src/HOL/Rat.thy
author wenzelm
Wed Aug 22 22:55:41 2012 +0200 (2012-08-22)
changeset 48891 c0eafbd55de3
parent 47952 36a8c477dae8
child 49962 a8cc904a6820
permissions -rw-r--r--
prefer ML_file over old uses;
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(*  Title:  HOL/Rat.thy
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    Author: Markus Wenzel, TU Muenchen
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*)
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header {* Rational numbers *}
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theory Rat
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imports GCD Archimedean_Field
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begin
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subsection {* Rational numbers as quotient *}
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subsubsection {* Construction of the type of rational numbers *}
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definition
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  ratrel :: "(int \<times> int) \<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)
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text{*
haftmann@35369
   308
  Decompose a fraction into normalized, i.e. coprime numerator and denominator:
haftmann@35369
   309
*}
haftmann@35369
   310
haftmann@35369
   311
definition quotient_of :: "rat \<Rightarrow> int \<times> int" where
haftmann@35369
   312
  "quotient_of x = (THE pair. x = Fract (fst pair) (snd pair) &
haftmann@35369
   313
                   snd pair > 0 & coprime (fst pair) (snd pair))"
haftmann@35369
   314
haftmann@35369
   315
lemma quotient_of_unique:
haftmann@35369
   316
  "\<exists>!p. r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
haftmann@35369
   317
proof (cases r)
haftmann@35369
   318
  case (Fract a b)
haftmann@35369
   319
  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
   320
  then show ?thesis proof (rule ex1I)
haftmann@35369
   321
    fix p
haftmann@35369
   322
    obtain c d :: int where p: "p = (c, d)" by (cases p)
haftmann@35369
   323
    assume "r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
haftmann@35369
   324
    with p have Fract': "r = Fract c d" "d > 0" "coprime c d" by simp_all
haftmann@35369
   325
    have "c = a \<and> d = b"
haftmann@35369
   326
    proof (cases "a = 0")
haftmann@35369
   327
      case True with Fract Fract' show ?thesis by (simp add: eq_rat)
haftmann@35369
   328
    next
haftmann@35369
   329
      case False
haftmann@35369
   330
      with Fract Fract' have *: "c * b = a * d" and "c \<noteq> 0" by (auto simp add: eq_rat)
haftmann@35369
   331
      then have "c * b > 0 \<longleftrightarrow> a * d > 0" by auto
haftmann@35369
   332
      with `b > 0` `d > 0` have "a > 0 \<longleftrightarrow> c > 0" by (simp add: zero_less_mult_iff)
haftmann@35369
   333
      with `a \<noteq> 0` `c \<noteq> 0` have sgn: "sgn a = sgn c" by (auto simp add: not_less)
haftmann@35369
   334
      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
   335
        by (simp add: coprime_crossproduct_int)
haftmann@35369
   336
      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
   337
      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
   338
      with sgn * show ?thesis by (auto simp add: sgn_0_0)
nipkow@33805
   339
    qed
haftmann@35369
   340
    with p show "p = (a, b)" by simp
nipkow@33805
   341
  qed
nipkow@33805
   342
qed
nipkow@33805
   343
haftmann@35369
   344
lemma quotient_of_Fract [code]:
haftmann@35369
   345
  "quotient_of (Fract a b) = normalize (a, b)"
haftmann@35369
   346
proof -
haftmann@35369
   347
  have "Fract a b = Fract (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?Fract)
haftmann@35369
   348
    by (rule sym) (auto intro: normalize_eq)
haftmann@35369
   349
  moreover have "0 < snd (normalize (a, b))" (is ?denom_pos) 
haftmann@35369
   350
    by (cases "normalize (a, b)") (rule normalize_denom_pos, simp)
haftmann@35369
   351
  moreover have "coprime (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?coprime)
haftmann@35369
   352
    by (rule normalize_coprime) simp
haftmann@35369
   353
  ultimately have "?Fract \<and> ?denom_pos \<and> ?coprime" by blast
haftmann@35369
   354
  with quotient_of_unique have
haftmann@35369
   355
    "(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
   356
    by (rule the1_equality)
haftmann@35369
   357
  then show ?thesis by (simp add: quotient_of_def)
haftmann@35369
   358
qed
haftmann@35369
   359
haftmann@35369
   360
lemma quotient_of_number [simp]:
haftmann@35369
   361
  "quotient_of 0 = (0, 1)"
haftmann@35369
   362
  "quotient_of 1 = (1, 1)"
huffman@47108
   363
  "quotient_of (numeral k) = (numeral k, 1)"
huffman@47108
   364
  "quotient_of (neg_numeral k) = (neg_numeral k, 1)"
haftmann@35369
   365
  by (simp_all add: rat_number_expand quotient_of_Fract)
nipkow@33805
   366
haftmann@35369
   367
lemma quotient_of_eq: "quotient_of (Fract a b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
haftmann@35369
   368
  by (simp add: quotient_of_Fract normalize_eq)
haftmann@35369
   369
haftmann@35369
   370
lemma quotient_of_denom_pos: "quotient_of r = (p, q) \<Longrightarrow> q > 0"
haftmann@35369
   371
  by (cases r) (simp add: quotient_of_Fract normalize_denom_pos)
haftmann@35369
   372
haftmann@35369
   373
lemma quotient_of_coprime: "quotient_of r = (p, q) \<Longrightarrow> coprime p q"
haftmann@35369
   374
  by (cases r) (simp add: quotient_of_Fract normalize_coprime)
nipkow@33805
   375
haftmann@35369
   376
lemma quotient_of_inject:
haftmann@35369
   377
  assumes "quotient_of a = quotient_of b"
