2 Author: Jacques D. Fleuriot, University of Edinburgh, 1998
3 Author: Larry Paulson, University of Cambridge
4 Author: Jeremy Avigad, Carnegie Mellon University
5 Author: Florian Zuleger, Johannes Hoelzl, and Simon Funke, TU Muenchen
6 Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
7 Construction of Cauchy Reals by Brian Huffman, 2010
10 header {* Development of the Reals using Cauchy Sequences *}
13 imports Rat Conditionally_Complete_Lattices
17 This theory contains a formalization of the real numbers as
18 equivalence classes of Cauchy sequences of rationals. See
19 @{file "~~/src/HOL/ex/Dedekind_Real.thy"} for an alternative
20 construction using Dedekind cuts.
23 subsection {* Preliminary lemmas *}
26 fixes a b c d :: "'a::ab_group_add"
27 shows "(a + c) - (b + d) = (a - b) + (c - d)"
30 lemma minus_diff_minus:
31 fixes a b :: "'a::ab_group_add"
32 shows "- a - - b = - (a - b)"
36 fixes x y a b :: "'a::ring"
37 shows "(x * y - a * b) = x * (y - b) + (x - a) * b"
38 by (simp add: algebra_simps)
40 lemma inverse_diff_inverse:
41 fixes a b :: "'a::division_ring"
42 assumes "a \<noteq> 0" and "b \<noteq> 0"
43 shows "inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
44 using assms by (simp add: algebra_simps)
47 fixes r :: rat assumes r: "0 < r"
48 obtains s t where "0 < s" and "0 < t" and "r = s + t"
50 from r show "0 < r/2" by simp
51 from r show "0 < r/2" by simp
52 show "r = r/2 + r/2" by simp
55 subsection {* Sequences that converge to zero *}
58 vanishes :: "(nat \<Rightarrow> rat) \<Rightarrow> bool"
60 "vanishes X = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r)"
62 lemma vanishesI: "(\<And>r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r) \<Longrightarrow> vanishes X"
63 unfolding vanishes_def by simp
65 lemma vanishesD: "\<lbrakk>vanishes X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r"
66 unfolding vanishes_def by simp
68 lemma vanishes_const [simp]: "vanishes (\<lambda>n. c) \<longleftrightarrow> c = 0"
69 unfolding vanishes_def
70 apply (cases "c = 0", auto)
71 apply (rule exI [where x="\<bar>c\<bar>"], auto)
74 lemma vanishes_minus: "vanishes X \<Longrightarrow> vanishes (\<lambda>n. - X n)"
75 unfolding vanishes_def by simp
78 assumes X: "vanishes X" and Y: "vanishes Y"
79 shows "vanishes (\<lambda>n. X n + Y n)"
80 proof (rule vanishesI)
81 fix r :: rat assume "0 < r"
82 then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
83 by (rule obtain_pos_sum)
84 obtain i where i: "\<forall>n\<ge>i. \<bar>X n\<bar> < s"
85 using vanishesD [OF X s] ..
86 obtain j where j: "\<forall>n\<ge>j. \<bar>Y n\<bar> < t"
87 using vanishesD [OF Y t] ..
88 have "\<forall>n\<ge>max i j. \<bar>X n + Y n\<bar> < r"
90 fix n assume n: "i \<le> n" "j \<le> n"
91 have "\<bar>X n + Y n\<bar> \<le> \<bar>X n\<bar> + \<bar>Y n\<bar>" by (rule abs_triangle_ineq)
92 also have "\<dots> < s + t" by (simp add: add_strict_mono i j n)
93 finally show "\<bar>X n + Y n\<bar> < r" unfolding r .
95 thus "\<exists>k. \<forall>n\<ge>k. \<bar>X n + Y n\<bar> < r" ..
99 assumes X: "vanishes X" and Y: "vanishes Y"
100 shows "vanishes (\<lambda>n. X n - Y n)"
101 unfolding diff_conv_add_uminus by (intro vanishes_add vanishes_minus X Y)
103 lemma vanishes_mult_bounded:
104 assumes X: "\<exists>a>0. \<forall>n. \<bar>X n\<bar> < a"
105 assumes Y: "vanishes (\<lambda>n. Y n)"
106 shows "vanishes (\<lambda>n. X n * Y n)"
107 proof (rule vanishesI)
108 fix r :: rat assume r: "0 < r"
109 obtain a where a: "0 < a" "\<forall>n. \<bar>X n\<bar> < a"
111 obtain b where b: "0 < b" "r = a * b"
113 show "0 < r / a" using r a by (simp add: divide_pos_pos)
114 show "r = a * (r / a)" using a by simp
116 obtain k where k: "\<forall>n\<ge>k. \<bar>Y n\<bar> < b"
117 using vanishesD [OF Y b(1)] ..
118 have "\<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r"
119 by (simp add: b(2) abs_mult mult_strict_mono' a k)
120 thus "\<exists>k. \<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r" ..
123 subsection {* Cauchy sequences *}
126 cauchy :: "(nat \<Rightarrow> rat) \<Rightarrow> bool"
128 "cauchy X \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r)"
131 "(\<And>r. 0 < r \<Longrightarrow> \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r) \<Longrightarrow> cauchy X"
132 unfolding cauchy_def by simp
135 "\<lbrakk>cauchy X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r"
136 unfolding cauchy_def by simp
138 lemma cauchy_const [simp]: "cauchy (\<lambda>n. x)"
139 unfolding cauchy_def by simp
141 lemma cauchy_add [simp]:
142 assumes X: "cauchy X" and Y: "cauchy Y"
143 shows "cauchy (\<lambda>n. X n + Y n)"
145 fix r :: rat assume "0 < r"
146 then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
147 by (rule obtain_pos_sum)
148 obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"
149 using cauchyD [OF X s] ..
150 obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>Y m - Y n\<bar> < t"
151 using cauchyD [OF Y t] ..
152 have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>(X m + Y m) - (X n + Y n)\<bar> < r"
154 fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"
155 have "\<bar>(X m + Y m) - (X n + Y n)\<bar> \<le> \<bar>X m - X n\<bar> + \<bar>Y m - Y n\<bar>"
156 unfolding add_diff_add by (rule abs_triangle_ineq)
157 also have "\<dots> < s + t"
158 by (rule add_strict_mono, simp_all add: i j *)
159 finally show "\<bar>(X m + Y m) - (X n + Y n)\<bar> < r" unfolding r .
161 thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>(X m + Y m) - (X n + Y n)\<bar> < r" ..
164 lemma cauchy_minus [simp]:
165 assumes X: "cauchy X"
166 shows "cauchy (\<lambda>n. - X n)"
167 using assms unfolding cauchy_def
168 unfolding minus_diff_minus abs_minus_cancel .
170 lemma cauchy_diff [simp]:
171 assumes X: "cauchy X" and Y: "cauchy Y"
172 shows "cauchy (\<lambda>n. X n - Y n)"
173 using assms unfolding diff_conv_add_uminus by (simp del: add_uminus_conv_diff)
175 lemma cauchy_imp_bounded:
176 assumes "cauchy X" shows "\<exists>b>0. \<forall>n. \<bar>X n\<bar> < b"
178 obtain k where k: "\<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < 1"
179 using cauchyD [OF assms zero_less_one] ..
180 show "\<exists>b>0. \<forall>n. \<bar>X n\<bar> < b"
181 proof (intro exI conjI allI)
182 have "0 \<le> \<bar>X 0\<bar>" by simp
183 also have "\<bar>X 0\<bar> \<le> Max (abs ` X ` {..k})" by simp
184 finally have "0 \<le> Max (abs ` X ` {..k})" .
185 thus "0 < Max (abs ` X ` {..k}) + 1" by simp
188 show "\<bar>X n\<bar> < Max (abs ` X ` {..k}) + 1"
189 proof (rule linorder_le_cases)
191 hence "\<bar>X n\<bar> \<le> Max (abs ` X ` {..k})" by simp
192 thus "\<bar>X n\<bar> < Max (abs ` X ` {..k}) + 1" by simp
195 have "\<bar>X n\<bar> = \<bar>X k + (X n - X k)\<bar>" by simp
196 also have "\<bar>X k + (X n - X k)\<bar> \<le> \<bar>X k\<bar> + \<bar>X n - X k\<bar>"
197 by (rule abs_triangle_ineq)
198 also have "\<dots> < Max (abs ` X ` {..k}) + 1"
199 by (rule add_le_less_mono, simp, simp add: k `k \<le> n`)
200 finally show "\<bar>X n\<bar> < Max (abs ` X ` {..k}) + 1" .
205 lemma cauchy_mult [simp]:
206 assumes X: "cauchy X" and Y: "cauchy Y"
207 shows "cauchy (\<lambda>n. X n * Y n)"
209 fix r :: rat assume "0 < r"
210 then obtain u v where u: "0 < u" and v: "0 < v" and "r = u + v"
211 by (rule obtain_pos_sum)
212 obtain a where a: "0 < a" "\<forall>n. \<bar>X n\<bar> < a"
213 using cauchy_imp_bounded [OF X] by fast
214 obtain b where b: "0 < b" "\<forall>n. \<bar>Y n\<bar> < b"
215 using cauchy_imp_bounded [OF Y] by fast
216 obtain s t where s: "0 < s" and t: "0 < t" and r: "r = a * t + s * b"
218 show "0 < v/b" using v b(1) by (rule divide_pos_pos)
219 show "0 < u/a" using u a(1) by (rule divide_pos_pos)
220 show "r = a * (u/a) + (v/b) * b"
221 using a(1) b(1) `r = u + v` by simp
223 obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"
224 using cauchyD [OF X s] ..
225 obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>Y m - Y n\<bar> < t"
226 using cauchyD [OF Y t] ..
227 have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>X m * Y m - X n * Y n\<bar> < r"
229 fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"
230 have "\<bar>X m * Y m - X n * Y n\<bar> = \<bar>X m * (Y m - Y n) + (X m - X n) * Y n\<bar>"
231 unfolding mult_diff_mult ..
232 also have "\<dots> \<le> \<bar>X m * (Y m - Y n)\<bar> + \<bar>(X m - X n) * Y n\<bar>"
233 by (rule abs_triangle_ineq)
234 also have "\<dots> = \<bar>X m\<bar> * \<bar>Y m - Y n\<bar> + \<bar>X m - X n\<bar> * \<bar>Y n\<bar>"
235 unfolding abs_mult ..
236 also have "\<dots> < a * t + s * b"
237 by (simp_all add: add_strict_mono mult_strict_mono' a b i j *)
238 finally show "\<bar>X m * Y m - X n * Y n\<bar> < r" unfolding r .
240 thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m * Y m - X n * Y n\<bar> < r" ..
243 lemma cauchy_not_vanishes_cases:
244 assumes X: "cauchy X"
245 assumes nz: "\<not> vanishes X"
246 shows "\<exists>b>0. \<exists>k. (\<forall>n\<ge>k. b < - X n) \<or> (\<forall>n\<ge>k. b < X n)"
248 obtain r where "0 < r" and r: "\<forall>k. \<exists>n\<ge>k. r \<le> \<bar>X n\<bar>"
249 using nz unfolding vanishes_def by (auto simp add: not_less)
250 obtain s t where s: "0 < s" and t: "0 < t" and "r = s + t"
251 using `0 < r` by (rule obtain_pos_sum)
252 obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"
253 using cauchyD [OF X s] ..
