src/HOL/Rat.thy
author haftmann
Fri Feb 26 10:48:20 2010 +0100 (2010-02-26)
changeset 35375 cb06a11a7955
parent 35373 f759782e35ac
child 35377 d84eec579695
permissions -rw-r--r--
implement quotient_of for odl SML code generator
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(*  Title:  HOL/Rat.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 Rat
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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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  "Fract a b = Abs_Rat (ratrel `` {if b = 0 then (0, 1) else (a, b)})"
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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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lemma Rat_cases [case_names Fract, cases type: rat]:
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  assumes "\<And>a b. q = Fract a b \<Longrightarrow> b > 0 \<Longrightarrow> coprime a b \<Longrightarrow> C"
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  shows C
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proof -
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  obtain a b :: int where "q = Fract a b" and "b \<noteq> 0"
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    by (cases q) (clarsimp simp add: Fract_def Rat_def quotient_def)
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  let ?a = "a div gcd a b"
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  let ?b = "b div gcd a b"
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  from `b \<noteq> 0` have "?b * gcd a b = b"
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    by (simp add: dvd_div_mult_self)
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  with `b \<noteq> 0` have "?b \<noteq> 0" by auto
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  from `q = Fract a b` `b \<noteq> 0` `?b \<noteq> 0` have q: "q = Fract ?a ?b"
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    by (simp add: eq_rat dvd_div_mult mult_commute [of a])
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  from `b \<noteq> 0` have coprime: "coprime ?a ?b"
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    by (auto intro: div_gcd_coprime_int)
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  show C proof (cases "b > 0")
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    case True
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    note assms
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    moreover note q
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    moreover from True have "?b > 0" by (simp add: nonneg1_imp_zdiv_pos_iff)
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    moreover note coprime
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    ultimately show C .
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  next
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    case False
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    note assms
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    moreover from q have "q = Fract (- ?a) (- ?b)" by (simp add: Fract_def)
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    moreover from False `b \<noteq> 0` have "- ?b > 0" by (simp add: pos_imp_zdiv_neg_iff)
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    moreover from coprime have "coprime (- ?a) (- ?b)" by simp
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    ultimately show C .
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  qed
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qed
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lemma Rat_induct [case_names Fract, induct type: rat]:
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  assumes "\<And>a b. b > 0 \<Longrightarrow> coprime a b \<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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instantiation rat :: comm_ring_1
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begin
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definition
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  Zero_rat_def: "0 = Fract 0 1"
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definition
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  One_rat_def: "1 = Fract 1 1"
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definition
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  add_rat_def:
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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:
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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]: "- 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: "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:
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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: "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:
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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 > 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> coprime a b \<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 > 0" and "coprime a b" 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 > 0` have "a \<noteq> 0" by (simp add: Zero_rat_def eq_rat)
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  with Fract `q = Fract a b` `b > 0` `coprime a b` show C by blast
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qed
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subsubsection {* Function @{text normalize} *}
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lemma Fract_coprime: "Fract (a div gcd a b) (b div gcd a b) = Fract a b"
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proof (cases "b = 0")
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  case True then show ?thesis by (simp add: eq_rat)
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next
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  case False
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  moreover have "b div gcd a b * gcd a b = b"
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    by (rule dvd_div_mult_self) simp
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  ultimately have "b div gcd a b \<noteq> 0" by auto
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  with False show ?thesis by (simp add: eq_rat dvd_div_mult mult_commute [of a])
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qed
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definition normalize :: "int \<times> int \<Rightarrow> int \<times> int" where
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  "normalize p = (if snd p > 0 then (let a = gcd (fst p) (snd p) in (fst p div a, snd p div a))
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    else if snd p = 0 then (0, 1)
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    else (let a = - gcd (fst p) (snd p) in (fst p div a, snd p div a)))"
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lemma normalize_crossproduct:
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  assumes "q \<noteq> 0" "s \<noteq> 0"
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  assumes "normalize (p, q) = normalize (r, s)"
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  shows "p * s = r * q"
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proof -
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  have aux: "p * gcd r s = sgn (q * s) * r * gcd p q \<Longrightarrow> q * gcd r s = sgn (q * s) * s * gcd p q \<Longrightarrow> p * s = q * r"
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  proof -
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    assume "p * gcd r s = sgn (q * s) * r * gcd p q" and "q * gcd r s = sgn (q * s) * s * gcd p q"
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    then have "(p * gcd r s) * (sgn (q * s) * s * gcd p q) = (q * gcd r s) * (sgn (q * s) * r * gcd p q)" by simp
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    with assms show "p * s = q * r" by (auto simp add: mult_ac sgn_times sgn_0_0)
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   299
  qed
haftmann@35369
   300
  from assms show ?thesis
haftmann@35369
   301
    by (auto simp add: normalize_def Let_def dvd_div_div_eq_mult mult_commute sgn_times split: if_splits intro: aux)
nipkow@33805
   302
qed
nipkow@33805
   303
haftmann@35369
   304
lemma normalize_eq: "normalize (a, b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
haftmann@35369
   305
  by (auto simp add: normalize_def Let_def Fract_coprime dvd_div_neg rat_number_collapse
haftmann@35369
   306
    split:split_if_asm)
haftmann@35369
   307
haftmann@35369
   308
lemma normalize_denom_pos: "normalize r = (p, q) \<Longrightarrow> q > 0"
haftmann@35369
   309
  by (auto simp add: normalize_def Let_def dvd_div_neg pos_imp_zdiv_neg_iff nonneg1_imp_zdiv_pos_iff
haftmann@35369
   310
    split:split_if_asm)
haftmann@35369
   311
haftmann@35369
   312
