awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
inst_2 : F.Final
inst_3 : IsFilteredOrEmpty.{v₁, u₁} C
X Y : D
f g : X ⟶ Y
P : StructuredArrow X F → Prop :=
  fun (h : StructuredArrow X F) ↦
    ∃ (Z : C) (q₁ : h.right ⟶ Z) (q₂ : Final.lift F Y ⟶ Z), h.hom ≫ F.map q₁ = f ≫ Final.homToLift F Y ≫ F.map q₂
h₀ : P (StructuredArrow.mk (Y := Final.lift F Y) (f ≫ Final.homToLift F Y))
U V : StructuredArrow X F
h : U ⟶ V
Z : C
q₁ : U.right ⟶ Z
q₂ : Final.lift F Y ⟶ Z
hq : U.hom ≫ F.map q₁ = f ≫ Final.homToLift F Y ≫ F.map q₂
⊢ P V
have refine_1 : ∀ (W : C) (q₃ : Z ⟶ W) (q₄ : V.right ⟶ W) (hq' : q₁ ≫ q₃ = StructuredArrow.Hom.right h ≫ q₄), P V
Tactic: obtain ⟨W, q₃, q₄, hq'⟩ := IsFiltered.span q₁ h.right
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
inst_2 : F.Final
inst_3 : IsFilteredOrEmpty.{v₁, u₁} C
X Y : D
f g : X ⟶ Y
P : StructuredArrow X F → Prop :=
  fun (h : StructuredArrow X F) ↦
    ∃ (Z : C) (q₁ : h.right ⟶ Z) (q₂ : Final.lift F Y ⟶ Z), h.hom ≫ F.map q₁ = f ≫ Final.homToLift F Y ≫ F.map q₂
h₀ : P (StructuredArrow.mk (Y := Final.lift F Y) (f ≫ Final.homToLift F Y))
U V : StructuredArrow X F
h : U ⟶ V
Z : C
q₁ : U.right ⟶ Z
q₂ : Final.lift F Y ⟶ Z
hq : U.hom ≫ F.map q₁ = f ≫ Final.homToLift F Y ≫ F.map q₂
W : C
q₃ : Z ⟶ W
q₄ : V.right ⟶ W
hq' : q₁ ≫ q₃ = StructuredArrow.Hom.right h ≫ q₄
⊢ P V
have refine_1 : V.hom ≫ F.map q₄ = f ≫ Final.homToLift F Y ≫ F.map (q₂ ≫ q₃)
Tactic: refine ⟨W, q₄, q₂ ≫ q₃, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
inst_2 : F.Final
inst_3 : IsFilteredOrEmpty.{v₁, u₁} C
X Y : D
f g : X ⟶ Y
P : StructuredArrow X F → Prop :=
  fun (h : StructuredArrow X F) ↦
    ∃ (Z : C) (q₁ : h.right ⟶ Z) (q₂ : Final.lift F Y ⟶ Z), h.hom ≫ F.map q₁ = f ≫ Final.homToLift F Y ≫ F.map q₂
h₀ : P (StructuredArrow.mk (Y := Final.lift F Y) (f ≫ Final.homToLift F Y))
U V : StructuredArrow X F
h : U ⟶ V
Z : C
q₁ : U.right ⟶ Z
q₂ : Final.lift F Y ⟶ Z
hq : U.hom ≫ F.map q₁ = f ≫ Final.homToLift F Y ≫ F.map q₂
W : C
q₃ : Z ⟶ W
q₄ : V.right ⟶ W
hq' : q₁ ≫ q₃ = StructuredArrow.Hom.right h ≫ q₄
⊢ V.hom ≫ F.map q₄ = f ≫ Final.homToLift F Y ≫ F.map (q₂ ≫ q₃)
have refine_1 : V.hom ≫ F.map q₄ = f ≫ Final.homToLift F Y ≫ F.map q₂ ≫ F.map q₃
Tactic: rw [F.map_comp, ← reassoc_of% hq, ← F.map_comp, hq', F.map_comp, StructuredArrow.w_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
inst_2 : F.Final
inst_3 : IsFilteredOrEmpty.{v₁, u₁} C
X Y : D
f g : X ⟶ Y
P : StructuredArrow X F → Prop :=
  fun (h : StructuredArrow X F) ↦
    ∃ (Z : C) (q₁ : h.right ⟶ Z) (q₂ : Final.lift F Y ⟶ Z), h.hom ≫ F.map q₁ = f ≫ Final.homToLift F Y ≫ F.map q₂
h₀ : P (StructuredArrow.mk (Y := Final.lift F Y) (f ≫ Final.homToLift F Y))
U V : StructuredArrow X F
h : U ⟶ V
Z : C
q₁ : U.right ⟶ Z
