library theories for debugging and parallel computing using code generation towards Isabelle/ML
2 Author: Markus Wenzel, TU Muenchen
5 header {* Rational numbers *}
8 imports GCD Archimedean_Field
9 uses ("Tools/float_syntax.ML")
12 subsection {* Rational numbers as quotient *}
14 subsubsection {* Construction of the type of rational numbers *}
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
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)"
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"
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"
77 obtain a b :: int where "q = Fract a b" and "b \<noteq> 0"
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")
92 moreover from True have "?b > 0" by (simp add: nonneg1_imp_zdiv_pos_iff)
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
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)"
108 using assms by (cases q) simp
110 instantiation rat :: field_inverse_zero
113 lift_definition zero_rat :: "rat" is "(0, 1)"
116 lift_definition one_rat :: "rat" is "(1, 1)"
119 lemma Zero_rat_def: "0 = Fract 0 1"
122 lemma One_rat_def: "1 = Fract 1 1"
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)"
137 lemma minus_rat [simp]: "- Fract a b = Fract (- a) b"
140 lemma minus_rat_cancel [simp]: "Fract (- a) (- b) = Fract a b"
141 by (cases "b = 0") (simp_all add: eq_rat)
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)"
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"
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)
178 show "(q * r) * s = q * (r * s)"
184 show "(q + r) + s = q + (r + s)"
185 by transfer (simp add: algebra_simps)
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"
198 { assume "q \<noteq> 0" thus "inverse q * q = 1"
200 show "q / r = q * inverse r"
201 by (fact divide_rat_def)
202 show "inverse 0 = (0::rat)"
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:
223 "Fract (numeral w) 1 = numeral w"
224 "Fract (neg_numeral w) 1 = neg_numeral w"
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:
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"
241 proof (cases "q = 0")
242 case True then show C using 0 by auto
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
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)
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])
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"
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"
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)
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)
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
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
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
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)
309 Decompose a fraction into normalized, i.e. coprime numerator and denominator:
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)"
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)
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)
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)
341 with p show "p = (a, b)" by simp
345 lemma quotient_of_Fract [code]:
346 "quotient_of (Fract a b) = normalize (a, b)"
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)
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"
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)
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 lift_definition positive :: "rat \<Rightarrow> bool"
404 is "\<lambda>x. 0 < fst x * snd x"
407 assume "b \<noteq> 0" and "d \<noteq> 0" and "a * d = c * b"
408 hence "a * d * b * d = c * b * b * d"
410 hence "a * b * d\<twosuperior> = c * d * b\<twosuperior>"
411 unfolding power2_eq_square by (simp add: mult_ac)
412 hence "0 < a * b * d\<twosuperior> \<longleftrightarrow> 0 < c * d * b\<twosuperior>"
414 thus "0 < a * b \<longleftrightarrow> 0 < c * d"
415 using `b \<noteq> 0` and `d \<noteq> 0`
416 by (simp add: zero_less_mult_iff)
419 lemma positive_zero: "\<not> positive 0"
423 "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x + y)"
425 apply (simp add: zero_less_mult_iff)
426 apply (elim disjE, simp_all add: add_pos_pos add_neg_neg
427 mult_pos_pos mult_pos_neg mult_neg_pos mult_neg_neg)
431 "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x * y)"
432 by transfer (drule (1) mult_pos_pos, simp add: mult_ac)
434 lemma positive_minus:
435 "\<not> positive x \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> positive (- x)"
436 by transfer (force simp: neq_iff zero_less_mult_iff mult_less_0_iff)
438 instantiation rat :: linordered_field_inverse_zero
442 "x < y \<longleftrightarrow> positive (y - x)"
445 "x \<le> (y::rat) \<longleftrightarrow> x < y \<or> x = y"
