author | nipkow |
Tue, 26 Aug 2008 12:07:06 +0200 | |
changeset 28001 | 4642317e0deb |
parent 27682 | 25aceefd4786 |
child 28010 | 8312edc51969 |
permissions | -rw-r--r-- |
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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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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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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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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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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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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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next |
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fix q r s :: rat show "(q + r) + s = q + (r + s)" |
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by (cases q, cases r, cases s) (simp add: 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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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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next |
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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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next |
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fix q r :: rat show "q - r = q + - r" |
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by (cases q, cases r) (simp add: eq_rat) |
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next |
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fix q r s :: rat show "(q + r) * s = q * s + r * s" |
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by (cases q, cases r, cases s) (simp add: eq_rat ring_simps) |
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next |
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show "(0::rat) \<noteq> 1" by (simp add: Zero_rat_def One_rat_def eq_rat) |
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next |
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fix q :: rat show "q * 1 = q" |
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by (cases q) (simp add: One_rat_def eq_rat) |
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next |
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fix q :: rat |
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fix n :: nat |
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show "q ^ 0 = 1" by simp |
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show "q ^ (Suc n) = q * (q ^ n)" by simp |
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qed |
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end |
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lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1" |
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by (induct k) (simp_all add: Zero_rat_def One_rat_def) |
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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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lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k" |
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by (rule of_nat_rat [symmetric]) |
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lemma Fract_of_int_eq: "Fract k 1 = of_int k" |
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by (rule of_int_rat [symmetric]) |
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instantiation rat :: number_ring |
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begin |
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definition |
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rat_number_of_def [code func del]: "number_of w = Fract w 1" |
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instance by intro_classes (simp add: rat_number_of_def of_int_rat) |
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end |
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lemma rat_number_collapse [code post]: |
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"Fract 0 k = 0" |
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"Fract 1 1 = 1" |
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"Fract (number_of k) 1 = number_of k" |
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"Fract k 0 = 0" |
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by (cases "k = 0") |
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(simp_all add: Zero_rat_def One_rat_def number_of_is_id number_of_eq of_int_rat eq_rat Fract_def) |
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lemma rat_number_expand [code unfold]: |
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"0 = Fract 0 1" |
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"1 = Fract 1 1" |
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"number_of k = Fract (number_of k) 1" |
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by (simp_all add: rat_number_collapse) |
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lemma iszero_rat [simp]: |
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"iszero (number_of k :: rat) \<longleftrightarrow> iszero (number_of k :: int)" |
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by (simp add: iszero_def rat_number_expand number_of_is_id eq_rat) |
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lemma Rat_cases_nonzero [case_names Fract 0]: |
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assumes Fract: "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> C" |
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assumes 0: "q = 0 \<Longrightarrow> C" |
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shows C |
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proof (cases "q = 0") |
|
251 |
case True then show C using 0 by auto |
|
252 |
next |
|
253 |
case False |
|
254 |
then obtain a b where "q = Fract a b" and "b \<noteq> 0" by (cases q) auto |
|
255 |
moreover with False have "0 \<noteq> Fract a b" by simp |
|
256 |
with `b \<noteq> 0` have "a \<noteq> 0" by (simp add: Zero_rat_def eq_rat) |
|
257 |
with Fract `q = Fract a b` `b \<noteq> 0` show C by auto |
|
258 |
qed |
|
259 |
||
260 |
||
261 |
||
262 |
subsubsection {* The field of rational numbers *} |
|
263 |
||
264 |
instantiation rat :: "{field, division_by_zero}" |
|
265 |
begin |
|
266 |
||
267 |
definition |
|
268 |
inverse_rat_def [code func del]: |
|
269 |
"inverse q = Abs_Rat (\<Union>x \<in> Rep_Rat q. |
|
270 |
ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)})" |
|
271 |
||
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272 |
lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a" |
27551 | 273 |
proof - |
274 |
have "(\<lambda>x. ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)}) respects ratrel" |
|
275 |
by (auto simp add: congruent_def mult_commute) |
|
276 |
then show ?thesis by (simp add: Fract_def inverse_rat_def UN_ratrel) |
|
27509 | 277 |
qed |
278 |
||
27551 | 279 |
definition |
280 |
divide_rat_def [code func del]: "q / r = q * inverse (r::rat)" |
|
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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|
283 |
by (simp add: divide_rat_def) |
27551 | 284 |
|
285 |
instance proof |
|
27652
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|
286 |
show "inverse 0 = (0::rat)" by (simp add: rat_number_expand) |
27551 | 287 |
(simp add: rat_number_collapse) |
288 |
next |
|
289 |
fix q :: rat |
|
290 |
assume "q \<noteq> 0" |
|
291 |
then show "inverse q * q = 1" by (cases q rule: Rat_cases_nonzero) |
|
292 |
(simp_all add: mult_rat inverse_rat rat_number_expand eq_rat) |
|
293 |
next |
|
294 |
fix q r :: rat |
|
295 |
show "q / r = q * inverse r" by (simp add: divide_rat_def) |
|
296 |
qed |
|
297 |
||
298 |
end |
|
299 |
||
300 |
||
301 |
subsubsection {* Various *} |
|
302 |
||
303 |
lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1" |
|
27652
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|
304 |
by (simp add: rat_number_expand) |
27551 | 305 |
|
306 |
lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l" |
|
27652
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changeset
|
307 |
by (simp add: Fract_of_int_eq [symmetric]) |
27551 | 308 |
|
309 |
lemma Fract_number_of_quotient [code post]: |
|
310 |
"Fract (number_of k) (number_of l) = number_of k / number_of l" |
|
311 |
unfolding Fract_of_int_quotient number_of_is_id number_of_eq .. |
|
312 |
||
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|
313 |
lemma Fract_1_number_of [code post]: |
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changeset
|
314 |
"Fract 1 (number_of k) = 1 / number_of k" |
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diff
changeset
|
315 |
unfolding Fract_of_int_quotient number_of_eq by simp |
27551 | 316 |
|
317 |
subsubsection {* The ordered field of rational numbers *} |
|
27509 | 318 |
|
319 |
instantiation rat :: linorder |
|
320 |
begin |
|
321 |
||
322 |
definition |
|
323 |
le_rat_def [code func del]: |
|
324 |
"q \<le> r \<longleftrightarrow> contents (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r. |
|
27551 | 325 |
{(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})" |
326 |
||
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|
327 |
lemma le_rat [simp]: |
27551 | 328 |
assumes "b \<noteq> 0" and "d \<noteq> 0" |
329 |
shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)" |
|
330 |
proof - |
|
331 |
have "(\<lambda>x y. {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)}) |
|
332 |
respects2 ratrel" |
|
333 |
proof (clarsimp simp add: congruent2_def) |
|
334 |
fix a b a' b' c d c' d'::int |
|
335 |
assume neq: "b \<noteq> 0" "b' \<noteq> 0" "d \<noteq> 0" "d' \<noteq> 0" |
|
336 |
assume eq1: "a * b' = a' * b" |
|
337 |
assume eq2: "c * d' = c' * d" |
|
338 |
||
339 |
let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))" |
|
340 |
{ |
|
341 |
fix a b c d x :: int assume x: "x \<noteq> 0" |
|
342 |
have "?le a b c d = ?le (a * x) (b * x) c d" |
|
343 |
proof - |
|
344 |
from x have "0 < x * x" by (auto simp add: zero_less_mult_iff) |
|
345 |
hence "?le a b c d = |
|
346 |
((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))" |
|
347 |
by (simp add: mult_le_cancel_right) |
|
348 |
also have "... = ?le (a * x) (b * x) c d" |
|
349 |
by (simp add: mult_ac) |
|
350 |
finally show ?thesis . |
|
351 |
qed |
|
352 |
} note le_factor = this |
|
353 |
||
354 |
let ?D = "b * d" and ?D' = "b' * d'" |
|
355 |
from neq have D: "?D \<noteq> 0" by simp |
|
356 |
from neq have "?D' \<noteq> 0" by simp |
|
357 |
hence "?le a b c d = ?le (a * ?D') (b * ?D') c d" |
|
358 |
by (rule le_factor) |
|
27668 | 359 |
also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')" |
27551 | 360 |
by (simp add: mult_ac) |
361 |
also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')" |
|
362 |
by (simp only: eq1 eq2) |
|
363 |
also have "... = ?le (a' * ?D) (b' * ?D) c' d'" |
|
364 |
by (simp add: mult_ac) |
|
365 |
also from D have "... = ?le a' b' c' d'" |
|
366 |
by (rule le_factor [symmetric]) |
|
367 |
finally show "?le a b c d = ?le a' b' c' d'" . |
|
368 |
qed |
|
369 |
with assms show ?thesis by (simp add: Fract_def le_rat_def UN_ratrel2) |
|
370 |
qed |
|
27509 | 371 |
|
372 |
definition |
|
373 |
less_rat_def [code func del]: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w" |
|
374 |
||
27652
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changeset
|
375 |
lemma less_rat [simp]: |
27551 | 376 |
assumes "b \<noteq> 0" and "d \<noteq> 0" |
377 |
shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)" |
|
27652
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haftmann
parents:
27551
diff
changeset
|
378 |
using assms by (simp add: less_rat_def eq_rat order_less_le) |
27509 | 379 |
|
380 |
instance proof |
|
14365
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paulson
parents:
diff
changeset
|
381 |
fix q r s :: rat |
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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 |
27668 | 399 |
also have "... = (c * f) * (d * f) * (b * b)" by algebra |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
400 |
also have "... \<le> (e * d) * (d * f) * (b * b)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
401 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
402 |
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
|
403 |
by simp |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
404 |
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
|
405 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
406 |
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
|
407 |
by (simp only: mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
408 |
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
|
409 |
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
|
410 |
with neq show ?thesis by simp |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
411 |
qed |
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 |
next |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
414 |
assume "q \<le> r" and "r \<le> q" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
