MSetProperties.v

(************************************************************************) (* * The Coq Proof Assistant / The Coq Development Team *) (* v * Copyright INRIA, CNRS and contributors *) (* <O___,, * (see version control and CREDITS file for authors & dates) *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) (* * (see LICENSE file for the text of the license) *) (************************************************************************) (** * Finite sets library *) (** This functor derives additional properties from [MSetInterface.S]. Contrary to the functor in [MSetEqProperties] it uses predicates over sets instead of sets operations, i.e. [In x s] instead of [mem x s=true], [Equal s s'] instead of [equal s s'=true], etc. *)

Require Export MSetInterface.

Require Import PeanoNat DecidableTypeEx OrdersLists MSetFacts MSetDecide.

Set Implicit Arguments.

Unset Strict Implicit.

Local Ltac Tauto.intuition_solver ::= auto with relations.

#[global] Hint Unfold transpose : core.

(** First, a functor for Weak Sets in functorial version. *)

Module WPropertiesOn (Import E : DecidableType)(M : WSetsOn E).

Module Import Dec := WDecideOn E M.

Module Import FM := Dec.F (* MSetFacts.WFactsOn E M *).

Lemma In_dec : forall x s, {In x s} + {~ In x s}.

intros; generalize (mem_iff s x); case (mem x s); intuition auto with bool.

Definition Add x s s' := forall y, In y s' <-> E.eq x y \/ In y s.

Lemma Add_Equal : forall x s s', Add x s s' <-> s' [=] add x s.

rewrite H; clear H.

Ltac expAdd := repeat rewrite Add_Equal.

Section BasicProperties.

Variable s s' s'' s1 s2 s3 : t.

Variable x x' : elt.

Lemma equal_refl : s[=]s.

Lemma equal_sym : s[=]s' -> s'[=]s.

Lemma equal_trans : s1[=]s2 -> s2[=]s3 -> s1[=]s3.

Lemma subset_refl : s[<=]s.

Lemma subset_trans : s1[<=]s2 -> s2[<=]s3 -> s1[<=]s3.

Lemma subset_antisym : s[<=]s' -> s'[<=]s -> s[=]s'.

Lemma subset_equal : s[=]s' -> s[<=]s'.

Lemma subset_empty : empty[<=]s.

Lemma subset_remove_3 : s1[<=]s2 -> remove x s1 [<=] s2.

Lemma subset_diff : s1[<=]s3 -> diff s1 s2 [<=] s3.

Lemma subset_add_3 : In x s2 -> s1[<=]s2 -> add x s1 [<=] s2.

Lemma subset_add_2 : s1[<=]s2 -> s1[<=] add x s2.

Lemma in_subset : In x s1 -> s1[<=]s2 -> In x s2.

Lemma double_inclusion : s1[=]s2 <-> s1[<=]s2 /\ s2[<=]s1.

intuition fsetdec.

Lemma empty_is_empty_1 : Empty s -> s[=]empty.

Lemma empty_is_empty_2 : s[=]empty -> Empty s.

Lemma add_equal : In x s -> add x s [=] s.

Lemma add_add : add x (add x' s) [=] add x' (add x s).

Lemma remove_equal : ~ In x s -> remove x s [=] s.

Lemma Equal_remove : s[=]s' -> remove x s [=] remove x s'.

Lemma add_remove : In x s -> add x (remove x s) [=] s.

Lemma remove_add : ~In x s -> remove x (add x s) [=] s.

Lemma singleton_equal_add : singleton x [=] add x empty.

Lemma remove_singleton_empty : In x s -> remove x s [=] empty -> singleton x [=] s.

Lemma union_sym : union s s' [=] union s' s.

Lemma union_subset_equal : s[<=]s' -> union s s' [=] s'.

Lemma union_equal_1 : s[=]s' -> union s s'' [=] union s' s''.

Lemma union_equal_2 : s'[=]s'' -> union s s' [=] union s s''.

Lemma union_assoc : union (union s s') s'' [=] union s (union s' s'').

Lemma add_union_singleton : add x s [=] union (singleton x) s.

Lemma union_add : union (add x s) s' [=] add x (union s s').

Lemma union_remove_add_1 : union (remove x s) (add x s') [=] union (add x s) (remove x s').

Lemma union_remove_add_2 : In x s -> union (remove x s) (add x s') [=] union s s'.

Lemma union_subset_1 : s [<=] union s s'.

Lemma union_subset_2 : s' [<=] union s s'.

Lemma union_subset_3 : s[<=]s'' -> s'[<=]s'' -> union s s' [<=] s''.

Lemma union_subset_4 : s[<=]s' -> union s s'' [<=] union s' s''.

