module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.AlgebraicTopology.ModelCategory.RightHomotopy | {
"line": 263,
"column": 2
} | {
"line": 265,
"column": 37
} | {
"line": 267,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\ninst✝¹ : ModelCategory C\nf₀ f₁ f₂ : X ⟶ Y\ninst✝ : IsFibrant Y\nh : RightHomotopyRel f₀ f₁\nh' : RightHomotopyRel f₁ f₂\n⊢ RightHomotopyRel f₀ f₂",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"HomotopicalAlgebra.PathObject"... | [] | obtain ⟨P, ⟨h⟩⟩ := h
obtain ⟨P', _, ⟨h'⟩⟩ := h'.exists_good_pathObject
exact (h.trans h').rightHomotopyRel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.BifunctorAssociator | {
"line": 760,
"column": 4
} | {
"line": 764,
"column": 23
} | {
"line": 765,
"column": 2
} | [
{
"pp": "case pos\nC₁ : Type u_1\nC₂ : Type u_2\nC₁₂ : Type u_3\nC₂₃ : Type u_4\nC₃ : Type u_5\nC₄ : Type u_6\ninst✝³³ : Category.{v_1, u_1} C₁\ninst✝³² : Category.{v_2, u_2} C₂\ninst✝³¹ : Category.{v_3, u_5} C₃\ninst✝³⁰ : Category.{v_4, u_6} C₄\ninst✝²⁹ : Category.{v_5, u_3} C₁₂\ninst✝²⁸ : Category.{v_6, u_4} ... | [] | rw [mapBifunctor₁₂.d₁_eq _ _ _ _ _ _ _ h₁, mapBifunctor₂₃.d₁_eq _ _ _ _ _ _ _ _ h₁,
Linear.comp_units_smul, Linear.units_smul_comp, assoc,
ComplexShape.associative_ε₁_eq_mul c₁ c₂ c₃ c₁₂ c₂₃ c₄,
ιOrZero_mapBifunctorAssociatorX_hom, smul_left_cancel_iff,
reassoc_of% this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory | {
"line": 285,
"column": 4
} | {
"line": 288,
"column": 84
} | {
"line": 289,
"column": 4
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝¹³ : Category.{v_1, u_1} C\ninst✝¹² : Category.{v_2, u_2} D\ninst✝¹¹ : Category.{v_3, u_3} D₁\ninst✝¹⁰ : Category.{v_4, u_4} D₂\nA B : ObjectProperty C\ninst✝⁹ : HasZeroObject C\ninst✝⁸ : HasShift C ℤ\ninst✝⁷ : Preadditive C\ninst✝⁶ : ∀ (n ... | [
"case refine_1\nC : Type u_1\nD : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝¹³ : Category.{v_1, u_1} C\ninst✝¹² : Category.{v_2, u_2} D\ninst✝¹¹ : Category.{v_3, u_3} D₁\ninst✝¹⁰ : Category.{v_4, u_4} D₂\nA B : ObjectProperty C\ninst✝⁹ : HasZeroObject C\ninst✝⁸ : HasShift C ℤ\ninst✝⁷ : Preadditive C\ninst✝⁶ : ∀ ... | refine Functor.IsLocalization.of_equivalence_source L₁.op (B.inverseImage A.ι).trW.op
_ _ A.opEquivalence.symm ?_ ?_
((Functor.associator _ _ _).symm ≪≫
Functor.isoWhiskerRight A.opEquivalence.counitIso _ ≪≫ Functor.leftUnitor _) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 60,
"column": 4
} | {
"line": 63,
"column": 41
} | {
"line": 64,
"column": 2
} | [
{
"pp": "case h₁\nC : 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\nT T' : Triangle C\nf₁ f₂ : T ⟶ T'\nhT : T ∈ distinguishedTriangles\nhT' : T' ∈ distingu... | [] | obtain ⟨g, hg⟩ := Triangle.coyoneda_exact₂ _ (inv_rot_of_distTriang _ hT')
f.hom₁ (by simp [← f.comm₁, hf])
simp [hg, t.zero_of_isLE_of_isGE g a (b + 1) (by lia)
h₀ (t.isGE_shift _ b (-1) (b + 1))] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 60,
"column": 4
} | {
"line": 63,
"column": 41
} | {
"line": 64,
"column": 2
} | [
{
"pp": "case h₁\nC : 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\nT T' : Triangle C\nf₁ f₂ : T ⟶ T'\nhT : T ∈ distinguishedTriangles\nhT' : T' ∈ distingu... | [] | obtain ⟨g, hg⟩ := Triangle.coyoneda_exact₂ _ (inv_rot_of_distTriang _ hT')
f.hom₁ (by simp [← f.comm₁, hf])
simp [hg, t.zero_of_isLE_of_isGE g a (b + 1) (by lia)
h₀ (t.isGE_shift _ b (-1) (b + 1))] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.BifunctorAssociator | {
"line": 802,
"column": 43
} | {
"line": 802,
"column": 69
} | {
"line": 803,
"column": 6
} | [
{
"pp": "case pos\nC₁ : Type u_1\nC₂ : Type u_2\nC₁₂ : Type u_3\nC₂₃ : Type u_4\nC₃ : Type u_5\nC₄ : Type u_6\ninst✝³³ : Category.{v_1, u_1} C₁\ninst✝³² : Category.{v_2, u_2} C₂\ninst✝³¹ : Category.{v_3, u_5} C₃\ninst✝³⁰ : Category.{v_4, u_6} C₄\ninst✝²⁹ : Category.{v_5, u_3} C₁₂\ninst✝²⁸ : Category.{v_6, u_4} ... | [
"case pos\nC₁ : Type u_1\nC₂ : Type u_2\nC₁₂ : Type u_3\nC₂₃ : Type u_4\nC₃ : Type u_5\nC₄ : Type u_6\ninst✝³³ : Category.{v_1, u_1} C₁\ninst✝³² : Category.{v_2, u_2} C₂\ninst✝³¹ : Category.{v_3, u_5} C₃\ninst✝³⁰ : Category.{v_4, u_6} C₄\ninst✝²⁹ : Category.{v_5, u_3} C₁₂\ninst✝²⁸ : Category.{v_6, u_4} C₂₃\ninst✝²⁷... | NatTrans.naturality_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 148,
"column": 16
} | {
"line": 148,
"column": 50
} | {
"line": 149,
"column": 4
} | [
{
"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\nn : ℤ\nX Y : C\nφ : X ⟶ Y\nH : ∃ f, f.hom₂ = φ\n⊢ (triangle t n X).mor₂ ≫ H.choose.hom₃ = φ ≫ (t... | [] | rw [H.choose.comm₂, H.choose_spec] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 148,
"column": 16
} | {
"line": 148,
"column": 50
} | {
"line": 149,
"column": 4
} | [
{
"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\nn : ℤ\nX Y : C\nφ : X ⟶ Y\nH : ∃ f, f.hom₂ = φ\n⊢ (triangle t n X).mor₂ ≫ H.choose.hom₃ = φ ≫ (t... | [] | rw [H.choose.comm₂, H.choose_spec] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 148,
"column": 16
} | {
"line": 148,
"column": 50
} | {
"line": 149,
"column": 4
} | [
{
"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\nn : ℤ\nX Y : C\nφ : X ⟶ Y\nH : ∃ f, f.hom₂ = φ\n⊢ (triangle t n X).mor₂ ≫ H.choose.hom₃ = φ ≫ (t... | [] | rw [H.choose.comm₂, H.choose_spec] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 183,
"column": 16
} | {
"line": 183,
"column": 50
} | {
"line": 184,
"column": 4
} | [
{
"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\nn : ℤ\nX Y : C\nφ : X ⟶ Y\nA : C\na b : ℤ\nh : a ≤ b\nH : ∃ f, f.hom₂ = 𝟙 (triangle t a A).obj₂... | [] | rw [H.choose.comm₂, H.choose_spec] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 183,
"column": 16
} | {
"line": 183,
"column": 50
} | {
"line": 184,
"column": 4
} | [
{
"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\nn : ℤ\nX Y : C\nφ : X ⟶ Y\nA : C\na b : ℤ\nh : a ≤ b\nH : ∃ f, f.hom₂ = 𝟙 (triangle t a A).obj₂... | [] | rw [H.choose.comm₂, H.choose_spec] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 183,
"column": 16
} | {
"line": 183,
"column": 50
} | {
"line": 184,
"column": 4
} | [
{
"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\nn : ℤ\nX Y : C\nφ : X ⟶ Y\nA : C\na b : ℤ\nh : a ≤ b\nH : ∃ f, f.hom₂ = 𝟙 (triangle t a A).obj₂... | [] | rw [H.choose.comm₂, H.choose_spec] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.ModelCategory.Homotopy | {
"line": 60,
"column": 35
} | {
"line": 60,
"column": 48
} | {
"line": 60,
"column": 48
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : ModelCategory C\nX Y Z : C\nf g : X ⟶ Y\ninst✝¹ : IsCofibrant X\nh✝ : LeftHomotopyRel f g\nQ : PathObject Y\ninst✝ : Q.IsGood\nP : Cylinder X := ⋯.choose\nh : ⋯.choose.LeftHomotopy f g\nh' : ⋯.choose.IsGood\nsq : CommSq (f ≫ Q.ι) P.i₀ Q.p (prod.lift (P.π... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : ModelCategory C\nX Y Z : C\nf g : X ⟶ Y\ninst✝¹ : IsCofibrant X\nh✝ : LeftHomotopyRel f g\nQ : PathObject Y\ninst✝ : Q.IsGood\nP : Cylinder X := ⋯.choose\nh : ⋯.choose.LeftHomotopy f g\nh' : ⋯.choose.IsGood\nsq : CommSq (f ≫ Q.ι) P.i₀ Q.p (prod.lift (P.π ≫ f) h.h)\n... | prod.lift_snd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.ModelCategory.Homotopy | {
