src/HOL/Rat.thy
author wenzelm
Sat, 07 Apr 2012 16:41:59 +0200
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explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions; tuned;
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(*  Title:  HOL/Rat.thy
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    Author: Markus Wenzel, TU Muenchen
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*)
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header {* Rational numbers *}
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theory Rat
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imports GCD Archimedean_Field
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uses ("Tools/float_syntax.ML")
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begin
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subsection {* Rational numbers as quotient *}
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subsubsection {* Construction of the type of rational numbers *}
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definition
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  ratrel :: "((int \<times> int) \<times> (int \<times> int)) set" where
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  "ratrel = {(x, y). snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x}"
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lemma ratrel_iff [simp]:
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  "(x, y) \<in> ratrel \<longleftrightarrow> snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x"
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  by (simp add: ratrel_def)
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lemma refl_on_ratrel: "refl_on {x. snd x \<noteq> 0} ratrel"
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  by (auto simp add: refl_on_def ratrel_def)
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lemma sym_ratrel: "sym ratrel"
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  by (simp add: ratrel_def sym_def)
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lemma trans_ratrel: "trans ratrel"
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proof (rule transI, unfold split_paired_all)
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  fix a b a' b' a'' b'' :: int
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  assume A: "((a, b), (a', b')) \<in> ratrel"
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  assume B: "((a', b'), (a'', b'')) \<in> ratrel"
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  have "b' * (a * b'') = b'' * (a * b')" by simp
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  also from A have "a * b' = a' * b" by auto
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  also have "b'' * (a' * b) = b * (a' * b'')" by simp
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  also from B have "a' * b'' = a'' * b'" by auto
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  also have "b * (a'' * b') = b' * (a'' * b)" by simp
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  finally have "b' * (a * b'') = b' * (a'' * b)" .
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  moreover from B have "b' \<noteq> 0" by auto
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  ultimately have "a * b'' = a'' * b" by simp
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  with A B show "((a, b), (a'', b'')) \<in> ratrel" by auto
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qed
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lemma equiv_ratrel: "equiv {x. snd x \<noteq> 0} ratrel"
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  by (rule equivI [OF refl_on_ratrel sym_ratrel trans_ratrel])
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lemmas UN_ratrel = UN_equiv_class [OF equiv_ratrel]
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lemmas UN_ratrel2 = UN_equiv_class2 [OF equiv_ratrel equiv_ratrel]
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lemma equiv_ratrel_iff [iff]: 
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  assumes "snd x \<noteq> 0" and "snd y \<noteq> 0"
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  shows "ratrel `` {x} = ratrel `` {y} \<longleftrightarrow> (x, y) \<in> ratrel"
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  by (rule eq_equiv_class_iff, rule equiv_ratrel) (auto simp add: assms)
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definition "Rat = {x. snd x \<noteq> 0} // ratrel"
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typedef (open) rat = Rat
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  morphisms Rep_Rat Abs_Rat
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  unfolding Rat_def
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proof
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  have "(0::int, 1::int) \<in> {x. snd x \<noteq> 0}" by simp
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  then show "ratrel `` {(0, 1)} \<in> {x. snd x \<noteq> 0} // ratrel" by (rule quotientI)
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qed
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lemma ratrel_in_Rat [simp]: "snd x \<noteq> 0 \<Longrightarrow> ratrel `` {x} \<in> Rat"
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  by (simp add: Rat_def quotientI)
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declare Abs_Rat_inject [simp] Abs_Rat_inverse [simp]
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subsubsection {* Representation and basic operations *}
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definition
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  Fract :: "int \<Rightarrow> int \<Rightarrow> rat" where
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  "Fract a b = Abs_Rat (ratrel `` {if b = 0 then (0, 1) else (a, b)})"
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lemma eq_rat:
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  shows "\<And>a b c d. b \<noteq> 0 \<Longrightarrow> d \<noteq> 0 \<Longrightarrow> Fract a b = Fract c d \<longleftrightarrow> a * d = c * b"
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  and "\<And>a. Fract a 0 = Fract 0 1"
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  and "\<And>a c. Fract 0 a = Fract 0 c"
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  by (simp_all add: Fract_def)
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lemma Rat_cases [case_names Fract, cases type: rat]:
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  assumes "\<And>a b. q = Fract a b \<Longrightarrow> b > 0 \<Longrightarrow> coprime a b \<Longrightarrow> C"
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  shows C
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proof -
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  obtain a b :: int where "q = Fract a b" and "b \<noteq> 0"
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    by (cases q) (clarsimp simp add: Fract_def Rat_def quotient_def)
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  let ?a = "a div gcd a b"
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  let ?b = "b div gcd a b"
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  from `b \<noteq> 0` have "?b * gcd a b = b"
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    by (simp add: dvd_div_mult_self)
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  with `b \<noteq> 0` have "?b \<noteq> 0" by auto
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  from `q = Fract a b` `b \<noteq> 0` `?b \<noteq> 0` have q: "q = Fract ?a ?b"
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    by (simp add: eq_rat dvd_div_mult mult_commute [of a])
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  from `b \<noteq> 0` have coprime: "coprime ?a ?b"
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    by (auto intro: div_gcd_coprime_int)
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  show C proof (cases "b > 0")
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    case True
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    note assms
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    moreover note q
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    moreover from True have "?b > 0" by (simp add: nonneg1_imp_zdiv_pos_iff)
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    moreover note coprime
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    ultimately show C .
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  next
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    case False
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    note assms
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    moreover from q have "q = Fract (- ?a) (- ?b)" by (simp add: Fract_def)
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    moreover from False `b \<noteq> 0` have "- ?b > 0" by (simp add: pos_imp_zdiv_neg_iff)
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    moreover from coprime have "coprime (- ?a) (- ?b)" by simp
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    ultimately show C .
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  qed
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qed
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lemma Rat_induct [case_names Fract, induct type: rat]:
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  assumes "\<And>a b. b > 0 \<Longrightarrow> coprime a b \<Longrightarrow> P (Fract a b)"
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  shows "P q"
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  using assms by (cases q) simp
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instantiation rat :: comm_ring_1
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begin
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definition
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  Zero_rat_def: "0 = Fract 0 1"
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definition
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  One_rat_def: "1 = Fract 1 1"
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definition
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  add_rat_def:
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  "q + r = Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
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    ratrel `` {(fst x * snd y + fst y * snd x, snd x * snd y)})"
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lemma add_rat [simp]:
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  assumes "b \<noteq> 0" and "d \<noteq> 0"
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  shows "Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
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proof -
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   140
  have "(\<lambda>x y. ratrel``{(fst x * snd y + fst y * snd x, snd x * snd y)})
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   141
    respects2 ratrel"
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  by (rule equiv_ratrel [THEN congruent2_commuteI]) (simp_all add: left_distrib)
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   143
  with assms show ?thesis by (simp add: Fract_def add_rat_def UN_ratrel2)
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   144
qed
18913
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parents: 18372
diff changeset
   145
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   146
definition
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  minus_rat_def:
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   148
  "- q = Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel `` {(- fst x, snd x)})"
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   149
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   150
lemma minus_rat [simp]: "- Fract a b = Fract (- a) b"
27551
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   151
proof -
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   152
  have "(\<lambda>x. ratrel `` {(- fst x, snd x)}) respects ratrel"
40819
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   153
    by (simp add: congruent_def split_paired_all)
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   154
  then show ?thesis by (simp add: Fract_def minus_rat_def UN_ratrel)
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parents: 27509
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   155
qed
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parents: 27509
diff changeset
   156
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   157
lemma minus_rat_cancel [simp]: "Fract (- a) (- b) = Fract a b"
27551
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   158
  by (cases "b = 0") (simp_all add: eq_rat)
25571
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   159
c9e39eafc7a0 instantiation target rather than legacy instance
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   160
definition
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   161
  diff_rat_def: "q - r = q + - (r::rat)"
18913
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huffman
parents: 18372
diff changeset
   162
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   163
lemma diff_rat [simp]:
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   164
