awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts (goal)
Goal:
J : Type v
inst : SmallCategory J
C : Type u
inst_1 : Category.{v, u} C
D : Type u'
inst_2 : Category.{v', u'} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Fan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryFan.{v, u} (f 0) c₁.pt
t₁ : IsLimit c₁
t₂ : IsLimit c₂
s : Cone.{0, v, 0, u} (Discrete.functor f)
x : Discrete (Fin (n + 1))
j : Fin (n + 1)
⊢ (↑(BinaryFan.IsLimit.lift' t₂ (s.π.app { as := 0 : Discrete (Fin (n + 1)) })
            (t₁.lift
              { pt := s.pt,
                π :=
                  Discrete.natTrans fun (x : Discrete (Fin n)) ↦
                    match (motive :=
                      (x : Discrete (Fin n)) →
                        Functor.obj.{0, v, 0, u} (Functor.obj (Functor.const (Discrete (Fin n))) s.pt) x ⟶
                          Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) x)
                      x with
                    | { as := i } => s.π.app { as := i.succ : Discrete (Fin (n + 1)) } :
                Cone.{0, v, 0, u} (J := Discrete (Fin n)) (C := C) (Discrete.functor fun (i : Fin n) ↦ f i.succ) })) :
        ((Functor.const (Discrete (Fin (n + 1)))).obj s.pt).obj { as := 0 } ⟶ c₂.pt) ≫
      (extendFan c₁ c₂).π.app { as := j } =
    s.π.app { as := j : Discrete (Fin (n + 1)) }
have refine_1 : (↑(BinaryFan.IsLimit.lift' t₂ (s.π.app { as := 0 : Discrete (Fin (n + 1)) })
          (t₁.lift
            { pt := s.pt,
              π :=
                Discrete.natTrans fun (x : Discrete (Fin n)) ↦
                  match (motive :=
                    (x : Discrete (Fin n)) →
                      Functor.obj.{0, v, 0, u} (Functor.obj (Functor.const (Discrete (Fin n))) s.pt) x ⟶
                        Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) x)
                    x with
                  | { as := i } => s.π.app { as := i.succ : Discrete (Fin (n + 1)) } :
              Cone.{0, v, 0, u} (J := Discrete (Fin n)) (C := C) (Discrete.functor fun (i : Fin n) ↦ f i.succ) })) :
      ((Functor.const (Discrete (Fin (n + 1)))).obj s.pt).obj { as := 0 } ⟶ c₂.pt) ≫
    (extendFan c₁ c₂).π.app { as := 0 } =
  s.π.app { as := 0 : Discrete (Fin (n + 1)) }
Tactic: refine Fin.inductionOn j ?_ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts (goal)
Goal:
J : Type v
inst : SmallCategory J
C : Type u
inst_1 : Category.{v, u} C
D : Type u'
inst_2 : Category.{v', u'} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Fan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryFan.{v, u} (f 0) c₁.pt
t₁ : IsLimit c₁
t₂ : IsLimit c₂
s : Cone.{0, v, 0, u} (Discrete.functor f)
x : Discrete (Fin (n + 1))
j : Fin (n + 1)
i : Fin n
⊢ (↑(BinaryFan.IsLimit.lift' t₂ (s.π.app { as := 0 : Discrete (Fin (n + 1)) })
            (t₁.lift
              { pt := s.pt,
                π :=
                  Discrete.natTrans fun (x : Discrete (Fin n)) ↦
                    match (motive :=
                      (x : Discrete (Fin n)) →
                        Functor.obj.{0, v, 0, u} (Functor.obj (Functor.const (Discrete (Fin n))) s.pt) x ⟶
                          Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) x)
                      x with
                    | { as := i } => s.π.app { as := i.succ : Discrete (Fin (n + 1)) } :
                Cone.{0, v, 0, u} (J := Discrete (Fin n)) (C := C) (Discrete.functor fun (i : Fin n) ↦ f i.succ) })) :
        ((Functor.const (Discrete (Fin (n + 1)))).obj s.pt).obj { as := 0 } ⟶ c₂.pt) ≫
      (extendFan c₁ c₂).π.app { as := i.succ } =
    s.π.app { as := i.succ : Discrete (Fin (n + 1)) }
have refine_2 : (↑(BinaryFan.IsLimit.lift' t₂ (s.π.app { as := 0 : Discrete (Fin (n + 1)) })
          (t₁.lift
            { pt := s.pt,
