src/HOL/Computational_Algebra/Fraction_Field.thy
 author wenzelm Wed Nov 01 20:46:23 2017 +0100 (22 months ago) changeset 66983 df83b66f1d94 parent 65435 378175f44328 permissions -rw-r--r--
proper merge (amending fb46c031c841);
```     1 (*  Title:      HOL/Computational_Algebra/Fraction_Field.thy
```
```     2     Author:     Amine Chaieb, University of Cambridge
```
```     3 *)
```
```     4
```
```     5 section\<open>A formalization of the fraction field of any integral domain;
```
```     6          generalization of theory Rat from int to any integral domain\<close>
```
```     7
```
```     8 theory Fraction_Field
```
```     9 imports Main
```
```    10 begin
```
```    11
```
```    12 subsection \<open>General fractions construction\<close>
```
```    13
```
```    14 subsubsection \<open>Construction of the type of fractions\<close>
```
```    15
```
```    16 context idom begin
```
```    17
```
```    18 definition fractrel :: "'a \<times> 'a \<Rightarrow> 'a * 'a \<Rightarrow> bool" where
```
```    19   "fractrel = (\<lambda>x y. snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x)"
```
```    20
```
```    21 lemma fractrel_iff [simp]:
```
```    22   "fractrel x y \<longleftrightarrow> snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x"
```
```    23   by (simp add: fractrel_def)
```
```    24
```
```    25 lemma symp_fractrel: "symp fractrel"
```
```    26   by (simp add: symp_def)
```
```    27
```
```    28 lemma transp_fractrel: "transp fractrel"
```
```    29 proof (rule transpI, unfold split_paired_all)
```
```    30   fix a b a' b' a'' b'' :: 'a
```
```    31   assume A: "fractrel (a, b) (a', b')"
```
```    32   assume B: "fractrel (a', b') (a'', b'')"
```
```    33   have "b' * (a * b'') = b'' * (a * b')" by (simp add: ac_simps)
```
```    34   also from A have "a * b' = a' * b" by auto
```
```    35   also have "b'' * (a' * b) = b * (a' * b'')" by (simp add: ac_simps)
```
```    36   also from B have "a' * b'' = a'' * b'" by auto
```
```    37   also have "b * (a'' * b') = b' * (a'' * b)" by (simp add: ac_simps)
```
```    38   finally have "b' * (a * b'') = b' * (a'' * b)" .
```
```    39   moreover from B have "b' \<noteq> 0" by auto
```
```    40   ultimately have "a * b'' = a'' * b" by simp
```
```    41   with A B show "fractrel (a, b) (a'', b'')" by auto
```
```    42 qed
```
```    43
```
```    44 lemma part_equivp_fractrel: "part_equivp fractrel"
```
```    45 using _ symp_fractrel transp_fractrel
```
```    46 by(rule part_equivpI)(rule exI[where x="(0, 1)"]; simp)
```
```    47
```
```    48 end
```
```    49
```
```    50 quotient_type (overloaded) 'a fract = "'a :: idom \<times> 'a" / partial: "fractrel"
```
```    51 by(rule part_equivp_fractrel)
```
```    52
```
```    53 subsubsection \<open>Representation and basic operations\<close>
```
```    54
```
```    55 lift_definition Fract :: "'a :: idom \<Rightarrow> 'a \<Rightarrow> 'a fract"
```
```    56   is "\<lambda>a b. if b = 0 then (0, 1) else (a, b)"
```
```    57   by simp
```
```    58
```
```    59 lemma Fract_cases [cases type: fract]:
```
```    60   obtains (Fract) a b where "q = Fract a b" "b \<noteq> 0"
```
```    61 by transfer simp
```
```    62
```
```    63 lemma Fract_induct [case_names Fract, induct type: fract]:
```
```    64   "(\<And>a b. b \<noteq> 0 \<Longrightarrow> P (Fract a b)) \<Longrightarrow> P q"
```
```    65   by (cases q) simp
```
```    66
```
```    67 lemma eq_fract:
```
```    68   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"
```
```    69     and "\<And>a. Fract a 0 = Fract 0 1"
```
```    70     and "\<And>a c. Fract 0 a = Fract 0 c"
```
```    71 by(transfer; simp)+
```
```    72
```
```    73 instantiation fract :: (idom) comm_ring_1
