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(* -*- coding: utf-8 -*- *)
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** * Decidable setoid equality theory.
Author: Matthieu Sozeau
Institution: LRI, CNRS UMR 8623 - University Paris Sud
*)
Set Implicit Arguments.
Unset Strict Implicit.
Generalizable Variables A B .
(** Export notations. *)
Require Export Coq.Classes.SetoidClass.
(** The [DecidableSetoid] class asserts decidability of a [Setoid].
It can be useful in proofs to reason more classically. *)
Require Import Coq.Logic.Decidable.
Class DecidableSetoid `(S : Setoid A) :=
setoid_decidable : forall x y : A, decidable (x == y).
(** The [EqDec] class gives a decision procedure for a particular setoid
equality. *)
Class EqDec `(S : Setoid A) :=
equiv_dec : forall x y : A, { x == y } + { x =/= y }.
(** We define the [==] overloaded notation for deciding equality. It does not
take precedence of [==] defined in the type scope, hence we can have both
at the same time. *)
Notation " x == y " := (equiv_dec (x :>) (y :>)) (no associativity, at level 70).
Definition swap_sumbool {A B} (x : { A } + { B }) : { B } + { A } :=
match x with
| left H => @right _ _ H
| right H => @left _ _ H
end.
Require Import Coq.Program.Program.
Local Open Scope program_scope.
(** Invert the branches. *)
Program Definition nequiv_dec `{EqDec A} (x y : A) : { x =/= y } + { x == y } := swap_sumbool (x == y).
(** Overloaded notation for inequality. *)
Infix "=/=" := nequiv_dec (no associativity, at level 70).
(** Define boolean versions, losing the logical information. *)
Definition equiv_decb `{EqDec A} (x y : A) : bool :=
if x == y then true else false.
Definition nequiv_decb `{EqDec A} (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. *)
Require Import Coq.Arith.Arith.
(** The equiv is burried inside the setoid, but we can recover
it by specifying which setoid we're talking about. *)
Program Instance eq_setoid A : Setoid A | 10 :=
{ equiv := eq ; setoid_equiv := eq_equivalence }.
Program Instance nat_eq_eqdec : EqDec (eq_setoid nat) :=
eq_nat_dec.
Require Import Coq.Bool.Bool.
Program Instance bool_eqdec : EqDec (eq_setoid bool) :=
bool_dec.
Program Instance unit_eqdec : EqDec (eq_setoid unit) :=
fun x y => in_left.
Next Obligation.
Proof.
destruct x ; destruct y.
reflexivity.
Qed.
Program Instance prod_eqdec `(! EqDec (eq_setoid A), ! EqDec (eq_setoid B))
: EqDec (eq_setoid (prod A B)) :=
fun x y =>
let '(x1, x2) := x in
let '(y1, y2) := y in
if x1 == y1 then
if x2 == y2 then in_left
else in_right
else in_right.
Solve Obligations with unfold complement ; program_simpl.
(** Objects of function spaces with countable domains like bool
have decidable equality. *)
Program Instance bool_function_eqdec `(! EqDec (eq_setoid A))
: EqDec (eq_setoid (bool -> A)) :=
fun f g =>
if f true == g true then
if f false == g false then in_left
else in_right
else in_right.
Solve Obligations with try red ; unfold complement ; program_simpl.
Next Obligation.
Proof.
extensionality x.
destruct x ; auto.
Qed.
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