have h2 : m = m * b / b := by
  exact Nat.eq_div_of_mul_eq_left (not_eq_zero_of_lt hb) rfl
case succ
b n : ℕ
hb : 1 < b
k : ℕ
ih :
  ∀ {m : ℕ}, 0 < m → b.digits n ++ List.replicate k 0 ++ b.digits m = b.digits (n + b ^ (k + (b.digits n).length) * m)
m : ℕ
hm : 0 < m
hmb : 0 < m * b
h1 : b.digits (m * b) = m * b % b :: b.digits (m * b / b) := digits_def' hb hmb
⊢ b.digits n ++ (List.replicate k 0 ++ 0 :: b.digits m) = b.digits (n + b ^ (k + 1 + (b.digits n).length) * m)
case succ
b n : ℕ
hb : 1 < b
k : ℕ
ih :
  ∀ {m : ℕ}, 0 < m → b.digits n ++ List.replicate k 0 ++ b.digits m = b.digits (n + b ^ (k + (b.digits n).length) * m)
m : ℕ
hm : 0 < m
hmb : 0 < m * b
h1 : b.digits (m * b) = m * b % b :: b.digits (m * b / b) := digits_def' hb hmb
h2 : m = m * b / b
⊢ b.digits n ++ (List.replicate k 0 ++ 0 :: b.digits m) = b.digits (n + b ^ (k + 1 + (b.digits n).length) * m)