  assumes "b \<noteq> 0" and "d \<noteq> 0"
9a5543d4cc24 Fract now total; improved code generator setup
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   165
  shows "Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
27652
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   166
  using assms by (simp add: diff_rat_def)
25571
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haftmann
parents: 25502
diff changeset
   167
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
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diff changeset
   168
definition
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   169
  mult_rat_def:
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   170
  "q * r = Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
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   171
    ratrel``{(fst x * fst y, snd x * snd y)})"
14365
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paulson
parents:
diff changeset
   172
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   173
lemma mult_rat [simp]: "Fract a b * Fract c d = Fract (a * c) (b * d)"
27551
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diff changeset
   174
proof -
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parents: 27509
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   175
  have "(\<lambda>x y. ratrel `` {(fst x * fst y, snd x * snd y)}) respects2 ratrel"
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haftmann
parents: 27509
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   176
    by (rule equiv_ratrel [THEN congruent2_commuteI]) simp_all
9a5543d4cc24 Fract now total; improved code generator setup
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parents: 27509
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   177
  then show ?thesis by (simp add: Fract_def mult_rat_def UN_ratrel2)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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   178
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   179
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   180
lemma mult_rat_cancel:
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   181
  assumes "c \<noteq> 0"
9a5543d4cc24 Fract now total; improved code generator setup
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   182
  shows "Fract (c * a) (c * b) = Fract a b"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   183
proof -
9a5543d4cc24 Fract now total; improved code generator setup
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   184
  from assms have "Fract c c = Fract 1 1" by (simp add: Fract_def)
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   185
  then show ?thesis by (simp add: mult_rat [symmetric])
27551
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parents: 27509
diff changeset
   186
qed
27509
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parents: 26732
diff changeset
   187
63161d5f8f29 rearrange instantiations
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   188
instance proof
27668
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chaieb
parents: 27652
diff changeset
   189
  fix q r s :: rat show "(q * r) * s = q * (r * s)" 
27652
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haftmann
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diff changeset
   190
    by (cases q, cases r, cases s) (simp add: eq_rat)
27551
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diff changeset
   191
next
9a5543d4cc24 Fract now total; improved code generator setup
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diff changeset
   192
  fix q r :: rat show "q * r = r * q"
27652
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   193
    by (cases q, cases r) (simp add: eq_rat)
27551
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diff changeset
   194
next
9a5543d4cc24 Fract now total; improved code generator setup
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diff changeset
   195
  fix q :: rat show "1 * q = q"
27652
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   196
    by (cases q) (simp add: One_rat_def eq_rat)
27551
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diff changeset
   197
next
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   198
  fix q r s :: rat show "(q + r) + s = q + (r + s)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29332
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   199
    by (cases q, cases r, cases s) (simp add: eq_rat algebra_simps)
27551
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parents: 27509
diff changeset
   200
next
9a5543d4cc24 Fract now total; improved code generator setup
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diff changeset
   201
  fix q r :: rat show "q + r = r + q"
27652
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haftmann
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   202
    by (cases q, cases r) (simp add: eq_rat)
27551
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haftmann
parents: 27509
diff changeset
   203
next
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   204
  fix q :: rat show "0 + q = q"
27652
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haftmann
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   205
    by (cases q) (simp add: Zero_rat_def eq_rat)
27551
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parents: 27509
diff changeset
   206
next
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   207
  fix q :: rat show "- q + q = 0"
27652
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diff changeset
   208
    by (cases q) (simp add: Zero_rat_def eq_rat)
27551
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diff changeset
   209
next
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   210
  fix q r :: rat show "q - r = q + - r"
27652
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diff changeset
   211
    by (cases q, cases r) (simp add: eq_rat)
27551
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haftmann
parents: 27509
diff changeset
   212
next
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   213
  fix q r s :: rat show "(q + r) * s = q * s + r * s"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29332
diff changeset
   214
    by (cases q, cases r, cases s) (simp add: eq_rat algebra_simps)
27551
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haftmann
parents: 27509
diff changeset
   215
next
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   216
  show "(0::rat) \<noteq> 1" by (simp add: Zero_rat_def One_rat_def eq_rat)
27509
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huffman
parents: 26732
diff changeset
   217
qed
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   218
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   219
end
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   220
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   221
lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1"
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   222
  by (induct k) (simp_all add: Zero_rat_def One_rat_def)
27551
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haftmann
parents: 27509
diff changeset
   223
9a5543d4cc24 Fract now total; improved code generator setup
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diff changeset
   224
lemma of_int_rat: "of_int k = Fract k 1"
27652
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diff changeset
   225
  by (cases k rule: int_diff_cases) (simp add: of_nat_rat)
27551
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parents: 27509
diff changeset
   226
9a5543d4cc24 Fract now total; improved code generator setup
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diff changeset
   227
lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
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diff changeset
   228
  by (rule of_nat_rat [symmetric])
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   229
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
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   230
lemma Fract_of_int_eq: "Fract k 1 = of_int k"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   231
  by (rule of_int_rat [symmetric])
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haftmann
parents: 27509
diff changeset
   232
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   233
lemma rat_number_collapse:
27551
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   234
  "Fract 0 k = 0"
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haftmann
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diff changeset
   235
  "Fract 1 1 = 1"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   236
  "Fract (numeral w) 1 = numeral w"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
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   237
  "Fract (neg_numeral w) 1 = neg_numeral w"
27551
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   238
  "Fract k 0 = 0"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   239
  using Fract_of_int_eq [of "numeral w"]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   240
  using Fract_of_int_eq [of "neg_numeral w"]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   241
  by (simp_all add: Zero_rat_def One_rat_def eq_rat)
27551
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parents: 27509
diff changeset
   242
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   243
lemma rat_number_expand:
27551
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   244
  "0 = Fract 0 1"
9a5543d4cc24 Fract now total; improved code generator setup
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   245
  "1 = Fract 1 1"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   246
  "numeral k = Fract (numeral k) 1"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   247
  "neg_numeral k = Fract (neg_numeral k) 1"
27551
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diff changeset
   248
  by (simp_all add: rat_number_collapse)
9a5543d4cc24 Fract now total; improved code generator setup
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parents: 27509
diff changeset
   249
9a5543d4cc24 Fract now total; improved code generator setup
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diff changeset
   250
lemma Rat_cases_nonzero [case_names Fract 0]:
35369
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parents: 35293
diff changeset
   251
  assumes Fract: "\<And>a b. q = Fract a b \<Longrightarrow> b > 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> coprime a b \<Longrightarrow> C"
27551
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   252
  assumes 0: "q = 0 \<Longrightarrow> C"
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   253
  shows C
9a5543d4cc24 Fract now total; improved code generator setup
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   254
proof (cases "q = 0")
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   255
  case True then show C using 0 by auto
9a5543d4cc24 Fract now total; improved code generator setup
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   256
next
9a5543d4cc24 Fract now total; improved code generator setup
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   257
  case False
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haftmann
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   258
  then obtain a b where "q = Fract a b" and "b > 0" and "coprime a b" by (cases q) auto
27551
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haftmann
parents: 27509
diff changeset
   259
  moreover with False have "0 \<noteq> Fract a b" by simp
35369
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haftmann
parents: 35293
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   260
  with `b > 0` have "a \<noteq> 0" by (simp add: Zero_rat_def eq_rat)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   261
  with Fract `q = Fract a b` `b > 0` `coprime a b` show C by blast
27551
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parents: 27509
diff changeset
   262
qed
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parents: 27509
diff changeset
   263
33805
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
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   264
subsubsection {* Function @{text normalize} *}
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
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   265
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   266
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
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   267
proof (cases "b = 0")
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
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   268
  case True then show ?thesis by (simp add: eq_rat)
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   269
next
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
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   270
  case False
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
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   271
  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
   272
    by (rule dvd_div_mult_self) simp
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
