module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.Finsupp.Weight | {
"line": 307,
"column": 46
} | {
"line": 307,
"column": 51
} | {
"line": 308,
"column": 4
} | [
{
"pp": "σ : Type u_5\ng f : σ →₀ ℕ\nhgf : g ≤ g + f\nIH : degree g ≤ degree (g + f) → ∃ g_1 ≤ g + f, degree g_1 = degree g\nhn : degree g + 1 ≤ degree (g + f)\n⊢ f.support.Nonempty",
"ppTerm": "?m.91",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finsupp.Weight | {
"line": 307,
"column": 46
} | {
"line": 307,
"column": 51
} | {
"line": 308,
"column": 4
} | [
{
"pp": "σ : Type u_5\ng f : σ →₀ ℕ\nhgf : g ≤ g + f\nIH : degree g ≤ degree (g + f) → ∃ g_1 ≤ g + f, degree g_1 = degree g\nhn : degree g + 1 ≤ degree (g + f)\n⊢ f.support.Nonempty",
"ppTerm": "?m.91",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finsupp.Weight | {
"line": 335,
"column": 46
} | {
"line": 335,
"column": 51
} | {
"line": 336,
"column": 4
} | [
{
"pp": "α : Type u_5\ns : Set α\nn : ℕ\nih : n • (fun x ↦ single x 1) '' s = {x | degree x = n ∧ ↑x.support ⊆ s}\nf : α →₀ ℕ\nx✝ : f ∈ {x | degree x = n + 1 ∧ ↑x.support ⊆ s}\nf_deg : degree f = n + 1\nf_supp : ↑f.support ⊆ s\n⊢ f.support.Nonempty",
"ppTerm": "?m.90",
"assigned": true,
"usedConstan... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Finsupp.Weight | {
"line": 335,
"column": 46
} | {
"line": 335,
"column": 51
} | {
"line": 336,
"column": 4
} | [
{
"pp": "α : Type u_5\ns : Set α\nn : ℕ\nih : n • (fun x ↦ single x 1) '' s = {x | degree x = n ∧ ↑x.support ⊆ s}\nf : α →₀ ℕ\nx✝ : f ∈ {x | degree x = n + 1 ∧ ↑x.support ⊆ s}\nf_deg : degree f = n + 1\nf_supp : ↑f.support ⊆ s\n⊢ f.support.Nonempty",
"ppTerm": "?m.90",
"assigned": true,
"usedConstan... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finsupp.Weight | {
"line": 335,
"column": 46
} | {
"line": 335,
"column": 51
} | {
"line": 336,
"column": 4
} | [
{
"pp": "α : Type u_5\ns : Set α\nn : ℕ\nih : n • (fun x ↦ single x 1) '' s = {x | degree x = n ∧ ↑x.support ⊆ s}\nf : α →₀ ℕ\nx✝ : f ∈ {x | degree x = n + 1 ∧ ↑x.support ⊆ s}\nf_deg : degree f = n + 1\nf_supp : ↑f.support ⊆ s\n⊢ f.support.Nonempty",
"ppTerm": "?m.90",
"assigned": true,
"usedConstan... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors | {
"line": 246,
"column": 54
} | {
"line": 246,
"column": 69
} | {
"line": 246,
"column": 69
} | [
{
"pp": "α : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\np x : α\nh : x ≠ 0\nx✝ : Irreducible p ∧ normalize p = p ∧ p ∣ x\nh₁ : Irreducible p\nh₂ : normalize p = p\nh₃ : p ∣ x\ny : α\nhy₁ : y ∈ factors x\nhy₂ : p ~ᵤ y\n⊢ y ~ᵤ p",
"ppTerm": "?... | [
"α : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\np x : α\nh : x ≠ 0\nx✝ : Irreducible p ∧ normalize p = p ∧ p ∣ x\nh₁ : Irreducible p\nh₂ : normalize p = p\nh₃ : p ∣ x\ny : α\nhy₁ : y ∈ factors x\nhy₂ : p ~ᵤ y\n⊢ p ~ᵤ y"
] | Associated.comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.UniqueFactorizationDomain.Basic | {
"line": 331,
"column": 6
} | {
"line": 354,
"column": 69
} | {
"line": 354,
"column": 69
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : IsCancelMulZero α\npf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a\na b : α\nane0 : a ≠ 0\nc : α\nhc : ¬IsUnit c\nb_eq : b = a * c\nh : ¬b = 0\n⊢ ↑(Classical.choose ⋯).card < if h : b = 0 then ⊤ else ↑(Classical.choose ⋯).card... | [] | · rw [dif_neg h, Nat.cast_lt]
have cne0 : c ≠ 0 := by
refine mt (fun con => ?_) h
rw [b_eq, con, mul_zero]
calc
Multiset.card (Classical.choose (pf a ane0)) <
_ + Multiset.card (Classical.choose (pf c cne0)) :=
lt_add_of_pos_right _
(... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.Roots | {
"line": 493,
"column": 7
} | {
"line": 493,
"column": 28
} | {
"line": 493,
"column": 29
} | [
{
"pp": "S : Type v\nT : Type w\ninst✝⁵ : CommRing T\ninst✝⁴ : IsDomain T\ninst✝³ : CommRing S\ninst✝² : IsDomain S\ninst✝¹ : Algebra T S\ninst✝ : Module.IsTorsionFree T S\np q : T[X]\nhpq : p * q ≠ 0\n⊢ map (algebraMap T S) p * map (algebraMap T S) q ≠ 0",
"ppTerm": "?m.66",
"assigned": true,
"used... | [
"S : Type v\nT : Type w\ninst✝⁵ : CommRing T\ninst✝⁴ : IsDomain T\ninst✝³ : CommRing S\ninst✝² : IsDomain S\ninst✝¹ : Algebra T S\ninst✝ : Module.IsTorsionFree T S\np q : T[X]\nhpq : p * q ≠ 0\n⊢ map (algebraMap T S) (p * q) ≠ 0"
] | ← Polynomial.map_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 374,
"column": 17
} | {
"line": 374,
"column": 28
} | {
"line": 374,
"column": 29
} | [
{
"pp": "R : Type u\ninst✝ : Field R\np q : R[X]\nhq0 : q ≠ 0\nh : p.degree < q.degree\nthis : ¬(q * C q.leadingCoeff⁻¹).degree ≤ p.degree\n⊢ p %ₘ (q * C q.leadingCoeff⁻¹) = p",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Polynomial.modByMonic.eq_1",
"Eq.mpr",
"Polynom... | [
"R : Type u\ninst✝ : Field R\np q : R[X]\nhq0 : q ≠ 0\nh : p.degree < q.degree\nthis : ¬(q * C q.leadingCoeff⁻¹).degree ≤ p.degree\n⊢ (if hq : (q * C q.leadingCoeff⁻¹).Monic then (p.divModByMonicAux hq).2 else p) = p"
] | modByMonic, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet | {
"line": 180,
"column": 2
} | {
"line": 185,
"column": 28
} | {
"line": 187,
"column": 0
} | [
{
"pp": "case coe.coe\nα : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Nontrivial α\na✝¹ a✝ : Multiset { a // Irreducible a }\nh : prod ↑a✝¹ = prod ↑a✝\n⊢ ↑a✝¹ = ↑a✝",
"ppTerm": "?coe.coe",
"assigned": true,
"usedConstants": [
"Subtype.coe_mk",
... | [] | · congr 1
rw [← Multiset.map_eq_map Subtype.coe_injective]
apply unique' _ _ h <;>
· intro a ha
obtain ⟨⟨a', irred⟩, -, rfl⟩ := Multiset.mem_map.mp ha
rwa [Subtype.coe_mk] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 604,
"column": 47
} | {
"line": 604,
"column": 68
} | {
"line": 605,
"column": 6
} | [
{
"pp": "case neg\nR : Type u\nk : Type y\ninst✝¹ : Field R\ninst✝ : Field k\nf : R →+* k\nx y : R[X]\nH : ¬x = 0\n⊢ map f x * map f (C x.leadingCoeff⁻¹) ∣ map f y ↔ x * C x.leadingCoeff⁻¹ ∣ y",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Poly... | [
"case neg\nR : Type u\nk : Type y\ninst✝¹ : Field R\ninst✝ : Field k\nf : R →+* k\nx y : R[X]\nH : ¬x = 0\n⊢ map f (x * C x.leadingCoeff⁻¹) ∣ map f y ↔ x * C x.leadingCoeff⁻¹ ∣ y"
] | ← Polynomial.map_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 639,
"column": 17
} | {
"line": 639,
"column": 90
} | {
