src/HOL/Rat.thy
 author huffman Thu May 10 21:18:41 2012 +0200 (2012-05-10) changeset 47906 09a896d295bd parent 47108 2a1953f0d20d child 47907 54e3847f1669 permissions -rw-r--r--
convert Rat.thy to use lift_definition/transfer
1 (*  Title:  HOL/Rat.thy
2     Author: Markus Wenzel, TU Muenchen
3 *)
5 header {* Rational numbers *}
7 theory Rat
8 imports GCD Archimedean_Field
9 uses ("Tools/float_syntax.ML")
10 begin
12 subsection {* Rational numbers as quotient *}
14 subsubsection {* Construction of the type of rational numbers *}
16 definition
17   ratrel :: "(int \<times> int) \<Rightarrow> (int \<times> int) \<Rightarrow> bool" where
18   "ratrel = (\<lambda>x y. snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x)"
20 lemma ratrel_iff [simp]:
21   "ratrel x y \<longleftrightarrow> snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x"
22   by (simp add: ratrel_def)
24 lemma exists_ratrel_refl: "\<exists>x. ratrel x x"
25   by (auto intro!: one_neq_zero)
27 lemma symp_ratrel: "symp ratrel"
28   by (simp add: ratrel_def symp_def)
30 lemma transp_ratrel: "transp ratrel"
31 proof (rule transpI, unfold split_paired_all)
32   fix a b a' b' a'' b'' :: int
33   assume A: "ratrel (a, b) (a', b')"
34   assume B: "ratrel (a', b') (a'', b'')"
35   have "b' * (a * b'') = b'' * (a * b')" by simp
36   also from A have "a * b' = a' * b" by auto
37   also have "b'' * (a' * b) = b * (a' * b'')" by simp
38   also from B have "a' * b'' = a'' * b'" by auto
39   also have "b * (a'' * b') = b' * (a'' * b)" by simp
40   finally have "b' * (a * b'') = b' * (a'' * b)" .
41   moreover from B have "b' \<noteq> 0" by auto
42   ultimately have "a * b'' = a'' * b" by simp
43   with A B show "ratrel (a, b) (a'', b'')" by auto
44 qed
46 lemma part_equivp_ratrel: "part_equivp ratrel"
47   by (rule part_equivpI [OF exists_ratrel_refl symp_ratrel transp_ratrel])
49 quotient_type rat = "int \<times> int" / partial: "ratrel"
50   morphisms Rep_Rat Abs_Rat
51   by (rule part_equivp_ratrel)
53 declare rat.forall_transfer [transfer_rule del]
55 lemma forall_rat_transfer [transfer_rule]: (* TODO: generate automatically *)
56   "(fun_rel (fun_rel cr_rat op =) op =)
57     (transfer_bforall (\<lambda>x. snd x \<noteq> 0)) transfer_forall"
58   using rat.forall_transfer by simp
61 subsubsection {* Representation and basic operations *}
63 lift_definition Fract :: "int \<Rightarrow> int \<Rightarrow> rat"
64   is "\<lambda>a b. if b = 0 then (0, 1) else (a, b)"
65   by simp
67 lemma eq_rat:
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)+
73 lemma Rat_cases [case_names Fract, cases type: rat]:
74   assumes "\<And>a b. q = Fract a b \<Longrightarrow> b > 0 \<Longrightarrow> coprime a b \<Longrightarrow> C"
75   shows C
76 proof -
77   obtain a b :: int where "q = Fract a b" and "b \<noteq> 0"
78     by transfer simp
79   let ?a = "a div gcd a b"
80   let ?b = "b div gcd a b"
81   from `b \<noteq> 0` have "?b * gcd a b = b"
82     by (simp add: dvd_div_mult_self)
83   with `b \<noteq> 0` have "?b \<noteq> 0" by auto
84   from `q = Fract a b` `b \<noteq> 0` `?b \<noteq> 0` have q: "q = Fract ?a ?b"
85     by (simp add: eq_rat dvd_div_mult mult_commute [of a])
86   from `b \<noteq> 0` have coprime: "coprime ?a ?b"
87     by (auto intro: div_gcd_coprime_int)
88   show C proof (cases "b > 0")
89     case True
90     note assms
91     moreover note q
92     moreover from True have "?b > 0" by (simp add: nonneg1_imp_zdiv_pos_iff)
93     moreover note coprime
94     ultimately show C .
95   next
96     case False
97     note assms
98     moreover have "q = Fract (- ?a) (- ?b)" unfolding q by transfer simp
99     moreover from False `b \<noteq> 0` have "- ?b > 0" by (simp add: pos_imp_zdiv_neg_iff)
100     moreover from coprime have "coprime (- ?a) (- ?b)" by simp
101     ultimately show C .
