src/HOL/Real/Rational.thy
author haftmann
Fri Jul 11 09:03:11 2008 +0200 (2008-07-11)
changeset 27551 9a5543d4cc24
parent 27509 63161d5f8f29
child 27652 818666de6c24
permissions -rw-r--r--
Fract now total; improved code generator setup
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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 "../Presburger" GCD Abstract_Rat
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uses ("rat_arith.ML")
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begin
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subsection {* Rational numbers as quotient *}
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subsubsection {* Construction of the type of rational numbers *}
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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 ratrel_iff [simp]:
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  "(x, y) \<in> ratrel \<longleftrightarrow> 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 {x. snd x \<noteq> 0} ratrel"
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  by (auto simp add: refl_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: "trans ratrel"
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proof (rule transI, unfold split_paired_all)
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  fix a b a' b' a'' b'' :: int
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  assume A: "((a, b), (a', b')) \<in> ratrel"
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  assume B: "((a', b'), (a'', b'')) \<in> ratrel"
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  have "b' * (a * b'') = b'' * (a * b')" by simp
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  also from A have "a * b' = a' * b" by auto
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  also have "b'' * (a' * b) = b * (a' * b'')" by simp
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  also from B have "a' * b'' = a'' * b'" by auto
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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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  moreover from B have "b' \<noteq> 0" by auto
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  ultimately have "a * b'' = a'' * b" by simp
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  with A B show "((a, b), (a'', b'')) \<in> ratrel" by auto
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qed
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lemma equiv_ratrel: "equiv {x. snd x \<noteq> 0} ratrel"
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  by (rule equiv.intro [OF refl_ratrel sym_ratrel trans_ratrel])
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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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lemma equiv_ratrel_iff [iff]: 
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  assumes "snd x \<noteq> 0" and "snd y \<noteq> 0"
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  shows "ratrel `` {x} = ratrel `` {y} \<longleftrightarrow> (x, y) \<in> ratrel"
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  by (rule eq_equiv_class_iff, rule equiv_ratrel) (auto simp add: assms)
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typedef (Rat) rat = "{x. snd x \<noteq> 0} // ratrel"
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proof
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  have "(0::int, 1::int) \<in> {x. snd x \<noteq> 0}" by simp
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  then show "ratrel `` {(0, 1)} \<in> {x. snd x \<noteq> 0} // 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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subsubsection {* Representation and basic operations *}
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definition
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  Fract :: "int \<Rightarrow> int \<Rightarrow> rat" where
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  [code func del]: "Fract a b = Abs_Rat (ratrel `` {if b = 0 then (0, 1) else (a, b)})"
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code_datatype Fract
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lemma Rat_cases [case_names Fract, cases type: rat]:
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  assumes "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> C"
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  shows C
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  using assms by (cases q) (clarsimp simp add: Fract_def Rat_def quotient_def)
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lemma Rat_induct [case_names Fract, induct type: rat]:
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  assumes "\<And>a b. b \<noteq> 0 \<Longrightarrow> P (Fract a b)"
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  shows "P q"
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  using assms by (cases q) simp
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lemma eq_rat:
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  shows "\<And>a b c d. b \<noteq> 0 \<Longrightarrow> d \<noteq> 0 \<Longrightarrow> Fract a b = Fract c d \<longleftrightarrow> a * d = c * b"
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  and "\<And>a c. Fract a 0 = Fract c 0"
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  by (simp_all add: Fract_def)
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instantiation rat :: "{comm_ring_1, recpower}"
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begin
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definition
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  Zero_rat_def [code, code unfold]: "0 = Fract 0 1"
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definition
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  One_rat_def [code, code unfold]: "1 = Fract 1 1"
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definition
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  add_rat_def [code func del]:
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  "q + r = 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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lemma add_rat:
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  assumes "b \<noteq> 0" and "d \<noteq> 0"
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  shows "Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
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proof -
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  have "(\<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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  by (rule equiv_ratrel [THEN congruent2_commuteI]) (simp_all add: left_distrib)
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  with assms show ?thesis by (simp add: Fract_def add_rat_def UN_ratrel2)
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qed
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definition
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  minus_rat_def [code func del]:
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  "- q = Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel `` {(- fst x, snd x)})"
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lemma minus_rat: "- Fract a b = Fract (- a) b"
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proof -
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  have "(\<lambda>x. ratrel `` {(- fst x, snd x)}) respects ratrel"
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    by (simp add: congruent_def)
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  then show ?thesis by (simp add: Fract_def minus_rat_def UN_ratrel)
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qed
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lemma minus_rat_cancel [simp]: 
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  "Fract (- a) (- b) = Fract a b"
