src/HOL/Rational.thy
 author wenzelm Thu Feb 11 23:00:22 2010 +0100 (2010-02-11) changeset 35115 446c5063e4fd parent 35063 893062359bec child 35216 7641e8d831d2 permissions -rw-r--r--
modernized translations;
formal markup of @{syntax_const} and @{const_syntax};
minor tuning;
```     1 (*  Title:  HOL/Rational.thy
```
```     2     Author: Markus Wenzel, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* Rational numbers *}
```
```     6
```
```     7 theory Rational
```
```     8 imports GCD Archimedean_Field
```
```     9 begin
```
```    10
```
```    11 subsection {* Rational numbers as quotient *}
```
```    12
```
```    13 subsubsection {* Construction of the type of rational numbers *}
```
```    14
```
```    15 definition
```
```    16   ratrel :: "((int \<times> int) \<times> (int \<times> int)) set" where
```
```    17   "ratrel = {(x, y). snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x}"
```
```    18
```
```    19 lemma ratrel_iff [simp]:
```
```    20   "(x, y) \<in> ratrel \<longleftrightarrow> snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x"
```
```    21   by (simp add: ratrel_def)
```
```    22
```
```    23 lemma refl_on_ratrel: "refl_on {x. snd x \<noteq> 0} ratrel"
```
```    24   by (auto simp add: refl_on_def ratrel_def)
```
```    25
```
```    26 lemma sym_ratrel: "sym ratrel"
```
```    27   by (simp add: ratrel_def sym_def)
```
```    28
```
```    29 lemma trans_ratrel: "trans ratrel"
```
```    30 proof (rule transI, unfold split_paired_all)
```
```    31   fix a b a' b' a'' b'' :: int
```
```    32   assume A: "((a, b), (a', b')) \<in> ratrel"
```
```    33   assume B: "((a', b'), (a'', b'')) \<in> ratrel"
```
```    34   have "b' * (a * b'') = b'' * (a * b')" by simp
```
```    35   also from A have "a * b' = a' * b" by auto
```
```    36   also have "b'' * (a' * b) = b * (a' * b'')" by simp
```
```    37   also from B have "a' * b'' = a'' * b'" by auto
```
```    38   also have "b * (a'' * b') = b' * (a'' * b)" by simp
```
```    39   finally have "b' * (a * b'') = b' * (a'' * b)" .
```
```    40   moreover from B have "b' \<noteq> 0" by auto
```
```    41   ultimately have "a * b'' = a'' * b" by simp
```
```    42   with A B show "((a, b), (a'', b'')) \<in> ratrel" by auto
```
```    43 qed
```
```    44
```
```    45 lemma equiv_ratrel: "equiv {x. snd x \<noteq> 0} ratrel"
```
```    46   by (rule equiv.intro [OF refl_on_ratrel sym_ratrel trans_ratrel])
```
```    47
```
```    48 lemmas UN_ratrel = UN_equiv_class [OF equiv_ratrel]
```
```    49 lemmas UN_ratrel2 = UN_equiv_class2 [OF equiv_ratrel equiv_ratrel]
```
```    50
```
```    51 lemma equiv_ratrel_iff [iff]:
```
```    52   assumes "snd x \<noteq> 0" and "snd y \<noteq> 0"
```
```    53   shows "ratrel `` {x} = ratrel `` {y} \<longleftrightarrow> (x, y) \<in> ratrel"
```
```    54   by (rule eq_equiv_class_iff, rule equiv_ratrel) (auto simp add: assms)
```
```    55
```
```    56 typedef (Rat) rat = "{x. snd x \<noteq> 0} // ratrel"
```
```    57 proof
```
```    58   have "(0::int, 1::int) \<in> {x. snd x \<noteq> 0}" by simp
```
```    59   then show "ratrel `` {(0, 1)} \<in> {x. snd x \<noteq> 0} // ratrel" by (rule quotientI)
```
```    60 qed
```
```    61
```
```    62 lemma ratrel_in_Rat [simp]: "snd x \<noteq> 0 \<Longrightarrow> ratrel `` {x} \<in> Rat"
```
```    63   by (simp add: Rat_def quotientI)
```
```    64
```
```    65 declare Abs_Rat_inject [simp] Abs_Rat_inverse [simp]
```
```    66
```
```    67
```
```    68 subsubsection {* Representation and basic operations *}
```
```    69
```
```    70 definition
```
```    71   Fract :: "int \<Rightarrow> int \<Rightarrow> rat" where
```
```    72   [code del]: "Fract a b = Abs_Rat (ratrel `` {if b = 0 then (0, 1) else (a, b)})"
```
```    73
```
```    74 code_datatype Fract
```
```    75
```
```    76 lemma Rat_cases [case_names Fract, cases type: rat]:
```
```    77   assumes "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> C"
```
```    78   shows C
```
```    79   using assms by (cases q) (clarsimp simp add: Fract_def Rat_def quotient_def)
```
```    80
```
```    81 lemma Rat_induct [case_names Fract, induct type: rat]:
```
```    82   assumes "\<And>a b. b \<noteq> 0 \<Longrightarrow> P (Fract a b)"
```
```    83   shows "P q"
```
```    84   using assms by (cases q) simp
```
```    85
```
```    86 lemma eq_rat:
```
```    87   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"
```
```    88   and "\<And>a. Fract a 0 = Fract 0 1"
```
```    89   and "\<And>a c. Fract 0 a = Fract 0 c"
```
```    90   by (simp_all add: Fract_def)
```
```    91
```
```    92 instantiation rat :: comm_ring_1
```
```    93 begin
```
```    94
```
```    95 definition
```
```    96   Zero_rat_def [code, code_unfold]: "0 = Fract 0 1"
```
```    97
```
```    98 definition
```
```    99   One_rat_def [code, code_unfold]: "1 = Fract 1 1"
```
```   100
```
```   101 definition
```
```   102   add_rat_def [code del]:
```
```   103   "q + r = Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
```
```   104     ratrel `` {(fst x * snd y + fst y * snd x, snd x * snd y)})"
```
```   105
```
```   106 lemma add_rat [simp]:
```
```   107   assumes "b \<noteq> 0" and "d \<noteq> 0"
```
```   108   shows "Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
```
```   109 proof -
```
```   110   have "(\<lambda>x y. ratrel``{(fst x * snd y + fst y * snd x, snd x * snd y)})
```
```   111     respects2 ratrel"
```
```   112   by (rule equiv_ratrel [THEN congruent2_commuteI]) (simp_all add: left_distrib)
```
```   113   with assms show ?thesis by (simp add: Fract_def add_rat_def UN_ratrel2)
```
```   114 qed
```
```   115
```
```   116 definition
```
```   117   minus_rat_def [code del]:
```
```   118   "- q = Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel `` {(- fst x, snd x)})"
```
```   119
```
```   120 lemma minus_rat [simp, code]: "- Fract a b = Fract (- a) b"
```
```   121 proof -
```
```   122   have "(\<lambda>x. ratrel `` {(- fst x, snd x)}) respects ratrel"
```
```   123     by (simp add: congruent_def)
```
```   124   then show ?thesis by (simp add: Fract_def minus_rat_def UN_ratrel)
```
```   125 qed
```
```   126
```
```   127 lemma minus_rat_cancel [simp]: "Fract (- a) (- b) = Fract a b"
```
```   128   by (cases "b = 0") (simp_all add: eq_rat)
```
```   129
```
```   130 definition
```
```   131   diff_rat_def [code del]: "q - r = q + - (r::rat)"
```
```   132
```
```   133 lemma diff_rat [simp]:
```
```   134   assumes "b \<noteq> 0" and "d \<noteq> 0"
```
```   135   shows "Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
```
```   136   using assms by (simp add: diff_rat_def)
```
```   137
```
```   138 definition
```
```   139   mult_rat_def [code del]:
```
```   140   "q * r = Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
```
```   141     ratrel``{(fst x * fst y, snd x * snd y)})"
```
```   142
```
```   143 lemma mult_rat [simp]: "Fract a b * Fract c d = Fract (a * c) (b * d)"
```
```   144 proof -
```
```   145   have "(\<lambda>x y. ratrel `` {(fst x * fst y, snd x * snd y)}) respects2 ratrel"
```
```   146     by (rule equiv_ratrel [THEN congruent2_commuteI]) simp_all
```
```   147   then show ?thesis by (simp add: Fract_def mult_rat_def UN_ratrel2)
```
```   148 qed
```
```   149
```
```   150 lemma mult_rat_cancel:
```
```   151   assumes "c \<noteq> 0"
```
```   152   shows "Fract (c * a) (c * b) = Fract a b"
```
```   153 proof -
```
```   154   from assms have "Fract c c = Fract 1 1" by (simp add: Fract_def)
```
```   155   then show ?thesis by (simp add: mult_rat [symmetric])
```
```   156 qed
```
```   157
```
```   158 instance proof
```
```   159   fix q r s :: rat show "(q * r) * s = q * (r * s)"
```
```   160     by (cases q, cases r, cases s) (simp add: eq_rat)
```
```   161 next
```
```   162   fix q r :: rat show "q * r = r * q"
```
```   163     by (cases q, cases r) (simp add: eq_rat)
```
```   164 next
```
```   165   fix q :: rat show "1 * q = q"
```
```   166     by (cases q) (simp add: One_rat_def eq_rat)
```
```   167 next
```
```   168   fix q r s :: rat show "(q + r) + s = q + (r + s)"
```
```   169     by (cases q, cases r, cases s) (simp add: eq_rat algebra_simps)
```
```   170 next
```
```   171   fix q r :: rat show "q + r = r + q"
```
```   172     by (cases q, cases r) (simp add: eq_rat)
```
```   173 next
```
```   174   fix q :: rat show "0 + q = q"
```
```   175     by (cases q) (simp add: Zero_rat_def eq_rat)
```
```   176 next
```
```   177   fix q :: rat show "- q + q = 0"
```
```   178     by (cases q) (simp add: Zero_rat_def eq_rat)
```
```   179 next
```
```   180   fix q r :: rat show "q - r = q + - r"
```
```   181     by (cases q, cases r) (simp add: eq_rat)
```
```   182 next
```
```   183   fix q r s :: rat show "(q + r) * s = q * s + r * s"
```
```   184     by (cases q, cases r, cases s) (simp add: eq_rat algebra_simps)
```
```   185 next
```
```   186   show "(0::rat) \<noteq> 1" by (simp add: Zero_rat_def One_rat_def eq_rat)
```
```   187 qed
```
```   188
```
```   189 end
```
```   190
```
```   191 lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1"
```
```   192   by (induct k) (simp_all add: Zero_rat_def One_rat_def)
```
```   193
```
```   194 lemma of_int_rat: "of_int k = Fract k 1"
```
```   195   by (cases k rule: int_diff_cases) (simp add: of_nat_rat)
```
```   196
```
```   197 lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
```
```   198   by (rule of_nat_rat [symmetric])
```
```   199
```
```   200 lemma Fract_of_int_eq: "Fract k 1 = of_int k"
```
```   201   by (rule of_int_rat [symmetric])
```
```   202
```
```   203 instantiation rat :: number_ring
```
```   204 begin
```
```   205
```
```   206 definition
```
```   207   rat_number_of_def [code del]: "number_of w = Fract w 1"
```
```   208
```
```   209 instance proof
```
```   210 qed (simp add: rat_number_of_def of_int_rat)
```
```   211
```
```   212 end
```
```   213
```
```   214 lemma rat_number_collapse [code_post]:
```
```   215   "Fract 0 k = 0"
```
```   216   "Fract 1 1 = 1"
```
```   217   "Fract (number_of k) 1 = number_of k"
```
```   218   "Fract k 0 = 0"
```
```   219   by (cases "k = 0")
```
```   220     (simp_all add: Zero_rat_def One_rat_def number_of_is_id number_of_eq of_int_rat eq_rat Fract_def)
```
```   221
```
```   222 lemma rat_number_expand [code_unfold]:
```
```   223   "0 = Fract 0 1"
```
```   224   "1 = Fract 1 1"
```
```   225   "number_of k = Fract (number_of k) 1"
```
```   226   by (simp_all add: rat_number_collapse)
```
```   227
```
```   228 lemma iszero_rat [simp]:
```
```   229   "iszero (number_of k :: rat) \<longleftrightarrow> iszero (number_of k :: int)"
```
```   230   by (simp add: iszero_def rat_number_expand number_of_is_id eq_rat)
```
```   231
```
```   232 lemma Rat_cases_nonzero [case_names Fract 0]:
```
```   233   assumes Fract: "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> C"
```
```   234   assumes 0: "q = 0 \<Longrightarrow> C"
```
```   235   shows C
```
```   236 proof (cases "q = 0")
```
```   237   case True then show C using 0 by auto
```
```   238 next
```
```   239   case False
```
```   240   then obtain a b where "q = Fract a b" and "b \<noteq> 0" by (cases q) auto
```
```   241   moreover with False have "0 \<noteq> Fract a b" by simp
```
```   242   with `b \<noteq> 0` have "a \<noteq> 0" by (simp add: Zero_rat_def eq_rat)
```
```   243   with Fract `q = Fract a b` `b \<noteq> 0` show C by auto
```
```   244 qed
```
```   245
```
```   246 subsubsection {* Function @{text normalize} *}
```
```   247
```
```   248 text{*
```
```   249 Decompose a fraction into normalized, i.e. coprime numerator and denominator:
```
```   250 *}
```
```   251
```
```   252 definition normalize :: "rat \<Rightarrow> int \<times> int" where
```
```   253 "normalize x \<equiv> THE pair. x = Fract (fst pair) (snd pair) &
```
```   254                    snd pair > 0 & gcd (fst pair) (snd pair) = 1"
```
```   255
```
```   256 declare normalize_def[code del]
```
```   257
```
```   258 lemma Fract_norm: "Fract (a div gcd a b) (b div gcd a b) = Fract a b"
```
```   259 proof (cases "a = 0 | b = 0")
```
```   260   case True then show ?thesis by (auto simp add: eq_rat)
```
```   261 next
```
```   262   let ?c = "gcd a b"
```
```   263   case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
```
```   264   then have "?c \<noteq> 0" by simp
```
```   265   then have "Fract ?c ?c = Fract 1 1" by (simp add: eq_rat)
```
```   266   moreover have "Fract (a div ?c * ?c + a mod ?c) (b div ?c * ?c + b mod ?c) = Fract a b"
```
```   267     by (simp add: semiring_div_class.mod_div_equality)
```
```   268   moreover have "a mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric])
```
```   269   moreover have "b mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric])
```
```   270   ultimately show ?thesis
```
```   271     by (simp add: mult_rat [symmetric])
```
```   272 qed
```
```   273
```
```   274 text{* Proof by Ren\'e Thiemann: *}
```
```   275 lemma normalize_code[code]:
```
```   276 "normalize (Fract a b) =
```
```   277  (if b > 0 then (let g = gcd a b in (a div g, b div g))
```
```   278   else if b = 0 then (0,1)
```
```   279   else (let g = - gcd a b in (a div g, b div g)))"
```
```   280 proof -
```
```   281   let ?cond = "% r p. r = Fract (fst p) (snd p) & snd p > 0 &
```
```   282                  gcd (fst p) (snd p) = 1"
```
```   283   show ?thesis
```
```   284   proof (cases "b = 0")
```
```   285     case True
```
```   286     thus ?thesis
```
```   287     proof (simp add: normalize_def)
```
```   288       show "(THE pair. ?cond (Fract a 0) pair) = (0,1)"
```
```   289       proof
```
```   290         show "?cond (Fract a 0) (0,1)"
```
```   291           by (simp add: rat_number_collapse)
```
```   292       next
```
```   293         fix pair
```
```   294         assume cond: "?cond (Fract a 0) pair"
```
```   295         show "pair = (0,1)"
```
```   296         proof (cases pair)
```
```   297           case (Pair den num)
```
```   298           with cond have num: "num > 0" by auto
```
```   299           with Pair cond have den: "den = 0" by (simp add: eq_rat)
```
```   300           show ?thesis
```
```   301           proof (cases "num = 1", simp add: Pair den)
```
```   302             case False
```
```   303             with num have gr: "num > 1" by auto
```
```   304             with den have "gcd den num = num" by auto
```
```   305             with Pair cond False gr show ?thesis by auto
```
```   306           qed
```
```   307         qed
```
```   308       qed
```
```   309     qed
```
```   310   next
```
```   311     { fix a b :: int assume b: "b > 0"
```
```   312       hence b0: "b \<noteq> 0" and "b >= 0" by auto
```
```   313       let ?g = "gcd a b"
```
```   314       from b0 have g0: "?g \<noteq> 0" by auto
```
```   315       then have gp: "?g > 0" by simp
```
```   316       then have gs: "?g <= b" by (metis b gcd_le2_int)
```
```   317       from gcd_dvd1_int[of a b] obtain a' where a': "a = ?g * a'"
```
```   318         unfolding dvd_def by auto
```
```   319       from gcd_dvd2_int[of a b] obtain b' where b': "b = ?g * b'"
```
```   320         unfolding dvd_def by auto
```
```   321       hence b'2: "b' * ?g = b" by (simp add: ring_simps)
```
```   322       with b0 have b'0: "b' \<noteq> 0" by auto
```
```   323       from b b' zero_less_mult_iff[of ?g b'] gp have b'p: "b' > 0" by arith
```
```   324       have "normalize (Fract a b) = (a div ?g, b div ?g)"
```
```   325       proof (simp add: normalize_def)
```
```   326         show "(THE pair. ?cond (Fract a b) pair) = (a div ?g, b div ?g)"
```
```   327         proof
```
```   328           have "1 = b div b" using b0 by auto
```
```   329           also have "\<dots> <= b div ?g" by (rule zdiv_mono2[OF `b >= 0` gp gs])
```
```   330           finally have div0: "b div ?g > 0" by simp
```
```   331           show "?cond (Fract a b) (a div ?g, b div ?g)"
```
```   332             by (simp add: b0 Fract_norm div_gcd_coprime_int div0)
```
```   333         next
```
```   334           fix pair assume cond: "?cond (Fract a b) pair"
```
```   335           show "pair = (a div ?g, b div ?g)"
```
```   336           proof (cases pair)
```
```   337             case (Pair den num)
```
```   338             with cond
```
```   339             have num: "num > 0" and num0: "num \<noteq> 0" and gcd: "gcd den num = 1"
```
```   340               by auto
```
```   341             obtain g where g: "g = ?g" by auto
```
```   342             with gp have gg0: "g > 0" by auto
```
```   343             from cond Pair eq_rat(1)[OF b0 num0]
```
```   344             have eq: "a * num = den * b" by auto
```
```   345             hence "a' * g * num = den * g * b'"
```
```   346               using a'[simplified g[symmetric]] b'[simplified g[symmetric]]
```
```   347               by simp
```
```   348             hence "a' * num * g = b' * den * g" by (simp add: algebra_simps)
```
```   349             hence eq2: "a' * num = b' * den" using gg0 by auto
```
```   350             have "a div ?g = ?g * a' div ?g" using a' by force
```
```   351             hence adiv: "a div ?g = a'" using g0 by auto
```
```   352             have "b div ?g = ?g * b' div ?g" using b' by force
```
```   353             hence bdiv: "b div ?g = b'" using g0 by auto
```
```   354             from div_gcd_coprime_int[of a b] b0
```
```   355             have "gcd (a div ?g) (b div ?g) = 1" by auto
```
```   356             with adiv bdiv have gcd2: "gcd a' b' = 1" by auto
```
```   357             from gcd have gcd3: "gcd num den = 1"
```
```   358               by (simp add: gcd_commute_int[of den num])
```
```   359             from gcd2 have gcd4: "gcd b' a' = 1"
```
```   360               by (simp add: gcd_commute_int[of a' b'])
```
```   361             have one: "num dvd b'"
```
```   362               by (metis coprime_dvd_mult_int[OF gcd3] dvd_triv_right eq2)
```
```   363             have two: "b' dvd num"
```
```   364               by (metis coprime_dvd_mult_int[OF gcd4] dvd_triv_left eq2 zmult_commute)
```
```   365             from zdvd_antisym_abs[OF one two] b'p num
```
```   366             have numb': "num = b'" by auto
```
```   367             with eq2 b'0 have "a' = den" by auto
```
```   368             with numb' adiv bdiv Pair show ?thesis by simp
```
```   369           qed
```
```   370         qed
```
```   371       qed
```
```   372     }
```
```   373     note main = this
```
```   374     assume "b \<noteq> 0"
```
```   375     hence "b > 0 | b < 0" by arith
```
```   376     thus ?thesis
```
```   377     proof
```
```   378       assume b: "b > 0" thus ?thesis by (simp add: Let_def main[OF b])
```
```   379     next
```
```   380       assume b: "b < 0"
```
```   381       thus ?thesis
```
```   382         by(simp add:main Let_def minus_rat_cancel[of a b, symmetric]
```
```   383                     zdiv_zminus2 del:minus_rat_cancel)
```
```   384     qed
```
```   385   qed
```
```   386 qed
```
```   387
```
```   388 lemma normalize_id: "normalize (Fract a b) = (p,q) \<Longrightarrow> Fract p q = Fract a b"
```
```   389 by(auto simp add: normalize_code Let_def Fract_norm dvd_div_neg rat_number_collapse
```
```   390         split:split_if_asm)
```
```   391
```
```   392 lemma normalize_denom_pos: "normalize (Fract a b) = (p,q) \<Longrightarrow> q > 0"
```
```   393 by(auto simp add: normalize_code Let_def dvd_div_neg pos_imp_zdiv_neg_iff nonneg1_imp_zdiv_pos_iff
```
```   394         split:split_if_asm)
```
```   395
```
```   396 lemma normalize_coprime: "normalize (Fract a b) = (p,q) \<Longrightarrow> coprime p q"
```
```   397 by(auto simp add: normalize_code Let_def dvd_div_neg div_gcd_coprime_int
```
```   398         split:split_if_asm)
```
```   399
```
```   400
```
```   401 subsubsection {* The field of rational numbers *}
```
```   402
```
```   403 instantiation rat :: "{field, division_by_zero}"
```
```   404 begin
```
```   405
```
```   406 definition
```
```   407   inverse_rat_def [code del]:
```
```   408   "inverse q = Abs_Rat (\<Union>x \<in> Rep_Rat q.
```
```   409      ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)})"
```
```   410
```
```   411 lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"
```
```   412 proof -
```
```   413   have "(\<lambda>x. ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)}) respects ratrel"
```
```   414     by (auto simp add: congruent_def mult_commute)
```
```   415   then show ?thesis by (simp add: Fract_def inverse_rat_def UN_ratrel)
```
```   416 qed
```
```   417
```
```   418 definition
```
```   419   divide_rat_def [code del]: "q / r = q * inverse (r::rat)"
```
```   420
```
```   421 lemma divide_rat [simp]: "Fract a b / Fract c d = Fract (a * d) (b * c)"
```
```   422   by (simp add: divide_rat_def)
```
```   423
```
```   424 instance proof
```
```   425   show "inverse 0 = (0::rat)" by (simp add: rat_number_expand)
```
```   426     (simp add: rat_number_collapse)
```
```   427 next
```
```   428   fix q :: rat
```
```   429   assume "q \<noteq> 0"
```
```   430   then show "inverse q * q = 1" by (cases q rule: Rat_cases_nonzero)
```
```   431    (simp_all add: mult_rat  inverse_rat rat_number_expand eq_rat)
```
```   432 next
```
```   433   fix q r :: rat
```
```   434   show "q / r = q * inverse r" by (simp add: divide_rat_def)
```
```   435 qed
```
```   436
```
```   437 end
```
```   438
```
```   439
```
```   440 subsubsection {* Various *}
```
```   441
```
```   442 lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
```
```   443   by (simp add: rat_number_expand)
```
```   444
```
```   445 lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"
```
```   446   by (simp add: Fract_of_int_eq [symmetric])
```
```   447
```
```   448 lemma Fract_number_of_quotient [code_post]:
```
```   449   "Fract (number_of k) (number_of l) = number_of k / number_of l"
```
```   450   unfolding Fract_of_int_quotient number_of_is_id number_of_eq ..
```
```   451
```
```   452 lemma Fract_1_number_of [code_post]:
```
```   453   "Fract 1 (number_of k) = 1 / number_of k"
```
```   454   unfolding Fract_of_int_quotient number_of_eq by simp
```
```   455
```
```   456 subsubsection {* The ordered field of rational numbers *}
```
```   457
```
```   458 instantiation rat :: linorder
```
```   459 begin
```
```   460
```
```   461 definition
```
```   462   le_rat_def [code del]:
```
```   463    "q \<le> r \<longleftrightarrow> contents (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
```
```   464       {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})"
```
```   465
```
```   466 lemma le_rat [simp]:
```
```   467   assumes "b \<noteq> 0" and "d \<noteq> 0"
```
```   468   shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   469 proof -
```
```   470   have "(\<lambda>x y. {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})
```
```   471     respects2 ratrel"
```
```   472   proof (clarsimp simp add: congruent2_def)
```
```   473     fix a b a' b' c d c' d'::int
```
```   474     assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
```
```   475     assume eq1: "a * b' = a' * b"
```
```   476     assume eq2: "c * d' = c' * d"
```
```   477
```
```   478     let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
```
```   479     {
```
```   480       fix a b c d x :: int assume x: "x \<noteq> 0"
```
```   481       have "?le a b c d = ?le (a * x) (b * x) c d"
```
```   482       proof -
```
```   483         from x have "0 < x * x" by (auto simp add: zero_less_mult_iff)
```
```   484         hence "?le a b c d =
```
```   485             ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
```
```   486           by (simp add: mult_le_cancel_right)
```
```   487         also have "... = ?le (a * x) (b * x) c d"
```
```   488           by (simp add: mult_ac)
```
```   489         finally show ?thesis .
```
```   490       qed
```
```   491     } note le_factor = this
```
```   492
```
```   493     let ?D = "b * d" and ?D' = "b' * d'"
```
```   494     from neq have D: "?D \<noteq> 0" by simp
```
```   495     from neq have "?D' \<noteq> 0" by simp
```
```   496     hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"
```
```   497       by (rule le_factor)
```
```   498     also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')"
```
```   499       by (simp add: mult_ac)
```
```   500     also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
```
```   501       by (simp only: eq1 eq2)
```
```   502     also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
```
```   503       by (simp add: mult_ac)
```
```   504     also from D have "... = ?le a' b' c' d'"
```
```   505       by (rule le_factor [symmetric])
```
```   506     finally show "?le a b c d = ?le a' b' c' d'" .
```
```   507   qed
```
```   508   with assms show ?thesis by (simp add: Fract_def le_rat_def UN_ratrel2)
```
```   509 qed
```
```   510
```
```   511 definition
```
```   512   less_rat_def [code del]: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
```
```   513
```
```   514 lemma less_rat [simp]:
```
```   515   assumes "b \<noteq> 0" and "d \<noteq> 0"
```
```   516   shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
```
```   517   using assms by (simp add: less_rat_def eq_rat order_less_le)
```
```   518
```
```   519 instance proof
```
```   520   fix q r s :: rat
```
```   521   {
```
```   522     assume "q \<le> r" and "r \<le> s"
```
```   523     show "q \<le> s"
```
```   524     proof (insert prems, induct q, induct r, induct s)
```
```   525       fix a b c d e f :: int
```
```   526       assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
```
```   527       assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"
```
```   528       show "Fract a b \<le> Fract e f"
```
```   529       proof -
```
```   530         from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
```
```   531           by (auto simp add: zero_less_mult_iff linorder_neq_iff)
```
```   532         have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
```
```   533         proof -
```
```   534           from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   535             by simp
```
```   536           with ff show ?thesis by (simp add: mult_le_cancel_right)
```
```   537         qed
```
```   538         also have "... = (c * f) * (d * f) * (b * b)" by algebra
```
```   539         also have "... \<le> (e * d) * (d * f) * (b * b)"
```
```   540         proof -
```
```   541           from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
```
```   542             by simp
```
```   543           with bb show ?thesis by (simp add: mult_le_cancel_right)
```
```   544         qed
```
```   545         finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
```
```   546           by (simp only: mult_ac)
```
```   547         with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
```
```   548           by (simp add: mult_le_cancel_right)
```
```   549         with neq show ?thesis by simp
```
```   550       qed
```
```   551     qed
```
```   552   next
```
```   553     assume "q \<le> r" and "r \<le> q"
```
```   554     show "q = r"
```
```   555     proof (insert prems, induct q, induct r)
```
```   556       fix a b c d :: int
```
```   557       assume neq: "b \<noteq> 0"  "d \<noteq> 0"
```
```   558       assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"
```
```   559       show "Fract a b = Fract c d"
```
```   560       proof -
```
```   561         from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   562           by simp
```
```   563         also have "... \<le> (a * d) * (b * d)"
```
```   564         proof -
```
```   565           from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
```
```   566             by simp
```
```   567           thus ?thesis by (simp only: mult_ac)
```
```   568         qed
```
```   569         finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
```
```   570         moreover from neq have "b * d \<noteq> 0" by simp
```
```   571         ultimately have "a * d = c * b" by simp
```
```   572         with neq show ?thesis by (simp add: eq_rat)
```
```   573       qed
```
```   574     qed
```
```   575   next
```
```   576     show "q \<le> q"
```
```   577       by (induct q) simp
```
```   578     show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)"
```
```   579       by (induct q, induct r) (auto simp add: le_less mult_commute)
```
```   580     show "q \<le> r \<or> r \<le> q"
```
```   581       by (induct q, induct r)
```
```   582          (simp add: mult_commute, rule linorder_linear)
```
```   583   }
```
```   584 qed
```
```   585
```
```   586 end
```
```   587
```
```   588 instantiation rat :: "{distrib_lattice, abs_if, sgn_if}"
```
```   589 begin
```
```   590
```
```   591 definition
```
```   592   abs_rat_def [code del]: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))"
```
```   593
```
```   594 lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
```
```   595   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)
```
```   596
```
```   597 definition
```
```   598   sgn_rat_def [code del]: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
```
```   599
```
```   600 lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)"
```
```   601   unfolding Fract_of_int_eq
```
```   602   by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat)
```
```   603     (auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)
```
```   604
```
```   605 definition
```
```   606   "(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min"
```
```   607
```
```   608 definition
```
```   609   "(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max"
```
```   610
```
```   611 instance by intro_classes
```
```   612   (auto simp add: abs_rat_def sgn_rat_def min_max.sup_inf_distrib1 inf_rat_def sup_rat_def)
```
```   613
```
```   614 end
```
```   615
```
```   616 instance rat :: linordered_field
```
```   617 proof
```
```   618   fix q r s :: rat
```
```   619   show "q \<le> r ==> s + q \<le> s + r"
```
```   620   proof (induct q, induct r, induct s)
```
```   621     fix a b c d e f :: int
```
```   622     assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
```
```   623     assume le: "Fract a b \<le> Fract c d"
```
```   624     show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
```
```   625     proof -
```
```   626       let ?F = "f * f" from neq have F: "0 < ?F"
```
```   627         by (auto simp add: zero_less_mult_iff)
```
```   628       from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   629         by simp
```
```   630       with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
```
```   631         by (simp add: mult_le_cancel_right)
```
```   632       with neq show ?thesis by (simp add: mult_ac int_distrib)
```
```   633     qed
```
```   634   qed
```
```   635   show "q < r ==> 0 < s ==> s * q < s * r"
```
```   636   proof (induct q, induct r, induct s)
```
```   637     fix a b c d e f :: int
```
```   638     assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
```
```   639     assume le: "Fract a b < Fract c d"
```
```   640     assume gt: "0 < Fract e f"
```
```   641     show "Fract e f * Fract a b < Fract e f * Fract c d"
```
```   642     proof -
```
```   643       let ?E = "e * f" and ?F = "f * f"
```
```   644       from neq gt have "0 < ?E"
```
```   645         by (auto simp add: Zero_rat_def order_less_le eq_rat)
```
```   646       moreover from neq have "0 < ?F"
```
```   647         by (auto simp add: zero_less_mult_iff)
```
```   648       moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
```
```   649         by simp
```
```   650       ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
```
```   651         by (simp add: mult_less_cancel_right)
```
```   652       with neq show ?thesis
```
```   653         by (simp add: mult_ac)
```
```   654     qed
```
```   655   qed
```
```   656 qed auto
```
```   657
```
```   658 lemma Rat_induct_pos [case_names Fract, induct type: rat]:
```
```   659   assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
```
```   660   shows "P q"
```
```   661 proof (cases q)
```
```   662   have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)"
```
```   663   proof -
```
```   664     fix a::int and b::int
```
```   665     assume b: "b < 0"
```
```   666     hence "0 < -b" by simp
```
```   667     hence "P (Fract (-a) (-b))" by (rule step)
```
```   668     thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
```
```   669   qed
```
```   670   case (Fract a b)
```
```   671   thus "P q" by (force simp add: linorder_neq_iff step step')
```
```   672 qed
```
```   673
```
```   674 lemma zero_less_Fract_iff:
```
```   675   "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a"
```
```   676   by (simp add: Zero_rat_def zero_less_mult_iff)
```
```   677
```
```   678 lemma Fract_less_zero_iff:
```
```   679   "0 < b \<Longrightarrow> Fract a b < 0 \<longleftrightarrow> a < 0"
```
```   680   by (simp add: Zero_rat_def mult_less_0_iff)
```
```   681
```
```   682 lemma zero_le_Fract_iff:
```
```   683   "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a"
```
```   684   by (simp add: Zero_rat_def zero_le_mult_iff)
```
```   685
```
```   686 lemma Fract_le_zero_iff:
```
```   687   "0 < b \<Longrightarrow> Fract a b \<le> 0 \<longleftrightarrow> a \<le> 0"
```
```   688   by (simp add: Zero_rat_def mult_le_0_iff)
```
```   689
```
```   690 lemma one_less_Fract_iff:
```
```   691   "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a"
```
```   692   by (simp add: One_rat_def mult_less_cancel_right_disj)
```
```   693
```
```   694 lemma Fract_less_one_iff:
```
```   695   "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b"
```
```   696   by (simp add: One_rat_def mult_less_cancel_right_disj)
```
```   697
```
```   698 lemma one_le_Fract_iff:
```
```   699   "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a"
```
```   700   by (simp add: One_rat_def mult_le_cancel_right)
```
```   701
```
```   702 lemma Fract_le_one_iff:
```
```   703   "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b"
```
```   704   by (simp add: One_rat_def mult_le_cancel_right)
```
```   705
```
```   706
```
```   707 subsubsection {* Rationals are an Archimedean field *}
```
```   708
```
```   709 lemma rat_floor_lemma:
```
```   710   assumes "0 < b"
```
```   711   shows "of_int (a div b) \<le> Fract a b \<and> Fract a b < of_int (a div b + 1)"
```
```   712 proof -
```
```   713   have "Fract a b = of_int (a div b) + Fract (a mod b) b"
```
```   714     using `0 < b` by (simp add: of_int_rat)
```
```   715   moreover have "0 \<le> Fract (a mod b) b \<and> Fract (a mod b) b < 1"
```
```   716     using `0 < b` by (simp add: zero_le_Fract_iff Fract_less_one_iff)
```
```   717   ultimately show ?thesis by simp
```
```   718 qed
```
```   719
```
```   720 instance rat :: archimedean_field
```
```   721 proof
```
```   722   fix r :: rat
```
```   723   show "\<exists>z. r \<le> of_int z"
```
```   724   proof (induct r)
```
```   725     case (Fract a b)
```
```   726     then have "Fract a b \<le> of_int (a div b + 1)"
```
```   727       using rat_floor_lemma [of b a] by simp
```
```   728     then show "\<exists>z. Fract a b \<le> of_int z" ..
```
```   729   qed
```
```   730 qed
```
```   731
```
```   732 lemma floor_Fract:
```
```   733   assumes "0 < b" shows "floor (Fract a b) = a div b"
```
```   734   using rat_floor_lemma [OF `0 < b`, of a]
```
```   735   by (simp add: floor_unique)
```
```   736
```
```   737
```
```   738 subsection {* Linear arithmetic setup *}
```
```   739
```
```   740 declaration {*
```
```   741   K (Lin_Arith.add_inj_thms [@{thm of_nat_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
```
```   742     (* not needed because x < (y::nat) can be rewritten as Suc x <= y: of_nat_less_iff RS iffD2 *)
```
```   743   #> Lin_Arith.add_inj_thms [@{thm of_int_le_iff} RS iffD2, @{thm of_int_eq_iff} RS iffD2]
```
```   744     (* not needed because x < (y::int) can be rewritten as x + 1 <= y: of_int_less_iff RS iffD2 *)
```
```   745   #> Lin_Arith.add_simps [@{thm neg_less_iff_less},
```
```   746       @{thm True_implies_equals},
```
```   747       read_instantiate @{context} [(("a", 0), "(number_of ?v)")] @{thm right_distrib},
```
```   748       @{thm divide_1}, @{thm divide_zero_left},
```
```   749       @{thm times_divide_eq_right}, @{thm times_divide_eq_left},
```
```   750       @{thm minus_divide_left} RS sym, @{thm minus_divide_right} RS sym,
```
```   751       @{thm of_int_minus}, @{thm of_int_diff},
```
```   752       @{thm of_int_of_nat_eq}]
```
```   753   #> Lin_Arith.add_simprocs Numeral_Simprocs.field_cancel_numeral_factors
```
```   754   #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat => rat"})
```
```   755   #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int => rat"}))
```
```   756 *}
```
```   757
```
```   758
```
```   759 subsection {* Embedding from Rationals to other Fields *}
```
```   760
```
```   761 class field_char_0 = field + ring_char_0
```
```   762
```
```   763 subclass (in linordered_field) field_char_0 ..
```
```   764
```
```   765 context field_char_0
```
```   766 begin
```
```   767
```
```   768 definition of_rat :: "rat \<Rightarrow> 'a" where
```
```   769   [code del]: "of_rat q = contents (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})"
```
```   770
```
```   771 end
```
```   772
```
```   773 lemma of_rat_congruent:
```
```   774   "(\<lambda>(a, b). {of_int a / of_int b :: 'a::field_char_0}) respects ratrel"
```
```   775 apply (rule congruent.intro)
```
```   776 apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
```
```   777 apply (simp only: of_int_mult [symmetric])
```
```   778 done
```
```   779
```
```   780 lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
```
```   781   unfolding Fract_def of_rat_def by (simp add: UN_ratrel of_rat_congruent)
```
```   782
```
```   783 lemma of_rat_0 [simp]: "of_rat 0 = 0"
```
```   784 by (simp add: Zero_rat_def of_rat_rat)
```
```   785
```
```   786 lemma of_rat_1 [simp]: "of_rat 1 = 1"
```
```   787 by (simp add: One_rat_def of_rat_rat)
```
```   788
```
```   789 lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
```
```   790 by (induct a, induct b, simp add: of_rat_rat add_frac_eq)
```
```   791
```
```   792 lemma of_rat_minus: "of_rat (- a) = - of_rat a"
```
```   793 by (induct a, simp add: of_rat_rat)
```
```   794
```
```   795 lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
```
```   796 by (simp only: diff_minus of_rat_add of_rat_minus)
```
```   797
```
```   798 lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
```
```   799 apply (induct a, induct b, simp add: of_rat_rat)
```
```   800 apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
```
```   801 done
```
```   802
```
```   803 lemma nonzero_of_rat_inverse:
```
```   804   "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
```
```   805 apply (rule inverse_unique [symmetric])
```
```   806 apply (simp add: of_rat_mult [symmetric])
```
```   807 done
```
```   808
```
```   809 lemma of_rat_inverse:
```
```   810   "(of_rat (inverse a)::'a::{field_char_0,division_by_zero}) =
```
```   811    inverse (of_rat a)"
```
```   812 by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
```
```   813
```
```   814 lemma nonzero_of_rat_divide:
```
```   815   "b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
```
```   816 by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
```
```   817
```
```   818 lemma of_rat_divide:
```
```   819   "(of_rat (a / b)::'a::{field_char_0,division_by_zero})
```
```   820    = of_rat a / of_rat b"
```
```   821 by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)
```
```   822
```
```   823 lemma of_rat_power:
```
```   824   "(of_rat (a ^ n)::'a::field_char_0) = of_rat a ^ n"
```
```   825 by (induct n) (simp_all add: of_rat_mult)
```
```   826
```
```   827 lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
```
```   828 apply (induct a, induct b)
```
```   829 apply (simp add: of_rat_rat eq_rat)
```
```   830 apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
```
```   831 apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
```
```   832 done
```
```   833
```
```   834 lemma of_rat_less:
```
```   835   "(of_rat r :: 'a::linordered_field) < of_rat s \<longleftrightarrow> r < s"
```
```   836 proof (induct r, induct s)
```
```   837   fix a b c d :: int
```
```   838   assume not_zero: "b > 0" "d > 0"
```
```   839   then have "b * d > 0" by (rule mult_pos_pos)
```
```   840   have of_int_divide_less_eq:
```
```   841     "(of_int a :: 'a) / of_int b < of_int c / of_int d
```
```   842       \<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b"
```
```   843     using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
```
```   844   show "(of_rat (Fract a b) :: 'a::linordered_field) < of_rat (Fract c d)
```
```   845     \<longleftrightarrow> Fract a b < Fract c d"
```
```   846     using not_zero `b * d > 0`
```
```   847     by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult)
```
```   848 qed
```
```   849
```
```   850 lemma of_rat_less_eq:
```
```   851   "(of_rat r :: 'a::linordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s"
```
```   852   unfolding le_less by (auto simp add: of_rat_less)
```
```   853
```
```   854 lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
```
```   855
```
```   856 lemma of_rat_eq_id [simp]: "of_rat = id"
```
```   857 proof
```
```   858   fix a
```
```   859   show "of_rat a = id a"
```
```   860   by (induct a)
```
```   861      (simp add: of_rat_rat Fract_of_int_eq [symmetric])
```
```   862 qed
```
```   863
```
```   864 text{*Collapse nested embeddings*}
```
```   865 lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
```
```   866 by (induct n) (simp_all add: of_rat_add)
```
```   867
```
```   868 lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
```
```   869 by (cases z rule: int_diff_cases) (simp add: of_rat_diff)
```
```   870
```
```   871 lemma of_rat_number_of_eq [simp]:
```
```   872   "of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})"
```
```   873 by (simp add: number_of_eq)
```
```   874
```
```   875 lemmas zero_rat = Zero_rat_def
```
```   876 lemmas one_rat = One_rat_def
```
```   877
```
```   878 abbreviation
```
```   879   rat_of_nat :: "nat \<Rightarrow> rat"
```
```   880 where
```
```   881   "rat_of_nat \<equiv> of_nat"
```
```   882
```
```   883 abbreviation
```
```   884   rat_of_int :: "int \<Rightarrow> rat"
```
```   885 where
```
```   886   "rat_of_int \<equiv> of_int"
```
```   887
```
```   888 subsection {* The Set of Rational Numbers *}
```
```   889
```
```   890 context field_char_0
```
```   891 begin
```
```   892
```
```   893 definition
```
```   894   Rats  :: "'a set" where
```
```   895   [code del]: "Rats = range of_rat"
```
```   896
```
```   897 notation (xsymbols)
```
```   898   Rats  ("\<rat>")
```
```   899
```
```   900 end
```
```   901
```
```   902 lemma Rats_of_rat [simp]: "of_rat r \<in> Rats"
```
```   903 by (simp add: Rats_def)
```
```   904
```
```   905 lemma Rats_of_int [simp]: "of_int z \<in> Rats"
```
```   906 by (subst of_rat_of_int_eq [symmetric], rule Rats_of_rat)
```
```   907
```
```   908 lemma Rats_of_nat [simp]: "of_nat n \<in> Rats"
```
```   909 by (subst of_rat_of_nat_eq [symmetric], rule Rats_of_rat)
```
```   910
```
```   911 lemma Rats_number_of [simp]:
```
```   912   "(number_of w::'a::{number_ring,field_char_0}) \<in> Rats"
```
```   913 by (subst of_rat_number_of_eq [symmetric], rule Rats_of_rat)
```
```   914
```
```   915 lemma Rats_0 [simp]: "0 \<in> Rats"
```
```   916 apply (unfold Rats_def)
```
```   917 apply (rule range_eqI)
```
```   918 apply (rule of_rat_0 [symmetric])
```
```   919 done
```
```   920
```
```   921 lemma Rats_1 [simp]: "1 \<in> Rats"
```
```   922 apply (unfold Rats_def)
```
```   923 apply (rule range_eqI)
```
```   924 apply (rule of_rat_1 [symmetric])
```
```   925 done
```
```   926
```
```   927 lemma Rats_add [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a + b \<in> Rats"
```
```   928 apply (auto simp add: Rats_def)
```
```   929 apply (rule range_eqI)
```
```   930 apply (rule of_rat_add [symmetric])
```
```   931 done
```
```   932
```
```   933 lemma Rats_minus [simp]: "a \<in> Rats \<Longrightarrow> - a \<in> Rats"
```
```   934 apply (auto simp add: Rats_def)
```
```   935 apply (rule range_eqI)
```
```   936 apply (rule of_rat_minus [symmetric])
```
```   937 done
```
```   938
```
```   939 lemma Rats_diff [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a - b \<in> Rats"
```
```   940 apply (auto simp add: Rats_def)
```
```   941 apply (rule range_eqI)
```
```   942 apply (rule of_rat_diff [symmetric])
```
```   943 done
```
```   944
```
```   945 lemma Rats_mult [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a * b \<in> Rats"
```
```   946 apply (auto simp add: Rats_def)
```
```   947 apply (rule range_eqI)
```
```   948 apply (rule of_rat_mult [symmetric])
```
```   949 done
```
```   950
```
```   951 lemma nonzero_Rats_inverse:
```
```   952   fixes a :: "'a::field_char_0"
```
```   953   shows "\<lbrakk>a \<in> Rats; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Rats"
```
```   954 apply (auto simp add: Rats_def)
```
```   955 apply (rule range_eqI)
```
```   956 apply (erule nonzero_of_rat_inverse [symmetric])
```
```   957 done
```
```   958
```
```   959 lemma Rats_inverse [simp]:
```
```   960   fixes a :: "'a::{field_char_0,division_by_zero}"
```
```   961   shows "a \<in> Rats \<Longrightarrow> inverse a \<in> Rats"
```
```   962 apply (auto simp add: Rats_def)
```
```   963 apply (rule range_eqI)
```
```   964 apply (rule of_rat_inverse [symmetric])
```
```   965 done
```
```   966
```
```   967 lemma nonzero_Rats_divide:
```
```   968   fixes a b :: "'a::field_char_0"
```
```   969   shows "\<lbrakk>a \<in> Rats; b \<in> Rats; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
```
```   970 apply (auto simp add: Rats_def)
```
```   971 apply (rule range_eqI)
```
```   972 apply (erule nonzero_of_rat_divide [symmetric])
```
```   973 done
```
```   974
```
```   975 lemma Rats_divide [simp]:
```
```   976   fixes a b :: "'a::{field_char_0,division_by_zero}"
```
```   977   shows "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
```
```   978 apply (auto simp add: Rats_def)
```
```   979 apply (rule range_eqI)
```
```   980 apply (rule of_rat_divide [symmetric])
```
```   981 done
```
```   982
```
```   983 lemma Rats_power [simp]:
```
```   984   fixes a :: "'a::field_char_0"
```
```   985   shows "a \<in> Rats \<Longrightarrow> a ^ n \<in> Rats"
```
```   986 apply (auto simp add: Rats_def)
```
```   987 apply (rule range_eqI)
```
```   988 apply (rule of_rat_power [symmetric])
```
```   989 done
```
```   990
```
```   991 lemma Rats_cases [cases set: Rats]:
```
```   992   assumes "q \<in> \<rat>"
```
```   993   obtains (of_rat) r where "q = of_rat r"
```
```   994   unfolding Rats_def
```
```   995 proof -
```
```   996   from `q \<in> \<rat>` have "q \<in> range of_rat" unfolding Rats_def .
```
```   997   then obtain r where "q = of_rat r" ..
```
```   998   then show thesis ..
```
```   999 qed
```
```  1000
```
```  1001 lemma Rats_induct [case_names of_rat, induct set: Rats]:
```
```  1002   "q \<in> \<rat> \<Longrightarrow> (\<And>r. P (of_rat r)) \<Longrightarrow> P q"
```
```  1003   by (rule Rats_cases) auto
```
```  1004
```
```  1005
```
```  1006 subsection {* Implementation of rational numbers as pairs of integers *}
```
```  1007
```
```  1008 definition Fract_norm :: "int \<Rightarrow> int \<Rightarrow> rat" where
```
```  1009   [simp, code del]: "Fract_norm a b = Fract a b"
```
```  1010
```
```  1011 lemma Fract_norm_code [code]: "Fract_norm a b = (if a = 0 \<or> b = 0 then 0 else let c = gcd a b in
```
```  1012   if b > 0 then Fract (a div c) (b div c) else Fract (- (a div c)) (- (b div c)))"
```
```  1013   by (simp add: eq_rat Zero_rat_def Let_def Fract_norm)
```
```  1014
```
```  1015 lemma [code]:
```
```  1016   "of_rat (Fract a b) = (if b \<noteq> 0 then of_int a / of_int b else 0)"
```
```  1017   by (cases "b = 0") (simp_all add: rat_number_collapse of_rat_rat)
```
```  1018
```
```  1019 instantiation rat :: eq
```
```  1020 begin
```
```  1021
```
```  1022 definition [code del]: "eq_class.eq (a\<Colon>rat) b \<longleftrightarrow> a - b = 0"
```
```  1023
```
```  1024 instance by default (simp add: eq_rat_def)
```
```  1025
```
```  1026 lemma rat_eq_code [code]:
```
```  1027   "eq_class.eq (Fract a b) (Fract c d) \<longleftrightarrow> (if b = 0
```
```  1028        then c = 0 \<or> d = 0
```
```  1029      else if d = 0
```
```  1030        then a = 0 \<or> b = 0
```
```  1031      else a * d = b * c)"
```
```  1032   by (auto simp add: eq eq_rat)
```
```  1033
```
```  1034 lemma rat_eq_refl [code nbe]:
```
```  1035   "eq_class.eq (r::rat) r \<longleftrightarrow> True"
```
```  1036   by (rule HOL.eq_refl)
```
```  1037
```
```  1038 end
```
```  1039
```
```  1040 lemma le_rat':
```
```  1041   assumes "b \<noteq> 0"
```
```  1042     and "d \<noteq> 0"
```
```  1043   shows "Fract a b \<le> Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d"
```
```  1044 proof -
```
```  1045   have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp
```
```  1046   have "a * d * (b * d) \<le> c * b * (b * d) \<longleftrightarrow> a * d * (sgn b * sgn d) \<le> c * b * (sgn b * sgn d)"
```
```  1047   proof (cases "b * d > 0")
```
```  1048     case True
```
```  1049     moreover from True have "sgn b * sgn d = 1"
```
```  1050       by (simp add: sgn_times [symmetric] sgn_1_pos)
```
```  1051     ultimately show ?thesis by (simp add: mult_le_cancel_right)
```
```  1052   next
```
```  1053     case False with assms have "b * d < 0" by (simp add: less_le)
```
```  1054     moreover from this have "sgn b * sgn d = - 1"
```
```  1055       by (simp only: sgn_times [symmetric] sgn_1_neg)
```
```  1056     ultimately show ?thesis by (simp add: mult_le_cancel_right)
```
```  1057   qed
```
```  1058   also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d"
```
```  1059     by (simp add: abs_sgn mult_ac)
```
```  1060   finally show ?thesis using assms by simp
```
```  1061 qed
```
```  1062
```
```  1063 lemma less_rat':
```
```  1064   assumes "b \<noteq> 0"
```
```  1065     and "d \<noteq> 0"
```
```  1066   shows "Fract a b < Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d"
```
```  1067 proof -
```
```  1068   have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp
```
```  1069   have "a * d * (b * d) < c * b * (b * d) \<longleftrightarrow> a * d * (sgn b * sgn d) < c * b * (sgn b * sgn d)"
```
```  1070   proof (cases "b * d > 0")
```
```  1071     case True
```
```  1072     moreover from True have "sgn b * sgn d = 1"
```
```  1073       by (simp add: sgn_times [symmetric] sgn_1_pos)
```
```  1074     ultimately show ?thesis by (simp add: mult_less_cancel_right)
```
```  1075   next
```
```  1076     case False with assms have "b * d < 0" by (simp add: less_le)
```
```  1077     moreover from this have "sgn b * sgn d = - 1"
```
```  1078       by (simp only: sgn_times [symmetric] sgn_1_neg)
```
```  1079     ultimately show ?thesis by (simp add: mult_less_cancel_right)
```
```  1080   qed
```
```  1081   also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d"
```
```  1082     by (simp add: abs_sgn mult_ac)
```
```  1083   finally show ?thesis using assms by simp
```
```  1084 qed
```
```  1085
```
```  1086 lemma rat_le_eq_code [code]:
```
```  1087   "Fract a b < Fract c d \<longleftrightarrow> (if b = 0
```
```  1088        then sgn c * sgn d > 0
```
```  1089      else if d = 0
```
```  1090        then sgn a * sgn b < 0
```
```  1091      else a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d)"
```
```  1092   by (auto simp add: sgn_times mult_less_0_iff zero_less_mult_iff less_rat' eq_rat simp del: less_rat)
```
```  1093
```
```  1094 lemma rat_less_eq_code [code]:
```
```  1095   "Fract a b \<le> Fract c d \<longleftrightarrow> (if b = 0
```
```  1096        then sgn c * sgn d \<ge> 0
```
```  1097      else if d = 0
```
```  1098        then sgn a * sgn b \<le> 0
```
```  1099      else a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d)"
```
```  1100   by (auto simp add: sgn_times mult_le_0_iff zero_le_mult_iff le_rat' eq_rat simp del: le_rat)
```
```  1101     (auto simp add: le_less not_less sgn_0_0)
```
```  1102
```
```  1103
```
```  1104 lemma rat_plus_code [code]:
```
```  1105   "Fract a b + Fract c d = (if b = 0
```
```  1106      then Fract c d
```
```  1107    else if d = 0
```
```  1108      then Fract a b
```
```  1109    else Fract_norm (a * d + c * b) (b * d))"
```
```  1110   by (simp add: eq_rat, simp add: Zero_rat_def)
```
```  1111
```
```  1112 lemma rat_times_code [code]:
```
```  1113   "Fract a b * Fract c d = Fract_norm (a * c) (b * d)"
```
```  1114   by simp
```
```  1115
```
```  1116 lemma rat_minus_code [code]:
```
```  1117   "Fract a b - Fract c d = (if b = 0
```
```  1118      then Fract (- c) d
```
```  1119    else if d = 0
```
```  1120      then Fract a b
```
```  1121    else Fract_norm (a * d - c * b) (b * d))"
```
```  1122   by (simp add: eq_rat, simp add: Zero_rat_def)
```
```  1123
```
```  1124 lemma rat_inverse_code [code]:
```
```  1125   "inverse (Fract a b) = (if b = 0 then Fract 1 0
```
```  1126     else if a < 0 then Fract (- b) (- a)
```
```  1127     else Fract b a)"
```
```  1128   by (simp add: eq_rat)
```
```  1129
```
```  1130 lemma rat_divide_code [code]:
```
```  1131   "Fract a b / Fract c d = Fract_norm (a * d) (b * c)"
```
```  1132   by simp
```
```  1133
```
```  1134 definition (in term_syntax)
```
```  1135   valterm_fract :: "int \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> int \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> rat \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
```
```  1136   [code_unfold]: "valterm_fract k l = Code_Evaluation.valtermify Fract {\<cdot>} k {\<cdot>} l"
```
```  1137
```
```  1138 notation fcomp (infixl "o>" 60)
```
```  1139 notation scomp (infixl "o\<rightarrow>" 60)
```
```  1140
```
```  1141 instantiation rat :: random
```
```  1142 begin
```
```  1143
```
```  1144 definition
```
```  1145   "Quickcheck.random i = Quickcheck.random i o\<rightarrow> (\<lambda>num. Random.range i o\<rightarrow> (\<lambda>denom. Pair (
```
```  1146      let j = Code_Numeral.int_of (denom + 1)
```
```  1147      in valterm_fract num (j, \<lambda>u. Code_Evaluation.term_of j))))"
```
```  1148
```
```  1149 instance ..
```
```  1150
```
```  1151 end
```
```  1152
```
```  1153 no_notation fcomp (infixl "o>" 60)
```
```  1154 no_notation scomp (infixl "o\<rightarrow>" 60)
```
```  1155
```
```  1156 hide (open) const Fract_norm
```
```  1157
```
```  1158 text {* Setup for SML code generator *}
```
```  1159
```
```  1160 types_code
```
```  1161   rat ("(int */ int)")
```
```  1162 attach (term_of) {*
```
```  1163 fun term_of_rat (p, q) =
```
```  1164   let
```
```  1165     val rT = Type ("Rational.rat", [])
```
```  1166   in
```
```  1167     if q = 1 orelse p = 0 then HOLogic.mk_number rT p
```
```  1168     else @{term "op / \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat"} \$
```
```  1169       HOLogic.mk_number rT p \$ HOLogic.mk_number rT q
```
```  1170   end;
```
```  1171 *}
```
```  1172 attach (test) {*
```
```  1173 fun gen_rat i =
```
```  1174   let
```
```  1175     val p = random_range 0 i;
```
```  1176     val q = random_range 1 (i + 1);
```
```  1177     val g = Integer.gcd p q;
```
```  1178     val p' = p div g;
```
```  1179     val q' = q div g;
```
```  1180     val r = (if one_of [true, false] then p' else ~ p',
```
```  1181       if p' = 0 then 1 else q')
```
```  1182   in
```
```  1183     (r, fn () => term_of_rat r)
```
```  1184   end;
```
```  1185 *}
```
```  1186
```
```  1187 consts_code
```
```  1188   Fract ("(_,/ _)")
```
```  1189
```
```  1190 consts_code
```
```  1191   "of_int :: int \<Rightarrow> rat" ("\<module>rat'_of'_int")
```
```  1192 attach {*
```
```  1193 fun rat_of_int i = (i, 1);
```
```  1194 *}
```
```  1195
```
```  1196 setup {*
```
```  1197   Nitpick.register_frac_type @{type_name rat}
```
```  1198    [(@{const_name zero_rat_inst.zero_rat}, @{const_name Nitpick.zero_frac}),
```
```  1199     (@{const_name one_rat_inst.one_rat}, @{const_name Nitpick.one_frac}),
```
```  1200     (@{const_name plus_rat_inst.plus_rat}, @{const_name Nitpick.plus_frac}),
```
```  1201     (@{const_name times_rat_inst.times_rat}, @{const_name Nitpick.times_frac}),
```
```  1202     (@{const_name uminus_rat_inst.uminus_rat}, @{const_name Nitpick.uminus_frac}),
```
```  1203     (@{const_name number_rat_inst.number_of_rat}, @{const_name Nitpick.number_of_frac}),
```
```  1204     (@{const_name inverse_rat_inst.inverse_rat}, @{const_name Nitpick.inverse_frac}),
```
```  1205     (@{const_name ord_rat_inst.less_eq_rat}, @{const_name Nitpick.less_eq_frac}),
```
```  1206     (@{const_name field_char_0_class.of_rat}, @{const_name Nitpick.of_frac}),
```
```  1207     (@{const_name field_char_0_class.Rats}, @{const_name UNIV})]
```
```  1208 *}
```
```  1209
```
```  1210 lemmas [nitpick_def] = inverse_rat_inst.inverse_rat
```
```  1211   number_rat_inst.number_of_rat one_rat_inst.one_rat ord_rat_inst.less_eq_rat
```
```  1212   plus_rat_inst.plus_rat times_rat_inst.times_rat uminus_rat_inst.uminus_rat
```
```  1213   zero_rat_inst.zero_rat
```
```  1214
```
```  1215 end
```