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+(*
+  Cours "Sémantique et Application à la Vérification de programmes"
+  
+  Antoine Miné 2014
+  Ecole normale supérieure, Paris, France / CNRS / INRIA
+*)
+  
+(*  
+  This file is derived from the map.ml file from the OCaml distribution.
+  Changes are marked with the [AM] symbol.
+  Based on rev. 10468 2010-05-25 13:29:43Z
+
+  Original copyright follows.
+*)
+
+(***********************************************************************)
+(*                                                                     *)
+(*                           Objective Caml                            *)
+(*                                                                     *)
+(*            Xavier Leroy, projet Cristal, INRIA Rocquencourt         *)
+(*                                                                     *)
+(*  Copyright 1996 Institut National de Recherche en Informatique et   *)
+(*  en Automatique.  All rights reserved.  This file is distributed    *)
+(*  under the terms of the GNU Library General Public License, with    *)
+(*  the special exception on linking described in file ../LICENSE.     *)
+(*                                                                     *)
+(***********************************************************************)
+
+module type OrderedType =
+  sig
+    type t
+    val compare: t -> t -> int
+  end
+
+module type S =
+  sig
+    type key
+    type +'a t
+    val empty: 'a t
+    val is_empty: 'a t -> bool
+    val mem:  key -> 'a t -> bool
+    val add: key -> 'a -> 'a t -> 'a t
+    val singleton: key -> 'a -> 'a t
+    val remove: key -> 'a t -> 'a t
+    val merge: (key -> 'a option -> 'b option -> 'c option) -> 'a t -> 'b t -> 'c t
+    val compare: ('a -> 'a -> int) -> 'a t -> 'a t -> int
+    val equal: ('a -> 'a -> bool) -> 'a t -> 'a t -> bool
+    val iter: (key -> 'a -> unit) -> 'a t -> unit
+    val fold: (key -> 'a -> 'b -> 'b) -> 'a t -> 'b -> 'b
+    val for_all: (key -> 'a -> bool) -> 'a t -> bool
+    val exists: (key -> 'a -> bool) -> 'a t -> bool
+    val filter: (key -> 'a -> bool) -> 'a t -> 'a t
+    val partition: (key -> 'a -> bool) -> 'a t -> 'a t * 'a t
+    val cardinal: 'a t -> int
+    val bindings: 'a t -> (key * 'a) list
+    val min_binding: 'a t -> (key * 'a)
+    val max_binding: 'a t -> (key * 'a)
+    val choose: 'a t -> (key * 'a)
+    val split: key -> 'a t -> 'a t * 'a option * 'a t
+    val find: key -> 'a t -> 'a
+    val map: ('a -> 'b) -> 'a t -> 'b t
+    val mapi: (key -> 'a -> 'b) -> 'a t -> 'b t
+
+
+    (* [AM] additions by Antoine Mine' *)
+
+    val of_list: (key * 'a) list -> 'a t
+
+    val map2: (key -> 'a -> 'b -> 'c) -> 'a t -> 'b t -> 'c t
+    val iter2: (key -> 'a -> 'b -> unit) -> 'a t -> 'b t -> unit
+    val fold2: (key -> 'a -> 'b -> 'c -> 'c) -> 'a t -> 'b t -> 'c -> 'c
+    val for_all2: (key -> 'a -> 'b -> bool) -> 'a t -> 'b t -> bool
+    val exists2: (key -> 'a -> 'b -> bool) -> 'a t -> 'b t -> bool
+
+    val map2z: (key -> 'a -> 'a -> 'a) -> 'a t -> 'a t -> 'a t
+    val iter2z: (key -> 'a -> 'a -> unit) -> 'a t -> 'a t -> unit
+    val fold2z: (key -> 'a -> 'a -> 'b -> 'b) -> 'a t -> 'a t -> 'b -> 'b
+    val for_all2z: (key -> 'a -> 'a -> bool) -> 'a t -> 'a t -> bool
+    val exists2z: (key -> 'a -> 'a -> bool) -> 'a t -> 'a t -> bool
+
+    val map2o: (key -> 'a -> 'c) -> (key -> 'b -> 'c) -> (key -> 'a -> 'b -> 'c) -> 'a t -> 'b t -> 'c t
+    val iter2o: (key -> 'a -> unit) -> (key -> 'b -> unit) -> (key -> 'a -> 'b -> unit) -> 'a t -> 'b t -> unit
+    val fold2o: (key -> 'a -> 'c -> 'c) -> (key -> 'b -> 'c -> 'c) -> (key -> 'a -> 'b -> 'c -> 'c) -> 'a t -> 'b t -> 'c -> 'c
+    val for_all2o: (key -> 'a -> bool) -> (key -> 'b -> bool) -> (key -> 'a -> 'b -> bool) -> 'a t -> 'b t -> bool
+    val exists2o: (key -> 'a -> bool) -> (key -> 'b -> bool) -> (key -> 'a -> 'b -> bool) -> 'a t -> 'b t -> bool
+
+    val map2zo: (key -> 'a -> 'a) -> (key -> 'a -> 'a) -> (key -> 'a -> 'a -> 'a) -> 'a t -> 'a t -> 'a t
+    val iter2zo: (key -> 'a -> unit) -> (key -> 'a -> unit) -> (key -> 'a -> 'a -> unit) -> 'a t -> 'a t -> unit
+    val fold2zo: (key -> 'a -> 'b -> 'b) -> (key -> 'a -> 'b -> 'b) -> (key -> 'a -> 'a -> 'b -> 'b) -> 'a t -> 'a t -> 'b -> 'b
+    val for_all2zo: (key -> 'a -> bool) -> (key -> 'a -> bool) -> (key -> 'a -> 'a -> bool) -> 'a t -> 'a t -> bool
+     val exists2zo: (key -> 'a -> bool) -> (key -> 'a -> bool) -> (key -> 'a -> 'a -> bool) -> 'a t -> 'a t -> bool
+
+    val map_slice: (key -> 'a -> 'a) -> 'a t -> key -> key -> 'a t
+    val iter_slice: (key -> 'a -> unit) -> 'a t -> key -> key -> unit
+    val fold_slice: (key -> 'a -> 'b -> 'b) -> 'a t -> key -> key -> 'b -> 'b
+    val for_all_slice: (key -> 'a -> bool) -> 'a t -> key -> key -> bool
+    val exists_slice: (key -> 'a -> bool) -> 'a t -> key -> key -> bool
+
+    val key_equal: 'a t -> 'a t -> bool
+    val key_subset: 'a t -> 'a t -> bool
+
+    val find_greater: key -> 'a t -> key * 'a
+    val find_less: key -> 'a t -> key * 'a
+    val find_greater_equal: key -> 'a t -> key * 'a
+    val find_less_equal: key -> 'a t -> key * 'a
+
+  end
+
+module Make(Ord: OrderedType) = (struct
+
+    type key = Ord.t
+
+    type 'a t =
+        Empty
+      | Node of 'a t * key * 'a * 'a t * int
+
+    let height = function
