src/HOL/Rational.thy
 author huffman Wed Feb 25 09:09:50 2009 -0800 (2009-02-25) changeset 30095 c6e184561159 parent 29940 83b373f61d41 child 30097 57df8626c23b permissions -rw-r--r--
add lemmas about comparisons of Fract a b with 0 and 1
```     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 subsubsection {* The field of rational numbers *}
```
```   261
```
```   262 instantiation rat :: "{field, division_by_zero}"
```
```   263 begin
```
```   264
```
```   265 definition
```
```   266   inverse_rat_def [code del]:
```
```   267   "inverse q = Abs_Rat (\<Union>x \<in> Rep_Rat q.
```
```   268      ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)})"
```
```   269
```
```   270 lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"
```
```   271 proof -
```
```   272   have "(\<lambda>x. ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)}) respects ratrel"
```
```   273     by (auto simp add: congruent_def mult_commute)
```
```   274   then show ?thesis by (simp add: Fract_def inverse_rat_def UN_ratrel)
```
```   275 qed
```
```   276
```
```   277 definition
```
```   278   divide_rat_def [code del]: "q / r = q * inverse (r::rat)"
```
```   279
```
```   280 lemma divide_rat [simp]: "Fract a b / Fract c d = Fract (a * d) (b * c)"
```
```   281   by (simp add: divide_rat_def)
```
```   282
```
```   283 instance proof
```
```   284   show "inverse 0 = (0::rat)" by (simp add: rat_number_expand)
```
```   285     (simp add: rat_number_collapse)
```
```   286 next
```
```   287   fix q :: rat
```
```   288   assume "q \<noteq> 0"
```
```   289   then show "inverse q * q = 1" by (cases q rule: Rat_cases_nonzero)
```
```   290    (simp_all add: mult_rat  inverse_rat rat_number_expand eq_rat)
```
```   291 next
```
```   292   fix q r :: rat
```
```   293   show "q / r = q * inverse r" by (simp add: divide_rat_def)
```
```   294 qed
```
```   295
```
```   296 end
```
```   297
```
```   298
```
```   299 subsubsection {* Various *}
```
```   300
```
```   301 lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
```
```   302   by (simp add: rat_number_expand)
```
```   303
```
```   304 lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"
```
```   305   by (simp add: Fract_of_int_eq [symmetric])
```
```   306
```
```   307 lemma Fract_number_of_quotient [code post]:
```
```   308   "Fract (number_of k) (number_of l) = number_of k / number_of l"
```
```   309   unfolding Fract_of_int_quotient number_of_is_id number_of_eq ..
```
```   310
```
```   311 lemma Fract_1_number_of [code post]:
```
```   312   "Fract 1 (number_of k) = 1 / number_of k"
```
```   313   unfolding Fract_of_int_quotient number_of_eq by simp
```
```   314
```
```   315 subsubsection {* The ordered field of rational numbers *}
```
```   316
```
```   317 instantiation rat :: linorder
```
```   318 begin
```
```   319
```
```   320 definition
```
```   321   le_rat_def [code del]:
```
```   322    "q \<le> r \<longleftrightarrow> contents (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
```
```   323       {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})"
```
```   324
```
```   325 lemma le_rat [simp]:
```
```   326   assumes "b \<noteq> 0" and "d \<noteq> 0"
```
```   327   shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   328 proof -
```
```   329   have "(\<lambda>x y. {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})
```
```   330     respects2 ratrel"
```
```   331   proof (clarsimp simp add: congruent2_def)
```
```   332     fix a b a' b' c d c' d'::int
```
```   333     assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
```
```   334     assume eq1: "a * b' = a' * b"
```
```   335     assume eq2: "c * d' = c' * d"
```
```   336
```
```   337     let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
```
```   338     {
```
```   339       fix a b c d x :: int assume x: "x \<noteq> 0"
```
```   340       have "?le a b c d = ?le (a * x) (b * x) c d"
```
```   341       proof -
```
```   342         from x have "0 < x * x" by (auto simp add: zero_less_mult_iff)
```
```   343         hence "?le a b c d =
```
```   344             ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
```
```   345           by (simp add: mult_le_cancel_right)
```
```   346         also have "... = ?le (a * x) (b * x) c d"
```
```   347           by (simp add: mult_ac)
```
```   348         finally show ?thesis .
