paulson@14365
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(* Title: HOL/Library/Rational.thy
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paulson@14365
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ID: $Id$
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paulson@14365
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Author: Markus Wenzel, TU Muenchen
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paulson@14365
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*)
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paulson@14365
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wenzelm@14691
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header {* Rational numbers *}
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paulson@14365
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nipkow@15131
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theory Rational
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berghofe@24533
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imports Abstract_Rat
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haftmann@16417
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uses ("rat_arith.ML")
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nipkow@15131
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begin
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huffman@18913
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subsection {* Rational numbers *}
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paulson@14365
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subsubsection {* Equivalence of fractions *}
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paulson@14365
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wenzelm@19765
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definition
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wenzelm@21404
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fraction :: "(int \<times> int) set" where
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wenzelm@19765
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"fraction = {x. snd x \<noteq> 0}"
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wenzelm@21404
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definition
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wenzelm@21404
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ratrel :: "((int \<times> int) \<times> (int \<times> int)) set" where
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wenzelm@19765
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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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huffman@18913
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lemma ratrel_iff [simp]:
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huffman@18913
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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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huffman@18913
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assumes 1: "a * b' = a' * b"
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huffman@18913
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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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huffman@18913
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have "b' * (a * b'') = b'' * (a * b')" by simp
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huffman@18913
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also note 1
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huffman@18913
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also have "b'' * (a' * b) = b * (a' * b'')" by simp
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huffman@18913
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also note 2
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huffman@18913
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also have "b * (a'' * b') = b' * (a'' * b)" by simp
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huffman@18913
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finally have "b' * (a * b'') = b' * (a'' * b)" .
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huffman@18913
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with 3 show "a * b'' = a'' * b" by simp
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qed
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paulson@14365
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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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wenzelm@21404
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Fract :: "int \<Rightarrow> int \<Rightarrow> rat" where
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haftmann@24198
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[code func del]: "Fract a b = Abs_Rat (ratrel``{(a,b)})"
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lemma Fract_zero:
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haftmann@24198
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"Fract k 0 = Fract l 0"
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by (simp add: Fract_def ratrel_def)
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huffman@18913
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theorem Rat_cases [case_names Fract, cases type: rat]:
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wenzelm@21404
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"(!!a b. q = Fract a b ==> b \<noteq> 0 ==> C) ==> C"
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wenzelm@21404
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by (cases q) (clarsimp simp add: Fract_def Rat_def fraction_def quotient_def)
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huffman@18913
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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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huffman@18913
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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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huffman@18913
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"(\<lambda>x y. ratrel``{(fst x * fst y, snd x * snd y)}) respects2 ratrel"
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huffman@18913
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by (rule equiv_ratrel [THEN congruent2_commuteI], simp_all)
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huffman@18913
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huffman@18913
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lemma inverse_congruent:
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huffman@18913
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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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huffman@18913
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by (auto simp add: congruent_def mult_commute)
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huffman@18913
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lemma le_congruent2:
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huffman@18982
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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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huffman@18913
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respects2 ratrel"
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proof (clarsimp simp add: congruent2_def)
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huffman@18913
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fix a b a' b' c d c' d'::int
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paulson@14365
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assume neq: "b \<noteq> 0" "b' \<noteq> 0" "d \<noteq> 0" "d' \<noteq> 0"
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huffman@18913
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assume eq1: "a * b' = a' * b"
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huffman@18913
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assume eq2: "c * d' = c' * d"
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paulson@14365
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paulson@14365
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let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
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paulson@14365
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{
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paulson@14365
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fix a b c d x :: int assume x: "x \<noteq> 0"
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paulson@14365
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have "?le a b c d = ?le (a * x) (b * x) c d"
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paulson@14365
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proof -
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paulson@14365
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from x have "0 < x * x" by (auto simp add: zero_less_mult_iff)
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paulson@14365
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hence "?le a b c d =
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paulson@14365
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((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
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paulson@14365
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by (simp add: mult_le_cancel_right)
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paulson@14365
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also have "... = ?le (a * x) (b * x) c d"
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paulson@14365
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by (simp add: mult_ac)
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paulson@14365
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finally show ?thesis .
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paulson@14365
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qed
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paulson@14365
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} note le_factor = this
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paulson@14365
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paulson@14365
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let ?D = "b * d" and ?D' = "b' * d'"
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paulson@14365
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from neq have D: "?D \<noteq> 0" by simp
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paulson@14365
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from neq have "?D' \<noteq> 0" by simp
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paulson@14365
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hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"
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paulson@14365
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by (rule le_factor)
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paulson@14365
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also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')"
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paulson@14365
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by (simp add: mult_ac)
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paulson@14365
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also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
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paulson@14365
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by (simp only: eq1 eq2)
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paulson@14365
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also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
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paulson@14365
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by (simp add: mult_ac)
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paulson@14365
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also from D have "... = ?le a' b' c' d'"
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paulson@14365
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by (rule le_factor [symmetric])
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huffman@18913
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finally show "?le a b c d = ?le a' b' c' d'" .
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paulson@14365
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qed
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paulson@14365
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huffman@18913
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lemmas UN_ratrel = UN_equiv_class [OF equiv_ratrel]
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huffman@18913
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lemmas UN_ratrel2 = UN_equiv_class2 [OF equiv_ratrel equiv_ratrel]
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paulson@14365
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paulson@14365
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paulson@14365
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subsubsection {* Standard operations on rational numbers *}
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paulson@14365
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haftmann@23879
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instance rat :: zero
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haftmann@23879
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Zero_rat_def: "0 == Fract 0 1" ..
