module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Abelian.EpiWithInjectiveKernel | {
"line": 91,
"column": 20
} | {
"line": 91,
"column": 53
} | {
"line": 92,
"column": 10
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nX Y Z : C\ng₁ : X ⟶ Y\ng₂ : Y ⟶ Z\nI₁ : C\nw✝¹ : Injective I₁\nf₁ : I₁ ⟶ X\nw₁ : f₁ ≫ g₁ = 0\nσ₁ : { X₁ := I₁, X₂ := X, X₃ := Y, f := f₁, g := g₁, zero := w₁ }.Splitting\nI₂ : C\nw✝ : Injective I₂\nf₂ : I₂ ⟶ Y\nw₂ : f₂ ≫ g₂ = 0\nσ₂ : { X₁... | [] | simp [reassoc_of% σ₁.s_g, σ₂.s_g] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Functor.OfSequence | {
"line": 42,
"column": 61
} | {
"line": 44,
"column": 6
} | {
"line": 46,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : ℕ → C\nf : (n : ℕ) → X n ⟶ X (n + 1)\ni j : ℕ\nh : i = j\n⊢ f i = eqToHom ⋯ ≫ f j ≫ eqToHom ⋯",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
... | [] | by
subst h
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 625,
"column": 53
} | {
"line": 625,
"column": 80
} | {
"line": 625,
"column": 80
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nn₀ n₁ : ℤ\nh : n₀ + 1 = n₁\nX : C\nhX : ∀ (Y : C) (f : Y ⟶ X), t.IsLE Y n₀ → f = 0\n⊢ t.IsLE ((t... | [] | by rw [← h]; infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.ModelCategory.Homotopy | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 40
} | {
"line": 136,
"column": 4
} | [
{
"pp": "case left\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : ModelCategory C\nX Y Z : C\ninst✝² : IsCofibrant X\ng : Y ⟶ Z\ninst✝¹ : Fibration g\ninst✝ : WeakEquivalence g\nf₁ : LeftHomotopyClass X Y\nf₀ : X ⟶ Y\nh : (fun f ↦ f.postcomp g) (mk f₀) = (fun f ↦ f.postcomp g) f₁\n⊢ mk f₀ = f₁",
"ppTerm"... | [
"case left\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : ModelCategory C\nX Y Z : C\ninst✝² : IsCofibrant X\ng : Y ⟶ Z\ninst✝¹ : Fibration g\ninst✝ : WeakEquivalence g\nf₀ f₁ : X ⟶ Y\nh : (fun f ↦ f.postcomp g) (mk f₀) = (fun f ↦ f.postcomp g) (mk f₁)\n⊢ mk f₀ = mk f₁"
] | obtain ⟨f₁, rfl⟩ := f₁.mk_surjective | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicTopology.ModelCategory.Homotopy | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 40
} | {
"line": 183,
"column": 4
} | [
{
"pp": "case left\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : ModelCategory C\nX Y Z : C\ninst✝² : IsFibrant Z\nf : X ⟶ Y\ninst✝¹ : Cofibration f\ninst✝ : WeakEquivalence f\nf₁ : RightHomotopyClass Y Z\nf₀ : Y ⟶ Z\nh : (fun g ↦ g.precomp f) (mk f₀) = (fun g ↦ g.precomp f) f₁\n⊢ mk f₀ = f₁",
"ppTerm":... | [
"case left\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : ModelCategory C\nX Y Z : C\ninst✝² : IsFibrant Z\nf : X ⟶ Y\ninst✝¹ : Cofibration f\ninst✝ : WeakEquivalence f\nf₀ f₁ : Y ⟶ Z\nh : (fun g ↦ g.precomp f) (mk f₀) = (fun g ↦ g.precomp f) (mk f₁)\n⊢ mk f₀ = mk f₁"
] | obtain ⟨f₁, rfl⟩ := f₁.mk_surjective | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.MorphismProperty.Quotient | {
"line": 45,
"column": 59
} | {
"line": 50,
"column": 46
} | {
"line": 52,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nW : MorphismProperty C\nhomRel : HomRel C\ninst✝² : HomRel.IsStableUnderPrecomp homRel\ninst✝¹ : HomRel.IsStableUnderPostcomp homRel\ninst✝ : W.HasQuotient homRel\nX Y : C\nf g : X ⟶ Y\nh : Relation.EqvGen homRel f g\n⊢ W f ↔ W g",
"ppTerm": "?m.22",
... | [] | by
induction h with
| rel _ _ h => exact iff W h
| refl => rfl
| symm _ _ _ h => exact h.symm
| trans _ _ _ _ _ h₁ h₂ => exact h₁.trans h₂ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.Factorizations.CM5a | {
"line": 238,
"column": 4
} | {
"line": 238,
"column": 54
} | {
"line": 238,
"column": 54
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\nK L : CochainComplex C ℤ\nf : K ⟶ L\ninst✝ : EnoughInjectives C\nn i : ℤ\nhi : i + 1 = n\n⊢ ((cokernel f).truncGE n).d i n ≫ Injective.ι (((cokernel f).truncGE n).X n) = 0",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants... | [] | exact (isZero_of_isStrictlyGE _ n _).eq_of_src _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Localization.DerivabilityStructure.Constructor | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 20
} | {
"line": 94,
"column": 2
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\ninst✝⁶ : Category.{v_1, u_1} C₁\ninst✝⁵ : Category.{v_2, u_2} C₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nΦ : LocalizerMorphism W₁ W₂\ninst✝⁴ : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution X₂)\ninst✝³ : Φ.arrow.HasRightResolutions\ninst✝² : W₂.ContainsIdentities... | [
"C₁ : Type u_1\nC₂ : Type u_2\ninst✝⁶ : Category.{v_1, u_1} C₁\ninst✝⁵ : Category.{v_2, u_2} C₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nΦ : LocalizerMorphism W₁ W₂\ninst✝⁴ : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution X₂)\ninst✝³ : Φ.arrow.HasRightResolutions\ninst✝² : W₂.ContainsIdentities\nD : Type u... | dsimp [w] at x fac | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Localization.DerivabilityStructure.Basic | {
"line": 163,
"column": 6
} | {
"line": 163,
"column": 36
} | {
"line": 164,
"column": 6
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\ninst✝¹ : Category.{v₁, u₁} C₁\ninst✝ : Category.{v₂, u₂} C₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nΦ : LocalizerMorphism W₁ W₂\nF : W₁.Localization ⥤ W₂.Localization := Φ.localizedFunctor W₁.Q W₂.Q\ne : Φ.functor ⋙ W₂.Q ≅ W₁.Q ⋙ F := CatCommSq.iso Φ.functor W₁.... | [
"C₁ : Type u₁\nC₂ : Type u₂\ninst✝¹ : Category.{v₁, u₁} C₁\ninst✝ : Category.{v₂, u₂} C₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nΦ : LocalizerMorphism W₁ W₂\nF : W₁.Localization ⥤ W₂.Localization := Φ.localizedFunctor W₁.Q W₂.Q\ne : Φ.functor ⋙ W₂.Q ≅ W₁.Q ⋙ F := CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.lo... | rw [hasLeftResolutions_iff_op] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Localization.DerivabilityStructure.Basic | {
"line": 189,
"column": 2
} | {
"line": 190,
"column": 17
} | {
"line": 191,
"column": 2
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\ninst✝⁶ : Category.{v₁, u₁} C₁\ninst✝⁵ : Category.{v₂, u₂} C₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nΦ : LocalizerMorphism W₁ W₂\nD₁ : Type u_1\nD₂ : Type u_2\ninst✝⁴ : Category.{v_1, u_1} D₁\ninst✝³ : Category.{v_2, u_2} D₂\nL₁ : C₁ ⥤ D₁\nL₂ : C₂ ⥤ D₂\ninst✝² :... | [
"C₁ : Type u₁\nC₂ : Type u₂\ninst✝⁶ : Category.{v₁, u₁} C₁\ninst✝⁵ : Category.{v₂, u₂} C₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nΦ : LocalizerMorphism W₁ W₂\nD₁ : Type u_1\nD₂ : Type u_2\ninst✝⁴ : Category.{v_1, u_1} D₁\ninst✝³ : Category.{v_2, u_2} D₂\nL₁ : C₁ ⥤ D₁\nL₂ : C₂ ⥤ D₂\ninst✝² : L₁.IsLocali... | rw [(LocalizerMorphism.id W₁).isLeftDerivabilityStructure_iff W₁.Q W₁.Q (𝟭 W₁.Localization)
