src/HOL/Rational.thy
 author huffman Wed Feb 25 11:29:59 2009 -0800 (2009-02-25) changeset 30097 57df8626c23b parent 30095 c6e184561159 child 30198 922f944f03b2 permissions -rw-r--r--
generalize floor/ceiling to work with real and rat; rename floor_mono2 to floor_mono
1 (*  Title:  HOL/Rational.thy
2     Author: Markus Wenzel, TU Muenchen
3 *)
5 header {* Rational numbers *}
7 theory Rational
8 imports GCD Archimedean_Field
9 uses ("Tools/rat_arith.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) \<times> (int \<times> int)) set" where
18   "ratrel = {(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   "(x, y) \<in> ratrel \<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 refl_ratrel: "refl {x. snd x \<noteq> 0} ratrel"
25   by (auto simp add: refl_def ratrel_def)
27 lemma sym_ratrel: "sym ratrel"
28   by (simp add: ratrel_def sym_def)
30 lemma trans_ratrel: "trans ratrel"
31 proof (rule transI, unfold split_paired_all)
32   fix a b a' b' a'' b'' :: int
33   assume A: "((a, b), (a', b')) \<in> ratrel"
34   assume B: "((a', b'), (a'', b'')) \<in> ratrel"
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 "((a, b), (a'', b'')) \<in> ratrel" by auto
44 qed
46 lemma equiv_ratrel: "equiv {x. snd x \<noteq> 0} ratrel"
47   by (rule equiv.intro [OF refl_ratrel sym_ratrel trans_ratrel])
49 lemmas UN_ratrel = UN_equiv_class [OF equiv_ratrel]
50 lemmas UN_ratrel2 = UN_equiv_class2 [OF equiv_ratrel equiv_ratrel]
52 lemma equiv_ratrel_iff [iff]:
53   assumes "snd x \<noteq> 0" and "snd y \<noteq> 0"
54   shows "ratrel `` {x} = ratrel `` {y} \<longleftrightarrow> (x, y) \<in> ratrel"
55   by (rule eq_equiv_class_iff, rule equiv_ratrel) (auto simp add: assms)
57 typedef (Rat) rat = "{x. snd x \<noteq> 0} // ratrel"
58 proof
59   have "(0::int, 1::int) \<in> {x. snd x \<noteq> 0}" by simp
60   then show "ratrel `` {(0, 1)} \<in> {x. snd x \<noteq> 0} // ratrel" by (rule quotientI)
61 qed
63 lemma ratrel_in_Rat [simp]: "snd x \<noteq> 0 \<Longrightarrow> ratrel `` {x} \<in> Rat"
64   by (simp add: Rat_def quotientI)
66 declare Abs_Rat_inject [simp] Abs_Rat_inverse [simp]
69 subsubsection {* Representation and basic operations *}
71 definition
72   Fract :: "int \<Rightarrow> int \<Rightarrow> rat" where
73   [code del]: "Fract a b = Abs_Rat (ratrel `` {if b = 0 then (0, 1) else (a, b)})"
75 code_datatype Fract
77 lemma Rat_cases [case_names Fract, cases type: rat]:
78   assumes "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> C"
79   shows C
80   using assms by (cases q) (clarsimp simp add: Fract_def Rat_def quotient_def)
82 lemma Rat_induct [case_names Fract, induct type: rat]:
83   assumes "\<And>a b. b \<noteq> 0 \<Longrightarrow> P (Fract a b)"
84   shows "P q"
85   using assms by (cases q) simp
87 lemma eq_rat:
88   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"
89   and "\<And>a. Fract a 0 = Fract 0 1"
90   and "\<And>a c. Fract 0 a = Fract 0 c"
91   by (simp_all add: Fract_def)
93 instantiation rat :: "{comm_ring_1, recpower}"
94 begin
96 definition
97   Zero_rat_def [code, code unfold]: "0 = Fract 0 1"
99 definition
100   One_rat_def [code, code unfold]: "1 = Fract 1 1"
102 definition
103   add_rat_def [code del]:
104   "q + r = Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
105     ratrel `` {(fst x * snd y + fst y * snd x, snd x * snd y)})"
107 lemma add_rat [simp]:
108   assumes "b \<noteq> 0" and "d \<noteq> 0"
109   shows "Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
110 proof -
111   have "(\<lambda>x y. ratrel``{(fst x * snd y + fst y * snd x, snd x * snd y)})
112     respects2 ratrel"
113   by (rule equiv_ratrel [THEN congruent2_commuteI]) (simp_all add: left_distrib)
114   with assms show ?thesis by (simp add: Fract_def add_rat_def UN_ratrel2)
115 qed
117 definition
118   minus_rat_def [code del]:
119   "- q = Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel `` {(- fst x, snd x)})"
121 lemma minus_rat [simp, code]: "- Fract a b = Fract (- a) b"
122 proof -
123   have "(\<lambda>x. ratrel `` {(- fst x, snd x)}) respects ratrel"
124     by (simp add: congruent_def)
125   then show ?thesis by (simp add: Fract_def minus_rat_def UN_ratrel)
126 qed
128 lemma minus_rat_cancel [simp]: "Fract (- a) (- b) = Fract a b"
129   by (cases "b = 0") (simp_all add: eq_rat)
131 definition
132   diff_rat_def [code del]: "q - r = q + - (r::rat)"
134 lemma diff_rat [simp]:
135   assumes "b \<noteq> 0" and "d \<noteq> 0"