Lemma union_subset_5 : s[<=]s' -> union s'' s [<=] union s'' s'.

Lemma empty_union_1 : Empty s -> union s s' [=] s'.

Lemma empty_union_2 : Empty s -> union s' s [=] s'.

Lemma not_in_union : ~In x s -> ~In x s' -> ~In x (union s s').

Lemma inter_sym : inter s s' [=] inter s' s.

Lemma inter_subset_equal : s[<=]s' -> inter s s' [=] s.

Lemma inter_equal_1 : s[=]s' -> inter s s'' [=] inter s' s''.

Lemma inter_equal_2 : s'[=]s'' -> inter s s' [=] inter s s''.

Lemma inter_assoc : inter (inter s s') s'' [=] inter s (inter s' s'').

Lemma union_inter_1 : inter (union s s') s'' [=] union (inter s s'') (inter s' s'').

Lemma union_inter_2 : union (inter s s') s'' [=] inter (union s s'') (union s' s'').

Lemma inter_add_1 : In x s' -> inter (add x s) s' [=] add x (inter s s').

Lemma inter_add_2 : ~ In x s' -> inter (add x s) s' [=] inter s s'.

Lemma empty_inter_1 : Empty s -> Empty (inter s s').

Lemma empty_inter_2 : Empty s' -> Empty (inter s s').

Lemma inter_subset_1 : inter s s' [<=] s.

Lemma inter_subset_2 : inter s s' [<=] s'.

Lemma inter_subset_3 : s''[<=]s -> s''[<=]s' -> s''[<=] inter s s'.

Lemma empty_diff_1 : Empty s -> Empty (diff s s').

Lemma empty_diff_2 : Empty s -> diff s' s [=] s'.

Lemma diff_subset : diff s s' [<=] s.

Lemma diff_subset_equal : s[<=]s' -> diff s s' [=] empty.

Lemma remove_diff_singleton : remove x s [=] diff s (singleton x).

Lemma diff_inter_empty : inter (diff s s') (inter s s') [=] empty.

Lemma diff_inter_all : union (diff s s') (inter s s') [=] s.

Lemma Add_add : Add x s (add x s).

Lemma Add_remove : In x s -> Add x (remove x s) s.

Lemma union_Add : Add x s s' -> Add x (union s s'') (union s' s'').

Lemma inter_Add : In x s'' -> Add x s s' -> Add x (inter s s'') (inter s' s'').

Lemma union_Equal : In x s'' -> Add x s s' -> union s s'' [=] union s' s''.

Lemma inter_Add_2 : ~In x s'' -> Add x s s' -> inter s s'' [=] inter s' s''.

End BasicProperties.

#[global] Hint Immediate equal_sym add_remove remove_add union_sym inter_sym: set.

#[global] Hint Resolve equal_refl equal_trans subset_refl subset_equal subset_antisym subset_trans subset_empty subset_remove_3 subset_diff subset_add_3 subset_add_2 in_subset empty_is_empty_1 empty_is_empty_2 add_equal remove_equal singleton_equal_add union_subset_equal union_equal_1 union_equal_2 union_assoc add_union_singleton union_add union_subset_1 union_subset_2 union_subset_3 inter_subset_equal inter_equal_1 inter_equal_2 inter_assoc union_inter_1 union_inter_2 inter_add_1 inter_add_2 empty_inter_1 empty_inter_2 empty_union_1 empty_union_2 empty_diff_1 empty_diff_2 union_Add inter_Add union_Equal inter_Add_2 not_in_union inter_subset_1 inter_subset_2 inter_subset_3 diff_subset diff_subset_equal remove_diff_singleton diff_inter_empty diff_inter_all Add_add Add_remove Equal_remove add_add : set.

(** * Properties of elements *)

Lemma elements_Empty : forall s, Empty s <-> elements s = nil.

assert (forall a, ~ List.In a (elements s)).

rewrite elements_iff.

rewrite InA_alt; exists a; auto with relations.

destruct (elements s); auto.

elim (H0 e); simpl; auto.

rewrite elements_iff in H0.

rewrite InA_alt in H0; destruct H0.

rewrite H in H0; destruct H0 as (_,H0); inversion H0.

Lemma elements_empty : elements empty = nil.

rewrite <-elements_Empty; auto with set.

(** * Conversions between lists and sets *)

Definition of_list (l : list elt) := List.fold_right add empty l.

Definition to_list := elements.

Lemma of_list_1 : forall l x, In x (of_list l) <-> InA E.eq x l.

induction l; simpl; intro x.

rewrite empty_iff, InA_nil.

rewrite add_iff, InA_cons, IHl.

Lemma of_list_2 : forall l, equivlistA E.eq (to_list (of_list l)) l.

unfold to_list; red; intros.

rewrite <- elements_iff; apply of_list_1.

