awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equalizer (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasEqualizers.{v, u} C
inst_2 : HasPullbacks.{v, u} C
X Y S T : C
f g : X ⟶ Y
s : X ⟶ S
t : Y ⟶ S
hf : f ≫ t = s
hg : g ≫ t = s
v : T ⟶ S
⊢ Eq (α :=
    equalizer (X := pullback s v) (Y := pullback t v) (pullback.map s v t v f (𝟙 T) (𝟙 S) ⋯ ⋯)
        (pullback.map s v t v g (𝟙 T) (𝟙 S) ⋯ ⋯) ⟶
      pullback s v)
    (CategoryStruct.comp (Y := pullback (X := equalizer f g) (Y := pullback s v) (equalizer.ι f g) (pullback.fst s v))
        (Iso.hom (equalizerPullbackMapIso hf hg v) :
          equalizer (X := pullback s v) (Y := pullback t v) (pullback.map s v t v f (𝟙 T) (𝟙 S) ⋯ ⋯)
              (pullback.map s v t v g (𝟙 T) (𝟙 S) ⋯ ⋯) ⟶
            pullback (X := equalizer f g) (Y := pullback s v) (equalizer.ι f g) (pullback.fst s v))
        (pullback.snd (equalizer.ι f g) (pullback.fst s v) :
          pullback (X := equalizer f g) (Y := pullback s v) (equalizer.ι f g) (pullback.fst s v) ⟶ pullback s v) :
      equalizer (X := pullback s v) (Y := pullback t v) (pullback.map s v t v f (𝟙 T) (𝟙 S) ⋯ ⋯)
          (pullback.map s v t v g (𝟙 T) (𝟙 S) ⋯ ⋯) ⟶
        pullback s v)
    (equalizer.ι (pullback.map s v t v f (𝟙 T) (𝟙 S) ⋯ ⋯) (pullback.map s v t v g (𝟙 T) (𝟙 S) ⋯ ⋯) :
      equalizer (X := pullback s v) (Y := pullback t v) (pullback.map s v t v f (𝟙 T) (𝟙 S) ⋯ ⋯)
          (pullback.map s v t v g (𝟙 T) (𝟙 S) ⋯ ⋯) ⟶
        pullback s v)
have h₀ : Eq (α :=
  equalizer (X := pullback s v) (Y := pullback t v) (pullback.map s v t v f (𝟙 T) (𝟙 S) sorry sorry)
      (pullback.map s v t v g (𝟙 T) (𝟙 S) sorry sorry) ⟶
    X)
  (CategoryStruct.comp (Y := pullback s v)
      (CategoryStruct.comp (Y := pullback (X := equalizer f g) (Y := pullback s v) (equalizer.ι f g) (pullback.fst s v))
          (Iso.hom (equalizerPullbackMapIso hf hg v) :
            equalizer (X := pullback s v) (Y := pullback t v) (pullback.map s v t v f (𝟙 T) (𝟙 S) sorry sorry)
                (pullback.map s v t v g (𝟙 T) (𝟙 S) sorry sorry) ⟶
              pullback (X := equalizer f g) (Y := pullback s v) (equalizer.ι f g) (pullback.fst s v))
          (pullback.snd (equalizer.ι f g) (pullback.fst s v) :
            pullback (X := equalizer f g) (Y := pullback s v) (equalizer.ι f g) (pullback.fst s v) ⟶ pullback s v) :
        equalizer (X := pullback s v) (Y := pullback t v) (pullback.map s v t v f (𝟙 T) (𝟙 S) sorry sorry)
            (pullback.map s v t v g (𝟙 T) (𝟙 S) sorry sorry) ⟶
          pullback s v)
      (pullback.fst s v : pullback s v ⟶ X) :
    equalizer (X := pullback s v) (Y := pullback t v) (pullback.map s v t v f (𝟙 T) (𝟙 S) sorry sorry)
        (pullback.map s v t v g (𝟙 T) (𝟙 S) sorry sorry) ⟶
      X)
