src/HOL/Rat.thy
 author wenzelm Mon Aug 31 21:28:08 2015 +0200 (2015-08-31) changeset 61070 b72a990adfe2 parent 60758 d8d85a8172b5 child 61144 5e94dfead1c2 permissions -rw-r--r--
prefer symbols;
1 (*  Title:  HOL/Rat.thy
2     Author: Markus Wenzel, TU Muenchen
3 *)
5 section \<open>Rational numbers\<close>
7 theory Rat
8 imports GCD Archimedean_Field
9 begin
11 subsection \<open>Rational numbers as quotient\<close>
13 subsubsection \<open>Construction of the type of rational numbers\<close>
15 definition
16   ratrel :: "(int \<times> int) \<Rightarrow> (int \<times> int) \<Rightarrow> bool" where
17   "ratrel = (\<lambda>x y. snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x)"
19 lemma ratrel_iff [simp]:
20   "ratrel x y \<longleftrightarrow> snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x"
23 lemma exists_ratrel_refl: "\<exists>x. ratrel x x"
24   by (auto intro!: one_neq_zero)
26 lemma symp_ratrel: "symp ratrel"
27   by (simp add: ratrel_def symp_def)
29 lemma transp_ratrel: "transp ratrel"
30 proof (rule transpI, unfold split_paired_all)
31   fix a b a' b' a'' b'' :: int
32   assume A: "ratrel (a, b) (a', b')"
33   assume B: "ratrel (a', b') (a'', b'')"
34   have "b' * (a * b'') = b'' * (a * b')" by simp
35   also from A have "a * b' = a' * b" by auto
36   also have "b'' * (a' * b) = b * (a' * b'')" by simp
37   also from B have "a' * b'' = a'' * b'" by auto
38   also have "b * (a'' * b') = b' * (a'' * b)" by simp
39   finally have "b' * (a * b'') = b' * (a'' * b)" .
40   moreover from B have "b' \<noteq> 0" by auto
41   ultimately have "a * b'' = a'' * b" by simp
42   with A B show "ratrel (a, b) (a'', b'')" by auto
43 qed
45 lemma part_equivp_ratrel: "part_equivp ratrel"
46   by (rule part_equivpI [OF exists_ratrel_refl symp_ratrel transp_ratrel])
48 quotient_type rat = "int \<times> int" / partial: "ratrel"
49   morphisms Rep_Rat Abs_Rat
50   by (rule part_equivp_ratrel)
52 lemma Domainp_cr_rat [transfer_domain_rule]: "Domainp pcr_rat = (\<lambda>x. snd x \<noteq> 0)"
55 subsubsection \<open>Representation and basic operations\<close>
57 lift_definition Fract :: "int \<Rightarrow> int \<Rightarrow> rat"
58   is "\<lambda>a b. if b = 0 then (0, 1) else (a, b)"
59   by simp
61 lemma eq_rat:
62   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"
63   and "\<And>a. Fract a 0 = Fract 0 1"
64   and "\<And>a c. Fract 0 a = Fract 0 c"
65   by (transfer, simp)+
67 lemma Rat_cases [case_names Fract, cases type: rat]:
68   assumes "\<And>a b. q = Fract a b \<Longrightarrow> b > 0 \<Longrightarrow> coprime a b \<Longrightarrow> C"
69   shows C
70 proof -
71   obtain a b :: int where "q = Fract a b" and "b \<noteq> 0"
72     by transfer simp
73   let ?a = "a div gcd a b"
74   let ?b = "b div gcd a b"
75   from \<open>b \<noteq> 0\<close> have "?b * gcd a b = b"
76     by simp
77   with \<open>b \<noteq> 0\<close> have "?b \<noteq> 0" by fastforce
78   from \<open>q = Fract a b\<close> \<open>b \<noteq> 0\<close> \<open>?b \<noteq> 0\<close> have q: "q = Fract ?a ?b"
79     by (simp add: eq_rat dvd_div_mult mult.commute [of a])
80   from \<open>b \<noteq> 0\<close> have coprime: "coprime ?a ?b"
81     by (auto intro: div_gcd_coprime_int)
82   show C proof (cases "b > 0")
83     case True
84     note assms
85     moreover note q
86     moreover from True have "?b > 0" by (simp add: nonneg1_imp_zdiv_pos_iff)
87     moreover note coprime
88     ultimately show C .
89   next
90     case False
91     note assms
92     moreover have "q = Fract (- ?a) (- ?b)" unfolding q by transfer simp
93     moreover from False \<open>b \<noteq> 0\<close> have "- ?b > 0" by (simp add: pos_imp_zdiv_neg_iff)
94     moreover from coprime have "coprime (- ?a) (- ?b)" by simp
95     ultimately show C .