```
```    74 begin
```
```    75
```
```    76 lift_definition zero_fract :: "'a fract" is "(0, 1)" by simp
```
```    77
```
```    78 lemma Zero_fract_def: "0 = Fract 0 1"
```
```    79 by transfer simp
```
```    80
```
```    81 lift_definition one_fract :: "'a fract" is "(1, 1)" by simp
```
```    82
```
```    83 lemma One_fract_def: "1 = Fract 1 1"
```
```    84 by transfer simp
```
```    85
```
```    86 lift_definition plus_fract :: "'a fract \<Rightarrow> 'a fract \<Rightarrow> 'a fract"
```
```    87   is "\<lambda>q r. (fst q * snd r + fst r * snd q, snd q * snd r)"
```
```    88 by(auto simp add: algebra_simps)
```
```    89
```
```    90 lemma add_fract [simp]:
```
```    91   "\<lbrakk> b \<noteq> 0; d \<noteq> 0 \<rbrakk> \<Longrightarrow> Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
```
```    92 by transfer simp
```
```    93
```
```    94 lift_definition uminus_fract :: "'a fract \<Rightarrow> 'a fract"
```
```    95   is "\<lambda>x. (- fst x, snd x)"
```
```    96 by simp
```
```    97
```
```    98 lemma minus_fract [simp]:
```
```    99   fixes a b :: "'a::idom"
```
```   100   shows "- Fract a b = Fract (- a) b"
```
```   101 by transfer simp
```
```   102
```
```   103 lemma minus_fract_cancel [simp]: "Fract (- a) (- b) = Fract a b"
```
```   104   by (cases "b = 0") (simp_all add: eq_fract)
```
```   105
```
```   106 definition diff_fract_def: "q - r = q + - (r::'a fract)"
```
```   107
```
```   108 lemma diff_fract [simp]:
```
```   109   "\<lbrakk> b \<noteq> 0; d \<noteq> 0 \<rbrakk> \<Longrightarrow> Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
```
```   110   by (simp add: diff_fract_def)
```
```   111
```
```   112 lift_definition times_fract :: "'a fract \<Rightarrow> 'a fract \<Rightarrow> 'a fract"
```
```   113   is "\<lambda>q r. (fst q * fst r, snd q * snd r)"
```
```   114 by(simp add: algebra_simps)
```
```   115
```
```   116 lemma mult_fract [simp]: "Fract (a::'a::idom) b * Fract c d = Fract (a * c) (b * d)"
```
```   117 by transfer simp
```
```   118
```
```   119 lemma mult_fract_cancel:
```
```   120   "c \<noteq> 0 \<Longrightarrow> Fract (c * a) (c * b) = Fract a b"
```
```   121 by transfer simp
```
```   122
```
```   123 instance
```
```   124 proof
```
```   125   fix q r s :: "'a fract"
```
```   126   show "(q * r) * s = q * (r * s)"
```
```   127     by (cases q, cases r, cases s) (simp add: eq_fract algebra_simps)
```
```   128   show "q * r = r * q"
```
```   129     by (cases q, cases r) (simp add: eq_fract algebra_simps)
```
```   130   show "1 * q = q"
```
```   131     by (cases q) (simp add: One_fract_def eq_fract)
```
```   132   show "(q + r) + s = q + (r + s)"
```
```   133     by (cases q, cases r, cases s) (simp add: eq_fract algebra_simps)
```
```   134   show "q + r = r + q"
```
```   135     by (cases q, cases r) (simp add: eq_fract algebra_simps)
```
```   136   show "0 + q = q"
```
```   137     by (cases q) (simp add: Zero_fract_def eq_fract)
```
```   138   show "- q + q = 0"
```
```   139     by (cases q) (simp add: Zero_fract_def eq_fract)
```
```   140   show "q - r = q + - r"
```
```   141     by (cases q, cases r) (simp add: eq_fract)
```
```   142   show "(q + r) * s = q * s + r * s"
```
```   143     by (cases q, cases r, cases s) (simp add: eq_fract algebra_simps)
```
```   144   show "(0::'a fract) \<noteq> 1"
```
```   145     by (simp add: Zero_fract_def One_fract_def eq_fract)
```
```   146 qed
```
```   147
```
```   148 end
```
```   149
```
```   150 lemma of_nat_fract: "of_nat k = Fract (of_nat k) 1"
```
```   151   by (induct k) (simp_all add: Zero_fract_def One_fract_def)
```
```   152
```
```   153 lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
```
