awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Preserves.Bifunctor (goal)
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesColimit₂ K₁ K₂ G
c₁ : Cocone K₂
hc₁ : IsColimit c₁
c₂ : Cocone K₁
hc₂ : IsColimit c₂
⊢ Nonempty (IsColimit (G.flip.mapCocone₂ c₁ c₂))
have val : IsColimit (G.flip.mapCocone₂ c₁ c₂)
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Preserves.Bifunctor (goal)
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesColimit₂ K₁ K₂ G
c₁ : Cocone K₂
hc₁ : IsColimit c₁
c₂ : Cocone K₁
hc₂ : IsColimit c₂
E₀ : obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip) ≅
  obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).flip :=
  Iso.refl (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip))
E₁ : obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip) ≅
  Prod.swap J₂ J₁ ⋙ obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  E₀ ≪≫ (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).uncurryObjFlip
⊢ IsColimit (G.flip.mapCocone₂ c₁ c₂)
have val : IsColimit ((Cocone.precompose E₁.inv).obj (G.flip.mapCocone₂ c₁ c₂))
Tactic: refine IsColimit.precomposeInvEquiv E₁ _ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Preserves.Bifunctor (goal)
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesColimit₂ K₁ K₂ G
c₁ : Cocone K₂
hc₁ : IsColimit c₁
c₂ : Cocone K₁
hc₂ : IsColimit c₂
E₀ : obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip) ≅
  obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).flip :=
  Iso.refl (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip))
E₁ : obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip) ≅
  Prod.swap J₂ J₁ ⋙ obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  E₀ ≪≫ (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).uncurryObjFlip
⊢ IsColimit ((Cocone.precompose E₁.inv).obj (G.flip.mapCocone₂ c₁ c₂))
have val : IsColimit (F :=
  Equivalence.functor (Prod.braiding J₁ J₂) ⋙
    Prod.swap J₂ J₁ ⋙ obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))
  (Cocone.whisker (Equivalence.functor (Prod.braiding J₁ J₂))
    ((Cocone.precompose E₁.inv).obj (G.flip.mapCocone₂ c₁ c₂)))
Tactic: apply IsColimit.ofWhiskerEquivalence (e := Prod.braiding _ _)
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Preserves.Bifunctor (goal)
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesColimit₂ K₁ K₂ G
c₁ : Cocone K₂
hc₁ : IsColimit c₁
c₂ : Cocone K₁
hc₂ : IsColimit c₂
E₀ : obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip) ≅
  obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).flip :=
  Iso.refl (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip))
E₁ : obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip) ≅
  Prod.swap J₂ J₁ ⋙ obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  E₀ ≪≫ (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).uncurryObjFlip
⊢ IsColimit (F :=
    Equivalence.functor (Prod.braiding J₁ J₂) ⋙
      Prod.swap J₂ J₁ ⋙ obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))
    (Cocone.whisker (Equivalence.functor (Prod.braiding J₁ J₂))
      ((Cocone.precompose E₁.inv).obj (G.flip.mapCocone₂ c₁ c₂)))
