module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Category.ModuleCat.Semi | {
"line": 267,
"column": 47
} | {
"line": 267,
"column": 52
} | {
"line": 267,
"column": 52
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nX₁ X₂ : Type v\nX Y : SemimoduleCat R\ni : X ≅ Y\n⊢ Hom.hom i.hom ∘ₗ Hom.hom i.inv = LinearMap.id",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"SemimoduleCat.isModule",
"LinearMap.id",
"SemimoduleCat.moduleCategory",
"Semi... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Semi | {
"line": 267,
"column": 58
} | {
"line": 267,
"column": 63
} | {
"line": 267,
"column": 63
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nX₁ X₂ : Type v\nX Y : SemimoduleCat R\ni : X ≅ Y\n⊢ Hom.hom i.inv ∘ₗ Hom.hom i.hom = LinearMap.id",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"SemimoduleCat.isModule",
"LinearMap.id",
"SemimoduleCat.moduleCategory",
"Semi... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Category.ModuleCat.Semi | {
"line": 267,
"column": 58
} | {
"line": 267,
"column": 63
} | {
"line": 267,
"column": 63
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nX₁ X₂ : Type v\nX Y : SemimoduleCat R\ni : X ≅ Y\n⊢ Hom.hom i.inv ∘ₗ Hom.hom i.hom = LinearMap.id",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"SemimoduleCat.isModule",
"LinearMap.id",
"SemimoduleCat.moduleCategory",
"Semi... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Semi | {
"line": 267,
"column": 58
} | {
"line": 267,
"column": 63
} | {
"line": 267,
"column": 63
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nX₁ X₂ : Type v\nX Y : SemimoduleCat R\ni : X ≅ Y\n⊢ Hom.hom i.inv ∘ₗ Hom.hom i.hom = LinearMap.id",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"SemimoduleCat.isModule",
"LinearMap.id",
"SemimoduleCat.moduleCategory",
"Semi... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits | {
"line": 146,
"column": 17
} | {
"line": 146,
"column": 22
} | {
"line": 147,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nj j' : WalkingParallelPair\n⊢ ∀ (x : WalkingParallelPairHom j j'),\n x ∈\n WalkingParallelPair.recOn j\n (WalkingParallelPair.recOn j' [WalkingParallelPairHom.id zero].toFinset [left, right].toFinset)\n (WalkingParallelPair.recOn j' ∅ [Walkin... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits | {
"line": 146,
"column": 17
} | {
"line": 146,
"column": 22
} | {
"line": 147,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nj j' : WalkingParallelPair\n⊢ ∀ (x : WalkingParallelPairHom j j'),\n x ∈\n WalkingParallelPair.recOn j\n (WalkingParallelPair.recOn j' [WalkingParallelPairHom.id zero].toFinset [left, right].toFinset)\n (WalkingParallelPair.recOn j' ∅ [Walkin... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits | {
"line": 146,
"column": 17
} | {
"line": 146,
"column": 22
} | {
"line": 147,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nj j' : WalkingParallelPair\n⊢ ∀ (x : WalkingParallelPairHom j j'),\n x ∈\n WalkingParallelPair.recOn j\n (WalkingParallelPair.recOn j' [WalkingParallelPairHom.id zero].toFinset [left, right].toFinset)\n (WalkingParallelPair.recOn j' ∅ [Walkin... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.ObjectProperty.CompleteLattice | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 25
} | {
"line": 87,
"column": 4
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝ : Category.{v, u} C\nα : Sort u_1\nP : α → ObjectProperty C\n⊢ (⨆ a, P a).isoClosure ≤ ⨆ a, (P a).isoClosure",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"iSup",
"Prop.instCompleteLattice",
"Exists",
"CategoryTheory.I... | [
"case refine_1\nC : Type u\ninst✝ : Category.{v, u} C\nα : Sort u_1\nP : α → ObjectProperty C\nX Y : C\nhY : (⨆ a, P a) Y\ne : X ≅ Y\n⊢ (⨆ a, (P a).isoClosure) X"
] | rintro X ⟨Y, hY, ⟨e⟩⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 51
} | {
"line": 85,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nG : C ⥤ D\nX Y : C\ninst✝¹ : HasBinaryProduct X Y\ninst✝ : HasBinaryProduct (G.obj X) (G.obj Y)\ni : IsIso (prodComparison G X Y)\n⊢ IsLimit (G.mapCone (BinaryFan.mk prod.fst prod.snd))",
"ppTerm": "?m.50",
"a... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nG : C ⥤ D\nX Y : C\ninst✝¹ : HasBinaryProduct X Y\ninst✝ : HasBinaryProduct (G.obj X) (G.obj Y)\ni : IsIso (prodComparison G X Y)\n⊢ IsLimit (BinaryFan.mk (G.map prod.fst) (G.map prod.snd))"
] | apply (isLimitMapConeBinaryFanEquiv _ _ _).symm _ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 107,
"column": 30
} | {
"line": 107,
"column": 35
} | {
"line": 109,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ns : Fork f 0\n⊢ ∀ (j : WalkingParallelPair), s.π.app j = (Iso.refl s.pt).hom ≫ (Fork.ofι s.ι ⋯).π.app j",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Lim... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 107,
"column": 30
} | {
"line": 107,
"column": 35
} | {
"line": 109,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ns : Fork f 0\n⊢ ∀ (j : WalkingParallelPair), s.π.app j = (Iso.refl s.pt).hom ≫ (Fork.ofι s.ι ⋯).π.app j",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Lim... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 107,
"column": 30