lemma normalize_coprime: "normalize r = (p, q) \<Longrightarrow> coprime p q"
haftmann@35369
   313
  by (auto simp add: normalize_def Let_def dvd_div_neg div_gcd_coprime_int
haftmann@35369
   314
    split:split_if_asm)
haftmann@35369
   315
haftmann@35369
   316
lemma normalize_stable [simp]:
haftmann@35369
   317
  "q > 0 \<Longrightarrow> coprime p q \<Longrightarrow> normalize (p, q) = (p, q)"
haftmann@35369
   318
  by (simp add: normalize_def)
haftmann@35369
   319
haftmann@35369
   320
lemma normalize_denom_zero [simp]:
haftmann@35369
   321
  "normalize (p, 0) = (0, 1)"
haftmann@35369
   322
  by (simp add: normalize_def)
haftmann@35369
   323
haftmann@35369
   324
lemma normalize_negative [simp]:
haftmann@35369
   325
  "q < 0 \<Longrightarrow> normalize (p, q) = normalize (- p, - q)"
haftmann@35369
   326
  by (simp add: normalize_def Let_def dvd_div_neg dvd_neg_div)
haftmann@35369
   327
haftmann@35369
   328
text{*
haftmann@35369
   329
  Decompose a fraction into normalized, i.e. coprime numerator and denominator:
haftmann@35369
   330
*}
haftmann@35369
   331
haftmann@35369
   332
definition quotient_of :: "rat \<Rightarrow> int \<times> int" where
haftmann@35369
   333
  "quotient_of x = (THE pair. x = Fract (fst pair) (snd pair) &
haftmann@35369
   334
                   snd pair > 0 & coprime (fst pair) (snd pair))"
haftmann@35369
   335
haftmann@35369
   336
lemma quotient_of_unique:
haftmann@35369
   337
  "\<exists>!p. r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
haftmann@35369
   338
proof (cases r)
haftmann@35369
   339
  case (Fract a b)
haftmann@35369
   340
  then have "r = Fract (fst (a, b)) (snd (a, b)) \<and> snd (a, b) > 0 \<and> coprime (fst (a, b)) (snd (a, b))" by auto
haftmann@35369
   341
  then show ?thesis proof (rule ex1I)
haftmann@35369
   342
    fix p
haftmann@35369
   343
    obtain c d :: int where p: "p = (c, d)" by (cases p)
haftmann@35369
   344
    assume "r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
haftmann@35369
   345
    with p have Fract': "r = Fract c d" "d > 0" "coprime c d" by simp_all
haftmann@35369
   346
    have "c = a \<and> d = b"
haftmann@35369
   347
    proof (cases "a = 0")
haftmann@35369
   348
      case True with Fract Fract' show ?thesis by (simp add: eq_rat)
haftmann@35369
   349
    next
haftmann@35369
   350
      case False
haftmann@35369
   351
      with Fract Fract' have *: "c * b = a * d" and "c \<noteq> 0" by (auto simp add: eq_rat)
haftmann@35369
   352
      then have "c * b > 0 \<longleftrightarrow> a * d > 0" by auto
haftmann@35369
   353
      with `b > 0` `d > 0` have "a > 0 \<longleftrightarrow> c > 0" by (simp add: zero_less_mult_iff)
haftmann@35369
   354
      with `a \<noteq> 0` `c \<noteq> 0` have sgn: "sgn a = sgn c" by (auto simp add: not_less)
haftmann@35369
   355
      from `coprime a b` `coprime c d` have "\<bar>a\<bar> * \<bar>d\<bar> = \<bar>c\<bar> * \<bar>b\<bar> \<longleftrightarrow> \<bar>a\<bar> = \<bar>c\<bar> \<and> \<bar>d\<bar> = \<bar>b\<bar>"
haftmann@35369
   356
        by (simp add: coprime_crossproduct_int)
haftmann@35369
   357
      with `b > 0` `d > 0` have "\<bar>a\<bar> * d = \<bar>c\<bar> * b \<longleftrightarrow> \<bar>a\<bar> = \<bar>c\<bar> \<and> d = b" by simp
haftmann@35369
   358
      then have "a * sgn a * d = c * sgn c * b \<longleftrightarrow> a * sgn a = c * sgn c \<and> d = b" by (simp add: abs_sgn)
haftmann@35369
   359
      with sgn * show ?thesis by (auto simp add: sgn_0_0)
nipkow@33805
   360
    qed
haftmann@35369
   361
    with p show "p = (a, b)" by simp
nipkow@33805
   362
  qed
nipkow@33805
   363
qed
nipkow@33805
   364
haftmann@35369
   365
lemma quotient_of_Fract [code]:
haftmann@35369
   366
  "quotient_of (Fract a b) = normalize (a, b)"
haftmann@35369
   367
proof -
haftmann@35369
   368
  have "Fract a b = Fract (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?Fract)
haftmann@35369
   369
    by (rule sym) (auto intro: normalize_eq)
haftmann@35369
   370
  moreover have "0 < snd (normalize (a, b))" (is ?denom_pos) 
haftmann@35369
   371
    by (cases "normalize (a, b)") (rule normalize_denom_pos, simp)
haftmann@35369
   372
  moreover have "coprime (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?coprime)
haftmann@35369
   373
    by (rule normalize_coprime) simp
haftmann@35369
   374
  ultimately have "?Fract \<and> ?denom_pos \<and> ?coprime" by blast
haftmann@35369
   375
  with quotient_of_unique have
haftmann@35369
   376
    "(THE p. Fract a b = Fract (fst p) (snd p) \<and> 0 < snd p \<and> coprime (fst p) (snd p)) = normalize (a, b)"
haftmann@35369
   377
    by (rule the1_equality)
haftmann@35369
   378
  then show ?thesis by (simp add: quotient_of_def)
haftmann@35369
   379
qed
haftmann@35369
   380
haftmann@35369
   381
lemma quotient_of_number [simp]:
haftmann@35369
   382
  "quotient_of 0 = (0, 1)"
haftmann@35369
   383
  "quotient_of 1 = (1, 1)"
haftmann@35369
   384
  "quotient_of (number_of k) = (number_of k, 1)"
haftmann@35369
   385
  by (simp_all add: rat_number_expand quotient_of_Fract)
nipkow@33805
   386
haftmann@35369
   387
lemma quotient_of_eq: "quotient_of (Fract a b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
haftmann@35369
   388
  by (simp add: quotient_of_Fract normalize_eq)
haftmann@35369
   389
haftmann@35369
   390
lemma quotient_of_denom_pos: "quotient_of r = (p, q) \<Longrightarrow> q > 0"
haftmann@35369
   391
  by (cases r) (simp add: quotient_of_Fract normalize_denom_pos)
haftmann@35369
   392
haftmann@35369
   393
lemma quotient_of_coprime: "quotient_of r = (p, q) \<Longrightarrow> coprime p q"
haftmann@35369
   394
  by (cases r) (simp add: quotient_of_Fract normalize_coprime)
nipkow@33805
   395
haftmann@35369
   396
lemma quotient_of_inject:
haftmann@35369
   397
  assumes "quotient_of a = quotient_of b"
haftmann@35369
   398
  shows "a = b"
haftmann@35369
   399
proof -
haftmann@35369
   400
  obtain p q r s where a: "a = Fract p q"
haftmann@35369
   401
    and b: "b = Fract r s"
haftmann@35369
   402
    and "q > 0" and "s > 0" by (cases a, cases b)
haftmann@35369
   403
  with assms show ?thesis by (simp add: eq_rat quotient_of_Fract normalize_crossproduct)
haftmann@35369
   404
qed
haftmann@35369
   405
haftmann@35369
   406
lemma quotient_of_inject_eq:
haftmann@35369
   407
  "quotient_of a = quotient_of b \<longleftrightarrow> a = b"
haftmann@35369
   408
  by (auto simp add: quotient_of_inject)
nipkow@33805
   409
haftmann@27551
   410
haftmann@27551
   411
subsubsection {* The field of rational numbers *}
haftmann@27551
   412
haftmann@27551
   413
instantiation rat :: "{field, division_by_zero}"
haftmann@27551
   414
begin
haftmann@27551
   415
haftmann@27551
   416
definition
haftmann@35369
   417
  inverse_rat_def:
haftmann@27551
   418
  "inverse q = Abs_Rat (\<Union>x \<in> Rep_Rat q.
haftmann@27551
   419
     ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)})"
haftmann@27551
   420
haftmann@27652
   421
lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"
haftmann@27551
   422
proof -
haftmann@27551
   423
  have "(\<lambda>x. ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)}) respects ratrel"
haftmann@27551
   424
    by (auto simp add: congruent_def mult_commute)
haftmann@27551
   425
  then show ?thesis by (simp add: Fract_def inverse_rat_def UN_ratrel)
huffman@27509
   426
qed
huffman@27509
   427
haftmann@27551
   428
definition
haftmann@35369
   429
  divide_rat_def: "q / r = q * inverse (r::rat)"
haftmann@27551
   430
haftmann@27652
   431
lemma divide_rat [simp]: "Fract a b / Fract c d = Fract (a * d) (b * c)"
haftmann@27652
   432
  by (simp add: divide_rat_def)
haftmann@27551
   433
haftmann@27551
   434
instance proof
haftmann@27652
   435
  show "inverse 0 = (0::rat)" by (simp add: rat_number_expand)
haftmann@27551
   436
    (simp add: rat_number_collapse)
haftmann@27551
   437
next
haftmann@27551
   438
  fix q :: rat
haftmann@27551
   439
  assume "q \<noteq> 0"
haftmann@27551
   440
  then show "inverse q * q = 1" by (cases q rule: Rat_cases_nonzero)
huffman@35216
   441
   (simp_all add: rat_number_expand eq_rat)
haftmann@27551
   442
next
haftmann@27551
   443
  fix q r :: rat
haftmann@27551
   444
  show "q / r = q * inverse r" by (simp add: divide_rat_def)
haftmann@27551
   445
qed
haftmann@27551
   446
haftmann@27551
   447
end
haftmann@27551
   448
haftmann@27551
   449
haftmann@27551
   450
subsubsection {* Various *}
haftmann@27551
   451
haftmann@27551
   452
lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
haftmann@27652
   453
  by (simp add: rat_number_expand)
haftmann@27551
   454
haftmann@27551
   455
lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"
haftmann@27652
   456
  by (simp add: Fract_of_int_eq [symmetric])
haftmann@27551
   457
haftmann@35369
   458
lemma Fract_number_of_quotient:
haftmann@27551
   459
  "Fract (number_of k) (number_of l) = number_of k / number_of l"
haftmann@27551
   460
  unfolding Fract_of_int_quotient number_of_is_id number_of_eq ..
