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(* Title: HOL/Library/ExecutableRat.thy
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ID: $Id$
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Author: Florian Haftmann, TU Muenchen
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*)
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header {* Executable implementation of rational numbers in HOL *}
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theory ExecutableRat
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imports "~~/src/HOL/Real/Rational" "~~/src/HOL/NumberTheory/IntPrimes"
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begin
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text {*
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Actually nothing is proved about the implementation.
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*}
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datatype erat = Rat bool int int
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instance erat :: zero by intro_classes
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instance erat :: one by intro_classes
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instance erat :: plus by intro_classes
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instance erat :: minus by intro_classes
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instance erat :: times by intro_classes
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instance erat :: inverse by intro_classes
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instance erat :: ord by intro_classes
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consts
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norm :: "erat \<Rightarrow> erat"
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common :: "(int * int) * (int * int) \<Rightarrow> (int * int) * int"
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of_quotient :: "int * int \<Rightarrow> erat"
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of_rat :: "rat \<Rightarrow> erat"
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to_rat :: "erat \<Rightarrow> rat"
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defs
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norm_def [simp]: "norm r == case r of (Rat a p q) \<Rightarrow>
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if p = 0 then Rat True 0 1
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else
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let
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absp = abs p
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in let
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m = zgcd (absp, q)
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in Rat (a = (0 <= p)) (absp div m) (q div m)"
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common_def [simp]: "common r == case r of ((p1, q1), (p2, q2)) \<Rightarrow>
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let q' = q1 * q2 div int (gcd (nat q1, nat q2))
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in ((p1 * (q' div q1), p2 * (q' div q2)), q')"
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of_quotient_def [simp]: "of_quotient r == case r of (a, b) \<Rightarrow>
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norm (Rat True a b)"
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of_rat_def [simp]: "of_rat r == of_quotient (THE s. s : Rep_Rat r)"
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to_rat_def [simp]: "to_rat r == case r of (Rat a p q) \<Rightarrow>
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if a then Fract p q else Fract (uminus p) q"
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consts
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zero :: erat
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one :: erat
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add :: "erat \<Rightarrow> erat \<Rightarrow> erat"
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neg :: "erat \<Rightarrow> erat"
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mult :: "erat \<Rightarrow> erat \<Rightarrow> erat"
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inv :: "erat \<Rightarrow> erat"
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le :: "erat \<Rightarrow> erat \<Rightarrow> bool"
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defs (overloaded)
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zero_rat_def [simp]: "0 == Rat False 0 1"
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one_rat_def [simp]: "1 == Rat False 1 1"
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add_rat_def [simp]: "r + s == case r of Rat a1 p1 q1 \<Rightarrow> case s of Rat a2 p2 q2 \<Rightarrow>
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let
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((r1, r2), den) = common ((p1, q1), (p2, q2))
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in let
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num = (if a1 then r1 else -r1) + (if a2 then r2 else -r2)
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in norm (Rat True num den)"
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uminus_rat_def [simp]: "- r == case r of Rat a p q \<Rightarrow>
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if p = 0 then Rat a p q
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else Rat (\<not> a) p q"
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times_rat_def [simp]: "r * s == case r of Rat a1 p1 q1 \<Rightarrow> case s of Rat a2 p2 q2 \<Rightarrow>
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norm (Rat (a1 = a2) (p1 * p2) (q1 * q2))"
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inverse_rat_def [simp]: "inverse r == case r of Rat a p q \<Rightarrow>
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if p = 0 then arbitrary
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else Rat a q p"
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le_rat_def [simp]: "r <= s == case r of Rat a1 p1 q1 \<Rightarrow> case s of Rat a2 p2 q2 \<Rightarrow>
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(\<not> a1 \<and> a2) \<or>
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(\<not> (a1 \<and> \<not> a2) \<and>
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(let
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((r1, r2), dummy) = common ((p1, q1), (p2, q2))
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in if a1 then r1 <= r2 else r2 <= r1))"
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code_syntax_tyco rat
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ml (target_atom "{*erat*}")
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haskell (target_atom "{*erat*}")
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code_alias
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(* an intermediate solution until name resolving of ad-hoc overloaded
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constants is refined *)
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"HOL.inverse" "Rational.inverse"
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"HOL.divide" "Rational.divide"
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code_syntax_const
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Fract
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ml ("{*of_quotient*}")
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haskell ("{*of_quotient*}")
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0 :: "rat"
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ml ("{*0::erat*}")
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haskell ("{*1::erat*}")
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1 :: "rat"
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ml ("{*1::erat*}")
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haskell ("{*1::erat*}")
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"op +" :: "rat \<Rightarrow> rat \<Rightarrow> rat"
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ml ("{*op + :: erat \<Rightarrow> erat \<Rightarrow> erat*}")
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haskell ("{*op + :: erat \<Rightarrow> erat \<Rightarrow> erat*}")
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uminus :: "rat \<Rightarrow> rat"
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ml ("{*uminus :: erat \<Rightarrow> erat*}")
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haskell ("{*uminus :: erat \<Rightarrow> erat*}")
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"op *" :: "rat \<Rightarrow> rat \<Rightarrow> rat"
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ml ("{*op * :: erat \<Rightarrow> erat \<Rightarrow> erat*}")
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haskell ("{*op * :: erat \<Rightarrow> erat \<Rightarrow> erat*}")
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inverse :: "rat \<Rightarrow> rat"
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ml ("{*inverse :: erat \<Rightarrow> erat*}")
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haskell ("{*inverse :: erat \<Rightarrow> erat*}")
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divide :: "rat \<Rightarrow> rat \<Rightarrow> rat"
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ml ("{*op * :: erat \<Rightarrow> erat \<Rightarrow> erat*}/ _/ ({*inverse :: erat \<Rightarrow> erat*}/ _)")
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haskell ("{*op * :: erat \<Rightarrow> erat \<Rightarrow> erat*}/ _/ ({*inverse :: erat \<Rightarrow> erat*}/ _)")
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"op <=" :: "rat \<Rightarrow> rat \<Rightarrow> bool"
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ml ("{*op <= :: erat \<Rightarrow> erat \<Rightarrow> bool*}")
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haskell ("{*op <= :: erat \<Rightarrow> erat \<Rightarrow> bool*}")
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end
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