src/HOL/Rat.thy
author huffman
Thu May 10 21:18:41 2012 +0200 (2012-05-10)
changeset 47906 09a896d295bd
parent 47108 2a1953f0d20d
child 47907 54e3847f1669
permissions -rw-r--r--
convert Rat.thy to use lift_definition/transfer
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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) \<Rightarrow> (int \<times> int) \<Rightarrow> bool" where
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  "ratrel = (\<lambda>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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  "ratrel x y \<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 exists_ratrel_refl: "\<exists>x. ratrel x x"
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  by (auto intro!: one_neq_zero)
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lemma symp_ratrel: "symp ratrel"
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  by (simp add: ratrel_def symp_def)
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lemma transp_ratrel: "transp ratrel"
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proof (rule transpI, unfold split_paired_all)
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  fix a b a' b' a'' b'' :: int
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  assume A: "ratrel (a, b) (a', b')"
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  assume B: "ratrel (a', b') (a'', b'')"
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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 "ratrel (a, b) (a'', b'')" by auto
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qed
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lemma part_equivp_ratrel: "part_equivp ratrel"
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  by (rule part_equivpI [OF exists_ratrel_refl symp_ratrel transp_ratrel])
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quotient_type rat = "int \<times> int" / partial: "ratrel"
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  morphisms Rep_Rat Abs_Rat
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  by (rule part_equivp_ratrel)
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declare rat.forall_transfer [transfer_rule del]
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lemma forall_rat_transfer [transfer_rule]: (* TODO: generate automatically *)
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  "(fun_rel (fun_rel cr_rat op =) op =)
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    (transfer_bforall (\<lambda>x. snd x \<noteq> 0)) transfer_forall"
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  using rat.forall_transfer by simp
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subsubsection {* Representation and basic operations *}
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lift_definition Fract :: "int \<Rightarrow> int \<Rightarrow> rat"
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  is "\<lambda>a b. if b = 0 then (0, 1) else (a, b)"
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  by simp
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lemma eq_rat:
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  shows "\<And>a b c d. b \<noteq> 0 \<Longrightarrow> d \<noteq> 0 \<Longrightarrow> Fract a b = Fract c d \<longleftrightarrow> a * d = c * b"
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  and "\<And>a. Fract a 0 = Fract 0 1"
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  and "\<And>a c. Fract 0 a = Fract 0 c"
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  by (transfer, simp)+
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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 transfer simp
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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 have "q = Fract (- ?a) (- ?b)" unfolding q by transfer simp
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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 :: field_inverse_zero
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begin
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lift_definition zero_rat :: "rat" is "(0, 1)"
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  by simp
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lift_definition one_rat :: "rat" is "(1, 1)"
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  by simp
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lemma Zero_rat_def: "0 = Fract 0 1"
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  by transfer simp
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lemma One_rat_def: "1 = Fract 1 1"
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  by transfer simp
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lift_definition plus_rat :: "rat \<Rightarrow> rat \<Rightarrow> rat"
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  is "\<lambda>x y. (fst x * snd y + fst y * snd x, snd x * snd y)"
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  by (clarsimp, simp add: left_distrib, simp add: mult_ac)
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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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  using assms by transfer simp
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lift_definition uminus_rat :: "rat \<Rightarrow> rat" is "\<lambda>x. (- fst x, snd x)"
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  by simp
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lemma minus_rat [simp]: "- Fract a b = Fract (- a) b"
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  by transfer simp
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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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lift_definition times_rat :: "rat \<Rightarrow> rat \<Rightarrow> rat"
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  is "\<lambda>x y. (fst x * fst y, snd x * snd y)"
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  by (simp add: mult_ac)
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lemma mult_rat [simp]: "Fract a b * Fract c d = Fract (a * c) (b * d)"
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  by transfer simp
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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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  using assms by transfer simp
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lift_definition inverse_rat :: "rat \<Rightarrow> rat"
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  is "\<lambda>x. if fst x = 0 then (0, 1) else (snd x, fst x)"
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  by (auto simp add: mult_commute)
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lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"
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  by transfer simp
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definition
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  divide_rat_def: "q / r = q * inverse (r::rat)"
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lemma divide_rat [simp]: "Fract a b / Fract c d = Fract (a * d) (b * c)"
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  by (simp add: divide_rat_def)
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instance proof
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  fix q r s :: rat
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  show "(q * r) * s = q * (r * s)"
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    by transfer simp
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  show "q * r = r * q"
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    by transfer simp
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  show "1 * q = q"
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    by transfer simp
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  show "(q + r) + s = q + (r + s)"
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    by transfer (simp add: algebra_simps)
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  show "q + r = r + q"
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    by transfer simp
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  show "0 + q = q"
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    by transfer simp
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  show "- q + q = 0"
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    by transfer simp
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  show "q - r = q + - r"
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    by (fact diff_rat_def)
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  show "(q + r) * s = q * s + r * s"
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    by transfer (simp add: algebra_simps)
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  show "(0::rat) \<noteq> 1"
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    by transfer simp
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  { assume "q \<noteq> 0" thus "inverse q * q = 1"
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    by transfer simp }
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  show "q / r = q * inverse r"
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    by (fact divide_rat_def)
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  show "inverse 0 = (0::rat)"
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    by transfer simp
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qed
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end
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lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1"
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  by (induct k) (simp_all add: Zero_rat_def One_rat_def)
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lemma of_int_rat: "of_int k = Fract k 1"
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  by (cases k rule: int_diff_cases) (simp add: of_nat_rat)
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lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
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  by (rule of_nat_rat [symmetric])
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lemma Fract_of_int_eq: "Fract k 1 = of_int k"
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  by (rule of_int_rat [symmetric])
