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

Goal state before:
pp: V : Type u₁
inst✝⁵ : Category.{v₁, u₁} V
inst✝⁴ : MonoidalCategory V
D : Type u₂
D' : Type u₃
E : Type u₄
inst✝³ : Category.{v₂, u₂} D
inst✝² : Category.{v₃, u₃} D'
inst✝¹ : Category.{v₄, u₄} E
H : D ⥤ V
L : D ⥤ D'
H' : D' ⥤ V
α : H ⟶ L ⋙ H'
K : E ⥤ V
d : D'
P : (Functor.LeftExtension.mk H' α).IsPointwiseLeftKanExtensionAt d
e : E
inst✝ : Limits.PreservesColimitsOfShape (CostructuredArrow L d) (tensorRight (K.obj e))
cone : Limits.Cocone (CostructuredArrow.proj (L.prod (𝟭 E)) (d, e) ⋙ H ⊠ K) :=
  (Functor.LeftExtension.mk (H' ⊠ K) (extensionUnitLeft H' α K)).coconeAt (d, e)
equiv : CostructuredArrow L d × CostructuredArrow (𝟭 E) e ≌ CostructuredArrow (L.prod (𝟭 E)) (d, e) :=
  (CostructuredArrow.prodEquivalence L (𝟭 E) d e).symm
I : CostructuredArrow L d ⥤ CostructuredArrow L d × CostructuredArrow (𝟭 E) e :=
  (prod.rightUnitorEquivalence (CostructuredArrow L d)).inverse ⋙
    (𝟭 (CostructuredArrow L d)).prod (Functor.fromPUnit (CostructuredArrow.mk (𝟙 ((𝟭 E).obj e))))
this : (Functor.fromPUnit (CostructuredArrow.mk (𝟙 e))).Final :=
  Functor.final_fromPUnit_of_isTerminal CostructuredArrow.mkIdTerminal
⊢ I.Final

Tactic:
apply Iff.mp <| Functor.final_iff_final_comp
      (F := (prod.rightUnitorEquivalence <| CostructuredArrow L d).inverse)
      (G := (𝟭 _).prod <| Functor.fromPUnit.{0} (.mk (𝟙 e) : CostructuredArrow (𝟭 E) e))

Goal state after:
V : Type u₁
inst✝⁵ : Category.{v₁, u₁} V
inst✝⁴ : MonoidalCategory V
D : Type u₂
D' : Type u₃
E : Type u₄
inst✝³ : Category.{v₂, u₂} D
inst✝² : Category.{v₃, u₃} D'
inst✝¹ : Category.{v₄, u₄} E
H : D ⥤ V
L : D ⥤ D'
H' : D' ⥤ V
α : H ⟶ L ⋙ H'
K : E ⥤ V
d : D'
P : (Functor.LeftExtension.mk H' α).IsPointwiseLeftKanExtensionAt d
e : E
inst✝ : Limits.PreservesColimitsOfShape (CostructuredArrow L d) (tensorRight (K.obj e))
cone : Limits.Cocone (CostructuredArrow.proj (L.prod (𝟭 E)) (d, e) ⋙ H ⊠ K) := ⋯
equiv : CostructuredArrow L d × CostructuredArrow (𝟭 E) e ≌ CostructuredArrow (L.prod (𝟭 E)) (d, e) := ⋯
I : CostructuredArrow L d ⥤ CostructuredArrow L d × CostructuredArrow (𝟭 E) e := ⋯
this : (Functor.fromPUnit (CostructuredArrow.mk (𝟙 e))).Final := ⋯
⊢ ((𝟭 (CostructuredArrow L d)).prod (Functor.fromPUnit (CostructuredArrow.mk (𝟙 e)))).Final
Module: Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed

Goal state before:
pp: C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
V : Type u₂
inst✝⁴ : Category.{v₂, u₂} V
inst✝³ : MonoidalCategory C
inst✝² : MonoidalCategory V
inst✝¹ : MonoidalClosed V
F G✝ H G : C ⥤ V
inst✝ : DayConvolution F G
ℌ : DayConvolutionInternalHom F (F ⊛ G) H
c c' : C
f : c ⟶ c'
j : C
⊢ G.map f ≫ MonoidalClosed.curry ((DayConvolution.unit F G).app (j, c')) =
    MonoidalClosed.curry ((DayConvolution.unit F G).app (j, c)) ≫ (ihom (F.obj j)).map ((F ⊛ G).map (j ◁ f))

