Register dec_lt as num.nat.dec_lt.
Register dec_ge as num.nat.dec_ge.
Register dec_gt as num.nat.dec_gt.
Register not_eq as num.nat.not_eq.
Register not_le as num.nat.not_le.
Register not_lt as num.nat.not_lt.
Register not_ge as num.nat.not_ge.
Register not_gt as num.nat.not_gt.
(** A ternary comparison function in the spirit of [Z.compare]. See now [Nat.compare] and its properties. In scope [nat_scope], the notation for [Nat.compare] is "?=" *)
Notation nat_compare_S := Nat.compare_succ (only parsing).

Lemma nat_compare_le n m : n<=m <-> (n ?= m) <> Gt.