102   qed
103 qed
105 lemma Rat_induct [case_names Fract, induct type: rat]:
106   assumes "\<And>a b. b > 0 \<Longrightarrow> coprime a b \<Longrightarrow> P (Fract a b)"
107   shows "P q"
108   using assms by (cases q) simp
110 instantiation rat :: field_inverse_zero
111 begin
113 lift_definition zero_rat :: "rat" is "(0, 1)"
114   by simp
116 lift_definition one_rat :: "rat" is "(1, 1)"
117   by simp
119 lemma Zero_rat_def: "0 = Fract 0 1"
120   by transfer simp
122 lemma One_rat_def: "1 = Fract 1 1"
123   by transfer simp
125 lift_definition plus_rat :: "rat \<Rightarrow> rat \<Rightarrow> rat"
126   is "\<lambda>x y. (fst x * snd y + fst y * snd x, snd x * snd y)"
127   by (clarsimp, simp add: left_distrib, simp add: mult_ac)
129 lemma add_rat [simp]:
130   assumes "b \<noteq> 0" and "d \<noteq> 0"
131   shows "Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
132   using assms by transfer simp
134 lift_definition uminus_rat :: "rat \<Rightarrow> rat" is "\<lambda>x. (- fst x, snd x)"
135   by simp
137 lemma minus_rat [simp]: "- Fract a b = Fract (- a) b"
138   by transfer simp
140 lemma minus_rat_cancel [simp]: "Fract (- a) (- b) = Fract a b"
141   by (cases "b = 0") (simp_all add: eq_rat)
143 definition
144   diff_rat_def: "q - r = q + - (r::rat)"
146 lemma diff_rat [simp]:
147   assumes "b \<noteq> 0" and "d \<noteq> 0"
148   shows "Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
149   using assms by (simp add: diff_rat_def)
151 lift_definition times_rat :: "rat \<Rightarrow> rat \<Rightarrow> rat"
152   is "\<lambda>x y. (fst x * fst y, snd x * snd y)"
153   by (simp add: mult_ac)
155 lemma mult_rat [simp]: "Fract a b * Fract c d = Fract (a * c) (b * d)"
156   by transfer simp
158 lemma mult_rat_cancel:
159   assumes "c \<noteq> 0"
160   shows "Fract (c * a) (c * b) = Fract a b"
161   using assms by transfer simp
163 lift_definition inverse_rat :: "rat \<Rightarrow> rat"
164   is "\<lambda>x. if fst x = 0 then (0, 1) else (snd x, fst x)"
165   by (auto simp add: mult_commute)
167 lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"
168   by transfer simp
170 definition
171   divide_rat_def: "q / r = q * inverse (r::rat)"
173 lemma divide_rat [simp]: "Fract a b / Fract c d = Fract (a * d) (b * c)"
174   by (simp add: divide_rat_def)
176 instance proof
177   fix q r s :: rat
178   show "(q * r) * s = q * (r * s)"
179     by transfer simp
180   show "q * r = r * q"
181     by transfer simp
182   show "1 * q = q"
183     by transfer simp
184   show "(q + r) + s = q + (r + s)"
185     by transfer (simp add: algebra_simps)
186   show "q + r = r + q"
187     by transfer simp
188   show "0 + q = q"
189     by transfer simp
190   show "- q + q = 0"
191     by transfer simp
192   show "q - r = q + - r"
193     by (fact diff_rat_def)
194   show "(q + r) * s = q * s + r * s"
195     by transfer (simp add: algebra_simps)
196   show "(0::rat) \<noteq> 1"
197     by transfer simp
198   { assume "q \<noteq> 0" thus "inverse q * q = 1"
199     by transfer simp }
200   show "q / r = q * inverse r"
201     by (fact divide_rat_def)
202   show "inverse 0 = (0::rat)"
203     by transfer simp
204 qed
206 end
208 lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1"
209   by (induct k) (simp_all add: Zero_rat_def One_rat_def)
211 lemma of_int_rat: "of_int k = Fract k 1"
212   by (cases k rule: int_diff_cases) (simp add: of_nat_rat)
214 lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
215   by (rule of_nat_rat [symmetric])
217 lemma Fract_of_int_eq: "Fract k 1 = of_int k"
218   by (rule of_int_rat [symmetric])
220 lemma rat_number_collapse:
221   "Fract 0 k = 0"
222   "Fract 1 1 = 1"
223   "Fract (numeral w) 1 = numeral w"
224   "Fract (neg_numeral w) 1 = neg_numeral w"
225   "Fract k 0 = 0"
226   using Fract_of_int_eq [of "numeral w"]
227   using Fract_of_int_eq [of "neg_numeral w"]
228   by (simp_all add: Zero_rat_def One_rat_def eq_rat)
230 lemma rat_number_expand:
231   "0 = Fract 0 1"
232   "1 = Fract 1 1"
233   "numeral k = Fract (numeral k) 1"
234   "neg_numeral k = Fract (neg_numeral k) 1"
235   by (simp_all add: rat_number_collapse)
237 lemma Rat_cases_nonzero [case_names Fract 0]:
238   assumes Fract: "\<And>a b. q = Fract a b \<Longrightarrow> b > 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> coprime a b \<Longrightarrow> C"
239   assumes 0: "q = 0 \<Longrightarrow> C"
240   shows C
241 proof (cases "q = 0")
242   case True then show C using 0 by auto
243 next
244   case False
245   then obtain a b where "q = Fract a b" and "b > 0" and "coprime a b" by (cases q) auto
246   moreover with False have "0 \<noteq> Fract a b" by simp
247   with `b > 0` have "a \<noteq> 0" by (simp add: Zero_rat_def eq_rat)
248   with Fract `q = Fract a b` `b > 0` `coprime a b` show C by blast
249 qed
251 subsubsection {* Function @{text normalize} *}
253 lemma Fract_coprime: "Fract (a div gcd a b) (b div gcd a b) = Fract a b"
254 proof (cases "b = 0")
255   case True then show ?thesis by (simp add: eq_rat)
256 next
257   case False
258   moreover have "b div gcd a b * gcd a b = b"
259     by (rule dvd_div_mult_self) simp
260   ultimately have "b div gcd a b \<noteq> 0" by auto
261   with False show ?thesis by (simp add: eq_rat dvd_div_mult mult_commute [of a])
262 qed
264 definition normalize :: "int \<times> int \<Rightarrow> int \<times> int" where
265   "normalize p = (if snd p > 0 then (let a = gcd (fst p) (snd p) in (fst p div a, snd p div a))
266     else if snd p = 0 then (0, 1)
267     else (let a = - gcd (fst p) (snd p) in (fst p div a, snd p div a)))"
269 lemma normalize_crossproduct:
270   assumes "q \<noteq> 0" "s \<noteq> 0"
271   assumes "normalize (p, q) = normalize (r, s)"
272   shows "p * s = r * q"
273 proof -
274   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"
275   proof -
276     assume "p * gcd r s = sgn (q * s) * r * gcd p q" and "q * gcd r s = sgn (q * s) * s * gcd p q"
277     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
278     with assms show "p * s = q * r" by (auto simp add: mult_ac sgn_times sgn_0_0)
279   qed
280   from assms show ?thesis
