Module Decidableplus

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

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.

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.

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.

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.

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.

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.

Program Instance Decidable_eq_nat : forall (x y : nat), Decidable (eq x y) := {
Decidable_witness := beq_nat x y
}.

Next Obligation.

apply beq_nat_true_iff.
Qed.

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.

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

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.

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.

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 omega.
Qed.

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 omega.
Qed.

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; omega. rewrite EQ; ring. }
  congruence.
Qed.

Deciding properties over lists

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.

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.

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.

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.

Program Instance Finite_bool : Finite bool := {
Finite_elements := false :: true :: nil
}.

Next Obligation.

destruct x; auto.
Qed.

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.

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.