src/HOL/Real/Rational.thy
author haftmann
Fri, 18 Jul 2008 18:25:56 +0200
changeset 27652 818666de6c24
parent 27551 9a5543d4cc24
child 27668 6eb20b2cecf8
permissions -rw-r--r--
refined code generator setup for rational numbers; more simplification rules for rational numbers
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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
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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. Fract a 0 = Fract 0 1"
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  and "\<And>a c. Fract 0 a = Fract 0 c"
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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 [simp]:
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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 [simp, code]: "- 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]: "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 [simp]:
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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)
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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 [simp]: "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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parents:
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lemma mult_rat_cancel:
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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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   155
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])
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qed
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   159
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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: eq_rat)
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   168
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: eq_rat)
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   171
next
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   172
  fix q :: rat show "1 * q = q"
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    by (cases q) (simp add: One_rat_def eq_rat)
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   174
next
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   175
  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: 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: eq_rat)
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next
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   181
  fix q :: rat show "0 + q = q"
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    by (cases q) (simp add: Zero_rat_def eq_rat)
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   183
next
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   184
  fix q :: rat show "- q + q = 0"
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    by (cases q) (simp add: Zero_rat_def eq_rat)
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   186
next
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   187
  fix q r :: rat show "q - r = q + - r"
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   188
    by (cases q, cases r) (simp add: eq_rat)
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   189
next
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   190
  fix q r s :: rat show "(q + r) * s = q * s + r * s"
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   191
    by (cases q, cases r, cases s) (simp add: eq_rat ring_simps)
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   192
next
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   193
  show "(0::rat) \<noteq> 1" by (simp add: Zero_rat_def One_rat_def eq_rat)
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   194
next
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   195
  fix q :: rat show "q * 1 = q"
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    by (cases q) (simp add: One_rat_def eq_rat)
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   197
next
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   198
  fix q :: rat
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   199
  fix n :: nat
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   200
  show "q ^ 0 = 1" by simp
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   201
  show "q ^ (Suc n) = q * (q ^ n)" by simp
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   202
qed
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   203
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   204
end
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   205
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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)
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   208
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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)
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   211
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   212
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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   214
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lemma Fract_of_int_eq: "Fract k 1 = of_int k"
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   216
  by (rule of_int_rat [symmetric])
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   217
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   218
instantiation rat :: number_ring
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begin
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   220
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   221
definition
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  rat_number_of_def [code func del]: "number_of w = Fract w 1"
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   223
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   224
instance by intro_classes (simp add: rat_number_of_def of_int_rat)
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   225
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   226
end
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   227
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   228
lemma rat_number_collapse [code post]:
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   229
  "Fract 0 k = 0"
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   230
  "Fract 1 1 = 1"
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   231
  "Fract (number_of k) 1 = number_of k"
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   232
  "Fract k 0 = 0"
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   233
  by (cases "k = 0")
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   234
    (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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   235
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   236
lemma rat_number_expand [code unfold]:
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   237
  "0 = Fract 0 1"
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   238
  "1 = Fract 1 1"
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   239
  "number_of k = Fract (number_of k) 1"
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   240
  by (simp_all add: rat_number_collapse)
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   241
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   242
lemma iszero_rat [simp]:
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   243
  "iszero (number_of k :: rat) \<longleftrightarrow> iszero (number_of k :: int)"
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   244
  by (simp add: iszero_def rat_number_expand number_of_is_id eq_rat)
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   245
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   246
lemma Rat_cases_nonzero [case_names Fract 0]:
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   247
  assumes Fract: "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> C"
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   248
  assumes 0: "q = 0 \<Longrightarrow> C"
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   249
  shows C
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   250
proof (cases "q = 0")
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   251
  case True then show C using 0 by auto
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   252
next
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   253
  case False
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   254
  then obtain a b where "q = Fract a b" and "b \<noteq> 0" by (cases q) auto
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   255
  moreover with False have "0 \<noteq> Fract a b" by simp
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   256
  with `b \<noteq> 0` have "a \<noteq> 0" by (simp add: Zero_rat_def eq_rat)
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diff changeset
   257
  with Fract `q = Fract a b` `b \<noteq> 0` show C by auto
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diff changeset
   258
qed
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parents: 27509
diff changeset
   259
  
9a5543d4cc24 Fract now total; improved code generator setup
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   260
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   261
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   262
subsubsection {* The field of rational numbers *}
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   263
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   264
instantiation rat :: "{field, division_by_zero}"
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   265
begin
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   266
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   267
definition
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   268
  inverse_rat_def [code func del]:
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   269
  "inverse q = Abs_Rat (\<Union>x \<in> Rep_Rat q.
