module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Homology.Factorizations.CM5b | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 10
} | {
"line": 84,
"column": 0
} | [
{
"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)",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
... | [] | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.Factorizations.CM5b | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 10
} | {
"line": 84,
"column": 0
} | [
{
"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)",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
... | [] | simp [i] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Factorizations.CM5b | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 10
} | {
"line": 84,
"column": 0
} | [
{
"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)",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
... | [] | simp [i] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Factorizations.CM5b | {
"line": 90,
"column": 34
} | {
"line": 90,
"column": 42
} | {
"line": 92,
"column": 0
} | [
{
"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",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"HomologicalComplex.instCategory... | [] | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.Factorizations.CM5b | {
"line": 90,
"column": 34
} | {
"line": 90,
"column": 42
} | {
"line": 92,
"column": 0
} | [
{
"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",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"HomologicalComplex.instCategory... | [] | simp [i] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Factorizations.CM5b | {
"line": 90,
"column": 34
} | {
"line": 90,
"column": 42
} | {
"line": 92,
"column": 0
} | [
{
"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",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"HomologicalComplex.instCategory... | [] | simp [i] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Generator.HomologicalComplex | {
"line": 49,
"column": 10
} | {
"line": 49,
"column": 11
} | {
"line": 49,
"column": 12
} | [
{
"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 :... | [
"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 L : HomologicalComplex C c\n⊢ ∀ (f g : K ⟶ L),\n (∀ (G : HomologicalComplex C c), ObjectPropert... | L | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.AlgebraicTopology.ModelCategory.CategoryWithCofibrations | {
"line": 282,
"column": 2
} | {
"line": 283,
"column": 16
} | {
"line": 285,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX✝ Y✝ : C\nf✝ : X✝ ⟶ Y✝\ninst✝¹ : CategoryWithWeakEquivalences C\nP : ObjectProperty C\nX Y : P.FullSubcategory\nf : X ⟶ Y\ninst✝ : WeakEquivalence f\n⊢ WeakEquivalence (P.ι.map f)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Categ... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.ModelCategory.CategoryWithCofibrations | {
"line": 282,
"column": 2
} | {
"line": 283,
"column": 16
} | {
"line": 285,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX✝ Y✝ : C\nf✝ : X✝ ⟶ Y✝\ninst✝¹ : CategoryWithWeakEquivalences C\nP : ObjectProperty C\nX Y : P.FullSubcategory\nf : X ⟶ Y\ninst✝ : WeakEquivalence f\n⊢ WeakEquivalence (P.ι.map f)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Categ... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 261,
"column": 35
} | {
"line": 261,
"column": 51
} | {
"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",
"ppTerm": "?m.96",
"assigned... | [] | by simp [hf, hg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 267,
"column": 35
} | {
"line": 267,
"column": 51
} | {
"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",
"ppTerm": "?m.96",
"assigned... | [] | by simp [hf, hg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 338,
"column": 33
} | {
"line": 338,
"column": 49
} | {
"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",
"ppTerm": "?m.96",
"assigned... | [] | by simp [hf, hg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 344,
"column": 33
} | {
"line": 344,
"column": 49
} | {
"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",
"ppTerm": "?m.96",
"assigned... | [] | by simp [hf, hg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.ModelCategory.IsCofibrant | {
"line": 98,
"column": 19
} | {
"line": 101,
"column": 16
} | {
"line": 103,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : CategoryWithFibrations C\ninst✝² : HasTerminal C\ninst✝¹ : (fibrations C).IsStableUnderComposition\nX Y : C\np : X ⟶ Y\ninst✝ : Fibration p\nhY : IsFibrant Y\n⊢ IsFibrant X",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
... | [] | by
rw [isFibrant_iff] at hY ⊢
rw [Subsingleton.elim (terminal.from X) (p ≫ terminal.from Y)]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.ModelCategory.Cylinder | {
"line": 98,
"column": 44
} | {
