-- Mathlib/Data/Set/UnionLift.lean
theorem iUnionLift_binary (dir : Directed (· ≤ ·) S) (op : T → T → T) (opi : ∀ i, S i → S i → S i)
    (hopi :
      ∀ i x y,
        Set.inclusion (show S i ⊆ T from hT'.symm ▸ Set.subset_iUnion S i) (opi i x y) =
          op (Set.inclusion (show S i ⊆ T from hT'.symm ▸ Set.subset_iUnion S i) x)
            (Set.inclusion (show S i ⊆ T from hT'.symm ▸ Set.subset_iUnion S i) y))
    (opβ : β → β → β) (h : ∀ (i) (x y : S i), f i (opi i x y) = opβ (f i x) (f i y)) (x y : T) :
    iUnionLift S f hf T (le_of_eq hT') (op x y) =
      opβ (iUnionLift S f hf T (le_of_eq hT') x) (iUnionLift S f hf T (le_of_eq hT') y)

Proof state:
α : Type u_1
ι : Sort u_2
β : Sort u_3
S : ι → Set α
f : (i : ι) → ↑(S i) → β
hf : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩
T : Set α
hT' : T = iUnion S
dir : Directed (fun x1 x2 => x1 ≤ x2) S
op : ↑T → ↑T → ↑T
opi : (i : ι) → ↑(S i) → ↑(S i) → ↑(S i)
hopi : ∀ (i : ι) (x y : ↑(S i)), inclusion ⋯ (opi i x y) = op (inclusion ⋯ x) (inclusion ⋯ y)
opβ : β → β → β
h : ∀ (i : ι) (x y : ↑(S i)), f i (opi i x y) = opβ (f i x) (f i y)
x y : ↑T
⊢ iUnionLift S f hf T ⋯ (op x y) = opβ (iUnionLift S f hf T ⋯ x) (iUnionLift S f hf T ⋯ y)

Tactic:
subst hT'

Result:
α : Type u_1
ι : Sort u_2
β : Sort u_3
S : ι → Set α
f : (i : ι) → ↑(S i) → β
hf : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩
dir : Directed (fun x1 x2 => x1 ≤ x2) S
opi : (i : ι) → ↑(S i) → ↑(S i) → ↑(S i)
opβ : β → β → β
h : ∀ (i : ι) (x y : ↑(S i)), f i (opi i x y) = opβ (f i x) (f i y)
op : ↑(iUnion S) → ↑(iUnion S) → ↑(iUnion S)
hopi : ∀ (i : ι) (x y : ↑(S i)), inclusion ⋯ (opi i x y) = op (inclusion ⋯ x) (inclusion ⋯ y)
x y : ↑(iUnion S)
⊢ iUnionLift S f hf (iUnion S) ⋯ (op x y) = opβ (iUnionLift S f hf (iUnion S) ⋯ x) (iUnionLift S f hf (iUnion S) ⋯ y)

Proof state:
case intro.intro.intro.intro
α : Type u_1
ι : Sort u_2
β : Sort u_3
S : ι → Set α
f : (i : ι) → ↑(S i) → β
hf : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩
dir : Directed (fun x1 x2 => x1 ≤ x2) S
opi : (i : ι) → ↑(S i) → ↑(S i) → ↑(S i)
opβ : β → β → β
h : ∀ (i : ι) (x y : ↑(S i)), f i (opi i x y) = opβ (f i x) (f i y)
op : ↑(iUnion S) → ↑(iUnion S) → ↑(iUnion S)
hopi : ∀ (i : ι) (x y : ↑(S i)), inclusion ⋯ (opi i x y) = op (inclusion ⋯ x) (inclusion ⋯ y)
x y : ↑(iUnion S)
i : ι
hi : ↑x ∈ S i
j : ι
hj : ↑y ∈ S j
k : ι
hik : S i ≤ S k
hjk : S j ≤ S k
⊢ iUnionLift S f hf (iUnion S) ⋯ (op x y) = f k (opi k ⟨↑x, ⋯⟩ ⟨↑y, ⋯⟩)

Tactic:
have hx : x = Set.inclusion (Set.subset_iUnion S k) ⟨x, hik hi⟩ := by
  cases x
  rfl

