# HG changeset patch
# User haftmann
# Date 1181049422 -7200
# Node ID 471b576aad25ae019e296a8baee482f741f7205e
# Parent  9886802cbbd67dd0d8eb2c89d0929b63d72e1c82
moved generic algebra modules

diff -r 9886802cbbd6 -r 471b576aad25 NEWS
--- a/NEWS	Tue Jun 05 15:16:11 2007 +0200
+++ b/NEWS	Tue Jun 05 15:17:02 2007 +0200
@@ -738,8 +738,7 @@
 * Renamed constants "0" to "HOL.zero_class.zero" and "1" to "HOL.one_class.one".
 INCOMPATIBILITY.
 
-* Added theory Code_Generator providing class 'eq', allowing for code
-generation with polymorphic equality.
+* Added class "HOL.eq", allowing for code generation with polymorphic equality.
 
 * Numeral syntax: type 'bin' which was a mere type copy of 'int' has
 been abandoned in favour of plain 'int'. INCOMPATIBILITY --
@@ -952,6 +951,8 @@
 
 *** ML ***
 
+* Generic arithmetic modules: Tools/integer.ML, Tools/rat.ML, Tools/float.ML
+
 * Context data interfaces (Theory/Proof/GenericDataFun): removed
 name/print, uninitialized data defaults to ad-hoc copy of empty value,
 init only required for impure data.  INCOMPATIBILITY: empty really
diff -r 9886802cbbd6 -r 471b576aad25 src/HOL/Real/Float.thy
--- a/src/HOL/Real/Float.thy	Tue Jun 05 15:16:11 2007 +0200
+++ b/src/HOL/Real/Float.thy	Tue Jun 05 15:17:02 2007 +0200
@@ -7,7 +7,7 @@
 
 theory Float
 imports Real Parity
-uses "~~/src/Pure/General/float.ML" ("float_arith.ML")
+uses "~~/src/Tools/float.ML" ("float_arith.ML")
 begin
 
