src/HOL/Real.thy
 author blanchet Thu Sep 11 18:54:36 2014 +0200 (2014-09-11) changeset 58306 117ba6cbe414 parent 58134 b563ec62d04e child 58788 d17b3844b726 permissions -rw-r--r--
renamed 'rep_datatype' to 'old_rep_datatype' (HOL)
1 (*  Title:      HOL/Real.thy
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
8 *)
10 header {* Development of the Reals using Cauchy Sequences *}
12 theory Real
13 imports Rat Conditionally_Complete_Lattices
14 begin
16 text {*
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.
21 *}
23 subsection {* Preliminary lemmas *}
26   fixes a b c d :: "'a::ab_group_add"
27   shows "(a + c) - (b + d) = (a - b) + (c - d)"
28   by simp
30 lemma minus_diff_minus:
31   fixes a b :: "'a::ab_group_add"
32   shows "- a - - b = - (a - b)"
33   by simp
35 lemma mult_diff_mult:
36   fixes x y a b :: "'a::ring"
37   shows "(x * y - a * b) = x * (y - b) + (x - a) * b"
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)
46 lemma obtain_pos_sum:
47   fixes r :: rat assumes r: "0 < r"
48   obtains s t where "0 < s" and "0 < t" and "r = s + t"
49 proof
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
53 qed
55 subsection {* Sequences that converge to zero *}
57 definition
58   vanishes :: "(nat \<Rightarrow> rat) \<Rightarrow> bool"
59 where
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)
72   done
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"
89   proof (clarsimp)
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 .
94   qed
95   thus "\<exists>k. \<forall>n\<ge>k. \<bar>X n + Y n\<bar> < r" ..
96 qed
98 lemma vanishes_diff:
99   assumes X: "vanishes X" and Y: "vanishes Y"
100   shows "vanishes (\<lambda>n. X n - Y n)"
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"
110     using X by fast
111   obtain b where b: "0 < b" "r = a * b"
112   proof
113     show "0 < r / a" using r a by simp
114     show "r = a * (r / a)" using a by simp
115   qed
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" ..
121 qed
123 subsection {* Cauchy sequences *}
125 definition
126   cauchy :: "(nat \<Rightarrow> rat) \<Rightarrow> bool"
127 where
128   "cauchy X \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r)"
130 lemma cauchyI:
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
134 lemma cauchyD:
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
142   assumes X: "cauchy X" and Y: "cauchy Y"
143   shows "cauchy (\<lambda>n. X n + Y n)"
144 proof (rule cauchyI)
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"
153   proof (clarsimp)
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>"
157     also have "\<dots> < s + t"
159     finally show "\<bar>(X m + Y m) - (X n + Y n)\<bar> < r" unfolding r .
160   qed
161   thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>(X m + Y m) - (X n + Y n)\<bar> < r" ..
162 qed
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)"
175 lemma cauchy_imp_bounded:
176   assumes "cauchy X" shows "\<exists>b>0. \<forall>n. \<bar>X n\<bar> < b"
177 proof -
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
186   next
187     fix n :: nat
188     show "\<bar>X n\<bar> < Max (abs  X  {..k}) + 1"
189     proof (rule linorder_le_cases)
190       assume "n \<le> k"
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
193     next
194       assume "k \<le> n"
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" .
201     qed
202   qed
203 qed
205 lemma cauchy_mult [simp]:
206   assumes X: "cauchy X" and Y: "cauchy Y"
207   shows "cauchy (\<lambda>n. X n * Y n)"
208 proof (rule cauchyI)
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"
217   proof
218     show "0 < v/b" using v b(1) by simp
219     show "0 < u/a" using u a(1) by simp
220     show "r = a * (u/a) + (v/b) * b"
221       using a(1) b(1) r = u + v by simp
222   qed
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"
228   proof (clarsimp)
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"
238     finally show "\<bar>X m * Y m - X n * Y n\<bar> < r" unfolding r .
239   qed
240   thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m * Y m - X n * Y n\<bar> < r" ..
241 qed
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)"
247 proof -
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>"
255     using r by fast
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)"
264     using t by auto
265 qed
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))"
278 proof (rule cauchyI)
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"
284   proof
285     show "0 < b * r * b" by (simp add: 0 < r b)
286     show "r = inverse b * (b * r * b) * inverse b"
287       using b by simp
288   qed
289   obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>X m - X n\<bar> < s"
290     using cauchyD [OF X s] ..
291   have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>inverse (X m) - inverse (X n)\<bar> < r"
292   proof (clarsimp)
293     fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"
294     have "\<bar>inverse (X m) - inverse (X n)\<bar> =
295           inverse \<bar>X m\<bar> * \<bar>X m - X n\<bar> * inverse \<bar>X n\<bar>"
296       by (simp add: inverse_diff_inverse nz * abs_mult)
297     also have "\<dots> < inverse b * s * inverse b"
298       by (simp add: mult_strict_mono less_imp_inverse_less
299                     i j b * s)
300     finally show "\<bar>inverse (X m) - inverse (X n)\<bar> < r" unfolding r .
301   qed
302   thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>inverse (X m) - inverse (X n)\<bar> < r" ..
303 qed
305 lemma vanishes_diff_inverse:
306   assumes X: "cauchy X" "\<not> vanishes X"
307   assumes Y: "cauchy Y" "\<not> vanishes Y"
308   assumes XY: "vanishes (\<lambda>n. X n - Y n)"
309   shows "vanishes (\<lambda>n. inverse (X n) - inverse (Y n))"
310 proof (rule vanishesI)
311   fix r :: rat assume r: "0 < r"
312   obtain a i where a: "0 < a" and i: "\<forall>n\<ge>i. a < \<bar>X n\<bar>"
313     using cauchy_not_vanishes [OF X] by fast
314   obtain b j where b: "0 < b" and j: "\<forall>n\<ge>j. b < \<bar>Y n\<bar>"
315     using cauchy_not_vanishes [OF Y] by fast
316   obtain s where s: "0 < s" and "inverse a * s * inverse b = r"
317   proof
318     show "0 < a * r * b"
319       using a r b by simp
320     show "inverse a * (a * r * b) * inverse b = r"
321       using a r b by simp
322   qed
323   obtain k where k: "\<forall>n\<ge>k. \<bar>X n - Y n\<bar> < s"
324     using vanishesD [OF XY s] ..
325   have "\<forall>n\<ge>max (max i j) k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r"
326   proof (clarsimp)
327     fix n assume n: "i \<le> n" "j \<le> n" "k \<le> n"
328     have "X n \<noteq> 0" and "Y n \<noteq> 0"
329       using i j a b n by auto
330     hence "\<bar>inverse (X n) - inverse (Y n)\<bar> =
331         inverse \<bar>X n\<bar> * \<bar>X n - Y n\<bar> * inverse \<bar>Y n\<bar>"
332       by (simp add: inverse_diff_inverse abs_mult)
333     also have "\<dots> < inverse a * s * inverse b"
334       apply (intro mult_strict_mono' less_imp_inverse_less)
335       apply (simp_all add: a b i j k n)
336       done
337     also note inverse a * s * inverse b = r
338     finally show "\<bar>inverse (X n) - inverse (Y n)\<bar> < r" .
339   qed
340   thus "\<exists>k. \<forall>n\<ge>k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r" ..
