src/HOL/Rat.thy
 author hoelzl Thu Jan 31 11:31:27 2013 +0100 (2013-01-31) changeset 50999 3de230ed0547 parent 50178 ad52ddd35c3a child 51126 df86080de4cb permissions -rw-r--r--
introduce order topology
```     1 (*  Title:  HOL/Rat.thy
```
```     2     Author: Markus Wenzel, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* Rational numbers *}
```
```     6
```
```     7 theory Rat
```
```     8 imports GCD Archimedean_Field
```
```     9 begin
```
```    10
```
```    11 subsection {* Rational numbers as quotient *}
```
```    12
```
```    13 subsubsection {* Construction of the type of rational numbers *}
```
```    14
```
```    15 definition
```
```    16   ratrel :: "(int \<times> int) \<Rightarrow> (int \<times> int) \<Rightarrow> bool" where
```
```    17   "ratrel = (\<lambda>x y. snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x)"
```
```    18
```
```    19 lemma ratrel_iff [simp]:
```
```    20   "ratrel x y \<longleftrightarrow> snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x"
```
```    21   by (simp add: ratrel_def)
```
```    22
```
```    23 lemma exists_ratrel_refl: "\<exists>x. ratrel x x"
```
```    24   by (auto intro!: one_neq_zero)
```
```    25
```
```    26 lemma symp_ratrel: "symp ratrel"
```
```    27   by (simp add: ratrel_def symp_def)
```
```    28
```
```    29 lemma transp_ratrel: "transp ratrel"
```
```    30 proof (rule transpI, unfold split_paired_all)
```
```    31   fix a b a' b' a'' b'' :: int
```
```    32   assume A: "ratrel (a, b) (a', b')"
```
```    33   assume B: "ratrel (a', b') (a'', b'')"
```
```    34   have "b' * (a * b'') = b'' * (a * b')" by simp
```
```    35   also from A have "a * b' = a' * b" by auto
```
```    36   also have "b'' * (a' * b) = b * (a' * b'')" by simp
```
```    37   also from B have "a' * b'' = a'' * b'" by auto
```
```    38   also have "b * (a'' * b') = b' * (a'' * b)" by simp
```
```    39   finally have "b' * (a * b'') = b' * (a'' * b)" .
```
```    40   moreover from B have "b' \<noteq> 0" by auto
```
```    41   ultimately have "a * b'' = a'' * b" by simp
```
```    42   with A B show "ratrel (a, b) (a'', b'')" by auto
```
```    43 qed
```
```    44
```
```    45 lemma part_equivp_ratrel: "part_equivp ratrel"
```
```    46   by (rule part_equivpI [OF exists_ratrel_refl symp_ratrel transp_ratrel])
```
```    47
```
```    48 quotient_type rat = "int \<times> int" / partial: "ratrel"
```
```    49   morphisms Rep_Rat Abs_Rat
```
```    50   by (rule part_equivp_ratrel)
```
```    51
```
```    52 declare rat.forall_transfer [transfer_rule del]
```
```    53
```
```    54 lemma forall_rat_transfer [transfer_rule]: (* TODO: generate automatically *)
```
```    55   "(fun_rel (fun_rel cr_rat op =) op =)
```
```    56     (transfer_bforall (\<lambda>x. snd x \<noteq> 0)) transfer_forall"
```
```    57   using rat.forall_transfer by simp
```
```    58
```
```    59
```
```    60 subsubsection {* Representation and basic operations *}
```
```    61
```
```    62 lift_definition Fract :: "int \<Rightarrow> int \<Rightarrow> rat"
```
```    63   is "\<lambda>a b. if b = 0 then (0, 1) else (a, b)"
```
```    64   by simp
```
```    65
```
```    66 lemma eq_rat:
```
```    67   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"
```
```    68   and "\<And>a. Fract a 0 = Fract 0 1"
```
```    69   and "\<And>a c. Fract 0 a = Fract 0 c"
```
```    70   by (transfer, simp)+
```
```    71
```
```    72 lemma Rat_cases [case_names Fract, cases type: rat]:
```
```    73   assumes "\<And>a b. q = Fract a b \<Longrightarrow> b > 0 \<Longrightarrow> coprime a b \<Longrightarrow> C"
```
```    74   shows C
```
```    75 proof -
```
```    76   obtain a b :: int where "q = Fract a b" and "b \<noteq> 0"
```
```    77     by transfer simp
```
```    78   let ?a = "a div gcd a b"
```
```    79   let ?b = "b div gcd a b"
```
```    80   from `b \<noteq> 0` have "?b * gcd a b = b"
```
```    81     by (simp add: dvd_div_mult_self)
```
```    82   with `b \<noteq> 0` have "?b \<noteq> 0" by auto
```
```    83   from `q = Fract a b` `b \<noteq> 0` `?b \<noteq> 0` have q: "q = Fract ?a ?b"
```
```    84     by (simp add: eq_rat dvd_div_mult mult_commute [of a])
```
```    85   from `b \<noteq> 0` have coprime: "coprime ?a ?b"
```
```    86     by (auto intro: div_gcd_coprime_int)
```
```    87   show C proof (cases "b > 0")
```
```    88     case True
```
```    89     note assms
```
```    90     moreover note q
```
```    91     moreover from True have "?b > 0" by (simp add: nonneg1_imp_zdiv_pos_iff)
```
```    92     moreover note coprime
```
```    93     ultimately show C .
```
```    94   next
```
```    95     case False
```
```    96     note assms
```
```    97     moreover have "q = Fract (- ?a) (- ?b)" unfolding q by transfer simp
```
```    98     moreover from False `b \<noteq> 0` have "- ?b > 0" by (simp add: pos_imp_zdiv_neg_iff)
```
```    99     moreover from coprime have "coprime (- ?a) (- ?b)" by simp
```
```   100     ultimately show C .
