haftmann@27551
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(* Title: HOL/Library/Rational.thy
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haftmann@27551
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ID: $Id$
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Author: Markus Wenzel, TU Muenchen
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paulson@14365
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*)
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paulson@14365
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wenzelm@14691
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header {* Rational numbers *}
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paulson@14365
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nipkow@15131
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theory Rational
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haftmann@27652
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imports "../Presburger" GCD
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haftmann@16417
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uses ("rat_arith.ML")
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nipkow@15131
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begin
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subsection {* Rational numbers as quotient *}
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subsubsection {* Construction of the type of rational numbers *}
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wenzelm@21404
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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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huffman@18913
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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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huffman@18913
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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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huffman@18913
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also have "b * (a'' * b') = b' * (a'' * b)" by simp
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huffman@18913
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finally have "b' * (a * b'') = b' * (a'' * b)" .
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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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huffman@18913
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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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huffman@18913
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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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paulson@14365
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qed
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paulson@14365
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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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chaieb@27668
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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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haftmann@27652
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by (cases q, cases r) (simp add: eq_rat)
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next
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haftmann@27551
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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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haftmann@27551
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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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huffman@27509
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show "q ^ (Suc n) = q * (q ^ n)" by simp
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huffman@27509
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qed
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huffman@27509
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huffman@27509
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end
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huffman@27509
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haftmann@27551
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lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1"
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haftmann@27652
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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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haftmann@27652
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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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haftmann@27551
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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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haftmann@27551
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end
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haftmann@27551
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haftmann@27551
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lemma rat_number_collapse [code post]:
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haftmann@27551
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"Fract 0 k = 0"
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haftmann@27551
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"Fract 1 1 = 1"
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haftmann@27551
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"Fract (number_of k) 1 = number_of k"
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haftmann@27551
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"Fract k 0 = 0"
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haftmann@27551
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by (cases "k = 0")
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haftmann@27551
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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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haftmann@27551
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haftmann@27551
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lemma rat_number_expand [code unfold]:
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haftmann@27551
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"0 = Fract 0 1"
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haftmann@27551
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"1 = Fract 1 1"
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haftmann@27551
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"number_of k = Fract (number_of k) 1"
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haftmann@27551
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by (simp_all add: rat_number_collapse)
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haftmann@27551
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haftmann@27551
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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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haftmann@27551
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by (simp add: iszero_def rat_number_expand number_of_is_id eq_rat)
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haftmann@27551
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haftmann@27551
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lemma Rat_cases_nonzero [case_names Fract 0]:
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haftmann@27551
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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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haftmann@27551
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assumes 0: "q = 0 \<Longrightarrow> C"
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haftmann@27551
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shows C
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haftmann@27551
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proof (cases "q = 0")
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haftmann@27551
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case True then show C using 0 by auto
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next
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case False
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haftmann@27551
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then obtain a b where "q = Fract a b" and "b \<noteq> 0" by (cases q) auto
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haftmann@27551
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moreover with False have "0 \<noteq> Fract a b" by simp
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haftmann@27551
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with `b \<noteq> 0` have "a \<noteq> 0" by (simp add: Zero_rat_def eq_rat)
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haftmann@27551
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with Fract `q = Fract a b` `b \<noteq> 0` show C by auto
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haftmann@27551
|
258 |
qed
|
haftmann@27551
|
259 |
|
haftmann@27551
|
260 |
|
haftmann@27551
|
261 |
|
haftmann@27551
|
262 |
subsubsection {* The field of rational numbers *}
|
haftmann@27551
|
263 |
|
haftmann@27551
|
264 |
instantiation rat :: "{field, division_by_zero}"
|
haftmann@27551
|
265 |
begin
|
haftmann@27551
|
266 |
|
haftmann@27551
|
267 |
definition
|
haftmann@27551
|
268 |
inverse_rat_def [code func del]:
|
haftmann@27551
|
269 |
"inverse q = Abs_Rat (\<Union>x \<in> Rep_Rat q.
|
haftmann@27551
|
270 |
ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)})"
|
haftmann@27551
|
271 |
|
haftmann@27652
|
272 |
lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"
|
haftmann@27551
|
273 |
proof -
|
haftmann@27551
|
274 |
have "(\<lambda>x. ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)}) respects ratrel"
|
haftmann@27551
|
275 |
by (auto simp add: congruent_def mult_commute)
|
haftmann@27551
|
276 |
then show ?thesis by (simp add: Fract_def inverse_rat_def UN_ratrel)
|
huffman@27509
|
277 |
qed
|
huffman@27509
|
278 |
|
haftmann@27551
|
279 |
definition
|
haftmann@27551
|
280 |
divide_rat_def [code func del]: "q / r = q * inverse (r::rat)"
|
haftmann@27551
|
281 |
|
haftmann@27652
|
282 |
lemma divide_rat [simp]: "Fract a b / Fract c d = Fract (a * d) (b * c)"
|
haftmann@27652
|
283 |
by (simp add: divide_rat_def)
|
haftmann@27551
|
284 |
|
haftmann@27551
|
285 |
instance proof
|
haftmann@27652
|
286 |
show "inverse 0 = (0::rat)" by (simp add: rat_number_expand)
|
haftmann@27551
|
287 |
(simp add: rat_number_collapse)
|
haftmann@27551
|
288 |
next
|
haftmann@27551
|
289 |
fix q :: rat
|
haftmann@27551
|
290 |
assume "q \<noteq> 0"
|
haftmann@27551
|
291 |
then show "inverse q * q = 1" by (cases q rule: Rat_cases_nonzero)
|
haftmann@27551
|
292 |
(simp_all add: mult_rat inverse_rat rat_number_expand eq_rat)
|
haftmann@27551
|
293 |
next
|
haftmann@27551
|
294 |
fix q r :: rat
|
haftmann@27551
|
295 |
show "q / r = q * inverse r" by (simp add: divide_rat_def)
|
haftmann@27551
|
296 |
qed
|
haftmann@27551
|
297 |
|
haftmann@27551
|
298 |
end
|
haftmann@27551
|
299 |
|
haftmann@27551
|
300 |
|
haftmann@27551
|
301 |
subsubsection {* Various *}
|
haftmann@27551
|
302 |
|
haftmann@27551
|
303 |
lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
|
haftmann@27652
|
304 |
by (simp add: rat_number_expand)
|
haftmann@27551
|
305 |
|
haftmann@27551
|
306 |
lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"
|
haftmann@27652
|
307 |
by (simp add: Fract_of_int_eq [symmetric])
|
haftmann@27551
|
308 |
|
haftmann@27551
|
309 |
lemma Fract_number_of_quotient [code post]:
|
haftmann@27551
|
310 |
"Fract (number_of k) (number_of l) = number_of k / number_of l"
|
haftmann@27551
|
311 |
unfolding Fract_of_int_quotient number_of_is_id number_of_eq ..