haftmann@27551
   461
haftmann@35369
   462
lemma Fract_1_number_of:
haftmann@27652
   463
  "Fract 1 (number_of k) = 1 / number_of k"
haftmann@27652
   464
  unfolding Fract_of_int_quotient number_of_eq by simp
haftmann@27551
   465
haftmann@27551
   466
subsubsection {* The ordered field of rational numbers *}
huffman@27509
   467
huffman@27509
   468
instantiation rat :: linorder
huffman@27509
   469
begin
huffman@27509
   470
huffman@27509
   471
definition
haftmann@35369
   472
  le_rat_def:
huffman@27509
   473
   "q \<le> r \<longleftrightarrow> contents (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
haftmann@27551
   474
      {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})"
haftmann@27551
   475
haftmann@27652
   476
lemma le_rat [simp]:
haftmann@27551
   477
  assumes "b \<noteq> 0" and "d \<noteq> 0"
haftmann@27551
   478
  shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27551
   479
proof -
haftmann@27551
   480
  have "(\<lambda>x y. {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})
haftmann@27551
   481
    respects2 ratrel"
haftmann@27551
   482
  proof (clarsimp simp add: congruent2_def)
haftmann@27551
   483
    fix a b a' b' c d c' d'::int
haftmann@27551
   484
    assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
haftmann@27551
   485
    assume eq1: "a * b' = a' * b"
haftmann@27551
   486
    assume eq2: "c * d' = c' * d"
haftmann@27551
   487
haftmann@27551
   488
    let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
haftmann@27551
   489
    {
haftmann@27551
   490
      fix a b c d x :: int assume x: "x \<noteq> 0"
haftmann@27551
   491
      have "?le a b c d = ?le (a * x) (b * x) c d"
haftmann@27551
   492
      proof -
haftmann@27551
   493
        from x have "0 < x * x" by (auto simp add: zero_less_mult_iff)
haftmann@27551
   494
        hence "?le a b c d =
haftmann@27551
   495
            ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
haftmann@27551
   496
          by (simp add: mult_le_cancel_right)
haftmann@27551
   497
        also have "... = ?le (a * x) (b * x) c d"
haftmann@27551
   498
          by (simp add: mult_ac)
haftmann@27551
   499
        finally show ?thesis .
haftmann@27551
   500
      qed
haftmann@27551
   501
    } note le_factor = this
haftmann@27551
   502
haftmann@27551
   503
    let ?D = "b * d" and ?D' = "b' * d'"
haftmann@27551
   504
    from neq have D: "?D \<noteq> 0" by simp
haftmann@27551
   505
    from neq have "?D' \<noteq> 0" by simp
haftmann@27551
   506
    hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"
haftmann@27551
   507
      by (rule le_factor)
chaieb@27668
   508
    also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')" 
haftmann@27551
   509
      by (simp add: mult_ac)
haftmann@27551
   510
    also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
haftmann@27551
   511
      by (simp only: eq1 eq2)
haftmann@27551
   512
    also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
haftmann@27551
   513
      by (simp add: mult_ac)
haftmann@27551
   514
    also from D have "... = ?le a' b' c' d'"
haftmann@27551
   515
      by (rule le_factor [symmetric])
haftmann@27551
   516
    finally show "?le a b c d = ?le a' b' c' d'" .
haftmann@27551
   517
  qed
haftmann@27551
   518
  with assms show ?thesis by (simp add: Fract_def le_rat_def UN_ratrel2)
haftmann@27551
   519
qed
huffman@27509
   520
huffman@27509
   521
definition
haftmann@35369
   522
  less_rat_def: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
huffman@27509
   523
haftmann@27652
   524
lemma less_rat [simp]:
haftmann@27551
   525
  assumes "b \<noteq> 0" and "d \<noteq> 0"
haftmann@27551
   526
  shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
haftmann@27652
   527
  using assms by (simp add: less_rat_def eq_rat order_less_le)
huffman@27509
   528
huffman@27509
   529
instance proof
paulson@14365
   530
  fix q r s :: rat
paulson@14365
   531
  {
paulson@14365
   532
    assume "q \<le> r" and "r \<le> s"
haftmann@35369
   533
    then show "q \<le> s" 
haftmann@35369
   534
    proof (induct q, induct r, induct s)
paulson@14365
   535
      fix a b c d e f :: int
haftmann@35369
   536
      assume neq: "b > 0"  "d > 0"  "f > 0"
paulson@14365
   537
      assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"
paulson@14365
   538
      show "Fract a b \<le> Fract e f"
paulson@14365
   539
      proof -
paulson@14365
   540
        from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
paulson@14365
   541
          by (auto simp add: zero_less_mult_iff linorder_neq_iff)
paulson@14365
   542
        have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
paulson@14365
   543
        proof -
paulson@14365
   544
          from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27652
   545
            by simp
paulson@14365
   546
          with ff show ?thesis by (simp add: mult_le_cancel_right)
paulson@14365
   547
        qed
chaieb@27668
   548
        also have "... = (c * f) * (d * f) * (b * b)" by algebra
paulson@14365
   549
        also have "... \<le> (e * d) * (d * f) * (b * b)"
paulson@14365
   550
        proof -
paulson@14365
   551
          from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
haftmann@27652
   552
            by simp
paulson@14365
   553
          with bb show ?thesis by (simp add: mult_le_cancel_right)
paulson@14365
   554
        qed
paulson@14365
   555
        finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
paulson@14365
   556
          by (simp only: mult_ac)
paulson@14365
   557
        with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
paulson@14365
   558
          by (simp add: mult_le_cancel_right)
haftmann@27652
   559
        with neq show ?thesis by simp
paulson@14365
   560
      qed
paulson@14365
   561
    qed
paulson@14365
   562
  next
paulson@14365
   563
    assume "q \<le> r" and "r \<le> q"
haftmann@35369
   564
    then show "q = r"
haftmann@35369
   565
    proof (induct q, induct r)
paulson@14365
   566
      fix a b c d :: int
haftmann@35369
   567
      assume neq: "b > 0"  "d > 0"
paulson@14365
   568
      assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"
paulson@14365
   569
      show "Fract a b = Fract c d"
paulson@14365
   570
      proof -
paulson@14365
   571
        from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27652
   572
          by simp
paulson@14365
   573
        also have "... \<le> (a * d) * (b * d)"
paulson@14365
   574
        proof -
paulson@14365
   575
          from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
haftmann@27652
   576
            by simp
paulson@14365
   577
          thus ?thesis by (simp only: mult_ac)
paulson@14365
   578
        qed
paulson@14365
   579
        finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
paulson@14365
   580
        moreover from neq have "b * d \<noteq> 0" by simp
paulson@14365
   581
        ultimately have "a * d = c * b" by simp
paulson@14365
   582
        with neq show ?thesis by (simp add: eq_rat)
paulson@14365
   583
      qed
paulson@14365
   584
    qed
paulson@14365
   585
  next
paulson@14365
   586
    show "q \<le> q"
haftmann@27652
   587
      by (induct q) simp
haftmann@27682
   588
    show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)"
haftmann@27682
   589
      by (induct q, induct r) (auto simp add: le_less mult_commute)
paulson@14365
   590
    show "q \<le> r \<or> r \<le> q"
huffman@18913
   591
      by (induct q, induct r)
haftmann@27652
   592
         (simp add: mult_commute, rule linorder_linear)
paulson@14365
   593
  }
paulson@14365
   594
qed
paulson@14365
   595
huffman@27509
   596
end
huffman@27509
   597
haftmann@27551