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  by (cases "b = 0") (simp_all add: eq_rat)
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definition
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  diff_rat_def [code func del]: "q - r = q + - (r::rat)"
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lemma diff_rat:
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  assumes "b \<noteq> 0" and "d \<noteq> 0"
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  shows "Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
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  using assms by (simp add: diff_rat_def add_rat minus_rat)
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definition
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  mult_rat_def [code func del]:
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  "q * r = 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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lemma mult_rat: "Fract a b * Fract c d = Fract (a * c) (b * d)"
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proof -
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  have "(\<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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  then show ?thesis by (simp add: Fract_def mult_rat_def UN_ratrel2)
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qed
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lemma mult_rat_cancel [simp]:
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  assumes "c \<noteq> 0"
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  shows "Fract (c * a) (c * b) = Fract a b"
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proof -
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  from assms have "Fract c c = Fract 1 1" by (simp add: Fract_def)
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  then show ?thesis by (simp add: mult_rat [symmetric] mult_rat)
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qed
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primrec power_rat
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where
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  rat_power_0:     "q ^ 0 = (1\<Colon>rat)"
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  | rat_power_Suc: "q ^ Suc n = (q\<Colon>rat) * (q ^ n)"
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instance proof
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  fix q r s :: rat show "(q * r) * s = q * (r * s)"
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    by (cases q, cases r, cases s) (simp add: mult_rat eq_rat)
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next
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  fix q r :: rat show "q * r = r * q"
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    by (cases q, cases r) (simp add: mult_rat eq_rat)
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next
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  fix q :: rat show "1 * q = q"
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    by (cases q) (simp add: One_rat_def mult_rat eq_rat)
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next
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  fix q r s :: rat show "(q + r) + s = q + (r + s)"
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    by (cases q, cases r, cases s) (simp add: add_rat eq_rat ring_simps)
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next
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  fix q r :: rat show "q + r = r + q"
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    by (cases q, cases r) (simp add: add_rat eq_rat)
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next
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  fix q :: rat show "0 + q = q"
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    by (cases q) (simp add: Zero_rat_def add_rat eq_rat)
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next
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  fix q :: rat show "- q + q = 0"
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    by (cases q) (simp add: Zero_rat_def add_rat minus_rat eq_rat)
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next
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  fix q r :: rat show "q - r = q + - r"
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    by (cases q, cases r) (simp add: diff_rat add_rat minus_rat eq_rat)
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next
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  fix q r s :: rat show "(q + r) * s = q * s + r * s"
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    by (cases q, cases r, cases s) (simp add: add_rat mult_rat eq_rat ring_simps)
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next
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  show "(0::rat) \<noteq> 1" by (simp add: Zero_rat_def One_rat_def eq_rat)
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next
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  fix q :: rat show "q * 1 = q"
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    by (cases q) (simp add: One_rat_def mult_rat eq_rat)
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next
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  fix q :: rat
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  fix n :: nat
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  show "q ^ 0 = 1" by simp
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  show "q ^ (Suc n) = q * (q ^ n)" by simp
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qed
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end
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lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1"
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  by (induct k) (simp_all add: Zero_rat_def One_rat_def add_rat)
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lemma of_int_rat: "of_int k = Fract k 1"
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  by (cases k rule: int_diff_cases, simp add: of_nat_rat diff_rat)
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lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
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  by (rule of_nat_rat [symmetric])
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lemma Fract_of_int_eq: "Fract k 1 = of_int k"
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  by (rule of_int_rat [symmetric])
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instantiation rat :: number_ring
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begin
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definition
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  rat_number_of_def [code func del]: "number_of w = Fract w 1"
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instance by intro_classes (simp add: rat_number_of_def of_int_rat)
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end
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lemma rat_number_collapse [code post]:
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  "Fract 0 k = 0"
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  "Fract 1 1 = 1"
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  "Fract (number_of k) 1 = number_of k"
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  "Fract k 0 = 0"
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  by (cases "k = 0")
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    (simp_all add: Zero_rat_def One_rat_def number_of_is_id number_of_eq of_int_rat eq_rat Fract_def)
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lemma rat_number_expand [code unfold]:
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  "0 = Fract 0 1"
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  "1 = Fract 1 1"
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  "number_of k = Fract (number_of k) 1"