(Iso.refl _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.GuitartExact.KanExtension | {
"line": 113,
"column": 16
} | {
"line": 113,
"column": 31
} | {
"line": 114,
"column": 2
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\nC₄ : Type u₄\nD : Type u₅\ninst✝⁶ : Category.{v₁, u₁} C₁\ninst✝⁵ : Category.{v₂, u₂} C₂\ninst✝⁴ : Category.{v₃, u₃} C₃\ninst✝³ : Category.{v₄, u₄} C₄\ninst✝² : Category.{v₅, u₅} D\nT : C₁ ⥤ C₂\nL : C₁ ⥤ C₃\nR : C₂ ⥤ C₄\nB : C₃ ⥤ C₄\nF : C₂ ⥤ D\nE : R.LeftExtens... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.GuitartExact.KanExtension | {
"line": 114,
"column": 17
} | {
"line": 114,
"column": 32
} | {
"line": 116,
"column": 0
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\nC₄ : Type u₄\nD : Type u₅\ninst✝⁶ : Category.{v₁, u₁} C₁\ninst✝⁵ : Category.{v₂, u₂} C₂\ninst✝⁴ : Category.{v₃, u₃} C₃\ninst✝³ : Category.{v₄, u₄} C₄\ninst✝² : Category.{v₅, u₅} D\nT : C₁ ⥤ C₂\nL : C₁ ⥤ C₃\nR : C₂ ⥤ C₄\nB : C₃ ⥤ C₄\nF : C₂ ⥤ D\nE : R.LeftExtens... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.DerivedCategory.DerivabilityStructureInjectives | {
"line": 70,
"column": 9
} | {
"line": 70,
"column": 77
} | {
"line": 70,
"column": 77
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : EnoughInjectives C\nK : Plus C\nn : ℤ\ninst✝ : K.obj.IsStrictlyGE n\nL : CochainComplex C ℤ\ni : K.obj ⟶ L\nw✝¹ : QuasiIso i\nw✝ : ∀ (n : ℤ), Injective (L.X n)\nh✝ : L.IsStrictlyGE n\nL' : CochainComplex (InjectiveObject C) ℤ :=... | [
"C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : EnoughInjectives C\nK : Plus C\nn : ℤ\ninst✝ : K.obj.IsStrictlyGE n\nL : CochainComplex C ℤ\ni : K.obj ⟶ L\nw✝¹ : QuasiIso i\nw✝ : ∀ (n : ℤ), Injective (L.X n)\nh✝ : L.IsStrictlyGE n\nL' : CochainComplex (InjectiveObject C) ℤ := Homological... | ← isStrictlyGE_mapHomologicalComplex_obj_iff _ (InjectiveObject.ι _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.DerivedCategory.DerivabilityStructureInjectives | {
"line": 147,
"column": 13
} | {
"line": 147,
"column": 81
} | {
"line": 147,
"column": 81
} | [
{
"pp": "C : Type u_1\nH : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Category.{v_2, u_2} H\ninst✝ : EnoughInjectives C\nobj✝ : CochainComplex C ℤ\nn : ℤ\nh✝ : obj✝.IsStrictlyGE n\nproperty✝ : fibrantObjects (Plus C) { obj := obj✝, property := ⋯ }\n⊢ ((HomologicalComplex.liftFunctorO... | [
"C : Type u_1\nH : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Category.{v_2, u_2} H\ninst✝ : EnoughInjectives C\nobj✝ : CochainComplex C ℤ\nn : ℤ\nh✝ : obj✝.IsStrictlyGE n\nproperty✝ : fibrantObjects (Plus C) { obj := obj✝, property := ⋯ }\n⊢ IsStrictlyGE\n (((InjectiveObject.ι C).map... | ← isStrictlyGE_mapHomologicalComplex_obj_iff _ (InjectiveObject.ι _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.DerivedCategory.Ext.MapBijective | {
"line": 61,
"column": 59
} | {
"line": 61,
"column": 74
} | {
"line": 61,
"column": 74
} | [
{
"pp": "C : Type u\ninst✝¹² : Category.{v, u} C\ninst✝¹¹ : Abelian C\nD : Type u'\ninst✝¹⁰ : Category.{v', u'} D\ninst✝⁹ : Abelian D\nF : C ⥤ D\ninst✝⁸ : F.Additive\ninst✝⁷ : PreservesFiniteLimits F\ninst✝⁶ : PreservesFiniteColimits F\ninst✝⁵ : F.Full\ninst✝⁴ : F.Faithful\ninst✝³ : HasExt C\ninst✝² : HasExt D\... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.DerivedCategory.Ext.MapBijective | {
"line": 62,
"column": 67
} | {
"line": 62,
"column": 82
} | {
"line": 62,
"column": 82
} | [
{
"pp": "C : Type u\ninst✝¹² : Category.{v, u} C\ninst✝¹¹ : Abelian C\nD : Type u'\ninst✝¹⁰ : Category.{v', u'} D\ninst✝⁹ : Abelian D\nF : C ⥤ D\ninst✝⁸ : F.Additive\ninst✝⁷ : PreservesFiniteLimits F\ninst✝⁶ : PreservesFiniteColimits F\ninst✝⁵ : F.Full\ninst✝⁴ : F.Faithful\ninst✝³ : HasExt C\ninst✝² : HasExt D\... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.DerivedCategory.Ext.MapBijective | {
"line": 83,
"column": 55
} | {
"line": 83,
"column": 70
} | {
"line": 83,
"column": 70
} | [
{
"pp": "C : Type u\ninst✝¹² : Category.{v, u} C\ninst✝¹¹ : Abelian C\nD : Type u'\ninst✝¹⁰ : Category.{v', u'} D\ninst✝⁹ : Abelian D\nF : C ⥤ D\ninst✝⁸ : F.Additive\ninst✝⁷ : PreservesFiniteLimits F\ninst✝⁶ : PreservesFiniteColimits F\ninst✝⁵ : F.Full\ninst✝⁴ : F.Faithful\ninst✝³ : HasExt C\ninst✝² : HasExt D\... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.DerivedCategory.Ext.MapBijective | {
"line": 84,
"column": 63
} | {
"line": 84,
"column": 78
} | {
"line": 84,
"column": 78
} | [
{
"pp": "C : Type u\ninst✝¹² : Category.{v, u} C\ninst✝¹¹ : Abelian C\nD : Type u'\ninst✝¹⁰ : Category.{v', u'} D\ninst✝⁹ : Abelian D\nF : C ⥤ D\ninst✝⁸ : F.Additive\ninst✝⁷ : PreservesFiniteLimits F\ninst✝⁶ : PreservesFiniteColimits F\ninst✝⁵ : F.Full\ninst✝⁴ : F.Faithful\ninst✝³ : HasExt C\ninst✝² : HasExt D\... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Map | {
"line": 89,
"column": 2
} | {
"line": 103,
"column": 6
} | {
"line": 105,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : Abelian C\nD : Type u'\ninst✝⁶ : Category.{v', u'} D\ninst✝⁵ : Abelian D\ninst✝⁴ : HasDerivedCategory C\ninst✝³ : HasDerivedCategory D\nS : ShortComplex C\nhS : S.ShortExact\nF : C ⥤ D\ninst✝² : F.Additive\ninst✝¹ : PreservesFiniteLimits F\ninst✝ : Prese... | [] | dsimp [ShiftedHom.map, ShortComplex.ShortExact.singleδ]
simp only [Functor.map_comp, Category.assoc, Functor.commShiftIso_hom_naturality,
DerivedCategory.map_triangleOfSESδ, singleFunctorsPostcompQIso_hom_hom,
singleFunctorsPostcompQIso_inv_hom]
generalize_proofs _ _ _ _ _ _ h1 _ _ h2
dsimp [CochainComple... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Map | {
"line": 89,
"column": 2
} | {
"line": 103,
"column": 6
} | {
"line": 105,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : Abelian C\nD : Type u'\ninst✝⁶ : Category.{v', u'} D\ninst✝⁵ : Abelian D\ninst✝⁴ : HasDerivedCategory C\ninst✝³ : HasDerivedCategory D\nS : ShortComplex C\nhS : S.ShortExact\nF : C ⥤ D\ninst✝² : F.Additive\ninst✝¹ : PreservesFiniteLimits F\ninst✝ : Prese... | [] | dsimp [ShiftedHom.map, ShortComplex.ShortExact.singleδ]
simp only [Functor.map_comp, Category.assoc, Functor.commShiftIso_hom_naturality,
DerivedCategory.map_triangleOfSESδ, singleFunctorsPostcompQIso_hom_hom,
singleFunctorsPostcompQIso_inv_hom]
generalize_proofs _ _ _ _ _ _ h1 _ _ h2
dsimp [CochainComple... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Map | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 7
} | {
