Ajout de theories/FSets contenant la partie "light" de FSets et FMap:

pas d'implementations par AVL, mais celles par lists, ainsi que les 
foncteurs de proprietes. 

Au passage, ajout de MoreList (complements de List) et SetoidList 
(quelques relations sur des listes considerees modulo un eq ou lt 
non standard.



git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@8628 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 750 insertions, 0 deletions

diff --git a/theories/FSets/FSetBridge.v b/theories/FSets/FSetBridge.v
new file mode 100644
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+++ b/theories/FSets/FSetBridge.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       *)
+(***********************************************************************)
+
+(* $Id: FSetBridge.v,v 1.6 2006/03/09 18:34:51 letouzey Exp $ *)
+
+(** * Finite sets library *)
+
+(** This module implements bridges (as functors) from dependent
+    to/from non-dependent set signature. *)
+
+Require Export FSetInterface.
+Set Implicit Arguments.
+Unset Strict Implicit.
+Set Firstorder Depth 2.
+
+(** * From non-dependent signature [S] to dependent signature [Sdep]. *)
+
+Module DepOfNodep (M: S) <: Sdep with Module E := M.E.
+  Import M.
+
+  Module ME := OrderedTypeFacts E.
+   
+  Definition empty : {s : t | Empty s}.
+  Proof. 
+    exists empty; auto.
+  Qed.
+
+  Definition is_empty : forall s : t, {Empty s} + {~ Empty s}.
+  Proof. 
+    intros; generalize (is_empty_1 (s:=s)) (is_empty_2 (s:=s)).
+    case (is_empty s); intuition.
+  Qed.
+
+
+  Definition mem : forall (x : elt) (s : t), {In x s} + {~ In x s}.
+  Proof. 
+    intros; generalize (mem_1 (s:=s) (x:=x)) (mem_2 (s:=s) (x:=x)).
+    case (mem x s); intuition.
+  Qed.
+ 
+  Definition Add (x : elt) (s s' : t) :=
+    forall y : elt, In y s' <-> E.eq x y \/ In y s.
+ 
+  Definition add : forall (x : elt) (s : t), {s' : t | Add x s s'}.
+  Proof.
+    intros; exists (add x s); auto.
+    unfold Add in |- *; intuition.
+    elim (ME.eq_dec x y); auto.
+    intros; right. 
+    eapply add_3; eauto.
+  Qed. 
+ 
+  Definition singleton :
+    forall x : elt, {s : t | forall y : elt, In y s <-> E.eq x y}.
+  Proof. 
+    intros; exists (singleton x); intuition.
+  Qed.
+ 
+  Definition remove :
+    forall (x : elt) (s : t),
+    {s' : t | forall y : elt, In y s' <-> ~ E.eq x y /\ In y s}.
+  Proof.
+    intros; exists (remove x s); intuition.
+    absurd (In x (remove x s)); auto.
+    apply In_1 with y; auto. 
+    elim (ME.eq_dec x y); intros; auto.
+    absurd (In x (remove x s)); auto.
+    apply In_1 with y; auto. 
+    eauto.
+  Qed.
+
+  Definition union :
+    forall s s' : t, {s'' : t | forall x : elt, In x s'' <-> In x s \/ In x s'}.
+  Proof.
+    intros; exists (union s s'); intuition.
+  Qed.    
+
+  Definition inter :
+    forall s s' : t, {s'' : t | forall x : elt, In x s'' <-> In x s /\ In x s'}.
+  Proof. 
+    intros; exists (inter s s'); intuition; eauto.
+  Qed.
+
+  Definition diff :
+    forall s s' : t, {s'' : t | forall x : elt, In x s'' <-> In x s /\ ~ In x s'}.
+  Proof. 
+    intros; exists (diff s s'); intuition; eauto. 
+    absurd (In x s'); eauto. 
+  Qed. 