```
```   349       qed
```
```   350     } note le_factor = this
```
```   351
```
```   352     let ?D = "b * d" and ?D' = "b' * d'"
```
```   353     from neq have D: "?D \<noteq> 0" by simp
```
```   354     from neq have "?D' \<noteq> 0" by simp
```
```   355     hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"
```
```   356       by (rule le_factor)
```
```   357     also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')"
```
```   358       by (simp add: mult_ac)
```
```   359     also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
```
```   360       by (simp only: eq1 eq2)
```
```   361     also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
```
```   362       by (simp add: mult_ac)
```
```   363     also from D have "... = ?le a' b' c' d'"
```
```   364       by (rule le_factor [symmetric])
```
```   365     finally show "?le a b c d = ?le a' b' c' d'" .
```
```   366   qed
```
```   367   with assms show ?thesis by (simp add: Fract_def le_rat_def UN_ratrel2)
```
```   368 qed
```
```   369
```
```   370 definition
```
```   371   less_rat_def [code del]: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
```
```   372
```
```   373 lemma less_rat [simp]:
```
```   374   assumes "b \<noteq> 0" and "d \<noteq> 0"
```
```   375   shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
```
```   376   using assms by (simp add: less_rat_def eq_rat order_less_le)
```
```   377
```
```   378 instance proof
```
```   379   fix q r s :: rat
```
```   380   {
```
```   381     assume "q \<le> r" and "r \<le> s"
```
```   382     show "q \<le> s"
```
```   383     proof (insert prems, induct q, induct r, induct s)
```
```   384       fix a b c d e f :: int
```
```   385       assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
```
```   386       assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"
```
```   387       show "Fract a b \<le> Fract e f"
```
```   388       proof -
```
```   389         from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
```
```   390           by (auto simp add: zero_less_mult_iff linorder_neq_iff)
```
```   391         have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
```
```   392         proof -
```
```   393           from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   394             by simp
```
```   395           with ff show ?thesis by (simp add: mult_le_cancel_right)
```
```   396         qed
```
```   397         also have "... = (c * f) * (d * f) * (b * b)" by algebra
```
```   398         also have "... \<le> (e * d) * (d * f) * (b * b)"
```
```   399         proof -
```
```   400           from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
```
```   401             by simp
```
```   402           with bb show ?thesis by (simp add: mult_le_cancel_right)
```
```   403         qed
```
```   404         finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
```
```   405           by (simp only: mult_ac)
```
```   406         with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
```
```   407           by (simp add: mult_le_cancel_right)
```
```   408         with neq show ?thesis by simp
```
```   409       qed
```
```   410     qed
```
```   411   next
```
```   412     assume "q \<le> r" and "r \<le> q"
```
```   413     show "q = r"
```
```   414     proof (insert prems, induct q, induct r)
```
```   415       fix a b c d :: int
```
```   416       assume neq: "b \<noteq> 0"  "d \<noteq> 0"
```
```   417       assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"
```
```   418       show "Fract a b = Fract c d"
```
```   419       proof -
```
```   420         from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   421           by simp
```
```   422         also have "... \<le> (a * d) * (b * d)"
```
```   423         proof -
```
```   424           from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
```
```   425             by simp
```
```   426           thus ?thesis by (simp only: mult_ac)
```
```   427         qed
```
```   428         finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
```
```   429         moreover from neq have "b * d \<noteq> 0" by simp
```
```   430         ultimately have "a * d = c * b" by simp
```
```   431         with neq show ?thesis by (simp add: eq_rat)
```
```   432       qed
```
```   433     qed
```
```   434   next
```
```   435     show "q \<le> q"
```
```   436       by (induct q) simp
```
```   437     show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)"
```
```   438       by (induct q, induct r) (auto simp add: le_less mult_commute)
```
```   439     show "q \<le> r \<or> r \<le> q"
```
```   440       by (induct q, induct r)
```
```   441          (simp add: mult_commute, rule linorder_linear)
```
```   442   }
```
```   443 qed
```
```   444
```
```   445 end
```
```   446
```
```   447 instantiation rat :: "{distrib_lattice, abs_if, sgn_if}"
```
```   448 begin
```
```   449
```
```   450 definition
```
```   451   abs_rat_def [code del]: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))"
```
```   452
```
```   453 lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
```
```   454   by (auto simp add: abs_rat_def zabs_def Zero_rat_def less_rat not_less le_less minus_rat eq_rat zero_compare_simps)
```
```   455
```
```   456 definition
```
```   457   sgn_rat_def [code del]: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
```
```   458
```
