src/HOL/Rational.thy
author haftmann
Fri Feb 05 14:33:50 2010 +0100 (2010-02-05)
changeset 35028 108662d50512
parent 33814 984fb2171607
child 35063 893062359bec
permissions -rw-r--r--
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
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(*  Title:  HOL/Rational.thy
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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 GCD Archimedean_Field
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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_on_ratrel: "refl_on {x. snd x \<noteq> 0} ratrel"
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  by (auto simp add: refl_on_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_on_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 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
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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 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 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 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 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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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 algebra_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 algebra_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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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 del]: "number_of w = Fract w 1"
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instance proof
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qed (simp add: rat_number_of_def of_int_rat)
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end
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lemma rat_number_collapse [code_post]:
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  "Fract 0 k = 0"
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  "Fract 1 1 = 1"
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  "Fract (number_of k) 1 = number_of k"
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  "Fract k 0 = 0"
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  by (cases "k = 0")
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    (simp_all add: Zero_rat_def One_rat_def number_of_is_id number_of_eq of_int_rat eq_rat Fract_def)
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lemma rat_number_expand [code_unfold]:
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  "0 = Fract 0 1"
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  "1 = Fract 1 1"
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  "number_of k = Fract (number_of k) 1"
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  by (simp_all add: rat_number_collapse)
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lemma iszero_rat [simp]:
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  "iszero (number_of k :: rat) \<longleftrightarrow> iszero (number_of k :: int)"
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  by (simp add: iszero_def rat_number_expand number_of_is_id eq_rat)
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lemma Rat_cases_nonzero [case_names Fract 0]:
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  assumes Fract: "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> C"
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  assumes 0: "q = 0 \<Longrightarrow> C"
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  shows C
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proof (cases "q = 0")
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  case True then show C using 0 by auto
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next
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  case False
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  then obtain a b where "q = Fract a b" and "b \<noteq> 0" by (cases q) auto
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  moreover with False have "0 \<noteq> Fract a b" by simp
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  with `b \<noteq> 0` have "a \<noteq> 0" by (simp add: Zero_rat_def eq_rat)
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  with Fract `q = Fract a b` `b \<noteq> 0` show C by auto
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qed
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subsubsection {* Function @{text normalize} *}
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text{*
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Decompose a fraction into normalized, i.e. coprime numerator and denominator:
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*}
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definition normalize :: "rat \<Rightarrow> int \<times> int" where
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"normalize x \<equiv> THE pair. x = Fract (fst pair) (snd pair) &
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                   snd pair > 0 & gcd (fst pair) (snd pair) = 1"
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declare normalize_def[code del]
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lemma Fract_norm: "Fract (a div gcd a b) (b div gcd a b) = Fract a b"
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proof (cases "a = 0 | b = 0")
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  case True then show ?thesis by (auto simp add: eq_rat)
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next
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  let ?c = "gcd a b"
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  case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
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  then have "?c \<noteq> 0" by simp
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  then have "Fract ?c ?c = Fract 1 1" by (simp add: eq_rat)
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  moreover have "Fract (a div ?c * ?c + a mod ?c) (b div ?c * ?c + b mod ?c) = Fract a b"
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    by (simp add: semiring_div_class.mod_div_equality)
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  moreover have "a mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric])
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  moreover have "b mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric])
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  ultimately show ?thesis
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    by (simp add: mult_rat [symmetric])
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qed
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text{* Proof by Ren\'e Thiemann: *}
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lemma normalize_code[code]:
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"normalize (Fract a b) =
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 (if b > 0 then (let g = gcd a b in (a div g, b div g))
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  else if b = 0 then (0,1)
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  else (let g = - gcd a b in (a div g, b div g)))"
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proof -
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  let ?cond = "% r p. r = Fract (fst p) (snd p) & snd p > 0 &
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                 gcd (fst p) (snd p) = 1"
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  show ?thesis
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  proof (cases "b = 0")
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    case True
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    thus ?thesis
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    proof (simp add: normalize_def)
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      show "(THE pair. ?cond (Fract a 0) pair) = (0,1)"
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      proof
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        show "?cond (Fract a 0) (0,1)"
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          by (simp add: rat_number_collapse)
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      next
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        fix pair
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        assume cond: "?cond (Fract a 0) pair"
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        show "pair = (0,1)"
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        proof (cases pair)
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          case (Pair den num)
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          with cond have num: "num > 0" by auto
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          with Pair cond have den: "den = 0" by (simp add: eq_rat)
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          show ?thesis
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          proof (cases "num = 1", simp add: Pair den)