  (equalizer.ι (pullback.map s v t v f (𝟙 T) (𝟙 S) sorry sorry) (pullback.map s v t v g (𝟙 T) (𝟙 S) sorry sorry) ≫ pullback.fst s v :
    equalizer (X := pullback s v) (Y := pullback t v) (pullback.map s v t v f (𝟙 T) (𝟙 S) sorry sorry)
        (pullback.map s v t v g (𝟙 T) (𝟙 S) sorry sorry) ⟶
      X)
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equifibered (goal)
Goal:
C : Type u_3
ι : Type u_5
inst : Category.{v_2, u_3} C
F G : Functor.{u_5, v_2, u_5, u_3} (Discrete ι) C
α : F ⟶ G
i : ι
⊢ IsPullback
    (F.map
      { down := { down := ⋯ : PLift ({ as := i : Discrete ι }.as = { as := i : Discrete ι }.as) } :
        ULift.{u_5, 0} (PLift ({ as := i : Discrete ι }.as = { as := i : Discrete ι }.as)) })
    (α.app { as := i }) (α.app { as := i })
    (G.map
      { down := { down := ⋯ : PLift ({ as := i : Discrete ι }.as = { as := i : Discrete ι }.as) } :
        ULift.{u_5, 0} (PLift ({ as := i : Discrete ι }.as = { as := i : Discrete ι }.as)) })
have simp : IsPullback (𝟙 (F.obj { as := i })) (α.app { as := i : Discrete ι }) (α.app { as := i : Discrete ι })
  (𝟙 (G.obj { as := i }))
Tactic: simp only [Discrete.functor_map_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equifibered (goal)
Goal:
C : Type u_3
ι : Type u_5
inst : Category.{v_2, u_3} C
F G : Functor.{u_5, v_2, u_5, u_3} (Discrete ι) C
α : F ⟶ G
i : ι
⊢ 𝟙 (F.obj { as := i : Discrete ι }) ≫ α.app { as := i : Discrete ι } =
    α.app { as := i : Discrete ι } ≫ 𝟙 (G.obj { as := i })
have rw : α.app { as := i } = α.app { as := i : Discrete ι } ≫ 𝟙 (G.obj { as := i : Discrete ι })
Tactic: rw [Category.id_comp, Category.comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equifibered (goal)
Goal:
C : Type u_3
ι : Type u_5
inst : Category.{v_2, u_3} C
F G : Functor.{u_5, v_2, u_5, u_3} (Discrete ι) C
α : F ⟶ G
i : ι
rw : α.app { as := i } = α.app { as := i : Discrete ι } ≫ 𝟙 (G.obj { as := i : Discrete ι })
⊢ 𝟙 (F.obj { as := i : Discrete ι }) ≫ α.app { as := i : Discrete ι } =
    α.app { as := i : Discrete ι } ≫ 𝟙 (G.obj { as := i })
have rw₁ : Eq (α := F.obj { as := i : Discrete ι } ⟶ G.obj { as := i }) (α.app { as := i }) (α.app { as := i })
Tactic: rw [Category.id_comp, Category.comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equifibered (goal)
Goal:
C : Type u_3
inst : Category.{v_2, u_3} C
ι : Type u_6
F G : Functor.{u_6, v_2, u_6, u_3} (Discrete ι) C
α : F ⟶ G
i : ι
⊢ IsPullback
    (F.map
      { down := { down := ⋯ : PLift ({ as := i : Discrete ι }.as = { as := i : Discrete ι }.as) } :
        ULift.{u_6, 0} (PLift ({ as := i : Discrete ι }.as = { as := i : Discrete ι }.as)) })
    (α.app { as := i }) (α.app { as := i })
    (G.map
      { down := { down := ⋯ : PLift ({ as := i : Discrete ι }.as = { as := i : Discrete ι }.as) } :