 definition
diff -r 9886802cbbd6 -r 471b576aad25 src/Pure/General/float.ML
--- a/src/Pure/General/float.ML	Tue Jun 05 15:16:11 2007 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,60 +0,0 @@
-(*  Title:      Pure/General/float.ML
-    ID:         $Id$
-    Author:     Steven Obua, Florian Haftmann, TU Muenchen
-
-Implementation of real numbers as mantisse-exponent pairs.
-*)
-
-signature FLOAT =
-sig
-  type float = Intt.int * Intt.int
-  val zero: float
-  val eq: float * float -> bool
-  val cmp: float * float -> order
-  val cmp_zero: float -> order
-  val min: float -> float -> float
-  val max: float -> float -> float
-  val add: float -> float -> float
-  val sub: float -> float -> float
-  val neg: float -> float
-  val mult: float -> float -> float
-  val positive_part: float -> float
-  val negative_part: float -> float
-end;
-
-structure Float : FLOAT =
-struct
-
-type float = Intt.int * Intt.int;
-
-val zero = (Intt.zero, Intt.zero);
-
-fun add (a1, b1) (a2, b2) =
-  if Intt.cmp (b1, b2) = LESS then
-    (Intt.add a1 (Intt.mult a2 (Intt.exp (Intt.sub b2 b1))), b1)
-  else
-    (Intt.add (Intt.mult a1 (Intt.exp (Intt.sub b1 b2))) a2, b2);
-
-fun sub (a1, b1) (a2, b2) =
-  if Intt.cmp (b1, b2) = LESS then
-    (Intt.sub a1 (Intt.mult a2 (Intt.exp (Intt.sub b2 b1))), b1)
-  else
-    (Intt.sub (Intt.mult a1 (Intt.exp (Intt.sub b1 b2))) a2, b2);
-
-fun neg (a, b) = (Intt.neg a, b);
-
-fun mult (a1, b1) (a2, b2) = (Intt.mult a1 a2, Intt.add b1 b2);
-
-fun cmp_zero (a, b) = Intt.cmp_zero a;
-
-fun cmp (r, s) = cmp_zero (sub r s);
-
-fun eq (r, s) = cmp (r, s) = EQUAL;
-
-fun min r s = case cmp (r, s) of LESS => r | _ => s;
-fun max r s = case cmp (r, s) of LESS => s | _ => r;
-
-fun positive_part (a, b) = (Intt.max Intt.zero a, b);
-fun negative_part (a, b) = (Intt.min Intt.zero a, b);
-
-end;
diff -r 9886802cbbd6 -r 471b576aad25 src/Pure/General/int.ML
--- a/src/Pure/General/int.ML	Tue Jun 05 15:16:11 2007 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,82 +0,0 @@
-(*  Title:      Pure/General/int.ML
-    ID:         $Id$
-    Author:     Florian Haftmann, TU Muenchen
-
-Unbounded integers.
-*)
-
-signature INTT =
-sig
-  type int
-  val zero: int
-  val one: int
-  val two: int
-  val int: Int.int -> int
-  val machine_int: int -> Int.int
-  val string_of_int: int -> string
-  val int_of_string: string -> int option
-  val eq: int * int -> bool
-  val cmp: int * int -> order
-  val le: int -> int -> bool
-  val cmp_zero: int -> order
-  val min: int -> int -> int
-  val max: int -> int -> int
-  val inc: int -> int
-  val add: int -> int -> int
-  val sub: int -> int -> int
-  val mult: int -> int -> int
-  val divmod: int -> int -> int * int
-  val div: int -> int -> int
-  val mod: int -> int -> int
-  val neg: int -> int
-  val exp: int -> int
-  val log: int -> int
-  val pow: int -> int -> int (* exponent -> base -> result *)
-end;
-
-structure Intt: INTT =
-struct
-
-open IntInf;
-
-val int = fromInt;
-
-val zero = int 0;
-val one = int 1;
-val two = int 2;
-
-val machine_int = toInt;
-val string_of_int = toString;
-val int_of_string = fromString;
-
-val eq = op = : int * int -> bool;
-val cmp = compare;
-val le = curry (op <=);
-val cmp_zero = curry cmp zero;
-
-val min = curry min;
-val max = curry max;
-
-val inc = curry (op +) one;
-
-val add = curry (op +);
-val sub = curry (op -);
-val mult = curry ( op * );
-val divmod = curry divMod;
-nonfix div val div = curry div;
-nonfix mod val mod = curry mod;
-val neg = ~;
-
-fun pow k l =
-  let
-    fun pw 0 = 1
-      | pw k = mult l (pw (sub k 1));
-  in if k < zero
-    then error "pow: negative exponent"
-    else pw k
-  end;
-
-fun exp k = pow k two;
-val log = int o log2;
-
-end;
diff -r 9886802cbbd6 -r 471b576aad25 src/Pure/General/rat.ML
--- a/src/Pure/General/rat.ML	Tue Jun 05 15:16:11 2007 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,140 +0,0 @@