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            case False
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            with num have gr: "num > 1" by auto
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            with den have "gcd den num = num" by auto
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            with Pair cond False gr show ?thesis by auto
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          qed
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        qed
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      qed
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    qed
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  next
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    { fix a b :: int assume b: "b > 0"
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      hence b0: "b \<noteq> 0" and "b >= 0" by auto
nipkow@33805
   313
      let ?g = "gcd a b"
nipkow@33805
   314
      from b0 have g0: "?g \<noteq> 0" by auto
nipkow@33805
   315
      then have gp: "?g > 0" by simp
nipkow@33805
   316
      then have gs: "?g <= b" by (metis b gcd_le2_int)
nipkow@33805
   317
      from gcd_dvd1_int[of a b] obtain a' where a': "a = ?g * a'"
nipkow@33805
   318
        unfolding dvd_def by auto
nipkow@33805
   319
      from gcd_dvd2_int[of a b] obtain b' where b': "b = ?g * b'"
nipkow@33805
   320
        unfolding dvd_def by auto
nipkow@33805
   321
      hence b'2: "b' * ?g = b" by (simp add: ring_simps)
nipkow@33805
   322
      with b0 have b'0: "b' \<noteq> 0" by auto
nipkow@33805
   323
      from b b' zero_less_mult_iff[of ?g b'] gp have b'p: "b' > 0" by arith
nipkow@33805
   324
      have "normalize (Fract a b) = (a div ?g, b div ?g)"
nipkow@33805
   325
      proof (simp add: normalize_def)
nipkow@33805
   326
        show "(THE pair. ?cond (Fract a b) pair) = (a div ?g, b div ?g)"
nipkow@33805
   327
        proof
nipkow@33805
   328
          have "1 = b div b" using b0 by auto
nipkow@33805
   329
          also have "\<dots> <= b div ?g" by (rule zdiv_mono2[OF `b >= 0` gp gs])
nipkow@33805
   330
          finally have div0: "b div ?g > 0" by simp
nipkow@33805
   331
          show "?cond (Fract a b) (a div ?g, b div ?g)"
nipkow@33805
   332
            by (simp add: b0 Fract_norm div_gcd_coprime_int div0)
nipkow@33805
   333
        next
nipkow@33805
   334
          fix pair assume cond: "?cond (Fract a b) pair"
nipkow@33805
   335
          show "pair = (a div ?g, b div ?g)"
nipkow@33805
   336
          proof (cases pair)
nipkow@33805
   337
            case (Pair den num)
nipkow@33805
   338
            with cond
nipkow@33805
   339
            have num: "num > 0" and num0: "num \<noteq> 0" and gcd: "gcd den num = 1"
nipkow@33805
   340
              by auto
nipkow@33805
   341
            obtain g where g: "g = ?g" by auto
nipkow@33805
   342
            with gp have gg0: "g > 0" by auto
nipkow@33805
   343
            from cond Pair eq_rat(1)[OF b0 num0]
nipkow@33805
   344
            have eq: "a * num = den * b" by auto
nipkow@33805
   345
            hence "a' * g * num = den * g * b'"
nipkow@33805
   346
              using a'[simplified g[symmetric]] b'[simplified g[symmetric]]
nipkow@33805
   347
              by simp
nipkow@33805
   348
            hence "a' * num * g = b' * den * g" by (simp add: algebra_simps)
nipkow@33805
   349
            hence eq2: "a' * num = b' * den" using gg0 by auto
nipkow@33805
   350
            have "a div ?g = ?g * a' div ?g" using a' by force
nipkow@33805
   351
            hence adiv: "a div ?g = a'" using g0 by auto
nipkow@33805
   352
            have "b div ?g = ?g * b' div ?g" using b' by force
nipkow@33805
   353
            hence bdiv: "b div ?g = b'" using g0 by auto
nipkow@33805
   354
            from div_gcd_coprime_int[of a b] b0
nipkow@33805
   355
            have "gcd (a div ?g) (b div ?g) = 1" by auto
nipkow@33805
   356
            with adiv bdiv have gcd2: "gcd a' b' = 1" by auto
nipkow@33805
   357
            from gcd have gcd3: "gcd num den = 1"
nipkow@33805
   358
              by (simp add: gcd_commute_int[of den num])
nipkow@33805
   359
            from gcd2 have gcd4: "gcd b' a' = 1"
nipkow@33805
   360
              by (simp add: gcd_commute_int[of a' b'])
nipkow@33805
   361
            have one: "num dvd b'"
nipkow@33814
   362
              by (metis coprime_dvd_mult_int[OF gcd3] dvd_triv_right eq2)
nipkow@33814
   363
            have two: "b' dvd num"
nipkow@33814
   364
              by (metis coprime_dvd_mult_int[OF gcd4] dvd_triv_left eq2 zmult_commute)
nipkow@33814
   365
            from zdvd_antisym_abs[OF one two] b'p num
nipkow@33814
   366
            have numb': "num = b'" by auto
nipkow@33805
   367
            with eq2 b'0 have "a' = den" by auto
nipkow@33805
   368
            with numb' adiv bdiv Pair show ?thesis by simp
nipkow@33805
   369
          qed
nipkow@33805
   370
        qed
nipkow@33805
   371
      qed
nipkow@33805
   372
    }
nipkow@33805
   373
    note main = this
nipkow@33805
   374
    assume "b \<noteq> 0"
nipkow@33805
   375
    hence "b > 0 | b < 0" by arith
nipkow@33805
   376
    thus ?thesis
nipkow@33805
   377
    proof
nipkow@33805
   378
      assume b: "b > 0" thus ?thesis by (simp add: Let_def main[OF b])
nipkow@33805
   379
    next
nipkow@33805
   380
      assume b: "b < 0"
nipkow@33805
   381
      thus ?thesis
nipkow@33805
   382
        by(simp add:main Let_def minus_rat_cancel[of a b, symmetric]
nipkow@33805
   383
                    zdiv_zminus2 del:minus_rat_cancel)
nipkow@33805
   384
    qed
nipkow@33805
   385
  qed
nipkow@33805
   386
qed
nipkow@33805
   387
nipkow@33805
   388
lemma normalize_id: "normalize (Fract a b) = (p,q) \<Longrightarrow> Fract p q = Fract a b"
nipkow@33805
   389
by(auto simp add: normalize_code Let_def Fract_norm dvd_div_neg rat_number_collapse
nipkow@33805
   390
        split:split_if_asm)
nipkow@33805
   391
nipkow@33805
   392
lemma normalize_denom_pos: "normalize (Fract a b) = (p,q) \<Longrightarrow> q > 0"
nipkow@33805
   393
by(auto simp add: normalize_code Let_def dvd_div_neg pos_imp_zdiv_neg_iff nonneg1_imp_zdiv_pos_iff
nipkow@33805
   394
        split:split_if_asm)
nipkow@33805
   395
nipkow@33805
   396
lemma normalize_coprime: "normalize (Fract a b) = (p,q) \<Longrightarrow> coprime p q"
nipkow@33805
   397
by(auto simp add: normalize_code Let_def dvd_div_neg div_gcd_coprime_int
nipkow@33805
   398
        split:split_if_asm)
nipkow@33805
   399
haftmann@27551
   400
haftmann@27551
   401
subsubsection {* The field of rational numbers *}
haftmann@27551
   402
haftmann@27551
   403
instantiation rat :: "{field, division_by_zero}"
haftmann@27551
   404
begin
haftmann@27551
   405
haftmann@27551
   406
definition
haftmann@28562
   407
  inverse_rat_def [code del]:
haftmann@27551
   408
  "inverse q = Abs_Rat (\<Union>x \<in> Rep_Rat q.
haftmann@27551
   409
     ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)})"
haftmann@27551
   410
haftmann@27652
   411
lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"
haftmann@27551
   412
proof -
haftmann@27551
   413
  have "(\<lambda>x. ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)}) respects ratrel"
haftmann@27551
   414
    by (auto simp add: congruent_def mult_commute)
haftmann@27551
   415
  then show ?thesis by (simp add: Fract_def inverse_rat_def UN_ratrel)
huffman@27509
   416
qed
huffman@27509
   417
haftmann@27551
   418
definition
haftmann@28562
   419
  divide_rat_def [code del]: "q / r = q * inverse (r::rat)"
haftmann@27551
   420
haftmann@27652
   421
lemma divide_rat [simp]: "Fract a b / Fract c d = Fract (a * d) (b * c)"
haftmann@27652
   422
  by (simp add: divide_rat_def)
haftmann@27551
   423
haftmann@27551
   424
instance proof
haftmann@27652
   425
  show "inverse 0 = (0::rat)" by (simp add: rat_number_expand)
haftmann@27551
   426
    (simp add: rat_number_collapse)
haftmann@27551
   427
next
haftmann@27551
   428
  fix q :: rat
haftmann@27551
   429
  assume "q \<noteq> 0"
haftmann@27551
   430
  then show "inverse q * q = 1" by (cases q rule: Rat_cases_nonzero)
haftmann@27551
   431
   (simp_all add: mult_rat  inverse_rat rat_number_expand eq_rat)
haftmann@27551
   432
next
haftmann@27551
   433
  fix q r :: rat
haftmann@27551
   434
  show "q / r = q * inverse r" by (simp add: divide_rat_def)
haftmann@27551
   435
qed
haftmann@27551
   436
haftmann@27551
   437
end
haftmann@27551
   438
haftmann@27551
   439
haftmann@27551
   440
subsubsection {* Various *}
haftmann@27551
   441
haftmann@27551
   442
lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
haftmann@27652
   443
  by (simp add: rat_number_expand)
haftmann@27551
   444
haftmann@27551
   445
lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"
haftmann@27652
   446
  by (simp add: Fract_of_int_eq [symmetric])
haftmann@27551
   447
haftmann@31998
   448
lemma Fract_number_of_quotient [code_post]:
haftmann@27551
   449
  "Fract (number_of k) (number_of l) = number_of k / number_of l"
haftmann@27551
   450
  unfolding Fract_of_int_quotient number_of_is_id number_of_eq ..
