module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Homology.Embedding.CochainComplex | {
"line": 233,
"column": 42
} | {
"line": 233,
"column": 47
} | {
"line": 233,
"column": 47
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nK L : CochainComplex C ℤ\nφ : K ⟶ L\ne : K ≅ L\ninst✝ : HasZeroObject C\nX : ChainComplex C ℕ\nx✝¹ : ℤ\nx✝ : ∀ (i : ℕ), (embeddingUpIntLE 0).f i ≠ x✝¹\n⊢ ∀ (i : ℕ), embeddingDownNat.f i ≠ x✝¹",
"ppTerm": "?m.54",
"assign... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Embedding.CochainComplex | {
"line": 233,
"column": 42
} | {
"line": 233,
"column": 47
} | {
"line": 233,
"column": 47
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nK L : CochainComplex C ℤ\nφ : K ⟶ L\ne : K ≅ L\ninst✝ : HasZeroObject C\nX : ChainComplex C ℕ\nx✝¹ : ℤ\nx✝ : ∀ (i : ℕ), (embeddingUpIntLE 0).f i ≠ x✝¹\n⊢ ∀ (i : ℕ), embeddingDownNat.f i ≠ x✝¹",
"ppTerm": "?m.54",
"assign... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Embedding.CochainComplex | {
"line": 250,
"column": 23
} | {
"line": 250,
"column": 28
} | {
"line": 250,
"column": 29
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nK : CochainComplex C ℤ\ninst✝² : HasZeroObject C\nn : ℤ\ninst✝¹ : K.IsStrictlyGE n\ninst✝ : K.IsStrictlyLE n\n⊢ mkHomFromSingle (𝟙 (K.X n)) ⋯ ≫ mkHomToSingle (𝟙 (K.X n)) ⋯ = 𝟙 ((single C (up ℤ) n).obj (K.X n))",
"ppTerm":... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.Embedding.CochainComplex | {
"line": 250,
"column": 23
} | {
"line": 250,
"column": 28
} | {
"line": 250,
"column": 29
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nK : CochainComplex C ℤ\ninst✝² : HasZeroObject C\nn : ℤ\ninst✝¹ : K.IsStrictlyGE n\ninst✝ : K.IsStrictlyLE n\n⊢ mkHomFromSingle (𝟙 (K.X n)) ⋯ ≫ mkHomToSingle (𝟙 (K.X n)) ⋯ = 𝟙 ((single C (up ℤ) n).obj (K.X n))",
"ppTerm":... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Embedding.CochainComplex | {
"line": 250,
"column": 23
} | {
"line": 250,
"column": 28
} | {
"line": 250,
"column": 29
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nK : CochainComplex C ℤ\ninst✝² : HasZeroObject C\nn : ℤ\ninst✝¹ : K.IsStrictlyGE n\ninst✝ : K.IsStrictlyLE n\n⊢ mkHomFromSingle (𝟙 (K.X n)) ⋯ ≫ mkHomToSingle (𝟙 (K.X n)) ⋯ = 𝟙 ((single C (up ℤ) n).obj (K.X n))",
"ppTerm":... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.DerivedCategory.Fractions | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 47
} | {
"line": 109,
"column": 0
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasDerivedCategory C\nX Y : CochainComplex C ℤ\nn : ℤ\ninst✝ : Y.IsStrictlyGE n\nY' : CochainComplex C ℤ\ng : X ⟶ Y'\ns : Y ⟶ Y'\nhs : IsIso (Q.map s)\nthis : IsIso (Q.map (CochainComplex.truncGEMap s n))\neq : Q.map (Y... | [] | simp only [eq, IsIso.inv_hom_id_assoc, eq'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic | {
"line": 406,
"column": 81
} | {
"line": 406,
"column": 86
} | {
"line": 406,
"column": 86
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y Z T : C\nn a b c : ℕ\nh : a + b = c\n⊢ ∀ (a_1 b_1 : Ext X Y a),\n AddMonoidHom.mk' (fun β ↦ (a_1 + b_1).comp β h) ⋯ =\n AddMonoidHom.mk' (fun β ↦ a_1.comp β h) ⋯ + AddMonoidHom.mk' (fun β ↦ b_1.comp β h) ⋯",
"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic | {
"line": 406,
"column": 81
} | {
"line": 406,
"column": 86
} | {
"line": 406,
"column": 86
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y Z T : C\nn a b c : ℕ\nh : a + b = c\n⊢ ∀ (a_1 b_1 : Ext X Y a),\n AddMonoidHom.mk' (fun β ↦ (a_1 + b_1).comp β h) ⋯ =\n AddMonoidHom.mk' (fun β ↦ a_1.comp β h) ⋯ + AddMonoidHom.mk' (fun β ↦ b_1.comp β h) ⋯",
"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic | {
"line": 406,
"column": 81
} | {
"line": 406,
"column": 86
} | {
"line": 406,
"column": 86
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y Z T : C\nn a b c : ℕ\nh : a + b = c\n⊢ ∀ (a_1 b_1 : Ext X Y a),\n AddMonoidHom.mk' (fun β ↦ (a_1 + b_1).comp β h) ⋯ =\n AddMonoidHom.mk' (fun β ↦ a_1.comp β h) ⋯ + AddMonoidHom.mk' (fun β ↦ b_1.comp β h) ⋯",
"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Shift.Adjunction | {
"line": 93,
"column": 37
} | {
"line": 106,
"column": 41
} | {
"line": 108,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nA : Type u_3\ninst✝² : AddMonoid A\ninst✝¹ : HasShift C A\ninst✝ : HasShift D A\na : A\ne₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a\ne₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftF... | [] | by
intro Y
have eq := h (G.obj Y)
simp only [← cancel_mono (e₂.inv.app _ ≫ G.map (e₁.inv.app _)),
assoc, Iso.hom_inv_id_app_assoc, comp_id, ← Functor.map_comp,
Iso.hom_inv_id_app, Functor.comp_obj, Functor.map_id] at eq
apply (adj.homEquiv _ _).injective
dsimp
rw [adj.homEquiv_unit, adj.homEquiv_uni... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Shift.Pullback | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 37
} | {
"line": 92,
"column": 38
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nA : Type u_2\nB : Type u_3\ninst✝² : AddMonoid A\ninst✝¹ : AddMonoid B\ninst✝ : HasShift C B\nφ : A →+ B\nX : PullbackShift C φ\n⊢ 𝟙 ((𝟭 (PullbackShift C φ)).obj X) =\n (shiftFunctorZero (PullbackShift C φ) A).inv.app X ≫\n (pullbackShiftIso C φ 0... | [
"C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nA : Type u_2\nB : Type u_3\ninst✝² : AddMonoid A\ninst✝¹ : AddMonoid B\ninst✝ : HasShift C B\nφ : A →+ B\nX : PullbackShift C φ\n⊢ 𝟙 ((𝟭 (PullbackShift C φ)).obj X) =\n ((shiftFunctorZero C B).inv.app X ≫ (pullbackShiftIso C φ 0 0 ⋯).inv.app X) ≫\n (pullbackS... | pullbackShiftFunctorZero_inv_app, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.Opposite.Basic | {
"line": 221,
"column": 46
} | {
"line": 221,
"column": 65
} | {
"line": 221,
"column": 66
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasShift C ℤ\nX : Cᵒᵖ\n⊢ (shiftFunctor C 0).map ((shiftFunctorOpIso C 0 0 ⋯).inv.app X).unop ≫\n (shiftFunctorCompIsoId C 0 0 ⋯).hom.app (Opposite.unop X) =\n (shiftFunctor C 0).map\n (((shiftFunctorOpIso C 0 0 ⋯).inv.app X).unop ≫ ((... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasShift C ℤ\nX : Cᵒᵖ\n⊢ (shiftFunctor C 0).map ((shiftFunctorOpIso C 0 0 ⋯).inv.app X).unop ≫\n (shiftFunctorCompIsoId C 0 0 ⋯).hom.app (Opposite.unop X) =\n (shiftFunctor C 0).map\n (((shiftFunctorOpIso C 0 0 ⋯).inv.app X).unop ≫ (shiftFunctorZ... | Quiver.Hom.unop_op, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.Opposite.Basic | {