"line": 166,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : Abelian C\nD : Type u'\ninst✝⁶ : Category.{v', u'} D\ninst✝⁵ : Abelian D\nF : C ⥤ D\ninst✝⁴ : F.Additive\ninst✝³ : PreservesFiniteLimits F\ninst✝² : PreservesFiniteColimits F\ninst✝¹ : HasExt C\ninst✝ : HasExt D\nX Y : C\nn : ℕ\n⊢ mapExactFunctor F 0 = 0... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Map | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 7
} | {
"line": 166,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : Abelian C\nD : Type u'\ninst✝⁶ : Category.{v', u'} D\ninst✝⁵ : Abelian D\nF : C ⥤ D\ninst✝⁴ : F.Additive\ninst✝³ : PreservesFiniteLimits F\ninst✝² : PreservesFiniteColimits F\ninst✝¹ : HasExt C\ninst✝ : HasExt D\nX Y : C\nn : ℕ\n⊢ mapExactFunctor F 0 = 0... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Map | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 7
} | {
"line": 166,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : Abelian C\nD : Type u'\ninst✝⁶ : Category.{v', u'} D\ninst✝⁵ : Abelian D\nF : C ⥤ D\ninst✝⁴ : F.Additive\ninst✝³ : PreservesFiniteLimits F\ninst✝² : PreservesFiniteColimits F\ninst✝¹ : HasExt C\ninst✝ : HasExt D\nX Y : C\nn : ℕ\n⊢ mapExactFunctor F 0 = 0... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Map | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 7
} | {
"line": 171,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : Abelian C\nD : Type u'\ninst✝⁶ : Category.{v', u'} D\ninst✝⁵ : Abelian D\nF : C ⥤ D\ninst✝⁴ : F.Additive\ninst✝³ : PreservesFiniteLimits F\ninst✝² : PreservesFiniteColimits F\ninst✝¹ : HasExt C\ninst✝ : HasExt D\nX Y : C\nn : ℕ\nf g : Ext X Y n\n⊢ mapExa... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Map | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 7
} | {
"line": 171,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : Abelian C\nD : Type u'\ninst✝⁶ : Category.{v', u'} D\ninst✝⁵ : Abelian D\nF : C ⥤ D\ninst✝⁴ : F.Additive\ninst✝³ : PreservesFiniteLimits F\ninst✝² : PreservesFiniteColimits F\ninst✝¹ : HasExt C\ninst✝ : HasExt D\nX Y : C\nn : ℕ\nf g : Ext X Y n\n⊢ mapExa... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Map | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 7
} | {
"line": 171,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : Abelian C\nD : Type u'\ninst✝⁶ : Category.{v', u'} D\ninst✝⁵ : Abelian D\nF : C ⥤ D\ninst✝⁴ : F.Additive\ninst✝³ : PreservesFiniteLimits F\ninst✝² : PreservesFiniteColimits F\ninst✝¹ : HasExt C\ninst✝ : HasExt D\nX Y : C\nn : ℕ\nf g : Ext X Y n\n⊢ mapExa... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Map | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 7
} | {
"line": 191,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹² : Category.{v, u} C\ninst✝¹¹ : Abelian C\nD : Type u'\ninst✝¹⁰ : Category.{v', u'} D\ninst✝⁹ : Abelian D\nF : C ⥤ D\ninst✝⁸ : F.Additive\ninst✝⁷ : PreservesFiniteLimits F\ninst✝⁶ : PreservesFiniteColimits F\ninst✝⁵ : HasExt C\ninst✝⁴ : HasExt D\nX Y : C\nn : ℕ\nR : Type u_1\ninst✝³ ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Map | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 7
} | {
"line": 191,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹² : Category.{v, u} C\ninst✝¹¹ : Abelian C\nD : Type u'\ninst✝¹⁰ : Category.{v', u'} D\ninst✝⁹ : Abelian D\nF : C ⥤ D\ninst✝⁸ : F.Additive\ninst✝⁷ : PreservesFiniteLimits F\ninst✝⁶ : PreservesFiniteColimits F\ninst✝⁵ : HasExt C\ninst✝⁴ : HasExt D\nX Y : C\nn : ℕ\nR : Type u_1\ninst✝³ ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Map | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 7
} | {
"line": 191,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹² : Category.{v, u} C\ninst✝¹¹ : Abelian C\nD : Type u'\ninst✝¹⁰ : Category.{v', u'} D\ninst✝⁹ : Abelian D\nF : C ⥤ D\ninst✝⁸ : F.Additive\ninst✝⁷ : PreservesFiniteLimits F\ninst✝⁶ : PreservesFiniteColimits F\ninst✝⁵ : HasExt C\ninst✝⁴ : HasExt D\nX Y : C\nn : ℕ\nR : Type u_1\ninst✝³ ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Double | {
"line": 45,
"column": 8
} | {
"line": 45,
"column": 13
} | {
"line": 45,
"column": 13
} | [
{
"pp": "case hnc\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nX₀ X₁ : C\nf : X₀ ⟶ X₁\nι : Type u_2\nc : ComplexShape ι\ni₀ i₁ : ι\nhi₀₁ : c.Rel i₀ i₁\nk k' : ι\nhk : k = i₀ ∧ k' = i₁ ∧ i₀ ≠ i₁\n⊢ ¬k' = i₀",
"ppTerm": "?hnc",
"assigned": true,
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.Double | {
"line": 58,
"column": 31
} | {
"line": 58,
"column": 36
} | {
"line": 58,
"column": 36
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nX₀ X₁ : C\nf : X₀ ⟶ X₁\nι : Type u_2\nc : ComplexShape ι\ni₀ i₁ : ι\nhi₀₁ : c.Rel i₀ i₁\ni j : ι\nhij : ¬c.Rel i j\n⊢ ¬(i = i₀ ∧ j = i₁ ∧ i₀ ≠ i₁)",
"ppTerm": "?m.94",
"assigned": true,
"used... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.Double | {
"line": 58,
"column": 31
} | {
"line": 58,
"column": 36
} | {
"line": 58,
"column": 36
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nX₀ X₁ : C\nf : X₀ ⟶ X₁\nι : Type u_2\nc : ComplexShape ι\ni₀ i₁ : ι\nhi₀₁ : c.Rel i₀ i₁\ni j : ι\nhij : ¬c.Rel i j\n⊢ ¬(i = i₀ ∧ j = i₁ ∧ i₀ ≠ i₁)",
"ppTerm": "?m.94",
"assigned": true,
"used... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Double | {
"line": 58,
"column": 31
} | {
"line": 58,
"column": 36
} | {
"line": 58,
"column": 36
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nX₀ X₁ : C\nf : X₀ ⟶ X₁\nι : Type u_2\nc : ComplexShape ι\ni₀ i₁ : ι\nhi₀₁ : c.Rel i₀ i₁\ni j : ι\nhij : ¬c.Rel i j\n⊢ ¬(i = i₀ ∧ j = i₁ ∧ i₀ ≠ i₁)",
"ppTerm": "?m.94",
"assigned": true,
"used... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.DifferentialObject | {
"line": 97,
"column": 2
} | {
"line": 99,
"column": 5
} | {
"line": 101,
"column": 0
} | [
{
"pp": "S : Type u_1\ninst✝³ : AddMonoidWithOne S\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasShift C S\nX Y : DifferentialObject S C\nh : X = Y\n⊢ (eqToHom h).f = eqToHom ⋯",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Categor... | [] | subst h
rw [eqToHom_refl, eqToHom_refl]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.DifferentialObject | {
"line": 97,
"column": 2
} | {
"line": 99,
"column": 5
} | {
"line": 101,
"column": 0
} | [
{
"pp": "S : Type u_1\ninst✝³ : AddMonoidWithOne S\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasShift C S\nX Y : DifferentialObject S C\nh : X = Y\n⊢ (eqToHom h).f = eqToHom ⋯",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Categor... | [] | subst h
rw [eqToHom_refl, eqToHom_refl]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Embedding.Connect | {
"line": 176,
"column": 57
} | {
"line": 176,
