mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Monoidal.Functor

Goal state before:
pp: C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : MonoidalCategory C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : MonoidalCategory D
E : Type u₃
inst✝³ : Category.{v₃, u₃} E
inst✝² : MonoidalCategory E
C' : Type u₁'
inst✝¹ : Category.{v₁', u₁'} C'
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝ : F.OplaxMonoidal
X Y Z : D
this :
  ∀ {X Y : D} (f : X ⟶ Y) (X' : D) {Z : D} (h : Y ⊗ X' ⟶ Z),
    F.map (G.map f) ▷ F.obj (G.obj X') ≫ (adj.counit.app Y ⊗ₘ adj.counit.app X') ≫ h =
      (adj.counit.app X ⊗ₘ adj.counit.app X') ≫ f ▷ X' ≫ h
⊢ F.map (α_ (G.obj X) (G.obj Y) (G.obj Z)).hom ≫
      δ F (G.obj X) (G.obj Y ⊗ G.obj Z) ≫
        F.obj (G.obj X) ◁ δ F (G.obj Y) (G.obj Z) ≫
          adj.counit.app X ▷ (F.obj (G.obj Y) ⊗ F.obj (G.obj Z)) ≫ X ◁ (adj.counit.app Y ⊗ₘ adj.counit.app Z) =
    F.map (α_ (G.obj X) (G.obj Y) (G.obj Z)).hom ≫
      δ F (G.obj X) (G.obj Y ⊗ G.obj Z) ≫
        F.obj (G.obj X) ◁ δ F (G.obj Y) (G.obj Z) ≫
          F.obj (G.obj X) ◁ (adj.counit.app Y ⊗ₘ adj.counit.app Z) ≫ adj.counit.app X ▷ (Y ⊗ Z)

Tactic:
whisker_exchange

Goal state after:
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : MonoidalCategory C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : MonoidalCategory D
E : Type u₃
inst✝³ : Category.{v₃, u₃} E
inst✝² : MonoidalCategory E
C' : Type u₁'
inst✝¹ : Category.{v₁', u₁'} C'
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝ : F.OplaxMonoidal
X Y Z : D
this :
  ∀ {X Y : D} (f : X ⟶ Y) (X' : D) {Z : D} (h : Y ⊗ X' ⟶ Z),
    F.map (G.map f) ▷ F.obj (G.obj X') ≫ (adj.counit.app Y ⊗ₘ adj.counit.app X') ≫ h =
      (adj.counit.app X ⊗ₘ adj.counit.app X') ≫ f ▷ X' ≫ h
⊢ F.map (α_ (G.obj X) (G.obj Y) (G.obj Z)).hom ≫
      δ F (G.obj X) (G.obj Y ⊗ G.obj Z) ≫
        F.obj (G.obj X) ◁ δ F (G.obj Y) (G.obj Z) ≫
          adj.counit.app X ▷ (F.obj (G.obj Y) ⊗ F.obj (G.obj Z)) ≫ X ◁ (adj.counit.app Y ⊗ₘ adj.counit.app Z) =
    F.map (α_ (G.obj X) (G.obj Y) (G.obj Z)).hom ≫
      δ F (G.obj X) (G.obj Y ⊗ G.obj Z) ≫
        F.obj (G.obj X) ◁ δ F (G.obj Y) (G.obj Z) ≫
          adj.counit.app X ▷ (F.obj (G.obj Y) ⊗ F.obj (G.obj Z)) ≫ X ◁ (adj.counit.app Y ⊗ₘ adj.counit.app Z)
Module: Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
G : C ⥤ D
W X Y : C
f : W ⟶ X
g : W ⟶ Y
inst✝ : PreservesColimit (span f g) G
c : Cocone (span g f)
hc : IsColimit c
⊢ IsColimit (G.mapCocone c)

Tactic:
apply (IsColimit.precomposeHomEquiv (diagramIsoSpan.{v₂} _).symm _).toFun
    apply IsColimit.ofIsoColimit _ (PushoutCocone.isoMk _).symm
    apply PushoutCocone.isColimitOfFlip
    apply (isColimitMapCoconePushoutCoconeEquiv _ _).toFun
    · -- Need to unfold these to allow the `PreservesColimit` instance to be found.
      dsimp only [span_map_fst, span_map_snd]
      exact isColimitOfPreserves _ (PushoutCocone.flipIsColimit hc)

