#[global]
Instance lt_alt_wd : Proper (eq==>eq==>iff) lt_alt.
(** With these alternative definition, the dichotomy:  [forall n m, n <= m \/ m <= n]  becomes:  [forall n m, (exists p, p + n == m) \/ (exists p, p + m == n)]  We will need this in the proof of induction principle for integers constructed as pairs of natural numbers. This formula can be proved from know properties of [<=]. However, it can also be done directly. *)

(* Specifications: *)
Theorem le_alt_dichotomy : forall n m, le_alt n m \/ le_alt m n.
Theorem add_dichotomy :
forall n m, (exists p, p + n == m) \/ (exists p, p + m == n).