"column": 76
} | {
"line": 176,
"column": 76
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nK K' K'' : ChainComplex C ℕ\nL L' L'' : CochainComplex C ℕ\nh : ConnectData K L\nh' : ConnectData K' L'\nh'' : ConnectData K'' L''\nn : ℕ\ninst✝² : NeZero n\nm : ℤ\nhm : m = ↑n\ninst✝¹ : HasHomology h.cochainComplex m\ninst✝ : HasHomo... | [] | by cases n <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.Embedding.Connect | {
"line": 188,
"column": 70
} | {
"line": 188,
"column": 89
} | {
"line": 188,
"column": 89
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nK K' K'' : ChainComplex C ℕ\nL L' L'' : CochainComplex C ℕ\nh : ConnectData K L\nh' : ConnectData K' L'\nh'' : ConnectData K'' L''\nn : ℕ\ninst✝² : NeZero n\nm : ℤ\nhm : m = -↑(n + 1)\ninst✝¹ : HasHomology h.cochainComplex m\ninst✝ : ... | [] | by cases n <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 11
} | {
"line": 85,
"column": 4
} | [
{
"pp": "case pos\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\ninst✝ : HasZeroObject C\nX : C\nK : CochainComplex C ℤ\np q n : ℤ\nh : p + n = q\nα : Cochain ((singleFunctor C p).obj X) K n\np' q' : ℤ\nhpq' : p' + n = q'\nhp : p' = p\n⊢ ((fun f ↦ fromSingleMk f h)\n ((fun α ↦ (Homolo... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 11
} | {
"line": 85,
"column": 4
} | [
{
"pp": "case pos\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\ninst✝ : HasZeroObject C\nX : C\nK : CochainComplex C ℤ\np q n : ℤ\nh : p + n = q\nα : Cochain ((singleFunctor C p).obj X) K n\np' q' : ℤ\nhpq' : p' + n = q'\nhp : p' = p\n⊢ ((fun f ↦ fromSingleMk f h)\n ((fun α ↦ (Homolo... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 11
} | {
"line": 85,
"column": 4
} | [
{
"pp": "case pos\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\ninst✝ : HasZeroObject C\nX : C\nK : CochainComplex C ℤ\np q n : ℤ\nh : p + n = q\nα : Cochain ((singleFunctor C p).obj X) K n\np' q' : ℤ\nhpq' : p' + n = q'\nhp : p' = p\n⊢ ((fun f ↦ fromSingleMk f h)\n ((fun α ↦ (Homolo... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 174,
"column": 6
} | {
"line": 174,
"column": 11
} | {
"line": 175,
"column": 4
} | [
{
"pp": "case pos\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\ninst✝ : HasZeroObject C\nX : C\nK : CochainComplex C ℤ\np q n : ℤ\nh : p + n = q\nα : Cochain K ((singleFunctor C q).obj X) n\np' q' : ℤ\nhpq' : p' + n = q'\nhq : q' = q\n⊢ ((fun f ↦ toSingleMk f h) ((fun α ↦ α.v p q h ≫ (Homolog... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 174,
"column": 6
} | {
"line": 174,
"column": 11
} | {
"line": 175,
"column": 4
} | [
{
"pp": "case pos\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\ninst✝ : HasZeroObject C\nX : C\nK : CochainComplex C ℤ\np q n : ℤ\nh : p + n = q\nα : Cochain K ((singleFunctor C q).obj X) n\np' q' : ℤ\nhpq' : p' + n = q'\nhq : q' = q\n⊢ ((fun f ↦ toSingleMk f h) ((fun α ↦ α.v p q h ≫ (Homolog... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 174,
"column": 6
} | {
"line": 174,
"column": 11
} | {
"line": 175,
"column": 4
} | [
{
"pp": "case pos\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\ninst✝ : HasZeroObject C\nX : C\nK : CochainComplex C ℤ\np q n : ℤ\nh : p + n = q\nα : Cochain K ((singleFunctor C q).obj X) n\np' q' : ℤ\nhpq' : p' + n = q'\nhq : q' = q\n⊢ ((fun f ↦ toSingleMk f h) ((fun α ↦ α.v p q h ≫ (Homolog... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 59
} | {
"line": 254,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\ninst✝ : HasZeroObject C\nX : C\nK : CochainComplex C ℤ\np n : ℤ\nα : Cocycle ((singleFunctor C p).obj X) K n\nq : ℤ\nh : p + n = q\nq' : ℤ\nhq' : q + 1 = q'\n⊢ ∃ f, ∃ (hf : f ≫ K.d q q' = 0), fromSingleMk f h q' hq' hf = α",
"ppTerm": ... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\ninst✝ : HasZeroObject C\nX : C\nK : CochainComplex C ℤ\np n : ℤ\nα : Cocycle ((singleFunctor C p).obj X) K n\nq : ℤ\nh : p + n = q\nq' : ℤ\nhq' : q + 1 = q'\nf : X ⟶ K.X q\nhf : Cochain.fromSingleMk f h = ↑α\n⊢ ∃ f, ∃ (hf : f ≫ K.d q q' = 0), fromSing... | obtain ⟨f, hf⟩ := Cochain.fromSingleMk_surjective α.1 q h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 304,
"column": 4
} | {
"line": 305,
"column": 8
} | {
"line": 305,
"column": 8
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\ninst✝ : HasZeroObject C\nX : C\nK : CochainComplex C ℤ\np q : ℤ\nf : K.X p ⟶ X\nn : ℤ\nh : p + n = q\np' : ℤ\nhp' : p' + 1 = p\nhf : K.d p' p ≫ f = 0\n⊢ δ n (n + 1) (Cochain.toSingleMk f h) = 0",
"ppTerm": "?m.68",
"assigned": true... | [] | rw [Cochain.δ_toSingleMk _ _ _ p' (by lia), hf]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 304,
"column": 4
} | {
"line": 305,
"column": 8
} | {
"line": 305,
"column": 8
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\ninst✝ : HasZeroObject C\nX : C\nK : CochainComplex C ℤ\np q : ℤ\nf : K.X p ⟶ X\nn : ℤ\nh : p + n = q\np' : ℤ\nhp' : p' + 1 = p\nhf : K.d p' p ≫ f = 0\n⊢ δ n (n + 1) (Cochain.toSingleMk f h) = 0",
"ppTerm": "?m.68",
"assigned": true... | [] | rw [Cochain.δ_toSingleMk _ _ _ p' (by lia), hf]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Idempotents.FunctorExtension | {
"line": 69,
"column": 22
} | {
"line": 78,
"column": 29
} | {
"line": 80,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nF G : C ⥤ Karoubi D\nφ : F ⟶ G\nx✝¹ x✝ : Karoubi C\nf : x✝¹ ⟶ x✝\n⊢ (obj F).map f ≫ { f := (F.map x✝.p).f ≫ (φ.app x✝.X).f, comm := ⋯ } =\n { f := (F.map x✝¹.p).f ... | [] | by
ext
dsimp [obj]
have h := φ.naturality f.f
have h' := F.congr_map (comp_p f)
have h'' := F.congr_map (p_comp f)
simp only [hom_ext_iff, Functor.map_comp, comp_f] at h h' h'' ⊢
slice_rhs 2 3 => rw [← h]
slice_lhs 1 2 => rw [h']
slice_rhs 1 2 => rw [h''] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.GradedObject.Single | {
"line": 53,
"column": 2
} | {
"line": 55,
"column": 24
} | {
"line": 57,
"column": 0
} | [
{
"pp": "J : Type u_1\nC : Type u_2\ninst✝² : Category.{v_1, u_2} C\ninst✝¹ : HasInitial C\ninst✝ : DecidableEq J\nj : J\nX : C\ni : J\nh : i ≠ j\n⊢ IsInitial ((single j).obj X i)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CategoryTheory.Limits... | [] | dsimp [single]
rw [if_neg h]
exact initialIsInitial | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.LocalCohomology | {
"line": 203,
"column": 6
} | {
"line": 204,
"column": 94
} | {
"line": 205,
"column": 4
} | [
{
"pp": "case inst\nR : Type u\ninst✝ : CommRing R\nI J K : Ideal R\nhR : IsNoetherian R R\nJ' : SelfLERadical J\n⊢ Nonempty (CostructuredArrow (idealPowersToSelfLERadical J) J')",
"ppTerm": "?inst✝",
"assigned": true,
"usedConstants": [