haftmann@27551
   451
haftmann@31998
   452
lemma Fract_1_number_of [code_post]:
haftmann@27652
   453
  "Fract 1 (number_of k) = 1 / number_of k"
haftmann@27652
   454
  unfolding Fract_of_int_quotient number_of_eq by simp
haftmann@27551
   455
haftmann@27551
   456
subsubsection {* The ordered field of rational numbers *}
huffman@27509
   457
huffman@27509
   458
instantiation rat :: linorder
huffman@27509
   459
begin
huffman@27509
   460
huffman@27509
   461
definition
haftmann@28562
   462
  le_rat_def [code del]:
huffman@27509
   463
   "q \<le> r \<longleftrightarrow> contents (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
haftmann@27551
   464
      {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})"
haftmann@27551
   465
haftmann@27652
   466
lemma le_rat [simp]:
haftmann@27551
   467
  assumes "b \<noteq> 0" and "d \<noteq> 0"
haftmann@27551
   468
  shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27551
   469
proof -
haftmann@27551
   470
  have "(\<lambda>x y. {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})
haftmann@27551
   471
    respects2 ratrel"
haftmann@27551
   472
  proof (clarsimp simp add: congruent2_def)
haftmann@27551
   473
    fix a b a' b' c d c' d'::int
haftmann@27551
   474
    assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
haftmann@27551
   475
    assume eq1: "a * b' = a' * b"
haftmann@27551
   476
    assume eq2: "c * d' = c' * d"
haftmann@27551
   477
haftmann@27551
   478
    let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
haftmann@27551
   479
    {
haftmann@27551
   480
      fix a b c d x :: int assume x: "x \<noteq> 0"
haftmann@27551
   481
      have "?le a b c d = ?le (a * x) (b * x) c d"
haftmann@27551
   482
      proof -
haftmann@27551
   483
        from x have "0 < x * x" by (auto simp add: zero_less_mult_iff)
haftmann@27551
   484
        hence "?le a b c d =
haftmann@27551
   485
            ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
haftmann@27551
   486
          by (simp add: mult_le_cancel_right)
haftmann@27551
   487
        also have "... = ?le (a * x) (b * x) c d"
haftmann@27551
   488
          by (simp add: mult_ac)
haftmann@27551
   489
        finally show ?thesis .
haftmann@27551
   490
      qed
haftmann@27551
   491
    } note le_factor = this
haftmann@27551
   492
haftmann@27551
   493
    let ?D = "b * d" and ?D' = "b' * d'"
haftmann@27551
   494
    from neq have D: "?D \<noteq> 0" by simp
haftmann@27551
   495
    from neq have "?D' \<noteq> 0" by simp
haftmann@27551
   496
    hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"
haftmann@27551
   497
      by (rule le_factor)
chaieb@27668
   498
    also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')" 
haftmann@27551
   499
      by (simp add: mult_ac)
haftmann@27551
   500
    also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
haftmann@27551
   501
      by (simp only: eq1 eq2)
haftmann@27551
   502
    also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
haftmann@27551
   503
      by (simp add: mult_ac)
haftmann@27551
   504
    also from D have "... = ?le a' b' c' d'"
haftmann@27551
   505
      by (rule le_factor [symmetric])
haftmann@27551
   506
    finally show "?le a b c d = ?le a' b' c' d'" .
haftmann@27551
   507
  qed
haftmann@27551
   508
  with assms show ?thesis by (simp add: Fract_def le_rat_def UN_ratrel2)
haftmann@27551
   509
qed
huffman@27509
   510
huffman@27509
   511
definition
haftmann@28562
   512
  less_rat_def [code del]: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
huffman@27509
   513
haftmann@27652
   514
lemma less_rat [simp]:
haftmann@27551
   515
  assumes "b \<noteq> 0" and "d \<noteq> 0"
haftmann@27551
   516
  shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
haftmann@27652
   517
  using assms by (simp add: less_rat_def eq_rat order_less_le)
huffman@27509
   518
huffman@27509
   519
instance proof
paulson@14365
   520
  fix q r s :: rat
paulson@14365
   521
  {
paulson@14365
   522
    assume "q \<le> r" and "r \<le> s"
paulson@14365
   523
    show "q \<le> s"
paulson@14365
   524
    proof (insert prems, induct q, induct r, induct s)
paulson@14365
   525
      fix a b c d e f :: int
paulson@14365
   526
      assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
paulson@14365
   527
      assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"
paulson@14365
   528
      show "Fract a b \<le> Fract e f"
paulson@14365
   529
      proof -
paulson@14365
   530
        from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
paulson@14365
   531
          by (auto simp add: zero_less_mult_iff linorder_neq_iff)
paulson@14365
   532
        have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
paulson@14365
   533
        proof -
paulson@14365
   534
          from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27652
   535
            by simp
paulson@14365
   536
          with ff show ?thesis by (simp add: mult_le_cancel_right)
paulson@14365
   537
        qed
chaieb@27668
   538
        also have "... = (c * f) * (d * f) * (b * b)" by algebra
paulson@14365
   539
        also have "... \<le> (e * d) * (d * f) * (b * b)"
paulson@14365
   540
        proof -
paulson@14365
   541
          from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
haftmann@27652
   542
            by simp
paulson@14365
   543
          with bb show ?thesis by (simp add: mult_le_cancel_right)
paulson@14365
   544
        qed
paulson@14365
   545
        finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
paulson@14365
   546
          by (simp only: mult_ac)
paulson@14365
   547
        with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
paulson@14365
   548
          by (simp add: mult_le_cancel_right)
haftmann@27652
   549
        with neq show ?thesis by simp
paulson@14365
   550
      qed
paulson@14365
   551
    qed
paulson@14365
   552
  next
paulson@14365
   553
    assume "q \<le> r" and "r \<le> q"
paulson@14365
   554
    show "q = r"
paulson@14365
   555
    proof (insert prems, induct q, induct r)
paulson@14365
   556
      fix a b c d :: int
paulson@14365
   557
      assume neq: "b \<noteq> 0"  "d \<noteq> 0"
paulson@14365
   558
      assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"
paulson@14365
   559
      show "Fract a b = Fract c d"
paulson@14365
   560
      proof -
paulson@14365
   561
        from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27652
   562
          by simp
paulson@14365
   563
        also have "... \<le> (a * d) * (b * d)"
paulson@14365
   564
        proof -
paulson@14365
   565
          from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
haftmann@27652
   566
            by simp
paulson@14365
   567
          thus ?thesis by (simp only: mult_ac)
paulson@14365
   568
        qed
paulson@14365
   569
        finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
paulson@14365
   570
        moreover from neq have "b * d \<noteq> 0" by simp
paulson@14365
   571
        ultimately have "a * d = c * b" by simp
paulson@14365
   572
        with neq show ?thesis by (simp add: eq_rat)
paulson@14365
   573
      qed
paulson@14365
   574
    qed
paulson@14365
   575
  next
paulson@14365
   576
    show "q \<le> q"
haftmann@27652
   577
      by (induct q) simp
haftmann@27682
   578
    show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)"
haftmann@27682
   579
      by (induct q, induct r) (auto simp add: le_less mult_commute)
paulson@14365
   580
    show "q \<le> r \<or> r \<le> q"
huffman@18913
   581
      by (induct q, induct r)
haftmann@27652
   582
         (simp add: mult_commute, rule linorder_linear)
paulson@14365
   583
  }
paulson@14365
   584
qed
