src/HOL/Rat.thy
author wenzelm
Wed Nov 30 16:27:10 2011 +0100 (2011-11-30)
changeset 45694 4a8743618257
parent 45507 6975db7fd6f0
child 45818 53a697f5454a
permissions -rw-r--r--
prefer typedef without extra definition and alternative name;
tuned proofs;
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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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uses ("Tools/float_syntax.ML")
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begin
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subsection {* Rational numbers as quotient *}
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subsubsection {* Construction of the type of rational numbers *}
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definition
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  ratrel :: "((int \<times> int) \<times> (int \<times> int)) set" where
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  "ratrel = {(x, y). snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x}"
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lemma ratrel_iff [simp]:
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  "(x, y) \<in> ratrel \<longleftrightarrow> snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x"
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  by (simp add: ratrel_def)
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lemma refl_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 equivI [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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definition "Rat = {x. snd x \<noteq> 0} // ratrel"
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typedef (open) rat = Rat
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  morphisms Rep_Rat Abs_Rat
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  unfolding Rat_def
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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 split_paired_all)
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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
haftmann@35369
   303
    with assms show "p * s = q * r" by (auto simp add: mult_ac sgn_times sgn_0_0)
haftmann@35369
   304
  qed
haftmann@35369
   305
  from assms show ?thesis
haftmann@35369
   306
    by (auto simp add: normalize_def Let_def dvd_div_div_eq_mult mult_commute sgn_times split: if_splits intro: aux)
nipkow@33805
   307
qed
nipkow@33805
   308
haftmann@35369
   309
lemma normalize_eq: "normalize (a, b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
haftmann@35369
   310
  by (auto simp add: normalize_def Let_def Fract_coprime dvd_div_neg rat_number_collapse
haftmann@35369
   311
    split:split_if_asm)
haftmann@35369
   312
haftmann@35369
   313
lemma normalize_denom_pos: "normalize r = (p, q) \<Longrightarrow> q > 0"
haftmann@35369
   314
  by (auto simp add: normalize_def Let_def dvd_div_neg pos_imp_zdiv_neg_iff nonneg1_imp_zdiv_pos_iff
haftmann@35369
   315
    split:split_if_asm)
haftmann@35369
   316
haftmann@35369
   317
lemma normalize_coprime: "normalize r = (p, q) \<Longrightarrow> coprime p q"
haftmann@35369
   318
  by (auto simp add: normalize_def Let_def dvd_div_neg div_gcd_coprime_int
haftmann@35369
   319
    split:split_if_asm)
haftmann@35369
   320
haftmann@35369
   321
lemma normalize_stable [simp]:
haftmann@35369
   322
  "q > 0 \<Longrightarrow> coprime p q \<Longrightarrow> normalize (p, q) = (p, q)"
haftmann@35369
   323
  by (simp add: normalize_def)
haftmann@35369
   324
haftmann@35369
   325
lemma normalize_denom_zero [simp]:
haftmann@35369
   326
  "normalize (p, 0) = (0, 1)"
haftmann@35369
   327
  by (simp add: normalize_def)
haftmann@35369
   328
haftmann@35369
   329
lemma normalize_negative [simp]:
haftmann@35369
   330
  "q < 0 \<Longrightarrow> normalize (p, q) = normalize (- p, - q)"
haftmann@35369
   331
  by (simp add: normalize_def Let_def dvd_div_neg dvd_neg_div)
haftmann@35369
   332
haftmann@35369
   333
text{*
haftmann@35369
   334
  Decompose a fraction into normalized, i.e. coprime numerator and denominator:
haftmann@35369
   335
*}
haftmann@35369
   336
haftmann@35369
   337
definition quotient_of :: "rat \<Rightarrow> int \<times> int" where
haftmann@35369
   338
  "quotient_of x = (THE pair. x = Fract (fst pair) (snd pair) &
haftmann@35369
   339
                   snd pair > 0 & coprime (fst pair) (snd pair))"
haftmann@35369
   340
haftmann@35369
   341
lemma quotient_of_unique:
haftmann@35369
   342
  "\<exists>!p. r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
haftmann@35369
   343
proof (cases r)
haftmann@35369
   344
  case (Fract a b)
haftmann@35369
   345
  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
   346
  then show ?thesis proof (rule ex1I)
haftmann@35369
   347
    fix p
haftmann@35369
   348
    obtain c d :: int where p: "p = (c, d)" by (cases p)
haftmann@35369
   349
    assume "r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
haftmann@35369
   350
    with p have Fract': "r = Fract c d" "d > 0" "coprime c d" by simp_all
haftmann@35369
   351
    have "c = a \<and> d = b"
haftmann@35369
   352
    proof (cases "a = 0")
haftmann@35369
   353
      case True with Fract Fract' show ?thesis by (simp add: eq_rat)
haftmann@35369
   354
    next
haftmann@35369
   355
      case False
haftmann@35369
   356
      with Fract Fract' have *: "c * b = a * d" and "c \<noteq> 0" by (auto simp add: eq_rat)
haftmann@35369
   357
      then have "c * b > 0 \<longleftrightarrow> a * d > 0" by auto
haftmann@35369
   358
      with `b > 0` `d > 0` have "a > 0 \<longleftrightarrow> c > 0" by (simp add: zero_less_mult_iff)
haftmann@35369
   359
      with `a \<noteq> 0` `c \<noteq> 0` have sgn: "sgn a = sgn c" by (auto simp add: not_less)
haftmann@35369
   360
      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
   361
        by (simp add: coprime_crossproduct_int)
haftmann@35369
   362
      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
   363
      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
   364
      with sgn * show ?thesis by (auto simp add: sgn_0_0)
nipkow@33805
   365
    qed
haftmann@35369
   366
    with p show "p = (a, b)" by simp
nipkow@33805
   367
  qed
nipkow@33805
   368
qed
nipkow@33805
   369
haftmann@35369
   370
lemma quotient_of_Fract [code]:
haftmann@35369
   371
  "quotient_of (Fract a b) = normalize (a, b)"
haftmann@35369
   372
proof -
haftmann@35369
   373
  have "Fract a b = Fract (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?Fract)
haftmann@35369
   374
    by (rule sym) (auto intro: normalize_eq)
haftmann@35369
   375
  moreover have "0 < snd (normalize (a, b))" (is ?denom_pos) 
haftmann@35369
   376
    by (cases "normalize (a, b)") (rule normalize_denom_pos, simp)
haftmann@35369
   377
  moreover have "coprime (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?coprime)
haftmann@35369
   378
    by (rule normalize_coprime) simp
haftmann@35369
   379
  ultimately have "?Fract \<and> ?denom_pos \<and> ?coprime" by blast
haftmann@35369
   380
  with quotient_of_unique have
haftmann@35369
   381
    "(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
   382
    by (rule the1_equality)
haftmann@35369
   383
  then show ?thesis by (simp add: quotient_of_def)
haftmann@35369
   384
qed
haftmann@35369
   385
haftmann@35369
   386
lemma quotient_of_number [simp]:
haftmann@35369
   387
  "quotient_of 0 = (0, 1)"
haftmann@35369
   388
  "quotient_of 1 = (1, 1)"
haftmann@35369
   389
  "quotient_of (number_of k) = (number_of k, 1)"
haftmann@35369
   390
  by (simp_all add: rat_number_expand quotient_of_Fract)
nipkow@33805
   391
haftmann@35369
   392
lemma quotient_of_eq: "quotient_of (Fract a b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
haftmann@35369
   393
  by (simp add: quotient_of_Fract normalize_eq)
haftmann@35369
   394
haftmann@35369
   395
lemma quotient_of_denom_pos: "quotient_of r = (p, q) \<Longrightarrow> q > 0"
haftmann@35369
   396
  by (cases r) (simp add: quotient_of_Fract normalize_denom_pos)
haftmann@35369
   397
haftmann@35369
   398
lemma quotient_of_coprime: "quotient_of r = (p, q) \<Longrightarrow> coprime p q"
haftmann@35369
   399
  by (cases r) (simp add: quotient_of_Fract normalize_coprime)
nipkow@33805
   400
haftmann@35369
   401
lemma quotient_of_inject:
haftmann@35369
   402
  assumes "quotient_of a = quotient_of b"
haftmann@35369
   403
  shows "a = b"
haftmann@35369
   404
proof -
haftmann@35369
   405
  obtain p q r s where a: "a = Fract p q"
haftmann@35369
   406
    and b: "b = Fract r s"
haftmann@35369
   407
    and "q > 0" and "s > 0" by (cases a, cases b)
haftmann@35369
   408
  with assms show ?thesis by (simp add: eq_rat quotient_of_Fract normalize_crossproduct)
haftmann@35369
   409
qed
haftmann@35369
   410
haftmann@35369
   411
lemma quotient_of_inject_eq:
haftmann@35369
   412
  "quotient_of a = quotient_of b \<longleftrightarrow> a = b"
haftmann@35369
   413
  by (auto simp add: quotient_of_inject)
nipkow@33805
   414
haftmann@27551
   415
haftmann@27551
   416
subsubsection {* The field of rational numbers *}
haftmann@27551
   417
haftmann@36409
   418
instantiation rat :: field_inverse_zero
haftmann@27551
   419
begin
haftmann@27551
   420
haftmann@27551
   421
definition
haftmann@35369
   422
  inverse_rat_def:
haftmann@27551
   423
  "inverse q = Abs_Rat (\<Union>x \<in> Rep_Rat q.
