awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
F_1 : CategoryTheory.Functor.{v, v', u, u'} C D
F : CategoryTheory.Functor.{0, v, 0, u} (Discrete WalkingPair) C
c : Cone F
X : C
hX : IsTerminal.{v, u} X
f : F.obj { as := WalkingPair.left : Discrete WalkingPair } ⟶ X
g : F.obj { as := WalkingPair.right : Discrete WalkingPair } ⟶ X
hc :
  IsLimit (F := cospan f g)
    (PullbackCone.mk (c.π.app { as := WalkingPair.left : Discrete WalkingPair }) (c.π.app { as := WalkingPair.right })
        ⋯ :
      PullbackCone f g)
s : Cone F
m : s.pt ⟶ c.pt
J : ∀ (j : Discrete WalkingPair), m ≫ c.π.app j = s.π.app j
c' : PullbackCone f g :=
  PullbackCone.mk (m ≫ c.π.app { as := WalkingPair.left : Discrete WalkingPair })
    (m ≫ c.π.app { as := WalkingPair.right }) ⋯
⊢ Eq (α := s.pt ⟶ (cospan f g).obj (some WalkingPair.left))
    (m ≫
      NatTrans.app
        (Cone.π
          (PullbackCone.mk (c.π.app { as := WalkingPair.left : Discrete WalkingPair })
            (c.π.app { as := WalkingPair.right }) ⋯))
        (some WalkingPair.left))
    (CategoryStruct.comp (Y :=
      Cone.pt (F := cospan f g)
        (PullbackCone.mk (c.π.app { as := WalkingPair.left : Discrete WalkingPair })
            (c.π.app { as := WalkingPair.right }) ⋯ :
          PullbackCone f g))
      (hc.lift
        (PullbackCone.mk (m ≫ c.π.app { as := WalkingPair.left : Discrete WalkingPair })
          (m ≫ c.π.app { as := WalkingPair.right }) ⋯))
      (NatTrans.app
        (Cone.π
          (PullbackCone.mk (c.π.app { as := WalkingPair.left : Discrete WalkingPair })
            (c.π.app { as := WalkingPair.right }) ⋯))
        (some WalkingPair.left)))
have some : Eq (α := s.pt ⟶ (cospan f g).obj (some WalkingPair.left)) (m ≫ c.π.app { as := WalkingPair.left })
  (hc.lift
      (PullbackCone.mk (m ≫ c.π.app { as := WalkingPair.left : Discrete WalkingPair })
        (m ≫ c.π.app { as := WalkingPair.right }) sorry) ≫
    c.π.app { as := WalkingPair.left : Discrete WalkingPair })
Tactic: simp only [PullbackCone.mk_π_app]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
F_1 : CategoryTheory.Functor.{v, v', u, u'} C D
F : CategoryTheory.Functor.{0, v, 0, u} (Discrete WalkingPair) C
c : Cone F
X : C
hX : IsTerminal.{v, u} X
f : F.obj { as := WalkingPair.left : Discrete WalkingPair } ⟶ X
g : F.obj { as := WalkingPair.right : Discrete WalkingPair } ⟶ X
hc :
  IsLimit (F := cospan f g)
    (PullbackCone.mk (c.π.app { as := WalkingPair.left : Discrete WalkingPair }) (c.π.app { as := WalkingPair.right })
        ⋯ :
      PullbackCone f g)
s : Cone F
m : s.pt ⟶ c.pt
J : ∀ (j : Discrete WalkingPair), m ≫ c.π.app j = s.π.app j
c' : PullbackCone f g :=
  PullbackCone.mk (m ≫ c.π.app { as := WalkingPair.left : Discrete WalkingPair })
    (m ≫ c.π.app { as := WalkingPair.right }) ⋯
⊢ Eq (α := s.pt ⟶ (cospan f g).obj (some WalkingPair.right))
    (m ≫
      NatTrans.app
        (Cone.π
          (PullbackCone.mk (c.π.app { as := WalkingPair.left : Discrete WalkingPair })
            (c.π.app { as := WalkingPair.right }) ⋯))
        (some WalkingPair.right))
    (CategoryStruct.comp (Y :=