"line": 98,
"column": 52
} | {
"line": 100,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : C\nP : Precylinder A\ninst✝ : HasBinaryCoproduct A A\n⊢ coprod.inl ≫ P.i = P.i₀",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"HomotopicalAlgeb... | [] | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.ModelCategory.Cylinder | {
"line": 98,
"column": 44
} | {
"line": 98,
"column": 52
} | {
"line": 100,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : C\nP : Precylinder A\ninst✝ : HasBinaryCoproduct A A\n⊢ coprod.inl ≫ P.i = P.i₀",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"HomotopicalAlgeb... | [] | simp [i] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.ModelCategory.Cylinder | {
"line": 98,
"column": 44
} | {
"line": 98,
"column": 52
} | {
"line": 100,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : C\nP : Precylinder A\ninst✝ : HasBinaryCoproduct A A\n⊢ coprod.inl ≫ P.i = P.i₀",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"HomotopicalAlgeb... | [] | simp [i] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.ModelCategory.Cylinder | {
"line": 102,
"column": 44
} | {
"line": 102,
"column": 52
} | {
"line": 104,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : C\nP : Precylinder A\ninst✝ : HasBinaryCoproduct A A\n⊢ coprod.inr ≫ P.i = P.i₁",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"HomotopicalAlgeb... | [] | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.ModelCategory.Cylinder | {
"line": 102,
"column": 44
} | {
"line": 102,
"column": 52
} | {
"line": 104,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : C\nP : Precylinder A\ninst✝ : HasBinaryCoproduct A A\n⊢ coprod.inr ≫ P.i = P.i₁",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"HomotopicalAlgeb... | [] | simp [i] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.ModelCategory.Cylinder | {
"line": 102,
"column": 44
} | {
"line": 102,
"column": 52
} | {
"line": 104,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : C\nP : Precylinder A\ninst✝ : HasBinaryCoproduct A A\n⊢ coprod.inr ≫ P.i = P.i₁",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"HomotopicalAlgeb... | [] | simp [i] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.ModelCategory.RightHomotopy | {
"line": 169,
"column": 2
} | {
"line": 176,
"column": 37
} | {
"line": 178,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\ninst✝ : ModelCategory C\nP : PathObject Y\nf g : X ⟶ Y\nh : P.RightHomotopy f g\nd : (trivialCofibrations C).MapFactorizationData (fibrations C) P.p :=\n (trivialCofibrations C).factorizationData (fibrations C) P.p\n⊢ ∃ P', P'.IsGood ∧ Nonempty (P'.Righ... | [] | exact
⟨{ P := d.Z
p₀ := d.p ≫ prod.fst
p₁ := d.p ≫ prod.snd
ι := P.ι ≫ d.i }, ⟨by
rw [fibration_iff]
convert! d.hp
aesop⟩, ⟨{ h := h.h ≫ d.i }⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.HomotopyCategory.Plus | {
"line": 242,
"column": 51
} | {
"line": 244,
"column": 16
} | {
"line": 246,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\nA : Type u_2\ninst✝³ : Category.{v_2, u_2} A\ninst✝² : Abelian A\ninst✝¹ : HasZeroObject C\ninst✝ : HasBinaryBiproducts C\nn : ℤ\n⊢ (singleFunctor C n).Additive",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
... | [] | by
dsimp [singleFunctor, singleFunctors]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Preadditive.Projective.Resolution | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 79
} | {
"line": 110,
"column": 2
} | [
{
"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",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Nat.instOne",
"CategoryTheory.CategoryStruct.toQuiver",
... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nZ : C\nP : ProjectiveResolution Z\n⊢ CokernelCofork.ofπ (pOpcycles P.complex 0) ⋯ ≅ P.cokernelCofork"
] | refine IsColimit.ofIsoColimit (P.complex.opcyclesIsCokernel 1 0 (by simp)) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Idempotents.FunctorExtension | {
"line": 78,
"column": 25
} | {
"line": 78,
"column": 28
} | {
"line": 78,
"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 ≫ (... | [
"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 ≫ (F.map x✝.p).... | h'' | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.GradedObject.Monoidal | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 49
} | {
"line": 149,
"column": 0
} | [
{
"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
} | {
"line": 149,
"column": 0
} | [
{
"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.Algebra.Homology.SpectralObject.Cycles | {
"line": 129,
"column": 2
} | {
"line": 130,
"column": 16
} | {
"line": 132,
"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\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\n⊢ Mono (X.kernelSequenceCycles f g n₀ n₁ hn₁).f",
"ppTerm": "?m.31",
"assigned": true,
