case neg hm =>
  by_cases n = 0
  case pos hn =>
    cases hn
    unfold bitwise; simp [hm]
    cases f true false <;> simp
    . exact Nat.pos_pow_of_pos k.succ (Nat.zero_lt_succ 1)
    . exact hmk
  case neg hn =>
    unfold bitwise; simp [hm, hn]
    rw [←Nat.mul_two, Nat.pow_succ]
    have : m/2 < 2^k := (Nat.div_lt_iff_lt_mul (Nat.zero_lt_succ 1)).mpr hmk
    have : n/2 < 2^k := (Nat.div_lt_iff_lt_mul (Nat.zero_lt_succ 1)).mpr hnk
    cases f (m%2=1) (n%2=1) <;> simp
    . apply Nat.mul_lt_mul_of_pos_right ?_ (Nat.zero_lt_succ 1)
      exact h_ind ‹_› ‹_›
    . calc
      bitwise f (m/2) (n/2) * 2 + 1 
        < bitwise f (m/2) (n/2) * 2 + 2 := Nat.lt_succ_self _
      _ = (bitwise f (m/2) (n/2)) * 2 + 1*2 := rfl
      _ = (bitwise f (m/2) (n/2) + 1) * 2 := (Nat.add_mul _ _ 2).symm
      _ ≤ 2^k * 2 := Nat.mul_le_mul_right 2 (h_ind ‹m/2 < 2^k› ‹n/2 < 2^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
⊢ bitwise f m n < 2 ^ succ k
no goals