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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 (numeral w) 1 = numeral w"
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  "Fract (neg_numeral w) 1 = neg_numeral w"
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  "Fract k 0 = 0"
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  using Fract_of_int_eq [of "numeral w"]
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  using Fract_of_int_eq [of "neg_numeral w"]
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  by (simp_all add: Zero_rat_def One_rat_def eq_rat)
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lemma rat_number_expand:
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  "0 = Fract 0 1"
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  "1 = Fract 1 1"
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  "numeral k = Fract (numeral k) 1"
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  "neg_numeral k = Fract (neg_numeral k) 1"
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  by (simp_all add: rat_number_collapse)
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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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  qed
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  from assms show ?thesis
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    by (auto simp add: normalize_def Let_def dvd_div_div_eq_mult mult_commute sgn_times split: if_splits intro: aux)
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qed
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lemma normalize_eq: "normalize (a, b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
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  by (auto simp add: normalize_def Let_def Fract_coprime dvd_div_neg rat_number_collapse
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    split:split_if_asm)
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lemma normalize_denom_pos: "normalize r = (p, q) \<Longrightarrow> q > 0"
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  by (auto simp add: normalize_def Let_def dvd_div_neg pos_imp_zdiv_neg_iff nonneg1_imp_zdiv_pos_iff
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    split:split_if_asm)
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lemma normalize_coprime: "normalize r = (p, q) \<Longrightarrow> coprime p q"
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  by (auto simp add: normalize_def Let_def dvd_div_neg div_gcd_coprime_int
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    split:split_if_asm)
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lemma normalize_stable [simp]:
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  "q > 0 \<Longrightarrow> coprime p q \<Longrightarrow> normalize (p, q) = (p, q)"
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  by (simp add: normalize_def)
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lemma normalize_denom_zero [simp]:
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  "normalize (p, 0) = (0, 1)"
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  by (simp add: normalize_def)
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lemma normalize_negative [simp]:
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  "q < 0 \<Longrightarrow> normalize (p, q) = normalize (- p, - q)"
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  by (simp add: normalize_def Let_def dvd_div_neg dvd_neg_div)
haftmann@35369
   307
haftmann@35369
   308
text{*
haftmann@35369
   309
  Decompose a fraction into normalized, i.e. coprime numerator and denominator:
haftmann@35369
   310
*}
haftmann@35369
   311
haftmann@35369
   312
definition quotient_of :: "rat \<Rightarrow> int \<times> int" where
haftmann@35369
   313
  "quotient_of x = (THE pair. x = Fract (fst pair) (snd pair) &
haftmann@35369
   314
                   snd pair > 0 & coprime (fst pair) (snd pair))"
haftmann@35369
   315
haftmann@35369
   316
lemma quotient_of_unique:
haftmann@35369
   317
  "\<exists>!p. r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
haftmann@35369
   318
proof (cases r)
haftmann@35369
   319
  case (Fract a b)
haftmann@35369
   320
  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
   321
  then show ?thesis proof (rule ex1I)
haftmann@35369
   322
    fix p
haftmann@35369
   323
    obtain c d :: int where p: "p = (c, d)" by (cases p)
haftmann@35369
   324
    assume "r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
haftmann@35369
   325
    with p have Fract': "r = Fract c d" "d > 0" "coprime c d" by simp_all
haftmann@35369
   326
    have "c = a \<and> d = b"
haftmann@35369
   327
    proof (cases "a = 0")
haftmann@35369
   328
      case True with Fract Fract' show ?thesis by (simp add: eq_rat)
haftmann@35369
   329
    next
haftmann@35369
   330
      case False
haftmann@35369
   331
      with Fract Fract' have *: "c * b = a * d" and "c \<noteq> 0" by (auto simp add: eq_rat)
haftmann@35369
   332
      then have "c * b > 0 \<longleftrightarrow> a * d > 0" by auto
haftmann@35369
   333
      with `b > 0` `d > 0` have "a > 0 \<longleftrightarrow> c > 0" by (simp add: zero_less_mult_iff)
haftmann@35369
   334
      with `a \<noteq> 0` `c \<noteq> 0` have sgn: "sgn a = sgn c" by (auto simp add: not_less)
haftmann@35369
   335
      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
   336
        by (simp add: coprime_crossproduct_int)
haftmann@35369
   337
      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
   338
      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
   339
      with sgn * show ?thesis by (auto simp add: sgn_0_0)
nipkow@33805
   340
    qed
haftmann@35369
   341
    with p show "p = (a, b)" by simp
nipkow@33805
   342
  qed
nipkow@33805
   343
qed
nipkow@33805
   344
haftmann@35369
   345
lemma quotient_of_Fract [code]:
haftmann@35369
   346
  "quotient_of (Fract a b) = normalize (a, b)"
haftmann@35369
   347
proof -
haftmann@35369
   348
  have "Fract a b = Fract (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?Fract)
haftmann@35369
   349
    by (rule sym) (auto intro: normalize_eq)
haftmann@35369
   350
  moreover have "0 < snd (normalize (a, b))" (is ?denom_pos) 
haftmann@35369
   351
    by (cases "normalize (a, b)") (rule normalize_denom_pos, simp)
haftmann@35369
   352
  moreover have "coprime (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?coprime)
haftmann@35369
   353
    by (rule normalize_coprime) simp
haftmann@35369
   354
  ultimately have "?Fract \<and> ?denom_pos \<and> ?coprime" by blast
haftmann@35369
   355
  with quotient_of_unique have
haftmann@35369
   356
    "(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
   357
    by (rule the1_equality)
haftmann@35369
   358
  then show ?thesis by (simp add: quotient_of_def)
haftmann@35369
   359
qed
haftmann@35369
   360
haftmann@35369
   361
lemma quotient_of_number [simp]:
haftmann@35369
   362
  "quotient_of 0 = (0, 1)"
haftmann@35369
   363
  "quotient_of 1 = (1, 1)"
huffman@47108
   364
  "quotient_of (numeral k) = (numeral k, 1)"
huffman@47108
   365
  "quotient_of (neg_numeral k) = (neg_numeral k, 1)"
haftmann@35369
   366
  by (simp_all add: rat_number_expand quotient_of_Fract)
nipkow@33805
   367
haftmann@35369
   368
lemma quotient_of_eq: "quotient_of (Fract a b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
haftmann@35369
   369
  by (simp add: quotient_of_Fract normalize_eq)
haftmann@35369
   370
haftmann@35369
   371
lemma quotient_of_denom_pos: "quotient_of r = (p, q) \<Longrightarrow> q > 0"
haftmann@35369
   372
  by (cases r) (simp add: quotient_of_Fract normalize_denom_pos)
haftmann@35369
   373
haftmann@35369
   374
lemma quotient_of_coprime: "quotient_of r = (p, q) \<Longrightarrow> coprime p q"
haftmann@35369
   375
  by (cases r) (simp add: quotient_of_Fract normalize_coprime)
nipkow@33805
   376
haftmann@35369
   377
lemma quotient_of_inject:
haftmann@35369
   378
  assumes "quotient_of a = quotient_of b"
haftmann@35369
   379
  shows "a = b"
haftmann@35369
   380
proof -
haftmann@35369
   381
  obtain p q r s where a: "a = Fract p q"
haftmann@35369
   382
    and b: "b = Fract r s"
haftmann@35369
   383
    and "q > 0" and "s > 0" by (cases a, cases b)
haftmann@35369
   384
  with assms show ?thesis by (simp add: eq_rat quotient_of_Fract normalize_crossproduct)
haftmann@35369
   385
qed
haftmann@35369
   386
haftmann@35369
   387
lemma quotient_of_inject_eq:
haftmann@35369
   388
  "quotient_of a = quotient_of b \<longleftrightarrow> a = b"
haftmann@35369
   389
  by (auto simp add: quotient_of_inject)
nipkow@33805
   390
haftmann@27551
   391
haftmann@27551
   392
subsubsection {* Various *}
haftmann@27551
   393
haftmann@27551
   394
lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"
haftmann@27652
   395
  by (simp add: Fract_of_int_eq [symmetric])
haftmann@27551
   396
huffman@47108
   397
lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
huffman@47108
   398
  by (simp add: rat_number_expand)
haftmann@27551
   399
haftmann@27551
   400
haftmann@27551
   401
subsubsection {* The ordered field of rational numbers *}
huffman@27509
   402
huffman@27509
   403
instantiation rat :: linorder
huffman@27509
   404
begin
huffman@27509
   405
huffman@47906
   406
lift_definition less_eq_rat :: "rat \<Rightarrow> rat \<Rightarrow> bool"
huffman@47906
   407
  is "\<lambda>x y. (fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)"
huffman@47906
   408
proof (clarsimp)
huffman@47906
   409
  fix a b a' b' c d c' d'::int
huffman@47906
   410
  assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
huffman@47906
   411
  assume eq1: "a * b' = a' * b"
huffman@47906
   412
  assume eq2: "c * d' = c' * d"
huffman@47906
   413
huffman@47906
   414
  let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
huffman@47906
   415
  {
huffman@47906
   416
    fix a b c d x :: int assume x: "x \<noteq> 0"
huffman@47906
   417
    have "?le a b c d = ?le (a * x) (b * x) c d"
huffman@47906
   418
    proof -
huffman@47906
   419
      from x have "0 < x * x" by (auto simp add: zero_less_mult_iff)
huffman@47906
   420
      hence "?le a b c d =
huffman@47906
   421
        ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
huffman@47906
   422
        by (simp add: mult_le_cancel_right)
huffman@47906
   423
      also have "... = ?le (a * x) (b * x) c d"
huffman@47906
   424
        by (simp add: mult_ac)
huffman@47906
   425
      finally show ?thesis .