} | {
"line": 107,
"column": 35
} | {
"line": 109,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ns : Fork f 0\n⊢ ∀ (j : WalkingParallelPair), s.π.app j = (Iso.refl s.pt).hom ≫ (Fork.ofι s.ι ⋯).π.app j",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Lim... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 124,
"column": 70
} | {
"line": 124,
"column": 75
} | {
"line": 126,
"column": 0
} | [
{
"pp": "case zero.zero.id\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\nD : Type u'\ninst✝² : Category.{v, u'} D\ninst✝¹ : HasZeroMorphisms D\nF : C ⥤ D\ninst✝ : F.IsEquivalence\napp : (j : WalkingParallelPair) → (parallelPair f 0 ⋙ F).obj j ≅ (parallelPair (F.map f)... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 124,
"column": 70
} | {
"line": 124,
"column": 75
} | {
"line": 126,
"column": 0
} | [
{
"pp": "case zero.one.left\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\nD : Type u'\ninst✝² : Category.{v, u'} D\ninst✝¹ : HasZeroMorphisms D\nF : C ⥤ D\ninst✝ : F.IsEquivalence\napp : (j : WalkingParallelPair) → (parallelPair f 0 ⋙ F).obj j ≅ (parallelPair (F.map f... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 124,
"column": 70
} | {
"line": 124,
"column": 75
} | {
"line": 126,
"column": 0
} | [
{
"pp": "case zero.one.right\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\nD : Type u'\ninst✝² : Category.{v, u'} D\ninst✝¹ : HasZeroMorphisms D\nF : C ⥤ D\ninst✝ : F.IsEquivalence\napp : (j : WalkingParallelPair) → (parallelPair f 0 ⋙ F).obj j ≅ (parallelPair (F.map ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 124,
"column": 70
} | {
"line": 124,
"column": 75
} | {
"line": 126,
"column": 0
} | [
{
"pp": "case one.one.id\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\nD : Type u'\ninst✝² : Category.{v, u'} D\ninst✝¹ : HasZeroMorphisms D\nF : C ⥤ D\ninst✝ : F.IsEquivalence\napp : (j : WalkingParallelPair) → (parallelPair f 0 ⋙ F).obj j ≅ (parallelPair (F.map f) 0... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 1075,
"column": 8
} | {
"line": 1075,
"column": 15
} | {
"line": 1075,
"column": 15
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : HasZeroMorphisms C\nX✝ Y✝ : C\nf✝ : X✝ ⟶ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝³ : HasImage f\ninst✝² : HasCokernel (image.ι f)\ninst✝¹ : HasCokernel f\ninst✝ : Epi (factorThruImage f)\nw : image.ι f ≫ cokernel.π f = 0\n⊢ image.ι f ≫ cokernel.π f = 0",
"ppTer... | [] | exact w | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Preadditive.AdditiveFunctor | {
"line": 146,
"column": 6
} | {
"line": 146,
"column": 65
} | {
"line": 146,
"column": 65
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nF : C ⥤ D\nhF : IsZero F\nx✝³ x✝² : C\nx✝¹ x✝ : x✝³ ⟶ x✝²\n⊢ 𝟙 (F.obj x✝²) = 0",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Categor... | [] | exact NatTrans.congr_app ((IsZero.iff_id_eq_zero _).1 hF) _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.Shapes.Biproducts | {
"line": 630,
"column": 36
} | {
"line": 630,
"column": 41
} | {
"line": 630,
"column": 41
} | [
{
"pp": "J✝ : Type w\nC✝ : Type uC\ninst✝⁸ : Category.{uC', uC} C✝\ninst✝⁷ : HasZeroMorphisms C✝\nD : Type uD\ninst✝⁶ : Category.{uD', uD} D\ninst✝⁵ : HasZeroMorphisms D\nF : J✝ → C✝\nJ : Type w\nK : Type u_1\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nf g : J → C\ninst✝² : HasBiproduc... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Limits.Shapes.Biproducts | {
"line": 630,
"column": 36
} | {
"line": 630,
"column": 41
} | {
"line": 630,
"column": 41
} | [
{
"pp": "J✝ : Type w\nC✝ : Type uC\ninst✝⁸ : Category.{uC', uC} C✝\ninst✝⁷ : HasZeroMorphisms C✝\nD : Type uD\ninst✝⁶ : Category.{uD', uD} D\ninst✝⁵ : HasZeroMorphisms D\nF : J✝ → C✝\nJ : Type w\nK : Type u_1\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nf g : J → C\ninst✝² : HasBiproduc... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Biproducts | {
"line": 630,
"column": 36
} | {
"line": 630,
"column": 41
} | {
"line": 630,
"column": 41
} | [
{
"pp": "J✝ : Type w\nC✝ : Type uC\ninst✝⁸ : Category.{uC', uC} C✝\ninst✝⁷ : HasZeroMorphisms C✝\nD : Type uD\ninst✝⁶ : Category.{uD', uD} D\ninst✝⁵ : HasZeroMorphisms D\nF : J✝ → C✝\nJ : Type w\nK : Type u_1\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nf g : J → C\ninst✝² : HasBiproduc... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Biproducts | {
"line": 637,
"column": 36
} | {
"line": 637,
"column": 41
} | {
"line": 637,
"column": 41
} | [
{
"pp": "J✝ : Type w\nC✝ : Type uC\ninst✝⁸ : Category.{uC', uC} C✝\ninst✝⁷ : HasZeroMorphisms C✝\nD : Type uD\ninst✝⁶ : Category.{uD', uD} D\ninst✝⁵ : HasZeroMorphisms D\nF : J✝ → C✝\nJ : Type w\nK : Type u_1\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nf g : J → C\ninst✝² : HasBiproduc... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Limits.Shapes.Biproducts | {
"line": 637,
"column": 36
} | {
"line": 637,
"column": 41
} | {
"line": 637,
"column": 41
} | [
{
"pp": "J✝ : Type w\nC✝ : Type uC\ninst✝⁸ : Category.{uC', uC} C✝\ninst✝⁷ : HasZeroMorphisms C✝\nD : Type uD\ninst✝⁶ : Category.{uD', uD} D\ninst✝⁵ : HasZeroMorphisms D\nF : J✝ → C✝\nJ : Type w\nK : Type u_1\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nf g : J → C\ninst✝² : HasBiproduc... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Biproducts | {