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  by (simp_all add: rat_number_collapse)
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lemma iszero_rat [simp]:
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  "iszero (number_of k :: rat) \<longleftrightarrow> iszero (number_of k :: int)"
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  by (simp add: iszero_def rat_number_expand number_of_is_id eq_rat)
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lemma Rat_cases_nonzero [case_names Fract 0]:
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  assumes Fract: "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> C"
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  assumes 0: "q = 0 \<Longrightarrow> C"
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  shows C
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proof (cases "q = 0")
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  case True then show C using 0 by auto
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next
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  case False
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  then obtain a b where "q = Fract a b" and "b \<noteq> 0" by (cases q) auto
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  moreover with False have "0 \<noteq> Fract a b" by simp
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  with `b \<noteq> 0` have "a \<noteq> 0" by (simp add: Zero_rat_def eq_rat)
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  with Fract `q = Fract a b` `b \<noteq> 0` show C by auto
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qed
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subsubsection {* The field of rational numbers *}
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instantiation rat :: "{field, division_by_zero}"
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begin
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definition
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  inverse_rat_def [code func del]:
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  "inverse q = 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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lemma inverse_rat: "inverse (Fract a b) = Fract b a"
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proof -
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  have "(\<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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  then show ?thesis by (simp add: Fract_def inverse_rat_def UN_ratrel)
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qed
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definition
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  divide_rat_def [code func del]: "q / r = q * inverse (r::rat)"
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lemma divide_rat: "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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instance proof
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  show "inverse 0 = (0::rat)" by (simp add: rat_number_expand inverse_rat)
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    (simp add: rat_number_collapse)
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next
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  fix q :: rat
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  assume "q \<noteq> 0"
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  then show "inverse q * q = 1" by (cases q rule: Rat_cases_nonzero)
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   (simp_all add: mult_rat  inverse_rat rat_number_expand eq_rat)
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next
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  fix q r :: rat
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  show "q / r = q * inverse r" by (simp add: divide_rat_def)
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qed
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end
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subsubsection {* Various *}
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lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
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  by (simp add: rat_number_expand add_rat)
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lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"
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  by (simp add: Fract_of_int_eq [symmetric] divide_rat)
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lemma Fract_number_of_quotient [code post]:
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  "Fract (number_of k) (number_of l) = number_of k / number_of l"
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  unfolding Fract_of_int_quotient number_of_is_id number_of_eq ..
haftmann@27551
   312
haftmann@27551
   313
haftmann@27551
   314
subsubsection {* The ordered field of rational numbers *}
huffman@27509
   315
huffman@27509
   316
instantiation rat :: linorder
huffman@27509
   317
begin
huffman@27509
   318
huffman@27509
   319
definition
huffman@27509
   320
  le_rat_def [code func del]:
huffman@27509
   321
   "q \<le> r \<longleftrightarrow> contents (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
haftmann@27551
   322
      {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})"
haftmann@27551
   323
haftmann@27551
   324
lemma le_rat:
haftmann@27551
   325
  assumes "b \<noteq> 0" and "d \<noteq> 0"
haftmann@27551
   326
  shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27551
   327
proof -
haftmann@27551
   328
  have "(\<lambda>x y. {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})
haftmann@27551
   329
    respects2 ratrel"
haftmann@27551
   330
  proof (clarsimp simp add: congruent2_def)
haftmann@27551
   331
    fix a b a' b' c d c' d'::int
haftmann@27551
   332
    assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
haftmann@27551
   333
    assume eq1: "a * b' = a' * b"
haftmann@27551
   334
    assume eq2: "c * d' = c' * d"
haftmann@27551
   335
haftmann@27551
   336
    let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
haftmann@27551
   337
    {
haftmann@27551
   338
      fix a b c d x :: int assume x: "x \<noteq> 0"
haftmann@27551
   339
      have "?le a b c d = ?le (a * x) (b * x) c d"
haftmann@27551
   340
      proof -
haftmann@27551
   341
        from x have "0 < x * x" by (auto simp add: zero_less_mult_iff)
haftmann@27551
   342
        hence "?le a b c d =
haftmann@27551
   343
            ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
haftmann@27551
   344
          by (simp add: mult_le_cancel_right)
haftmann@27551
   345
        also have "... = ?le (a * x) (b * x) c d"
haftmann@27551
   346
          by (simp add: mult_ac)
haftmann@27551
   347
        finally show ?thesis .
haftmann@27551
   348
      qed
haftmann@27551
   349
    } note le_factor = this
haftmann@27551
   350
haftmann@27551
   351
    let ?D = "b * d" and ?D' = "b' * d'"
haftmann@27551
   352
    from neq have D: "?D \<noteq> 0" by simp
haftmann@27551
   353
    from neq have "?D' \<noteq> 0" by simp
haftmann@27551
   354
    hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"
haftmann@27551
   355
      by (rule le_factor)
haftmann@27551
   356
    also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')"
haftmann@27551
   357
      by (simp add: mult_ac)
haftmann@27551
   358
    also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
haftmann@27551
   359
      by (simp only: eq1 eq2)
haftmann@27551
   360
    also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
haftmann@27551
   361
      by (simp add: mult_ac)
haftmann@27551
   362
    also from D have "... = ?le a' b' c' d'"
haftmann@27551
   363
      by (rule le_factor [symmetric])
haftmann@27551
   364
    finally show "?le a b c d = ?le a' b' c' d'" .
