src/HOL/Rat.thy
author hoelzl
Tue, 12 Nov 2013 19:28:51 +0100
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permissions -rw-r--r--
support of_rat with 0 or 1 on order relations
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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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lemma Domainp_cr_rat [transfer_domain_rule]: "Domainp pcr_rat = (\<lambda>x. snd x \<noteq> 0)"
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by (simp add: rat.domain_eq)
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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: distrib_right, 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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   148
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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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   155
  using assms by transfer simp
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   156
09a896d295bd convert Rat.thy to use lift_definition/transfer
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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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   159
  by (auto simp add: mult_commute)
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   160
09a896d295bd convert Rat.thy to use lift_definition/transfer
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lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"
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   162
  by transfer simp
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   163
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   164
definition
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  divide_rat_def: "q / r = q * inverse (r::rat)"
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   166
09a896d295bd convert Rat.thy to use lift_definition/transfer
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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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   173
    by transfer simp
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  show "q * r = r * q"
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   175
    by transfer simp
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  show "1 * q = q"
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   177
    by transfer simp
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   178
  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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   181
    by transfer simp
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   182
  show "0 + q = q"
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   183
    by transfer simp
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   184
  show "- q + q = 0"
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   185
    by transfer simp
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   186
  show "q - r = q + - r"
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    by (fact diff_rat_def)
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   188
  show "(q + r) * s = q * s + r * s"
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   189
    by transfer (simp add: algebra_simps)
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  show "(0::rat) \<noteq> 1"
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   191
    by transfer simp
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  { assume "q \<noteq> 0" thus "inverse q * q = 1"
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   193
    by transfer simp }
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  show "q / r = q * inverse r"
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   195
    by (fact divide_rat_def)
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   196
  show "inverse 0 = (0::rat)"
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    by transfer simp
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qed
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63161d5f8f29 rearrange instantiations
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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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   204
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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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   207
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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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   213
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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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   221
  using Fract_of_int_eq [of "neg_numeral w"]
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   222
  by (simp_all add: Zero_rat_def One_rat_def eq_rat)
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   223
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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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   229
  by (simp_all add: rat_number_collapse)
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   230
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   231
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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   233
  assumes 0: "q = 0 \<Longrightarrow> C"
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   234
  shows C
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   235
proof (cases "q = 0")
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   236
  case True then show C using 0 by auto
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   237
next
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   238
  case False
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   239
  then obtain a b where "q = Fract a b" and "b > 0" and "coprime a b" by (cases q) auto
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   240
  with False have "0 \<noteq> Fract a b" by simp
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   241
  with `b > 0` have "a \<noteq> 0" by (simp add: Zero_rat_def eq_rat)
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   242
  with Fract `q = Fract a b` `b > 0` `coprime a b` show C by blast
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   243
qed
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   244
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subsubsection {* Function @{text normalize} *}
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   246
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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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   248
proof (cases "b = 0")
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  case True then show ?thesis by (simp add: eq_rat)
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   250
next
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  case False
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   252
  moreover have "b div gcd a b * gcd a b = b"
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   253
    by (rule dvd_div_mult_self) simp
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   254
  ultimately have "b div gcd a b \<noteq> 0" by auto
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   255
  with False show ?thesis by (simp add: eq_rat dvd_div_mult mult_commute [of a])
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   256
qed
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   257
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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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   261
    else (let a = - gcd (fst p) (snd p) in (fst p div a, snd p div a)))"
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   262
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   263
lemma normalize_crossproduct:
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   264
  assumes "q \<noteq> 0" "s \<noteq> 0"
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   265
  assumes "normalize (p, q) = normalize (r, s)"
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   266
  shows "p * s = r * q"
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   267
proof -
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   268
  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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   269
  proof -
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   270
    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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   271
    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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   272
    with assms show "p * s = q * r" by (auto simp add: mult_ac sgn_times sgn_0_0)
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   273
  qed
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   274
  from assms show ?thesis
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   275
    by (auto simp add: normalize_def Let_def dvd_div_div_eq_mult mult_commute sgn_times split: if_splits intro: aux)
33805
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   276
qed
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nipkow
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diff changeset
   277
35369
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   278
lemma normalize_eq: "normalize (a, b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
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   279
  by (auto simp add: normalize_def Let_def Fract_coprime dvd_div_neg rat_number_collapse
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   280
    split:split_if_asm)
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   281
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lemma normalize_denom_pos: "normalize r = (p, q) \<Longrightarrow> q > 0"
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haftmann
parents: 35293
diff changeset
   283
  by (auto simp add: normalize_def Let_def dvd_div_neg pos_imp_zdiv_neg_iff nonneg1_imp_zdiv_pos_iff
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   284
    split:split_if_asm)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   285
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   286
lemma normalize_coprime: "normalize r = (p, q) \<Longrightarrow> coprime p q"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   287
  by (auto simp add: normalize_def Let_def dvd_div_neg div_gcd_coprime_int
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   288
    split:split_if_asm)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   289
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   290
lemma normalize_stable [simp]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   291
  "q > 0 \<Longrightarrow> coprime p q \<Longrightarrow> normalize (p, q) = (p, q)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   292
  by (simp add: normalize_def)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   293
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   294
lemma normalize_denom_zero [simp]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   295
  "normalize (p, 0) = (0, 1)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   296
  by (simp add: normalize_def)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   297
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   298
lemma normalize_negative [simp]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   299
  "q < 0 \<Longrightarrow> normalize (p, q) = normalize (- p, - q)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   300
  by (simp add: normalize_def Let_def dvd_div_neg dvd_neg_div)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   301
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   302
text{*
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   303
  Decompose a fraction into normalized, i.e. coprime numerator and denominator:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   304
*}
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   305
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   306
definition quotient_of :: "rat \<Rightarrow> int \<times> int" where
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   307
  "quotient_of x = (THE pair. x = Fract (fst pair) (snd pair) &
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   308
                   snd pair > 0 & coprime (fst pair) (snd pair))"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   309
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   310
lemma quotient_of_unique:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   311
  "\<exists>!p. r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   312
