module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Comma.StructuredArrow.Basic | {
"line": 643,
"column": 39
} | {
"line": 643,
"column": 50
} | {
"line": 643,
"column": 51
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nT T' T'' : D\nY Y' Y'' : C\nS S' : C ⥤ D\nf f' : CostructuredArrow S T\ng : f.left ≅ f'.left\nw : S.map g.hom ≫ f'.hom = f.hom\n⊢ S.map g.hom ≫ f'.hom = f.hom ≫ (Functor.fromPUnit T).map (eqToIso ⋯).hom",
"ppTerm":... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nT T' T'' : D\nY Y' Y'' : C\nS S' : C ⥤ D\nf f' : CostructuredArrow S T\ng : f.left ≅ f'.left\nw : S.map g.hom ≫ f'.hom = f.hom\n⊢ S.map g.hom ≫ f'.hom = f.hom"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Comma.StructuredArrow.Basic | {
"line": 775,
"column": 34
} | {
"line": 775,
"column": 55
} | {
"line": 775,
"column": 56
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nT T' T'' : D\nY Y' Y'' : C\nS✝ S' : C ⥤ D\nA : Type u₃\ninst✝¹ : Category.{v₃, u₃} A\nB : Type u₄\ninst✝ : Category.{v₄, u₄} B\nF : B ⥤ C\nG : C ⥤ D\nS : C\nx✝³ x✝² : CostructuredArrow F S\nx✝¹ x✝ : x✝³ ⟶ x✝²\nh : (po... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nT T' T'' : D\nY Y' Y'' : C\nS✝ S' : C ⥤ D\nA : Type u₃\ninst✝¹ : Category.{v₃, u₃} A\nB : Type u₄\ninst✝ : Category.{v₄, u₄} B\nF : B ⥤ C\nG : C ⥤ D\nS : C\nx✝³ x✝² : CostructuredArrow F S\nx✝¹ x✝ : x✝³ ⟶ x✝²\nh : (post F G S).ma... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Comma.StructuredArrow.Basic | {
"line": 779,
"column": 57
} | {
"line": 779,
"column": 68
} | {
"line": 779,
"column": 69
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nT T' T'' : D\nY Y' Y'' : C\nS✝ S' : C ⥤ D\nA : Type u₃\ninst✝² : Category.{v₃, u₃} A\nB : Type u₄\ninst✝¹ : Category.{v₄, u₄} B\nF : B ⥤ C\nG : C ⥤ D\nS : C\ninst✝ : G.Faithful\nX✝ Y✝ : CostructuredArrow F S\nf : (pos... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nT T' T'' : D\nY Y' Y'' : C\nS✝ S' : C ⥤ D\nA : Type u₃\ninst✝² : Category.{v₃, u₃} A\nB : Type u₄\ninst✝¹ : Category.{v₄, u₄} B\nF : B ⥤ C\nG : C ⥤ D\nS : C\ninst✝ : G.Faithful\nX✝ Y✝ : CostructuredArrow F S\nf : (post F G S).obj... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackCone | {
"line": 332,
"column": 14
} | {
"line": 332,
"column": 25
} | {
"line": 332,
"column": 26
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\nt : PushoutCocone f g\nw : f ≫ t.ι.app WalkingSpan.left = t.ι.app WalkingSpan.zero ≫ 𝟙 t.pt\n⊢ t.ι.app WalkingSpan.zero = f ≫ t.inl",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functo... | [
"C : Type u\ninst✝ : Category.{v, u} C\nX Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\nt : PushoutCocone f g\nw : f ≫ t.ι.app WalkingSpan.left = t.ι.app WalkingSpan.zero ≫ 𝟙 t.pt\n⊢ t.ι.app WalkingSpan.zero = f ≫ t.inl"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Comma.StructuredArrow.Basic | {
"line": 886,
"column": 4
} | {
"line": 886,
"column": 15
} | {
"line": 886,
"column": 16
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nT T' T'' : D\nY Y' Y'' : C\nS S' : C ⥤ D\nA : Type u₃\ninst✝¹ : Category.{v₃, u₃} A\nB : Type u₄\ninst✝ : Category.{v₄, u₄} B\nU : A ⥤ B\nV : B\nF✝ : C ⥤ A\nG✝ : D ⥤ B\nα✝ : F✝ ⋙ U ⟶ S ⋙ G✝\nβ✝ : G✝.obj T ⟶ V\nF : C ≌... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nT T' T'' : D\nY Y' Y'' : C\nS S' : C ⥤ D\nA : Type u₃\ninst✝¹ : Category.{v₃, u₃} A\nB : Type u₄\ninst✝ : Category.{v₄, u₄} B\nU : A ⥤ B\nV : B\nF✝ : C ⥤ A\nG✝ : D ⥤ B\nα✝ : F✝ ⋙ U ⟶ S ⋙ G✝\nβ✝ : G✝.obj T ⟶ V\nF : C ≌ A\nG : D ≌ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Comma.StructuredArrow.Basic | {
"line": 889,
"column": 4
} | {
"line": 889,
"column": 15
} | {
"line": 889,
"column": 16
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nT T' T'' : D\nY Y' Y'' : C\nS S' : C ⥤ D\nA : Type u₃\ninst✝¹ : Category.{v₃, u₃} A\nB : Type u₄\ninst✝ : Category.{v₄, u₄} B\nU : A ⥤ B\nV : B\nF✝ : C ⥤ A\nG✝ : D ⥤ B\nα✝ : F✝ ⋙ U ⟶ S ⋙ G✝\nβ✝ : G✝.obj T ⟶ V\nF : C ≌... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nT T' T'' : D\nY Y' Y'' : C\nS S' : C ⥤ D\nA : Type u₃\ninst✝¹ : Category.{v₃, u₃} A\nB : Type u₄\ninst✝ : Category.{v₄, u₄} B\nU : A ⥤ B\nV : B\nF✝ : C ⥤ A\nG✝ : D ⥤ B\nα✝ : F✝ ⋙ U ⟶ S ⋙ G✝\nβ✝ : G✝.obj T ⟶ V\nF : C ≌ A\nG : D ≌ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Comma.Over.Basic | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 13
} | {
"line": 74,
"column": 14
} | [
{
"pp": "T : Type u₁\ninst✝ : Category.{v₁, u₁} T\nX : T\nf g : Over X\nφ : f ⟶ g\n⊢ Hom.left φ ≫ g.hom = f.hom",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"T : Type u₁\ninst✝ : Category.{v₁, u₁} T\nX : T\nf g : Over X\nφ : f ⟶ g\n⊢ Hom.left φ ≫ g.hom = f.hom"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 15
} | {
"line": 70,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf✝ : X ⟶ Z\ng : Y ⟶ Z\nf : X ⟶ Y\ninst✝ : Mono f\ns : PullbackCone f f\nm : s.pt ⟶ X\nm₁ : m ≫ 𝟙 X = s.fst\nx✝ : m ≫ 𝟙 X = s.snd\n⊢ m = s.fst",
"ppTerm": "?m.93",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoa... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf✝ : X ⟶ Z\ng : Y ⟶ Z\nf : X ⟶ Y\ninst✝ : Mono f\ns : PullbackCone f f\nm : s.pt ⟶ X\nm₁ : m ≫ 𝟙 X = s.fst\nx✝ : m ≫ 𝟙 X = s.snd\n⊢ m = s.fst"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Comma.Over.Basic | {
"line": 477,
"column": 76
} | {
"line": 477,
"column": 87
} | {
"line": 477,
"column": 88
} | [
{
"pp": "T : Type u₁\ninst✝¹ : Category.{v₁, u₁} T\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nX : T\nf✝ g : Over X\nφ : f✝ ⟶ g\nF : T ⥤ D\nh : F.FullyFaithful\nA B : Over X\nf : (post F).obj A ⟶ (post F).obj B\n⊢ F.map (h.preimage (Hom.left f) ≫ B.hom) = F.map A.hom",
"ppTerm": "?m.59",
"assigned": true... | [
"T : Type u₁\ninst✝¹ : Category.{v₁, u₁} T\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nX : T\nf✝ g : Over X\nφ : f✝ ⟶ g\nF : T ⥤ D\nh : F.FullyFaithful\nA B : Over X\nf : (post F).obj A ⟶ (post F).obj B\n⊢ Hom.left f ≫ F.map B.hom = F.map A.hom"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Comma.Over.Basic | {
"line": 535,
"column": 31
} | {
"line": 535,
"column": 42
} | {
"line": 535,
"column": 43
} | [
{
"pp": "T : Type u₁\ninst✝² : Category.{v₁, u₁} T\nD✝ : Type u₂\ninst✝¹ : Category.{v₂, u₂} D✝\nX✝¹ : T\nf✝ g : Over X✝¹\nφ : f✝ ⟶ g\nF : T ⥤ D✝\nJ : Type u_1\ninst✝ : Category.{v_1, u_1} J\nD : J ⥤ T\nX : T\ns : D ⟶ (Functor.const J).obj X\nX✝ Y✝ : J\nf : X✝ ⟶ Y✝\n⊢ D.map f ≫ (mk (s.app Y✝)).hom = (mk (s.app ... | [
"T : Type u₁\ninst✝² : Category.{v₁, u₁} T\nD✝ : Type u₂\ninst✝¹ : Category.{v₂, u₂} D✝\nX✝¹ : T\nf✝ g : Over X✝¹\nφ : f✝ ⟶ g\nF : T ⥤ D✝\nJ : Type u_1\ninst✝ : Category.{v_1, u_1} J\nD : J ⥤ T\nX : T\ns : D ⟶ (Functor.const J).obj X\nX✝ Y✝ : J\nf : X✝ ⟶ Y✝\n⊢ D.map f ≫ s.app Y✝ = s.app X✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Comma.Over.Basic | {
"line": 636,
"column": 12
} | {
"line": 636,
"column": 23
} | {
"line": 636,
"column": 24
} | [
{
"pp": "T : Type u₁\ninst✝¹ : Category.{v₁, u₁} T\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : D ⥤ T\nX : T\nY : Over X\nf : CostructuredArrow (toOver F X) Y\n⊢ F.map (Iso.refl ((𝟭 (CostructuredArrow (toOver F X) Y)).obj f).left.left).hom ≫\n ((costructuredArrowToOverEquivalence.functor F Y ⋙ costructu... | [
"T : Type u₁\ninst✝¹ : Category.{v₁, u₁} T\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : D ⥤ T\nX : T\nY : Over X\nf : CostructuredArrow (toOver F X) Y\n⊢ Over.Hom.left f.hom ≫ Y.hom = f.left.hom"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Comma.Over.Basic | {
"line": 678,
"column": 2
} | {
"line": 678,
"column": 13
} | {
"line": 678,
"column": 14
} | [
{