"line": 640,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝ : CommRing K\nf : K[X]\na : K\n⊢ ?m.57",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Polynomial.derivative",
"Polynomial.C",
"Semiring.toModule",
"HMul.hMul",
"congrArg",
"LinearMap.instFunLike",
"HSub.hSub",
"... | [] | apply congrArg derivative <| X_sub_C_mul_divByMonic_eq_sub_modByMonic f a | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 639,
"column": 17
} | {
"line": 639,
"column": 90
} | {
"line": 640,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝ : CommRing K\nf : K[X]\na : K\n⊢ ?m.57",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Polynomial.derivative",
"Polynomial.C",
"Semiring.toModule",
"HMul.hMul",
"congrArg",
"LinearMap.instFunLike",
"HSub.hSub",
"... | [] | apply congrArg derivative <| X_sub_C_mul_divByMonic_eq_sub_modByMonic f a | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 639,
"column": 17
} | {
"line": 639,
"column": 90
} | {
"line": 640,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝ : CommRing K\nf : K[X]\na : K\n⊢ ?m.57",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Polynomial.derivative",
"Polynomial.C",
"Semiring.toModule",
"HMul.hMul",
"congrArg",
"LinearMap.instFunLike",
"HSub.hSub",
"... | [] | apply congrArg derivative <| X_sub_C_mul_divByMonic_eq_sub_modByMonic f a | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 139,
"column": 28
} | {
"line": 139,
"column": 68
} | {
"line": 139,
"column": 68
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nx : R\n⊢ (aeval ((algebraMap R A) x)) (X - C x) = 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"MonoidHom.instFunLike",
... | [] | rw [map_sub, aeval_X, aeval_C, sub_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 139,
"column": 28
} | {
"line": 139,
"column": 68
} | {
"line": 139,
"column": 68
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nx : R\n⊢ (aeval ((algebraMap R A) x)) (X - C x) = 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"MonoidHom.instFunLike",
... | [] | rw [map_sub, aeval_X, aeval_C, sub_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 139,
"column": 28
} | {
"line": 139,
"column": 68
} | {
"line": 139,
"column": 68
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nx : R\n⊢ (aeval ((algebraMap R A) x)) (X - C x) = 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"MonoidHom.instFunLike",
... | [] | rw [map_sub, aeval_X, aeval_C, sub_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 230,
"column": 6
} | {
"line": 230,
"column": 48
} | {
"line": 230,
"column": 48
} | [
{
"pp": "R : Type u\nS : Type u_1\nA : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing S\ninst✝⁷ : Ring A\ninst✝⁶ : Algebra R A\nB : Type u_2\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra S B\nFRS : Type u_3\nFAB : Type u_4\ninst✝³ : FunLike FRS R S\ninst✝² : RingHomClass FRS R S\ninst✝¹ : FunLike FAB A B\ninst✝ : RingHomC... | [
"R : Type u\nS : Type u_1\nA : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing S\ninst✝⁷ : Ring A\ninst✝⁶ : Algebra R A\nB : Type u_2\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra S B\nFRS : Type u_3\nFAB : Type u_4\ninst✝³ : FunLike FRS R S\ninst✝² : RingHomClass FRS R S\ninst✝¹ : FunLike FAB A B\ninst✝ : RingHomClass FAB A B... | Algebra.transcendental_iff_not_isAlgebraic | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 259,
"column": 6
} | {
"line": 259,
"column": 48
} | {
"line": 259,
"column": 48
} | [
{
"pp": "R : Type u\nS : Type u_1\nA : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing S\ninst✝⁷ : Ring A\ninst✝⁶ : Algebra R A\nB : Type u_2\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra S B\nFRS : Type u_3\nFAB : Type u_4\ninst✝³ : FunLike FRS R S\ninst✝² : RingHomClass FRS R S\ninst✝¹ : FunLike FAB A B\ninst✝ : RingHomC... | [
"R : Type u\nS : Type u_1\nA : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing S\ninst✝⁷ : Ring A\ninst✝⁶ : Algebra R A\nB : Type u_2\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra S B\nFRS : Type u_3\nFAB : Type u_4\ninst✝³ : FunLike FRS R S\ninst✝² : RingHomClass FRS R S\ninst✝¹ : FunLike FAB A B\ninst✝ : RingHomClass FAB A B... | Algebra.transcendental_iff_not_isAlgebraic | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Central.End | {
"line": 39,
"column": 9
} | {
"line": 39,
"column": 14
} | {
"line": 39,
"column": 14
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_3\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : Free R M\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Module S M\ninst✝³ : SMulCommClass R S M\ninst✝² : Algebra S R\ninst✝¹ : IsScalarTower S R M\ninst✝ : IsCentral S R\nT : End R M\nhT : T ∈ S... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Central.End | {
"line": 39,
"column": 9
} | {
"line": 39,
"column": 14
} | {
"line": 39,
"column": 14
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_3\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : Free R M\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Module S M\ninst✝³ : SMulCommClass R S M\ninst✝² : Algebra S R\ninst✝¹ : IsScalarTower S R M\ninst✝ : IsCentral S R\nT : End R M\nhT : T ∈ S... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Central.End | {
"line": 39,
"column": 9
} | {
"line": 39,
"column": 14
} | {
"line": 39,
"column": 14
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_3\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : Free R M\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Module S M\ninst✝³ : SMulCommClass R S M\ninst✝² : Algebra S R\ninst✝¹ : IsScalarTower S R M\ninst✝ : IsCentral S R\nT : End R M\nhT : T ∈ S... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 610,
"column": 2
} | {
"line": 611,
"column": 6
} | {
"line": 613,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\ns : S\nh : Transcendental R s\nf : R[X]\n⊢ (algEquivOfTranscendental R s h).symm ((aeval ⟨s, ⋯⟩) f) = f",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Equ... | [] | apply (algEquivOfTranscendental R s h).toEquiv.injective
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 610,
"column": 2
} | {
"line": 611,
"column": 6
} | {
"line": 613,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\ns : S\nh : Transcendental R s\nf : R[X]\n⊢ (algEquivOfTranscendental R s h).symm ((aeval ⟨s, ⋯⟩) f) = f",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Equ... | [] | apply (algEquivOfTranscendental R s h).toEquiv.injective