```   459 lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)"
```
```   460   unfolding Fract_of_int_eq
```
```   461   by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat)
```
```   462     (auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)
```
```   463
```
```   464 definition
```
```   465   "(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min"
```
```   466
```
```   467 definition
```
```   468   "(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max"
```
```   469
```
```   470 instance by intro_classes
```
```   471   (auto simp add: abs_rat_def sgn_rat_def min_max.sup_inf_distrib1 inf_rat_def sup_rat_def)
```
```   472
```
```   473 end
```
```   474
```
```   475 instance rat :: ordered_field
```
```   476 proof
```
```   477   fix q r s :: rat
```
```   478   show "q \<le> r ==> s + q \<le> s + r"
```
```   479   proof (induct q, induct r, induct s)
```
```   480     fix a b c d e f :: int
```
```   481     assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
```
```   482     assume le: "Fract a b \<le> Fract c d"
```
```   483     show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
```
```   484     proof -
```
```   485       let ?F = "f * f" from neq have F: "0 < ?F"
```
```   486         by (auto simp add: zero_less_mult_iff)
```
```   487       from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   488         by simp
```
```   489       with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
```
```   490         by (simp add: mult_le_cancel_right)
```
```   491       with neq show ?thesis by (simp add: mult_ac int_distrib)
```
```   492     qed
```
```   493   qed
```
```   494   show "q < r ==> 0 < s ==> s * q < s * r"
```
```   495   proof (induct q, induct r, induct s)
```
```   496     fix a b c d e f :: int
```
```   497     assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
```
```   498     assume le: "Fract a b < Fract c d"
```
```   499     assume gt: "0 < Fract e f"
```
```   500     show "Fract e f * Fract a b < Fract e f * Fract c d"
```
```   501     proof -
```
```   502       let ?E = "e * f" and ?F = "f * f"
```
```   503       from neq gt have "0 < ?E"
```
```   504         by (auto simp add: Zero_rat_def order_less_le eq_rat)
```
```   505       moreover from neq have "0 < ?F"
```
```   506         by (auto simp add: zero_less_mult_iff)
```
```   507       moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
```
```   508         by simp
```
```   509       ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
```
```   510         by (simp add: mult_less_cancel_right)
```
```   511       with neq show ?thesis
```
```   512         by (simp add: mult_ac)
```
```   513     qed
```
```   514   qed
```
```   515 qed auto
```
```   516
```
```   517 lemma Rat_induct_pos [case_names Fract, induct type: rat]:
```
```   518   assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
```
```   519   shows "P q"
```
```   520 proof (cases q)
```
```   521   have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)"
```
```   522   proof -
```
```   523     fix a::int and b::int
```
```   524     assume b: "b < 0"
```
```   525     hence "0 < -b" by simp
```
```   526     hence "P (Fract (-a) (-b))" by (rule step)
```
```   527     thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
```
```   528   qed
```
```   529   case (Fract a b)
```
```   530   thus "P q" by (force simp add: linorder_neq_iff step step')
```
```   531 qed
```
```   532
```
```   533 lemma zero_less_Fract_iff:
```
```   534   "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a"
```
```   535   by (simp add: Zero_rat_def zero_less_mult_iff)
```
```   536
```
```   537 lemma Fract_less_zero_iff:
```
```   538   "0 < b \<Longrightarrow> Fract a b < 0 \<longleftrightarrow> a < 0"
```
```   539   by (simp add: Zero_rat_def mult_less_0_iff)
```
```   540
```
```   541 lemma zero_le_Fract_iff:
```
```   542   "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a"
```
```   543   by (simp add: Zero_rat_def zero_le_mult_iff)
```
```   544
```
```   545 lemma Fract_le_zero_iff:
```
```   546   "0 < b \<Longrightarrow> Fract a b \<le> 0 \<longleftrightarrow> a \<le> 0"
```
```   547   by (simp add: Zero_rat_def mult_le_0_iff)
```
```   548
```
```   549 lemma one_less_Fract_iff:
```
```   550   "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a"
```
```   551   by (simp add: One_rat_def mult_less_cancel_right_disj)
```
```   552
```
```   553 lemma Fract_less_one_iff:
```
```   554   "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b"
```
```   555   by (simp add: One_rat_def mult_less_cancel_right_disj)
```
```   556
```
```   557 lemma one_le_Fract_iff:
```
```   558   "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a"
```
```   559   by (simp add: One_rat_def mult_le_cancel_right)
```
```   560
```
```   561 lemma Fract_le_one_iff:
```
```   562   "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b"
```
```   563   by (simp add: One_rat_def mult_le_cancel_right)
```
```   564
```
```   565
```
```   566 subsection {* Arithmetic setup *}
```
```   567
```
```   568 use "Tools/rat_arith.ML"
```
```   569 declaration {* K rat_arith_setup *}
```
```   570
```
```   571
```
```   572 subsection {* Embedding from Rationals to other Fields *}
```
```   573
```
```   574 class field_char_0 = field + ring_char_0
```
```   575
```
```   576 subclass (in ordered_field) field_char_0 ..
