path: root/theories/FSets/FSetProperties.v

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(*  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       *)
(***********************************************************************)

(** * Finite sets library *)

(** This functor derives additional properties from [FSetInterface.S].
    Contrary to the functor in [FSetEqProperties] 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 FSetInterface.
Require Import DecidableTypeEx FSetFacts FSetDecide.
Set Implicit Arguments.
Unset Strict Implicit.

Hint Unfold transpose compat_op Proper respectful.
Hint Extern 1 (Equivalence _) => constructor; congruence.

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

Module WProperties_fun (Import E : DecidableType)(M : WSfun E).
  Module Import Dec := WDecide_fun E M.
  Module Import FM := Dec.F (* FSetFacts.WFacts_fun E M *).
  Import M.

  Lemma In_dec : forall x s, {In x s} + {~ In x s}.
  Proof.
  intros; generalize (mem_iff s x); case (mem x s); intuition.
  Qed.

  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.
  Proof.
  unfold Add.
  split; intros.
  red; intros.
  rewrite H; clear H.
  fsetdec.
  fsetdec.
  Qed.

  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.
  Proof. fsetdec. Qed.

  Lemma equal_sym : s[=]s' -> s'[=]s.
  Proof. fsetdec. Qed.

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

  Lemma subset_refl : s[<=]s.
  Proof. fsetdec. Qed.

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

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

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

  Lemma subset_empty : empty[<=]s.
  Proof. fsetdec. Qed.

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

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

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

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

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

  Lemma double_inclusion : s1[=]s2 <-> s1[<=]s2 /\ s2[<=]s1.
  Proof. intuition fsetdec. Qed.

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

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

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

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

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

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

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

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

  Lemma singleton_equal_add : singleton x [=] add x empty.
  Proof. fsetdec. Qed.

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

  Lemma union_sym : union s s' [=] union s' s.
  Proof. fsetdec. Qed.

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

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

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

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

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

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

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

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

  Lemma union_subset_1 : s [<=] union s s'.
  Proof. fsetdec. Qed.

  Lemma union_subset_2 : s' [<=] union s s'.
  Proof. fsetdec. Qed.

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

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

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

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

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

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

  Lemma inter_sym : inter s s' [=] inter s' s.
  Proof. fsetdec. Qed.

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

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

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

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

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

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

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

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

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

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

  Lemma inter_subset_1 : inter s s' [<=] s.
  Proof. fsetdec. Qed.

  Lemma inter_subset_2 : inter s s' [<=] s'.
  Proof. fsetdec. Qed.

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

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

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

  Lemma diff_subset : diff s s' [<=] s.
  Proof. fsetdec. Qed.

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

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

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

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

  Lemma Add_add : Add x s (add x s).
  Proof. expAdd; fsetdec. Qed.

  Lemma Add_remove : In x s -> Add x (remove x s) s.
  Proof. expAdd; fsetdec. Qed.

  Lemma union_Add : Add x s s' -> Add x (union s s'') (union s' s'').
  Proof. expAdd; fsetdec. Qed.

  Lemma inter_Add :
   In x s'' -> Add x s s' -> Add x (inter s s'') (inter s' s'').
  Proof. expAdd; fsetdec. Qed.

  Lemma union_Equal :
   In x s'' -> Add x s s' -> union s s'' [=] union s' s''.
  Proof. expAdd; fsetdec. Qed.

  Lemma inter_Add_2 :
   ~In x s'' -> Add x s s' -> inter s s'' [=] inter s' s''.
  Proof. expAdd; fsetdec. Qed.

  End BasicProperties.

  Hint Immediate equal_sym add_remove remove_add union_sym inter_sym: set.
  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.
  Proof.
  intros.
  unfold Empty.
  split; intros.
  assert (forall a, ~ List.In a (elements s)).
   red; intros.
   apply (H a).
   rewrite elements_iff.
   rewrite InA_alt; exists a; auto.
  destruct (elements s); auto.
  elim (H0 e); simpl; auto.
  red; intros.
  rewrite elements_iff in H0.
  rewrite InA_alt in H0; destruct H0.
  rewrite H in H0; destruct H0 as (_,H0); inversion H0.
  Qed.