Lemma of_list_3 : forall s, of_list (to_list s) [=] s.

unfold to_list; red; intros.

rewrite of_list_1; symmetry; apply elements_iff.

Notation NoDup := (NoDupA E.eq).

Notation InA := (InA E.eq).

(** Alternative specification via [fold_right] *)

Lemma fold_spec_right (s:t)(A:Type)(i:A)(f : elt -> A -> A) : fold f s i = List.fold_right f i (rev (elements s)).

rewrite fold_spec.

apply fold_left_rev_right.

(** ** Induction principles for fold (contributed by S. Lescuyer) *) (** In the following lemma, the step hypothesis is deliberately restricted to the precise set s we are considering. *)

Theorem fold_rec : forall (A:Type)(P : t -> A -> Type)(f : elt -> A -> A)(i:A)(s:t), (forall s', Empty s' -> P s' i) -> (forall x a s' s'', In x s -> ~In x s' -> Add x s' s'' -> P s' a -> P s'' (f x a)) -> P s (fold f s i).

intros A P f i s Pempty Pstep.

rewrite fold_spec_right.

set (l:=rev (elements s)).

assert (Pstep' : forall x a s' s'', InA x l -> ~In x s' -> Add x s' s'' -> P s' a -> P s'' (f x a)).

intros; eapply Pstep; eauto.

rewrite elements_iff, <- InA_rev; auto with *.

assert (Hdup : NoDup l) by (unfold l; eauto using elements_3w, NoDupA_rev with *).

assert (Hsame : forall x, In x s <-> InA x l) by (unfold l; intros; rewrite elements_iff, InA_rev; intuition).

clear Pstep; clearbody l; revert s Hsame; induction l.

intros s Hsame; simpl.

rewrite Hsame, InA_nil; intuition.

intros s Hsame; simpl.

apply Pstep' with (of_list l); auto with relations.

inversion_clear Hdup; rewrite of_list_1; auto.

rewrite Hsame, of_list_1, InA_cons; intuition.

intros; eapply Pstep'; eauto.

inversion_clear Hdup; auto.

exact (of_list_1 l).

(** Same, with [empty] and [add] instead of [Empty] and [Add]. In this case, [P] must be compatible with equality of sets *)

Theorem fold_rec_bis : forall (A:Type)(P : t -> A -> Type)(f : elt -> A -> A)(i:A)(s:t), (forall s s' a, s[=]s' -> P s a -> P s' a) -> (P empty i) -> (forall x a s', In x s -> ~In x s' -> P s' a -> P (add x s') (f x a)) -> P s (fold f s i).

intros A P f i s Pmorphism Pempty Pstep.

apply fold_rec; intros.

apply Pmorphism with empty; auto with set.

rewrite Add_Equal in H1; auto with set.

apply Pmorphism with (add x s'); auto with set.

Lemma fold_rec_nodep : forall (A:Type)(P : A -> Type)(f : elt -> A -> A)(i:A)(s:t), P i -> (forall x a, In x s -> P a -> P (f x a)) -> P (fold f s i).

intros; apply fold_rec_bis with (P:=fun _ => P); auto.

(** [fold_rec_weak] is a weaker principle than [fold_rec_bis] : the step hypothesis must here be applicable to any [x]. At the same time, it looks more like an induction principle, and hence can be easier to use. *)

Lemma fold_rec_weak : forall (A:Type)(P : t -> A -> Type)(f : elt -> A -> A)(i:A), (forall s s' a, s[=]s' -> P s a -> P s' a) -> P empty i -> (forall x a s, ~In x s -> P s a -> P (add x s) (f x a)) -> forall s, P s (fold f s i).

intros; apply fold_rec_bis; auto.

Lemma fold_rel : forall (A B:Type)(R : A -> B -> Type) (f : elt -> A -> A)(g : elt -> B -> B)(i : A)(j : B)(s : t), R i j -> (forall x a b, In x s -> R a b -> R (f x a) (g x b)) -> R (fold f s i) (fold g s j).

intros A B R f g i j s Rempty Rstep.

rewrite 2 fold_spec_right.

set (l:=rev (elements s)).

assert (Rstep' : forall x a b, InA x l -> R a b -> R (f x a) (g x b)) by (intros; apply Rstep; auto; rewrite elements_iff, <- InA_rev; auto with *).

clearbody l; clear Rstep s.

induction l; simpl; auto with relations.

(** From the induction principle on [fold], we can deduce some general induction principles on sets. *)

Lemma set_induction : forall P : t -> Type, (forall s, Empty s -> P s) -> (forall s s', P s -> forall x, ~In x s -> Add x s s' -> P s') -> forall s, P s.

apply (@fold_rec _ (fun s _ => P s) (fun _ _ => tt) tt s); eauto.