haftmann@35369
   378
  shows "a = b"
haftmann@35369
   379
proof -
haftmann@35369
   380
  obtain p q r s where a: "a = Fract p q"
haftmann@35369
   381
    and b: "b = Fract r s"
haftmann@35369
   382
    and "q > 0" and "s > 0" by (cases a, cases b)
haftmann@35369
   383
  with assms show ?thesis by (simp add: eq_rat quotient_of_Fract normalize_crossproduct)
haftmann@35369
   384
qed
haftmann@35369
   385
haftmann@35369
   386
lemma quotient_of_inject_eq:
haftmann@35369
   387
  "quotient_of a = quotient_of b \<longleftrightarrow> a = b"
haftmann@35369
   388
  by (auto simp add: quotient_of_inject)
nipkow@33805
   389
haftmann@27551
   390
haftmann@27551
   391
subsubsection {* Various *}
haftmann@27551
   392
haftmann@27551
   393
lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"
haftmann@27652
   394
  by (simp add: Fract_of_int_eq [symmetric])
haftmann@27551
   395
huffman@47108
   396
lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
huffman@47108
   397
  by (simp add: rat_number_expand)
haftmann@27551
   398
haftmann@27551
   399
haftmann@27551
   400
subsubsection {* The ordered field of rational numbers *}
huffman@27509
   401
huffman@47907
   402
lift_definition positive :: "rat \<Rightarrow> bool"
huffman@47907
   403
  is "\<lambda>x. 0 < fst x * snd x"
huffman@47907
   404
proof (clarsimp)
huffman@47907
   405
  fix a b c d :: int
huffman@47907
   406
  assume "b \<noteq> 0" and "d \<noteq> 0" and "a * d = c * b"
huffman@47907
   407
  hence "a * d * b * d = c * b * b * d"
huffman@47907
   408
    by simp
huffman@47907
   409
  hence "a * b * d\<twosuperior> = c * d * b\<twosuperior>"
huffman@47907
   410
    unfolding power2_eq_square by (simp add: mult_ac)
huffman@47907
   411
  hence "0 < a * b * d\<twosuperior> \<longleftrightarrow> 0 < c * d * b\<twosuperior>"
huffman@47907
   412
    by simp
huffman@47907
   413
  thus "0 < a * b \<longleftrightarrow> 0 < c * d"
huffman@47907
   414
    using `b \<noteq> 0` and `d \<noteq> 0`
huffman@47907
   415
    by (simp add: zero_less_mult_iff)
huffman@47907
   416
qed
huffman@47907
   417
huffman@47907
   418
lemma positive_zero: "\<not> positive 0"
huffman@47907
   419
  by transfer simp
huffman@47907
   420
huffman@47907
   421
lemma positive_add:
huffman@47907
   422
  "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x + y)"
huffman@47907
   423
apply transfer
huffman@47907
   424
apply (simp add: zero_less_mult_iff)
huffman@47907
   425
apply (elim disjE, simp_all add: add_pos_pos add_neg_neg
huffman@47907
   426
  mult_pos_pos mult_pos_neg mult_neg_pos mult_neg_neg)
huffman@47907
   427
done
huffman@47907
   428
huffman@47907
   429
lemma positive_mult:
huffman@47907
   430
  "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x * y)"
huffman@47907
   431
by transfer (drule (1) mult_pos_pos, simp add: mult_ac)
huffman@47907
   432
huffman@47907
   433
lemma positive_minus:
huffman@47907
   434
  "\<not> positive x \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> positive (- x)"
huffman@47907
   435
by transfer (force simp: neq_iff zero_less_mult_iff mult_less_0_iff)
huffman@47907
   436
huffman@47907
   437
instantiation rat :: linordered_field_inverse_zero
huffman@27509
   438
begin
huffman@27509
   439
huffman@47907
   440
definition
huffman@47907
   441
  "x < y \<longleftrightarrow> positive (y - x)"
huffman@47907
   442
huffman@47907
   443
definition
huffman@47907
   444
  "x \<le> (y::rat) \<longleftrightarrow> x < y \<or> x = y"
huffman@47907
   445
huffman@47907
   446
definition
huffman@47907
   447
  "abs (a::rat) = (if a < 0 then - a else a)"
huffman@47907
   448
huffman@47907
   449
definition
huffman@47907
   450
  "sgn (a::rat) = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
huffman@47906
   451
huffman@47907
   452
instance proof
huffman@47907
   453
  fix a b c :: rat
huffman@47907
   454
  show "\<bar>a\<bar> = (if a < 0 then - a else a)"
huffman@47907
   455
    by (rule abs_rat_def)
huffman@47907
   456
  show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a"
huffman@47907
   457
    unfolding less_eq_rat_def less_rat_def
huffman@47907
   458
    by (auto, drule (1) positive_add, simp_all add: positive_zero)
huffman@47907
   459
  show "a \<le> a"
huffman@47907
   460
    unfolding less_eq_rat_def by simp
huffman@47907
   461
  show "a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
huffman@47907
   462
    unfolding less_eq_rat_def less_rat_def
huffman@47907
   463
    by (auto, drule (1) positive_add, simp add: algebra_simps)
huffman@47907
   464
  show "a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> a = b"
huffman@47907
   465
    unfolding less_eq_rat_def less_rat_def
huffman@47907
   466
    by (auto, drule (1) positive_add, simp add: positive_zero)
huffman@47907
   467
  show "a \<le> b \<Longrightarrow> c + a \<le> c + b"
huffman@47907
   468
    unfolding less_eq_rat_def less_rat_def by (auto simp: diff_minus)
huffman@47907
   469
  show "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
huffman@47907
   470
    by (rule sgn_rat_def)
huffman@47907
   471
  show "a \<le> b \<or> b \<le> a"
huffman@47907
   472
    unfolding less_eq_rat_def less_rat_def
huffman@47907
   473
    by (auto dest!: positive_minus)
huffman@47907
   474
  show "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
huffman@47907
   475
    unfolding less_rat_def
huffman@47907
   476
    by (drule (1) positive_mult, simp add: algebra_simps)
huffman@47906
   477
qed
haftmann@27551
   478
huffman@47907
   479
end
huffman@47907
   480
huffman@47907
   481
instantiation rat :: distrib_lattice
huffman@47907
   482
begin
huffman@47907
   483
