module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape | {
"line": 122,
"column": 8
} | {
"line": 122,
"column": 19
} | {
"line": 122,
"column": 20
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nP : ObjectProperty C\nJ : Type u'\ninst✝¹ : Category.{v', u'} J\nJ' : Type u''\ninst✝ : Category.{v'', u''} J'\nX : C\np : P.LimitOfShape J X\nX✝ Y✝ : J\nf : X✝ ⟶ Y✝\n⊢ (StructuredArrow.mk (p.π.app X✝)).hom ≫ P.... | [
"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nP : ObjectProperty C\nJ : Type u'\ninst✝¹ : Category.{v', u'} J\nJ' : Type u''\ninst✝ : Category.{v'', u''} J'\nX : C\np : P.LimitOfShape J X\nX✝ Y✝ : J\nf : X✝ ⟶ Y✝\n⊢ p.π.app X✝ ≫ p.diag.map f = p.π.app Y✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape | {
"line": 272,
"column": 2
} | {
"line": 272,
"column": 13
} | {
"line": 272,
"column": 14
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nJ : Type u'\ninst✝¹ : Category.{v', u'} J\nP : ObjectProperty D\ninst✝ : P.IsClosedUnderIsomorphisms\ne : C ≌ D\nH : (P.inverseImage e.functor).IsClosedUnderLimitsOfShape J\nX : D\n⊢ P X ↔ (P.inverseImage e.func... | [
"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nJ : Type u'\ninst✝¹ : Category.{v', u'} J\nP : ObjectProperty D\ninst✝ : P.IsClosedUnderIsomorphisms\ne : C ≌ D\nH : (P.inverseImage e.functor).IsClosedUnderLimitsOfShape J\nX : D\n⊢ P X ↔ P (e.functor.obj (e.inverse.obj X)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.ObjectProperty.ColimitsOfShape | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 13
} | {
"line": 271,
"column": 14
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nJ : Type u'\ninst✝¹ : Category.{v', u'} J\nP : ObjectProperty D\ninst✝ : P.IsClosedUnderIsomorphisms\ne : C ≌ D\nH : (P.inverseImage e.functor).IsClosedUnderColimitsOfShape J\nX : D\n⊢ P X ↔ (P.inverseImage e.fu... | [
"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nJ : Type u'\ninst✝¹ : Category.{v', u'} J\nP : ObjectProperty D\ninst✝ : P.IsClosedUnderIsomorphisms\ne : C ≌ D\nH : (P.inverseImage e.functor).IsClosedUnderColimitsOfShape J\nX : D\n⊢ P X ↔ P (e.functor.obj (e.inverse.obj ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Limits | {
"line": 831,
"column": 10
} | {
"line": 831,
"column": 55
} | {
"line": 831,
"column": 56
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasPullbacks C\nP : MorphismProperty C\ninst✝² : P.IsStableUnderComposition\ninst✝¹ : P.RespectsIso\ninst✝ : P.IsStableUnderBaseChange\nX✝ Y✝ Z✝ : C\nx✝¹ : X✝ ⟶ Y✝\nx✝ : Y✝ ⟶ Z✝\nh₁ : P.diagonal x✝¹\nh₂ : P.diagonal x✝\n⊢ P\n ((pullbackDiagonalMapIdIs... | [
"C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasPullbacks C\nP : MorphismProperty C\ninst✝² : P.IsStableUnderComposition\ninst✝¹ : P.RespectsIso\ninst✝ : P.IsStableUnderBaseChange\nX✝ Y✝ Z✝ : C\nx✝¹ : X✝ ⟶ Y✝\nx✝ : Y✝ ⟶ Z✝\nh₁ : P.diagonal x✝¹\nh₂ : P.diagonal x✝\n⊢ P (pullback.snd (pullback.diagonal x✝) (pull... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Limits | {
"line": 829,
"column": 4
} | {
"line": 831,
"column": 78
} | {
"line": 833,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasPullbacks C\nP : MorphismProperty C\ninst✝² : P.IsStableUnderComposition\ninst✝¹ : P.RespectsIso\ninst✝ : P.IsStableUnderBaseChange\nX✝ Y✝ Z✝ : C\nx✝¹ : X✝ ⟶ Y✝\nx✝ : Y✝ ⟶ Z✝\nh₁ : P.diagonal x✝¹\nh₂ : P.diagonal x✝\n⊢ P.diagonal (x✝¹ ≫ x✝)",
"ppT... | [] | rw [diagonal_iff, pullback.diagonal_comp]
exact P.comp_mem _ _ h₁
(by simpa only [cancel_left_of_respectsIso] using P.pullback_snd _ _ h₂) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.MorphismProperty.Limits | {
"line": 829,
"column": 4
} | {
"line": 831,
"column": 78
} | {
"line": 833,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasPullbacks C\nP : MorphismProperty C\ninst✝² : P.IsStableUnderComposition\ninst✝¹ : P.RespectsIso\ninst✝ : P.IsStableUnderBaseChange\nX✝ Y✝ Z✝ : C\nx✝¹ : X✝ ⟶ Y✝\nx✝ : Y✝ ⟶ Z✝\nh₁ : P.diagonal x✝¹\nh₂ : P.diagonal x✝\n⊢ P.diagonal (x✝¹ ≫ x✝)",
"ppT... | [] | rw [diagonal_iff, pullback.diagonal_comp]
exact P.comp_mem _ _ h₁
(by simpa only [cancel_left_of_respectsIso] using P.pullback_snd _ _ h₂) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.MorphismProperty.Limits | {
"line": 858,
"column": 57
} | {
"line": 858,
"column": 68
} | {
"line": 858,
"column": 69
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasPullbacks C\nP : MorphismProperty C\ninst✝² : P.IsStableUnderBaseChange\ninst✝¹ : P.IsMultiplicative\nQ : MorphismProperty C\ninst✝ : Q.IsStableUnderBaseChange\nhP : P.HasOfPostcompProperty Q\nX Y : C\nf : X ⟶ Y\nhf : Q f\n⊢ P (pullback.diagonal f ≫ p... | [
"C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasPullbacks C\nP : MorphismProperty C\ninst✝² : P.IsStableUnderBaseChange\ninst✝¹ : P.IsMultiplicative\nQ : MorphismProperty C\ninst✝ : Q.IsStableUnderBaseChange\nhP : P.HasOfPostcompProperty Q\nX Y : C\nf : X ⟶ Y\nhf : Q f\n⊢ P (𝟙 X)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Limits | {
"line": 881,
"column": 10
} | {
"line": 881,
"column": 67
} | {
"line": 881,
"column": 68
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nP : MorphismProperty C\nX Y Z : C\ne : X ≅ Y\nf : Y ⟶ Z\nhf : P.universally f\nX' Z' : C\ni₁ : X' ⟶ X\ni₂ : Z' ⟶ Z\nf' : X' ⟶ Z'\nH : IsPullback f' i₁ i₂ (e.hom ≫ f)\nthis : IsPullback (𝟙 X') (i₁ ≫ e.hom) i₁ e.inv\n⊢ IsPullback f' ?m.112 ?m.113 f",
"ppTerm": ... | [
"C : Type u\ninst✝ : Category.{v, u} C\nP : MorphismProperty C\nX Y Z : C\ne : X ≅ Y\nf : Y ⟶ Z\nhf : P.universally f\nX' Z' : C\ni₁ : X' ⟶ X\ni₂ : Z' ⟶ Z\nf' : X' ⟶ Z'\nH : IsPullback f' i₁ i₂ (e.hom ≫ f)\nthis : IsPullback (𝟙 X') (i₁ ≫ e.hom) i₁ e.inv\n⊢ IsPullback f' ?m.112 ?m.113 f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Limits | {
"line": 885,
"column": 23
} | {
"line": 886,
"column": 47
} | {
"line": 886,
"column": 48
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nP : MorphismProperty C\nX Y Z : C\ne : Y ≅ Z\nf : X ⟶ Y\nhf : P.universally f\nX' Z' : C\ni₁ : X' ⟶ X\ni₂ : Z' ⟶ Z\nf' : X' ⟶ Z'\nH : IsPullback f' i₁ i₂ (f ≫ e.hom)\nthis : IsPullback (𝟙 Z') i₂ (i₂ ≫ e.inv) e.inv\n⊢ IsPullback f' ?m.217 ?m.218 f",
"ppTerm": ... | [
"C : Type u\ninst✝ : Category.{v, u} C\nP : MorphismProperty C\nX Y Z : C\ne : Y ≅ Z\nf : X ⟶ Y\nhf : P.universally f\nX' Z' : C\ni₁ : X' ⟶ X\ni₂ : Z' ⟶ Z\nf' : X' ⟶ Z'\nH : IsPullback f' i₁ i₂ (f ≫ e.hom)\nthis : IsPullback (𝟙 Z') i₂ (i₂ ≫ e.inv) e.inv\n⊢ IsPullback f' ?m.217 ?m.218 f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Limits | {
"line": 898,
"column": 4
} | {
"line": 898,
"column": 90
} | {
"line": 900,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : P.IsStableUnderComposition\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : P.universally f\nhg : P.universally g\nX' Z' : C\ni₁ : X' ⟶ X\ni₂ : Z' ⟶ Z\nf' : X' ⟶ Z'\nH✝ : IsPullback f' i₁ i₂ (f ≫ g)\nH : IsPullback (pull... | [] | exact hf _ _ _ (H.of_right (pullback.lift_snd _ _ _) (IsPullback.of_hasPullback i₂ g)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Monoidal.Cartesian.Basic | {
"line": 218,
"column": 52
} | {
"line": 218,
"column": 63
} | {
"line": 218,
"column": 64
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX₁ X₂ X₃ Y₁ Y₂ Y₃ Z₁ Z₂ : C\nthis : MonoidalCategoryStruct C :=\n { tensorObj := ofChosenFiniteProducts.tensorObj ℬ,\n whiskerLeft := fun X {x x_1} g ↦ ofChosenFiniteProducts.tensorHom ℬ (... | [
"C : Type u\ninst✝ : Category.{v, u} C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX₁ X₂ X₃ Y₁ Y₂ Y₃ Z₁ Z₂ : C\nthis : MonoidalCategoryStruct C :=\n { tensorObj := ofChosenFiniteProducts.tensorObj ℬ,\n whiskerLeft := fun X {x x_1} g ↦ ofChosenFiniteProducts.tensorHom ℬ (𝟙 X) g,\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.Cartesian.Basic | {
"line": 218,
"column": 72
} | {
"line": 218,
"column": 83
} | {
"line": 218,
