module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Order.Interval.Finset.Basic | {
"line": 618,
"column": 68
} | {
"line": 618,
"column": 90
} | {
"line": 620,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : LocallyFiniteOrder α\na b : α\ninst✝ : DecidableEq α\nh : a < b\n⊢ Ioc a b \\ Ioo a b = {b}",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Set.Ioc",
"Finset.coe_singleton",
"Set.Ioc_sdiff_Ioo_same",
"congrAr... | [] | by simp [← coe_inj, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.WellFoundedSet | {
"line": 298,
"column": 6
} | {
"line": 298,
"column": 42
} | {
"line": 298,
"column": 42
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nr : α → α → Prop\nr' : β → β → Prop\nf : α → β\ns : Set α\nhs : s.PartiallyWellOrderedOn r\nhf : ∀ a₁ ∈ s, ∀ a₂ ∈ s, r a₁ a₂ → r' (f a₁) (f a₂)\n⊢ (f '' s).PartiallyWellOrderedOn r'",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"α : Type u_2\nβ : Type u_3\nr : α → α → Prop\nr' : β → β → Prop\nf : α → β\ns : Set α\nhs : ∀ (f : ℕ → α), (∀ (n : ℕ), f n ∈ s) → ∃ m n, m < n ∧ r (f m) (f n)\nhf : ∀ a₁ ∈ s, ∀ a₂ ∈ s, r a₁ a₂ → r' (f a₁) (f a₂)\n⊢ ∀ (f_1 : ℕ → β), (∀ (n : ℕ), f_1 n ∈ f '' s) → ∃ m n, m < n ∧ r' (f_1 m) (f_1 n)"
] | partiallyWellOrderedOn_iff_exists_lt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.WellFoundedSet | {
"line": 344,
"column": 6
} | {
"line": 344,
"column": 42
} | {
"line": 344,
"column": 42
} | [
{
"pp": "α : Type u_2\nr : α → α → Prop\ns : Set α\ninst✝¹ : Std.Refl r\ninst✝ : Std.Symm r\n⊢ (∀ t ⊆ s, IsAntichain r t → t.Finite) → s.PartiallyWellOrderedOn r",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.partiallyWellOrderedOn_iff_exists_lt",
"Set.Par... | [
"α : Type u_2\nr : α → α → Prop\ns : Set α\ninst✝¹ : Std.Refl r\ninst✝ : Std.Symm r\n⊢ (∀ t ⊆ s, IsAntichain r t → t.Finite) → ∀ (f : ℕ → α), (∀ (n : ℕ), f n ∈ s) → ∃ m n, m < n ∧ r (f m) (f n)"
] | partiallyWellOrderedOn_iff_exists_lt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.WellFoundedSet | {
"line": 380,
"column": 6
} | {
"line": 380,
"column": 42
} | {
"line": 380,
"column": 42
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nr : α → α → Prop\nr' : β → β → Prop\ns : Set α\ninst✝ : IsPreorder α r\nt : Set β\nhs : s.PartiallyWellOrderedOn r\nht : t.PartiallyWellOrderedOn r'\n⊢ (s ×ˢ t).PartiallyWellOrderedOn fun x y ↦ r x.1 y.1 ∧ r' x.2 y.2",
"ppTerm": "?m.16",
"assigned": true,
"usedCo... | [
"α : Type u_2\nβ : Type u_3\nr : α → α → Prop\nr' : β → β → Prop\ns : Set α\ninst✝ : IsPreorder α r\nt : Set β\nhs : s.PartiallyWellOrderedOn r\nht : t.PartiallyWellOrderedOn r'\n⊢ ∀ (f : ℕ → α × β), (∀ (n : ℕ), f n ∈ s ×ˢ t) → ∃ m n, m < n ∧ r (f m).1 (f n).1 ∧ r' (f m).2 (f n).2"
] | partiallyWellOrderedOn_iff_exists_lt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.Defs | {
"line": 452,
"column": 4
} | {
"line": 452,
"column": 96
} | {
"line": 453,
"column": 2
} | [
{
"pp": "G : Type u_1\nα : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G α\nH : Subgroup G\nx : Quotient G α\na b : ↑x.orbit\nc : α\nh : ⟦a⟧ = ⟦b⟧\nhb : b ∈ MulAction.orbit ↥H ⟦a⟧.out\n⊢ MulAction.orbit (↥H) a = MulAction.orbit ↥H ⟦a⟧.out",
"ppTerm": "?m.296",
"assigned": true,
"usedConstants": [
... | [] | rw [orbit_eq_iff, ← orbitRel_apply, ← Quotient.eq'', Quotient.out_eq', @Quotient.mk''_eq_mk] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.WellFoundedSet | {
"line": 502,
"column": 34
} | {
"line": 509,
"column": 15
} | {
"line": 511,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : Preorder α\ns : Set α\na : α\nhs : s.IsPWO\nha : a ∈ s\n⊢ ∃ b ≤ a, Minimal (fun x ↦ x ∈ s) b",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"WellQuasiOrdered.wellFounded",
"WellFounded.min_mem",
"le_rfl",
"Classical.byContradiction",
... | [] | by
let t : Set s := {x | x ≤ a}
let h : t.Nonempty := ⟨⟨a, ha⟩, le_rfl⟩
refine ⟨hs.wellFounded.min t h, hs.wellFounded.min_mem t h,
(hs.wellFounded.min t h).2, fun y hy hle => ?_⟩
by_contra hnle
exact hs.wellFounded.not_lt_min t (x := ⟨y, hy⟩) (hle.trans (hs.wellFounded.min_mem t h))
⟨hle, hnle⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.WellFoundedSet | {
"line": 757,
"column": 6
} | {
"line": 757,
"column": 42
} | {
"line": 757,
"column": 42
} | [
{
"pp": "α : Type u_2\nr : α → α → Prop\ns : Set α\nhs : s.PartiallyWellOrderedOn r\nf : ℕ → α\nhf : ∀ (x : ℕ), ∃ y ∈ f ⁻¹' s, x < y\nφ : ℕ → ℕ\nhφm : StrictMono φ\nhφs : ∀ (n : ℕ), φ n ∈ f ⁻¹' s\n⊢ ∃ m n, m < n ∧ r (f m) (f n)",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Set.par... | [
"α : Type u_2\nr : α → α → Prop\ns : Set α\nhs : ∀ (f : ℕ → α), (∀ (n : ℕ), f n ∈ s) → ∃ m n, m < n ∧ r (f m) (f n)\nf : ℕ → α\nhf : ∀ (x : ℕ), ∃ y ∈ f ⁻¹' s, x < y\nφ : ℕ → ℕ\nhφm : StrictMono φ\nhφs : ∀ (n : ℕ), φ n ∈ f ⁻¹' s\n⊢ ∃ m n, m < n ∧ r (f m) (f n)"
] | partiallyWellOrderedOn_iff_exists_lt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.WellFoundedSet | {
"line": 779,
"column": 6
} | {
"line": 779,
"column": 42
} | {
"line": 779,
"column": 42
} | [
{
"pp": "α : Type u_2\nr : α → α → Prop\ns : Set α\n⊢ s.PartiallyWellOrderedOn r ↔ ∀ (f : ℕ → α), ¬IsBadSeq r s f",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.partiallyWellOrderedOn_iff_exists_lt",
"Set.PartiallyWellOrderedOn",
"congrArg",
"Me... | [
"α : Type u_2\nr : α → α → Prop\ns : Set α\n⊢ (∀ (f : ℕ → α), (∀ (n : ℕ), f n ∈ s) → ∃ m n, m < n ∧ r (f m) (f n)) ↔ ∀ (f : ℕ → α), ¬IsBadSeq r s f"
] | partiallyWellOrderedOn_iff_exists_lt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Submonoid.Center | {
"line": 188,
"column": 88
} | {
"line": 188,
"column": 93
} | {
"line": 190,
"column": 0
} | [
{
"pp": "M : Type u_2\ninst✝ : Mul M\nx : M\n⊢ ((∀ (a : M), Commute (op x) (op a)) ∧\n (∀ (a a_1 : M), a_1 * a * x = a_1 * (a * x)) ∧ ∀ (a a_1 : M), x * (a_1 * a) = x * a_1 * a) ↔\n (∀ (a : M), Commute x a) ∧ (∀ (b c : M), x * (b * c) = x * b * c) ∧ ∀ (a b : M), a * b * x = a * (b * x)",
"ppTerm": "... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Group.Center | {
"line": 266,
"column": 2
} | {
"line": 267,
"column": 37
} | {
"line": 269,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝¹ : Monoid M\na : M\ninst✝ : Invertible a\nha : a ∈ center M\n⊢ ⅟a ∈ center M",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Semigroup.toMul",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
... | [] | rw [Semigroup.mem_center_iff] at *
exact (Commute.invOf_right <| ha ·) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Center | {