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Functor

Goal state before:
pp: C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : MonoidalCategory C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : MonoidalCategory D
E : Type u₃
inst✝³ : Category.{v₃, u₃} E
inst✝² : MonoidalCategory E
C' : Type u₁'
inst✝¹ : Category.{v₁', u₁'} C'
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝ : F.OplaxMonoidal
X Y Z : D
⊢ (adj.homEquiv ((G.obj X ⊗ G.obj Y) ⊗ G.obj Z) (X ⊗ Y ⊗ Z)).symm
      ((adj.homEquiv (G.obj X ⊗ G.obj Y) (X ⊗ Y)) (δ F (G.obj X) (G.obj Y) ≫ (adj.counit.app X ⊗ₘ adj.counit.app Y)) ▷
          G.obj Z ≫
        (adj.homEquiv (G.obj (X ⊗ Y) ⊗ G.obj Z) ((X ⊗ Y) ⊗ Z))
            (δ F (G.obj (X ⊗ Y)) (G.obj Z) ≫ (adj.counit.app (X ⊗ Y) ⊗ₘ adj.counit.app Z)) ≫
          G.map (α_ X Y Z).hom) =
    (adj.homEquiv ((G.obj X ⊗ G.obj Y) ⊗ G.obj Z) (X ⊗ Y ⊗ Z)).symm
      ((α_ (G.obj X) (G.obj Y) (G.obj Z)).hom ≫
        G.obj X ◁
            (adj.homEquiv (G.obj Y ⊗ G.obj Z) (Y ⊗ Z))
              (δ F (G.obj Y) (G.obj Z) ≫ (adj.counit.app Y ⊗ₘ adj.counit.app Z)) ≫
          (adj.homEquiv (G.obj X ⊗ G.obj (Y ⊗ Z)) (X ⊗ Y ⊗ Z))
            (δ F (G.obj X) (G.obj (Y ⊗ Z)) ≫ (adj.counit.app X ⊗ₘ adj.counit.app (Y ⊗ Z))))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
G : C ⥤ D
W X Y : C
f : W ⟶ X
g : W ⟶ Y
inst✝ : PreservesColimit (span f g) G
c : Cocone (span g f)
hc : IsColimit c
⊢ IsColimit (G.mapCocone c)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
G : C ⥤ D
W X Y : C
f : W ⟶ X
g : W ⟶ Y
inst✝² : PreservesColimit (span f g) G
inst✝¹ : HasPushout f g
inst✝ : HasPushout (G.map f) (G.map g)
⊢ pushout.inl (G.map f) (G.map g) ≫ (iso G f g).hom = G.map (pushout.inl f g)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
G : C ⥤ D
W X Y : C
f : W ⟶ X
g : W ⟶ Y
inst✝² : PreservesColimit (span f g) G
inst✝¹ : HasPushout f g
inst✝ : HasPushout (G.map f) (G.map g)
⊢ pushout.inr (G.map f) (G.map g) ≫ (iso G f g).hom = G.map (pushout.inr f g)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
G : C ⥤ D
W X Y : C
f : W ⟶ X
g : W ⟶ Y
inst✝² : PreservesColimit (span f g) G
inst✝¹ : HasPushout f g
inst✝ : HasPushout (G.map f) (G.map g)
⊢ G.map (pushout.inl f g) ≫ (iso G f g).inv = pushout.inl (G.map f) (G.map g)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
G : C ⥤ D
W X Y : C
f : W ⟶ X
g : W ⟶ Y
inst✝² : PreservesColimit (span f g) G
inst✝¹ : HasPushout f g
inst✝ : HasPushout (G.map f) (G.map g)
⊢ G.map (pushout.inr f g) ≫ (iso G f g).inv = pushout.inr (G.map f) (G.map g)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Braided.Basic

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X Y Z : C
⊢ (β_ (X ⊗ Y) Z).hom = (α_ X Y Z).hom ≫ X ◁ (β_ Y Z).hom ≫ (α_ X Z Y).inv ≫ (β_ X Z).hom ▷ Y ≫ (α_ Z X Y).hom

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
inst✝¹ : HasPullback f g
inst✝ : HasPullback (G.map f) (G.map g)
i : IsIso (pullbackComparison G f g)
⊢ PreservesLimit (cospan f g) G

Tactic:
apply preservesLimit_of_preserves_limit_cone (pullbackIsPullback f g)
  apply (isLimitMapConePullbackConeEquiv _ _).symm _
  exact @IsLimit.ofPointIso _ _ _ _ _ _ _ (limit.isLimit (cospan (G.map f) (G.map g))) i

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
inst✝¹ : HasPullback f g
inst✝ : HasPullback (G.map f) (G.map g)
i : IsIso (pullbackComparison G f g)
⊢ PreservesLimit (cospan f g) G

