src/Tools/rat.ML
 author haftmann Fri, 29 Jun 2007 21:23:05 +0200 changeset 23520 483fe92f00c1 parent 23297 06f108974fa1 child 24521 9565ac68c3cd permissions -rw-r--r--
tuned arithmetic modules
```
(*  Title:      Pure/General/rat.ML
ID:         \$Id\$
Author:     Tobias Nipkow, Florian Haftmann, TU Muenchen

Canonical implementation of exact rational numbers.
*)

signature RAT =
sig
eqtype rat
exception DIVZERO
val zero: rat
val one: rat
val two: rat
val rat_of_int: integer -> rat
val rat_of_quotient: integer * integer -> rat
val quotient_of_rat: rat -> integer * integer
val string_of_rat: rat -> string
val eq: rat * rat -> bool
val ord: rat * rat -> order
val le: rat -> rat -> bool
val lt: rat -> rat -> bool
val signabs: rat -> order * rat
val sign: rat -> order
val abs: rat -> rat
val add: rat -> rat -> rat
val mult: rat -> rat -> rat
val neg: rat -> rat
val inv: rat -> rat
val rounddown: rat -> rat
val roundup: rat -> rat
end;

structure Rat : RAT =
struct

fun common (p1, q1) (p2, q2) =
let
val m = Integer.lcm q1 q2;
in ((Integer.mult p1 (Integer.div m q1), Integer.mult p2 (Integer.div m q2)), m) end;

datatype rat = Rat of integer * integer; (* nominator, denominator (positive!) *)

exception DIVZERO;

fun rat_of_quotient (p, q) =
let
val m = Integer.gcd (Integer.abs p) q
in Rat (Integer.div p m, Integer.div q m) end handle Div => raise DIVZERO;

fun quotient_of_rat (Rat r) = r;

fun rat_of_int i = Rat (i, 1);

val zero = rat_of_int 0;
val one = rat_of_int 1;
val two = rat_of_int 2;

fun string_of_rat (Rat (p, q)) =
Integer.string_of_int p ^ "/" ^ Integer.string_of_int q;

fun eq (Rat (p1, q1), Rat (p2, q2)) = (p1 = p2 andalso q1 = q2);

fun ord (Rat (p1, q1), Rat (p2, q2)) = case (Integer.sign p1, Integer.sign p2)
of (LESS, EQUAL) => LESS
| (LESS, GREATER) => LESS
| (EQUAL, LESS) => GREATER
| (EQUAL, EQUAL) => EQUAL
| (EQUAL, GREATER) => LESS
| (GREATER, LESS) => GREATER
| (GREATER, EQUAL) => GREATER
| _ => Integer.ord (fst (common (p1, q1) (p2, q2)));

fun le a b = not (ord (a, b) = GREATER);
fun lt a b = (ord (a, b) = LESS);

fun sign (Rat (p, _)) = Integer.sign p;

fun abs (Rat (p, q)) = Rat (Integer.abs p, q);

fun signabs (Rat (p, q)) =
let
val (s, p') = Integer.signabs p;
in (s, Rat (p', q)) end;

fun add (Rat (p1, q1)) (Rat (p2, q2)) =
let
val ((m1, m2), n) = common (p1, q1) (p2, q2);
in rat_of_quotient (Integer.add m1 m2, n) end;

fun mult (Rat (p1, q1)) (Rat (p2, q2)) =
rat_of_quotient (Integer.mult p1 p2, Integer.mult q1 q2);

fun neg (Rat (p, q)) = Rat (Integer.neg p, q);

fun inv (Rat (p, 0)) = raise DIVZERO
| inv (Rat (p, q)) = Rat (q, p);

fun rounddown (Rat (p, q)) = Rat (Integer.div p q, 1);

fun roundup (Rat (p, q)) = case Integer.divmod p q
of (m, 0) => Rat (m, 1)
| (m, _) => Rat (m + 1, 1);

end;

infix 7 */ //;
infix 6 +/ -/;
infix 4 =/ </ <=/ >/ >=/ <>/;

fun a +/ b = Rat.add a b;
fun a -/ b = a +/ Rat.neg b;
fun a */ b = Rat.mult a b;
fun a // b = a */ Rat.inv b;
fun a =/ b = Rat.eq (a, b);
fun a </ b = Rat.lt a b;
fun a <=/ b = Rat.le a b;
fun a >/ b = b </ a;
fun a >=/ b = b <=/ a;
fun a <>/ b = not (a =/ b);
```