  Lemma elements_empty : elements empty = nil.
  Proof.
  rewrite <-elements_Empty; auto with set.
  Qed.

  (** * 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.
  Proof.
  induction l; simpl; intro x.
  rewrite empty_iff, InA_nil. intuition.
  rewrite add_iff, InA_cons, IHl. intuition.
  Qed.

  Lemma of_list_2 : forall l, equivlistA E.eq (to_list (of_list l)) l.
  Proof.
  unfold to_list; red; intros.
  rewrite <- elements_iff; apply of_list_1.
  Qed.

  Lemma of_list_3 : forall s, of_list (to_list s) [=] s.
  Proof.
  unfold to_list; red; intros.
  rewrite of_list_1; symmetry; apply elements_iff.
  Qed.

  (** * Fold *)

  Section Fold.

  (** 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)).
  Proof.
   rewrite fold_1. symmetry. apply fold_left_rev_right.
  Qed.

  Notation NoDup := (NoDupA E.eq).
  Notation InA := (InA E.eq).

  (** ** 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).
  Proof.
  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.
  (* empty *)
  intros s Hsame; simpl.
  apply Pempty. intro x. rewrite Hsame, InA_nil; intuition.
  (* step *)
  intros s Hsame; simpl.
  apply Pstep' with (of_list l); auto.
   inversion_clear Hdup; rewrite of_list_1; auto.
   red. intros. rewrite Hsame, of_list_1, InA_cons; intuition.
  apply IHl.
   intros; eapply Pstep'; eauto.
   inversion_clear Hdup; auto.
   exact (of_list_1 l).
  Qed.

  (** 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).
  Proof.
  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.
  Qed.

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

  (** [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).
  Proof.
  intros; apply fold_rec_bis; auto.
  Qed.

  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).
  Proof.
  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.
  Qed.

  (** 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.
  Proof.
  intros. apply (@fold_rec _ (fun s _ => P s) (fun _ _ => tt) tt s); eauto.
  Qed.

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

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

  Lemma fold_identity : forall s, fold add s empty [=] s.
  Proof.
  intros.
  apply fold_rec with (P:=fun s acc => acc[=]s); auto with set.
  intros. rewrite H2; rewrite Add_Equal in H1; auto with set.
  Qed.

  (** ** Alternative (weaker) specifications for [fold] *)

  (** When [FSets] 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.
  Proof.
  intros; exists (rev (elements s)); split.
  apply NoDupA_rev; auto with *.
  split; intros.
  rewrite elements_iff; do 2 rewrite InA_alt.
  split; destruct 1; generalize (In_rev (elements s) x0); exists x0; intuition.
  apply fold_spec_right.
  Qed.

  (** 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.
  Proof.
  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.
  reflexivity.
  elim (H e).
  elim (H2 e); intuition.
  Qed.

  Lemma fold_2 :
   forall s s' x (A : Type) (eqA : A -> A -> Prop)
     (st : Equivalence eqA) (i : A) (f : elt -> A -> A),
   compat_op E.eq eqA f ->
   transpose eqA f ->
   ~ In x s -> Add x s s' -> eqA (fold f s' i) (f x (fold f s i)).
  Proof.
  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 with *.
  rewrite <- Hl1; auto.
  intros a; rewrite InA_cons; rewrite <- Hl1; rewrite <- Hl'1;
   rewrite (H2 a); intuition.
  Qed.

  (** 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.
  Proof.
  intros.
  rewrite M.fold_1.
  rewrite elements_Empty in H; rewrite H; simpl; auto.
  Qed.

  Section Fold_More.

  Variables (A:Type)(eqA:A->A->Prop)(st:Equivalence eqA).
  Variables (f:elt->A->A)(Comp:compat_op E.eq 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)).
  Proof.
  intros.
  apply fold_rel with (R:=fun u v => eqA u (f x v)); intros.
  reflexivity.
  transitivity (f x0 (f x b)); auto. apply Comp; auto with *.
  Qed.