-(*  Title:      Pure/General/rat.ML
-    ID:         $Id$
-    Author:     Tobias Nipkow, TU Muenchen
-
-Canonical implementation of exact rational numbers.
-*)
-
-signature RAT =
-sig
-  type rat
-  exception DIVZERO
-  val zero: rat
-  val one: rat
-  val two: rat
-  val rat_of_int: Intt.int -> rat
-  val rat_of_quotient: Intt.int * Intt.int -> rat
-  val quotient_of_rat: rat -> Intt.int * Intt.int
-  val string_of_rat: rat -> string
-  val eq: rat * rat -> bool
-  val cmp: rat * rat -> order
-  val le: rat -> rat -> bool
-  val lt: rat -> rat -> bool
-  val cmp_zero: rat -> order
-  val add: rat -> rat -> rat
-  val mult: rat -> rat -> rat
-  val neg: rat -> rat
-  val inv: rat -> rat
-  val roundup: rat -> rat
-  val rounddown: rat -> rat
-end;
-
-structure Rat : RAT =
-struct
-
-datatype rat = Rat of bool * Intt.int * Intt.int;
-
-exception DIVZERO;
-
-val zero = Rat (true, Intt.int 0, Intt.int 1);
-val one = Rat (true, Intt.int 1, Intt.int 1);
-val two = Rat (true, Intt.int 2, Intt.int 1);
-
-fun rat_of_int i =
-  if i < Intt.int 0
-  then Rat (false, ~i, Intt.int 1)
-  else Rat (true, i, Intt.int 1);
-
-fun norm (a, p, q) =
-  if p = Intt.int 0 then Rat (true, Intt.int 0, Intt.int 1)
-  else
-    let
-      val absp = abs p
-      val m = gcd (absp, q)
-    in Rat(a = (Intt.int 0 <= p), absp div m, q div m) end;
-
-fun common (p1, q1, p2, q2) =
-  let val q' = lcm (q1, q2)
-  in (p1 * (q' div q1), p2 * (q' div q2), q') end
-
-fun rat_of_quotient (p, q) =
-  if q = Intt.int 0 then raise DIVZERO
-  else norm (Intt.int 0 <= q, p, abs q);
-
-fun quotient_of_rat (Rat (a, p, q)) = (if a then p else ~p, q);
-
-fun string_of_rat r =
-  let val (p, q) = quotient_of_rat r
-  in Intt.string_of_int p ^ "/" ^ Intt.string_of_int q end;
-
-fun eq (Rat (false, _, _), Rat (true, _, _)) = false
-  | eq (Rat (true, _, _), Rat (false, _, _)) = false
-  | eq (Rat (_, p1, q1), Rat (_, p2, q2)) = p1 = p2 andalso q1 = q2
-
-fun cmp (Rat (false, _, _), Rat (true, _, _)) = LESS
-  | cmp (Rat (true, _, _), Rat (false, _, _)) = GREATER
-  | cmp (Rat (a, p1, q1), Rat (_, p2, q2)) =
-      let val (r1, r2, _) = common (p1, q1, p2, q2)
-      in if a then Intt.cmp (r1, r2) else Intt.cmp (r2, r1) end;
-
-fun le a b = let val order = cmp (a, b) in order = LESS orelse order = EQUAL end;
-fun lt a b = cmp (a, b) = LESS;
-
-fun cmp_zero (Rat (false, _, _)) = LESS
-  | cmp_zero (Rat (_, 0, _)) = EQUAL
-  | cmp_zero (Rat (_, _, _)) = GREATER;
-
-fun add (Rat (a1, p1, q1)) (Rat(a2, p2, q2)) =
-  let
-    val (r1, r2, den) = common (p1, q1, p2, q2)
-    val num = (if a1 then r1 else ~r1) + (if a2 then r2 else ~r2)
-  in norm (true, num, den) end;
-
-fun mult (Rat (a1, p1, q1)) (Rat (a2, p2, q2)) =
-  norm (a1=a2, p1*p2, q1*q2);
-
-fun neg (r as Rat (b, p, q)) =
-  if p = Intt.int 0 then r
-  else Rat (not b, p, q);
-
-fun inv (Rat (a, p, q)) =
-  if p = Intt.int 0 then raise DIVZERO
-  else Rat (a, q, p);
-
-fun roundup (r as Rat (a, p, q)) =
-  if q = Intt.int 1 then r
-  else
-    let
-      fun round true q = Rat (true, q + Intt.int 1, Intt.int 1)
-        | round false q =
-            if q = Intt.int 0
-            then Rat (true, Intt.int 0, Intt.int 1)
-            else Rat (false, q, Intt.int 1);
-    in round a (p div q) end;
-
-fun rounddown (r as Rat (a, p, q)) =
-  if q = Intt.int 1 then r
-  else
-    let
-      fun round true q = Rat (true, q, Intt.int 1)
-        | round false q = Rat (false, q + Intt.int 1, Intt.int 1)
-    in round a (p div q) end;
-
-end;
-
-infix 5 +/; 
-infix 5 -/;
-infix 7 */;
-infix 7 //; 
-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);
diff -r 9886802cbbd6 -r 471b576aad25 src/Pure/library.ML
--- a/src/Pure/library.ML	Tue Jun 05 15:16:11 2007 +0200
+++ b/src/Pure/library.ML	Tue Jun 05 15:17:02 2007 +0200
@@ -124,8 +124,6 @@
   val suffixes: 'a list -> 'a list list
 