```
```   101   qed
```
```   102 qed
```
```   103
```
```   104 lemma Rat_induct [case_names Fract, induct type: rat]:
```
```   105   assumes "\<And>a b. b > 0 \<Longrightarrow> coprime a b \<Longrightarrow> P (Fract a b)"
```
```   106   shows "P q"
```
```   107   using assms by (cases q) simp
```
```   108
```
```   109 instantiation rat :: field_inverse_zero
```
```   110 begin
```
```   111
```
```   112 lift_definition zero_rat :: "rat" is "(0, 1)"
```
```   113   by simp
```
```   114
```
```   115 lift_definition one_rat :: "rat" is "(1, 1)"
```
```   116   by simp
```
```   117
```
```   118 lemma Zero_rat_def: "0 = Fract 0 1"
```
```   119   by transfer simp
```
```   120
```
```   121 lemma One_rat_def: "1 = Fract 1 1"
```
```   122   by transfer simp
```
```   123
```
```   124 lift_definition plus_rat :: "rat \<Rightarrow> rat \<Rightarrow> rat"
```
```   125   is "\<lambda>x y. (fst x * snd y + fst y * snd x, snd x * snd y)"
```
```   126   by (clarsimp, simp add: distrib_right, simp add: mult_ac)
```
```   127
```
```   128 lemma add_rat [simp]:
```
```   129   assumes "b \<noteq> 0" and "d \<noteq> 0"
```
```   130   shows "Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
```
```   131   using assms by transfer simp
```
```   132
```
```   133 lift_definition uminus_rat :: "rat \<Rightarrow> rat" is "\<lambda>x. (- fst x, snd x)"
```
```   134   by simp
```
```   135
```
```   136 lemma minus_rat [simp]: "- Fract a b = Fract (- a) b"
```
```   137   by transfer simp
```
```   138
```
```   139 lemma minus_rat_cancel [simp]: "Fract (- a) (- b) = Fract a b"
```
```   140   by (cases "b = 0") (simp_all add: eq_rat)
```
```   141
```
```   142 definition
```
```   143   diff_rat_def: "q - r = q + - (r::rat)"
```
```   144
```
```   145 lemma diff_rat [simp]:
```
```   146   assumes "b \<noteq> 0" and "d \<noteq> 0"
```
```   147   shows "Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
```
```   148   using assms by (simp add: diff_rat_def)
```
```   149
```
```   150 lift_definition times_rat :: "rat \<Rightarrow> rat \<Rightarrow> rat"
```
```   151   is "\<lambda>x y. (fst x * fst y, snd x * snd y)"
```
```   152   by (simp add: mult_ac)
```
```   153
```
```   154 lemma mult_rat [simp]: "Fract a b * Fract c d = Fract (a * c) (b * d)"
```
```   155   by transfer simp
```
```   156
```
```   157 lemma mult_rat_cancel:
```
```   158   assumes "c \<noteq> 0"
```
```   159   shows "Fract (c * a) (c * b) = Fract a b"
```
```   160   using assms by transfer simp
```
```   161
```
```   162 lift_definition inverse_rat :: "rat \<Rightarrow> rat"
```
```   163   is "\<lambda>x. if fst x = 0 then (0, 1) else (snd x, fst x)"
```
```   164   by (auto simp add: mult_commute)
```
```   165
```
```   166 lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"
```
```   167   by transfer simp
```
```   168
```
```   169 definition
```
```   170   divide_rat_def: "q / r = q * inverse (r::rat)"
```
```   171
```
```   172 lemma divide_rat [simp]: "Fract a b / Fract c d = Fract (a * d) (b * c)"
```
```   173   by (simp add: divide_rat_def)
```
```   174
```
```   175 instance proof
```
```   176   fix q r s :: rat
```
```   177   show "(q * r) * s = q * (r * s)"
```
```   178     by transfer simp
```
```   179   show "q * r = r * q"
```
```   180     by transfer simp
```
```   181   show "1 * q = q"
```
```   182     by transfer simp
```
```   183   show "(q + r) + s = q + (r + s)"
```
```   184     by transfer (simp add: algebra_simps)
```
```   185   show "q + r = r + q"
```
```   186     by transfer simp
```
```   187   show "0 + q = q"
```
```   188     by transfer simp
```
```   189   show "- q + q = 0"
```
```   190     by transfer simp
```
```   191   show "q - r = q + - r"
```
```   192     by (fact diff_rat_def)
```
```   193   show "(q + r) * s = q * s + r * s"
```
```   194     by transfer (simp add: algebra_simps)
```
```   195   show "(0::rat) \<noteq> 1"
```
```   196     by transfer simp
```
```   197   { assume "q \<noteq> 0" thus "inverse q * q = 1"
```
```   198     by transfer simp }
```
```   199   show "q / r = q * inverse r"
```
```   200     by (fact divide_rat_def)
```
```   201   show "inverse 0 = (0::rat)"
```
```   202     by transfer simp
```
```   203 qed
```
```   204
```
```   205 end
```
```   206
```
```   207 lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1"
```
```   208   by (induct k) (simp_all add: Zero_rat_def One_rat_def)
```
```   209
```
```   210 lemma of_int_rat: "of_int k = Fract k 1"
```
```   211   by (cases k rule: int_diff_cases) (simp add: of_nat_rat)
```
```   212
```
```   213 lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
```
```   214   by (rule of_nat_rat [symmetric])
```
```   215
```
```   216 lemma Fract_of_int_eq: "Fract k 1 = of_int k"
```
```   217   by (rule of_int_rat [symmetric])
```
```   218
```
```   219 lemma rat_number_collapse:
```
```   220   "Fract 0 k = 0"
```
```   221   "Fract 1 1 = 1"
```
```   222   "Fract (numeral w) 1 = numeral w"
```
```   223   "Fract (neg_numeral w) 1 = neg_numeral w"
```
```   224   "Fract k 0 = 0"
```
```   225   using Fract_of_int_eq [of "numeral w"]
```
```   226   using Fract_of_int_eq [of "neg_numeral w"]
```
```   227   by (simp_all add: Zero_rat_def One_rat_def eq_rat)
```
```   228
```
```   229 lemma rat_number_expand:
```
```   230   "0 = Fract 0 1"
```
```   231   "1 = Fract 1 1"
```
```   232   "numeral k = Fract (numeral k) 1"
```
```   233   "neg_numeral k = Fract (neg_numeral k) 1"
```
```   234   by (simp_all add: rat_number_collapse)
```
```   235
```
```   236 lemma Rat_cases_nonzero [case_names Fract 0]:
```
```   237   assumes Fract: "\<And>a b. q = Fract a b \<Longrightarrow> b > 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> coprime a b \<Longrightarrow> C"
```
```   238   assumes 0: "q = 0 \<Longrightarrow> C"
```
```   239   shows C
```
```   240 proof (cases "q = 0")
```
```   241   case True then show C using 0 by auto
```
```   242 next
```
```   243   case False
```
```   244   then obtain a b where "q = Fract a b" and "b > 0" and "coprime a b" by (cases q) auto
```
```   245   moreover with False have "0 \<noteq> Fract a b" by simp
```
```   246   with `b > 0` have "a \<noteq> 0" by (simp add: Zero_rat_def eq_rat)
```
```   247   with Fract `q = Fract a b` `b > 0` `coprime a b` show C by blast
```
```   248 qed
```
```   249
```
```   250 subsubsection {* Function @{text normalize} *}
```
```   251
```
```   252 lemma Fract_coprime: "Fract (a div gcd a b) (b div gcd a b) = Fract a b"
```
```   253 proof (cases "b = 0")
```
```   254   case True then show ?thesis by (simp add: eq_rat)
```
```   255 next
```
```   256   case False
```
```   257   moreover have "b div gcd a b * gcd a b = b"
```
```   258     by (rule dvd_div_mult_self) simp
```
```   259   ultimately have "b div gcd a b \<noteq> 0" by auto
```
```   260   with False show ?thesis by (simp add: eq_rat dvd_div_mult mult_commute [of a])
```
```   261 qed
```
```   262
```
```   263 definition normalize :: "int \<times> int \<Rightarrow> int \<times> int" where
```
```   264   "normalize p = (if snd p > 0 then (let a = gcd (fst p) (snd p) in (fst p div a, snd p div a))
```
```   265     else if snd p = 0 then (0, 1)
```
```   266     else (let a = - gcd (fst p) (snd p) in (fst p div a, snd p div a)))"