448 "abs (a::rat) = (if a < 0 then - a else a)"
451 "sgn (a::rat) = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
455 show "\<bar>a\<bar> = (if a < 0 then - a else a)"
456 by (rule abs_rat_def)
457 show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a"
458 unfolding less_eq_rat_def less_rat_def
459 by (auto, drule (1) positive_add, simp_all add: positive_zero)
461 unfolding less_eq_rat_def by simp
462 show "a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
463 unfolding less_eq_rat_def less_rat_def
464 by (auto, drule (1) positive_add, simp add: algebra_simps)
465 show "a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> a = b"
466 unfolding less_eq_rat_def less_rat_def
467 by (auto, drule (1) positive_add, simp add: positive_zero)
468 show "a \<le> b \<Longrightarrow> c + a \<le> c + b"
469 unfolding less_eq_rat_def less_rat_def by (auto simp: diff_minus)
470 show "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
471 by (rule sgn_rat_def)
472 show "a \<le> b \<or> b \<le> a"
473 unfolding less_eq_rat_def less_rat_def
474 by (auto dest!: positive_minus)
475 show "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
476 unfolding less_rat_def
477 by (drule (1) positive_mult, simp add: algebra_simps)
482 instantiation rat :: distrib_lattice
486 "(inf :: rat \<Rightarrow> rat \<Rightarrow> rat) = min"
489 "(sup :: rat \<Rightarrow> rat \<Rightarrow> rat) = max"
492 qed (auto simp add: inf_rat_def sup_rat_def min_max.sup_inf_distrib1)
496 lemma positive_rat: "positive (Fract a b) \<longleftrightarrow> 0 < a * b"
499 lemma less_rat [simp]:
500 assumes "b \<noteq> 0" and "d \<noteq> 0"
501 shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
502 using assms unfolding less_rat_def
503 by (simp add: positive_rat algebra_simps)
506 assumes "b \<noteq> 0" and "d \<noteq> 0"
507 shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
508 using assms unfolding le_less by (simp add: eq_rat)
510 lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
511 by (auto simp add: abs_rat_def zabs_def Zero_rat_def not_less le_less eq_rat zero_less_mult_iff)
513 lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)"
514 unfolding Fract_of_int_eq
515 by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat)
516 (auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)
518 lemma Rat_induct_pos [case_names Fract, induct type: rat]:
519 assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
522 have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)"
524 fix a::int and b::int
526 hence "0 < -b" by simp
527 hence "P (Fract (-a) (-b))" by (rule step)
528 thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
531 thus "P q" by (force simp add: linorder_neq_iff step step')
534 lemma zero_less_Fract_iff:
535 "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a"
536 by (simp add: Zero_rat_def zero_less_mult_iff)
538 lemma Fract_less_zero_iff:
539 "0 < b \<Longrightarrow> Fract a b < 0 \<longleftrightarrow> a < 0"
540 by (simp add: Zero_rat_def mult_less_0_iff)
542 lemma zero_le_Fract_iff:
543 "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a"
544 by (simp add: Zero_rat_def zero_le_mult_iff)
546 lemma Fract_le_zero_iff:
547 "0 < b \<Longrightarrow> Fract a b \<le> 0 \<longleftrightarrow> a \<le> 0"
548 by (simp add: Zero_rat_def mult_le_0_iff)
550 lemma one_less_Fract_iff:
551 "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a"
552 by (simp add: One_rat_def mult_less_cancel_right_disj)
554 lemma Fract_less_one_iff:
555 "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b"
556 by (simp add: One_rat_def mult_less_cancel_right_disj)
558 lemma one_le_Fract_iff:
559 "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a"
560 by (simp add: One_rat_def mult_le_cancel_right)
562 lemma Fract_le_one_iff:
563 "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b"
564 by (simp add: One_rat_def mult_le_cancel_right)
567 subsubsection {* Rationals are an Archimedean field *}
569 lemma rat_floor_lemma:
570 shows "of_int (a div b) \<le> Fract a b \<and> Fract a b < of_int (a div b + 1)"
572 have "Fract a b = of_int (a div b) + Fract (a mod b) b"
573 by (cases "b = 0", simp, simp add: of_int_rat)
574 moreover have "0 \<le> Fract (a mod b) b \<and> Fract (a mod b) b < 1"
575 unfolding Fract_of_int_quotient
576 by (rule linorder_cases [of b 0]) (simp add: divide_nonpos_neg, simp, simp add: divide_nonneg_pos)
577 ultimately show ?thesis by simp
580 instance rat :: archimedean_field
583 show "\<exists>z. r \<le> of_int z"
586 have "Fract a b \<le> of_int (a div b + 1)"
587 using rat_floor_lemma [of a b] by simp
588 then show "\<exists>z. Fract a b \<le> of_int z" ..
