(* Title: Pure/library.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
Basic library: booleans, lists, pairs, input/output, etc.
*)
(**** Booleans: 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) ;
fun orl [] = false
| orl (x::l) = x orelse orl l;
fun andl [] = true
| andl (x::l) = x andalso andl l;
(*exists pred [x1,...,xn] ======> pred(x1) orelse ... orelse pred(xn)*)
fun exists (pred: 'a -> bool) : 'a list -> bool =
let fun boolf [] = false
| boolf (x::l) = (pred x) orelse boolf l
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::l) = (pred x) andalso (boolf l)
in boolf end;
(** curried equality **)
fun equal x y = (x = y);
fun not_equal x y = x <> y;
(*** Lists ***)
exception LIST of string;
(*discriminator and selectors for lists. *)
fun null [] = true
| null (_::_) = false;
fun hd [] = raise LIST "hd"
| hd (a::_) = a;
fun tl [] = raise LIST "tl"
| tl (_::l) = l;
(*curried cons and reverse cons*)
fun cons x xs = x :: xs;
fun rcons xs x = x :: xs;
(*curried functions for pairing and reversed pairing*)
fun pair x y = (x,y);
fun rpair x y = (y,x);
fun fst(x,y) = x and snd(x,y) = y;
(*Handy combinators*)
fun curry f x y = f(x,y);
fun uncurry f(x,y) = f x y;
fun I x = x and K x y = 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);
(*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);
fun square (n: int) = n*n;
fun fact 0 = 1
| fact n = n * fact(n-1);
(*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;
(*Length of a list. Should unquestionably be a standard function*)
local fun length1 (n, [ ]) = n (*TAIL RECURSIVE*)
| length1 (n, x::l) = length1 (n+1, l)
in fun length l = length1 (0,l) end;
(*Take the first n elements from l.*)
fun take (n, []) = []
| take (n, x::xs) = if n>0 then x::take(n-1,xs)
else [];
(*Drop the first n elements from l.*)
fun drop (_, []) = []
| drop (n, x::xs) = if n>0 then drop (n-1, xs)
else x::xs;
(*Return nth element of l, 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::l => x;
(*Last element of a list*)
fun last_elem [] = raise LIST "last_elem"
| last_elem [x] = x
| last_elem (_ :: xs) = last_elem xs;
(*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;
(*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::l) = (proc x; seqf l)
in seqf end;
(*** Balanced folding; access to 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 ();
(*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""));
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;
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;
(*** 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 ^ " ****")));
(*** Misc functions ***)
(*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;
(*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);
(*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;
(*Flatten a list of lists to a list.*)
fun flat (ls: 'c list list) : 'c list = foldr (op @) (ls,[]);
(*** polymorphic set operations ***)
(*membership in a list*)
infix mem;
fun x mem [] = false
| x mem (y::l) = (x=y) orelse (x mem l);
(*insertion into list if not already there*)
infix ins;
fun x ins xs = if x mem 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);
infix inter;
fun [] inter ys = []
| (x::xs) inter ys = if x mem ys then x::(xs inter ys)
else xs inter ys;
infix subset;
fun [] subset ys = true
| (x::xs) subset ys = x mem ys andalso xs subset ys;
(*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 \);
(*** option stuff ***)
datatype 'a option = None | Some of 'a;
exception OPTION of string;
fun the (Some x) = x
| the None = raise OPTION "the";
fun is_some (Some _) = true
| is_some None = false;
fun is_none (Some _) = false
| is_none None = true;
(*** 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;
(*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;
(*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);
(*** List operations, generalized to an arbitrary equality function "eq"
-- so what good are equality types?? ***)
(*removing an element from a list -- possibly WITH duplicates*)
fun gen_rem eq (xs,y) = filter_out (fn x => eq(x,y)) xs;
(*generalized membership test*)
fun gen_mem eq (x, []) = false
| gen_mem eq (x, y::ys) = eq(x,y) orelse gen_mem eq (x,ys);
(*generalized insertion*)
fun gen_ins eq (x,xs) = if gen_mem eq (x,xs) then xs else x::xs;
(*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));
(*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);
(** Finding list elements and duplicates **)
(* 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;
(*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;
fun distinct1 (seen, []) = rev seen
| distinct1 (seen, x::xs) =
if x mem seen then distinct1 (seen, xs)
else distinct1 (x::seen, xs);
(*Makes a list of the distinct members of the input*)
fun distinct xs = distinct1([],xs);
(*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;
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;
(*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;
(*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;
(*** Converting integers to strings, generating identifiers, etc. ***)
(*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);
val print_int = prs o string_of_int;
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),"")));
(*** 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";
(*** Lexical scanning ***)
(* [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));
infix prefix;
fun [] prefix _ = true
| (x::xs) prefix (y::ys) = (x=y) andalso (xs prefix ys)
| _ prefix _ = false;
(* [x1, x2, ..., xn] ---> [x1, s, x2, s, ..., s, xn] *)
fun separate s (x :: (xs as _ :: _)) = x :: s :: separate s xs
| separate _ xs = xs;
(*space_implode "..." (explode "hello"); gives "h...e...l...l...o" *)
fun space_implode a bs = implode (separate a bs);
(*simple quoting (does not escape special chars) *)
fun quote s = "\"" ^ s ^ "\"";
(*Concatenate messages, one per line, into a string*)
val cat_lines = implode o (map (apr(op^,"\n")));
(*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;
(*** Operations on 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;