huffman@47906
   426
    qed
huffman@47906
   427
  } note le_factor = this
huffman@47906
   428
  
huffman@47906
   429
  let ?D = "b * d" and ?D' = "b' * d'"
huffman@47906
   430
  from neq have D: "?D \<noteq> 0" by simp
huffman@47906
   431
  from neq have "?D' \<noteq> 0" by simp
huffman@47906
   432
  hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"
huffman@47906
   433
    by (rule le_factor)
huffman@47906
   434
  also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')" 
huffman@47906
   435
    by (simp add: mult_ac)
huffman@47906
   436
  also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
huffman@47906
   437
    by (simp only: eq1 eq2)
huffman@47906
   438
  also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
huffman@47906
   439
    by (simp add: mult_ac)
huffman@47906
   440
  also from D have "... = ?le a' b' c' d'"
huffman@47906
   441
    by (rule le_factor [symmetric])
huffman@47906
   442
  finally show "?le a b c d = ?le a' b' c' d'" .
huffman@47906
   443
qed
haftmann@27551
   444
haftmann@27652
   445
lemma le_rat [simp]:
haftmann@27551
   446
  assumes "b \<noteq> 0" and "d \<noteq> 0"
haftmann@27551
   447
  shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
huffman@47906
   448
  using assms by transfer simp
huffman@27509
   449
huffman@27509
   450
definition
haftmann@35369
   451
  less_rat_def: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
huffman@27509
   452
haftmann@27652
   453
lemma less_rat [simp]:
haftmann@27551
   454
  assumes "b \<noteq> 0" and "d \<noteq> 0"
haftmann@27551
   455
  shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
haftmann@27652
   456
  using assms by (simp add: less_rat_def eq_rat order_less_le)
huffman@27509
   457
huffman@27509
   458
instance proof
paulson@14365
   459
  fix q r s :: rat
paulson@14365
   460
  {
paulson@14365
   461
    assume "q \<le> r" and "r \<le> s"
haftmann@35369
   462
    then show "q \<le> s" 
haftmann@35369
   463
    proof (induct q, induct r, induct s)
paulson@14365
   464
      fix a b c d e f :: int
haftmann@35369
   465
      assume neq: "b > 0"  "d > 0"  "f > 0"
paulson@14365
   466
      assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"
paulson@14365
   467
      show "Fract a b \<le> Fract e f"
paulson@14365
   468
      proof -
paulson@14365
   469
        from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
paulson@14365
   470
          by (auto simp add: zero_less_mult_iff linorder_neq_iff)
paulson@14365
   471
        have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
paulson@14365
   472
        proof -
paulson@14365
   473
          from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27652
   474
            by simp
paulson@14365
   475
          with ff show ?thesis by (simp add: mult_le_cancel_right)
paulson@14365
   476
        qed
chaieb@27668
   477
        also have "... = (c * f) * (d * f) * (b * b)" by algebra
paulson@14365
   478
        also have "... \<le> (e * d) * (d * f) * (b * b)"
paulson@14365
   479
        proof -
paulson@14365
   480
          from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
haftmann@27652
   481
            by simp
paulson@14365
   482
          with bb show ?thesis by (simp add: mult_le_cancel_right)
paulson@14365
   483
        qed
paulson@14365
   484
        finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
paulson@14365
   485
          by (simp only: mult_ac)
paulson@14365
   486
        with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
paulson@14365
   487
          by (simp add: mult_le_cancel_right)
haftmann@27652
   488
        with neq show ?thesis by simp
paulson@14365
   489
      qed
paulson@14365
   490
    qed
paulson@14365
   491
  next
paulson@14365
   492
    assume "q \<le> r" and "r \<le> q"
haftmann@35369
   493
    then show "q = r"
haftmann@35369
   494
    proof (induct q, induct r)
paulson@14365
   495
      fix a b c d :: int
haftmann@35369
   496
      assume neq: "b > 0"  "d > 0"
paulson@14365
   497
      assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"
paulson@14365
   498
      show "Fract a b = Fract c d"
paulson@14365
   499
      proof -
paulson@14365
   500
        from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27652
   501
          by simp
paulson@14365
   502
        also have "... \<le> (a * d) * (b * d)"
paulson@14365
   503
        proof -
paulson@14365
   504
          from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
haftmann@27652
   505
            by simp
paulson@14365
   506
          thus ?thesis by (simp only: mult_ac)
paulson@14365
   507
        qed
paulson@14365
   508
        finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
paulson@14365
   509
        moreover from neq have "b * d \<noteq> 0" by simp
paulson@14365
   510
        ultimately have "a * d = c * b" by simp
paulson@14365
   511
        with neq show ?thesis by (simp add: eq_rat)
paulson@14365
   512
      qed
paulson@14365
   513
    qed
paulson@14365
   514
  next
paulson@14365
   515
    show "q \<le> q"
haftmann@27652
   516
      by (induct q) simp
haftmann@27682
   517
    show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)"
haftmann@27682
   518
      by (induct q, induct r) (auto simp add: le_less mult_commute)
paulson@14365
   519
    show "q \<le> r \<or> r \<le> q"
huffman@18913
   520
      by (induct q, induct r)
haftmann@27652
   521
         (simp add: mult_commute, rule linorder_linear)
paulson@14365
   522
  }
paulson@14365
   523
qed
paulson@14365
   524
huffman@27509
   525
end
huffman@27509
   526
haftmann@27551
   527
instantiation rat :: "{distrib_lattice, abs_if, sgn_if}"
haftmann@25571
   528
begin
haftmann@25571
   529
haftmann@25571
   530
definition
haftmann@35369
   531
  abs_rat_def: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))"
haftmann@27551
   532
haftmann@27652
   533
lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
huffman@35216
   534
  by (auto simp add: abs_rat_def zabs_def Zero_rat_def not_less le_less eq_rat zero_less_mult_iff)
haftmann@27551
   535
haftmann@27551
   536
definition
haftmann@35369
   537
  sgn_rat_def: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
haftmann@27551
   538
haftmann@27652
   539
lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)"
haftmann@27551
   540
  unfolding Fract_of_int_eq
haftmann@27652
   541
  by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat)
haftmann@27551
   542
    (auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)
haftmann@27551
   543
haftmann@27551
   544
definition
haftmann@25571
   545
  "(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min"
haftmann@25571
   546
haftmann@25571
   547
definition
haftmann@25571
   548
  "(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max"
haftmann@25571
   549
haftmann@27551
   550
instance by intro_classes
haftmann@27551
   551
  (auto simp add: abs_rat_def sgn_rat_def min_max.sup_inf_distrib1 inf_rat_def sup_rat_def)
haftmann@22456
   552
haftmann@25571
   553
end
haftmann@25571
   554
haftmann@36409
   555
instance rat :: linordered_field_inverse_zero
haftmann@27551
   556
proof
paulson@14365
   557
  fix q r s :: rat
paulson@14365
   558
  show "q \<le> r ==> s + q \<le> s + r"
paulson@14365
   559
  proof (induct q, induct r, induct s)
paulson@14365
   560
    fix a b c d e f :: int
haftmann@35369
   561
    assume neq: "b > 0"  "d > 0"  "f > 0"
paulson@14365
   562
    assume le: "Fract a b \<le> Fract c d"
paulson@14365
   563
    show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
paulson@14365
   564
    proof -
paulson@14365
   565
      let ?F = "f * f" from neq have F: "0 < ?F"
paulson@14365
   566
        by (auto simp add: zero_less_mult_iff)
paulson@14365
   567
      from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
haftmann@27652
   568
        by simp
paulson@14365
   569
      with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
paulson@14365
   570
        by (simp add: mult_le_cancel_right)
haftmann@27652
   571
