diff options
Diffstat (limited to 'theories/FSets/FSetFacts.v')
| -rw-r--r-- | theories/FSets/FSetFacts.v | 100 |
1 files changed, 44 insertions, 56 deletions
diff --git a/theories/FSets/FSetFacts.v b/theories/FSets/FSetFacts.v index 674caaac..b750edfc 100644 --- a/theories/FSets/FSetFacts.v +++ b/theories/FSets/FSetFacts.v @@ -6,13 +6,13 @@ (* * GNU Lesser General Public License Version 2.1 *) (***********************************************************************) -(* $Id: FSetFacts.v 12187 2009-06-13 19:36:59Z msozeau $ *) +(* $Id$ *) (** * 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. + 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. *) @@ -30,7 +30,7 @@ Definition eqb x y := if eq_dec x y then true else false. (** * Specifications written using equivalences *) -Section IffSpec. +Section IffSpec. Variable s s' s'' : t. Variable x y z : elt. @@ -50,12 +50,12 @@ rewrite mem_iff; destruct (mem x s); intuition. Qed. Lemma equal_iff : s[=]s' <-> equal s s' = true. -Proof. +Proof. split; [apply equal_1|apply equal_2]. Qed. Lemma subset_iff : s[<=]s' <-> subset s s' = true. -Proof. +Proof. split; [apply subset_1|apply subset_2]. Qed. @@ -64,8 +64,8 @@ Proof. intuition; apply (empty_1 H). Qed. -Lemma is_empty_iff : Empty s <-> is_empty s = true. -Proof. +Lemma is_empty_iff : Empty s <-> is_empty s = true. +Proof. split; [apply is_empty_1|apply is_empty_2]. Qed. @@ -75,7 +75,7 @@ split; [apply singleton_1|apply singleton_2]. Qed. Lemma add_iff : In y (add x s) <-> E.eq x y \/ In y s. -Proof. +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). @@ -116,8 +116,8 @@ 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. +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 -> @@ -125,7 +125,7 @@ Lemma for_all_iff : compat_bool E.eq f -> 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. @@ -133,17 +133,17 @@ split; [apply exists_1 | apply exists_2]; auto. Qed. Lemma elements_iff : In x s <-> InA E.eq x (elements s). -Proof. +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 := + +Ltac set_iff := repeat (progress ( - rewrite add_iff || rewrite remove_iff || rewrite singleton_iff + rewrite add_iff || rewrite remove_iff || rewrite singleton_iff || rewrite union_iff || rewrite inter_iff || rewrite diff_iff || rewrite empty_iff)). @@ -154,7 +154,7 @@ Variable s s' s'' : t. Variable x y z : elt. Lemma mem_b : E.eq x y -> mem x s = mem y s. -Proof. +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. @@ -191,7 +191,7 @@ destruct (mem y s); destruct (mem y (remove x s)); intuition. Qed. Lemma singleton_b : mem y (singleton x) = eqb x y. -Proof. +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. @@ -236,7 +236,7 @@ Qed. Variable f : elt->bool. Lemma filter_b : compat_bool E.eq f -> mem x (filter f s) = mem x s && f x. -Proof. +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. @@ -264,7 +264,7 @@ rewrite H2. rewrite InA_alt; eauto. Qed. -Lemma exists_b : compat_bool E.eq f -> +Lemma exists_b : compat_bool E.eq f -> exists_ f s = existsb f (elements s). Proof. intros. @@ -291,39 +291,27 @@ End BoolSpec. (** * [E.eq] and [Equal] are setoid equalities *) -Definition E_ST : Equivalence E.eq. +Instance E_ST : Equivalence E.eq. Proof. constructor ; red; [apply E.eq_refl|apply E.eq_sym|apply E.eq_trans]. Qed. -Definition Equal_ST : Equivalence Equal. -Proof. +Instance Equal_ST : Equivalence Equal. +Proof. constructor ; red; [apply eq_refl | apply eq_sym | apply eq_trans]. Qed. -Add Relation elt E.eq - reflexivity proved by E.eq_refl - symmetry proved by E.eq_sym - transitivity proved by E.eq_trans - as EltSetoid. - -Add Relation t Equal - reflexivity proved by eq_refl - symmetry proved by eq_sym - transitivity proved by eq_trans - as EqualSetoid. - -Add Morphism In with signature E.eq ==> Equal ==> iff as In_m. +Instance In_m : Proper (E.eq ==> Equal ==> iff) In. 