136   shows "Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
137   using assms by (simp add: diff_rat_def)
139 definition
140   mult_rat_def [code del]:
141   "q * r = Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
142     ratrel``{(fst x * fst y, snd x * snd y)})"
144 lemma mult_rat [simp]: "Fract a b * Fract c d = Fract (a * c) (b * d)"
145 proof -
146   have "(\<lambda>x y. ratrel `` {(fst x * fst y, snd x * snd y)}) respects2 ratrel"
147     by (rule equiv_ratrel [THEN congruent2_commuteI]) simp_all
148   then show ?thesis by (simp add: Fract_def mult_rat_def UN_ratrel2)
149 qed
151 lemma mult_rat_cancel:
152   assumes "c \<noteq> 0"
153   shows "Fract (c * a) (c * b) = Fract a b"
154 proof -
155   from assms have "Fract c c = Fract 1 1" by (simp add: Fract_def)
156   then show ?thesis by (simp add: mult_rat [symmetric])
157 qed
159 primrec power_rat
160 where
161   rat_power_0:     "q ^ 0 = (1\<Colon>rat)"
162   | rat_power_Suc: "q ^ Suc n = (q\<Colon>rat) * (q ^ n)"
164 instance proof
165   fix q r s :: rat show "(q * r) * s = q * (r * s)"
166     by (cases q, cases r, cases s) (simp add: eq_rat)
167 next
168   fix q r :: rat show "q * r = r * q"
169     by (cases q, cases r) (simp add: eq_rat)
170 next
171   fix q :: rat show "1 * q = q"
172     by (cases q) (simp add: One_rat_def eq_rat)
173 next
174   fix q r s :: rat show "(q + r) + s = q + (r + s)"
175     by (cases q, cases r, cases s) (simp add: eq_rat algebra_simps)
176 next
177   fix q r :: rat show "q + r = r + q"
178     by (cases q, cases r) (simp add: eq_rat)
179 next
180   fix q :: rat show "0 + q = q"
181     by (cases q) (simp add: Zero_rat_def eq_rat)
182 next
183   fix q :: rat show "- q + q = 0"
184     by (cases q) (simp add: Zero_rat_def eq_rat)
185 next
186   fix q r :: rat show "q - r = q + - r"
187     by (cases q, cases r) (simp add: eq_rat)
188 next
189   fix q r s :: rat show "(q + r) * s = q * s + r * s"
190     by (cases q, cases r, cases s) (simp add: eq_rat algebra_simps)
191 next
192   show "(0::rat) \<noteq> 1" by (simp add: Zero_rat_def One_rat_def eq_rat)
193 next
194   fix q :: rat show "q * 1 = q"
195     by (cases q) (simp add: One_rat_def eq_rat)
196 next
197   fix q :: rat
198   fix n :: nat
199   show "q ^ 0 = 1" by simp
200   show "q ^ (Suc n) = q * (q ^ n)" by simp
201 qed
203 end
205 lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1"
206   by (induct k) (simp_all add: Zero_rat_def One_rat_def)
208 lemma of_int_rat: "of_int k = Fract k 1"
209   by (cases k rule: int_diff_cases) (simp add: of_nat_rat)
211 lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
212   by (rule of_nat_rat [symmetric])
214 lemma Fract_of_int_eq: "Fract k 1 = of_int k"
215   by (rule of_int_rat [symmetric])
217 instantiation rat :: number_ring
218 begin
220 definition
221   rat_number_of_def [code del]: "number_of w = Fract w 1"
223 instance by intro_classes (simp add: rat_number_of_def of_int_rat)
225 end
227 lemma rat_number_collapse [code post]:
228   "Fract 0 k = 0"
229   "Fract 1 1 = 1"
230   "Fract (number_of k) 1 = number_of k"
231   "Fract k 0 = 0"
232   by (cases "k = 0")
233     (simp_all add: Zero_rat_def One_rat_def number_of_is_id number_of_eq of_int_rat eq_rat Fract_def)
235 lemma rat_number_expand [code unfold]:
236   "0 = Fract 0 1"
237   "1 = Fract 1 1"
238   "number_of k = Fract (number_of k) 1"
239   by (simp_all add: rat_number_collapse)
241 lemma iszero_rat [simp]:
242   "iszero (number_of k :: rat) \<longleftrightarrow> iszero (number_of k :: int)"
243   by (simp add: iszero_def rat_number_expand number_of_is_id eq_rat)
245 lemma Rat_cases_nonzero [case_names Fract 0]:
246   assumes Fract: "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> C"
247   assumes 0: "q = 0 \<Longrightarrow> C"
248   shows C
249 proof (cases "q = 0")
250   case True then show C using 0 by auto
251 next
252   case False
253   then obtain a b where "q = Fract a b" and "b \<noteq> 0" by (cases q) auto
254   moreover with False have "0 \<noteq> Fract a b" by simp
255   with `b \<noteq> 0` have "a \<noteq> 0" by (simp add: Zero_rat_def eq_rat)
256   with Fract `q = Fract a b` `b \<noteq> 0` show C by auto
257 qed
260 subsubsection {* The field of rational numbers *}
262 instantiation rat :: "{field, division_by_zero}"
263 begin
265 definition
266   inverse_rat_def [code del]:
267   "inverse q = Abs_Rat (\<Union>x \<in> Rep_Rat q.
