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diff --git a/theories/FSets/FSetFacts.v b/theories/FSets/FSetFacts.v new file mode 100644 index 00000000..d8c0b802 --- /dev/null +++ b/theories/FSets/FSetFacts.v @@ -0,0 +1,409 @@ +(***********************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) +(* \VV/ *************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(***********************************************************************) + +(* $Id: FSetFacts.v 8681 2006-04-05 11:56:14Z letouzey $ *) + +(** * Finite sets library *) + +(** This functor derives additional facts from [FSetInterface.S]. These + facts are mainly the specifications of [FSetInterface.S] written using + different styles: equivalence and boolean equalities. + Moreover, we prove that [E.Eq] and [Equal] are setoid equalities. +*) + +Require Export FSetInterface. +Set Implicit Arguments. +Unset Strict Implicit. + +Module Facts (M: S). +Module ME := OrderedTypeFacts M.E. +Import ME. +Import M. +Import Logic. (* to unmask [eq] *) +Import Peano. (* to unmask [lt] *) + +(** * Specifications written using equivalences *) + +Section IffSpec. +Variable s s' s'' : t. +Variable x y z : elt. + +Lemma In_eq_iff : E.eq x y -> (In x s <-> In y s). +Proof. +split; apply In_1; auto. +Qed. + +Lemma mem_iff : In x s <-> mem x s = true. +Proof. +split; [apply mem_1|apply mem_2]. +Qed. + +Lemma not_mem_iff : ~In x s <-> mem x s = false. +Proof. +rewrite mem_iff; destruct (mem x s); intuition. +Qed. + +Lemma equal_iff : s[=]s' <-> equal s s' = true. +Proof. +split; [apply equal_1|apply equal_2]. +Qed. + +Lemma subset_iff : s[<=]s' <-> subset s s' = true. +Proof. +split; [apply subset_1|apply subset_2]. +Qed. + +Lemma empty_iff : In x empty <-> False. +Proof. +intuition; apply (empty_1 H). +Qed. + +Lemma is_empty_iff : Empty s <-> is_empty s = true. +Proof. +split; [apply is_empty_1|apply is_empty_2]. +Qed. + +Lemma singleton_iff : In y (singleton x) <-> E.eq x y. +Proof. +split; [apply singleton_1|apply singleton_2]. +Qed. + +Lemma add_iff : In y (add x s) <-> E.eq x y \/ In y s. +Proof. +split; [ | destruct 1; [apply add_1|apply add_2]]; auto. +destruct (eq_dec x y) as [E|E]; auto. +intro H; right; exact (add_3 E H). +Qed. + +Lemma add_neq_iff : ~ E.eq x y -> (In y (add x s) <-> In y s). +Proof. +split; [apply add_3|apply add_2]; auto. +Qed. + +Lemma remove_iff : In y (remove x s) <-> In y s /\ ~E.eq x y. +Proof. +split; [split; [apply remove_3 with x |] | destruct 1; apply remove_2]; auto. +intro. +apply (remove_1 H0 H). +Qed. + +Lemma remove_neq_iff : ~ E.eq x y -> (In y (remove x s) <-> In y s). +Proof. +split; [apply remove_3|apply remove_2]; auto. +Qed. + +Lemma union_iff : In x (union s s') <-> In x s \/ In x s'. +Proof. +split; [apply union_1 | destruct 1; [apply union_2|apply union_3]]; auto. +Qed. + +Lemma inter_iff : In x (inter s s') <-> In x s /\ In x s'. +Proof. +split; [split; [apply inter_1 with s' | apply inter_2 with s] | destruct 1; apply inter_3]; auto. +Qed. + +Lemma diff_iff : In x (diff s s') <-> In x s /\ ~ In x s'. +Proof. +split; [split; [apply diff_1 with s' | apply diff_2 with s] | destruct 1; apply diff_3]; auto. +Qed. + +Variable f : elt->bool. + +Lemma filter_iff : compat_bool E.eq f -> (In x (filter f s) <-> In x s /\ f x = true). +Proof. +split; [split; [apply filter_1 with f | apply filter_2 with s] | destruct 1; apply filter_3]; auto. +Qed. + +Lemma for_all_iff : compat_bool E.eq f -> + (For_all (fun x => f x = true) s <-> for_all f s = true). +Proof. +split; [apply for_all_1 | apply for_all_2]; auto. +Qed. + +Lemma exists_iff : compat_bool E.eq f -> + (Exists (fun x => f x = true) s <-> exists_ f s = true). +Proof. +split; [apply exists_1 | apply exists_2]; auto. +Qed. + +Lemma elements_iff : In x s <-> ME.In x (elements s). +Proof. +split; [apply elements_1 | apply elements_2]. +Qed. + +End IffSpec. + +(** Useful tactic for simplifying expressions like [In