module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Algebra.Homology.BifunctorAssociator | {
"line": 698,
"column": 56
} | {
"line": 698,
"column": 89
} | [
{
"pp": "case pos.h\nC₁ : Type u_1\nC₂ : Type u_2\nC₂₃ : Type u_4\nC₃ : Type u_5\nC₄ : Type u_6\ninst✝²³ : Category.{v_1, u_1} C₁\ninst✝²² : Category.{v_2, u_2} C₂\ninst✝²¹ : Category.{v_3, u_5} C₃\ninst✝²⁰ : Category.{v_4, u_6} C₄\ninst✝¹⁹ : Category.{v_6, u_4} C₂₃\ninst✝¹⁸ : HasZeroMorphisms C₁\ninst✝¹⁷ : Has... | ComplexShape.next_π₂ c₂ c₂₃ i₂ h₃ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.EulerCharacteristic | {
"line": 75,
"column": 31
} | {
"line": 75,
"column": 54
} | [
{
"pp": "ι : Type u_1\nc : ComplexShape ι\ninst✝ : c.EulerCharSigns\ni✝ : ℤ\n⊢ (i✝ + 1).negOnePow = -i✝.negOnePow",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
... | rw [Int.negOnePow_succ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Embedding.ExtendHomotopy | {
"line": 202,
"column": 4
} | {
"line": 203,
"column": 55
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\ninst✝³ : e.IsRelIff\nC : Type u_3\ninst✝² : Category.{v_1, u_3} C\ninst✝¹ : HasZeroObject C\ninst✝ : Preadditive C\nK L : HomologicalComplex C c\nφ :\n (e.extendHomotopyFunctor C).obj ((HomotopyCategory.quotient... | obtain ⟨φ : K.extend e ⟶ L.extend e, rfl⟩ :=
(HomotopyCategory.quotient C c').map_surjective φ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Homology.Embedding.Connect | {
"line": 221,
"column": 91
} | {
"line": 229,
"column": 6
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : HasZeroMorphisms C\nK K' : ChainComplex C ℕ\nL L' : CochainComplex C ℕ\nh : ConnectData K L\nh' : ConnectData K' L'\nfK : K ⟶ K'\nfL : L ⟶ L'\nf_comm : fK.f 0 ≫ h'.d₀ = h.d₀ ≫ fL.f 0\nn : ℕ\ninst✝⁴ : NeZero n\nm : ℤ\nhmn : m = ↑n\ninst✝³ : HasHomology h.... | by
rw [← cancel_mono (HomologicalComplex.homologyι ..)]
dsimp [homologyIsoPos]
simp only [homologyι_naturality, Category.assoc, restrictionHomologyIso_hom_homologyι,
homologyι_naturality_assoc, restrictionHomologyIso_inv_homologyι_assoc]
congr 1
rw [← cancel_epi (HomologicalComplex.pOpcycles ..)]
subst ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Generator.HomologicalComplex | {
"line": 49,
"column": 10
} | {
"line": 49,
"column": 11
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nι : Type w\nc : ComplexShape ι\ninst✝² : c.HasNoLoop\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nα : Type t\nX : α → C\nhX : (ObjectProperty.ofObj X).IsSeparating\nK : HomologicalComplex C c\n⊢ ∀ ⦃Y : HomologicalComplex C c⦄ (f g : K ⟶ Y),\n (∀ (G :... | L | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Homology.Factorizations.CM5b | {
"line": 45,
"column": 2
} | {
"line": 46,
"column": 16
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nK L : CochainComplex C ℤ\nf : K ⟶ L\nn : ℤ\n⊢ Injective ((I K).X n)",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.Injective",
"CategoryTheory.Injective.inje... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Factorizations.CM5b | {
"line": 45,
"column": 2
} | {
"line": 46,
"column": 16
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nK L : CochainComplex C ℤ\nf : K ⟶ L\nn : ℤ\n⊢ Injective ((I K).X n)",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.Injective",
"CategoryTheory.Injective.inje... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Factorizations.CM5b | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 10
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nK L : CochainComplex C ℤ\nf : K ⟶ L\nn : ℤ\n⊢ (i f).f n ≫ biprod.fst.f n ≫ (mappingCone.snd (𝟙 (I K))).v n n ⋯ = Injective.ι (K.X n)",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
... | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.Factorizations.CM5b | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 10
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nK L : CochainComplex C ℤ\nf : K ⟶ L\nn : ℤ\n⊢ (i f).f n ≫ biprod.fst.f n ≫ (mappingCone.snd (𝟙 (I K))).v n n ⋯ = Injective.ι (K.X n)",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
... | simp [i] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Factorizations.CM5b | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 10
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nK L : CochainComplex C ℤ\nf : K ⟶ L\nn : ℤ\n⊢ (i f).f n ≫ biprod.fst.f n ≫ (mappingCone.snd (𝟙 (I K))).v n n ⋯ = Injective.ι (K.X n)",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
... | simp [i] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Factorizations.CM5b | {
"line": 90,
"column": 34
} | {
"line": 90,
"column": 42
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nK L : CochainComplex C ℤ\nf : K ⟶ L\n⊢ i f ≫ p K L = f",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"HomologicalComplex.instCategory",
"CochainComplex.cm5b.i._proof_4",
... | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.Factorizations.CM5b | {
"line": 90,
"column": 34
} | {
"line": 90,
"column": 42
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nK L : CochainComplex C ℤ\nf : K ⟶ L\n⊢ i f ≫ p K L = f",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"HomologicalComplex.instCategory",
"CochainComplex.cm5b.i._proof_4",
... | simp [i] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Factorizations.CM5b | {
"line": 90,
"column": 34
} | {
"line": 90,
"column": 42
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nK L : CochainComplex C ℤ\nf : K ⟶ L\n⊢ i f ≫ p K L = f",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"HomologicalComplex.instCategory",
