(*
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)
