simp [hm, hn]
f : Bool → Bool → Bool
k : Nat
h_ind : ∀ {m n : Nat}, m < 2 ^ k → n < 2 ^ k → bitwise f m n < 2 ^ k
m n : Nat
hmk : m < 2 ^ succ k
hnk : n < 2 ^ succ k
hm : ¬m = 0
hn : ¬n = 0
⊢ (if m = 0 then if f false true = true then n else 0
    else
      if n = 0 then if f true false = true then m else 0
      else
        let n' := m / 2;
        let m' := n / 2;
        let b₁ := m % 2 = 1;
        let b₂ := n % 2 = 1;
        let r := bitwise f n' m';
        if f (decide b₁) (decide b₂) = true then r + r + 1 else r + r) <
    2 ^ succ k
f : Bool → Bool → Bool
k : Nat
h_ind : ∀ {m n : Nat}, m < 2 ^ k → n < 2 ^ k → bitwise f m n < 2 ^ k
m n : Nat
hmk : m < 2 ^ succ k
hnk : n < 2 ^ succ k
hm : ¬m = 0
hn : ¬n = 0
⊢ (if f (decide (m % 2 = 1)) (decide (n % 2 = 1)) = true then bitwise f (m / 2) (n / 2) + bitwise f (m / 2) (n / 2) + 1
    else bitwise f (m / 2) (n / 2) + bitwise f (m / 2) (n / 2)) <
    2 ^ succ k