huffman@47907
   484
definition
huffman@47907
   485
  "(inf :: rat \<Rightarrow> rat \<Rightarrow> rat) = min"
huffman@27509
   486
huffman@27509
   487
definition
huffman@47907
   488
  "(sup :: rat \<Rightarrow> rat \<Rightarrow> rat) = max"
huffman@47907
   489
huffman@47907
   490
instance proof
huffman@47907
   491
qed (auto simp add: inf_rat_def sup_rat_def min_max.sup_inf_distrib1)
huffman@47907
   492
huffman@47907
   493
end
huffman@47907
   494
huffman@47907
   495
lemma positive_rat: "positive (Fract a b) \<longleftrightarrow> 0 < a * b"
huffman@47907
   496
  by transfer simp
huffman@27509
   497
haftmann@27652
   498
lemma less_rat [simp]:
haftmann@27551
   499
  assumes "b \<noteq> 0" and "d \<noteq> 0"
haftmann@27551
   500
  shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
huffman@47907
   501
  using assms unfolding less_rat_def
huffman@47907
   502
  by (simp add: positive_rat algebra_simps)
huffman@27509
   503
huffman@47907
   504
lemma le_rat [simp]:
huffman@47907
   505
  assumes "b \<noteq> 0" and "d \<noteq> 0"
huffman@47907
   506
  shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
huffman@47907
   507
  using assms unfolding le_less by (simp add: eq_rat)
haftmann@27551
   508
haftmann@27652
   509
lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
huffman@35216
   510
  by (auto simp add: abs_rat_def zabs_def Zero_rat_def not_less le_less eq_rat zero_less_mult_iff)
haftmann@27551
   511
haftmann@27652
   512
lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)"
haftmann@27551
   513
  unfolding Fract_of_int_eq
haftmann@27652
   514
  by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat)
haftmann@27551
   515
    (auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)
haftmann@27551
   516
haftmann@27551
   517
lemma Rat_induct_pos [case_names Fract, induct type: rat]:
haftmann@27551
   518
  assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
haftmann@27551
   519
  shows "P q"
paulson@14365
   520
proof (cases q)
haftmann@27551
   521
  have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)"
paulson@14365
   522
  proof -
paulson@14365
   523
    fix a::int and b::int
paulson@14365
   524
    assume b: "b < 0"
paulson@14365
   525
    hence "0 < -b" by simp
paulson@14365
   526
    hence "P (Fract (-a) (-b))" by (rule step)
paulson@14365
   527
    thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
paulson@14365
   528
  qed
paulson@14365
   529
  case (Fract a b)
paulson@14365
   530
  thus "P q" by (force simp add: linorder_neq_iff step step')
paulson@14365
   531
qed
paulson@14365
   532
paulson@14365
   533
lemma zero_less_Fract_iff:
huffman@30095
   534
  "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a"
huffman@30095
   535
  by (simp add: Zero_rat_def zero_less_mult_iff)
huffman@30095
   536
huffman@30095
   537
lemma Fract_less_zero_iff:
huffman@30095
   538
  "0 < b \<Longrightarrow> Fract a b < 0 \<longleftrightarrow> a < 0"
huffman@30095
   539
  by (simp add: Zero_rat_def mult_less_0_iff)
huffman@30095
   540
huffman@30095
   541
lemma zero_le_Fract_iff:
huffman@30095
   542
  "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a"
huffman@30095
   543
  by (simp add: Zero_rat_def zero_le_mult_iff)
huffman@30095
   544
huffman@30095
   545
lemma Fract_le_zero_iff:
huffman@30095
   546
  "0 < b \<Longrightarrow> Fract a b \<le> 0 \<longleftrightarrow> a \<le> 0"
huffman@30095
   547
  by (simp add: Zero_rat_def mult_le_0_iff)
huffman@30095
   548
huffman@30095
   549
lemma one_less_Fract_iff:
huffman@30095
   550
  "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a"
huffman@30095
   551
  by (simp add: One_rat_def mult_less_cancel_right_disj)
huffman@30095
   552
huffman@30095
   553
lemma Fract_less_one_iff:
huffman@30095
   554
  "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b"
huffman@30095
   555
  by (simp add: One_rat_def mult_less_cancel_right_disj)
huffman@30095
   556
huffman@30095
   557
lemma one_le_Fract_iff:
huffman@30095
   558
  "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a"
huffman@30095
   559
  by (simp add: One_rat_def mult_le_cancel_right)
huffman@30095
   560
huffman@30095
   561
lemma Fract_le_one_iff:
huffman@30095
   562
  "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b"
huffman@30095
   563
  by (simp add: One_rat_def mult_le_cancel_right)
paulson@14365
   564
paulson@14378
   565
huffman@30097
   566
subsubsection {* Rationals are an Archimedean field *}
huffman@30097
   567
huffman@30097
   568
lemma rat_floor_lemma:
huffman@30097
   569
  shows "of_int (a div b) \<le> Fract a b \<and> Fract a b < of_int (a div b + 1)"
huffman@30097
   570
proof -
huffman@30097
   571
  have "Fract a b = of_int (a div b) + Fract (a mod b) b"
huffman@35293
   572
    by (cases "b = 0", simp, simp add: of_int_rat)
huffman@30097
   573
  moreover have "0 \<le> Fract (a mod b) b \<and> Fract (a mod b) b < 1"
huffman@35293
   574
    unfolding Fract_of_int_quotient
haftmann@36409
   575
    by (rule linorder_cases [of b 0]) (simp add: divide_nonpos_neg, simp, simp add: divide_nonneg_pos)
huffman@30097
   576
  ultimately show ?thesis by simp
huffman@30097
   577
qed
huffman@30097
   578
huffman@30097
   579
instance rat :: archimedean_field
huffman@30097
   580
proof
huffman@30097
   581
  fix r :: rat
huffman@30097
   582
  show "\<exists>z. r \<le> of_int z"
huffman@30097
   583
  proof (induct r)
huffman@30097
   584
    case (Fract a b)
huffman@35293
   585
    have "Fract a b \<le> of_int (a div b + 1)"
huffman@35293
   586
      using rat_floor_lemma [of a b] by simp
huffman@30097
   587
    then show "\<exists>z. Fract a b \<le> of_int z" ..