      with neq show ?thesis by (simp add: mult_ac int_distrib)
paulson@14365
   572
    qed
paulson@14365
   573
  qed
paulson@14365
   574
  show "q < r ==> 0 < s ==> s * q < s * r"
paulson@14365
   575
  proof (induct q, induct r, induct s)
paulson@14365
   576
    fix a b c d e f :: int
haftmann@35369
   577
    assume neq: "b > 0"  "d > 0"  "f > 0"
paulson@14365
   578
    assume le: "Fract a b < Fract c d"
paulson@14365
   579
    assume gt: "0 < Fract e f"
paulson@14365
   580
    show "Fract e f * Fract a b < Fract e f * Fract c d"
paulson@14365
   581
    proof -
paulson@14365
   582
      let ?E = "e * f" and ?F = "f * f"
paulson@14365
   583
      from neq gt have "0 < ?E"
haftmann@27652
   584
        by (auto simp add: Zero_rat_def order_less_le eq_rat)
paulson@14365
   585
      moreover from neq have "0 < ?F"
paulson@14365
   586
        by (auto simp add: zero_less_mult_iff)
paulson@14365
   587
      moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
haftmann@27652
   588
        by simp
paulson@14365
   589
      ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
paulson@14365
   590
        by (simp add: mult_less_cancel_right)
paulson@14365
   591
      with neq show ?thesis
haftmann@27652
   592
        by (simp add: mult_ac)
paulson@14365
   593
    qed
paulson@14365
   594
  qed
haftmann@27551
   595
qed auto
paulson@14365
   596
haftmann@27551
   597
lemma Rat_induct_pos [case_names Fract, induct type: rat]:
haftmann@27551
   598
  assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
haftmann@27551
   599
  shows "P q"
paulson@14365
   600
proof (cases q)
haftmann@27551
   601
  have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)"
paulson@14365
   602
  proof -
paulson@14365
   603
    fix a::int and b::int
paulson@14365
   604
    assume b: "b < 0"
paulson@14365
   605
    hence "0 < -b" by simp
paulson@14365
   606
    hence "P (Fract (-a) (-b))" by (rule step)
paulson@14365
   607
    thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
paulson@14365
   608
  qed
paulson@14365
   609
  case (Fract a b)
paulson@14365
   610
  thus "P q" by (force simp add: linorder_neq_iff step step')
paulson@14365
   611
qed
paulson@14365
   612
paulson@14365
   613
lemma zero_less_Fract_iff:
huffman@30095
   614
  "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a"
huffman@30095
   615
  by (simp add: Zero_rat_def zero_less_mult_iff)
huffman@30095
   616
huffman@30095
   617
lemma Fract_less_zero_iff:
huffman@30095
   618
  "0 < b \<Longrightarrow> Fract a b < 0 \<longleftrightarrow> a < 0"
huffman@30095
   619
  by (simp add: Zero_rat_def mult_less_0_iff)
huffman@30095
   620
huffman@30095
   621
lemma zero_le_Fract_iff:
huffman@30095
   622
  "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a"
huffman@30095
   623
  by (simp add: Zero_rat_def zero_le_mult_iff)
huffman@30095
   624
huffman@30095
   625
lemma Fract_le_zero_iff:
huffman@30095
   626
  "0 < b \<Longrightarrow> Fract a b \<le> 0 \<longleftrightarrow> a \<le> 0"
huffman@30095
   627
  by (simp add: Zero_rat_def mult_le_0_iff)
huffman@30095
   628
huffman@30095
   629
lemma one_less_Fract_iff:
huffman@30095
   630
  "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a"
huffman@30095
   631
  by (simp add: One_rat_def mult_less_cancel_right_disj)
huffman@30095
   632
huffman@30095
   633
lemma Fract_less_one_iff:
huffman@30095
   634
  "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b"
huffman@30095
   635
  by (simp add: One_rat_def mult_less_cancel_right_disj)
huffman@30095
   636
huffman@30095
   637
lemma one_le_Fract_iff:
huffman@30095
   638
  "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a"
huffman@30095
   639
  by (simp add: One_rat_def mult_le_cancel_right)
huffman@30095
   640
huffman@30095
   641
lemma Fract_le_one_iff:
huffman@30095
   642
  "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b"
huffman@30095
   643
  by (simp add: One_rat_def mult_le_cancel_right)
paulson@14365
   644
paulson@14378
   645
huffman@30097
   646
subsubsection {* Rationals are an Archimedean field *}
huffman@30097
   647
huffman@30097
   648
lemma rat_floor_lemma:
huffman@30097
   649
  shows "of_int (a div b) \<le> Fract a b \<and> Fract a b < of_int (a div b + 1)"
huffman@30097
   650
proof -
huffman@30097
   651
  have "Fract a b = of_int (a div b) + Fract (a mod b) b"
huffman@35293
   652
    by (cases "b = 0", simp, simp add: of_int_rat)
huffman@30097
   653
  moreover have "0 \<le> Fract (a mod b) b \<and> Fract (a mod b) b < 1"
huffman@35293
   654
    unfolding Fract_of_int_quotient
haftmann@36409
   655
    by (rule linorder_cases [of b 0]) (simp add: divide_nonpos_neg, simp, simp add: divide_nonneg_pos)
huffman@30097
   656
  ultimately show ?thesis by simp
huffman@30097
   657
qed
huffman@30097
   658
huffman@30097
   659
instance rat :: archimedean_field
huffman@30097
   660
proof
huffman@30097
   661
  fix r :: rat
huffman@30097
   662
  show "\<exists>z. r \<le> of_int z"
huffman@30097
   663
  proof (induct r)
huffman@30097
   664
    case (Fract a b)
huffman@35293
   665
    have "Fract a b \<le> of_int (a div b + 1)"
huffman@35293
   666
      using rat_floor_lemma [of a b] by simp
huffman@30097
   667
    then show "\<exists>z. Fract a b \<le> of_int z" ..
huffman@30097
   668
  qed
huffman@30097
   669
qed
huffman@30097
   670
bulwahn@43732
   671
instantiation rat :: floor_ceiling
bulwahn@43732
   672
begin
bulwahn@43732
   673
bulwahn@43732
   674
definition [code del]:
bulwahn@43732
   675
  "floor (x::rat) = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))"
bulwahn@43732
   676
bulwahn@43732
   677
instance proof
bulwahn@43732
   678
  fix x :: rat
bulwahn@43732
   679
  show "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
bulwahn@43732
   680
    unfolding floor_rat_def using floor_exists1 by (rule theI')
bulwahn@43732
   681
qed
bulwahn@43732
   682
bulwahn@43732
   683
end
bulwahn@43732
   684
huffman@35293
   685
lemma floor_Fract: "floor (Fract a b) = a div b"
huffman@35293
   686
  using rat_floor_lemma [of a b]
huffman@30097
   687
  by (simp add: floor_unique)
huffman@30097
   688
huffman@30097
   689
haftmann@31100
   690
subsection {* Linear arithmetic setup *}
paulson@14387
   691
haftmann@31100
   692
declaration {*
haftmann@31100
   693
  K (Lin_Arith.add_inj_thms [@{thm of_nat_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
haftmann@31100
   694
    (* not needed because x < (y::nat) can be rewritten as Suc x <= y: of_nat_less_iff RS iffD2 *)
haftmann@31100
   695
  #> Lin_Arith.add_inj_thms [@{thm of_int_le_iff} RS iffD2, @{thm of_int_eq_iff} RS iffD2]
haftmann@31100
   696
    (* not needed because x < (y::int) can be rewritten as x + 1 <= y: of_int_less_iff RS iffD2 *)
haftmann@31100
   697
  #> Lin_Arith.add_simps [@{thm neg_less_iff_less},
haftmann@31100
   698
      @{thm True_implies_equals},
huffman@47108
   699
      read_instantiate @{context} [(("a", 0), "(numeral ?v)")] @{thm right_distrib},
huffman@47108
   700
      read_instantiate @{context} [(("a", 0), "(neg_numeral ?v)")] @{thm right_distrib},
haftmann@31100
   701
      @{thm divide_1}, @{thm divide_zero_left},
haftmann@31100
   702
      @{thm times_divide_eq_right}, @{thm times_divide_eq_left},
haftmann@31100
   703
      @{thm minus_divide_left} RS sym, @{thm minus_divide_right} RS sym,
haftmann@31100
   704
      @{thm of_int_minus}, @{thm of_int_diff},
haftmann@31100
   705
      @{thm of_int_of_nat_eq}]
haftmann@31100
   706
  #> Lin_Arith.add_simprocs Numeral_Simprocs.field_cancel_numeral_factors
haftmann@31100
   707
  #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat => rat"})
haftmann@31100
   708
  #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int => rat"}))
haftmann@31100
   709
*}
paulson@14387
   710
huffman@23342
   711
huffman@23342
   712
subsection {* Embedding from Rationals to other Fields *}
huffman@23342
   713
haftmann@24198
   714
class field_char_0 = field + ring_char_0
huffman@23342
   715
haftmann@35028
   716
subclass (in linordered_field) field_char_0 ..