proof (cases r)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   313
  case (Fract a b)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   314
  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
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   315
  then show ?thesis proof (rule ex1I)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   316
    fix p
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   317
    obtain c d :: int where p: "p = (c, d)" by (cases p)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   318
    assume "r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   319
    with p have Fract': "r = Fract c d" "d > 0" "coprime c d" by simp_all
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   320
    have "c = a \<and> d = b"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   321
    proof (cases "a = 0")
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   322
      case True with Fract Fract' show ?thesis by (simp add: eq_rat)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   323
    next
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   324
      case False
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   325
      with Fract Fract' have *: "c * b = a * d" and "c \<noteq> 0" by (auto simp add: eq_rat)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   326
      then have "c * b > 0 \<longleftrightarrow> a * d > 0" by auto
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   327
      with `b > 0` `d > 0` have "a > 0 \<longleftrightarrow> c > 0" by (simp add: zero_less_mult_iff)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   328
      with `a \<noteq> 0` `c \<noteq> 0` have sgn: "sgn a = sgn c" by (auto simp add: not_less)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   329
      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>"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   330
        by (simp add: coprime_crossproduct_int)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   331
      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
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   332
      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)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   333
      with sgn * show ?thesis by (auto simp add: sgn_0_0)
33805
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   334
    qed
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   335
    with p show "p = (a, b)" by simp
33805
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   336
  qed
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   337
qed
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   338
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   339
lemma quotient_of_Fract [code]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   340
  "quotient_of (Fract a b) = normalize (a, b)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   341
proof -
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   342
  have "Fract a b = Fract (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?Fract)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   343
    by (rule sym) (auto intro: normalize_eq)
52146
wenzelm
parents: 51956
diff changeset
   344
  moreover have "0 < snd (normalize (a, b))" (is ?denom_pos)
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   345
    by (cases "normalize (a, b)") (rule normalize_denom_pos, simp)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   346
  moreover have "coprime (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?coprime)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   347
    by (rule normalize_coprime) simp
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   348
  ultimately have "?Fract \<and> ?denom_pos \<and> ?coprime" by blast
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   349
  with quotient_of_unique have
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   350
    "(THE p. Fract a b = Fract (fst p) (snd p) \<and> 0 < snd p \<and> coprime (fst p) (snd p)) = normalize (a, b)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   351
    by (rule the1_equality)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   352
  then show ?thesis by (simp add: quotient_of_def)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   353
qed
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   354
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   355
lemma quotient_of_number [simp]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   356
  "quotient_of 0 = (0, 1)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   357
  "quotient_of 1 = (1, 1)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   358
  "quotient_of (numeral k) = (numeral k, 1)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   359
  "quotient_of (neg_numeral k) = (neg_numeral k, 1)"
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   360
  by (simp_all add: rat_number_expand quotient_of_Fract)
33805
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   361
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   362
lemma quotient_of_eq: "quotient_of (Fract a b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   363
  by (simp add: quotient_of_Fract normalize_eq)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   364
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   365
lemma quotient_of_denom_pos: "quotient_of r = (p, q) \<Longrightarrow> q > 0"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   366
  by (cases r) (simp add: quotient_of_Fract normalize_denom_pos)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   367
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   368
lemma quotient_of_coprime: "quotient_of r = (p, q) \<Longrightarrow> coprime p q"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   369
  by (cases r) (simp add: quotient_of_Fract normalize_coprime)
33805
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   370
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   371
lemma quotient_of_inject:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   372
  assumes "quotient_of a = quotient_of b"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   373
  shows "a = b"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   374
proof -
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   375
  obtain p q r s where a: "a = Fract p q"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   376
    and b: "b = Fract r s"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   377
    and "q > 0" and "s > 0" by (cases a, cases b)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   378
  with assms show ?thesis by (simp add: eq_rat quotient_of_Fract normalize_crossproduct)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   379
qed
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   380
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   381
lemma quotient_of_inject_eq:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   382
  "quotient_of a = quotient_of b \<longleftrightarrow> a = b"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   383
  by (auto simp add: quotient_of_inject)
33805
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   384
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   385
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   386
subsubsection {* Various *}
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   387
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   388
lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   389
  by (simp add: Fract_of_int_eq [symmetric])
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   390
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   391
lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   392
  by (simp add: rat_number_expand)
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   393
50178
ad52ddd35c3a add quotient_of_div
hoelzl
parents: 49962
diff changeset
   394
lemma quotient_of_div:
ad52ddd35c3a add quotient_of_div
hoelzl
parents: 49962
diff changeset
   395
  assumes r: "quotient_of r = (n,d)"
ad52ddd35c3a add quotient_of_div
hoelzl
parents: 49962
diff changeset
   396
  shows "r = of_int n / of_int d"
ad52ddd35c3a add quotient_of_div
hoelzl
parents: 49962
diff changeset
   397
proof -
ad52ddd35c3a add quotient_of_div
hoelzl
parents: 49962
diff changeset
   398
  from theI'[OF quotient_of_unique[of r], unfolded r[unfolded quotient_of_def]]
ad52ddd35c3a add quotient_of_div
hoelzl
parents: 49962
diff changeset
   399
  have "r = Fract n d" by simp
ad52ddd35c3a add quotient_of_div
hoelzl
parents: 49962
diff changeset
   400
  thus ?thesis using Fract_of_int_quotient by simp
ad52ddd35c3a add quotient_of_div
hoelzl
parents: 49962
diff changeset
   401
qed
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   402
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   403
subsubsection {* The ordered field of rational numbers *}
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   404
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   405
lift_definition positive :: "rat \<Rightarrow> bool"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   406
  is "\<lambda>x. 0 < fst x * snd x"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   407
proof (clarsimp)
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   408
  fix a b c d :: int
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   409
  assume "b \<noteq> 0" and "d \<noteq> 0" and "a * d = c * b"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   410
  hence "a * d * b * d = c * b * b * d"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   411
    by simp
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52146
diff changeset
   412
  hence "a * b * d\<^sup>2 = c * d * b\<^sup>2"
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   413
    unfolding power2_eq_square by (simp add: mult_ac)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 52146
diff changeset
   414
  hence "0 < a * b * d\<^sup>2 \<longleftrightarrow> 0 < c * d * b\<^sup>2"
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   415
    by simp
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   416
  thus "0 < a * b \<longleftrightarrow> 0 < c * d"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   417
    using `b \<noteq> 0` and `d \<noteq> 0`
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   418
    by (simp add: zero_less_mult_iff)
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   419
qed
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   420
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   421
lemma positive_zero: "\<not> positive 0"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   422
  by transfer simp
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   423
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   424
lemma positive_add:
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   425
  "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x + y)"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   426
apply transfer
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   427
apply (simp add: zero_less_mult_iff)
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   428
apply (elim disjE, simp_all add: add_pos_pos add_neg_neg
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   429
  mult_pos_pos mult_pos_neg mult_neg_pos mult_neg_neg)
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   430
done
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   431
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   432
lemma positive_mult:
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   433
  "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x * y)"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   434
by transfer (drule (1) mult_pos_pos, simp add: mult_ac)
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   435
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   436
lemma positive_minus:
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   437
  "\<not> positive x \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> positive (- x)"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   438
by transfer (force simp: neq_iff zero_less_mult_iff mult_less_0_iff)
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   439
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   440
instantiation rat :: linordered_field_inverse_zero
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   441
begin
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   442
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   443
definition
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   444
  "x < y \<longleftrightarrow> positive (y - x)"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   445
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   446
definition
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   447
