Added function unique_strings.
(* Title: Pure/library.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Author: Markus Wenzel, TU Munich
License: GPL (GNU GENERAL PUBLIC LICENSE)
Basic library: functions, options, pairs, booleans, lists, integers,
strings, lists as sets, association lists, generic tables, balanced
trees, orders, I/O and diagnostics, timing, 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 3 oo ooo oooo;
signature LIBRARY =
sig
(*functions*)
val curry: ('a * 'b -> 'c) -> 'a -> 'b -> 'c
val uncurry: ('a -> 'b -> 'c) -> 'a * 'b -> 'c
val I: 'a -> 'a
val K: 'a -> 'b -> 'a
val |> : 'a * ('a -> 'b) -> 'b
val |>> : ('a * 'b) * ('a -> 'c) -> 'c * 'b
val |>>> : ('a * 'b) * ('a -> 'c * 'd) -> 'c * ('b * 'd)
val apl: 'a * ('a * 'b -> 'c) -> 'b -> 'c
val apr: ('a * 'b -> 'c) * 'b -> 'a -> 'c
val funpow: int -> ('a -> 'a) -> 'a -> 'a
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
(*stamps*)
type stamp
val stamp: unit -> stamp
(*options*)
datatype 'a option = None | Some of 'a
exception OPTION
val the: 'a option -> 'a
val if_none: 'a option -> 'a -> 'a
val is_some: 'a option -> bool
val is_none: 'a option -> bool
val apsome: ('a -> 'b) -> 'a option -> 'b option
val try: ('a -> 'b) -> 'a -> 'b option
val can: ('a -> 'b) -> 'a -> bool
(*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
(*lists*)
exception LIST of string
val null: 'a list -> bool
val hd: 'a list -> 'a
val tl: 'a list -> 'a list
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 foldl: ('a * 'b -> 'a) -> 'a * 'b list -> 'a
val foldr: ('a * 'b -> 'b) -> 'a list * 'b -> 'b
val foldr1: ('a * 'a -> 'a) -> 'a list -> 'a
val foldl_map: ('a * 'b -> 'a * 'c) -> 'a * 'b list -> 'a * 'c list
val foldln: ('a * int -> 'b -> 'b) -> 'a list -> 'b -> 'b
val length: 'a list -> int
val take: int * 'a list -> 'a list
val drop: int * 'a list -> 'a list
val dropwhile: ('a -> bool) -> 'a list -> 'a list
val nth_elem: int * 'a list -> 'a
val last_elem: 'a list -> 'a
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 flat: 'a list list -> 'a list
val seq: ('a -> unit) -> 'a list -> unit
val separate: 'a -> 'a list -> 'a list
val replicate: int -> 'a -> 'a list
val multiply: 'a list * 'a list list -> 'a list list
val filter: ('a -> bool) -> 'a list -> 'a list
val filter_out: ('a -> bool) -> 'a list -> 'a list
val mapfilter: ('a -> 'b option) -> 'a list -> 'b 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
(*integers*)
val gcd: int * int -> int
val lcm: int * int -> 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
(*rational numbers*)
type rat
val rep_rat: rat -> int * int
val ratadd: rat * rat -> rat
val ratmul: rat * rat -> rat
val ratinv: rat -> rat
val int_ratdiv: int * int -> rat
val ratneg: rat -> rat
val rat_of_int: int -> rat
(*strings*)
val nth_elem_string: int * string -> string
val foldl_string: ('a * string -> 'a) -> 'a * string -> 'a
val exists_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 std_output: string -> unit
val prefix_lines: string -> string -> string
val split_lines: string -> string list
val untabify: string list -> string list
val suffix: string -> string -> string
val unsuffix: string -> string -> string
val replicate_string: int -> string -> string
(*lists as sets*)
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 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
(*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 overwrite_warn: (''a * 'b) list * (''a * 'b) -> string -> (''a * 'b) list
val gen_overwrite: ('a * 'a -> bool) -> ('a * 'b) list * ('a * 'b) -> ('a * 'b) list
(*generic tables*)
val generic_extend: ('a * 'a -> bool)
-> ('b -> 'a list) -> ('a list -> 'b) -> 'b -> 'a list -> 'b
val generic_merge: ('a * 'a -> bool) -> ('b -> 'a list) -> ('a list -> 'b) -> 'b -> 'b -> 'b
