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

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon C
M : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
R S T U : Mon C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
⊢ (α_ P.X Q.X T.X).hom ▷ L.X ≫
      (α_ P.X (Q.X ⊗ T.X) L.X).hom ≫
        ((P.X ◁ (α_ Q.X T.X L.X).hom ≫ P.X ◁ Q.X ◁ L.actLeft) ≫
            P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ≫
          coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))
            ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫
              P.X ◁ TensorBimod.actLeft Q L) =
    (α_ (P.X ⊗ Q.X) T.X L.X).hom ≫
      (((P.X ⊗ Q.X) ◁ L.actLeft ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ L.X) ≫
          (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X)
              ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv) ≫
        coequalizer.desc
          ((α_ P.X Q.X L.X).hom ≫
            P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
              coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))
                ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫
                  P.X ◁ TensorBimod.actLeft Q L))
          ⋯

Tactic:
slice_rhs 3 5 => rw [π_tensor_id_preserves_coequalizer_inv_desc]

Goal state after:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon C
M : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
R S T U : Mon C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
⊢ (α_ P.X Q.X T.X).hom ▷ L.X ≫
      (α_ P.X (Q.X ⊗ T.X) L.X).hom ≫
        ((P.X ◁ (α_ Q.X T.X L.X).hom ≫ P.X ◁ Q.X ◁ L.actLeft) ≫
            P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ≫
          coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))
            ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫
              P.X ◁ TensorBimod.actLeft Q L) =
    (α_ (P.X ⊗ Q.X) T.X L.X).hom ≫
      (P.X ⊗ Q.X) ◁ L.actLeft ≫
        (α_ P.X Q.X L.X).hom ≫
          P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
            coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))
              ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫
                P.X ◁ TensorBimod.actLeft Q L)
Module: Mathlib.CategoryTheory.Monoidal.Bimod

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon C
M : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
R S T U : Mon C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
⊢ (((α_ P.X (Q.X ⊗ T.X) L.X).inv ≫ (P.X ◁ Q.actRight) ▷ L.X) ≫
        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ L.X) ≫
      coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
        ((α_ (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X L.X).hom ≫
          coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ◁ L.actLeft) =
    P.X ◁ ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
      (α_ P.X Q.X L.X).inv ≫
        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ L.X ≫
          coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
            ((α_ (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X L.X).hom ≫
              coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ◁ L.actLeft)

Tactic:
← Iso.inv_hom_id_assoc (α_ _ _ _) (P.X ◁ Q.actRight),

Goal state after:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon C
M : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
R S T U : Mon C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
⊢ (((α_ P.X (Q.X ⊗ T.X) L.X).inv ≫ ((α_ P.X Q.X T.X).inv ≫ (α_ P.X Q.X T.X).hom ≫ P.X ◁ Q.actRight) ▷ L.X) ≫
        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ L.X) ≫
      coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
        ((α_ (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X L.X).hom ≫
          coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ◁ L.actLeft) =
    P.X ◁ ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
      (α_ P.X Q.X L.X).inv ≫
        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ L.X ≫
          coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
            ((α_ (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X L.X).hom ≫
              coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ◁ L.actLeft)
Module: Mathlib.CategoryTheory.Monoidal.Hopf_

Goal state before:
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 ◁ ((β_ A A).hom ≫ ε ▷ A)

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 ◁ (λ_ 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
| μ

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 ▷ A

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
| (α_ A A A).hom

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
| A ◁ 𝒮 ▷ A

Tactic:
rw [← BraidedCategory.braiding_naturality_right]

Goal state after:
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 ◁ ε ≫ (β_ A (𝟙_ C)).hom)

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 ◁ (λ_ 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
| μ

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 ▷ A

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
| (α_ A A A).hom

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
| A ◁ 𝒮 ▷ A
Module: Mathlib.CategoryTheory.Monoidal.Limits.Basic

Goal state before:
pp: J : Type w
inst✝³ : SmallCategory J
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasLimitsOfShape J C
inst✝ : MonoidalCategory C
F G H : J ⥤ C
j : J
| limit.lift (F ⊗ G) { pt := limit F ⊗ limit G, π := { app := fun j ↦ limit.π F j ⊗ₘ limit.π G j, naturality := ⋯ } } ▷
      limit H ≫
    (limit.π (F ⊗ G) j ⊗ₘ limit.π H j) ≫ (α_ (F.obj j) (G.obj j) (H.obj j)).hom

