src/HOL/Real/Rational.thy
author wenzelm
Fri Nov 17 02:20:03 2006 +0100 (2006-11-17)
changeset 21404 eb85850d3eb7
parent 20522 05072ae0d435
child 22456 6070e48ecb78
permissions -rw-r--r--
more robust syntax for definition/abbreviation/notation;
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(*  Title: HOL/Library/Rational.thy
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    ID:    $Id$
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    Author: Markus Wenzel, TU Muenchen
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*)
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header {* Rational numbers *}
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theory Rational
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imports Main
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uses ("rat_arith.ML")
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begin
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subsection {* Rational numbers *}
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subsubsection {* Equivalence of fractions *}
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definition
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  fraction :: "(int \<times> int) set" where
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  "fraction = {x. snd x \<noteq> 0}"
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definition
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  ratrel :: "((int \<times> int) \<times> (int \<times> int)) set" where
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  "ratrel = {(x,y). snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x}"
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lemma fraction_iff [simp]: "(x \<in> fraction) = (snd x \<noteq> 0)"
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by (simp add: fraction_def)
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lemma ratrel_iff [simp]:
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  "((x,y) \<in> ratrel) =
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   (snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x)"
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by (simp add: ratrel_def)
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lemma refl_ratrel: "refl fraction ratrel"
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by (auto simp add: refl_def fraction_def ratrel_def)
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lemma sym_ratrel: "sym ratrel"
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by (simp add: ratrel_def sym_def)
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lemma trans_ratrel_lemma:
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  assumes 1: "a * b' = a' * b"
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  assumes 2: "a' * b'' = a'' * b'"
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  assumes 3: "b' \<noteq> (0::int)"
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  shows "a * b'' = a'' * b"
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proof -
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  have "b' * (a * b'') = b'' * (a * b')" by simp
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  also note 1
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  also have "b'' * (a' * b) = b * (a' * b'')" by simp
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  also note 2
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  also have "b * (a'' * b') = b' * (a'' * b)" by simp
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  finally have "b' * (a * b'') = b' * (a'' * b)" .
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  with 3 show "a * b'' = a'' * b" by simp
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qed
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lemma trans_ratrel: "trans ratrel"
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by (auto simp add: trans_def elim: trans_ratrel_lemma)
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lemma equiv_ratrel: "equiv fraction ratrel"
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by (rule equiv.intro [OF refl_ratrel sym_ratrel trans_ratrel])
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lemmas equiv_ratrel_iff [iff] = eq_equiv_class_iff [OF equiv_ratrel]
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lemma equiv_ratrel_iff2:
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  "\<lbrakk>snd x \<noteq> 0; snd y \<noteq> 0\<rbrakk>
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    \<Longrightarrow> (ratrel `` {x} = ratrel `` {y}) = ((x,y) \<in> ratrel)"
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by (rule eq_equiv_class_iff [OF equiv_ratrel], simp_all)
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subsubsection {* The type of rational numbers *}
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typedef (Rat) rat = "fraction//ratrel"
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proof
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  have "(0,1) \<in> fraction" by (simp add: fraction_def)
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  thus "ratrel``{(0,1)} \<in> fraction//ratrel" by (rule quotientI)
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qed
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lemma ratrel_in_Rat [simp]: "snd x \<noteq> 0 \<Longrightarrow> ratrel``{x} \<in> Rat"
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by (simp add: Rat_def quotientI)
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declare Abs_Rat_inject [simp]  Abs_Rat_inverse [simp]
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definition
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  Fract :: "int \<Rightarrow> int \<Rightarrow> rat" where
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  "Fract a b = Abs_Rat (ratrel``{(a,b)})"
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theorem Rat_cases [case_names Fract, cases type: rat]:
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    "(!!a b. q = Fract a b ==> b \<noteq> 0 ==> C) ==> C"
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  by (cases q) (clarsimp simp add: Fract_def Rat_def fraction_def quotient_def)
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theorem Rat_induct [case_names Fract, induct type: rat]:
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    "(!!a b. b \<noteq> 0 ==> P (Fract a b)) ==> P q"