+ 
+  Definition equal : forall s s' : t, {Equal s s'} + {~ Equal s s'}.
+  Proof. 
+    intros. 
+    generalize (equal_1 (s:=s) (s':=s')) (equal_2 (s:=s) (s':=s')).
+    case (equal s s'); intuition.
+  Qed.
+
+  Definition subset : forall s s' : t, {Subset s s'} + {~Subset s s'}.
+  Proof. 
+    intros. 
+    generalize (subset_1 (s:=s) (s':=s')) (subset_2 (s:=s) (s':=s')).
+    case (subset s s'); intuition.
+  Qed.   
+
+  Definition elements :
+    forall s : t,
+    {l : list elt | ME.Sort l /\ (forall x : elt, In x s <-> ME.In x l)}.
+   Proof.
+     intros; exists (elements s); intuition.   
+   Defined. 
+
+  Definition fold :
+    forall (A : Set) (f : elt -> A -> A) (s : t) (i : A),
+    {r : A |   let (l,_) := elements s in 
+                  r = fold_left (fun a e => f e a) l i}.
+  Proof. 
+  intros; exists (fold (A:=A) f s i); exact (fold_1 s i f).
+  Qed.
+
+  Definition cardinal :
+      forall s : t,
+      {r : nat | let (l,_) := elements s in r = length l }.
+  Proof.
+    intros; exists (cardinal s); exact (cardinal_1 s).
+  Qed.    
+
+  Definition fdec (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x})
+    (x : elt) := if Pdec x then true else false. 
+
+  Lemma compat_P_aux :
+   forall (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x}),
+   compat_P E.eq P -> compat_bool E.eq (fdec Pdec).
+  Proof.
+    unfold compat_P, compat_bool, fdec in |- *; intros.
+    generalize (E.eq_sym H0); case (Pdec x); case (Pdec y); firstorder.
+  Qed.
+
+  Hint Resolve compat_P_aux.
+
+  Definition filter :
+    forall (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x}) (s : t),
+    {s' : t | compat_P E.eq P -> forall x : elt, In x s' <-> In x s /\ P x}.
+  Proof.
+    intros. 
+    exists (filter (fdec Pdec) s).
+    intro H; assert (compat_bool E.eq (fdec Pdec)); auto.
+    intuition.
+    eauto.
+    generalize (filter_2 H0 H1).
+    unfold fdec in |- *.
+    case (Pdec x); intuition.
+    inversion H2.
+    apply filter_3; auto.
+    unfold fdec in |- *; simpl in |- *.
+    case (Pdec x); intuition.
+  Qed.
+
+  Definition for_all :
+    forall (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x}) (s : t),
+    {compat_P E.eq P -> For_all P s} + {compat_P E.eq P -> ~ For_all P s}.
+  Proof. 
+    intros. 
+    generalize (for_all_1 (s:=s) (f:=fdec Pdec))
+     (for_all_2 (s:=s) (f:=fdec Pdec)).
+    case (for_all (fdec Pdec) s); unfold For_all in |- *; [ left | right ];
+     intros.
+    assert (compat_bool E.eq (fdec Pdec)); auto.
+    generalize (H0 H3 (refl_equal _) _ H2).
+    unfold fdec in |- *. 
+    case (Pdec x); intuition.
+    inversion H4.
+    intuition.    
+    absurd (false = true); [ auto with bool | apply H; auto ].
+    intro.
+    unfold fdec in |- *. 
+    case (Pdec x); intuition.
+  Qed.
+
+  Definition exists_ :
+    forall (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x}) (s : t),
+    {compat_P E.eq P -> Exists P s} + {compat_P E.eq P -> ~ Exists P s}.
+  Proof. 
+    intros. 
+    generalize (exists_1 (s:=s) (f:=fdec Pdec))
+     (exists_2 (s:=s) (f:=fdec Pdec)).
+    case (exists_ (fdec Pdec) s); unfold Exists in |- *; [ left | right ];
+     intros.