"pp": "T : Type u₁\ninst✝ : Category.{v₁, u₁} T\nX : T\nf g : Under X\nφ : f ⟶ g\n⊢ f.hom ≫ Hom.right φ = g.hom",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"T : Type u₁\ninst✝ : Category.{v₁, u₁} T\nX : T\nf g : Under X\nφ : f ⟶ g\n⊢ f.hom ≫ Hom.right φ = g.hom"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono | {
"line": 114,
"column": 30
} | {
"line": 114,
"column": 41
} | {
"line": 114,
"column": 42
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf✝ : X ⟶ Z\ng✝ : Y ⟶ Z\nf : X ⟶ W\ng : Y ⟶ W\ni : W ⟶ Z\ninst✝ : Mono i\ns✝ : PullbackCone f g\nH : IsLimit s✝\ns : PullbackCone (f ≫ i) (g ≫ i)\n⊢ (s.fst ≫ f) ≫ i = (s.snd ≫ g) ≫ i",
"ppTerm": "?m.155",
"assigned": true,
"usedConstants":... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf✝ : X ⟶ Z\ng✝ : Y ⟶ Z\nf : X ⟶ W\ng : Y ⟶ W\ni : W ⟶ Z\ninst✝ : Mono i\ns✝ : PullbackCone f g\nH : IsLimit s✝\ns : PullbackCone (f ≫ i) (g ≫ i)\n⊢ s.fst ≫ f ≫ i = s.snd ≫ g ≫ i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono | {
"line": 141,
"column": 6
} | {
"line": 141,
"column": 17
} | {
"line": 141,
"column": 18
} | [
{
"pp": "case h₀\nC✝ : Type u\ninst✝³ : Category.{v, u} C✝\nW✝ X✝ Y✝ Z✝ : C✝\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\nX Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝¹ : HasPullback f g\ninst✝ : HasBinaryProduct X Y\nW : C\ni₁ i₂ : W ⟶ pullback f g\nh : i₁ ≫ prod.lift (pullback.fst f g) (pullback.snd f g) = i₂ ≫ pr... | [
"case h₀\nC✝ : Type u\ninst✝³ : Category.{v, u} C✝\nW✝ X✝ Y✝ Z✝ : C✝\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\nX Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝¹ : HasPullback f g\ninst✝ : HasBinaryProduct X Y\nW : C\ni₁ i₂ : W ⟶ pullback f g\nh : i₁ ≫ prod.lift (pullback.fst f g) (pullback.snd f g) = i₂ ≫ prod.lift (pul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono | {
"line": 142,
"column": 6
} | {
"line": 142,
"column": 17
} | {
"line": 142,
"column": 18
} | [
{
"pp": "case h₁\nC✝ : Type u\ninst✝³ : Category.{v, u} C✝\nW✝ X✝ Y✝ Z✝ : C✝\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\nX Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝¹ : HasPullback f g\ninst✝ : HasBinaryProduct X Y\nW : C\ni₁ i₂ : W ⟶ pullback f g\nh : i₁ ≫ prod.lift (pullback.fst f g) (pullback.snd f g) = i₂ ≫ pr... | [
"case h₁\nC✝ : Type u\ninst✝³ : Category.{v, u} C✝\nW✝ X✝ Y✝ Z✝ : C✝\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\nX Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝¹ : HasPullback f g\ninst✝ : HasBinaryProduct X Y\nW : C\ni₁ i₂ : W ⟶ pullback f g\nh : i₁ ≫ prod.lift (pullback.fst f g) (pullback.snd f g) = i₂ ≫ prod.lift (pul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 37
} | {
"line": 163,
"column": 38
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf : X ⟶ Z\ni : Z ⟶ W\ninst✝ : Mono i\n⊢ HasPullback i (f ≫ i)",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf : X ⟶ Z\ni : Z ⟶ W\ninst✝ : Mono i\n⊢ HasPullback i (f ≫ i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Comma.StructuredArrow.Basic | {
"line": 1023,
"column": 26
} | {
"line": 1023,
"column": 51
} | {
"line": 1023,
"column": 51
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nd : D\nX✝ Y✝ : (StructuredArrow (op d) F.op)ᵒᵖ\nf : X✝ ⟶ Y✝\n⊢ (F.map (Hom.right f.unop).unop ≫ (CostructuredArrow.mk (unop Y✝).hom.unop).hom).op =\n (CostructuredArrow.mk (unop X✝).hom.unop).hom.op",
... | [] | by simp [dsimp% f.unop.w] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 37
} | {
"line": 175,
"column": 38
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf : X ⟶ Z\ni : Z ⟶ W\ninst✝ : Mono i\n⊢ HasPullback (f ≫ i) i",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf : X ⟶ Z\ni : Z ⟶ W\ninst✝ : Mono i\n⊢ HasPullback (f ≫ i) i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Comma.Over.Basic | {
"line": 989,
"column": 78
} | {
"line": 989,
"column": 89
} | {
"line": 989,
"column": 90
} | [
{
"pp": "T : Type u₁\ninst✝¹ : Category.{v₁, u₁} T\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nX : T\nf✝ g : Under X\nφ : f✝ ⟶ g\nF : T ⥤ D\nh : F.FullyFaithful\nA B : Under X\nf : (post F).obj A ⟶ (post F).obj B\n⊢ F.map (A.hom ≫ h.preimage (Hom.right f)) = F.map B.hom",
"ppTerm": "?m.59",
"assigned": t... | [
"T : Type u₁\ninst✝¹ : Category.{v₁, u₁} T\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nX : T\nf✝ g : Under X\nφ : f✝ ⟶ g\nF : T ⥤ D\nh : F.FullyFaithful\nA B : Under X\nf : (post F).obj A ⟶ (post F).obj B\n⊢ F.map A.hom ≫ Hom.right f = F.map B.hom"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Comma.StructuredArrow.Basic | {
"line": 1051,
"column": 26
} | {
"line": 1051,
"column": 51
} | {
"line": 1051,
"column": 51
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nd : D\nX✝ Y✝ : (CostructuredArrow F.op (op d))ᵒᵖ\nf : X✝ ⟶ Y✝\n⊢ ((StructuredArrow.mk (unop X✝).hom.unop).hom ≫ F.map f.unop.left.unop).op =\n (StructuredArrow.mk (unop Y✝).hom.unop).hom.op",
"ppTerm"... | [] | by simp [dsimp% f.unop.w] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Comma.Over.Basic | {
"line": 1044,
"column": 37
} | {
"line": 1044,
"column": 48
} | {
"line": 1044,
"column": 49
} | [
{
"pp": "T : Type u₁\ninst✝² : Category.{v₁, u₁} T\nD✝ : Type u₂\ninst✝¹ : Category.{v₂, u₂} D✝\nX✝¹ : T\nf✝ g : Under X✝¹\nφ : f✝ ⟶ g\nF : T ⥤ D✝\nJ : Type u_1\ninst✝ : Category.{v_1, u_1} J\nD : J ⥤ T\nX : T\ns : (Functor.const J).obj X ⟶ D\nX✝ Y✝ : J\nf : X✝ ⟶ Y✝\n⊢ (mk (s.app X✝)).hom ≫ D.map f = (mk (s.app... | [
"T : Type u₁\ninst✝² : Category.{v₁, u₁} T\nD✝ : Type u₂\ninst✝¹ : Category.{v₂, u₂} D✝\nX✝¹ : T\nf✝ g : Under X✝¹\nφ : f✝ ⟶ g\nF : T ⥤ D✝\nJ : Type u_1\ninst✝ : Category.{v_1, u_1} J\nD : J ⥤ T\nX : T\ns : (Functor.const J).obj X ⟶ D\nX✝ Y✝ : J\nf : X✝ ⟶ Y✝\n⊢ s.app X✝ ≫ D.map f = s.app Y✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono | {
"line": 247,
"column": 4
} | {
"line": 247,
"column": 15
} | {
"line": 247,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf✝ : X ⟶ Y\ng : X ⟶ Z\nf : X ⟶ Y\ninst✝ : Epi f\ns : PushoutCocone f f\nm : Y ⟶ s.pt\nm₁ : 𝟙 Y ≫ m = s.inl\nx✝ : 𝟙 Y ≫ m = s.inr\n⊢ m = s.inl",
"ppTerm": "?m.93",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoa... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf✝ : X ⟶ Y\ng : X ⟶ Z\nf : X ⟶ Y\ninst✝ : Epi f\ns : PushoutCocone f f\nm : Y ⟶ s.pt\nm₁ : 𝟙 Y ≫ m = s.inl\nx✝ : 𝟙 Y ≫ m = s.inr\n⊢ m = s.inl"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Comma.StructuredArrow.Basic | {
"line": 1125,
"column": 73
} | {
"line": 1125,
"column": 84
} | {
"line": 1125,
"column": 85
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\nF : C ⥤ D\nG : D ⥤ E\ne : E\nf : StructuredArrow e G\nX : StructuredArrow f (pre e F G)\n⊢ ((𝟭 (StructuredArrow f (pre e F G))).obj X).right.hom ≫\n (F ⋙ G).map (Iso.ref... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\nF : C ⥤ D\nG : D ⥤ E\ne : E\nf : StructuredArrow e G\nX : StructuredArrow f (pre e F G)\n⊢ X.right.hom = f.hom ≫ G.map (Hom.right X.hom)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono | {
"line": 280,
"column": 10
} | {
"line": 280,
"column": 25
} | {
"line": 281,
"column": 10
} | [
{
"pp": "case h₁\nC : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf✝ : X ⟶ Y\ng✝ : X ⟶ Z\nf : X ⟶ Y\ng : X ⟶ Z\nh : X ⟶ W\ninst✝ : Epi h\nx : W ⟶ Y\ny : W ⟶ Z\nhhx : h ≫ x = f\nhhy : h ≫ y = g\ns : PushoutCocone f g\nhs : IsColimit s\nt : PushoutCocone x y\nm✝ : (mk s.inl s.inr ⋯).pt ⟶ t.pt\nhr : s.inl ≫ m... | [
"case h₁\nC : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf✝ : X ⟶ Y\ng✝ : X ⟶ Z\nf : X ⟶ Y\ng : X ⟶ Z\nh : X ⟶ W\ninst✝ : Epi h\nx : W ⟶ Y\ny : W ⟶ Z\nhhx : h ≫ x = f\nhhy : h ≫ y = g\ns : PushoutCocone f g\nhs : IsColimit s\nt : PushoutCocone x y\nm✝ : (mk s.inl s.inr ⋯).pt ⟶ t.pt\nhr : s.inl ≫ m✝ = t.inl\nh... | simp only [hr'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Comma.Over.Basic | {
"line": 1370,
"column": 39
} | {