"column": 84
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX₁ X₂ X₃ Y₁ Y₂ Y₃ Z₁ Z₂ : C\nthis : MonoidalCategoryStruct C :=\n { tensorObj := ofChosenFiniteProducts.tensorObj ℬ,\n whiskerLeft := fun X {x x_1} g ↦ ofChosenFiniteProducts.tensorHom ℬ (... | [
"C : Type u\ninst✝ : Category.{v, u} C\n𝒯 : LimitCone (Functor.empty C)\nℬ : (X Y : C) → LimitCone (pair X Y)\nX₁ X₂ X₃ Y₁ Y₂ Y₃ Z₁ Z₂ : C\nthis : MonoidalCategoryStruct C :=\n { tensorObj := ofChosenFiniteProducts.tensorObj ℬ,\n whiskerLeft := fun X {x x_1} g ↦ ofChosenFiniteProducts.tensorHom ℬ (𝟙 X) g,\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HopfAlgebra.Basic | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 45
} | {
"line": 99,
"column": 46
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\n⊢ (antipode R) 1 = 1",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\n⊢ (antipode R) 1 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HopfAlgebra.Basic | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 34
} | {
"line": 106,
"column": 35
} | [
{
"pp": "R : Type u\nA : Type v\nι : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\na : A\nrepr : Coalgebra.Repr R a ι\n⊢ ∑ i ∈ repr.index, (antipode R) (repr.left i) * repr.right i = (algebraMap R A) (counit a)",
"ppTerm": "?m.58",
"assigned": false,
"usedConstants... | [
"R : Type u\nA : Type v\nι : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\na : A\nrepr : Coalgebra.Repr R a ι\n⊢ ∑ i ∈ repr.index, (antipode R) (repr.left i) * repr.right i = (algebraMap R A) (counit a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HopfAlgebra.Basic | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 34
} | {
"line": 111,
"column": 35
} | [
{
"pp": "R : Type u\nA : Type v\nι : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\na : A\nrepr : Coalgebra.Repr R a ι\n⊢ ∑ i ∈ repr.index, repr.left i * (antipode R) (repr.right i) = (algebraMap R A) (counit a)",
"ppTerm": "?m.58",
"assigned": false,
"usedConstants... | [
"R : Type u\nA : Type v\nι : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\na : A\nrepr : Coalgebra.Repr R a ι\n⊢ ∑ i ∈ repr.index, repr.left i * (antipode R) (repr.right i) = (algebraMap R A) (counit a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HopfAlgebra.Basic | {
"line": 128,
"column": 23
} | {
"line": 128,
"column": 34
} | {
"line": 128,
"column": 35
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\na : A\n⊢ counit (∑ i ∈ (ℛ R a).index, (ℛ R a).left i * (antipode R) ((ℛ R a).right i)) = counit a",
"ppTerm": "?m.115",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Coalgebra.toCoal... | [
"R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\na : A\n⊢ ∑ x ∈ (ℛ R a).index, counit ((ℛ R a).left x) * counit ((antipode R) ((ℛ R a).right x)) = counit a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.Cartesian.Basic | {
"line": 934,
"column": 52
} | {
"line": 934,
"column": 63
} | {
"line": 934,
"column": 64
} | [
{
"pp": "C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\ninst✝⁴ : CartesianMonoidalCategory C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\ninst✝² : CartesianMonoidalCategory D\nF : C ⥤ D\ninst✝¹ : F.Monoidal\ninst✝ : PreservesFiniteProducts F\nX Y : C\n⊢ (μIso F X Y).symm.hom = (prodComparisonIso F X Y).hom",
"pp... | [
"C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\ninst✝⁴ : CartesianMonoidalCategory C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\ninst✝² : CartesianMonoidalCategory D\nF : C ⥤ D\ninst✝¹ : F.Monoidal\ninst✝ : PreservesFiniteProducts F\nX Y : C\n⊢ δ F X Y = CartesianMonoidalCategory.prodComparison F X Y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Adjunction.Parametrized | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 42
} | {
"line": 84,
"column": 43
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\ninst✝² : Category.{v₁, u₁} C₁\ninst✝¹ : Category.{v₂, u₂} C₂\ninst✝ : Category.{v₃, u₃} C₃\nF : C₁ ⥤ C₂ ⥤ C₃\nG : C₁ᵒᵖ ⥤ C₃ ⥤ C₂\nadj : (X₁ : C₁) → F.obj X₁ ⊣ G.obj (op X₁)\nh :\n ∀ {X₁ Y₁ : C₁} (f : X₁ ⟶ Y₁) {X₂ : C₂} {X₃ : C₃} (g : (F.obj Y₁).obj X₂ ⟶ X₃),\n... | [
"C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\ninst✝² : Category.{v₁, u₁} C₁\ninst✝¹ : Category.{v₂, u₂} C₂\ninst✝ : Category.{v₃, u₃} C₃\nF : C₁ ⥤ C₂ ⥤ C₃\nG : C₁ᵒᵖ ⥤ C₃ ⥤ C₂\nadj : (X₁ : C₁) → F.obj X₁ ⊣ G.obj (op X₁)\nh :\n ∀ {X₁ Y₁ : C₁} (f : X₁ ⟶ Y₁) {X₂ : C₂} {X₃ : C₃} (g : (F.obj Y₁).obj X₂ ⟶ X₃),\n ((adj X₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Adjunction.Parametrized | {
"line": 121,
"column": 27
} | {
"line": 121,
"column": 60
} | {
"line": 121,
"column": 60
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\ninst✝² : Category.{v₁, u₁} C₁\ninst✝¹ : Category.{v₂, u₂} C₂\ninst✝ : Category.{v₃, u₃} C₃\nF : C₁ ⥤ C₂ ⥤ C₃\nG : C₁ᵒᵖ ⥤ C₃ ⥤ C₂\nadj₂ : F ⊣₂ G\nX₁ Y₁ : C₁\nX₂ : C₂\nX₃ : C₃\nf₁ : X₁ ⟶ Y₁\ng : X₂ ⟶ (G.obj (op Y₁)).obj X₃\n⊢ adj₂.homEquiv (adj₂.homEquiv.symm (g ... | [] | by simp [homEquiv_naturality_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.HopfAlgebra.Convolution | {
"line": 45,
"column": 2
} | {
"line": 46,
"column": 11
} | {
"line": 47,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\n⊢ toConv (antipode R ∘ₗ LinearMap.mul' R A ∘ₗ ↑(TensorProduct.comm R A A)) =\n toConv (LinearMap.mul' R A ∘ₗ (antipode R ⊗ₘ antipode R))",
"ppTerm": "?m.132",
"assigned": true,
"usedConstan... | [
"case hba\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\na b : A\n⊢ ((AlgebraTensorModule.curry\n (toConv (antipode R ∘ₗ LinearMap.mul' R A ∘ₗ ↑(TensorProduct.comm R A A)) *\n toConv (LinearMap.mul' R A ∘ₗ ↑(TensorProduct.comm R A A))).ofCon... | apply left_inv_eq_right_inv (a := toConv <| LinearMap.mul' R A ∘ₗ TensorProduct.comm R A A) <;>
ext a b | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Monoidal.Grp | {
"line": 196,
"column": 23
} | {
"line": 196,
"column": 59
} | {
"line": 196,
"column": 59
} | [
{
"pp": "case refine_1\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : CartesianMonoidalCategory C\nA B : C\ninst✝ : GrpObj B\nf g h : A ⟶ B\n⊢ f = lift (g ≫ ι) h ≫ μ → lift g f ≫ μ = h",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"CategoryTheory.GrpObj.inv",
"Categ... | [] | rintro rfl; simp [← lift_lift_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Grp | {
"line": 196,
"column": 23
} | {
"line": 196,
"column": 59
} | {
"line": 196,
"column": 59
} | [
{
"pp": "case refine_1\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : CartesianMonoidalCategory C\nA B : C\ninst✝ : GrpObj B\nf g h : A ⟶ B\n⊢ f = lift (g ≫ ι) h ≫ μ → lift g f ≫ μ = h",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"CategoryTheory.GrpObj.inv",
"Categ... | [] | rintro rfl; simp [← lift_lift_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Grp | {
"line": 196,
"column": 23
} | {
"line": 196,
"column": 59
} | {
"line": 196,
"column": 59
} | [
{
"pp": "case refine_2\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : CartesianMonoidalCategory C\nA B : C\ninst✝ : GrpObj B\nf g h : A ⟶ B\n⊢ lift g f ≫ μ = h → f = lift (g ≫ ι) h ≫ μ",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"CategoryTheory.GrpObj.inv",
"Categ... | [] | rintro rfl; simp [← lift_lift_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Grp | {
"line": 196,
"column": 23
} | {
"line": 196,
"column": 59
} | {
"line": 196,
"column": 59
} | [
{
"pp": "case refine_2\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : CartesianMonoidalCategory C\nA B : C\ninst✝ : GrpObj B\nf g h : A ⟶ B\n⊢ lift g f ≫ μ = h → f = lift (g ≫ ι) h ≫ μ",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"CategoryTheory.GrpObj.inv",
"Categ... | [] | rintro rfl; simp [← lift_lift_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Grp | {
"line": 318,
"column": 8
} | {
"line": 318,
"column": 19
} | {
"line": 318,
