src/HOL/Rat.thy
author wenzelm
Mon, 20 Jun 2016 22:30:23 +0200
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permissions -rw-r--r--
misc tuning and modernization;
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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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section \<open>Rational numbers\<close>
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theory Rat
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imports GCD Archimedean_Field
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begin
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subsection \<open>Rational numbers as quotient\<close>
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subsubsection \<open>Construction of the type of rational numbers\<close>
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definition ratrel :: "(int \<times> int) \<Rightarrow> (int \<times> int) \<Rightarrow> bool"
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  where "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]: "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 \<open>Representation and basic operations\<close>
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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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  "\<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>a. Fract a 0 = Fract 0 1"
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  "\<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 that: "\<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: "q = Fract a b" and b: "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 have "?b * gcd a b = b"
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    by simp
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  with b have "?b \<noteq> 0"
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    by fastforce
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  with q b have q2: "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 have coprime: "coprime ?a ?b"
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    by (auto intro: div_gcd_coprime)
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  show C
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  proof (cases "b > 0")
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    case True
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    then have "?b > 0"
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      by (simp add: nonneg1_imp_zdiv_pos_iff)
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    from q2 this coprime show C by (rule that)
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  next
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    case False
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    have "q = Fract (- ?a) (- ?b)"
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      unfolding q2 by transfer simp
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    moreover from False b have "- ?b > 0"
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      by (simp add: pos_imp_zdiv_neg_iff)
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    moreover from coprime have "coprime (- ?a) (- ?b)"
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      by simp
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    ultimately show C
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      by (rule that)
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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
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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: ac_simps)
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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 diff_rat_def: "q - r = q + - r" for 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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   145
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huffman
parents: 47108
diff changeset
   146
lift_definition times_rat :: "rat \<Rightarrow> rat \<Rightarrow> rat"
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   147
  is "\<lambda>x y. (fst x * fst y, snd x * snd y)"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   148
  by (simp add: ac_simps)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   149
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   150
lemma mult_rat [simp]: "Fract a b * Fract c d = Fract (a * c) (b * d)"
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   151
  by transfer simp
14365
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paulson
parents:
diff changeset
   152
27652
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haftmann
parents: 27551
diff changeset
   153
lemma mult_rat_cancel:
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   154
  assumes "c \<noteq> 0"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   155
  shows "Fract (c * a) (c * b) = Fract a b"
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   156
  using assms by transfer simp
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   157
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   158
lift_definition inverse_rat :: "rat \<Rightarrow> rat"
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   159
  is "\<lambda>x. if fst x = 0 then (0, 1) else (snd x, fst x)"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57275
diff changeset
   160
  by (auto simp add: mult.commute)
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   161
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   162
lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   163
  by transfer simp
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   164
63326
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wenzelm
parents: 62390
diff changeset
   165
definition divide_rat_def: "q div r = q * inverse r" for q r :: rat
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   166
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60352
diff changeset
   167
lemma divide_rat [simp]: "Fract a b div Fract c d = Fract (a * d) (b * c)"
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   168
  by (simp add: divide_rat_def)
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   169
63326
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wenzelm
parents: 62390
diff changeset
   170
instance
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   171
proof
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   172
  fix q r s :: rat
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   173
  show "(q * r) * s = q * (r * s)"
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   174
    by transfer simp
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   175
  show "q * r = r * q"
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   176
    by transfer simp
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   177
  show "1 * q = q"
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   178
    by transfer simp
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   179
  show "(q + r) + s = q + (r + s)"
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   180
    by transfer (simp add: algebra_simps)
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   181
  show "q + r = r + q"
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   182
    by transfer simp
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   183
  show "0 + q = q"
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   184
    by transfer simp
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   185
  show "- q + q = 0"
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   186
    by transfer simp
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   187
  show "q - r = q + - r"
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   188
    by (fact diff_rat_def)
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   189
  show "(q + r) * s = q * s + r * s"
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   190
    by transfer (simp add: algebra_simps)
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   191
  show "(0::rat) \<noteq> 1"
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   192
    by transfer simp
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   193
  show "inverse q * q = 1" if "q \<noteq> 0"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   194
    using that by transfer simp
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60352
diff changeset
   195
  show "q div r = q * inverse r"
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   196
    by (fact divide_rat_def)
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   197
  show "inverse 0 = (0::rat)"
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   198
    by transfer simp
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   199
qed
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   200
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   201
end
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   202
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   203
lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   204
  by (induct k) (simp_all add: Zero_rat_def One_rat_def)
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   205
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   206
lemma of_int_rat: "of_int k = Fract k 1"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   207
  by (cases k rule: int_diff_cases) (simp add: of_nat_rat)
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   208
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   209
lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   210
  by (rule of_nat_rat [symmetric])
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   211
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   212
lemma Fract_of_int_eq: "Fract k 1 = of_int k"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   213
  by (rule of_int_rat [symmetric])
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   214
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   215
lemma rat_number_collapse:
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   216
  "Fract 0 k = 0"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   217
  "Fract 1 1 = 1"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   218
  "Fract (numeral w) 1 = numeral w"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54409
diff changeset
   219
  "Fract (- numeral w) 1 = - numeral w"
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54409
diff changeset
   220
  "Fract (- 1) 1 = - 1"
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   221
  "Fract k 0 = 0"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   222
  using Fract_of_int_eq [of "numeral w"]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54409
diff changeset
   223
  using Fract_of_int_eq [of "- numeral w"]
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   224
  by (simp_all add: Zero_rat_def One_rat_def eq_rat)
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   225
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   226
lemma rat_number_expand:
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   227
  "0 = Fract 0 1"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   228
  "1 = Fract 1 1"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   229
  "numeral k = Fract (numeral k) 1"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54409
diff changeset
   230
  "- 1 = Fract (- 1) 1"
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54409
diff changeset
   231
  "- numeral k = Fract (- numeral k) 1"
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   232
  by (simp_all add: rat_number_collapse)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   233
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   234
lemma Rat_cases_nonzero [case_names Fract 0]:
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   235
  assumes Fract: "\<And>a b. q = Fract a b \<Longrightarrow> b > 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> coprime a b \<Longrightarrow> C"
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   236
    and 0: "q = 0 \<Longrightarrow> C"
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   237
  shows C
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   238
proof (cases "q = 0")
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   239
  case True
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   240
  then show C using 0 by auto
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   241
next
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   242
  case False
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   243
  then obtain a b where *: "q = Fract a b" "b > 0" "coprime a b"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   244
    by (cases q) auto
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   245
  with False have "0 \<noteq> Fract a b"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   246
    by simp
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   247
  with \<open>b > 0\<close> have "a \<noteq> 0"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   248