        ULift.{u_6, 0} (PLift ({ as := i : Discrete ι }.as = { as := i : Discrete ι }.as)) })
have simp : IsPullback (𝟙 (F.obj { as := i })) (α.app { as := i : Discrete ι }) (α.app { as := i : Discrete ι })
  (𝟙 (G.obj { as := i }))
Tactic: simp only [Discrete.functor_map_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equifibered (goal)
Goal:
C : Type u_3
inst : Category.{v_2, u_3} C
ι : Type u_6
F G : Functor.{u_6, v_2, u_6, u_3} (Discrete ι) C
α : F ⟶ G
i : ι
⊢ 𝟙 (F.obj { as := i : Discrete ι }) ≫ α.app { as := i : Discrete ι } =
    α.app { as := i : Discrete ι } ≫ 𝟙 (G.obj { as := i })
have rw : α.app { as := i } = α.app { as := i : Discrete ι } ≫ 𝟙 (G.obj { as := i : Discrete ι })
Tactic: rw [Category.id_comp, Category.comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equifibered (goal)
Goal:
C : Type u_3
inst : Category.{v_2, u_3} C
ι : Type u_6
F G : Functor.{u_6, v_2, u_6, u_3} (Discrete ι) C
α : F ⟶ G
i : ι
rw : α.app { as := i } = α.app { as := i : Discrete ι } ≫ 𝟙 (G.obj { as := i : Discrete ι })
⊢ 𝟙 (F.obj { as := i : Discrete ι }) ≫ α.app { as := i : Discrete ι } =
    α.app { as := i : Discrete ι } ≫ 𝟙 (G.obj { as := i })
have rw₁ : Eq (α := F.obj { as := i : Discrete ι } ⟶ G.obj { as := i }) (α.app { as := i }) (α.app { as := i })
Tactic: rw [Category.id_comp, Category.comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Pullback.EquifiberedLimits (goal)
Goal:
J : Type u_1
K : Type u_2
C : Type u_3
D : Type u_4
ι : Type u_5
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_3} C
inst_2 : Category.{v_3, u_2} K
inst_3 : Category.{v_4, u_4} D
F : Functor.{v_2, v_4, u_3, u_4} C D
inst_4 : ∀ (a b : C), HasCoproductsOfShape.{v_2, u_4, v_4} (a ⟶ b) D
⊢ IsClosedUnderLimitsOfShape (C := Over F) (fun (f : Over F) ↦ Equifibered f.hom) J
have inr : ∀
  (this :
    ∀ {J : Type u_1} [inst : Category.{v_1, u_1} J] (hJ : IsConnected.{v_1, u_1} J),
      IsClosedUnderLimitsOfShape (C := Over F) (fun (f : Over F) ↦ Equifibered f.hom) J)
  (hJ : ¬IsConnected.{v_1, u_1} J), IsClosedUnderLimitsOfShape (C := Over F) (fun (f : Over F) ↦ Equifibered f.hom) J
Tactic: wlog hJ : IsConnected J generalizing J
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Pullback.EquifiberedLimits (goal)
Goal:
J : Type u_1
K : Type u_2
C : Type u_3
D : Type u_4
ι : Type u_5
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_3} C
inst_2 : Category.{v_3, u_2} K
inst_3 : Category.{v_4, u_4} D
F : Functor.{v_2, v_4, u_3, u_4} C D
inst_4 : ∀ (a b : C), HasCoproductsOfShape.{v_2, u_4, v_4} (a ⟶ b) D
inr :
  ∀
    (this :
      ∀ {J : Type u_1} [inst : Category.{v_1, u_1} J] (hJ : IsConnected.{v_1, u_1} J),
        IsClosedUnderLimitsOfShape (C := Over F) (fun (f : Over F) ↦ Equifibered f.hom) J)