Tactic:
have := DayConvolution.unit_naturality F G (𝟙 j) f

Goal state after:
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
V : Type u₂
inst✝⁴ : Category.{v₂, u₂} V
inst✝³ : MonoidalCategory C
inst✝² : MonoidalCategory V
inst✝¹ : MonoidalClosed V
F G✝ H G : C ⥤ V
inst✝ : DayConvolution F G
ℌ : DayConvolutionInternalHom F (F ⊛ G) H
c c' : C
f : c ⟶ c'
j : C
this :
  (F.map (𝟙 j) ⊗ₘ G.map f) ≫ (DayConvolution.unit F G).app (j, c') =
    (DayConvolution.unit F G).app (j, c) ≫ (F ⊛ G).map (𝟙 j ⊗ₘ f)
⊢ G.map f ≫ MonoidalClosed.curry ((DayConvolution.unit F G).app (j, c')) =
    MonoidalClosed.curry ((DayConvolution.unit F G).app (j, c)) ≫ (ihom (F.obj j)).map ((F ⊛ G).map (j ◁ f))
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 ≫
      (α_ P.X Q.X T.X).inv ▷ L.X ≫
        (((α_ (P.X ⊗ Q.X) T.X L.X).hom ≫
              coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ (T.X ⊗ L.X)) ≫
            coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ◁ L.actLeft) ≫
          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 ≫
      (((α_ P.X Q.X (T.X ⊗ L.X)).inv ≫ (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) ≫
        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:
slice_rhs 3 4 => rw [whisker_exchange]

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 ▷ L.X ≫
        (((α_ (P.X ⊗ Q.X) T.X L.X).hom ≫
              coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ (T.X ⊗ L.X)) ≫
            coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ◁ L.actLeft) ≫
          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 ≫
      (α_ P.X Q.X (T.X ⊗ L.X)).inv ≫
        (coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ (T.X ⊗ L.X) ≫
            coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ◁ L.actLeft) ≫
          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 hba
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : BraidedCategory C
A : C
inst✝ : HopfObj A
comm : (β_ A A).hom ≫ μ = μ
⊢ 𝒮 ≫ 𝒮 * 𝒮 = 1

Tactic:
rw [Conv.mul_eq, Conv.one_eq]
    simp only [comp_whiskerRight, Category.assoc]
    rw [← comm, ← tensorHom_def_assoc, ← mul_antipode]
    simp

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)
X Y : Mon C
M N : Bimod X Y
f : M ⟶ N
⊢ X.X ◁ f.hom ≫ colimit.ι (parallelPair (μ ▷ N.X) ((α_ X.X X.X N.X).hom ≫ X.X ◁ N.actLeft)) WalkingParallelPair.one =
    X.X ◁ f.hom ≫
      (((λ_ (X.X ⊗ N.X)).inv ≫ 𝟙_ C ◁ N.actLeft) ≫ η ▷ N.X) ≫
        coequalizer.π (μ ▷ N.X) ((α_ X.X X.X N.X).hom ≫ X.X ◁ N.actLeft)

Tactic:
slice_rhs 3 4 => rw [whisker_exchange]

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)
X Y : Mon C
M N : Bimod X Y
f : M ⟶ N
⊢ X.X ◁ f.hom ≫ colimit.ι (parallelPair (μ ▷ N.X) ((α_ X.X X.X N.X).hom ≫ X.X ◁ N.actLeft)) WalkingParallelPair.one =
    X.X ◁ f.hom ≫
      (λ_ (X.X ⊗ N.X)).inv ≫
        (η ▷ (X.X ⊗ N.X) ≫ X.X ◁ N.actLeft) ≫ coequalizer.π (μ ▷ N.X) ((α_ X.X X.X N.X).hom ≫ X.X ◁ N.actLeft)
Module: Mathlib.CategoryTheory.Presentable.ColimitPresentation