   598
instantiation rat :: "{distrib_lattice, abs_if, sgn_if}"
haftmann@25571
   599
begin
haftmann@25571
   600
haftmann@25571
   601
definition
haftmann@35369
   602
  abs_rat_def: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))"
haftmann@27551
   603
haftmann@27652
   604
lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
huffman@35216
   605
  by (auto simp add: abs_rat_def zabs_def Zero_rat_def not_less le_less eq_rat zero_less_mult_iff)
haftmann@27551
   606
haftmann@27551
   607
definition
haftmann@35369
   608
  sgn_rat_def: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
haftmann@27551
   609
haftmann@27652
   610
lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)"
haftmann@27551
   611
  unfolding Fract_of_int_eq
haftmann@27652
   612
  by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat)
haftmann@27551
   613
    (auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)
haftmann@27551
   614
haftmann@27551
   615
definition
haftmann@25571
   616
  "(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min"
haftmann@25571
   617
haftmann@25571
   618
definition
haftmann@25571
   619
  "(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max"
haftmann@25571
   620
haftmann@27551
   621
instance by intro_classes
haftmann@27551
   622
  (auto simp add: abs_rat_def sgn_rat_def min_max.sup_inf_distrib1 inf_rat_def sup_rat_def)
haftmann@22456
   623
haftmann@25571
   624
end
haftmann@25571
   625
haftmann@35028
   626
instance rat :: linordered_field
haftmann@27551
   627
proof
paulson@14365
   628
  fix q r s :: rat
paulson@14365
   629
  show "q \<le> r ==> s + q \<le> s + r"
paulson@14365
   630
  proof (induct q, induct r, induct s)
paulson@14365
   631
    fix a b c d e f :: int
haftmann@35369
   632
    assume neq: "b > 0"  "d > 0"  "f > 0"
paulson@14365
   633
    assume le: "Fract a b \<le> Fract c d"
paulson@14365
   634
    show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
paulson@14365
   635
    proof -
paulson@14365
   636
      let ?F = "f * f" from neq have F: "0 < ?F"
paulson@14365
   637
        by (auto simp add: zero_less_mult_iff)
paulson@14365
   638
      from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27652
   639
        by simp
paulson@14365
   640
      with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
paulson@14365
   641
        by (simp add: mult_le_cancel_right)
haftmann@27652
   642
      with neq show ?thesis by (simp add: mult_ac int_distrib)
paulson@14365
   643
    qed
paulson@14365
   644
  qed
paulson@14365
   645
  show "q < r ==> 0 < s ==> s * q < s * r"
paulson@14365
   646
  proof (induct q, induct r, induct s)
paulson@14365
   647
    fix a b c d e f :: int
haftmann@35369
   648
    assume neq: "b > 0"  "d > 0"  "f > 0"
paulson@14365
   649
    assume le: "Fract a b < Fract c d"
paulson@14365
   650
    assume gt: "0 < Fract e f"
paulson@14365
   651
    show "Fract e f * Fract a b < Fract e f * Fract c d"
paulson@14365
   652
    proof -
paulson@14365
   653
      let ?E = "e * f" and ?F = "f * f"
paulson@14365
   654
      from neq gt have "0 < ?E"
haftmann@27652
   655
        by (auto simp add: Zero_rat_def order_less_le eq_rat)
paulson@14365
   656
      moreover from neq have "0 < ?F"
paulson@14365
   657
        by (auto simp add: zero_less_mult_iff)
paulson@14365
   658
      moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
haftmann@27652
   659
        by simp
paulson@14365
   660
      ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
paulson@14365
   661
        by (simp add: mult_less_cancel_right)
paulson@14365
   662
      with neq show ?thesis
haftmann@27652
   663
        by (simp add: mult_ac)
paulson@14365
   664
    qed
paulson@14365
   665
  qed
haftmann@27551
   666
qed auto
paulson@14365
   667
haftmann@27551
   668
lemma Rat_induct_pos [case_names Fract, induct type: rat]:
haftmann@27551
   669
  assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
haftmann@27551
   670
  shows "P q"
paulson@14365
   671
proof (cases q)
haftmann@27551
   672
  have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)"
paulson@14365
   673
  proof -
paulson@14365
   674
    fix a::int and b::int
paulson@14365
   675
    assume b: "b < 0"
paulson@14365
   676
    hence "0 < -b" by simp
paulson@14365
   677
    hence "P (Fract (-a) (-b))" by (rule step)
paulson@14365
   678
    thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
paulson@14365
   679
  qed
paulson@14365
   680
  case (Fract a b)
paulson@14365
   681
  thus "P q" by (force simp add: linorder_neq_iff step step')
paulson@14365
   682
qed
paulson@14365
   683
paulson@14365
   684
lemma zero_less_Fract_iff:
huffman@30095
   685
  "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a"
huffman@30095
   686
  by (simp add: Zero_rat_def zero_less_mult_iff)
huffman@30095
   687
huffman@30095
   688
lemma Fract_less_zero_iff:
huffman@30095
   689
  "0 < b \<Longrightarrow> Fract a b < 0 \<longleftrightarrow> a < 0"
huffman@30095
   690
  by (simp add: Zero_rat_def mult_less_0_iff)
huffman@30095
   691
huffman@30095
   692
lemma zero_le_Fract_iff:
huffman@30095
   693
  "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a"
huffman@30095
   694
  by (simp add: Zero_rat_def zero_le_mult_iff)
huffman@30095
   695
huffman@30095
   696
lemma Fract_le_zero_iff:
huffman@30095
   697
  "0 < b \<Longrightarrow> Fract a b \<le> 0 \<longleftrightarrow> a \<le> 0"
huffman@30095
   698
  by (simp add: Zero_rat_def mult_le_0_iff)
huffman@30095
   699
huffman@30095
   700
lemma one_less_Fract_iff:
huffman@30095
   701
  "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a"
huffman@30095
   702
  by (simp add: One_rat_def mult_less_cancel_right_disj)
huffman@30095
   703
huffman@30095
   704
lemma Fract_less_one_iff:
huffman@30095
   705
  "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b"
huffman@30095
   706
  by (simp add: One_rat_def mult_less_cancel_right_disj)
huffman@30095
   707
huffman@30095
   708
lemma one_le_Fract_iff:
huffman@30095
   709
  "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a"
huffman@30095
   710
  by (simp add: One_rat_def mult_le_cancel_right)
huffman@30095
   711
huffman@30095
   712
lemma Fract_le_one_iff:
huffman@30095
   713
  "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b"
huffman@30095
   714
  by (simp add: One_rat_def mult_le_cancel_right)
paulson@14365
   715
paulson@14378
   716
huffman@30097
   717
subsubsection {* Rationals are an Archimedean field *}
huffman@30097
   718
huffman@30097
   719
lemma rat_floor_lemma:
huffman@30097
   720
  shows "of_int (a div b) \<le> Fract a b \<and> Fract a b < of_int (a div b + 1)"
huffman@30097
   721
proof -
huffman@30097
   722
  have "Fract a b = of_int (a div b) + Fract (a mod b) b"
huffman@35293
   723
    by (cases "b = 0", simp, simp add: of_int_rat)
huffman@30097
   724
  moreover have "0 \<le> Fract (a mod b) b \<and> Fract (a mod b) b < 1"
huffman@35293
   725
    unfolding Fract_of_int_quotient
huffman@35293
   726
    by (rule linorder_cases [of b 0])
huffman@35293
   727
       (simp add: divide_nonpos_neg, simp, simp add: divide_nonneg_pos)
huffman@30097
   728
  ultimately show ?thesis by simp
huffman@30097
   729
qed
huffman@30097
   730
huffman@30097
   731
instance rat :: archimedean_field
huffman@30097
   732
proof
huffman@30097
   733
  fix r :: rat
huffman@30097
   734
  show "\<exists>z. r \<le> of_int z"
huffman@30097
   735
  proof (induct r)
huffman@30097
   736
    case (Fract a b)
huffman@35293
   737
    have "Fract a b \<le> of_int (a div b + 1)"
huffman@35293
   738
      using rat_floor_lemma [of a b] by simp
huffman@30097
   739
    then show "\<exists>z. Fract a b \<le> of_int z" ..