"line": 1370,
"column": 50
} | {
"line": 1370,
"column": 51
} | [
{
"pp": "T : Type u₁\ninst✝² : Category.{v₁, u₁} T\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF : C ⥤ T\nG : D ⥤ T\nc : C\nX✝ Y✝ : CostructuredArrow (Comma.fst F G) c\nf : X✝ ⟶ Y✝\n⊢ f.left.left ≫ { left := Over.mk Y✝.hom, right := Y✝.left.right, hom := Y✝.left.hom }.l... | [
"T : Type u₁\ninst✝² : Category.{v₁, u₁} T\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF : C ⥤ T\nG : D ⥤ T\nc : C\nX✝ Y✝ : CostructuredArrow (Comma.fst F G) c\nf : X✝ ⟶ Y✝\n⊢ f.left.left ≫ Y✝.hom = X✝.hom"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Comma.StructuredArrow.Basic | {
"line": 1167,
"column": 8
} | {
"line": 1167,
"column": 19
} | {
"line": 1167,
"column": 20
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\nF : C ⥤ D\nG : D ⥤ E\ne : E\nf : CostructuredArrow G e\nX : CostructuredArrow (pre F G e) f\n⊢ (F ⋙ G).map (Iso.refl ((𝟭 (CostructuredArrow (pre F G e) f)).obj X).left.left).... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\nF : C ⥤ D\nG : D ⥤ E\ne : E\nf : CostructuredArrow G e\nX : CostructuredArrow (pre F G e) f\n⊢ G.map X.hom.left ≫ f.hom = X.left.hom"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono | {
"line": 295,
"column": 29
} | {
"line": 295,
"column": 40
} | {
"line": 295,
"column": 41
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf✝ : X ⟶ Y\ng✝ : X ⟶ Z\nf : X ⟶ Y\ng : X ⟶ Z\nh : W ⟶ X\ninst✝ : Epi h\ns✝ : PushoutCocone f g\nH : IsColimit s✝\ns : PushoutCocone (h ≫ f) (h ≫ g)\n⊢ h ≫ f ≫ s.inl = h ≫ g ≫ s.inr",
"ppTerm": "?m.155",
"assigned": false,
"usedConstants":... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf✝ : X ⟶ Y\ng✝ : X ⟶ Z\nf : X ⟶ Y\ng : X ⟶ Z\nh : W ⟶ X\ninst✝ : Epi h\ns✝ : PushoutCocone f g\nH : IsColimit s✝\ns : PushoutCocone (h ≫ f) (h ≫ g)\n⊢ h ≫ f ≫ s.inl = h ≫ g ≫ s.inr"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono | {
"line": 320,
"column": 6
} | {
"line": 320,
"column": 17
} | {
"line": 320,
"column": 18
} | [
{
"pp": "case h₀\nC✝ : Type u\ninst✝³ : Category.{v, u} C✝\nW✝ X✝ Y✝ Z✝ : C✝\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\nX Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\ninst✝¹ : HasPushout f g\ninst✝ : HasBinaryCoproduct Y Z\nW : C\ni₁ i₂ : pushout f g ⟶ W\nh : coprod.desc (pushout.inl f g) (pushout.inr f g) ≫ i₁ = coprod.... | [
"case h₀\nC✝ : Type u\ninst✝³ : Category.{v, u} C✝\nW✝ X✝ Y✝ Z✝ : C✝\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\nX Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\ninst✝¹ : HasPushout f g\ninst✝ : HasBinaryCoproduct Y Z\nW : C\ni₁ i₂ : pushout f g ⟶ W\nh : coprod.desc (pushout.inl f g) (pushout.inr f g) ≫ i₁ = coprod.desc (pushou... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono | {
"line": 321,
"column": 6
} | {
"line": 321,
"column": 17
} | {
"line": 321,
"column": 18
} | [
{
"pp": "case h₁\nC✝ : Type u\ninst✝³ : Category.{v, u} C✝\nW✝ X✝ Y✝ Z✝ : C✝\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\nX Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\ninst✝¹ : HasPushout f g\ninst✝ : HasBinaryCoproduct Y Z\nW : C\ni₁ i₂ : pushout f g ⟶ W\nh : coprod.desc (pushout.inl f g) (pushout.inr f g) ≫ i₁ = coprod.... | [
"case h₁\nC✝ : Type u\ninst✝³ : Category.{v, u} C✝\nW✝ X✝ Y✝ Z✝ : C✝\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\nX Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\ninst✝¹ : HasPushout f g\ninst✝ : HasBinaryCoproduct Y Z\nW : C\ni₁ i₂ : pushout f g ⟶ W\nh : coprod.desc (pushout.inl f g) (pushout.inr f g) ≫ i₁ = coprod.desc (pushou... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono | {
"line": 341,
"column": 2
} | {
"line": 341,
"column": 37
} | {
"line": 341,
"column": 38
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf : X ⟶ Z\nh : W ⟶ X\ninst✝ : Epi h\n⊢ HasPushout h (h ≫ f)",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf : X ⟶ Z\nh : W ⟶ X\ninst✝ : Epi h\n⊢ HasPushout h (h ≫ f)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono | {
"line": 358,
"column": 2
} | {
"line": 358,
"column": 37
} | {
"line": 358,
"column": 38
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf✝ : X ⟶ Z\nh : W ⟶ X\ninst✝ : Epi h\nf : X ⟶ Y\n⊢ HasPushout (h ≫ f) h",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf✝ : X ⟶ Z\nh : W ⟶ X\ninst✝ : Epi h\nf : X ⟶ Y\n⊢ HasPushout (h ≫ f) h"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Images | {
"line": 109,
"column": 30
} | {
"line": 109,
"column": 41
} | {
"line": 109,
"column": 42
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nf : X ⟶ Y\nI✝ : C\nFm : I✝ ⟶ Y\nm_mono✝¹ : Mono Fm\ne✝¹ : X ⟶ I✝\nFfac : e✝¹ ≫ Fm = f\nFm' : I✝ ⟶ Y\nm_mono✝ : Mono Fm'\ne✝ : X ⟶ I✝\nFfac' : e✝ ≫ Fm' = f\nhm :\n { I := I✝, m := Fm, m_mono := m_mono✝¹, e := e✝¹, fac := Ffac }.m =\n eqToHom ⋯ ≫ { I :=... | [
"C : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nf : X ⟶ Y\nI✝ : C\nFm : I✝ ⟶ Y\nm_mono✝¹ : Mono Fm\ne✝¹ : X ⟶ I✝\nFfac : e✝¹ ≫ Fm = f\nFm' : I✝ ⟶ Y\nm_mono✝ : Mono Fm'\ne✝ : X ⟶ I✝\nFfac' : e✝ ≫ Fm' = f\nhm :\n { I := I✝, m := Fm, m_mono := m_mono✝¹, e := e✝¹, fac := Ffac }.m =\n eqToHom ⋯ ≫ { I := I✝, m := Fm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Images | {
"line": 249,
"column": 4
} | {
"line": 250,
"column": 30
} | {
"line": 250,
"column": 31
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\nf✝ : X ⟶ Y\nF✝ F'✝ : MonoFactorisation f✝\nhF✝ : IsImage F✝\nhF' : IsImage F'✝\nf g : Arrow C\nF : MonoFactorisation f.hom\nhF : IsImage F\nsq : f ⟶ g\ninst✝ : IsIso sq\nF' : MonoFactorisation g.hom\n⊢ hF.lift (F'.ofArrowIso (inv sq)) ≫ F'.m = (F.ofArrow... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\nf✝ : X ⟶ Y\nF✝ F'✝ : MonoFactorisation f✝\nhF✝ : IsImage F✝\nhF' : IsImage F'✝\nf g : Arrow C\nF : MonoFactorisation f.hom\nhF : IsImage F\nsq : f ⟶ g\ninst✝ : IsIso sq\nF' : MonoFactorisation g.hom\n⊢ hF.lift (F'.ofArrowIso (inv sq)) ≫ F'.m = F.m ≫ Arrow.Hom.right ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Images | {
"line": 479,
"column": 67
} | {
"line": 502,
"column": 37
} | {
"line": 504,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf : X ⟶ Y\ninst✝¹ : HasImage f\nW : C\ng h : image f ⟶ W\ninst✝ : HasLimit (parallelPair g h)\nw : factorThruImage f ≫ g = factorThruImage f ≫ h\n⊢ g = h",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.M... | [] | by
let q := equalizer.ι g h
let e' := equalizer.lift _ w
let F' : MonoFactorisation f :=
{ I := equalizer g h
m := q ≫ image.ι f
m_mono := mono_comp _ _
e := e' }
let v := image.lift F'
have t₀ : v ≫ q ≫ image.ι f = image.ι f := image.lift_fac F'
have t : v ≫ q = 𝟙 (image f) :=
(c... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Images | {
"line": 536,
"column": 36
} | {
"line": 538,
"column": 64
} | {
"line": 538,
"column": 64
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf f' : X ⟶ Y\ninst✝¹ : HasImage f\ninst✝ : HasImage f'\nh : f = f'\n⊢ (image.eqToHom h ≫ image.eqToHom ⋯) ≫ image.ι f = 𝟙 (image f) ≫ image.ι f",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.factorThru... | [] | by
subst h
simp [image.eqToHom, Category.assoc, Category.id_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | {
"line": 174,
"column": 30
} | {
"line": 175,
"column": 83
} | {
"line": 175,
"column": 83
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nX : C\nh : 𝟙 X = 0\nY : C\nf : X ⟶ Y\n⊢ f = default",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Inhabited.default",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | [] | by
rw [← id_comp f, ← id_comp (0 : X ⟶ Y), h, zero_comp, zero_comp]; simp only | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | {
"line": 200,
"column": 4
} | {
"line": 200,
"column": 28
} | {
"line": 202,
"column": 0
} | [
{