"line": 232,
"column": 46
} | {
"line": 232,
"column": 65
} | {
"line": 232,
"column": 66
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasShift C ℤ\nX : Cᵒᵖ\n⊢ (shiftFunctorCompIsoId C 0 0 ⋯).inv.app (Opposite.unop X) ≫\n (shiftFunctor C 0).map ((shiftFunctorOpIso C 0 0 ⋯).hom.app X).unop =\n (shiftFunctorZero C ℤ).inv.app (Opposite.unop X) ≫\n (shiftFunctor C 0).map\n... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasShift C ℤ\nX : Cᵒᵖ\n⊢ (shiftFunctorCompIsoId C 0 0 ⋯).inv.app (Opposite.unop X) ≫\n (shiftFunctor C 0).map ((shiftFunctorOpIso C 0 0 ⋯).hom.app X).unop =\n (shiftFunctorZero C ℤ).inv.app (Opposite.unop X) ≫\n (shiftFunctor C 0).map\n ((sh... | Quiver.Hom.unop_op, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Shift.Pullback | {
"line": 267,
"column": 4
} | {
"line": 274,
"column": 8
} | {
"line": 276,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹⁰ : Category.{v_1, u_1} C\nA : Type u_2\nB : Type u_3\ninst✝⁹ : AddMonoid A\ninst✝⁸ : AddMonoid B\ninst✝⁷ : HasShift C B\nφ : A →+ B\nX : PullbackShift C φ\na₁ a₂ a₃ : A\nh : a₁ + a₂ = a₃\nb₁ b₂ b₃ : B\nh₁ : b₁ = φ a₁\nh₂ : b₂ = φ a₂\nh₃ : b₃ = φ a₃\nD : Type u_4\ninst✝⁶ : Category.... | [] | ext
dsimp [PullbackShift.natIsoComp]
simp only [commShiftPullback_iso_eq φ _ _ _ rfl, Iso.trans_hom, isoWhiskerRight_hom,
isoWhiskerLeft_hom, Iso.symm_hom, comp_app, comp_obj, whiskerRight_app, Functor.comp_map,
commShiftIso_comp_hom_app, whiskerLeft_app, assoc, map_id, comp_id, map_comp, id_comp]
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Shift.Pullback | {
"line": 267,
"column": 4
} | {
"line": 274,
"column": 8
} | {
"line": 276,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹⁰ : Category.{v_1, u_1} C\nA : Type u_2\nB : Type u_3\ninst✝⁹ : AddMonoid A\ninst✝⁸ : AddMonoid B\ninst✝⁷ : HasShift C B\nφ : A →+ B\nX : PullbackShift C φ\na₁ a₂ a₃ : A\nh : a₁ + a₂ = a₃\nb₁ b₂ b₃ : B\nh₁ : b₁ = φ a₁\nh₂ : b₂ = φ a₂\nh₃ : b₃ = φ a₃\nD : Type u_4\ninst✝⁶ : Category.... | [] | ext
dsimp [PullbackShift.natIsoComp]
simp only [commShiftPullback_iso_eq φ _ _ _ rfl, Iso.trans_hom, isoWhiskerRight_hom,
isoWhiskerLeft_hom, Iso.symm_hom, comp_app, comp_obj, whiskerRight_app, Functor.comp_map,
commShiftIso_comp_hom_app, whiskerLeft_app, assoc, map_id, comp_id, map_comp, id_comp]
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Triangulated.Opposite.Pretriangulated | {
"line": 158,
"column": 4
} | {
"line": 158,
"column": 23
} | {
"line": 158,
"column": 24
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasZeroObject C\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nT₁ T₂ : Triangle Cᵒᵖ\nhT₁ : Opposite.unop ((triangleOpEquivalence C).inverse.obj T₁) ∈ Pretriangulated.disti... | [
"C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasZeroObject C\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nT₁ T₂ : Triangle Cᵒᵖ\nhT₁ : Opposite.unop ((triangleOpEquivalence C).inverse.obj T₁) ∈ Pretriangulated.distinguishedTria... | Quiver.Hom.unop_op, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Linear | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 7
} | {
"line": 64,
"column": 0
} | [
{
"pp": "R : Type t\ninst✝⁴ : Ring R\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : Linear R C\ninst✝ : HasExt C\nX Y Z : C\na b : ℕ\nα : Ext X Y a\nβ : Ext Y Z b\nc : ℕ\nh : a + b = c\nr : R\nthis : HasDerivedCategory C := HasDerivedCategory.standard C\n⊢ α.comp (r • β) h = r • α.comp β ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Linear | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 7
} | {
"line": 71,
"column": 0
} | [
{
"pp": "R : Type t\ninst✝⁴ : Ring R\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : Linear R C\ninst✝ : HasExt C\nX Y Z : C\na b : ℕ\nα : Ext X Y a\nβ : Ext Y Z b\nc : ℕ\nh : a + b = c\nr : R\nthis : HasDerivedCategory C := HasDerivedCategory.standard C\n⊢ (r • α).comp β h = r • α.comp β ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Linear | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 7
} | {
"line": 86,
"column": 0
} | [
{
"pp": "R : Type t\ninst✝⁴ : Ring R\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : Linear R C\ninst✝ : HasExt C\nX Y : C\nr : R\nf : X ⟶ Y\nthis : HasDerivedCategory C := HasDerivedCategory.standard C\n⊢ mk₀ (r • f) = r • mk₀ f",
"ppTerm": "?m.34",
"assigned": true,
"usedCons... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Linear | {
"line": 113,
"column": 17
} | {
"line": 113,
"column": 22
} | {
"line": 114,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : HasExt C\nR : Type t\ninst✝¹ : CommRing R\ninst✝ : Linear R C\nX Y Z : C\na b c : ℕ\nh : a + b = c\n⊢ ∀ (x y : Ext X Y a),\n { toFun := fun β ↦ (x + y).comp β h, map_add' := ⋯, map_smul' := ⋯ } =\n { toFun := fun β ↦ x.comp β ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Linear | {
"line": 113,
"column": 17
} | {
"line": 113,
"column": 22
} | {
"line": 114,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : HasExt C\nR : Type t\ninst✝¹ : CommRing R\ninst✝ : Linear R C\nX Y Z : C\na b c : ℕ\nh : a + b = c\n⊢ ∀ (x y : Ext X Y a),\n { toFun := fun β ↦ (x + y).comp β h, map_add' := ⋯, map_smul' := ⋯ } =\n { toFun := fun β ↦ x.comp β ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Linear | {
"line": 113,
"column": 17
} | {
"line": 113,
"column": 22
} | {
"line": 114,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : HasExt C\nR : Type t\ninst✝¹ : CommRing R\ninst✝ : Linear R C\nX Y Z : C\na b c : ℕ\nh : a + b = c\n⊢ ∀ (x y : Ext X Y a),\n { toFun := fun β ↦ (x + y).comp β h, map_add' := ⋯, map_smul' := ⋯ } =\n { toFun := fun β ↦ x.comp β ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Linear | {
"line": 114,
"column": 18
} | {
"line": 114,
"column": 23
} | {
"line": 116,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : HasExt C\nR : Type t\ninst✝¹ : CommRing R\ninst✝ : Linear R C\nX Y Z : C\na b c : ℕ\nh : a + b = c\n⊢ ∀ (m : R) (x : Ext X Y a),\n { toFun := fun β ↦ (m • x).comp β h, map_add' := ⋯, map_smul' := ⋯ } =\n (RingHom.id R) m • { t... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Linear | {