q₂ : Final.lift F Y ⟶ Z
hq : U.hom ≫ F.map q₁ = f ≫ Final.homToLift F Y ≫ F.map q₂
W : C
q₃ : Z ⟶ W
q₄ : V.right ⟶ W
hq' : q₁ ≫ q₃ = StructuredArrow.Hom.right h ≫ q₄
refine_1 : V.hom ≫ F.map q₄ = f ≫ Final.homToLift F Y ≫ F.map q₂ ≫ F.map q₃
⊢ V.hom ≫ F.map q₄ = f ≫ Final.homToLift F Y ≫ F.map (q₂ ≫ q₃)
have refine_1₁ : V.hom ≫ F.map q₄ = U.hom ≫ F.map q₁ ≫ F.map q₃
Tactic: rw [F.map_comp, ← reassoc_of% hq, ← F.map_comp, hq', F.map_comp, StructuredArrow.w_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
inst_2 : F.Final
inst_3 : IsFilteredOrEmpty.{v₁, u₁} C
X Y : D
f g : X ⟶ Y
P : StructuredArrow X F → Prop :=
  fun (h : StructuredArrow X F) ↦
    ∃ (Z : C) (q₁ : h.right ⟶ Z) (q₂ : Final.lift F Y ⟶ Z), h.hom ≫ F.map q₁ = f ≫ Final.homToLift F Y ≫ F.map q₂
h₀ : P (StructuredArrow.mk (Y := Final.lift F Y) (f ≫ Final.homToLift F Y))
U V : StructuredArrow X F
h : U ⟶ V
Z : C
q₁ : U.right ⟶ Z
q₂ : Final.lift F Y ⟶ Z
hq : U.hom ≫ F.map q₁ = f ≫ Final.homToLift F Y ≫ F.map q₂
W : C
q₃ : Z ⟶ W
q₄ : V.right ⟶ W
hq' : q₁ ≫ q₃ = StructuredArrow.Hom.right h ≫ q₄
refine_1 : V.hom ≫ F.map q₄ = f ≫ Final.homToLift F Y ≫ F.map q₂ ≫ F.map q₃
refine_1₁ : V.hom ≫ F.map q₄ = U.hom ≫ F.map q₁ ≫ F.map q₃
⊢ V.hom ≫ F.map q₄ = f ≫ Final.homToLift F Y ≫ F.map (q₂ ≫ q₃)
have refine_1₂ : V.hom ≫ F.map q₄ = U.hom ≫ F.map (q₁ ≫ q₃)
Tactic: rw [F.map_comp, ← reassoc_of% hq, ← F.map_comp, hq', F.map_comp, StructuredArrow.w_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
inst_2 : F.Final
inst_3 : IsFilteredOrEmpty.{v₁, u₁} C
X Y : D
f g : X ⟶ Y
P : StructuredArrow X F → Prop :=
  fun (h : StructuredArrow X F) ↦
    ∃ (Z : C) (q₁ : h.right ⟶ Z) (q₂ : Final.lift F Y ⟶ Z), h.hom ≫ F.map q₁ = f ≫ Final.homToLift F Y ≫ F.map q₂
h₀ : P (StructuredArrow.mk (Y := Final.lift F Y) (f ≫ Final.homToLift F Y))
U V : StructuredArrow X F
h : U ⟶ V
Z : C
q₁ : U.right ⟶ Z
q₂ : Final.lift F Y ⟶ Z
hq : U.hom ≫ F.map q₁ = f ≫ Final.homToLift F Y ≫ F.map q₂
W : C
q₃ : Z ⟶ W
q₄ : V.right ⟶ W
hq' : q₁ ≫ q₃ = StructuredArrow.Hom.right h ≫ q₄
refine_1 : V.hom ≫ F.map q₄ = f ≫ Final.homToLift F Y ≫ F.map q₂ ≫ F.map q₃
refine_1₁ : V.hom ≫ F.map q₄ = U.hom ≫ F.map q₁ ≫ F.map q₃
refine_1₂ : V.hom ≫ F.map q₄ = U.hom ≫ F.map (q₁ ≫ q₃)
⊢ V.hom ≫ F.map q₄ = f ≫ Final.homToLift F Y ≫ F.map (q₂ ≫ q₃)
have refine_1₃ : V.hom ≫ F.map q₄ = U.hom ≫ F.map (StructuredArrow.Hom.right h ≫ q₄)
Tactic: rw [F.map_comp, ← reassoc_of% hq, ← F.map_comp, hq', F.map_comp, StructuredArrow.w_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
inst_2 : F.Final
inst_3 : IsFilteredOrEmpty.{v₁, u₁} C
X Y : D
f g : X ⟶ Y
P : StructuredArrow X F → Prop :=
  fun (h : StructuredArrow X F) ↦
    ∃ (Z : C) (q₁ : h.right ⟶ Z) (q₂ : Final.lift F Y ⟶ Z), h.hom ≫ F.map q₁ = f ≫ Final.homToLift F Y ≫ F.map q₂
