src/HOL/Rational.thy
 author nipkow Sun Feb 15 22:58:02 2009 +0100 (2009-02-15) changeset 29925 17d1e32ef867 parent 29880 3dee8ff45d3d child 29940 83b373f61d41 permissions -rw-r--r--
dvd and setprod lemmas
```     1 (*  Title:  HOL/Rational.thy
```
```     2     Author: Markus Wenzel, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* Rational numbers *}
```
```     6
```
```     7 theory Rational
```
```     8 imports GCD
```
```     9 uses ("Tools/rat_arith.ML")
```
```    10 begin
```
```    11
```
```    12 subsection {* Rational numbers as quotient *}
```
```    13
```
```    14 subsubsection {* Construction of the type of rational numbers *}
```
```    15
```
```    16 definition
```
```    17   ratrel :: "((int \<times> int) \<times> (int \<times> int)) set" where
```
```    18   "ratrel = {(x, y). snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x}"
```
```    19
```
```    20 lemma ratrel_iff [simp]:
```
```    21   "(x, y) \<in> ratrel \<longleftrightarrow> snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x"
```
```    22   by (simp add: ratrel_def)
```
```    23
```
```    24 lemma refl_ratrel: "refl {x. snd x \<noteq> 0} ratrel"
```
```    25   by (auto simp add: refl_def ratrel_def)
```
```    26
```
```    27 lemma sym_ratrel: "sym ratrel"
```
```    28   by (simp add: ratrel_def sym_def)
```
```    29
```
```    30 lemma trans_ratrel: "trans ratrel"
```
```    31 proof (rule transI, unfold split_paired_all)
```
```    32   fix a b a' b' a'' b'' :: int
```
```    33   assume A: "((a, b), (a', b')) \<in> ratrel"
```
```    34   assume B: "((a', b'), (a'', b'')) \<in> ratrel"
```
```    35   have "b' * (a * b'') = b'' * (a * b')" by simp
```
```    36   also from A have "a * b' = a' * b" by auto
```
```    37   also have "b'' * (a' * b) = b * (a' * b'')" by simp
```
```    38   also from B have "a' * b'' = a'' * b'" by auto
```
```    39   also have "b * (a'' * b') = b' * (a'' * b)" by simp
```
```    40   finally have "b' * (a * b'') = b' * (a'' * b)" .
```
```    41   moreover from B have "b' \<noteq> 0" by auto
```
```    42   ultimately have "a * b'' = a'' * b" by simp
```
```    43   with A B show "((a, b), (a'', b'')) \<in> ratrel" by auto
```
```    44 qed
```
```    45
```
```    46 lemma equiv_ratrel: "equiv {x. snd x \<noteq> 0} ratrel"
```
```    47   by (rule equiv.intro [OF refl_ratrel sym_ratrel trans_ratrel])
```
```    48
```
```    49 lemmas UN_ratrel = UN_equiv_class [OF equiv_ratrel]
```
```    50 lemmas UN_ratrel2 = UN_equiv_class2 [OF equiv_ratrel equiv_ratrel]
```
```    51
```
```    52 lemma equiv_ratrel_iff [iff]:
```
```    53   assumes "snd x \<noteq> 0" and "snd y \<noteq> 0"
```
```    54   shows "ratrel `` {x} = ratrel `` {y} \<longleftrightarrow> (x, y) \<in> ratrel"
```
```    55   by (rule eq_equiv_class_iff, rule equiv_ratrel) (auto simp add: assms)
```
```    56
```
```    57 typedef (Rat) rat = "{x. snd x \<noteq> 0} // ratrel"
```
```    58 proof
```
```    59   have "(0::int, 1::int) \<in> {x. snd x \<noteq> 0}" by simp
```
```    60   then show "ratrel `` {(0, 1)} \<in> {x. snd x \<noteq> 0} // ratrel" by (rule quotientI)
```
```    61 qed
```
```    62
```
```    63 lemma ratrel_in_Rat [simp]: "snd x \<noteq> 0 \<Longrightarrow> ratrel `` {x} \<in> Rat"
```
```    64   by (simp add: Rat_def quotientI)
```
```    65
```
```    66 declare Abs_Rat_inject [simp] Abs_Rat_inverse [simp]
```
```    67
```
```    68
```
```    69 subsubsection {* Representation and basic operations *}
```
```    70
```
```    71 definition
```
```    72   Fract :: "int \<Rightarrow> int \<Rightarrow> rat" where
```
```    73   [code del]: "Fract a b = Abs_Rat (ratrel `` {if b = 0 then (0, 1) else (a, b)})"
```
```    74
```
```    75 code_datatype Fract
```
```    76
```
```    77 lemma Rat_cases [case_names Fract, cases type: rat]:
```
```    78   assumes "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> C"
```
```    79   shows C
```
```    80   using assms by (cases q) (clarsimp simp add: Fract_def Rat_def quotient_def)
```
```    81
```
```    82 lemma Rat_induct [case_names Fract, induct type: rat]:
```
```    83   assumes "\<And>a b. b \<noteq> 0 \<Longrightarrow> P (Fract a b)"
```
```    84   shows "P q"
```
```    85   using assms by (cases q) simp
```
```    86
```
```    87 lemma eq_rat:
```
```    88   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"
```
```    89   and "\<And>a. Fract a 0 = Fract 0 1"
```
```    90   and "\<And>a c. Fract 0 a = Fract 0 c"
```
```    91   by (simp_all add: Fract_def)
```
```    92
```
```    93 instantiation rat :: "{comm_ring_1, recpower}"
```
```    94 begin
```
```    95
```
```    96 definition
```
```    97   Zero_rat_def [code, code unfold]: "0 = Fract 0 1"
```
```    98
```
```    99 definition
```
```   100   One_rat_def [code, code unfold]: "1 = Fract 1 1"
```
```   101
```
```   102 definition
```
```   103   add_rat_def [code del]:
```
```   104   "q + r = Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
```
```   105     ratrel `` {(fst x * snd y + fst y * snd x, snd x * snd y)})"
```
```   106
```
```   107 lemma add_rat [simp]:
```
```   108   assumes "b \<noteq> 0" and "d \<noteq> 0"
```
```   109   shows "Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
```
```   110 proof -
```
```   111   have "(\<lambda>x y. ratrel``{(fst x * snd y + fst y * snd x, snd x * snd y)})