haftmann@27551
   365
  qed
haftmann@27551
   366
  with assms show ?thesis by (simp add: Fract_def le_rat_def UN_ratrel2)
haftmann@27551
   367
qed
huffman@27509
   368
huffman@27509
   369
definition
huffman@27509
   370
  less_rat_def [code func del]: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
huffman@27509
   371
haftmann@27551
   372
lemma less_rat:
haftmann@27551
   373
  assumes "b \<noteq> 0" and "d \<noteq> 0"
haftmann@27551
   374
  shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
haftmann@27551
   375
  using assms by (simp add: less_rat_def le_rat eq_rat order_less_le)
huffman@27509
   376
huffman@27509
   377
instance proof
paulson@14365
   378
  fix q r s :: rat
paulson@14365
   379
  {
paulson@14365
   380
    assume "q \<le> r" and "r \<le> s"
paulson@14365
   381
    show "q \<le> s"
paulson@14365
   382
    proof (insert prems, induct q, induct r, induct s)
paulson@14365
   383
      fix a b c d e f :: int
paulson@14365
   384
      assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
paulson@14365
   385
      assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"
paulson@14365
   386
      show "Fract a b \<le> Fract e f"
paulson@14365
   387
      proof -
paulson@14365
   388
        from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
paulson@14365
   389
          by (auto simp add: zero_less_mult_iff linorder_neq_iff)
paulson@14365
   390
        have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
paulson@14365
   391
        proof -
paulson@14365
   392
          from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
paulson@14365
   393
            by (simp add: le_rat)
paulson@14365
   394
          with ff show ?thesis by (simp add: mult_le_cancel_right)
paulson@14365
   395
        qed
paulson@14365
   396
        also have "... = (c * f) * (d * f) * (b * b)"
paulson@14365
   397
          by (simp only: mult_ac)
paulson@14365
   398
        also have "... \<le> (e * d) * (d * f) * (b * b)"
paulson@14365
   399
        proof -
paulson@14365
   400
          from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
paulson@14365
   401
            by (simp add: le_rat)
paulson@14365
   402
          with bb show ?thesis by (simp add: mult_le_cancel_right)
paulson@14365
   403
        qed
paulson@14365
   404
        finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
paulson@14365
   405
          by (simp only: mult_ac)
paulson@14365
   406
        with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
paulson@14365
   407
          by (simp add: mult_le_cancel_right)
paulson@14365
   408
        with neq show ?thesis by (simp add: le_rat)
paulson@14365
   409
      qed
paulson@14365
   410
    qed
paulson@14365
   411
  next
paulson@14365
   412
    assume "q \<le> r" and "r \<le> q"
paulson@14365
   413
    show "q = r"
paulson@14365
   414
    proof (insert prems, induct q, induct r)
paulson@14365
   415
      fix a b c d :: int
paulson@14365
   416
      assume neq: "b \<noteq> 0"  "d \<noteq> 0"
paulson@14365
   417
      assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"
paulson@14365
   418
      show "Fract a b = Fract c d"
paulson@14365
   419
      proof -
paulson@14365
   420
        from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
paulson@14365
   421
          by (simp add: le_rat)
paulson@14365
   422
        also have "... \<le> (a * d) * (b * d)"
paulson@14365
   423
        proof -
paulson@14365
   424
          from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
paulson@14365
   425
            by (simp add: le_rat)
paulson@14365
   426
          thus ?thesis by (simp only: mult_ac)
paulson@14365
   427
        qed
paulson@14365
   428
        finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
paulson@14365
   429
        moreover from neq have "b * d \<noteq> 0" by simp
paulson@14365
   430
        ultimately have "a * d = c * b" by simp
paulson@14365
   431
        with neq show ?thesis by (simp add: eq_rat)
paulson@14365
   432
      qed
paulson@14365
   433
    qed
paulson@14365
   434
  next
paulson@14365
   435
    show "q \<le> q"
paulson@14365
   436
      by (induct q) (simp add: le_rat)
paulson@14365
   437
    show "(q < r) = (q \<le> r \<and> q \<noteq> r)"
paulson@14365
   438
      by (simp only: less_rat_def)
paulson@14365
   439
    show "q \<le> r \<or> r \<le> q"
huffman@18913
   440
      by (induct q, induct r)
huffman@18913
   441
         (simp add: le_rat mult_commute, rule linorder_linear)
paulson@14365
   442
  }
paulson@14365
   443
qed
paulson@14365
   444
huffman@27509
   445
end
huffman@27509
   446
haftmann@27551
   447
instantiation rat :: "{distrib_lattice, abs_if, sgn_if}"
haftmann@25571
   448
begin
haftmann@25571
   449
haftmann@25571
   450
definition
haftmann@27551
   451
  abs_rat_def: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))"
haftmann@27551
   452
haftmann@27551
   453
lemma abs_rat: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
haftmann@27551
   454