    by (simp add: Zero_rat_def eq_rat)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   249
  with Fract * show C by blast
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   250
qed
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   251
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   252
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61144
diff changeset
   253
subsubsection \<open>Function \<open>normalize\<close>\<close>
33805
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   254
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   255
lemma Fract_coprime: "Fract (a div gcd a b) (b div gcd a b) = Fract a b"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   256
proof (cases "b = 0")
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   257
  case True
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   258
  then show ?thesis by (simp add: eq_rat)
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   259
next
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   260
  case False
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   261
  moreover have "b div gcd a b * gcd a b = b"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   262
    by (rule dvd_div_mult_self) simp
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   263
  ultimately have "b div gcd a b * gcd a b \<noteq> 0"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   264
    by simp
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   265
  then have "b div gcd a b \<noteq> 0"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   266
    by fastforce
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   267
  with False show ?thesis
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   268
    by (simp add: eq_rat dvd_div_mult mult.commute [of a])
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   269
qed
33805
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   270
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   271
definition normalize :: "int \<times> int \<Rightarrow> int \<times> int"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   272
  where "normalize p =
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   273
   (if snd p > 0 then (let a = gcd (fst p) (snd p) in (fst p div a, snd p div a))
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   274
    else if snd p = 0 then (0, 1)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   275
    else (let a = - gcd (fst p) (snd p) in (fst p div a, snd p div a)))"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   276
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   277
lemma normalize_crossproduct:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   278
  assumes "q \<noteq> 0" "s \<noteq> 0"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   279
  assumes "normalize (p, q) = normalize (r, s)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   280
  shows "p * s = r * q"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   281
proof -
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   282
  have *: "p * s = q * r"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   283
    if "p * gcd r s = sgn (q * s) * r * gcd p q" and "q * gcd r s = sgn (q * s) * s * gcd p q"
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   284
  proof -
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   285
    from that
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   286
    have "(p * gcd r s) * (sgn (q * s) * s * gcd p q) = (q * gcd r s) * (sgn (q * s) * r * gcd p q)"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   287
      by simp
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   288
    with assms show ?thesis
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   289
      by (auto simp add: ac_simps sgn_times sgn_0_0)
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   290
  qed
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   291
  from assms show ?thesis
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   292
    by (auto simp add: normalize_def Let_def dvd_div_div_eq_mult mult.commute sgn_times
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   293
        split: if_splits intro: *)
33805
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   294
qed
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   295
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   296
lemma normalize_eq: "normalize (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
   297
  by (auto simp add: normalize_def Let_def Fract_coprime dvd_div_neg rat_number_collapse
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   298
      split: if_split_asm)
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   299
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   300
lemma normalize_denom_pos: "normalize 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
   301
  by (auto simp add: normalize_def Let_def dvd_div_neg pos_imp_zdiv_neg_iff nonneg1_imp_zdiv_pos_iff
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   302
      split: if_split_asm)
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   303
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   304
lemma normalize_coprime: "normalize r = (p, q) \<Longrightarrow> coprime p q"
62348
9a5f43dac883 dropped various legacy fact bindings
haftmann
parents: 62079
diff changeset
   305
  by (auto simp add: normalize_def Let_def dvd_div_neg div_gcd_coprime
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   306
      split: if_split_asm)
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   307
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   308
lemma normalize_stable [simp]: "q > 0 \<Longrightarrow> coprime p q \<Longrightarrow> normalize (p, q) = (p, q)"
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   309
  by (simp add: normalize_def)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   310
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   311
lemma normalize_denom_zero [simp]: "normalize (p, 0) = (0, 1)"
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   312
  by (simp add: normalize_def)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   313
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   314
lemma normalize_negative [simp]: "q < 0 \<Longrightarrow> normalize (p, q) = normalize (- p, - q)"
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   315
  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
   316
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
   317
text\<open>
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   318
  Decompose a fraction into normalized, i.e. coprime numerator and denominator:
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
   319
\<close>
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   320
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   321
definition quotient_of :: "rat \<Rightarrow> int \<times> int"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   322
  where "quotient_of x =
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   323
    (THE pair. x = Fract (fst pair) (snd pair) \<and> snd pair > 0 \<and> coprime (fst pair) (snd pair))"
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   324
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   325
lemma quotient_of_unique: "\<exists>!p. r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   326
proof (cases r)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   327
  case (Fract a b)
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   328
  then have "r = Fract (fst (a, b)) (snd (a, b)) \<and> snd (a, b) > 0 \<and> coprime (fst (a, b)) (snd (a, b))"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   329
    by auto
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   330
  then show ?thesis
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   331
  proof (rule ex1I)
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   332
    fix p
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   333
    obtain c d :: int where p: "p = (c, d)"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   334
      by (cases p)
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   335
    assume "r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   336
    with p have Fract': "r = Fract c d" "d > 0" "coprime c d"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   337
      by simp_all
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   338
    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
   339
    proof (cases "a = 0")
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   340
      case True
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   341
      with Fract Fract' show ?thesis
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   342
        by (simp add: eq_rat)
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   343
    next
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   344
      case False
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   345
      with Fract Fract' have *: "c * b = a * d" and "c \<noteq> 0"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   346
        by (auto simp add: eq_rat)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   347
      then have "c * b > 0 \<longleftrightarrow> a * d > 0"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   348
        by auto
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   349
      with \<open>b > 0\<close> \<open>d > 0\<close> have "a > 0 \<longleftrightarrow> c > 0"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   350
        by (simp add: zero_less_mult_iff)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   351
      with \<open>a \<noteq> 0\<close> \<open>c \<noteq> 0\<close> have sgn: "sgn a = sgn c"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   352
        by (auto simp add: not_less)
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
   353
      from \<open>coprime a b\<close> \<open>coprime c d\<close> 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>"
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   354
        by (simp add: coprime_crossproduct_int)
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   355
      with \<open>b > 0\<close> \<open>d > 0\<close> have "\<bar>a\<bar> * d = \<bar>c\<bar> * b \<longleftrightarrow> \<bar>a\<bar> = \<bar>c\<bar> \<and> d = b"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   356
        by simp
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   357
      then have "a * sgn a * d = c * sgn c * b \<longleftrightarrow> a * sgn a = c * sgn c \<and> d = b"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   358
        by (simp add: abs_sgn)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   359
      with sgn * show ?thesis
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   360
        by (auto simp add: sgn_0_0)
33805
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   361
    qed
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   362
    with p show "p = (a, b)"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   363
      by simp
33805
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   364
  qed
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   365
qed
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   366
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   367
lemma quotient_of_Fract [code]: "quotient_of (Fract a b) = normalize (a, b)"
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   368
proof -
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   369
  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
   370
    by (rule sym) (auto intro: normalize_eq)
52146
wenzelm
parents: 51956
diff changeset
   371
  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
   372
    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
   373
  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
   374
    by (rule normalize_coprime) simp
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   375
  ultimately have "?Fract \<and> ?denom_pos \<and> ?coprime" by blast
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   376
  with quotient_of_unique
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   377
  have "(THE p. Fract a b = Fract (fst p) (snd p) \<and> 0 < snd p \<and>
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   378
    coprime (fst p) (snd p)) = normalize (a, b)" by (rule the1_equality)
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   379
  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
   380
qed
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   381
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   382
lemma quotient_of_number [simp]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   383
  "quotient_of 0 = (0, 1)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   384
  "quotient_of 1 = (1, 1)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   385
  "quotient_of (numeral k) = (numeral k, 1)"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54409
diff changeset
   386
  "quotient_of (- 1) = (- 1, 1)"
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54409
