induction' n with pn hpn
n : ℕ
⊢ ∃ k, (Nat.digits 10 (2 ^ n.succ * ↑k)).length = n.succ ∧ all_one_or_two (Nat.digits 10 (2 ^ n.succ * ↑k))
case zero

⊢ ∃ k, (Nat.digits 10 (2 ^ Nat.succ 0 * ↑k)).length = Nat.succ 0 ∧ all_one_or_two (Nat.digits 10 (2 ^ Nat.succ 0 * ↑k))

case succ
pn : ℕ
hpn : ∃ k, (Nat.digits 10 (2 ^ pn.succ * ↑k)).length = pn.succ ∧ all_one_or_two (Nat.digits 10 (2 ^ pn.succ * ↑k))
⊢ ∃ k,
    (Nat.digits 10 (2 ^ (pn + 1).succ * ↑k)).length = (pn + 1).succ ∧
      all_one_or_two (Nat.digits 10 (2 ^ (pn + 1).succ * ↑k))