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haftmann@24198
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lemmas [code func del] = Zero_rat_def
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paulson@14365
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haftmann@23879
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instance rat :: one
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haftmann@23879
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One_rat_def: "1 == Fract 1 1" ..
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haftmann@24198
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lemmas [code func del] = One_rat_def
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huffman@18913
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haftmann@23879
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instance rat :: plus
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huffman@18913
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add_rat_def:
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huffman@18913
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"q + r ==
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huffman@18913
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Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
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haftmann@23879
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ratrel``{(fst x * snd y + fst y * snd x, snd x * snd y)})" ..
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haftmann@23879
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lemmas [code func del] = add_rat_def
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huffman@18913
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haftmann@23879
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instance rat :: minus
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huffman@18913
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minus_rat_def:
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huffman@18913
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"- q == Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel``{(- fst x, snd x)})"
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haftmann@23879
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diff_rat_def: "q - r == q + - (r::rat)" ..
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haftmann@24198
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lemmas [code func del] = minus_rat_def diff_rat_def
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huffman@18913
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haftmann@23879
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instance rat :: times
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huffman@18913
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mult_rat_def:
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huffman@18913
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"q * r ==
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huffman@18913
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Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
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haftmann@23879
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ratrel``{(fst x * fst y, snd x * snd y)})" ..
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haftmann@23879
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lemmas [code func del] = mult_rat_def
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huffman@18913
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haftmann@23879
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instance rat :: inverse
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huffman@18913
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inverse_rat_def:
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huffman@18913
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"inverse q ==
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huffman@18913
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Abs_Rat (\<Union>x \<in> Rep_Rat q.
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huffman@18913
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ratrel``{if fst x=0 then (0,1) else (snd x, fst x)})"
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haftmann@23879
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divide_rat_def: "q / r == q * inverse (r::rat)" ..
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haftmann@24198
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lemmas [code func del] = inverse_rat_def divide_rat_def
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huffman@18913
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haftmann@23879
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instance rat :: ord
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huffman@18913
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le_rat_def:
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huffman@18982
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"q \<le> r == contents (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
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huffman@18982
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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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haftmann@23879
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less_rat_def: "(z < (w::rat)) == (z \<le> w & z \<noteq> w)" ..
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haftmann@23879
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lemmas [code func del] = le_rat_def less_rat_def
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huffman@18913
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haftmann@23879
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instance rat :: abs
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haftmann@23879
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abs_rat_def: "\<bar>q\<bar> == if q < 0 then -q else (q::rat)" ..
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huffman@18913
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nipkow@24506
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instance rat :: sgn
