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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* Evgeny Makarov, INRIA, 2007 *)
(************************************************************************)
Require Import ZAxioms ZProperties BinInt.
Local Open Scope Z_scope.
(** BinInt.Z is already implementing [ZAxiomsMiniSig] *)
Module Z
<: ZAxiomsSig <: UsualOrderedTypeFull <: TotalOrder
<: UsualDecidableTypeFull
:= BinInt.Z.
(** * An [order] tactic for integers *)
Ltac z_order := Z.order.
(** Note that [z_order] is domain-agnostic: it will not prove
[1<=2] or [x<=x+x], but rather things like [x<=y -> y<=x -> x=y]. *)
Section TestOrder.
Let test : forall x y, x<=y -> y<=x -> x=y.
Proof.
z_order.
Qed.
End TestOrder.
(** Z forms a ring *)
(*Lemma Zring : ring_theory 0 1 NZadd NZmul NZsub Z.opp NZeq.
Proof.
constructor.
exact Zadd_0_l.
exact Zadd_comm.
exact Zadd_assoc.
exact Zmul_1_l.
exact Zmul_comm.
exact Zmul_assoc.
exact Zmul_add_distr_r.
intros; now rewrite Zadd_opp_minus.
exact Zadd_opp_r.
Qed.
Add Ring ZR : Zring.*)
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