h₀ : P (StructuredArrow.mk (Y := Final.lift F Y) (f ≫ Final.homToLift F Y))
U V : StructuredArrow X F
h : U ⟶ V
Z : C
q₁ : U.right ⟶ Z
q₂ : Final.lift F Y ⟶ Z
hq : U.hom ≫ F.map q₁ = f ≫ Final.homToLift F Y ≫ F.map q₂
W : C
q₃ : Z ⟶ W
q₄ : V.right ⟶ W
hq' : q₁ ≫ q₃ = StructuredArrow.Hom.right h ≫ q₄
refine_1 : V.hom ≫ F.map q₄ = f ≫ Final.homToLift F Y ≫ F.map q₂ ≫ F.map q₃
refine_1₁ : V.hom ≫ F.map q₄ = U.hom ≫ F.map q₁ ≫ F.map q₃
refine_1₂ : V.hom ≫ F.map q₄ = U.hom ≫ F.map (q₁ ≫ q₃)
refine_1₃ : V.hom ≫ F.map q₄ = U.hom ≫ F.map (StructuredArrow.Hom.right h ≫ q₄)
⊢ V.hom ≫ F.map q₄ = f ≫ Final.homToLift F Y ≫ F.map (q₂ ≫ q₃)
have refine_1₄ : V.hom ≫ F.map q₄ = U.hom ≫ F.map (StructuredArrow.Hom.right h) ≫ F.map q₄
Tactic: rw [F.map_comp, ← reassoc_of% hq, ← F.map_comp, hq', F.map_comp, StructuredArrow.w_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
inst_2 : F.Final
inst_3 : IsFilteredOrEmpty.{v₁, u₁} C
X Y : D
f g : X ⟶ Y
P : StructuredArrow X F → Prop :=
  fun (h : StructuredArrow X F) ↦
    ∃ (Z : C) (q₁ : h.right ⟶ Z) (q₂ : Final.lift F Y ⟶ Z), h.hom ≫ F.map q₁ = f ≫ Final.homToLift F Y ≫ F.map q₂
h₀ : P (StructuredArrow.mk (Y := Final.lift F Y) (f ≫ Final.homToLift F Y))
U V : StructuredArrow X F
h : U ⟶ V
Z : C
q₁ : U.right ⟶ Z
q₂ : Final.lift F Y ⟶ Z
hq : U.hom ≫ F.map q₁ = f ≫ Final.homToLift F Y ≫ F.map q₂
W : C
q₃ : Z ⟶ W
q₄ : V.right ⟶ W
hq' : q₁ ≫ q₃ = StructuredArrow.Hom.right h ≫ q₄
refine_1 : V.hom ≫ F.map q₄ = f ≫ Final.homToLift F Y ≫ F.map q₂ ≫ F.map q₃
refine_1₁ : V.hom ≫ F.map q₄ = U.hom ≫ F.map q₁ ≫ F.map q₃
refine_1₂ : V.hom ≫ F.map q₄ = U.hom ≫ F.map (q₁ ≫ q₃)
refine_1₃ : V.hom ≫ F.map q₄ = U.hom ≫ F.map (StructuredArrow.Hom.right h ≫ q₄)
refine_1₄ : V.hom ≫ F.map q₄ = U.hom ≫ F.map (StructuredArrow.Hom.right h) ≫ F.map q₄
⊢ V.hom ≫ F.map q₄ = f ≫ Final.homToLift F Y ≫ F.map (q₂ ≫ q₃)
have refine_1₅ : V.hom ≫ F.map q₄ = V.hom ≫ F.map q₄
Tactic: rw [F.map_comp, ← reassoc_of% hq, ← F.map_comp, hq', F.map_comp, StructuredArrow.w_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
E : Type u₃
inst_2 : Category.{v₃, u₃} E
T : Functor.{v₁, v₂, u₁, u₂} C D
inst_3 : T.Final
S : Functor.{v₂, v₃, u₂, u₃} D E
X : E
⊢ (pre X T S).Final
have refine : ∀ (f : StructuredArrow X S), IsConnected (StructuredArrow f (pre X T S))
Tactic: refine ⟨fun f => ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
E : Type u₃
inst_2 : Category.{v₃, u₃} E
T : Functor.{v₁, v₂, u₁, u₂} C D
inst_3 : T.Final
S : Functor.{v₂, v₃, u₂, u₃} D E
X : E
f : StructuredArrow X S
⊢ IsConnected (StructuredArrow f (pre X T S))
have rw : IsConnected (StructuredArrow f.right T)
Tactic: rw [isConnected_iff_of_equivalence (StructuredArrow.preEquivalence T f)]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