|
haftmann@27551
|
312 |
|
haftmann@27652
|
313 |
lemma Fract_1_number_of [code post]:
|
haftmann@27652
|
314 |
"Fract 1 (number_of k) = 1 / number_of k"
|
haftmann@27652
|
315 |
unfolding Fract_of_int_quotient number_of_eq by simp
|
haftmann@27551
|
316 |
|
haftmann@27551
|
317 |
subsubsection {* The ordered field of rational numbers *}
|
huffman@27509
|
318 |
|
huffman@27509
|
319 |
instantiation rat :: linorder
|
huffman@27509
|
320 |
begin
|
huffman@27509
|
321 |
|
huffman@27509
|
322 |
definition
|
huffman@27509
|
323 |
le_rat_def [code func del]:
|
huffman@27509
|
324 |
"q \<le> r \<longleftrightarrow> contents (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
|
haftmann@27551
|
325 |
{(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})"
|
haftmann@27551
|
326 |
|
haftmann@27652
|
327 |
lemma le_rat [simp]:
|
haftmann@27551
|
328 |
assumes "b \<noteq> 0" and "d \<noteq> 0"
|
haftmann@27551
|
329 |
shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
|
haftmann@27551
|
330 |
proof -
|
haftmann@27551
|
331 |
have "(\<lambda>x y. {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})
|
haftmann@27551
|
332 |
respects2 ratrel"
|
haftmann@27551
|
333 |
proof (clarsimp simp add: congruent2_def)
|
haftmann@27551
|
334 |
fix a b a' b' c d c' d'::int
|
haftmann@27551
|
335 |
assume neq: "b \<noteq> 0" "b' \<noteq> 0" "d \<noteq> 0" "d' \<noteq> 0"
|
haftmann@27551
|
336 |
assume eq1: "a * b' = a' * b"
|
haftmann@27551
|
337 |
assume eq2: "c * d' = c' * d"
|
haftmann@27551
|
338 |
|
haftmann@27551
|
339 |
let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
|
haftmann@27551
|
340 |
{
|
haftmann@27551
|
341 |
fix a b c d x :: int assume x: "x \<noteq> 0"
|
haftmann@27551
|
342 |
have "?le a b c d = ?le (a * x) (b * x) c d"
|
haftmann@27551
|
343 |
proof -
|
haftmann@27551
|
344 |
from x have "0 < x * x" by (auto simp add: zero_less_mult_iff)
|
haftmann@27551
|
345 |
hence "?le a b c d =
|
haftmann@27551
|
346 |
((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
|
haftmann@27551
|
347 |
by (simp add: mult_le_cancel_right)
|
haftmann@27551
|
348 |
also have "... = ?le (a * x) (b * x) c d"
|
haftmann@27551
|
349 |
by (simp add: mult_ac)
|
haftmann@27551
|
350 |
finally show ?thesis .
|
haftmann@27551
|
351 |
qed
|
haftmann@27551
|
352 |
} note le_factor = this
|
haftmann@27551
|
353 |
|
haftmann@27551
|
354 |
let ?D = "b * d" and ?D' = "b' * d'"
|
haftmann@27551
|
355 |
from neq have D: "?D \<noteq> 0" by simp
|
haftmann@27551
|
356 |
from neq have "?D' \<noteq> 0" by simp
|
haftmann@27551
|
357 |
hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"
|
haftmann@27551
|
358 |
by (rule le_factor)
|
chaieb@27668
|
359 |
also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')"
|
haftmann@27551
|
360 |
by (simp add: mult_ac)
|
haftmann@27551
|
361 |
also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
|
haftmann@27551
|
362 |
by (simp only: eq1 eq2)
|
haftmann@27551
|
363 |
also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
|
haftmann@27551
|
364 |
by (simp add: mult_ac)
|
haftmann@27551
|
365 |
also from D have "... = ?le a' b' c' d'"
|
haftmann@27551
|
366 |
by (rule le_factor [symmetric])
|
haftmann@27551
|
367 |
finally show "?le a b c d = ?le a' b' c' d'" .
|
haftmann@27551
|
368 |
qed
|
haftmann@27551
|
369 |
with assms show ?thesis by (simp add: Fract_def le_rat_def UN_ratrel2)
|
haftmann@27551
|
370 |
qed
|
huffman@27509
|
371 |
|
huffman@27509
|
372 |
definition
|
huffman@27509
|
373 |
less_rat_def [code func del]: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
|
huffman@27509
|
374 |
|
haftmann@27652
|
375 |
lemma less_rat [simp]:
|
haftmann@27551
|
376 |
assumes "b \<noteq> 0" and "d \<noteq> 0"
|
haftmann@27551
|
377 |
shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
|
haftmann@27652
|
378 |
using assms by (simp add: less_rat_def eq_rat order_less_le)
|
huffman@27509
|
379 |
|
huffman@27509
|
380 |
instance proof
|
paulson@14365
|
381 |
fix q r s :: rat
|
paulson@14365
|
382 |
{
|
paulson@14365
|
383 |
assume "q \<le> r" and "r \<le> s"
|
paulson@14365
|
384 |
show "q \<le> s"
|
paulson@14365
|
385 |
proof (insert prems, induct q, induct r, induct s)
|
paulson@14365
|
386 |
fix a b c d e f :: int
|
paulson@14365
|
387 |
assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0"
|
paulson@14365
|
388 |
assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"
|
paulson@14365
|
389 |
show "Fract a b \<le> Fract e f"
|
paulson@14365
|
390 |
proof -
|
paulson@14365
|
391 |
from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
|
paulson@14365
|
392 |
by (auto simp add: zero_less_mult_iff linorder_neq_iff)
|
paulson@14365
|
393 |
have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
|
paulson@14365
|
394 |
proof -
|
paulson@14365
|
395 |
from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
|
haftmann@27652
|
396 |
by simp
|
paulson@14365
|
397 |
with ff show ?thesis by (simp add: mult_le_cancel_right)
|
paulson@14365
|
398 |
qed
|
chaieb@27668
|
399 |
also have "... = (c * f) * (d * f) * (b * b)" by algebra
|
paulson@14365
|
400 |
also have "... \<le> (e * d) * (d * f) * (b * b)"