341 qed
343 subsection {* Equivalence relation on Cauchy sequences *}
345 definition realrel :: "(nat \<Rightarrow> rat) \<Rightarrow> (nat \<Rightarrow> rat) \<Rightarrow> bool"
346   where "realrel = (\<lambda>X Y. cauchy X \<and> cauchy Y \<and> vanishes (\<lambda>n. X n - Y n))"
348 lemma realrelI [intro?]:
349   assumes "cauchy X" and "cauchy Y" and "vanishes (\<lambda>n. X n - Y n)"
350   shows "realrel X Y"
351   using assms unfolding realrel_def by simp
353 lemma realrel_refl: "cauchy X \<Longrightarrow> realrel X X"
354   unfolding realrel_def by simp
356 lemma symp_realrel: "symp realrel"
357   unfolding realrel_def
358   by (rule sympI, clarify, drule vanishes_minus, simp)
360 lemma transp_realrel: "transp realrel"
361   unfolding realrel_def
362   apply (rule transpI, clarify)
365   done
367 lemma part_equivp_realrel: "part_equivp realrel"
368   by (fast intro: part_equivpI symp_realrel transp_realrel
369     realrel_refl cauchy_const)
371 subsection {* The field of real numbers *}
373 quotient_type real = "nat \<Rightarrow> rat" / partial: realrel
374   morphisms rep_real Real
375   by (rule part_equivp_realrel)
377 lemma cr_real_eq: "pcr_real = (\<lambda>x y. cauchy x \<and> Real x = y)"
378   unfolding real.pcr_cr_eq cr_real_def realrel_def by auto
380 lemma Real_induct [induct type: real]: (* TODO: generate automatically *)
381   assumes "\<And>X. cauchy X \<Longrightarrow> P (Real X)" shows "P x"
382 proof (induct x)
383   case (1 X)
384   hence "cauchy X" by (simp add: realrel_def)
385   thus "P (Real X)" by (rule assms)
386 qed
388 lemma eq_Real:
389   "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> Real X = Real Y \<longleftrightarrow> vanishes (\<lambda>n. X n - Y n)"
390   using real.rel_eq_transfer
391   unfolding real.pcr_cr_eq cr_real_def rel_fun_def realrel_def by simp
393 lemma Domainp_pcr_real [transfer_domain_rule]: "Domainp pcr_real = cauchy"
394 by (simp add: real.domain_eq realrel_def)
396 instantiation real :: field_inverse_zero
397 begin
399 lift_definition zero_real :: "real" is "\<lambda>n. 0"
402 lift_definition one_real :: "real" is "\<lambda>n. 1"
405 lift_definition plus_real :: "real \<Rightarrow> real \<Rightarrow> real" is "\<lambda>X Y n. X n + Y n"
409 lift_definition uminus_real :: "real \<Rightarrow> real" is "\<lambda>X n. - X n"
410   unfolding realrel_def minus_diff_minus
411   by (simp only: cauchy_minus vanishes_minus simp_thms)
413 lift_definition times_real :: "real \<Rightarrow> real \<Rightarrow> real" is "\<lambda>X Y n. X n * Y n"
414   unfolding realrel_def mult_diff_mult
415   by (subst (4) mult.commute, simp only: cauchy_mult vanishes_add
416     vanishes_mult_bounded cauchy_imp_bounded simp_thms)
418 lift_definition inverse_real :: "real \<Rightarrow> real"
419   is "\<lambda>X. if vanishes X then (\<lambda>n. 0) else (\<lambda>n. inverse (X n))"
420 proof -
421   fix X Y assume "realrel X Y"
422   hence X: "cauchy X" and Y: "cauchy Y" and XY: "vanishes (\<lambda>n. X n - Y n)"
423     unfolding realrel_def by simp_all
424   have "vanishes X \<longleftrightarrow> vanishes Y"
425   proof
426     assume "vanishes X"
427     from vanishes_diff [OF this XY] show "vanishes Y" by simp
428   next
429     assume "vanishes Y"
430     from vanishes_add [OF this XY] show "vanishes X" by simp
431   qed
432   thus "?thesis X Y"
433     unfolding realrel_def
434     by (simp add: vanishes_diff_inverse X Y XY)
435 qed
437 definition
438   "x - y = (x::real) + - y"
440 definition
441   "x / y = (x::real) * inverse y"
444   assumes X: "cauchy X" and Y: "cauchy Y"
445   shows "Real X + Real Y = Real (\<lambda>n. X n + Y n)"
446   using assms plus_real.transfer
447   unfolding cr_real_eq rel_fun_def by simp
449 lemma minus_Real:
450   assumes X: "cauchy X"
451   shows "- Real X = Real (\<lambda>n. - X n)"
452   using assms uminus_real.transfer
453   unfolding cr_real_eq rel_fun_def by simp
455 lemma diff_Real:
456   assumes X: "cauchy X" and Y: "cauchy Y"
457   shows "Real X - Real Y = Real (\<lambda>n. X n - Y n)"
458   unfolding minus_real_def
461 lemma mult_Real:
462   assumes X: "cauchy X" and Y: "cauchy Y"
463   shows "Real X * Real Y = Real (\<lambda>n. X n * Y n)"
464   using assms times_real.transfer
465   unfolding cr_real_eq rel_fun_def by simp
467 lemma inverse_Real:
468   assumes X: "cauchy X"
469   shows "inverse (Real X) =
470     (if vanishes X then 0 else Real (\<lambda>n. inverse (X n)))"
471   using assms inverse_real.transfer zero_real.transfer
472   unfolding cr_real_eq rel_fun_def by (simp split: split_if_asm, metis)
474 instance proof
475   fix a b c :: real
476   show "a + b = b + a"
477     by transfer (simp add: ac_simps realrel_def)
478   show "(a + b) + c = a + (b + c)"
479     by transfer (simp add: ac_simps realrel_def)
480   show "0 + a = a"
481     by transfer (simp add: realrel_def)
482   show "- a + a = 0"
483     by transfer (simp add: realrel_def)
484   show "a - b = a + - b"
485     by (rule minus_real_def)
486   show "(a * b) * c = a * (b * c)"
487     by transfer (simp add: ac_simps realrel_def)
488   show "a * b = b * a"
489     by transfer (simp add: ac_simps realrel_def)
490   show "1 * a = a"
491     by transfer (simp add: ac_simps realrel_def)
492   show "(a + b) * c = a * c + b * c"
493     by transfer (simp add: distrib_right realrel_def)
494   show "(0\<Colon>real) \<noteq> (1\<Colon>real)"
495     by transfer (simp add: realrel_def)
496   show "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
497     apply transfer
499     apply (rule vanishesI)
500     apply (frule (1) cauchy_not_vanishes, clarify)
501     apply (rule_tac x=k in exI, clarify)
502     apply (drule_tac x=n in spec, simp)
503     done
504   show "a / b = a * inverse b"
505     by (rule divide_real_def)
506   show "inverse (0::real) = 0"
507     by transfer (simp add: realrel_def)
508 qed
510 end
512 subsection {* Positive reals *}
514 lift_definition positive :: "real \<Rightarrow> bool"
515   is "\<lambda>X. \<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n"
516 proof -
517   { fix X Y
518     assume "realrel X Y"
519     hence XY: "vanishes (\<lambda>n. X n - Y n)"
520       unfolding realrel_def by simp_all
521     assume "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n"
522     then obtain r i where "0 < r" and i: "\<forall>n\<ge>i. r < X n"
523       by fast
524     obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
525       using 0 < r by (rule obtain_pos_sum)
526     obtain j where j: "\<forall>n\<ge>j. \<bar>X n - Y n\<bar> < s"
527       using vanishesD [OF XY s] ..
528     have "\<forall>n\<ge>max i j. t < Y n"
529     proof (clarsimp)
530       fix n assume n: "i \<le> n" "j \<le> n"
531       have "\<bar>X n - Y n\<bar> < s" and "r < X n"
532         using i j n by simp_all
533       thus "t < Y n" unfolding r by simp
534     qed
535     hence "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < Y n" using t by fast
536   } note 1 = this
537   fix X Y assume "realrel X Y"
538   hence "realrel X Y" and "realrel Y X"
539     using symp_realrel unfolding symp_def by auto
540   thus "?thesis X Y"
541     by (safe elim!: 1)
542 qed
544 lemma positive_Real:
545   assumes X: "cauchy X"
546   shows "positive (Real X) \<longleftrightarrow> (\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n)"
547   using assms positive.transfer
548   unfolding cr_real_eq rel_fun_def by simp
550 lemma positive_zero: "\<not> positive 0"
551   by transfer auto
554   "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x + y)"
555 apply transfer
556 apply (clarify, rename_tac a b i j)
557 apply (rule_tac x="a + b" in exI, simp)
558 apply (rule_tac x="max i j" in exI, clarsimp)
560 done
562 lemma positive_mult:
563   "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x * y)"
564 apply transfer