"pp": "case mpr\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nh : f = 0\n⊢ f ≫ retraction f = 0",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"CategoryTheory.retraction.congr_simp",
"CategoryTheory.CategorySt... | [] | simp only [h, zero_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | {
"line": 202,
"column": 90
} | {
"line": 209,
"column": 12
} | {
"line": 211,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitEpi f\n⊢ IsZero Y ↔ f = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
... | [] | by
rw [iff_id_eq_zero]
constructor
· intro h
rw [← Category.comp_id f, h, comp_zero]
· intro h
rw [← IsSplitEpi.id f]
simp [h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Images | {
"line": 765,
"column": 2
} | {
"line": 765,
"column": 76
} | {
"line": 766,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nf g : Arrow C\ninst✝¹ : HasImage f.hom\ninst✝ : HasImage g.hom\nsq : f ⟶ g\nmap : image f.hom ⟶ image g.hom\nmap_ι : map ≫ image.ι g.hom = image.ι f.hom ≫ Arrow.Hom.right sq\nmap' : image f.hom ⟶ image g.hom\nmap_ι' : map' ≫ image.ι g.hom = image.ι f.hom ≫ Arrow.... | [
"C : Type u\ninst✝² : Category.{v, u} C\nf g : Arrow C\ninst✝¹ : HasImage f.hom\ninst✝ : HasImage g.hom\nsq : f ⟶ g\nmap : image f.hom ⟶ image g.hom\nmap_ι : map ≫ image.ι g.hom = image.ι f.hom ≫ Arrow.Hom.right sq\nmap' : image f.hom ⟶ image g.hom\nmap_ι' : map' ≫ image.ι g.hom = image.ι f.hom ≫ Arrow.Hom.right sq... | have : map ≫ image.ι g.hom = map' ≫ image.ι g.hom := by rw [map_ι, map_ι'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | {
"line": 342,
"column": 2
} | {
"line": 342,
"column": 13
} | {
"line": 342,
"column": 14
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ni : Y ≅ 0\nh : f = f ≫ i.hom ≫ 𝟙 0 ≫ i.inv\n⊢ f = 0",
"ppTerm": "?m.42",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ni : Y ≅ 0\nh : f = f ≫ i.hom ≫ 𝟙 0 ≫ i.inv\n⊢ f = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | {
"line": 349,
"column": 2
} | {
"line": 349,
"column": 13
} | {
"line": 349,
"column": 14
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ni : X ≅ 0\nh : f = i.hom ≫ 𝟙 0 ≫ i.inv ≫ f\n⊢ f = 0",
"ppTerm": "?m.42",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ni : X ≅ 0\nh : f = i.hom ≫ 𝟙 0 ≫ i.inv ≫ f\n⊢ f = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | {
"line": 403,
"column": 76
} | {
"line": 405,
"column": 37
} | {
"line": 407,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nD : Type u'\ninst✝³ : Category.{v', u'} D\ninst✝² : HasZeroObject C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : Epi f\nh : f = 0\n⊢ Y ≅ 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"CategoryTheory.Epi",
"Catego... | [] | by
subst h
apply isoZeroOfEpiZero (X := X) ‹_› | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | {
"line": 442,
"column": 67
} | {
"line": 442,
"column": 78
} | {
"line": 442,
"column": 79
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\ninst✝ : HasZeroMorphisms C\nX : C\n⊢ IsIso 0 ≃ 𝟙 X = 0",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"C : Type u\ninst✝² : Category.{v, u} C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\ninst✝ : HasZeroMorphisms C\nX : C\n⊢ IsIso 0 ≃ 𝟙 X = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | {
"line": 553,
"column": 2
} | {
"line": 553,
"column": 51
} | {
"line": 554,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX Y : C\ninst✝ : HasImage 0\n⊢ ι 0 = 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.MonoFactorisation.I",
"Eq.mpr",
"CategoryTheory.CategorySt... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX Y : C\ninst✝ : HasImage 0\n⊢ lift (monoFactorisationZero X Y) ≫ (monoFactorisationZero X Y).m = 0"
] | rw [← image.lift_fac (monoFactorisationZero X Y)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | {
"line": 689,
"column": 33
} | {
"line": 689,
"column": 44
} | {
"line": 689,
"column": 45
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nD : Type u'\ninst✝³ : Category.{v', u'} D\ninst✝² : HasZeroMorphisms C\nβ : Type w\ninst✝¹ : DecidableEq β\nf : β → C\ninst✝ : HasProduct f\nb : β\nZ✝ : C\nx✝¹ x✝ : Z✝ ⟶ f b\ne : x✝¹ ≫ Pi.ι f b = x✝ ≫ Pi.ι f b\n⊢ x✝¹ = x✝",
"ppTerm": "?m.23",
"assigned": ... | [
"C : Type u\ninst✝⁴ : Category.{v, u} C\nD : Type u'\ninst✝³ : Category.{v', u'} D\ninst✝² : HasZeroMorphisms C\nβ : Type w\ninst✝¹ : DecidableEq β\nf : β → C\ninst✝ : HasProduct f\nb : β\nZ✝ : C\nx✝¹ x✝ : Z✝ ⟶ f b\ne : x✝¹ ≫ Pi.ι f b = x✝ ≫ Pi.ι f b\n⊢ x✝¹ = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | {
"line": 717,
"column": 32
} | {
"line": 717,
"column": 43
} | {
"line": 717,
"column": 44
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nD : Type u'\ninst✝³ : Category.{v', u'} D\ninst✝² : HasZeroMorphisms C\nβ : Type w\ninst✝¹ : DecidableEq β\nf : β → C\ninst✝ : HasCoproduct f\nb : β\nZ✝ : C\nx✝¹ x✝ : f b ⟶ Z✝\ne : Sigma.π f b ≫ x✝¹ = Sigma.π f b ≫ x✝\n⊢ x✝¹ = x✝",
"ppTerm": "?m.23",
"ass... | [
"C : Type u\ninst✝⁴ : Category.{v, u} C\nD : Type u'\ninst✝³ : Category.{v', u'} D\ninst✝² : HasZeroMorphisms C\nβ : Type w\ninst✝¹ : DecidableEq β\nf : β → C\ninst✝ : HasCoproduct f\nb : β\nZ✝ : C\nx✝¹ x✝ : f b ⟶ Z✝\ne : Sigma.π f b ≫ x✝¹ = Sigma.π f b ≫ x✝\n⊢ x✝¹ = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | {
"line": 756,
"column": 33
} | {
"line": 756,
"column": 44
} | {
"line": 756,
"column": 45
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : HasBinaryProduct X Y\nZ✝ : C\nx✝¹ x✝ : Z✝ ⟶ X\ne : x✝¹ ≫ prod.inl X Y = x✝ ≫ prod.inl X Y\n⊢ x✝¹ = x✝",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : HasBinaryProduct X Y\nZ✝ : C\nx✝¹ x✝ : Z✝ ⟶ X\ne : x✝¹ ≫ prod.inl X Y = x✝ ≫ prod.inl X Y\n⊢ x✝¹ = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | {
"line": 759,
"column": 33
} | {
"line": 759,
"column": 44
} | {
"line": 759,
"column": 45
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : HasBinaryProduct X Y\nZ✝ : C\nx✝¹ x✝ : Z✝ ⟶ Y\ne : x✝¹ ≫ prod.inr X Y = x✝ ≫ prod.inr X Y\n⊢ x✝¹ = x✝",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : HasBinaryProduct X Y\nZ✝ : C\nx✝¹ x✝ : Z✝ ⟶ Y\ne : x✝¹ ≫ prod.inr X Y = x✝ ≫ prod.inr X Y\n⊢ x✝¹ = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | {
"line": 779,
"column": 60
} | {
"line": 780,
"column": 19
} | {
"line": 782,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : HasBinaryCoproduct X Y\n⊢ inl ≫ fst X Y = 𝟙 X",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"Categor... | [] | by
simp [coprod.fst] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | {
"line": 784,
"column": 58
} | {
"line": 785,
"column": 19
} | {
"line": 787,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : HasBinaryCoproduct X Y\n⊢ inr ≫ fst X Y = 0",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTh... | [] | by
simp [coprod.fst] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | {
"line": 798,
"column": 32
} | {
"line": 798,
"column": 43
} | {
"line": 798,
"column": 44
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : HasBinaryCoproduct X Y\nZ✝ : C\nx✝¹ x✝ : X ⟶ Z✝\ne : coprod.fst X Y ≫ x✝¹ = coprod.fst X Y ≫ x✝\n⊢ x✝¹ = x✝",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : HasBinaryCoproduct X Y\nZ✝ : C\nx✝¹ x✝ : X ⟶ Z✝\ne : coprod.fst X Y ≫ x✝¹ = coprod.fst X Y ≫ x✝\n⊢ x✝¹ = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | {
"line": 801,
"column": 32
} | {
"line": 801,
"column": 43
} | {
"line": 801,
"column": 44
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : HasBinaryCoproduct X Y\nZ✝ : C\nx✝¹ x✝ : Y ⟶ Z✝\ne : coprod.snd X Y ≫ x✝¹ = coprod.snd X Y ≫ x✝\n⊢ x✝¹ = x✝",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : HasBinaryCoproduct X Y\nZ✝ : C\nx✝¹ x✝ : Y ⟶ Z✝\ne : coprod.snd X Y ≫ x✝¹ = coprod.snd X Y ≫ x✝\n⊢ x✝¹ = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Equalizers | {