have val : (Cocone.precompose (Iso.inv (Iso.refl (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))))).obj
    (G.mapCocone₂ c₂ c₁) ≅
  Cocone.whisker (Equivalence.functor (Prod.braiding J₁ J₂)) ((Cocone.precompose E₁.inv).obj (G.flip.mapCocone₂ c₁ c₂))
Tactic: refine IsColimit.equivOfNatIsoOfIso (Iso.refl _) (G.mapCocone₂ c₂ c₁) _ ?_ |>.toFun <|
      isColimitOfPreserves₂ G hc₂ hc₁
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Preserves.Bifunctor (goal)
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
c₁ : Cone K₁
hc₁ : IsLimit c₁
c₂ : Cone K₂
hc₂ : IsLimit c₂
c₃ : Cone (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))
hc₃ : IsLimit c₃
j : J₁ × J₂
⊢ (isoObjConePointsOfIsLimit G hc₁ hc₂ hc₃).hom ≫ c₃.π.app j =
    NatTrans.app (G.map (c₁.π.app j.1)) c₂.pt ≫ (G.obj (K₁.obj j.1)).map (c₂.π.app j.2)
have dsimp : ((isLimitOfPreserves₂ G hc₁ hc₂).conePointUniqueUpToIso hc₃).hom ≫ c₃.π.app j =
  NatTrans.app (G.map (c₁.π.app j.1)) c₂.pt ≫ (G.obj (K₁.obj j.1)).map (c₂.π.app j.2)
Tactic: dsimp [isoObjConePointsOfIsLimit, Functor.mapCocone₂]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Preserves.Bifunctor (goal)
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
c₁ : Cone K₁
hc₁ : IsLimit c₁
c₂ : Cone K₂
hc₂ : IsLimit c₂
c₃ : Cone (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))
hc₃ : IsLimit c₃
j : J₁ × J₂
⊢ (isoObjConePointsOfIsLimit G hc₁ hc₂ hc₃).inv ≫
      NatTrans.app (G.map (c₁.π.app j.1)) c₂.pt ≫ (G.obj (K₁.obj j.1)).map (c₂.π.app j.2) =
    c₃.π.app j
have rw : NatTrans.app (G.map (c₁.π.app j.1)) c₂.pt ≫ (G.obj (K₁.obj j.1)).map (c₂.π.app j.2) =
  (isoObjConePointsOfIsLimit G hc₁ hc₂ hc₃).hom ≫ c₃.π.app j
Tactic: rw [Iso.inv_comp_eq]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Preserves.Bifunctor (goal)
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
inst_6 : ∀ (x : C₂), PreservesLimit K₁ (G.flip.obj x)
inst_7 : ∀ (x : C₁), PreservesLimit K₂ (G.obj x)
c₁ : Cone K₁
hc₁ : IsLimit c₁
c₂ : Cone K₂
hc₂ : IsLimit c₂
Q₀ : DiagramOfCones (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  { obj := fun (j₁ : J₁) ↦ (G.obj (K₁.obj j₁)).mapCone c₂,
    map := fun {j j' : J₁} (f : j ⟶ j') ↦ { hom := (G.map (K₁.map f)).app c₂.pt, w := ⋯ }, id := ⋯, comp := ⋯ }
P : (j₁ : J₁) → IsLimit (Q₀.obj j₁) := fun (x : J₁) ↦ isLimitOfPreserves (G.obj (K₁.obj x)) hc₂
E₀ : Q₀.conePoints ≅ K₁ ⋙ G.flip.obj c₂.pt := NatIso.ofComponents (fun (x : J₁) ↦ Iso.refl (Q₀.conePoints.obj x)) ⋯
j₁ : J₁
⊢ ((Cone.postcompose E₀.hom).obj (coneOfConeUncurry P (G.mapCone₂ c₁ c₂))).π.app j₁ =
    Iso.hom (Iso.refl ((Cone.postcompose E₀.hom).obj (coneOfConeUncurry P (G.mapCone₂ c₁ c₂))).pt) ≫
      ((G.flip.obj c₂.pt).mapCone c₁).π.app j₁
have dsimp : (P j₁).lift
      { pt := (G.obj c₁.pt).obj c₂.pt,
        π :=
          { app := fun (k : J₂) ↦ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app k),
            naturality := sorry } } ≫
    𝟙 ((G.obj (K₁.obj j₁)).obj c₂.pt) =
  𝟙 ((G.obj c₁.pt).obj c₂.pt) ≫ (G.map (c₁.π.app j₁)).app c₂.pt
Tactic: dsimp [E₀, Q₀]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Preserves.Bifunctor (goal)
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