565 apply (clarify, rename_tac a b i j)
566 apply (rule_tac x="a * b" in exI, simp)
567 apply (rule_tac x="max i j" in exI, clarsimp)
568 apply (rule mult_strict_mono, auto)
569 done
571 lemma positive_minus:
572   "\<not> positive x \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> positive (- x)"
573 apply transfer
575 apply (drule (1) cauchy_not_vanishes_cases, safe, fast, fast)
576 done
578 instantiation real :: linordered_field_inverse_zero
579 begin
581 definition
582   "x < y \<longleftrightarrow> positive (y - x)"
584 definition
585   "x \<le> (y::real) \<longleftrightarrow> x < y \<or> x = y"
587 definition
588   "abs (a::real) = (if a < 0 then - a else a)"
590 definition
591   "sgn (a::real) = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
593 instance proof
594   fix a b c :: real
595   show "\<bar>a\<bar> = (if a < 0 then - a else a)"
596     by (rule abs_real_def)
597   show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a"
598     unfolding less_eq_real_def less_real_def
600   show "a \<le> a"
601     unfolding less_eq_real_def by simp
602   show "a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
603     unfolding less_eq_real_def less_real_def
605   show "a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> a = b"
606     unfolding less_eq_real_def less_real_def
608   show "a \<le> b \<Longrightarrow> c + a \<le> c + b"
609     unfolding less_eq_real_def less_real_def by auto
610     (* FIXME: Procedure int_combine_numerals: c + b - (c + a) \<equiv> b + - a *)
611     (* Should produce c + b - (c + a) \<equiv> b - a *)
612   show "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
613     by (rule sgn_real_def)
614   show "a \<le> b \<or> b \<le> a"
615     unfolding less_eq_real_def less_real_def
616     by (auto dest!: positive_minus)
617   show "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
618     unfolding less_real_def
619     by (drule (1) positive_mult, simp add: algebra_simps)
620 qed
622 end
624 instantiation real :: distrib_lattice
625 begin
627 definition
628   "(inf :: real \<Rightarrow> real \<Rightarrow> real) = min"
630 definition
631   "(sup :: real \<Rightarrow> real \<Rightarrow> real) = max"
633 instance proof
634 qed (auto simp add: inf_real_def sup_real_def max_min_distrib2)
636 end
638 lemma of_nat_Real: "of_nat x = Real (\<lambda>n. of_nat x)"
639 apply (induct x)
642 done
644 lemma of_int_Real: "of_int x = Real (\<lambda>n. of_int x)"
645 apply (cases x rule: int_diff_cases)
646 apply (simp add: of_nat_Real diff_Real)
647 done
649 lemma of_rat_Real: "of_rat x = Real (\<lambda>n. x)"
650 apply (induct x)
651 apply (simp add: Fract_of_int_quotient of_rat_divide)
652 apply (simp add: of_int_Real divide_inverse)
653 apply (simp add: inverse_Real mult_Real)
654 done
656 instance real :: archimedean_field
657 proof
658   fix x :: real
659   show "\<exists>z. x \<le> of_int z"
660     apply (induct x)
661     apply (frule cauchy_imp_bounded, clarify)
662     apply (rule_tac x="ceiling b + 1" in exI)
663     apply (rule less_imp_le)
664     apply (simp add: of_int_Real less_real_def diff_Real positive_Real)
665     apply (rule_tac x=1 in exI, simp add: algebra_simps)
666     apply (rule_tac x=0 in exI, clarsimp)
667     apply (rule le_less_trans [OF abs_ge_self])
668     apply (rule less_le_trans [OF _ le_of_int_ceiling])
669     apply simp
670     done
671 qed
673 instantiation real :: floor_ceiling
674 begin
676 definition [code del]:
677   "floor (x::real) = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))"
679 instance proof
680   fix x :: real
681   show "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
682     unfolding floor_real_def using floor_exists1 by (rule theI')
683 qed
685 end
687 subsection {* Completeness *}
689 lemma not_positive_Real:
690   assumes X: "cauchy X"
691   shows "\<not> positive (Real X) \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> r)"
692 unfolding positive_Real [OF X]
693 apply (auto, unfold not_less)
694 apply (erule obtain_pos_sum)
695 apply (drule_tac x=s in spec, simp)
696 apply (drule_tac r=t in cauchyD [OF X], clarify)
697 apply (drule_tac x=k in spec, clarsimp)
698 apply (rule_tac x=n in exI, clarify, rename_tac m)
699 apply (drule_tac x=m in spec, simp)
700 apply (drule_tac x=n in spec, simp)
701 apply (drule spec, drule (1) mp, clarify, rename_tac i)
702 apply (rule_tac x="max i k" in exI, simp)
703 done
705 lemma le_Real:
706   assumes X: "cauchy X" and Y: "cauchy Y"
707   shows "Real X \<le> Real Y = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r)"
708 unfolding not_less [symmetric, where 'a=real] less_real_def
709 apply (simp add: diff_Real not_positive_Real X Y)
710 apply (simp add: diff_le_eq ac_simps)
711 done
713 lemma le_RealI:
714   assumes Y: "cauchy Y"
715   shows "\<forall>n. x \<le> of_rat (Y n) \<Longrightarrow> x \<le> Real Y"
716 proof (induct x)
717   fix X assume X: "cauchy X" and "\<forall>n. Real X \<le> of_rat (Y n)"
718   hence le: "\<And>m r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. X n \<le> Y m + r"
719     by (simp add: of_rat_Real le_Real)
720   {
721     fix r :: rat assume "0 < r"
722     then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
723       by (rule obtain_pos_sum)
724     obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>Y m - Y n\<bar> < s"
725       using cauchyD [OF Y s] ..
726     obtain j where j: "\<forall>n\<ge>j. X n \<le> Y i + t"
727       using le [OF t] ..
728     have "\<forall>n\<ge>max i j. X n \<le> Y n + r"
729     proof (clarsimp)
730       fix n assume n: "i \<le> n" "j \<le> n"
731       have "X n \<le> Y i + t" using n j by simp
732       moreover have "\<bar>Y i - Y n\<bar> < s" using n i by simp
733       ultimately show "X n \<le> Y n + r" unfolding r by simp
734     qed
735     hence "\<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r" ..
736   }
737   thus "Real X \<le> Real Y"
738     by (simp add: of_rat_Real le_Real X Y)
739 qed
741 lemma Real_leI:
742   assumes X: "cauchy X"
743   assumes le: "\<forall>n. of_rat (X n) \<le> y"
744   shows "Real X \<le> y"
745 proof -
746   have "- y \<le> - Real X"
747     by (simp add: minus_Real X le_RealI of_rat_minus le)
748   thus ?thesis by simp
749 qed
751 lemma less_RealD:
752   assumes Y: "cauchy Y"
753   shows "x < Real Y \<Longrightarrow> \<exists>n. x < of_rat (Y n)"
754 by (erule contrapos_pp, simp add: not_less, erule Real_leI [OF Y])
756 lemma of_nat_less_two_power:
757   "of_nat n < (2::'a::linordered_idom) ^ n"
758 apply (induct n)
759 apply simp
760 apply (subgoal_tac "(1::'a) \<le> 2 ^ n")
761 apply (drule (1) add_le_less_mono, simp)
762 apply simp
763 done
765 lemma complete_real:
766   fixes S :: "real set"
767   assumes "\<exists>x. x \<in> S" and "\<exists>z. \<forall>x\<in>S. x \<le> z"
768   shows "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)"
769 proof -
770   obtain x where x: "x \<in> S" using assms(1) ..
771   obtain z where z: "\<forall>x\<in>S. x \<le> z" using assms(2) ..
773   def P \<equiv> "\<lambda>x. \<forall>y\<in>S. y \<le> of_rat x"
774   obtain a where a: "\<not> P a"
775   proof
776     have "of_int (floor (x - 1)) \<le> x - 1" by (rule of_int_floor_le)
777     also have "x - 1 < x" by simp
778     finally have "of_int (floor (x - 1)) < x" .
779     hence "\<not> x \<le> of_int (floor (x - 1))" by (simp only: not_le)
780     then show "\<not> P (of_int (floor (x - 1)))"
781       unfolding P_def of_rat_of_int_eq using x by fast
782   qed
783   obtain b where b: "P b"
784   proof
785     show "P (of_int (ceiling z))"
786     unfolding P_def of_rat_of_int_eq
787     proof
788       fix y assume "y \<in> S"
789       hence "y \<le> z" using z by simp
790       also have "z \<le> of_int (ceiling z)" by (rule le_of_int_ceiling)
791       finally show "y \<le> of_int (ceiling z)" .
