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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: DecidableType.v 8933 2006-06-09 14:08:38Z herbelin $ *)
+
+Require Export SetoidList.
+Set Implicit Arguments.
+Unset Strict Implicit.
+
+(** * Types with decidable Equalities (but no ordering) *)
+
+Module Type DecidableType. 
+
+  Parameter t : Set.
+
+  Parameter eq : t -> t -> Prop.
+
+  Axiom eq_refl : forall x : t, eq x x.
+  Axiom eq_sym : forall x y : t, eq x y -> eq y x.
+  Axiom eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z.
+
+  Parameter eq_dec : forall x y : t, { eq x y } + { ~ eq x y }.
+
+  Hint Immediate eq_sym.
+  Hint Resolve eq_refl eq_trans.
+
+End DecidableType. 
+
+(** * Additional notions about keys and datas used in FMap *)
+
+Module KeyDecidableType(D:DecidableType).
+ Import D.
+
+ Section Elt.
+ Variable elt : Set.
+ Notation key:=t.
+
+  Definition eqk (p p':key*elt) := eq (fst p) (fst p').
+  Definition eqke (p p':key*elt) := 
+          eq (fst p) (fst p') /\ (snd p) = (snd p').
+
+  Hint Unfold eqk eqke.
+  Hint Extern 2 (eqke ?a ?b) => split.
+
+   (* eqke is stricter than eqk *)
+ 
+   Lemma eqke_eqk : forall x x', eqke x x' -> eqk x x'.
+   Proof.
+     unfold eqk, eqke; intuition.
+   Qed.
+
+  (* eqk, eqke are equalities *)
+ 
+  Lemma eqk_refl : forall e, eqk e e.
+  Proof. auto. Qed.
+
+  Lemma eqke_refl : forall e, eqke e e.
+  Proof. auto. Qed.
+
+  Lemma eqk_sym : forall e e', eqk e e' -> eqk e' e.
+  Proof. auto. Qed.
+
+  Lemma eqke_sym : forall e e', eqke e e' -> eqke e' e.
+  Proof. unfold eqke; intuition. Qed.
+
+  Lemma eqk_trans : forall e e' e'', eqk e e' -> eqk e' e'' -> eqk e e''.
+  Proof. eauto. Qed.
+
+  Lemma eqke_trans : forall e e' e'', eqke e e' -> eqke e' e'' -> eqke e e''.
+  Proof.
+    unfold eqke; intuition; [ eauto | congruence ].
+  Qed.
+
+  Hint Resolve eqk_trans eqke_trans eqk_refl eqke_refl.
+  Hint Immediate eqk_sym eqke_sym.
+
+  Lemma InA_eqke_eqk : 
+     forall x m, InA eqke x m -> InA eqk x m.
+  Proof.
+    unfold eqke; induction 1; intuition.
+  Qed.
+  Hint Resolve InA_eqke_eqk.
+
+  Lemma InA_eqk : forall p q m, eqk p q -> InA eqk p m -> InA eqk q m.
+  Proof.
+   intros; apply InA_eqA with p; auto; apply eqk_trans; auto.
+  Qed.
+
+  Definition MapsTo (k:key)(e:elt):= InA eqke (k,e).
+  Definition In k m := exists e:elt, MapsTo k e m.
+
+  Hint Unfold MapsTo In.
+
+  (* An alternative formulation for [In k l] is [exists e, InA eqk (k,e) l] *)
+
+  Lemma In_alt : forall k l, In k l <-> exists e, InA eqk (k,e) l.
+  Proof.
+  firstorder.
+  exists x; auto.
+  induction H.
+  destruct y.
+  exists e; auto.
+  destruct IHInA as [e H0].
+  exists e; auto.
+  Qed.
+
+  Lemma MapsTo_eq : forall l x y e, eq x y -> MapsTo x e l -> MapsTo y e l.
+  Proof.
+  intros; unfold MapsTo in *; apply InA_eqA with (x,e); eauto.
+  Qed.
+
+  Lemma In_eq : forall l x y, eq x y -> In x l -> In y l.
+  Proof.
+  destruct 2 as (e,E); exists e; eapply MapsTo_eq; eauto.
+  Qed. 
+
+  Lemma In_inv : forall k k' e l, In k ((k',e) :: l) -> eq k k' \/ In k l.
+  Proof.
+    inversion 1.
+    inversion_clear H0; eauto.
+    destruct H1; simpl in *; intuition.
+  Qed.      
+
+  Lemma In_inv_2 : forall k k' e e' l, 
+      InA eqk (k, e) ((k', e') :: l) -> ~ eq k k' -> InA eqk (k, e) l.
+  Proof.  
+   inversion_clear 1; compute in H0; intuition.
+  Qed.
+
+  Lemma In_inv_3 : forall x x' l, 
+      InA eqke x (x' :: l) -> ~ eqk x x' -> InA eqke x l.
+  Proof.
+   inversion_clear 1; compute in H0; intuition.
+  Qed.
+
+ End Elt.
+
+ Hint Unfold eqk eqke.
+ Hint Extern 2 (eqke ?a ?b) => split.
+ Hint Resolve eqk_trans eqke_trans eqk_refl eqke_refl.
+ Hint Immediate eqk_sym eqke_sym.
+ Hint Resolve InA_eqke_eqk.
+ Hint Unfold MapsTo In.
+ Hint Resolve In_inv_2 In_inv_3.
+
+End KeyDecidableType.
+
+
+
+
+