Goal state before:
pp: case refine_2
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
a : J := IsFiltered.max k.fst l.fst
a' : I (IsFiltered.max k.fst l.fst)
f : Total.mk P k.fst k.snd ⟶ Total.mk P (IsFiltered.max k.fst l.fst) a'
hf : f.base = IsFiltered.leftToMax k.fst l.fst
b' : I (IsFiltered.max k.fst l.fst)
g : Total.mk P l.fst l.snd ⟶ Total.mk P (IsFiltered.max k.fst l.fst) b'
hg : g.base = IsFiltered.rightToMax k.fst l.fst
⊢ l ⟶ ⟨a, IsFiltered.max a' b'⟩

Tactic:
exact g ≫ { base := 𝟙 _, hom := (P _).diag.map (IsFiltered.rightToMax _ _) }

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

Goal state before:
pp: case refine_1.refine_1
C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
X Y : C
f : X ⟶ Y
x✝ : P.ind f
J : Type w
w✝¹ : SmallCategory J
w✝ : IsFiltered J
D : J ⥤ C
t : (Functor.const J).obj X ⟶ D
s : D ⟶ (Functor.const J).obj Y
hs : IsColimit { pt := Y, ι := s }
hst : ∀ (j : J), P (t.app j) ∧ t.app j ≫ s.app j = f
⊢ Under.lift D t ⟶ (Functor.const J).obj (CategoryTheory.Under.mk f)

pp: case refine_1.refine_2
C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
X Y : C
f : X ⟶ Y
x✝ : P.ind f
J : Type w
w✝¹ : SmallCategory J
w✝ : IsFiltered J
D : J ⥤ C
t : (Functor.const J).obj X ⟶ D
s : D ⟶ (Functor.const J).obj Y
hs : IsColimit { pt := Y, ι := s }
hst : ∀ (j : J), P (t.app j) ∧ t.app j ≫ s.app j = f
⊢ IsColimit { pt := CategoryTheory.Under.mk f, ι := ?refine_1.refine_1 }

pp: case refine_1.refine_3
C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
X Y : C
f : X ⟶ Y
x✝ : P.ind f
J : Type w
w✝¹ : SmallCategory J
w✝ : IsFiltered J
D : J ⥤ C
t : (Functor.const J).obj X ⟶ D
s : D ⟶ (Functor.const J).obj Y
hs : IsColimit { pt := Y, ι := s }
hst : ∀ (j : J), P (t.app j) ∧ t.app j ≫ s.app j = f
⊢ ∀ (i : J),
    P.underObj ({ diag := Under.lift D t, ι := ?refine_1.refine_1, isColimit := ?refine_1.refine_2 }.diag.obj i)

Tactic:
· exact { app j := CategoryTheory.Under.homMk (s.app j) (by simp [hst]) }

Goal state after:
case refine_1.refine_2
C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
X Y : C
f : X ⟶ Y
x✝ : P.ind f
J : Type w
w✝¹ : SmallCategory J
w✝ : IsFiltered J
D : J ⥤ C
t : (Functor.const J).obj X ⟶ D
s : D ⟶ (Functor.const J).obj Y
hs : IsColimit { pt := Y, ι := s }
hst : ∀ (j : J), P (t.app j) ∧ t.app j ≫ s.app j = f
⊢ IsColimit
    { pt := CategoryTheory.Under.mk f, ι := { app := fun j ↦ CategoryTheory.Under.homMk (s.app j) ⋯, naturality := ⋯ } }

case refine_1.refine_3
C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
X Y : C
f : X ⟶ Y
x✝ : P.ind f
J : Type w
w✝¹ : SmallCategory J
w✝ : IsFiltered J
D : J ⥤ C
t : (Functor.const J).obj X ⟶ D
s : D ⟶ (Functor.const J).obj Y
hs : IsColimit { pt := Y, ι := s }
hst : ∀ (j : J), P (t.app j) ∧ t.app j ≫ s.app j = f
⊢ ∀ (i : J),
    P.underObj
      ({ diag := Under.lift D t, ι := { app := fun j ↦ CategoryTheory.Under.homMk (s.app j) ⋯, naturality := ⋯ },
              isColimit := ?refine_1.refine_2 }.diag.obj
        i)
Module: Mathlib.CategoryTheory.MorphismProperty.Ind