y (add x (union s s'))] *) + +Ltac set_iff := + repeat (progress ( + rewrite add_iff || rewrite remove_iff || rewrite singleton_iff + || rewrite union_iff || rewrite inter_iff || rewrite diff_iff + || rewrite empty_iff)). + +(** * Specifications written using boolean predicates *) + +Section BoolSpec. +Variable s s' s'' : t. +Variable x y z : elt. + +Lemma mem_b : E.eq x y -> mem x s = mem y s. +Proof. +intros. +generalize (mem_iff s x) (mem_iff s y)(In_eq_iff s H). +destruct (mem x s); destruct (mem y s); intuition. +Qed. + +Lemma add_b : mem y (add x s) = eqb x y || mem y s. +Proof. +generalize (mem_iff (add x s) y)(mem_iff s y)(add_iff s x y); unfold eqb. +destruct (eq_dec x y); destruct (mem y s); destruct (mem y (add x s)); intuition. +Qed. + +Lemma add_neq_b : ~ E.eq x y -> mem y (add x s) = mem y s. +Proof. +intros; generalize (mem_iff (add x s) y)(mem_iff s y)(add_neq_iff s H). +destruct (mem y s); destruct (mem y (add x s)); intuition. +Qed. + +Lemma remove_b : mem y (remove x s) = mem y s && negb (eqb x y). +Proof. +generalize (mem_iff (remove x s) y)(mem_iff s y)(remove_iff s x y); unfold eqb. +destruct (eq_dec x y); destruct (mem y s); destruct (mem y (remove x s)); simpl; intuition. +Qed. + +Lemma remove_neq_b : ~ E.eq x y -> mem y (remove x s) = mem y s. +Proof. +intros; generalize (mem_iff (remove x s) y)(mem_iff s y)(remove_neq_iff s H). +destruct (mem y s); destruct (mem y (remove x s)); intuition. +Qed. + +Lemma singleton_b : mem y (singleton x) = eqb x y. +Proof. +generalize (mem_iff (singleton x) y)(singleton_iff x y); unfold eqb. +destruct (eq_dec x y); destruct (mem y (singleton x)); intuition. +Qed. + +Lemma union_b : mem x (union s s') = mem x s || mem x s'. +Proof. +generalize (mem_iff (union s s') x)(mem_iff s x)(mem_iff s' x)(union_iff s s' x). +destruct (mem x s); destruct (mem x s'); destruct (mem x (union s s')); intuition. +Qed. + +Lemma inter_b : mem x (inter s s') = mem x s && mem x s'. +Proof. +generalize (mem_iff (inter s s') x)(mem_iff s x)(mem_iff s' x)(inter_iff s s' x). +destruct (mem x s); destruct (mem x s'); destruct (mem x (inter s s')); intuition. +Qed. + +Lemma diff_b : mem x (diff s s') = mem x s && negb (mem x s'). +Proof. +generalize (mem_iff (diff s s') x)(mem_iff s x)(mem_iff s' x)(diff_iff s s' x). +destruct (mem x s); destruct (mem x s'); destruct (mem x (diff s s')); simpl; intuition. +Qed. + +Lemma elements_b : mem x s = existsb (eqb x) (elements s). +Proof. +generalize (mem_iff s x)(elements_iff s x)(existsb_exists (eqb x) (elements s)). +rewrite InA_alt. +destruct (mem x s); destruct (existsb (eqb x) (elements s)); auto; intros. +symmetry. +rewrite H1. +destruct H0 as (H0,_). +destruct H0 as (a,(Ha1,Ha2)); [ intuition |]. +exists a; intuition. +unfold eqb; destruct (eq_dec x a); auto. +rewrite <- H. +rewrite H0. +destruct H1 as (H1,_). +destruct H1 as (a,(Ha1,Ha2)); [intuition|]. +exists a; intuition. +unfold eqb in *; destruct (eq_dec x a); auto; discriminate. +Qed. + +Variable f : elt->bool. + +Lemma filter_b : compat_bool E.eq f -> mem x (filter f s) = mem x s && f x. +Proof. +intros. +generalize (mem_iff (filter f s) x)(mem_iff s x)(filter_iff s x H). +destruct (mem x s); destruct (mem x (filter f s)); destruct (f x); simpl; intuition. +Qed. + +Lemma for_all_b : compat_bool E.eq f -> + for_all f s = forallb f (elements s). +Proof. +intros. +generalize (forallb_forall f (elements s))(for_all_iff s H)(elements_iff s). +unfold For_all. +destruct (forallb f (elements s)); destruct (for_all f s); auto; intros. +rewrite <- H1; intros. +destruct H0 as (H0,_). +rewrite (H2 x0) in H3. +rewrite (InA_alt E.eq x0 (elements s)) in H3. +destruct H3 as (a,(Ha1,Ha2)). +rewrite (H _ _ Ha1). +apply H0; auto. +symmetry. +rewrite H0; intros. +destruct H1 as (_,H1). +apply H1; auto. +Qed. + +Lemma exists_b : compat_bool E.eq f -> + exists_ f s = existsb f (elements s). +Proof. +intros. +generalize (existsb_exists f (elements