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  by (cases q) simp
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subsubsection {* Congruence lemmas *}
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lemma add_congruent2:
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     "(\<lambda>x y. ratrel``{(fst x * snd y + fst y * snd x, snd x * snd y)})
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      respects2 ratrel"
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apply (rule equiv_ratrel [THEN congruent2_commuteI])
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apply (simp_all add: left_distrib)
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done
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lemma minus_congruent:
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  "(\<lambda>x. ratrel``{(- fst x, snd x)}) respects ratrel"
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by (simp add: congruent_def)
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lemma mult_congruent2:
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  "(\<lambda>x y. ratrel``{(fst x * fst y, snd x * snd y)}) respects2 ratrel"
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by (rule equiv_ratrel [THEN congruent2_commuteI], simp_all)
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lemma inverse_congruent:
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  "(\<lambda>x. ratrel``{if fst x=0 then (0,1) else (snd x, fst x)}) respects ratrel"
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by (auto simp add: congruent_def mult_commute)
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lemma le_congruent2:
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  "(\<lambda>x y. {(fst x * snd y)*(snd x * snd y) \<le> (fst y * snd x)*(snd x * snd y)})
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   respects2 ratrel"
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proof (clarsimp simp add: congruent2_def)
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  fix a b a' b' c d c' d'::int
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  assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
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  assume eq1: "a * b' = a' * b"
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  assume eq2: "c * d' = c' * d"
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  let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
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  {
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    fix a b c d x :: int assume x: "x \<noteq> 0"
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    have "?le a b c d = ?le (a * x) (b * x) c d"
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    proof -
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      from x have "0 < x * x" by (auto simp add: zero_less_mult_iff)
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      hence "?le a b c d =
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          ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
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        by (simp add: mult_le_cancel_right)
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      also have "... = ?le (a * x) (b * x) c d"
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        by (simp add: mult_ac)
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      finally show ?thesis .
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    qed
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  } note le_factor = this
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  let ?D = "b * d" and ?D' = "b' * d'"
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  from neq have D: "?D \<noteq> 0" by simp
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  from neq have "?D' \<noteq> 0" by simp
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  hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"
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    by (rule le_factor)
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  also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')"
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    by (simp add: mult_ac)
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  also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
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    by (simp only: eq1 eq2)
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  also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
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    by (simp add: mult_ac)
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  also from D have "... = ?le a' b' c' d'"
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    by (rule le_factor [symmetric])
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  finally show "?le a b c d = ?le a' b' c' d'" .
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qed
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lemmas UN_ratrel = UN_equiv_class [OF equiv_ratrel]
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lemmas UN_ratrel2 = UN_equiv_class2 [OF equiv_ratrel equiv_ratrel]
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subsubsection {* Standard operations on rational numbers *}
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instance rat :: "{ord, zero, one, plus, times, minus, inverse, power}" ..
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defs (overloaded)
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  Zero_rat_def:  "0 == Fract 0 1"
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  One_rat_def:   "1 == Fract 1 1"
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  add_rat_def:
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   "q + r ==
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       Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
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           ratrel``{(fst x * snd y + fst y * snd x, snd x * snd y)})"
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  minus_rat_def:
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    "- q == Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel``{(- fst x, snd x)})"
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  diff_rat_def:  "q - r == q + - (r::rat)"