Lemma set_induction_bis : forall P : t -> Type, (forall s s', s [=] s' -> P s -> P s') -> P empty -> (forall x s, ~In x s -> P s -> P (add x s)) -> forall s, P s.

apply (@fold_rec_bis _ (fun s _ => P s) (fun _ _ => tt) tt s); eauto.

(** [fold] can be used to reconstruct the same initial set. *)

Lemma fold_identity : forall s, fold add s empty [=] s.

apply fold_rec with (P:=fun s acc => acc[=]s); auto with set.

rewrite H2; rewrite Add_Equal in H1; auto with set.

(** ** Alternative (weaker) specifications for [fold] *) (** When [MSets] was first designed, the order in which Ocaml's [Set.fold] takes the set elements was unspecified. This specification reflects this fact: *)

Lemma fold_0 : forall s (A : Type) (i : A) (f : elt -> A -> A), exists l : list elt, NoDup l /\ (forall x : elt, In x s <-> InA x l) /\ fold f s i = fold_right f i l.

intros; exists (rev (elements s)); split.

apply NoDupA_rev; auto with *.

rewrite elements_iff; do 2 rewrite InA_alt.

split; destruct 1; generalize (In_rev (elements s) x0); exists x0; intuition.

apply fold_spec_right.

(** An alternate (and previous) specification for [fold] was based on the recursive structure of a set. It is now lemmas [fold_1] and [fold_2]. *)

Lemma fold_1 : forall s (A : Type) (eqA : A -> A -> Prop) (st : Equivalence eqA) (i : A) (f : elt -> A -> A), Empty s -> eqA (fold f s i) i.

unfold Empty; intros; destruct (fold_0 s i f) as (l,(H1, (H2, H3))).

rewrite H3; clear H3.

generalize H H2; clear H H2; case l; simpl; intros.

elim (H2 e); intuition.

Lemma fold_2 : forall s s' x (A : Type) (eqA : A -> A -> Prop) (st : Equivalence eqA) (i : A) (f : elt -> A -> A), Proper (E.eq==>eqA==>eqA) f -> transpose eqA f -> ~ In x s -> Add x s s' -> eqA (fold f s' i) (f x (fold f s i)).

intros; destruct (fold_0 s i f) as (l,(Hl, (Hl1, Hl2))); destruct (fold_0 s' i f) as (l',(Hl', (Hl'1, Hl'2))).

rewrite Hl2; rewrite Hl'2; clear Hl2 Hl'2.

apply fold_right_add with (eqA:=E.eq)(eqB:=eqA); auto.

rewrite <- Hl1; auto.

intros a; rewrite InA_cons; rewrite <- Hl1; rewrite <- Hl'1; rewrite (H2 a); intuition.

(** In fact, [fold] on empty sets is more than equivalent to the initial element, it is Leibniz-equal to it. *)

Lemma fold_1b : forall s (A : Type)(i : A) (f : elt -> A -> A), Empty s -> (fold f s i) = i.

rewrite FM.fold_1.

rewrite elements_Empty in H; rewrite H; simpl; auto.

Section Fold_More.

Variables (A:Type)(eqA:A->A->Prop)(st:Equivalence eqA).

Variables (f:elt->A->A)(Comp:Proper (E.eq==>eqA==>eqA) f)(Ass:transpose eqA f).

Lemma fold_commutes : forall i s x, eqA (fold f s (f x i)) (f x (fold f s i)).

apply fold_rel with (R:=fun u v => eqA u (f x v)); intros.

transitivity (f x0 (f x b)); auto.

apply Comp; auto with relations.

(** ** Fold is a morphism *)

Lemma fold_init : forall i i' s, eqA i i' -> eqA (fold f s i) (fold f s i').

apply fold_rel with (R:=eqA); auto.

intros; apply Comp; auto with relations.

Lemma fold_equal : forall i s s', s[=]s' -> eqA (fold f s i) (fold f s' i).

intros i s; pattern s; apply set_induction; clear s; intros.

apply fold_1; auto.

symmetry; apply fold_1; auto.

rewrite <- H0; auto.

transitivity (f x (fold f s i)).

apply fold_2 with (eqA := eqA); auto.

symmetry; apply fold_2 with (eqA := eqA); auto.

unfold Add in *; intros.

rewrite <- H2; auto.

(** ** Fold and other set operators *)

Lemma fold_empty : forall i, fold f empty i = i.

intros i; apply fold_1b; auto with set.

Lemma fold_add : forall i s x, ~In x s -> eqA (fold f (add x s) i) (f x (fold f s i)).

intros; apply fold_2 with (eqA := eqA); auto with set.