"line": 637,
"column": 36
} | {
"line": 637,
"column": 41
} | {
"line": 637,
"column": 41
} | [
{
"pp": "J✝ : Type w\nC✝ : Type uC\ninst✝⁸ : Category.{uC', uC} C✝\ninst✝⁷ : HasZeroMorphisms C✝\nD : Type uD\ninst✝⁶ : Category.{uD', uD} D\ninst✝⁵ : HasZeroMorphisms D\nF : J✝ → C✝\nJ : Type w\nK : Type u_1\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nf g : J → C\ninst✝² : HasBiproduc... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Biproducts | {
"line": 644,
"column": 38
} | {
"line": 644,
"column": 43
} | {
"line": 644,
"column": 43
} | [
{
"pp": "J✝ : Type w\nC✝ : Type uC\ninst✝⁸ : Category.{uC', uC} C✝\ninst✝⁷ : HasZeroMorphisms C✝\nD : Type uD\ninst✝⁶ : Category.{uD', uD} D\ninst✝⁵ : HasZeroMorphisms D\nF : J✝ → C✝\nJ : Type w\nK : Type u_1\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nf g : J → C\ninst✝² : HasBiproduc... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Limits.Shapes.Biproducts | {
"line": 644,
"column": 38
} | {
"line": 644,
"column": 43
} | {
"line": 644,
"column": 43
} | [
{
"pp": "J✝ : Type w\nC✝ : Type uC\ninst✝⁸ : Category.{uC', uC} C✝\ninst✝⁷ : HasZeroMorphisms C✝\nD : Type uD\ninst✝⁶ : Category.{uD', uD} D\ninst✝⁵ : HasZeroMorphisms D\nF : J✝ → C✝\nJ : Type w\nK : Type u_1\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nf g : J → C\ninst✝² : HasBiproduc... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Biproducts | {
"line": 644,
"column": 38
} | {
"line": 644,
"column": 43
} | {
"line": 644,
"column": 43
} | [
{
"pp": "J✝ : Type w\nC✝ : Type uC\ninst✝⁸ : Category.{uC', uC} C✝\ninst✝⁷ : HasZeroMorphisms C✝\nD : Type uD\ninst✝⁶ : Category.{uD', uD} D\ninst✝⁵ : HasZeroMorphisms D\nF : J✝ → C✝\nJ : Type w\nK : Type u_1\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nf g : J → C\ninst✝² : HasBiproduc... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts | {
"line": 246,
"column": 64
} | {
"line": 246,
"column": 72
} | {
"line": 246,
"column": 73
} | [
{
"pp": "J : Type w\nC : Type uC\ninst✝³ : Category.{uC', uC} C\ninst✝² : HasZeroMorphisms C\nD : Type uD\ninst✝¹ : Category.{uD', uD} D\ninst✝ : HasZeroMorphisms D\nP P' Q Q' X Y : C\nb : BinaryBicone X Y\nx✝ : Discrete WalkingPair\nas : WalkingPair\n⊢ b.toBicone.toCone.π.app { as := as } = (Iso.refl b.toBicon... | [
"case left\nJ : Type w\nC : Type uC\ninst✝³ : Category.{uC', uC} C\ninst✝² : HasZeroMorphisms C\nD : Type uD\ninst✝¹ : Category.{uD', uD} D\ninst✝ : HasZeroMorphisms D\nP P' Q Q' X Y : C\nb : BinaryBicone X Y\nx✝ : Discrete WalkingPair\n⊢ b.toBicone.toCone.π.app { as := WalkingPair.left } =\n (Iso.refl b.toBicon... | cases as | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts | {
"line": 253,
"column": 70
} | {
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{
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Mathlib.CategoryTheory.Preadditive.Biproducts | {
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Mathlib.CategoryTheory.Preadditive.Biproducts | {
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Mathlib.CategoryTheory.Preadditive.Biproducts | {
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Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts | {
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Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts | {
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Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts | {
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Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts | {
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Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts | {
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Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts | {
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Mathlib.CategoryTheory.Preadditive.Biproducts | {
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Mathlib.CategoryTheory.Preadditive.Biproducts | {
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Mathlib.CategoryTheory.Preadditive.Biproducts | {
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Mathlib.CategoryTheory.Preadditive.Biproducts | {
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Mathlib.CategoryTheory.Preadditive.Biproducts | {
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Mathlib.CategoryTheory.Preadditive.Biproducts | {
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Mathlib.CategoryTheory.Preadditive.Biproducts | {
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Mathlib.CategoryTheory.Preadditive.Biproducts | {
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Mathlib.CategoryTheory.Preadditive.Biproducts | {
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Mathlib.CategoryTheory.Preadditive.Biproducts | {
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} | {
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Mathlib.CategoryTheory.Preadditive.Biproducts | {
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} | {
