < (sqrt.iter (n + 2) ((n + 2) / 2) + 1) * (sqrt.iter (n + 2) ((n + 2) / 2) + 1)",
"state_before": "n✝ n : ℕ\nh : ¬n + 2 ≤ 1\n⊢ Nat.IsSqrt (n + 2) (sqrt (n + 2))",
"tactic": "simp only [IsSqrt, sqrt, h, ite_false]"
},
{
"state_after": "n✝ n : ℕ\nh : ¬n + 2 ≤ 1\n⊢ n + 2 < ((n + 2) / 2 + 1) * ((n + 2) / 2 + 1)",
"state_before": "n✝ n : ℕ\nh : ¬n + 2 ≤ 1\n⊢ sqrt.iter (n + 2) ((n + 2) / 2) * sqrt.iter (n + 2) ((n + 2) / 2) ≤ n + 2 ∧\n n + 2 < (sqrt.iter (n + 2) ((n + 2) / 2) + 1) * (sqrt.iter (n + 2) ((n + 2) / 2) + 1)",
"tactic": "refine ⟨sqrt.iter_sq_le _ _, sqrt.lt_iter_succ_sq _ _ ?_⟩"
},
{
"state_after": "n✝ n : ℕ\nh : ¬n + 2 ≤ 1\n⊢ n + 2 < (n + 2) / 2 * ((n + 2) / 2) + (n + 2) / 2 + (n + 2) / 2 + 1",
"state_before": "n✝ n : ℕ\nh : ¬n + 2 ≤ 1\n⊢ n + 2 < ((n + 2) / 2 + 1) * ((n + 2) / 2 + 1)",
"tactic": "simp only [mul_add, add_mul, one_mul, mul_one, ← add_assoc]"
},
{
"state_after": "n✝ n : ℕ\nh : ¬n + 2 ≤ 1\n⊢ n + 2 ≤ (n + 2) / 2 * ((n + 2) / 2) + (n + 2) / 2 * 2",
"state_before": "n✝ n : ℕ\nh : ¬n + 2 ≤ 1\n⊢ n + 2 < (n + 2) / 2 * ((n + 2) / 2) + (n + 2) / 2 + (n + 2) / 2 + 1",
"tactic": "rw [lt_add_one_iff, add_assoc, ← mul_two]"
},
{
"state_after": "n✝ n : ℕ\nh : ¬n + 2 ≤ 1\n⊢ (n + 2) / 2 * 2 + (n + 2) % 2 ≤ (n + 2) / 2 * ((n + 2) / 2) + (n + 2) / 2 * 2",
"state_before": "n✝ n : ℕ\nh : ¬n + 2 ≤ 1\n⊢ n + 2 ≤ (n + 2) / 2 * ((n + 2) / 2) + (n + 2) / 2 * 2",
"tactic": "refine le_trans (div_add_mod' (n + 2) 2).ge ?_"
},
{
"state_after": "n✝ n : ℕ\nh : ¬n + 2 ≤ 1\n⊢ n % 2 ≤ (n + 2) / 2 * ((n + 2) / 2)",
"state_before": "n✝ n : ℕ\nh : ¬n + 2 ≤ 1\n⊢ (n + 2) / 2 * 2 + (n + 2) % 2 ≤ (n + 2) / 2 * ((n + 2) / 2) + (n + 2) / 2 * 2",
"tactic": "rw [add_comm, add_le_add_iff_right, add_mod_right]"
},
{
"state_after": "n✝ n : ℕ\nh : ¬n + 2 ≤ 1\n⊢ n % 2 ≤ n / 2 * (n / 2) + n / 2 + n / 2 + 1",
"state_before": "n✝ n : ℕ\nh : ¬n + 2 ≤ 1\n⊢ n % 2 ≤ (n + 2) / 2 * ((n + 2) / 2)",
"tactic": "simp only [zero_lt_two, add_div_right, succ_mul_succ_eq]"
},
{
"state_after": "case refine_1\nn✝ n : ℕ\nh : ¬n + 2 ≤ 1\n⊢ n % 2 ≤ 1\n\ncase refine_2\nn✝ n : ℕ\nh : ¬n + 2 ≤ 1\n⊢ 1 ≤ n / 2 * (n / 2) + n / 2 + n / 2 + 1",
"state_before": "n✝ n : ℕ\nh : ¬n + 2 ≤ 1\n⊢ n % 2 ≤ n / 2 * (n / 2) + n / 2 + n / 2 + 1",
"tactic": "refine le_trans (b := 1) ?_ ?_"
},
{
"state_after": "no goals",
"state_before": "n✝ n : ℕ\n⊢ ¬n + 2 ≤ 1",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nn✝ n : ℕ\nh : ¬n + 2 ≤ 1\n⊢ n % 2 ≤ 1",
"tactic": "exact (lt_succ.1 $ mod_lt n zero_lt_two)"
},
{
"state_after": "case refine_2\nn✝ n : ℕ\nh : ¬n + 2 ≤ 1\n⊢ 0 ≤ n / 2 * (n / 2) + n / 2 + n / 2",
"state_before": "case refine_2\nn✝ n : ℕ\nh : ¬n + 2 ≤ 1\n⊢ 1 ≤ n / 2 * (n / 2) + n / 2 + n / 2 + 1",
"tactic": "simp only [le_add_iff_nonneg_left]"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nn✝ n : ℕ\nh : ¬n + 2 ≤ 1\n⊢ 0 ≤ n / 2 * (n / 2) + n / 2 + n / 2",
"tactic": "exact zero_le _"
}
] |
[
65,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
9
] |
Mathlib/Topology/Separation.lean
|
not_preirreducible_nontrivial_t2
|
[] |
[
1454,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1452,
1
] |
Mathlib/Algebra/AddTorsor.lean
|
Equiv.coe_constVAdd
|
[] |
[
412,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
411,
1
] |
Mathlib/Algebra/Order/Sub/Defs.lean
|
AddLECancellable.le_tsub_of_add_le_left
|
[] |
[
212,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
211,
11
] |
Mathlib/Algebra/IndicatorFunction.lean
|
Set.inter_indicator_mul
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.124894\nι : Type ?u.124897\nM : Type u_2\nN : Type ?u.124903\ninst✝ : MulZeroClass M\ns t : Set α\nf✝ g✝ : α → M\na : α\nt1 t2 : Set α\nf g : α → M\nx : α\n⊢ indicator t1 (indicator t2 fun x => f x * g x) x = indicator t1 f x * indicator t2 g x",
"state_before": "α : Type u_1\nβ : Type ?u.124894\nι : Type ?u.124897\nM : Type u_2\nN : Type ?u.124903\ninst✝ : MulZeroClass M\ns t : Set α\nf✝ g✝ : α → M\na : α\nt1 t2 : Set α\nf g : α → M\nx : α\n⊢ indicator (t1 ∩ t2) (fun x => f x * g x) x = indicator t1 f x * indicator t2 g x",
"tactic": "rw [← Set.indicator_indicator]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.124894\nι : Type ?u.124897\nM : Type u_2\nN : Type ?u.124903\ninst✝ : MulZeroClass M\ns t : Set α\nf✝ g✝ : α → M\na : α\nt1 t2 : Set α\nf g : α → M\nx : α\n⊢ (if x ∈ t1 then if x ∈ t2 then f x * g x else 0 else 0) = (if x ∈ t1 then f x else 0) * if x ∈ t2 then g x else 0",
"state_before": "α : Type u_1\nβ : Type ?u.124894\nι : Type ?u.124897\nM : Type u_2\nN : Type ?u.124903\ninst✝ : MulZeroClass M\ns t : Set α\nf✝ g✝ : α → M\na : α\nt1 t2 : Set α\nf g : α → M\nx : α\n⊢ indicator t1 (indicator t2 fun x => f x * g x) x = indicator t1 f x * indicator t2 g x",
"tactic": "simp_rw [indicator]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.124894\nι : Type ?u.124897\nM : Type u_2\nN : Type ?u.124903\ninst✝ : MulZeroClass M\ns t : Set α\nf✝ g✝ : α → M\na : α\nt1 t2 : Set α\nf g : α → M\nx : α\n⊢ (if x ∈ t1 then if x ∈ t2 then f x * g x else 0 else 0) = (if x ∈ t1 then f x else 0) * if x ∈ t2 then g x else 0",
"tactic": "split_ifs <;> simp"
}
] |
[
707,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
703,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean
|
expMapCircle_two_pi
|
[] |
[
102,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/GroupTheory/Schreier.lean
|
Subgroup.closure_mul_image_eq_top'
|
[
{
"state_after": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR✝ S✝ : Set G\ninst✝ : DecidableEq G\nR S : Finset G\nhR : ↑R ∈ rightTransversals ↑H\nhR1 : 1 ∈ R\nhS : closure ↑S = ⊤\n⊢ closure ((fun g => { val := g * (↑(toFun hR g))⁻¹, property := (_ : g * (↑(toFun hR g))⁻¹ ∈ H) }) '' (↑R * ↑S)) = ⊤",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR✝ S✝ : Set G\ninst✝ : DecidableEq G\nR S : Finset G\nhR : ↑R ∈ rightTransversals ↑H\nhR1 : 1 ∈ R\nhS : closure ↑S = ⊤\n⊢ closure\n ↑(Finset.image (fun g => { val := g * (↑(toFun hR g))⁻¹, property := (_ : g * (↑(toFun hR g))⁻¹ ∈ H) }) (R * S)) =\n ⊤",
"tactic": "rw [Finset.coe_image, Finset.coe_mul]"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR✝ S✝ : Set G\ninst✝ : DecidableEq G\nR S : Finset G\nhR : ↑R ∈ rightTransversals ↑H\nhR1 : 1 ∈ R\nhS : closure ↑S = ⊤\n⊢ closure ((fun g => { val := g * (↑(toFun hR g))⁻¹, property := (_ : g * (↑(toFun hR g))⁻¹ ∈ H) }) '' (↑R * ↑S)) = ⊤",
"tactic": "exact closure_mul_image_eq_top hR hR1 hS"
}
] |
[
103,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearEquiv.symm_preimage_preimage
|
[] |
[
2127,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2125,
11
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Subalgebra.mem_toSubmodule
|
[] |
[
344,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
344,
1
] |
Mathlib/GroupTheory/Coset.lean
|
one_leftCoset
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Monoid α\ns : Set α\n⊢ ∀ (x : α), x ∈ 1 *l s ↔ x ∈ s",
"tactic": "simp [leftCoset]"
}
] |
[
152,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/GroupTheory/Perm/Support.lean
|
Equiv.Perm.support_pow_le
|
[] |
[
350,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
349,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Bilinear.lean
|
IsBoundedBilinearMap.differentiableAt
|
[] |
[
99,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Prod.lean
|
fderiv.snd
|
[] |
[
321,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
319,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.sup_eq_union
|
[] |
[
113,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.PreservesPullback.iso_hom_snd
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nG : C ⥤ D\nW X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\nh : W ⟶ X\nk : W ⟶ Y\ncomm : h ≫ f = k ≫ g\ninst✝² : PreservesLimit (cospan f g) G\ninst✝¹ : HasPullback f g\ninst✝ : HasPullback (G.map f) (G.map g)\n⊢ (iso G f g).hom ≫ pullback.snd = G.map pullback.snd",
"tactic": "simp [PreservesPullback.iso]"
}
] |
[
129,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/ZeroObjects.lean
|
CategoryTheory.Limits.isZero_zero
|
[] |
[
206,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/Order/Hom/Bounded.lean
|
TopHom.top_apply
|
[] |
[
320,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
319,
1
] |
Mathlib/CategoryTheory/Preadditive/Basic.lean
|
CategoryTheory.Preadditive.coforkOfCokernelCofork_π
|
[] |
[
376,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
374,
1
] |
Mathlib/Data/List/Pairwise.lean
|
List.Pairwise.forall_of_forall_of_flip
|
[
{
"state_after": "case nil\nα : Type u_1\nβ : Type ?u.5128\nR S T : α → α → Prop\na : α\nl : List α\nh₁✝ : ∀ (x : α), x ∈ l → R x x\nh₂✝ : Pairwise R l\nh₃✝ : Pairwise (flip R) l\nh₁ : ∀ (x : α), x ∈ [] → R x x\nh₂ : Pairwise R []\nh₃ : Pairwise (flip R) []\n⊢ ∀ ⦃x : α⦄, x ∈ [] → ∀ ⦃y : α⦄, y ∈ [] → R x y\n\ncase cons\nα : Type u_1\nβ : Type ?u.5128\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\nh₁✝ : ∀ (x : α), x ∈ l✝ → R x x\nh₂✝ : Pairwise R l✝\nh₃✝ : Pairwise (flip R) l✝\na : α\nl : List α\nih : (∀ (x : α), x ∈ l → R x x) → Pairwise R l → Pairwise (flip R) l → ∀ ⦃x : α⦄, x ∈ l → ∀ ⦃y : α⦄, y ∈ l → R x y\nh₁ : ∀ (x : α), x ∈ a :: l → R x x\nh₂ : Pairwise R (a :: l)\nh₃ : Pairwise (flip R) (a :: l)\n⊢ ∀ ⦃x : α⦄, x ∈ a :: l → ∀ ⦃y : α⦄, y ∈ a :: l → R x y",
"state_before": "α : Type u_1\nβ : Type ?u.5128\nR S T : α → α → Prop\na : α\nl : List α\nh₁ : ∀ (x : α), x ∈ l → R x x\nh₂ : Pairwise R l\nh₃ : Pairwise (flip R) l\n⊢ ∀ ⦃x : α⦄, x ∈ l → ∀ ⦃y : α⦄, y ∈ l → R x y",
"tactic": "induction' l with a l ih"
},
{
"state_after": "case cons\nα : Type u_1\nβ : Type ?u.5128\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\nh₁✝ : ∀ (x : α), x ∈ l✝ → R x x\nh₂✝ : Pairwise R l✝\nh₃✝ : Pairwise (flip R) l✝\na : α\nl : List α\nih : (∀ (x : α), x ∈ l → R x x) → Pairwise R l → Pairwise (flip R) l → ∀ ⦃x : α⦄, x ∈ l → ∀ ⦃y : α⦄, y ∈ l → R x y\nh₁ : ∀ (x : α), x ∈ a :: l → R x x\nh₂ : (∀ (a' : α), a' ∈ l → R a a') ∧ Pairwise R l\nh₃ : (∀ (a' : α), a' ∈ l → flip R a a') ∧ Pairwise (flip R) l\n⊢ ∀ ⦃x : α⦄, x ∈ a :: l → ∀ ⦃y : α⦄, y ∈ a :: l → R x y",
"state_before": "case cons\nα : Type u_1\nβ : Type ?u.5128\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\nh₁✝ : ∀ (x : α), x ∈ l✝ → R x x\nh₂✝ : Pairwise R l✝\nh₃✝ : Pairwise (flip R) l✝\na : α\nl : List α\nih : (∀ (x : α), x ∈ l → R x x) → Pairwise R l → Pairwise (flip R) l → ∀ ⦃x : α⦄, x ∈ l → ∀ ⦃y : α⦄, y ∈ l → R x y\nh₁ : ∀ (x : α), x ∈ a :: l → R x x\nh₂ : Pairwise R (a :: l)\nh₃ : Pairwise (flip R) (a :: l)\n⊢ ∀ ⦃x : α⦄, x ∈ a :: l → ∀ ⦃y : α⦄, y ∈ a :: l → R x y",
"tactic": "rw [pairwise_cons] at h₂ h₃"