```
```   112     respects2 ratrel"
```
```   113   by (rule equiv_ratrel [THEN congruent2_commuteI]) (simp_all add: left_distrib)
```
```   114   with assms show ?thesis by (simp add: Fract_def add_rat_def UN_ratrel2)
```
```   115 qed
```
```   116
```
```   117 definition
```
```   118   minus_rat_def [code del]:
```
```   119   "- q = Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel `` {(- fst x, snd x)})"
```
```   120
```
```   121 lemma minus_rat [simp, code]: "- Fract a b = Fract (- a) b"
```
```   122 proof -
```
```   123   have "(\<lambda>x. ratrel `` {(- fst x, snd x)}) respects ratrel"
```
```   124     by (simp add: congruent_def)
```
```   125   then show ?thesis by (simp add: Fract_def minus_rat_def UN_ratrel)
```
```   126 qed
```
```   127
```
```   128 lemma minus_rat_cancel [simp]: "Fract (- a) (- b) = Fract a b"
```
```   129   by (cases "b = 0") (simp_all add: eq_rat)
```
```   130
```
```   131 definition
```
```   132   diff_rat_def [code del]: "q - r = q + - (r::rat)"
```
```   133
```
```   134 lemma diff_rat [simp]:
```
```   135   assumes "b \<noteq> 0" and "d \<noteq> 0"
```
```   136   shows "Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
```
```   137   using assms by (simp add: diff_rat_def)
```
```   138
```
```   139 definition
```
```   140   mult_rat_def [code del]:
```
```   141   "q * r = Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
```
```   142     ratrel``{(fst x * fst y, snd x * snd y)})"
```
```   143
```
```   144 lemma mult_rat [simp]: "Fract a b * Fract c d = Fract (a * c) (b * d)"
```
```   145 proof -
```
```   146   have "(\<lambda>x y. ratrel `` {(fst x * fst y, snd x * snd y)}) respects2 ratrel"
```
```   147     by (rule equiv_ratrel [THEN congruent2_commuteI]) simp_all
```
```   148   then show ?thesis by (simp add: Fract_def mult_rat_def UN_ratrel2)
```
```   149 qed
```
```   150
```
```   151 lemma mult_rat_cancel:
```
```   152   assumes "c \<noteq> 0"
```
```   153   shows "Fract (c * a) (c * b) = Fract a b"
```
```   154 proof -
```
```   155   from assms have "Fract c c = Fract 1 1" by (simp add: Fract_def)
```
```   156   then show ?thesis by (simp add: mult_rat [symmetric])
```
```   157 qed
```
```   158
```
```   159 primrec power_rat
```
```   160 where
```
```   161   rat_power_0:     "q ^ 0 = (1\<Colon>rat)"
```
```   162   | rat_power_Suc: "q ^ Suc n = (q\<Colon>rat) * (q ^ n)"
```
```   163
```
```   164 instance proof
```
```   165   fix q r s :: rat show "(q * r) * s = q * (r * s)"
```
```   166     by (cases q, cases r, cases s) (simp add: eq_rat)
```
```   167 next
```
```   168   fix q r :: rat show "q * r = r * q"
```
```   169     by (cases q, cases r) (simp add: eq_rat)
```
```   170 next
```
```   171   fix q :: rat show "1 * q = q"
```
```   172     by (cases q) (simp add: One_rat_def eq_rat)
```
```   173 next
```
```   174   fix q r s :: rat show "(q + r) + s = q + (r + s)"
```
```   175     by (cases q, cases r, cases s) (simp add: eq_rat algebra_simps)
```
```   176 next
```
```   177   fix q r :: rat show "q + r = r + q"
```
```   178     by (cases q, cases r) (simp add: eq_rat)
```
```   179 next
```
```   180   fix q :: rat show "0 + q = q"
```
```   181     by (cases q) (simp add: Zero_rat_def eq_rat)
```
```   182 next
```
```   183   fix q :: rat show "- q + q = 0"
```
```   184     by (cases q) (simp add: Zero_rat_def eq_rat)
```
```   185 next
```
```   186   fix q r :: rat show "q - r = q + - r"
```
```   187     by (cases q, cases r) (simp add: eq_rat)
```
```   188 next
```
```   189   fix q r s :: rat show "(q + r) * s = q * s + r * s"
```
```   190     by (cases q, cases r, cases s) (simp add: eq_rat algebra_simps)
```
```   191 next
```
```   192   show "(0::rat) \<noteq> 1" by (simp add: Zero_rat_def One_rat_def eq_rat)
```
```   193 next
```
```   194   fix q :: rat show "q * 1 = q"
```
```   195     by (cases q) (simp add: One_rat_def eq_rat)
```
```   196 next
```
```   197   fix q :: rat
```
```   198   fix n :: nat
```
```   199   show "q ^ 0 = 1" by simp
```
```   200   show "q ^ (Suc n) = q * (q ^ n)" by simp
```
```   201 qed
```
```   202
```
```   203 end
```
```   204
```
```   205 lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1"
```
```   206   by (induct k) (simp_all add: Zero_rat_def One_rat_def)
```
```   207
```
```   208 lemma of_int_rat: "of_int k = Fract k 1"
```
```   209   by (cases k rule: int_diff_cases) (simp add: of_nat_rat)
```
```   210
```
```   211 lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
```
```   212   by (rule of_nat_rat [symmetric])
```
```   213
```
```   214 lemma Fract_of_int_eq: "Fract k 1 = of_int k"
```
```   215   by (rule of_int_rat [symmetric])
```
```   216
```
```   217 instantiation rat :: number_ring
```
```   218 begin
```
```   219
```
```   220 definition
```
```   221   rat_number_of_def [code del]: "number_of w = Fract w 1"
```
```   222
```
```   223 instance by intro_classes (simp add: rat_number_of_def of_int_rat)
```
```   224
```
```   225 end
```
```   226
```
```   227 lemma rat_number_collapse [code post]:
```
```   228   "Fract 0 k = 0"
```
```   229   "Fract 1 1 = 1"
```
```   230   "Fract (number_of k) 1 = number_of k"
```
```   231   "Fract k 0 = 0"
```
```   232   by (cases "k = 0")
```
```   233     (simp_all add: Zero_rat_def One_rat_def number_of_is_id number_of_eq of_int_rat eq_rat Fract_def)