592 instantiation rat :: floor_ceiling
595 definition [code del]:
596 "floor (x::rat) = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))"
600 show "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
601 unfolding floor_rat_def using floor_exists1 by (rule theI')
606 lemma floor_Fract: "floor (Fract a b) = a div b"
607 using rat_floor_lemma [of a b]
608 by (simp add: floor_unique)
611 subsection {* Linear arithmetic setup *}
614 K (Lin_Arith.add_inj_thms [@{thm of_nat_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
615 (* not needed because x < (y::nat) can be rewritten as Suc x <= y: of_nat_less_iff RS iffD2 *)
616 #> Lin_Arith.add_inj_thms [@{thm of_int_le_iff} RS iffD2, @{thm of_int_eq_iff} RS iffD2]
617 (* not needed because x < (y::int) can be rewritten as x + 1 <= y: of_int_less_iff RS iffD2 *)
618 #> Lin_Arith.add_simps [@{thm neg_less_iff_less},
619 @{thm True_implies_equals},
620 read_instantiate @{context} [(("a", 0), "(numeral ?v)")] @{thm right_distrib},
621 read_instantiate @{context} [(("a", 0), "(neg_numeral ?v)")] @{thm right_distrib},
622 @{thm divide_1}, @{thm divide_zero_left},
623 @{thm times_divide_eq_right}, @{thm times_divide_eq_left},
624 @{thm minus_divide_left} RS sym, @{thm minus_divide_right} RS sym,
625 @{thm of_int_minus}, @{thm of_int_diff},
626 @{thm of_int_of_nat_eq}]
627 #> Lin_Arith.add_simprocs Numeral_Simprocs.field_cancel_numeral_factors
628 #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat => rat"})
629 #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int => rat"}))
633 subsection {* Embedding from Rationals to other Fields *}
635 class field_char_0 = field + ring_char_0
637 subclass (in linordered_field) field_char_0 ..
642 lift_definition of_rat :: "rat \<Rightarrow> 'a"
643 is "\<lambda>x. of_int (fst x) / of_int (snd x)"
644 apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
645 apply (simp only: of_int_mult [symmetric])
650 lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"
653 lemma of_rat_0 [simp]: "of_rat 0 = 0"
656 lemma of_rat_1 [simp]: "of_rat 1 = 1"
659 lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"
660 by transfer (simp add: add_frac_eq)
662 lemma of_rat_minus: "of_rat (- a) = - of_rat a"
665 lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"
666 by (simp only: diff_minus of_rat_add of_rat_minus)
668 lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"
670 apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)
673 lemma nonzero_of_rat_inverse:
674 "a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)"
675 apply (rule inverse_unique [symmetric])
676 apply (simp add: of_rat_mult [symmetric])
679 lemma of_rat_inverse:
680 "(of_rat (inverse a)::'a::{field_char_0, field_inverse_zero}) =
682 by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)
684 lemma nonzero_of_rat_divide:
685 "b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b"
686 by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)
689 "(of_rat (a / b)::'a::{field_char_0, field_inverse_zero})
690 = of_rat a / of_rat b"
691 by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)
694 "(of_rat (a ^ n)::'a::field_char_0) = of_rat a ^ n"
695 by (induct n) (simp_all add: of_rat_mult)
697 lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"
699 apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
700 apply (simp only: of_int_mult [symmetric] of_int_eq_iff)
704 "(of_rat r :: 'a::linordered_field) < of_rat s \<longleftrightarrow> r < s"