268      ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)})"
270 lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"
271 proof -
272   have "(\<lambda>x. ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)}) respects ratrel"
273     by (auto simp add: congruent_def mult_commute)
274   then show ?thesis by (simp add: Fract_def inverse_rat_def UN_ratrel)
275 qed
277 definition
278   divide_rat_def [code del]: "q / r = q * inverse (r::rat)"
280 lemma divide_rat [simp]: "Fract a b / Fract c d = Fract (a * d) (b * c)"
281   by (simp add: divide_rat_def)
283 instance proof
284   show "inverse 0 = (0::rat)" by (simp add: rat_number_expand)
285     (simp add: rat_number_collapse)
286 next
287   fix q :: rat
288   assume "q \<noteq> 0"
289   then show "inverse q * q = 1" by (cases q rule: Rat_cases_nonzero)
290    (simp_all add: mult_rat  inverse_rat rat_number_expand eq_rat)
291 next
292   fix q r :: rat
293   show "q / r = q * inverse r" by (simp add: divide_rat_def)
294 qed
296 end
299 subsubsection {* Various *}
301 lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"
302   by (simp add: rat_number_expand)
304 lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"
305   by (simp add: Fract_of_int_eq [symmetric])
307 lemma Fract_number_of_quotient [code post]:
308   "Fract (number_of k) (number_of l) = number_of k / number_of l"
309   unfolding Fract_of_int_quotient number_of_is_id number_of_eq ..
311 lemma Fract_1_number_of [code post]:
312   "Fract 1 (number_of k) = 1 / number_of k"
313   unfolding Fract_of_int_quotient number_of_eq by simp
315 subsubsection {* The ordered field of rational numbers *}
317 instantiation rat :: linorder
318 begin
320 definition
321   le_rat_def [code del]:
322    "q \<le> r \<longleftrightarrow> contents (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
323       {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})"
325 lemma le_rat [simp]:
326   assumes "b \<noteq> 0" and "d \<noteq> 0"
327   shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
328 proof -
329   have "(\<lambda>x y. {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})
330     respects2 ratrel"
331   proof (clarsimp simp add: congruent2_def)
332     fix a b a' b' c d c' d'::int
333     assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
334     assume eq1: "a * b' = a' * b"
335     assume eq2: "c * d' = c' * d"
337     let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
338     {
339       fix a b c d x :: int assume x: "x \<noteq> 0"
340       have "?le a b c d = ?le (a * x) (b * x) c d"
341       proof -
342         from x have "0 < x * x" by (auto simp add: zero_less_mult_iff)
343         hence "?le a b c d =
344             ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
345           by (simp add: mult_le_cancel_right)
346         also have "... = ?le (a * x) (b * x) c d"
347           by (simp add: mult_ac)
348         finally show ?thesis .
349       qed
350     } note le_factor = this
352     let ?D = "b * d" and ?D' = "b' * d'"
353     from neq have D: "?D \<noteq> 0" by simp
354     from neq have "?D' \<noteq> 0" by simp
355     hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"
356       by (rule le_factor)
357     also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')"
358       by (simp add: mult_ac)
359     also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
360       by (simp only: eq1 eq2)
361     also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
362       by (simp add: mult_ac)
363     also from D have "... = ?le a' b' c' d'"
364       by (rule le_factor [symmetric])
365     finally show "?le a b c d = ?le a' b' c' d'" .