Lemma add_fold : forall i s x, In x s -> eqA (fold f (add x s) i) (fold f s i).

intros; apply fold_equal; auto with set.

Lemma remove_fold_1: forall i s x, In x s -> eqA (f x (fold f (remove x s) i)) (fold f s i).

apply fold_2 with (eqA:=eqA); auto with set relations.

Lemma remove_fold_2: forall i s x, ~In x s -> eqA (fold f (remove x s) i) (fold f s i).

apply fold_equal; auto with set.

Lemma fold_union_inter : forall i s s', eqA (fold f (union s s') (fold f (inter s s') i)) (fold f s (fold f s' i)).

intros; pattern s; apply set_induction; clear s; intros.

transitivity (fold f s' (fold f (inter s s') i)).

apply fold_equal; auto with set.

transitivity (fold f s' i).

apply fold_init; auto.

apply fold_1; auto with set.

symmetry; apply fold_1; auto.

rename s'0 into s''.

destruct (In_dec x s').

transitivity (fold f (union s'' s') (f x (fold f (inter s s') i))); auto with set.

apply fold_init; auto.

apply fold_2 with (eqA:=eqA); auto with set.

rewrite inter_iff; intuition.

transitivity (f x (fold f s (fold f s' i))).

1:transitivity (fold f (union s s') (f x (fold f (inter s s') i))).

2:transitivity (f x (fold f (union s s') (fold f (inter s s') i))).

apply fold_equal; auto.

apply equal_sym; apply union_Equal with x; auto with set.

apply fold_commutes; auto.

apply Comp; auto with relations.

symmetry; apply fold_2 with (eqA:=eqA); auto.

transitivity (f x (fold f (union s s') (fold f (inter s'' s') i))).

apply fold_2 with (eqA:=eqA); auto with set.

transitivity (f x (fold f (union s s') (fold f (inter s s') i))).

apply Comp;auto with relations.

apply fold_init;auto.

apply fold_equal;auto.

apply equal_sym; apply inter_Add_2 with x; auto with set.

transitivity (f x (fold f s (fold f s' i))).

apply Comp; auto with relations.

symmetry; apply fold_2 with (eqA:=eqA); auto.

Lemma fold_diff_inter : forall i s s', eqA (fold f (diff s s') (fold f (inter s s') i)) (fold f s i).

transitivity (fold f (union (diff s s') (inter s s')) (fold f (inter (diff s s') (inter s s')) i)).

1:symmetry; apply fold_union_inter; auto.

transitivity (fold f s (fold f (inter (diff s s') (inter s s')) i)).

apply fold_equal; auto with set.

apply fold_init; auto.

apply fold_1; auto with set.

Lemma fold_union: forall i s s', (forall x, ~(In x s/\In x s')) -> eqA (fold f (union s s') i) (fold f s (fold f s' i)).

transitivity (fold f (union s s') (fold f (inter s s') i)).

apply fold_init; auto.

symmetry; apply fold_1; auto with set.

unfold Empty; intro a; generalize (H a); set_iff; tauto.

apply fold_union_inter; auto.

Lemma fold_plus : forall s p, fold (fun _ => S) s p = fold (fun _ => S) s 0 + p.

apply fold_rel with (R:=fun u v => u = v + p); simpl; auto.

(** * Cardinal *) (** ** Characterization of cardinal in terms of fold *)

Lemma cardinal_fold : forall s, cardinal s = fold (fun _ => S) s 0.

intros; rewrite cardinal_1; rewrite FM.fold_1.

symmetry; apply fold_left_length; auto.

(** ** Old specifications for [cardinal]. *)

Lemma cardinal_0 : forall s, exists l : list elt, NoDupA E.eq l /\ (forall x : elt, In x s <-> InA E.eq x l) /\ cardinal s = length l.

intros; exists (elements s); intuition auto with set; apply cardinal_1.

Lemma cardinal_1 : forall s, Empty s -> cardinal s = 0.

intros; rewrite cardinal_fold; apply fold_1; auto with *.

Lemma cardinal_2 : forall s s' x, ~ In x s -> Add x s s' -> cardinal s' = S (cardinal s).

intros; do 2 rewrite cardinal_fold.

change S with ((fun _ => S) x).

apply fold_2; auto.

split; congruence.

(** ** Cardinal and (non-)emptiness *)

Lemma cardinal_Empty : forall s, Empty s <-> cardinal s = 0.

rewrite elements_Empty, FM.cardinal_1.

destruct (elements s); intuition; discriminate.

Lemma cardinal_inv_1 : forall s, cardinal s = 0 -> Empty s.

intros; rewrite cardinal_Empty; auto.