  (** ** Fold is a morphism *)

  Lemma fold_init : forall i i' s, eqA i i' ->
   eqA (fold f s i) (fold f s i').
  Proof.
  intros. apply fold_rel with (R:=eqA); auto.
   intros; apply Comp; auto with *.
  Qed.

  Lemma fold_equal :
   forall i s s', s[=]s' -> eqA (fold f s i) (fold f s' i).
  Proof.
  intros i s; pattern s; apply set_induction; clear s; intros.
  transitivity i.
  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.
  Qed.

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

  Lemma fold_empty : forall i, fold f empty i = i.
  Proof.
  intros i; apply fold_1b; auto with set.
  Qed.

  Lemma fold_add : forall i s x, ~In x s ->
   eqA (fold f (add x s) i) (f x (fold f s i)).
  Proof.
  intros; apply fold_2 with (eqA := eqA); auto with set.
  Qed.

  Lemma add_fold : forall i s x, In x s ->
   eqA (fold f (add x s) i) (fold f s i).
  Proof.
  intros; apply fold_equal; auto with set.
  Qed.

  Lemma remove_fold_1: forall i s x, In x s ->
   eqA (f x (fold f (remove x s) i)) (fold f s i).
  Proof.
  intros.
  symmetry.
  apply fold_2 with (eqA:=eqA); auto with set.
  Qed.

  Lemma remove_fold_2: forall i s x, ~In x s ->
   eqA (fold f (remove x s) i) (fold f s i).
  Proof.
  intros.
  apply fold_equal; auto with set.
  Qed.

  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)).
  Proof.
  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').
  (* In 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))).
  transitivity (fold f (union s s') (f x (fold f (inter s s') i))).
  apply fold_equal; auto.
  apply equal_sym; apply union_Equal with x; auto with set.
  transitivity (f x (fold f (union s s') (fold f (inter s s') i))).
  apply fold_commutes; auto.
  apply Comp; auto.
  symmetry; apply fold_2 with (eqA:=eqA); auto.
  (* ~(In x s') *)
  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.
  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.
  symmetry; apply fold_2 with (eqA:=eqA); auto.
  Qed.

  Lemma fold_diff_inter : forall i s s',
   eqA (fold f (diff s s') (fold f (inter s s') i)) (fold f s i).
  Proof.
  intros.
  transitivity (fold f (union (diff s s') (inter s s'))
               (fold f (inter (diff s s') (inter s s')) i)).
  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.
  Qed.

  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)).
  Proof.
  intros.
  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.
  Qed.

  End Fold_More.

  Lemma fold_plus :
   forall s p, fold (fun _ => S) s p = fold (fun _ => S) s 0 + p.
  Proof.
  intros. apply fold_rel with (R:=fun u v => u = v + p); simpl; auto.
  Qed.

  End Fold.

  (** * Cardinal *)

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

  Lemma cardinal_fold : forall s, cardinal s = fold (fun _ => S) s 0.
  Proof.
  intros; rewrite cardinal_1; rewrite M.fold_1.
  symmetry; apply fold_left_length; auto.
  Qed.

  (** ** 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.
  Proof.
  intros; exists (elements s); intuition; apply cardinal_1.
  Qed.

  Lemma cardinal_1 : forall s, Empty s -> cardinal s = 0.
  Proof.
  intros; rewrite cardinal_fold; apply fold_1; auto.
  Qed.

  Lemma cardinal_2 :
    forall s s' x, ~ In x s -> Add x s s' -> cardinal s' = S (cardinal s).
  Proof.
  intros; do 2 rewrite cardinal_fold.
  change S with ((fun _ => S) x).
  apply fold_2; auto.
  Qed.

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

  Lemma cardinal_Empty : forall s, Empty s <-> cardinal s = 0.
  Proof.
  intros.
  rewrite elements_Empty, M.cardinal_1.
  destruct (elements s); intuition; discriminate.
  Qed.

  Lemma cardinal_inv_1 : forall s, cardinal s = 0 -> Empty s.
  Proof.
  intros; rewrite cardinal_Empty; auto.
  Qed.
  Hint Resolve cardinal_inv_1.