inst_6 : ∀ (x : C₂), PreservesLimit K₁ (G.flip.obj x)
inst_7 : ∀ (x : C₁), PreservesLimit K₂ (G.obj x)
c₁ : Cone K₁
hc₁ : IsLimit c₁
c₂ : Cone K₂
hc₂ : IsLimit c₂
Q₀ : DiagramOfCones (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  { obj := fun (j₁ : J₁) ↦ (G.obj (K₁.obj j₁)).mapCone c₂,
    map := fun {j j' : J₁} (f : j ⟶ j') ↦ { hom := (G.map (K₁.map f)).app c₂.pt, w := ⋯ }, id := ⋯, comp := ⋯ }
P : (j₁ : J₁) → IsLimit (Q₀.obj j₁) := fun (x : J₁) ↦ isLimitOfPreserves (G.obj (K₁.obj x)) hc₂
E₀ : Q₀.conePoints ≅ K₁ ⋙ G.flip.obj c₂.pt := NatIso.ofComponents (fun (x : J₁) ↦ Iso.refl (Q₀.conePoints.obj x)) ⋯
j₁ : J₁
⊢ (P j₁).lift
        { pt := (G.obj c₁.pt).obj c₂.pt,
          π :=
            { app := fun (k : J₂) ↦ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app k),
              naturality := ⋯ } } ≫
      𝟙 ((G.obj (K₁.obj j₁)).obj c₂.pt) =
    𝟙 ((G.obj c₁.pt).obj c₂.pt) ≫ (G.map (c₁.π.app j₁)).app c₂.pt
have simp : Eq (α := (G.obj c₁.pt).obj c₂.pt ⟶ (G.obj (K₁.obj j₁)).obj c₂.pt)
  ((P j₁).lift
      { pt := (G.obj c₁.pt).obj c₂.pt,
        π :=
          { app := fun (k : J₂) ↦ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app k),
            naturality := sorry } } :
    Cone.pt.{v_2, v_5, u_2, u_5} (J := J₂) (F := (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁)
        { pt := (G.obj c₁.pt).obj c₂.pt,
          π :=
            { app := fun (k : J₂) ↦ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app k),
              naturality := sorry :
              NatTrans (obj (const J₂) ((G.obj c₁.pt).obj c₂.pt))
                ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁) } :
          Cone ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁) } ⟶
      ((G.obj (K₁.obj j₁)).mapCone c₂).pt)
  ((G.map (c₁.π.app j₁)).app c₂.pt)
Tactic: simp only [id_comp, comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Preserves.Bifunctor (goal)
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
inst_6 : ∀ (x : C₂), PreservesLimit K₁ (G.flip.obj x)
inst_7 : ∀ (x : C₁), PreservesLimit K₂ (G.obj x)
c₁ : Cone K₁
hc₁ : IsLimit c₁
c₂ : Cone K₂
hc₂ : IsLimit c₂
Q₀ : DiagramOfCones (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  { obj := fun (j₁ : J₁) ↦ (G.obj (K₁.obj j₁)).mapCone c₂,
    map := fun {j j' : J₁} (f : j ⟶ j') ↦ { hom := (G.map (K₁.map f)).app c₂.pt, w := ⋯ }, id := ⋯, comp := ⋯ }
P : (j₁ : J₁) → IsLimit (Q₀.obj j₁) := fun (x : J₁) ↦ isLimitOfPreserves (G.obj (K₁.obj x)) hc₂
E₀ : Q₀.conePoints ≅ K₁ ⋙ G.flip.obj c₂.pt := NatIso.ofComponents (fun (x : J₁) ↦ Iso.refl (Q₀.conePoints.obj x)) ⋯
j₁ : J₁
refine_2 :
  Eq (α := (G.obj c₁.pt).obj c₂.pt ⟶ (G.obj (K₁.obj j₁)).obj c₂.pt)
    ((P j₁).lift
        { pt := (G.obj c₁.pt).obj c₂.pt,
          π :=
            { app := fun (k : J₂) ↦ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app k),
              naturality := ⋯ } } :
      Cone.pt.{v_2, v_5, u_2, u_5} (J := J₂) (F := (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁)
          { pt := (G.obj c₁.pt).obj c₂.pt,
            π :=
              { app := fun (k : J₂) ↦ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app k),
                naturality := ⋯ :