"usedC... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.GradedObject.Monoidal | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 49
} | {
"line": 149,
"column": 0
} | [
{
"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.Algebra.Homology.SpectralObject.Cycles | {
"line": 129,
"column": 2
} | {
"line": 130,
"column": 16
} | {
"line": 132,
"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\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\n⊢ Mono (X.kernelSequenceCycles f g n₀ n₁ hn₁).f",
"ppTerm": "?m.31",
"assigned": true,
"usedC... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.SpectralObject.Cycles | {
"line": 135,
"column": 2
} | {
"line": 136,
"column": 16
} | {
"line": 138,
"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\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\n⊢ Epi (X.cokernelSequenceOpcycles f g n₀ n₁ hn₁).g",
"ppTerm": "?m.31",
"assigned": true,
"us... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.SpectralObject.Cycles | {
"line": 135,
"column": 2
} | {
"line": 136,
"column": 16
} | {
"line": 138,
"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\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\n⊢ Epi (X.cokernelSequenceOpcycles f g n₀ n₁ hn₁).g",
"ppTerm": "?m.31",
"assigned": true,
"us... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.GradedObject.Monoidal | {
"line": 391,
"column": 17
} | {
"line": 391,
"column": 20
} | {
"line": 392,
"column": 2
} | [
{
"pp": "I : 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 X₁ (tenso... | [
"I : 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 X₁ (tensorObj X₂ (ten... | h'' | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Homology.SpectralObject.Cycles | {
"line": 377,
"column": 2
} | {
"line": 378,
"column": 16
} | {
"line": 380,
"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\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": 377,
"column": 2
} | {
"line": 378,
"column": 16
} | {
"line": 380,
"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\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": 382,
"column": 2
} | {
"line": 383,
"column": 16
} | {
"line": 385,
"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\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": 382,
"column": 2
} | {
"line": 383,
"column": 16
} | {
"line": 385,
"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\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.CategoryTheory.GradedObject.Monoidal | {
"line": 425,
"column": 2
} | {
"line": 439,
"column": 84
} | {
"line": 440,
"column": 2
} | [
{
"pp": "I : 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 (tensorObj X₂... | [
"I : 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 (tensorObj X₂ X₃)\ninst✝¹... | 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.Homology | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 65
} | {
"line": 76,
"column": 2
} | [
{
"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... | [
"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 : n₁ + 1 = ... | 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": 133,
"column": 2
} | {
"line": 134,
"column": 60
} | {
"line": 136,
"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 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": 133,
"column": 2
} | {
"line": 134,
"column": 60
} | {
"line": 136,
"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 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.Page | {
"line": 633,
"column": 2
} | {
"line": 634,
"column": 24
} | {
"line": 638,
"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\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": 722,
"column": 69
} | {
"line": 724,
"column": 26
} | {
"line": 725,
"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 : 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.Order.Filter.EventuallyConst | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 47
} | {
"line": 171,
"column": 4
} | [
{
"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",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"instDistribLatticeNat",
"PartialOrder.toPreorder",
"Preord... | [] | induction m, hm using Nat.le_induction with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 270,
"column": 2
} | {
"line": 271,
"column": 16
} | {
"line": 273,
"column": 0
} | [
{
"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": 270,
"column": 2
} | {
"line": 271,
"column": 16
} | {
"line": 273,
"column": 0
} | [
{
"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": 303,
"column": 14
} | {
"line": 303,
"column": 18
} | {
"line": 303,
"column": 19
} | [
{
"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... | [
"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 i₁ i₂ i₃ : ... | hi₀, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 365,