  "x \<le> (y::rat) \<longleftrightarrow> x < y \<or> x = y"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   448
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   449
definition
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   450
  "abs (a::rat) = (if a < 0 then - a else a)"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   451
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   452
definition
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   453
  "sgn (a::rat) = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   454
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   455
instance proof
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   456
  fix a b c :: rat
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   457
  show "\<bar>a\<bar> = (if a < 0 then - a else a)"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   458
    by (rule abs_rat_def)
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   459
  show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   460
    unfolding less_eq_rat_def less_rat_def
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   461
    by (auto, drule (1) positive_add, simp_all add: positive_zero)
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   462
  show "a \<le> a"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   463
    unfolding less_eq_rat_def by simp
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   464
  show "a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   465
    unfolding less_eq_rat_def less_rat_def
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   466
    by (auto, drule (1) positive_add, simp add: algebra_simps)
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   467
  show "a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> a = b"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   468
    unfolding less_eq_rat_def less_rat_def
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   469
    by (auto, drule (1) positive_add, simp add: positive_zero)
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   470
  show "a \<le> b \<Longrightarrow> c + a \<le> c + b"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53652
diff changeset
   471
    unfolding less_eq_rat_def less_rat_def by auto
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   472
  show "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   473
    by (rule sgn_rat_def)
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   474
  show "a \<le> b \<or> b \<le> a"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   475
    unfolding less_eq_rat_def less_rat_def
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   476
    by (auto dest!: positive_minus)
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   477
  show "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   478
    unfolding less_rat_def
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   479
    by (drule (1) positive_mult, simp add: algebra_simps)
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   480
qed
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   481
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   482
end
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   483
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   484
instantiation rat :: distrib_lattice
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   485
begin
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   486
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   487
definition
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   488
  "(inf :: rat \<Rightarrow> rat \<Rightarrow> rat) = min"
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   489
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   490
definition
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   491
  "(sup :: rat \<Rightarrow> rat \<Rightarrow> rat) = max"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   492
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   493
instance proof
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   494
qed (auto simp add: inf_rat_def sup_rat_def min_max.sup_inf_distrib1)
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   495
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   496
end
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   497
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   498
lemma positive_rat: "positive (Fract a b) \<longleftrightarrow> 0 < a * b"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   499
  by transfer simp
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   500
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   501
lemma less_rat [simp]:
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   502
  assumes "b \<noteq> 0" and "d \<noteq> 0"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   503
  shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   504
  using assms unfolding less_rat_def
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   505
  by (simp add: positive_rat algebra_simps)
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   506
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   507
lemma le_rat [simp]:
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   508
  assumes "b \<noteq> 0" and "d \<noteq> 0"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   509
  shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   510
  using assms unfolding le_less by (simp add: eq_rat)
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   511
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   512
lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35063
diff changeset
   513
  by (auto simp add: abs_rat_def zabs_def Zero_rat_def not_less le_less eq_rat zero_less_mult_iff)
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   514
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   515
lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)"
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   516
  unfolding Fract_of_int_eq
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   517
  by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat)
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   518
    (auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   519
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   520
lemma Rat_induct_pos [case_names Fract, induct type: rat]:
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   521
  assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   522
  shows "P q"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   523
proof (cases q)
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   524
  have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   525
  proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   526
    fix a::int and b::int
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   527
    assume b: "b < 0"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   528
    hence "0 < -b" by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   529
    hence "P (Fract (-a) (-b))" by (rule step)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   530
    thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   531
  qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   532
  case (Fract a b)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   533
  thus "P q" by (force simp add: linorder_neq_iff step step')
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   534
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   535
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   536
lemma zero_less_Fract_iff:
30095
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   537
  "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a"
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   538
  by (simp add: Zero_rat_def zero_less_mult_iff)
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   539
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   540
lemma Fract_less_zero_iff:
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   541
  "0 < b \<Longrightarrow> Fract a b < 0 \<longleftrightarrow> a < 0"
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   542
  by (simp add: Zero_rat_def mult_less_0_iff)
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   543
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   544
lemma zero_le_Fract_iff:
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   545
  "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a"
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   546
  by (simp add: Zero_rat_def zero_le_mult_iff)
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   547
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   548
lemma Fract_le_zero_iff:
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   549
  "0 < b \<Longrightarrow> Fract a b \<le> 0 \<longleftrightarrow> a \<le> 0"
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   550
  by (simp add: Zero_rat_def mult_le_0_iff)
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   551
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   552
lemma one_less_Fract_iff:
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   553
  "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a"
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   554
  by (simp add: One_rat_def mult_less_cancel_right_disj)
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   555
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   556
lemma Fract_less_one_iff:
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   557
  "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b"
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   558
  by (simp add: One_rat_def mult_less_cancel_right_disj)
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   559
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   560
lemma one_le_Fract_iff:
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   561
  "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a"
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   562
  by (simp add: One_rat_def mult_le_cancel_right)
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   563
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   564
lemma Fract_le_one_iff:
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   565
  "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b"
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   566
  by (simp add: One_rat_def mult_le_cancel_right)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   567
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   568
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   569
subsubsection {* Rationals are an Archimedean field *}
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   570
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   571
lemma rat_floor_lemma:
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   572
  shows "of_int (a div b) \<le> Fract a b \<and> Fract a b < of_int (a div b + 1)"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   573
proof -
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   574
  have "Fract a b = of_int (a div b) + Fract (a mod b) b"
35293
06a98796453e remove unneeded premise from rat_floor_lemma and floor_Fract
huffman
parents: 35216
diff changeset
   575
    by (cases "b = 0", simp, simp add: of_int_rat)
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   576
  moreover have "0 \<le> Fract (a mod b) b \<and> Fract (a mod b) b < 1"
35293
06a98796453e remove unneeded premise from rat_floor_lemma and floor_Fract
huffman
parents: 35216
diff changeset
   577
    unfolding Fract_of_int_quotient
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   578
    by (rule linorder_cases [of b 0]) (simp add: divide_nonpos_neg, simp, simp add: divide_nonneg_pos)
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   579
  ultimately show ?thesis by simp
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   580
qed
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   581
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   582
instance rat :: archimedean_field
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   583
proof
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   584
  fix r :: rat
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   585
  show "\<exists>z. r \<le> of_int z"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   586
  proof (induct r)
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   587
    case (Fract a b)
35293
06a98796453e remove unneeded premise from rat_floor_lemma and floor_Fract
huffman
parents: 35216
diff changeset
   588
    have "Fract a b \<le> of_int (a div b + 1)"
06a98796453e remove unneeded premise from rat_floor_lemma and floor_Fract
huffman
parents: 35216
diff changeset
   589
      using rat_floor_lemma [of a b] by simp
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   590
    then show "\<exists>z. Fract a b \<le> of_int z" ..
