(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)
Local Open Scope nat_scope.
Implicit Types m n x y : nat.
Definition zerop n : {n = 0} + {0 < n}.
Register dec_le as num.nat.dec_le.
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).
(** Some projections of the above equivalences. *)
(** A previous definition of [nat_compare] in terms of [lt_eq_lt_dec]. The new version avoids the creation of proof parts. *)
Definition nat_compare_alt (n m:nat) :=
match lt_eq_lt_dec n m with
| inleft (left _) => Lt
| inleft (right _) => Eq
| inright _ => Gt
end.
(** A boolean version of [le] over [nat]. See now [Nat.leb] and its properties. In scope [nat_scope], the notation for [Nat.leb] is "<=?" *)
Notation leb := Nat.leb (only parsing).
Notation leb_iff := Nat.leb_le (only parsing).

Lemma leb_complete m n : (m <=? n) = true -> m <= n.