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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : MonoidalCategory C
D : Type u_2
inst✝¹ : Category.{v_2, u_2} D
inst✝ : MonoidalCategory D
F₁ F₂ : C ⥤ D
f : F₁ ⟶ F₂
x✝² x✝¹ : C
x✝ : x✝² ⟶ x✝¹
⊢ (curriedTensorPre F₁).map x✝ ≫ { app := fun X₂ ↦ f.app x✝¹ ⊗ₘ f.app X₂, naturality := ⋯ } =
    { app := fun X₂ ↦ f.app x✝² ⊗ₘ f.app X₂, naturality := ⋯ } ≫ (curriedTensorPre F₂).map x✝

Tactic:
by
        ext
        simp [← tensorHom_id]

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

Goal state before:
pp: case h₀
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : HasPushouts C
inst✝³ : HasInitial C
inst✝² : CartesianMonoidalCategory C
inst✝¹ : MonoidalClosed C
inst✝ : BraidedCategory C
X₁ X₂ X₃ Y₁ Y₂ Y₃ : Arrow C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
f₃ : X₃ ⟶ Y₃
⊢ Arrow.Hom.right f₁ ▷ (X₂.left ⊗ X₃.right) ≫
      (α_ Y₁.right X₂.left X₃.right).inv ≫
        (α_ Y₁.right X₂.left X₃.right).hom ≫
          Y₁.right ◁ X₂.left ◁ Arrow.Hom.right f₃ ≫
            Y₁.right ◁ Arrow.Hom.left f₂ ▷ Y₃.right ≫ Y₁.right ◁ pushout.inr (Y₂.hom ▷ Y₃.left) (Y₂.left ◁ Y₃.hom) =
    Arrow.Hom.right f₁ ▷ (X₂.left ⊗ X₃.right) ≫
      Y₁.right ◁ (X₂.left ◁ Arrow.Hom.right f₃ ≫ Arrow.Hom.left f₂ ▷ Y₃.right) ≫
        Y₁.right ◁ pushout.inr (Y₂.hom ▷ Y₃.left) (Y₂.left ◁ Y₃.hom)

Tactic:
Iso.inv_hom_id_assoc,

Goal state after:
case h₀
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : HasPushouts C
inst✝³ : HasInitial C
inst✝² : CartesianMonoidalCategory C
inst✝¹ : MonoidalClosed C
inst✝ : BraidedCategory C
X₁ X₂ X₃ Y₁ Y₂ Y₃ : Arrow C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
f₃ : X₃ ⟶ Y₃
⊢ Arrow.Hom.right f₁ ▷ (X₂.left ⊗ X₃.right) ≫
      Y₁.right ◁ X₂.left ◁ Arrow.Hom.right f₃ ≫
        Y₁.right ◁ Arrow.Hom.left f₂ ▷ Y₃.right ≫ Y₁.right ◁ pushout.inr (Y₂.hom ▷ Y₃.left) (Y₂.left ◁ Y₃.hom) =
    Arrow.Hom.right f₁ ▷ (X₂.left ⊗ X₃.right) ≫
      Y₁.right ◁ (X₂.left ◁ Arrow.Hom.right f₃ ≫ Arrow.Hom.left f₂ ▷ Y₃.right) ≫
        Y₁.right ◁ pushout.inr (Y₂.hom ▷ Y₃.left) (Y₂.left ◁ Y₃.hom)
Module: Mathlib.CategoryTheory.Monoidal.Bimod

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ R.X ◁ (α_ P.X S.X Q.X).hom ≫ ((α_ R.X P.X (S.X ⊗ Q.X)).inv ≫ (R.X ⊗ P.X) ◁ Q.actLeft) ≫ P.actLeft ▷ Q.X =
    ((α_ R.X (P.X ⊗ S.X) Q.X).inv ≫ (α_ R.X P.X S.X).inv ▷ Q.X ≫ P.actLeft ▷ S.X ▷ Q.X) ≫
      (α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft

Tactic:
slice_lhs 3 4 => rw [whisker_exchange]