haftmann@27551
   424
     ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)})"
haftmann@27551
   425
haftmann@27652
   426
lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"
haftmann@27551
   427
proof -
haftmann@27551
   428
  have "(\<lambda>x. ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)}) respects ratrel"
haftmann@27551
   429
    by (auto simp add: congruent_def mult_commute)
haftmann@27551
   430
  then show ?thesis by (simp add: Fract_def inverse_rat_def UN_ratrel)
huffman@27509
   431
qed
huffman@27509
   432
haftmann@27551
   433
definition
haftmann@35369
   434
  divide_rat_def: "q / r = q * inverse (r::rat)"
haftmann@27551
   435
haftmann@27652
   436
lemma divide_rat [simp]: "Fract a b / Fract c d = Fract (a * d) (b * c)"
haftmann@27652
   437
  by (simp add: divide_rat_def)
haftmann@27551
   438
haftmann@27551
   439
instance proof
haftmann@27551
   440
  fix q :: rat
haftmann@27551
   441
  assume "q \<noteq> 0"
haftmann@27551
   442
  then show "inverse q * q = 1" by (cases q rule: Rat_cases_nonzero)
huffman@35216
   443
   (simp_all add: rat_number_expand eq_rat)
haftmann@27551
   444
next
haftmann@27551
   445
  fix q r :: rat
haftmann@27551
   446
  show "q / r = q * inverse r" by (simp add: divide_rat_def)
haftmann@36415
   447
next
haftmann@36415
   448
  show "inverse 0 = (0::rat)" by (simp add: rat_number_expand, simp add: rat_number_collapse)
haftmann@36415
   449
qed
haftmann@27551
   450
haftmann@27551
   451
end
haftmann@27551
   452
haftmann@27551
   453
haftmann@27551
   454
subsubsection {* Various *}
haftmann@27551
   455
haftmann@27551
   456
lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
haftmann@27652
   457
  by (simp add: rat_number_expand)
haftmann@27551
   458
haftmann@27551
   459
lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"
haftmann@27652
   460
  by (simp add: Fract_of_int_eq [symmetric])
haftmann@27551
   461
haftmann@35369
   462
lemma Fract_number_of_quotient:
haftmann@27551
   463
  "Fract (number_of k) (number_of l) = number_of k / number_of l"
haftmann@27551
   464
  unfolding Fract_of_int_quotient number_of_is_id number_of_eq ..
haftmann@27551
   465
haftmann@35369
   466
lemma Fract_1_number_of:
haftmann@27652
   467
  "Fract 1 (number_of k) = 1 / number_of k"
haftmann@27652
   468
  unfolding Fract_of_int_quotient number_of_eq by simp
haftmann@27551
   469
haftmann@27551
   470
subsubsection {* The ordered field of rational numbers *}
huffman@27509
   471
huffman@27509
   472
instantiation rat :: linorder
huffman@27509
   473
begin
huffman@27509
   474
huffman@27509
   475
definition
haftmann@35369
   476
  le_rat_def:
haftmann@39910
   477
   "q \<le> r \<longleftrightarrow> the_elem (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
haftmann@27551
   478
      {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})"
haftmann@27551
   479
haftmann@27652
   480
lemma le_rat [simp]:
haftmann@27551
   481
  assumes "b \<noteq> 0" and "d \<noteq> 0"
haftmann@27551
   482
  shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27551
   483
proof -
haftmann@27551
   484
  have "(\<lambda>x y. {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})
haftmann@27551
   485
    respects2 ratrel"
haftmann@27551
   486
  proof (clarsimp simp add: congruent2_def)
haftmann@27551
   487
    fix a b a' b' c d c' d'::int
haftmann@27551
   488
    assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
haftmann@27551
   489
    assume eq1: "a * b' = a' * b"
haftmann@27551
   490
    assume eq2: "c * d' = c' * d"
haftmann@27551
   491
haftmann@27551
   492
    let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
haftmann@27551
   493
    {
haftmann@27551
   494
      fix a b c d x :: int assume x: "x \<noteq> 0"
haftmann@27551
   495
      have "?le a b c d = ?le (a * x) (b * x) c d"
haftmann@27551
   496
      proof -
haftmann@27551
   497
        from x have "0 < x * x" by (auto simp add: zero_less_mult_iff)
haftmann@27551
   498
        hence "?le a b c d =
haftmann@27551
   499
            ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
haftmann@27551
   500
          by (simp add: mult_le_cancel_right)
haftmann@27551
   501
        also have "... = ?le (a * x) (b * x) c d"
haftmann@27551
   502
          by (simp add: mult_ac)
haftmann@27551
   503
        finally show ?thesis .
haftmann@27551
   504
      qed
haftmann@27551
   505
    } note le_factor = this
haftmann@27551
   506
haftmann@27551
   507
    let ?D = "b * d" and ?D' = "b' * d'"
haftmann@27551
   508
    from neq have D: "?D \<noteq> 0" by simp
haftmann@27551
   509
    from neq have "?D' \<noteq> 0" by simp
haftmann@27551
   510
    hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"
haftmann@27551
   511
      by (rule le_factor)
chaieb@27668
   512
    also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')" 
haftmann@27551
   513
      by (simp add: mult_ac)
haftmann@27551
   514
    also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
haftmann@27551
   515
      by (simp only: eq1 eq2)
haftmann@27551
   516
    also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
haftmann@27551
   517
      by (simp add: mult_ac)
haftmann@27551
   518
    also from D have "... = ?le a' b' c' d'"
haftmann@27551
   519
      by (rule le_factor [symmetric])
haftmann@27551
   520
    finally show "?le a b c d = ?le a' b' c' d'" .