96   qed
97 qed
99 lemma Rat_induct [case_names Fract, induct type: rat]:
100   assumes "\<And>a b. b > 0 \<Longrightarrow> coprime a b \<Longrightarrow> P (Fract a b)"
101   shows "P q"
102   using assms by (cases q) simp
104 instantiation rat :: field
105 begin
107 lift_definition zero_rat :: "rat" is "(0, 1)"
108   by simp
110 lift_definition one_rat :: "rat" is "(1, 1)"
111   by simp
113 lemma Zero_rat_def: "0 = Fract 0 1"
114   by transfer simp
116 lemma One_rat_def: "1 = Fract 1 1"
117   by transfer simp
119 lift_definition plus_rat :: "rat \<Rightarrow> rat \<Rightarrow> rat"
120   is "\<lambda>x y. (fst x * snd y + fst y * snd x, snd x * snd y)"
124   assumes "b \<noteq> 0" and "d \<noteq> 0"
125   shows "Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
126   using assms by transfer simp
128 lift_definition uminus_rat :: "rat \<Rightarrow> rat" is "\<lambda>x. (- fst x, snd x)"
129   by simp
131 lemma minus_rat [simp]: "- Fract a b = Fract (- a) b"
132   by transfer simp
134 lemma minus_rat_cancel [simp]: "Fract (- a) (- b) = Fract a b"
135   by (cases "b = 0") (simp_all add: eq_rat)
137 definition
138   diff_rat_def: "q - r = q + - (r::rat)"
140 lemma diff_rat [simp]:
141   assumes "b \<noteq> 0" and "d \<noteq> 0"
142   shows "Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
143   using assms by (simp add: diff_rat_def)
145 lift_definition times_rat :: "rat \<Rightarrow> rat \<Rightarrow> rat"
146   is "\<lambda>x y. (fst x * fst y, snd x * snd y)"
149 lemma mult_rat [simp]: "Fract a b * Fract c d = Fract (a * c) (b * d)"
150   by transfer simp
152 lemma mult_rat_cancel:
153   assumes "c \<noteq> 0"
154   shows "Fract (c * a) (c * b) = Fract a b"
155   using assms by transfer simp
157 lift_definition inverse_rat :: "rat \<Rightarrow> rat"
158   is "\<lambda>x. if fst x = 0 then (0, 1) else (snd x, fst x)"
159   by (auto simp add: mult.commute)
161 lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"
162   by transfer simp
164 definition
165   divide_rat_def: "q div r = q * inverse (r::rat)"
167 lemma divide_rat [simp]: "Fract a b div Fract c d = Fract (a * d) (b * c)"
170 instance proof
171   fix q r s :: rat
172   show "(q * r) * s = q * (r * s)"
173     by transfer simp
174   show "q * r = r * q"
175     by transfer simp
176   show "1 * q = q"
177     by transfer simp
178   show "(q + r) + s = q + (r + s)"
179     by transfer (simp add: algebra_simps)
180   show "q + r = r + q"
181     by transfer simp
182   show "0 + q = q"
183     by transfer simp
184   show "- q + q = 0"
185     by transfer simp
186   show "q - r = q + - r"
187     by (fact diff_rat_def)
188   show "(q + r) * s = q * s + r * s"
189     by transfer (simp add: algebra_simps)
190   show "(0::rat) \<noteq> 1"
191     by transfer simp
192   { assume "q \<noteq> 0" thus "inverse q * q = 1"
193     by transfer simp }
194   show "q div r = q * inverse r"
195     by (fact divide_rat_def)
196   show "inverse 0 = (0::rat)"
197     by transfer simp
198 qed
200 end
202 lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1"
203   by (induct k) (simp_all add: Zero_rat_def One_rat_def)
205 lemma of_int_rat: "of_int k = Fract k 1"
206   by (cases k rule: int_diff_cases) (simp add: of_nat_rat)
208 lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
209   by (rule of_nat_rat [symmetric])
211 lemma Fract_of_int_eq: "Fract k 1 = of_int k"
212   by (rule of_int_rat [symmetric])
214 lemma rat_number_collapse:
215   "Fract 0 k = 0"
216   "Fract 1 1 = 1"
217   "Fract (numeral w) 1 = numeral w"
218   "Fract (- numeral w) 1 = - numeral w"
219   "Fract (- 1) 1 = - 1"
220   "Fract k 0 = 0"
221   using Fract_of_int_eq [of "numeral w"]
222   using Fract_of_int_eq [of "- numeral w"]
223   by (simp_all add: Zero_rat_def One_rat_def eq_rat)
225 lemma rat_number_expand:
226   "0 = Fract 0 1"
227   "1 = Fract 1 1"
228   "numeral k = Fract (numeral k) 1"
229   "- 1 = Fract (- 1) 1"
230   "- numeral k = Fract (- numeral k) 1"
233 lemma Rat_cases_nonzero [case_names Fract 0]:
234   assumes Fract: "\<And>a b. q = Fract a b \<Longrightarrow> b > 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> coprime a b \<Longrightarrow> C"
235   assumes 0: "q = 0 \<Longrightarrow> C"
236   shows C
237 proof (cases "q = 0")
238   case True then show C using 0 by auto
239 next
240   case False
241   then obtain a b where "q = Fract a b" and "b > 0" and "coprime a b" by (cases q) auto
242   with False have "0 \<noteq> Fract a b" by simp
243   with \<open>b > 0\<close> have "a \<noteq> 0" by (simp add: Zero_rat_def eq_rat)
244   with Fract \<open>q = Fract a b\<close> \<open>b > 0\<close> \<open>coprime a b\<close> show C by blast
245 qed
247 subsubsection \<open>Function @{text normalize}\<close>
249 lemma Fract_coprime: "Fract (a div gcd a b) (b div gcd a b) = Fract a b"
250 proof (cases "b = 0")
251   case True then show ?thesis by (simp add: eq_rat)
252 next
253   case False
254   moreover have "b div gcd a b * gcd a b = b"
255     by (rule dvd_div_mult_self) simp
256   ultimately have "b div gcd a b * gcd a b \<noteq> 0" by simp
257   then have "b div gcd a b \<noteq> 0" by fastforce
258   with False show ?thesis by (simp add: eq_rat dvd_div_mult mult.commute [of a])
259 qed
261 definition normalize :: "int \<times> int \<Rightarrow> int \<times> int" where
262   "normalize p = (if snd p > 0 then (let a = gcd (fst p) (snd p) in (fst p div a, snd p div a))
263     else if snd p = 0 then (0, 1)
264     else (let a = - gcd (fst p) (snd p) in (fst p div a, snd p div a)))"
266 lemma normalize_crossproduct:
267   assumes "q \<noteq> 0" "s \<noteq> 0"
268   assumes "normalize (p, q) = normalize (r, s)"
269   shows "p * s = r * q"
270 proof -
271   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"
272   proof -
273     assume "p * gcd r s = sgn (q * s) * r * gcd p q" and "q * gcd r s = sgn (q * s) * s * gcd p q"
274     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
275     with assms show "p * s = q * r" by (auto simp add: ac_simps sgn_times sgn_0_0)
276   qed
277   from assms show ?thesis