```
```   577
```
```   578 context field_char_0
```
```   579 begin
```
```   580
```
```   581 definition of_rat :: "rat \<Rightarrow> 'a" where
```
```   582   [code del]: "of_rat q = contents (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})"
```
```   583
```
```   584 end
```
```   585
```
```   586 lemma of_rat_congruent:
```
```   587   "(\<lambda>(a, b). {of_int a / of_int b :: 'a::field_char_0}) respects ratrel"
```
```   588 apply (rule congruent.intro)
```
```   589 apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
```
```   590 apply (simp only: of_int_mult [symmetric])
```
```   591 done
```
```   592
```
```   593 lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
```
```   594   unfolding Fract_def of_rat_def by (simp add: UN_ratrel of_rat_congruent)
```
```   595
```
```   596 lemma of_rat_0 [simp]: "of_rat 0 = 0"
```
```   597 by (simp add: Zero_rat_def of_rat_rat)
```
```   598
```
```   599 lemma of_rat_1 [simp]: "of_rat 1 = 1"
```
```   600 by (simp add: One_rat_def of_rat_rat)
```
```   601
```
```   602 lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
```
```   603 by (induct a, induct b, simp add: of_rat_rat add_frac_eq)
```
```   604
```
```   605 lemma of_rat_minus: "of_rat (- a) = - of_rat a"
```
```   606 by (induct a, simp add: of_rat_rat)
```
```   607
```
```   608 lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
```
```   609 by (simp only: diff_minus of_rat_add of_rat_minus)
```
```   610
```
```   611 lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
```
```   612 apply (induct a, induct b, simp add: of_rat_rat)
```
```   613 apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
```
```   614 done
```
```   615
```
```   616 lemma nonzero_of_rat_inverse:
```
```   617   "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
```
```   618 apply (rule inverse_unique [symmetric])
```
```   619 apply (simp add: of_rat_mult [symmetric])
```
```   620 done
```
```   621
```
```   622 lemma of_rat_inverse:
```
```   623   "(of_rat (inverse a)::'a::{field_char_0,division_by_zero}) =
```
```   624    inverse (of_rat a)"
```
```   625 by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
```
```   626
```
```   627 lemma nonzero_of_rat_divide:
```
```   628   "b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
```
```   629 by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
```
```   630
```
```   631 lemma of_rat_divide:
```
```   632   "(of_rat (a / b)::'a::{field_char_0,division_by_zero})
```
```   633    = of_rat a / of_rat b"
```
```   634 by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)
```
```   635
```
```   636 lemma of_rat_power:
```
```   637   "(of_rat (a ^ n)::'a::{field_char_0,recpower}) = of_rat a ^ n"
```
```   638 by (induct n) (simp_all add: of_rat_mult power_Suc)
```
```   639
```
```   640 lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
```
```   641 apply (induct a, induct b)
```
```   642 apply (simp add: of_rat_rat eq_rat)
```
```   643 apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
```
```   644 apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
```
```   645 done
```
```   646
```
```   647 lemma of_rat_less:
```
```   648   "(of_rat r :: 'a::ordered_field) < of_rat s \<longleftrightarrow> r < s"
```
```   649 proof (induct r, induct s)
```
```   650   fix a b c d :: int
```
```   651   assume not_zero: "b > 0" "d > 0"
```
```   652   then have "b * d > 0" by (rule mult_pos_pos)
```
```   653   have of_int_divide_less_eq:
```
```   654     "(of_int a :: 'a) / of_int b < of_int c / of_int d
```
```   655       \<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b"
```
```   656     using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
```
```   657   show "(of_rat (Fract a b) :: 'a::ordered_field) < of_rat (Fract c d)
```
```   658     \<longleftrightarrow> Fract a b < Fract c d"
```