huffman@30097
   588
  qed
huffman@30097
   589
qed
huffman@30097
   590
bulwahn@43732
   591
instantiation rat :: floor_ceiling
bulwahn@43732
   592
begin
bulwahn@43732
   593
bulwahn@43732
   594
definition [code del]:
bulwahn@43732
   595
  "floor (x::rat) = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))"
bulwahn@43732
   596
bulwahn@43732
   597
instance proof
bulwahn@43732
   598
  fix x :: rat
bulwahn@43732
   599
  show "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
bulwahn@43732
   600
    unfolding floor_rat_def using floor_exists1 by (rule theI')
bulwahn@43732
   601
qed
bulwahn@43732
   602
bulwahn@43732
   603
end
bulwahn@43732
   604
huffman@35293
   605
lemma floor_Fract: "floor (Fract a b) = a div b"
huffman@35293
   606
  using rat_floor_lemma [of a b]
huffman@30097
   607
  by (simp add: floor_unique)
huffman@30097
   608
huffman@30097
   609
haftmann@31100
   610
subsection {* Linear arithmetic setup *}
paulson@14387
   611
haftmann@31100
   612
declaration {*
haftmann@31100
   613
  K (Lin_Arith.add_inj_thms [@{thm of_nat_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
haftmann@31100
   614
    (* not needed because x < (y::nat) can be rewritten as Suc x <= y: of_nat_less_iff RS iffD2 *)
haftmann@31100
   615
  #> Lin_Arith.add_inj_thms [@{thm of_int_le_iff} RS iffD2, @{thm of_int_eq_iff} RS iffD2]
haftmann@31100
   616
    (* not needed because x < (y::int) can be rewritten as x + 1 <= y: of_int_less_iff RS iffD2 *)
haftmann@31100
   617
  #> Lin_Arith.add_simps [@{thm neg_less_iff_less},
haftmann@31100
   618
      @{thm True_implies_equals},
huffman@47108
   619
      read_instantiate @{context} [(("a", 0), "(numeral ?v)")] @{thm right_distrib},
huffman@47108
   620
      read_instantiate @{context} [(("a", 0), "(neg_numeral ?v)")] @{thm right_distrib},
haftmann@31100
   621
      @{thm divide_1}, @{thm divide_zero_left},
haftmann@31100
   622
      @{thm times_divide_eq_right}, @{thm times_divide_eq_left},
haftmann@31100
   623
      @{thm minus_divide_left} RS sym, @{thm minus_divide_right} RS sym,
haftmann@31100
   624
      @{thm of_int_minus}, @{thm of_int_diff},
haftmann@31100
   625
      @{thm of_int_of_nat_eq}]
haftmann@31100
   626
  #> Lin_Arith.add_simprocs Numeral_Simprocs.field_cancel_numeral_factors
haftmann@31100
   627
  #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat => rat"})
haftmann@31100
   628
  #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int => rat"}))
haftmann@31100
   629
*}
paulson@14387
   630
huffman@23342
   631
huffman@23342
   632
subsection {* Embedding from Rationals to other Fields *}
huffman@23342
   633
haftmann@24198
   634
class field_char_0 = field + ring_char_0
huffman@23342
   635
haftmann@35028
   636
subclass (in linordered_field) field_char_0 ..
huffman@23342
   637
haftmann@27551
   638
context field_char_0
haftmann@27551
   639
begin
haftmann@27551
   640
huffman@47906
   641
lift_definition of_rat :: "rat \<Rightarrow> 'a"
huffman@47906
   642
  is "\<lambda>x. of_int (fst x) / of_int (snd x)"
huffman@23342
   643
apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
huffman@23342
   644
apply (simp only: of_int_mult [symmetric])
huffman@23342
   645
done
huffman@23342
   646
huffman@47906
   647
end
huffman@47906
   648
haftmann@27551
   649
lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
huffman@47906
   650
  by transfer simp
huffman@23342
   651
huffman@23342
   652
lemma of_rat_0 [simp]: "of_rat 0 = 0"
huffman@47906
   653
  by transfer simp
huffman@23342
   654
huffman@23342
   655
lemma of_rat_1 [simp]: "of_rat 1 = 1"
huffman@47906
   656
  by transfer simp
huffman@23342
   657
huffman@23342
   658
lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
huffman@47906
   659
  by transfer (simp add: add_frac_eq)
huffman@23342
   660
huffman@23343
   661
lemma of_rat_minus: "of_rat (- a) = - of_rat a"
huffman@47906
   662
  by transfer simp
huffman@23343
   663
huffman@23343
   664
lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
huffman@23343
   665
by (simp only: diff_minus of_rat_add of_rat_minus)
huffman@23343
   666
huffman@23342
   667
lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
huffman@47906
   668
apply transfer
huffman@23342
   669
apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
huffman@23342
   670
done
huffman@23342
   671
huffman@23342
   672
lemma nonzero_of_rat_inverse:
huffman@23342
   673
  "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
huffman@23343
   674
apply (rule inverse_unique [symmetric])
huffman@23343
   675
apply (simp add: of_rat_mult [symmetric])
huffman@23342
   676
done
huffman@23342
   677
huffman@23342
   678
lemma of_rat_inverse:
haftmann@36409
   679
  "(of_rat (inverse a)::'a::{field_char_0, field_inverse_zero}) =
huffman@23342
   680
   inverse (of_rat a)"
huffman@23342
   681
by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
huffman@23342
   682
huffman@23342
   683
lemma nonzero_of_rat_divide:
huffman@23342
   684
  "b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
huffman@23342
   685
by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
huffman@23342
   686
huffman@23342
   687
lemma of_rat_divide:
haftmann@36409
   688
  "(of_rat (a / b)::'a::{field_char_0, field_inverse_zero})
huffman@23342
   689
   = of_rat a / of_rat b"
haftmann@27652
   690
by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)
huffman@23342
   691
huffman@23343
   692
lemma of_rat_power:
haftmann@31017
   693
  "(of_rat (a ^ n)::'a::field_char_0) = of_rat a ^ n"
huffman@30273
   694
by (induct n) (simp_all add: of_rat_mult)
huffman@23343
   695
huffman@23343
   696
lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