val extend_list: ''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_rev_lists: ''a list -> ''a list -> ''a 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*)
datatype order = LESS | EQUAL | GREATER
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 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
(*I/O and diagnostics*)
val cd: string -> unit
val pwd: unit -> string
val writeln_fn: (string -> unit) ref
val priority_fn: (string -> unit) ref
val warning_fn: (string -> unit) ref
val error_fn: (string -> unit) ref
val writeln: string -> unit
val priority: string -> unit
val warning: string -> unit
exception ERROR
val error_msg: string -> unit
val error: string -> 'a
val sys_error: string -> 'a
val assert: bool -> string -> unit
val deny: bool -> string -> unit
val assert_all: ('a -> bool) -> 'a list -> ('a -> string) -> unit
datatype 'a error = Error of string | OK of 'a
val get_error: 'a error -> string option
val get_ok: 'a error -> 'a option
val handle_error: ('a -> 'b) -> 'a -> 'b error
exception ERROR_MESSAGE of string
val transform_error: ('a -> 'b) -> 'a -> 'b
val transform_failure: (exn -> exn) -> ('a -> 'b) -> 'a -> 'b
(*timing*)
val cond_timeit: bool -> (unit -> 'a) -> 'a
val timeit: (unit -> 'a) -> 'a
val timeap: ('a -> 'b) -> 'a -> 'b
val timing: bool ref
(*misc*)
val make_keylist: ('a -> 'b) -> 'a list -> ('a * 'b) list
val keyfilter: ('a * ''b) list -> ''b -> 'a list
val partition: ('a -> bool) -> 'a list -> 'a list * '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 transitive_closure: (string * string list) list -> (string * string list) list
val init_gensym: unit -> unit (* FIXME !!??! *)
val gensym: string -> string
val bump_int_list: string list -> string list
val bump_list: string list * string -> string list
val bump_string: string -> string
val scanwords: (string -> bool) -> string list -> string list
datatype 'a mtree = Join of 'a * 'a mtree list
end;
structure Library: LIBRARY =
struct
(** functions **)
(*handy combinators*)
fun curry f x y = f (x, y);
fun uncurry f (x, y) = f x y;
fun I x = x;
fun K x y = x;
(*reverse apply*)
fun (x |> f) = f x;
fun ((x, y) |>> f) = (f x, y);
fun ((x, y) |>>> f) = let val (x', z) = f x in (x', (y, z)) 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);
(*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;
(*concatenation: 2 and 3 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);
(** stamps **)
type stamp = unit ref;
val stamp: unit -> stamp = ref;
(** options **)
datatype 'a option = None | Some of 'a;
exception OPTION;
fun the (Some x) = x
| the None = raise OPTION;
(*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;
fun apsome f (Some x) = Some (f x)
| apsome _ None = None;
(* 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);
(** 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 errors*)
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;
(** lists **)
exception LIST of string;
fun null [] = true
| null (_ :: _) = false;
fun hd [] = raise LIST "hd"
| hd (x :: _) = x;
fun tl [] = raise LIST "tl"
| tl (_ :: xs) = xs;
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 *)
(*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 foldl_map _ (x, []) = (x, [])
| foldl_map f (x, y :: ys) =
let
val (x', y') = f (x, y);
val (x'', ys') = foldl_map f (x', ys);
in (x'', y' :: ys') end;
fun foldln f xs e = fst (foldl (fn ((e,i), x) => (f (x,i) e, i+1)) ((e,1),xs));
(* basic list functions *)
(*length of a list, should unquestionably be a standard function*)
local fun length1 (n, []) = n (*TAIL RECURSIVE*)
| length1 (n, x :: xs) = length1 (n + 1, xs)
in fun length l = length1 (0, l) end;
(*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 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 NL =
(case drop NL of
[] => raise LIST "nth_elem"
| x :: _ => x);
(*last element of a list*)