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 616,
"column": 2
} | {
"line": 617,
"column": 6
} | {
"line": 619,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\ns : S\nh : Transcendental R s\n⊢ (algEquivOfTranscendental R s h).symm ⟨s, ⋯⟩ = X",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Equiv.instEquivLike",
... | [] | apply (algEquivOfTranscendental R s h).toEquiv.injective
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 616,
"column": 2
} | {
"line": 617,
"column": 6
} | {
"line": 619,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\ns : S\nh : Transcendental R s\n⊢ (algEquivOfTranscendental R s h).symm ⟨s, ⋯⟩ = X",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Equiv.instEquivLike",
... | [] | apply (algEquivOfTranscendental R s h).toEquiv.injective
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorProduct.DirectLimit | {
"line": 50,
"column": 56
} | {
"line": 50,
"column": 61
} | {
"line": 52,
"column": 0
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝⁶ : CommSemiring R\nι : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Preorder ι\nG : ι → Type u_3\ninst✝³ : (i : ι) → AddCommMonoid (G i)\ninst✝² : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\nM : Type u_4\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.TensorProduct.DirectLimit | {
"line": 50,
"column": 56
} | {
"line": 50,
"column": 61
} | {
"line": 52,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝⁶ : CommSemiring R\nι : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Preorder ι\nG : ι → Type u_3\ninst✝³ : (i : ι) → AddCommMonoid (G i)\ninst✝² : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\nM : Type u_4\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.TensorProduct.DirectLimit | {
"line": 50,
"column": 56
} | {
"line": 50,
"column": 61
} | {
"line": 52,
"column": 0
} | [
{
"pp": "case refine_3\nR : Type u_1\ninst✝⁶ : CommSemiring R\nι : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Preorder ι\nG : ι → Type u_3\ninst✝³ : (i : ι) → AddCommMonoid (G i)\ninst✝² : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\nM : Type u_4\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.TensorProduct.DirectLimit | {
"line": 101,
"column": 63
} | {
"line": 101,
"column": 68
} | {
"line": 101,
"column": 68
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommSemiring R\nι : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Preorder ι\nG : ι → Type u_3\ninst✝³ : (i : ι) → AddCommMonoid (G i)\ninst✝² : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\nM : Type u_4\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ni j : ι\nh : ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.TensorProduct.DirectLimit | {
"line": 101,
"column": 63
} | {
"line": 101,
"column": 68
} | {
"line": 101,
"column": 68
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommSemiring R\nι : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Preorder ι\nG : ι → Type u_3\ninst✝³ : (i : ι) → AddCommMonoid (G i)\ninst✝² : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\nM : Type u_4\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ni j : ι\nh : ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.DirectLimit | {
"line": 101,
"column": 63
} | {
"line": 101,
"column": 68
} | {
"line": 101,
"column": 68
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommSemiring R\nι : Type u_2\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Preorder ι\nG : ι → Type u_3\ninst✝³ : (i : ι) → AddCommMonoid (G i)\ninst✝² : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\nM : Type u_4\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ni j : ι\nh : ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.DirectedInverseSystem | {
"line": 364,
"column": 23
} | {
"line": 364,
"column": 49
} | {
"line": 365,
"column": 8
} | [
{
"pp": "case inl.inr\nι : Type u_6\nF : ι → Type u_7\nX : ι → Type u_8\ni : ι\ninst✝² : LinearOrder ι\nf : ⦃i j : ι⦄ → i ≤ j → F j → F i\ninst✝¹ : SuccOrder ι\nequiv : (j : ↑(Iic i)) → F ↑j ≃ piLT X ↑j\ne : F i⁺ ≃ F i × X i\nhi : ¬IsMax i\ninst✝ : InverseSystem f\nH : ∀ (x : F i⁺), (e x).1 = f ⋯ x\nnat : IsNat... | [
"case inl.inr\nι : Type u_6\nF : ι → Type u_7\nX : ι → Type u_8\ni : ι\ninst✝² : LinearOrder ι\nf : ⦃i j : ι⦄ → i ≤ j → F j → F i\ninst✝¹ : SuccOrder ι\nequiv : (j : ↑(Iic i)) → F ↑j ≃ piLT X ↑j\ne : F i⁺ ≃ F i × X i\nhi : ¬IsMax i\ninst✝ : InverseSystem f\nH : ∀ (x : F i⁺), (e x).1 = f ⋯ x\nnat : IsNatEquiv f equi... | piSplitLE_lt (lt_succ hk), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.DirectedInverseSystem | {
"line": 378,
"column": 84
} | {
"line": 378,
"column": 93
} | {
"line": 378,
"column": 93
} | [
{
"pp": "ι✝ : Type u_1\ninst✝¹ : Preorder ι✝\nF₁ : ι✝ → Type u_2\nF₂ : ι✝ → Type u_3\nF✝ : ι✝ → Type u_4\nX✝ : ι✝ → Type u_5\nf✝ : ⦃i j : ι✝⦄ → i ≤ j → F✝ j → F✝ i\ni✝ j : ι✝\nh✝ : i✝ ≤ j\nι : Type u_6\nF : ι → Type u_7\nX : ι → Type u_8\ni : ι\ninst✝ : LinearOrder ι\nf : ⦃i j : ι⦄ → i ≤ j → F j → F i\nequiv : ... | [] | apply nat | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Order.DirectedInverseSystem | {
"line": 400,
"column": 68
} | {
"line": 400,
"column": 77
} | {
"line": 402,
"column": 0
} | [
{
"pp": "case inr\nι : Type u_6\nF : ι → Type u_7\nX : ι → Type u_8\ni : ι\ninst✝¹ : LinearOrder ι\nf : ⦃i j : ι⦄ → i ≤ j → F j → F i\nequiv : (j : ↑(Iio i)) → F ↑j ≃ piLT X ↑j\nnat : IsNatEquiv f equiv\nequivLim : F i ≃ ↑(limit f i)\nhi : IsSuccPrelimit i\ninst✝ : InverseSystem f\nH : ∀ (x : F i) (l : ↑(Iio i)... | [] | apply nat | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.MvPolynomial.Basic | {
"line": 157,
"column": 65
} | {
"line": 157,
"column": 70
} | {
"line": 157,
"column": 70
} | [
{
"pp": "σ : Type u\nR : Type v\ninst✝ : CommSemiring R\np m✝ : ℕ\ns : Set (σ →₀ ℕ)\nhs : IsUpperSet s\nx y : MvPolynomial σ R\nhy : y ∈ (restrictSupport R s).carrier\nm : σ →₀ ℕ\nhm : ¬∑ x_1 ∈ Finset.antidiagonal m, coeff x_1.1 x * coeff x_1.2 y = 0\ni j : σ →₀ ℕ\nhij : (i, j) ∈ Finset.antidiagonal m\ne : coef... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.MvPolynomial.Basic | {
"line": 157,
"column": 65
} | {
"line": 157,
"column": 70
} | {
"line": 157,
"column": 70
} | [
{