},
{
"state_after": "case cons\nα : Type u_1\nβ : Type ?u.5128\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\nh₁✝ : ∀ (x : α), x ∈ l✝ → R x x\nh₂✝ : Pairwise R l✝\nh₃✝ : Pairwise (flip R) l✝\na : α\nl : List α\nih : (∀ (x : α), x ∈ l → R x x) → Pairwise R l → Pairwise (flip R) l → ∀ ⦃x : α⦄, x ∈ l → ∀ ⦃y : α⦄, y ∈ l → R x y\nh₁ : ∀ (x : α), x ∈ a :: l → R x x\nh₂ : (∀ (a' : α), a' ∈ l → R a a') ∧ Pairwise R l\nh₃ : (∀ (a' : α), a' ∈ l → flip R a a') ∧ Pairwise (flip R) l\n⊢ ∀ ⦃x : α⦄, x = a ∨ x ∈ l → ∀ ⦃y : α⦄, y = a ∨ y ∈ l → R x y",
"state_before": "case cons\nα : Type u_1\nβ : Type ?u.5128\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\nh₁✝ : ∀ (x : α), x ∈ l✝ → R x x\nh₂✝ : Pairwise R l✝\nh₃✝ : Pairwise (flip R) l✝\na : α\nl : List α\nih : (∀ (x : α), x ∈ l → R x x) → Pairwise R l → Pairwise (flip R) l → ∀ ⦃x : α⦄, x ∈ l → ∀ ⦃y : α⦄, y ∈ l → R x y\nh₁ : ∀ (x : α), x ∈ a :: l → R x x\nh₂ : (∀ (a' : α), a' ∈ l → R a a') ∧ Pairwise R l\nh₃ : (∀ (a' : α), a' ∈ l → flip R a a') ∧ Pairwise (flip R) l\n⊢ ∀ ⦃x : α⦄, x ∈ a :: l → ∀ ⦃y : α⦄, y ∈ a :: l → R x y",
"tactic": "simp only [mem_cons]"
},
{
"state_after": "case cons.inl.inl\nα : Type u_1\nβ : Type ?u.5128\nR S T : α → α → Prop\na : α\nl✝ : List α\nh₁✝ : ∀ (x : α), x ∈ l✝ → R x x\nh₂✝ : Pairwise R l✝\nh₃✝ : Pairwise (flip R) l✝\nl : List α\nih : (∀ (x : α), x ∈ l → R x x) → Pairwise R l → Pairwise (flip R) l → ∀ ⦃x : α⦄, x ∈ l → ∀ ⦃y : α⦄, y ∈ l → R x y\ny : α\nh₁ : ∀ (x : α), x ∈ y :: l → R x x\nh₂ : (∀ (a' : α), a' ∈ l → R y a') ∧ Pairwise R l\nh₃ : (∀ (a' : α), a' ∈ l → flip R y a') ∧ Pairwise (flip R) l\n⊢ R y y\n\ncase cons.inl.inr\nα : Type u_1\nβ : Type ?u.5128\nR S T : α → α → Prop\na : α\nl✝ : List α\nh₁✝ : ∀ (x : α), x ∈ l✝ → R x x\nh₂✝ : Pairwise R l✝\nh₃✝ : Pairwise (flip R) l✝\nl : List α\nih : (∀ (x : α), x ∈ l → R x x) → Pairwise R l → Pairwise (flip R) l → ∀ ⦃x : α⦄, x ∈ l → ∀ ⦃y : α⦄, y ∈ l → R x y\nx : α\nh₁ : ∀ (x_1 : α), x_1 ∈ x :: l → R x_1 x_1\nh₂ : (∀ (a' : α), a' ∈ l → R x a') ∧ Pairwise R l\nh₃ : (∀ (a' : α), a' ∈ l → flip R x a') ∧ Pairwise (flip R) l\ny : α\nhy : y ∈ l\n⊢ R x y\n\ncase cons.inr.inl\nα : Type u_1\nβ : Type ?u.5128\nR S T : α → α → Prop\na : α\nl✝ : List α\nh₁✝ : ∀ (x : α), x ∈ l✝ → R x x\nh₂✝ : Pairwise R l✝\nh₃✝ : Pairwise (flip R) l✝\nl : List α\nih : (∀ (x : α), x ∈ l → R x x) → Pairwise R l → Pairwise (flip R) l → ∀ ⦃x : α⦄, x ∈ l → ∀ ⦃y : α⦄, y ∈ l → R x y\nx : α\nhx : x ∈ l\ny : α\nh₁ : ∀ (x : α), x ∈ y :: l → R x x\nh₂ : (∀ (a' : α), a' ∈ l → R y a') ∧ Pairwise R l\nh₃ : (∀ (a' : α), a' ∈ l → flip R y a') ∧ Pairwise (flip R) l\n⊢ R x y\n\ncase cons.inr.inr\nα : Type u_1\nβ : Type ?u.5128\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\nh₁✝ : ∀ (x : α), x ∈ l✝ → R x x\nh₂✝ : Pairwise R l✝\nh₃✝ : Pairwise (flip R) l✝\na : α\nl : List α\nih : (∀ (x : α), x ∈ l → R x x) → Pairwise R l → Pairwise (flip R) l → ∀ ⦃x : α⦄, x ∈ l → ∀ ⦃y : α⦄, y ∈ l → R x y\nh₁ : ∀ (x : α), x ∈ a :: l → R x x\nh₂ : (∀ (a' : α), a' ∈ l → R a a') ∧ Pairwise R l\nh₃ : (∀ (a' : α), a' ∈ l → flip R a a') ∧ Pairwise (flip R) l\nx : α\nhx : x ∈ l\ny : α\nhy : y ∈ l\n⊢ R x y",
"state_before": "case cons\nα : Type u_1\nβ : Type ?u.5128\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\nh₁✝ : ∀ (x : α), x ∈ l✝ → R x x\nh₂✝ : Pairwise R l✝\nh₃✝ : Pairwise (flip R) l✝\na : α\nl : List α\nih : (∀ (x : α), x ∈ l → R x x) → Pairwise R l → Pairwise (flip R) l → ∀ ⦃x : α⦄, x ∈ l → ∀ ⦃y : α⦄, y ∈ l → R x y\nh₁ : ∀ (x : α), x ∈ a :: l → R x x\nh₂ : (∀ (a' : α), a' ∈ l → R a a') ∧ Pairwise R l\nh₃ : (∀ (a' : α), a' ∈ l → flip R a a') ∧ Pairwise (flip R) l\n⊢ ∀ ⦃x : α⦄, x = a ∨ x ∈ l → ∀ ⦃y : α⦄, y = a ∨ y ∈ l → R x y",
"tactic": "rintro x (rfl | hx) y (rfl | hy)"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\nβ : Type ?u.5128\nR S T : α → α → Prop\na : α\nl : List α\nh₁✝ : ∀ (x : α), x ∈ l → R x x\nh₂✝ : Pairwise R l\nh₃✝ : Pairwise (flip R) l\nh₁ : ∀ (x : α), x ∈ [] → R x x\nh₂ : Pairwise R []\nh₃ : Pairwise (flip R) []\n⊢ ∀ ⦃x : α⦄, x ∈ [] → ∀ ⦃y : α⦄, y ∈ [] → R x y",
"tactic": "exact forall_mem_nil _"
},
{
"state_after": "no goals",
"state_before": "case cons.inl.inl\nα : Type u_1\nβ : Type ?u.5128\nR S T : α → α → Prop\na : α\nl✝ : List α\nh₁✝ : ∀ (x : α), x ∈ l✝ → R x x\nh₂✝ : Pairwise R l✝\nh₃✝ : Pairwise (flip R) l✝\nl : List α\nih : (∀ (x : α), x ∈ l → R x x) → Pairwise R l → Pairwise (flip R) l → ∀ ⦃x : α⦄, x ∈ l → ∀ ⦃y : α⦄, y ∈ l → R x y\ny : α\nh₁ : ∀ (x : α), x ∈ y :: l → R x x\nh₂ : (∀ (a' : α), a' ∈ l → R y a') ∧ Pairwise R l\nh₃ : (∀ (a' : α), a' ∈ l → flip R y a') ∧ Pairwise (flip R) l\n⊢ R y y",
"tactic": "exact h₁ _ (l.mem_cons_self _)"
},
{
"state_after": "no goals",
"state_before": "case cons.inl.inr\nα : Type u_1\nβ : Type ?u.5128\nR S T : α → α → Prop\na : α\nl✝ : List α\nh₁✝ : ∀ (x : α), x ∈ l✝ → R x x\nh₂✝ : Pairwise R l✝\nh₃✝ : Pairwise (flip R) l✝\nl : List α\nih : (∀ (x : α), x ∈ l → R x x) → Pairwise R l → Pairwise (flip R) l → ∀ ⦃x : α⦄, x ∈ l → ∀ ⦃y : α⦄, y ∈ l → R x y\nx : α\nh₁ : ∀ (x_1 : α), x_1 ∈ x :: l → R x_1 x_1\nh₂ : (∀ (a' : α), a' ∈ l → R x a') ∧ Pairwise R l\nh₃ : (∀ (a' : α), a' ∈ l → flip R x a') ∧ Pairwise (flip R) l\ny : α\nhy : y ∈ l\n⊢ R x y",
"tactic": "exact h₂.1 _ hy"
},
{
"state_after": "no goals",
"state_before": "case cons.inr.inl\nα : Type u_1\nβ : Type ?u.5128\nR S T : α → α → Prop\na : α\nl✝ : List α\nh₁✝ : ∀ (x : α), x ∈ l✝ → R x x\nh₂✝ : Pairwise R l✝\nh₃✝ : Pairwise (flip R) l✝\nl : List α\nih : (∀ (x : α), x ∈ l → R x x) → Pairwise R l → Pairwise (flip R) l → ∀ ⦃x : α⦄, x ∈ l → ∀ ⦃y : α⦄, y ∈ l → R x y\nx : α\nhx : x ∈ l\ny : α\nh₁ : ∀ (x : α), x ∈ y :: l → R x x\nh₂ : (∀ (a' : α), a' ∈ l → R y a') ∧ Pairwise R l\nh₃ : (∀ (a' : α), a' ∈ l → flip R y a') ∧ Pairwise (flip R) l\n⊢ R x y",
"tactic": "exact h₃.1 _ hx"
},
{
"state_after": "no goals",
"state_before": "case cons.inr.inr\nα : Type u_1\nβ : Type ?u.5128\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\nh₁✝ : ∀ (x : α), x ∈ l✝ → R x x\nh₂✝ : Pairwise R l✝\nh₃✝ : Pairwise (flip R) l✝\na : α\nl : List α\nih : (∀ (x : α), x ∈ l → R x x) → Pairwise R l → Pairwise (flip R) l → ∀ ⦃x : α⦄, x ∈ l → ∀ ⦃y : α⦄, y ∈ l → R x y\nh₁ : ∀ (x : α), x ∈ a :: l → R x x\nh₂ : (∀ (a' : α), a' ∈ l → R a a') ∧ Pairwise R l\nh₃ : (∀ (a' : α), a' ∈ l → flip R a a') ∧ Pairwise (flip R) l\nx : α\nhx : x ∈ l\ny : α\nhy : y ∈ l\n⊢ R x y",
"tactic": "exact ih (fun x hx => h₁ _ <| mem_cons_of_mem _ hx) h₂.2 h₃.2 hx hy"
}
] |
[
135,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Topology/Sets/Opens.lean
|
TopologicalSpace.Opens.ne_bot_iff_nonempty
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.28417\nα : Type u_1\nβ : Type ?u.28423\nγ : Type ?u.28426\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nU : Opens α\n⊢ U ≠ ⊥ ↔ Set.Nonempty ↑U",
"tactic": "rw [Ne.def, ← not_nonempty_iff_eq_bot, not_not]"
}
] |
[
275,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
274,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearEquiv.subsingleton_or_nnnorm_symm_pos
|
[] |
[
1981,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1979,
1
] |
Mathlib/GroupTheory/Commutator.lean
|
Subgroup.commutator_le_inf
|
[] |
[
169,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.ae_dirac_eq
|
[
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.600866\nγ : Type ?u.600869\nδ : Type ?u.600872\nι : Type ?u.600875\nR : Type ?u.600878\nR' : Type ?u.600881\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\na : α\ns : Set α\n⊢ s ∈ ae (dirac a) ↔ s ∈ pure a",
"state_before": "α : Type u_1\nβ : Type ?u.600866\nγ : Type ?u.600869\nδ : Type ?u.600872\nι : Type ?u.600875\nR : Type ?u.600878\nR' : Type ?u.600881\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\na : α\n⊢ ae (dirac a) = pure a",
"tactic": "ext s"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.600866\nγ : Type ?u.600869\nδ : Type ?u.600872\nι : Type ?u.600875\nR : Type ?u.600878\nR' : Type ?u.600881\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\na : α\ns : Set α\n⊢ s ∈ ae (dirac a) ↔ s ∈ pure a",
"tactic": "simp [mem_ae_iff, imp_false]"
}
] |
[
3017,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3015,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.isPath_iff_eq_nil
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu : V\np : Walk G u u\n⊢ IsPath p ↔ p = nil",
"tactic": "cases p <;> simp [IsPath.nil]"
}
] |
[
980,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
979,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.support_mapRange
|
[] |
[
794,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
792,
1
] |
Mathlib/Logic/Equiv/Defs.lean
|
Equiv.arrowCongr_refl
|
[] |
[
530,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
529,
9
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.coeIdeal_eq_zero
|
[] |
[
1012,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1011,
1
] |
Mathlib/Data/Setoid/Partition.lean
|
Setoid.exists_of_mem_partition
|
[] |
[
241,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
238,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.exists_mem_forall_mem_nhds_within_pos
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.10003\nR : Type ?u.10006\nR' : Type ?u.10009\nms : Set (OuterMeasure α)\nm✝ : OuterMeasure α\ninst✝¹ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nm : OuterMeasure α\ns : Set α\nhs : ∀ (x : α), x ∈ s → ∃ t, t ∈ 𝓝[s] x ∧ ↑m t ≤ 0\n⊢ ↑m s = 0",
"state_before": "α : Type u_1\nβ : Type ?u.10003\nR : Type ?u.10006\nR' : Type ?u.10009\nms : Set (OuterMeasure α)\nm✝ : OuterMeasure α\ninst✝¹ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nm : OuterMeasure α\ns : Set α\nhs : ↑m s ≠ 0\n⊢ ∃ x, x ∈ s ∧ ∀ (t : Set α), t ∈ 𝓝[s] x → 0 < ↑m t",
"tactic": "contrapose! hs"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.10003\nR : Type ?u.10006\nR' : Type ?u.10009\nms : Set (OuterMeasure α)\nm✝ : OuterMeasure α\ninst✝¹ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nm : OuterMeasure α\ns : Set α\nhs : ∀ (x : α), x ∈ s → ∃ t _h, ↑m t = 0\n⊢ ↑m s = 0",
"state_before": "α : Type u_1\nβ : Type ?u.10003\nR : Type ?u.10006\nR' : Type ?u.10009\nms : Set (OuterMeasure α)\nm✝ : OuterMeasure α\ninst✝¹ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nm : OuterMeasure α\ns : Set α\nhs : ∀ (x : α), x ∈ s → ∃ t, t ∈ 𝓝[s] x ∧ ↑m t ≤ 0\n⊢ ↑m s = 0",
"tactic": "simp only [nonpos_iff_eq_zero, ← exists_prop] at hs"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.10003\nR : Type ?u.10006\nR' : Type ?u.10009\nms : Set (OuterMeasure α)\nm✝ : OuterMeasure α\ninst✝¹ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nm : OuterMeasure α\ns : Set α\nhs : ∀ (x : α), x ∈ s → ∃ t _h, ↑m t = 0\n⊢ ∀ (x : α), x ∈ s → ∃ u, u ∈ 𝓝[s] x ∧ ↑m u = 0",