diff changeset
   387
  "quotient_of (- numeral k) = (- numeral k, 1)"
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   388
  by (simp_all add: rat_number_expand quotient_of_Fract)
33805
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   389
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   390
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
   391
  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
   392
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   393
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
   394
  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
   395
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   396
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
   397
  by (cases r) (simp add: quotient_of_Fract normalize_coprime)
33805
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   398
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   399
lemma quotient_of_inject:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   400
  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
   401
  shows "a = b"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   402
proof -
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   403
  obtain p q r s where a: "a = Fract p q" and b: "b = Fract r s" and "q > 0" and "s > 0"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   404
    by (cases a, cases b)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   405
  with assms show ?thesis
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   406
    by (simp add: eq_rat quotient_of_Fract normalize_crossproduct)
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   407
qed
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   408
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   409
lemma quotient_of_inject_eq: "quotient_of a = quotient_of b \<longleftrightarrow> a = b"
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   410
  by (auto simp add: quotient_of_inject)
33805
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   411
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   412
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
   413
subsubsection \<open>Various\<close>
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   414
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   415
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
   416
  by (simp add: Fract_of_int_eq [symmetric])
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   417
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   418
lemma Fract_add_one: "n \<noteq> 0 \<Longrightarrow> Fract (m + n) n = Fract m n + 1"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   419
  by (simp add: rat_number_expand)
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   420
50178
ad52ddd35c3a add quotient_of_div
hoelzl
parents: 49962
diff changeset
   421
lemma quotient_of_div:
ad52ddd35c3a add quotient_of_div
hoelzl
parents: 49962
diff changeset
   422
  assumes r: "quotient_of r = (n,d)"
ad52ddd35c3a add quotient_of_div
hoelzl
parents: 49962
diff changeset
   423
  shows "r = of_int n / of_int d"
ad52ddd35c3a add quotient_of_div
hoelzl
parents: 49962
diff changeset
   424
proof -
ad52ddd35c3a add quotient_of_div
hoelzl
parents: 49962
diff changeset
   425
  from theI'[OF quotient_of_unique[of r], unfolded r[unfolded quotient_of_def]]
ad52ddd35c3a add quotient_of_div
hoelzl
parents: 49962
diff changeset
   426
  have "r = Fract n d" by simp
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   427
  then show ?thesis using Fract_of_int_quotient
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   428
    by simp
50178
ad52ddd35c3a add quotient_of_div
hoelzl
parents: 49962
diff changeset
   429
qed
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   430
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   431
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
   432
subsubsection \<open>The ordered field of rational numbers\<close>
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   433
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   434
lift_definition positive :: "rat \<Rightarrow> bool"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   435
  is "\<lambda>x. 0 < fst x * snd x"
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   436
proof clarsimp
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   437
  fix a b c d :: int
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   438
  assume "b \<noteq> 0" and "d \<noteq> 0" and "a * d = c * b"
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   439
  then have "a * d * b * d = c * b * b * d"
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   440
    by simp
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   441
  then have "a * b * d\<^sup>2 = c * d * b\<^sup>2"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   442
    unfolding power2_eq_square by (simp add: ac_simps)
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   443
  then have "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
   444
    by simp
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   445
  then show "0 < a * b \<longleftrightarrow> 0 < c * d"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
   446
    using \<open>b \<noteq> 0\<close> and \<open>d \<noteq> 0\<close>
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   447
    by (simp add: zero_less_mult_iff)
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   448
qed
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   449
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   450
lemma positive_zero: "\<not> positive 0"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   451
  by transfer simp
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   452
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   453
lemma positive_add: "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x + y)"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   454
  apply transfer
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   455
  apply (simp add: zero_less_mult_iff)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   456
  apply (elim disjE, simp_all add: add_pos_pos add_neg_neg mult_pos_neg mult_neg_pos mult_neg_neg)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   457
  done
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   458
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   459
lemma positive_mult: "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x * y)"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   460
  apply transfer
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   461
  apply (drule (1) mult_pos_pos)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   462
  apply (simp add: ac_simps)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   463
  done
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   464
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   465
lemma positive_minus: "\<not> positive x \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> positive (- x)"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   466
  by transfer (auto simp: neq_iff zero_less_mult_iff mult_less_0_iff)
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   467
59867
58043346ca64 given up separate type classes demanding `inverse 0 = 0`
haftmann
parents: 59667
diff changeset
   468
instantiation rat :: linordered_field
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   469
begin
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   470
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   471
definition "x < y \<longleftrightarrow> positive (y - x)"
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   472
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   473
definition "x \<le> y \<longleftrightarrow> x < y \<or> x = y" for x y :: rat
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   474
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   475
definition "\<bar>a\<bar> = (if a < 0 then - a else a)" for a :: rat
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   476
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   477
definition "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)" for a :: rat
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   478
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   479
instance
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   480
proof
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   481
  fix a b c :: rat
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   482
  show "\<bar>a\<bar> = (if a < 0 then - a else a)"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   483
    by (rule abs_rat_def)
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   484
  show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   485
    unfolding less_eq_rat_def less_rat_def
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   486
    apply auto
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   487
    apply (drule (1) positive_add)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   488
    apply (simp_all add: positive_zero)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   489
    done
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   490
  show "a \<le> a"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   491
    unfolding less_eq_rat_def by simp
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   492
  show "a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   493
    unfolding less_eq_rat_def less_rat_def
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   494
    apply auto
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   495
    apply (drule (1) positive_add)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   496
    apply (simp add: algebra_simps)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   497
    done
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   498
  show "a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> a = b"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   499
    unfolding less_eq_rat_def less_rat_def
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   500
    apply auto
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   501
    apply (drule (1) positive_add)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   502
    apply (simp add: positive_zero)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   503
    done
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   504
  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
   505
    unfolding less_eq_rat_def less_rat_def by auto
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   506
  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
   507
    by (rule sgn_rat_def)
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   508
  show "a \<le> b \<or> b \<le> a"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   509
    unfolding less_eq_rat_def less_rat_def
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   510
    by (auto dest!: positive_minus)
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   511
  show "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   512
    unfolding less_rat_def
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   513
    apply (drule (1) positive_mult)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   514
    apply (simp add: algebra_simps)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   515
    done
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   516
qed
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   517
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   518
end
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   519
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   520
instantiation rat :: distrib_lattice
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   521
begin
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   522
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   523
definition "(inf :: rat \<Rightarrow> rat \<Rightarrow> rat) = min"
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   524
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   525
definition "(sup :: rat \<Rightarrow> rat \<Rightarrow> rat) = max"
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   526
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   527
instance
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   528
  by standard (auto simp add: inf_rat_def sup_rat_def max_min_distrib2)
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   529
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   530
end
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   531
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   532
lemma positive_rat: "positive (Fract a b) \<longleftrightarrow> 0 < a * b"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   533
  by transfer simp
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   534
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   535
lemma less_rat [simp]:
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   536
  assumes "b \<noteq> 0" and "d \<noteq> 0"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   537