huffman@30097
   740
  qed
huffman@30097
   741
qed
huffman@30097
   742
huffman@35293
   743
lemma floor_Fract: "floor (Fract a b) = a div b"
huffman@35293
   744
  using rat_floor_lemma [of a b]
huffman@30097
   745
  by (simp add: floor_unique)
huffman@30097
   746
huffman@30097
   747
haftmann@31100
   748
subsection {* Linear arithmetic setup *}
paulson@14387
   749
haftmann@31100
   750
declaration {*
haftmann@31100
   751
  K (Lin_Arith.add_inj_thms [@{thm of_nat_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
haftmann@31100
   752
    (* not needed because x < (y::nat) can be rewritten as Suc x <= y: of_nat_less_iff RS iffD2 *)
haftmann@31100
   753
  #> Lin_Arith.add_inj_thms [@{thm of_int_le_iff} RS iffD2, @{thm of_int_eq_iff} RS iffD2]
haftmann@31100
   754
    (* not needed because x < (y::int) can be rewritten as x + 1 <= y: of_int_less_iff RS iffD2 *)
haftmann@31100
   755
  #> Lin_Arith.add_simps [@{thm neg_less_iff_less},
haftmann@31100
   756
      @{thm True_implies_equals},
haftmann@31100
   757
      read_instantiate @{context} [(("a", 0), "(number_of ?v)")] @{thm right_distrib},
haftmann@31100
   758
      @{thm divide_1}, @{thm divide_zero_left},
haftmann@31100
   759
      @{thm times_divide_eq_right}, @{thm times_divide_eq_left},
haftmann@31100
   760
      @{thm minus_divide_left} RS sym, @{thm minus_divide_right} RS sym,
haftmann@31100
   761
      @{thm of_int_minus}, @{thm of_int_diff},
haftmann@31100
   762
      @{thm of_int_of_nat_eq}]
haftmann@31100
   763
  #> Lin_Arith.add_simprocs Numeral_Simprocs.field_cancel_numeral_factors
haftmann@31100
   764
  #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat => rat"})
haftmann@31100
   765
  #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int => rat"}))
haftmann@31100
   766
*}
paulson@14387
   767
huffman@23342
   768
huffman@23342
   769
subsection {* Embedding from Rationals to other Fields *}
huffman@23342
   770
haftmann@24198
   771
class field_char_0 = field + ring_char_0
huffman@23342
   772
haftmann@35028
   773
subclass (in linordered_field) field_char_0 ..
huffman@23342
   774
haftmann@27551
   775
context field_char_0
haftmann@27551
   776
begin
haftmann@27551
   777
haftmann@27551
   778
definition of_rat :: "rat \<Rightarrow> 'a" where
haftmann@35369
   779
  "of_rat q = contents (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})"
huffman@23342
   780
haftmann@27551
   781
end
haftmann@27551
   782
huffman@23342
   783
lemma of_rat_congruent:
haftmann@27551
   784
  "(\<lambda>(a, b). {of_int a / of_int b :: 'a::field_char_0}) respects ratrel"
huffman@23342
   785
apply (rule congruent.intro)
huffman@23342
   786
apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
huffman@23342
   787
apply (simp only: of_int_mult [symmetric])
huffman@23342
   788
done
huffman@23342
   789
haftmann@27551
   790
lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
haftmann@27551
   791
  unfolding Fract_def of_rat_def by (simp add: UN_ratrel of_rat_congruent)
huffman@23342
   792
huffman@23342
   793
lemma of_rat_0 [simp]: "of_rat 0 = 0"
huffman@23342
   794
by (simp add: Zero_rat_def of_rat_rat)
huffman@23342
   795
huffman@23342
   796
lemma of_rat_1 [simp]: "of_rat 1 = 1"
huffman@23342
   797
by (simp add: One_rat_def of_rat_rat)
huffman@23342
   798
huffman@23342
   799
lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
haftmann@27652
   800
by (induct a, induct b, simp add: of_rat_rat add_frac_eq)
huffman@23342
   801
huffman@23343
   802
lemma of_rat_minus: "of_rat (- a) = - of_rat a"
haftmann@27652
   803
by (induct a, simp add: of_rat_rat)
huffman@23343
   804
huffman@23343
   805
lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
huffman@23343
   806
by (simp only: diff_minus of_rat_add of_rat_minus)
huffman@23343
   807
huffman@23342
   808
lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
haftmann@27652
   809
apply (induct a, induct b, simp add: of_rat_rat)
huffman@23342
   810
apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
huffman@23342
   811
done
huffman@23342
   812
huffman@23342
   813
lemma nonzero_of_rat_inverse:
huffman@23342
   814
  "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
huffman@23343
   815
apply (rule inverse_unique [symmetric])
huffman@23343
   816
apply (simp add: of_rat_mult [symmetric])
huffman@23342
   817
done
huffman@23342
   818
huffman@23342
   819
lemma of_rat_inverse:
huffman@23342
   820
  "(of_rat (inverse a)::'a::{field_char_0,division_by_zero}) =
huffman@23342
   821
   inverse (of_rat a)"
huffman@23342
   822
by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
huffman@23342
   823
huffman@23342
   824
lemma nonzero_of_rat_divide:
huffman@23342
   825
  "b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
huffman@23342
   826
by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
huffman@23342
   827
huffman@23342
   828
lemma of_rat_divide:
huffman@23342
   829
  "(of_rat (a / b)::'a::{field_char_0,division_by_zero})
huffman@23342
   830
   = of_rat a / of_rat b"
haftmann@27652
   831
by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)
huffman@23342
   832
huffman@23343
   833
lemma of_rat_power:
haftmann@31017
   834
  "(of_rat (a ^ n)::'a::field_char_0) = of_rat a ^ n"
huffman@30273
   835
by (induct n) (simp_all add: of_rat_mult)
huffman@23343
   836
huffman@23343
   837
lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
huffman@23343
   838
apply (induct a, induct b)
huffman@23343
   839
apply (simp add: of_rat_rat eq_rat)
huffman@23343
   840
apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
huffman@23343
   841
apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
huffman@23343
   842
done
huffman@23343
   843
haftmann@27652
   844
lemma of_rat_less:
haftmann@35028
   845
  "(of_rat r :: 'a::linordered_field) < of_rat s \<longleftrightarrow> r < s"
haftmann@27652
   846
proof (induct r, induct s)
haftmann@27652
   847
  fix a b c d :: int
haftmann@27652
   848
  assume not_zero: "b > 0" "d > 0"
haftmann@27652
   849
  then have "b * d > 0" by (rule mult_pos_pos)
haftmann@27652
   850
  have of_int_divide_less_eq:
haftmann@27652
   851
    "(of_int a :: 'a) / of_int b < of_int c / of_int d
haftmann@27652
   852
      \<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b"
haftmann@27652
   853
    using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
haftmann@35028
   854
  show "(of_rat (Fract a b) :: 'a::linordered_field) < of_rat (Fract c d)
haftmann@27652
   855
    \<longleftrightarrow> Fract a b < Fract c d"
haftmann@27652
   856
    using not_zero `b * d > 0`
haftmann@27652
   857
    by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult)
haftmann@27652
   858
qed
haftmann@27652
   859
haftmann@27652
   860
lemma of_rat_less_eq:
haftmann@35028
   861
  "(of_rat r :: 'a::linordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s"
haftmann@27652
   862