fun last_elem [] = raise LIST "last_elem"
| last_elem [x] = x
| last_elem (_ :: xs) = last_elem xs;
(*rear decomposition*)
fun split_last [] = raise LIST "split_last"
| split_last [x] = ([], x)
| split_last (x :: xs) = apfst (cons x) (split_last xs);
(*update nth element*)
fun nth_update x (n, xs) =
let
val prfx = take (n, xs);
val sffx = drop (n, xs);
in
(case sffx of
[] => raise LIST "nth_update"
| _ :: sffx' => prfx @ (x :: sffx'))
end;
(*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*)
fun flat (ls: 'c list list) : 'c list = foldr (op @) (ls, []);
(*like Lisp's MAPC -- seq proc [x1, ..., xn] evaluates
(proc x1; ...; proc xn) for side effects*)
fun seq (proc: 'a -> unit) : 'a list -> unit =
let fun seqf [] = ()
| seqf (x :: xs) = (proc x; seqf xs)
in seqf end;
(*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 LIST "replicate"
else rep (n, [])
end;
(*multiply [a, b, c, ...] * [xs, ys, zs, ...]*)
fun multiply ([], _) = []
| multiply (x :: xs, yss) = map (cons x) yss @ multiply (xs, yss);
(* filter *)
(*copy the list preserving elements that satisfy the predicate*)
fun filter (pred: 'a->bool) : 'a list -> 'a list =
let fun filt [] = []
| filt (x :: xs) = if pred x then x :: filt xs else filt xs
in filt end;
fun filter_out f = filter (not o f);
fun mapfilter (f: 'a -> 'b option) ([]: 'a list) = [] : 'b list
| mapfilter f (x :: xs) =
(case f x of
None => mapfilter f xs
| Some y => y :: mapfilter f xs);
(* lists of pairs *)
fun map2 _ ([], []) = []
| map2 f (x :: xs, y :: ys) = (f (x, y) :: map2 f (xs, ys))
| map2 _ _ = raise LIST "map2";
fun exists2 _ ([], []) = false
| exists2 pred (x :: xs, y :: ys) = pred (x, y) orelse exists2 pred (xs, ys)
| exists2 _ _ = raise LIST "exists2";
fun forall2 _ ([], []) = true
| forall2 pred (x :: xs, y :: ys) = pred (x, y) andalso forall2 pred (xs, ys)
| forall2 _ _ = raise LIST "forall2";
fun seq2 _ ([], []) = ()
| seq2 f (x :: xs, y :: ys) = (f (x, y); seq2 f (xs, ys))
| seq2 _ _ = raise LIST "seq2";
(*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 LIST "~~";
(*inverse of ~~; the old 'split':
[(x1, y1), ..., (xn, yn)] ===> ([x1, ..., xn], [y1, ..., yn])*)
fun split_list (l: ('a * 'b) list) = (map #1 l, map #2 l);
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));
(** integers **)
fun gcd(x,y) =
let fun gxd x y =
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;
(** 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 LIST "nth_elem_string");
fun foldl_string f (x0, str) =
let
val n = size str;
fun fold (x, i) = if i < n then fold (f (x, String.substring (str, i, 1)), i + 1) else x
in fold (x0, 0) end;
fun exists_string pred str = foldl_string (fn (b, s) => b orelse pred s) (false, str);
(*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";
(*untabify*)
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;
(*append suffix*)
fun suffix sfx s = s ^ sfx;
(*remove suffix*)
fun unsuffix sfx s =
let
val cs = explode s;
val prfx_len = size s - size sfx;
in
if prfx_len >= 0 andalso implode (drop (prfx_len, cs)) = sfx then
implode (take (prfx_len, cs))
else raise LIST "unsuffix"
end;
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 **)
(*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);
(*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) = if gen_mem eq (x, xs) then xs else 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 = foldl (gen_rem eq);
(*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;
(** 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;
(** generic tables **)
(*Tables are supposed to be 'efficient' encodings of lists of elements distinct
wrt. an equality "eq". The extend and merge operations below are optimized
for long-term space efficiency.*)
(*append (new) elements to a table*)
fun generic_extend _ _ _ tab [] = tab