"line": 266,
"column": 2
} | {
"line": 267,
"column": 37
} | {
"line": 269,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝¹ : Monoid M\na : M\ninst✝ : Invertible a\nha : a ∈ center M\n⊢ ⅟a ∈ center M",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Semigroup.toMul",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
... | [] | rw [Semigroup.mem_center_iff] at *
exact (Commute.invOf_right <| ha ·) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Subgroup.Centralizer | {
"line": 54,
"column": 2
} | {
"line": 55,
"column": 15
} | {
"line": 57,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\ng k : G\n⊢ k ∈ centralizer {g} ↔ k * g = g * k",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"_private.Mathlib.GroupTheory.Subgroup.Centralizer.0.Subgroup.mem_centralizer_singleton_iff._simp_1_1",
"Mon... | [] | simp only [mem_centralizer_iff, Set.mem_singleton_iff, forall_eq]
exact eq_comm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Subgroup.Centralizer | {
"line": 54,
"column": 2
} | {
"line": 55,
"column": 15
} | {
"line": 57,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\ng k : G\n⊢ k ∈ centralizer {g} ↔ k * g = g * k",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"_private.Mathlib.GroupTheory.Subgroup.Centralizer.0.Subgroup.mem_centralizer_singleton_iff._simp_1_1",
"Mon... | [] | simp only [mem_centralizer_iff, Set.mem_singleton_iff, forall_eq]
exact eq_comm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.WellFoundedSet | {
"line": 875,
"column": 13
} | {
"line": 875,
"column": 49
} | {
"line": 875,
"column": 49
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : PartialOrder α\ninst✝ : Preorder β\ns : Set (Lex (α × β))\nhα : ((fun x ↦ (ofLex x).1) '' s).IsPWO\nhβ : ∀ (a : α), {y | toLex (a, y) ∈ s}.IsPWO\n⊢ s.PartiallyWellOrderedOn fun x1 x2 ↦ x1 ≤ x2",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_2\nβ : Type u_3\ninst✝¹ : PartialOrder α\ninst✝ : Preorder β\ns : Set (Lex (α × β))\nhα : ((fun x ↦ (ofLex x).1) '' s).IsPWO\nhβ : ∀ (a : α), {y | toLex (a, y) ∈ s}.IsPWO\n⊢ ∀ (f : ℕ → Lex (α × β)), (∀ (n : ℕ), f n ∈ s) → ∃ m n, m < n ∧ f m ≤ f n"
] | partiallyWellOrderedOn_iff_exists_lt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Subgroup.Pointwise | {
"line": 224,
"column": 18
} | {
"line": 224,
"column": 36
} | {
"line": 226,
"column": 0
} | [
{
"pp": "case zero\nG : Type u_2\ninst✝ : Group G\ns : Set G\nH : Subgroup G\nhs : s ⊆ ↑H\n⊢ s ^ 0 ⊆ ↑H",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"SetLike.mem_coe._simp_1",
"MulOne.toOne",
"Subgroup.instSubgroupClass",
"InvOneClass.toOne",
"DivInvOneMono... | [] | simp [pow_succ, *] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Group.Subgroup.Pointwise | {
"line": 224,
"column": 18
} | {
"line": 224,
"column": 36
} | {
"line": 226,
"column": 0
} | [
{
"pp": "case succ\nG : Type u_2\ninst✝ : Group G\ns : Set G\nH : Subgroup G\nhs : s ⊆ ↑H\nn✝ : ℕ\na✝ : s ^ n✝ ⊆ ↑H\n⊢ s ^ (n✝ + 1) ⊆ ↑H",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulO... | [] | simp [pow_succ, *] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Group.Subgroup.Pointwise | {
"line": 337,
"column": 61
} | {
"line": 337,
"column": 66
} | {
"line": 337,
"column": 67
} | [
{
"pp": "G : Type u_2\ninst✝ : Group G\nH K : Subgroup G\ng : G\nhg : map (↑(MulAut.conj g)) H = H ∧ map (↑(MulAut.conj g)) K = K\n⊢ map (↑(MulAut.conj g)) H ⊔ map (↑(MulAut.conj g)) K = H ⊔ K",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulEquiv.instEquivLike",
... | [
"G : Type u_2\ninst✝ : Group G\nH K : Subgroup G\ng : G\nhg : True ∧ map (↑(MulAut.conj g)) K = K\n⊢ H ⊔ map (↑(MulAut.conj g)) K = H ⊔ K"
] | hg.1, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Setoid.Basic | {
"line": 419,
"column": 2
} | {
"line": 419,
"column": 20
} | {
"line": 419,
"column": 21
} | [
{
"pp": "case symm\nα : Type u_1\nβ : Type u_2\nr : Setoid α\nf : α → β\nhf : ker f ≤ r\ni✝¹ i✝ x✝ y✝ : β\na✝ : Relation.EqvGen (Relation.Map (⇑r) f f) x✝ y✝\nih : (Relation.Map (⇑r) f f ⊔ fun x1 x2 ↦ x1 = x2) x✝ y✝\n⊢ (Relation.Map (⇑r) f f ⊔ fun x1 x2 ↦ x1 = x2) y✝ x✝",
"ppTerm": "?symm",
"assigned": ... | [] | | symm _ _ _ ih => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.GroupTheory.Congruence.Defs | {
"line": 726,
"column": 2
} | {
"line": 726,
"column": 29
} | {
"line": 727,
"column": 2
} | [
{
"pp": "case refine_1\nM : Type u_1\nN : Type u_2\nP : Type u_3\nα : Type u_4\ninst✝ : Monoid M\nc : Con M\nu : c.Quotientˣ\nf : (x y : M) → c (x * y) 1 → c (y * x) 1 → α\nHf :\n ∀ (x y : M) (hxy : c (x * y) 1) (hyx : c (y * x) 1) (x' y' : M) (hxy' : c (x' * y') 1) (hyx' : c (y' * x') 1),\n c x x' → c y y'... | [
"case refine_2\nM : Type u_1\nN : Type u_2\nP : Type u_3\nα : Type u_4\ninst✝ : Monoid M\nc : Con M\nu : c.Quotientˣ\nf : (x y : M) → c (x * y) 1 → c (y * x) 1 → α\nHf :\n ∀ (x y : M) (hxy : c (x * y) 1) (hyx : c (y * x) 1) (x' y' : M) (hxy' : c (x' * y') 1) (hyx' : c (y' * x') 1),\n c x x' → c y y' → f x y hxy... | · rw [c.eq.2 hx, c.eq.2 hy] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 47
} | {
"line": 145,
"column": 4
} | [
{
"pp": "case mp\nα : Type u\nL₁ L₂ : List (α × Bool)\na : α\nb : Bool\n⊢ Step ((a, b) :: L₁) L₂ → (∃ L, Step L₁ L ∧ L₂ = (a, b) :: L) ∨ L₁ = (a, !b) :: L₂",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Prod.mk",
"List.cons",
"List",
"Bool",
"Eq.refl",
"... | [
"case mp\nα : Type u\nL₁ L₂ : List (α × Bool)\na : α\nb : Bool\nL : List (α × Bool)\nhL : (a, b) :: L₁ = L\n⊢ Step L L₂ → (∃ L, Step L₁ L ∧ L₂ = (a, b) :: L) ∨ L₁ = (a, !b) :: L₂"
] | generalize hL : ((a, b) :: L₁ : List _) = L | Lean.Elab.Tactic.evalGeneralize | Lean.Parser.Tactic.generalize |
Mathlib.GroupTheory.Congruence.Defs | {
"line": 729,
"column": 4
} | {
"line": 729,
"column": 31
} | {
"line": 730,
"column": 4
} | [
{
"pp": "case refine_2.refine_1\nM : Type u_1\nN : Type u_2\nP : Type u_3\nα : Type u_4\ninst✝ : Monoid M\nc : Con M\nu : c.Quotientˣ\nf : (x y : M) → c (x * y) 1 → c (y * x) 1 → α\nHf :\n ∀ (x y : M) (hxy : c (x * y) 1) (hyx : c (y * x) 1) (x' y' : M) (hxy' : c (x' * y') 1) (hyx' : c (y' * x') 1),\n c x x'... | [
"case refine_2.refine_2\nM : Type u_1\nN : Type u_2\nP : Type u_3\nα : Type u_4\ninst✝ : Monoid M\nc : Con M\nu : c.Quotientˣ\nf : (x y : M) → c (x * y) 1 → c (y * x) 1 → α\nHf :\n ∀ (x y : M) (hxy : c (x * y) 1) (hyx : c (y * x) 1) (x' y' : M) (hxy' : c (x' * y') 1) (hyx' : c (y' * x') 1),\n c x x' → c y y' → ... | · rw [c.eq.2 hx, c.eq.2 hy] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 389,