s))(exists_iff s H)(elements_iff s). +unfold Exists. +destruct (existsb f (elements s)); destruct (exists_ f s); auto; intros. +rewrite <- H1; intros. +destruct H0 as (H0,_). +destruct H0 as (a,(Ha1,Ha2)); auto. +exists a; auto. +symmetry. +rewrite H0. +destruct H1 as (_,H1). +destruct H1 as (a,(Ha1,Ha2)); auto. +rewrite (H2 a) in Ha1. +rewrite (InA_alt E.eq a (elements s)) in Ha1. +destruct Ha1 as (b,(Hb1,Hb2)). +exists b; auto. +rewrite <- (H _ _ Hb1); auto. +Qed. + +End BoolSpec. + +(** * [E.eq] and [Equal] are setoid equalities *) + +Definition E_ST : Setoid_Theory elt E.eq. +Proof. +constructor; [apply E.eq_refl|apply E.eq_sym|apply E.eq_trans]. +Qed. + +Add Setoid elt E.eq E_ST as EltSetoid. + +Definition Equal_ST : Setoid_Theory t Equal. +Proof. +constructor; [apply eq_refl | apply eq_sym | apply eq_trans]. +Qed. + +Add Setoid t Equal Equal_ST as EqualSetoid. + +Add Morphism In with signature E.eq ==> Equal ==> iff as In_m. +Proof. +unfold Equal; intros x y H s s' H0. +rewrite (In_eq_iff s H); auto. +Qed. + +Add Morphism is_empty : is_empty_m. +Proof. +unfold Equal; intros s s' H. +generalize (is_empty_iff s)(is_empty_iff s'). +destruct (is_empty s); destruct (is_empty s'); + unfold Empty; auto; intros. +symmetry. +rewrite <- H1; intros a Ha. +rewrite <- (H a) in Ha. +destruct H0 as (_,H0). +exact (H0 (refl_equal true) _ Ha). +rewrite <- H0; intros a Ha. +rewrite (H a) in Ha. +destruct H1 as (_,H1). +exact (H1 (refl_equal true) _ Ha). +Qed. + +Add Morphism Empty with signature Equal ==> iff as Empty_m. +Proof. +intros; do 2 rewrite is_empty_iff; rewrite H; intuition. +Qed. + +Add Morphism mem : mem_m. +Proof. +unfold Equal; intros x y H s s' H0. +generalize (H0 x); clear H0; rewrite (In_eq_iff s' H). +generalize (mem_iff s x)(mem_iff s' y). +destruct (mem x s); destruct (mem y s'); intuition. +Qed. + +Add Morphism singleton : singleton_m. +Proof. +unfold Equal; intros x y H a. +do 2 rewrite singleton_iff; split; order. +Qed. + +Add Morphism add : add_m. +Proof. +unfold Equal; intros x y H s s' H0 a. +do 2 rewrite add_iff; rewrite H; rewrite H0; intuition. +Qed. + +Add Morphism remove : remove_m. +Proof. +unfold Equal; intros x y H s s' H0 a. +do 2 rewrite remove_iff; rewrite H; rewrite H0; intuition. +Qed. + +Add Morphism union : union_m. +Proof. +unfold Equal; intros s s' H s'' s''' H0 a. +do 2 rewrite union_iff; rewrite H; rewrite H0; intuition. +Qed. + +Add Morphism inter : inter_m. +Proof. +unfold Equal; intros s s' H s'' s''' H0 a. +do 2 rewrite inter_iff; rewrite H; rewrite H0; intuition. +Qed. + +Add Morphism diff : diff_m. +Proof. +unfold Equal; intros s s' H s'' s''' H0 a. +do 2 rewrite diff_iff; rewrite H; rewrite H0; intuition. +Qed. + +Add Morphism Subset with signature Equal ==> Equal ==> iff as Subset_m. +Proof. +unfold Equal, Subset; firstorder. +Qed. + +Add Morphism subset : subset_m. +Proof. +intros s s' H s'' s''' H0. +generalize (subset_iff s s'') (subset_iff s' s'''). +destruct (subset s s''); destruct (subset s' s'''); auto; intros. +rewrite H in H1; rewrite H0 in H1; intuition. +rewrite H in H1; rewrite H0 in H1; intuition. +Qed. + +Add Morphism equal : equal_m. +Proof. +intros s s' H s'' s''' H0. +generalize (equal_iff s s'') (equal_iff s' s'''). +destruct (equal s s''); destruct (equal s' s'''); auto; intros. +rewrite H in H1; rewrite H0 in H1; intuition. +rewrite H in H1; rewrite H0 in H1; intuition. +Qed. + +(* [fold], [filter], [for_all], [exists_] and [partition] cannot be proved morphism + without additional hypothesis on [f]. For instance: *) + +Lemma filter_equal : forall f, compat_bool E.eq f -> + forall s s', s[=]s' -> filter f s [=] filter f s'. +Proof. +unfold Equal; intros; repeat rewrite filter_iff; auto; rewrite H0; tauto. +Qed. + +(* For [elements], [min_elt], [max_elt] and [choose], we would need setoid + structures on [list elt] and [option elt]. *) + +(* Later: +Add Morphism cardinal ; cardinal_m. +*) + +End Facts. |