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diff changeset
   273
  ultimately have "b div gcd a b \<noteq> 0" by auto
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   274
  with False show ?thesis by (simp add: eq_rat dvd_div_mult mult_commute [of a])
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   275
qed
33805
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   276
35369
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   277
definition normalize :: "int \<times> int \<Rightarrow> int \<times> int" where
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   278
  "normalize p = (if snd p > 0 then (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
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diff changeset
   279
    else if snd p = 0 then (0, 1)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
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diff changeset
   280
    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
   281
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   282
lemma normalize_crossproduct:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   283
  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
   284
  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
   285
  shows "p * s = r * q"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   286
proof -
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   287
  have aux: "p * gcd r s = sgn (q * s) * r * gcd p q \<Longrightarrow> q * gcd r s = sgn (q * s) * s * gcd p q \<Longrightarrow> p * s = q * r"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   288
  proof -
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   289
    assume "p * gcd r s = sgn (q * s) * r * gcd p q" and "q * gcd r s = sgn (q * s) * s * gcd p q"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   290
    then have "(p * gcd r s) * (sgn (q * s) * s * gcd p q) = (q * gcd r s) * (sgn (q * s) * r * gcd p q)" by simp
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   291
    with assms show "p * s = q * r" by (auto simp add: mult_ac sgn_times sgn_0_0)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   292
  qed
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   293
  from assms show ?thesis
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   294
    by (auto simp add: normalize_def Let_def dvd_div_div_eq_mult mult_commute sgn_times split: if_splits intro: aux)
33805
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   295
qed
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   296
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   297
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
   298
  by (auto simp add: normalize_def Let_def Fract_coprime dvd_div_neg rat_number_collapse
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   299
    split:split_if_asm)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   300
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   301
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
   302
  by (auto simp add: normalize_def Let_def dvd_div_neg pos_imp_zdiv_neg_iff nonneg1_imp_zdiv_pos_iff
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   303
    split:split_if_asm)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   304
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   305
lemma normalize_coprime: "normalize r = (p, q) \<Longrightarrow> coprime p q"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   306
  by (auto simp add: normalize_def Let_def dvd_div_neg div_gcd_coprime_int
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   307
    split:split_if_asm)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   308
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   309
lemma normalize_stable [simp]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   310
  "q > 0 \<Longrightarrow> coprime p q \<Longrightarrow> normalize (p, q) = (p, q)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   311
  by (simp add: normalize_def)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   312
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   313
lemma normalize_denom_zero [simp]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   314
  "normalize (p, 0) = (0, 1)"
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)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   316
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   317
lemma normalize_negative [simp]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   318
  "q < 0 \<Longrightarrow> normalize (p, q) = normalize (- p, - q)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   319
  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
   320
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   321
text{*
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   322
  Decompose a fraction into normalized, i.e. coprime numerator and denominator:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   323
*}
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   324
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   325
definition quotient_of :: "rat \<Rightarrow> int \<times> int" where
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   326
  "quotient_of x = (THE pair. x = Fract (fst pair) (snd pair) &
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   327
                   snd pair > 0 & coprime (fst pair) (snd pair))"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   328
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   329
lemma quotient_of_unique:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   330
  "\<exists>!p. r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   331
proof (cases r)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   332
  case (Fract a b)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   333
  then have "r = Fract (fst (a, b)) (snd (a, b)) \<and> snd (a, b) > 0 \<and> coprime (fst (a, b)) (snd (a, b))" by auto
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   334
  then show ?thesis proof (rule ex1I)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   335
    fix p
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   336
    obtain c d :: int where p: "p = (c, d)" by (cases p)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   337
    assume "r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   338
    with p have Fract': "r = Fract c d" "d > 0" "coprime c d" by simp_all
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   339
    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
   340
    proof (cases "a = 0")
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   341
      case True with Fract Fract' show ?thesis by (simp add: eq_rat)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   342
    next
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   343
      case False
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   344
      with Fract Fract' have *: "c * b = a * d" and "c \<noteq> 0" by (auto simp add: eq_rat)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   345
      then have "c * b > 0 \<longleftrightarrow> a * d > 0" by auto
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   346
      with `b > 0` `d > 0` have "a > 0 \<longleftrightarrow> c > 0" by (simp add: zero_less_mult_iff)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   347
      with `a \<noteq> 0` `c \<noteq> 0` have sgn: "sgn a = sgn c" by (auto simp add: not_less)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   348
      from `coprime a b` `coprime c d` have "\<bar>a\<bar> * \<bar>d\<bar> = \<bar>c\<bar> * \<bar>b\<bar> \<longleftrightarrow> \<bar>a\<bar> = \<bar>c\<bar> \<and> \<bar>d\<bar> = \<bar>b\<bar>"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   349
        by (simp add: coprime_crossproduct_int)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   350
      with `b > 0` `d > 0` have "\<bar>a\<bar> * d = \<bar>c\<bar> * b \<longleftrightarrow> \<bar>a\<bar> = \<bar>c\<bar> \<and> d = b" by simp
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   351
      then have "a * sgn a * d = c * sgn c * b \<longleftrightarrow> a * sgn a = c * sgn c \<and> d = b" by (simp add: abs_sgn)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   352
      with sgn * show ?thesis by (auto simp add: sgn_0_0)
33805
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   353
    qed
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   354
    with p show "p = (a, b)" by simp
33805
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   355
  qed
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   356
qed
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   357
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   358
lemma quotient_of_Fract [code]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   359
  "quotient_of (Fract a b) = normalize (a, b)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   360
proof -
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   361
  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
   362
    by (rule sym) (auto intro: normalize_eq)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   363
  moreover have "0 < snd (normalize (a, b))" (is ?denom_pos) 
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   364
    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
   365
  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
   366
    by (rule normalize_coprime) simp
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   367
  ultimately have "?Fract \<and> ?denom_pos \<and> ?coprime" by blast
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   368
  with quotient_of_unique have
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   369
    "(THE p. Fract a b = Fract (fst p) (snd p) \<and> 0 < snd p \<and> coprime (fst p) (snd p)) = normalize (a, b)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   370
    by (rule the1_equality)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   371
  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
   372
qed
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   373
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   374
lemma quotient_of_number [simp]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   375
  "quotient_of 0 = (0, 1)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   376
  "quotient_of 1 = (1, 1)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   377
  "quotient_of (numeral k) = (numeral k, 1)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   378
  "quotient_of (neg_numeral k) = (neg_numeral k, 1)"
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   379
  by (simp_all add: rat_number_expand quotient_of_Fract)
33805
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   380
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   381
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
   382
  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
   383
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   384
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
   385
  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
   386
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   387
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
   388
  by (cases r) (simp add: quotient_of_Fract normalize_coprime)
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_inject:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   391
  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
   392
  shows "a = b"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   393
proof -
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   394
  obtain p q r s where a: "a = Fract p q"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   395
    and b: "b = Fract r s"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   396
    and "q > 0" and "s > 0" by (cases a, cases b)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   397
  with assms show ?thesis by (simp add: eq_rat quotient_of_Fract normalize_crossproduct)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   398
qed
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   399
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   400
lemma quotient_of_inject_eq:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   401
  "quotient_of a = quotient_of b \<longleftrightarrow> a = b"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   402
  by (auto simp add: quotient_of_inject)
33805
0461a101e27e added Rene Thiemann's normalize function
nipkow
parents: 33209
diff changeset
   403
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   404
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   405
subsubsection {* The field of rational numbers *}
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   406
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   407
instantiation rat :: field_inverse_zero
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   408
begin
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   409
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   410
definition
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   411
  inverse_rat_def:
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   412
  "inverse q = Abs_Rat (\<Union>x \<in> Rep_Rat q.