+        Empty -> 0
+      | Node(_,_,_,_,h) -> h
+
+    let create l x d r =
+      let hl = height l and hr = height r in
+      Node(l, x, d, r, (if hl >= hr then hl + 1 else hr + 1))
+
+    let singleton x d = Node(Empty, x, d, Empty, 1)
+
+    let bal l x d r =
+      let hl = match l with Empty -> 0 | Node(_,_,_,_,h) -> h in
+      let hr = match r with Empty -> 0 | Node(_,_,_,_,h) -> h in
+      if hl > hr + 2 then begin
+        match l with
+          Empty -> invalid_arg "Mapext.bal"
+        | Node(ll, lv, ld, lr, _) ->
+            if height ll >= height lr then
+              create ll lv ld (create lr x d r)
+            else begin
+              match lr with
+                Empty -> invalid_arg "Mapext.bal"
+              | Node(lrl, lrv, lrd, lrr, _)->
+                  create (create ll lv ld lrl) lrv lrd (create lrr x d r)
+            end
+      end else if hr > hl + 2 then begin
+        match r with
+          Empty -> invalid_arg "Mapext.bal"
+        | Node(rl, rv, rd, rr, _) ->
+            if height rr >= height rl then
+              create (create l x d rl) rv rd rr
+            else begin
+              match rl with
+                Empty -> invalid_arg "Mapext.bal"
+              | Node(rll, rlv, rld, rlr, _) ->
+                  create (create l x d rll) rlv rld (create rlr rv rd rr)
+            end
+      end else
+        Node(l, x, d, r, (if hl >= hr then hl + 1 else hr + 1))
+
+    let empty = Empty
+
+    let is_empty = function Empty -> true | _ -> false
+
+    let rec add x data = function
+        Empty ->
+          Node(Empty, x, data, Empty, 1)
+      | Node(l, v, d, r, h) ->
+          let c = Ord.compare x v in
+          if c = 0 then
+            Node(l, x, data, r, h)
+          else if c < 0 then
+            bal (add x data l) v d r
+          else
+            bal l v d (add x data r)
+
+    let rec find x = function
+        Empty ->
+          raise Not_found
+      | Node(l, v, d, r, _) ->
+          let c = Ord.compare x v in
+          if c = 0 then d
+          else find x (if c < 0 then l else r)
+
+    let rec mem x = function
+        Empty ->
+          false
+      | Node(l, v, d, r, _) ->
+          let c = Ord.compare x v in
+          c = 0 || mem x (if c < 0 then l else r)
+
+    let rec min_binding = function
+        Empty -> raise Not_found
+      | Node(Empty, x, d, r, _) -> (x, d)
+      | Node(l, x, d, r, _) -> min_binding l
+
+    let rec max_binding = function
+        Empty -> raise Not_found
+      | Node(l, x, d, Empty, _) -> (x, d)
+      | Node(l, x, d, r, _) -> max_binding r
+
+    let rec remove_min_binding = function
+        Empty -> invalid_arg "Mapext.remove_min_elt"
+      | Node(Empty, x, d, r, _) -> r
+      | Node(l, x, d, r, _) -> bal (remove_min_binding l) x d r
+
+    let merge t1 t2 =
+      match (t1, t2) with
+        (Empty, t) -> t
+      | (t, Empty) -> t
+      | (_, _) ->
+          let (x, d) = min_binding t2 in
+          bal t1 x d (remove_min_binding t2)
+
+    let rec remove x = function
+        Empty ->
+          Empty
+      | Node(l, v, d, r, h) ->
+          let c = Ord.compare x v in
+          if c = 0 then
+            merge l r
+          else if c < 0 then
+            bal (remove x l) v d r
+          else
+            bal l v d (remove x r)
+
+    let rec iter f = function
+        Empty -> ()
+      | Node(l, v, d, r, _) ->
+          iter f l; f v d; iter f r
+
+    let rec map f = function
+        Empty ->
+          Empty
+      | Node(l, v, d, r, h) ->
+          let l' = map f l in
+          let d' = f d in
+          let r' = map f r in
+          Node(l', v, d', r', h)
+
+    let rec mapi f = function
+        Empty ->
+          Empty
+      | Node(l, v, d, r, h) ->
+          let l' = mapi f l in
+          let d' = f v d in
+          let r' = mapi f r in
+          Node(l', v, d', r', h)
+
+    let rec fold f m accu =
+      match m with
+        Empty -> accu
+      | Node(l, v, d, r, _) ->
+          fold f r (f v d (fold f l accu))
+
+    (* [AM] changed to call p in the key order *)
+    let rec for_all p = function
+        Empty -> true
+      | Node(l, v, d, r, _) -> for_all p l && p v d && for_all p r
+
+    (* [AM] changed to call p in the key order *)
+    let rec exists p = function
+        Empty -> false
+      | Node(l, v, d, r, _) -> exists p l || p v d || exists p r
+
+    (* [AM] changed to call p in the key order *)
+    let filter p s =
+      fold (fun k d a -> if p k d then add k d a else a) s Empty
+
+    let partition p s =
+      let rec part (t, f as accu) = function
+        | Empty -> accu
+        | Node(l, v, d, r, _) ->
+            part (part (if p v d then (add v d t, f) else (t, add v d f)) l) r in
+      part (Empty, Empty) s
+
+    (* Same as create and bal, but no assumptions are made on the
+       relative heights of l and r. *)
+
+    let rec join l v d r =
+      match (l, r) with
+        (Empty, _) -> add v d r
+      | (_, Empty) -> add v d l
+      | (Node(ll, lv, ld, lr, lh), Node(rl, rv, rd, rr, rh)) ->
+          if lh > rh + 2 then bal ll lv ld (join lr v d r) else
+          if rh > lh + 2 then bal (join l v d rl) rv rd rr else
+          create l v d r
+
+    (* Merge two trees l and r into one.
+       All elements of l must precede the elements of r.