705 proof (induct r, induct s)
707 assume not_zero: "b > 0" "d > 0"
708 then have "b * d > 0" by (rule mult_pos_pos)
709 have of_int_divide_less_eq:
710 "(of_int a :: 'a) / of_int b < of_int c / of_int d
711 \<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b"
712 using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)
713 show "(of_rat (Fract a b) :: 'a::linordered_field) < of_rat (Fract c d)
714 \<longleftrightarrow> Fract a b < Fract c d"
715 using not_zero `b * d > 0`
716 by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult)
719 lemma of_rat_less_eq:
720 "(of_rat r :: 'a::linordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s"
721 unfolding le_less by (auto simp add: of_rat_less)
723 lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]
725 lemma of_rat_eq_id [simp]: "of_rat = id"
728 show "of_rat a = id a"
730 (simp add: of_rat_rat Fract_of_int_eq [symmetric])
733 text{*Collapse nested embeddings*}
734 lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
735 by (induct n) (simp_all add: of_rat_add)
737 lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"
738 by (cases z rule: int_diff_cases) (simp add: of_rat_diff)
740 lemma of_rat_numeral_eq [simp]:
741 "of_rat (numeral w) = numeral w"
742 using of_rat_of_int_eq [of "numeral w"] by simp
744 lemma of_rat_neg_numeral_eq [simp]:
745 "of_rat (neg_numeral w) = neg_numeral w"
746 using of_rat_of_int_eq [of "neg_numeral w"] by simp
748 lemmas zero_rat = Zero_rat_def
749 lemmas one_rat = One_rat_def
752 rat_of_nat :: "nat \<Rightarrow> rat"
754 "rat_of_nat \<equiv> of_nat"
757 rat_of_int :: "int \<Rightarrow> rat"
759 "rat_of_int \<equiv> of_int"
761 subsection {* The Set of Rational Numbers *}
767 Rats :: "'a set" where
768 "Rats = range of_rat"
775 lemma Rats_of_rat [simp]: "of_rat r \<in> Rats"
776 by (simp add: Rats_def)
778 lemma Rats_of_int [simp]: "of_int z \<in> Rats"
779 by (subst of_rat_of_int_eq [symmetric], rule Rats_of_rat)
781 lemma Rats_of_nat [simp]: "of_nat n \<in> Rats"
782 by (subst of_rat_of_nat_eq [symmetric], rule Rats_of_rat)
784 lemma Rats_number_of [simp]: "numeral w \<in> Rats"
785 by (subst of_rat_numeral_eq [symmetric], rule Rats_of_rat)
787 lemma Rats_neg_number_of [simp]: "neg_numeral w \<in> Rats"
788 by (subst of_rat_neg_numeral_eq [symmetric], rule Rats_of_rat)
790 lemma Rats_0 [simp]: "0 \<in> Rats"
791 apply (unfold Rats_def)
792 apply (rule range_eqI)
793 apply (rule of_rat_0 [symmetric])
796 lemma Rats_1 [simp]: "1 \<in> Rats"
797 apply (unfold Rats_def)
798 apply (rule range_eqI)
799 apply (rule of_rat_1 [symmetric])
802 lemma Rats_add [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a + b \<in> Rats"
803 apply (auto simp add: Rats_def)
804 apply (rule range_eqI)
805 apply (rule of_rat_add [symmetric])
808 lemma Rats_minus [simp]: "a \<in> Rats \<Longrightarrow> - a \<in> Rats"
809 apply (auto simp add: Rats_def)
810 apply (rule range_eqI)
811 apply (rule of_rat_minus [symmetric])
814 lemma Rats_diff [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a - b \<in> Rats"
815 apply (auto simp add: Rats_def)
816 apply (rule range_eqI)
817 apply (rule of_rat_diff [symmetric])
820 lemma Rats_mult [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a * b \<in> Rats"
821 apply (auto simp add: Rats_def)
822 apply (rule range_eqI)
823 apply (rule of_rat_mult [symmetric])
826 lemma nonzero_Rats_inverse:
827 fixes a :: "'a::field_char_0"