792     qed
793   qed
795   def avg \<equiv> "\<lambda>x y :: rat. x/2 + y/2"
796   def bisect \<equiv> "\<lambda>(x, y). if P (avg x y) then (x, avg x y) else (avg x y, y)"
797   def A \<equiv> "\<lambda>n. fst ((bisect ^^ n) (a, b))"
798   def B \<equiv> "\<lambda>n. snd ((bisect ^^ n) (a, b))"
799   def C \<equiv> "\<lambda>n. avg (A n) (B n)"
800   have A_0 [simp]: "A 0 = a" unfolding A_def by simp
801   have B_0 [simp]: "B 0 = b" unfolding B_def by simp
802   have A_Suc [simp]: "\<And>n. A (Suc n) = (if P (C n) then A n else C n)"
803     unfolding A_def B_def C_def bisect_def split_def by simp
804   have B_Suc [simp]: "\<And>n. B (Suc n) = (if P (C n) then C n else B n)"
805     unfolding A_def B_def C_def bisect_def split_def by simp
807   have width: "\<And>n. B n - A n = (b - a) / 2^n"
809     apply (induct_tac n, simp)
810     apply (simp add: C_def avg_def algebra_simps)
811     done
813   have twos: "\<And>y r :: rat. 0 < r \<Longrightarrow> \<exists>n. y / 2 ^ n < r"
815     apply (subst mult.commute)
816     apply (frule_tac y=y in ex_less_of_nat_mult)
817     apply clarify
818     apply (rule_tac x=n in exI)
819     apply (erule less_trans)
820     apply (rule mult_strict_right_mono)
821     apply (rule le_less_trans [OF _ of_nat_less_two_power])
822     apply simp
823     apply assumption
824     done
826   have PA: "\<And>n. \<not> P (A n)"
827     by (induct_tac n, simp_all add: a)
828   have PB: "\<And>n. P (B n)"
829     by (induct_tac n, simp_all add: b)
830   have ab: "a < b"
831     using a b unfolding P_def
832     apply (clarsimp simp add: not_le)
833     apply (drule (1) bspec)
834     apply (drule (1) less_le_trans)
836     done
837   have AB: "\<And>n. A n < B n"
839   have A_mono: "\<And>i j. i \<le> j \<Longrightarrow> A i \<le> A j"
840     apply (auto simp add: le_less [where 'a=nat])
841     apply (erule less_Suc_induct)
842     apply (clarsimp simp add: C_def avg_def)
844     apply (rule AB [THEN less_imp_le])
845     apply simp
846     done
847   have B_mono: "\<And>i j. i \<le> j \<Longrightarrow> B j \<le> B i"
848     apply (auto simp add: le_less [where 'a=nat])
849     apply (erule less_Suc_induct)
850     apply (clarsimp simp add: C_def avg_def)
852     apply (rule AB [THEN less_imp_le])
853     apply simp
854     done
855   have cauchy_lemma:
856     "\<And>X. \<forall>n. \<forall>i\<ge>n. A n \<le> X i \<and> X i \<le> B n \<Longrightarrow> cauchy X"
857     apply (rule cauchyI)
858     apply (drule twos [where y="b - a"])
859     apply (erule exE)
860     apply (rule_tac x=n in exI, clarify, rename_tac i j)
861     apply (rule_tac y="B n - A n" in le_less_trans) defer
863     apply (drule_tac x=n in spec)
864     apply (frule_tac x=i in spec, drule (1) mp)
865     apply (frule_tac x=j in spec, drule (1) mp)
866     apply (frule A_mono, drule B_mono)
867     apply (frule A_mono, drule B_mono)
868     apply arith
869     done
870   have "cauchy A"
871     apply (rule cauchy_lemma [rule_format])
873     apply (erule order_trans [OF less_imp_le [OF AB] B_mono])
874     done
875   have "cauchy B"
876     apply (rule cauchy_lemma [rule_format])
878     apply (erule order_trans [OF A_mono less_imp_le [OF AB]])
879     done
880   have 1: "\<forall>x\<in>S. x \<le> Real B"
881   proof
882     fix x assume "x \<in> S"
883     then show "x \<le> Real B"
884       using PB [unfolded P_def] cauchy B
886   qed
887   have 2: "\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> Real A \<le> z"
888     apply clarify
889     apply (erule contrapos_pp)
891     apply (drule less_RealD [OF cauchy A], clarify)
892     apply (subgoal_tac "\<not> P (A n)")
893     apply (simp add: P_def not_le, clarify)
894     apply (erule rev_bexI)
895     apply (erule (1) less_trans)
897     done
898   have "vanishes (\<lambda>n. (b - a) / 2 ^ n)"
899   proof (rule vanishesI)
900     fix r :: rat assume "0 < r"
901     then obtain k where k: "\<bar>b - a\<bar> / 2 ^ k < r"
902       using twos by fast
903     have "\<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r"
904     proof (clarify)
905       fix n assume n: "k \<le> n"
906       have "\<bar>(b - a) / 2 ^ n\<bar> = \<bar>b - a\<bar> / 2 ^ n"
907         by simp
908       also have "\<dots> \<le> \<bar>b - a\<bar> / 2 ^ k"
909         using n by (simp add: divide_left_mono)
910       also note k
911       finally show "\<bar>(b - a) / 2 ^ n\<bar> < r" .
912     qed
913     thus "\<exists>k. \<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r" ..
914   qed
915   hence 3: "Real B = Real A"
916     by (simp add: eq_Real cauchy A cauchy B width)
917   show "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)"
918     using 1 2 3 by (rule_tac x="Real B" in exI, simp)
919 qed
921 instantiation real :: linear_continuum
922 begin
924 subsection{*Supremum of a set of reals*}
926 definition "Sup X = (LEAST z::real. \<forall>x\<in>X. x \<le> z)"
927 definition "Inf (X::real set) = - Sup (uminus  X)"
929 instance
930 proof
931   { fix x :: real and X :: "real set"
932     assume x: "x \<in> X" "bdd_above X"
933     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"
934       using complete_real[of X] unfolding bdd_above_def by blast
935     then show "x \<le> Sup X"
936       unfolding Sup_real_def by (rule LeastI2_order) (auto simp: x) }
937   note Sup_upper = this
939   { fix z :: real and X :: "real set"
940     assume x: "X \<noteq> {}" and z: "\<And>x. x \<in> X \<Longrightarrow> x \<le> z"
941     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"
942       using complete_real[of X] by blast
943     then have "Sup X = s"
944       unfolding Sup_real_def by (best intro: Least_equality)
945     also from s z have "... \<le> z"
946       by blast
947     finally show "Sup X \<le> z" . }
948   note Sup_least = this
950   { fix x :: real and X :: "real set" assume x: "x \<in> X" "bdd_below X" then show "Inf X \<le> x"
951       using Sup_upper[of "-x" "uminus  X"] by (auto simp: Inf_real_def) }
952   { fix z :: real and X :: "real set" assume "X \<noteq> {}" "\<And>x. x \<in> X \<Longrightarrow> z \<le> x" then show "z \<le> Inf X"
953       using Sup_least[of "uminus  X" "- z"] by (force simp: Inf_real_def) }
954   show "\<exists>a b::real. a \<noteq> b"
955     using zero_neq_one by blast
956 qed
957 end
960 subsection {* Hiding implementation details *}
962 hide_const (open) vanishes cauchy positive Real
964 declare Real_induct [induct del]
965 declare Abs_real_induct [induct del]
966 declare Abs_real_cases [cases del]
968 lifting_update real.lifting
969 lifting_forget real.lifting
971 subsection{*More Lemmas*}
973 text {* BH: These lemmas should not be necessary; they should be
974 covered by existing simp rules and simplification procedures. *}
976 lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
977 by simp (* solved by linordered_ring_less_cancel_factor simproc *)
979 lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)"
980 by simp (* solved by linordered_ring_le_cancel_factor simproc *)
982 lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
983 by simp (* solved by linordered_ring_le_cancel_factor simproc *)
986 subsection {* Embedding numbers into the Reals *}
988 abbreviation
989   real_of_nat :: "nat \<Rightarrow> real"
990 where
991   "real_of_nat \<equiv> of_nat"
993 abbreviation
994   real_of_int :: "int \<Rightarrow> real"
995 where
996   "real_of_int \<equiv> of_int"
998 abbreviation
999   real_of_rat :: "rat \<Rightarrow> real"
1000 where
1001   "real_of_rat \<equiv> of_rat"
1003 class real_of =
1004   fixes real :: "'a \<Rightarrow> real"
1006 instantiation nat :: real_of
1007 begin
1009 definition real_nat :: "nat \<Rightarrow> real" where real_of_nat_def [code_unfold]: "real \<equiv> of_nat"
1011 instance ..
1012 end
1014 instantiation int :: real_of
1015 begin
1017 definition real_int :: "int \<Rightarrow> real" where real_of_int_def [code_unfold]: "real \<equiv> of_int"
1019 instance ..
1020 end
1022 declare [[coercion_enabled]]
1023 declare [[coercion "real::nat\<Rightarrow>real"]]
1024 declare [[coercion "real::int\<Rightarrow>real"]]
1025 declare [[coercion "int"]]
1027 declare [[coercion_map map]]
1028 declare [[coercion_map "% f g h x. g (h (f x))"]]
1029 declare [[coercion_map "% f g (x,y) . (f x, g y)"]]
1031 lemma real_eq_of_nat: "real = of_nat"
1032   unfolding real_of_nat_def ..
1034 lemma real_eq_of_int: "real = of_int"
1035   unfolding real_of_int_def ..
1037 lemma real_of_int_zero [simp]: "real (0::int) = 0"
1040 lemma real_of_one [simp]: "real (1::int) = (1::real)"
1043 lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y"
1046 lemma real_of_int_minus [simp]: "real(-x) = -real (x::int)"
1049 lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y"
1052 lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y"
1055 lemma real_of_int_power [simp]: "real (x ^ n) = real (x::int) ^ n"
1056 by (simp add: real_of_int_def of_int_power)
1058 lemmas power_real_of_int = real_of_int_power [symmetric]
1060 lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))"
1061   apply (subst real_eq_of_int)+
1062   apply (rule of_int_setsum)
1063 done
1065 lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) =
1066     (PROD x:A. real(f x))"
1067   apply (subst real_eq_of_int)+
1068   apply (rule of_int_setprod)
1069 done
1071 lemma real_of_int_zero_cancel [simp, algebra, presburger]: "(real x = 0) = (x = (0::int))"
1074 lemma real_of_int_inject [iff, algebra, presburger]: "(real (x::int) = real y) = (x = y)"
1077 lemma real_of_int_less_iff [iff, presburger]: "(real (x::int) < real y) = (x < y)"
1080 lemma real_of_int_le_iff [simp, presburger]: "(real (x::int) \<le> real y) = (x \<le> y)"
1083 lemma real_of_int_gt_zero_cancel_iff [simp, presburger]: "(0 < real (n::int)) = (0 < n)"
1086 lemma real_of_int_ge_zero_cancel_iff [simp, presburger]: "(0 <= real (n::int)) = (0 <= n)"
1089 lemma real_of_int_lt_zero_cancel_iff [simp, presburger]: "(real (n::int) < 0) = (n < 0)"
1092 lemma real_of_int_le_zero_cancel_iff [simp, presburger]: "(real (n::int) <= 0) = (n <= 0)"
1095 lemma one_less_real_of_int_cancel_iff: "1 < real (i :: int) \<longleftrightarrow> 1 < i"
1096   unfolding real_of_one[symmetric] real_of_int_less_iff ..
