(* Copyright (c) 2012-2013, Robbert Krebbers. *) (* This file is distributed under the terms of the BSD license. *) (** This file collects theorems, definitions, tactics, related to propositions with a decidable equality. Such propositions are collected by the [Decision] type class. *) Require Export base tactics. Hint Extern 200 (Decision _) => progress (lazy beta) : typeclass_instances. Lemma dec_stable `{Decision P} : ¬¬P → P. Proof. firstorder. Qed. Lemma Is_true_reflect (b : bool) : reflect b b. Proof. destruct b. by left. right. intros []. Qed. (** We introduce [decide_rel] to avoid inefficienct computation due to eager evaluation of propositions by [vm_compute]. This inefficiency occurs if [(x = y) := (f x = f y)] as [decide (x = y)] evaluates to [decide (f x = f y)] which then might lead to evaluation of [f x] and [f y]. Using [decide_rel] we hide [f] under a lambda abstraction to avoid this unnecessary evaluation. *) Definition decide_rel {A B} (R : A → B → Prop) {dec : ∀ x y, Decision (R x y)} (x : A) (y : B) : Decision (R x y) := dec x y. Lemma decide_rel_correct {A B} (R : A → B → Prop) `{∀ x y, Decision (R x y)} (x : A) (y : B) : decide_rel R x y = decide (R x y). Proof. done. Qed. (** The tactic [destruct_decide] destructs a sumbool [dec]. If one of the components is double negated, it will try to remove the double negation. *) Ltac destruct_decide dec := let H := fresh in destruct dec as [H|H]; try match type of H with | ¬¬_ => apply dec_stable in H end. (** The tactic [case_decide] performs case analysis on an arbitrary occurrence of [decide] or [decide_rel] in the conclusion or hypotheses. *) Ltac case_decide := match goal with | H : context [@decide ?P ?dec] |- _ => destruct_decide (@decide P dec) | H : context [@decide_rel _ _ ?R ?x ?y ?dec] |- _ => destruct_decide (@decide_rel _ _ R x y dec) | |- context [@decide ?P ?dec] => destruct_decide (@decide P dec) | |- context [@decide_rel _ _ ?R ?x ?y ?dec] => destruct_decide (@decide_rel _ _ R x y dec) end. (** The tactic [solve_decision] uses Coq's [decide equality] tactic together with instance resolution to automatically generate decision procedures. *) Ltac solve_trivial_decision := match goal with | |- Decision (?P) => apply _ | |- sumbool ?P (¬?P) => change (Decision P); apply _ end. Ltac solve_decision := intros; first [ solve_trivial_decision | unfold Decision; decide equality; solve_trivial_decision ]. (** The following combinators are useful to create Decision proofs in combination with the [refine] tactic. *) Notation cast_if S := (if S then left _ else right _). Notation cast_if_and S1 S2 := (if S1 then cast_if S2 else right _). Notation cast_if_and3 S1 S2 S3 := (if S1 then cast_if_and S2 S3 else right _). Notation cast_if_and4 S1 S2 S3 S4 := (if S1 then cast_if_and3 S2 S3 S4 else right _). Notation cast_if_or S1 S2 := (if S1 then left _ else cast_if S2). Notation cast_if_not_or S1 S2 := (if S1 then cast_if S2 else left _). Notation cast_if_not S := (if S then right _ else left _). (** We can convert decidable propositions to booleans. *) Definition bool_decide (P : Prop) {dec : Decision P} : bool := if dec then true else false. Lemma bool_decide_reflect P `{dec : Decision P} : reflect P (bool_decide P). Proof. unfold bool_decide. destruct dec. by left. by right. Qed. Ltac case_bool_decide := match goal with | H : context [@bool_decide ?P ?dec] |- _ => destruct_decide (@bool_decide_reflect P