  by (auto simp add: abs_rat_def zabs_def Zero_rat_def less_rat not_less le_less minus_rat eq_rat zero_compare_simps)
haftmann@27551
   455
haftmann@27551
   456
definition
haftmann@27551
   457
  sgn_rat_def: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
haftmann@27551
   458
haftmann@27551
   459
lemma sgn_rat: "sgn (Fract a b) = Fract (sgn a * sgn b) 1"
haftmann@27551
   460
  unfolding Fract_of_int_eq
haftmann@27551
   461
  by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat less_rat)
haftmann@27551
   462
    (auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)
haftmann@27551
   463
haftmann@27551
   464
definition
haftmann@25571
   465
  "(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min"
haftmann@25571
   466
haftmann@25571
   467
definition
haftmann@25571
   468
  "(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max"
haftmann@25571
   469
haftmann@27551
   470
instance by intro_classes
haftmann@27551
   471
  (auto simp add: abs_rat_def sgn_rat_def min_max.sup_inf_distrib1 inf_rat_def sup_rat_def)
haftmann@22456
   472
haftmann@25571
   473
end
haftmann@25571
   474
haftmann@27551
   475
instance rat :: ordered_field
haftmann@27551
   476
proof
paulson@14365
   477
  fix q r s :: rat
paulson@14365
   478
  show "q \<le> r ==> s + q \<le> s + r"
paulson@14365
   479
  proof (induct q, induct r, induct s)
paulson@14365
   480
    fix a b c d e f :: int
paulson@14365
   481
    assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
paulson@14365
   482
    assume le: "Fract a b \<le> Fract c d"
paulson@14365
   483
    show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
paulson@14365
   484
    proof -
paulson@14365
   485
      let ?F = "f * f" from neq have F: "0 < ?F"
paulson@14365
   486
        by (auto simp add: zero_less_mult_iff)
paulson@14365
   487
      from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
paulson@14365
   488
        by (simp add: le_rat)
paulson@14365
   489
      with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
paulson@14365
   490
        by (simp add: mult_le_cancel_right)
paulson@14365
   491
      with neq show ?thesis by (simp add: add_rat le_rat mult_ac int_distrib)
paulson@14365
   492
    qed
paulson@14365
   493
  qed
paulson@14365
   494
  show "q < r ==> 0 < s ==> s * q < s * r"
paulson@14365
   495
  proof (induct q, induct r, induct s)
paulson@14365
   496
    fix a b c d e f :: int
paulson@14365
   497
    assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
paulson@14365
   498
    assume le: "Fract a b < Fract c d"
paulson@14365
   499
    assume gt: "0 < Fract e f"
paulson@14365
   500
    show "Fract e f * Fract a b < Fract e f * Fract c d"
paulson@14365
   501
    proof -
paulson@14365
   502
      let ?E = "e * f" and ?F = "f * f"
paulson@14365
   503
      from neq gt have "0 < ?E"
haftmann@23879
   504
        by (auto simp add: Zero_rat_def less_rat le_rat order_less_le eq_rat)
paulson@14365
   505
      moreover from neq have "0 < ?F"
paulson@14365
   506
        by (auto simp add: zero_less_mult_iff)
paulson@14365
   507
      moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
paulson@14365
   508
        by (simp add: less_rat)
paulson@14365
   509
      ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
paulson@14365
   510
        by (simp add: mult_less_cancel_right)
paulson@14365
   511
      with neq show ?thesis
paulson@14365
   512
        by (simp add: less_rat mult_rat mult_ac)
paulson@14365
   513
    qed
paulson@14365
   514
  qed
haftmann@27551
   515
qed auto
paulson@14365
   516
haftmann@27551
   517
lemma Rat_induct_pos [case_names Fract, induct type: rat]:
haftmann@27551
   518
  assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
haftmann@27551
   519
  shows "P q"
paulson@14365
   520
proof (cases q)
haftmann@27551
   521
  have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)"
paulson@14365
   522
  proof -
paulson@14365
   523
    fix a::int and b::int
paulson@14365
   524
    assume b: "b < 0"
paulson@14365
   525
    hence "0 < -b" by simp
paulson@14365
   526
    hence "P (Fract (-a) (-b))" by (rule step)
paulson@14365
   527
    thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
paulson@14365
   528
  qed
paulson@14365
   529
  case (Fract a b)
paulson@14365
   530
  thus "P q" by (force simp add: linorder_neq_iff step step')
paulson@14365
   531
qed
paulson@14365
   532
paulson@14365
   533
lemma zero_less_Fract_iff:
paulson@14365
   534
     "0 < b ==> (0 < Fract a b) = (0 < a)"
haftmann@23879
   535
by (simp add: Zero_rat_def less_rat order_less_imp_not_eq2 zero_less_mult_iff)
paulson@14365
   536
paulson@14378
   537
haftmann@27551
   538
subsection {* Arithmetic setup *}
paulson@14387
   539
paulson@14387
   540
use "rat_arith.ML"
wenzelm@24075
   541
declaration {* K rat_arith_setup *}
paulson@14387
   542
huffman@23342
   543
huffman@23342
   544
subsection {* Embedding from Rationals to other Fields *}
huffman@23342
   545
haftmann@24198
   546
class field_char_0 = field + ring_char_0
huffman@23342
   547
haftmann@27551
   548
subclass (in ordered_field) field_char_0 ..