```
```   267
```
```   268 lemma normalize_crossproduct:
```
```   269   assumes "q \<noteq> 0" "s \<noteq> 0"
```
```   270   assumes "normalize (p, q) = normalize (r, s)"
```
```   271   shows "p * s = r * q"
```
```   272 proof -
```
```   273   have aux: "p * gcd r s = sgn (q * s) * r * gcd p q \<Longrightarrow> q * gcd r s = sgn (q * s) * s * gcd p q \<Longrightarrow> p * s = q * r"
```
```   274   proof -
```
```   275     assume "p * gcd r s = sgn (q * s) * r * gcd p q" and "q * gcd r s = sgn (q * s) * s * gcd p q"
```
```   276     then have "(p * gcd r s) * (sgn (q * s) * s * gcd p q) = (q * gcd r s) * (sgn (q * s) * r * gcd p q)" by simp
```
```   277     with assms show "p * s = q * r" by (auto simp add: mult_ac sgn_times sgn_0_0)
```
```   278   qed
```
```   279   from assms show ?thesis
```
```   280     by (auto simp add: normalize_def Let_def dvd_div_div_eq_mult mult_commute sgn_times split: if_splits intro: aux)
```
```   281 qed
```
```   282
```
```   283 lemma normalize_eq: "normalize (a, b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
```
```   284   by (auto simp add: normalize_def Let_def Fract_coprime dvd_div_neg rat_number_collapse
```
```   285     split:split_if_asm)
```
```   286
```
```   287 lemma normalize_denom_pos: "normalize r = (p, q) \<Longrightarrow> q > 0"
```
```   288   by (auto simp add: normalize_def Let_def dvd_div_neg pos_imp_zdiv_neg_iff nonneg1_imp_zdiv_pos_iff
```
```   289     split:split_if_asm)
```
```   290
```
```   291 lemma normalize_coprime: "normalize r = (p, q) \<Longrightarrow> coprime p q"
```
```   292   by (auto simp add: normalize_def Let_def dvd_div_neg div_gcd_coprime_int
```
```   293     split:split_if_asm)
```
```   294
```
```   295 lemma normalize_stable [simp]:
```
```   296   "q > 0 \<Longrightarrow> coprime p q \<Longrightarrow> normalize (p, q) = (p, q)"
```
```   297   by (simp add: normalize_def)
```
```   298
```
```   299 lemma normalize_denom_zero [simp]:
```
```   300   "normalize (p, 0) = (0, 1)"
```
```   301   by (simp add: normalize_def)
```
```   302
```
```   303 lemma normalize_negative [simp]:
```
```   304   "q < 0 \<Longrightarrow> normalize (p, q) = normalize (- p, - q)"
```
```   305   by (simp add: normalize_def Let_def dvd_div_neg dvd_neg_div)
```
```   306
```
```   307 text{*
```
```   308   Decompose a fraction into normalized, i.e. coprime numerator and denominator:
```
```   309 *}
```
```   310
```
```   311 definition quotient_of :: "rat \<Rightarrow> int \<times> int" where
```
```   312   "quotient_of x = (THE pair. x = Fract (fst pair) (snd pair) &
```
```   313                    snd pair > 0 & coprime (fst pair) (snd pair))"
```
```   314
```
```   315 lemma quotient_of_unique:
```
```   316   "\<exists>!p. r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
```
```   317 proof (cases r)
```
```   318   case (Fract a b)
```
```   319   then have "r = Fract (fst (a, b)) (snd (a, b)) \<and> snd (a, b) > 0 \<and> coprime (fst (a, b)) (snd (a, b))" by auto
```
```   320   then show ?thesis proof (rule ex1I)
```
```   321     fix p
```
```   322     obtain c d :: int where p: "p = (c, d)" by (cases p)
```
```   323     assume "r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
```
```   324     with p have Fract': "r = Fract c d" "d > 0" "coprime c d" by simp_all
```
```   325     have "c = a \<and> d = b"
```
```   326     proof (cases "a = 0")
```
```   327       case True with Fract Fract' show ?thesis by (simp add: eq_rat)
```
```   328     next
```
```   329       case False
```
```   330       with Fract Fract' have *: "c * b = a * d" and "c \<noteq> 0" by (auto simp add: eq_rat)
```
```   331       then have "c * b > 0 \<longleftrightarrow> a * d > 0" by auto
```
```   332       with `b > 0` `d > 0` have "a > 0 \<longleftrightarrow> c > 0" by (simp add: zero_less_mult_iff)
```
```   333       with `a \<noteq> 0` `c \<noteq> 0` have sgn: "sgn a = sgn c" by (auto simp add: not_less)
```
```   334       from `coprime a b` `coprime c d` have "\<bar>a\<bar> * \<bar>d\<bar> = \<bar>c\<bar> * \<bar>b\<bar> \<longleftrightarrow> \<bar>a\<bar> = \<bar>c\<bar> \<and> \<bar>d\<bar> = \<bar>b\<bar>"
```
```   335         by (simp add: coprime_crossproduct_int)
```
```   336       with `b > 0` `d > 0` have "\<bar>a\<bar> * d = \<bar>c\<bar> * b \<longleftrightarrow> \<bar>a\<bar> = \<bar>c\<bar> \<and> d = b" by simp
```
```   337       then have "a * sgn a * d = c * sgn c * b \<longleftrightarrow> a * sgn a = c * sgn c \<and> d = b" by (simp add: abs_sgn)
```
```   338       with sgn * show ?thesis by (auto simp add: sgn_0_0)
```
```   339     qed
```
```   340     with p show "p = (a, b)" by simp
```
```   341   qed
```
```   342 qed
```
```   343
```
```   344 lemma quotient_of_Fract [code]:
```
```   345   "quotient_of (Fract a b) = normalize (a, b)"
```
```   346 proof -
```
```   347   have "Fract a b = Fract (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?Fract)
```
```   348     by (rule sym) (auto intro: normalize_eq)
```
```   349   moreover have "0 < snd (normalize (a, b))" (is ?denom_pos)
```
```   350     by (cases "normalize (a, b)") (rule normalize_denom_pos, simp)
```
```   351   moreover have "coprime (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?coprime)
```
```   352     by (rule normalize_coprime) simp
```
```   353   ultimately have "?Fract \<and> ?denom_pos \<and> ?coprime" by blast
```
```   354   with quotient_of_unique have
```
```   355     "(THE p. Fract a b = Fract (fst p) (snd p) \<and> 0 < snd p \<and> coprime (fst p) (snd p)) = normalize (a, b)"
```
```   356     by (rule the1_equality)
```
```   357   then show ?thesis by (simp add: quotient_of_def)
```
```   358 qed
```
```   359
```
```   360 lemma quotient_of_number [simp]:
```
```   361   "quotient_of 0 = (0, 1)"
```
```   362   "quotient_of 1 = (1, 1)"
```
```   363   "quotient_of (numeral k) = (numeral k, 1)"
```
```   364   "quotient_of (neg_numeral k) = (neg_numeral k, 1)"
```
```   365   by (simp_all add: rat_number_expand quotient_of_Fract)
```
```   366
```
```   367 lemma quotient_of_eq: "quotient_of (Fract a b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
```
```   368   by (simp add: quotient_of_Fract normalize_eq)
```
```   369
```
```   370 lemma quotient_of_denom_pos: "quotient_of r = (p, q) \<Longrightarrow> q > 0"
```
```   371   by (cases r) (simp add: quotient_of_Fract normalize_denom_pos)
```
```   372
```
```   373 lemma quotient_of_coprime: "quotient_of r = (p, q) \<Longrightarrow> coprime p q"
```
```   374   by (cases r) (simp add: quotient_of_Fract normalize_coprime)
```
```   375
```
```   376 lemma quotient_of_inject:
```
```   377   assumes "quotient_of a = quotient_of b"
```
```   378   shows "a = b"
```
```   379 proof -
```
```   380   obtain p q r s where a: "a = Fract p q"
```
```   381     and b: "b = Fract r s"
```
```   382     and "q > 0" and "s > 0" by (cases a, cases b)
```
```   383   with assms show ?thesis by (simp add: eq_rat quotient_of_Fract normalize_crossproduct)
```
```   384 qed
```
```   385
```
```   386 lemma quotient_of_inject_eq:
```
```   387   "quotient_of a = quotient_of b \<longleftrightarrow> a = b"
```
```   388   by (auto simp add: quotient_of_inject)
```
```   389
```
```   390