254 obtain k where "i \<le> k" and "r \<le> \<bar>X k\<bar>"
256 have k: "\<forall>n\<ge>k. \<bar>X n - X k\<bar> < s"
257 using i `i \<le> k` by auto
258 have "X k \<le> - r \<or> r \<le> X k"
259 using `r \<le> \<bar>X k\<bar>` by auto
260 hence "(\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)"
261 unfolding `r = s + t` using k by auto
262 hence "\<exists>k. (\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)" ..
263 thus "\<exists>t>0. \<exists>k. (\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)"
267 lemma cauchy_not_vanishes:
268 assumes X: "cauchy X"
269 assumes nz: "\<not> vanishes X"
270 shows "\<exists>b>0. \<exists>k. \<forall>n\<ge>k. b < \<bar>X n\<bar>"
271 using cauchy_not_vanishes_cases [OF assms]
272 by clarify (rule exI, erule conjI, rule_tac x=k in exI, auto)
274 lemma cauchy_inverse [simp]:
275 assumes X: "cauchy X"
276 assumes nz: "\<not> vanishes X"
277 shows "cauchy (\<lambda>n. inverse (X n))"
279 fix r :: rat assume "0 < r"
280 obtain b i where b: "0 < b" and i: "\<forall>n\<ge>i. b < \<bar>X n\<bar>"
281 using cauchy_not_vanishes [OF X nz] by fast
282 from b i have nz: "\<forall>n\<ge>i. X n \<noteq> 0" by auto
283 obtain s where s: "0 < s" and r: "r = inverse b * s * inverse b"
286 by (simp add: `0 < r` b mult_pos_pos)
287 show "r = inverse b * (b * r * b) * inverse b"
290 obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>X m - X n\<bar> < s"
291 using cauchyD [OF X s] ..
292 have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>inverse (X m) - inverse (X n)\<bar> < r"
294 fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"
295 have "\<bar>inverse (X m) - inverse (X n)\<bar> =
296 inverse \<bar>X m\<bar> * \<bar>X m - X n\<bar> * inverse \<bar>X n\<bar>"
297 by (simp add: inverse_diff_inverse nz * abs_mult)
298 also have "\<dots> < inverse b * s * inverse b"
299 by (simp add: mult_strict_mono less_imp_inverse_less
300 mult_pos_pos i j b * s)
301 finally show "\<bar>inverse (X m) - inverse (X n)\<bar> < r" unfolding r .
303 thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>inverse (X m) - inverse (X n)\<bar> < r" ..
306 lemma vanishes_diff_inverse:
307 assumes X: "cauchy X" "\<not> vanishes X"
308 assumes Y: "cauchy Y" "\<not> vanishes Y"
309 assumes XY: "vanishes (\<lambda>n. X n - Y n)"
310 shows "vanishes (\<lambda>n. inverse (X n) - inverse (Y n))"
311 proof (rule vanishesI)
312 fix r :: rat assume r: "0 < r"
313 obtain a i where a: "0 < a" and i: "\<forall>n\<ge>i. a < \<bar>X n\<bar>"
314 using cauchy_not_vanishes [OF X] by fast
315 obtain b j where b: "0 < b" and j: "\<forall>n\<ge>j. b < \<bar>Y n\<bar>"
316 using cauchy_not_vanishes [OF Y] by fast
317 obtain s where s: "0 < s" and "inverse a * s * inverse b = r"
320 using a r b by (simp add: mult_pos_pos)
321 show "inverse a * (a * r * b) * inverse b = r"
324 obtain k where k: "\<forall>n\<ge>k. \<bar>X n - Y n\<bar> < s"
325 using vanishesD [OF XY s] ..
326 have "\<forall>n\<ge>max (max i j) k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r"
328 fix n assume n: "i \<le> n" "j \<le> n" "k \<le> n"
329 have "X n \<noteq> 0" and "Y n \<noteq> 0"
330 using i j a b n by auto
331 hence "\<bar>inverse (X n) - inverse (Y n)\<bar> =
332 inverse \<bar>X n\<bar> * \<bar>X n - Y n\<bar> * inverse \<bar>Y n\<bar>"
333 by (simp add: inverse_diff_inverse abs_mult)
334 also have "\<dots> < inverse a * s * inverse b"
335 apply (intro mult_strict_mono' less_imp_inverse_less)
336 apply (simp_all add: a b i j k n mult_nonneg_nonneg)
338 also note `inverse a * s * inverse b = r`
339 finally show "\<bar>inverse (X n) - inverse (Y n)\<bar> < r" .
341 thus "\<exists>k. \<forall>n\<ge>k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r" ..
344 subsection {* Equivalence relation on Cauchy sequences *}
346 definition realrel :: "(nat \<Rightarrow> rat) \<Rightarrow> (nat \<Rightarrow> rat) \<Rightarrow> bool"
347 where "realrel = (\<lambda>X Y. cauchy X \<and> cauchy Y \<and> vanishes (\<lambda>n. X n - Y n))"
349 lemma realrelI [intro?]:
350 assumes "cauchy X" and "cauchy Y" and "vanishes (\<lambda>n. X n - Y n)"
352 using assms unfolding realrel_def by simp
354 lemma realrel_refl: "cauchy X \<Longrightarrow> realrel X X"
355 unfolding realrel_def by simp
357 lemma symp_realrel: "symp realrel"
358 unfolding realrel_def
359 by (rule sympI, clarify, drule vanishes_minus, simp)
361 lemma transp_realrel: "transp realrel"
362 unfolding realrel_def
363 apply (rule transpI, clarify)
364 apply (drule (1) vanishes_add)
365 apply (simp add: algebra_simps)
368 lemma part_equivp_realrel: "part_equivp realrel"
369 by (fast intro: part_equivpI symp_realrel transp_realrel
370 realrel_refl cauchy_const)
372 subsection {* The field of real numbers *}
374 quotient_type real = "nat \<Rightarrow> rat" / partial: realrel
375 morphisms rep_real Real
376 by (rule part_equivp_realrel)
378 lemma cr_real_eq: "pcr_real = (\<lambda>x y. cauchy x \<and> Real x = y)"
379 unfolding real.pcr_cr_eq cr_real_def realrel_def by auto
381 lemma Real_induct [induct type: real]: (* TODO: generate automatically *)
382 assumes "\<And>X. cauchy X \<Longrightarrow> P (Real X)" shows "P x"
385 hence "cauchy X" by (simp add: realrel_def)
386 thus "P (Real X)" by (rule assms)
390 "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> Real X = Real Y \<longleftrightarrow> vanishes (\<lambda>n. X n - Y n)"
391 using real.rel_eq_transfer
392 unfolding real.pcr_cr_eq cr_real_def fun_rel_def realrel_def by simp
394 lemma Domainp_pcr_real [transfer_domain_rule]: "Domainp pcr_real = cauchy"
395 by (simp add: real.domain_eq realrel_def)
397 instantiation real :: field_inverse_zero
400 lift_definition zero_real :: "real" is "\<lambda>n. 0"
401 by (simp add: realrel_refl)
403 lift_definition one_real :: "real" is "\<lambda>n. 1"
404 by (simp add: realrel_refl)
406 lift_definition plus_real :: "real \<Rightarrow> real \<Rightarrow> real" is "\<lambda>X Y n. X n + Y n"
407 unfolding realrel_def add_diff_add
408 by (simp only: cauchy_add vanishes_add simp_thms)
410 lift_definition uminus_real :: "real \<Rightarrow> real" is "\<lambda>X n. - X n"
411 unfolding realrel_def minus_diff_minus
412 by (simp only: cauchy_minus vanishes_minus simp_thms)
414 lift_definition times_real :: "real \<Rightarrow> real \<Rightarrow> real" is "\<lambda>X Y n. X n * Y n"
415 unfolding realrel_def mult_diff_mult
416 by (subst (4) mult_commute, simp only: cauchy_mult vanishes_add
417 vanishes_mult_bounded cauchy_imp_bounded simp_thms)
419 lift_definition inverse_real :: "real \<Rightarrow> real"
420 is "\<lambda>X. if vanishes X then (\<lambda>n. 0) else (\<lambda>n. inverse (X n))"
422 fix X Y assume "realrel X Y"
423 hence X: "cauchy X" and Y: "cauchy Y" and XY: "vanishes (\<lambda>n. X n - Y n)"
424 unfolding realrel_def by simp_all
425 have "vanishes X \<longleftrightarrow> vanishes Y"
428 from vanishes_diff [OF this XY] show "vanishes Y" by simp
431 from vanishes_add [OF this XY] show "vanishes X" by simp
434 unfolding realrel_def
435 by (simp add: vanishes_diff_inverse X Y XY)
439 "x - y = (x::real) + - y"
442 "x / y = (x::real) * inverse y"
445 assumes X: "cauchy X" and Y: "cauchy Y"
446 shows "Real X + Real Y = Real (\<lambda>n. X n + Y n)"
447 using assms plus_real.transfer
448 unfolding cr_real_eq fun_rel_def by simp
451 assumes X: "cauchy X"
452 shows "- Real X = Real (\<lambda>n. - X n)"
453 using assms uminus_real.transfer
454 unfolding cr_real_eq fun_rel_def by simp
457 assumes X: "cauchy X" and Y: "cauchy Y"
458 shows "Real X - Real Y = Real (\<lambda>n. X n - Y n)"
459 unfolding minus_real_def
460 by (simp add: minus_Real add_Real X Y)
463 assumes X: "cauchy X" and Y: "cauchy Y"
464 shows "Real X * Real Y = Real (\<lambda>n. X n * Y n)"
465 using assms times_real.transfer
466 unfolding cr_real_eq fun_rel_def by simp
469 assumes X: "cauchy X"
470 shows "inverse (Real X) =
471 (if vanishes X then 0 else Real (\<lambda>n. inverse (X n)))"
472 using assms inverse_real.transfer zero_real.transfer
473 unfolding cr_real_eq fun_rel_def by (simp split: split_if_asm, metis)
478 by transfer (simp add: add_ac realrel_def)
479 show "(a + b) + c = a + (b + c)"
480 by transfer (simp add: add_ac realrel_def)
482 by transfer (simp add: realrel_def)
484 by transfer (simp add: realrel_def)
485 show "a - b = a + - b"
486 by (rule minus_real_def)
487 show "(a * b) * c = a * (b * c)"
488 by transfer (simp add: mult_ac realrel_def)
490 by transfer (simp add: mult_ac realrel_def)
492 by transfer (simp add: mult_ac realrel_def)
493 show "(a + b) * c = a * c + b * c"
494 by transfer (simp add: distrib_right realrel_def)
495 show "(0\<Colon>real) \<noteq> (1\<Colon>real)"
496 by transfer (simp add: realrel_def)
497 show "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
499 apply (simp add: realrel_def)
500 apply (rule vanishesI)
501 apply (frule (1) cauchy_not_vanishes, clarify)
502 apply (rule_tac x=k in exI, clarify)
503 apply (drule_tac x=n in spec, simp)
505 show "a / b = a * inverse b"
506 by (rule divide_real_def)
507 show "inverse (0::real) = 0"
508 by transfer (simp add: realrel_def)
513 subsection {* Positive reals *}
515 lift_definition positive :: "real \<Rightarrow> bool"
516 is "\<lambda>X. \<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n"
520 hence XY: "vanishes (\<lambda>n. X n - Y n)"
521 unfolding realrel_def by simp_all
522 assume "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n"
523 then obtain r i where "0 < r" and i: "\<forall>n\<ge>i. r < X n"
525 obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
526 using `0 < r` by (rule obtain_pos_sum)
527 obtain j where j: "\<forall>n\<ge>j. \<bar>X n - Y n\<bar> < s"
528 using vanishesD [OF XY s] ..