haftmann@27551
   521
  qed
haftmann@27551
   522
  with assms show ?thesis by (simp add: Fract_def le_rat_def UN_ratrel2)
haftmann@27551
   523
qed
huffman@27509
   524
huffman@27509
   525
definition
haftmann@35369
   526
  less_rat_def: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
huffman@27509
   527
haftmann@27652
   528
lemma less_rat [simp]:
haftmann@27551
   529
  assumes "b \<noteq> 0" and "d \<noteq> 0"
haftmann@27551
   530
  shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
haftmann@27652
   531
  using assms by (simp add: less_rat_def eq_rat order_less_le)
huffman@27509
   532
huffman@27509
   533
instance proof
paulson@14365
   534
  fix q r s :: rat
paulson@14365
   535
  {
paulson@14365
   536
    assume "q \<le> r" and "r \<le> s"
haftmann@35369
   537
    then show "q \<le> s" 
haftmann@35369
   538
    proof (induct q, induct r, induct s)
paulson@14365
   539
      fix a b c d e f :: int
haftmann@35369
   540
      assume neq: "b > 0"  "d > 0"  "f > 0"
paulson@14365
   541
      assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"
paulson@14365
   542
      show "Fract a b \<le> Fract e f"
paulson@14365
   543
      proof -
paulson@14365
   544
        from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
paulson@14365
   545
          by (auto simp add: zero_less_mult_iff linorder_neq_iff)
paulson@14365
   546
        have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
paulson@14365
   547
        proof -
paulson@14365
   548
          from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27652
   549
            by simp
paulson@14365
   550
          with ff show ?thesis by (simp add: mult_le_cancel_right)
paulson@14365
   551
        qed
chaieb@27668
   552
        also have "... = (c * f) * (d * f) * (b * b)" by algebra
paulson@14365
   553
        also have "... \<le> (e * d) * (d * f) * (b * b)"
paulson@14365
   554
        proof -
paulson@14365
   555
          from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
haftmann@27652
   556
            by simp
paulson@14365
   557
          with bb show ?thesis by (simp add: mult_le_cancel_right)
paulson@14365
   558
        qed
paulson@14365
   559
        finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
paulson@14365
   560
          by (simp only: mult_ac)
paulson@14365
   561
        with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
paulson@14365
   562
          by (simp add: mult_le_cancel_right)
haftmann@27652
   563
        with neq show ?thesis by simp
paulson@14365
   564
      qed
paulson@14365
   565
    qed
paulson@14365
   566
  next
paulson@14365
   567
    assume "q \<le> r" and "r \<le> q"
haftmann@35369
   568
    then show "q = r"
haftmann@35369
   569
    proof (induct q, induct r)
paulson@14365
   570
      fix a b c d :: int
haftmann@35369
   571
      assume neq: "b > 0"  "d > 0"
paulson@14365
   572
      assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"
paulson@14365
   573
      show "Fract a b = Fract c d"
paulson@14365
   574
      proof -
paulson@14365
   575
        from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27652
   576
          by simp
paulson@14365
   577
        also have "... \<le> (a * d) * (b * d)"
paulson@14365
   578
        proof -
paulson@14365
   579
          from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
haftmann@27652
   580
            by simp
paulson@14365
   581
          thus ?thesis by (simp only: mult_ac)
paulson@14365
   582
        qed
paulson@14365
   583
        finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
paulson@14365
   584
        moreover from neq have "b * d \<noteq> 0" by simp
paulson@14365
   585
        ultimately have "a * d = c * b" by simp
paulson@14365
   586
        with neq show ?thesis by (simp add: eq_rat)
paulson@14365
   587
      qed
paulson@14365
   588
    qed
paulson@14365
   589
  next
paulson@14365
   590
    show "q \<le> q"
haftmann@27652
   591
      by (induct q) simp
haftmann@27682
   592
    show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)"
haftmann@27682
   593
      by (induct q, induct r) (auto simp add: le_less mult_commute)
paulson@14365
   594
    show "q \<le> r \<or> r \<le> q"
huffman@18913
   595
      by (induct q, induct r)
haftmann@27652
   596
         (simp add: mult_commute, rule linorder_linear)
paulson@14365
   597
  }
paulson@14365
   598
qed
paulson@14365
   599
huffman@27509
   600
end
huffman@27509
   601
haftmann@27551
   602
instantiation rat :: "{distrib_lattice, abs_if, sgn_if}"
haftmann@25571
   603
begin
haftmann@25571
   604
haftmann@25571
   605
definition
haftmann@35369
   606
  abs_rat_def: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))"
haftmann@27551
   607
haftmann@27652
   608
lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
huffman@35216
   609
  by (auto simp add: abs_rat_def zabs_def Zero_rat_def not_less le_less eq_rat zero_less_mult_iff)
haftmann@27551
   610
haftmann@27551
   611
definition
haftmann@35369
   612
  sgn_rat_def: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
haftmann@27551
   613
haftmann@27652
   614
lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)"
haftmann@27551
   615
  unfolding Fract_of_int_eq
haftmann@27652
   616
  by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat)
haftmann@27551
   617
    (auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)
haftmann@27551
   618
haftmann@27551
   619
definition
haftmann@25571
   620
  "(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min"
haftmann@25571
   621
haftmann@25571
   622
definition
haftmann@25571
   623
  "(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max"
haftmann@25571
   624
haftmann@27551
   625
instance by intro_classes
haftmann@27551
   626
  (auto simp add: abs_rat_def sgn_rat_def min_max.sup_inf_distrib1 inf_rat_def sup_rat_def)
haftmann@22456
   627
haftmann@25571
   628
end
haftmann@25571
   629
haftmann@36409
   630
instance rat :: linordered_field_inverse_zero
haftmann@27551
   631
proof
paulson@14365
   632
  fix q r s :: rat
paulson@14365
   633
  show "q \<le> r ==> s + q \<le> s + r"
paulson@14365
   634
  proof (induct q, induct r, induct s)
paulson@14365
   635
    fix a b c d e f :: int
haftmann@35369
   636
    assume neq: "b > 0"  "d > 0"  "f > 0"
paulson@14365
   637
    assume le: "Fract a b \<le> Fract c d"
paulson@14365
   638
    show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
paulson@14365
   639
    proof -
paulson@14365
   640
      let ?F = "f * f" from neq have F: "0 < ?F"
paulson@14365
   641
        by (auto simp add: zero_less_mult_iff)
paulson@14365
   642
      from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27652
   643
        by simp
paulson@14365
   644
      with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
paulson@14365
   645
        by (simp add: mult_le_cancel_right)
haftmann@27652
   646
      with neq show ?thesis by (simp add: mult_ac int_distrib)
paulson@14365
   647
    qed
paulson@14365
   648
  qed
paulson@14365
   649
  show "q < r ==> 0 < s ==> s * q < s * r"
paulson@14365
   650
  proof (induct q, induct r, induct s)
paulson@14365
   651
    fix a b c d e f :: int
haftmann@35369
   652
    assume neq: "b > 0"  "d > 0"  "f > 0"
paulson@14365
   653
    assume le: "Fract a b < Fract c d"
paulson@14365
   654
    assume gt: "0 < Fract e f"
paulson@14365
   655
    show "Fract e f * Fract a b < Fract e f * Fract c d"