```
```   234
```
```   235 lemma rat_number_expand [code unfold]:
```
```   236   "0 = Fract 0 1"
```
```   237   "1 = Fract 1 1"
```
```   238   "number_of k = Fract (number_of k) 1"
```
```   239   by (simp_all add: rat_number_collapse)
```
```   240
```
```   241 lemma iszero_rat [simp]:
```
```   242   "iszero (number_of k :: rat) \<longleftrightarrow> iszero (number_of k :: int)"
```
```   243   by (simp add: iszero_def rat_number_expand number_of_is_id eq_rat)
```
```   244
```
```   245 lemma Rat_cases_nonzero [case_names Fract 0]:
```
```   246   assumes Fract: "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> C"
```
```   247   assumes 0: "q = 0 \<Longrightarrow> C"
```
```   248   shows C
```
```   249 proof (cases "q = 0")
```
```   250   case True then show C using 0 by auto
```
```   251 next
```
```   252   case False
```
```   253   then obtain a b where "q = Fract a b" and "b \<noteq> 0" by (cases q) auto
```
```   254   moreover with False have "0 \<noteq> Fract a b" by simp
```
```   255   with `b \<noteq> 0` have "a \<noteq> 0" by (simp add: Zero_rat_def eq_rat)
```
```   256   with Fract `q = Fract a b` `b \<noteq> 0` show C by auto
```
```   257 qed
```
```   258
```
```   259
```
```   260
```
```   261 subsubsection {* The field of rational numbers *}
```
```   262
```
```   263 instantiation rat :: "{field, division_by_zero}"
```
```   264 begin
```
```   265
```
```   266 definition
```
```   267   inverse_rat_def [code del]:
```
```   268   "inverse q = Abs_Rat (\<Union>x \<in> Rep_Rat q.
```
```   269      ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)})"
```
```   270
```
```   271 lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"
```
```   272 proof -
```
```   273   have "(\<lambda>x. ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)}) respects ratrel"
```
```   274     by (auto simp add: congruent_def mult_commute)
```
```   275   then show ?thesis by (simp add: Fract_def inverse_rat_def UN_ratrel)
```
```   276 qed
```
```   277
```
```   278 definition
```
```   279   divide_rat_def [code del]: "q / r = q * inverse (r::rat)"
```
```   280
```
```   281 lemma divide_rat [simp]: "Fract a b / Fract c d = Fract (a * d) (b * c)"
```
```   282   by (simp add: divide_rat_def)
```
```   283
```
```   284 instance proof
```
```   285   show "inverse 0 = (0::rat)" by (simp add: rat_number_expand)
```
```   286     (simp add: rat_number_collapse)
```
```   287 next
```
```   288   fix q :: rat
```
```   289   assume "q \<noteq> 0"
```
```   290   then show "inverse q * q = 1" by (cases q rule: Rat_cases_nonzero)
```
```   291    (simp_all add: mult_rat  inverse_rat rat_number_expand eq_rat)
```
```   292 next
```
```   293   fix q r :: rat
```
```   294   show "q / r = q * inverse r" by (simp add: divide_rat_def)
```
```   295 qed
```
```   296
```
```   297 end
```
```   298
```
```   299
```
```   300 subsubsection {* Various *}
```
```   301
```
```   302 lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
```
```   303   by (simp add: rat_number_expand)
```
```   304
```
```   305 lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"
```
```   306   by (simp add: Fract_of_int_eq [symmetric])
```
```   307
```
```   308 lemma Fract_number_of_quotient [code post]:
```
```   309   "Fract (number_of k) (number_of l) = number_of k / number_of l"
```
```   310   unfolding Fract_of_int_quotient number_of_is_id number_of_eq ..
```
```   311
```
```   312 lemma Fract_1_number_of [code post]:
```
```   313   "Fract 1 (number_of k) = 1 / number_of k"
```
```   314   unfolding Fract_of_int_quotient number_of_eq by simp
```
```   315
```
```   316 subsubsection {* The ordered field of rational numbers *}
```
```   317
```
```   318 instantiation rat :: linorder
```
```   319 begin
```
```   320
```
```   321 definition
```
```   322   le_rat_def [code del]:
```
```   323    "q \<le> r \<longleftrightarrow> contents (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
```
```   324       {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})"
```
```   325
```
```   326 lemma le_rat [simp]:
```
```   327   assumes "b \<noteq> 0" and "d \<noteq> 0"
```
```   328   shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   329 proof -
```
```   330   have "(\<lambda>x y. {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})
```
```   331     respects2 ratrel"
```
```   332   proof (clarsimp simp add: congruent2_def)
```
```   333     fix a b a' b' c d c' d'::int
```
```   334     assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
```
```   335     assume eq1: "a * b' = a' * b"
```
```   336     assume eq2: "c * d' = c' * d"
```
```   337
```
```   338     let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
```
```   339     {
```
```   340       fix a b c d x :: int assume x: "x \<noteq> 0"
```
```   341       have "?le a b c d = ?le (a * x) (b * x) c d"
```
```   342       proof -
```
```   343         from x have "0 < x * x" by (auto simp add: zero_less_mult_iff)
```
```   344         hence "?le a b c d =
```
```   345             ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
```
```   346           by (simp add: mult_le_cancel_right)
```
```   347         also have "... = ?le (a * x) (b * x) c d"
```
```   348           by (simp add: mult_ac)
```
```   349         finally show ?thesis .