"state_before": "α : Type u_1\nβ : Type ?u.10003\nR : Type ?u.10006\nR' : Type ?u.10009\nms : Set (OuterMeasure α)\nm✝ : OuterMeasure α\ninst✝¹ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nm : OuterMeasure α\ns : Set α\nhs : ∀ (x : α), x ∈ s → ∃ t _h, ↑m t = 0\n⊢ ↑m s = 0",
"tactic": "apply m.null_of_locally_null s"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.10003\nR : Type ?u.10006\nR' : Type ?u.10009\nms : Set (OuterMeasure α)\nm✝ : OuterMeasure α\ninst✝¹ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nm : OuterMeasure α\ns : Set α\nhs : ∀ (x : α), x ∈ s → ∃ t _h, ↑m t = 0\nx : α\nhx : x ∈ s\n⊢ ∃ u, u ∈ 𝓝[s] x ∧ ↑m u = 0",
"state_before": "α : Type u_1\nβ : Type ?u.10003\nR : Type ?u.10006\nR' : Type ?u.10009\nms : Set (OuterMeasure α)\nm✝ : OuterMeasure α\ninst✝¹ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nm : OuterMeasure α\ns : Set α\nhs : ∀ (x : α), x ∈ s → ∃ t _h, ↑m t = 0\n⊢ ∀ (x : α), x ∈ s → ∃ u, u ∈ 𝓝[s] x ∧ ↑m u = 0",
"tactic": "intro x hx"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.10003\nR : Type ?u.10006\nR' : Type ?u.10009\nms : Set (OuterMeasure α)\nm✝ : OuterMeasure α\ninst✝¹ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nm : OuterMeasure α\ns : Set α\nx : α\nhx : x ∈ s\nhs : ∃ t _h, ↑m t = 0\n⊢ ∃ u, u ∈ 𝓝[s] x ∧ ↑m u = 0",
"state_before": "α : Type u_1\nβ : Type ?u.10003\nR : Type ?u.10006\nR' : Type ?u.10009\nms : Set (OuterMeasure α)\nm✝ : OuterMeasure α\ninst✝¹ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nm : OuterMeasure α\ns : Set α\nhs : ∀ (x : α), x ∈ s → ∃ t _h, ↑m t = 0\nx : α\nhx : x ∈ s\n⊢ ∃ u, u ∈ 𝓝[s] x ∧ ↑m u = 0",
"tactic": "specialize hs x hx"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.10003\nR : Type ?u.10006\nR' : Type ?u.10009\nms : Set (OuterMeasure α)\nm✝ : OuterMeasure α\ninst✝¹ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nm : OuterMeasure α\ns : Set α\nx : α\nhx : x ∈ s\nhs : ∃ t _h, ↑m t = 0\n⊢ ∃ u, u ∈ 𝓝[s] x ∧ ↑m u = 0",
"tactic": "exact Iff.mp bex_def hs"
}
] |
[
176,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Order/Compare.lean
|
Ordering.Compares.le_total
|
[] |
[
114,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/LinearAlgebra/Matrix/Basis.lean
|
Basis.toMatrix_apply
|
[] |
[
66,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Std/Data/List/Init/Lemmas.lean
|
List.forall_mem_cons
|
[] |
[
116,
65
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
113,
1
] |
Mathlib/Data/MvPolynomial/Division.lean
|
MvPolynomial.support_divMonomial
|
[] |
[
62,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
div_le_div_iff'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : CommGroup α\ninst✝¹ : LE α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c d : α\n⊢ a / b ≤ c / d ↔ a * d ≤ c * b",
"tactic": "simpa only [div_eq_mul_inv] using mul_inv_le_mul_inv_iff'"
}
] |
[
805,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
804,
1
] |
Mathlib/LinearAlgebra/Quotient.lean
|
Submodule.Quotient.mk'_eq_mk'
|
[] |
[
68,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/CategoryTheory/Subobject/Limits.lean
|
CategoryTheory.Limits.kernelSubobjectMap_comp
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁴ : Category C\nX Y Z : C\ninst✝³ : HasZeroMorphisms C\nf : X ⟶ Y\ninst✝² : HasKernel f\nX' Y' : C\nf' : X' ⟶ Y'\ninst✝¹ : HasKernel f'\nX'' Y'' : C\nf'' : X'' ⟶ Y''\ninst✝ : HasKernel f''\nsq : Arrow.mk f ⟶ Arrow.mk f'\nsq' : Arrow.mk f' ⟶ Arrow.mk f''\n⊢ kernelSubobjectMap (sq ≫ sq') = kernelSubobjectMap sq ≫ kernelSubobjectMap sq'",
"tactic": "aesop_cat"
}
] |
[
174,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean
|
Ideal.homogeneousCore_mono
|
[] |
[
192,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/Logic/Equiv/Set.lean
|
Equiv.image_preimage
|
[] |
[
94,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.map_smul_nnreal
|
[] |
[
1215,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1213,
11
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.Lifts.exists_max_three
|
[
{
"state_after": "case intro.intro.intro\nF : Type u_3\nE : Type u_2\nK : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nS : Set E\nc : Set (Lifts F E K)\nx y z : Lifts F E K\nhc : IsChain (fun x x_1 => x ≤ x_1) c\nhx : x ∈ insert ⊥ c\nhy : y ∈ insert ⊥ c\nhz : z ∈ insert ⊥ c\nv : Lifts F E K\nhv : v ∈ insert ⊥ c\nhxv : x ≤ v\nhyv : y ≤ v\n⊢ ∃ w, w ∈ insert ⊥ c ∧ x ≤ w ∧ y ≤ w ∧ z ≤ w",
"state_before": "F : Type u_3\nE : Type u_2\nK : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nS : Set E\nc : Set (Lifts F E K)\nx y z : Lifts F E K\nhc : IsChain (fun x x_1 => x ≤ x_1) c\nhx : x ∈ insert ⊥ c\nhy : y ∈ insert ⊥ c\nhz : z ∈ insert ⊥ c\n⊢ ∃ w, w ∈ insert ⊥ c ∧ x ≤ w ∧ y ≤ w ∧ z ≤ w",
"tactic": "obtain ⟨v, hv, hxv, hyv⟩ := Lifts.exists_max_two hc hx hy"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nF : Type u_3\nE : Type u_2\nK : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nS : Set E\nc : Set (Lifts F E K)\nx y z : Lifts F E K\nhc : IsChain (fun x x_1 => x ≤ x_1) c\nhx : x ∈ insert ⊥ c\nhy : y ∈ insert ⊥ c\nhz : z ∈ insert ⊥ c\nv : Lifts F E K\nhv : v ∈ insert ⊥ c\nhxv : x ≤ v\nhyv : y ≤ v\nw : Lifts F E K\nhw : w ∈ insert ⊥ c\nhzw : z ≤ w\nhvw : v ≤ w\n⊢ ∃ w, w ∈ insert ⊥ c ∧ x ≤ w ∧ y ≤ w ∧ z ≤ w",
"state_before": "case intro.intro.intro\nF : Type u_3\nE : Type u_2\nK : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nS : Set E\nc : Set (Lifts F E K)\nx y z : Lifts F E K\nhc : IsChain (fun x x_1 => x ≤ x_1) c\nhx : x ∈ insert ⊥ c\nhy : y ∈ insert ⊥ c\nhz : z ∈ insert ⊥ c\nv : Lifts F E K\nhv : v ∈ insert ⊥ c\nhxv : x ≤ v\nhyv : y ≤ v\n⊢ ∃ w, w ∈ insert ⊥ c ∧ x ≤ w ∧ y ≤ w ∧ z ≤ w",
"tactic": "obtain ⟨w, hw, hzw, hvw⟩ := Lifts.exists_max_two hc hz hv"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro\nF : Type u_3\nE : Type u_2\nK : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nS : Set E\nc : Set (Lifts F E K)\nx y z : Lifts F E K\nhc : IsChain (fun x x_1 => x ≤ x_1) c\nhx : x ∈ insert ⊥ c\nhy : y ∈ insert ⊥ c\nhz : z ∈ insert ⊥ c\nv : Lifts F E K\nhv : v ∈ insert ⊥ c\nhxv : x ≤ v\nhyv : y ≤ v\nw : Lifts F E K\nhw : w ∈ insert ⊥ c\nhzw : z ≤ w\nhvw : v ≤ w\n⊢ ∃ w, w ∈ insert ⊥ c ∧ x ≤ w ∧ y ≤ w ∧ z ≤ w",
"tactic": "exact ⟨w, hw, le_trans hxv hvw, le_trans hyv hvw, hzw⟩"
}
] |
[
1022,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1016,
1
] |
Mathlib/Analysis/NormedSpace/WeakDual.lean
|
WeakDual.isCompact_closedBall
|
[] |
[
256,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/Algebra/Regular/Pow.lean
|
IsRightRegular.pow
|
[
{
"state_after": "R : Type u_1\na b : R\ninst✝ : Monoid R\nn : ℕ\nrra : IsRightRegular a\n⊢ Function.Injective ((fun x => x * a)^[n])",
"state_before": "R : Type u_1\na b : R\ninst✝ : Monoid R\nn : ℕ\nrra : IsRightRegular a\n⊢ IsRightRegular (a ^ n)",
"tactic": "rw [IsRightRegular, ← mul_right_iterate]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\na b : R\ninst✝ : Monoid R\nn : ℕ\nrra : IsRightRegular a\n⊢ Function.Injective ((fun x => x * a)^[n])",
"tactic": "exact rra.iterate n"
}
] |
[
40,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
38,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.StronglyMeasurable.aemeasurable
|
[] |
[
366,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
363,
11
] |
Mathlib/Data/List/Basic.lean
|
List.length_eq_two
|
[] |
[
220,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
Cubic.degree_of_a_eq_zero'
|
[] |
[
318,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
317,
1
] |
Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean
|
GromovHausdorff.isCompact_candidatesB
|
[
{
"state_after": "case refine'_1\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ ∀ (f : (X ⊕ Y) × (X ⊕ Y) →ᵇ ℝ) (x : (X ⊕ Y) × (X ⊕ Y)),\n f ∈ GromovHausdorff.candidatesB X Y → ↑f x ∈ Icc 0 ↑(GromovHausdorff.maxVar X Y)\n\ncase refine'_2\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ Equicontinuous fun x => ↑↑x",
"state_before": "X : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ IsCompact (GromovHausdorff.candidatesB X Y)",
"tactic": "refine' arzela_ascoli₂\n (Icc 0 (maxVar X Y) : Set ℝ) isCompact_Icc (candidatesB X Y) closed_candidatesB _ _"
},
{
"state_after": "case refine'_1.mk\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf : (X ⊕ Y) × (X ⊕ Y) →ᵇ ℝ\nx1 x2 : X ⊕ Y\nhf : f ∈ GromovHausdorff.candidatesB X Y\n⊢ ↑f (x1, x2) ∈ Icc 0 ↑(GromovHausdorff.maxVar X Y)",
"state_before": "case refine'_1\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ ∀ (f : (X ⊕ Y) × (X ⊕ Y) →ᵇ ℝ) (x : (X ⊕ Y) × (X ⊕ Y)),\n f ∈ GromovHausdorff.candidatesB X Y → ↑f x ∈ Icc 0 ↑(GromovHausdorff.maxVar X Y)",
"tactic": "rintro f ⟨x1, x2⟩ hf"
},
{
"state_after": "case refine'_1.mk\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf : (X ⊕ Y) × (X ⊕ Y) →ᵇ ℝ\nx1 x2 : X ⊕ Y\nhf : f ∈ GromovHausdorff.candidatesB X Y\n⊢ 0 ≤ ↑f (x1, x2) ∧ ↑f (x1, x2) ≤ ↑(GromovHausdorff.maxVar X Y)",
"state_before": "case refine'_1.mk\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf : (X ⊕ Y) × (X ⊕ Y) →ᵇ ℝ\nx1 x2 : X ⊕ Y\nhf : f ∈ GromovHausdorff.candidatesB X Y\n⊢ ↑f (x1, x2) ∈ Icc 0 ↑(GromovHausdorff.maxVar X Y)",
"tactic": "simp only [Set.mem_Icc]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.mk\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf : (X ⊕ Y) × (X ⊕ Y) →ᵇ ℝ\nx1 x2 : X ⊕ Y\nhf : f ∈ GromovHausdorff.candidatesB X Y\n⊢ 0 ≤ ↑f (x1, x2) ∧ ↑f (x1, x2) ≤ ↑(GromovHausdorff.maxVar X Y)",
"tactic": "exact ⟨candidates_nonneg hf, candidates_le_maxVar hf⟩"
},
{
"state_after": "case refine'_2.refine'_1\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ Tendsto (fun t => 2 * ↑(GromovHausdorff.maxVar X Y) * t) (𝓝 0) (𝓝 0)\n\ncase refine'_2.refine'_2\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ ∀ (x y : (X ⊕ Y) × (X ⊕ Y)) (i : ↑(GromovHausdorff.candidatesB X Y)),\n dist (↑↑i x) (↑↑i y) ≤ (fun t => 2 * ↑(GromovHausdorff.maxVar X Y) * t) (dist x y)",
"state_before": "case refine'_2\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ Equicontinuous fun x => ↑↑x",
"tactic": "refine' equicontinuous_of_continuity_modulus (fun t => 2 * maxVar X Y * t) _ _ _"
},
{
"state_after": "case refine'_2.refine'_1\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nthis : Tendsto (fun t => 2 * ↑(GromovHausdorff.maxVar X Y) * t) (𝓝 0) (𝓝 (2 * ↑(GromovHausdorff.maxVar X Y) * 0))\n⊢ Tendsto (fun t => 2 * ↑(GromovHausdorff.maxVar X Y) * t) (𝓝 0) (𝓝 0)",
