Module Z
<: ZAxiomsSig
<: UsualOrderedTypeFull
<: UsualDecidableTypeFull
<: TotalOrder.
(** * Definitions of operations, now in a separate file *)
Include BinIntDef.Z.
Register add as num.Z.add.
Register opp as num.Z.opp.
Register succ as num.Z.succ.
Register pred as num.Z.pred.
Register sub as num.Z.sub.
Register mul as num.Z.mul.
Register of_nat as num.Z.of_nat.
(** When including property functors, only inline t eq zero one two *)
Set Inline Level 30.
(** * Logic Predicates *)
Definition eq := @Logic.eq Z.
Definition eq_equiv := @eq_equivalence Z.
Definition lt x y := (x ?= y) = Lt.
Definition gt x y := (x ?= y) = Gt.
Definition le x y := (x ?= y) <> Gt.
Definition ge x y := (x ?= y) <> Lt.
Infix "<=" := le : Z_scope.
Infix "<" := lt : Z_scope.
Infix ">=" := ge : Z_scope.
Infix ">" := gt : Z_scope.
Notation "x <= y <= z" := (x <= y /\ y <= z) : Z_scope.
Notation "x <= y < z" := (x <= y /\ y < z) : Z_scope.
Notation "x < y < z" := (x < y /\ y < z) : Z_scope.
Notation "x < y <= z" := (x < y /\ y <= z) : Z_scope.
Definition divide x y := exists z, y = z*x.
Notation "( x | y )" := (divide x y) (at level 0).
Definition Even a := exists b, a = 2*b.
Definition Odd a := exists b, a = 2*b+1.
Register le as num.Z.le.
Register lt as num.Z.lt.
Register ge as num.Z.ge.
Register gt as num.Z.gt.
(** * Decidability of equality. *)
Definition eq_dec (x y : Z) : {x = y} + {x <> y}.
(** Particular cases of the previous result *)
(** The opposite of [pos_sub] is [pos_sub] with reversed arguments *)

Lemma pos_sub_opp p q : - pos_sub p q = pos_sub q p.