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nipkow@24506
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sgn_rat_def: "sgn(q::rat) == (if q=0 then 0 else if 0<q then 1 else - 1)" ..
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nipkow@24506
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haftmann@23879
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instance rat :: power ..
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paulson@14365
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huffman@20522
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primrec (rat)
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huffman@20522
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rat_power_0: "q ^ 0 = 1"
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huffman@20522
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rat_power_Suc: "q ^ (Suc n) = (q::rat) * (q ^ n)"
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huffman@20522
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huffman@18913
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theorem eq_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
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huffman@18913
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(Fract a b = Fract c d) = (a * d = c * b)"
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huffman@18913
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by (simp add: Fract_def)
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paulson@14365
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paulson@14365
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theorem add_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
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paulson@14365
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Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
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huffman@18913
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by (simp add: Fract_def add_rat_def add_congruent2 UN_ratrel2)
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paulson@14365
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paulson@14365
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theorem minus_rat: "b \<noteq> 0 ==> -(Fract a b) = Fract (-a) b"
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huffman@18913
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by (simp add: Fract_def minus_rat_def minus_congruent UN_ratrel)
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paulson@14365
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paulson@14365
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theorem diff_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
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paulson@14365
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Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
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huffman@18913
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by (simp add: diff_rat_def add_rat minus_rat)
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paulson@14365
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paulson@14365
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theorem mult_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
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paulson@14365
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Fract a b * Fract c d = Fract (a * c) (b * d)"
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huffman@18913
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by (simp add: Fract_def mult_rat_def mult_congruent2 UN_ratrel2)
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paulson@14365
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huffman@18913
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theorem inverse_rat: "a \<noteq> 0 ==> b \<noteq> 0 ==>
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paulson@14365
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inverse (Fract a b) = Fract b a"
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huffman@18913
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by (simp add: Fract_def inverse_rat_def inverse_congruent UN_ratrel)
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paulson@14365
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huffman@18913
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theorem divide_rat: "c \<noteq> 0 ==> b \<noteq> 0 ==> d \<noteq> 0 ==>
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paulson@14365
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Fract a b / Fract c d = Fract (a * d) (b * c)"
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huffman@18913
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by (simp add: divide_rat_def inverse_rat mult_rat)
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paulson@14365
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paulson@14365
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theorem le_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
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paulson@14365
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(Fract a b \<le> Fract c d) = ((a * d) * (b * d) \<le> (c * b) * (b * d))"
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huffman@18982
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by (simp add: Fract_def le_rat_def le_congruent2 UN_ratrel2)
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paulson@14365
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paulson@14365
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theorem less_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
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paulson@14365
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(Fract a b < Fract c d) = ((a * d) * (b * d) < (c * b) * (b * d))"
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huffman@18913
|
254 |
by (simp add: less_rat_def le_rat eq_rat order_less_le)
|
paulson@14365
|
255 |
|
paulson@14365
|
256 |
theorem abs_rat: "b \<noteq> 0 ==> \<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
|
haftmann@23879
|
257 |
by (simp add: abs_rat_def minus_rat Zero_rat_def less_rat eq_rat)