1098 lemma one_le_real_of_int_cancel_iff: "1 \<le> real (i :: int) \<longleftrightarrow> 1 \<le> i"
1099   unfolding real_of_one[symmetric] real_of_int_le_iff ..
1101 lemma real_of_int_less_one_cancel_iff: "real (i :: int) < 1 \<longleftrightarrow> i < 1"
1102   unfolding real_of_one[symmetric] real_of_int_less_iff ..
1104 lemma real_of_int_le_one_cancel_iff: "real (i :: int) \<le> 1 \<longleftrightarrow> i \<le> 1"
1105   unfolding real_of_one[symmetric] real_of_int_le_iff ..
1107 lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))"
1108 by (auto simp add: abs_if)
1110 lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)"
1111   apply (subgoal_tac "real n + 1 = real (n + 1)")
1113   apply auto
1114 done
1116 lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)"
1117   apply (subgoal_tac "real m + 1 = real (m + 1)")
1119   apply simp
1120 done
1122 lemma real_of_int_div_aux: "(real (x::int)) / (real d) =
1123     real (x div d) + (real (x mod d)) / (real d)"
1124 proof -
1125   have "x = (x div d) * d + x mod d"
1126     by auto
1127   then have "real x = real (x div d) * real d + real(x mod d)"
1128     by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym])
1129   then have "real x / real d = ... / real d"
1130     by simp
1131   then show ?thesis
1133 qed
1135 lemma real_of_int_div: "(d :: int) dvd n ==>
1136     real(n div d) = real n / real d"
1137   apply (subst real_of_int_div_aux)
1138   apply simp
1140 done
1142 lemma real_of_int_div2:
1143   "0 <= real (n::int) / real (x) - real (n div x)"
1144   apply (case_tac "x = 0")
1145   apply simp
1146   apply (case_tac "0 < x")
1148   apply (subst real_of_int_div_aux)
1149   apply simp
1151   apply (subst real_of_int_div_aux)
1152   apply simp
1153   apply (subst zero_le_divide_iff)
1154   apply auto
1155 done
1157 lemma real_of_int_div3:
1158   "real (n::int) / real (x) - real (n div x) <= 1"
1160   apply (subst real_of_int_div_aux)
1161   apply (auto simp add: divide_le_eq intro: order_less_imp_le)
1162 done
1164 lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x"
1165 by (insert real_of_int_div2 [of n x], simp)
1167 lemma Ints_real_of_int [simp]: "real (x::int) \<in> Ints"
1168 unfolding real_of_int_def by (rule Ints_of_int)
1171 subsection{*Embedding the Naturals into the Reals*}
1173 lemma real_of_nat_zero [simp]: "real (0::nat) = 0"
1176 lemma real_of_nat_1 [simp]: "real (1::nat) = 1"
1179 lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"
1182 lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"
1185 (*Not for addsimps: often the LHS is used to represent a positive natural*)
1186 lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"
1189 lemma real_of_nat_less_iff [iff]:
1190      "(real (n::nat) < real m) = (n < m)"
1193 lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"
1196 lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"
1199 lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
1200 by (simp add: real_of_nat_def del: of_nat_Suc)
1202 lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
1203 by (simp add: real_of_nat_def of_nat_mult)
1205 lemma real_of_nat_power [simp]: "real (m ^ n) = real (m::nat) ^ n"
1206 by (simp add: real_of_nat_def of_nat_power)
1208 lemmas power_real_of_nat = real_of_nat_power [symmetric]
1210 lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) =
1211     (SUM x:A. real(f x))"
1212   apply (subst real_eq_of_nat)+
1213   apply (rule of_nat_setsum)
1214 done
1216 lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) =
1217     (PROD x:A. real(f x))"
1218   apply (subst real_eq_of_nat)+
1219   apply (rule of_nat_setprod)
1220 done
1222 lemma real_of_card: "real (card A) = setsum (%x.1) A"
1223   apply (subst card_eq_setsum)
1224   apply (subst real_of_nat_setsum)
1225   apply simp
1226 done
1228 lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
1231 lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"
1234 lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"
1237 lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"
1238 by (auto simp: real_of_nat_def)
1240 lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"
1243 lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"
1246 lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)"
1247   apply (subgoal_tac "real n + 1 = real (Suc n)")
1248   apply simp
1249   apply (auto simp add: real_of_nat_Suc)
1250 done
1252 lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"
1253   apply (subgoal_tac "real m + 1 = real (Suc m)")
1256 done
1258 lemma real_of_nat_div_aux: "(real (x::nat)) / (real d) =
1259     real (x div d) + (real (x mod d)) / (real d)"
1260 proof -
1261   have "x = (x div d) * d + x mod d"
1262     by auto
1263   then have "real x = real (x div d) * real d + real(x mod d)"
1264     by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym])
1265   then have "real x / real d = \<dots> / real d"
1266     by simp
1267   then show ?thesis
1269 qed
1271 lemma real_of_nat_div: "(d :: nat) dvd n ==>
1272     real(n div d) = real n / real d"
1273   by (subst real_of_nat_div_aux)
1274     (auto simp add: dvd_eq_mod_eq_0 [symmetric])
1276 lemma real_of_nat_div2:
1277   "0 <= real (n::nat) / real (x) - real (n div x)"
1279 apply (subst real_of_nat_div_aux)
1280 apply simp
1281 done
1283 lemma real_of_nat_div3:
1284   "real (n::nat) / real (x) - real (n div x) <= 1"
1285 apply(case_tac "x = 0")
1286 apply (simp)
1288 apply (subst real_of_nat_div_aux)
1289 apply simp
1290 done
1292 lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x"
1293 by (insert real_of_nat_div2 [of n x], simp)
1295 lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"
1296 by (simp add: real_of_int_def real_of_nat_def)
1298 lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x"
1299   apply (subgoal_tac "real(int(nat x)) = real(nat x)")
1300   apply force
1301   apply (simp only: real_of_int_of_nat_eq)
1302 done
1304 lemma Nats_real_of_nat [simp]: "real (n::nat) \<in> Nats"
1305 unfolding real_of_nat_def by (rule of_nat_in_Nats)
1307 lemma Ints_real_of_nat [simp]: "real (n::nat) \<in> Ints"
1308 unfolding real_of_nat_def by (rule Ints_of_nat)
1310 subsection {* The Archimedean Property of the Reals *}
1312 theorem reals_Archimedean:
1313   assumes x_pos: "0 < x"
1314   shows "\<exists>n. inverse (real (Suc n)) < x"
1315   unfolding real_of_nat_def using x_pos
1316   by (rule ex_inverse_of_nat_Suc_less)
1318 lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"
1319   unfolding real_of_nat_def by (rule ex_less_of_nat)
1321 lemma reals_Archimedean3:
1322   assumes x_greater_zero: "0 < x"
1323   shows "\<forall>(y::real). \<exists>(n::nat). y < real n * x"
1324   unfolding real_of_nat_def using 0 < x
1325   by (auto intro: ex_less_of_nat_mult)
1328 subsection{* Rationals *}
1330 lemma Rats_real_nat[simp]: "real(n::nat) \<in> \<rat>"
1333 lemma Rats_eq_int_div_int:
1334   "\<rat> = { real(i::int)/real(j::int) |i j. j \<noteq> 0}" (is "_ = ?S")
1335 proof
1336   show "\<rat> \<subseteq> ?S"
1337   proof
1338     fix x::real assume "x : \<rat>"
1339     then obtain r where "x = of_rat r" unfolding Rats_def ..