415 |
show "q = r" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
416 |
proof (insert prems, induct q, induct r) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
417 |
fix a b c d :: int |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
418 |
assume neq: "b \<noteq> 0" "d \<noteq> 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
419 |
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
|
420 |
show "Fract a b = Fract c d" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
421 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
422 |
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
|
423 |
by simp |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
424 |
also have "... \<le> (a * d) * (b * d)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
425 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
426 |
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
|
427 |
by simp |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
428 |
thus ?thesis by (simp only: mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
429 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
430 |
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
|
431 |
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
|
432 |
ultimately have "a * d = c * b" by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
433 |
with neq show ?thesis by (simp add: eq_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
434 |
qed |
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 |
next |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
437 |
show "q \<le> q" |
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
438 |
by (induct q) simp |
27682 | 439 |
show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)" |
440 |
by (induct q, induct r) (auto simp add: le_less mult_commute) |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
441 |
show "q \<le> r \<or> r \<le> q" |
18913 | 442 |
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
|
443 |
(simp add: mult_commute, rule linorder_linear) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
444 |
} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
445 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
446 |
|
27509 | 447 |
end |
448 |
||
27551 | 449 |
instantiation rat :: "{distrib_lattice, abs_if, sgn_if}" |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
450 |
begin |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
451 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
452 |
definition |
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
453 |
abs_rat_def [code func del]: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))" |
27551 | 454 |
|
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
455 |
lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>" |
27551 | 456 |
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) |
457 |
||
458 |
definition |
|
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
459 |
sgn_rat_def [code func del]: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)" |
27551 | 460 |
|
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
461 |
lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)" |
27551 | 462 |
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
|
463 |
by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat) |
27551 | 464 |
(auto simp: rat_number_collapse not_less le_less zero_less_mult_iff) |
465 |
||
466 |
definition |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
467 |
"(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min" |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
468 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
469 |
definition |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
470 |
"(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max" |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
471 |
|
27551 | 472 |
instance by intro_classes |
473 |
(auto simp add: abs_rat_def sgn_rat_def min_max.sup_inf_distrib1 inf_rat_def sup_rat_def) |
|
22456 | 474 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
475 |
end |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
476 |
|
27551 | 477 |
instance rat :: ordered_field |
478 |
proof |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
479 |
fix q r s :: rat |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
480 |
show "q \<le> r ==> s + q \<le> s + r" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
481 |
proof (induct q, induct r, induct s) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
482 |
fix a b c d e f :: int |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
483 |
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
|
484 |
assume le: "Fract a b \<le> Fract c d" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
485 |
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
|
486 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
487 |
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
|
488 |
by (auto simp add: zero_less_mult_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
489 |
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
|
490 |
by simp |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
491 |
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
|
492 |
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
|
493 |
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
|
494 |
qed |
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 |
show "q < r ==> 0 < s ==> s * q < s * r" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
497 |
proof (induct q, induct r, induct s) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
498 |
fix a b c d e f :: int |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
499 |
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
|
500 |
assume le: "Fract a b < Fract c d" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
501 |
assume gt: "0 < Fract e f" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
502 |
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
|
503 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
504 |
let ?E = "e * f" and ?F = "f * f" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
505 |
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
|
506 |
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
|
507 |
moreover from neq have "0 < ?F" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
508 |
by (auto simp add: zero_less_mult_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
509 |
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
|
510 |
by simp |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
511 |
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