Goal state before:
pp: case refine_1
C : Type u
inst✝¹ : Category.{v, u} C
P Q : MorphismProperty C
inst✝ : P.RespectsIso
X Y Z : C
i : Y ⟶ Z
hi : IsIso i
f : X ⟶ Y
x✝ : P.ind f
J : Type u_1
w✝¹ : SmallCategory J
w✝ : IsFiltered J
D : J ⥤ C
t : (Functor.const J).obj X ⟶ D
s : D ⟶ (Functor.const J).obj Y
hs : IsColimit { pt := Y, ι := s }
hst : ∀ (j : J), P (t.app j) ∧ t.app j ≫ s.app j = f
⊢ IsColimit { pt := Z, ι := s ≫ (Functor.const J).map i }

pp: case refine_2
C : Type u
inst✝¹ : Category.{v, u} C
P Q : MorphismProperty C
inst✝ : P.RespectsIso
X Y Z : C
i : Y ⟶ Z
hi : IsIso i
f : X ⟶ Y
x✝ : P.ind f
J : Type u_1
w✝¹ : SmallCategory J
w✝ : IsFiltered J
D : J ⥤ C
t : (Functor.const J).obj X ⟶ D
s : D ⟶ (Functor.const J).obj Y
hs : IsColimit { pt := Y, ι := s }
hst : ∀ (j : J), P (t.app j) ∧ t.app j ≫ s.app j = f
j : J
⊢ t.app j ≫ (s ≫ (Functor.const J).map i).app j = f ≫ i

Tactic:
· exact (IsColimit.equivIsoColimit (Cocone.ext (asIso i))) hs

Goal state after:
case refine_2
C : Type u
inst✝¹ : Category.{v, u} C
P Q : MorphismProperty C
inst✝ : P.RespectsIso
X Y Z : C
i : Y ⟶ Z
hi : IsIso i
f : X ⟶ Y
x✝ : P.ind f
J : Type u_1
w✝¹ : SmallCategory J
w✝ : IsFiltered J
D : J ⥤ C
t : (Functor.const J).obj X ⟶ D
s : D ⟶ (Functor.const J).obj Y
hs : IsColimit { pt := Y, ι := s }
hst : ∀ (j : J), P (t.app j) ∧ t.app j ≫ s.app j = f
j : J
⊢ t.app j ≫ (s ≫ (Functor.const J).map i).app j = f ≫ i
Module: Mathlib.CategoryTheory.MorphismProperty.Ind

Goal state before:
pp: case H
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)
⊢ P.underObj ≤ ObjectProperty.isFinitelyPresentable (Under X)

Tactic:
· intro Y hY
    exact H _ hY

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
s : Cocone F
j : J
x' y' : (F ⋙ forget PartOrdEmb).obj j
hx :
  (ConcreteCategory.hom (hc.desc ((forget PartOrdEmb).mapCocone s))) ((ConcreteCategory.hom (c.ι.app j)) x') =
    (ConcreteCategory.hom (s.ι.app j)) x'
hy :
  (ConcreteCategory.hom (hc.desc ((forget PartOrdEmb).mapCocone s))) ((ConcreteCategory.hom (c.ι.app j)) y') =
    (ConcreteCategory.hom (s.ι.app j)) y'
x✝ : (ConcreteCategory.hom (c.ι.app j)) x' ≤ (ConcreteCategory.hom (c.ι.app j)) y'
k : J
x y : ↑(F.obj k)
hx' : (ConcreteCategory.hom (c.ι.app k)) x = (ConcreteCategory.hom (c.ι.app j)) x'
hy' : (ConcreteCategory.hom (c.ι.app k)) y = (ConcreteCategory.hom (c.ι.app j)) y'
h : x ≤ y
l : J
f : k ⟶ l
g : j ⟶ l
hl : (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map g)) x'
l' : J
f' : k ⟶ l'
g' : j ⟶ l'
hl' : (ConcreteCategory.hom (F.map f')) y = (ConcreteCategory.hom (F.map g')) y'
m : J
a : l ⟶ m
b : l' ⟶ m
h₁ : f ≫ a = f' ≫ b
h₂ : g ≫ a = g' ≫ b
⊢ x' ≤ y'