```
```   391 subsubsection {* Various *}
```
```   392
```
```   393 lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"
```
```   394   by (simp add: Fract_of_int_eq [symmetric])
```
```   395
```
```   396 lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
```
```   397   by (simp add: rat_number_expand)
```
```   398
```
```   399 lemma quotient_of_div:
```
```   400   assumes r: "quotient_of r = (n,d)"
```
```   401   shows "r = of_int n / of_int d"
```
```   402 proof -
```
```   403   from theI'[OF quotient_of_unique[of r], unfolded r[unfolded quotient_of_def]]
```
```   404   have "r = Fract n d" by simp
```
```   405   thus ?thesis using Fract_of_int_quotient by simp
```
```   406 qed
```
```   407
```
```   408 subsubsection {* The ordered field of rational numbers *}
```
```   409
```
```   410 lift_definition positive :: "rat \<Rightarrow> bool"
```
```   411   is "\<lambda>x. 0 < fst x * snd x"
```
```   412 proof (clarsimp)
```
```   413   fix a b c d :: int
```
```   414   assume "b \<noteq> 0" and "d \<noteq> 0" and "a * d = c * b"
```
```   415   hence "a * d * b * d = c * b * b * d"
```
```   416     by simp
```
```   417   hence "a * b * d\<twosuperior> = c * d * b\<twosuperior>"
```
```   418     unfolding power2_eq_square by (simp add: mult_ac)
```
```   419   hence "0 < a * b * d\<twosuperior> \<longleftrightarrow> 0 < c * d * b\<twosuperior>"
```
```   420     by simp
```
```   421   thus "0 < a * b \<longleftrightarrow> 0 < c * d"
```
```   422     using `b \<noteq> 0` and `d \<noteq> 0`
```
```   423     by (simp add: zero_less_mult_iff)
```
```   424 qed
```
```   425
```
```   426 lemma positive_zero: "\<not> positive 0"
```
```   427   by transfer simp
```
```   428
```
```   429 lemma positive_add:
```
```   430   "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x + y)"
```
```   431 apply transfer
```
```   432 apply (simp add: zero_less_mult_iff)
```
```   433 apply (elim disjE, simp_all add: add_pos_pos add_neg_neg
```
```   434   mult_pos_pos mult_pos_neg mult_neg_pos mult_neg_neg)
```
```   435 done
```
```   436
```
```   437 lemma positive_mult:
```
```   438   "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x * y)"
```
```   439 by transfer (drule (1) mult_pos_pos, simp add: mult_ac)
```
```   440
```
```   441 lemma positive_minus:
```
```   442   "\<not> positive x \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> positive (- x)"
```
```   443 by transfer (force simp: neq_iff zero_less_mult_iff mult_less_0_iff)
```
```   444
```
```   445 instantiation rat :: linordered_field_inverse_zero
```
```   446 begin
```
```   447
```
```   448 definition
```
```   449   "x < y \<longleftrightarrow> positive (y - x)"
```
```   450
```
```   451 definition
```
```   452   "x \<le> (y::rat) \<longleftrightarrow> x < y \<or> x = y"
```
```   453
```
```   454 definition
```
```   455   "abs (a::rat) = (if a < 0 then - a else a)"
```
```   456
```
```   457 definition
```
```   458   "sgn (a::rat) = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
```
```   459
```
```   460 instance proof
```
```   461   fix a b c :: rat
```
```   462   show "\<bar>a\<bar> = (if a < 0 then - a else a)"
```
```   463     by (rule abs_rat_def)
```
```   464   show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a"
```
```   465     unfolding less_eq_rat_def less_rat_def
```
```   466     by (auto, drule (1) positive_add, simp_all add: positive_zero)
```
```   467   show "a \<le> a"
```
```   468     unfolding less_eq_rat_def by simp
```
```   469   show "a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
```
```   470     unfolding less_eq_rat_def less_rat_def
```
```   471     by (auto, drule (1) positive_add, simp add: algebra_simps)
```
```   472   show "a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> a = b"
```
```   473     unfolding less_eq_rat_def less_rat_def
```
```   474     by (auto, drule (1) positive_add, simp add: positive_zero)
```
```   475   show "a \<le> b \<Longrightarrow> c + a \<le> c + b"
```
```   476     unfolding less_eq_rat_def less_rat_def by (auto simp: diff_minus)
```
```   477   show "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
```
```   478     by (rule sgn_rat_def)
```
```   479   show "a \<le> b \<or> b \<le> a"
```
```   480     unfolding less_eq_rat_def less_rat_def
```
```   481     by (auto dest!: positive_minus)
```
```   482   show "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
```
```   483     unfolding less_rat_def
```
```   484     by (drule (1) positive_mult, simp add: algebra_simps)
```
```   485 qed
```
```   486
```
```   487 end
```
```   488
```
```   489 instantiation rat :: distrib_lattice
```
```   490 begin
```
```   491
```
```   492 definition
```
```   493   "(inf :: rat \<Rightarrow> rat \<Rightarrow> rat) = min"
```
```   494
```
```   495 definition
```
```   496   "(sup :: rat \<Rightarrow> rat \<Rightarrow> rat) = max"
```
```   497
```
```   498 instance proof
```
```   499 qed (auto simp add: inf_rat_def sup_rat_def min_max.sup_inf_distrib1)
```
```   500
```
```   501 end
```
```   502
```
```   503 lemma positive_rat: "positive (Fract a b) \<longleftrightarrow> 0 < a * b"
```
```   504   by transfer simp
```
```   505
```
```   506 lemma less_rat [simp]:
```
```   507   assumes "b \<noteq> 0" and "d \<noteq> 0"
```
```   508   shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
```
```   509   using assms unfolding less_rat_def
```
```   510   by (simp add: positive_rat algebra_simps)
```
```   511
```
```   512 lemma le_rat [simp]:
```
```   513   assumes "b \<noteq> 0" and "d \<noteq> 0"
```
```   514   shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   515   using assms unfolding le_less by (simp add: eq_rat)
```
```   516
```
```   517 lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
```
```   518   by (auto simp add: abs_rat_def zabs_def Zero_rat_def not_less le_less eq_rat zero_less_mult_iff)
```
```   519
```
```   520 lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)"
```
```   521   unfolding Fract_of_int_eq
```
```   522   by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat)
```
```   523     (auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)
```
```   524
```
```   525 lemma Rat_induct_pos [case_names Fract, induct type: rat]:
```
```   526   assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
```
```   527   shows "P q"
```
```   528 proof (cases q)
```
```   529   have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)"
```
```   530   proof -
```
```   531     fix a::int and b::int
```
```   532     assume b: "b < 0"
```
```   533     hence "0 < -b" by simp
```
```   534     hence "P (Fract (-a) (-b))" by (rule step)
```
```   535     thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
```
```   536   qed
```
```   537   case (Fract a b)
```
```   538   thus "P q" by (force simp add: linorder_neq_iff step step')
```
```   539 qed
```
```   540
```
```   541 lemma zero_less_Fract_iff:
```
```   542   "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a"
```
```   543   by (simp add: Zero_rat_def zero_less_mult_iff)
```
```   544
```
```   545 lemma Fract_less_zero_iff:
```
```   546   "0 < b \<Longrightarrow> Fract a b < 0 \<longleftrightarrow> a < 0"
```
```   547   by (simp add: Zero_rat_def mult_less_0_iff)
```
```   548
```
```   549 lemma zero_le_Fract_iff:
```
```   550   "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a"
```
```   551   by (simp add: Zero_rat_def zero_le_mult_iff)
```
```   552
```
```   553 lemma Fract_le_zero_iff:
```
```   554   "0 < b \<Longrightarrow> Fract a b \<le> 0 \<longleftrightarrow> a \<le> 0"
```
```   555   by (simp add: Zero_rat_def mult_le_0_iff)
```
```   556
```
```   557 lemma one_less_Fract_iff:
```
```   558   "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a"
```
```   559   by (simp add: One_rat_def mult_less_cancel_right_disj)
```
```   560
```
```   561 lemma Fract_less_one_iff:
```
```   562   "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b"
```
```   563   by (simp add: One_rat_def mult_less_cancel_right_disj)
```
```   564
```
```   565 lemma one_le_Fract_iff:
```
```   566   "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a"
```
```   567   by (simp add: One_rat_def mult_le_cancel_right)
```
```   568
```
```   569 lemma Fract_le_one_iff:
```
```   570   "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b"
```
```   571   by (simp add: One_rat_def mult_le_cancel_right)
```
```   572
```
```   573
```
```   574 subsubsection {* Rationals are an Archimedean field *}
```
```   575
```
```   576 lemma rat_floor_lemma:
```
```   577   shows "of_int (a div b) \<le> Fract a b \<and> Fract a b < of_int (a div b + 1)"
```
```   578 proof -
```
```   579   have "Fract a b = of_int (a div b) + Fract (a mod b) b"
```
```   580     by (cases "b = 0", simp, simp add: of_int_rat)
```
```   581   moreover have "0 \<le> Fract (a mod b) b \<and> Fract (a mod b) b < 1"
```
```   582     unfolding Fract_of_int_quotient
```
```   583     by (rule linorder_cases [of b 0]) (simp add: divide_nonpos_neg, simp, simp add: divide_nonneg_pos)
```
```   584   ultimately show ?thesis by simp
```
```   585 qed
```
```   586
```
```   587 instance rat :: archimedean_field
```
```   588 proof
```
```   589   fix r :: rat
```
```   590   show "\<exists>z. r \<le> of_int z"
```
```   591   proof (induct r)
```
```   592     case (Fract a b)
```
```   593     have "Fract a b \<le> of_int (a div b + 1)"
```
```   594       using rat_floor_lemma [of a b] by simp
```
```   595     then show "\<exists>z. Fract a b \<le> of_int z" ..
```
```   596   qed
```
```   597 qed
```
```   598
```
```   599 instantiation rat :: floor_ceiling
```
```   600 begin
```
```   601
```
```   602 definition [code del]:
```
```   603   "floor (x::rat) = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))"
```
```   604
```
```   605 instance proof
```
```   606   fix x :: rat
```
```   607   show "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
```
```   608     unfolding floor_rat_def using floor_exists1 by (rule theI')
```
```   609 qed
```
```   610
```
```   611 end
```
```   612
```
```   613 lemma floor_Fract: "floor (Fract a b) = a div b"
```
```   614   using rat_floor_lemma [of a b]
```
```   615   by (simp add: floor_unique)
```
```   616
```
```   617
```
```   618 subsection {* Linear arithmetic setup *}
```
```   619
```
```   620 declaration {*
```
```   621   K (Lin_Arith.add_inj_thms [@{thm of_nat_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
```
```   622     (* not needed because x < (y::nat) can be rewritten as Suc x <= y: of_nat_less_iff RS iffD2 *)
```
```   623   #> Lin_Arith.add_inj_thms [@{thm of_int_le_iff} RS iffD2, @{thm of_int_eq_iff} RS iffD2]
```
```   624     (* not needed because x < (y::int) can be rewritten as x + 1 <= y: of_int_less_iff RS iffD2 *)
```
```   625   #> Lin_Arith.add_simps [@{thm neg_less_iff_less},
```
```   626       @{thm True_implies_equals},
```
```   627       read_instantiate @{context} [(("a", 0), "(numeral ?v)")] @{thm distrib_left},
```
```   628       read_instantiate @{context} [(("a", 0), "(neg_numeral ?v)")] @{thm distrib_left},
```
```   629       @{thm divide_1}, @{thm divide_zero_left},
```
```   630       @{thm times_divide_eq_right}, @{thm times_divide_eq_left},
```
```   631       @{thm minus_divide_left} RS sym, @{thm minus_divide_right} RS sym,
```
```   632       @{thm of_int_minus}, @{thm of_int_diff},
```
```   633       @{thm of_int_of_nat_eq}]
```
```   634   #> Lin_Arith.add_simprocs Numeral_Simprocs.field_cancel_numeral_factors
```
```   635   #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat => rat"})
```
```   636   #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int => rat"}))
```
```   637 *}
```
```   638
```
```   639
```
```   640 subsection {* Embedding from Rationals to other Fields *}
```
```   641
```
```   642 class field_char_0 = field + ring_char_0
```
```   643
```
```   644 subclass (in linordered_field) field_char_0 ..
```
```   645
```
```   646 context field_char_0
```
```   647 begin
```
```   648
```
```   649 lift_definition of_rat :: "rat \<Rightarrow> 'a"
```
```   650   is "\<lambda>x. of_int (fst x) / of_int (snd x)"
```
```   651 apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
```
```   652 apply (simp only: of_int_mult [symmetric])
```
```   653 done
```
```   654
```
```   655 end
```
```   656
```
```   657 lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
```
```   658   by transfer simp
```
```   659
```
```   660 lemma of_rat_0 [simp]: "of_rat 0 = 0"
```
```   661   by transfer simp
```
```   662
```
```   663 lemma of_rat_1 [simp]: "of_rat 1 = 1"
```
```   664   by transfer simp
```
```   665
```
```   666 lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
```
```   667   by transfer (simp add: add_frac_eq)
```
```   668
```
```   669 lemma of_rat_minus: "of_rat (- a) = - of_rat a"
```
```   670   by transfer simp
```
```   671
```
```   672 lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
```
```   673 by (simp only: diff_minus of_rat_add of_rat_minus)
```
```   674
```
```   675 lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
```
```   676 apply transfer
```
```   677 apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
```
```   678 done
```
```   679
```
```   680 lemma nonzero_of_rat_inverse:
```
```   681   "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
```
```   682 apply (rule inverse_unique [symmetric])
```
```   683 apply (simp add: of_rat_mult [symmetric])
```
```   684 done
```
```   685
```
```   686 lemma of_rat_inverse:
```
```   687   "(of_rat (inverse a)::'a::{field_char_0, field_inverse_zero}) =
```
```   688    inverse (of_rat a)"
```
```   689 by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
```
```   690
```
```   691 lemma nonzero_of_rat_divide:
```
```   692   "b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
```
```   693 by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
```
```   694
```
```   695 lemma of_rat_divide:
```
```   696   "(of_rat (a / b)::'a::{field_char_0, field_inverse_zero})
```
```   697    = of_rat a / of_rat b"
```
```   698 by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)
```
```   699
```
```   700 lemma of_rat_power:
```
```   701   "(of_rat (a ^ n)::'a::field_char_0) = of_rat a ^ n"
```
```   702 by (induct n) (simp_all add: of_rat_mult)
```
```   703
```
```   704 lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
```
```   705 apply transfer
```