"state_before": "case refine'_2.refine'_1\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ Tendsto (fun t => 2 * ↑(GromovHausdorff.maxVar X Y) * t) (𝓝 0) (𝓝 0)",
"tactic": "have : Tendsto (fun t : ℝ => 2 * (maxVar X Y : ℝ) * t) (𝓝 0) (𝓝 (2 * maxVar X Y * 0)) :=\n tendsto_const_nhds.mul tendsto_id"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.refine'_1\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nthis : Tendsto (fun t => 2 * ↑(GromovHausdorff.maxVar X Y) * t) (𝓝 0) (𝓝 (2 * ↑(GromovHausdorff.maxVar X Y) * 0))\n⊢ Tendsto (fun t => 2 * ↑(GromovHausdorff.maxVar X Y) * t) (𝓝 0) (𝓝 0)",
"tactic": "simpa using this"
},
{
"state_after": "case refine'_2.refine'_2.mk\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx✝ y✝ z t : X ⊕ Y\nx y : (X ⊕ Y) × (X ⊕ Y)\nf : (X ⊕ Y) × (X ⊕ Y) →ᵇ ℝ\nhf : f ∈ GromovHausdorff.candidatesB X Y\n⊢ dist (↑↑{ val := f, property := hf } x) (↑↑{ val := f, property := hf } y) ≤\n (fun t => 2 * ↑(GromovHausdorff.maxVar X Y) * t) (dist x y)",
"state_before": "case refine'_2.refine'_2\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ ∀ (x y : (X ⊕ Y) × (X ⊕ Y)) (i : ↑(GromovHausdorff.candidatesB X Y)),\n dist (↑↑i x) (↑↑i y) ≤ (fun t => 2 * ↑(GromovHausdorff.maxVar X Y) * t) (dist x y)",
"tactic": "rintro x y ⟨f, hf⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.refine'_2.mk\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx✝ y✝ z t : X ⊕ Y\nx y : (X ⊕ Y) × (X ⊕ Y)\nf : (X ⊕ Y) × (X ⊕ Y) →ᵇ ℝ\nhf : f ∈ GromovHausdorff.candidatesB X Y\n⊢ dist (↑↑{ val := f, property := hf } x) (↑↑{ val := f, property := hf } y) ≤\n (fun t => 2 * ↑(GromovHausdorff.maxVar X Y) * t) (dist x y)",
"tactic": "exact (candidates_lipschitz hf).dist_le_mul _ _"
}
] |
[
282,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
271,
9
] |
Mathlib/CategoryTheory/Limits/Shapes/Products.lean
|
CategoryTheory.Limits.ι_comp_sigmaComparison
|
[] |
[
270,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
268,
1
] |
Mathlib/Logic/Equiv/Defs.lean
|
Equiv.self_trans_symm
|
[
{
"state_after": "no goals",
"state_before": "α : Sort u\nβ : Sort v\nγ : Sort w\ne : α ≃ β\n⊢ ∀ (x : α), ↑(e.trans e.symm) x = ↑(Equiv.refl α) x",
"tactic": "simp"
}
] |
[
353,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
353,
9
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.zero_sub
|
[
{
"state_after": "α : Type ?u.131620\nβ : Type ?u.131623\nγ : Type ?u.131626\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na : Ordinal\n⊢ 0 - a ≤ 0",
"state_before": "α : Type ?u.131620\nβ : Type ?u.131623\nγ : Type ?u.131626\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na : Ordinal\n⊢ 0 - a = 0",
"tactic": "rw [← Ordinal.le_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.131620\nβ : Type ?u.131623\nγ : Type ?u.131626\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na : Ordinal\n⊢ 0 - a ≤ 0",
"tactic": "apply sub_le_self"
}
] |
[
591,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
591,
1
] |
Mathlib/Analysis/Calculus/LHopital.lean
|
deriv.lhopital_zero_left_on_Ioo
|
[
{
"state_after": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf✝ : DifferentiableOn ℝ f (Ioo a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → deriv g x ≠ 0\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 0)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[Iio b] b) l\nhdf : ∀ (x : ℝ), x ∈ Ioo a b → DifferentiableAt ℝ f x\n⊢ Tendsto (fun x => f x / g x) (𝓝[Iio b] b) l",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : DifferentiableOn ℝ f (Ioo a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → deriv g x ≠ 0\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 0)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[Iio b] b) l\n⊢ Tendsto (fun x => f x / g x) (𝓝[Iio b] b) l",
"tactic": "have hdf : ∀ x ∈ Ioo a b, DifferentiableAt ℝ f x := fun x hx =>\n (hdf x hx).differentiableAt (Ioo_mem_nhds hx.1 hx.2)"
},
{
"state_after": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf✝ : DifferentiableOn ℝ f (Ioo a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → deriv g x ≠ 0\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 0)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[Iio b] b) l\nhdf : ∀ (x : ℝ), x ∈ Ioo a b → DifferentiableAt ℝ f x\nhdg : ∀ (x : ℝ), x ∈ Ioo a b → DifferentiableAt ℝ g x\n⊢ Tendsto (fun x => f x / g x) (𝓝[Iio b] b) l",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf✝ : DifferentiableOn ℝ f (Ioo a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → deriv g x ≠ 0\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 0)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[Iio b] b) l\nhdf : ∀ (x : ℝ), x ∈ Ioo a b → DifferentiableAt ℝ f x\n⊢ Tendsto (fun x => f x / g x) (𝓝[Iio b] b) l",
"tactic": "have hdg : ∀ x ∈ Ioo a b, DifferentiableAt ℝ g x := fun x hx =>\n by_contradiction fun h => hg' x hx (deriv_zero_of_not_differentiableAt h)"
},
{
"state_after": "no goals",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf✝ : DifferentiableOn ℝ f (Ioo a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → deriv g x ≠ 0\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 0)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[Iio b] b) l\nhdf : ∀ (x : ℝ), x ∈ Ioo a b → DifferentiableAt ℝ f x\nhdg : ∀ (x : ℝ), x ∈ Ioo a b → DifferentiableAt ℝ g x\n⊢ Tendsto (fun x => f x / g x) (𝓝[Iio b] b) l",
"tactic": "exact HasDerivAt.lhopital_zero_left_on_Ioo hab (fun x hx => (hdf x hx).hasDerivAt)\n (fun x hx => (hdg x hx).hasDerivAt) hg' hfb hgb hdiv"
}
] |
[
245,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
235,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.equiv_of_eq
|
[
{
"state_after": "x : PGame\n⊢ x ≈ x",
"state_before": "x y : PGame\nh : x = y\n⊢ x ≈ y",
"tactic": "subst h"
},
{
"state_after": "no goals",
"state_before": "x : PGame\n⊢ x ≈ x",
"tactic": "rfl"
}
] |
[
775,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
775,
1
] |
Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean
|
SimpleGraph.dotProduct_adjMatrix
|
[
{
"state_after": "no goals",
"state_before": "V : Type u_2\nα : Type u_1\nβ : Type ?u.37315\nG : SimpleGraph V\ninst✝² : DecidableRel G.Adj\ninst✝¹ : Fintype V\ninst✝ : NonAssocSemiring α\nv : V\nvec : V → α\n⊢ vec ⬝ᵥ adjMatrix α G v = ∑ u in neighborFinset G v, vec u",
"tactic": "simp [neighborFinset_eq_filter, dotProduct, sum_filter, Finset.sum_apply]"
}
] |
[
210,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/Data/Polynomial/Lifts.lean
|
Polynomial.lifts_and_natDegree_eq_and_monic
|
[
{
"state_after": "case inl\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nhlifts : p ∈ lifts f\nhp : Monic p\nhR : Subsingleton S\n⊢ ∃ q, map f q = p ∧ natDegree q = natDegree p ∧ Monic q\n\ncase inr\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nhlifts : p ∈ lifts f\nhp : Monic p\nhR : Nontrivial S\n⊢ ∃ q, map f q = p ∧ natDegree q = natDegree p ∧ Monic q",
"state_before": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nhlifts : p ∈ lifts f\nhp : Monic p\n⊢ ∃ q, map f q = p ∧ natDegree q = natDegree p ∧ Monic q",
"tactic": "cases' subsingleton_or_nontrivial S with hR hR"
},
{
"state_after": "case inr.intro.intro.intro\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nhlifts : p ∈ lifts f\nhp : Monic p\nhR : Nontrivial S\np' : R[X]\nh₁ : map f p' = p\nh₂ : degree p' = degree p\nh₃ : Monic p'\n⊢ ∃ q, map f q = p ∧ natDegree q = natDegree p ∧ Monic q",
"state_before": "case inr\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nhlifts : p ∈ lifts f\nhp : Monic p\nhR : Nontrivial S\n⊢ ∃ q, map f q = p ∧ natDegree q = natDegree p ∧ Monic q",
"tactic": "obtain ⟨p', h₁, h₂, h₃⟩ := lifts_and_degree_eq_and_monic hlifts hp"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro.intro\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nhlifts : p ∈ lifts f\nhp : Monic p\nhR : Nontrivial S\np' : R[X]\nh₁ : map f p' = p\nh₂ : degree p' = degree p\nh₃ : Monic p'\n⊢ ∃ q, map f q = p ∧ natDegree q = natDegree p ∧ Monic q",
"tactic": "exact ⟨p', h₁, natDegree_eq_of_degree_eq h₂, h₃⟩"
},
{
"state_after": "case inl\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nhR : Subsingleton S\nhlifts : 1 ∈ lifts f\nhp : Monic 1\n⊢ ∃ q, map f q = 1 ∧ natDegree q = natDegree 1 ∧ Monic q",
"state_before": "case inl\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nhlifts : p ∈ lifts f\nhp : Monic p\nhR : Subsingleton S\n⊢ ∃ q, map f q = p ∧ natDegree q = natDegree p ∧ Monic q",
"tactic": "obtain rfl : p = 1 := Subsingleton.elim _ _"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nhR : Subsingleton S\nhlifts : 1 ∈ lifts f\nhp : Monic 1\n⊢ ∃ q, map f q = 1 ∧ natDegree q = natDegree 1 ∧ Monic q",
"tactic": "refine' ⟨1, Subsingleton.elim _ _, by simp, by simp⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nhR : Subsingleton S\nhlifts : 1 ∈ lifts f\nhp : Monic 1\n⊢ natDegree 1 = natDegree 1",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nhR : Subsingleton S\nhlifts : 1 ∈ lifts f\nhp : Monic 1\n⊢ Monic 1",
"tactic": "simp"
}
] |
[
242,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean
|
CategoryTheory.tensorRightHomEquiv_tensor_id_comp_evaluation
|
[
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX Y : C\ninst✝ : HasRightDual X\nf : Y ⟶ Xᘁ\n⊢ (ρ_ Y).inv ≫ (𝟙 Y ⊗ η_ X Xᘁ) ≫ (α_ Y X Xᘁ).inv ≫ ((f ⊗ 𝟙 X) ≫ ε_ X Xᘁ ⊗ 𝟙 Xᘁ) = f ≫ (λ_ Xᘁ).inv",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX Y : C\ninst✝ : HasRightDual X\nf : Y ⟶ Xᘁ\n⊢ ↑(tensorRightHomEquiv Y X Xᘁ (𝟙_ C)) ((f ⊗ 𝟙 X) ≫ ε_ X Xᘁ) = f ≫ (λ_ Xᘁ).inv",
"tactic": "dsimp [tensorRightHomEquiv]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX Y : C\ninst✝ : HasRightDual X\nf : Y ⟶ Xᘁ\n⊢ (ρ_ Y).inv ≫ (𝟙 Y ⊗ η_ X Xᘁ) ≫ (α_ Y X Xᘁ).inv ≫ ((f ⊗ 𝟙 X) ⊗ 𝟙 Xᘁ) ≫ (ε_ X Xᘁ ⊗ 𝟙 Xᘁ) = f ≫ (λ_ Xᘁ).inv",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX Y : C\ninst✝ : HasRightDual X\nf : Y ⟶ Xᘁ\n⊢ (ρ_ Y).inv ≫ (𝟙 Y ⊗ η_ X Xᘁ) ≫ (α_ Y X Xᘁ).inv ≫ ((f ⊗ 𝟙 X) ≫ ε_ X Xᘁ ⊗ 𝟙 Xᘁ) = f ≫ (λ_ Xᘁ).inv",
"tactic": "rw [comp_tensor_id]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX Y : C\ninst✝ : HasRightDual X\nf : Y ⟶ Xᘁ\n⊢ (ρ_ Y).inv ≫ (𝟙 Y ⊗ η_ X Xᘁ) ≫ ((f ⊗ 𝟙 X ⊗ 𝟙 Xᘁ) ≫ (α_ Xᘁ X Xᘁ).inv) ≫ (ε_ X Xᘁ ⊗ 𝟙 Xᘁ) = f ≫ (λ_ Xᘁ).inv",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX Y : C\ninst✝ : HasRightDual X\nf : Y ⟶ Xᘁ\n⊢ (ρ_ Y).inv ≫ (𝟙 Y ⊗ η_ X Xᘁ) ≫ (α_ Y X Xᘁ).inv ≫ ((f ⊗ 𝟙 X) ⊗ 𝟙 Xᘁ) ≫ (ε_ X Xᘁ ⊗ 𝟙 Xᘁ) = f ≫ (λ_ Xᘁ).inv",