Goal state after:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ R.X ◁ (α_ P.X S.X Q.X).hom ≫ (α_ R.X P.X (S.X ⊗ Q.X)).inv ≫ P.actLeft ▷ (S.X ⊗ Q.X) ≫ P.X ◁ Q.actLeft =
    ((α_ R.X (P.X ⊗ S.X) Q.X).inv ≫ (α_ R.X P.X S.X).inv ▷ Q.X ≫ P.actLeft ▷ S.X ▷ Q.X) ≫
      (α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft
Module: Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Basic

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : MonoidalClosed C
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
inst✝ : ∀ (F₁ F₂ : J ⥤ C), HasFunctorEnrichedHom C F₁ F₂
F₁ F₂ F₂' F₃ F₃' : J ⥤ C
g : F₂ ⟶ functorEnrichedHom C F₁ F₃
j j' : J
φ : j ⟶ j'
⊢ (F₁.map φ ⊗ₘ F₂.map φ) ≫
      uncurry (g.app j' ≫ enrichedHomπ C (Under.forget j' ⋙ F₁) (Under.forget j' ⋙ F₃) (Under.mk (𝟙 j'))) =
    uncurry
      ((g.app j ≫ enrichedHomπ C (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃) (Under.mk (𝟙 j))) ≫
        (ihom (F₁.obj j)).map (F₃.map φ))

Tactic:
tensorHom_def'_assoc,

Goal state after:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : MonoidalClosed C
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
inst✝ : ∀ (F₁ F₂ : J ⥤ C), HasFunctorEnrichedHom C F₁ F₂
F₁ F₂ F₂' F₃ F₃' : J ⥤ C
g : F₂ ⟶ functorEnrichedHom C F₁ F₃
j j' : J
φ : j ⟶ j'
⊢ F₁.obj j ◁ F₂.map φ ≫
      F₁.map φ ▷ F₂.obj j' ≫
        uncurry (g.app j' ≫ enrichedHomπ C (Under.forget j' ⋙ F₁) (Under.forget j' ⋙ F₃) (Under.mk (𝟙 j'))) =
    uncurry
      ((g.app j ≫ enrichedHomπ C (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃) (Under.mk (𝟙 j))) ≫
        (ihom (F₁.obj j)).map (F₃.map φ))
Module: Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete

Goal state before:
pp: I : Type u₂
inst✝⁴ : Category.{v₂, u₂} I
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : MonoidalClosed C
inst✝ : ∀ (F : Discrete I ⥤ C), (Discrete.functor _root_.id).HasRightKanExtension F
A✝ B✝ : I ⥤ C
f : A✝ ⟶ B✝
h : ∀ (X : Discrete I), IsIso ((((whiskeringLeft (Discrete I) I C).obj (incl I)).map f).app X)
X : I
⊢ IsIso (f.app X)

Tactic:
exact h ⟨X⟩

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Basic

Goal state before:
pp: C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : MonoidalCategory C
inst✝³ : MonoidalClosed C
J : Type u₂
inst✝² : Category.{v₂, u₂} J
inst✝¹ : ∀ (F₁ F₂ : J ⥤ C), HasFunctorEnrichedHom C F₁ F₂
F₁ F₂ F₃ F₃' : J ⥤ C
inst✝ : ∀ (F₁ F₂ : J ⥤ C), HasEnrichedHom C F₁ F₂
f : F₁ ⊗ F₂ ⟶ F₃
f₃ : F₃ ⟶ F₃'
j : J
k : Under j
⊢ F₂.map k.hom ≫ curry (f.app k.right ≫ f₃.app k.right) =
    end_.lift (fun k ↦ F₂.map k.hom ≫ curry (f.app k.right)) ⋯ ≫
      (ρ_ (enrichedHom C (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃))).inv ≫
        (ρ_ (enrichedHom C (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃))).hom ≫
          end_.π (diagram C (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃)) k ≫
            (ρ_ (((diagram C (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃)).obj (Opposite.op k)).obj k)).inv ≫
              ((diagram C (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃)).obj (Opposite.op k)).obj k ◁
                  ((Enriched.FunctorCategory.homEquiv C) f₃ ≫ precompEnrichedHom C F₃ F₃' (Under.forget j)) ≫
                ((diagram C (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃)).obj (Opposite.op k)).obj k ◁
                    end_.π (diagram C (Under.forget j ⋙ F₃) (Under.forget j ⋙ F₃')) k ≫
                  eComp C (Opposite.unop ((Under.forget j ⋙ F₁).op.obj (Opposite.op k))) ((Under.forget j ⋙ F₃).obj k)
                    ((Under.forget j ⋙ F₃').obj k)