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   413
     ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)})"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   414
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   415
lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   416
proof -
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   417
  have "(\<lambda>x. ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)}) respects ratrel"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   418
    by (auto simp add: congruent_def mult_commute)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   419
  then show ?thesis by (simp add: Fract_def inverse_rat_def UN_ratrel)
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   420
qed
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   421
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   422
definition
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   423
  divide_rat_def: "q / r = q * inverse (r::rat)"
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   424
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   425
lemma divide_rat [simp]: "Fract a b / Fract c d = Fract (a * d) (b * c)"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   426
  by (simp add: divide_rat_def)
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   427
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   428
instance proof
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   429
  fix q :: rat
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   430
  assume "q \<noteq> 0"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   431
  then show "inverse q * q = 1" by (cases q rule: Rat_cases_nonzero)
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35063
diff changeset
   432
   (simp_all add: rat_number_expand eq_rat)
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   433
next
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   434
  fix q r :: rat
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   435
  show "q / r = q * inverse r" by (simp add: divide_rat_def)
36415
a168ac750096 explicit is better than implicit
haftmann
parents: 36409
diff changeset
   436
next
a168ac750096 explicit is better than implicit
haftmann
parents: 36409
diff changeset
   437
  show "inverse 0 = (0::rat)" by (simp add: rat_number_expand, simp add: rat_number_collapse)
a168ac750096 explicit is better than implicit
haftmann
parents: 36409
diff changeset
   438
qed
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   439
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   440
end
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   441
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   442
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   443
subsubsection {* Various *}
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   444
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   445
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
   446
  by (simp add: Fract_of_int_eq [symmetric])
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   447
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   448
lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   449
  by (simp add: rat_number_expand)
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   450
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   451
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   452
subsubsection {* The ordered field of rational numbers *}
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   453
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   454
instantiation rat :: linorder
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   455
begin
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   456
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   457
definition
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   458
  le_rat_def:
39910
10097e0a9dbd constant `contents` renamed to `the_elem`
haftmann
parents: 38857
diff changeset
   459
   "q \<le> r \<longleftrightarrow> the_elem (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   460
      {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   461
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   462
lemma le_rat [simp]:
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   463
  assumes "b \<noteq> 0" and "d \<noteq> 0"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   464
  shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   465
proof -
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   466
  have "(\<lambda>x y. {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   467
    respects2 ratrel"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   468
  proof (clarsimp simp add: congruent2_def)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   469
    fix a b a' b' c d c' d'::int
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   470
    assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   471
    assume eq1: "a * b' = a' * b"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   472
    assume eq2: "c * d' = c' * d"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   473
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   474
    let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   475
    {
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   476
      fix a b c d x :: int assume x: "x \<noteq> 0"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   477
      have "?le a b c d = ?le (a * x) (b * x) c d"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   478
      proof -
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   479
        from x have "0 < x * x" by (auto simp add: zero_less_mult_iff)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   480
        hence "?le a b c d =
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   481
            ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   482
          by (simp add: mult_le_cancel_right)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   483
        also have "... = ?le (a * x) (b * x) c d"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   484
          by (simp add: mult_ac)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   485
        finally show ?thesis .
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   486
      qed
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   487
    } note le_factor = this
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   488
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   489
    let ?D = "b * d" and ?D' = "b' * d'"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   490
    from neq have D: "?D \<noteq> 0" by simp
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   491
    from neq have "?D' \<noteq> 0" by simp
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   492
    hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   493
      by (rule le_factor)
27668
6eb20b2cecf8 Tuned and simplified proofs
chaieb
parents: 27652
diff changeset
   494
    also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')" 
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   495
      by (simp add: mult_ac)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   496
    also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   497
      by (simp only: eq1 eq2)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   498
    also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   499
      by (simp add: mult_ac)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   500
    also from D have "... = ?le a' b' c' d'"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   501
      by (rule le_factor [symmetric])
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   502
    finally show "?le a b c d = ?le a' b' c' d'" .
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   503
  qed
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   504
  with assms show ?thesis by (simp add: Fract_def le_rat_def UN_ratrel2)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   505
qed
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   506
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   507
definition
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   508
  less_rat_def: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   509
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   510
lemma less_rat [simp]:
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   511
  assumes "b \<noteq> 0" and "d \<noteq> 0"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   512
  shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   513
  using assms by (simp add: less_rat_def eq_rat order_less_le)
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   514
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   515
instance proof
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   516
  fix q r s :: rat
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   517
  {
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   518
    assume "q \<le> r" and "r \<le> s"
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   519
    then show "q \<le> s" 
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   520
    proof (induct q, induct r, induct s)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   521
      fix a b c d e f :: int
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   522
      assume neq: "b > 0"  "d > 0"  "f > 0"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   523
      assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   524
      show "Fract a b \<le> Fract e f"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   525
      proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   526
        from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   527
          by (auto simp add: zero_less_mult_iff linorder_neq_iff)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   528
        have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   529
        proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   530
          from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   531
            by simp
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   532
          with ff show ?thesis by (simp add: mult_le_cancel_right)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   533
        qed
27668
6eb20b2cecf8 Tuned and simplified proofs
chaieb
parents: 27652
diff changeset
   534
        also have "... = (c * f) * (d * f) * (b * b)" by algebra
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   535
        also have "... \<le> (e * d) * (d * f) * (b * b)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   536
        proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   537
          from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   538
            by simp
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   539
          with bb show ?thesis by (simp add: mult_le_cancel_right)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   540
        qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   541
        finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   542
          by (simp only: mult_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   543
        with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   544
          by (simp add: mult_le_cancel_right)
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   545
        with neq show ?thesis by simp
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   546
      qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   547
    qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   548
  next
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   549
    assume "q \<le> r" and "r \<le> q"
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   550
    then show "q = r"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   551
    proof (induct q, induct r)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   552
      fix a b c d :: int
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   553
      assume neq: "b > 0"  "d > 0"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   554
      assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   555
      show "Fract a b = Fract c d"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   556
      proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   557
        from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   558
          by simp
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   559
        also have "... \<le> (a * d) * (b * d)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   560
        proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   561
          from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   562
            by simp
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   563
          thus ?thesis by (simp only: mult_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   564
        qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   565
        finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   566
        moreover from neq have "b * d \<noteq> 0" by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   567
        ultimately have "a * d = c * b" by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   568
        with neq show ?thesis by (simp add: eq_rat)
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
    qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   571
  next
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   572
    show "q \<le> q"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   573
      by (induct q) simp
27682
25aceefd4786 added class preorder
haftmann
parents: 27668
diff changeset
   574
    show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)"
25aceefd4786 added class preorder
haftmann
parents: 27668
diff changeset
   575
      by (induct q, induct r) (auto simp add: le_less mult_commute)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   576
    show "q \<le> r \<or> r \<le> q"
18913
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   577
      by (induct q, induct r)
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   578
         (simp add: mult_commute, rule linorder_linear)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   579
  }
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   580
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   581
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   582
end
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   583
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   584
instantiation rat :: "{distrib_lattice, abs_if, sgn_if}"
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   585
begin
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   586
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   587
definition
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   588
  abs_rat_def: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))"
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   589
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   590
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
   591
  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
   592
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   593
definition
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   594
  sgn_rat_def: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   595
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   596
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
   597
  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
   598
  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
   599
    (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
   600
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   601
definition
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   602
  "(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min"
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   603
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   604
definition
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   605
  "(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max"
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   606
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   607
instance by intro_classes
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   608
  (auto simp add: abs_rat_def sgn_rat_def min_max.sup_inf_distrib1 inf_rat_def sup_rat_def)
22456
6070e48ecb78 added lattice definitions
haftmann
parents: 21404
diff changeset
   609
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   610
end
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   611
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   612
instance rat :: linordered_field_inverse_zero
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   613
proof
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   614
  fix q r s :: rat
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   615
  show "q \<le> r ==> s + q \<le> s + r"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   616
  proof (induct q, induct r, induct s)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   617