"column": 14
} | {
"line": 389,
"column": 79
} | {
"line": 390,
"column": 2
} | [
{
"pp": "ι : Type u_5\nM : Type u_6\ns₁ s₂ : Finset ι\nf g : ι → M\ninst✝ : CommMonoid M\nh₁ : ∀ a ∈ s₁, a ∉ s₂ → f a = 1\nh₂ : ∀ a ∈ s₂, a ∉ s₁ → g a = 1\nh : ∀ a ∈ s₁, a ∈ s₂ → f a = g a\n| ∏ a ∈ s₁, f a",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"False",... | [
"ι : Type u_5\nM : Type u_6\ns₁ s₂ : Finset ι\nf g : ι → M\ninst✝ : CommMonoid M\nh₁ : ∀ a ∈ s₁, a ∉ s₂ → f a = 1\nh₂ : ∀ a ∈ s₂, a ∉ s₁ → g a = 1\nh : ∀ a ∈ s₁, a ∈ s₂ → f a = g a\n| ∏ a ∈ s₁ ∩ s₂, f a"
] | rw [← sdiff_union_inter s₁ s₂, prod_union_eq_right (by simp_all)] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 389,
"column": 14
} | {
"line": 389,
"column": 79
} | {
"line": 390,
"column": 2
} | [
{
"pp": "ι : Type u_5\nM : Type u_6\ns₁ s₂ : Finset ι\nf g : ι → M\ninst✝ : CommMonoid M\nh₁ : ∀ a ∈ s₁, a ∉ s₂ → f a = 1\nh₂ : ∀ a ∈ s₂, a ∉ s₁ → g a = 1\nh : ∀ a ∈ s₁, a ∈ s₂ → f a = g a\n| ∏ a ∈ s₁, f a",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"False",... | [
"ι : Type u_5\nM : Type u_6\ns₁ s₂ : Finset ι\nf g : ι → M\ninst✝ : CommMonoid M\nh₁ : ∀ a ∈ s₁, a ∉ s₂ → f a = 1\nh₂ : ∀ a ∈ s₂, a ∉ s₁ → g a = 1\nh : ∀ a ∈ s₁, a ∈ s₂ → f a = g a\n| ∏ a ∈ s₁ ∩ s₂, f a"
] | rw [← sdiff_union_inter s₁ s₂, prod_union_eq_right (by simp_all)] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 389,
"column": 14
} | {
"line": 389,
"column": 79
} | {
"line": 390,
"column": 2
} | [
{
"pp": "ι : Type u_5\nM : Type u_6\ns₁ s₂ : Finset ι\nf g : ι → M\ninst✝ : CommMonoid M\nh₁ : ∀ a ∈ s₁, a ∉ s₂ → f a = 1\nh₂ : ∀ a ∈ s₂, a ∉ s₁ → g a = 1\nh : ∀ a ∈ s₁, a ∈ s₂ → f a = g a\n| ∏ a ∈ s₁, f a",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"False",... | [
"ι : Type u_5\nM : Type u_6\ns₁ s₂ : Finset ι\nf g : ι → M\ninst✝ : CommMonoid M\nh₁ : ∀ a ∈ s₁, a ∉ s₂ → f a = 1\nh₂ : ∀ a ∈ s₂, a ∉ s₁ → g a = 1\nh : ∀ a ∈ s₁, a ∈ s₂ → f a = g a\n| ∏ a ∈ s₁ ∩ s₂, f a"
] | rw [← sdiff_union_inter s₁ s₂, prod_union_eq_right (by simp_all)] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 212,
"column": 6
} | {
"line": 212,
"column": 47
} | {
"line": 213,
"column": 6
} | [
{
"pp": "α : Type u\nL₁ L₂ : List (α × Bool)\np : α × Bool\n⊢ Red (p :: L₁) (p :: L₂) → Red L₁ L₂",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"List.cons",
"List",
"Bool",
"Eq.refl",
"Prod"
],
"usedFVars": [
"α",
"p",
"L₁"
],
... | [
"α : Type u\nL₁ L₂ : List (α × Bool)\np : α × Bool\nLL₁ : List (α × Bool)\neq₁ : p :: L₁ = LL₁\n⊢ Red LL₁ (p :: L₂) → Red L₁ L₂"
] | generalize eq₁ : (p :: L₁ : List _) = LL₁ | Lean.Elab.Tactic.evalGeneralize | Lean.Parser.Tactic.generalize |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 408,
"column": 8
} | {
"line": 408,
"column": 14
} | {
"line": 408,
"column": 15
} | [
{
"pp": "case neg\nι : Type u_1\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset ι\nf : ι → M\na b : ι\nhn : a ≠ b\nh₀ : ∀ c ∈ s, c ≠ a ∧ c ≠ b → f c = 1\nha : a ∉ s → f a = 1\nhb : b ∉ s → f b = 1\nthis : DecidableEq ι\nh₁ : a ∈ s\nh₂ : b ∉ s\n⊢ ∏ x ∈ s, f x = f a * f b",
"ppTerm": "?neg✝",
"assigned": ... | [
"case neg\nι : Type u_1\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset ι\nf : ι → M\na b : ι\nhn : a ≠ b\nh₀ : ∀ c ∈ s, c ≠ a ∧ c ≠ b → f c = 1\nha : a ∉ s → f a = 1\nhb : b ∉ s → f b = 1\nthis : DecidableEq ι\nh₁ : a ∈ s\nh₂ : b ∉ s\n⊢ ∏ x ∈ s, f x = f a * 1"
] | hb h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 414,
"column": 15
} | {
"line": 414,
"column": 21
} | {
"line": 414,
"column": 22
} | [
{
"pp": "case neg\nι : Type u_1\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset ι\nf : ι → M\na b : ι\nhn : a ≠ b\nh₀ : ∀ c ∈ s, c ≠ a ∧ c ≠ b → f c = 1\nha : a ∉ s → f a = 1\nhb : b ∉ s → f b = 1\nthis : DecidableEq ι\nh₁ : a ∉ s\nh₂ : b ∉ s\n⊢ ∏ x ∈ s, f x = 1 * f b",
"ppTerm": "?neg✝",
"assigned": tr... | [
"case neg\nι : Type u_1\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset ι\nf : ι → M\na b : ι\nhn : a ≠ b\nh₀ : ∀ c ∈ s, c ≠ a ∧ c ≠ b → f c = 1\nha : a ∉ s → f a = 1\nhb : b ∉ s → f b = 1\nthis : DecidableEq ι\nh₁ : a ∉ s\nh₂ : b ∉ s\n⊢ ∏ x ∈ s, f x = 1 * 1"
] | hb h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 305,
"column": 2
} | {
"line": 305,
"column": 24
} | {
"line": 306,
"column": 2
} | [
{
"pp": "case nil\nα : Type u\nL₁ L₂ : List (α × Bool)\nx1 : α\nb1 : Bool\nx2 : α\nb2 : Bool\nH1 : (x1, b1) ≠ (x2, b2)\nH2 : Red ((x1, b1) :: L₁) ((x2, b2) :: L₂)\nthis : Red ((x1, b1) :: L₁) ([(x2, b2)] ++ L₂)\nL₄ : List (α × Bool)\neq : (x1, b1) :: L₁ = [] ++ L₄\nh₁ : Red [] [(x2, b2)]\nh₂ : Red L₄ L₂\n⊢ Red ... | [
"case cons\nα : Type u\nL₁ L₂ : List (α × Bool)\nx1 : α\nb1 : Bool\nx2 : α\nb2 : Bool\nH1 : (x1, b1) ≠ (x2, b2)\nH2 : Red ((x1, b1) :: L₁) ((x2, b2) :: L₂)\nthis : Red ((x1, b1) :: L₁) ([(x2, b2)] ++ L₂)\np : α × Bool\nL₃ L₄ : List (α × Bool)\neq : (x1, b1) :: L₁ = p :: L₃ ++ L₄\nh₁ : Red (p :: L₃) [(x2, b2)]\nh₂ :... | · simp [nil_iff] at h₁ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 603,
"column": 18
} | {
"line": 603,
"column": 45
} | {
"line": 604,
"column": 2
} | [
{
"pp": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\n⊢ ∀ (a b c : FreeGroup α), a * b * c = a * (b * c)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"FreeGroup.Red.Step",
"HMul.hMul",
"List.append_assoc",
"congrArg",
"Quot.ind",
"instHAppendOfAppend"... | [] | rintro ⟨L₁⟩ ⟨L₂⟩ ⟨L₃⟩; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 603,
"column": 18
} | {
"line": 603,
"column": 45
} | {
"line": 604,
"column": 2
} | [
{
"pp": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\n⊢ ∀ (a b c : FreeGroup α), a * b * c = a * (b * c)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"FreeGroup.Red.Step",
"HMul.hMul",
"List.append_assoc",
"congrArg",
"Quot.ind",
"instHAppendOfAppend"... | [] | rintro ⟨L₁⟩ ⟨L₂⟩ ⟨L₃⟩; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Finiteness | {
"line": 550,
"column": 4
} | {
"line": 550,
"column": 36
} | {
"line": 551,
"column": 4
} | [
{
"pp": "M✝ : Type u_1\nN : Type u_2\ninst✝⁵ : Monoid M✝\nG : Type u_3\nH : Type u_4\ninst✝⁴ : Group G\ninst✝³ : AddGroup H\nι : Type u_5\ninst✝² : Finite ι\nM : ι → Type u_6\ninst✝¹ : (i : ι) → Monoid (M i)\ninst✝ : ∀ (i : ι), Monoid.FG (M i)\n⊢ ⊤.FG",
"ppTerm": "?m.8",
"assigned": true,
"usedConst... | [
