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src/Pure/library.ML

author | lcp |

Tue, 21 Jun 1994 17:20:34 +0200 | |

changeset 435 | ca5356bd315a |

parent 410 | c8171ee32744 |

child 512 | 55755ed9fab9 |

permissions | -rw-r--r-- |

Addition of cardinals and order types, various tidying

(* Title: Pure/library.ML ID: $Id$ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1992 University of Cambridge Basic library: functions, options, pairs, booleans, lists, integers, strings, lists as sets, association lists, generic tables, balanced trees, input / output, timing, filenames, misc functions. *) (** 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*) infix |>; fun (x |> f) = f x; (*combine two functions forming the union of their domains*) infix orelf; fun f orelf g = fn x => f x handle Match => g x; (*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); (*functional for pairs*) fun pairself f (x, y) = (f x, f 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; (** options **) datatype 'a option = None | Some of 'a; exception OPTION of string; fun the (Some x) = x | the None = raise OPTION "the"; 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; (** 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 the function to a component of a pair*) fun apfst f (x, y) = (f x, y); fun apsnd f (x, y) = (x, f y); (** booleans **) (* equality *) fun equal x y = x = y; fun not_equal x y = x <> y; (* operators for combining predicates *) infix orf; fun p orf q = fn x => p x orelse q x; infix andf; fun p andf q = fn x => p x andalso q x; fun notf p x = not (p x); (* predicates on lists *) fun orl [] = false | orl (x :: xs) = x orelse orl xs; fun andl [] = true | andl (x :: xs) = x andalso andl xs; (*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); (** 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; (* 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 (* FIXME [] case: elim warn (?) *) | itr (x::l) = f(x, itr l) in itr l end; (* 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; (*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; (*find the position of an element in a list*) fun find (x, ys) = let fun f (y :: ys, i) = if x = y then i else f (ys, i + 1) | f (_, _) = raise LIST "find" in f (ys, 0) end; (*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; (* 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); fun find_first _ [] = None | find_first pred (x :: xs) = if pred x then Some x else find_first pred 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"; (*combine two lists forming a list of pairs: [x1, ..., xn] ~~ [y1, ..., yn] ===> [(x1, y1), ..., (xn, yn)]*) infix ~~; 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); (* prefixes, suffixes *) infix prefix; 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 inc i = i := ! i + 1; fun dec i = i := ! i - 1; (* lists of integers *) (*make the list [from, from + 1, ..., to]*) infix upto; fun from upto to = if from > to then [] else from :: ((from + 1) upto to); (*make the list [from, from - 1, ..., to]*) infix downto; 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; (* operations on integer lists *) fun sum [] = 0 | sum (n :: ns) = n + sum ns; fun max [m:int] = m | max (m :: n :: ns) = if m > n then max (m :: ns) else max (n :: ns) | max [] = raise LIST "max"; fun min [m:int] = m | min (m :: n :: ns) = if m < n then min (m :: ns) else min (n :: ns) | min [] = raise LIST "min"; (* 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; fun string_of_int n = if n < 0 then "~" ^ radixstring (10, "0", ~n) else radixstring (10, "0", n); (** strings **) fun is_letter ch = ord "A" <= ord ch andalso ord ch <= ord "Z" orelse ord "a" <= ord ch andalso ord ch <= ord "z"; fun is_digit ch = ord "0" <= ord ch andalso ord ch <= ord "9"; (*letter or _ or prime (')*) fun is_quasi_letter "_" = true | is_quasi_letter "'" = true | is_quasi_letter ch = is_letter ch; (*white space: blanks, tabs, newlines*) val is_blank : string -> bool = fn " " => true | "\t" => true | "\n" => true | _ => false; val is_letdig = is_quasi_letter orf is_digit; (*lower all chars of string*) val to_lower = let fun lower ch = if ch >= "A" andalso ch <= "Z" then chr (ord ch - ord "A" + ord "a") else ch; in implode o (map lower) o explode end; (*parentesize*) fun parents lpar rpar str = lpar ^ str ^ rpar; (*simple quoting (does not escape special chars)*) val quote = parents "\"" "\""; (*space_implode "..." (explode "hello"); gives "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"; (** lists as sets **) (*membership in a list*) infix mem; fun x mem [] = false | x mem (y :: ys) = x = y orelse