1(*  Title:      Pure/General/rat.ML
2    Author:     Tobias Nipkow, Florian Haftmann, TU Muenchen
3    Author:     Makarius
4
5Canonical implementation of exact rational numbers.
6*)
7
8signature RAT =
9sig
10  eqtype rat
11  val of_int: int -> rat
12  val make: int * int -> rat
13  val dest: rat -> int * int
14  val string_of_rat: rat -> string
15  val signed_string_of_rat: rat -> string
16  val ord: rat ord
17  val le: rat -> rat -> bool
18  val lt: rat -> rat -> bool
19  val sign: rat -> order
20  val abs: rat -> rat
21  val add: rat -> rat -> rat
22  val mult: rat -> rat -> rat
23  val neg: rat -> rat
24  val inv: rat -> rat
25  val floor: rat -> int
26  val ceil: rat -> int
27end;
28
29structure Rat : RAT =
30struct
31
32datatype rat = Rat of int * int;  (*numerator, positive (!) denominator*)
33
34fun of_int i = Rat (i, 1);
35
36fun common (p1, q1) (p2, q2) =
37  let val m = Integer.lcm q1 q2
38  in ((p1 * (m div q1), p2 * (m div q2)), m) end;
39
40fun make (_, 0) = raise Div
41  | make (p, q) =
42    let
43      val m = Integer.gcd p q;
44      val (p', q') = (p div m, q div m);
45    in Rat (if q' < 0 then (~ p', ~ q') else (p', q')) end
46
47fun dest (Rat r) = r;
48
49fun string_of_rat (Rat (p, 1)) = string_of_int p
50  | string_of_rat (Rat (p, q)) = string_of_int p ^ "/" ^ string_of_int q;
51
52fun signed_string_of_rat (Rat (p, 1)) = signed_string_of_int p
53  | signed_string_of_rat (Rat (p, q)) = signed_string_of_int p ^ "/" ^ string_of_int q;
54
55fun ord (Rat (p1, q1), Rat (p2, q2)) =
56  (case (Integer.sign p1, Integer.sign p2) of
57    (LESS, EQUAL) => LESS
58  | (LESS, GREATER) => LESS
59  | (EQUAL, LESS) => GREATER
60  | (EQUAL, EQUAL) => EQUAL
61  | (EQUAL, GREATER) => LESS
62  | (GREATER, LESS) => GREATER
63  | (GREATER, EQUAL) => GREATER
64  | _ => int_ord (fst (common (p1, q1) (p2, q2))));
65
66fun le a b = ord (a, b) <> GREATER;
67fun lt a b = ord (a, b) = LESS;
68
69fun sign (Rat (p, _)) = Integer.sign p;
70
71fun abs (r as Rat (p, q)) = if p < 0 then Rat (~ p, q) else r;
72
73fun add (Rat r1) (Rat r2) =
74  let val ((m1, m2), n) = common r1 r2
75  in make (m1 + m2, n) end;
76
77fun mult (Rat (p1, q1)) (Rat (p2, q2)) = make (p1 * p2, q1 * q2);
78
79fun neg (Rat (p, q)) = Rat (~ p, q);
80
81fun inv (Rat (p, q)) =
82  (case Integer.sign p of
83    LESS => Rat (~ q, ~ p)
84  | EQUAL => raise Div
85  | GREATER => Rat (q, p));
86
87fun floor (Rat (p, q)) = p div q;
88
89fun ceil (Rat (p, q)) =
90  (case Integer.div_mod p q of
91    (m, 0) => m
92  | (m, _) => m + 1);
93
94end;
95
96ML_system_overload (uncurry Rat.add) "+";
97ML_system_overload (fn (a, b) => Rat.add a (Rat.neg b)) "-";
98ML_system_overload (uncurry Rat.mult) "*";
99ML_system_overload (fn (a, b) => Rat.mult a (Rat.inv b)) "/";
100ML_system_overload (uncurry Rat.lt) "<";
101ML_system_overload (uncurry Rat.le) "<=";
102ML_system_overload (fn (a, b) => Rat.lt b a) ">";
103ML_system_overload (fn (a, b) => Rat.le b a) ">=";
104ML_system_overload Rat.neg "~";
105ML_system_overload Rat.abs "abs";
106
107ML_system_pp (fn _ => fn _ => fn x => Pretty.to_polyml (Pretty.str ("@" ^ Rat.string_of_rat x)));
108