   (*integers*)
-  val gcd: IntInf.int * IntInf.int -> IntInf.int
-  val lcm: IntInf.int * IntInf.int -> IntInf.int
   val inc: int ref -> int
   val dec: int ref -> int
   val upto: int * int -> int list
@@ -639,13 +637,6 @@
 
 (** integers **)
 
-fun gcd (x, y) =
-  let fun gxd x y : IntInf.int =
-    if y = IntInf.fromInt 0 then x else gxd y (x mod y)
-  in if x < y then gxd y x else gxd x y end;
-
-fun lcm (x, y) = (x * y) div gcd (x, y);
-
 fun inc i = (i := ! i + 1; ! i);
 fun dec i = (i := ! i - 1; ! i);
 
diff -r 9886802cbbd6 -r 471b576aad25 src/Tools/float.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Tools/float.ML	Tue Jun 05 15:17:02 2007 +0200
@@ -0,0 +1,60 @@
+(*  Title:      Pure/General/float.ML
+    ID:         $Id$
+    Author:     Steven Obua, Florian Haftmann, TU Muenchen
+
+Implementation of real numbers as mantisse-exponent pairs.
+*)
+
+signature FLOAT =
+sig
+  type float = Intt.int * Intt.int
+  val zero: float
+  val eq: float * float -> bool
+  val cmp: float * float -> order
+  val cmp_zero: float -> order
+  val min: float -> float -> float
+  val max: float -> float -> float
+  val add: float -> float -> float
+  val sub: float -> float -> float
+  val neg: float -> float
+  val mult: float -> float -> float
+  val positive_part: float -> float
+  val negative_part: float -> float
+end;
+
+structure Float : FLOAT =
+struct
+
+type float = Intt.int * Intt.int;
+
+val zero = (Intt.zero, Intt.zero);
+
+fun add (a1, b1) (a2, b2) =
+  if Intt.cmp (b1, b2) = LESS then
+    (Intt.add a1 (Intt.mult a2 (Intt.exp (Intt.sub b2 b1))), b1)
+  else
+    (Intt.add (Intt.mult a1 (Intt.exp (Intt.sub b1 b2))) a2, b2);
+
+fun sub (a1, b1) (a2, b2) =
+  if Intt.cmp (b1, b2) = LESS then
+    (Intt.sub a1 (Intt.mult a2 (Intt.exp (Intt.sub b2 b1))), b1)
+  else
+    (Intt.sub (Intt.mult a1 (Intt.exp (Intt.sub b1 b2))) a2, b2);
+
+fun neg (a, b) = (Intt.neg a, b);
+
+fun mult (a1, b1) (a2, b2) = (Intt.mult a1 a2, Intt.add b1 b2);
+
+fun cmp_zero (a, b) = Intt.cmp_zero a;
+
+fun cmp (r, s) = cmp_zero (sub r s);
+
+fun eq (r, s) = cmp (r, s) = EQUAL;
+
+fun min r s = case cmp (r, s) of LESS => r | _ => s;
+fun max r s = case cmp (r, s) of LESS => s | _ => r;
+
+fun positive_part (a, b) = (Intt.max Intt.zero a, b);
+fun negative_part (a, b) = (Intt.min Intt.zero a, b);
+
+end;
diff -r 9886802cbbd6 -r 471b576aad25 src/Tools/integer.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Tools/integer.ML	Tue Jun 05 15:17:02 2007 +0200
@@ -0,0 +1,116 @@
+(*  Title:      Pure/General/int.ML
+    ID:         $Id$
+    Author:     Florian Haftmann, TU Muenchen
+
+Unbounded integers.
+*)
+
+signature INTEGER =
+sig
+  type int
+  val zero: int
+  val one: int
+  val two: int
+  val int: Int.int -> int
+  val machine_int: int -> Int.int
+  val string_of_int: int -> string
+  val int_of_string: string -> int option
+  val eq: int * int -> bool
+  val cmp: int * int -> order
+  val le: int -> int -> bool