"pp": "σ : Type u\nR : Type v\ninst✝ : CommSemiring R\np m✝ : ℕ\ns : Set (σ →₀ ℕ)\nhs : IsUpperSet s\nx y : MvPolynomial σ R\nhy : y ∈ (restrictSupport R s).carrier\nm : σ →₀ ℕ\nhm : ¬∑ x_1 ∈ Finset.antidiagonal m, coeff x_1.1 x * coeff x_1.2 y = 0\ni j : σ →₀ ℕ\nhij : (i, j) ∈ Finset.antidiagonal m\ne : coef... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.Basic | {
"line": 157,
"column": 65
} | {
"line": 157,
"column": 70
} | {
"line": 157,
"column": 70
} | [
{
"pp": "σ : Type u\nR : Type v\ninst✝ : CommSemiring R\np m✝ : ℕ\ns : Set (σ →₀ ℕ)\nhs : IsUpperSet s\nx y : MvPolynomial σ R\nhy : y ∈ (restrictSupport R s).carrier\nm : σ →₀ ℕ\nhm : ¬∑ x_1 ∈ Finset.antidiagonal m, coeff x_1.1 x * coeff x_1.2 y = 0\ni j : σ →₀ ℕ\nhij : (i, j) ∈ Finset.antidiagonal m\ne : coef... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 294,
"column": 6
} | {
"line": 294,
"column": 16
} | {
"line": 294,
"column": 17
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : AddCommGroup P\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nQ : Type u_5\ninst✝¹ : AddCommGroup Q\ninst✝ : Module R Q\... | [
"R : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : AddCommGroup P\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nQ : Type u_5\ninst✝¹ : AddCommGroup Q\ninst✝ : Module R Q\nhfg : Exact... | exact_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.Finiteness | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 53
} | {
"line": 146,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R M\ninst✝ : Module R N\ns : Set (M ⊗[R] N)\nhs : s.Finite\nw✝ : Submodule R M\nN' : Submodule R N\nleft✝ : Module.Finite R ↥w✝\nhfin : Module.Finite R ↥N'\nh : s ⊆ ↑(m... | [
"R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R M\ninst✝ : Module R N\ns : Set (M ⊗[R] N)\nhs : s.Finite\nw✝ : Submodule R M\nN' : Submodule R N\nleft✝ : Module.Finite R ↥w✝\nhfin : Module.Finite R ↥N'\nh : s ⊆ ↑(LinearMap.lTe... | rw [mapIncl, ← LinearMap.lTensor_comp_rTensor] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Nilpotent.Defs | {
"line": 158,
"column": 6
} | {
"line": 158,
"column": 15
} | {
"line": 158,
"column": 15
} | [
{
"pp": "R : Type u_1\nx y : R\ninst✝ : Semiring R\nh_comm : Commute x y\nh : IsNilpotent y\n⊢ IsNilpotent (x * y)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"NPow.toPow",
"instDistribOfSemiring",
"Dist... | [
"R : Type u_1\nx y : R\ninst✝ : Semiring R\nh_comm : Commute x y\nh : IsNilpotent y\n⊢ IsNilpotent (y * x)"
] | h_comm.eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Nilpotent.Defs | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 43
} | {
"line": 161,
"column": 0
} | [
{
"pp": "R : Type u_1\nx y : R\ninst✝ : Semiring R\nh_comm : Commute x y\nh : IsNilpotent y\n⊢ IsNilpotent (y * x)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"instDistribOfSemiring",
"Commute.symm",
"Distrib.toMul",
"Commute.isNilpotent_mul_right"
],
"u... | [] | exact h_comm.symm.isNilpotent_mul_right h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 473,
"column": 6
} | {
"line": 473,
"column": 22
} | {
"line": 474,
"column": 6
} | [
{
"pp": "case a.refine_4\nR : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nI : Ideal A\nx✝ : A ⊗[R] B\nhx : x✝ ∈ ↑(Submodule.restrictScalars R (Submodule.span (A ⊗[R] B) (⇑includeLeft '' ↑I)))\na : A ⊗[R] B\nx... | [] | induction a with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 537,
"column": 6
} | {
"line": 537,
"column": 22
} | {
"line": 538,
"column": 6
} | [
{
"pp": "case a.refine_4\nR : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nI : Ideal B\nx✝ : A ⊗[R] B\nhx : x✝ ∈ ↑(Submodule.restrictScalars R (Submodule.span (A ⊗[R] B) (⇑includeRight '' ↑I)))\na : A ⊗[R] B\n... | [] | induction a with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Flat.Basic | {
"line": 361,
"column": 2
} | {
"line": 363,
"column": 30
} | {
"line": 364,
"column": 2
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Small.{v', u} R\n⊢ Flat R M ↔\n ∀ ⦃N N' N'' : Type v'⦄ [inst : AddCommGroup N] [inst_1 : AddCommGroup N'] [inst_2 : AddCommGroup N'']\n [inst_3 : Module R N] [inst_4 : Module R N'] [inst_5 : Modul... | [
"R : Type u\nM : Type v\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Small.{v', u} R\nH :\n ∀ ⦃N N' N'' : Type v'⦄ [inst : AddCommGroup N] [inst_1 : AddCommGroup N'] [inst_2 : AddCommGroup N'']\n [inst_3 : Module R N] [inst_4 : Module R N'] [inst_5 : Module R N''] ⦃f : N →ₗ[R] N'⦄... | refine ⟨fun _ ↦ lTensor_exact _, fun H ↦ iff_lTensor_preserves_injective_linearMap'.mpr
fun N' N'' _ _ _ _ L hL ↦ LinearMap.ker_eq_bot |>.mp <| eq_bot_iff |>.mpr
fun x (hx : _ = 0) ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Central.TensorProduct | {
"line": 53,
"column": 2
} | {
"line": 59,
"column": 46
} | {
"line": 61,
"column": 0
} | [
{
"pp": "K : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝⁴ : CommSemiring K\ninst✝³ : Semiring B\ninst✝² : Semiring C\ninst✝¹ : Algebra K B\ninst✝ : Algebra K C\nx : B ⊗[K] C\nhx : x ∈ Subalgebra.map includeRight (Subalgebra.center K C)\n⊢ x ∈ Subalgebra.center K (B ⊗[K] C)",
"ppTerm": "?m.36",
"assigned... | [] | simp only [Subalgebra.mem_map, Subalgebra.mem_center_iff] at hx ⊢
obtain ⟨c, hc0, rfl⟩ := hx
intro bc
induction bc using TensorProduct.induction_on with
| zero => simp
| tmul b c' => simp [hc0]
| add _ _ _ _ => simp_all [add_mul, mul_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Central.TensorProduct | {
"line": 53,
"column": 2
} | {
"line": 59,
"column": 46
} | {
"line": 61,
"column": 0
} | [
{
"pp": "K : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝⁴ : CommSemiring K\ninst✝³ : Semiring B\ninst✝² : Semiring C\ninst✝¹ : Algebra K B\ninst✝ : Algebra K C\nx : B ⊗[K] C\nhx : x ∈ Subalgebra.map includeRight (Subalgebra.center K C)\n⊢ x ∈ Subalgebra.center K (B ⊗[K] C)",
"ppTerm": "?m.36",
"assigned... | [] | simp only [Subalgebra.mem_map, Subalgebra.mem_center_iff] at hx ⊢
obtain ⟨c, hc0, rfl⟩ := hx
intro bc
induction bc using TensorProduct.induction_on with
| zero => simp
| tmul b c' => simp [hc0]
| add _ _ _ _ => simp_all [add_mul, mul_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Balance | {
"line": 57,
"column": 46
} | {
"line": 57,
"column": 72
} | {
"line": 59,
"column": 0
} | [
{
"pp": "ι : Type u_1\nH : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : Fintype ι\ninst✝⁵ : AddCommGroup G\ninst✝⁴ : Module ℚ≥0 G\ninst✝³ : AddCommGroup H\ninst✝² : Module ℚ≥0 H\ninst✝¹ : FunLike F G H\ninst✝ : LinearMapClass F ℚ≥0 G H\ng : F\nf : ι → G\na : ι\n⊢ g (balance f a) = balance (⇑g ∘ f) a",