```   659     using not_zero `b * d > 0`
```
```   660     by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult)
```
```   661       (auto intro: mult_strict_right_mono mult_right_less_imp_less)
```
```   662 qed
```
```   663
```
```   664 lemma of_rat_less_eq:
```
```   665   "(of_rat r :: 'a::ordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s"
```
```   666   unfolding le_less by (auto simp add: of_rat_less)
```
```   667
```
```   668 lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
```
```   669
```
```   670 lemma of_rat_eq_id [simp]: "of_rat = id"
```
```   671 proof
```
```   672   fix a
```
```   673   show "of_rat a = id a"
```
```   674   by (induct a)
```
```   675      (simp add: of_rat_rat Fract_of_int_eq [symmetric])
```
```   676 qed
```
```   677
```
```   678 text{*Collapse nested embeddings*}
```
```   679 lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
```
```   680 by (induct n) (simp_all add: of_rat_add)
```
```   681
```
```   682 lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
```
```   683 by (cases z rule: int_diff_cases) (simp add: of_rat_diff)
```
```   684
```
```   685 lemma of_rat_number_of_eq [simp]:
```
```   686   "of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})"
```
```   687 by (simp add: number_of_eq)
```
```   688
```
```   689 lemmas zero_rat = Zero_rat_def
```
```   690 lemmas one_rat = One_rat_def
```
```   691
```
```   692 abbreviation
```
```   693   rat_of_nat :: "nat \<Rightarrow> rat"
```
```   694 where
```
```   695   "rat_of_nat \<equiv> of_nat"
```
```   696
```
```   697 abbreviation
```
```   698   rat_of_int :: "int \<Rightarrow> rat"
```
```   699 where
```
```   700   "rat_of_int \<equiv> of_int"
```
```   701
```
```   702 subsection {* The Set of Rational Numbers *}
```
```   703
```
```   704 context field_char_0
```
```   705 begin
```
```   706
```
```   707 definition
```
```   708   Rats  :: "'a set" where
```
```   709   [code del]: "Rats = range of_rat"
```
```   710
```
```   711 notation (xsymbols)
```
```   712   Rats  ("\<rat>")
```
```   713
```
```   714 end
```
```   715
```
```   716 lemma Rats_of_rat [simp]: "of_rat r \<in> Rats"
```
```   717 by (simp add: Rats_def)
```
```   718
```
```   719 lemma Rats_of_int [simp]: "of_int z \<in> Rats"
```
```   720 by (subst of_rat_of_int_eq [symmetric], rule Rats_of_rat)
```
```   721
```
```   722 lemma Rats_of_nat [simp]: "of_nat n \<in> Rats"
```
```   723 by (subst of_rat_of_nat_eq [symmetric], rule Rats_of_rat)
```
```   724
```
```   725 lemma Rats_number_of [simp]:
```
```   726   "(number_of w::'a::{number_ring,field_char_0}) \<in> Rats"
```
```   727 by (subst of_rat_number_of_eq [symmetric], rule Rats_of_rat)
```
```   728
```
```   729 lemma Rats_0 [simp]: "0 \<in> Rats"
```
```   730 apply (unfold Rats_def)
```
```   731 apply (rule range_eqI)
```
```   732 apply (rule of_rat_0 [symmetric])
```
```   733 done
```
```   734
```
```   735 lemma Rats_1 [simp]: "1 \<in> Rats"
```
```   736 apply (unfold Rats_def)
```
```   737 apply (rule range_eqI)
```
```   738 apply (rule of_rat_1 [symmetric])
```
```   739 done
```
```   740
```
```   741 lemma Rats_add [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a + b \<in> Rats"
```
```   742 apply (auto simp add: Rats_def)
```
```   743 apply (rule range_eqI)
```
```   744 apply (rule of_rat_add [symmetric])
```
```   745 done
```
```   746
```
```   747 lemma Rats_minus [simp]: "a \<in> Rats \<Longrightarrow> - a \<in> Rats"
```
```   748 apply (auto simp add: Rats_def)
```
```   749 apply (rule range_eqI)
```
```   750 apply (rule of_rat_minus [symmetric])
```
```   751 done
```
```   752
```
```   753 lemma Rats_diff [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a - b \<in> Rats"
```
```   754 apply (auto simp add: Rats_def)
```
```   755 apply (rule range_eqI)
```
```   756 apply (rule of_rat_diff [symmetric])