"line": 114,
"column": 18
} | {
"line": 114,
"column": 23
} | {
"line": 116,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : HasExt C\nR : Type t\ninst✝¹ : CommRing R\ninst✝ : Linear R C\nX Y Z : C\na b c : ℕ\nh : a + b = c\n⊢ ∀ (m : R) (x : Ext X Y a),\n { toFun := fun β ↦ (m • x).comp β h, map_add' := ⋯, map_smul' := ⋯ } =\n (RingHom.id R) m • { t... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Linear | {
"line": 114,
"column": 18
} | {
"line": 114,
"column": 23
} | {
"line": 116,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : HasExt C\nR : Type t\ninst✝¹ : CommRing R\ninst✝ : Linear R C\nX Y Z : C\na b c : ℕ\nh : a + b = c\n⊢ ∀ (m : R) (x : Ext X Y a),\n { toFun := fun β ↦ (m • x).comp β h, map_add' := ⋯, map_smul' := ⋯ } =\n (RingHom.id R) m • { t... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Ext.DimensionShifting | {
"line": 39,
"column": 43
} | {
"line": 42,
"column": 34
} | {
"line": 44,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : Small.{v, u} R\nM : ModuleCat R\n⊢ M.projectiveShortComplex.ShortExact",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Pi.Function.module",
"Module.Basis.ofRepr",
"instHSMul",
"Semiring.toModule",
"Pi.addCommM... | [] | by
apply LinearMap.shortExact_shortComplexKer
refine fun m ↦ ⟨Finsupp.single m 1, ?_⟩
simp [Module.Basis.constr_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 116,
"column": 73
} | {
"line": 116,
"column": 78
} | {
"line": 118,
"column": 0
} | [
{
"pp": "A : Type u\ninst✝ : Ring A\nrelations : Relations A\n⊢ relations.toQuotient ∘ₗ relations.map = 0",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"Finsupp.module",
"Module.Relations.Quotient",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 116,
"column": 73
} | {
"line": 116,
"column": 78
} | {
"line": 118,
"column": 0
} | [
{
"pp": "A : Type u\ninst✝ : Ring A\nrelations : Relations A\n⊢ relations.toQuotient ∘ₗ relations.map = 0",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"Finsupp.module",
"Module.Relations.Quotient",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 116,
"column": 73
} | {
"line": 116,
"column": 78
} | {
"line": 118,
"column": 0
} | [
{
"pp": "A : Type u\ninst✝ : Ring A\nrelations : Relations A\n⊢ relations.toQuotient ∘ₗ relations.map = 0",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"Finsupp.module",
"Module.Relations.Quotient",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 154,
"column": 59
} | {
"line": 154,
"column": 64
} | {
"line": 156,
"column": 0
} | [
{
"pp": "A : Type u\ninst✝² : Ring A\nrelations : Relations A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nsolution : relations.Solution M\n⊢ solution.π ∘ₗ relations.map = 0",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 154,
"column": 59
} | {
"line": 154,
"column": 64
} | {
"line": 156,
"column": 0
} | [
{
"pp": "A : Type u\ninst✝² : Ring A\nrelations : Relations A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nsolution : relations.Solution M\n⊢ solution.π ∘ₗ relations.map = 0",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 154,
"column": 59
} | {
"line": 154,
"column": 64
} | {
"line": 156,
"column": 0
} | [
{
"pp": "A : Type u\ninst✝² : Ring A\nrelations : Relations A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nsolution : relations.Solution M\n⊢ solution.π ∘ₗ relations.map = 0",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 190,
"column": 94
} | {
"line": 190,
"column": 99
} | {
"line": 191,
"column": 4
} | [
{
"pp": "A : Type u\ninst✝⁴ : Ring A\nrelations : Relations A\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Module A M\nsolution : relations.Solution M\nN : Type v'\ninst✝¹ : AddCommGroup N\ninst✝ : Module A N\nf : M →ₗ[A] N\nr : relations.R\n⊢ (Finsupp.linearCombination A fun g ↦ f (solution.var g)) = f ∘ₗ so... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 331,
"column": 86
} | {
"line": 331,
"column": 91
} | {
"line": 333,
"column": 0
} | [
{
"pp": "A : Type u\ninst✝⁴ : Ring A\nrelations : Relations A\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Module A M\nsolution : relations.Solution M\nh : solution.IsPresentation\nN : Type v'\ninst✝¹ : AddCommGroup N\ninst✝ : Module A N\ns : relations.Solution N\n⊢ h.desc s ∘ₗ solution.π = s.π",
"ppTerm"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 331,
"column": 86
} | {
"line": 331,
"column": 91
} | {
"line": 333,
"column": 0
} | [
{
"pp": "A : Type u\ninst✝⁴ : Ring A\nrelations : Relations A\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Module A M\nsolution : relations.Solution M\nh : solution.IsPresentation\nN : Type v'\ninst✝¹ : AddCommGroup N\ninst✝ : Module A N\ns : relations.Solution N\n⊢ h.desc s ∘ₗ solution.π = s.π",
"ppTerm"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 331,
"column": 86
} | {
"line": 331,
"column": 91
} | {
"line": 333,
"column": 0
} | [
{
"pp": "A : Type u\ninst✝⁴ : Ring A\nrelations : Relations A\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Module A M\nsolution : relations.Solution M\nh : solution.IsPresentation\nN : Type v'\ninst✝¹ : AddCommGroup N\ninst✝ : Module A N\ns : relations.Solution N\n⊢ h.desc s ∘ₗ solution.π = s.π",
"ppTerm"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 340,
"column": 43
} | {
"line": 340,
"column": 48
} | {
"line": 342,
"column": 0
} | [
{
"pp": "A : Type u\ninst✝⁴ : Ring A\nrelations : Relations A\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Module A M\nsolution : relations.Solution M\nh : solution.IsPresentation\nN : Type v'\ninst✝¹ : AddCommGroup N\ninst✝ : Module A N\ns : relations.Solution N\n⊢ solution.postcomp (h.desc s) = s",
"ppT... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 340,
"column": 43
} | {
"line": 340,
"column": 48
} | {
"line": 342,
"column": 0
} | [
{
"pp": "A : Type u\ninst✝⁴ : Ring A\nrelations : Relations A\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Module A M\nsolution : relations.Solution M\nh : solution.IsPresentation\nN : Type v'\ninst✝¹ : AddCommGroup N\ninst✝ : Module A N\ns : relations.Solution N\n⊢ solution.postcomp (h.desc s) = s",
"ppT... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 340,
"column": 43
} | {
"line": 340,
"column": 48
} | {
"line": 342,