"column": 20
} | [
{
"pp": "case refine_1\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : CartesianMonoidalCategory C\nA : C\ninst✝ : GrpObj A\ns : PullbackCone μ μ\nm : s.pt ⟶ (A ⊗ A) ⊗ A\nhm₁ : m ≫ μ ▷ A = s.fst\nhm₂ : m ≫ (α_ A A A).hom ≫ A ◁ μ = s.snd\n⊢ (m ≫ fst (A ⊗ A) A) ≫ fst A A =\n (lift (lift (s.snd ≫ fst A A) ... | [
"case refine_1\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : CartesianMonoidalCategory C\nA : C\ninst✝ : GrpObj A\ns : PullbackCone μ μ\nm : s.pt ⟶ (A ⊗ A) ⊗ A\nhm₁ : m ≫ μ ▷ A = s.fst\nhm₂ : m ≫ (α_ A A A).hom ≫ A ◁ μ = s.snd\n⊢ m ≫ fst (A ⊗ A) A ≫ fst A A = s.snd ≫ fst A A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.Grp | {
"line": 319,
"column": 63
} | {
"line": 319,
"column": 74
} | {
"line": 319,
"column": 75
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : CartesianMonoidalCategory C\nA : C\ninst✝ : GrpObj A\ns : PullbackCone μ μ\nm : s.pt ⟶ (A ⊗ A) ⊗ A\nhm₁ : m ≫ μ ▷ A = s.fst\nhm₂ : m ≫ (α_ A A A).hom ≫ A ◁ μ = s.snd\n⊢ m ≫ fst (A ⊗ A) A ≫ fst A A = s.snd ≫ fst A A",
"ppTerm": "?m.589",
"assig... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : CartesianMonoidalCategory C\nA : C\ninst✝ : GrpObj A\ns : PullbackCone μ μ\nm : s.pt ⟶ (A ⊗ A) ⊗ A\nhm₁ : m ≫ μ ▷ A = s.fst\nhm₂ : m ≫ (α_ A A A).hom ≫ A ◁ μ = s.snd\n⊢ m ≫ fst (A ⊗ A) A ≫ fst A A = s.snd ≫ fst A A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.Grp | {
"line": 324,
"column": 8
} | {
"line": 324,
"column": 19
} | {
"line": 324,
"column": 20
} | [
{
"pp": "case refine_3\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : CartesianMonoidalCategory C\nA : C\ninst✝ : GrpObj A\ns : PullbackCone μ μ\nm : s.pt ⟶ (A ⊗ A) ⊗ A\nhm₁ : m ≫ μ ▷ A = s.fst\nhm₂ : m ≫ (α_ A A A).hom ≫ A ◁ μ = s.snd\n⊢ m ≫ snd (A ⊗ A) A =\n lift (lift (s.snd ≫ fst A A) (lift (s.snd ... | [
"case refine_3\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : CartesianMonoidalCategory C\nA : C\ninst✝ : GrpObj A\ns : PullbackCone μ μ\nm : s.pt ⟶ (A ⊗ A) ⊗ A\nhm₁ : m ≫ μ ▷ A = s.fst\nhm₂ : m ≫ (α_ A A A).hom ≫ A ◁ μ = s.snd\n⊢ m ≫ snd (A ⊗ A) A = s.fst ≫ snd A A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.Grp | {
"line": 331,
"column": 33
} | {
"line": 331,
"column": 44
} | {
"line": 331,
"column": 45
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : CartesianMonoidalCategory C\nA B : C\ninst✝² : GrpObj A\ninst✝¹ : GrpObj B\nf : A ⟶ B\ninst✝ : IsMonHom f\nthis : lift (lift f (ι ≫ f)) f = lift (lift f (f ≫ ι)) f\n⊢ ι ≫ f = f ≫ ι",
"ppTerm": "?m.88",
"assigned": false,
"usedConstants": [... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : CartesianMonoidalCategory C\nA B : C\ninst✝² : GrpObj A\ninst✝¹ : GrpObj B\nf : A ⟶ B\ninst✝ : IsMonHom f\nthis : lift (lift f (ι ≫ f)) f = lift (lift f (f ≫ ι)) f\n⊢ ι ≫ f = f ≫ ι"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.LinearMap.DivisionRing | {
"line": 31,
"column": 38
} | {
"line": 31,
"column": 73
} | {
"line": 31,
"column": 74
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : DivisionSemiring R\ninst✝ : Module R M\nf : M →ₗ[R] R\nhf : f ≠ 0\nz : R\n⊢ ∃ y, f y ≠ 0",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"LinearMap.instFunLike",
"Exists",
"... | [
"R : Type u_1\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : DivisionSemiring R\ninst✝ : Module R M\nf : M →ₗ[R] R\nhf : f ≠ 0\nz : R\n⊢ ∃ y, ¬f y = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.W.Basic | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 89
} | {
"line": 96,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : α → Type u_2\na b : α\nha : Nonempty (β a)\nhe : IsEmpty (β b)\nhf : Finite (WType β)\n⊢ False",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"False",
"Nonempty.elim",
"Eq.rec",
"Ne",
"IsEmpty",
"IsEmpty.elim'",
"Eq"
... | [
"α : Type u_1\nβ : α → Type u_2\na b : α\nha : Nonempty (β a)\nhe : IsEmpty (β b)\nhf : Finite (WType β)\nhba : b ≠ a\n⊢ False"
] | have hba : b ≠ a := fun h => ha.elim (IsEmpty.elim' (show IsEmpty (β a) from h ▸ he)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.SetTheory.Cardinal.Subfield | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 54
} | {
"line": 71,
"column": 0
} | [
{
"pp": "α : Type u\ns : Set α\ninst✝ : DivisionRing α\n⊢ Set.range (WType.elim (↥(closure s)) (operate s)) = Set.univ",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Subfield.toDivisionRing",
"Subfield.closure",
"Membership.mem",
"Subtype",
... | [] | exact SetLike.coe_set_eq.mpr (rangeOfWType_eq_top s) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Dimension.ErdosKaplansky | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 19
} | {
"line": 55,
"column": 4
} | [
{
"pp": "K : Type u\ninst✝ : DivisionRing K\naleph0_le : ℵ₀ ≤ Module.rank K (ℕ → K)\ncard_K : ℵ₀ < #K\nthis✝ : Module.rank K (ℕ → K) < #K\nιK : Type u\nbK : Module.Basis ιK K (ℕ → K)\nL : Subfield K := Subfield.closure (Set.range fun i ↦ bK i.1 i.2)\nhLK : #↥L < #K\nthis : Module (↥L)ᵐᵒᵖ K := Module.compHom K (... | [
"K : Type u\ninst✝ : DivisionRing K\naleph0_le : ℵ₀ ≤ Module.rank K (ℕ → K)\ncard_K : ℵ₀ < #K\nthis✝ : Module.rank K (ℕ → K) < #K\nιK : Type u\nbK : Module.Basis ιK K (ℕ → K)\nL : Subfield K := Subfield.closure (Set.range fun i ↦ bK i.1 i.2)\nthis : Module (↥L)ᵐᵒᵖ K := Module.compHom K (RingHom.op L.subtype)\nιL : ... | contrapose! hLK | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.LinearAlgebra.Dual.Basis | {
"line": 144,
"column": 39
} | {
"line": 144,
"column": 59
} | {
"line": 144,
"column": 60
} | [
{
"pp": "case intro\nR : Type uR\nM : Type uM\nι : Type uι\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : DecidableEq ι\nb : Basis ι R M\ninst✝ : Finite ι\nf : ι →₀ R\ni : ι\nval✝ : Fintype ι\n⊢ (f.sum fun i a ↦ a • b.dualBasis i) (b i) = f i",
"ppTerm": "?intro",
"ass... | [
"case intro\nR : Type uR\nM : Type uM\nι : Type uι\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : DecidableEq ι\nb : Basis ι R M\ninst✝ : Finite ι\nf : ι →₀ R\ni : ι\nval✝ : Fintype ι\n⊢ (∑ i, f i • b.dualBasis i) (b i) = f i",
"case intro.h\nR : Type uR\nM : Type uM\nι : Type u... | Finsupp.sum_fintype, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.ErdosKaplansky | {
"line": 138,
"column": 54
} | {
"line": 139,
"column": 50
} | {
"line": 141,
"column": 0
} | [
{
"pp": "K : Type u\ninst✝² : DivisionRing K\nV : Type u\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nh : ℵ₀ ≤ Module.rank K V\n⊢ Module.rank K V < Module.rank Kᵐᵒᵖ (V →ₗ[K] K)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Semiring.toModule"... | [] | by
convert! lift_rank_lt_rank_dual' h; rw [lift_id] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dual.Basis | {
"line": 285,
"column": 8
} | {
"line": 286,
"column": 31
} | {
"line": 287,
"column": 6
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nι : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ne : ι → M\nε : ι → Dual R M\nh : DualBases e ε\nv w : M\n⊢ h.coeffs (v + w) = h.coeffs v + h.coeffs w",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"LinearMap.m... | [] | ext i
exact (ε i).map_add v w | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Dual.Basis | {
"line": 285,
"column": 8
} | {
"line": 286,
"column": 31
} | {
"line": 287,
"column": 6
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nι : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ne : ι → M\nε : ι → Dual R M\nh : DualBases e ε\nv w : M\n⊢ h.coeffs (v + w) = h.coeffs v + h.coeffs w",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"LinearMap.m... | [] | ext i
exact (ε i).map_add v w | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dimension.DivisionRing | {
"line": 92,
"column": 6
} | {
"line": 92,
"column": 42
} | {
"line": 92,
"column": 43
} | [
{