huffman@47906
   697
apply transfer
huffman@23343
   698
apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
huffman@23343
   699
apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
huffman@23343
   700
done
huffman@23343
   701
haftmann@27652
   702
lemma of_rat_less:
haftmann@35028
   703
  "(of_rat r :: 'a::linordered_field) < of_rat s \<longleftrightarrow> r < s"
haftmann@27652
   704
proof (induct r, induct s)
haftmann@27652
   705
  fix a b c d :: int
haftmann@27652
   706
  assume not_zero: "b > 0" "d > 0"
haftmann@27652
   707
  then have "b * d > 0" by (rule mult_pos_pos)
haftmann@27652
   708
  have of_int_divide_less_eq:
haftmann@27652
   709
    "(of_int a :: 'a) / of_int b < of_int c / of_int d
haftmann@27652
   710
      \<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b"
haftmann@27652
   711
    using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
haftmann@35028
   712
  show "(of_rat (Fract a b) :: 'a::linordered_field) < of_rat (Fract c d)
haftmann@27652
   713
    \<longleftrightarrow> Fract a b < Fract c d"
haftmann@27652
   714
    using not_zero `b * d > 0`
haftmann@27652
   715
    by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult)
haftmann@27652
   716
qed
haftmann@27652
   717
haftmann@27652
   718
lemma of_rat_less_eq:
haftmann@35028
   719
  "(of_rat r :: 'a::linordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s"
haftmann@27652
   720
  unfolding le_less by (auto simp add: of_rat_less)
haftmann@27652
   721
huffman@23343
   722
lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
huffman@23343
   723
haftmann@27652
   724
lemma of_rat_eq_id [simp]: "of_rat = id"
huffman@23343
   725
proof
huffman@23343
   726
  fix a
huffman@23343
   727
  show "of_rat a = id a"
huffman@23343
   728
  by (induct a)
haftmann@27652
   729
     (simp add: of_rat_rat Fract_of_int_eq [symmetric])
huffman@23343
   730
qed
huffman@23343
   731
huffman@23343
   732
text{*Collapse nested embeddings*}
huffman@23343
   733
lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
huffman@23343
   734
by (induct n) (simp_all add: of_rat_add)
huffman@23343
   735
huffman@23343
   736
lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
haftmann@27652
   737
by (cases z rule: int_diff_cases) (simp add: of_rat_diff)
huffman@23343
   738
huffman@47108
   739
lemma of_rat_numeral_eq [simp]:
huffman@47108
   740
  "of_rat (numeral w) = numeral w"
huffman@47108
   741
using of_rat_of_int_eq [of "numeral w"] by simp
huffman@47108
   742
huffman@47108
   743
lemma of_rat_neg_numeral_eq [simp]:
huffman@47108
   744
  "of_rat (neg_numeral w) = neg_numeral w"
huffman@47108
   745
using of_rat_of_int_eq [of "neg_numeral w"] by simp
huffman@23343
   746
haftmann@23879
   747
lemmas zero_rat = Zero_rat_def
haftmann@23879
   748
lemmas one_rat = One_rat_def
haftmann@23879
   749
haftmann@24198
   750
abbreviation
haftmann@24198
   751
  rat_of_nat :: "nat \<Rightarrow> rat"
haftmann@24198
   752
where
haftmann@24198
   753
  "rat_of_nat \<equiv> of_nat"
haftmann@24198
   754
haftmann@24198
   755
abbreviation
haftmann@24198
   756
  rat_of_int :: "int \<Rightarrow> rat"
haftmann@24198
   757
where
haftmann@24198
   758
  "rat_of_int \<equiv> of_int"
haftmann@24198
   759
huffman@28010
   760
subsection {* The Set of Rational Numbers *}
berghofe@24533
   761
nipkow@28001
   762
context field_char_0
nipkow@28001
   763
begin
nipkow@28001
   764
nipkow@28001
   765
definition
nipkow@28001
   766
  Rats  :: "'a set" where
haftmann@35369
   767
  "Rats = range of_rat"
nipkow@28001
   768
nipkow@28001
   769
notation (xsymbols)
nipkow@28001
   770
  Rats  ("\<rat>")
nipkow@28001
   771
nipkow@28001
   772
end
nipkow@28001
   773
huffman@28010
   774
lemma Rats_of_rat [simp]: "of_rat r \<in> Rats"
huffman@28010
   775
by (simp add: Rats_def)
huffman@28010
   776
huffman@28010
   777
lemma Rats_of_int [simp]: "of_int z \<in> Rats"
huffman@28010
   778
by (subst of_rat_of_int_eq [symmetric], rule Rats_of_rat)
huffman@28010
   779
huffman@28010
   780
lemma Rats_of_nat [simp]: "of_nat n \<in> Rats"
huffman@28010
   781
by (subst of_rat_of_nat_eq [symmetric], rule Rats_of_rat)
huffman@28010
   782
huffman@47108
   783
lemma Rats_number_of [simp]: "numeral w \<in> Rats"
huffman@47108
   784
by (subst of_rat_numeral_eq [symmetric], rule Rats_of_rat)
huffman@47108
   785
huffman@47108
   786
lemma Rats_neg_number_of [simp]: "neg_numeral w \<in> Rats"
huffman@47108
   787
by (subst of_rat_neg_numeral_eq [symmetric], rule Rats_of_rat)
huffman@28010
   788
huffman@28010
   789
lemma Rats_0 [simp]: "0 \<in> Rats"
huffman@28010
   790
apply (unfold Rats_def)
huffman@28010
   791
apply (rule range_eqI)
huffman@28010
   792
apply (rule of_rat_0 [symmetric])
huffman@28010
   793
done
huffman@28010
   794
huffman@28010
   795
lemma Rats_1 [simp]: "1 \<in> Rats"
huffman@28010
   796
apply (unfold Rats_def)
huffman@28010
   797
apply (rule range_eqI)
huffman@28010
   798
apply (rule of_rat_1 [symmetric])
huffman@28010
   799
done
huffman@28010
   800
huffman@28010
   801
lemma Rats_add [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a + b \<in> Rats"
huffman@28010
   802
apply (auto simp add: Rats_def)
huffman@28010
   803
apply (rule range_eqI)
huffman@28010
   804
apply (rule of_rat_add [symmetric])
huffman@28010
   805
done
huffman@28010
   806
huffman@28010
   807
lemma Rats_minus [simp]: "a \<in> Rats \<Longrightarrow> - a \<in> Rats"
huffman@28010
   808
apply (auto simp add: Rats_def)
huffman@28010
   809
apply (rule range_eqI)
huffman@28010
   810
apply (rule of_rat_minus [symmetric])
huffman@28010
   811
done
huffman@28010
   812
huffman@28010
   813