paulson@14365
   585
huffman@27509
   586
end
huffman@27509
   587
haftmann@27551
   588
instantiation rat :: "{distrib_lattice, abs_if, sgn_if}"
haftmann@25571
   589
begin
haftmann@25571
   590
haftmann@25571
   591
definition
haftmann@28562
   592
  abs_rat_def [code del]: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))"
haftmann@27551
   593
haftmann@27652
   594
lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
haftmann@27551
   595
  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
   596
haftmann@27551
   597
definition
haftmann@28562
   598
  sgn_rat_def [code del]: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
haftmann@27551
   599
haftmann@27652
   600
lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)"
haftmann@27551
   601
  unfolding Fract_of_int_eq
haftmann@27652
   602
  by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat)
haftmann@27551
   603
    (auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)
haftmann@27551
   604
haftmann@27551
   605
definition
haftmann@25571
   606
  "(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min"
haftmann@25571
   607
haftmann@25571
   608
definition
haftmann@25571
   609
  "(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max"
haftmann@25571
   610
haftmann@27551
   611
instance by intro_classes
haftmann@27551
   612
  (auto simp add: abs_rat_def sgn_rat_def min_max.sup_inf_distrib1 inf_rat_def sup_rat_def)
haftmann@22456
   613
haftmann@25571
   614
end
haftmann@25571
   615
haftmann@35028
   616
instance rat :: linordered_field
haftmann@27551
   617
proof
paulson@14365
   618
  fix q r s :: rat
paulson@14365
   619
  show "q \<le> r ==> s + q \<le> s + r"
paulson@14365
   620
  proof (induct q, induct r, induct s)
paulson@14365
   621
    fix a b c d e f :: int
paulson@14365
   622
    assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
paulson@14365
   623
    assume le: "Fract a b \<le> Fract c d"
paulson@14365
   624
    show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
paulson@14365
   625
    proof -
paulson@14365
   626
      let ?F = "f * f" from neq have F: "0 < ?F"
paulson@14365
   627
        by (auto simp add: zero_less_mult_iff)
paulson@14365
   628
      from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27652
   629
        by simp
paulson@14365
   630
      with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
paulson@14365
   631
        by (simp add: mult_le_cancel_right)
haftmann@27652
   632
      with neq show ?thesis by (simp add: mult_ac int_distrib)
paulson@14365
   633
    qed
paulson@14365
   634
  qed
paulson@14365
   635
  show "q < r ==> 0 < s ==> s * q < s * r"
paulson@14365
   636
  proof (induct q, induct r, induct s)
paulson@14365
   637
    fix a b c d e f :: int
paulson@14365
   638
    assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
paulson@14365
   639
    assume le: "Fract a b < Fract c d"
paulson@14365
   640
    assume gt: "0 < Fract e f"
paulson@14365
   641
    show "Fract e f * Fract a b < Fract e f * Fract c d"
paulson@14365
   642
    proof -
paulson@14365
   643
      let ?E = "e * f" and ?F = "f * f"
paulson@14365
   644
      from neq gt have "0 < ?E"
haftmann@27652
   645
        by (auto simp add: Zero_rat_def order_less_le eq_rat)
paulson@14365
   646
      moreover from neq have "0 < ?F"
paulson@14365
   647
        by (auto simp add: zero_less_mult_iff)
paulson@14365
   648
      moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
haftmann@27652
   649
        by simp
paulson@14365
   650
      ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
paulson@14365
   651
        by (simp add: mult_less_cancel_right)
paulson@14365
   652
      with neq show ?thesis
haftmann@27652
   653
        by (simp add: mult_ac)
paulson@14365
   654
    qed
paulson@14365
   655
  qed
haftmann@27551
   656
qed auto
paulson@14365
   657
haftmann@27551
   658
lemma Rat_induct_pos [case_names Fract, induct type: rat]:
haftmann@27551
   659
  assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
haftmann@27551
   660
  shows "P q"
paulson@14365
   661
proof (cases q)
haftmann@27551
   662
  have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)"
paulson@14365
   663
  proof -
paulson@14365
   664
    fix a::int and b::int
paulson@14365
   665
    assume b: "b < 0"
paulson@14365
   666
    hence "0 < -b" by simp
paulson@14365
   667
    hence "P (Fract (-a) (-b))" by (rule step)
paulson@14365
   668
    thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
paulson@14365
   669
  qed
paulson@14365
   670
  case (Fract a b)
paulson@14365
   671
  thus "P q" by (force simp add: linorder_neq_iff step step')
paulson@14365
   672
qed
paulson@14365
   673
paulson@14365
   674
lemma zero_less_Fract_iff:
huffman@30095
   675
  "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a"
huffman@30095
   676
  by (simp add: Zero_rat_def zero_less_mult_iff)
huffman@30095
   677
huffman@30095
   678
lemma Fract_less_zero_iff:
huffman@30095
   679
  "0 < b \<Longrightarrow> Fract a b < 0 \<longleftrightarrow> a < 0"
huffman@30095
   680
  by (simp add: Zero_rat_def mult_less_0_iff)
huffman@30095
   681
huffman@30095
   682
lemma zero_le_Fract_iff:
huffman@30095
   683
  "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a"
huffman@30095
   684
  by (simp add: Zero_rat_def zero_le_mult_iff)
huffman@30095
   685
huffman@30095
   686
lemma Fract_le_zero_iff:
huffman@30095
   687
  "0 < b \<Longrightarrow> Fract a b \<le> 0 \<longleftrightarrow> a \<le> 0"
huffman@30095
   688
  by (simp add: Zero_rat_def mult_le_0_iff)
huffman@30095
   689
huffman@30095
   690
lemma one_less_Fract_iff:
huffman@30095
   691
  "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a"
huffman@30095
   692
  by (simp add: One_rat_def mult_less_cancel_right_disj)
huffman@30095
   693
huffman@30095
   694
lemma Fract_less_one_iff:
huffman@30095
   695
  "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b"
huffman@30095
   696
  by (simp add: One_rat_def mult_less_cancel_right_disj)
huffman@30095
   697
huffman@30095
   698
lemma one_le_Fract_iff:
huffman@30095
   699
  "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a"
huffman@30095
   700
  by (simp add: One_rat_def mult_le_cancel_right)
huffman@30095
   701
huffman@30095
   702
lemma Fract_le_one_iff:
huffman@30095
   703
  "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b"
huffman@30095
   704
  by (simp add: One_rat_def mult_le_cancel_right)
paulson@14365
   705
paulson@14378
   706
huffman@30097
   707
subsubsection {* Rationals are an Archimedean field *}
huffman@30097
   708
huffman@30097
   709
lemma rat_floor_lemma:
huffman@30097
   710
  assumes "0 < b"
huffman@30097
   711
  shows "of_int (a div b) \<le> Fract a b \<and> Fract a b < of_int (a div b + 1)"
huffman@30097
   712
proof -
huffman@30097
   713
  have "Fract a b = of_int (a div b) + Fract (a mod b) b"
huffman@30097
   714
    using `0 < b` by (simp add: of_int_rat)
huffman@30097
   715
  moreover have "0 \<le> Fract (a mod b) b \<and> Fract (a mod b) b < 1"
huffman@30097
   716
    using `0 < b` by (simp add: zero_le_Fract_iff Fract_less_one_iff)
huffman@30097
   717
  ultimately show ?thesis by simp
huffman@30097
   718
qed
huffman@30097
   719
huffman@30097
   720
instance rat :: archimedean_field
huffman@30097
   721
proof
huffman@30097
   722
  fix r :: rat
huffman@30097
   723
  show "\<exists>z. r \<le> of_int z"
huffman@30097
   724
  proof (induct r)
huffman@30097
   725
    case (Fract a b)
huffman@30097
   726
    then have "Fract a b \<le> of_int (a div b + 1)"
huffman@30097
   727
      using rat_floor_lemma [of b a] by simp
huffman@30097
   728
    then show "\<exists>z. Fract a b \<le> of_int z" ..