"line": 827,
"column": 53
} | {
"line": 827,
"column": 91
} | {
"line": 827,
"column": 91
} | [
{
"pp": "C : Type u\nX✝ Y✝ : C\ninst✝ : Category.{v, u} C\nf✝ g✝ : X✝ ⟶ Y✝\nX Y : C\nf g : X ⟶ Y\nX' Y' : C\nc : Cofork f g\nf' g' : X' ⟶ Y'\nc' : Cofork f' g'\ne₀ : X ≅ X'\ne₁ : Y ≅ Y'\ne : c.pt ≅ c'.pt\ncomm₁ : e₀.hom ≫ f' = f ≫ e₁.hom\ncomm₂ : e₀.hom ≫ g' = g ≫ e₁.hom\ncomm₃ : e₁.inv ≫ c.π ≫ e.hom = c'.π\ni ... | [] | by rw [← comm₃, ← Category.assoc]; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Linear.Basic | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 27
} | {
"line": 139,
"column": 28
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\nR : Type w\ninst✝³ : Semiring R\ninst✝² : Linear R C\nX Y : C\nf : X ⟶ Y\ninst✝¹ : Epi f\nr : R\ninst✝ : Invertible r\nZ✝ : C\ng g' : Y ⟶ Z✝\nH : r • g = r • g'\n⊢ g = g'",
"ppTerm": "?m.109",
"assigned": false,
"usedConstants"... | [
"C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\nR : Type w\ninst✝³ : Semiring R\ninst✝² : Linear R C\nX Y : C\nf : X ⟶ Y\ninst✝¹ : Epi f\nr : R\ninst✝ : Invertible r\nZ✝ : C\ng g' : Y ⟶ Z✝\nH : r • g = r • g'\n⊢ g = g'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Linear.Basic | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 27
} | {
"line": 144,
"column": 28
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\nR : Type w\ninst✝³ : Semiring R\ninst✝² : Linear R C\nX Y : C\nf : X ⟶ Y\ninst✝¹ : Mono f\nr : R\ninst✝ : Invertible r\nZ✝ : C\ng g' : Z✝ ⟶ X\nH : r • g = r • g'\n⊢ g = g'",
"ppTerm": "?m.109",
"assigned": false,
"usedConstants... | [
"C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\nR : Type w\ninst✝³ : Semiring R\ninst✝² : Linear R C\nX Y : C\nf : X ⟶ Y\ninst✝¹ : Mono f\nr : R\ninst✝ : Invertible r\nZ✝ : C\ng g' : Z✝ ⟶ X\nH : r • g = r • g'\n⊢ g = g'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Preserves.Finite | {
"line": 217,
"column": 4
} | {
"line": 218,
"column": 73
} | {
"line": 221,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : PreservesColimitsOfSize.{w, w₂, v₁, v₂, u₁, u₂} F\nJ : Type\nsJ : SmallCategory J\nfJ : FinCategory J\n⊢ PreservesColimitsOfShape J F",
"ppTerm": "?m.20",
"assigned": true,
"usedConstant... | [] | haveI := preservesSmallestColimits_of_preservesColimits F
exact preservesColimitsOfShape_of_equiv (FinCategory.equivAsType J) F | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Preserves.Finite | {
"line": 217,
"column": 4
} | {
"line": 218,
"column": 73
} | {
"line": 221,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : PreservesColimitsOfSize.{w, w₂, v₁, v₂, u₁, u₂} F\nJ : Type\nsJ : SmallCategory J\nfJ : FinCategory J\n⊢ PreservesColimitsOfShape J F",
"ppTerm": "?m.20",
"assigned": true,
"usedConstant... | [] | haveI := preservesSmallestColimits_of_preservesColimits F
exact preservesColimitsOfShape_of_equiv (FinCategory.equivAsType J) F | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Ext | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 15
} | {
"line": 104,
"column": 2
} | [
{
"pp": "M : Type u\n⊢ Injective (@toRightCancelMonoid M)",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"CancelMonoid.toRightCancelMonoid",
"RightCancelMonoid",
"CancelMonoid",
"Eq"
],
"usedFVars": [
"M"
],
"usedGoals": [
{
"new"... | [
"M : Type u\nm₁ m₂ : CancelMonoid M\nh : m₁.toRightCancelMonoid = m₂.toRightCancelMonoid\n⊢ m₁ = m₂"
] | intro m₁ m₂ h | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 188,
"column": 34
} | {
"line": 188,
"column": 45
} | {
"line": 188,
"column": 46
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\nW : C\ng : Y ⟶ W\nh✝ : X ⟶ W\nc : KernelFork h✝\ni : IsLimit c\nhf : Fork.ι c ≫ f = 0\nhfg : f ≫ g = h✝\ns : Fork f 0\nm : s.pt ⟶ (KernelFork.ofι (Fork.ι c) hf).pt\nh : m ≫ Fork.ι (KernelFork.ofι (Fork.ι c) hf) = s.... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\nW : C\ng : Y ⟶ W\nh✝ : X ⟶ W\nc : KernelFork h✝\ni : IsLimit c\nhf : Fork.ι c ≫ f = 0\nhfg : f ≫ g = h✝\ns : Fork f 0\nm : s.pt ⟶ (KernelFork.ofι (Fork.ι c) hf).pt\nh : m ≫ Fork.ι (KernelFork.ofι (Fork.ι c) hf) = s.ι\n⊢ m ≫ For... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 217,
"column": 49
} | {
"line": 217,
"column": 82
} | {
"line": 217,
"column": 83
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nX Y : C\ng : X ⟶ Y\nc : KernelFork g\nhc : IsLimit c\nX' Y' : C\ng' : X' ⟶ Y'\ne : X ≅ X'\niff : ∀ ⦃W : C⦄ (φ : W ⟶ X), φ ≫ g = 0 ↔ φ ≫ e.hom ≫ g' = 0\nW'✝ : C\ns : W'✝ ⟶ X'\nhs : s ≫ g' = 0\nm : W'✝ ⟶ c.pt\nhm ... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nX Y : C\ng : X ⟶ Y\nc : KernelFork g\nhc : IsLimit c\nX' Y' : C\ng' : X' ⟶ Y'\ne : X ≅ X'\niff : ∀ ⦃W : C⦄ (φ : W ⟶ X), φ ≫ g = 0 ↔ φ ≫ e.hom ≫ g' = 0\nW'✝ : C\ns : W'✝ ⟶ X'\nhs : s ≫ g' = 0\nm : W'✝ ⟶ c.pt\nhm : m ≫ Fork.ι... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 227,
"column": 67
} | {
"line": 227,
"column": 78
} | {
"line": 227,
"column": 79
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nX Y : C\ng : X ⟶ Y\nc : KernelFork g\nhc : IsLimit c\nY' : C\ng' : X ⟶ Y'\niff : ∀ ⦃W : C⦄ (φ : W ⟶ X), φ ≫ g = 0 ↔ φ ≫ g' = 0\n⊢ ∀ ⦃W : C⦄ (φ : W ⟶ X), φ ≫ g = 0 ↔ φ ≫ (Iso.refl X).hom ≫ g' = 0",
"ppTerm": ... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nX Y : C\ng : X ⟶ Y\nc : KernelFork g\nhc : IsLimit c\nY' : C\ng' : X ⟶ Y'\niff : ∀ ⦃W : C⦄ (φ : W ⟶ X), φ ≫ g = 0 ↔ φ ≫ g' = 0\n⊢ ∀ ⦃W : C⦄ (φ : W ⟶ X), φ ≫ g = 0 ↔ φ ≫ g' = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 548,
"column": 6
} | {
"line": 548,
"column": 17
} | {
"line": 548,
"column": 18
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\nZ : C\nl : X ⟶ Z\ni : Z ≅ Y\nh✝ : l ≫ i.hom = f\ns✝ : KernelFork f\nhs : IsLimit s✝\ns : Fork l 0\nm : s.pt ⟶ (KernelFork.ofι (Fork.ι s✝) ⋯).pt\nh : m ≫ Fork.ι (KernelFork.ofι (Fork.ι s✝) ⋯) = s.ι\n⊢ m ≫ Fork.ι s✝ =... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\nZ : C\nl : X ⟶ Z\ni : Z ≅ Y\nh✝ : l ≫ i.hom = f\ns✝ : KernelFork f\nhs : IsLimit s✝\ns : Fork l 0\nm : s.pt ⟶ (KernelFork.ofι (Fork.ι s✝) ⋯).pt\nh : m ≫ Fork.ι (KernelFork.ofι (Fork.ι s✝) ⋯) = s.ι\n⊢ m ≫ Fork.ι s✝ = s.ι"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 692,
"column": 6
} | {
"line": 692,
"column": 17
} | {
"line": 692,
"column": 18
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\nW : C\ng : W ⟶ X\nh✝ : W ⟶ Y\nc : CokernelCofork h✝\ni : IsColimit c\nhf : f ≫ Cofork.π c = 0\nhfg : g ≫ f = h✝\ns : Cofork f 0\nm : (CokernelCofork.ofπ (Cofork.π c) hf).pt ⟶ s.pt\nh : Cofork.π (CokernelCofork.ofπ (... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\nW : C\ng : W ⟶ X\nh✝ : W ⟶ Y\nc : CokernelCofork h✝\ni : IsColimit c\nhf : f ≫ Cofork.π c = 0\nhfg : g ≫ f = h✝\ns : Cofork f 0\nm : (CokernelCofork.ofπ (Cofork.π c) hf).pt ⟶ s.pt\nh : Cofork.π (CokernelCofork.ofπ (Cofork.π c) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 722,
"column": 53
} | {
"line": 722,
"column": 85
} | {
"line": 722,
"column": 86
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nX✝ Y✝ : C\nf✝ : X✝ ⟶ Y✝\nX Y : C\nf : X ⟶ Y\nc : CokernelCofork f\nhc : IsColimit c\nX' Y' : C\nf' : X' ⟶ Y'\ne : Y' ≅ Y\niff : ∀ ⦃W : C⦄ (φ : Y ⟶ W), f ≫ φ = 0 ↔ f' ≫ e.hom ≫ φ = 0\nZ'✝ : C\ns : Y' ⟶ Z'✝\nhs : f' ≫ s = 0\nm : c.pt ⟶ Z... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nX✝ Y✝ : C\nf✝ : X✝ ⟶ Y✝\nX Y : C\nf : X ⟶ Y\nc : CokernelCofork f\nhc : IsColimit c\nX' Y' : C\nf' : X' ⟶ Y'\ne : Y' ≅ Y\niff : ∀ ⦃W : C⦄ (φ : Y ⟶ W), f ≫ φ = 0 ↔ f' ≫ e.hom ≫ φ = 0\nZ'✝ : C\ns : Y' ⟶ Z'✝\nhs : f' ≫ s = 0\nm : c.pt ⟶ Z'✝\nhm : (e.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 732,
"column": 75
} | {
"line": 732,
"column": 86
} | {
"line": 732,
"column": 87