"line": 194,
"column": 40
} | {
"line": 194,
"column": 53
} | {
"line": 194,
"column": 53
} | [
{
"pp": "C : 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✝ : RightHomotopyRel (f ≫ f₀) (f ≫ f₁)\nP : PathObject Z\nleft✝ : P.IsGood\nh : P.RightHomotopy (f ≫ f₀) (f ≫ f₁)\nsq : CommSq... | [
"C : 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✝ : RightHomotopyRel (f ≫ f₀) (f ≫ f₁)\nP : PathObject Z\nleft✝ : P.IsGood\nh : P.RightHomotopy (f ≫ f₀) (f ≫ f₁)\nsq : CommSq h.h f P.p (... | prod.lift_snd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.ModelCategory.FibrantObjectHomotopy | {
"line": 67,
"column": 49
} | {
"line": 67,
"column": 80
} | {
"line": 69,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : ModelCategory C\n⊢ toHoCat.Full",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"HomotopicalAlgebra.ModelCategory.cm1a",
"HomotopicalAlgebra.FibrantObject.toHoCat",
"CategoryTheory.Quotient.full_functor",
... | [] | dsimp [toHoCat]; infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.ModelCategory.FibrantObjectHomotopy | {
"line": 67,
"column": 49
} | {
"line": 67,
"column": 80
} | {
"line": 69,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : ModelCategory C\n⊢ toHoCat.Full",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"HomotopicalAlgebra.ModelCategory.cm1a",
"HomotopicalAlgebra.FibrantObject.toHoCat",
"CategoryTheory.Quotient.full_functor",
... | [] | dsimp [toHoCat]; infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Factorizations.CM5a | {
"line": 317,
"column": 4
} | {
"line": 317,
"column": 39
} | {
"line": 318,
"column": 4
} | [
{
"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 : ℤ\nthis : IsIso ((S f n).homologyπ n)\n⊢ Mono (homologyπ ((cokernel f).truncGE n) n ≫ homologyMap (p f n) n)",
"ppTerm": "?m.97",
"assigned": true,
"usedCon... | [
"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 : ℤ\nthis : IsIso ((S f n).homologyπ n)\n⊢ Mono (cyclesMap (p f n) n ≫ (S f n).homologyπ n)"
] | rw [homologyπ_naturality (p f n) n] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Functor.Derived.PointwiseRightDerived | {
"line": 127,
"column": 45
} | {
"line": 127,
"column": 71
} | {
"line": 127,
"column": 72
} | [
{
"pp": "case refine_1\nC : Type u₁\nD : Type u₂\nH : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} H\nF' : D ⥤ H\nF : C ⥤ H\nL : C ⥤ D\nα : F ⟶ L ⋙ F'\nW : MorphismProperty C\nG : D ⥤ H\ne : F ≅ L ⋙ G\ninst✝ : L.IsLocalization W\nY : C\ns : Cocone (Costructured... | [
"case refine_1\nC : Type u₁\nD : Type u₂\nH : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} H\nF' : D ⥤ H\nF : C ⥤ H\nL : C ⥤ D\nα : F ⟶ L ⋙ F'\nW : MorphismProperty C\nG : D ⥤ H\ne : F ≅ L ⋙ G\ninst✝ : L.IsLocalization W\nY : C\ns : Cocone (CostructuredArrow.proj L... | NatTrans.naturality_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Localization.DerivabilityStructure.Derives | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 73
} | {
"line": 71,
"column": 2
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\nH : Type u₃\ninst✝⁶ : Category.{v₁, u₁} C₁\ninst✝⁵ : Category.{v₂, u₂} C₂\ninst✝⁴ : Category.{v₃, u₃} H\nD₂ : Type u₄\ninst✝³ : Category.{v₄, u₄} D₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nΦ : LocalizerMorphism W₁ W₂\nF : C₂ ⥤ H\nh : Φ.Derives F\ninst✝² : Φ.IsRi... | [
"C₁ : Type u₁\nC₂ : Type u₂\nH : Type u₃\ninst✝⁶ : Category.{v₁, u₁} C₁\ninst✝⁵ : Category.{v₂, u₂} C₂\ninst✝⁴ : Category.{v₃, u₃} H\nD₂ : Type u₄\ninst✝³ : Category.{v₄, u₄} D₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nΦ : LocalizerMorphism W₁ W₂\nF : C₂ ⥤ H\nh : Φ.Derives F\ninst✝² : Φ.IsRightDerivabil... | let G : W₁.Localization ⥤ H := Localization.lift (Φ.functor ⋙ F) h W₁.Q | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Functor.Derived.PointwiseRightDerived | {
"line": 140,
"column": 8
} | {
"line": 140,
"column": 34
} | {
"line": 140,
"column": 35
} | [
{
"pp": "case refine_2\nC : Type u₁\nD : Type u₂\nH : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} H\nF' : D ⥤ H\nF : C ⥤ H\nL : C ⥤ D\nα : F ⟶ L ⋙ F'\nW : MorphismProperty C\nG : D ⥤ H\ne : F ≅ L ⋙ G\ninst✝ : L.IsLocalization W\nY : C\ns : Cocone (Costructured... | [
"case refine_2\nC : Type u₁\nD : Type u₂\nH : Type u₃\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} H\nF' : D ⥤ H\nF : C ⥤ H\nL : C ⥤ D\nα : F ⟶ L ⋙ F'\nW : MorphismProperty C\nG : D ⥤ H\ne : F ≅ L ⋙ G\ninst✝ : L.IsLocalization W\nY : C\ns : Cocone (CostructuredArrow.proj L... | NatTrans.naturality_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.DifferentialObject | {
"line": 53,
"column": 74
} | {
"line": 53,
"column": 87
} | {
"line": 55,
"column": 0
} | [
{
"pp": "β : Type u_1\ninst✝² : AddCommGroup β\nb : β\nV : Type u_2\ninst✝¹ : Category.{v_1, u_2} V\ninst✝ : HasZeroMorphisms V\nX : DifferentialObject ℤ (GradedObjectWithShift b V)\nx y : β\nh : x = y\n⊢ X.objEqToHom h ≫ X.d y = X.d x ≫ X.objEqToHom ⋯",
"ppTerm": "?m.83",
"assigned": true,
"usedCon... | [] | cases h; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.DifferentialObject | {
"line": 53,
"column": 74
} | {
"line": 53,
"column": 87
} | {
"line": 55,
"column": 0
} | [
{
"pp": "β : Type u_1\ninst✝² : AddCommGroup β\nb : β\nV : Type u_2\ninst✝¹ : Category.{v_1, u_2} V\ninst✝ : HasZeroMorphisms V\nX : DifferentialObject ℤ (GradedObjectWithShift b V)\nx y : β\nh : x = y\n⊢ X.objEqToHom h ≫ X.d y = X.d x ≫ X.objEqToHom ⋯",
"ppTerm": "?m.83",
"assigned": true,
"usedCon... | [] | cases h; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.DifferentialObject | {
"line": 62,
"column": 72
} | {
"line": 62,
"column": 85
} | {
"line": 64,
"column": 0
} | [
{
"pp": "β : Type u_1\ninst✝² : AddCommGroup β\nb : β\nV : Type u_2\ninst✝¹ : Category.{v_1, u_2} V\ninst✝ : HasZeroMorphisms V\nX Y : DifferentialObject ℤ (GradedObjectWithShift b V)\nf : X ⟶ Y\nx y : β\nh : x = y\n⊢ X.objEqToHom h ≫ f.f y = f.f x ≫ Y.objEqToHom h",
"ppTerm": "?m.66",
"assigned": true,... | [] | cases h; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.DifferentialObject | {
"line": 62,
"column": 72
} | {
"line": 62,
"column": 85
} | {
"line": 64,
"column": 0
} | [
{
"pp": "β : Type u_1\ninst✝² : AddCommGroup β\nb : β\nV : Type u_2\ninst✝¹ : Category.{v_1, u_2} V\ninst✝ : HasZeroMorphisms V\nX Y : DifferentialObject ℤ (GradedObjectWithShift b V)\nf : X ⟶ Y\nx y : β\nh : x = y\n⊢ X.objEqToHom h ≫ f.f y = f.f x ≫ Y.objEqToHom h",
"ppTerm": "?m.66",
"assigned": true,... | [] | cases h; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.DifferentialObject | {
"line": 75,
"column": 56
} | {
"line": 75,
"column": 69
} | {
"line": 77,
"column": 0
} | [
{