|
wenzelm@14691
|
258 |
(auto simp add: mult_less_0_iff zero_less_mult_iff order_le_less
|
paulson@14365
|
259 |
split: abs_split)
|
paulson@14365
|
260 |
|
paulson@14365
|
261 |
|
paulson@14365
|
262 |
subsubsection {* The ordered field of rational numbers *}
|
paulson@14365
|
263 |
|
paulson@14365
|
264 |
instance rat :: field
|
paulson@14365
|
265 |
proof
|
paulson@14365
|
266 |
fix q r s :: rat
|
paulson@14365
|
267 |
show "(q + r) + s = q + (r + s)"
|
huffman@18913
|
268 |
by (induct q, induct r, induct s)
|
huffman@18913
|
269 |
(simp add: add_rat add_ac mult_ac int_distrib)
|
paulson@14365
|
270 |
show "q + r = r + q"
|
paulson@14365
|
271 |
by (induct q, induct r) (simp add: add_rat add_ac mult_ac)
|
paulson@14365
|
272 |
show "0 + q = q"
|
haftmann@23879
|
273 |
by (induct q) (simp add: Zero_rat_def add_rat)
|
paulson@14365
|
274 |
show "(-q) + q = 0"
|
haftmann@23879
|
275 |
by (induct q) (simp add: Zero_rat_def minus_rat add_rat eq_rat)
|
paulson@14365
|
276 |
show "q - r = q + (-r)"
|
paulson@14365
|
277 |
by (induct q, induct r) (simp add: add_rat minus_rat diff_rat)
|
paulson@14365
|
278 |
show "(q * r) * s = q * (r * s)"
|
paulson@14365
|
279 |
by (induct q, induct r, induct s) (simp add: mult_rat mult_ac)
|
paulson@14365
|
280 |
show "q * r = r * q"
|
paulson@14365
|
281 |
by (induct q, induct r) (simp add: mult_rat mult_ac)
|
paulson@14365
|
282 |
show "1 * q = q"
|
haftmann@23879
|
283 |
by (induct q) (simp add: One_rat_def mult_rat)
|
paulson@14365
|
284 |
show "(q + r) * s = q * s + r * s"
|
wenzelm@14691
|
285 |
by (induct q, induct r, induct s)
|
paulson@14365
|
286 |
(simp add: add_rat mult_rat eq_rat int_distrib)
|
paulson@14365
|
287 |
show "q \<noteq> 0 ==> inverse q * q = 1"
|
haftmann@23879
|
288 |
by (induct q) (simp add: inverse_rat mult_rat One_rat_def Zero_rat_def eq_rat)
|
paulson@14430
|
289 |
show "q / r = q * inverse r"
|
wenzelm@14691
|
290 |
by (simp add: divide_rat_def)
|
paulson@14365
|
291 |
show "0 \<noteq> (1::rat)"
|
haftmann@23879
|
292 |
by (simp add: Zero_rat_def One_rat_def eq_rat)
|
paulson@14365
|
293 |
qed
|
paulson@14365
|
294 |
|
paulson@14365
|
295 |
instance rat :: linorder
|
paulson@14365
|
296 |
proof
|
paulson@14365
|
297 |
fix q r s :: rat
|
paulson@14365
|
298 |
{
|
paulson@14365
|
299 |
assume "q \<le> r" and "r \<le> s"
|
paulson@14365
|
300 |
show "q \<le> s"
|
paulson@14365
|
301 |
proof (insert prems, induct q, induct r, induct s)
|
paulson@14365
|
302 |
fix a b c d e f :: int
|
paulson@14365
|
303 |
assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0"
|
paulson@14365
|
304 |
assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"
|
paulson@14365
|
305 |
show "Fract a b \<le> Fract e f"
|
paulson@14365
|
306 |
proof -
|
paulson@14365
|
307 |
from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
|
paulson@14365
|
308 |
by (auto simp add: zero_less_mult_iff linorder_neq_iff)
|
paulson@14365
|
309 |
have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
|
paulson@14365
|
310 |
proof -
|
paulson@14365
|
311 |
from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
|
paulson@14365
|
312 |
by (simp add: le_rat)
|
paulson@14365
|
313 |
with ff show ?thesis by (simp add: mult_le_cancel_right)
|
paulson@14365
|
314 |
qed
|
paulson@14365
|
315 |
also have "... = (c * f) * (d * f) * (b * b)"
|
paulson@14365
|
316 |
by (simp only: mult_ac)
|
paulson@14365
|
317 |
also have "... \<le> (e * d) * (d * f) * (b * b)"
|
paulson@14365
|
318 |
proof -
|
paulson@14365
|
319 |
from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
|
paulson@14365
|
320 |
by (simp add: le_rat)
|
paulson@14365
|
321 |
with bb show ?thesis by (simp add: mult_le_cancel_right)
|
paulson@14365
|
322 |
qed
|
paulson@14365
|
323 |
finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
|
paulson@14365
|
324 |
by (simp only: mult_ac)
|
paulson@14365
|
325 |
with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
|
paulson@14365
|
326 |
by (simp add: mult_le_cancel_right)
|
paulson@14365
|
327 |
with neq show ?thesis by (simp add: le_rat)
|
paulson@14365
|
328 |
qed
|
paulson@14365
|
329 |
qed
|
paulson@14365
|
330 |
next
|
paulson@14365
|
331 |
assume "q \<le> r" and "r \<le> q"
|
paulson@14365
|
332 |
show "q = r"
|
paulson@14365
|
333 |
proof (insert prems, induct q, induct r)
|
paulson@14365
|
334 |
fix a b c d :: int
|
paulson@14365
|
335 |
assume neq: "b \<noteq> 0" "d \<noteq> 0"
|
paulson@14365
|
336 |
assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"
|
paulson@14365
|
337 |
show "Fract a b = Fract c d"
|
paulson@14365
|
338 |
proof -
|
paulson@14365
|
339 |
from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
|
paulson@14365
|
340 |
by (simp add: le_rat)
|
paulson@14365
|
341 |
also have "... \<le> (a * d) * (b * d)"
|
paulson@14365
|
342 |
proof -
|
paulson@14365
|
343 |
from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
|
paulson@14365
|
344 |
by (simp add: le_rat)
|
paulson@14365
|
345 |
thus ?thesis by (simp only: mult_ac)
|
paulson@14365
|
346 |
qed
|
paulson@14365
|
347 |
finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
|
paulson@14365
|
348 |
moreover from neq have "b * d \<noteq> 0" by simp
|
paulson@14365
|
349 |
ultimately have "a * d = c * b" by simp
|
paulson@14365
|
350 |
with neq show ?thesis by (simp add: eq_rat)
|
paulson@14365
|
351 |
qed
|
paulson@14365
|
352 |
qed
|
paulson@14365
|
353 |
next
|
paulson@14365
|
354 |
show "q \<le> q"
|
paulson@14365
|
355 |
by (induct q) (simp add: le_rat)
|
paulson@14365
|
356 |
show "(q < r) = (q \<le> r \<and> q \<noteq> r)"
|
paulson@14365
|
357 |
by (simp only: less_rat_def)
|
paulson@14365
|
358 |
show "q \<le> r \<or> r \<le> q"
|
huffman@18913
|
359 |
by (induct q, induct r)
|
huffman@18913
|
360 |
(simp add: le_rat mult_commute, rule linorder_linear)
|
paulson@14365
|
361 |
}
|
paulson@14365
|
362 |
qed
|
paulson@14365
|
363 |
|
haftmann@22456
|
364 |
instance rat :: distrib_lattice
|
haftmann@22456
|
365 |
"inf r s \<equiv> min r s"
|
haftmann@22456
|
366 |
"sup r s \<equiv> max r s"
|
haftmann@22456
|
367 |
by default (auto simp add: min_max.sup_inf_distrib1 inf_rat_def sup_rat_def)
|
haftmann@22456
|
368 |
|
paulson@14365
|
369 |
instance rat :: ordered_field
|
paulson@14365
|
370 |
proof
|
paulson@14365
|
371 |
fix q r s :: rat
|
paulson@14365
|
372 |
show "q \<le> r ==> s + q \<le> s + r"
|
