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(* -*- coq-prog-args: ("-emacs-U" "-nois") -*- *)
(************************************************************************)
(* 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 *)
(************************************************************************)
(* Certified Haskell Prelude.
* Author: Matthieu Sozeau
* Institution: LRI, CNRS UMR 8623 - Universit�copyright Paris Sud
* 91405 Orsay, France *)
(* $Id: FSetAVL_prog.v 616 2007-08-08 12:28:10Z msozeau $ *)
Set Implicit Arguments.
Unset Strict Implicit.
Require Import Coq.Classes.SetoidClass.
Class [ Setoid A R ] => EqDec :=
equiv_dec : forall x y : A, { x == y } + { x =!= y }.
Infix "==" := equiv_dec (no associativity, at level 70).
Require Import Coq.Program.Program.
Program Definition nequiv_dec [ EqDec A R ] (x y : A) : { x =!= y } + { x == y } :=
if x == y then right else left.
Infix "<>" := nequiv_dec (no associativity, at level 70).
Definition equiv_decb [ EqDec A R ] (x y : A) : bool :=
if x == y then true else false.
Definition nequiv_decb [ EqDec A R ] (x y : A) : bool :=
negb (equiv_decb x y).
Infix "==b" := equiv_decb (no associativity, at level 70).
Infix "<>b" := nequiv_decb (no associativity, at level 70).
(** Decidable leibniz equality instances. *)
Implicit Arguments eq [[A]].
Require Import Coq.Arith.Arith.
Program Instance nat_eqdec : EqDec nat eq :=
equiv_dec := eq_nat_dec.
Require Import Coq.Bool.Bool.
Program Instance bool_eqdec : EqDec bool eq :=
equiv_dec := bool_dec.
Program Instance unit_eqdec : EqDec unit eq :=
equiv_dec x y := left.
Next Obligation.
Proof.
destruct x ; destruct y.
reflexivity.
Qed.
Program Instance [ EqDec A eq, EqDec B eq ] => prod_eqdec : EqDec (prod A B) eq :=
equiv_dec x y :=
dest x as (x1, x2) in
dest y as (y1, y2) in
if x1 == y1 then
if x2 == y2 then left
else right
else right.
Solve Obligations using unfold equiv ; program_simpl ; try red ; intros ; autoinjections ; discriminates.
Program Instance [ EqDec A eq ] => bool_function_eqdec : EqDec (bool -> A) eq :=
equiv_dec f g :=
if f true == g true then
if f false == g false then left
else right
else right.
Solve Obligations using unfold equiv ; program_simpl.
Require Import Coq.Program.FunctionalExtensionality.
Next Obligation.
Proof.
unfold equiv.
extensionality x.
destruct x ; auto.
Qed.
Next Obligation.
Proof.
unfold equiv in *.
red ; intro ; subst.
discriminates.
Qed.
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