"M✝ : Type u_1\nN : Type u_2\ninst✝⁵ : Monoid M✝\nG : Type u_3\nH : Type u_4\ninst✝⁴ : Group G\ninst✝³ : AddGroup H\nι : Type u_5\ninst✝² : Finite ι\nM : ι → Type u_6\ninst✝¹ : (i : ι) → Monoid (M i)\ninst✝ : ∀ (i : ι), Monoid.FG (M i)\n⊢ (Submonoid.pi Set.univ fun i ↦ ⊤).FG"
] | rw [← Submonoid.pi_top Set.univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.QuotientGroup.Defs | {
"line": 181,
"column": 75
} | {
"line": 181,
"column": 80
} | {
"line": 183,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nN : Subgroup G\nnN : N.Normal\n⊢ Subgroup.map (mk' N) N = ⊥",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Subgroup.instSubgroupClass",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.GroupTheory.QuotientGroup.Defs | {
"line": 181,
"column": 75
} | {
"line": 181,
"column": 80
} | {
"line": 183,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nN : Subgroup G\nnN : N.Normal\n⊢ Subgroup.map (mk' N) N = ⊥",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Subgroup.instSubgroupClass",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.QuotientGroup.Defs | {
"line": 181,
"column": 75
} | {
"line": 181,
"column": 80
} | {
"line": 183,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nN : Subgroup G\nnN : N.Normal\n⊢ Subgroup.map (mk' N) N = ⊥",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Subgroup.instSubgroupClass",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Hom.Monoid | {
"line": 191,
"column": 31
} | {
"line": 191,
"column": 50
} | {
"line": 193,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝⁶ : Group α\ninst✝⁵ : Monoid β\nF : Type u_6\ninst✝⁴ : FunLike F α β\ninst✝³ : MonoidHomClass F α β\ninst✝² : LE β\ninst✝¹ : MulRightMono β\ninst✝ : MulLeftMono β\nf✝ g✝ : F\nx✝ : α\nf g : F\nx : α\nhfg : f x⁻¹ ≤ g x⁻¹\n⊢ f x * (g x⁻¹ * g x) = f x",
"ppTerm": "?m.94... | [] | by simp [← map_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Ring.Equiv | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 35
} | {
"line": 138,
"column": 2
} | [
{
"pp": "case mk.mk\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nR : Type u_4\nS : Type u_5\nS' : Type u_6\ninst✝⁵ : Mul R\ninst✝⁴ : Mul S\ninst✝³ : Add R\ninst✝² : Add S\ninst✝¹ : Mul S'\ninst✝ : Add S'\ntoEquiv✝¹ : R ≃ S\nmap_mul'✝¹ : ∀ (x y : R), toEquiv✝¹.toFun (x * y) = toEquiv✝¹.toFun x * toEquiv✝¹.toFun y\... | [] | apply Equiv.coe_fn_injective h₁ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Order.Ring.WithTop | {
"line": 60,
"column": 26
} | {
"line": 60,
"column": 31
} | {
"line": 62,
"column": 0
} | [
{
"pp": "case top.top\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\n⊢ ⊤ * ⊤ = if ⊤ = 0 ∨ ⊤ = 0 then 0 else map₂ (fun x1 x2 ↦ x1 * x2) ⊤ ⊤",
"ppTerm": "?top.top",
"assigned": true,
"usedConstants": [
"False",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.WithTop | {
"line": 60,
"column": 26
} | {
"line": 60,
"column": 31
} | {
"line": 62,
"column": 0
} | [
{
"pp": "case top.coe\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\na✝ : α\n⊢ ⊤ * ↑a✝ = if ⊤ = 0 ∨ ↑a✝ = 0 then 0 else map₂ (fun x1 x2 ↦ x1 * x2) ⊤ ↑a✝",
"ppTerm": "?top.coe",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"HMul.hMul",
"WithTop.map₂_... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.WithTop | {
"line": 60,
"column": 26
} | {
"line": 60,
"column": 31
} | {
"line": 62,
"column": 0
} | [
{
"pp": "case coe.top\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\na✝ : α\n⊢ ↑a✝ * ⊤ = if ↑a✝ = 0 ∨ ⊤ = 0 then 0 else map₂ (fun x1 x2 ↦ x1 * x2) ↑a✝ ⊤",
"ppTerm": "?coe.top",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"HMul.hMul",
"MulZeroClass.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.WithTop | {
"line": 60,
"column": 26
} | {
"line": 60,
"column": 31
} | {
"line": 62,
"column": 0
} | [
{
"pp": "case coe.coe\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\na✝¹ a✝ : α\n⊢ ↑a✝¹ * ↑a✝ = if ↑a✝¹ = 0 ∨ ↑a✝ = 0 then 0 else map₂ (fun x1 x2 ↦ x1 * x2) ↑a✝¹ ↑a✝",
"ppTerm": "?coe.coe",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"WithTop.map₂_co... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.WithTop | {
"line": 62,
"column": 85
} | {
"line": 62,
"column": 90
} | {
"line": 64,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\na b : WithTop α\n⊢ (if a = 0 ∨ b = 0 then 0 else map₂ (fun x1 x2 ↦ x1 * x2) a b) = ⊤ ↔ a ≠ 0 ∧ b = ⊤ ∨ a = ⊤ ∧ b ≠ 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Decidable.casesO... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.GroupTheory.MonoidLocalization.Maps | {
"line": 371,
"column": 4
} | {
"line": 371,
"column": 23
} | {
"line": 372,
"column": 2
} | [
{
"pp": "case inl\nM : Type u_1\ninst✝³ : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝² : CommMonoid N\nP : Type u_3\ninst✝¹ : CommMonoid P\nf : S.LocalizationMap N\ng : M →* P\nT : Submonoid P\nhy : ∀ (y : ↥S), g ↑y ∈ T\nQ : Type u_4\ninst✝ : CommMonoid Q\nk : T.LocalizationMap Q\nhg : Injective ⇑g\nz w ... | [] | exact ⟨⟨c, hc⟩, eq⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.Ring.WithTop | {
"line": 176,
"column": 50
} | {
"line": 176,
"column": 66
} | {
"line": 177,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : MonoidWithZero α\ninst✝¹ : NoZeroDivisors α\ninst✝ : Nontrivial α\nx : WithTop α\nn : ℕ\na : WithTop α\n⊢ (match a, 0 with\n | Option.some a, n => ↑(a ^ n)\n | none, 0 => 1\n | none, _n.succ => ⊤) =\n 1",
"ppTerm": "?m.71",
"assigned": ... | [] | cases a <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.GroupTheory.MonoidLocalization.Basic | {
"line": 870,
"column": 52
} | {
"line": 870,
"column": 70
} | {
"line": 870,
"column": 71
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\nS : Submonoid M\ninst✝ : CommMonoid N\nf : S.LocalizationMap N\nm : M\nhm : IsRegular m\nn₁ n₂ : N\nms₁ : M × ↥S\neq₁ : n₁ * f ↑ms₁.2 = f ms₁.1\nms₂ : M × ↥S\neq₂ : n₂ * f ↑ms₂.2 = f ms₂.1\neq : f m * (f ms₁.1 * f ↑ms₂.2) = f m * (n₂ * f ↑ms₁.2 * f ↑ms... | [
"M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\nS : Submonoid M\ninst✝ : CommMonoid N\nf : S.LocalizationMap N\nm : M\nhm : IsRegular m\nn₁ n₂ : N\nms₁ : M × ↥S\neq₁ : n₁ * f ↑ms₁.2 = f ms₁.1\nms₂ : M × ↥S\neq₂ : n₂ * f ↑ms₂.2 = f ms₂.1\neq : f m * (f ms₁.1 * f ↑ms₂.2) = f m * (n₂ * f ↑ms₂.2 * f ↑ms₁.2)\n⊢ f ms... | mul_right_comm n₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Ring.WithTop | {
"line": 267,
"column": 2
} | {
"line": 267,
"column": 33
} | {