huffman@23342
   717
haftmann@27551
   718
context field_char_0
haftmann@27551
   719
begin
haftmann@27551
   720
huffman@47906
   721
lift_definition of_rat :: "rat \<Rightarrow> 'a"
huffman@47906
   722
  is "\<lambda>x. of_int (fst x) / of_int (snd x)"
huffman@23342
   723
apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
huffman@23342
   724
apply (simp only: of_int_mult [symmetric])
huffman@23342
   725
done
huffman@23342
   726
huffman@47906
   727
end
huffman@47906
   728
haftmann@27551
   729
lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
huffman@47906
   730
  by transfer simp
huffman@23342
   731
huffman@23342
   732
lemma of_rat_0 [simp]: "of_rat 0 = 0"
huffman@47906
   733
  by transfer simp
huffman@23342
   734
huffman@23342
   735
lemma of_rat_1 [simp]: "of_rat 1 = 1"
huffman@47906
   736
  by transfer simp
huffman@23342
   737
huffman@23342
   738
lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
huffman@47906
   739
  by transfer (simp add: add_frac_eq)
huffman@23342
   740
huffman@23343
   741
lemma of_rat_minus: "of_rat (- a) = - of_rat a"
huffman@47906
   742
  by transfer simp
huffman@23343
   743
huffman@23343
   744
lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
huffman@23343
   745
by (simp only: diff_minus of_rat_add of_rat_minus)
huffman@23343
   746
huffman@23342
   747
lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
huffman@47906
   748
apply transfer
huffman@23342
   749
apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
huffman@23342
   750
done
huffman@23342
   751
huffman@23342
   752
lemma nonzero_of_rat_inverse:
huffman@23342
   753
  "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
huffman@23343
   754
apply (rule inverse_unique [symmetric])
huffman@23343
   755
apply (simp add: of_rat_mult [symmetric])
huffman@23342
   756
done
huffman@23342
   757
huffman@23342
   758
lemma of_rat_inverse:
haftmann@36409
   759
  "(of_rat (inverse a)::'a::{field_char_0, field_inverse_zero}) =
huffman@23342
   760
   inverse (of_rat a)"
huffman@23342
   761
by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
huffman@23342
   762
huffman@23342
   763
lemma nonzero_of_rat_divide:
huffman@23342
   764
  "b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
huffman@23342
   765
by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
huffman@23342
   766
huffman@23342
   767
lemma of_rat_divide:
haftmann@36409
   768
  "(of_rat (a / b)::'a::{field_char_0, field_inverse_zero})
huffman@23342
   769
   = of_rat a / of_rat b"
haftmann@27652
   770
by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)
huffman@23342
   771
huffman@23343
   772
lemma of_rat_power:
haftmann@31017
   773
  "(of_rat (a ^ n)::'a::field_char_0) = of_rat a ^ n"
huffman@30273
   774
by (induct n) (simp_all add: of_rat_mult)
huffman@23343
   775
huffman@23343
   776
lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
huffman@47906
   777
apply transfer
huffman@23343
   778
apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
huffman@23343
   779
apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
huffman@23343
   780
done
huffman@23343
   781
haftmann@27652
   782
lemma of_rat_less:
haftmann@35028
   783
  "(of_rat r :: 'a::linordered_field) < of_rat s \<longleftrightarrow> r < s"
haftmann@27652
   784
proof (induct r, induct s)
haftmann@27652
   785
  fix a b c d :: int
haftmann@27652
   786
  assume not_zero: "b > 0" "d > 0"
haftmann@27652
   787
  then have "b * d > 0" by (rule mult_pos_pos)
haftmann@27652
   788
  have of_int_divide_less_eq:
haftmann@27652
   789
    "(of_int a :: 'a) / of_int b < of_int c / of_int d
haftmann@27652
   790
      \<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b"
haftmann@27652
   791
    using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
haftmann@35028
   792
  show "(of_rat (Fract a b) :: 'a::linordered_field) < of_rat (Fract c d)
haftmann@27652
   793
    \<longleftrightarrow> Fract a b < Fract c d"
haftmann@27652
   794
    using not_zero `b * d > 0`
haftmann@27652
   795
    by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult)
haftmann@27652
   796
qed
haftmann@27652
   797
haftmann@27652
   798
lemma of_rat_less_eq:
haftmann@35028
   799
  "(of_rat r :: 'a::linordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s"
haftmann@27652
   800
  unfolding le_less by (auto simp add: of_rat_less)
haftmann@27652
   801
huffman@23343
   802
lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
huffman@23343
   803
haftmann@27652
   804
lemma of_rat_eq_id [simp]: "of_rat = id"
huffman@23343
   805
proof
huffman@23343
   806
  fix a
huffman@23343
   807
  show "of_rat a = id a"
huffman@23343
   808
  by (induct a)
haftmann@27652
   809
     (simp add: of_rat_rat Fract_of_int_eq [symmetric])
huffman@23343
   810
qed
huffman@23343
   811
huffman@23343
   812
text{*Collapse nested embeddings*}
huffman@23343
   813
lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
huffman@23343
   814
by (induct n) (simp_all add: of_rat_add)
huffman@23343
   815
huffman@23343
   816
lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
haftmann@27652
   817
by (cases z rule: int_diff_cases) (simp add: of_rat_diff)
huffman@23343
   818
huffman@47108
   819
lemma of_rat_numeral_eq [simp]:
huffman@47108
   820
  "of_rat (numeral w) = numeral w"
huffman@47108
   821
using of_rat_of_int_eq [of "numeral w"] by simp
huffman@47108
   822
huffman@47108
   823
lemma of_rat_neg_numeral_eq [simp]:
huffman@47108
   824
  "of_rat (neg_numeral w) = neg_numeral w"
huffman@47108
   825
using of_rat_of_int_eq [of "neg_numeral w"] by simp
huffman@23343
   826
haftmann@23879
   827
lemmas zero_rat = Zero_rat_def
haftmann@23879
   828
lemmas one_rat = One_rat_def
haftmann@23879
   829
haftmann@24198
   830
abbreviation
haftmann@24198
   831