    (hJ : ¬IsConnected.{v_1, u_1} J), IsClosedUnderLimitsOfShape (C := Over F) (fun (f : Over F) ↦ Equifibered f.hom) J
⊢ IsClosedUnderLimitsOfShape (C := Over F) (fun (f : Over F) ↦ Equifibered f.hom) J
have wlog : ∀ {J : Type u_1} [inst : Category.{v_1, u_1} J] (hJ : IsConnected.{v_1, u_1} J),
  IsClosedUnderLimitsOfShape (C := Over F) (fun (f : Over F) ↦ Equifibered f.hom) J
Tactic: wlog hJ : IsConnected J generalizing J
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Pullback.EquifiberedLimits (goal)
Goal:
J : Type u_1
K : Type u_2
C : Type u_3
D : Type u_4
ι : Type u_5
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_3} C
inst_2 : Category.{v_3, u_2} K
inst_3 : Category.{v_4, u_4} D
F : Functor.{v_2, v_4, u_3, u_4} C D
inst_4 : ∀ (a b : C), HasCoproductsOfShape.{v_2, u_4, v_4} (a ⟶ b) D
this :
  ∀ {J : Type u_1} [inst : Category.{v_1, u_1} J] (hJ : IsConnected.{v_1, u_1} J),
    IsClosedUnderLimitsOfShape (C := Over F) (fun (f : Over F) ↦ Equifibered f.hom) J
hJ : ¬IsConnected.{v_1, u_1} J
⊢ IsClosedUnderLimitsOfShape (C := Over F) (fun (f : Over F) ↦ Equifibered f.hom) J
have inr : ∀ (G : Over F),
  limitsOfShape (fun (f : Over F) ↦ Equifibered f.hom) J G →
    ∀ (U V : C) (f : U ⟶ V) (c : J ⥤ Over F) (α : obj (const J) G ⟶ c) (hc : IsLimit { pt := G, π := α : Cone c })
      (H : ∀ (j : J), Equifibered ({ diag := c, π := α, isLimit := hc : LimitPresentation J G }.diag.obj j).hom),
      IsPullback (G.left.map f) (G.hom.app U) (G.hom.app V) (F.map f)
Tactic: refine ⟨fun G ⟨⟨c, α, hc⟩, H⟩ U V f ↦ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Pullback.EquifiberedLimits (hypothesis)
Goal:
J : Type u_1
K : Type u_2
C : Type u_3
D : Type u_4
ι : Type u_5
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_3} C
inst_2 : Category.{v_3, u_2} K
inst_3 : Category.{v_4, u_4} D
F : Functor.{v_2, v_4, u_3, u_4} C D
inst_4 : ∀ (a b : C), HasCoproductsOfShape.{v_2, u_4, v_4} (a ⟶ b) D
this :
  ∀ {J : Type u_1} [inst : Category.{v_1, u_1} J] (hJ : IsConnected.{v_1, u_1} J),
    IsClosedUnderLimitsOfShape (C := Over F) (fun (f : Over F) ↦ Equifibered f.hom) J
hJ : ¬IsConnected.{v_1, u_1} J
G : Over F
h : limitsOfShape (fun (f : Over F) ↦ Equifibered f.hom) J G
U V : C
f : U ⟶ V
c : J ⥤ Over F
α : obj (const J) G ⟶ c
hc : IsLimit { pt := G, π := α : Cone c }
H : ∀ (j : J), Equifibered ({ diag := c, π := α, isLimit := hc : LimitPresentation J G }.diag.obj j).hom
⊢ IsPullback (G.left.map f) (G.hom.app U) (G.hom.app V) (F.map f)
have hα : ∀ (i j : J) (f : i ⟶ j), α.app i ≫ c.map f = α.app j
Tactic: have hα (i j) (f : i ⟶ j) : α.app i ≫ c.map f = α.app j := by simp [← NatTrans.naturality]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Pullback.EquifiberedLimits (goal)
Goal:
J : Type u_1
K : Type u_2
C : Type u_3
D : Type u_4
ι : Type u_5