      Cone.pt (F := cospan f g)
        (PullbackCone.mk (c.π.app { as := WalkingPair.left : Discrete WalkingPair })
            (c.π.app { as := WalkingPair.right }) ⋯ :
          PullbackCone f g))
      (hc.lift
        (PullbackCone.mk (m ≫ c.π.app { as := WalkingPair.left : Discrete WalkingPair })
          (m ≫ c.π.app { as := WalkingPair.right }) ⋯))
      (NatTrans.app
        (Cone.π
          (PullbackCone.mk (c.π.app { as := WalkingPair.left : Discrete WalkingPair })
            (c.π.app { as := WalkingPair.right }) ⋯))
        (some WalkingPair.right)))
have some : Eq (α := s.pt ⟶ (cospan f g).obj (some WalkingPair.right)) (m ≫ c.π.app { as := WalkingPair.right })
  (hc.lift
      (PullbackCone.mk (m ≫ c.π.app { as := WalkingPair.left : Discrete WalkingPair })
        (m ≫ c.π.app { as := WalkingPair.right }) sorry) ≫
    c.π.app { as := WalkingPair.right : Discrete WalkingPair })
Tactic: simp only [PullbackCone.mk_π_app]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
F : CategoryTheory.Functor.{v, v', u, u'} C D
W X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
h : W ⟶ X
k : W ⟶ Y
H₁ : IsTerminal.{v, u} Z
H₂ : IsLimit (F := cospan f g) (PullbackCone.mk h k ⋯)
⊢ IsLimit (BinaryFan.mk h k)
have hc : IsLimit (F := cospan f g)
  (PullbackCone.mk ((BinaryFan.mk h k).π.app { as := WalkingPair.left : Discrete WalkingPair })
      ((BinaryFan.mk h k).π.app { as := WalkingPair.right }) sorry :
    PullbackCone f g)
Tactic: apply isBinaryProductOfIsTerminalIsPullback _ _ H₁
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
F : CategoryTheory.Functor.{v, v', u, u'} C D
W X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
h : W ⟶ X
k : W ⟶ Y
H₁ : IsTerminal.{v, u} Z
H₂ : IsLimit (F := cospan f g) (PullbackCone.mk h k ⋯)
hc :
  IsLimit (F := cospan f g)
    (PullbackCone.mk ((BinaryFan.mk h k).π.app { as := WalkingPair.left : Discrete WalkingPair })
        ((BinaryFan.mk h k).π.app { as := WalkingPair.right }) ⋯ :
      PullbackCone f g)
⊢ IsLimit (BinaryFan.mk h k)
have f₁ : Functor.obj.{0, v, 0, u} (pair X Y) { as := WalkingPair.left : Discrete WalkingPair } ⟶ Z
Tactic: apply isBinaryProductOfIsTerminalIsPullback _ _ H₁
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
F : CategoryTheory.Functor.{v, v', u, u'} C D
W X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
h : W ⟶ X
k : W ⟶ Y
H₁ : IsTerminal.{v, u} Z
H₂ : IsLimit (F := cospan f g) (PullbackCone.mk h k ⋯)
hc :
  IsLimit (F := cospan f g)
    (PullbackCone.mk ((BinaryFan.mk h k).π.app { as := WalkingPair.left : Discrete WalkingPair })
        ((BinaryFan.mk h k).π.app { as := WalkingPair.right }) ⋯ :
      PullbackCone f g)
f₁ : Functor.obj.{0, v, 0, u} (pair X Y) { as := WalkingPair.left : Discrete WalkingPair } ⟶ Z
⊢ IsLimit (BinaryFan.mk h k)
have g₁ : Functor.obj.{0, v, 0, u} (pair X Y) { as := WalkingPair.right : Discrete WalkingPair } ⟶ Z
Tactic: apply isBinaryProductOfIsTerminalIsPullback _ _ H₁
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
F : CategoryTheory.Functor.{v, v', u, u'} C D
W X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