  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
   538
  using assms unfolding less_rat_def
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   539
  by (simp add: positive_rat algebra_simps)
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   540
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   541
lemma le_rat [simp]:
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   542
  assumes "b \<noteq> 0" and "d \<noteq> 0"
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
   543
  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
   544
  using assms unfolding le_less by (simp add: eq_rat)
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   545
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   546
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
   547
  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
   548
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   549
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
   550
  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
   551
  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
   552
    (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
   553
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   554
lemma Rat_induct_pos [case_names Fract, induct type: rat]:
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   555
  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
   556
  shows "P q"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   557
proof (cases q)
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   558
  case (Fract a b)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   559
  have step': "P (Fract a b)" if b: "b < 0" for a b :: int
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   560
  proof -
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   561
    from b have "0 < - b"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   562
      by simp
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   563
    then have "P (Fract (- a) (- b))"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   564
      by (rule step)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   565
    then show "P (Fract a b)"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   566
      by (simp add: order_less_imp_not_eq [OF b])
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   567
  qed
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   568
  from Fract show "P q" by (auto simp add: linorder_neq_iff step step')
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   569
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   570
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   571
lemma zero_less_Fract_iff: "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a"
30095
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   572
  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
   573
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   574
lemma Fract_less_zero_iff: "0 < b \<Longrightarrow> Fract a b < 0 \<longleftrightarrow> a < 0"
30095
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   575
  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
   576
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   577
lemma zero_le_Fract_iff: "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a"
30095
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   578
  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
   579
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   580
lemma Fract_le_zero_iff: "0 < b \<Longrightarrow> Fract a b \<le> 0 \<longleftrightarrow> a \<le> 0"
30095
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   581
  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
   582
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   583
lemma one_less_Fract_iff: "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a"
30095
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   584
  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
   585
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   586
lemma Fract_less_one_iff: "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b"
30095
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   587
  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
   588
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   589
lemma one_le_Fract_iff: "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a"
30095
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   590
  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
   591
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   592
lemma Fract_le_one_iff: "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b"
30095
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   593
  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
   594
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   595
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
   596
subsubsection \<open>Rationals are an Archimedean field\<close>
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   597
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   598
lemma rat_floor_lemma: "of_int (a div b) \<le> Fract a b \<and> Fract a b < of_int (a div b + 1)"
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   599
proof -
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   600
  have "Fract a b = of_int (a div b) + Fract (a mod b) b"
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   601
    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
   602
  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
   603
    unfolding Fract_of_int_quotient
56571
f4635657d66f added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents: 56544
diff changeset
   604
    by (rule linorder_cases [of b 0]) (simp_all add: divide_nonpos_neg)
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   605
  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
   606
qed
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   607
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   608
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
   609
proof
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   610
  show "\<exists>z. r \<le> of_int z" for r :: rat
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   611
  proof (induct r)
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   612
    case (Fract a b)
35293
06a98796453e remove unneeded premise from rat_floor_lemma and floor_Fract
huffman
parents: 35216
diff changeset
   613
    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
   614
      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
   615
    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
   616
  qed
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   617
qed
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   618
43732
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   619
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
   620
begin
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   621
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   622
definition [code del]: "\<lfloor>x\<rfloor> = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))" for x :: rat
43732
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   623
61942
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61799
diff changeset
   624
instance
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61799
diff changeset
   625
proof
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   626
  show "of_int \<lfloor>x\<rfloor> \<le> x \<and> x < of_int (\<lfloor>x\<rfloor> + 1)" for x :: rat
43732
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   627
    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
   628
qed
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   629
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   630
end
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   631
61942
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61799
diff changeset
   632
lemma floor_Fract: "\<lfloor>Fract a b\<rfloor> = a div b"
59984
4f1eccec320c conversion between division on nat/int and division in archmedean fields
haftmann
parents: 59867
diff changeset
   633
  by (simp add: Fract_of_int_quotient floor_divide_of_int_eq)
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   634
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   635
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
   636
subsection \<open>Linear arithmetic setup\<close>
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   637
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
   638
declaration \<open>
31100
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   639
  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
   640
    (* 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
   641
  #> 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
   642
    (* 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
   643
  #> Lin_Arith.add_simps [@{thm neg_less_iff_less},
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   644
      @{thm True_implies_equals},
55143
04448228381d explicit eigen-context for attributes "where", "of", and corresponding read_instantiate, instantiate_tac;
wenzelm
parents: 54863
diff changeset
   645
      @{thm distrib_left [where a = "numeral v" for v]},
04448228381d explicit eigen-context for attributes "where", "of", and corresponding read_instantiate, instantiate_tac;
wenzelm
parents: 54863
diff changeset
   646
      @{thm distrib_left [where a = "- numeral v" for v]},
31100
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   647
      @{thm divide_1}, @{thm divide_zero_left},
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   648
      @{thm times_divide_eq_right}, @{thm times_divide_eq_left},
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   649
      @{thm minus_divide_left} RS sym, @{thm minus_divide_right} RS sym,
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   650
      @{thm of_int_minus}, @{thm of_int_diff},
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   651
      @{thm of_int_of_nat_eq}]
61144
5e94dfead1c2 simplified simproc programming interfaces;
wenzelm
parents: 61070
diff changeset
   652
  #> Lin_Arith.add_simprocs [Numeral_Simprocs.field_divide_cancel_numeral_factor]
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   653
  #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat \<Rightarrow> rat"})
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   654
  #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int \<Rightarrow> rat"}))
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
   655
\<close>
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   656
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   657
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
   658
subsection \<open>Embedding from Rationals to other Fields\<close>
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   659
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   660
context field_char_0
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   661
begin
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   662
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   663
lift_definition of_rat :: "rat \<Rightarrow> 'a"
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   664
  is "\<lambda>x. of_int (fst x) / of_int (snd x)"
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   665
  apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   666
  apply (simp only: of_int_mult [symmetric])
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   667
  done
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   668
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   669
end
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   670
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   671
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
   672
  by transfer simp
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   673
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   674
lemma of_rat_0 [simp]: "of_rat 0 = 0"
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   675
  by transfer simp
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   676
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   677
lemma of_rat_1 [simp]: "of_rat 1 = 1"
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
   678
  by transfer simp
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   679
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   680
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
   681
  by transfer (simp add: add_frac_eq)
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   682
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   683
lemma of_rat_minus: "of_rat (- a) = - of_rat a"
56479
91958d4b30f7 revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents: 56409
diff changeset
   684
  by transfer simp
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   685
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   686
lemma of_rat_neg_one [simp]: "of_rat (- 1) = - 1"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54409
diff changeset
   687