"pp... | [] | simp [balance, map_expect] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.BigOperators.Balance | {
"line": 57,
"column": 46
} | {
"line": 57,
"column": 72
} | {
"line": 59,
"column": 0
} | [
{
"pp": "ι : Type u_1\nH : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : Fintype ι\ninst✝⁵ : AddCommGroup G\ninst✝⁴ : Module ℚ≥0 G\ninst✝³ : AddCommGroup H\ninst✝² : Module ℚ≥0 H\ninst✝¹ : FunLike F G H\ninst✝ : LinearMapClass F ℚ≥0 G H\ng : F\nf : ι → G\na : ι\n⊢ g (balance f a) = balance (⇑g ∘ f) a",
"pp... | [] | simp [balance, map_expect] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Balance | {
"line": 57,
"column": 46
} | {
"line": 57,
"column": 72
} | {
"line": 59,
"column": 0
} | [
{
"pp": "ι : Type u_1\nH : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : Fintype ι\ninst✝⁵ : AddCommGroup G\ninst✝⁴ : Module ℚ≥0 G\ninst✝³ : AddCommGroup H\ninst✝² : Module ℚ≥0 H\ninst✝¹ : FunLike F G H\ninst✝ : LinearMapClass F ℚ≥0 G H\ng : F\nf : ι → G\na : ι\n⊢ g (balance f a) = balance (⇑g ∘ f) a",
"pp... | [] | simp [balance, map_expect] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Field | {
"line": 36,
"column": 53
} | {
"line": 37,
"column": 22
} | {
"line": 39,
"column": 0
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝ : Fintype β\ns : Finset α\nt : α → Finset β\nh : (↑s).PairwiseDisjoint t\n⊢ (s.disjiUnion t h).dens = ∑ a ∈ s, (t a).dens",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHDiv",
"Gro... | [] | by
simp [dens, sum_div] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.BigOperators.Ring.Nat | {
"line": 45,
"column": 47
} | {
"line": 45,
"column": 52
} | {
"line": 45,
"column": 52
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nf : ι → M\ns : Finset M\nhb : ∀ b ∈ s, {a | f a = b}.Finite\nt : Finset M := ⋯.toFinset\n⊢ ∀ x ∈ s, x ∉ t → Nat.card { a // f a = x } = 0",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"not_exists._simp_1",
"SetLike.mem_coe._simp_1",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.BigOperators.Ring.Nat | {
"line": 45,
"column": 47
} | {
"line": 45,
"column": 52
} | {
"line": 45,
"column": 52
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nf : ι → M\ns : Finset M\nhb : ∀ b ∈ s, {a | f a = b}.Finite\nt : Finset M := ⋯.toFinset\n⊢ ∀ x ∈ s, x ∉ t → Nat.card { a // f a = x } = 0",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"not_exists._simp_1",
"SetLike.mem_coe._simp_1",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Ring.Nat | {
"line": 45,
"column": 47
} | {
"line": 45,
"column": 52
} | {
"line": 45,
"column": 52
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nf : ι → M\ns : Finset M\nhb : ∀ b ∈ s, {a | f a = b}.Finite\nt : Finset M := ⋯.toFinset\n⊢ ∀ x ∈ s, x ∉ t → Nat.card { a // f a = x } = 0",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"not_exists._simp_1",
"SetLike.mem_coe._simp_1",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Ring.Nat | {
"line": 46,
"column": 78
} | {
"line": 46,
"column": 83
} | {
"line": 46,
"column": 83
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nf : ι → M\ns : Finset M\nhb : ∀ b ∈ s, {a | f a = b}.Finite\nt : Finset M := ⋯.toFinset\n⊢ ∀ b ∈ ↑t, (f ⁻¹' {b}).Finite",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"SetLike.mem_coe._simp_1",
"congrArg",
"Finset",
"Classical.p... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.BigOperators.Ring.Nat | {
"line": 46,
"column": 78
} | {
"line": 46,
"column": 83
} | {
"line": 46,
"column": 83
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nf : ι → M\ns : Finset M\nhb : ∀ b ∈ s, {a | f a = b}.Finite\nt : Finset M := ⋯.toFinset\n⊢ ∀ b ∈ ↑t, (f ⁻¹' {b}).Finite",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"SetLike.mem_coe._simp_1",
"congrArg",
"Finset",
"Classical.p... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Ring.Nat | {
"line": 46,
"column": 78
} | {
"line": 46,
"column": 83
} | {
"line": 46,
"column": 83
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nf : ι → M\ns : Finset M\nhb : ∀ b ∈ s, {a | f a = b}.Finite\nt : Finset M := ⋯.toFinset\n⊢ ∀ b ∈ ↑t, (f ⁻¹' {b}).Finite",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"SetLike.mem_coe._simp_1",
"congrArg",
"Finset",
"Classical.p... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Ring.Nat | {
"line": 50,
"column": 51
} | {
"line": 50,
"column": 56
} | {
"line": 50,
"column": 56
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nf : ι → M\ns : Finset M\nhb : ∀ b ∈ s, {a | f a = b}.Finite\nt : Finset M := ⋯.toFinset\nht : (f ⁻¹' ↑t).Finite\nm : M\nhm : m ∈ t\n⊢ m ∈ s",
"ppTerm": "?m.119",
"assigned": true,
"usedConstants": [
"SetLike.mem_coe._simp_1",
"congrArg",
"Finset... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.BigOperators.Ring.Nat | {
"line": 50,
"column": 51
} | {
"line": 50,
"column": 56
} | {
"line": 50,
"column": 56
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nf : ι → M\ns : Finset M\nhb : ∀ b ∈ s, {a | f a = b}.Finite\nt : Finset M := ⋯.toFinset\nht : (f ⁻¹' ↑t).Finite\nm : M\nhm : m ∈ t\n⊢ m ∈ s",
"ppTerm": "?m.119",
"assigned": true,
"usedConstants": [
"SetLike.mem_coe._simp_1",
"congrArg",
"Finset... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Ring.Nat | {
"line": 50,
"column": 51
} | {
"line": 50,
"column": 56
} | {
"line": 50,
"column": 56
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nf : ι → M\ns : Finset M\nhb : ∀ b ∈ s, {a | f a = b}.Finite\nt : Finset M := ⋯.toFinset\nht : (f ⁻¹' ↑t).Finite\nm : M\nhm : m ∈ t\n⊢ m ∈ s",
"ppTerm": "?m.119",
"assigned": true,
"usedConstants": [
"SetLike.mem_coe._simp_1",
"congrArg",
"Finset... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.IsTensorProduct | {
"line": 83,
"column": 2
} | {
"line": 85,
"column": 6
} | {
"line": 87,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommSemiring R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\ninst✝⁵ : AddCommMonoid M₁\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M₁\ninst✝¹ : Module R M₂\ninst✝ : Module R M\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nh : IsTensorProduct f\nx₁ : M₁\nx₂ : M₂\n⊢ h.equiv.... | [] | apply h.equiv.injective
refine (h.equiv.apply_symm_apply _).trans ?_