|
paulson@14365
|
401 |
proof -
|
paulson@14365
|
402 |
from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
|
haftmann@27652
|
403 |
by simp
|
paulson@14365
|
404 |
with bb show ?thesis by (simp add: mult_le_cancel_right)
|
paulson@14365
|
405 |
qed
|
paulson@14365
|
406 |
finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
|
paulson@14365
|
407 |
by (simp only: mult_ac)
|
paulson@14365
|
408 |
with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
|
paulson@14365
|
409 |
by (simp add: mult_le_cancel_right)
|
haftmann@27652
|
410 |
with neq show ?thesis by simp
|
paulson@14365
|
411 |
qed
|
paulson@14365
|
412 |
qed
|
paulson@14365
|
413 |
next
|
paulson@14365
|
414 |
assume "q \<le> r" and "r \<le> q"
|
paulson@14365
|
415 |
show "q = r"
|
paulson@14365
|
416 |
proof (insert prems, induct q, induct r)
|
paulson@14365
|
417 |
fix a b c d :: int
|
paulson@14365
|
418 |
assume neq: "b \<noteq> 0" "d \<noteq> 0"
|
paulson@14365
|
419 |
assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"
|
paulson@14365
|
420 |
show "Fract a b = Fract c d"
|
paulson@14365
|
421 |
proof -
|
paulson@14365
|
422 |
from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
|
haftmann@27652
|
423 |
by simp
|
paulson@14365
|
424 |
also have "... \<le> (a * d) * (b * d)"
|
paulson@14365
|
425 |
proof -
|
paulson@14365
|
426 |
from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
|
haftmann@27652
|
427 |
by simp
|
paulson@14365
|
428 |
thus ?thesis by (simp only: mult_ac)
|
paulson@14365
|
429 |
qed
|
paulson@14365
|
430 |
finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
|
paulson@14365
|
431 |
moreover from neq have "b * d \<noteq> 0" by simp
|
paulson@14365
|
432 |
ultimately have "a * d = c * b" by simp
|
paulson@14365
|
433 |
with neq show ?thesis by (simp add: eq_rat)
|
paulson@14365
|
434 |
qed
|
paulson@14365
|
435 |
qed
|
paulson@14365
|
436 |
next
|
paulson@14365
|
437 |
show "q \<le> q"
|
haftmann@27652
|
438 |
by (induct q) simp
|
haftmann@27682
|
439 |
show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)"
|
haftmann@27682
|
440 |
by (induct q, induct r) (auto simp add: le_less mult_commute)
|
paulson@14365
|
441 |
show "q \<le> r \<or> r \<le> q"
|
huffman@18913
|
442 |
by (induct q, induct r)
|
haftmann@27652
|
443 |
(simp add: mult_commute, rule linorder_linear)
|
paulson@14365
|
444 |
}
|
paulson@14365
|
445 |
qed
|
paulson@14365
|
446 |
|
huffman@27509
|
447 |
end
|
huffman@27509
|
448 |
|
haftmann@27551
|
449 |
instantiation rat :: "{distrib_lattice, abs_if, sgn_if}"
|
haftmann@25571
|
450 |
begin
|
haftmann@25571
|
451 |
|
haftmann@25571
|
452 |
definition
|
haftmann@27652
|
453 |
abs_rat_def [code func del]: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))"
|
haftmann@27551
|
454 |
|
haftmann@27652
|
455 |
lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
|
haftmann@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)
|
haftmann@27551
|
457 |
|
haftmann@27551
|
458 |
definition
|
haftmann@27652
|
459 |
sgn_rat_def [code func del]: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
|
haftmann@27551
|
460 |
|
haftmann@27652
|
461 |
lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)"
|
haftmann@27551
|
462 |
unfolding Fract_of_int_eq
|
haftmann@27652
|
463 |
by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat)
|
haftmann@27551
|
464 |
(auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)
|
haftmann@27551
|
465 |
|
haftmann@27551
|
466 |
definition
|
haftmann@25571
|
467 |
"(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min"
|
haftmann@25571
|
468 |
|
haftmann@25571
|
469 |
definition
|
haftmann@25571
|
470 |
"(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max"
|
haftmann@25571
|
471 |
|
haftmann@27551
|
472 |
instance by intro_classes
|
haftmann@27551
|
473 |
(auto simp add: abs_rat_def sgn_rat_def min_max.sup_inf_distrib1 inf_rat_def sup_rat_def)
|
haftmann@22456
|
474 |
|
haftmann@25571
|
475 |
end
|
haftmann@25571
|
476 |
|
haftmann@27551
|
477 |
instance rat :: ordered_field
|
haftmann@27551
|
478 |
proof
|
paulson@14365
|
479 |
fix q r s :: rat
|
paulson@14365
|
480 |
show "q \<le> r ==> s + q \<le> s + r"
|
paulson@14365
|
481 |
proof (induct q, induct r, induct s)
|
paulson@14365
|
482 |
fix a b c d e f :: int
|
paulson@14365
|
483 |
assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0"
|
paulson@14365
|
484 |
assume le: "Fract a b \<le> Fract c d"
|
paulson@14365
|
485 |
show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
|
paulson@14365
|
486 |
proof -
|
paulson@14365
|
487 |
let ?F = "f * f" from neq have F: "0 < ?F"
|
paulson@14365
|
488 |
by (auto simp add: zero_less_mult_iff)
|
paulson@14365
|
489 |
from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
|
haftmann@27652
|
490 |
by simp
|
paulson@14365
|
491 |
with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
|
paulson@14365
|
492 |
by (simp add: mult_le_cancel_right)
|
haftmann@27652
|
493 |
with neq show ?thesis by (simp add: mult_ac int_distrib)
|
paulson@14365
|
494 |
qed
|
paulson@14365
|
495 |
qed
|
paulson@14365
|
496 |
show "q < r ==> 0 < s ==> s * q < s * r"