h : W ⟶ X
k : W ⟶ Y
H₁ : IsTerminal.{v, u} Z
H₂ : IsLimit (BinaryFan.mk h k)
⊢ IsLimit (F := cospan f g) (PullbackCone.mk h k ⋯)
have create : (s : PullbackCone f g) →
  { l : s.pt ⟶ Cone.pt (F := cospan f g) (PullbackCone.mk h k sorry) //
    l ≫ PullbackCone.fst (PullbackCone.mk h k sorry) = s.fst ∧
      l ≫ PullbackCone.snd (PullbackCone.mk h k sorry) = s.snd ∧
        ∀ {m : s.pt ⟶ Cone.pt (F := cospan f g) (PullbackCone.mk h k sorry)},
          m ≫ PullbackCone.fst (PullbackCone.mk h k sorry) = s.fst →
            m ≫ PullbackCone.snd (PullbackCone.mk h k sorry) = s.snd → m = l }
Tactic: apply PullbackCone.isLimitAux'
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
F : CategoryTheory.Functor.{v, v', u, u'} C D
W X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
h : W ⟶ X
k : W ⟶ Y
H₁ : IsTerminal.{v, u} Z
H₂ : IsLimit (BinaryFan.mk h k)
s : PullbackCone f g
⊢ { l : s.pt ⟶ Cone.pt (F := cospan f g) (PullbackCone.mk h k ⋯) //
    l ≫ PullbackCone.fst (PullbackCone.mk h k ⋯) = s.fst ∧
      l ≫ PullbackCone.snd (PullbackCone.mk h k ⋯) = s.snd ∧
        ∀ {m : s.pt ⟶ Cone.pt (F := cospan f g) (PullbackCone.mk h k ⋯)},
          m ≫ PullbackCone.fst (PullbackCone.mk h k ⋯) = s.fst →
            m ≫ PullbackCone.snd (PullbackCone.mk h k ⋯) = s.snd → m = l }
have property : BinaryFan.IsLimit.lift H₂ s.fst s.snd ≫ PullbackCone.fst (PullbackCone.mk h k sorry) = s.fst ∧
  BinaryFan.IsLimit.lift H₂ s.fst s.snd ≫ PullbackCone.snd (PullbackCone.mk h k sorry) = s.snd ∧
    ∀ {m : s.pt ⟶ Cone.pt (F := cospan f g) (PullbackCone.mk h k sorry)},
      m ≫ PullbackCone.fst (PullbackCone.mk h k sorry) = s.fst →
        m ≫ PullbackCone.snd (PullbackCone.mk h k sorry) = s.snd → m = BinaryFan.IsLimit.lift H₂ s.fst s.snd
Tactic: use BinaryFan.IsLimit.lift H₂ s.fst s.snd
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
F : CategoryTheory.Functor.{v, v', u, u'} C D
W X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
h : W ⟶ X
k : W ⟶ Y
H₁ : IsTerminal.{v, u} Z
H₂ : IsLimit (BinaryFan.mk h k)
s : PullbackCone f g
⊢ BinaryFan.IsLimit.lift H₂ s.fst s.snd ≫ PullbackCone.fst (PullbackCone.mk h k ⋯) = s.fst ∧
    BinaryFan.IsLimit.lift H₂ s.fst s.snd ≫ PullbackCone.snd (PullbackCone.mk h k ⋯) = s.snd ∧
      ∀ {m : s.pt ⟶ Cone.pt (F := cospan f g) (PullbackCone.mk h k ⋯)},
        m ≫ PullbackCone.fst (PullbackCone.mk h k ⋯) = s.fst →
          m ≫ PullbackCone.snd (PullbackCone.mk h k ⋯) = s.snd → m = BinaryFan.IsLimit.lift H₂ s.fst s.snd
have right : BinaryFan.IsLimit.lift H₂ s.fst s.snd ≫ PullbackCone.snd (PullbackCone.mk h k sorry) = s.snd ∧
  ∀ {m : s.pt ⟶ Cone.pt (F := cospan f g) (PullbackCone.mk h k sorry)},
    m ≫ PullbackCone.fst (PullbackCone.mk h k sorry) = s.fst →
      m ≫ PullbackCone.snd (PullbackCone.mk h k sorry) = s.snd → m = BinaryFan.IsLimit.lift H₂ s.fst s.snd
Tactic: use BinaryFan.IsLimit.lift_fst _ _ _
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
F : CategoryTheory.Functor.{v, v', u, u'} C D
W X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
h : W ⟶ X
k : W ⟶ Y
H₁ : IsTerminal.{v, u} Z
H₂ : IsLimit (BinaryFan.mk h k)