```   706 apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
```
```   707 apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
```
```   708 done
```
```   709
```
```   710 lemma of_rat_less:
```
```   711   "(of_rat r :: 'a::linordered_field) < of_rat s \<longleftrightarrow> r < s"
```
```   712 proof (induct r, induct s)
```
```   713   fix a b c d :: int
```
```   714   assume not_zero: "b > 0" "d > 0"
```
```   715   then have "b * d > 0" by (rule mult_pos_pos)
```
```   716   have of_int_divide_less_eq:
```
```   717     "(of_int a :: 'a) / of_int b < of_int c / of_int d
```
```   718       \<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b"
```
```   719     using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
```
```   720   show "(of_rat (Fract a b) :: 'a::linordered_field) < of_rat (Fract c d)
```
```   721     \<longleftrightarrow> Fract a b < Fract c d"
```
```   722     using not_zero `b * d > 0`
```
```   723     by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult)
```
```   724 qed
```
```   725
```
```   726 lemma of_rat_less_eq:
```
```   727   "(of_rat r :: 'a::linordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s"
```
```   728   unfolding le_less by (auto simp add: of_rat_less)
```
```   729
```
```   730 lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
```
```   731
```
```   732 lemma of_rat_eq_id [simp]: "of_rat = id"
```
```   733 proof
```
```   734   fix a
```
```   735   show "of_rat a = id a"
```
```   736   by (induct a)
```
```   737      (simp add: of_rat_rat Fract_of_int_eq [symmetric])
```
```   738 qed
```
```   739
```
```   740 text{*Collapse nested embeddings*}
```
```   741 lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
```
```   742 by (induct n) (simp_all add: of_rat_add)
```
```   743
```
```   744 lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
```
```   745 by (cases z rule: int_diff_cases) (simp add: of_rat_diff)
```
```   746
```
```   747 lemma of_rat_numeral_eq [simp]:
```
```   748   "of_rat (numeral w) = numeral w"
```
```   749 using of_rat_of_int_eq [of "numeral w"] by simp
```
```   750
```
```   751 lemma of_rat_neg_numeral_eq [simp]:
```
```   752   "of_rat (neg_numeral w) = neg_numeral w"
```
```   753 using of_rat_of_int_eq [of "neg_numeral w"] by simp
```
```   754
```
```   755 lemmas zero_rat = Zero_rat_def
```
```   756 lemmas one_rat = One_rat_def
```
```   757
```
```   758 abbreviation
```
```   759   rat_of_nat :: "nat \<Rightarrow> rat"
```
```   760 where
```
```   761   "rat_of_nat \<equiv> of_nat"
```
```   762
```
```   763 abbreviation
```
```   764   rat_of_int :: "int \<Rightarrow> rat"
```
```   765 where
```
```   766   "rat_of_int \<equiv> of_int"
```
```   767
```
```   768 subsection {* The Set of Rational Numbers *}
```
```   769
```
```   770 context field_char_0
```
```   771 begin
```
```   772
```
```   773 definition
```
```   774   Rats  :: "'a set" where
```
```   775   "Rats = range of_rat"
```
```   776
```
```   777 notation (xsymbols)
```
```   778   Rats  ("\<rat>")
```
```   779
```
```   780 end
```
```   781
```
```   782 lemma Rats_of_rat [simp]: "of_rat r \<in> Rats"
```
```   783 by (simp add: Rats_def)
```
```   784
```
```   785 lemma Rats_of_int [simp]: "of_int z \<in> Rats"
```
```   786 by (subst of_rat_of_int_eq [symmetric], rule Rats_of_rat)
```
```   787
```
```   788 lemma Rats_of_nat [simp]: "of_nat n \<in> Rats"
```
```   789 by (subst of_rat_of_nat_eq [symmetric], rule Rats_of_rat)
```
```   790
```
```   791 lemma Rats_number_of [simp]: "numeral w \<in> Rats"
```
```   792 by (subst of_rat_numeral_eq [symmetric], rule Rats_of_rat)
```
```   793
```
```   794 lemma Rats_neg_number_of [simp]: "neg_numeral w \<in> Rats"
```
```   795 by (subst of_rat_neg_numeral_eq [symmetric], rule Rats_of_rat)
```
```   796
```
```   797 lemma Rats_0 [simp]: "0 \<in> Rats"
```
```   798 apply (unfold Rats_def)
```
```   799 apply (rule range_eqI)
```
```   800 apply (rule of_rat_0 [symmetric])
```
```   801 done
```
```   802
```
```   803 lemma Rats_1 [simp]: "1 \<in> Rats"
```
```   804 apply (unfold Rats_def)
```
```   805 apply (rule range_eqI)
```
```   806 apply (rule of_rat_1 [symmetric])
```
```   807 done
```
```   808
```
```   809 lemma Rats_add [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a + b \<in> Rats"
```
```   810 apply (auto simp add: Rats_def)
```
```   811 apply (rule range_eqI)
```
```   812 apply (rule of_rat_add [symmetric])
```
```   813 done
```
```   814
```
```   815 lemma Rats_minus [simp]: "a \<in> Rats \<Longrightarrow> - a \<in> Rats"
```
```   816 apply (auto simp add: Rats_def)
```
```   817 apply (rule range_eqI)
```
```   818 apply (rule of_rat_minus [symmetric])
```
```   819 done
```
```   820
```
```   821 lemma Rats_diff [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a - b \<in> Rats"
```
```   822 apply (auto simp add: Rats_def)
```
```   823 apply (rule range_eqI)
```
```   824 apply (rule of_rat_diff [symmetric])
```
```   825 done
```
```   826
```
```   827 lemma Rats_mult [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a * b \<in> Rats"
```
```   828 apply (auto simp add: Rats_def)
```
```   829 apply (rule range_eqI)
```
```   830 apply (rule of_rat_mult [symmetric])
```
```   831 done
```
```   832
```
```   833 lemma nonzero_Rats_inverse:
```
```   834   fixes a :: "'a::field_char_0"
```
```   835   shows "\<lbrakk>a \<in> Rats; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Rats"
```
```   836 apply (auto simp add: Rats_def)
```
```   837 apply (rule range_eqI)
```
```   838 apply (erule nonzero_of_rat_inverse [symmetric])
```
```   839 done
```
```   840
```
```   841 lemma Rats_inverse [simp]:
```
```   842   fixes a :: "'a::{field_char_0, field_inverse_zero}"
```
```   843   shows "a \<in> Rats \<Longrightarrow> inverse a \<in> Rats"
```
```   844 apply (auto simp add: Rats_def)
```
```   845 apply (rule range_eqI)
```
```   846 apply (rule of_rat_inverse [symmetric])
```
```   847 done
```
```   848
```
```   849 lemma nonzero_Rats_divide:
```
```   850   fixes a b :: "'a::field_char_0"
```
```   851   shows "\<lbrakk>a \<in> Rats; b \<in> Rats; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
```
```   852 apply (auto simp add: Rats_def)
```
```   853 apply (rule range_eqI)
```
```   854 apply (erule nonzero_of_rat_divide [symmetric])
```
```   855 done
```
```   856
```
```   857 lemma Rats_divide [simp]:
```
```   858   fixes a b :: "'a::{field_char_0, field_inverse_zero}"
```
```   859   shows "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
```
```   860 apply (auto simp add: Rats_def)
```
```   861 apply (rule range_eqI)
```
```   862 apply (rule of_rat_divide [symmetric])
```
```   863 done
```
```   864
```
```   865 lemma Rats_power [simp]:
```
```   866   fixes a :: "'a::field_char_0"
```
```   867   shows "a \<in> Rats \<Longrightarrow> a ^ n \<in> Rats"
```
```   868 apply (auto simp add: Rats_def)
```
```   869 apply (rule range_eqI)
```
```   870 apply (rule of_rat_power [symmetric])
```
```   871 done
```
```   872
```
```   873 lemma Rats_cases [cases set: Rats]:
```
```   874   assumes "q \<in> \<rat>"
```
```   875   obtains (of_rat) r where "q = of_rat r"
```
```   876 proof -
```
```   877   from `q \<in> \<rat>` have "q \<in> range of_rat" unfolding Rats_def .