paulson@14365
   656
    proof -
paulson@14365
   657
      let ?E = "e * f" and ?F = "f * f"
paulson@14365
   658
      from neq gt have "0 < ?E"
haftmann@27652
   659
        by (auto simp add: Zero_rat_def order_less_le eq_rat)
paulson@14365
   660
      moreover from neq have "0 < ?F"
paulson@14365
   661
        by (auto simp add: zero_less_mult_iff)
paulson@14365
   662
      moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
haftmann@27652
   663
        by simp
paulson@14365
   664
      ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
paulson@14365
   665
        by (simp add: mult_less_cancel_right)
paulson@14365
   666
      with neq show ?thesis
haftmann@27652
   667
        by (simp add: mult_ac)
paulson@14365
   668
    qed
paulson@14365
   669
  qed
haftmann@27551
   670
qed auto
paulson@14365
   671
haftmann@27551
   672
lemma Rat_induct_pos [case_names Fract, induct type: rat]:
haftmann@27551
   673
  assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
haftmann@27551
   674
  shows "P q"
paulson@14365
   675
proof (cases q)
haftmann@27551
   676
  have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)"
paulson@14365
   677
  proof -
paulson@14365
   678
    fix a::int and b::int
paulson@14365
   679
    assume b: "b < 0"
paulson@14365
   680
    hence "0 < -b" by simp
paulson@14365
   681
    hence "P (Fract (-a) (-b))" by (rule step)
paulson@14365
   682
    thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
paulson@14365
   683
  qed
paulson@14365
   684
  case (Fract a b)
paulson@14365
   685
  thus "P q" by (force simp add: linorder_neq_iff step step')
paulson@14365
   686
qed
paulson@14365
   687
paulson@14365
   688
lemma zero_less_Fract_iff:
huffman@30095
   689
  "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a"
huffman@30095
   690
  by (simp add: Zero_rat_def zero_less_mult_iff)
huffman@30095
   691
huffman@30095
   692
lemma Fract_less_zero_iff:
huffman@30095
   693
  "0 < b \<Longrightarrow> Fract a b < 0 \<longleftrightarrow> a < 0"
huffman@30095
   694
  by (simp add: Zero_rat_def mult_less_0_iff)
huffman@30095
   695
huffman@30095
   696
lemma zero_le_Fract_iff:
huffman@30095
   697
  "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a"
huffman@30095
   698
  by (simp add: Zero_rat_def zero_le_mult_iff)
huffman@30095
   699
huffman@30095
   700
lemma Fract_le_zero_iff:
huffman@30095
   701
  "0 < b \<Longrightarrow> Fract a b \<le> 0 \<longleftrightarrow> a \<le> 0"
huffman@30095
   702
  by (simp add: Zero_rat_def mult_le_0_iff)
huffman@30095
   703
huffman@30095
   704
lemma one_less_Fract_iff:
huffman@30095
   705
  "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a"
huffman@30095
   706
  by (simp add: One_rat_def mult_less_cancel_right_disj)
huffman@30095
   707
huffman@30095
   708
lemma Fract_less_one_iff:
huffman@30095
   709
  "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b"
huffman@30095
   710
  by (simp add: One_rat_def mult_less_cancel_right_disj)
huffman@30095
   711
huffman@30095
   712
lemma one_le_Fract_iff:
huffman@30095
   713
  "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a"
huffman@30095
   714
  by (simp add: One_rat_def mult_le_cancel_right)
huffman@30095
   715
huffman@30095
   716
lemma Fract_le_one_iff:
huffman@30095
   717
  "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b"
huffman@30095
   718
  by (simp add: One_rat_def mult_le_cancel_right)
paulson@14365
   719
paulson@14378
   720
huffman@30097
   721
subsubsection {* Rationals are an Archimedean field *}
huffman@30097
   722
huffman@30097
   723
lemma rat_floor_lemma:
huffman@30097
   724
  shows "of_int (a div b) \<le> Fract a b \<and> Fract a b < of_int (a div b + 1)"
huffman@30097
   725
proof -
huffman@30097
   726
  have "Fract a b = of_int (a div b) + Fract (a mod b) b"
huffman@35293
   727
    by (cases "b = 0", simp, simp add: of_int_rat)
huffman@30097
   728
  moreover have "0 \<le> Fract (a mod b) b \<and> Fract (a mod b) b < 1"
huffman@35293
   729
    unfolding Fract_of_int_quotient
haftmann@36409
   730
    by (rule linorder_cases [of b 0]) (simp add: divide_nonpos_neg, simp, simp add: divide_nonneg_pos)
huffman@30097
   731
  ultimately show ?thesis by simp
huffman@30097
   732
qed
huffman@30097
   733
huffman@30097
   734
instance rat :: archimedean_field
huffman@30097
   735
proof
huffman@30097
   736
  fix r :: rat
huffman@30097
   737
  show "\<exists>z. r \<le> of_int z"
huffman@30097
   738
  proof (induct r)
huffman@30097
   739
    case (Fract a b)
huffman@35293
   740
    have "Fract a b \<le> of_int (a div b + 1)"
huffman@35293
   741
      using rat_floor_lemma [of a b] by simp
huffman@30097
   742
    then show "\<exists>z. Fract a b \<le> of_int z" ..
huffman@30097
   743
  qed
huffman@30097
   744
qed
huffman@30097
   745
bulwahn@43732
   746
instantiation rat :: floor_ceiling
bulwahn@43732
   747
begin
bulwahn@43732
   748
bulwahn@43732
   749
definition [code del]:
bulwahn@43732
   750
  "floor (x::rat) = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))"
bulwahn@43732
   751
bulwahn@43732
   752
instance proof
bulwahn@43732
   753
  fix x :: rat
bulwahn@43732
   754
  show "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
bulwahn@43732
   755
    unfolding floor_rat_def using floor_exists1 by (rule theI')
bulwahn@43732
   756
qed
bulwahn@43732
   757
bulwahn@43732
   758
end
bulwahn@43732
   759
huffman@35293
   760
lemma floor_Fract: "floor (Fract a b) = a div b"
huffman@35293
   761
  using rat_floor_lemma [of a b]
huffman@30097
   762
  by (simp add: floor_unique)
huffman@30097
   763
huffman@30097
   764
haftmann@31100
   765
subsection {* Linear arithmetic setup *}
paulson@14387
   766
haftmann@31100
   767
declaration {*
haftmann@31100
   768
  K (Lin_Arith.add_inj_thms [@{thm of_nat_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
haftmann@31100
   769
    (* not needed because x < (y::nat) can be rewritten as Suc x <= y: of_nat_less_iff RS iffD2 *)
haftmann@31100
   770
  #> Lin_Arith.add_inj_thms [@{thm of_int_le_iff} RS iffD2, @{thm of_int_eq_iff} RS iffD2]
haftmann@31100
   771
    (* not needed because x < (y::int) can be rewritten as x + 1 <= y: of_int_less_iff RS iffD2 *)
haftmann@31100
   772
  #> Lin_Arith.add_simps [@{thm neg_less_iff_less},
haftmann@31100
   773
      @{thm True_implies_equals},
haftmann@31100
   774
      read_instantiate @{context} [(("a", 0), "(number_of ?v)")] @{thm right_distrib},
haftmann@31100
   775
      @{thm divide_1}, @{thm divide_zero_left},
haftmann@31100
   776
      @{thm times_divide_eq_right}, @{thm times_divide_eq_left},
haftmann@31100
   777
      @{thm minus_divide_left} RS sym, @{thm minus_divide_right} RS sym,
haftmann@31100
   778
      @{thm of_int_minus}, @{thm of_int_diff},
haftmann@31100
   779
      @{thm of_int_of_nat_eq}]
haftmann@31100
   780
  #> Lin_Arith.add_simprocs Numeral_Simprocs.field_cancel_numeral_factors
haftmann@31100
   781
  #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat => rat"})
haftmann@31100
   782
  #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int => rat"}))
haftmann@31100
   783
*}
paulson@14387
   784
huffman@23342
   785
huffman@23342
   786
subsection {* Embedding from Rationals to other Fields *}
huffman@23342
   787
haftmann@24198
   788
class field_char_0 = field + ring_char_0
huffman@23342
   789
haftmann@35028
   790
subclass (in linordered_field) field_char_0 ..