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Mathlib.CategoryTheory.Preadditive.Biproducts | {
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} | {
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} | {
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Mathlib.Algebra.Category.AlgCat.Basic | {
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} | {
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} | {
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{
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"usedConstants": [
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Mathlib.Algebra.Category.AlgCat.Basic | {
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} | {
"line": 212,
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} | {
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} | [
{
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"usedConstants": [
"Fr... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.AlgCat.Basic | {
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} | {
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} | {
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Mathlib.Algebra.Category.AlgCat.Basic | {
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} | {
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} | {
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} | [
{
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"usedConstants": ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Category.AlgCat.Basic | {
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"usedConstants": ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.AlgCat.Basic | {
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} | {
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} | {
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Mathlib.Algebra.Category.ModuleCat.Basic | {
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} | {
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} | {
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} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nX₁ X₂ : Type v\nX Y : ModuleCat R\ni : X ≅ Y\n⊢ Hom.hom i.hom ∘ₗ Hom.hom i.inv = LinearMap.id",
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Mathlib.Algebra.Category.ModuleCat.Basic | {
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} | {
"line": 291,
"column": 41
} | {
"line": 291,
"column": 41
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nX₁ X₂ : Type v\nX Y : ModuleCat R\ni : X ≅ Y\n⊢ Hom.hom i.hom ∘ₗ Hom.hom i.inv = LinearMap.id",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"ModuleCat",
"LinearMap.ext",
"congrArg",
"AddCommGroup.toAddComm... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Basic | {
"line": 291,
"column": 36
} | {
"line": 291,
"column": 41
} | {
"line": 291,
"column": 41
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nX₁ X₂ : Type v\nX Y : ModuleCat R\ni : X ≅ Y\n⊢ Hom.hom i.hom ∘ₗ Hom.hom i.inv = LinearMap.id",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"ModuleCat",
"LinearMap.ext",
"congrArg",
"AddCommGroup.toAddComm... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Basic | {
"line": 291,
"column": 47
} | {
"line": 291,
"column": 52
} | {
"line": 291,
"column": 52
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nX₁ X₂ : Type v\nX Y : ModuleCat R\ni : X ≅ Y\n⊢ Hom.hom i.inv ∘ₗ Hom.hom i.hom = LinearMap.id",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"ModuleCat",
"LinearMap.ext",
"congrArg",
"AddCommGroup.toAddComm... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Category.ModuleCat.Basic | {
"line": 291,
"column": 47
} | {
"line": 291,
"column": 52
} | {
"line": 291,
"column": 52
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nX₁ X₂ : Type v\nX Y : ModuleCat R\ni : X ≅ Y\n⊢ Hom.hom i.inv ∘ₗ Hom.hom i.hom = LinearMap.id",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"ModuleCat",
"LinearMap.ext",
"congrArg",
"AddCommGroup.toAddComm... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Basic | {
"line": 291,
"column": 47
} | {
"line": 291,
"column": 52
} | {
"line": 291,
"column": 52
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nX₁ X₂ : Type v\nX Y : ModuleCat R\ni : X ≅ Y\n⊢ Hom.hom i.inv ∘ₗ Hom.hom i.hom = LinearMap.id",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"ModuleCat",
"LinearMap.ext",
"congrArg",
"AddCommGroup.toAddComm... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Preadditive.Biproducts | {
"line": 817,
"column": 4
} | {
"line": 817,
"column": 67
} | {
"line": 818,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\ninst✝¹ : HasBinaryBiproducts C\nW X Y Z : C\nf : W ⊞ X ⟶ Y ⊞ Z\ninst✝ : IsIso f\nnz : 𝟙 W ≠ 0\na₁ : biprod.inl ≫ f ≫ biprod.fst = 0\na₂ : biprod.inl ≫ f ≫ biprod.snd = 0\nx : W ⟶ W := biprod.inl ≫ f ≫ inv f ≫ biprod.fst\nh₁ : x = 𝟙 W\n⊢ ... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\ninst✝¹ : HasBinaryBiproducts C\nW X Y Z : C\nf : W ⊞ X ⟶ Y ⊞ Z\ninst✝ : IsIso f\nnz : 𝟙 W ≠ 0\na₁ : biprod.inl ≫ f ≫ biprod.fst = 0\na₂ : biprod.inl ≫ f ≫ biprod.snd = 0\nx : W ⟶ W := biprod.inl ≫ f ≫ inv f ≫ biprod.fst\nh₁ : x = 𝟙 W\n⊢ biprod.inl ≫... | rw [← Category.id_comp (inv f), Category.assoc, ← biprod.total] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.Antidiag.Pi | {