"pp": "β : Type u_1\ninst✝² : AddCommGroup β\nb : β\nV : Type u_2\ninst✝¹ : Category.{v_1, u_2} V\ninst✝ : HasZeroMorphisms V\nX : HomologicalComplex V (ComplexShape.up' b)\nx y z : β\nh : y = z\n⊢ X.d x y ≫ eqToHom ⋯ = X.d x z",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Categ... | [] | cases h; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.DifferentialObject | {
"line": 75,
"column": 56
} | {
"line": 75,
"column": 69
} | {
"line": 77,
"column": 0
} | [
{
"pp": "β : Type u_1\ninst✝² : AddCommGroup β\nb : β\nV : Type u_2\ninst✝¹ : Category.{v_1, u_2} V\ninst✝ : HasZeroMorphisms V\nX : HomologicalComplex V (ComplexShape.up' b)\nx y z : β\nh : y = z\n⊢ X.d x y ≫ eqToHom ⋯ = X.d x z",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Categ... | [] | cases h; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Double | {
"line": 120,
"column": 15
} | {
"line": 120,
"column": 26
} | {
"line": 120,
"column": 26
} | [
{
"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\nh : i₀ ≠ i₁\nK : HomologicalComplex C c\nφ₀ : X₀ ⟶ K.X i₀\nφ₁ : X₁ ⟶ K.X i₁\ncomm : φ₀ ≫ K.d i₀ i₁ = f ≫ φ₁\nhφ : ... | [] | by rw [hk₀] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.Double | {
"line": 120,
"column": 71
} | {
"line": 120,
"column": 82
} | {
"line": 120,
"column": 82
} | [
{
"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\nh : i₀ ≠ i₁\nK : HomologicalComplex C c\nφ₀ : X₀ ⟶ K.X i₀\nφ₁ : X₁ ⟶ K.X i₁\ncomm : φ₀ ≫ K.d i₀ i₁ = f ≫ φ₁\nhφ : ... | [] | by rw [hk₀] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.EulerCharacteristic | {
"line": 86,
"column": 31
} | {
"line": 86,
"column": 45
} | {
"line": 88,
"column": 0
} | [
{
"pp": "ι : Type u_1\nc : ComplexShape ι\ninst✝ : c.EulerCharSigns\ni✝ : ℕ\n⊢ (-1) ^ (i✝ + 1) = -(-1) ^ i✝",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Int.instCommMonoid",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommR... | [] | simp [pow_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.EulerCharacteristic | {
"line": 92,
"column": 31
} | {
"line": 92,
"column": 45
} | {
"line": 94,
"column": 0
} | [
{
"pp": "ι : Type u_1\nc : ComplexShape ι\ninst✝ : c.EulerCharSigns\nj✝ : ℕ\n⊢ (-1) ^ j✝ = -(-1) ^ (j✝ + 1)",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Int.instCommMonoid",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommR... | [] | simp [pow_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.Functor | {
"line": 63,
"column": 4
} | {
"line": 65,
"column": 25
} | {
"line": 68,
"column": 0
} | [
{
"pp": "T : Type u_1\ninst✝² : Category.{v_1, u_1} T\nV : Type u_2\ninst✝¹ : Category.{v_2, u_2} V\ninst✝ : HasZeroMorphisms V\nι : Type u_3\nc : ComplexShape ι\nC : HomologicalComplex (T ⥤ V) c\nX✝ Y✝ Z✝ : T\nh₁ : X✝ ⟶ Y✝\nh₂ : Y✝ ⟶ Z✝\n⊢ { f := fun i ↦ (C.X i).map (h₁ ≫ h₂), comm' := ⋯ } =\n { f := fun i ... | [] | ext i
dsimp
rw [Functor.map_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Functor | {
"line": 63,
"column": 4
} | {
"line": 65,
"column": 25
} | {
"line": 68,
"column": 0
} | [
{
"pp": "T : Type u_1\ninst✝² : Category.{v_1, u_1} T\nV : Type u_2\ninst✝¹ : Category.{v_2, u_2} V\ninst✝ : HasZeroMorphisms V\nι : Type u_3\nc : ComplexShape ι\nC : HomologicalComplex (T ⥤ V) c\nX✝ Y✝ Z✝ : T\nh₁ : X✝ ⟶ Y✝\nh₂ : Y✝ ⟶ Z✝\n⊢ { f := fun i ↦ (C.X i).map (h₁ ≫ h₂), comm' := ⋯ } =\n { f := fun i ... | [] | ext i
dsimp
rw [Functor.map_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCategory.SpectralObject | {
"line": 59,
"column": 22
} | {
"line": 61,
"column": 69
} | {
"line": 63,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preadditive C\ninst✝¹ : HasZeroObject C\ninst✝ : HasBinaryBiproducts C\nD : ComposableArrows (CochainComplex C ℤ) 2\n⊢ Pretriangulated.Triangle.mk\n ((composableArrowsFunctor C ⋙ quotient C (ComplexShape.up ℤ)).map\n ((ComposableArrows.... | [] | by
obtain ⟨_, _, _, f, g, rfl⟩ := ComposableArrows.mk₂_surjective D
exact HomotopyCategory.mappingConeCompTriangleh_distinguished f g | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Idempotents.FunctorExtension | {
"line": 212,
"column": 6
} | {
"line": 212,
"column": 36
} | {
"line": 212,
"column": 36
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : Category.{v_3, u_3} E\ninst✝ : IsIdempotentComplete D\n⊢ (functorExtension₂ C D).IsEquivalence",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"C... | [
"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : Category.{v_3, u_3} E\ninst✝ : IsIdempotentComplete D\n⊢ (karoubiUniversal₂ C D).functor.IsEquivalence"
] | ← karoubiUniversal₂_functor_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.GradedObject.Unitor | {
"line": 251,
"column": 2
} | {
"line": 258,
"column": 5
} | {
"line": 260,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : Category.{v_2, u_2} D\ninst✝⁵ : Zero I\ninst✝⁴ : DecidableEq I\ninst✝³ : HasInitial C\nF : D ⥤ C ⥤ D\nY : C\ne : F.flip.obj Y ≅ 𝟭 D\ninst✝² : ∀ (X : D), PreservesColimit (Functor.empty C) (F.obj X)\np : J ... | [] | ext j
dsimp
rw [mapBifunctorRightUnitor_inv_apply, mapBifunctorRightUnitor_inv_apply, assoc, assoc,
ι_mapBifunctorMapMap]
dsimp
rw [Functor.map_id, id_comp, NatTrans.naturality_assoc]
erw [← NatTrans.naturality_assoc e.inv]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.GradedObject.Unitor | {
"line": 251,
"column": 2
} | {
"line": 258,
"column": 5
} | {
"line": 260,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : Category.{v_2, u_2} D\ninst✝⁵ : Zero I\ninst✝⁴ : DecidableEq I\ninst✝³ : HasInitial C\nF : D ⥤ C ⥤ D\nY : C\ne : F.flip.obj Y ≅ 𝟭 D\ninst✝² : ∀ (X : D), PreservesColimit (Functor.empty C) (F.obj X)\np : J ... | [] | ext j
dsimp
rw [mapBifunctorRightUnitor_inv_apply, mapBifunctorRightUnitor_inv_apply, assoc, assoc,
ι_mapBifunctorMapMap]
dsimp
rw [Functor.map_id, id_comp, NatTrans.naturality_assoc]
erw [← NatTrans.naturality_assoc e.inv]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.SpectralObject.Cycles | {
"line": 251,
"column": 2
} | {
"line": 253,
"column": 55
} | {
"line": 255,
"column": 0
} | [
{
"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 : ι\nf : i ⟶ j\ng : j ⟶ k\nn : ℤ\n⊢ X.opcyclesMap f g f g (𝟙 (mk₂ f g)) n = 𝟙 (X.opcycles f g n)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstan... | [] | rw [← cancel_epi (X.pOpcycles f g n),
X.p_opcyclesMap f g f g (𝟙 _) (𝟙 _),
Functor.map_id, Category.comp_id, Category.id_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.SpectralObject.Cycles | {
"line": 251,
"column": 2
} | {
"line": 253,
"column": 55
} | {
"line": 255,
"column": 0
} | [
{
"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 : ι\nf : i ⟶ j\ng : j ⟶ k\nn : ℤ\n⊢ X.opcyclesMap f g f g (𝟙 (mk₂ f g)) n = 𝟙 (X.opcycles f g n)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstan... | [] | rw [← cancel_epi (X.pOpcycles f g n),
X.p_opcyclesMap f g f g (𝟙 _) (𝟙 _),
Functor.map_id, Category.comp_id, Category.id_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.SpectralObject.Cycles | {
"line": 251,
"column": 2
} | {
"line": 253,
"column": 55
} | {
"line": 255,
"column": 0
} | [
{