huffman@23342
   791
haftmann@27551
   792
context field_char_0
haftmann@27551
   793
begin
haftmann@27551
   794
haftmann@27551
   795
definition of_rat :: "rat \<Rightarrow> 'a" where
haftmann@39910
   796
  "of_rat q = the_elem (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})"
huffman@23342
   797
haftmann@27551
   798
end
haftmann@27551
   799
huffman@23342
   800
lemma of_rat_congruent:
haftmann@27551
   801
  "(\<lambda>(a, b). {of_int a / of_int b :: 'a::field_char_0}) respects ratrel"
haftmann@40816
   802
apply (rule congruentI)
huffman@23342
   803
apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
huffman@23342
   804
apply (simp only: of_int_mult [symmetric])
huffman@23342
   805
done
huffman@23342
   806
haftmann@27551
   807
lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
haftmann@27551
   808
  unfolding Fract_def of_rat_def by (simp add: UN_ratrel of_rat_congruent)
huffman@23342
   809
huffman@23342
   810
lemma of_rat_0 [simp]: "of_rat 0 = 0"
huffman@23342
   811
by (simp add: Zero_rat_def of_rat_rat)
huffman@23342
   812
huffman@23342
   813
lemma of_rat_1 [simp]: "of_rat 1 = 1"
huffman@23342
   814
by (simp add: One_rat_def of_rat_rat)
huffman@23342
   815
huffman@23342
   816
lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
haftmann@27652
   817
by (induct a, induct b, simp add: of_rat_rat add_frac_eq)
huffman@23342
   818
huffman@23343
   819
lemma of_rat_minus: "of_rat (- a) = - of_rat a"
haftmann@27652
   820
by (induct a, simp add: of_rat_rat)
huffman@23343
   821
huffman@23343
   822
lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
huffman@23343
   823
by (simp only: diff_minus of_rat_add of_rat_minus)
huffman@23343
   824
huffman@23342
   825
lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
haftmann@27652
   826
apply (induct a, induct b, simp add: of_rat_rat)
huffman@23342
   827
apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
huffman@23342
   828
done
huffman@23342
   829
huffman@23342
   830
lemma nonzero_of_rat_inverse:
huffman@23342
   831
  "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
huffman@23343
   832
apply (rule inverse_unique [symmetric])
huffman@23343
   833
apply (simp add: of_rat_mult [symmetric])
huffman@23342
   834
done
huffman@23342
   835
huffman@23342
   836
lemma of_rat_inverse:
haftmann@36409
   837
  "(of_rat (inverse a)::'a::{field_char_0, field_inverse_zero}) =
huffman@23342
   838
   inverse (of_rat a)"
huffman@23342
   839
by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
huffman@23342
   840
huffman@23342
   841
lemma nonzero_of_rat_divide:
huffman@23342
   842
  "b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
huffman@23342
   843
by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
huffman@23342
   844
huffman@23342
   845
lemma of_rat_divide:
haftmann@36409
   846
  "(of_rat (a / b)::'a::{field_char_0, field_inverse_zero})
huffman@23342
   847
   = of_rat a / of_rat b"
haftmann@27652
   848
by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)
huffman@23342
   849
huffman@23343
   850
lemma of_rat_power:
haftmann@31017
   851
  "(of_rat (a ^ n)::'a::field_char_0) = of_rat a ^ n"
huffman@30273
   852
by (induct n) (simp_all add: of_rat_mult)
huffman@23343
   853
huffman@23343
   854
lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
huffman@23343
   855
apply (induct a, induct b)
huffman@23343
   856
apply (simp add: of_rat_rat eq_rat)
huffman@23343
   857
apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
huffman@23343
   858
apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
huffman@23343
   859
done
huffman@23343
   860
haftmann@27652
   861
lemma of_rat_less:
haftmann@35028
   862
  "(of_rat r :: 'a::linordered_field) < of_rat s \<longleftrightarrow> r < s"
haftmann@27652
   863
proof (induct r, induct s)
haftmann@27652
   864
  fix a b c d :: int
haftmann@27652
   865
  assume not_zero: "b > 0" "d > 0"
haftmann@27652
   866
  then have "b * d > 0" by (rule mult_pos_pos)
haftmann@27652
   867
  have of_int_divide_less_eq:
haftmann@27652
   868
    "(of_int a :: 'a) / of_int b < of_int c / of_int d
haftmann@27652
   869
      \<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b"
haftmann@27652
   870
    using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
haftmann@35028
   871
  show "(of_rat (Fract a b) :: 'a::linordered_field) < of_rat (Fract c d)
haftmann@27652
   872
    \<longleftrightarrow> Fract a b < Fract c d"
haftmann@27652
   873
    using not_zero `b * d > 0`
haftmann@27652
   874
    by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult)
haftmann@27652
   875
qed
haftmann@27652
   876
haftmann@27652
   877
lemma of_rat_less_eq:
haftmann@35028
   878
  "(of_rat r :: 'a::linordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s"
haftmann@27652
   879
  unfolding le_less by (auto simp add: of_rat_less)
haftmann@27652
   880
huffman@23343
   881
lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
huffman@23343
   882
haftmann@27652
   883
lemma of_rat_eq_id [simp]: "of_rat = id"
huffman@23343
   884
proof
huffman@23343
   885
  fix a
huffman@23343
   886
  show "of_rat a = id a"
huffman@23343
   887
  by (induct a)
haftmann@27652
   888
     (simp add: of_rat_rat Fract_of_int_eq [symmetric])
huffman@23343
   889
qed
huffman@23343
   890
huffman@23343
   891
text{*Collapse nested embeddings*}
huffman@23343
   892
lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
huffman@23343
   893
by (induct n) (simp_all add: of_rat_add)
huffman@23343
   894
huffman@23343
   895
lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
haftmann@27652
   896
by (cases z rule: int_diff_cases) (simp add: of_rat_diff)
huffman@23343
   897
huffman@23343
   898
lemma of_rat_number_of_eq [simp]:
huffman@23343
   899
  "of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})"
huffman@23343
   900
by (simp add: number_of_eq)
huffman@23343
   901
