src/HOL/Library/Rational_Numbers.thy
 author oheimb Wed Jan 31 10:15:55 2001 +0100 (2001-01-31) changeset 11008 f7333f055ef6 parent 10681 ec76e17f73c5 child 11549 e7265e70fd7c permissions -rw-r--r--
improved theory reference in comment
```     1 (*  Title:      HOL/Library/Rational_Numbers.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Markus Wenzel, TU Muenchen
```
```     4     License:    GPL (GNU GENERAL PUBLIC LICENSE)
```
```     5 *)
```
```     6
```
```     7 header {*
```
```     8   \title{Rational numbers}
```
```     9   \author{Markus Wenzel}
```
```    10 *}
```
```    11
```
```    12 theory Rational_Numbers = Quotient + Ring_and_Field:
```
```    13
```
```    14 subsection {* Fractions *}
```
```    15
```
```    16 subsubsection {* The type of fractions *}
```
```    17
```
```    18 typedef fraction = "{(a, b) :: int \<times> int | a b. b \<noteq> 0}"
```
```    19 proof
```
```    20   show "(0, #1) \<in> ?fraction" by simp
```
```    21 qed
```
```    22
```
```    23 constdefs
```
```    24   fract :: "int => int => fraction"
```
```    25   "fract a b == Abs_fraction (a, b)"
```
```    26   num :: "fraction => int"
```
```    27   "num Q == fst (Rep_fraction Q)"
```
```    28   den :: "fraction => int"
```
```    29   "den Q == snd (Rep_fraction Q)"
```
```    30
```
```    31 lemma fract_num [simp]: "b \<noteq> 0 ==> num (fract a b) = a"
```
```    32   by (simp add: fract_def num_def fraction_def Abs_fraction_inverse)
```
```    33
```
```    34 lemma fract_den [simp]: "b \<noteq> 0 ==> den (fract a b) = b"
```
```    35   by (simp add: fract_def den_def fraction_def Abs_fraction_inverse)
```
```    36
```
```    37 lemma fraction_cases [case_names fract, cases type: fraction]:
```
```    38   "(!!a b. Q = fract a b ==> b \<noteq> 0 ==> C) ==> C"
```
```    39 proof -
```
```    40   assume r: "!!a b. Q = fract a b ==> b \<noteq> 0 ==> C"
```
```    41   obtain a b where "Q = fract a b" and "b \<noteq> 0"
```
```    42     by (cases Q) (auto simp add: fract_def fraction_def)
```
```    43   thus C by (rule r)
```
```    44 qed
```
```    45
```
```    46 lemma fraction_induct [case_names fract, induct type: fraction]:
```
```    47     "(!!a b. b \<noteq> 0 ==> P (fract a b)) ==> P Q"
```
```    48   by (cases Q) simp
```
```    49
```
```    50
```
```    51 subsubsection {* Equivalence of fractions *}
```
```    52
```
```    53 instance fraction :: eqv ..
```
```    54
```
```    55 defs (overloaded)
```
```    56   equiv_fraction_def: "Q \<sim> R == num Q * den R = num R * den Q"
```
```    57
```
```    58 lemma equiv_fraction_iff:
```
```    59     "b \<noteq> 0 ==> b' \<noteq> 0 ==> (fract a b \<sim> fract a' b') = (a * b' = a' * b)"
```
```    60   by (simp add: equiv_fraction_def)
```
```    61
```
```    62 lemma equiv_fractionI [intro]:
```
```    63     "a * b' = a' * b ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> fract a b \<sim> fract a' b'"
```
```    64   by (insert equiv_fraction_iff) blast
```
```    65
```
```    66 lemma equiv_fractionD [dest]:
```
```    67     "fract a b \<sim> fract a' b' ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> a * b' = a' * b"
```
```    68   by (insert equiv_fraction_iff) blast
```
```    69
```
```    70 instance fraction :: equiv
```
```    71 proof
```
```    72   fix Q R S :: fraction
```
```    73   {
```
```    74     show "Q \<sim> Q"
```
```    75     proof (induct Q)
```
```    76       fix a b :: int
```
```    77       assume "b \<noteq> 0" and "b \<noteq> 0"
```
```    78       with refl show "fract a b \<sim> fract a b" ..
```
```    79     qed
```
```    80   next
```
```    81     assume "Q \<sim> R" and "R \<sim> S"
```
```    82     show "Q \<sim> S"
```
```    83     proof (insert prems, induct Q, induct R, induct S)
```
```    84       fix a b a' b' a'' b'' :: int
```
```    85       assume b: "b \<noteq> 0" and b': "b' \<noteq> 0" and b'': "b'' \<noteq> 0"
```
```    86       assume "fract a b \<sim> fract a' b'" hence eq1: "a * b' = a' * b" ..