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   270
     ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)})"
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   271
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   272
lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"
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   273
proof -
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   274
  have "(\<lambda>x. ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)}) respects ratrel"
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   275
    by (auto simp add: congruent_def mult_commute)
9a5543d4cc24 Fract now total; improved code generator setup
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   276
  then show ?thesis by (simp add: Fract_def inverse_rat_def UN_ratrel)
27509
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   277
qed
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diff changeset
   278
27551
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   279
definition
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   280
  divide_rat_def [code func del]: "q / r = q * inverse (r::rat)"
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diff changeset
   281
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   282
lemma divide_rat [simp]: "Fract a b / Fract c d = Fract (a * d) (b * c)"
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diff changeset
   283
  by (simp add: divide_rat_def)
27551
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diff changeset
   284
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diff changeset
   285
instance proof
27652
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haftmann
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diff changeset
   286
  show "inverse 0 = (0::rat)" by (simp add: rat_number_expand)
27551
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diff changeset
   287
    (simp add: rat_number_collapse)
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diff changeset
   288
next
9a5543d4cc24 Fract now total; improved code generator setup
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diff changeset
   289
  fix q :: rat
9a5543d4cc24 Fract now total; improved code generator setup
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diff changeset
   290
  assume "q \<noteq> 0"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   291
  then show "inverse q * q = 1" by (cases q rule: Rat_cases_nonzero)
9a5543d4cc24 Fract now total; improved code generator setup
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parents: 27509
diff changeset
   292
   (simp_all add: mult_rat  inverse_rat rat_number_expand eq_rat)
9a5543d4cc24 Fract now total; improved code generator setup
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parents: 27509
diff changeset
   293
next
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   294
  fix q r :: rat
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   295
  show "q / r = q * inverse r" by (simp add: divide_rat_def)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   296
qed
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   297
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   298
end
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   299
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   300
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   301
subsubsection {* Various *}
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   302
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   303
lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   304
  by (simp add: rat_number_expand)
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   305
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   306
lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   307
  by (simp add: Fract_of_int_eq [symmetric])
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   308
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   309
lemma Fract_number_of_quotient [code post]:
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   310
  "Fract (number_of k) (number_of l) = number_of k / number_of l"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   311
  unfolding Fract_of_int_quotient number_of_is_id number_of_eq ..
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   312
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   313
lemma Fract_1_number_of [code post]:
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   314
  "Fract 1 (number_of k) = 1 / number_of k"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   315
  unfolding Fract_of_int_quotient number_of_eq by simp
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   316
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   317
subsubsection {* The ordered field of rational numbers *}
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   318
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   319
instantiation rat :: linorder
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   320
begin
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   321
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   322
definition
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   323
  le_rat_def [code func del]:
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   324
   "q \<le> r \<longleftrightarrow> contents (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   325
      {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   326
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   327
lemma le_rat [simp]:
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   328
  assumes "b \<noteq> 0" and "d \<noteq> 0"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   329
  shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   330
proof -
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   331
  have "(\<lambda>x y. {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   332
    respects2 ratrel"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   333
  proof (clarsimp simp add: congruent2_def)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   334
    fix a b a' b' c d c' d'::int
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   335
    assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   336
    assume eq1: "a * b' = a' * b"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   337
    assume eq2: "c * d' = c' * d"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   338
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   339
    let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   340
    {
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   341
      fix a b c d x :: int assume x: "x \<noteq> 0"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   342
      have "?le a b c d = ?le (a * x) (b * x) c d"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   343
      proof -
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   344
        from x have "0 < x * x" by (auto simp add: zero_less_mult_iff)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   345
        hence "?le a b c d =
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   346
            ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   347
          by (simp add: mult_le_cancel_right)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   348
        also have "... = ?le (a * x) (b * x) c d"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   349
          by (simp add: mult_ac)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   350
        finally show ?thesis .