Tactic:
← (F.map (f ≫ a)).le_iff_le

Goal state after:
J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F : J ⥤ PartOrdEmb
c : Cocone (F ⋙ forget PartOrdEmb)
hc : IsColimit c
s : Cocone F
j : J
x' y' : (F ⋙ forget PartOrdEmb).obj j
hx :
  (ConcreteCategory.hom (hc.desc ((forget PartOrdEmb).mapCocone s))) ((ConcreteCategory.hom (c.ι.app j)) x') =
    (ConcreteCategory.hom (s.ι.app j)) x'
hy :
  (ConcreteCategory.hom (hc.desc ((forget PartOrdEmb).mapCocone s))) ((ConcreteCategory.hom (c.ι.app j)) y') =
    (ConcreteCategory.hom (s.ι.app j)) y'
x✝ : (ConcreteCategory.hom (c.ι.app j)) x' ≤ (ConcreteCategory.hom (c.ι.app j)) y'
k : J
x y : ↑(F.obj k)
hx' : (ConcreteCategory.hom (c.ι.app k)) x = (ConcreteCategory.hom (c.ι.app j)) x'
hy' : (ConcreteCategory.hom (c.ι.app k)) y = (ConcreteCategory.hom (c.ι.app j)) y'
l : J
f : k ⟶ l
g : j ⟶ l
hl : (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map g)) x'
l' : J
f' : k ⟶ l'
g' : j ⟶ l'
hl' : (ConcreteCategory.hom (F.map f')) y = (ConcreteCategory.hom (F.map g')) y'
m : J
a : l ⟶ m
h : (ConcreteCategory.hom (F.map (f ≫ a))) x ≤ (ConcreteCategory.hom (F.map (f ≫ a))) y
b : l' ⟶ m
h₁ : f ≫ a = f' ≫ b
h₂ : g ≫ a = g' ≫ b
⊢ x' ≤ y'
Module: Mathlib.CategoryTheory.Presentable.OrthogonalReflection

Goal state before:
pp: case refine_1
C : Type u
inst✝⁷ : Category.{v, u} C
W : MorphismProperty C
Z : C
inst✝⁶ : HasPushouts C
inst✝⁵ : ∀ (Z : C), HasCoproduct D₁.obj₁
inst✝⁴ : ∀ (Z : C), HasCoproduct D₁.obj₂
inst✝³ : ∀ (Z : C), HasMulticoequalizer (D₂.multispanIndex W Z)
κ : Cardinal.{w}
inst✝² : OrderBot κ.ord.ToType
inst✝¹ : HasIterationOfShape κ.ord.ToType C
inst✝ : Fact κ.IsRegular
hW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ
H : W.isLocal.isLocal.TransfiniteCompositionOfShape κ.ord.ToType (reflection W Z κ) :=
  transfiniteCompositionOfShapeReflection W Z κ
X Y : C
f : X ⟶ Y
hf : W f
left✝ : IsCardinalPresentable X κ
right✝ : IsCardinalPresentable Y κ
j : κ.ord.ToType
g₁ g₂ : Y ⟶ H.F.obj j
h : f ≫ g₁ ≫ H.incl.app j = f ≫ g₂ ≫ H.incl.app j
k : κ.ord.ToType
u : j ⟶ k
hk : (f ≫ g₁) ≫ H.F.map u = (f ≫ g₂) ≫ H.F.map u
hg : g₁ ≫ H.F.map u ≫ (iteration W Z κ).map (homOfLE ⋯) = g₂ ≫ H.F.map u ≫ (iteration W Z κ).map (homOfLE ⋯)
this :
  H.F.map (u ≫ homOfLE ⋯) ≫ H.incl.app (Order.succ k) =
    H.incl.app j ≫ ((Functor.const κ.ord.ToType).obj (reflectionObj W Z κ)).map (u ≫ homOfLE ⋯)
⊢ g₁ ≫ H.incl.app j = g₂ ≫ H.incl.app j

