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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: FMapInterface.v 8671 2006-03-29 08:31:28Z letouzey $ *)
+
+(** * Finite map library *)  
+
+(** This file proposes an interface for finite maps *)
+
+(* begin hide *)
+Set Implicit Arguments.
+Unset Strict Implicit.
+Require Import FSetInterface. 
+(* end hide *)
+
+(** When compared with Ocaml Map, this signature has been split in two: 
+   - The first part [S] contains the usual operators (add, find, ...)
+     It only requires a ordered key type, the data type can be arbitrary. 
+     The only function that asks more is [equal], whose first argument should 
+     be an equality on data. 
+   - Then, [Sord] extends [S] with a complete comparison function. For 
+     that, the data type should have a decidable total ordering. 
+*)
+
+
+Module Type S.
+
+  Declare Module E : OrderedType.
+
+  Definition key := E.t.
+
+  Parameter t : Set -> Set. (** the abstract type of maps *)
+ 
+  Section Types. 
+
+    Variable elt:Set.
+
+    Parameter empty : t elt.
+      (** The empty map. *)
+
+    Parameter is_empty : t elt -> bool.
+    (** Test whether a map is empty or not. *)
+
+    Parameter add : key -> elt -> t elt -> t elt.
+    (** [add x y m] returns a map containing the same bindings as [m], 
+	plus a binding of [x] to [y]. If [x] was already bound in [m], 
+	its previous binding disappears. *)
+
+    Parameter find : key -> t elt -> option elt. 
+    (** [find x m] returns the current binding of [x] in [m], 
+	or raises [Not_found] if no such binding exists.
+	NB: in Coq, the exception mechanism becomes a option type. *)
+
+    Parameter remove : key -> t elt -> t elt.
+    (** [remove x m] returns a map containing the same bindings as [m], 
+	except for [x] which is unbound in the returned map. *)
+
+    Parameter mem : key -> t elt -> bool.
+    (** [mem x m] returns [true] if [m] contains a binding for [x], 
+	and [false] otherwise. *)
+
+    (** Coq comment: [iter] is useless in a purely functional world *)
+    (** val iter : (key -> 'a -> unit) -> 'a t -> unit *)
+    (** iter f m applies f to all bindings in map m. f receives the key as 
+	first argument, and the associated value as second argument. 
+	The bindings are passed to f in increasing order with respect to the 
+	ordering over the type of the keys. Only current bindings are 
+	presented to f: bindings hidden by more recent bindings are not 
+	passed to f. *)
+
+    Variable elt' : Set. 
+    Variable elt'': Set.
+
+    Parameter map : (elt -> elt') -> t elt -> t elt'.
+    (** [map f m] returns a map with same domain as [m], where the associated 
+	value a of all bindings of [m] has been replaced by the result of the
+	application of [f] to [a]. The bindings are passed to [f] in 
+	increasing order with respect to the ordering over the type of the 
+	keys. *)
+
+    Parameter mapi : (key -> elt -> elt') -> t elt -> t elt'.
+    (** Same as [S.map], but the function receives as arguments both the 
+	key and the associated value for each binding of the map. *)
+
+    Parameter map2 : (option elt -> option elt' -> option elt'') -> t elt -> t elt' ->  t elt''.
+    (** Not present in Ocaml. 
+         [map f m m'] creates a new map whose bindings belong to the ones of either 
+         [m] or [m']. The presence and value for a key [k] is determined by [f e e'] 
+         where [e] and [e'] are the (optional) bindings of [k] in [m] and [m']. *)
+
+    Parameter elements : t elt -> list (key*elt).
+    (** Not present in Ocaml. 
+         [elements m] returns an assoc list corresponding to the bindings of [m].
+         Elements of this list are sorted with respect to their first components. 
+         Useful to specify [fold] ... *)
+
+    Parameter fold : forall A: Set, (key -> elt -> A -> A) -> t elt -> A -> A.
+    (** [fold f m a] computes [(f kN dN ... (f k1 d1 a)...)], 
+	where [k1] ... [kN] are the keys of all bindings in [m] 
+	(in increasing order), and [d1] ... [dN] are the associated data. *)
+
+    Parameter equal : (elt -> elt -> bool) -> t elt -> t elt -> bool.
+    (** [equal cmp m1 m2] tests whether the maps [m1] and [m2] are equal, 
+      that is, contain equal keys and associate them with equal data. 
+      [cmp] is the equality predicate used to compare the data associated 
+      with the keys. *)
+
+    Section Spec. 
+      
+      Variable m m' m'' : t elt.
+      Variable x y z : key.
+      Variable e e' : elt.
+
+      Parameter MapsTo : key -> elt -> t elt -> Prop.
+
+      Definition In (k:key)(m: t elt) : Prop := exists e:elt, MapsTo k e m.
+
+      Definition Empty m := forall (a : key)(e:elt) , ~ MapsTo a e m.
+
+      Definition eq_key (p p':key*elt) := E.eq (fst p) (fst p').
+      
+      Definition eq_key_elt (p p':key*elt) := 
+          E.eq (fst p) (fst p') /\ (snd p) = (snd p').
+
+      Definition lt_key (p p':key*elt) := E.lt (fst p) (fst p').
+
+    (** Specification of [MapsTo] *)
+      Parameter MapsTo_1 : E.eq x y -> MapsTo x e m -> MapsTo y e m.
+      
+    (** Specification of [mem] *)
+      Parameter mem_1 : In x m -> mem x m = true.
+      Parameter mem_2 : mem x m = true -> In x m. 
