(** A few helper functions used during proofs *)

Definition hd d :=
  match d with
  | Nil => 0
  | D0 _ => 0
  | D1 _ => 1
  | D2 _ => 2
  | D3 _ => 3
  | D4 _ => 4
  | D5 _ => 5
  | D6 _ => 6
  | D7 _ => 7
  | D8 _ => 8
  | D9 _ => 9
 end.Definition tl d :=
  match d with
  | Nil => d
  | D0 d | D1 d | D2 d | D3 d | D4 d | D5 d | D6 d | D7 d | D8 d | D9 d => d
  end.Fixpoint usize (d:uint) : nat :=
  match d with
  | Nil => 0
  | D0 d => S (usize d)
  | D1 d => S (usize d)
  | D2 d => S (usize d)
  | D3 d => S (usize d)
  | D4 d => S (usize d)
  | D5 d => S (usize d)
  | D6 d => S (usize d)
  | D7 d => S (usize d)
  | D8 d => S (usize d)
  | D9 d => S (usize d)
  end.(** A direct version of [to_little_uint], not tail-recursive *)
Fixpoint to_lu n :=
  match n with
  | 0 => Decimal.zero
  | S n => Little.succ (to_lu n)
  end.(** A direct version of [of_little_uint] *)
Fixpoint of_lu (d:uint) : nat :=
  match d with
  | Nil => 0
  | D0 d => 10 * of_lu d
  | D1 d => 1 + 10 * of_lu d
  | D2 d => 2 + 10 * of_lu d
  | D3 d => 3 + 10 * of_lu d
  | D4 d => 4 + 10 * of_lu d
  | D5 d => 5 + 10 * of_lu d
  | D6 d => 6 + 10 * of_lu d
  | D7 d => 7 + 10 * of_lu d
  | D8 d => 8 + 10 * of_lu d
  | D9 d => 9 + 10 * of_lu d
  end.

Lemma to_little_uint_succ n d : Nat.to_little_uint n (Little.succ d) = Little.succ (Nat.to_little_uint n d).