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.IsTensorProduct | {
"line": 83,
"column": 2
} | {
"line": 85,
"column": 6
} | {
"line": 87,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommSemiring R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\ninst✝⁵ : AddCommMonoid M₁\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M₁\ninst✝¹ : Module R M₂\ninst✝ : Module R M\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nh : IsTensorProduct f\nx₁ : M₁\nx₂ : M₂\n⊢ h.equiv.... | [] | apply h.equiv.injective
refine (h.equiv.apply_symm_apply _).trans ?_
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.List.Sym | {
"line": 64,
"column": 4
} | {
"line": 64,
"column": 17
} | {
"line": 65,
"column": 4
} | [
{
"pp": "case cons\nα : Type u_1\na b x : α\nxs : List α\nih : s(a, b) ∈ xs.sym2 → a ∈ xs\nh : s(a, b) ∈ (x :: xs).sym2\n⊢ a ∈ x :: xs",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"id",
"List.cons",
"List",
... | [
"case cons\nα : Type u_1\na b x : α\nxs : List α\nih : s(a, b) ∈ xs.sym2 → a ∈ xs\nh : s(a, b) ∈ (x :: xs).sym2\n⊢ a = x ∨ a ∈ xs"
] | rw [mem_cons] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.IsTensorProduct | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 28
} | {
"line": 123,
"column": 0
} | [
{
"pp": "case add\nR : Type u_1\ninst✝⁶ : CommSemiring R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\ninst✝⁵ : AddCommMonoid M₁\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M₁\ninst✝¹ : Module R M₂\ninst✝ : Module R M\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nh : IsTensorProduct f\nmotive : M → Prop\... | [] | apply add <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.Finset.Sym | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 48
} | {
"line": 69,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : DecidableEq α\na : α\ns : Finset α\nha : a ∉ s\n⊢ (insert a s).sym2 = image (fun b ↦ s(a, b)) (insert a s) ∪ s.sym2",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.instUnion",
"Sym2.mk",
"Finset.cons",
... | [] | simpa [map_eq_image] using! sym2_cons a s ha | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.Data.Finset.Sym | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 48
} | {
"line": 69,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : DecidableEq α\na : α\ns : Finset α\nha : a ∉ s\n⊢ (insert a s).sym2 = image (fun b ↦ s(a, b)) (insert a s) ∪ s.sym2",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.instUnion",
"Sym2.mk",
"Finset.cons",
... | [] | simpa [map_eq_image] using! sym2_cons a s ha | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.Sym | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 48
} | {
"line": 69,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : DecidableEq α\na : α\ns : Finset α\nha : a ∉ s\n⊢ (insert a s).sym2 = image (fun b ↦ s(a, b)) (insert a s) ∪ s.sym2",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.instUnion",
"Sym2.mk",
"Finset.cons",
... | [] | simpa [map_eq_image] using! sym2_cons a s ha | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.List.Sym | {
"line": 214,
"column": 2
} | {
"line": 217,
"column": 76
} | {
"line": 219,
"column": 0
} | [
{
"pp": "α : Type u_1\nxs : List α\n⊢ List.sym 1 xs = map (fun x ↦ x ::ₛ Sym.nil) xs",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Sym.nil",
"List.map_cons",
"congrArg",
"Sym.cons",
"List.map",
"False.elim",
"noCon... | [] | induction xs with
| nil => simp only [List.sym, Nat.succ_eq_add_one, Nat.reduceAdd, map_nil]
| cons x xs ih =>
rw [map_cons, ← ih, List.sym, List.sym, map_singleton, singleton_append] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Data.List.Sym | {
"line": 214,
"column": 2
} | {
"line": 217,
"column": 76
} | {
"line": 219,
"column": 0
} | [
{
"pp": "α : Type u_1\nxs : List α\n⊢ List.sym 1 xs = map (fun x ↦ x ::ₛ Sym.nil) xs",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Sym.nil",
"List.map_cons",
"congrArg",
"Sym.cons",
"List.map",
"False.elim",
"noCon... | [] | induction xs with
| nil => simp only [List.sym, Nat.succ_eq_add_one, Nat.reduceAdd, map_nil]
| cons x xs ih =>
rw [map_cons, ← ih, List.sym, List.sym, map_singleton, singleton_append] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.List.Sym | {
"line": 214,
"column": 2
} | {
"line": 217,
"column": 76
} | {
"line": 219,
"column": 0
} | [
{
"pp": "α : Type u_1\nxs : List α\n⊢ List.sym 1 xs = map (fun x ↦ x ::ₛ Sym.nil) xs",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Sym.nil",
"List.map_cons",
"congrArg",
"Sym.cons",
"List.map",
"False.elim",
"noCon... | [] | induction xs with
| nil => simp only [List.sym, Nat.succ_eq_add_one, Nat.reduceAdd, map_nil]
| cons x xs ih =>
rw [map_cons, ← ih, List.sym, List.sym, map_singleton, singleton_append] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.List.Sym | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 76
} | {
"line": 224,
"column": 4
} | [
{
"pp": "case cons\nα : Type u_1\nxs✝ : List α\nx : α\nxs : List α\nih : map (⇑(Sym2.equivSym α)) xs.sym2 = List.sym 2 xs\n⊢ map (⇑(Sym2.equivSym α)) (x :: xs).sym2 = List.sym 2 (x :: xs)",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"Sy... | [
"case cons\nα : Type u_1\nxs✝ : List α\nx : α\nxs : List α\nih : map (⇑(Sym2.equivSym α)) xs.sym2 = List.sym 2 xs\n⊢ map (⇑(Sym2.equivSym α) ∘ fun y ↦ s(x, y)) (x :: xs) ++ map (⇑(Sym2.equivSym α)) xs.sym2 =\n map ((fun p ↦ x ::ₛ p) ∘ fun x ↦ x ::ₛ Sym.nil) (x :: xs) ++ map (⇑(Sym2.equivSym α)) xs.sym2"
] | rw [List.sym, ← ih, sym_one_eq, map_map, List.sym2, map_append, map_map] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.BigOperators.Sym | {
"line": 28,
"column": 12
} | {
"line": 28,
"column": 17
} | {
"line": 30,
"column": 0
} | [
{
"pp": "case refine_1\nι : Type u_1\nM : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid M\ns : Finset ι\np : Sym2 ι → M\n⊢ ∀ (i : { p // p.1 ≤ p.2 }),\n i ∈ Finset.subtype (fun i ↦ i.1 ≤ i.2) s.offDiag ↔ Sym2.sortEquiv.symm i ∈ {i ∈ s.sym2 | ¬i.IsDiag}",
"ppTerm": "?refine_1",
"assigned": t... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.BigOperators.Sym | {
"line": 28,
"column": 12
} | {
"line": 28,
"column": 17
} | {
"line": 30,
"column": 0
} | [
{
"pp": "case refine_1\nι : Type u_1\nM : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid M\ns : Finset ι\np : Sym2 ι → M\n⊢ ∀ (i : { p // p.1 ≤ p.2 }),\n i ∈ Finset.subtype (fun i ↦ i.1 ≤ i.2) s.offDiag ↔ Sym2.sortEquiv.symm i ∈ {i ∈ s.sym2 | ¬i.IsDiag}",