huffman@30097
   729
  qed
huffman@30097
   730
qed
huffman@30097
   731
huffman@30097
   732
lemma floor_Fract:
huffman@30097
   733
  assumes "0 < b" shows "floor (Fract a b) = a div b"
huffman@30097
   734
  using rat_floor_lemma [OF `0 < b`, of a]
huffman@30097
   735
  by (simp add: floor_unique)
huffman@30097
   736
huffman@30097
   737
haftmann@31100
   738
subsection {* Linear arithmetic setup *}
paulson@14387
   739
haftmann@31100
   740
declaration {*
haftmann@31100
   741
  K (Lin_Arith.add_inj_thms [@{thm of_nat_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
haftmann@31100
   742
    (* not needed because x < (y::nat) can be rewritten as Suc x <= y: of_nat_less_iff RS iffD2 *)
haftmann@31100
   743
  #> Lin_Arith.add_inj_thms [@{thm of_int_le_iff} RS iffD2, @{thm of_int_eq_iff} RS iffD2]
haftmann@31100
   744
    (* not needed because x < (y::int) can be rewritten as x + 1 <= y: of_int_less_iff RS iffD2 *)
haftmann@31100
   745
  #> Lin_Arith.add_simps [@{thm neg_less_iff_less},
haftmann@31100
   746
      @{thm True_implies_equals},
haftmann@31100
   747
      read_instantiate @{context} [(("a", 0), "(number_of ?v)")] @{thm right_distrib},
haftmann@31100
   748
      @{thm divide_1}, @{thm divide_zero_left},
haftmann@31100
   749
      @{thm times_divide_eq_right}, @{thm times_divide_eq_left},
haftmann@31100
   750
      @{thm minus_divide_left} RS sym, @{thm minus_divide_right} RS sym,
haftmann@31100
   751
      @{thm of_int_minus}, @{thm of_int_diff},
haftmann@31100
   752
      @{thm of_int_of_nat_eq}]
haftmann@31100
   753
  #> Lin_Arith.add_simprocs Numeral_Simprocs.field_cancel_numeral_factors
haftmann@31100
   754
  #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat => rat"})
haftmann@31100
   755
  #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int => rat"}))
haftmann@31100
   756
*}
paulson@14387
   757
huffman@23342
   758
huffman@23342
   759
subsection {* Embedding from Rationals to other Fields *}
huffman@23342
   760
haftmann@24198
   761
class field_char_0 = field + ring_char_0
huffman@23342
   762
haftmann@35028
   763
subclass (in linordered_field) field_char_0 ..
huffman@23342
   764
haftmann@27551
   765
context field_char_0
haftmann@27551
   766
begin
haftmann@27551
   767
haftmann@27551
   768
definition of_rat :: "rat \<Rightarrow> 'a" where
haftmann@28562
   769
  [code del]: "of_rat q = contents (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})"
huffman@23342
   770
haftmann@27551
   771
end
haftmann@27551
   772
huffman@23342
   773
lemma of_rat_congruent:
haftmann@27551
   774
  "(\<lambda>(a, b). {of_int a / of_int b :: 'a::field_char_0}) respects ratrel"
huffman@23342
   775
apply (rule congruent.intro)
huffman@23342
   776
apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
huffman@23342
   777
apply (simp only: of_int_mult [symmetric])
huffman@23342
   778
done
huffman@23342
   779
haftmann@27551
   780
lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
haftmann@27551
   781
  unfolding Fract_def of_rat_def by (simp add: UN_ratrel of_rat_congruent)
huffman@23342
   782
huffman@23342
   783
lemma of_rat_0 [simp]: "of_rat 0 = 0"
huffman@23342
   784
by (simp add: Zero_rat_def of_rat_rat)
huffman@23342
   785
huffman@23342
   786
lemma of_rat_1 [simp]: "of_rat 1 = 1"
huffman@23342
   787
by (simp add: One_rat_def of_rat_rat)
huffman@23342
   788
huffman@23342
   789
lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
haftmann@27652
   790
by (induct a, induct b, simp add: of_rat_rat add_frac_eq)
huffman@23342
   791
huffman@23343
   792
lemma of_rat_minus: "of_rat (- a) = - of_rat a"
haftmann@27652
   793
by (induct a, simp add: of_rat_rat)
huffman@23343
   794
huffman@23343
   795
lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
huffman@23343
   796
by (simp only: diff_minus of_rat_add of_rat_minus)
huffman@23343
   797
huffman@23342
   798
lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
haftmann@27652
   799
apply (induct a, induct b, simp add: of_rat_rat)
huffman@23342
   800
apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
huffman@23342
   801
done
huffman@23342
   802
huffman@23342
   803
lemma nonzero_of_rat_inverse:
huffman@23342
   804
  "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
huffman@23343
   805
apply (rule inverse_unique [symmetric])
huffman@23343
   806
apply (simp add: of_rat_mult [symmetric])
huffman@23342
   807
done
huffman@23342
   808
huffman@23342
   809
lemma of_rat_inverse:
huffman@23342
   810
  "(of_rat (inverse a)::'a::{field_char_0,division_by_zero}) =
huffman@23342
   811
   inverse (of_rat a)"
huffman@23342
   812
by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
huffman@23342
   813
huffman@23342
   814
lemma nonzero_of_rat_divide:
huffman@23342
   815
  "b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
huffman@23342
   816
by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
huffman@23342
   817
huffman@23342
   818
lemma of_rat_divide:
huffman@23342
   819
  "(of_rat (a / b)::'a::{field_char_0,division_by_zero})
huffman@23342
   820
   = of_rat a / of_rat b"
haftmann@27652
   821
by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)
huffman@23342
   822
huffman@23343
   823
lemma of_rat_power:
haftmann@31017
   824
  "(of_rat (a ^ n)::'a::field_char_0) = of_rat a ^ n"
huffman@30273
   825
by (induct n) (simp_all add: of_rat_mult)
huffman@23343
   826
huffman@23343
   827
lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
huffman@23343
   828
apply (induct a, induct b)
huffman@23343
   829
apply (simp add: of_rat_rat eq_rat)
huffman@23343
   830
apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
huffman@23343
   831
apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
huffman@23343
   832
done
huffman@23343
   833
haftmann@27652
   834
lemma of_rat_less:
haftmann@35028
   835
  "(of_rat r :: 'a::linordered_field) < of_rat s \<longleftrightarrow> r < s"
haftmann@27652
   836
proof (induct r, induct s)
haftmann@27652
   837
  fix a b c d :: int
haftmann@27652
   838
  assume not_zero: "b > 0" "d > 0"
haftmann@27652
   839
  then have "b * d > 0" by (rule mult_pos_pos)
haftmann@27652
   840
  have of_int_divide_less_eq:
haftmann@27652
   841
    "(of_int a :: 'a) / of_int b < of_int c / of_int d
haftmann@27652
   842
      \<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b"
haftmann@27652
   843
    using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
haftmann@35028
   844
  show "(of_rat (Fract a b) :: 'a::linordered_field) < of_rat (Fract c d)
haftmann@27652
   845
    \<longleftrightarrow> Fract a b < Fract c d"
haftmann@27652
   846
    using not_zero `b * d > 0`
haftmann@27652
   847
    by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult)
haftmann@27652
   848
qed
haftmann@27652
   849
haftmann@27652
   850
lemma of_rat_less_eq:
haftmann@35028
   851
  "(of_rat r :: 'a::linordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s"
haftmann@27652
   852