828 shows "\<lbrakk>a \<in> Rats; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Rats"
829 apply (auto simp add: Rats_def)
830 apply (rule range_eqI)
831 apply (erule nonzero_of_rat_inverse [symmetric])
834 lemma Rats_inverse [simp]:
835 fixes a :: "'a::{field_char_0, field_inverse_zero}"
836 shows "a \<in> Rats \<Longrightarrow> inverse a \<in> Rats"
837 apply (auto simp add: Rats_def)
838 apply (rule range_eqI)
839 apply (rule of_rat_inverse [symmetric])
842 lemma nonzero_Rats_divide:
843 fixes a b :: "'a::field_char_0"
844 shows "\<lbrakk>a \<in> Rats; b \<in> Rats; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
845 apply (auto simp add: Rats_def)
846 apply (rule range_eqI)
847 apply (erule nonzero_of_rat_divide [symmetric])
850 lemma Rats_divide [simp]:
851 fixes a b :: "'a::{field_char_0, field_inverse_zero}"
852 shows "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a / b \<in> Rats"
853 apply (auto simp add: Rats_def)
854 apply (rule range_eqI)
855 apply (rule of_rat_divide [symmetric])
858 lemma Rats_power [simp]:
859 fixes a :: "'a::field_char_0"
860 shows "a \<in> Rats \<Longrightarrow> a ^ n \<in> Rats"
861 apply (auto simp add: Rats_def)
862 apply (rule range_eqI)
863 apply (rule of_rat_power [symmetric])
866 lemma Rats_cases [cases set: Rats]:
867 assumes "q \<in> \<rat>"
868 obtains (of_rat) r where "q = of_rat r"
870 from `q \<in> \<rat>` have "q \<in> range of_rat" unfolding Rats_def .
871 then obtain r where "q = of_rat r" ..
875 lemma Rats_induct [case_names of_rat, induct set: Rats]:
876 "q \<in> \<rat> \<Longrightarrow> (\<And>r. P (of_rat r)) \<Longrightarrow> P q"
877 by (rule Rats_cases) auto
880 subsection {* Implementation of rational numbers as pairs of integers *}
882 text {* Formal constructor *}
884 definition Frct :: "int \<times> int \<Rightarrow> rat" where
885 [simp]: "Frct p = Fract (fst p) (snd p)"
887 lemma [code abstype]:
888 "Frct (quotient_of q) = q"
889 by (cases q) (auto intro: quotient_of_eq)
894 declare quotient_of_Fract [code abstract]
896 definition of_int :: "int \<Rightarrow> rat"
898 [code_abbrev]: "of_int = Int.of_int"
899 hide_const (open) of_int
901 lemma quotient_of_int [code abstract]:
902 "quotient_of (Rat.of_int a) = (a, 1)"
903 by (simp add: of_int_def of_int_rat quotient_of_Fract)
906 "numeral k = Rat.of_int (numeral k)"
907 by (simp add: Rat.of_int_def)
910 "neg_numeral k = Rat.of_int (neg_numeral k)"
911 by (simp add: Rat.of_int_def)
913 lemma Frct_code_post [code_post]:
917 "Frct (numeral k, 1) = numeral k"
918 "Frct (neg_numeral k, 1) = neg_numeral k"
919 "Frct (1, numeral k) = 1 / numeral k"
920 "Frct (1, neg_numeral k) = 1 / neg_numeral k"
921 "Frct (numeral k, numeral l) = numeral k / numeral l"
922 "Frct (numeral k, neg_numeral l) = numeral k / neg_numeral l"
923 "Frct (neg_numeral k, numeral l) = neg_numeral k / numeral l"
924 "Frct (neg_numeral k, neg_numeral l) = neg_numeral k / neg_numeral l"
925 by (simp_all add: Fract_of_int_quotient)
928 text {* Operations *}
930 lemma rat_zero_code [code abstract]:
931 "quotient_of 0 = (0, 1)"
932 by (simp add: Zero_rat_def quotient_of_Fract normalize_def)
934 lemma rat_one_code [code abstract]:
935 "quotient_of 1 = (1, 1)"
936 by (simp add: One_rat_def quotient_of_Fract normalize_def)
938 lemma rat_plus_code [code abstract]:
939 "quotient_of (p + q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
940 in normalize (a * d + b * c, c * d))"