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Braided.Basic

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X Y Z : C
⊢ (β_ X (Y ⊗ Z)).hom = (α_ X Y Z).inv ≫ (β_ X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫ Y ◁ (β_ X Z).hom ≫ (α_ Y Z X).inv

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
G : C ⥤ D
X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
inst✝² : HasPullback f g
inst✝¹ : HasPullback (G.map f) (G.map g)
inst✝ : PreservesLimit (cospan f g) G
⊢ IsIso (pullbackComparison G f g)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Braided.Basic

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X Y Z : C
⊢ inv (β_ (X ⊗ Y) Z).inv = inv ((α_ Z X Y).inv ≫ (β_ X Z).inv ▷ Y ≫ (α_ X Z Y).hom ≫ X ◁ (β_ Y Z).inv ≫ (α_ X Y Z).inv)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
X Y Z : C
f : X ⟶ Y
g : X ⟶ Z
inst✝¹ : HasPushout f g
inst✝ : HasPushout (G.map f) (G.map g)
i : IsIso (pushoutComparison G f g)
⊢ PreservesColimit (span f g) G

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Braided.Basic

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X Y Z : C
⊢ inv (β_ X (Y ⊗ Z)).inv = inv ((α_ Y Z X).hom ≫ Y ◁ (β_ X Z).inv ≫ (α_ Y X Z).inv ≫ (β_ X Y).inv ▷ Z ≫ (α_ X Y Z).hom)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
G : C ⥤ D
X Y Z : C
f : X ⟶ Y
g : X ⟶ Z
inst✝² : HasPushout f g
inst✝¹ : HasPushout (G.map f) (G.map g)
inst✝ : PreservesColimit (span f g) G
⊢ IsIso (pushoutComparison G f g)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Braided.Basic

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X X' Y Y' : C
f : X ⟶ Y
g : X' ⟶ Y'
⊢ (f ⊗ₘ g) ≫ (β_ Y Y').hom = (β_ X X').hom ≫ (g ⊗ₘ f)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Opposites.Pullbacks

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
J : Type u₂
inst✝ : Category.{v₂, u₂} J
X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
⊢ ∀ {X_1 Y_1 : WalkingSpan} (f_1 : X_1 ⟶ Y_1),
    (span f.op g.op).map f_1 ≫
        (match Y_1 with
          | none => Iso.refl ((span f.op g.op).obj none)
          | some WalkingPair.left => Iso.refl ((span f.op g.op).obj WalkingSpan.left)
          | some WalkingPair.right => Iso.refl ((span f.op g.op).obj WalkingSpan.right)).hom =
      (match X_1 with
          | none => Iso.refl ((span f.op g.op).obj none)
          | some WalkingPair.left => Iso.refl ((span f.op g.op).obj WalkingSpan.left)
          | some WalkingPair.right => Iso.refl ((span f.op g.op).obj WalkingSpan.right)).hom ≫
        (walkingCospanOpEquiv.inverse ⋙ (cospan f g).op).map f_1

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
G : C ⥤ D
X✝ Y Z : C
f : X✝ ⟶ Y
g : X✝ ⟶ Z
inst✝² : HasPushout f g
inst✝¹ : HasPushout (G.map f) (G.map g)
inst✝ : PreservesColimit (span f g) G
c : PushoutCocone f g
X : C
⊢ ∀ (j : WalkingCospan),
    ((Cone.postcompose (isoWhiskerRight (cospanOp f g).symm (yoneda.obj X)).hom).obj
              (Cone.whisker walkingSpanOpEquiv.symm.functor ((yoneda.obj X).mapCone (Cocone.op c)))).π.app
        j =
      (Iso.refl
            ((Cone.postcompose (isoWhiskerRight (cospanOp f g).symm (yoneda.obj X)).hom).obj
                (Cone.whisker walkingSpanOpEquiv.symm.functor ((yoneda.obj X).mapCone (Cocone.op c)))).pt).hom ≫
        ((yoneda.obj X).mapCone c.op).π.app j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
G : C ⥤ D
X✝ Y Z : C
f : X✝ ⟶ Y
g : X✝ ⟶ Z
inst✝² : HasPushout f g
inst✝¹ : HasPushout (G.map f) (G.map g)
inst✝ : PreservesColimit (span f g) G
c : PushoutCocone f g
X : C
⊢ (Cone.postcompose (isoWhiskerRight (cospanOp f g).symm (yoneda.obj X)).hom).obj
      (Cone.whisker walkingSpanOpEquiv.symm.functor ((yoneda.obj X).mapCone (Cocone.op c))) ≅
    (yoneda.obj X).mapCone c.op

