induction t <;> simp[bind, he]
L : Language
μ : Type v
μ₂ : Type u_1
μ₁ : Type u_2
b₁ b₂ : ℕ → UTerm L μ₂
e₁ e₂ : μ₁ → UTerm L μ₂
t : UTerm L μ₁
hb : ∀ x < t.bv, b₁ x = b₂ x
he : ∀ (x : μ₁), e₁ x = e₂ x
⊢ bind b₁ e₁ t = bind b₂ e₂ t
case bvar
L : Language
μ : Type v
μ₂ : Type u_1
μ₁ : Type u_2
b₁ b₂ : ℕ → UTerm L μ₂
e₁ e₂ : μ₁ → UTerm L μ₂
he : ∀ (x : μ₁), e₁ x = e₂ x
a✝ : ℕ
hb : ∀ x < (bvar a✝).bv, b₁ x = b₂ x
⊢ b₁ a✝ = b₂ a✝

case func
L : Language
μ : Type v
μ₂ : Type u_1
μ₁ : Type u_2
b₁ b₂ : ℕ → UTerm L μ₂
e₁ e₂ : μ₁ → UTerm L μ₂
he : ∀ (x : μ₁), e₁ x = e₂ x
arity✝ : ℕ
a✝¹ : L.Func arity✝
a✝ : Fin arity✝ → UTerm L μ₁
a_ih✝ : ∀ (a : Fin arity✝), (∀ x < (a✝ a).bv, b₁ x = b₂ x) → bind b₁ e₁ (a✝ a) = bind b₂ e₂ (a✝ a)
hb : ∀ x < (func a✝¹ a✝).bv, b₁ x = b₂ x
⊢ (fun i => bind b₁ e₁ (a✝ i)) = fun i => bind b₂ e₂ (a✝ i)