Result:
case intro.intro.intro.intro
α : Type u_1
ι : Sort u_2
β : Sort u_3
S : ι → Set α
f : (i : ι) → ↑(S i) → β
hf : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩
dir : Directed (fun x1 x2 => x1 ≤ x2) S
opi : (i : ι) → ↑(S i) → ↑(S i) → ↑(S i)
opβ : β → β → β
h : ∀ (i : ι) (x y : ↑(S i)), f i (opi i x y) = opβ (f i x) (f i y)
op : ↑(iUnion S) → ↑(iUnion S) → ↑(iUnion S)
hopi : ∀ (i : ι) (x y : ↑(S i)), inclusion ⋯ (opi i x y) = op (inclusion ⋯ x) (inclusion ⋯ y)
x y : ↑(iUnion S)
i : ι
hi : ↑x ∈ S i
j : ι
hj : ↑y ∈ S j
k : ι
hik : S i ≤ S k
hjk : S j ≤ S k
hx : x = inclusion ⋯ ⟨↑x, ⋯⟩
⊢ iUnionLift S f hf (iUnion S) ⋯ (op x y) = f k (opi k ⟨↑x, ⋯⟩ ⟨↑y, ⋯⟩)

Proof state:
case intro
α : Type u_1
ι : Sort u_2
β : Sort u_3
S : ι → Set α
f : (i : ι) → ↑(S i) → β
hf : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩
dir : Directed (fun x1 x2 => x1 ≤ x2) S
opi : (i : ι) → ↑(S i) → ↑(S i) → ↑(S i)
opβ : β → β → β
h : ∀ (i : ι) (x y : ↑(S i)), f i (opi i x y) = opβ (f i x) (f i y)
op : ↑(iUnion S) → ↑(iUnion S) → ↑(iUnion S)
hopi : ∀ (i : ι) (x y : ↑(S i)), inclusion ⋯ (opi i x y) = op (inclusion ⋯ x) (inclusion ⋯ y)
x y : ↑(iUnion S)
i : ι
hi : ↑x ∈ S i
⊢ iUnionLift S f hf (iUnion S) ⋯ (op x y) = opβ (iUnionLift S f hf (iUnion S) ⋯ x) (iUnionLift S f hf (iUnion S) ⋯ y)

Tactic:
obtain ⟨j, hj⟩ := Set.mem_iUnion.1 y.prop

Result:
case intro.intro
α : Type u_1
ι : Sort u_2
β : Sort u_3
S : ι → Set α
f : (i : ι) → ↑(S i) → β
hf : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩
dir : Directed (fun x1 x2 => x1 ≤ x2) S
opi : (i : ι) → ↑(S i) → ↑(S i) → ↑(S i)
opβ : β → β → β
h : ∀ (i : ι) (x y : ↑(S i)), f i (opi i x y) = opβ (f i x) (f i y)
op : ↑(iUnion S) → ↑(iUnion S) → ↑(iUnion S)
hopi : ∀ (i : ι) (x y : ↑(S i)), inclusion ⋯ (opi i x y) = op (inclusion ⋯ x) (inclusion ⋯ y)
x y : ↑(iUnion S)
i : ι
hi : ↑x ∈ S i
j : ι
hj : ↑y ∈ S j
⊢ iUnionLift S f hf (iUnion S) ⋯ (op x y) = opβ (iUnionLift S f hf (iUnion S) ⋯ x) (iUnionLift S f hf (iUnion S) ⋯ y)

Proof state:
case intro.intro.intro.intro
α : Type u_1
ι : Sort u_2
β : Sort u_3
S : ι → Set α
f : (i : ι) → ↑(S i) → β
hf : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩
dir : Directed (fun x1 x2 => x1 ≤ x2) S
opi : (i : ι) → ↑(S i) → ↑(S i) → ↑(S i)
opβ : β → β → β
h : ∀ (i : ι) (x y : ↑(S i)), f i (opi i x y) = opβ (f i x) (f i y)
op : ↑(iUnion S) → ↑(iUnion S) → ↑(iUnion S)
hopi : ∀ (i : ι) (x y : ↑(S i)), inclusion ⋯ (opi i x y) = op (inclusion ⋯ x) (inclusion ⋯ y)
x y : ↑(iUnion S)
i : ι
hi : ↑x ∈ S i
j : ι
hj : ↑y ∈ S j
k : ι
hik : S i ≤ S k
hjk : S j ≤ S k
hx : x = inclusion ⋯ ⟨↑x, ⋯⟩
hy : y = inclusion ⋯ ⟨↑y, ⋯⟩
⊢ iUnionLift S f hf (iUnion S) ⋯ (op x y) = f k (opi k ⟨↑x, ⋯⟩ ⟨↑y, ⋯⟩)

