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, 237 insertions, 0 deletions

diff --git a/theories/Lists/SetoidList.v b/theories/Lists/SetoidList.v
new file mode 100644
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+++ b/theories/Lists/SetoidList.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: Lib.v,v 1.2 2006/02/26 15:59:48 letouzey Exp $ *)
+
+Require Export List.
+Require Export MoreList.
+Require Export Sorting.
+Require Export Setoid.
+Set Implicit Arguments.
+Unset Strict Implicit.
+
+(** * Logical relations over lists with respect to a setoid equality 
+      or ordering. *) 
+
+(** This can be seen as a complement of predicate [lelistA] and [sort] 
+    found in [Sorting]. *)
+
+Section Type_with_equality.
+Variable A : Set.
+Variable eqA : A -> A -> Prop. 
+
+(** Being in a list modulo an equality relation over type [A]. *)
+
+Inductive InA (x : A) : list A -> Prop :=
+  | InA_cons_hd : forall y l, eqA x y -> InA x (y :: l)
+  | InA_cons_tl : forall y l, InA x l -> InA x (y :: l).
+
+Hint Constructors InA.
+
+(** An alternative definition of [InA]. *)
+
+Lemma InA_alt : forall x l, InA x l <-> exists y, eqA x y /\ In y l.
+Proof. 
+ induction l; intuition.
+ inversion H.
+ firstorder.
+ inversion H1; firstorder.
+ firstorder; subst; auto.
+Qed.
+
+(** A list without redundancy. *)
+
+Inductive noredun : list A -> Prop := 
+ | noredun_nil : noredun nil 
+ | noredun_cons : forall x l, ~ In x l -> noredun l -> noredun (x::l). 
+
+
+(** Similarly, a list without redundancy modulo the equality over [A]. *)
+
+Inductive noredunA : list A -> Prop :=
+  | noredunA_nil : noredunA nil
+  | noredunA_cons : forall x l, ~ InA x l -> noredunA l -> noredunA (x::l).
+
+Hint Constructors noredunA.
+
+
+(** Results concerning lists modulo [eqA] *)
+
+Hypothesis eqA_refl : forall x, eqA x x.
+Hypothesis eqA_sym : forall x y, eqA x y -> eqA y x.
+Hypothesis eqA_trans : forall x y z, eqA x y -> eqA y z -> eqA x z.
+
+Hint Resolve eqA_refl eqA_trans.
+Hint Immediate eqA_sym.
+
+Lemma InA_eqA : forall l x y, eqA x y -> InA x l -> InA y l.
+Proof. 
+ intros s x y.
+ do 2 rewrite InA_alt.
+ intros H (z,(U,V)).  
+ exists z; split; eauto.
+Qed.
+Hint Immediate InA_eqA.
+
+Lemma In_InA : forall l x, In x l -> InA x l.
+Proof.
+ simple induction l; simpl in |- *; intuition.
+ subst; auto.  
+Qed.
+Hint Resolve In_InA.
+
+(** Results concerning lists modulo [eqA] and [ltA] *)
+
+Variable ltA : A -> A -> Prop.
+
+Hypothesis ltA_trans : forall x y z, ltA x y -> ltA y z -> ltA x z.
+Hypothesis ltA_not_eqA : forall x y, ltA x y -> ~ eqA x y.
+Hypothesis ltA_eqA : forall x y z, ltA x y -> eqA y z -> ltA x z.
+Hypothesis eqA_ltA : forall x y z, eqA x y -> ltA y z -> ltA x z.
+
+Hint Resolve ltA_trans.
+Hint Immediate ltA_eqA eqA_ltA.
+
+Notation InfA:=(lelistA ltA).
+Notation SortA:=(sort ltA).
+
+Lemma InfA_ltA :
+ forall l x y, ltA x y -> InfA y l -> InfA x l.
+Proof.
+ intro s; case s; constructor; inversion_clear H0.
+ eapply ltA_trans; eauto.
+Qed.
+ 
+Lemma InfA_eqA :
+ forall l x y, eqA x y -> InfA y l -> InfA x l.