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   351
      qed
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   352
    } note le_factor = this
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   353
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   354
    let ?D = "b * d" and ?D' = "b' * d'"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   355
    from neq have D: "?D \<noteq> 0" by simp
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   356
    from neq have "?D' \<noteq> 0" by simp
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   357
    hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   358
      by (rule le_factor)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   359
    also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   360
      by (simp add: mult_ac)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   361
    also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   362
      by (simp only: eq1 eq2)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   363
    also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   364
      by (simp add: mult_ac)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   365
    also from D have "... = ?le a' b' c' d'"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   366
      by (rule le_factor [symmetric])
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   367
    finally show "?le a b c d = ?le a' b' c' d'" .
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   368
  qed
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   369
  with assms show ?thesis by (simp add: Fract_def le_rat_def UN_ratrel2)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   370
qed
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   371
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   372
definition
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   373
  less_rat_def [code func del]: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   374
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   375
lemma less_rat [simp]:
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   376
  assumes "b \<noteq> 0" and "d \<noteq> 0"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   377
  shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   378
  using assms by (simp add: less_rat_def eq_rat order_less_le)
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   379
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   380
instance proof
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   381
  fix q r s :: rat
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   382
  {
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   383
    assume "q \<le> r" and "r \<le> s"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   384
    show "q \<le> s"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   385
    proof (insert prems, induct q, induct r, induct s)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   386
      fix a b c d e f :: int
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   387
      assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   388
      assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   389
      show "Fract a b \<le> Fract e f"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   390
      proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   391
        from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   392
          by (auto simp add: zero_less_mult_iff linorder_neq_iff)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   393
        have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   394
        proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   395
          from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   396
            by simp
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   397
          with ff show ?thesis by (simp add: mult_le_cancel_right)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   398
        qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   399
        also have "... = (c * f) * (d * f) * (b * b)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   400
          by (simp only: mult_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   401
        also have "... \<le> (e * d) * (d * f) * (b * b)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   402
        proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   403
          from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   404
            by simp
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   405
          with bb show ?thesis by (simp add: mult_le_cancel_right)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   406
        qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   407
        finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   408
          by (simp only: mult_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   409
        with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   410
          by (simp add: mult_le_cancel_right)
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   411
        with neq show ?thesis by simp
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   412
      qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   413
    qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   414
  next
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   415
    assume "q \<le> r" and "r \<le> q"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   416
    show "q = r"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   417
    proof (insert prems, induct q, induct r)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   418
      fix a b c d :: int
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   419
      assume neq: "b \<noteq> 0"  "d \<noteq> 0"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   420
      assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   421
      show "Fract a b = Fract c d"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   422
      proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   423
        from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   424
          by simp
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   425
        also have "... \<le> (a * d) * (b * d)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   426
        proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   427
          from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   428
            by simp
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   429
          thus ?thesis by (simp only: mult_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   430
        qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   431
        finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   432
        moreover from neq have "b * d \<noteq> 0" by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   433
        ultimately have "a * d = c * b" by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   434
        with neq show ?thesis by (simp add: eq_rat)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   435
      qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   436
    qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   437
  next
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   438
    show "q \<le> q"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   439
      by (induct q) simp
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   440
    show "(q < r) = (q \<le> r \<and> q \<noteq> r)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   441
      by (simp only: less_rat_def)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   442
    show "q \<le> r \<or> r \<le> q"
18913
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   443
      by (induct q, induct r)
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   444
         (simp add: mult_commute, rule linorder_linear)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   445
  }
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   446
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   447
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   448
end
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   449
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   450
instantiation rat :: "{distrib_lattice, abs_if, sgn_if}"
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   451
begin
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   452
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   453
definition
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   454
  abs_rat_def [code func del]: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))"
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   455
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   456
lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   457
  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)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   458
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   459
definition
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   460
  sgn_rat_def [code func del]: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   461
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   462
lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)"
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   463
  unfolding Fract_of_int_eq
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   464
  by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat)
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   465
    (auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   466
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   467
definition
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   468
  "(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min"
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   469
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   470
definition
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   471
  "(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max"
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   472
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   473
instance by intro_classes
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   474
  (auto simp add: abs_rat_def sgn_rat_def min_max.sup_inf_distrib1 inf_rat_def sup_rat_def)
22456
6070e48ecb78 added lattice definitions
haftmann
parents: 21404
diff changeset
   475
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   476
end
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   477
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   478
instance rat :: ordered_field
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   479
proof
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   480
  fix q r s :: rat
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   481
  show "q \<le> r ==> s + q \<le> s + r"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   482
  proof (induct q, induct r, induct s)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   483
    fix a b c d e f :: int
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   484
    assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   485
    assume le: "Fract a b \<le> Fract c d"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   486
    show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   487
    proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   488
      let ?F = "f * f" from neq have F: "0 < ?F"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   489
        by (auto simp add: zero_less_mult_iff)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   490
      from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   491
        by simp
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   492
      with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   493
        by (simp add: mult_le_cancel_right)
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   494
      with neq show ?thesis by (simp add: mult_ac int_distrib)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   495
    qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   496
  qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   497
  show "q < r ==> 0 < s ==> s * q < s * r"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   498
  proof (induct q, induct r, induct s)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   499
    fix a b c d e f :: int
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   500
    assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   501
    assume le: "Fract a b < Fract c d"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   502
    assume gt: "0 < Fract e f"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   503
    show "Fract e f * Fract a b < Fract e f * Fract c d"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   504
    proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   505
      let ?E = "e * f" and ?F = "f * f"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   506
      from neq gt have "0 < ?E"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   507
        by (auto simp add: Zero_rat_def order_less_le eq_rat)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   508
      moreover from neq have "0 < ?F"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   509
        by (auto simp add: zero_less_mult_iff)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   510
      moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   511
        by simp
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   512
      ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   513
        by (simp add: mult_less_cancel_right)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   514
      with neq show ?thesis
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   515
        by (simp add: mult_ac)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   516
    qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   517
  qed
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   518
qed auto
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   519
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   520
lemma Rat_induct_pos [case_names Fract, induct type: rat]:
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   521
  assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   522
  shows "P q"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   523
proof (cases q)
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   524
  have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   525
  proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   526
    fix a::int and b::int
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   527
    assume b: "b < 0"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   528
    hence "0 < -b" by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   529
    hence "P (Fract (-a) (-b))" by (rule step)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   530
    thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   531
  qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   532
  case (Fract a b)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   533
  thus "P q" by (force simp add: linorder_neq_iff step step')
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   534
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   535
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   536
lemma zero_less_Fract_iff:
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   537
  "0 < b ==> (0 < Fract a b) = (0 < a)"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   538
by (simp add: Zero_rat_def order_less_imp_not_eq2 zero_less_mult_iff)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   539
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   540
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   541
subsection {* Arithmetic setup *}
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   542
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   543
use "rat_arith.ML"
24075
366d4d234814 arith method setup: proper context;
wenzelm
parents: 23879
diff changeset
   544
declaration {* K rat_arith_setup *}
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   545
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   546
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   547
subsection {* Embedding from Rationals to other Fields *}
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   548
24198
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   549
class field_char_0 = field + ring_char_0
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   550
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   551
subclass (in ordered_field) field_char_0 ..
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   552
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   553
context field_char_0
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   554
begin
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   555
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   556
definition of_rat :: "rat \<Rightarrow> 'a" where
24198
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   557
  [code func del]: "of_rat q = contents (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})"
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   558
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   559
end
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   560
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   561
lemma of_rat_congruent:
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   562
  "(\<lambda>(a, b). {of_int a / of_int b :: 'a::field_char_0}) respects ratrel"
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   563
apply (rule congruent.intro)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   564
apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   565
apply (simp only: of_int_mult [symmetric])
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   566
done
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   567
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   568
lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   569
  unfolding Fract_def of_rat_def by (simp add: UN_ratrel of_rat_congruent)
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   570
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   571
lemma of_rat_0 [simp]: "of_rat 0 = 0"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   572
by (simp add: Zero_rat_def of_rat_rat)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   573
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   574
lemma of_rat_1 [simp]: "of_rat 1 = 1"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   575
by (simp add: One_rat_def of_rat_rat)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   576
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   577
lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   578
by (induct a, induct b, simp add: of_rat_rat add_frac_eq)
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   579
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   580
lemma of_rat_minus: "of_rat (- a) = - of_rat a"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   581
by (induct a, simp add: of_rat_rat)
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   582
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   583
lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   584
by (simp only: diff_minus of_rat_add of_rat_minus)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   585
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   586
lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   587
apply (induct a, induct b, simp add: of_rat_rat)
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   588
apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   589
done
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   590
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   591
lemma nonzero_of_rat_inverse:
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   592
  "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   593
apply (rule inverse_unique [symmetric])
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   594
apply (simp add: of_rat_mult [symmetric])
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   595
done
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   596
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   597
lemma of_rat_inverse:
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   598
  "(of_rat (inverse a)::'a::{field_char_0,division_by_zero}) =
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   599
   inverse (of_rat a)"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   600
by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   601
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   602
lemma nonzero_of_rat_divide:
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   603
  "b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   604
by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   605
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   606
lemma of_rat_divide:
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   607
  "(of_rat (a / b)::'a::{field_char_0,division_by_zero})
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   608
   = of_rat a / of_rat b"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   609
by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   610
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   611
lemma of_rat_power:
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   612
  "(of_rat (a ^ n)::'a::{field_char_0,recpower}) = of_rat a ^ n"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   613
by (induct n) (simp_all add: of_rat_mult power_Suc)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   614
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   615
lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   616
apply (induct a, induct b)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   617
apply (simp add: of_rat_rat eq_rat)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   618
apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   619
apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   620
done
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   621
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   622
lemma of_rat_less:
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   623
  "(of_rat r :: 'a::ordered_field) < of_rat s \<longleftrightarrow> r < s"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   624
proof (induct r, induct s)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   625
  fix a b c d :: int
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   626
  assume not_zero: "b > 0" "d > 0"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   627
  then have "b * d > 0" by (rule mult_pos_pos)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   628
  have of_int_divide_less_eq:
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   629
    "(of_int a :: 'a) / of_int b < of_int c / of_int d
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   630
      \<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   631
    using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   632
  show "(of_rat (Fract a b) :: 'a::ordered_field) < of_rat (Fract c d)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   633
    \<longleftrightarrow> Fract a b < Fract c d"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   634
    using not_zero `b * d > 0`
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   635
    by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   636
      (auto intro: mult_strict_right_mono mult_right_less_imp_less)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   637
qed
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   638
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   639
lemma of_rat_less_eq:
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   640
  "(of_rat r :: 'a::ordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   641
  unfolding le_less by (auto simp add: of_rat_less)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   642
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   643
lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   644
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   645
lemma of_rat_eq_id [simp]: "of_rat = id"
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   646
proof
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   647
  fix a
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   648
  show "of_rat a = id a"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   649
  by (induct a)
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   650
     (simp add: of_rat_rat Fract_of_int_eq [symmetric])
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   651
qed
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   652
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   653
text{*Collapse nested embeddings*}
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   654
lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   655
by (induct n) (simp_all add: of_rat_add)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   656
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   657
lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   658
by (cases z rule: int_diff_cases) (simp add: of_rat_diff)
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   659
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   660
lemma of_rat_number_of_eq [simp]:
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   661
  "of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   662
by (simp add: number_of_eq)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   663
23879
4776af8be741 split class abs from class minus
haftmann
parents: 23429
diff changeset
   664
lemmas zero_rat = Zero_rat_def
4776af8be741 split class abs from class minus
haftmann
parents: 23429
diff changeset
   665
lemmas one_rat = One_rat_def
4776af8be741 split class abs from class minus
haftmann
parents: 23429
diff changeset
   666
24198
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   667
abbreviation
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   668
  rat_of_nat :: "nat \<Rightarrow> rat"
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   669
where