Tactic:
Iso.inv_hom_id_assoc,

Goal state after:
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : MonoidalCategory C
inst✝³ : MonoidalClosed C
J : Type u₂
inst✝² : Category.{v₂, u₂} J
inst✝¹ : ∀ (F₁ F₂ : J ⥤ C), HasFunctorEnrichedHom C F₁ F₂
F₁ F₂ F₃ F₃' : J ⥤ C
inst✝ : ∀ (F₁ F₂ : J ⥤ C), HasEnrichedHom C F₁ F₂
f : F₁ ⊗ F₂ ⟶ F₃
f₃ : F₃ ⟶ F₃'
j : J
k : Under j
⊢ F₂.map k.hom ≫ curry (f.app k.right ≫ f₃.app k.right) =
    end_.lift (fun k ↦ F₂.map k.hom ≫ curry (f.app k.right)) ⋯ ≫
      end_.π (diagram C (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃)) k ≫
        (ρ_ (((diagram C (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃)).obj (Opposite.op k)).obj k)).inv ≫
          ((diagram C (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃)).obj (Opposite.op k)).obj k ◁
              ((Enriched.FunctorCategory.homEquiv C) f₃ ≫ precompEnrichedHom C F₃ F₃' (Under.forget j)) ≫
            ((diagram C (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃)).obj (Opposite.op k)).obj k ◁
                end_.π (diagram C (Under.forget j ⋙ F₃) (Under.forget j ⋙ F₃')) k ≫
              eComp C (Opposite.unop ((Under.forget j ⋙ F₁).op.obj (Opposite.op k))) ((Under.forget j ⋙ F₃).obj k)
                ((Under.forget j ⋙ F₃').obj k)
Module: Mathlib.CategoryTheory.Monoidal.Hopf_

Goal state before:
pp: case a.a.a.a.a.a.a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : BraidedCategory C
A : C
inst✝ : HopfObj A
| A ◁ ((α_ A A A).inv ≫ (β_ (A ⊗ A) A).hom ≫ (α_ A A A).inv ≫ (α_ A A A).hom ≫ A ◁ μ)

pp: case a.a.a.a.a.a.a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : BraidedCategory C
A : C
inst✝ : HopfObj A
| A ◁ 𝒮 ▷ A

pp: case a.a.a.a.a.a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : BraidedCategory C
A : C
inst✝ : HopfObj A
| (α_ A A A).inv

pp: case a.a.a.a.a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : BraidedCategory C
A : C
inst✝ : HopfObj A
| μ ▷ A

pp: case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : BraidedCategory C
A : C
inst✝ : HopfObj A
| Δ

pp: case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : BraidedCategory C
A : C
inst✝ : HopfObj A
| Δ ▷ A

pp: case a.a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : BraidedCategory C
A : C
inst✝ : HopfObj A
| (α_ A A A).hom

pp: case a.a.a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : BraidedCategory C
A : C
inst✝ : HopfObj A
| A ◁ A ◁ Δ

pp: case a.a.a.a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : BraidedCategory C
A : C
inst✝ : HopfObj A
| A ◁ A ◁ 𝒮 ▷ A

Tactic:
Iso.inv_hom_id_assoc,

Goal state after:
case a.a.a.a.a.a.a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : BraidedCategory C
A : C
inst✝ : HopfObj A
| A ◁ ((α_ A A A).inv ≫ (β_ (A ⊗ A) A).hom ≫ A ◁ μ)

case a.a.a.a.a.a.a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : BraidedCategory C
A : C
inst✝ : HopfObj A
| A ◁ 𝒮 ▷ A

case a.a.a.a.a.a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : BraidedCategory C
A : C
inst✝ : HopfObj A
| (α_ A A A).inv

case a.a.a.a.a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : BraidedCategory C
A : C
inst✝ : HopfObj A
| μ ▷ A

case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : BraidedCategory C
A : C
inst✝ : HopfObj A
| Δ

case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : BraidedCategory C
A : C
inst✝ : HopfObj A
| Δ ▷ A

case a.a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : BraidedCategory C
A : C
inst✝ : HopfObj A
| (α_ A A A).hom

case a.a.a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : BraidedCategory C
A : C
inst✝ : HopfObj A
| A ◁ A ◁ Δ

case a.a.a.a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : BraidedCategory C
A : C
inst✝ : HopfObj A
| A ◁ A ◁ 𝒮 ▷ A
Module: Mathlib.CategoryTheory.Monoidal.Bimod

