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(* *********************************************************************)
(* *)
(* The Compcert verified compiler *)
(* *)
(* Xavier Leroy, INRIA Paris *)
(* *)
(* Copyright Institut National de Recherche en Informatique et en *)
(* Automatique. All rights reserved. This file is distributed *)
(* under the terms of the GNU Lesser General Public License as *)
(* published by the Free Software Foundation, either version 2.1 of *)
(* the License, or (at your option) any later version. *)
(* This file is also distributed under the terms of the *)
(* INRIA Non-Commercial License Agreement. *)
(* *)
(* *********************************************************************)
(** This library provides type classes and tactics to decide logical
propositions by reflection into computable Boolean equalities.
It extends the [DecidableClass] module from the standard library
of Coq 8.5 with more instances of decidable properties, including
universal and existential quantification over finite types. *)
Require Export DecidableClass.
Require Import Coqlib.
Ltac decide_goal := eapply Decidable_sound; reflexivity.
(** Deciding logical connectives *)
Global Program Instance Decidable_not (P: Prop) (dP: Decidable P) : Decidable (~ P) := {
Decidable_witness := negb (@Decidable_witness P dP)
}.
Next Obligation.
rewrite negb_true_iff. split. apply Decidable_complete_alt. apply Decidable_sound_alt.
Qed.
Global Program Instance Decidable_equiv (P Q: Prop) (dP: Decidable P) (dQ: Decidable Q) : Decidable (P <-> Q) := {
Decidable_witness := Bool.eqb (@Decidable_witness P dP) (@Decidable_witness Q dQ)
}.
Next Obligation.
rewrite eqb_true_iff.
split; intros.
split; intros; eapply Decidable_sound; [rewrite <- H | rewrite H]; eapply Decidable_complete; eauto.
destruct (@Decidable_witness Q dQ) eqn:D.
eapply Decidable_complete; rewrite H; eapply Decidable_sound; eauto.
eapply Decidable_sound_alt; rewrite H; eapply Decidable_complete_alt; eauto.
Qed.
Global Program Instance Decidable_and (P Q: Prop) (dP: Decidable P) (dQ: Decidable Q) : Decidable (P /\ Q) := {
Decidable_witness := @Decidable_witness P dP && @Decidable_witness Q dQ
}.
Next Obligation.
rewrite andb_true_iff. rewrite ! Decidable_spec. tauto.
Qed.
Global Program Instance Decidable_or (P Q: Prop) (dP: Decidable P) (dQ: Decidable Q) : Decidable (P \/ Q) := {
Decidable_witness := @Decidable_witness P dP || @Decidable_witness Q dQ
}.
Next Obligation.
rewrite orb_true_iff. rewrite ! Decidable_spec. tauto.
Qed.
Global Program Instance Decidable_implies (P Q: Prop) (dP: Decidable P) (dQ: Decidable Q) : Decidable (P -> Q) := {
Decidable_witness := if @Decidable_witness P dP then @Decidable_witness Q dQ else true
}.
Next Obligation.
split.
- intros. rewrite Decidable_complete in H by auto. eapply Decidable_sound; eauto.
- intros. destruct (@Decidable_witness P dP) eqn:WP; auto.
eapply Decidable_complete. apply H. eapply Decidable_sound; eauto.
Qed.
(** Deciding equalities. *)
Program Definition Decidable_eq {A: Type} (eqdec: forall (x y: A), {x=y} + {x<>y}) (x y: A) : Decidable (eq x y) := {|
Decidable_witness := proj_sumbool (eqdec x y)
|}.
Next Obligation.
split; intros. InvBooleans. auto. subst y. apply dec_eq_true.
Qed.
Global Program Instance Decidable_eq_bool : forall (x y : bool), Decidable (eq x y) := {
Decidable_witness := Bool.eqb x y
}.
Next Obligation.
apply eqb_true_iff.
Qed.
Global Program Instance Decidable_eq_nat : forall (x y : nat), Decidable (eq x y) := {
Decidable_witness := Nat.eqb x y
}.
Next Obligation.
apply Nat.eqb_eq.
Qed.
Global Program Instance Decidable_eq_positive : forall (x y : positive), Decidable (eq x y) := {
Decidable_witness := Pos.eqb x y
}.
Next Obligation.
apply Pos.eqb_eq.
Qed.
Global Program Instance Decidable_eq_Z : forall (x y : Z), Decidable (eq x y) := {
Decidable_witness := Z.eqb x y
}.
Next Obligation.
apply Z.eqb_eq.
Qed.
(** Deciding order on Z *)
Global Program Instance Decidable_le_Z : forall (x y: Z), Decidable (x <= y) := {
Decidable_witness := Z.leb x y
}.