+Proof.
+ intro s; case s; constructor; inversion_clear H0; eauto.
+Qed.
+Hint Immediate InfA_ltA InfA_eqA. 
+
+Lemma SortA_InfA_InA :
+ forall l x a, SortA l -> InfA a l -> InA x l -> ltA a x.
+Proof. 
+ simple induction l.
+ intros; inversion H1.
+ intros.
+ inversion_clear H0; inversion_clear H1; inversion_clear H2.
+ eapply ltA_eqA; eauto.
+ eauto.
+Qed.
+  
+Lemma In_InfA :
+ forall l x, (forall y, In y l -> ltA x y) -> InfA x l.
+Proof.
+ simple induction l; simpl in |- *; intros; constructor; auto.
+Qed.
+ 
+Lemma InA_InfA :
+ forall l x, (forall y, InA y l -> ltA x y) -> InfA x l.
+Proof.
+ simple induction l; simpl in |- *; intros; constructor; auto.
+Qed.
+
+(* In fact, this may be used as an alternative definition for InfA: *)
+
+Lemma InfA_alt : 
+ forall l x, SortA l -> (InfA x l <-> (forall y, InA y l -> ltA x y)).
+Proof. 
+split.
+intros; eapply SortA_InfA_InA; eauto.
+apply InA_InfA.
+Qed.
+
+Lemma SortA_noredunA : forall l, SortA l -> noredunA l.
+Proof.
+ simple induction l; auto.
+ intros x l' H H0.
+ inversion_clear H0.
+ constructor; auto.  
+ intro.
+ assert (ltA x x) by eapply SortA_InfA_InA; eauto.
+ elim (ltA_not_eqA H3); auto.
+Qed.
+
+Lemma noredunA_app : forall l l', noredunA l -> noredunA l' -> 
+  (forall x, InA x l -> InA x l' -> False) -> 
+  noredunA (l++l').
+Proof.
+induction l; simpl; auto; intros.
+inversion_clear H.
+constructor.
+rewrite InA_alt; intros (y,(H4,H5)).
+destruct (in_app_or _ _ _ H5).
+elim H2.
+rewrite InA_alt.
+exists y; auto.
+apply (H1 a).
+auto.
+rewrite InA_alt.
+exists y; auto.
+apply IHl; auto.
+intros.
+apply (H1 x); auto.
+Qed.
+
+
+Lemma noredunA_rev : forall l, noredunA l -> noredunA (rev l).
+Proof.
+induction l.
+simpl; auto.
+simpl; intros.
+inversion_clear H.
+apply noredunA_app; auto.
+constructor; auto.
+intro H2; inversion H2.
+intros x.
+rewrite InA_alt.
+intros (x1,(H2,H3)).
+inversion_clear 1.
+destruct H0.
+apply InA_eqA with x1; eauto.
+apply In_InA.
+rewrite In_rev; auto.
+inversion H4.
+Qed.
+
+
+Lemma InA_app : forall l1 l2 x,
+ InA x (l1 ++ l2) -> InA x l1 \/ InA x l2.
+Proof.
+ induction l1; simpl in *; intuition.
+ inversion_clear H; auto.
+ elim (IHl1 l2 x H0); auto.
+Qed.
+
+ Hint Constructors lelistA sort.
+
+Lemma InfA_app : forall l1 l2 a, InfA a l1 -> InfA a l2 -> InfA a (l1++l2).
+Proof.
+ induction l1; simpl; auto.
+ inversion_clear 1; auto.
+Qed.
+
+Lemma SortA_app :
+ forall l1 l2, SortA l1 -> SortA l2 ->
+ (forall x y, InA x l1 -> InA y l2 -> ltA x y) ->
+ SortA (l1 ++ l2).
+Proof.
+ induction l1; simpl in *; intuition.
+ inversion_clear H.
+ constructor; auto.
+ apply InfA_app; auto.
+ destruct l2; auto.
+Qed.
+
+End Type_with_equality.
+
+Hint Constructors InA.
+Hint Constructors noredunA. 
+Hint Constructors sort.
+Hint Constructors lelistA.