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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 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. *)