                NatTrans (obj (const J₂) ((G.obj c₁.pt).obj c₂.pt))
                  ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁) } :
            Cone ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁) } ⟶
        ((G.obj (K₁.obj j₁)).mapCone c₂).pt)
    ((G.map (c₁.π.app j₁)).app c₂.pt)
⊢ Eq (α := (G.obj c₁.pt).obj c₂.pt ⟶ (G.obj (K₁.obj j₁)).obj c₂.pt)
    ((P j₁).lift
        { pt := (G.obj c₁.pt).obj c₂.pt,
          π :=
            { app := fun (k : J₂) ↦ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app k),
              naturality := ⋯ } } :
      Cone.pt.{v_2, v_5, u_2, u_5} (J := J₂) (F := (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁)
          { pt := (G.obj c₁.pt).obj c₂.pt,
            π :=
              { app := fun (k : J₂) ↦ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app k),
                naturality := ⋯ :
                NatTrans (obj (const J₂) ((G.obj c₁.pt).obj c₂.pt))
                  ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁) } :
            Cone ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁) } ⟶
        ((G.obj (K₁.obj j₁)).mapCone c₂).pt)
    ((G.map (c₁.π.app j₁)).app c₂.pt)
have refine_1 : Cone ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁)
Tactic: let s : Cone (whiskeringLeft₂ C |>.obj K₁ |>.obj K₂ |>.obj G |>.obj j₁) := ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Preserves.Bifunctor (goal)
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
inst_6 : ∀ (x : C₂), PreservesLimit K₁ (G.flip.obj x)
inst_7 : ∀ (x : C₁), PreservesLimit K₂ (G.obj x)
c₁ : Cone K₁
hc₁ : IsLimit c₁
c₂ : Cone K₂
hc₂ : IsLimit c₂
Q₀ : DiagramOfCones (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  { obj := fun (j₁ : J₁) ↦ (G.obj (K₁.obj j₁)).mapCone c₂,
    map := fun {j j' : J₁} (f : j ⟶ j') ↦ { hom := (G.map (K₁.map f)).app c₂.pt, w := ⋯ }, id := ⋯, comp := ⋯ }
P : (j₁ : J₁) → IsLimit (Q₀.obj j₁) := fun (x : J₁) ↦ isLimitOfPreserves (G.obj (K₁.obj x)) hc₂
E₀ : Q₀.conePoints ≅ K₁ ⋙ G.flip.obj c₂.pt := NatIso.ofComponents (fun (x : J₁) ↦ Iso.refl (Q₀.conePoints.obj x)) ⋯
j₁ : J₁
s : Cone ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁) :=
  { pt := (G.obj c₁.pt).obj c₂.pt,
    π :=
      { app := fun (k : J₂) ↦ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app k),
        naturality := ⋯ } }
⊢ (P j₁).lift s = (G.map (c₁.π.app j₁)).app c₂.pt
have refine_2 : (G.map (c₁.π.app j₁)).app c₂.pt = (P j₁).lift s
Tactic: symm
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Preserves.Bifunctor (goal)
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
inst_6 : ∀ (x : C₂), PreservesLimit K₁ (G.flip.obj x)
inst_7 : ∀ (x : C₁), PreservesLimit K₂ (G.obj x)
c₁ : Cone K₁
hc₁ : IsLimit c₁
c₂ : Cone K₂
hc₂ : IsLimit c₂
Q₀ : DiagramOfCones (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  { obj := fun (j₁ : J₁) ↦ (G.obj (K₁.obj j₁)).mapCone c₂,
    map := fun {j j' : J₁} (f : j ⟶ j') ↦ { hom := (G.map (K₁.map f)).app c₂.pt, w := ⋯ }, id := ⋯, comp := ⋯ }
P : (j₁ : J₁) → IsLimit (Q₀.obj j₁) := fun (x : J₁) ↦ isLimitOfPreserves (G.obj (K₁.obj x)) hc₂