|
paulson@14365
|
497 |
proof (induct q, induct r, induct s)
|
paulson@14365
|
498 |
fix a b c d e f :: int
|
paulson@14365
|
499 |
assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0"
|
paulson@14365
|
500 |
assume le: "Fract a b < Fract c d"
|
paulson@14365
|
501 |
assume gt: "0 < Fract e f"
|
paulson@14365
|
502 |
show "Fract e f * Fract a b < Fract e f * Fract c d"
|
paulson@14365
|
503 |
proof -
|
paulson@14365
|
504 |
let ?E = "e * f" and ?F = "f * f"
|
paulson@14365
|
505 |
from neq gt have "0 < ?E"
|
haftmann@27652
|
506 |
by (auto simp add: Zero_rat_def order_less_le eq_rat)
|
paulson@14365
|
507 |
moreover from neq have "0 < ?F"
|
paulson@14365
|
508 |
by (auto simp add: zero_less_mult_iff)
|
paulson@14365
|
509 |
moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
|
haftmann@27652
|
510 |
by simp
|
paulson@14365
|
511 |
ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
|
paulson@14365
|
512 |
by (simp add: mult_less_cancel_right)
|
paulson@14365
|
513 |
with neq show ?thesis
|
haftmann@27652
|
514 |
by (simp add: mult_ac)
|
paulson@14365
|
515 |
qed
|
paulson@14365
|
516 |
qed
|
haftmann@27551
|
517 |
qed auto
|
paulson@14365
|
518 |
|
haftmann@27551
|
519 |
lemma Rat_induct_pos [case_names Fract, induct type: rat]:
|
haftmann@27551
|
520 |
assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
|
haftmann@27551
|
521 |
shows "P q"
|
paulson@14365
|
522 |
proof (cases q)
|
haftmann@27551
|
523 |
have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)"
|
paulson@14365
|
524 |
proof -
|
paulson@14365
|
525 |
fix a::int and b::int
|
paulson@14365
|
526 |
assume b: "b < 0"
|
paulson@14365
|
527 |
hence "0 < -b" by simp
|
paulson@14365
|
528 |
hence "P (Fract (-a) (-b))" by (rule step)
|
paulson@14365
|
529 |
thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
|
paulson@14365
|
530 |
qed
|
paulson@14365
|
531 |
case (Fract a b)
|
paulson@14365
|
532 |
thus "P q" by (force simp add: linorder_neq_iff step step')
|
paulson@14365
|
533 |
qed
|
paulson@14365
|
534 |
|
paulson@14365
|
535 |
lemma zero_less_Fract_iff:
|
haftmann@27652
|
536 |
"0 < b ==> (0 < Fract a b) = (0 < a)"
|
haftmann@27652
|
537 |
by (simp add: Zero_rat_def order_less_imp_not_eq2 zero_less_mult_iff)
|
paulson@14365
|
538 |
|
paulson@14378
|
539 |
|
haftmann@27551
|
540 |
subsection {* Arithmetic setup *}
|
paulson@14387
|
541 |
|
paulson@14387
|
542 |
use "rat_arith.ML"
|
wenzelm@24075
|
543 |
declaration {* K rat_arith_setup *}
|
paulson@14387
|
544 |
|
huffman@23342
|
545 |
|
huffman@23342
|
546 |
subsection {* Embedding from Rationals to other Fields *}
|
huffman@23342
|
547 |
|
haftmann@24198
|
548 |
class field_char_0 = field + ring_char_0
|
huffman@23342
|
549 |
|
haftmann@27551
|
550 |
subclass (in ordered_field) field_char_0 ..
|
huffman@23342
|
551 |
|
haftmann@27551
|
552 |
context field_char_0
|
haftmann@27551
|
553 |
begin
|
haftmann@27551
|
554 |
|
haftmann@27551
|
555 |
definition of_rat :: "rat \<Rightarrow> 'a" where
|
haftmann@24198
|
556 |
[code func del]: "of_rat q = contents (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})"
|
huffman@23342
|
557 |
|
haftmann@27551
|
558 |
end
|
haftmann@27551
|
559 |
|
huffman@23342
|
560 |
lemma of_rat_congruent:
|
haftmann@27551
|
561 |
"(\<lambda>(a, b). {of_int a / of_int b :: 'a::field_char_0}) respects ratrel"
|
huffman@23342
|
562 |
apply (rule congruent.intro)
|
huffman@23342
|
563 |
apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
|
huffman@23342
|
564 |
apply (simp only: of_int_mult [symmetric])
|
huffman@23342
|
565 |
done
|
huffman@23342
|
566 |
|
haftmann@27551
|
567 |
lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
|
haftmann@27551
|
568 |
unfolding Fract_def of_rat_def by (simp add: UN_ratrel of_rat_congruent)
|
huffman@23342
|
569 |
|
huffman@23342
|
570 |
lemma of_rat_0 [simp]: "of_rat 0 = 0"
|
huffman@23342
|
571 |
by (simp add: Zero_rat_def of_rat_rat)
|
huffman@23342
|
572 |
|
huffman@23342
|
573 |
lemma of_rat_1 [simp]: "of_rat 1 = 1"
|
huffman@23342
|
574 |
by (simp add: One_rat_def of_rat_rat)
|
huffman@23342
|
575 |
|
huffman@23342
|
576 |
lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
|
haftmann@27652
|
577 |
by (induct a, induct b, simp add: of_rat_rat add_frac_eq)
|
huffman@23342
|
578 |
|
huffman@23343
|
579 |
lemma of_rat_minus: "of_rat (- a) = - of_rat a"
|
haftmann@27652
|
580 |
by (induct a, simp add: of_rat_rat)
|
huffman@23343
|
581 |
|
huffman@23343
|
582 |
lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
|
huffman@23343
|
583 |
by (simp only: diff_minus of_rat_add of_rat_minus)
|
huffman@23343
|
584 |
|
huffman@23342
|
585 |
lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
|
haftmann@27652
|
586 |
apply (induct a, induct b, simp add: of_rat_rat)
|
huffman@23342
|
587 |
apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
|
huffman@23342
|
588 |
done
|
huffman@23342
|
589 |
|
huffman@23342
|
590 |
lemma nonzero_of_rat_inverse:
|
huffman@23342
|
591 |
"a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
|
huffman@23343
|
592 |
apply (rule inverse_unique [symmetric])
|