"line": 268,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : DecidableEq α\ninst✝⁴ : CommSemiring α\ninst✝³ : PartialOrder α\ninst✝² : OrderBot α\ninst✝¹ : CanonicallyOrderedAdd α\ninst✝ : PosMulStrictMono α\na₂ b₁ b₂ : WithTop α\nhb : b₁ < b₂\nthis : MulPosStrictMono α\na₁ : α\nha : ↑a₁ < a₂\n⊢ ↑a₁ * b₁ < a₂ * b₂",
"ppTerm": "?m.63",
... | [
"α : Type u_1\ninst✝⁵ : DecidableEq α\ninst✝⁴ : CommSemiring α\ninst✝³ : PartialOrder α\ninst✝² : OrderBot α\ninst✝¹ : CanonicallyOrderedAdd α\ninst✝ : PosMulStrictMono α\na₂ b₂ : WithTop α\nthis : MulPosStrictMono α\na₁ : α\nha : ↑a₁ < a₂\nb₁ : α\nhb : ↑b₁ < b₂\n⊢ ↑a₁ * ↑b₁ < a₂ * b₂"
] | lift b₁ to α using hb.lt_top.ne | Mathlib.Tactic._aux_Mathlib_Tactic_Lift___elabRules_Mathlib_Tactic_lift_1 | Mathlib.Tactic.lift |
Mathlib.Algebra.Order.Ring.WithTop | {
"line": 320,
"column": 26
} | {
"line": 320,
"column": 31
} | {
"line": 322,
"column": 0
} | [
{
"pp": "case bot.bot\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\n⊢ ⊥ * ⊥ = if ⊥ = 0 ∨ ⊥ = 0 then 0 else map₂ (fun x1 x2 ↦ x1 * x2) ⊥ ⊥",
"ppTerm": "?bot.bot",
"assigned": true,
"usedConstants": [
"False",
"WithBot",
"HMul.hMul",
"MulZeroClass.toMul",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.WithTop | {
"line": 320,
"column": 26
} | {
"line": 320,
"column": 31
} | {
"line": 322,
"column": 0
} | [
{
"pp": "case bot.coe\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\na✝ : α\n⊢ ⊥ * ↑a✝ = if ⊥ = 0 ∨ ↑a✝ = 0 then 0 else map₂ (fun x1 x2 ↦ x1 * x2) ⊥ ↑a✝",
"ppTerm": "?bot.coe",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"WithBot.some",
"WithBot",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.WithTop | {
"line": 320,
"column": 26
} | {
"line": 320,
"column": 31
} | {
"line": 322,
"column": 0
} | [
{
"pp": "case coe.bot\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\na✝ : α\n⊢ ↑a✝ * ⊥ = if ↑a✝ = 0 ∨ ⊥ = 0 then 0 else map₂ (fun x1 x2 ↦ x1 * x2) ↑a✝ ⊥",
"ppTerm": "?coe.bot",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"WithBot.some",
"WithBot",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.WithTop | {
"line": 320,
"column": 26
} | {
"line": 320,
"column": 31
} | {
"line": 322,
"column": 0
} | [
{
"pp": "case coe.coe\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\na✝¹ a✝ : α\n⊢ ↑a✝¹ * ↑a✝ = if ↑a✝¹ = 0 ∨ ↑a✝ = 0 then 0 else map₂ (fun x1 x2 ↦ x1 * x2) ↑a✝¹ ↑a✝",
"ppTerm": "?coe.coe",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"WithBot.some",
"WithBot",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.WithTop | {
"line": 322,
"column": 85
} | {
"line": 322,
"column": 90
} | {
"line": 324,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\na b : WithBot α\n⊢ (if a = 0 ∨ b = 0 then 0 else map₂ (fun x1 x2 ↦ x1 * x2) a b) = ⊥ ↔ a ≠ 0 ∧ b = ⊥ ∨ a = ⊥ ∧ b ≠ 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Decidable.casesO... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.SuccPred | {
"line": 149,
"column": 2
} | {
"line": 150,
"column": 30
} | {
"line": 152,
"column": 0
} | [
{
"pp": "α : Type u_1\nx y : α\ninst✝⁴ : PartialOrder α\ninst✝³ : Sub α\ninst✝² : One α\ninst✝¹ : PredSubOrder α\ninst✝ : NoMinOrder α\n⊢ x ⋖ y ↔ y - 1 = x",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PredSubOrder.toPredOrder",
"Preorder.toLT",
"congrA... | [] | rw [← pred_eq_sub_one]
exact pred_eq_iff_covBy.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.SuccPred | {
"line": 149,
"column": 2
} | {
"line": 150,
"column": 30
} | {
"line": 152,
"column": 0
} | [
{
"pp": "α : Type u_1\nx y : α\ninst✝⁴ : PartialOrder α\ninst✝³ : Sub α\ninst✝² : One α\ninst✝¹ : PredSubOrder α\ninst✝ : NoMinOrder α\n⊢ x ⋖ y ↔ y - 1 = x",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PredSubOrder.toPredOrder",
"Preorder.toLT",
"congrA... | [] | rw [← pred_eq_sub_one]
exact pred_eq_iff_covBy.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SuccPred.Basic | {
"line": 891,
"column": 2
} | {
"line": 891,
"column": 34
} | {
"line": 892,
"column": 2
} | [
{
"pp": "α : Type u_3\ninst✝² : PartialOrder α\ns : Set α\ninst✝¹ : s.OrdConnected\ninst✝ : PredOrder α\na : ↑s\nh : pred ↑a ∈ s\n⊢ ↑(pred a) = pred ↑a",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"Membership.mem",
"Set.Elem",
"id",
... | [
"α : Type u_3\ninst✝² : PartialOrder α\ns : Set α\ninst✝¹ : s.OrdConnected\ninst✝ : PredOrder α\na : ↑s\nh : pred ↑a ∈ s\n⊢ ↑(if h : pred ↑a ∈ s then ⟨pred ↑a, h⟩ else a) = pred ↑a"
] | change Subtype.val (dite ..) = _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Order.SuccPred.Basic | {
"line": 915,
"column": 2
} | {
"line": 915,
"column": 34
} | {
"line": 916,
"column": 2
} | [
{
"pp": "α : Type u_3\ninst✝² : PartialOrder α\ns : Set α\ninst✝¹ : s.OrdConnected\ninst✝ : SuccOrder α\na : ↑s\nh : succ ↑a ∈ s\n⊢ ↑(succ a) = succ ↑a",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Set.OrdConnected.succOrder",
"Order.succ",
"PartialOrder.toPreorder",
... | [
"α : Type u_3\ninst✝² : PartialOrder α\ns : Set α\ninst✝¹ : s.OrdConnected\ninst✝ : SuccOrder α\na : ↑s\nh : succ ↑a ∈ s\n⊢ ↑(if h : pred ↑(ofDual a) ∈ ⇑ofDual ⁻¹' s then ⟨pred ↑(ofDual a), h⟩ else ofDual a) = succ ↑a"
] | change Subtype.val (dite ..) = _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Data.ENat.Basic | {
"line": 265,
"column": 16
} | {
"line": 265,
"column": 32
} | {
"line": 266,
"column": 2
} | [
{
"pp": "case coe.top\na : ℕ\n⊢ (↑a * ⊤).toNat = (↑a).toNat * ⊤.toNat",
"ppTerm": "?coe.top",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"NeZero.one",
"instCharZeroENat",
"instAddMonoidWithOneENat",
"HMul.hMul",
"AddMonoid.toAddSem... | [] | cases a <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.ENat.Basic | {
"line": 546,
"column": 67
} | {
"line": 546,
"column": 86
} | {
"line": 548,
"column": 0
} | [
{
"pp": "n : ℕ∞\nα : Type u_1\ninst✝³ : AddMonoidWithOne α\ninst✝² : PartialOrder α\ninst✝¹ : AddLeftMono α\ninst✝ : ZeroLEOneClass α\n⊢ 0 ≤ map Nat.cast n",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"WithTop.instPartialOrder",
"AddMonoid.toAddSemigroup",
"Nat.cast_no... | [] | by cases n <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finset.Piecewise | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 21
} | {
"line": 75,
"column": 22
} | [
{
"pp": "ι : Type u_1\nπ : ι → Sort u_2\ns : Finset ι\nf g : (i : ι) → π i\ninst✝ : (j : ι) → Decidable (j ∈ s)\ni : ι\np : π i → Prop\nhf : p (f i)\nhg : p (g i)\n⊢ p (s.piecewise f g i)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Finset",
"Membership.mem",
"Finset.... | [