              π := Discrete.natTrans fun (x : Discrete (Fin n)) ↦ s.π.app { as := x.as.succ : Discrete (Fin (n + 1)) } :
              Cone.{0, v, 0, u} (J := Discrete (Fin n)) (C := C) (Discrete.functor fun (i : Fin n) ↦ f i.succ) })) :
      ((Functor.const (Discrete (Fin (n + 1)))).obj s.pt).obj { as := 0 } ⟶ c₂.pt) ≫
    Fin.cases (motive := fun (x : Fin (n + 1)) ↦ c₂.pt ⟶ f x) c₂.fst (fun (i : Fin n) ↦ c₂.snd ≫ c₁.π.app { as := i })
      i.succ =
  s.π.app { as := i.succ : Discrete (Fin (n + 1)) }
Tactic: dsimp only [extendFan_π_app]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts (goal)
Goal:
J : Type v
inst : SmallCategory J
C : Type u
inst_1 : Category.{v, u} C
D : Type u'
inst_2 : Category.{v', u'} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Fan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryFan.{v, u} (f 0) c₁.pt
t₁ : IsLimit c₁
t₂ : IsLimit c₂
s : Cone.{0, v, 0, u} (Discrete.functor f)
x : Discrete (Fin (n + 1))
j : Fin (n + 1)
i : Fin n
⊢ (↑(BinaryFan.IsLimit.lift' t₂ (s.π.app { as := 0 : Discrete (Fin (n + 1)) })
            (t₁.lift
              { pt := s.pt,
                π :=
                  Discrete.natTrans fun (x : Discrete (Fin n)) ↦ s.π.app { as := x.as.succ : Discrete (Fin (n + 1)) } :
                Cone.{0, v, 0, u} (J := Discrete (Fin n)) (C := C) (Discrete.functor fun (i : Fin n) ↦ f i.succ) })) :
        ((Functor.const (Discrete (Fin (n + 1)))).obj s.pt).obj { as := 0 } ⟶ c₂.pt) ≫
      Fin.cases (motive := fun (x : Fin (n + 1)) ↦ c₂.pt ⟶ f x) c₂.fst (fun (i : Fin n) ↦ c₂.snd ≫ c₁.π.app { as := i })
        i.succ =
    s.π.app { as := i.succ : Discrete (Fin (n + 1)) }
have refine_2 : Eq.{v + 1} (α := s.pt ⟶ Functor.obj.{0, v, 0, u} (Discrete.functor f) { as := i.succ : Discrete (Fin (n + 1)) })
  (NatTrans.app (Cone.π { pt := s.pt, π := Discrete.natTrans fun (x : Discrete (Fin n)) ↦ s.π.app { as := x.as.succ } })
      { as := i } :
    Functor.obj.{0, v, 0, u}
        (Functor.obj (Functor.const (Discrete (Fin n)))
          (Cone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
            { pt := s.pt,
              π := Discrete.natTrans fun (x : Discrete (Fin n)) ↦ s.π.app { as := x.as.succ : Discrete (Fin (n + 1)) } :
              Cone.{0, v, 0, u} (J := Discrete (Fin n)) (C := C) (Discrete.functor fun (i : Fin n) ↦ f i.succ) }))
        { as := i : Discrete (Fin n) } ⟶
      Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) { as := i : Discrete (Fin n) })
  (s.π.app { as := i.succ })
Tactic: rw [Fin.cases_succ, ← assoc, (BinaryFan.IsLimit.lift' t₂ _ _).2.2, t₁.fac]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts (goal)
Goal:
J : Type v
inst : SmallCategory J
C : Type u
inst_1 : Category.{v, u} C
D : Type u'
inst_2 : Category.{v', u'} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Fan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryFan.{v, u} (f 0) c₁.pt
t₁ : IsLimit c₁
t₂ : IsLimit c₂
s : Cone.{0, v, 0, u} (Discrete.functor f)
m : s.pt ⟶ (extendFan c₁ c₂).pt
w : ∀ (j : Discrete (Fin (n + 1))), m ≫ (extendFan c₁ c₂).π.app j = s.π.app j
⊢ m =
    (↑(BinaryFan.IsLimit.lift' t₂ (s.π.app { as := 0 })
          (t₁.lift
            { pt := s.pt,
              π :=
                Discrete.natTrans fun (x : Discrete (Fin n)) ↦
                  match (motive :=
                    (x : Discrete (Fin n)) →
                      Functor.obj.{0, v, 0, u} (Functor.obj (Functor.const (Discrete (Fin n))) s.pt) x ⟶
                        Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) x)
                    x with
                  | { as := i } => s.π.app { as := i.succ } })) :
      ((Functor.const (Discrete (Fin (n + 1)))).obj s.pt).obj { as := 0 } ⟶ c₂.pt)
have h₁ : m ≫ c₂.fst =
  (↑(BinaryFan.IsLimit.lift' t₂ (s.π.app { as := 0 : Discrete (Fin (n + 1)) })