lemma Rats_diff [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a - b \<in> Rats"
huffman@28010
   814
apply (auto simp add: Rats_def)
huffman@28010
   815
apply (rule range_eqI)
huffman@28010
   816
apply (rule of_rat_diff [symmetric])
huffman@28010
   817
done
huffman@28010
   818
huffman@28010
   819
lemma Rats_mult [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a * b \<in> Rats"
huffman@28010
   820
apply (auto simp add: Rats_def)
huffman@28010
   821
apply (rule range_eqI)
huffman@28010
   822
apply (rule of_rat_mult [symmetric])
huffman@28010
   823
done
huffman@28010
   824
huffman@28010
   825
lemma nonzero_Rats_inverse:
huffman@28010
   826
  fixes a :: "'a::field_char_0"
huffman@28010
   827
  shows "\<lbrakk>a \<in> Rats; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Rats"
huffman@28010
   828
apply (auto simp add: Rats_def)
huffman@28010
   829
apply (rule range_eqI)
huffman@28010
   830
apply (erule nonzero_of_rat_inverse [symmetric])
huffman@28010
   831
done
huffman@28010
   832
huffman@28010
   833
lemma Rats_inverse [simp]:
haftmann@36409
   834
  fixes a :: "'a::{field_char_0, field_inverse_zero}"
huffman@28010
   835
  shows "a \<in> Rats \<Longrightarrow> inverse a \<in> Rats"
huffman@28010
   836
apply (auto simp add: Rats_def)
huffman@28010
   837
apply (rule range_eqI)
huffman@28010
   838
apply (rule of_rat_inverse [symmetric])
huffman@28010
   839
done
huffman@28010
   840
huffman@28010
   841
lemma nonzero_Rats_divide:
huffman@28010
   842
  fixes a b :: "'a::field_char_0"
huffman@28010
   843
  shows "\<lbrakk>a \<in> Rats; b \<in> Rats; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
huffman@28010
   844
apply (auto simp add: Rats_def)
huffman@28010
   845
apply (rule range_eqI)
huffman@28010
   846
apply (erule nonzero_of_rat_divide [symmetric])
huffman@28010
   847
done
huffman@28010
   848
huffman@28010
   849
lemma Rats_divide [simp]:
haftmann@36409
   850
  fixes a b :: "'a::{field_char_0, field_inverse_zero}"
huffman@28010
   851
  shows "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
huffman@28010
   852
apply (auto simp add: Rats_def)
huffman@28010
   853
apply (rule range_eqI)
huffman@28010
   854
apply (rule of_rat_divide [symmetric])
huffman@28010
   855
done
huffman@28010
   856
huffman@28010
   857
lemma Rats_power [simp]:
haftmann@31017
   858
  fixes a :: "'a::field_char_0"
huffman@28010
   859
  shows "a \<in> Rats \<Longrightarrow> a ^ n \<in> Rats"
huffman@28010
   860
apply (auto simp add: Rats_def)
huffman@28010
   861
apply (rule range_eqI)
huffman@28010
   862
apply (rule of_rat_power [symmetric])
huffman@28010
   863
done
huffman@28010
   864
huffman@28010
   865
lemma Rats_cases [cases set: Rats]:
huffman@28010
   866
  assumes "q \<in> \<rat>"
huffman@28010
   867
  obtains (of_rat) r where "q = of_rat r"
huffman@28010
   868
proof -
huffman@28010
   869
  from `q \<in> \<rat>` have "q \<in> range of_rat" unfolding Rats_def .
huffman@28010
   870
  then obtain r where "q = of_rat r" ..
huffman@28010
   871
  then show thesis ..
huffman@28010
   872
qed
huffman@28010
   873
huffman@28010
   874
lemma Rats_induct [case_names of_rat, induct set: Rats]:
huffman@28010
   875
  "q \<in> \<rat> \<Longrightarrow> (\<And>r. P (of_rat r)) \<Longrightarrow> P q"
huffman@28010
   876
  by (rule Rats_cases) auto
huffman@28010
   877
nipkow@28001
   878
berghofe@24533
   879
subsection {* Implementation of rational numbers as pairs of integers *}
berghofe@24533
   880
huffman@47108
   881
text {* Formal constructor *}
huffman@47108
   882
haftmann@35369
   883
definition Frct :: "int \<times> int \<Rightarrow> rat" where
haftmann@35369
   884
  [simp]: "Frct p = Fract (fst p) (snd p)"
haftmann@35369
   885
haftmann@36112
   886
lemma [code abstype]:
haftmann@36112
   887
  "Frct (quotient_of q) = q"
haftmann@36112
   888
  by (cases q) (auto intro: quotient_of_eq)
haftmann@35369
   889
huffman@47108
   890
huffman@47108
   891
text {* Numerals *}
haftmann@35369
   892
haftmann@35369
   893
declare quotient_of_Fract [code abstract]
haftmann@35369
   894
huffman@47108
   895
definition of_int :: "int \<Rightarrow> rat"
huffman@47108
   896
where
huffman@47108
   897
  [code_abbrev]: "of_int = Int.of_int"
huffman@47108
   898
hide_const (open) of_int
huffman@47108
   899
huffman@47108
   900
lemma quotient_of_int [code abstract]:
huffman@47108
   901
  "quotient_of (Rat.of_int a) = (a, 1)"
huffman@47108
   902
  by (simp add: of_int_def of_int_rat quotient_of_Fract)
huffman@47108
   903
huffman@47108
   904
lemma [code_unfold]:
huffman@47108
   905
  "numeral k = Rat.of_int (numeral k)"
huffman@47108
   906
  by (simp add: Rat.of_int_def)
huffman@47108
   907
huffman@47108
   908
lemma [code_unfold]:
huffman@47108
   909
  "neg_numeral k = Rat.of_int (neg_numeral k)"
huffman@47108
   910
  by (simp add: Rat.of_int_def)
huffman@47108
   911
huffman@47108
   912
lemma Frct_code_post [code_post]:
huffman@47108
   913
  "Frct (0, a) = 0"
huffman@47108
   914
  "Frct (a, 0) = 0"
huffman@47108
   915
  "Frct (1, 1) = 1"
huffman@47108
   916
  "Frct (numeral k, 1) = numeral k"
huffman@47108
   917
  "Frct (neg_numeral k, 1) = neg_numeral k"
huffman@47108
   918
  "Frct (1, numeral k) = 1 / numeral k"
huffman@47108
   919
  "Frct (1, neg_numeral k) = 1 / neg_numeral k"
huffman@47108
   920
  "Frct (numeral k, numeral l) = numeral k / numeral l"
huffman@47108
   921
  "Frct (numeral k, neg_numeral l) = numeral k / neg_numeral l"
huffman@47108
   922
  "Frct (neg_numeral k, numeral l) = neg_numeral k / numeral l"
huffman@47108
   923
  "Frct (neg_numeral k, neg_numeral l) = neg_numeral k / neg_numeral l"
huffman@47108
   924
  by (simp_all add: Fract_of_int_quotient)
