(* Uniqueleaf is decidable *)
Definition distinct_leaves_dec :
  forall a, {distinct_leaves a} + {~ distinct_leaves a}.(* Compute the code associated with a binary tree *)
Fixpoint compute_code (a : btree) : list (A * list bool) :=
  match a with
  | leaf b => (b, nil) :: nil
  | node l1 l2 =>
      map
        (fun v : A * list bool =>
         match v with
         | (a1, b1) => (a1, false :: b1)
         end) (compute_code l1) ++
      map
        (fun v : A * list bool =>
         match v with
         | (a1, b1) => (a1, true :: b1)
         end) (compute_code l2)
  end.

Theorem btree_unique_prefix1 : forall (t : btree) (a1 a2 : A) (lb1 lb2 : list bool), In (a1, lb1) (compute_code t) -> In (a2, lb2) (compute_code t) -> is_prefix lb1 lb2 -> a1 = a2.