278     by (auto simp add: normalize_def Let_def dvd_div_div_eq_mult mult.commute sgn_times split: if_splits intro: aux)
279 qed
281 lemma normalize_eq: "normalize (a, b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
282   by (auto simp add: normalize_def Let_def Fract_coprime dvd_div_neg rat_number_collapse
283     split:split_if_asm)
285 lemma normalize_denom_pos: "normalize r = (p, q) \<Longrightarrow> q > 0"
286   by (auto simp add: normalize_def Let_def dvd_div_neg pos_imp_zdiv_neg_iff nonneg1_imp_zdiv_pos_iff
287     split:split_if_asm)
289 lemma normalize_coprime: "normalize r = (p, q) \<Longrightarrow> coprime p q"
290   by (auto simp add: normalize_def Let_def dvd_div_neg div_gcd_coprime_int
291     split:split_if_asm)
293 lemma normalize_stable [simp]:
294   "q > 0 \<Longrightarrow> coprime p q \<Longrightarrow> normalize (p, q) = (p, q)"
297 lemma normalize_denom_zero [simp]:
298   "normalize (p, 0) = (0, 1)"
301 lemma normalize_negative [simp]:
302   "q < 0 \<Longrightarrow> normalize (p, q) = normalize (- p, - q)"
303   by (simp add: normalize_def Let_def dvd_div_neg dvd_neg_div)
305 text\<open>
306   Decompose a fraction into normalized, i.e. coprime numerator and denominator:
307 \<close>
309 definition quotient_of :: "rat \<Rightarrow> int \<times> int" where
310   "quotient_of x = (THE pair. x = Fract (fst pair) (snd pair) &
311                    snd pair > 0 & coprime (fst pair) (snd pair))"
313 lemma quotient_of_unique:
314   "\<exists>!p. r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
315 proof (cases r)
316   case (Fract a b)
317   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
318   then show ?thesis proof (rule ex1I)
319     fix p
320     obtain c d :: int where p: "p = (c, d)" by (cases p)
321     assume "r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
322     with p have Fract': "r = Fract c d" "d > 0" "coprime c d" by simp_all
323     have "c = a \<and> d = b"
324     proof (cases "a = 0")
325       case True with Fract Fract' show ?thesis by (simp add: eq_rat)
326     next
327       case False
328       with Fract Fract' have *: "c * b = a * d" and "c \<noteq> 0" by (auto simp add: eq_rat)
329       then have "c * b > 0 \<longleftrightarrow> a * d > 0" by auto
330       with \<open>b > 0\<close> \<open>d > 0\<close> have "a > 0 \<longleftrightarrow> c > 0" by (simp add: zero_less_mult_iff)
331       with \<open>a \<noteq> 0\<close> \<open>c \<noteq> 0\<close> have sgn: "sgn a = sgn c" by (auto simp add: not_less)
332       from \<open>coprime a b\<close> \<open>coprime c d\<close> 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>"
334       with \<open>b > 0\<close> \<open>d > 0\<close> have "\<bar>a\<bar> * d = \<bar>c\<bar> * b \<longleftrightarrow> \<bar>a\<bar> = \<bar>c\<bar> \<and> d = b" by simp
335       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)
336       with sgn * show ?thesis by (auto simp add: sgn_0_0)
337     qed
338     with p show "p = (a, b)" by simp
339   qed
340 qed
342 lemma quotient_of_Fract [code]:
343   "quotient_of (Fract a b) = normalize (a, b)"
344 proof -
345   have "Fract a b = Fract (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?Fract)
346     by (rule sym) (auto intro: normalize_eq)
347   moreover have "0 < snd (normalize (a, b))" (is ?denom_pos)
348     by (cases "normalize (a, b)") (rule normalize_denom_pos, simp)
349   moreover have "coprime (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?coprime)
350     by (rule normalize_coprime) simp
351   ultimately have "?Fract \<and> ?denom_pos \<and> ?coprime" by blast
352   with quotient_of_unique have
353     "(THE p. Fract a b = Fract (fst p) (snd p) \<and> 0 < snd p \<and> coprime (fst p) (snd p)) = normalize (a, b)"
354     by (rule the1_equality)
355   then show ?thesis by (simp add: quotient_of_def)
356 qed
358 lemma quotient_of_number [simp]:
359   "quotient_of 0 = (0, 1)"
360   "quotient_of 1 = (1, 1)"
361   "quotient_of (numeral k) = (numeral k, 1)"
362   "quotient_of (- 1) = (- 1, 1)"
363   "quotient_of (- numeral k) = (- numeral k, 1)"
364   by (simp_all add: rat_number_expand quotient_of_Fract)
366 lemma quotient_of_eq: "quotient_of (Fract a b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
367   by (simp add: quotient_of_Fract normalize_eq)
369 lemma quotient_of_denom_pos: "quotient_of r = (p, q) \<Longrightarrow> q > 0"
370   by (cases r) (simp add: quotient_of_Fract normalize_denom_pos)
372 lemma quotient_of_coprime: "quotient_of r = (p, q) \<Longrightarrow> coprime p q"
373   by (cases r) (simp add: quotient_of_Fract normalize_coprime)
375 lemma quotient_of_inject:
376   assumes "quotient_of a = quotient_of b"
377   shows "a = b"
378 proof -
379   obtain p q r s where a: "a = Fract p q"
380     and b: "b = Fract r s"
381     and "q > 0" and "s > 0" by (cases a, cases b)
382   with assms show ?thesis by (simp add: eq_rat quotient_of_Fract normalize_crossproduct)
383 qed
385 lemma quotient_of_inject_eq:
386   "quotient_of a = quotient_of b \<longleftrightarrow> a = b"
387   by (auto simp add: quotient_of_inject)
390 subsubsection \<open>Various\<close>
392 lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"
393   by (simp add: Fract_of_int_eq [symmetric])
395 lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
398 lemma quotient_of_div:
399   assumes r: "quotient_of r = (n,d)"
400   shows "r = of_int n / of_int d"
401 proof -
402   from theI'[OF quotient_of_unique[of r], unfolded r[unfolded quotient_of_def]]
403   have "r = Fract n d" by simp
404   thus ?thesis using Fract_of_int_quotient by simp
405 qed
407 subsubsection \<open>The ordered field of rational numbers\<close>
409 lift_definition positive :: "rat \<Rightarrow> bool"
410   is "\<lambda>x. 0 < fst x * snd x"
411 proof (clarsimp)
412   fix a b c d :: int
413   assume "b \<noteq> 0" and "d \<noteq> 0" and "a * d = c * b"
414   hence "a * d * b * d = c * b * b * d"
415     by simp
416   hence "a * b * d\<^sup>2 = c * d * b\<^sup>2"
417     unfolding power2_eq_square by (simp add: ac_simps)