529 have "\<forall>n\<ge>max i j. t < Y n"
531 fix n assume n: "i \<le> n" "j \<le> n"
532 have "\<bar>X n - Y n\<bar> < s" and "r < X n"
533 using i j n by simp_all
534 thus "t < Y n" unfolding r by simp
536 hence "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < Y n" using t by fast
538 fix X Y assume "realrel X Y"
539 hence "realrel X Y" and "realrel Y X"
540 using symp_realrel unfolding symp_def by auto
546 assumes X: "cauchy X"
547 shows "positive (Real X) \<longleftrightarrow> (\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n)"
548 using assms positive.transfer
549 unfolding cr_real_eq fun_rel_def by simp
551 lemma positive_zero: "\<not> positive 0"
555 "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x + y)"
557 apply (clarify, rename_tac a b i j)
558 apply (rule_tac x="a + b" in exI, simp)
559 apply (rule_tac x="max i j" in exI, clarsimp)
560 apply (simp add: add_strict_mono)
564 "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x * y)"
566 apply (clarify, rename_tac a b i j)
567 apply (rule_tac x="a * b" in exI, simp add: mult_pos_pos)
568 apply (rule_tac x="max i j" in exI, clarsimp)
569 apply (rule mult_strict_mono, auto)
572 lemma positive_minus:
573 "\<not> positive x \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> positive (- x)"
575 apply (simp add: realrel_def)
576 apply (drule (1) cauchy_not_vanishes_cases, safe, fast, fast)
579 instantiation real :: linordered_field_inverse_zero
583 "x < y \<longleftrightarrow> positive (y - x)"
586 "x \<le> (y::real) \<longleftrightarrow> x < y \<or> x = y"
589 "abs (a::real) = (if a < 0 then - a else a)"
592 "sgn (a::real) = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
596 show "\<bar>a\<bar> = (if a < 0 then - a else a)"
597 by (rule abs_real_def)
598 show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a"
599 unfolding less_eq_real_def less_real_def
600 by (auto, drule (1) positive_add, simp_all add: positive_zero)
602 unfolding less_eq_real_def by simp
603 show "a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
604 unfolding less_eq_real_def less_real_def
605 by (auto, drule (1) positive_add, simp add: algebra_simps)
606 show "a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> a = b"
607 unfolding less_eq_real_def less_real_def
608 by (auto, drule (1) positive_add, simp add: positive_zero)
609 show "a \<le> b \<Longrightarrow> c + a \<le> c + b"
610 unfolding less_eq_real_def less_real_def by auto
611 (* FIXME: Procedure int_combine_numerals: c + b - (c + a) \<equiv> b + - a *)
612 (* Should produce c + b - (c + a) \<equiv> b - a *)
613 show "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
614 by (rule sgn_real_def)
615 show "a \<le> b \<or> b \<le> a"
616 unfolding less_eq_real_def less_real_def
617 by (auto dest!: positive_minus)
618 show "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
619 unfolding less_real_def
620 by (drule (1) positive_mult, simp add: algebra_simps)
625 instantiation real :: distrib_lattice
629 "(inf :: real \<Rightarrow> real \<Rightarrow> real) = min"
632 "(sup :: real \<Rightarrow> real \<Rightarrow> real) = max"
635 qed (auto simp add: inf_real_def sup_real_def min_max.sup_inf_distrib1)
639 lemma of_nat_Real: "of_nat x = Real (\<lambda>n. of_nat x)"
641 apply (simp add: zero_real_def)
642 apply (simp add: one_real_def add_Real)
645 lemma of_int_Real: "of_int x = Real (\<lambda>n. of_int x)"
646 apply (cases x rule: int_diff_cases)
647 apply (simp add: of_nat_Real diff_Real)
650 lemma of_rat_Real: "of_rat x = Real (\<lambda>n. x)"
652 apply (simp add: Fract_of_int_quotient of_rat_divide)
653 apply (simp add: of_int_Real divide_inverse)
654 apply (simp add: inverse_Real mult_Real)
657 instance real :: archimedean_field
660 show "\<exists>z. x \<le> of_int z"
662 apply (frule cauchy_imp_bounded, clarify)
663 apply (rule_tac x="ceiling b + 1" in exI)
664 apply (rule less_imp_le)
665 apply (simp add: of_int_Real less_real_def diff_Real positive_Real)
666 apply (rule_tac x=1 in exI, simp add: algebra_simps)
667 apply (rule_tac x=0 in exI, clarsimp)
668 apply (rule le_less_trans [OF abs_ge_self])
669 apply (rule less_le_trans [OF _ le_of_int_ceiling])
674 instantiation real :: floor_ceiling
677 definition [code del]:
678 "floor (x::real) = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))"
682 show "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
683 unfolding floor_real_def using floor_exists1 by (rule theI')
688 subsection {* Completeness *}
690 lemma not_positive_Real:
691 assumes X: "cauchy X"
692 shows "\<not> positive (Real X) \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> r)"
693 unfolding positive_Real [OF X]
694 apply (auto, unfold not_less)
695 apply (erule obtain_pos_sum)
696 apply (drule_tac x=s in spec, simp)
697 apply (drule_tac r=t in cauchyD [OF X], clarify)
698 apply (drule_tac x=k in spec, clarsimp)
699 apply (rule_tac x=n in exI, clarify, rename_tac m)
700 apply (drule_tac x=m in spec, simp)
701 apply (drule_tac x=n in spec, simp)
702 apply (drule spec, drule (1) mp, clarify, rename_tac i)
703 apply (rule_tac x="max i k" in exI, simp)
707 assumes X: "cauchy X" and Y: "cauchy Y"
708 shows "Real X \<le> Real Y = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r)"
709 unfolding not_less [symmetric, where 'a=real] less_real_def
710 apply (simp add: diff_Real not_positive_Real X Y)
711 apply (simp add: diff_le_eq add_ac)
715 assumes Y: "cauchy Y"
716 shows "\<forall>n. x \<le> of_rat (Y n) \<Longrightarrow> x \<le> Real Y"
718 fix X assume X: "cauchy X" and "\<forall>n. Real X \<le> of_rat (Y n)"
719 hence le: "\<And>m r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. X n \<le> Y m + r"
720 by (simp add: of_rat_Real le_Real)
722 fix r :: rat assume "0 < r"
723 then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
724 by (rule obtain_pos_sum)
725 obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>Y m - Y n\<bar> < s"
726 using cauchyD [OF Y s] ..
727 obtain j where j: "\<forall>n\<ge>j. X n \<le> Y i + t"
729 have "\<forall>n\<ge>max i j. X n \<le> Y n + r"
731 fix n assume n: "i \<le> n" "j \<le> n"
732 have "X n \<le> Y i + t" using n j by simp
733 moreover have "\<bar>Y i - Y n\<bar> < s" using n i by simp
734 ultimately show "X n \<le> Y n + r" unfolding r by simp
736 hence "\<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r" ..
738 thus "Real X \<le> Real Y"
739 by (simp add: of_rat_Real le_Real X Y)
743 assumes X: "cauchy X"
744 assumes le: "\<forall>n. of_rat (X n) \<le> y"
745 shows "Real X \<le> y"
747 have "- y \<le> - Real X"
748 by (simp add: minus_Real X le_RealI of_rat_minus le)
753 assumes Y: "cauchy Y"
754 shows "x < Real Y \<Longrightarrow> \<exists>n. x < of_rat (Y n)"
755 by (erule contrapos_pp, simp add: not_less, erule Real_leI [OF Y])
757 lemma of_nat_less_two_power:
758 "of_nat n < (2::'a::linordered_idom) ^ n"
761 apply (subgoal_tac "(1::'a) \<le> 2 ^ n")
762 apply (drule (1) add_le_less_mono, simp)
767 fixes S :: "real set"
768 assumes "\<exists>x. x \<in> S" and "\<exists>z. \<forall>x\<in>S. x \<le> z"
769 shows "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)"
771 obtain x where x: "x \<in> S" using assms(1) ..
772 obtain z where z: "\<forall>x\<in>S. x \<le> z" using assms(2) ..
774 def P \<equiv> "\<lambda>x. \<forall>y\<in>S. y \<le> of_rat x"
775 obtain a where a: "\<not> P a"
777 have "of_int (floor (x - 1)) \<le> x - 1" by (rule of_int_floor_le)
778 also have "x - 1 < x" by simp
779 finally have "of_int (floor (x - 1)) < x" .
780 hence "\<not> x \<le> of_int (floor (x - 1))" by (simp only: not_le)
781 then show "\<not> P (of_int (floor (x - 1)))"
782 unfolding P_def of_rat_of_int_eq using x by fast
784 obtain b where b: "P b"
786 show "P (of_int (ceiling z))"
787 unfolding P_def of_rat_of_int_eq
789 fix y assume "y \<in> S"
790 hence "y \<le> z" using z by simp
791 also have "z \<le> of_int (ceiling z)" by (rule le_of_int_ceiling)
792 finally show "y \<le> of_int (ceiling z)" .