  unfolding le_less by (auto simp add: of_rat_less)
haftmann@27652
   853
huffman@23343
   854
lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
huffman@23343
   855
haftmann@27652
   856
lemma of_rat_eq_id [simp]: "of_rat = id"
huffman@23343
   857
proof
huffman@23343
   858
  fix a
huffman@23343
   859
  show "of_rat a = id a"
huffman@23343
   860
  by (induct a)
haftmann@27652
   861
     (simp add: of_rat_rat Fract_of_int_eq [symmetric])
huffman@23343
   862
qed
huffman@23343
   863
huffman@23343
   864
text{*Collapse nested embeddings*}
huffman@23343
   865
lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
huffman@23343
   866
by (induct n) (simp_all add: of_rat_add)
huffman@23343
   867
huffman@23343
   868
lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
haftmann@27652
   869
by (cases z rule: int_diff_cases) (simp add: of_rat_diff)
huffman@23343
   870
huffman@23343
   871
lemma of_rat_number_of_eq [simp]:
huffman@23343
   872
  "of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})"
huffman@23343
   873
by (simp add: number_of_eq)
huffman@23343
   874
haftmann@23879
   875
lemmas zero_rat = Zero_rat_def
haftmann@23879
   876
lemmas one_rat = One_rat_def
haftmann@23879
   877
haftmann@24198
   878
abbreviation
haftmann@24198
   879
  rat_of_nat :: "nat \<Rightarrow> rat"
haftmann@24198
   880
where
haftmann@24198
   881
  "rat_of_nat \<equiv> of_nat"
haftmann@24198
   882
haftmann@24198
   883
abbreviation
haftmann@24198
   884
  rat_of_int :: "int \<Rightarrow> rat"
haftmann@24198
   885
where
haftmann@24198
   886
  "rat_of_int \<equiv> of_int"
haftmann@24198
   887
huffman@28010
   888
subsection {* The Set of Rational Numbers *}
berghofe@24533
   889
nipkow@28001
   890
context field_char_0
nipkow@28001
   891
begin
nipkow@28001
   892
nipkow@28001
   893
definition
nipkow@28001
   894
  Rats  :: "'a set" where
haftmann@28562
   895
  [code del]: "Rats = range of_rat"
nipkow@28001
   896
nipkow@28001
   897
notation (xsymbols)
nipkow@28001
   898
  Rats  ("\<rat>")
nipkow@28001
   899
nipkow@28001
   900
end
nipkow@28001
   901
huffman@28010
   902
lemma Rats_of_rat [simp]: "of_rat r \<in> Rats"
huffman@28010
   903
by (simp add: Rats_def)
huffman@28010
   904
huffman@28010
   905
lemma Rats_of_int [simp]: "of_int z \<in> Rats"
huffman@28010
   906
by (subst of_rat_of_int_eq [symmetric], rule Rats_of_rat)
huffman@28010
   907
huffman@28010
   908
lemma Rats_of_nat [simp]: "of_nat n \<in> Rats"
huffman@28010
   909
by (subst of_rat_of_nat_eq [symmetric], rule Rats_of_rat)
huffman@28010
   910
huffman@28010
   911
lemma Rats_number_of [simp]:
huffman@28010
   912
  "(number_of w::'a::{number_ring,field_char_0}) \<in> Rats"
huffman@28010
   913
by (subst of_rat_number_of_eq [symmetric], rule Rats_of_rat)
huffman@28010
   914
huffman@28010
   915
lemma Rats_0 [simp]: "0 \<in> Rats"
huffman@28010
   916
apply (unfold Rats_def)
huffman@28010
   917
apply (rule range_eqI)
huffman@28010
   918
apply (rule of_rat_0 [symmetric])
huffman@28010
   919
done
huffman@28010
   920
huffman@28010
   921
lemma Rats_1 [simp]: "1 \<in> Rats"
huffman@28010
   922
apply (unfold Rats_def)
huffman@28010
   923
apply (rule range_eqI)
huffman@28010
   924
apply (rule of_rat_1 [symmetric])
huffman@28010
   925
done
huffman@28010
   926
huffman@28010
   927
lemma Rats_add [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a + b \<in> Rats"
huffman@28010
   928
apply (auto simp add: Rats_def)
huffman@28010
   929
apply (rule range_eqI)
huffman@28010
   930
apply (rule of_rat_add [symmetric])
huffman@28010
   931
done
huffman@28010
   932
huffman@28010
   933
lemma Rats_minus [simp]: "a \<in> Rats \<Longrightarrow> - a \<in> Rats"
huffman@28010
   934
apply (auto simp add: Rats_def)
huffman@28010
   935
apply (rule range_eqI)
huffman@28010
   936
apply (rule of_rat_minus [symmetric])
huffman@28010
   937
done
huffman@28010
   938
huffman@28010
   939
lemma Rats_diff [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a - b \<in> Rats"
huffman@28010
   940
apply (auto simp add: Rats_def)
huffman@28010
   941
apply (rule range_eqI)
huffman@28010
   942
apply (rule of_rat_diff [symmetric])
huffman@28010
   943
done
huffman@28010
   944
huffman@28010
   945
lemma Rats_mult [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a * b \<in> Rats"
huffman@28010
   946
apply (auto simp add: Rats_def)
huffman@28010
   947
apply (rule range_eqI)
huffman@28010
   948
apply (rule of_rat_mult [symmetric])
huffman@28010
   949
done
huffman@28010
   950
huffman@28010
   951
lemma nonzero_Rats_inverse:
huffman@28010
   952
  fixes a :: "'a::field_char_0"
huffman@28010
   953
  shows "\<lbrakk>a \<in> Rats; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Rats"
huffman@28010
   954
apply (auto simp add: Rats_def)
huffman@28010
   955
apply (rule range_eqI)
huffman@28010
   956
apply (erule nonzero_of_rat_inverse [symmetric])
huffman@28010
   957
done
huffman@28010
   958
huffman@28010
   959
lemma Rats_inverse [simp]:
huffman@28010
   960
  fixes a :: "'a::{field_char_0,division_by_zero}"
huffman@28010
   961
  shows "a \<in> Rats \<Longrightarrow> inverse a \<in> Rats"
huffman@28010
   962
apply (auto simp add: Rats_def)
huffman@28010
   963
apply (rule range_eqI)
huffman@28010
   964
apply (rule of_rat_inverse [symmetric])
huffman@28010
   965
done
huffman@28010
   966
huffman@28010
   967
lemma nonzero_Rats_divide:
huffman@28010
   968
  fixes a b :: "'a::field_char_0"
huffman@28010
   969
  shows "\<lbrakk>a \<in> Rats; b \<in> Rats; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
huffman@28010
   970
apply (auto simp add: Rats_def)
huffman@28010
   971
apply (rule range_eqI)
huffman@28010
   972
apply (erule nonzero_of_rat_divide [symmetric])
huffman@28010
   973
done
huffman@28010
   974
huffman@28010
   975
lemma Rats_divide [simp]:
huffman@28010
   976
  fixes a b :: "'a::{field_char_0,division_by_zero}"
huffman@28010
   977
  shows "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
huffman@28010
   978
apply (auto simp add: Rats_def)
huffman@28010
   979
apply (rule range_eqI)
huffman@28010
   980
apply (rule of_rat_divide [symmetric])
huffman@28010
   981
done
huffman@28010
   982
huffman@28010
   983
lemma Rats_power [simp]:
haftmann@31017
   984
  fixes a :: "'a::field_char_0"
huffman@28010
   985
  shows "a \<in> Rats \<Longrightarrow> a ^ n \<in> Rats"
huffman@28010
   986
apply (auto simp add: Rats_def)
huffman@28010
   987
apply (rule range_eqI)
huffman@28010
   988
apply (rule of_rat_power [symmetric])
huffman@28010
   989
done
huffman@28010
   990
huffman@28010
   991
lemma Rats_cases [cases set: Rats]:
huffman@28010
   992
  assumes "q \<in> \<rat>"
huffman@28010
   993
  obtains (of_rat) r where "q = of_rat r"
huffman@28010
   994
  unfolding Rats_def
huffman@28010
   995
proof -
huffman@28010
   996
  from `q \<in> \<rat>` have "q \<in> range of_rat" unfolding Rats_def .
huffman@28010
   997
  then obtain r where "q = of_rat r" ..
huffman@28010
   998
  then show thesis ..