E₀ : Q₀.conePoints ≅ K₁ ⋙ G.flip.obj c₂.pt := NatIso.ofComponents (fun (x : J₁) ↦ Iso.refl (Q₀.conePoints.obj x)) ⋯
j₁ : J₁
s : Cone ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁) :=
  { pt := (G.obj c₁.pt).obj c₂.pt,
    π :=
      { app := fun (k : J₂) ↦ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app k),
        naturality := ⋯ } }
⊢ (G.map (c₁.π.app j₁)).app c₂.pt = (P j₁).lift s
have refine_2 : ∀ (j : J₂), (G.map (c₁.π.app j₁)).app c₂.pt ≫ (Q₀.obj j₁).π.app j = (P j₁).lift s ≫ (Q₀.obj j₁).π.app j
Tactic: apply (P j₁).hom_ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Preserves.Bifunctor (hypothesis)
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
inst_6 : ∀ (x : C₂), PreservesLimit K₁ (G.flip.obj x)
inst_7 : ∀ (x : C₁), PreservesLimit K₂ (G.obj x)
c₁ : Cone K₁
hc₁ : IsLimit c₁
c₂ : Cone K₂
hc₂ : IsLimit c₂
Q₀ : DiagramOfCones (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  { obj := fun (j₁ : J₁) ↦ (G.obj (K₁.obj j₁)).mapCone c₂,
    map := fun {j j' : J₁} (f : j ⟶ j') ↦ { hom := (G.map (K₁.map f)).app c₂.pt, w := ⋯ }, id := ⋯, comp := ⋯ }
P : (j₁ : J₁) → IsLimit (Q₀.obj j₁) := fun (x : J₁) ↦ isLimitOfPreserves (G.obj (K₁.obj x)) hc₂
E₀ : Q₀.conePoints ≅ K₁ ⋙ G.flip.obj c₂.pt := NatIso.ofComponents (fun (x : J₁) ↦ Iso.refl (Q₀.conePoints.obj x)) ⋯
j₁ : J₁
s : Cone ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁) :=
  { pt := (G.obj c₁.pt).obj c₂.pt,
    π :=
      { app := fun (k : J₂) ↦ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app k),
        naturality := ⋯ } }
j₂ : J₂
⊢ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (Q₀.obj j₁).π.app j₂ = (P j₁).lift s ≫ (Q₀.obj j₁).π.app j₂
have this : (P j₁).lift s ≫ (Q₀.obj j₁).π.app j₂ = s.π.app j₂
Tactic: haveI := (P j₁).fac s j₂
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Preserves.Bifunctor (goal)
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
c₁ : Cone K₂
hc₁ : IsLimit c₁
c₂ : Cone K₁
hc₂ : IsLimit c₂
⊢ Nonempty (IsLimit (G.flip.mapCone₂ c₁ c₂))
have val : IsLimit (G.flip.mapCone₂ c₁ c₂)
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Preserves.Bifunctor (goal)
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
c₁ : Cone K₂
hc₁ : IsLimit c₁
c₂ : Cone K₁
hc₂ : IsLimit c₂
E₀ : obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip) ≅
  obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).flip :=
  Iso.refl (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip))
E₁ : obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip) ≅
  Prod.swap J₂ J₁ ⋙ obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  E₀ ≪≫ (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).uncurryObjFlip
⊢ IsLimit (G.flip.mapCone₂ c₁ c₂)
have val : IsLimit ((Cone.postcompose E₁.hom).obj (G.flip.mapCone₂ c₁ c₂))
Tactic: refine IsLimit.postcomposeHomEquiv E₁ _ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Preserves.Bifunctor (goal)
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
c₁ : Cone K₂
hc₁ : IsLimit c₁
c₂ : Cone K₁
hc₂ : IsLimit c₂
E₀ : obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip) ≅
  obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).flip :=
  Iso.refl (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip))