"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 : ι\nf : i ⟶ j\ng : j ⟶ k\nn : ℤ\n⊢ X.opcyclesMap f g f g (𝟙 (mk₂ f g)) n = 𝟙 (X.opcycles f g n)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstan... | [] | rw [← cancel_epi (X.pOpcycles f g n),
X.p_opcyclesMap f g f g (𝟙 _) (𝟙 _),
Functor.map_id, Category.comp_id, Category.id_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.GradedObject.Unitor | {
"line": 365,
"column": 2
} | {
"line": 365,
"column": 6
} | {
"line": 366,
"column": 2
} | [
{
"pp": "case e_a.e_a.e_a\nC₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nD : Type u_4\nI₁ : Type u_5\nI₂ : Type u_6\nI₃ : Type u_7\nJ : Type u_8\ninst✝¹⁵ : Category.{v_1, u_1} C₁\ninst✝¹⁴ : Category.{v_2, u_2} C₂\ninst✝¹³ : Category.{v_3, u_3} C₃\ninst✝¹² : Category.{v_4, u_4} D\ninst✝¹¹ : Zero I₂\ninst✝¹⁰ : Dec... | [
"case e_a.e_a.e_a\nC₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nD : Type u_4\nI₁ : Type u_5\nI₂ : Type u_6\nI₃ : Type u_7\nJ : Type u_8\ninst✝¹⁵ : Category.{v_1, u_1} C₁\ninst✝¹⁴ : Category.{v_2, u_2} C₂\ninst✝¹³ : Category.{v_3, u_3} C₃\ninst✝¹² : Category.{v_4, u_4} D\ninst✝¹¹ : Zero I₂\ninst✝¹⁰ : DecidableEq I₂\... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 80
} | {
"line": 158,
"column": 0
} | [
{
"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 : ι\nf : i ⟶ j\ng : j ⟶ k\nhf : IsIso f\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\n⊢ X.δ f g n₀ n₁ hn₁ = 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants":... | [] | simpa only [Preadditive.IsIso.comp_left_eq_zero] using X.zero₃ f g _ rfl n₀ n₁ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 80
} | {
"line": 158,
"column": 0
} | [
{
"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 : ι\nf : i ⟶ j\ng : j ⟶ k\nhf : IsIso f\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\n⊢ X.δ f g n₀ n₁ hn₁ = 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants":... | [] | simpa only [Preadditive.IsIso.comp_left_eq_zero] using X.zero₃ f g _ rfl n₀ n₁ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 80
} | {
"line": 158,
"column": 0
} | [
{
"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 : ι\nf : i ⟶ j\ng : j ⟶ k\nhf : IsIso f\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\n⊢ X.δ f g n₀ n₁ hn₁ = 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants":... | [] | simpa only [Preadditive.IsIso.comp_left_eq_zero] using X.zero₃ f g _ rfl n₀ n₁ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 291,
"column": 2
} | {
"line": 291,
"column": 72
} | {
"line": 292,
"column": 2
} | [
{
"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\nf₁₂ : i ⟶ k\nh₁₂ : f₁ ≫ f₂ = f₁₂\nf₂₃ : j ⟶ l\nh₂₃ : f₂ ≫ f₃ = f₂₃\nn₀ n₁ n₂ : ℤ\nhn₁ : n₀ + 1 = n₁\nhn₂ : n₁ + 1 = n₂\... | [
"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\nf₁₂ : i ⟶ k\nh₁₂ : f₁ ≫ f₂ = f₁₂\nf₂₃ : j ⟶ l\nh₂₃ : f₂ ≫ f₃ = f₂₃\nn₀ n₁ n₂ : ℤ\nhn₁ : n₀ + 1 = n₁\nhn₂ : n₁ + 1 = n₂\nhp : IsColi... | let hp := (X.cokernelSequenceOpcycles_exact f₂ f₃ _ _ hn₁).gIsCokernel | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 315,
"column": 2
} | {
"line": 315,
"column": 72
} | {
"line": 316,
"column": 2
} | [
{
"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\nn₀ n₁ n₂ : ℤ\nhn₁ : n₀ + 1 = n₁\nhn₂ : n₁ + 1 = n₂\n⊢ (X.rightHomologyDataShortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g' = ... | [
"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\nn₀ n₁ n₂ : ℤ\nhn₁ : n₀ + 1 = n₁\nhn₂ : n₁ + 1 = n₂\nhp : IsColimit (CokernelCofork.ofπ (X.cokernelSequenceOpcycles f₂ f₃ n₀ n₁ hn₁)... | let hp := (X.cokernelSequenceOpcycles_exact f₂ f₃ _ _ hn₁).gIsCokernel | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence | {
"line": 223,
"column": 16
} | {
"line": 223,
"column": 44
} | {
"line": 224,
"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\nl : ℕ\npq : ℤ × Fin l\n⊢ pq.2.castSucc ≤ pq.2.succ",
"ppTerm": "?m.148",
"assigned": true,
"usedConstants": [
"Nat.le_add_right._simp_1",
... | [] | simp [Fin.le_iff_val_le_val] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence | {
"line": 223,
"column": 16
} | {
"line": 223,
"column": 44
} | {
"line": 224,
"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\nl : ℕ\npq : ℤ × Fin l\n⊢ pq.2.castSucc ≤ pq.2.succ",
"ppTerm": "?m.148",
"assigned": true,
"usedConstants": [
"Nat.le_add_right._simp_1",
... | [] | simp [Fin.le_iff_val_le_val] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence | {
"line": 223,
"column": 16
} | {
"line": 223,
"column": 44
} | {
"line": 224,
"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\nl : ℕ\npq : ℤ × Fin l\n⊢ pq.2.castSucc ≤ pq.2.succ",
"ppTerm": "?m.148",
"assigned": true,
"usedConstants": [
"Nat.le_add_right._simp_1",
... | [] | simp [Fin.le_iff_val_le_val] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Artinian.Module | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 66
} | {
"line": 105,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u_1\nM : Type u_2\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\nS : Type u_5\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra S R\ninst✝² : Module S M\ninst✝¹ : IsArtinian R M\ninst✝ : IsScalarTower S R M\nH : Function.Surjective ⇑(algebraMap S R)\nN : Submodule... | [
"case refine_1\nR : Type u_1\nM : Type u_2\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\nS : Type u_5\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra S R\ninst✝² : Module S M\ninst✝¹ : IsArtinian R M\ninst✝ : IsScalarTower S R M\nH : Function.Surjective ⇑(algebraMap S R)\nN : Submodule S M\n⊢ ∀ (c... | refine { toAddSubmonoid := N.toAddSubmonoid, smul_mem' := ?_ } | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Artinian.Module | {
"line": 358,
"column": 2
} | {
"line": 358,
"column": 33
} | {
"line": 359,
"column": 2
} | [
{
"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\nx : M\nn : ℕ\nhn : ∀ (m : ℕ), n ≤ m → (r ^ n • LinearMap.id).range = (r ^ m • LinearMap.id).range\n⊢ ∃ n y, r ^ n.succ • y = r ^ n • x",
"ppTerm": "?m.39",
"assigne... | [
"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\nx : M\nn : ℕ\nhn : ∀ (m : ℕ), n ≤ m → ∀ (x : M), x ∈ (r ^ n • LinearMap.id).range ↔ x ∈ (r ^ m • LinearMap.id).range\n⊢ ∃ n y, r ^ n.succ • y = r ^ n • x"
] | simp_rw [SetLike.ext_iff] at hn | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 623,
"column": 2
} | {
"line": 623,
"column": 12
} | {
"line": 624,
"column": 2
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\ns : Set L\n⊢ s ⊆ ↑(lieSpan R L s)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Set.instMembership",
"Set"
],
"usedFVars": [
"L"
],
... | [
"R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\ns : Set L\nm : L\nhm : m ∈ s\n⊢ m ∈ ↑(lieSpan R L s)"
] | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Lie.Ideal | {
"line": 322,
"column": 2
} | {
"line": 322,
"column": 30
} | {
"line": 324,
"column": 0
} | [
{
"pp": "case mpr\nR : 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'\nI : LieIdeal R L\nh : ∀ x ∈ I, f x = 0\nx : L\nhx : x ∈ I\n⊢ x ∈ f.ker",