haftmann@23879
   902
lemmas zero_rat = Zero_rat_def
haftmann@23879
   903
lemmas one_rat = One_rat_def
haftmann@23879
   904
haftmann@24198
   905
abbreviation
haftmann@24198
   906
  rat_of_nat :: "nat \<Rightarrow> rat"
haftmann@24198
   907
where
haftmann@24198
   908
  "rat_of_nat \<equiv> of_nat"
haftmann@24198
   909
haftmann@24198
   910
abbreviation
haftmann@24198
   911
  rat_of_int :: "int \<Rightarrow> rat"
haftmann@24198
   912
where
haftmann@24198
   913
  "rat_of_int \<equiv> of_int"
haftmann@24198
   914
huffman@28010
   915
subsection {* The Set of Rational Numbers *}
berghofe@24533
   916
nipkow@28001
   917
context field_char_0
nipkow@28001
   918
begin
nipkow@28001
   919
nipkow@28001
   920
definition
nipkow@28001
   921
  Rats  :: "'a set" where
haftmann@35369
   922
  "Rats = range of_rat"
nipkow@28001
   923
nipkow@28001
   924
notation (xsymbols)
nipkow@28001
   925
  Rats  ("\<rat>")
nipkow@28001
   926
nipkow@28001
   927
end
nipkow@28001
   928
huffman@28010
   929
lemma Rats_of_rat [simp]: "of_rat r \<in> Rats"
huffman@28010
   930
by (simp add: Rats_def)
huffman@28010
   931
huffman@28010
   932
lemma Rats_of_int [simp]: "of_int z \<in> Rats"
huffman@28010
   933
by (subst of_rat_of_int_eq [symmetric], rule Rats_of_rat)
huffman@28010
   934
huffman@28010
   935
lemma Rats_of_nat [simp]: "of_nat n \<in> Rats"
huffman@28010
   936
by (subst of_rat_of_nat_eq [symmetric], rule Rats_of_rat)
huffman@28010
   937
huffman@28010
   938
lemma Rats_number_of [simp]:
huffman@28010
   939
  "(number_of w::'a::{number_ring,field_char_0}) \<in> Rats"
huffman@28010
   940
by (subst of_rat_number_of_eq [symmetric], rule Rats_of_rat)
huffman@28010
   941
huffman@28010
   942
lemma Rats_0 [simp]: "0 \<in> Rats"
huffman@28010
   943
apply (unfold Rats_def)
huffman@28010
   944
apply (rule range_eqI)
huffman@28010
   945
apply (rule of_rat_0 [symmetric])
huffman@28010
   946
done
huffman@28010
   947
huffman@28010
   948
lemma Rats_1 [simp]: "1 \<in> Rats"
huffman@28010
   949
apply (unfold Rats_def)
huffman@28010
   950
apply (rule range_eqI)
huffman@28010
   951
apply (rule of_rat_1 [symmetric])
huffman@28010
   952
done
huffman@28010
   953
huffman@28010
   954
lemma Rats_add [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a + b \<in> Rats"
huffman@28010
   955
apply (auto simp add: Rats_def)
huffman@28010
   956
apply (rule range_eqI)
huffman@28010
   957
apply (rule of_rat_add [symmetric])
huffman@28010
   958
done
huffman@28010
   959
huffman@28010
   960
lemma Rats_minus [simp]: "a \<in> Rats \<Longrightarrow> - a \<in> Rats"
huffman@28010
   961
apply (auto simp add: Rats_def)
huffman@28010
   962
apply (rule range_eqI)
huffman@28010
   963
apply (rule of_rat_minus [symmetric])
huffman@28010
   964
done
huffman@28010
   965
huffman@28010
   966
lemma Rats_diff [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a - b \<in> Rats"
huffman@28010
   967
apply (auto simp add: Rats_def)
huffman@28010
   968
apply (rule range_eqI)
huffman@28010
   969
apply (rule of_rat_diff [symmetric])
huffman@28010
   970
done
huffman@28010
   971
huffman@28010
   972
lemma Rats_mult [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a * b \<in> Rats"
huffman@28010
   973
apply (auto simp add: Rats_def)
huffman@28010
   974
apply (rule range_eqI)
huffman@28010
   975
apply (rule of_rat_mult [symmetric])
huffman@28010
   976
done
huffman@28010
   977
huffman@28010
   978
lemma nonzero_Rats_inverse:
huffman@28010
   979
  fixes a :: "'a::field_char_0"
huffman@28010
   980
  shows "\<lbrakk>a \<in> Rats; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Rats"
huffman@28010
   981
apply (auto simp add: Rats_def)
huffman@28010
   982
apply (rule range_eqI)
huffman@28010
   983
apply (erule nonzero_of_rat_inverse [symmetric])
huffman@28010
   984
done
huffman@28010
   985
huffman@28010
   986
lemma Rats_inverse [simp]:
haftmann@36409
   987
  fixes a :: "'a::{field_char_0, field_inverse_zero}"
huffman@28010
   988
  shows "a \<in> Rats \<Longrightarrow> inverse a \<in> Rats"
huffman@28010
   989
apply (auto simp add: Rats_def)
huffman@28010
   990
apply (rule range_eqI)
huffman@28010
   991
apply (rule of_rat_inverse [symmetric])
huffman@28010
   992
done
huffman@28010
   993
huffman@28010
   994
lemma nonzero_Rats_divide:
huffman@28010
   995
  fixes a b :: "'a::field_char_0"
huffman@28010
   996
  shows "\<lbrakk>a \<in> Rats; b \<in> Rats; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
huffman@28010
   997
apply (auto simp add: Rats_def)
huffman@28010
   998
apply (rule range_eqI)
huffman@28010
   999
apply (erule nonzero_of_rat_divide [symmetric])
huffman@28010
  1000
done
huffman@28010
  1001
huffman@28010
  1002
lemma Rats_divide [simp]:
haftmann@36409
  1003
  fixes a b :: "'a::{field_char_0, field_inverse_zero}"
huffman@28010
  1004
  shows "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
huffman@28010
  1005
apply (auto simp add: Rats_def)
huffman@28010
  1006
apply (rule range_eqI)
huffman@28010
  1007
apply (rule of_rat_divide [symmetric])
huffman@28010
  1008
done
huffman@28010
  1009
huffman@28010
  1010
lemma Rats_power [simp]:
haftmann@31017
  1011
  fixes a :: "'a::field_char_0"
huffman@28010
  1012
  shows "a \<in> Rats \<Longrightarrow> a ^ n \<in> Rats"
huffman@28010
  1013
apply (auto simp add: Rats_def)
huffman@28010
  1014
apply (rule range_eqI)
huffman@28010
  1015
apply (rule of_rat_power [symmetric])
huffman@28010
  1016
done
huffman@28010
  1017
huffman@28010
  1018
lemma Rats_cases [cases set: Rats]:
huffman@28010
  1019
  assumes "q \<in> \<rat>"
huffman@28010
  1020
  obtains (of_rat) r where "q = of_rat r"
huffman@28010
  1021
  unfolding Rats_def
huffman@28010
  1022
proof -
huffman@28010
  1023
  from `q \<in> \<rat>` have "q \<in> range of_rat" unfolding Rats_def .
huffman@28010
  1024
  then obtain r where "q = of_rat r" ..
huffman@28010
  1025
  then show thesis ..