366   qed
367   with assms show ?thesis by (simp add: Fract_def le_rat_def UN_ratrel2)
368 qed
370 definition
371   less_rat_def [code del]: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
373 lemma less_rat [simp]:
374   assumes "b \<noteq> 0" and "d \<noteq> 0"
375   shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
376   using assms by (simp add: less_rat_def eq_rat order_less_le)
378 instance proof
379   fix q r s :: rat
380   {
381     assume "q \<le> r" and "r \<le> s"
382     show "q \<le> s"
383     proof (insert prems, induct q, induct r, induct s)
384       fix a b c d e f :: int
385       assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
386       assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"
387       show "Fract a b \<le> Fract e f"
388       proof -
389         from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
390           by (auto simp add: zero_less_mult_iff linorder_neq_iff)
391         have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
392         proof -
393           from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
394             by simp
395           with ff show ?thesis by (simp add: mult_le_cancel_right)
396         qed
397         also have "... = (c * f) * (d * f) * (b * b)" by algebra
398         also have "... \<le> (e * d) * (d * f) * (b * b)"
399         proof -
400           from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
401             by simp
402           with bb show ?thesis by (simp add: mult_le_cancel_right)
403         qed
404         finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
405           by (simp only: mult_ac)
406         with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
407           by (simp add: mult_le_cancel_right)
408         with neq show ?thesis by simp
409       qed
410     qed
411   next
412     assume "q \<le> r" and "r \<le> q"
413     show "q = r"
414     proof (insert prems, induct q, induct r)
415       fix a b c d :: int
416       assume neq: "b \<noteq> 0"  "d \<noteq> 0"
417       assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"
418       show "Fract a b = Fract c d"
419       proof -
420         from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
421           by simp
422         also have "... \<le> (a * d) * (b * d)"
423         proof -
424           from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
425             by simp
426           thus ?thesis by (simp only: mult_ac)
427         qed
428         finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
429         moreover from neq have "b * d \<noteq> 0" by simp
430         ultimately have "a * d = c * b" by simp
431         with neq show ?thesis by (simp add: eq_rat)
432       qed
433     qed
434   next
435     show "q \<le> q"
436       by (induct q) simp
437     show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)"
438       by (induct q, induct r) (auto simp add: le_less mult_commute)
439     show "q \<le> r \<or> r \<le> q"
440       by (induct q, induct r)
441          (simp add: mult_commute, rule linorder_linear)
442   }
443 qed
445 end
447 instantiation rat :: "{distrib_lattice, abs_if, sgn_if}"
448 begin
450 definition
451   abs_rat_def [code del]: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))"
453 lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
454   by (auto simp add: abs_rat_def zabs_def Zero_rat_def less_rat not_less le_less minus_rat eq_rat zero_compare_simps)
456 definition
457   sgn_rat_def [code del]: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
459 lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)"
460   unfolding Fract_of_int_eq
461   by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat)
462     (auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)
464 definition
465   "(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min"
467 definition
468   "(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max"
470 instance by intro_classes
471   (auto simp add: abs_rat_def sgn_rat_def min_max.sup_inf_distrib1 inf_rat_def sup_rat_def)
473 end
475 instance rat :: ordered_field
476 proof
477   fix q r s :: rat
478   show "q \<le> r ==> s + q \<le> s + r"
479   proof (induct q, induct r, induct s)
480     fix a b c d e f :: int
481     assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
482     assume le: "Fract a b \<le> Fract c d"
483     show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
484     proof -
485       let ?F = "f * f" from neq have F: "0 < ?F"
486         by (auto simp add: zero_less_mult_iff)
487       from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
488         by simp
489       with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
490         by (simp add: mult_le_cancel_right)
491       with neq show ?thesis by (simp add: mult_ac int_distrib)
492     qed
493   qed
494   show "q < r ==> 0 < s ==> s * q < s * r"
495   proof (induct q, induct r, induct s)
496     fix a b c d e f :: int
497     assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
498     assume le: "Fract a b < Fract c d"
499     assume gt: "0 < Fract e f"
500     show "Fract e f * Fract a b < Fract e f * Fract c d"
501     proof -
502       let ?E = "e * f" and ?F = "f * f"
503       from neq gt have "0 < ?E"
504         by (auto simp add: Zero_rat_def order_less_le eq_rat)
505       moreover from neq have "0 < ?F"
506         by (auto simp add: zero_less_mult_iff)