"ppTerm": "?mpr",
"assigned": true,
"usedC... | [] | · rw [mem_ker]; apply h x hx | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.Ideal | {
"line": 376,
"column": 24
} | {
"line": 380,
"column": 29
} | {
"line": 382,
"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'\nI : LieIdeal R L\ny : L'\nh₁ : Function.Surjective ⇑f\nh₂ : y ∈ map f I\n⊢ ∃ x, f ↑x = y",
"ppTerm": "?m.57",
"assigned": true,
... | [] | by
rw [← LieSubmodule.mem_toSubmodule, coe_map_of_surjective h₁, Submodule.mem_map] at h₂
obtain ⟨x, hx, rfl⟩ := h₂
use ⟨x, hx⟩
rw [LieHom.coe_toLinearMap] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Submodule | {
"line": 613,
"column": 2
} | {
"line": 613,
"column": 12
} | {
"line": 614,
"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\ns : Set M\n⊢ s ⊆ ↑(lieSpan R L s)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Set.instMembership... | [
"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\ns : Set M\nm : M\nhm : m ∈ s\n⊢ m ∈ ↑(lieSpan R L s)"
] | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Lie.IdealOperations | {
"line": 145,
"column": 58
} | {
"line": 145,
"column": 93
} | {
"line": 147,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieAlgebra R L\nI : LieIdeal R L\n⊢ ⁅I, ⊥⁆ = ⊥",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"LieSubmodule.instSet... | [] | rw [eq_bot_iff]; apply lie_le_right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.IdealOperations | {
"line": 145,
"column": 58
} | {
"line": 145,
"column": 93
} | {
"line": 147,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieAlgebra R L\nI : LieIdeal R L\n⊢ ⁅I, ⊥⁆ = ⊥",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"LieSubmodule.instSet... | [] | rw [eq_bot_iff]; apply lie_le_right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.IdealOperations | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 37
} | {
"line": 151,
"column": 37
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nN : LieSubmodule R L M\ninst✝ : LieAlgebra R L\nm : M\nx : L\nhx : x ∈ ⊥\nn : ↥N\nhn : ⁅↑⟨x, hx⟩, ↑n⁆ = m\n⊢ ⁅↑⟨x, hx⟩, ↑n⁆ ∈ ↑⊥",
"ppTerm": "?m.129... | [
"R : Type u\nL : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nN : LieSubmodule R L M\ninst✝ : LieAlgebra R L\nm : M\nx : L\nn : ↥N\nhx : x ∈ ⊥\nhn : ⁅↑⟨x, hx⟩, ↑n⁆ = m\n⊢ ⁅↑⟨x, hx⟩, ↑n⁆ ∈ ↑⊥"
] | change x ∈ (⊥ : LieIdeal R L) at hx | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 6
} | {
"line": 58,
"column": 6
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝² : Field A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\npmonic : p.Monic\nhp : (Polynomial.aeval x) p = 0\npmin : ∀ (q : A[X]), q.Monic → (Polynomial.aeval x) q = 0 → p.degree ≤ q.degree\nhx : IsIntegral A x\n⊢ p = minpoly A x",
"ppTerm": "?m.47",
"a... | [
"A : Type u_1\nB : Type u_2\ninst✝² : Field A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\npmonic : p.Monic\nhp : (Polynomial.aeval x) p = 0\npmin : ∀ (q : A[X]), q.Monic → (Polynomial.aeval x) q = 0 → p.degree ≤ q.degree\nhx : IsIntegral A x\n⊢ minpoly A x = p"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Jordan.Basic | {
"line": 235,
"column": 2
} | {
"line": 235,
"column": 6
} | {
"line": 236,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : NonUnitalNonAssocCommRing A\ninst✝ : IsCommJordan A\na b c : A\n⊢ 2 • (⁅L a, L (b * c)⁆ + ⁅L b, L (c * a)⁆ + ⁅L c, L (a * b)⁆) = 0",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"AddMonoid.End.mulLeft",
"instHSMul",
"HMul.hMul",
"Lie... | [
"A : Type u_1\ninst✝¹ : NonUnitalNonAssocCommRing A\ninst✝ : IsCommJordan A\na b c : A\n⊢ 0 = 2 • (⁅L a, L (b * c)⁆ + ⁅L b, L (c * a)⁆ + ⁅L c, L (a * b)⁆)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Jordan.Basic | {
"line": 237,
"column": 58
} | {
"line": 238,
"column": 52
} | {
"line": 239,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝¹ : NonUnitalNonAssocCommRing A\ninst✝ : IsCommJordan A\na b c : A\n⊢ 0 = ⁅L (a + b + c), L ((a + b + c) * (a + b + c))⁆",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
"AddMonoid.End.mulLeft",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemir... | [] | by
rw [(commute_lmul_lmul_sq (a + b + c)).lie_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Separable | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 33
} | {
"line": 177,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\np q : R[X]\nhsep : p.Separable\nhq : 1 < emultiplicity q p\n⊢ ↑2 ≤ emultiplicity q p",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"instAddMonoidWithOneENat",
"CommSemiring.toSemiring",
"instAddENat",
"instPreorderENat"... | [] | exact Order.add_one_le_of_lt hq | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.Separable | {
"line": 295,
"column": 71
} | {
"line": 297,
"column": 69
} | {
"line": 299,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\nhsep : p.Separable\n⊢ p.roots.Nodup",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Polynomial.roots",
"Multiset.nodup_iff_count_le_one",
"Multiset.Nodup",
"Classical.propDecida... | [] | by
classical
exact Multiset.nodup_iff_count_le_one.mpr (count_roots_le_one hsep) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Determinant | {
"line": 215,
"column": 6
} | {
"line": 215,
"column": 57
} | {
"line": 216,
"column": 4
} | [
{
"pp": "M : Type u_2\ninst✝⁴ : AddCommGroup M\nι : Type u_4\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nA : Type u_5\ninst✝¹ : CommRing A\ninst✝ : Module A M\nb : Basis ι A M\nf : M →ₗ[A] M\nthis : DecidableEq M\n⊢ ((toMatrix b b) f).det = LinearMap.det f",
"ppTerm": "?m.50",
"assigned": true,
"us... | [
"M : Type u_2\ninst✝⁴ : AddCommGroup M\nι : Type u_4\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nA : Type u_5\ninst✝¹ : CommRing A\ninst✝ : Module A M\nb : Basis ι A M\nf : M →ₗ[A] M\nthis : DecidableEq M\n⊢ ((toMatrix b b) f).det = ((toMatrix b.reindexFinsetRange b.reindexFinsetRange) f).det"
] | det_eq_det_toMatrix_of_finset b.reindexFinsetRange, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Separable | {
"line": 436,
"column": 4
} | {
"line": 436,
"column": 78
} | {
"line": 440,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nn : ℕ\nx : F\nhn : 0 < n\nhx : x ≠ 0\nh : (X ^ n - C x).Separable\nhn' : ↑n = 0\n⊢ IsUnit (X ^ n - C x)",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Polynomial.derivative",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne"... | [] | simpa [separable_def, derivative_X_pow, hn', isCoprime_zero_right] using h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.FieldTheory.Separable | {
"line": 436,
"column": 4
} | {
"line": 436,
"column": 78
} | {
"line": 440,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nn : ℕ\nx : F\nhn : 0 < n\nhx : x ≠ 0\nh : (X ^ n - C x).Separable\nhn' : ↑n = 0\n⊢ IsUnit (X ^ n - C x)",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Polynomial.derivative",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne"... | [] | simpa [separable_def, derivative_X_pow, hn', isCoprime_zero_right] using h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Separable | {