E : Type u₃
inst_2 : Category.{v₃, u₃} E
T : Functor.{v₁, v₂, u₁, u₂} C D
inst_3 : T.Initial
S : Functor.{v₂, v₃, u₂, u₃} D E
X : E
⊢ (pre T S X).Initial
have refine : ∀ (f : CostructuredArrow S X), IsConnected (CostructuredArrow (pre T S X) f)
Tactic: refine ⟨fun f => ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
E : Type u₃
inst_2 : Category.{v₃, u₃} E
T : Functor.{v₁, v₂, u₁, u₂} C D
inst_3 : T.Initial
S : Functor.{v₂, v₃, u₂, u₃} D E
X : E
f : CostructuredArrow S X
⊢ IsConnected (CostructuredArrow (pre T S X) f)
have rw : IsConnected (CostructuredArrow T f.left)
Tactic: rw [isConnected_iff_of_equivalence (CostructuredArrow.preEquivalence T f)]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : D ⥤ Cat
G : Functor.{v₁, v₂, u₁, u₂} C D
hG : G.Final
⊢ (pre F G).Final
have out : ∀ (d : Grothendieck F), IsConnected (StructuredArrow d (pre F G))
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : D ⥤ Cat
G : Functor.{v₁, v₂, u₁, u₂} C D
hG : G.Final
d : D
f : (↑(F.obj d) : Type u_2)
⊢ IsConnected (StructuredArrow { base := d, fiber := f : Grothendieck F } (pre F G))
have out : ∀ (u : Discrete PUnit.{1}) (c : C) (g : obj.{0, v₂, 0, u₂} (fromPUnit d) u ⟶ G.obj c),
  IsConnected (StructuredArrow { base := d, fiber := f : Grothendieck F } (pre F G))
Tactic: let ⟨u, c, g⟩ : Nonempty (StructuredArrow d G) := inferInstance
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : D ⥤ Cat
G : Functor.{v₁, v₂, u₁, u₂} C D
hG : G.Final
d : D
f : (↑(F.obj d) : Type u_2)
u : Discrete PUnit.{1}
c : C
g : obj.{0, v₂, 0, u₂} (fromPUnit d) u ⟶ G.obj c
this : Nonempty (StructuredArrow { base := d, fiber := f : Grothendieck F } (pre F G)) := ⋯
⊢ IsConnected (StructuredArrow { base := d, fiber := f : Grothendieck F } (pre F G))
have out : ∀ (j₁ j₂ : StructuredArrow { base := d, fiber := f : Grothendieck F } (pre F G)), Zigzag j₁ j₂
Tactic: apply zigzag_isConnected
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : D ⥤ Cat
G : Functor.{v₁, v₂, u₁, u₂} C D
hG : G.Final
d : D
f : (↑(F.obj d) : Type u_2)
u : Discrete PUnit.{1}
c : C
g : obj.{0, v₂, 0, u₂} (fromPUnit d) u ⟶ G.obj c
this : Nonempty (StructuredArrow { base := d, fiber := f : Grothendieck F } (pre F G)) := ⋯
bi : C
fi : (↑(F.obj (G.obj bi)) : Type u_2)
gbi : d ⟶ G.obj bi
gfi : (F.map gbi).toFunctor.obj f ⟶ fi
bj : C
fj : (↑(F.obj (G.obj bj)) : Type u_2)
gbj : d ⟶ G.obj bj
gfj : (F.map gbj).toFunctor.obj f ⟶ fj
⊢ base.{u₁, v₁, u_2, u_1} (F := G ⋙ F)
      (StructuredArrow.right (D := Grothendieck F) (S := { base := d, fiber := f }) (T := pre F G)
        (StructuredArrow.mk (Y := { base := bi, fiber := (F.map gbi).toFunctor.obj f })