+    elim H0; auto; intros.
+    exists x; intuition.
+    generalize H4.
+    unfold fdec in |- *. 
+    case (Pdec x); intuition.
+    inversion H2.
+    intuition. 
+    elim H2; intros.    
+    absurd (false = true); [ auto with bool | apply H; auto ].
+    exists x; intuition.
+    unfold fdec in |- *. 
+    case (Pdec x); intuition.
+  Qed.
+
+  Definition partition :
+    forall (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x}) (s : t),
+    {partition : t * t |
+    let (s1, s2) := partition in
+    compat_P E.eq P ->
+    For_all P s1 /\
+    For_all (fun x => ~ P x) s2 /\
+    (forall x : elt, In x s <-> In x s1 \/ In x s2)}.
+  Proof.
+    intros.
+    exists (partition (fdec Pdec) s).
+    generalize (partition_1 s (f:=fdec Pdec)) (partition_2 s (f:=fdec Pdec)).
+    case (partition (fdec Pdec) s).
+    intros s1 s2; simpl in |- *.
+    intros; assert (compat_bool E.eq (fdec Pdec)); auto.
+    intros; assert (compat_bool E.eq (fun x => negb (fdec Pdec x))).
+    generalize H2; unfold compat_bool in |- *; intuition;
+     apply (f_equal negb); auto.
+    intuition.
+    generalize H4; unfold For_all, Equal in |- *; intuition.
+    elim (H0 x); intros.
+    assert (fdec Pdec x = true).
+     eauto.      
+    generalize H8; unfold fdec in |- *; case (Pdec x); intuition.
+    inversion H9.
+    generalize H; unfold For_all, Equal in |- *; intuition.
+    elim (H0 x); intros.
+    cut ((fun x => negb (fdec Pdec x)) x = true). 
+    unfold fdec in |- *; case (Pdec x); intuition.
+      change ((fun x => negb (fdec Pdec x)) x = true) in |- *.
+      apply (filter_2 (s:=s) (x:=x)); auto.
+    set (b := fdec Pdec x) in *; generalize (refl_equal b);
+     pattern b at -1 in |- *; case b; unfold b in |- *; 
+     [ left | right ].
+    elim (H4 x); intros _ B; apply B; auto.
+    elim (H x); intros _ B; apply B; auto.
+    apply filter_3; auto.
+    rewrite H5; auto.
+    eapply (filter_1 (s:=s) (x:=x) H2); elim (H4 x); intros B _; apply B;
+     auto.
+    eapply (filter_1 (s:=s) (x:=x) H3); elim (H x); intros B _; apply B; auto.
+  Qed. 
+
+  Definition choose : forall s : t, {x : elt | In x s} + {Empty s}.
+  Proof.  
+    intros.
+    generalize (choose_1 (s:=s)) (choose_2 (s:=s)).
+    case (choose s); [ left | right ]; auto.
+    exists e; auto.    
+  Qed.
+
+  Definition min_elt :
+    forall s : t,
+    {x : elt | In x s /\ For_all (fun y => ~ E.lt y x) s} + {Empty s}.
+  Proof. 
+    intros;
+     generalize (min_elt_1 (s:=s)) (min_elt_2 (s:=s)) (min_elt_3 (s:=s)).
+    case (min_elt s); [ left | right ]; auto.    
+    exists e; unfold For_all in |- *; eauto.
+  Qed. 
+
+  Definition max_elt :
+    forall s : t,
+    {x : elt | In x s /\ For_all (fun y => ~ E.lt x y) s} + {Empty s}.
+  Proof. 
+    intros;
+     generalize (max_elt_1 (s:=s)) (max_elt_2 (s:=s)) (max_elt_3 (s:=s)).
+    case (max_elt s); [ left | right ]; auto.    
+    exists e; unfold For_all in |- *; eauto.
+  Qed. 