1340     have "of_rat r : ?S"
1341       by (cases r)(auto simp add:of_rat_rat real_eq_of_int)
1342     thus "x : ?S" using x = of_rat r by simp
1343   qed
1344 next
1345   show "?S \<subseteq> \<rat>"
1346   proof(auto simp:Rats_def)
1347     fix i j :: int assume "j \<noteq> 0"
1348     hence "real i / real j = of_rat(Fract i j)"
1350     thus "real i / real j \<in> range of_rat" by blast
1351   qed
1352 qed
1354 lemma Rats_eq_int_div_nat:
1355   "\<rat> = { real(i::int)/real(n::nat) |i n. n \<noteq> 0}"
1356 proof(auto simp:Rats_eq_int_div_int)
1357   fix i j::int assume "j \<noteq> 0"
1358   show "EX (i'::int) (n::nat). real i/real j = real i'/real n \<and> 0<n"
1359   proof cases
1360     assume "j>0"
1361     hence "real i/real j = real i/real(nat j) \<and> 0<nat j"
1362       by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)
1363     thus ?thesis by blast
1364   next
1365     assume "~ j>0"
1366     hence "real i/real j = real(-i)/real(nat(-j)) \<and> 0<nat(-j)" using j\<noteq>0
1367       by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)
1368     thus ?thesis by blast
1369   qed
1370 next
1371   fix i::int and n::nat assume "0 < n"
1372   hence "real i/real n = real i/real(int n) \<and> int n \<noteq> 0" by simp
1373   thus "\<exists>(i'::int) j::int. real i/real n = real i'/real j \<and> j \<noteq> 0" by blast
1374 qed
1376 lemma Rats_abs_nat_div_natE:
1377   assumes "x \<in> \<rat>"
1378   obtains m n :: nat
1379   where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "gcd m n = 1"
1380 proof -
1381   from x \<in> \<rat> obtain i::int and n::nat where "n \<noteq> 0" and "x = real i / real n"
1383   hence "\<bar>x\<bar> = real(nat(abs i)) / real n" by simp
1384   then obtain m :: nat where x_rat: "\<bar>x\<bar> = real m / real n" by blast
1385   let ?gcd = "gcd m n"
1386   from n\<noteq>0 have gcd: "?gcd \<noteq> 0" by simp
1387   let ?k = "m div ?gcd"
1388   let ?l = "n div ?gcd"
1389   let ?gcd' = "gcd ?k ?l"
1390   have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m"
1391     by (rule dvd_mult_div_cancel)
1392   have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n"
1393     by (rule dvd_mult_div_cancel)
1394   from n\<noteq>0 and gcd_l have "?l \<noteq> 0" by (auto iff del: neq0_conv)
1395   moreover
1396   have "\<bar>x\<bar> = real ?k / real ?l"
1397   proof -
1398     from gcd have "real ?k / real ?l =
1399         real (?gcd * ?k) / real (?gcd * ?l)" by simp
1400     also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp
1401     also from x_rat have "\<dots> = \<bar>x\<bar>" ..
1402     finally show ?thesis ..
1403   qed
1404   moreover
1405   have "?gcd' = 1"
1406   proof -
1407     have "?gcd * ?gcd' = gcd (?gcd * ?k) (?gcd * ?l)"
1408       by (rule gcd_mult_distrib_nat)
1409     with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp
1410     with gcd show ?thesis by auto
1411   qed
1412   ultimately show ?thesis ..
1413 qed
1415 subsection{*Density of the Rational Reals in the Reals*}
1417 text{* This density proof is due to Stefan Richter and was ported by TN.  The
1418 original source is \emph{Real Analysis} by H.L. Royden.
1419 It employs the Archimedean property of the reals. *}
1421 lemma Rats_dense_in_real:
1422   fixes x :: real
1423   assumes "x < y" shows "\<exists>r\<in>\<rat>. x < r \<and> r < y"
1424 proof -
1425   from x<y have "0 < y-x" by simp
1426   with reals_Archimedean obtain q::nat
1427     where q: "inverse (real q) < y-x" and "0 < q" by auto
1428   def p \<equiv> "ceiling (y * real q) - 1"
1429   def r \<equiv> "of_int p / real q"
1430   from q have "x < y - inverse (real q)" by simp
1431   also have "y - inverse (real q) \<le> r"
1432     unfolding r_def p_def
1433     by (simp add: le_divide_eq left_diff_distrib le_of_int_ceiling 0 < q)
1434   finally have "x < r" .
1435   moreover have "r < y"
1436     unfolding r_def p_def
1437     by (simp add: divide_less_eq diff_less_eq 0 < q
1438       less_ceiling_iff [symmetric])
1439   moreover from r_def have "r \<in> \<rat>" by simp
1440   ultimately show ?thesis by fast
1441 qed
1443 lemma of_rat_dense:
1444   fixes x y :: real
1445   assumes "x < y"
1446   shows "\<exists>q :: rat. x < of_rat q \<and> of_rat q < y"
1447 using Rats_dense_in_real [OF x < y]
1448 by (auto elim: Rats_cases)
1451 subsection{*Numerals and Arithmetic*}
1453 lemma [code_abbrev]:
1454   "real_of_int (numeral k) = numeral k"
1455   "real_of_int (- numeral k) = - numeral k"
1456   by simp_all
1458 text{*Collapse applications of @{const real} to @{const numeral}*}
1459 lemma real_numeral [simp]:
1460   "real (numeral v :: int) = numeral v"
1461   "real (- numeral v :: int) = - numeral v"
1464 lemma  real_of_nat_numeral [simp]:
1465   "real (numeral v :: nat) = numeral v"
1468 declaration {*
1469   K (Lin_Arith.add_inj_thms [@{thm real_of_nat_le_iff} RS iffD2, @{thm real_of_nat_inject} RS iffD2]
1470     (* not needed because x < (y::nat) can be rewritten as Suc x <= y: real_of_nat_less_iff RS iffD2 *)
1471   #> Lin_Arith.add_inj_thms [@{thm real_of_int_le_iff} RS iffD2, @{thm real_of_int_inject} RS iffD2]
1472     (* not needed because x < (y::int) can be rewritten as x + 1 <= y: real_of_int_less_iff RS iffD2 *)
1474       @{thm real_of_nat_mult}, @{thm real_of_int_zero}, @{thm real_of_one},
1475       @{thm real_of_int_add}, @{thm real_of_int_minus}, @{thm real_of_int_diff},
1476       @{thm real_of_int_mult}, @{thm real_of_int_of_nat_eq},
1477       @{thm real_of_nat_numeral}, @{thm real_numeral(1)}, @{thm real_numeral(2)},
1478       @{thm real_of_int_def[symmetric]}, @{thm real_of_nat_def[symmetric]}]
1479   #> Lin_Arith.add_inj_const (@{const_name real}, @{typ "nat \<Rightarrow> real"})
1480   #> Lin_Arith.add_inj_const (@{const_name real}, @{typ "int \<Rightarrow> real"})
1481   #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat \<Rightarrow> real"})
1482   #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int \<Rightarrow> real"}))
1483 *}
1485 subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}
1487 lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)"
1488 by arith
1490 text {* FIXME: redundant with @{text add_eq_0_iff} below *}
1491 lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)"
1492 by auto
1494 lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"
1495 by auto
1497 lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)"
1498 by auto
1500 lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> -x)"
1501 by auto
1503 lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (-x \<le> y)"
1504 by auto
1506 subsection {* Lemmas about powers *}
1508 text {* FIXME: declare this in Rings.thy or not at all *}
1509 declare abs_mult_self [simp]
1511 (* used by Import/HOL/real.imp *)
1512 lemma two_realpow_ge_one: "(1::real) \<le> 2 ^ n"
1513 by simp
1515 lemma two_realpow_gt [simp]: "real (n::nat) < 2 ^ n"
1516 apply (induct "n")
1517 apply (auto simp add: real_of_nat_Suc)
1518 apply (subst mult_2)
1520 apply (rule two_realpow_ge_one)
1521 done
1523 text {* TODO: no longer real-specific; rename and move elsewhere *}
1524 lemma realpow_Suc_le_self:
1525   fixes r :: "'a::linordered_semidom"
1526   shows "[| 0 \<le> r; r \<le> 1 |] ==> r ^ Suc n \<le> r"
1527 by (insert power_decreasing [of 1 "Suc n" r], simp)
1529 text {* TODO: no longer real-specific; rename and move elsewhere *}
1530 lemma realpow_minus_mult:
1531   fixes x :: "'a::monoid_mult"
1532   shows "0 < n \<Longrightarrow> x ^ (n - 1) * x = x ^ n"
1535 text {* FIXME: declare this [simp] for all types, or not at all *}
1537   "(x * x + y * y = 0) = ((x::real) = 0 \<and> y = 0)"
1538 by (rule sum_squares_eq_zero_iff)
1540 text {* FIXME: declare this [simp] for all types, or not at all *}
1541 lemma realpow_two_sum_zero_iff [simp]:
1542      "(x\<^sup>2 + y\<^sup>2 = (0::real)) = (x = 0 & y = 0)"
1543 by (rule sum_power2_eq_zero_iff)
1545 lemma real_minus_mult_self_le [simp]: "-(u * u) \<le> (x * (x::real))"
1546 by (rule_tac y = 0 in order_trans, auto)
1548 lemma realpow_square_minus_le [simp]: "- u\<^sup>2 \<le> (x::real)\<^sup>2"
1549 by (auto simp add: power2_eq_square)
1552 lemma numeral_power_le_real_of_nat_cancel_iff[simp]:
1553   "(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::nat) ^ n \<le> a"
1554   unfolding real_of_nat_le_iff[symmetric] by simp
1556 lemma real_of_nat_le_numeral_power_cancel_iff[simp]:
1557   "real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::nat) ^ n"
1558   unfolding real_of_nat_le_iff[symmetric] by simp
1560 lemma numeral_power_le_real_of_int_cancel_iff[simp]:
1561   "(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::int) ^ n \<le> a"
1562   unfolding real_of_int_le_iff[symmetric] by simp
1564 lemma real_of_int_le_numeral_power_cancel_iff[simp]:
1565   "real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::int) ^ n"
1566   unfolding real_of_int_le_iff[symmetric] by simp
1568 lemma neg_numeral_power_le_real_of_int_cancel_iff[simp]:
1569   "(- numeral x::real) ^ n \<le> real a \<longleftrightarrow> (- numeral x::int) ^ n \<le> a"
1570   unfolding real_of_int_le_iff[symmetric] by simp
1572 lemma real_of_int_le_neg_numeral_power_cancel_iff[simp]:
1573   "real a \<le> (- numeral x::real) ^ n \<longleftrightarrow> a \<le> (- numeral x::int) ^ n"
1574   unfolding real_of_int_le_iff[symmetric] by simp
1577 subsection{*Density of the Reals*}
1579 lemma real_lbound_gt_zero:
1580      "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"
1581 apply (rule_tac x = " (min d1 d2) /2" in exI)
1583 done
1586 text{*Similar results are proved in @{text Fields}*}
1587 lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"
1588   by auto
1590 lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"
1591   by auto
1593 lemma real_sum_of_halves: "x/2 + x/2 = (x::real)"
1594   by simp
1596 subsection{*Absolute Value Function for the Reals*}
1598 lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"
1601 (* FIXME: redundant, but used by Integration/RealRandVar.thy in AFP *)
1602 lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))"
1603 by (force simp add: abs_le_iff)
1605 lemma abs_add_one_gt_zero: "(0::real) < 1 + abs(x)"
1608 lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)"
1609 by (rule abs_of_nonneg [OF real_of_nat_ge_zero])
1611 lemma abs_add_one_not_less_self: "~ abs(x) + (1::real) < x"
1612 by simp
1614 lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"
1615 by simp
1618 subsection{*Floor and Ceiling Functions from the Reals to the Integers*}
1620 (* FIXME: theorems for negative numerals *)
1621 lemma numeral_less_real_of_int_iff [simp]:
1622      "((numeral n) < real (m::int)) = (numeral n < m)"
1623 apply auto
1624 apply (rule real_of_int_less_iff [THEN iffD1])
1625 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
1626 done
1628 lemma numeral_less_real_of_int_iff2 [simp]:
1629      "(real (m::int) < (numeral n)) = (m < numeral n)"
1630 apply auto
1631 apply (rule real_of_int_less_iff [THEN iffD1])
1632 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
1633 done
1635 lemma real_of_nat_less_numeral_iff [simp]:
1636   "real (n::nat) < numeral w \<longleftrightarrow> n < numeral w"
1637   using real_of_nat_less_iff[of n "numeral w"] by simp
1639 lemma numeral_less_real_of_nat_iff [simp]:
1640   "numeral w < real (n::nat) \<longleftrightarrow> numeral w < n"
1641   using real_of_nat_less_iff[of "numeral w" n] by simp
1643 lemma numeral_le_real_of_int_iff [simp]:
1644      "((numeral n) \<le> real (m::int)) = (numeral n \<le> m)"
1645 by (simp add: linorder_not_less [symmetric])
1647 lemma numeral_le_real_of_int_iff2 [simp]:
1648      "(real (m::int) \<le> (numeral n)) = (m \<le> numeral n)"
1649 by (simp add: linorder_not_less [symmetric])
1651 lemma floor_real_of_nat [simp]: "floor (real (n::nat)) = int n"
1652 unfolding real_of_nat_def by simp
1654 lemma floor_minus_real_of_nat [simp]: "floor (- real (n::nat)) = - int n"
1655 unfolding real_of_nat_def by (simp add: floor_minus)
1657 lemma floor_real_of_int [simp]: "floor (real (n::int)) = n"
1658 unfolding real_of_int_def by simp
1660 lemma floor_minus_real_of_int [simp]: "floor (- real (n::int)) = - n"
1661 unfolding real_of_int_def by (simp add: floor_minus)
1663 lemma real_lb_ub_int: " \<exists>n::int. real n \<le> r & r < real (n+1)"
1664 unfolding real_of_int_def by (rule floor_exists)
1666 lemma lemma_floor: "real m \<le> r \<Longrightarrow> r < real n + 1 \<Longrightarrow> m \<le> (n::int)"
1667   by simp
1669 lemma real_of_int_floor_le [simp]: "real (floor r) \<le> r"
1670 unfolding real_of_int_def by (rule of_int_floor_le)
1672 lemma lemma_floor2: "real n < real (x::int) + 1 ==> n \<le> x"
1673   by simp
1675 lemma real_of_int_floor_cancel [simp]:
1676     "(real (floor x) = x) = (\<exists>n::int. x = real n)"
1677   using floor_real_of_int by metis
1679 lemma floor_eq: "[| real n < x; x < real n + 1 |] ==> floor x = n"
1680   by linarith
1682 lemma floor_eq2: "[| real n \<le> x; x < real n + 1 |] ==> floor x = n"
1683   by linarith
1685 lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"
1686   by linarith
1688 lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"
1689   by linarith
1691 lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real(floor r)"
1692   by linarith
1694 lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real(floor r)"
1695   by linarith
1697 lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real(floor r) + 1"
1698   by linarith
1700 lemma real_of_int_floor_add_one_gt [simp]: "r < real(floor r) + 1"
1701   by linarith
1703 lemma le_floor: "real a <= x ==> a <= floor x"
1704   by linarith
1706 lemma real_le_floor: "a <= floor x ==> real a <= x"
1707   by linarith
1709 lemma le_floor_eq: "(a <= floor x) = (real a <= x)"
1710   by linarith
1712 lemma floor_less_eq: "(floor x < a) = (x < real a)"
1713   by linarith
1715 lemma less_floor_eq: "(a < floor x) = (real a + 1 <= x)"
1716   by linarith
1718 lemma floor_le_eq: "(floor x <= a) = (x < real a + 1)"
1719   by linarith
1721 lemma floor_add [simp]: "floor (x + real a) = floor x + a"
1722   by linarith
1724 lemma floor_subtract [simp]: "floor (x - real a) = floor x - a"
1725   by linarith
1727 lemma floor_divide_eq_div:
1728   "floor (real a / real b) = a div b"
1729 proof cases
1730   assume "b \<noteq> 0 \<or> b dvd a"
1731   with real_of_int_div3[of a b] show ?thesis
1732     by (auto simp: real_of_int_div[symmetric] intro!: floor_eq2 real_of_int_div4 neq_le_trans)
1733        (metis add_left_cancel zero_neq_one real_of_int_div_aux real_of_int_inject
1734               real_of_int_zero_cancel right_inverse_eq div_self mod_div_trivial)
1735 qed (auto simp: real_of_int_div)
1737 lemma floor_divide_eq_div_numeral[simp]: "\<lfloor>numeral a / numeral b::real\<rfloor> = numeral a div numeral b"
1738   using floor_divide_eq_div[of "numeral a" "numeral b"] by simp
1740 lemma floor_minus_divide_eq_div_numeral[simp]: "\<lfloor>- (numeral a / numeral b)::real\<rfloor> = - numeral a div numeral b"
1741   using floor_divide_eq_div[of "- numeral a" "numeral b"] by simp
1743 lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"
1744   by linarith
1746 lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)"
1747   by linarith
1749 lemma ceiling_real_of_int [simp]: "ceiling (real (n::int)) = n"
1750   by linarith
1752 lemma real_of_int_ceiling_cancel [simp]:
1753      "(real (ceiling x) = x) = (\<exists>n::int. x = real n)"
1754   using ceiling_real_of_int by metis
1756 lemma ceiling_eq: "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1"
1757   by linarith