huffman@23343
|
593 |
apply (simp add: of_rat_mult [symmetric])
|
huffman@23342
|
594 |
done
|
huffman@23342
|
595 |
|
huffman@23342
|
596 |
lemma of_rat_inverse:
|
huffman@23342
|
597 |
"(of_rat (inverse a)::'a::{field_char_0,division_by_zero}) =
|
huffman@23342
|
598 |
inverse (of_rat a)"
|
huffman@23342
|
599 |
by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
|
huffman@23342
|
600 |
|
huffman@23342
|
601 |
lemma nonzero_of_rat_divide:
|
huffman@23342
|
602 |
"b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
|
huffman@23342
|
603 |
by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
|
huffman@23342
|
604 |
|
huffman@23342
|
605 |
lemma of_rat_divide:
|
huffman@23342
|
606 |
"(of_rat (a / b)::'a::{field_char_0,division_by_zero})
|
huffman@23342
|
607 |
= of_rat a / of_rat b"
|
haftmann@27652
|
608 |
by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)
|
huffman@23342
|
609 |
|
huffman@23343
|
610 |
lemma of_rat_power:
|
huffman@23343
|
611 |
"(of_rat (a ^ n)::'a::{field_char_0,recpower}) = of_rat a ^ n"
|
huffman@23343
|
612 |
by (induct n) (simp_all add: of_rat_mult power_Suc)
|
huffman@23343
|
613 |
|
huffman@23343
|
614 |
lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
|
huffman@23343
|
615 |
apply (induct a, induct b)
|
huffman@23343
|
616 |
apply (simp add: of_rat_rat eq_rat)
|
huffman@23343
|
617 |
apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
|
huffman@23343
|
618 |
apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
|
huffman@23343
|
619 |
done
|
huffman@23343
|
620 |
|
haftmann@27652
|
621 |
lemma of_rat_less:
|
haftmann@27652
|
622 |
"(of_rat r :: 'a::ordered_field) < of_rat s \<longleftrightarrow> r < s"
|
haftmann@27652
|
623 |
proof (induct r, induct s)
|
haftmann@27652
|
624 |
fix a b c d :: int
|
haftmann@27652
|
625 |
assume not_zero: "b > 0" "d > 0"
|
haftmann@27652
|
626 |
then have "b * d > 0" by (rule mult_pos_pos)
|
haftmann@27652
|
627 |
have of_int_divide_less_eq:
|
haftmann@27652
|
628 |
"(of_int a :: 'a) / of_int b < of_int c / of_int d
|
haftmann@27652
|
629 |
\<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b"
|
haftmann@27652
|
630 |
using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
|
haftmann@27652
|
631 |
show "(of_rat (Fract a b) :: 'a::ordered_field) < of_rat (Fract c d)
|
haftmann@27652
|
632 |
\<longleftrightarrow> Fract a b < Fract c d"
|
haftmann@27652
|
633 |
using not_zero `b * d > 0`
|
haftmann@27652
|
634 |
by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult)
|
haftmann@27652
|
635 |
(auto intro: mult_strict_right_mono mult_right_less_imp_less)
|
haftmann@27652
|
636 |
qed
|
haftmann@27652
|
637 |
|
haftmann@27652
|
638 |
lemma of_rat_less_eq:
|
haftmann@27652
|
639 |
"(of_rat r :: 'a::ordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s"
|
haftmann@27652
|
640 |
unfolding le_less by (auto simp add: of_rat_less)
|
haftmann@27652
|
641 |
|
huffman@23343
|
642 |
lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
|
huffman@23343
|
643 |
|
haftmann@27652
|
644 |
lemma of_rat_eq_id [simp]: "of_rat = id"
|
huffman@23343
|
645 |
proof
|
huffman@23343
|
646 |
fix a
|
huffman@23343
|
647 |
show "of_rat a = id a"
|
huffman@23343
|
648 |
by (induct a)
|
haftmann@27652
|
649 |
(simp add: of_rat_rat Fract_of_int_eq [symmetric])
|
huffman@23343
|
650 |
qed
|
huffman@23343
|
651 |
|
huffman@23343
|
652 |
text{*Collapse nested embeddings*}
|
huffman@23343
|
653 |
lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
|
huffman@23343
|
654 |
by (induct n) (simp_all add: of_rat_add)
|
huffman@23343
|
655 |
|
huffman@23343
|
656 |
lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
|
haftmann@27652
|
657 |
by (cases z rule: int_diff_cases) (simp add: of_rat_diff)
|
huffman@23343
|
658 |
|
huffman@23343
|
659 |
lemma of_rat_number_of_eq [simp]:
|
huffman@23343
|
660 |
"of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})"
|
huffman@23343
|
661 |
by (simp add: number_of_eq)
|
huffman@23343
|
662 |
|
haftmann@23879
|
663 |
lemmas zero_rat = Zero_rat_def
|
haftmann@23879
|
664 |
lemmas one_rat = One_rat_def
|
haftmann@23879
|
665 |
|
haftmann@24198
|
666 |
abbreviation
|
haftmann@24198
|
667 |
rat_of_nat :: "nat \<Rightarrow> rat"
|
haftmann@24198
|
668 |
where
|
haftmann@24198
|
669 |
"rat_of_nat \<equiv> of_nat"
|
haftmann@24198
|
670 |
|
haftmann@24198
|
671 |
abbreviation
|
haftmann@24198
|
672 |
rat_of_int :: "int \<Rightarrow> rat"
|
haftmann@24198
|
673 |
where
|
haftmann@24198
|
674 |
"rat_of_int \<equiv> of_int"
|
haftmann@24198
|
675 |
|
huffman@28010
|
676 |
subsection {* The Set of Rational Numbers *}
|
berghofe@24533
|
677 |
|
nipkow@28001
|
678 |
context field_char_0
|
nipkow@28001
|
679 |
begin
|
nipkow@28001
|
680 |
|
nipkow@28001
|
681 |
definition
|
nipkow@28001
|
682 |
Rats :: "'a set" where
|
nipkow@28001
|
683 |
[code func del]: "Rats = range of_rat"
|
nipkow@28001
|
684 |
|
nipkow@28001
|
685 |
notation (xsymbols)
|
nipkow@28001
|
686 |
Rats ("\<rat>")
|
nipkow@28001
|
687 |
|
nipkow@28001
|
688 |
end
|
nipkow@28001
|
689 |
|
huffman@28010
|
690 |
lemma Rats_of_rat [simp]: "of_rat r \<in> Rats"
|
huffman@28010