941 by (cases p, cases q) (simp add: quotient_of_Fract)
943 lemma rat_uminus_code [code abstract]:
944 "quotient_of (- p) = (let (a, b) = quotient_of p in (- a, b))"
945 by (cases p) (simp add: quotient_of_Fract)
947 lemma rat_minus_code [code abstract]:
948 "quotient_of (p - q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
949 in normalize (a * d - b * c, c * d))"
950 by (cases p, cases q) (simp add: quotient_of_Fract)
952 lemma rat_times_code [code abstract]:
953 "quotient_of (p * q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
954 in normalize (a * b, c * d))"
955 by (cases p, cases q) (simp add: quotient_of_Fract)
957 lemma rat_inverse_code [code abstract]:
958 "quotient_of (inverse p) = (let (a, b) = quotient_of p
959 in if a = 0 then (0, 1) else (sgn a * b, \<bar>a\<bar>))"
961 case (Fract a b) then show ?thesis
962 by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract gcd_int.commute)
965 lemma rat_divide_code [code abstract]:
966 "quotient_of (p / q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
967 in normalize (a * d, c * b))"
968 by (cases p, cases q) (simp add: quotient_of_Fract)
970 lemma rat_abs_code [code abstract]:
971 "quotient_of \<bar>p\<bar> = (let (a, b) = quotient_of p in (\<bar>a\<bar>, b))"
972 by (cases p) (simp add: quotient_of_Fract)
974 lemma rat_sgn_code [code abstract]:
975 "quotient_of (sgn p) = (sgn (fst (quotient_of p)), 1)"
977 case (Fract a b) then show ?thesis
978 by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract)
981 lemma rat_floor_code [code]:
982 "floor p = (let (a, b) = quotient_of p in a div b)"
983 by (cases p) (simp add: quotient_of_Fract floor_Fract)
985 instantiation rat :: equal
989 "HOL.equal a b \<longleftrightarrow> quotient_of a = quotient_of b"
992 qed (simp add: equal_rat_def quotient_of_inject_eq)
994 lemma rat_eq_refl [code nbe]:
995 "HOL.equal (r::rat) r \<longleftrightarrow> True"
1000 lemma rat_less_eq_code [code]:
1001 "p \<le> q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d \<le> c * b)"
1002 by (cases p, cases q) (simp add: quotient_of_Fract mult.commute)
1004 lemma rat_less_code [code]:
1005 "p < q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d < c * b)"
1006 by (cases p, cases q) (simp add: quotient_of_Fract mult.commute)
1009 "of_rat p = (let (a, b) = quotient_of p in of_int a / of_int b)"
1010 by (cases p) (simp add: quotient_of_Fract of_rat_rat)
1013 text {* Quickcheck *}
1015 definition (in term_syntax)
1016 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
1017 [code_unfold]: "valterm_fract k l = Code_Evaluation.valtermify Fract {\<cdot>} k {\<cdot>} l"
1019 notation fcomp (infixl "\<circ>>" 60)
1020 notation scomp (infixl "\<circ>\<rightarrow>" 60)
1022 instantiation rat :: random
1026 "Quickcheck.random i = Quickcheck.random i \<circ>\<rightarrow> (\<lambda>num. Random.range i \<circ>\<rightarrow> (\<lambda>denom. Pair (
1027 let j = Code_Numeral.int_of (denom + 1)
1028 in valterm_fract num (j, \<lambda>u. Code_Evaluation.term_of j))))"
1034 no_notation fcomp (infixl "\<circ>>" 60)
1035 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
1037 instantiation rat :: exhaustive
1041 "exhaustive_rat f d = Quickcheck_Exhaustive.exhaustive (%l. Quickcheck_Exhaustive.exhaustive (%k. f (Fract k (Code_Numeral.int_of l + 1))) d) d"
1047 instantiation rat :: full_exhaustive
1051 "full_exhaustive_rat f d = Quickcheck_Exhaustive.full_exhaustive (%(l, _). Quickcheck_Exhaustive.full_exhaustive (%k.
