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Diffstat (limited to 'theories/FSets/FSetWeakFacts.v')
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diff --git a/theories/FSets/FSetWeakFacts.v b/theories/FSets/FSetWeakFacts.v deleted file mode 100644 index 61797a95..00000000 --- a/theories/FSets/FSetWeakFacts.v +++ /dev/null @@ -1,421 +0,0 @@ -(***********************************************************************) -(* 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: FSetWeakFacts.v 8882 2006-05-31 21:55:30Z 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 FSetWeakInterface. -Set Implicit Arguments. -Unset Strict Implicit. - -Module Facts (M: S). -Import M.E. -Import M. -Import Logic. (* to unmask [eq] *) - -(** * 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 *) - -Definition eqb x y := if eq_dec x y then true else false. - -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. -rewrite H2. -rewrite InA_alt; eauto. -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. -split; auto. -rewrite H2; rewrite InA_alt; eauto. -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; unfold Equal; firstorder. -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. -intros; apply E.eq_trans with x; auto. -intros; apply E.eq_trans with y; auto. -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. |