|
691 |
by (simp add: Rats_def)
|
huffman@28010
|
692 |
|
huffman@28010
|
693 |
lemma Rats_of_int [simp]: "of_int z \<in> Rats"
|
huffman@28010
|
694 |
by (subst of_rat_of_int_eq [symmetric], rule Rats_of_rat)
|
huffman@28010
|
695 |
|
huffman@28010
|
696 |
lemma Rats_of_nat [simp]: "of_nat n \<in> Rats"
|
huffman@28010
|
697 |
by (subst of_rat_of_nat_eq [symmetric], rule Rats_of_rat)
|
huffman@28010
|
698 |
|
huffman@28010
|
699 |
lemma Rats_number_of [simp]:
|
huffman@28010
|
700 |
"(number_of w::'a::{number_ring,field_char_0}) \<in> Rats"
|
huffman@28010
|
701 |
by (subst of_rat_number_of_eq [symmetric], rule Rats_of_rat)
|
huffman@28010
|
702 |
|
huffman@28010
|
703 |
lemma Rats_0 [simp]: "0 \<in> Rats"
|
huffman@28010
|
704 |
apply (unfold Rats_def)
|
huffman@28010
|
705 |
apply (rule range_eqI)
|
huffman@28010
|
706 |
apply (rule of_rat_0 [symmetric])
|
huffman@28010
|
707 |
done
|
huffman@28010
|
708 |
|
huffman@28010
|
709 |
lemma Rats_1 [simp]: "1 \<in> Rats"
|
huffman@28010
|
710 |
apply (unfold Rats_def)
|
huffman@28010
|
711 |
apply (rule range_eqI)
|
huffman@28010
|
712 |
apply (rule of_rat_1 [symmetric])
|
huffman@28010
|
713 |
done
|
huffman@28010
|
714 |
|
huffman@28010
|
715 |
lemma Rats_add [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a + b \<in> Rats"
|
huffman@28010
|
716 |
apply (auto simp add: Rats_def)
|
huffman@28010
|
717 |
apply (rule range_eqI)
|
huffman@28010
|
718 |
apply (rule of_rat_add [symmetric])
|
huffman@28010
|
719 |
done
|
huffman@28010
|
720 |
|
huffman@28010
|
721 |
lemma Rats_minus [simp]: "a \<in> Rats \<Longrightarrow> - a \<in> Rats"
|
huffman@28010
|
722 |
apply (auto simp add: Rats_def)
|
huffman@28010
|
723 |
apply (rule range_eqI)
|
huffman@28010
|
724 |
apply (rule of_rat_minus [symmetric])
|
huffman@28010
|
725 |
done
|
huffman@28010
|
726 |
|
huffman@28010
|
727 |
lemma Rats_diff [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a - b \<in> Rats"
|
huffman@28010
|
728 |
apply (auto simp add: Rats_def)
|
huffman@28010
|
729 |
apply (rule range_eqI)
|
huffman@28010
|
730 |
apply (rule of_rat_diff [symmetric])
|
huffman@28010
|
731 |
done
|
huffman@28010
|
732 |
|
huffman@28010
|
733 |
lemma Rats_mult [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a * b \<in> Rats"
|
huffman@28010
|
734 |
apply (auto simp add: Rats_def)
|
huffman@28010
|
735 |
apply (rule range_eqI)
|
huffman@28010
|
736 |
apply (rule of_rat_mult [symmetric])
|
huffman@28010
|
737 |
done
|
huffman@28010
|
738 |
|
huffman@28010
|
739 |
lemma nonzero_Rats_inverse:
|
huffman@28010
|
740 |
fixes a :: "'a::field_char_0"
|
huffman@28010
|
741 |
shows "\<lbrakk>a \<in> Rats; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Rats"
|
huffman@28010
|
742 |
apply (auto simp add: Rats_def)
|
huffman@28010
|
743 |
apply (rule range_eqI)
|
huffman@28010
|
744 |
apply (erule nonzero_of_rat_inverse [symmetric])
|
huffman@28010
|
745 |
done
|
huffman@28010
|
746 |
|
huffman@28010
|
747 |
lemma Rats_inverse [simp]:
|
huffman@28010
|
748 |
fixes a :: "'a::{field_char_0,division_by_zero}"
|
huffman@28010
|
749 |
shows "a \<in> Rats \<Longrightarrow> inverse a \<in> Rats"
|
huffman@28010
|
750 |
apply (auto simp add: Rats_def)
|
huffman@28010
|
751 |
apply (rule range_eqI)
|
huffman@28010
|
752 |
apply (rule of_rat_inverse [symmetric])
|
huffman@28010
|
753 |
done
|
huffman@28010
|
754 |
|
huffman@28010
|
755 |
lemma nonzero_Rats_divide:
|
huffman@28010
|
756 |
fixes a b :: "'a::field_char_0"
|
huffman@28010
|
757 |
shows "\<lbrakk>a \<in> Rats; b \<in> Rats; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
|
huffman@28010
|
758 |
apply (auto simp add: Rats_def)
|
huffman@28010
|
759 |
apply (rule range_eqI)
|
huffman@28010
|
760 |
apply (erule nonzero_of_rat_divide [symmetric])
|
huffman@28010
|
761 |
done
|
huffman@28010
|
762 |
|
huffman@28010
|
763 |
lemma Rats_divide [simp]:
|
huffman@28010
|
764 |
fixes a b :: "'a::{field_char_0,division_by_zero}"
|
huffman@28010
|
765 |
shows "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
|
huffman@28010
|
766 |
apply (auto simp add: Rats_def)
|
huffman@28010
|
767 |
apply (rule range_eqI)
|
huffman@28010
|
768 |
apply (rule of_rat_divide [symmetric])
|
huffman@28010
|
769 |
done
|
huffman@28010
|
770 |
|
huffman@28010
|
771 |
lemma Rats_power [simp]:
|
huffman@28010
|
772 |
fixes a :: "'a::{field_char_0,recpower}"
|
huffman@28010
|
773 |
shows "a \<in> Rats \<Longrightarrow> a ^ n \<in> Rats"
|
huffman@28010
|
774 |
apply (auto simp add: Rats_def)
|
huffman@28010
|
775 |
apply (rule range_eqI)
|
huffman@28010
|
776 |
apply (rule of_rat_power [symmetric])
|
huffman@28010
|
777 |
done
|
huffman@28010
|
778 |
|
huffman@28010
|
779 |
lemma Rats_cases [cases set: Rats]:
|
huffman@28010
|
780 |
assumes "q \<in> \<rat>"
|
huffman@28010
|
781 |
obtains (of_rat) r where "q = of_rat r"
|
huffman@28010
|
782 |
unfolding Rats_def
|
huffman@28010
|
783 |
proof -
|
huffman@28010
|
784 |
from `q \<in> \<rat>` have "q \<in> range of_rat" unfolding Rats_def .
|
huffman@28010
|
785 |
then obtain r where "q = of_rat r" ..
|
huffman@28010
|
786 |
then show thesis ..