"line": 179,
"column": 82
} | {
"line": 179,
"column": 87
} | {
"line": 179,
"column": 87
} | [
{
"pp": "ι : Type u_1\nμ : Type u_2\ninst✝³ : DecidableEq ι\ninst✝² : AddCancelCommMonoid μ\ninst✝¹ : HasAntidiagonal μ\ninst✝ : DecidableEq μ\ni : ι\ns : Finset ι\nhi : i ∉ s\nn : μ\nf : ι → μ\nhn : f i + ∑ x ∈ s, f x = n\nhf : ∀ (i_1 : ι), ¬f i_1 = 0 → i_1 = i ∨ i_1 ∈ s\n⊢ (addRightEmbedding fun t ↦ if t = i ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Antidiag.Pi | {
"line": 179,
"column": 82
} | {
"line": 179,
"column": 87
} | {
"line": 179,
"column": 87
} | [
{
"pp": "ι : Type u_1\nμ : Type u_2\ninst✝³ : DecidableEq ι\ninst✝² : AddCancelCommMonoid μ\ninst✝¹ : HasAntidiagonal μ\ninst✝ : DecidableEq μ\ni : ι\ns : Finset ι\nhi : i ∉ s\nn : μ\nf : ι → μ\nhn : f i + ∑ x ∈ s, f x = n\nhf : ∀ (i_1 : ι), ¬f i_1 = 0 → i_1 = i ∨ i_1 ∈ s\n⊢ (addRightEmbedding fun t ↦ if t = i ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Antidiag.Pi | {
"line": 179,
"column": 82
} | {
"line": 179,
"column": 87
} | {
"line": 179,
"column": 87
} | [
{
"pp": "ι : Type u_1\nμ : Type u_2\ninst✝³ : DecidableEq ι\ninst✝² : AddCancelCommMonoid μ\ninst✝¹ : HasAntidiagonal μ\ninst✝ : DecidableEq μ\ni : ι\ns : Finset ι\nhi : i ∉ s\nn : μ\nf : ι → μ\nhn : f i + ∑ x ∈ s, f x = n\nhf : ∀ (i_1 : ι), ¬f i_1 = 0 → i_1 = i ∨ i_1 ∈ s\n⊢ (addRightEmbedding fun t ↦ if t = i ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Enumerative.Composition | {
"line": 366,
"column": 8
} | {
"line": 366,
"column": 17
} | {
"line": 367,
"column": 4
} | [
{
"pp": "case mpr.refine_1\nn : ℕ\nc : Composition n\nj : Fin n\ni : Fin c.length\nh : c.sizeUpTo ↑i ≤ ↑j ∧ ↑j < c.sizeUpTo (↑i).succ\n⊢ c.sizeUpTo ↑i ≤ ↑j",
"ppTerm": "?mpr.refine_1✝",
"assigned": true,
"usedConstants": [
"Composition.length",
"LE.le",
"instLENat",
"Fin.val"... | [] | exact h.1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Enumerative.Composition | {
"line": 366,
"column": 8
} | {
"line": 366,
"column": 17
} | {
"line": 367,
"column": 4
} | [
{
"pp": "case mpr.refine_1\nn : ℕ\nc : Composition n\nj : Fin n\ni : Fin c.length\nh : c.sizeUpTo ↑i ≤ ↑j ∧ ↑j < c.sizeUpTo (↑i).succ\n⊢ c.sizeUpTo ↑i ≤ ↑j",
"ppTerm": "?mpr.refine_1✝",
"assigned": true,
"usedConstants": [
"Composition.length",
"LE.le",
"instLENat",
"Fin.val"... | [] | exact h.1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Enumerative.Composition | {
"line": 366,
"column": 8
} | {
"line": 366,
"column": 17
} | {
"line": 367,
"column": 4
} | [
{
"pp": "case mpr.refine_1\nn : ℕ\nc : Composition n\nj : Fin n\ni : Fin c.length\nh : c.sizeUpTo ↑i ≤ ↑j ∧ ↑j < c.sizeUpTo (↑i).succ\n⊢ c.sizeUpTo ↑i ≤ ↑j",
"ppTerm": "?mpr.refine_1✝",
"assigned": true,
"usedConstants": [
"Composition.length",
"LE.le",
"instLENat",
"Fin.val"... | [] | exact h.1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Antidiag.Pi | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 23
} | {
"line": 224,
"column": 4
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : DecidableEq (ι → ℕ)\ns : Finset ι\nm n : ℕ\nhn : n ≠ 0\nf : ι → ℕ\nhfsum : s.sum f = n * m\nhfsup : ∀ (i : ι), f i ≠ 0 → i ∈ s\nhfdvd : ∀ i ∈ s, n ∣ f i\ni : ι\n⊢ n ∣ f i",
"ppTerm": "?m.134",
"assigned": true,
"usedConstants": [
"Dvd.dvd"... | [
"case pos\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : DecidableEq (ι → ℕ)\ns : Finset ι\nm n : ℕ\nhn : n ≠ 0\nf : ι → ℕ\nhfsum : s.sum f = n * m\nhfsup : ∀ (i : ι), f i ≠ 0 → i ∈ s\nhfdvd : ∀ i ∈ s, n ∣ f i\ni : ι\nhi : i ∈ s\n⊢ n ∣ f i",
"case neg\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : DecidableEq (ι ... | by_cases hi : i ∈ s | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.GroupTheory.Perm.Support | {
"line": 272,
"column": 53
} | {
"line": 272,
"column": 78
} | {
"line": 274,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n⊢ support 1 = ∅",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.support",
"Equiv.Perm.support_eq_empty_iff",
"Equiv.Perm.instOne",
"congrArg",
"Finset",
"id",
... | [] | rw [support_eq_empty_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Perm.Support | {
"line": 272,
"column": 53
} | {
"line": 272,
"column": 78
} | {
"line": 274,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n⊢ support 1 = ∅",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.support",
"Equiv.Perm.support_eq_empty_iff",
"Equiv.Perm.instOne",
"congrArg",
"Finset",
"id",
... | [] | rw [support_eq_empty_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.Support | {
"line": 272,
"column": 53
} | {
"line": 272,
"column": 78
} | {
"line": 274,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n⊢ support 1 = ∅",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.support",
"Equiv.Perm.support_eq_empty_iff",
"Equiv.Perm.instOne",
"congrArg",
"Finset",
"id",
... | [] | rw [support_eq_empty_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Enumerative.Composition | {
"line": 605,
"column": 4
} | {
"line": 605,
"column": 25
} | {
"line": 606,
"column": 4
} | [
{