paulson@14365
|
373 |
proof (induct q, induct r, induct s)
|
paulson@14365
|
374 |
fix a b c d e f :: int
|
paulson@14365
|
375 |
assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0"
|
paulson@14365
|
376 |
assume le: "Fract a b \<le> Fract c d"
|
paulson@14365
|
377 |
show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
|
paulson@14365
|
378 |
proof -
|
paulson@14365
|
379 |
let ?F = "f * f" from neq have F: "0 < ?F"
|
paulson@14365
|
380 |
by (auto simp add: zero_less_mult_iff)
|
paulson@14365
|
381 |
from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
|
paulson@14365
|
382 |
by (simp add: le_rat)
|
paulson@14365
|
383 |
with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
|
paulson@14365
|
384 |
by (simp add: mult_le_cancel_right)
|
paulson@14365
|
385 |
with neq show ?thesis by (simp add: add_rat le_rat mult_ac int_distrib)
|
paulson@14365
|
386 |
qed
|
paulson@14365
|
387 |
qed
|
paulson@14365
|
388 |
show "q < r ==> 0 < s ==> s * q < s * r"
|
paulson@14365
|
389 |
proof (induct q, induct r, induct s)
|
paulson@14365
|
390 |
fix a b c d e f :: int
|
paulson@14365
|
391 |
assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0"
|
paulson@14365
|
392 |
assume le: "Fract a b < Fract c d"
|
paulson@14365
|
393 |
assume gt: "0 < Fract e f"
|
paulson@14365
|
394 |
show "Fract e f * Fract a b < Fract e f * Fract c d"
|
paulson@14365
|
395 |
proof -
|
paulson@14365
|
396 |
let ?E = "e * f" and ?F = "f * f"
|
paulson@14365
|
397 |
from neq gt have "0 < ?E"
|
haftmann@23879
|
398 |
by (auto simp add: Zero_rat_def less_rat le_rat order_less_le eq_rat)
|
paulson@14365
|
399 |
moreover from neq have "0 < ?F"
|
paulson@14365
|
400 |
by (auto simp add: zero_less_mult_iff)
|
paulson@14365
|
401 |
moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
|
paulson@14365
|
402 |
by (simp add: less_rat)
|
paulson@14365
|
403 |
ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
|
paulson@14365
|
404 |
by (simp add: mult_less_cancel_right)
|
paulson@14365
|
405 |
with neq show ?thesis
|
paulson@14365
|
406 |
by (simp add: less_rat mult_rat mult_ac)
|
paulson@14365
|
407 |
qed
|
paulson@14365
|
408 |
qed
|
paulson@14365
|
409 |
show "\<bar>q\<bar> = (if q < 0 then -q else q)"
|
paulson@14365
|
410 |
by (simp only: abs_rat_def)
|
nipkow@24506
|
411 |
qed (auto simp: sgn_rat_def)
|
paulson@14365
|
412 |
|
paulson@14365
|
413 |
instance rat :: division_by_zero
|
paulson@14365
|
414 |
proof
|
huffman@18913
|
415 |
show "inverse 0 = (0::rat)"
|
haftmann@23879
|
416 |
by (simp add: Zero_rat_def Fract_def inverse_rat_def
|
huffman@18913
|
417 |
inverse_congruent UN_ratrel)
|
paulson@14365
|
418 |
qed
|
paulson@14365
|
419 |
|
huffman@20522
|
420 |
instance rat :: recpower
|
huffman@20522
|
421 |
proof
|
huffman@20522
|
422 |
fix q :: rat
|
huffman@20522
|
423 |
fix n :: nat
|
huffman@20522
|
424 |
show "q ^ 0 = 1" by simp
|
huffman@20522
|
425 |
show "q ^ (Suc n) = q * (q ^ n)" by simp
|
huffman@20522
|
426 |
qed
|
huffman@20522
|
427 |
|
paulson@14365
|
428 |
|
paulson@14365
|
429 |
subsection {* Various Other Results *}
|
paulson@14365
|
430 |
|
paulson@14365
|
431 |
lemma minus_rat_cancel [simp]: "b \<noteq> 0 ==> Fract (-a) (-b) = Fract a b"
|
huffman@18913
|
432 |
by (simp add: eq_rat)
|
paulson@14365
|
433 |
|
paulson@14365
|
434 |
theorem Rat_induct_pos [case_names Fract, induct type: rat]:
|
paulson@14365
|
435 |
assumes step: "!!a b. 0 < b ==> P (Fract a b)"
|
paulson@14365
|
436 |
shows "P q"
|
paulson@14365
|
437 |
proof (cases q)
|
paulson@14365
|
438 |
have step': "!!a b. b < 0 ==> P (Fract a b)"
|
paulson@14365
|
439 |
proof -
|
paulson@14365
|
440 |
fix a::int and b::int
|
paulson@14365
|
441 |
assume b: "b < 0"
|
paulson@14365
|
442 |
hence "0 < -b" by simp
|
paulson@14365
|
443 |
hence "P (Fract (-a) (-b))" by (rule step)
|
paulson@14365
|
444 |
thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
|
paulson@14365
|
445 |
qed
|
paulson@14365
|
446 |
case (Fract a b)
|
paulson@14365
|
447 |
thus "P q" by (force simp add: linorder_neq_iff step step')
|
paulson@14365
|
448 |
qed
|
paulson@14365
|
449 |
|
paulson@14365
|
450 |
lemma zero_less_Fract_iff:
|
paulson@14365
|
451 |
"0 < b ==> (0 < Fract a b) = (0 < a)"
|
haftmann@23879
|
452 |
by (simp add: Zero_rat_def less_rat order_less_imp_not_eq2 zero_less_mult_iff)
|
paulson@14365
|
453 |
|
paulson@14378
|
454 |
lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
|
paulson@14378
|
455 |
apply (insert add_rat [of concl: m n 1 1])
|
haftmann@23879
|
456 |
apply (simp add: One_rat_def [symmetric])
|
paulson@14378
|
457 |
done
|
paulson@14378
|
458 |
|
huffman@23429
|
459 |
lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1"
|
haftmann@23879
|
460 |
by (induct k) (simp_all add: Zero_rat_def One_rat_def add_rat)
|
huffman@23429
|
461 |
|
huffman@23429
|
462 |
lemma of_int_rat: "of_int k = Fract k 1"
|
huffman@23429
|
463 |
by (cases k rule: int_diff_cases, simp add: of_nat_rat diff_rat)
|
huffman@23429
|
464 |
|
paulson@14378
|
465 |
lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
|
huffman@23429
|
466 |
by (rule of_nat_rat [symmetric])
|
paulson@14378
|
467 |
|
paulson@14378
|
468 |
lemma Fract_of_int_eq: "Fract k 1 = of_int k"
|
huffman@23429
|
469 |
by (rule of_int_rat [symmetric])
|
paulson@14378
|
470 |
|
haftmann@24198
|
471 |
lemma Fract_of_int_quotient: "Fract k l = (if l = 0 then Fract 1 0 else of_int k / of_int l)"
|
haftmann@24198
|
472 |
by (auto simp add: Fract_zero Fract_of_int_eq [symmetric] divide_rat)
|
haftmann@24198
|
473 |
|
paulson@14378
|
474 |
|
wenzelm@14691
|
475 |
subsection {* Numerals and Arithmetic *}
|
paulson@14387
|
476 |
|
haftmann@22456
|
477 |
instance rat :: number
|
haftmann@22456
|
478 |
rat_number_of_def: "(number_of w :: rat) \<equiv> of_int w" ..
|
paulson@14387
|
479 |
|
paulson@14387
|
480 |
instance rat :: number_ring
|
wenzelm@19765
|
481 |
by default (simp add: rat_number_of_def)
|
paulson@14387
|
482 |
|
paulson@14387
|
483 |
use "rat_arith.ML"
|
wenzelm@24075
|
484 |
declaration {* K rat_arith_setup *}
|
paulson@14387
|
485 |
|
huffman@23342
|
486 |
|
huffman@23342
|
487 |
subsection {* Embedding from Rationals to other Fields *}
|
huffman@23342
|
488 |
|
haftmann@24198
|
489 |
class field_char_0 = field + ring_char_0
|
huffman@23342
|
490 |
|
huffman@23342
|
491 |
instance ordered_field < field_char_0 ..