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  mult_rat_def:
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   "q * r ==
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       Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
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           ratrel``{(fst x * fst y, snd x * snd y)})"
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  inverse_rat_def:
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    "inverse q ==
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        Abs_Rat (\<Union>x \<in> Rep_Rat q.
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            ratrel``{if fst x=0 then (0,1) else (snd x, fst x)})"
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  divide_rat_def:  "q / r == q * inverse (r::rat)"
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  le_rat_def:
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   "q \<le> r == contents (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
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      {(fst x * snd y)*(snd x * snd y) \<le> (fst y * snd x)*(snd x * snd y)})"
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  less_rat_def: "(z < (w::rat)) == (z \<le> w & z \<noteq> w)"
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  abs_rat_def: "\<bar>q\<bar> == if q < 0 then -q else (q::rat)"
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primrec (rat)
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  rat_power_0:   "q ^ 0       = 1"
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  rat_power_Suc: "q ^ (Suc n) = (q::rat) * (q ^ n)"
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lemma zero_rat: "0 = Fract 0 1"
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by (simp add: Zero_rat_def)
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lemma one_rat: "1 = Fract 1 1"
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by (simp add: One_rat_def)
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theorem eq_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
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  (Fract a b = Fract c d) = (a * d = c * b)"
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by (simp add: Fract_def)
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theorem add_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
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  Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
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by (simp add: Fract_def add_rat_def add_congruent2 UN_ratrel2)
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theorem minus_rat: "b \<noteq> 0 ==> -(Fract a b) = Fract (-a) b"
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by (simp add: Fract_def minus_rat_def minus_congruent UN_ratrel)
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theorem diff_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
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    Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
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by (simp add: diff_rat_def add_rat minus_rat)
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theorem mult_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
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  Fract a b * Fract c d = Fract (a * c) (b * d)"
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by (simp add: Fract_def mult_rat_def mult_congruent2 UN_ratrel2)
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theorem inverse_rat: "a \<noteq> 0 ==> b \<noteq> 0 ==>
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  inverse (Fract a b) = Fract b a"
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by (simp add: Fract_def inverse_rat_def inverse_congruent UN_ratrel)
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theorem divide_rat: "c \<noteq> 0 ==> b \<noteq> 0 ==> d \<noteq> 0 ==>
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  Fract a b / Fract c d = Fract (a * d) (b * c)"
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by (simp add: divide_rat_def inverse_rat mult_rat)
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theorem le_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
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  (Fract a b \<le> Fract c d) = ((a * d) * (b * d) \<le> (c * b) * (b * d))"
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by (simp add: Fract_def le_rat_def le_congruent2 UN_ratrel2)
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theorem less_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
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    (Fract a b < Fract c d) = ((a * d) * (b * d) < (c * b) * (b * d))"
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by (simp add: less_rat_def le_rat eq_rat order_less_le)
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theorem abs_rat: "b \<noteq> 0 ==> \<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
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  by (simp add: abs_rat_def minus_rat zero_rat less_rat eq_rat)
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     (auto simp add: mult_less_0_iff zero_less_mult_iff order_le_less
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                split: abs_split)
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subsubsection {* The ordered field of rational numbers *}
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instance rat :: field
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proof
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  fix q r s :: rat
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  show "(q + r) + s = q + (r + s)"
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    by (induct q, induct r, induct s)
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       (simp add: add_rat add_ac mult_ac int_distrib)
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  show "q + r = r + q"
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    by (induct q, induct r) (simp add: add_rat add_ac mult_ac)
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  show "0 + q = q"
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    by (induct q) (simp add: zero_rat add_rat)
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  show "(-q) + q = 0"