Tactic:
have hxy : (Set.inclusion (Set.subset_iUnion S k) (opi k ⟨x, hik hi⟩ ⟨y, hjk hj⟩) : α) ∈ S k :=
  (opi k ⟨x, hik hi⟩ ⟨y, hjk hj⟩).prop

Result:
case intro.intro.intro.intro
α : Type u_1
ι : Sort u_2
β : Sort u_3
S : ι → Set α
f : (i : ι) → ↑(S i) → β
hf : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩
dir : Directed (fun x1 x2 => x1 ≤ x2) S
opi : (i : ι) → ↑(S i) → ↑(S i) → ↑(S i)
opβ : β → β → β
h : ∀ (i : ι) (x y : ↑(S i)), f i (opi i x y) = opβ (f i x) (f i y)
op : ↑(iUnion S) → ↑(iUnion S) → ↑(iUnion S)
hopi : ∀ (i : ι) (x y : ↑(S i)), inclusion ⋯ (opi i x y) = op (inclusion ⋯ x) (inclusion ⋯ y)
x y : ↑(iUnion S)
i : ι
hi : ↑x ∈ S i
j : ι
hj : ↑y ∈ S j
k : ι
hik : S i ≤ S k
hjk : S j ≤ S k
hx : x = inclusion ⋯ ⟨↑x, ⋯⟩
hy : y = inclusion ⋯ ⟨↑y, ⋯⟩
hxy : ↑(inclusion ⋯ (opi k ⟨↑x, ⋯⟩ ⟨↑y, ⋯⟩)) ∈ S k
⊢ iUnionLift S f hf (iUnion S) ⋯ (op x y) = f k (opi k ⟨↑x, ⋯⟩ ⟨↑y, ⋯⟩)

Proof state:
α : Type u_1
ι : Sort u_2
β : Sort u_3
S : ι → Set α
f : (i : ι) → ↑(S i) → β
hf : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩
dir : Directed (fun x1 x2 => x1 ≤ x2) S
opi : (i : ι) → ↑(S i) → ↑(S i) → ↑(S i)
opβ : β → β → β
h : ∀ (i : ι) (x y : ↑(S i)), f i (opi i x y) = opβ (f i x) (f i y)
op : ↑(iUnion S) → ↑(iUnion S) → ↑(iUnion S)
hopi : ∀ (i : ι) (x y : ↑(S i)), inclusion ⋯ (opi i x y) = op (inclusion ⋯ x) (inclusion ⋯ y)
x y : ↑(iUnion S)
⊢ iUnionLift S f hf (iUnion S) ⋯ (op x y) = opβ (iUnionLift S f hf (iUnion S) ⋯ x) (iUnionLift S f hf (iUnion S) ⋯ y)

Tactic:
obtain ⟨i, hi⟩ := Set.mem_iUnion.1 x.prop

Result:
case intro
α : Type u_1
ι : Sort u_2
β : Sort u_3
S : ι → Set α
f : (i : ι) → ↑(S i) → β
hf : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩
dir : Directed (fun x1 x2 => x1 ≤ x2) S
opi : (i : ι) → ↑(S i) → ↑(S i) → ↑(S i)
opβ : β → β → β
h : ∀ (i : ι) (x y : ↑(S i)), f i (opi i x y) = opβ (f i x) (f i y)
op : ↑(iUnion S) → ↑(iUnion S) → ↑(iUnion S)
hopi : ∀ (i : ι) (x y : ↑(S i)), inclusion ⋯ (opi i x y) = op (inclusion ⋯ x) (inclusion ⋯ y)
x y : ↑(iUnion S)
i : ι
hi : ↑x ∈ S i
⊢ iUnionLift S f hf (iUnion S) ⋯ (op x y) = opβ (iUnionLift S f hf (iUnion S) ⋯ x) (iUnionLift S f hf (iUnion S) ⋯ y)