```   154   by (rule of_nat_fract [symmetric])
```
```   155
```
```   156 lemma fract_collapse:
```
```   157   "Fract 0 k = 0"
```
```   158   "Fract 1 1 = 1"
```
```   159   "Fract k 0 = 0"
```
```   160 by(transfer; simp)+
```
```   161
```
```   162 lemma fract_expand:
```
```   163   "0 = Fract 0 1"
```
```   164   "1 = Fract 1 1"
```
```   165   by (simp_all add: fract_collapse)
```
```   166
```
```   167 lemma Fract_cases_nonzero:
```
```   168   obtains (Fract) a b where "q = Fract a b" and "b \<noteq> 0" and "a \<noteq> 0"
```
```   169     | (0) "q = 0"
```
```   170 proof (cases "q = 0")
```
```   171   case True
```
```   172   then show thesis using 0 by auto
```
```   173 next
```
```   174   case False
```
```   175   then obtain a b where "q = Fract a b" and "b \<noteq> 0" by (cases q) auto
```
```   176   with False have "0 \<noteq> Fract a b" by simp
```
```   177   with \<open>b \<noteq> 0\<close> have "a \<noteq> 0" by (simp add: Zero_fract_def eq_fract)
```
```   178   with Fract \<open>q = Fract a b\<close> \<open>b \<noteq> 0\<close> show thesis by auto
```
```   179 qed
```
```   180
```
```   181
```
```   182 subsubsection \<open>The field of rational numbers\<close>
```
```   183
```
```   184 context idom
```
```   185 begin
```
```   186
```
```   187 subclass ring_no_zero_divisors ..
```
```   188
```
```   189 end
```
```   190
```
```   191 instantiation fract :: (idom) field
```
```   192 begin
```
```   193
```
```   194 lift_definition inverse_fract :: "'a fract \<Rightarrow> 'a fract"
```
```   195   is "\<lambda>x. if fst x = 0 then (0, 1) else (snd x, fst x)"
```
```   196 by(auto simp add: algebra_simps)
```
```   197
```
```   198 lemma inverse_fract [simp]: "inverse (Fract a b) = Fract (b::'a::idom) a"
```
```   199 by transfer simp
```
```   200
```
```   201 definition divide_fract_def: "q div r = q * inverse (r:: 'a fract)"
```
```   202
```
```   203 lemma divide_fract [simp]: "Fract a b div Fract c d = Fract (a * d) (b * c)"
```
```   204   by (simp add: divide_fract_def)
```
```   205
```
```   206 instance
```
```   207 proof
```
```   208   fix q :: "'a fract"
```
```   209   assume "q \<noteq> 0"
```
```   210   then show "inverse q * q = 1"
```
```   211     by (cases q rule: Fract_cases_nonzero)
```
```   212       (simp_all add: fract_expand eq_fract mult.commute)
```
```   213 next
```
```   214   fix q r :: "'a fract"
```
```   215   show "q div r = q * inverse r" by (simp add: divide_fract_def)
```
```   216 next
```
```   217   show "inverse 0 = (0:: 'a fract)"
```
```   218     by (simp add: fract_expand) (simp add: fract_collapse)
```
```   219 qed
```
```   220
```
```   221 end
```
```   222
```
```   223
```
```   224 subsubsection \<open>The ordered field of fractions over an ordered idom\<close>
```
```   225
```
```   226 instantiation fract :: (linordered_idom) linorder
```
```   227 begin
```
```   228
```
```   229 lemma less_eq_fract_respect:
```
```   230   fixes a b a' b' c d c' d' :: 'a
```
```   231   assumes neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
```
```   232   assumes eq1: "a * b' = a' * b"
```
```   233   assumes eq2: "c * d' = c' * d"
```
```   234   shows "((a * d) * (b * d) \<le> (c * b) * (b * d)) \<longleftrightarrow> ((a' * d') * (b' * d') \<le> (c' * b') * (b' * d'))"
```
```   235 proof -
```
```   236   let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
```
```   237   {
```
```   238     fix a b c d x :: 'a
```
```   239     assume x: "x \<noteq> 0"
```
```   240     have "?le a b c d = ?le (a * x) (b * x) c d"
```
```   241     proof -
```
```   242       from x have "0 < x * x"
```
```   243         by (auto simp add: zero_less_mult_iff)
```
```   244       then have "?le a b c d =