+
+  Module E := E. 
+
+  Definition elt := elt.
+  Definition t := t.
+
+  Definition In := In. 
+  Definition Equal s s' := forall a : elt, In a s <-> In a s'.
+  Definition Subset s s' := forall a : elt, In a s -> In a s'.
+  Definition Empty s := forall a : elt, ~ In a s.
+  Definition For_all (P : elt -> Prop) (s : t) :=
+    forall x : elt, In x s -> P x.
+  Definition Exists (P : elt -> Prop) (s : t) :=
+    exists x : elt, In x s /\ P x.
+  
+  Definition eq_In := In_1.
+
+  Definition eq := Equal.
+  Definition lt := lt.
+  Definition eq_refl := eq_refl.
+  Definition eq_sym := eq_sym.
+  Definition eq_trans := eq_trans.
+  Definition lt_trans := lt_trans. 
+  Definition lt_not_eq := lt_not_eq.
+  Definition compare := compare.
+
+End DepOfNodep.
+
+
+(** * From dependent signature [Sdep] to non-dependent signature [S]. *)
+
+Module NodepOfDep (M: Sdep) <: S with Module E := M.E.
+  Import M.
+
+  Module ME := OrderedTypeFacts E.
+
+  Definition empty : t := let (s, _) := empty in s.
+
+  Lemma empty_1 : Empty empty.
+  Proof.
+    unfold empty in |- *; case M.empty; auto.
+  Qed.
+
+  Definition is_empty (s : t) : bool :=
+    if is_empty s then true else false.
+
+  Lemma is_empty_1 : forall s : t, Empty s -> is_empty s = true.
+  Proof.
+    intros; unfold is_empty in |- *; case (M.is_empty s); auto.
+  Qed.
+
+  Lemma is_empty_2 : forall s : t, is_empty s = true -> Empty s.
+  Proof.
+    intro s; unfold is_empty in |- *; case (M.is_empty s); auto.
+    intros; discriminate H.
+  Qed.
+
+  Definition mem (x : elt) (s : t) : bool :=
+    if mem x s then true else false.
+
+  Lemma mem_1 : forall (s : t) (x : elt), In x s -> mem x s = true.
+  Proof.
+    intros; unfold mem in |- *; case (M.mem x s); auto.
+  Qed.
+   
+  Lemma mem_2 : forall (s : t) (x : elt), mem x s = true -> In x s.
+  Proof.
+    intros s x; unfold mem in |- *; case (M.mem x s); auto.
+    intros; discriminate H.
+  Qed.
+
+  Definition equal (s s' : t) : bool :=
+    if equal s s' then true else false.
+
+  Lemma equal_1 : forall s s' : t, Equal s s' -> equal s s' = true.
+  Proof. 
+    intros; unfold equal in |- *; case M.equal; intuition.
+  Qed.    
+ 
+  Lemma equal_2 : forall s s' : t, equal s s' = true -> Equal s s'.
+  Proof. 
+    intros s s'; unfold equal in |- *; case (M.equal s s'); intuition;
+     inversion H.
+  Qed.
+  
+  Definition subset (s s' : t) : bool :=
+    if subset s s' then true else false.
+
+  Lemma subset_1 : forall s s' : t, Subset s s' -> subset s s' = true.
+  Proof. 
+    intros; unfold subset in |- *; case M.subset; intuition.
+  Qed.    
+ 
+  Lemma subset_2 : forall s s' : t, subset s s' = true -> Subset s s'.
+  Proof. 
+    intros s s'; unfold subset in |- *; case (M.subset s s'); intuition;
+     inversion H.
+  Qed.
+
+  Definition choose (s : t) : option elt :=
+    match choose s with
+    | inleft (exist x _) => Some x
+    | inright _ => None
+    end.
+
+  Lemma choose_1 : forall (s : t) (x : elt), choose s = Some x -> In x s.
+  Proof.