+       No assumption on the heights of l and r. *)
+
+    let concat t1 t2 =
+      match (t1, t2) with
+        (Empty, t) -> t
+      | (t, Empty) -> t
+      | (_, _) ->
+          let (x, d) = min_binding t2 in
+          join t1 x d (remove_min_binding t2)
+
+    let concat_or_join t1 v d t2 =
+      match d with
+      | Some d -> join t1 v d t2
+      | None -> concat t1 t2
+
+    let rec split x = function
+        Empty ->
+          (Empty, None, Empty)
+      | Node(l, v, d, r, _) ->
+          let c = Ord.compare x v in
+          if c = 0 then (l, Some d, r)
+          else if c < 0 then
+            let (ll, pres, rl) = split x l in (ll, pres, join rl v d r)
+          else
+            let (lr, pres, rr) = split x r in (join l v d lr, pres, rr)
+
+    let rec merge f s1 s2 =
+      match (s1, s2) with
+        (Empty, Empty) -> Empty
+      | (Node (l1, v1, d1, r1, h1), _) when h1 >= height s2 ->
+          let (l2, d2, r2) = split v1 s2 in
+          concat_or_join (merge f l1 l2) v1 (f v1 (Some d1) d2) (merge f r1 r2)
+      | (_, Node (l2, v2, d2, r2, h2)) ->
+          let (l1, d1, r1) = split v2 s1 in
+          concat_or_join (merge f l1 l2) v2 (f v2 d1 (Some d2)) (merge f r1 r2)
+      | _ ->
+          assert false
+
+    type 'a enumeration = End | More of key * 'a * 'a t * 'a enumeration
+
+    let rec cons_enum m e =
+      match m with
+        Empty -> e
+      | Node(l, v, d, r, _) -> cons_enum l (More(v, d, r, e))
+
+    let compare cmp m1 m2 =
+      let rec compare_aux e1 e2 =
+          match (e1, e2) with
+          (End, End) -> 0
+        | (End, _)  -> -1
+        | (_, End) -> 1
+        | (More(v1, d1, r1, e1), More(v2, d2, r2, e2)) ->
+            let c = Ord.compare v1 v2 in
+            if c <> 0 then c else
+            let c = cmp d1 d2 in
+            if c <> 0 then c else
+            compare_aux (cons_enum r1 e1) (cons_enum r2 e2)
+      in compare_aux (cons_enum m1 End) (cons_enum m2 End)
+
+    let equal cmp m1 m2 =
+      let rec equal_aux e1 e2 =
+          match (e1, e2) with
+          (End, End) -> true
+        | (End, _)  -> false
+        | (_, End) -> false
+        | (More(v1, d1, r1, e1), More(v2, d2, r2, e2)) ->
+            Ord.compare v1 v2 = 0 && cmp d1 d2 &&
+            equal_aux (cons_enum r1 e1) (cons_enum r2 e2)
+      in equal_aux (cons_enum m1 End) (cons_enum m2 End)
+
+    let rec cardinal = function
+        Empty -> 0
+      | Node(l, _, _, r, _) -> cardinal l + 1 + cardinal r
+
+    let rec bindings_aux accu = function
+        Empty -> accu
+      | Node(l, v, d, r, _) -> bindings_aux ((v, d) :: bindings_aux accu r) l
+
+    let bindings s =
+      bindings_aux [] s
+
+    let choose = min_binding
+
+
+
+    (* [AM] additions by Antoine Mine' *)
+    (* ******************************* *)
+
+
+    let of_list l =
+      List.fold_left (fun acc (k,x) -> add k x acc) empty l
+
+
+    (* similar to split, but returns unbalanced trees *)
+    let rec cut k = function
+      Empty -> Empty,None,Empty
+    | Node (l1,k1,d1,r1,h1) ->
+        let c = Ord.compare k k1 in
+        if c < 0 then 
+          let l2,d2,r2 = cut k l1 in (l2,d2,Node (r2,k1,d1,r1,h1))
+        else if c > 0 then
+          let l2,d2,r2 = cut k r1 in (Node (l1,k1,d1,l2,h1),d2,r2)
+        else (l1,Some d1,r1)
+
+
+    (* binary operations that fail on maps with different keys *)
+
+    (* functions are called in increasing key order *)
+
+    let rec map2 f m1 m2 =
+      match m1 with
+      | Empty -> if m2 = Empty then Empty else invalid_arg "Mapext.map2"
+      | Node (l1,k,d1,r1,h1) ->
+          match cut k m2 with 
+          | l2, Some d2, r2 ->
+              Node (map2 f l1 l2, k, f k d1 d2, map2 f r1 r2, h1)
+          | _, None, _ -> invalid_arg "Mapext.map2"
+            
+    let rec iter2 f m1 m2 =
+      match m1 with
+      | Empty -> if m2 = Empty then () else invalid_arg "Mapext.iter2"
+      | Node (l1,k,d1,r1,h1) ->
+          match cut k m2 with 
+          | l2, Some d2, r2 -> iter2 f l1 l2; f k d1 d2; iter2 f r1 r2
+          | _, None, _ -> invalid_arg "Mapext.iter2"
+            
+    let rec fold2 f m1 m2 acc =
+      match m1 with
+      | Empty -> if m2 = Empty then acc else invalid_arg "Mapext.fold2"
+      | Node (l1,k,d1,r1,h1) ->
+          match cut k m2 with 
+          | l2, Some d2, r2 -> 
+              fold2 f r1 r2 (f k d1 d2 (fold2 f l1 l2 acc))
+          | _, None, _ -> invalid_arg "Mapext.fold2"
+            
+    let rec for_all2 f m1 m2 =
+      match m1 with
+      | Empty -> if m2 = Empty then true else invalid_arg "Mapext.for_all2"
+      | Node (l1,k,d1,r1,h1) ->
+          match cut k m2 with 
+          | l2, Some d2, r2 -> 
+              for_all2 f l1 l2 && f k d1 d2 && for_all2 f r1 r2
+          | _, None, _ -> invalid_arg "Mapext.for_all2"
+            
+    let rec exists2 f m1 m2 =
+      match m1 with
+      | Empty -> if m2 = Empty then false else invalid_arg "Mapext.exists2"
+      | Node (l1,k,d1,r1,h1) ->
+          match cut k m2 with 
+          | l2, Some d2, r2 -> 
+              exists2 f l1 l2 || f k d1 d2 || exists2 f r1 r2
+          | _, None, _ -> invalid_arg "Mapext.exists2"
+            
+
+    (* as above, but ignore physically equal subtrees
+       - for map, assumes: f k d d = d
+       - for iter, assumes: f k d d has no effect
+       - for fold, assumes: k f d d acc = acc
+       - for for_all, assumes: f k d d = true
+       - for exists, assumes: f k d d = false
+     *)
+
+    let rec map2z f m1 m2 =
+      if m1 == m2 then m1 else
+      match m1 with
+      | Empty -> if m2 = Empty then Empty else invalid_arg "Mapext.map2z"
+      | Node (l1,k,d1,r1,h1) ->
+          match cut k m2 with 
+          | l2, Some d2, r2 ->