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   670
  "rat_of_nat \<equiv> of_nat"
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   671
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   672
abbreviation
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   673
  rat_of_int :: "int \<Rightarrow> rat"
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   674
where
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   675
  "rat_of_int \<equiv> of_int"
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   676
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   677
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   678
subsection {* Implementation of rational numbers as pairs of integers *}
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   679
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   680
lemma Fract_norm: "Fract (a div zgcd a b) (b div zgcd a b) = Fract a b"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   681
proof (cases "a = 0 \<or> b = 0")
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   682
  case True then show ?thesis by (auto simp add: eq_rat)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   683
next
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   684
  let ?c = "zgcd a b"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   685
  case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   686
  then have "?c \<noteq> 0" by simp
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   687
  then have "Fract ?c ?c = Fract 1 1" by (simp add: eq_rat)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   688
  moreover have "Fract (a div ?c * ?c + a mod ?c) (b div ?c * ?c + b mod ?c) = Fract a b"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   689
   by (simp add: times_div_mod_plus_zero_one.mod_div_equality)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   690
  moreover have "a mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric])
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   691
  moreover have "b mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric])
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   692
  ultimately show ?thesis
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   693
    by (simp add: mult_rat [symmetric])
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   694
qed
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   695
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   696
definition Fract_norm :: "int \<Rightarrow> int \<Rightarrow> rat" where
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   697
  [simp, code func del]: "Fract_norm a b = Fract a b"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   698
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   699
lemma [code func]: "Fract_norm a b = (if a = 0 \<or> b = 0 then 0 else let c = zgcd a b in
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   700
  if b > 0 then Fract (a div c) (b div c) else Fract (- (a div c)) (- (b div c)))"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   701
  by (simp add: eq_rat Zero_rat_def Let_def Fract_norm)
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   702
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   703
lemma [code]:
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   704
  "of_rat (Fract a b) = (if b \<noteq> 0 then of_int a / of_int b else 0)"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   705
  by (cases "b = 0") (simp_all add: rat_number_collapse of_rat_rat)
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   706
26513
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
   707
instantiation rat :: eq
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
   708
begin
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
   709
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   710
definition [code func del]: "eq_class.eq (a\<Colon>rat) b \<longleftrightarrow> a - b = 0"
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   711
26513
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
   712
instance by default (simp add: eq_rat_def)
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
   713
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   714
lemma rat_eq_code [code]:
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   715
  "eq_class.eq (Fract a b) (Fract c d) \<longleftrightarrow> (if b = 0
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   716
       then c = 0 \<or> d = 0
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   717
     else if d = 0
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   718
       then a = 0 \<or> b = 0
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   719
     else a * d =  b * c)"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   720
  by (auto simp add: eq eq_rat)
26513
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
   721
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
   722
end
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   723
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   724
lemma le_rat':
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   725
  assumes "b \<noteq> 0"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   726
    and "d \<noteq> 0"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   727
  shows "Fract a b \<le> Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d"
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   728
proof -
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   729
  have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   730
  have "a * d * (b * d) \<le> c * b * (b * d) \<longleftrightarrow> a * d * (sgn b * sgn d) \<le> c * b * (sgn b * sgn d)"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   731
  proof (cases "b * d > 0")
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   732
    case True
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   733
    moreover from True have "sgn b * sgn d = 1"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   734
      by (simp add: sgn_times [symmetric] sgn_1_pos)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   735
    ultimately show ?thesis by (simp add: mult_le_cancel_right)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   736
  next
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   737
    case False with assms have "b * d < 0" by (simp add: less_le)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   738
    moreover from this have "sgn b * sgn d = - 1"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   739
      by (simp only: sgn_times [symmetric] sgn_1_neg)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   740
    ultimately show ?thesis by (simp add: mult_le_cancel_right)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   741
  qed
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   742
  also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   743
    by (simp add: abs_sgn mult_ac)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   744
  finally show ?thesis using assms by simp
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   745
qed
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   746
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   747
lemma less_rat': 
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   748
  assumes "b \<noteq> 0"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   749
    and "d \<noteq> 0"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   750
  shows "Fract a b < Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d"
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   751
proof -
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   752
  have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   753
  have "a * d * (b * d) < c * b * (b * d) \<longleftrightarrow> a * d * (sgn b * sgn d) < c * b * (sgn b * sgn d)"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   754
  proof (cases "b * d > 0")
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   755
    case True
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   756
    moreover from True have "sgn b * sgn d = 1"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   757
      by (simp add: sgn_times [symmetric] sgn_1_pos)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   758
    ultimately show ?thesis by (simp add: mult_less_cancel_right)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   759
  next
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   760
    case False with assms have "b * d < 0" by (simp add: less_le)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   761
    moreover from this have "sgn b * sgn d = - 1"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   762
      by (simp only: sgn_times [symmetric] sgn_1_neg)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   763