1052 f (let j = Code_Numeral.int_of l + 1
1053 in valterm_fract k (j, %_. Code_Evaluation.term_of j))) d) d"
1059 instantiation rat :: partial_term_of
1067 "partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Narrowing_variable p tt) == Code_Evaluation.Free (STR ''_'') (Typerep.Typerep (STR ''Rat.rat'') [])"
1068 "partial_term_of (ty :: rat itself) (Quickcheck_Narrowing.Narrowing_constructor 0 [l, k]) ==
1069 Code_Evaluation.App (Code_Evaluation.Const (STR ''Rat.Frct'')
1070 (Typerep.Typerep (STR ''fun'') [Typerep.Typerep (STR ''Product_Type.prod'') [Typerep.Typerep (STR ''Int.int'') [], Typerep.Typerep (STR ''Int.int'') []],
1071 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))"
1072 by (rule partial_term_of_anything)+
1074 instantiation rat :: narrowing
1078 "narrowing = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.apply
1079 (Quickcheck_Narrowing.cons (%nom denom. Fract nom denom)) narrowing) narrowing"
1086 subsection {* Setup for Nitpick *}
1089 Nitpick_HOL.register_frac_type @{type_name rat}
1090 [(@{const_name zero_rat_inst.zero_rat}, @{const_name Nitpick.zero_frac}),
1091 (@{const_name one_rat_inst.one_rat}, @{const_name Nitpick.one_frac}),
1092 (@{const_name plus_rat_inst.plus_rat}, @{const_name Nitpick.plus_frac}),
1093 (@{const_name times_rat_inst.times_rat}, @{const_name Nitpick.times_frac}),
1094 (@{const_name uminus_rat_inst.uminus_rat}, @{const_name Nitpick.uminus_frac}),
1095 (@{const_name inverse_rat_inst.inverse_rat}, @{const_name Nitpick.inverse_frac}),
1096 (@{const_name ord_rat_inst.less_rat}, @{const_name Nitpick.less_frac}),
1097 (@{const_name ord_rat_inst.less_eq_rat}, @{const_name Nitpick.less_eq_frac}),
1098 (@{const_name field_char_0_class.of_rat}, @{const_name Nitpick.of_frac})]
1101 lemmas [nitpick_unfold] = inverse_rat_inst.inverse_rat
1102 one_rat_inst.one_rat ord_rat_inst.less_rat
1103 ord_rat_inst.less_eq_rat plus_rat_inst.plus_rat times_rat_inst.times_rat
1104 uminus_rat_inst.uminus_rat zero_rat_inst.zero_rat
1106 subsection{* Float syntax *}
1108 syntax "_Float" :: "float_const \<Rightarrow> 'a" ("_")
1110 use "Tools/float_syntax.ML"
1111 setup Float_Syntax.setup
1114 lemma "123.456 = -111.111 + 200 + 30 + 4 + 5/10 + 6/100 + (7/1000::rat)"
1118 hide_const (open) normalize positive
1120 lemmas [transfer_rule del] =
1121 rat.All_transfer rat.Ex_transfer rat.rel_eq_transfer forall_rat_transfer
1122 Fract.transfer zero_rat.transfer one_rat.transfer plus_rat.transfer
1123 uminus_rat.transfer times_rat.transfer inverse_rat.transfer
1124 positive.transfer of_rat.transfer
1126 text {* De-register @{text "rat"} as a quotient type: *}
1128 declare Quotient_rat[quot_del]