"CochainComplex.cm5b.i._proof_4",
... | simp [i] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Idempotents.FunctorExtension | {
"line": 77,
"column": 25
} | {
"line": 77,
"column": 28
} | [
{
"pp": "case a\nC : 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✝\nh : (F.map f.f).f ≫ (φ.app x✝.X).f = (φ.app x✝¹.X).f ≫ (G.map f.f).f\nh' : (F.map f.f).f ≫ (... | h'' | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 261,
"column": 35
} | {
"line": 261,
"column": 51
} | [
{
"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 g : X ⟶ K.X q\nn : ℤ\nh : p + n = q\nq' : ℤ\nhq' : q + 1 = q'\nhf : f ≫ K.d q q' = 0\nhg : g ≫ K.d q q' = 0\n⊢ (f + g) ≫ K.d q q' = 0",
"usedConstants": [
"Catego... | by simp [hf, hg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 267,
"column": 35
} | {
"line": 267,
"column": 51
} | [
{
"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 g : X ⟶ K.X q\nn : ℤ\nh : p + n = q\nq' : ℤ\nhq' : q + 1 = q'\nhf : f ≫ K.d q q' = 0\nhg : g ≫ K.d q q' = 0\n⊢ (f - g) ≫ K.d q q' = 0",
"usedConstants": [
"Catego... | by simp [hf, hg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Preadditive.Projective.Resolution | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 79
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nZ : C\nP : ProjectiveResolution Z\n⊢ IsColimit P.cokernelCofork",
"usedConstants": [
"Nat.instOne",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.Limits.Wa... | refine IsColimit.ofIsoColimit (P.complex.opcyclesIsCokernel 1 0 (by simp)) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 338,
"column": 33
} | {
"line": 338,
"column": 49
} | [
{
"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 g : K.X p ⟶ X\nn : ℤ\nh : p + n = q\np' : ℤ\nhp' : p' + 1 = p\nhf : K.d p' p ≫ f = 0\nhg : K.d p' p ≫ g = 0\n⊢ K.d p' p ≫ (f + g) = 0",
"usedConstants": [
"Catego... | by simp [hf, hg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 344,
"column": 33
} | {
"line": 344,
"column": 49
} | [
{
"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 g : K.X p ⟶ X\nn : ℤ\nh : p + n = q\np' : ℤ\nhp' : p' + 1 = p\nhf : K.d p' p ≫ f = 0\nhg : K.d p' p ≫ g = 0\n⊢ K.d p' p ≫ (f - g) = 0",
"usedConstants": [
"Catego... | by simp [hf, hg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.GradedObject.Monoidal | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 49
} | [
{
"pp": "I : Type u\ninst✝⁵ : AddMonoid I\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : MonoidalCategory C\nX₁ X₂ Y₁ Y₂ : GradedObject I C\nf : X₁ ⟶ X₂\ng : Y₁ ⟶ Y₂\ninst✝² : X₁.HasTensor Y₁\ninst✝¹ : X₂.HasTensor Y₂\ninst✝ : X₂.HasTensor Y₁\n⊢ tensorHom f g = whiskerRight f Y₁ ≫ whiskerLeft X₂ g",
... | rw [tensorHom_comp_tensorHom, id_comp, comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.GradedObject.Monoidal | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 49
} | [
{
"pp": "I : Type u\ninst✝⁵ : AddMonoid I\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : MonoidalCategory C\nX₁ X₂ Y₁ Y₂ : GradedObject I C\nf : X₁ ⟶ X₂\ng : Y₁ ⟶ Y₂\ninst✝² : X₁.HasTensor Y₁\ninst✝¹ : X₂.HasTensor Y₂\ninst✝ : X₂.HasTensor Y₁\n⊢ tensorHom f g = whiskerRight f Y₁ ≫ whiskerLeft X₂ g",
... | rw [tensorHom_comp_tensorHom, id_comp, comp_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.GradedObject.Monoidal | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 49
} | [
{
"pp": "I : Type u\ninst✝⁵ : AddMonoid I\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : MonoidalCategory C\nX₁ X₂ Y₁ Y₂ : GradedObject I C\nf : X₁ ⟶ X₂\ng : Y₁ ⟶ Y₂\ninst✝² : X₁.HasTensor Y₁\ninst✝¹ : X₂.HasTensor Y₂\ninst✝ : X₂.HasTensor Y₁\n⊢ tensorHom f g = whiskerRight f Y₁ ≫ whiskerLeft X₂ g",
... | rw [tensorHom_comp_tensorHom, id_comp, comp_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.GradedObject.Monoidal | {
"line": 391,
"column": 17
} | {
"line": 391,
"column": 20
} | [
{
"pp": "case h.h\nI : Type u\ninst✝⁶ : AddMonoid I\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : MonoidalCategory C\nX₁ X₂ X₃ X₄ : GradedObject I C\ninst✝³ : X₃.HasTensor X₄\ninst✝² : X₂.HasTensor (tensorObj X₃ X₄)\ninst✝¹ : X₁.HasTensor (tensorObj X₂ (tensorObj X₃ X₄))\nj : I\nA : C\nf g : tensorObj... | h'' | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.GradedObject.Monoidal | {
"line": 425,
"column": 2
} | {
"line": 439,
"column": 84
} | [
{
"pp": "case h.h\nI : Type u\ninst✝²⁵ : AddMonoid I\nC : Type u_1\ninst✝²⁴ : Category.{v_1, u_1} C\ninst✝²³ : MonoidalCategory C\nX₁ X₂ X₃ X₄ : GradedObject I C\ninst✝²² : X₁.HasTensor X₂\ninst✝²¹ : X₂.HasTensor X₃\ninst✝²⁰ : X₃.HasTensor X₄\ninst✝¹⁹ : (tensorObj X₁ X₂).HasTensor X₃\ninst✝¹⁸ : X₁.HasTensor (te... | conv_lhs =>
rw [ιTensorObj₄_eq X₁ X₂ X₃ X₄ i₁ i₂ i₃ i₄ j h _ rfl, assoc, ι_tensorHom_assoc]
dsimp only [categoryOfGradedObjects_id, id_eq, eq_mpr_eq_cast, cast_eq]
rw [id_tensorHom, ← MonoidalCategory.whiskerLeft_comp_assoc, ιTensorObj₃_associator_inv,
ιTensorObj₃'_eq X₂ X₃ X₄ i₂ i₃ i₄ _ rfl _ rfl, Mo... | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convLHS_1 | Mathlib.Tactic.Conv.convLHS |
Mathlib.Algebra.Homology.SpectralObject.Cycles | {
"line": 128,
"column": 2
} | {
"line": 129,
"column": 16
} | [
{
"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\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\n⊢ Mono (X.kernelSequenceCycles f g n₀ n₁ hn₁).f",
"usedConstants": [
"CategoryTheory.Abelian.to... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.SpectralObject.Cycles | {
"line": 128,
"column": 2
} | {
"line": 129,
"column": 16
} | [
{