  rat_of_nat :: "nat \<Rightarrow> rat"
haftmann@24198
   832
where
haftmann@24198
   833
  "rat_of_nat \<equiv> of_nat"
haftmann@24198
   834
haftmann@24198
   835
abbreviation
haftmann@24198
   836
  rat_of_int :: "int \<Rightarrow> rat"
haftmann@24198
   837
where
haftmann@24198
   838
  "rat_of_int \<equiv> of_int"
haftmann@24198
   839
huffman@28010
   840
subsection {* The Set of Rational Numbers *}
berghofe@24533
   841
nipkow@28001
   842
context field_char_0
nipkow@28001
   843
begin
nipkow@28001
   844
nipkow@28001
   845
definition
nipkow@28001
   846
  Rats  :: "'a set" where
haftmann@35369
   847
  "Rats = range of_rat"
nipkow@28001
   848
nipkow@28001
   849
notation (xsymbols)
nipkow@28001
   850
  Rats  ("\<rat>")
nipkow@28001
   851
nipkow@28001
   852
end
nipkow@28001
   853
huffman@28010
   854
lemma Rats_of_rat [simp]: "of_rat r \<in> Rats"
huffman@28010
   855
by (simp add: Rats_def)
huffman@28010
   856
huffman@28010
   857
lemma Rats_of_int [simp]: "of_int z \<in> Rats"
huffman@28010
   858
by (subst of_rat_of_int_eq [symmetric], rule Rats_of_rat)
huffman@28010
   859
huffman@28010
   860
lemma Rats_of_nat [simp]: "of_nat n \<in> Rats"
huffman@28010
   861
by (subst of_rat_of_nat_eq [symmetric], rule Rats_of_rat)
huffman@28010
   862
huffman@47108
   863
lemma Rats_number_of [simp]: "numeral w \<in> Rats"
huffman@47108
   864
by (subst of_rat_numeral_eq [symmetric], rule Rats_of_rat)
huffman@47108
   865
huffman@47108
   866
lemma Rats_neg_number_of [simp]: "neg_numeral w \<in> Rats"
huffman@47108
   867
by (subst of_rat_neg_numeral_eq [symmetric], rule Rats_of_rat)
huffman@28010
   868
huffman@28010
   869
lemma Rats_0 [simp]: "0 \<in> Rats"
huffman@28010
   870
apply (unfold Rats_def)
huffman@28010
   871
apply (rule range_eqI)
huffman@28010
   872
apply (rule of_rat_0 [symmetric])
huffman@28010
   873
done
huffman@28010
   874
huffman@28010
   875
lemma Rats_1 [simp]: "1 \<in> Rats"
huffman@28010
   876
apply (unfold Rats_def)
huffman@28010
   877
apply (rule range_eqI)
huffman@28010
   878
apply (rule of_rat_1 [symmetric])
huffman@28010
   879
done
huffman@28010
   880
huffman@28010
   881
lemma Rats_add [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a + b \<in> Rats"
huffman@28010
   882
apply (auto simp add: Rats_def)
huffman@28010
   883
apply (rule range_eqI)
huffman@28010
   884
apply (rule of_rat_add [symmetric])
huffman@28010
   885
done
huffman@28010
   886
huffman@28010
   887
lemma Rats_minus [simp]: "a \<in> Rats \<Longrightarrow> - a \<in> Rats"
huffman@28010
   888
apply (auto simp add: Rats_def)
huffman@28010
   889
apply (rule range_eqI)
huffman@28010
   890
apply (rule of_rat_minus [symmetric])
huffman@28010
   891
done
huffman@28010
   892
huffman@28010
   893
lemma Rats_diff [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a - b \<in> Rats"
huffman@28010
   894
apply (auto simp add: Rats_def)
huffman@28010
   895
apply (rule range_eqI)
huffman@28010
   896
apply (rule of_rat_diff [symmetric])
huffman@28010
   897
done
huffman@28010
   898
huffman@28010
   899
lemma Rats_mult [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a * b \<in> Rats"
huffman@28010
   900
apply (auto simp add: Rats_def)
huffman@28010
   901
apply (rule range_eqI)
huffman@28010
   902
apply (rule of_rat_mult [symmetric])
huffman@28010
   903
done
huffman@28010
   904
huffman@28010
   905
lemma nonzero_Rats_inverse:
huffman@28010
   906
  fixes a :: "'a::field_char_0"
huffman@28010
   907
  shows "\<lbrakk>a \<in> Rats; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Rats"
huffman@28010
   908
apply (auto simp add: Rats_def)
huffman@28010
   909
apply (rule range_eqI)
huffman@28010
   910
apply (erule nonzero_of_rat_inverse [symmetric])
huffman@28010
   911
done
huffman@28010
   912
huffman@28010
   913
lemma Rats_inverse [simp]:
haftmann@36409
   914
  fixes a :: "'a::{field_char_0, field_inverse_zero}"
huffman@28010
   915
  shows "a \<in> Rats \<Longrightarrow> inverse a \<in> Rats"
huffman@28010
   916
apply (auto simp add: Rats_def)
huffman@28010
   917
apply (rule range_eqI)
huffman@28010
   918
apply (rule of_rat_inverse [symmetric])
huffman@28010
   919
done
huffman@28010
   920
huffman@28010
   921
lemma nonzero_Rats_divide:
huffman@28010
   922
  fixes a b :: "'a::field_char_0"
huffman@28010
   923
  shows "\<lbrakk>a \<in> Rats; b \<in> Rats; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
huffman@28010
   924
apply (auto simp add: Rats_def)
huffman@28010
   925
apply (rule range_eqI)
huffman@28010
   926
apply (erule nonzero_of_rat_divide [symmetric])
huffman@28010
   927
done
huffman@28010
   928
huffman@28010
   929
lemma Rats_divide [simp]:
haftmann@36409
   930
  fixes a b :: "'a::{field_char_0, field_inverse_zero}"
huffman@28010
   931
  shows "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
huffman@28010
   932
apply (auto simp add: Rats_def)
huffman@28010
   933
apply (rule range_eqI)
huffman@28010
   934
apply (rule of_rat_divide [symmetric])
huffman@28010
   935
done
huffman@28010
   936
huffman@28010
   937
lemma Rats_power [simp]:
haftmann@31017
   938
  fixes a :: "'a::field_char_0"
huffman@28010
   939
  shows "a \<in> Rats \<Longrightarrow> a ^ n \<in> Rats"
huffman@28010
   940
apply (auto simp add: Rats_def)
huffman@28010
   941
apply (rule range_eqI)
huffman@28010
   942
apply (rule of_rat_power [symmetric])
huffman@28010
   943
done
huffman@28010
   944
huffman@28010
   945
lemma Rats_cases [cases set: Rats]:
huffman@28010
   946
  assumes "q \<in> \<rat>"
huffman@28010
   947
  obtains (of_rat) r where "q = of_rat r"
huffman@28010
   948
proof -
huffman@28010
   949
  from `q \<in> \<rat>` have "q \<in> range of_rat" unfolding Rats_def .
huffman@28010
   950
  then obtain r where "q = of_rat r" ..
huffman@28010
   951
  then show thesis ..