dec) | |- context [@bool_decide ?P ?dec] => destruct_decide (@bool_decide_reflect P dec) end. Lemma bool_decide_unpack (P : Prop) {dec : Decision P} : bool_decide P → P. Proof. unfold bool_decide. by destruct dec. Qed. Lemma bool_decide_pack (P : Prop) {dec : Decision P} : P → bool_decide P. Proof. unfold bool_decide. by destruct dec. Qed. (** * Decidable Sigma types *) (** Leibniz equality on Sigma types requires the equipped proofs to be equal as Coq does not support proof irrelevance. For decidable we propositions we define the type [dsig P] whose Leibniz equality is proof irrelevant. That is [∀ x y : dsig P, x = y ↔ `x = `y]. Due to the absence of universe polymorpic definitions we also define a variant [dsigS] for types in [Set]. *) Definition dsig `(P : A → Prop) `{∀ x : A, Decision (P x)} := { x | bool_decide (P x) }. Definition dsigS {A : Set} (P : A → Prop) `{∀ x : A, Decision (P x)} : Set := { x | bool_decide (P x) }. Definition proj2_dsig `{∀ x : A, Decision (P x)} (x : dsig P) : P (`x) := bool_decide_unpack _ (proj2_sig x). Definition dexist `{∀ x : A, Decision (P x)} (x : A) (p : P x) : dsig P := x↾bool_decide_pack _ p. Lemma dsig_eq `(P : A → Prop) `{∀ x, Decision (P x)} (x y : dsig P) : x = y ↔ `x = `y. Proof. split. * destruct x, y. apply proj1_sig_inj. * intro. destruct x as [x Hx], y as [y Hy]. simpl in *. subst. f_equal. revert Hx Hy. case (bool_decide (P y)). + by intros [] []. + done. Qed. Lemma dexists_proj1 `(P : A → Prop) `{∀ x, Decision (P x)} (x : dsig P) p : dexist (`x) p = x. Proof. by apply dsig_eq. Qed. Global Instance dsig_eq_dec `(P : A → Prop) `{∀ x, Decision (P x)} `{∀ x y : A, Decision (x = y)} (x y : dsig P) : Decision (x = y). Proof. refine (cast_if (decide (`x = `y))); by rewrite dsig_eq. Defined. (** * Instances of Decision *) (** Instances of [Decision] for operators of propositional logic. *) Instance True_dec: Decision True := left I. Instance False_dec: Decision False := right (False_rect False). Section prop_dec. Context `(P_dec : Decision P) `(Q_dec : Decision Q). Global Instance not_dec: Decision (¬P). Proof. refine (cast_if_not P_dec); intuition. Defined. Global Instance and_dec: Decision (P ∧ Q). Proof. refine (cast_if_and P_dec Q_dec); intuition. Defined. Global Instance or_dec: Decision (P ∨ Q). Proof. refine (cast_if_or P_dec Q_dec); intuition. Defined. Global Instance impl_dec: Decision (P → Q). Proof. refine (if P_dec then cast_if Q_dec else left _); intuition. Defined. End prop_dec. (** Instances of [Decision] for common data types. *) Instance bool_eq_dec (x y : bool) : Decision (x = y). Proof. solve_decision. Defined. Instance unit_eq_dec (x y : unit) : Decision (x = y). Proof. refine (left _); by destruct x, y. Defined. Instance prod_eq_dec `(A_dec : ∀ x y : A, Decision (x = y)) `(B_dec : ∀ x y : B, Decision (x = y)) (x y : A * B) : Decision (x = y). Proof. refine (cast_if_and (A_dec (fst x) (fst y)) (B_dec (snd x) (snd y))); abstract (destruct x, y; simpl in *; congruence). Defined. Instance sum_eq_dec `(A_dec : ∀ x y : A, Decision (x = y)) `(B_dec : ∀ x y : B, Decision (x = y)) (x y : A + B) : Decision (x = y). Proof. solve_decision. Defined. Instance curry_dec `(P_dec : ∀ (x : A) (y : B), Decision (P x y)) p : Decision (curry P p) := match p as p return Decision (curry P p) with | (x,y) => P_dec x y end. Instance uncurry_dec `(P_dec : ∀ (p : A * B), Decision (P p)) x y : Decision (uncurry P x y) := P_dec (x,y).