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nX✝ Y✝ : C\nf✝ : X✝ ⟶ Y✝\nX Y : C\nf : X ⟶ Y\nc : CokernelCofork f\nhc : IsColimit c\nX' : C\nf' : X' ⟶ Y\niff : ∀ ⦃W : C⦄ (φ : Y ⟶ W), f ≫ φ = 0 ↔ f' ≫ φ = 0\n⊢ ∀ ⦃W : C⦄ (φ : Y ⟶ W), f ≫ φ = 0 ↔ f' ≫ (Iso.refl Y).hom ≫ φ = 0",
"pp... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nX✝ Y✝ : C\nf✝ : X✝ ⟶ Y✝\nX Y : C\nf : X ⟶ Y\nc : CokernelCofork f\nhc : IsColimit c\nX' : C\nf' : X' ⟶ Y\niff : ∀ ⦃W : C⦄ (φ : Y ⟶ W), f ≫ φ = 0 ↔ f' ≫ φ = 0\n⊢ ∀ ⦃W : C⦄ (φ : Y ⟶ W), f ≫ φ = 0 ↔ f' ≫ φ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 1136,
"column": 83
} | {
"line": 1137,
"column": 39
} | {
"line": 1139,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nX✝ Y✝ : C\nf✝ : X✝ ⟶ Y✝\ninst✝ : HasZeroObject C\nX Y : C\nf : X ⟶ Y\nhf : ∀ (Z : C) (g : Y ⟶ Z), f ≫ g = 0 → g = 0\ns : Cofork f 0\nm : (CokernelCofork.ofπ 0 ⋯).pt ⟶ s.pt\nx✝ : Cofork.π (CokernelCofork.ofπ 0 ⋯) ≫ m = s.π\n⊢ m = 0",
... | [] | by
apply HasZeroObject.from_zero_ext | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 1153,
"column": 6
} | {
"line": 1153,
"column": 17
} | {
"line": 1153,
"column": 18
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\nZ : C\nl : Z ⟶ Y\ni : X ≅ Z\nh✝ : i.hom ≫ l = f\ns✝ : CokernelCofork f\nhs : IsColimit s✝\ns : Cofork l 0\nm : (CokernelCofork.ofπ (Cofork.π s✝) ⋯).pt ⟶ s.pt\nh : Cofork.π (CokernelCofork.ofπ (Cofork.π s✝) ⋯) ≫ m = ... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\nZ : C\nl : Z ⟶ Y\ni : X ≅ Z\nh✝ : i.hom ≫ l = f\ns✝ : CokernelCofork f\nhs : IsColimit s✝\ns : Cofork l 0\nm : (CokernelCofork.ofπ (Cofork.π s✝) ⋯).pt ⟶ s.pt\nh : Cofork.π (CokernelCofork.ofπ (Cofork.π s✝) ⋯) ≫ m = s.π\n⊢ Cofor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Biproducts | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 13
} | {
"line": 74,
"column": 14
} | [
{
"pp": "J : Type w\nC : Type uC\ninst✝¹ : Category.{uC', uC} C\ninst✝ : HasZeroMorphisms C\nF : J → C\nB : Bicone F\nj : J\n⊢ B.ι j ≫ B.π j = 𝟙 (F j)",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"J : Type w\nC : Type uC\ninst✝¹ : Category.{uC', uC} C\ninst✝ : HasZeroMorphisms C\nF : J → C\nB : Bicone F\nj : J\n⊢ B.ι j ≫ B.π j = 𝟙 (F j)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Biproducts | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 17
} | {
"line": 78,
"column": 18
} | [
{
"pp": "J : Type w\nC : Type uC\ninst✝¹ : Category.{uC', uC} C\ninst✝ : HasZeroMorphisms C\nF : J → C\nB : Bicone F\nj j' : J\nh : j ≠ j'\n⊢ B.ι j ≫ B.π j' = 0",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"J : Type w\nC : Type uC\ninst✝¹ : Category.{uC', uC} C\ninst✝ : HasZeroMorphisms C\nF : J → C\nB : Bicone F\nj j' : J\nh : j ≠ j'\n⊢ B.ι j ≫ B.π j' = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Biproducts | {
"line": 388,
"column": 6
} | {
"line": 388,
"column": 62
} | {
"line": 389,
"column": 8
} | [
{
"pp": "J : Type w\nC : Type uC\ninst✝² : Category.{uC', uC} C\ninst✝¹ : HasZeroMorphisms C\nK : Type w'\ninst✝ : HasBiproductsOfShape K C\ne : J ≃ K\nF : J → C\nh : LimitBicone (F ∘ ⇑e.symm)\nc : Bicone (F ∘ ⇑e.symm)\nhc : c.IsBilimit\n⊢ LimitBicone F",
"ppTerm": "?m.43",
"assigned": false,
"usedC... | [
"J : Type w\nC : Type uC\ninst✝² : Category.{uC', uC} C\ninst✝¹ : HasZeroMorphisms C\nK : Type w'\ninst✝ : HasBiproductsOfShape K C\ne : J ≃ K\nF : J → C\nh : LimitBicone (F ∘ ⇑e.symm)\nc : Bicone (F ∘ ⇑e.symm)\nhc : c.IsBilimit\n⊢ LimitBicone F"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 1256,
"column": 5
} | {
"line": 1256,
"column": 75
} | {
"line": 1256,
"column": 75
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\nD : Type u₂\ninst✝⁶ : Category.{v₂, u₂} D\ninst✝⁵ : HasZeroMorphisms D\nG : C ⥤ D\ninst✝⁴ : G.PreservesZeroMorphisms\nX' Y' : C\ninst✝³ : HasCokernel f\ninst✝² : HasCokernel (G.map f)\ng : X' ⟶ Y'\ninst✝¹ : HasCoke... | [] | by simp only [← G.map_comp]; exact G.congr_map (cokernel.π_desc _ _ _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Preadditive.AdditiveFunctor | {
"line": 137,
"column": 20
} | {
"line": 137,
"column": 82
} | {
"line": 137,
"column": 82
} | [
{
"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 : Preadditive C\ninst✝² : Preadditive D\nF : C ⥤ D\ninst✝¹ : F.Additive\ninst✝ : HasZeroObject C\n⊢ IsZero (F.obj 0)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by rw [IsZero.iff_id_eq_zero, ← F.map_id, id_zero, F.map_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Preadditive.Biproducts | {
"line": 101,
"column": 10
} | {
"line": 101,
"column": 21
} | {
"line": 101,
"column": 22
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\nJ : Type u_1\ninst✝ : Fintype J\nf : J → C\nb : Bicone f\ntotal : ∑ j, b.π j ≫ b.ι j = 𝟙 b.pt\ns : Cone (Discrete.functor f)\nm : s.pt ⟶ b.toCone.pt\nh : ∀ (j : Discrete J), m ≫ b.toCone.π.app j = s.π.app j\nj : J\na✝ : j ∈ Finset.univ\n⊢... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\nJ : Type u_1\ninst✝ : Fintype J\nf : J → C\nb : Bicone f\ntotal : ∑ j, b.π j ≫ b.ι j = 𝟙 b.pt\ns : Cone (Discrete.functor f)\nm : s.pt ⟶ b.toCone.pt\nh : ∀ (j : Discrete J), m ≫ b.toCone.π.app j = s.π.app j\nj : J\na✝ : j ∈ Finset.univ\n⊢ m ≫ b.π j ≫... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Preadditive.Biproducts | {
"line": 116,
"column": 8
} | {
"line": 116,
"column": 19
} | {
"line": 116,
"column": 20
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\nJ : Type u_1\ninst✝ : Fintype J\nf : J → C\nb : Bicone f\ntotal : ∑ j, b.π j ≫ b.ι j = 𝟙 b.pt\ns : Cocone (Discrete.functor f)\nm : b.toCocone.pt ⟶ s.pt\nh : ∀ (j : Discrete J), b.toCocone.ι.app j ≫ m = s.ι.app j\nj : J\na✝ : j ∈ Finset.u... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\nJ : Type u_1\ninst✝ : Fintype J\nf : J → C\nb : Bicone f\ntotal : ∑ j, b.π j ≫ b.ι j = 𝟙 b.pt\ns : Cocone (Discrete.functor f)\nm : b.toCocone.pt ⟶ s.pt\nh : ∀ (j : Discrete J), b.toCocone.ι.app j ≫ m = s.ι.app j\nj : J\na✝ : j ∈ Finset.univ\n⊢ b.π j... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Preadditive.Biproducts | {
"line": 319,
"column": 8
} | {
"line": 319,
"column": 32
} | {
"line": 319,
"column": 33
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nX Y : C\nb : BinaryBicone X Y\ntotal : b.fst ≫ b.inl + b.snd ≫ b.inr = 𝟙 b.pt\ns : Cone (pair X Y)\nm : s.pt ⟶ b.toCone.pt\nh : ∀ (j : Discrete WalkingPair), m ≫ b.toCone.π.app j = s.π.app j\nhₗ : m ≫ b.fst = s.π.app { as := WalkingPair.le... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nX Y : C\nb : BinaryBicone X Y\ntotal : b.fst ≫ b.inl + b.snd ≫ b.inr = 𝟙 b.pt\ns : Cone (pair X Y)\nm : s.pt ⟶ b.toCone.pt\nh : ∀ (j : Discrete WalkingPair), m ≫ b.toCone.π.app j = s.π.app j\nhₗ : m ≫ b.fst = s.π.app { as := WalkingPair.left }\nhᵣ : m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Preadditive.Biproducts | {
"line": 328,
"column": 8
} | {
"line": 328,
"column": 32
} | {
"line": 328,
"column": 33
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nX Y : C\nb : BinaryBicone X Y\ntotal : b.fst ≫ b.inl + b.snd ≫ b.inr = 𝟙 b.pt\ns : Cocone (pair X Y)\nm : b.toCocone.pt ⟶ s.pt\nh : ∀ (j : Discrete WalkingPair), b.toCocone.ι.app j ≫ m = s.ι.app j\nhₗ : b.inl ≫ m = s.ι.app { as := WalkingP... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nX Y : C\nb : BinaryBicone X Y\ntotal : b.fst ≫ b.inl + b.snd ≫ b.inr = 𝟙 b.pt\ns : Cocone (pair X Y)\nm : b.toCocone.pt ⟶ s.pt\nh : ∀ (j : Discrete WalkingPair), b.toCocone.ι.app j ≫ m = s.ι.app j\nhₗ : b.inl ≫ m = s.ι.app { as := WalkingPair.left }\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Biproducts | {
"line": 920,
"column": 10
} | {
"line": 920,
"column": 21
} | {
"line": 920,
"column": 22
} | [
{
"pp": "case neg\nJ✝ : Type w\nC✝ : Type uC\ninst✝⁷ : Category.{uC', uC} C✝\ninst✝⁶ : HasZeroMorphisms C✝\nD : Type uD\ninst✝⁵ : Category.{uD', uD} D\ninst✝⁴ : HasZeroMorphisms D\nF : J✝ → C✝\nJ : Type w\nK✝ : Type u_1\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nK : Type\ninst✝¹ : Fin... | [