"column": 2
} | {
"line": 366,
"column": 16
} | {
"line": 368,
"column": 0
} | [
{
"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": 365,
"column": 2
} | {
"line": 366,
"column": 16
} | {
"line": 368,
"column": 0
} | [
{
"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.Lie.OfAssociative | {
"line": 251,
"column": 4
} | {
"line": 251,
"column": 25
} | {
"line": 252,
"column": 4
} | [
{
"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... | [
"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 R L M) y\nm... | refine h _ fun m ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Lie.OfAssociative | {
"line": 312,
"column": 8
} | {
"line": 312,
"column": 24
} | {
"line": 312,
"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... | [
"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 R L) x ^ ij... | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 467,
"column": 2
} | {
"line": 467,
"column": 25
} | {
"line": 468,
"column": 2
} | [
{
"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)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"LieSubalgebra.instPartialOrder_1",
... | [
"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 (SetLike.coe '' S) ↑(sInf S)"
] | 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
} | {
"line": 471,
"column": 0
} | [
{
"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)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LieSubalgebra.instPartialOrder_1",
"IsGLB.of_image",
"ChainCo... | [] | 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
} | {
"line": 471,
"column": 0
} | [
{
"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)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LieSubalgebra.instPartialOrder_1",
"IsGLB.of_image",
"ChainCo... | [] | 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.Algebra.Lie.Ideal | {
"line": 279,
"column": 97
} | {
"line": 287,
"column": 67
} | {
"line": 289,
"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'\n⊢ f.IsIdealMorphism ↔ ∀ (x : L') (y : L), ∃ z, ⁅x, f y⁆ = f z",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
... | [] | 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.FieldTheory.Minpoly.Basic | {
"line": 232,
"column": 32
} | {
"line": 232,
"column": 39
} | {
"line": 232,
"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",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLi... | [
"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⊢ ¬?m.38 ↔ x ∉ S",
"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 : Subalgebr... | Iff.not | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.SModEq.Basic | {
"line": 46,
"column": 48
} | {
"line": 46,
"column": 89
} | {
"line": 48,
"column": 0
} | [
{
"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",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Submodule.Quotient.eq",
"Eq.mpr",
"Submodule",
"congrArg",
... | [] | by rw [SModEq.def, Submodule.Quotient.eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.IdealOperations | {
"line": 289,
"column": 2
} | {
"line": 289,
"column": 59
} | {
"line": 291,
"column": 0
} | [
{
"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",
"ppTerm": "?m.120",
"assi... | [] | exact le_trans (LieSubmodule.lie_le_left _ _) inf_le_left | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.PowerBasis | {
"line": 183,
"column": 6
} | {
"line": 183,
"column": 18
} | {
"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",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Finsupp.inst... | [
"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 < ↑pb.dim"
] | degree_X_pow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerBasis | {
"line": 203,
"column": 47
} | {
"line": 203,
"column": 59
} | {
"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",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"WithBot",
"PowerBasis.dim",
"congrA... | [
"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⊢ ↑pb.dim = ↑pb.dim"
] | degree_X_pow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerBasis | {
"line": 203,
"column": 47
} | {
"line": 203,
"column": 59
} | {
"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",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Finsupp.... | [
"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 < ↑pb.dim"
] | degree_X_pow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerBasis | {
"line": 294,
"column": 16
} | {
"line": 294,
"column": 82
} | {
"line": 295,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring S\ninst✝⁷ : Algebra R S\nA : Type u_4\nB : Type u_5\ninst✝⁶ : CommRing A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra A B\nK : Type u_6\ninst✝³ : Field K\ninst✝² : Algebra A S\nS' : Type u_7\ninst✝¹ : Ring S'\ninst✝ : Algebra A S'\n... | [] | by convert! pb.constr_pow_algebraMap hy 1 using 2 <;> rw [map_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Norm.Defs | {