418   hence "0 < a * b * d\<^sup>2 \<longleftrightarrow> 0 < c * d * b\<^sup>2"
419     by simp
420   thus "0 < a * b \<longleftrightarrow> 0 < c * d"
421     using \<open>b \<noteq> 0\<close> and \<open>d \<noteq> 0\<close>
423 qed
425 lemma positive_zero: "\<not> positive 0"
426   by transfer simp
429   "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x + y)"
430 apply transfer
433   mult_pos_neg mult_neg_pos mult_neg_neg)
434 done
436 lemma positive_mult:
437   "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x * y)"
438 by transfer (drule (1) mult_pos_pos, simp add: ac_simps)
440 lemma positive_minus:
441   "\<not> positive x \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> positive (- x)"
442 by transfer (force simp: neq_iff zero_less_mult_iff mult_less_0_iff)
444 instantiation rat :: linordered_field
445 begin
447 definition
448   "x < y \<longleftrightarrow> positive (y - x)"
450 definition
451   "x \<le> (y::rat) \<longleftrightarrow> x < y \<or> x = y"
453 definition
454   "abs (a::rat) = (if a < 0 then - a else a)"
456 definition
457   "sgn (a::rat) = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
459 instance proof
460   fix a b c :: rat
461   show "\<bar>a\<bar> = (if a < 0 then - a else a)"
462     by (rule abs_rat_def)
463   show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a"
464     unfolding less_eq_rat_def less_rat_def
466   show "a \<le> a"
467     unfolding less_eq_rat_def by simp
468   show "a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
469     unfolding less_eq_rat_def less_rat_def
471   show "a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> a = b"
472     unfolding less_eq_rat_def less_rat_def
474   show "a \<le> b \<Longrightarrow> c + a \<le> c + b"
475     unfolding less_eq_rat_def less_rat_def by auto
476   show "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
477     by (rule sgn_rat_def)
478   show "a \<le> b \<or> b \<le> a"
479     unfolding less_eq_rat_def less_rat_def
480     by (auto dest!: positive_minus)
481   show "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
482     unfolding less_rat_def
483     by (drule (1) positive_mult, simp add: algebra_simps)
484 qed
486 end
488 instantiation rat :: distrib_lattice
489 begin
491 definition
492   "(inf :: rat \<Rightarrow> rat \<Rightarrow> rat) = min"
494 definition
495   "(sup :: rat \<Rightarrow> rat \<Rightarrow> rat) = max"
497 instance proof
498 qed (auto simp add: inf_rat_def sup_rat_def max_min_distrib2)
500 end
502 lemma positive_rat: "positive (Fract a b) \<longleftrightarrow> 0 < a * b"
503   by transfer simp
505 lemma less_rat [simp]:
506   assumes "b \<noteq> 0" and "d \<noteq> 0"
507   shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
508   using assms unfolding less_rat_def
509   by (simp add: positive_rat algebra_simps)
511 lemma le_rat [simp]:
512   assumes "b \<noteq> 0" and "d \<noteq> 0"
513   shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
514   using assms unfolding le_less by (simp add: eq_rat)
516 lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
517   by (auto simp add: abs_rat_def zabs_def Zero_rat_def not_less le_less eq_rat zero_less_mult_iff)
519 lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)"
520   unfolding Fract_of_int_eq
521   by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat)
522     (auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)
524 lemma Rat_induct_pos [case_names Fract, induct type: rat]:
525   assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
526   shows "P q"
527 proof (cases q)
528   have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)"
529   proof -
530     fix a::int and b::int
531     assume b: "b < 0"
532     hence "0 < -b" by simp
533     hence "P (Fract (-a) (-b))" by (rule step)
534     thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
535   qed
536   case (Fract a b)
537   thus "P q" by (force simp add: linorder_neq_iff step step')
538 qed
540 lemma zero_less_Fract_iff:
541   "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a"
542   by (simp add: Zero_rat_def zero_less_mult_iff)
544 lemma Fract_less_zero_iff:
545   "0 < b \<Longrightarrow> Fract a b < 0 \<longleftrightarrow> a < 0"
546   by (simp add: Zero_rat_def mult_less_0_iff)
548 lemma zero_le_Fract_iff:
549   "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a"
550   by (simp add: Zero_rat_def zero_le_mult_iff)
552 lemma Fract_le_zero_iff:
553   "0 < b \<Longrightarrow> Fract a b \<le> 0 \<longleftrightarrow> a \<le> 0"
554   by (simp add: Zero_rat_def mult_le_0_iff)
556 lemma one_less_Fract_iff:
557   "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a"
558   by (simp add: One_rat_def mult_less_cancel_right_disj)
560 lemma Fract_less_one_iff:
561   "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b"
562   by (simp add: One_rat_def mult_less_cancel_right_disj)
564 lemma one_le_Fract_iff:
565   "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a"
566   by (simp add: One_rat_def mult_le_cancel_right)
568 lemma Fract_le_one_iff:
569   "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b"
570   by (simp add: One_rat_def mult_le_cancel_right)
573 subsubsection \<open>Rationals are an Archimedean field\<close>
575 lemma rat_floor_lemma:
576   shows "of_int (a div b) \<le> Fract a b \<and> Fract a b < of_int (a div b + 1)"
577 proof -
578   have "Fract a b = of_int (a div b) + Fract (a mod b) b"
579     by (cases "b = 0", simp, simp add: of_int_rat)
580   moreover have "0 \<le> Fract (a mod b) b \<and> Fract (a mod b) b < 1"
581     unfolding Fract_of_int_quotient
582     by (rule linorder_cases [of b 0]) (simp_all add: divide_nonpos_neg)
583   ultimately show ?thesis by simp
584 qed
586 instance rat :: archimedean_field
587 proof
588   fix r :: rat
589   show "\<exists>z. r \<le> of_int z"
590   proof (induct r)
591     case (Fract a b)
592     have "Fract a b \<le> of_int (a div b + 1)"
593       using rat_floor_lemma [of a b] by simp
594     then show "\<exists>z. Fract a b \<le> of_int z" ..