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   591
  qed
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   592
qed
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   593
43732
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   594
instantiation rat :: floor_ceiling
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   595
begin
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   596
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   597
definition [code del]:
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   598
  "floor (x::rat) = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))"
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   599
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   600
instance proof
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   601
  fix x :: rat
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   602
  show "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   603
    unfolding floor_rat_def using floor_exists1 by (rule theI')
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   604
qed
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   605
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   606
end
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   607
35293
06a98796453e remove unneeded premise from rat_floor_lemma and floor_Fract
huffman
parents: 35216
diff changeset
   608
lemma floor_Fract: "floor (Fract a b) = a div b"
06a98796453e remove unneeded premise from rat_floor_lemma and floor_Fract
huffman
parents: 35216
diff changeset
   609
  using rat_floor_lemma [of a b]
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   610
  by (simp add: floor_unique)
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   611
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   612
31100
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   613
subsection {* Linear arithmetic setup *}
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   614
31100
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   615
declaration {*
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   616
  K (Lin_Arith.add_inj_thms [@{thm of_nat_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   617
    (* not needed because x < (y::nat) can be rewritten as Suc x <= y: of_nat_less_iff RS iffD2 *)
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   618
  #> Lin_Arith.add_inj_thms [@{thm of_int_le_iff} RS iffD2, @{thm of_int_eq_iff} RS iffD2]
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   619
    (* not needed because x < (y::int) can be rewritten as x + 1 <= y: of_int_less_iff RS iffD2 *)
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   620
  #> Lin_Arith.add_simps [@{thm neg_less_iff_less},
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   621
      @{thm True_implies_equals},
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 48891
diff changeset
   622
      read_instantiate @{context} [(("a", 0), "(numeral ?v)")] @{thm distrib_left},
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 48891
diff changeset
   623
      read_instantiate @{context} [(("a", 0), "(neg_numeral ?v)")] @{thm distrib_left},
31100
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   624
      @{thm divide_1}, @{thm divide_zero_left},
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   625
      @{thm times_divide_eq_right}, @{thm times_divide_eq_left},
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   626
      @{thm minus_divide_left} RS sym, @{thm minus_divide_right} RS sym,
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   627
      @{thm of_int_minus}, @{thm of_int_diff},
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   628
      @{thm of_int_of_nat_eq}]
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   629
  #> Lin_Arith.add_simprocs Numeral_Simprocs.field_cancel_numeral_factors
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   630
  #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat => rat"})
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   631
  #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int => rat"}))
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   632
*}
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   633
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   634
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   635
subsection {* Embedding from Rationals to other Fields *}
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   636
24198
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   637
class field_char_0 = field + ring_char_0
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   638
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 33814
diff changeset
   639
subclass (in linordered_field) field_char_0 ..
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   640
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   641
context field_char_0
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   642
begin
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   643
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   644
lift_definition of_rat :: "rat \<Rightarrow> 'a"
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   645
  is "\<lambda>x. of_int (fst x) / of_int (snd x)"
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   646
apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   647
apply (simp only: of_int_mult [symmetric])
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   648
done
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   649
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   650
end
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   651
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   652
lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   653
  by transfer simp
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   654
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   655
lemma of_rat_0 [simp]: "of_rat 0 = 0"
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   656
  by transfer simp
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   657
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   658
lemma of_rat_1 [simp]: "of_rat 1 = 1"
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   659
  by transfer simp
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   660
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   661
lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   662
  by transfer (simp add: add_frac_eq)
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   663
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   664
lemma of_rat_minus: "of_rat (- a) = - of_rat a"
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   665
  by transfer simp
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   666
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   667
lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53652
diff changeset
   668
  using of_rat_add [of a "- b"] by (simp add: of_rat_minus)
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   669
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   670
lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   671
apply transfer
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   672
apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   673
done
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   674
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   675
lemma nonzero_of_rat_inverse:
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   676
  "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   677
apply (rule inverse_unique [symmetric])
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   678
apply (simp add: of_rat_mult [symmetric])
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   679
done
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   680
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   681
lemma of_rat_inverse:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   682
  "(of_rat (inverse a)::'a::{field_char_0, field_inverse_zero}) =
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   683
   inverse (of_rat a)"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   684
by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   685
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   686
lemma nonzero_of_rat_divide:
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   687
  "b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   688
by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   689
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   690
lemma of_rat_divide:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   691
  "(of_rat (a / b)::'a::{field_char_0, field_inverse_zero})
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   692
   = of_rat a / of_rat b"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   693
by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   694
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   695
lemma of_rat_power:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30960
diff changeset
   696
  "(of_rat (a ^ n)::'a::field_char_0) = of_rat a ^ n"
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30242
diff changeset
   697
by (induct n) (simp_all add: of_rat_mult)
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   698
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   699
lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   700
apply transfer
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   701
apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   702
apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   703
done
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   704
54409
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   705
lemma of_rat_eq_0_iff [simp]: "(of_rat a = 0) = (a = 0)"
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   706
  using of_rat_eq_iff [of _ 0] by simp
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   707
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   708
lemma zero_eq_of_rat_iff [simp]: "(0 = of_rat a) = (0 = a)"
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   709
  by simp
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   710
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   711
lemma of_rat_eq_1_iff [simp]: "(of_rat a = 1) = (a = 1)"
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   712
  using of_rat_eq_iff [of _ 1] by simp
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   713
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   714
lemma one_eq_of_rat_iff [simp]: "(1 = of_rat a) = (1 = a)"
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   715
  by simp
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   716
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   717
lemma of_rat_less:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 33814
diff changeset
   718
  "(of_rat r :: 'a::linordered_field) < of_rat s \<longleftrightarrow> r < s"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   719
proof (induct r, induct s)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   720
  fix a b c d :: int
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   721
  assume not_zero: "b > 0" "d > 0"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   722
  then have "b * d > 0" by (rule mult_pos_pos)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   723
  have of_int_divide_less_eq:
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   724
    "(of_int a :: 'a) / of_int b < of_int c / of_int d
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   725
      \<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   726
    using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 33814
diff changeset
   727
  show "(of_rat (Fract a b) :: 'a::linordered_field) < of_rat (Fract c d)
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   728
    \<longleftrightarrow> Fract a b < Fract c d"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   729
    using not_zero `b * d > 0`
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   730
    by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   731
qed
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   732
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   733
lemma of_rat_less_eq:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 33814
diff changeset
   734
  "(of_rat r :: 'a::linordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   735
  unfolding le_less by (auto simp add: of_rat_less)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   736
54409
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   737
lemma of_rat_le_0_iff [simp]: "((of_rat r :: 'a::linordered_field) \<le> 0) = (r \<le> 0)"
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   738
  using of_rat_less_eq [of r 0, where 'a='a] by simp
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   739
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   740
lemma zero_le_of_rat_iff [simp]: "(0 \<le> (of_rat r :: 'a::linordered_field)) = (0 \<le> r)"
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   741
  using of_rat_less_eq [of 0 r, where 'a='a] by simp
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   742
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   743
lemma of_rat_le_1_iff [simp]: "((of_rat r :: 'a::linordered_field) \<le> 1) = (r \<le> 1)"
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   744
  using of_rat_less_eq [of r 1] by simp
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   745
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   746
lemma one_le_of_rat_iff [simp]: "(1 \<le> (of_rat r :: 'a::linordered_field)) = (1 \<le> r)"
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   747