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.EpiMono

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
F : C ⥤ D
X Y : C
f : X ⟶ Y
inst✝¹ : PreservesLimit (cospan f f) F
inst✝ : Mono f
⊢ Mono (F.map f)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.EpiMono

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
F : C ⥤ D
X Y : C
f : X ⟶ Y
inst✝¹ : ReflectsLimit (cospan f f) F
inst✝ : Mono (F.map f)
⊢ Mono f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.EpiMono

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
F : C ⥤ D
X Y : C
f : X ⟶ Y
inst✝¹ : PreservesColimit (span f f) F
inst✝ : Epi f
⊢ Epi (F.map f)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.EpiMono

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
F : C ⥤ D
X Y : C
f : X ⟶ Y
inst✝¹ : ReflectsColimit (span f f) F
inst✝ : Epi (F.map f)
⊢ Epi f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X Y : Cᵒᵖ
f : X ⟶ Y
⊢ identities Cᵒᵖ f ↔ identities C f.unop

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
W : MorphismProperty C
inst✝ : W.ContainsIdentities
x y : C
h : x = y
⊢ W (CategoryTheory.eqToHom h)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
P : MorphismProperty D
inst✝ : P.ContainsIdentities
F : C ⥤ D
X : C
⊢ P.inverseImage F (𝟙 X)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
ι : Type u_1
W : ι → MorphismProperty C
inst✝ : ∀ (i : ι), (W i).ContainsIdentities
⊢ ∀ W' ∈ Set.range fun i ↦ W i, W'.ContainsIdentities

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
ι : Type u_1
W : ι → MorphismProperty C
inst✝ : ∀ (i : ι), (W i).ContainsIdentities
⊢ (⨅ i, W i).ContainsIdentities

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
W : MorphismProperty C
x✝ : W.ContainsIdentities
⊢ identities C ≤ W

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
⊢ identities C ≤ identities C

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
W : Set (MorphismProperty C)
h : ∀ W' ∈ W, W'.IsStableUnderComposition
X✝ Y✝ Z✝ : C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
hf : InfSet.sInf W f
hg : InfSet.sInf W g
⊢ InfSet.sInf W (f ≫ g)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
ι : Type u_1
W : ι → MorphismProperty C
inst✝ : ∀ (i : ι), (W i).IsStableUnderComposition
⊢ ∀ W' ∈ Set.range fun i ↦ W i, W'.IsStableUnderComposition

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
ι : Type u_1
W : ι → MorphismProperty C
inst✝ : ∀ (i : ι), (W i).IsStableUnderComposition
⊢ (⨅ i, W i).IsStableUnderComposition

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
P : MorphismProperty D
inst✝ : P.IsStableUnderComposition
F : C ⥤ D
X✝ Y✝ Z✝ : C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
hf : P.inverseImage F f
hg : P.inverseImage F g
⊢ P.inverseImage F (f ≫ g)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
F₁ F₂ : C ⥤ D
app : (X : C) → F₁.obj X ⟶ F₂.obj X
X✝ Y✝ Z✝ : C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
hf : naturalityProperty app f
hg : naturalityProperty app g
⊢ naturalityProperty app (f ≫ g)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Functor

Goal state before:
pp: C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : MonoidalCategory C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : MonoidalCategory D
E : Type u₃
inst✝³ : Category.{v₃, u₃} E
inst✝² : MonoidalCategory E
C' : Type u₁'
inst✝¹ : Category.{v₁', u₁'} C'
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝ : F.OplaxMonoidal
X : D
⊢ (adj.homEquiv (𝟙_ C ⊗ G.obj X) X).symm (λ_ (G.obj X)).hom =
    (adj.homEquiv (𝟙_ C ⊗ G.obj X) X).symm
      ((adj.homEquiv (𝟙_ C) (𝟙_ D)) (η F) ▷ G.obj X ≫
        (adj.homEquiv (G.obj (𝟙_ D) ⊗ G.obj X) (𝟙_ D ⊗ X))
            (δ F (G.obj (𝟙_ D)) (G.obj X) ≫ (adj.counit.app (𝟙_ D) ⊗ₘ adj.counit.app X)) ≫
          G.map (λ_ X).hom)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Braided.Basic

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X Y Z : C
⊢ (α_ X Y Z).inv ≫
      (β_ X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫ Y ◁ (β_ X Z).hom ≫ (α_ Y Z X).inv ≫ (β_ Y Z).hom ▷ X ≫ (α_ Z Y X).hom =
    X ◁ (β_ Y Z).hom ≫ (α_ X Z Y).inv ≫ (β_ X Z).hom ▷ Y ≫ (α_ Z X Y).hom ≫ Z ◁ (β_ X Y).hom

