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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import BinInt Zcompare Zorder Zbool ZArith_dec
Equalities Orders OrdersTac.
Local Open Scope Z_scope.
(** * DecidableType structure for binary integers *)
Module Z_as_UBE <: UsualBoolEq.
Definition t := Z.
Definition eq := @eq Z.
Definition eqb := Zeq_bool.
Definition eqb_eq x y := iff_sym (Zeq_is_eq_bool x y).
End Z_as_UBE.
Module Z_as_DT <: UsualDecidableTypeFull := Make_UDTF Z_as_UBE.
(** Note that the last module fulfills by subtyping many other
interfaces, such as [DecidableType] or [EqualityType]. *)
(** * OrderedType structure for binary integers *)
Module Z_as_OT <: OrderedTypeFull.
Include Z_as_DT.
Definition lt := Zlt.
Definition le := Zle.
Definition compare := Zcompare.
Instance lt_strorder : StrictOrder Zlt.
Proof. split; [ exact Zlt_irrefl | exact Zlt_trans ]. Qed.
Instance lt_compat : Proper (Logic.eq==>Logic.eq==>iff) Zlt.
Proof. repeat red; intros; subst; auto. Qed.
Definition le_lteq := Zle_lt_or_eq_iff.
Definition compare_spec := Zcompare_spec.
End Z_as_OT.
(** Note that [Z_as_OT] can also be seen as a [UsualOrderedType]
and a [OrderedType] (and also as a [DecidableType]). *)
(** * An [order] tactic for integers *)
Module ZOrder := OTF_to_OrderTac Z_as_OT.
Ltac z_order := ZOrder.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]. *)
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