"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\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\n⊢ Mono (X.kernelSequenceCycles f g n₀ n₁ hn₁).f",
"usedConstants": [
"CategoryTheory.Abelian.to... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.SpectralObject.Cycles | {
"line": 133,
"column": 2
} | {
"line": 134,
"column": 16
} | [
{
"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\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\n⊢ Epi (X.cokernelSequenceOpcycles f g n₀ n₁ hn₁).g",
"usedConstants": [
"CategoryTheory.Abelian... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.SpectralObject.Cycles | {
"line": 133,
"column": 2
} | {
"line": 134,
"column": 16
} | [
{
"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\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\n⊢ Epi (X.cokernelSequenceOpcycles f g n₀ n₁ hn₁).g",
"usedConstants": [
"CategoryTheory.Abelian... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.SpectralObject.Cycles | {
"line": 372,
"column": 2
} | {
"line": 373,
"column": 16
} | [
{
"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\ni' j' k' : ι\nf' : i' ⟶ j'\ng' : j' ⟶ k'\ni'' j'' k'' : ι\nf'' : i'' ⟶ j''\ng'' : j'' ⟶ k''\nfg : i ⟶ k\nh : f ≫ g = fg\nfg' : i' ⟶ k'\... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.SpectralObject.Cycles | {
"line": 372,
"column": 2
} | {
"line": 373,
"column": 16
} | [
{
"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\ni' j' k' : ι\nf' : i' ⟶ j'\ng' : j' ⟶ k'\ni'' j'' k'' : ι\nf'' : i'' ⟶ j''\ng'' : j'' ⟶ k''\nfg : i ⟶ k\nh : f ≫ g = fg\nfg' : i' ⟶ k'\... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.SpectralObject.Cycles | {
"line": 376,
"column": 2
} | {
"line": 377,
"column": 16
} | [
{
"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\ni' j' k' : ι\nf' : i' ⟶ j'\ng' : j' ⟶ k'\ni'' j'' k'' : ι\nf'' : i'' ⟶ j''\ng'' : j'' ⟶ k''\nfg : i ⟶ k\nh : f ≫ g = fg\nfg' : i' ⟶ k'\... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.SpectralObject.Cycles | {
"line": 376,
"column": 2
} | {
"line": 377,
"column": 16
} | [
{
"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\ni' j' k' : ι\nf' : i' ⟶ j'\ng' : j' ⟶ k'\ni'' j'' k'' : ι\nf'' : i'' ⟶ j''\ng'' : j'' ⟶ k''\nfg : i ⟶ k\nh : f ≫ g = fg\nfg' : i' ⟶ k'\... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.SpectralObject.Homology | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 65
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : Category.{v_2, u_2} ι\nX : SpectralObject C ι\ni₀ i₁ i₂ i₃ i₄ i₅ : ι\nf₁ : i₀ ⟶ i₁\nf₂ : i₁ ⟶ i₂\nf₃ : i₂ ⟶ i₃\nf₄ : i₃ ⟶ i₄\nf₅ : i₄ ⟶ i₅\nf₃₄ : i₂ ⟶ i₄\nh₃₄ : f₃ ≫ f₄ = f₃₄\nn₀ n₁ n₂ n₃ : ℤ\nhn₁ : n₀ + 1 = n₁\nhn₂... | refine ⟨A₁, π₁, inferInstance, x₁ ≫ X.πE f₃ f₄ f₅ n₀ n₁ n₂, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 131,
"column": 2
} | {
"line": 132,
"column": 60
} | [
{
"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\ni' j' k' l' : ι\nf₁' : i' ⟶ j'\nf₂' : j' ⟶ k'\nf₃' : k' ⟶ l'\ni'' j'' k'' l'' : ι\nf₁'' : i'' ⟶ j''\nf₂'' : j'' ⟶ k''\n... | dsimp only [map]
simp [shortComplexMap_comp, ShortComplex.homologyMap_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 131,
"column": 2
} | {
"line": 132,
"column": 60
} | [
{
"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\ni' j' k' l' : ι\nf₁' : i' ⟶ j'\nf₂' : j' ⟶ k'\nf₃' : k' ⟶ l'\ni'' j'' k'' l'' : ι\nf₁'' : i'' ⟶ j''\nf₂'' : j'' ⟶ k''\n... | dsimp only [map]
simp [shortComplexMap_comp, ShortComplex.homologyMap_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 252,
"column": 2
} | {
"line": 253,
"column": 16
} | [
{
"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\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq pq' pq'' : κ\nhpq : (c r).prev pq' = pq\nhpq' : ... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 252,
"column": 2
} | {
"line": 253,
"column": 16
} | [
{
"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\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq pq' pq'' : κ\nhpq : (c r).prev pq' = pq\nhpq' : ... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 284,
"column": 14
} | {
"line": 284,
"column": 18
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq' pq'' : κ\nhpq' : (c r).next pq' = pq''\ni₀' i₀... | hi₀, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.EventuallyConst | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 47
} | [
{
"pp": "case refine_2\nα : Type u_1\nf : ℕ → α\nn : ℕ\nh : ∀ (m : ℕ), n ≤ m → f (m + 1) = f m\nm : ℕ\nhm : n ≤ m\n⊢ f m = f n",
"usedConstants": [
"Lattice.toSemilatticeSup",
"instDistribLatticeNat",
"PartialOrder.toPreorder",
"Preorder.toLE",
"DistribLattice.toLattice",
... | induction m, hm using Nat.le_induction with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 616,
"column": 2
} | {
"line": 617,
"column": 24
} | [
{
"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\ni₀ i₁ i₂ i₃ : ι\nf₁ : i₀ ⟶ i₁\nf₂ : i₁ ⟶ i₂\nf₃ : i₂ ⟶ i₃\nf₁₂ : i₀ ⟶ i₂\nh₁₂ : f₁ ≫ f₂ = f₁₂\nn₀ n₁ n₂ : ℤ\nhn₁ : n₀ + 1 = n₁\nhn₂ : n₁ + 1 = n₂\nA : C\nx₂ : A ⟶ X.opc... | simp [Category.assoc, hy₂, reassoc_of% hy₁, Preadditive.add_comp, δ_pOpcycles,
comp_zero, add_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 702,
"column": 69
} | {
"line": 704,
"column": 26
} | [
{
"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 : ι\nf : i ⟶ j\ni' j' : ι\nf' : i' ⟶ j'\nα : mk₁ f ⟶ mk₁ f'\nβ : mk₃ (𝟙 i) f (𝟙 j) ⟶ mk₃ (𝟙 i') f' (𝟙 j')\nn₀ n₁ n₂ : ℤ\nhβ : β = homMk₃ (α.app 0) (α.app 0) (α.... | by
subst hβ
exact hom_ext₁ rfl rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 467,
"column": 2
} | {
"line": 467,
"column": 25