```
```   350       qed
```
```   351     } note le_factor = this
```
```   352
```
```   353     let ?D = "b * d" and ?D' = "b' * d'"
```
```   354     from neq have D: "?D \<noteq> 0" by simp
```
```   355     from neq have "?D' \<noteq> 0" by simp
```
```   356     hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"
```
```   357       by (rule le_factor)
```
```   358     also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')"
```
```   359       by (simp add: mult_ac)
```
```   360     also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
```
```   361       by (simp only: eq1 eq2)
```
```   362     also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
```
```   363       by (simp add: mult_ac)
```
```   364     also from D have "... = ?le a' b' c' d'"
```
```   365       by (rule le_factor [symmetric])
```
```   366     finally show "?le a b c d = ?le a' b' c' d'" .
```
```   367   qed
```
```   368   with assms show ?thesis by (simp add: Fract_def le_rat_def UN_ratrel2)
```
```   369 qed
```
```   370
```
```   371 definition
```
```   372   less_rat_def [code del]: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
```
```   373
```
```   374 lemma less_rat [simp]:
```
```   375   assumes "b \<noteq> 0" and "d \<noteq> 0"
```
```   376   shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
```
```   377   using assms by (simp add: less_rat_def eq_rat order_less_le)
```
```   378
```
```   379 instance proof
```
```   380   fix q r s :: rat
```
```   381   {
```
```   382     assume "q \<le> r" and "r \<le> s"
```
```   383     show "q \<le> s"
```
```   384     proof (insert prems, induct q, induct r, induct s)
```
```   385       fix a b c d e f :: int
```
```   386       assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
```
```   387       assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"
```
```   388       show "Fract a b \<le> Fract e f"
```
```   389       proof -
```
```   390         from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
```
```   391           by (auto simp add: zero_less_mult_iff linorder_neq_iff)
```
```   392         have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
```
```   393         proof -
```
```   394           from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   395             by simp
```
```   396           with ff show ?thesis by (simp add: mult_le_cancel_right)
```
```   397         qed
```
```   398         also have "... = (c * f) * (d * f) * (b * b)" by algebra
```
```   399         also have "... \<le> (e * d) * (d * f) * (b * b)"
```
```   400         proof -
```
```   401           from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
```
```   402             by simp
```
```   403           with bb show ?thesis by (simp add: mult_le_cancel_right)
```
```   404         qed
```
```   405         finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
```
```   406           by (simp only: mult_ac)
```
```   407         with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
```
```   408           by (simp add: mult_le_cancel_right)
```
```   409         with neq show ?thesis by simp
```
```   410       qed
```
```   411     qed
```
```   412   next
```
```   413     assume "q \<le> r" and "r \<le> q"
```
```   414     show "q = r"
```
```   415     proof (insert prems, induct q, induct r)
```
```   416       fix a b c d :: int
```
```   417       assume neq: "b \<noteq> 0"  "d \<noteq> 0"
```
```   418       assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"
```
```   419       show "Fract a b = Fract c d"
```
```   420       proof -
```
```   421         from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   422           by simp
```
```   423         also have "... \<le> (a * d) * (b * d)"
```
```   424         proof -
```
```   425           from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
```
```   426             by simp
```
```   427           thus ?thesis by (simp only: mult_ac)
```
```   428         qed
```
```   429         finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
```
```   430         moreover from neq have "b * d \<noteq> 0" by simp
```
```   431         ultimately have "a * d = c * b" by simp
```
```   432         with neq show ?thesis by (simp add: eq_rat)
```
```   433       qed
```
```   434     qed
```
```   435   next
```
```   436     show "q \<le> q"
```
```   437       by (induct q) simp
```
```   438     show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)"
```
```   439       by (induct q, induct r) (auto simp add: le_less mult_commute)
```
```   440     show "q \<le> r \<or> r \<le> q"
```
```   441       by (induct q, induct r)
```
```   442          (simp add: mult_commute, rule linorder_linear)
```
```   443   }
```
```   444 qed
```
```   445
```
```   446 end
```
```   447
```
```   448 instantiation rat :: "{distrib_lattice, abs_if, sgn_if}"
```
```   449 begin
```
```   450
```
```   451 definition
```
```   452   abs_rat_def [code del]: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))"
```
```   453
```
```   454 lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
```
```   455   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)
```
```   456
```
```   457 definition
```
```   458   sgn_rat_def [code del]: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
```
```   459
```
```   460 lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)"
```
```   461   unfolding Fract_of_int_eq
```
```   462   by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat)
```
```   463     (auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)
```
```   464
```
```   465 definition
```
```   466   "(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min"
```
```   467
```
```   468 definition
```
```   469   "(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max"
```
```   470
```
```   471 instance by intro_classes
```
```   472   (auto simp add: abs_rat_def sgn_rat_def min_max.sup_inf_distrib1 inf_rat_def sup_rat_def)
```
```   473
```
```   474 end
```
```   475
```
```   476 instance rat :: ordered_field
```
```   477 proof
```
```   478   fix q r s :: rat
```
```   479   show "q \<le> r ==> s + q \<le> s + r"
```
```   480   proof (induct q, induct r, induct s)
```
```   481     fix a b c d e f :: int
```
```   482     assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
```
```   483     assume le: "Fract a b \<le> Fract c d"
```
```   484     show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
```
```   485     proof -