+              let d = if d1 == d2 then d1 else f k d1 d2 in
+              Node (map2z f l1 l2, k, d, map2z f r1 r2, h1)
+          | _, None, _ -> invalid_arg "Mapext.map2z"
+            
+    let rec iter2z f m1 m2 =
+      if m1 == m2 then () else
+      match m1 with
+      | Empty -> if m2 = Empty then () else invalid_arg "Mapext.iter2z"
+      | Node (l1,k,d1,r1,h1) ->
+          match cut k m2 with 
+          | l2, Some d2, r2 -> 
+              iter2z f l1 l2; (if d1 != d2 then f k d1 d2); iter2z f r1 r2
+          | _, None, _ -> invalid_arg "Mapext.iter2z"
+            
+    let rec fold2z f m1 m2 acc =
+      if m1 == m2 then acc else
+      match m1 with
+      | Empty -> if m2 = Empty then acc else invalid_arg "Mapext.fold2z"
+      | Node (l1,k,d1,r1,h1) ->
+          match cut k m2 with 
+          | l2, Some d2, r2 -> 
+              let acc = fold2z f l1 l2 acc in
+              let acc = if d1 == d2 then acc else f k d1 d2 acc in
+              fold2z f r1 r2 acc
+          | _, None, _ -> invalid_arg "Mapext.fold2z"
+            
+    let rec for_all2z f m1 m2 =
+      (m1 == m2) ||
+      (match m1 with
+      | Empty -> if m2 = Empty then true else invalid_arg "Mapext.for_all2z"
+      | Node (l1,k,d1,r1,h1) ->
+          match cut k m2 with 
+          | l2, Some d2, r2 -> 
+              (for_all2z f l1 l2) && 
+              (d1 == d2 || f k d1 d2) && 
+              (for_all2z f r1 r2)
+          | _, None, _ -> invalid_arg "Mapext.for_all2z"
+      )     
+            
+    let rec exists2z f m1 m2 =
+      (m1 != m2) &&
+      (match m1 with
+      | Empty -> if m2 = Empty then false else invalid_arg "Mapext.exists2z"
+      | Node (l1,k,d1,r1,h1) ->
+          match cut k m2 with 
+          | l2, Some d2, r2 -> 
+              (exists2z f l1 l2) ||
+              (d1 != d2 && f k d1 d2) ||
+              (exists2z f r1 r2)
+          | _, None, _ -> invalid_arg "Mapext.exists2z"
+      )     
+
+
+    (* as above, but allow maps with different keys *)
+
+    let rec map2o f1 f2 f m1 m2 =
+      match m1 with
+      | Empty -> mapi f2 m2
+      | Node (l1,k,d1,r1,h1) ->
+          let l2, d2, r2 = cut k m2 in
+          let l = map2o f1 f2 f l1 l2 in
+          let d = match d2 with None -> f1 k d1 | Some d2 -> f k d1 d2 in
+          let r = map2o f1 f2 f r1 r2 in
+          join l k d r
+            
+    let rec iter2o f1 f2 f m1 m2 =
+      match m1 with
+      | Empty -> iter f2 m2
+      | Node (l1,k,d1,r1,h1) ->
+          let l2, d2, r2 = cut k m2 in
+          iter2o f1 f2 f l1 l2;
+          (match d2 with None -> f1 k d1 | Some d2 -> f k d1 d2);
+          iter2o f1 f2 f r1 r2
+            
+    let rec fold2o f1 f2 f m1 m2 acc =
+      match m1 with
+      | Empty -> fold f2 m2 acc
+      | Node (l1,k,d1,r1,h1) ->
+          let l2, d2, r2 = cut k m2 in
+          let acc = fold2o f1 f2 f l1 l2 acc in
+          let acc = match d2 with 
+          | None -> f1 k d1 acc | Some d2 -> f k d1 d2 acc
+          in
+          fold2o f1 f2 f r1 r2 acc
+            
+    let rec for_all2o f1 f2 f m1 m2 =
+      match m1 with
+      | Empty -> for_all f2 m2 
+      | Node (l1,k,d1,r1,h1) ->
+          let l2, d2, r2 = cut k m2 in
+          (for_all2o f1 f2 f l1 l2) &&
+          (match d2 with None -> f1 k d1 | Some d2 -> f k d1 d2) &&
+          (for_all2o f1 f2 f r1 r2)
+
+    let rec exists2o f1 f2 f m1 m2 =
+      match m1 with
+      | Empty -> exists f2 m2 
+      | Node (l1,k,d1,r1,h1) ->
+          let l2, d2, r2 = cut k m2 in
+          (exists2o f1 f2 f l1 l2) ||
+          (match d2 with None -> f1 k d1 | Some d2 -> f k d1 d2) ||
+          (exists2o f1 f2 f r1 r2)
+
+
+    (* all together now *)
+
+    let rec map2zo f1 f2 f m1 m2 =
+      if m1 == m2 then m1 else
+      match m1 with
+      | Empty -> mapi f2 m2
+      | Node (l1,k,d1,r1,h1) ->
+          let l2, d2, r2 = cut k m2 in
+          let l = map2zo f1 f2 f l1 l2 in
+          let d = match d2 with 
+          | None -> f1 k d1 
+          | Some d2 -> if d1 == d2 then d1 else f k d1 d2 
+          in
+          let r = map2zo f1 f2 f r1 r2 in
+          join l k d r
+            
+    let rec iter2zo f1 f2 f m1 m2 =
+      if m1 == m2 then () else
+      match m1 with
+      | Empty -> iter f2 m2
+      | Node (l1,k,d1,r1,h1) ->
+          let l2, d2, r2 = cut k m2 in
+          iter2zo f1 f2 f l1 l2;
+          (match d2 with 
+          | None -> f1 k d1 
+          | Some d2 -> if d1 != d2 then f k d1 d2);
+          iter2zo f1 f2 f r1 r2
+            
+    let rec fold2zo f1 f2 f m1 m2 acc =
+      if m1 == m2 then acc else
+      match m1 with
+      | Empty -> fold f2 m2 acc
+      | Node (l1,k,d1,r1,h1) ->
+          let l2, d2, r2 = cut k m2 in
+          let acc = fold2zo f1 f2 f l1 l2 acc in
+          let acc = match d2 with 
+          | None -> f1 k d1 acc 
+          | Some d2 -> if d1 == d2 then acc else f k d1 d2 acc
+          in
+          fold2zo f1 f2 f r1 r2 acc
+            
+    let rec for_all2zo f1 f2 f m1 m2 =
+      (m1 == m2) ||
+      (match m1 with
+      | Empty -> for_all f2 m2 
+      | Node (l1,k,d1,r1,h1) ->
+          let l2, d2, r2 = cut k m2 in
+          (for_all2zo f1 f2 f l1 l2) &&
+          (match d2 with None -> f1 k d1 | Some d2 -> d1 == d2 || f k d1 d2) &&
+          (for_all2zo f1 f2 f r1 r2)
+      )
+
+    let rec exists2zo f1 f2 f m1 m2 =
+      (m1 != m2) &&
+      (match m1 with
+      | Empty -> exists f2 m2 
+      | Node (l1,k,d1,r1,h1) ->
+          let l2, d2, r2 = cut k m2 in
+          (exists2zo f1 f2 f l1 l2) ||
+          (match d2 with None -> f1 k d1 | Some d2 -> d1 != d2 && f k d1 d2) ||
+          (exists2zo f1 f2 f r1 r2)
+      )
+
+
+    (* iterators limited to keys between two bounds *)
+
+    let rec map_slice f m lo hi =