huffman@28010
   999
qed
huffman@28010
  1000
huffman@28010
  1001
lemma Rats_induct [case_names of_rat, induct set: Rats]:
huffman@28010
  1002
  "q \<in> \<rat> \<Longrightarrow> (\<And>r. P (of_rat r)) \<Longrightarrow> P q"
huffman@28010
  1003
  by (rule Rats_cases) auto
huffman@28010
  1004
nipkow@28001
  1005
berghofe@24533
  1006
subsection {* Implementation of rational numbers as pairs of integers *}
berghofe@24533
  1007
haftmann@27652
  1008
definition Fract_norm :: "int \<Rightarrow> int \<Rightarrow> rat" where
haftmann@28562
  1009
  [simp, code del]: "Fract_norm a b = Fract a b"
haftmann@27652
  1010
huffman@31706
  1011
lemma Fract_norm_code [code]: "Fract_norm a b = (if a = 0 \<or> b = 0 then 0 else let c = gcd a b in
haftmann@27652
  1012
  if b > 0 then Fract (a div c) (b div c) else Fract (- (a div c)) (- (b div c)))"
haftmann@27652
  1013
  by (simp add: eq_rat Zero_rat_def Let_def Fract_norm)
berghofe@24533
  1014
berghofe@24533
  1015
lemma [code]:
haftmann@27652
  1016
  "of_rat (Fract a b) = (if b \<noteq> 0 then of_int a / of_int b else 0)"
haftmann@27652
  1017
  by (cases "b = 0") (simp_all add: rat_number_collapse of_rat_rat)
berghofe@24533
  1018
haftmann@26513
  1019
instantiation rat :: eq
haftmann@26513
  1020
begin
haftmann@26513
  1021
haftmann@28562
  1022
definition [code del]: "eq_class.eq (a\<Colon>rat) b \<longleftrightarrow> a - b = 0"
berghofe@24533
  1023
haftmann@26513
  1024
instance by default (simp add: eq_rat_def)
haftmann@26513
  1025
haftmann@27652
  1026
lemma rat_eq_code [code]:
haftmann@27652
  1027
  "eq_class.eq (Fract a b) (Fract c d) \<longleftrightarrow> (if b = 0
haftmann@27652
  1028
       then c = 0 \<or> d = 0
haftmann@27652
  1029
     else if d = 0
haftmann@27652
  1030
       then a = 0 \<or> b = 0
haftmann@29332
  1031
     else a * d = b * c)"
haftmann@27652
  1032
  by (auto simp add: eq eq_rat)
haftmann@26513
  1033
haftmann@28351
  1034
lemma rat_eq_refl [code nbe]:
haftmann@28351
  1035
  "eq_class.eq (r::rat) r \<longleftrightarrow> True"
haftmann@28351
  1036
  by (rule HOL.eq_refl)
haftmann@28351
  1037
haftmann@26513
  1038
end
berghofe@24533
  1039
haftmann@27652
  1040
lemma le_rat':
haftmann@27652
  1041
  assumes "b \<noteq> 0"
haftmann@27652
  1042
    and "d \<noteq> 0"
haftmann@27652
  1043
  shows "Fract a b \<le> Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d"
berghofe@24533
  1044
proof -
haftmann@27652
  1045
  have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp
haftmann@27652
  1046
  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
  1047
  proof (cases "b * d > 0")
haftmann@27652
  1048
    case True
haftmann@27652
  1049
    moreover from True have "sgn b * sgn d = 1"
haftmann@27652
  1050
      by (simp add: sgn_times [symmetric] sgn_1_pos)
haftmann@27652
  1051
    ultimately show ?thesis by (simp add: mult_le_cancel_right)
haftmann@27652
  1052
  next
haftmann@27652
  1053
    case False with assms have "b * d < 0" by (simp add: less_le)
haftmann@27652
  1054
    moreover from this have "sgn b * sgn d = - 1"
haftmann@27652
  1055
      by (simp only: sgn_times [symmetric] sgn_1_neg)
haftmann@27652
  1056
    ultimately show ?thesis by (simp add: mult_le_cancel_right)
haftmann@27652
  1057
  qed
haftmann@27652
  1058
  also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d"
haftmann@27652
  1059
    by (simp add: abs_sgn mult_ac)
haftmann@27652
  1060
  finally show ?thesis using assms by simp
berghofe@24533
  1061
qed
berghofe@24533
  1062
haftmann@27652
  1063
lemma less_rat': 
haftmann@27652
  1064
  assumes "b \<noteq> 0"
haftmann@27652
  1065
    and "d \<noteq> 0"
haftmann@27652
  1066
  shows "Fract a b < Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d"
berghofe@24533
  1067
proof -
haftmann@27652
  1068
  have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp
haftmann@27652
  1069
  have "a * d * (b * d) < c * b * (b * d) \<longleftrightarrow> a * d * (sgn b * sgn d) < c * b * (sgn b * sgn d)"
haftmann@27652
  1070
  proof (cases "b * d > 0")
haftmann@27652
  1071
    case True
haftmann@27652
  1072
    moreover from True have "sgn b * sgn d = 1"
haftmann@27652
  1073
      by (simp add: sgn_times [symmetric] sgn_1_pos)
haftmann@27652
  1074
    ultimately show ?thesis by (simp add: mult_less_cancel_right)
haftmann@27652
  1075
  next
haftmann@27652
  1076
    case False with assms have "b * d < 0" by (simp add: less_le)
haftmann@27652
  1077
    moreover from this have "sgn b * sgn d = - 1"
haftmann@27652
  1078
      by (simp only: sgn_times [symmetric] sgn_1_neg)
haftmann@27652
  1079
    ultimately show ?thesis by (simp add: mult_less_cancel_right)
haftmann@27652
  1080
  qed
haftmann@27652
  1081
  also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d"
haftmann@27652
  1082
    by (simp add: abs_sgn mult_ac)
haftmann@27652
  1083
  finally show ?thesis using assms by simp
berghofe@24533
  1084
qed
berghofe@24533
  1085
haftmann@35028
  1086
lemma (in linordered_idom) sgn_greater [simp]:
haftmann@29940
  1087
  "0 < sgn a \<longleftrightarrow> 0 < a"
haftmann@29940
  1088
  unfolding sgn_if by auto
haftmann@29940
  1089
haftmann@35028
  1090
lemma (in linordered_idom) sgn_less [simp]:
haftmann@29940
  1091
  "sgn a < 0 \<longleftrightarrow> a < 0"
haftmann@29940
  1092
  unfolding sgn_if by auto
berghofe@24533
  1093
haftmann@27652
  1094
lemma rat_le_eq_code [code]:
haftmann@27652
  1095
  "Fract a b < Fract c d \<longleftrightarrow> (if b = 0
haftmann@27652
  1096
       then sgn c * sgn d > 0
haftmann@27652
  1097
     else if d = 0
haftmann@27652
  1098
       then sgn a * sgn b < 0
haftmann@27652
  1099
     else a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d)"
haftmann@29940
  1100
  by (auto simp add: sgn_times mult_less_0_iff zero_less_mult_iff less_rat' eq_rat simp del: less_rat)
haftmann@29940
  1101
haftmann@29940
  1102
lemma rat_less_eq_code [code]:
haftmann@29940
  1103
  "Fract a b \<le> Fract c d \<longleftrightarrow> (if b = 0
haftmann@29940
  1104
       then sgn c * sgn d \<ge> 0
haftmann@29940
  1105
     else if d = 0
haftmann@29940
  1106
       then sgn a * sgn b \<le> 0
haftmann@29940
  1107
     else a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d)"