281     by (auto simp add: normalize_def Let_def dvd_div_div_eq_mult mult_commute sgn_times split: if_splits intro: aux)
282 qed
284 lemma normalize_eq: "normalize (a, b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
285   by (auto simp add: normalize_def Let_def Fract_coprime dvd_div_neg rat_number_collapse
286     split:split_if_asm)
288 lemma normalize_denom_pos: "normalize r = (p, q) \<Longrightarrow> q > 0"
289   by (auto simp add: normalize_def Let_def dvd_div_neg pos_imp_zdiv_neg_iff nonneg1_imp_zdiv_pos_iff
290     split:split_if_asm)
292 lemma normalize_coprime: "normalize r = (p, q) \<Longrightarrow> coprime p q"
293   by (auto simp add: normalize_def Let_def dvd_div_neg div_gcd_coprime_int
294     split:split_if_asm)
296 lemma normalize_stable [simp]:
297   "q > 0 \<Longrightarrow> coprime p q \<Longrightarrow> normalize (p, q) = (p, q)"
298   by (simp add: normalize_def)
300 lemma normalize_denom_zero [simp]:
301   "normalize (p, 0) = (0, 1)"
302   by (simp add: normalize_def)
304 lemma normalize_negative [simp]:
305   "q < 0 \<Longrightarrow> normalize (p, q) = normalize (- p, - q)"
306   by (simp add: normalize_def Let_def dvd_div_neg dvd_neg_div)
308 text{*
309   Decompose a fraction into normalized, i.e. coprime numerator and denominator:
310 *}
312 definition quotient_of :: "rat \<Rightarrow> int \<times> int" where
313   "quotient_of x = (THE pair. x = Fract (fst pair) (snd pair) &
314                    snd pair > 0 & coprime (fst pair) (snd pair))"
316 lemma quotient_of_unique:
317   "\<exists>!p. r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
318 proof (cases r)
319   case (Fract a b)
320   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
321   then show ?thesis proof (rule ex1I)
322     fix p
323     obtain c d :: int where p: "p = (c, d)" by (cases p)
324     assume "r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
325     with p have Fract': "r = Fract c d" "d > 0" "coprime c d" by simp_all
326     have "c = a \<and> d = b"
327     proof (cases "a = 0")
328       case True with Fract Fract' show ?thesis by (simp add: eq_rat)
329     next
330       case False
331       with Fract Fract' have *: "c * b = a * d" and "c \<noteq> 0" by (auto simp add: eq_rat)
332       then have "c * b > 0 \<longleftrightarrow> a * d > 0" by auto
333       with `b > 0` `d > 0` have "a > 0 \<longleftrightarrow> c > 0" by (simp add: zero_less_mult_iff)
334       with `a \<noteq> 0` `c \<noteq> 0` have sgn: "sgn a = sgn c" by (auto simp add: not_less)
335       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>"
336         by (simp add: coprime_crossproduct_int)
337       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
338       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)
339       with sgn * show ?thesis by (auto simp add: sgn_0_0)
340     qed
341     with p show "p = (a, b)" by simp
342   qed
343 qed
345 lemma quotient_of_Fract [code]:
346   "quotient_of (Fract a b) = normalize (a, b)"
347 proof -
348   have "Fract a b = Fract (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?Fract)
349     by (rule sym) (auto intro: normalize_eq)
350   moreover have "0 < snd (normalize (a, b))" (is ?denom_pos)
351     by (cases "normalize (a, b)") (rule normalize_denom_pos, simp)
352   moreover have "coprime (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?coprime)
353     by (rule normalize_coprime) simp
354   ultimately have "?Fract \<and> ?denom_pos \<and> ?coprime" by blast
355   with quotient_of_unique have
356     "(THE p. Fract a b = Fract (fst p) (snd p) \<and> 0 < snd p \<and> coprime (fst p) (snd p)) = normalize (a, b)"
357     by (rule the1_equality)
358   then show ?thesis by (simp add: quotient_of_def)
359 qed
361 lemma quotient_of_number [simp]:
362   "quotient_of 0 = (0, 1)"
363   "quotient_of 1 = (1, 1)"
364   "quotient_of (numeral k) = (numeral k, 1)"
365   "quotient_of (neg_numeral k) = (neg_numeral k, 1)"
366   by (simp_all add: rat_number_expand quotient_of_Fract)
368 lemma quotient_of_eq: "quotient_of (Fract a b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
369   by (simp add: quotient_of_Fract normalize_eq)
371 lemma quotient_of_denom_pos: "quotient_of r = (p, q) \<Longrightarrow> q > 0"
372   by (cases r) (simp add: quotient_of_Fract normalize_denom_pos)
374 lemma quotient_of_coprime: "quotient_of r = (p, q) \<Longrightarrow> coprime p q"
375   by (cases r) (simp add: quotient_of_Fract normalize_coprime)
377 lemma quotient_of_inject:
378   assumes "quotient_of a = quotient_of b"
379   shows "a = b"
380 proof -
381   obtain p q r s where a: "a = Fract p q"
382     and b: "b = Fract r s"
383     and "q > 0" and "s > 0" by (cases a, cases b)
384   with assms show ?thesis by (simp add: eq_rat quotient_of_Fract normalize_crossproduct)
385 qed
387 lemma quotient_of_inject_eq:
388   "quotient_of a = quotient_of b \<longleftrightarrow> a = b"
389   by (auto simp add: quotient_of_inject)
392 subsubsection {* Various *}
394 lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"
395   by (simp add: Fract_of_int_eq [symmetric])
397 lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
398   by (simp add: rat_number_expand)
401 subsubsection {* The ordered field of rational numbers *}
403 instantiation rat :: linorder
404 begin
406 lift_definition less_eq_rat :: "rat \<Rightarrow> rat \<Rightarrow> bool"
407   is "\<lambda>x y. (fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)"
408 proof (clarsimp)
409   fix a b a' b' c d c' d'::int
410   assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
411   assume eq1: "a * b' = a' * b"
412   assume eq2: "c * d' = c' * d"
414   let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
415   {
416     fix a b c d x :: int assume x: "x \<noteq> 0"
417     have "?le a b c d = ?le (a * x) (b * x) c d"
418     proof -
419       from x have "0 < x * x" by (auto simp add: zero_less_mult_iff)
420       hence "?le a b c d =
421         ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
422         by (simp add: mult_le_cancel_right)
423       also have "... = ?le (a * x) (b * x) c d"
424         by (simp add: mult_ac)
425       finally show ?thesis .