+    intros s x; unfold choose in |- *; case (M.choose s).
+    simple destruct s0; intros; injection H; intros; subst; auto.
+    intros; discriminate H.
+  Qed.
+
+  Lemma choose_2 : forall s : t, choose s = None -> Empty s.
+  Proof.
+    intro s; unfold choose in |- *; case (M.choose s); auto.
+    simple destruct s0; intros; discriminate H.
+  Qed.
+
+  Definition elements (s : t) : list elt := let (l, _) := elements s in l. 
+ 
+  Lemma elements_1 : forall (s : t) (x : elt), In x s -> ME.In x (elements s).
+  Proof. 
+    intros; unfold elements in |- *; case (M.elements s); firstorder.
+  Qed.
+
+  Lemma elements_2 : forall (s : t) (x : elt), ME.In x (elements s) -> In x s.
+  Proof. 
+    intros s x; unfold elements in |- *; case (M.elements s); firstorder.
+  Qed.
+
+  Lemma elements_3 : forall s : t, ME.Sort (elements s).  
+  Proof. 
+    intros; unfold elements in |- *; case (M.elements s); firstorder.
+  Qed.
+
+  Definition min_elt (s : t) : option elt :=
+    match min_elt s with
+    | inleft (exist x _) => Some x
+    | inright _ => None
+    end.
+
+  Lemma min_elt_1 : forall (s : t) (x : elt), min_elt s = Some x -> In x s. 
+  Proof.
+    intros s x; unfold min_elt in |- *; case (M.min_elt s).
+    simple destruct s0; intros; injection H; intros; subst; intuition.
+    intros; discriminate H.
+  Qed. 
+
+  Lemma min_elt_2 :
+   forall (s : t) (x y : elt), min_elt s = Some x -> In y s -> ~ E.lt y x. 
+  Proof.
+    intros s x y; unfold min_elt in |- *; case (M.min_elt s).
+    unfold For_all in |- *; simple destruct s0; intros; injection H; intros;
+     subst; firstorder.
+    intros; discriminate H.
+  Qed. 
+
+  Lemma min_elt_3 : forall s : t, min_elt s = None -> Empty s.
+  Proof.
+    intros s; unfold min_elt in |- *; case (M.min_elt s); auto.
+    simple destruct s0; intros; discriminate H.
+  Qed. 
+
+  Definition max_elt (s : t) : option elt :=
+    match max_elt s with
+    | inleft (exist x _) => Some x
+    | inright _ => None
+    end.
+
+  Lemma max_elt_1 : forall (s : t) (x : elt), max_elt s = Some x -> In x s. 
+  Proof.
+    intros s x; unfold max_elt in |- *; case (M.max_elt s).
+    simple destruct s0; intros; injection H; intros; subst; intuition.
+    intros; discriminate H.
+  Qed. 
+
+  Lemma max_elt_2 :
+   forall (s : t) (x y : elt), max_elt s = Some x -> In y s -> ~ E.lt x y. 
+  Proof.
+    intros s x y; unfold max_elt in |- *; case (M.max_elt s).
+    unfold For_all in |- *; simple destruct s0; intros; injection H; intros;
+     subst; firstorder.
+    intros; discriminate H.
+  Qed. 
+
+  Lemma max_elt_3 : forall s : t, max_elt s = None -> Empty s.
+  Proof.
+    intros s; unfold max_elt in |- *; case (M.max_elt s); auto.
+    simple destruct s0; intros; discriminate H.
+  Qed. 
+
+  Definition add (x : elt) (s : t) : t := let (s', _) := add x s in s'.
+
+  Lemma add_1 : forall (s : t) (x y : elt), E.eq x y -> In y (add x s).
+  Proof.
+    intros; unfold add in |- *; case (M.add x s); unfold Add in |- *;
+     firstorder.
+  Qed.
+
+  Lemma add_2 : forall (s : t) (x y : elt), In y s -> In y (add x s).