huffman@47108
   925
huffman@47108
   926
huffman@47108
   927
text {* Operations *}
huffman@47108
   928
haftmann@35369
   929
lemma rat_zero_code [code abstract]:
haftmann@35369
   930
  "quotient_of 0 = (0, 1)"
haftmann@35369
   931
  by (simp add: Zero_rat_def quotient_of_Fract normalize_def)
haftmann@35369
   932
haftmann@35369
   933
lemma rat_one_code [code abstract]:
haftmann@35369
   934
  "quotient_of 1 = (1, 1)"
haftmann@35369
   935
  by (simp add: One_rat_def quotient_of_Fract normalize_def)
haftmann@35369
   936
haftmann@35369
   937
lemma rat_plus_code [code abstract]:
haftmann@35369
   938
  "quotient_of (p + q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369
   939
     in normalize (a * d + b * c, c * d))"
haftmann@35369
   940
  by (cases p, cases q) (simp add: quotient_of_Fract)
haftmann@27652
   941
haftmann@35369
   942
lemma rat_uminus_code [code abstract]:
haftmann@35369
   943
  "quotient_of (- p) = (let (a, b) = quotient_of p in (- a, b))"
haftmann@35369
   944
  by (cases p) (simp add: quotient_of_Fract)
haftmann@35369
   945
haftmann@35369
   946
lemma rat_minus_code [code abstract]:
haftmann@35369
   947
  "quotient_of (p - q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369
   948
     in normalize (a * d - b * c, c * d))"
haftmann@35369
   949
  by (cases p, cases q) (simp add: quotient_of_Fract)
haftmann@35369
   950
haftmann@35369
   951
lemma rat_times_code [code abstract]:
haftmann@35369
   952
  "quotient_of (p * q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369
   953
     in normalize (a * b, c * d))"
haftmann@35369
   954
  by (cases p, cases q) (simp add: quotient_of_Fract)
berghofe@24533
   955
haftmann@35369
   956
lemma rat_inverse_code [code abstract]:
haftmann@35369
   957
  "quotient_of (inverse p) = (let (a, b) = quotient_of p
haftmann@35369
   958
    in if a = 0 then (0, 1) else (sgn a * b, \<bar>a\<bar>))"
haftmann@35369
   959
proof (cases p)
haftmann@35369
   960
  case (Fract a b) then show ?thesis
haftmann@35369
   961
    by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract gcd_int.commute)
haftmann@35369
   962
qed
haftmann@35369
   963
haftmann@35369
   964
lemma rat_divide_code [code abstract]:
haftmann@35369
   965
  "quotient_of (p / q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
haftmann@35369
   966
     in normalize (a * d, c * b))"
haftmann@35369
   967
  by (cases p, cases q) (simp add: quotient_of_Fract)
haftmann@35369
   968
haftmann@35369
   969
lemma rat_abs_code [code abstract]:
haftmann@35369
   970
  "quotient_of \<bar>p\<bar> = (let (a, b) = quotient_of p in (\<bar>a\<bar>, b))"
haftmann@35369
   971
  by (cases p) (simp add: quotient_of_Fract)
haftmann@35369
   972
haftmann@35369
   973
lemma rat_sgn_code [code abstract]:
haftmann@35369
   974
  "quotient_of (sgn p) = (sgn (fst (quotient_of p)), 1)"
haftmann@35369
   975
proof (cases p)
haftmann@35369
   976
  case (Fract a b) then show ?thesis
haftmann@35369
   977
  by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract)
haftmann@35369
   978
qed
berghofe@24533
   979
bulwahn@43733
   980
lemma rat_floor_code [code]:
bulwahn@43733
   981
  "floor p = (let (a, b) = quotient_of p in a div b)"
bulwahn@43733
   982
by (cases p) (simp add: quotient_of_Fract floor_Fract)
bulwahn@43733
   983
haftmann@38857
   984
instantiation rat :: equal
haftmann@26513
   985
begin
haftmann@26513
   986
haftmann@35369
   987
definition [code]:
haftmann@38857
   988
  "HOL.equal a b \<longleftrightarrow> quotient_of a = quotient_of b"
haftmann@26513
   989
haftmann@35369
   990
instance proof
haftmann@38857
   991
qed (simp add: equal_rat_def quotient_of_inject_eq)
haftmann@26513
   992
haftmann@28351
   993
lemma rat_eq_refl [code nbe]:
haftmann@38857
   994
  "HOL.equal (r::rat) r \<longleftrightarrow> True"
haftmann@38857
   995
  by (rule equal_refl)
haftmann@28351
   996
haftmann@26513
   997
end
berghofe@24533
   998
haftmann@35369
   999
lemma rat_less_eq_code [code]:
haftmann@35369
  1000
  "p \<le> q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d \<le> c * b)"
haftmann@35726
  1001
  by (cases p, cases q) (simp add: quotient_of_Fract mult.commute)
berghofe@24533
  1002
haftmann@35369
  1003
lemma rat_less_code [code]:
haftmann@35369
  1004
  "p < q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d < c * b)"
haftmann@35726
  1005
  by (cases p, cases q) (simp add: quotient_of_Fract mult.commute)
berghofe@24533
  1006
haftmann@35369
  1007
lemma [code]:
haftmann@35369
  1008
  "of_rat p = (let (a, b) = quotient_of p in of_int a / of_int b)"
haftmann@35369
  1009
  by (cases p) (simp add: quotient_of_Fract of_rat_rat)
haftmann@27652
  1010
huffman@47108
  1011
huffman@47108
  1012
text {* Quickcheck *}
huffman@47108
  1013
haftmann@31203
  1014
definition (in term_syntax)
haftmann@32657
  1015
  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
  1016
  [code_unfold]: "valterm_fract k l = Code_Evaluation.valtermify Fract {\<cdot>} k {\<cdot>} l"
haftmann@31203
  1017
haftmann@37751
  1018
notation fcomp (infixl "\<circ>>" 60)
haftmann@37751
  1019
notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@31203
  1020
haftmann@31203
  1021
instantiation rat :: random
haftmann@31203
  1022
begin
haftmann@31203
  1023
haftmann@31203
  1024
definition
haftmann@37751
  1025
  "Quickcheck.random i = Quickcheck.random i \<circ>\<rightarrow> (\<lambda>num. Random.range i \<circ>\<rightarrow> (\<lambda>denom. Pair (
haftmann@31205
  1026
     let j = Code_Numeral.int_of (denom + 1)
haftmann@32657
  1027
     in valterm_fract num (j, \<lambda>u. Code_Evaluation.term_of j))))"
haftmann@31203
  1028
haftmann@31203
  1029
instance ..