  unfolding le_less by (auto simp add: of_rat_less)
haftmann@27652
   863
huffman@23343
   864
lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
huffman@23343
   865
haftmann@27652
   866
lemma of_rat_eq_id [simp]: "of_rat = id"
huffman@23343
   867
proof
huffman@23343
   868
  fix a
huffman@23343
   869
  show "of_rat a = id a"
huffman@23343
   870
  by (induct a)
haftmann@27652
   871
     (simp add: of_rat_rat Fract_of_int_eq [symmetric])
huffman@23343
   872
qed
huffman@23343
   873
huffman@23343
   874
text{*Collapse nested embeddings*}
huffman@23343
   875
lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
huffman@23343
   876
by (induct n) (simp_all add: of_rat_add)
huffman@23343
   877
huffman@23343
   878
lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
haftmann@27652
   879
by (cases z rule: int_diff_cases) (simp add: of_rat_diff)
huffman@23343
   880
huffman@23343
   881
lemma of_rat_number_of_eq [simp]:
huffman@23343
   882
  "of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})"
huffman@23343
   883
by (simp add: number_of_eq)
huffman@23343
   884
haftmann@23879
   885
lemmas zero_rat = Zero_rat_def
haftmann@23879
   886
lemmas one_rat = One_rat_def
haftmann@23879
   887
haftmann@24198
   888
abbreviation
haftmann@24198
   889
  rat_of_nat :: "nat \<Rightarrow> rat"
haftmann@24198
   890
where
haftmann@24198
   891
  "rat_of_nat \<equiv> of_nat"
haftmann@24198
   892
haftmann@24198
   893
abbreviation
haftmann@24198
   894
  rat_of_int :: "int \<Rightarrow> rat"
haftmann@24198
   895
where
haftmann@24198
   896
  "rat_of_int \<equiv> of_int"
haftmann@24198
   897
huffman@28010
   898
subsection {* The Set of Rational Numbers *}
berghofe@24533
   899
nipkow@28001
   900
context field_char_0
nipkow@28001
   901
begin
nipkow@28001
   902
nipkow@28001
   903
definition
nipkow@28001
   904
  Rats  :: "'a set" where
haftmann@35369
   905
  "Rats = range of_rat"
nipkow@28001
   906
nipkow@28001
   907
notation (xsymbols)
nipkow@28001
   908
  Rats  ("\<rat>")
nipkow@28001
   909
nipkow@28001
   910
end
nipkow@28001
   911
huffman@28010
   912
lemma Rats_of_rat [simp]: "of_rat r \<in> Rats"
huffman@28010
   913
by (simp add: Rats_def)
huffman@28010
   914
huffman@28010
   915
lemma Rats_of_int [simp]: "of_int z \<in> Rats"
huffman@28010
   916
by (subst of_rat_of_int_eq [symmetric], rule Rats_of_rat)
huffman@28010
   917
huffman@28010
   918
lemma Rats_of_nat [simp]: "of_nat n \<in> Rats"
huffman@28010
   919
by (subst of_rat_of_nat_eq [symmetric], rule Rats_of_rat)
huffman@28010
   920
huffman@28010
   921
lemma Rats_number_of [simp]:
huffman@28010
   922
  "(number_of w::'a::{number_ring,field_char_0}) \<in> Rats"
huffman@28010
   923
by (subst of_rat_number_of_eq [symmetric], rule Rats_of_rat)
huffman@28010
   924
huffman@28010
   925
lemma Rats_0 [simp]: "0 \<in> Rats"
huffman@28010
   926
apply (unfold Rats_def)
huffman@28010
   927
apply (rule range_eqI)
huffman@28010
   928
apply (rule of_rat_0 [symmetric])
huffman@28010
   929
done
huffman@28010
   930
huffman@28010
   931
lemma Rats_1 [simp]: "1 \<in> Rats"
huffman@28010
   932
apply (unfold Rats_def)
huffman@28010
   933
apply (rule range_eqI)
huffman@28010
   934
apply (rule of_rat_1 [symmetric])
huffman@28010
   935
done
huffman@28010
   936
huffman@28010
   937
lemma Rats_add [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a + b \<in> Rats"
huffman@28010
   938
apply (auto simp add: Rats_def)
huffman@28010
   939
apply (rule range_eqI)
huffman@28010
   940
apply (rule of_rat_add [symmetric])
huffman@28010
   941
done
huffman@28010
   942
huffman@28010
   943
lemma Rats_minus [simp]: "a \<in> Rats \<Longrightarrow> - a \<in> Rats"
huffman@28010
   944
apply (auto simp add: Rats_def)
huffman@28010
   945
apply (rule range_eqI)
huffman@28010
   946
apply (rule of_rat_minus [symmetric])
huffman@28010
   947
done
huffman@28010
   948
huffman@28010
   949
lemma Rats_diff [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a - b \<in> Rats"
huffman@28010
   950
apply (auto simp add: Rats_def)
huffman@28010
   951
apply (rule range_eqI)
huffman@28010
   952
apply (rule of_rat_diff [symmetric])
huffman@28010
   953
done
huffman@28010
   954
huffman@28010
   955
lemma Rats_mult [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a * b \<in> Rats"
huffman@28010
   956
apply (auto simp add: Rats_def)
huffman@28010
   957
apply (rule range_eqI)
huffman@28010
   958
apply (rule of_rat_mult [symmetric])
huffman@28010
   959
done
huffman@28010
   960
huffman@28010
   961
lemma nonzero_Rats_inverse:
huffman@28010
   962
  fixes a :: "'a::field_char_0"
huffman@28010
   963
  shows "\<lbrakk>a \<in> Rats; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Rats"
huffman@28010
   964
apply (auto simp add: Rats_def)
huffman@28010
   965
apply (rule range_eqI)
huffman@28010
   966
apply (erule nonzero_of_rat_inverse [symmetric])
huffman@28010
   967
done
huffman@28010
   968
huffman@28010
   969
lemma Rats_inverse [simp]:
huffman@28010
   970
  fixes a :: "'a::{field_char_0,division_by_zero}"
huffman@28010
   971
  shows "a \<in> Rats \<Longrightarrow> inverse a \<in> Rats"
huffman@28010
   972
apply (auto simp add: Rats_def)
huffman@28010
   973
apply (rule range_eqI)
huffman@28010
   974
apply (rule of_rat_inverse [symmetric])
huffman@28010
   975
done
huffman@28010
   976
huffman@28010
   977
lemma nonzero_Rats_divide:
huffman@28010
   978
  fixes a b :: "'a::field_char_0"
huffman@28010
   979
  shows "\<lbrakk>a \<in> Rats; b \<in> Rats; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
huffman@28010
   980
apply (auto simp add: Rats_def)
huffman@28010
   981
apply (rule range_eqI)
huffman@28010
   982
apply (erule nonzero_of_rat_divide [symmetric])
huffman@28010
   983
done
huffman@28010
   984
huffman@28010
   985
lemma Rats_divide [simp]:
huffman@28010
   986
  fixes a b :: "'a::{field_char_0,division_by_zero}"
huffman@28010
   987
  shows "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
huffman@28010
   988
apply (auto simp add: Rats_def)
huffman@28010
   989
apply (rule range_eqI)
huffman@28010
   990
apply (rule of_rat_divide [symmetric])
huffman@28010
   991
done
huffman@28010
   992
huffman@28010
   993
lemma Rats_power [simp]:
haftmann@31017
   994
  fixes a :: "'a::field_char_0"
huffman@28010
   995
  shows "a \<in> Rats \<Longrightarrow> a ^ n \<in> Rats"
huffman@28010
   996
apply (auto simp add: Rats_def)
huffman@28010
   997
apply (rule range_eqI)
huffman@28010
   998
apply (rule of_rat_power [symmetric])
huffman@28010
   999
done
huffman@28010
  1000
huffman@28010
  1001
lemma Rats_cases [cases set: Rats]:
huffman@28010
  1002
  assumes "q \<in> \<rat>"
huffman@28010
  1003
  obtains (of_rat) r where "q = of_rat r"
huffman@28010
  1004
  unfolding Rats_def
huffman@28010
  1005
proof -
huffman@28010
  1006
  from `q \<in> \<rat>` have "q \<in> range of_rat" unfolding Rats_def .