  using of_rat_less_eq [of 1 r] by simp
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   748
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   749
lemma of_rat_less_0_iff [simp]: "((of_rat r :: 'a::linordered_field) < 0) = (r < 0)"
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   750
  using of_rat_less [of r 0, where 'a='a] by simp
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   751
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   752
lemma zero_less_of_rat_iff [simp]: "(0 < (of_rat r :: 'a::linordered_field)) = (0 < r)"
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   753
  using of_rat_less [of 0 r, where 'a='a] by simp
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   754
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   755
lemma of_rat_less_1_iff [simp]: "((of_rat r :: 'a::linordered_field) < 1) = (r < 1)"
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   756
  using of_rat_less [of r 1] by simp
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   757
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   758
lemma one_less_of_rat_iff [simp]: "(1 < (of_rat r :: 'a::linordered_field)) = (1 < r)"
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   759
  using of_rat_less [of 1 r] by simp
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   760
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   761
lemma of_rat_eq_id [simp]: "of_rat = id"
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   762
proof
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   763
  fix a
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   764
  show "of_rat a = id a"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   765
  by (induct a)
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   766
     (simp add: of_rat_rat Fract_of_int_eq [symmetric])
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   767
qed
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   768
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   769
text{*Collapse nested embeddings*}
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   770
lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   771
by (induct n) (simp_all add: of_rat_add)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   772
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   773
lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   774
by (cases z rule: int_diff_cases) (simp add: of_rat_diff)
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   775
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   776
lemma of_rat_numeral_eq [simp]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   777
  "of_rat (numeral w) = numeral w"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   778
using of_rat_of_int_eq [of "numeral w"] by simp
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   779
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   780
lemma of_rat_neg_numeral_eq [simp]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   781
  "of_rat (neg_numeral w) = neg_numeral w"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   782
using of_rat_of_int_eq [of "neg_numeral w"] by simp
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   783
23879
4776af8be741 split class abs from class minus
haftmann
parents: 23429
diff changeset
   784
lemmas zero_rat = Zero_rat_def
4776af8be741 split class abs from class minus
haftmann
parents: 23429
diff changeset
   785
lemmas one_rat = One_rat_def
4776af8be741 split class abs from class minus
haftmann
parents: 23429
diff changeset
   786
24198
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   787
abbreviation
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   788
  rat_of_nat :: "nat \<Rightarrow> rat"
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   789
where
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   790
  "rat_of_nat \<equiv> of_nat"
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   791
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   792
abbreviation
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   793
  rat_of_int :: "int \<Rightarrow> rat"
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   794
where
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   795
  "rat_of_int \<equiv> of_int"
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   796
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   797
subsection {* The Set of Rational Numbers *}
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   798
28001
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   799
context field_char_0
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   800
begin
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   801
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   802
definition
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   803
  Rats  :: "'a set" where
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   804
  "Rats = range of_rat"
28001
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   805
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   806
notation (xsymbols)
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   807
  Rats  ("\<rat>")
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   808
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   809
end
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   810
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   811
lemma Rats_of_rat [simp]: "of_rat r \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   812
by (simp add: Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   813
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   814
lemma Rats_of_int [simp]: "of_int z \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   815
by (subst of_rat_of_int_eq [symmetric], rule Rats_of_rat)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   816
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   817
lemma Rats_of_nat [simp]: "of_nat n \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   818
by (subst of_rat_of_nat_eq [symmetric], rule Rats_of_rat)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   819
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   820
lemma Rats_number_of [simp]: "numeral w \<in> Rats"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   821
by (subst of_rat_numeral_eq [symmetric], rule Rats_of_rat)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   822
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   823
lemma Rats_neg_number_of [simp]: "neg_numeral w \<in> Rats"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   824
by (subst of_rat_neg_numeral_eq [symmetric], rule Rats_of_rat)
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   825
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   826
lemma Rats_0 [simp]: "0 \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   827
apply (unfold Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   828
apply (rule range_eqI)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   829
apply (rule of_rat_0 [symmetric])
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   830
done
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   831
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   832
lemma Rats_1 [simp]: "1 \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   833
apply (unfold Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   834
apply (rule range_eqI)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   835
apply (rule of_rat_1 [symmetric])
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   836
done
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   837
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   838
lemma Rats_add [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a + b \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   839
apply (auto simp add: Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   840
apply (rule range_eqI)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   841
apply (rule of_rat_add [symmetric])
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   842
done
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   843
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   844
lemma Rats_minus [simp]: "a \<in> Rats \<Longrightarrow> - a \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   845
apply (auto simp add: Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   846
apply (rule range_eqI)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   847
apply (rule of_rat_minus [symmetric])
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   848
done
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   849
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   850
lemma Rats_diff [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a - b \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   851
apply (auto simp add: Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   852
apply (rule range_eqI)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   853
apply (rule of_rat_diff [symmetric])
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   854
done
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   855
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   856
lemma Rats_mult [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a * b \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   857
apply (auto simp add: Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   858
apply (rule range_eqI)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   859
apply (rule of_rat_mult [symmetric])
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   860
done
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   861
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   862
lemma nonzero_Rats_inverse:
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   863
  fixes a :: "'a::field_char_0"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   864
  shows "\<lbrakk>a \<in> Rats; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   865
apply (auto simp add: Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   866
apply (rule range_eqI)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   867
apply (erule nonzero_of_rat_inverse [symmetric])
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   868
done
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   869
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   870
lemma Rats_inverse [simp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   871
  fixes a :: "'a::{field_char_0, field_inverse_zero}"
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   872
  shows "a \<in> Rats \<Longrightarrow> inverse a \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   873
apply (auto simp add: Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   874
apply (rule range_eqI)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   875
apply (rule of_rat_inverse [symmetric])
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   876
done
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   877
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   878
lemma nonzero_Rats_divide:
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   879
  fixes a b :: "'a::field_char_0"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   880
  shows "\<lbrakk>a \<in> Rats; b \<in> Rats; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   881
apply (auto simp add: Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   882
apply (rule range_eqI)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   883
apply (erule nonzero_of_rat_divide [symmetric])
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   884
done
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   885
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   886
lemma Rats_divide [simp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   887
  fixes a b :: "'a::{field_char_0, field_inverse_zero}"
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   888
  shows "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   889
apply (auto simp add: Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   890
apply (rule range_eqI)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   891
apply (rule of_rat_divide [symmetric])
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   892
done
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   893
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   894
lemma Rats_power [simp]:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30960
diff changeset
   895
  fixes a :: "'a::field_char_0"
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   896
  shows "a \<in> Rats \<Longrightarrow> a ^ n \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   897
apply (auto simp add: Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   898
apply (rule range_eqI)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   899
apply (rule of_rat_power [symmetric])
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   900
done
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   901
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   902
lemma Rats_cases [cases set: Rats]:
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   903
  assumes "q \<in> \<rat>"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   904
  obtains (of_rat) r where "q = of_rat r"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   905
proof -
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   906
  from `q \<in> \<rat>` have "q \<in> range of_rat" unfolding Rats_def .
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   907
  then obtain r where "q = of_rat r" ..
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   908
  then show thesis ..