#[global] Hint Resolve cardinal_inv_1 : core.

Lemma cardinal_inv_2 : forall s n, cardinal s = S n -> { x : elt | In x s }.

intros; rewrite FM.cardinal_1 in H.

generalize (elements_2 (s:=s)).

destruct (elements s); try discriminate.

exists e; auto with relations.

Lemma cardinal_inv_2b : forall s, cardinal s <> 0 -> { x : elt | In x s }.

intro; generalize (@cardinal_inv_2 s); destruct cardinal; [intuition|eauto].

(** ** Cardinal is a morphism *)

Lemma Equal_cardinal : forall s s', s[=]s' -> cardinal s = cardinal s'.

remember (cardinal s) as n; symmetry in Heqn; revert s s' Heqn H.

induction n; intros.

apply cardinal_1; rewrite <- H; auto.

destruct (cardinal_inv_2 Heqn) as (x,H2).

rewrite (cardinal_2 (s:=remove x s) (s':=s) (x:=x)); auto with set relations.

rewrite (cardinal_2 (s:=remove x s') (s':=s') (x:=x)); eauto with set relations.

#[global] Instance cardinal_m : Proper (Equal==>Logic.eq) cardinal.

exact Equal_cardinal.

#[global] Hint Resolve Add_add Add_remove Equal_remove cardinal_inv_1 Equal_cardinal : core.

(** ** Cardinal and set operators *)

Lemma empty_cardinal : cardinal empty = 0.

rewrite cardinal_fold; apply fold_1; auto with *.

#[global] Hint Immediate empty_cardinal cardinal_1 : set.

Lemma singleton_cardinal : forall x, cardinal (singleton x) = 1.

rewrite (singleton_equal_add x).

replace 0 with (cardinal empty); auto with set.

apply cardinal_2 with x; auto with set.

#[global] Hint Resolve singleton_cardinal: set.

Lemma diff_inter_cardinal : forall s s', cardinal (diff s s') + cardinal (inter s s') = cardinal s .

intros; do 3 rewrite cardinal_fold.

rewrite <- fold_plus.

apply fold_diff_inter with (eqA:=@Logic.eq nat); auto with *.

Lemma union_cardinal: forall s s', (forall x, ~(In x s/\In x s')) -> cardinal (union s s')=cardinal s+cardinal s'.

intros; do 3 rewrite cardinal_fold.

rewrite <- fold_plus.

apply fold_union; auto.

split; congruence.

Lemma subset_cardinal : forall s s', s[<=]s' -> cardinal s <= cardinal s' .

rewrite <- (diff_inter_cardinal s' s).

rewrite (inter_sym s' s).

rewrite (inter_subset_equal H).

apply Nat.le_add_l.

Lemma subset_cardinal_lt : forall s s' x, s[<=]s' -> In x s' -> ~In x s -> cardinal s < cardinal s'.

rewrite <- (diff_inter_cardinal s' s).

rewrite (inter_sym s' s).

rewrite (inter_subset_equal H).

generalize (@cardinal_inv_1 (diff s' s)).

destruct (cardinal (diff s' s)).

intro H2; destruct (H2 (eq_refl _) x).

change (0 + cardinal s < S n + cardinal s).

apply Nat.add_lt_le_mono; [ apply Nat.lt_0_succ | reflexivity ].

Theorem union_inter_cardinal : forall s s', cardinal (union s s') + cardinal (inter s s') = cardinal s + cardinal s' .

do 4 rewrite cardinal_fold.

do 2 rewrite <- fold_plus.

apply fold_union_inter with (eqA:=@Logic.eq nat); auto with *.

Lemma union_cardinal_inter : forall s s', cardinal (union s s') = cardinal s + cardinal s' - cardinal (inter s s').

rewrite <- union_inter_cardinal, Nat.add_sub.

Lemma union_cardinal_le : forall s s', cardinal (union s s') <= cardinal s + cardinal s'.

intros; generalize (union_inter_cardinal s s').

intros; rewrite <- H; auto with arith.

Lemma add_cardinal_1 : forall s x, In x s -> cardinal (add x s) = cardinal s.

Lemma add_cardinal_2 : forall s x, ~In x s -> cardinal (add x s) = S (cardinal s).

do 2 rewrite cardinal_fold.

change S with ((fun _ => S) x); apply fold_add with (eqA:=@Logic.eq nat); auto with *.

Lemma remove_cardinal_1 : forall s x, In x s -> S (cardinal (remove x s)) = cardinal s.

do 2 rewrite cardinal_fold.

change S with ((fun _ =>S) x).

apply remove_fold_1 with (eqA:=@Logic.eq nat); auto with *.

Lemma remove_cardinal_2 : forall s x, ~In x s -> cardinal (remove x s) = cardinal s.