Goal state before:
pp: case a.a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : HasCoequalizers C
R S : Mon C
P : Bimod R S
| ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft) ≫ coequalizer.π (μ ▷ P.X) ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft)

pp: case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : HasCoequalizers C
R S : Mon C
P : Bimod R S
| (λ_ (R.X ⊗ P.X)).inv

pp: case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : HasCoequalizers C
R S : Mon C
P : Bimod R S
| η ▷ (R.X ⊗ P.X)

pp: case a.a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : HasCoequalizers C
R S : Mon C
P : Bimod R S
| (α_ R.X R.X P.X).inv

Tactic:
← coequalizer.condition

Goal state after:
case a.a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : HasCoequalizers C
R S : Mon C
P : Bimod R S
| μ ▷ P.X ≫ coequalizer.π (μ ▷ P.X) ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft)

case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : HasCoequalizers C
R S : Mon C
P : Bimod R S
| (λ_ (R.X ⊗ P.X)).inv

case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : HasCoequalizers C
R S : Mon C
P : Bimod R S
| η ▷ (R.X ⊗ P.X)

case a.a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : HasCoequalizers C
R S : Mon C
P : Bimod R S
| (α_ R.X R.X P.X).inv
Module: Mathlib.CategoryTheory.ObjectProperty.Ind

Goal state before:
pp: case refine_1
C : Type u
inst✝² : Category.{v, u} C
D : Type u_1
inst✝¹ : Category.{u_2, u_1} D
P : ObjectProperty D
e : C ≌ D
inst✝ : P.IsClosedUnderIsomorphisms
X : D
x✝³ : Type w
x✝² : SmallCategory x✝³
x✝¹ : IsFiltered x✝³
x✝ : ∃ pres, ∀ (i : x✝³), P.inverseImage e.functor (pres.diag.obj i)
pres : ColimitPresentation x✝³ (e.inverse.obj X)
h : ∀ (i : x✝³), P.inverseImage e.functor (pres.diag.obj i)
⊢ ∃ pres, ∀ (i : x✝³), P (pres.diag.obj i)

pp: case refine_2
C : Type u
inst✝² : Category.{v, u} C
D : Type u_1
inst✝¹ : Category.{u_2, u_1} D
P : ObjectProperty D
e : C ≌ D
inst✝ : P.IsClosedUnderIsomorphisms
X : D
x✝³ : Type w
x✝² : SmallCategory x✝³
x✝¹ : IsFiltered x✝³
x✝ : ∃ pres, ∀ (i : x✝³), P (pres.diag.obj i)
pres : ColimitPresentation x✝³ X
h : ∀ (i : x✝³), P (pres.diag.obj i)
⊢ ∃ pres, ∀ (i : x✝³), P.inverseImage e.functor (pres.diag.obj i)

Tactic:
· exact ⟨.ofIso (pres.map e.functor) (e.counitIso.app X), fun i ↦ h i⟩

Goal state after:
case refine_2
C : Type u
inst✝² : Category.{v, u} C
D : Type u_1
inst✝¹ : Category.{u_2, u_1} D
P : ObjectProperty D
e : C ≌ D
inst✝ : P.IsClosedUnderIsomorphisms
X : D
x✝³ : Type w
x✝² : SmallCategory x✝³
x✝¹ : IsFiltered x✝³
x✝ : ∃ pres, ∀ (i : x✝³), P (pres.diag.obj i)
pres : ColimitPresentation x✝³ X
h : ∀ (i : x✝³), P (pres.diag.obj i)
⊢ ∃ pres, ∀ (i : x✝³), P.inverseImage e.functor (pres.diag.obj i)
Module: Mathlib.CategoryTheory.Preadditive.OfBiproducts