    fix a b c d e f :: int
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   618
    assume neq: "b > 0"  "d > 0"  "f > 0"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   619
    assume le: "Fract a b \<le> Fract c d"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   620
    show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   621
    proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   622
      let ?F = "f * f" from neq have F: "0 < ?F"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   623
        by (auto simp add: zero_less_mult_iff)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   624
      from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   625
        by simp
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   626
      with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   627
        by (simp add: mult_le_cancel_right)
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   628
      with neq show ?thesis by (simp add: mult_ac int_distrib)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   629
    qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   630
  qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   631
  show "q < r ==> 0 < s ==> s * q < s * r"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   632
  proof (induct q, induct r, induct s)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   633
    fix a b c d e f :: int
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   634
    assume neq: "b > 0"  "d > 0"  "f > 0"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   635
    assume le: "Fract a b < Fract c d"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   636
    assume gt: "0 < Fract e f"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   637
    show "Fract e f * Fract a b < Fract e f * Fract c d"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   638
    proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   639
      let ?E = "e * f" and ?F = "f * f"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   640
      from neq gt have "0 < ?E"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   641
        by (auto simp add: Zero_rat_def order_less_le eq_rat)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   642
      moreover from neq have "0 < ?F"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   643
        by (auto simp add: zero_less_mult_iff)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   644
      moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   645
        by simp
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   646
      ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   647
        by (simp add: mult_less_cancel_right)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   648
      with neq show ?thesis
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   649
        by (simp add: mult_ac)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   650
    qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   651
  qed
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   652
qed auto
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   653
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   654
lemma Rat_induct_pos [case_names Fract, induct type: rat]:
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   655
  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
   656
  shows "P q"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   657
proof (cases q)
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   658
  have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   659
  proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   660
    fix a::int and b::int
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   661
    assume b: "b < 0"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   662
    hence "0 < -b" by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   663
    hence "P (Fract (-a) (-b))" by (rule step)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   664
    thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   665
  qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   666
  case (Fract a b)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   667
  thus "P q" by (force simp add: linorder_neq_iff step step')
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   668
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   669
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   670
lemma zero_less_Fract_iff:
30095
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   671
  "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a"
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   672
  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
   673
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   674
lemma Fract_less_zero_iff:
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   675
  "0 < b \<Longrightarrow> Fract a b < 0 \<longleftrightarrow> a < 0"
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   676
  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
   677
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   678
lemma zero_le_Fract_iff:
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   679
  "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a"
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   680
  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
   681
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   682
lemma Fract_le_zero_iff:
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   683
  "0 < b \<Longrightarrow> Fract a b \<le> 0 \<longleftrightarrow> a \<le> 0"
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   684
  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
   685
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   686
lemma one_less_Fract_iff:
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   687
  "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a"
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   688
  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
   689
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   690
lemma Fract_less_one_iff:
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   691
  "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b"
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   692
  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
   693
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   694
lemma one_le_Fract_iff:
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   695
  "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a"
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   696
  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
   697
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   698
lemma Fract_le_one_iff:
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   699
  "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b"
c6e184561159 add lemmas about comparisons of Fract a b with 0 and 1
huffman
parents: 29940
diff changeset
   700
  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
   701
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   702
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   703
subsubsection {* Rationals are an Archimedean field *}
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   704
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   705
lemma rat_floor_lemma:
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   706
  shows "of_int (a div b) \<le> Fract a b \<and> Fract a b < of_int (a div b + 1)"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   707
proof -
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   708
  have "Fract a b = of_int (a div b) + Fract (a mod b) b"
35293
06a98796453e remove unneeded premise from rat_floor_lemma and floor_Fract
huffman
parents: 35216
diff changeset
   709
    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
   710
  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
   711
    unfolding Fract_of_int_quotient
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   712
    by (rule linorder_cases [of b 0]) (simp add: divide_nonpos_neg, simp, simp add: divide_nonneg_pos)
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   713
  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
   714
qed
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   715
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   716
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
   717
proof
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   718
  fix r :: rat
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   719
  show "\<exists>z. r \<le> of_int z"
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   720
  proof (induct r)
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   721
    case (Fract a b)
35293
06a98796453e remove unneeded premise from rat_floor_lemma and floor_Fract
huffman
parents: 35216
diff changeset
   722
    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
   723
      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
   724
    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
   725
  qed
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   726
qed
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   727
43732
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   728
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
   729
begin
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   730
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   731
definition [code del]:
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   732
  "floor (x::rat) = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))"
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   733
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   734
instance proof
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   735
  fix x :: rat
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   736
  show "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   737
    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
   738
qed
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   739
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   740
end
6b2bdc57155b adding a floor_ceiling type class for different instantiations of floor (changeset from Brian Huffman)
bulwahn
parents: 42311
diff changeset
   741
35293
06a98796453e remove unneeded premise from rat_floor_lemma and floor_Fract
huffman
parents: 35216
diff changeset
   742
lemma floor_Fract: "floor (Fract a b) = a div b"
06a98796453e remove unneeded premise from rat_floor_lemma and floor_Fract
huffman
parents: 35216
diff changeset
   743
  using rat_floor_lemma [of a b]
30097
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   744
  by (simp add: floor_unique)
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   745
57df8626c23b generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
huffman
parents: 30095
diff changeset
   746
31100
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   747
subsection {* Linear arithmetic setup *}
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   748
31100
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   749
declaration {*
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   750
  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
   751
    (* 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
   752
  #> 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
   753
    (* 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
   754
  #> Lin_Arith.add_simps [@{thm neg_less_iff_less},
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   755
      @{thm True_implies_equals},
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   756
      read_instantiate @{context} [(("a", 0), "(numeral ?v)")] @{thm right_distrib},
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   757
      read_instantiate @{context} [(("a", 0), "(neg_numeral ?v)")] @{thm right_distrib},
31100
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   758
      @{thm divide_1}, @{thm divide_zero_left},
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   759
      @{thm times_divide_eq_right}, @{thm times_divide_eq_left},
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   760
      @{thm minus_divide_left} RS sym, @{thm minus_divide_right} RS sym,
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   761
      @{thm of_int_minus}, @{thm of_int_diff},
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   762
      @{thm of_int_of_nat_eq}]
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   763
  #> Lin_Arith.add_simprocs Numeral_Simprocs.field_cancel_numeral_factors
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   764
  #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat => rat"})
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   765
  #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int => rat"}))
6a2e67fe4488 tuned interface of Lin_Arith
haftmann
parents: 31021
diff changeset
   766
*}
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   767
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   768
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   769
subsection {* Embedding from Rationals to other Fields *}
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   770
24198
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   771
class field_char_0 = field + ring_char_0
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   772
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 33814
diff changeset
   773
subclass (in linordered_field) field_char_0 ..
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   774
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   775
context field_char_0
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   776
begin
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   777
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   778
definition of_rat :: "rat \<Rightarrow> 'a" where
39910
10097e0a9dbd constant `contents` renamed to `the_elem`
haftmann
parents: 38857
diff changeset
   779
  "of_rat q = the_elem (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})"
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   780
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   781
end
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   782
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   783
lemma of_rat_congruent:
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   784
  "(\<lambda>(a, b). {of_int a / of_int b :: 'a::field_char_0}) respects ratrel"
40816
19c492929756 replaced slightly odd locale congruent by plain definition
haftmann
parents: 40815
diff changeset
   785
apply (rule congruentI)
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   786
apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   787
apply (simp only: of_int_mult [symmetric])
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   788
done
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   789
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   790
lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   791
  unfolding Fract_def of_rat_def by (simp add: UN_ratrel of_rat_congruent)
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   792
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   793
lemma of_rat_0 [simp]: "of_rat 0 = 0"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   794
by (simp add: Zero_rat_def of_rat_rat)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   795
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   796
lemma of_rat_1 [simp]: "of_rat 1 = 1"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   797
by (simp add: One_rat_def of_rat_rat)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   798
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   799
lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   800
by (induct a, induct b, simp add: of_rat_rat add_frac_eq)
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   801
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   802
lemma of_rat_minus: "of_rat (- a) = - of_rat a"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   803
by (induct a, simp add: of_rat_rat)
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   804
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   805
lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   806
by (simp only: diff_minus of_rat_add of_rat_minus)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   807
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   808
lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   809
apply (induct a, induct b, simp add: of_rat_rat)
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   810
apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   811
done
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   812
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   813
lemma nonzero_of_rat_inverse:
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   814
  "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   815
apply (rule inverse_unique [symmetric])
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   816
apply (simp add: of_rat_mult [symmetric])