"line": 72,
"column": 23
} | {
"line": 72,
"column": 76
} | {
"line": 74,
"column": 0
} | [
{
"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",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Algebra.lmul",
"MonoidHom.instFunLike",
"instSMulO... | [] | 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
} | {
"line": 74,
"column": 0
} | [
{
"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",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Algebra.lmul",
"MonoidHom.instFunLike",
"instSMulO... | [] | 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
} | {
"line": 114,
"column": 53
} | [
{
"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",
"ppTerm": "?m.233",
"... | [
"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) ((algebraMap K (AdjoinRoot g)) a) = a ^ g.natDegree"
] | ← AdjoinRoot.algebraMap_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Quotient | {
"line": 232,
"column": 4
} | {
"line": 233,
"column": 16
} | {
"line": 234,
"column": 4
} | [
{
"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... | [
"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 ↦ (Ideal.Quotient.m... | simp only [map_add, MvPolynomial.eval₂_add]
at hp hq ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Squarefree.Basic | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 19
} | {
"line": 176,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoidWithZero R\ninst✝ : IsCancelMulZero R\nx : R\nh0 : x ≠ 0\nh : IsRadical x\n⊢ Squarefree x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"Dvd.dvd",
"HMul.hMul",
"IsRadical",
"Monoid.toMulOneClass",
... | [
"R : Type u_1\ninst✝¹ : CommMonoidWithZero R\ninst✝ : IsCancelMulZero R\nz w : R\nh0 : z * z * w ≠ 0\nh : IsRadical (z * z * w)\n⊢ IsUnit z"
] | rintro z ⟨w, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Squarefree.Basic | {
"line": 298,
"column": 2
} | {
"line": 299,
"column": 41
} | {
"line": 301,
"column": 0
} | [
{
"pp": "n : ℤ\n⊢ Squarefree n.natAbs ↔ Squarefree n",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"HMul.hMul",
"Function.Surjective.forall",
"Monoid.toMulOneClass",
"congrArg",
"Nat.instMonoid",
"_private.Mathlib.Algebr... | [] | 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
} | {
"line": 301,
"column": 0
} | [
{
"pp": "n : ℤ\n⊢ Squarefree n.natAbs ↔ Squarefree n",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"HMul.hMul",
"Function.Surjective.forall",
"Monoid.toMulOneClass",
"congrArg",
"Nat.instMonoid",
"_private.Mathlib.Algebr... | [] | 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
} | {
"line": 301,
"column": 0
} | [
{
"pp": "n : ℤ\n⊢ Squarefree n.natAbs ↔ Squarefree n",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"HMul.hMul",
"Function.Surjective.forall",
"Monoid.toMulOneClass",
"congrArg",
"Nat.instMonoid",
"_private.Mathlib.Algebr... | [] | 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
} | {
"line": 244,
"column": 2
} | [
{
"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",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Polynomial.derivative",
"Units.val",
"Polynomial.C",
"Polyn... | [
"case inr\nR : Type u\ninst✝ : CommRing R\nn : ℕ\nu : Rˣ\nhn : IsUnit ↑n\na✝ : Nontrivial R\nhpos : n > 0\n⊢ ∃ a b, a * (X ^ n - C ↑u) + b * derivative (X ^ n - C ↑u) = 1"
] | 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
} | {
"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",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"congrArg",
"Finset",
"HSub.hSu... | [] | 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
} | {
"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",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"congrArg",
"Finset",
"HSub.hSu... | [] | 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.LinearAlgebra.Determinant | {
"line": 287,
"column": 41
} | {
"line": 290,
"column": 26
} | {
"line": 292,
"column": 0
} | [
{
"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",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.instFunLike",
"Semiring.toModule",
... | [] | 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": 389,
"column": 2
} | {
"line": 390,
"column": 93
} | {
"line": 391,
"column": 2
} | [
{
"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",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero",
"Polynomial.derivative",
"Eq.mpr",
"NonAssoc... | [
"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\nhf2 : derivative ((expand F (p ^ n)) f) = 0\n⊢ IsUnit f"
] | 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.AnnihilatingPolynomial | {
"line": 103,
"column": 4
} | {
"line": 103,
"column": 49
} | {
"line": 104,
"column": 4
} | [
{