426     qed
427   } note le_factor = this
429   let ?D = "b * d" and ?D' = "b' * d'"
430   from neq have D: "?D \<noteq> 0" by simp
431   from neq have "?D' \<noteq> 0" by simp
432   hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"
433     by (rule le_factor)
434   also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')"
435     by (simp add: mult_ac)
436   also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
437     by (simp only: eq1 eq2)
438   also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
439     by (simp add: mult_ac)
440   also from D have "... = ?le a' b' c' d'"
441     by (rule le_factor [symmetric])
442   finally show "?le a b c d = ?le a' b' c' d'" .
443 qed
445 lemma le_rat [simp]:
446   assumes "b \<noteq> 0" and "d \<noteq> 0"
447   shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
448   using assms by transfer simp
450 definition
451   less_rat_def: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
453 lemma less_rat [simp]:
454   assumes "b \<noteq> 0" and "d \<noteq> 0"
455   shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
456   using assms by (simp add: less_rat_def eq_rat order_less_le)
458 instance proof
459   fix q r s :: rat
460   {
461     assume "q \<le> r" and "r \<le> s"
462     then show "q \<le> s"
463     proof (induct q, induct r, induct s)
464       fix a b c d e f :: int
465       assume neq: "b > 0"  "d > 0"  "f > 0"
466       assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"
467       show "Fract a b \<le> Fract e f"
468       proof -
469         from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
470           by (auto simp add: zero_less_mult_iff linorder_neq_iff)
471         have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
472         proof -
473           from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
474             by simp
475           with ff show ?thesis by (simp add: mult_le_cancel_right)
476         qed
477         also have "... = (c * f) * (d * f) * (b * b)" by algebra
478         also have "... \<le> (e * d) * (d * f) * (b * b)"
479         proof -
480           from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
481             by simp
482           with bb show ?thesis by (simp add: mult_le_cancel_right)
483         qed
484         finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
485           by (simp only: mult_ac)
486         with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
487           by (simp add: mult_le_cancel_right)
488         with neq show ?thesis by simp
489       qed
490     qed
491   next
492     assume "q \<le> r" and "r \<le> q"
493     then show "q = r"
494     proof (induct q, induct r)
495       fix a b c d :: int
496       assume neq: "b > 0"  "d > 0"
497       assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"
498       show "Fract a b = Fract c d"
499       proof -
500         from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
501           by simp
502         also have "... \<le> (a * d) * (b * d)"
503         proof -
504           from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
505             by simp
506           thus ?thesis by (simp only: mult_ac)
507         qed
508         finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
509         moreover from neq have "b * d \<noteq> 0" by simp
510         ultimately have "a * d = c * b" by simp
511         with neq show ?thesis by (simp add: eq_rat)
512       qed
513     qed
514   next
515     show "q \<le> q"
516       by (induct q) simp
517     show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)"
518       by (induct q, induct r) (auto simp add: le_less mult_commute)
519     show "q \<le> r \<or> r \<le> q"
520       by (induct q, induct r)
521          (simp add: mult_commute, rule linorder_linear)
522   }
523 qed
525 end
527 instantiation rat :: "{distrib_lattice, abs_if, sgn_if}"
528 begin
530 definition
531   abs_rat_def: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))"
533 lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
534   by (auto simp add: abs_rat_def zabs_def Zero_rat_def not_less le_less eq_rat zero_less_mult_iff)
536 definition
537   sgn_rat_def: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
539 lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)"
540   unfolding Fract_of_int_eq
541   by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat)
542     (auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)
544 definition
545   "(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min"
547 definition
548   "(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max"
550 instance by intro_classes
551   (auto simp add: abs_rat_def sgn_rat_def min_max.sup_inf_distrib1 inf_rat_def sup_rat_def)
553 end
555 instance rat :: linordered_field_inverse_zero
556 proof
557   fix q r s :: rat
558   show "q \<le> r ==> s + q \<le> s + r"
559   proof (induct q, induct r, induct s)
560     fix a b c d e f :: int
561     assume neq: "b > 0"  "d > 0"  "f > 0"
562     assume le: "Fract a b \<le> Fract c d"
563     show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
564     proof -
565       let ?F = "f * f" from neq have F: "0 < ?F"
566         by (auto simp add: zero_less_mult_iff)
567       from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
568         by simp
569       with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
570         by (simp add: mult_le_cancel_right)