796 def avg \<equiv> "\<lambda>x y :: rat. x/2 + y/2"
797 def bisect \<equiv> "\<lambda>(x, y). if P (avg x y) then (x, avg x y) else (avg x y, y)"
798 def A \<equiv> "\<lambda>n. fst ((bisect ^^ n) (a, b))"
799 def B \<equiv> "\<lambda>n. snd ((bisect ^^ n) (a, b))"
800 def C \<equiv> "\<lambda>n. avg (A n) (B n)"
801 have A_0 [simp]: "A 0 = a" unfolding A_def by simp
802 have B_0 [simp]: "B 0 = b" unfolding B_def by simp
803 have A_Suc [simp]: "\<And>n. A (Suc n) = (if P (C n) then A n else C n)"
804 unfolding A_def B_def C_def bisect_def split_def by simp
805 have B_Suc [simp]: "\<And>n. B (Suc n) = (if P (C n) then C n else B n)"
806 unfolding A_def B_def C_def bisect_def split_def by simp
808 have width: "\<And>n. B n - A n = (b - a) / 2^n"
809 apply (simp add: eq_divide_eq)
810 apply (induct_tac n, simp)
811 apply (simp add: C_def avg_def algebra_simps)
814 have twos: "\<And>y r :: rat. 0 < r \<Longrightarrow> \<exists>n. y / 2 ^ n < r"
815 apply (simp add: divide_less_eq)
816 apply (subst mult_commute)
817 apply (frule_tac y=y in ex_less_of_nat_mult)
819 apply (rule_tac x=n in exI)
820 apply (erule less_trans)
821 apply (rule mult_strict_right_mono)
822 apply (rule le_less_trans [OF _ of_nat_less_two_power])
827 have PA: "\<And>n. \<not> P (A n)"
828 by (induct_tac n, simp_all add: a)
829 have PB: "\<And>n. P (B n)"
830 by (induct_tac n, simp_all add: b)
832 using a b unfolding P_def
833 apply (clarsimp simp add: not_le)
834 apply (drule (1) bspec)
835 apply (drule (1) less_le_trans)
836 apply (simp add: of_rat_less)
838 have AB: "\<And>n. A n < B n"
839 by (induct_tac n, simp add: ab, simp add: C_def avg_def)
840 have A_mono: "\<And>i j. i \<le> j \<Longrightarrow> A i \<le> A j"
841 apply (auto simp add: le_less [where 'a=nat])
842 apply (erule less_Suc_induct)
843 apply (clarsimp simp add: C_def avg_def)
844 apply (simp add: add_divide_distrib [symmetric])
845 apply (rule AB [THEN less_imp_le])
848 have B_mono: "\<And>i j. i \<le> j \<Longrightarrow> B j \<le> B i"
849 apply (auto simp add: le_less [where 'a=nat])
850 apply (erule less_Suc_induct)
851 apply (clarsimp simp add: C_def avg_def)
852 apply (simp add: add_divide_distrib [symmetric])
853 apply (rule AB [THEN less_imp_le])
857 "\<And>X. \<forall>n. \<forall>i\<ge>n. A n \<le> X i \<and> X i \<le> B n \<Longrightarrow> cauchy X"
859 apply (drule twos [where y="b - a"])
861 apply (rule_tac x=n in exI, clarify, rename_tac i j)
862 apply (rule_tac y="B n - A n" in le_less_trans) defer
863 apply (simp add: width)
864 apply (drule_tac x=n in spec)
865 apply (frule_tac x=i in spec, drule (1) mp)
866 apply (frule_tac x=j in spec, drule (1) mp)
867 apply (frule A_mono, drule B_mono)
868 apply (frule A_mono, drule B_mono)
872 apply (rule cauchy_lemma [rule_format])
873 apply (simp add: A_mono)
874 apply (erule order_trans [OF less_imp_le [OF AB] B_mono])
877 apply (rule cauchy_lemma [rule_format])
878 apply (simp add: B_mono)
879 apply (erule order_trans [OF A_mono less_imp_le [OF AB]])
881 have 1: "\<forall>x\<in>S. x \<le> Real B"
883 fix x assume "x \<in> S"
884 then show "x \<le> Real B"
885 using PB [unfolded P_def] `cauchy B`
886 by (simp add: le_RealI)
888 have 2: "\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> Real A \<le> z"
890 apply (erule contrapos_pp)
891 apply (simp add: not_le)
892 apply (drule less_RealD [OF `cauchy A`], clarify)
893 apply (subgoal_tac "\<not> P (A n)")
894 apply (simp add: P_def not_le, clarify)
895 apply (erule rev_bexI)
896 apply (erule (1) less_trans)
899 have "vanishes (\<lambda>n. (b - a) / 2 ^ n)"
900 proof (rule vanishesI)
901 fix r :: rat assume "0 < r"
902 then obtain k where k: "\<bar>b - a\<bar> / 2 ^ k < r"
904 have "\<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r"
906 fix n assume n: "k \<le> n"
907 have "\<bar>(b - a) / 2 ^ n\<bar> = \<bar>b - a\<bar> / 2 ^ n"
909 also have "\<dots> \<le> \<bar>b - a\<bar> / 2 ^ k"
910 using n by (simp add: divide_left_mono mult_pos_pos)
912 finally show "\<bar>(b - a) / 2 ^ n\<bar> < r" .
914 thus "\<exists>k. \<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r" ..
916 hence 3: "Real B = Real A"
917 by (simp add: eq_Real `cauchy A` `cauchy B` width)
918 show "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)"
919 using 1 2 3 by (rule_tac x="Real B" in exI, simp)
922 (* TODO: generalize to ordered group *)
923 lemma bdd_above_uminus[simp]: "bdd_above (uminus ` X) \<longleftrightarrow> bdd_below (X::real set)"
924 by (auto simp: bdd_above_def bdd_below_def intro: le_imp_neg_le) (metis le_imp_neg_le minus_minus)
926 lemma bdd_below_uminus[simp]: "bdd_below (uminus ` X) \<longleftrightarrow> bdd_above (X::real set)"
927 by (auto simp: bdd_above_def bdd_below_def intro: le_imp_neg_le) (metis le_imp_neg_le minus_minus)
929 instantiation real :: linear_continuum
932 subsection{*Supremum of a set of reals*}
935 Sup_real_def: "Sup X \<equiv> LEAST z::real. \<forall>x\<in>X. x\<le>z"
938 Inf_real_def: "Inf (X::real set) \<equiv> - Sup (uminus ` X)"
942 { fix x :: real and X :: "real set"
943 assume x: "x \<in> X" "bdd_above X"
944 then obtain s where s: "\<forall>y\<in>X. y \<le> s" "\<And>z. \<forall>y\<in>X. y \<le> z \<Longrightarrow> s \<le> z"
945 using complete_real[of X] unfolding bdd_above_def by blast
946 then show "x \<le> Sup X"
947 unfolding Sup_real_def by (rule LeastI2_order) (auto simp: x) }
948 note Sup_upper = this
950 { fix z :: real and X :: "real set"
951 assume x: "X \<noteq> {}" and z: "\<And>x. x \<in> X \<Longrightarrow> x \<le> z"
952 then obtain s where s: "\<forall>y\<in>X. y \<le> s" "\<And>z. \<forall>y\<in>X. y \<le> z \<Longrightarrow> s \<le> z"
953 using complete_real[of X] by blast
954 then have "Sup X = s"
955 unfolding Sup_real_def by (best intro: Least_equality)
956 also from s z have "... \<le> z"
958 finally show "Sup X \<le> z" . }
959 note Sup_least = this
961 { fix x z :: real and X :: "real set"
962 assume x: "x \<in> X" and [simp]: "bdd_below X"
963 have "-x \<le> Sup (uminus ` X)"
964 by (rule Sup_upper) (auto simp add: image_iff x)
965 then show "Inf X \<le> x"
966 by (auto simp add: Inf_real_def) }
968 { fix z :: real and X :: "real set"
969 assume x: "X \<noteq> {}" and z: "\<And>x. x \<in> X \<Longrightarrow> z \<le> x"
970 have "Sup (uminus ` X) \<le> -z"
971 using x z by (force intro: Sup_least)
972 then show "z \<le> Inf X"
973 by (auto simp add: Inf_real_def) }
975 show "\<exists>a b::real. a \<noteq> b"
976 using zero_neq_one by blast
981 \medskip Completeness properties using @{text "isUb"}, @{text "isLub"}:
984 lemma reals_complete: "\<exists>X. X \<in> S \<Longrightarrow> \<exists>Y. isUb (UNIV::real set) S Y \<Longrightarrow> \<exists>t. isLub (UNIV :: real set) S t"
985 by (intro exI[of _ "Sup S"] isLub_cSup) (auto simp: setle_def isUb_def intro!: cSup_upper)
988 subsection {* Hiding implementation details *}
990 hide_const (open) vanishes cauchy positive Real
992 declare Real_induct [induct del]
993 declare Abs_real_induct [induct del]
994 declare Abs_real_cases [cases del]
996 lifting_update real.lifting
997 lifting_forget real.lifting
999 subsection{*More Lemmas*}
1001 text {* BH: These lemmas should not be necessary; they should be
1002 covered by existing simp rules and simplification procedures. *}
1004 lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
1005 by simp (* redundant with mult_cancel_left *)
1007 lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
1008 by simp (* redundant with mult_cancel_right *)
1010 lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
1011 by simp (* solved by linordered_ring_less_cancel_factor simproc *)
1013 lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)"
1014 by simp (* solved by linordered_ring_le_cancel_factor simproc *)
1016 lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
1017 by simp (* solved by linordered_ring_le_cancel_factor simproc *)
1020 subsection {* Embedding numbers into the Reals *}
1023 real_of_nat :: "nat \<Rightarrow> real"
1025 "real_of_nat \<equiv> of_nat"
1028 real_of_int :: "int \<Rightarrow> real"
1030 "real_of_int \<equiv> of_int"
1033 real_of_rat :: "rat \<Rightarrow> real"
1035 "real_of_rat \<equiv> of_rat"
1038 (*overloaded constant for injecting other types into "real"*)
1039 real :: "'a => real"
1042 real_of_nat_def [code_unfold]: "real == real_of_nat"
1043 real_of_int_def [code_unfold]: "real == real_of_int"
1045 declare [[coercion_enabled]]
1046 declare [[coercion "real::nat\<Rightarrow>real"]]
1047 declare [[coercion "real::int\<Rightarrow>real"]]
1048 declare [[coercion "int"]]
1050 declare [[coercion_map map]]
1051 declare [[coercion_map "% f g h x. g (h (f x))"]]
1052 declare [[coercion_map "% f g (x,y) . (f x, g y)"]]
1054 lemma real_eq_of_nat: "real = of_nat"
1055 unfolding real_of_nat_def ..
1057 lemma real_eq_of_int: "real = of_int"
1058 unfolding real_of_int_def ..
1060 lemma real_of_int_zero [simp]: "real (0::int) = 0"
1061 by (simp add: real_of_int_def)
1063 lemma real_of_one [simp]: "real (1::int) = (1::real)"
1064 by (simp add: real_of_int_def)
1066 lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y"
1067 by (simp add: real_of_int_def)
1069 lemma real_of_int_minus [simp]: "real(-x) = -real (x::int)"
1070 by (simp add: real_of_int_def)
1072 lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y"
1073 by (simp add: real_of_int_def)
1075 lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y"
1076 by (simp add: real_of_int_def)
1078 lemma real_of_int_power [simp]: "real (x ^ n) = real (x::int) ^ n"
1079 by (simp add: real_of_int_def of_int_power)
1081 lemmas power_real_of_int = real_of_int_power [symmetric]
1083 lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))"
1084 apply (subst real_eq_of_int)+
1085 apply (rule of_int_setsum)
1088 lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) =
1089 (PROD x:A. real(f x))"
1090 apply (subst real_eq_of_int)+
1091 apply (rule of_int_setprod)
1094 lemma real_of_int_zero_cancel [simp, algebra, presburger]: "(real x = 0) = (x = (0::int))"
1095 by (simp add: real_of_int_def)
1097 lemma real_of_int_inject [iff, algebra, presburger]: "(real (x::int) = real y) = (x = y)"
1098 by (simp add: real_of_int_def)
1100 lemma real_of_int_less_iff [iff, presburger]: "(real (x::int) < real y) = (x < y)"
1101 by (simp add: real_of_int_def)
1103 lemma real_of_int_le_iff [simp, presburger]: "(real (x::int) \<le> real y) = (x \<le> y)"
1104 by (simp add: real_of_int_def)
1106 lemma real_of_int_gt_zero_cancel_iff [simp, presburger]: "(0 < real (n::int)) = (0 < n)"
1107 by (simp add: real_of_int_def)
1109 lemma real_of_int_ge_zero_cancel_iff [simp, presburger]: "(0 <= real (n::int)) = (0 <= n)"
1110 by (simp add: real_of_int_def)
1112 lemma real_of_int_lt_zero_cancel_iff [simp, presburger]: "(real (n::int) < 0) = (n < 0)"
1113 by (simp add: real_of_int_def)
1115 lemma real_of_int_le_zero_cancel_iff [simp, presburger]: "(real (n::int) <= 0) = (n <= 0)"
1116 by (simp add: real_of_int_def)
1118 lemma one_less_real_of_int_cancel_iff: "1 < real (i :: int) \<longleftrightarrow> 1 < i"
1119 unfolding real_of_one[symmetric] real_of_int_less_iff ..