```
```    87       assume "fract a' b' \<sim> fract a'' b''" hence eq2: "a' * b'' = a'' * b'" ..
```
```    88       have "a * b'' = a'' * b"
```
```    89       proof cases
```
```    90         assume "a' = 0"
```
```    91         with b' eq1 eq2 have "a = 0 \<and> a'' = 0" by auto
```
```    92         thus ?thesis by simp
```
```    93       next
```
```    94         assume a': "a' \<noteq> 0"
```
```    95         from eq1 eq2 have "(a * b') * (a' * b'') = (a' * b) * (a'' * b')" by simp
```
```    96         hence "(a * b'') * (a' * b') = (a'' * b) * (a' * b')" by (simp only: zmult_ac)
```
```    97         with a' b' show ?thesis by simp
```
```    98       qed
```
```    99       thus "fract a b \<sim> fract a'' b''" ..
```
```   100     qed
```
```   101   next
```
```   102     show "Q \<sim> R ==> R \<sim> Q"
```
```   103     proof (induct Q, induct R)
```
```   104       fix a b a' b' :: int
```
```   105       assume b: "b \<noteq> 0" and b': "b' \<noteq> 0"
```
```   106       assume "fract a b \<sim> fract a' b'"
```
```   107       hence "a * b' = a' * b" ..
```
```   108       hence "a' * b = a * b'" ..
```
```   109       thus "fract a' b' \<sim> fract a b" ..
```
```   110     qed
```
```   111   }
```
```   112 qed
```
```   113
```
```   114 lemma eq_fraction_iff:
```
```   115     "b \<noteq> 0 ==> b' \<noteq> 0 ==> (\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>) = (a * b' = a' * b)"
```
```   116   by (simp add: equiv_fraction_iff quot_equality)
```
```   117
```
```   118 lemma eq_fractionI [intro]:
```
```   119     "a * b' = a' * b ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> \<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>"
```
```   120   by (insert eq_fraction_iff) blast
```
```   121
```
```   122 lemma eq_fractionD [dest]:
```
```   123     "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> a * b' = a' * b"
```
```   124   by (insert eq_fraction_iff) blast
```
```   125
```
```   126
```
```   127 subsubsection {* Operations on fractions *}
```
```   128
```
```   129 text {*
```
```   130  We define the basic arithmetic operations on fractions and
```
```   131  demonstrate their ``well-definedness'', i.e.\ congruence with respect
```
```   132  to equivalence of fractions.
```
```   133 *}
```
```   134
```
```   135 instance fraction :: zero ..
```
```   136 instance fraction :: plus ..
```
```   137 instance fraction :: minus ..
```
```   138 instance fraction :: times ..
```
```   139 instance fraction :: inverse ..
```
```   140 instance fraction :: ord ..
```
```   141
```
```   142 defs (overloaded)
```
```   143   zero_fraction_def: "0 == fract 0 #1"
```
```   144   add_fraction_def: "Q + R ==
```
```   145     fract (num Q * den R + num R * den Q) (den Q * den R)"
```
```   146   minus_fraction_def: "-Q == fract (-(num Q)) (den Q)"
```
```   147   mult_fraction_def: "Q * R == fract (num Q * num R) (den Q * den R)"
```
```   148   inverse_fraction_def: "inverse Q == fract (den Q) (num Q)"
```
```   149   le_fraction_def: "Q \<le> R ==
```
```   150     (num Q * den R) * (den Q * den R) \<le> (num R * den Q) * (den Q * den R)"
```
```   151
```
```   152 lemma is_zero_fraction_iff: "b \<noteq> 0 ==> (\<lfloor>fract a b\<rfloor> = \<lfloor>0\<rfloor>) = (a = 0)"
```
```   153   by (simp add: zero_fraction_def eq_fraction_iff)
```
```   154
```
```   155 theorem add_fraction_cong:
```
```   156   "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>
```
```   157     ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0
```
```   158     ==> \<lfloor>fract a b + fract c d\<rfloor> = \<lfloor>fract a' b' + fract c' d'\<rfloor>"
```
```   159 proof -
```
```   160   assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
```
```   161   assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>" hence eq1: "a * b' = a' * b" ..
```
```   162   assume "\<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>" hence eq2: "c * d' = c' * d" ..
```
```   163   have "\<lfloor>fract (a * d + c * b) (b * d)\<rfloor> = \<lfloor>fract (a' * d' + c' * b') (b' * d')\<rfloor>"
```
```   164   proof
```
```   165     show "(a * d + c * b) * (b' * d') = (a' * d' + c' * b') * (b * d)"
```
```   166       (is "?lhs = ?rhs")
```
```   167     proof -
```
```   168       have "?lhs = (a * b') * (d * d') + (c * d') * (b * b')"
```
```   169         by (simp add: int_distrib zmult_ac)
```
```   170       also have "... = (a' * b) * (d * d') + (c' * d) * (b * b')"
```
```   171         by (simp only: eq1 eq2)
```
```   172       also have "... = ?rhs"
```
```   173         by (simp add: int_distrib zmult_ac)
```
```   174       finally show "?lhs = ?rhs" .
