(** 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_Lt_lt n m : (n ?= m) = Lt -> n<m.