| generic_extend eq dest_tab mk_tab tab1 lst2 =
let
val lst1 = dest_tab tab1;
val new_lst2 = gen_rems eq (lst2, lst1);
in
if null new_lst2 then tab1
else mk_tab (lst1 @ new_lst2)
end;
(*append (new) elements of 2nd table to 1st table*)
fun generic_merge eq dest_tab mk_tab tab1 tab2 =
let
val lst1 = dest_tab tab1;
val lst2 = dest_tab tab2;
val new_lst2 = gen_rems eq (lst2, lst1);
in
if null new_lst2 then tab1
else if gen_subset eq (lst1, lst2) then tab2
else mk_tab (lst1 @ new_lst2)
end;
(*lists as tables*)
fun extend_list tab = generic_extend (op =) I I tab;
fun merge_lists tab = generic_merge (op =) I I tab;
fun merge_alists tab = generic_merge eq_fst I I tab;
fun merge_rev_lists xs [] = xs
| merge_rev_lists [] ys = ys
| merge_rev_lists xs (y :: ys) =
(if y mem xs then I else cons y) (merge_rev_lists xs ys);
(** 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 k = length xs div 2
in f (fold_bal f (take(k, xs)),
fold_bal f (drop(k, xs)))
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 **)
datatype order = LESS | EQUAL | GREATER;
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;
fun int_ord (i, j: int) =
if i < j then LESS
else if i = j then EQUAL
else GREATER;
fun string_ord (a, b: string) =
if a < b then LESS
else if a = b then EQUAL
else 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 _ ([], []) = EQUAL
| dict_ord _ ([], _ :: _) = LESS
| dict_ord _ (_ :: _, []) = GREATER
| dict_ord elem_ord (x :: xs, y :: ys) =
(case elem_ord (x, y) of EQUAL => dict_ord elem_ord (xs, ys) | ord => ord);
(*lexicographic product of lists*)
fun list_ord elem_ord (xs, ys) =
prod_ord int_ord (dict_ord elem_ord) ((length xs, xs), (length ys, ys));
(* sorting *)
(*quicksort (stable, i.e. does not reorder equal elements)*)
fun sort ord =
let
fun qsort xs =
let val len = length xs in
if len <= 1 then xs
else
let val (lts, eqs, gts) = part (nth_elem (len div 2, xs)) xs in
qsort lts @ eqs @ qsort gts
end
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);
(** input / output and diagnostics **)
val cd = OS.FileSys.chDir;
val pwd = OS.FileSys.getDir;
fun std_output s =
(TextIO.output (TextIO.stdOut, s); TextIO.flushOut TextIO.stdOut);
fun prefix_lines prfx txt =
txt |> split_lines |> map (fn s => prfx ^ s) |> cat_lines;
(*hooks for output channels: normal, warning, error*)
val writeln_fn = ref (std_output o suffix "\n");
val priority_fn = ref (fn s => ! writeln_fn s);
val warning_fn = ref (std_output o suffix "\n" o prefix_lines "### ");
val error_fn = ref (std_output o suffix "\n" o prefix_lines "*** ");
fun writeln s = ! writeln_fn s;
fun priority s = ! priority_fn s;
fun warning s = ! warning_fn s;
(*print error message and abort to top level*)
fun error_msg s = ! error_fn s;
fun error s = (error_msg s; raise ERROR);
fun sys_error msg = error ("## SYSTEM ERROR ##\n" ^ msg);
fun assert p msg = if p then () else error msg;
fun deny p msg = if p then error msg else ();
(*Assert pred for every member of l, generating a message if pred fails*)
fun assert_all pred l msg_fn =
let fun asl [] = ()
| asl (x::xs) = if pred x then asl xs else error (msg_fn x)
in asl l end;
(* handle errors capturing messages *)
datatype 'a error =
Error of string |
OK of 'a;
fun get_error (Error msg) = Some msg
| get_error _ = None;
fun get_ok (OK x) = Some x
| get_ok _ = None;
datatype 'a result =
Result of 'a |
Exn of exn;
fun handle_error f x =
let
val buffer = ref ([]: string list);
fun capture s = buffer := ! buffer @ [s];
fun err_msg () = if not (null (! buffer)) then error_msg (cat_lines (! buffer)) else ();
in
(case Result (setmp error_fn capture f x) handle exn => Exn exn of
Result y => (err_msg (); OK y)
| Exn ERROR => Error (cat_lines (! buffer))
| Exn exn => (err_msg (); raise exn))
end;
(* transform ERROR into ERROR_MESSAGE *)
exception ERROR_MESSAGE of string;
fun transform_error f x =