s : PullbackCone f g
⊢ BinaryFan.IsLimit.lift H₂ s.fst s.snd ≫ PullbackCone.snd (PullbackCone.mk h k ⋯) = s.snd ∧
    ∀ {m : s.pt ⟶ Cone.pt (F := cospan f g) (PullbackCone.mk h k ⋯)},
      m ≫ PullbackCone.fst (PullbackCone.mk h k ⋯) = s.fst →
        m ≫ PullbackCone.snd (PullbackCone.mk h k ⋯) = s.snd → m = BinaryFan.IsLimit.lift H₂ s.fst s.snd
have right : ∀ {m : s.pt ⟶ Cone.pt (F := cospan f g) (PullbackCone.mk h k sorry)},
  m ≫ PullbackCone.fst (PullbackCone.mk h k sorry) = s.fst →
    m ≫ PullbackCone.snd (PullbackCone.mk h k sorry) = s.snd → m = BinaryFan.IsLimit.lift H₂ s.fst s.snd
Tactic: use BinaryFan.IsLimit.lift_snd _ _ _
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
F : CategoryTheory.Functor.{v, v', u, u'} C D
W X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
h : W ⟶ X
k : W ⟶ Y
H₁ : IsTerminal.{v, u} Z
H₂ : IsLimit (BinaryFan.mk h k)
s : PullbackCone f g
m : s.pt ⟶ Cone.pt (F := cospan f g) (PullbackCone.mk h k ⋯)
h₁ : m ≫ PullbackCone.fst (PullbackCone.mk h k ⋯) = s.fst
h₂ : m ≫ PullbackCone.snd (PullbackCone.mk h k ⋯) = s.snd
⊢ m = BinaryFan.IsLimit.lift H₂ s.fst s.snd
have right : ∀ (j : Discrete WalkingPair),
  m ≫ (BinaryFan.mk h k).π.app j = BinaryFan.IsLimit.lift H₂ s.fst s.snd ≫ (BinaryFan.mk h k).π.app j
Tactic: apply H₂.hom_ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
F : CategoryTheory.Functor.{v, v', u, u'} C D
inst_2 : HasTerminal.{v, u} C
inst_3 : HasPullbacks.{v, u} C
inst_4 : PreservesLimitsOfShape (Discrete PEmpty.{1}) F
inst_5 : PreservesLimitsOfShape WalkingCospan F
K : CategoryTheory.Functor.{0, v, 0, u} (Discrete WalkingPair) C
⊢ IsLimit (F.mapCone (limitConeOfTerminalAndPullbacks K).cone)
have hc : IsLimit (F :=
  cospan (X := (K ⋙ F).obj { as := WalkingPair.left }) (Z := F.obj (terminal.{v, u} C))
    (F.2 (terminal.from (K.obj { as := WalkingPair.left })))
    (F.map (terminal.from (K.obj { as := WalkingPair.right }))))
  (PullbackCone.mk
      ((F.mapCone (limitConeOfTerminalAndPullbacks K).cone).π.app { as := WalkingPair.left : Discrete WalkingPair })
      ((F.mapCone (limitConeOfTerminalAndPullbacks K).cone).π.app { as := WalkingPair.right }) sorry :
    PullbackCone (X := (K ⋙ F).obj { as := WalkingPair.left }) (Z := F.obj (terminal.{v, u} C))
      (F.2 (terminal.from (K.obj { as := WalkingPair.left })))
      (F.map (terminal.from (K.obj { as := WalkingPair.right }))))
Tactic: apply
          isBinaryProductOfIsTerminalIsPullback _ _ (isLimitOfHasTerminalOfPreservesLimit F)
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
F : CategoryTheory.Functor.{v, v', u, u'} C D
inst_2 : HasTerminal.{v, u} C
inst_3 : HasPullbacks.{v, u} C
inst_4 : PreservesLimitsOfShape (Discrete PEmpty.{1}) F
inst_5 : PreservesLimitsOfShape WalkingCospan F
K : CategoryTheory.Functor.{0, v, 0, u} (Discrete WalkingPair) C
hc :
  IsLimit (F :=
    cospan (X := (K ⋙ F).obj { as := WalkingPair.left }) (Z := F.obj (terminal.{v, u} C))
      (F.2 (terminal.from (K.obj { as := WalkingPair.left })))
      (F.map (terminal.from (K.obj { as := WalkingPair.right }))))
    (PullbackCone.mk