"case pos\nι : Type u_1\nπ : ι → Sort u_2\ns : Finset ι\nf g : (i : ι) → π i\ninst✝ : (j : ι) → Decidable (j ∈ s)\ni : ι\np : π i → Prop\nhf : p (f i)\nhg : p (g i)\nhi : i ∈ s\n⊢ p (s.piecewise f g i)",
"case neg\nι : Type u_1\nπ : ι → Sort u_2\ns : Finset ι\nf g : (i : ι) → π i\ninst✝ : (j : ι) → Decidable (j ∈... | by_cases hi : i ∈ s | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Data.Fintype.Sigma | {
"line": 29,
"column": 76
} | {
"line": 29,
"column": 81
} | {
"line": 31,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nκ : ι → Type u_3\ninst✝ : (i : ι) → Fintype (κ i)\ns : Finset ι\nf : Sigma κ → Set α\n⊢ ⋃ ij ∈ s.sigma fun x ↦ Finset.univ, f ij = ⋃ i ∈ s, ⋃ j, f ⟨i, j⟩",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Sigma.exist... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Fintype.Sigma | {
"line": 29,
"column": 76
} | {
"line": 29,
"column": 81
} | {
"line": 31,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nκ : ι → Type u_3\ninst✝ : (i : ι) → Fintype (κ i)\ns : Finset ι\nf : Sigma κ → Set α\n⊢ ⋃ ij ∈ s.sigma fun x ↦ Finset.univ, f ij = ⋃ i ∈ s, ⋃ j, f ⟨i, j⟩",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Sigma.exist... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Fintype.Sigma | {
"line": 29,
"column": 76
} | {
"line": 29,
"column": 81
} | {
"line": 31,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nκ : ι → Type u_3\ninst✝ : (i : ι) → Fintype (κ i)\ns : Finset ι\nf : Sigma κ → Set α\n⊢ ⋃ ij ∈ s.sigma fun x ↦ Finset.univ, f ij = ⋃ i ∈ s, ⋃ j, f ⟨i, j⟩",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Sigma.exist... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Fintype.Sigma | {
"line": 32,
"column": 80
} | {
"line": 32,
"column": 85
} | {
"line": 34,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nκ : ι → Type u_3\ninst✝ : (i : ι) → Fintype (κ i)\ns : Finset ι\nf : (i : ι) → κ i → Set α\n⊢ ⋃ i ∈ s, ⋃ j, f i j = ⋃ ij ∈ s.sigma fun x ↦ Finset.univ, f ij.fst ij.snd",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Fintype.Sigma | {
"line": 32,
"column": 80
} | {
"line": 32,
"column": 85
} | {
"line": 34,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nκ : ι → Type u_3\ninst✝ : (i : ι) → Fintype (κ i)\ns : Finset ι\nf : (i : ι) → κ i → Set α\n⊢ ⋃ i ∈ s, ⋃ j, f i j = ⋃ ij ∈ s.sigma fun x ↦ Finset.univ, f ij.fst ij.snd",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Fintype.Sigma | {
"line": 32,
"column": 80
} | {
"line": 32,
"column": 85
} | {
"line": 34,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nκ : ι → Type u_3\ninst✝ : (i : ι) → Fintype (κ i)\ns : Finset ι\nf : (i : ι) → κ i → Set α\n⊢ ⋃ i ∈ s, ⋃ j, f i j = ⋃ ij ∈ s.sigma fun x ↦ Finset.univ, f ij.fst ij.snd",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Fintype.Sigma | {
"line": 35,
"column": 76
} | {
"line": 35,
"column": 81
} | {
"line": 37,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nκ : ι → Type u_3\ninst✝ : (i : ι) → Fintype (κ i)\ns : Finset ι\nf : Sigma κ → Set α\n⊢ ⋂ ij ∈ s.sigma fun x ↦ Finset.univ, f ij = ⋂ i ∈ s, ⋂ j, f ⟨i, j⟩",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Finset.univ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Fintype.Sigma | {
"line": 35,
"column": 76
} | {
"line": 35,
"column": 81
} | {
"line": 37,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nκ : ι → Type u_3\ninst✝ : (i : ι) → Fintype (κ i)\ns : Finset ι\nf : Sigma κ → Set α\n⊢ ⋂ ij ∈ s.sigma fun x ↦ Finset.univ, f ij = ⋂ i ∈ s, ⋂ j, f ⟨i, j⟩",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Finset.univ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Fintype.Sigma | {
"line": 35,
"column": 76
} | {
"line": 35,
"column": 81
} | {
"line": 37,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nκ : ι → Type u_3\ninst✝ : (i : ι) → Fintype (κ i)\ns : Finset ι\nf : Sigma κ → Set α\n⊢ ⋂ ij ∈ s.sigma fun x ↦ Finset.univ, f ij = ⋂ i ∈ s, ⋂ j, f ⟨i, j⟩",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Finset.univ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Fintype.Sigma | {
"line": 39,
"column": 80
} | {
"line": 39,
"column": 85
} | {
"line": 41,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nκ : ι → Type u_3\ninst✝ : (i : ι) → Fintype (κ i)\ns : Finset ι\nf : (i : ι) → κ i → Set α\n⊢ ⋂ i ∈ s, ⋂ j, f i j = ⋂ ij ∈ s.sigma fun x ↦ Finset.univ, f ij.fst ij.snd",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Fintype.Sigma | {
"line": 39,
"column": 80
} | {
"line": 39,
"column": 85
} | {
"line": 41,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nκ : ι → Type u_3\ninst✝ : (i : ι) → Fintype (κ i)\ns : Finset ι\nf : (i : ι) → κ i → Set α\n⊢ ⋂ i ∈ s, ⋂ j, f i j = ⋂ ij ∈ s.sigma fun x ↦ Finset.univ, f ij.fst ij.snd",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Fintype.Sigma | {
"line": 39,
"column": 80
} | {
"line": 39,
"column": 85
} | {
"line": 41,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nκ : ι → Type u_3\ninst✝ : (i : ι) → Fintype (κ i)\ns : Finset ι\nf : (i : ι) → κ i → Set α\n⊢ ⋂ i ∈ s, ⋂ j, f i j = ⋂ ij ∈ s.sigma fun x ↦ Finset.univ, f ij.fst ij.snd",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Group.Finset.Piecewise | {
"line": 71,
"column": 74
} | {
"line": 71,
"column": 79
} | {
"line": 73,
"column": 0
} | [
{
"pp": "case refine_1\nι : Type u_1\nM : Type u_3\ns : Finset ι\ninst✝¹ : CommMonoid M\np : ι → Prop\ninst✝ : DecidablePred p\nh : ∀ i ∈ s, ¬p i\nf : (i : ι) → p i → M\ng : (i : ι) → ¬p i → M\n⊢ ∀ (a : ι) (ha : a ∈ s), ⟨a, ha⟩ ∈ univ",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.BigOperators.Group.Finset.Piecewise | {
"line": 71,
"column": 74
} | {
"line": 71,
"column": 79
} | {
"line": 73,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u_1\nM : Type u_3\ns : Finset ι\ninst✝¹ : CommMonoid M\np : ι → Prop\ninst✝ : DecidablePred p\nh : ∀ i ∈ s, ¬p i\nf : (i : ι) → p i → M\ng : (i : ι) → ¬p i → M\n⊢ ∀ a ∈ univ, ↑a ∈ s",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Finset.univ",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.BigOperators.Group.Finset.Piecewise | {
"line": 71,
"column": 74
} | {
"line": 71,
"column": 79
} | {
"line": 73,
"column": 0
} | [
{
"pp": "case refine_3\nι : Type u_1\nM : Type u_3\ns : Finset ι\ninst✝¹ : CommMonoid M\np : ι → Prop\ninst✝ : DecidablePred p\nh : ∀ i ∈ s, ¬p i\nf : (i : ι) → p i → M\ng : (i : ι) → ¬p i → M\n⊢ ∀ (a : ι) (ha : a ∈ s), ↑⟨a, ha⟩ = a",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.BigOperators.Group.Finset.Piecewise | {
"line": 71,
"column": 74
} | {
"line": 71,
"column": 79
} | {