```
```   486       let ?F = "f * f" from neq have F: "0 < ?F"
```
```   487         by (auto simp add: zero_less_mult_iff)
```
```   488       from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   489         by simp
```
```   490       with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
```
```   491         by (simp add: mult_le_cancel_right)
```
```   492       with neq show ?thesis by (simp add: mult_ac int_distrib)
```
```   493     qed
```
```   494   qed
```
```   495   show "q < r ==> 0 < s ==> s * q < s * r"
```
```   496   proof (induct q, induct r, induct s)
```
```   497     fix a b c d e f :: int
```
```   498     assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
```
```   499     assume le: "Fract a b < Fract c d"
```
```   500     assume gt: "0 < Fract e f"
```
```   501     show "Fract e f * Fract a b < Fract e f * Fract c d"
```
```   502     proof -
```
```   503       let ?E = "e * f" and ?F = "f * f"
```
```   504       from neq gt have "0 < ?E"
```
```   505         by (auto simp add: Zero_rat_def order_less_le eq_rat)
```
```   506       moreover from neq have "0 < ?F"
```
```   507         by (auto simp add: zero_less_mult_iff)
```
```   508       moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
```
```   509         by simp
```
```   510       ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
```
```   511         by (simp add: mult_less_cancel_right)
```
```   512       with neq show ?thesis
```
```   513         by (simp add: mult_ac)
```
```   514     qed
```
```   515   qed
```
```   516 qed auto
```
```   517
```
```   518 lemma Rat_induct_pos [case_names Fract, induct type: rat]:
```
```   519   assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
```
```   520   shows "P q"
```
```   521 proof (cases q)
```
```   522   have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)"
```
```   523   proof -
```
```   524     fix a::int and b::int
```
```   525     assume b: "b < 0"
```
```   526     hence "0 < -b" by simp
```
```   527     hence "P (Fract (-a) (-b))" by (rule step)
```
```   528     thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
```
```   529   qed
```
```   530   case (Fract a b)
```
```   531   thus "P q" by (force simp add: linorder_neq_iff step step')
```
```   532 qed
```
```   533
```
```   534 lemma zero_less_Fract_iff:
```
```   535   "0 < b ==> (0 < Fract a b) = (0 < a)"
```
```   536 by (simp add: Zero_rat_def order_less_imp_not_eq2 zero_less_mult_iff)
```
```   537
```
```   538
```
```   539 subsection {* Arithmetic setup *}
```
```   540
```
```   541 use "Tools/rat_arith.ML"
```
```   542 declaration {* K rat_arith_setup *}
```
```   543
```
```   544
```
```   545 subsection {* Embedding from Rationals to other Fields *}
```
```   546
```
```   547 class field_char_0 = field + ring_char_0
```
```   548
```
```   549 subclass (in ordered_field) field_char_0 ..
```
```   550
```
```   551 context field_char_0
```
```   552 begin
```
```   553
```
```   554 definition of_rat :: "rat \<Rightarrow> 'a" where
```
```   555   [code del]: "of_rat q = contents (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})"
```
```   556
```
```   557 end
```
```   558
```
```   559 lemma of_rat_congruent:
```
```   560   "(\<lambda>(a, b). {of_int a / of_int b :: 'a::field_char_0}) respects ratrel"
```
```   561 apply (rule congruent.intro)
```
```   562 apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
```
```   563 apply (simp only: of_int_mult [symmetric])
```
```   564 done
```
```   565
```
```   566 lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
```
```   567   unfolding Fract_def of_rat_def by (simp add: UN_ratrel of_rat_congruent)
```
```   568
```
```   569 lemma of_rat_0 [simp]: "of_rat 0 = 0"
```
```   570 by (simp add: Zero_rat_def of_rat_rat)
```
```   571
```
```   572 lemma of_rat_1 [simp]: "of_rat 1 = 1"
```
```   573 by (simp add: One_rat_def of_rat_rat)
```
```   574
```
```   575 lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
```
```   576 by (induct a, induct b, simp add: of_rat_rat add_frac_eq)
```
```   577
```
```   578 lemma of_rat_minus: "of_rat (- a) = - of_rat a"
```
```   579 by (induct a, simp add: of_rat_rat)
```
```   580
```
```   581 lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
```
```   582 by (simp only: diff_minus of_rat_add of_rat_minus)
```
```   583
```
```   584 lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
```
```   585 apply (induct a, induct b, simp add: of_rat_rat)
```
```   586 apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
```
```   587 done
```
```   588
```
```   589 lemma nonzero_of_rat_inverse:
```
```   590   "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
```
```   591 apply (rule inverse_unique [symmetric])
```
```   592 apply (simp add: of_rat_mult [symmetric])
```
```   593 done
```
```   594
```
```   595 lemma of_rat_inverse:
```
```   596   "(of_rat (inverse a)::'a::{field_char_0,division_by_zero}) =
```
```   597    inverse (of_rat a)"
```
```   598 by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
```
```   599
```
```   600 lemma nonzero_of_rat_divide:
```
```   601   "b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
```
```   602 by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
```
```   603
```
```   604 lemma of_rat_divide:
```
```   605   "(of_rat (a / b)::'a::{field_char_0,division_by_zero})
```
```   606    = of_rat a / of_rat b"
```
```   607 by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)
```
```   608
```
```   609 lemma of_rat_power:
```
```   610   "(of_rat (a ^ n)::'a::{field_char_0,recpower}) = of_rat a ^ n"
```
```   611 by (induct n) (simp_all add: of_rat_mult power_Suc)
```
```   612
```
```   613 lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
```
```   614 apply (induct a, induct b)
```
```   615 apply (simp add: of_rat_rat eq_rat)
```
```   616 apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
```
```   617 apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
```
```   618 done
```
```   619
```
```   620 lemma of_rat_less:
```
```   621   "(of_rat r :: 'a::ordered_field) < of_rat s \<longleftrightarrow> r < s"
```
```   622 proof (induct r, induct s)
```
```   623   fix a b c d :: int
```
```   624   assume not_zero: "b > 0" "d > 0"
```
```   625   then have "b * d > 0" by (rule mult_pos_pos)
```
```   626   have of_int_divide_less_eq:
```
```   627     "(of_int a :: 'a) / of_int b < of_int c / of_int d
```
```   628       \<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b"