x mem 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*) infix ins; fun x ins xs = if x mem 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*) infix union; fun xs union [] = xs | [] union ys = ys | (x :: xs) union ys = xs union (x ins 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*) infix inter; fun [] inter ys = [] | (x :: xs) inter ys = if x mem ys then x :: (xs inter ys) else xs inter ys; (*subset*) infix subset; fun [] subset ys = true | (x :: xs) subset ys = x mem ys andalso xs subset ys; fun gen_subset eq (xs, ys) = forall (fn x => gen_mem eq (x, ys)) xs; (*eq_set*) fun eq_set (xs, ys) = xs = ys orelse (xs subset ys andalso ys subset xs); (*removing an element from a list WITHOUT duplicates*) infix \; fun (y :: ys) \ x = if x = y then ys else y :: (ys \ x) | [] \ x = []; infix \\; val op \\ = foldl (op \); (*removing an element from a list -- possibly WITH duplicates*) fun gen_rem eq (xs, y) = filter_out (fn x => eq (x, y)) xs; val gen_rems = foldl o gen_rem; (*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; val distinct = gen_distinct (op =); (*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; val duplicates = gen_duplicates (op =); (** association lists **) (*association list lookup*) fun assoc ([], key) = None | assoc ((keyi, xi) :: pairs, key) = if key = keyi then Some xi else assoc (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; (** 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*) val extend_list = generic_extend (op =) I I; val merge_lists = generic_merge (op =) I I; 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; (** input / output **) fun prs s = output (std_out, s); fun writeln s = prs (s ^ "\n"); (*print error message and abort to top level*) exception ERROR; fun error msg = (writeln msg; raise ERROR); fun sys_error msg = (writeln "*** SYSTEM ERROR ***"; error msg); fun assert p msg = if p then () else error msg; fun deny p msg = if p then error msg else (); (* FIXME close file (?) *) (*for the "test" target in Makefiles -- signifies successful termination*) fun maketest msg = (writeln msg; output (open_out "test", "Test examples ran successfully\n")); (*print a list surrounded by the brackets lpar and rpar, with comma separator print nothing for empty list*) fun print_list (lpar, rpar, pre: 'a -> unit) (l : 'a list) = let fun prec x = (prs ","; pre x) in (case l of [] => () | x::l => (prs lpar; pre x; seq prec l; prs rpar)) end; (*print a list of items separated by newlines*) fun print_list_ln (pre: 'a -> unit) : 'a list -> unit = seq (fn x => (pre x; writeln "")); val print_int = prs o string_of_int; (** timing **) (*unconditional timing function*) val timeit = cond_timeit true; (*timed application function*) fun timeap f x = timeit (fn () => f x); (*timed "use" function, printing filenames*) fun time_use fname = timeit (fn () => (writeln ("\n**** Starting " ^ fname ^ " ****"); use fname; writeln ("\n**** Finished " ^ fname ^ " ****"))); (** filenames **) (*convert UNIX filename of the form "path/file" to "path/" and "file"; if filename contains no slash, then it returns "" and "file"*) val split_filename = (pairself implode) o take_suffix (not_equal "/") o explode; val base_name = #2 o split_filename; (*merge splitted filename (path and file); if path does not end with one a slash is appended*) fun tack_on "" name = name | tack_on path name = if last_elem (explode path) = "/" then path ^ name else path ^ "/" ^ name; (*remove the extension of a filename, i.e. the part after the last '.'*) val remove_ext = implode o #1 o take_suffix (not_equal ".") o explode; (** misc functions **) (*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; (* sorting *) (*insertion sort; stable (does not reorder equal elements) 'less' is less-than test on type 'a*) fun sort (less: 'a*'a -> bool) = let fun insert (x, []) = [x] | insert (x, y::ys) = if less(y, x) then y :: insert (x, ys) else x::y::ys; fun sort1 [] = [] | sort1 (x::xs) = insert (x, sort1 xs) in sort1 end; (*sort strings*) val sort_strings = sort (op <= : string * string -> bool); (* 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 zs) (ys, ys) fun step(u, us) = (u, if x mem us then zs union us else us) in (x, zs) :: map step qs end; (* generating identifiers *) local val a = ord "a" and z = ord "z" and A = ord "A" and Z = ord "Z" and k0 = ord "0" and k9 = ord "9" 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 k0 <= ord(c) andalso ord(c) < k9 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 (a <= k andalso k < z) orelse (A <= k andalso k < Z) orelse (k0 <= k andalso k < k9) 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;