awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Extensive (goal)
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
D : Type u''
inst_2 : Category.{v'', u''} D
X Y : C
X' Y' : Type u
αX : X' ⟶ PUnit.{u + 1}
αY : Y' ⟶ PUnit.{u + 1}
f : (Types.binaryCoproductCocone X' Y').pt ⟶ (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt
hαX :
  αX ≫ BinaryCofan.inl (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}) =
    BinaryCofan.inl (Types.binaryCoproductCocone X' Y') ≫ f
hαY :
  αY ≫ BinaryCofan.inr (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}) =
    BinaryCofan.inr (Types.binaryCoproductCocone X' Y') ≫ f
s : PullbackCone f (BinaryCofan.inr (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}))
l : (x : s.pt) → Y'
hl :
  ∀ (x : s.pt),
    (fun (y : Y') ↦
        Eq (α :=
          (fun (X : Type u) ↦ X)
            (Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) (Types.binaryCoproductCocone X' Y').pt)
              { as := WalkingPair.right : Discrete WalkingPair }))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) s.fst :
                (a : (fun (X : Type u) ↦ X) s.pt) →
                  (fun (X : Type u) ↦ X)
                    (((Functor.const (Discrete WalkingPair)).obj (Types.binaryCoproductCocone X' Y').pt).obj
                      { as := WalkingPair.right }))
              x :
            (fun (X : Type u) ↦ X)
              (Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) (Types.binaryCoproductCocone X' Y').pt)
                { as := WalkingPair.right : Discrete WalkingPair }))
          (Sum.inr y : X' ⊕ Y'))
      (l x)
hl' :
  ∀ (x : s.pt) (y : Y'),
    (fun (y : Y') ↦
          Eq (α :=
            (fun (X : Type u) ↦ X)
              (Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) (Types.binaryCoproductCocone X' Y').pt)
                { as := WalkingPair.right : Discrete WalkingPair }))
            ((ConcreteCategory.hom (FC := TypeCat.Fun) s.fst :
                  (a : (fun (X : Type u) ↦ X) s.pt) →
                    (fun (X : Type u) ↦ X)
                      (((Functor.const (Discrete WalkingPair)).obj (Types.binaryCoproductCocone X' Y').pt).obj
                        { as := WalkingPair.right }))
                x :
              (fun (X : Type u) ↦ X)
                (Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) (Types.binaryCoproductCocone X' Y').pt)
                  { as := WalkingPair.right : Discrete WalkingPair }))
            (Sum.inr y : X' ⊕ Y'))
        y →
      y = l x
l' : s.pt ⟶ (PullbackCone.mk (BinaryCofan.inr (Types.binaryCoproductCocone X' Y')) αY ⋯).pt
h₁ : l' ≫ (PullbackCone.mk (BinaryCofan.inr (Types.binaryCoproductCocone X' Y')) αY ⋯).fst = s.fst
h : l' ≫ (PullbackCone.mk (BinaryCofan.inr (Types.binaryCoproductCocone X' Y')) αY ⋯).snd = s.snd
⊢ l' = ↾l
have h₂ : ∀ (x : s.pt),
  (ConcreteCategory.hom (FC := TypeCat.Fun) l').toFun x = (ConcreteCategory.hom (FC := TypeCat.Fun) (↾l)).toFun x
Tactic: ext x
-- Lean 4 proof step from Mathlib.CategoryTheory.Extensive (goal)
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
D : Type u''
inst_2 : Category.{v'', u''} D
X Y : C
Z : Type u
f : Z ⟶ (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt
⊢ IsColimit
    (BinaryCofan.mk
      (PullbackCone.fst
        (Types.pullbackLimitCone f (BinaryCofan.inl (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}))).cone)
      (PullbackCone.fst
        (Types.pullbackLimitCone f (BinaryCofan.inr (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}))).cone))