```
```   878   then obtain r where "q = of_rat r" ..
```
```   879   then show thesis ..
```
```   880 qed
```
```   881
```
```   882 lemma Rats_induct [case_names of_rat, induct set: Rats]:
```
```   883   "q \<in> \<rat> \<Longrightarrow> (\<And>r. P (of_rat r)) \<Longrightarrow> P q"
```
```   884   by (rule Rats_cases) auto
```
```   885
```
```   886
```
```   887 subsection {* Implementation of rational numbers as pairs of integers *}
```
```   888
```
```   889 text {* Formal constructor *}
```
```   890
```
```   891 definition Frct :: "int \<times> int \<Rightarrow> rat" where
```
```   892   [simp]: "Frct p = Fract (fst p) (snd p)"
```
```   893
```
```   894 lemma [code abstype]:
```
```   895   "Frct (quotient_of q) = q"
```
```   896   by (cases q) (auto intro: quotient_of_eq)
```
```   897
```
```   898
```
```   899 text {* Numerals *}
```
```   900
```
```   901 declare quotient_of_Fract [code abstract]
```
```   902
```
```   903 definition of_int :: "int \<Rightarrow> rat"
```
```   904 where
```
```   905   [code_abbrev]: "of_int = Int.of_int"
```
```   906 hide_const (open) of_int
```
```   907
```
```   908 lemma quotient_of_int [code abstract]:
```
```   909   "quotient_of (Rat.of_int a) = (a, 1)"
```
```   910   by (simp add: of_int_def of_int_rat quotient_of_Fract)
```
```   911
```
```   912 lemma [code_unfold]:
```
```   913   "numeral k = Rat.of_int (numeral k)"
```
```   914   by (simp add: Rat.of_int_def)
```
```   915
```
```   916 lemma [code_unfold]:
```
```   917   "neg_numeral k = Rat.of_int (neg_numeral k)"
```
```   918   by (simp add: Rat.of_int_def)
```
```   919
```
```   920 lemma Frct_code_post [code_post]:
```
```   921   "Frct (0, a) = 0"
```
```   922   "Frct (a, 0) = 0"
```
```   923   "Frct (1, 1) = 1"
```
```   924   "Frct (numeral k, 1) = numeral k"
```
```   925   "Frct (neg_numeral k, 1) = neg_numeral k"
```
```   926   "Frct (1, numeral k) = 1 / numeral k"
```
```   927   "Frct (1, neg_numeral k) = 1 / neg_numeral k"
```
```   928   "Frct (numeral k, numeral l) = numeral k / numeral l"
```
```   929   "Frct (numeral k, neg_numeral l) = numeral k / neg_numeral l"
```
```   930   "Frct (neg_numeral k, numeral l) = neg_numeral k / numeral l"
```
```   931   "Frct (neg_numeral k, neg_numeral l) = neg_numeral k / neg_numeral l"
```
```   932   by (simp_all add: Fract_of_int_quotient)
```
```   933
```
```   934
```
```   935 text {* Operations *}
```
```   936
```
```   937 lemma rat_zero_code [code abstract]:
```
```   938   "quotient_of 0 = (0, 1)"
```
```   939   by (simp add: Zero_rat_def quotient_of_Fract normalize_def)
```
```   940
```
```   941 lemma rat_one_code [code abstract]:
```
```   942   "quotient_of 1 = (1, 1)"
```
```   943   by (simp add: One_rat_def quotient_of_Fract normalize_def)
```
```   944
```
```   945 lemma rat_plus_code [code abstract]:
```
```   946   "quotient_of (p + q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
```
```   947      in normalize (a * d + b * c, c * d))"
```
```   948   by (cases p, cases q) (simp add: quotient_of_Fract)
```
```   949
```
```   950 lemma rat_uminus_code [code abstract]:
```
```   951   "quotient_of (- p) = (let (a, b) = quotient_of p in (- a, b))"
```
```   952   by (cases p) (simp add: quotient_of_Fract)
```
```   953
```
```   954 lemma rat_minus_code [code abstract]:
```
```   955   "quotient_of (p - q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
```
```   956      in normalize (a * d - b * c, c * d))"
```
```   957   by (cases p, cases q) (simp add: quotient_of_Fract)
```
```   958
```
```   959 lemma rat_times_code [code abstract]:
```
```   960   "quotient_of (p * q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
```
```   961      in normalize (a * b, c * d))"
```
```   962   by (cases p, cases q) (simp add: quotient_of_Fract)
```
```   963
```
```   964 lemma rat_inverse_code [code abstract]:
```
```   965   "quotient_of (inverse p) = (let (a, b) = quotient_of p
```
```   966     in if a = 0 then (0, 1) else (sgn a * b, \<bar>a\<bar>))"
```
```   967 proof (cases p)
```
```   968   case (Fract a b) then show ?thesis
```
```   969     by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract gcd_int.commute)
```
```   970 qed
```
```   971
```
```   972 lemma rat_divide_code [code abstract]:
```
```   973   "quotient_of (p / q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
```
```   974      in normalize (a * d, c * b))"
```
```   975   by (cases p, cases q) (simp add: quotient_of_Fract)
```
```   976
```
```   977 lemma rat_abs_code [code abstract]:
```
```   978   "quotient_of \<bar>p\<bar> = (let (a, b) = quotient_of p in (\<bar>a\<bar>, b))"
```
```   979   by (cases p) (simp add: quotient_of_Fract)
```
```   980
```
```   981 lemma rat_sgn_code [code abstract]:
```
```   982   "quotient_of (sgn p) = (sgn (fst (quotient_of p)), 1)"
```
```   983 proof (cases p)
```
```   984   case (Fract a b) then show ?thesis
```
```   985   by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract)
```
```   986 qed
```
```   987
```
```   988 lemma rat_floor_code [code]:
```
```   989   "floor p = (let (a, b) = quotient_of p in a div b)"
```
```   990 by (cases p) (simp add: quotient_of_Fract floor_Fract)
```
```   991
```
```   992 instantiation rat :: equal
```
```   993 begin
```
```   994
```
```   995 definition [code]:
```
```   996   "HOL.equal a b \<longleftrightarrow> quotient_of a = quotient_of b"
```
```   997
```
```   998 instance proof
```
```   999 qed (simp add: equal_rat_def quotient_of_inject_eq)
```
```  1000
```
```  1001 lemma rat_eq_refl [code nbe]:
```
```  1002   "HOL.equal (r::rat) r \<longleftrightarrow> True"
```
```  1003   by (rule equal_refl)
```
```  1004
```
```  1005 end
```
```  1006
```
```  1007 lemma rat_less_eq_code [code]:
```
```  1008   "p \<le> q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d \<le> c * b)"
```
```  1009   by (cases p, cases q) (simp add: quotient_of_Fract mult.commute)
```
```  1010
```
```  1011 lemma rat_less_code [code]:
```
```  1012   "p < q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d < c * b)"
```
```  1013   by (cases p, cases q) (simp add: quotient_of_Fract mult.commute)
```
```  1014
```
```  1015 lemma [code]:
```
```  1016   "of_rat p = (let (a, b) = quotient_of p in of_int a / of_int b)"
```
```  1017   by (cases p) (simp add: quotient_of_Fract of_rat_rat)
```
```  1018
```
```  1019
```
```  1020 text {* Quickcheck *}
```
```  1021
```
```  1022 definition (in term_syntax)
```
```  1023   valterm_fract :: "int \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> int \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> rat \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
```
```  1024   [code_unfold]: "valterm_fract k l = Code_Evaluation.valtermify Fract {\<cdot>} k {\<cdot>} l"
```
```  1025
```
```  1026 notation fcomp (infixl "\<circ>>" 60)
```
```  1027 notation scomp (infixl "\<circ>\<rightarrow>" 60)
```
```  1028
```
```  1029 instantiation rat :: random
```
```  1030 begin
```
```  1031
```
```  1032 definition
```
```  1033   "Quickcheck.random i = Quickcheck.random i \<circ>\<rightarrow> (\<lambda>num. Random.range i \<circ>\<rightarrow> (\<lambda>denom. Pair (
```
```  1034      let j = Code_Numeral.int_of (denom + 1)
```
```  1035      in valterm_fract num (j, \<lambda>u. Code_Evaluation.term_of j))))"
```
```  1036
```
```  1037 instance ..