571       with neq show ?thesis by (simp add: mult_ac int_distrib)
572     qed
573   qed
574   show "q < r ==> 0 < s ==> s * q < s * r"
575   proof (induct q, induct r, induct s)
576     fix a b c d e f :: int
577     assume neq: "b > 0"  "d > 0"  "f > 0"
578     assume le: "Fract a b < Fract c d"
579     assume gt: "0 < Fract e f"
580     show "Fract e f * Fract a b < Fract e f * Fract c d"
581     proof -
582       let ?E = "e * f" and ?F = "f * f"
583       from neq gt have "0 < ?E"
584         by (auto simp add: Zero_rat_def order_less_le eq_rat)
585       moreover from neq have "0 < ?F"
586         by (auto simp add: zero_less_mult_iff)
587       moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
588         by simp
589       ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
590         by (simp add: mult_less_cancel_right)
591       with neq show ?thesis
592         by (simp add: mult_ac)
593     qed
594   qed
595 qed auto
597 lemma Rat_induct_pos [case_names Fract, induct type: rat]:
598   assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
599   shows "P q"
600 proof (cases q)
601   have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)"
602   proof -
603     fix a::int and b::int
604     assume b: "b < 0"
605     hence "0 < -b" by simp
606     hence "P (Fract (-a) (-b))" by (rule step)
607     thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
608   qed
609   case (Fract a b)
610   thus "P q" by (force simp add: linorder_neq_iff step step')
611 qed
613 lemma zero_less_Fract_iff:
614   "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a"
615   by (simp add: Zero_rat_def zero_less_mult_iff)
617 lemma Fract_less_zero_iff:
618   "0 < b \<Longrightarrow> Fract a b < 0 \<longleftrightarrow> a < 0"
619   by (simp add: Zero_rat_def mult_less_0_iff)
621 lemma zero_le_Fract_iff:
622   "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a"
623   by (simp add: Zero_rat_def zero_le_mult_iff)
625 lemma Fract_le_zero_iff:
626   "0 < b \<Longrightarrow> Fract a b \<le> 0 \<longleftrightarrow> a \<le> 0"
627   by (simp add: Zero_rat_def mult_le_0_iff)
629 lemma one_less_Fract_iff:
630   "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a"
631   by (simp add: One_rat_def mult_less_cancel_right_disj)
633 lemma Fract_less_one_iff:
634   "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b"
635   by (simp add: One_rat_def mult_less_cancel_right_disj)
637 lemma one_le_Fract_iff:
638   "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a"
639   by (simp add: One_rat_def mult_le_cancel_right)
641 lemma Fract_le_one_iff:
642   "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b"
643   by (simp add: One_rat_def mult_le_cancel_right)
646 subsubsection {* Rationals are an Archimedean field *}
648 lemma rat_floor_lemma:
649   shows "of_int (a div b) \<le> Fract a b \<and> Fract a b < of_int (a div b + 1)"
650 proof -
651   have "Fract a b = of_int (a div b) + Fract (a mod b) b"
652     by (cases "b = 0", simp, simp add: of_int_rat)
653   moreover have "0 \<le> Fract (a mod b) b \<and> Fract (a mod b) b < 1"
654     unfolding Fract_of_int_quotient
655     by (rule linorder_cases [of b 0]) (simp add: divide_nonpos_neg, simp, simp add: divide_nonneg_pos)
656   ultimately show ?thesis by simp
657 qed
659 instance rat :: archimedean_field
660 proof
661   fix r :: rat
662   show "\<exists>z. r \<le> of_int z"
663   proof (induct r)
664     case (Fract a b)
665     have "Fract a b \<le> of_int (a div b + 1)"
666       using rat_floor_lemma [of a b] by simp
667     then show "\<exists>z. Fract a b \<le> of_int z" ..
668   qed
669 qed
671 instantiation rat :: floor_ceiling
672 begin
674 definition [code del]:
675   "floor (x::rat) = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))"
677 instance proof
678   fix x :: rat
679   show "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
680     unfolding floor_rat_def using floor_exists1 by (rule theI')
681 qed
683 end
685 lemma floor_Fract: "floor (Fract a b) = a div b"
686   using rat_floor_lemma [of a b]
687   by (simp add: floor_unique)
690 subsection {* Linear arithmetic setup *}
692 declaration {*
693   K (Lin_Arith.add_inj_thms [@{thm of_nat_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
694     (* not needed because x < (y::nat) can be rewritten as Suc x <= y: of_nat_less_iff RS iffD2 *)
695   #> Lin_Arith.add_inj_thms [@{thm of_int_le_iff} RS iffD2, @{thm of_int_eq_iff} RS iffD2]
696     (* not needed because x < (y::int) can be rewritten as x + 1 <= y: of_int_less_iff RS iffD2 *)
697   #> Lin_Arith.add_simps [@{thm neg_less_iff_less},
698       @{thm True_implies_equals},
699       read_instantiate @{context} [(("a", 0), "(numeral ?v)")] @{thm right_distrib},
700       read_instantiate @{context} [(("a", 0), "(neg_numeral ?v)")] @{thm right_distrib},
701       @{thm divide_1}, @{thm divide_zero_left},
702       @{thm times_divide_eq_right}, @{thm times_divide_eq_left},
703       @{thm minus_divide_left} RS sym, @{thm minus_divide_right} RS sym,
704       @{thm of_int_minus}, @{thm of_int_diff},
705       @{thm of_int_of_nat_eq}]
706   #> Lin_Arith.add_simprocs Numeral_Simprocs.field_cancel_numeral_factors
707   #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat => rat"})
708   #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int => rat"}))
709 *}
712 subsection {* Embedding from Rationals to other Fields *}
714 class field_char_0 = field + ring_char_0
716 subclass (in linordered_field) field_char_0 ..