#[global] Hint Resolve subset_cardinal union_cardinal add_cardinal_1 add_cardinal_2 : core.

End WPropertiesOn.

(** Now comes variants for self-contained weak sets and for full sets. For these variants, only one argument is necessary. Thanks to the subtyping [WS<=S], the [Properties] functor which is meant to be used on modules [(M:S)] can simply be an alias of [WProperties]. *)

Module WProperties (M:WSets) := WPropertiesOn M.E M.

Module Properties := WProperties.

(** Now comes some properties specific to the element ordering, invalid for Weak Sets. *)

Module OrdProperties (M:Sets).

Module Import ME:=OrderedTypeFacts(M.E).

Module Import ML:=OrderedTypeLists(M.E).

Module Import P := Properties M.

#[global] Hint Resolve elements_spec2 : core.

#[global] Hint Immediate min_elt_spec1 min_elt_spec2 min_elt_spec3 max_elt_spec1 max_elt_spec2 max_elt_spec3 : set.

(** First, a specialized version of SortA_equivlistA_eqlistA: *)

Lemma sort_equivlistA_eqlistA : forall l l' : list elt, sort E.lt l -> sort E.lt l' -> equivlistA E.eq l l' -> eqlistA E.eq l l'.

apply SortA_equivlistA_eqlistA; eauto with *.

Definition gtb x y := match E.compare x y with Gt => true | _ => false end.

Definition leb x := fun y => negb (gtb x y).

Definition elements_lt x s := List.filter (gtb x) (elements s).

Definition elements_ge x s := List.filter (leb x) (elements s).

Lemma gtb_1 : forall x y, gtb x y = true <-> E.lt y x.

intros; rewrite <- compare_gt_iff.

destruct E.compare; intuition; try discriminate.

Lemma leb_1 : forall x y, leb x y = true <-> ~E.lt y x.

intros; rewrite <- compare_gt_iff.

destruct E.compare; intuition; try discriminate.

#[global] Instance gtb_compat x : Proper (E.eq==>Logic.eq) (gtb x).

#[global] Instance leb_compat x : Proper (E.eq==>Logic.eq) (leb x).

intros a b H; unfold leb.

#[global] Hint Resolve gtb_compat leb_compat : core.

Lemma elements_split : forall x s, elements s = elements_lt x s ++ elements_ge x s.

unfold elements_lt, elements_ge, leb; intros.

eapply (@filter_split _ E.eq); eauto with *.

rewrite gtb_1 in H.

assert (~E.lt y x).

unfold gtb in *; elim_compare x y; intuition; try discriminate; order.

Lemma elements_Add : forall s s' x, ~In x s -> Add x s s' -> eqlistA E.eq (elements s') (elements_lt x s ++ x :: elements_ge x s).

intros; unfold elements_ge, elements_lt.

apply sort_equivlistA_eqlistA; auto with set.

apply (@SortA_app _ E.eq); auto with *.

apply (@filter_sort _ E.eq); auto with *; eauto with *.

constructor; auto.

apply (@filter_sort _ E.eq); auto with *; eauto with *.

rewrite Inf_alt by (apply (@filter_sort _ E.eq); eauto with *).

rewrite filter_InA in H1; auto with *; destruct H1.

rewrite leb_1 in H2.

rewrite <- elements_iff in H1.

assert (~E.eq x y).

contradict H; rewrite H; auto.

rewrite filter_InA in H1; auto with *; destruct H1.

rewrite gtb_1 in H3.

inversion_clear H2.

rewrite filter_InA in H4; auto with *; destruct H4.

rewrite leb_1 in H4.

rewrite InA_app_iff, InA_cons, !filter_InA, <-!elements_iff, leb_1, gtb_1, (H0 a) by (auto with *).

elim_compare a x; intuition.

right; right; split; auto.

Definition Above x s := forall y, In y s -> E.lt y x.

Definition Below x s := forall y, In y s -> E.lt x y.

Lemma elements_Add_Above : forall s s' x, Above x s -> Add x s s' -> eqlistA E.eq (elements s') (elements s ++ x::nil).

apply sort_equivlistA_eqlistA; auto with set.

apply (@SortA_app _ E.eq); auto with *.

rewrite <- elements_iff in H1.

setoid_replace y with x; auto.

rewrite InA_app_iff, InA_cons, InA_nil, <-!elements_iff, (H0 a) by (auto with *).

Lemma elements_Add_Below : forall s s' x, Below x s -> Add x s s' -> eqlistA E.eq (elements s') (x::elements s).

apply sort_equivlistA_eqlistA; auto with set.

change (sort E.lt ((x::nil) ++ elements s)).

apply (@SortA_app _ E.eq); auto with *.

rewrite <- elements_iff in H2.

setoid_replace x0 with x; auto.

rewrite InA_cons, <- !elements_iff, (H0 a); intuition.