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. +Instance is_empty_m : Proper (Equal==> Logic.eq) is_empty. Proof. unfold Equal; intros s s' H. generalize (is_empty_iff s)(is_empty_iff s'). -destruct (is_empty s); destruct (is_empty s'); +destruct (is_empty s); destruct (is_empty s'); unfold Empty; auto; intros. symmetry. rewrite <- H1; intros a Ha. @@ -336,12 +324,12 @@ destruct H1 as (_,H1). exact (H1 (refl_equal true) _ Ha). Qed. -Add Morphism Empty with signature Equal ==> iff as Empty_m. +Instance Empty_m : Proper (Equal ==> iff) Empty. Proof. -intros; do 2 rewrite is_empty_iff; rewrite H; intuition. +repeat red; intros; do 2 rewrite is_empty_iff; rewrite H; intuition. Qed. -Add Morphism mem : mem_m. +Instance mem_m : Proper (E.eq ==> Equal ==> Logic.eq) mem. Proof. unfold Equal; intros x y H s s' H0. generalize (H0 x); clear H0; rewrite (In_eq_iff s' H). @@ -349,7 +337,7 @@ generalize (mem_iff s x)(mem_iff s' y). destruct (mem x s); destruct (mem y s'); intuition. Qed. -Add Morphism singleton : singleton_m. +Instance singleton_m : Proper (E.eq ==> Equal) singleton. Proof. unfold Equal; intros x y H a. do 2 rewrite singleton_iff; split; intros. @@ -357,51 +345,51 @@ apply E.eq_trans with x; auto. apply E.eq_trans with y; auto. Qed. -Add Morphism add : add_m. +Instance add_m : Proper (E.eq==>Equal==>Equal) add. 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. +Instance remove_m : Proper (E.eq==>Equal==>Equal) remove. 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. +Instance union_m : Proper (Equal==>Equal==>Equal) union. 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. +Instance inter_m : Proper (Equal==>Equal==>Equal) inter. 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. +Instance diff_m : Proper (Equal==>Equal==>Equal) diff. 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. +Instance Subset_m : Proper (Equal==>Equal==>iff) Subset. +Proof. unfold Equal, Subset; firstorder. Qed. -Add Morphism subset : subset_m. +Instance subset_m : Proper (Equal ==> Equal ==> Logic.eq) subset. Proof. intros s s' H s'' s''' H0. -generalize (subset_iff s s'') (subset_iff s' s'''). +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. +Instance equal_m : Proper (Equal ==> Equal ==> Logic.eq) equal. Proof. intros s s' H s'' s''' H0. generalize (equal_iff s s'') (equal_iff s' s'''). @@ -424,7 +412,7 @@ Add Relation t Subset transitivity proved by Subset_trans as SubsetSetoid. -Instance In_s_m : Morphisms.Morphism (E.eq ==> Subset ++> Basics.impl) In | 1. +Instance In_s_m : Morphisms.Proper (E.eq ==> Subset ++> Basics.impl) In | 1. Proof. simpl_relation. eauto with set. Qed. @@ -467,7 +455,7 @@ 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 -> +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. @@ -478,10 +466,10 @@ Lemma filter_ext : forall f f', compat_bool E.eq f -> (forall x, f x = f' x) -> Proof. intros f f' Hf Hff' s s' Hss' x. do 2 (rewrite filter_iff; auto). rewrite Hff', Hss'; intuition. -red; intros; rewrite <- 2 Hff'; auto. +repeat red; intros; rewrite <- 2 Hff'; auto. Qed. -Lemma filter_subset : forall f, compat_bool E.eq f -> +Lemma filter_subset : forall f, compat_bool E.eq f -> forall s s', s[<=]s' -> filter f s [<=] filter f s'. Proof. unfold Subset; intros; rewrite filter_iff in *; intuition. |