        ((F.mapCone (limitConeOfTerminalAndPullbacks K).cone).π.app { as := WalkingPair.left : Discrete WalkingPair })
        ((F.mapCone (limitConeOfTerminalAndPullbacks K).cone).π.app { as := WalkingPair.right }) ⋯ :
      PullbackCone (X := (K ⋙ F).obj { as := WalkingPair.left }) (Z := F.obj (terminal.{v, u} C))
        (F.2 (terminal.from (K.obj { as := WalkingPair.left })))
        (F.map (terminal.from (K.obj { as := WalkingPair.right }))))
⊢ IsLimit (F.mapCone (limitConeOfTerminalAndPullbacks K).cone)
have f : (K ⋙ F).obj { as := WalkingPair.left : Discrete WalkingPair } ⟶ F.obj (terminal.{v, u} C)
Tactic: apply
          isBinaryProductOfIsTerminalIsPullback _ _ (isLimitOfHasTerminalOfPreservesLimit F)
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
F : CategoryTheory.Functor.{v, v', u, u'} C D
inst_2 : HasTerminal.{v, u} C
inst_3 : HasPullbacks.{v, u} C
inst_4 : PreservesLimitsOfShape (Discrete PEmpty.{1}) F
inst_5 : PreservesLimitsOfShape WalkingCospan F
K : CategoryTheory.Functor.{0, v, 0, u} (Discrete WalkingPair) C
hc :
  IsLimit (F :=
    cospan (X := (K ⋙ F).obj { as := WalkingPair.left }) (Z := F.obj (terminal.{v, u} C))
      (F.2 (terminal.from (K.obj { as := WalkingPair.left })))
      (F.map (terminal.from (K.obj { as := WalkingPair.right }))))
    (PullbackCone.mk
        ((F.mapCone (limitConeOfTerminalAndPullbacks K).cone).π.app { as := WalkingPair.left : Discrete WalkingPair })
        ((F.mapCone (limitConeOfTerminalAndPullbacks K).cone).π.app { as := WalkingPair.right }) ⋯ :
      PullbackCone (X := (K ⋙ F).obj { as := WalkingPair.left }) (Z := F.obj (terminal.{v, u} C))
        (F.2 (terminal.from (K.obj { as := WalkingPair.left })))
        (F.map (terminal.from (K.obj { as := WalkingPair.right }))))
f : (K ⋙ F).obj { as := WalkingPair.left : Discrete WalkingPair } ⟶ F.obj (terminal.{v, u} C)
⊢ IsLimit (F.mapCone (limitConeOfTerminalAndPullbacks K).cone)
have g : (K ⋙ F).obj { as := WalkingPair.right : Discrete WalkingPair } ⟶ F.obj (terminal.{v, u} C)
Tactic: apply
          isBinaryProductOfIsTerminalIsPullback _ _ (isLimitOfHasTerminalOfPreservesLimit F)
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
F_1 : CategoryTheory.Functor.{v, v', u, u'} C D
F : CategoryTheory.Functor.{0, v, 0, u} (Discrete WalkingPair) C
c : Cocone F
X : C
hX : IsInitial.{v, u} X
f : X ⟶ F.obj { as := WalkingPair.left }
g : X ⟶ F.obj { as := WalkingPair.right }
hc :
  IsColimit (F := span f g)
    (PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair }) (c.ι.app { as := WalkingPair.right })
        ⋯ :
      PushoutCocone f g)
s : Cocone F
m : c.pt ⟶ s.pt
J : ∀ (j : Discrete WalkingPair), c.ι.app j ≫ m = s.ι.app j
c' : PushoutCocone f g :=
  PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair } ≫ m)
    (c.ι.app { as := WalkingPair.right } ≫ m) ⋯
⊢ m =
    hc.desc
      (PushoutCocone.mk (s.ι.app { as := WalkingPair.left : Discrete WalkingPair })
        (s.ι.app { as := WalkingPair.right }) ⋯)
have rw : m =
  hc.desc
    (PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair } ≫ m)
      (c.ι.app { as := WalkingPair.right } ≫ m) sorry)
