src/HOL/Rational.thy
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```     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
```
```   247 subsubsection {* The field of rational numbers *}
```
```   248
```
```   249 instantiation rat :: "{field, division_by_zero}"
```
```   250 begin
```
```   251
```
```   252 definition
```
```   253   inverse_rat_def [code del]:
```
```   254   "inverse q = Abs_Rat (\<Union>x \<in> Rep_Rat q.
```
```   255      ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)})"
```
```   256
```
```   257 lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"
```
```   258 proof -
```
```   259   have "(\<lambda>x. ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)}) respects ratrel"
```
```   260     by (auto simp add: congruent_def mult_commute)
```
```   261   then show ?thesis by (simp add: Fract_def inverse_rat_def UN_ratrel)
```
```   262 qed
```
```   263
```
```   264 definition
```
```   265   divide_rat_def [code del]: "q / r = q * inverse (r::rat)"
```
```   266
```
```   267 lemma divide_rat [simp]: "Fract a b / Fract c d = Fract (a * d) (b * c)"
```
```   268   by (simp add: divide_rat_def)
```
```   269
```
```   270 instance proof
```
```   271   show "inverse 0 = (0::rat)" by (simp add: rat_number_expand)
```
```   272     (simp add: rat_number_collapse)
```
```   273 next
```
```   274   fix q :: rat
```
```   275   assume "q \<noteq> 0"
```
```   276   then show "inverse q * q = 1" by (cases q rule: Rat_cases_nonzero)
```
```   277    (simp_all add: mult_rat  inverse_rat rat_number_expand eq_rat)
```
```   278 next
```
```   279   fix q r :: rat
```
```   280   show "q / r = q * inverse r" by (simp add: divide_rat_def)
```
```   281 qed
```
```   282
```
```   283 end
```
```   284
```
```   285
```
```   286 subsubsection {* Various *}
```
```   287
```
```   288 lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
```
```   289   by (simp add: rat_number_expand)
```
```   290
```
```   291 lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"
```
```   292   by (simp add: Fract_of_int_eq [symmetric])
```
```   293
```
```   294 lemma Fract_number_of_quotient [code post]:
```
```   295   "Fract (number_of k) (number_of l) = number_of k / number_of l"
```
```   296   unfolding Fract_of_int_quotient number_of_is_id number_of_eq ..
```
```   297
```
```   298 lemma Fract_1_number_of [code post]:
```
```   299   "Fract 1 (number_of k) = 1 / number_of k"
```
```   300   unfolding Fract_of_int_quotient number_of_eq by simp
```
```   301
```
```   302 subsubsection {* The ordered field of rational numbers *}
```
```   303
```
```   304 instantiation rat :: linorder
```
```   305 begin
```
```   306
```
```   307 definition
```
```   308   le_rat_def [code del]:
```
```   309    "q \<le> r \<longleftrightarrow> contents (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
```
```   310       {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})"
```
```   311
```
```   312 lemma le_rat [simp]:
```
```   313   assumes "b \<noteq> 0" and "d \<noteq> 0"
```
```   314   shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   315 proof -
```
```   316   have "(\<lambda>x y. {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})
```
```   317     respects2 ratrel"
```
```   318   proof (clarsimp simp add: congruent2_def)
```
```   319     fix a b a' b' c d c' d'::int
```
```   320     assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
```
```   321     assume eq1: "a * b' = a' * b"
```
```   322     assume eq2: "c * d' = c' * d"
```
```   323
```
```   324     let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
```
```   325     {
```
```   326       fix a b c d x :: int assume x: "x \<noteq> 0"
```
```   327       have "?le a b c d = ?le (a * x) (b * x) c d"
```
```   328       proof -
```
```   329         from x have "0 < x * x" by (auto simp add: zero_less_mult_iff)
```
```   330         hence "?le a b c d =
```
```   331             ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
```
```   332           by (simp add: mult_le_cancel_right)
```
```   333         also have "... = ?le (a * x) (b * x) c d"
```
```   334           by (simp add: mult_ac)
```
```   335         finally show ?thesis .
```
```   336       qed
```
```   337     } note le_factor = this
```
```   338
```
```   339     let ?D = "b * d" and ?D' = "b' * d'"
```
```   340     from neq have D: "?D \<noteq> 0" by simp
```
```   341     from neq have "?D' \<noteq> 0" by simp
```
```   342     hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"
```
```   343       by (rule le_factor)
```
```   344     also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')"
```
```   345       by (simp add: mult_ac)
```
```   346     also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
```
```   347       by (simp only: eq1 eq2)
```
```   348     also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
```
```   349       by (simp add: mult_ac)
```
```   350     also from D have "... = ?le a' b' c' d'"
```
```   351       by (rule le_factor [symmetric])
```
```   352     finally show "?le a b c d = ?le a' b' c' d'" .
```
```   353   qed
```
```   354   with assms show ?thesis by (simp add: Fract_def le_rat_def UN_ratrel2)
```
```   355 qed
```
```   356
```
```   357 definition
```
```   358   less_rat_def [code del]: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
```
```   359
```
```   360 lemma less_rat [simp]:
```
```   361   assumes "b \<noteq> 0" and "d \<noteq> 0"
```
```   362   shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
```
```   363   using assms by (simp add: less_rat_def eq_rat order_less_le)
```
```   364
```
```   365 instance proof
```
```   366   fix q r s :: rat
```
```   367   {
```
```   368     assume "q \<le> r" and "r \<le> s"
```
```   369     show "q \<le> s"
```
```   370     proof (insert prems, induct q, induct r, induct s)
```
```   371       fix a b c d e f :: int
```
```   372       assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
```
```   373       assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"
```
```   374       show "Fract a b \<le> Fract e f"
```
```   375       proof -
```
```   376         from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
```
```   377           by (auto simp add: zero_less_mult_iff linorder_neq_iff)
```
```   378         have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
```
```   379         proof -
```
```   380           from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   381             by simp
```
```   382           with ff show ?thesis by (simp add: mult_le_cancel_right)
```
```   383         qed
```
```   384         also have "... = (c * f) * (d * f) * (b * b)" by algebra
```
```   385         also have "... \<le> (e * d) * (d * f) * (b * b)"
```
```   386         proof -
```
```   387           from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
```
```   388             by simp
```
```   389           with bb show ?thesis by (simp add: mult_le_cancel_right)
```
```   390         qed
```
```   391         finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
```
```   392           by (simp only: mult_ac)
```
```   393         with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
```
```   394           by (simp add: mult_le_cancel_right)
```
```   395         with neq show ?thesis by simp
```
```   396       qed
```
```   397     qed
```
```   398   next
```
```   399     assume "q \<le> r" and "r \<le> q"
```
```   400     show "q = r"
```
```   401     proof (insert prems, induct q, induct r)
```
```   402       fix a b c d :: int
```
```   403       assume neq: "b \<noteq> 0"  "d \<noteq> 0"
```
```   404       assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"
```
```   405       show "Fract a b = Fract c d"
```
```   406       proof -
```
```   407         from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   408           by simp
```
```   409         also have "... \<le> (a * d) * (b * d)"
```
```   410         proof -
```
```   411           from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
```
```   412             by simp
```
```   413           thus ?thesis by (simp only: mult_ac)
```
```   414         qed
```
```   415         finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
```
```   416         moreover from neq have "b * d \<noteq> 0" by simp
```
```   417         ultimately have "a * d = c * b" by simp
```
```   418         with neq show ?thesis by (simp add: eq_rat)
```
```   419       qed
```
```   420     qed
```
```   421   next
```
```   422     show "q \<le> q"
```
```   423       by (induct q) simp
```
```   424     show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)"
```
```   425       by (induct q, induct r) (auto simp add: le_less mult_commute)
```
```   426     show "q \<le> r \<or> r \<le> q"
```
```   427       by (induct q, induct r)
```
```   428          (simp add: mult_commute, rule linorder_linear)
```
```   429   }
```
```   430 qed
```
```   431
```
```   432 end
```
```   433
```
```   434 instantiation rat :: "{distrib_lattice, abs_if, sgn_if}"
```
```   435 begin
```
```   436
```
```   437 definition
```
```   438   abs_rat_def [code del]: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))"
```
```   439
```
```   440 lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
```
```   441   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)
```
```   442
```
```   443 definition
```
```   444   sgn_rat_def [code del]: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
```
```   445
```
```   446 lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)"
```
```   447   unfolding Fract_of_int_eq
```
```   448   by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat)
```
```   449     (auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)
```
```   450
```
```   451 definition
```
```   452   "(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min"
```
```   453
```
```   454 definition
```
```   455   "(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max"
```
```   456
```
```   457 instance by intro_classes
```
```   458   (auto simp add: abs_rat_def sgn_rat_def min_max.sup_inf_distrib1 inf_rat_def sup_rat_def)
```
```   459
```
```   460 end
```
```   461
```
```   462 instance rat :: ordered_field
```
```   463 proof
```
```   464   fix q r s :: rat
```
```   465   show "q \<le> r ==> s + q \<le> s + r"
```
```   466   proof (induct q, induct r, induct s)
```
```   467     fix a b c d e f :: int
```
```   468     assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
```
```   469     assume le: "Fract a b \<le> Fract c d"
```
```   470     show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
```
```   471     proof -
```
```   472       let ?F = "f * f" from neq have F: "0 < ?F"
```
```   473         by (auto simp add: zero_less_mult_iff)
```
```   474       from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   475         by simp
```
```   476       with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
```
```   477         by (simp add: mult_le_cancel_right)
```
```   478       with neq show ?thesis by (simp add: mult_ac int_distrib)
```
```   479     qed
```
```   480   qed
```
```   481   show "q < r ==> 0 < s ==> s * q < s * r"
```
```   482   proof (induct q, induct r, induct s)
```
```   483     fix a b c d e f :: int
```
```   484     assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
```
```   485     assume le: "Fract a b < Fract c d"
```
```   486     assume gt: "0 < Fract e f"
```
```   487     show "Fract e f * Fract a b < Fract e f * Fract c d"
```
```   488     proof -
```
```   489       let ?E = "e * f" and ?F = "f * f"
```
```   490       from neq gt have "0 < ?E"
```
```   491         by (auto simp add: Zero_rat_def order_less_le eq_rat)
```
```   492       moreover from neq have "0 < ?F"
```
```   493         by (auto simp add: zero_less_mult_iff)
```
```   494       moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
```
```   495         by simp
```
```   496       ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
```
```   497         by (simp add: mult_less_cancel_right)
```
```   498       with neq show ?thesis
```
```   499         by (simp add: mult_ac)
```
```   500     qed
```
```   501   qed
```
```   502 qed auto
```
```   503
```
```   504 lemma Rat_induct_pos [case_names Fract, induct type: rat]:
```
```   505   assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
```
```   506   shows "P q"
```
```   507 proof (cases q)
```
```   508   have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)"
```
```   509   proof -
```
```   510     fix a::int and b::int
```
```   511     assume b: "b < 0"
```
```   512     hence "0 < -b" by simp
```
```   513     hence "P (Fract (-a) (-b))" by (rule step)
```
```   514     thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
```
```   515   qed
```
```   516   case (Fract a b)
```
```   517   thus "P q" by (force simp add: linorder_neq_iff step step')
```
```   518 qed
```
```   519
```
```   520 lemma zero_less_Fract_iff:
```
```   521   "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a"
```
```   522   by (simp add: Zero_rat_def zero_less_mult_iff)
```
```   523
```
```   524 lemma Fract_less_zero_iff:
```
```   525   "0 < b \<Longrightarrow> Fract a b < 0 \<longleftrightarrow> a < 0"
```
```   526   by (simp add: Zero_rat_def mult_less_0_iff)
```
```   527
```
```   528 lemma zero_le_Fract_iff:
```
```   529   "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a"
```
```   530   by (simp add: Zero_rat_def zero_le_mult_iff)
```
```   531
```
```   532 lemma Fract_le_zero_iff:
```
```   533   "0 < b \<Longrightarrow> Fract a b \<le> 0 \<longleftrightarrow> a \<le> 0"
```
```   534   by (simp add: Zero_rat_def mult_le_0_iff)
```
```   535
```
```   536 lemma one_less_Fract_iff:
```
```   537   "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a"
```
```   538   by (simp add: One_rat_def mult_less_cancel_right_disj)
```
```   539
```
```   540 lemma Fract_less_one_iff:
```
```   541   "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b"
```
```   542   by (simp add: One_rat_def mult_less_cancel_right_disj)
```
```   543
```
```   544 lemma one_le_Fract_iff:
```
```   545   "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a"
```
```   546   by (simp add: One_rat_def mult_le_cancel_right)
```
```   547
```
```   548 lemma Fract_le_one_iff:
```
```   549   "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b"
```
```   550   by (simp add: One_rat_def mult_le_cancel_right)
```
```   551
```
```   552
```
```   553 subsubsection {* Rationals are an Archimedean field *}
```
```   554
```
```   555 lemma rat_floor_lemma:
```
```   556   assumes "0 < b"
```
```   557   shows "of_int (a div b) \<le> Fract a b \<and> Fract a b < of_int (a div b + 1)"
```
```   558 proof -
```
```   559   have "Fract a b = of_int (a div b) + Fract (a mod b) b"
```
```   560     using `0 < b` by (simp add: of_int_rat)
```
```   561   moreover have "0 \<le> Fract (a mod b) b \<and> Fract (a mod b) b < 1"
```
```   562     using `0 < b` by (simp add: zero_le_Fract_iff Fract_less_one_iff)
```
```   563   ultimately show ?thesis by simp
```
```   564 qed
```
```   565
```
```   566 instance rat :: archimedean_field
```
```   567 proof
```
```   568   fix r :: rat
```
```   569   show "\<exists>z. r \<le> of_int z"
```
```   570   proof (induct r)
```
```   571     case (Fract a b)
```
```   572     then have "Fract a b \<le> of_int (a div b + 1)"
```
```   573       using rat_floor_lemma [of b a] by simp
```
```   574     then show "\<exists>z. Fract a b \<le> of_int z" ..
```
```   575   qed
```
```   576 qed
```
```   577
```
```   578 lemma floor_Fract:
```
```   579   assumes "0 < b" shows "floor (Fract a b) = a div b"
```
```   580   using rat_floor_lemma [OF `0 < b`, of a]
```
```   581   by (simp add: floor_unique)
```
```   582
```
```   583
```
```   584 subsection {* Linear arithmetic setup *}
```
```   585
```
```   586 declaration {*
```
```   587   K (Lin_Arith.add_inj_thms [@{thm of_nat_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
```
```   588     (* not needed because x < (y::nat) can be rewritten as Suc x <= y: of_nat_less_iff RS iffD2 *)
```
```   589   #> Lin_Arith.add_inj_thms [@{thm of_int_le_iff} RS iffD2, @{thm of_int_eq_iff} RS iffD2]
```
```   590     (* not needed because x < (y::int) can be rewritten as x + 1 <= y: of_int_less_iff RS iffD2 *)
```
```   591   #> Lin_Arith.add_simps [@{thm neg_less_iff_less},
```
```   592       @{thm True_implies_equals},
```
```   593       read_instantiate @{context} [(("a", 0), "(number_of ?v)")] @{thm right_distrib},
```
```   594       @{thm divide_1}, @{thm divide_zero_left},
```
```   595       @{thm times_divide_eq_right}, @{thm times_divide_eq_left},
```
```   596       @{thm minus_divide_left} RS sym, @{thm minus_divide_right} RS sym,
```
```   597       @{thm of_int_minus}, @{thm of_int_diff},
```
```   598       @{thm of_int_of_nat_eq}]
```
```   599   #> Lin_Arith.add_simprocs Numeral_Simprocs.field_cancel_numeral_factors
```
```   600   #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat => rat"})
```
```   601   #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int => rat"}))
```
```   602 *}
```
```   603
```
```   604
```
```   605 subsection {* Embedding from Rationals to other Fields *}
```
```   606
```
```   607 class field_char_0 = field + ring_char_0
```
```   608
```
```   609 subclass (in ordered_field) field_char_0 ..
```
```   610
```
```   611 context field_char_0
```
```   612 begin
```
```   613
```
```   614 definition of_rat :: "rat \<Rightarrow> 'a" where
```
```   615   [code del]: "of_rat q = contents (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})"
```
```   616
```
```   617 end
```
```   618
```
```   619 lemma of_rat_congruent:
```
```   620   "(\<lambda>(a, b). {of_int a / of_int b :: 'a::field_char_0}) respects ratrel"
```
```   621 apply (rule congruent.intro)
```
```   622 apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
```
```   623 apply (simp only: of_int_mult [symmetric])
```
```   624 done
```
```   625
```
```   626 lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
```
```   627   unfolding Fract_def of_rat_def by (simp add: UN_ratrel of_rat_congruent)
```
```   628
```
```   629 lemma of_rat_0 [simp]: "of_rat 0 = 0"
```
```   630 by (simp add: Zero_rat_def of_rat_rat)
```
```   631
```
```   632 lemma of_rat_1 [simp]: "of_rat 1 = 1"
```
```   633 by (simp add: One_rat_def of_rat_rat)
```
```   634
```
```   635 lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
```
```   636 by (induct a, induct b, simp add: of_rat_rat add_frac_eq)
```
```   637
```
```   638 lemma of_rat_minus: "of_rat (- a) = - of_rat a"
```
```   639 by (induct a, simp add: of_rat_rat)
```
```   640
```
```   641 lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
```
```   642 by (simp only: diff_minus of_rat_add of_rat_minus)
```
```   643
```
```   644 lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
```
```   645 apply (induct a, induct b, simp add: of_rat_rat)
```
```   646 apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
```
```   647 done
```
```   648
```
```   649 lemma nonzero_of_rat_inverse:
```
```   650   "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
```
```   651 apply (rule inverse_unique [symmetric])
```
```   652 apply (simp add: of_rat_mult [symmetric])
```
```   653 done
```
```   654
```
```   655 lemma of_rat_inverse:
```
```   656   "(of_rat (inverse a)::'a::{field_char_0,division_by_zero}) =
```
```   657    inverse (of_rat a)"
```
```   658 by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
```
```   659
```
```   660 lemma nonzero_of_rat_divide:
```
```   661   "b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
```
```   662 by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
```
```   663
```
```   664 lemma of_rat_divide:
```
```   665   "(of_rat (a / b)::'a::{field_char_0,division_by_zero})
```
```   666    = of_rat a / of_rat b"
```
```   667 by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)
```
```   668
```
```   669 lemma of_rat_power:
```
```   670   "(of_rat (a ^ n)::'a::field_char_0) = of_rat a ^ n"
```
```   671 by (induct n) (simp_all add: of_rat_mult)
```
```   672
```
```   673 lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
```
```   674 apply (induct a, induct b)
```
```   675 apply (simp add: of_rat_rat eq_rat)
```
```   676 apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
```
```   677 apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
```
```   678 done
```
```   679
```
```   680 lemma of_rat_less:
```
```   681   "(of_rat r :: 'a::ordered_field) < of_rat s \<longleftrightarrow> r < s"
```
```   682 proof (induct r, induct s)
```
```   683   fix a b c d :: int
```
```   684   assume not_zero: "b > 0" "d > 0"
```
```   685   then have "b * d > 0" by (rule mult_pos_pos)
```
```   686   have of_int_divide_less_eq:
```
```   687     "(of_int a :: 'a) / of_int b < of_int c / of_int d
```
```   688       \<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b"
```
```   689     using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
```
```   690   show "(of_rat (Fract a b) :: 'a::ordered_field) < of_rat (Fract c d)
```
```   691     \<longleftrightarrow> Fract a b < Fract c d"
```
```   692     using not_zero `b * d > 0`
```
```   693     by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult)
```
```   694 qed
```
```   695
```
```   696 lemma of_rat_less_eq:
```
```   697   "(of_rat r :: 'a::ordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s"
```
```   698   unfolding le_less by (auto simp add: of_rat_less)
```
```   699
```
```   700 lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
```
```   701
```
```   702 lemma of_rat_eq_id [simp]: "of_rat = id"
```
```   703 proof
```
```   704   fix a
```
```   705   show "of_rat a = id a"
```
```   706   by (induct a)
```
```   707      (simp add: of_rat_rat Fract_of_int_eq [symmetric])
```
```   708 qed
```
```   709
```
```   710 text{*Collapse nested embeddings*}
```
```   711 lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
```
```   712 by (induct n) (simp_all add: of_rat_add)
```
```   713
```
```   714 lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
```
```   715 by (cases z rule: int_diff_cases) (simp add: of_rat_diff)
```
```   716
```
```   717 lemma of_rat_number_of_eq [simp]:
```
```   718   "of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})"
```
```   719 by (simp add: number_of_eq)
```
```   720
```
```   721 lemmas zero_rat = Zero_rat_def
```
```   722 lemmas one_rat = One_rat_def
```
```   723
```
```   724 abbreviation
```
```   725   rat_of_nat :: "nat \<Rightarrow> rat"
```
```   726 where
```
```   727   "rat_of_nat \<equiv> of_nat"
```
```   728
```
```   729 abbreviation
```
```   730   rat_of_int :: "int \<Rightarrow> rat"
```
```   731 where
```
```   732   "rat_of_int \<equiv> of_int"
```
```   733
```
```   734 subsection {* The Set of Rational Numbers *}
```
```   735
```
```   736 context field_char_0
```
```   737 begin
```
```   738
```
```   739 definition
```
```   740   Rats  :: "'a set" where
```
```   741   [code del]: "Rats = range of_rat"
```
```   742
```
```   743 notation (xsymbols)
```
```   744   Rats  ("\<rat>")
```
```   745
```
```   746 end
```
```   747
```
```   748 lemma Rats_of_rat [simp]: "of_rat r \<in> Rats"
```
```   749 by (simp add: Rats_def)
```
```   750
```
```   751 lemma Rats_of_int [simp]: "of_int z \<in> Rats"
```
```   752 by (subst of_rat_of_int_eq [symmetric], rule Rats_of_rat)
```
```   753
```
```   754 lemma Rats_of_nat [simp]: "of_nat n \<in> Rats"
```
```   755 by (subst of_rat_of_nat_eq [symmetric], rule Rats_of_rat)
```
```   756
```
```   757 lemma Rats_number_of [simp]:
```
```   758   "(number_of w::'a::{number_ring,field_char_0}) \<in> Rats"
```
```   759 by (subst of_rat_number_of_eq [symmetric], rule Rats_of_rat)
```
```   760
```
```   761 lemma Rats_0 [simp]: "0 \<in> Rats"
```
```   762 apply (unfold Rats_def)
```
```   763 apply (rule range_eqI)
```
```   764 apply (rule of_rat_0 [symmetric])
```
```   765 done
```
```   766
```
```   767 lemma Rats_1 [simp]: "1 \<in> Rats"
```
```   768 apply (unfold Rats_def)
```
```   769 apply (rule range_eqI)
```
```   770 apply (rule of_rat_1 [symmetric])
```
```   771 done
```
```   772
```
```   773 lemma Rats_add [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a + b \<in> Rats"
```
```   774 apply (auto simp add: Rats_def)
```
```   775 apply (rule range_eqI)
```
```   776 apply (rule of_rat_add [symmetric])
```
```   777 done
```
```   778
```
```   779 lemma Rats_minus [simp]: "a \<in> Rats \<Longrightarrow> - a \<in> Rats"
```
```   780 apply (auto simp add: Rats_def)
```
```   781 apply (rule range_eqI)
```
```   782 apply (rule of_rat_minus [symmetric])
```
```   783 done
```
```   784
```
```   785 lemma Rats_diff [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a - b \<in> Rats"
```
```   786 apply (auto simp add: Rats_def)
```
```   787 apply (rule range_eqI)
```
```   788 apply (rule of_rat_diff [symmetric])
```
```   789 done
```
```   790
```
```   791 lemma Rats_mult [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a * b \<in> Rats"
```
```   792 apply (auto simp add: Rats_def)
```
```   793 apply (rule range_eqI)
```
```   794 apply (rule of_rat_mult [symmetric])
```
```   795 done
```
```   796
```
```   797 lemma nonzero_Rats_inverse:
```
```   798   fixes a :: "'a::field_char_0"
```
```   799   shows "\<lbrakk>a \<in> Rats; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Rats"
```
```   800 apply (auto simp add: Rats_def)
```
```   801 apply (rule range_eqI)
```
```   802 apply (erule nonzero_of_rat_inverse [symmetric])
```
```   803 done
```
```   804
```
```   805 lemma Rats_inverse [simp]:
```
```   806   fixes a :: "'a::{field_char_0,division_by_zero}"
```
```   807   shows "a \<in> Rats \<Longrightarrow> inverse a \<in> Rats"
```
```   808 apply (auto simp add: Rats_def)
```
```   809 apply (rule range_eqI)
```
```   810 apply (rule of_rat_inverse [symmetric])
```
```   811 done
```
```   812
```
```   813 lemma nonzero_Rats_divide:
```
```   814   fixes a b :: "'a::field_char_0"
```
```   815   shows "\<lbrakk>a \<in> Rats; b \<in> Rats; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
```
```   816 apply (auto simp add: Rats_def)
```
```   817 apply (rule range_eqI)
```
```   818 apply (erule nonzero_of_rat_divide [symmetric])
```
```   819 done
```
```   820
```
```   821 lemma Rats_divide [simp]:
```
```   822   fixes a b :: "'a::{field_char_0,division_by_zero}"
```
```   823   shows "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
```
```   824 apply (auto simp add: Rats_def)
```
```   825 apply (rule range_eqI)
```
```   826 apply (rule of_rat_divide [symmetric])
```
```   827 done
```
```   828
```
```   829 lemma Rats_power [simp]:
```
```   830   fixes a :: "'a::field_char_0"
```
```   831   shows "a \<in> Rats \<Longrightarrow> a ^ n \<in> Rats"
```
```   832 apply (auto simp add: Rats_def)
```
```   833 apply (rule range_eqI)
```
```   834 apply (rule of_rat_power [symmetric])
```
```   835 done
```
```   836
```
```   837 lemma Rats_cases [cases set: Rats]:
```
```   838   assumes "q \<in> \<rat>"
```
```   839   obtains (of_rat) r where "q = of_rat r"
```
```   840   unfolding Rats_def
```
```   841 proof -
```
```   842   from `q \<in> \<rat>` have "q \<in> range of_rat" unfolding Rats_def .
```
```   843   then obtain r where "q = of_rat r" ..
```
```   844   then show thesis ..
```
```   845 qed
```
```   846
```
```   847 lemma Rats_induct [case_names of_rat, induct set: Rats]:
```
```   848   "q \<in> \<rat> \<Longrightarrow> (\<And>r. P (of_rat r)) \<Longrightarrow> P q"
```
```   849   by (rule Rats_cases) auto
```
```   850
```
```   851
```
```   852 subsection {* Implementation of rational numbers as pairs of integers *}
```
```   853
```
```   854 lemma Fract_norm: "Fract (a div zgcd a b) (b div zgcd a b) = Fract a b"
```
```   855 proof (cases "a = 0 \<or> b = 0")
```
```   856   case True then show ?thesis by (auto simp add: eq_rat)
```
```   857 next
```
```   858   let ?c = "zgcd a b"
```
```   859   case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
```
```   860   then have "?c \<noteq> 0" by simp
```
```   861   then have "Fract ?c ?c = Fract 1 1" by (simp add: eq_rat)
```
```   862   moreover have "Fract (a div ?c * ?c + a mod ?c) (b div ?c * ?c + b mod ?c) = Fract a b"
```
```   863     by (simp add: semiring_div_class.mod_div_equality)
```
```   864   moreover have "a mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric])
```
```   865   moreover have "b mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric])
```
```   866   ultimately show ?thesis
```
```   867     by (simp add: mult_rat [symmetric])
```
```   868 qed
```
```   869
```
```   870 definition Fract_norm :: "int \<Rightarrow> int \<Rightarrow> rat" where
```
```   871   [simp, code del]: "Fract_norm a b = Fract a b"
```
```   872
```
```   873 lemma Fract_norm_code [code]: "Fract_norm a b = (if a = 0 \<or> b = 0 then 0 else let c = zgcd a b in
```
```   874   if b > 0 then Fract (a div c) (b div c) else Fract (- (a div c)) (- (b div c)))"
```
```   875   by (simp add: eq_rat Zero_rat_def Let_def Fract_norm)
```
```   876
```
```   877 lemma [code]:
```
```   878   "of_rat (Fract a b) = (if b \<noteq> 0 then of_int a / of_int b else 0)"
```
```   879   by (cases "b = 0") (simp_all add: rat_number_collapse of_rat_rat)
```
```   880
```
```   881 instantiation rat :: eq
```
```   882 begin
```
```   883
```
```   884 definition [code del]: "eq_class.eq (a\<Colon>rat) b \<longleftrightarrow> a - b = 0"
```
```   885
```
```   886 instance by default (simp add: eq_rat_def)
```
```   887
```
```   888 lemma rat_eq_code [code]:
```
```   889   "eq_class.eq (Fract a b) (Fract c d) \<longleftrightarrow> (if b = 0
```
```   890        then c = 0 \<or> d = 0
```
```   891      else if d = 0
```
```   892        then a = 0 \<or> b = 0
```
```   893      else a * d = b * c)"
```
```   894   by (auto simp add: eq eq_rat)
```
```   895
```
```   896 lemma rat_eq_refl [code nbe]:
```
```   897   "eq_class.eq (r::rat) r \<longleftrightarrow> True"
```
```   898   by (rule HOL.eq_refl)
```
```   899
```
```   900 end
```
```   901
```
```   902 lemma le_rat':
```
```   903   assumes "b \<noteq> 0"
```
```   904     and "d \<noteq> 0"
```
```   905   shows "Fract a b \<le> Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d"
```
```   906 proof -
```
```   907   have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp
```
```   908   have "a * d * (b * d) \<le> c * b * (b * d) \<longleftrightarrow> a * d * (sgn b * sgn d) \<le> c * b * (sgn b * sgn d)"
```
```   909   proof (cases "b * d > 0")
```
```   910     case True
```
```   911     moreover from True have "sgn b * sgn d = 1"
```
```   912       by (simp add: sgn_times [symmetric] sgn_1_pos)
```
```   913     ultimately show ?thesis by (simp add: mult_le_cancel_right)
```
```   914   next
```
```   915     case False with assms have "b * d < 0" by (simp add: less_le)
```
```   916     moreover from this have "sgn b * sgn d = - 1"
```
```   917       by (simp only: sgn_times [symmetric] sgn_1_neg)
```
```   918     ultimately show ?thesis by (simp add: mult_le_cancel_right)
```
```   919   qed
```
```   920   also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d"
```
```   921     by (simp add: abs_sgn mult_ac)
```
```   922   finally show ?thesis using assms by simp
```
```   923 qed
```
```   924
```
```   925 lemma less_rat':
```
```   926   assumes "b \<noteq> 0"
```
```   927     and "d \<noteq> 0"
```
```   928   shows "Fract a b < Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d"
```
```   929 proof -
```
```   930   have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp
```
```   931   have "a * d * (b * d) < c * b * (b * d) \<longleftrightarrow> a * d * (sgn b * sgn d) < c * b * (sgn b * sgn d)"
```
```   932   proof (cases "b * d > 0")
```
```   933     case True
```
```   934     moreover from True have "sgn b * sgn d = 1"
```
```   935       by (simp add: sgn_times [symmetric] sgn_1_pos)
```
```   936     ultimately show ?thesis by (simp add: mult_less_cancel_right)
```
```   937   next
```
```   938     case False with assms have "b * d < 0" by (simp add: less_le)
```
```   939     moreover from this have "sgn b * sgn d = - 1"
```
```   940       by (simp only: sgn_times [symmetric] sgn_1_neg)
```
```   941     ultimately show ?thesis by (simp add: mult_less_cancel_right)
```
```   942   qed
```
```   943   also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d"
```
```   944     by (simp add: abs_sgn mult_ac)
```
```   945   finally show ?thesis using assms by simp
```
```   946 qed
```
```   947
```
```   948 lemma (in ordered_idom) sgn_greater [simp]:
```
```   949   "0 < sgn a \<longleftrightarrow> 0 < a"
```
```   950   unfolding sgn_if by auto
```
```   951
```
```   952 lemma (in ordered_idom) sgn_less [simp]:
```
```   953   "sgn a < 0 \<longleftrightarrow> a < 0"
```
```   954   unfolding sgn_if by auto
```
```   955
```
```   956 lemma rat_le_eq_code [code]:
```
```   957   "Fract a b < Fract c d \<longleftrightarrow> (if b = 0
```
```   958        then sgn c * sgn d > 0
```
```   959      else if d = 0
```
```   960        then sgn a * sgn b < 0
```