have h₂ : IsColimit (F := pair (Types.PullbackObj f (↾Sum.inl)) (Types.PullbackObj f (↾Sum.inr)))
  (BinaryCofan.mk (P := Z) (↾fun (p : Types.PullbackObj f (↾Sum.inl)) ↦ (↑p : Z × PUnit.{u + 1}).1)
    (↾fun (p : Types.PullbackObj f (↾Sum.inr)) ↦ (↑p : Z × PUnit.{u + 1}).1))
Tactic: dsimp [Limits.Types.binaryCoproductCocone]
-- Lean 4 proof step from Mathlib.CategoryTheory.Extensive (goal)
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
D : Type u''
inst_2 : Category.{v'', u''} D
X Y : C
Z : Type u
f : Z ⟶ (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt
x : Z
⊢ Eq (α := (fun (X : Type u) ↦ X) (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt)
      ((ConcreteCategory.hom (FC := TypeCat.Fun) f :
            (a : (fun (X : Type u) ↦ X) Z) →
              (fun (X : Type u) ↦ X) (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt)
          x :
        (fun (X : Type u) ↦ X) (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt)
      (Sum.inl PUnit.unit : PUnit.{u + 1} ⊕ PUnit.{u + 1}) ∨
    Eq (α := (fun (X : Type u) ↦ X) (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt)
      ((ConcreteCategory.hom (FC := TypeCat.Fun) f :
            (a : (fun (X : Type u) ↦ X) Z) →
              (fun (X : Type u) ↦ X) (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt)
          x :
        (fun (X : Type u) ↦ X) (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt)
      (Sum.inr PUnit.unit : PUnit.{u + 1} ⊕ PUnit.{u + 1})
have inl : Eq (α := (fun (X : Type u) ↦ X) (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt) (Sum.inl PUnit.unit)
    (Sum.inl PUnit.unit) ∨
  Sum.inl PUnit.unit = Sum.inr PUnit.unit
Tactic: rcases f x with (⟨⟨⟩⟩ | ⟨⟨⟩⟩)
-- Lean 4 proof step from Mathlib.CategoryTheory.Extensive (goal)
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
D : Type u''
inst_2 : Category.{v'', u''} D
X Y : C
Z : Type u
f : Z ⟶ (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt
x : Z
inl :
  Eq (α := (fun (X : Type u) ↦ X) (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt) (Sum.inl PUnit.unit)
      (Sum.inl PUnit.unit) ∨
    Sum.inl PUnit.unit = Sum.inr PUnit.unit
⊢ Eq (α := (fun (X : Type u) ↦ X) (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt)
      ((ConcreteCategory.hom (FC := TypeCat.Fun) f :
            (a : (fun (X : Type u) ↦ X) Z) →
              (fun (X : Type u) ↦ X) (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt)
          x :
        (fun (X : Type u) ↦ X) (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt)
      (Sum.inl PUnit.unit : PUnit.{u + 1} ⊕ PUnit.{u + 1}) ∨
    Eq (α := (fun (X : Type u) ↦ X) (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt)
      ((ConcreteCategory.hom (FC := TypeCat.Fun) f :
            (a : (fun (X : Type u) ↦ X) Z) →
              (fun (X : Type u) ↦ X) (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt)
          x :
        (fun (X : Type u) ↦ X) (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt)
      (Sum.inr PUnit.unit : PUnit.{u + 1} ⊕ PUnit.{u + 1})
have inr : Sum.inr PUnit.unit = Sum.inl PUnit.unit ∨
  Eq (α := (fun (X : Type u) ↦ X) (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt) (Sum.inr PUnit.unit)
    (Sum.inr PUnit.unit)
Tactic: rcases f x with (⟨⟨⟩⟩ | ⟨⟨⟩⟩)
-- Lean 4 proof step from Mathlib.CategoryTheory.Extensive (hypothesis)
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
D : Type u''
inst_2 : Category.{v'', u''} D
X Y : C
Z : Type u
f : Z ⟶ (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt
⊢ IsColimit (F := pair (Types.PullbackObj f (↾Sum.inl)) (Types.PullbackObj f (↾Sum.inr)))