  Lemma cardinal_inv_2 :
   forall s n, cardinal s = S n -> { x : elt | In x s }.
  Proof.
  intros; rewrite M.cardinal_1 in H.
  generalize (elements_2 (s:=s)).
  destruct (elements s); try discriminate.
  exists e; auto.
  Qed.

  Lemma cardinal_inv_2b :
   forall s, cardinal s <> 0 -> { x : elt | In x s }.
  Proof.
  intro; generalize (@cardinal_inv_2 s); destruct cardinal;
   [intuition|eauto].
  Qed.

  (** ** Cardinal is a morphism *)

  Lemma Equal_cardinal : forall s s', s[=]s' -> cardinal s = cardinal s'.
  Proof.
  symmetry.
  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).
  revert Heqn.
  rewrite (cardinal_2 (s:=remove x s) (s':=s) (x:=x)); auto with set.
  rewrite (cardinal_2 (s:=remove x s') (s':=s') (x:=x)); eauto with set.
  Qed.

  Add Morphism cardinal : cardinal_m.
  Proof.
  exact Equal_cardinal.
  Qed.

  Hint Resolve Add_add Add_remove Equal_remove cardinal_inv_1 Equal_cardinal.

  (** ** Cardinal and set operators *)

  Lemma empty_cardinal : cardinal empty = 0.
  Proof.
  rewrite cardinal_fold; apply fold_1; auto with set.
  Qed.

  Hint Immediate empty_cardinal cardinal_1 : set.

  Lemma singleton_cardinal : forall x, cardinal (singleton x) = 1.
  Proof.
  intros.
  rewrite (singleton_equal_add x).
  replace 0 with (cardinal empty); auto with set.
  apply cardinal_2 with x; auto with set.
  Qed.

  Hint Resolve singleton_cardinal: set.

  Lemma diff_inter_cardinal :
   forall s s', cardinal (diff s s') + cardinal (inter s s') = cardinal s .
  Proof.
  intros; do 3 rewrite cardinal_fold.
  rewrite <- fold_plus.
  apply fold_diff_inter with (eqA:=@Logic.eq nat); auto.
  Qed.

  Lemma union_cardinal:
   forall s s', (forall x, ~(In x s/\In x s')) ->
   cardinal (union s s')=cardinal s+cardinal s'.
  Proof.
  intros; do 3 rewrite cardinal_fold.
  rewrite <- fold_plus.
  apply fold_union; auto.
  Qed.

  Lemma subset_cardinal :
   forall s s', s[<=]s' -> cardinal s <= cardinal s' .
  Proof.
  intros.
  rewrite <- (diff_inter_cardinal s' s).
  rewrite (inter_sym s' s).
  rewrite (inter_subset_equal H); auto with arith.
  Qed.

  Lemma subset_cardinal_lt :
   forall s s' x, s[<=]s' -> In x s' -> ~In x s -> cardinal s < cardinal s'.
  Proof.
  intros.
  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 (refl_equal _) x).
  set_iff; auto.
  intros _.
  change (0 + cardinal s < S n + cardinal s).
  apply Plus.plus_lt_le_compat; auto with arith.
  Qed.

  Theorem union_inter_cardinal :
   forall s s', cardinal (union s s') + cardinal (inter s s')  = cardinal s  + cardinal s' .
  Proof.
  intros.
  do 4 rewrite cardinal_fold.
  do 2 rewrite <- fold_plus.
  apply fold_union_inter with (eqA:=@Logic.eq nat); auto.
  Qed.

  Lemma union_cardinal_inter :
   forall s s', cardinal (union s s') = cardinal s + cardinal s' - cardinal (inter s s').
  Proof.
  intros.
  rewrite <- union_inter_cardinal.
  rewrite Plus.plus_comm.
  auto with arith.
  Qed.

  Lemma union_cardinal_le :
   forall s s', cardinal (union s s') <= cardinal s  + cardinal s'.
  Proof.
   intros; generalize (union_inter_cardinal s s').
   intros; rewrite <- H; auto with arith.
  Qed.