|
huffman@23342
|
492 |
|
huffman@23342
|
493 |
definition
|
huffman@23342
|
494 |
of_rat :: "rat \<Rightarrow> 'a::field_char_0"
|
huffman@23342
|
495 |
where
|
haftmann@24198
|
496 |
[code func del]: "of_rat q = contents (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})"
|
huffman@23342
|
497 |
|
huffman@23342
|
498 |
lemma of_rat_congruent:
|
huffman@23342
|
499 |
"(\<lambda>(a, b). {of_int a / of_int b::'a::field_char_0}) respects ratrel"
|
huffman@23342
|
500 |
apply (rule congruent.intro)
|
huffman@23342
|
501 |
apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
|
huffman@23342
|
502 |
apply (simp only: of_int_mult [symmetric])
|
huffman@23342
|
503 |
done
|
huffman@23342
|
504 |
|
huffman@23342
|
505 |
lemma of_rat_rat:
|
huffman@23342
|
506 |
"b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
|
huffman@23342
|
507 |
unfolding Fract_def of_rat_def
|
huffman@23342
|
508 |
by (simp add: UN_ratrel of_rat_congruent)
|
huffman@23342
|
509 |
|
huffman@23342
|
510 |
lemma of_rat_0 [simp]: "of_rat 0 = 0"
|
huffman@23342
|
511 |
by (simp add: Zero_rat_def of_rat_rat)
|
huffman@23342
|
512 |
|
huffman@23342
|
513 |
lemma of_rat_1 [simp]: "of_rat 1 = 1"
|
huffman@23342
|
514 |
by (simp add: One_rat_def of_rat_rat)
|
huffman@23342
|
515 |
|
huffman@23342
|
516 |
lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
|
huffman@23342
|
517 |
by (induct a, induct b, simp add: add_rat of_rat_rat add_frac_eq)
|
huffman@23342
|
518 |
|
huffman@23343
|
519 |
lemma of_rat_minus: "of_rat (- a) = - of_rat a"
|
huffman@23343
|
520 |
by (induct a, simp add: minus_rat of_rat_rat)
|
huffman@23343
|
521 |
|
huffman@23343
|
522 |
lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
|
huffman@23343
|
523 |
by (simp only: diff_minus of_rat_add of_rat_minus)
|
huffman@23343
|
524 |
|
huffman@23342
|
525 |
lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
|
huffman@23342
|
526 |
apply (induct a, induct b, simp add: mult_rat of_rat_rat)
|
huffman@23342
|
527 |
apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
|
huffman@23342
|
528 |
done
|
huffman@23342
|
529 |
|
huffman@23342
|
530 |
lemma nonzero_of_rat_inverse:
|
huffman@23342
|
531 |
"a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
|
huffman@23343
|
532 |
apply (rule inverse_unique [symmetric])
|
huffman@23343
|
533 |
apply (simp add: of_rat_mult [symmetric])
|
huffman@23342
|
534 |
done
|
huffman@23342
|
535 |
|
huffman@23342
|
536 |
lemma of_rat_inverse:
|
huffman@23342
|
537 |
"(of_rat (inverse a)::'a::{field_char_0,division_by_zero}) =
|
huffman@23342
|
538 |
inverse (of_rat a)"
|
huffman@23342
|
539 |
by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
|
huffman@23342
|
540 |
|
huffman@23342
|
541 |
lemma nonzero_of_rat_divide:
|
huffman@23342
|
542 |
"b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
|
huffman@23342
|
543 |
by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
|
huffman@23342
|
544 |
|
huffman@23342
|
545 |
lemma of_rat_divide:
|
huffman@23342
|
546 |
"(of_rat (a / b)::'a::{field_char_0,division_by_zero})
|
huffman@23342
|
547 |
= of_rat a / of_rat b"
|
huffman@23342
|
548 |
by (cases "b = 0", simp_all add: nonzero_of_rat_divide)
|
huffman@23342
|
549 |
|
huffman@23343
|
550 |
lemma of_rat_power:
|
huffman@23343
|
551 |
"(of_rat (a ^ n)::'a::{field_char_0,recpower}) = of_rat a ^ n"
|
huffman@23343
|
552 |
by (induct n) (simp_all add: of_rat_mult power_Suc)
|
huffman@23343
|
553 |
|
huffman@23343
|
554 |
lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
|
huffman@23343
|
555 |
apply (induct a, induct b)
|
huffman@23343
|
556 |
apply (simp add: of_rat_rat eq_rat)
|
huffman@23343
|
557 |
apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
|
huffman@23343
|
558 |
apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
|
huffman@23343
|
559 |
done
|
huffman@23343
|
560 |
|
huffman@23343
|
561 |
lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
|
huffman@23343
|
562 |
|
huffman@23343
|
563 |
lemma of_rat_eq_id [simp]: "of_rat = (id :: rat \<Rightarrow> rat)"
|
huffman@23343
|
564 |
proof
|
huffman@23343