"pp": "n : ℕ\nc : Composition n\nm : ℕ\nc₁ : Composition m\nc₂ : Composition n\ni : ℕ\nhi : i ∈ c₁.blocks ++ c₂.blocks\n⊢ 0 < i",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"congrArg",
"Composition.blocks",
"Membership.mem",
"Eq.mp",
"instHAppendOfAppend"... | [
"n : ℕ\nc : Composition n\nm : ℕ\nc₁ : Composition m\nc₂ : Composition n\ni : ℕ\nhi : i ∈ c₁.blocks ∨ i ∈ c₂.blocks\n⊢ 0 < i"
] | rw [mem_append] at hi | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Fintype.Perm | {
"line": 107,
"column": 8
} | {
"line": 107,
"column": 59
} | {
"line": 108,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\na : α\nl : List α\nhl : (a :: l).Nodup\nhl' : l.Nodup\nhln' : (permsOfList l).Nodup\nhmeml : ∀ {f : Equiv.Perm α}, f ∈ permsOfList l → f a = a\ni j : ℕ\nhi : i < l.length\nhj : j < l.length\nhij : i < j\nx : Equiv.Perm α\nhx₁ : x ∈ (fun b ↦ List.map (fun f ↦ Equiv.s... | [] | rw [← hg.2, mul_apply, hmeml hg.1, swap_apply_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Perm.List | {
"line": 279,
"column": 6
} | {
"line": 279,
"column": 30
} | {
"line": 280,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nx y x' y' : α\nl l' : List α\nhd : (x :: y :: l).Nodup\nh : ∀ (x_1 : α), (x :: y :: l).formPerm x_1 = (x' :: y' :: l').formPerm x_1\nhd' : (x' ≠ y' ∧ x' ∉ l') ∧ y' ∉ l' ∧ l'.Nodup\n⊢ y' ≠ x'",
"ppTerm": "?m.78",
"assigned": true,
"usedConstants": [
... | [] | exact hd'.left.left.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Fintype.Perm | {
"line": 110,
"column": 6
} | {
"line": 115,
"column": 88
} | {
"line": 117,
"column": 0
} | [
{
"pp": "case refine_4\nα : Type u_1\ninst✝ : DecidableEq α\na : α\nl : List α\nhl : (a :: l).Nodup\nhl' : l.Nodup\nhln' : (permsOfList l).Nodup\nhmeml : ∀ {f : Equiv.Perm α}, f ∈ permsOfList l → f a = a\n⊢ (permsOfList l).Disjoint (flatMap (fun b ↦ List.map (fun f ↦ Equiv.swap a b * f) (permsOfList l)) l)",
... | [] | intro f hf₁ hf₂
let ⟨x, hx, hx'⟩ := List.mem_flatMap.1 hf₂
let ⟨g, hg⟩ := List.mem_map.1 hx'
obtain rfl : g.symm x = a := f.injective <| by rw [hmeml hf₁, ← hg.2]; simp
have hxa : x ≠ g.symm x := fun h => (List.nodup_cons.1 hl).1 (h ▸ hx)
exact (List.nodup_cons.1 hl).1 <| mem_of_mem_permsO... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Fintype.Perm | {
"line": 110,
"column": 6
} | {
"line": 115,
"column": 88
} | {
"line": 117,
"column": 0
} | [
{
"pp": "case refine_4\nα : Type u_1\ninst✝ : DecidableEq α\na : α\nl : List α\nhl : (a :: l).Nodup\nhl' : l.Nodup\nhln' : (permsOfList l).Nodup\nhmeml : ∀ {f : Equiv.Perm α}, f ∈ permsOfList l → f a = a\n⊢ (permsOfList l).Disjoint (flatMap (fun b ↦ List.map (fun f ↦ Equiv.swap a b * f) (permsOfList l)) l)",
... | [] | intro f hf₁ hf₂
let ⟨x, hx, hx'⟩ := List.mem_flatMap.1 hf₂
let ⟨g, hg⟩ := List.mem_map.1 hx'
obtain rfl : g.symm x = a := f.injective <| by rw [hmeml hf₁, ← hg.2]; simp
have hxa : x ≠ g.symm x := fun h => (List.nodup_cons.1 hl).1 (h ▸ hx)
exact (List.nodup_cons.1 hl).1 <| mem_of_mem_permsO... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Fintype.Perm | {
"line": 150,
"column": 92
} | {
"line": 150,
"column": 97
} | {
"line": 150,
"column": 97
} | [
{
"pp": "α✝ : Type u_1\nβ✝ : Type u_2\nγ : Type u_3\ninst✝⁷ : DecidableEq α✝\ninst✝⁶ : DecidableEq β✝\nα : Type u_4\nβ : Type u_5\ninst✝⁵ : Mul α\ninst✝⁴ : Mul β\ninst✝³ : DecidableEq α\ninst✝² : DecidableEq β\ninst✝¹ : Fintype α\ninst✝ : Fintype β\n⊢ ∀ (a a' : α ≃ β),\n ∀ b ∈ if h : ∀ (a_1 b : α), a (a_1 * ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Fintype.Perm | {
"line": 150,
"column": 92
} | {
"line": 150,
"column": 97
} | {
"line": 150,
"column": 97
} | [
{
"pp": "α✝ : Type u_1\nβ✝ : Type u_2\nγ : Type u_3\ninst✝⁷ : DecidableEq α✝\ninst✝⁶ : DecidableEq β✝\nα : Type u_4\nβ : Type u_5\ninst✝⁵ : Mul α\ninst✝⁴ : Mul β\ninst✝³ : DecidableEq α\ninst✝² : DecidableEq β\ninst✝¹ : Fintype α\ninst✝ : Fintype β\n⊢ ∀ (a a' : α ≃ β),\n ∀ b ∈ if h : ∀ (a_1 b : α), a (a_1 * ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Fintype.Perm | {
"line": 150,
"column": 92
} | {
"line": 150,
"column": 97
} | {
"line": 150,
"column": 97
} | [
{
"pp": "α✝ : Type u_1\nβ✝ : Type u_2\nγ : Type u_3\ninst✝⁷ : DecidableEq α✝\ninst✝⁶ : DecidableEq β✝\nα : Type u_4\nβ : Type u_5\ninst✝⁵ : Mul α\ninst✝⁴ : Mul β\ninst✝³ : DecidableEq α\ninst✝² : DecidableEq β\ninst✝¹ : Fintype α\ninst✝ : Fintype β\n⊢ ∀ (a a' : α ≃ β),\n ∀ b ∈ if h : ∀ (a_1 b : α), a (a_1 * ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Closure | {
"line": 38,
"column": 68
} | {
"line": 42,
"column": 95
} | {
"line": 44,
"column": 0
} | [
{
"pp": "β : Type u_3\ninst✝ : Finite β\n⊢ closure {σ | σ.IsCycle} = ⊤",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Equiv.Perm.closure_isSwap",
"Subgroup.closure",
"PartialOrder.toPreorder",
"ge_of_eq",
"setOf",
"Equiv.Perm.IsSwap",
"Classical.p... | [] | by
classical
cases nonempty_fintype β
exact
top_le_iff.mp (le_trans (ge_of_eq closure_isSwap) (closure_mono fun _ => IsSwap.isCycle)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Perm.Cycle.Basic | {
"line": 318,
"column": 4
} | {
"line": 318,
"column": 9
} | {
"line": 319,
"column": 2
} | [
{
"pp": "case mp\nα : Type u_2\ninst✝ : DecidableEq α\na b : α\nh : (swap a b).IsSwap\nhab : a = b\n⊢ swap a b = 1",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.instOne",