  by (simp add: of_rat_minus)
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54409
diff changeset
   688
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   689
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
   690
  using of_rat_add [of a "- b"] by (simp add: of_rat_minus)
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   691
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   692
lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   693
  by transfer (simp add: divide_inverse nonzero_inverse_mult_distrib ac_simps)
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   694
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   695
lemma of_rat_setsum: "of_rat (\<Sum>a\<in>A. f a) = (\<Sum>a\<in>A. of_rat (f a))"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   696
  by (induct rule: infinite_finite_induct) (auto simp: of_rat_add)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   697
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   698
lemma of_rat_setprod: "of_rat (\<Prod>a\<in>A. f a) = (\<Prod>a\<in>A. of_rat (f a))"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   699
  by (induct rule: infinite_finite_induct) (auto simp: of_rat_mult)
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   700
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   701
lemma nonzero_of_rat_inverse: "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   702
  by (rule inverse_unique [symmetric]) (simp add: of_rat_mult [symmetric])
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   703
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   704
lemma of_rat_inverse: "(of_rat (inverse a) :: 'a::{field_char_0,field}) = inverse (of_rat a)"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   705
  by (cases "a = 0") (simp_all add: nonzero_of_rat_inverse)
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   706
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   707
lemma nonzero_of_rat_divide: "b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   708
  by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   709
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   710
lemma of_rat_divide: "(of_rat (a / b) :: 'a::{field_char_0,field}) = of_rat a / of_rat b"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   711
  by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   712
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   713
lemma of_rat_power: "(of_rat (a ^ n) :: 'a::field_char_0) = of_rat a ^ n"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   714
  by (induct n) (simp_all add: of_rat_mult)
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   715
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   716
lemma of_rat_eq_iff [simp]: "of_rat a = of_rat b \<longleftrightarrow> a = b"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   717
  apply transfer
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   718
  apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   719
  apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   720
  done
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   721
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   722
lemma of_rat_eq_0_iff [simp]: "of_rat a = 0 \<longleftrightarrow> a = 0"
54409
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   723
  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
   724
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   725
lemma zero_eq_of_rat_iff [simp]: "0 = of_rat a \<longleftrightarrow> 0 = a"
54409
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   726
  by simp
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   727
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   728
lemma of_rat_eq_1_iff [simp]: "of_rat a = 1 \<longleftrightarrow> a = 1"
54409
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   729
  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
   730
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   731
lemma one_eq_of_rat_iff [simp]: "1 = of_rat a \<longleftrightarrow> 1 = a"
54409
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   732
  by simp
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   733
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   734
lemma of_rat_less: "(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
   735
proof (induct r, induct s)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   736
  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
   737
  assume not_zero: "b > 0" "d > 0"
56544
b60d5d119489 made mult_pos_pos a simp rule
nipkow
parents: 56479
diff changeset
   738
  then have "b * d > 0" by simp
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   739
  have of_int_divide_less_eq:
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   740
    "(of_int a :: 'a) / of_int b < of_int c / of_int d \<longleftrightarrow>
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   741
      (of_int a :: 'a) * of_int d < of_int c * of_int b"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   742
    using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   743
  show "(of_rat (Fract a b) :: 'a::linordered_field) < of_rat (Fract c d) \<longleftrightarrow>
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   744
      Fract a b < Fract c d"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
   745
    using not_zero \<open>b * d > 0\<close>
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   746
    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
   747
qed
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   748
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   749
lemma of_rat_less_eq: "(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
   750
  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
   751
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   752
lemma of_rat_le_0_iff [simp]: "(of_rat r :: 'a::linordered_field) \<le> 0 \<longleftrightarrow> r \<le> 0"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   753
  using of_rat_less_eq [of r 0, where 'a = 'a] by simp
54409
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   754
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   755
lemma zero_le_of_rat_iff [simp]: "0 \<le> (of_rat r :: 'a::linordered_field) \<longleftrightarrow> 0 \<le> r"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   756
  using of_rat_less_eq [of 0 r, where 'a = 'a] by simp
54409
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   757
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   758
lemma of_rat_le_1_iff [simp]: "(of_rat r :: 'a::linordered_field) \<le> 1 \<longleftrightarrow> r \<le> 1"
54409
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   759
  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
   760
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   761
lemma one_le_of_rat_iff [simp]: "1 \<le> (of_rat r :: 'a::linordered_field) \<longleftrightarrow> 1 \<le> r"
54409
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   762
  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
   763
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   764
lemma of_rat_less_0_iff [simp]: "(of_rat r :: 'a::linordered_field) < 0 \<longleftrightarrow> r < 0"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   765
  using of_rat_less [of r 0, where 'a = 'a] by simp
54409
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   766
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   767
lemma zero_less_of_rat_iff [simp]: "0 < (of_rat r :: 'a::linordered_field) \<longleftrightarrow> 0 < r"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   768
  using of_rat_less [of 0 r, where 'a = 'a] by simp
54409
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   769
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   770
lemma of_rat_less_1_iff [simp]: "(of_rat r :: 'a::linordered_field) < 1 \<longleftrightarrow> r < 1"
54409
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   771
  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
   772
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   773
lemma one_less_of_rat_iff [simp]: "1 < (of_rat r :: 'a::linordered_field) \<longleftrightarrow> 1 < r"
54409
2e501a90dec7 support of_rat with 0 or 1 on order relations
hoelzl
parents: 54230
diff changeset
   774
  using of_rat_less [of 1 r] by simp
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   775
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   776
lemma of_rat_eq_id [simp]: "of_rat = id"
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   777
proof
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   778
  show "of_rat a = id a" for a
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   779
    by (induct a) (simp add: of_rat_rat Fract_of_int_eq [symmetric])
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   780
qed
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   781
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   782
text \<open>Collapse nested embeddings\<close>
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   783
lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   784
  by (induct n) (simp_all add: of_rat_add)
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   785
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   786
lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   787
  by (cases z rule: int_diff_cases) (simp add: of_rat_diff)
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   788
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   789
lemma of_rat_numeral_eq [simp]: "of_rat (numeral w) = numeral w"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   790
  using of_rat_of_int_eq [of "numeral w"] by simp
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   791
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   792
lemma of_rat_neg_numeral_eq [simp]: "of_rat (- numeral w) = - numeral w"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   793
  using of_rat_of_int_eq [of "- numeral w"] by simp
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   794
23879
4776af8be741 split class abs from class minus
haftmann
parents: 23429
diff changeset
   795
lemmas zero_rat = Zero_rat_def
4776af8be741 split class abs from class minus
haftmann
parents: 23429
diff changeset
   796
lemmas one_rat = One_rat_def
4776af8be741 split class abs from class minus
haftmann
parents: 23429
diff changeset
   797
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   798
abbreviation rat_of_nat :: "nat \<Rightarrow> rat"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   799
  where "rat_of_nat \<equiv> of_nat"
24198
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   800
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   801
abbreviation rat_of_int :: "int \<Rightarrow> rat"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   802
  where "rat_of_int \<equiv> of_int"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   803
24198
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   804
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
   805
subsection \<open>The Set of Rational Numbers\<close>
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   806
28001
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   807
context field_char_0
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   808
begin
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   809
61070
b72a990adfe2 prefer symbols;
wenzelm
parents: 60758
diff changeset
   810
definition Rats :: "'a set" ("\<rat>")
b72a990adfe2 prefer symbols;
wenzelm
parents: 60758
diff changeset
   811
  where "\<rat> = range of_rat"
28001
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   812
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   813
end
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   814
61070
b72a990adfe2 prefer symbols;
wenzelm
parents: 60758
diff changeset
   815
lemma Rats_of_rat [simp]: "of_rat r \<in> \<rat>"
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   816
  by (simp add: Rats_def)
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   817
61070
b72a990adfe2 prefer symbols;
wenzelm
parents: 60758
diff changeset
   818
lemma Rats_of_int [simp]: "of_int z \<in> \<rat>"
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   819
  by (subst of_rat_of_int_eq [symmetric]) (rule Rats_of_rat)
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   820
61070
b72a990adfe2 prefer symbols;
wenzelm
parents: 60758
diff changeset
   821
lemma Rats_of_nat [simp]: "of_nat n \<in> \<rat>"
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   822
  by (subst of_rat_of_nat_eq [symmetric]) (rule Rats_of_rat)
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   823
61070
b72a990adfe2 prefer symbols;
wenzelm
parents: 60758
diff changeset
   824
lemma Rats_number_of [simp]: "numeral w \<in> \<rat>"
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   825
  by (subst of_rat_numeral_eq [symmetric]) (rule Rats_of_rat)
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   826
61070
b72a990adfe2 prefer symbols;
wenzelm
parents: 60758
diff changeset
   827
lemma Rats_0 [simp]: "0 \<in> \<rat>"
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   828