```
```   757 done
```
```   758
```
```   759 lemma Rats_mult [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a * b \<in> Rats"
```
```   760 apply (auto simp add: Rats_def)
```
```   761 apply (rule range_eqI)
```
```   762 apply (rule of_rat_mult [symmetric])
```
```   763 done
```
```   764
```
```   765 lemma nonzero_Rats_inverse:
```
```   766   fixes a :: "'a::field_char_0"
```
```   767   shows "\<lbrakk>a \<in> Rats; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Rats"
```
```   768 apply (auto simp add: Rats_def)
```
```   769 apply (rule range_eqI)
```
```   770 apply (erule nonzero_of_rat_inverse [symmetric])
```
```   771 done
```
```   772
```
```   773 lemma Rats_inverse [simp]:
```
```   774   fixes a :: "'a::{field_char_0,division_by_zero}"
```
```   775   shows "a \<in> Rats \<Longrightarrow> inverse a \<in> Rats"
```
```   776 apply (auto simp add: Rats_def)
```
```   777 apply (rule range_eqI)
```
```   778 apply (rule of_rat_inverse [symmetric])
```
```   779 done
```
```   780
```
```   781 lemma nonzero_Rats_divide:
```
```   782   fixes a b :: "'a::field_char_0"
```
```   783   shows "\<lbrakk>a \<in> Rats; b \<in> Rats; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
```
```   784 apply (auto simp add: Rats_def)
```
```   785 apply (rule range_eqI)
```
```   786 apply (erule nonzero_of_rat_divide [symmetric])
```
```   787 done
```
```   788
```
```   789 lemma Rats_divide [simp]:
```
```   790   fixes a b :: "'a::{field_char_0,division_by_zero}"
```
```   791   shows "\<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_divide [symmetric])
```
```   795 done
```
```   796
```
```   797 lemma Rats_power [simp]:
```
```   798   fixes a :: "'a::{field_char_0,recpower}"
```
```   799   shows "a \<in> Rats \<Longrightarrow> a ^ n \<in> Rats"
```
```   800 apply (auto simp add: Rats_def)
```
```   801 apply (rule range_eqI)
```
```   802 apply (rule of_rat_power [symmetric])
```
```   803 done
```
```   804
```
```   805 lemma Rats_cases [cases set: Rats]:
```
```   806   assumes "q \<in> \<rat>"
```
```   807   obtains (of_rat) r where "q = of_rat r"
```
```   808   unfolding Rats_def
```
```   809 proof -
```
```   810   from `q \<in> \<rat>` have "q \<in> range of_rat" unfolding Rats_def .
```
```   811   then obtain r where "q = of_rat r" ..
```
```   812   then show thesis ..
```
```   813 qed
```
```   814
```
```   815 lemma Rats_induct [case_names of_rat, induct set: Rats]:
```
```   816   "q \<in> \<rat> \<Longrightarrow> (\<And>r. P (of_rat r)) \<Longrightarrow> P q"
```
```   817   by (rule Rats_cases) auto
```
```   818
```
```   819
```
```   820 subsection {* Implementation of rational numbers as pairs of integers *}
```
```   821
```
```   822 lemma Fract_norm: "Fract (a div zgcd a b) (b div zgcd a b) = Fract a b"
```
```   823 proof (cases "a = 0 \<or> b = 0")
```
```   824   case True then show ?thesis by (auto simp add: eq_rat)
```
```   825 next
```
```   826   let ?c = "zgcd a b"
```
```   827   case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
```
```   828   then have "?c \<noteq> 0" by simp
```
```   829   then have "Fract ?c ?c = Fract 1 1" by (simp add: eq_rat)
```
```   830   moreover have "Fract (a div ?c * ?c + a mod ?c) (b div ?c * ?c + b mod ?c) = Fract a b"
```
```   831     by (simp add: semiring_div_class.mod_div_equality)
```
```   832   moreover have "a mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric])
```
```   833   moreover have "b mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric])
```
```   834   ultimately show ?thesis
```
```   835     by (simp add: mult_rat [symmetric])
```
```   836 qed
```
```   837
```
```   838 definition Fract_norm :: "int \<Rightarrow> int \<Rightarrow> rat" where
```
```   839   [simp, code del]: "Fract_norm a b = Fract a b"
```
```   840
```