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nS : Set (LieSubalgebra R L)\nh : ∀ (K K' : LieSubalgebra R L), ↑K ≤ ↑K' ↔ K ≤ K'\n⊢ IsGLB S (sInf S)",
"usedConstants": [
"LieSubalgebra.instPartialOrder_1",
"IsGLB.of_image",
"CompleteLattice... | apply IsGLB.of_image @h | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 464,
"column": 2
} | {
"line": 469,
"column": 19
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nS : Set (LieSubalgebra R L)\n⊢ IsGLB S (sInf S)",
"usedConstants": [
"Eq.mpr",
"LieSubalgebra.instPartialOrder_1",
"IsGLB.of_image",
"CompleteLattice.instOmegaCompletePartialOrder",
... | have h : ∀ K K' : LieSubalgebra R L, (K : Set L) ≤ K' ↔ K ≤ K' := by
intros
exact Iff.rfl
apply IsGLB.of_image @h
simp only [coe_sInf]
exact isGLB_biInf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 464,
"column": 2
} | {
"line": 469,
"column": 19
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nS : Set (LieSubalgebra R L)\n⊢ IsGLB S (sInf S)",
"usedConstants": [
"Eq.mpr",
"LieSubalgebra.instPartialOrder_1",
"IsGLB.of_image",
"CompleteLattice.instOmegaCompletePartialOrder",
... | have h : ∀ K K' : LieSubalgebra R L, (K : Set L) ≤ K' ↔ K ≤ K' := by
intros
exact Iff.rfl
apply IsGLB.of_image @h
simp only [coe_sInf]
exact isGLB_biInf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Artinian.Module | {
"line": 635,
"column": 6
} | {
"line": 635,
"column": 86
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsArtinianRing R\nf : MaximalSpectrum R → Ideal R := MaximalSpectrum.asIdeal\nx : R\n⊢ x ∈ nilradical R ↔ x ∈ ⨅ i, f i",
"usedConstants": [
"MaximalSpectrum.asIdeal",
"Eq.mpr",
"Submodule",
"iInf",
"IsArtinianRing.primeSpectru... | rw [PrimeSpectrum.nilradical_eq_iInf, iInf, primeSpectrum_asIdeal_range_eq]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Artinian.Module | {
"line": 635,
"column": 6
} | {
"line": 635,
"column": 86
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsArtinianRing R\nf : MaximalSpectrum R → Ideal R := MaximalSpectrum.asIdeal\nx : R\n⊢ x ∈ nilradical R ↔ x ∈ ⨅ i, f i",
"usedConstants": [
"MaximalSpectrum.asIdeal",
"Eq.mpr",
"Submodule",
"iInf",
"IsArtinianRing.primeSpectru... | rw [PrimeSpectrum.nilradical_eq_iInf, iInf, primeSpectrum_asIdeal_range_eq]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.OfAssociative | {
"line": 238,
"column": 4
} | {
"line": 238,
"column": 25
} | [
{
"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\nh : ∀ (x : L), (∀ (m : M), ⁅x, m⁆ = 0) → x = 0\nx y : L\nhxy : (toEnd R L M) x = (toEnd... | refine h _ fun m ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Lie.OfAssociative | {
"line": 299,
"column": 8
} | {
"line": 299,
"column": 24
} | [
{
"pp": "case succ\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\nx y : L\nz : M\nn : ℕ\nih : (φ x ^ n) ⁅y, z⁆ = ∑ ij ∈ antidiagonal n, n.choose ij.1 • ⁅((ad... | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Ideal | {
"line": 273,
"column": 97
} | {
"line": 281,
"column": 67
} | [
{
"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'\n⊢ f.IsIdealMorphism ↔ ∀ (x : L') (y : L), ∃ z, ⁅x, f y⁆ = f z",
"usedConstants": [
"LieHom",
"LieAlgebra.toModule",
... | by
simp only [isIdealMorphism_def, idealRange_eq_lieSpan_range, ←
LieSubalgebra.toSubmodule_inj, ← f.range.coe_toSubmodule,
LieIdeal.toLieSubalgebra_toSubmodule, LieSubmodule.coe_lieSpan_submodule_eq_iff,
LieSubalgebra.mem_toSubmodule, mem_range, exists_imp,
Submodule.exists_lieSubmodule_coe_eq_iff]
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.IdealOperations | {
"line": 289,
"column": 2
} | {
"line": 289,
"column": 59
} | [
{
"pp": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nJ₁ J₂ : LieIdeal R L'\nh : f.IsIdealMorphism\n⊢ ⁅f.idealRange ⊓ J₁, f.idealRange ⊓ J₂⁆ ≤ f.idealRange",
"usedConstants": [
"Lie... | exact le_trans (LieSubmodule.lie_le_left _ _) inf_le_left | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.Minpoly.Basic | {
"line": 236,
"column": 32
} | {
"line": 236,
"column": 39
} | [
{
"pp": "A : Type u_1\ninst✝³ : CommRing A\nB : Type u_4\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : Nontrivial B\nS : Subalgebra A B\nx : B\nint : IsIntegral (↥S) x\n⊢ x ∉ (algebraMap (↥S) B).range ↔ x ∉ S",
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"Subring.instSetLi... | Iff.not | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.SModEq.Basic | {
"line": 45,
"column": 48
} | {
"line": 45,
"column": 89
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Submodule R M\nx y : M\n⊢ x ≡ y [SMOD U] ↔ x - y ∈ U",
"usedConstants": [
"Submodule.Quotient.eq",
"Eq.mpr",
"Submodule",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"Su... | by rw [SModEq.def, Submodule.Quotient.eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.PowerBasis | {
"line": 183,
"column": 6
} | {
"line": 183,
"column": 18
} | [
{
"pp": "S : Type u_2\ninst✝² : Ring S\nA : Type u_4\ninst✝¹ : CommRing A\ninst✝ : Algebra A S\npb : PowerBasis A S\na✝ : Nontrivial A\n⊢ (∑ i, C ((pb.basis.repr (pb.gen ^ pb.dim)) i) * X ^ ↑i).degree < (X ^ pb.dim).degree",
"usedConstants": [
"Finsupp.instFunLike",
"WithBot.instPreorder",
... | degree_X_pow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerBasis | {
"line": 203,
"column": 47
} | {
"line": 203,
"column": 59
} | [
{
"pp": "S : Type u_2\ninst✝³ : Ring S\nA : Type u_4\ninst✝² : CommRing A\ninst✝¹ : Algebra A S\ninst✝ : Nontrivial A\npb : PowerBasis A S\n⊢ (X ^ pb.dim).degree = ↑pb.dim",
"usedConstants": [
"Eq.mpr",
"WithBot",
"PowerBasis.dim",
"congrArg",
"CommSemiring.toSemiring",