1121 lemma one_le_real_of_int_cancel_iff: "1 \<le> real (i :: int) \<longleftrightarrow> 1 \<le> i"
1122 unfolding real_of_one[symmetric] real_of_int_le_iff ..
1124 lemma real_of_int_less_one_cancel_iff: "real (i :: int) < 1 \<longleftrightarrow> i < 1"
1125 unfolding real_of_one[symmetric] real_of_int_less_iff ..
1127 lemma real_of_int_le_one_cancel_iff: "real (i :: int) \<le> 1 \<longleftrightarrow> i \<le> 1"
1128 unfolding real_of_one[symmetric] real_of_int_le_iff ..
1130 lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))"
1131 by (auto simp add: abs_if)
1133 lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)"
1134 apply (subgoal_tac "real n + 1 = real (n + 1)")
1135 apply (simp del: real_of_int_add)
1139 lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)"
1140 apply (subgoal_tac "real m + 1 = real (m + 1)")
1141 apply (simp del: real_of_int_add)
1145 lemma real_of_int_div_aux: "(real (x::int)) / (real d) =
1146 real (x div d) + (real (x mod d)) / (real d)"
1148 have "x = (x div d) * d + x mod d"
1150 then have "real x = real (x div d) * real d + real(x mod d)"
1151 by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym])
1152 then have "real x / real d = ... / real d"
1155 by (auto simp add: add_divide_distrib algebra_simps)
1158 lemma real_of_int_div: "(d :: int) dvd n ==>
1159 real(n div d) = real n / real d"
1160 apply (subst real_of_int_div_aux)
1162 apply (simp add: dvd_eq_mod_eq_0)
1165 lemma real_of_int_div2:
1166 "0 <= real (n::int) / real (x) - real (n div x)"
1167 apply (case_tac "x = 0")
1169 apply (case_tac "0 < x")
1170 apply (simp add: algebra_simps)
1171 apply (subst real_of_int_div_aux)
1173 apply (subst zero_le_divide_iff)
1175 apply (simp add: algebra_simps)
1176 apply (subst real_of_int_div_aux)
1178 apply (subst zero_le_divide_iff)
1182 lemma real_of_int_div3:
1183 "real (n::int) / real (x) - real (n div x) <= 1"
1184 apply (simp add: algebra_simps)
1185 apply (subst real_of_int_div_aux)
1186 apply (auto simp add: divide_le_eq intro: order_less_imp_le)
1189 lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x"
1190 by (insert real_of_int_div2 [of n x], simp)
1192 lemma Ints_real_of_int [simp]: "real (x::int) \<in> Ints"
1193 unfolding real_of_int_def by (rule Ints_of_int)
1196 subsection{*Embedding the Naturals into the Reals*}
1198 lemma real_of_nat_zero [simp]: "real (0::nat) = 0"
1199 by (simp add: real_of_nat_def)
1201 lemma real_of_nat_1 [simp]: "real (1::nat) = 1"
1202 by (simp add: real_of_nat_def)
1204 lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"
1205 by (simp add: real_of_nat_def)
1207 lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"
1208 by (simp add: real_of_nat_def)
1210 (*Not for addsimps: often the LHS is used to represent a positive natural*)
1211 lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"
1212 by (simp add: real_of_nat_def)
1214 lemma real_of_nat_less_iff [iff]:
1215 "(real (n::nat) < real m) = (n < m)"
1216 by (simp add: real_of_nat_def)
1218 lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"
1219 by (simp add: real_of_nat_def)
1221 lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"
1222 by (simp add: real_of_nat_def)
1224 lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
1225 by (simp add: real_of_nat_def del: of_nat_Suc)
1227 lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
1228 by (simp add: real_of_nat_def of_nat_mult)
1230 lemma real_of_nat_power [simp]: "real (m ^ n) = real (m::nat) ^ n"
1231 by (simp add: real_of_nat_def of_nat_power)
1233 lemmas power_real_of_nat = real_of_nat_power [symmetric]
1235 lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) =
1236 (SUM x:A. real(f x))"
1237 apply (subst real_eq_of_nat)+
1238 apply (rule of_nat_setsum)
1241 lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) =
1242 (PROD x:A. real(f x))"
1243 apply (subst real_eq_of_nat)+
1244 apply (rule of_nat_setprod)
1247 lemma real_of_card: "real (card A) = setsum (%x.1) A"
1248 apply (subst card_eq_setsum)
1249 apply (subst real_of_nat_setsum)
1253 lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
1254 by (simp add: real_of_nat_def)
1256 lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"
1257 by (simp add: real_of_nat_def)
1259 lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"
1260 by (simp add: add: real_of_nat_def of_nat_diff)
1262 lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"
1263 by (auto simp: real_of_nat_def)
1265 lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"
1266 by (simp add: add: real_of_nat_def)
1268 lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"
1269 by (simp add: add: real_of_nat_def)
1271 lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)"
1272 apply (subgoal_tac "real n + 1 = real (Suc n)")
1274 apply (auto simp add: real_of_nat_Suc)
1277 lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"
1278 apply (subgoal_tac "real m + 1 = real (Suc m)")
1279 apply (simp add: less_Suc_eq_le)
1280 apply (simp add: real_of_nat_Suc)
1283 lemma real_of_nat_div_aux: "(real (x::nat)) / (real d) =
1284 real (x div d) + (real (x mod d)) / (real d)"
1286 have "x = (x div d) * d + x mod d"
1288 then have "real x = real (x div d) * real d + real(x mod d)"
1289 by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym])
1290 then have "real x / real d = \<dots> / real d"
1293 by (auto simp add: add_divide_distrib algebra_simps)
1296 lemma real_of_nat_div: "(d :: nat) dvd n ==>
1297 real(n div d) = real n / real d"
1298 by (subst real_of_nat_div_aux)
1299 (auto simp add: dvd_eq_mod_eq_0 [symmetric])
1301 lemma real_of_nat_div2:
1302 "0 <= real (n::nat) / real (x) - real (n div x)"
1303 apply (simp add: algebra_simps)
1304 apply (subst real_of_nat_div_aux)
1306 apply (subst zero_le_divide_iff)
1310 lemma real_of_nat_div3:
1311 "real (n::nat) / real (x) - real (n div x) <= 1"
1312 apply(case_tac "x = 0")
1314 apply (simp add: algebra_simps)
1315 apply (subst real_of_nat_div_aux)
1319 lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x"
1320 by (insert real_of_nat_div2 [of n x], simp)
1322 lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"
1323 by (simp add: real_of_int_def real_of_nat_def)
1325 lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x"
1326 apply (subgoal_tac "real(int(nat x)) = real(nat x)")
1328 apply (simp only: real_of_int_of_nat_eq)
1331 lemma Nats_real_of_nat [simp]: "real (n::nat) \<in> Nats"
1332 unfolding real_of_nat_def by (rule of_nat_in_Nats)
1334 lemma Ints_real_of_nat [simp]: "real (n::nat) \<in> Ints"
1335 unfolding real_of_nat_def by (rule Ints_of_nat)
1337 subsection {* The Archimedean Property of the Reals *}
1339 theorem reals_Archimedean:
1340 assumes x_pos: "0 < x"
1341 shows "\<exists>n. inverse (real (Suc n)) < x"
1342 unfolding real_of_nat_def using x_pos
1343 by (rule ex_inverse_of_nat_Suc_less)
1345 lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"
1346 unfolding real_of_nat_def by (rule ex_less_of_nat)
1348 lemma reals_Archimedean3:
1349 assumes x_greater_zero: "0 < x"
1350 shows "\<forall>(y::real). \<exists>(n::nat). y < real n * x"
1351 unfolding real_of_nat_def using `0 < x`
1352 by (auto intro: ex_less_of_nat_mult)
1355 subsection{* Rationals *}
1357 lemma Rats_real_nat[simp]: "real(n::nat) \<in> \<rat>"
1358 by (simp add: real_eq_of_nat)
1361 lemma Rats_eq_int_div_int:
1362 "\<rat> = { real(i::int)/real(j::int) |i j. j \<noteq> 0}" (is "_ = ?S")
1364 show "\<rat> \<subseteq> ?S"
1366 fix x::real assume "x : \<rat>"
1367 then obtain r where "x = of_rat r" unfolding Rats_def ..
1368 have "of_rat r : ?S"
1369 by (cases r)(auto simp add:of_rat_rat real_eq_of_int)
1370 thus "x : ?S" using `x = of_rat r` by simp
1373 show "?S \<subseteq> \<rat>"
1374 proof(auto simp:Rats_def)
1375 fix i j :: int assume "j \<noteq> 0"
1376 hence "real i / real j = of_rat(Fract i j)"
1377 by (simp add:of_rat_rat real_eq_of_int)
1378 thus "real i / real j \<in> range of_rat" by blast
1382 lemma Rats_eq_int_div_nat:
1383 "\<rat> = { real(i::int)/real(n::nat) |i n. n \<noteq> 0}"
1384 proof(auto simp:Rats_eq_int_div_int)
1385 fix i j::int assume "j \<noteq> 0"
1386 show "EX (i'::int) (n::nat). real i/real j = real i'/real n \<and> 0<n"
1389 hence "real i/real j = real i/real(nat j) \<and> 0<nat j"
1390 by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)
1391 thus ?thesis by blast
1394 hence "real i/real j = real(-i)/real(nat(-j)) \<and> 0<nat(-j)" using `j\<noteq>0`
1395 by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)
1396 thus ?thesis by blast
1399 fix i::int and n::nat assume "0 < n"
1400 hence "real i/real n = real i/real(int n) \<and> int n \<noteq> 0" by simp
1401 thus "\<exists>(i'::int) j::int. real i/real n = real i'/real j \<and> j \<noteq> 0" by blast
1404 lemma Rats_abs_nat_div_natE:
1405 assumes "x \<in> \<rat>"
1407 where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "gcd m n = 1"
1409 from `x \<in> \<rat>` obtain i::int and n::nat where "n \<noteq> 0" and "x = real i / real n"
1410 by(auto simp add: Rats_eq_int_div_nat)
1411 hence "\<bar>x\<bar> = real(nat(abs i)) / real n" by simp
1412 then obtain m :: nat where x_rat: "\<bar>x\<bar> = real m / real n" by blast
1413 let ?gcd = "gcd m n"
1414 from `n\<noteq>0` have gcd: "?gcd \<noteq> 0" by simp
1415 let ?k = "m div ?gcd"
1416 let ?l = "n div ?gcd"
1417 let ?gcd' = "gcd ?k ?l"
1418 have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m"
1419 by (rule dvd_mult_div_cancel)
1420 have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n"
1421 by (rule dvd_mult_div_cancel)
1422 from `n\<noteq>0` and gcd_l have "?l \<noteq> 0" by (auto iff del: neq0_conv)
1424 have "\<bar>x\<bar> = real ?k / real ?l"
1426 from gcd have "real ?k / real ?l =
1427 real (?gcd * ?k) / real (?gcd * ?l)" by simp
1428 also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp
1429 also from x_rat have "\<dots> = \<bar>x\<bar>" ..
