Definition If (T : Set) (b : bool) (t e : T) :=
   (* If T b t e  =  if b then t else e *)
  match b with
  | true => t
  | false => e
  end.

Lemma lt_pow_lt : forall i j : Z, 0 <= i -> 0 <= j -> 2 ^ i < 2 ^ j -> i < j.