          { base := gbi, fiber := 𝟙 ((F.map gbi).toFunctor.obj { base := d, fiber := f : Grothendieck F }.fiber) })) ⟶
    base.{u₁, v₁, u_2, u_1} (F := G ⋙ F)
      (StructuredArrow.right (D := Grothendieck F) (S := { base := d, fiber := f : Grothendieck F }) (T := pre F G)
        { left := { as := PUnit.unit : Discrete PUnit.{1} },
          right := { base := bi, fiber := fi : Grothendieck (G ⋙ F) },
          hom :=
            { base := gbi, fiber := gfi :
              (obj (fromPUnit { base := d, fiber := f : Grothendieck F }) { as := PUnit.unit : Discrete PUnit.{1} }).Hom
                ((pre F G).obj { base := bi, fiber := fi : Grothendieck (G ⋙ F) }) } :
          Comma (fromPUnit { base := d, fiber := f : Grothendieck F }) (pre F G) })
have dsimp : bi ⟶ bi
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : D ⥤ Cat
G : Functor.{v₁, v₂, u₁, u₂} C D
hG : G.Final
d : D
f : (↑(F.obj d) : Type u_2)
u : Discrete PUnit.{1}
c : C
g : obj.{0, v₂, 0, u₂} (fromPUnit d) u ⟶ G.obj c
this : Nonempty (StructuredArrow { base := d, fiber := f : Grothendieck F } (pre F G)) := ⋯
bi : C
fi : (↑(F.obj (G.obj bi)) : Type u_2)
gbi : d ⟶ G.obj bi
gfi : (F.map gbi).toFunctor.obj f ⟶ fi
bj : C
fj : (↑(F.obj (G.obj bj)) : Type u_2)
gbj : d ⟶ G.obj bj
gfj : (F.map gbj).toFunctor.obj f ⟶ fj
⊢ Zigzag (J := StructuredArrow { base := d, fiber := f : Grothendieck F } (pre F G))
    { left := { as := PUnit.unit }, right := { base := bi, fiber := fi }, hom := { base := gbi, fiber := gfi } }
    { left := { as := PUnit.unit }, right := { base := bj, fiber := fj }, hom := { base := gbj, fiber := gfj } }
have out : Zigzag (J := StructuredArrow { base := d, fiber := f : Grothendieck F } (pre F G))
  (StructuredArrow.mk (Y := { base := bi, fiber := (F.map gbi).toFunctor.obj f })
    { base := gbi, fiber := 𝟙 ((F.map gbi).toFunctor.obj { base := d, fiber := f : Grothendieck F }.fiber) })
  { left := { as := PUnit.unit }, right := { base := bj, fiber := fj }, hom := { base := gbj, fiber := gfj } }
Tactic: apply Zigzag.trans (j₂ := StructuredArrow.mk (Y := ⟨bi, ((F.map gbi).toFunctor.obj f)⟩)
      (Grothendieck.Hom.mk gbi (𝟙 _)))
    (.of_zag (.inr ⟨StructuredArrow.homMk (Grothendieck.Hom.mk (by dsimp; exact 𝟙 _)
      (eqToHom (by simp) ≫ gfi)) (by apply Grothendieck.ext <;> simp)⟩))
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : D ⥤ Cat
G : Functor.{v₁, v₂, u₁, u₂} C D
hG : G.Final
d : D
f : (↑(F.obj d) : Type u_2)
u : Discrete PUnit.{1}
c : C
g : obj.{0, v₂, 0, u₂} (fromPUnit d) u ⟶ G.obj c
this : Nonempty (StructuredArrow { base := d, fiber := f : Grothendieck F } (pre F G)) := ⋯
bi : C
fi : (↑(F.obj (G.obj bi)) : Type u_2)
gbi : d ⟶ G.obj bi
gfi : (F.map gbi).toFunctor.obj f ⟶ fi
bj : C
fj : (↑(F.obj (G.obj bj)) : Type u_2)
gbj : d ⟶ G.obj bj