1430 finally show ?thesis ..
1435 have "?gcd * ?gcd' = gcd (?gcd * ?k) (?gcd * ?l)"
1436 by (rule gcd_mult_distrib_nat)
1437 with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp
1438 with gcd show ?thesis by auto
1440 ultimately show ?thesis ..
1443 subsection{*Density of the Rational Reals in the Reals*}
1445 text{* This density proof is due to Stefan Richter and was ported by TN. The
1446 original source is \emph{Real Analysis} by H.L. Royden.
1447 It employs the Archimedean property of the reals. *}
1449 lemma Rats_dense_in_real:
1451 assumes "x < y" shows "\<exists>r\<in>\<rat>. x < r \<and> r < y"
1453 from `x<y` have "0 < y-x" by simp
1454 with reals_Archimedean obtain q::nat
1455 where q: "inverse (real q) < y-x" and "0 < q" by auto
1456 def p \<equiv> "ceiling (y * real q) - 1"
1457 def r \<equiv> "of_int p / real q"
1458 from q have "x < y - inverse (real q)" by simp
1459 also have "y - inverse (real q) \<le> r"
1460 unfolding r_def p_def
1461 by (simp add: le_divide_eq left_diff_distrib le_of_int_ceiling `0 < q`)
1462 finally have "x < r" .
1463 moreover have "r < y"
1464 unfolding r_def p_def
1465 by (simp add: divide_less_eq diff_less_eq `0 < q`
1466 less_ceiling_iff [symmetric])
1467 moreover from r_def have "r \<in> \<rat>" by simp
1468 ultimately show ?thesis by fast
1473 subsection{*Numerals and Arithmetic*}
1475 lemma [code_abbrev]:
1476 "real_of_int (numeral k) = numeral k"
1477 "real_of_int (neg_numeral k) = neg_numeral k"
1480 text{*Collapse applications of @{term real} to @{term number_of}*}
1481 lemma real_numeral [simp]:
1482 "real (numeral v :: int) = numeral v"
1483 "real (neg_numeral v :: int) = neg_numeral v"
1484 by (simp_all add: real_of_int_def)
1486 lemma real_of_nat_numeral [simp]:
1487 "real (numeral v :: nat) = numeral v"
1488 by (simp add: real_of_nat_def)
1491 K (Lin_Arith.add_inj_thms [@{thm real_of_nat_le_iff} RS iffD2, @{thm real_of_nat_inject} RS iffD2]
1492 (* not needed because x < (y::nat) can be rewritten as Suc x <= y: real_of_nat_less_iff RS iffD2 *)
1493 #> Lin_Arith.add_inj_thms [@{thm real_of_int_le_iff} RS iffD2, @{thm real_of_int_inject} RS iffD2]
1494 (* not needed because x < (y::int) can be rewritten as x + 1 <= y: real_of_int_less_iff RS iffD2 *)
1495 #> Lin_Arith.add_simps [@{thm real_of_nat_zero}, @{thm real_of_nat_Suc}, @{thm real_of_nat_add},
1496 @{thm real_of_nat_mult}, @{thm real_of_int_zero}, @{thm real_of_one},
1497 @{thm real_of_int_add}, @{thm real_of_int_minus}, @{thm real_of_int_diff},
1498 @{thm real_of_int_mult}, @{thm real_of_int_of_nat_eq},
1499 @{thm real_of_nat_numeral}, @{thm real_numeral(1)}, @{thm real_numeral(2)}]
1500 #> Lin_Arith.add_inj_const (@{const_name real}, @{typ "nat \<Rightarrow> real"})
1501 #> Lin_Arith.add_inj_const (@{const_name real}, @{typ "int \<Rightarrow> real"}))
1505 subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}
1507 lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)"
1510 text {* FIXME: redundant with @{text add_eq_0_iff} below *}
1511 lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)"
1514 lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"
1517 lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)"
1520 lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> -x)"
1523 lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (-x \<le> y)"
1526 subsection {* Lemmas about powers *}
1528 text {* FIXME: declare this in Rings.thy or not at all *}
1529 declare abs_mult_self [simp]
1531 (* used by Import/HOL/real.imp *)
1532 lemma two_realpow_ge_one: "(1::real) \<le> 2 ^ n"
1535 lemma two_realpow_gt [simp]: "real (n::nat) < 2 ^ n"
1537 apply (auto simp add: real_of_nat_Suc)
1538 apply (subst mult_2)
1539 apply (erule add_less_le_mono)
1540 apply (rule two_realpow_ge_one)
1543 text {* TODO: no longer real-specific; rename and move elsewhere *}
1544 lemma realpow_Suc_le_self:
1545 fixes r :: "'a::linordered_semidom"
1546 shows "[| 0 \<le> r; r \<le> 1 |] ==> r ^ Suc n \<le> r"
1547 by (insert power_decreasing [of 1 "Suc n" r], simp)
1549 text {* TODO: no longer real-specific; rename and move elsewhere *}
1550 lemma realpow_minus_mult:
1551 fixes x :: "'a::monoid_mult"
1552 shows "0 < n \<Longrightarrow> x ^ (n - 1) * x = x ^ n"
1553 by (simp add: power_commutes split add: nat_diff_split)
1555 text {* FIXME: declare this [simp] for all types, or not at all *}
1556 lemma real_two_squares_add_zero_iff [simp]:
1557 "(x * x + y * y = 0) = ((x::real) = 0 \<and> y = 0)"
1558 by (rule sum_squares_eq_zero_iff)
1560 text {* FIXME: declare this [simp] for all types, or not at all *}
1561 lemma realpow_two_sum_zero_iff [simp]:
1562 "(x\<^sup>2 + y\<^sup>2 = (0::real)) = (x = 0 & y = 0)"
1563 by (rule sum_power2_eq_zero_iff)
1565 lemma real_minus_mult_self_le [simp]: "-(u * u) \<le> (x * (x::real))"
1566 by (rule_tac y = 0 in order_trans, auto)
1568 lemma realpow_square_minus_le [simp]: "- u\<^sup>2 \<le> (x::real)\<^sup>2"
1569 by (auto simp add: power2_eq_square)
1572 lemma numeral_power_le_real_of_nat_cancel_iff[simp]:
1573 "(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::nat) ^ n \<le> a"
1574 unfolding real_of_nat_le_iff[symmetric] by simp
1576 lemma real_of_nat_le_numeral_power_cancel_iff[simp]:
1577 "real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::nat) ^ n"
1578 unfolding real_of_nat_le_iff[symmetric] by simp
1580 lemma numeral_power_le_real_of_int_cancel_iff[simp]:
1581 "(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::int) ^ n \<le> a"
1582 unfolding real_of_int_le_iff[symmetric] by simp
1584 lemma real_of_int_le_numeral_power_cancel_iff[simp]:
1585 "real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::int) ^ n"
1586 unfolding real_of_int_le_iff[symmetric] by simp
1588 lemma neg_numeral_power_le_real_of_int_cancel_iff[simp]:
1589 "(neg_numeral x::real) ^ n \<le> real a \<longleftrightarrow> (neg_numeral x::int) ^ n \<le> a"
1590 unfolding real_of_int_le_iff[symmetric] by simp
1592 lemma real_of_int_le_neg_numeral_power_cancel_iff[simp]:
1593 "real a \<le> (neg_numeral x::real) ^ n \<longleftrightarrow> a \<le> (neg_numeral x::int) ^ n"
1594 unfolding real_of_int_le_iff[symmetric] by simp
1596 subsection{*Density of the Reals*}
1598 lemma real_lbound_gt_zero:
1599 "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"
1600 apply (rule_tac x = " (min d1 d2) /2" in exI)
1601 apply (simp add: min_def)
1605 text{*Similar results are proved in @{text Fields}*}
1606 lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"
1609 lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"
1612 lemma real_sum_of_halves: "x/2 + x/2 = (x::real)"
1615 subsection{*Absolute Value Function for the Reals*}
1617 lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"
1618 by (simp add: abs_if)
1620 (* FIXME: redundant, but used by Integration/RealRandVar.thy in AFP *)
1621 lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))"
1622 by (force simp add: abs_le_iff)
1624 lemma abs_add_one_gt_zero: "(0::real) < 1 + abs(x)"
1625 by (simp add: abs_if)
1627 lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)"
1628 by (rule abs_of_nonneg [OF real_of_nat_ge_zero])
1630 lemma abs_add_one_not_less_self: "~ abs(x) + (1::real) < x"
1633 lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"
1637 subsection{*Floor and Ceiling Functions from the Reals to the Integers*}
1639 (* FIXME: theorems for negative numerals *)
1640 lemma numeral_less_real_of_int_iff [simp]:
1641 "((numeral n) < real (m::int)) = (numeral n < m)"
1643 apply (rule real_of_int_less_iff [THEN iffD1])
1644 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
1647 lemma numeral_less_real_of_int_iff2 [simp]:
1648 "(real (m::int) < (numeral n)) = (m < numeral n)"
1650 apply (rule real_of_int_less_iff [THEN iffD1])
1651 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
1654 lemma numeral_le_real_of_int_iff [simp]:
1655 "((numeral n) \<le> real (m::int)) = (numeral n \<le> m)"
1656 by (simp add: linorder_not_less [symmetric])
1658 lemma numeral_le_real_of_int_iff2 [simp]:
1659 "(real (m::int) \<le> (numeral n)) = (m \<le> numeral n)"
1660 by (simp add: linorder_not_less [symmetric])
1662 lemma floor_real_of_nat [simp]: "floor (real (n::nat)) = int n"
1663 unfolding real_of_nat_def by simp
1665 lemma floor_minus_real_of_nat [simp]: "floor (- real (n::nat)) = - int n"
1666 unfolding real_of_nat_def by (simp add: floor_minus)
1668 lemma floor_real_of_int [simp]: "floor (real (n::int)) = n"
1669 unfolding real_of_int_def by simp
1671 lemma floor_minus_real_of_int [simp]: "floor (- real (n::int)) = - n"
1672 unfolding real_of_int_def by (simp add: floor_minus)
1674 lemma real_lb_ub_int: " \<exists>n::int. real n \<le> r & r < real (n+1)"
1675 unfolding real_of_int_def by (rule floor_exists)
1678 assumes a1: "real m \<le> r" and a2: "r < real n + 1"
1679 shows "m \<le> (n::int)"
1681 have "real m < real n + 1" using a1 a2 by (rule order_le_less_trans)
1682 also have "... = real (n + 1)" by simp
1683 finally have "m < n + 1" by (simp only: real_of_int_less_iff)
1684 thus ?thesis by arith
1687 lemma real_of_int_floor_le [simp]: "real (floor r) \<le> r"
1688 unfolding real_of_int_def by (rule of_int_floor_le)
1690 lemma lemma_floor2: "real n < real (x::int) + 1 ==> n \<le> x"
1691 by (auto intro: lemma_floor)
1693 lemma real_of_int_floor_cancel [simp]:
1694 "(real (floor x) = x) = (\<exists>n::int. x = real n)"
1695 using floor_real_of_int by metis
1697 lemma floor_eq: "[| real n < x; x < real n + 1 |] ==> floor x = n"
1698 unfolding real_of_int_def using floor_unique [of n x] by simp
1700 lemma floor_eq2: "[| real n \<le> x; x < real n + 1 |] ==> floor x = n"
1701 unfolding real_of_int_def by (rule floor_unique)
1703 lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"
1704 apply (rule inj_int [THEN injD])
1705 apply (simp add: real_of_nat_Suc)
1706 apply (simp add: real_of_nat_Suc floor_eq floor_eq [where n = "int n"])
1709 lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"
1710 apply (drule order_le_imp_less_or_eq)
1711 apply (auto intro: floor_eq3)
1714 lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real(floor r)"
1715 unfolding real_of_int_def using floor_correct [of r] by simp
1717 lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real(floor r)"
1718 unfolding real_of_int_def using floor_correct [of r] by simp
1720 lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real(floor r) + 1"
1721 unfolding real_of_int_def using floor_correct [of r] by simp
1723 lemma real_of_int_floor_add_one_gt [simp]: "r < real(floor r) + 1"
1724 unfolding real_of_int_def using floor_correct [of r] by simp
1726 lemma le_floor: "real a <= x ==> a <= floor x"
1727 unfolding real_of_int_def by (simp add: le_floor_iff)
1729 lemma real_le_floor: "a <= floor x ==> real a <= x"
1730 unfolding real_of_int_def by (simp add: le_floor_iff)
1732 lemma le_floor_eq: "(a <= floor x) = (real a <= x)"
1733 unfolding real_of_int_def by (rule le_floor_iff)
1735 lemma floor_less_eq: "(floor x < a) = (x < real a)"
1736 unfolding real_of_int_def by (rule floor_less_iff)
1738 lemma less_floor_eq: "(a < floor x) = (real a + 1 <= x)"
1739 unfolding real_of_int_def by (rule less_floor_iff)
1741 lemma floor_le_eq: "(floor x <= a) = (x < real a + 1)"
1742 unfolding real_of_int_def by (rule floor_le_iff)
1744 lemma floor_add [simp]: "floor (x + real a) = floor x + a"
1745 unfolding real_of_int_def by (rule floor_add_of_int)
1747 lemma floor_subtract [simp]: "floor (x - real a) = floor x - a"
1748 unfolding real_of_int_def by (rule floor_diff_of_int)
1750 lemma le_mult_floor:
1751 assumes "0 \<le> (a :: real)" and "0 \<le> b"
1752 shows "floor a * floor b \<le> floor (a * b)"
1754 have "real (floor a) \<le> a"
1755 and "real (floor b) \<le> b" by auto
1756 hence "real (floor a * floor b) \<le> a * b"
1757 using assms by (auto intro!: mult_mono)
1758 also have "a * b < real (floor (a * b) + 1)" by auto
1759 finally show ?thesis unfolding real_of_int_less_iff by simp
1762 lemma floor_divide_eq_div:
1763 "floor (real a / real b) = a div b"
1765 assume "b \<noteq> 0 \<or> b dvd a"
1766 with real_of_int_div3[of a b] show ?thesis
1767 by (auto simp: real_of_int_div[symmetric] intro!: floor_eq2 real_of_int_div4 neq_le_trans)
1768 (metis add_left_cancel zero_neq_one real_of_int_div_aux real_of_int_inject
1769 real_of_int_zero_cancel right_inverse_eq div_self mod_div_trivial)
1770 qed (auto simp: real_of_int_div)
1772 lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"
1773 unfolding real_of_nat_def by simp
1775 lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)"
1776 unfolding real_of_int_def by (rule le_of_int_ceiling)
1778 lemma ceiling_real_of_int [simp]: "ceiling (real (n::int)) = n"
1779 unfolding real_of_int_def by simp
1781 lemma real_of_int_ceiling_cancel [simp]:
1782 "(real (ceiling x) = x) = (\<exists>n::int. x = real n)"
1783 using ceiling_real_of_int by metis
1785 lemma ceiling_eq: "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1"
1786 unfolding real_of_int_def using ceiling_unique [of "n + 1" x] by simp
1788 lemma ceiling_eq2: "[| real n < x; x \<le> real n + 1 |] ==> ceiling x = n + 1"
1789 unfolding real_of_int_def using ceiling_unique [of "n + 1" x] by simp
1791 lemma ceiling_eq3: "[| real n - 1 < x; x \<le> real n |] ==> ceiling x = n"
1792 unfolding real_of_int_def using ceiling_unique [of n x] by simp
1794 lemma real_of_int_ceiling_diff_one_le [simp]: "real (ceiling r) - 1 \<le> r"
1795 unfolding real_of_int_def using ceiling_correct [of r] by simp
1797 lemma real_of_int_ceiling_le_add_one [simp]: "real (ceiling r) \<le> r + 1"
1798 unfolding real_of_int_def using ceiling_correct [of r] by simp
1800 lemma ceiling_le: "x <= real a ==> ceiling x <= a"
1801 unfolding real_of_int_def by (simp add: ceiling_le_iff)
1803 lemma ceiling_le_real: "ceiling x <= a ==> x <= real a"
1804 unfolding real_of_int_def by (simp add: ceiling_le_iff)
1806 lemma ceiling_le_eq: "(ceiling x <= a) = (x <= real a)"
1807 unfolding real_of_int_def by (rule ceiling_le_iff)
1809 lemma less_ceiling_eq: "(a < ceiling x) = (real a < x)"
1810 unfolding real_of_int_def by (rule less_ceiling_iff)
1812 lemma ceiling_less_eq: "(ceiling x < a) = (x <= real a - 1)"
1813 unfolding real_of_int_def by (rule ceiling_less_iff)
1815 lemma le_ceiling_eq: "(a <= ceiling x) = (real a - 1 < x)"
1816 unfolding real_of_int_def by (rule le_ceiling_iff)
1818 lemma ceiling_add [simp]: "ceiling (x + real a) = ceiling x + a"
1819 unfolding real_of_int_def by (rule ceiling_add_of_int)
1821 lemma ceiling_subtract [simp]: "ceiling (x - real a) = ceiling x - a"
1822 unfolding real_of_int_def by (rule ceiling_diff_of_int)
1825 subsubsection {* Versions for the natural numbers *}
1828 natfloor :: "real => nat" where
1829 "natfloor x = nat(floor x)"
1832 natceiling :: "real => nat" where
1833 "natceiling x = nat(ceiling x)"
1835 lemma natfloor_zero [simp]: "natfloor 0 = 0"
1836 by (unfold natfloor_def, simp)
1838 lemma natfloor_one [simp]: "natfloor 1 = 1"
1839 by (unfold natfloor_def, simp)
1841 lemma zero_le_natfloor [simp]: "0 <= natfloor x"
1842 by (unfold natfloor_def, simp)
1844 lemma natfloor_numeral_eq [simp]: "natfloor (numeral n) = numeral n"
1845 by (unfold natfloor_def, simp)
1847 lemma natfloor_real_of_nat [simp]: "natfloor(real n) = n"
1848 by (unfold natfloor_def, simp)
1850 lemma real_natfloor_le: "0 <= x ==> real(natfloor x) <= x"
1851 by (unfold natfloor_def, simp)
1853 lemma natfloor_neg: "x <= 0 ==> natfloor x = 0"
1854 unfolding natfloor_def by simp
1856 lemma natfloor_mono: "x <= y ==> natfloor x <= natfloor y"
1857 unfolding natfloor_def by (intro nat_mono floor_mono)
1859 lemma le_natfloor: "real x <= a ==> x <= natfloor a"
1860 apply (unfold natfloor_def)
1861 apply (subst nat_int [THEN sym])
1862 apply (rule nat_mono)
1863 apply (rule le_floor)
1867 lemma natfloor_less_iff: "0 \<le> x \<Longrightarrow> natfloor x < n \<longleftrightarrow> x < real n"
1868 unfolding natfloor_def real_of_nat_def
1869 by (simp add: nat_less_iff floor_less_iff)
1871 lemma less_natfloor:
1872 assumes "0 \<le> x" and "x < real (n :: nat)"
1873 shows "natfloor x < n"
1874 using assms by (simp add: natfloor_less_iff)
1876 lemma le_natfloor_eq: "0 <= x ==> (a <= natfloor x) = (real a <= x)"
1878 apply (rule order_trans)
1880 apply (erule real_natfloor_le)
1881 apply (subst real_of_nat_le_iff)
1883 apply (erule le_natfloor)
1886 lemma le_natfloor_eq_numeral [simp]:
1887 "~ neg((numeral n)::int) ==> 0 <= x ==>
1888 (numeral n <= natfloor x) = (numeral n <= x)"
1889 apply (subst le_natfloor_eq, assumption)
1893 lemma le_natfloor_eq_one [simp]: "(1 <= natfloor x) = (1 <= x)"
1894 apply (case_tac "0 <= x")
1895 apply (subst le_natfloor_eq, assumption, simp)
1897 apply (subgoal_tac "natfloor x <= natfloor 0")
1899 apply (rule natfloor_mono)
1904 lemma natfloor_eq: "real n <= x ==> x < real n + 1 ==> natfloor x = n"
1905 unfolding natfloor_def by (simp add: floor_eq2 [where n="int n"])
1907 lemma real_natfloor_add_one_gt: "x < real(natfloor x) + 1"
1908 apply (case_tac "0 <= x")
1909 apply (unfold natfloor_def)
1914 lemma real_natfloor_gt_diff_one: "x - 1 < real(natfloor x)"
1915 using real_natfloor_add_one_gt by (simp add: algebra_simps)
1917 lemma ge_natfloor_plus_one_imp_gt: "natfloor z + 1 <= n ==> z < real n"
1918 apply (subgoal_tac "z < real(natfloor z) + 1")
1920 apply (rule real_natfloor_add_one_gt)
1923 lemma natfloor_add [simp]: "0 <= x ==> natfloor (x + real a) = natfloor x + a"
1924 unfolding natfloor_def
1925 unfolding real_of_int_of_nat_eq [symmetric] floor_add
1926 by (simp add: nat_add_distrib)
1928 lemma natfloor_add_numeral [simp]:
1929 "~neg ((numeral n)::int) ==> 0 <= x ==>
1930 natfloor (x + numeral n) = natfloor x + numeral n"
1931 by (simp add: natfloor_add [symmetric])
1933 lemma natfloor_add_one: "0 <= x ==> natfloor(x + 1) = natfloor x + 1"
1934 by (simp add: natfloor_add [symmetric] del: One_nat_def)
1936 lemma natfloor_subtract [simp]:
1937 "natfloor(x - real a) = natfloor x - a"
1938 unfolding natfloor_def real_of_int_of_nat_eq [symmetric] floor_subtract
1941 lemma natfloor_div_nat:
1942 assumes "1 <= x" and "y > 0"
1943 shows "natfloor (x / real y) = natfloor x div y"
1944 proof (rule natfloor_eq)
1945 have "(natfloor x) div y * y \<le> natfloor x"
1946 by (rule add_leD1 [where k="natfloor x mod y"], simp)
1947 thus "real (natfloor x div y) \<le> x / real y"
1948 using assms by (simp add: le_divide_eq le_natfloor_eq)
1949 have "natfloor x < (natfloor x) div y * y + y"
1950 apply (subst mod_div_equality [symmetric])
1951 apply (rule add_strict_left_mono)
1952 apply (rule mod_less_divisor)
1955 thus "x / real y < real (natfloor x div y) + 1"
1957 by (simp add: divide_less_eq natfloor_less_iff distrib_right)
1960 lemma le_mult_natfloor:
1961 shows "natfloor a * natfloor b \<le> natfloor (a * b)"
1962 by (cases "0 <= a & 0 <= b")
1963 (auto simp add: le_natfloor_eq mult_nonneg_nonneg mult_mono' real_natfloor_le natfloor_neg)
1965 lemma natceiling_zero [simp]: "natceiling 0 = 0"
1966 by (unfold natceiling_def, simp)
1968 lemma natceiling_one [simp]: "natceiling 1 = 1"
1969 by (unfold natceiling_def, simp)
1971 lemma zero_le_natceiling [simp]: "0 <= natceiling x"
1972 by (unfold natceiling_def, simp)
1974 lemma natceiling_numeral_eq [simp]: "natceiling (numeral n) = numeral n"
1975 by (unfold natceiling_def, simp)
1977 lemma natceiling_real_of_nat [simp]: "natceiling(real n) = n"
1978 by (unfold natceiling_def, simp)
1980 lemma real_natceiling_ge: "x <= real(natceiling x)"
1981 unfolding natceiling_def by (cases "x < 0", simp_all)
1983 lemma natceiling_neg: "x <= 0 ==> natceiling x = 0"
1984 unfolding natceiling_def by simp
1986 lemma natceiling_mono: "x <= y ==> natceiling x <= natceiling y"
1987 unfolding natceiling_def by (intro nat_mono ceiling_mono)
1989 lemma natceiling_le: "x <= real a ==> natceiling x <= a"
1990 unfolding natceiling_def real_of_nat_def
1991 by (simp add: nat_le_iff ceiling_le_iff)
1993 lemma natceiling_le_eq: "(natceiling x <= a) = (x <= real a)"
1994 unfolding natceiling_def real_of_nat_def
1995 by (simp add: nat_le_iff ceiling_le_iff)
1997 lemma natceiling_le_eq_numeral [simp]:
1998 "~ neg((numeral n)::int) ==>
1999 (natceiling x <= numeral n) = (x <= numeral n)"
2000 by (simp add: natceiling_le_eq)
2002 lemma natceiling_le_eq_one: "(natceiling x <= 1) = (x <= 1)"
2003 unfolding natceiling_def
2004 by (simp add: nat_le_iff ceiling_le_iff)
2006 lemma natceiling_eq: "real n < x ==> x <= real n + 1 ==> natceiling x = n + 1"
2007 unfolding natceiling_def
2008 by (simp add: ceiling_eq2 [where n="int n"])
2010 lemma natceiling_add [simp]: "0 <= x ==>
2011 natceiling (x + real a) = natceiling x + a"
2012 unfolding natceiling_def
2013 unfolding real_of_int_of_nat_eq [symmetric] ceiling_add
2014 by (simp add: nat_add_distrib)
2016 lemma natceiling_add_numeral [simp]:
2017 "~ neg ((numeral n)::int) ==> 0 <= x ==>
2018 natceiling (x + numeral n) = natceiling x + numeral n"
2019 by (simp add: natceiling_add [symmetric])
2021 lemma natceiling_add_one: "0 <= x ==> natceiling(x + 1) = natceiling x + 1"
2022 by (simp add: natceiling_add [symmetric] del: One_nat_def)
2024 lemma natceiling_subtract [simp]: "natceiling(x - real a) = natceiling x - a"
2025 unfolding natceiling_def real_of_int_of_nat_eq [symmetric] ceiling_subtract
2028 subsection {* Exponentiation with floor *}
2031 assumes "x = real (floor x)"
2032 shows "floor (x ^ n) = floor x ^ n"
2034 have *: "x ^ n = real (floor x ^ n)"
2035 using assms by (induct n arbitrary: x) simp_all
2036 show ?thesis unfolding real_of_int_inject[symmetric]
2037 unfolding * floor_real_of_int ..
2040 lemma natfloor_power:
2041 assumes "x = real (natfloor x)"
2042 shows "natfloor (x ^ n) = natfloor x ^ n"
2044 from assms have "0 \<le> floor x" by auto
2045 note assms[unfolded natfloor_def real_nat_eq_real[OF `0 \<le> floor x`]]
2046 from floor_power[OF this]
2047 show ?thesis unfolding natfloor_def nat_power_eq[OF `0 \<le> floor x`, symmetric]
2052 subsection {* Implementation of rational real numbers *}
2054 text {* Formal constructor *}
2056 definition Ratreal :: "rat \<Rightarrow> real" where
2057 [code_abbrev, simp]: "Ratreal = of_rat"
2059 code_datatype Ratreal
2064 lemma [code_abbrev]:
2065 "(of_rat (of_int a) :: real) = of_int a"
2068 lemma [code_abbrev]:
2069 "(of_rat 0 :: real) = 0"
2072 lemma [code_abbrev]:
2073 "(of_rat 1 :: real) = 1"
2076 lemma [code_abbrev]:
2077 "(of_rat (numeral k) :: real) = numeral k"
2080 lemma [code_abbrev]:
2081 "(of_rat (neg_numeral k) :: real) = neg_numeral k"
2085 "(of_rat (0 / r) :: real) = 0"
2086 "(of_rat (r / 0) :: real) = 0"
2087 "(of_rat (1 / 1) :: real) = 1"
2088 "(of_rat (numeral k / 1) :: real) = numeral k"
2089 "(of_rat (neg_numeral k / 1) :: real) = neg_numeral k"
2090 "(of_rat (1 / numeral k) :: real) = 1 / numeral k"
2091 "(of_rat (1 / neg_numeral k) :: real) = 1 / neg_numeral k"
2092 "(of_rat (numeral k / numeral l) :: real) = numeral k / numeral l"
2093 "(of_rat (numeral k / neg_numeral l) :: real) = numeral k / neg_numeral l"
2094 "(of_rat (neg_numeral k / numeral l) :: real) = neg_numeral k / numeral l"
2095 "(of_rat (neg_numeral k / neg_numeral l) :: real) = neg_numeral k / neg_numeral l"
2096 by (simp_all add: of_rat_divide)
2099 text {* Operations *}
2101 lemma zero_real_code [code]:
2105 lemma one_real_code [code]:
2109 instantiation real :: equal
2112 definition "HOL.equal (x\<Colon>real) y \<longleftrightarrow> x - y = 0"
2115 qed (simp add: equal_real_def)
2117 lemma real_equal_code [code]:
2118 "HOL.equal (Ratreal x) (Ratreal y) \<longleftrightarrow> HOL.equal x y"
2119 by (simp add: equal_real_def equal)
2122 "HOL.equal (x::real) x \<longleftrightarrow> True"
2123 by (rule equal_refl)
2127 lemma real_less_eq_code [code]: "Ratreal x \<le> Ratreal y \<longleftrightarrow> x \<le> y"
2128 by (simp add: of_rat_less_eq)
2130 lemma real_less_code [code]: "Ratreal x < Ratreal y \<longleftrightarrow> x < y"
2131 by (simp add: of_rat_less)
2133 lemma real_plus_code [code]: "Ratreal x + Ratreal y = Ratreal (x + y)"
2134 by (simp add: of_rat_add)
2136 lemma real_times_code [code]: "Ratreal x * Ratreal y = Ratreal (x * y)"
2137 by (simp add: of_rat_mult)
2139 lemma real_uminus_code [code]: "- Ratreal x = Ratreal (- x)"
2140 by (simp add: of_rat_minus)
2142 lemma real_minus_code [code]: "Ratreal x - Ratreal y = Ratreal (x - y)"
2143 by (simp add: of_rat_diff)
2145 lemma real_inverse_code [code]: "inverse (Ratreal x) = Ratreal (inverse x)"
2146 by (simp add: of_rat_inverse)
2148 lemma real_divide_code [code]: "Ratreal x / Ratreal y = Ratreal (x / y)"
2149 by (simp add: of_rat_divide)
2151 lemma real_floor_code [code]: "floor (Ratreal x) = floor x"
2152 by (metis Ratreal_def floor_le_iff floor_unique le_floor_iff of_int_floor_le of_rat_of_int_eq real_less_eq_code)
2155 text {* Quickcheck *}
2157 definition (in term_syntax)
2158 valterm_ratreal :: "rat \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> real \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
2159 [code_unfold]: "valterm_ratreal k = Code_Evaluation.valtermify Ratreal {\<cdot>} k"
2161 notation fcomp (infixl "\<circ>>" 60)
2162 notation scomp (infixl "\<circ>\<rightarrow>" 60)
2164 instantiation real :: random
2168 "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>r. Pair (valterm_ratreal r))"
2174 no_notation fcomp (infixl "\<circ>>" 60)
2175 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
2177 instantiation real :: exhaustive
2181 "exhaustive_real f d = Quickcheck_Exhaustive.exhaustive (%r. f (Ratreal r)) d"
2187 instantiation real :: full_exhaustive
2191 "full_exhaustive_real f d = Quickcheck_Exhaustive.full_exhaustive (%r. f (valterm_ratreal r)) d"
2197 instantiation real :: narrowing
2201 "narrowing = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.cons Ratreal) narrowing"
2208 subsection {* Setup for Nitpick *}
2211 Nitpick_HOL.register_frac_type @{type_name real}
2212 [(@{const_name zero_real_inst.zero_real}, @{const_name Nitpick.zero_frac}),
2213 (@{const_name one_real_inst.one_real}, @{const_name Nitpick.one_frac}),
2214 (@{const_name plus_real_inst.plus_real}, @{const_name Nitpick.plus_frac}),
2215 (@{const_name times_real_inst.times_real}, @{const_name Nitpick.times_frac}),
2216 (@{const_name uminus_real_inst.uminus_real}, @{const_name Nitpick.uminus_frac}),
2217 (@{const_name inverse_real_inst.inverse_real}, @{const_name Nitpick.inverse_frac}),
2218 (@{const_name ord_real_inst.less_real}, @{const_name Nitpick.less_frac}),
2219 (@{const_name ord_real_inst.less_eq_real}, @{const_name Nitpick.less_eq_frac})]
2222 lemmas [nitpick_unfold] = inverse_real_inst.inverse_real one_real_inst.one_real
2223 ord_real_inst.less_real ord_real_inst.less_eq_real plus_real_inst.plus_real
2224 times_real_inst.times_real uminus_real_inst.uminus_real
2225 zero_real_inst.zero_real
2227 ML_file "Tools/SMT/smt_real.ML"
2228 setup SMT_Real.setup