          (t₁.lift
            { pt := s.pt,
              π :=
                Discrete.natTrans fun (x : Discrete (Fin n)) ↦
                  match (motive :=
                    (x : Discrete (Fin n)) →
                      Functor.obj.{0, v, 0, u} (Functor.obj (Functor.const (Discrete (Fin n))) s.pt) x ⟶
                        Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) x)
                    x with
                  | { as := i } => s.π.app { as := i.succ : Discrete (Fin (n + 1)) } :
              Cone.{0, v, 0, u} (J := Discrete (Fin n)) (C := C) (Discrete.functor fun (i : Fin n) ↦ f i.succ) })) :
      ((Functor.const (Discrete (Fin (n + 1)))).obj s.pt).obj { as := 0 } ⟶ c₂.pt) ≫
    c₂.fst
Tactic: apply BinaryFan.IsLimit.hom_ext t₂
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts (goal)
Goal:
J : Type v
inst : SmallCategory J
C : Type u
inst_1 : Category.{v, u} C
D : Type u'
inst_2 : Category.{v', u'} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Fan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryFan.{v, u} (f 0) c₁.pt
t₁ : IsLimit c₁
t₂ : IsLimit c₂
s : Cone.{0, v, 0, u} (Discrete.functor f)
m : s.pt ⟶ (extendFan c₁ c₂).pt
w : ∀ (j : Discrete (Fin (n + 1))), m ≫ (extendFan c₁ c₂).π.app j = s.π.app j
h₁ :
  m ≫ c₂.fst =
    (↑(BinaryFan.IsLimit.lift' t₂ (s.π.app { as := 0 : Discrete (Fin (n + 1)) })
            (t₁.lift
              { pt := s.pt,
                π :=
                  Discrete.natTrans fun (x : Discrete (Fin n)) ↦
                    match (motive :=
                      (x : Discrete (Fin n)) →
                        Functor.obj.{0, v, 0, u} (Functor.obj (Functor.const (Discrete (Fin n))) s.pt) x ⟶
                          Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) x)
                      x with
                    | { as := i } => s.π.app { as := i.succ : Discrete (Fin (n + 1)) } :
                Cone.{0, v, 0, u} (J := Discrete (Fin n)) (C := C) (Discrete.functor fun (i : Fin n) ↦ f i.succ) })) :
        ((Functor.const (Discrete (Fin (n + 1)))).obj s.pt).obj { as := 0 } ⟶ c₂.pt) ≫
      c₂.fst
⊢ m =
    (↑(BinaryFan.IsLimit.lift' t₂ (s.π.app { as := 0 })
          (t₁.lift
            { pt := s.pt,
              π :=
                Discrete.natTrans fun (x : Discrete (Fin n)) ↦
                  match (motive :=
                    (x : Discrete (Fin n)) →
                      Functor.obj.{0, v, 0, u} (Functor.obj (Functor.const (Discrete (Fin n))) s.pt) x ⟶
                        Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) x)
                    x with
                  | { as := i } => s.π.app { as := i.succ } })) :
      ((Functor.const (Discrete (Fin (n + 1)))).obj s.pt).obj { as := 0 } ⟶ c₂.pt)
have h₂ : m ≫ c₂.snd =
  (↑(BinaryFan.IsLimit.lift' t₂ (s.π.app { as := 0 : Discrete (Fin (n + 1)) })
          (t₁.lift
            { pt := s.pt,
              π :=
                Discrete.natTrans fun (x : Discrete (Fin n)) ↦
                  match (motive :=
                    (x : Discrete (Fin n)) →
                      Functor.obj.{0, v, 0, u} (Functor.obj (Functor.const (Discrete (Fin n))) s.pt) x ⟶
                        Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) x)
                    x with
                  | { as := i } => s.π.app { as := i.succ : Discrete (Fin (n + 1)) } :
              Cone.{0, v, 0, u} (J := Discrete (Fin n)) (C := C) (Discrete.functor fun (i : Fin n) ↦ f i.succ) })) :
      ((Functor.const (Discrete (Fin (n + 1)))).obj s.pt).obj { as := 0 } ⟶ c₂.pt) ≫
    c₂.snd
Tactic: apply BinaryFan.IsLimit.hom_ext t₂
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts (goal)
Goal:
J : Type v
inst : SmallCategory J
C : Type u
inst_1 : Category.{v, u} C
D : Type u'
