src/Pure/General/rat.ML
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(*  Title:      Pure/General/rat.ML
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    ID:         $Id$
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    Author:     Tobias Nipkow, 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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  type 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: Intt.int -> rat
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  val rat_of_quotient: Intt.int * Intt.int -> rat
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  val quotient_of_rat: rat -> Intt.int * Intt.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 cmp: 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 cmp_zero: rat -> order
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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 roundup: rat -> rat
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  val rounddown: rat -> rat
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end;
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structure Rat : RAT =
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struct
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datatype rat = Rat of bool * Intt.int * Intt.int;
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exception DIVZERO;
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val zero = Rat (true, Intt.int 0, Intt.int 1);
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val one = Rat (true, Intt.int 1, Intt.int 1);
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val two = Rat (true, Intt.int 2, Intt.int 1);
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fun rat_of_int i =
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  if i < Intt.int 0
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  then Rat (false, ~i, Intt.int 1)
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  else Rat (true, i, Intt.int 1);
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fun norm (a, p, q) =
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  if p = Intt.int 0 then Rat (true, Intt.int 0, Intt.int 1)
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  else
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    let
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      val absp = abs p
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      val m = gcd (absp, q)
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    in Rat(a = (Intt.int 0 <= p), absp div m, q div m) end;
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fun common (p1, q1, p2, q2) =
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  let val q' = lcm (q1, q2)
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  in (p1 * (q' div q1), p2 * (q' div q2), q') end
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fun rat_of_quotient (p, q) =
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  if q = Intt.int 0 then raise DIVZERO
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  else norm (Intt.int 0 <= q, p, abs q);
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fun quotient_of_rat (Rat (a, p, q)) = (if a then p else ~p, q);
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fun string_of_rat r =
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  let val (p, q) = quotient_of_rat r
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  in Intt.string_of_int p ^ "/" ^ Intt.string_of_int q end;
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fun eq (Rat (false, _, _), Rat (true, _, _)) = false
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  | eq (Rat (true, _, _), Rat (false, _, _)) = false
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  | eq (Rat (_, p1, q1), Rat (_, p2, q2)) = p1 = p2 andalso q1 = q2
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fun cmp (Rat (false, _, _), Rat (true, _, _)) = LESS
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  | cmp (Rat (true, _, _), Rat (false, _, _)) = GREATER
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  | cmp (Rat (a, p1, q1), Rat (_, p2, q2)) =
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      let val (r1, r2, _) = common (p1, q1, p2, q2)
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      in if a then Intt.cmp (r1, r2) else Intt.cmp (r2, r1) end;
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fun le a b = let val order = cmp (a, b) in order = LESS orelse order = EQUAL end;
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fun lt a b = cmp (a, b) = LESS;
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fun cmp_zero (Rat (false, _, _)) = LESS
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  | cmp_zero (Rat (_, 0, _)) = EQUAL
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  | cmp_zero (Rat (_, _, _)) = GREATER;
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fun add (Rat (a1, p1, q1)) (Rat(a2, p2, q2)) =
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  let
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    val (r1, r2, den) = common (p1, q1, p2, q2)
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    val num = (if a1 then r1 else ~r1) + (if a2 then r2 else ~r2)
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  in norm (true, num, den) end;
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fun mult (Rat (a1, p1, q1)) (Rat (a2, p2, q2)) =
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  norm (a1=a2, p1*p2, q1*q2);
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fun neg (r as Rat (b, p, q)) =
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  if p = Intt.int 0 then r
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  else Rat (not b, p, q);
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fun inv (Rat (a, p, q)) =
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  if p = Intt.int 0 then raise DIVZERO
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  else Rat (a, q, p);
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fun roundup (r as Rat (a, p, q)) =
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  if q = Intt.int 1 then r
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  else
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    let
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      fun round true q = Rat (true, q + Intt.int 1, Intt.int 1)
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        | round false q =
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            if q = Intt.int 0
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            then Rat (true, Intt.int 0, Intt.int 1)
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            else Rat (false, q, Intt.int 1);
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    in round a (p div q) end;
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fun rounddown (r as Rat (a, p, q)) =
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  if q = Intt.int 1 then r
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  else
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    let
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      fun round true q = Rat (true, q, Intt.int 1)
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        | round false q = Rat (false, q + Intt.int 1, Intt.int 1)
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    in round a (p div q) end;
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end;
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infix 5 +/; 
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infix 5 -/;
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infix 7 */;
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infix 7 //; 
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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);