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    by (induct q) (simp add: zero_rat minus_rat add_rat eq_rat)
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  show "q - r = q + (-r)"
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    by (induct q, induct r) (simp add: add_rat minus_rat diff_rat)
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  show "(q * r) * s = q * (r * s)"
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    by (induct q, induct r, induct s) (simp add: mult_rat mult_ac)
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  show "q * r = r * q"
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    by (induct q, induct r) (simp add: mult_rat mult_ac)
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  show "1 * q = q"
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    by (induct q) (simp add: one_rat mult_rat)
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  show "(q + r) * s = q * s + r * s"
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    by (induct q, induct r, induct s)
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       (simp add: add_rat mult_rat eq_rat int_distrib)
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  show "q \<noteq> 0 ==> inverse q * q = 1"
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    by (induct q) (simp add: inverse_rat mult_rat one_rat zero_rat eq_rat)
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  show "q / r = q * inverse r"
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    by (simp add: divide_rat_def)
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  show "0 \<noteq> (1::rat)"
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    by (simp add: zero_rat one_rat eq_rat)
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qed
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instance rat :: linorder
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proof
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  fix q r s :: rat
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  {
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    assume "q \<le> r" and "r \<le> s"
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    show "q \<le> s"
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    proof (insert prems, induct q, induct r, induct s)
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      fix a b c d e f :: int
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      assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
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      assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"
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      show "Fract a b \<le> Fract e f"
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      proof -
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        from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
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          by (auto simp add: zero_less_mult_iff linorder_neq_iff)
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        have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
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        proof -
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          from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
paulson@14365
   299
            by (simp add: le_rat)
paulson@14365
   300
          with ff show ?thesis by (simp add: mult_le_cancel_right)
paulson@14365
   301
        qed
paulson@14365
   302
        also have "... = (c * f) * (d * f) * (b * b)"
paulson@14365
   303
          by (simp only: mult_ac)
paulson@14365
   304
        also have "... \<le> (e * d) * (d * f) * (b * b)"
paulson@14365
   305
        proof -
paulson@14365
   306
          from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
paulson@14365
   307
            by (simp add: le_rat)
paulson@14365
   308
          with bb show ?thesis by (simp add: mult_le_cancel_right)
paulson@14365
   309
        qed
paulson@14365
   310
        finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
paulson@14365
   311
          by (simp only: mult_ac)
paulson@14365
   312
        with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
paulson@14365
   313
          by (simp add: mult_le_cancel_right)
paulson@14365
   314
        with neq show ?thesis by (simp add: le_rat)
paulson@14365
   315
      qed
paulson@14365
   316
    qed
paulson@14365
   317
  next
paulson@14365
   318
    assume "q \<le> r" and "r \<le> q"
paulson@14365
   319
    show "q = r"
paulson@14365
   320
    proof (insert prems, induct q, induct r)
paulson@14365
   321
      fix a b c d :: int
paulson@14365
   322
      assume neq: "b \<noteq> 0"  "d \<noteq> 0"
paulson@14365
   323
      assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"
paulson@14365
   324
      show "Fract a b = Fract c d"
paulson@14365
   325
      proof -
paulson@14365
   326
        from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
paulson@14365
   327
          by (simp add: le_rat)
paulson@14365
   328
        also have "... \<le> (a * d) * (b * d)"
paulson@14365
   329
        proof -
paulson@14365
   330
          from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
paulson@14365
   331
            by (simp add: le_rat)
paulson@14365
   332
          thus ?thesis by (simp only: mult_ac)
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   333
        qed
paulson@14365
   334
        finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
paulson@14365
   335
        moreover from neq have "b * d \<noteq> 0" by simp
paulson@14365
   336
        ultimately have "a * d = c * b" by simp
paulson@14365
   337
        with neq show ?thesis by (simp add: eq_rat)
paulson@14365
   338
      qed
paulson@14365
   339
    qed
paulson@14365
   340
  next
paulson@14365
   341
    show "q \<le> q"
paulson@14365
   342
      by (induct q) (simp add: le_rat)
paulson@14365
   343
    show "(q < r) = (q \<le> r \<and> q \<noteq> r)"
paulson@14365
   344
      by (simp only: less_rat_def)
paulson@14365
   345
    show "q \<le> r \<or> r \<le> q"
huffman@18913
   346
      by (induct q, induct r)
huffman@18913
   347
         (simp add: le_rat mult_commute, rule linorder_linear)
paulson@14365
   348
  }
paulson@14365
   349
qed
paulson@14365