507       moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
508         by simp
509       ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
510         by (simp add: mult_less_cancel_right)
511       with neq show ?thesis
512         by (simp add: mult_ac)
513     qed
514   qed
515 qed auto
517 lemma Rat_induct_pos [case_names Fract, induct type: rat]:
518   assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
519   shows "P q"
520 proof (cases q)
521   have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)"
522   proof -
523     fix a::int and b::int
524     assume b: "b < 0"
525     hence "0 < -b" by simp
526     hence "P (Fract (-a) (-b))" by (rule step)
527     thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
528   qed
529   case (Fract a b)
530   thus "P q" by (force simp add: linorder_neq_iff step step')
531 qed
533 lemma zero_less_Fract_iff:
534   "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a"
535   by (simp add: Zero_rat_def zero_less_mult_iff)
537 lemma Fract_less_zero_iff:
538   "0 < b \<Longrightarrow> Fract a b < 0 \<longleftrightarrow> a < 0"
539   by (simp add: Zero_rat_def mult_less_0_iff)
541 lemma zero_le_Fract_iff:
542   "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a"
543   by (simp add: Zero_rat_def zero_le_mult_iff)
545 lemma Fract_le_zero_iff:
546   "0 < b \<Longrightarrow> Fract a b \<le> 0 \<longleftrightarrow> a \<le> 0"
547   by (simp add: Zero_rat_def mult_le_0_iff)
549 lemma one_less_Fract_iff:
550   "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a"
551   by (simp add: One_rat_def mult_less_cancel_right_disj)
553 lemma Fract_less_one_iff:
554   "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b"
555   by (simp add: One_rat_def mult_less_cancel_right_disj)
557 lemma one_le_Fract_iff:
558   "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a"
559   by (simp add: One_rat_def mult_le_cancel_right)
561 lemma Fract_le_one_iff:
562   "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b"
563   by (simp add: One_rat_def mult_le_cancel_right)
566 subsubsection {* Rationals are an Archimedean field *}
568 lemma rat_floor_lemma:
569   assumes "0 < b"
570   shows "of_int (a div b) \<le> Fract a b \<and> Fract a b < of_int (a div b + 1)"
571 proof -
572   have "Fract a b = of_int (a div b) + Fract (a mod b) b"
573     using `0 < b` by (simp add: of_int_rat)
574   moreover have "0 \<le> Fract (a mod b) b \<and> Fract (a mod b) b < 1"
575     using `0 < b` by (simp add: zero_le_Fract_iff Fract_less_one_iff)
576   ultimately show ?thesis by simp
577 qed
579 instance rat :: archimedean_field
580 proof
581   fix r :: rat
582   show "\<exists>z. r \<le> of_int z"
583   proof (induct r)
584     case (Fract a b)
585     then have "Fract a b \<le> of_int (a div b + 1)"
586       using rat_floor_lemma [of b a] by simp
587     then show "\<exists>z. Fract a b \<le> of_int z" ..
588   qed
589 qed
591 lemma floor_Fract:
592   assumes "0 < b" shows "floor (Fract a b) = a div b"
593   using rat_floor_lemma [OF `0 < b`, of a]
594   by (simp add: floor_unique)
597 subsection {* Arithmetic setup *}
599 use "Tools/rat_arith.ML"
600 declaration {* K rat_arith_setup *}
603 subsection {* Embedding from Rationals to other Fields *}
605 class field_char_0 = field + ring_char_0
607 subclass (in ordered_field) field_char_0 ..
609 context field_char_0
610 begin
612 definition of_rat :: "rat \<Rightarrow> 'a" where
613   [code del]: "of_rat q = contents (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})"
615 end
617 lemma of_rat_congruent:
618   "(\<lambda>(a, b). {of_int a / of_int b :: 'a::field_char_0}) respects ratrel"
619 apply (rule congruent.intro)
620 apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
621 apply (simp only: of_int_mult [symmetric])
622 done
624 lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
625   unfolding Fract_def of_rat_def by (simp add: UN_ratrel of_rat_congruent)
627 lemma of_rat_0 [simp]: "of_rat 0 = 0"
628 by (simp add: Zero_rat_def of_rat_rat)
630 lemma of_rat_1 [simp]: "of_rat 1 = 1"
631 by (simp add: One_rat_def of_rat_rat)
633 lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
634 by (induct a, induct b, simp add: of_rat_rat add_frac_eq)
636 lemma of_rat_minus: "of_rat (- a) = - of_rat a"
637 by (induct a, simp add: of_rat_rat)
639 lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
640 by (simp only: diff_minus of_rat_add of_rat_minus)
642 lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
643 apply (induct a, induct b, simp add: of_rat_rat)
644 apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
645 done
647 lemma nonzero_of_rat_inverse:
648   "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
649 apply (rule inverse_unique [symmetric])
650 apply (simp add: of_rat_mult [symmetric])
651 done
653 lemma of_rat_inverse:
654   "(of_rat (inverse a)::'a::{field_char_0,division_by_zero}) =
655    inverse (of_rat a)"
656 by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
658 lemma nonzero_of_rat_divide:
659   "b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
660 by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
662 lemma of_rat_divide:
663   "(of_rat (a / b)::'a::{field_char_0,division_by_zero})
664    = of_rat a / of_rat b"
665 by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)
667 lemma of_rat_power:
668   "(of_rat (a ^ n)::'a::{field_char_0,recpower}) = of_rat a ^ n"
669 by (induct n) (simp_all add: of_rat_mult power_Suc)
671 lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
672 apply (induct a, induct b)
673 apply (simp add: of_rat_rat eq_rat)
674 apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
675 apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
676 done
678 lemma of_rat_less:
679   "(of_rat r :: 'a::ordered_field) < of_rat s \<longleftrightarrow> r < s"
680 proof (induct r, induct s)
681   fix a b c d :: int
682   assume not_zero: "b > 0" "d > 0"
683   then have "b * d > 0" by (rule mult_pos_pos)
684   have of_int_divide_less_eq:
685     "(of_int a :: 'a) / of_int b < of_int c / of_int d
686       \<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b"
687     using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
688   show "(of_rat (Fract a b) :: 'a::ordered_field) < of_rat (Fract c d)
689     \<longleftrightarrow> Fract a b < Fract c d"
690     using not_zero `b * d > 0`
691     by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult)
692       (auto intro: mult_strict_right_mono mult_right_less_imp_less)
693 qed
695 lemma of_rat_less_eq:
696   "(of_rat r :: 'a::ordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s"
697   unfolding le_less by (auto simp add: of_rat_less)
699 lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
701 lemma of_rat_eq_id [simp]: "of_rat = id"
702 proof
703   fix a
704   show "of_rat a = id a"
705   by (induct a)
706      (simp add: of_rat_rat Fract_of_int_eq [symmetric])
707 qed
709 text{*Collapse nested embeddings*}
710 lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
711 by (induct n) (simp_all add: of_rat_add)
713 lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
714 by (cases z rule: int_diff_cases) (simp add: of_rat_diff)
716 lemma of_rat_number_of_eq [simp]:
717   "of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})"
718 by (simp add: number_of_eq)
720 lemmas zero_rat = Zero_rat_def
721 lemmas one_rat = One_rat_def
723 abbreviation
724   rat_of_nat :: "nat \<Rightarrow> rat"
725 where
726   "rat_of_nat \<equiv> of_nat"
728 abbreviation
729   rat_of_int :: "int \<Rightarrow> rat"
730 where
731   "rat_of_int \<equiv> of_int"
733 subsection {* The Set of Rational Numbers *}
735 context field_char_0
736 begin
738 definition
739   Rats  :: "'a set" where
740   [code del]: "Rats = range of_rat"
742 notation (xsymbols)
743   Rats  ("\<rat>")
745 end
747 lemma Rats_of_rat [simp]: "of_rat r \<in> Rats"
748 by (simp add: Rats_def)
750 lemma Rats_of_int [simp]: "of_int z \<in> Rats"
751 by (subst of_rat_of_int_eq [symmetric], rule Rats_of_rat)
753 lemma Rats_of_nat [simp]: "of_nat n \<in> Rats"
754 by (subst of_rat_of_nat_eq [symmetric], rule Rats_of_rat)
756 lemma Rats_number_of [simp]:
757   "(number_of w::'a::{number_ring,field_char_0}) \<in> Rats"
758 by (subst of_rat_number_of_eq [symmetric], rule Rats_of_rat)
760 lemma Rats_0 [simp]: "0 \<in> Rats"
761 apply (unfold Rats_def)
762 apply (rule range_eqI)
763 apply (rule of_rat_0 [symmetric])
764 done
766 lemma Rats_1 [simp]: "1 \<in> Rats"
767 apply (unfold Rats_def)
768 apply (rule range_eqI)
769 apply (rule of_rat_1 [symmetric])
770 done
772 lemma Rats_add [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a + b \<in> Rats"
773 apply (auto simp add: Rats_def)
774 apply (rule range_eqI)
775 apply (rule of_rat_add [symmetric])
776 done
778 lemma Rats_minus [simp]: "a \<in> Rats \<Longrightarrow> - a \<in> Rats"
779 apply (auto simp add: Rats_def)
780 apply (rule range_eqI)
781 apply (rule of_rat_minus [symmetric])
782 done
784 lemma Rats_diff [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a - b \<in> Rats"
785 apply (auto simp add: Rats_def)
786 apply (rule range_eqI)
787 apply (rule of_rat_diff [symmetric])
788 done
790 lemma Rats_mult [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a * b \<in> Rats"
791 apply (auto simp add: Rats_def)
792 apply (rule range_eqI)
793 apply (rule of_rat_mult [symmetric])
794 done
796 lemma nonzero_Rats_inverse:
797   fixes a :: "'a::field_char_0"
798   shows "\<lbrakk>a \<in> Rats; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Rats"
799 apply (auto simp add: Rats_def)
800 apply (rule range_eqI)
801 apply (erule nonzero_of_rat_inverse [symmetric])
802 done
804 lemma Rats_inverse [simp]:
805   fixes a :: "'a::{field_char_0,division_by_zero}"
806   shows "a \<in> Rats \<Longrightarrow> inverse a \<in> Rats"
807 apply (auto simp add: Rats_def)
808 apply (rule range_eqI)
809 apply (rule of_rat_inverse [symmetric])
810 done
812 lemma nonzero_Rats_divide:
813   fixes a b :: "'a::field_char_0"
814   shows "\<lbrakk>a \<in> Rats; b \<in> Rats; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
815 apply (auto simp add: Rats_def)
816 apply (rule range_eqI)
817 apply (erule nonzero_of_rat_divide [symmetric])
818 done
820 lemma Rats_divide [simp]:
821   fixes a b :: "'a::{field_char_0,division_by_zero}"
822   shows "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
823 apply (auto simp add: Rats_def)
824 apply (rule range_eqI)
825 apply (rule of_rat_divide [symmetric])
826 done
828 lemma Rats_power [simp]:
829   fixes a :: "'a::{field_char_0,recpower}"
830   shows "a \<in> Rats \<Longrightarrow> a ^ n \<in> Rats"
831 apply (auto simp add: Rats_def)
832 apply (rule range_eqI)
833 apply (rule of_rat_power [symmetric])
834 done
836 lemma Rats_cases [cases set: Rats]:
837   assumes "q \<in> \<rat>"
838   obtains (of_rat) r where "q = of_rat r"
839   unfolding Rats_def
840 proof -
841   from `q \<in> \<rat>` have "q \<in> range of_rat" unfolding Rats_def .
842   then obtain r where "q = of_rat r" ..
843   then show thesis ..
844 qed
846 lemma Rats_induct [case_names of_rat, induct set: Rats]:
847   "q \<in> \<rat> \<Longrightarrow> (\<And>r. P (of_rat r)) \<Longrightarrow> P q"
848   by (rule Rats_cases) auto
851 subsection {* Implementation of rational numbers as pairs of integers *}
853 lemma Fract_norm: "Fract (a div zgcd a b) (b div zgcd a b) = Fract a b"
854 proof (cases "a = 0 \<or> b = 0")
855   case True then show ?thesis by (auto simp add: eq_rat)
856 next
857   let ?c = "zgcd a b"
858   case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
859   then have "?c \<noteq> 0" by simp
860   then have "Fract ?c ?c = Fract 1 1" by (simp add: eq_rat)
861   moreover have "Fract (a div ?c * ?c + a mod ?c) (b div ?c * ?c + b mod ?c) = Fract a b"
862     by (simp add: semiring_div_class.mod_div_equality)
863   moreover have "a mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric])
864   moreover have "b mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric])
865   ultimately show ?thesis
866     by (simp add: mult_rat [symmetric])
867 qed
869 definition Fract_norm :: "int \<Rightarrow> int \<Rightarrow> rat" where
870   [simp, code del]: "Fract_norm a b = Fract a b"
872 lemma Fract_norm_code [code]: "Fract_norm a b = (if a = 0 \<or> b = 0 then 0 else let c = zgcd a b in
873   if b > 0 then Fract (a div c) (b div c) else Fract (- (a div c)) (- (b div c)))"
874   by (simp add: eq_rat Zero_rat_def Let_def Fract_norm)
876 lemma [code]:
877   "of_rat (Fract a b) = (if b \<noteq> 0 then of_int a / of_int b else 0)"
878   by (cases "b = 0") (simp_all add: rat_number_collapse of_rat_rat)
880 instantiation rat :: eq
881 begin
883 definition [code del]: "eq_class.eq (a\<Colon>rat) b \<longleftrightarrow> a - b = 0"
885 instance by default (simp add: eq_rat_def)
887 lemma rat_eq_code [code]:
888   "eq_class.eq (Fract a b) (Fract c d) \<longleftrightarrow> (if b = 0
889        then c = 0 \<or> d = 0
890      else if d = 0
891        then a = 0 \<or> b = 0
892      else a * d = b * c)"
893   by (auto simp add: eq eq_rat)
895 lemma rat_eq_refl [code nbe]:
896   "eq_class.eq (r::rat) r \<longleftrightarrow> True"
897   by (rule HOL.eq_refl)
899 end
901 lemma le_rat':
902   assumes "b \<noteq> 0"
903     and "d \<noteq> 0"
904   shows "Fract a b \<le> Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d"
905 proof -
906   have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp
907   have "a * d * (b * d) \<le> c * b * (b * d) \<longleftrightarrow> a * d * (sgn b * sgn d) \<le> c * b * (sgn b * sgn d)"
908   proof (cases "b * d > 0")
909     case True
910     moreover from True have "sgn b * sgn d = 1"
911       by (simp add: sgn_times [symmetric] sgn_1_pos)
912     ultimately show ?thesis by (simp add: mult_le_cancel_right)
913   next
914     case False with assms have "b * d < 0" by (simp add: less_le)
915     moreover from this have "sgn b * sgn d = - 1"
916       by (simp only: sgn_times [symmetric] sgn_1_neg)
917     ultimately show ?thesis by (simp add: mult_le_cancel_right)
918   qed
919   also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d"
920     by (simp add: abs_sgn mult_ac)
921   finally show ?thesis using assms by simp
922 qed
924 lemma less_rat':
925   assumes "b \<noteq> 0"
926     and "d \<noteq> 0"
927   shows "Fract a b < Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d"
928 proof -
929   have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp
930   have "a * d * (b * d) < c * b * (b * d) \<longleftrightarrow> a * d * (sgn b * sgn d) < c * b * (sgn b * sgn d)"
931   proof (cases "b * d > 0")