"line": 73,
"column": 0
} | [
{
"pp": "case refine_4\nι : Type u_1\nM : Type u_3\ns : Finset ι\ninst✝¹ : CommMonoid M\np : ι → Prop\ninst✝ : DecidablePred p\nh : ∀ i ∈ s, ¬p i\nf : (i : ι) → p i → M\ng : (i : ι) → ¬p i → M\n⊢ ∀ a ∈ univ, ⟨↑a, ⋯⟩ = a",
"ppTerm": "?refine_4",
"assigned": true,
"usedConstants": [
"Finset.univ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.BigOperators.Group.Finset.Piecewise | {
"line": 71,
"column": 74
} | {
"line": 71,
"column": 79
} | {
"line": 73,
"column": 0
} | [
{
"pp": "case refine_5\nι : Type u_1\nM : Type u_3\ns : Finset ι\ninst✝¹ : CommMonoid M\np : ι → Prop\ninst✝ : DecidablePred p\nh : ∀ i ∈ s, ¬p i\nf : (i : ι) → p i → M\ng : (i : ι) → ¬p i → M\n⊢ ∀ (a : ι) (ha : a ∈ s), (if hi : p a then f a hi else g a hi) = g ↑⟨a, ha⟩ ⋯",
"ppTerm": "?refine_5",
"assig... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Fintype.Sum | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 57
} | {
"line": 117,
"column": 0
} | [
{
"pp": "case h.inr.a\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Fintype α\ninst✝ : DecidableEq β\nt : Finset β\nhαt : Fintype.card α = #t\na : α\ns : Finset α\nhas : a ∉ s\nH : ∀ {f : α → β}, image f s ⊆ t → Set.InjOn f ↑s → ∃ g, ∀ i ∈ s, ↑(g i) = f i\nf : α → β\nhfst : image f (insert a s) ⊆ t\nhfs : Set.InjOn f ↑... | [] | · exact g'.injective.ne (ne_of_mem_of_not_mem hi has) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.BigOperators.Group.Finset.Piecewise | {
"line": 199,
"column": 32
} | {
"line": 199,
"column": 37
} | {
"line": 201,
"column": 0
} | [
{
"pp": "case neg.h\nι : Type u_1\nM : Type u_3\ninst✝¹ : CommMonoid M\ninst✝ : DecidableEq ι\ns : Finset ι\ni : ι\nf : ι → M\nh : f i = 1\nhs : i ∉ s\n⊢ s = s \\ {i}",
"ppTerm": "?neg.h✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"False",
"eq_false",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.BigOperators.Group.Finset.Piecewise | {
"line": 199,
"column": 32
} | {
"line": 199,
"column": 37
} | {
"line": 201,
"column": 0
} | [
{
"pp": "case neg.a\nι : Type u_1\nM : Type u_3\ninst✝¹ : CommMonoid M\ninst✝ : DecidableEq ι\ns : Finset ι\ni : ι\nf : ι → M\nh : f i = 1\nhs : i ∉ s\n⊢ ∀ x ∈ s \\ {i}, f x = f x",
"ppTerm": "?neg.a✝",
"assigned": true,
"usedConstants": [
"Finset",
"Finset.instSDiff",
"Membership.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Fintype.BigOperators | {
"line": 236,
"column": 37
} | {
"line": 236,
"column": 58
} | {
"line": 236,
"column": 58
} | [
{
"pp": "β : Type u_2\ninst✝ : CommMonoid β\nn : ℕ\nc : Fin n → β\n⊢ ∏ i, c i = ∏ i, if h : ↑i < n then c ⟨↑i, h⟩ else 1",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Finset.univ",
"Monoid.toMulOneClass",
"congrArg",
"Finset"... | [
"β : Type u_2\ninst✝ : CommMonoid β\nn : ℕ\nc : Fin n → β\n⊢ univ.prod ?m.36 = ∏ i, if h : ↑i < n then c ⟨↑i, h⟩ else 1",
"β : Type u_2\ninst✝ : CommMonoid β\nn : ℕ\nc : Fin n → β\n⊢ ∀ x ∈ univ, c x = ?m.36 x",
"β : Type u_2\ninst✝ : CommMonoid β\nn : ℕ\nc : Fin n → β\n⊢ Fin n → β"
] | Finset.prod_congr rfl | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.InitialSeg | {
"line": 524,
"column": 2
} | {
"line": 524,
"column": 33
} | {
"line": 526,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nr : α → α → Prop\ns : β → β → Prop\ninst✝ : IsWellOrder β s\nf : r ↪r s\na : α\n⊢ ⋯.min\n {b |\n ∀ (a_1 : α),\n r a_1 a →\n s (↑(IsWellFounded.fix r (fun a IH ↦ ⟨⋯.min {b | ∀ (a_2 : α) (h : r a_2 a), s (↑(IH a_2 h)) b} ⋯, ⋯⟩) a_1))\n ... | [] | exact WellFounded.min_mem _ _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Sum.Order | {
"line": 524,
"column": 21
} | {
"line": 524,
"column": 26
} | {
"line": 526,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nα₁ : Type u_4\nα₂ : Type u_5\nβ₁ : Type u_6\nβ₂ : Type u_7\nγ₁ : Type u_8\nγ₂ : Type u_9\ninst✝⁸ : LE α\ninst✝⁷ : LE β\ninst✝⁶ : LE γ\ninst✝⁵ : LE α₁\ninst✝⁴ : LE α₂\ninst✝³ : LE β₁\ninst✝² : LE β₂\ninst✝¹ : LE γ₁\ninst✝ : LE γ₂\na : α\nb : β\nc : γ\nea : α₁ ≃o... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Sum.Order | {
"line": 524,
"column": 21
} | {
"line": 524,
"column": 26
} | {
"line": 526,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nα₁ : Type u_4\nα₂ : Type u_5\nβ₁ : Type u_6\nβ₂ : Type u_7\nγ₁ : Type u_8\nγ₂ : Type u_9\ninst✝⁸ : LE α\ninst✝⁷ : LE β\ninst✝⁶ : LE γ\ninst✝⁵ : LE α₁\ninst✝⁴ : LE α₂\ninst✝³ : LE β₁\ninst✝² : LE β₂\ninst✝¹ : LE γ₁\ninst✝ : LE γ₂\na : α\nb : β\nc : γ\nea : α₁ ≃o... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Sum.Order | {
"line": 524,
"column": 21
} | {
"line": 524,
"column": 26
} | {
"line": 526,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nα₁ : Type u_4\nα₂ : Type u_5\nβ₁ : Type u_6\nβ₂ : Type u_7\nγ₁ : Type u_8\nγ₂ : Type u_9\ninst✝⁸ : LE α\ninst✝⁷ : LE β\ninst✝⁶ : LE γ\ninst✝⁵ : LE α₁\ninst✝⁴ : LE α₂\ninst✝³ : LE β₁\ninst✝² : LE β₂\ninst✝¹ : LE γ₁\ninst✝ : LE γ₂\na : α\nb : β\nc : γ\nea : α₁ ≃o... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.InitialSeg | {
"line": 561,
"column": 8
} | {
"line": 563,
"column": 66
} | {
"line": 564,
"column": 6
} | [
{
"pp": "case inr.inl\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nr✝ : α → α → Prop\ns✝ : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ns : β → β → Prop\ninst✝¹ : IsWellOrder α r\ninst✝ : IsWellOrder β s\nf : r ≺i Sum.Lex r s\ng : s ≺i Sum.Lex r s\nh : f.top = g.top\n⊢ Nonempty ((r ≼i s) ⊕ (s ≼i r))",
"p... | [] | let f := f.subrelIso
rw [h] at f
exact ⟨Sum.inl <| (f.symm.trans g.subrelIso).toInitialSeg⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.InitialSeg | {
"line": 561,
"column": 8
} | {
"line": 563,
"column": 66
} | {
"line": 564,
"column": 6
} | [
{
"pp": "case inr.inl\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nr✝ : α → α → Prop\ns✝ : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ns : β → β → Prop\ninst✝¹ : IsWellOrder α r\ninst✝ : IsWellOrder β s\nf : r ≺i Sum.Lex r s\ng : s ≺i Sum.Lex r s\nh : f.top = g.top\n⊢ Nonempty ((r ≼i s) ⊕ (s ≼i r))",
"p... | [] | let f := f.subrelIso
rw [h] at f
exact ⟨Sum.inl <| (f.symm.trans g.subrelIso).toInitialSeg⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.UpperLower.Basic | {
"line": 100,
"column": 75
} | {
"line": 100,
"column": 80
} | {
"line": 102,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : LE α\ns t : Set α\nhs : IsUpperSet s\n⊢ IsLowerSet (Subtype.val ⁻¹' t) ↔ ∀ b ∈ s, ∀ c ∈ t, b ≤ c → b ∈ t",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Subtype.forall._simp_1",
"Membership.mem",