```
```   175     qed
```
```   176     from neq show "b * d \<noteq> 0" by simp
```
```   177     from neq show "b' * d' \<noteq> 0" by simp
```
```   178   qed
```
```   179   with neq show ?thesis by (simp add: add_fraction_def)
```
```   180 qed
```
```   181
```
```   182 theorem minus_fraction_cong:
```
```   183   "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> b \<noteq> 0 ==> b' \<noteq> 0
```
```   184     ==> \<lfloor>-(fract a b)\<rfloor> = \<lfloor>-(fract a' b')\<rfloor>"
```
```   185 proof -
```
```   186   assume neq: "b \<noteq> 0"  "b' \<noteq> 0"
```
```   187   assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>"
```
```   188   hence "a * b' = a' * b" ..
```
```   189   hence "-a * b' = -a' * b" by simp
```
```   190   hence "\<lfloor>fract (-a) b\<rfloor> = \<lfloor>fract (-a') b'\<rfloor>" ..
```
```   191   with neq show ?thesis by (simp add: minus_fraction_def)
```
```   192 qed
```
```   193
```
```   194 theorem mult_fraction_cong:
```
```   195   "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>
```
```   196     ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0
```
```   197     ==> \<lfloor>fract a b * fract c d\<rfloor> = \<lfloor>fract a' b' * fract c' d'\<rfloor>"
```
```   198 proof -
```
```   199   assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
```
```   200   assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>" hence eq1: "a * b' = a' * b" ..
```
```   201   assume "\<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>" hence eq2: "c * d' = c' * d" ..
```
```   202   have "\<lfloor>fract (a * c) (b * d)\<rfloor> = \<lfloor>fract (a' * c') (b' * d')\<rfloor>"
```
```   203   proof
```
```   204     from eq1 eq2 have "(a * b') * (c * d') = (a' * b) * (c' * d)" by simp
```
```   205     thus "(a * c) * (b' * d') = (a' * c') * (b * d)" by (simp add: zmult_ac)
```
```   206     from neq show "b * d \<noteq> 0" by simp
```
```   207     from neq show "b' * d' \<noteq> 0" by simp
```
```   208   qed
```
```   209   with neq show "\<lfloor>fract a b * fract c d\<rfloor> = \<lfloor>fract a' b' * fract c' d'\<rfloor>"
```
```   210     by (simp add: mult_fraction_def)
```
```   211 qed
```
```   212
```
```   213 theorem inverse_fraction_cong:
```
```   214   "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract a b\<rfloor> \<noteq> \<lfloor>0\<rfloor> ==> \<lfloor>fract a' b'\<rfloor> \<noteq> \<lfloor>0\<rfloor>
```
```   215     ==> b \<noteq> 0 ==> b' \<noteq> 0
```
```   216     ==> \<lfloor>inverse (fract a b)\<rfloor> = \<lfloor>inverse (fract a' b')\<rfloor>"
```
```   217 proof -
```
```   218   assume neq: "b \<noteq> 0"  "b' \<noteq> 0"
```
```   219   assume "\<lfloor>fract a b\<rfloor> \<noteq> \<lfloor>0\<rfloor>" and "\<lfloor>fract a' b'\<rfloor> \<noteq> \<lfloor>0\<rfloor>"
```
```   220   with neq obtain "a \<noteq> 0" and "a' \<noteq> 0" by (simp add: is_zero_fraction_iff)
```
```   221   assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>"
```
```   222   hence "a * b' = a' * b" ..
```
```   223   hence "b * a' = b' * a" by (simp only: zmult_ac)
```
```   224   hence "\<lfloor>fract b a\<rfloor> = \<lfloor>fract b' a'\<rfloor>" ..
```
```   225   with neq show ?thesis by (simp add: inverse_fraction_def)
```
```   226 qed
```
```   227
```
```   228 theorem le_fraction_cong:
```
```   229   "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>
```
```   230     ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0
```
```   231     ==> (fract a b \<le> fract c d) = (fract a' b' \<le> fract c' d')"
```
```   232 proof -
```
```   233   assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
```
```   234   assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>" hence eq1: "a * b' = a' * b" ..
```
```   235   assume "\<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>" hence eq2: "c * d' = c' * d" ..