gfj : (F.map gbj).toFunctor.obj f ⟶ fj
⊢ base.{u₁, v₁, u_2, u_1} (F := G ⋙ F)
      (StructuredArrow.right (D := Grothendieck F) (S := { base := d, fiber := f }) (T := pre F G)
        (StructuredArrow.mk (Y := { base := bj, fiber := (F.map gbj).toFunctor.obj f })
          { base := gbj, fiber := 𝟙 ((F.map gbj).toFunctor.obj { base := d, fiber := f : Grothendieck F }.fiber) })) ⟶
    base.{u₁, v₁, u_2, u_1} (F := G ⋙ F)
      (StructuredArrow.right (D := Grothendieck F) (S := { base := d, fiber := f : Grothendieck F }) (T := pre F G)
        { left := { as := PUnit.unit : Discrete PUnit.{1} },
          right := { base := bj, fiber := fj : Grothendieck (G ⋙ F) },
          hom :=
            { base := gbj, fiber := gfj :
              (obj (fromPUnit { base := d, fiber := f : Grothendieck F }) { as := PUnit.unit : Discrete PUnit.{1} }).Hom
                ((pre F G).obj { base := bj, fiber := fj : Grothendieck (G ⋙ F) }) } :
          Comma (fromPUnit { base := d, fiber := f : Grothendieck F }) (pre F G) })
have dsimp : bj ⟶ bj
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : D ⥤ Cat
G : Functor.{v₁, v₂, u₁, u₂} C D
hG : G.Final
d : D
f : (↑(F.obj d) : Type u_2)
u : Discrete PUnit.{1}
c : C
g : obj.{0, v₂, 0, u₂} (fromPUnit d) u ⟶ G.obj c
this : Nonempty (StructuredArrow { base := d, fiber := f : Grothendieck F } (pre F G)) := ⋯
bi : C
fi : (↑(F.obj (G.obj bi)) : Type u_2)
gbi : d ⟶ G.obj bi
gfi : (F.map gbi).toFunctor.obj f ⟶ fi
bj : C
fj : (↑(F.obj (G.obj bj)) : Type u_2)
gbj : d ⟶ G.obj bj
gfj : (F.map gbj).toFunctor.obj f ⟶ fj
⊢ Zigzag (J := StructuredArrow { base := d, fiber := f : Grothendieck F } (pre F G))
    (StructuredArrow.mk (Y := { base := bi, fiber := (F.map gbi).toFunctor.obj f })
      { base := gbi, fiber := 𝟙 ((F.map gbi).toFunctor.obj { base := d, fiber := f : Grothendieck F }.fiber) })
    { left := { as := PUnit.unit }, right := { base := bj, fiber := fj }, hom := { base := gbj, fiber := gfj } }
have out : Zigzag (J := StructuredArrow { base := d, fiber := f : Grothendieck F } (pre F G))
  (StructuredArrow.mk (Y := { base := bi, fiber := (F.map gbi).toFunctor.obj f })
    { base := gbi, fiber := 𝟙 ((F.map gbi).toFunctor.obj { base := d, fiber := f : Grothendieck F }.fiber) })
  (StructuredArrow.mk (Y := { base := bj, fiber := (F.map gbj).toFunctor.obj f })
    { base := gbj, fiber := 𝟙 ((F.map gbj).toFunctor.obj { base := d, fiber := f : Grothendieck F }.fiber) })
Tactic: refine Zigzag.trans (j₂ := StructuredArrow.mk (Y := ⟨bj, ((F.map gbj).toFunctor.obj f)⟩)
      (Grothendieck.Hom.mk gbj (𝟙 _))) ?_
    (.of_zag (.inl ⟨StructuredArrow.homMk (Grothendieck.Hom.mk (by dsimp; exact 𝟙 _)
      (eqToHom (by simp) ≫ gfj)) (by apply Grothendieck.ext <;> simp)⟩))