595   qed
596 qed
598 instantiation rat :: floor_ceiling
599 begin
601 definition [code del]:
602   "floor (x::rat) = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))"
604 instance proof
605   fix x :: rat
606   show "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
607     unfolding floor_rat_def using floor_exists1 by (rule theI')
608 qed
610 end
612 lemma floor_Fract: "floor (Fract a b) = a div b"
613   by (simp add: Fract_of_int_quotient floor_divide_of_int_eq)
616 subsection \<open>Linear arithmetic setup\<close>
618 declaration \<open>
619   K (Lin_Arith.add_inj_thms [@{thm of_nat_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
620     (* not needed because x < (y::nat) can be rewritten as Suc x <= y: of_nat_less_iff RS iffD2 *)
621   #> Lin_Arith.add_inj_thms [@{thm of_int_le_iff} RS iffD2, @{thm of_int_eq_iff} RS iffD2]
622     (* not needed because x < (y::int) can be rewritten as x + 1 <= y: of_int_less_iff RS iffD2 *)
624       @{thm True_implies_equals},
625       @{thm distrib_left [where a = "numeral v" for v]},
626       @{thm distrib_left [where a = "- numeral v" for v]},
627       @{thm divide_1}, @{thm divide_zero_left},
628       @{thm times_divide_eq_right}, @{thm times_divide_eq_left},
629       @{thm minus_divide_left} RS sym, @{thm minus_divide_right} RS sym,
630       @{thm of_int_minus}, @{thm of_int_diff},
631       @{thm of_int_of_nat_eq}]
633   #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat => rat"})
634   #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int => rat"}))
635 \<close>
638 subsection \<open>Embedding from Rationals to other Fields\<close>
640 context field_char_0
641 begin
643 lift_definition of_rat :: "rat \<Rightarrow> 'a"
644   is "\<lambda>x. of_int (fst x) / of_int (snd x)"
645 apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
646 apply (simp only: of_int_mult [symmetric])
647 done
649 end
651 lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
652   by transfer simp
654 lemma of_rat_0 [simp]: "of_rat 0 = 0"
655   by transfer simp
657 lemma of_rat_1 [simp]: "of_rat 1 = 1"
658   by transfer simp
660 lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
663 lemma of_rat_minus: "of_rat (- a) = - of_rat a"
664   by transfer simp
666 lemma of_rat_neg_one [simp]:
667   "of_rat (- 1) = - 1"
670 lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
673 lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
674 apply transfer
675 apply (simp add: divide_inverse nonzero_inverse_mult_distrib ac_simps)
676 done
678 lemma of_rat_setsum: "of_rat (\<Sum>a\<in>A. f a) = (\<Sum>a\<in>A. of_rat (f a))"
679   by (induct rule: infinite_finite_induct) (auto simp: of_rat_add)
681 lemma of_rat_setprod: "of_rat (\<Prod>a\<in>A. f a) = (\<Prod>a\<in>A. of_rat (f a))"
682   by (induct rule: infinite_finite_induct) (auto simp: of_rat_mult)
684 lemma nonzero_of_rat_inverse:
685   "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
686 apply (rule inverse_unique [symmetric])
687 apply (simp add: of_rat_mult [symmetric])
688 done
690 lemma of_rat_inverse:
691   "(of_rat (inverse a)::'a::{field_char_0, field}) =
692    inverse (of_rat a)"
693 by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
695 lemma nonzero_of_rat_divide:
696   "b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
697 by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
699 lemma of_rat_divide:
700   "(of_rat (a / b)::'a::{field_char_0, field})
701    = of_rat a / of_rat b"
702 by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)
704 lemma of_rat_power:
705   "(of_rat (a ^ n)::'a::field_char_0) = of_rat a ^ n"
706 by (induct n) (simp_all add: of_rat_mult)
708 lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
709 apply transfer
710 apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
711 apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
712 done
714 lemma of_rat_eq_0_iff [simp]: "(of_rat a = 0) = (a = 0)"
715   using of_rat_eq_iff [of _ 0] by simp
717 lemma zero_eq_of_rat_iff [simp]: "(0 = of_rat a) = (0 = a)"
718   by simp
720 lemma of_rat_eq_1_iff [simp]: "(of_rat a = 1) = (a = 1)"
721   using of_rat_eq_iff [of _ 1] by simp
723 lemma one_eq_of_rat_iff [simp]: "(1 = of_rat a) = (1 = a)"
724   by simp
726 lemma of_rat_less:
727   "(of_rat r :: 'a::linordered_field) < of_rat s \<longleftrightarrow> r < s"
728 proof (induct r, induct s)
729   fix a b c d :: int
730   assume not_zero: "b > 0" "d > 0"
731   then have "b * d > 0" by simp
732   have of_int_divide_less_eq:
733     "(of_int a :: 'a) / of_int b < of_int c / of_int d
734       \<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b"
735     using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
736   show "(of_rat (Fract a b) :: 'a::linordered_field) < of_rat (Fract c d)
737     \<longleftrightarrow> Fract a b < Fract c d"
738     using not_zero \<open>b * d > 0\<close>
739     by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult)
740 qed
742 lemma of_rat_less_eq:
743   "(of_rat r :: 'a::linordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s"
744   unfolding le_less by (auto simp add: of_rat_less)
746 lemma of_rat_le_0_iff [simp]: "((of_rat r :: 'a::linordered_field) \<le> 0) = (r \<le> 0)"
747   using of_rat_less_eq [of r 0, where 'a='a] by simp
749 lemma zero_le_of_rat_iff [simp]: "(0 \<le> (of_rat r :: 'a::linordered_field)) = (0 \<le> r)"
750   using of_rat_less_eq [of 0 r, where 'a='a] by simp
752 lemma of_rat_le_1_iff [simp]: "((of_rat r :: 'a::linordered_field) \<le> 1) = (r \<le> 1)"
753   using of_rat_less_eq [of r 1] by simp
755 lemma one_le_of_rat_iff [simp]: "(1 \<le> (of_rat r :: 'a::linordered_field)) = (1 \<le> r)"
756   using of_rat_less_eq [of 1 r] by simp
758 lemma of_rat_less_0_iff [simp]: "((of_rat r :: 'a::linordered_field) < 0) = (r < 0)"
759   using of_rat_less [of r 0, where 'a='a] by simp
761 lemma zero_less_of_rat_iff [simp]: "(0 < (of_rat r :: 'a::linordered_field)) = (0 < r)"
762   using of_rat_less [of 0 r, where 'a='a] by simp
764 lemma of_rat_less_1_iff [simp]: "((of_rat r :: 'a::linordered_field) < 1) = (r < 1)"
765   using of_rat_less [of r 1] by simp
767 lemma one_less_of_rat_iff [simp]: "(1 < (of_rat r :: 'a::linordered_field)) = (1 < r)"
768   using of_rat_less [of 1 r] by simp
770 lemma of_rat_eq_id [simp]: "of_rat = id"
771 proof
772   fix a
773   show "of_rat a = id a"
774   by (induct a)
775      (simp add: of_rat_rat Fract_of_int_eq [symmetric])
776 qed
778 text\<open>Collapse nested embeddings\<close>
779 lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
782 lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
783 by (cases z rule: int_diff_cases) (simp add: of_rat_diff)
785 lemma of_rat_numeral_eq [simp]:
786   "of_rat (numeral w) = numeral w"
787 using of_rat_of_int_eq [of "numeral w"] by simp
789 lemma of_rat_neg_numeral_eq [simp]:
790   "of_rat (- numeral w) = - numeral w"
791 using of_rat_of_int_eq [of "- numeral w"] by simp
793 lemmas zero_rat = Zero_rat_def
794 lemmas one_rat = One_rat_def
796 abbreviation
797   rat_of_nat :: "nat \<Rightarrow> rat"
798 where
799   "rat_of_nat \<equiv> of_nat"
801 abbreviation
802   rat_of_int :: "int \<Rightarrow> rat"
803 where
804   "rat_of_int \<equiv> of_int"
806 subsection \<open>The Set of Rational Numbers\<close>
808 context field_char_0
809 begin
811 definition Rats :: "'a set" ("\<rat>")
812   where "\<rat> = range of_rat"
814 end
816 lemma Rats_of_rat [simp]: "of_rat r \<in> \<rat>"
819 lemma Rats_of_int [simp]: "of_int z \<in> \<rat>"
820 by (subst of_rat_of_int_eq [symmetric], rule Rats_of_rat)
822 lemma Rats_of_nat [simp]: "of_nat n \<in> \<rat>"
823 by (subst of_rat_of_nat_eq [symmetric], rule Rats_of_rat)
825 lemma Rats_number_of [simp]: "numeral w \<in> \<rat>"
826 by (subst of_rat_numeral_eq [symmetric], rule Rats_of_rat)
828 lemma Rats_0 [simp]: "0 \<in> \<rat>"
829 apply (unfold Rats_def)
830 apply (rule range_eqI)
831 apply (rule of_rat_0 [symmetric])
832 done
834 lemma Rats_1 [simp]: "1 \<in> \<rat>"
835 apply (unfold Rats_def)
836 apply (rule range_eqI)
837 apply (rule of_rat_1 [symmetric])
838 done
840 lemma Rats_add [simp]: "\<lbrakk>a \<in> \<rat>; b \<in> \<rat>\<rbrakk> \<Longrightarrow> a + b \<in> \<rat>"
841 apply (auto simp add: Rats_def)
842 apply (rule range_eqI)
844 done
846 lemma Rats_minus [simp]: "a \<in> \<rat> \<Longrightarrow> - a \<in> \<rat>"
847 apply (auto simp add: Rats_def)
848 apply (rule range_eqI)
849 apply (rule of_rat_minus [symmetric])
850 done
852 lemma Rats_diff [simp]: "\<lbrakk>a \<in> \<rat>; b \<in> \<rat>\<rbrakk> \<Longrightarrow> a - b \<in> \<rat>"
853 apply (auto simp add: Rats_def)
854 apply (rule range_eqI)
855 apply (rule of_rat_diff [symmetric])
856 done
858 lemma Rats_mult [simp]: "\<lbrakk>a \<in> \<rat>; b \<in> \<rat>\<rbrakk> \<Longrightarrow> a * b \<in> \<rat>"
859 apply (auto simp add: Rats_def)
860 apply (rule range_eqI)
861 apply (rule of_rat_mult [symmetric])
862 done
864 lemma nonzero_Rats_inverse:
865   fixes a :: "'a::field_char_0"
866   shows "\<lbrakk>a \<in> \<rat>; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> \<rat>"
867 apply (auto simp add: Rats_def)
868 apply (rule range_eqI)
869 apply (erule nonzero_of_rat_inverse [symmetric])
870 done
872 lemma Rats_inverse [simp]:
873   fixes a :: "'a::{field_char_0, field}"
874   shows "a \<in> \<rat> \<Longrightarrow> inverse a \<in> \<rat>"
875 apply (auto simp add: Rats_def)
876 apply (rule range_eqI)
877 apply (rule of_rat_inverse [symmetric])
878 done
880 lemma nonzero_Rats_divide:
881   fixes a b :: "'a::field_char_0"
882   shows "\<lbrakk>a \<in> \<rat>; b \<in> \<rat>; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> \<rat>"
883 apply (auto simp add: Rats_def)
884 apply (rule range_eqI)
885 apply (erule nonzero_of_rat_divide [symmetric])
886 done
888 lemma Rats_divide [simp]:
889   fixes a b :: "'a::{field_char_0, field}"
890   shows "\<lbrakk>a \<in> \<rat>; b \<in> \<rat>\<rbrakk> \<Longrightarrow> a / b \<in> \<rat>"
891 apply (auto simp add: Rats_def)
892 apply (rule range_eqI)
893 apply (rule of_rat_divide [symmetric])
894 done
896 lemma Rats_power [simp]:
897   fixes a :: "'a::field_char_0"
898   shows "a \<in> \<rat> \<Longrightarrow> a ^ n \<in> \<rat>"
899 apply (auto simp add: Rats_def)
900 apply (rule range_eqI)
901 apply (rule of_rat_power [symmetric])
902 done
904 lemma Rats_cases [cases set: Rats]:
905   assumes "q \<in> \<rat>"
906   obtains (of_rat) r where "q = of_rat r"
907 proof -
908   from \<open>q \<in> \<rat>\<close> have "q \<in> range of_rat" unfolding Rats_def .
909   then obtain r where "q = of_rat r" ..
910   then show thesis ..
911 qed
913 lemma Rats_induct [case_names of_rat, induct set: Rats]:
914   "q \<in> \<rat> \<Longrightarrow> (\<And>r. P (of_rat r)) \<Longrightarrow> P q"
915   by (rule Rats_cases) auto
917 lemma Rats_infinite: "\<not> finite \<rat>"
918   by (auto dest!: finite_imageD simp: inj_on_def infinite_UNIV_char_0 Rats_def)
920 subsection \<open>Implementation of rational numbers as pairs of integers\<close>
922 text \<open>Formal constructor\<close>
924 definition Frct :: "int \<times> int \<Rightarrow> rat" where
925   [simp]: "Frct p = Fract (fst p) (snd p)"
927 lemma [code abstype]:
928   "Frct (quotient_of q) = q"
929   by (cases q) (auto intro: quotient_of_eq)
932 text \<open>Numerals\<close>
934 declare quotient_of_Fract [code abstract]
936 definition of_int :: "int \<Rightarrow> rat"
937 where
938   [code_abbrev]: "of_int = Int.of_int"
939 hide_const (open) of_int
941 lemma quotient_of_int [code abstract]:
942   "quotient_of (Rat.of_int a) = (a, 1)"
943   by (simp add: of_int_def of_int_rat quotient_of_Fract)
945 lemma [code_unfold]:
946   "numeral k = Rat.of_int (numeral k)"
949 lemma [code_unfold]:
950   "- numeral k = Rat.of_int (- numeral k)"
953 lemma Frct_code_post [code_post]:
954   "Frct (0, a) = 0"
955   "Frct (a, 0) = 0"
956   "Frct (1, 1) = 1"
957   "Frct (numeral k, 1) = numeral k"
958   "Frct (1, numeral k) = 1 / numeral k"
959   "Frct (numeral k, numeral l) = numeral k / numeral l"
960   "Frct (- a, b) = - Frct (a, b)"
961   "Frct (a, - b) = - Frct (a, b)"
962   "- (- Frct q) = Frct q"
966 text \<open>Operations\<close>
968 lemma rat_zero_code [code abstract]:
969   "quotient_of 0 = (0, 1)"
970   by (simp add: Zero_rat_def quotient_of_Fract normalize_def)
972 lemma rat_one_code [code abstract]:
973   "quotient_of 1 = (1, 1)"
974   by (simp add: One_rat_def quotient_of_Fract normalize_def)
976 lemma rat_plus_code [code abstract]:
977   "quotient_of (p + q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
978      in normalize (a * d + b * c, c * d))"
979   by (cases p, cases q) (simp add: quotient_of_Fract)
981 lemma rat_uminus_code [code abstract]:
982   "quotient_of (- p) = (let (a, b) = quotient_of p in (- a, b))"
983   by (cases p) (simp add: quotient_of_Fract)
985 lemma rat_minus_code [code abstract]:
986   "quotient_of (p - q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
987      in normalize (a * d - b * c, c * d))"
988   by (cases p, cases q) (simp add: quotient_of_Fract)
990 lemma rat_times_code [code abstract]:
991   "quotient_of (p * q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
992      in normalize (a * b, c * d))"
993   by (cases p, cases q) (simp add: quotient_of_Fract)
995 lemma rat_inverse_code [code abstract]:
996   "quotient_of (inverse p) = (let (a, b) = quotient_of p
997     in if a = 0 then (0, 1) else (sgn a * b, \<bar>a\<bar>))"
998 proof (cases p)
999   case (Fract a b) then show ?thesis
1000     by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract gcd.commute)
1001 qed
1003 lemma rat_divide_code [code abstract]:
1004   "quotient_of (p / q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
1005      in normalize (a * d, c * b))"
1006   by (cases p, cases q) (simp add: quotient_of_Fract)
1008 lemma rat_abs_code [code abstract]:
1009   "quotient_of \<bar>p\<bar> = (let (a, b) = quotient_of p in (\<bar>a\<bar>, b))"
1010   by (cases p) (simp add: quotient_of_Fract)
1012 lemma rat_sgn_code [code abstract]:
1013   "quotient_of (sgn p) = (sgn (fst (quotient_of p)), 1)"
1014 proof (cases p)
1015   case (Fract a b) then show ?thesis
1016   by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract)
1017 qed
1019 lemma rat_floor_code [code]:
1020   "floor p = (let (a, b) = quotient_of p in a div b)"
1021 by (cases p) (simp add: quotient_of_Fract floor_Fract)
1023 instantiation rat :: equal
1024 begin
1026 definition [code]:
1027   "HOL.equal a b \<longleftrightarrow> quotient_of a = quotient_of b"
1029 instance proof
1030 qed (simp add: equal_rat_def quotient_of_inject_eq)
1032 lemma rat_eq_refl [code nbe]:
1033   "HOL.equal (r::rat) r \<longleftrightarrow> True"
1034   by (rule equal_refl)
1036 end
1038 lemma rat_less_eq_code [code]:
1039   "p \<le> q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d \<le> c * b)"
1040   by (cases p, cases q) (simp add: quotient_of_Fract mult.commute)
1042 lemma rat_less_code [code]:
1043   "p < q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d < c * b)"
1044   by (cases p, cases q) (simp add: quotient_of_Fract mult.commute)
1046 lemma [code]:
1047   "of_rat p = (let (a, b) = quotient_of p in of_int a / of_int b)"
1048   by (cases p) (simp add: quotient_of_Fract of_rat_rat)
1051 text \<open>Quickcheck\<close>
1053 definition (in term_syntax)
1054   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
1055   [code_unfold]: "valterm_fract k l = Code_Evaluation.valtermify Fract {\<cdot>} k {\<cdot>} l"
1057 notation fcomp (infixl "\<circ>>" 60)
1058 notation scomp (infixl "\<circ>\<rightarrow>" 60)
1060 instantiation rat :: random
1061 begin
1063 definition
1064   "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>num. Random.range i \<circ>\<rightarrow> (\<lambda>denom. Pair (
1065      let j = int_of_integer (integer_of_natural (denom + 1))
1066      in valterm_fract num (j, \<lambda>u. Code_Evaluation.term_of j))))"
1068 instance ..