718 context field_char_0
719 begin
721 lift_definition of_rat :: "rat \<Rightarrow> 'a"
722   is "\<lambda>x. of_int (fst x) / of_int (snd x)"
723 apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
724 apply (simp only: of_int_mult [symmetric])
725 done
727 end
729 lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
730   by transfer simp
732 lemma of_rat_0 [simp]: "of_rat 0 = 0"
733   by transfer simp
735 lemma of_rat_1 [simp]: "of_rat 1 = 1"
736   by transfer simp
738 lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
741 lemma of_rat_minus: "of_rat (- a) = - of_rat a"
742   by transfer simp
744 lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
745 by (simp only: diff_minus of_rat_add of_rat_minus)
747 lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
748 apply transfer
749 apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
750 done
752 lemma nonzero_of_rat_inverse:
753   "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
754 apply (rule inverse_unique [symmetric])
755 apply (simp add: of_rat_mult [symmetric])
756 done
758 lemma of_rat_inverse:
759   "(of_rat (inverse a)::'a::{field_char_0, field_inverse_zero}) =
760    inverse (of_rat a)"
761 by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
763 lemma nonzero_of_rat_divide:
764   "b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
765 by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
767 lemma of_rat_divide:
768   "(of_rat (a / b)::'a::{field_char_0, field_inverse_zero})
769    = of_rat a / of_rat b"
770 by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)
772 lemma of_rat_power:
773   "(of_rat (a ^ n)::'a::field_char_0) = of_rat a ^ n"
774 by (induct n) (simp_all add: of_rat_mult)
776 lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
777 apply transfer
778 apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
779 apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
780 done
782 lemma of_rat_less:
783   "(of_rat r :: 'a::linordered_field) < of_rat s \<longleftrightarrow> r < s"
784 proof (induct r, induct s)
785   fix a b c d :: int
786   assume not_zero: "b > 0" "d > 0"
787   then have "b * d > 0" by (rule mult_pos_pos)
788   have of_int_divide_less_eq:
789     "(of_int a :: 'a) / of_int b < of_int c / of_int d
790       \<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b"
791     using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
792   show "(of_rat (Fract a b) :: 'a::linordered_field) < of_rat (Fract c d)
793     \<longleftrightarrow> Fract a b < Fract c d"
794     using not_zero `b * d > 0`
795     by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult)
796 qed
798 lemma of_rat_less_eq:
799   "(of_rat r :: 'a::linordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s"
800   unfolding le_less by (auto simp add: of_rat_less)
802 lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
804 lemma of_rat_eq_id [simp]: "of_rat = id"
805 proof
806   fix a
807   show "of_rat a = id a"
808   by (induct a)
809      (simp add: of_rat_rat Fract_of_int_eq [symmetric])
810 qed
812 text{*Collapse nested embeddings*}
813 lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
814 by (induct n) (simp_all add: of_rat_add)
816 lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
817 by (cases z rule: int_diff_cases) (simp add: of_rat_diff)
819 lemma of_rat_numeral_eq [simp]:
820   "of_rat (numeral w) = numeral w"
821 using of_rat_of_int_eq [of "numeral w"] by simp
823 lemma of_rat_neg_numeral_eq [simp]:
824   "of_rat (neg_numeral w) = neg_numeral w"
825 using of_rat_of_int_eq [of "neg_numeral w"] by simp
827 lemmas zero_rat = Zero_rat_def
828 lemmas one_rat = One_rat_def
830 abbreviation
831   rat_of_nat :: "nat \<Rightarrow> rat"
832 where
833   "rat_of_nat \<equiv> of_nat"
835 abbreviation
836   rat_of_int :: "int \<Rightarrow> rat"
837 where
838   "rat_of_int \<equiv> of_int"
840 subsection {* The Set of Rational Numbers *}
842 context field_char_0
843 begin
845 definition
846   Rats  :: "'a set" where
847   "Rats = range of_rat"
849 notation (xsymbols)
850   Rats  ("\<rat>")
852 end
854 lemma Rats_of_rat [simp]: "of_rat r \<in> Rats"
855 by (simp add: Rats_def)
857 lemma Rats_of_int [simp]: "of_int z \<in> Rats"
858 by (subst of_rat_of_int_eq [symmetric], rule Rats_of_rat)
860 lemma Rats_of_nat [simp]: "of_nat n \<in> Rats"
861 by (subst of_rat_of_nat_eq [symmetric], rule Rats_of_rat)
863 lemma Rats_number_of [simp]: "numeral w \<in> Rats"
864 by (subst of_rat_numeral_eq [symmetric], rule Rats_of_rat)
866 lemma Rats_neg_number_of [simp]: "neg_numeral w \<in> Rats"
867 by (subst of_rat_neg_numeral_eq [symmetric], rule Rats_of_rat)
869 lemma Rats_0 [simp]: "0 \<in> Rats"
870 apply (unfold Rats_def)
871 apply (rule range_eqI)
872 apply (rule of_rat_0 [symmetric])
873 done
875 lemma Rats_1 [simp]: "1 \<in> Rats"
876 apply (unfold Rats_def)
877 apply (rule range_eqI)
878 apply (rule of_rat_1 [symmetric])
879 done
881 lemma Rats_add [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a + b \<in> Rats"
882 apply (auto simp add: Rats_def)
883 apply (rule range_eqI)
884 apply (rule of_rat_add [symmetric])
885 done
887 lemma Rats_minus [simp]: "a \<in> Rats \<Longrightarrow> - a \<in> Rats"
888 apply (auto simp add: Rats_def)
889 apply (rule range_eqI)
890 apply (rule of_rat_minus [symmetric])
891 done
893 lemma Rats_diff [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a - b \<in> Rats"
894 apply (auto simp add: Rats_def)
895 apply (rule range_eqI)
896 apply (rule of_rat_diff [symmetric])
897 done
899 lemma Rats_mult [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a * b \<in> Rats"
900 apply (auto simp add: Rats_def)
901 apply (rule range_eqI)
902 apply (rule of_rat_mult [symmetric])
903 done
905 lemma nonzero_Rats_inverse:
906   fixes a :: "'a::field_char_0"
907   shows "\<lbrakk>a \<in> Rats; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Rats"
908 apply (auto simp add: Rats_def)
909 apply (rule range_eqI)
910 apply (erule nonzero_of_rat_inverse [symmetric])
911 done
913 lemma Rats_inverse [simp]:
914   fixes a :: "'a::{field_char_0, field_inverse_zero}"
915   shows "a \<in> Rats \<Longrightarrow> inverse a \<in> Rats"
916 apply (auto simp add: Rats_def)
917 apply (rule range_eqI)
918 apply (rule of_rat_inverse [symmetric])
919 done
921 lemma nonzero_Rats_divide:
922   fixes a b :: "'a::field_char_0"
923   shows "\<lbrakk>a \<in> Rats; b \<in> Rats; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
924 apply (auto simp add: Rats_def)
925 apply (rule range_eqI)
926 apply (erule nonzero_of_rat_divide [symmetric])
927 done
929 lemma Rats_divide [simp]:
930   fixes a b :: "'a::{field_char_0, field_inverse_zero}"
931   shows "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
932 apply (auto simp add: Rats_def)
933 apply (rule range_eqI)
934 apply (rule of_rat_divide [symmetric])
935 done
937 lemma Rats_power [simp]:
938   fixes a :: "'a::field_char_0"
939   shows "a \<in> Rats \<Longrightarrow> a ^ n \<in> Rats"
940 apply (auto simp add: Rats_def)
941 apply (rule range_eqI)
942 apply (rule of_rat_power [symmetric])
943 done
945 lemma Rats_cases [cases set: Rats]:
946   assumes "q \<in> \<rat>"
947   obtains (of_rat) r where "q = of_rat r"
948 proof -
949   from `q \<in> \<rat>` have "q \<in> range of_rat" unfolding Rats_def .