```
```   629     using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
```
```   630   show "(of_rat (Fract a b) :: 'a::ordered_field) < of_rat (Fract c d)
```
```   631     \<longleftrightarrow> Fract a b < Fract c d"
```
```   632     using not_zero `b * d > 0`
```
```   633     by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult)
```
```   634       (auto intro: mult_strict_right_mono mult_right_less_imp_less)
```
```   635 qed
```
```   636
```
```   637 lemma of_rat_less_eq:
```
```   638   "(of_rat r :: 'a::ordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s"
```
```   639   unfolding le_less by (auto simp add: of_rat_less)
```
```   640
```
```   641 lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
```
```   642
```
```   643 lemma of_rat_eq_id [simp]: "of_rat = id"
```
```   644 proof
```
```   645   fix a
```
```   646   show "of_rat a = id a"
```
```   647   by (induct a)
```
```   648      (simp add: of_rat_rat Fract_of_int_eq [symmetric])
```
```   649 qed
```
```   650
```
```   651 text{*Collapse nested embeddings*}
```
```   652 lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
```
```   653 by (induct n) (simp_all add: of_rat_add)
```
```   654
```
```   655 lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
```
```   656 by (cases z rule: int_diff_cases) (simp add: of_rat_diff)
```
```   657
```
```   658 lemma of_rat_number_of_eq [simp]:
```
```   659   "of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})"
```
```   660 by (simp add: number_of_eq)
```
```   661
```
```   662 lemmas zero_rat = Zero_rat_def
```
```   663 lemmas one_rat = One_rat_def
```
```   664
```
```   665 abbreviation
```
```   666   rat_of_nat :: "nat \<Rightarrow> rat"
```
```   667 where
```
```   668   "rat_of_nat \<equiv> of_nat"
```
```   669
```
```   670 abbreviation
```
```   671   rat_of_int :: "int \<Rightarrow> rat"
```
```   672 where
```
```   673   "rat_of_int \<equiv> of_int"
```
```   674
```
```   675 subsection {* The Set of Rational Numbers *}
```
```   676
```
```   677 context field_char_0
```
```   678 begin
```
```   679
```
```   680 definition
```
```   681   Rats  :: "'a set" where
```
```   682   [code del]: "Rats = range of_rat"
```
```   683
```
```   684 notation (xsymbols)
```
```   685   Rats  ("\<rat>")
```
```   686
```
```   687 end
```
```   688
```
```   689 lemma Rats_of_rat [simp]: "of_rat r \<in> Rats"
```
```   690 by (simp add: Rats_def)
```
```   691
```
```   692 lemma Rats_of_int [simp]: "of_int z \<in> Rats"
```
```   693 by (subst of_rat_of_int_eq [symmetric], rule Rats_of_rat)
```
```   694
```
```   695 lemma Rats_of_nat [simp]: "of_nat n \<in> Rats"
```
```   696 by (subst of_rat_of_nat_eq [symmetric], rule Rats_of_rat)
```
```   697
```
```   698 lemma Rats_number_of [simp]:
```
```   699   "(number_of w::'a::{number_ring,field_char_0}) \<in> Rats"
```
```   700 by (subst of_rat_number_of_eq [symmetric], rule Rats_of_rat)
```
```   701
```
```   702 lemma Rats_0 [simp]: "0 \<in> Rats"
```
```   703 apply (unfold Rats_def)
```
```   704 apply (rule range_eqI)
```
```   705 apply (rule of_rat_0 [symmetric])
```
```   706 done
```
```   707
```
```   708 lemma Rats_1 [simp]: "1 \<in> Rats"
```
```   709 apply (unfold Rats_def)
```
```   710 apply (rule range_eqI)
```
```   711 apply (rule of_rat_1 [symmetric])
```
```   712 done
```
```   713
```
```   714 lemma Rats_add [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a + b \<in> Rats"
```
```   715 apply (auto simp add: Rats_def)
```
```   716 apply (rule range_eqI)
```
```   717 apply (rule of_rat_add [symmetric])
```
```   718 done
```
```   719
```
```   720 lemma Rats_minus [simp]: "a \<in> Rats \<Longrightarrow> - a \<in> Rats"
```
```   721 apply (auto simp add: Rats_def)
```
```   722 apply (rule range_eqI)
```
```   723 apply (rule of_rat_minus [symmetric])
```
```   724 done
```
```   725
```
```   726 lemma Rats_diff [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a - b \<in> Rats"
```
```   727 apply (auto simp add: Rats_def)
```
```   728 apply (rule range_eqI)
```
```   729 apply (rule of_rat_diff [symmetric])
```
```   730 done
```
```   731
```
```   732 lemma Rats_mult [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a * b \<in> Rats"
```
```   733 apply (auto simp add: Rats_def)
```
```   734 apply (rule range_eqI)
```
```   735 apply (rule of_rat_mult [symmetric])
```
```   736 done
```
```   737
```
```   738 lemma nonzero_Rats_inverse:
```
```   739   fixes a :: "'a::field_char_0"
```
```   740   shows "\<lbrakk>a \<in> Rats; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Rats"
```
```   741 apply (auto simp add: Rats_def)
```
```   742 apply (rule range_eqI)
```
```   743 apply (erule nonzero_of_rat_inverse [symmetric])
```
```   744 done
```
```   745
```
```   746 lemma Rats_inverse [simp]:
```
```   747   fixes a :: "'a::{field_char_0,division_by_zero}"
```
```   748   shows "a \<in> Rats \<Longrightarrow> inverse a \<in> Rats"
```
```   749 apply (auto simp add: Rats_def)
```
```   750 apply (rule range_eqI)
```
```   751 apply (rule of_rat_inverse [symmetric])
```
```   752 done
```
```   753
```
```   754 lemma nonzero_Rats_divide:
```
```   755   fixes a b :: "'a::field_char_0"
```
```   756   shows "\<lbrakk>a \<in> Rats; b \<in> Rats; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
```
```   757 apply (auto simp add: Rats_def)
```
```   758 apply (rule range_eqI)
```
```   759 apply (erule nonzero_of_rat_divide [symmetric])
```
```   760 done
```
```   761
```
```   762 lemma Rats_divide [simp]:
```
```   763   fixes a b :: "'a::{field_char_0,division_by_zero}"
```
```   764   shows "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
```
```   765 apply (auto simp add: Rats_def)
```
```   766 apply (rule range_eqI)
```
```   767 apply (rule of_rat_divide [symmetric])
```
```   768 done
```
```   769
```
```   770 lemma Rats_power [simp]:
```
```   771   fixes a :: "'a::{field_char_0,recpower}"
```
```   772   shows "a \<in> Rats \<Longrightarrow> a ^ n \<in> Rats"
```
```   773 apply (auto simp add: Rats_def)
```
```   774 apply (rule range_eqI)
```
```   775 apply (rule of_rat_power [symmetric])
```
```   776 done
```
```   777
```
```   778 lemma Rats_cases [cases set: Rats]:
```
```   779   assumes "q \<in> \<rat>"
```
```   780   obtains (of_rat) r where "q = of_rat r"
```
```   781   unfolding Rats_def
```
```   782 proof -
```
```   783   from `q \<in> \<rat>` have "q \<in> range of_rat" unfolding Rats_def .
