src/HOL/Library/ExecutableRat.thy

select_goals: split original conjunctions;

(*  Title:      HOL/Library/ExecutableRat.thy
    ID:         $Id$
    Author:     Florian Haftmann, TU Muenchen
*)

header {* Executable implementation of rational numbers in HOL *}

theory ExecutableRat
imports "~~/src/HOL/Real/Rational" "~~/src/HOL/NumberTheory/IntPrimes"
begin

text {*
  Actually nothing is proved about the implementation.
*}

datatype erat = Rat bool int int

instance erat :: zero by intro_classes
instance erat :: one by intro_classes
instance erat :: plus by intro_classes
instance erat :: minus by intro_classes
instance erat :: times by intro_classes
instance erat :: inverse by intro_classes
instance erat :: ord by intro_classes

consts
  norm :: "erat \<Rightarrow> erat"
  common :: "(int * int) * (int * int) \<Rightarrow> (int * int) * int"
  of_quotient :: "int * int \<Rightarrow> erat"
  of_rat :: "rat \<Rightarrow> erat"
  to_rat :: "erat \<Rightarrow> rat"

defs
  norm_def [simp]: "norm r == case r of (Rat a p q) \<Rightarrow>
     if p = 0 then Rat True 0 1
     else
       let
         absp = abs p
       in let
         m = zgcd (absp, q)
       in Rat (a = (0 <= p)) (absp div m) (q div m)"
  common_def [simp]: "common r == case r of ((p1, q1), (p2, q2)) \<Rightarrow>
       let q' = q1 * q2 div int (gcd (nat q1, nat q2))
       in ((p1 * (q' div q1), p2 * (q' div q2)), q')"
  of_quotient_def [simp]: "of_quotient r == case r of (a, b) \<Rightarrow>
       norm (Rat True a b)"
  of_rat_def [simp]: "of_rat r == of_quotient (THE s. s : Rep_Rat r)"
  to_rat_def [simp]: "to_rat r == case r of (Rat a p q) \<Rightarrow>
       if a then Fract p q else Fract (uminus p) q"

consts
  zero :: erat
  one :: erat
  add :: "erat \<Rightarrow> erat \<Rightarrow> erat"
  neg :: "erat \<Rightarrow> erat"
  mult :: "erat \<Rightarrow> erat \<Rightarrow> erat"
  inv :: "erat \<Rightarrow> erat"
  le :: "erat \<Rightarrow> erat \<Rightarrow> bool"

defs (overloaded)
  zero_rat_def [simp]: "0 == Rat False 0 1"
  one_rat_def [simp]: "1 == Rat False 1 1"
  add_rat_def [simp]: "r + s == case r of Rat a1 p1 q1 \<Rightarrow> case s of Rat a2 p2 q2 \<Rightarrow>
        let
          ((r1, r2), den) = common ((p1, q1), (p2, q2))
        in let
          num = (if a1 then r1 else -r1) + (if a2 then r2 else -r2)
        in norm (Rat True num den)"
  uminus_rat_def [simp]: "- r == case r of Rat a p q \<Rightarrow>
        if p = 0 then Rat a p q
        else Rat (\<not> a) p q"
  times_rat_def [simp]: "r * s == case r of Rat a1 p1 q1 \<Rightarrow> case s of Rat a2 p2 q2 \<Rightarrow>
        norm (Rat (a1 = a2) (p1 * p2) (q1 * q2))"
  inverse_rat_def [simp]: "inverse r == case r of Rat a p q \<Rightarrow>
        if p = 0 then arbitrary
        else Rat a q p"
  le_rat_def [simp]: "r <= s == case r of Rat a1 p1 q1 \<Rightarrow> case s of Rat a2 p2 q2 \<Rightarrow>
        (\<not> a1 \<and> a2) \<or>
        (\<not> (a1 \<and> \<not> a2) \<and>
          (let
            ((r1, r2), dummy) = common ((p1, q1), (p2, q2))
          in if a1 then r1 <= r2 else r2 <= r1))"

code_syntax_tyco rat
  ml (target_atom "{*erat*}")
  haskell (target_atom "{*erat*}")

code_alias
  (* an intermediate solution until name resolving of ad-hoc overloaded
     constants is refined *)
  "HOL.inverse" "Rational.inverse"
  "HOL.divide" "Rational.divide"

code_syntax_const
  Fract
    ml ("{*of_quotient*}")
    haskell ("{*of_quotient*}")
  "0 :: rat"
    ml ("{*0::erat*}")
    haskell ("{*1::erat*}")
  "1 :: rat"
    ml ("{*1::erat*}")
    haskell ("{*1::erat*}")
  "op + :: rat \<Rightarrow> rat \<Rightarrow> rat"
    ml ("{*op + :: erat \<Rightarrow> erat \<Rightarrow> erat*}")
    haskell ("{*op + :: erat \<Rightarrow> erat \<Rightarrow> erat*}")
  "uminus :: rat \<Rightarrow> rat"
    ml ("{*uminus :: erat \<Rightarrow> erat*}")
    haskell ("{*uminus :: erat \<Rightarrow> erat*}")
  "op * :: rat \<Rightarrow> rat \<Rightarrow> rat"
    ml ("{*op * :: erat \<Rightarrow> erat \<Rightarrow> erat*}")
    haskell ("{*op * :: erat \<Rightarrow> erat \<Rightarrow> erat*}")
  "inverse :: rat \<Rightarrow> rat"
    ml ("{*inverse :: erat \<Rightarrow> erat*}")
    haskell ("{*inverse :: erat \<Rightarrow> erat*}")
  "divide :: rat \<Rightarrow> rat \<Rightarrow> rat"
    ml ("{*op * :: erat \<Rightarrow> erat \<Rightarrow> erat*}/ _/ ({*inverse :: erat \<Rightarrow> erat*}/ _)")
    haskell ("{*op * :: erat \<Rightarrow> erat \<Rightarrow> erat*}/ _/ ({*inverse :: erat \<Rightarrow> erat*}/ _)")
  "op <= :: rat \<Rightarrow> rat \<Rightarrow> bool"
    ml ("{*op <= :: erat \<Rightarrow> erat \<Rightarrow> bool*}")
    haskell ("{*op <= :: erat \<Rightarrow> erat \<Rightarrow> bool*}")

end