"column": 0
} | [
{
"pp": "A : Type u\ninst✝⁴ : Ring A\nrelations : Relations A\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Module A M\nsolution : relations.Solution M\nh : solution.IsPresentation\nN : Type v'\ninst✝¹ : AddCommGroup N\ninst✝ : Module A N\ns : relations.Solution N\n⊢ solution.postcomp (h.desc s) = s",
"ppT... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 393,
"column": 55
} | {
"line": 393,
"column": 60
} | {
"line": 394,
"column": 4
} | [
{
"pp": "A : Type u\ninst✝⁴ : Ring A\nrelations : Relations A\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Module A M\nsolution : relations.Solution M\nh : solution.IsPresentation\nN : Type v'\ninst✝¹ : AddCommGroup N\ninst✝ : Module A N\ne : M ≃ₗ[A] N\n⊢ (solution.postcomp ↑e).fromQuotient = ↑e ∘ₗ solution.f... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 411,
"column": 80
} | {
"line": 411,
"column": 85
} | {
"line": 413,
"column": 0
} | [
{
"pp": "A : Type u\ninst✝ : Ring A\nrelations : Relations A\n⊢ (ofQuotient relations).fromQuotient = LinearMap.id",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"Eq.mpr",
"Semiring.toModule",
"Finsupp.module",
"Module.Relations.Solution.ofQuo... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 411,
"column": 80
} | {
"line": 411,
"column": 85
} | {
"line": 413,
"column": 0
} | [
{
"pp": "A : Type u\ninst✝ : Ring A\nrelations : Relations A\n⊢ (ofQuotient relations).fromQuotient = LinearMap.id",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"Eq.mpr",
"Semiring.toModule",
"Finsupp.module",
"Module.Relations.Solution.ofQuo... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 411,
"column": 80
} | {
"line": 411,
"column": 85
} | {
"line": 413,
"column": 0
} | [
{
"pp": "A : Type u\ninst✝ : Ring A\nrelations : Relations A\n⊢ (ofQuotient relations).fromQuotient = LinearMap.id",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"Eq.mpr",
"Semiring.toModule",
"Finsupp.module",
"Module.Relations.Solution.ofQuo... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 464,
"column": 50
} | {
"line": 464,
"column": 55
} | {
"line": 464,
"column": 55
} | [
{
"pp": "A : Type u\ninst✝² : Ring A\nrelations : Relations A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nsolution : relations.Solution M\nh : solution.IsPresentationCore\n⊢ solution.postcomp (solution.fromQuotient ∘ₗ h.down.desc (ofQuotient relations)) = solution.postcomp LinearMap.id",
"ppTe... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 464,
"column": 50
} | {
"line": 464,
"column": 55
} | {
"line": 464,
"column": 55
} | [
{
"pp": "A : Type u\ninst✝² : Ring A\nrelations : Relations A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nsolution : relations.Solution M\nh : solution.IsPresentationCore\n⊢ solution.postcomp (solution.fromQuotient ∘ₗ h.down.desc (ofQuotient relations)) = solution.postcomp LinearMap.id",
"ppTe... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 464,
"column": 50
} | {
"line": 464,
"column": 55
} | {
"line": 464,
"column": 55
} | [
{
"pp": "A : Type u\ninst✝² : Ring A\nrelations : Relations A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nsolution : relations.Solution M\nh : solution.IsPresentationCore\n⊢ solution.postcomp (solution.fromQuotient ∘ₗ h.down.desc (ofQuotient relations)) = solution.postcomp LinearMap.id",
"ppTe... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 464,
"column": 62
} | {
"line": 464,
"column": 67
} | {
"line": 464,
"column": 67
} | [
{
"pp": "A : Type u\ninst✝² : Ring A\nrelations : Relations A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nsolution : relations.Solution M\nh : solution.IsPresentationCore\n⊢ h.down.desc (ofQuotient relations) ∘ₗ solution.fromQuotient = LinearMap.id",
"ppTerm": "?m.105",
"assigned": true,
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 464,
"column": 62
} | {
"line": 464,
"column": 67
} | {
"line": 464,
"column": 67
} | [
{
"pp": "A : Type u\ninst✝² : Ring A\nrelations : Relations A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nsolution : relations.Solution M\nh : solution.IsPresentationCore\n⊢ h.down.desc (ofQuotient relations) ∘ₗ solution.fromQuotient = LinearMap.id",
"ppTerm": "?m.105",
"assigned": true,
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 464,
"column": 62
} | {
"line": 464,
"column": 67
} | {
"line": 464,
"column": 67
} | [
{
"pp": "A : Type u\ninst✝² : Ring A\nrelations : Relations A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nsolution : relations.Solution M\nh : solution.IsPresentationCore\n⊢ h.down.desc (ofQuotient relations) ∘ₗ solution.fromQuotient = LinearMap.id",
"ppTerm": "?m.105",
"assigned": true,
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Basis | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 83
} | {
"line": 207,
"column": 0
} | [
{
"pp": "case intro\nι : Type u_1\nι' : Type u_2\nκ : Type u_3\nκ' : Type u_4\nR : Type u_5\nM : Type u_6\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nN : Type u_9\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nb : Basis ι R M\nb' : Basis ι' R M\nc : Basis κ R N\nc' : Basis κ' R N\... | [] | rw [basis_toMatrix_mul_linearMap_toMatrix, linearMap_toMatrix_mul_basis_toMatrix] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.Nondegenerate | {
"line": 195,
"column": 20
} | {
"line": 195,
"column": 25
} | {
"line": 196,
"column": 2
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nR : Type u_3\nM : Type u_4\ninst✝⁴ : Fintype ι\ninst✝³ : Finite κ\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nhv : ∀ (g : ι → R), ∑ i, g i • v i = 0 → ∀ (i : ι), g i = 0\nA : Matrix κ ι R\nhA : A.Nondegenerate\nthis✝ : Fintype κ\nw : κ → R\n... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Matrix.Nondegenerate | {
"line": 195,
"column": 20
} | {
"line": 195,
"column": 25
} | {
"line": 196,
"column": 2
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nR : Type u_3\nM : Type u_4\ninst✝⁴ : Fintype ι\ninst✝³ : Finite κ\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nhv : ∀ (g : ι → R), ∑ i, g i • v i = 0 → ∀ (i : ι), g i = 0\nA : Matrix κ ι R\nhA : A.Nondegenerate\nthis✝ : Fintype κ\nw : κ → R\n... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Nondegenerate | {
"line": 195,
