(** ** Addition is associative *)

(* Specifications: *)
Theorem add_assoc p q r : p + (q + r) = p + q + r.
(** ** Commutation of addition and double *)
Lemma add_xO p q : (p + q)~0 = p~0 + q~0.
Lemma add_xI_pred_double p q :
(p + q)~0 = p~1 + pred_double q.
Lemma add_xO_pred_double p q :
pred_double (p + q) = p~0 + pred_double q.