+  val lt: int -> int -> bool
+  val cmp_zero: int -> order
+  val min: int -> int -> int
+  val max: int -> int -> int
+  val inc: int -> int
+  val add: int -> int -> int
+  val sub: int -> int -> int
+  val mult: int -> int -> int
+  val divmod: int -> int -> int * int
+  val div: int -> int -> int
+  val mod: int -> int -> int
+  val neg: int -> int
+  val signabs: int -> bool * int
+  val exp: int -> int
+  val log: int -> int
+  val pow: int -> int -> int (* exponent -> base -> result *)
+  val gcd: int -> int -> int
+  val lcm: int -> int -> int
+end;
+
+structure Integer : INTEGER =
+struct
+
+open IntInf;
+
+val int = fromInt;
+
+val zero = int 0;
+val one = int 1;
+val two = int 2;
+
+val machine_int = toInt;
+val string_of_int = toString;
+val int_of_string = fromString;
+
+val eq = op = : int * int -> bool;
+val cmp = compare;
+val le = curry (op <=);
+val lt = curry (op <);
+fun cmp_zero k = cmp (k, zero);
+
+val min = curry min;
+val max = curry max;
+
+val inc = curry (op +) one;
+
+val add = curry (op +);
+val sub = curry (op -);
+val mult = curry ( op * );
+val divmod = curry divMod;
+nonfix div val div = curry div;
+nonfix mod val mod = curry mod;
+val neg = ~;
+
+fun signabs k = if cmp_zero k = LESS then (false, neg k) else (true, k);
+
+fun pow k l =
+  let
+    fun pw 0 = 1
+      | pw k = mult l (pw (sub k 1));
+  in if k < zero
+    then error "pow: negative exponent"
+    else pw k
+  end;
+
+fun exp k = pow k two;
+val log = int o log2;
+
+fun gcd x y =
+  let
+    fun gxd x y = if y = zero then x else gxd y (mod x y)
+  in if lt x y then gxd y x else gxd x y end;
+
+fun lcm x y = div (mult x y) (gcd x y);
+
+end;
+
+type integer = Integer.int;
+
+infix 7 *%;
+infix 6 +% -%; 
+infix 4 =% <% <=% >% >=% <>%;
+
+fun a +% b = Integer.add a b;
+fun a -% b = a +% Integer.neg b;
+fun a *% b = Integer.mult a b;
+fun a =% b = Integer.eq (a, b);
+fun a <% b = Integer.lt a b;
+fun a <=% b = Integer.le a b;
+fun a >% b = b <% a;
+fun a >=% b = b <=% a;
+fun a <>% b = not (a =% b);
+
+structure Intt = Integer; (*FIXME*)
+val gcd = uncurry Integer.gcd; (*FIXME*)
+val lcm = uncurry Integer.lcm; (*FIXME*)
diff -r 9886802cbbd6 -r 471b576aad25 src/Tools/rat.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Tools/rat.ML	Tue Jun 05 15:17:02 2007 +0200
@@ -0,0 +1,140 @@
+(*  Title:      Pure/General/rat.ML
+    ID:         $Id$
+    Author:     Tobias Nipkow, TU Muenchen
+
+Canonical implementation of exact rational numbers.
+*)
+
+signature RAT =
+sig
+  type rat
+  exception DIVZERO
+  val zero: rat
+  val one: rat
+  val two: rat
+  val rat_of_int: Intt.int -> rat
+  val rat_of_quotient: Intt.int * Intt.int -> rat
+  val quotient_of_rat: rat -> Intt.int * Intt.int
+  val string_of_rat: rat -> string
+  val eq: rat * rat -> bool
+  val cmp: rat * rat -> order
+  val le: rat -> rat -> bool
+  val lt: rat -> rat -> bool
+  val cmp_zero: rat -> order
+  val add: rat -> rat -> rat
+  val mult: rat -> rat -> rat
