src/HOL/Real/Rational.thy
author huffman
Wed, 20 Jun 2007 05:06:16 +0200
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parent 23365 f31794033ae1
child 23879 4776af8be741
permissions -rw-r--r--
simplify some proofs
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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 Main
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uses ("rat_arith.ML")
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begin
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subsection {* Rational numbers *}
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subsubsection {* Equivalence of fractions *}
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definition
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  fraction :: "(int \<times> int) set" where
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  "fraction = {x. snd x \<noteq> 0}"
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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 fraction_iff [simp]: "(x \<in> fraction) = (snd x \<noteq> 0)"
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by (simp add: fraction_def)
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lemma ratrel_iff [simp]:
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  "((x,y) \<in> ratrel) =
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   (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 fraction ratrel"
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by (auto simp add: refl_def fraction_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_lemma:
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  assumes 1: "a * b' = a' * b"
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  assumes 2: "a' * b'' = a'' * b'"
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  assumes 3: "b' \<noteq> (0::int)"
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  shows "a * b'' = a'' * b"
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proof -
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  have "b' * (a * b'') = b'' * (a * b')" by simp
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  also note 1
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  also have "b'' * (a' * b) = b * (a' * b'')" by simp
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  also note 2
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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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  with 3 show "a * b'' = a'' * b" by simp
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qed
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lemma trans_ratrel: "trans ratrel"
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by (auto simp add: trans_def elim: trans_ratrel_lemma)
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lemma equiv_ratrel: "equiv fraction ratrel"
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by (rule equiv.intro [OF refl_ratrel sym_ratrel trans_ratrel])
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lemmas equiv_ratrel_iff [iff] = eq_equiv_class_iff [OF equiv_ratrel]
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lemma equiv_ratrel_iff2:
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  "\<lbrakk>snd x \<noteq> 0; snd y \<noteq> 0\<rbrakk>
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    \<Longrightarrow> (ratrel `` {x} = ratrel `` {y}) = ((x,y) \<in> ratrel)"
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by (rule eq_equiv_class_iff [OF equiv_ratrel], simp_all)
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subsubsection {* The type of rational numbers *}
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typedef (Rat) rat = "fraction//ratrel"
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proof
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  have "(0,1) \<in> fraction" by (simp add: fraction_def)
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  thus "ratrel``{(0,1)} \<in> fraction//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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definition
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  Fract :: "int \<Rightarrow> int \<Rightarrow> rat" where
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  "Fract a b = Abs_Rat (ratrel``{(a,b)})"
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theorem Rat_cases [case_names Fract, cases type: rat]:
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    "(!!a b. q = Fract a b ==> b \<noteq> 0 ==> C) ==> C"
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  by (cases q) (clarsimp simp add: Fract_def Rat_def fraction_def quotient_def)
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theorem Rat_induct [case_names Fract, induct type: rat]:
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    "(!!a b. b \<noteq> 0 ==> P (Fract a b)) ==> P q"
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  by (cases q) simp
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subsubsection {* Congruence lemmas *}
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lemma add_congruent2:
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     "(\<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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apply (rule equiv_ratrel [THEN congruent2_commuteI])
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apply (simp_all add: left_distrib)
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done
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lemma minus_congruent:
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  "(\<lambda>x. ratrel``{(- fst x, snd x)}) respects ratrel"
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by (simp add: congruent_def)
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lemma mult_congruent2:
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  "(\<lambda>x y. ratrel``{(fst x * fst y, snd x * snd y)}) respects2 ratrel"
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by (rule equiv_ratrel [THEN congruent2_commuteI], simp_all)
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lemma inverse_congruent:
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  "(\<lambda>x. ratrel``{if fst x=0 then (0,1) else (snd x, fst x)}) respects ratrel"
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by (auto simp add: congruent_def mult_commute)
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lemma le_congruent2:
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  "(\<lambda>x y. {(fst x * snd y)*(snd x * snd y) \<le> (fst y * snd x)*(snd x * snd y)})
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   respects2 ratrel"
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proof (clarsimp simp add: congruent2_def)
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  fix a b a' b' c d c' d'::int
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  assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
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  assume eq1: "a * b' = a' * b"
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  assume eq2: "c * d' = c' * d"
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  let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
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  {
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    fix a b c d x :: int assume x: "x \<noteq> 0"
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    have "?le a b c d = ?le (a * x) (b * x) c d"
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    proof -
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      from x have "0 < x * x" by (auto simp add: zero_less_mult_iff)
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      hence "?le a b c d =
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          ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
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        by (simp add: mult_le_cancel_right)
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      also have "... = ?le (a * x) (b * x) c d"
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        by (simp add: mult_ac)
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      finally show ?thesis .
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    qed
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  } note le_factor = this
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  let ?D = "b * d" and ?D' = "b' * d'"
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  from neq have D: "?D \<noteq> 0" by simp
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  from neq have "?D' \<noteq> 0" by simp
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   143
  hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"
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paulson
parents:
diff changeset
   144
    by (rule le_factor)
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paulson
parents:
diff changeset
   145
  also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   146
    by (simp add: mult_ac)
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paulson
parents:
diff changeset
   147
  also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   148
    by (simp only: eq1 eq2)
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paulson
parents:
diff changeset
   149
  also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   150
    by (simp add: mult_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   151
  also from D have "... = ?le a' b' c' d'"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   152
    by (rule le_factor [symmetric])
18913
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huffman
parents: 18372
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   153
  finally show "?le a b c d = ?le a' b' c' d'" .
14365
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paulson
parents:
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   154
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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   155
18913
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   156
lemmas UN_ratrel = UN_equiv_class [OF equiv_ratrel]
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parents: 18372
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   157
lemmas UN_ratrel2 = UN_equiv_class2 [OF equiv_ratrel equiv_ratrel]
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   158
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   159
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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   160
subsubsection {* Standard operations on rational numbers *}
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paulson
parents:
diff changeset
   161
20522
05072ae0d435 added instance rat :: recpower
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parents: 20485
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   162
instance rat :: "{ord, zero, one, plus, times, minus, inverse, power}" ..
14365
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paulson
parents:
diff changeset
   163
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   164
defs (overloaded)
18913
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huffman
parents: 18372
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   165
  Zero_rat_def:  "0 == Fract 0 1"
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   166
  One_rat_def:   "1 == Fract 1 1"
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   167
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   168
  add_rat_def:
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   169
   "q + r ==
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
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   170
       Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
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   171
           ratrel``{(fst x * snd y + fst y * snd x, snd x * snd y)})"
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
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   172
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
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   173
  minus_rat_def:
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   174
    "- q == Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel``{(- fst x, snd x)})"
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   175
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
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   176
  diff_rat_def:  "q - r == q + - (r::rat)"
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
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   177
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
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   178
  mult_rat_def:
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
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   179
   "q * r ==
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
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   180
       Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
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   181
           ratrel``{(fst x * fst y, snd x * snd y)})"
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   182
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
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   183
  inverse_rat_def:
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   184
    "inverse q ==
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   185
        Abs_Rat (\<Union>x \<in> Rep_Rat q.
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
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   186
            ratrel``{if fst x=0 then (0,1) else (snd x, fst x)})"
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   187
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   188
  divide_rat_def:  "q / r == q * inverse (r::rat)"
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   189
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   190
  le_rat_def:
18982
a2950f748445 no longer need All_equiv lemmas
huffman
parents: 18913
diff changeset
   191
   "q \<le> r == contents (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
a2950f748445 no longer need All_equiv lemmas
huffman
parents: 18913
diff changeset
   192
      {(fst x * snd y)*(snd x * snd y) \<le> (fst y * snd x)*(snd x * snd y)})"
18913
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   193
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   194
  less_rat_def: "(z < (w::rat)) == (z \<le> w & z \<noteq> w)"
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   195
14365
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paulson
parents:
diff changeset
   196
  abs_rat_def: "\<bar>q\<bar> == if q < 0 then -q else (q::rat)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   197
20522
05072ae0d435 added instance rat :: recpower
huffman
parents: 20485
diff changeset
   198
primrec (rat)
05072ae0d435 added instance rat :: recpower
huffman
parents: 20485
diff changeset
   199
  rat_power_0:   "q ^ 0       = 1"
05072ae0d435 added instance rat :: recpower
huffman
parents: 20485
diff changeset
   200
  rat_power_Suc: "q ^ (Suc n) = (q::rat) * (q ^ n)"
05072ae0d435 added instance rat :: recpower
huffman
parents: 20485
diff changeset
   201
18913
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
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   202
lemma zero_rat: "0 = Fract 0 1"
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   203
by (simp add: Zero_rat_def)
14365
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paulson
parents:
diff changeset
   204
18913
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huffman
parents: 18372
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   205
lemma one_rat: "1 = Fract 1 1"
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   206
by (simp add: One_rat_def)
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   207
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   208
theorem eq_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   209
  (Fract a b = Fract c d) = (a * d = c * b)"
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   210
by (simp add: Fract_def)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   211
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   212
theorem add_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   213
  Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
18913
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   214
by (simp add: Fract_def add_rat_def add_congruent2 UN_ratrel2)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   215
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   216
theorem minus_rat: "b \<noteq> 0 ==> -(Fract a b) = Fract (-a) b"
18913
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   217
by (simp add: Fract_def minus_rat_def minus_congruent UN_ratrel)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   218
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   219
theorem diff_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   220
    Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
18913
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   221
by (simp add: diff_rat_def add_rat minus_rat)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   222
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   223
theorem mult_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   224
  Fract a b * Fract c d = Fract (a * c) (b * d)"
18913
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   225
by (simp add: Fract_def mult_rat_def mult_congruent2 UN_ratrel2)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   226
18913
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   227
theorem inverse_rat: "a \<noteq> 0 ==> b \<noteq> 0 ==>
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   228
  inverse (Fract a b) = Fract b a"
18913
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   229
by (simp add: Fract_def inverse_rat_def inverse_congruent UN_ratrel)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   230
18913
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   231
theorem divide_rat: "c \<noteq> 0 ==> b \<noteq> 0 ==> d \<noteq> 0 ==>
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   232
  Fract a b / Fract c d = Fract (a * d) (b * c)"
18913
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   233
by (simp add: divide_rat_def inverse_rat mult_rat)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   234
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   235
theorem le_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   236
  (Fract a b \<le> Fract c d) = ((a * d) * (b * d) \<le> (c * b) * (b * d))"
18982
a2950f748445 no longer need All_equiv lemmas
huffman
parents: 18913
diff changeset
   237
by (simp add: Fract_def le_rat_def le_congruent2 UN_ratrel2)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   238
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   239
theorem less_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   240
    (Fract a b < Fract c d) = ((a * d) * (b * d) < (c * b) * (b * d))"
18913
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   241
by (simp add: less_rat_def le_rat eq_rat order_less_le)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   242
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   243
theorem abs_rat: "b \<noteq> 0 ==> \<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   244
  by (simp add: abs_rat_def minus_rat zero_rat less_rat eq_rat)
14691
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14430
diff changeset
   245
     (auto simp add: mult_less_0_iff zero_less_mult_iff order_le_less
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   246
                split: abs_split)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   247
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   248
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   249
subsubsection {* The ordered field of rational numbers *}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   250
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   251
instance rat :: field
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   252
proof
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   253
  fix q r s :: rat
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   254
  show "(q + r) + s = q + (r + s)"
18913
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   255
    by (induct q, induct r, induct s)
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   256
       (simp add: add_rat add_ac mult_ac int_distrib)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   257
  show "q + r = r + q"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   258
    by (induct q, induct r) (simp add: add_rat add_ac mult_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   259
  show "0 + q = q"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   260
    by (induct q) (simp add: zero_rat add_rat)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   261
  show "(-q) + q = 0"
18913
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   262
    by (induct q) (simp add: zero_rat minus_rat add_rat eq_rat)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   263
  show "q - r = q + (-r)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   264
    by (induct q, induct r) (simp add: add_rat minus_rat diff_rat)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   265
  show "(q * r) * s = q * (r * s)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   266
    by (induct q, induct r, induct s) (simp add: mult_rat mult_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   267
  show "q * r = r * q"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   268
    by (induct q, induct r) (simp add: mult_rat mult_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   269
  show "1 * q = q"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   270
    by (induct q) (simp add: one_rat mult_rat)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   271
  show "(q + r) * s = q * s + r * s"
14691
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14430
diff changeset
   272
    by (induct q, induct r, induct s)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   273
       (simp add: add_rat mult_rat eq_rat int_distrib)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   274
  show "q \<noteq> 0 ==> inverse q * q = 1"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   275
    by (induct q) (simp add: inverse_rat mult_rat one_rat zero_rat eq_rat)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   276
  show "q / r = q * inverse r"
14691
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14430
diff changeset
   277
    by (simp add: divide_rat_def)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   278
  show "0 \<noteq> (1::rat)"
14691
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14430
diff changeset
   279
    by (simp add: zero_rat one_rat eq_rat)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   280
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   281
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   282
instance rat :: linorder
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   283
proof
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   284
  fix q r s :: rat
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   285
  {
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   286
    assume "q \<le> r" and "r \<le> s"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   287
    show "q \<le> s"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   288
    proof (insert prems, induct q, induct r, induct s)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   289
      fix a b c d e f :: int
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   290
      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
   291
      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
   292
      show "Fract a b \<le> Fract e f"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   293
      proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   294
        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
   295
          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
   296
        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
   297
        proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   298
          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
   299
            by (simp add: le_rat)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   300
          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
   301
        qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   302
        also have "... = (c * f) * (d * f) * (b * b)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   303
          by (simp only: mult_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   304
        also have "... \<le> (e * d) * (d * f) * (b * b)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   305
        proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   306
          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
   307
            by (simp add: le_rat)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   308
          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
   309
        qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   310
        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
   311
          by (simp only: mult_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   312
        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
   313
          by (simp add: mult_le_cancel_right)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   314
        with neq show ?thesis by (simp add: le_rat)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   315
      qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   316
    qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   317
  next
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   318
    assume "q \<le> r" and "r \<le> q"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   319
    show "q = r"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   320
    proof (insert prems, induct q, induct r)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   321
      fix a b c d :: int
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   322
      assume neq: "b \<noteq> 0"  "d \<noteq> 0"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   323
      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
   324
      show "Fract a b = Fract c d"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   325
      proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   326
        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
   327
          by (simp add: le_rat)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   328
        also have "... \<le> (a * d) * (b * d)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   329
        proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   330
          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
   331
            by (simp add: le_rat)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   332
          thus ?thesis by (simp only: mult_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   333
        qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   334
        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
   335
        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
   336
        ultimately have "a * d = c * b" by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   337
        with neq show ?thesis by (simp add: eq_rat)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   338
      qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   339
    qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   340
  next
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   341
    show "q \<le> q"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   342
      by (induct q) (simp add: le_rat)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   343
    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
   344
      by (simp only: less_rat_def)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   345
    show "q \<le> r \<or> r \<le> q"
18913
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   346
      by (induct q, induct r)
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   347
         (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
   348
  }
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   349
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   350
22456
6070e48ecb78 added lattice definitions
haftmann
parents: 21404
diff changeset
   351
instance rat :: distrib_lattice
6070e48ecb78 added lattice definitions
haftmann
parents: 21404
diff changeset
   352
  "inf r s \<equiv> min r s"
6070e48ecb78 added lattice definitions
haftmann
parents: 21404
diff changeset
   353
  "sup r s \<equiv> max r s"
6070e48ecb78 added lattice definitions
haftmann
parents: 21404
diff changeset
   354
  by default (auto simp add: min_max.sup_inf_distrib1 inf_rat_def sup_rat_def)
6070e48ecb78 added lattice definitions
haftmann
parents: 21404
diff changeset
   355
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   356
instance rat :: ordered_field
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   357
proof
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   358
  fix q r s :: rat
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   359
  show "q \<le> r ==> s + q \<le> s + r"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   360
  proof (induct q, induct r, induct s)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   361
    fix a b c d e f :: int
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   362
    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
   363
    assume le: "Fract a b \<le> Fract c d"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   364
    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
   365
    proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   366
      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
   367
        by (auto simp add: zero_less_mult_iff)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   368
      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
   369
        by (simp add: le_rat)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   370
      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
   371
        by (simp add: mult_le_cancel_right)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   372
      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
   373
    qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   374
  qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   375
  show "q < r ==> 0 < s ==> s * q < s * r"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   376
  proof (induct q, induct r, induct s)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   377
    fix a b c d e f :: int
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   378
    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
   379
    assume le: "Fract a b < Fract c d"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   380
    assume gt: "0 < Fract e f"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   381
    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
   382
    proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   383
      let ?E = "e * f" and ?F = "f * f"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   384
      from neq gt have "0 < ?E"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   385
        by (auto simp add: zero_rat 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
   386
      moreover from neq have "0 < ?F"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   387
        by (auto simp add: zero_less_mult_iff)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   388
      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
   389
        by (simp add: less_rat)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   390
      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
   391
        by (simp add: mult_less_cancel_right)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   392
      with neq show ?thesis
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   393
        by (simp add: less_rat mult_rat mult_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   394
    qed
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
  show "\<bar>q\<bar> = (if q < 0 then -q else q)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   397
    by (simp only: abs_rat_def)
22456
6070e48ecb78 added lattice definitions
haftmann
parents: 21404
diff changeset
   398
qed auto
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   399
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   400
instance rat :: division_by_zero
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   401
proof
18913
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   402
  show "inverse 0 = (0::rat)"
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   403
    by (simp add: zero_rat Fract_def inverse_rat_def
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   404
                  inverse_congruent UN_ratrel)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   405
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   406
20522
05072ae0d435 added instance rat :: recpower
huffman
parents: 20485
diff changeset
   407
instance rat :: recpower
05072ae0d435 added instance rat :: recpower
huffman
parents: 20485
diff changeset
   408
proof
05072ae0d435 added instance rat :: recpower
huffman
parents: 20485
diff changeset
   409
  fix q :: rat
05072ae0d435 added instance rat :: recpower
huffman
parents: 20485
diff changeset
   410
  fix n :: nat
05072ae0d435 added instance rat :: recpower
huffman
parents: 20485
diff changeset
   411
  show "q ^ 0 = 1" by simp
05072ae0d435 added instance rat :: recpower
huffman
parents: 20485
diff changeset
   412
  show "q ^ (Suc n) = q * (q ^ n)" by simp
05072ae0d435 added instance rat :: recpower
huffman
parents: 20485
diff changeset
   413
qed
05072ae0d435 added instance rat :: recpower
huffman
parents: 20485
diff changeset
   414
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   415
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   416
subsection {* Various Other Results *}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   417
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   418
lemma minus_rat_cancel [simp]: "b \<noteq> 0 ==> Fract (-a) (-b) = Fract a b"
18913
57f19fad8c2a reimplemented using Equiv_Relations.thy
huffman
parents: 18372
diff changeset
   419
by (simp add: eq_rat)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   420
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   421
theorem Rat_induct_pos [case_names Fract, induct type: rat]:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   422
  assumes step: "!!a b. 0 < b ==> P (Fract a b)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   423
    shows "P q"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   424
proof (cases q)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   425
  have step': "!!a b. b < 0 ==> P (Fract a b)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   426
  proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   427
    fix a::int and b::int
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   428
    assume b: "b < 0"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   429
    hence "0 < -b" by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   430
    hence "P (Fract (-a) (-b))" by (rule step)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   431
    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
   432
  qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   433
  case (Fract a b)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   434
  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
   435
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   436
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   437
lemma zero_less_Fract_iff:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   438
     "0 < b ==> (0 < Fract a b) = (0 < a)"
14691
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14430
diff changeset
   439
by (simp add: zero_rat 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
   440
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   441
lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   442
apply (insert add_rat [of concl: m n 1 1])
14691
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14430
diff changeset
   443
apply (simp add: one_rat  [symmetric])
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   444
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   445
23429
5a55a9409e57 simplify some proofs
huffman
parents: 23365
diff changeset
   446
lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1"
5a55a9409e57 simplify some proofs
huffman
parents: 23365
diff changeset
   447
by (induct k) (simp_all add: zero_rat one_rat add_rat)
5a55a9409e57 simplify some proofs
huffman
parents: 23365
diff changeset
   448
5a55a9409e57 simplify some proofs
huffman
parents: 23365
diff changeset
   449
lemma of_int_rat: "of_int k = Fract k 1"
5a55a9409e57 simplify some proofs
huffman
parents: 23365
diff changeset
   450
by (cases k rule: int_diff_cases, simp add: of_nat_rat diff_rat)
5a55a9409e57 simplify some proofs
huffman
parents: 23365
diff changeset
   451
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   452
lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
23429
5a55a9409e57 simplify some proofs
huffman
parents: 23365
diff changeset
   453
by (rule of_nat_rat [symmetric])
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   454
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   455
lemma Fract_of_int_eq: "Fract k 1 = of_int k"
23429
5a55a9409e57 simplify some proofs
huffman
parents: 23365
diff changeset
   456
by (rule of_int_rat [symmetric])
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   457
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   458
14691
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14430
diff changeset
   459
subsection {* Numerals and Arithmetic *}
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   460
22456
6070e48ecb78 added lattice definitions
haftmann
parents: 21404
diff changeset
   461
instance rat :: number
6070e48ecb78 added lattice definitions
haftmann
parents: 21404
diff changeset
   462
  rat_number_of_def: "(number_of w :: rat) \<equiv> of_int w" ..
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   463
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   464
instance rat :: number_ring
19765
dfe940911617 misc cleanup;
wenzelm
parents: 18983
diff changeset
   465
  by default (simp add: rat_number_of_def) 
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   466
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   467
use "rat_arith.ML"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   468
setup rat_arith_setup
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   469
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   470
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   471
subsection {* Embedding from Rationals to other Fields *}
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   472
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   473
axclass field_char_0 < field, ring_char_0
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   474
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   475
instance ordered_field < field_char_0 ..
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   476
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   477
definition
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   478
  of_rat :: "rat \<Rightarrow> 'a::field_char_0"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   479
where
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   480
  "of_rat q = contents (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   481
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   482
lemma of_rat_congruent:
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   483
  "(\<lambda>(a, b). {of_int a / of_int b::'a::field_char_0}) respects ratrel"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   484
apply (rule congruent.intro)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   485
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
   486
apply (simp only: of_int_mult [symmetric])
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   487
done
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   488
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   489
lemma of_rat_rat:
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   490
  "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   491
unfolding Fract_def of_rat_def
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   492
by (simp add: UN_ratrel of_rat_congruent)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   493
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   494
lemma of_rat_0 [simp]: "of_rat 0 = 0"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   495
by (simp add: Zero_rat_def of_rat_rat)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   496
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   497
lemma of_rat_1 [simp]: "of_rat 1 = 1"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   498
by (simp add: One_rat_def of_rat_rat)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   499
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   500
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
   501
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
   502
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   503
lemma of_rat_minus: "of_rat (- a) = - of_rat a"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   504
by (induct a, simp add: minus_rat of_rat_rat)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   505
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   506
lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   507
by (simp only: diff_minus of_rat_add of_rat_minus)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   508
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   509
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
   510
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
   511
apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   512
done
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   513
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   514
lemma nonzero_of_rat_inverse:
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   515
  "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   516
apply (rule inverse_unique [symmetric])
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   517
apply (simp add: of_rat_mult [symmetric])
23342
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   518
done
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   519
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   520
lemma of_rat_inverse:
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   521
  "(of_rat (inverse a)::'a::{field_char_0,division_by_zero}) =
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   522
   inverse (of_rat a)"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   523
by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   524
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   525
lemma nonzero_of_rat_divide:
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   526
  "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
   527
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
   528
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   529
lemma of_rat_divide:
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   530
  "(of_rat (a / b)::'a::{field_char_0,division_by_zero})
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   531
   = of_rat a / of_rat b"
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   532
by (cases "b = 0", simp_all add: nonzero_of_rat_divide)
0261d2da0b1c add function of_rat and related lemmas
huffman
parents: 22456
diff changeset
   533
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   534
lemma of_rat_power:
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   535
  "(of_rat (a ^ n)::'a::{field_char_0,recpower}) = of_rat a ^ n"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   536
by (induct n) (simp_all add: of_rat_mult power_Suc)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   537
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   538
lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   539
apply (induct a, induct b)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   540
apply (simp add: of_rat_rat eq_rat)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   541
apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   542
apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   543
done
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   544
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   545
lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   546
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   547
lemma of_rat_eq_id [simp]: "of_rat = (id :: rat \<Rightarrow> rat)"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   548
proof
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   549
  fix a
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   550
  show "of_rat a = id a"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   551
  by (induct a)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   552
     (simp add: of_rat_rat divide_rat Fract_of_int_eq [symmetric])
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   553
qed
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   554
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   555
text{*Collapse nested embeddings*}
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   556
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
   557
by (induct n) (simp_all add: of_rat_add)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   558
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   559
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
   560
by (cases z rule: int_diff_cases, simp add: of_rat_diff)
23343
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   561
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   562
lemma of_rat_number_of_eq [simp]:
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   563
  "of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})"
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   564
by (simp add: number_of_eq)
6a83ca5fe282 more of_rat lemmas
huffman
parents: 23342
diff changeset
   565
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff changeset
   566
end