"Submodule",
"Semiring.toModule",
"Opposite"... | [] | obtain ⟨k, hk⟩ := Ideal.exists_pow_le_of_le_radical_of_fg J'.2 (isNoetherian_def.mp hR _)
exact ⟨CostructuredArrow.mk (⟨⟨⟨hk⟩⟩⟩ : (idealPowersToSelfLERadical J).obj (op k) ⟶ J')⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.GradedObject.Single | {
"line": 53,
"column": 2
} | {
"line": 55,
"column": 24
} | {
"line": 57,
"column": 0
} | [
{
"pp": "J : Type u_1\nC : Type u_2\ninst✝² : Category.{v_1, u_2} C\ninst✝¹ : HasInitial C\ninst✝ : DecidableEq J\nj : J\nX : C\ni : J\nh : i ≠ j\n⊢ IsInitial ((single j).obj X i)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CategoryTheory.Limits... | [] | dsimp [single]
rw [if_neg h]
exact initialIsInitial | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.LocalCohomology | {
"line": 203,
"column": 6
} | {
"line": 204,
"column": 94
} | {
"line": 205,
"column": 4
} | [
{
"pp": "case inst\nR : Type u\ninst✝ : CommRing R\nI J K : Ideal R\nhR : IsNoetherian R R\nJ' : SelfLERadical J\n⊢ Nonempty (CostructuredArrow (idealPowersToSelfLERadical J) J')",
"ppTerm": "?inst✝",
"assigned": true,
"usedConstants": [
"Submodule",
"Semiring.toModule",
"Opposite"... | [] | obtain ⟨k, hk⟩ := Ideal.exists_pow_le_of_le_radical_of_fg J'.2 (isNoetherian_def.mp hR _)
exact ⟨CostructuredArrow.mk (⟨⟨⟨hk⟩⟩⟩ : (idealPowersToSelfLERadical J).obj (op k) ⟶ J')⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.GradedObject.Unitor | {
"line": 64,
"column": 68
} | {
"line": 64,
"column": 73
} | {
"line": 64,
"column": 73
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\ninst✝³ : Zero I\ninst✝² : DecidableEq I\ninst✝¹ : HasInitial C\nF : C ⥤ D ⥤ D\nX : C\ne : F.obj X ≅ 𝟭 D\ninst✝ : ∀ (Y : D), PreservesColimit (Functor.empty C) (F.flip.obj Y)\np : I ×... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.GradedObject.Unitor | {
"line": 64,
"column": 68
} | {
"line": 64,
"column": 73
} | {
"line": 64,
"column": 73
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\ninst✝³ : Zero I\ninst✝² : DecidableEq I\ninst✝¹ : HasInitial C\nF : C ⥤ D ⥤ D\nX : C\ne : F.obj X ≅ 𝟭 D\ninst✝ : ∀ (Y : D), PreservesColimit (Functor.empty C) (F.flip.obj Y)\np : I ×... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.GradedObject.Unitor | {
"line": 64,
"column": 68
} | {
"line": 64,
"column": 73
} | {
"line": 64,
"column": 73
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\ninst✝³ : Zero I\ninst✝² : DecidableEq I\ninst✝¹ : HasInitial C\nF : C ⥤ D ⥤ D\nX : C\ne : F.obj X ≅ 𝟭 D\ninst✝ : ∀ (Y : D), PreservesColimit (Functor.empty C) (F.flip.obj Y)\np : I ×... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.GradedObject.Unitor | {
"line": 181,
"column": 68
} | {
"line": 181,
"column": 73
} | {
"line": 181,
"column": 73
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\ninst✝³ : Zero I\ninst✝² : DecidableEq I\ninst✝¹ : HasInitial C\nF : D ⥤ C ⥤ D\nY : C\ne : F.flip.obj Y ≅ 𝟭 D\ninst✝ : ∀ (X : D), PreservesColimit (Functor.empty C) (F.obj X)\np : J ×... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.GradedObject.Unitor | {
"line": 181,
"column": 68
} | {
"line": 181,
"column": 73
} | {
"line": 181,
"column": 73
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\ninst✝³ : Zero I\ninst✝² : DecidableEq I\ninst✝¹ : HasInitial C\nF : D ⥤ C ⥤ D\nY : C\ne : F.flip.obj Y ≅ 𝟭 D\ninst✝ : ∀ (X : D), PreservesColimit (Functor.empty C) (F.obj X)\np : J ×... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.GradedObject.Unitor | {
"line": 181,
"column": 68
} | {
"line": 181,
"column": 73
} | {
"line": 181,
"column": 73
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\ninst✝³ : Zero I\ninst✝² : DecidableEq I\ninst✝¹ : HasInitial C\nF : D ⥤ C ⥤ D\nY : C\ne : F.flip.obj Y ≅ 𝟭 D\ninst✝ : ∀ (X : D), PreservesColimit (Functor.empty C) (F.obj X)\np : J ×... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Monoidal | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 53
} | {
"line": 216,
"column": 0
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝⁹ : Category.{v_1, u_1} C\ninst✝⁸ : MonoidalCategory C\ninst✝⁷ : Preadditive C\ninst✝⁶ : HasZeroObject C\ninst✝⁵ : (curriedTensor C).Additive\ninst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive\nI : Type u_2\ninst✝³ : AddMonoid I\nc : ComplexShape I\ninst✝² : c.Tensor... | [] | simp only [shape _ _ _ hij, comp_zero, zero_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.Monoidal | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 53
} | {
"line": 216,
"column": 0
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝⁹ : Category.{v_1, u_1} C\ninst✝⁸ : MonoidalCategory C\ninst✝⁷ : Preadditive C\ninst✝⁶ : HasZeroObject C\ninst✝⁵ : (curriedTensor C).Additive\ninst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive\nI : Type u_2\ninst✝³ : AddMonoid I\nc : ComplexShape I\ninst✝² : c.Tensor... | [] | simp only [shape _ _ _ hij, comp_zero, zero_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Monoidal | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 53
} | {
"line": 216,
"column": 0
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝⁹ : Category.{v_1, u_1} C\ninst✝⁸ : MonoidalCategory C\ninst✝⁷ : Preadditive C\ninst✝⁶ : HasZeroObject C\ninst✝⁵ : (curriedTensor C).Additive\ninst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive\nI : Type u_2\ninst✝³ : AddMonoid I\nc : ComplexShape I\ninst✝² : c.Tensor... | [] | simp only [shape _ _ _ hij, comp_zero, zero_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Monoidal | {
"line": 262,
"column": 4
} | {
"line": 262,
"column": 53
} | {
"line": 264,
"column": 0
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝⁹ : Category.{v_1, u_1} C\ninst✝⁸ : MonoidalCategory C\ninst✝⁷ : Preadditive C\ninst✝⁶ : HasZeroObject C\ninst✝⁵ : (curriedTensor C).Additive\ninst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive\nI : Type u_2\ninst✝³ : AddMonoid I\nc : ComplexShape I\ninst✝² : c.Tensor... | [] | simp only [shape _ _ _ hij, comp_zero, zero_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.Monoidal | {
"line": 262,
"column": 4
} | {
"line": 262,
"column": 53
} | {
"line": 264,
"column": 0
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝⁹ : Category.{v_1, u_1} C\ninst✝⁸ : MonoidalCategory C\ninst✝⁷ : Preadditive C\ninst✝⁶ : HasZeroObject C\ninst✝⁵ : (curriedTensor C).Additive\ninst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive\nI : Type u_2\ninst✝³ : AddMonoid I\nc : ComplexShape I\ninst✝² : c.Tensor... | [] | simp only [shape _ _ _ hij, comp_zero, zero_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Monoidal | {