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Opposites.Pullbacks

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
J : Type u₂
inst✝ : Category.{v₂, u₂} J
X Y Z : Cᵒᵖ
f : X ⟶ Z
g : Y ⟶ Z
⊢ ∀ {X_1 Y_1 : WalkingSpan} (f_1 : X_1 ⟶ Y_1),
    (span f.unop g.unop).map f_1 ≫
        (match Y_1 with
          | none => Iso.refl ((span f.unop g.unop).obj none)
          | some WalkingPair.left => Iso.refl ((span f.unop g.unop).obj WalkingSpan.left)
          | some WalkingPair.right => Iso.refl ((span f.unop g.unop).obj WalkingSpan.right)).hom =
      (match X_1 with
          | none => Iso.refl ((span f.unop g.unop).obj none)
          | some WalkingPair.left => Iso.refl ((span f.unop g.unop).obj WalkingSpan.left)
          | some WalkingPair.right => Iso.refl ((span f.unop g.unop).obj WalkingSpan.right)).hom ≫
        (walkingCospanOpEquiv.inverse ⋙ (cospan f g).leftOp).map f_1

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
F₁ F₂ : C ⥤ D
app : (X : C) → F₁.obj X ⟶ F₂.obj X
X Y : C
e : X ≅ Y
he : naturalityProperty app e.hom
⊢ naturalityProperty app e.inv

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
X✝ Y✝ Z✝ : C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
hf : isomorphisms C f
hg : isomorphisms C g
⊢ isomorphisms C (f ≫ g)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
X✝ Y✝ Z✝ : C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
hf : monomorphisms C f
hg : monomorphisms C g
⊢ monomorphisms C (f ≫ g)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
X✝ Y✝ Z✝ : C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
hf : epimorphisms C f
hg : epimorphisms C g
⊢ epimorphisms C (f ≫ g)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
⊢ ∀ {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z), identities C f → identities C g → identities C (f ≫ g)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
W : Set (MorphismProperty C)
h : ∀ W' ∈ W, W'.IsMultiplicative
⊢ (InfSet.sInf W).IsMultiplicative

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
ι : Type u_1
W : ι → MorphismProperty C
inst✝ : ∀ (i : ι), (W i).IsMultiplicative
⊢ ∀ W' ∈ Set.range fun i ↦ W i, W'.IsMultiplicative

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
ι : Type u_1
W : ι → MorphismProperty C
inst✝ : ∀ (i : ι), (W i).IsMultiplicative
⊢ (⨅ i, W i).IsMultiplicative

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
F₁ F₂ : C ⥤ D
app : (X : C) → F₁.obj X ⟶ F₂.obj X
x✝ : C
⊢ MorphismProperty.naturalityProperty app (𝟙 x✝)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
W : MorphismProperty C
X✝ Y✝ Z✝ : C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
hf : W.multiplicativeClosure f
hg : W.multiplicativeClosure g
⊢ W.multiplicativeClosure (f ≫ g)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
W : MorphismProperty C
X✝ Y✝ Z✝ : C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
hf : W.multiplicativeClosure' f
hg : W.multiplicativeClosure' g
⊢ W.multiplicativeClosure' (f ≫ g)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
W : MorphismProperty C
inst✝ : W.IsMultiplicative
⊢ W.multiplicativeClosure = W

Tactic:
apply le_antisymm _ <| le_multiplicativeClosure W
  intro _ _ _ hf
  induction hf with
  | of _ hf₀ => exact hf₀
  | id x => exact W.id_mem x
  | comp_of _ _ _ hg hf => exact W.comp_mem _ _ hf hg

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
W : MorphismProperty C
inst✝ : W.IsMultiplicative
⊢ W.multiplicativeClosure = W

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
W : MorphismProperty C
h : W.multiplicativeClosure = W
⊢ W.IsMultiplicative

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
W W' : MorphismProperty C
inst✝ : W'.IsMultiplicative
h : W ≤ W'
⊢ W.multiplicativeClosure ≤ W'

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
x✝ W' : MorphismProperty C
h : x✝ ≤ W'
⊢ x✝.multiplicativeClosure ≤ W'.multiplicativeClosure

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Composition

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
W : MorphismProperty C
x y : C
f : x ⟶ y
hf : W.multiplicativeClosure' f
⊢ W.multiplicativeClosure f

Tactic:
by

Goal state after: