src/HOL/Real/Rational.thy
author haftmann
Fri, 11 Jul 2008 09:03:11 +0200
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child 27652 818666de6c24
permissions -rw-r--r--
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(*  Title:  HOL/Library/Rational.thy
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    ID:     $Id$
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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 Rational
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imports "../Presburger" GCD Abstract_Rat
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uses ("rat_arith.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_ratrel: "refl {x. snd x \<noteq> 0} ratrel"
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  by (auto simp add: refl_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 equiv.intro [OF refl_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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typedef (Rat) rat = "{x. snd x \<noteq> 0} // ratrel"
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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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  [code func del]: "Fract a b = Abs_Rat (ratrel `` {if b = 0 then (0, 1) else (a, b)})"
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code_datatype Fract
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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 \<noteq> 0 \<Longrightarrow> C"
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  shows C
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  using assms by (cases q) (clarsimp simp add: Fract_def Rat_def quotient_def)
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lemma Rat_induct [case_names Fract, induct type: rat]:
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  assumes "\<And>a b. b \<noteq> 0 \<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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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 c. Fract a 0 = Fract c 0"
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  by (simp_all add: Fract_def)
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instantiation rat :: "{comm_ring_1, recpower}"
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begin
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definition
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  Zero_rat_def [code, code unfold]: "0 = Fract 0 1"
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definition
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  One_rat_def [code, code unfold]: "1 = Fract 1 1"
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definition
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  add_rat_def [code func del]:
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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:
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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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  have "(\<lambda>x y. ratrel``{(fst x * snd y + fst y * snd x, snd x * snd y)})
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    respects2 ratrel"
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  by (rule equiv_ratrel [THEN congruent2_commuteI]) (simp_all add: left_distrib)
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  with assms show ?thesis by (simp add: Fract_def add_rat_def UN_ratrel2)
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qed
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definition
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  minus_rat_def [code func del]:
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  "- q = Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel `` {(- fst x, snd x)})"
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lemma minus_rat: "- Fract a b = Fract (- a) b"
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proof -
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  have "(\<lambda>x. ratrel `` {(- fst x, snd x)}) respects ratrel"
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    by (simp add: congruent_def)
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  then show ?thesis by (simp add: Fract_def minus_rat_def UN_ratrel)
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qed
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lemma minus_rat_cancel [simp]: 
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  "Fract (- a) (- b) = Fract a b"
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  by (cases "b = 0") (simp_all add: eq_rat)
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definition
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  diff_rat_def [code func del]: "q - r = q + - (r::rat)"
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lemma diff_rat:
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  assumes "b \<noteq> 0" and "d \<noteq> 0"
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  shows "Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
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  using assms by (simp add: diff_rat_def add_rat minus_rat)
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definition
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  mult_rat_def [code func del]:
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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 * fst y, snd x * snd y)})"
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lemma mult_rat: "Fract a b * Fract c d = Fract (a * c) (b * d)"
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   146
proof -
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   147
  have "(\<lambda>x y. ratrel `` {(fst x * fst y, snd x * snd y)}) respects2 ratrel"
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   148
    by (rule equiv_ratrel [THEN congruent2_commuteI]) simp_all
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   149
  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
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parents:
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qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
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parents:
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   151
27551
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lemma mult_rat_cancel [simp]:
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   153
  assumes "c \<noteq> 0"
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   154
  shows "Fract (c * a) (c * b) = Fract a b"
9a5543d4cc24 Fract now total; improved code generator setup
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   155
proof -
9a5543d4cc24 Fract now total; improved code generator setup
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   156
  from assms have "Fract c c = Fract 1 1" by (simp add: Fract_def)
9a5543d4cc24 Fract now total; improved code generator setup
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   157
  then show ?thesis by (simp add: mult_rat [symmetric] mult_rat)
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   158
qed
27509
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   159
63161d5f8f29 rearrange instantiations
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   160
primrec power_rat
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   161
where
27551
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   162
  rat_power_0:     "q ^ 0 = (1\<Colon>rat)"
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   163
  | rat_power_Suc: "q ^ Suc n = (q\<Colon>rat) * (q ^ n)"
27509
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   164
63161d5f8f29 rearrange instantiations
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   165
instance proof
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   166
  fix q r s :: rat show "(q * r) * s = q * (r * s)"
9a5543d4cc24 Fract now total; improved code generator setup
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   167
    by (cases q, cases r, cases s) (simp add: mult_rat eq_rat)
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   168
next
9a5543d4cc24 Fract now total; improved code generator setup
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   169
  fix q r :: rat show "q * r = r * q"
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   170
    by (cases q, cases r) (simp add: mult_rat eq_rat)
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   171
next
9a5543d4cc24 Fract now total; improved code generator setup
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   172
  fix q :: rat show "1 * q = q"
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   173
    by (cases q) (simp add: One_rat_def mult_rat eq_rat)
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   174
next
9a5543d4cc24 Fract now total; improved code generator setup
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   175
  fix q r s :: rat show "(q + r) + s = q + (r + s)"
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   176
    by (cases q, cases r, cases s) (simp add: add_rat eq_rat ring_simps)
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   177
next
9a5543d4cc24 Fract now total; improved code generator setup
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   178
  fix q r :: rat show "q + r = r + q"
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   179
    by (cases q, cases r) (simp add: add_rat eq_rat)
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   180
next
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   181
  fix q :: rat show "0 + q = q"
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   182
    by (cases q) (simp add: Zero_rat_def add_rat eq_rat)
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   183
next
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   184
  fix q :: rat show "- q + q = 0"
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   185
    by (cases q) (simp add: Zero_rat_def add_rat minus_rat eq_rat)
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   186
next
9a5543d4cc24 Fract now total; improved code generator setup
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parents: 27509
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   187
  fix q r :: rat show "q - r = q + - r"
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   188
    by (cases q, cases r) (simp add: diff_rat add_rat minus_rat eq_rat)
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parents: 27509
diff changeset
   189
next
9a5543d4cc24 Fract now total; improved code generator setup
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parents: 27509
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   190
  fix q r s :: rat show "(q + r) * s = q * s + r * s"
9a5543d4cc24 Fract now total; improved code generator setup
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parents: 27509
diff changeset
   191
    by (cases q, cases r, cases s) (simp add: add_rat mult_rat eq_rat ring_simps)
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   192
next
9a5543d4cc24 Fract now total; improved code generator setup
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parents: 27509
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   193
  show "(0::rat) \<noteq> 1" by (simp add: Zero_rat_def One_rat_def eq_rat)
9a5543d4cc24 Fract now total; improved code generator setup
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parents: 27509
diff changeset
   194
next
9a5543d4cc24 Fract now total; improved code generator setup
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parents: 27509
diff changeset
   195
  fix q :: rat show "q * 1 = q"
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parents: 27509
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   196
    by (cases q) (simp add: One_rat_def mult_rat eq_rat)
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   197
next
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   198
  fix q :: rat
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   199
  fix n :: nat
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   200
  show "q ^ 0 = 1" by simp
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   201
  show "q ^ (Suc n) = q * (q ^ n)" by simp
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   202
qed
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   203
63161d5f8f29 rearrange instantiations
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   204
end
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   205
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   206
lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1"
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   207
  by (induct k) (simp_all add: Zero_rat_def One_rat_def add_rat)
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   208
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   209
lemma of_int_rat: "of_int k = Fract k 1"
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   210
  by (cases k rule: int_diff_cases, simp add: of_nat_rat diff_rat)
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   211
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   212
lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
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   213
  by (rule of_nat_rat [symmetric])
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   214
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   215
lemma Fract_of_int_eq: "Fract k 1 = of_int k"
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   216
  by (rule of_int_rat [symmetric])
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   217
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   218
instantiation rat :: number_ring
9a5543d4cc24 Fract now total; improved code generator setup
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   219
begin
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   220
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   221
definition
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   222
  rat_number_of_def [code func del]: "number_of w = Fract w 1"
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diff changeset
   223
9a5543d4cc24 Fract now total; improved code generator setup
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   224
instance by intro_classes (simp add: rat_number_of_def of_int_rat)
9a5543d4cc24 Fract now total; improved code generator setup
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   225
9a5543d4cc24 Fract now total; improved code generator setup
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   226
end
9a5543d4cc24 Fract now total; improved code generator setup
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parents: 27509
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   227
9a5543d4cc24 Fract now total; improved code generator setup
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   228
lemma rat_number_collapse [code post]:
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   229
  "Fract 0 k = 0"
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   230
  "Fract 1 1 = 1"
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parents: 27509
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   231
  "Fract (number_of k) 1 = number_of k"
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parents: 27509
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   232
  "Fract k 0 = 0"
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   233
  by (cases "k = 0")
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   234
    (simp_all add: Zero_rat_def One_rat_def number_of_is_id number_of_eq of_int_rat eq_rat Fract_def)
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parents: 27509
diff changeset
   235
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   236
lemma rat_number_expand [code unfold]:
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   237
  "0 = Fract 0 1"
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   238
  "1 = Fract 1 1"
9a5543d4cc24 Fract now total; improved code generator setup
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   239
  "number_of k = Fract (number_of k) 1"
9a5543d4cc24 Fract now total; improved code generator setup
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   240
  by (simp_all add: rat_number_collapse)
9a5543d4cc24 Fract now total; improved code generator setup
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parents: 27509
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   241
9a5543d4cc24 Fract now total; improved code generator setup
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   242
lemma iszero_rat [simp]:
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   243
  "iszero (number_of k :: rat) \<longleftrightarrow> iszero (number_of k :: int)"
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parents: 27509
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   244
  by (simp add: iszero_def rat_number_expand number_of_is_id eq_rat)
9a5543d4cc24 Fract now total; improved code generator setup
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parents: 27509
diff changeset
   245
9a5543d4cc24 Fract now total; improved code generator setup
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diff changeset
   246
lemma Rat_cases_nonzero [case_names Fract 0]:
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diff changeset
   247
  assumes Fract: "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> C"
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parents: 27509
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   248
  assumes 0: "q = 0 \<Longrightarrow> C"
9a5543d4cc24 Fract now total; improved code generator setup
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parents: 27509
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   249
  shows C
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   250
proof (cases "q = 0")
9a5543d4cc24 Fract now total; improved code generator setup
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parents: 27509
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   251
  case True then show C using 0 by auto
9a5543d4cc24 Fract now total; improved code generator setup
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parents: 27509
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   252
next
9a5543d4cc24 Fract now total; improved code generator setup
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parents: 27509
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   253
  case False
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   254
  then obtain a b where "q = Fract a b" and "b \<noteq> 0" by (cases q) auto
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   255
  moreover with False have "0 \<noteq> Fract a b" by simp
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   256
  with `b \<noteq> 0` have "a \<noteq> 0" by (simp add: Zero_rat_def eq_rat)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   257
  with Fract `q = Fract a b` `b \<noteq> 0` show C by auto
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   258
qed
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   259
  
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   260
9a5543d4cc24 Fract now total; improved code generator setup
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   261
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haftmann
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   262
subsubsection {* The field of rational numbers *}
9a5543d4cc24 Fract now total; improved code generator setup
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parents: 27509
diff changeset
   263
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haftmann
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diff changeset
   264
instantiation rat :: "{field, division_by_zero}"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
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diff changeset
   265
begin
9a5543d4cc24 Fract now total; improved code generator setup
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parents: 27509
diff changeset
   266
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parents: 27509
diff changeset
   267
definition
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parents: 27509
diff changeset
   268
  inverse_rat_def [code func del]:
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parents: 27509
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   269
  "inverse q = Abs_Rat (\<Union>x \<in> Rep_Rat q.
9a5543d4cc24 Fract now total; improved code generator setup
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parents: 27509
diff changeset
   270
     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
   271
9a5543d4cc24 Fract now total; improved code generator setup
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   272
lemma inverse_rat: "inverse (Fract a b) = Fract b a"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   273
proof -
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   274
  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
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   275
    by (auto simp add: congruent_def mult_commute)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   276
  then show ?thesis by (simp add: Fract_def inverse_rat_def UN_ratrel)
27509
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parents: 26732
diff changeset
   277
qed
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   278
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   279
definition
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   280
  divide_rat_def [code func del]: "q / r = q * inverse (r::rat)"
9a5543d4cc24 Fract now total; improved code generator setup
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parents: 27509
diff changeset
   281
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   282
lemma divide_rat: "Fract a b / Fract c d = Fract (a * d) (b * c)"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   283
  by (simp add: divide_rat_def inverse_rat mult_rat)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   284
9a5543d4cc24 Fract now total; improved code generator setup
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parents: 27509
diff changeset
   285
instance proof
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   286
  show "inverse 0 = (0::rat)" by (simp add: rat_number_expand inverse_rat)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   287
    (simp add: rat_number_collapse)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   288
next
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   289
  fix q :: rat
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   290
  assume "q \<noteq> 0"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   291
  then show "inverse q * q = 1" by (cases q rule: Rat_cases_nonzero)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   292
   (simp_all add: mult_rat  inverse_rat rat_number_expand eq_rat)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   293
next
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   294
  fix q r :: rat
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   295
  show "q / r = q * inverse r" by (simp add: divide_rat_def)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   296
qed
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   297
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   298
end
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   299
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   300
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   301
subsubsection {* Various *}
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   302
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   303
lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   304
  by (simp add: rat_number_expand add_rat)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   305
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   306
lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   307
  by (simp add: Fract_of_int_eq [symmetric] divide_rat)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   308
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   309
lemma Fract_number_of_quotient [code post]:
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   310
  "Fract (number_of k) (number_of l) = number_of k / number_of l"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   311
  unfolding Fract_of_int_quotient number_of_is_id number_of_eq ..
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   312
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   313
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   314
subsubsection {* The ordered field of rational numbers *}
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   315
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   316
instantiation rat :: linorder
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   317
begin
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   318
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   319
definition
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   320
  le_rat_def [code func del]:
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   321
   "q \<le> r \<longleftrightarrow> contents (\<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
   322
      {(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
   323
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   324
lemma le_rat:
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   325
  assumes "b \<noteq> 0" and "d \<noteq> 0"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   326
  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
   327
proof -
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   328
  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
   329
    respects2 ratrel"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   330
  proof (clarsimp simp add: congruent2_def)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   331
    fix a b a' b' c d c' d'::int
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   332
    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
   333
    assume eq1: "a * b' = a' * b"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   334
    assume eq2: "c * d' = c' * d"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   335
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   336
    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
   337
    {
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   338
      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
   339
      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
   340
      proof -
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   341
        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
   342
        hence "?le a b c d =
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   343
            ((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
   344
          by (simp add: mult_le_cancel_right)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   345
        also have "... = ?le (a * x) (b * x) c d"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   346
          by (simp add: mult_ac)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   347
        finally show ?thesis .
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   348
      qed
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   349
    } note le_factor = this
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   350
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   351
    let ?D = "b * d" and ?D' = "b' * d'"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   352
    from neq have D: "?D \<noteq> 0" by simp
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   353
    from neq have "?D' \<noteq> 0" by simp
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   354
    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
   355
      by (rule le_factor)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   356
    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
   357
      by (simp add: mult_ac)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   358
    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
   359
      by (simp only: eq1 eq2)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   360
    also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   361
      by (simp add: mult_ac)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   362
    also from D have "... = ?le a' b' c' d'"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   363
      by (rule le_factor [symmetric])
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   364
    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
   365
  qed
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   366
  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
   367
qed
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   368
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   369
definition
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   370
  less_rat_def [code func del]: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   371
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   372
lemma less_rat:
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   373
  assumes "b \<noteq> 0" and "d \<noteq> 0"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   374
  shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   375
  using assms by (simp add: less_rat_def le_rat eq_rat order_less_le)
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   376
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   377
instance proof
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   378
  fix q r s :: rat
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   379
  {
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   380
    assume "q \<le> r" and "r \<le> s"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   381
    show "q \<le> s"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   382
    proof (insert prems, induct q, induct r, induct s)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   383
      fix a b c d e f :: int
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   384
      assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   385
      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
   386
      show "Fract a b \<le> Fract e f"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   387
      proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   388
        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
   389
          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
   390
        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
   391
        proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   392
          from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   393
            by (simp add: le_rat)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   394
          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
   395
        qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   396
        also have "... = (c * f) * (d * f) * (b * b)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   397
          by (simp only: mult_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   398
        also have "... \<le> (e * d) * (d * f) * (b * b)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   399
        proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   400
          from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   401
            by (simp add: le_rat)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   402
          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
   403
        qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   404
        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
   405
          by (simp only: mult_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   406
        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
   407
          by (simp add: mult_le_cancel_right)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   408
        with neq show ?thesis by (simp add: le_rat)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   409
      qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   410
    qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   411
  next
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   412
    assume "q \<le> r" and "r \<le> q"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   413
    show "q = r"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   414
    proof (insert prems, induct q, induct r)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   415
      fix a b c d :: int
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   416
      assume neq: "b \<noteq> 0"  "d \<noteq> 0"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   417
      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
   418
      show "Fract a b = Fract c d"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   419
      proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   420
        from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   421
          by (simp add: le_rat)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   422
        also have "... \<le> (a * d) * (b * d)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   423
        proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   424
          from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   425
            by (simp add: le_rat)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   426
          thus ?thesis by (simp only: mult_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   427
        qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   428
        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
   429
        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
   430
        ultimately have "a * d = c * b" by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   431
        with neq show ?thesis by (simp add: eq_rat)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   432
      qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   433
    qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   434
  next
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   435
    show "q \<le> q"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   436
      by (induct q) (simp add: le_rat)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   437
    show "(q < r) = (q \<le> r \<and> q \<noteq> r)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   438
      by (simp only: less_rat_def)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   439
    show "q \<le> r \<or> r \<le> q"
18913
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   440
      by (induct q, induct r)
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   441
         (simp add: le_rat mult_commute, rule linorder_linear)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   442
  }
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   443
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   444
27509
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   445
end
63161d5f8f29 rearrange instantiations
huffman
parents: 26732
diff changeset
   446
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   447
instantiation rat :: "{distrib_lattice, abs_if, sgn_if}"
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   448
begin
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   449
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   450
definition
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   451
  abs_rat_def: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   452
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   453
lemma abs_rat: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   454
  by (auto simp add: abs_rat_def zabs_def Zero_rat_def less_rat not_less le_less minus_rat eq_rat zero_compare_simps)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   455
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   456
definition
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   457
  sgn_rat_def: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   458
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   459
lemma sgn_rat: "sgn (Fract a b) = Fract (sgn a * sgn b) 1"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   460
  unfolding Fract_of_int_eq
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   461
  by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat less_rat)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   462
    (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
   463
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   464
definition
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   465
  "(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min"
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   466
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   467
definition
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   468
  "(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max"
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   469
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   470
instance by intro_classes
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   471
  (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
   472
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   473
end
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25502
diff changeset
   474
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   475
instance rat :: ordered_field
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   476
proof
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   477
  fix q r s :: rat
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   478
  show "q \<le> r ==> s + q \<le> s + r"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   479
  proof (induct q, induct r, induct s)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   480
    fix a b c d e f :: int
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   481
    assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   482
    assume le: "Fract a b \<le> Fract c d"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   483
    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
   484
    proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   485
      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
   486
        by (auto simp add: zero_less_mult_iff)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   487
      from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   488
        by (simp add: le_rat)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   489
      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
   490
        by (simp add: mult_le_cancel_right)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   491
      with neq show ?thesis by (simp add: add_rat le_rat mult_ac int_distrib)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   492
    qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   493
  qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   494
  show "q < r ==> 0 < s ==> s * q < s * r"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   495
  proof (induct q, induct r, induct s)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   496
    fix a b c d e f :: int
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   497
    assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   498
    assume le: "Fract a b < Fract c d"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   499
    assume gt: "0 < Fract e f"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   500
    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
   501
    proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   502
      let ?E = "e * f" and ?F = "f * f"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   503
      from neq gt have "0 < ?E"
23879
4776af8be741 split class abs from class minus
haftmann
parents: 23429
diff changeset
   504
        by (auto simp add: Zero_rat_def less_rat le_rat order_less_le eq_rat)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   505
      moreover from neq have "0 < ?F"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   506
        by (auto simp add: zero_less_mult_iff)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   507
      moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   508
        by (simp add: less_rat)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   509
      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
   510
        by (simp add: mult_less_cancel_right)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   511
      with neq show ?thesis
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   512
        by (simp add: less_rat mult_rat mult_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   513
    qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   514
  qed
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   515
qed auto
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   516
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   517
lemma Rat_induct_pos [case_names Fract, induct type: rat]:
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   518
  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
   519
  shows "P q"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   520
proof (cases q)
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   521
  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
   522
  proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   523
    fix a::int and b::int
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   524
    assume b: "b < 0"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   525
    hence "0 < -b" by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   526
    hence "P (Fract (-a) (-b))" by (rule step)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   527
    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
   528
  qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   529
  case (Fract a b)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   530
  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
   531
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   532
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   533
lemma zero_less_Fract_iff:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   534
     "0 < b ==> (0 < Fract a b) = (0 < a)"
23879
4776af8be741 split class abs from class minus
haftmann
parents: 23429
diff changeset
   535
by (simp add: Zero_rat_def less_rat order_less_imp_not_eq2 zero_less_mult_iff)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   536
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   537
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   538
subsection {* Arithmetic setup *}
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   539
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   540
use "rat_arith.ML"
24075
366d4d234814 arith method setup: proper context;
wenzelm
parents: 23879
diff changeset
   541
declaration {* K rat_arith_setup *}
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   542
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   543
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   544
subsection {* Embedding from Rationals to other Fields *}
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   545
24198
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   546
class field_char_0 = field + ring_char_0
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   547
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   548
subclass (in ordered_field) field_char_0 ..
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   549
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   550
context field_char_0
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   551
begin
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   552
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   553
definition of_rat :: "rat \<Rightarrow> 'a" where
24198
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   554
  [code func del]: "of_rat q = contents (\<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
   555
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   556
end
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   557
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   558
lemma of_rat_congruent:
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   559
  "(\<lambda>(a, b). {of_int a / of_int b :: 'a::field_char_0}) respects ratrel"
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   560
apply (rule congruent.intro)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   561
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
   562
apply (simp only: of_int_mult [symmetric])
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   563
done
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   564
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   565
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
   566
  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
   567
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   568
lemma of_rat_0 [simp]: "of_rat 0 = 0"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   569
by (simp add: Zero_rat_def of_rat_rat)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   570
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   571
lemma of_rat_1 [simp]: "of_rat 1 = 1"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   572
by (simp add: One_rat_def of_rat_rat)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   573
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   574
lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   575
by (induct a, induct b, simp add: add_rat of_rat_rat add_frac_eq)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   576
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   577
lemma of_rat_minus: "of_rat (- a) = - of_rat a"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   578
by (induct a, simp add: minus_rat of_rat_rat)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   579
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   580
lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   581
by (simp only: diff_minus of_rat_add of_rat_minus)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   582
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   583
lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   584
apply (induct a, induct b, simp add: mult_rat of_rat_rat)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   585
apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   586
done
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   587
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   588
lemma nonzero_of_rat_inverse:
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   589
  "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   590
apply (rule inverse_unique [symmetric])
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   591
apply (simp add: of_rat_mult [symmetric])
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   592
done
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   593
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   594
lemma of_rat_inverse:
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   595
  "(of_rat (inverse a)::'a::{field_char_0,division_by_zero}) =
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   596
   inverse (of_rat a)"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   597
by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   598
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   599
lemma nonzero_of_rat_divide:
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   600
  "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
   601
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
   602
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   603
lemma of_rat_divide:
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   604
  "(of_rat (a / b)::'a::{field_char_0,division_by_zero})
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   605
   = of_rat a / of_rat b"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   606
by (cases "b = 0", simp_all add: nonzero_of_rat_divide)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   607
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   608
lemma of_rat_power:
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   609
  "(of_rat (a ^ n)::'a::{field_char_0,recpower}) = of_rat a ^ n"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   610
by (induct n) (simp_all add: of_rat_mult power_Suc)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   611
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   612
lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   613
apply (induct a, induct b)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   614
apply (simp add: of_rat_rat eq_rat)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   615
apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   616
apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   617
done
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   618
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   619
lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   620
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   621
lemma of_rat_eq_id [simp]: "of_rat = (id :: rat \<Rightarrow> rat)"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   622
proof
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   623
  fix a
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   624
  show "of_rat a = id a"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   625
  by (induct a)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   626
     (simp add: of_rat_rat divide_rat Fract_of_int_eq [symmetric])
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   627
qed
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   628
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   629
text{*Collapse nested embeddings*}
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   630
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
   631
by (induct n) (simp_all add: of_rat_add)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   632
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   633
lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
23365
f31794033ae1 removed constant int :: nat => int;
huffman
parents: 23343
diff changeset
   634
by (cases z rule: int_diff_cases, simp add: of_rat_diff)
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   635
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   636
lemma of_rat_number_of_eq [simp]:
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   637
  "of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   638
by (simp add: number_of_eq)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   639
23879
4776af8be741 split class abs from class minus
haftmann
parents: 23429
diff changeset
   640
lemmas zero_rat = Zero_rat_def
4776af8be741 split class abs from class minus
haftmann
parents: 23429
diff changeset
   641
lemmas one_rat = One_rat_def
4776af8be741 split class abs from class minus
haftmann
parents: 23429
diff changeset
   642
24198
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   643
abbreviation
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   644
  rat_of_nat :: "nat \<Rightarrow> rat"
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   645
where
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   646
  "rat_of_nat \<equiv> of_nat"
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   647
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   648
abbreviation
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   649
  rat_of_int :: "int \<Rightarrow> rat"
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   650
where
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   651
  "rat_of_int \<equiv> of_int"
4031da6d8ba3 adaptions for code generation
haftmann
parents: 24075
diff changeset
   652
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   653
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   654
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
   655
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   656
lemma INum_Fract [simp]: "INum = split Fract"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   657
  by (auto simp add: expand_fun_eq INum_def Fract_of_int_quotient)
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   658
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   659
lemma split_Fract_normNum [simp]: "split Fract (normNum (k, l)) = Fract k l"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   660
  unfolding INum_Fract [symmetric] normNum by simp
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   661
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   662
lemma [code]:
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   663
  "of_rat (Fract k l) = (if l \<noteq> 0 then of_int k / of_int l else 0)"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   664
  by (cases "l = 0") (simp_all add: rat_number_collapse of_rat_rat)
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   665
26513
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
   666
instantiation rat :: eq
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
   667
begin
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
   668
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   669
definition [code func del]: "eq_class.eq (r\<Colon>rat) s \<longleftrightarrow> r - s = 0"
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   670
26513
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
   671
instance by default (simp add: eq_rat_def)
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
   672
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   673
lemma rat_eq_code [code]: "eq_class.eq (Fract k l) (Fract r s) \<longleftrightarrow> eq_class.eq (normNum (k, l)) (normNum (r, s))"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   674
  by (simp add: eq INum_normNum_iff [where ?'a = rat, symmetric])
26513
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
   675
6f306c8c2c54 explicit class "eq" for operational equality
haftmann
parents: 25965
diff changeset
   676
end
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   677
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   678
lemma rat_less_eq_code [code]: "Fract k l \<le> Fract r s \<longleftrightarrow> normNum (k, l) \<le>\<^sub>N normNum (r, s)"
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   679
proof -
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   680
  have "normNum (k, l) \<le>\<^sub>N normNum (r, s) \<longleftrightarrow> split Fract (normNum (k, l)) \<le> split Fract (normNum (r, s))" 
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   681
    by (simp add: INum_Fract [symmetric] del: INum_Fract normNum)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   682
  also have "\<dots> = (Fract k l \<le> Fract r s)" by simp
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   683
  finally show ?thesis by simp
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   684
qed
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   685
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   686
lemma rat_less_code [code]: "Fract k l < Fract r s \<longleftrightarrow> normNum (k, l) <\<^sub>N normNum (r, s)"
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   687
proof -
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   688
  have "normNum (k, l) <\<^sub>N normNum (r, s) \<longleftrightarrow> split Fract (normNum (k, l)) < split Fract (normNum (r, s))" 
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   689
    by (simp add: INum_Fract [symmetric] del: INum_Fract normNum)
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   690
  also have "\<dots> = (Fract k l < Fract r s)" by simp
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   691
  finally show ?thesis by simp
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   692
qed
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   693
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   694
lemma rat_add_code [code]: "Fract k l + Fract r s = split Fract ((k, l) +\<^sub>N (r, s))"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   695
  by (simp add: INum_Fract [symmetric] del: INum_Fract, simp)
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   696
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   697
lemma rat_mul_code [code]: "Fract k l * Fract r s = split Fract ((k, l) *\<^sub>N (r, s))"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   698
  by (simp add: INum_Fract [symmetric] del: INum_Fract, simp)
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   699
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   700
lemma rat_neg_code [code]: "- Fract k l = split Fract (~\<^sub>N (k, l))"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   701
  by (simp add: INum_Fract [symmetric] del: INum_Fract, simp)
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   702
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   703
lemma rat_sub_code [code]: "Fract k l - Fract r s = split Fract ((k, l) -\<^sub>N (r, s))"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   704
  by (simp add: INum_Fract [symmetric] del: INum_Fract, simp)
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   705
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   706
lemma rat_inv_code [code]: "inverse (Fract k l) = split Fract (Ninv (k, l))"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   707
  by (simp add: INum_Fract [symmetric] del: INum_Fract, simp add: divide_rat_def)
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   708
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   709
lemma rat_div_code [code]: "Fract k l / Fract r s = split Fract ((k, l) \<div>\<^sub>N (r, s))"
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   710
  by (simp add: INum_Fract [symmetric] del: INum_Fract, simp)
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   711
24622
8116eb022282 renamed constructor RatC to Rational
haftmann
parents: 24533
diff changeset
   712
text {* Setup for SML code generator *}
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   713
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   714
types_code
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   715
  rat ("(int */ int)")
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   716
attach (term_of) {*
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   717
fun term_of_rat (p, q) =
24622
8116eb022282 renamed constructor RatC to Rational
haftmann
parents: 24533
diff changeset
   718
  let
24661
a705b9834590 fixed cg setup
haftmann
parents: 24630
diff changeset
   719
    val rT = Type ("Rational.rat", [])
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   720
  in
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   721
    if q = 1 orelse p = 0 then HOLogic.mk_number rT p
25885
6fbc3f54f819 New interface for test data generators.
berghofe
parents: 25762
diff changeset
   722
    else @{term "op / \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat"} $
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   723
      HOLogic.mk_number rT p $ HOLogic.mk_number rT q
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   724
  end;
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   725
*}
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   726
attach (test) {*
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   727
fun gen_rat i =
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   728
  let
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   729
    val p = random_range 0 i;
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   730
    val q = random_range 1 (i + 1);
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   731
    val g = Integer.gcd p q;
24630
351a308ab58d simplified type int (eliminated IntInf.int, integer);
wenzelm
parents: 24622
diff changeset
   732
    val p' = p div g;
351a308ab58d simplified type int (eliminated IntInf.int, integer);
wenzelm
parents: 24622
diff changeset
   733
    val q' = q div g;
25885
6fbc3f54f819 New interface for test data generators.
berghofe
parents: 25762
diff changeset
   734
    val r = (if one_of [true, false] then p' else ~ p',
6fbc3f54f819 New interface for test data generators.
berghofe
parents: 25762
diff changeset
   735
      if p' = 0 then 0 else q')
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   736
  in
25885
6fbc3f54f819 New interface for test data generators.
berghofe
parents: 25762
diff changeset
   737
    (r, fn () => term_of_rat r)
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   738
  end;
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   739
*}
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   740
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   741
consts_code
27551
9a5543d4cc24 Fract now total; improved code generator setup
haftmann
parents: 27509
diff changeset
   742
  Fract ("(_,/ _)")
24533
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   743
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   744
consts_code
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   745
  "of_int :: int \<Rightarrow> rat" ("\<module>rat'_of'_int")
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   746
attach {*
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   747
fun rat_of_int 0 = (0, 0)
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   748
  | rat_of_int i = (i, 1);
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   749
*}
fe1f93f6a15a Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents: 24506
diff changeset
   750
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   751
end