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_3} C
inst_2 : Category.{v_3, u_2} K
inst_3 : Category.{v_4, u_4} D
F : Functor.{v_2, v_4, u_3, u_4} C D
inst_4 : ∀ (a b : C), HasCoproductsOfShape.{v_2, u_4, v_4} (a ⟶ b) D
this :
  ∀ {J : Type u_1} [inst : Category.{v_1, u_1} J] (hJ : IsConnected.{v_1, u_1} J),
    IsClosedUnderLimitsOfShape (C := Over F) (fun (f : Over F) ↦ Equifibered f.hom) J
hJ : ¬IsConnected.{v_1, u_1} J
G : Over F
h : limitsOfShape (fun (f : Over F) ↦ Equifibered f.hom) J G
U V : C
f : U ⟶ V
c : J ⥤ Over F
α : obj (const J) G ⟶ c
hc : IsLimit { pt := G, π := α : Cone c }
H : ∀ (j : J), Equifibered ({ diag := c, π := α, isLimit := hc : LimitPresentation J G }.diag.obj j).hom
hα : ∀ (i j : J) (f : i ⟶ j), α.app i ≫ c.map f = α.app j
⊢ IsPullback (G.left.map f) (G.hom.app U) (G.hom.app V) (F.map f)
have inr : ∀ (hc' : IsLimit (obj (Equivalence.functor WithTerminal.coneEquiv) { pt := G, π := α : Cone c })),
  IsPullback (G.left.map f) (G.hom.app U) (G.hom.app V) (F.map f)
Tactic: have hc' := WithTerminal.isLimitEquiv.symm hc
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Pullback.EquifiberedLimits (hypothesis)
Goal:
J : Type u_1
K : Type u_2
C : Type u_3
D : Type u_4
ι : Type u_5
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_3} C
inst_2 : Category.{v_3, u_2} K
inst_3 : Category.{v_4, u_4} D
F : Functor.{v_2, v_4, u_3, u_4} C D
inst_4 : ∀ (a b : C), HasCoproductsOfShape.{v_2, u_4, v_4} (a ⟶ b) D
this :
  ∀ {J : Type u_1} [inst : Category.{v_1, u_1} J] (hJ : IsConnected.{v_1, u_1} J),
    IsClosedUnderLimitsOfShape (C := Over F) (fun (f : Over F) ↦ Equifibered f.hom) J
hJ : ¬IsConnected.{v_1, u_1} J
G : Over F
h : limitsOfShape (fun (f : Over F) ↦ Equifibered f.hom) J G
U V : C
f : U ⟶ V
c : J ⥤ Over F
α : obj (const J) G ⟶ c
hc : IsLimit { pt := G, π := α : Cone c }
H : ∀ (j : J), Equifibered ({ diag := c, π := α, isLimit := hc : LimitPresentation J G }.diag.obj j).hom
hα : ∀ (i j : J) (f : i ⟶ j), α.app i ≫ c.map f = α.app j
hc' : IsLimit (obj (Equivalence.functor WithTerminal.coneEquiv) { pt := G, π := α : Cone c })
⊢ IsPullback (G.left.map f) (G.hom.app U) (G.hom.app V) (F.map f)
have inst : IsConnected.{v_1, u_1} (WithTerminal J)
Tactic: have inst : IsConnected (WithTerminal J) := isConnected_of_hasTerminal _
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Pullback.EquifiberedLimits (goal)
Goal:
J : Type u_1
K : Type u_2
C : Type u_3
D : Type u_4
ι : Type u_5
inst_1 : Category.{v_1, u_1} J
inst_2 : Category.{v_2, u_3} C
inst_3 : Category.{v_3, u_2} K
inst_4 : Category.{v_4, u_4} D
F : Functor.{v_2, v_4, u_3, u_4} C D
inst_5 : ∀ (a b : C), HasCoproductsOfShape.{v_2, u_4, v_4} (a ⟶ b) D
this :
  ∀ {J : Type u_1} [inst : Category.{v_1, u_1} J] (hJ : IsConnected.{v_1, u_1} J),