   350
paulson@14365
   351
instance rat :: ordered_field
paulson@14365
   352
proof
paulson@14365
   353
  fix q r s :: rat
paulson@14365
   354
  show "q \<le> r ==> s + q \<le> s + r"
paulson@14365
   355
  proof (induct q, induct r, induct s)
paulson@14365
   356
    fix a b c d e f :: int
paulson@14365
   357
    assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
paulson@14365
   358
    assume le: "Fract a b \<le> Fract c d"
paulson@14365
   359
    show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
paulson@14365
   360
    proof -
paulson@14365
   361
      let ?F = "f * f" from neq have F: "0 < ?F"
paulson@14365
   362
        by (auto simp add: zero_less_mult_iff)
paulson@14365
   363
      from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
paulson@14365
   364
        by (simp add: le_rat)
paulson@14365
   365
      with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
paulson@14365
   366
        by (simp add: mult_le_cancel_right)
paulson@14365
   367
      with neq show ?thesis by (simp add: add_rat le_rat mult_ac int_distrib)
paulson@14365
   368
    qed
paulson@14365
   369
  qed
paulson@14365
   370
  show "q < r ==> 0 < s ==> s * q < s * r"
paulson@14365
   371
  proof (induct q, induct r, induct s)
paulson@14365
   372
    fix a b c d e f :: int
paulson@14365
   373
    assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
paulson@14365
   374
    assume le: "Fract a b < Fract c d"
paulson@14365
   375
    assume gt: "0 < Fract e f"
paulson@14365
   376
    show "Fract e f * Fract a b < Fract e f * Fract c d"
paulson@14365
   377
    proof -
paulson@14365
   378
      let ?E = "e * f" and ?F = "f * f"
paulson@14365
   379
      from neq gt have "0 < ?E"
paulson@14378
   380
        by (auto simp add: zero_rat less_rat le_rat order_less_le eq_rat)
paulson@14365
   381
      moreover from neq have "0 < ?F"
paulson@14365
   382
        by (auto simp add: zero_less_mult_iff)
paulson@14365
   383
      moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
paulson@14365
   384
        by (simp add: less_rat)
paulson@14365
   385
      ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
paulson@14365
   386
        by (simp add: mult_less_cancel_right)
paulson@14365
   387
      with neq show ?thesis
paulson@14365
   388
        by (simp add: less_rat mult_rat mult_ac)
paulson@14365
   389
    qed
paulson@14365
   390
  qed
paulson@14365
   391
  show "\<bar>q\<bar> = (if q < 0 then -q else q)"
paulson@14365
   392
    by (simp only: abs_rat_def)
paulson@14365
   393
qed
paulson@14365
   394
paulson@14365
   395
instance rat :: division_by_zero
paulson@14365
   396
proof
huffman@18913
   397
  show "inverse 0 = (0::rat)"
huffman@18913
   398
    by (simp add: zero_rat Fract_def inverse_rat_def
huffman@18913
   399
                  inverse_congruent UN_ratrel)
paulson@14365
   400
qed
paulson@14365
   401
huffman@20522
   402
instance rat :: recpower
huffman@20522
   403
proof
huffman@20522
   404
  fix q :: rat
huffman@20522
   405
  fix n :: nat
huffman@20522
   406
  show "q ^ 0 = 1" by simp
huffman@20522
   407
  show "q ^ (Suc n) = q * (q ^ n)" by simp
huffman@20522
   408
qed
huffman@20522
   409
paulson@14365
   410
paulson@14365
   411
subsection {* Various Other Results *}
paulson@14365
   412
paulson@14365
   413
lemma minus_rat_cancel [simp]: "b \<noteq> 0 ==> Fract (-a) (-b) = Fract a b"
huffman@18913
   414
by (simp add: eq_rat)
paulson@14365
   415
paulson@14365
   416
theorem Rat_induct_pos [case_names Fract, induct type: rat]:
paulson@14365
   417
  assumes step: "!!a b. 0 < b ==> P (Fract a b)"
paulson@14365
   418
    shows "P q"
paulson@14365
   419
proof (cases q)
paulson@14365
   420
  have step': "!!a b. b < 0 ==> P (Fract a b)"
paulson@14365
   421
  proof -
paulson@14365
   422
    fix a::int and b::int
paulson@14365
   423
    assume b: "b < 0"
paulson@14365
   424
    hence "0 < -b" by simp
paulson@14365
   425
    hence "P (Fract (-a) (-b))" by (rule step)
paulson@14365
   426
    thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
paulson@14365
   427
  qed
paulson@14365
   428
  case (Fract a b)
paulson@14365
   429
  thus "P q" by (force simp add: linorder_neq_iff step step')
paulson@14365
   430
qed
paulson@14365
   431
paulson@14365
   432
lemma zero_less_Fract_iff:
paulson@14365
   433
     "0 < b ==> (0 < Fract a b) = (0 < a)"
wenzelm@14691
   434
by (simp add: zero_rat less_rat order_less_imp_not_eq2 zero_less_mult_iff)
paulson@14365
   435
paulson@14378
   436
lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
paulson@14378
   437
apply (insert add_rat [of concl: m n 1 1])
wenzelm@14691
   438
apply (simp add: one_rat  [symmetric])
paulson@14378
   439
done
paulson@14378
   440
paulson@14378
   441
lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
wenzelm@14691
   442
apply (induct k)
wenzelm@14691
   443
apply (simp add: zero_rat)
wenzelm@14691
   444
apply (simp add: Fract_add_one)
paulson@14378
   445
done
paulson@14378
   446
paulson@14378
   447
lemma Fract_of_int_eq: "Fract k 1 = of_int k"
wenzelm@14691
   448
proof (cases k rule: int_cases)
paulson@14378
   449
  case (nonneg n)
paulson@14378
   450
    thus ?thesis by (simp add: int_eq_of_nat Fract_of_nat_eq)
wenzelm@14691
   451
next
paulson@14378
   452
  case (neg n)
paulson@14378
   453
    hence "Fract k 1 = - (Fract (of_nat (Suc n)) 1)"
wenzelm@14691
   454
      by (simp only: minus_rat int_eq_of_nat)
paulson@14378
   455
    also have "... =  - (of_nat (Suc n))"
paulson@14378
   456
      by (simp only: Fract_of_nat_eq)
wenzelm@14691
   457
    finally show ?thesis
wenzelm@14691
   458
      by (simp add: only: prems int_eq_of_nat of_int_minus of_int_of_nat_eq)
wenzelm@14691
   459
qed
paulson@14378
   460
paulson@14378
   461
wenzelm@14691
   462
subsection {* Numerals and Arithmetic *}
paulson@14387
   463
paulson@14387
   464
instance rat :: number ..
paulson@14387
   465
paulson@15013
   466
defs (overloaded)
haftmann@20485
   467
  rat_number_of_def: "(number_of w :: rat) == of_int w"
paulson@15013
   468
    --{*the type constraint is essential!*}
paulson@14387
   469
paulson@14387
   470
instance rat :: number_ring
wenzelm@19765
   471
  by default (simp add: rat_number_of_def) 
paulson@14387
   472
paulson@14387
   473
use "rat_arith.ML"
paulson@14387
   474
setup rat_arith_setup
paulson@14387
   475
paulson@14365
   476
end