|
huffman@28010
|
787 |
qed
|
huffman@28010
|
788 |
|
huffman@28010
|
789 |
lemma Rats_induct [case_names of_rat, induct set: Rats]:
|
huffman@28010
|
790 |
"q \<in> \<rat> \<Longrightarrow> (\<And>r. P (of_rat r)) \<Longrightarrow> P q"
|
huffman@28010
|
791 |
by (rule Rats_cases) auto
|
huffman@28010
|
792 |
|
nipkow@28001
|
793 |
|
berghofe@24533
|
794 |
subsection {* Implementation of rational numbers as pairs of integers *}
|
berghofe@24533
|
795 |
|
haftmann@27652
|
796 |
lemma Fract_norm: "Fract (a div zgcd a b) (b div zgcd a b) = Fract a b"
|
haftmann@27652
|
797 |
proof (cases "a = 0 \<or> b = 0")
|
haftmann@27652
|
798 |
case True then show ?thesis by (auto simp add: eq_rat)
|
haftmann@27652
|
799 |
next
|
haftmann@27652
|
800 |
let ?c = "zgcd a b"
|
haftmann@27652
|
801 |
case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
|
haftmann@27652
|
802 |
then have "?c \<noteq> 0" by simp
|
haftmann@27652
|
803 |
then have "Fract ?c ?c = Fract 1 1" by (simp add: eq_rat)
|
haftmann@27652
|
804 |
moreover have "Fract (a div ?c * ?c + a mod ?c) (b div ?c * ?c + b mod ?c) = Fract a b"
|
haftmann@28053
|
805 |
by (simp add: semiring_div_class.mod_div_equality)
|
haftmann@27652
|
806 |
moreover have "a mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric])
|
haftmann@27652
|
807 |
moreover have "b mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric])
|
haftmann@27652
|
808 |
ultimately show ?thesis
|
haftmann@27652
|
809 |
by (simp add: mult_rat [symmetric])
|
haftmann@27652
|
810 |
qed
|
berghofe@24533
|
811 |
|
haftmann@27652
|
812 |
definition Fract_norm :: "int \<Rightarrow> int \<Rightarrow> rat" where
|
haftmann@27652
|
813 |
[simp, code func del]: "Fract_norm a b = Fract a b"
|
haftmann@27652
|
814 |
|
haftmann@27652
|
815 |
lemma [code func]: "Fract_norm a b = (if a = 0 \<or> b = 0 then 0 else let c = zgcd a b in
|
haftmann@27652
|
816 |
if b > 0 then Fract (a div c) (b div c) else Fract (- (a div c)) (- (b div c)))"
|
haftmann@27652
|
817 |
by (simp add: eq_rat Zero_rat_def Let_def Fract_norm)
|
berghofe@24533
|
818 |
|
berghofe@24533
|
819 |
lemma [code]:
|
haftmann@27652
|
820 |
"of_rat (Fract a b) = (if b \<noteq> 0 then of_int a / of_int b else 0)"
|
haftmann@27652
|
821 |
by (cases "b = 0") (simp_all add: rat_number_collapse of_rat_rat)
|
berghofe@24533
|
822 |
|
haftmann@26513
|
823 |
instantiation rat :: eq
|
haftmann@26513
|
824 |
begin
|
berghofe@24533
|
825 |
|
haftmann@27652
|
826 |
definition [code func del]: "eq_class.eq (a\<Colon>rat) b \<longleftrightarrow> a - b = 0"
|
haftmann@26513
|
827 |
|
haftmann@26513
|
828 |
instance by default (simp add: eq_rat_def)
|
haftmann@26513
|
829 |
|
haftmann@27652
|
830 |
lemma rat_eq_code [code]:
|
haftmann@27652
|
831 |
"eq_class.eq (Fract a b) (Fract c d) \<longleftrightarrow> (if b = 0
|
haftmann@27652
|
832 |
then c = 0 \<or> d = 0
|
haftmann@27652
|
833 |
else if d = 0
|
haftmann@27652
|
834 |
then a = 0 \<or> b = 0
|
haftmann@27652
|
835 |
else a * d = b * c)"
|
haftmann@27652
|
836 |
by (auto simp add: eq eq_rat)
|
haftmann@26513
|
837 |
|
haftmann@26513
|
838 |
end
|
berghofe@24533
|
839 |
|
haftmann@27652
|
840 |
lemma le_rat':
|
haftmann@27652
|
841 |
assumes "b \<noteq> 0"
|
haftmann@27652
|
842 |
and "d \<noteq> 0"
|
haftmann@27652
|
843 |
shows "Fract a b \<le> Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d"
|
berghofe@24533
|
844 |
proof -
|
haftmann@27652
|
845 |
have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp
|
haftmann@27652
|
846 |
have "a * d * (b * d) \<le> c * b * (b * d) \<longleftrightarrow> a * d * (sgn b * sgn d) \<le> c * b * (sgn b * sgn d)"
|
haftmann@27652
|
847 |
proof (cases "b * d > 0")
|
haftmann@27652
|
848 |
case True
|
haftmann@27652
|
849 |
moreover from True have "sgn b * sgn d = 1"
|
haftmann@27652
|
850 |
by (simp add: sgn_times [symmetric] sgn_1_pos)
|
haftmann@27652
|
851 |
ultimately show ?thesis by (simp add: mult_le_cancel_right)
|
haftmann@27652
|
852 |
next
|
haftmann@27652
|
853 |
case False with assms have "b * d < 0" by (simp add: less_le)
|
haftmann@27652
|
854 |
moreover from this have "sgn b * sgn d = - 1"
|
haftmann@27652
|
855 |
by (simp only: sgn_times [symmetric] sgn_1_neg)
|
haftmann@27652
|
856 |
ultimately show ?thesis by (simp add: mult_le_cancel_right)
|
haftmann@27652
|
857 |
qed
|
haftmann@27652
|
858 |
also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d"
|
haftmann@27652
|
859 |
by (simp add: abs_sgn mult_ac)
|
haftmann@27652
|
860 |
finally show ?thesis using assms by simp
|
berghofe@24533
|
861 |
qed
|
berghofe@24533
|
862 |
|
haftmann@27652
|
863 |
lemma less_rat':
|
haftmann@27652
|
864 |
assumes "b \<noteq> 0"
|
haftmann@27652
|
865 |
and "d \<noteq> 0"
|
haftmann@27652
|
866 |
shows "Fract a b < Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d"
|
berghofe@24533
|
867 |
proof -
|
haftmann@27652
|
868 |
have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp
|
haftmann@27652
|
869 |
have "a * d * (b * d) < c * b * (b * d) \<longleftrightarrow> a * d * (sgn b * sgn d) < c * b * (sgn b * sgn d)"
|
haftmann@27652
|
870 |
proof (cases "b * d > 0")
|
haftmann@27652
|
871 |
case True
|
haftmann@27652
|
872 |
moreover from True have "sgn b * sgn d = 1"
|
haftmann@27652
|
873 |
by (simp add: sgn_times [symmetric] sgn_1_pos)
|
haftmann@27652
|
874 |