(case handle_error f x of
OK y => y
| Error msg => raise ERROR_MESSAGE msg);
(* transform any exception, including ERROR *)
fun transform_failure exn f x =
transform_error f x handle e => raise exn e;
(** timing **)
(*a conditional timing function: applies f to () and, if the flag is true,
prints its runtime on warning channel*)
fun cond_timeit flag f =
if flag then
let val start = startTiming()
val result = f ()
in warning (endTiming start); result end
else f ();
(*unconditional timing function*)
fun timeit x = cond_timeit true x;
(*timed application function*)
fun timeap f x = timeit (fn () => f x);
(*global timing mode*)
val timing = ref false;
(** rational numbers **)
datatype rat = Rat of bool * int * int
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 ratadd(Rat(a,p,q),Rat(b,r,s)) =
let val den = lcm(q,s)
val p = p*(den div q) and r = r*(den div 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 error("ratinv") else Rat(a,q,p)
fun int_ratdiv(p,q) =
if q=0 then error("int_ratdiv") else ratnorm(0<=q, p, abs q)
fun ratneg(Rat(b,p,q)) = Rat(not b,p,q);
fun rat_of_int i = if i < 0 then Rat(false,abs i,1) else Rat(true,i,1);
(** misc **)
fun overwrite_warn (args as (alist, (a, _))) msg =
(if is_none (assoc (alist, a)) then () else warning msg;
overwrite args);
(*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.*)
fun partition (pred: 'a->bool) (ys: 'a list) : ('a list * 'a list) =
let fun part ([], answer) = answer
| part (x::xs, (ys, ns)) = if pred(x)
then part (xs, (x::ys, ns))
else part (xs, (ys, x::ns))
in part (rev ys, ([], [])) end;
fun partition_eq (eq:'a * 'a -> bool) =
let fun part [] = []
| part (x::ys) = let val (xs, xs') = partition (apl(x, eq)) 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 LIST "partition_list")
else
let val (ns, rest) = partition (p k) xs;
in ns :: part(k+1)rest end
in part i end;
(* transitive closure (not Warshall's algorithm) *)
fun transitive_closure [] = []
| transitive_closure ((x, ys)::ps) =
let val qs = transitive_closure ps
val zs = foldl (fn (zs, y) => assocs qs y union_string zs) (ys, ys)
fun step(u, us) = (u, if x mem_string us then zs union_string us
else us)
in (x, zs) :: map step qs 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 init_gensym() = (seedr := 0); (* FIXME !!??! *)
fun gensym pre = pre ^ (#1(newid (!seedr), inc seedr));
end;
local
(*Identifies those character codes legal in identifiers.
chould use Basis Library character functions if Poly/ML provided characters*)
fun idCode k = (ord "a" <= k andalso k < ord "z") orelse
(ord "A" <= k andalso k < ord "Z") orelse
(ord "0" <= k andalso k < ord "9");
val idCodeVec = Vector.tabulate (256, idCode);
in
(*Increment a list of letters like a reversed base 26 number.
If head is "z", bumps chars in tail.
Digits are incremented as if they were integers.
"_" and "'" are not changed.
For making variants of identifiers.*)
fun bump_int_list(c::cs) =
if c="9" then "0" :: bump_int_list cs
else
if "0" <= c andalso c < "9" then chr(ord(c)+1) :: cs
else "1" :: c :: cs
| bump_int_list([]) = error("bump_int_list: not an identifier");
fun bump_list([], d) = [d]
| bump_list(["'"], d) = [d, "'"]
| bump_list("z"::cs, _) = "a" :: bump_list(cs, "a")
| bump_list("Z"::cs, _) = "A" :: bump_list(cs, "A")
| bump_list("9"::cs, _) = "0" :: bump_int_list cs
| bump_list(c::cs, _) =
let val k = ord(c)
in if Vector.sub(idCodeVec,k) then chr(k+1) :: cs
else
if c="'" orelse c="_" then c :: bump_list(cs, "")
else error("bump_list: not legal in identifier: " ^
implode(rev(c::cs)))
end;
end;
fun bump_string s : string = implode (rev (bump_list(rev(explode s), "")));
(* 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;
(* Variable-branching trees: for proof terms etc. *)
datatype 'a mtree = Join of 'a * 'a mtree list;
end;
open Library;