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   909
qed
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   910
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   911
lemma Rats_induct [case_names of_rat, induct set: Rats]:
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   912
  "q \<in> \<rat> \<Longrightarrow> (\<And>r. P (of_rat r)) \<Longrightarrow> P q"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   913
  by (rule Rats_cases) auto
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   914
28001
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   915
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   916
subsection {* Implementation of rational numbers as pairs of integers *}
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   917
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   918
text {* Formal constructor *}
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   919
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   920
definition Frct :: "int \<times> int \<Rightarrow> rat" where
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   921
  [simp]: "Frct p = Fract (fst p) (snd p)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   922
36112
7fa17a225852 user interface for abstract datatypes is an attribute, not a command
haftmann
parents: 35726
diff changeset
   923
lemma [code abstype]:
7fa17a225852 user interface for abstract datatypes is an attribute, not a command
haftmann
parents: 35726
diff changeset
   924
  "Frct (quotient_of q) = q"
7fa17a225852 user interface for abstract datatypes is an attribute, not a command
haftmann
parents: 35726
diff changeset
   925
  by (cases q) (auto intro: quotient_of_eq)
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   926
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   927
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   928
text {* Numerals *}
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   929
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   930
declare quotient_of_Fract [code abstract]
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   931
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   932
definition of_int :: "int \<Rightarrow> rat"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   933
where
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   934
  [code_abbrev]: "of_int = Int.of_int"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   935
hide_const (open) of_int
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   936
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   937
lemma quotient_of_int [code abstract]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   938
  "quotient_of (Rat.of_int a) = (a, 1)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   939
  by (simp add: of_int_def of_int_rat quotient_of_Fract)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   940
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   941
lemma [code_unfold]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   942
  "numeral k = Rat.of_int (numeral k)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   943
  by (simp add: Rat.of_int_def)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   944
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   945
lemma [code_unfold]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   946
  "neg_numeral k = Rat.of_int (neg_numeral k)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   947
  by (simp add: Rat.of_int_def)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   948
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   949
lemma Frct_code_post [code_post]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   950
  "Frct (0, a) = 0"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   951
  "Frct (a, 0) = 0"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   952
  "Frct (1, 1) = 1"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   953
  "Frct (numeral k, 1) = numeral k"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   954
  "Frct (neg_numeral k, 1) = neg_numeral k"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   955
  "Frct (1, numeral k) = 1 / numeral k"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   956
  "Frct (1, neg_numeral k) = 1 / neg_numeral k"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   957
  "Frct (numeral k, numeral l) = numeral k / numeral l"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   958
  "Frct (numeral k, neg_numeral l) = numeral k / neg_numeral l"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   959
  "Frct (neg_numeral k, numeral l) = neg_numeral k / numeral l"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   960
  "Frct (neg_numeral k, neg_numeral l) = neg_numeral k / neg_numeral l"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   961
  by (simp_all add: Fract_of_int_quotient)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   962
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   963
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   964
text {* Operations *}
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   965
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   966
lemma rat_zero_code [code abstract]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   967
  "quotient_of 0 = (0, 1)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   968
  by (simp add: Zero_rat_def quotient_of_Fract normalize_def)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   969
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   970
lemma rat_one_code [code abstract]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   971
  "quotient_of 1 = (1, 1)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   972
  by (simp add: One_rat_def quotient_of_Fract normalize_def)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   973
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   974
lemma rat_plus_code [code abstract]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   975
  "quotient_of (p + q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   976
     in normalize (a * d + b * c, c * d))"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   977
  by (cases p, cases q) (simp add: quotient_of_Fract)
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   978
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   979
lemma rat_uminus_code [code abstract]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   980
  "quotient_of (- p) = (let (a, b) = quotient_of p in (- a, b))"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   981
  by (cases p) (simp add: quotient_of_Fract)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   982
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   983
lemma rat_minus_code [code abstract]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   984
  "quotient_of (p - q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   985
     in normalize (a * d - b * c, c * d))"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   986
  by (cases p, cases q) (simp add: quotient_of_Fract)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   987
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   988
lemma rat_times_code [code abstract]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   989
  "quotient_of (p * q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   990
     in normalize (a * b, c * d))"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   991
  by (cases p, cases q) (simp add: quotient_of_Fract)
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   992
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   993
lemma rat_inverse_code [code abstract]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   994
  "quotient_of (inverse p) = (let (a, b) = quotient_of p
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   995
    in if a = 0 then (0, 1) else (sgn a * b, \<bar>a\<bar>))"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   996
proof (cases p)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   997
  case (Fract a b) then show ?thesis
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   998
    by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract gcd_int.commute)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   999
qed
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1000
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1001
lemma rat_divide_code [code abstract]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1002
  "quotient_of (p / q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1003
     in normalize (a * d, c * b))"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1004
  by (cases p, cases q) (simp add: quotient_of_Fract)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1005
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1006
lemma rat_abs_code [code abstract]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1007
  "quotient_of \<bar>p\<bar> = (let (a, b) = quotient_of p in (\<bar>a\<bar>, b))"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1008
  by (cases p) (simp add: quotient_of_Fract)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1009
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1010
lemma rat_sgn_code [code abstract]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1011
  "quotient_of (sgn p) = (sgn (fst (quotient_of p)), 1)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1012
proof (cases p)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1013
  case (Fract a b) then show ?thesis
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1014
  by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1015
qed
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
  1016
43733
a6ca7b83612f adding code equations to execute floor and ceiling on rational and real numbers
bulwahn
parents: 43732
diff changeset
  1017
lemma rat_floor_code [code]:
a6ca7b83612f adding code equations to execute floor and ceiling on rational and real numbers
bulwahn
parents: 43732
diff changeset
  1018
  "floor p = (let (a, b) = quotient_of p in a div b)"
a6ca7b83612f adding code equations to execute floor and ceiling on rational and real numbers
bulwahn
parents: 43732
diff changeset
  1019
by (cases p) (simp add: quotient_of_Fract floor_Fract)
a6ca7b83612f adding code equations to execute floor and ceiling on rational and real numbers
bulwahn
parents: 43732
diff changeset
  1020
38857
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38287
diff changeset
  1021
instantiation rat :: equal
26513
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
  1022
begin
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
  1023
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1024
definition [code]:
38857
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38287
diff changeset
  1025
  "HOL.equal a b \<longleftrightarrow> quotient_of a = quotient_of b"
26513
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
  1026
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1027
instance proof
38857
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38287
diff changeset
  1028
qed (simp add: equal_rat_def quotient_of_inject_eq)
26513
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
  1029
28351
abfc66969d1f non left-linear equations for nbe
haftmann
parents: 28313
diff changeset
  1030
lemma rat_eq_refl [code nbe]:
38857
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38287
diff changeset
  1031
  "HOL.equal (r::rat) r \<longleftrightarrow> True"
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38287
diff changeset
  1032
  by (rule equal_refl)
28351
abfc66969d1f non left-linear equations for nbe
haftmann
parents: 28313
diff changeset
  1033
26513
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
  1034
end
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
  1035
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1036
lemma rat_less_eq_code [code]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1037
  "p \<le> q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d \<le> c * b)"
35726
059d2f7b979f tuned prefixes of ac interpretations
haftmann
parents: 35402
diff changeset
  1038
  by (cases p, cases q) (simp add: quotient_of_Fract mult.commute)
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
  1039
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1040
lemma rat_less_code [code]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1041
  "p < q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d < c * b)"
35726
059d2f7b979f tuned prefixes of ac interpretations
haftmann
parents: 35402
diff changeset
  1042
  by (cases p, cases q) (simp add: quotient_of_Fract mult.commute)
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
  1043
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1044
lemma [code]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1045
  "of_rat p = (let (a, b) = quotient_of p in of_int a / of_int b)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1046
  by (cases p) (simp add: quotient_of_Fract of_rat_rat)
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
  1047
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1048
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1049
text {* Quickcheck *}
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1050
31203
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1051
definition (in term_syntax)
32657
5f13912245ff Code_Eval(uation)
haftmann
parents: 32069
diff changeset
  1052
  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
5f13912245ff Code_Eval(uation)
haftmann
parents: 32069
diff changeset
  1053
  [code_unfold]: "valterm_fract k l = Code_Evaluation.valtermify Fract {\<cdot>} k {\<cdot>} l"
31203
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1054
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37397
diff changeset
  1055
notation fcomp (infixl "\<circ>>" 60)
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37397
diff changeset
  1056
notation scomp (infixl "\<circ>\<rightarrow>" 60)
31203
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1057
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1058
instantiation rat :: random
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1059
begin
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1060
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1061
definition
51126
df86080de4cb reform of predicate compiler / quickcheck theories:
haftmann
parents: 50178
diff changeset
  1062
  "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>num. Random.range i \<circ>\<rightarrow> (\<lambda>denom. Pair (
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 51126
diff changeset
  1063
     let j = int_of_integer (integer_of_natural (denom + 1))
32657
5f13912245ff Code_Eval(uation)
haftmann
parents: 32069
diff changeset
  1064
     in valterm_fract num (j, \<lambda>u. Code_Evaluation.term_of j))))"
31203
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1065
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1066
instance ..