Goal state before:
pp: case h₀
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := ⋯
hd₁ : biprod.inl ≫ diag = biprod.lift f h
hd₂ : biprod.inr ≫ diag = biprod.lift g k
h₁ : biprod.lift (rightAdd X Y f g) (rightAdd X Y h k) = biprod.lift (𝟙 X) (𝟙 X) ≫ diag
⊢ biprod.inl ≫ diag ≫ biprod.desc (𝟙 Y) (𝟙 Y) = biprod.inl ≫ biprod.desc (leftAdd X Y f h) (leftAdd X Y g k)

Tactic:
simp [reassoc_of% hd₁, reassoc_of% hd₂]

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.OfBiproducts

Goal state before:
pp: case h₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := ⋯
hd₁ : biprod.inl ≫ diag = biprod.lift f h
hd₂ : biprod.inr ≫ diag = biprod.lift g k
h₁ : biprod.lift (rightAdd X Y f g) (rightAdd X Y h k) = biprod.lift (𝟙 X) (𝟙 X) ≫ diag
⊢ biprod.inr ≫ diag ≫ biprod.desc (𝟙 Y) (𝟙 Y) = biprod.inr ≫ biprod.desc (leftAdd X Y f h) (leftAdd X Y g k)

Tactic:
simp [reassoc_of% hd₁, reassoc_of% hd₂]

Goal state after:
Module: Mathlib.Order.Category.PartOrdEmb

Goal state before:
pp: J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F : J ⥤ PartOrdEmb
c : Cocone (F ⋙ forget PartOrdEmb)
hc : IsColimit c
j : J
x y : ↑(F.1 j)
k : J
x' y' : ↑(F.obj k)
hx : (ConcreteCategory.hom (c.ι.app k)) x' = { toFun := ⇑(ConcreteCategory.hom (c.ι.app j)), inj' := ⋯ } x
hy : (ConcreteCategory.hom (c.ι.app k)) y' = { toFun := ⇑(ConcreteCategory.hom (c.ι.app j)), inj' := ⋯ } y
h : x' ≤ y'
l₁ : J
a₁ : k ⟶ l₁
b₁ : j ⟶ l₁
hl₁ : (ConcreteCategory.hom (F.map a₁)) x' = (ConcreteCategory.hom (F.map b₁)) x
l₂ : J
a₂ : k ⟶ l₂
b₂ : j ⟶ l₂
hl₂ : (ConcreteCategory.hom (F.map a₂)) y' = (ConcreteCategory.hom (F.map b₂)) y
m : J
d : l₁ ⟶ m
d' : l₂ ⟶ m
h₁ : a₁ ≫ d = a₂ ≫ d'
h₂ : b₁ ≫ d = b₂ ≫ d'
⊢ x ≤ y

Tactic:
rw [← (F.map (a₁ ≫ d)).le_iff_le] at h

Goal state after:
J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F : J ⥤ PartOrdEmb
c : Cocone (F ⋙ forget PartOrdEmb)
hc : IsColimit c
j : J
x y : ↑(F.1 j)
k : J
x' y' : ↑(F.obj k)
hx : (ConcreteCategory.hom (c.ι.app k)) x' = { toFun := ⇑(ConcreteCategory.hom (c.ι.app j)), inj' := ⋯ } x
hy : (ConcreteCategory.hom (c.ι.app k)) y' = { toFun := ⇑(ConcreteCategory.hom (c.ι.app j)), inj' := ⋯ } y
l₁ : J
a₁ : k ⟶ l₁
b₁ : j ⟶ l₁
hl₁ : (ConcreteCategory.hom (F.map a₁)) x' = (ConcreteCategory.hom (F.map b₁)) x
l₂ : J
a₂ : k ⟶ l₂
b₂ : j ⟶ l₂
hl₂ : (ConcreteCategory.hom (F.map a₂)) y' = (ConcreteCategory.hom (F.map b₂)) y
m : J
d : l₁ ⟶ m
h : (ConcreteCategory.hom (F.map (a₁ ≫ d))) x' ≤ (ConcreteCategory.hom (F.map (a₁ ≫ d))) y'
d' : l₂ ⟶ m
h₁ : a₁ ≫ d = a₂ ≫ d'
h₂ : b₁ ≫ d = b₂ ≫ d'
⊢ x ≤ y
Module: Mathlib.CategoryTheory.Presentable.OrthogonalReflection