"column": 20
} | {
"line": 195,
"column": 25
} | {
"line": 196,
"column": 2
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nR : Type u_3\nM : Type u_4\ninst✝⁴ : Fintype ι\ninst✝³ : Finite κ\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nhv : ∀ (g : ι → R), ∑ i, g i • v i = 0 → ∀ (i : ι), g i = 0\nA : Matrix κ ι R\nhA : A.Nondegenerate\nthis✝ : Fintype κ\nw : κ → R\n... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.IntegralDomain | {
"line": 67,
"column": 26
} | {
"line": 72,
"column": 48
} | {
"line": 74,
"column": 0
} | [
{
"pp": "R : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : GCDMonoid R\ninst✝ : Subsingleton Rˣ\na b c : R\nn : ℕ\ncp : IsCoprime a b\nh : a * b = c ^ n\n⊢ ∃ d, a = d ^ n",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Distrib.leftDistribClass"... | [] | by
refine exists_eq_pow_of_mul_eq_pow (isUnit_of_dvd_one ?_) h
obtain ⟨x, y, hxy⟩ := cp
rw [← hxy]
exact dvd_add (dvd_mul_of_dvd_right (gcd_dvd_left _ _) _)
(dvd_mul_of_dvd_right (gcd_dvd_right _ _) _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs | {
"line": 95,
"column": 2
} | {
"line": 99,
"column": 37
} | {
"line": 101,
"column": 0
} | [
{
"pp": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nR : Type v\ninst✝¹ : CommRing R\ninst✝ : Nonempty n\nr : Rˣ\ni : n\n⊢ ∃ a, det a = r",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"M... | [] | refine ⟨⟨diagonal fun j ↦ if j = i then r else 1, diagonal fun j ↦ if j = i then r⁻¹.1 else 1,
?_, ?_⟩, by simp [det]⟩
<;> simp only [diagonal_mul_diagonal, mul_ite, ite_mul, Units.mul_inv, one_mul, mul_one,
diagonal_eq_one]
<;> funext j <;> split_ifs <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 61
} | {
"line": 174,
"column": 2
} | [
{
"pp": "K : Type u_3\nV : Type u_8\ninst✝⁴ : Field K\ninst✝³ : Invertible 2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nQ : QuadraticForm K V\nhQ : LinearMap.SeparatingLeft (associated Q)\nv : Basis (Fin (finrank K V)) K V\nhv₁ : LinearMap.IsOrthoᵢ ((associatedHom K) Q) ⇑v\n⊢ ... | [
"K : Type u_3\nV : Type u_8\ninst✝⁴ : Field K\ninst✝³ : Invertible 2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nQ : QuadraticForm K V\nhQ : LinearMap.SeparatingLeft (associated Q)\nv : Basis (Fin (finrank K V)) K V\nhv₁ : LinearMap.IsOrthoᵢ ((associatedHom K) Q) ⇑v\nhv₂ : ∀ (i : F... | have hv₂ := hv₁.not_isOrtho_basis_self_of_separatingLeft hQ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 110,
"column": 98
} | {
"line": 115,
"column": 43
} | {
"line": 117,
"column": 0
} | [
{
"pp": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nR : Type v\ninst✝¹ : CommRing R\ninst✝ : Subsingleton n\n⊢ Subsingleton (SpecialLinearGroup n R)",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Matrix.SpecialLinearGroup",
"Matrix.SpecialLinearGroup.ext",
"... | [] | by
refine ⟨fun ⟨A, hA⟩ ⟨B, hB⟩ ↦ ?_⟩
ext i j
rcases isEmpty_or_nonempty n with hn | hn; · exfalso; exact IsEmpty.false i
rw [det_eq_elem_of_subsingleton _ i] at hA hB
simp only [Subsingleton.elim j i, hA, hB] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 258,
"column": 60
} | {
"line": 258,
"column": 65
} | {
"line": 258,
"column": 65
} | [
{
"pp": "n : Type u\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type v\ninst✝ : CommRing R\nA : SpecialLinearGroup n R\nhn : IsEmpty n\n⊢ A ∈ center (SpecialLinearGroup n R) → ∃ r, r ^ Fintype.card n = 1 ∧ (scalar n) r = ↑A",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 258,
"column": 60
} | {
"line": 258,
"column": 65
} | {
"line": 258,
"column": 65
} | [
{
"pp": "n : Type u\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type v\ninst✝ : CommRing R\nA : SpecialLinearGroup n R\nhn : IsEmpty n\n⊢ A ∈ center (SpecialLinearGroup n R) → ∃ r, r ^ Fintype.card n = 1 ∧ (scalar n) r = ↑A",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 258,
"column": 60
} | {
"line": 258,
"column": 65
} | {
"line": 258,
"column": 65
} | [
{
"pp": "n : Type u\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type v\ninst✝ : CommRing R\nA : SpecialLinearGroup n R\nhn : IsEmpty n\n⊢ A ∈ center (SpecialLinearGroup n R) → ∃ r, r ^ Fintype.card n = 1 ∧ (scalar n) r = ↑A",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 276,
"column": 43
} | {
"line": 276,
"column": 48
} | {
"line": 276,
"column": 48
} | [
{
"pp": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nR : Type v\ninst✝¹ : CommRing R\nS : Type u_1\ninst✝ : CommRing S\ni : n\na : ↥(rootsOfUnity (Fintype.card n) R)\n⊢ (a • 1).det = 1",
"ppTerm": "?m.98",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matrix.smul",
"U... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 276,
"column": 43
} | {
"line": 276,
"column": 48
} | {
"line": 276,
"column": 48
} | [
{
"pp": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nR : Type v\ninst✝¹ : CommRing R\nS : Type u_1\ninst✝ : CommRing S\ni : n\na : ↥(rootsOfUnity (Fintype.card n) R)\n⊢ (a • 1).det = 1",
"ppTerm": "?m.98",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matrix.smul",
"U... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 276,
"column": 43
} | {
"line": 276,
"column": 48
} | {
"line": 276,
"column": 48
} | [
{
"pp": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nR : Type v\ninst✝¹ : CommRing R\nS : Type u_1\ninst✝ : CommRing S\ni : n\na : ↥(rootsOfUnity (Fintype.card n) R)\n⊢ (a • 1).det = 1",
"ppTerm": "?m.98",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matrix.smul",
"U... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.CliffordAlgebra.Basic | {
"line": 237,
"column": 6
} | {
"line": 237,
"column": 21
} | {
"line": 238,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nQ : QuadraticForm R M\nA : Type u_4\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\nhf : ∀ (x : M), f x * f x = (algebraMap R A) (Q x)\na b : M\n⊢ f (a + b) * f (a + b) - f a * f a - f b * f b =\n (a... | [] | rw [hf, hf, hf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.CliffordAlgebra.Basic | {
"line": 237,
"column": 6
} | {
"line": 237,
"column": 21
} | {
"line": 238,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nQ : QuadraticForm R M\nA : Type u_4\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\nhf : ∀ (x : M), f x * f x = (algebraMap R A) (Q x)\na b : M\n⊢ f (a + b) * f (a + b) - f a * f a - f b * f b =\n (a... | [] | rw [hf, hf, hf] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.CliffordAlgebra.Basic | {
"line": 237,
"column": 6
} | {
"line": 237,
"column": 21
} | {
"line": 238,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nQ : QuadraticForm R M\nA : Type u_4\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\nhf : ∀ (x : M), f x * f x = (algebraMap R A) (Q x)\na b : M\n⊢ f (a + b) * f (a + b) - f a * f a - f b * f b =\n (a... | [] | rw [hf, hf, hf] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.Associator | {
"line": 96,
"column": 14
} | {
"line": 96,
"column": 53
} | {
"line": 98,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalNonAssocSemiring R\nh : Std.Associative fun x y ↦ x * y\n⊢ mulLeft₃ = mulRight₃",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"AddMonoidHom.mulLeft₃",
"HMul.hMul",
"Std.Associative.assoc",
"congrArg",
"AddMonoid.toAddZ... | [] | ext x y z; simp [Std.Associative.assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.Associator | {
"line": 96,
"column": 14
} | {
"line": 96,
"column": 53
} | {
"line": 98,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalNonAssocSemiring R\nh : Std.Associative fun x y ↦ x * y\n⊢ mulLeft₃ = mulRight₃",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"AddMonoidHom.mulLeft₃",
"HMul.hMul",
"Std.Associative.assoc",
"congrArg",
"AddMonoid.toAddZ... | [] | ext x y z; simp [Std.Associative.assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 682,
"column": 2
} | {
"line": 683,
"column": 92
} | {
"line": 684,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\ni₀ : ι\nD : ι → F\nhD : (diagonal D).det = 1\ng : ι → ι → F := fun i k ↦ if k = i then D i else if k = i₀ then (D i)⁻¹ else 1\nhg_def : g = fun i k ↦ if k = i then D i else if k = i₀ then (D i)⁻¹ else 1\n⊢ diagonal... | [
"F : Type u_1\ninst✝² : Field F\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\ni₀ : ι\nD : ι → F\nhD : (diagonal D).det = 1\ng : ι → ι → F := fun i k ↦ if k = i then D i else if k = i₀ then (D i)⁻¹ else 1\nhg_def : g = fun i k ↦ if k = i then D i else if k = i₀ then (D i)⁻¹ else 1\n⊢ diagonal D =\n {i... | rw [Finset.noncommProd_congr (s₂ := {i | i ≠ i₀}) rfl (fun i hi ↦
(dif_pos (Finset.mem_filter.1 hi).2 : _ = (diag2n (Finset.mem_filter.1 hi).2 _ _).1))] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 686,
"column": 2
} | {
"line": 686,
"column": 97
} | {
"line": 687,
"column": 2
} | [
{
"pp": "case convert_3\nF : Type u_1\ninst✝² : Field F\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\ni₀ : ι\nD : ι → F\nhD : (diagonal D).det = 1\ng : ι → ι → F := fun i k ↦ if k = i then D i else if k = i₀ then (D i)⁻¹ else 1\nhg_def : g = fun i k ↦ if k = i then D i else if k = i₀ then (D i)⁻¹ el... | [
"case convert_3\nF : Type u_1\ninst✝² : Field F\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\ni₀ : ι\nD : ι → F\nhD : (diagonal D).det = 1\ng : ι → ι → F := fun i k ↦ if k = i then D i else if k = i₀ then (D i)⁻¹ else 1\nhg_def : g = fun i k ↦ if k = i then D i else if k = i₀ then (D i)⁻¹ else 1\n⊢ (dia... | rw [← Finset.map_noncommProd _ _ (fun _ _ _ _ _ ↦ Commute.all _ _), Finset.noncommProd_eq_prod] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 77,
"column": 75
} | {
"line": 77,
"column": 80
} | {
"line": 79,
"column": 0
} | [
{
"pp": "R₁ : Type u_2\nS₁ : Type u_3\nR₂ : Type u_4\nS₂ : Type u_5\nN₂ : Type u_10\nn : Type u_11\nm : Type u_12\ninst✝¹¹ : Semiring R₁\ninst✝¹⁰ : Semiring S₁\ninst✝⁹ : Semiring R₂\ninst✝⁸ : Semiring S₂\ninst✝⁷ : AddCommMonoid N₂\ninst✝⁶ : Module S₁ N₂\ninst✝⁵ : Module S₂ N₂\ninst✝⁴ : SMulCommClass S₂ S₁ N₂\ni... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 77,
"column": 75
} | {
"line": 77,
"column": 80
} | {
"line": 79,
"column": 0
} | [
{
"pp": "R₁ : Type u_2\nS₁ : Type u_3\nR₂ : Type u_4\nS₂ : Type u_5\nN₂ : Type u_10\nn : Type u_11\nm : Type u_12\ninst✝¹¹ : Semiring R₁\ninst✝¹⁰ : Semiring S₁\ninst✝⁹ : Semiring R₂\ninst✝⁸ : Semiring S₂\ninst✝⁷ : AddCommMonoid N₂\ninst✝⁶ : Module S₁ N₂\ninst✝⁵ : Module S₂ N₂\ninst✝⁴ : SMulCommClass S₂ S₁ N₂\ni... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 77,
"column": 75
} | {
"line": 77,
"column": 80
} | {
"line": 79,
"column": 0
} | [
{
"pp": "R₁ : Type u_2\nS₁ : Type u_3\nR₂ : Type u_4\nS₂ : Type u_5\nN₂ : Type u_10\nn : Type u_11\nm : Type u_12\ninst✝¹¹ : Semiring R₁\ninst✝¹⁰ : Semiring S₁\ninst✝⁹ : Semiring R₂\ninst✝⁸ : Semiring S₂\ninst✝⁷ : AddCommMonoid N₂\ninst✝⁶ : Module S₁ N₂\ninst✝⁵ : Module S₂ N₂\ninst✝⁴ : SMulCommClass S₂ S₁ N₂\ni... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 841,
"column": 4
} | {
"line": 841,
"column": 26
} | {
"line": 842,
"column": 4
} | [
{
"pp": "case succ\nn : ℕ\nh : ↑(T ^ ↑n) = !![1, ↑n; 0, 1]\n⊢ !![1 * 1 + ↑n * 0, 1 * 1 + ↑n * 1; 0 * 1 + 1 * 0, 0 * 1 + 1 * 1] = !![1, ↑n + 1; 0, 1]",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemir... | [
"case succ\nn : ℕ\nh : ↑(T ^ ↑n) = !![1, ↑n; 0, 1]\n⊢ 1 * 1 + ↑n * 1 = ↑n + 1"
] | congrm !![_, ?_; _, _] | Mathlib.Tactic._aux_Mathlib_Tactic_CongrM___elabRules_Mathlib_Tactic_congrM_1 | Mathlib.Tactic.congrM |
Mathlib.RingTheory.GradedAlgebra.Basic | {
"line": 130,
"column": 83
} | {
"line": 132,
"column": 57
} | {
"line": 134,
"column": 0
} | [
{
"pp": "ι : Type u_1\nA : Type u_3\nσ : Type u_4\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring A\ninst✝³ : SetLike σ A\ninst✝² : AddSubmonoidClass σ A\n𝒜 : ι → σ\ni j : ι\ninst✝¹ : AddLeftCancelMonoid ι\ninst✝ : GradedRing 𝒜\na b : A\na_mem : a ∈ 𝒜 i\n⊢ ↑(((decompose 𝒜) (a * b)) (i + j)) = a * ↑(((decompose �... | [] | by
lift a to 𝒜 i using a_mem
rw [decompose_mul, decompose_coe, coe_of_mul_apply_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.GradedAlgebra.Basic | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 27
} | {
"line": 157,
"column": 2
} | [
{
"pp": "ι : Type u_1\nA : Type u_3\nσ : Type u_4\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring A\ninst✝³ : SetLike σ A\ninst✝² : AddSubmonoidClass σ A\n𝒜 : ι → σ\ni : ι\ninst✝¹ : AddMonoid ι\ninst✝ : GradedRing 𝒜\na b : A\nb_mem : b ∈ 𝒜 0\n⊢ ↑(((decompose 𝒜) (a * b)) i) = ↑(((decompose 𝒜) a) i) * b",
"pp... | [