Tactic:
tensorHom_def,

Goal state after:
J : Type w
inst✝³ : SmallCategory J
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasLimitsOfShape J C
inst✝ : MonoidalCategory C
F G H : J ⥤ C
j : J
| limit.lift (F ⊗ G) { pt := limit F ⊗ limit G, π := { app := fun j ↦ limit.π F j ⊗ₘ limit.π G j, naturality := ⋯ } } ▷
      limit H ≫
    (limit.π (F ⊗ G) j ▷ limit H ≫ (F.obj j ⊗ G.obj j) ◁ limit.π H j) ≫ (α_ (F.obj j) (G.obj j) (H.obj j)).hom
Module: Mathlib.CategoryTheory.Monoidal.Limits.Basic

Goal state before:
pp: J : Type w
inst✝³ : SmallCategory J
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasLimitsOfShape J C
inst✝ : MonoidalCategory C
F G H : J ⥤ C
j : J
| (α_ (limit F) (limit G) (limit H)).hom ≫
    limit F ◁
        limit.lift (G ⊗ H)
          { pt := limit G ⊗ limit H, π := { app := fun j ↦ limit.π G j ⊗ₘ limit.π H j, naturality := ⋯ } } ≫
      (limit.π F j ⊗ₘ limit.π (G ⊗ H) j)

Tactic:
tensorHom_def,

Goal state after:
J : Type w
inst✝³ : SmallCategory J
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasLimitsOfShape J C
inst✝ : MonoidalCategory C
F G H : J ⥤ C
j : J
| (α_ (limit F) (limit G) (limit H)).hom ≫
    limit F ◁
        limit.lift (G ⊗ H)
          { pt := limit G ⊗ limit H, π := { app := fun j ↦ limit.π G j ⊗ₘ limit.π H j, naturality := ⋯ } } ≫
      limit.π F j ▷ limit (G ⊗ H) ≫ F.obj j ◁ limit.π (G ⊗ H) j
Module: Mathlib.CategoryTheory.Monoidal.Limits.Basic

Goal state before:
pp: J : Type w
inst✝³ : SmallCategory J
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasLimitsOfShape J C
inst✝ : MonoidalCategory C
F G H : J ⥤ C
j : J
| limit.π F j ▷ limit G ▷ limit H ≫ (α_ (F.obj j) (limit G) (limit H)).hom ≫ F.obj j ◁ (limit.π G j ⊗ₘ limit.π H j)

Tactic:
tensorHom_def,

Goal state after:
J : Type w
inst✝³ : SmallCategory J
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasLimitsOfShape J C
inst✝ : MonoidalCategory C
F G H : J ⥤ C
j : J
| limit.π F j ▷ limit G ▷ limit H ≫
    (α_ (F.obj j) (limit G) (limit H)).hom ≫ F.obj j ◁ (limit.π G j ▷ limit H ≫ G.obj j ◁ limit.π H j)
Module: Mathlib.CategoryTheory.Monoidal.Opposite.Mon

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : MonoidalCategory C
M : C
inst✝² : MonObj M
N : C
inst✝¹ : MonObj N
f : M ⟶ N
inst✝ : IsMonHom f
⊢ MonObj.one ≫ f.mop = MonObj.one

Tactic:
apply mopEquiv C |>.fullyFaithfulInverse.map_injective
    simpa [-IsMonHom.one_hom] using! IsMonHom.one_hom f

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

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
R S : Mon C
P : Bimod R S
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ (P.tensorBimod (regular S)).actRight ≫ hom P = hom P ▷ S.X ≫ P.actRight

Tactic:
by
  dsimp; dsimp [hom, TensorBimod.actRight, regular]
  refine (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 ?_
  dsimp
  slice_lhs 1 4 => rw [π_tensor_id_preserves_coequalizer_inv_colimMap_desc]
  slice_lhs 2 3 => rw [right_assoc]
  slice_rhs 1 2 => rw [← comp_whiskerRight, coequalizer.π_desc]
  rw [Iso.hom_inv_id_assoc]