Tactic:
simp only [Functor.const_obj_obj, Functor.const_obj_map, Category.comp_id] at this

Goal state after:
case refine_1
C : Type u
inst✝⁷ : Category.{v, u} C
W : MorphismProperty C
Z : C
inst✝⁶ : HasPushouts C
inst✝⁵ : ∀ (Z : C), HasCoproduct D₁.obj₁
inst✝⁴ : ∀ (Z : C), HasCoproduct D₁.obj₂
inst✝³ : ∀ (Z : C), HasMulticoequalizer (D₂.multispanIndex W Z)
κ : Cardinal.{w}
inst✝² : OrderBot κ.ord.ToType
inst✝¹ : HasIterationOfShape κ.ord.ToType C
inst✝ : Fact κ.IsRegular
hW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ
H : W.isLocal.isLocal.TransfiniteCompositionOfShape κ.ord.ToType (reflection W Z κ) :=
  transfiniteCompositionOfShapeReflection W Z κ
X Y : C
f : X ⟶ Y
hf : W f
left✝ : IsCardinalPresentable X κ
right✝ : IsCardinalPresentable Y κ
j : κ.ord.ToType
g₁ g₂ : Y ⟶ H.F.obj j
h : f ≫ g₁ ≫ H.incl.app j = f ≫ g₂ ≫ H.incl.app j
k : κ.ord.ToType
u : j ⟶ k
hk : (f ≫ g₁) ≫ H.F.map u = (f ≫ g₂) ≫ H.F.map u
hg : g₁ ≫ H.F.map u ≫ (iteration W Z κ).map (homOfLE ⋯) = g₂ ≫ H.F.map u ≫ (iteration W Z κ).map (homOfLE ⋯)
this : H.F.map (u ≫ homOfLE ⋯) ≫ H.incl.app (Order.succ k) = H.incl.app j
⊢ g₁ ≫ H.incl.app j = g₂ ≫ H.incl.app j
Module: Mathlib.CategoryTheory.Sites.Coherent.Basic

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preregular C
X Y : C
f : Y ⟶ X
S : Presieve X
Z : C
π : Z ⟶ X
hπ : S = Presieve.ofArrows (fun x ↦ Z) fun x ↦ π
h_epi : EffectiveEpi π
W : C
h : W ⟶ Y
w✝ : EffectiveEpi h
i : W ⟶ Z
this : i ≫ π = h ≫ f
⊢ Presieve.singleton h = Presieve.ofArrows (fun x ↦ W) fun x ↦ h

Tactic:
{rw [Presieve.ofArrows_pUnit h]}

Goal state after:
Module: Mathlib.CategoryTheory.RegularCategory.Basic

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Regular C
X Y : C
f : X ⟶ Y
⊢ pullback.fst (coequalizer.desc f ⋯) (coequalizer.desc f ⋯) =
    pullback.snd (coequalizer.desc f ⋯) (coequalizer.desc f ⋯)

Tactic:
let m := (coequalizer.desc f pullback.condition)

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Regular C
X Y : C
f : X ⟶ Y
m : coequalizer (pullback.fst f f) (pullback.snd f f) ⟶ Y := coequalizer.desc f ⋯
⊢ pullback.fst (coequalizer.desc f ⋯) (coequalizer.desc f ⋯) =
    pullback.snd (coequalizer.desc f ⋯) (coequalizer.desc f ⋯)
Module: Mathlib.CategoryTheory.RegularCategory.Basic

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Regular C
X Y : C
f : X ⟶ Y
m : coequalizer (pullback.fst f f) (pullback.snd f f) ⟶ Y := coequalizer.desc f ⋯
e : X ⟶ coequalizer (pullback.fst f f) (pullback.snd f f) := coequalizer.π (pullback.fst f f) (pullback.snd f f)
k₁ : pullback m m ⟶ coequalizer (pullback.fst f f) (pullback.snd f f) := pullback.fst m m
k₂ : pullback m m ⟶ coequalizer (pullback.fst f f) (pullback.snd f f) := pullback.snd m m
⊢ pullback.fst (coequalizer.desc f ⋯) (coequalizer.desc f ⋯) =
    pullback.snd (coequalizer.desc f ⋯) (coequalizer.desc f ⋯)