+      match m with
+      | Empty -> Empty
+      | Node (l,k,d,r,h) ->
+          let c1, c2 = Ord.compare k lo, Ord.compare k hi in
+          let l = if c1 > 0 then map_slice f l lo k else l in
+          let d = if c1 >= 0 && c2 <= 0 then f k d else d in
+          let r = if c2 < 0 then map_slice f r k hi else r in
+          Node (l,k,d,r,h)
+
+    let rec iter_slice f m lo hi =
+      match m with
+      | Empty -> ()
+      | Node (l,k,d,r,_) ->
+          let c1, c2 = Ord.compare k lo, Ord.compare k hi in
+          if c1 > 0 then iter_slice f l lo k;
+          if c1 >= 0 && c2 <= 0 then f k d;
+          if c2 < 0 then iter_slice f r k hi
+
+    let rec fold_slice f m lo hi acc =
+      match m with
+      | Empty -> acc
+      | Node (l,k,d,r,_) ->
+          let c1, c2 = Ord.compare k lo, Ord.compare k hi in
+          let acc = if c1 > 0 then fold_slice f l lo k acc else acc in
+          let acc = if c1 >= 0 && c2 <= 0 then f k d acc else acc in
+          if c2 < 0 then fold_slice f r k hi acc else acc
+
+    let rec for_all_slice f m lo hi =
+      match m with
+      | Empty -> true
+      | Node (l,k,d,r,_) ->
+          let c1, c2 = Ord.compare k lo, Ord.compare k hi in
+          (c1 <= 0 || for_all_slice f l lo k) &&
+          (c1 < 0 || c2 > 0 || f k d) &&
+          (c2 >= 0 || for_all_slice f r k hi)
+
+    let rec exists_slice f m lo hi =
+      match m with
+      | Empty -> false
+      | Node (l,k,d,r,_) ->
+          let c1, c2 = Ord.compare k lo, Ord.compare k hi in
+          (c1 > 0 && exists_slice f l lo k) ||
+          (c1 >= 0 && c2 <= 0 && f k d) ||
+          (c2 < 0 && exists_slice f r k hi)
+
+
+    (* key set comparison *)
+
+    let rec key_equal m1 m2 =
+      (m1 == m2) || 
+      (match m1 with
+      | Empty -> m2 = Empty
+      | Node (l1, k, _, r1, _) ->
+          match cut k m2 with
+          | _, None, _ -> false
+          | l2, Some _, r2 -> key_equal l1 l2 && key_equal r1 r2
+      )
+
+    let rec key_subset m1 m2 =
+      (m1 == m2) || 
+      (match m1 with
+      | Empty -> true
+      | Node (l1, k, _, r1, _) ->
+          match cut k m2 with
+          | _, None, _ -> false
+          | l2, Some _, r2 -> key_subset l1 l2 && key_subset r1 r2
+      )
+
+
+    (* nagivation *)
+
+    let find_greater_equal k m =
+      let rec aux m found = match m with
+      | Empty -> (match found with None -> raise Not_found | Some x -> x)
+      | Node (l, kk, d, r, _) ->
+          let c = Ord.compare k kk in
+          if c = 0 then kk, d else
+          if c > 0 then aux r found else
+          aux l (Some (kk, d))
+      in
+      aux m None 
+
+    let find_greater k m =
+      let rec aux m found = match m with
+      | Empty -> (match found with None -> raise Not_found | Some x -> x)
+      | Node (l, kk, d, r, _) ->
+          let c = Ord.compare k kk in
+          if c >= 0 then aux r found else
+          aux l (Some (kk, d))
+      in
+      aux m None 
+
+    let find_less_equal k m =
+      let rec aux m found = match m with
+      | Empty -> (match found with None -> raise Not_found | Some x -> x)
+      | Node (l, kk, d, r, _) ->
+          let c = Ord.compare k kk in
+          if c = 0 then kk, d else
+          if c < 0 then aux l found else
+          aux r (Some (kk, d))
+      in
+      aux m None 
+
+    let find_less k m =
+      let rec aux m found = match m with
+      | Empty -> (match found with None -> raise Not_found | Some x -> x)
+      | Node (l, kk, d, r, _) ->
+          let c = Ord.compare k kk in
+          if c <= 0 then aux l found else
+          aux r (Some (kk, d))
+      in
+      aux m None 
+
+
+end: S with type key = Ord.t)
diff --git a/libs/mapext.mli b/libs/mapext.mli
new file mode 100644
index 0000000..abee2c0
--- /dev/null
+++ b/libs/mapext.mli
@@ -0,0 +1,473 @@
+(*
+  Cours "Sémantique et Application à la Vérification de programmes"
+  
+  Antoine Miné 2014
+  Ecole normale supérieure, Paris, France / CNRS / INRIA
+*)
+  
+(*  
+  This file is derived from the mapi.ml file from the OCaml distribution.
+  Changes are marked with the [AM] symbol.
+  Based on rev. 10632 2010-07-24 14:16:58Z.
+
+  Original copyright follows.
+*)
+
+
+(***********************************************************************)
+(*                                                                     *)
+(*                           Objective Caml                            *)
+(*                                                                     *)
+(*            Xavier Leroy, projet Cristal, INRIA Rocquencourt         *)
+(*                                                                     *)
+(*  Copyright 1996 Institut National de Recherche en Informatique et   *)
+(*  en Automatique.  All rights reserved.  This file is distributed    *)
+(*  under the terms of the GNU Library General Public License, with    *)
+(*  the special exception on linking described in file ../LICENSE.     *)
+(*                                                                     *)
+(***********************************************************************)
+
+(* $Id: mapext.mli,v 1.1 2014-02-24 16:25:06 mine Exp $ *)
+
+(** Association tables over ordered types.
+
+   This module implements applicative association tables, also known as
+   finite maps or dictionaries, given a total ordering function
+   over the keys.
+   All operations over maps are purely applicative (no side-effects).
+   The implementation uses balanced binary trees, and therefore searching
+   and insertion take time logarithmic in the size of the map.
+*)
+
+module type OrderedType =
+  sig
+    type t
+      (** The type of the map keys. *)
+    val compare : t -> t -> int
+      (** A total ordering function over the keys.