```
```   236
```
```   237   let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
```
```   238   {
```
```   239     fix a b c d x :: int 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" by (auto simp add: int_less_le)
```
```   243       hence "?le a b c d =
```
```   244           ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
```
```   245         by (simp add: zmult_zle_cancel2)
```
```   246       also have "... = ?le (a * x) (b * x) c d"
```
```   247         by (simp add: zmult_ac)
```
```   248       finally show ?thesis .
```
```   249     qed
```
```   250   } note le_factor = this
```
```   251
```
```   252   let ?D = "b * d" and ?D' = "b' * d'"
```
```   253   from neq have D: "?D \<noteq> 0" by simp
```
```   254   from neq have "?D' \<noteq> 0" by simp
```
```   255   hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"
```
```   256     by (rule le_factor)
```
```   257   also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')"
```
```   258     by (simp add: zmult_ac)
```
```   259   also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
```
```   260     by (simp only: eq1 eq2)
```
```   261   also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
```
```   262     by (simp add: zmult_ac)
```
```   263   also from D have "... = ?le a' b' c' d'"
```
```   264     by (rule le_factor [symmetric])
```
```   265   finally have "?le a b c d = ?le a' b' c' d'" .
```
```   266   with neq show ?thesis by (simp add: le_fraction_def)
```
```   267 qed
```
```   268
```
```   269
```
```   270 subsection {* Rational numbers *}
```
```   271
```
```   272 subsubsection {* The type of rational numbers *}
```
```   273
```
```   274 typedef (Rat)
```
```   275   rat = "UNIV :: fraction quot set" ..
```
```   276
```
```   277 lemma RatI [intro, simp]: "Q \<in> Rat"
```
```   278   by (simp add: Rat_def)
```
```   279
```
```   280 constdefs
```
```   281   fraction_of :: "rat => fraction"
```
```   282   "fraction_of q == pick (Rep_Rat q)"
```
```   283   rat_of :: "fraction => rat"
```
```   284   "rat_of Q == Abs_Rat \<lfloor>Q\<rfloor>"
```
```   285
```
```   286 theorem rat_of_equality [iff?]: "(rat_of Q = rat_of Q') = (\<lfloor>Q\<rfloor> = \<lfloor>Q'\<rfloor>)"
```
```   287   by (simp add: rat_of_def Abs_Rat_inject)
```
```   288
```
```   289 lemma rat_of: "\<lfloor>Q\<rfloor> = \<lfloor>Q'\<rfloor> ==> rat_of Q = rat_of Q'" ..
```
```   290
```
```   291 constdefs
```
```   292   Fract :: "int => int => rat"
```
```   293   "Fract a b == rat_of (fract a b)"
```
```   294
```
```   295 theorem Fract_inverse: "\<lfloor>fraction_of (Fract a b)\<rfloor> = \<lfloor>fract a b\<rfloor>"
```
```   296   by (simp add: fraction_of_def rat_of_def Fract_def Abs_Rat_inverse pick_inverse)
```
```   297
```
```   298 theorem Fract_equality [iff?]:
```
```   299     "(Fract a b = Fract c d) = (\<lfloor>fract a b\<rfloor> = \<lfloor>fract c d\<rfloor>)"
```
```   300   by (simp add: Fract_def rat_of_equality)
```
```   301
```
```   302 theorem eq_rat:
```
```   303     "b \<noteq> 0 ==> d \<noteq> 0 ==> (Fract a b = Fract c d) = (a * d = c * b)"
```
```   304   by (simp add: Fract_equality eq_fraction_iff)
```
```   305
```
```   306 theorem Rat_cases [case_names Fract, cases type: rat]:
```
```   307   "(!!a b. q = Fract a b ==> b \<noteq> 0 ==> C) ==> C"
```
```   308 proof -
```
```   309   assume r: "!!a b. q = Fract a b ==> b \<noteq> 0 ==> C"
```
```   310   obtain x where "q = Abs_Rat x" by (cases q)
```
```   311   moreover obtain Q where "x = \<lfloor>Q\<rfloor>" by (cases x)
```
```   312   moreover obtain a b where "Q = fract a b" and "b \<noteq> 0" by (cases Q)
```
```   313   ultimately have "q = Fract a b" by (simp only: Fract_def rat_of_def)
```
```   314   thus ?thesis by (rule r)
```
```   315 qed
```
```   316
```
```   317 theorem Rat_induct [case_names Fract, induct type: rat]:
```
```   318     "(!!a b. b \<noteq> 0 ==> P (Fract a b)) ==> P q"
```
```   319   by (cases q) simp
```
```   320
```
```   321
```
```   322 subsubsection {* Canonical function definitions *}
```
```   323
```
```   324 text {*
```
```   325   Note that the unconditional version below is much easier to read.
```
```   326 *}
```
```   327
```
```   328 theorem rat_cond_function:
```
```   329   "(!!q r. P \<lfloor>fraction_of q\<rfloor> \<lfloor>fraction_of r\<rfloor> ==>
```
```   330       f q r == g (fraction_of q) (fraction_of r)) ==>
```
```   331     (!!a b a' b' c d c' d'.