huffman@23342
   549
haftmann@27551
   550
context field_char_0
haftmann@27551
   551
begin
haftmann@27551
   552
haftmann@27551
   553
definition of_rat :: "rat \<Rightarrow> 'a" where
haftmann@24198
   554
  [code func del]: "of_rat q = contents (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})"
huffman@23342
   555
haftmann@27551
   556
end
haftmann@27551
   557
huffman@23342
   558
lemma of_rat_congruent:
haftmann@27551
   559
  "(\<lambda>(a, b). {of_int a / of_int b :: 'a::field_char_0}) respects ratrel"
huffman@23342
   560
apply (rule congruent.intro)
huffman@23342
   561
apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
huffman@23342
   562
apply (simp only: of_int_mult [symmetric])
huffman@23342
   563
done
huffman@23342
   564
haftmann@27551
   565
lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
haftmann@27551
   566
  unfolding Fract_def of_rat_def by (simp add: UN_ratrel of_rat_congruent)
huffman@23342
   567
huffman@23342
   568
lemma of_rat_0 [simp]: "of_rat 0 = 0"
huffman@23342
   569
by (simp add: Zero_rat_def of_rat_rat)
huffman@23342
   570
huffman@23342
   571
lemma of_rat_1 [simp]: "of_rat 1 = 1"
huffman@23342
   572
by (simp add: One_rat_def of_rat_rat)
huffman@23342
   573
huffman@23342
   574
lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
huffman@23342
   575
by (induct a, induct b, simp add: add_rat of_rat_rat add_frac_eq)
huffman@23342
   576
huffman@23343
   577
lemma of_rat_minus: "of_rat (- a) = - of_rat a"
huffman@23343
   578
by (induct a, simp add: minus_rat of_rat_rat)
huffman@23343
   579
huffman@23343
   580
lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
huffman@23343
   581
by (simp only: diff_minus of_rat_add of_rat_minus)
huffman@23343
   582
huffman@23342
   583
lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
huffman@23342
   584
apply (induct a, induct b, simp add: mult_rat of_rat_rat)
huffman@23342
   585
apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
huffman@23342
   586
done
huffman@23342
   587
huffman@23342
   588
lemma nonzero_of_rat_inverse:
huffman@23342
   589
  "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
huffman@23343
   590
apply (rule inverse_unique [symmetric])
huffman@23343
   591
apply (simp add: of_rat_mult [symmetric])
huffman@23342
   592
done
huffman@23342
   593
huffman@23342
   594
lemma of_rat_inverse:
huffman@23342
   595
  "(of_rat (inverse a)::'a::{field_char_0,division_by_zero}) =
huffman@23342
   596
   inverse (of_rat a)"
huffman@23342
   597
by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
huffman@23342
   598
huffman@23342
   599
lemma nonzero_of_rat_divide:
huffman@23342
   600
  "b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
huffman@23342
   601
by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
huffman@23342
   602
huffman@23342
   603
lemma of_rat_divide:
huffman@23342
   604
  "(of_rat (a / b)::'a::{field_char_0,division_by_zero})
huffman@23342
   605
   = of_rat a / of_rat b"
huffman@23342
   606
by (cases "b = 0", simp_all add: nonzero_of_rat_divide)
huffman@23342
   607
huffman@23343
   608
lemma of_rat_power:
huffman@23343
   609
  "(of_rat (a ^ n)::'a::{field_char_0,recpower}) = of_rat a ^ n"
huffman@23343
   610
by (induct n) (simp_all add: of_rat_mult power_Suc)
huffman@23343
   611
huffman@23343
   612
lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
huffman@23343
   613
apply (induct a, induct b)
huffman@23343
   614
apply (simp add: of_rat_rat eq_rat)
huffman@23343
   615
apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
huffman@23343
   616
apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