+          This is a two-argument function [f] such that
+          [f e1 e2] is zero if the keys [e1] and [e2] are equal,
+          [f e1 e2] is strictly negative if [e1] is smaller than [e2],
+          and [f e1 e2] is strictly positive if [e1] is greater than [e2].
+          Example: a suitable ordering function is the generic structural
+          comparison function {!Pervasives.compare}. *)
+  end
+(** Input signature of the functor {!Map.Make}. *)
+
+module type S =
+  sig
+    type key
+    (** The type of the map keys. *)
+
+    type (+'a) t
+    (** The type of maps from type [key] to type ['a]. *)
+
+    val empty: 'a t
+    (** The empty map. *)
+
+    val is_empty: 'a t -> bool
+    (** Test whether a map is empty or not. *)
+
+    val mem: key -> 'a t -> bool
+    (** [mem x m] returns [true] if [m] contains a binding for [x],
+       and [false] otherwise. *)
+
+    val add: key -> 'a -> 'a t -> 'a t
+    (** [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. *)
+
+    val singleton: key -> 'a -> 'a t
+    (** [singleton x y] returns the one-element map that contains a binding [y]
+        for [x].
+        @since 3.12.0
+     *)
+
+    val remove: key -> 'a t -> 'a t
+    (** [remove x m] returns a map containing the same bindings as
+       [m], except for [x] which is unbound in the returned map. *)
+
+    val merge:
+         (key -> 'a option -> 'b option -> 'c option) -> 'a t -> 'b t -> 'c t
+    (** [merge f m1 m2] computes a map whose keys is a subset of keys of [m1]
+        and of [m2]. The presence of each such binding, and the corresponding
+        value, is determined with the function [f].
+        @since 3.12.0
+     *)
+
+    val compare: ('a -> 'a -> int) -> 'a t -> 'a t -> int
+    (** Total ordering between maps.  The first argument is a total ordering
+        used to compare data associated with equal keys in the two maps. *)
+
+    val equal: ('a -> 'a -> bool) -> 'a t -> 'a t -> 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. *)
+
+    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. *)
+
+    val fold: (key -> 'a -> 'b -> 'b) -> 'a t -> 'b -> 'b
+    (** [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. *)
+
+    val for_all: (key -> 'a -> bool) -> 'a t -> bool        
+    (* [AM] now guarantees the evaluation order *)
+    (** [for_all p m] checks if all the bindings of the map
+        satisfy the predicate [p].
+        The predicate [p] is tested on bindings according to the key order.
+        @since 3.12.0
+     *)
+
+    val exists: (key -> 'a -> bool) -> 'a t -> bool
+    (* [AM] now guarantees the evaluation order *)
+    (** [exists p m] checks if at least one binding of the map
+        satisfy the predicate [p].
+        The predicate [p] is tested on bindings according to the key order.
+        @since 3.12.0
+     *)
+
+    val filter: (key -> 'a -> bool) -> 'a t -> 'a t
+    (* [AM] now guarantees the evaluation order *)
+    (** [filter p m] returns the map with all the bindings in [m]
+        that satisfy predicate [p].
+        The predicate [p] is tested on bindings according to the key order.
+        @since 3.12.0
+     *)
+
+    val partition: (key -> 'a -> bool) -> 'a t -> 'a t * 'a t
+    (** [partition p m] returns a pair of maps [(m1, m2)], where
+        [m1] contains all the bindings of [s] that satisfy the
+        predicate [p], and [m2] is the map with all the bindings of
+        [s] that do not satisfy [p].
+        @since 3.12.0
+     *)
+
+    val cardinal: 'a t -> int
+    (** Return the number of bindings of a map.
+        @since 3.12.0
+     *)
+
+    val bindings: 'a t -> (key * 'a) list
+    (** Return the list of all bindings of the given map.
+       The returned list is sorted in increasing order with respect
+       to the ordering [Ord.compare], where [Ord] is the argument
+       given to {!Map.Make}.
+        @since 3.12.0
+     *)
+
+    val min_binding: 'a t -> (key * 'a)
+    (** Return the smallest binding of the given map
+       (with respect to the [Ord.compare] ordering), or raise
+       [Not_found] if the map is empty.
+        @since 3.12.0
+     *)
+
+    val max_binding: 'a t -> (key * 'a)
+    (** Same as {!Map.S.min_binding}, but returns the largest binding
+        of the given map.
+        @since 3.12.0
+     *)
+
+    val choose: 'a t -> (key * 'a)
+    (** Return one binding of the given map, or raise [Not_found] if
+       the map is empty. Which binding is chosen is unspecified,
+       but equal bindings will be chosen for equal maps.
+        @since 3.12.0
+     *)
+
+    val split: key -> 'a t -> 'a t * 'a option * 'a t
+    (** [split x m] returns a triple [(l, data, r)], where
+          [l] is the map with all the bindings of [m] whose key
+        is strictly less than [x];
+          [r] is the map with all the bindings of [m] whose key
+        is strictly greater than [x];
+          [data] is [None] if [m] contains no binding for [x],
+          or [Some v] if [m] binds [v] to [x].
+        @since 3.12.0
+     *)
+
+    val find: key -> 'a t -> 'a
+    (** [find x m] returns the current binding of [x] in [m],
+       or raises [Not_found] if no such binding exists. *)
+
+    val map: ('a -> 'b) -> 'a t -> 'b t
+    (** [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. *)
+
+    val mapi: (key -> 'a -> 'b) -> 'a t -> 'b t
+    (** Same as {!Map.S.map}, but the function receives as arguments both the
+       key and the associated value for each binding of the map. *)
+
+
+    (* [AM] additions *)
+   
+    (** {2 Additional functions} *)
+
+    val of_list: (key * 'a) list -> 'a t
+    (** [of_list l] converts an association list to a map. *)
+
+    val map2: (key -> 'a -> 'b -> 'c) -> 'a t -> 'b t -> 'c t
+    (** [map2 f m1 m2] is similar to [map] but applies [f] to pairs
+       of bindings [a1] from [m1] and [a2] from [m2] corresponding to
+       the same key to construct a new map with the same key set.
+       [m1] and [m2] must have the same key sets.
+       The binging are passed to [f] in increasing order of key. *)
+
+    val iter2: (key -> 'a -> 'b -> unit) -> 'a t -> 'b t -> unit
+    (** [iter2 f m1 m2] is similar to [map] but applies [f] to pairs
+       of bindings [a1] from [m1] and [a2] from [m2] corresponding to
+       the same key.
+       [m1] and [m2] must have the same key sets.
+       The binging are passed to [f] in increasing order of key. *)
+
+    val fold2: (key -> 'a -> 'b -> 'c -> 'c) -> 'a t -> 'b t -> 'c -> 'c
+    (** [fold2 f m1 m2 x] is similar to [fold] but applies [f] to pairs
+       of bindings [a1] from [m1] and [a2] from [m2] corresponding to
+       the same key.