+  Proof.
+    intros; unfold add in |- *; case (M.add x s); unfold Add in |- *;
+     firstorder.
+  Qed.
+
+  Lemma add_3 :
+   forall (s : t) (x y : elt), ~ E.eq x y -> In y (add x s) -> In y s.
+  Proof.
+    intros s x y; unfold add in |- *; case (M.add x s); unfold Add in |- *;
+     firstorder.
+  Qed.
+
+  Definition remove (x : elt) (s : t) : t := let (s', _) := remove x s in s'.
+
+  Lemma remove_1 : forall (s : t) (x y : elt), E.eq x y -> ~ In y (remove x s).
+  Proof.
+    intros; unfold remove in |- *; case (M.remove x s); firstorder.
+  Qed.
+
+  Lemma remove_2 :
+   forall (s : t) (x y : elt), ~ E.eq x y -> In y s -> In y (remove x s).
+  Proof.
+    intros; unfold remove in |- *; case (M.remove x s); firstorder.
+  Qed.
+
+  Lemma remove_3 : forall (s : t) (x y : elt), In y (remove x s) -> In y s.
+  Proof.
+    intros s x y; unfold remove in |- *; case (M.remove x s); firstorder.
+  Qed.
+  
+  Definition singleton (x : elt) : t := let (s, _) := singleton x in s. 
+ 
+  Lemma singleton_1 : forall x y : elt, In y (singleton x) -> E.eq x y. 
+  Proof.
+    intros x y; unfold singleton in |- *; case (M.singleton x); firstorder.
+  Qed.
+
+  Lemma singleton_2 : forall x y : elt, E.eq x y -> In y (singleton x). 
+  Proof.
+    intros x y; unfold singleton in |- *; case (M.singleton x); firstorder.
+  Qed.
+  
+  Definition union (s s' : t) : t := let (s'', _) := union s s' in s''.
+ 
+  Lemma union_1 :
+   forall (s s' : t) (x : elt), In x (union s s') -> In x s \/ In x s'.
+  Proof. 
+    intros s s' x; unfold union in |- *; case (M.union s s'); firstorder.
+  Qed.
+
+  Lemma union_2 : forall (s s' : t) (x : elt), In x s -> In x (union s s'). 
+  Proof. 
+    intros s s' x; unfold union in |- *; case (M.union s s'); firstorder.
+  Qed.
+
+  Lemma union_3 : forall (s s' : t) (x : elt), In x s' -> In x (union s s').
+  Proof. 
+    intros s s' x; unfold union in |- *; case (M.union s s'); firstorder.
+  Qed.
+
+  Definition inter (s s' : t) : t := let (s'', _) := inter s s' in s''.
+ 
+  Lemma inter_1 : forall (s s' : t) (x : elt), In x (inter s s') -> In x s.
+  Proof. 
+    intros s s' x; unfold inter in |- *; case (M.inter s s'); firstorder.
+  Qed.
+
+  Lemma inter_2 : forall (s s' : t) (x : elt), In x (inter s s') -> In x s'.
+  Proof. 
+    intros s s' x; unfold inter in |- *; case (M.inter s s'); firstorder.
+  Qed.
+
+  Lemma inter_3 :
+   forall (s s' : t) (x : elt), In x s -> In x s' -> In x (inter s s').
+  Proof. 
+    intros s s' x; unfold inter in |- *; case (M.inter s s'); firstorder.
+  Qed.
+
+  Definition diff (s s' : t) : t := let (s'', _) := diff s s' in s''.
+ 
+  Lemma diff_1 : forall (s s' : t) (x : elt), In x (diff s s') -> In x s.
+  Proof. 
+    intros s s' x; unfold diff in |- *; case (M.diff s s'); firstorder.
+  Qed.
+
+  Lemma diff_2 : forall (s s' : t) (x : elt), In x (diff s s') -> ~ In x s'.
+  Proof. 