1070 end
1072 no_notation fcomp (infixl "\<circ>>" 60)
1073 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
1075 instantiation rat :: exhaustive
1076 begin
1078 definition
1079   "exhaustive_rat f d = Quickcheck_Exhaustive.exhaustive
1080     (\<lambda>l. Quickcheck_Exhaustive.exhaustive (\<lambda>k. f (Fract k (int_of_integer (integer_of_natural l) + 1))) d) d"
1082 instance ..
1084 end
1086 instantiation rat :: full_exhaustive
1087 begin
1089 definition
1090   "full_exhaustive_rat f d = Quickcheck_Exhaustive.full_exhaustive (%(l, _). Quickcheck_Exhaustive.full_exhaustive (%k.
1091      f (let j = int_of_integer (integer_of_natural l) + 1
1092         in valterm_fract k (j, %_. Code_Evaluation.term_of j))) d) d"
1094 instance ..
1096 end
1098 instantiation rat :: partial_term_of
1099 begin
1101 instance ..
1103 end
1105 lemma [code]:
1106   "partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Narrowing_variable p tt) == Code_Evaluation.Free (STR ''_'') (Typerep.Typerep (STR ''Rat.rat'') [])"
1107   "partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Narrowing_constructor 0 [l, k]) ==
1108      Code_Evaluation.App (Code_Evaluation.Const (STR ''Rat.Frct'')
1109      (Typerep.Typerep (STR ''fun'') [Typerep.Typerep (STR ''Product_Type.prod'') [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''Int.int'') []],
1110         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))"
1111 by (rule partial_term_of_anything)+
1113 instantiation rat :: narrowing
1114 begin
1116 definition
1117   "narrowing = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.apply
1118     (Quickcheck_Narrowing.cons (%nom denom. Fract nom denom)) narrowing) narrowing"
1120 instance ..
1122 end
1125 subsection \<open>Setup for Nitpick\<close>
1127 declaration \<open>
1128   Nitpick_HOL.register_frac_type @{type_name rat}
1129    [(@{const_name zero_rat_inst.zero_rat}, @{const_name Nitpick.zero_frac}),
1130     (@{const_name one_rat_inst.one_rat}, @{const_name Nitpick.one_frac}),
1131     (@{const_name plus_rat_inst.plus_rat}, @{const_name Nitpick.plus_frac}),
1132     (@{const_name times_rat_inst.times_rat}, @{const_name Nitpick.times_frac}),
1133     (@{const_name uminus_rat_inst.uminus_rat}, @{const_name Nitpick.uminus_frac}),
1134     (@{const_name inverse_rat_inst.inverse_rat}, @{const_name Nitpick.inverse_frac}),
1135     (@{const_name ord_rat_inst.less_rat}, @{const_name Nitpick.less_frac}),
1136     (@{const_name ord_rat_inst.less_eq_rat}, @{const_name Nitpick.less_eq_frac}),
1137     (@{const_name field_char_0_class.of_rat}, @{const_name Nitpick.of_frac})]
1138 \<close>
1140 lemmas [nitpick_unfold] = inverse_rat_inst.inverse_rat
1141   one_rat_inst.one_rat ord_rat_inst.less_rat
1142   ord_rat_inst.less_eq_rat plus_rat_inst.plus_rat times_rat_inst.times_rat
1143   uminus_rat_inst.uminus_rat zero_rat_inst.zero_rat
1146 subsection \<open>Float syntax\<close>
1148 syntax "_Float" :: "float_const \<Rightarrow> 'a"    ("_")
1150 parse_translation \<open>
1151   let
1152     fun mk_frac str =
1153       let
1154         val {mant = i, exp = n} = Lexicon.read_float str;
1155         val exp = Syntax.const @{const_syntax Power.power};
1156         val ten = Numeral.mk_number_syntax 10;
1157         val exp10 = if n = 1 then ten else exp \$ ten \$ Numeral.mk_number_syntax n;
1158       in Syntax.const @{const_syntax Fields.inverse_divide} \$ Numeral.mk_number_syntax i \$ exp10 end;
1160     fun float_tr [(c as Const (@{syntax_const "_constrain"}, _)) \$ t \$ u] = c \$ float_tr [t] \$ u
1161       | float_tr [t as Const (str, _)] = mk_frac str
1162       | float_tr ts = raise TERM ("float_tr", ts);
1163   in [(@{syntax_const "_Float"}, K float_tr)] end
1164 \<close>
1166 text\<open>Test:\<close>
1167 lemma "123.456 = -111.111 + 200 + 30 + 4 + 5/10 + 6/100 + (7/1000::rat)"
1168   by simp
1171 subsection \<open>Hiding implementation details\<close>
1173 hide_const (open) normalize positive
1175 lifting_update rat.lifting
1176 lifting_forget rat.lifting
1178 end