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (hypothesis)
Goal:
C_1 : Type u₁
inst : Category.{v₁, u₁} C_1
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F_1 : Functor.{v₂, max ?u.411012 ?u.411013, u₂, max (?u.411012 + 1) (?u.411013 + 1)} D Cat
G_1 : Functor.{v₁, v₂, u₁, u₂} C_1 D
C : Type u₁
inst_2 : Category.{v₁, u₁} C
F G : C ⥤ Cat
α : F ⟶ G
inst_3 : ∀ (X : C), (α.app X).toFunctor.Final
H : Grothendieck G ⥤ Type u₂
X Y : C
f : X ⟶ Y
d : (↑(F.obj X) : Type u₂)
⊢ Eq (α := H.obj { base := X, fiber := (α.app X).toFunctor.obj d : Grothendieck G } ⟶ (fiberwiseColimit H).obj Y)
    (CategoryStruct.comp (Y := H.obj { base := Y, fiber := (α.app Y).toFunctor.obj ((F.map f).toFunctor.obj d) })
      (H.map { base := f, fiber := eqToHom ⋯ })
      (colimit.ι (ι G Y ⋙ H) ((α.app Y).toFunctor.obj ((F.map f).toFunctor.obj d))))
    (CategoryStruct.comp (Y := H.obj { base := Y, fiber := (G.map f).toFunctor.obj ((α.app X).toFunctor.obj d) })
      (H.map { base := f, fiber := 𝟙 ((G.map f).toFunctor.obj ((α.app X).toFunctor.obj d)) })
      (colimit.ι (ι G Y ⋙ H) ((G.map f).toFunctor.obj ((α.app X).toFunctor.obj d))))
have this : Eq (α := (↑(G.obj Y) : Type u₂)) (obj (Cat.Hom.toFunctor (F.map f ≫ α.app Y)) d)
  (obj (Cat.Hom.toFunctor (α.app X ≫ G.map f)) d)
Tactic: have := Functor.congr_obj congr($(α.naturality f).toFunctor) d
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C_1 : Type u₁
inst : Category.{v₁, u₁} C_1
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F_1 : Functor.{v₂, max ?u.411012 ?u.411013, u₂, max (?u.411012 + 1) (?u.411013 + 1)} D Cat
G_1 : Functor.{v₁, v₂, u₁, u₂} C_1 D
C : Type u₁
inst_2 : Category.{v₁, u₁} C
F G : C ⥤ Cat
α : F ⟶ G
inst_3 : ∀ (X : C), (α.app X).toFunctor.Final
H : Grothendieck G ⥤ Type u₂
X Y : C
f : X ⟶ Y
d : (↑(F.obj X) : Type u₂)
this : (α.app Y).toFunctor.obj ((F.map f).toFunctor.obj d) = (G.map f).toFunctor.obj ((α.app X).toFunctor.obj d)
⊢ Eq (α := H.obj { base := X, fiber := (α.app X).toFunctor.obj d : Grothendieck G } ⟶ (fiberwiseColimit H).obj Y)
    (CategoryStruct.comp (Y := H.obj { base := Y, fiber := (α.app Y).toFunctor.obj ((F.map f).toFunctor.obj d) })
      (H.map { base := f, fiber := eqToHom ⋯ })
      (colimit.ι (ι G Y ⋙ H) ((α.app Y).toFunctor.obj ((F.map f).toFunctor.obj d))))
    (CategoryStruct.comp (Y := H.obj { base := Y, fiber := (G.map f).toFunctor.obj ((α.app X).toFunctor.obj d) })
      (H.map { base := f, fiber := 𝟙 ((G.map f).toFunctor.obj ((α.app X).toFunctor.obj d)) })
      (colimit.ι (ι G Y ⋙ H) ((G.map f).toFunctor.obj ((α.app X).toFunctor.obj d))))
have h : HEq.{v₂ + 1} (α :=
  (G.map f).toFunctor.obj { base := X, fiber := (α.app X).toFunctor.obj d : Grothendieck G }.fiber ⟶