```
```   784   then obtain r where "q = of_rat r" ..
```
```   785   then show thesis ..
```
```   786 qed
```
```   787
```
```   788 lemma Rats_induct [case_names of_rat, induct set: Rats]:
```
```   789   "q \<in> \<rat> \<Longrightarrow> (\<And>r. P (of_rat r)) \<Longrightarrow> P q"
```
```   790   by (rule Rats_cases) auto
```
```   791
```
```   792
```
```   793 subsection {* Implementation of rational numbers as pairs of integers *}
```
```   794
```
```   795 lemma Fract_norm: "Fract (a div zgcd a b) (b div zgcd a b) = Fract a b"
```
```   796 proof (cases "a = 0 \<or> b = 0")
```
```   797   case True then show ?thesis by (auto simp add: eq_rat)
```
```   798 next
```
```   799   let ?c = "zgcd a b"
```
```   800   case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
```
```   801   then have "?c \<noteq> 0" by simp
```
```   802   then have "Fract ?c ?c = Fract 1 1" by (simp add: eq_rat)
```
```   803   moreover have "Fract (a div ?c * ?c + a mod ?c) (b div ?c * ?c + b mod ?c) = Fract a b"
```
```   804     by (simp add: semiring_div_class.mod_div_equality)
```
```   805   moreover have "a mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric])
```
```   806   moreover have "b mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric])
```
```   807   ultimately show ?thesis
```
```   808     by (simp add: mult_rat [symmetric])
```
```   809 qed
```
```   810
```
```   811 definition Fract_norm :: "int \<Rightarrow> int \<Rightarrow> rat" where
```
```   812   [simp, code del]: "Fract_norm a b = Fract a b"
```
```   813
```
```   814 lemma Fract_norm_code [code]: "Fract_norm a b = (if a = 0 \<or> b = 0 then 0 else let c = zgcd a b in
```
```   815   if b > 0 then Fract (a div c) (b div c) else Fract (- (a div c)) (- (b div c)))"
```
```   816   by (simp add: eq_rat Zero_rat_def Let_def Fract_norm)
```
```   817
```
```   818 lemma [code]:
```
```   819   "of_rat (Fract a b) = (if b \<noteq> 0 then of_int a / of_int b else 0)"
```
```   820   by (cases "b = 0") (simp_all add: rat_number_collapse of_rat_rat)
```
```   821
```
```   822 instantiation rat :: eq
```
```   823 begin
```
```   824
```
```   825 definition [code del]: "eq_class.eq (a\<Colon>rat) b \<longleftrightarrow> a - b = 0"
```
```   826
```
```   827 instance by default (simp add: eq_rat_def)
```
```   828
```
```   829 lemma rat_eq_code [code]:
```
```   830   "eq_class.eq (Fract a b) (Fract c d) \<longleftrightarrow> (if b = 0
```
```   831        then c = 0 \<or> d = 0
```
```   832      else if d = 0
```
```   833        then a = 0 \<or> b = 0
```
```   834      else a * d = b * c)"
```
```   835   by (auto simp add: eq eq_rat)
```
```   836
```
```   837 lemma rat_eq_refl [code nbe]:
```
```   838   "eq_class.eq (r::rat) r \<longleftrightarrow> True"
```
```   839   by (rule HOL.eq_refl)
```
```   840
```
```   841 end
```
```   842
```
```   843 lemma le_rat':
```
```   844   assumes "b \<noteq> 0"
```
```   845     and "d \<noteq> 0"
```
```   846   shows "Fract a b \<le> Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d"
```
```   847 proof -
```
```   848   have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp
```
```   849   have "a * d * (b * d) \<le> c * b * (b * d) \<longleftrightarrow> a * d * (sgn b * sgn d) \<le> c * b * (sgn b * sgn d)"
```
```   850   proof (cases "b * d > 0")
```
```   851     case True
```
```   852     moreover from True have "sgn b * sgn d = 1"
```
```   853       by (simp add: sgn_times [symmetric] sgn_1_pos)
```
```   854     ultimately show ?thesis by (simp add: mult_le_cancel_right)
```
```   855   next
```
```   856     case False with assms have "b * d < 0" by (simp add: less_le)
```
```   857     moreover from this have "sgn b * sgn d = - 1"
```
```   858       by (simp only: sgn_times [symmetric] sgn_1_neg)
```
```   859     ultimately show ?thesis by (simp add: mult_le_cancel_right)
```
```   860   qed
```
```   861   also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d"
```
```   862     by (simp add: abs_sgn mult_ac)
```
```   863   finally show ?thesis using assms by simp
```
```   864 qed
```
```   865
```
```   866 lemma less_rat':
```
```   867   assumes "b \<noteq> 0"
```
```   868     and "d \<noteq> 0"
```
```   869   shows "Fract a b < Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d"
```
```   870 proof -
```
```   871   have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp
```
```   872   have "a * d * (b * d) < c * b * (b * d) \<longleftrightarrow> a * d * (sgn b * sgn d) < c * b * (sgn b * sgn d)"
```
```   873   proof (cases "b * d > 0")
```