E₁ : obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip) ≅
  Prod.swap J₂ J₁ ⋙ obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  E₀ ≪≫ (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).uncurryObjFlip
⊢ IsLimit ((Cone.postcompose E₁.hom).obj (G.flip.mapCone₂ c₁ c₂))
have val : IsLimit (F :=
  Equivalence.functor (Prod.braiding J₁ J₂) ⋙
    Prod.swap J₂ J₁ ⋙ obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))
  (Cone.whisker (Equivalence.functor (Prod.braiding J₁ J₂)) ((Cone.postcompose E₁.hom).obj (G.flip.mapCone₂ c₁ c₂)))
Tactic: apply IsLimit.ofWhiskerEquivalence (e := Prod.braiding _ _)
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Preserves.Bifunctor (goal)
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
c₁ : Cone K₂
hc₁ : IsLimit c₁
c₂ : Cone K₁
hc₂ : IsLimit c₂
E₀ : obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip) ≅
  obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).flip :=
  Iso.refl (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip))
E₁ : obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip) ≅
  Prod.swap J₂ J₁ ⋙ obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  E₀ ≪≫ (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).uncurryObjFlip
⊢ IsLimit (F :=
    Equivalence.functor (Prod.braiding J₁ J₂) ⋙
      Prod.swap J₂ J₁ ⋙ obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))
    (Cone.whisker (Equivalence.functor (Prod.braiding J₁ J₂)) ((Cone.postcompose E₁.hom).obj (G.flip.mapCone₂ c₁ c₂)))
have val : (Cone.postcompose (Iso.hom (Iso.refl (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))))).obj
    (G.mapCone₂ c₂ c₁) ≅
  Cone.whisker (Equivalence.functor (Prod.braiding J₁ J₂)) ((Cone.postcompose E₁.hom).obj (G.flip.mapCone₂ c₁ c₂))
Tactic: refine IsLimit.equivOfNatIsoOfIso (Iso.refl _) (G.mapCone₂ c₂ c₁) _ ?_ |>.toFun <|
      isLimitOfPreserves₂ G hc₂ hc₁
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Preserves.BifunctorCokernel (goal)
Goal:
C₁ : Type u_1
C₂ : Type u_2
C : Type u_3
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
inst_2 : Category.{v_3, u_3} C
inst_3 : HasZeroMorphisms.{v_1, u_1} C₁
inst_4 : HasZeroMorphisms.{v_2, u_2} C₂
inst_5 : HasZeroMorphisms.{v_3, u_3} C
X₁ Y₁ : C₁
f₁ : X₁ ⟶ Y₁
c₁ : CokernelCofork f₁
hc₁ : IsColimit c₁
X₂ Y₂ : C₂
f₂ : X₂ ⟶ Y₂
c₂ : CokernelCofork f₂
hc₂ : IsColimit c₂
F : C₁ ⥤ Functor.{v_2, v_3, u_2, u_3} C₂ C
inst_6 : (F.obj c₁.pt).PreservesZeroMorphisms
inst_7 : F.PreservesZeroMorphisms
inst_8 : HasBinaryCoproduct.{v_3, u_3} ((F.obj X₁).obj Y₂) ((F.obj Y₁).obj X₂)
⊢ Eq (α := coprod.{v_3, u_3} ((F.obj X₁).obj Y₂) ((F.obj Y₁).obj X₂) ⟶ (F.obj c₁.pt).obj c₂.pt)
    (coprod.desc ((F.map f₁).app Y₂) ((F.obj Y₁).map f₂) ≫
      (F.map (Cofork.π c₁)).app Y₂ ≫ (F.obj c₁.pt).map (Cofork.π c₂))
    0
have h₁ : CategoryStruct.comp (Y := coprod.{v_3, u_3} ((F.obj X₁).obj Y₂) ((F.obj Y₁).obj X₂)) coprod.inl
    (coprod.desc ((F.map f₁).app Y₂) ((F.obj Y₁).map f₂) ≫
      (F.map (Cofork.π c₁)).app Y₂ ≫ (F.obj c₁.pt).map (Cofork.π c₂)) =
  CategoryStruct.comp (Y := coprod.{v_3, u_3} ((F.obj X₁).obj Y₂) ((F.obj Y₁).obj X₂)) coprod.inl 0
Tactic: ext