induction' k with k ih
b k n : ℕ
hb : 1 < b
⊢ ∀ {m : ℕ}, 0 < m → b.digits n ++ List.replicate k 0 ++ b.digits m = b.digits (n + b ^ (k + (b.digits n).length) * m)
case zero
b n : ℕ
hb : 1 < b
⊢ ∀ {m : ℕ}, 0 < m → b.digits n ++ List.replicate 0 0 ++ b.digits m = b.digits (n + b ^ (0 + (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},
    0 < m → b.digits n ++ List.replicate (k + 1) 0 ++ b.digits m = b.digits (n + b ^ (k + 1 + (b.digits n).length) * m)