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

Goal state before:
pp: C : Type u
inst✝⁷ : Category.{v, u} C
J✝ : Type u_1
I✝ : J✝ → Type u_2
inst✝⁶ : Category.{v_1, u_1} J✝
inst✝⁵ : (j : J✝) → Category.{?u.14, u_2} (I✝ j)
D✝ : J✝ ⥤ C
P✝ : (j : J✝) → ColimitPresentation (I✝ j) (D✝.obj j)
J : Type w
I : J → Type w
inst✝⁴ : SmallCategory J
inst✝³ : (j : J) → SmallCategory (I j)
D : J ⥤ C
P : (j : J) → ColimitPresentation (I j) (D.obj j)
inst✝² : IsFiltered J
inst✝¹ : ∀ (j : J), IsFiltered (I j)
inst✝ : ∀ (j : J) (i : I j), IsFinitelyPresentable ((P j).diag.obj i)
k l : Total P
f g : k ⟶ l
a : J := IsFiltered.coeq f.base g.base
a' : I (IsFiltered.coeq f.base g.base)
u : Total.mk P l.fst l.snd ⟶ Total.mk P (IsFiltered.coeq f.base g.base) a'
hu : u.base = IsFiltered.coeqHom f.base g.base
⊢ (P k.fst).ι.app k.snd ≫ D.map f.base ≫ D.map u.base = (P k.fst).ι.app k.snd ≫ D.map g.base ≫ D.map u.base

Tactic:
← Functor.map_comp,

Goal state after:
C : Type u
inst✝⁷ : Category.{v, u} C
J✝ : Type u_1
I✝ : J✝ → Type u_2
inst✝⁶ : Category.{v_1, u_1} J✝
inst✝⁵ : (j : J✝) → Category.{?u.14, u_2} (I✝ j)
D✝ : J✝ ⥤ C
P✝ : (j : J✝) → ColimitPresentation (I✝ j) (D✝.obj j)
J : Type w
I : J → Type w
inst✝⁴ : SmallCategory J
inst✝³ : (j : J) → SmallCategory (I j)
D : J ⥤ C
P : (j : J) → ColimitPresentation (I j) (D.obj j)
inst✝² : IsFiltered J
inst✝¹ : ∀ (j : J), IsFiltered (I j)
inst✝ : ∀ (j : J) (i : I j), IsFinitelyPresentable ((P j).diag.obj i)
k l : Total P
f g : k ⟶ l
a : J := IsFiltered.coeq f.base g.base
a' : I (IsFiltered.coeq f.base g.base)
u : Total.mk P l.fst l.snd ⟶ Total.mk P (IsFiltered.coeq f.base g.base) a'
hu : u.base = IsFiltered.coeqHom f.base g.base
⊢ (P k.fst).ι.app k.snd ≫ D.map (f.base ≫ u.base) = (P k.fst).ι.app k.snd ≫ D.map g.base ≫ D.map u.base
Module: Mathlib.CategoryTheory.ObjectProperty.Ind

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
D : Type u_1
inst✝² : Category.{u_2, u_1} D
P : ObjectProperty D
F : C ⥤ D
inst✝¹ : P.IsClosedUnderIsomorphisms
inst✝ : F.IsEquivalence
X : C
x✝ : P.ind.inverseImage F X
J : Type w
w✝¹ : SmallCategory J
w✝ : IsFiltered J
pres : ColimitPresentation J (F.obj X)
h : ∀ (i : J), P (pres.diag.obj i)
⊢ (P.inverseImage F).ind X

Tactic:
refine ⟨J, ‹_›, ‹_›, .ofIso (pres.map F.asEquivalence.inverse) ?_, fun j ↦ ?_⟩

Goal state after:
case refine_1
C : Type u
inst✝³ : Category.{v, u} C
D : Type u_1
inst✝² : Category.{u_2, u_1} D
P : ObjectProperty D
F : C ⥤ D
inst✝¹ : P.IsClosedUnderIsomorphisms
inst✝ : F.IsEquivalence
X : C
x✝ : P.ind.inverseImage F X
J : Type w
w✝¹ : SmallCategory J
w✝ : IsFiltered J
pres : ColimitPresentation J (F.obj X)
h : ∀ (i : J), P (pres.diag.obj i)
⊢ F.asEquivalence.inverse.obj (F.obj X) ≅ X