"line": 262,
"column": 4
} | {
"line": 262,
"column": 53
} | {
"line": 264,
"column": 0
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝⁹ : Category.{v_1, u_1} C\ninst✝⁸ : MonoidalCategory C\ninst✝⁷ : Preadditive C\ninst✝⁶ : HasZeroObject C\ninst✝⁵ : (curriedTensor C).Additive\ninst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive\nI : Type u_2\ninst✝³ : AddMonoid I\nc : ComplexShape I\ninst✝² : c.Tensor... | [] | simp only [shape _ _ _ hij, comp_zero, zero_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.SpectralObject.Basic | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 46
} | {
"line": 93,
"column": 0
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Category.{u_4, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni j k : ι\nf : i ⟶ j\ng : j ⟶ k\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\n⊢ X.δ f g n₀ n₁ hn₁ ≫ (X.H n₁).map (twoδ₂Toδ₁ f g (f ≫ g) ⋯) = 0",
"ppTerm": "?m.63",
"assigned":... | [] | exact (X.exact₁' n₀ n₁ hn₁ (mk₂ f g)).zero 0 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence | {
"line": 154,
"column": 13
} | {
"line": 154,
"column": 44
} | {
"line": 155,
"column": 2
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : Preorder ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\n⊢ ∀ (r : ℤ) (pq pq' : ℤ × ℤ),\n (ComplexShape.up' (r, 1 - r)).Rel pq pq' →\n autoParam (2 ≤ r) SpectralSequenceDataCore._auto_16 → WithBotTop.coe (pq... | [] | rintro r pq hr rfl _; simp; lia | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence | {
"line": 154,
"column": 13
} | {
"line": 154,
"column": 44
} | {
"line": 155,
"column": 2
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : Preorder ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\n⊢ ∀ (r : ℤ) (pq pq' : ℤ × ℤ),\n (ComplexShape.up' (r, 1 - r)).Rel pq pq' →\n autoParam (2 ≤ r) SpectralSequenceDataCore._auto_16 → WithBotTop.coe (pq... | [] | rintro r pq hr rfl _; simp; lia | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence | {
"line": 155,
"column": 13
} | {
"line": 155,
"column": 44
} | {
"line": 156,
"column": 2
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : Preorder ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\n⊢ ∀ (r : ℤ) (pq pq' : ℤ × ℤ),\n (ComplexShape.up' (r, 1 - r)).Rel pq pq' →\n autoParam (2 ≤ r) SpectralSequenceDataCore._auto_20 → WithBotTop.coe pq.... | [] | rintro r pq hr rfl _; simp; lia | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence | {
"line": 155,
"column": 13
} | {
"line": 155,
"column": 44
} | {
"line": 156,
"column": 2
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : Preorder ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\n⊢ ∀ (r : ℤ) (pq pq' : ℤ × ℤ),\n (ComplexShape.up' (r, 1 - r)).Rel pq pq' →\n autoParam (2 ≤ r) SpectralSequenceDataCore._auto_20 → WithBotTop.coe pq.... | [] | rintro r pq hr rfl _; simp; lia | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 476,
"column": 8
} | {
"line": 479,
"column": 37
} | {
"line": 479,
"column": 37
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni j k l : ι\nf₁ : i ⟶ j\nf₂ : j ⟶ k\nf₃ : k ⟶ l\nf₂₃ : j ⟶ l\nh₂₃ : f₂ ≫ f₃ = f₂₃\nn₀ n₁ n₂ : ℤ\nhn₁ : n₀ + 1 = n₁\nhn₂ : n₁ + 1 = n₂\nA : C\nx₂ : A ⟶ (X.H n₁).obj (mk₁... | [] | dsimp
rw [← X.fromOpcyles_δ f₁ f₂ f₃ f₂₃ h₂₃ n₁ n₂,
X.liftOpcycles_fromOpcycles_assoc]
simpa using hx₂ =≫ biprod.snd | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 476,
"column": 8
} | {
"line": 479,
"column": 37
} | {
"line": 479,
"column": 37
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni j k l : ι\nf₁ : i ⟶ j\nf₂ : j ⟶ k\nf₃ : k ⟶ l\nf₂₃ : j ⟶ l\nh₂₃ : f₂ ≫ f₃ = f₂₃\nn₀ n₁ n₂ : ℤ\nhn₁ : n₀ + 1 = n₁\nhn₂ : n₁ + 1 = n₂\nA : C\nx₂ : A ⟶ (X.H n₁).obj (mk₁... | [] | dsimp
rw [← X.fromOpcyles_δ f₁ f₂ f₃ f₂₃ h₂₃ n₁ n₂,
X.liftOpcycles_fromOpcycles_assoc]
simpa using hx₂ =≫ biprod.snd | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Artinian.Module | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 73
} | {
"line": 184,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : ℕ → Submodule R M\nh : ∀ (n : ℕ), Disjoint ((partialSups (⇑OrderDual.toDual ∘ f)) n) (OrderDual.toDual (f (n + 1)))\n⊢ ∃ n, ∀ (m : ℕ), n ≤ m → f m = ⊤",
"ppTerm": "?m.44",
... | [
"R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : ℕ → Submodule R M\nh : ∀ (n : ℕ), Disjoint ((partialSups (⇑OrderDual.toDual ∘ f)) n) (OrderDual.toDual (f (n + 1)))\nn : ℕ\nw : ∀ (m : ℕ), n ≤ m → OrderDual.toDual f (m + 1) = ⊤\n⊢ ∃ n, ∀ (m ... | rsuffices ⟨n, w⟩ : ∃ n : ℕ, ∀ m, n ≤ m → OrderDual.toDual f (m + 1) = ⊤ | Mathlib.Tactic._aux_Mathlib_Tactic_RSuffices___macroRules_Mathlib_Tactic_rsuffices_1 | Mathlib.Tactic.rsuffices |
Mathlib.RingTheory.Artinian.Module | {
"line": 351,
"column": 8
} | {
"line": 351,
"column": 80
} | {
"line": 351,
"column": 80
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nr : R\nn m : ℕ\nh : n ≤ m\nx : M\nx✝ : x ∈ (fun n ↦ (r ^ n • LinearMap.id).range) m\ny : M\nhy : r ^ m • y = x\n⊢ r ^ n • r ^ (m - n) • y = x",
"ppTerm": "?m.213",
"assign... | [] | rw [← smul_assoc, smul_eq_mul, ← pow_add, ← hy, add_tsub_cancel_of_le h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Artinian.Module | {
"line": 535,
"column": 2
} | {
"line": 535,
"column": 63
} | {
"line": 536,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsArtinianRing R\ninst✝ : IsDomain R\n⊢ IsField R",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.to... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsArtinianRing R\ninst✝ : IsDomain R\nx : R\nhx : x ≠ 0\n⊢ ∃ b, x * b = 1"
] | refine ⟨Nontrivial.exists_pair_ne, mul_comm, fun {x} hx ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 721,
"column": 2
} | {
"line": 726,
"column": 46
} | {
"line": 728,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\ns✝ s : Set L\nx : L\nhx : x ∈ lieSpan R L (-s)\n⊢ x ∈ lieSpan R L s",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"SMulMemClass.smul_mem",
"LieAlgebra.toModule",
"Set.mem_n... | [] | induction hx using lieSpan_induction with
| mem y h => exact neg_mem_iff.mp <| subset_lieSpan <| Set.mem_neg.mp h
| zero => exact zero_mem _
| add _ _ _ _ hu hv => exact add_mem hu hv
| smul t _ _ hu => exact SMulMemClass.smul_mem t hu
| lie _ _ _ _ hu hv => exact lie_mem _ hu hv | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Algebra.Lie.OfAssociative | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 90
} | {
"line": 57,
"column": 0
} | [
{
"pp": "A : Type v\ninst✝ : Ring A\nx✝² x✝¹ x✝ : A\n⊢ ⁅x✝², ⁅x✝¹, x✝⁆⁆ = ⁅⁅x✝², x✝¹⁆, x✝⁆ + ⁅x✝¹, ⁅x✝², x✝⁆⁆",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"Ring.toNonAssocRing",