    IsClosedUnderLimitsOfShape (C := Over F) (fun (f : Over F) ↦ Equifibered f.hom) J
hJ : ¬IsConnected.{v_1, u_1} J
G : Over F
h : limitsOfShape (fun (f : Over F) ↦ Equifibered f.hom) J G
U V : C
f : U ⟶ V
c : J ⥤ Over F
α : obj (const J) G ⟶ c
hc : IsLimit { pt := G, π := α : Cone c }
H : ∀ (j : J), Equifibered ({ diag := c, π := α, isLimit := hc : LimitPresentation J G }.diag.obj j).hom
hα : ∀ (i j : J) (f : i ⟶ j), α.app i ≫ c.map f = α.app j
hc' : IsLimit (obj (Equivalence.functor WithTerminal.coneEquiv) { pt := G, π := α : Cone c })
inst : IsConnected.{v_1, u_1} (WithTerminal J)
⊢ IsLimit (J := WithTerminal J) (F := WithTerminal.liftToTerminal c Over.mkIdTerminal ⋙ Over.forget F)
    (mapCone (Over.forget F)
      { pt := G,
        π :=
          {
            app := fun (i : WithTerminal J) ↦
              WithTerminal.casesOn (motive := fun (x : WithTerminal J) ↦
                obj (obj (const (WithTerminal J)) G) x ⟶ (WithTerminal.liftToTerminal c Over.mkIdTerminal).obj x) i
                α.app
                (IsTerminal.from Over.mkIdTerminal
                  (obj (C := WithTerminal J) (obj (const (WithTerminal J)) G) WithTerminal.star)),
            naturality := ⋯ } })
have refine_1 : obj WithTerminal.liftFromOver c ≅ WithTerminal.liftToTerminal c Over.mkIdTerminal ⋙ Over.forget F
Tactic: refine IsLimit.equivOfNatIsoOfIso ?_ _ _ ?_ (WithTerminal.isLimitEquiv.symm hc)
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Pullback.EquifiberedLimits (goal)
Goal:
J : Type u_1
K : Type u_2
C : Type u_3
D : Type u_4
ι : Type u_5
inst_1 : Category.{v_1, u_1} J
inst_2 : Category.{v_2, u_3} C
inst_3 : Category.{v_3, u_2} K
inst_4 : Category.{v_4, u_4} D
F : Functor.{v_2, v_4, u_3, u_4} C D
inst_5 : ∀ (a b : C), HasCoproductsOfShape.{v_2, u_4, v_4} (a ⟶ b) D
this :
  ∀ {J : Type u_1} [inst : Category.{v_1, u_1} J] (hJ : IsConnected.{v_1, u_1} J),
    IsClosedUnderLimitsOfShape (C := Over F) (fun (f : Over F) ↦ Equifibered f.hom) J
hJ : ¬IsConnected.{v_1, u_1} J
G : Over F
h : limitsOfShape (fun (f : Over F) ↦ Equifibered f.hom) J G
U V : C
f : U ⟶ V
c : J ⥤ Over F
α : obj (const J) G ⟶ c
hc : IsLimit { pt := G, π := α : Cone c }
H : ∀ (j : J), Equifibered ({ diag := c, π := α, isLimit := hc : LimitPresentation J G }.diag.obj j).hom
hα : ∀ (i j : J) (f : i ⟶ j), α.app i ≫ c.map f = α.app j
hc' : IsLimit (obj (Equivalence.functor WithTerminal.coneEquiv) { pt := G, π := α : Cone c })
inst : IsConnected.{v_1, u_1} (WithTerminal J)
refine_1 : obj WithTerminal.liftFromOver c ≅ WithTerminal.liftToTerminal c Over.mkIdTerminal ⋙ Over.forget F
⊢ IsLimit (J := WithTerminal J) (F := WithTerminal.liftToTerminal c Over.mkIdTerminal ⋙ Over.forget F)
    (mapCone (Over.forget F)
      { pt := G,
        π :=
          {
            app := fun (i : WithTerminal J) ↦