inst_2 : Category.{v', u'} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Fan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryFan.{v, u} (f 0) c₁.pt
t₁ : IsLimit c₁
t₂ : IsLimit c₂
s : Cone.{0, v, 0, u} (Discrete.functor f)
m : s.pt ⟶ (extendFan c₁ c₂).pt
w : ∀ (j : Discrete (Fin (n + 1))), m ≫ (extendFan c₁ c₂).π.app j = s.π.app j
⊢ m ≫ c₂.fst =
    (↑(BinaryFan.IsLimit.lift' t₂ (s.π.app { as := 0 : Discrete (Fin (n + 1)) })
            (t₁.lift
              { pt := s.pt,
                π :=
                  Discrete.natTrans fun (x : Discrete (Fin n)) ↦
                    match (motive :=
                      (x : Discrete (Fin n)) →
                        Functor.obj.{0, v, 0, u} (Functor.obj (Functor.const (Discrete (Fin n))) s.pt) x ⟶
                          Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) x)
                      x with
                    | { as := i } => s.π.app { as := i.succ : Discrete (Fin (n + 1)) } :
                Cone.{0, v, 0, u} (J := Discrete (Fin n)) (C := C) (Discrete.functor fun (i : Fin n) ↦ f i.succ) })) :
        ((Functor.const (Discrete (Fin (n + 1)))).obj s.pt).obj { as := 0 } ⟶ c₂.pt) ≫
      c₂.fst
have h₁ : m ≫ c₂.fst = s.π.app { as := 0 }
Tactic: rw [(BinaryFan.IsLimit.lift' t₂ _ _).2.1]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts (goal)
Goal:
J : Type v
inst : SmallCategory J
C : Type u
inst_1 : Category.{v, u} C
D : Type u'
inst_2 : Category.{v', u'} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Fan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryFan.{v, u} (f 0) c₁.pt
t₁ : IsLimit c₁
t₂ : IsLimit c₂
s : Cone.{0, v, 0, u} (Discrete.functor f)
m : s.pt ⟶ (extendFan c₁ c₂).pt
w : ∀ (j : Discrete (Fin (n + 1))), m ≫ (extendFan c₁ c₂).π.app j = s.π.app j
⊢ m ≫ c₂.snd =
    (↑(BinaryFan.IsLimit.lift' t₂ (s.π.app { as := 0 : Discrete (Fin (n + 1)) })
            (t₁.lift
              { pt := s.pt,
                π :=
                  Discrete.natTrans fun (x : Discrete (Fin n)) ↦
                    match (motive :=
                      (x : Discrete (Fin n)) →
                        Functor.obj.{0, v, 0, u} (Functor.obj (Functor.const (Discrete (Fin n))) s.pt) x ⟶
                          Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) x)
                      x with
                    | { as := i } => s.π.app { as := i.succ : Discrete (Fin (n + 1)) } :
                Cone.{0, v, 0, u} (J := Discrete (Fin n)) (C := C) (Discrete.functor fun (i : Fin n) ↦ f i.succ) })) :
        ((Functor.const (Discrete (Fin (n + 1)))).obj s.pt).obj { as := 0 } ⟶ c₂.pt) ≫
      c₂.snd
have h₂ : m ≫ c₂.snd =
  t₁.lift
    { pt := s.pt,
      π :=
        Discrete.natTrans fun (x : Discrete (Fin n)) ↦
          match (motive :=
            (x : Discrete (Fin n)) →
              Functor.obj.{0, v, 0, u} (Functor.obj (Functor.const (Discrete (Fin n))) s.pt) x ⟶
                Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) x)
            x with
          | { as := i } => s.π.app { as := i.succ } }
Tactic: rw [(BinaryFan.IsLimit.lift' t₂ _ _).2.2]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts (goal)
Goal:
J : Type v
inst : SmallCategory J
C : Type u
inst_1 : Category.{v, u} C
D : Type u'
inst_2 : Category.{v', u'} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Fan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryFan.{v, u} (f 0) c₁.pt
t₁ : IsLimit c₁
t₂ : IsLimit c₂
s : Cone.{0, v, 0, u} (Discrete.functor f)
m : s.pt ⟶ (extendFan c₁ c₂).pt
w : ∀ (j : Discrete (Fin (n + 1))), m ≫ (extendFan c₁ c₂).π.app j = s.π.app j
⊢ m ≫ c₂.snd =
    t₁.lift
      { pt := s.pt,
        π :=
          Discrete.natTrans fun (x : Discrete (Fin n)) ↦
            match (motive :=
              (x : Discrete (Fin n)) →