  unfolding Rats_def by (rule range_eqI) (rule of_rat_0 [symmetric])
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   829
61070
b72a990adfe2 prefer symbols;
wenzelm
parents: 60758
diff changeset
   830
lemma Rats_1 [simp]: "1 \<in> \<rat>"
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   831
  unfolding Rats_def by (rule range_eqI) (rule of_rat_1 [symmetric])
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   832
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   833
lemma Rats_add [simp]: "a \<in> \<rat> \<Longrightarrow> b \<in> \<rat> \<Longrightarrow> a + b \<in> \<rat>"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   834
  apply (auto simp add: Rats_def)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   835
  apply (rule range_eqI)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   836
  apply (rule of_rat_add [symmetric])
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   837
  done
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   838
61070
b72a990adfe2 prefer symbols;
wenzelm
parents: 60758
diff changeset
   839
lemma Rats_minus [simp]: "a \<in> \<rat> \<Longrightarrow> - a \<in> \<rat>"
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   840
  apply (auto simp add: Rats_def)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   841
  apply (rule range_eqI)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   842
  apply (rule of_rat_minus [symmetric])
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   843
  done
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   844
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   845
lemma Rats_diff [simp]: "a \<in> \<rat> \<Longrightarrow> b \<in> \<rat> \<Longrightarrow> a - b \<in> \<rat>"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   846
  apply (auto simp add: Rats_def)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   847
  apply (rule range_eqI)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   848
  apply (rule of_rat_diff [symmetric])
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   849
  done
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   850
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   851
lemma Rats_mult [simp]: "a \<in> \<rat> \<Longrightarrow> b \<in> \<rat> \<Longrightarrow> a * b \<in> \<rat>"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   852
  apply (auto simp add: Rats_def)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   853
  apply (rule range_eqI)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   854
  apply (rule of_rat_mult [symmetric])
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   855
  done
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   856
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   857
lemma nonzero_Rats_inverse: "a \<in> \<rat> \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> inverse a \<in> \<rat>" for a :: "'a::field_char_0"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   858
  apply (auto simp add: Rats_def)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   859
  apply (rule range_eqI)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   860
  apply (erule nonzero_of_rat_inverse [symmetric])
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   861
  done
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   862
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   863
lemma Rats_inverse [simp]: "a \<in> \<rat> \<Longrightarrow> inverse a \<in> \<rat>" for a :: "'a::{field_char_0,field}"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   864
  apply (auto simp add: Rats_def)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   865
  apply (rule range_eqI)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   866
  apply (rule of_rat_inverse [symmetric])
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   867
  done
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   868
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   869
lemma nonzero_Rats_divide: "a \<in> \<rat> \<Longrightarrow> b \<in> \<rat> \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a / b \<in> \<rat>" for a b :: "'a::field_char_0"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   870
  apply (auto simp add: Rats_def)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   871
  apply (rule range_eqI)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   872
  apply (erule nonzero_of_rat_divide [symmetric])
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   873
  done
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   874
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   875
lemma Rats_divide [simp]: "a \<in> \<rat> \<Longrightarrow> b \<in> \<rat> \<Longrightarrow> a / b \<in> \<rat>" for a b :: "'a::{field_char_0, field}"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   876
  apply (auto simp add: Rats_def)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   877
  apply (rule range_eqI)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   878
  apply (rule of_rat_divide [symmetric])
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   879
  done
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   880
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   881
lemma Rats_power [simp]: "a \<in> \<rat> \<Longrightarrow> a ^ n \<in> \<rat>" for a :: "'a::field_char_0"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   882
  apply (auto simp add: Rats_def)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   883
  apply (rule range_eqI)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   884
  apply (rule of_rat_power [symmetric])
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   885
  done
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   886
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   887
lemma Rats_cases [cases set: Rats]:
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   888
  assumes "q \<in> \<rat>"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   889
  obtains (of_rat) r where "q = of_rat r"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   890
proof -
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
   891
  from \<open>q \<in> \<rat>\<close> have "q \<in> range of_rat" unfolding Rats_def .
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   892
  then obtain r where "q = of_rat r" ..
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   893
  then show thesis ..
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   894
qed
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   895
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   896
lemma Rats_induct [case_names of_rat, induct set: Rats]: "q \<in> \<rat> \<Longrightarrow> (\<And>r. P (of_rat r)) \<Longrightarrow> P q"
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   897
  by (rule Rats_cases) auto
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   898
57275
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57136
diff changeset
   899
lemma Rats_infinite: "\<not> finite \<rat>"
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 57136
diff changeset
   900
  by (auto dest!: finite_imageD simp: inj_on_def infinite_UNIV_char_0 Rats_def)
28001
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   901
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   902
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
   903
subsection \<open>Implementation of rational numbers as pairs of integers\<close>
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   904
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
   905
text \<open>Formal constructor\<close>
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   906
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   907
definition Frct :: "int \<times> int \<Rightarrow> rat"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   908
  where [simp]: "Frct p = Fract (fst p) (snd p)"
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   909
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   910
lemma [code abstype]: "Frct (quotient_of q) = q"
36112
7fa17a225852 user interface for abstract datatypes is an attribute, not a command
haftmann
parents: 35726
diff changeset
   911
  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
   912
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   913
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
   914
text \<open>Numerals\<close>
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   915
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   916
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
   917
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   918
definition of_int :: "int \<Rightarrow> rat"
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   919
  where [code_abbrev]: "of_int = Int.of_int"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   920
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   921
hide_const (open) of_int
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   922
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   923
lemma quotient_of_int [code abstract]: "quotient_of (Rat.of_int a) = (a, 1)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   924
  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
   925
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   926
lemma [code_unfold]: "numeral k = Rat.of_int (numeral k)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   927
  by (simp add: Rat.of_int_def)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   928
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   929
lemma [code_unfold]: "- numeral k = Rat.of_int (- numeral k)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   930
  by (simp add: Rat.of_int_def)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   931
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   932
lemma Frct_code_post [code_post]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   933
  "Frct (0, a) = 0"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   934
  "Frct (a, 0) = 0"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   935
  "Frct (1, 1) = 1"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   936
  "Frct (numeral k, 1) = numeral k"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   937
  "Frct (1, numeral k) = 1 / numeral k"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   938
  "Frct (numeral k, numeral l) = numeral k / numeral l"
57576
083dfad2727c more appropriate postprocessing of rational numbers: extract sign to front of fraction
haftmann
parents: 57514
diff changeset
   939
  "Frct (- a, b) = - Frct (a, b)"
083dfad2727c more appropriate postprocessing of rational numbers: extract sign to front of fraction
haftmann
parents: 57514
diff changeset
   940
  "Frct (a, - b) = - Frct (a, b)"
083dfad2727c more appropriate postprocessing of rational numbers: extract sign to front of fraction
haftmann
parents: 57514
diff changeset
   941
  "- (- Frct q) = Frct q"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   942
  by (simp_all add: Fract_of_int_quotient)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   943
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   944
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
   945
text \<open>Operations\<close>
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   946
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   947
lemma rat_zero_code [code abstract]: "quotient_of 0 = (0, 1)"
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   948
  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
   949
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   950
lemma rat_one_code [code abstract]: "quotient_of 1 = (1, 1)"
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   951
  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
   952
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   953
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
   954
  "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
   955
     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
   956
  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
   957
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   958
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
   959
  "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
   960
  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
   961
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   962
lemma rat_minus_code [code abstract]:
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   963
  "quotient_of (p - q) =
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   964
    (let (a, c) = quotient_of p; (b, d) = quotient_of q
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   965
     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
   966
  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
   967
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   968
lemma rat_times_code [code abstract]:
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   969
  "quotient_of (p * q) =
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   970
    (let (a, c) = quotient_of p; (b, d) = quotient_of q
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   971