"case neg\nJ✝ : Type w\nC✝ : Type uC\ninst✝⁷ : Category.{uC', uC} C✝\ninst✝⁶ : HasZeroMorphisms C✝\nD : Type uD\ninst✝⁵ : Category.{uD', uD} D\ninst✝⁴ : HasZeroMorphisms D\nF : J✝ → C✝\nJ : Type w\nK✝ : Type u_1\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nK : Type\ninst✝¹ : Finite K\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Preadditive.Biproducts | {
"line": 551,
"column": 8
} | {
"line": 551,
"column": 19
} | {
"line": 551,
"column": 20
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\nX✝ Y✝ : C\ninst✝ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nb : BinaryBicone X Y\nhb : IsLimit b.sndKernelFork\nT : C\nf : T ⟶ X\ng : T ⟶ Y\nm : T ⟶ b.pt\nh₁ : m ≫ BinaryFan.fst b.toCone = f\nh₂ : m ≫ BinaryFan.snd b.toCone = g\n⊢ (m - (f ≫ b.in... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\nX✝ Y✝ : C\ninst✝ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nb : BinaryBicone X Y\nhb : IsLimit b.sndKernelFork\nT : C\nf : T ⟶ X\ng : T ⟶ Y\nm : T ⟶ b.pt\nh₁ : m ≫ BinaryFan.fst b.toCone = f\nh₂ : m ≫ BinaryFan.snd b.toCone = g\n⊢ m ≫ b.fst - f = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Preadditive.Biproducts | {
"line": 552,
"column": 65
} | {
"line": 552,
"column": 76
} | {
"line": 552,
"column": 77
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\nX✝ Y✝ : C\ninst✝ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nb : BinaryBicone X Y\nhb : IsLimit b.sndKernelFork\nT : C\nf : T ⟶ X\ng : T ⟶ Y\nm : T ⟶ b.pt\nh₁ : m ≫ BinaryFan.fst b.toCone = f\nh₂ : m ≫ BinaryFan.snd b.toCone = g\nh₁' : (m - (f ≫ ... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\nX✝ Y✝ : C\ninst✝ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nb : BinaryBicone X Y\nhb : IsLimit b.sndKernelFork\nT : C\nf : T ⟶ X\ng : T ⟶ Y\nm : T ⟶ b.pt\nh₁ : m ≫ BinaryFan.fst b.toCone = f\nh₂ : m ≫ BinaryFan.snd b.toCone = g\nh₁' : (m - (f ≫ b.inl + g ≫ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Finset.Scalar | {
"line": 341,
"column": 96
} | {
"line": 343,
"column": 37
} | {
"line": 345,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝⁴ : DecidableEq β\ninst✝³ : DecidableEq γ\ninst✝² : SMul αᵐᵒᵖ β\ninst✝¹ : SMul β γ\ninst✝ : SMul α γ\na : α\ns : Finset β\nt : Finset γ\nh : ∀ (a : α) (b : β) (c : γ), (op a • b) • c = b • a • c\n⊢ (op a • s) • t = s • a • t",
"ppTerm": "?m.37",
"a... | [] | by
ext
simp [mem_smul, mem_smul_finset, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Biproducts | {
"line": 958,
"column": 10
} | {
"line": 958,
"column": 21
} | {
"line": 958,
"column": 22
} | [
{
"pp": "case neg\nJ✝ : Type w\nC✝ : Type uC\ninst✝⁷ : Category.{uC', uC} C✝\ninst✝⁶ : HasZeroMorphisms C✝\nD : Type uD\ninst✝⁵ : Category.{uD', uD} D\ninst✝⁴ : HasZeroMorphisms D\nF : J✝ → C✝\nJ : Type w\nK✝ : Type u_1\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nK : Type\ninst✝¹ : Fin... | [
"case neg\nJ✝ : Type w\nC✝ : Type uC\ninst✝⁷ : Category.{uC', uC} C✝\ninst✝⁶ : HasZeroMorphisms C✝\nD : Type uD\ninst✝⁵ : Category.{uD', uD} D\ninst✝⁴ : HasZeroMorphisms D\nF : J✝ → C✝\nJ : Type w\nK✝ : Type u_1\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nK : Type\ninst✝¹ : Finite K\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Fin.Tuple | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 48
} | {
"line": 180,
"column": 49
} | [
{
"pp": "n : ℕ\np : Fin (n + 1)\nh : p ≠ last n\nx : { x // x ≠ p }\n⊢ some ((finSuccAboveEquiv p).symm x) = some ((p.castLT ⋯).predAbove ↑x)",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Fin.succAbove",
"Eq.mpr",
"Equiv.instEquivLike",
"congrArg",
"instDec... | [
"n : ℕ\np : Fin (n + 1)\nh : p ≠ last n\nx : { x // x ≠ p }\n⊢ ↑x = p.succAbove ((p.castLT ⋯).predAbove ↑x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Preadditive.Biproducts | {
"line": 567,
"column": 65
} | {
"line": 567,
"column": 76
} | {
"line": 567,
"column": 77
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\nX✝ Y✝ : C\ninst✝ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nb : BinaryBicone X Y\nhb : IsLimit b.fstKernelFork\nT : C\nf : T ⟶ X\ng : T ⟶ Y\nm : T ⟶ b.pt\nh₁ : m ≫ BinaryFan.fst b.toCone = f\nh₂ : m ≫ BinaryFan.snd b.toCone = g\n⊢ (m - (f ≫ b.in... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\nX✝ Y✝ : C\ninst✝ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nb : BinaryBicone X Y\nhb : IsLimit b.fstKernelFork\nT : C\nf : T ⟶ X\ng : T ⟶ Y\nm : T ⟶ b.pt\nh₁ : m ≫ BinaryFan.fst b.toCone = f\nh₂ : m ≫ BinaryFan.snd b.toCone = g\n⊢ m ≫ b.fst - f = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Preadditive.Biproducts | {
"line": 568,
"column": 65
} | {
"line": 568,
"column": 76
} | {
"line": 568,
"column": 77
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\nX✝ Y✝ : C\ninst✝ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nb : BinaryBicone X Y\nhb : IsLimit b.fstKernelFork\nT : C\nf : T ⟶ X\ng : T ⟶ Y\nm : T ⟶ b.pt\nh₁ : m ≫ BinaryFan.fst b.toCone = f\nh₂ : m ≫ BinaryFan.snd b.toCone = g\nh₁' : (m - (f ≫ ... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\nX✝ Y✝ : C\ninst✝ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nb : BinaryBicone X Y\nhb : IsLimit b.fstKernelFork\nT : C\nf : T ⟶ X\ng : T ⟶ Y\nm : T ⟶ b.pt\nh₁ : m ≫ BinaryFan.fst b.toCone = f\nh₂ : m ≫ BinaryFan.snd b.toCone = g\nh₁' : (m - (f ≫ b.inl + g ≫ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Preadditive.Biproducts | {
"line": 583,
"column": 65
} | {
"line": 583,
"column": 76
} | {
"line": 583,
"column": 77
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\nX✝ Y✝ : C\ninst✝ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nb : BinaryBicone X Y\nhb : IsColimit b.inrCokernelCofork\nT : C\nf : X ⟶ T\ng : Y ⟶ T\nm : b.pt ⟶ T\nh₁ : BinaryCofan.inl b.toCocone ≫ m = f\nh₂ : BinaryCofan.inr b.toCocone ≫ m = g\n⊢ ... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\nX✝ Y✝ : C\ninst✝ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nb : BinaryBicone X Y\nhb : IsColimit b.inrCokernelCofork\nT : C\nf : X ⟶ T\ng : Y ⟶ T\nm : b.pt ⟶ T\nh₁ : BinaryCofan.inl b.toCocone ≫ m = f\nh₂ : BinaryCofan.inr b.toCocone ≫ m = g\n⊢ b.inl ≫ m - ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Preadditive.Biproducts | {
"line": 584,
"column": 65
} | {
"line": 584,
"column": 76
} | {
"line": 584,
"column": 77
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\nX✝ Y✝ : C\ninst✝ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nb : BinaryBicone X Y\nhb : IsColimit b.inrCokernelCofork\nT : C\nf : X ⟶ T\ng : Y ⟶ T\nm : b.pt ⟶ T\nh₁ : BinaryCofan.inl b.toCocone ≫ m = f\nh₂ : BinaryCofan.inr b.toCocone ≫ m = g\nh₁... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\nX✝ Y✝ : C\ninst✝ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nb : BinaryBicone X Y\nhb : IsColimit b.inrCokernelCofork\nT : C\nf : X ⟶ T\ng : Y ⟶ T\nm : b.pt ⟶ T\nh₁ : BinaryCofan.inl b.toCocone ≫ m = f\nh₂ : BinaryCofan.inr b.toCocone ≫ m = g\nh₁' : b.inl ≫ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Preadditive.Biproducts | {
"line": 599,
"column": 65
} | {
"line": 599,
"column": 76
} | {
"line": 599,
"column": 77
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\nX✝ Y✝ : C\ninst✝ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nb : BinaryBicone X Y\nhb : IsColimit b.inlCokernelCofork\nT : C\nf : X ⟶ T\ng : Y ⟶ T\nm : b.pt ⟶ T\nh₁ : BinaryCofan.inl b.toCocone ≫ m = f\nh₂ : BinaryCofan.inr b.toCocone ≫ m = g\n⊢ ... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\nX✝ Y✝ : C\ninst✝ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nb : BinaryBicone X Y\nhb : IsColimit b.inlCokernelCofork\nT : C\nf : X ⟶ T\ng : Y ⟶ T\nm : b.pt ⟶ T\nh₁ : BinaryCofan.inl b.toCocone ≫ m = f\nh₂ : BinaryCofan.inr b.toCocone ≫ m = g\n⊢ b.inl ≫ m - ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Preadditive.Biproducts | {