                Functor.obj.{0, v, 0, u} (Functor.obj (Functor.const (Discrete (Fin n))) s.pt) x ⟶
                  Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) x)
              x with
            | { as := i } => s.π.app { as := i.succ } }
have h₂ : ∀ (j : Discrete (Fin n)),
  (m ≫ c₂.snd) ≫ c₁.π.app j =
    NatTrans.app
      (Cone.π
        { pt := s.1,
          π :=
            Discrete.natTrans fun (x : Discrete (Fin n)) ↦
              match (motive :=
                (x : Discrete (Fin n)) →
                  Functor.obj.{0, v, 0, u} (Functor.obj (Functor.const (Discrete (Fin n))) s.pt) x ⟶
                    Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) x)
                x with
              | { as := i } => s.π.app { as := i.succ } })
      j
Tactic: apply t₁.uniq ⟨_, _⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts (goal)
Goal:
J : Type v
inst : SmallCategory J
C : Type u
inst_1 : Category.{v, u} C
D : Type u'
inst_2 : Category.{v', u'} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Fan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryFan.{v, u} (f 0) c₁.pt
t₁ : IsLimit c₁
t₂ : IsLimit c₂
s : Cone.{0, v, 0, u} (Discrete.functor f)
m : s.pt ⟶ (extendFan c₁ c₂).pt
w : ∀ (j : Discrete (Fin (n + 1))), m ≫ (extendFan c₁ c₂).π.app j = s.π.app j
j : Fin n
⊢ Eq (α :=
    Quiver.Hom.{v, u} (V := C)
      (Cone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
        { pt := s.1,
          π :=
            Discrete.natTrans fun (x : Discrete (Fin n)) ↦
              match (motive :=
                (x : Discrete (Fin n)) →
                  Functor.obj.{0, v, 0, u} (Functor.obj (Functor.const (Discrete (Fin n))) s.pt) x ⟶
                    Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) x)
                x with
              | { as := i } => s.π.app { as := i.succ } })
      (Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) { as := j : Discrete (Fin n) }))
    ((m ≫ c₂.snd) ≫ c₁.π.app { as := j })
    (NatTrans.app
      (Cone.π
        { pt := s.1,
          π :=
            Discrete.natTrans fun (x : Discrete (Fin n)) ↦
              match (motive :=
                (x : Discrete (Fin n)) →
                  Functor.obj.{0, v, 0, u} (Functor.obj (Functor.const (Discrete (Fin n))) s.pt) x ⟶
                    Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) x)
                x with
              | { as := i } => s.π.app { as := i.succ } })
      { as := j })
have h₂ : Eq (α :=
  Quiver.Hom.{v, u} (V := C)
    (Cone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
      { pt := s.1,
        π :=
          Discrete.natTrans fun (x : Discrete (Fin n)) ↦
            match (motive :=
              (x : Discrete (Fin n)) →
                Functor.obj.{0, v, 0, u} (Functor.obj (Functor.const (Discrete (Fin n))) s.pt) x ⟶
                  Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) x)
              x with
            | { as := i } => s.π.app { as := i.succ } })
    (Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) { as := j : Discrete (Fin n) }))
  (m ≫ c₂.snd ≫ c₁.π.app { as := j })
  (NatTrans.app
    (Cone.π
      { pt := s.1,
        π :=
          Discrete.natTrans fun (x : Discrete (Fin n)) ↦
            match (motive :=
              (x : Discrete (Fin n)) →
                Functor.obj.{0, v, 0, u} (Functor.obj (Functor.const (Discrete (Fin n))) s.pt) x ⟶
                  Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) x)
              x with
            | { as := i } => s.π.app { as := i.succ } })
    { as := j })
Tactic: rw [assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts (goal)
Goal:
J : Type v
inst : SmallCategory J
C : Type u
inst_1 : Category.{v, u} C
D : Type u'