"tactic": "slice_lhs 3 4 => rw [← associator_inv_naturality]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX Y : C\ninst✝ : HasRightDual X\nf : Y ⟶ Xᘁ\n⊢ (ρ_ Y).inv ≫ (((f ⊗ 𝟙 tensorUnit') ≫ (𝟙 Xᘁ ⊗ η_ X Xᘁ)) ≫ (α_ Xᘁ X Xᘁ).inv) ≫ (ε_ X Xᘁ ⊗ 𝟙 Xᘁ) = f ≫ (λ_ Xᘁ).inv",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX Y : C\ninst✝ : HasRightDual X\nf : Y ⟶ Xᘁ\n⊢ (ρ_ Y).inv ≫ (𝟙 Y ⊗ η_ X Xᘁ) ≫ ((f ⊗ 𝟙 X ⊗ 𝟙 Xᘁ) ≫ (α_ Xᘁ X Xᘁ).inv) ≫ (ε_ X Xᘁ ⊗ 𝟙 Xᘁ) = f ≫ (λ_ Xᘁ).inv",
"tactic": "slice_lhs 2 3 => rw [tensor_id, id_tensor_comp_tensor_id, ← tensor_id_comp_id_tensor]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX Y : C\ninst✝ : HasRightDual X\nf : Y ⟶ Xᘁ\n⊢ (ρ_ Y).inv ≫ (f ⊗ 𝟙 tensorUnit') ≫ (ρ_ Xᘁ).hom ≫ (λ_ Xᘁ).inv = f ≫ (λ_ Xᘁ).inv",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX Y : C\ninst✝ : HasRightDual X\nf : Y ⟶ Xᘁ\n⊢ (ρ_ Y).inv ≫ (((f ⊗ 𝟙 tensorUnit') ≫ (𝟙 Xᘁ ⊗ η_ X Xᘁ)) ≫ (α_ Xᘁ X Xᘁ).inv) ≫ (ε_ X Xᘁ ⊗ 𝟙 Xᘁ) = f ≫ (λ_ Xᘁ).inv",
"tactic": "slice_lhs 3 5 => rw [coevaluation_evaluation]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\nX Y : C\ninst✝ : HasRightDual X\nf : Y ⟶ Xᘁ\n⊢ (ρ_ Y).inv ≫ (f ⊗ 𝟙 tensorUnit') ≫ (ρ_ Xᘁ).hom ≫ (λ_ Xᘁ).inv = f ≫ (λ_ Xᘁ).inv",
"tactic": "simp"
}
] |
[
541,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
534,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.filter_cons
|
[
{
"state_after": "case inl\nα : Type ?u.367165\nβ : Type ?u.367168\nγ : Type ?u.367171\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\na : α\ns : Finset α\nha : ¬a ∈ s\nh✝ : p a\n⊢ _root_.Disjoint {a} (filter p s)\n\ncase inr\nα : Type ?u.367165\nβ : Type ?u.367168\nγ : Type ?u.367171\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\na : α\ns : Finset α\nha : ¬a ∈ s\nh✝ : ¬p a\n⊢ _root_.Disjoint ∅ (filter p s)",
"state_before": "α : Type ?u.367165\nβ : Type ?u.367168\nγ : Type ?u.367171\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\na : α\ns : Finset α\nha : ¬a ∈ s\n⊢ _root_.Disjoint (if p a then {a} else ∅) (filter p s)",
"tactic": "split_ifs"
},
{
"state_after": "case inl\nα : Type ?u.367165\nβ : Type ?u.367168\nγ : Type ?u.367171\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\na : α\ns : Finset α\nha : ¬a ∈ s\nh✝ : p a\n⊢ ¬a ∈ filter p s",
"state_before": "case inl\nα : Type ?u.367165\nβ : Type ?u.367168\nγ : Type ?u.367171\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\na : α\ns : Finset α\nha : ¬a ∈ s\nh✝ : p a\n⊢ _root_.Disjoint {a} (filter p s)",
"tactic": "rw [disjoint_singleton_left]"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type ?u.367165\nβ : Type ?u.367168\nγ : Type ?u.367171\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\na : α\ns : Finset α\nha : ¬a ∈ s\nh✝ : p a\n⊢ ¬a ∈ filter p s",
"tactic": "exact mem_filter.not.mpr <| mt And.left ha"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type ?u.367165\nβ : Type ?u.367168\nγ : Type ?u.367171\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\na : α\ns : Finset α\nha : ¬a ∈ s\nh✝ : ¬p a\n⊢ _root_.Disjoint ∅ (filter p s)",
"tactic": "exact disjoint_empty_left _"
},
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.367168\nγ : Type ?u.367171\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\na : α\ns : Finset α\nha : ¬a ∈ s\nh : p a\n⊢ filter p (cons a s ha) = disjUnion {a} (filter p s) (_ : _root_.Disjoint {a} (filter p s))\n\ncase inr\nα : Type u_1\nβ : Type ?u.367168\nγ : Type ?u.367171\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\na : α\ns : Finset α\nha : ¬a ∈ s\nh : ¬p a\n⊢ filter p (cons a s ha) = disjUnion ∅ (filter p s) (_ : _root_.Disjoint ∅ (filter p s))",
"state_before": "α : Type u_1\nβ : Type ?u.367168\nγ : Type ?u.367171\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\na : α\ns : Finset α\nha : ¬a ∈ s\n⊢ filter p (cons a s ha) =\n disjUnion (if p a then {a} else ∅) (filter p s) (_ : _root_.Disjoint (if p a then {a} else ∅) (filter p s))",
"tactic": "split_ifs with h"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.367168\nγ : Type ?u.367171\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\na : α\ns : Finset α\nha : ¬a ∈ s\nh : p a\n⊢ filter p (cons a s ha) = disjUnion {a} (filter p s) (_ : _root_.Disjoint {a} (filter p s))",
"tactic": "rw [filter_cons_of_pos _ _ _ ha h, singleton_disjUnion]"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.367168\nγ : Type ?u.367171\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\na : α\ns : Finset α\nha : ¬a ∈ s\nh : ¬p a\n⊢ filter p (cons a s ha) = disjUnion ∅ (filter p s) (_ : _root_.Disjoint ∅ (filter p s))",
"tactic": "rw [filter_cons_of_neg _ _ _ ha h, empty_disjUnion]"
}
] |
[
2807,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2797,
1
] |
Mathlib/Algebra/Invertible.lean
|
invOf_two_add_invOf_two
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝¹ : NonAssocSemiring α\ninst✝ : Invertible 2\n⊢ ⅟2 + ⅟2 = 1",
"tactic": "rw [← two_mul, mul_invOf_self]"
}
] |
[
270,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
269,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.TM2to1.tr_supports
|
[
{
"state_after": "K : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nl' : Λ'\nh : l' ∈ trSupp M S\n⊢ ∀ (q : Stmt₂),\n TM2.SupportsStmt S q →\n (∀ (x : Λ'), x ∈ trStmts₁ q → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q → TM1.SupportsStmt (trSupp M S) (tr M l')",
"state_before": "K : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nl' : Λ'\nh : l' ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "suffices ∀ (q) (_ : TM2.SupportsStmt S q) (_ : ∀ x ∈ trStmts₁ q, x ∈ trSupp M S),\n TM1.SupportsStmt (trSupp M S) (trNormal q) ∧\n ∀ l' ∈ trStmts₁ q, TM1.SupportsStmt (trSupp M S) (tr M l') by\n rcases Finset.mem_biUnion.1 h with ⟨l, lS, h⟩\n have :=\n this _ (ss.2 l lS) fun x hx ↦ Finset.mem_biUnion.2 ⟨_, lS, Finset.mem_insert_of_mem hx⟩\n rcases Finset.mem_insert.1 h with (rfl | h) <;> [exact this.1; exact this.2 _ h]"
},
{
"state_after": "K : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\n⊢ ∀ (q : Stmt₂),\n TM2.SupportsStmt S q →\n (∀ (x : Λ'), x ∈ trStmts₁ q → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q → TM1.SupportsStmt (trSupp M S) (tr M l')",
"state_before": "K : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nl' : Λ'\nh : l' ∈ trSupp M S\n⊢ ∀ (q : Stmt₂),\n TM2.SupportsStmt S q →\n (∀ (x : Λ'), x ∈ trStmts₁ q → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q → TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "clear h l'"
},
{
"state_after": "case refine'_1\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\n⊢ ∀ (k : K) (s : StAct k) (q : Stmt₂),\n (TM2.SupportsStmt S q →\n (∀ (x : Λ'), x ∈ trStmts₁ q → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q → TM1.SupportsStmt (trSupp M S) (tr M l')) →\n TM2.SupportsStmt S (stRun s q) →\n (∀ (x : Λ'), x ∈ trStmts₁ (stRun s q) → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (stRun s q) → TM1.SupportsStmt (trSupp M S) (tr M l')\n\ncase refine'_2\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\n⊢ ∀ (a : σ → σ) (q : Stmt₂),\n (TM2.SupportsStmt S q →\n (∀ (x : Λ'), x ∈ trStmts₁ q → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q → TM1.SupportsStmt (trSupp M S) (tr M l')) →\n TM2.SupportsStmt S (TM2.Stmt.load a q) →\n (∀ (x : Λ'), x ∈ trStmts₁ (TM2.Stmt.load a q) → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.load a q)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (TM2.Stmt.load a q) → TM1.SupportsStmt (trSupp M S) (tr M l')\n\ncase refine'_3\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\n⊢ ∀ (p : σ → Bool) (q₁ q₂ : Stmt₂),\n (TM2.SupportsStmt S q₁ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₁ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₁) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₁ → TM1.SupportsStmt (trSupp M S) (tr M l')) →\n (TM2.SupportsStmt S q₂ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₂ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₂) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₂ → TM1.SupportsStmt (trSupp M S) (tr M l')) →\n TM2.SupportsStmt S (TM2.Stmt.branch p q₁ q₂) →\n (∀ (x : Λ'), x ∈ trStmts₁ (TM2.Stmt.branch p q₁ q₂) → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p q₁ q₂)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (TM2.Stmt.branch p q₁ q₂) → TM1.SupportsStmt (trSupp M S) (tr M l')\n\ncase refine'_4\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\n⊢ ∀ (l : σ → Λ),\n TM2.SupportsStmt S (TM2.Stmt.goto l) →\n (∀ (x : Λ'), x ∈ trStmts₁ (TM2.Stmt.goto l) → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (TM2.Stmt.goto l) → TM1.SupportsStmt (trSupp M S) (tr M l')\n\ncase refine'_5\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\n⊢ TM2.SupportsStmt S TM2.Stmt.halt →\n (∀ (x : Λ'), x ∈ trStmts₁ TM2.Stmt.halt → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal TM2.Stmt.halt) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ TM2.Stmt.halt → TM1.SupportsStmt (trSupp M S) (tr M l')",
"state_before": "K : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\n⊢ ∀ (q : Stmt₂),\n TM2.SupportsStmt S q →\n (∀ (x : Λ'), x ∈ trStmts₁ q → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q → TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "refine' stmtStRec _ _ _ _ _"
},
{
"state_after": "case intro.intro\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nl' : Λ'\nh✝ : l' ∈ trSupp M S\nthis :\n ∀ (q : Stmt₂),\n TM2.SupportsStmt S q →\n (∀ (x : Λ'), x ∈ trStmts₁ q → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q → TM1.SupportsStmt (trSupp M S) (tr M l')\nl : Λ\nlS : l ∈ S\nh : l' ∈ insert (normal l) (trStmts₁ (M l))\n⊢ TM1.SupportsStmt (trSupp M S) (tr M l')",
"state_before": "K : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nl' : Λ'\nh : l' ∈ trSupp M S\nthis :\n ∀ (q : Stmt₂),\n TM2.SupportsStmt S q →\n (∀ (x : Λ'), x ∈ trStmts₁ q → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q → TM1.SupportsStmt (trSupp M S) (tr M l')\n⊢ TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "rcases Finset.mem_biUnion.1 h with ⟨l, lS, h⟩"
},
{
"state_after": "case intro.intro\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nl' : Λ'\nh✝ : l' ∈ trSupp M S\nthis✝ :\n ∀ (q : Stmt₂),\n TM2.SupportsStmt S q →\n (∀ (x : Λ'), x ∈ trStmts₁ q → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q → TM1.SupportsStmt (trSupp M S) (tr M l')\nl : Λ\nlS : l ∈ S\nh : l' ∈ insert (normal l) (trStmts₁ (M l))\nthis :\n TM1.SupportsStmt (trSupp M S) (trNormal (M l)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (M l) → TM1.SupportsStmt (trSupp M S) (tr M l')\n⊢ TM1.SupportsStmt (trSupp M S) (tr M l')",