"... | degree_X_pow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerBasis | {
"line": 203,
"column": 47
} | {
"line": 203,
"column": 59
} | [
{
"pp": "S : Type u_2\ninst✝³ : Ring S\nA : Type u_4\ninst✝² : CommRing A\ninst✝¹ : Algebra A S\ninst✝ : Nontrivial A\npb : PowerBasis A S\n⊢ (∑ i, C ((pb.basis.repr (pb.gen ^ pb.dim)) i) * X ^ ↑i).degree < (X ^ pb.dim).degree",
"usedConstants": [
"Finsupp.instFunLike",
"WithBot.instPreorder",
... | degree_X_pow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Norm.Defs | {
"line": 72,
"column": 23
} | {
"line": 72,
"column": 76
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nh : ¬∃ s, Nonempty (Basis (↥s) R S)\nx : S\n⊢ (norm R) x = 1",
"usedConstants": [
"Eq.mpr",
"Algebra.lmul",
"MonoidHom.instFunLike",
"MonoidHom",
"congrArg",
"CommSemiring.toSe... | rw [norm_apply, LinearMap.det]; split_ifs <;> trivial | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Norm.Defs | {
"line": 72,
"column": 23
} | {
"line": 72,
"column": 76
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nh : ¬∃ s, Nonempty (Basis (↥s) R S)\nx : S\n⊢ (norm R) x = 1",
"usedConstants": [
"Eq.mpr",
"Algebra.lmul",
"MonoidHom.instFunLike",
"MonoidHom",
"congrArg",
"CommSemiring.toSe... | rw [norm_apply, LinearMap.det]; split_ifs <;> trivial | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.KummerPolynomial | {
"line": 114,
"column": 25
} | {
"line": 114,
"column": 52
} | [
{
"pp": "K : Type u\ninst✝ : Field K\np : ℕ\nhp : Nat.Prime p\na : K\nha : ∀ (b : K), b ^ p ≠ a\nthis✝ : ¬IsUnit (X ^ p - C a)\ng : K[X]\nhg : Irreducible g\nhg' : g ∣ X ^ p - C a\nh : ¬g.natDegree = p\nthis : root g ^ p = (of g) a\n⊢ (Algebra.norm K) ((of g) a) = a ^ g.natDegree",
"usedConstants": [
... | ← AdjoinRoot.algebraMap_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Squarefree.Basic | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 19
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoidWithZero R\ninst✝ : IsCancelMulZero R\nx : R\nh0 : x ≠ 0\nh : IsRadical x\n⊢ Squarefree x",
"usedConstants": []
}
] | rintro z ⟨w, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.RingTheory.Polynomial.Quotient | {
"line": 232,
"column": 4
} | {
"line": 233,
"column": 16
} | [
{
"pp": "case add\nR : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nI : Ideal R\nf p q : MvPolynomial σ (R ⧸ I)\nhp :\n (Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯)\n (eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯)\n (fun i ↦ (Idea... | simp only [map_add, MvPolynomial.eval₂_add]
at hp hq ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Squarefree.Basic | {
"line": 298,
"column": 2
} | {
"line": 299,
"column": 41
} | [
{
"pp": "n : ℤ\n⊢ Squarefree n.natAbs ↔ Squarefree n",
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"HMul.hMul",
"Function.Surjective.forall",
"Monoid.toMulOneClass",
"congrArg",
"Nat.instMonoid",
"_private.Mathlib.Algebra.Squarefree.Basic.0.Int.squarefree_natAbs._... | simp_rw [Squarefree, natAbs_surjective.forall, ← natAbs_mul, natAbs_dvd_natAbs,
isUnit_iff_natAbs_eq, Nat.isUnit_iff] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.Squarefree.Basic | {
"line": 298,
"column": 2
} | {
"line": 299,
"column": 41
} | [
{
"pp": "n : ℤ\n⊢ Squarefree n.natAbs ↔ Squarefree n",
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"HMul.hMul",
"Function.Surjective.forall",
"Monoid.toMulOneClass",
"congrArg",
"Nat.instMonoid",
"_private.Mathlib.Algebra.Squarefree.Basic.0.Int.squarefree_natAbs._... | simp_rw [Squarefree, natAbs_surjective.forall, ← natAbs_mul, natAbs_dvd_natAbs,
isUnit_iff_natAbs_eq, Nat.isUnit_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Squarefree.Basic | {
"line": 298,
"column": 2
} | {
"line": 299,
"column": 41
} | [
{
"pp": "n : ℤ\n⊢ Squarefree n.natAbs ↔ Squarefree n",
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"HMul.hMul",
"Function.Surjective.forall",
"Monoid.toMulOneClass",
"congrArg",
"Nat.instMonoid",
"_private.Mathlib.Algebra.Squarefree.Basic.0.Int.squarefree_natAbs._... | simp_rw [Squarefree, natAbs_surjective.forall, ← natAbs_mul, natAbs_dvd_natAbs,
isUnit_iff_natAbs_eq, Nat.isUnit_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Separable | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 46
} | [
{
"pp": "case inr\nR : Type u\ninst✝ : CommRing R\nn : ℕ\nu : Rˣ\nhn : IsUnit ↑n\na✝ : Nontrivial R\nhpos : n > 0\n⊢ (X ^ n - C ↑u).Separable",
"usedConstants": [
"Iff.mpr",
"Polynomial.derivative",
"Units.val",
"Polynomial.C",
"Polynomial.instOne",
"Semiring.toModule",
... | apply (separable_def' (X ^ n - C (u : R))).2 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.FieldTheory.Separable | {
"line": 327,
"column": 4
} | {
"line": 333,
"column": 36
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nι : Type u_1\nf : ι → F\ns : Finset ι\nH : ∀ x ∈ s, ∀ y ∈ s, f x = f y → x = y\n⊢ (∏ i ∈ s, (X - C (f i))).Separable",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"congrArg",
"Finset",
"HSub.hSub",
"RingHom",
"Membership.mem",
... | rw [← prod_attach]
exact
separable_prod'
(fun x _hx y _hy hxy =>
@pairwise_coprime_X_sub_C _ _ { x // x ∈ s } (fun x => f x)
(fun x y hxy => Subtype.ext <| H x.1 x.2 y.1 y.2 hxy) _ _ hxy)
fun _ _ => separable_X_sub_C | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Separable | {
"line": 327,
"column": 4
} | {
"line": 333,
"column": 36
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nι : Type u_1\nf : ι → F\ns : Finset ι\nH : ∀ x ∈ s, ∀ y ∈ s, f x = f y → x = y\n⊢ (∏ i ∈ s, (X - C (f i))).Separable",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"congrArg",