haftmann@31203
  1030
haftmann@31203
  1031
end
haftmann@31203
  1032
haftmann@37751
  1033
no_notation fcomp (infixl "\<circ>>" 60)
haftmann@37751
  1034
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@31203
  1035
bulwahn@41920
  1036
instantiation rat :: exhaustive
bulwahn@41231
  1037
begin
bulwahn@41231
  1038
bulwahn@41231
  1039
definition
bulwahn@45818
  1040
  "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
  1041
bulwahn@42311
  1042
instance ..
bulwahn@42311
  1043
bulwahn@42311
  1044
end
bulwahn@42311
  1045
bulwahn@42311
  1046
instantiation rat :: full_exhaustive
bulwahn@42311
  1047
begin
bulwahn@42311
  1048
bulwahn@42311
  1049
definition
bulwahn@45818
  1050
  "full_exhaustive_rat f d = Quickcheck_Exhaustive.full_exhaustive (%(l, _). Quickcheck_Exhaustive.full_exhaustive (%k.
bulwahn@45507
  1051
     f (let j = Code_Numeral.int_of l + 1
bulwahn@45507
  1052
        in valterm_fract k (j, %_. Code_Evaluation.term_of j))) d) d"
bulwahn@41231
  1053
bulwahn@41231
  1054
instance ..
bulwahn@41231
  1055
bulwahn@41231
  1056
end
bulwahn@41231
  1057
bulwahn@43889
  1058
instantiation rat :: partial_term_of
bulwahn@43889
  1059
begin
bulwahn@43889
  1060
bulwahn@43889
  1061
instance ..
bulwahn@43889
  1062
bulwahn@43889
  1063
end
bulwahn@43889
  1064
bulwahn@43889
  1065
lemma [code]:
bulwahn@46758
  1066
  "partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Narrowing_variable p tt) == Code_Evaluation.Free (STR ''_'') (Typerep.Typerep (STR ''Rat.rat'') [])"
bulwahn@46758
  1067
  "partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Narrowing_constructor 0 [l, k]) ==
bulwahn@45507
  1068
     Code_Evaluation.App (Code_Evaluation.Const (STR ''Rat.Frct'')
bulwahn@45507
  1069
     (Typerep.Typerep (STR ''fun'') [Typerep.Typerep (STR ''Product_Type.prod'') [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''Int.int'') []],
bulwahn@45507
  1070
        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
  1071
by (rule partial_term_of_anything)+
bulwahn@43889
  1072
bulwahn@43887
  1073
instantiation rat :: narrowing
bulwahn@43887
  1074
begin
bulwahn@43887
  1075
bulwahn@43887
  1076
definition
bulwahn@45507
  1077
  "narrowing = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.apply
bulwahn@45507
  1078
    (Quickcheck_Narrowing.cons (%nom denom. Fract nom denom)) narrowing) narrowing"
bulwahn@43887
  1079
bulwahn@43887
  1080
instance ..
bulwahn@43887
  1081
bulwahn@43887
  1082
end
bulwahn@43887
  1083
bulwahn@43887
  1084
bulwahn@45183
  1085
subsection {* Setup for Nitpick *}
berghofe@24533
  1086
blanchet@38287
  1087
declaration {*
blanchet@38287
  1088
  Nitpick_HOL.register_frac_type @{type_name rat}
wenzelm@33209
  1089
   [(@{const_name zero_rat_inst.zero_rat}, @{const_name Nitpick.zero_frac}),
wenzelm@33209
  1090
    (@{const_name one_rat_inst.one_rat}, @{const_name Nitpick.one_frac}),
wenzelm@33209
  1091
    (@{const_name plus_rat_inst.plus_rat}, @{const_name Nitpick.plus_frac}),
wenzelm@33209
  1092
    (@{const_name times_rat_inst.times_rat}, @{const_name Nitpick.times_frac}),
wenzelm@33209
  1093
    (@{const_name uminus_rat_inst.uminus_rat}, @{const_name Nitpick.uminus_frac}),
wenzelm@33209
  1094
    (@{const_name inverse_rat_inst.inverse_rat}, @{const_name Nitpick.inverse_frac}),
blanchet@37397
  1095
    (@{const_name ord_rat_inst.less_rat}, @{const_name Nitpick.less_frac}),
wenzelm@33209
  1096
    (@{const_name ord_rat_inst.less_eq_rat}, @{const_name Nitpick.less_eq_frac}),
blanchet@45478
  1097
    (@{const_name field_char_0_class.of_rat}, @{const_name Nitpick.of_frac})]
blanchet@33197
  1098
*}
blanchet@33197
  1099
blanchet@41792
  1100
lemmas [nitpick_unfold] = inverse_rat_inst.inverse_rat
huffman@47108
  1101
  one_rat_inst.one_rat ord_rat_inst.less_rat
blanchet@37397
  1102
  ord_rat_inst.less_eq_rat plus_rat_inst.plus_rat times_rat_inst.times_rat
blanchet@37397
  1103
  uminus_rat_inst.uminus_rat zero_rat_inst.zero_rat
blanchet@33197
  1104
huffman@35343
  1105
subsection{* Float syntax *}
huffman@35343
  1106
huffman@35343
  1107
syntax "_Float" :: "float_const \<Rightarrow> 'a"    ("_")
huffman@35343
  1108
wenzelm@48891
  1109
ML_file "Tools/float_syntax.ML"
huffman@35343
  1110
setup Float_Syntax.setup
huffman@35343
  1111
huffman@35343
  1112
text{* Test: *}
huffman@35343
  1113
lemma "123.456 = -111.111 + 200 + 30 + 4 + 5/10 + 6/100 + (7/1000::rat)"
huffman@35343
  1114
by simp
huffman@35343
  1115
wenzelm@37143
  1116
huffman@47907
  1117
hide_const (open) normalize positive
wenzelm@37143
  1118
huffman@47906
  1119
lemmas [transfer_rule del] =
huffman@47906
  1120
  rat.All_transfer rat.Ex_transfer rat.rel_eq_transfer forall_rat_transfer
huffman@47906
  1121
  Fract.transfer zero_rat.transfer one_rat.transfer plus_rat.transfer
huffman@47906
  1122
  uminus_rat.transfer times_rat.transfer inverse_rat.transfer
huffman@47907
  1123
  positive.transfer of_rat.transfer
huffman@47906
  1124
huffman@47906
  1125
text {* De-register @{text "rat"} as a quotient type: *}
huffman@47906
  1126
kuncar@47952
  1127
declare Quotient_rat[quot_del]
huffman@47906
  1128
huffman@29880
  1129
end