"ι : Type u_1\nA : Type u_3\nσ : Type u_4\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring A\ninst✝³ : SetLike σ A\ninst✝² : AddSubmonoidClass σ A\n𝒜 : ι → σ\ni : ι\ninst✝¹ : AddMonoid ι\ninst✝ : GradedRing 𝒜\na : A\nb : ↥(𝒜 0)\n⊢ ↑(((decompose 𝒜) (a * ↑b)) i) = ↑(((decompose 𝒜) a) i) * ↑b"
] | lift b to 𝒜 0 using b_mem | Mathlib.Tactic._aux_Mathlib_Tactic_Lift___elabRules_Mathlib_Tactic_lift_1 | Mathlib.Tactic.lift |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 691,
"column": 2
} | {
"line": 692,
"column": 75
} | {
"line": 694,
"column": 0
} | [
{
"pp": "R : Type u_1\nn : Type u_11\nm : Type u_12\ninst✝⁴ : CommRing R\ninst✝³ : DecidableEq m\ninst✝² : Fintype m\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix m n R\n⊢ ((toLinearMap₂' R) M).SeparatingLeft ↔ M.SeparatingLeft",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
... | [] | refine ⟨fun h ↦ separatingLeft_def.mpr ?_, SeparatingLeft.toLinearMap₂'⟩
exact fun v hv => h v fun w => (M.toLinearMap₂'_apply' _ _).trans <| hv w | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 691,
"column": 2
} | {
"line": 692,
"column": 75
} | {
"line": 694,
"column": 0
} | [
{
"pp": "R : Type u_1\nn : Type u_11\nm : Type u_12\ninst✝⁴ : CommRing R\ninst✝³ : DecidableEq m\ninst✝² : Fintype m\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix m n R\n⊢ ((toLinearMap₂' R) M).SeparatingLeft ↔ M.SeparatingLeft",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
... | [] | refine ⟨fun h ↦ separatingLeft_def.mpr ?_, SeparatingLeft.toLinearMap₂'⟩
exact fun v hv => h v fun w => (M.toLinearMap₂'_apply' _ _).trans <| hv w | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.PowersetCard | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 46
} | {
"line": 62,
"column": 0
} | [
{
"pp": "α : Type u_1\nn : ℕ\ns : ↑(powersetCard α n)\n⊢ (↑s).ncard = n",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.powersetCard.coe_coe",
"Finset",
"Membership.mem",
"Set.Elem",
"id",
"SetLike.coe",
"Finse... | [] | rw [← coe_coe, Set.ncard_coe_finset, s.prop] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Set.PowersetCard | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 46
} | {
"line": 62,
"column": 0
} | [
{
"pp": "α : Type u_1\nn : ℕ\ns : ↑(powersetCard α n)\n⊢ (↑s).ncard = n",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.powersetCard.coe_coe",
"Finset",
"Membership.mem",
"Set.Elem",
"id",
"SetLike.coe",
"Finse... | [] | rw [← coe_coe, Set.ncard_coe_finset, s.prop] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.PowersetCard | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 46
} | {
"line": 62,
"column": 0
} | [
{
"pp": "α : Type u_1\nn : ℕ\ns : ↑(powersetCard α n)\n⊢ (↑s).ncard = n",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.powersetCard.coe_coe",
"Finset",
"Membership.mem",
"Set.Elem",
"id",
"SetLike.coe",
"Finse... | [] | rw [← coe_coe, Set.ncard_coe_finset, s.prop] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.CliffordAlgebra.Grading | {
"line": 125,
"column": 2
} | {
"line": 133,
"column": 94
} | {
"line": 135,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\n⊢ ⨆ i, (ι Q).range ^ i = ⊤",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"CliffordAlgebra.ι",
"RingHomSurjective.id... | [] | rw [← (DirectSum.Decomposition.isInternal (evenOdd Q)).submodule_iSup_eq_top, eq_comm]
calc
-- Porting note: needs extra annotations, no longer unifies against the goal in the face of
-- ambiguity
⨆ (i : ZMod 2) (j : { n : ℕ // ↑n = i }), LinearMap.range (ι Q) ^ (j : ℕ) =
⨆ i : Σ i : ZMod 2, { n :... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.CliffordAlgebra.Grading | {
"line": 125,
"column": 2
} | {
"line": 133,
"column": 94
} | {
"line": 135,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\n⊢ ⨆ i, (ι Q).range ^ i = ⊤",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"CliffordAlgebra.ι",
"RingHomSurjective.id... | [] | rw [← (DirectSum.Decomposition.isInternal (evenOdd Q)).submodule_iSup_eq_top, eq_comm]
calc
-- Porting note: needs extra annotations, no longer unifies against the goal in the face of
-- ambiguity
⨆ (i : ZMod 2) (j : { n : ℕ // ↑n = i }), LinearMap.range (ι Q) ^ (j : ℕ) =
⨆ i : Σ i : ZMod 2, { n :... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.PowersetCard | {
"line": 235,
"column": 4
} | {
"line": 235,
"column": 9
} | {
"line": 236,
"column": 2
} | [
{
"pp": "α : Type u_1\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : Infinite α\nfinite : Finite ↑(powersetCard α n)\nthis : ⋃₀ (SetLike.coe '' powersetCard α n) = univ\n⊢ ∀ t ∈ SetLike.coe '' powersetCard α n, t.Finite",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.PowersetCard | {
"line": 267,
"column": 6
} | {
"line": 267,
"column": 11
} | {
"line": 268,
"column": 2
} | [
{
"pp": "case inl.inr\nα : Type u_1\nn : ℕ\nh1 : 0 < n\nval✝ : Fintype α\nh2 : n < Nat.card α\nh : n = 0 ∨ Nat.card α = n\n⊢ False",
"ppTerm": "?inl.inr",
"assigned": true,
"usedConstants": [
"False",
"congrArg",
"False.elim",
"lt_self_iff_false._simp_1",
"Eq.mp",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Multilinear.Curry | {
"line": 332,
"column": 17
} | {
"line": 332,
"column": 22
} | {
"line": 333,
"column": 2
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nι' : Type uι'\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁸ : CommSemiring R\ninst✝⁷ : (i : Fin n.succ) → AddCommMonoid (M i)\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : (i : Fin n.succ) → M... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Multilinear.Curry | {
"line": 332,
"column": 17
} | {
"line": 332,
"column": 22
} | {
"line": 333,
"column": 2
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nι' : Type uι'\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁸ : CommSemiring R\ninst✝⁷ : (i : Fin n.succ) → AddCommMonoid (M i)\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : (i : Fin n.succ) → M... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Multilinear.Curry | {
"line": 332,
"column": 17
} | {
"line": 332,
"column": 22
} | {
"line": 333,
"column": 2
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nι' : Type uι'\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁸ : CommSemiring R\ninst✝⁷ : (i : Fin n.succ) → AddCommMonoid (M i)\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : (i : Fin n.succ) → M... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Multilinear.Curry | {
