(** Alternative definitions of [<=] and [<] based on [+] *)
Definition lt_alt n m := exists p, S p + n == m.

(* Specifications: *)
Lemma le_equiv : forall n m, le_alt n m <-> n <= m.
Lemma lt_equiv : forall n m, lt_alt n m <-> n < m.
#[global]
Instance le_alt_wd : Proper (eq==>eq==>iff) le_alt.