```   841 lemma Fract_norm_code [code]: "Fract_norm a b = (if a = 0 \<or> b = 0 then 0 else let c = zgcd a b in
```
```   842   if b > 0 then Fract (a div c) (b div c) else Fract (- (a div c)) (- (b div c)))"
```
```   843   by (simp add: eq_rat Zero_rat_def Let_def Fract_norm)
```
```   844
```
```   845 lemma [code]:
```
```   846   "of_rat (Fract a b) = (if b \<noteq> 0 then of_int a / of_int b else 0)"
```
```   847   by (cases "b = 0") (simp_all add: rat_number_collapse of_rat_rat)
```
```   848
```
```   849 instantiation rat :: eq
```
```   850 begin
```
```   851
```
```   852 definition [code del]: "eq_class.eq (a\<Colon>rat) b \<longleftrightarrow> a - b = 0"
```
```   853
```
```   854 instance by default (simp add: eq_rat_def)
```
```   855
```
```   856 lemma rat_eq_code [code]:
```
```   857   "eq_class.eq (Fract a b) (Fract c d) \<longleftrightarrow> (if b = 0
```
```   858        then c = 0 \<or> d = 0
```
```   859      else if d = 0
```
```   860        then a = 0 \<or> b = 0
```
```   861      else a * d = b * c)"
```
```   862   by (auto simp add: eq eq_rat)
```
```   863
```
```   864 lemma rat_eq_refl [code nbe]:
```
```   865   "eq_class.eq (r::rat) r \<longleftrightarrow> True"
```
```   866   by (rule HOL.eq_refl)
```
```   867
```
```   868 end
```
```   869
```
```   870 lemma le_rat':
```
```   871   assumes "b \<noteq> 0"
```
```   872     and "d \<noteq> 0"
```
```   873   shows "Fract a b \<le> Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d"
```
```   874 proof -
```
```   875   have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp
```
```   876   have "a * d * (b * d) \<le> c * b * (b * d) \<longleftrightarrow> a * d * (sgn b * sgn d) \<le> c * b * (sgn b * sgn d)"
```
```   877   proof (cases "b * d > 0")
```
```   878     case True
```
```   879     moreover from True have "sgn b * sgn d = 1"
```
```   880       by (simp add: sgn_times [symmetric] sgn_1_pos)
```
```   881     ultimately show ?thesis by (simp add: mult_le_cancel_right)
```
```   882   next
```
```   883     case False with assms have "b * d < 0" by (simp add: less_le)
```
```   884     moreover from this have "sgn b * sgn d = - 1"
```
```   885       by (simp only: sgn_times [symmetric] sgn_1_neg)
```
```   886     ultimately show ?thesis by (simp add: mult_le_cancel_right)
```
```   887   qed
```
```   888   also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d"
```
```   889     by (simp add: abs_sgn mult_ac)
```
```   890   finally show ?thesis using assms by simp
```
```   891 qed
```
```   892
```
```   893 lemma less_rat':
```
```   894   assumes "b \<noteq> 0"
```
```   895     and "d \<noteq> 0"
```
```   896   shows "Fract a b < Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d"
```
```   897 proof -
```
```   898   have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp
```
```   899   have "a * d * (b * d) < c * b * (b * d) \<longleftrightarrow> a * d * (sgn b * sgn d) < c * b * (sgn b * sgn d)"
```
```   900   proof (cases "b * d > 0")
```
```   901     case True
```
```   902     moreover from True have "sgn b * sgn d = 1"
```
```   903       by (simp add: sgn_times [symmetric] sgn_1_pos)
```
```   904     ultimately show ?thesis by (simp add: mult_less_cancel_right)
```
```   905   next
```
```   906     case False with assms have "b * d < 0" by (simp add: less_le)
```
```   907     moreover from this have "sgn b * sgn d = - 1"
```
```   908       by (simp only: sgn_times [symmetric] sgn_1_neg)
```
```   909     ultimately show ?thesis by (simp add: mult_less_cancel_right)
```
```   910   qed
```
```   911   also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d"
```
```   912     by (simp add: abs_sgn mult_ac)
```
```   913   finally show ?thesis using assms by simp
```
```   914 qed
```
```   915
```
```   916 lemma (in ordered_idom) sgn_greater [simp]:
```
```   917   "0 < sgn a \<longleftrightarrow> 0 < a"