    ultimately show ?thesis by (simp add: mult_less_cancel_right)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   764
  qed
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   765
  also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   766
    by (simp add: abs_sgn mult_ac)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   767
  finally show ?thesis using assms by simp
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   768
qed
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   769
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   770
lemma rat_less_eq_code [code]:
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   771
  "Fract a b \<le> Fract c d \<longleftrightarrow> (if b = 0
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   772
       then sgn c * sgn d \<ge> 0
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   773
     else if d = 0
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   774
       then sgn a * sgn b \<le> 0
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   775
     else a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d)"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   776
by (auto simp add: sgn_times mult_le_0_iff zero_le_mult_iff le_rat' eq_rat simp del: le_rat)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   777
  (auto simp add: sgn_times sgn_0_0 le_less sgn_1_pos [symmetric] sgn_1_neg [symmetric])
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   778
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   779
lemma rat_le_eq_code [code]:
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   780
  "Fract a b < Fract c d \<longleftrightarrow> (if b = 0
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   781
       then sgn c * sgn d > 0
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   782
     else if d = 0
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   783
       then sgn a * sgn b < 0
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   784
     else a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d)"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   785
by (auto simp add: sgn_times mult_less_0_iff zero_less_mult_iff less_rat' eq_rat simp del: less_rat)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   786
   (auto simp add: sgn_times sgn_0_0 sgn_1_pos [symmetric] sgn_1_neg [symmetric],
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   787
     auto simp add: sgn_1_pos)
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   788
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   789
lemma rat_plus_code [code]:
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   790
  "Fract a b + Fract c d = (if b = 0
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   791
     then Fract c d
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   792
   else if d = 0
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   793
     then Fract a b
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   794
   else Fract_norm (a * d + c * b) (b * d))"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   795
  by (simp add: eq_rat, simp add: Zero_rat_def)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   796
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   797
lemma rat_times_code [code]:
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   798
  "Fract a b * Fract c d = Fract_norm (a * c) (b * d)"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   799
  by simp
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   800
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   801
lemma rat_minus_code [code]:
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   802
  "Fract a b - Fract c d = (if b = 0
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   803
     then Fract (- c) d
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   804
   else if d = 0
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   805
     then Fract a b
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   806
   else Fract_norm (a * d - c * b) (b * d))"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   807
  by (simp add: eq_rat, simp add: Zero_rat_def)
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   808
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   809
lemma rat_inverse_code [code]:
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   810
  "inverse (Fract a b) = (if b = 0 then Fract 1 0
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   811
    else if a < 0 then Fract (- b) (- a)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   812
    else Fract b a)"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   813
  by (simp add: eq_rat)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   814
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   815
lemma rat_divide_code [code]:
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   816
  "Fract a b / Fract c d = Fract_norm (a * d) (b * c)"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   817
  by simp
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   818
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   819
hide (open) const Fract_norm
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   820
24622
8116eb022282 renamed constructor RatC to Rational
haftmann
parents: 24533
diff changeset
   821
text {* Setup for SML code generator *}
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   822
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   823
types_code
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   824
  rat ("(int */ int)")
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   825
attach (term_of) {*
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   826
fun term_of_rat (p, q) =
24622
8116eb022282 renamed constructor RatC to Rational
haftmann
parents: 24533
diff changeset
   827
  let
24661
a705b9834590 fixed cg setup
haftmann
parents: 24630
diff changeset
   828
    val rT = Type ("Rational.rat", [])
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   829
  in
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   830
    if q = 1 orelse p = 0 then HOLogic.mk_number rT p
25885
6fbc3f54f819 New interface for test data generators.
berghofe
parents: 25762
diff changeset
   831
    else @{term "op / \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat"} $
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   832
      HOLogic.mk_number rT p $ HOLogic.mk_number rT q
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   833
  end;
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   834
*}
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   835
attach (test) {*
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   836
fun gen_rat i =
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   837
  let
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   838
    val p = random_range 0 i;
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   839
    val q = random_range 1 (i + 1);
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   840
    val g = Integer.gcd p q;
24630
351a308ab58d simplified type int (eliminated IntInf.int, integer);
wenzelm
parents: 24622
diff changeset
   841
    val p' = p div g;
351a308ab58d simplified type int (eliminated IntInf.int, integer);
wenzelm
parents: 24622
diff changeset
   842
    val q' = q div g;
25885
6fbc3f54f819 New interface for test data generators.
berghofe
parents: 25762
diff changeset
   843
    val r = (if one_of [true, false] then p' else ~ p',
6fbc3f54f819 New interface for test data generators.
berghofe
parents: 25762
diff changeset
   844
      if p' = 0 then 0 else q')
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   845
  in
25885
6fbc3f54f819 New interface for test data generators.
berghofe
parents: 25762
diff changeset
   846
    (r, fn () => term_of_rat r)
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   847
  end;
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   848
*}
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   849
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   850
consts_code
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   851
  Fract ("(_,/ _)")
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   852
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   853
consts_code
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   854
  "of_int :: int \<Rightarrow> rat" ("\<module>rat'_of'_int")
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   855
attach {*
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   856
fun rat_of_int 0 = (0, 0)
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   857
  | rat_of_int i = (i, 1);
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   858
*}
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   859
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   860
end