"pp": "K : Type u\nV V₁ V₂ V₃ : Type v\ninst✝⁸ : DivisionRing K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\ninst✝⁵ : AddCommGroup V₁\ninst✝⁴ : Module K V₁\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\ndb : V₂ →ₗ[K] V\neb : V₃ →ₗ[K] V\ncd : V₁ →ₗ[K] V₂\nce :... | [
"K : Type u\nV V₁ V₂ V₃ : Type v\ninst✝⁸ : DivisionRing K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\ninst✝⁵ : AddCommGroup V₁\ninst✝⁴ : Module K V₁\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\ndb : V₂ →ₗ[K] V\neb : V₃ →ₗ[K] V\ncd : V₁ →ₗ[K] V₂\nce : V₁ →ₗ[K] V₃... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.RankNullity | {
"line": 119,
"column": 37
} | {
"line": 119,
"column": 52
} | {
"line": 119,
"column": 53
} | [
{
"pp": "R : Type u_1\nM : Type u\nM' : Type v\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : HasRankNullity.{v, u_1} R\nf : M →ₗ[R] M'\np : Submodule R M\nf' : M' ⧸ Submodule.map f p →ₗ[R] M' ⧸ f.range := factor ⋯\ng : ↥f.range →ₗ[R] ↥f'.... | [
"R : Type u_1\nM : Type u\nM' : Type v\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : HasRankNullity.{v, u_1} R\nf : M →ₗ[R] M'\np : Submodule R M\nf' : M' ⧸ Submodule.map f p →ₗ[R] M' ⧸ f.range := factor ⋯\ng : ↥f.range →ₗ[R] ↥f'.ker :=\n ↑(... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FiniteDimensional.Lemmas | {
"line": 99,
"column": 59
} | {
"line": 99,
"column": 91
} | {
"line": 99,
"column": 92
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\nW : Submodule K V\nv : V\nhv : v ∉ W\nhW : W ⊔ K ∙ v = ⊤\nw : V\nhw : w ∈ W ⊓ K ∙ v\n⊢ w ∈ W ∧ ∃ t, t • v = w",
"ppTerm": "?m.151",
"assigned": false,
"usedConstants": [... | [
"K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\nW : Submodule K V\nv : V\nhv : v ∉ W\nhW : W ⊔ K ∙ v = ⊤\nw : V\nhw : w ∈ W ⊓ K ∙ v\n⊢ w ∈ W ∧ ∃ t, t • v = w"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.RankNullity | {
"line": 166,
"column": 4
} | {
"line": 166,
"column": 75
} | {
"line": 166,
"column": 76
} | [
{
"pp": "R : Type u_1\nM : Type u\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : HasRankNullity.{u, u_1} R\ninst✝ : StrongRankCondition R\nn : ℕ\nv : Fin n → M\nhv : LinearIndependent R v\nh : ↑n < Module.rank R M\nt : Set M\nh₁ : Set.range v ⊆ t\nh₂ : lift.{u, 0} #(Fin n) = lift.{0, u... | [
"R : Type u_1\nM : Type u\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : HasRankNullity.{u, u_1} R\ninst✝ : StrongRankCondition R\nn : ℕ\nv : Fin n → M\nhv : LinearIndependent R v\nh : ↑n < Module.rank R M\nt : Set M\nh₁ : Set.range v ⊆ t\nh₂ : lift.{u, 0} #(Fin n) = lift.{0, u} (Module.ra... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FiniteDimensional.Lemmas | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 33
} | {
"line": 119,
"column": 34
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\np : Submodule K V\nv x : V\na : K\nhx : a • v ∈ p\nhx' : a • v = x\nhv : ¬a = 0\n⊢ v ∈ p",
"ppTerm": "?m.123",
"assigned": false,
"usedConstants": [],
"usedFVars": [... | [
"K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\np : Submodule K V\nv x : V\na : K\nhx : a • v ∈ p\nhx' : a • v = x\nhv : ¬a = 0\n⊢ v ∈ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Contraction | {
"line": 155,
"column": 70
} | {
"line": 157,
"column": 73
} | {
"line": 159,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nb : Basis ι R M\nx : Dual R M ⊗[R] N\n⊢ (dualTensorHomEquivOfBasis b).symm ((dualTens... | [] | by
rw [← dualTensorHomEquivOfBasis_apply b,
LinearEquiv.symm_apply_apply <| dualTensorHomEquivOfBasis (N := N) b] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Contraction | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 75
} | {
"line": 245,
"column": 0
} | [
{
"pp": "R : Type u_2\nM : Type u_3\nP : Type u_5\nQ : Type u_6\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid P\ninst✝⁵ : AddCommMonoid Q\ninst✝⁴ : Module R M\ninst✝³ : Module R P\ninst✝² : Module R Q\ninst✝¹ : Free R M\ninst✝ : Module.Finite R M\nx : (M →ₗ[R] P) ⊗[R] Q\n⊢ (rTensorH... | [] | rw [← LinearEquiv.coe_toLinearMap, rTensorHomEquivHomRTensor_toLinearMap] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Contraction | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 75
} | {
"line": 245,
"column": 0
} | [
{
"pp": "R : Type u_2\nM : Type u_3\nP : Type u_5\nQ : Type u_6\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid P\ninst✝⁵ : AddCommMonoid Q\ninst✝⁴ : Module R M\ninst✝³ : Module R P\ninst✝² : Module R Q\ninst✝¹ : Free R M\ninst✝ : Module.Finite R M\nx : (M →ₗ[R] P) ⊗[R] Q\n⊢ (rTensorH... | [] | rw [← LinearEquiv.coe_toLinearMap, rTensorHomEquivHomRTensor_toLinearMap] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Contraction | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 75
} | {
"line": 245,
"column": 0
} | [
{
"pp": "R : Type u_2\nM : Type u_3\nP : Type u_5\nQ : Type u_6\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid P\ninst✝⁵ : AddCommMonoid Q\ninst✝⁴ : Module R M\ninst✝³ : Module R P\ninst✝² : Module R Q\ninst✝¹ : Free R M\ninst✝ : Module.Finite R M\nx : (M →ₗ[R] P) ⊗[R] Q\n⊢ (rTensorH... | [] | rw [← LinearEquiv.coe_toLinearMap, rTensorHomEquivHomRTensor_toLinearMap] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Contraction | {
"line": 316,
"column": 4
} | {
"line": 316,
"column": 15
} | {
"line": 316,
"column": 16
} | [
{
"pp": "R : Type u_7\nM : Type u_8\nN : Type u_9\nι : Type u_10\nκ : Type u_11\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : DecidableEq κ\ninst✝⁶ : Fintype ι\ninst✝⁵ : Fintype κ\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R M\ninst✝ : Module R N\nb : Basis ι R M\nc : Basi... | [
"R : Type u_7\nM : Type u_8\nN : Type u_9\nι : Type u_10\nκ : Type u_11\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : DecidableEq κ\ninst✝⁶ : Fintype ι\ninst✝⁵ : Fintype κ\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R M\ninst✝ : Module R N\nb : Basis ι R M\nc : Basis κ R N\ni :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 315,
"column": 2
} | {
"line": 315,
"column": 63
} | {
"line": 315,
"column": 64
} | [
{
"pp": "R : Type u_1\nR₁ : Type u_2\nM : Type u_5\nM₁ : Type u_6\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M\nH : B.IsAlt\nx y : M₁\nH1 : (B (y + x)) (y + x) = 0\... | [
"R : Type u_1\nR₁ : Type u_2\nM : Type u_5\nM₁ : Type u_6\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M\nH : B.IsAlt\nx y : M₁\nH1 : (B (y + x)) (y + x) = 0\n⊢ -(B x) y ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Constructions.Equalizers | {
"line": 78,
"column": 6
} | {
"line": 78,
"column": 32
} | {
"line": 78,
"column": 33
} | [
{
"pp": "case h₀\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nG : C ⥤ D\ninst✝¹ : HasBinaryProducts C\ninst✝ : HasPullbacks C\nF : WalkingParallelPair ⥤ C\nc : Cone F\nm✝ : c.pt ⟶ (equalizerCone F).pt\nJ : ∀ (j : WalkingParallelPair), m✝ ≫ (equalizerCone F).π.app j = c.π.a... | [
"case h₀\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nG : C ⥤ D\ninst✝¹ : HasBinaryProducts C\ninst✝ : HasPullbacks C\nF : WalkingParallelPair ⥤ C\nc : Cone F\nm✝ : c.pt ⟶ (equalizerCone F).pt\nJ : ∀ (j : WalkingParallelPair), m✝ ≫ (equalizerCone F).π.app j = c.π.app j\nJ0 : m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 446,
"column": 19
} | {
"line": 446,
"column": 30
} | {
"line": 446,
"column": 31
} | [
{
"pp": "R : Type u_20\nM : Type u_21\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nf : M ≃ M\nhf : IsAdjointPair B B ⇑f ⇑f.symm\nx y : M\n⊢ (B (f x)) (f y) = (B x) y",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"R : Type u_20\nM : Type u_21\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nf : M ≃ M\nhf : IsAdjointPair B B ⇑f ⇑f.symm\nx y : M\n⊢ (B (f x)) (f y) = (B x) y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 446,
"column": 59
} | {
"line": 446,
"column": 70
} | {
"line": 446,
"column": 71
} | [
{