```
```   245           ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
```
```   246         by (simp add: mult_le_cancel_right)
```
```   247       also have "... = ?le (a * x) (b * x) c d"
```
```   248         by (simp add: ac_simps)
```
```   249       finally show ?thesis .
```
```   250     qed
```
```   251   } note le_factor = this
```
```   252
```
```   253   let ?D = "b * d" and ?D' = "b' * d'"
```
```   254   from neq have D: "?D \<noteq> 0" by simp
```
```   255   from neq have "?D' \<noteq> 0" by simp
```
```   256   then have "?le a b c d = ?le (a * ?D') (b * ?D') c d"
```
```   257     by (rule le_factor)
```
```   258   also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')"
```
```   259     by (simp add: ac_simps)
```
```   260   also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
```
```   261     by (simp only: eq1 eq2)
```
```   262   also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
```
```   263     by (simp add: ac_simps)
```
```   264   also from D have "... = ?le a' b' c' d'"
```
```   265     by (rule le_factor [symmetric])
```
```   266   finally show "?le a b c d = ?le a' b' c' d'" .
```
```   267 qed
```
```   268
```
```   269 lift_definition less_eq_fract :: "'a fract \<Rightarrow> 'a fract \<Rightarrow> bool"
```
```   270   is "\<lambda>q r. (fst q * snd r) * (snd q * snd r) \<le> (fst r * snd q) * (snd q * snd r)"
```
```   271 by (clarsimp simp add: less_eq_fract_respect)
```
```   272
```
```   273 definition less_fract_def: "z < (w::'a fract) \<longleftrightarrow> z \<le> w \<and> \<not> w \<le> z"
```
```   274
```
```   275 lemma le_fract [simp]:
```
```   276   "\<lbrakk> b \<noteq> 0; d \<noteq> 0 \<rbrakk> \<Longrightarrow> Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   277   by transfer simp
```
```   278
```
```   279 lemma less_fract [simp]:
```
```   280   "\<lbrakk> b \<noteq> 0; d \<noteq> 0 \<rbrakk> \<Longrightarrow> Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
```
```   281   by (simp add: less_fract_def less_le_not_le ac_simps)
```
```   282
```
```   283 instance
```
```   284 proof
```
```   285   fix q r s :: "'a fract"
```
```   286   assume "q \<le> r" and "r \<le> s"
```
```   287   then show "q \<le> s"
```
```   288   proof (induct q, induct r, induct s)
```
```   289     fix a b c d e f :: 'a
```
```   290     assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0"
```
```   291     assume 1: "Fract a b \<le> Fract c d"
```
```   292     assume 2: "Fract c d \<le> Fract e f"
```
```   293     show "Fract a b \<le> Fract e f"
```
```   294     proof -
```
```   295       from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
```
```   296         by (auto simp add: zero_less_mult_iff linorder_neq_iff)
```
```   297       have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
```
```   298       proof -
```
```   299         from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   300           by simp
```
```   301         with ff show ?thesis by (simp add: mult_le_cancel_right)
```
```   302       qed
```
```   303       also have "... = (c * f) * (d * f) * (b * b)"
```
```   304         by (simp only: ac_simps)
```
```   305       also have "... \<le> (e * d) * (d * f) * (b * b)"
```
```   306       proof -
```
```   307         from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
```
```   308           by simp
```
```   309         with bb show ?thesis by (simp add: mult_le_cancel_right)
```
```   310       qed
```
```   311       finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
```
```   312         by (simp only: ac_simps)
```
```   313       with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
```
```   314         by (simp add: mult_le_cancel_right)
```
```   315       with neq show ?thesis by simp
```
```   316     qed
```
```   317   qed
```
```   318 next
```
```   319   fix q r :: "'a fract"
```
```   320   assume "q \<le> r" and "r \<le> q"
```
```   321   then show "q = r"
```
```   322   proof (induct q, induct r)
```
```   323     fix a b c d :: 'a
```
```   324     assume neq: "b \<noteq> 0" "d \<noteq> 0"
```
```   325     assume 1: "Fract a b \<le> Fract c d"
```
```   326     assume 2: "Fract c d \<le> Fract a b"
```
```   327     show "Fract a b = Fract c d"
```
```   328     proof -
```
```   329       from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   330         by simp
```
```   331       also have "... \<le> (a * d) * (b * d)"
```
```   332       proof -
```
```   333         from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
```
```   334           by simp
```
```   335         then show ?thesis by (simp only: ac_simps)
```
```   336       qed
```
```   337       finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
```
```   338       moreover from neq have "b * d \<noteq> 0" by simp
```
```   339       ultimately have "a * d = c * b" by simp
```
```   340       with neq show ?thesis by (simp add: eq_fract)
```
```   341     qed
```
```   342   qed
```
```   343 next
```
```   344   fix q r :: "'a fract"
```
```   345   show "q \<le> q"
```
```   346     by (induct q) simp
```
```   347   show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)"
```
```   348     by (simp only: less_fract_def)
```
```   349   show "q \<le> r \<or> r \<le> q"
```
```   350     by (induct q, induct r)
```
```   351        (simp add: mult.commute, rule linorder_linear)
```
```   352 qed
```
```   353
```
```   354 end
```
```   355
```
```   356 instantiation fract :: (linordered_idom) linordered_field
```
```   357 begin
```
```   358
```
```   359 definition abs_fract_def2:
```
```   360   "\<bar>q\<bar> = (if q < 0 then -q else (q::'a fract))"
```
```   361
```
```   362 definition sgn_fract_def:
```
```   363   "sgn (q::'a fract) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
```
```   364
```
```   365 theorem abs_fract [simp]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
```
```   366   unfolding abs_fract_def2 not_le [symmetric]
```
```   367   by transfer (auto simp add: zero_less_mult_iff le_less)
```
```   368
```
```   369 instance proof
```
```   370   fix q r s :: "'a fract"
```
```   371   assume "q \<le> r"
```
```   372   then show "s + q \<le> s + r"
```
```   373   proof (induct q, induct r, induct s)
```
```   374     fix a b c d e f :: 'a
```
```   375     assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0"
```
```   376     assume le: "Fract a b \<le> Fract c d"
```
```   377     show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
```
```   378     proof -
```
```   379       let ?F = "f * f" from neq have F: "0 < ?F"
```
```   380         by (auto simp add: zero_less_mult_iff)
```
```   381       from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   382         by simp
```
```   383       with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
```
```   384         by (simp add: mult_le_cancel_right)
```
```   385       with neq show ?thesis by (simp add: field_simps)
```
```   386     qed
```
```   387   qed
```
```   388 next
```
```   389   fix q r s :: "'a fract"
```
```   390   assume "q < r" and "0 < s"
```
```   391   then show "s * q < s * r"
```
```   392   proof (induct q, induct r, induct s)
```
```   393     fix a b c d e f :: 'a
```
```   394     assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0"
```
```   395     assume le: "Fract a b < Fract c d"
```
```   396     assume gt: "0 < Fract e f"
```