+  val neg: rat -> rat
+  val inv: rat -> rat
+  val roundup: rat -> rat
+  val rounddown: rat -> rat
+end;
+
+structure Rat : RAT =
+struct
+
+datatype rat = Rat of bool * Intt.int * Intt.int;
+
+exception DIVZERO;
+
+val zero = Rat (true, Intt.int 0, Intt.int 1);
+val one = Rat (true, Intt.int 1, Intt.int 1);
+val two = Rat (true, Intt.int 2, Intt.int 1);
+
+fun rat_of_int i =
+  if i < Intt.int 0
+  then Rat (false, ~i, Intt.int 1)
+  else Rat (true, i, Intt.int 1);
+
+fun norm (a, p, q) =
+  if p = Intt.int 0 then Rat (true, Intt.int 0, Intt.int 1)
+  else
+    let
+      val absp = abs p
+      val m = gcd (absp, q)
+    in Rat(a = (Intt.int 0 <= p), absp div m, q div m) end;
+
+fun common (p1, q1, p2, q2) =
+  let val q' = lcm (q1, q2)
+  in (p1 * (q' div q1), p2 * (q' div q2), q') end
+
+fun rat_of_quotient (p, q) =
+  if q = Intt.int 0 then raise DIVZERO
+  else norm (Intt.int 0 <= q, p, abs q);
+
+fun quotient_of_rat (Rat (a, p, q)) = (if a then p else ~p, q);
+
+fun string_of_rat r =
+  let val (p, q) = quotient_of_rat r
+  in Intt.string_of_int p ^ "/" ^ Intt.string_of_int q end;
+
+fun eq (Rat (false, _, _), Rat (true, _, _)) = false
+  | eq (Rat (true, _, _), Rat (false, _, _)) = false
+  | eq (Rat (_, p1, q1), Rat (_, p2, q2)) = p1 = p2 andalso q1 = q2
+
+fun cmp (Rat (false, _, _), Rat (true, _, _)) = LESS
+  | cmp (Rat (true, _, _), Rat (false, _, _)) = GREATER
+  | cmp (Rat (a, p1, q1), Rat (_, p2, q2)) =
+      let val (r1, r2, _) = common (p1, q1, p2, q2)
+      in if a then Intt.cmp (r1, r2) else Intt.cmp (r2, r1) end;
+
+fun le a b = let val order = cmp (a, b) in order = LESS orelse order = EQUAL end;
+fun lt a b = cmp (a, b) = LESS;
+
+fun cmp_zero (Rat (false, _, _)) = LESS
+  | cmp_zero (Rat (_, 0, _)) = EQUAL
+  | cmp_zero (Rat (_, _, _)) = GREATER;
+
+fun add (Rat (a1, p1, q1)) (Rat(a2, p2, q2)) =
+  let
+    val (r1, r2, den) = common (p1, q1, p2, q2)
+    val num = (if a1 then r1 else ~r1) + (if a2 then r2 else ~r2)
+  in norm (true, num, den) end;
+
+fun mult (Rat (a1, p1, q1)) (Rat (a2, p2, q2)) =
+  norm (a1=a2, p1*p2, q1*q2);
+
+fun neg (r as Rat (b, p, q)) =
+  if p = Intt.int 0 then r
+  else Rat (not b, p, q);
+
+fun inv (Rat (a, p, q)) =
+  if p = Intt.int 0 then raise DIVZERO
+  else Rat (a, q, p);
+
+fun roundup (r as Rat (a, p, q)) =
+  if q = Intt.int 1 then r
+  else
+    let
+      fun round true q = Rat (true, q + Intt.int 1, Intt.int 1)
+        | round false q =
+            if q = Intt.int 0
+            then Rat (true, Intt.int 0, Intt.int 1)
+            else Rat (false, q, Intt.int 1);
+    in round a (p div q) end;
+
+fun rounddown (r as Rat (a, p, q)) =
+  if q = Intt.int 1 then r
+  else
+    let
+      fun round true q = Rat (true, q, Intt.int 1)
+        | round false q = Rat (false, q + Intt.int 1, Intt.int 1)
+    in round a (p div q) end;
+
+end;
+
+infix 5 +/; 
+infix 5 -/;
+infix 7 */;
+infix 7 //; 
+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);