case refine_2
C : Type u
inst✝³ : Category.{v, u} C
D : Type u_1
inst✝² : Category.{u_2, u_1} D
P : ObjectProperty D
F : C ⥤ D
inst✝¹ : P.IsClosedUnderIsomorphisms
inst✝ : F.IsEquivalence
X : C
x✝ : P.ind.inverseImage F X
J : Type w
w✝¹ : SmallCategory J
w✝ : IsFiltered J
pres : ColimitPresentation J (F.obj X)
h : ∀ (i : J), P (pres.diag.obj i)
j : J
⊢ P.inverseImage F (((pres.map F.asEquivalence.inverse).ofIso ?refine_1).diag.obj j)
Module: Mathlib.CategoryTheory.ObjectProperty.Ind

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
P : ObjectProperty C
ι : Type u_1
inst✝³ : Small.{w, u_1} ι
inst✝² : P.IsClosedUnderLimitsOfShape (Discrete ι)
inst✝¹ : HasProductsOfShape ι C
inst✝ : IsIPCOfShape.{w, u_1, v, u} ι C
X : C
x✝ : P.ind.strictLimitsOfShape (Discrete ι) X
Y : Discrete ι ⥤ C
h : ∀ (j : Discrete ι), P.ind (Y.obj j)
e : ι ≃ Shrink.{w, u_1} ι := equivShrink ι
this : HasProductsOfShape (Shrink.{w, u_1} ι) C
⊢ P.ind (limit Y)

Tactic:
have : IsIPCOfShape.{w} (Shrink.{w} ι) C := .of_equiv e

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
P : ObjectProperty C
ι : Type u_1
inst✝³ : Small.{w, u_1} ι
inst✝² : P.IsClosedUnderLimitsOfShape (Discrete ι)
inst✝¹ : HasProductsOfShape ι C
inst✝ : IsIPCOfShape.{w, u_1, v, u} ι C
X : C
x✝ : P.ind.strictLimitsOfShape (Discrete ι) X
Y : Discrete ι ⥤ C
h : ∀ (j : Discrete ι), P.ind (Y.obj j)
e : ι ≃ Shrink.{w, u_1} ι := equivShrink ι
this✝ : HasProductsOfShape (Shrink.{w, u_1} ι) C
this : IsIPCOfShape.{w, w, v, u} (Shrink.{w, u_1} ι) C
⊢ P.ind (limit Y)
Module: Mathlib.CategoryTheory.Presentable.ColimitPresentation

Goal state before:
pp: C : Type u
inst✝⁷ : Category.{v, u} C
J✝ : Type u_1
I✝ : J✝ → Type u_2
inst✝⁶ : Category.{v_1, u_1} J✝
inst✝⁵ : (j : J✝) → Category.{?u.14, u_2} (I✝ j)
D✝ : J✝ ⥤ C
P✝ : (j : J✝) → ColimitPresentation (I✝ j) (D✝.obj j)
J : Type w
I : J → Type w
inst✝⁴ : SmallCategory J
inst✝³ : (j : J) → SmallCategory (I j)
D : J ⥤ C
P : (j : J) → ColimitPresentation (I j) (D.obj j)
inst✝² : IsFiltered J
inst✝¹ : ∀ (j : J), IsFiltered (I j)
inst✝ : ∀ (j : J) (i : I j), IsFinitelyPresentable ((P j).diag.obj i)
k l : Total P
f g : k ⟶ l
⊢ ∃ Z h, f ≫ h = g ≫ h

Tactic:
let a := IsFiltered.coeq f.base g.base
    obtain ⟨a', u, hu⟩ := Total.exists_hom_of_hom (P := P) l.2 (IsFiltered.coeqHom f.base g.base)
    have : (f.hom ≫ u.hom) ≫ (P _).ι.app _ = (g.hom ≫ u.hom) ≫ (P _).ι.app _ := by
      simp only [Category.assoc, Functor.const_obj_obj, ← u.w, ← f.w_assoc, ← g.w_assoc]
      rw [← Functor.map_comp, hu, IsFiltered.coeq_condition f.base g.base]
      simp
    obtain ⟨j, p, q, hpq⟩ := IsFinitelyPresentable.exists_eq_of_isColimit (P _).isColimit _ _ this
    dsimp at p q
    refine ⟨⟨a, IsFiltered.coeq p q⟩,
      u ≫ { base := 𝟙 _, hom := (P _).diag.map (p ≫ IsFiltered.coeqHom p q) }, ?_⟩
    apply Total.Hom.ext
    · simp [hu, IsFiltered.coeq_condition f.base g.base]
    · rw [Category.assoc] at hpq
      simp only [Functor.map_comp, comp_hom, reassoc_of% hpq]
      simp [← Functor.map_comp, ← IsFiltered.coeq_condition]