"line": 436,
"column": 4
} | {
"line": 436,
"column": 78
} | {
"line": 440,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nn : ℕ\nx : F\nhn : 0 < n\nhx : x ≠ 0\nh : (X ^ n - C x).Separable\nhn' : ↑n = 0\n⊢ IsUnit (X ^ n - C x)",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Polynomial.derivative",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne"... | [] | simpa [separable_def, derivative_X_pow, hn', isCoprime_zero_right] using h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Semisimple | {
"line": 170,
"column": 2
} | {
"line": 175,
"column": 85
} | {
"line": 177,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\np : Submodule R M\nhp : p ∈ f.invtSubmodule\nhf : f.IsSemisimple\n⊢ IsSemisimple (LinearMap.restrict f hp)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"OrderIso.com... | [] | rw [IsSemisimple] at hf ⊢
let e : Submodule R[X] (AEval' (LinearMap.restrict f hp)) ≃o
Iic (AEval.mapSubmodule R M f ⟨p, hp⟩) :=
(Submodule.orderIsoMapComap <| AEval.restrict_equiv_mapSubmodule f p hp).trans <|
Submodule.mapIic _
exact (isSemisimpleModule_iff ..).mpr (e.complementedLattice_iff.mpr i... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Semisimple | {
"line": 170,
"column": 2
} | {
"line": 175,
"column": 85
} | {
"line": 177,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\np : Submodule R M\nhp : p ∈ f.invtSubmodule\nhf : f.IsSemisimple\n⊢ IsSemisimple (LinearMap.restrict f hp)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"OrderIso.com... | [] | rw [IsSemisimple] at hf ⊢
let e : Submodule R[X] (AEval' (LinearMap.restrict f hp)) ≃o
Iic (AEval.mapSubmodule R M f ⟨p, hp⟩) :=
(Submodule.orderIsoMapComap <| AEval.restrict_equiv_mapSubmodule f p hp).trans <|
Submodule.mapIic _
exact (isSemisimpleModule_iff ..).mpr (e.complementedLattice_iff.mpr i... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.BilinearForm.Hom | {
"line": 185,
"column": 2
} | {
"line": 189,
"column": 38
} | {
"line": 190,
"column": 2
} | [
{
"pp": "case mp\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nM' : Type w\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nB₁ B₂ : BilinForm R M'\nl r : M →ₗ[R] M'\nhₗ : Function.Surjective ⇑l\nhᵣ : Function.Surjective ⇑r\nh : B₁.comp l r = B₂.comp l r\n⊢ ... | [
"case mpr\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nM' : Type w\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nB₁ B₂ : BilinForm R M'\nl r : M →ₗ[R] M'\nhₗ : Function.Surjective ⇑l\nhᵣ : Function.Surjective ⇑r\nh : B₁ = B₂\n⊢ B₁.comp l r = B₂.comp l r"
] | · -- B₁.comp l r = B₂.comp l r → B₁ = B₂
ext x y
obtain ⟨x', rfl⟩ := hₗ x
obtain ⟨y', rfl⟩ := hᵣ y
rw [← comp_apply, ← comp_apply, h] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.BilinearForm.Properties | {
"line": 447,
"column": 54
} | {
"line": 450,
"column": 51
} | {
"line": 452,
"column": 0
} | [
{
"pp": "V : Type u_5\nK : Type u_6\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nB₁ B₂ : BilinForm K V\nb₂ : B₂.Nondegenerate\nv : V\n⊢ B₂ ((B₁.symmCompOfNondegenerate B₂ b₂) v) = B₁ v",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
... | [] | by
rw [symmCompOfNondegenerate]
simp only [coe_comp, LinearEquiv.coe_coe, Function.comp_apply]
erw [LinearEquiv.apply_symm_apply (B₂.toDual b₂)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 156,
"column": 10
} | {
"line": 159,
"column": 72
} | {
"line": 159,
"column": 73
} | [
{
"pp": "case a\nR : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R ↥I\ny : ↥I\nhy : y ∈ ↑↑J\na : L\nha : a ∈ I\nb : L\nhb : b ∈ sSup ({I' | IsAtom I'} \\ {I})\n⊢ ⁅b, ↑I.incl y⁆ ∈ Submodule.map ↑... | [] | suffices ⁅b, y.val⁆ = 0 by erw [this]; simp only [zero_mem]
rw [← LieSubmodule.mem_bot (R := R) (L := L),
← (IsSemisimple.sSupIndep_isAtom hI).eq_bot]
exact ⟨lie_mem_right R L I b y y.2, lie_mem_left _ _ _ _ _ hb⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 156,
"column": 10
} | {
"line": 159,
"column": 72
} | {
"line": 159,
"column": 73
} | [
{
"pp": "case a\nR : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R ↥I\ny : ↥I\nhy : y ∈ ↑↑J\na : L\nha : a ∈ I\nb : L\nhb : b ∈ sSup ({I' | IsAtom I'} \\ {I})\n⊢ ⁅b, ↑I.incl y⁆ ∈ Submodule.map ↑... | [] | suffices ⁅b, y.val⁆ = 0 by erw [this]; simp only [zero_mem]
rw [← LieSubmodule.mem_bot (R := R) (L := L),
← (IsSemisimple.sSupIndep_isAtom hI).eq_bot]
exact ⟨lie_mem_right R L I b y y.2, lie_mem_left _ _ _ _ _ hb⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Normalizer | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 12
} | {
"line": 66,
"column": 2
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\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\n⊢ N ≤ N.normalizer",
"ppTerm": "?m.50",
"assigned": true,
"u... | [
"R : Type u_1\nL : Type u_2\nM : Type u_3\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\nm : M\nhm : m ∈ N\n⊢ m ∈ N.normalizer"
] | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Lie.Normalizer | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 12
} | {
"line": 75,
"column": 2
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\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\nN₁ N₂ : LieSubmodule R L M\nh : N₁ ≤ N₂\n⊢ N₁.normalizer ≤ N₂.normalizer",
"ppTerm": "?m.62"... | [
"R : Type u_1\nL : Type u_2\nM : Type u_3\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\nN₁ N₂ : LieSubmodule R L M\nh : N₁ ≤ N₂\nm : M\nhm : m ∈ N₁.normalizer\n⊢ m ∈ N₂.normalizer"
] | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Ring.Divisibility.Lemmas | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 47
} | {
"line": 50,
"column": 2
} | [
{
"pp": "R : Type u_1\nx y : R\nn m p : ℕ\ninst✝ : Semiring R\nhp : n + m ≤ p + 1\nh_comm : Commute x y\nhy : y ^ n = 0\nx✝ : ℕ × ℕ\ni j : ℕ\nhij : (i, j) ∈ Finset.antidiagonal p\n⊢ x ^ m ∣ p.choose (i, j).1 • (x ^ (i, j).1 * y ^ (i, j).2)",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_1\nx y : R\nn m p : ℕ\ninst✝ : Semiring R\nhp : n + m ≤ p + 1\nh_comm : Commute x y\nhy : y ^ n = 0\nx✝ : ℕ × ℕ\ni j : ℕ\nhij : i + j = p\n⊢ x ^ m ∣ p.choose (i, j).1 • (x ^ (i, j).1 * y ^ (i, j).2)"
] | replace hij : i + j = p := by simpa using hij | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Algebra.Lie.CartanSubalgebra | {
"line": 82,
"column": 4
} | {
"line": 95,
"column": 52
} | {
"line": 97,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nH : LieSubalgebra R L\nk : ℕ\nhk : ∀ (l : ℕ), k ≤ l → LieSubmodule.ucs l ⊥ = H.toLieSubmodule\n⊢ H.IsCartanSubalgebra",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"LieAlgebra... | [] | exact
{ nilpotent := by
dsimp only [LieRing.IsNilpotent]
-- The instance for the second `H` in the goal is `lieRingSelfModule`
-- but `rw` expects it to be `H.toLieSubmodule.instLieRingModuleSubtypeMem`,
-- and these are not reducibly defeq.