Goal state before:
pp: case h₁
C : Type u
inst✝⁴ : Category.{v, u} C
W : MorphismProperty C
Z : C
inst✝³ : HasCoproduct D₁.obj₁
inst✝² : HasCoproduct D₁.obj₂
inst✝¹ : HasPushouts C
inst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)
hZ : W.isLocal Z
f : succ W Z ⟶ Z
hf : toSucc W Z ≫ f = 𝟙 Z
⊢ pushout.inr (D₁.t W Z) (D₁.l W Z) ≫ Multicoequalizer.π (D₂.multispanIndex W Z) PUnit.unit ≫ f ≫ toSucc W Z =
    pushout.inr (D₁.t W Z) (D₁.l W Z) ≫ Multicoequalizer.π (D₂.multispanIndex W Z) PUnit.unit ≫ 𝟙 (succ W Z)

Tactic:
· simp [reassoc_of% hf]

Goal state after:
Module: Mathlib.CategoryTheory.Presentable.Directed

Goal state before:
pp: case inr
J : Type w
inst✝¹ : SmallCategory J
κ : Cardinal.{w}
inst✝ : Fact κ.IsRegular
ι : Type w
D : ι → DiagramWithUniqueTerminal J κ
hι : HasCardinalLT ι κ
m : J
u : (i : ι) → (D i).top ⟶ m
hD : ∀ {i : ι}, ¬(D i).P m
f : m ⟶ m
i : ι
hj : (D i).P m
hi : Arrow.mk f = Arrow.mk ((D ⟨i, ⟨m, hj⟩⟩.fst).isTerminal.lift ⋯ ≫ u ⟨i, ⟨m, hj⟩⟩.fst)
⊢ f = 𝟙 m

Tactic:
exact (hD hj).elim

Goal state after:
Module: Mathlib.CategoryTheory.Presentable.Directed

Goal state before:
pp: case inl.inr
J : Type w
inst✝² : SmallCategory J
κ : Cardinal.{w}
inst✝¹ : Fact κ.IsRegular
inst✝ : IsCardinalFiltered J κ
hJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)
ι : Type w
D : ι → DiagramWithUniqueTerminal J κ
hι : HasCardinalLT ι κ
m : J
u : (i : ι) → (D i).top ⟶ m
hm₀ : ∀ (i : ι), IsEmpty (m ⟶ (D i).top)
hm :
  ∀ (i₁ i₂ : ι) (j : J) (hj₁ : (D i₁).P j) (hj₂ : (D i₂).P j),
    (D i₁).isTerminal.lift hj₁ ≫ u i₁ = (D i₂).isTerminal.lift hj₂ ≫ u i₂
φ : (x : (i : ι) × Subtype (D i).P) → ↑x.snd ⟶ m := fun x ↦ (D x.fst).isTerminal.lift ⋯ ≫ u x.fst
hD : ∀ {i : ι}, ¬(D i).P m
i : ι
X Y : J
i✝ : Unit
hj : (D i).P m
⊢ 𝟙 m = φ ⟨i, ⟨m, hj⟩⟩

Tactic:
exact (hD hj).elim

Goal state after:
Module: Mathlib.CategoryTheory.RepresentedBy

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type w
X : C
x : F.obj (op X)
Y : C
e : Y ≅ X
⊢ F.IsRepresentedBy ((ConcreteCategory.hom (F.map e.hom.op)) x) ↔ F.IsRepresentedBy x

Tactic:
refine ⟨fun h ↦ ?_, fun h ↦ h.of_isoObj e⟩

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type w
X : C
x : F.obj (op X)
Y : C
e : Y ≅ X
h : F.IsRepresentedBy ((ConcreteCategory.hom (F.map e.hom.op)) x)
⊢ F.IsRepresentedBy x
Module: Mathlib.CategoryTheory.Presentable.Directed