huffman@28010
  1007
  then obtain r where "q = of_rat r" ..
huffman@28010
  1008
  then show thesis ..
huffman@28010
  1009
qed
huffman@28010
  1010
huffman@28010
  1011
lemma Rats_induct [case_names of_rat, induct set: Rats]:
huffman@28010
  1012
  "q \<in> \<rat> \<Longrightarrow> (\<And>r. P (of_rat r)) \<Longrightarrow> P q"
huffman@28010
  1013
  by (rule Rats_cases) auto
huffman@28010
  1014
nipkow@28001
  1015
berghofe@24533
  1016
subsection {* Implementation of rational numbers as pairs of integers *}
berghofe@24533
  1017
haftmann@35369
  1018
definition Frct :: "int \<times> int \<Rightarrow> rat" where
haftmann@35369
  1019
  [simp]: "Frct p = Fract (fst p) (snd p)"
haftmann@35369
  1020
haftmann@35369
  1021
code_abstype Frct quotient_of
haftmann@35369
  1022
proof (rule eq_reflection)
haftmann@35373
  1023
  fix r :: rat
haftmann@35373
  1024
  show "Frct (quotient_of r) = r" by (cases r) (auto intro: quotient_of_eq)
haftmann@35369
  1025
qed
haftmann@35369
  1026
haftmann@35369
  1027
lemma Frct_code_post [code_post]:
haftmann@35369
  1028
  "Frct (0, k) = 0"
haftmann@35369
  1029
  "Frct (k, 0) = 0"
haftmann@35369
  1030
  "Frct (1, 1) = 1"
haftmann@35369
  1031
  "Frct (number_of k, 1) = number_of k"
haftmann@35369
  1032
  "Frct (1, number_of k) = 1 / number_of k"
haftmann@35369
  1033
  "Frct (number_of k, number_of l) = number_of k / number_of l"
haftmann@35369
  1034
  by (simp_all add: rat_number_collapse Fract_number_of_quotient Fract_1_number_of)
haftmann@35369
  1035
haftmann@35369
  1036
declare quotient_of_Fract [code abstract]
haftmann@35369
  1037
haftmann@35369
  1038
lemma rat_zero_code [code abstract]:
haftmann@35369
  1039
  "quotient_of 0 = (0, 1)"
haftmann@35369
  1040
  by (simp add: Zero_rat_def quotient_of_Fract normalize_def)
haftmann@35369
  1041
haftmann@35369
  1042
lemma rat_one_code [code abstract]:
haftmann@35369
  1043
  "quotient_of 1 = (1, 1)"
haftmann@35369
  1044
  by (simp add: One_rat_def quotient_of_Fract normalize_def)
haftmann@35369
  1045
haftmann@35369
  1046
lemma rat_plus_code [code abstract]:
haftmann@35369
  1047
  "quotient_of (p + q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369
  1048
     in normalize (a * d + b * c, c * d))"
haftmann@35369
  1049
  by (cases p, cases q) (simp add: quotient_of_Fract)
haftmann@27652
  1050
haftmann@35369
  1051
lemma rat_uminus_code [code abstract]:
haftmann@35369
  1052
  "quotient_of (- p) = (let (a, b) = quotient_of p in (- a, b))"
haftmann@35369
  1053
  by (cases p) (simp add: quotient_of_Fract)
haftmann@35369
  1054
haftmann@35369
  1055
lemma rat_minus_code [code abstract]:
haftmann@35369
  1056
  "quotient_of (p - q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369
  1057
     in normalize (a * d - b * c, c * d))"
haftmann@35369
  1058
  by (cases p, cases q) (simp add: quotient_of_Fract)
haftmann@35369
  1059
haftmann@35369
  1060
lemma rat_times_code [code abstract]:
haftmann@35369
  1061
  "quotient_of (p * q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369
  1062
     in normalize (a * b, c * d))"
haftmann@35369
  1063
  by (cases p, cases q) (simp add: quotient_of_Fract)
berghofe@24533
  1064
haftmann@35369
  1065
lemma rat_inverse_code [code abstract]:
haftmann@35369
  1066
  "quotient_of (inverse p) = (let (a, b) = quotient_of p
haftmann@35369
  1067
    in if a = 0 then (0, 1) else (sgn a * b, \<bar>a\<bar>))"
haftmann@35369
  1068
proof (cases p)
haftmann@35369
  1069
  case (Fract a b) then show ?thesis
haftmann@35369
  1070
    by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract gcd_int.commute)
haftmann@35369
  1071
qed
haftmann@35369
  1072
haftmann@35369
  1073
lemma rat_divide_code [code abstract]:
haftmann@35369
  1074
  "quotient_of (p / q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369
  1075
     in normalize (a * d, c * b))"
haftmann@35369
  1076
  by (cases p, cases q) (simp add: quotient_of_Fract)
haftmann@35369
  1077
haftmann@35369
  1078
lemma rat_abs_code [code abstract]:
haftmann@35369
  1079
  "quotient_of \<bar>p\<bar> = (let (a, b) = quotient_of p in (\<bar>a\<bar>, b))"
haftmann@35369
  1080
  by (cases p) (simp add: quotient_of_Fract)
haftmann@35369
  1081
haftmann@35369
  1082
lemma rat_sgn_code [code abstract]:
haftmann@35369
  1083
  "quotient_of (sgn p) = (sgn (fst (quotient_of p)), 1)"
haftmann@35369
  1084
proof (cases p)
haftmann@35369
  1085
  case (Fract a b) then show ?thesis
haftmann@35369
  1086
  by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract)
haftmann@35369
  1087
qed
berghofe@24533
  1088
haftmann@26513
  1089
instantiation rat :: eq
haftmann@26513
  1090
begin
haftmann@26513
  1091
haftmann@35369
  1092
definition [code]:
haftmann@35369
  1093
  "eq_class.eq a b \<longleftrightarrow> quotient_of a = quotient_of b"
haftmann@26513
  1094
haftmann@35369
  1095
instance proof
haftmann@35369
  1096
qed (simp add: eq_rat_def quotient_of_inject_eq)
haftmann@26513
  1097
haftmann@28351
  1098
lemma rat_eq_refl [code nbe]:
haftmann@28351
  1099
  "eq_class.eq (r::rat) r \<longleftrightarrow> True"
haftmann@28351
  1100
  by (rule HOL.eq_refl)
haftmann@28351
  1101
haftmann@26513