inst_2 : Category.{v', u'} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Fan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryFan.{v, u} (f 0) c₁.pt
t₁ : IsLimit c₁
t₂ : IsLimit c₂
s : Cone.{0, v, 0, u} (Discrete.functor f)
m : s.pt ⟶ (extendFan c₁ c₂).pt
w : ∀ (j : Discrete (Fin (n + 1))), m ≫ (extendFan c₁ c₂).π.app j = s.π.app j
j : Fin n
⊢ Eq (α :=
    Quiver.Hom.{v, u} (V := C)
      (Cone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
        { pt := s.1,
          π :=
            Discrete.natTrans fun (x : Discrete (Fin n)) ↦
              match (motive :=
                (x : Discrete (Fin n)) →
                  Functor.obj.{0, v, 0, u} (Functor.obj (Functor.const (Discrete (Fin n))) s.pt) x ⟶
                    Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) x)
                x with
              | { as := i } => s.π.app { as := i.succ } })
      (Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) { as := j : Discrete (Fin n) }))
    (m ≫ c₂.snd ≫ c₁.π.app { as := j })
    (NatTrans.app
      (Cone.π
        { pt := s.1,
          π :=
            Discrete.natTrans fun (x : Discrete (Fin n)) ↦
              match (motive :=
                (x : Discrete (Fin n)) →
                  Functor.obj.{0, v, 0, u} (Functor.obj (Functor.const (Discrete (Fin n))) s.pt) x ⟶
                    Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) x)
                x with
              | { as := i } => s.π.app { as := i.succ } })
      { as := j })
have h₂ : Eq (α :=
  Quiver.Hom.{v, u} (V := C) s.1
    (Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) { as := j : Discrete (Fin n) }))
  (m ≫ c₂.snd ≫ c₁.π.app { as := j }) (s.π.app { as := j.succ })
Tactic: dsimp only [Discrete.natTrans_app]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts (goal)
Goal:
J : Type v
inst : SmallCategory J
C : Type u
inst_1 : Category.{v, u} C
D : Type u'
inst_2 : Category.{v', u'} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Fan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryFan.{v, u} (f 0) c₁.pt
t₁ : IsLimit c₁
t₂ : IsLimit c₂
s : Cone.{0, v, 0, u} (Discrete.functor f)
m : s.pt ⟶ (extendFan c₁ c₂).pt
w : ∀ (j : Discrete (Fin (n + 1))), m ≫ (extendFan c₁ c₂).π.app j = s.π.app j
j : Fin n
⊢ Eq (α :=
    Quiver.Hom.{v, u} (V := C) s.1
      (Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) { as := j : Discrete (Fin n) }))
    (m ≫ c₂.snd ≫ c₁.π.app { as := j }) (s.π.app { as := j.succ })
have h₂ : Eq (α :=
  Quiver.Hom.{v, u} (V := C) s.1
    (Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) { as := j : Discrete (Fin n) }))
  (m ≫ c₂.snd ≫ c₁.π.app { as := j }) (m ≫ (extendFan c₁ c₂).π.app { as := j.succ })
Tactic: rw [← w ⟨j.succ⟩]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts (goal)
Goal:
J : Type v
inst : SmallCategory J
C : Type u
inst_1 : Category.{v, u} C
D : Type u'
inst_2 : Category.{v', u'} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Fan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryFan.{v, u} (f 0) c₁.pt
t₁ : IsLimit c₁
t₂ : IsLimit c₂
s : Cone.{0, v, 0, u} (Discrete.functor f)
m : s.pt ⟶ (extendFan c₁ c₂).pt
w : ∀ (j : Discrete (Fin (n + 1))), m ≫ (extendFan c₁ c₂).π.app j = s.π.app j
j : Fin n
⊢ Eq (α :=
    Quiver.Hom.{v, u} (V := C) s.1
      (Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) { as := j : Discrete (Fin n) }))
    (m ≫ c₂.snd ≫ c₁.π.app { as := j }) (m ≫ (extendFan c₁ c₂).π.app { as := j.succ })
have h₂ : Eq (α :=
  Quiver.Hom.{v, u} (V := C) s.1
    (Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) { as := j : Discrete (Fin n) }))
  (m ≫ c₂.snd ≫ c₁.π.app { as := j })
  (m ≫
    Fin.cases (motive := fun (x : Fin (n + 1)) ↦ c₂.pt ⟶ f x) c₂.fst (fun (i : Fin n) ↦ c₂.snd ≫ c₁.π.app { as := i })
      j.succ)
Tactic: dsimp only [extendFan_π_app]