"co... | [] | simp only [Ring.lie_def, mul_sub_left_distrib, mul_sub_right_distrib, mul_assoc]; abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.OfAssociative | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 90
} | {
"line": 57,
"column": 0
} | [
{
"pp": "A : Type v\ninst✝ : Ring A\nx✝² x✝¹ x✝ : A\n⊢ ⁅x✝², ⁅x✝¹, x✝⁆⁆ = ⁅⁅x✝², x✝¹⁆, x✝⁆ + ⁅x✝¹, ⁅x✝², x✝⁆⁆",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"Ring.toNonAssocRing",
"co... | [] | simp only [Ring.lie_def, mul_sub_left_distrib, mul_sub_right_distrib, mul_assoc]; abel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 744,
"column": 55
} | {
"line": 744,
"column": 60
} | {
"line": 745,
"column": 4
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nK : LieSubalgebra R L\nι : Type u_1\nf : ι → ↥K\nx : L\nhx : x ∈ K\nhx' : x ∈ lieSpan R L (range (Subtype.val ∘ f))\nthis : x ∈ map K.incl (lieSpan R (↥K) (range f))\n⊢ ⟨x, hx⟩ ∈ lieSpan R (↥K) (range f)",
"ppT... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 744,
"column": 55
} | {
"line": 744,
"column": 60
} | {
"line": 745,
"column": 4
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nK : LieSubalgebra R L\nι : Type u_1\nf : ι → ↥K\nx : L\nhx : x ∈ K\nhx' : x ∈ lieSpan R L (range (Subtype.val ∘ f))\nthis : x ∈ map K.incl (lieSpan R (↥K) (range f))\n⊢ ⟨x, hx⟩ ∈ lieSpan R (↥K) (range f)",
"ppT... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 744,
"column": 55
} | {
"line": 744,
"column": 60
} | {
"line": 745,
"column": 4
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nK : LieSubalgebra R L\nι : Type u_1\nf : ι → ↥K\nx : L\nhx : x ∈ K\nhx' : x ∈ lieSpan R L (range (Subtype.val ∘ f))\nthis : x ∈ map K.incl (lieSpan R (↥K) (range f))\n⊢ ⟨x, hx⟩ ∈ lieSpan R (↥K) (range f)",
"ppT... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 749,
"column": 6
} | {
"line": 749,
"column": 11
} | {
"line": 750,
"column": 4
} | [
{
"pp": "case a.mem\nR : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nK : LieSubalgebra R L\nι : Type u_1\nf : ι → ↥K\nx u : L\nhu : u ∈ range (Subtype.val ∘ f)\nthis : ∀ (i : ι), f i ∈ lieSpan R (↥K) (range f)\n⊢ u ∈ map K.incl (lieSpan R (↥K) (range f))",
"ppTerm": ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Lie.Submodule | {
"line": 71,
"column": 34
} | {
"line": 71,
"column": 64
} | {
"line": 71,
"column": 64
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nN N' : LieSubmodule R L M\nx : L\nm : M\nh : m ∈ (Submodule.toAddSubmonoid 0).carrier\n⊢ ⁅x, m⁆ ∈ (Submodule.toAddSubmonoid 0).carrier",
"ppTerm": "?... | [
"R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nN N' : LieSubmodule R L M\nx : L\nm : M\nh : m ∈ (Submodule.toAddSubmonoid 0).carrier\n⊢ ⁅x, 0⁆ ∈ (Submodule.toAddSubmonoid 0).carrier"
] | rw [(Submodule.mem_bot R).1 h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.OfAssociative | {
"line": 458,
"column": 25
} | {
"line": 458,
"column": 53
} | {
"line": 458,
"column": 53
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nL' : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\ne : L ≃ₗ⁅R⁆ L'\nx : L\ny' : L'\n⊢ ⁅e.toLinearEquiv x, e.toLinearEquiv (e.toLinearEquiv.symm y')⁆ = ⁅e.toLinearEquiv x, y'⁆",
"ppTerm": "?m.125",... | [
"R : Type u_1\nL : Type u_2\nL' : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\ne : L ≃ₗ⁅R⁆ L'\nx : L\ny' : L'\n⊢ ⁅e.toLinearEquiv x, y'⁆ = ⁅e.toLinearEquiv x, y'⁆"
] | LinearEquiv.apply_symm_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Ideal | {
"line": 338,
"column": 2
} | {
"line": 343,
"column": 36
} | {
"line": 345,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieRing L'\ninst✝¹ : LieAlgebra R L'\ninst✝ : LieAlgebra R L\nf : L →ₗ⁅R⁆ L'\nh : Function.Surjective ⇑f\n⊢ f.idealRange = ⊤",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"LieHom",
... | [] | rw [← f.range_eq_top] at h
rw [idealRange_eq_lieSpan_range, h, ← LieSubalgebra.coe_toSubmodule, ←
LieSubmodule.toSubmodule_inj, LieSubmodule.top_toSubmodule,
LieSubalgebra.top_toSubmodule, LieSubmodule.coe_lieSpan_submodule_eq_iff]
use ⊤
exact LieSubmodule.top_toSubmodule | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Ideal | {
"line": 338,
"column": 2
} | {
"line": 343,
"column": 36
} | {
"line": 345,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieRing L'\ninst✝¹ : LieAlgebra R L'\ninst✝ : LieAlgebra R L\nf : L →ₗ⁅R⁆ L'\nh : Function.Surjective ⇑f\n⊢ f.idealRange = ⊤",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"LieHom",
... | [] | rw [← f.range_eq_top] at h
rw [idealRange_eq_lieSpan_range, h, ← LieSubalgebra.coe_toSubmodule, ←
LieSubmodule.toSubmodule_inj, LieSubmodule.top_toSubmodule,
LieSubalgebra.top_toSubmodule, LieSubmodule.coe_lieSpan_submodule_eq_iff]
use ⊤
exact LieSubmodule.top_toSubmodule | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Submodule | {
"line": 387,
"column": 8
} | {
"line": 398,
"column": 36
} | {
"line": 398,
"column": 37
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nN N' : LieSubmodule R L M\nS : Set (LieSubmodule R L M)\nx : L\nm : M\ns : Finset (Submodule R M)\nhs : ↑s ⊆ {x | ∃ p ∈ S, ↑p = x}\nhsm : m ∈ ⨆ i ∈ s, i\... | [] | induction s using Finset.induction_on generalizing m with
| empty =>
replace hsm : m = 0 := by simpa using hsm
simp [hsm]
| insert q t hqt ih =>
rw [Finset.iSup_insert] at hsm
obtain ⟨m', hm', u, hu, rfl⟩ := Submodule.mem_sup.mp hsm
rw [lie_add]
... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Algebra.Lie.Submodule | {
"line": 387,
"column": 8
} | {
"line": 398,
"column": 36
} | {
"line": 398,
"column": 37
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nN N' : LieSubmodule R L M\nS : Set (LieSubmodule R L M)\nx : L\nm : M\ns : Finset (Submodule R M)\nhs : ↑s ⊆ {x | ∃ p ∈ S, ↑p = x}\nhsm : m ∈ ⨆ i ∈ s, i\... | [] | induction s using Finset.induction_on generalizing m with
| empty =>
replace hsm : m = 0 := by simpa using hsm
simp [hsm]
| insert q t hqt ih =>
rw [Finset.iSup_insert] at hsm
obtain ⟨m', hm', u, hu, rfl⟩ := Submodule.mem_sup.mp hsm
rw [lie_add]
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Submodule | {
"line": 387,
"column": 8
} | {
"line": 398,
"column": 36
} | {
"line": 398,
"column": 37
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nN N' : LieSubmodule R L M\nS : Set (LieSubmodule R L M)\nx : L\nm : M\ns : Finset (Submodule R M)\nhs : ↑s ⊆ {x | ∃ p ∈ S, ↑p = x}\nhsm : m ∈ ⨆ i ∈ s, i\... | [] | induction s using Finset.induction_on generalizing m with
| empty =>
replace hsm : m = 0 := by simpa using hsm
simp [hsm]
| insert q t hqt ih =>
rw [Finset.iSup_insert] at hsm