|
565 |
fix a
|
huffman@23343
|
566 |
show "of_rat a = id a"
|
huffman@23343
|
567 |
by (induct a)
|
huffman@23343
|
568 |
(simp add: of_rat_rat divide_rat Fract_of_int_eq [symmetric])
|
huffman@23343
|
569 |
qed
|
huffman@23343
|
570 |
|
huffman@23343
|
571 |
text{*Collapse nested embeddings*}
|
huffman@23343
|
572 |
lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
|
huffman@23343
|
573 |
by (induct n) (simp_all add: of_rat_add)
|
huffman@23343
|
574 |
|
huffman@23343
|
575 |
lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
|
huffman@23365
|
576 |
by (cases z rule: int_diff_cases, simp add: of_rat_diff)
|
huffman@23343
|
577 |
|
huffman@23343
|
578 |
lemma of_rat_number_of_eq [simp]:
|
huffman@23343
|
579 |
"of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})"
|
huffman@23343
|
580 |
by (simp add: number_of_eq)
|
huffman@23343
|
581 |
|
haftmann@23879
|
582 |
lemmas zero_rat = Zero_rat_def
|
haftmann@23879
|
583 |
lemmas one_rat = One_rat_def
|
haftmann@23879
|
584 |
|
haftmann@24198
|
585 |
abbreviation
|
haftmann@24198
|
586 |
rat_of_nat :: "nat \<Rightarrow> rat"
|
haftmann@24198
|
587 |
where
|
haftmann@24198
|
588 |
"rat_of_nat \<equiv> of_nat"
|
haftmann@24198
|
589 |
|
haftmann@24198
|
590 |
abbreviation
|
haftmann@24198
|
591 |
rat_of_int :: "int \<Rightarrow> rat"
|
haftmann@24198
|
592 |
where
|
haftmann@24198
|
593 |
"rat_of_int \<equiv> of_int"
|
haftmann@24198
|
594 |
|
berghofe@24533
|
595 |
|
berghofe@24533
|
596 |
subsection {* Implementation of rational numbers as pairs of integers *}
|
berghofe@24533
|
597 |
|
berghofe@24533
|
598 |
definition
|
haftmann@24622
|
599 |
Rational :: "int \<times> int \<Rightarrow> rat"
|
berghofe@24533
|
600 |
where
|
haftmann@24622
|
601 |
"Rational = INum"
|
berghofe@24533
|
602 |
|
haftmann@24622
|
603 |
code_datatype Rational
|
berghofe@24533
|
604 |
|
haftmann@24622
|
605 |
lemma Rational_simp:
|
haftmann@24622
|
606 |
"Rational (k, l) = rat_of_int k / rat_of_int l"
|
haftmann@24622
|
607 |
unfolding Rational_def INum_def by simp
|
berghofe@24533
|
608 |
|
haftmann@24622
|
609 |
lemma Rational_zero [simp]: "Rational 0\<^sub>N = 0"
|
haftmann@24622
|
610 |
by (simp add: Rational_simp)
|
berghofe@24533
|
611 |
|
haftmann@24622
|
612 |
lemma Rational_lit [simp]: "Rational i\<^sub>N = rat_of_int i"
|
haftmann@24622
|
613 |
by (simp add: Rational_simp)
|
berghofe@24533
|
614 |
|
berghofe@24533
|
615 |
lemma zero_rat_code [code, code unfold]:
|
haftmann@24622
|
616 |
"0 = Rational 0\<^sub>N" by simp
|
berghofe@24533
|
617 |
|
berghofe@24533
|
618 |
lemma zero_rat_code [code, code unfold]:
|
haftmann@24622
|
619 |
"1 = Rational 1\<^sub>N" by simp
|
berghofe@24533
|
620 |
|
berghofe@24533
|
621 |
lemma [code, code unfold]:
|
berghofe@24533
|
622 |
"number_of k = rat_of_int (number_of k)"
|
berghofe@24533
|
623 |
by (simp add: number_of_is_id rat_number_of_def)
|
berghofe@24533
|
624 |
|
berghofe@24533
|
625 |
definition
|
berghofe@24533
|
626 |
[code func del]: "Fract' (b\<Colon>bool) k l = Fract k l"
|
berghofe@24533
|
627 |
|
berghofe@24533
|
628 |
lemma [code]:
|
berghofe@24533
|
629 |
"Fract k l = Fract' (l \<noteq> 0) k l"
|
berghofe@24533
|
630 |
unfolding Fract'_def ..
|
berghofe@24533
|
631 |
|
berghofe@24533
|
632 |
lemma [code]:
|
haftmann@24622
|
633 |
"Fract' True k l = (if l \<noteq> 0 then Rational (k, l) else Fract 1 0)"
|
haftmann@24622
|
634 |
by (simp add: Fract'_def Rational_simp Fract_of_int_quotient [of k l])
|
berghofe@24533
|
635 |
|
berghofe@24533
|
636 |
lemma [code]:
|
haftmann@24622
|
637 |
"of_rat (Rational (k, l)) = (if l \<noteq> 0 then of_int k / of_int l else 0)"
|
berghofe@24533
|
638 |
by (cases "l = 0")
|
haftmann@24622
|
639 |
(auto simp add: Rational_simp of_rat_rat [simplified Fract_of_int_quotient [of k l], symmetric])
|
berghofe@24533
|
640 |
|
berghofe@24533
|
641 |
instance rat :: eq ..
|
berghofe@24533
|
642 |
|
haftmann@24622
|
643 |
lemma rat_eq_code [code]: "Rational x = Rational y \<longleftrightarrow> normNum x = normNum y"
|
haftmann@24622
|
644 |
unfolding Rational_def INum_normNum_iff ..