```   961      else a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d)"
```
```   962   by (auto simp add: sgn_times mult_less_0_iff zero_less_mult_iff less_rat' eq_rat simp del: less_rat)
```
```   963
```
```   964 lemma rat_less_eq_code [code]:
```
```   965   "Fract a b \<le> Fract c d \<longleftrightarrow> (if b = 0
```
```   966        then sgn c * sgn d \<ge> 0
```
```   967      else if d = 0
```
```   968        then sgn a * sgn b \<le> 0
```
```   969      else a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d)"
```
```   970   by (auto simp add: sgn_times mult_le_0_iff zero_le_mult_iff le_rat' eq_rat simp del: le_rat)
```
```   971     (auto simp add: le_less not_less sgn_0_0)
```
```   972
```
```   973
```
```   974 lemma rat_plus_code [code]:
```
```   975   "Fract a b + Fract c d = (if b = 0
```
```   976      then Fract c d
```
```   977    else if d = 0
```
```   978      then Fract a b
```
```   979    else Fract_norm (a * d + c * b) (b * d))"
```
```   980   by (simp add: eq_rat, simp add: Zero_rat_def)
```
```   981
```
```   982 lemma rat_times_code [code]:
```
```   983   "Fract a b * Fract c d = Fract_norm (a * c) (b * d)"
```
```   984   by simp
```
```   985
```
```   986 lemma rat_minus_code [code]:
```
```   987   "Fract a b - Fract c d = (if b = 0
```
```   988      then Fract (- c) d
```
```   989    else if d = 0
```
```   990      then Fract a b
```
```   991    else Fract_norm (a * d - c * b) (b * d))"
```
```   992   by (simp add: eq_rat, simp add: Zero_rat_def)
```
```   993
```
```   994 lemma rat_inverse_code [code]:
```
```   995   "inverse (Fract a b) = (if b = 0 then Fract 1 0
```
```   996     else if a < 0 then Fract (- b) (- a)
```
```   997     else Fract b a)"
```
```   998   by (simp add: eq_rat)
```
```   999
```
```  1000 lemma rat_divide_code [code]:
```
```  1001   "Fract a b / Fract c d = Fract_norm (a * d) (b * c)"
```
```  1002   by simp
```
```  1003
```
```  1004 definition (in term_syntax)
```
```  1005   valterm_fract :: "int \<times> (unit \<Rightarrow> Code_Eval.term) \<Rightarrow> int \<times> (unit \<Rightarrow> Code_Eval.term) \<Rightarrow> rat \<times> (unit \<Rightarrow> Code_Eval.term)" where
```
```  1006   [code inline]: "valterm_fract k l = Code_Eval.valtermify Fract {\<cdot>} k {\<cdot>} l"
```
```  1007
```
```  1008 notation fcomp (infixl "o>" 60)
```
```  1009 notation scomp (infixl "o\<rightarrow>" 60)
```
```  1010
```
```  1011 instantiation rat :: random
```
```  1012 begin
```
```  1013
```
```  1014 definition
```
```  1015   "Quickcheck.random i = Quickcheck.random i o\<rightarrow> (\<lambda>num. Random.range i o\<rightarrow> (\<lambda>denom. Pair (
```
```  1016      let j = Code_Numeral.int_of (denom + 1)
```
```  1017      in valterm_fract num (j, \<lambda>u. Code_Eval.term_of j))))"
```
```  1018
```
```  1019 instance ..
```
```  1020
```
```  1021 end
```
```  1022
```
```  1023 no_notation fcomp (infixl "o>" 60)
```
```  1024 no_notation scomp (infixl "o\<rightarrow>" 60)
```
```  1025
```
```  1026 hide (open) const Fract_norm
```
```  1027
```
```  1028 text {* Setup for SML code generator *}
```
```  1029
```
```  1030 types_code
```
```  1031   rat ("(int */ int)")
```
```  1032 attach (term_of) {*
```
```  1033 fun term_of_rat (p, q) =
```
```  1034   let
```
```  1035     val rT = Type ("Rational.rat", [])
```
```  1036   in
```
```  1037     if q = 1 orelse p = 0 then HOLogic.mk_number rT p
```
```  1038     else @{term "op / \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat"} \$
```
```  1039       HOLogic.mk_number rT p \$ HOLogic.mk_number rT q
```
```  1040   end;
```
```  1041 *}
```
```  1042 attach (test) {*
```
```  1043 fun gen_rat i =
```
```  1044   let
```
```  1045     val p = random_range 0 i;
```
```  1046     val q = random_range 1 (i + 1);
```
```  1047     val g = Integer.gcd p q;
```
```  1048     val p' = p div g;
```
```  1049     val q' = q div g;
```
```  1050     val r = (if one_of [true, false] then p' else ~ p',
```
```  1051       if p' = 0 then 0 else q')
```
```  1052   in
```
```  1053     (r, fn () => term_of_rat r)
```
```  1054   end;
```
```  1055 *}
```
```  1056
```
```  1057 consts_code
```
```  1058   Fract ("(_,/ _)")
```
```  1059
```
```  1060 consts_code
```
```  1061   "of_int :: int \<Rightarrow> rat" ("\<module>rat'_of'_int")
```
```  1062 attach {*
```
```  1063 fun rat_of_int 0 = (0, 0)
```
```  1064   | rat_of_int i = (i, 1);
```
```  1065 *}
```
```  1066
```
```  1067 end
```