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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 *)
(************************************************************************)
(* Decidable equivalences.
*
* Author: Matthieu Sozeau
* Institution: LRI, CNRS UMR 8623 - UniversitÃcopyright Paris Sud
* 91405 Orsay, France *)
(* $Id$ *)
(** Export notations. *)
Require Export Coq.Classes.Equivalence.
(** The [DecidableSetoid] class asserts decidability of a [Setoid]. It can be useful in proofs to reason more
classically. *)
Require Import Coq.Logic.Decidable.
Require Import Coq.Bool.Bool.
Require Import Coq.Arith.Peano_dec.
Require Import Coq.Program.Program.
Open Scope equiv_scope.
Class DecidableEquivalence `(equiv : Equivalence A) :=
setoid_decidable : forall x y : A, decidable (x === y).
(** The [EqDec] class gives a decision procedure for a particular setoid equality. *)
Class EqDec A R {equiv : Equivalence R} :=
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) : equiv_scope.
Definition swap_sumbool {A B} (x : { A } + { B }) : { B } + { A } :=
match x with
| left H => @right _ _ H
| right H => @left _ _ H
end.
Open Local 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) : equiv_scope.
(** 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. *)
(** The equiv is burried inside the setoid, but we can recover it by specifying which setoid we're talking about. *)
Program Instance nat_eq_eqdec : EqDec nat eq := eq_nat_dec.
Program Instance bool_eqdec : EqDec bool eq := bool_dec.
Program Instance unit_eqdec : EqDec unit eq := λ x y, in_left.
Next Obligation.
Proof.
destruct x ; destruct y.
reflexivity.
Qed.
Obligation Tactic := unfold complement, equiv ; program_simpl.
Program Instance prod_eqdec `(EqDec A eq, EqDec B eq) :
! EqDec (prod A B) eq :=
{ equiv_dec 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 }.
Program Instance sum_eqdec `(EqDec A eq, EqDec B eq) :
EqDec (sum A B) eq := {
equiv_dec x y :=
match x, y with
| inl a, inl b => if a == b then in_left else in_right
| inr a, inr b => if a == b then in_left else in_right
| inl _, inr _ | inr _, inl _ => in_right
end }.
(** Objects of function spaces with countable domains like bool have decidable equality.
Proving the reflection requires functional extensionality though. *)
Program Instance bool_function_eqdec `(EqDec A eq) : ! EqDec (bool -> A) eq :=
{ equiv_dec f g :=
if f true == g true then
if f false == g false then in_left
else in_right
else in_right }.
Next Obligation.
Proof.
extensionality x.
destruct x ; auto.
Qed.
Require Import List.
Program Instance list_eqdec `(eqa : EqDec A eq) : ! EqDec (list A) eq :=
{ equiv_dec :=
fix aux (x : list A) y { struct x } :=
match x, y with
| nil, nil => in_left
| cons hd tl, cons hd' tl' =>
if hd == hd' then
if aux tl tl' then in_left else in_right
else in_right
| _, _ => in_right
end }.
Solve Obligations using unfold equiv, complement in * ; program_simpl ; intuition (discriminate || eauto).
Next Obligation. destruct y ; intuition eauto. Defined.
Solve Obligations using unfold equiv, complement in * ; program_simpl ; intuition (discriminate || eauto).
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