|
berghofe@24533
|
645 |
|
haftmann@24622
|
646 |
lemma rat_less_eq_code [code]: "Rational x \<le> Rational y \<longleftrightarrow> normNum x \<le>\<^sub>N normNum y"
|
berghofe@24533
|
647 |
proof -
|
haftmann@24622
|
648 |
have "normNum x \<le>\<^sub>N normNum y \<longleftrightarrow> Rational (normNum x) \<le> Rational (normNum y)"
|
haftmann@24622
|
649 |
by (simp add: Rational_def del: normNum)
|
haftmann@24622
|
650 |
also have "\<dots> = (Rational x \<le> Rational y)" by (simp add: Rational_def)
|
berghofe@24533
|
651 |
finally show ?thesis by simp
|
berghofe@24533
|
652 |
qed
|
berghofe@24533
|
653 |
|
haftmann@24622
|
654 |
lemma rat_less_code [code]: "Rational x < Rational y \<longleftrightarrow> normNum x <\<^sub>N normNum y"
|
berghofe@24533
|
655 |
proof -
|
haftmann@24622
|
656 |
have "normNum x <\<^sub>N normNum y \<longleftrightarrow> Rational (normNum x) < Rational (normNum y)"
|
haftmann@24622
|
657 |
by (simp add: Rational_def del: normNum)
|
haftmann@24622
|
658 |
also have "\<dots> = (Rational x < Rational y)" by (simp add: Rational_def)
|
berghofe@24533
|
659 |
finally show ?thesis by simp
|
berghofe@24533
|
660 |
qed
|
berghofe@24533
|
661 |
|
haftmann@24622
|
662 |
lemma rat_add_code [code]: "Rational x + Rational y = Rational (x +\<^sub>N y)"
|
haftmann@24622
|
663 |
unfolding Rational_def by simp
|
berghofe@24533
|
664 |
|
haftmann@24622
|
665 |
lemma rat_mul_code [code]: "Rational x * Rational y = Rational (x *\<^sub>N y)"
|
haftmann@24622
|
666 |
unfolding Rational_def by simp
|
berghofe@24533
|
667 |
|
haftmann@24622
|
668 |
lemma rat_neg_code [code]: "- Rational x = Rational (~\<^sub>N x)"
|
haftmann@24622
|
669 |
unfolding Rational_def by simp
|
berghofe@24533
|
670 |
|
haftmann@24622
|
671 |
lemma rat_sub_code [code]: "Rational x - Rational y = Rational (x -\<^sub>N y)"
|
haftmann@24622
|
672 |
unfolding Rational_def by simp
|
berghofe@24533
|
673 |
|
haftmann@24622
|
674 |
lemma rat_inv_code [code]: "inverse (Rational x) = Rational (Ninv x)"
|
haftmann@24622
|
675 |
unfolding Rational_def Ninv divide_rat_def by simp
|
berghofe@24533
|
676 |
|
haftmann@24622
|
677 |
lemma rat_div_code [code]: "Rational x / Rational y = Rational (x \<div>\<^sub>N y)"
|
haftmann@24622
|
678 |
unfolding Rational_def by simp
|
berghofe@24533
|
679 |
|
haftmann@24622
|
680 |
text {* Setup for SML code generator *}
|
berghofe@24533
|
681 |
|
berghofe@24533
|
682 |
types_code
|
berghofe@24533
|
683 |
rat ("(int */ int)")
|
berghofe@24533
|
684 |
attach (term_of) {*
|
berghofe@24533
|
685 |
fun term_of_rat (p, q) =
|
haftmann@24622
|
686 |
let
|
haftmann@24622
|
687 |
val rT = @{typ rat}
|
berghofe@24533
|
688 |
in
|
berghofe@24533
|
689 |
if q = 1 orelse p = 0 then HOLogic.mk_number rT p
|
haftmann@24622
|
690 |
else @{term "op / \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat"} $
|
berghofe@24533
|
691 |
HOLogic.mk_number rT p $ HOLogic.mk_number rT q
|
berghofe@24533
|
692 |
end;
|
berghofe@24533
|
693 |
*}
|
berghofe@24533
|
694 |
attach (test) {*
|
berghofe@24533
|
695 |
fun gen_rat i =
|
berghofe@24533
|
696 |
let
|
berghofe@24533
|
697 |
val p = random_range 0 i;
|
berghofe@24533
|
698 |
val q = random_range 1 (i + 1);
|
berghofe@24533
|
699 |
val g = Integer.gcd p q;
|
wenzelm@24630
|
700 |
val p' = p div g;
|
wenzelm@24630
|
701 |
val q' = q div g;
|
berghofe@24533
|
702 |
in
|
berghofe@24533
|
703 |
(if one_of [true, false] then p' else ~ p',
|
berghofe@24533
|
704 |
if p' = 0 then 0 else q')
|
berghofe@24533
|
705 |
end;
|
berghofe@24533
|
706 |
*}
|
berghofe@24533
|
707 |
|
berghofe@24533
|
708 |
consts_code
|
haftmann@24622
|
709 |
Rational ("(_)")
|
berghofe@24533
|
710 |
|
berghofe@24533
|
711 |
consts_code
|
berghofe@24533
|
712 |
"of_int :: int \<Rightarrow> rat" ("\<module>rat'_of'_int")
|
berghofe@24533
|
713 |
attach {*
|
berghofe@24533
|
714 |
fun rat_of_int 0 = (0, 0)
|
berghofe@24533
|
715 |
| rat_of_int i = (i, 1);
|
berghofe@24533
|
716 |
*}
|
berghofe@24533
|
717 |
|
paulson@14365
|
718 |
end
|