  1102
end
berghofe@24533
  1103
haftmann@35369
  1104
lemma rat_less_eq_code [code]:
haftmann@35369
  1105
  "p \<le> q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d \<le> c * b)"
haftmann@35369
  1106
  by (cases p, cases q) (simp add: quotient_of_Fract times.commute)
berghofe@24533
  1107
haftmann@35369
  1108
lemma rat_less_code [code]:
haftmann@35369
  1109
  "p < q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d < c * b)"
haftmann@35369
  1110
  by (cases p, cases q) (simp add: quotient_of_Fract times.commute)
berghofe@24533
  1111
haftmann@35369
  1112
lemma [code]:
haftmann@35369
  1113
  "of_rat p = (let (a, b) = quotient_of p in of_int a / of_int b)"
haftmann@35369
  1114
  by (cases p) (simp add: quotient_of_Fract of_rat_rat)
haftmann@27652
  1115
haftmann@31203
  1116
definition (in term_syntax)
haftmann@32657
  1117
  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
  1118
  [code_unfold]: "valterm_fract k l = Code_Evaluation.valtermify Fract {\<cdot>} k {\<cdot>} l"
haftmann@31203
  1119
haftmann@31203
  1120
notation fcomp (infixl "o>" 60)
haftmann@31203
  1121
notation scomp (infixl "o\<rightarrow>" 60)
haftmann@31203
  1122
haftmann@31203
  1123
instantiation rat :: random
haftmann@31203
  1124
begin
haftmann@31203
  1125
haftmann@31203
  1126
definition
haftmann@31641
  1127
  "Quickcheck.random i = Quickcheck.random i o\<rightarrow> (\<lambda>num. Random.range i o\<rightarrow> (\<lambda>denom. Pair (
haftmann@31205
  1128
     let j = Code_Numeral.int_of (denom + 1)
haftmann@32657
  1129
     in valterm_fract num (j, \<lambda>u. Code_Evaluation.term_of j))))"
haftmann@31203
  1130
haftmann@31203
  1131
instance ..
haftmann@31203
  1132
haftmann@31203
  1133
end
haftmann@31203
  1134
haftmann@31203
  1135
no_notation fcomp (infixl "o>" 60)
haftmann@31203
  1136
no_notation scomp (infixl "o\<rightarrow>" 60)
haftmann@31203
  1137
haftmann@24622
  1138
text {* Setup for SML code generator *}
berghofe@24533
  1139
berghofe@24533
  1140
types_code
berghofe@24533
  1141
  rat ("(int */ int)")
berghofe@24533
  1142
attach (term_of) {*
berghofe@24533
  1143
fun term_of_rat (p, q) =
haftmann@24622
  1144
  let
haftmann@35372
  1145
    val rT = Type ("Rat.rat", [])
berghofe@24533
  1146
  in
berghofe@24533
  1147
    if q = 1 orelse p = 0 then HOLogic.mk_number rT p
berghofe@25885
  1148
    else @{term "op / \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat"} $
berghofe@24533
  1149
      HOLogic.mk_number rT p $ HOLogic.mk_number rT q
berghofe@24533
  1150
  end;
berghofe@24533
  1151
*}
berghofe@24533
  1152
attach (test) {*
berghofe@24533
  1153
fun gen_rat i =
berghofe@24533
  1154
  let
berghofe@24533
  1155
    val p = random_range 0 i;
berghofe@24533
  1156
    val q = random_range 1 (i + 1);
berghofe@24533
  1157
    val g = Integer.gcd p q;
wenzelm@24630
  1158
    val p' = p div g;
wenzelm@24630
  1159
    val q' = q div g;
berghofe@25885
  1160
    val r = (if one_of [true, false] then p' else ~ p',
haftmann@31666
  1161
      if p' = 0 then 1 else q')
berghofe@24533
  1162
  in
berghofe@25885
  1163
    (r, fn () => term_of_rat r)
berghofe@24533
  1164
  end;
berghofe@24533
  1165
*}
berghofe@24533
  1166
berghofe@24533
  1167
consts_code
haftmann@27551
  1168
  Fract ("(_,/ _)")
berghofe@24533
  1169
berghofe@24533
  1170
consts_code
haftmann@35375
  1171
  quotient_of ("{*normalize*}")
haftmann@35375
  1172
haftmann@35375
  1173
consts_code
berghofe@24533
  1174
  "of_int :: int \<Rightarrow> rat" ("\<module>rat'_of'_int")
berghofe@24533
  1175
attach {*
haftmann@31674
  1176
fun rat_of_int i = (i, 1);
berghofe@24533
  1177
*}
berghofe@24533
  1178
blanchet@33197
  1179
setup {*
wenzelm@33209
  1180
  Nitpick.register_frac_type @{type_name rat}
wenzelm@33209
  1181
   [(@{const_name zero_rat_inst.zero_rat}, @{const_name Nitpick.zero_frac}),
wenzelm@33209
  1182
    (@{const_name one_rat_inst.one_rat}, @{const_name Nitpick.one_frac}),
wenzelm@33209
  1183
    (@{const_name plus_rat_inst.plus_rat}, @{const_name Nitpick.plus_frac}),
wenzelm@33209
  1184
    (@{const_name times_rat_inst.times_rat}, @{const_name Nitpick.times_frac}),
wenzelm@33209
  1185
    (@{const_name uminus_rat_inst.uminus_rat}, @{const_name Nitpick.uminus_frac}),
wenzelm@33209
  1186
    (@{const_name number_rat_inst.number_of_rat}, @{const_name Nitpick.number_of_frac}),
wenzelm@33209
  1187
    (@{const_name inverse_rat_inst.inverse_rat}, @{const_name Nitpick.inverse_frac}),
wenzelm@33209
  1188
    (@{const_name ord_rat_inst.less_eq_rat}, @{const_name Nitpick.less_eq_frac}),
wenzelm@33209
  1189
    (@{const_name field_char_0_class.of_rat}, @{const_name Nitpick.of_frac}),
wenzelm@33209
  1190
    (@{const_name field_char_0_class.Rats}, @{const_name UNIV})]
blanchet@33197
  1191
*}
blanchet@33197
  1192
blanchet@33197
  1193
lemmas [nitpick_def] = inverse_rat_inst.inverse_rat
wenzelm@33209
  1194
  number_rat_inst.number_of_rat one_rat_inst.one_rat ord_rat_inst.less_eq_rat
wenzelm@33209
  1195
  plus_rat_inst.plus_rat times_rat_inst.times_rat uminus_rat_inst.uminus_rat
wenzelm@33209
  1196
  zero_rat_inst.zero_rat
blanchet@33197
  1197
huffman@29880
  1198
end