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[sledge] Add Map and Set from Stdlib

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Summary: 4.11.1 version

Reviewed By: ngorogiannis

Differential Revision: D26250520

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In the following, "the OCaml Core System" refers to all files marked
"Copyright INRIA" in this distribution.
The OCaml Core System is distributed under the terms of the
GNU Lesser General Public License (LGPL) version 2.1 (included below).
As a special exception to the GNU Lesser General Public License, you
may link, statically or dynamically, a "work that uses the OCaml Core
System" with a publicly distributed version of the OCaml Core System
to produce an executable file containing portions of the OCaml Core
System, and distribute that executable file under terms of your
choice, without any of the additional requirements listed in clause 6
of the GNU Lesser General Public License. By "a publicly distributed
version of the OCaml Core System", we mean either the unmodified OCaml
Core System as distributed by INRIA, or a modified version of the
OCaml Core System that is distributed under the conditions defined in
clause 2 of the GNU Lesser General Public License. This exception
does not however invalidate any other reasons why the executable file
might be covered by the GNU Lesser General Public License.
----------------------------------------------------------------------
GNU LESSER GENERAL PUBLIC LICENSE
Version 2.1, February 1999
Copyright (C) 1991, 1999 Free Software Foundation, Inc.
51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
Everyone is permitted to copy and distribute verbatim copies
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as the successor of the GNU Library Public License, version 2, hence
the version number 2.1.]
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The files in this directory are derived from code in the OCaml
standard library. This code is licensed under the LGPL 2.1 with OCaml
linking exception. See the LICENSE file which is copied verbatim from
the [original](https://github.com/ocaml/ocaml/blob/4.11.1/LICENSE).

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(**************************************************************************)
(* *)
(* OCaml *)
(* *)
(* 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 Lesser General Public License version 2.1, with the *)
(* special exception on linking described in the 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 update: key -> ('a option -> 'a option) -> '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 union: (key -> 'a -> 'a -> 'a option) -> 'a t -> 'a t -> 'a 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 filter_map: (key -> 'a -> 'b option) -> 'a t -> 'b 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 min_binding_opt: 'a t -> (key * 'a) option
val max_binding: 'a t -> (key * 'a)
val max_binding_opt: 'a t -> (key * 'a) option
val choose: 'a t -> (key * 'a)
val choose_opt: 'a t -> (key * 'a) option
val split: key -> 'a t -> 'a t * 'a option * 'a t
val find: key -> 'a t -> 'a
val find_opt: key -> 'a t -> 'a option
val find_first: (key -> bool) -> 'a t -> key * 'a
val find_first_opt: (key -> bool) -> 'a t -> (key * 'a) option
val find_last: (key -> bool) -> 'a t -> key * 'a
val find_last_opt: (key -> bool) -> 'a t -> (key * 'a) option
val map: ('a -> 'b) -> 'a t -> 'b t
val mapi: (key -> 'a -> 'b) -> 'a t -> 'b t
val to_seq : 'a t -> (key * 'a) Seq.t
val to_seq_from : key -> 'a t -> (key * 'a) Seq.t
val add_seq : (key * 'a) Seq.t -> 'a t -> 'a t
val of_seq : (key * 'a) Seq.t -> 'a t
end
module Make(Ord: OrderedType) = struct
type key = Ord.t
type 'a t =
Empty
| Node of {l:'a t; v:key; d:'a; r:'a t; h: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; v=x; d; r; h=(if hl >= hr then hl + 1 else hr + 1)}
let singleton x d = Node{l=Empty; v=x; d; r=Empty; h=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 "Map.bal"
| Node{l=ll; v=lv; d=ld; r=lr} ->
if height ll >= height lr then
create ll lv ld (create lr x d r)
else begin
match lr with
Empty -> invalid_arg "Map.bal"
| Node{l=lrl; v=lrv; d=lrd; r=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 "Map.bal"
| Node{l=rl; v=rv; d=rd; r=rr} ->
if height rr >= height rl then
create (create l x d rl) rv rd rr
else begin
match rl with
Empty -> invalid_arg "Map.bal"
| Node{l=rll; v=rlv; d=rld; r=rlr} ->
create (create l x d rll) rlv rld (create rlr rv rd rr)
end
end else
Node{l; v=x; d; r; h=(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{l=Empty; v=x; d=data; r=Empty; h=1}
| Node {l; v; d; r; h} as m ->
let c = Ord.compare x v in
if c = 0 then
if d == data then m else Node{l; v=x; d=data; r; h}
else if c < 0 then
let ll = add x data l in
if l == ll then m else bal ll v d r
else
let rr = add x data r in
if r == rr then m else bal l v d rr
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 find_first_aux v0 d0 f = function
Empty ->
(v0, d0)
| Node {l; v; d; r} ->
if f v then
find_first_aux v d f l
else
find_first_aux v0 d0 f r
let rec find_first f = function
Empty ->
raise Not_found
| Node {l; v; d; r} ->
if f v then
find_first_aux v d f l
else
find_first f r
let rec find_first_opt_aux v0 d0 f = function
Empty ->
Some (v0, d0)
| Node {l; v; d; r} ->
if f v then
find_first_opt_aux v d f l
else
find_first_opt_aux v0 d0 f r
let rec find_first_opt f = function
Empty ->
None
| Node {l; v; d; r} ->
if f v then
find_first_opt_aux v d f l
else
find_first_opt f r
let rec find_last_aux v0 d0 f = function
Empty ->
(v0, d0)
| Node {l; v; d; r} ->
if f v then
find_last_aux v d f r
else
find_last_aux v0 d0 f l
let rec find_last f = function
Empty ->
raise Not_found
| Node {l; v; d; r} ->
if f v then
find_last_aux v d f r
else
find_last f l
let rec find_last_opt_aux v0 d0 f = function
Empty ->
Some (v0, d0)
| Node {l; v; d; r} ->
if f v then
find_last_opt_aux v d f r
else
find_last_opt_aux v0 d0 f l
let rec find_last_opt f = function
Empty ->
None
| Node {l; v; d; r} ->
if f v then
find_last_opt_aux v d f r
else
find_last_opt f l
let rec find_opt x = function
Empty ->
None
| Node {l; v; d; r} ->
let c = Ord.compare x v in
if c = 0 then Some d
else find_opt x (if c < 0 then l else r)
let rec mem x = function
Empty ->
false
| Node {l; v; 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 {l=Empty; v; d} -> (v, d)
| Node {l} -> min_binding l
let rec min_binding_opt = function
Empty -> None
| Node {l=Empty; v; d} -> Some (v, d)
| Node {l}-> min_binding_opt l
let rec max_binding = function
Empty -> raise Not_found
| Node {v; d; r=Empty} -> (v, d)
| Node {r} -> max_binding r
let rec max_binding_opt = function
Empty -> None
| Node {v; d; r=Empty} -> Some (v, d)
| Node {r} -> max_binding_opt r
let rec remove_min_binding = function
Empty -> invalid_arg "Map.remove_min_elt"
| Node {l=Empty; r} -> r
| Node {l; v; d; r} -> bal (remove_min_binding l) v 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} as m) ->
let c = Ord.compare x v in
if c = 0 then merge l r
else if c < 0 then
let ll = remove x l in if l == ll then m else bal ll v d r
else
let rr = remove x r in if r == rr then m else bal l v d rr
let rec update x f = function
Empty ->
begin match f None with
| None -> Empty
| Some data -> Node{l=Empty; v=x; d=data; r=Empty; h=1}
end
| Node {l; v; d; r; h} as m ->
let c = Ord.compare x v in
if c = 0 then begin
match f (Some d) with
| None -> merge l r
| Some data ->
if d == data then m else Node{l; v=x; d=data; r; h}
end else if c < 0 then
let ll = update x f l in
if l == ll then m else bal ll v d r
else
let rr = update x f r in
if r == rr then m else bal l v d rr
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=l'; v; d=d'; r=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=l'; v; d=d'; r=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))
let rec for_all p = function
Empty -> true
| Node {l; v; d; r} -> p v d && for_all p l && for_all p r
let rec exists p = function
Empty -> false
| Node {l; v; d; r} -> p v d || exists p l || exists p r
(* Beware: those two functions assume that the added k is *strictly*
smaller (or bigger) than all the present keys in the tree; it
does not test for equality with the current min (or max) key.
Indeed, they are only used during the "join" operation which
respects this precondition.
*)
let rec add_min_binding k x = function
| Empty -> singleton k x
| Node {l; v; d; r} ->
bal (add_min_binding k x l) v d r
let rec add_max_binding k x = function
| Empty -> singleton k x
| Node {l; v; d; r} ->
bal l v d (add_max_binding k x r)
(* 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_min_binding v d r
| (_, Empty) -> add_max_binding v d l
| (Node{l=ll; v=lv; d=ld; r=lr; h=lh},
Node{l=rl; v=rv; d=rd; r=rr; h=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 {l=l1; v=v1; d=d1; r=r1; h=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 {l=l2; v=v2; d=d2; r=r2}) ->
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
let rec union f s1 s2 =
match (s1, s2) with
| (Empty, s) | (s, Empty) -> s
| (Node {l=l1; v=v1; d=d1; r=r1; h=h1},
Node {l=l2; v=v2; d=d2; r=r2; h=h2}) ->
if h1 >= h2 then
let (l2, d2, r2) = split v1 s2 in
let l = union f l1 l2 and r = union f r1 r2 in
match d2 with
| None -> join l v1 d1 r
| Some d2 -> concat_or_join l v1 (f v1 d1 d2) r
else
let (l1, d1, r1) = split v2 s1 in
let l = union f l1 l2 and r = union f r1 r2 in
match d1 with
| None -> join l v2 d2 r
| Some d1 -> concat_or_join l v2 (f v2 d1 d2) r
let rec filter p = function
Empty -> Empty
| Node {l; v; d; r} as m ->
(* call [p] in the expected left-to-right order *)
let l' = filter p l in
let pvd = p v d in
let r' = filter p r in
if pvd then if l==l' && r==r' then m else join l' v d r'
else concat l' r'
let rec filter_map f = function
Empty -> Empty
| Node {l; v; d; r} ->
(* call [f] in the expected left-to-right order *)
let l' = filter_map f l in
let fvd = f v d in
let r' = filter_map f r in
begin match fvd with
| Some d' -> join l' v d' r'
| None -> concat l' r'
end
let rec partition p = function
Empty -> (Empty, Empty)
| Node {l; v; d; r} ->
(* call [p] in the expected left-to-right order *)
let (lt, lf) = partition p l in
let pvd = p v d in
let (rt, rf) = partition p r in
if pvd
then (join lt v d rt, concat lf rf)
else (concat lt rt, join lf v d rf)
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
let choose_opt = min_binding_opt
let add_seq i m =
Seq.fold_left (fun m (k,v) -> add k v m) m i
let of_seq i = add_seq i empty
let rec seq_of_enum_ c () = match c with
| End -> Seq.Nil
| More (k,v,t,rest) -> Seq.Cons ((k,v), seq_of_enum_ (cons_enum t rest))
let to_seq m =
seq_of_enum_ (cons_enum m End)
let to_seq_from low m =
let rec aux low m c = match m with
| Empty -> c
| Node {l; v; d; r; _} ->
begin match Ord.compare v low with
| 0 -> More (v, d, r, c)
| n when n<0 -> aux low r c
| _ -> aux low l (More (v, d, r, c))
end
in
seq_of_enum_ (aux low m End)
end

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(**************************************************************************)
(* *)
(* OCaml *)
(* *)
(* 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 Lesser General Public License version 2.1, with the *)
(* special exception on linking described in the file LICENSE. *)
(* *)
(**************************************************************************)
(** 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.
For instance:
{[
module IntPairs =
struct
type t = int * int
let compare (x0,y0) (x1,y1) =
match Stdlib.compare x0 x1 with
0 -> Stdlib.compare y0 y1
| c -> c
end
module PairsMap = Map.Make(IntPairs)
let m = PairsMap.(empty |> add (0,1) "hello" |> add (1,0) "world")
]}
This creates a new module [PairsMap], with a new type ['a PairsMap.t]
of maps from [int * int] to ['a]. In this example, [m] contains [string]
values so its type is [string PairsMap.t].
*)
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 {!Stdlib.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] to a value that is physically equal to [y],
[m] is returned unchanged (the result of the function is
then physically equal to [m]). Otherwise, the previous binding
of [x] in [m] disappears.
@before 4.03 Physical equality was not ensured. *)
val update: key -> ('a option -> 'a option) -> 'a t -> 'a t
(** [update x f m] returns a map containing the same bindings as
[m], except for the binding of [x]. Depending on the value of
[y] where [y] is [f (find_opt x m)], the binding of [x] is
added, removed or updated. If [y] is [None], the binding is
removed if it exists; otherwise, if [y] is [Some z] then [x]
is associated to [z] in the resulting map. If [x] was already
bound in [m] to a value that is physically equal to [z], [m]
is returned unchanged (the result of the function is then
physically equal to [m]).
@since 4.06.0
*)
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.
If [x] was not in [m], [m] is returned unchanged
(the result of the function is then physically equal to [m]).
@before 4.03 Physical equality was not ensured. *)
val merge:
(key -> 'a option -> 'b option -> 'c option) -> 'a t -> 'b t -> 'c t
(** [merge f m1 m2] computes a map whose keys are a subset of the keys of
[m1] and of [m2]. The presence of each such binding, and the
corresponding value, is determined with the function [f].
In terms of the [find_opt] operation, we have
[find_opt x (merge f m1 m2) = f x (find_opt x m1) (find_opt x m2)]
for any key [x], provided that [f x None None = None].
@since 3.12.0
*)
val union: (key -> 'a -> 'a -> 'a option) -> 'a t -> 'a t -> 'a t
(** [union f m1 m2] computes a map whose keys are a subset of the keys
of [m1] and of [m2]. When the same binding is defined in both
arguments, the function [f] is used to combine them.
This is a special case of [merge]: [union f m1 m2] is equivalent
to [merge f' m1 m2], where
- [f' _key None None = None]
- [f' _key (Some v) None = Some v]
- [f' _key None (Some v) = Some v]
- [f' key (Some v1) (Some v2) = f key v1 v2]
@since 4.03.0
*)
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
(** [for_all p m] checks if all the bindings of the map
satisfy the predicate [p].
@since 3.12.0
*)
val exists: (key -> 'a -> bool) -> 'a t -> bool
(** [exists p m] checks if at least one binding of the map
satisfies the predicate [p].
@since 3.12.0
*)
val filter: (key -> 'a -> bool) -> 'a t -> 'a t
(** [filter p m] returns the map with all the bindings in [m]
that satisfy predicate [p]. If every binding in [m] satisfies [p],
[m] is returned unchanged (the result of the function is then
physically equal to [m])
@since 3.12.0
@before 4.03 Physical equality was not ensured.
*)
val filter_map: (key -> 'a -> 'b option) -> 'a t -> 'b t
(** [filter_map f m] applies the function [f] to every binding of
[m], and builds a map from the results. For each binding
[(k, v)] in the input map:
- if [f k v] is [None] then [k] is not in the result,
- if [f k v] is [Some v'] then the binding [(k, v')]
is in the output map.
For example, the following function on maps whose values are lists
{[
filter_map
(fun _k li -> match li with [] -> None | _::tl -> Some tl)
m
]}
drops all bindings of [m] whose value is an empty list, and pops
the first element of each value that is non-empty.
@since 4.11.0
*)
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 [m] that satisfy the
predicate [p], and [m2] is the map with all the bindings of
[m] 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 of keys 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 binding with the smallest key in a given map
(with respect to the [Ord.compare] ordering), or raise
[Not_found] if the map is empty.
@since 3.12.0
*)
val min_binding_opt: 'a t -> (key * 'a) option
(** Return the binding with the smallest key in the given map
(with respect to the [Ord.compare] ordering), or [None]
if the map is empty.
@since 4.05
*)
val max_binding: 'a t -> (key * 'a)
(** Same as {!Map.S.min_binding}, but returns the binding with
the largest key in the given map.
@since 3.12.0
*)
val max_binding_opt: 'a t -> (key * 'a) option
(** Same as {!Map.S.min_binding_opt}, but returns the binding with
the largest key in the given map.
@since 4.05
*)
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 choose_opt: 'a t -> (key * 'a) option
(** Return one binding of the given map, or [None] if
the map is empty. Which binding is chosen is unspecified,
but equal bindings will be chosen for equal maps.
@since 4.05
*)
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 value of [x] in [m],
or raises [Not_found] if no binding for [x] exists. *)
val find_opt: key -> 'a t -> 'a option
(** [find_opt x m] returns [Some v] if the current value of [x]
in [m] is [v], or [None] if no binding for [x] exists.
@since 4.05
*)
val find_first: (key -> bool) -> 'a t -> key * 'a
(** [find_first f m], where [f] is a monotonically increasing function,
returns the binding of [m] with the lowest key [k] such that [f k],
or raises [Not_found] if no such key exists.
For example, [find_first (fun k -> Ord.compare k x >= 0) m] will return
the first binding [k, v] of [m] where [Ord.compare k x >= 0]
(intuitively: [k >= x]), or raise [Not_found] if [x] is greater than any
element of [m].
@since 4.05
*)
val find_first_opt: (key -> bool) -> 'a t -> (key * 'a) option
(** [find_first_opt f m], where [f] is a monotonically increasing function,
returns an option containing the binding of [m] with the lowest key [k]
such that [f k], or [None] if no such key exists.
@since 4.05
*)
val find_last: (key -> bool) -> 'a t -> key * 'a
(** [find_last f m], where [f] is a monotonically decreasing function,
returns the binding of [m] with the highest key [k] such that [f k],
or raises [Not_found] if no such key exists.
@since 4.05
*)
val find_last_opt: (key -> bool) -> 'a t -> (key * 'a) option
(** [find_last_opt f m], where [f] is a monotonically decreasing function,
returns an option containing the binding of [m] with the highest key [k]
such that [f k], or [None] if no such key exists.
@since 4.05
*)
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. *)
(** {1 Iterators} *)
val to_seq : 'a t -> (key * 'a) Seq.t
(** Iterate on the whole map, in ascending order of keys
@since 4.07 *)
val to_seq_from : key -> 'a t -> (key * 'a) Seq.t
(** [to_seq_from k m] iterates on a subset of the bindings of [m],
in ascending order of keys, from key [k] or above.
@since 4.07 *)
val add_seq : (key * 'a) Seq.t -> 'a t -> 'a t
(** Add the given bindings to the map, in order.
@since 4.07 *)
val of_seq : (key * 'a) Seq.t -> 'a t
(** Build a map from the given bindings
@since 4.07 *)
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. *)

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(**************************************************************************)
(* *)
(* OCaml *)
(* *)
(* 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 Lesser General Public License version 2.1, with the *)
(* special exception on linking described in the file LICENSE. *)
(* *)
(**************************************************************************)
(* Sets over ordered types *)
module type OrderedType =
sig
type t
val compare: t -> t -> int
end
module type S =
sig
type elt
type t
val empty: t
val is_empty: t -> bool
val mem: elt -> t -> bool
val add: elt -> t -> t
val singleton: elt -> t
val remove: elt -> t -> t
val union: t -> t -> t
val inter: t -> t -> t
val disjoint: t -> t -> bool
val diff: t -> t -> t
val compare: t -> t -> int
val equal: t -> t -> bool
val subset: t -> t -> bool
val iter: (elt -> unit) -> t -> unit
val map: (elt -> elt) -> t -> t
val fold: (elt -> 'a -> 'a) -> t -> 'a -> 'a
val for_all: (elt -> bool) -> t -> bool
val exists: (elt -> bool) -> t -> bool
val filter: (elt -> bool) -> t -> t
val filter_map: (elt -> elt option) -> t -> t
val partition: (elt -> bool) -> t -> t * t
val cardinal: t -> int
val elements: t -> elt list
val min_elt: t -> elt
val min_elt_opt: t -> elt option
val max_elt: t -> elt
val max_elt_opt: t -> elt option
val choose: t -> elt
val choose_opt: t -> elt option
val split: elt -> t -> t * bool * t
val find: elt -> t -> elt
val find_opt: elt -> t -> elt option
val find_first: (elt -> bool) -> t -> elt
val find_first_opt: (elt -> bool) -> t -> elt option
val find_last: (elt -> bool) -> t -> elt
val find_last_opt: (elt -> bool) -> t -> elt option
val of_list: elt list -> t
val to_seq_from : elt -> t -> elt Seq.t
val to_seq : t -> elt Seq.t
val add_seq : elt Seq.t -> t -> t
val of_seq : elt Seq.t -> t
end
module Make(Ord: OrderedType) =
struct
type elt = Ord.t
type t = Empty | Node of {l:t; v:elt; r:t; h:int}
(* Sets are represented by balanced binary trees (the heights of the
children differ by at most 2 *)
let height = function
Empty -> 0
| Node {h} -> h
(* Creates a new node with left son l, value v and right son r.
We must have all elements of l < v < all elements of r.
l and r must be balanced and | height l - height r | <= 2.
Inline expansion of height for better speed. *)
let create l v r =
let hl = match l with Empty -> 0 | Node {h} -> h in
let hr = match r with Empty -> 0 | Node {h} -> h in
Node{l; v; r; h=(if hl >= hr then hl + 1 else hr + 1)}
(* Same as create, but performs one step of rebalancing if necessary.
Assumes l and r balanced and | height l - height r | <= 3.
Inline expansion of create for better speed in the most frequent case
where no rebalancing is required. *)
let bal l v 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 "Set.bal"
| Node{l=ll; v=lv; r=lr} ->
if height ll >= height lr then
create ll lv (create lr v r)
else begin
match lr with
Empty -> invalid_arg "Set.bal"
| Node{l=lrl; v=lrv; r=lrr}->
create (create ll lv lrl) lrv (create lrr v r)
end
end else if hr > hl + 2 then begin
match r with
Empty -> invalid_arg "Set.bal"
| Node{l=rl; v=rv; r=rr} ->
if height rr >= height rl then
create (create l v rl) rv rr
else begin
match rl with
Empty -> invalid_arg "Set.bal"
| Node{l=rll; v=rlv; r=rlr} ->
create (create l v rll) rlv (create rlr rv rr)
end
end else
Node{l; v; r; h=(if hl >= hr then hl + 1 else hr + 1)}
(* Insertion of one element *)
let rec add x = function
Empty -> Node{l=Empty; v=x; r=Empty; h=1}
| Node{l; v; r} as t ->
let c = Ord.compare x v in
if c = 0 then t else
if c < 0 then
let ll = add x l in
if l == ll then t else bal ll v r
else
let rr = add x r in
if r == rr then t else bal l v rr
let singleton x = Node{l=Empty; v=x; r=Empty; h=1}
(* Beware: those two functions assume that the added v is *strictly*
smaller (or bigger) than all the present elements in the tree; it
does not test for equality with the current min (or max) element.
Indeed, they are only used during the "join" operation which
respects this precondition.
*)
let rec add_min_element x = function
| Empty -> singleton x
| Node {l; v; r} ->
bal (add_min_element x l) v r
let rec add_max_element x = function
| Empty -> singleton x
| Node {l; v; r} ->
bal l v (add_max_element x r)
(* Same as create and bal, but no assumptions are made on the
relative heights of l and r. *)
let rec join l v r =
match (l, r) with
(Empty, _) -> add_min_element v r
| (_, Empty) -> add_max_element v l
| (Node{l=ll; v=lv; r=lr; h=lh}, Node{l=rl; v=rv; r=rr; h=rh}) ->
if lh > rh + 2 then bal ll lv (join lr v r) else
if rh > lh + 2 then bal (join l v rl) rv rr else
create l v r
(* Smallest and greatest element of a set *)
let rec min_elt = function
Empty -> raise Not_found
| Node{l=Empty; v} -> v
| Node{l} -> min_elt l
let rec min_elt_opt = function
Empty -> None
| Node{l=Empty; v} -> Some v
| Node{l} -> min_elt_opt l
let rec max_elt = function
Empty -> raise Not_found
| Node{v; r=Empty} -> v
| Node{r} -> max_elt r
let rec max_elt_opt = function
Empty -> None
| Node{v; r=Empty} -> Some v
| Node{r} -> max_elt_opt r
(* Remove the smallest element of the given set *)
let rec remove_min_elt = function
Empty -> invalid_arg "Set.remove_min_elt"
| Node{l=Empty; r} -> r
| Node{l; v; r} -> bal (remove_min_elt l) v r
(* Merge two trees l and r into one.
All elements of l must precede the elements of r.
Assume | height l - height r | <= 2. *)
let merge t1 t2 =
match (t1, t2) with
(Empty, t) -> t
| (t, Empty) -> t
| (_, _) -> bal t1 (min_elt t2) (remove_min_elt t2)
(* 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
| (_, _) -> join t1 (min_elt t2) (remove_min_elt t2)
(* Splitting. split x s returns a triple (l, present, r) where
- l is the set of elements of s that are < x
- r is the set of elements of s that are > x
- present is false if s contains no element equal to x,
or true if s contains an element equal to x. *)
let rec split x = function
Empty ->
(Empty, false, Empty)
| Node{l; v; r} ->
let c = Ord.compare x v in
if c = 0 then (l, true, r)
else if c < 0 then
let (ll, pres, rl) = split x l in (ll, pres, join rl v r)
else
let (lr, pres, rr) = split x r in (join l v lr, pres, rr)
(* Implementation of the set operations *)
let empty = Empty
let is_empty = function Empty -> true | _ -> false
let rec mem x = function
Empty -> false
| Node{l; v; r} ->
let c = Ord.compare x v in
c = 0 || mem x (if c < 0 then l else r)
let rec remove x = function
Empty -> Empty
| (Node{l; v; r} as t) ->
let c = Ord.compare x v in
if c = 0 then merge l r
else
if c < 0 then
let ll = remove x l in
if l == ll then t
else bal ll v r
else
let rr = remove x r in
if r == rr then t
else bal l v rr
let rec union s1 s2 =
match (s1, s2) with
(Empty, t2) -> t2
| (t1, Empty) -> t1
| (Node{l=l1; v=v1; r=r1; h=h1}, Node{l=l2; v=v2; r=r2; h=h2}) ->
if h1 >= h2 then
if h2 = 1 then add v2 s1 else begin
let (l2, _, r2) = split v1 s2 in
join (union l1 l2) v1 (union r1 r2)
end
else
if h1 = 1 then add v1 s2 else begin
let (l1, _, r1) = split v2 s1 in
join (union l1 l2) v2 (union r1 r2)
end
let rec inter s1 s2 =
match (s1, s2) with
(Empty, _) -> Empty
| (_, Empty) -> Empty
| (Node{l=l1; v=v1; r=r1}, t2) ->
match split v1 t2 with
(l2, false, r2) ->
concat (inter l1 l2) (inter r1 r2)
| (l2, true, r2) ->
join (inter l1 l2) v1 (inter r1 r2)
(* Same as split, but compute the left and right subtrees
only if the pivot element is not in the set. The right subtree
is computed on demand. *)
type split_bis =
| Found
| NotFound of t * (unit -> t)
let rec split_bis x = function
Empty ->
NotFound (Empty, (fun () -> Empty))
| Node{l; v; r; _} ->
let c = Ord.compare x v in
if c = 0 then Found
else if c < 0 then
match split_bis x l with
| Found -> Found
| NotFound (ll, rl) -> NotFound (ll, (fun () -> join (rl ()) v r))
else
match split_bis x r with
| Found -> Found
| NotFound (lr, rr) -> NotFound (join l v lr, rr)
let rec disjoint s1 s2 =
match (s1, s2) with
(Empty, _) | (_, Empty) -> true
| (Node{l=l1; v=v1; r=r1}, t2) ->
if s1 == s2 then false
else match split_bis v1 t2 with
NotFound(l2, r2) -> disjoint l1 l2 && disjoint r1 (r2 ())
| Found -> false
let rec diff s1 s2 =
match (s1, s2) with
(Empty, _) -> Empty
| (t1, Empty) -> t1
| (Node{l=l1; v=v1; r=r1}, t2) ->
match split v1 t2 with
(l2, false, r2) ->
join (diff l1 l2) v1 (diff r1 r2)
| (l2, true, r2) ->
concat (diff l1 l2) (diff r1 r2)
type enumeration = End | More of elt * t * enumeration
let rec cons_enum s e =
match s with
Empty -> e
| Node{l; v; r} -> cons_enum l (More(v, r, e))
let rec compare_aux e1 e2 =
match (e1, e2) with
(End, End) -> 0
| (End, _) -> -1
| (_, End) -> 1
| (More(v1, r1, e1), More(v2, r2, e2)) ->
let c = Ord.compare v1 v2 in
if c <> 0
then c
else compare_aux (cons_enum r1 e1) (cons_enum r2 e2)
let compare s1 s2 =
compare_aux (cons_enum s1 End) (cons_enum s2 End)
let equal s1 s2 =
compare s1 s2 = 0
let rec subset s1 s2 =
match (s1, s2) with
Empty, _ ->
true
| _, Empty ->
false
| Node {l=l1; v=v1; r=r1}, (Node {l=l2; v=v2; r=r2} as t2) ->
let c = Ord.compare v1 v2 in
if c = 0 then
subset l1 l2 && subset r1 r2
else if c < 0 then
subset (Node {l=l1; v=v1; r=Empty; h=0}) l2 && subset r1 t2
else
subset (Node {l=Empty; v=v1; r=r1; h=0}) r2 && subset l1 t2
let rec iter f = function
Empty -> ()
| Node{l; v; r} -> iter f l; f v; iter f r
let rec fold f s accu =
match s with
Empty -> accu
| Node{l; v; r} -> fold f r (f v (fold f l accu))
let rec for_all p = function
Empty -> true
| Node{l; v; r} -> p v && for_all p l && for_all p r
let rec exists p = function
Empty -> false
| Node{l; v; r} -> p v || exists p l || exists p r
let rec filter p = function
Empty -> Empty
| (Node{l; v; r}) as t ->
(* call [p] in the expected left-to-right order *)
let l' = filter p l in
let pv = p v in
let r' = filter p r in
if pv then
if l==l' && r==r' then t else join l' v r'
else concat l' r'
let rec partition p = function
Empty -> (Empty, Empty)
| Node{l; v; r} ->
(* call [p] in the expected left-to-right order *)
let (lt, lf) = partition p l in
let pv = p v in
let (rt, rf) = partition p r in
if pv
then (join lt v rt, concat lf rf)
else (concat lt rt, join lf v rf)
let rec cardinal = function
Empty -> 0
| Node{l; r} -> cardinal l + 1 + cardinal r
let rec elements_aux accu = function
Empty -> accu
| Node{l; v; r} -> elements_aux (v :: elements_aux accu r) l
let elements s =
elements_aux [] s
let choose = min_elt
let choose_opt = min_elt_opt
let rec find x = function
Empty -> raise Not_found
| Node{l; v; r} ->
let c = Ord.compare x v in
if c = 0 then v
else find x (if c < 0 then l else r)
let rec find_first_aux v0 f = function
Empty ->
v0
| Node{l; v; r} ->
if f v then
find_first_aux v f l
else
find_first_aux v0 f r
let rec find_first f = function
Empty ->
raise Not_found
| Node{l; v; r} ->
if f v then
find_first_aux v f l
else
find_first f r
let rec find_first_opt_aux v0 f = function
Empty ->
Some v0
| Node{l; v; r} ->
if f v then
find_first_opt_aux v f l
else
find_first_opt_aux v0 f r
let rec find_first_opt f = function
Empty ->
None
| Node{l; v; r} ->
if f v then
find_first_opt_aux v f l
else
find_first_opt f r
let rec find_last_aux v0 f = function
Empty ->
v0
| Node{l; v; r} ->
if f v then
find_last_aux v f r
else
find_last_aux v0 f l
let rec find_last f = function
Empty ->
raise Not_found
| Node{l; v; r} ->
if f v then
find_last_aux v f r
else
find_last f l
let rec find_last_opt_aux v0 f = function
Empty ->
Some v0
| Node{l; v; r} ->
if f v then
find_last_opt_aux v f r
else
find_last_opt_aux v0 f l
let rec find_last_opt f = function
Empty ->
None
| Node{l; v; r} ->
if f v then
find_last_opt_aux v f r
else
find_last_opt f l
let rec find_opt x = function
Empty -> None
| Node{l; v; r} ->
let c = Ord.compare x v in
if c = 0 then Some v
else find_opt x (if c < 0 then l else r)
let try_join l v r =
(* [join l v r] can only be called when (elements of l < v <
elements of r); use [try_join l v r] when this property may
not hold, but you hope it does hold in the common case *)
if (l = Empty || Ord.compare (max_elt l) v < 0)
&& (r = Empty || Ord.compare v (min_elt r) < 0)
then join l v r
else union l (add v r)
let rec map f = function
| Empty -> Empty
| Node{l; v; r} as t ->
(* enforce left-to-right evaluation order *)
let l' = map f l in
let v' = f v in
let r' = map f r in
if l == l' && v == v' && r == r' then t
else try_join l' v' r'
let try_concat t1 t2 =
match (t1, t2) with
(Empty, t) -> t
| (t, Empty) -> t
| (_, _) -> try_join t1 (min_elt t2) (remove_min_elt t2)
let rec filter_map f = function
| Empty -> Empty
| Node{l; v; r} as t ->
(* enforce left-to-right evaluation order *)
let l' = filter_map f l in
let v' = f v in
let r' = filter_map f r in
begin match v' with
| Some v' ->
if l == l' && v == v' && r == r' then t
else try_join l' v' r'
| None ->
try_concat l' r'
end
let of_sorted_list l =
let rec sub n l =
match n, l with
| 0, l -> Empty, l
| 1, x0 :: l -> Node {l=Empty; v=x0; r=Empty; h=1}, l
| 2, x0 :: x1 :: l ->
Node{l=Node{l=Empty; v=x0; r=Empty; h=1}; v=x1; r=Empty; h=2}, l
| 3, x0 :: x1 :: x2 :: l ->
Node{l=Node{l=Empty; v=x0; r=Empty; h=1}; v=x1;
r=Node{l=Empty; v=x2; r=Empty; h=1}; h=2}, l
| n, l ->
let nl = n / 2 in
let left, l = sub nl l in
match l with
| [] -> assert false
| mid :: l ->
let right, l = sub (n - nl - 1) l in
create left mid right, l
in
fst (sub (List.length l) l)
let of_list l =
match l with
| [] -> empty
| [x0] -> singleton x0
| [x0; x1] -> add x1 (singleton x0)
| [x0; x1; x2] -> add x2 (add x1 (singleton x0))
| [x0; x1; x2; x3] -> add x3 (add x2 (add x1 (singleton x0)))
| [x0; x1; x2; x3; x4] -> add x4 (add x3 (add x2 (add x1 (singleton x0))))
| _ -> of_sorted_list (List.sort_uniq Ord.compare l)
let add_seq i m =
Seq.fold_left (fun s x -> add x s) m i
let of_seq i = add_seq i empty
let rec seq_of_enum_ c () = match c with
| End -> Seq.Nil
| More (x, t, rest) -> Seq.Cons (x, seq_of_enum_ (cons_enum t rest))
let to_seq c = seq_of_enum_ (cons_enum c End)
let to_seq_from low s =
let rec aux low s c = match s with
| Empty -> c
| Node {l; r; v; _} ->
begin match Ord.compare v low with
| 0 -> More (v, r, c)
| n when n<0 -> aux low r c
| _ -> aux low l (More (v, r, c))
end
in
seq_of_enum_ (aux low s End)
end

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(**************************************************************************)
(* *)
(* OCaml *)
(* *)
(* 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 Lesser General Public License version 2.1, with the *)
(* special exception on linking described in the file LICENSE. *)
(* *)
(**************************************************************************)
(** Sets over ordered types.
This module implements the set data structure, given a total ordering
function over the set elements. All operations over sets
are purely applicative (no side-effects).
The implementation uses balanced binary trees, and is therefore
reasonably efficient: insertion and membership take time
logarithmic in the size of the set, for instance.
The {!Make} functor constructs implementations for any type, given a
[compare] function.
For instance:
{[
module IntPairs =
struct
type t = int * int
let compare (x0,y0) (x1,y1) =
match Stdlib.compare x0 x1 with
0 -> Stdlib.compare y0 y1
| c -> c
end
module PairsSet = Set.Make(IntPairs)
let m = PairsSet.(empty |> add (2,3) |> add (5,7) |> add (11,13))
]}
This creates a new module [PairsSet], with a new type [PairsSet.t]
of sets of [int * int].
*)
module type OrderedType =
sig
type t
(** The type of the set elements. *)
val compare : t -> t -> int
(** A total ordering function over the set elements.
This is a two-argument function [f] such that
[f e1 e2] is zero if the elements [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 {!Stdlib.compare}. *)
end
(** Input signature of the functor {!Set.Make}. *)
module type S =
sig
type elt
(** The type of the set elements. *)
type t
(** The type of sets. *)
val empty: t
(** The empty set. *)
val is_empty: t -> bool
(** Test whether a set is empty or not. *)
val mem: elt -> t -> bool
(** [mem x s] tests whether [x] belongs to the set [s]. *)
val add: elt -> t -> t
(** [add x s] returns a set containing all elements of [s],
plus [x]. If [x] was already in [s], [s] is returned unchanged
(the result of the function is then physically equal to [s]).
@before 4.03 Physical equality was not ensured. *)
val singleton: elt -> t
(** [singleton x] returns the one-element set containing only [x]. *)
val remove: elt -> t -> t
(** [remove x s] returns a set containing all elements of [s],
except [x]. If [x] was not in [s], [s] is returned unchanged
(the result of the function is then physically equal to [s]).
@before 4.03 Physical equality was not ensured. *)
val union: t -> t -> t
(** Set union. *)
val inter: t -> t -> t
(** Set intersection. *)
val disjoint: t -> t -> bool
(** Test if two sets are disjoint.
@since 4.08.0 *)
val diff: t -> t -> t
(** Set difference: [diff s1 s2] contains the elements of [s1]
that are not in [s2]. *)
val compare: t -> t -> int
(** Total ordering between sets. Can be used as the ordering function
for doing sets of sets. *)
val equal: t -> t -> bool
(** [equal s1 s2] tests whether the sets [s1] and [s2] are
equal, that is, contain equal elements. *)
val subset: t -> t -> bool
(** [subset s1 s2] tests whether the set [s1] is a subset of
the set [s2]. *)
val iter: (elt -> unit) -> t -> unit
(** [iter f s] applies [f] in turn to all elements of [s].
The elements of [s] are presented to [f] in increasing order
with respect to the ordering over the type of the elements. *)
val map: (elt -> elt) -> t -> t
(** [map f s] is the set whose elements are [f a0],[f a1]... [f
aN], where [a0],[a1]...[aN] are the elements of [s].
The elements are passed to [f] in increasing order
with respect to the ordering over the type of the elements.
If no element of [s] is changed by [f], [s] is returned
unchanged. (If each output of [f] is physically equal to its
input, the returned set is physically equal to [s].)
@since 4.04.0 *)
val fold: (elt -> 'a -> 'a) -> t -> 'a -> 'a
(** [fold f s a] computes [(f xN ... (f x2 (f x1 a))...)],
where [x1 ... xN] are the elements of [s], in increasing order. *)
val for_all: (elt -> bool) -> t -> bool
(** [for_all p s] checks if all elements of the set
satisfy the predicate [p]. *)
val exists: (elt -> bool) -> t -> bool
(** [exists p s] checks if at least one element of
the set satisfies the predicate [p]. *)
val filter: (elt -> bool) -> t -> t
(** [filter p s] returns the set of all elements in [s]
that satisfy predicate [p]. If [p] satisfies every element in [s],
[s] is returned unchanged (the result of the function is then
physically equal to [s]).
@before 4.03 Physical equality was not ensured.*)
val filter_map: (elt -> elt option) -> t -> t
(** [filter_map f s] returns the set of all [v] such that
[f x = Some v] for some element [x] of [s].
For example,
{[filter_map (fun n -> if n mod 2 = 0 then Some (n / 2) else None) s]}
is the set of halves of the even elements of [s].
If no element of [s] is changed or dropped by [f] (if
[f x = Some x] for each element [x]), then
[s] is returned unchanged: the result of the function
is then physically equal to [s].
@since 4.11.0
*)
val partition: (elt -> bool) -> t -> t * t
(** [partition p s] returns a pair of sets [(s1, s2)], where
[s1] is the set of all the elements of [s] that satisfy the
predicate [p], and [s2] is the set of all the elements of
[s] that do not satisfy [p]. *)
val cardinal: t -> int
(** Return the number of elements of a set. *)
val elements: t -> elt list
(** Return the list of all elements of the given set.
The returned list is sorted in increasing order with respect
to the ordering [Ord.compare], where [Ord] is the argument
given to {!Set.Make}. *)
val min_elt: t -> elt
(** Return the smallest element of the given set
(with respect to the [Ord.compare] ordering), or raise
[Not_found] if the set is empty. *)
val min_elt_opt: t -> elt option
(** Return the smallest element of the given set
(with respect to the [Ord.compare] ordering), or [None]
if the set is empty.
@since 4.05
*)
val max_elt: t -> elt
(** Same as {!Set.S.min_elt}, but returns the largest element of the
given set. *)
val max_elt_opt: t -> elt option
(** Same as {!Set.S.min_elt_opt}, but returns the largest element of the
given set.
@since 4.05
*)
val choose: t -> elt
(** Return one element of the given set, or raise [Not_found] if
the set is empty. Which element is chosen is unspecified,
but equal elements will be chosen for equal sets. *)
val choose_opt: t -> elt option
(** Return one element of the given set, or [None] if
the set is empty. Which element is chosen is unspecified,
but equal elements will be chosen for equal sets.
@since 4.05
*)
val split: elt -> t -> t * bool * t
(** [split x s] returns a triple [(l, present, r)], where
[l] is the set of elements of [s] that are
strictly less than [x];
[r] is the set of elements of [s] that are
strictly greater than [x];
[present] is [false] if [s] contains no element equal to [x],
or [true] if [s] contains an element equal to [x]. *)
val find: elt -> t -> elt
(** [find x s] returns the element of [s] equal to [x] (according
to [Ord.compare]), or raise [Not_found] if no such element
exists.
@since 4.01.0 *)
val find_opt: elt -> t -> elt option
(** [find_opt x s] returns the element of [s] equal to [x] (according
to [Ord.compare]), or [None] if no such element
exists.
@since 4.05 *)
val find_first: (elt -> bool) -> t -> elt
(** [find_first f s], where [f] is a monotonically increasing function,
returns the lowest element [e] of [s] such that [f e],
or raises [Not_found] if no such element exists.
For example, [find_first (fun e -> Ord.compare e x >= 0) s] will return
the first element [e] of [s] where [Ord.compare e x >= 0] (intuitively:
[e >= x]), or raise [Not_found] if [x] is greater than any element of
[s].
@since 4.05
*)
val find_first_opt: (elt -> bool) -> t -> elt option
(** [find_first_opt f s], where [f] is a monotonically increasing function,
returns an option containing the lowest element [e] of [s] such that
[f e], or [None] if no such element exists.
@since 4.05
*)
val find_last: (elt -> bool) -> t -> elt
(** [find_last f s], where [f] is a monotonically decreasing function,
returns the highest element [e] of [s] such that [f e],
or raises [Not_found] if no such element exists.
@since 4.05
*)
val find_last_opt: (elt -> bool) -> t -> elt option
(** [find_last_opt f s], where [f] is a monotonically decreasing function,
returns an option containing the highest element [e] of [s] such that
[f e], or [None] if no such element exists.
@since 4.05
*)
val of_list: elt list -> t
(** [of_list l] creates a set from a list of elements.
This is usually more efficient than folding [add] over the list,
except perhaps for lists with many duplicated elements.
@since 4.02.0 *)
(** {1 Iterators} *)
val to_seq_from : elt -> t -> elt Seq.t
(** [to_seq_from x s] iterates on a subset of the elements of [s]
in ascending order, from [x] or above.
@since 4.07 *)
val to_seq : t -> elt Seq.t
(** Iterate on the whole set, in ascending order
@since 4.07 *)
val add_seq : elt Seq.t -> t -> t
(** Add the given elements to the set, in order.
@since 4.07 *)
val of_seq : elt Seq.t -> t
(** Build a set from the given bindings
@since 4.07 *)
end
(** Output signature of the functor {!Set.Make}. *)
module Make (Ord : OrderedType) : S with type elt = Ord.t
(** Functor building an implementation of the set structure
given a totally ordered type. *)
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