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   817
done
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   818
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   819
lemma of_rat_inverse:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   820
  "(of_rat (inverse a)::'a::{field_char_0, field_inverse_zero}) =
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   821
   inverse (of_rat a)"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   822
by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   823
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   824
lemma nonzero_of_rat_divide:
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   825
  "b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   826
by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   827
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   828
lemma of_rat_divide:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   829
  "(of_rat (a / b)::'a::{field_char_0, field_inverse_zero})
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   830
   = of_rat a / of_rat b"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   831
by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   832
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   833
lemma of_rat_power:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30960
diff changeset
   834
  "(of_rat (a ^ n)::'a::field_char_0) = of_rat a ^ n"
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30242
diff changeset
   835
by (induct n) (simp_all add: of_rat_mult)
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   836
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   837
lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   838
apply (induct a, induct b)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   839
apply (simp add: of_rat_rat eq_rat)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   840
apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   841
apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   842
done
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   843
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   844
lemma of_rat_less:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 33814
diff changeset
   845
  "(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
   846
proof (induct r, induct s)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   847
  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
   848
  assume not_zero: "b > 0" "d > 0"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   849
  then have "b * d > 0" by (rule mult_pos_pos)
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   850
  have of_int_divide_less_eq:
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   851
    "(of_int a :: 'a) / of_int b < of_int c / of_int d
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   852
      \<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   853
    using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 33814
diff changeset
   854
  show "(of_rat (Fract a b) :: 'a::linordered_field) < of_rat (Fract c d)
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   855
    \<longleftrightarrow> Fract a b < Fract c d"
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   856
    using not_zero `b * d > 0`
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   857
    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
   858
qed
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   859
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   860
lemma of_rat_less_eq:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 33814
diff changeset
   861
  "(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
   862
  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
   863
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   864
lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   865
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   866
lemma of_rat_eq_id [simp]: "of_rat = id"
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   867
proof
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   868
  fix a
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   869
  show "of_rat a = id a"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   870
  by (induct a)
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   871
     (simp add: of_rat_rat Fract_of_int_eq [symmetric])
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   872
qed
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   873
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   874
text{*Collapse nested embeddings*}
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   875
lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   876
by (induct n) (simp_all add: of_rat_add)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   877
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   878
lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
27652
818666de6c24 refined code generator setup for rational numbers; more simplification rules for rational numbers
haftmann
parents: 27551
diff changeset
   879
by (cases z rule: int_diff_cases) (simp add: of_rat_diff)
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   880
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   881
lemma of_rat_numeral_eq [simp]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   882
  "of_rat (numeral w) = numeral w"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   883
using of_rat_of_int_eq [of "numeral w"] by simp
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   884
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   885
lemma of_rat_neg_numeral_eq [simp]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   886
  "of_rat (neg_numeral w) = neg_numeral w"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   887
using of_rat_of_int_eq [of "neg_numeral w"] by simp
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   888
23879
4776af8be741 split class abs from class minus
haftmann
parents: 23429
diff changeset
   889
lemmas zero_rat = Zero_rat_def
4776af8be741 split class abs from class minus
haftmann
parents: 23429
diff changeset
   890
lemmas one_rat = One_rat_def
4776af8be741 split class abs from class minus
haftmann
parents: 23429
diff changeset
   891
24198
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   892
abbreviation
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   893
  rat_of_nat :: "nat \<Rightarrow> rat"
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   894
where
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   895
  "rat_of_nat \<equiv> of_nat"
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   896
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   897
abbreviation
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   898
  rat_of_int :: "int \<Rightarrow> rat"
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   899
where
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   900
  "rat_of_int \<equiv> of_int"
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   901
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   902
subsection {* The Set of Rational Numbers *}
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   903
28001
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   904
context field_char_0
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   905
begin
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   906
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   907
definition
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   908
  Rats  :: "'a set" where
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
   909
  "Rats = range of_rat"
28001
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   910
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   911
notation (xsymbols)
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   912
  Rats  ("\<rat>")
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   913
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   914
end
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
   915
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   916
lemma Rats_of_rat [simp]: "of_rat r \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   917
by (simp add: Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   918
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   919
lemma Rats_of_int [simp]: "of_int z \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   920
by (subst of_rat_of_int_eq [symmetric], rule Rats_of_rat)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   921
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   922
lemma Rats_of_nat [simp]: "of_nat n \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   923
by (subst of_rat_of_nat_eq [symmetric], rule Rats_of_rat)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   924
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   925
lemma Rats_number_of [simp]: "numeral w \<in> Rats"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   926
by (subst of_rat_numeral_eq [symmetric], rule Rats_of_rat)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   927
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   928
lemma Rats_neg_number_of [simp]: "neg_numeral w \<in> Rats"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
   929
by (subst of_rat_neg_numeral_eq [symmetric], rule Rats_of_rat)
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   930
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   931
lemma Rats_0 [simp]: "0 \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   932
apply (unfold Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   933
apply (rule range_eqI)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   934
apply (rule of_rat_0 [symmetric])
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   935
done
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   936
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   937
lemma Rats_1 [simp]: "1 \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   938
apply (unfold Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   939
apply (rule range_eqI)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   940
apply (rule of_rat_1 [symmetric])
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   941
done
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   942
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   943
lemma Rats_add [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a + b \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   944
apply (auto simp add: Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   945
apply (rule range_eqI)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   946
apply (rule of_rat_add [symmetric])
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   947
done
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   948
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   949
lemma Rats_minus [simp]: "a \<in> Rats \<Longrightarrow> - a \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   950
apply (auto simp add: Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   951
apply (rule range_eqI)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   952
apply (rule of_rat_minus [symmetric])
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   953
done
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   954
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   955
lemma Rats_diff [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a - b \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   956
apply (auto simp add: Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   957
apply (rule range_eqI)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   958
apply (rule of_rat_diff [symmetric])
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   959
done
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   960
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   961
lemma Rats_mult [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a * b \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   962
apply (auto simp add: Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   963
apply (rule range_eqI)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   964
apply (rule of_rat_mult [symmetric])
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   965
done
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   966
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   967
lemma nonzero_Rats_inverse:
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   968
  fixes a :: "'a::field_char_0"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   969
  shows "\<lbrakk>a \<in> Rats; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   970
apply (auto simp add: Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   971
apply (rule range_eqI)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   972
apply (erule nonzero_of_rat_inverse [symmetric])
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   973
done
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   974
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   975
lemma Rats_inverse [simp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   976
  fixes a :: "'a::{field_char_0, field_inverse_zero}"
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   977
  shows "a \<in> Rats \<Longrightarrow> inverse a \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   978
apply (auto simp add: Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   979
apply (rule range_eqI)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   980
apply (rule of_rat_inverse [symmetric])
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   981
done
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   982
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   983
lemma nonzero_Rats_divide:
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   984
  fixes a b :: "'a::field_char_0"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   985
  shows "\<lbrakk>a \<in> Rats; b \<in> Rats; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   986
apply (auto simp add: Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   987
apply (rule range_eqI)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   988
apply (erule nonzero_of_rat_divide [symmetric])
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   989
done
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   990
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   991
lemma Rats_divide [simp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   992
  fixes a b :: "'a::{field_char_0, field_inverse_zero}"
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   993
  shows "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   994
apply (auto simp add: Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   995
apply (rule range_eqI)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   996
apply (rule of_rat_divide [symmetric])
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   997
done
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   998
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
   999
lemma Rats_power [simp]:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30960
diff changeset
  1000
  fixes a :: "'a::field_char_0"
28010
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
  1001
  shows "a \<in> Rats \<Longrightarrow> a ^ n \<in> Rats"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
  1002
apply (auto simp add: Rats_def)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
  1003
apply (rule range_eqI)
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
  1004
apply (rule of_rat_power [symmetric])
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
  1005
done
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
  1006
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
  1007
lemma Rats_cases [cases set: Rats]:
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
  1008
  assumes "q \<in> \<rat>"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
  1009
  obtains (of_rat) r where "q = of_rat r"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
  1010
  unfolding Rats_def
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
  1011
proof -
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
  1012
  from `q \<in> \<rat>` have "q \<in> range of_rat" unfolding Rats_def .
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
  1013
  then obtain r where "q = of_rat r" ..
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
  1014
  then show thesis ..
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
  1015
qed
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
  1016
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
  1017
lemma Rats_induct [case_names of_rat, induct set: Rats]:
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
  1018
  "q \<in> \<rat> \<Longrightarrow> (\<And>r. P (of_rat r)) \<Longrightarrow> P q"
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
  1019
  by (rule Rats_cases) auto
8312edc51969 add lemmas about Rats similar to those about Reals
huffman
parents: 28001
diff changeset
  1020
28001
4642317e0deb Defined rationals (Rats) globally in Rational.