huffman@23343
   617
done
huffman@23343
   618
huffman@23343
   619
lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
huffman@23343
   620
huffman@23343
   621
lemma of_rat_eq_id [simp]: "of_rat = (id :: rat \<Rightarrow> rat)"
huffman@23343
   622
proof
huffman@23343
   623
  fix a
huffman@23343
   624
  show "of_rat a = id a"
huffman@23343
   625
  by (induct a)
huffman@23343
   626
     (simp add: of_rat_rat divide_rat Fract_of_int_eq [symmetric])
huffman@23343
   627
qed
huffman@23343
   628
huffman@23343
   629
text{*Collapse nested embeddings*}
huffman@23343
   630
lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
huffman@23343
   631
by (induct n) (simp_all add: of_rat_add)
huffman@23343
   632
huffman@23343
   633
lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
huffman@23365
   634
by (cases z rule: int_diff_cases, simp add: of_rat_diff)
huffman@23343
   635
huffman@23343
   636
lemma of_rat_number_of_eq [simp]:
huffman@23343
   637
  "of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})"
huffman@23343
   638
by (simp add: number_of_eq)
huffman@23343
   639
haftmann@23879
   640
lemmas zero_rat = Zero_rat_def
haftmann@23879
   641
lemmas one_rat = One_rat_def
haftmann@23879
   642
haftmann@24198
   643
abbreviation
haftmann@24198
   644
  rat_of_nat :: "nat \<Rightarrow> rat"
haftmann@24198
   645
where
haftmann@24198
   646
  "rat_of_nat \<equiv> of_nat"
haftmann@24198
   647
haftmann@24198
   648
abbreviation
haftmann@24198
   649
  rat_of_int :: "int \<Rightarrow> rat"
haftmann@24198
   650
where
haftmann@24198
   651
  "rat_of_int \<equiv> of_int"
haftmann@24198
   652
berghofe@24533
   653
berghofe@24533
   654
subsection {* Implementation of rational numbers as pairs of integers *}
berghofe@24533
   655
haftmann@27551
   656
lemma INum_Fract [simp]: "INum = split Fract"
haftmann@27551
   657
  by (auto simp add: expand_fun_eq INum_def Fract_of_int_quotient)
berghofe@24533
   658
haftmann@27551
   659
lemma split_Fract_normNum [simp]: "split Fract (normNum (k, l)) = Fract k l"
haftmann@27551
   660
  unfolding INum_Fract [symmetric] normNum by simp
berghofe@24533
   661
berghofe@24533
   662
lemma [code]:
haftmann@27551
   663
  "of_rat (Fract k l) = (if l \<noteq> 0 then of_int k / of_int l else 0)"
haftmann@27551
   664
  by (cases "l = 0") (simp_all add: rat_number_collapse of_rat_rat)
berghofe@24533
   665
haftmann@26513
   666
instantiation rat :: eq
haftmann@26513
   667
begin
haftmann@26513
   668
haftmann@27551
   669
definition [code func del]: "eq_class.eq (r\<Colon>rat) s \<longleftrightarrow> r - s = 0"
berghofe@24533
   670
haftmann@26513
   671
instance by default (simp add: eq_rat_def)
haftmann@26513
   672
haftmann@27551
   673
lemma rat_eq_code [code]: "eq_class.eq (Fract k l) (Fract r s) \<longleftrightarrow> eq_class.eq (normNum (k, l)) (normNum (r, s))"
haftmann@27551
   674
  by (simp add: eq INum_normNum_iff [where ?'a = rat, symmetric])
haftmann@26513
   675
haftmann@26513
   676
end
berghofe@24533
   677
haftmann@27551
   678
lemma rat_less_eq_code [code]: "Fract k l \<le> Fract r s \<longleftrightarrow> normNum (k, l) \<le>\<^sub>N normNum (r, s)"
berghofe@24533
   679
proof -
haftmann@27551
   680
  have "normNum (k, l) \<le>\<^sub>N normNum (r, s) \<longleftrightarrow> split Fract (normNum (k, l)) \<le> split Fract (normNum (r, s))" 
haftmann@27551
   681
    by (simp add: INum_Fract [symmetric] del: INum_Fract normNum)
haftmann@27551
   682
  also have "\<dots> = (Fract k l \<le> Fract r s)" by simp
berghofe@24533
   683
  finally show ?thesis by simp
berghofe@24533
   684
qed
berghofe@24533
   685