"line": 600,
"column": 65
} | {
"line": 600,
"column": 76
} | {
"line": 600,
"column": 77
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\nX✝ Y✝ : C\ninst✝ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nb : BinaryBicone X Y\nhb : IsColimit b.inlCokernelCofork\nT : C\nf : X ⟶ T\ng : Y ⟶ T\nm : b.pt ⟶ T\nh₁ : BinaryCofan.inl b.toCocone ≫ m = f\nh₂ : BinaryCofan.inr b.toCocone ≫ m = g\nh₁... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\nX✝ Y✝ : C\ninst✝ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nb : BinaryBicone X Y\nhb : IsColimit b.inlCokernelCofork\nT : C\nf : X ⟶ T\ng : Y ⟶ T\nm : b.pt ⟶ T\nh₁ : BinaryCofan.inl b.toCocone ≫ m = f\nh₂ : BinaryCofan.inr b.toCocone ≫ m = g\nh₁' : b.inl ≫ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Antidiag.Finsupp | {
"line": 47,
"column": 4
} | {
"line": 47,
"column": 15
} | {
"line": 47,
"column": 16
} | [
{
"pp": "ι : Type u_1\nμ : Type u_2\nμ' : Type u_3\ninst✝³ : DecidableEq ι\ninst✝² : AddCommMonoid μ\ninst✝¹ : HasAntidiagonal μ\ninst✝ : DecidableEq μ\ns✝ : Finset ι\nn✝ : μ\nf✝ : ι →₀ μ\ns : Finset ι\nn : μ\nf : ↥(s.piAntidiag n)\n⊢ ∀ (a : ι), a ∈ {x ∈ s | ↑f x ≠ 0} ↔ ↑f a ≠ 0",
"ppTerm": "?m.62",
"as... | [
"ι : Type u_1\nμ : Type u_2\nμ' : Type u_3\ninst✝³ : DecidableEq ι\ninst✝² : AddCommMonoid μ\ninst✝¹ : HasAntidiagonal μ\ninst✝ : DecidableEq μ\ns✝ : Finset ι\nn✝ : μ\nf✝ : ι →₀ μ\ns : Finset ι\nn : μ\nf : ↥(s.piAntidiag n)\n⊢ ∀ (a : ι), ¬↑f a = 0 → a ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Antidiag.Finsupp | {
"line": 105,
"column": 12
} | {
"line": 105,
"column": 70
} | {
"line": 105,
"column": 71
} | [
{
"pp": "case inr\nι : Type u_1\nμ : Type u_2\nμ' : Type u_3\ninst✝³ : DecidableEq ι\ninst✝² : AddCommMonoid μ\ninst✝¹ : HasAntidiagonal μ\ninst✝ : DecidableEq μ\ns✝ : Finset ι\nn✝ : μ\nf✝ : ι →₀ μ\na : ι\ns : Finset ι\nh : a ∉ s\nn : μ\np : μ × μ\nx✝¹ x✝ : ↥(s.finsuppAntidiag p.2)\nf : ι →₀ μ\nhf✝ : f ∈ s.fins... | [
"case inr\nι : Type u_1\nμ : Type u_2\nμ' : Type u_3\ninst✝³ : DecidableEq ι\ninst✝² : AddCommMonoid μ\ninst✝¹ : HasAntidiagonal μ\ninst✝ : DecidableEq μ\ns✝ : Finset ι\nn✝ : μ\nf✝ : ι →₀ μ\na : ι\ns : Finset ι\nh : a ∉ s\nn : μ\np : μ × μ\nx✝¹ x✝ : ↥(s.finsuppAntidiag p.2)\nf : ι →₀ μ\nhf✝ : f ∈ s.finsuppAntidiag ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Antidiag.Pi | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 15
} | {
"line": 117,
"column": 16
} | [
{
"pp": "case refine_1\nι : Type u_1\nμ : Type u_2\nμ' : Type u_3\ninst✝³ : DecidableEq ι\ninst✝² : AddCommMonoid μ\ninst✝¹ : HasAntidiagonal μ\ninst✝ : DecidableEq μ\nn✝ : μ\ns : Finset ι\nn : μ\ne : ↥s ≃ Fin #s\nf g : Fin #s → μ\nhfg : (fun f i ↦ if hi : i ∈ s then f (e ⟨i, hi⟩) else 0) f = (fun f i ↦ if hi :... | [
"case refine_1\nι : Type u_1\nμ : Type u_2\nμ' : Type u_3\ninst✝³ : DecidableEq ι\ninst✝² : AddCommMonoid μ\ninst✝¹ : HasAntidiagonal μ\ninst✝ : DecidableEq μ\nn✝ : μ\ns : Finset ι\nn : μ\ne : ↥s ≃ Fin #s\nf g : Fin #s → μ\nhfg : (fun f i ↦ if hi : i ∈ s then f (e ⟨i, hi⟩) else 0) f = (fun f i ↦ if hi : i ∈ s then ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Fin.Tuple.NatAntidiagonal | {
"line": 153,
"column": 4
} | {
"line": 159,
"column": 11
} | {
"line": 161,
"column": 0
} | [
{
"pp": "k n : ℕ\n⊢ Pairwise\n (fun a₁ a₂ ↦\n ∀ a ∈ antidiagonalTuple k a₁.2,\n ∀ a_2 ∈ antidiagonalTuple k a₂.2,\n a₁.1 < a₂.1 ∨ a₁.1 = a₂.1 ∧ Pi.Lex (fun x1 x2 ↦ x1 < x2) (fun i x1 x2 ↦ x1 < x2) a a_2)\n (antidiagonal n)",
"ppTerm": "?m.87",
"assigned": true,
"usedConsta... | [] | induction n with
| zero =>
rw [antidiagonal_zero]
exact List.pairwise_singleton _ _
| succ n n_ih =>
simp
grind | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.Enumerative.Partition.Basic | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 27
} | {
"line": 72,
"column": 28
} | [
{
"pp": "n : ℕ\np : n.Partition\nm : ℕ\nh : m ∈ p.parts\n⊢ m ≤ n",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\np : n.Partition\nm : ℕ\nh : m ∈ p.parts\n⊢ m ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.Partition.Basic | {
"line": 84,
"column": 21
} | {
"line": 84,
"column": 32
} | {
"line": 84,
"column": 33
} | [
{
"pp": "n : ℕ\nb : List ℕ\nhb₁ : ∀ {i : ℕ}, i ∈ ⟦b⟧ → 0 < i\nhb₂ : sum ⟦b⟧ = n\n⊢ b.sum = n",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nb : List ℕ\nhb₁ : ∀ {i : ℕ}, i ∈ ⟦b⟧ → 0 < i\nhb₂ : sum ⟦b⟧ = n\n⊢ b.sum = n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.Composition | {
"line": 173,
"column": 6
} | {
"line": 173,
"column": 20
} | {
"line": 173,
"column": 20
} | [
{
"pp": "n : ℕ\nc : Composition n\ni : ℕ\nh : i ∈ c.blocks\n⊢ i ≤ n",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"congrArg",
"Composition.blocks",
"List.sum",
"id",
"LE.le",
"instLENat",
"Composition.... | [
"n : ℕ\nc : Composition n\ni : ℕ\nh : i ∈ c.blocks\n⊢ i ≤ c.blocks.sum"
] | ← c.blocks_sum | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Enumerative.Composition | {
"line": 195,
"column": 18
} | {
"line": 195,
"column": 32
} | {
"line": 195,
"column": 32
} | [
{
"pp": "n : ℕ\nc : Composition n\n| n",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"congrArg",
"Composition.blocks",
"List.sum",
"Composition.blocks_sum",
"Nat",
"instAddNat",
"Eq.symm",
"Eq.trans",
"Mul... | [
"n : ℕ\nc : Composition n\n| c.blocks.sum"
] | ← c.blocks_sum | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Combinatorics.Enumerative.Composition | {
"line": 202,
"column": 4
} | {
"line": 202,
"column": 15
} | {
"line": 202,
"column": 16
} | [
{
"pp": "case mp\nn : ℕ\nc : Composition n\nh : c.blocks = []\n⊢ n = 0",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mp\nn : ℕ\nc : Composition n\nh : c.blocks = []\n⊢ n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Antidiag.Pi | {
"line": 168,
"column": 18
} | {
"line": 168,
"column": 51
} | {
"line": 168,
"column": 52
} | [
{
"pp": "ι : Type u_1\nμ : Type u_2\ninst✝³ : DecidableEq ι\ninst✝² : AddCancelCommMonoid μ\ninst✝¹ : HasAntidiagonal μ\ninst✝ : DecidableEq μ\ni : ι\ns : Finset ι\nhi : i ∉ s\nn a c : μ\nf : ι → μ\nhab : (a, s.sum f) ∈ ↑(antidiagonal n)\ng : ι → μ\nhcd : (c, s.sum g) ∈ ↑(antidiagonal n)\nhfg : ¬s.sum f = s.sum... | [
"ι : Type u_1\nμ : Type u_2\ninst✝³ : DecidableEq ι\ninst✝² : AddCancelCommMonoid μ\ninst✝¹ : HasAntidiagonal μ\ninst✝ : DecidableEq μ\ni : ι\ns : Finset ι\nhi : i ∉ s\nn a c : μ\nf : ι → μ\nhab : (a, s.sum f) ∈ ↑(antidiagonal n)\ng : ι → μ\nhcd : (c, s.sum g) ∈ ↑(antidiagonal n)\nhfg : ¬s.sum f = s.sum g\nhgf : (g... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.Partition.Basic | {
"line": 180,
"column": 35
} | {
"line": 180,
"column": 46
} | {
"line": 180,
"column": 47
} | [
{
"pp": "p : Partition 1\nh : p.parts = replicate p.parts.card 1\n⊢ p.parts.card = 1",
"ppTerm": "?m.50",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : Partition 1\nh : p.parts = replicate p.parts.card 1\n⊢ p.parts.card = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.Partition.Basic | {
"line": 242,
"column": 8
} | {
"line": 242,
"column": 35
} | {
"line": 242,
"column": 36
} | [
{
"pp": "n✝ : ℕ\nσ : Type u_1\nτ : Type u_2\ninst✝¹ : DecidableEq σ\ninst✝ : DecidableEq τ\nn a : ℕ\nha1 : 1 ≤ a\nha : a ≤ n\np : { p // a ∈ p.parts }\n⊢ a + ((↑p).parts.erase a).sum = n",
"ppTerm": "?m.94",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n✝ : ℕ\nσ : Type u_1\nτ : Type u_2\ninst✝¹ : DecidableEq σ\ninst✝ : DecidableEq τ\nn a : ℕ\nha1 : 1 ≤ a\nha : a ≤ n\np : { p // a ∈ p.parts }\n⊢ a + ((↑p).parts.erase a).sum = n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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