"Finset",
"HSub.hSub",
"RingHom",
"Membership.mem",
... | rw [← prod_attach]
exact
separable_prod'
(fun x _hx y _hy hxy =>
@pairwise_coprime_X_sub_C _ _ { x // x ∈ s } (fun x => f x)
(fun x y hxy => Subtype.ext <| H x.1 x.2 y.1 y.2 hxy) _ _ hxy)
fun _ _ => separable_X_sub_C | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Separable | {
"line": 389,
"column": 2
} | {
"line": 390,
"column": 93
} | [
{
"pp": "F : Type u\ninst✝ : Field F\np : ℕ\nHF : CharP F p\nf : F[X]\nn : ℕ\nhp : 0 < p\nhf : ((expand F (p ^ n)) f).Separable\nhn : n ≠ 0\n⊢ IsUnit f",
"usedConstants": [
"CharP.cast_eq_zero",
"Polynomial.derivative",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"NonU... | have hf2 : derivative (expand F (p ^ n) f) = 0 := by
rw [derivative_expand, Nat.cast_pow, CharP.cast_eq_zero, zero_pow hn, zero_mul, mul_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Determinant | {
"line": 281,
"column": 41
} | {
"line": 284,
"column": 26
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Subsingleton M\nf : M →ₗ[R] M\n⊢ LinearMap.det f = 1",
"usedConstants": [
"Eq.mpr",
"MonoidHom.instFunLike",
"Semiring.toModule",
"Matrix.module",
"MonoidHom",
... | by
have b : Basis (Fin 0) R M := Basis.empty M
rw [← f.det_toMatrix b]
exact Matrix.det_isEmpty | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Separable | {
"line": 442,
"column": 2
} | {
"line": 442,
"column": 48
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nn : ℕ\n⊢ (X ^ n - 1).Separable ↔ ↑n ≠ 0",
"usedConstants": [
"Nat.eq_zero_or_pos"
]
}
] | rcases (Nat.eq_zero_or_pos n) with (hz | hpos) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.LinearAlgebra.AnnihilatingPolynomial | {
"line": 103,
"column": 4
} | {
"line": 103,
"column": 49
} | [
{
"pp": "case neg.h2.hx\n𝕜 : Type u_1\nA : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nh : ¬annIdealGenerator 𝕜 a = 0\n⊢ ¬(IsPrincipal.generator (annIdeal 𝕜 a)).leadingCoeff = 0",
"usedConstants": [
"Iff.mpr",
"Semiring.toModule",
"CommSemiring.toSemiring"... | apply Polynomial.leadingCoeff_eq_zero.not.mpr | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.AnnihilatingPolynomial | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 34
} | [
{
"pp": "case neg.h2.hx\n𝕜 : Type u_1\nA : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nh : ¬annIdealGenerator 𝕜 a = 0\n⊢ ¬IsPrincipal.generator (annIdeal 𝕜 a) = 0",
"usedConstants": [
"Polynomial.C",
"IsDomain.to_noZeroDivisors",
"NonUnitalCommRing.toNonUn... | apply (mul_ne_zero_iff.mp h).1 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.Determinant | {
"line": 567,
"column": 4
} | {
"line": 569,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ne : M ≃ₗ[R] M\nf f' : M →ₗ[R] M\nh : ∀ (x : M), f x = f' (e x)\nthis : Associated (LinearMap.det (f' ∘ₗ ↑e)) (LinearMap.det f')\n⊢ Associated (LinearMap.det f) (LinearMap.det f')",
"usedConstants": [
... | convert this using 2
ext x
exact h x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Determinant | {
"line": 567,
"column": 4
} | {
"line": 569,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ne : M ≃ₗ[R] M\nf f' : M →ₗ[R] M\nh : ∀ (x : M), f x = f' (e x)\nthis : Associated (LinearMap.det (f' ∘ₗ ↑e)) (LinearMap.det f')\n⊢ Associated (LinearMap.det f) (LinearMap.det f')",
"usedConstants": [
... | convert this using 2
ext x
exact h x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Determinant | {
"line": 746,
"column": 31
} | {
"line": 746,
"column": 40
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_4\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nhli : LinearIndependent R v\nhsp : ⊤ ≤ span R (Set.range v)\ni k : ι\nhik : k ≠ i\n⊢ e.det v * 0 = ↑e.det (update... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AdjoinRoot | {
"line": 706,
"column": 4
} | {
"line": 707,
"column": 9
} | [
{
"pp": "case h₁.a\nK : Type u_5\ninst✝ : Field K\nf : K[X]\nhf : f ≠ 0\nf'_monic : (f * C f.leadingCoeff⁻¹).Monic\nq : K[X]\nq_monic : q.Monic\nq_aeval : (aeval (root f)) q = 0\nx✝ : K\n⊢ (((lift (algebraMap K (AdjoinRoot f)) (root f) q_aeval).comp (mk q)).comp C) x✝ = ((mk f).comp C) x✝",
"usedConstants":... | · simp only [RingHom.comp_apply, mk_C, lift_of]
rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 79,
"column": 6
} | {
"line": 79,
"column": 17
} | [
{
"pp": "case inr\nA : Type u_1\ninst✝⁴ : Ring A\ninst✝³ : Module ℚ A\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module A M\ninst✝ : Module ℚ M\na : A\nm : M\nk : ℕ\nh : a ^ k • m = 0\nhn : IsNilpotent a\nh₀ : k < nilpotencyClass a\nr : ℕ\nh₂ : r ∈ Ico k (nilpotencyClass a)\n⊢ ((↑r !)⁻¹ • a ^ r) • m = 0",... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AdjoinRoot | {
"line": 795,
"column": 2
} | {
"line": 795,
"column": 77
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\n⊢ Function.Surjective ⇑(toAdjoin R x)",
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"Lattice.toSemilatticeSup",
"eq_top_iff",
"AdjoinRoot",
"CompleteLatti... | rw [← AlgHom.range_eq_top, _root_.eq_top_iff, ← adjoin_adjoin_coe_preimage] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 131,
"column": 59
} | {
"line": 131,
"column": 68
} | [
{
"pp": "case inr\nA : Type u_1\ninst✝¹ : Ring A\ninst✝ : Module ℚ A\na b : A\nh₁ : Commute a b\nn₁ : ℕ\nhn₁ : a ^ n₁ = 0\nn₂ : ℕ\nhn₂ : b ^ n₂ = 0\nN : ℕ := max n₁ n₂\nh₄ : a ^ (N + 1) = 0\nh₅ : b ^ (N + 1) = 0\nR2N : Finset ℕ := range (2 * N + 1)\nhR2N : R2N = range (2 * N + 1)\nRN : Finset ℕ := range (N + 1)... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 132,
"column": 2
} | {
"line": 134,