"ppTerm": "?refine_1",
"assigned": t... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Sym | {
"line": 28,
"column": 12
} | {
"line": 28,
"column": 17
} | {
"line": 30,
"column": 0
} | [
{
"pp": "case refine_1\nι : Type u_1\nM : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid M\ns : Finset ι\np : Sym2 ι → M\n⊢ ∀ (i : { p // p.1 ≤ p.2 }),\n i ∈ Finset.subtype (fun i ↦ i.1 ≤ i.2) s.offDiag ↔ Sym2.sortEquiv.symm i ∈ {i ∈ s.sym2 | ¬i.IsDiag}",
"ppTerm": "?refine_1",
"assigned": t... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Sym | {
"line": 28,
"column": 12
} | {
"line": 28,
"column": 17
} | {
"line": 30,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u_1\nM : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid M\ns : Finset ι\np : Sym2 ι → M\n⊢ ∀ i ∈ Finset.subtype (fun i ↦ i.1 ≤ i.2) s.offDiag, p s((↑i).1, (↑i).2) = p (Sym2.sortEquiv.symm i)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.BigOperators.Sym | {
"line": 28,
"column": 12
} | {
"line": 28,
"column": 17
} | {
"line": 30,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u_1\nM : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid M\ns : Finset ι\np : Sym2 ι → M\n⊢ ∀ i ∈ Finset.subtype (fun i ↦ i.1 ≤ i.2) s.offDiag, p s((↑i).1, (↑i).2) = p (Sym2.sortEquiv.symm i)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Sym | {
"line": 28,
"column": 12
} | {
"line": 28,
"column": 17
} | {
"line": 30,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u_1\nM : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid M\ns : Finset ι\np : Sym2 ι → M\n⊢ ∀ i ∈ Finset.subtype (fun i ↦ i.1 ≤ i.2) s.offDiag, p s((↑i).1, (↑i).2) = p (Sym2.sortEquiv.symm i)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Sym | {
"line": 24,
"column": 89
} | {
"line": 28,
"column": 17
} | {
"line": 30,
"column": 0
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid M\ns : Finset ι\np : Sym2 ι → M\n⊢ ∑ i ∈ s.sym2 with ¬i.IsDiag, p i = ∑ i ∈ s.offDiag with i.1 < i.2, p s(i.1, i.2)",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
... | [] | by
rw [Finset.offDiag_filter_lt_eq_filter_le]
conv_rhs => rw [← Finset.sum_subtype_eq_sum_filter]
refine (Finset.sum_equiv Sym2.sortEquiv.symm ?_ ?_).symm
all_goals aesop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.List.Sym | {
"line": 269,
"column": 6
} | {
"line": 269,
"column": 19
} | {
"line": 270,
"column": 6
} | [
{
"pp": "case inr\nα : Type u_1\nn✝ : ℕ\nxs✝ : List α\na : α\nn : ℕ\nx : α\nxs : List α\nz : Sym α (n + 1)\nha : a ∈ z\nhz : z ∈ List.sym (n + 1) xs\n⊢ a ∈ x :: xs",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"id",
"Li... | [
"case inr\nα : Type u_1\nn✝ : ℕ\nxs✝ : List α\na : α\nn : ℕ\nx : α\nxs : List α\nz : Sym α (n + 1)\nha : a ∈ z\nhz : z ∈ List.sym (n + 1) xs\n⊢ a = x ∨ a ∈ xs"
] | rw [mem_cons] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Sym.Sym2 | {
"line": 266,
"column": 15
} | {
"line": 266,
"column": 20
} | {
"line": 268,
"column": 0
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ng : β → γ\nf : α → β\nx✝ y✝ : α\n⊢ map g (map f s(x✝, y✝)) = map (g ∘ f) s(x✝, y✝)",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Sym2.map",
"Sym2.mk",
"Eq.refl",
"Sym2"
],
"usedFVars": [
"γ",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Sym.Sym2 | {
"line": 317,
"column": 6
} | {
"line": 317,
"column": 11
} | {
"line": 318,
"column": 4
} | [
{
"pp": "α : Type u_1\na b c w✝ : α\nh : b = a ∧ c = w✝ ∨ b = w✝ ∧ c = a\n⊢ a = b ∨ a = c",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"congrArg",
"true_or",
"Or.casesOn",
"And",
"congr",
"True",
"eq_self",
"of_eq_true",
"Or",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Sym.Sym2 | {
"line": 332,
"column": 4
} | {
"line": 332,
"column": 9
} | {
"line": 334,
"column": 0
} | [
{
"pp": "case mk.mk\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nz z' : Sym2 α\nx y x' y' : α\nh : ∀ (x_1 : α), Sym2.Mem x_1 (Quot.mk (Rel α) (x, y)) ↔ Sym2.Mem x_1 (Quot.mk (Rel α) (x', y'))\nhx : True ∨ x = y ↔ x = x' ∨ x = y'\nhy : y = x ∨ True ↔ y = x' ∨ y = y'\nhx' : x' = x ∨ x' = y ↔ True ∨ x' = y'\nhy' : y... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Sym.Sym2 | {
"line": 390,
"column": 4
} | {
"line": 390,
"column": 9
} | {
"line": 391,
"column": 2
} | [
{
"pp": "case mp.h\nα : Type u_1\nx y : α\nhne : x ≠ y\nx✝ y✝ : α\n⊢ (x = x✝ ∨ x = y✝) ∧ (y = x✝ ∨ y = y✝) → s(x✝, y✝) = s(x, y)",
"ppTerm": "?mp.h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Sym2.Rel",
"Sym2.eq._simp_1",
"eq_false",
"Sym2.mk",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Sym.Sym2 | {
"line": 406,
"column": 2
} | {
"line": 406,
"column": 7
} | {
"line": 408,
"column": 0
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\nf : α → β\nb : β\nx✝ y✝ : α\n⊢ b ∈ map f s(x✝, y✝) ↔ ∃ a ∈ s(x✝, y✝), f a = b",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Sym2.map",
"Sym2.mem_iff._simp_1",
"Sym2.mk",
"congrArg",
"true_or",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Sym.Sym2 | {
"line": 437,
"column": 4
} | {
"line": 437,
"column": 34
} | {
"line": 438,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nP : α → Prop\nf : (a : α) → P a → β\ns : Sym2 α\ng : (p : α × α) → (∀ a ∈ s(p.1, p.2), P a) → Sym2 β := fun p H ↦ s(f p.1 ⋯, f p.2 ⋯)\np q : α × α\nhpq✝ : Rel α p q\nhpq : p = q ∨ p = q.swap\nHq : ∀ a ∈ Quot.mk (Rel α) q, P a\nHp : ∀ a ∈ s(p.1, p.2), P a\nh :\n... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nP : α → Prop\nf : (a : α) → P a → β\ns : Sym2 α\ng : (p : α × α) → (∀ a ∈ s(p.1, p.2), P a) → Sym2 β := fun p H ↦ s(f p.1 ⋯, f p.2 ⋯)\np q : α × α\nhpq✝ : Rel α p q\nhpq : p = q ∨ p = q.swap\nHq : ∀ a ∈ Quot.mk (Rel α) q, P a\nHp : ∀ a ∈ s(p.1, p.2), P a\nh :\n ∀ {s₂ : Sy... | refine h.trans (Quot.sound ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Data.Sym.Sym2 | {
"line": 455,
"column": 2
} | {
"line": 455,
"column": 7
} | {
"line": 457,
"column": 0
} | [
{
"pp": "case mk\nα : Type u_1\nβ : Type u_2\nP : α → Prop\nf : (a : α) → P a → β\nz : Sym2 α\nb : β\nx y : α\nh : ∀ a ∈ Quot.mk (Rel α) (x, y), P a\n⊢ b ∈ s(f x ⋯, f y ⋯) ↔ ∃ a, ∃ (ha : a ∈ Quot.mk (Rel α) (x, y)), b = f a ⋯",
"ppTerm": "?mk",
"assigned": true,
"usedConstants": [