|
512 |
by (simp add: mult_less_cancel_right) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
513 |
with neq show ?thesis |
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
514 |
by (simp add: mult_ac) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
515 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
516 |
qed |
27551 | 517 |
qed auto |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
518 |
|
27551 | 519 |
lemma Rat_induct_pos [case_names Fract, induct type: rat]: |
520 |
assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)" |
|
521 |
shows "P q" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
522 |
proof (cases q) |
27551 | 523 |
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
|
524 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
525 |
fix a::int and b::int |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
526 |
assume b: "b < 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
527 |
hence "0 < -b" by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
528 |
hence "P (Fract (-a) (-b))" by (rule step) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
529 |
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
|
530 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
531 |
case (Fract a b) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
532 |
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
|
533 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
534 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
535 |
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
|
536 |
"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
|
537 |
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
|
538 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
539 |
|
27551 | 540 |
subsection {* Arithmetic setup *} |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
541 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
542 |
use "rat_arith.ML" |
24075 | 543 |
declaration {* K rat_arith_setup *} |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
544 |
|
23342 | 545 |
|
546 |
subsection {* Embedding from Rationals to other Fields *} |
|
547 |
||
24198 | 548 |
class field_char_0 = field + ring_char_0 |
23342 | 549 |
|
27551 | 550 |
subclass (in ordered_field) field_char_0 .. |
23342 | 551 |
|
27551 | 552 |
context field_char_0 |
553 |
begin |
|
554 |
||
555 |
definition of_rat :: "rat \<Rightarrow> 'a" where |
|
24198 | 556 |
[code func del]: "of_rat q = contents (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})" |
23342 | 557 |
|
27551 | 558 |
end |
559 |
||
23342 | 560 |
lemma of_rat_congruent: |
27551 | 561 |
"(\<lambda>(a, b). {of_int a / of_int b :: 'a::field_char_0}) respects ratrel" |
23342 | 562 |
apply (rule congruent.intro) |
563 |
apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq) |
|
564 |
apply (simp only: of_int_mult [symmetric]) |
|
565 |
done |
|
566 |
||
27551 | 567 |
lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b" |
568 |
unfolding Fract_def of_rat_def by (simp add: UN_ratrel of_rat_congruent) |
|
23342 | 569 |
|
570 |
lemma of_rat_0 [simp]: "of_rat 0 = 0" |
|
571 |
by (simp add: Zero_rat_def of_rat_rat) |
|
572 |
||
573 |
lemma of_rat_1 [simp]: "of_rat 1 = 1" |
|
574 |
by (simp add: One_rat_def of_rat_rat) |
|
575 |
||
576 |
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
|
577 |
by (induct a, induct b, simp add: of_rat_rat add_frac_eq) |
23342 | 578 |
|
23343 | 579 |
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
|
580 |
by (induct a, simp add: of_rat_rat) |
23343 | 581 |
|
582 |
lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b" |
|
583 |
by (simp only: diff_minus of_rat_add of_rat_minus) |
|
584 |
||
23342 | 585 |
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
|
586 |
apply (induct a, induct b, simp add: of_rat_rat) |
23342 | 587 |
apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac) |
588 |
done |
|
589 |
||
590 |
lemma nonzero_of_rat_inverse: |
|
591 |
"a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)" |
|
23343 | 592 |
apply (rule inverse_unique [symmetric]) |
593 |
apply (simp add: of_rat_mult [symmetric]) |
|
23342 | 594 |
done |
595 |
||
596 |
lemma of_rat_inverse: |
|
597 |
"(of_rat (inverse a)::'a::{field_char_0,division_by_zero}) = |
|
598 |
inverse (of_rat a)" |
|
599 |
by (cases "a = 0", simp_all add: nonzero_of_rat_inverse) |
|
600 |
||
601 |
lemma nonzero_of_rat_divide: |
|
602 |
"b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b" |
|
603 |
by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse) |
|
604 |
||
605 |
lemma of_rat_divide: |
|
606 |
"(of_rat (a / b)::'a::{field_char_0,division_by_zero}) |
|
607 |
= 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
|
608 |
by (cases "b = 0") (simp_all add: nonzero_of_rat_divide) |
23342 | 609 |
|
23343 | 610 |
lemma of_rat_power: |
611 |
"(of_rat (a ^ n)::'a::{field_char_0,recpower}) = of_rat a ^ n" |
|
612 |
by (induct n) (simp_all add: of_rat_mult power_Suc) |
|
613 |
||
614 |
lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)" |
|
615 |
apply (induct a, induct b) |
|
616 |
apply (simp add: of_rat_rat eq_rat) |
|
617 |
apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq) |
|
618 |
apply (simp only: of_int_mult [symmetric] of_int_eq_iff) |
|
619 |
done |
|
620 |
||
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
621 |
lemma of_rat_less: |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
622 |
"(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
|
623 |
proof (induct r, induct s) |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
624 |
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
|
625 |
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
|
626 |
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
|
627 |
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
|
628 |
"(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
|
629 |
\<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
|
630 |
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
|
631 |
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
|
632 |
\<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
|
633 |
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
|
634 |
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
|
635 |
(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
|
636 |
qed |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
637 |
|
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
638 |
lemma of_rat_less_eq: |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
639 |
"(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
|
640 |
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