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1067
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1068
end
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1069
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37397
diff changeset
  1070
no_notation fcomp (infixl "\<circ>>" 60)
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37397
diff changeset
  1071
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
31203
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1072
41920
d4fb7a418152 moving exhaustive_generators.ML to Quickcheck directory
bulwahn
parents: 41792
diff changeset
  1073
instantiation rat :: exhaustive
41231
2e901158675e adding exhaustive tester instances for numeric types: code_numeral, nat, rat and real
bulwahn
parents: 40819
diff changeset
  1074
begin
2e901158675e adding exhaustive tester instances for numeric types: code_numeral, nat, rat and real
bulwahn
parents: 40819
diff changeset
  1075
2e901158675e adding exhaustive tester instances for numeric types: code_numeral, nat, rat and real
bulwahn
parents: 40819
diff changeset
  1076
definition
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 51126
diff changeset
  1077
  "exhaustive_rat f d = Quickcheck_Exhaustive.exhaustive
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 51126
diff changeset
  1078
    (\<lambda>l. Quickcheck_Exhaustive.exhaustive (\<lambda>k. f (Fract k (int_of_integer (integer_of_natural l) + 1))) d) d"
42311
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1079
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1080
instance ..
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1081
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1082
end
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1083
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1084
instantiation rat :: full_exhaustive
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1085
begin
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1086
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1087
definition
45818
53a697f5454a hiding constants and facts in the Quickcheck_Exhaustive and Quickcheck_Narrowing theory;
bulwahn
parents: 45694
diff changeset
  1088
  "full_exhaustive_rat f d = Quickcheck_Exhaustive.full_exhaustive (%(l, _). Quickcheck_Exhaustive.full_exhaustive (%k.
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 51126
diff changeset
  1089
     f (let j = int_of_integer (integer_of_natural l) + 1
45507
6975db7fd6f0 improved generators for rational numbers to generate negative numbers;
bulwahn
parents: 45478
diff changeset
  1090
        in valterm_fract k (j, %_. Code_Evaluation.term_of j))) d) d"
41231
2e901158675e adding exhaustive tester instances for numeric types: code_numeral, nat, rat and real
bulwahn
parents: 40819
diff changeset
  1091
2e901158675e adding exhaustive tester instances for numeric types: code_numeral, nat, rat and real
bulwahn
parents: 40819
diff changeset
  1092
instance ..
2e901158675e adding exhaustive tester instances for numeric types: code_numeral, nat, rat and real
bulwahn
parents: 40819
diff changeset
  1093
2e901158675e adding exhaustive tester instances for numeric types: code_numeral, nat, rat and real
bulwahn
parents: 40819
diff changeset
  1094
end
2e901158675e adding exhaustive tester instances for numeric types: code_numeral, nat, rat and real
bulwahn
parents: 40819
diff changeset
  1095
43889
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1096
instantiation rat :: partial_term_of
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1097
begin
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1098
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1099
instance ..
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1100
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1101
end
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1102
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1103
lemma [code]:
46758
4106258260b3 choosing longer constant names in Quickcheck_Narrowing to reduce the chances of name clashes in Quickcheck-Narrowing
bulwahn
parents: 45818
diff changeset
  1104
  "partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Narrowing_variable p tt) == Code_Evaluation.Free (STR ''_'') (Typerep.Typerep (STR ''Rat.rat'') [])"
4106258260b3 choosing longer constant names in Quickcheck_Narrowing to reduce the chances of name clashes in Quickcheck-Narrowing
bulwahn
parents: 45818
diff changeset
  1105
  "partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Narrowing_constructor 0 [l, k]) ==
45507
6975db7fd6f0 improved generators for rational numbers to generate negative numbers;
bulwahn
parents: 45478
diff changeset
  1106
     Code_Evaluation.App (Code_Evaluation.Const (STR ''Rat.Frct'')
6975db7fd6f0 improved generators for rational numbers to generate negative numbers;
bulwahn
parents: 45478
diff changeset
  1107
     (Typerep.Typerep (STR ''fun'') [Typerep.Typerep (STR ''Product_Type.prod'') [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''Int.int'') []],
6975db7fd6f0 improved generators for rational numbers to generate negative numbers;
bulwahn
parents: 45478
diff changeset
  1108
        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))"
43889
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1109
by (rule partial_term_of_anything)+
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1110
43887
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1111
instantiation rat :: narrowing
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1112
begin
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1113
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1114
definition
45507
6975db7fd6f0 improved generators for rational numbers to generate negative numbers;
bulwahn
parents: 45478
diff changeset
  1115
  "narrowing = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.apply
6975db7fd6f0 improved generators for rational numbers to generate negative numbers;
bulwahn
parents: 45478
diff changeset
  1116
    (Quickcheck_Narrowing.cons (%nom denom. Fract nom denom)) narrowing) narrowing"
43887
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1117
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1118
instance ..