+    intros s s' x; unfold diff in |- *; case (M.diff s s'); firstorder.
+  Qed.
+
+  Lemma diff_3 :
+   forall (s s' : t) (x : elt), In x s -> ~ In x s' -> In x (diff s s').
+  Proof. 
+    intros s s' x; unfold diff in |- *; case (M.diff s s'); firstorder.
+  Qed.
+
+  Definition cardinal (s : t) : nat := let (f, _) := cardinal s in f.
+
+  Lemma cardinal_1 : forall s, cardinal s = length (elements s).
+  Proof.
+    intros; unfold cardinal in |- *; case (M.cardinal s); unfold elements in *; 
+    destruct (M.elements s); auto.
+  Qed.
+
+  Definition fold (B : Set) (f : elt -> B -> B) (i : t) 
+    (s : B) : B := let (fold, _) := fold f i s in fold.
+
+  Lemma fold_1 :
+   forall (s : t) (A : Set) (i : A) (f : elt -> A -> A),
+   fold f s i = fold_left (fun a e => f e a) (elements s) i.
+  Proof.
+    intros; unfold fold in |- *; case (M.fold f s i); unfold elements in *; 
+    destruct (M.elements s); auto.
+  Qed.  
+
+  Definition f_dec :
+    forall (f : elt -> bool) (x : elt), {f x = true} + {f x <> true}.
+  Proof.
+    intros; case (f x); auto with bool.
+  Defined. 
+
+  Lemma compat_P_aux :
+   forall f : elt -> bool,
+   compat_bool E.eq f -> compat_P E.eq (fun x => f x = true).
+  Proof.
+     unfold compat_bool, compat_P in |- *; intros; rewrite <- H1; firstorder.
+  Qed.
+
+  Hint Resolve compat_P_aux.
+    
+  Definition filter (f : elt -> bool) (s : t) : t :=
+    let (s', _) := filter (P:=fun x => f x = true) (f_dec f) s in s'.
+
+  Lemma filter_1 :
+   forall (s : t) (x : elt) (f : elt -> bool),
+   compat_bool E.eq f -> In x (filter f s) -> In x s.
+  Proof.
+    intros s x f; unfold filter in |- *; case M.filter; intuition.
+    generalize (i (compat_P_aux H)); firstorder.
+  Qed.
+
+  Lemma filter_2 :
+   forall (s : t) (x : elt) (f : elt -> bool),
+   compat_bool E.eq f -> In x (filter f s) -> f x = true. 
+  Proof.
+    intros s x f; unfold filter in |- *; case M.filter; intuition.
+    generalize (i (compat_P_aux H)); firstorder.
+  Qed.
+
+  Lemma filter_3 :
+   forall (s : t) (x : elt) (f : elt -> bool),
+   compat_bool E.eq f -> In x s -> f x = true -> In x (filter f s).     
+  Proof.
+    intros s x f; unfold filter in |- *; case M.filter; intuition.
+    generalize (i (compat_P_aux H)); firstorder.
+  Qed.
+
+  Definition for_all (f : elt -> bool) (s : t) : bool :=
+    if for_all (P:=fun x => f x = true) (f_dec f) s
+    then true
+    else false. 
+
+  Lemma for_all_1 :
+   forall (s : t) (f : elt -> bool),
+   compat_bool E.eq f ->
+   For_all (fun x => f x = true) s -> for_all f s = true.
+  Proof. 
+    intros s f; unfold for_all in |- *; case M.for_all; intuition; elim n;
+     auto.
+  Qed.
+ 
+  Lemma for_all_2 :
+   forall (s : t) (f : elt -> bool),
+   compat_bool E.eq f ->
+   for_all f s = true -> For_all (fun x => f x = true) s.
+  Proof. 
+    intros s f; unfold for_all in |- *; case M.for_all; intuition;
+     inversion H0.
+  Qed.