    (BinaryCofan.mk (P := Z) (↾fun (p : Types.PullbackObj f (↾Sum.inl)) ↦ (↑p : Z × PUnit.{u + 1}).1)
      (↾fun (p : Types.PullbackObj f (↾Sum.inr)) ↦ (↑p : Z × PUnit.{u + 1}).1))
have this : ∀ (x : (fun (X : Type u) ↦ X) Z),
  Eq (α := (fun (X : Type u) ↦ X) (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt)
      ((ConcreteCategory.hom (FC := TypeCat.Fun) f :
            (a : (fun (X : Type u) ↦ X) Z) →
              (fun (X : Type u) ↦ X) (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt)
          x :
        (fun (X : Type u) ↦ X) (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt)
      (Sum.inl PUnit.unit : PUnit.{u + 1} ⊕ PUnit.{u + 1}) ∨
    Eq (α := (fun (X : Type u) ↦ X) (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt)
      ((ConcreteCategory.hom (FC := TypeCat.Fun) f :
            (a : (fun (X : Type u) ↦ X) Z) →
              (fun (X : Type u) ↦ X) (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt)
          x :
        (fun (X : Type u) ↦ X) (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt)
      (Sum.inr PUnit.unit : PUnit.{u + 1} ⊕ PUnit.{u + 1})
Tactic: have : ∀ x, f x = Sum.inl PUnit.unit ∨ f x = Sum.inr PUnit.unit := by
      intro x
      rcases f x with (⟨⟨⟩⟩ | ⟨⟨⟩⟩)
      exacts [Or.inl rfl, Or.inr rfl]
-- Lean 4 proof step from Mathlib.CategoryTheory.Extensive (goal)
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
D : Type u''
inst_2 : Category.{v'', u''} D
X Y : C
⊢ FinitaryExtensive TopCat
have rw : IsVanKampenColimit.{0, 0, u, u + 1} (F := pair (TopCat.of PUnit.{u + 1}) (TopCat.of PUnit.{u + 1}))
  ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1}))
Tactic: rw [finitaryExtensive_iff_of_isTerminal TopCat.{u} _ TopCat.isTerminalPUnit _
      (TopCat.binaryCofanIsColimit _ _)]
-- Lean 4 proof step from Mathlib.CategoryTheory.Extensive (goal)
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
D : Type u''
inst_2 : Category.{v'', u''} D
X Y : C
⊢ IsVanKampenColimit.{0, 0, u, u + 1} (F := pair (TopCat.of PUnit.{u + 1}) (TopCat.of PUnit.{u + 1}))
    ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1}))
have h₁ : ∀ {X' Y' : TopCat.{u}} (αX : X' ⟶ TopCat.of PUnit.{u + 1}) (αY : Y' ⟶ TopCat.of PUnit.{u + 1})
  (f :
    (X'.binaryCofan Y').pt ⟶
      Cocone.pt (F := pair (TopCat.of PUnit.{u + 1}) (TopCat.of PUnit.{u + 1}))
        ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1}))),
  αX ≫ BinaryCofan.inl ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})) = (X'.binaryCofan Y').inl ≫ f →
    αY ≫ BinaryCofan.inr ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})) =
        (X'.binaryCofan Y').inr ≫ f →
      IsPullback (X'.binaryCofan Y').inl αX f
          (BinaryCofan.inl ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1}))) ∧
        IsPullback (X'.binaryCofan Y').inr αY f
          (BinaryCofan.inr ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})))
Tactic: apply BinaryCofan.isVanKampen_mk _ _ (fun X Y => TopCat.binaryCofanIsColimit X Y) _
      fun f g => TopCat.pullbackConeIsLimit f g
-- Lean 4 proof step from Mathlib.CategoryTheory.Extensive (goal)
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
D : Type u''
inst_2 : Category.{v'', u''} D
X Y : C
h₁ :
  ∀ {X' Y' : TopCat.{u}} (αX : X' ⟶ TopCat.of PUnit.{u + 1}) (αY : Y' ⟶ TopCat.of PUnit.{u + 1})
    (f :
      (X'.binaryCofan Y').pt ⟶
        Cocone.pt (F := pair (TopCat.of PUnit.{u + 1}) (TopCat.of PUnit.{u + 1}))
          ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1}))),