"state_before": "case intro.intro\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nl' : Λ'\nh✝ : l' ∈ trSupp M S\nthis :\n ∀ (q : Stmt₂),\n TM2.SupportsStmt S q →\n (∀ (x : Λ'), x ∈ trStmts₁ q → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q → TM1.SupportsStmt (trSupp M S) (tr M l')\nl : Λ\nlS : l ∈ S\nh : l' ∈ insert (normal l) (trStmts₁ (M l))\n⊢ TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "have :=\n this _ (ss.2 l lS) fun x hx ↦ Finset.mem_biUnion.2 ⟨_, lS, Finset.mem_insert_of_mem hx⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nl' : Λ'\nh✝ : l' ∈ trSupp M S\nthis✝ :\n ∀ (q : Stmt₂),\n TM2.SupportsStmt S q →\n (∀ (x : Λ'), x ∈ trStmts₁ q → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q → TM1.SupportsStmt (trSupp M S) (tr M l')\nl : Λ\nlS : l ∈ S\nh : l' ∈ insert (normal l) (trStmts₁ (M l))\nthis :\n TM1.SupportsStmt (trSupp M S) (trNormal (M l)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (M l) → TM1.SupportsStmt (trSupp M S) (tr M l')\n⊢ TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "rcases Finset.mem_insert.1 h with (rfl | h) <;> [exact this.1; exact this.2 _ h]"
},
{
"state_after": "case refine'_1\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (stRun s q✝)\nsub : ∀ (x : Λ'), x ∈ trStmts₁ (stRun s q✝) → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (stRun s q✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"state_before": "case refine'_1\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\n⊢ ∀ (k : K) (s : StAct k) (q : Stmt₂),\n (TM2.SupportsStmt S q →\n (∀ (x : Λ'), x ∈ trStmts₁ q → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q → TM1.SupportsStmt (trSupp M S) (tr M l')) →\n TM2.SupportsStmt S (stRun s q) →\n (∀ (x : Λ'), x ∈ trStmts₁ (stRun s q) → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (stRun s q) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "intro _ s _ IH ss' sub"
},
{
"state_after": "case refine'_1\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), x ∈ trStmts₁ (stRun s q✝) → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (stRun s q✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"state_before": "case refine'_1\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (stRun s q✝)\nsub : ∀ (x : Λ'), x ∈ trStmts₁ (stRun s q✝) → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (stRun s q✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "rw [TM2to1.supports_run] at ss'"
},
{
"state_after": "case refine'_1\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (stRun s q✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"state_before": "case refine'_1\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), x ∈ trStmts₁ (stRun s q✝) → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (stRun s q✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "simp only [TM2to1.trStmts₁_run, Finset.mem_union, Finset.mem_insert, Finset.mem_singleton]\n at sub"
},
{
"state_after": "case refine'_1\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (stRun s q✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"state_before": "case refine'_1\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (stRun s q✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "have hgo := sub _ (Or.inl <| Or.inl rfl)"
},
{
"state_after": "case refine'_1\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\nhret : ret q✝ ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (stRun s q✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"state_before": "case refine'_1\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (stRun s q✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "have hret := sub _ (Or.inl <| Or.inr rfl)"
},
{
"state_after": "case refine'_1.intro\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (stRun s q✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"state_before": "case refine'_1\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\nhret : ret q✝ ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (stRun s q✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "cases' IH ss' fun x hx ↦ sub x <| Or.inr hx with IH₁ IH₂"
},
{
"state_after": "case refine'_1.intro\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nl : Λ'\nh : l ∈ trStmts₁ (stRun s q✝)\n⊢ TM1.SupportsStmt (trSupp M S) (tr M l)",
"state_before": "case refine'_1.intro\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (stRun s q✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "refine' ⟨by simp only [trNormal_run, TM1.SupportsStmt]; intros; exact hgo, fun l h ↦ _⟩"
},
{
"state_after": "case refine'_1.intro\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nl : Λ'\nh : l ∈ {go k✝ s q✝, ret q✝} ∪ trStmts₁ q✝\n⊢ TM1.SupportsStmt (trSupp M S) (tr M l)",
"state_before": "case refine'_1.intro\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nl : Λ'\nh : l ∈ trStmts₁ (stRun s q✝)\n⊢ TM1.SupportsStmt (trSupp M S) (tr M l)",
"tactic": "rw [trStmts₁_run] at h"
},
{
"state_after": "case refine'_1.intro\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nl : Λ'\nh : (l = go k✝ s q✝ ∨ l = ret q✝) ∨ l ∈ trStmts₁ q✝\n⊢ TM1.SupportsStmt (trSupp M S) (tr M l)",
"state_before": "case refine'_1.intro\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nl : Λ'\nh : l ∈ {go k✝ s q✝, ret q✝} ∪ trStmts₁ q✝\n⊢ TM1.SupportsStmt (trSupp M S) (tr M l)",
"tactic": "simp only [TM2to1.trStmts₁_run, Finset.mem_union, Finset.mem_insert, Finset.mem_singleton]\n at h"
},
{
"state_after": "case refine'_1.intro.inl.inl\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\n⊢ TM1.SupportsStmt (trSupp M S) (tr M (go k✝ s q✝))\n\ncase refine'_1.intro.inl.inr\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\n⊢ TM1.SupportsStmt (trSupp M S) (tr M (ret q✝))\n\ncase refine'_1.intro.inr\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nl : Λ'\nh : l ∈ trStmts₁ q✝\n⊢ TM1.SupportsStmt (trSupp M S) (tr M l)",
"state_before": "case refine'_1.intro\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nl : Λ'\nh : (l = go k✝ s q✝ ∨ l = ret q✝) ∨ l ∈ trStmts₁ q✝\n⊢ TM1.SupportsStmt (trSupp M S) (tr M l)",
"tactic": "rcases h with (⟨rfl | rfl⟩ | h)"
},
{
"state_after": "K : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\n⊢ Γ' → σ → go k✝ s q✝ ∈ trSupp M S",
"state_before": "K : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q✝))",
"tactic": "simp only [trNormal_run, TM1.SupportsStmt]"
},
{
"state_after": "K : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\na✝ : Γ'\nv✝ : σ\n⊢ go k✝ s q✝ ∈ trSupp M S",
"state_before": "K : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\n⊢ Γ' → σ → go k✝ s q✝ ∈ trSupp M S",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "K : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\na✝ : Γ'\nv✝ : σ\n⊢ go k✝ s q✝ ∈ trSupp M S",
"tactic": "exact hgo"
},
{
"state_after": "case refine'_1.intro.inl.inl.push\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\na✝ : σ → Γ k✝\nsub : ∀ (x : Λ'), (x = go k✝ (StAct.push a✝) q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ (StAct.push a✝) q✝ ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (tr M (go k✝ (StAct.push a✝) q✝))\n\ncase refine'_1.intro.inl.inl.peek\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\na✝ : σ → Option (Γ k✝) → σ\nsub : ∀ (x : Λ'), (x = go k✝ (StAct.peek a✝) q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ (StAct.peek a✝) q✝ ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (tr M (go k✝ (StAct.peek a✝) q✝))\n\ncase refine'_1.intro.inl.inl.pop\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\na✝ : σ → Option (Γ k✝) → σ\nsub : ∀ (x : Λ'), (x = go k✝ (StAct.pop a✝) q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ (StAct.pop a✝) q✝ ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (tr M (go k✝ (StAct.pop a✝) q✝))",
"state_before": "case refine'_1.intro.inl.inl\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\n⊢ TM1.SupportsStmt (trSupp M S) (tr M (go k✝ s q✝))",
"tactic": "cases s"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.inl.inl.push\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\na✝ : σ → Γ k✝\nsub : ∀ (x : Λ'), (x = go k✝ (StAct.push a✝) q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ (StAct.push a✝) q✝ ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (tr M (go k✝ (StAct.push a✝) q✝))",
"tactic": "exact ⟨fun _ _ ↦ hret, fun _ _ ↦ hgo⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.inl.inl.peek\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\na✝ : σ → Option (Γ k✝) → σ\nsub : ∀ (x : Λ'), (x = go k✝ (StAct.peek a✝) q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ (StAct.peek a✝) q✝ ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (tr M (go k✝ (StAct.peek a✝) q✝))",
"tactic": "exact ⟨fun _ _ ↦ hret, fun _ _ ↦ hgo⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.inl.inl.pop\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\na✝ : σ → Option (Γ k✝) → σ\nsub : ∀ (x : Λ'), (x = go k✝ (StAct.pop a✝) q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ (StAct.pop a✝) q✝ ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (tr M (go k✝ (StAct.pop a✝) q✝))",
"tactic": "exact ⟨⟨fun _ _ ↦ hret, fun _ _ ↦ hret⟩, fun _ _ ↦ hgo⟩"
},
{
"state_after": "case refine'_1.intro.inl.inr\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\n⊢ match\n match ret q✝ with\n | normal q => trNormal (M q)\n | go k s q =>\n branch (fun a x => Option.isNone (Prod.snd a k)) (trStAct (goto fun x x => ret q) s)\n (move Dir.right (goto fun x x => go k s q))\n | ret q => branch (fun a x => a.fst) (trNormal q) (move Dir.left (goto fun x x => ret q)) with\n | move a q => TM1.SupportsStmt (trSupp M S) q\n | write a q => TM1.SupportsStmt (trSupp M S) q\n | load a q => TM1.SupportsStmt (trSupp M S) q\n | branch a q₁ q₂ => TM1.SupportsStmt (trSupp M S) q₁ ∧ TM1.SupportsStmt (trSupp M S) q₂\n | goto l => ∀ (a : Γ') (v : σ), l a v ∈ trSupp M S\n | halt => True",
"state_before": "case refine'_1.intro.inl.inr\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\n⊢ TM1.SupportsStmt (trSupp M S) (tr M (ret q✝))",
"tactic": "unfold TM1.SupportsStmt TM2to1.tr"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.inl.inr\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\n⊢ match\n match ret q✝ with\n | normal q => trNormal (M q)\n | go k s q =>\n branch (fun a x => Option.isNone (Prod.snd a k)) (trStAct (goto fun x x => ret q) s)\n (move Dir.right (goto fun x x => go k s q))\n | ret q => branch (fun a x => a.fst) (trNormal q) (move Dir.left (goto fun x x => ret q)) with\n | move a q => TM1.SupportsStmt (trSupp M S) q\n | write a q => TM1.SupportsStmt (trSupp M S) q\n | load a q => TM1.SupportsStmt (trSupp M S) q\n | branch a q₁ q₂ => TM1.SupportsStmt (trSupp M S) q₁ ∧ TM1.SupportsStmt (trSupp M S) q₂\n | goto l => ∀ (a : Γ') (v : σ), l a v ∈ trSupp M S\n | halt => True",