huffman@28010
   952
qed
huffman@28010
   953
huffman@28010
   954
lemma Rats_induct [case_names of_rat, induct set: Rats]:
huffman@28010
   955
  "q \<in> \<rat> \<Longrightarrow> (\<And>r. P (of_rat r)) \<Longrightarrow> P q"
huffman@28010
   956
  by (rule Rats_cases) auto
huffman@28010
   957
nipkow@28001
   958
berghofe@24533
   959
subsection {* Implementation of rational numbers as pairs of integers *}
berghofe@24533
   960
huffman@47108
   961
text {* Formal constructor *}
huffman@47108
   962
haftmann@35369
   963
definition Frct :: "int \<times> int \<Rightarrow> rat" where
haftmann@35369
   964
  [simp]: "Frct p = Fract (fst p) (snd p)"
haftmann@35369
   965
haftmann@36112
   966
lemma [code abstype]:
haftmann@36112
   967
  "Frct (quotient_of q) = q"
haftmann@36112
   968
  by (cases q) (auto intro: quotient_of_eq)
haftmann@35369
   969
huffman@47108
   970
huffman@47108
   971
text {* Numerals *}
haftmann@35369
   972
haftmann@35369
   973
declare quotient_of_Fract [code abstract]
haftmann@35369
   974
huffman@47108
   975
definition of_int :: "int \<Rightarrow> rat"
huffman@47108
   976
where
huffman@47108
   977
  [code_abbrev]: "of_int = Int.of_int"
huffman@47108
   978
hide_const (open) of_int
huffman@47108
   979
huffman@47108
   980
lemma quotient_of_int [code abstract]:
huffman@47108
   981
  "quotient_of (Rat.of_int a) = (a, 1)"
huffman@47108
   982
  by (simp add: of_int_def of_int_rat quotient_of_Fract)
huffman@47108
   983
huffman@47108
   984
lemma [code_unfold]:
huffman@47108
   985
  "numeral k = Rat.of_int (numeral k)"
huffman@47108
   986
  by (simp add: Rat.of_int_def)
huffman@47108
   987
huffman@47108
   988
lemma [code_unfold]:
huffman@47108
   989
  "neg_numeral k = Rat.of_int (neg_numeral k)"
huffman@47108
   990
  by (simp add: Rat.of_int_def)
huffman@47108
   991
huffman@47108
   992
lemma Frct_code_post [code_post]:
huffman@47108
   993
  "Frct (0, a) = 0"
huffman@47108
   994
  "Frct (a, 0) = 0"
huffman@47108
   995
  "Frct (1, 1) = 1"
huffman@47108
   996
  "Frct (numeral k, 1) = numeral k"
huffman@47108
   997
  "Frct (neg_numeral k, 1) = neg_numeral k"
huffman@47108
   998
  "Frct (1, numeral k) = 1 / numeral k"
huffman@47108
   999
  "Frct (1, neg_numeral k) = 1 / neg_numeral k"
huffman@47108
  1000
  "Frct (numeral k, numeral l) = numeral k / numeral l"
huffman@47108
  1001
  "Frct (numeral k, neg_numeral l) = numeral k / neg_numeral l"
huffman@47108
  1002
  "Frct (neg_numeral k, numeral l) = neg_numeral k / numeral l"
huffman@47108
  1003
  "Frct (neg_numeral k, neg_numeral l) = neg_numeral k / neg_numeral l"
huffman@47108
  1004
  by (simp_all add: Fract_of_int_quotient)
huffman@47108
  1005
huffman@47108
  1006
huffman@47108
  1007
text {* Operations *}
huffman@47108
  1008
haftmann@35369
  1009
lemma rat_zero_code [code abstract]:
haftmann@35369
  1010
  "quotient_of 0 = (0, 1)"
haftmann@35369
  1011
  by (simp add: Zero_rat_def quotient_of_Fract normalize_def)
haftmann@35369
  1012
haftmann@35369
  1013
lemma rat_one_code [code abstract]:
haftmann@35369
  1014
  "quotient_of 1 = (1, 1)"
haftmann@35369
  1015
  by (simp add: One_rat_def quotient_of_Fract normalize_def)
haftmann@35369
  1016
haftmann@35369
  1017
lemma rat_plus_code [code abstract]:
haftmann@35369
  1018
  "quotient_of (p + q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369
  1019
     in normalize (a * d + b * c, c * d))"
haftmann@35369
  1020
  by (cases p, cases q) (simp add: quotient_of_Fract)
haftmann@27652
  1021
haftmann@35369
  1022
lemma rat_uminus_code [code abstract]:
haftmann@35369
  1023
  "quotient_of (- p) = (let (a, b) = quotient_of p in (- a, b))"
haftmann@35369
  1024
  by (cases p) (simp add: quotient_of_Fract)
haftmann@35369
  1025
haftmann@35369
  1026
lemma rat_minus_code [code abstract]:
haftmann@35369
  1027
  "quotient_of (p - q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369
  1028
     in normalize (a * d - b * c, c * d))"
haftmann@35369
  1029
  by (cases p, cases q) (simp add: quotient_of_Fract)
haftmann@35369
  1030
haftmann@35369
  1031
lemma rat_times_code [code abstract]:
haftmann@35369
  1032
  "quotient_of (p * q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369
  1033
     in normalize (a * b, c * d))"
haftmann@35369
  1034
  by (cases p, cases q) (simp add: quotient_of_Fract)
berghofe@24533
  1035
haftmann@35369
  1036
lemma rat_inverse_code [code abstract]:
haftmann@35369
  1037
  "quotient_of (inverse p) = (let (a, b) = quotient_of p
haftmann@35369
  1038
    in if a = 0 then (0, 1) else (sgn a * b, \<bar>a\<bar>))"
haftmann@35369
  1039
proof (cases p)
haftmann@35369
  1040
  case (Fract a b) then show ?thesis
haftmann@35369
  1041
    by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract gcd_int.commute)
haftmann@35369
  1042
qed
haftmann@35369
  1043
haftmann@35369
  1044
lemma rat_divide_code [code abstract]:
haftmann@35369
  1045
  "quotient_of (p / q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369
  1046
     in normalize (a * d, c * b))"
haftmann@35369
  1047
  by (cases p, cases q) (simp add: quotient_of_Fract)
haftmann@35369
  1048
haftmann@35369
  1049
lemma rat_abs_code [code abstract]:
haftmann@35369
  1050
  "quotient_of \<bar>p\<bar> = (let (a, b) = quotient_of p in (\<bar>a\<bar>, b))"
haftmann@35369
  1051
  by (cases p) (simp add: quotient_of_Fract)
haftmann@35369
  1052
haftmann@35369
  1053
lemma rat_sgn_code [code abstract]:
haftmann@35369
  1054
  "quotient_of (sgn p) = (sgn (fst (quotient_of p)), 1)"
haftmann@35369
  1055
proof (cases p)
haftmann@35369
  1056
  case (Fract a b) then show ?thesis
haftmann@35369
  1057
  by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract)
haftmann@35369
  1058
qed
berghofe@24533
  1059
bulwahn@43733
  1060
lemma rat_floor_code [code]:
bulwahn@43733
  1061
  "floor p = (let (a, b) = quotient_of p in a div b)"
bulwahn@43733
  1062
by (cases p) (simp add: quotient_of_Fract floor_Fract)
bulwahn@43733
  1063
haftmann@38857
  1064
instantiation rat :: equal
haftmann@26513
  1065
begin
haftmann@26513
  1066
haftmann@35369
  1067
definition [code]:
haftmann@38857
  1068
  "HOL.equal a b \<longleftrightarrow> quotient_of a = quotient_of b"
haftmann@26513
  1069
haftmann@35369
  1070
instance proof
haftmann@38857
  1071
qed (simp add: equal_rat_def quotient_of_inject_eq)
haftmann@26513
  1072
haftmann@28351
  1073
lemma rat_eq_refl [code nbe]:
haftmann@38857
  1074
  "HOL.equal (r::rat) r \<longleftrightarrow> True"
haftmann@38857
  1075
  by (rule equal_refl)
haftmann@28351
  1076
haftmann@26513
  1077
end
berghofe@24533
  1078
haftmann@35369
  1079
lemma rat_less_eq_code [code]:
haftmann@35369
  1080
  "p \<le> q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d \<le> c * b)"
haftmann@35726
  1081
  by (cases p, cases q) (simp add: quotient_of_Fract mult.commute)
berghofe@24533
  1082
haftmann@35369
  1083
lemma rat_less_code [code]:
haftmann@35369
  1084
  "p < q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d < c * b)"
haftmann@35726
  1085
  by (cases p, cases q) (simp add: quotient_of_Fract mult.commute)
berghofe@24533
  1086
haftmann@35369
  1087
lemma [code]:
haftmann@35369
  1088
  "of_rat p = (let (a, b) = quotient_of p in of_int a / of_int b)"
haftmann@35369
  1089
  by (cases p) (simp add: quotient_of_Fract of_rat_rat)
haftmann@27652
  1090
huffman@47108
  1091
huffman@47108
  1092
text {* Quickcheck *}
huffman@47108
  1093
haftmann@31203
  1094
definition (in term_syntax)
haftmann@32657
  1095
  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
  1096
  [code_unfold]: "valterm_fract k l = Code_Evaluation.valtermify Fract {\<cdot>} k {\<cdot>} l"
haftmann@31203
  1097
haftmann@37751
  1098
notation fcomp (infixl "\<circ>>" 60)
haftmann@37751
  1099
notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@31203
  1100
haftmann@31203
  1101
instantiation rat :: random
haftmann@31203
  1102
begin
haftmann@31203
  1103
haftmann@31203
  1104
definition
haftmann@37751
  1105
  "Quickcheck.random i = Quickcheck.random i \<circ>\<rightarrow> (\<lambda>num. Random.range i \<circ>\<rightarrow> (\<lambda>denom. Pair (
haftmann@31205
  1106
     let j = Code_Numeral.int_of (denom + 1)
haftmann@32657
  1107
     in valterm_fract num (j, \<lambda>u. Code_Evaluation.term_of j))))"
haftmann@31203
  1108
haftmann@31203
  1109
instance ..