"column": 72
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : Module ℚ A\na b : A\nh₁ : Commute a b\nn₁ : ℕ\nhn₁ : a ^ n₁ = 0\nn₂ : ℕ\nhn₂ : b ^ n₂ = 0\nN : ℕ := max n₁ n₂\nh₄ : a ^ (N + 1) = 0\nh₅ : b ^ (N + 1) = 0\nR2N : Finset ℕ := range (2 * N + 1)\nhR2N : R2N = range (2 * N + 1)\nRN : Finset ℕ := range (N + 1)\nhRN : RN... | have split₁ := sum_filter_add_sum_filter_not (R2N ×ˢ R2N)
(fun ij ↦ ij.1 + ij.2 ≤ 2 * N)
(fun ij ↦ ((ij.1 ! : ℚ)⁻¹ * (ij.2 ! : ℚ)⁻¹) • (a ^ ij.1 * b ^ ij.2)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.AdjoinRoot | {
"line": 1036,
"column": 22
} | {
"line": 1036,
"column": 54
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nU : Type u_4\nK : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\npb : PowerBasis R S\nI : Ideal R\nx : R\n⊢ Ideal.map (of (minpoly R pb.gen)) I = Ideal.map ((↑↑(equiv' (minpoly R pb.gen) pb ⋯ ⋯).symm).comp (algebraMap R S)) I",
"us... | ← AlgEquiv.coe_ringHom_commutes, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AdjoinRoot | {
"line": 1036,
"column": 55
} | {
"line": 1036,
"column": 82
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nU : Type u_4\nK : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\npb : PowerBasis R S\nI : Ideal R\nx : R\n⊢ Ideal.map (of (minpoly R pb.gen)) I = Ideal.map ((↑↑(equiv' (minpoly R pb.gen) pb ⋯ ⋯).symm).comp (algebraMap R S)) I",
"us... | ← AdjoinRoot.algebraMap_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 143,
"column": 72
} | {
"line": 143,
"column": 81
} | [
{
"pp": "case inr\nA : Type u_1\ninst✝¹ : Ring A\ninst✝ : Module ℚ A\na b : A\nh₁ : Commute a b\nn₁ : ℕ\nhn₁ : a ^ n₁ = 0\nn₂ : ℕ\nhn₂ : b ^ n₂ = 0\nN : ℕ := max n₁ n₂\nh₄ : a ^ (N + 1) = 0\nh₅ : b ^ (N + 1) = 0\nR2N : Finset ℕ := range (2 * N + 1)\nhR2N : R2N = range (2 * N + 1)\nRN : Finset ℕ := range (N + 1)... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 139,
"column": 27
} | {
"line": 143,
"column": 92
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : Module ℚ A\na b : A\nh₁ : Commute a b\nn₁ : ℕ\nhn₁ : a ^ n₁ = 0\nn₂ : ℕ\nhn₂ : b ^ n₂ = 0\nN : ℕ := max n₁ n₂\nh₄ : a ^ (N + 1) = 0\nh₅ : b ^ (N + 1) = 0\nR2N : Finset ℕ := range (2 * N + 1)\nhR2N : R2N = range (2 * N + 1)\nRN : Finset ℕ := range (N + 1)\nhRN : RN... | by
simp only [not_and, not_le, mem_filter] at hi
cases le_or_gt (N + 1) i.1 with
| inl h => rw [pow_eq_zero_of_le h h₄, zero_mul, smul_zero]
| inr h => rw [pow_eq_zero_of_le (hi.2 (Nat.le_of_lt_succ h)) h₅, mul_zero, smul_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Derivation.Basic | {
"line": 298,
"column": 4
} | {
"line": 299,
"column": 89
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nD1 D2 : LieDerivation R L L\na b : L\n⊢ ⁅↑D1, ↑D2⁆ ⁅a, b⁆ = ⁅a, ⁅↑D1, ↑D2⁆ b⁆ - ⁅b, ⁅↑D1, ↑D2⁆ a⁆",
"usedConstants": [
"LieAlgebra.toModule",
"Module.End.instRing",
"Eq.mpr",
"NegZer... | simp only [Ring.lie_def, apply_lie_eq_add, coeFn_coe,
LinearMap.sub_apply, Module.End.mul_apply, map_add, sub_lie, lie_sub, ← lie_skew b] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Derivation.Basic | {
"line": 323,
"column": 37
} | {
"line": 323,
"column": 97
} | [
{
"pp": "case H\nR : Type u_1\nL : Type u_2\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nD1 D2 : LieDerivation R L L\nr : R\nd e : LieDerivation R L L\na : L\n⊢ ⁅d, r • e⁆ a = (r • ⁅d, e⁆) a",
"usedConstants": [
"LieAlgebra.toModule",
"instHSMul",
"SemilinearMapClass.t... | simp only [commutator_apply, map_smul, smul_sub, smul_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Solvable | {
"line": 94,
"column": 6
} | {
"line": 95,
"column": 67
} | [
{
"pp": "case succ.inl\nR : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nI J : LieIdeal R L\nh₁ : I ≤ J\nk : ℕ\nih : ∀ {l : ℕ}, l ≤ k → D k I ≤ D l J\nl : ℕ\nh₂ : l ≤ k + 1\nh : l = k.succ\n⊢ D (k + 1) I ≤ D l J",
"usedConstants": [
"LieAlgebra.toModule",
... | rw [h, derivedSeriesOfIdeal_succ, derivedSeriesOfIdeal_succ]
exact LieSubmodule.mono_lie (ih (le_refl k)) (ih (le_refl k)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Solvable | {
"line": 94,
"column": 6
} | {
"line": 95,
"column": 67
} | [
{
"pp": "case succ.inl\nR : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nI J : LieIdeal R L\nh₁ : I ≤ J\nk : ℕ\nih : ∀ {l : ℕ}, l ≤ k → D k I ≤ D l J\nl : ℕ\nh₂ : l ≤ k + 1\nh : l = k.succ\n⊢ D (k + 1) I ≤ D l J",
"usedConstants": [
"LieAlgebra.toModule",
... | rw [h, derivedSeriesOfIdeal_succ, derivedSeriesOfIdeal_succ]
exact LieSubmodule.mono_lie (ih (le_refl k)) (ih (le_refl k)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.BaseChange | {
"line": 64,
"column": 64
} | {
"line": 84,
"column": 84
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nL : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nx : A ⊗[R] L\n⊢ ⁅x, x⁆ = 0",
"usedConstants": [
"LieAlgebra.toModule",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
... | by
simp only [bracket_def]
refine x.induction_on ?_ ?_ ?_
· simp only [map_zero]
· intro a l
simp only [bracket'_tmul, TensorProduct.tmul_zero, lie_self]
· intro z₁ z₂ h₁ h₂
suffices bracket' R A L L z₁ z₂ + bracket' R A L L z₂ z₁ = 0 by
rw [map_add, map_add, LinearMap.add_apply, LinearMap.add_a... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.BilinearForm.Orthogonal | {
"line": 228,
"column": 25
} | {
"line": 228,
"column": 48
} | [
{
"pp": "case h.mp\nV : Type u_5\nK : Type u_6\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nB : BilinForm K V\nW : Subspace K V\nx : V\n⊢ x ∈ (domRestrict B W).range.dualCoannihilator → x ∈ B.orthogonal W",
