cases mod_two_eq_zero_or_one (Nat.succ x) with
| inl even =>
  have h : 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 :=
  And.intro (zero_lt_succ x) even;
  show (if 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 then
  let vbn_half := val_bin_nat (Nat.succ x / 2);
  (vbn_half.fst + 1, vbn_half.snd)
  else if 0 < Nat.succ x then
  (0, Nat.succ x)
  else
  (0, 0)).snd ≤ Nat.succ x
  rw [if_pos h]
  have h' := ih (Nat.succ x / 2) (div2_succ_le_self x);
  exact le_trans h' (le_succ x)
| inr odd =>
  show (if 0 < Nat.succ x ∧ Nat.succ x % 2 = 0 then
  let vbn_half := val_bin_nat (Nat.succ x / 2);
  (vbn_half.fst + 1, vbn_half.snd)
  else if 0 < Nat.succ x then
  (0, Nat.succ x)
  else
  (0, 0)).snd ≤ Nat.succ x
  have not_even : Nat.succ x % 2 ≠ 0 := by
    rw [odd]
    simp;
  rw [if_neg (not_and_of_not_right (0 < Nat.succ x) not_even)]
  rw [if_pos (zero_lt_succ x)]
case succ.refl
x : ℕ
ih : ∀ (y : ℕ), y ≤ x → (val_bin_nat y).snd ≤ x
⊢ (val_bin_nat (Nat.succ x)).snd ≤ Nat.succ x
no goals