haftmann@31203
  1110
haftmann@31203
  1111
end
haftmann@31203
  1112
haftmann@37751
  1113
no_notation fcomp (infixl "\<circ>>" 60)
haftmann@37751
  1114
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@31203
  1115
bulwahn@41920
  1116
instantiation rat :: exhaustive
bulwahn@41231
  1117
begin
bulwahn@41231
  1118
bulwahn@41231
  1119
definition
bulwahn@45818
  1120
  "exhaustive_rat f d = Quickcheck_Exhaustive.exhaustive (%l. Quickcheck_Exhaustive.exhaustive (%k. f (Fract k (Code_Numeral.int_of l + 1))) d) d"
bulwahn@42311
  1121
bulwahn@42311
  1122
instance ..
bulwahn@42311
  1123
bulwahn@42311
  1124
end
bulwahn@42311
  1125
bulwahn@42311
  1126
instantiation rat :: full_exhaustive
bulwahn@42311
  1127
begin
bulwahn@42311
  1128
bulwahn@42311
  1129
definition
bulwahn@45818
  1130
  "full_exhaustive_rat f d = Quickcheck_Exhaustive.full_exhaustive (%(l, _). Quickcheck_Exhaustive.full_exhaustive (%k.
bulwahn@45507
  1131
     f (let j = Code_Numeral.int_of l + 1
bulwahn@45507
  1132
        in valterm_fract k (j, %_. Code_Evaluation.term_of j))) d) d"
bulwahn@41231
  1133
bulwahn@41231
  1134
instance ..
bulwahn@41231
  1135
bulwahn@41231
  1136
end
bulwahn@41231
  1137
bulwahn@43889
  1138
instantiation rat :: partial_term_of
bulwahn@43889
  1139
begin
bulwahn@43889
  1140
bulwahn@43889
  1141
instance ..
bulwahn@43889
  1142
bulwahn@43889
  1143
end
bulwahn@43889
  1144
bulwahn@43889
  1145
lemma [code]:
bulwahn@46758
  1146
  "partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Narrowing_variable p tt) == Code_Evaluation.Free (STR ''_'') (Typerep.Typerep (STR ''Rat.rat'') [])"
bulwahn@46758
  1147
  "partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Narrowing_constructor 0 [l, k]) ==
bulwahn@45507
  1148
     Code_Evaluation.App (Code_Evaluation.Const (STR ''Rat.Frct'')
bulwahn@45507
  1149
     (Typerep.Typerep (STR ''fun'') [Typerep.Typerep (STR ''Product_Type.prod'') [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''Int.int'') []],
bulwahn@45507
  1150
        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
  1151
by (rule partial_term_of_anything)+
bulwahn@43889
  1152
bulwahn@43887
  1153
instantiation rat :: narrowing
bulwahn@43887
  1154
begin
bulwahn@43887
  1155
bulwahn@43887
  1156
definition
bulwahn@45507
  1157
  "narrowing = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.apply
bulwahn@45507
  1158
    (Quickcheck_Narrowing.cons (%nom denom. Fract nom denom)) narrowing) narrowing"
bulwahn@43887
  1159
bulwahn@43887
  1160
instance ..
bulwahn@43887
  1161
bulwahn@43887
  1162
end
bulwahn@43887
  1163
bulwahn@43887
  1164
bulwahn@45183
  1165
subsection {* Setup for Nitpick *}
berghofe@24533
  1166
blanchet@38287
  1167
declaration {*
blanchet@38287
  1168
  Nitpick_HOL.register_frac_type @{type_name rat}
wenzelm@33209
  1169
   [(@{const_name zero_rat_inst.zero_rat}, @{const_name Nitpick.zero_frac}),
wenzelm@33209
  1170
    (@{const_name one_rat_inst.one_rat}, @{const_name Nitpick.one_frac}),
wenzelm@33209
  1171
    (@{const_name plus_rat_inst.plus_rat}, @{const_name Nitpick.plus_frac}),
wenzelm@33209
  1172
    (@{const_name times_rat_inst.times_rat}, @{const_name Nitpick.times_frac}),
wenzelm@33209
  1173
    (@{const_name uminus_rat_inst.uminus_rat}, @{const_name Nitpick.uminus_frac}),
wenzelm@33209
  1174
    (@{const_name inverse_rat_inst.inverse_rat}, @{const_name Nitpick.inverse_frac}),
blanchet@37397
  1175
    (@{const_name ord_rat_inst.less_rat}, @{const_name Nitpick.less_frac}),
wenzelm@33209
  1176
    (@{const_name ord_rat_inst.less_eq_rat}, @{const_name Nitpick.less_eq_frac}),
blanchet@45478
  1177
    (@{const_name field_char_0_class.of_rat}, @{const_name Nitpick.of_frac})]
blanchet@33197
  1178
*}
blanchet@33197
  1179
blanchet@41792
  1180
lemmas [nitpick_unfold] = inverse_rat_inst.inverse_rat
huffman@47108
  1181
  one_rat_inst.one_rat ord_rat_inst.less_rat
blanchet@37397
  1182
  ord_rat_inst.less_eq_rat plus_rat_inst.plus_rat times_rat_inst.times_rat
blanchet@37397
  1183
  uminus_rat_inst.uminus_rat zero_rat_inst.zero_rat
blanchet@33197
  1184
huffman@35343
  1185
subsection{* Float syntax *}
huffman@35343
  1186
huffman@35343
  1187
syntax "_Float" :: "float_const \<Rightarrow> 'a"    ("_")
huffman@35343
  1188
huffman@35343
  1189
use "Tools/float_syntax.ML"
huffman@35343
  1190
setup Float_Syntax.setup
huffman@35343
  1191
huffman@35343
  1192
text{* Test: *}
huffman@35343
  1193
lemma "123.456 = -111.111 + 200 + 30 + 4 + 5/10 + 6/100 + (7/1000::rat)"
huffman@35343
  1194
by simp
huffman@35343
  1195
wenzelm@37143
  1196
wenzelm@37143
  1197
hide_const (open) normalize
wenzelm@37143
  1198
huffman@47906
  1199
lemmas [transfer_rule del] =
huffman@47906
  1200
  rat.All_transfer rat.Ex_transfer rat.rel_eq_transfer forall_rat_transfer
huffman@47906
  1201
  Fract.transfer zero_rat.transfer one_rat.transfer plus_rat.transfer
huffman@47906
  1202
  uminus_rat.transfer times_rat.transfer inverse_rat.transfer
huffman@47906
  1203
  less_eq_rat.transfer of_rat.transfer
huffman@47906
  1204
huffman@47906
  1205
text {* De-register @{text "rat"} as a quotient type: *}
huffman@47906
  1206
huffman@47906
  1207
setup {* Context.theory_map (Lifting_Info.update_quotients @{type_name rat}
huffman@47906
  1208
  {quot_thm = @{thm identity_quotient [where 'a=rat]}}) *}
huffman@47906
  1209
huffman@29880
  1210
end