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

Goal state before:
pp: case refine_2.refine_1
C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
X Y : C
f : X ⟶ Y
x✝ : P.underObj.ind (CategoryTheory.Under.mk f)
J : Type w
w✝¹ : SmallCategory J
w✝ : IsFiltered J
pres : ColimitPresentation J (CategoryTheory.Under.mk f)
hpres : ∀ (i : J), P.underObj (pres.diag.obj i)
⊢ (Functor.const J).obj X ⟶ pres.diag ⋙ CategoryTheory.Under.forget X

Tactic:
exact { app j := (pres.diag.obj j).hom }

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

Goal state before:
pp: case right
C : Type u
inst✝¹ : Category.{v, u} C
P : MorphismProperty C
H✝ : P ≤ isFinitelyPresentable C
X Y : C
f : X ⟶ Y
inst✝ : IsFinitelyAccessibleCategory (Under X)
H : ∀ {Z : Under X} (g : Z ⟶ CategoryTheory.Under.mk f) [IsFinitelyPresentable Z], ∃ W u v, u ≫ v = g ∧ P.underObj W
Z : C
p : X ⟶ Z
g : Z ⟶ Y
hp : isFinitelyPresentable C p
hpg : p ≫ g = f
this : IsFinitelyPresentable (CategoryTheory.Under.mk p)
W : Under X
u : CategoryTheory.Under.mk p ⟶ W
v : W ⟶ CategoryTheory.Under.mk f
huv : u ≫ v = CategoryTheory.Under.homMk g hpg
hW : P.underObj W
⊢ P (p ≫ Under.Hom.right u)

Tactic:
show p ≫ u.right = W.hom from CategoryTheory.Under.w u

Goal state after:
case right
C : Type u
inst✝¹ : Category.{v, u} C
P : MorphismProperty C
H✝ : P ≤ isFinitelyPresentable C
X Y : C
f : X ⟶ Y
inst✝ : IsFinitelyAccessibleCategory (Under X)
H : ∀ {Z : Under X} (g : Z ⟶ CategoryTheory.Under.mk f) [IsFinitelyPresentable Z], ∃ W u v, u ≫ v = g ∧ P.underObj W
Z : C
p : X ⟶ Z
g : Z ⟶ Y
hp : isFinitelyPresentable C p
hpg : p ≫ g = f
this : IsFinitelyPresentable (CategoryTheory.Under.mk p)
W : Under X
u : CategoryTheory.Under.mk p ⟶ W
v : W ⟶ CategoryTheory.Under.mk f
huv : u ≫ v = CategoryTheory.Under.homMk g hpg
hW : P.underObj W
⊢ P W.hom
Module: Mathlib.CategoryTheory.Subobject.ArtinianObject

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X : C
hX : IsZero X
this : Subsingleton (Subobject X)
⊢ ∀ (f : ℕ →o (Subobject X)ᵒᵈ), ∃ n, ∀ (m : ℕ), n ≤ m → f n = f m

Tactic:
intro f

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
X : C
hX : IsZero X
this : Subsingleton (Subobject X)
f : ℕ →o (Subobject X)ᵒᵈ
⊢ ∃ n, ∀ (m : ℕ), n ≤ m → f n = f m
Module: Mathlib.CategoryTheory.Subobject.ArtinianObject

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
X Y : C
i : X ⟶ Y
inst✝¹ : Mono i
inst✝ : IsArtinianObject Y
⊢ ∀ (f : ℕ →o (Subobject X)ᵒᵈ), ∃ n, ∀ (m : ℕ), n ≤ m → f n = f m

Tactic:
intro f

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
X Y : C
i : X ⟶ Y
inst✝¹ : Mono i
inst✝ : IsArtinianObject Y
f : ℕ →o (Subobject X)ᵒᵈ
⊢ ∃ n, ∀ (m : ℕ), n ≤ m → f n = f m
Module: Mathlib.CategoryTheory.Subobject.NoetherianObject