              WithTerminal.casesOn (motive := fun (x : WithTerminal J) ↦
                obj (obj (const (WithTerminal J)) G) x ⟶ (WithTerminal.liftToTerminal c Over.mkIdTerminal).obj x) i
                α.app
                (IsTerminal.from Over.mkIdTerminal
                  (obj (C := WithTerminal J) (obj (const (WithTerminal J)) G) WithTerminal.star)),
            naturality := ⋯ } })
have refine_2 : Iso (C := Cone (WithTerminal.liftToTerminal c Over.mkIdTerminal ⋙ Over.forget F))
  (obj
    (Cone.postcompose
      (Iso.hom
        (NatIso.ofComponents
          (fun (i : WithTerminal J) ↦
            WithTerminal.casesOn (motive := fun (x : WithTerminal J) ↦
              (obj WithTerminal.liftFromOver c).obj x ≅
                (WithTerminal.liftToTerminal c Over.mkIdTerminal ⋙ Over.forget F).obj x)
              i (fun (x : J) ↦ Iso.refl ((obj WithTerminal.liftFromOver c).obj (WithTerminal.of x)))
              (Iso.refl ((obj WithTerminal.liftFromOver c).obj WithTerminal.star)))
          sorry)))
    (obj (Equivalence.functor WithTerminal.coneEquiv) { pt := G, π := α : Cone c }))
  (mapCone (Over.forget F)
    { pt := G,
      π :=
        {
          app := fun (i : WithTerminal J) ↦
            WithTerminal.casesOn (motive := fun (x : WithTerminal J) ↦
              obj (obj (const (WithTerminal J)) G) x ⟶ (WithTerminal.liftToTerminal c Over.mkIdTerminal).obj x) i α.app
              (IsTerminal.from Over.mkIdTerminal
                (obj (C := WithTerminal J) (obj (const (WithTerminal J)) G) WithTerminal.star)),
          naturality := sorry } })
Tactic: refine IsLimit.equivOfNatIsoOfIso ?_ _ _ ?_ (WithTerminal.isLimitEquiv.symm hc)
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Pullback.EquifiberedLimits (goal)
Goal:
J_1 : Type u_1
K : Type u_2
C : Type u_3
D : Type u_4
ι : Type u_5
inst : Category.{v_1, u_1} J_1
inst_1 : Category.{v_2, u_3} C
inst_2 : Category.{v_3, u_2} K
inst_3 : Category.{v_4, u_4} D
F : Functor.{v_2, v_4, u_3, u_4} C D
inst_4 : ∀ (a b : C), HasCoproductsOfShape.{v_2, u_4, v_4} (a ⟶ b) D
J : Type u_1
inst_5 : Category.{v_1, u_1} J
hJ : IsConnected.{v_1, u_1} J
⊢ IsClosedUnderLimitsOfShape (C := Over F) (fun (f : Over F) ↦ Equifibered f.hom) J
have refine : (G : Over F) →
  limitsOfShape (fun (f : Over F) ↦ Equifibered f.hom) J G →
    (i j : C) →
      (f : i ⟶ j) →
        (c : J ⥤ Over F) →
          (α : obj (const J) G ⟶ c) →
            (hc : IsLimit { pt := G, π := α : Cone c }) →
              (H :
                  ∀ (j : J),
                    Equifibered ({ diag := c, π := α, isLimit := hc : LimitPresentation J G }.diag.obj j).hom) →
                (s : PullbackCone (G.hom.app j) (F.map f)) →
                  { l : s.pt ⟶ (PullbackCone.mk (G.left.map f) (G.hom.app i) sorry).pt //