Tactic:
let d : pullback f f ⟶ (pullback m m) :=
    pullback.lift (pullback.fst f f ≫ e) (pullback.snd f f ≫ e) (by simp [m, e, pullback.condition])

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Regular C
X Y : C
f : X ⟶ Y
m : coequalizer (pullback.fst f f) (pullback.snd f f) ⟶ Y := coequalizer.desc f ⋯
e : X ⟶ coequalizer (pullback.fst f f) (pullback.snd f f) := coequalizer.π (pullback.fst f f) (pullback.snd f f)
k₁ : pullback m m ⟶ coequalizer (pullback.fst f f) (pullback.snd f f) := pullback.fst m m
k₂ : pullback m m ⟶ coequalizer (pullback.fst f f) (pullback.snd f f) := pullback.snd m m
d : pullback f f ⟶ pullback m m := pullback.lift (pullback.fst f f ≫ e) (pullback.snd f f ≫ e) ⋯
⊢ pullback.fst (coequalizer.desc f ⋯) (coequalizer.desc f ⋯) =
    pullback.snd (coequalizer.desc f ⋯) (coequalizer.desc f ⋯)
Module: Mathlib.CategoryTheory.Presentable.Directed

Goal state before:
pp: 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
⊢ ∃ j, ∀ (k : ι), D k ≤ j

Tactic:
let he : (D₂ D hι u).IsTerminal m := by
      have H {i : ι} {j : J} (hj : (D i).P j) {f : j ⟶ m} (hf : (D₂ D hι u).W f) :
          f = φ ⟨i, ⟨_, hj⟩⟩ := by
        simp only [D₂_W] at hf
        obtain ((hf | ⟨⟨⟩⟩) | ⟨⟨i', j, hj'⟩⟩) := hf
        · simp only [MorphismProperty.iSup_iff] at hf
          obtain ⟨i, hf⟩ := hf
          exact (hD ((D i).tgt hf)).elim
        · exact (hD hj).elim
        · apply hm
      refine .ofExistsUnique ?_ ?_ ?_ ?_
      · exact Or.inl (Or.inr ⟨⟨⟩⟩)
      · simp only [D₂_P, D₂_W]
        rintro j (⟨i, hi⟩ | rfl)
        · exact ⟨φ ⟨i, _, hi⟩, Or.inr (.mk _)⟩
        · exact ⟨𝟙 _, Or.inl (Or.inr ⟨⟨⟩⟩)⟩
      · intro j hj l₁ l₂ hl₁ hl₂
        simp only [D₂_P] at hj
        obtain (⟨i, hj⟩ | rfl) := hj
        · obtain rfl := H hj hl₁
          obtain rfl := H hj hl₂
          rfl
        · rw [eq_id_of_D₂_W D hι u hD hl₁, eq_id_of_D₂_W D hι u hD hl₂]
      · rintro j k f ((hf | ⟨⟨⟩⟩) | ⟨⟨i, j, hj⟩⟩)
        · simp only [Diagram.iSup_W, MorphismProperty.iSup_iff] at hf
          obtain ⟨i, hf⟩ := hf
          exact ⟨φ ⟨i, j, (D i).src hf⟩, φ ⟨i, k, (D i).tgt hf⟩, Or.inr ⟨_⟩, Or.inr ⟨_⟩,
            by simp [φ, (D i).isTerminal.comm_assoc _ hf]⟩
        · exact ⟨𝟙 _, 𝟙 _, Or.inl (Or.inr ⟨⟨⟩⟩), Or.inl (Or.inr ⟨⟨⟩⟩), by simp⟩
        · refine ⟨φ ⟨i, j, hj⟩, 𝟙 _, Or.inr ⟨_⟩, Or.inl (Or.inr ⟨⟨⟩⟩), by simp [φ]⟩

Goal state after:
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
he : (D₂ D hι u).IsTerminal m :=
  have H := ⋯;
  Diagram.IsTerminal.ofExistsUnique ⋯ ⋯ ⋯ ⋯
⊢ ∃ j, ∀ (k : ι), D k ≤ j