     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
   972
  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
   973
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   974
lemma rat_inverse_code [code abstract]:
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   975
  "quotient_of (inverse p) =
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   976
    (let (a, b) = quotient_of p
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   977
     in if a = 0 then (0, 1) else (sgn a * b, \<bar>a\<bar>))"
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   978
proof (cases p)
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   979
  case (Fract a b)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   980
  then show ?thesis
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60429
diff changeset
   981
    by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract gcd.commute)
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   982
qed
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   983
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   984
lemma rat_divide_code [code abstract]:
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   985
  "quotient_of (p / q) =
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   986
    (let (a, c) = quotient_of p; (b, d) = quotient_of q
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   987
     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
   988
  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
   989
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   990
lemma rat_abs_code [code abstract]: "quotient_of \<bar>p\<bar> = (let (a, b) = quotient_of p in (\<bar>a\<bar>, b))"
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   991
  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
   992
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   993
lemma rat_sgn_code [code abstract]: "quotient_of (sgn p) = (sgn (fst (quotient_of p)), 1)"
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   994
proof (cases p)
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   995
  case (Fract a b)
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   996
  then show ?thesis
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
   997
    by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract)
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   998
qed
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   999
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1000
lemma rat_floor_code [code]: "\<lfloor>p\<rfloor> = (let (a, b) = quotient_of p in a div b)"
61942
f02b26f7d39d prefer symbols for "floor", "ceiling";
wenzelm
parents: 61799
diff changeset
  1001
  by (cases p) (simp add: quotient_of_Fract floor_Fract)
43733
a6ca7b83612f adding code equations to execute floor and ceiling on rational and real numbers
bulwahn
parents: 43732
diff changeset
  1002
38857
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38287
diff changeset
  1003
instantiation rat :: equal
26513
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
  1004
begin
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
  1005
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1006
definition [code]: "HOL.equal a b \<longleftrightarrow> quotient_of a = quotient_of b"
26513
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
  1007
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1008
instance
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1009
  by standard (simp add: equal_rat_def quotient_of_inject_eq)
26513
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
  1010
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1011
lemma rat_eq_refl [code nbe]: "HOL.equal (r::rat) r \<longleftrightarrow> True"
38857
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38287
diff changeset
  1012
  by (rule equal_refl)
28351
abfc66969d1f non left-linear equations for nbe
haftmann
parents: 28313
diff changeset
  1013
26513
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
  1014
end
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
  1015
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1016
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
  1017
  "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
  1018
  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
  1019
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1020
lemma rat_less_code [code]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1021
  "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
  1022
  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
  1023
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1024
lemma [code]: "of_rat p = (let (a, b) = quotient_of p in of_int a / of_int b)"
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1025
  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
  1026
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1027
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
  1028
text \<open>Quickcheck\<close>
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1029
31203
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1030
definition (in term_syntax)
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1031
  valterm_fract :: "int \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> int \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow>
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1032
    rat \<times> (unit \<Rightarrow> Code_Evaluation.term)"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1033
  where [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
  1034
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37397
diff changeset
  1035
notation fcomp (infixl "\<circ>>" 60)
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37397
diff changeset
  1036
notation scomp (infixl "\<circ>\<rightarrow>" 60)
31203
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1037
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1038
instantiation rat :: random
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1039
begin
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1040
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1041
definition
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1042
  "Quickcheck_Random.random i =
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1043
    Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>num. Random.range i \<circ>\<rightarrow> (\<lambda>denom. Pair
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1044
      (let j = int_of_integer (integer_of_natural (denom + 1))
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1045
       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
  1046
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1047
instance ..
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1048
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1049
end
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1050
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37397
diff changeset
  1051
no_notation fcomp (infixl "\<circ>>" 60)
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37397
diff changeset
  1052
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
31203
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1053
41920
d4fb7a418152 moving exhaustive_generators.ML to Quickcheck directory
bulwahn
parents: 41792
diff changeset
  1054
instantiation rat :: exhaustive
41231
2e901158675e adding exhaustive tester instances for numeric types: code_numeral, nat, rat and real
bulwahn
parents: 40819
diff changeset
  1055
begin
2e901158675e adding exhaustive tester instances for numeric types: code_numeral, nat, rat and real
bulwahn
parents: 40819
diff changeset
  1056
2e901158675e adding exhaustive tester instances for numeric types: code_numeral, nat, rat and real
bulwahn
parents: 40819
diff changeset
  1057
definition
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1058
  "exhaustive_rat f d =
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1059
    Quickcheck_Exhaustive.exhaustive
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1060
      (\<lambda>l. Quickcheck_Exhaustive.exhaustive
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1061
        (\<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
  1062
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1063
instance ..
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1064
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1065
end
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1066
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1067
instantiation rat :: full_exhaustive
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1068
begin
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1069
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1070
definition
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1071
  "full_exhaustive_rat f d =
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1072
    Quickcheck_Exhaustive.full_exhaustive
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1073
      (\<lambda>(l, _). Quickcheck_Exhaustive.full_exhaustive
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1074
        (\<lambda>k. f
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1075
          (let j = int_of_integer (integer_of_natural l) + 1
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1076
           in valterm_fract k (j, \<lambda>_. Code_Evaluation.term_of j))) d) d"
43889
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1077
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1078
instance ..
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1079
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1080
end
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1081
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1082
instance rat :: partial_term_of ..
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1083
43889
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1084
lemma [code]:
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1085
  "partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Narrowing_variable p tt) \<equiv>
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1086
    Code_Evaluation.Free (STR ''_'') (Typerep.Typerep (STR ''Rat.rat'') [])"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1087
  "partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Narrowing_constructor 0 [l, k]) \<equiv>
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1088
    Code_Evaluation.App
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1089
      (Code_Evaluation.Const (STR ''Rat.Frct'')
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1090
        (Typerep.Typerep (STR ''fun'')
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1091
          [Typerep.Typerep (STR ''Product_Type.prod'')
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1092
           [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''Int.int'') []],
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1093
           Typerep.Typerep (STR ''Rat.rat'') []]))
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1094
      (Code_Evaluation.App
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1095
        (Code_Evaluation.App
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1096
          (Code_Evaluation.Const (STR ''Product_Type.Pair'')
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1097
            (Typerep.Typerep (STR ''fun'')
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1098
              [Typerep.Typerep (STR ''Int.int'') [],
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1099
               Typerep.Typerep (STR ''fun'')
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1100
                [Typerep.Typerep (STR ''Int.int'') [],
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1101
                 Typerep.Typerep (STR ''Product_Type.prod'')
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1102
                 [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''Int.int'') []]]]))
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1103
          (partial_term_of (TYPE(int)) l)) (partial_term_of (TYPE(int)) k))"
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1104
  by (rule partial_term_of_anything)+
43889
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1105
43887
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1106
instantiation rat :: narrowing
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1107
begin
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1108
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1109
definition
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1110
  "narrowing =
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1111
    Quickcheck_Narrowing.apply
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1112
      (Quickcheck_Narrowing.apply
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1113
        (Quickcheck_Narrowing.cons (\<lambda>nom denom. Fract nom denom)) narrowing) narrowing"
43887
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1114
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1115
instance ..