(** Two other induction principles on sets: we can be more restrictive on the element we add at each step. *)

Lemma set_induction_max : forall P : t -> Type, (forall s : t, Empty s -> P s) -> (forall s s', P s -> forall x, Above x s -> Add x s s' -> P s') -> forall s : t, P s.

intros; remember (cardinal s) as n; revert s Heqn; induction n; intros; auto.

case_eq (max_elt s); intros.

apply X0 with (remove e s) e; auto with set.

assert (S n = S (cardinal (remove e s))).

rewrite Heqn; apply cardinal_2 with e; auto with set relations.

inversion H0; auto.

rewrite remove_iff in H0; destruct H0.

generalize (@max_elt_spec2 s e y H H0); order.

assert (H0:=max_elt_spec3 H).

rewrite cardinal_Empty in H0; rewrite H0 in Heqn; inversion Heqn.

Lemma set_induction_min : forall P : t -> Type, (forall s : t, Empty s -> P s) -> (forall s s', P s -> forall x, Below x s -> Add x s s' -> P s') -> forall s : t, P s.

intros; remember (cardinal s) as n; revert s Heqn; induction n; intros; auto.

case_eq (min_elt s); intros.

apply X0 with (remove e s) e; auto with set.

assert (S n = S (cardinal (remove e s))).

rewrite Heqn; apply cardinal_2 with e; auto with set relations.

inversion H0; auto.

rewrite remove_iff in H0; destruct H0.

generalize (@min_elt_spec2 s e y H H0); order.

assert (H0:=min_elt_spec3 H).

rewrite cardinal_Empty in H0; auto; rewrite H0 in Heqn; inversion Heqn.

(** More properties of [fold] : behavior with respect to Above/Below *)

Lemma fold_3 : forall s s' x (A : Type) (eqA : A -> A -> Prop) (st : Equivalence eqA) (i : A) (f : elt -> A -> A), Proper (E.eq==>eqA==>eqA) f -> Above x s -> Add x s s' -> eqA (fold f s' i) (f x (fold f s i)).

rewrite 2 fold_spec_right.

change (f x (fold_right f i (rev (elements s)))) with (fold_right f i (rev (x::nil)++rev (elements s))).

apply (@fold_right_eqlistA E.t E.eq A eqA st); auto with *.

rewrite <- distr_rev.

apply eqlistA_rev.

apply elements_Add_Above; auto.

Lemma fold_4 : forall s s' x (A : Type) (eqA : A -> A -> Prop) (st : Equivalence eqA) (i : A) (f : elt -> A -> A), Proper (E.eq==>eqA==>eqA) f -> Below x s -> Add x s s' -> eqA (fold f s' i) (fold f s (f x i)).

rewrite !fold_spec.

change (eqA (fold_left (flip f) (elements s') i) (fold_left (flip f) (x::elements s) i)).

unfold flip; rewrite <-!fold_left_rev_right.

apply (@fold_right_eqlistA E.t E.eq A eqA st); auto.

apply eqlistA_rev.

apply elements_Add_Below; auto.

(** The following results have already been proved earlier, but we can now prove them with one hypothesis less: no need for [(transpose eqA f)]. *)

Variables (A:Type)(eqA:A->A->Prop)(st:Equivalence eqA).

Variables (f:elt->A->A)(Comp:Proper (E.eq==>eqA==>eqA) f).

Lemma fold_equal : forall i s s', s[=]s' -> eqA (fold f s i) (fold f s' i).

rewrite 2 fold_spec_right.

apply (@fold_right_eqlistA E.t E.eq A eqA st); auto.

apply eqlistA_rev.

apply sort_equivlistA_eqlistA; auto with set.

red; intro a; do 2 rewrite <- elements_iff; auto.

Lemma add_fold : forall i s x, In x s -> eqA (fold f (add x s) i) (fold f s i).

intros; apply fold_equal; auto with set.

Lemma remove_fold_2: forall i s x, ~In x s -> eqA (fold f (remove x s) i) (fold f s i).

apply fold_equal; auto with set.

(** An alternative version of [choose_3] *)

Lemma choose_equal : forall s s', Equal s s' -> match choose s, choose s' with | Some x, Some x' => E.eq x x' | None, None => True | _, _ => False end.

intros s s' H; generalize (@choose_spec1 s)(@choose_spec2 s) (@choose_spec1 s')(@choose_spec2 s')(@choose_spec3 s s'); destruct (choose s); destruct (choose s'); simpl; intuition.

Timings for MSetProperties.v