```
```  1038
```
```  1039 end
```
```  1040
```
```  1041 no_notation fcomp (infixl "\<circ>>" 60)
```
```  1042 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
```
```  1043
```
```  1044 instantiation rat :: exhaustive
```
```  1045 begin
```
```  1046
```
```  1047 definition
```
```  1048   "exhaustive_rat f d = Quickcheck_Exhaustive.exhaustive (%l. Quickcheck_Exhaustive.exhaustive (%k. f (Fract k (Code_Numeral.int_of l + 1))) d) d"
```
```  1049
```
```  1050 instance ..
```
```  1051
```
```  1052 end
```
```  1053
```
```  1054 instantiation rat :: full_exhaustive
```
```  1055 begin
```
```  1056
```
```  1057 definition
```
```  1058   "full_exhaustive_rat f d = Quickcheck_Exhaustive.full_exhaustive (%(l, _). Quickcheck_Exhaustive.full_exhaustive (%k.
```
```  1059      f (let j = Code_Numeral.int_of l + 1
```
```  1060         in valterm_fract k (j, %_. Code_Evaluation.term_of j))) d) d"
```
```  1061
```
```  1062 instance ..
```
```  1063
```
```  1064 end
```
```  1065
```
```  1066 instantiation rat :: partial_term_of
```
```  1067 begin
```
```  1068
```
```  1069 instance ..
```
```  1070
```
```  1071 end
```
```  1072
```
```  1073 lemma [code]:
```
```  1074   "partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Narrowing_variable p tt) == Code_Evaluation.Free (STR ''_'') (Typerep.Typerep (STR ''Rat.rat'') [])"
```
```  1075   "partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Narrowing_constructor 0 [l, k]) ==
```
```  1076      Code_Evaluation.App (Code_Evaluation.Const (STR ''Rat.Frct'')
```
```  1077      (Typerep.Typerep (STR ''fun'') [Typerep.Typerep (STR ''Product_Type.prod'') [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''Int.int'') []],
```
```  1078         Typerep.Typerep (STR ''Rat.rat'') []])) (Code_Evaluation.App (Code_Evaluation.App (Code_Evaluation.Const (STR ''Product_Type.Pair'') (Typerep.Typerep (STR ''fun'') [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''fun'') [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''Product_Type.prod'') [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''Int.int'') []]]])) (partial_term_of (TYPE(int)) l)) (partial_term_of (TYPE(int)) k))"
```
```  1079 by (rule partial_term_of_anything)+
```
```  1080
```
```  1081 instantiation rat :: narrowing
```
```  1082 begin
```
```  1083
```
```  1084 definition
```
```  1085   "narrowing = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.apply
```
```  1086     (Quickcheck_Narrowing.cons (%nom denom. Fract nom denom)) narrowing) narrowing"
```
```  1087
```
```  1088 instance ..
```
```  1089
```
```  1090 end
```
```  1091
```
```  1092
```
```  1093 subsection {* Setup for Nitpick *}
```
```  1094
```
```  1095 declaration {*
```
```  1096   Nitpick_HOL.register_frac_type @{type_name rat}
```
```  1097    [(@{const_name zero_rat_inst.zero_rat}, @{const_name Nitpick.zero_frac}),
```
```  1098     (@{const_name one_rat_inst.one_rat}, @{const_name Nitpick.one_frac}),
```
```  1099     (@{const_name plus_rat_inst.plus_rat}, @{const_name Nitpick.plus_frac}),
```
```  1100     (@{const_name times_rat_inst.times_rat}, @{const_name Nitpick.times_frac}),
```
```  1101     (@{const_name uminus_rat_inst.uminus_rat}, @{const_name Nitpick.uminus_frac}),
```
```  1102     (@{const_name inverse_rat_inst.inverse_rat}, @{const_name Nitpick.inverse_frac}),
```
```  1103     (@{const_name ord_rat_inst.less_rat}, @{const_name Nitpick.less_frac}),
```
```  1104     (@{const_name ord_rat_inst.less_eq_rat}, @{const_name Nitpick.less_eq_frac}),
```
```  1105     (@{const_name field_char_0_class.of_rat}, @{const_name Nitpick.of_frac})]
```
```  1106 *}
```
```  1107
```
```  1108 lemmas [nitpick_unfold] = inverse_rat_inst.inverse_rat
```
```  1109   one_rat_inst.one_rat ord_rat_inst.less_rat
```
```  1110   ord_rat_inst.less_eq_rat plus_rat_inst.plus_rat times_rat_inst.times_rat
```
```  1111   uminus_rat_inst.uminus_rat zero_rat_inst.zero_rat
```
```  1112
```
```  1113 subsection{* Float syntax *}
```
```  1114
```
```  1115 syntax "_Float" :: "float_const \<Rightarrow> 'a"    ("_")
```
```  1116
```
```  1117 ML_file "Tools/float_syntax.ML"
```
```  1118 setup Float_Syntax.setup
```
```  1119
```
```  1120 text{* Test: *}
```
```  1121 lemma "123.456 = -111.111 + 200 + 30 + 4 + 5/10 + 6/100 + (7/1000::rat)"
```
```  1122 by simp
```
```  1123
```
```  1124
```
```  1125 hide_const (open) normalize positive
```
```  1126
```
```  1127 lemmas [transfer_rule del] =
```
```  1128   rat.All_transfer rat.Ex_transfer rat.rel_eq_transfer forall_rat_transfer
```
```  1129   Fract.transfer zero_rat.transfer one_rat.transfer plus_rat.transfer
```
```  1130   uminus_rat.transfer times_rat.transfer inverse_rat.transfer
```
```  1131   positive.transfer of_rat.transfer
```
```  1132
```
```  1133 text {* De-register @{text "rat"} as a quotient type: *}
```
```  1134
```
```  1135 declare Quotient_rat[quot_del]
```
```  1136
```
```  1137 end
```