induction' y with y ih generalizing z
α : Type ?u.25506
σ : Stream' α
y z x : ℕ
⊢ (fun x_1 => x_1 + x)^[y] z = x * y + z
case zero
α : Type ?u.25506
σ : Stream' α
z✝ x z : ℕ
⊢ (fun x_1 => x_1 + x)^[Nat.zero] z = x * Nat.zero + z

case succ
α : Type ?u.25506
σ : Stream' α
z✝ x y : ℕ
ih : ∀ {z : ℕ}, (fun x_1 => x_1 + x)^[y] z = x * y + z
z : ℕ
⊢ (fun x_1 => x_1 + x)^[Nat.succ y] z = x * Nat.succ y + z