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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \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)         *)
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(*                      Evgeny Makarov, INRIA, 2007                     *)
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Attributes deprecated(since="8.20", note="Use Coq.ZArith.BinInt instead.").
Require Export Ring.
Declare Scope pair_scope.
Local Open Scope pair_scope.
Notation "s #1" := (fst s) (at level 9, format "s '#1'") : pair_scope.
Notation "s #2" := (snd s) (at level 9, format "s '#2'") : pair_scope.
Module ZPairsAxiomsMod (Import N : NAxiomsMiniSig) <: ZAxiomsMiniSig.
Module Import NProp.
Include NSubProp N.
End NProp.
Declare Scope NScope.
Delimit Scope NScope with N.
Bind Scope NScope with N.t.
Infix "=="  := N.eq (at level 70) : NScope.
Notation "x ~= y" := (~ N.eq x y) (at level 70) : NScope.
Notation "0" := N.zero : NScope.
Notation "1" := N.one : NScope.
Notation "2" := N.two : NScope.
Infix "+" := N.add : NScope.
Infix "-" := N.sub : NScope.
Infix "*" := N.mul : NScope.
Infix "<" := N.lt : NScope.
Infix "<=" := N.le : NScope.
Local Open Scope NScope.
(** The definitions of functions ([add], [mul], etc.) will be unfolded by the properties functor. Since we don't want [add_comm] to refer to unfolded definitions of equality: [fun p1 p2 => (fst p1 + snd p2) = (fst p2 + snd p1)], we will provide an extra layer of definitions. *)
Module Z.
Definition t := (N.t * N.t)%type.
Definition zero : t := (0, 0).
Definition one : t := (1,0).
Definition two : t := (2,0).
Definition eq (p q : t) := (p#1 + q#2 == q#1 + p#2).
Definition succ (n : t) : t := (N.succ n#1, n#2).
Definition pred (n : t) : t := (n#1, N.succ n#2).
Definition opp (n : t) : t := (n#2, n#1).
Definition add (n m : t) : t := (n#1 + m#1, n#2 + m#2).
Definition sub (n m : t) : t := (n#1 + m#2, n#2 + m#1).
Definition mul (n m : t) : t :=
(n#1 * m#1 + n#2 * m#2, n#1 * m#2 + n#2 * m#1).
Definition lt (n m : t) := n#1 + m#2 < m#1 + n#2.
Definition le (n m : t) := n#1 + m#2 <= m#1 + n#2.
Definition min (n m : t) : t := (min (n#1 + m#2) (m#1 + n#2), n#2 + m#2).
Definition max (n m : t) : t := (max (n#1 + m#2) (m#1 + n#2), n#2 + m#2).
(** NB : We do not have [Z.pred (Z.succ n) = n] but only [Z.pred (Z.succ n) == n]. It could be possible to consider as canonical only pairs where one of the elements is 0, and make all operations convert canonical values into other canonical values. In that case, we could get rid of setoids and arrive at integers as signed natural numbers. *)
(** NB : Unfortunately, the elements of the pair keep increasing during many operations, even during subtraction. *)
End Z.
Declare Scope ZScope.
Delimit Scope ZScope with Z.
Bind Scope ZScope with Z.t.
Infix "=="  := Z.eq (at level 70) : ZScope.
Notation "x ~= y" := (~ Z.eq x y) (at level 70) : ZScope.
Notation "0" := Z.zero : ZScope.
Notation "1" := Z.one : ZScope.
Notation "2" := Z.two : ZScope.
Infix "+" := Z.add : ZScope.
Infix "-" := Z.sub : ZScope.
Infix "*" := Z.mul : ZScope.
Notation "- x" := (Z.opp x) : ZScope.
Infix "<" := Z.lt : ZScope.
Infix "<=" := Z.le : ZScope.
Local Open Scope ZScope.

Lemma sub_add_opp : forall n m, Z.sub n m = Z.add n (Z.opp m).
#[global]
Instance pair_wd : Proper (N.eq==>N.eq==>Z.eq) (@pair N.t N.t).