haftmann@27551
   686
lemma rat_less_code [code]: "Fract k l < Fract r s \<longleftrightarrow> normNum (k, l) <\<^sub>N normNum (r, s)"
berghofe@24533
   687
proof -
haftmann@27551
   688
  have "normNum (k, l) <\<^sub>N normNum (r, s) \<longleftrightarrow> split Fract (normNum (k, l)) < split Fract (normNum (r, s))" 
haftmann@27551
   689
    by (simp add: INum_Fract [symmetric] del: INum_Fract normNum)
haftmann@27551
   690
  also have "\<dots> = (Fract k l < Fract r s)" by simp
berghofe@24533
   691
  finally show ?thesis by simp
berghofe@24533
   692
qed
berghofe@24533
   693
haftmann@27551
   694
lemma rat_add_code [code]: "Fract k l + Fract r s = split Fract ((k, l) +\<^sub>N (r, s))"
haftmann@27551
   695
  by (simp add: INum_Fract [symmetric] del: INum_Fract, simp)
berghofe@24533
   696
haftmann@27551
   697
lemma rat_mul_code [code]: "Fract k l * Fract r s = split Fract ((k, l) *\<^sub>N (r, s))"
haftmann@27551
   698
  by (simp add: INum_Fract [symmetric] del: INum_Fract, simp)
berghofe@24533
   699
haftmann@27551
   700
lemma rat_neg_code [code]: "- Fract k l = split Fract (~\<^sub>N (k, l))"
haftmann@27551
   701
  by (simp add: INum_Fract [symmetric] del: INum_Fract, simp)
berghofe@24533
   702
haftmann@27551
   703
lemma rat_sub_code [code]: "Fract k l - Fract r s = split Fract ((k, l) -\<^sub>N (r, s))"
haftmann@27551
   704
  by (simp add: INum_Fract [symmetric] del: INum_Fract, simp)
berghofe@24533
   705
haftmann@27551
   706
lemma rat_inv_code [code]: "inverse (Fract k l) = split Fract (Ninv (k, l))"
haftmann@27551
   707
  by (simp add: INum_Fract [symmetric] del: INum_Fract, simp add: divide_rat_def)
berghofe@24533
   708
haftmann@27551
   709
lemma rat_div_code [code]: "Fract k l / Fract r s = split Fract ((k, l) \<div>\<^sub>N (r, s))"
haftmann@27551
   710
  by (simp add: INum_Fract [symmetric] del: INum_Fract, simp)
berghofe@24533
   711
haftmann@24622
   712
text {* Setup for SML code generator *}
berghofe@24533
   713
berghofe@24533
   714
types_code
berghofe@24533
   715
  rat ("(int */ int)")
berghofe@24533
   716
attach (term_of) {*
berghofe@24533
   717
fun term_of_rat (p, q) =
haftmann@24622
   718
  let
haftmann@24661
   719
    val rT = Type ("Rational.rat", [])
berghofe@24533
   720
  in
berghofe@24533
   721
    if q = 1 orelse p = 0 then HOLogic.mk_number rT p
berghofe@25885
   722
    else @{term "op / \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat"} $
berghofe@24533
   723
      HOLogic.mk_number rT p $ HOLogic.mk_number rT q
berghofe@24533
   724
  end;
berghofe@24533
   725
*}
berghofe@24533
   726
attach (test) {*
berghofe@24533
   727
fun gen_rat i =
berghofe@24533
   728
  let
berghofe@24533
   729
    val p = random_range 0 i;
berghofe@24533
   730
    val q = random_range 1 (i + 1);
berghofe@24533
   731
    val g = Integer.gcd p q;
wenzelm@24630
   732
    val p' = p div g;
wenzelm@24630
   733
    val q' = q div g;
berghofe@25885
   734
    val r = (if one_of [true, false] then p' else ~ p',
berghofe@25885
   735
      if p' = 0 then 0 else q')
berghofe@24533
   736
  in
berghofe@25885
   737
    (r, fn () => term_of_rat r)
berghofe@24533
   738
  end;
berghofe@24533
   739
*}
berghofe@24533
   740
berghofe@24533
   741
consts_code
haftmann@27551
   742
  Fract ("(_,/ _)")
berghofe@24533
   743
berghofe@24533
   744
consts_code
berghofe@24533
   745
  "of_int :: int \<Rightarrow> rat" ("\<module>rat'_of'_int")
berghofe@24533
   746
attach {*
berghofe@24533
   747
fun rat_of_int 0 = (0, 0)
berghofe@24533
   748
  | rat_of_int i = (i, 1);
berghofe@24533
   749
*}
berghofe@24533
   750
paulson@14365
   751
end