huffman@28010
  1026
qed
huffman@28010
  1027
huffman@28010
  1028
lemma Rats_induct [case_names of_rat, induct set: Rats]:
huffman@28010
  1029
  "q \<in> \<rat> \<Longrightarrow> (\<And>r. P (of_rat r)) \<Longrightarrow> P q"
huffman@28010
  1030
  by (rule Rats_cases) auto
huffman@28010
  1031
nipkow@28001
  1032
berghofe@24533
  1033
subsection {* Implementation of rational numbers as pairs of integers *}
berghofe@24533
  1034
haftmann@35369
  1035
definition Frct :: "int \<times> int \<Rightarrow> rat" where
haftmann@35369
  1036
  [simp]: "Frct p = Fract (fst p) (snd p)"
haftmann@35369
  1037
haftmann@36112
  1038
lemma [code abstype]:
haftmann@36112
  1039
  "Frct (quotient_of q) = q"
haftmann@36112
  1040
  by (cases q) (auto intro: quotient_of_eq)
haftmann@35369
  1041
haftmann@35369
  1042
lemma Frct_code_post [code_post]:
haftmann@35369
  1043
  "Frct (0, k) = 0"
haftmann@35369
  1044
  "Frct (k, 0) = 0"
haftmann@35369
  1045
  "Frct (1, 1) = 1"
haftmann@35369
  1046
  "Frct (number_of k, 1) = number_of k"
haftmann@35369
  1047
  "Frct (1, number_of k) = 1 / number_of k"
haftmann@35369
  1048
  "Frct (number_of k, number_of l) = number_of k / number_of l"
haftmann@35369
  1049
  by (simp_all add: rat_number_collapse Fract_number_of_quotient Fract_1_number_of)
haftmann@35369
  1050
haftmann@35369
  1051
declare quotient_of_Fract [code abstract]
haftmann@35369
  1052
haftmann@35369
  1053
lemma rat_zero_code [code abstract]:
haftmann@35369
  1054
  "quotient_of 0 = (0, 1)"
haftmann@35369
  1055
  by (simp add: Zero_rat_def quotient_of_Fract normalize_def)
haftmann@35369
  1056
haftmann@35369
  1057
lemma rat_one_code [code abstract]:
haftmann@35369
  1058
  "quotient_of 1 = (1, 1)"
haftmann@35369
  1059
  by (simp add: One_rat_def quotient_of_Fract normalize_def)
haftmann@35369
  1060
haftmann@35369
  1061
lemma rat_plus_code [code abstract]:
haftmann@35369
  1062
  "quotient_of (p + q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369
  1063
     in normalize (a * d + b * c, c * d))"
haftmann@35369
  1064
  by (cases p, cases q) (simp add: quotient_of_Fract)
haftmann@27652
  1065
haftmann@35369
  1066
lemma rat_uminus_code [code abstract]:
haftmann@35369
  1067
  "quotient_of (- p) = (let (a, b) = quotient_of p in (- a, b))"
haftmann@35369
  1068
  by (cases p) (simp add: quotient_of_Fract)
haftmann@35369
  1069
haftmann@35369
  1070
lemma rat_minus_code [code abstract]:
haftmann@35369
  1071
  "quotient_of (p - q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369
  1072
     in normalize (a * d - b * c, c * d))"
haftmann@35369
  1073
  by (cases p, cases q) (simp add: quotient_of_Fract)
haftmann@35369
  1074
haftmann@35369
  1075
lemma rat_times_code [code abstract]:
haftmann@35369
  1076
  "quotient_of (p * q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369
  1077
     in normalize (a * b, c * d))"
haftmann@35369
  1078
  by (cases p, cases q) (simp add: quotient_of_Fract)
berghofe@24533
  1079
haftmann@35369
  1080
lemma rat_inverse_code [code abstract]:
haftmann@35369
  1081
  "quotient_of (inverse p) = (let (a, b) = quotient_of p
haftmann@35369
  1082
    in if a = 0 then (0, 1) else (sgn a * b, \<bar>a\<bar>))"
haftmann@35369
  1083
proof (cases p)
haftmann@35369
  1084
  case (Fract a b) then show ?thesis
haftmann@35369
  1085
    by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract gcd_int.commute)
haftmann@35369
  1086
qed
haftmann@35369
  1087
haftmann@35369
  1088
lemma rat_divide_code [code abstract]:
haftmann@35369
  1089
  "quotient_of (p / q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369
  1090
     in normalize (a * d, c * b))"
haftmann@35369
  1091
  by (cases p, cases q) (simp add: quotient_of_Fract)
haftmann@35369
  1092
haftmann@35369
  1093
lemma rat_abs_code [code abstract]:
haftmann@35369
  1094
  "quotient_of \<bar>p\<bar> = (let (a, b) = quotient_of p in (\<bar>a\<bar>, b))"
haftmann@35369
  1095
  by (cases p) (simp add: quotient_of_Fract)
haftmann@35369
  1096
haftmann@35369
  1097
lemma rat_sgn_code [code abstract]:
haftmann@35369
  1098
  "quotient_of (sgn p) = (sgn (fst (quotient_of p)), 1)"
haftmann@35369
  1099
proof (cases p)
haftmann@35369
  1100
  case (Fract a b) then show ?thesis
haftmann@35369
  1101
  by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract)
haftmann@35369
  1102
qed
berghofe@24533
  1103
bulwahn@43733
  1104
lemma rat_floor_code [code]:
bulwahn@43733
  1105
  "floor p = (let (a, b) = quotient_of p in a div b)"
bulwahn@43733
  1106
by (cases p) (simp add: quotient_of_Fract floor_Fract)
bulwahn@43733
  1107
haftmann@38857
  1108
instantiation rat :: equal
haftmann@26513
  1109
begin
haftmann@26513
  1110
haftmann@35369
  1111
definition [code]:
haftmann@38857
  1112
  "HOL.equal a b \<longleftrightarrow> quotient_of a = quotient_of b"
haftmann@26513
  1113
haftmann@35369
  1114
instance proof
haftmann@38857
  1115
qed (simp add: equal_rat_def quotient_of_inject_eq)
haftmann@26513
  1116
haftmann@28351
  1117
lemma rat_eq_refl [code nbe]:
haftmann@38857
  1118
  "HOL.equal (r::rat) r \<longleftrightarrow> True"
haftmann@38857
  1119
  by (rule equal_refl)
haftmann@28351
  1120
haftmann@26513
  1121
end
berghofe@24533
  1122
haftmann@35369
  1123
lemma rat_less_eq_code [code]:
haftmann@35369
  1124
  "p \<le> q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d \<le> c * b)"
haftmann@35726
  1125
  by (cases p, cases q) (simp add: quotient_of_Fract mult.commute)
berghofe@24533
  1126
haftmann@35369
  1127
lemma rat_less_code [code]:
haftmann@35369
  1128
  "p < q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d < c * b)"
haftmann@35726
  1129
  by (cases p, cases q) (simp add: quotient_of_Fract mult.commute)
berghofe@24533
  1130
haftmann@35369
  1131
lemma [code]:
haftmann@35369
  1132
  "of_rat p = (let (a, b) = quotient_of p in of_int a / of_int b)"
haftmann@35369
  1133
  by (cases p) (simp add: quotient_of_Fract of_rat_rat)
haftmann@27652
  1134
haftmann@31203
  1135
definition (in term_syntax)
haftmann@32657
  1136
  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
  1137
  [code_unfold]: "valterm_fract k l = Code_Evaluation.valtermify Fract {\<cdot>} k {\<cdot>} l"
haftmann@31203
  1138
haftmann@37751
  1139
notation fcomp (infixl "\<circ>>" 60)
haftmann@37751
  1140
notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@31203
  1141
haftmann@31203
  1142
instantiation rat :: random
haftmann@31203
  1143
begin
haftmann@31203
  1144
haftmann@31203
  1145
definition
haftmann@37751
  1146
  "Quickcheck.random i = Quickcheck.random i \<circ>\<rightarrow> (\<lambda>num. Random.range i \<circ>\<rightarrow> (\<lambda>denom. Pair (
haftmann@31205
  1147
     let j = Code_Numeral.int_of (denom + 1)
haftmann@32657
  1148
     in valterm_fract num (j, \<lambda>u. Code_Evaluation.term_of j))))"
haftmann@31203
  1149
haftmann@31203
  1150
instance ..