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHomSurjective.ids... | rw [mem_orthogonal_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.BilinearForm.Orthogonal | {
"line": 228,
"column": 25
} | {
"line": 228,
"column": 48
} | [
{
"pp": "case h.mpr\nV : Type u_5\nK : Type u_6\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nB : BilinForm K V\nW : Subspace K V\nx : V\n⊢ x ∈ B.orthogonal W → x ∈ (domRestrict B W).range.dualCoannihilator",
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHomSurjective.id... | rw [mem_orthogonal_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 154,
"column": 8
} | {
"line": 157,
"column": 48
} | [
{
"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⊢ ⁅a, ↑I.incl y⁆ ∈ Submodule.map ↑... | · simp only [Submodule.mem_map, LieSubmodule.mem_toSubmodule, Subtype.exists]
erw [Submodule.coe_subtype]
simp only [exists_and_right, exists_eq_right, ha, lie_mem_left, exists_true_left]
exact lie_mem_right R I J ⟨a, ha⟩ y hy | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.BilinearForm.Properties | {
"line": 335,
"column": 50
} | {
"line": 335,
"column": 65
} | [
{
"pp": "R₁ : Type u_3\nM₁ : Type u_4\ninst✝² : CommRing R₁\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R₁ M₁\nB : BilinForm R₁ M₁\nb : B.Nondegenerate\nφ ψ₁ ψ₂ : M₁ →ₗ[R₁] M₁\nhψ₁ : IsAdjointPair B B ⇑ψ₁ ⇑φ\nhψ₂ : IsAdjointPair B B ⇑ψ₂ ⇑φ\nv w : M₁\n⊢ ((B.compLeft ψ₁) v) w = ((B.compLeft ψ₂) v) w",
"usedCons... | compLeft_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.BilinearForm.Properties | {
"line": 335,
"column": 66
} | {
"line": 335,
"column": 81
} | [
{
"pp": "R₁ : Type u_3\nM₁ : Type u_4\ninst✝² : CommRing R₁\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R₁ M₁\nB : BilinForm R₁ M₁\nb : B.Nondegenerate\nφ ψ₁ ψ₂ : M₁ →ₗ[R₁] M₁\nhψ₁ : IsAdjointPair B B ⇑ψ₁ ⇑φ\nhψ₂ : IsAdjointPair B B ⇑ψ₂ ⇑φ\nv w : M₁\n⊢ (B (ψ₁ v)) w = ((B.compLeft ψ₂) v) w",
"usedConstants": [... | compLeft_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.BilinearForm.Properties | {
"line": 392,
"column": 55
} | {
"line": 392,
"column": 71
} | [
{
"pp": "V : Type u_5\nK : Type u_6\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nι : Type u_9\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nB : BilinForm K V\nhB : B.Nondegenerate\nb : Basis ι K V\ni j : ι\nthis : FiniteDimensional K V\n⊢ (B ((B.toDual hB).symm (b.coord i))) (b j) = if j = i th... | ← toDual_def hB, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.Divisibility.Lemmas | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 50
} | [
{
"pp": "case h.inl\nR : 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\nhi : m ≤ i\n⊢ x ^ m ∣ x ^ (i, j).1 * y ^ (i, j).2",
"usedConstants": [
"dvd_mul_of_dvd_left",
"pow_dvd_pow",
"Prod.mk",... | · exact dvd_mul_of_dvd_left (pow_dvd_pow x hi) _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.Engel | {
"line": 222,
"column": 2
} | {
"line": 258,
"column": 11
} | [
{
"pp": "R : Type u₁\nL : Type u₂\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsNoetherian R L\n⊢ IsEngelian R L",
"usedConstants": [
"LieHom",
"LieAlgebra.toModule",
"Nontrivial",
"Module.End.instRing",
"Iff.mpr",
"LieSubalgebra.lieAlgebra"... | intro M _i1 _i2 _i3 _i4 h
rw [← isNilpotent_range_toEnd_iff R]
let L' := (toEnd R L M).range
replace h : ∀ y : L', IsNilpotent (y : Module.End R M) := by
rintro ⟨-, ⟨y, rfl⟩⟩
simp [h]
change LieModule.IsNilpotent L' M
let s := {K : LieSubalgebra R L' | LieAlgebra.IsEngelian R K}
have hs : s.Nonempty... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Engel | {
"line": 222,
"column": 2
} | {
"line": 258,
"column": 11
} | [
{
"pp": "R : Type u₁\nL : Type u₂\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsNoetherian R L\n⊢ IsEngelian R L",
"usedConstants": [
"LieHom",
"LieAlgebra.toModule",
"Nontrivial",
"Module.End.instRing",
"Iff.mpr",
"LieSubalgebra.lieAlgebra"... | intro M _i1 _i2 _i3 _i4 h
rw [← isNilpotent_range_toEnd_iff R]
let L' := (toEnd R L M).range
replace h : ∀ y : L', IsNilpotent (y : Module.End R M) := by
rintro ⟨-, ⟨y, rfl⟩⟩
simp [h]
change LieModule.IsNilpotent L' M
let s := {K : LieSubalgebra R L' | LieAlgebra.IsEngelian R K}
have hs : s.Nonempty... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 65
} | [
{
"pp": "k : Type u\ninst✝¹ : Field k\ninst✝ : IsAlgClosed k\np : k[X]\n⊢ p.roots.card = p.natDegree",
"usedConstants": [
"Polynomial.C",
"Polynomial.roots",
"HMul.hMul",
"Multiset.map",
"CommSemiring.toSemiring",
"Multiset.prod",
"HSub.hSub",
"RingHom",
... | have ⟨_, _, hdeg, hroots⟩ := exists_prod_multiset_X_sub_C_mul p | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 569,
"column": 4
} | {
"line": 569,
"column": 48
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\ninst✝ : CharZero K\nf g : K[X]\nhf0 : f ≠ 0\nhg0 : ¬g = 0\nhdf0 : ¬derivative f = 0\nhdg : derivative f * g ≠ 0\na : K\nhaf : eval a f = 0\nh0 : 0 < rootMultiplicity a f\n⊢ (f.IsRoot a → g.IsRoot a) → rootMultiplicity a f ≤ rootMultiplic... | rw [derivative_rootMultiplicity_of_root haf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 575,
"column": 4
} | {
"line": 575,
"column": 44
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\ninst✝ : CharZero K\nf g : K[X]\nhf0 : f ≠ 0\nhg0 : ¬g = 0\nhdf0 : ¬derivative f = 0\nhdg : derivative f * g ≠ 0\na : K\nhaf : ¬eval a f = 0\n⊢ (f.IsRoot a → g.IsRoot a) → rootMultiplicity a f ≤ rootMultiplicity a (derivative f) + rootMul... | simp [haf, rootMultiplicity_eq_zero haf] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
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