(** ** Bitwise operations *)

Definition double_S : forall n, double (S n) = S (S (double n))
                    := fun n => add_succ_r (S n) n.Definition double_add : forall n m, double (n + m) = double n + double m
                      := fun n m => add_shuffle1 n m n m.

Lemma odd_bitwise : forall op n a b, odd (bitwise op (S n) a b) = op (odd a) (odd b).