"congrArg",
"Equiv.swap",
"Equiv.Perm.IsSwap",
"id",
"E... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.GroupTheory.Perm.Cycle.Basic | {
"line": 402,
"column": 10
} | {
"line": 402,
"column": 19
} | {
"line": 402,
"column": 20
} | [
{
"pp": "case e'_2\nα : Type u_4\ninst✝ : DecidableEq α\nn : ℕ\nb x : α\nf : Perm α\nh : (f ^ Int.negSucc n) (f x) = b\nhfxb : f x ≠ b\nhfb : f b ≠ b\nhbx : b ≠ x\nhb : (swap x ((Equiv.symm f) x) * f⁻¹) ((Equiv.symm f) b) ≠ (Equiv.symm f) b\ni : ℤ\nhi : ((swap x ((Equiv.symm f) x) * Equiv.symm f) ^ i) ((Equiv.s... | [
"case e'_2\nα : Type u_4\ninst✝ : DecidableEq α\nn : ℕ\nb x : α\nf : Perm α\nh : (f ^ Int.negSucc n) (f x) = b\nhfxb : f x ≠ b\nhfb : f b ≠ b\nhbx : b ≠ x\nhb : (swap x ((Equiv.symm f) x) * f⁻¹) ((Equiv.symm f) b) ≠ (Equiv.symm f) b\ni : ℤ\nhi : ((swap x ((Equiv.symm f) x) * Equiv.symm f) ^ i) ((Equiv.symm f) x) = ... | zpow_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Finite | {
"line": 264,
"column": 4
} | {
"line": 264,
"column": 65
} | {
"line": 265,
"column": 2
} | [
{
"pp": "case mem\nα : Type u\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nS : Set (Perm α)\n⊢ ∀ x ∈ S, ↑x.support ⊆ ⋃ b ∈ S, ↑b.support",
"ppTerm": "?mem",
"assigned": true,
"usedConstants": [
"Equiv.Perm.support",
"ChainCompletePartialOrder.instOfCompleteLattice",
"Finset",
... | [] | exact fun x hx ↦ Set.subset_iUnion₂_of_subset x hx subset_rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.Perm.Finite | {
"line": 264,
"column": 4
} | {
"line": 264,
"column": 65
} | {
"line": 265,
"column": 2
} | [
{
"pp": "case mem\nα : Type u\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nS : Set (Perm α)\n⊢ ∀ x ∈ S, ↑x.support ⊆ ⋃ b ∈ S, ↑b.support",
"ppTerm": "?mem",
"assigned": true,
"usedConstants": [
"Equiv.Perm.support",
"ChainCompletePartialOrder.instOfCompleteLattice",
"Finset",
... | [] | exact fun x hx ↦ Set.subset_iUnion₂_of_subset x hx subset_rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.Finite | {
"line": 264,
"column": 4
} | {
"line": 264,
"column": 65
} | {
"line": 265,
"column": 2
} | [
{
"pp": "case mem\nα : Type u\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nS : Set (Perm α)\n⊢ ∀ x ∈ S, ↑x.support ⊆ ⋃ b ∈ S, ↑b.support",
"ppTerm": "?mem",
"assigned": true,
"usedConstants": [
"Equiv.Perm.support",
"ChainCompletePartialOrder.instOfCompleteLattice",
"Finset",
... | [] | exact fun x hx ↦ Set.subset_iUnion₂_of_subset x hx subset_rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Sign | {
"line": 298,
"column": 4
} | {
"line": 316,
"column": 43
} | {
"line": 318,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : DecidableEq α\nn : ℕ\nx : α\nl : List α\nf : Perm α\ne : α ≃ Fin n\nh : ∀ (x_1 : α), f x_1 ≠ x_1 → x_1 ∈ x :: l\n⊢ signAux ((e.symm.trans f).trans e) = signAux2 (x :: l) f",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.signAux... | [] | rw [signAux2]
by_cases hfx : x = f x
· rw [if_pos hfx]
exact
signAux_eq_signAux2 l f _ fun y (hy : f y ≠ y) =>
List.mem_of_ne_of_mem (fun h : y = x => by simp [h, hfx.symm] at hy) (h y hy)
· have hy : ∀ y : α, (swap x (f x) * f) y ≠ y → y ∈ l := fun y hy =>
have : f y ≠ y ∧ y... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.Sign | {
"line": 298,
"column": 4
} | {
"line": 316,
"column": 43
} | {
"line": 318,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : DecidableEq α\nn : ℕ\nx : α\nl : List α\nf : Perm α\ne : α ≃ Fin n\nh : ∀ (x_1 : α), f x_1 ≠ x_1 → x_1 ∈ x :: l\n⊢ signAux ((e.symm.trans f).trans e) = signAux2 (x :: l) f",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.signAux... | [] | rw [signAux2]
by_cases hfx : x = f x
· rw [if_pos hfx]
exact
signAux_eq_signAux2 l f _ fun y (hy : f y ≠ y) =>
List.mem_of_ne_of_mem (fun h : y = x => by simp [h, hfx.symm] at hy) (h y hy)
· have hy : ∀ y : α, (swap x (f x) * f) y ≠ y → y ∈ l := fun y hy =>
have : f y ≠ y ∧ y... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.NoncommPiCoprod | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 37
} | {
"line": 178,
"column": 2
} | [
{
"pp": "case hom\nM : Type u_1\ninst✝² : Monoid M\nι : Type u_2\ninst✝¹ : Fintype ι\nN : ι → Type u_3\ninst✝ : (i : ι) → Monoid (N i)\nϕ : (i : ι) → N i →* M\nhcomm : Pairwise fun i j ↦ ∀ (x : N i) (y : N j), Commute ((ϕ i) x) ((ϕ j) y)\nm : M\ncomm : ∀ (i : ι) (x : N i), Commute m ((ϕ i) x)\nh : (i : ι) → N i... | [] | exact fun x y ↦ Commute.mul_right | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.NoncommPiCoprod | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 37
} | {
"line": 178,
"column": 2
} | [
{
"pp": "case hom\nM : Type u_1\ninst✝² : Monoid M\nι : Type u_2\ninst✝¹ : Fintype ι\nN : ι → Type u_3\ninst✝ : (i : ι) → Monoid (N i)\nϕ : (i : ι) → N i →* M\nhcomm : Pairwise fun i j ↦ ∀ (x : N i) (y : N j), Commute ((ϕ i) x) ((ϕ j) y)\nm : M\ncomm : ∀ (i : ι) (x : N i), Commute m ((ϕ i) x)\nh : (i : ι) → N i... | [] | exact fun x y ↦ Commute.mul_right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.NoncommPiCoprod | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 37
} | {
"line": 178,