"pp": "R : Type u_20\nM : Type u_21\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nf : M ≃ M\nhf : IsOrthogonal B ⇑f\nx y : M\n⊢ (B (f x)) y = (B x) (f.symm y)",
"ppTerm": "?m.38",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"us... | [
"R : Type u_20\nM : Type u_21\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nf : M ≃ M\nhf : IsOrthogonal B ⇑f\nx y : M\n⊢ (B (f x)) y = (B x) (f.symm y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 458,
"column": 2
} | {
"line": 458,
"column": 30
} | {
"line": 458,
"column": 31
} | [
{
"pp": "R : Type u_20\nM : Type u_21\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nB : LinearMap.BilinForm R M\nF : Type u_22\ninst✝¹ : FunLike F M M\ninst✝ : LinearMapClass F R M M\nf : F\nh : IsLeftRegular 2\nhB : IsSymm B\nhf : ∀ (x : M), (B (f x)) (f x) = (B x) x\nx y : M\nthis : (B (... | [
"R : Type u_20\nM : Type u_21\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nB : LinearMap.BilinForm R M\nF : Type u_22\ninst✝¹ : FunLike F M M\ninst✝ : LinearMapClass F R M M\nf : F\nh : IsLeftRegular 2\nhB : IsSymm B\nhf : ∀ (x : M), (B (f x)) (f x) = (B x) x\nx y : M\nthis : (B (f x)) (f y) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Constructions.Equalizers | {
"line": 125,
"column": 10
} | {
"line": 131,
"column": 70
} | {
"line": 131,
"column": 71
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nG : C ⥤ D\ninst✝³ : HasBinaryProducts C\ninst✝² : HasPullbacks C\ninst✝¹ : PreservesLimitsOfShape (Discrete WalkingPair) G\ninst✝ : PreservesLimitsOfShape WalkingCospan G\nK : WalkingParallelPair ⥤ C\ns : Cone (K ⋙ G)\nm... | [] | rw [Iso.eq_comp_inv]
have := h WalkingParallelPair.zero
dsimp [equalizerCone] at this
ext <;>
simp only [PreservesPullback.iso_hom_snd, Category.assoc,
PreservesPullback.iso_hom_fst, pullback.lift_fst, pullback.lift_snd,
Category.comp_id, ← pullbackF... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Constructions.Equalizers | {
"line": 125,
"column": 10
} | {
"line": 131,
"column": 70
} | {
"line": 131,
"column": 71
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nG : C ⥤ D\ninst✝³ : HasBinaryProducts C\ninst✝² : HasPullbacks C\ninst✝¹ : PreservesLimitsOfShape (Discrete WalkingPair) G\ninst✝ : PreservesLimitsOfShape WalkingCospan G\nK : WalkingParallelPair ⥤ C\ns : Cone (K ⋙ G)\nm... | [] | rw [Iso.eq_comp_inv]
have := h WalkingParallelPair.zero
dsimp [equalizerCone] at this
ext <;>
simp only [PreservesPullback.iso_hom_snd, Category.assoc,
PreservesPullback.iso_hom_fst, pullback.lift_fst, pullback.lift_snd,
Category.comp_id, ← pullbackF... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 663,
"column": 4
} | {
"line": 663,
"column": 69
} | {
"line": 663,
"column": 70
} | [
{
"pp": "case mp\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : CommSemiring R₁\ninst✝⁴ : AddCommMonoid M₁\ninst✝³ : Module R₁ M₁\ninst✝² : CommSemiring R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝... | [
"case mp\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : CommSemiring R₁\ninst✝⁴ : AddCommMonoid M₁\ninst✝³ : Module R₁ M₁\ninst✝² : CommSemiring R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Constructions.Equalizers | {
"line": 175,
"column": 6
} | {
"line": 175,
"column": 17
} | {
"line": 175,
"column": 18
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nG : C ⥤ D\ninst✝¹ : HasBinaryCoproducts C\ninst✝ : HasPushouts C\nF : WalkingParallelPair ⥤ C\nc : Cocone F\nm : (coequalizerCocone F).pt ⟶ c.pt\nJ : ∀ (j : WalkingParallelPair), (coequalizerCocone F).ι.app j ≫ m = c.ι.a... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nG : C ⥤ D\ninst✝¹ : HasBinaryCoproducts C\ninst✝ : HasPushouts C\nF : WalkingParallelPair ⥤ C\nc : Cocone F\nm : (coequalizerCocone F).pt ⟶ c.pt\nJ : ∀ (j : WalkingParallelPair), (coequalizerCocone F).ι.app j ≫ m = c.ι.app j\n⊢ push... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 693,
"column": 2
} | {
"line": 693,
"column": 80
} | {
"line": 694,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_5\nM₁ : Type u_6\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R M₁\nB : M →ₗ[R] M →ₗ[R] M₁\nhB : B.IsRefl\nhB' : B.SeparatingLeft\n⊢ B.SeparatingRight",
"ppTerm": "?m.82",
"assigned": true,
"usedConstan... | [
"R : Type u_1\nM : Type u_5\nM₁ : Type u_6\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R M₁\nB : M →ₗ[R] M →ₗ[R] M₁\nhB : B.IsRefl\nhB' : B.SeparatingLeft\n⊢ B.ker = ⊥"
] | rw [separatingRight_iff_flip_ker_eq_bot, hB.ker_eq_bot_iff_ker_flip_eq_bot.mp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 836,
"column": 34
} | {
"line": 836,
"column": 60
} | {
"line": 836,
"column": 61
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nx y : M\naux : ∀ (x y : M), 0 ≤ (B x) x * ((B x) x * (B y) y - (B x) y * (B y) x)\nhx : (B x) x... | [
"R : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nx y : M\naux : ∀ (x y : M), 0 ≤ (B x) x * ((B x) x * (B y) y - (B x) y * (B y) x)\nhx : (B x) x = 0\nhy : (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 839,
"column": 8
} | {
"line": 839,
"column": 56
} | {
"line": 839,
"column": 57
} | [
{
"pp": "case inr.inr.a\nR : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nx y : M\naux : ∀ (x y : M), 0 ≤ (B x) x * ((B x) x * (B y) y - (B x) y * (B y) ... | [
"case inr.inr.a\nR : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nx y : M\naux : ∀ (x y : M), 0 ≤ (B x) x * ((B x) x * (B y) y - (B x) y * (B y) x)\nhx : (B ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 840,
"column": 8
} | {
"line": 840,
"column": 28
} | {
"line": 840,
"column": 29
} | [
{
"pp": "case a\nR : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nx y : M\naux : ∀ (x y : M), 0 ≤ (B x) x * ((B x) x * (B y) y - (B x) y * (B y) x)\nhx :... | [
"case a\nR : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nx y : M\naux : ∀ (x y : M), 0 ≤ (B x) x * ((B x) x * (B y) y - (B x) y * (B y) x)\nhx : (B x) x = 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 859,
"column": 4
} | {
"line": 859,
"column": 20
} | {
"line": 859,
"column": 21
} | [
{
"pp": "case pos\nR : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhp : ∀ (x : M), x ≠ 0 → 0 < (B x) x\nx y : M\nhe : (B x) y * (B y) x = (B x) x * (B y) y\nhz : (B x) y • x - (B x)... | [
"case pos\nR : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhp : ∀ (x : M), x ≠ 0 → 0 < (B x) x\nx y : M\nhe : (B x) y * (B y) x = (B x) x * (B y) y\nhz : (B x) y • x - (B x) x • y = 0\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 875,
"column": 2
} | {
"line": 877,
"column": 38
} | {
"line": 879,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nhB : IsSymm B\n⊢ Nondegenerate B ↔ ∀ (x : M), (B x) x = 0 ↔ x = 0",
"ppTerm": "?m.47",
... | [] | simp_rw [hB.isRefl.nondegenerate_iff_separatingLeft, separatingLeft_iff_ker_eq_bot,
Submodule.eq_bot_iff, B.apply_apply_same_eq_zero_iff hs hB, mem_ker]
exact forall_congr' fun x ↦ by aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 875,
"column": 2
} | {
"line": 877,
"column": 38
} | {
"line": 879,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nhB : IsSymm B\n⊢ Nondegenerate B ↔ ∀ (x : M), (B x) x = 0 ↔ x = 0",
"ppTerm": "?m.47",
... | [] | simp_rw [hB.isRefl.nondegenerate_iff_separatingLeft, separatingLeft_iff_ker_eq_bot,
Submodule.eq_bot_iff, B.apply_apply_same_eq_zero_iff hs hB, mem_ker]
exact forall_congr' fun x ↦ by aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 898,
"column": 2
} | {
"line": 898,
"column": 25
} | {
"line": 898,
"column": 26
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nhB : IsSymm B\nW : Submodule R M\nhW : Disjoint W (ker B)\nhB' : (domRestrict₁₂ B W W).IsRefl\n... | [
"R : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nhB : IsSymm B\nW : Submodule R M\nhW : Disjoint W (ker B)\nhB' : (domRestrict₁₂ B W W).IsRefl\nx : M\nhx : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers | {
"line": 90,
"column": 68
} | {
"line": 90,
"column": 79
} | {
"line": 90,
"column": 80
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : Type w\ninst✝ : SmallCategory J\nF : J ⥤ C\nc₁ : Fan F.obj\nc₂ : Fan fun f ↦ F.obj f.fst.2\ns t : c₁.pt ⟶ c₂.pt\nhs : ∀ (f : (p : J × J) × (p.1 ⟶ p.2)), s ≫ c₂.π.app { as := f } = c₁.π.app { as := f.fst.1 } ≫ F.map f.snd\nht : ∀ (f : (p : J × J) × (p.1 ⟶ p.2)... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nJ : Type w\ninst✝ : SmallCategory J\nF : J ⥤ C\nc₁ : Fan F.obj\nc₂ : Fan fun f ↦ F.obj f.fst.2\ns t : c₁.pt ⟶ c₂.pt\nhs : ∀ (f : (p : J × J) × (p.1 ⟶ p.2)), s ≫ c₂.π.app { as := f } = c₁.π.app { as := f.fst.1 } ≫ F.map f.snd\nht : ∀ (f : (p : J × J) × (p.1 ⟶ p.2)), t ≫ c₂.π.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Constructions.Pullbacks | {
"line": 49,
"column": 12
} | {
"line": 49,
"column": 23
} | {
"line": 49,
"column": 24
} | [
{
"pp": "case h₁\nC : Type u\n𝒞 : Category.{v, u} C\nX Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝¹ : HasLimit (pair X Y)\ninst✝ : HasLimit (parallelPair (prod.fst ≫ f) (prod.snd ≫ g))\nπ₁ : X ⨯ Y ⟶ X := ⋯\nπ₂ : X ⨯ Y ⟶ Y := ⋯\ne : equalizer (π₁ ≫ f) (π₂ ≫ g) ⟶ X ⨯ Y := ⋯\ns : PullbackCone f g\nm : s.pt ⟶ equalizer (... | [
"case h₁\nC : Type u\n𝒞 : Category.{v, u} C\nX Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝¹ : HasLimit (pair X Y)\ninst✝ : HasLimit (parallelPair (prod.fst ≫ f) (prod.snd ≫ g))\nπ₁ : X ⨯ Y ⟶ X := prod.fst\nπ₂ : X ⨯ Y ⟶ Y := prod.snd\ne : equalizer (π₁ ≫ f) (π₂ ≫ g) ⟶ X ⨯ Y := equalizer.ι (π₁ ≫ f) (π₂ ≫ g)\ns : PullbackCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Constructions.Pullbacks | {
"line": 50,
"column": 12
} | {
"line": 50,
"column": 23
} | {
"line": 50,
"column": 24
} | [
{
"pp": "case h₂\nC : Type u\n𝒞 : Category.{v, u} C\nX Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝¹ : HasLimit (pair X Y)\ninst✝ : HasLimit (parallelPair (prod.fst ≫ f) (prod.snd ≫ g))\nπ₁ : X ⨯ Y ⟶ X := ⋯\nπ₂ : X ⨯ Y ⟶ Y := ⋯\ne : equalizer (π₁ ≫ f) (π₂ ≫ g) ⟶ X ⨯ Y := ⋯\ns : PullbackCone f g\nm : s.pt ⟶ equalizer (... | [
"case h₂\nC : Type u\n𝒞 : Category.{v, u} C\nX Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝¹ : HasLimit (pair X Y)\ninst✝ : HasLimit (parallelPair (prod.fst ≫ f) (prod.snd ≫ g))\nπ₁ : X ⨯ Y ⟶ X := prod.fst\nπ₂ : X ⨯ Y ⟶ Y := prod.snd\ne : equalizer (π₁ ≫ f) (π₂ ≫ g) ⟶ X ⨯ Y := equalizer.ι (π₁ ≫ f) (π₂ ≫ g)\ns : PullbackCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Constructions.Pullbacks | {
"line": 86,
"column": 12
} | {
"line": 86,
"column": 23
} | {
"line": 86,
"column": 24
} | [
{
"pp": "case h₁\nC : Type u\n𝒞 : Category.{v, u} C\nX Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\ninst✝¹ : HasColimit (pair Y Z)\ninst✝ : HasColimit (parallelPair (f ≫ coprod.inl) (g ≫ coprod.inr))\nι₁ : Y ⟶ Y ⨿ Z := ⋯\nι₂ : Z ⟶ Y ⨿ Z := ⋯\nc : Y ⨿ Z ⟶ coequalizer (f ≫ ι₁) (g ≫ ι₂) := ⋯\ns : PushoutCocone f g\nm : coequal... | [
"case h₁\nC : Type u\n𝒞 : Category.{v, u} C\nX Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\ninst✝¹ : HasColimit (pair Y Z)\ninst✝ : HasColimit (parallelPair (f ≫ coprod.inl) (g ≫ coprod.inr))\nι₁ : Y ⟶ Y ⨿ Z := coprod.inl\nι₂ : Z ⟶ Y ⨿ Z := coprod.inr\nc : Y ⨿ Z ⟶ coequalizer (f ≫ ι₁) (g ≫ ι₂) := coequalizer.π (f ≫ ι₁) (g ≫ ι₂)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Constructions.Pullbacks | {
"line": 87,
"column": 12
} | {
"line": 87,
"column": 23
} | {
"line": 87,
"column": 24
} | [
{
"pp": "case h₂\nC : Type u\n𝒞 : Category.{v, u} C\nX Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\ninst✝¹ : HasColimit (pair Y Z)\ninst✝ : HasColimit (parallelPair (f ≫ coprod.inl) (g ≫ coprod.inr))\nι₁ : Y ⟶ Y ⨿ Z := ⋯\nι₂ : Z ⟶ Y ⨿ Z := ⋯\nc : Y ⨿ Z ⟶ coequalizer (f ≫ ι₁) (g ≫ ι₂) := ⋯\ns : PushoutCocone f g\nm : coequal... | [
"case h₂\nC : Type u\n𝒞 : Category.{v, u} C\nX Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\ninst✝¹ : HasColimit (pair Y Z)\ninst✝ : HasColimit (parallelPair (f ≫ coprod.inl) (g ≫ coprod.inr))\nι₁ : Y ⟶ Y ⨿ Z := coprod.inl\nι₂ : Z ⟶ Y ⨿ Z := coprod.inr\nc : Y ⨿ Z ⟶ coequalizer (f ≫ ι₁) (g ≫ ι₂) := coequalizer.π (f ≫ ι₁) (g ≫ ι₂)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Basic | {
"line": 81,
"column": 12
} | {
"line": 81,
"column": 23
} | {
"line": 81,
"column": 24
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y : C\ninst✝ : HasZeroMorphisms C\nf : X ⟶ Y\nI : NormalMono f\n⊢ f ≫ g = f ≫ 0",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y : C\ninst✝ : HasZeroMorphisms C\nf : X ⟶ Y\nI : NormalMono f\n⊢ f ≫ g = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Basic | {
"line": 202,
"column": 12
} | {
"line": 202,
"column": 23
} | {
"line": 202,
"column": 24
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y : C\ninst✝ : HasZeroMorphisms C\nf : X ⟶ Y\nI : NormalEpi f\n⊢ g ≫ f = 0 ≫ f",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y : C\ninst✝ : HasZeroMorphisms C\nf : X ⟶ Y\nI : NormalEpi f\n⊢ g ≫ f = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 324,
"column": 28
} | {
"line": 324,
"column": 82
} | {
"line": 325,
"column": 2
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np : Submodule R M\nx : M\nhx : x ∉ p\nhp' : Projective R (M ⧸ p)\nf : Dual R (M ⧸ p)\nhf : f (p.mkQ x) ≠ 0\n⊢ ∃ f, f x ≠ 0 ∧ map f p = ⊥",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
... | [] | exact ⟨f.comp p.mkQ, hf, by simp [Submodule.map_comp]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 459,
"column": 2
} | {
"line": 459,
"column": 9
} | {
"line": 460,
"column": 2
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nW : Subspace K V\n⊢ dualRestrict W ∘ₗ W.dualLift = 1",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Submodule",
"Algebra.to_smulCommClass",
"Semiring.toModule",
"Li... | [
"K : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nW : Subspace K V\nφ : Dual K ↥W\nx : ↥W\n⊢ ((dualRestrict W ∘ₗ W.dualLift) φ) x = (1 φ) x"
] | ext φ x | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 463,
"column": 63
} | {
"line": 464,
"column": 72
} | {
"line": 466,
"column": 0
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nW : Subspace K V\nx : Dual K ↥W\n⊢ (dualRestrict W) (W.dualLift x) = x",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Subspace.dualRestrict_comp_du... | [] | by
rw [← LinearMap.comp_apply, dualRestrict_comp_dualLift, End.one_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers | {
"line": 367,
"column": 70
} | {
"line": 367,
"column": 81
} | {
"line": 367,
"column": 82
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nJ : Type w\ninst✝¹ : SmallCategory J\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : J ⥤ C\nc₁ : Cofan fun f ↦ F.obj f.fst.1\nc₂ : Cofan F.obj\ns t : c₁.pt ⟶ c₂.pt\nhs : ∀ (f : (p : J × J) × (p.1 ⟶ p.2)), c₁.ι.app { as := f } ≫ s = F.map f.snd ≫ c₂.ι.app { as := f... | [
"C : Type u\ninst✝² : Category.{v, u} C\nJ : Type w\ninst✝¹ : SmallCategory J\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : J ⥤ C\nc₁ : Cofan fun f ↦ F.obj f.fst.1\nc₂ : Cofan F.obj\ns t : c₁.pt ⟶ c₂.pt\nhs : ∀ (f : (p : J × J) × (p.1 ⟶ p.2)), c₁.ι.app { as := f } ≫ s = F.map f.snd ≫ c₂.ι.app { as := f.fst.2 }\nht... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 612,
"column": 4
} | {
"line": 612,
"column": 30
} | {
"line": 612,
"column": 31
} | [
{
"pp": "case mp\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nW : Submodule R M\nψ : Dual R (M ⧸ W)\nthis : W.mkQ.dualMap ψ ∈ W.mkQ.dualMap.range\n⊢ W.mkQ.dualMap ψ ∈ W.dualAnnihilator",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"u... | [
"case mp\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nW : Submodule R M\nψ : Dual R (M ⧸ W)\nthis : W.mkQ.dualMap ψ ∈ W.mkQ.dualMap.range\n⊢ W.mkQ.dualMap ψ ∈ W.dualAnnihilator"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 708,