+       [m1] and [m2] must have the same key sets.
+       The bindings are passed to [f] in increasing order of keys. *)
+
+    val for_all2: (key -> 'a -> 'b -> bool) -> 'a t -> 'b t -> bool
+    (** [for_all2 f m1 m2] is similar to [for_all] but applies [f] to pairs
+       of bindings [a1] from [m1] and [a2] from [m2] corresponding to
+       the same key.
+       [m1] and [m2] must have the same key sets.
+       The bindings are passed to [f] in increasing order of keys. *)
+
+    val exists2: (key -> 'a -> 'b -> bool) -> 'a t -> 'b t -> bool
+    (** [exists2 f m1 m2] is similar to [exists] but applies [f] to pairs
+       of bindings [a1] from [m1] and [a2] from [m2] corresponding to
+       the same key.
+       [m1] and [m2] must have the same key sets.
+       The bindings are passed to [f] in increasing order of keys. *)
+
+
+
+
+    val map2z: (key -> 'a -> 'a -> 'a) -> 'a t -> 'a t -> 'a t
+    (** [map2z f m1 m2] is similar to [map2 f m1 m2], but physically
+        equal subtrees are put unchanged into the result instead of
+        being traversed.
+        This is more efficient than [map2], and equivalent if [f] is
+        side-effect free and idem-potent ([f k a a = a]).
+        [m1] and [m2] must have the same key sets.
+        The bindings are passed to [f] in increasing order of keys. *)
+  
+    val iter2z: (key -> 'a -> 'a -> unit) -> 'a t -> 'a t -> unit
+     (** [iter2z f m1 m2] is similar to [iter2 f m1 m2], but physically
+        equal subtrees are ignored.
+        This is more efficient than [iter2], and equivalent if 
+        [f k a a] has no effect.
+        [m1] and [m2] must have the same key sets.
+        The bindings are passed to [f] in increasing order of keys. *)
+
+    val fold2z: (key -> 'a -> 'a -> 'b -> 'b) -> 'a t -> 'a t -> 'b -> 'b
+    (** [fold2z f m1 m2 a] is similar to [fold2 f m1 m2 a], but physically
+        equal subtrees are ignored.
+        This is more efficient than [fold2], and equivalent if 
+        [f k a a x = x] and has no effect.
+        [m1] and [m2] must have the same key sets.
+        The bindings are passed to [f] in increasing order of keys. *)
+
+    val for_all2z: (key -> 'a -> 'a -> bool) -> 'a t -> 'a t -> bool
+    (** [for_all2z f m1 m2] is similar to [for_all2 f m1 m2], but returns
+        [true] for physically equal subtrees without traversing them.
+        This is more efficient than [for_all2z], and equivalent if
+        [f k a a = true] and has no effect.
+        [m1] and [m2] must have the same key sets.
+        The bindings are passed to [f] in increasing order of keys. *)
+
+    val exists2z: (key -> 'a -> 'a -> bool) -> 'a t -> 'a t -> bool
+    (** [exists2z f m1 m2] is similar to [exists2 f m1 m2], but returns
+        [false] for physically equal subtrees without traversing them.
+        This is more efficient than [exists2z], and equivalent if
+        [f k a a = false] and has no effect.
+        [m1] and [m2] must have the same key sets.
+        The bindings are passed to [f] in increasing order of keys. *)
+
+
+
+
+    val map2o: (key -> 'a -> 'c) -> (key -> 'b -> 'c) -> (key -> 'a -> 'b -> 'c) -> 'a t -> 'b t -> 'c t
+    (** [map2o f1 f2 f m1 m2] is similar to [map2 f m1 m2], but 
+        accepts maps defined over different sets of keys.
+        To get a new binding, [f1] is used for keys appearing only
+        in [m1], [f2] for keys appearing only in [m2], and [f] for
+        keys appearing in both maps.
+        The returned map has bindings for all keys appearing in either
+        [m1] or [m2].
+        The bindings are passed to [f], [f1], [f2] in increasing order of keys. *)
+
+    val iter2o: (key -> 'a -> unit) -> (key -> 'b -> unit) -> (key -> 'a -> 'b -> unit) -> 'a t -> 'b t -> unit
+    (** [iter2o f1 f2 f m1 m2] is similar to [iter2 f m1 m2], but 
+        accepts maps defined over different sets of keys.
+        [f1] is called for keys appearing only in [m1], 
+        [f2] for keys appearing only in [m2], 
+        and [f] for keys appearing in both maps.
+        The bindings are passed to [f], [f1], [f2] in increasing order of keys. *)
+
+    val fold2o: (key -> 'a -> 'c -> 'c) -> (key -> 'b -> 'c -> 'c) -> (key -> 'a -> 'b -> 'c -> 'c) -> 'a t -> 'b t -> 'c -> 'c
+    (** [fold2o f1 f2 f m1 m2 a] is similar to [fold2 f m1 m2 a], but 
+        accepts maps defined over different sets of keys.
+        [f1] is called for keys appearing only in [m1], 
+        [f2] for keys appearing only in [m2], 
+        and [f] for keys appearing in both maps.
+        The bindings are passed to [f], [f1], [f2] in increasing order of keys. *)
+
+    val for_all2o: (key -> 'a -> bool) -> (key -> 'b -> bool) -> (key -> 'a -> 'b -> bool) -> 'a t -> 'b t -> bool
+    (** [for_all2o f1 f2 f m1 m2] is similar to [for_all2 f m1 m2], but 
+        accepts maps defined over different sets of keys.
+        [f1] is called for keys appearing only in [m1], 
+        [f2] for keys appearing only in [m2], 
+        and [f] for keys appearing in both maps.
+        The bindings are passed to [f], [f1], [f2] in increasing order of keys. *)
+
+    val exists2o: (key -> 'a -> bool) -> (key -> 'b -> bool) -> (key -> 'a -> 'b -> bool) -> 'a t -> 'b t -> bool
+    (** [fexists2o f1 f2 f m1 m2] is similar to [fexists2 f m1 m2], but 
+        accepts maps defined over different sets of keys.
+        [f1] is called for keys appearing only in [m1], 
+        [f2] for keys appearing only in [m2], 
+        and [f] for keys appearing in both maps.
+        The bindings are passed to [f], [f1], [f2] in increasing order of keys. *)
+
+
+
+    val map2zo: (key -> 'a -> 'a) -> (key -> 'a -> 'a) -> (key -> 'a -> 'a -> 'a) -> 'a t -> 'a t -> 'a t
+    (** [map2zo f1 f2 f m1 m2] is similar to [map2o f1 f2 f m1 m2] but,
+        similary to [map2z], [f] is not called on physically equal
+        subtrees.
+        This is more efficient than [map2o], and equivalent if [f] is
+        side-effect free and idem-potent ([f k a a = a]).
+        The returned map has bindings for all keys appearing in either
+        [m1] or [m2].
+        The bindings are passed to [f], [f1], [f2] in increasing order of keys. *)
+        
+    val iter2zo: (key -> 'a -> unit) -> (key -> 'a -> unit) -> (key -> 'a -> 'a -> unit) -> 'a t -> 'a t -> unit
+    (** [iter2zo f1 f2 f m1 m2] is similar to [iter2o f1 f2 f m1 m2] but,
+        similary to [iter2z], [f] is not called on physically equal
+        subtrees.
+        This is more efficient than [iter2o], and equivalent if [f] is
+        side-effect free.
+        The bindings are passed to [f], [f1], [f2] in increasing order of keys. *)
+        
+    val fold2zo: (key -> 'a -> 'b -> 'b) -> (key -> 'a -> 'b -> 'b) -> (key -> 'a -> 'a -> 'b -> 'b) -> 'a t -> 'a t -> 'b -> 'b
+    (** [fold2zo f1 f2 f m1 m2 a] is similar to [fold2o f1 f2 f m1 m2 a] but,
+        similary to [fold2z], [f] is not called on physically equal
+        subtrees.
+        This is more efficient than [fold2o], and equivalent if 
+        [f k a a x = x] and has no side-effect.
+        The bindings are passed to [f], [f1], [f2] in increasing order of keys. *)
+        
+    val for_all2zo: (key -> 'a -> bool) -> (key -> 'a -> bool) -> (key -> 'a -> 'a -> bool) -> 'a t -> 'a t -> bool
+    (** [for_all2zo f1 f2 f m1 m2] is similar to [for_all2o f1 f2 f m1 m2] but,
+        similary to [for_all2z], [f] is not called on physically equal
+        subtrees.
+        This is more efficient than [for_all2o], and equivalent if 
+        [f k a a = true] and has no side-effect.
+        The bindings are passed to [f], [f1], [f2] in increasing order of keys. *)
+
+    val exists2zo: (key -> 'a -> bool) -> (key -> 'a -> bool) -> (key -> 'a -> 'a -> bool) -> 'a t -> 'a t -> bool
+    (** [exists2zo f1 f2 f m1 m2] is similar to [exists2o f1 f2 f m1 m2] but,
+        similary to [exists2z], [f] is not called on physically equal
+        subtrees.
+        This is more efficient than [exists2o], and equivalent if 
+        [f k a a = false] and has no side-effect.
+        The bindings are passed to [f], [f1], [f2] in increasing order of keys. *)
+
+    val map_slice: (key -> 'a -> 'a) -> 'a t -> key -> key -> 'a t
+    (** [map_slice f m k1 k2] is similar to [map f m], but only applies
+        [f] to bindings with key greater or equal to [k1] and smaller
+        or equal to [k2] to construct the returned map. Bindings with
+        keys outside this range in [m] are put unchanged in the result.
+        It is as if, outside this range, [f k a = a] and has no effect.
+        The result has the same key set as [m].
+        The bindings are passed to [f] in increasing order of keys,
+        between [k1] and [k2]. *)
+
+    val iter_slice: (key -> 'a -> unit) -> 'a t -> key -> key -> unit
+    (** [iter_slice f m k1 k2] is similar to [iter f m], but only calls
+        [f] on bindings with key greater or equal to [k1] and smaller
+        or equal to [k2].
+        It is as if, outside this range, [f k a] has no effect.
+        The bindings are passed to [f] in increasing order of keys,
+        between [k1] and [k2]. *)
+
+    val fold_slice: (key -> 'a -> 'b -> 'b) -> 'a t -> key -> key -> 'b -> 'b
+    (** [fold_slice f m k1 k2 a] is similar to [fold f m], but only calls
+        [f] on bindings with key greater or equal to [k1] and smaller
+        or equal to [k2].
+        It is as if, outside this range, [f k a x = x] and has no effect.
+        The bindings are passed to [f] in increasing order of keys,
+        between [k1] and [k2]. *)
+
+    val for_all_slice: (key -> 'a -> bool) -> 'a t -> key -> key -> bool
+    (** [for_all_slice f m k1 k2 a] is similar to [for_all f m], but only calls
+        [f] on bindings with key greater or equal to [k1] and smaller
+        or equal to [k2].
+        It is as if, outside this range, [f k a = true] and has no effect.
+        The bindings are passed to [f] in increasing order of keys,
+        between [k1] and [k2]. *)
+
+    val exists_slice: (key -> 'a -> bool) -> 'a t -> key -> key -> bool
+    (** [exists_slice f m k1 k2 a] is similar to [exists f m], but only calls
+        [f] on bindings with key greater or equal to [k1] and smaller
+        or equal to [k2].
+        It is as if, outside this range, [f k a = false] and has no effect.
+        The bindings are passed to [f] in increasing order of keys,
+        between [k1] and [k2]. *)
+
+    val key_equal: 'a t -> 'a t -> bool
+    (** [key_equal m1 m2] returns true if [m1] and [m2] are defined
+        over exactly the same set of keys (but with possibly different
+        values).
+     *)
+
+    val key_subset: 'a t -> 'a t -> bool
+    (** [key_equal m1 m2] returns true if [m1] is defined on a subset of
+        the keys of [m2] (but with possibly different values).
+     *)
+
+    val find_greater: key -> 'a t -> key * 'a
+    (** [find_greater k m] returns the binding (key and value) in [m]
+        with key strictly greater than [k] and as small as possible.
+        Raises [Not_found] if [m] has no binding for a key strictly greater
+        than [k].
+     *)
+
+    val find_less: key -> 'a t -> key * 'a
+    (** [find_less k m] returns the binding (key and value) in [m]
+        with key strictly less than [k] and as large as possible.
+        Raises [Not_found] if [m] has no binding for a key strictly less
+        than [k].
+     *)
+
+    val find_greater_equal: key -> 'a t -> key * 'a
+    (** [find_greater_euql k m] returns the binding (key and value) in [m]
+        with key greater or equal to [k] and as small as possible.
+        Raises [Not_found] if [m] has no binding for a key greater or equal
+        to [k].
+     *)
+
+    val find_less_equal: key -> 'a t -> key * 'a
+    (** [find_less_equal k m] returns the binding (key and value) in [m]
+        with key less or equal to [k] and as large as possible.
+        Raises [Not_found] if [m] has no binding for a key less or equal
+        to [k].
+     *)
+
+
+  end
+(** Output signature of the functor {!Map.Make}. *)
+
+module Make (Ord : OrderedType) : S with type key = Ord.t
+(** Functor building an implementation of the map structure
+   given a totally ordered type. *)