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X : C
hX : IsZero X
this : Subsingleton (Subobject X)
⊢ ∀ (f : ℕ →o Subobject X), ∃ n, ∀ (m : ℕ), n ≤ m → f n = f m

Tactic:
intro f

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
X : C
hX : IsZero X
this : Subsingleton (Subobject X)
f : ℕ →o Subobject X
⊢ ∃ n, ∀ (m : ℕ), n ≤ m → f n = f m
Module: Mathlib.CategoryTheory.Subobject.NoetherianObject

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
X Y : C
i : X ⟶ Y
inst✝¹ : Mono i
inst✝ : IsNoetherianObject Y
⊢ ∀ (f : ℕ →o Subobject X), ∃ n, ∀ (m : ℕ), n ≤ m → f n = f m

Tactic:
intro f

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
X Y : C
i : X ⟶ Y
inst✝¹ : Mono i
inst✝ : IsNoetherianObject Y
f : ℕ →o Subobject X
⊢ ∃ n, ∀ (m : ℕ), n ≤ m → f n = f m
Module: Mathlib.CategoryTheory.Pi.Monoidal

Goal state before:
pp: I : Type w₁
C : I → Type u₁
inst✝² : (i : I) → Category.{v₁, u₁} (C i)
inst✝¹ : (i : I) → MonoidalCategory (C i)
inst✝ : (i : I) → BraidedCategory (C i)
X Y Z : (i : I) → C i
⊢ (α_ X Y Z).hom ≫ (isoMk fun i ↦ β_ (X i) ((Y ⊗ Z) i)).hom ≫ (α_ Y Z X).hom =
    (isoMk fun i ↦ β_ (X i) (Y i)).hom ▷ Z ≫ (α_ Y X Z).hom ≫ Y ◁ (isoMk fun i ↦ β_ (X i) (Z i)).hom

Tactic:
by ext i; apply hexagon_forward

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

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon C
M M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
⊢ M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
      f.hom ▷ coequalizer (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
        colimit.ι
          (parallelPair (M'.actRight ▷ coequalizer (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft))
            ((α_ M'.X X.X (coequalizer (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft))).hom ≫
              M'.X ◁ TensorBimod.actLeft N P))
          WalkingParallelPair.one =
    (α_ M.X N.X P.X).inv ≫
      ((f.hom ▷ N.X ▷ P.X ≫
            colimit.ι (parallelPair (M'.actRight ▷ N.X) ((α_ M'.X X.X N.X).hom ≫ M'.X ◁ N.actLeft))
                WalkingParallelPair.one ▷
              P.X) ≫
          (PreservesCoequalizer.iso (tensorRight P.X) (M'.actRight ▷ N.X)
              ((α_ M'.X X.X N.X).hom ≫ M'.X ◁ N.actLeft)).inv) ≫
        coequalizer.desc
          ((α_ M'.X N.X P.X).hom ≫
            M'.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
              coequalizer.π (M'.actRight ▷ TensorBimod.X N P)
                ((α_ M'.X X.X (TensorBimod.X N P)).hom ≫ M'.X ◁ TensorBimod.actLeft N P))
          ⋯

Tactic:
slice_rhs 3 5 => rw [π_tensor_id_preserves_coequalizer_inv_desc]

Goal state after:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon C
M M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
⊢ M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
      f.hom ▷ coequalizer (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
        colimit.ι
          (parallelPair (M'.actRight ▷ coequalizer (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft))
            ((α_ M'.X X.X (coequalizer (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft))).hom ≫
              M'.X ◁ TensorBimod.actLeft N P))
          WalkingParallelPair.one =
    (α_ M.X N.X P.X).inv ≫
      f.hom ▷ N.X ▷ P.X ≫
        (α_ M'.X N.X P.X).hom ≫
          M'.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
            coequalizer.π (M'.actRight ▷ TensorBimod.X N P)
              ((α_ M'.X X.X (TensorBimod.X N P)).hom ≫ M'.X ◁ TensorBimod.actLeft N P)
Module: Mathlib.CategoryTheory.Preadditive.OfBiproducts

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
⊢ ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr

Tactic:
intro f

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f : X ⟶ Y
⊢ biprod.lift 0 f = f ≫ biprod.inr