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1116
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1117
end
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1118
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1119
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
  1120
subsection \<open>Setup for Nitpick\<close>
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
  1121
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
  1122
declaration \<open>
38287
796302ca3611 replace "setup" with "declaration"
blanchet
parents: 38242
diff changeset
  1123
  Nitpick_HOL.register_frac_type @{type_name rat}
62079
3a21fddf0328 more complete setup for 'Rat' in Nitpick
blanchet
parents: 61944
diff changeset
  1124
    [(@{const_name Abs_Rat}, @{const_name Nitpick.Abs_Frac}),
3a21fddf0328 more complete setup for 'Rat' in Nitpick
blanchet
parents: 61944
diff changeset
  1125
     (@{const_name zero_rat_inst.zero_rat}, @{const_name Nitpick.zero_frac}),
3a21fddf0328 more complete setup for 'Rat' in Nitpick
blanchet
parents: 61944
diff changeset
  1126
     (@{const_name one_rat_inst.one_rat}, @{const_name Nitpick.one_frac}),
3a21fddf0328 more complete setup for 'Rat' in Nitpick
blanchet
parents: 61944
diff changeset
  1127
     (@{const_name plus_rat_inst.plus_rat}, @{const_name Nitpick.plus_frac}),
3a21fddf0328 more complete setup for 'Rat' in Nitpick
blanchet
parents: 61944
diff changeset
  1128
     (@{const_name times_rat_inst.times_rat}, @{const_name Nitpick.times_frac}),
3a21fddf0328 more complete setup for 'Rat' in Nitpick
blanchet
parents: 61944
diff changeset
  1129
     (@{const_name uminus_rat_inst.uminus_rat}, @{const_name Nitpick.uminus_frac}),
3a21fddf0328 more complete setup for 'Rat' in Nitpick
blanchet
parents: 61944
diff changeset
  1130
     (@{const_name inverse_rat_inst.inverse_rat}, @{const_name Nitpick.inverse_frac}),
3a21fddf0328 more complete setup for 'Rat' in Nitpick
blanchet
parents: 61944
diff changeset
  1131
     (@{const_name ord_rat_inst.less_rat}, @{const_name Nitpick.less_frac}),
3a21fddf0328 more complete setup for 'Rat' in Nitpick
blanchet
parents: 61944
diff changeset
  1132
     (@{const_name ord_rat_inst.less_eq_rat}, @{const_name Nitpick.less_eq_frac}),
3a21fddf0328 more complete setup for 'Rat' in Nitpick
blanchet
parents: 61944
diff changeset
  1133
     (@{const_name field_char_0_class.of_rat}, @{const_name Nitpick.of_frac})]
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
  1134
\<close>
33197
de6285ebcc05 continuation of Nitpick's integration into Isabelle;
blanchet
parents: 32657
diff changeset
  1135
63326
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1136
lemmas [nitpick_unfold] =
9d2470571719 misc tuning and modernization;
wenzelm
parents: 62390
diff changeset
  1137
  inverse_rat_inst.inverse_rat
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1138
  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
  1139
  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
  1140
  uminus_rat_inst.uminus_rat zero_rat_inst.zero_rat
33197
de6285ebcc05 continuation of Nitpick's integration into Isabelle;
blanchet
parents: 32657
diff changeset
  1141
52146
wenzelm
parents: 51956
diff changeset
  1142
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
  1143
subsection \<open>Float syntax\<close>
35343
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1144
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1145
syntax "_Float" :: "float_const \<Rightarrow> 'a"    ("_")
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1146
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
  1147
parse_translation \<open>
52146
wenzelm
parents: 51956
diff changeset
  1148
  let
wenzelm
parents: 51956
diff changeset
  1149
    fun mk_frac str =
wenzelm
parents: 51956
diff changeset
  1150
      let
wenzelm
parents: 51956
diff changeset
  1151
        val {mant = i, exp = n} = Lexicon.read_float str;
wenzelm
parents: 51956
diff changeset
  1152
        val exp = Syntax.const @{const_syntax Power.power};
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 57576
diff changeset
  1153
        val ten = Numeral.mk_number_syntax 10;
60352
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59984
diff changeset
  1154
        val exp10 = if n = 1 then ten else exp $ ten $ Numeral.mk_number_syntax n;
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59984
diff changeset
  1155
      in Syntax.const @{const_syntax Fields.inverse_divide} $ Numeral.mk_number_syntax i $ exp10 end;
52146
wenzelm
parents: 51956
diff changeset
  1156
wenzelm
parents: 51956
diff changeset
  1157
    fun float_tr [(c as Const (@{syntax_const "_constrain"}, _)) $ t $ u] = c $ float_tr [t] $ u
wenzelm
parents: 51956
diff changeset
  1158
      | float_tr [t as Const (str, _)] = mk_frac str
wenzelm
parents: 51956
diff changeset
  1159
      | float_tr ts = raise TERM ("float_tr", ts);
wenzelm
parents: 51956
diff changeset
  1160
  in [(@{syntax_const "_Float"}, K float_tr)] end
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
  1161
\<close>
35343
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1162
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
  1163
text\<open>Test:\<close>
35343
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1164
lemma "123.456 = -111.111 + 200 + 30 + 4 + 5/10 + 6/100 + (7/1000::rat)"
52146
wenzelm
parents: 51956
diff changeset
  1165
  by simp
35343
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1166
55974
c835a9379026 more official const syntax: avoid educated guessing by Syntax_Phases.decode_term;
wenzelm
parents: 55143
diff changeset
  1167
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60688
diff changeset
  1168
subsection \<open>Hiding implementation details\<close>
37143
2a5182751151 constant Rat.normalize needs to be qualified;
wenzelm
parents: 36415
diff changeset
  1169
47907
54e3847f1669 simplify instance proofs for rat
huffman
parents: 47906
diff changeset
  1170
hide_const (open) normalize positive
37143
2a5182751151 constant Rat.normalize needs to be qualified;
wenzelm
parents: 36415
diff changeset
  1171
53652
18fbca265e2e use lifting_forget for deregistering numeric types as a quotient type
kuncar
parents: 53374
diff changeset
  1172
lifting_update rat.lifting
18fbca265e2e use lifting_forget for deregistering numeric types as a quotient type
kuncar
parents: 53374
diff changeset
  1173
lifting_forget rat.lifting
47906
09a896d295bd convert Rat.thy to use lift_definition/transfer
huffman
parents: 47108
diff changeset
  1174
29880
3dee8ff45d3d move countability proof from Rational to Countable; add instance rat :: countable
huffman
parents: 29667
diff changeset
  1175
end