932     case True
933     moreover from True have "sgn b * sgn d = 1"
934       by (simp add: sgn_times [symmetric] sgn_1_pos)
935     ultimately show ?thesis by (simp add: mult_less_cancel_right)
936   next
937     case False with assms have "b * d < 0" by (simp add: less_le)
938     moreover from this have "sgn b * sgn d = - 1"
939       by (simp only: sgn_times [symmetric] sgn_1_neg)
940     ultimately show ?thesis by (simp add: mult_less_cancel_right)
941   qed
942   also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d"
943     by (simp add: abs_sgn mult_ac)
944   finally show ?thesis using assms by simp
945 qed
947 lemma (in ordered_idom) sgn_greater [simp]:
948   "0 < sgn a \<longleftrightarrow> 0 < a"
949   unfolding sgn_if by auto
951 lemma (in ordered_idom) sgn_less [simp]:
952   "sgn a < 0 \<longleftrightarrow> a < 0"
953   unfolding sgn_if by auto
955 lemma rat_le_eq_code [code]:
956   "Fract a b < Fract c d \<longleftrightarrow> (if b = 0
957        then sgn c * sgn d > 0
958      else if d = 0
959        then sgn a * sgn b < 0
960      else a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d)"
961   by (auto simp add: sgn_times mult_less_0_iff zero_less_mult_iff less_rat' eq_rat simp del: less_rat)
963 lemma rat_less_eq_code [code]:
964   "Fract a b \<le> Fract c d \<longleftrightarrow> (if b = 0
965        then sgn c * sgn d \<ge> 0
966      else if d = 0
967        then sgn a * sgn b \<le> 0
968      else a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d)"
969   by (auto simp add: sgn_times mult_le_0_iff zero_le_mult_iff le_rat' eq_rat simp del: le_rat)
970     (auto simp add: le_less not_less sgn_0_0)
973 lemma rat_plus_code [code]:
974   "Fract a b + Fract c d = (if b = 0
975      then Fract c d
976    else if d = 0
977      then Fract a b
978    else Fract_norm (a * d + c * b) (b * d))"
979   by (simp add: eq_rat, simp add: Zero_rat_def)
981 lemma rat_times_code [code]:
982   "Fract a b * Fract c d = Fract_norm (a * c) (b * d)"
983   by simp
985 lemma rat_minus_code [code]:
986   "Fract a b - Fract c d = (if b = 0
987      then Fract (- c) d
988    else if d = 0
989      then Fract a b
990    else Fract_norm (a * d - c * b) (b * d))"
991   by (simp add: eq_rat, simp add: Zero_rat_def)
993 lemma rat_inverse_code [code]:
994   "inverse (Fract a b) = (if b = 0 then Fract 1 0
995     else if a < 0 then Fract (- b) (- a)
996     else Fract b a)"
997   by (simp add: eq_rat)
999 lemma rat_divide_code [code]:
1000   "Fract a b / Fract c d = Fract_norm (a * d) (b * c)"
1001   by simp
1003 hide (open) const Fract_norm
1005 text {* Setup for SML code generator *}
1007 types_code
1008   rat ("(int */ int)")
1009 attach (term_of) {*
1010 fun term_of_rat (p, q) =
1011   let
1012     val rT = Type ("Rational.rat", [])
1013   in
1014     if q = 1 orelse p = 0 then HOLogic.mk_number rT p
1015     else @{term "op / \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat"} \$
1016       HOLogic.mk_number rT p \$ HOLogic.mk_number rT q
1017   end;
1018 *}
1019 attach (test) {*
1020 fun gen_rat i =
1021   let
1022     val p = random_range 0 i;
1023     val q = random_range 1 (i + 1);
1024     val g = Integer.gcd p q;
1025     val p' = p div g;
1026     val q' = q div g;
1027     val r = (if one_of [true, false] then p' else ~ p',
1028       if p' = 0 then 0 else q')
1029   in
1030     (r, fn () => term_of_rat r)
1031   end;
1032 *}
1034 consts_code
1035   Fract ("(_,/ _)")
1037 consts_code
1038   "of_int :: int \<Rightarrow> rat" ("\<module>rat'_of'_int")
1039 attach {*
1040 fun rat_of_int 0 = (0, 0)
1041   | rat_of_int i = (i, 1);
1042 *}
1044 end