``` ```

# 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. Import Peano. ```

# 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 <-> InA E.eq 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 empty_b : mem y empty = false. Proof. generalize (empty_iff y)(mem_iff empty y). destruct (mem y empty); 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. 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. End Facts. ```