950   then obtain r where "q = of_rat r" ..
951   then show thesis ..
952 qed
954 lemma Rats_induct [case_names of_rat, induct set: Rats]:
955   "q \<in> \<rat> \<Longrightarrow> (\<And>r. P (of_rat r)) \<Longrightarrow> P q"
956   by (rule Rats_cases) auto
959 subsection {* Implementation of rational numbers as pairs of integers *}
961 text {* Formal constructor *}
963 definition Frct :: "int \<times> int \<Rightarrow> rat" where
964   [simp]: "Frct p = Fract (fst p) (snd p)"
966 lemma [code abstype]:
967   "Frct (quotient_of q) = q"
968   by (cases q) (auto intro: quotient_of_eq)
971 text {* Numerals *}
973 declare quotient_of_Fract [code abstract]
975 definition of_int :: "int \<Rightarrow> rat"
976 where
977   [code_abbrev]: "of_int = Int.of_int"
978 hide_const (open) of_int
980 lemma quotient_of_int [code abstract]:
981   "quotient_of (Rat.of_int a) = (a, 1)"
982   by (simp add: of_int_def of_int_rat quotient_of_Fract)
984 lemma [code_unfold]:
985   "numeral k = Rat.of_int (numeral k)"
986   by (simp add: Rat.of_int_def)
988 lemma [code_unfold]:
989   "neg_numeral k = Rat.of_int (neg_numeral k)"
990   by (simp add: Rat.of_int_def)
992 lemma Frct_code_post [code_post]:
993   "Frct (0, a) = 0"
994   "Frct (a, 0) = 0"
995   "Frct (1, 1) = 1"
996   "Frct (numeral k, 1) = numeral k"
997   "Frct (neg_numeral k, 1) = neg_numeral k"
998   "Frct (1, numeral k) = 1 / numeral k"
999   "Frct (1, neg_numeral k) = 1 / neg_numeral k"
1000   "Frct (numeral k, numeral l) = numeral k / numeral l"
1001   "Frct (numeral k, neg_numeral l) = numeral k / neg_numeral l"
1002   "Frct (neg_numeral k, numeral l) = neg_numeral k / numeral l"
1003   "Frct (neg_numeral k, neg_numeral l) = neg_numeral k / neg_numeral l"
1004   by (simp_all add: Fract_of_int_quotient)
1007 text {* Operations *}
1009 lemma rat_zero_code [code abstract]:
1010   "quotient_of 0 = (0, 1)"
1011   by (simp add: Zero_rat_def quotient_of_Fract normalize_def)
1013 lemma rat_one_code [code abstract]:
1014   "quotient_of 1 = (1, 1)"
1015   by (simp add: One_rat_def quotient_of_Fract normalize_def)
1017 lemma rat_plus_code [code abstract]:
1018   "quotient_of (p + q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
1019      in normalize (a * d + b * c, c * d))"
1020   by (cases p, cases q) (simp add: quotient_of_Fract)
1022 lemma rat_uminus_code [code abstract]:
1023   "quotient_of (- p) = (let (a, b) = quotient_of p in (- a, b))"
1024   by (cases p) (simp add: quotient_of_Fract)
1026 lemma rat_minus_code [code abstract]:
1027   "quotient_of (p - q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
1028      in normalize (a * d - b * c, c * d))"
1029   by (cases p, cases q) (simp add: quotient_of_Fract)
1031 lemma rat_times_code [code abstract]:
1032   "quotient_of (p * q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
1033      in normalize (a * b, c * d))"
1034   by (cases p, cases q) (simp add: quotient_of_Fract)
1036 lemma rat_inverse_code [code abstract]:
1037   "quotient_of (inverse p) = (let (a, b) = quotient_of p
1038     in if a = 0 then (0, 1) else (sgn a * b, \<bar>a\<bar>))"
1039 proof (cases p)
1040   case (Fract a b) then show ?thesis
1041     by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract gcd_int.commute)
1042 qed
1044 lemma rat_divide_code [code abstract]:
1045   "quotient_of (p / q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
1046      in normalize (a * d, c * b))"
1047   by (cases p, cases q) (simp add: quotient_of_Fract)
1049 lemma rat_abs_code [code abstract]:
1050   "quotient_of \<bar>p\<bar> = (let (a, b) = quotient_of p in (\<bar>a\<bar>, b))"
1051   by (cases p) (simp add: quotient_of_Fract)
1053 lemma rat_sgn_code [code abstract]:
1054   "quotient_of (sgn p) = (sgn (fst (quotient_of p)), 1)"
1055 proof (cases p)
1056   case (Fract a b) then show ?thesis
1057   by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract)
1058 qed
1060 lemma rat_floor_code [code]:
1061   "floor p = (let (a, b) = quotient_of p in a div b)"
1062 by (cases p) (simp add: quotient_of_Fract floor_Fract)
1064 instantiation rat :: equal
1065 begin
1067 definition [code]:
1068   "HOL.equal a b \<longleftrightarrow> quotient_of a = quotient_of b"
1070 instance proof
1071 qed (simp add: equal_rat_def quotient_of_inject_eq)
1073 lemma rat_eq_refl [code nbe]:
1074   "HOL.equal (r::rat) r \<longleftrightarrow> True"
1075   by (rule equal_refl)
1077 end
1079 lemma rat_less_eq_code [code]:
1080   "p \<le> q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d \<le> c * b)"
1081   by (cases p, cases q) (simp add: quotient_of_Fract mult.commute)
1083 lemma rat_less_code [code]:
1084   "p < q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d < c * b)"
1085   by (cases p, cases q) (simp add: quotient_of_Fract mult.commute)
1087 lemma [code]:
1088   "of_rat p = (let (a, b) = quotient_of p in of_int a / of_int b)"
1089   by (cases p) (simp add: quotient_of_Fract of_rat_rat)
1092 text {* Quickcheck *}
1094 definition (in term_syntax)
1095   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
1096   [code_unfold]: "valterm_fract k l = Code_Evaluation.valtermify Fract {\<cdot>} k {\<cdot>} l"
1098 notation fcomp (infixl "\<circ>>" 60)
1099 notation scomp (infixl "\<circ>\<rightarrow>" 60)
1101 instantiation rat :: random
1102 begin
1104 definition
1105   "Quickcheck.random i = Quickcheck.random i \<circ>\<rightarrow> (\<lambda>num. Random.range i \<circ>\<rightarrow> (\<lambda>denom. Pair (
1106      let j = Code_Numeral.int_of (denom + 1)
1107      in valterm_fract num (j, \<lambda>u. Code_Evaluation.term_of j))))"
1109 instance ..