Goal state before:
pp: case inr
J : Type w
inst✝² : SmallCategory J
κ : Cardinal.{w}
inst✝¹ : Fact κ.IsRegular
inst✝ : IsCardinalFiltered J κ
hJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)
this✝¹ : IsCardinalFiltered (DiagramWithUniqueTerminal J κ) κ
this✝ : IsFiltered J
this : IsFiltered (DiagramWithUniqueTerminal J κ)
j : J
D : DiagramWithUniqueTerminal J κ
f₁ f₂ : j ⟶ D.top
m₀ : J
t : D.top ⟶ m₀
hm₀ : IsEmpty (m₀ ⟶ D.top)
m₁ : J
u : m₀ ⟶ m₁
hu : f₁ ≫ t ≫ u = f₂ ≫ t ≫ u
h₁ : ¬D.P m₁
φ : (x : Subtype D.P) → ↑x ⟶ m₁ := fun x ↦ D.isTerminal.lift ⋯ ≫ t ≫ u
h₂ : ∀ {j : J} (hj : D.P j) {f : j ⟶ m₁}, (D₄ D φ).W f → f = φ ⟨j, hj⟩
f : m₁ ⟶ m₁
hf : ∃ i, Arrow.mk f = Arrow.mk (φ i)
⊢ f = 𝟙 m₁

Tactic:
obtain ⟨⟨j, hj⟩, hf⟩ := hf

Goal state after:
case inr
J : Type w
inst✝² : SmallCategory J
κ : Cardinal.{w}
inst✝¹ : Fact κ.IsRegular
inst✝ : IsCardinalFiltered J κ
hJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)
this✝¹ : IsCardinalFiltered (DiagramWithUniqueTerminal J κ) κ
this✝ : IsFiltered J
this : IsFiltered (DiagramWithUniqueTerminal J κ)
j✝ : J
D : DiagramWithUniqueTerminal J κ
f₁ f₂ : j✝ ⟶ D.top
m₀ : J
t : D.top ⟶ m₀
hm₀ : IsEmpty (m₀ ⟶ D.top)
m₁ : J
u : m₀ ⟶ m₁
hu : f₁ ≫ t ≫ u = f₂ ≫ t ≫ u
h₁ : ¬D.P m₁
φ : (x : Subtype D.P) → ↑x ⟶ m₁ := fun x ↦ D.isTerminal.lift ⋯ ≫ t ≫ u
h₂ : ∀ {j : J} (hj : D.P j) {f : j ⟶ m₁}, (D₄ D φ).W f → f = φ ⟨j, hj⟩
f : m₁ ⟶ m₁
j : J
hj : D.P j
hf : Arrow.mk f = Arrow.mk (φ ⟨j, hj⟩)
⊢ f = 𝟙 m₁
Module: Mathlib.CategoryTheory.Sites.Adjunction

Goal state before:
pp: case e'_3
C : Type u₁
inst✝² : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
E : Type u_1
inst✝ : Category.{v_1, u_1} E
F : D ⥤ E
G : E ⥤ D
adj : G ⊣ F
P Q : Cᵒᵖ ⥤ E
f : P ⟶ Q
hf : J.W f
this : F.IsRightAdjoint
R : Cᵒᵖ ⥤ D
hR : Presheaf.IsSheaf J R
e_1✝ : (Q ⋙ G ⟶ R) = (((whiskeringRight Cᵒᵖ E D).obj G).obj Q ⟶ R)
e_2✝ : (P ⟶ ((whiskeringRight Cᵒᵖ D E).obj F).obj R) = (P ⟶ R ⋙ F)
g : Q ⋙ G ⟶ R
X : Cᵒᵖ
⊢ (((Adjunction.whiskerRight Cᵒᵖ adj).homEquiv P R) (whiskerRight f G ≫ g)).app X =
    f.app X ≫ (((Adjunction.whiskerRight Cᵒᵖ adj).homEquiv Q R) g).app X

Tactic:
rw [← NatTrans.comp_app]