"tactic": "exact ⟨IH₁, fun _ _ ↦ hret⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.inr\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nk✝ : K\ns : StAct k✝\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q✝\nsub : ∀ (x : Λ'), (x = go k✝ s q✝ ∨ x = ret q✝) ∨ x ∈ trStmts₁ q✝ → x ∈ trSupp M S\nhgo : go k✝ s q✝ ∈ trSupp M S\nhret : ret q✝ ∈ trSupp M S\nIH₁ : TM1.SupportsStmt (trSupp M S) (trNormal q✝)\nIH₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nl : Λ'\nh : l ∈ trStmts₁ q✝\n⊢ TM1.SupportsStmt (trSupp M S) (tr M l)",
"tactic": "exact IH₂ _ h"
},
{
"state_after": "case refine'_2\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\na✝ : σ → σ\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.load a✝ q✝)\nsub : ∀ (x : Λ'), x ∈ trStmts₁ (TM2.Stmt.load a✝ q✝) → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.load a✝ q✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (TM2.Stmt.load a✝ q✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"state_before": "case refine'_2\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\n⊢ ∀ (a : σ → σ) (q : Stmt₂),\n (TM2.SupportsStmt S q →\n (∀ (x : Λ'), x ∈ trStmts₁ q → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q → TM1.SupportsStmt (trSupp M S) (tr M l')) →\n TM2.SupportsStmt S (TM2.Stmt.load a q) →\n (∀ (x : Λ'), x ∈ trStmts₁ (TM2.Stmt.load a q) → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.load a q)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (TM2.Stmt.load a q) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "intro _ _ IH ss' sub"
},
{
"state_after": "case refine'_2\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\na✝ : σ → σ\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.load a✝ q✝)\nsub : ∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.load a✝ q✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')",
"state_before": "case refine'_2\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\na✝ : σ → σ\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.load a✝ q✝)\nsub : ∀ (x : Λ'), x ∈ trStmts₁ (TM2.Stmt.load a✝ q✝) → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.load a✝ q✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (TM2.Stmt.load a✝ q✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "unfold TM2to1.trStmts₁ at ss' sub⊢"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\na✝ : σ → σ\nq✝ : Stmt₂\nIH :\n TM2.SupportsStmt S q✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.load a✝ q✝)\nsub : ∀ (x : Λ'), x ∈ trStmts₁ q✝ → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.load a✝ q✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q✝ → TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "exact IH ss' sub"
},
{
"state_after": "case refine'_3\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\np✝ : σ → Bool\nq₁✝ q₂✝ : Stmt₂\nIH₁ :\n TM2.SupportsStmt S q₁✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₁✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₁✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₁✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nIH₂ :\n TM2.SupportsStmt S q₂✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₂✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₂✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₂✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p✝ q₁✝ q₂✝)\nsub : ∀ (x : Λ'), x ∈ trStmts₁ (TM2.Stmt.branch p✝ q₁✝ q₂✝) → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p✝ q₁✝ q₂✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (TM2.Stmt.branch p✝ q₁✝ q₂✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"state_before": "case refine'_3\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\n⊢ ∀ (p : σ → Bool) (q₁ q₂ : Stmt₂),\n (TM2.SupportsStmt S q₁ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₁ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₁) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₁ → TM1.SupportsStmt (trSupp M S) (tr M l')) →\n (TM2.SupportsStmt S q₂ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₂ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₂) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₂ → TM1.SupportsStmt (trSupp M S) (tr M l')) →\n TM2.SupportsStmt S (TM2.Stmt.branch p q₁ q₂) →\n (∀ (x : Λ'), x ∈ trStmts₁ (TM2.Stmt.branch p q₁ q₂) → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p q₁ q₂)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (TM2.Stmt.branch p q₁ q₂) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "intro _ _ _ IH₁ IH₂ ss' sub"
},
{
"state_after": "case refine'_3\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\np✝ : σ → Bool\nq₁✝ q₂✝ : Stmt₂\nIH₁ :\n TM2.SupportsStmt S q₁✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₁✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₁✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₁✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nIH₂ :\n TM2.SupportsStmt S q₂✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₂✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₂✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₂✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p✝ q₁✝ q₂✝)\nsub : ∀ (x : Λ'), x ∈ trStmts₁ q₁✝ ∪ trStmts₁ q₂✝ → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p✝ q₁✝ q₂✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (TM2.Stmt.branch p✝ q₁✝ q₂✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"state_before": "case refine'_3\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\np✝ : σ → Bool\nq₁✝ q₂✝ : Stmt₂\nIH₁ :\n TM2.SupportsStmt S q₁✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₁✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₁✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₁✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nIH₂ :\n TM2.SupportsStmt S q₂✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₂✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₂✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₂✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p✝ q₁✝ q₂✝)\nsub : ∀ (x : Λ'), x ∈ trStmts₁ (TM2.Stmt.branch p✝ q₁✝ q₂✝) → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p✝ q₁✝ q₂✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (TM2.Stmt.branch p✝ q₁✝ q₂✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "unfold TM2to1.trStmts₁ at sub"
},
{
"state_after": "case refine'_3.intro\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\np✝ : σ → Bool\nq₁✝ q₂✝ : Stmt₂\nIH₁ :\n TM2.SupportsStmt S q₁✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₁✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₁✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₁✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nIH₂ :\n TM2.SupportsStmt S q₂✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₂✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₂✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₂✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p✝ q₁✝ q₂✝)\nsub : ∀ (x : Λ'), x ∈ trStmts₁ q₁✝ ∪ trStmts₁ q₂✝ → x ∈ trSupp M S\nIH₁₁ : TM1.SupportsStmt (trSupp M S) (trNormal q₁✝)\nIH₁₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q₁✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p✝ q₁✝ q₂✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (TM2.Stmt.branch p✝ q₁✝ q₂✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"state_before": "case refine'_3\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\np✝ : σ → Bool\nq₁✝ q₂✝ : Stmt₂\nIH₁ :\n TM2.SupportsStmt S q₁✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₁✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₁✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₁✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nIH₂ :\n TM2.SupportsStmt S q₂✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₂✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₂✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₂✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p✝ q₁✝ q₂✝)\nsub : ∀ (x : Λ'), x ∈ trStmts₁ q₁✝ ∪ trStmts₁ q₂✝ → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p✝ q₁✝ q₂✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (TM2.Stmt.branch p✝ q₁✝ q₂✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "cases' IH₁ ss'.1 fun x hx ↦ sub x <| Finset.mem_union_left _ hx with IH₁₁ IH₁₂"
},
{
"state_after": "case refine'_3.intro.intro\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\np✝ : σ → Bool\nq₁✝ q₂✝ : Stmt₂\nIH₁ :\n TM2.SupportsStmt S q₁✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₁✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₁✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₁✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nIH₂ :\n TM2.SupportsStmt S q₂✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₂✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₂✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₂✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p✝ q₁✝ q₂✝)\nsub : ∀ (x : Λ'), x ∈ trStmts₁ q₁✝ ∪ trStmts₁ q₂✝ → x ∈ trSupp M S\nIH₁₁ : TM1.SupportsStmt (trSupp M S) (trNormal q₁✝)\nIH₁₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q₁✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nIH₂₁ : TM1.SupportsStmt (trSupp M S) (trNormal q₂✝)\nIH₂₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q₂✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p✝ q₁✝ q₂✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (TM2.Stmt.branch p✝ q₁✝ q₂✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"state_before": "case refine'_3.intro\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\np✝ : σ → Bool\nq₁✝ q₂✝ : Stmt₂\nIH₁ :\n TM2.SupportsStmt S q₁✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₁✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₁✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₁✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nIH₂ :\n TM2.SupportsStmt S q₂✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₂✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₂✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₂✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p✝ q₁✝ q₂✝)\nsub : ∀ (x : Λ'), x ∈ trStmts₁ q₁✝ ∪ trStmts₁ q₂✝ → x ∈ trSupp M S\nIH₁₁ : TM1.SupportsStmt (trSupp M S) (trNormal q₁✝)\nIH₁₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q₁✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p✝ q₁✝ q₂✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (TM2.Stmt.branch p✝ q₁✝ q₂✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "cases' IH₂ ss'.2 fun x hx ↦ sub x <| Finset.mem_union_right _ hx with IH₂₁ IH₂₂"