    { base := Y, fiber := (α.app Y).toFunctor.obj ((F.map f).toFunctor.obj d) }.fiber)
  (eqToHom sorry :
    (G.map f).toFunctor.obj ((α.app X).toFunctor.obj d) ⟶ (α.app Y).toFunctor.obj ((F.map f).toFunctor.obj d))
  (β :=
  (G.map f).toFunctor.obj { base := X, fiber := (α.app X).toFunctor.obj d : Grothendieck G }.fiber ⟶
    { base := Y, fiber := (G.map f).toFunctor.obj ((α.app X).toFunctor.obj d) }.fiber)
  (𝟙 ((G.map f).toFunctor.obj ((α.app X).toFunctor.obj d)))
Tactic: congr
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : SmallCategory C
F G : C ⥤ Cat
α : F ⟶ G
hα : ∀ (X : C), (α.app X).toFunctor.Final
⊢ (map α).Final
have rw : ∀ (G_1 : Grothendieck G ⥤ Type u₁), IsIso (X := colimit (map α ⋙ G_1)) (Y := colimit G_1) (colimit.pre G_1 (map α))
Tactic: rw [final_iff_isIso_colimit_pre]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : SmallCategory C
F G : C ⥤ Cat
α : F ⟶ G
hα : ∀ (X : C), (α.app X).toFunctor.Final
H : Grothendieck G ⥤ Type u₁
i : colimit (map α ⋙ H) ≅ colimit H :=
  Iso.trans (Y := colimit (fiberwiseColimit (map α ⋙ H))) (Iso.symm (colimitFiberwiseColimitIso (map α ⋙ H)))
    (Iso.trans (Y := colimit (fiberwiseColimit H)) (HasColimit.isoOfNatIso (fiberwiseColimitMapCompEquivalence α H))
      (colimitFiberwiseColimitIso H))
⊢ IsIso (X := colimit (map α ⋙ H)) (Y := colimit H) (colimit.pre H (map α))
have h : colimit.pre H (map α) = i.hom
Tactic: convert! Iso.isIso_hom i
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (goal)
Goal:
C : Type u₁
inst : SmallCategory C
F G : C ⥤ Cat
α : F ⟶ G
hα : ∀ (X : C), (α.app X).toFunctor.Final
H : Grothendieck G ⥤ Type u₁
i : colimit (map α ⋙ H) ≅ colimit H :=
  Iso.trans (Y := colimit (fiberwiseColimit (map α ⋙ H))) (Iso.symm (colimitFiberwiseColimitIso (map α ⋙ H)))
    (Iso.trans (Y := colimit (fiberwiseColimit H)) (HasColimit.isoOfNatIso (fiberwiseColimitMapCompEquivalence α H))
      (colimitFiberwiseColimitIso H))
⊢ colimit.pre H (map α) = i.hom
have h : ∀ (j : Grothendieck F),
  CategoryStruct.comp (Y := colimit (map α ⋙ H)) (colimit.ι (map α ⋙ H) j) (colimit.pre H (map α)) =
    colimit.ι (map α ⋙ H) j ≫ i.hom
Tactic: apply colimit.hom_ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Final (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
F G : C ⥤ Cat
α : F ⟶ G
hα : ∀ (X : C), (α.app X).toFunctor.Final
sC : C ≌ AsSmall C := AsSmall.equiv
F' : AsSmall C ⥤ Cat := sC.inverse ⋙ F ⋙ Cat.asSmallFunctor
G' : AsSmall C ⥤ Cat := sC.inverse ⋙ G ⋙ Cat.asSmallFunctor
α' : F' ⟶ G' := sC.inverse.whiskerLeft (whiskerRight α Cat.asSmallFunctor)
⊢ (map α).Final
have this : ∀ (X : AsSmall C), (α'.app X).toFunctor.Final
Tactic: have : ∀ X, Final (α'.app X).toFunctor := fun X =>
    inferInstanceAs (AsSmall.equiv.inverse ⋙ _ ⋙ AsSmall.equiv.functor).Final