nipkow
parents: 27682
diff changeset
  1021
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
  1022
subsection {* Implementation of rational numbers as pairs of integers *}
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
  1023
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1024
text {* Formal constructor *}
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1025
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1026
definition Frct :: "int \<times> int \<Rightarrow> rat" where
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1027
  [simp]: "Frct p = Fract (fst p) (snd p)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1028
36112
7fa17a225852 user interface for abstract datatypes is an attribute, not a command
haftmann
parents: 35726
diff changeset
  1029
lemma [code abstype]:
7fa17a225852 user interface for abstract datatypes is an attribute, not a command
haftmann
parents: 35726
diff changeset
  1030
  "Frct (quotient_of q) = q"
7fa17a225852 user interface for abstract datatypes is an attribute, not a command
haftmann
parents: 35726
diff changeset
  1031
  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
  1032
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1033
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1034
text {* Numerals *}
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1035
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1036
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
  1037
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1038
definition of_int :: "int \<Rightarrow> rat"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1039
where
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1040
  [code_abbrev]: "of_int = Int.of_int"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1041
hide_const (open) of_int
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1042
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1043
lemma quotient_of_int [code abstract]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1044
  "quotient_of (Rat.of_int a) = (a, 1)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1045
  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
  1046
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1047
lemma [code_unfold]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1048
  "numeral k = Rat.of_int (numeral k)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1049
  by (simp add: Rat.of_int_def)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1050
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1051
lemma [code_unfold]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1052
  "neg_numeral k = Rat.of_int (neg_numeral k)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1053
  by (simp add: Rat.of_int_def)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1054
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1055
lemma Frct_code_post [code_post]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1056
  "Frct (0, a) = 0"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1057
  "Frct (a, 0) = 0"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1058
  "Frct (1, 1) = 1"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1059
  "Frct (numeral k, 1) = numeral k"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1060
  "Frct (neg_numeral k, 1) = neg_numeral k"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1061
  "Frct (1, numeral k) = 1 / numeral k"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1062
  "Frct (1, neg_numeral k) = 1 / neg_numeral k"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1063
  "Frct (numeral k, numeral l) = numeral k / numeral l"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1064
  "Frct (numeral k, neg_numeral l) = numeral k / neg_numeral l"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1065
  "Frct (neg_numeral k, numeral l) = neg_numeral k / numeral l"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1066
  "Frct (neg_numeral k, neg_numeral l) = neg_numeral k / neg_numeral l"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1067
  by (simp_all add: Fract_of_int_quotient)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1068
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1069
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1070
text {* Operations *}
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1071
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1072
lemma rat_zero_code [code abstract]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1073
  "quotient_of 0 = (0, 1)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1074
  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
  1075
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1076
lemma rat_one_code [code abstract]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1077
  "quotient_of 1 = (1, 1)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1078
  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
  1079
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1080
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
  1081
  "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
  1082
     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
  1083
  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
  1084
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1085
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
  1086
  "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
  1087
  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
  1088
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1089
lemma rat_minus_code [code abstract]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1090
  "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
  1091
     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
  1092
  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
  1093
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1094
lemma rat_times_code [code abstract]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1095
  "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
  1096
     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
  1097
  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
  1098
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1099
lemma rat_inverse_code [code abstract]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1100
  "quotient_of (inverse p) = (let (a, b) = quotient_of p
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1101
    in if a = 0 then (0, 1) else (sgn a * b, \<bar>a\<bar>))"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1102
proof (cases p)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1103
  case (Fract a b) then show ?thesis
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1104
    by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract gcd_int.commute)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1105
qed
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1106
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1107
lemma rat_divide_code [code abstract]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1108
  "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
  1109
     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
  1110
  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
  1111
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1112
lemma rat_abs_code [code abstract]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1113
  "quotient_of \<bar>p\<bar> = (let (a, b) = quotient_of p in (\<bar>a\<bar>, b))"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1114
  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
  1115
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1116
lemma rat_sgn_code [code abstract]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1117
  "quotient_of (sgn p) = (sgn (fst (quotient_of p)), 1)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1118
proof (cases p)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1119
  case (Fract a b) then show ?thesis
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1120
  by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract)
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1121
qed
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
  1122
43733
a6ca7b83612f adding code equations to execute floor and ceiling on rational and real numbers
bulwahn
parents: 43732
diff changeset
  1123
lemma rat_floor_code [code]:
a6ca7b83612f adding code equations to execute floor and ceiling on rational and real numbers
bulwahn
parents: 43732
diff changeset
  1124
  "floor p = (let (a, b) = quotient_of p in a div b)"
a6ca7b83612f adding code equations to execute floor and ceiling on rational and real numbers
bulwahn
parents: 43732
diff changeset
  1125
by (cases p) (simp add: quotient_of_Fract floor_Fract)
a6ca7b83612f adding code equations to execute floor and ceiling on rational and real numbers
bulwahn
parents: 43732
diff changeset
  1126
38857
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38287
diff changeset
  1127
instantiation rat :: equal
26513
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
  1128
begin
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
  1129
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1130
definition [code]:
38857
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38287
diff changeset
  1131
  "HOL.equal a b \<longleftrightarrow> quotient_of a = quotient_of b"
26513
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
  1132
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1133
instance proof
38857
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38287
diff changeset
  1134
qed (simp add: equal_rat_def quotient_of_inject_eq)
26513
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
  1135
28351
abfc66969d1f non left-linear equations for nbe
haftmann
parents: 28313
diff changeset
  1136
lemma rat_eq_refl [code nbe]:
38857
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38287
diff changeset
  1137
  "HOL.equal (r::rat) r \<longleftrightarrow> True"
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38287
diff changeset
  1138
  by (rule equal_refl)
28351
abfc66969d1f non left-linear equations for nbe
haftmann
parents: 28313
diff changeset
  1139
26513
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
  1140
end
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
  1141
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1142
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
  1143
  "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
  1144
  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
  1145
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1146
lemma rat_less_code [code]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1147
  "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
  1148
  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
  1149
35369
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1150
lemma [code]:
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1151
  "of_rat p = (let (a, b) = quotient_of p in of_int a / of_int b)"
e4a7947e02b8 more general case and induct rules; normalize and quotient_of; abstract code generation
haftmann
parents: 35293
diff changeset
  1152
  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
  1153
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1154
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1155
text {* Quickcheck *}
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1156
31203
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1157
definition (in term_syntax)
32657
5f13912245ff Code_Eval(uation)
haftmann
parents: 32069
diff changeset
  1158
  valterm_fract :: "int \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> int \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> rat \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
5f13912245ff Code_Eval(uation)
haftmann
parents: 32069
diff changeset
  1159
  [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
  1160
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37397
diff changeset
  1161
notation fcomp (infixl "\<circ>>" 60)
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37397
diff changeset
  1162
notation scomp (infixl "\<circ>\<rightarrow>" 60)
31203
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1163
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1164
instantiation rat :: random
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1165
begin
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1166
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1167
definition
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37397
diff changeset
  1168
  "Quickcheck.random i = Quickcheck.random i \<circ>\<rightarrow> (\<lambda>num. Random.range i \<circ>\<rightarrow> (\<lambda>denom. Pair (
31205
98370b26c2ce String.literal replaces message_string, code_numeral replaces (code_)index
haftmann
parents: 31203
diff changeset
  1169
     let j = Code_Numeral.int_of (denom + 1)
32657
5f13912245ff Code_Eval(uation)
haftmann
parents: 32069
diff changeset
  1170
     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
  1171
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1172
instance ..
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1173
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1174
end
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1175
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37397
diff changeset
  1176
no_notation fcomp (infixl "\<circ>>" 60)
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37397
diff changeset
  1177
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
31203
5c8fb4fd67e0 moved Code_Index, Random and Quickcheck before Main
haftmann
parents: 31100
diff changeset
  1178
41920
d4fb7a418152 moving exhaustive_generators.ML to Quickcheck directory
bulwahn
parents: 41792
diff changeset
  1179
instantiation rat :: exhaustive
41231
2e901158675e adding exhaustive tester instances for numeric types: code_numeral, nat, rat and real
bulwahn
parents: 40819
diff changeset
  1180
begin
2e901158675e adding exhaustive tester instances for numeric types: code_numeral, nat, rat and real
bulwahn
parents: 40819
diff changeset
  1181
2e901158675e adding exhaustive tester instances for numeric types: code_numeral, nat, rat and real
bulwahn
parents: 40819
diff changeset
  1182
definition
45818
53a697f5454a hiding constants and facts in the Quickcheck_Exhaustive and Quickcheck_Narrowing theory;
bulwahn
parents: 45694
diff changeset
  1183
  "exhaustive_rat f d = Quickcheck_Exhaustive.exhaustive (%l. Quickcheck_Exhaustive.exhaustive (%k. f (Fract k (Code_Numeral.int_of l + 1))) d) d"
42311
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1184
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1185
instance ..
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1186
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1187
end
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1188
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1189
instantiation rat :: full_exhaustive
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1190
begin
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1191
eb32a8474a57 rational and real instances for new compilation scheme for exhaustive quickcheck
bulwahn
parents: 41920
diff changeset
  1192
definition
45818
53a697f5454a hiding constants and facts in the Quickcheck_Exhaustive and Quickcheck_Narrowing theory;
bulwahn
parents: 45694
diff changeset
  1193
  "full_exhaustive_rat f d = Quickcheck_Exhaustive.full_exhaustive (%(l, _). Quickcheck_Exhaustive.full_exhaustive (%k.
