author | wenzelm |
Sat, 29 Mar 2008 19:14:03 +0100 | |
changeset 26481 | 92e901171cc8 |
parent 24630 | 351a308ab58d |
child 30161 | c26e515f1c29 |
permissions | -rw-r--r-- |
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(* Title: Tools/rat.ML |
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ID: $Id$ |
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Author: Tobias Nipkow, Florian Haftmann, TU Muenchen |
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Canonical implementation of exact rational numbers. |
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*) |
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signature RAT = |
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sig |
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eqtype rat |
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exception DIVZERO |
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val zero: rat |
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val one: rat |
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val two: rat |
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val rat_of_int: int -> rat |
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val rat_of_quotient: int * int -> rat |
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val quotient_of_rat: rat -> int * int |
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val string_of_rat: rat -> string |
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val eq: rat * rat -> bool |
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val ord: rat * rat -> order |
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val le: rat -> rat -> bool |
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val lt: rat -> rat -> bool |
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val sign: rat -> order |
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val abs: rat -> rat |
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val add: rat -> rat -> rat |
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val mult: rat -> rat -> rat |
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val neg: rat -> rat |
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val inv: rat -> rat |
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val rounddown: rat -> rat |
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val roundup: rat -> rat |
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end; |
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structure Rat : RAT = |
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struct |
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fun common (p1, q1) (p2, q2) = |
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let |
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val m = Integer.lcm q1 q2; |
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in ((p1 * (m div q1), p2 * (m div q2)), m) end; |
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datatype rat = Rat of int * int; (*nominator, denominator (positive!)*) |
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exception DIVZERO; |
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fun rat_of_quotient (p, q) = |
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let |
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val m = Integer.gcd (Int.abs p) q |
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simplified type int (eliminated IntInf.int, integer);
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parents:
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in Rat (p div m, q div m) end handle Div => raise DIVZERO; |
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fun quotient_of_rat (Rat r) = r; |
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fun rat_of_int i = Rat (i, 1); |
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val zero = rat_of_int 0; |
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val one = rat_of_int 1; |
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val two = rat_of_int 2; |
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fun string_of_rat (Rat (p, q)) = |
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string_of_int p ^ "/" ^ string_of_int q; |
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fun eq (Rat (p1, q1), Rat (p2, q2)) = (p1 = p2 andalso q1 = q2); |
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fun ord (Rat (p1, q1), Rat (p2, q2)) = |
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case (Integer.sign p1, Integer.sign p2) |
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of (LESS, EQUAL) => LESS |
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| (LESS, GREATER) => LESS |
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| (EQUAL, LESS) => GREATER |
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| (EQUAL, EQUAL) => EQUAL |
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| (EQUAL, GREATER) => LESS |
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| (GREATER, LESS) => GREATER |
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| (GREATER, EQUAL) => GREATER |
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| _ => int_ord (fst (common (p1, q1) (p2, q2))); |
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fun le a b = not (ord (a, b) = GREATER); |
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fun lt a b = (ord (a, b) = LESS); |
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fun sign (Rat (p, _)) = Integer.sign p; |
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fun abs (Rat (p, q)) = Rat (Int.abs p, q); |
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fun add (Rat (p1, q1)) (Rat (p2, q2)) = |
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let |
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val ((m1, m2), n) = common (p1, q1) (p2, q2); |
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in rat_of_quotient (m1 + m2, n) end; |
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fun mult (Rat (p1, q1)) (Rat (p2, q2)) = |
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simplified type int (eliminated IntInf.int, integer);
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rat_of_quotient (p1 * p2, q1 * q2); |
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simplified type int (eliminated IntInf.int, integer);
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parents:
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fun neg (Rat (p, q)) = Rat (~ p, q); |
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simplified type int (eliminated IntInf.int, integer);
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fun inv (Rat (p, q)) = |
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case Integer.sign p |
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of LESS => Rat (~ q, ~ p) |
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| EQUAL => raise DIVZERO |
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| GREATER => Rat (q, p); |
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simplified type int (eliminated IntInf.int, integer);
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parents:
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fun rounddown (Rat (p, q)) = Rat (p div q, 1); |
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simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
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fun roundup (Rat (p, q)) = |
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
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case Integer.div_mod p q |
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of (m, 0) => Rat (m, 1) |
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| (m, _) => Rat (m + 1, 1); |
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end; |
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infix 7 */ //; |
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infix 6 +/ -/; |
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infix 4 =/ </ <=/ >/ >=/ <>/; |
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fun a +/ b = Rat.add a b; |
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fun a -/ b = a +/ Rat.neg b; |
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fun a */ b = Rat.mult a b; |
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fun a // b = a */ Rat.inv b; |
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fun a =/ b = Rat.eq (a, b); |
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fun a </ b = Rat.lt a b; |
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fun a <=/ b = Rat.le a b; |
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fun a >/ b = b </ a; |
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fun a >=/ b = b <=/ a; |
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fun a <>/ b = not (a =/ b); |