},
{
"state_after": "case refine'_3.intro.intro\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\np✝ : σ → Bool\nq₁✝ q₂✝ : Stmt₂\nIH₁ :\n TM2.SupportsStmt S q₁✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₁✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₁✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₁✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nIH₂ :\n TM2.SupportsStmt S q₂✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₂✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₂✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₂✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p✝ q₁✝ q₂✝)\nsub : ∀ (x : Λ'), x ∈ trStmts₁ q₁✝ ∪ trStmts₁ q₂✝ → x ∈ trSupp M S\nIH₁₁ : TM1.SupportsStmt (trSupp M S) (trNormal q₁✝)\nIH₁₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q₁✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nIH₂₁ : TM1.SupportsStmt (trSupp M S) (trNormal q₂✝)\nIH₂₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q₂✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nl : Λ'\nh : l ∈ trStmts₁ (TM2.Stmt.branch p✝ q₁✝ q₂✝)\n⊢ TM1.SupportsStmt (trSupp M S) (tr M l)",
"state_before": "case refine'_3.intro.intro\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\np✝ : σ → Bool\nq₁✝ q₂✝ : Stmt₂\nIH₁ :\n TM2.SupportsStmt S q₁✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₁✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₁✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₁✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nIH₂ :\n TM2.SupportsStmt S q₂✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₂✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₂✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₂✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p✝ q₁✝ q₂✝)\nsub : ∀ (x : Λ'), x ∈ trStmts₁ q₁✝ ∪ trStmts₁ q₂✝ → x ∈ trSupp M S\nIH₁₁ : TM1.SupportsStmt (trSupp M S) (trNormal q₁✝)\nIH₁₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q₁✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nIH₂₁ : TM1.SupportsStmt (trSupp M S) (trNormal q₂✝)\nIH₂₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q₂✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p✝ q₁✝ q₂✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (TM2.Stmt.branch p✝ q₁✝ q₂✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "refine' ⟨⟨IH₁₁, IH₂₁⟩, fun l h ↦ _⟩"
},
{
"state_after": "case refine'_3.intro.intro\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\np✝ : σ → Bool\nq₁✝ q₂✝ : Stmt₂\nIH₁ :\n TM2.SupportsStmt S q₁✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₁✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₁✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₁✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nIH₂ :\n TM2.SupportsStmt S q₂✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₂✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₂✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₂✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p✝ q₁✝ q₂✝)\nsub : ∀ (x : Λ'), x ∈ trStmts₁ q₁✝ ∪ trStmts₁ q₂✝ → x ∈ trSupp M S\nIH₁₁ : TM1.SupportsStmt (trSupp M S) (trNormal q₁✝)\nIH₁₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q₁✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nIH₂₁ : TM1.SupportsStmt (trSupp M S) (trNormal q₂✝)\nIH₂₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q₂✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nl : Λ'\nh : l ∈ trStmts₁ q₁✝ ∪ trStmts₁ q₂✝\n⊢ TM1.SupportsStmt (trSupp M S) (tr M l)",
"state_before": "case refine'_3.intro.intro\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\np✝ : σ → Bool\nq₁✝ q₂✝ : Stmt₂\nIH₁ :\n TM2.SupportsStmt S q₁✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₁✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₁✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₁✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nIH₂ :\n TM2.SupportsStmt S q₂✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₂✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₂✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₂✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p✝ q₁✝ q₂✝)\nsub : ∀ (x : Λ'), x ∈ trStmts₁ q₁✝ ∪ trStmts₁ q₂✝ → x ∈ trSupp M S\nIH₁₁ : TM1.SupportsStmt (trSupp M S) (trNormal q₁✝)\nIH₁₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q₁✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nIH₂₁ : TM1.SupportsStmt (trSupp M S) (trNormal q₂✝)\nIH₂₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q₂✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nl : Λ'\nh : l ∈ trStmts₁ (TM2.Stmt.branch p✝ q₁✝ q₂✝)\n⊢ TM1.SupportsStmt (trSupp M S) (tr M l)",
"tactic": "rw [trStmts₁] at h"
},
{
"state_after": "no goals",
"state_before": "case refine'_3.intro.intro\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\np✝ : σ → Bool\nq₁✝ q₂✝ : Stmt₂\nIH₁ :\n TM2.SupportsStmt S q₁✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₁✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₁✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₁✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nIH₂ :\n TM2.SupportsStmt S q₂✝ →\n (∀ (x : Λ'), x ∈ trStmts₁ q₂✝ → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal q₂✝) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ q₂✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p✝ q₁✝ q₂✝)\nsub : ∀ (x : Λ'), x ∈ trStmts₁ q₁✝ ∪ trStmts₁ q₂✝ → x ∈ trSupp M S\nIH₁₁ : TM1.SupportsStmt (trSupp M S) (trNormal q₁✝)\nIH₁₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q₁✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nIH₂₁ : TM1.SupportsStmt (trSupp M S) (trNormal q₂✝)\nIH₂₂ : ∀ (l' : Λ'), l' ∈ trStmts₁ q₂✝ → TM1.SupportsStmt (trSupp M S) (tr M l')\nl : Λ'\nh : l ∈ trStmts₁ q₁✝ ∪ trStmts₁ q₂✝\n⊢ TM1.SupportsStmt (trSupp M S) (tr M l)",
"tactic": "rcases Finset.mem_union.1 h with (h | h) <;> [exact IH₁₂ _ h; exact IH₂₂ _ h]"
},
{
"state_after": "case refine'_4\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nl✝ : σ → Λ\nss' : TM2.SupportsStmt S (TM2.Stmt.goto l✝)\nx✝ : ∀ (x : Λ'), x ∈ trStmts₁ (TM2.Stmt.goto l✝) → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (TM2.Stmt.goto l✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"state_before": "case refine'_4\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\n⊢ ∀ (l : σ → Λ),\n TM2.SupportsStmt S (TM2.Stmt.goto l) →\n (∀ (x : Λ'), x ∈ trStmts₁ (TM2.Stmt.goto l) → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (TM2.Stmt.goto l) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "intro _ ss' _"
},
{
"state_after": "case refine'_4\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nl✝ : σ → Λ\nss' : TM2.SupportsStmt S (TM2.Stmt.goto l✝)\nx✝ : ∀ (x : Λ'), x ∈ trStmts₁ (TM2.Stmt.goto l✝) → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l✝)) ∧\n ∀ (l' : Λ'), False → TM1.SupportsStmt (trSupp M S) (tr M l')",
"state_before": "case refine'_4\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nl✝ : σ → Λ\nss' : TM2.SupportsStmt S (TM2.Stmt.goto l✝)\nx✝ : ∀ (x : Λ'), x ∈ trStmts₁ (TM2.Stmt.goto l✝) → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l✝)) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ (TM2.Stmt.goto l✝) → TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "simp only [trStmts₁, Finset.not_mem_empty]"
},
{
"state_after": "case refine'_4\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nl✝ : σ → Λ\nss' : TM2.SupportsStmt S (TM2.Stmt.goto l✝)\nx✝ : ∀ (x : Λ'), x ∈ trStmts₁ (TM2.Stmt.goto l✝) → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l✝))",
"state_before": "case refine'_4\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nl✝ : σ → Λ\nss' : TM2.SupportsStmt S (TM2.Stmt.goto l✝)\nx✝ : ∀ (x : Λ'), x ∈ trStmts₁ (TM2.Stmt.goto l✝) → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l✝)) ∧\n ∀ (l' : Λ'), False → TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "refine' ⟨_, fun _ ↦ False.elim⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_4\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nl✝ : σ → Λ\nss' : TM2.SupportsStmt S (TM2.Stmt.goto l✝)\nx✝ : ∀ (x : Λ'), x ∈ trStmts₁ (TM2.Stmt.goto l✝) → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l✝))",
"tactic": "exact fun _ v ↦ Finset.mem_biUnion.2 ⟨_, ss' v, Finset.mem_insert_self _ _⟩"
},
{
"state_after": "case refine'_5\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nx✝¹ : TM2.SupportsStmt S TM2.Stmt.halt\nx✝ : ∀ (x : Λ'), x ∈ trStmts₁ TM2.Stmt.halt → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal TM2.Stmt.halt) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ TM2.Stmt.halt → TM1.SupportsStmt (trSupp M S) (tr M l')",
"state_before": "case refine'_5\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\n⊢ TM2.SupportsStmt S TM2.Stmt.halt →\n (∀ (x : Λ'), x ∈ trStmts₁ TM2.Stmt.halt → x ∈ trSupp M S) →\n TM1.SupportsStmt (trSupp M S) (trNormal TM2.Stmt.halt) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ TM2.Stmt.halt → TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "intro _ _"
},
{
"state_after": "case refine'_5\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nx✝¹ : TM2.SupportsStmt S TM2.Stmt.halt\nx✝ : ∀ (x : Λ'), x ∈ trStmts₁ TM2.Stmt.halt → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal TM2.Stmt.halt) ∧ ∀ (l' : Λ'), False → TM1.SupportsStmt (trSupp M S) (tr M l')",
"state_before": "case refine'_5\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nx✝¹ : TM2.SupportsStmt S TM2.Stmt.halt\nx✝ : ∀ (x : Λ'), x ∈ trStmts₁ TM2.Stmt.halt → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal TM2.Stmt.halt) ∧\n ∀ (l' : Λ'), l' ∈ trStmts₁ TM2.Stmt.halt → TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "simp only [trStmts₁, Finset.not_mem_empty]"
},
{
"state_after": "no goals",
"state_before": "case refine'_5\nK : Type u_2\ninst✝² : DecidableEq K\nΓ : K → Type u_3\nΛ : Type u_1\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nS : Finset Λ\nss : TM2.Supports M S\nx✝¹ : TM2.SupportsStmt S TM2.Stmt.halt\nx✝ : ∀ (x : Λ'), x ∈ trStmts₁ TM2.Stmt.halt → x ∈ trSupp M S\n⊢ TM1.SupportsStmt (trSupp M S) (trNormal TM2.Stmt.halt) ∧ ∀ (l' : Λ'), False → TM1.SupportsStmt (trSupp M S) (tr M l')",
"tactic": "exact ⟨trivial, fun _ ↦ False.elim⟩"
}
] |
[
2817,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2772,
1
] |
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