"column": 6
} | {
"line": 709,
"column": 17
} | {
"line": 710,
"column": 4
} | [] | [
"case e'_2\nR : Type u_1\nM : Type u_2\nM' : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M'\ninst✝ : Module R M'\nf : M →ₗ[R] M'\nhf : Function.Surjective ⇑f.range.subtype.dualMap\nrr_surj : Function.Surjective ⇑f.rangeRestrict\nthis : f.rangeRestrict.dualMap.r... | _ = range ((range f).subtype.comp f.rangeRestrict).dualMap := by simp
_ = _ := ?_ | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 767,
"column": 26
} | {
"line": 767,
"column": 42
} | {
"line": 767,
"column": 43
} | [
{
"pp": "K : Type u_1\nV₁ : Type u_2\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V₁\ninst✝¹ : Module K V₁\ninst✝ : FiniteDimensional K V₁\nf g : Dual K V₁\nx : V₁\nh : LinearMap.ker f = LinearMap.ker g\nh' : f x = g x\nhx : f x ≠ 0\np : Submodule K V₁ := K ∙ x\nhp : p ≠ ⊥\nt : K\nhfy : f (t • x) = 0\nhpy : ... | [
"K : Type u_1\nV₁ : Type u_2\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V₁\ninst✝¹ : Module K V₁\ninst✝ : FiniteDimensional K V₁\nf g : Dual K V₁\nx : V₁\nh : LinearMap.ker f = LinearMap.ker g\nh' : f x = g x\nhx : f x ≠ 0\np : Submodule K V₁ := K ∙ x\nhp : p ≠ ⊥\nt : K\nhfy : f (t • x) = 0\nhpy : t • x ∈ p\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 849,
"column": 4
} | {
"line": 849,
"column": 71
} | {
"line": 850,
"column": 6
} | [
{
"pp": "case right\nK : Type u_1\nV₁ : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K V₁\nW : Subspace K V₁\nx : ↥W\nh : ∀ (x_1 : Dual K V₁ ⧸ dualAnnihilator W), ((Submodule.dualPairing W) x_1) x = 0\nφ : Dual K ↥W\n⊢ φ x = 0",
"ppTerm": "?right",
"assigned": false,
"usedCon... | [
"case right\nK : Type u_1\nV₁ : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K V₁\nW : Subspace K V₁\nx : ↥W\nh : ∀ (x_1 : Dual K V₁ ⧸ dualAnnihilator W), ((Submodule.dualPairing W) x_1) x = 0\nφ : Dual K ↥W\n⊢ φ x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 892,
"column": 8
} | {
"line": 892,
"column": 23
} | {
"line": 892,
"column": 24
} | [
{
"pp": "case h_empty\nK : Type u_1\nV₁ : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K V₁\nW : PEmpty.{u_4 + 1} → Subspace K V₁\n⊢ dualAnnihilator (⨅ i, W i) = ⨆ i, dualAnnihilator (W i)",
"ppTerm": "?h_empty",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Sub... | [
"case h_empty\nK : Type u_1\nV₁ : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K V₁\nW : PEmpty.{u_4 + 1} → Subspace K V₁\n⊢ dualAnnihilator (⨅ i, W i) = sSup ∅"
] | iSup_of_empty', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Equalizers | {
"line": 218,
"column": 60
} | {
"line": 218,
"column": 83
} | {
"line": 218,
"column": 83
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : HasZeroMorphisms C\ninst✝⁴ : HasFiniteCoproducts C\ninst✝³ : HasCokernels C\ninst✝² : IsNormalEpiCategory C\nX Y Z : C\na : X ⟶ Y\nb : X ⟶ Z\ninst✝¹ : Epi a\ninst✝ : Epi b\nP : C\nf : P ⟶ X\nhfa : f ≫ a = 0\ni : IsColimit (CokernelCofork.ofπ a hfa)... | [
"C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : HasZeroMorphisms C\ninst✝⁴ : HasFiniteCoproducts C\ninst✝³ : HasCokernels C\ninst✝² : IsNormalEpiCategory C\nX Y Z : C\na : X ⟶ Y\nb : X ⟶ Z\ninst✝¹ : Epi a\ninst✝ : Epi b\nP : C\nf : P ⟶ X\nhfa : f ≫ a = 0\ni : IsColimit (CokernelCofork.ofπ a hfa)\nQ : C\ng :... | PushoutCocone.condition | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 1033,
"column": 2
} | {
"line": 1033,
"column": 72
} | {
"line": 1034,
"column": 2
} | [
{
"pp": "K : Type u_4\nV : Type u_5\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nW : Subspace K V\ninst✝ : FiniteDimensional K ↥W\ne1 : ↥W ≃ₗ[K] Dual K V ⧸ dualAnnihilator W := (Free.chooseBasis K ↥W).toDualEquiv ≪≫ₗ W.quotAnnihilatorEquiv.symm\nthis✝ : FiniteDimensional K (Dual K V ⧸ dualAn... | [
"K : Type u_4\nV : Type u_5\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nW : Subspace K V\ninst✝ : FiniteDimensional K ↥W\ne1 : ↥W ≃ₗ[K] Dual K V ⧸ dualAnnihilator W := (Free.chooseBasis K ↥W).toDualEquiv ≪≫ₗ W.quotAnnihilatorEquiv.symm\nthis✝ : FiniteDimensional K (Dual K V ⧸ dualAnnihilator W)... | rw [eq_of_le_of_finrank_eq (map_le_dualAnnihilator_dualAnnihilator W)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 1061,
"column": 4
} | {
"line": 1061,
"column": 15
} | {
"line": 1061,
"column": 16
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nV : Type u_3\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Set ι\nv : ι → V\nhli : LinearIndepOn K id (v '' s)\n⊢ ⊤ ≤ span K (Set.range fun x ↦ id ↑x)",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"LinearIndepOn.extend",
... | [
"ι : Type u_1\nK : Type u_2\nV : Type u_3\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Set ι\nv : ι → V\nhli : LinearIndepOn K id (v '' s)\n⊢ span K (hli.extend ⋯) = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Basic | {
"line": 166,
"column": 8
} | {
"line": 170,
"column": 12
} | {
"line": 170,
"column": 13
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Preadditive C\ninst✝² : HasKernels C\ninst✝¹ : HasCokernels C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsIso (coimageImageComparison f)\nF : MonoFactorisation f\n⊢ (inv (coimageImageComparison f) ≫ cokernel.desc (kernel.ι f) F.e ⋯) ≫ F.m = (imageMonoFactorisation f)... | [] | rw [imageMonoFactorisation_m]
simp only [Category.assoc]
rw [IsIso.inv_comp_eq]
ext
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.Basic | {
"line": 166,
"column": 8
} | {
"line": 170,
"column": 12
} | {
"line": 170,
"column": 13
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Preadditive C\ninst✝² : HasKernels C\ninst✝¹ : HasCokernels C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsIso (coimageImageComparison f)\nF : MonoFactorisation f\n⊢ (inv (coimageImageComparison f) ≫ cokernel.desc (kernel.ι f) F.e ⋯) ≫ F.m = (imageMonoFactorisation f)... | [] | rw [imageMonoFactorisation_m]
simp only [Category.assoc]
rw [IsIso.inv_comp_eq]
ext
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.Grp.EpiMono | {
"line": 42,
"column": 20
} | {
"line": 42,
"column": 31
} | {
"line": 42,
"column": 32
} | [
{
"pp": "A : Type u\nB : Type v\ninst✝¹ : Group A\ninst✝ : Group B\nf : A →* B\nh : ∀ (u v : ↥f.ker →* A), f.comp u = f.comp v → u = v\n⊢ f.ker = ⊥",
"ppTerm": "?m.36",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"A : Type u\nB : Type v\ninst✝¹ : Group A\ninst✝ : Group B\nf : A →* B\nh : ∀ (u v : ↥f.ker →* A), f.comp u = f.comp v → u = v\n⊢ f.ker = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Category.Grp.EpiMono | {
"line": 228,
"column": 2
} | {
"line": 228,
"column": 18
} | {
"line": 228,
"column": 19
} | [
{
"pp": "A B : GrpCat\nf : A ⟶ B\nx : ↑B\nhx : x ∈ (Hom.hom f).range\n⊢ τ (fromCoset ⟨x • ↑⟨↑(Hom.hom f).range, ⋯⟩, ⋯⟩) = ∞",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"MonoidHom.range",
"instHSMul",
"instSMulOfMul",
"InvOneClass.toOne",
"Equiv.instEquivLi... | [
"A B : GrpCat\nf : A ⟶ B\nx : ↑B\nhx : x ∈ (Hom.hom f).range\n⊢ τ (fromCoset ⟨x • ↑(Hom.hom f).range, ⋯⟩) = ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Category.Grp.EpiMono | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 61
} | {
"line": 288,
"column": 62
} | [
{
"pp": "A B : GrpCat\nf : A ⟶ B\nx : ↑B\nhx : x ∉ (Hom.hom f).range\nr : (g x) (fromCoset ⟨↑(Hom.hom f).range, ⋯⟩) = (h x) (fromCoset ⟨↑(Hom.hom f).range, ⋯⟩)\n⊢ fromCoset ⟨x • ↑(Hom.hom f).range, ⋯⟩ = fromCoset ⟨↑(Hom.hom f).range, ⋯⟩",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
... | [
"A B : GrpCat\nf : A ⟶ B\nx : ↑B\nhx : x ∉ (Hom.hom f).range\nr : (g x) (fromCoset ⟨↑(Hom.hom f).range, ⋯⟩) = (h x) (fromCoset ⟨↑(Hom.hom f).range, ⋯⟩)\n⊢ x • Set.range ⇑(Hom.hom f) = Set.range ⇑(Hom.hom f)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 678,