Goal state after:
case e'_3
C : Type u₁
inst✝² : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
E : Type u_1
inst✝ : Category.{v_1, u_1} E
F : D ⥤ E
G : E ⥤ D
adj : G ⊣ F
P Q : Cᵒᵖ ⥤ E
f : P ⟶ Q
hf : J.W f
this : F.IsRightAdjoint
R : Cᵒᵖ ⥤ D
hR : Presheaf.IsSheaf J R
e_1✝ : (Q ⋙ G ⟶ R) = (((whiskeringRight Cᵒᵖ E D).obj G).obj Q ⟶ R)
e_2✝ : (P ⟶ ((whiskeringRight Cᵒᵖ D E).obj F).obj R) = (P ⟶ R ⋙ F)
g : Q ⋙ G ⟶ R
X : Cᵒᵖ
⊢ (((Adjunction.whiskerRight Cᵒᵖ adj).homEquiv P R) (whiskerRight f G ≫ g)).app X =
    (f ≫ ((Adjunction.whiskerRight Cᵒᵖ adj).homEquiv Q R) g).app X
Module: Mathlib.CategoryTheory.Sites.Coherent.RegularTopology

Goal state before:
pp: case mpr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preregular C
X : C
S : Sieve X
⊢ (∃ Y π, EffectiveEpi π ∧ S.arrows π) → S ∈ (regularTopology C) X

Tactic:
exact regularTopology.mem_sieves_of_hasEffectiveEpi S

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Coherent.Comparison

Goal state before:
pp: case refine_2.of.a
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y✝ : C
I : Type
w✝ : Finite I
X : I → C
f : (a : I) → X a ⟶ Y✝
hT : EffectiveEpiFamily X f
R Y : C
i✝ : Unit
ψ : R ⟶ ∐ fun i ↦ X i
this :
  generate (Presieve.ofArrows X fun i ↦ Sigma.ι X i) ≤ Sieve.pullback (Sigma.desc f) (generate (Presieve.ofArrows X f))
⊢ IsIso (Sigma.desc fun i ↦ Sigma.ι X i)

Tactic:
convert! IsIso.id _

Goal state after:
case e'_5
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preregular C
inst✝ : FinitaryPreExtensive C
B : C
S : Sieve B
Y✝ : C
I : Type
w✝ : Finite I
X : I → C
f : (a : I) → X a ⟶ Y✝
hT : EffectiveEpiFamily X f
R Y : C
i✝ : Unit
ψ : R ⟶ ∐ fun i ↦ X i
this :
  generate (Presieve.ofArrows X fun i ↦ Sigma.ι X i) ≤ Sieve.pullback (Sigma.desc f) (generate (Presieve.ofArrows X f))
e_4✝ : ∐ X = ∐ X
⊢ (Sigma.desc fun i ↦ Sigma.ι X i) = 𝟙 (∐ X)
Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology

Goal state before:
pp: case mp.top
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : FinitaryPreExtensive C
X✝ : C
S : Sieve X✝
X : C
this : IsIso (Sigma.desc (Cofan.mk X fun x ↦ 𝟙 X).inj)
⊢ IsColimit (Cofan.mk X fun x ↦ 𝟙 X)

Tactic:
exact Cofan.isColimitOfIsIsoSigmaDesc (Cofan.mk X _)

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves

Goal state before:
pp: case refine_1
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
X B : C
π : X ⟶ B
inst✝ : EffectiveEpi π
c : PullbackCone π π
hc : IsLimit c
this : HasPullback π π
hP :
  ∀
    (b :
      (fun X ↦ X)
        ↑{x |
            (ConcreteCategory.hom (P.map (pullback.fst π π).op)) x =
              (ConcreteCategory.hom (P.map (pullback.snd π π).op)) x}),
    ∃! a, (ConcreteCategory.hom (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯)) a = b
b : P.obj (op X)
hb : (ConcreteCategory.hom (P.map c.fst.op)) b = (ConcreteCategory.hom (P.map c.snd.op)) b
h₁ : ((pullbackIsPullback π π).conePointUniqueUpToIso hc).hom ≫ c.fst = pullback.fst π π
hb' : (ConcreteCategory.hom (P.map (pullback.fst π π).op)) b = (ConcreteCategory.hom (P.map (pullback.snd π π).op)) b
a : P.obj (op B)
ha₁ : (ConcreteCategory.hom (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯)) a = ⟨b, hb'⟩
ha₂ :
  ∀ (y : (fun X ↦ X) (P.obj (op B))),
    (fun a ↦ (ConcreteCategory.hom (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯)) a = ⟨b, hb'⟩) y → y = a
⊢ (fun x ↦ (ConcreteCategory.hom (P.map π.op)) x = b) a

Tactic:
simpa [mapToEqualizer] using ha₁

Goal state after: