(* Title: Pure/library.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Author: Markus Wenzel, TU Muenchen
Basic library: functions, options, pairs, booleans, lists, integers,
rational numbers, strings, lists as sets, association lists, generic
tables, balanced trees, orders, current directory, misc.
*)
infix |> |-> ||> ||>> |>> |>>> #> #-> ~~ \ \\ ins ins_string ins_int
orf andf prefix upto downto mem mem_int mem_string union union_int
union_string inter inter_int inter_string subset subset_int
subset_string;
infix 2 ?;
infix 3 oo ooo oooo;
signature BASIC_LIBRARY =
sig
(*functions*)
val I: 'a -> 'a
val K: 'a -> 'b -> 'a
val curry: ('a * 'b -> 'c) -> 'a -> 'b -> 'c
val uncurry: ('a -> 'b -> 'c) -> 'a * 'b -> 'c
val |> : 'a * ('a -> 'b) -> 'b
val |-> : ('c * 'a) * ('c -> 'a -> 'b) -> 'b
val ||> : ('c * 'a) * ('a -> 'b) -> 'c * 'b
val ||>> : ('c * 'a) * ('a -> 'd * 'b) -> ('c * 'd) * 'b
val |>> : ('a * 'c) * ('a -> 'b) -> 'b * 'c
val |>>> : ('a * 'c) * ('a -> 'b * 'd) -> 'b * ('c * 'd)
val #> : ('a -> 'b) * ('b -> 'c) -> 'a -> 'c
val #-> : ('a -> 'c * 'b) * ('c -> 'b -> 'd) -> 'a -> 'd
val ? : ('a -> bool) * ('a -> 'a) -> 'a -> 'a
val ` : ('b -> 'a) -> 'b -> 'a * 'b
val tap: ('b -> 'a) -> 'b -> 'b
val oo: ('a -> 'b) * ('c -> 'd -> 'a) -> 'c -> 'd -> 'b
val ooo: ('a -> 'b) * ('c -> 'd -> 'e -> 'a) -> 'c -> 'd -> 'e -> 'b
val oooo: ('a -> 'b) * ('c -> 'd -> 'e -> 'f -> 'a) -> 'c -> 'd -> 'e -> 'f -> 'b
val funpow: int -> ('a -> 'a) -> 'a -> 'a
val apl: 'a * ('a * 'b -> 'c) -> 'b -> 'c
val apr: ('a * 'b -> 'c) * 'b -> 'a -> 'c
(*old options -- invalidated*)
datatype invalid = None of invalid
exception OPTION of invalid
(*options*)
val the: 'a option -> 'a
val if_none: 'a option -> 'a -> 'a
val is_some: 'a option -> bool
val is_none: 'a option -> bool
exception ERROR
val try: ('a -> 'b) -> 'a -> 'b option
val can: ('a -> 'b) -> 'a -> bool
datatype 'a result = Result of 'a | Exn of exn
val capture: ('a -> 'b) -> 'a -> 'b result
val release: 'a result -> 'a
val get_result: 'a result -> 'a option
val get_exn: 'a result -> exn option
(*pairs*)
val pair: 'a -> 'b -> 'a * 'b
val rpair: 'a -> 'b -> 'b * 'a
val fst: 'a * 'b -> 'a
val snd: 'a * 'b -> 'b
val eq_fst: (''a * 'b) * (''a * 'c) -> bool
val eq_snd: ('a * ''b) * ('c * ''b) -> bool
val swap: 'a * 'b -> 'b * 'a
val apfst: ('a -> 'b) -> 'a * 'c -> 'b * 'c
val apsnd: ('a -> 'b) -> 'c * 'a -> 'c * 'b
val pairself: ('a -> 'b) -> 'a * 'a -> 'b * 'b
(*booleans*)
val equal: ''a -> ''a -> bool
val not_equal: ''a -> ''a -> bool
val orf: ('a -> bool) * ('a -> bool) -> 'a -> bool
val andf: ('a -> bool) * ('a -> bool) -> 'a -> bool
val exists: ('a -> bool) -> 'a list -> bool
val forall: ('a -> bool) -> 'a list -> bool
val set: bool ref -> bool
val reset: bool ref -> bool
val toggle: bool ref -> bool
val change: 'a ref -> ('a -> 'a) -> unit
val setmp: 'a ref -> 'a -> ('b -> 'c) -> 'b -> 'c
val conditional: bool -> (unit -> unit) -> unit
(*lists*)
exception UnequalLengths
val cons: 'a -> 'a list -> 'a list
val single: 'a -> 'a list
val append: 'a list -> 'a list -> 'a list
val apply: ('a -> 'a) list -> 'a -> 'a
val fold: ('a -> 'b -> 'b) -> 'a list -> 'b -> 'b
val fold_rev: ('a -> 'b -> 'b) -> 'a list -> 'b -> 'b
val fold_map: ('a -> 'b -> 'c * 'b) -> 'a list -> 'b -> 'c list * 'b
val foldl_map: ('a * 'b -> 'a * 'c) -> 'a * 'b list -> 'a * 'c list
val foldr1: ('a * 'a -> 'a) -> 'a list -> 'a
val foldln: ('a * int -> 'b -> 'b) -> 'a list -> 'b -> 'b
val unflat: 'a list list -> 'b list -> 'b list list
val splitAt: int * 'a list -> 'a list * 'a list
val dropwhile: ('a -> bool) -> 'a list -> 'a list
val map_nth_elem: int -> ('a -> 'a) -> 'a list -> 'a list
val split_last: 'a list -> 'a list * 'a
val nth_update: 'a -> int * 'a list -> 'a list
val find_index: ('a -> bool) -> 'a list -> int
val find_index_eq: ''a -> ''a list -> int
val find_first: ('a -> bool) -> 'a list -> 'a option
val get_first: ('a -> 'b option) -> 'a list -> 'b option
val separate: 'a -> 'a list -> 'a list
val replicate: int -> 'a -> 'a list
val multiply: 'a list * 'a list list -> 'a list list
val product: 'a list -> 'b list -> ('a * 'b) list
val filter: ('a -> bool) -> 'a list -> 'a list
val filter_out: ('a -> bool) -> 'a list -> 'a list
val map2: ('a * 'b -> 'c) -> 'a list * 'b list -> 'c list
val exists2: ('a * 'b -> bool) -> 'a list * 'b list -> bool
val forall2: ('a * 'b -> bool) -> 'a list * 'b list -> bool
val seq2: ('a * 'b -> unit) -> 'a list * 'b list -> unit
val ~~ : 'a list * 'b list -> ('a * 'b) list
val split_list: ('a * 'b) list -> 'a list * 'b list
val equal_lists: ('a * 'b -> bool) -> 'a list * 'b list -> bool
val prefix: ''a list * ''a list -> bool
val take_prefix: ('a -> bool) -> 'a list -> 'a list * 'a list
val take_suffix: ('a -> bool) -> 'a list -> 'a list * 'a list
val prefixes1: 'a list -> 'a list list
val suffixes1: 'a list -> 'a list list
(*integers*)
val gcd: IntInf.int * IntInf.int -> IntInf.int
val lcm: IntInf.int * IntInf.int -> IntInf.int
val inc: int ref -> int
val dec: int ref -> int
val upto: int * int -> int list
val downto: int * int -> int list
val downto0: int list * int -> bool
val radixpand: int * int -> int list
val radixstring: int * string * int -> string
val string_of_int: int -> string
val string_of_indexname: string * int -> string
val read_radixint: int * string list -> int * string list
val read_int: string list -> int * string list
val oct_char: string -> string
(*rational numbers*)
type rat
exception RAT of string
val rep_rat: rat -> IntInf.int * IntInf.int
val ratge0: rat -> bool
val ratgt0: rat -> bool
val ratle: rat * rat -> bool
val ratlt: rat * rat -> bool
val ratadd: rat * rat -> rat
val ratmul: rat * rat -> rat
val ratinv: rat -> rat
val int_ratdiv: IntInf.int * IntInf.int -> rat
val ratneg: rat -> rat
val rat_of_int: int -> rat
val rat_of_intinf: IntInf.int -> rat
val rat0: rat
(*strings*)
val nth_elem_string: int * string -> string
val fold_string: (string -> 'a -> 'a) -> string -> 'a -> 'a
val exists_string: (string -> bool) -> string -> bool
val forall_string: (string -> bool) -> string -> bool
val enclose: string -> string -> string -> string
val unenclose: string -> string
val quote: string -> string
val space_implode: string -> string list -> string
val commas: string list -> string
val commas_quote: string list -> string
val cat_lines: string list -> string
val space_explode: string -> string -> string list
val split_lines: string -> string list
val prefix_lines: string -> string -> string
val untabify: string list -> string list
val suffix: string -> string -> string
val unsuffix: string -> string -> string
val unprefix: string -> string -> string
val replicate_string: int -> string -> string
val translate_string: (string -> string) -> string -> string
(*lists as sets -- see also Pure/General/ord_list.ML*)
val mem: ''a * ''a list -> bool
val mem_int: int * int list -> bool
val mem_string: string * string list -> bool
val gen_mem: ('a * 'b -> bool) -> 'a * 'b list -> bool
val ins: ''a * ''a list -> ''a list
val ins_int: int * int list -> int list
val ins_string: string * string list -> string list
val gen_ins: ('a * 'a -> bool) -> 'a * 'a list -> 'a list
val member: ('b * 'a -> bool) -> 'a list -> 'b -> bool
val insert: ('a * 'a -> bool) -> 'a -> 'a list -> 'a list
val remove: ('b * 'a -> bool) -> 'b -> 'a list -> 'a list
val union: ''a list * ''a list -> ''a list
val union_int: int list * int list -> int list
val union_string: string list * string list -> string list
val gen_union: ('a * 'a -> bool) -> 'a list * 'a list -> 'a list
val gen_inter: ('a * 'b -> bool) -> 'a list * 'b list -> 'a list
val inter: ''a list * ''a list -> ''a list
val inter_int: int list * int list -> int list
val inter_string: string list * string list -> string list
val subset: ''a list * ''a list -> bool
val subset_int: int list * int list -> bool
val subset_string: string list * string list -> bool
val eq_set: ''a list * ''a list -> bool
val eq_set_string: string list * string list -> bool
val gen_subset: ('a * 'b -> bool) -> 'a list * 'b list -> bool
val \ : ''a list * ''a -> ''a list
val \\ : ''a list * ''a list -> ''a list
val gen_rem: ('a * 'b -> bool) -> 'a list * 'b -> 'a list
val gen_rems: ('a * 'b -> bool) -> 'a list * 'b list -> 'a list
val gen_distinct: ('a * 'a -> bool) -> 'a list -> 'a list
val distinct: ''a list -> ''a list
val findrep: ''a list -> ''a list
val gen_duplicates: ('a * 'a -> bool) -> 'a list -> 'a list
val duplicates: ''a list -> ''a list
val has_duplicates: ('a * 'a -> bool) -> 'a list -> bool
(*association lists*)
val assoc: (''a * 'b) list * ''a -> 'b option
val assoc_int: (int * 'a) list * int -> 'a option
val assoc_string: (string * 'a) list * string -> 'a option
val assoc_string_int: ((string * int) * 'a) list * (string * int) -> 'a option
val assocs: (''a * 'b list) list -> ''a -> 'b list
val assoc2: (''a * (''b * 'c) list) list * (''a * ''b) -> 'c option
val gen_assoc: ('a * 'b -> bool) -> ('b * 'c) list * 'a -> 'c option
val overwrite: (''a * 'b) list * (''a * 'b) -> (''a * 'b) list
val gen_overwrite: ('a * 'a -> bool) -> ('a * 'b) list * ('a * 'b) -> ('a * 'b) list
(*lists as tables*)
val gen_merge_lists: ('a * 'a -> bool) -> 'a list -> 'a list -> 'a list
val gen_merge_lists': ('a * 'a -> bool) -> 'a list -> 'a list -> 'a list
val merge_lists: ''a list -> ''a list -> ''a list
val merge_lists': ''a list -> ''a list -> ''a list
val merge_alists: (''a * 'b) list -> (''a * 'b) list -> (''a * 'b) list
val merge_alists': (''a * 'b) list -> (''a * 'b) list -> (''a * 'b) list
(*balanced trees*)
exception Balance
val fold_bal: ('a * 'a -> 'a) -> 'a list -> 'a
val access_bal: ('a -> 'a) * ('a -> 'a) * 'a -> int -> int -> 'a
val accesses_bal: ('a -> 'a) * ('a -> 'a) * 'a -> int -> 'a list
(*orders*)
val rev_order: order -> order
val make_ord: ('a * 'a -> bool) -> 'a * 'a -> order
val int_ord: int * int -> order
val string_ord: string * string -> order
val fast_string_ord: string * string -> order
val option_ord: ('a * 'b -> order) -> 'a option * 'b option -> order
val prod_ord: ('a * 'b -> order) -> ('c * 'd -> order) -> ('a * 'c) * ('b * 'd) -> order
val dict_ord: ('a * 'b -> order) -> 'a list * 'b list -> order
val list_ord: ('a * 'b -> order) -> 'a list * 'b list -> order
val sort: ('a * 'a -> order) -> 'a list -> 'a list
val sort_strings: string list -> string list
val sort_wrt: ('a -> string) -> 'a list -> 'a list
val unique_strings: string list -> string list
(*random numbers*)
exception RANDOM
val random: unit -> real
val random_range: int -> int -> int
val one_of: 'a list -> 'a
val frequency: (int * 'a) list -> 'a
(*current directory*)
val cd: string -> unit
val pwd: unit -> string
(*misc*)
val make_keylist: ('a -> 'b) -> 'a list -> ('a * 'b) list
val keyfilter: ('a * ''b) list -> ''b -> 'a list
val partition_eq: ('a * 'a -> bool) -> 'a list -> 'a list list
val partition_list: (int -> 'a -> bool) -> int -> int -> 'a list -> 'a list list
val gensym: string -> string
val scanwords: (string -> bool) -> string list -> string list
type stamp
val stamp: unit -> stamp
type serial
val serial: unit -> serial
structure Object: sig type T end
end;
signature LIBRARY =
sig
include BASIC_LIBRARY
val foldl: ('a * 'b -> 'a) -> 'a * 'b list -> 'a
val foldr: ('a * 'b -> 'b) -> 'a list * 'b -> 'b
val take: int * 'a list -> 'a list
val drop: int * 'a list -> 'a list
val nth_elem: int * 'a list -> 'a
val last_elem: 'a list -> 'a
val flat: 'a list list -> 'a list
val seq: ('a -> unit) -> 'a list -> unit
val partition: ('a -> bool) -> 'a list -> 'a list * 'a list
val mapfilter: ('a -> 'b option) -> 'a list -> 'b list
end;
structure Library: LIBRARY =
struct
(** functions **)
fun I x = x;
fun K x = fn _ => x;
fun curry f x y = f (x, y);
fun uncurry f (x, y) = f x y;
(*reverse application and structured results*)
fun x |> f = f x;
fun (x, y) |-> f = f x y;
fun (x, y) |>> f = (f x, y);
fun (x, y) ||> f = (x, f y);
fun (x, y) |>>> f = let val (x', z) = f x in (x', (y, z)) end;
fun (x, y) ||>> f = let val (z, y') = f y in ((x, z), y') end;
(*reverse composition*)
fun f #> g = g o f;
fun f #-> g = uncurry g o f;
(*conditional application*)
fun b ? f = fn x => if b x then f x else x
(*view results*)
fun `f = fn x => (f x, x);
fun tap f x = (f x; x);
(*composition with multiple args*)
fun (f oo g) x y = f (g x y);
fun (f ooo g) x y z = f (g x y z);
fun (f oooo g) x y z w = f (g x y z w);
(*function exponentiation: f(...(f x)...) with n applications of f*)
fun funpow n f x =
let fun rep (0, x) = x
| rep (n, x) = rep (n - 1, f x)
in rep (n, x) end;
(*application of (infix) operator to its left or right argument*)
fun apl (x, f) y = f (x, y);
fun apr (f, y) x = f (x, y);
(** options **)
(*invalidate former constructors to prevent accidental use as match-all patterns!*)
datatype invalid = None of invalid;
exception OPTION of invalid;
val the = Option.valOf;
(*strict!*)
fun if_none NONE y = y
| if_none (SOME x) _ = x;
fun is_some (SOME _) = true
| is_some NONE = false;
fun is_none (SOME _) = false
| is_none NONE = true;
(* exception handling *)
exception ERROR;
fun try f x = SOME (f x)
handle Interrupt => raise Interrupt | ERROR => raise ERROR | _ => NONE;
fun can f x = is_some (try f x);
datatype 'a result =
Result of 'a |
Exn of exn;
fun capture f x = Result (f x) handle e => Exn e;
fun release (Result y) = y
| release (Exn e) = raise e;
fun get_result (Result x) = SOME x
| get_result _ = NONE;
fun get_exn (Exn exn) = SOME exn
| get_exn _ = NONE;
(** pairs **)
fun pair x y = (x, y);
fun rpair x y = (y, x);
fun fst (x, y) = x;
fun snd (x, y) = y;
fun eq_fst ((x1, _), (x2, _)) = x1 = x2;
fun eq_snd ((_, y1), (_, y2)) = y1 = y2;
fun swap (x, y) = (y, x);
(*apply function to components*)
fun apfst f (x, y) = (f x, y);
fun apsnd f (x, y) = (x, f y);
fun pairself f (x, y) = (f x, f y);
(** booleans **)
(* equality *)
fun equal x y = x = y;
fun not_equal x y = x <> y;
(* operators for combining predicates *)
fun p orf q = fn x => p x orelse q x;
fun p andf q = fn x => p x andalso q x;
(* predicates on lists *)
(*exists pred [x1, ..., xn] ===> pred x1 orelse ... orelse pred xn*)
fun exists (pred: 'a -> bool) : 'a list -> bool =
let fun boolf [] = false
| boolf (x :: xs) = pred x orelse boolf xs
in boolf end;
(*forall pred [x1, ..., xn] ===> pred x1 andalso ... andalso pred xn*)
fun forall (pred: 'a -> bool) : 'a list -> bool =
let fun boolf [] = true
| boolf (x :: xs) = pred x andalso boolf xs
in boolf end;
(* flags *)
fun set flag = (flag := true; true);
fun reset flag = (flag := false; false);
fun toggle flag = (flag := not (! flag); ! flag);
fun change r f = r := f (! r);
(*temporarily set flag, handling exceptions*)
fun setmp flag value f x =
let
val orig_value = ! flag;
fun return y = (flag := orig_value; y);
in
flag := value;
return (f x handle exn => (return (); raise exn))
end;
(* conditional execution *)
fun conditional b f = if b then f () else ();
(** lists **)
exception UnequalLengths;
fun cons x xs = x :: xs;
fun single x = [x];
fun append xs ys = xs @ ys;
fun apply [] x = x
| apply (f :: fs) x = apply fs (f x);
(* fold *)
fun fold f =
let
fun fold_aux [] y = y
| fold_aux (x :: xs) y = fold_aux xs (f x y);
in fold_aux end;
fun fold_rev f =
let
fun fold_aux [] y = y
| fold_aux (x :: xs) y = f x (fold_aux xs y);
in fold_aux end;
fun fold_map f =
let
fun fold_aux [] y = ([], y)
| fold_aux (x :: xs) y =
let
val (x', y') = f x y;
val (xs', y'') = fold_aux xs y';
in (x' :: xs', y'') end;
in fold_aux end;
(*the following versions of fold are designed to fit nicely with infixes*)
(* (op @) (e, [x1, ..., xn]) ===> ((e @ x1) @ x2) ... @ xn
for operators that associate to the left (TAIL RECURSIVE)*)
fun foldl (f: 'a * 'b -> 'a) : 'a * 'b list -> 'a =
let fun itl (e, []) = e
| itl (e, a::l) = itl (f(e, a), l)
in itl end;
(* (op @) ([x1, ..., xn], e) ===> x1 @ (x2 ... @ (xn @ e))
for operators that associate to the right (not tail recursive)*)
fun foldr f (l, e) =
let fun itr [] = e
| itr (a::l) = f(a, itr l)
in itr l end;
(* (op @) [x1, ..., xn] ===> x1 @ (x2 ... @ (x[n-1] @ xn))
for n > 0, operators that associate to the right (not tail recursive)*)
fun foldr1 f l =
let fun itr [x] = x
| itr (x::l) = f(x, itr l)
in itr l end;
fun foldln f xs e = fst (foldl (fn ((e,i), x) => (f (x,i) e, i+1)) ((e,1),xs));
fun foldl_map f =
let
fun fold_aux (x, []) = (x, [])
| fold_aux (x, y :: ys) =
let
val (x', y') = f (x, y);
val (x'', ys') = fold_aux (x', ys);
in (x'', y' :: ys') end;
in fold_aux end;
(* basic list functions *)
(*take the first n elements from a list*)
fun take (n, []) = []
| take (n, x :: xs) =
if n > 0 then x :: take (n - 1, xs) else [];
(*drop the first n elements from a list*)
fun drop (n, []) = []
| drop (n, x :: xs) =
if n > 0 then drop (n - 1, xs) else x :: xs;
fun splitAt(n,[]) = ([],[])
| splitAt(n,xs as x::ys) =
if n>0 then let val (ps,qs) = splitAt(n-1,ys) in (x::ps,qs) end
else ([],xs)
fun dropwhile P [] = []
| dropwhile P (ys as x::xs) = if P x then dropwhile P xs else ys;
(*return nth element of a list, where 0 designates the first element;
raise EXCEPTION if list too short*)
fun nth_elem (i,xs) = List.nth(xs,i);
fun map_nth_elem 0 f (x :: xs) = f x :: xs
| map_nth_elem n f (x :: xs) = x :: map_nth_elem (n - 1) f xs
| map_nth_elem _ _ [] = raise Subscript;
(*last element of a list*)
val last_elem = List.last;
(*rear decomposition*)
fun split_last [] = raise Empty
| split_last [x] = ([], x)
| split_last (x :: xs) = apfst (cons x) (split_last xs);
(*update nth element*)
fun nth_update x n_xs =
(case splitAt n_xs of
(_,[]) => raise Subscript
| (prfx, _ :: sffx') => prfx @ (x :: sffx'))
(*find the position of an element in a list*)
fun find_index pred =
let fun find _ [] = ~1
| find n (x :: xs) = if pred x then n else find (n + 1) xs;
in find 0 end;
fun find_index_eq x = find_index (equal x);
(*find first element satisfying predicate*)
fun find_first _ [] = NONE
| find_first pred (x :: xs) =
if pred x then SOME x else find_first pred xs;
(*get first element by lookup function*)
fun get_first _ [] = NONE
| get_first f (x :: xs) =
(case f x of
NONE => get_first f xs
| some => some);
(*flatten a list of lists to a list*)
val flat = List.concat;
fun unflat (xs :: xss) ys =
let val (ps,qs) = splitAt(length xs,ys)
in ps :: unflat xss qs end
| unflat [] [] = []
| unflat _ _ = raise UnequalLengths;
(*like Lisp's MAPC -- seq proc [x1, ..., xn] evaluates
(proc x1; ...; proc xn) for side effects*)
val seq = List.app;
(*separate s [x1, x2, ..., xn] ===> [x1, s, x2, s, ..., s, xn]*)
fun separate s (x :: (xs as _ :: _)) = x :: s :: separate s xs
| separate _ xs = xs;
(*make the list [x, x, ..., x] of length n*)
fun replicate n (x: 'a) : 'a list =
let fun rep (0, xs) = xs
| rep (n, xs) = rep (n - 1, x :: xs)
in
if n < 0 then raise Subscript
else rep (n, [])
end;
fun translate_string f = String.translate (f o String.str);
(*multiply [a, b, c, ...] * [xs, ys, zs, ...]*)
fun multiply ([], _) = []
| multiply (x :: xs, yss) = map (cons x) yss @ multiply (xs, yss);
(*direct product*)
fun product _ [] = []
| product [] _ = []
| product (x :: xs) ys = map (pair x) ys @ product xs ys;
(* filter *)
(*copy the list preserving elements that satisfy the predicate*)
val filter = List.filter;
fun filter_out f = filter (not o f);
val mapfilter = List.mapPartial;
(* lists of pairs *)
exception UnequalLengths;
fun map2 _ ([], []) = []
| map2 f (x :: xs, y :: ys) = f (x, y) :: map2 f (xs, ys)
| map2 _ _ = raise UnequalLengths;
fun exists2 _ ([], []) = false
| exists2 pred (x :: xs, y :: ys) = pred (x, y) orelse exists2 pred (xs, ys)
| exists2 _ _ = raise UnequalLengths;
fun forall2 _ ([], []) = true
| forall2 pred (x :: xs, y :: ys) = pred (x, y) andalso forall2 pred (xs, ys)
| forall2 _ _ = raise UnequalLengths;
fun seq2 _ ([], []) = ()
| seq2 f (x :: xs, y :: ys) = (f (x, y); seq2 f (xs, ys))
| seq2 _ _ = raise UnequalLengths;
(*combine two lists forming a list of pairs:
[x1, ..., xn] ~~ [y1, ..., yn] ===> [(x1, y1), ..., (xn, yn)]*)
fun [] ~~ [] = []
| (x :: xs) ~~ (y :: ys) = (x, y) :: (xs ~~ ys)
| _ ~~ _ = raise UnequalLengths;
(*inverse of ~~; the old 'split':
[(x1, y1), ..., (xn, yn)] ===> ([x1, ..., xn], [y1, ..., yn])*)
val split_list = ListPair.unzip;
fun equal_lists eq (xs, ys) = length xs = length ys andalso forall2 eq (xs, ys);
(* prefixes, suffixes *)
fun [] prefix _ = true
| (x :: xs) prefix (y :: ys) = x = y andalso (xs prefix ys)
| _ prefix _ = false;
(* [x1, ..., xi, ..., xn] ---> ([x1, ..., x(i-1)], [xi, ..., xn])
where xi is the first element that does not satisfy the predicate*)
fun take_prefix (pred : 'a -> bool) (xs: 'a list) : 'a list * 'a list =
let fun take (rxs, []) = (rev rxs, [])
| take (rxs, x :: xs) =
if pred x then take(x :: rxs, xs) else (rev rxs, x :: xs)
in take([], xs) end;
(* [x1, ..., xi, ..., xn] ---> ([x1, ..., xi], [x(i+1), ..., xn])
where xi is the last element that does not satisfy the predicate*)
fun take_suffix _ [] = ([], [])
| take_suffix pred (x :: xs) =
(case take_suffix pred xs of
([], sffx) => if pred x then ([], x :: sffx) else ([x], sffx)
| (prfx, sffx) => (x :: prfx, sffx));
fun prefixes1 [] = []
| prefixes1 (x :: xs) = map (cons x) ([] :: prefixes1 xs);
fun suffixes1 xs = map rev (prefixes1 (rev xs));
(** integers **)
fun gcd (x, y) =
let fun gxd x y : IntInf.int =
if y = 0 then x else gxd y (x mod y)
in if x < y then gxd y x else gxd x y end;
fun lcm (x, y) = (x * y) div gcd (x, y);
fun inc i = (i := ! i + 1; ! i);
fun dec i = (i := ! i - 1; ! i);
(* lists of integers *)
(*make the list [from, from + 1, ..., to]*)
fun (from upto to) =
if from > to then [] else from :: ((from + 1) upto to);
(*make the list [from, from - 1, ..., to]*)
fun (from downto to) =
if from < to then [] else from :: ((from - 1) downto to);
(*predicate: downto0 (is, n) <=> is = [n, n - 1, ..., 0]*)
fun downto0 (i :: is, n) = i = n andalso downto0 (is, n - 1)
| downto0 ([], n) = n = ~1;
(* convert integers to strings *)
(*expand the number in the given base;
example: radixpand (2, 8) gives [1, 0, 0, 0]*)
fun radixpand (base, num) : int list =
let
fun radix (n, tail) =
if n < base then n :: tail
else radix (n div base, (n mod base) :: tail)
in radix (num, []) end;
(*expands a number into a string of characters starting from "zerochar";
example: radixstring (2, "0", 8) gives "1000"*)
fun radixstring (base, zerochar, num) =
let val offset = ord zerochar;
fun chrof n = chr (offset + n)
in implode (map chrof (radixpand (base, num))) end;
val string_of_int = Int.toString;
fun string_of_indexname (a,0) = a
| string_of_indexname (a,i) = a ^ "_" ^ Int.toString i;
(* read integers *)
fun read_radixint (radix: int, cs) : int * string list =
let val zero = ord"0"
val limit = zero+radix
fun scan (num,[]) = (num,[])
| scan (num, c::cs) =
if zero <= ord c andalso ord c < limit
then scan(radix*num + ord c - zero, cs)
else (num, c::cs)
in scan(0,cs) end;
fun read_int cs = read_radixint (10, cs);
fun oct_char s = chr (#1 (read_radixint (8, explode s)));
(** strings **)
(* functions tuned for strings, avoiding explode *)
fun nth_elem_string (i, str) =
(case try String.substring (str, i, 1) of
SOME s => s
| NONE => raise Subscript);
fun fold_string f str x0 =
let
val n = size str;
fun iter (x, i) =
if i < n then iter (f (String.substring (str, i, 1)) x, i + 1) else x;
in iter (x0, 0) end;
fun exists_string pred str =
let
val n = size str;
fun ex i = i < n andalso (pred (String.substring (str, i, 1)) orelse ex (i + 1));
in ex 0 end;
fun forall_string pred = not o exists_string (not o pred);
(*enclose in brackets*)
fun enclose lpar rpar str = lpar ^ str ^ rpar;
fun unenclose str = String.substring (str, 1, size str - 2);
(*simple quoting (does not escape special chars)*)
val quote = enclose "\"" "\"";
(*space_implode "..." (explode "hello") = "h...e...l...l...o"*)
fun space_implode a bs = implode (separate a bs);
val commas = space_implode ", ";
val commas_quote = commas o map quote;
(*concatenate messages, one per line, into a string*)
val cat_lines = space_implode "\n";
(*space_explode "." "h.e..l.lo" = ["h", "e", "", "l", "lo"]*)
fun space_explode _ "" = []
| space_explode sep str =
let
fun expl chs =
(case take_prefix (not_equal sep) chs of
(cs, []) => [implode cs]
| (cs, _ :: cs') => implode cs :: expl cs');
in expl (explode str) end;
val split_lines = space_explode "\n";
fun prefix_lines "" txt = txt
| prefix_lines prfx txt = txt |> split_lines |> map (fn s => prfx ^ s) |> cat_lines;
fun untabify chs =
let
val tab_width = 8;
fun untab (_, "\n") = (0, ["\n"])
| untab (pos, "\t") =
let val d = tab_width - (pos mod tab_width) in (pos + d, replicate d " ") end
| untab (pos, c) = (pos + 1, [c]);
in
if not (exists (equal "\t") chs) then chs
else flat (#2 (foldl_map untab (0, chs)))
end;
fun suffix sffx s = s ^ sffx;
fun unsuffix sffx s =
if String.isSuffix sffx s then String.substring (s, 0, size s - size sffx)
else raise Fail "unsuffix";
fun unprefix prfx s =
if String.isPrefix prfx s then String.substring (s, size prfx, size s - size prfx)
else raise Fail "unprefix";
fun replicate_string 0 _ = ""
| replicate_string 1 a = a
| replicate_string k a =
if k mod 2 = 0 then replicate_string (k div 2) (a ^ a)
else replicate_string (k div 2) (a ^ a) ^ a;
(** lists as sets -- see also Pure/General/ord_list.ML **)
(*membership in a list*)
fun x mem [] = false
| x mem (y :: ys) = x = y orelse x mem ys;
(*membership in a list, optimized version for ints*)
fun (x:int) mem_int [] = false
| x mem_int (y :: ys) = x = y orelse x mem_int ys;
(*membership in a list, optimized version for strings*)
fun (x:string) mem_string [] = false
| x mem_string (y :: ys) = x = y orelse x mem_string ys;
(*generalized membership test*)
fun gen_mem eq (x, []) = false
| gen_mem eq (x, y :: ys) = eq (x, y) orelse gen_mem eq (x, ys);
(*member, insert, and remove -- with canonical argument order*)
fun member eq xs x = gen_mem eq (x, xs);
fun insert eq x xs = if gen_mem eq (x, xs) then xs else x :: xs;
fun remove eq x xs = if gen_mem eq (x, xs) then filter_out (fn y => eq (x, y)) xs else xs;
(*insertion into list if not already there*)
fun (x ins xs) = if x mem xs then xs else x :: xs;
(*insertion into list, optimized version for ints*)
fun (x ins_int xs) = if x mem_int xs then xs else x :: xs;
(*insertion into list, optimized version for strings*)
fun (x ins_string xs) = if x mem_string xs then xs else x :: xs;
(*generalized insertion*)
fun gen_ins eq (x, xs) = insert eq x xs;
(*union of sets represented as lists: no repetitions*)
fun xs union [] = xs
| [] union ys = ys
| (x :: xs) union ys = xs union (x ins ys);
(*union of sets, optimized version for ints*)
fun (xs:int list) union_int [] = xs
| [] union_int ys = ys
| (x :: xs) union_int ys = xs union_int (x ins_int ys);
(*union of sets, optimized version for strings*)
fun (xs:string list) union_string [] = xs
| [] union_string ys = ys
| (x :: xs) union_string ys = xs union_string (x ins_string ys);
(*generalized union*)
fun gen_union eq (xs, []) = xs
| gen_union eq ([], ys) = ys
| gen_union eq (x :: xs, ys) = gen_union eq (xs, gen_ins eq (x, ys));
(*intersection*)
fun [] inter ys = []
| (x :: xs) inter ys =
if x mem ys then x :: (xs inter ys) else xs inter ys;
(*intersection, optimized version for ints*)
fun ([]:int list) inter_int ys = []
| (x :: xs) inter_int ys =
if x mem_int ys then x :: (xs inter_int ys) else xs inter_int ys;
(*intersection, optimized version for strings *)
fun ([]:string list) inter_string ys = []
| (x :: xs) inter_string ys =
if x mem_string ys then x :: (xs inter_string ys) else xs inter_string ys;
(*generalized intersection*)
fun gen_inter eq ([], ys) = []
| gen_inter eq (x::xs, ys) =
if gen_mem eq (x,ys) then x :: gen_inter eq (xs, ys)
else gen_inter eq (xs, ys);
(*subset*)
fun [] subset ys = true
| (x :: xs) subset ys = x mem ys andalso xs subset ys;
(*subset, optimized version for ints*)
fun ([]: int list) subset_int ys = true
| (x :: xs) subset_int ys = x mem_int ys andalso xs subset_int ys;
(*subset, optimized version for strings*)
fun ([]: string list) subset_string ys = true
| (x :: xs) subset_string ys = x mem_string ys andalso xs subset_string ys;
(*set equality*)
fun eq_set (xs, ys) =
xs = ys orelse (xs subset ys andalso ys subset xs);
(*set equality for strings*)
fun eq_set_string ((xs: string list), ys) =
xs = ys orelse (xs subset_string ys andalso ys subset_string xs);
fun gen_subset eq (xs, ys) = forall (fn x => gen_mem eq (x, ys)) xs;
(*removing an element from a list WITHOUT duplicates*)
fun (y :: ys) \ x = if x = y then ys else y :: (ys \ x)
| [] \ x = [];
fun ys \\ xs = foldl (op \) (ys,xs);
(*removing an element from a list -- possibly WITH duplicates*)
fun gen_rem eq (xs, y) = filter_out (fn x => eq (x, y)) xs;
fun gen_rems eq (xs, ys) = filter_out (fn x => gen_mem eq (x, ys)) xs;
(*makes a list of the distinct members of the input; preserves order, takes
first of equal elements*)
fun gen_distinct eq lst =
let
val memb = gen_mem eq;
fun dist (rev_seen, []) = rev rev_seen
| dist (rev_seen, x :: xs) =
if memb (x, rev_seen) then dist (rev_seen, xs)
else dist (x :: rev_seen, xs);
in
dist ([], lst)
end;
fun distinct l = gen_distinct (op =) l;
(*returns the tail beginning with the first repeated element, or []*)
fun findrep [] = []
| findrep (x :: xs) = if x mem xs then x :: xs else findrep xs;
(*returns a list containing all repeated elements exactly once; preserves
order, takes first of equal elements*)
fun gen_duplicates eq lst =
let
val memb = gen_mem eq;
fun dups (rev_dups, []) = rev rev_dups
| dups (rev_dups, x :: xs) =
if memb (x, rev_dups) orelse not (memb (x, xs)) then
dups (rev_dups, xs)
else dups (x :: rev_dups, xs);
in
dups ([], lst)
end;
fun duplicates l = gen_duplicates (op =) l;
fun has_duplicates eq =
let
fun dups [] = false
| dups (x :: xs) = member eq xs x orelse dups xs;
in dups end;
(** association lists **)
(*association list lookup*)
fun assoc ([], key) = NONE
| assoc ((keyi, xi) :: pairs, key) =
if key = keyi then SOME xi else assoc (pairs, key);
(*association list lookup, optimized version for ints*)
fun assoc_int ([], (key:int)) = NONE
| assoc_int ((keyi, xi) :: pairs, key) =
if key = keyi then SOME xi else assoc_int (pairs, key);
(*association list lookup, optimized version for strings*)
fun assoc_string ([], (key:string)) = NONE
| assoc_string ((keyi, xi) :: pairs, key) =
if key = keyi then SOME xi else assoc_string (pairs, key);
(*association list lookup, optimized version for string*ints*)
fun assoc_string_int ([], (key:string*int)) = NONE
| assoc_string_int ((keyi, xi) :: pairs, key) =
if key = keyi then SOME xi else assoc_string_int (pairs, key);
fun assocs ps x =
(case assoc (ps, x) of
NONE => []
| SOME ys => ys);
(*two-fold association list lookup*)
fun assoc2 (aal, (key1, key2)) =
(case assoc (aal, key1) of
SOME al => assoc (al, key2)
| NONE => NONE);
(*generalized association list lookup*)
fun gen_assoc eq ([], key) = NONE
| gen_assoc eq ((keyi, xi) :: pairs, key) =
if eq (key, keyi) then SOME xi else gen_assoc eq (pairs, key);
(*association list update*)
fun overwrite (al, p as (key, _)) =
let fun over ((q as (keyi, _)) :: pairs) =
if keyi = key then p :: pairs else q :: (over pairs)
| over [] = [p]
in over al end;
fun gen_overwrite eq (al, p as (key, _)) =
let fun over ((q as (keyi, _)) :: pairs) =
if eq (keyi, key) then p :: pairs else q :: (over pairs)
| over [] = [p]
in over al end;
(* lists as tables *)
fun gen_merge_lists _ xs [] = xs
| gen_merge_lists _ [] ys = ys
| gen_merge_lists eq xs ys = xs @ gen_rems eq (ys, xs);
fun gen_merge_lists' _ xs [] = xs
| gen_merge_lists' _ [] ys = ys
| gen_merge_lists' eq xs ys = gen_rems eq (ys, xs) @ xs;
fun merge_lists xs ys = gen_merge_lists (op =) xs ys;
fun merge_lists' xs ys = gen_merge_lists' (op =) xs ys;
fun merge_alists al = gen_merge_lists eq_fst al;
fun merge_alists' al = gen_merge_lists' eq_fst al;
(** balanced trees **)
exception Balance; (*indicates non-positive argument to balancing fun*)
(*balanced folding; avoids deep nesting*)
fun fold_bal f [x] = x
| fold_bal f [] = raise Balance
| fold_bal f xs =
let val (ps,qs) = splitAt(length xs div 2, xs)
in f (fold_bal f ps, fold_bal f qs) end;
(*construct something of the form f(...g(...(x)...)) for balanced access*)
fun access_bal (f, g, x) n i =
let fun acc n i = (*1<=i<=n*)
if n=1 then x else
let val n2 = n div 2
in if i<=n2 then f (acc n2 i)
else g (acc (n-n2) (i-n2))
end
in if 1<=i andalso i<=n then acc n i else raise Balance end;
(*construct ALL such accesses; could try harder to share recursive calls!*)
fun accesses_bal (f, g, x) n =
let fun acc n =
if n=1 then [x] else
let val n2 = n div 2
val acc2 = acc n2
in if n-n2=n2 then map f acc2 @ map g acc2
else map f acc2 @ map g (acc (n-n2)) end
in if 1<=n then acc n else raise Balance end;
(** orders **)
fun rev_order LESS = GREATER
| rev_order EQUAL = EQUAL
| rev_order GREATER = LESS;
(*assume rel is a linear strict order*)
fun make_ord rel (x, y) =
if rel (x, y) then LESS
else if rel (y, x) then GREATER
else EQUAL;
val int_ord = Int.compare;
val string_ord = String.compare;
fun fast_string_ord (s1, s2) =
(case int_ord (size s1, size s2) of EQUAL => string_ord (s1, s2) | ord => ord);
fun option_ord ord (SOME x, SOME y) = ord (x, y)
| option_ord _ (NONE, NONE) = EQUAL
| option_ord _ (NONE, SOME _) = LESS
| option_ord _ (SOME _, NONE) = GREATER;
(*lexicographic product*)
fun prod_ord a_ord b_ord ((x, y), (x', y')) =
(case a_ord (x, x') of EQUAL => b_ord (y, y') | ord => ord);
(*dictionary order -- in general NOT well-founded!*)
fun dict_ord elem_ord (x :: xs, y :: ys) =
(case elem_ord (x, y) of EQUAL => dict_ord elem_ord (xs, ys) | ord => ord)
| dict_ord _ ([], []) = EQUAL
| dict_ord _ ([], _ :: _) = LESS
| dict_ord _ (_ :: _, []) = GREATER;
(*lexicographic product of lists*)
fun list_ord elem_ord (xs, ys) =
(case int_ord (length xs, length ys) of EQUAL => dict_ord elem_ord (xs, ys) | ord => ord);
(* sorting *)
(*quicksort (stable, i.e. does not reorder equal elements)*)
fun sort ord =
let
fun qsort [] = []
| qsort (xs as [_]) = xs
| qsort (xs as [x, y]) = if ord (x, y) = GREATER then [y, x] else xs
| qsort xs =
let val (lts, eqs, gts) = part (nth_elem (length xs div 2, xs)) xs
in qsort lts @ eqs @ qsort gts end
and part _ [] = ([], [], [])
| part pivot (x :: xs) = add (ord (x, pivot)) x (part pivot xs)
and add LESS x (lts, eqs, gts) = (x :: lts, eqs, gts)
| add EQUAL x (lts, eqs, gts) = (lts, x :: eqs, gts)
| add GREATER x (lts, eqs, gts) = (lts, eqs, x :: gts);
in qsort end;
(*sort strings*)
val sort_strings = sort string_ord;
fun sort_wrt sel xs = sort (string_ord o pairself sel) xs;
fun unique_strings ([]: string list) = []
| unique_strings [x] = [x]
| unique_strings (x :: y :: ys) =
if x = y then unique_strings (y :: ys)
else x :: unique_strings (y :: ys);
(** random numbers **)
exception RANDOM;
fun rmod x y = x - y * Real.realFloor (x / y);
local
val a = 16807.0;
val m = 2147483647.0;
val random_seed = ref 1.0;
in
fun random () =
let val r = rmod (a * !random_seed) m
in (random_seed := r; r) end;
end;
fun random_range l h =
if h < l orelse l < 0 then raise RANDOM
else l + Real.floor (rmod (random ()) (real (h - l + 1)));
fun one_of xs = nth_elem (random_range 0 (length xs - 1), xs);
fun frequency xs =
let
val sum = foldl op + (0, map fst xs);
fun pick n ((k, x) :: xs) =
if n <= k then x else pick (n - k) xs
in pick (random_range 1 sum) xs end;
(** current directory **)
val cd = OS.FileSys.chDir;
val pwd = OS.FileSys.getDir;
(** rational numbers **)
(* Keep them normalized! *)
datatype rat = Rat of bool * IntInf.int * IntInf.int
exception RAT of string;
fun rep_rat(Rat(a,p,q)) = (if a then p else ~p,q)
fun ratnorm(a,p,q) = if p=0 then Rat(a,0,1) else
let val absp = abs p
val m = gcd(absp,q)
in Rat(a = (0 <= p), absp div m, q div m) end;
fun ratcommon(p,q,r,s) =
let val den = lcm(q,s)
in (p*(den div q), r*(den div s), den) end
fun ratge0(Rat(a,p,q)) = a;
fun ratgt0(Rat(a,p,q)) = a andalso p > 0;
fun ratle(Rat(a,p,q),Rat(b,r,s)) =
not a andalso b orelse
a = b andalso
let val (p,r,_) = ratcommon(p,q,r,s)
in if a then p <= r else r <= p end
fun ratlt(Rat(a,p,q),Rat(b,r,s)) =
not a andalso b orelse
a = b andalso
let val (p,r,_) = ratcommon(p,q,r,s)
in if a then p < r else r < p end
fun ratadd(Rat(a,p,q),Rat(b,r,s)) =
let val (p,r,den) = ratcommon(p,q,r,s)
val num = (if a then p else ~p) + (if b then r else ~r)
in ratnorm(true,num,den) end;
fun ratmul(Rat(a,p,q),Rat(b,r,s)) = ratnorm(a=b,p*r,q*s)
fun ratinv(Rat(a,p,q)) = if p=0 then raise RAT "ratinv" else Rat(a,q,p)
fun int_ratdiv(p,q) =
if q=0 then raise RAT "int_ratdiv" else ratnorm(0<=q, p, abs q)
fun ratneg(Rat(b,p,q)) = Rat(not b,p,q);
fun rat_of_intinf i = if i < 0 then Rat(false,abs i,1) else Rat(true,i,1);
fun rat_of_int i = rat_of_intinf (IntInf.fromInt i);
val rat0 = rat_of_int 0;
(** misc **)
(*use the keyfun to make a list of (x, key) pairs*)
fun make_keylist (keyfun: 'a->'b) : 'a list -> ('a * 'b) list =
let fun keypair x = (x, keyfun x)
in map keypair end;
(*given a list of (x, key) pairs and a searchkey
return the list of xs from each pair whose key equals searchkey*)
fun keyfilter [] searchkey = []
| keyfilter ((x, key) :: pairs) searchkey =
if key = searchkey then x :: keyfilter pairs searchkey
else keyfilter pairs searchkey;
(*Partition list into elements that satisfy predicate and those that don't.
Preserves order of elements in both lists.*)
val partition = List.partition;
fun partition_eq (eq:'a * 'a -> bool) =
let
fun part [] = []
| part (x :: ys) =
let val (xs, xs') = partition (fn y => eq (x, y)) ys
in (x::xs)::(part xs') end
in part end;
(*Partition a list into buckets [ bi, b(i+1), ..., bj ]
putting x in bk if p(k)(x) holds. Preserve order of elements if possible.*)
fun partition_list p i j =
let fun part k xs =
if k>j then
(case xs of [] => []
| _ => raise Fail "partition_list")
else
let val (ns, rest) = partition (p k) xs;
in ns :: part(k+1)rest end
in part i end;
(* generating identifiers *)
(** Freshly generated identifiers; supplied prefix MUST start with a letter **)
local
(*Maps 0-63 to A-Z, a-z, 0-9 or _ or ' for generating random identifiers*)
fun char i = if i<26 then chr (ord "A" + i)
else if i<52 then chr (ord "a" + i - 26)
else if i<62 then chr (ord"0" + i - 52)
else if i=62 then "_"
else (*i=63*) "'";
val charVec = Vector.tabulate (64, char);
fun newid n =
let
in implode (map (fn i => Vector.sub(charVec,i)) (radixpand (64,n))) end;
val seedr = ref 0;
in
fun gensym pre = pre ^ (#1(newid (!seedr), inc seedr));
end;
(* lexical scanning *)
(*scan a list of characters into "words" composed of "letters" (recognized by
is_let) and separated by any number of non-"letters"*)
fun scanwords is_let cs =
let fun scan1 [] = []
| scan1 cs =
let val (lets, rest) = take_prefix is_let cs
in implode lets :: scanwords is_let rest end;
in scan1 (#2 (take_prefix (not o is_let) cs)) end;
(* stamps and serial numbers *)
type stamp = unit ref;
val stamp: unit -> stamp = ref;
type serial = int;
local val count = ref 0
in fun serial () = inc count end;
(* generic objects *)
(*note that the builtin exception datatype may be extended by new
constructors at any time*)
structure Object = struct type T = exn end;
end;
structure BasicLibrary: BASIC_LIBRARY = Library;
open BasicLibrary;