haftmann@29940
  1108
  by (auto simp add: sgn_times mult_le_0_iff zero_le_mult_iff le_rat' eq_rat simp del: le_rat)
haftmann@29940
  1109
    (auto simp add: le_less not_less sgn_0_0)
haftmann@29940
  1110
berghofe@24533
  1111
haftmann@27652
  1112
lemma rat_plus_code [code]:
haftmann@27652
  1113
  "Fract a b + Fract c d = (if b = 0
haftmann@27652
  1114
     then Fract c d
haftmann@27652
  1115
   else if d = 0
haftmann@27652
  1116
     then Fract a b
haftmann@27652
  1117
   else Fract_norm (a * d + c * b) (b * d))"
haftmann@27652
  1118
  by (simp add: eq_rat, simp add: Zero_rat_def)
haftmann@27652
  1119
haftmann@27652
  1120
lemma rat_times_code [code]:
haftmann@27652
  1121
  "Fract a b * Fract c d = Fract_norm (a * c) (b * d)"
haftmann@27652
  1122
  by simp
berghofe@24533
  1123
haftmann@27652
  1124
lemma rat_minus_code [code]:
haftmann@27652
  1125
  "Fract a b - Fract c d = (if b = 0
haftmann@27652
  1126
     then Fract (- c) d
haftmann@27652
  1127
   else if d = 0
haftmann@27652
  1128
     then Fract a b
haftmann@27652
  1129
   else Fract_norm (a * d - c * b) (b * d))"
haftmann@27652
  1130
  by (simp add: eq_rat, simp add: Zero_rat_def)
berghofe@24533
  1131
haftmann@27652
  1132
lemma rat_inverse_code [code]:
haftmann@27652
  1133
  "inverse (Fract a b) = (if b = 0 then Fract 1 0
haftmann@27652
  1134
    else if a < 0 then Fract (- b) (- a)
haftmann@27652
  1135
    else Fract b a)"
haftmann@27652
  1136
  by (simp add: eq_rat)
haftmann@27652
  1137
haftmann@27652
  1138
lemma rat_divide_code [code]:
haftmann@27652
  1139
  "Fract a b / Fract c d = Fract_norm (a * d) (b * c)"
haftmann@27652
  1140
  by simp
haftmann@27652
  1141
haftmann@31203
  1142
definition (in term_syntax)
haftmann@32657
  1143
  valterm_fract :: "int \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> int \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> rat \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
haftmann@32657
  1144
  [code_unfold]: "valterm_fract k l = Code_Evaluation.valtermify Fract {\<cdot>} k {\<cdot>} l"
haftmann@31203
  1145
haftmann@31203
  1146
notation fcomp (infixl "o>" 60)
haftmann@31203
  1147
notation scomp (infixl "o\<rightarrow>" 60)
haftmann@31203
  1148
haftmann@31203
  1149
instantiation rat :: random
haftmann@31203
  1150
begin
haftmann@31203
  1151
haftmann@31203
  1152
definition
haftmann@31641
  1153
  "Quickcheck.random i = Quickcheck.random i o\<rightarrow> (\<lambda>num. Random.range i o\<rightarrow> (\<lambda>denom. Pair (
haftmann@31205
  1154
     let j = Code_Numeral.int_of (denom + 1)
haftmann@32657
  1155
     in valterm_fract num (j, \<lambda>u. Code_Evaluation.term_of j))))"
haftmann@31203
  1156
haftmann@31203
  1157
instance ..
haftmann@31203
  1158
haftmann@31203
  1159
end
haftmann@31203
  1160
haftmann@31203
  1161
no_notation fcomp (infixl "o>" 60)
haftmann@31203
  1162
no_notation scomp (infixl "o\<rightarrow>" 60)
haftmann@31203
  1163
haftmann@27652
  1164
hide (open) const Fract_norm
berghofe@24533
  1165
haftmann@24622
  1166
text {* Setup for SML code generator *}
berghofe@24533
  1167
berghofe@24533
  1168
types_code
berghofe@24533
  1169
  rat ("(int */ int)")
berghofe@24533
  1170
attach (term_of) {*
berghofe@24533
  1171
fun term_of_rat (p, q) =
haftmann@24622
  1172
  let
haftmann@24661
  1173
    val rT = Type ("Rational.rat", [])
berghofe@24533
  1174
  in
berghofe@24533
  1175
    if q = 1 orelse p = 0 then HOLogic.mk_number rT p
berghofe@25885
  1176
    else @{term "op / \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat"} $
berghofe@24533
  1177
      HOLogic.mk_number rT p $ HOLogic.mk_number rT q
berghofe@24533
  1178
  end;
berghofe@24533
  1179
*}
berghofe@24533
  1180
attach (test) {*
berghofe@24533
  1181
fun gen_rat i =
berghofe@24533
  1182
  let
berghofe@24533
  1183
    val p = random_range 0 i;
berghofe@24533
  1184
    val q = random_range 1 (i + 1);
berghofe@24533
  1185
    val g = Integer.gcd p q;
wenzelm@24630
  1186
    val p' = p div g;
wenzelm@24630
  1187
    val q' = q div g;
berghofe@25885
  1188
    val r = (if one_of [true, false] then p' else ~ p',
haftmann@31666
  1189
      if p' = 0 then 1 else q')
berghofe@24533
  1190
  in
berghofe@25885
  1191
    (r, fn () => term_of_rat r)
berghofe@24533
  1192
  end;
berghofe@24533
  1193
*}
berghofe@24533
  1194
berghofe@24533
  1195
consts_code
haftmann@27551
  1196
  Fract ("(_,/ _)")
berghofe@24533
  1197
berghofe@24533
  1198
consts_code
berghofe@24533
  1199
  "of_int :: int \<Rightarrow> rat" ("\<module>rat'_of'_int")
berghofe@24533
  1200
attach {*
haftmann@31674
  1201
fun rat_of_int i = (i, 1);
berghofe@24533
  1202
*}
berghofe@24533
  1203
blanchet@33197
  1204
setup {*
wenzelm@33209
  1205
  Nitpick.register_frac_type @{type_name rat}
wenzelm@33209
  1206
   [(@{const_name zero_rat_inst.zero_rat}, @{const_name Nitpick.zero_frac}),
wenzelm@33209
  1207
    (@{const_name one_rat_inst.one_rat}, @{const_name Nitpick.one_frac}),
wenzelm@33209
  1208
    (@{const_name plus_rat_inst.plus_rat}, @{const_name Nitpick.plus_frac}),
wenzelm@33209
  1209
    (@{const_name times_rat_inst.times_rat}, @{const_name Nitpick.times_frac}),
wenzelm@33209
  1210
    (@{const_name uminus_rat_inst.uminus_rat}, @{const_name Nitpick.uminus_frac}),
wenzelm@33209
  1211
    (@{const_name number_rat_inst.number_of_rat}, @{const_name Nitpick.number_of_frac}),
wenzelm@33209
  1212
    (@{const_name inverse_rat_inst.inverse_rat}, @{const_name Nitpick.inverse_frac}),
wenzelm@33209
  1213
    (@{const_name ord_rat_inst.less_eq_rat}, @{const_name Nitpick.less_eq_frac}),
wenzelm@33209
  1214
    (@{const_name field_char_0_class.of_rat}, @{const_name Nitpick.of_frac}),
wenzelm@33209
  1215
    (@{const_name field_char_0_class.Rats}, @{const_name UNIV})]
blanchet@33197
  1216
*}
blanchet@33197
  1217
blanchet@33197
  1218
lemmas [nitpick_def] = inverse_rat_inst.inverse_rat
wenzelm@33209
  1219
  number_rat_inst.number_of_rat one_rat_inst.one_rat ord_rat_inst.less_eq_rat
wenzelm@33209
  1220
  plus_rat_inst.plus_rat times_rat_inst.times_rat uminus_rat_inst.uminus_rat
wenzelm@33209
  1221
  zero_rat_inst.zero_rat
blanchet@33197
  1222
huffman@29880
  1223
end