```
```   332       \<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor> ==>
```
```   333       P \<lfloor>fract a b\<rfloor> \<lfloor>fract c d\<rfloor> ==> P \<lfloor>fract a' b'\<rfloor> \<lfloor>fract c' d'\<rfloor> ==>
```
```   334       b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0 ==>
```
```   335       g (fract a b) (fract c d) = g (fract a' b') (fract c' d')) ==>
```
```   336     P \<lfloor>fract a b\<rfloor> \<lfloor>fract c d\<rfloor> ==>
```
```   337       f (Fract a b) (Fract c d) = g (fract a b) (fract c d)"
```
```   338   (is "PROP ?eq ==> PROP ?cong ==> ?P ==> _")
```
```   339 proof -
```
```   340   assume eq: "PROP ?eq" and cong: "PROP ?cong" and P: ?P
```
```   341   have "f (Abs_Rat \<lfloor>fract a b\<rfloor>) (Abs_Rat \<lfloor>fract c d\<rfloor>) = g (fract a b) (fract c d)"
```
```   342   proof (rule quot_cond_function)
```
```   343     fix X Y assume "P X Y"
```
```   344     with eq show "f (Abs_Rat X) (Abs_Rat Y) == g (pick X) (pick Y)"
```
```   345       by (simp add: fraction_of_def pick_inverse Abs_Rat_inverse)
```
```   346   next
```
```   347     fix Q Q' R R' :: fraction
```
```   348     show "\<lfloor>Q\<rfloor> = \<lfloor>Q'\<rfloor> ==> \<lfloor>R\<rfloor> = \<lfloor>R'\<rfloor> ==>
```
```   349         P \<lfloor>Q\<rfloor> \<lfloor>R\<rfloor> ==> P \<lfloor>Q'\<rfloor> \<lfloor>R'\<rfloor> ==> g Q R = g Q' R'"
```
```   350       by (induct Q, induct Q', induct R, induct R') (rule cong)
```
```   351   qed
```
```   352   thus ?thesis by (unfold Fract_def rat_of_def)
```
```   353 qed
```
```   354
```
```   355 theorem rat_function:
```
```   356   "(!!q r. f q r == g (fraction_of q) (fraction_of r)) ==>
```
```   357     (!!a b a' b' c d c' d'.
```
```   358       \<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor> ==>
```
```   359       b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0 ==>
```
```   360       g (fract a b) (fract c d) = g (fract a' b') (fract c' d')) ==>
```
```   361     f (Fract a b) (Fract c d) = g (fract a b) (fract c d)"
```
```   362 proof -
```
```   363   case antecedent from this TrueI
```
```   364   show ?thesis by (rule rat_cond_function)
```
```   365 qed
```
```   366
```
```   367
```
```   368 subsubsection {* Standard operations on rational numbers *}
```
```   369
```
```   370 instance rat :: zero ..
```
```   371 instance rat :: plus ..
```
```   372 instance rat :: minus ..
```
```   373 instance rat :: times ..
```
```   374 instance rat :: inverse ..
```
```   375 instance rat :: ord ..
```
```   376 instance rat :: number ..
```
```   377
```
```   378 defs (overloaded)
```
```   379   zero_rat_def: "0 == rat_of 0"
```
```   380   add_rat_def: "q + r == rat_of (fraction_of q + fraction_of r)"
```
```   381   minus_rat_def: "-q == rat_of (-(fraction_of q))"
```
```   382   diff_rat_def: "q - r == q + (-(r::rat))"
```
```   383   mult_rat_def: "q * r == rat_of (fraction_of q * fraction_of r)"
```
```   384   inverse_rat_def: "q \<noteq> 0 ==> inverse q == rat_of (inverse (fraction_of q))"
```
```   385   divide_rat_def: "r \<noteq> 0 ==> q / r == q * inverse (r::rat)"
```
```   386   le_rat_def: "q \<le> r == fraction_of q \<le> fraction_of r"
```
```   387   less_rat_def: "q < r == q \<le> r \<and> q \<noteq> (r::rat)"
```
```   388   abs_rat_def: "\<bar>q\<bar> == if q < 0 then -q else (q::rat)"
```
```   389   number_of_rat_def: "number_of b == Fract (number_of b) #1"
```
```   390
```
```   391 theorem zero_rat: "0 = Fract 0 #1"
```
```   392   by (simp add: zero_rat_def zero_fraction_def rat_of_def Fract_def)
```
```   393
```
```   394 theorem add_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
```
```   395   Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
```
```   396 proof -
```
```   397   have "Fract a b + Fract c d = rat_of (fract a b + fract c d)"
```
```   398     by (rule rat_function, rule add_rat_def, rule rat_of, rule add_fraction_cong)
```
```   399   also
```