```   397     show "Fract e f * Fract a b < Fract e f * Fract c d"
```
```   398     proof -
```
```   399       let ?E = "e * f" and ?F = "f * f"
```
```   400       from neq gt have "0 < ?E"
```
```   401         by (auto simp add: Zero_fract_def order_less_le eq_fract)
```
```   402       moreover from neq have "0 < ?F"
```
```   403         by (auto simp add: zero_less_mult_iff)
```
```   404       moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
```
```   405         by simp
```
```   406       ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
```
```   407         by (simp add: mult_less_cancel_right)
```
```   408       with neq show ?thesis
```
```   409         by (simp add: ac_simps)
```
```   410     qed
```
```   411   qed
```
```   412 qed (fact sgn_fract_def abs_fract_def2)+
```
```   413
```
```   414 end
```
```   415
```
```   416 instantiation fract :: (linordered_idom) distrib_lattice
```
```   417 begin
```
```   418
```
```   419 definition inf_fract_def:
```
```   420   "(inf :: 'a fract \<Rightarrow> 'a fract \<Rightarrow> 'a fract) = min"
```
```   421
```
```   422 definition sup_fract_def:
```
```   423   "(sup :: 'a fract \<Rightarrow> 'a fract \<Rightarrow> 'a fract) = max"
```
```   424
```
```   425 instance
```
```   426   by standard (simp_all add: inf_fract_def sup_fract_def max_min_distrib2)
```
```   427
```
```   428 end
```
```   429
```
```   430 lemma fract_induct_pos [case_names Fract]:
```
```   431   fixes P :: "'a::linordered_idom fract \<Rightarrow> bool"
```
```   432   assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
```
```   433   shows "P q"
```
```   434 proof (cases q)
```
```   435   case (Fract a b)
```
```   436   {
```
```   437     fix a b :: 'a
```
```   438     assume b: "b < 0"
```
```   439     have "P (Fract a b)"
```
```   440     proof -
```
```   441       from b have "0 < - b" by simp
```
```   442       then have "P (Fract (- a) (- b))"
```
```   443         by (rule step)
```
```   444       then show "P (Fract a b)"
```
```   445         by (simp add: order_less_imp_not_eq [OF b])
```
```   446     qed
```
```   447   }
```
```   448   with Fract show "P q"
```
```   449     by (auto simp add: linorder_neq_iff step)
```
```   450 qed
```
```   451
```
```   452 lemma zero_less_Fract_iff: "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a"
```
```   453   by (auto simp add: Zero_fract_def zero_less_mult_iff)
```
```   454
```
```   455 lemma Fract_less_zero_iff: "0 < b \<Longrightarrow> Fract a b < 0 \<longleftrightarrow> a < 0"
```
```   456   by (auto simp add: Zero_fract_def mult_less_0_iff)
```
```   457
```
```   458 lemma zero_le_Fract_iff: "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a"
```
```   459   by (auto simp add: Zero_fract_def zero_le_mult_iff)
```
```   460
```
```   461 lemma Fract_le_zero_iff: "0 < b \<Longrightarrow> Fract a b \<le> 0 \<longleftrightarrow> a \<le> 0"
```
```   462   by (auto simp add: Zero_fract_def mult_le_0_iff)
```
```   463
```
```   464 lemma one_less_Fract_iff: "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a"
```
```   465   by (auto simp add: One_fract_def mult_less_cancel_right_disj)
```
```   466
```
```   467 lemma Fract_less_one_iff: "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b"
```
```   468   by (auto simp add: One_fract_def mult_less_cancel_right_disj)
```
```   469
```
```   470 lemma one_le_Fract_iff: "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a"
```
```   471   by (auto simp add: One_fract_def mult_le_cancel_right)
```
```   472
```
```   473 lemma Fract_le_one_iff: "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b"
```
```   474   by (auto simp add: One_fract_def mult_le_cancel_right)
```
```   475
```
```   476 end
```