Tactic: rw [← J, ← J]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
F_1 : CategoryTheory.Functor.{v, v', u, u'} C D
F : CategoryTheory.Functor.{0, v, 0, u} (Discrete WalkingPair) C
c : Cocone F
X : C
hX : IsInitial.{v, u} X
f : X ⟶ F.obj { as := WalkingPair.left }
g : X ⟶ F.obj { as := WalkingPair.right }
hc :
  IsColimit (F := span f g)
    (PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair }) (c.ι.app { as := WalkingPair.right })
        ⋯ :
      PushoutCocone f g)
s : Cocone F
m : c.pt ⟶ s.pt
J : ∀ (j : Discrete WalkingPair), c.ι.app j ≫ m = s.ι.app j
c' : PushoutCocone f g :=
  PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair } ≫ m)
    (c.ι.app { as := WalkingPair.right } ≫ m) ⋯
⊢ m =
    hc.desc
      (PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair } ≫ m)
        (c.ι.app { as := WalkingPair.right } ≫ m) ⋯)
have apply : ∀ (j : WalkingSpan),
  Eq (α := (span f g).obj j ⟶ s.pt)
    (NatTrans.app
          (Cocone.ι
            (PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair })
              (c.ι.app { as := WalkingPair.right }) sorry))
          j ≫
        m :
      (span f g).obj j ⟶ s.pt)
    (CategoryStruct.comp (Y :=
        Functor.obj.{0, v, 0, u}
          (Functor.obj (Functor.const WalkingSpan)
            (Cocone.pt (F := span f g)
              (PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair })
                  (c.ι.app { as := WalkingPair.right : Discrete WalkingPair }) sorry :
                PushoutCocone f g)))
          j)
        (NatTrans.app
          (Cocone.ι
            (PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair })
              (c.ι.app { as := WalkingPair.right }) sorry))
          j)
        (hc.desc
          (PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair } ≫ m)
            (c.ι.app { as := WalkingPair.right } ≫ m) sorry)) :
      (span f g).obj j ⟶ s.pt)
Tactic: apply hc.hom_ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
F_1 : CategoryTheory.Functor.{v, v', u, u'} C D
F : CategoryTheory.Functor.{0, v, 0, u} (Discrete WalkingPair) C
c : Cocone F
X : C
hX : IsInitial.{v, u} X
f : X ⟶ F.obj { as := WalkingPair.left }
g : X ⟶ F.obj { as := WalkingPair.right }
hc :
  IsColimit (F := span f g)
    (PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair }) (c.ι.app { as := WalkingPair.right })
        ⋯ :
      PushoutCocone f g)
s : Cocone F
m : c.pt ⟶ s.pt
J : ∀ (j : Discrete WalkingPair), c.ι.app j ≫ m = s.ι.app j
c' : PushoutCocone f g :=
  PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair } ≫ m)
    (c.ι.app { as := WalkingPair.right } ≫ m) ⋯
⊢ Eq (α := (span f g).obj none ⟶ s.pt)
    (NatTrans.app
          (Cocone.ι
            (PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair })
              (c.ι.app { as := WalkingPair.right }) ⋯))
          none ≫
        m :
      (span f g).obj none ⟶ s.pt)
    (CategoryStruct.comp (Y :=
        Functor.obj.{0, v, 0, u}
          (Functor.obj (Functor.const WalkingSpan)
            (Cocone.pt (F := span f g)
              (PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair })
                  (c.ι.app { as := WalkingPair.right : Discrete WalkingPair }) ⋯ :
                PushoutCocone f g)))
          none)
        (NatTrans.app
          (Cocone.ι
            (PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair })
              (c.ι.app { as := WalkingPair.right }) ⋯))
          none)
        (hc.desc
          (PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair } ≫ m)
            (c.ι.app { as := WalkingPair.right } ≫ m) ⋯)) :
      (span f g).obj none ⟶ s.pt)
have none : f ≫ c.ι.app { as := WalkingPair.left : Discrete WalkingPair } ≫ m =
  f ≫
    c.ι.app { as := WalkingPair.left : Discrete WalkingPair } ≫
      hc.desc
        (PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair } ≫ m)
          (c.ι.app { as := WalkingPair.right } ≫ m) sorry)
Tactic: simp only [PushoutCocone.mk_ι_app, Category.assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
F_1 : CategoryTheory.Functor.{v, v', u, u'} C D
F : CategoryTheory.Functor.{0, v, 0, u} (Discrete WalkingPair) C
c : Cocone F
X : C
hX : IsInitial.{v, u} X
f : X ⟶ F.obj { as := WalkingPair.left }
g : X ⟶ F.obj { as := WalkingPair.right }
hc :
  IsColimit (F := span f g)
    (PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair }) (c.ι.app { as := WalkingPair.right })
        ⋯ :
      PushoutCocone f g)
s : Cocone F
m : c.pt ⟶ s.pt
J : ∀ (j : Discrete WalkingPair), c.ι.app j ≫ m = s.ι.app j
c' : PushoutCocone f g :=
  PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair } ≫ m)
    (c.ι.app { as := WalkingPair.right } ≫ m) ⋯
⊢ Eq (α := (span f g).obj (some WalkingPair.left) ⟶ s.pt)
    (NatTrans.app
          (Cocone.ι
            (PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair })
              (c.ι.app { as := WalkingPair.right }) ⋯))
          (some WalkingPair.left) ≫
        m :
      (span f g).obj (some WalkingPair.left) ⟶ s.pt)
    (CategoryStruct.comp (Y :=
        Functor.obj.{0, v, 0, u}
          (Functor.obj (Functor.const WalkingSpan)
            (Cocone.pt (F := span f g)
              (PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair })
                  (c.ι.app { as := WalkingPair.right : Discrete WalkingPair }) ⋯ :
                PushoutCocone f g)))
          (some WalkingPair.left))
        (NatTrans.app
          (Cocone.ι
            (PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair })
              (c.ι.app { as := WalkingPair.right }) ⋯))
          (some WalkingPair.left))
        (hc.desc
          (PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair } ≫ m)
            (c.ι.app { as := WalkingPair.right } ≫ m) ⋯)) :
      (span f g).obj (some WalkingPair.left) ⟶ s.pt)
have some : Eq (α := (span f g).obj (some WalkingPair.left) ⟶ s.pt) (c.ι.app { as := WalkingPair.left } ≫ m)
  (c.ι.app { as := WalkingPair.left : Discrete WalkingPair } ≫
    hc.desc
      (PushoutCocone.mk (c.ι.app { as := WalkingPair.left : Discrete WalkingPair } ≫ m)
        (c.ι.app { as := WalkingPair.right } ≫ m) sorry))
Tactic: simp only [PushoutCocone.mk_ι_app, Category.assoc]