```
```   918   unfolding sgn_if by auto
```
```   919
```
```   920 lemma (in ordered_idom) sgn_less [simp]:
```
```   921   "sgn a < 0 \<longleftrightarrow> a < 0"
```
```   922   unfolding sgn_if by auto
```
```   923
```
```   924 lemma rat_le_eq_code [code]:
```
```   925   "Fract a b < Fract c d \<longleftrightarrow> (if b = 0
```
```   926        then sgn c * sgn d > 0
```
```   927      else if d = 0
```
```   928        then sgn a * sgn b < 0
```
```   929      else a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d)"
```
```   930   by (auto simp add: sgn_times mult_less_0_iff zero_less_mult_iff less_rat' eq_rat simp del: less_rat)
```
```   931
```
```   932 lemma rat_less_eq_code [code]:
```
```   933   "Fract a b \<le> Fract c d \<longleftrightarrow> (if b = 0
```
```   934        then sgn c * sgn d \<ge> 0
```
```   935      else if d = 0
```
```   936        then sgn a * sgn b \<le> 0
```
```   937      else a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d)"
```
```   938   by (auto simp add: sgn_times mult_le_0_iff zero_le_mult_iff le_rat' eq_rat simp del: le_rat)
```
```   939     (auto simp add: le_less not_less sgn_0_0)
```
```   940
```
```   941
```
```   942 lemma rat_plus_code [code]:
```
```   943   "Fract a b + Fract c d = (if b = 0
```
```   944      then Fract c d
```
```   945    else if d = 0
```
```   946      then Fract a b
```
```   947    else Fract_norm (a * d + c * b) (b * d))"
```
```   948   by (simp add: eq_rat, simp add: Zero_rat_def)
```
```   949
```
```   950 lemma rat_times_code [code]:
```
```   951   "Fract a b * Fract c d = Fract_norm (a * c) (b * d)"
```
```   952   by simp
```
```   953
```
```   954 lemma rat_minus_code [code]:
```
```   955   "Fract a b - Fract c d = (if b = 0
```
```   956      then Fract (- c) d
```
```   957    else if d = 0
```
```   958      then Fract a b
```
```   959    else Fract_norm (a * d - c * b) (b * d))"
```
```   960   by (simp add: eq_rat, simp add: Zero_rat_def)
```
```   961
```
```   962 lemma rat_inverse_code [code]:
```
```   963   "inverse (Fract a b) = (if b = 0 then Fract 1 0
```
```   964     else if a < 0 then Fract (- b) (- a)
```
```   965     else Fract b a)"
```
```   966   by (simp add: eq_rat)
```
```   967
```
```   968 lemma rat_divide_code [code]:
```
```   969   "Fract a b / Fract c d = Fract_norm (a * d) (b * c)"
```
```   970   by simp
```
```   971
```
```   972 hide (open) const Fract_norm
```
```   973
```
```   974 text {* Setup for SML code generator *}
```
```   975
```
```   976 types_code
```
```   977   rat ("(int */ int)")
```
```   978 attach (term_of) {*
```
```   979 fun term_of_rat (p, q) =
```
```   980   let
```
```   981     val rT = Type ("Rational.rat", [])
```
```   982   in
```
```   983     if q = 1 orelse p = 0 then HOLogic.mk_number rT p
```
```   984     else @{term "op / \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat"} \$
```
```   985       HOLogic.mk_number rT p \$ HOLogic.mk_number rT q
```
```   986   end;
```
```   987 *}
```
```   988 attach (test) {*
```
```   989 fun gen_rat i =
```
```   990   let
```
```   991     val p = random_range 0 i;
```
```   992     val q = random_range 1 (i + 1);
```
```   993     val g = Integer.gcd p q;
```
```   994     val p' = p div g;
```
```   995     val q' = q div g;
```
```   996     val r = (if one_of [true, false] then p' else ~ p',
```
```   997       if p' = 0 then 0 else q')
```
```   998   in
```
```   999     (r, fn () => term_of_rat r)
```
```  1000   end;
```
```  1001 *}
```
```  1002
```
```  1003 consts_code
```
```  1004   Fract ("(_,/ _)")
```
```  1005
```
```  1006 consts_code
```
```  1007   "of_int :: int \<Rightarrow> rat" ("\<module>rat'_of'_int")
```
```  1008 attach {*
```
```  1009 fun rat_of_int 0 = (0, 0)
```
```  1010   | rat_of_int i = (i, 1);
```
```  1011 *}
```
```  1012
```
```  1013 end
```