+  
+  Definition exists_ (f : elt -> bool) (s : t) : bool :=
+    if exists_ (P:=fun x => f x = true) (f_dec f) s
+    then true
+    else false. 
+
+  Lemma exists_1 :
+   forall (s : t) (f : elt -> bool),
+   compat_bool E.eq f -> Exists (fun x => f x = true) s -> exists_ f s = true.
+  Proof. 
+    intros s f; unfold exists_ in |- *; case M.exists_; intuition; elim n;
+     auto.
+  Qed.
+ 
+  Lemma exists_2 :
+   forall (s : t) (f : elt -> bool),
+   compat_bool E.eq f -> exists_ f s = true -> Exists (fun x => f x = true) s.
+  Proof. 
+    intros s f; unfold exists_ in |- *; case M.exists_; intuition;
+     inversion H0.
+  Qed.
+     
+  Definition partition (f : elt -> bool) (s : t) : 
+    t * t :=
+    let (p, _) := partition (P:=fun x => f x = true) (f_dec f) s in p.
+  
+  Lemma partition_1 :
+   forall (s : t) (f : elt -> bool),
+   compat_bool E.eq f -> Equal (fst (partition f s)) (filter f s).
+  Proof.
+    intros s f; unfold partition in |- *; case M.partition. 
+    intro p; case p; clear p; intros s1 s2 H C. 
+    generalize (H (compat_P_aux C)); clear H; intro H.
+    simpl in |- *; unfold Equal in |- *; intuition.
+    apply filter_3; firstorder. 
+    elim (H2 a); intros. 
+    assert (In a s). 
+     eapply filter_1; eauto.
+    elim H3; intros; auto.
+    absurd (f a = true).
+    exact (H a H6).
+    eapply filter_2; eauto. 
+  Qed.    
+    
+  Lemma partition_2 :
+   forall (s : t) (f : elt -> bool),
+   compat_bool E.eq f -> Equal (snd (partition f s)) (filter (fun x => negb (f x)) s).
+  Proof.
+    intros s f; unfold partition in |- *; case M.partition. 
+    intro p; case p; clear p; intros s1 s2 H C. 
+    generalize (H (compat_P_aux C)); clear H; intro H.
+    assert (D : compat_bool E.eq (fun x => negb (f x))).
+    generalize C; unfold compat_bool in |- *; intros; apply (f_equal negb);
+     auto.
+    simpl in |- *; unfold Equal in |- *; intuition.
+    apply filter_3; firstorder.
+    elim (H2 a); intros. 
+    assert (In a s). 
+     eapply filter_1; eauto.
+    elim H3; intros; auto.
+    absurd (f a = true).
+    intro.
+    generalize (filter_2 D H1). 
+    rewrite H7; intros H8; inversion H8.
+    exact (H0 a H6).
+  Qed. 
+
+
+  Module E := E. 
+  Definition elt := elt.
+  Definition t := t.
+
+  Definition In := In. 
+  Definition Equal s s' := forall a : elt, In a s <-> In a s'.
+  Definition Subset s s' := forall a : elt, In a s -> In a s'.
+  Definition Add (x : elt) (s s' : t) :=
+    forall y : elt, In y s' <-> E.eq y x \/ In y s.
+  Definition Empty s := forall a : elt, ~ In a s.
+  Definition For_all (P : elt -> Prop) (s : t) :=
+    forall x : elt, In x s -> P x.
+  Definition Exists (P : elt -> Prop) (s : t) :=
+    exists x : elt, In x s /\ P x.
+
+  Definition In_1 := eq_In.
+
+  Definition eq := Equal.
+  Definition lt := lt.
+  Definition eq_refl := eq_refl.
+  Definition eq_sym := eq_sym.
+  Definition eq_trans := eq_trans.
+  Definition lt_trans := lt_trans. 
+  Definition lt_not_eq := lt_not_eq.
+  Definition compare := compare.
+
+End NodepOfDep.