```   400   assume "b \<noteq> 0"  "d \<noteq> 0"
```
```   401   hence "fract a b + fract c d = fract (a * d + c * b) (b * d)"
```
```   402     by (simp add: add_fraction_def)
```
```   403   finally show ?thesis by (unfold Fract_def)
```
```   404 qed
```
```   405
```
```   406 theorem minus_rat: "b \<noteq> 0 ==> -(Fract a b) = Fract (-a) b"
```
```   407 proof -
```
```   408   have "-(Fract a b) = rat_of (-(fract a b))"
```
```   409     by (rule rat_function, rule minus_rat_def, rule rat_of, rule minus_fraction_cong)
```
```   410   also assume "b \<noteq> 0" hence "-(fract a b) = fract (-a) b"
```
```   411     by (simp add: minus_fraction_def)
```
```   412   finally show ?thesis by (unfold Fract_def)
```
```   413 qed
```
```   414
```
```   415 theorem diff_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
```
```   416     Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
```
```   417   by (simp add: diff_rat_def add_rat minus_rat)
```
```   418
```
```   419 theorem mult_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
```
```   420   Fract a b * Fract c d = Fract (a * c) (b * d)"
```
```   421 proof -
```
```   422   have "Fract a b * Fract c d = rat_of (fract a b * fract c d)"
```
```   423     by (rule rat_function, rule mult_rat_def, rule rat_of, rule mult_fraction_cong)
```
```   424   also
```
```   425   assume "b \<noteq> 0"  "d \<noteq> 0"
```
```   426   hence "fract a b * fract c d = fract (a * c) (b * d)"
```
```   427     by (simp add: mult_fraction_def)
```
```   428   finally show ?thesis by (unfold Fract_def)
```
```   429 qed
```
```   430
```
```   431 theorem inverse_rat: "Fract a b \<noteq> 0 ==> b \<noteq> 0 ==>
```
```   432   inverse (Fract a b) = Fract b a"
```
```   433 proof -
```
```   434   assume neq: "b \<noteq> 0" and nonzero: "Fract a b \<noteq> 0"
```
```   435   hence "\<lfloor>fract a b\<rfloor> \<noteq> \<lfloor>0\<rfloor>"
```
```   436     by (simp add: zero_rat eq_rat is_zero_fraction_iff)
```
```   437   with _ inverse_fraction_cong [THEN rat_of]
```
```   438   have "inverse (Fract a b) = rat_of (inverse (fract a b))"
```
```   439   proof (rule rat_cond_function)
```
```   440     fix q assume cond: "\<lfloor>fraction_of q\<rfloor> \<noteq> \<lfloor>0\<rfloor>"
```
```   441     have "q \<noteq> 0"
```
```   442     proof (cases q)
```
```   443       fix a b assume "b \<noteq> 0" and "q = Fract a b"
```
```   444       from this cond show ?thesis
```
```   445         by (simp add: Fract_inverse is_zero_fraction_iff zero_rat eq_rat)
```
```   446     qed
```
```   447     thus "inverse q == rat_of (inverse (fraction_of q))"
```
```   448       by (rule inverse_rat_def)
```
```   449   qed
```
```   450   also from neq nonzero have "inverse (fract a b) = fract b a"
```
```   451     by (simp add: inverse_fraction_def)
```
```   452   finally show ?thesis by (unfold Fract_def)
```
```   453 qed
```
```   454
```
```   455 theorem divide_rat: "Fract c d \<noteq> 0 ==> b \<noteq> 0 ==> d \<noteq> 0 ==>
```
```   456   Fract a b / Fract c d = Fract (a * d) (b * c)"
```
```   457 proof -
```
```   458   assume neq: "b \<noteq> 0"  "d \<noteq> 0" and nonzero: "Fract c d \<noteq> 0"
```
```   459   hence "c \<noteq> 0" by (simp add: zero_rat eq_rat)
```
```   460   with neq nonzero show ?thesis
```
```   461     by (simp add: divide_rat_def inverse_rat mult_rat)
```
```   462 qed
```
```   463
```
```   464 theorem le_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
```
```   465   (Fract a b \<le> Fract c d) = ((a * d) * (b * d) \<le> (c * b) * (b * d))"
```
```   466 proof -
```
```   467   have "(Fract a b \<le> Fract c d) = (fract a b \<le> fract c d)"
```
```   468     by (rule rat_function, rule le_rat_def, rule le_fraction_cong)
```
```   469   also
```
```   470   assume "b \<noteq> 0"  "d \<noteq> 0"
```
```   471   hence "(fract a b \<le> fract c d) = ((a * d) * (b * d) \<le> (c * b) * (b * d))"
```
```   472     by (simp add: le_fraction_def)
```
```   473   finally show ?thesis .