"Iff.mpr",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Sym.Sym2 | {
"line": 453,
"column": 2
} | {
"line": 455,
"column": 7
} | {
"line": 457,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nP : α → Prop\nf : (a : α) → P a → β\nz : Sym2 α\nh : ∀ a ∈ z, P a\nb : β\n⊢ b ∈ pmap f z h ↔ ∃ a, ∃ (ha : a ∈ z), b = f a ⋯",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Sym2.mem_mk_right",
"Sym2.Rel",
... | [] | obtain ⟨x, y⟩ := z
rw [pmap_pair f x y h]
aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Sym.Sym2 | {
"line": 453,
"column": 2
} | {
"line": 455,
"column": 7
} | {
"line": 457,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nP : α → Prop\nf : (a : α) → P a → β\nz : Sym2 α\nh : ∀ a ∈ z, P a\nb : β\n⊢ b ∈ pmap f z h ↔ ∃ a, ∃ (ha : a ∈ z), b = f a ⋯",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Sym2.mem_mk_right",
"Sym2.Rel",
... | [] | obtain ⟨x, y⟩ := z
rw [pmap_pair f x y h]
aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Functor.Basic | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 55
} | {
"line": 151,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nobj : C → D\nmap : {X Y : C} → (X ⟶ Y) → (obj X ⟶ obj Y)\nmap_id✝¹ : ∀ (X : C), map (𝟙 X) = 𝟙 (obj X)\nmap_comp✝¹ : ∀ {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z), map (f ≫ g) = map f ≫ map g\nobj' : C → D\nmap' : {X Y : C} →... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nobj : C → D\nmap : {X Y : C} → (X ⟶ Y) → (obj X ⟶ obj Y)\nmap_id✝¹ : ∀ (X : C), map (𝟙 X) = 𝟙 (obj X)\nmap_comp✝¹ : ∀ {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z), map (f ≫ g) = map f ≫ map g\nmap' : {X Y : C} → (X ⟶ Y) → (obj X ⟶ obj Y)... | obtain rfl : obj = obj' := congr_arg Prefunctor.obj h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Data.Sym.Sym2 | {
"line": 660,
"column": 38
} | {
"line": 660,
"column": 43
} | {
"line": 661,
"column": 2
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\nsym : Std.Symm r\nh : ∀ (a : α), ¬r a a\n⊢ ∀ (x y : α), s(x, y) ∈ fromRel sym → ¬s(x, y).IsDiag",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Sym2.fromRel_prop._simp_1",
"eq_false",
"Sym2.mk",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Category.Basic | {
"line": 312,
"column": 66
} | {
"line": 312,
"column": 71
} | {
"line": 314,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nP : Prop\ninst✝ : Decidable P\nX Y Z : C\nf : X ⟶ Y\ng g' : Y ⟶ Z\n⊢ (f ≫ if P then g else g') = if P then f ≫ g else f ≫ g'",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Decidable.casesOn",
"CategoryTheory.Cat... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Category.Basic | {
"line": 312,
"column": 66
} | {
"line": 312,
"column": 71
} | {
"line": 314,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nP : Prop\ninst✝ : Decidable P\nX Y Z : C\nf : X ⟶ Y\ng g' : Y ⟶ Z\n⊢ (f ≫ if P then g else g') = if P then f ≫ g else f ≫ g'",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Decidable.casesOn",
"CategoryTheory.Cat... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Category.Basic | {
"line": 312,
"column": 66
} | {
"line": 312,
"column": 71
} | {
"line": 314,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nP : Prop\ninst✝ : Decidable P\nX Y Z : C\nf : X ⟶ Y\ng g' : Y ⟶ Z\n⊢ (f ≫ if P then g else g') = if P then f ≫ g else f ≫ g'",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Decidable.casesOn",
"CategoryTheory.Cat... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Category.Basic | {
"line": 317,
"column": 82
} | {
"line": 317,
"column": 87
} | {
"line": 319,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nP : Prop\ninst✝ : Decidable P\nX Y Z : C\nf : X ⟶ Y\ng : P → (Y ⟶ Z)\ng' : ¬P → (Y ⟶ Z)\n⊢ (f ≫ if h : P then g h else g' h) = if h : P then f ≫ g h else f ≫ g' h",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Decidab... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Category.Basic | {
"line": 317,
"column": 82
} | {
"line": 317,
"column": 87
} | {
"line": 319,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nP : Prop\ninst✝ : Decidable P\nX Y Z : C\nf : X ⟶ Y\ng : P → (Y ⟶ Z)\ng' : ¬P → (Y ⟶ Z)\n⊢ (f ≫ if h : P then g h else g' h) = if h : P then f ≫ g h else f ≫ g' h",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Decidab... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Category.Basic | {
"line": 317,
"column": 82
} | {
"line": 317,
"column": 87
} | {
"line": 319,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nP : Prop\ninst✝ : Decidable P\nX Y Z : C\nf : X ⟶ Y\ng : P → (Y ⟶ Z)\ng' : ¬P → (Y ⟶ Z)\n⊢ (f ≫ if h : P then g h else g' h) = if h : P then f ≫ g h else f ≫ g' h",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Decidab... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Category.Basic | {
"line": 334,
"column": 19
} | {
"line": 334,
"column": 24
} | {
"line": 334,
"column": 24
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX✝ Y Z X Z✝ : C\ng h : X ⟶ Z✝\nw : 𝟙 X ≫ g = 𝟙 X ≫ h\n⊢ g = h",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.CategoryStruct.id",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Category.Basic | {
"line": 334,
"column": 19
} | {
"line": 334,
"column": 24
} | {
"line": 334,
"column": 24
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX✝ Y Z X Z✝ : C\ng h : X ⟶ Z✝\nw : 𝟙 X ≫ g = 𝟙 X ≫ h\n⊢ g = h",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.CategoryStruct.id",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Category.Basic | {
"line": 334,
"column": 19
} | {
"line": 334,
"column": 24
} | {
"line": 334,
"column": 24
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX✝ Y Z X Z✝ : C\ng h : X ⟶ Z✝\nw : 𝟙 X ≫ g = 𝟙 X ≫ h\n⊢ g = h",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.CategoryStruct.id",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Sym.Sym2 | {
"line": 952,
"column": 4
} | {
"line": 952,
"column": 9
} | {
"line": 952,
"column": 9
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nz : Sym2 α\nf : α → β\ninst✝ : DecidableEq α\na b c d e : α\nh : Rel α (b, c) (d, e)\nhy : a ∈ s(d, e)\nthis :\n ∀ {f : Sym2 α} {g : s(b, c) = f} {h : a ∈ f},\n Eq.ndrec (motive := fun x ↦ a ∈ x → α) (fun x ↦ if a = b then c else b) g h = if a = b then c el... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
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