+      
+    (** Specification of [empty] *)
+      Parameter empty_1 : Empty empty.
+
+    (** Specification of [is_empty] *)
+      Parameter is_empty_1 : Empty m -> is_empty m = true. 
+      Parameter is_empty_2 : is_empty m = true -> Empty m.
+      
+    (** Specification of [add] *)
+      Parameter add_1 : E.eq x y -> MapsTo y e (add x e m).
+      Parameter add_2 : ~ E.eq x y -> MapsTo y e m -> MapsTo y e (add x e' m).
+      Parameter add_3 : ~ E.eq x y -> MapsTo y e (add x e' m) -> MapsTo y e m.
+
+    (** Specification of [remove] *)
+      Parameter remove_1 : E.eq x y -> ~ In y (remove x m).
+      Parameter remove_2 : ~ E.eq x y -> MapsTo y e m -> MapsTo y e (remove x m).
+      Parameter remove_3 : MapsTo y e (remove x m) -> MapsTo y e m.
+
+    (** Specification of [find] *)
+      Parameter find_1 : MapsTo x e m -> find x m = Some e. 
+      Parameter find_2 : find x m = Some e -> MapsTo x e m.
+
+    (** Specification of [elements] *)
+      Parameter elements_1 : 
+        MapsTo x e m -> InA eq_key_elt (x,e) (elements m).
+      Parameter elements_2 : 
+        InA eq_key_elt (x,e) (elements m) -> MapsTo x e m.
+      Parameter elements_3 : sort lt_key (elements m).  
+
+    (** Specification of [fold] *)  
+      Parameter fold_1 :
+	forall (A : Set) (i : A) (f : key -> elt -> A -> A),
+        fold f m i = fold_left (fun a p => f (fst p) (snd p) a) (elements m) i.
+ 
+   Definition Equal cmp m m' := 
+         (forall k, In k m <-> In k m') /\ 
+         (forall k e e', MapsTo k e m -> MapsTo k e' m' -> cmp e e' = true).  
+
+   Variable cmp : elt -> elt -> bool. 
+
+   (** Specification of [equal] *)
+     Parameter equal_1 : Equal cmp m m' -> equal cmp m m' = true. 
+     Parameter equal_2 : equal cmp m m' = true -> Equal cmp m m'.
+
+    End Spec. 
+   End Types. 
+
+    (** Specification of [map] *)
+      Parameter map_1 : forall (elt elt':Set)(m: t elt)(x:key)(e:elt)(f:elt->elt'), 
+        MapsTo x e m -> MapsTo x (f e) (map f m).
+      Parameter map_2 : forall (elt elt':Set)(m: t elt)(x:key)(f:elt->elt'), 
+        In x (map f m) -> In x m.
+ 
+    (** Specification of [mapi] *)
+      Parameter mapi_1 : forall (elt elt':Set)(m: t elt)(x:key)(e:elt)
+        (f:key->elt->elt'), MapsTo x e m -> 
+        exists y, E.eq y x /\ MapsTo x (f y e) (mapi f m).
+      Parameter mapi_2 : forall (elt elt':Set)(m: t elt)(x:key)
+        (f:key->elt->elt'), In x (mapi f m) -> In x m.
+
+    (** Specification of [map2] *)
+      Parameter map2_1 : forall (elt elt' elt'':Set)(m: t elt)(m': t elt')
+	(x:key)(f:option elt->option elt'->option elt''), 
+	In x m \/ In x m' -> 
+        find x (map2 f m m') = f (find x m) (find x m').       
+
+     Parameter map2_2 : forall (elt elt' elt'':Set)(m: t elt)(m': t elt')
+	(x:key)(f:option elt->option elt'->option elt''), 
+        In x (map2 f m m') -> In x m \/ In x m'.
+
+  (* begin hide *)
+  Hint Immediate MapsTo_1 mem_2 is_empty_2.
+  
+  Hint Resolve mem_1 is_empty_1 is_empty_2 add_1 add_2 add_3 remove_1
+    remove_2 remove_3 find_1 find_2 fold_1 map_1 map_2 mapi_1 mapi_2. 
+  (* end hide *)
+
+End S.
+
+
+Module Type Sord.
+ 
+  Declare Module Data : OrderedType.   
+  Declare Module MapS : S. 
+  Import MapS.
+  
+  Definition t := MapS.t Data.t. 
+
+  Parameter eq : t -> t -> Prop.
+  Parameter lt : t -> t -> Prop. 
+ 
+  Axiom eq_refl : forall m : t, eq m m.
+  Axiom eq_sym : forall m1 m2 : t, eq m1 m2 -> eq m2 m1.
+  Axiom eq_trans : forall m1 m2 m3 : t, eq m1 m2 -> eq m2 m3 -> eq m1 m3.
+  Axiom lt_trans : forall m1 m2 m3 : t, lt m1 m2 -> lt m2 m3 -> lt m1 m3.
+  Axiom lt_not_eq : forall m1 m2 : t, lt m1 m2 -> ~ eq m1 m2.
+
+  Definition cmp e e' := match Data.compare e e' with EQ _ => true | _ => false end.	
+
+  Parameter eq_1 : forall m m', Equal cmp m m' -> eq m m'.
+  Parameter eq_2 : forall m m', eq m m' -> Equal cmp m m'.
+
+  Parameter compare : forall m1 m2, Compare lt eq m1 m2.
+  (** Total ordering between maps. The first argument (in Coq: Data.compare) 
+      is a total ordering used to compare data associated with equal keys 
+      in the two maps. *)
+
+End Sord.
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