1111 end
1113 no_notation fcomp (infixl "\<circ>>" 60)
1114 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
1116 instantiation rat :: exhaustive
1117 begin
1119 definition
1120   "exhaustive_rat f d = Quickcheck_Exhaustive.exhaustive (%l. Quickcheck_Exhaustive.exhaustive (%k. f (Fract k (Code_Numeral.int_of l + 1))) d) d"
1122 instance ..
1124 end
1126 instantiation rat :: full_exhaustive
1127 begin
1129 definition
1130   "full_exhaustive_rat f d = Quickcheck_Exhaustive.full_exhaustive (%(l, _). Quickcheck_Exhaustive.full_exhaustive (%k.
1131      f (let j = Code_Numeral.int_of l + 1
1132         in valterm_fract k (j, %_. Code_Evaluation.term_of j))) d) d"
1134 instance ..
1136 end
1138 instantiation rat :: partial_term_of
1139 begin
1141 instance ..
1143 end
1145 lemma [code]:
1146   "partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Narrowing_variable p tt) == Code_Evaluation.Free (STR ''_'') (Typerep.Typerep (STR ''Rat.rat'') [])"
1147   "partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Narrowing_constructor 0 [l, k]) ==
1148      Code_Evaluation.App (Code_Evaluation.Const (STR ''Rat.Frct'')
1149      (Typerep.Typerep (STR ''fun'') [Typerep.Typerep (STR ''Product_Type.prod'') [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''Int.int'') []],
1150         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))"
1151 by (rule partial_term_of_anything)+
1153 instantiation rat :: narrowing
1154 begin
1156 definition
1157   "narrowing = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.apply
1158     (Quickcheck_Narrowing.cons (%nom denom. Fract nom denom)) narrowing) narrowing"
1160 instance ..
1162 end
1165 subsection {* Setup for Nitpick *}
1167 declaration {*
1168   Nitpick_HOL.register_frac_type @{type_name rat}
1169    [(@{const_name zero_rat_inst.zero_rat}, @{const_name Nitpick.zero_frac}),
1170     (@{const_name one_rat_inst.one_rat}, @{const_name Nitpick.one_frac}),
1171     (@{const_name plus_rat_inst.plus_rat}, @{const_name Nitpick.plus_frac}),
1172     (@{const_name times_rat_inst.times_rat}, @{const_name Nitpick.times_frac}),
1173     (@{const_name uminus_rat_inst.uminus_rat}, @{const_name Nitpick.uminus_frac}),
1174     (@{const_name inverse_rat_inst.inverse_rat}, @{const_name Nitpick.inverse_frac}),
1175     (@{const_name ord_rat_inst.less_rat}, @{const_name Nitpick.less_frac}),
1176     (@{const_name ord_rat_inst.less_eq_rat}, @{const_name Nitpick.less_eq_frac}),
1177     (@{const_name field_char_0_class.of_rat}, @{const_name Nitpick.of_frac})]
1178 *}
1180 lemmas [nitpick_unfold] = inverse_rat_inst.inverse_rat
1181   one_rat_inst.one_rat ord_rat_inst.less_rat
1182   ord_rat_inst.less_eq_rat plus_rat_inst.plus_rat times_rat_inst.times_rat
1183   uminus_rat_inst.uminus_rat zero_rat_inst.zero_rat
1185 subsection{* Float syntax *}
1187 syntax "_Float" :: "float_const \<Rightarrow> 'a"    ("_")
1189 use "Tools/float_syntax.ML"
1190 setup Float_Syntax.setup
1192 text{* Test: *}
1193 lemma "123.456 = -111.111 + 200 + 30 + 4 + 5/10 + 6/100 + (7/1000::rat)"
1194 by simp
1197 hide_const (open) normalize
1199 lemmas [transfer_rule del] =
1200   rat.All_transfer rat.Ex_transfer rat.rel_eq_transfer forall_rat_transfer
1201   Fract.transfer zero_rat.transfer one_rat.transfer plus_rat.transfer
1202   uminus_rat.transfer times_rat.transfer inverse_rat.transfer
1203   less_eq_rat.transfer of_rat.transfer
1205 text {* De-register @{text "rat"} as a quotient type: *}
1207 setup {* Context.theory_map (Lifting_Info.update_quotients @{type_name rat}
1208   {quot_thm = @{thm identity_quotient [where 'a=rat]}}) *}
1210 end