                    l ≫ (PullbackCone.mk (G.left.map f) (G.hom.app i) sorry).fst = s.fst ∧
                      l ≫ (PullbackCone.mk (G.left.map f) (G.hom.app i) sorry).snd = s.snd ∧
                        ∀ {m : s.pt ⟶ (PullbackCone.mk (G.left.map f) (G.hom.app i) sorry).pt},
                          m ≫ (PullbackCone.mk (G.left.map f) (G.hom.app i) sorry).fst = s.fst →
                            m ≫ (PullbackCone.mk (G.left.map f) (G.hom.app i) sorry).snd = s.snd → m = l }
Tactic: refine ⟨fun G ⟨⟨c, α, hc⟩, H⟩ i j f ↦ ⟨⟨by simp⟩, ⟨Limits.PullbackCone.isLimitAux' _ fun s ↦ ?_⟩⟩⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.Pullback.EquifiberedLimits (goal)
Goal:
J_1 : Type u_1
K : Type u_2
C : Type u_3
D : Type u_4
ι : Type u_5
inst : Category.{v_1, u_1} J_1
inst_1 : Category.{v_2, u_3} C
inst_2 : Category.{v_3, u_2} K
inst_3 : Category.{v_4, u_4} D
F : Functor.{v_2, v_4, u_3, u_4} C D
inst_4 : ∀ (a b : C), HasCoproductsOfShape.{v_2, u_4, v_4} (a ⟶ b) D
J : Type u_1
inst_5 : Category.{v_1, u_1} J
hJ : IsConnected.{v_1, u_1} J
G : Over F
h : limitsOfShape (fun (f : Over F) ↦ Equifibered f.hom) J G
i j : C
f : i ⟶ j
c : J ⥤ Over F
α : obj (const J) G ⟶ c
hc : IsLimit { pt := G, π := α : Cone c }
H : ∀ (j : J), Equifibered ({ diag := c, π := α, isLimit := hc : LimitPresentation J G }.diag.obj j).hom
s : PullbackCone (G.hom.app j) (F.map f)
hcᵢ : IsLimit ((Over.forget F ⋙ obj (evaluation C D) i).mapCone { pt := G, π := α : Cone c }) :=
  isLimitOfPreserves (Over.forget F ⋙ obj (evaluation C D) i) hc
hcⱼ : IsLimit ((Over.forget F ⋙ obj (evaluation C D) j).mapCone { pt := G, π := α : Cone c }) :=
  isLimitOfPreserves (Over.forget F ⋙ obj (evaluation C D) j) hc
l : s.pt ⟶ G.left.obj i :=
  hcᵢ.lift
    { pt := s.1,
      π := { app := fun (k : J) ↦ ⋯.lift (s.fst ≫ (Over.Hom.left (α.app k)).app j) s.snd ⋯, naturality := ⋯ } }
⊢ { l : s.pt ⟶ (PullbackCone.mk (G.left.map f) (G.hom.app i) ⋯).pt //
    l ≫ (PullbackCone.mk (G.left.map f) (G.hom.app i) ⋯).fst = s.fst ∧
      l ≫ (PullbackCone.mk (G.left.map f) (G.hom.app i) ⋯).snd = s.snd ∧
        ∀ {m : s.pt ⟶ (PullbackCone.mk (G.left.map f) (G.hom.app i) ⋯).pt},
          m ≫ (PullbackCone.mk (G.left.map f) (G.hom.app i) ⋯).fst = s.fst →
            m ≫ (PullbackCone.mk (G.left.map f) (G.hom.app i) ⋯).snd = s.snd → m = l }
have refine_1 : ∀ (k : J),
  (l ≫ (PullbackCone.mk (G.left.map f) (G.hom.app i) sorry).fst) ≫
      ((Over.forget F ⋙ obj (evaluation C D) j).mapCone { pt := G, π := α : Cone c }).π.app k =
    s.fst ≫ ((Over.forget F ⋙ obj (evaluation C D) j).mapCone { pt := G, π := α }).π.app k
Tactic: refine ⟨l, hcⱼ.hom_ext fun k ↦ ?_, ?_, fun {m} hm₁ hm₂ ↦ hcᵢ.hom_ext
    fun k ↦ ((H k f).hom_ext ?_ ?_).trans (hcᵢ.fac ⟨_, _⟩ k).symm⟩