```   874     case True
```
```   875     moreover from True have "sgn b * sgn d = 1"
```
```   876       by (simp add: sgn_times [symmetric] sgn_1_pos)
```
```   877     ultimately show ?thesis by (simp add: mult_less_cancel_right)
```
```   878   next
```
```   879     case False with assms have "b * d < 0" by (simp add: less_le)
```
```   880     moreover from this have "sgn b * sgn d = - 1"
```
```   881       by (simp only: sgn_times [symmetric] sgn_1_neg)
```
```   882     ultimately show ?thesis by (simp add: mult_less_cancel_right)
```
```   883   qed
```
```   884   also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d"
```
```   885     by (simp add: abs_sgn mult_ac)
```
```   886   finally show ?thesis using assms by simp
```
```   887 qed
```
```   888
```
```   889 lemma rat_less_eq_code [code]:
```
```   890   "Fract a b \<le> Fract c d \<longleftrightarrow> (if b = 0
```
```   891        then sgn c * sgn d \<ge> 0
```
```   892      else if d = 0
```
```   893        then sgn a * sgn b \<le> 0
```
```   894      else a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d)"
```
```   895 by (auto simp add: sgn_times mult_le_0_iff zero_le_mult_iff le_rat' eq_rat simp del: le_rat)
```
```   896   (auto simp add: sgn_times sgn_0_0 le_less sgn_1_pos [symmetric] sgn_1_neg [symmetric])
```
```   897
```
```   898 lemma rat_le_eq_code [code]:
```
```   899   "Fract a b < Fract c d \<longleftrightarrow> (if b = 0
```
```   900        then sgn c * sgn d > 0
```
```   901      else if d = 0
```
```   902        then sgn a * sgn b < 0
```
```   903      else a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d)"
```
```   904 by (auto simp add: sgn_times mult_less_0_iff zero_less_mult_iff less_rat' eq_rat simp del: less_rat)
```
```   905    (auto simp add: sgn_times sgn_0_0 sgn_1_pos [symmetric] sgn_1_neg [symmetric],
```
```   906      auto simp add: sgn_1_pos)
```
```   907
```
```   908 lemma rat_plus_code [code]:
```
```   909   "Fract a b + Fract c d = (if b = 0
```
```   910      then Fract c d
```
```   911    else if d = 0
```
```   912      then Fract a b
```
```   913    else Fract_norm (a * d + c * b) (b * d))"
```
```   914   by (simp add: eq_rat, simp add: Zero_rat_def)
```
```   915
```
```   916 lemma rat_times_code [code]:
```
```   917   "Fract a b * Fract c d = Fract_norm (a * c) (b * d)"
```
```   918   by simp
```
```   919
```
```   920 lemma rat_minus_code [code]:
```
```   921   "Fract a b - Fract c d = (if b = 0
```
```   922      then Fract (- c) d
```
```   923    else if d = 0
```
```   924      then Fract a b
```
```   925    else Fract_norm (a * d - c * b) (b * d))"
```
```   926   by (simp add: eq_rat, simp add: Zero_rat_def)
```
```   927
```
```   928 lemma rat_inverse_code [code]:
```
```   929   "inverse (Fract a b) = (if b = 0 then Fract 1 0
```
```   930     else if a < 0 then Fract (- b) (- a)
```
```   931     else Fract b a)"
```
```   932   by (simp add: eq_rat)
```
```   933
```
```   934 lemma rat_divide_code [code]:
```
```   935   "Fract a b / Fract c d = Fract_norm (a * d) (b * c)"
```
```   936   by simp
```
```   937
```
```   938 hide (open) const Fract_norm
```
```   939
```
```   940 text {* Setup for SML code generator *}
```
```   941
```
```   942 types_code
```
```   943   rat ("(int */ int)")
```
```   944 attach (term_of) {*
```
```   945 fun term_of_rat (p, q) =
```
```   946   let
```
```   947     val rT = Type ("Rational.rat", [])
```
```   948   in
```
```   949     if q = 1 orelse p = 0 then HOLogic.mk_number rT p
```
```   950     else @{term "op / \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat"} \$
```
```   951       HOLogic.mk_number rT p \$ HOLogic.mk_number rT q
```
```   952   end;
```
```   953 *}
```
```   954 attach (test) {*
```
```   955 fun gen_rat i =
```
```   956   let
```
```   957     val p = random_range 0 i;
```
```   958     val q = random_range 1 (i + 1);
```
```   959     val g = Integer.gcd p q;
```
```   960     val p' = p div g;
```
```   961     val q' = q div g;
```
```   962     val r = (if one_of [true, false] then p' else ~ p',
```
```   963       if p' = 0 then 0 else q')
```
```   964   in
```
```   965     (r, fn () => term_of_rat r)
```
```   966   end;
```
```   967 *}
```
```   968
```
```   969 consts_code
```
```   970   Fract ("(_,/ _)")
```
```   971
```
```   972 consts_code
```
```   973   "of_int :: int \<Rightarrow> rat" ("\<module>rat'_of'_int")
```
```   974 attach {*
```
```   975 fun rat_of_int 0 = (0, 0)
```
```   976   | rat_of_int i = (i, 1);
```
```   977 *}
```
```   978
```
```   979 end
```