ultimately show ?thesis by (simp add: mult_less_cancel_right)
|
haftmann@27652
|
875 |
next
|
haftmann@27652
|
876 |
case False with assms have "b * d < 0" by (simp add: less_le)
|
haftmann@27652
|
877 |
moreover from this have "sgn b * sgn d = - 1"
|
haftmann@27652
|
878 |
by (simp only: sgn_times [symmetric] sgn_1_neg)
|
haftmann@27652
|
879 |
ultimately show ?thesis by (simp add: mult_less_cancel_right)
|
haftmann@27652
|
880 |
qed
|
haftmann@27652
|
881 |
also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d"
|
haftmann@27652
|
882 |
by (simp add: abs_sgn mult_ac)
|
haftmann@27652
|
883 |
finally show ?thesis using assms by simp
|
berghofe@24533
|
884 |
qed
|
berghofe@24533
|
885 |
|
haftmann@27652
|
886 |
lemma rat_less_eq_code [code]:
|
haftmann@27652
|
887 |
"Fract a b \<le> Fract c d \<longleftrightarrow> (if b = 0
|
haftmann@27652
|
888 |
then sgn c * sgn d \<ge> 0
|
haftmann@27652
|
889 |
else if d = 0
|
haftmann@27652
|
890 |
then sgn a * sgn b \<le> 0
|
haftmann@27652
|
891 |
else a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d)"
|
haftmann@27652
|
892 |
by (auto simp add: sgn_times mult_le_0_iff zero_le_mult_iff le_rat' eq_rat simp del: le_rat)
|
haftmann@27652
|
893 |
(auto simp add: sgn_times sgn_0_0 le_less sgn_1_pos [symmetric] sgn_1_neg [symmetric])
|
berghofe@24533
|
894 |
|
haftmann@27652
|
895 |
lemma rat_le_eq_code [code]:
|
haftmann@27652
|
896 |
"Fract a b < Fract c d \<longleftrightarrow> (if b = 0
|
haftmann@27652
|
897 |
then sgn c * sgn d > 0
|
haftmann@27652
|
898 |
else if d = 0
|
haftmann@27652
|
899 |
then sgn a * sgn b < 0
|
haftmann@27652
|
900 |
else a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d)"
|
haftmann@27652
|
901 |
by (auto simp add: sgn_times mult_less_0_iff zero_less_mult_iff less_rat' eq_rat simp del: less_rat)
|
haftmann@27652
|
902 |
(auto simp add: sgn_times sgn_0_0 sgn_1_pos [symmetric] sgn_1_neg [symmetric],
|
haftmann@27652
|
903 |
auto simp add: sgn_1_pos)
|
berghofe@24533
|
904 |
|
haftmann@27652
|
905 |
lemma rat_plus_code [code]:
|
haftmann@27652
|
906 |
"Fract a b + Fract c d = (if b = 0
|
haftmann@27652
|
907 |
then Fract c d
|
haftmann@27652
|
908 |
else if d = 0
|
haftmann@27652
|
909 |
then Fract a b
|
haftmann@27652
|
910 |
else Fract_norm (a * d + c * b) (b * d))"
|
haftmann@27652
|
911 |
by (simp add: eq_rat, simp add: Zero_rat_def)
|
berghofe@24533
|
912 |
|
haftmann@27652
|
913 |
lemma rat_times_code [code]:
|
haftmann@27652
|
914 |
"Fract a b * Fract c d = Fract_norm (a * c) (b * d)"
|
haftmann@27652
|
915 |
by simp
|
berghofe@24533
|
916 |
|
haftmann@27652
|
917 |
lemma rat_minus_code [code]:
|
haftmann@27652
|
918 |
"Fract a b - Fract c d = (if b = 0
|
haftmann@27652
|
919 |
then Fract (- c) d
|
haftmann@27652
|
920 |
else if d = 0
|
haftmann@27652
|
921 |
then Fract a b
|
haftmann@27652
|
922 |
else Fract_norm (a * d - c * b) (b * d))"
|
haftmann@27652
|
923 |
by (simp add: eq_rat, simp add: Zero_rat_def)
|
berghofe@24533
|
924 |
|
haftmann@27652
|
925 |
lemma rat_inverse_code [code]:
|
haftmann@27652
|
926 |
"inverse (Fract a b) = (if b = 0 then Fract 1 0
|
haftmann@27652
|
927 |
else if a < 0 then Fract (- b) (- a)
|
haftmann@27652
|
928 |
else Fract b a)"
|
haftmann@27652
|
929 |
by (simp add: eq_rat)
|
haftmann@27652
|
930 |
|
haftmann@27652
|
931 |
lemma rat_divide_code [code]:
|
haftmann@27652
|
932 |
"Fract a b / Fract c d = Fract_norm (a * d) (b * c)"
|
haftmann@27652
|
933 |
by simp
|
haftmann@27652
|
934 |
|
haftmann@27652
|
935 |
hide (open) const Fract_norm
|
berghofe@24533
|
936 |
|
haftmann@24622
|
937 |
text {* Setup for SML code generator *}
|
berghofe@24533
|
938 |
|
berghofe@24533
|
939 |
types_code
|
berghofe@24533
|
940 |
rat ("(int */ int)")
|
berghofe@24533
|
941 |
attach (term_of) {*
|
berghofe@24533
|
942 |
fun term_of_rat (p, q) =
|
haftmann@24622
|
943 |
let
|
haftmann@24661
|
944 |
val rT = Type ("Rational.rat", [])
|
berghofe@24533
|
945 |
in
|
berghofe@24533
|
946 |
if q = 1 orelse p = 0 then HOLogic.mk_number rT p
|
berghofe@25885
|
947 |
else @{term "op / \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat"} $
|
berghofe@24533
|
948 |
HOLogic.mk_number rT p $ HOLogic.mk_number rT q
|
berghofe@24533
|
949 |
end;
|
berghofe@24533
|
950 |
*}
|
berghofe@24533
|
951 |
attach (test) {*
|
berghofe@24533
|
952 |
fun gen_rat i =
|
berghofe@24533
|
953 |
let
|
berghofe@24533
|
954 |
val p = random_range 0 i;
|
berghofe@24533
|
955 |
val q = random_range 1 (i + 1);
|
berghofe@24533
|
956 |
val g = Integer.gcd p q;
|
wenzelm@24630
|
957 |
val p' = p div g;
|
wenzelm@24630
|
958 |
val q' = q div g;
|
berghofe@25885
|
959 |
val r = (if one_of [true, false] then p' else ~ p',
|
berghofe@25885
|
960 |
if p' = 0 then 0 else q')
|
berghofe@24533
|
961 |
in
|
berghofe@25885
|
962 |
(r, fn () => term_of_rat r)
|
berghofe@24533
|
963 |
end;
|
berghofe@24533
|
964 |
*}
|
berghofe@24533
|
965 |
|
berghofe@24533
|
966 |
consts_code
|
haftmann@27551
|
967 |
Fract ("(_,/ _)")
|
berghofe@24533
|
968 |
|
berghofe@24533
|
969 |
consts_code
|
berghofe@24533
|
970 |
"of_int :: int \<Rightarrow> rat" ("\<module>rat'_of'_int")
|
berghofe@24533
|
971 |
attach {*
|
berghofe@24533
|
972 |
fun rat_of_int 0 = (0, 0)
|
berghofe@24533
|
973 |
| rat_of_int i = (i, 1);
|
berghofe@24533
|
974 |
*}
|
berghofe@24533
|
975 |
|
haftmann@27652
|
976 |
end |