    αX ≫ BinaryCofan.inl ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})) =
        (X'.binaryCofan Y').inl ≫ f →
      αY ≫ BinaryCofan.inr ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})) =
          (X'.binaryCofan Y').inr ≫ f →
        IsPullback (X'.binaryCofan Y').inl αX f
            (BinaryCofan.inl ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1}))) ∧
          IsPullback (X'.binaryCofan Y').inr αY f
            (BinaryCofan.inr ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})))
⊢ IsVanKampenColimit.{0, 0, u, u + 1} (F := pair (TopCat.of PUnit.{u + 1}) (TopCat.of PUnit.{u + 1}))
    ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1}))
have h₂ : {Z : TopCat.{u}} →
  (f :
      Z ⟶
        Cocone.pt (F := pair (TopCat.of PUnit.{u + 1}) (TopCat.of PUnit.{u + 1}))
          ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1}))) →
    IsColimit (F :=
      pair
        (Cone.pt (F :=
          cospan (Y :=
            Functor.obj (pair (TopCat.of PUnit.{u + 1}) (TopCat.of PUnit.{u + 1}))
              { as := WalkingPair.left : Discrete WalkingPair })
            f (BinaryCofan.inl ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1}))))
          (TopCat.pullbackCone f (BinaryCofan.inl ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})))))
        (Cone.pt (F :=
          cospan (Y :=
            Functor.obj (pair (TopCat.of PUnit.{u + 1}) (TopCat.of PUnit.{u + 1}))
              { as := WalkingPair.right : Discrete WalkingPair })
            f (BinaryCofan.inr ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1}))))
          (TopCat.pullbackCone f (BinaryCofan.inr ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1}))))))
      (BinaryCofan.mk
        (PullbackCone.fst
          (TopCat.pullbackCone f (BinaryCofan.inl ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})))))
        (PullbackCone.fst
          (TopCat.pullbackCone f (BinaryCofan.inr ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1}))))))
Tactic: apply BinaryCofan.isVanKampen_mk _ _ (fun X Y => TopCat.binaryCofanIsColimit X Y) _
      fun f g => TopCat.pullbackConeIsLimit f g
-- Lean 4 proof step from Mathlib.CategoryTheory.Extensive (goal)
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
D : Type u''
inst_2 : Category.{v'', u''} D
X Y : C
X' Y' : TopCat
αX : X' ⟶ TopCat.of PUnit.{u + 1}
αY : Y' ⟶ TopCat.of PUnit.{u + 1}
f :
  (X'.binaryCofan Y').pt ⟶
    Cocone.pt (F := pair (TopCat.of PUnit.{u + 1}) (TopCat.of PUnit.{u + 1}))
      ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1}))
hαX :
  αX ≫ BinaryCofan.inl ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})) = (X'.binaryCofan Y').inl ≫ f
hαY :
  αY ≫ BinaryCofan.inr ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})) = (X'.binaryCofan Y').inr ≫ f
⊢ IsPullback (X'.binaryCofan Y').inl αX f
      (BinaryCofan.inl ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1}))) ∧
    IsPullback (X'.binaryCofan Y').inr αY f
      (BinaryCofan.inr ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})))
have h₁ : IsPullback (X'.binaryCofan Y').inl αX f
  (BinaryCofan.inl ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})))
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Extensive (goal)
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
D : Type u''
inst_2 : Category.{v'', u''} D
X Y : C
X' Y' : TopCat
αX : X' ⟶ TopCat.of PUnit.{u + 1}
αY : Y' ⟶ TopCat.of PUnit.{u + 1}
f :
  (X'.binaryCofan Y').pt ⟶
    Cocone.pt (F := pair (TopCat.of PUnit.{u + 1}) (TopCat.of PUnit.{u + 1}))
      ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1}))
hαX :
  αX ≫ BinaryCofan.inl ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})) = (X'.binaryCofan Y').inl ≫ f
hαY :
  αY ≫ BinaryCofan.inr ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})) = (X'.binaryCofan Y').inr ≫ f
h₁ :
  IsPullback (X'.binaryCofan Y').inl αX f
    (BinaryCofan.inl ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})))
⊢ IsPullback (X'.binaryCofan Y').inl αX f
      (BinaryCofan.inl ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1}))) ∧
    IsPullback (X'.binaryCofan Y').inr αY f
      (BinaryCofan.inr ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})))
have h₂ : IsPullback (X'.binaryCofan Y').inr αY f
  (BinaryCofan.inr ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})))
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Extensive (goal)
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
D : Type u''
inst_2 : Category.{v'', u''} D
X Y : C
X' Y' : TopCat
αX : X' ⟶ TopCat.of PUnit.{u + 1}
αY : Y' ⟶ TopCat.of PUnit.{u + 1}
f :
  (X'.binaryCofan Y').pt ⟶
    Cocone.pt (F := pair (TopCat.of PUnit.{u + 1}) (TopCat.of PUnit.{u + 1}))
      ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1}))
hαX :
  αX ≫ BinaryCofan.inl ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})) = (X'.binaryCofan Y').inl ≫ f
hαY :
  αY ≫ BinaryCofan.inr ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})) = (X'.binaryCofan Y').inr ≫ f
⊢ IsPullback (X'.binaryCofan Y').inl αX f
    (BinaryCofan.inl ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})))
have h₁ : (s :
    PullbackCone (Y := TopCat.of PUnit.{u + 1}) f
      (BinaryCofan.inl ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})))) →
  { l : s.pt ⟶ (PullbackCone.mk (X'.binaryCofan Y').inl αX sorry).pt //
    l ≫ (PullbackCone.mk (X'.binaryCofan Y').inl αX sorry).fst = s.fst ∧
      l ≫ (PullbackCone.mk (X'.binaryCofan Y').inl αX sorry).snd = s.snd ∧
        ∀ {m : s.pt ⟶ (PullbackCone.mk (X'.binaryCofan Y').inl αX sorry).pt},
          m ≫ (PullbackCone.mk (X'.binaryCofan Y').inl αX sorry).fst = s.fst →
            m ≫ (PullbackCone.mk (X'.binaryCofan Y').inl αX sorry).snd = s.snd → m = l }
Tactic: refine ⟨⟨hαX.symm⟩, ⟨PullbackCone.isLimitAux' _ ?_⟩⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Extensive (goal)
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
D : Type u''
inst_2 : Category.{v'', u''} D
X Y : C
X' Y' : TopCat
αX : X' ⟶ TopCat.of PUnit.{u + 1}
αY : Y' ⟶ TopCat.of PUnit.{u + 1}
f :
  (X'.binaryCofan Y').pt ⟶
    Cocone.pt (F := pair (TopCat.of PUnit.{u + 1}) (TopCat.of PUnit.{u + 1}))
      ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1}))
hαX :
  αX ≫ BinaryCofan.inl ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})) = (X'.binaryCofan Y').inl ≫ f
hαY :
  αY ≫ BinaryCofan.inr ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})) = (X'.binaryCofan Y').inr ≫ f
s :
  PullbackCone (Y := TopCat.of PUnit.{u + 1}) f
    (BinaryCofan.inl ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})))
x : (↑s.pt : Type u)
⊢ ∃! y : (↑X' : Type u),
    Eq (α :=
      (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) (X'.binaryCofan Y').pt)
            { as := WalkingPair.left : Discrete WalkingPair }) :
        Type u))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
              s.fst :
            (a : (↑s.pt : Type u)) →
              (↑(((Functor.const (Discrete WalkingPair)).obj (X'.binaryCofan Y').pt).obj { as := WalkingPair.left }) :
                Type u))
          x :