|
641 |
|
23343 | 642 |
lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified] |
643 |
||
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
644 |
lemma of_rat_eq_id [simp]: "of_rat = id" |
23343 | 645 |
proof |
646 |
fix a |
|
647 |
show "of_rat a = id a" |
|
648 |
by (induct a) |
|
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
649 |
(simp add: of_rat_rat Fract_of_int_eq [symmetric]) |
23343 | 650 |
qed |
651 |
||
652 |
text{*Collapse nested embeddings*} |
|
653 |
lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n" |
|
654 |
by (induct n) (simp_all add: of_rat_add) |
|
655 |
||
656 |
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
|
657 |
by (cases z rule: int_diff_cases) (simp add: of_rat_diff) |
23343 | 658 |
|
659 |
lemma of_rat_number_of_eq [simp]: |
|
660 |
"of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})" |
|
661 |
by (simp add: number_of_eq) |
|
662 |
||
23879 | 663 |
lemmas zero_rat = Zero_rat_def |
664 |
lemmas one_rat = One_rat_def |
|
665 |
||
24198 | 666 |
abbreviation |
667 |
rat_of_nat :: "nat \<Rightarrow> rat" |
|
668 |
where |
|
669 |
"rat_of_nat \<equiv> of_nat" |
|
670 |
||
671 |
abbreviation |
|
672 |
rat_of_int :: "int \<Rightarrow> rat" |
|
673 |
where |
|
674 |
"rat_of_int \<equiv> of_int" |
|
675 |
||
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
676 |
|
28001 | 677 |
context field_char_0 |
678 |
begin |
|
679 |
||
680 |
definition |
|
681 |
Rats :: "'a set" where |
|
682 |
[code func del]: "Rats = range of_rat" |
|
683 |
||
684 |
notation (xsymbols) |
|
685 |
Rats ("\<rat>") |
|
686 |
||
687 |
end |
|
688 |
||
689 |
||
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
690 |
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
|
691 |
|
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
692 |
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
|
693 |
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
|
694 |
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
|
695 |
next |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
696 |
let ?c = "zgcd a b" |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
697 |
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
|
698 |
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
|
699 |
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
|
700 |
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
|
701 |
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
|
702 |
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
|
703 |
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
|
704 |
ultimately show ?thesis |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
705 |
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
|
706 |
qed |
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
707 |
|
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
708 |
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
|
709 |
[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
|
710 |
|
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
711 |
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
|
712 |
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
|
713 |
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
|
714 |
|
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
715 |
lemma [code]: |
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
716 |
"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
|
717 |
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
|
718 |
|
26513 | 719 |
instantiation rat :: eq |
720 |
begin |
|
721 |
||
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
722 |
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
|
723 |
|
26513 | 724 |
instance by default (simp add: eq_rat_def) |
725 |
||
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
726 |
lemma rat_eq_code [code]: |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
727 |
"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
|
728 |
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
|
729 |
else if d = 0 |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
730 |
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
|
731 |
else a * d = b * c)" |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
732 |
by (auto simp add: eq eq_rat) |
26513 | 733 |
|
734 |
end |
|
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
735 |
|
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
736 |
lemma le_rat': |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
737 |
assumes "b \<noteq> 0" |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
738 |
and "d \<noteq> 0" |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
739 |
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
|
740 |
proof - |
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
741 |
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
|
742 |
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
|
743 |
proof (cases "b * d > 0") |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
744 |
case True |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
745 |
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
|
746 |
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
|
747 |
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
|
748 |
next |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
749 |
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
|
750 |
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
|
751 |
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
|
752 |
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
|
753 |
qed |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
754 |
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
|
755 |
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
|
756 |
finally show ?thesis using assms by simp |
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
757 |
qed |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
758 |
|
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
759 |
lemma less_rat': |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
760 |
assumes "b \<noteq> 0" |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
761 |
and "d \<noteq> 0" |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
762 |
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
|
763 |
proof - |
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
764 |
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
|
765 |
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
|
766 |
proof (cases "b * d > 0") |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
767 |
case True |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
768 |