  Lemma add_cardinal_1 :
   forall s x, In x s -> cardinal (add x s) = cardinal s.
  Proof.
  auto with set.
  Qed.

  Lemma add_cardinal_2 :
   forall s x, ~In x s -> cardinal (add x s) = S (cardinal s).
  Proof.
  intros.
  do 2 rewrite cardinal_fold.
  change S with ((fun _ => S) x);
   apply fold_add with (eqA:=@Logic.eq nat); auto.
  Qed.

  Lemma remove_cardinal_1 :
   forall s x, In x s -> S (cardinal (remove x s)) = cardinal s.
  Proof.
  intros.
  do 2 rewrite cardinal_fold.
  change S with ((fun _ =>S) x).
  apply remove_fold_1 with (eqA:=@Logic.eq nat); auto.
  Qed.

  Lemma remove_cardinal_2 :
   forall s x, ~In x s -> cardinal (remove x s) = cardinal s.
  Proof.
  auto with set.
  Qed.

  Hint Resolve subset_cardinal union_cardinal add_cardinal_1 add_cardinal_2.

End WProperties_fun.

(** 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:WS) := WProperties_fun M.E M.
Module Properties := WProperties.


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

Module OrdProperties (M:S).
  Module ME:=OrderedTypeFacts(M.E).
  Module Import P := Properties M.
  Import FM.
  Import M.E.
  Import M.

  (** 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'.
  Proof.
  apply SortA_equivlistA_eqlistA; eauto with *.
  Qed.

  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.
  Proof.
   intros; unfold gtb; destruct (E.compare x y); intuition; try discriminate; ME.order.
  Qed.

  Lemma leb_1 : forall x y, leb x y = true <-> ~E.lt y x.
  Proof.
   intros; unfold leb, gtb; destruct (E.compare x y); intuition; try discriminate; ME.order.
  Qed.

  Lemma gtb_compat : forall x, Proper (E.eq==>Logic.eq) (gtb x).
  Proof.
   red; intros x a b H.
   generalize (gtb_1 x a)(gtb_1 x b); destruct (gtb x a); destruct (gtb x b); auto.
   intros.
   symmetry; rewrite H1.
   apply ME.eq_lt with a; auto.
   rewrite <- H0; auto.
   intros.
   rewrite H0.
   apply ME.eq_lt with b; auto.
   rewrite <- H1; auto.
  Qed.

  Lemma leb_compat : forall x, Proper (E.eq==>Logic.eq) (leb x).
  Proof.
   red; intros x a b H; unfold leb.
   f_equal; apply gtb_compat; auto.
  Qed.
  Hint Resolve gtb_compat leb_compat.

  Lemma elements_split : forall x s,
   elements s = elements_lt x s ++ elements_ge x s.
  Proof.
  unfold elements_lt, elements_ge, leb; intros.
  eapply (@filter_split _ E.eq _ E.lt); auto with *.
  intros.
  rewrite gtb_1 in H.
  assert (~E.lt y x).
   unfold gtb in *; destruct (E.compare x y); intuition;
   try discriminate; ME.order.
  ME.order.
  Qed.

  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).
  Proof.
  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 *.
  constructor; auto.
  apply (@filter_sort _ E.eq); auto with *.
  rewrite ME.Inf_alt by (apply (@filter_sort _ E.eq); eauto with *).
  intros.
  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.
  ME.order.
  intros.
  rewrite filter_InA in H1; auto with *; destruct H1.
  rewrite gtb_1 in H3.
  inversion_clear H2.
  ME.order.
  rewrite filter_InA in H4; auto with *; destruct H4.
  rewrite leb_1 in H4.
  ME.order.
  red; intros a.
  rewrite InA_app_iff, InA_cons, !filter_InA, <-elements_iff,
   leb_1, gtb_1, (H0 a) by auto with *.
  intuition.
  destruct (E.compare a x); intuition.
  right; right; split; auto with *.
  ME.order.
  Qed.