Next Obligation.
apply Z.leb_le.
Qed.
Global Program Instance Decidable_lt_Z : forall (x y: Z), Decidable (x < y) := {
Decidable_witness := Z.ltb x y
}.
Next Obligation.
apply Z.ltb_lt.
Qed.
Global Program Instance Decidable_ge_Z : forall (x y: Z), Decidable (x >= y) := {
Decidable_witness := Z.geb x y
}.
Next Obligation.
rewrite Z.geb_le. intuition lia.
Qed.
Global Program Instance Decidable_gt_Z : forall (x y: Z), Decidable (x > y) := {
Decidable_witness := Z.gtb x y
}.
Next Obligation.
rewrite Z.gtb_lt. intuition lia.
Qed.
Global Program Instance Decidable_divides : forall (x y: Z), Decidable (x | y) := {
Decidable_witness := Z.eqb y ((y / x) * x)%Z
}.
Next Obligation.
split.
intros. apply Z.eqb_eq in H. exists (y / x). auto.
intros [k EQ]. apply Z.eqb_eq.
destruct (Z.eq_dec x 0).
subst x. rewrite Z.mul_0_r in EQ. subst y. reflexivity.
assert (k = y / x).
{ apply Zdiv_unique_full with 0. red; lia. rewrite EQ; ring. }
congruence.
Qed.
(** Deciding properties over lists *)
Global Program Instance Decidable_forall_in_list :
forall (A: Type) (l: list A) (P: A -> Prop) (dP: forall x:A, Decidable (P x)),
Decidable (forall x:A, In x l -> P x) := {
Decidable_witness := List.forallb (fun x => @Decidable_witness (P x) (dP x)) l
}.
Next Obligation.
rewrite List.forallb_forall. split; intros.
- eapply Decidable_sound; eauto.
- eapply Decidable_complete; eauto.
Qed.
Global Program Instance Decidable_exists_in_list :
forall (A: Type) (l: list A) (P: A -> Prop) (dP: forall x:A, Decidable (P x)),
Decidable (exists x:A, In x l /\ P x) := {
Decidable_witness := List.existsb (fun x => @Decidable_witness (P x) (dP x)) l
}.
Next Obligation.
rewrite List.existsb_exists. split.
- intros (x & U & V). exists x; split; auto. eapply Decidable_sound; eauto.
- intros (x & U & V). exists x; split; auto. eapply Decidable_complete; eauto.
Qed.
(** Types with finitely many elements. *)
Class Finite (T: Type) := {
Finite_elements: list T;
Finite_elements_spec: forall x:T, In x Finite_elements
}.
(** Deciding forall and exists quantification over finite types. *)
Global Program Instance Decidable_forall : forall (T: Type) (fT: Finite T) (P: T -> Prop) (dP: forall x:T, Decidable (P x)), Decidable (forall x, P x) := {
Decidable_witness := List.forallb (fun x => @Decidable_witness (P x) (dP x)) (@Finite_elements T fT)
}.
Next Obligation.
rewrite List.forallb_forall. split; intros.
- eapply Decidable_sound; eauto. apply H. apply Finite_elements_spec.
- eapply Decidable_complete; eauto.
Qed.
Global Program Instance Decidable_exists : forall (T: Type) (fT: Finite T) (P: T -> Prop) (dP: forall x:T, Decidable (P x)), Decidable (exists x, P x) := {
Decidable_witness := List.existsb (fun x => @Decidable_witness (P x) (dP x)) (@Finite_elements T fT)
}.
Next Obligation.
rewrite List.existsb_exists. split.
- intros (x & A & B). exists x. eapply Decidable_sound; eauto.
- intros (x & A). exists x; split. eapply Finite_elements_spec. eapply Decidable_complete; eauto.
Qed.
(** Some examples of finite types. *)
Global Program Instance Finite_bool : Finite bool := {
Finite_elements := false :: true :: nil
}.
Next Obligation.
destruct x; auto.
Qed.
Global Program Instance Finite_pair : forall (A B: Type) (fA: Finite A) (fB: Finite B), Finite (A * B) := {
Finite_elements := list_prod (@Finite_elements A fA) (@Finite_elements B fB)
}.
Next Obligation.
apply List.in_prod; eapply Finite_elements_spec.
Qed.
Global Program Instance Finite_option : forall (A: Type) (fA: Finite A), Finite (option A) := {
Finite_elements := None :: List.map (@Some A) (@Finite_elements A fA)
}.
Next Obligation.
destruct x; auto. right; apply List.in_map; eapply Finite_elements_spec.
Qed.
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