"line": 333,
"column": 18
} | {
"line": 333,
"column": 23
} | {
"line": 335,
"column": 0
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nι' : Type uι'\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁸ : CommSemiring R\ninst✝⁷ : (i : Fin n.succ) → AddCommMonoid (M i)\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : (i : Fin n.succ) → M... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Multilinear.Curry | {
"line": 333,
"column": 18
} | {
"line": 333,
"column": 23
} | {
"line": 335,
"column": 0
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nι' : Type uι'\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁸ : CommSemiring R\ninst✝⁷ : (i : Fin n.succ) → AddCommMonoid (M i)\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : (i : Fin n.succ) → M... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Multilinear.Curry | {
"line": 333,
"column": 18
} | {
"line": 333,
"column": 23
} | {
"line": 335,
"column": 0
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nι' : Type uι'\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁸ : CommSemiring R\ninst✝⁷ : (i : Fin n.succ) → AddCommMonoid (M i)\ninst✝⁶ : AddCommMonoid M'\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : (i : Fin n.succ) → M... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.ExteriorAlgebra.Basic | {
"line": 276,
"column": 6
} | {
"line": 276,
"column": 15
} | {
"line": 277,
"column": 6
} | [
{
"pp": "R✝ : Type u1\ninst✝⁵ : CommRing R✝\nM✝ : Type u2\ninst✝⁴ : AddCommGroup M✝\ninst✝³ : Module R✝ M✝\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nn : ℕ\nf : Fin n → M\nx y : Fin n\nhfxy : f x = f y\nhxy : x ≠ y\nh : x < y\n⊢ (List.ofFn fun i ↦ (ι R) (f i)).p... | [
"R✝ : Type u1\ninst✝⁵ : CommRing R✝\nM✝ : Type u2\ninst✝⁴ : AddCommGroup M✝\ninst✝³ : Module R✝ M✝\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nn : ℕ\nf : Fin n → M\nx y : Fin n\nhfxy : f x = f y\nh : x < y\n⊢ (List.ofFn fun i ↦ (ι R) (f i)).prod = 0"
] | clear hxy | Lean.Elab.Tactic.evalClear | Lean.Parser.Tactic.clear |
Mathlib.LinearAlgebra.ExteriorAlgebra.Basic | {
"line": 387,
"column": 4
} | {
"line": 387,
"column": 9
} | {
"line": 388,
"column": 2
} | [
{
"pp": "case pos\nR : Type u1\ninst✝³ : CommRing R\nM : Type u2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nm n : ℕ\nI : Type u_1\ninst✝ : LinearOrder I\nv : I → M\ns : ↑(powersetCard I m)\nt : ↑(powersetCard I n)\nh : Disjoint ↑s ↑t\ni : Fin (m + n)\ne : Fin (m + n) ≃o ↥↑(disjUnion h) := orderIsoOfFin (dis... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.ExteriorAlgebra.Basic | {
"line": 390,
"column": 4
} | {
"line": 390,
"column": 9
} | {
"line": 392,
"column": 0
} | [
{
"pp": "case neg\nR : Type u1\ninst✝³ : CommRing R\nM : Type u2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nm n : ℕ\nI : Type u_1\ninst✝ : LinearOrder I\nv : I → M\ns : ↑(powersetCard I m)\nt : ↑(powersetCard I n)\nh : Disjoint ↑s ↑t\ni : Fin (m + n)\ne : Fin (m + n) ≃o ↥↑(disjUnion h) := orderIsoOfFin (dis... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.ExteriorPower.Basic | {
"line": 98,
"column": 6
} | {
"line": 100,
"column": 11
} | {
"line": 101,
"column": 4
} | [
{
"pp": "case a.refine_1\nR : Type u\ninst✝² : CommRing R\nn : ℕ\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns : Set M\nhs : span R s = ⊤\nf : Fin n → ↑(⇑(ExteriorAlgebra.ι R) '' s)\nhx : (List.ofFn fun i ↦ ↑(f i)).prod ∈ (⇑(ExteriorAlgebra.ι R) '' s) ^ n\n⊢ ⇑ExteriorAlgebra.ιInv ∘ Subtype.val ∘... | [] | rw [Set.mem_setOf_eq, Set.range_comp, Set.image_subset_iff]
apply Subset.trans ?_ (s.image_subset_preimage_of_inverse ExteriorAlgebra.ι_leftInverse)
grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.ExteriorPower.Basic | {
"line": 98,
"column": 6
} | {
"line": 100,
"column": 11
} | {
"line": 101,
"column": 4
} | [
{
"pp": "case a.refine_1\nR : Type u\ninst✝² : CommRing R\nn : ℕ\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns : Set M\nhs : span R s = ⊤\nf : Fin n → ↑(⇑(ExteriorAlgebra.ι R) '' s)\nhx : (List.ofFn fun i ↦ ↑(f i)).prod ∈ (⇑(ExteriorAlgebra.ι R) '' s) ^ n\n⊢ ⇑ExteriorAlgebra.ιInv ∘ Subtype.val ∘... | [] | rw [Set.mem_setOf_eq, Set.range_comp, Set.image_subset_iff]
apply Subset.trans ?_ (s.image_subset_preimage_of_inverse ExteriorAlgebra.ι_leftInverse)
grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.ExteriorPower.Basic | {
"line": 187,
"column": 24
} | {
"line": 187,
"column": 29
} | {
"line": 188,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝⁸ : CommRing R\nn : ℕ\nM : Type u_1\nN : Type u_2\nN' : Type u_3\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : AddCommGroup N'\ninst✝² : Module R N'\nN✝ : Type ?u.42\ninst✝¹ : AddCommGroup N✝\ninst✝ : Module R N✝\ns : (relations R... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.ExteriorPower.Basic | {
"line": 187,
"column": 24
} | {
"line": 187,
"column": 29
} | {
"line": 188,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝⁸ : CommRing R\nn : ℕ\nM : Type u_1\nN : Type u_2\nN' : Type u_3\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : AddCommGroup N'\ninst✝² : Module R N'\nN✝ : Type ?u.42\ninst✝¹ : AddCommGroup N✝\ninst✝ : Module R N✝\ns : (relations R... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.ExteriorPower.Basic | {
"line": 187,
"column": 24
} | {
"line": 187,
"column": 29
} | {
"line": 188,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝⁸ : CommRing R\nn : ℕ\nM : Type u_1\nN : Type u_2\nN' : Type u_3\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : AddCommGroup N'\ninst✝² : Module R N'\nN✝ : Type ?u.42\ninst✝¹ : AddCommGroup N✝\ninst✝ : Module R N✝\ns : (relations R... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.ExteriorPower.Basic | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 7
} | {
"line": 296,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nn : ℕ\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\n⊢ map n LinearMap.id = LinearMap.id",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"AlternatingMap",
"Submodule",
"Semiring.toModule",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.