  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).
  Proof.
  intros.
  apply sort_equivlistA_eqlistA; auto with *.
  apply (@SortA_app _ E.eq); auto with *.
  intros.
  inversion_clear H2.
  rewrite <- elements_iff in H1.
  apply ME.lt_eq with x; auto.
  inversion H3.
  red; intros a.
  rewrite InA_app_iff, InA_cons, InA_nil by auto with *.
  do 2 rewrite <- elements_iff; rewrite (H0 a); intuition.
  Qed.

  Lemma elements_Add_Below : forall s s' x,
   Below x s -> Add x s s' ->
     eqlistA E.eq (elements s') (x::elements s).
  Proof.
  intros.
  apply sort_equivlistA_eqlistA; auto with *.
  change (sort E.lt ((x::nil) ++ elements s)).
  apply (@SortA_app _ E.eq); auto with *.
  intros.
  inversion_clear H1.
  rewrite <- elements_iff in H2.
  apply ME.eq_lt with x; auto.
  inversion H3.
  red; intros a.
  rewrite InA_cons.
  do 2 rewrite <- elements_iff; rewrite (H0 a); intuition.
  Qed.

  (** 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.
  Proof.
  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.
  apply IHn.
  assert (S n = S (cardinal (remove e s))).
   rewrite Heqn; apply cardinal_2 with e; auto with set.
  inversion H0; auto.
  red; intros.
  rewrite remove_iff in H0; destruct H0.
  generalize (@max_elt_2 s e y H H0); ME.order.

  assert (H0:=max_elt_3 H).
  rewrite cardinal_Empty in H0; rewrite H0 in Heqn; inversion Heqn.
  Qed.

  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.
  Proof.
  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.
  apply IHn.
  assert (S n = S (cardinal (remove e s))).
   rewrite Heqn; apply cardinal_2 with e; auto with set.
  inversion H0; auto.
  red; intros.
  rewrite remove_iff in H0; destruct H0.
  generalize (@min_elt_2 s e y H H0); ME.order.

  assert (H0:=min_elt_3 H).
  rewrite cardinal_Empty in H0; auto; rewrite H0 in Heqn; inversion Heqn.
  Qed.

  (** 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),
   compat_op E.eq eqA f ->
   Above x s -> Add x s s' -> eqA (fold f s' i) (f x (fold f s i)).
  Proof.
  intros.
  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.
  rewrite <- distr_rev.
  apply eqlistA_rev.
  apply elements_Add_Above; auto.
  Qed.

  Lemma fold_4 :
   forall s s' x (A : Type) (eqA : A -> A -> Prop)
     (st : Equivalence eqA) (i : A) (f : elt -> A -> A),
   compat_op E.eq eqA f ->
   Below x s -> Add x s s' -> eqA (fold f s' i) (fold f s (f x i)).
  Proof.
  intros.
  rewrite 2 M.fold_1.
  set (g:=fun (a : A) (e : elt) => f e a).
  change (eqA (fold_left g (elements s') i) (fold_left g (x::elements s) i)).
  unfold g.
  rewrite <- 2 fold_left_rev_right.
  apply (@fold_right_eqlistA E.t E.eq A eqA st); auto.
  apply eqlistA_rev.
  apply elements_Add_Below; auto.
  Qed.

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

  Section FoldOpt.
  Variables (A:Type)(eqA:A->A->Prop)(st:Equivalence eqA).
  Variables (f:elt->A->A)(Comp:compat_op E.eq eqA f).

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

  Lemma add_fold : forall i s x, In x s ->
   eqA (fold f (add x s) i) (fold f s i).
  Proof.
  intros; apply fold_equal; auto with set.
  Qed.

  Lemma remove_fold_2: forall i s x, ~In x s ->
   eqA (fold f (remove x s) i) (fold f s i).
  Proof.
  intros.
  apply fold_equal; auto with set.
  Qed.

  End FoldOpt.

  (** 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.
  Proof.
  intros s s' H;
  generalize (@choose_1 s)(@choose_2 s)
             (@choose_1 s')(@choose_2 s')(@choose_3 s s');
  destruct (choose s); destruct (choose s'); simpl; intuition.
  apply H5 with e; rewrite <-H; auto.
  apply H5 with e; rewrite H; auto.
  Qed.

End OrdProperties.