"column": 2
} | [
{
"pp": "case hom\nM : Type u_1\ninst✝² : Monoid M\nι : Type u_2\ninst✝¹ : Fintype ι\nN : ι → Type u_3\ninst✝ : (i : ι) → Monoid (N i)\nϕ : (i : ι) → N i →* M\nhcomm : Pairwise fun i j ↦ ∀ (x : N i) (y : N j), Commute ((ϕ i) x) ((ϕ j) y)\nm : M\ncomm : ∀ (i : ι) (x : N i), Commute m ((ϕ i) x)\nh : (i : ι) → N i... | [] | exact fun x y ↦ Commute.mul_right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Tactic.NormNum.GCD | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 18
} | {
"line": 50,
"column": 0
} | [
{
"pp": "d x y a b : ℕ\nhu : x % d = 0\nhv : y % d = 0\nh : x * a = y * b + d\n⊢ ↑x * ↑a = ↑y * ↑b + ↑d",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Nat.cast_mul._simp_1",
"NonAssocSemiring.toAddCommMonoidWithOne",
"NonUnitalCommRing.toNo... | [] | exact mod_cast h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 421,
"column": 63
} | {
"line": 422,
"column": 87
} | {
"line": 423,
"column": 2
} | [
{
"pp": "α : Type u_4\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ σ ∈ l, σ.IsCycle\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nthis : σ.IsCycle → (σ ∈ l ↔ ∀ (a : α), σ a ≠ a → σ a = l.prod a)\n⊢ σ ∈ l ↔ σ.IsCycle ∧ ∀ (a : α), σ a ≠ a → σ a = l.prod a",
"ppTerm": "?m.35",
"assigned": true,
"usedCons... | [] | by
exact ⟨fun hσ => ⟨h1 σ hσ, (this (h1 σ hσ)).mp hσ⟩, fun hσ => (this hσ.1).mpr hσ.2⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Perm.Cycle.Type | {
"line": 160,
"column": 9
} | {
"line": 160,
"column": 14
} | {
"line": 162,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\n⊢ Finset.filter (Membership.mem (Function.fixedPoints ⇑σ)) Finset.univ = σ.supportᶜ",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.support",
"Finset.mem_filter._simp_1",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.GroupTheory.Perm.Cycle.Type | {
"line": 208,
"column": 69
} | {
"line": 214,
"column": 38
} | {
"line": 216,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nhσ : Nat.Prime (orderOf σ)\n⊢ ∃ n, σ.cycleType = Multiset.replicate (n + 1) (orderOf σ)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
... | [] | by
refine ⟨Multiset.card σ.cycleType - 1, eq_replicate.2 ⟨?_, fun n hn ↦ ?_⟩⟩
· rw [tsub_add_cancel_of_le]
rw [Nat.succ_le_iff, card_cycleType_pos, Ne, ← orderOf_eq_one_iff]
exact hσ.ne_one
· exact (hσ.eq_one_or_self_of_dvd n (dvd_of_mem_cycleType hn)).resolve_left
(one_lt_of_mem_cycleType hn).ne' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 710,
"column": 2
} | {
"line": 712,
"column": 37
} | {
"line": 714,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng c : Perm α\nhc : c ∈ g.cycleFactorsFinset\n⊢ g.IsCycleOn ↑c.support",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Equiv.Perm.instDecidableRelSameCycle",
"Eq.mpr",
"Equiv.Perm.support",
"Equiv.Per... | [] | obtain ⟨x, hx⟩ := IsCycle.nonempty_support (mem_cycleFactorsFinset_iff.mp hc).1
rw [cycle_is_cycleOf hx hc]
exact isCycleOn_support_cycleOf g x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 710,
"column": 2
} | {
"line": 712,
"column": 37
} | {
"line": 714,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng c : Perm α\nhc : c ∈ g.cycleFactorsFinset\n⊢ g.IsCycleOn ↑c.support",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Equiv.Perm.instDecidableRelSameCycle",
"Eq.mpr",
"Equiv.Perm.support",
"Equiv.Per... | [] | obtain ⟨x, hx⟩ := IsCycle.nonempty_support (mem_cycleFactorsFinset_iff.mp hc).1
rw [cycle_is_cycleOf hx hc]
exact isCycleOn_support_cycleOf g x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 831,
"column": 12
} | {
"line": 831,
"column": 26
} | {
"line": 832,
"column": 12
} | [
{
"pp": "case neg\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng f✝ σ τ : Perm α\nhd : σ.Disjoint τ\na✝ : σ.IsCycle\nhσ : ∀ {f : Perm α}, f ∈ σ.cycleFactorsFinset → (σ * f⁻¹).cycleFactorsFinset = σ.cycleFactorsFinset \\ {f}\nhτ : ∀ {f : Perm α}, f ∈ τ.cycleFactorsFinset → (τ * f⁻¹).cycleFactorsFins... | [
"case neg\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng f✝ σ τ : Perm α\nhd : σ.Disjoint τ\na✝ : σ.IsCycle\nhσ : ∀ {f : Perm α}, f ∈ σ.cycleFactorsFinset → (σ * f⁻¹).cycleFactorsFinset = σ.cycleFactorsFinset \\ {f}\nhτ : ∀ {f : Perm α}, f ∈ τ.cycleFactorsFinset → (τ * f⁻¹).cycleFactorsFinset = τ.cycle... | rw [mul_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 846,
"column": 12
} | {
"line": 846,
"column": 26
} | {
"line": 847,
"column": 12
} | [
{
"pp": "case neg\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng f✝ σ τ : Perm α\nhd : σ.Disjoint τ\na✝ : σ.IsCycle\nhσ : ∀ {f : Perm α}, f ∈ σ.cycleFactorsFinset → (σ * f⁻¹).cycleFactorsFinset = σ.cycleFactorsFinset \\ {f}\nhτ : ∀ {f : Perm α}, f ∈ τ.cycleFactorsFinset → (τ * f⁻¹).cycleFactorsFins... | [
"case neg\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng f✝ σ τ : Perm α\nhd : σ.Disjoint τ\na✝ : σ.IsCycle\nhσ : ∀ {f : Perm α}, f ∈ σ.cycleFactorsFinset → (σ * f⁻¹).cycleFactorsFinset = σ.cycleFactorsFinset \\ {f}\nhτ : ∀ {f : Perm α}, f ∈ τ.cycleFactorsFinset → (τ * f⁻¹).cycleFactorsFinset = τ.cycle... | rw [mul_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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