haftmann@31203
  1151
haftmann@31203
  1152
end
haftmann@31203
  1153
haftmann@37751
  1154
no_notation fcomp (infixl "\<circ>>" 60)
haftmann@37751
  1155
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@31203
  1156
bulwahn@41920
  1157
instantiation rat :: exhaustive
bulwahn@41231
  1158
begin
bulwahn@41231
  1159
bulwahn@41231
  1160
definition
bulwahn@45507
  1161
  "exhaustive f d = exhaustive (%l. exhaustive (%k. f (Fract k (Code_Numeral.int_of l + 1))) d) d"
bulwahn@42311
  1162
bulwahn@42311
  1163
instance ..
bulwahn@42311
  1164
bulwahn@42311
  1165
end
bulwahn@42311
  1166
bulwahn@42311
  1167
instantiation rat :: full_exhaustive
bulwahn@42311
  1168
begin
bulwahn@42311
  1169
bulwahn@42311
  1170
definition
bulwahn@45507
  1171
  "full_exhaustive f d = full_exhaustive (%(l, _). full_exhaustive (%k.
bulwahn@45507
  1172
     f (let j = Code_Numeral.int_of l + 1
bulwahn@45507
  1173
        in valterm_fract k (j, %_. Code_Evaluation.term_of j))) d) d"
bulwahn@41231
  1174
bulwahn@41231
  1175
instance ..
bulwahn@41231
  1176
bulwahn@41231
  1177
end
bulwahn@41231
  1178
bulwahn@43889
  1179
instantiation rat :: partial_term_of
bulwahn@43889
  1180
begin
bulwahn@43889
  1181
bulwahn@43889
  1182
instance ..
bulwahn@43889
  1183
bulwahn@43889
  1184
end
bulwahn@43889
  1185
bulwahn@43889
  1186
lemma [code]:
bulwahn@43889
  1187
  "partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Var p tt) == Code_Evaluation.Free (STR ''_'') (Typerep.Typerep (STR ''Rat.rat'') [])"
bulwahn@43889
  1188
  "partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Ctr 0 [l, k]) ==
bulwahn@45507
  1189
     Code_Evaluation.App (Code_Evaluation.Const (STR ''Rat.Frct'')
bulwahn@45507
  1190
     (Typerep.Typerep (STR ''fun'') [Typerep.Typerep (STR ''Product_Type.prod'') [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''Int.int'') []],
bulwahn@45507
  1191
        Typerep.Typerep (STR ''Rat.rat'') []])) (Code_Evaluation.App (Code_Evaluation.App (Code_Evaluation.Const (STR ''Product_Type.Pair'') (Typerep.Typerep (STR ''fun'') [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''fun'') [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''Product_Type.prod'') [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''Int.int'') []]]])) (partial_term_of (TYPE(int)) l)) (partial_term_of (TYPE(int)) k))"
bulwahn@43889
  1192
by (rule partial_term_of_anything)+
bulwahn@43889
  1193
bulwahn@43887
  1194
instantiation rat :: narrowing
bulwahn@43887
  1195
begin
bulwahn@43887
  1196
bulwahn@43887
  1197
definition
bulwahn@45507
  1198
  "narrowing = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.apply
bulwahn@45507
  1199
    (Quickcheck_Narrowing.cons (%nom denom. Fract nom denom)) narrowing) narrowing"
bulwahn@43887
  1200
bulwahn@43887
  1201
instance ..
bulwahn@43887
  1202
bulwahn@43887
  1203
end
bulwahn@43887
  1204
bulwahn@43887
  1205
bulwahn@45183
  1206
subsection {* Setup for Nitpick *}
berghofe@24533
  1207
blanchet@38287
  1208
declaration {*
blanchet@38287
  1209
  Nitpick_HOL.register_frac_type @{type_name rat}
wenzelm@33209
  1210
   [(@{const_name zero_rat_inst.zero_rat}, @{const_name Nitpick.zero_frac}),
wenzelm@33209
  1211
    (@{const_name one_rat_inst.one_rat}, @{const_name Nitpick.one_frac}),
wenzelm@33209
  1212
    (@{const_name plus_rat_inst.plus_rat}, @{const_name Nitpick.plus_frac}),
wenzelm@33209
  1213
    (@{const_name times_rat_inst.times_rat}, @{const_name Nitpick.times_frac}),
wenzelm@33209
  1214
    (@{const_name uminus_rat_inst.uminus_rat}, @{const_name Nitpick.uminus_frac}),
wenzelm@33209
  1215
    (@{const_name number_rat_inst.number_of_rat}, @{const_name Nitpick.number_of_frac}),
wenzelm@33209
  1216
    (@{const_name inverse_rat_inst.inverse_rat}, @{const_name Nitpick.inverse_frac}),
blanchet@37397
  1217
    (@{const_name ord_rat_inst.less_rat}, @{const_name Nitpick.less_frac}),
wenzelm@33209
  1218
    (@{const_name ord_rat_inst.less_eq_rat}, @{const_name Nitpick.less_eq_frac}),
blanchet@45478
  1219
    (@{const_name field_char_0_class.of_rat}, @{const_name Nitpick.of_frac})]
blanchet@33197
  1220
*}
blanchet@33197
  1221
blanchet@41792
  1222
lemmas [nitpick_unfold] = inverse_rat_inst.inverse_rat
blanchet@37397
  1223
  number_rat_inst.number_of_rat one_rat_inst.one_rat ord_rat_inst.less_rat
blanchet@37397
  1224
  ord_rat_inst.less_eq_rat plus_rat_inst.plus_rat times_rat_inst.times_rat
blanchet@37397
  1225
  uminus_rat_inst.uminus_rat zero_rat_inst.zero_rat
blanchet@33197
  1226
huffman@35343
  1227
subsection{* Float syntax *}
huffman@35343
  1228
huffman@35343
  1229
syntax "_Float" :: "float_const \<Rightarrow> 'a"    ("_")
huffman@35343
  1230
huffman@35343
  1231
use "Tools/float_syntax.ML"
huffman@35343
  1232
setup Float_Syntax.setup
huffman@35343
  1233
huffman@35343
  1234
text{* Test: *}
huffman@35343
  1235
lemma "123.456 = -111.111 + 200 + 30 + 4 + 5/10 + 6/100 + (7/1000::rat)"
huffman@35343
  1236
by simp
huffman@35343
  1237
wenzelm@37143
  1238
wenzelm@37143
  1239
hide_const (open) normalize
wenzelm@37143
  1240
huffman@29880
  1241
end