erw [H.toLieSubmodule.... | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Lie.Engel | {
"line": 115,
"column": 7
} | {
"line": 115,
"column": 88
} | {
"line": 116,
"column": 2
} | [
{
"pp": "R : Type u₁\nL : Type u₂\nM : Type u₄\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\nI : LieIdeal R L\nx : L\nhxI : R ∙ x ⊔ (LieIdeal.toLieSubalgebra R L I).toSubmodule = ⊤\nn i j : ℕ\... | [] | simpa only [bot_sup_eq, LieIdeal.incl_coe, Submodule.map_zero, hxn] using! this n | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.Algebra.Lie.Engel | {
"line": 115,
"column": 7
} | {
"line": 115,
"column": 88
} | {
"line": 116,
"column": 2
} | [
{
"pp": "R : Type u₁\nL : Type u₂\nM : Type u₄\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\nI : LieIdeal R L\nx : L\nhxI : R ∙ x ⊔ (LieIdeal.toLieSubalgebra R L I).toSubmodule = ⊤\nn i j : ℕ\... | [] | simpa only [bot_sup_eq, LieIdeal.incl_coe, Submodule.map_zero, hxn] using! this n | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Engel | {
"line": 115,
"column": 7
} | {
"line": 115,
"column": 88
} | {
"line": 116,
"column": 2
} | [
{
"pp": "R : Type u₁\nL : Type u₂\nM : Type u₄\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\nI : LieIdeal R L\nx : L\nhxI : R ∙ x ⊔ (LieIdeal.toLieSubalgebra R L I).toSubmodule = ⊤\nn i j : ℕ\... | [] | simpa only [bot_sup_eq, LieIdeal.incl_coe, Submodule.map_zero, hxn] using! this n | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Tower | {
"line": 58,
"column": 39
} | {
"line": 58,
"column": 50
} | {
"line": 58,
"column": 51
} | [
{
"pp": "F : Type u\nK : Type v\nA : Type w\ninst✝⁶ : Semiring F\ninst✝⁵ : Semiring K\ninst✝⁴ : Module F K\ninst✝³ : AddCommMonoid A\ninst✝² : Module K A\ninst✝¹ : Module F A\ninst✝ : IsScalarTower F K A\nhf : Module.Finite F A\nb : Finset A\nhb : span F ↑b = ⊤\n⊢ restrictScalars F (span K ↑b) = ⊤",
"ppTerm... | [
"F : Type u\nK : Type v\nA : Type w\ninst✝⁶ : Semiring F\ninst✝⁵ : Semiring K\ninst✝⁴ : Module F K\ninst✝³ : AddCommMonoid A\ninst✝² : Module K A\ninst✝¹ : Module F A\ninst✝ : IsScalarTower F K A\nhf : Module.Finite F A\nb : Finset A\nhb : span F ↑b = ⊤\n⊢ ⊤ ≤ restrictScalars F (span K ↑b)"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Fixed | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 57
} | {
"line": 147,
"column": 4
} | [
{
"pp": "G : Type u\ninst✝² : Group G\nF : Type v\ninst✝¹ : Field F\ninst✝ : MulSemiringAction G F\ns✝ : Finset F\nthis✝ : IsEmpty ↑↑∅\na : F\ns : Finset F\nhas : a ∉ s\nih : LinearIndepOn (↥(subfield G F)) id ↑s → LinearIndepOn F ⇑(toFun G F) ↑s\nhs : LinearIndepOn (↥(subfield G F)) id ↑s ∧ id a ∉ Submodule.sp... | [
"G : Type u\ninst✝² : Group G\nF : Type v\ninst✝¹ : Field F\ninst✝ : MulSemiringAction G F\ns✝ : Finset F\nthis✝ : IsEmpty ↑↑∅\na : F\ns : Finset F\nhas : a ∉ s\nih : LinearIndepOn (↥(subfield G F)) id ↑s → LinearIndepOn F ⇑(toFun G F) ↑s\nhs : LinearIndepOn (↥(subfield G F)) id ↑s ∧ id a ∉ Submodule.span (↥(subfie... | simp_rw [Pi.smul_apply, toFun_apply, one_smul] at hla | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 586,
"column": 6
} | {
"line": 586,
"column": 17
} | {
"line": 586,
"column": 18
} | [
{
"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\nN : LieSubmodule R L M\ninst✝ : LieModule R L M\nk : ℕ\n⊢ ucs k N = ⊤ ↔ LieModule.lowerCentralSeries R L M k ≤ N",
"ppTerm"... | [
"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\nN : LieSubmodule R L M\ninst✝ : LieModule R L M\nk : ℕ\n⊢ ⊤ ≤ ucs k N ↔ LieModule.lowerCentralSeries R L M k ≤ N"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.SplittingField.IsSplittingField | {
"line": 70,
"column": 8
} | {
"line": 70,
"column": 19
} | {
"line": 70,
"column": 20
} | [
{
"pp": "F : Type u\nK : Type v\nL : Type w\ninst✝⁷ : Field K\ninst✝⁶ : Field L\ninst✝⁵ : Field F\ninst✝⁴ : Algebra K L\ninst✝³ : Algebra F K\ninst✝² : Algebra F L\ninst✝¹ : IsScalarTower F K L\nf : F[X]\ninst✝ : IsSplittingField F L f\n⊢ Subalgebra.restrictScalars F (Algebra.adjoin K ↑(Polynomial.map (algebraM... | [
"F : Type u\nK : Type v\nL : Type w\ninst✝⁷ : Field K\ninst✝⁶ : Field L\ninst✝⁵ : Field F\ninst✝⁴ : Algebra K L\ninst✝³ : Algebra F K\ninst✝² : Algebra F L\ninst✝¹ : IsScalarTower F K L\nf : F[X]\ninst✝ : IsSplittingField F L f\n⊢ ⊤ ≤ Subalgebra.restrictScalars F (Algebra.adjoin K ↑(Polynomial.map (algebraMap F L) ... | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | {
"line": 380,
"column": 2
} | {
"line": 380,
"column": 73
} | {
"line": 381,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS T : Set E\n⊢ ↑(adjoin (↥(adjoin F S)) T) = ↑(adjoin F (S ∪ T))",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"Alg... | [
"case a.left\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS T : Set E\n⊢ Set.range ⇑(algebraMap (↥(adjoin F S)) E) ⊆ ↑(adjoin F (S ∪ T))",
"case a.right\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS T : Set E\n⊢ T ⊆ ↑(adjoin F (S ∪... | apply subset_antisymm <;> rw [adjoin_subset_adjoin_iff] <;> constructor | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.FieldTheory.Extension | {
"line": 148,
"column": 4
} | {
"line": 148,
"column": 38
} | {
"line": 149,
"column": 4
} | [
{
"pp": "F : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Field K\ninst✝² : Algebra F E\ninst✝¹ : Algebra F K\nc : Set (Lifts F E K)\nhc : IsChain (fun x1 x2 ↦ x1 ≤ x2) c\nalg : Algebra.IsAlgebraic F E\ninst✝ : Nonempty ↑c\nhext : ∀ σ ∈ c, σ.IsExtendible\nS : Finset E\nΩ : ... | [
"F : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Field K\ninst✝² : Algebra F E\ninst✝¹ : Algebra F K\nc : Set (Lifts F E K)\nhc : IsChain (fun x1 x2 ↦ x1 ≤ x2) c\nalg : Algebra.IsAlgebraic F E\ninst✝ : Nonempty ↑c\nhext : ∀ σ ∈ c, σ.IsExtendible\nS : Finset E\nΩ : Type (max u_... | have ⟨θ, hθπ, hθ⟩ := hext _ π₀.2 S | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 383,
"column": 2
} | {
"line": 384,
"column": 54
} | {
"line": 386,
"column": 0
} | [
{
"pp": "k : Type u\ninst✝¹⁷ : Field k\nK : Type u\ninst✝¹⁶ : Field K\nL : Type v\nM : Type w\ninst✝¹⁵ : Field L\ninst✝¹⁴ : Algebra K L\ninst✝¹³ : Field M\ninst✝¹² : Algebra K M\ninst✝¹¹ : IsAlgClosed M\ninst✝¹⁰ : Algebra.IsAlgebraic K L\nR : Type u\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDomain R\nS : Type v\ninst✝⁷ ... | [] | obtain _ | ⟨p, _, _⟩ := CharP.exists' k
exacts [.ofCharZero, PerfectRing.toPerfectField k p] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 383,
"column": 2
} | {
"line": 384,
"column": 54
} | {
"line": 386,
"column": 0
} | [
{
"pp": "k : Type u\ninst✝¹⁷ : Field k\nK : Type u\ninst✝¹⁶ : Field K\nL : Type v\nM : Type w\ninst✝¹⁵ : Field L\ninst✝¹⁴ : Algebra K L\ninst✝¹³ : Field M\ninst✝¹² : Algebra K M\ninst✝¹¹ : IsAlgClosed M\ninst✝¹⁰ : Algebra.IsAlgebraic K L\nR : Type u\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDomain R\nS : Type v\ninst✝⁷ ... | [] | obtain _ | ⟨p, _, _⟩ := CharP.exists' k
exacts [.ofCharZero, PerfectRing.toPerfectField k p] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Extension | {
"line": 217,
"column": 6
} | {
"line": 217,
"column": 61
} | {
"line": 217,
"column": 61
} | [
{
