haftmann@24584: (*  Title:      Tools/rat.ML
haftmann@23520:     Author:     Tobias Nipkow, Florian Haftmann, TU Muenchen
haftmann@23251: 
haftmann@23251: Canonical implementation of exact rational numbers.
haftmann@23251: *)
haftmann@23251: 
haftmann@23251: signature RAT =
haftmann@23251: sig
wenzelm@23297:   eqtype rat
haftmann@23251:   exception DIVZERO
haftmann@23251:   val zero: rat
haftmann@23251:   val one: rat
haftmann@23251:   val two: rat
wenzelm@24630:   val rat_of_int: int -> rat
wenzelm@24630:   val rat_of_quotient: int * int -> rat
wenzelm@24630:   val quotient_of_rat: rat -> int * int
haftmann@23251:   val string_of_rat: rat -> string
haftmann@23251:   val eq: rat * rat -> bool
haftmann@23520:   val ord: rat * rat -> order
haftmann@23251:   val le: rat -> rat -> bool
haftmann@23251:   val lt: rat -> rat -> bool
haftmann@23520:   val sign: rat -> order
haftmann@23520:   val abs: rat -> rat
haftmann@23251:   val add: rat -> rat -> rat
haftmann@23251:   val mult: rat -> rat -> rat
haftmann@23251:   val neg: rat -> rat
haftmann@23251:   val inv: rat -> rat
haftmann@23520:   val rounddown: rat -> rat
haftmann@23251:   val roundup: rat -> rat
haftmann@23251: end;
haftmann@23251: 
haftmann@23251: structure Rat : RAT =
haftmann@23251: struct
haftmann@23251: 
haftmann@23520: fun common (p1, q1) (p2, q2) =
haftmann@23520:   let
haftmann@23520:     val m = Integer.lcm q1 q2;
wenzelm@24630:   in ((p1 * (m div q1), p2 * (m div q2)), m) end;
haftmann@23520: 
wenzelm@24630: datatype rat = Rat of int * int;  (*nominator, denominator (positive!)*)
haftmann@23251: 
haftmann@23251: exception DIVZERO;
haftmann@23251: 
haftmann@23251: fun rat_of_quotient (p, q) =
haftmann@23261:   let
wenzelm@24630:     val m = Integer.gcd (Int.abs p) q
wenzelm@24630:   in Rat (p div m, q div m) end handle Div => raise DIVZERO;
haftmann@23520: 
haftmann@23520: fun quotient_of_rat (Rat r) = r;
haftmann@23520: 
haftmann@23520: fun rat_of_int i = Rat (i, 1);
haftmann@23251: 
haftmann@23520: val zero = rat_of_int 0;
haftmann@23520: val one = rat_of_int 1;
haftmann@23520: val two = rat_of_int 2;
haftmann@23251: 
haftmann@23520: fun string_of_rat (Rat (p, q)) =
wenzelm@24630:   string_of_int p ^ "/" ^ string_of_int q;
haftmann@23520: 
haftmann@23520: fun eq (Rat (p1, q1), Rat (p2, q2)) = (p1 = p2 andalso q1 = q2);
haftmann@23251: 
wenzelm@24630: fun ord (Rat (p1, q1), Rat (p2, q2)) =
wenzelm@24630:  case (Integer.sign p1, Integer.sign p2)
haftmann@23520:  of (LESS, EQUAL) => LESS
haftmann@23520:   | (LESS, GREATER) => LESS
haftmann@23520:   | (EQUAL, LESS) => GREATER
haftmann@23520:   | (EQUAL, EQUAL) => EQUAL
haftmann@23520:   | (EQUAL, GREATER) => LESS
haftmann@23520:   | (GREATER, LESS) => GREATER
haftmann@23520:   | (GREATER, EQUAL) => GREATER
wenzelm@24630:   | _ => int_ord (fst (common (p1, q1) (p2, q2)));
haftmann@23251: 
haftmann@23520: fun le a b = not (ord (a, b) = GREATER);
haftmann@23520: fun lt a b = (ord (a, b) = LESS);
haftmann@23251: 
haftmann@23520: fun sign (Rat (p, _)) = Integer.sign p;
haftmann@23251: 
wenzelm@24630: fun abs (Rat (p, q)) = Rat (Int.abs p, q);
haftmann@23251: 
haftmann@23520: fun add (Rat (p1, q1)) (Rat (p2, q2)) =
haftmann@23520:   let
haftmann@23520:     val ((m1, m2), n) = common (p1, q1) (p2, q2);
wenzelm@24630:   in rat_of_quotient (m1 + m2, n) end;
haftmann@23251: 
haftmann@23520: fun mult (Rat (p1, q1)) (Rat (p2, q2)) =
wenzelm@24630:   rat_of_quotient (p1 * p2, q1 * q2);
wenzelm@24630: 
wenzelm@24630: fun neg (Rat (p, q)) = Rat (~ p, q);
haftmann@23251: 
wenzelm@24630: fun inv (Rat (p, q)) =
wenzelm@24630:  case Integer.sign p
wenzelm@24630:  of LESS => Rat (~ q, ~ p)
haftmann@24522:   | EQUAL => raise DIVZERO
haftmann@24521:   | GREATER => Rat (q, p);
haftmann@23251: 
wenzelm@24630: fun rounddown (Rat (p, q)) = Rat (p div q, 1);
haftmann@23520: 
wenzelm@24630: fun roundup (Rat (p, q)) =
wenzelm@24630:  case Integer.div_mod p q
haftmann@23520:  of (m, 0) => Rat (m, 1)
haftmann@23520:   | (m, _) => Rat (m + 1, 1);
haftmann@23251: 
haftmann@23251: end;
haftmann@23251: 
haftmann@23261: infix 7 */ //;
wenzelm@23297: infix 6 +/ -/;
haftmann@23251: infix 4 =/ </ <=/ >/ >=/ <>/;
haftmann@23251: 
haftmann@23251: fun a +/ b = Rat.add a b;
haftmann@23251: fun a -/ b = a +/ Rat.neg b;
haftmann@23251: fun a */ b = Rat.mult a b;
wenzelm@23297: fun a // b = a */ Rat.inv b;
haftmann@23261: fun a =/ b = Rat.eq (a, b);
haftmann@23251: fun a </ b = Rat.lt a b;
haftmann@23251: fun a <=/ b = Rat.le a b;
haftmann@23251: fun a >/ b = b </ a;
haftmann@23251: fun a >=/ b = b <=/ a;
haftmann@23251: fun a <>/ b = not (a =/ b);