        (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) (X'.binaryCofan Y').pt)
              { as := WalkingPair.left : Discrete WalkingPair }) :
          Type u))
      (Sum.inl y : (↑X' : Type u) ⊕ (↑Y' : Type u))
have inl : ∀ (val : (↑X' : Type u))
  (h :
    Eq (α :=
      (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) (X'.binaryCofan Y').pt)
            { as := WalkingPair.left : Discrete WalkingPair }) :
        Type u))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
              s.fst :
            (a : (↑s.pt : Type u)) →
              (↑(((Functor.const (Discrete WalkingPair)).obj (X'.binaryCofan Y').pt).obj { as := WalkingPair.left }) :
                Type u))
          x :
        (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) (X'.binaryCofan Y').pt)
              { as := WalkingPair.left : Discrete WalkingPair }) :
          Type u))
      (Sum.inl val : (↑X' : Type u) ⊕ (↑Y' : Type u))),
  ∃! y : (↑X' : Type u),
    Eq (α :=
      (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) (X'.binaryCofan Y').pt)
            { as := WalkingPair.left : Discrete WalkingPair }) :
        Type u))
      (Sum.inl val) (Sum.inl y)
Tactic: rcases h : s.fst x with val | val
-- Lean 4 proof step from Mathlib.CategoryTheory.Extensive (hypothesis)
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
D : Type u''
inst_2 : Category.{v'', u''} D
X Y : C
X' Y' : TopCat
αX : X' ⟶ TopCat.of PUnit.{u + 1}
αY : Y' ⟶ TopCat.of PUnit.{u + 1}
f :
  (X'.binaryCofan Y').pt ⟶
    Cocone.pt (F := pair (TopCat.of PUnit.{u + 1}) (TopCat.of PUnit.{u + 1}))
      ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1}))
hαX :
  αX ≫ BinaryCofan.inl ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})) = (X'.binaryCofan Y').inl ≫ f
hαY :
  αY ≫ BinaryCofan.inr ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})) = (X'.binaryCofan Y').inr ≫ f
s :
  PullbackCone (Y := TopCat.of PUnit.{u + 1}) f
    (BinaryCofan.inl ((TopCat.of PUnit.{u + 1}).binaryCofan (TopCat.of PUnit.{u + 1})))
⊢ { l : s.pt ⟶ (PullbackCone.mk (X'.binaryCofan Y').inl αX ⋯).pt //
    l ≫ (PullbackCone.mk (X'.binaryCofan Y').inl αX ⋯).fst = s.fst ∧
      l ≫ (PullbackCone.mk (X'.binaryCofan Y').inl αX ⋯).snd = s.snd ∧
        ∀ {m : s.pt ⟶ (PullbackCone.mk (X'.binaryCofan Y').inl αX ⋯).pt},
          m ≫ (PullbackCone.mk (X'.binaryCofan Y').inl αX ⋯).fst = s.fst →
            m ≫ (PullbackCone.mk (X'.binaryCofan Y').inl αX ⋯).snd = s.snd → m = l }
have this : ∀ (x : (↑s.pt : Type u)),
  ∃! y : (↑X' : Type u),
    Eq (α :=
      (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) (X'.binaryCofan Y').pt)
            { as := WalkingPair.left : Discrete WalkingPair }) :
        Type u))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
              s.fst :
            (a : (↑s.pt : Type u)) →
              (↑(((Functor.const (Discrete WalkingPair)).obj (X'.binaryCofan Y').pt).obj { as := WalkingPair.left }) :
                Type u))
          x :
        (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) (X'.binaryCofan Y').pt)
              { as := WalkingPair.left : Discrete WalkingPair }) :
          Type u))
      (Sum.inl y : (↑X' : Type u) ⊕ (↑Y' : Type u))
Tactic: have : ∀ x, ∃! y, s.fst x = Sum.inl y := by
        intro x
        rcases h : s.fst x with val | val
        · exact ⟨val, rfl, fun y h => Sum.inl_injective h.symm⟩
        · apply_fun f at h
          cases ((ConcreteCategory.congr_hom s.condition x).symm.trans h).trans
            (ConcreteCategory.congr_hom hαY val :).symm