"pp": "F : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nx : Lifts F E K\ns : E\nh1 : IsIntegral (↥x.carrier) s\nh2 : (Polynomial.map x.emb.toRingHom (minpoly (↥x.carrier) s)).Splits\nI2 : (minpoly (↥x.carrier) s).degree ... | [] | exact (eval_map _ _).symm.trans (eval_rootOfSplits _ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 12
} | {
"line": 147,
"column": 2
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝⁴ : LieRing.IsNilpotent ↥H\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nα χ : ↥H → R\nx : L\nhx : x ∈ rootSpace ... | [
"R : Type u_1\nL : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝⁴ : LieRing.IsNilpotent ↥H\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nα χ : ↥H → R\nx : L\nhx : x ∈ rootSpace H α\nm : M\n... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Lie.Weights.Linear | {
"line": 135,
"column": 2
} | {
"line": 136,
"column": 23
} | {
"line": 138,
"column": 0
} | [
{
"pp": "R : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : LieRing L\ninst✝⁹ : LieAlgebra R L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : LieRingModule L M\ninst✝⁵ : LieModule R L M\ninst✝⁴ : IsDomain R\ninst✝³ : IsPrincipalIdealRing R\ninst✝² : Free R M\ninst✝¹ : Module.Fini... | [] | rwa [← LieSubmodule.nontrivial_iff_ne_bot, ← rank_pos_iff_nontrivial (R := R), ← finrank_eq_rank,
Nat.cast_pos] at hχ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Algebra.Lie.Weights.Linear | {
"line": 135,
"column": 2
} | {
"line": 136,
"column": 23
} | {
"line": 138,
"column": 0
} | [
{
"pp": "R : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : LieRing L\ninst✝⁹ : LieAlgebra R L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : LieRingModule L M\ninst✝⁵ : LieModule R L M\ninst✝⁴ : IsDomain R\ninst✝³ : IsPrincipalIdealRing R\ninst✝² : Free R M\ninst✝¹ : Module.Fini... | [] | rwa [← LieSubmodule.nontrivial_iff_ne_bot, ← rank_pos_iff_nontrivial (R := R), ← finrank_eq_rank,
Nat.cast_pos] at hχ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Weights.Linear | {
"line": 135,
"column": 2
} | {
"line": 136,
"column": 23
} | {
"line": 138,
"column": 0
} | [
{
"pp": "R : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : LieRing L\ninst✝⁹ : LieAlgebra R L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : LieRingModule L M\ninst✝⁵ : LieModule R L M\ninst✝⁴ : IsDomain R\ninst✝³ : IsPrincipalIdealRing R\ninst✝² : Free R M\ninst✝¹ : Module.Fini... | [] | rwa [← LieSubmodule.nontrivial_iff_ne_bot, ← rank_pos_iff_nontrivial (R := R), ← finrank_eq_rank,
Nat.cast_pos] at hχ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 293,
"column": 35
} | {
"line": 293,
"column": 52
} | {
"line": 293,
"column": 53
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝² : LieRing.IsNilpotent ↥H\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsNoetherian R L\nα : ↥H → R\nx : ↥H\nthis : x ∈ corootSpace α ↔ ↑x ∈ LieSubmodule.map H.toLieSubmodule.incl (coro... | [
"R : Type u_1\nL : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝² : LieRing.IsNilpotent ↥H\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsNoetherian R L\nα : ↥H → R\nx : ↥H\nthis : x ∈ corootSpace α ↔ ↑x ∈ LieSubmodule.map H.toLieSubmodule.incl (corootSpace α)\n... | Set.mem_setOf_eq, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 319,
"column": 2
} | {
"line": 319,
"column": 12
} | {
"line": 320,
"column": 2
} | [
{
"pp": "case h\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : LieRing L\ninst✝⁶ : LieAlgebra R L\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : LieModule R L M\ninst✝¹ : LieRing.IsNilpotent L\ninst✝ : IsNoetherian R M\nχ : L → R\nx : L\n⊢ ↑(genWeig... | [
"case h\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : LieRing L\ninst✝⁶ : LieAlgebra R L\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : LieModule R L M\ninst✝¹ : LieRing.IsNilpotent L\ninst✝ : IsNoetherian R M\nχ : L → R\nx : L\nm : M\nhm : m ∈ ↑(genWe... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 360,
"column": 2
} | {
"line": 360,
"column": 12
} | {
"line": 361,
"column": 2
} | [
{
"pp": "R : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\ninst✝ : LieRing.IsNilpotent L\n⊢ (genWeightSpace M 0).normalizer ≤ genWeightSpace M 0",
"pp... | [
"R : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\ninst✝ : LieRing.IsNilpotent L\nm : M\nhm : m ∈ (genWeightSpace M 0).normalizer\n⊢ m ∈ genWeightSpace M 0"
] | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 339,
"column": 6
} | {
"line": 339,
"column": 17
} | {
"line": 339,
"column": 18
} | [
{
"pp": "L : Type u_2\ninst✝⁵ : LieRing L\nK : Type u_4\ninst✝⁴ : Field K\ninst✝³ : LieAlgebra K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\n⊢ H.toLieSubmodule ⊔ ⨆ α, ⨆ (_ : α.IsNonZero), rootSpace H ⇑α = ⊤",
"ppTerm": "?m.88"... | [
"L : Type u_2\ninst✝⁵ : LieRing L\nK : Type u_4\ninst✝⁴ : Field K\ninst✝³ : LieAlgebra K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\n⊢ ⊤ ≤ H.toLieSubmodule ⊔ ⨆ α, ⨆ (_ : α.IsNonZero), rootSpace H ⇑α"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 421,
"column": 4
} | {
"line": 421,
"column": 14
} | {
"line": 422,
"column": 4
} | [
{
"pp": "R : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\ninst✝ : LieRing.IsNilpotent L\nx : L\nk : ℕ\nthis : ∀ (m : M) (l : ℕ), ((toEnd R L M) x ^ l) m ... | [
"R : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\ninst✝ : LieRing.IsNilpotent L\nx : L\nk : ℕ\nthis : ∀ (m : M) (l : ℕ), ((toEnd R L M) x ^ l) m ∈ lowerCentr... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 508,
"column": 2
} | {
"line": 508,
"column": 12
} | {
"line": 509,
"column": 2
} | [
{
"pp": "R : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : LieRing L\ninst✝⁹ : LieAlgebra R L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : LieRingModule L M\ninst✝⁵ : LieModule R L M\ninst✝⁴ : LieRing.IsNilpotent L\nM₂ : Type u_5\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂... | [
"R : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : LieRing L\ninst✝⁹ : LieAlgebra R L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : LieRingModule L M\ninst✝⁵ : LieModule R L M\ninst✝⁴ : LieRing.IsNilpotent L\nM₂ : Type u_5\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : L... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure | {
"line": 185,
"column": 8
} | {
"line": 185,
"column": 14
} | {
"line": 185,
"column": 15
} | [
{
"pp": "case C\nk : Type u\ninst✝ : Field k\na✝ : k\nz : AlgebraicClosure k\nhp : (Ideal.Quotient.mk (maxIdeal k)) (MvPolynomial.C a✝) = z\n⊢ IsIntegral k ((Ideal.Quotient.mk (maxIdeal k)) (MvPolynomial.C a✝))",
"ppTerm": "?C",
"assigned": true,
"usedConstants": [
"AlgebraicClosure.instAlgebr... | [] | | C => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 639,
"column": 2
} | {
"line": 639,
"column": 58
} | {
"line": 641,
"column": 0
} | [
{
"pp": "case ind\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : LieRing L\ninst✝⁷ : LieAlgebra R L\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : LieRingModule L M\ninst✝³ : LieModule R L M\ninst✝² : LieRing.IsNilpotent L\ninst✝¹ : IsNoetherian R M\ninst✝ : IsArtinian R M\nP ... | [] | exact hN _ (LieSubmodule.map_incl_lt_iff_lt_top.mpr hN') | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
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