```
```   474 qed
```
```   475
```
```   476 theorem less_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
```
```   477     (Fract a b < Fract c d) = ((a * d) * (b * d) < (c * b) * (b * d))"
```
```   478   by (simp add: less_rat_def le_rat eq_rat int_less_le)
```
```   479
```
```   480 theorem abs_rat: "b \<noteq> 0 ==> \<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
```
```   481   by (simp add: abs_rat_def minus_rat zero_rat less_rat eq_rat)
```
```   482     (auto simp add: zmult_less_0_iff int_0_less_mult_iff int_le_less split: zabs_split)
```
```   483
```
```   484
```
```   485 subsubsection {* The ordered field of rational numbers *}
```
```   486
```
```   487 instance rat :: field
```
```   488 proof
```
```   489   fix q r s :: rat
```
```   490   show "(q + r) + s = q + (r + s)"
```
```   491     by (induct q, induct r, induct s) (simp add: add_rat zadd_ac zmult_ac int_distrib)
```
```   492   show "q + r = r + q"
```
```   493     by (induct q, induct r) (simp add: add_rat zadd_ac zmult_ac)
```
```   494   show "0 + q = q"
```
```   495     by (induct q) (simp add: zero_rat add_rat)
```
```   496   show "(-q) + q = 0"
```
```   497     by (induct q) (simp add: zero_rat minus_rat add_rat eq_rat)
```
```   498   show "q - r = q + (-r)"
```
```   499     by (induct q, induct r) (simp add: add_rat minus_rat diff_rat)
```
```   500   show "(0::rat) = #0"
```
```   501     by (simp add: zero_rat number_of_rat_def)
```
```   502   show "(q * r) * s = q * (r * s)"
```
```   503     by (induct q, induct r, induct s) (simp add: mult_rat zmult_ac)
```
```   504   show "q * r = r * q"
```
```   505     by (induct q, induct r) (simp add: mult_rat zmult_ac)
```
```   506   show "#1 * q = q"
```
```   507     by (induct q) (simp add: number_of_rat_def mult_rat)
```
```   508   show "(q + r) * s = q * s + r * s"
```
```   509     by (induct q, induct r, induct s) (simp add: add_rat mult_rat eq_rat int_distrib)
```
```   510   show "q \<noteq> 0 ==> inverse q * q = #1"
```
```   511     by (induct q) (simp add: inverse_rat mult_rat number_of_rat_def zero_rat eq_rat)
```
```   512   show "r \<noteq> 0 ==> q / r = q * inverse r"
```
```   513     by (induct q, induct r) (simp add: mult_rat divide_rat inverse_rat zero_rat eq_rat)
```
```   514 qed
```
```   515
```
```   516 instance rat :: linorder
```
```   517 proof
```
```   518   fix q r s :: rat
```
```   519   {
```
```   520     assume "q \<le> r" and "r \<le> s"
```
```   521     show "q \<le> s"
```
```   522     proof (insert prems, induct q, induct r, induct s)
```
```   523       fix a b c d e f :: int
```
```   524       assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
```
```   525       assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"
```
```   526       show "Fract a b \<le> Fract e f"
```
```   527       proof -
```
```   528         from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
```
```   529           by (auto simp add: int_less_le)
```
```   530         have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
```
```   531         proof -
```
```   532           from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   533             by (simp add: le_rat)
```
```   534           with ff show ?thesis by (simp add: zmult_zle_cancel2)
```
```   535         qed
```
```   536         also have "... = (c * f) * (d * f) * (b * b)"
```
```   537           by (simp only: zmult_ac)
```
```   538         also have "... \<le> (e * d) * (d * f) * (b * b)"
```
```   539         proof -
```
```   540           from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
```
```   541             by (simp add: le_rat)
```
```   542           with bb show ?thesis by (simp add: zmult_zle_cancel2)
```
```   543         qed
```
```   544         finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
```
```   545           by (simp only: zmult_ac)
```
```   546         with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
```
```   547           by (simp add: zmult_zle_cancel2)
```
```   548         with neq show ?thesis by (simp add: le_rat)
```
```   549       qed
```
```   550     qed
```
```   551   next
```
```   552     assume "q \<le> r" and "r \<le> q"
```
```   553     show "q = r"
```
```   554     proof (insert prems, induct q, induct r)
```
```   555       fix a b c d :: int
```
```   556       assume neq: "b \<noteq> 0"  "d \<noteq> 0"
```
```   557       assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"
```
```   558       show "Fract a b = Fract c d"
```
```   559       proof -
```
```   560         from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   561           by (simp add: le_rat)
```
```   562         also have "... \<le> (a * d) * (b * d)"