"Set.Elem"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.UpperLower.Basic | {
"line": 100,
"column": 75
} | {
"line": 100,
"column": 80
} | {
"line": 102,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : LE α\ns t : Set α\nhs : IsUpperSet s\n⊢ IsLowerSet (Subtype.val ⁻¹' t) ↔ ∀ b ∈ s, ∀ c ∈ t, b ≤ c → b ∈ t",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Subtype.forall._simp_1",
"Membership.mem",
"Set.Elem"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.UpperLower.Basic | {
"line": 100,
"column": 75
} | {
"line": 100,
"column": 80
} | {
"line": 102,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : LE α\ns t : Set α\nhs : IsUpperSet s\n⊢ IsLowerSet (Subtype.val ⁻¹' t) ↔ ∀ b ∈ s, ∀ c ∈ t, b ≤ c → b ∈ t",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Subtype.forall._simp_1",
"Membership.mem",
"Set.Elem"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.UpperLower.Basic | {
"line": 109,
"column": 17
} | {
"line": 109,
"column": 22
} | {
"line": 111,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : LE α\ns t : Set α\nhs : IsUpperSet s\nht : IsLowerSet t\n⊢ ∀ b ∈ s, ∀ c ∈ t, b ≤ c → b ∈ t",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Membership.mem",
"LE.le",
"Set.instMembership",
"Set"
],
"usedFVars": [
"α",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.UpperLower.Basic | {
"line": 109,
"column": 17
} | {
"line": 109,
"column": 22
} | {
"line": 111,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : LE α\ns t : Set α\nhs : IsUpperSet s\nht : IsLowerSet t\n⊢ ∀ b ∈ s, ∀ c ∈ t, b ≤ c → b ∈ t",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Membership.mem",
"LE.le",
"Set.instMembership",
"Set"
],
"usedFVars": [
"α",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.UpperLower.Basic | {
"line": 109,
"column": 17
} | {
"line": 109,
"column": 22
} | {
"line": 111,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : LE α\ns t : Set α\nhs : IsUpperSet s\nht : IsLowerSet t\n⊢ ∀ b ∈ s, ∀ c ∈ t, b ≤ c → b ∈ t",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Membership.mem",
"LE.le",
"Set.instMembership",
"Set"
],
"usedFVars": [
"α",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Part | {
"line": 662,
"column": 18
} | {
"line": 662,
"column": 23
} | {
"line": 664,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Inv α\na : Part α\nma : α\nha : ma ∈ a\n⊢ ∃ a_1, a_1 ∈ a ∧ a_1⁻¹ = ma⁻¹",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Part",
"Membership.mem",
"Part.instMembership",
"And",
"Inv.inv",
"And.intro",
"Exists.intro",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Part | {
"line": 670,
"column": 42
} | {
"line": 670,
"column": 47
} | {
"line": 672,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Div α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ ∃ a_1, a_1 ∈ a ∧ ∃ a, a ∈ b ∧ a_1 / a = ma / mb",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Part",
"instHDiv",
"Membership.mem",
"Exists",
"HDiv.hDiv",
"Part.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Part | {
"line": 681,
"column": 18
} | {
"line": 681,
"column": 23
} | {
"line": 683,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Div α\na b : Part α\nhab : (a / b).Dom\n⊢ (a.bind fun y ↦ map (fun x ↦ y / x) b).get ⋯ = a.get ⋯ / b.get ⋯",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Part",
"instHDiv",
"Part.bind",
"id",
"HDiv.hDiv",
"Part.get",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Part | {
"line": 687,
"column": 42
} | {
"line": 687,
"column": 47
} | {
"line": 689,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Mod α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ ∃ a_1, a_1 ∈ a ∧ ∃ a, a ∈ b ∧ a_1 % a = ma % mb",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Part",
"Membership.mem",
"Exists",
"instHMod",
"Part.instMembership",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Part | {
"line": 696,
"column": 18
} | {
"line": 696,
"column": 23
} | {
"line": 698,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Mod α\na b : Part α\nhab : (a % b).Dom\n⊢ (a.bind fun y ↦ map (fun x ↦ y % x) b).get ⋯ = a.get ⋯ % b.get ⋯",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Part",
"Part.bind",
"id",
"instHMod",
"Part.get",
"HMod.hMod",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Part | {
"line": 701,
"column": 47
} | {
"line": 701,
"column": 52
} | {
"line": 703,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Append α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ ∃ a_1, a_1 ∈ a ∧ ∃ a, a ∈ b ∧ a_1 ++ a = ma ++ mb",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Part",
"Membership.mem",
"Exists",
"Part.instMembership",
"instHApp... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Part | {
"line": 710,
"column": 21
} | {
"line": 710,
"column": 26
} | {
"line": 712,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Append α\na b : Part α\nhab : (a ++ b).Dom\n⊢ (a.bind fun y ↦ map (fun x ↦ y ++ x) b).get ⋯ = a.get ⋯ ++ b.get ⋯",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Part",
"Part.bind",
"Part.instAppend",
"id",
"Part.get",
"ins... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Part | {
"line": 716,
"column": 44
} | {
"line": 716,
"column": 49
} | {
"line": 718,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Inter α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ ∃ a_1, a_1 ∈ a ∧ ∃ a, a ∈ b ∧ a_1 ∩ a = ma ∩ mb",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Part",
"Membership.mem",
"Exists",
"Part.instMembership",
"Inter.inter... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Part | {
"line": 725,
"column": 20
} | {
"line": 725,
"column": 25
} | {
"line": 727,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Inter α\na b : Part α\nhab : (a ∩ b).Dom\n⊢ (a.bind fun y ↦ map (fun x ↦ y ∩ x) b).get ⋯ = a.get ⋯ ∩ b.get ⋯",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Part",
"Part.bind",
"Part.instInter",
"id",
"Part.get",
"Inter.in... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Part | {
"line": 731,
"column": 44
} | {
"line": 731,
"column": 49
} | {
"line": 733,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Union α\na b : Part α\nma mb : α\nha : ma ∈ a\nhb : mb ∈ b\n⊢ ∃ a_1, a_1 ∈ a ∧ ∃ a, a ∈ b ∧ a_1 ∪ a = ma ∪ mb",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Part",
"Membership.mem",
"Exists",
"Part.instMembership",
"And",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
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