45507
6975db7fd6f0 improved generators for rational numbers to generate negative numbers;
bulwahn
parents: 45478
diff changeset
  1194
     f (let j = Code_Numeral.int_of l + 1
6975db7fd6f0 improved generators for rational numbers to generate negative numbers;
bulwahn
parents: 45478
diff changeset
  1195
        in valterm_fract k (j, %_. Code_Evaluation.term_of j))) d) d"
41231
2e901158675e adding exhaustive tester instances for numeric types: code_numeral, nat, rat and real
bulwahn
parents: 40819
diff changeset
  1196
2e901158675e adding exhaustive tester instances for numeric types: code_numeral, nat, rat and real
bulwahn
parents: 40819
diff changeset
  1197
instance ..
2e901158675e adding exhaustive tester instances for numeric types: code_numeral, nat, rat and real
bulwahn
parents: 40819
diff changeset
  1198
2e901158675e adding exhaustive tester instances for numeric types: code_numeral, nat, rat and real
bulwahn
parents: 40819
diff changeset
  1199
end
2e901158675e adding exhaustive tester instances for numeric types: code_numeral, nat, rat and real
bulwahn
parents: 40819
diff changeset
  1200
43889
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1201
instantiation rat :: partial_term_of
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1202
begin
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1203
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1204
instance ..
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1205
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1206
end
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1207
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1208
lemma [code]:
46758
4106258260b3 choosing longer constant names in Quickcheck_Narrowing to reduce the chances of name clashes in Quickcheck-Narrowing
bulwahn
parents: 45818
diff changeset
  1209
  "partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Narrowing_variable p tt) == Code_Evaluation.Free (STR ''_'') (Typerep.Typerep (STR ''Rat.rat'') [])"
4106258260b3 choosing longer constant names in Quickcheck_Narrowing to reduce the chances of name clashes in Quickcheck-Narrowing
bulwahn
parents: 45818
diff changeset
  1210
  "partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Narrowing_constructor 0 [l, k]) ==
45507
6975db7fd6f0 improved generators for rational numbers to generate negative numbers;
bulwahn
parents: 45478
diff changeset
  1211
     Code_Evaluation.App (Code_Evaluation.Const (STR ''Rat.Frct'')
6975db7fd6f0 improved generators for rational numbers to generate negative numbers;
bulwahn
parents: 45478
diff changeset
  1212
     (Typerep.Typerep (STR ''fun'') [Typerep.Typerep (STR ''Product_Type.prod'') [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''Int.int'') []],
6975db7fd6f0 improved generators for rational numbers to generate negative numbers;
bulwahn
parents: 45478
diff changeset
  1213
        Typerep.Typerep (STR ''Rat.rat'') []])) (Code_Evaluation.App (Code_Evaluation.App (Code_Evaluation.Const (STR ''Product_Type.Pair'') (Typerep.Typerep (STR ''fun'') [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''fun'') [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''Product_Type.prod'') [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''Int.int'') []]]])) (partial_term_of (TYPE(int)) l)) (partial_term_of (TYPE(int)) k))"
43889
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1214
by (rule partial_term_of_anything)+
90d24cafb05d adding code equations for partial_term_of for rational numbers
bulwahn
parents: 43887
diff changeset
  1215
43887
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1216
instantiation rat :: narrowing
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1217
begin
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1218
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1219
definition
45507
6975db7fd6f0 improved generators for rational numbers to generate negative numbers;
bulwahn
parents: 45478
diff changeset
  1220
  "narrowing = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.apply
6975db7fd6f0 improved generators for rational numbers to generate negative numbers;
bulwahn
parents: 45478
diff changeset
  1221
    (Quickcheck_Narrowing.cons (%nom denom. Fract nom denom)) narrowing) narrowing"
43887
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1222
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1223
instance ..
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1224
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1225
end
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1226
442aceb54969 adding narrowing instances for real and rational
bulwahn
parents: 43733
diff changeset
  1227
45183
2e1ad4a54189 removing old code generator setup for rational numbers; tuned
bulwahn
parents: 43889
diff changeset
  1228
subsection {* Setup for Nitpick *}
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
  1229
38287
796302ca3611 replace "setup" with "declaration"
blanchet
parents: 38242
diff changeset
  1230
declaration {*
796302ca3611 replace "setup" with "declaration"
blanchet
parents: 38242
diff changeset
  1231
  Nitpick_HOL.register_frac_type @{type_name rat}
33209
d36ca3960e33 tuned white space;
wenzelm
parents: 33197
diff changeset
  1232
   [(@{const_name zero_rat_inst.zero_rat}, @{const_name Nitpick.zero_frac}),
d36ca3960e33 tuned white space;
wenzelm
parents: 33197
diff changeset
  1233
    (@{const_name one_rat_inst.one_rat}, @{const_name Nitpick.one_frac}),
d36ca3960e33 tuned white space;
wenzelm
parents: 33197
diff changeset
  1234
    (@{const_name plus_rat_inst.plus_rat}, @{const_name Nitpick.plus_frac}),
d36ca3960e33 tuned white space;
wenzelm
parents: 33197
diff changeset
  1235
    (@{const_name times_rat_inst.times_rat}, @{const_name Nitpick.times_frac}),
d36ca3960e33 tuned white space;
wenzelm
parents: 33197
diff changeset
  1236
    (@{const_name uminus_rat_inst.uminus_rat}, @{const_name Nitpick.uminus_frac}),
d36ca3960e33 tuned white space;
wenzelm
parents: 33197
diff changeset
  1237
    (@{const_name inverse_rat_inst.inverse_rat}, @{const_name Nitpick.inverse_frac}),
37397
18000f9d783e adjust Nitpick's handling of "<" on "rat"s and "reals"
blanchet
parents: 37143
diff changeset
  1238
    (@{const_name ord_rat_inst.less_rat}, @{const_name Nitpick.less_frac}),
33209
d36ca3960e33 tuned white space;
wenzelm
parents: 33197
diff changeset
  1239
    (@{const_name ord_rat_inst.less_eq_rat}, @{const_name Nitpick.less_eq_frac}),
45478
8e299034eab4 remove unsound line in Nitpick's "rat" setup
blanchet
parents: 45183
diff changeset
  1240
    (@{const_name field_char_0_class.of_rat}, @{const_name Nitpick.of_frac})]
33197
de6285ebcc05 continuation of Nitpick's integration into Isabelle;
blanchet
parents: 32657
diff changeset
  1241
*}
de6285ebcc05 continuation of Nitpick's integration into Isabelle;
blanchet
parents: 32657
diff changeset
  1242
41792
ff3cb0c418b7 renamed "nitpick\_def" to "nitpick_unfold" to reflect its new semantics
blanchet
parents: 41231
diff changeset
  1243
lemmas [nitpick_unfold] = inverse_rat_inst.inverse_rat
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46758
diff changeset
  1244
  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
  1245
  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
  1246
  uminus_rat_inst.uminus_rat zero_rat_inst.zero_rat
33197
de6285ebcc05 continuation of Nitpick's integration into Isabelle;
blanchet
parents: 32657
diff changeset
  1247
35343
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1248
subsection{* Float syntax *}
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1249
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1250
syntax "_Float" :: "float_const \<Rightarrow> 'a"    ("_")
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1251
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1252
use "Tools/float_syntax.ML"
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1253
setup Float_Syntax.setup
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1254
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1255
text{* Test: *}
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1256
lemma "123.456 = -111.111 + 200 + 30 + 4 + 5/10 + 6/100 + (7/1000::rat)"
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1257
by simp
523124691b3a move float syntax from RealPow to Rational
huffman
parents: 35293
diff changeset
  1258
37143
2a5182751151 constant Rat.normalize needs to be qualified;
wenzelm
parents: 36415
diff changeset
  1259
2a5182751151 constant Rat.normalize needs to be qualified;
wenzelm
parents: 36415
diff changeset
  1260
hide_const (open) normalize
2a5182751151 constant Rat.normalize needs to be qualified;
wenzelm
parents: 36415
diff changeset
  1261
29880
3dee8ff45d3d move countability proof from Rational to Countable; add instance rat :: countable
huffman
parents: 29667
diff changeset
  1262
end