```
```   563         proof -
```
```   564           from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
```
```   565             by (simp add: le_rat)
```
```   566           thus ?thesis by (simp only: zmult_ac)
```
```   567         qed
```
```   568         finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
```
```   569         moreover from neq have "b * d \<noteq> 0" by simp
```
```   570         ultimately have "a * d = c * b" by simp
```
```   571         with neq show ?thesis by (simp add: eq_rat)
```
```   572       qed
```
```   573     qed
```
```   574   next
```
```   575     show "q \<le> q"
```
```   576       by (induct q) (simp add: le_rat)
```
```   577     show "(q < r) = (q \<le> r \<and> q \<noteq> r)"
```
```   578       by (simp only: less_rat_def)
```
```   579     show "q \<le> r \<or> r \<le> q"
```
```   580       by (induct q, induct r) (simp add: le_rat zmult_ac, arith)
```
```   581   }
```
```   582 qed
```
```   583
```
```   584 instance rat :: ordered_field
```
```   585 proof
```
```   586   fix q r s :: rat
```
```   587   show "q \<le> r ==> s + q \<le> s + r"
```
```   588   proof (induct q, induct r, induct s)
```
```   589     fix a b c d e f :: int
```
```   590     assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
```
```   591     assume le: "Fract a b \<le> Fract c d"
```
```   592     show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
```
```   593     proof -
```
```   594       let ?F = "f * f" from neq have F: "0 < ?F"
```
```   595         by (auto simp add: int_less_le)
```
```   596       from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
```
```   597         by (simp add: le_rat)
```
```   598       with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
```
```   599         by (simp add: zmult_zle_cancel2)
```
```   600       with neq show ?thesis by (simp add: add_rat le_rat zmult_ac int_distrib)
```
```   601     qed
```
```   602   qed
```
```   603   show "q < r ==> 0 < s ==> s * q < s * r"
```
```   604   proof (induct q, induct r, induct s)
```
```   605     fix a b c d e f :: int
```
```   606     assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
```
```   607     assume le: "Fract a b < Fract c d"
```
```   608     assume gt: "0 < Fract e f"
```
```   609     show "Fract e f * Fract a b < Fract e f * Fract c d"
```
```   610     proof -
```
```   611       let ?E = "e * f" and ?F = "f * f"
```
```   612       from neq gt have "0 < ?E"
```
```   613         by (auto simp add: zero_rat less_rat le_rat int_less_le eq_rat)
```
```   614       moreover from neq have "0 < ?F"
```
```   615         by (auto simp add: int_less_le)
```
```   616       moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
```
```   617         by (simp add: less_rat)
```
```   618       ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
```
```   619         by (simp add: zmult_zless_cancel2)
```
```   620       with neq show ?thesis
```
```   621         by (simp add: less_rat mult_rat zmult_ac)
```
```   622     qed
```
```   623   qed
```
```   624   show "\<bar>q\<bar> = (if q < 0 then -q else q)"
```
```   625     by (simp only: abs_rat_def)
```
```   626 qed
```
```   627
```
```   628
```
```   629 subsection {* Embedding integers *}
```
```   630
```
```   631 constdefs
```
```   632   rat :: "int => rat"    (* FIXME generalize int to any numeric subtype *)
```
```   633   "rat z == Fract z #1"
```
```   634   int_set :: "rat set"    ("\<int>")    (* FIXME generalize rat to any numeric supertype *)
```
```   635   "\<int> == range rat"
```
```   636
```
```   637 lemma rat_inject: "(rat z = rat w) = (z = w)"
```
```   638 proof
```
```   639   assume "rat z = rat w"
```
```   640   hence "Fract z #1 = Fract w #1" by (unfold rat_def)
```
```   641   hence "\<lfloor>fract z #1\<rfloor> = \<lfloor>fract w #1\<rfloor>" ..
```
```   642   thus "z = w" by auto
```
```   643 next
```
```   644   assume "z = w"
```
```   645   thus "rat z = rat w" by simp
```
```   646 qed
```
```   647
```
```   648 lemma int_set_cases [case_names rat, cases set: int_set]:
```
```   649   "q \<in> \<int> ==> (!!z. q = rat z ==> C) ==> C"
```
```   650 proof (unfold int_set_def)
```
```   651   assume "!!z. q = rat z ==> C"
```
```   652   assume "q \<in> range rat" thus C ..
```
```   653 qed
```
```   654
```
```   655 lemma int_set_induct [case_names rat, induct set: int_set]:
```
```   656   "q \<in> \<int> ==> (!!z. P (rat z)) ==> P q"
```
```   657   by (rule int_set_cases) auto
```
```   658
```
```   659 theorem number_of_rat: "number_of b = rat (number_of b)"
```
```   660   by (simp only: number_of_rat_def rat_def)
```
```   661
```
```   662 end
```