"pp": "case neg.h2\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",
"ppTerm": "?neg.h2✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Semiri... | [
"case neg.h2\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"
] | 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
} | {
"line": 106,
"column": 0
} | [
{
"pp": "case neg.h2\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",
"ppTerm": "?neg.h2✝",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"IsDomain.to_noZe... | [] | apply (mul_ne_zero_iff.mp h).1 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.FieldTheory.Separable | {
"line": 442,
"column": 2
} | {
"line": 442,
"column": 48
} | {
"line": 443,
"column": 2
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nn : ℕ\n⊢ (X ^ n - 1).Separable ↔ ↑n ≠ 0",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Polynomial.instOne",
"AddGroupWithOne.toAddMonoidWithOne",
"HSub.hSub",
"Field.toDivisionRing",
"AddMonoidWithOne.toNatCast",
"... | [
"case inl\nF : Type u\ninst✝ : Field F\nn : ℕ\nhz : n = 0\n⊢ (X ^ n - 1).Separable ↔ ↑n ≠ 0",
"case inr\nF : Type u\ninst✝ : Field F\nn : ℕ\nhpos : n > 0\n⊢ (X ^ n - 1).Separable ↔ ↑n ≠ 0"
] | 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.Determinant | {
"line": 761,
"column": 31
} | {
"line": 761,
"column": 40
} | {
"line": 761,
"column": 41
} | [
{
"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... | [
"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⊢ 0 = ↑e.det (update v i (v k))"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AdjoinRoot | {
"line": 668,
"column": 4
} | {
"line": 669,
"column": 9
} | {
"line": 670,
"column": 4
} | [
{
"pp": "case h₁\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✝",
"ppTerm": "?h₁",
... | [
"case h₂\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\n⊢ ((lift (algebraMap K (AdjoinRoot f)) (root f) q_aeval).comp (mk q)) X = (mk f) X"
] | · simp only [RingHom.comp_apply, mk_C, lift_of]
rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.AdjoinRoot | {
"line": 757,
"column": 2
} | {
"line": 757,
"column": 77
} | {
"line": 758,
"column": 2
} | [
{
"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)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"Lattice.toSemilatticeSup",
"eq_top_i... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\n⊢ adjoin R (Subtype.val ⁻¹' {x}) ≤ (toAdjoin R x).range"
] | 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.AdjoinRoot | {
"line": 997,
"column": 22
} | {
"line": 997,
"column": 54
} | {
"line": 997,
"column": 55
} | [
{
"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 =\n Ideal.map ((↑(equiv' (minpoly R pb.gen) pb ⋯ ⋯).symm.toRingEquiv).comp (algebraMap R... | [
"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"
] | ← AlgEquiv.coe_ringHom_commutes, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AdjoinRoot | {
"line": 997,
"column": 55
} | {
"line": 997,
"column": 82
} | {
"line": 998,
"column": 22
} | [
{
"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",
"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 (algebraMap R (AdjoinRoot (minpoly R pb.gen))) I =\n Ideal.map ((↑↑(equiv' (minpoly R pb.gen) pb ⋯ ⋯).symm).comp (algebraMap R... | ← AdjoinRoot.algebraMap_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Dynamics.Newton | {
"line": 73,
"column": 37
} | {
"line": 73,
"column": 46
} | {
"line": 73,
"column": 47
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : R[X]\nx : S\nh : (aeval x) P = 0\n⊢ x - Ring.inverse ((aeval x) (derivative P)) * 0 = x",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Polynomial.derivative",
"Eq.mpr",
... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : R[X]\nx : S\nh : (aeval x) P = 0\n⊢ x - 0 = x"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 79,
"column": 6
} | {
"line": 79,
"column": 17
} | {
"line": 79,
"column": 18
} | [
{
"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",... | [
"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.Algebra.Lie.Derivation.Basic | {
"line": 297,
"column": 4
} | {
"line": 298,
"column": 89
} | {
"line": 299,
"column": 4
} | [
{
"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⁆",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"LieAlgebra.toModule",
"Module... | [
"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⊢ ⁅a, D1 (D2 b)⁆ + ⁅D1 a, D2 b⁆ + (⁅D2 a, D1 b⁆ + ⁅D1 (D2 a), b⁆) -\n (⁅a, D2 (D1 b)⁆ + ⁅D2 a, D1 b⁆ + (⁅D1 a, D2 b⁆ + ⁅D2 (D1 a), b⁆)) =\n ⁅a, D1 (D2 b)⁆ - ⁅a, D2 (D1 b)⁆ - -(... | 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": 322,
"column": 37
} | {
"line": 322,
"column": 97
} | {
"line": 324,
"column": 0
} | [
{
"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\nr : R\nd e : LieDerivation R L L\na : L\n⊢ ⁅d, r • e⁆ a = (r • ⁅d, e⁆) a",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"LieAlgebra.toModule",