obtain ⟨m', hm', u, hu, rfl⟩ := Submodule.mem_sup.mp hsm
rw [lie_add]
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Jordan.Basic | {
"line": 196,
"column": 15
} | {
"line": 196,
"column": 22
} | {
"line": 196,
"column": 22
} | [
{
"pp": "A : Type u_1\ninst✝ : NonUnitalNonAssocCommRing A\na b c : A\n⊢ ⁅L a + L b, L (a * a) + L (b * b) + L (c * c) + 2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆ +\n ⁅L c, L (a * a) + L (b * b) + L (c * c) + 2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆ =\n ⁅L a, L (a * a)⁆ + ⁅L a, L (b * b)⁆ + ⁅... | [
"A : Type u_1\ninst✝ : NonUnitalNonAssocCommRing A\na b c : A\n⊢ ⁅L a, L (a * a) + L (b * b) + L (c * c) + 2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆ +\n ⁅L b, L (a * a) + L (b * b) + L (c * c) + 2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆ +\n ⁅L c, L (a * a) + L (b * b) + L (c * c) + 2 • L (a * ... | add_lie | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Submodule | {
"line": 694,
"column": 2
} | {
"line": 694,
"column": 56
} | {
"line": 695,
"column": 2
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nm : M\ns : Set (LieSubmodule R L M)\nhne : s.Nonempty\nhdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s\nhsup : m ∈ ↑(sSup s)\n⊢ ∃ x ∈ s, lieSpan R L {m} ≤ x",
... | [
"R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nm : M\ns : Set (LieSubmodule R L M)\nhne : s.Nonempty\nhdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s\nhsup : m ∈ ↑(sSup s)\n⊢ ↑(sSup s) = ⋃ N ∈ s, ↑N"
] | suffices (↑(sSup s) : Set M) = ⋃ N ∈ s, ↑N by simp_all | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Algebra.Lie.Submodule | {
"line": 703,
"column": 59
} | {
"line": 703,
"column": 73
} | {
"line": 703,
"column": 74
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nN : LieSubmodule R L M\n⊢ ↑N ≤ sSup {x | ∃ s ∈ (fun x ↦ lieSpan R L {x}) '' ↑N, ↑s = x}",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants... | [
"R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nN : LieSubmodule R L M\n⊢ ↑N ≤ sSup {x | ∃ s, (∃ x ∈ ↑N, lieSpan R L {x} = s) ∧ ↑s = x}"
] | Set.mem_image, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Lie.Submodule | {
"line": 922,
"column": 2
} | {
"line": 922,
"column": 75
} | {
"line": 924,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\nN : Type w₁\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : LieRingModule L N\nf : M →ₗ⁅R,L⁆ N\n⊢ f.range = ⊤ ↔ Function.Surjective ⇑f",
"ppT... | [] | rw [SetLike.ext'_iff, coe_range, LieSubmodule.top_coe, Set.range_eq_univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Submodule | {
"line": 922,
"column": 2
} | {
"line": 922,
"column": 75
} | {
"line": 924,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\nN : Type w₁\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : LieRingModule L N\nf : M →ₗ⁅R,L⁆ N\n⊢ f.range = ⊤ ↔ Function.Surjective ⇑f",
"ppT... | [] | rw [SetLike.ext'_iff, coe_range, LieSubmodule.top_coe, Set.range_eq_univ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Submodule | {
"line": 922,
"column": 2
} | {
"line": 922,
"column": 75
} | {
"line": 924,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\nN : Type w₁\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : LieRingModule L N\nf : M →ₗ⁅R,L⁆ N\n⊢ f.range = ⊤ ↔ Function.Surjective ⇑f",
"ppT... | [] | rw [SetLike.ext'_iff, coe_range, LieSubmodule.top_coe, Set.range_eq_univ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Abelian | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 7
} | {
"line": 160,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\n⊢ (∀ (x : L), (∀ (m : M), ⁅x, m⁆ = 0) → x = 0) ↔ ∀ (m : L), m ∈ LieModule.ker R L M ↔ m ∈ ⊥",
"ppT... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Lie.Abelian | {
"line": 193,
"column": 4
} | {
"line": 193,
"column": 52
} | {
"line": 195,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN : LieSubmodule R L M\nh : ∀ x ∈ ⊤, ∀ m ∈ N, ⁅x, m⁆ = 0\nm : M\nhm : m ∈ N\n⊢ ∀ (x : L... | [] | exact fun x => h x (LieSubmodule.mem_top x) m hm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.Minpoly.Basic | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 30
} | {
"line": 189,
"column": 0
} | [
{
"pp": "case inr\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Subsingleton B\na✝ : Nontrivial A\nthis : (minpoly A x).degree ≤ 0\nh : 0 < (minpoly A x).degree\n⊢ minpoly A x = 1",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
... | [] | · exact (this.not_gt h).elim | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.Minpoly.Basic | {
"line": 227,
"column": 2
} | {
"line": 227,
"column": 51
} | {
"line": 228,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\nint : IsIntegral A x\n⊢ 2 ≤ (minpoly A x).natDegree ↔ x ∉ (algebraMap A B).range",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"not_le",
"Eq.mpr",
... | [
"A : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\nint : IsIntegral A x\n⊢ (minpoly A x).natDegree = 1 ↔ (minpoly A x).natDegree < 2"
] | rw [iff_not_comm, ← natDegree_eq_one_iff, not_le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 240,
"column": 2
} | {
"line": 241,
"column": 34
} | {
"line": 243,
"column": 0
} | [
{
"pp": "F : Type u_3\nE : Type u_4\nK : Type u_5\ninst✝⁶ : Field F\ninst✝⁵ : Ring E\ninst✝⁴ : CommRing K\ninst✝³ : IsDomain K\ninst✝² : Algebra F E\ninst✝¹ : Algebra F K\ninst✝ : FiniteDimensional F E\nf g : E →ₐ[F] K\nh : ∀ (x : ↑(range ⇑(Module.finBasis F E))), rootsOfMinPolyPiType F E K f x = rootsOfMinPoly... | [] | exact LinearMap.ext_on (Module.finBasis F E).span_eq fun e he =>
Subtype.ext_iff.mp (h ⟨e, he⟩) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.PowerBasis | {
"line": 100,
"column": 28
} | {
"line": 100,
"column": 42
} | {
"line": 100,
"column": 43
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nx y : S\nd : ℕ\nn : S\n⊢ (∃ y, x ^ ↑y = n) ↔ n ∈ (fun i ↦ x ^ i) '' ↑(Finset.range d)",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Mem... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nx y : S\nd : ℕ\nn : S\n⊢ (∃ y, x ^ ↑y = n) ↔ ∃ x_1 ∈ ↑(Finset.range d), x ^ x_1 = n"
] | Set.mem_image, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Polynomial.Ideal | {
"line": 70,
"column": 43
} | {
"line": 70,
"column": 48
} | {
"line": 70,
"column": 48
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nx : S\nI : Ideal R\ny : ↥R[x]\na : R\nha : a ∈ ↑I\ni : ℕ\n⊢ (if i = 0 then a else 0) ∈ I",
"ppTerm": "?m.120",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
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