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
|
769 |
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
|
770 |
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
|
771 |
next |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
772 |
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
|
773 |
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
|
774 |
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
|
775 |
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
|
776 |
qed |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
777 |
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
|
778 |
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
|
779 |
finally show ?thesis using assms by simp |
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
780 |
qed |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
781 |
|
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
782 |
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
|
783 |
"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
|
784 |
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
|
785 |
else if d = 0 |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
786 |
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
|
787 |
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
|
788 |
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
|
789 |
(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
|
790 |
|
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
791 |
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
|
792 |
"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
|
793 |
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
|
794 |
else if d = 0 |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
795 |
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
|
796 |
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
|
797 |
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
|
798 |
(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
|
799 |
auto simp add: sgn_1_pos) |
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_plus_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) |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
808 |
|
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
809 |
lemma rat_times_code [code]: |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
810 |
"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
|
811 |
by simp |
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
812 |
|
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
813 |
lemma rat_minus_code [code]: |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
814 |
"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
|
815 |
then Fract (- c) d |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
816 |
else if d = 0 |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
817 |
then Fract a b |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
818 |
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
|
819 |
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
|
820 |
|
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
821 |
lemma rat_inverse_code [code]: |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
822 |
"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
|
823 |
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
|
824 |
else Fract b a)" |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
825 |
by (simp add: eq_rat) |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
826 |
|
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
827 |
lemma rat_divide_code [code]: |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
828 |
"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
|
829 |
by simp |
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
830 |
|
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
831 |
hide (open) const Fract_norm |
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
832 |
|
24622 | 833 |
text {* Setup for SML code generator *} |
24533
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 |
types_code |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
836 |
rat ("(int */ int)") |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
837 |
attach (term_of) {* |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
838 |
fun term_of_rat (p, q) = |
24622 | 839 |
let |
24661 | 840 |
val rT = Type ("Rational.rat", []) |
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
841 |
in |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
842 |
if q = 1 orelse p = 0 then HOLogic.mk_number rT p |
25885 | 843 |
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
|
844 |
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
|
845 |
end; |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
846 |
*} |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
847 |
attach (test) {* |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
848 |
fun gen_rat i = |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
849 |
let |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
850 |
val p = random_range 0 i; |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
851 |
val q = random_range 1 (i + 1); |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
852 |
val g = Integer.gcd p q; |
24630
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
24622
diff
changeset
|
853 |
val p' = p div g; |
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
24622
diff
changeset
|
854 |
val q' = q div g; |
25885 | 855 |
val r = (if one_of [true, false] then p' else ~ p', |
856 |
if p' = 0 then 0 else q') |
|
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
857 |
in |
25885 | 858 |
(r, fn () => term_of_rat r) |
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
859 |
end; |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
860 |
*} |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
861 |
|
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
862 |
consts_code |
27551 | 863 |
Fract ("(_,/ _)") |
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
864 |
|
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
865 |
consts_code |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
866 |
"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
|
867 |
attach {* |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
868 |
fun rat_of_int 0 = (0, 0) |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
869 |
| rat_of_int i = (i, 1); |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
870 |
*} |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
871 |
|
27652
818666de6c24
refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents:
27551
diff
changeset
|
872 |
end |