"in... | [] | simp only [commutator_apply, map_smul, smul_sub, smul_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.JordanChevalley | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 61
} | {
"line": 49,
"column": 2
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nP : K[X]\nk : ℕ\nsep : P.Separable\nnil : minpoly K f ∣ P ^ k\n⊢ ∃ n ∈ K[f], ∃ s ∈ K[f], IsNilpotent n ∧ s.IsSemisimple ∧ f = n + s",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants":... | [
"K : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nP : K[X]\nk : ℕ\nsep : P.Separable\nnil : minpoly K f ∣ P ^ k\nff : ↥K[f] := ⟨f, ⋯⟩\n⊢ ∃ n ∈ K[f], ∃ s ∈ K[f], IsNilpotent n ∧ s.IsSemisimple ∧ f = n + s"
] | set ff : adjoin K {f} := ⟨f, self_mem_adjoin_singleton K f⟩ | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 131,
"column": 59
} | {
"line": 131,
"column": 68
} | {
"line": 131,
"column": 69
} | [
{
"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)... | [
"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)\nhRN : RN =... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 132,
"column": 2
} | {
"line": 134,
"column": 72
} | {
"line": 135,
"column": 2
} | [
{
"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... | [
"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 = range (N ... | 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.Nilpotent.Exp | {
"line": 143,
"column": 72
} | {
"line": 143,
"column": 81
} | {
"line": 143,
"column": 82
} | [
{
"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)... | [
"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)\nhRN : RN =... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 139,
"column": 27
} | {
"line": 143,
"column": 92
} | {
"line": 144,
"column": 2
} | [
{
"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.Solvable | {
"line": 101,
"column": 6
} | {
"line": 102,
"column": 67
} | {
"line": 103,
"column": 4
} | [
{
"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",
"ppTerm": "?succ.inl",
"assigned": true,
"used... | [] | 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": 101,
"column": 6
} | {
"line": 102,
"column": 67
} | {
"line": 103,
"column": 4
} | [
{
"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",
"ppTerm": "?succ.inl",
"assigned": true,
"used... | [] | 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
} | {
"line": 86,
"column": 0
} | [
{
"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",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"LieAlgebra.toModule",
"NonUnitalNonAssocCom... | [] | 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.Properties | {
"line": 335,
"column": 50
} | {
"line": 335,
"column": 65
} | {
"line": 335,
"column": 66
} | [
{
"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",
"ppTerm":... | [
"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"
] | compLeft_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.BilinearForm.Properties | {
"line": 335,
"column": 66
} | {
"line": 335,
"column": 81
} | {
"line": 335,
"column": 82
} | [
{
"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",
"ppTerm": "?m.108"... | [
"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 (ψ₂ v)) w"
] | compLeft_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.BilinearForm.Properties | {
"line": 392,
"column": 55
} | {
"line": 392,
"column": 71
} | {
"line": 393,
"column": 4
} | [
{
"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... | [
"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.toDual hB) ((B.toDual hB).symm (b.coord i))) (b j) = if j = i th... | ← toDual_def hB, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.BilinearForm.Orthogonal | {
"line": 222,
"column": 25
} | {
"line": 222,
"column": 48
} | {
"line": 222,
"column": 49
} | [
{
"pp": "case 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",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case 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 → ∀ n ∈ W, B.IsOrtho n x"
] | 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": 222,
"column": 25
} | {
"line": 222,
"column": 48
} | {
"line": 222,
"column": 49
} | [
{
"pp": "case 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",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case 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⊢ (∀ n ∈ W, B.IsOrtho n x) → x ∈ (domRestrict B W).range.dualCoannihilator"
] | 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": 151,
"column": 8
} | {
"line": 154,
"column": 48
} | {
"line": 156,
"column": 8
} | [
{
"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 ↑... | [
"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 ↑I.incl ↑J"
] | · 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 |
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