"column": 2
} | {
"line": 678,
"column": 54
} | {
"line": 679,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.HomologyData\nh₁ : S.LeftHomologyData\nh₂ : S.RightHomologyData\n⊢ leftRightHomologyComparison' h₁ h₂ = leftHomologyMap' (𝟙 S) h₁ h.left ≫ h.iso.hom ≫ rightHomologyMap' (𝟙 S) h.right h₂",
"ppTerm": "?m.7... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.HomologyData\nh₁ : S.LeftHomologyData\nh₂ : S.RightHomologyData\n⊢ leftRightHomologyComparison' h₁ h₂ = leftHomologyMap' (𝟙 S) h₁ h.left ≫ h.iso.hom ≫ rightHomologyMap' (𝟙 S) h.right h₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 956,
"column": 2
} | {
"line": 956,
"column": 61
} | {
"line": 956,
"column": 62
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\ninst✝ : S.HasHomology\n⊢ (S.leftHomologyπ ≫ S.leftHomologyIso.hom) ≫ S.rightHomologyIso.inv ≫ S.rightHomologyι = S.iCycles ≫ S.pOpcycles",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\ninst✝ : S.HasHomology\n⊢ S.leftHomologyπ ≫ S.leftHomologyIso.hom ≫ S.rightHomologyIso.inv ≫ S.rightHomologyι = S.iCycles ≫ S.pOpcycles"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 1000,
"column": 73
} | {
"line": 1002,
"column": 73
} | {
"line": 1004,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\ninst✝ : S.HasHomology\nh : S.LeftHomologyData\n⊢ h.homologyIso.hom ≫ h.leftHomologyIso.inv = S.leftHomologyIso.inv",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"CategoryTheory.Categor... | [] | by
dsimp only [homologyIso]
simp only [Iso.trans_hom, Iso.symm_hom, assoc, Iso.hom_inv_id, comp_id] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 1012,
"column": 75
} | {
"line": 1014,
"column": 73
} | {
"line": 1016,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\ninst✝ : S.HasHomology\nh : S.RightHomologyData\n⊢ h.homologyIso.hom ≫ h.rightHomologyIso.inv = S.rightHomologyIso.inv",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"CategoryTheory.Cate... | [] | by
dsimp only [homologyIso]
simp only [Iso.trans_hom, Iso.symm_hom, assoc, Iso.hom_inv_id, comp_id] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 335,
"column": 42
} | {
"line": 335,
"column": 67
} | {
"line": 336,
"column": 6
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.LeftHomologyData\nh₂ : S₂.LeftHomologyData\nφK : h₁.K ⟶ h₂.K := h₂.liftK (h₁.i ≫ φ.τ₂) ⋯\n⊢ h₁.f' ≫ φK ≫ h₂.i = φ.τ₁ ≫ h₂.f' ≫ h₂.i",
"ppTerm": "?m.148",
"assigned": true,... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.LeftHomologyData\nh₂ : S₂.LeftHomologyData\nφK : h₁.K ⟶ h₂.K := h₂.liftK (h₁.i ≫ φ.τ₂) ⋯\n⊢ h₁.f' ≫ h₁.i ≫ φ.τ₂ = φ.τ₁ ≫ h₂.f' ≫ h₂.i"
] | LeftHomologyData.liftK_i, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 472,
"column": 2
} | {
"line": 472,
"column": 41
} | {
"line": 472,
"column": 42
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\ninst✝ : S.HasLeftHomology\nA : C\nf₁ f₂ : S.leftHomology ⟶ A\nh : S.leftHomologyπ ≫ f₁ = S.leftHomologyπ ≫ f₂\n⊢ f₁ = f₂",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\ninst✝ : S.HasLeftHomology\nA : C\nf₁ f₂ : S.leftHomology ⟶ A\nh : S.leftHomologyπ ≫ f₁ = S.leftHomologyπ ≫ f₂\n⊢ S.leftHomologyπ ≫ f₁ = S.leftHomologyπ ≫ f₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 481,
"column": 2
} | {
"line": 481,
"column": 35
} | {
"line": 481,
"column": 36
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\ninst✝ : S.HasLeftHomology\nA : C\nf₁ f₂ : A ⟶ S.cycles\nh : f₁ ≫ S.iCycles = f₂ ≫ S.iCycles\n⊢ f₁ = f₂",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheo... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\ninst✝ : S.HasLeftHomology\nA : C\nf₁ f₂ : A ⟶ S.cycles\nh : f₁ ≫ S.iCycles = f₂ ≫ S.iCycles\n⊢ f₁ ≫ S.iCycles = f₂ ≫ S.iCycles"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 807,
"column": 16
} | {
"line": 807,
"column": 82
} | {
"line": 808,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\ne : S₁ ≅ S₂\ninst✝¹ : S₁.HasRightHomology\ninst✝ : S₂.HasRightHomology\n⊢ rightHomologyMap e.hom ≫ rightHomologyMap e.inv = 𝟙 S₁.rightHomology",
"ppTerm": "?m.56",
"assigned": true,
"use... | [] | by rw [← rightHomologyMap_comp, e.hom_inv_id, rightHomologyMap_id] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 882,
"column": 2
} | {
"line": 883,
"column": 47
} | {
"line": 884,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.RightHomologyData\nh₂ : S₂.RightHomologyData\nγ : RightHomologyMapData φ h₁ h₂\ninst✝¹ : S₁.HasRightHomology\ninst✝ : S₂.HasRightHomology\n⊢ rightHomologyMap φ =\n rightHomologyMap... | [
"C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.RightHomologyData\nh₂ : S₂.RightHomologyData\nγ : RightHomologyMapData φ h₁ h₂\ninst✝¹ : S₁.HasRightHomology\ninst✝ : S₂.HasRightHomology\n⊢ rightHomologyMap φ = rightHomologyMap' φ S₁.rightHomol... | rw [← γ.rightHomologyMap'_eq, ← rightHomologyMap'_comp,
← rightHomologyMap'_comp, id_comp, comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.ShortComplex.PreservesHomology | {
"line": 116,
"column": 53
} | {
"line": 116,
"column": 64
} | {
"line": 116,
"column": 65
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasZeroMorphisms D\nS S₁ S₂ : ShortComplex C\nh : S.LeftHomologyData\nF : C ⥤ D\ninst✝¹ : F.PreservesZeroMorphisms\ninst✝ : h.IsPreservedBy F\nthis✝ : PreservesLimit (parall... | [
"C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasZeroMorphisms D\nS S₁ S₂ : ShortComplex C\nh : S.LeftHomologyData\nF : C ⥤ D\ninst✝¹ : F.PreservesZeroMorphisms\ninst✝ : h.IsPreservedBy F\nthis✝ : PreservesLimit (parallelPair S.g 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 873,
"column": 2
} | {
"line": 875,
"column": 82
} | {
"line": 876,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₁.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₂.X₂ := h.i ≫ φ.τ₂\nwi : i ≫ S₂.g = 0\nhi : IsLimit (KernelFork.ofι i wi)\nf' : (KernelFork... | [
"C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₁.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₂.X₂ := h.i ≫ φ.τ₂\nwi : i ≫ S₂.g = 0\nhi : IsLimit (KernelFork.ofι i wi)\nf' : (KernelFork.ofι S₂.f ⋯)... | have hπ : IsColimit (CokernelCofork.ofπ h.π wπ) := CokernelCofork.IsColimit.ofπ _ _
(fun x hx => h.descH x (by rw [← hf', assoc, hx, comp_zero]))
(fun x hx => by simp) (fun x hx b hb => by rw [← cancel_epi h.π, π_descH, hb]) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Homology.ShortComplex.PreservesHomology | {
"line": 201,
"column": 53
} | {
"line": 201,
"column": 64
} | {
"line": 201,
"column": 65
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasZeroMorphisms D\nS S₁ S₂ : ShortComplex C\nh : S.RightHomologyData\nF : C ⥤ D\ninst✝¹ : F.PreservesZeroMorphisms\ninst✝ : h.IsPreservedBy F\nthis✝ : PreservesColimit (par... | [
"C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasZeroMorphisms D\nS S₁ S₂ : ShortComplex C\nh : S.RightHomologyData\nF : C ⥤ D\ninst✝¹ : F.PreservesZeroMorphisms\ninst✝ : h.IsPreservedBy F\nthis✝ : PreservesColimit (parallelPair S.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ShortComplex.Abelian | {
"line": 103,
"column": 6
} | {
"line": 104,
"column": 36
} | {
"line": 104,
"column": 37
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\nS : ShortComplex C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\ninst✝ : HasZeroMorphisms D\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\nf' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯\nhf' : f' = kernel.lift γ f' ⋯ ≫ kernel.... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\nS : ShortComplex C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\ninst✝ : HasZeroMorphisms D\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\nf' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯\nhf' : f' = kernel.lift γ f' ⋯ ≫ kernel.ι γ\nwπ : f'... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.