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1119
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1120
end
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1121
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1122
45183
2e1ad4a54189 removing old code generator setup for rational numbers; tuned
bulwahn
parents: 43889
diff changeset
  1123
subsection {* Setup for Nitpick *}
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
  1124
38287
796302ca3611 replace "setup" with "declaration"
blanchet
parents: 38242
diff changeset
  1125
declaration {*
796302ca3611 replace "setup" with "declaration"
blanchet
parents: 38242
diff changeset
  1126
  Nitpick_HOL.register_frac_type @{type_name rat}
33209
d36ca3960e33 tuned white space;
wenzelm
parents: 33197
diff changeset
  1127
   [(@{const_name zero_rat_inst.zero_rat}, @{const_name Nitpick.zero_frac}),
d36ca3960e33 tuned white space;
wenzelm
parents: 33197
diff changeset
  1128
    (@{const_name one_rat_inst.one_rat}, @{const_name Nitpick.one_frac}),
d36ca3960e33 tuned white space;
wenzelm
parents: 33197
diff changeset
  1129
    (@{const_name plus_rat_inst.plus_rat}, @{const_name Nitpick.plus_frac}),
d36ca3960e33 tuned white space;
wenzelm
parents: 33197
diff changeset
  1130
    (@{const_name times_rat_inst.times_rat}, @{const_name Nitpick.times_frac}),
d36ca3960e33 tuned white space;
wenzelm
parents: 33197
diff changeset
  1131
    (@{const_name uminus_rat_inst.uminus_rat}, @{const_name Nitpick.uminus_frac}),
d36ca3960e33 tuned white space;
wenzelm
parents: 33197
diff changeset
  1132
    (@{const_name inverse_rat_inst.inverse_rat}, @{const_name Nitpick.inverse_frac}),
37397
18000f9d783e adjust Nitpick's handling of "<" on "rat"s and "reals"
blanchet
parents: 37143
diff changeset
  1133
    (@{const_name ord_rat_inst.less_rat}, @{const_name Nitpick.less_frac}),
33209
d36ca3960e33 tuned white space;
wenzelm
parents: 33197
diff changeset
  1134
    (@{const_name ord_rat_inst.less_eq_rat}, @{const_name Nitpick.less_eq_frac}),
45478
8e299034eab4 remove unsound line in Nitpick's "rat" setup
blanchet
parents: 45183
diff changeset
  1135
    (@{const_name field_char_0_class.of_rat}, @{const_name Nitpick.of_frac})]
33197
de6285ebcc05 continuation of Nitpick's integration into Isabelle;
blanchet
parents: 32657
diff changeset
  1136
*}
de6285ebcc05 continuation of Nitpick's integration into Isabelle;
blanchet
parents: 32657
diff changeset
  1137
41792
ff3cb0c418b7 renamed "nitpick\_def" to "nitpick_unfold" to reflect its new semantics
blanchet
parents: 41231
diff changeset
  1138
lemmas [nitpick_unfold] = inverse_rat_inst.inverse_rat
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1139
  one_rat_inst.one_rat ord_rat_inst.less_rat
37397
18000f9d783e adjust Nitpick's handling of "<" on "rat"s and "reals"
blanchet
parents: 37143
diff changeset
  1140
  ord_rat_inst.less_eq_rat plus_rat_inst.plus_rat times_rat_inst.times_rat
18000f9d783e adjust Nitpick's handling of "<" on "rat"s and "reals"
blanchet
parents: 37143
diff changeset
  1141
  uminus_rat_inst.uminus_rat zero_rat_inst.zero_rat
33197
de6285ebcc05 continuation of Nitpick's integration into Isabelle;
blanchet
parents: 32657
diff changeset
  1142
52146
wenzelm
parents: 51956
diff changeset
  1143
wenzelm
parents: 51956
diff changeset
  1144
subsection {* Float syntax *}
35343
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1145
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1146
syntax "_Float" :: "float_const \<Rightarrow> 'a"    ("_")
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1147
52146
wenzelm
parents: 51956
diff changeset
  1148
parse_translation {*
wenzelm
parents: 51956
diff changeset
  1149
  let
wenzelm
parents: 51956
diff changeset
  1150
    fun mk_number i =
wenzelm
parents: 51956
diff changeset
  1151
      let
wenzelm
parents: 51956
diff changeset
  1152
        fun mk 1 = Syntax.const @{const_syntax Num.One}
wenzelm
parents: 51956
diff changeset
  1153
          | mk i =
wenzelm
parents: 51956
diff changeset
  1154
              let val (q, r) = Integer.div_mod i 2
wenzelm
parents: 51956
diff changeset
  1155
              in HOLogic.mk_bit r $ (mk q) end;
wenzelm
parents: 51956
diff changeset
  1156
      in
wenzelm
parents: 51956
diff changeset
  1157
        if i = 0 then Syntax.const @{const_syntax Groups.zero}
wenzelm
parents: 51956
diff changeset
  1158
        else if i > 0 then Syntax.const @{const_syntax Num.numeral} $ mk i
wenzelm
parents: 51956
diff changeset
  1159
        else Syntax.const @{const_syntax Num.neg_numeral} $ mk (~i)
wenzelm
parents: 51956
diff changeset
  1160
      end;
wenzelm
parents: 51956
diff changeset
  1161
wenzelm
parents: 51956
diff changeset
  1162
    fun mk_frac str =
wenzelm
parents: 51956
diff changeset
  1163
      let
wenzelm
parents: 51956
diff changeset
  1164
        val {mant = i, exp = n} = Lexicon.read_float str;
wenzelm
parents: 51956
diff changeset
  1165
        val exp = Syntax.const @{const_syntax Power.power};
wenzelm
parents: 51956
diff changeset
  1166
        val ten = mk_number 10;
wenzelm
parents: 51956
diff changeset
  1167
        val exp10 = if n = 1 then ten else exp $ ten $ mk_number n;
wenzelm
parents: 51956
diff changeset
  1168
      in Syntax.const @{const_syntax divide} $ mk_number i $ exp10 end;
wenzelm
parents: 51956
diff changeset
  1169
wenzelm
parents: 51956
diff changeset
  1170
    fun float_tr [(c as Const (@{syntax_const "_constrain"}, _)) $ t $ u] = c $ float_tr [t] $ u
wenzelm
parents: 51956
diff changeset
  1171
      | float_tr [t as Const (str, _)] = mk_frac str
wenzelm
parents: 51956
diff changeset
  1172
      | float_tr ts = raise TERM ("float_tr", ts);
wenzelm
parents: 51956
diff changeset
  1173
  in [(@{syntax_const "_Float"}, K float_tr)] end
wenzelm
parents: 51956
diff changeset
  1174
*}
35343
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1175
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1176
text{* Test: *}
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1177
lemma "123.456 = -111.111 + 200 + 30 + 4 + 5/10 + 6/100 + (7/1000::rat)"
52146
wenzelm
parents: 51956
diff changeset
  1178
  by simp
35343
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1179
53652
18fbca265e2e use lifting_forget for deregistering numeric types as a quotient type
kuncar
parents: 53374
diff changeset
  1180
subsection {* Hiding implementation details *}
37143
2a5182751151 constant Rat.normalize needs to be qualified;
wenzelm
parents: 36415
diff changeset
  1181
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
  1182
hide_const (open) normalize positive
37143
2a5182751151 constant Rat.normalize needs to be qualified;
wenzelm
parents: 36415
diff changeset
  1183
53652
18fbca265e2e use lifting_forget for deregistering numeric types as a quotient type
kuncar
parents: 53374
diff changeset
  1184
lifting_update rat.lifting
18fbca265e2e use lifting_forget for deregistering numeric types as a quotient type
kuncar
parents: 53374
diff changeset
  1185
lifting_forget rat.lifting
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
  1186
29880
3dee8ff45d3d move countability proof from Rational to Countable; add instance rat :: countable
huffman
parents: 29667
diff changeset
  1187
end
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 51126
diff changeset
  1188