1759 lemma ceiling_eq2: "[| real n < x; x \<le> real n + 1 |] ==> ceiling x = n + 1"
1760   by linarith
1762 lemma ceiling_eq3: "[| real n - 1 < x; x \<le> real n  |] ==> ceiling x = n"
1763   by linarith
1765 lemma real_of_int_ceiling_diff_one_le [simp]: "real (ceiling r) - 1 \<le> r"
1766   by linarith
1768 lemma real_of_int_ceiling_le_add_one [simp]: "real (ceiling r) \<le> r + 1"
1769   by linarith
1771 lemma ceiling_le: "x <= real a ==> ceiling x <= a"
1772   by linarith
1774 lemma ceiling_le_real: "ceiling x <= a ==> x <= real a"
1775   by linarith
1777 lemma ceiling_le_eq: "(ceiling x <= a) = (x <= real a)"
1778   by linarith
1780 lemma less_ceiling_eq: "(a < ceiling x) = (real a < x)"
1781   by linarith
1783 lemma ceiling_less_eq: "(ceiling x < a) = (x <= real a - 1)"
1784   by linarith
1786 lemma le_ceiling_eq: "(a <= ceiling x) = (real a - 1 < x)"
1787   by linarith
1789 lemma ceiling_add [simp]: "ceiling (x + real a) = ceiling x + a"
1790   by linarith
1792 lemma ceiling_subtract [simp]: "ceiling (x - real a) = ceiling x - a"
1793   by linarith
1795 lemma ceiling_divide_eq_div: "\<lceil>real a / real b\<rceil> = - (- a div b)"
1796   unfolding ceiling_def minus_divide_left real_of_int_minus[symmetric] floor_divide_eq_div by simp_all
1798 lemma ceiling_divide_eq_div_numeral [simp]:
1799   "\<lceil>numeral a / numeral b :: real\<rceil> = - (- numeral a div numeral b)"
1800   using ceiling_divide_eq_div[of "numeral a" "numeral b"] by simp
1802 lemma ceiling_minus_divide_eq_div_numeral [simp]:
1803   "\<lceil>- (numeral a / numeral b :: real)\<rceil> = - (numeral a div numeral b)"
1804   using ceiling_divide_eq_div[of "- numeral a" "numeral b"] by simp
1806 subsubsection {* Versions for the natural numbers *}
1808 definition
1809   natfloor :: "real => nat" where
1810   "natfloor x = nat(floor x)"
1812 definition
1813   natceiling :: "real => nat" where
1814   "natceiling x = nat(ceiling x)"
1816 lemma natfloor_split[arith_split]: "P (natfloor t) \<longleftrightarrow> (t < 0 \<longrightarrow> P 0) \<and> (\<forall>n. of_nat n \<le> t \<and> t < of_nat n + 1 \<longrightarrow> P n)"
1817 proof -
1818   have [dest]: "\<And>n m::nat. real n \<le> t \<Longrightarrow> t < real n + 1 \<Longrightarrow> real m \<le> t \<Longrightarrow> t < real m + 1 \<Longrightarrow> n = m"
1819     by simp
1820   show ?thesis
1821     by (auto simp: natfloor_def real_of_nat_def[symmetric] split: split_nat floor_split)
1822 qed
1824 lemma natceiling_split[arith_split]:
1825   "P (natceiling t) \<longleftrightarrow> (t \<le> - 1 \<longrightarrow> P 0) \<and> (\<forall>n. of_nat n - 1 < t \<and> t \<le> of_nat n \<longrightarrow> P n)"
1826 proof -
1827   have [dest]: "\<And>n m::nat. real n - 1 < t \<Longrightarrow> t \<le> real n \<Longrightarrow> real m - 1 < t \<Longrightarrow> t \<le> real m \<Longrightarrow> n = m"
1828     by simp
1829   show ?thesis
1830     by (auto simp: natceiling_def real_of_nat_def[symmetric] split: split_nat ceiling_split)
1831 qed
1833 lemma natfloor_zero [simp]: "natfloor 0 = 0"
1834   by linarith
1836 lemma natfloor_one [simp]: "natfloor 1 = 1"
1837   by linarith
1839 lemma natfloor_numeral_eq [simp]: "natfloor (numeral n) = numeral n"
1840   by (unfold natfloor_def, simp)
1842 lemma natfloor_real_of_nat [simp]: "natfloor(real n) = n"
1843   by linarith
1845 lemma real_natfloor_le: "0 <= x ==> real(natfloor x) <= x"
1846   by linarith
1848 lemma natfloor_neg: "x <= 0 ==> natfloor x = 0"
1849   by linarith
1851 lemma natfloor_mono: "x <= y ==> natfloor x <= natfloor y"
1852   by linarith
1854 lemma le_natfloor: "real x <= a ==> x <= natfloor a"
1855   by linarith
1857 lemma natfloor_less_iff: "0 \<le> x \<Longrightarrow> natfloor x < n \<longleftrightarrow> x < real n"
1858   by linarith
1860 lemma less_natfloor: "0 \<le> x \<Longrightarrow> x < real (n :: nat) \<Longrightarrow> natfloor x < n"
1861   by linarith
1863 lemma le_natfloor_eq: "0 <= x ==> (a <= natfloor x) = (real a <= x)"
1864   by linarith
1866 lemma le_natfloor_eq_numeral [simp]:
1867     "0 \<le> x \<Longrightarrow> (numeral n \<le> natfloor x) = (numeral n \<le> x)"
1868   by (subst le_natfloor_eq, assumption) simp
1870 lemma le_natfloor_eq_one [simp]: "(1 \<le> natfloor x) = (1 \<le> x)"
1871   by linarith
1873 lemma natfloor_eq: "real n \<le> x \<Longrightarrow> x < real n + 1 \<Longrightarrow> natfloor x = n"
1874   by linarith
1876 lemma real_natfloor_add_one_gt: "x < real (natfloor x) + 1"
1877   by linarith
1879 lemma real_natfloor_gt_diff_one: "x - 1 < real(natfloor x)"
1880   by linarith
1882 lemma ge_natfloor_plus_one_imp_gt: "natfloor z + 1 <= n ==> z < real n"
1883   by linarith
1885 lemma natfloor_add [simp]: "0 <= x ==> natfloor (x + real a) = natfloor x + a"
1886   by linarith
1889     "0 <= x \<Longrightarrow> natfloor (x + numeral n) = natfloor x + numeral n"
1892 lemma natfloor_add_one: "0 <= x ==> natfloor(x + 1) = natfloor x + 1"
1893   by linarith
1895 lemma natfloor_subtract [simp]:
1896     "natfloor(x - real a) = natfloor x - a"
1897   by linarith
1899 lemma natfloor_div_nat:
1900   assumes "1 <= x" and "y > 0"
1901   shows "natfloor (x / real y) = natfloor x div y"
1902 proof (rule natfloor_eq)
1903   have "(natfloor x) div y * y \<le> natfloor x"
1904     by (rule add_leD1 [where k="natfloor x mod y"], simp)
1905   thus "real (natfloor x div y) \<le> x / real y"
1906     using assms by (simp add: le_divide_eq le_natfloor_eq)
1907   have "natfloor x < (natfloor x) div y * y + y"
1908     apply (subst mod_div_equality [symmetric])
1910     apply (rule mod_less_divisor)
1911     apply fact
1912     done
1913   thus "x / real y < real (natfloor x div y) + 1"
1914     using assms
1915     by (simp add: divide_less_eq natfloor_less_iff distrib_right)
1916 qed
1918 lemma natfloor_div_numeral[simp]:
1919   "natfloor (numeral x / numeral y) = numeral x div numeral y"
1920   using natfloor_div_nat[of "numeral x" "numeral y"] by simp
1922 lemma le_mult_natfloor:
1923   shows "natfloor a * natfloor b \<le> natfloor (a * b)"
1924   by (cases "0 <= a & 0 <= b")
1925     (auto simp add: le_natfloor_eq mult_mono' real_natfloor_le natfloor_neg)
1927 lemma natceiling_zero [simp]: "natceiling 0 = 0"
1928   by linarith
1930 lemma natceiling_one [simp]: "natceiling 1 = 1"
1931   by linarith
1933 lemma zero_le_natceiling [simp]: "0 <= natceiling x"
1934   by linarith
1936 lemma natceiling_numeral_eq [simp]: "natceiling (numeral n) = numeral n"
1939 lemma natceiling_real_of_nat [simp]: "natceiling(real n) = n"
1940   by linarith
1942 lemma real_natceiling_ge: "x <= real(natceiling x)"
1943   by linarith
1945 lemma natceiling_neg: "x <= 0 ==> natceiling x = 0"
1946   by linarith
1948 lemma natceiling_mono: "x <= y ==> natceiling x <= natceiling y"
1949   by linarith
1951 lemma natceiling_le: "x <= real a ==> natceiling x <= a"
1952   by linarith
1954 lemma natceiling_le_eq: "(natceiling x <= a) = (x <= real a)"
1955   by linarith
1957 lemma natceiling_le_eq_numeral [simp]:
1958     "(natceiling x <= numeral n) = (x <= numeral n)"
1961 lemma natceiling_le_eq_one: "(natceiling x <= 1) = (x <= 1)"
1962   by linarith
1964 lemma natceiling_eq: "real n < x ==> x <= real n + 1 ==> natceiling x = n + 1"
1965   by linarith
1967 lemma natceiling_add [simp]: "0 <= x ==> natceiling (x + real a) = natceiling x + a"
1968   by linarith
1971     "0 <= x ==> natceiling (x + numeral n) = natceiling x + numeral n"
1974 lemma natceiling_add_one: "0 <= x ==> natceiling(x + 1) = natceiling x + 1"
1975   by linarith
1977 lemma natceiling_subtract [simp]: "natceiling(x - real a) = natceiling x - a"
1978   by linarith
1980 lemma Rats_no_top_le: "\<exists> q \<in> \<rat>. (x :: real) \<le> q"
1981   by (auto intro!: bexI[of _ "of_nat (natceiling x)"]) (metis real_natceiling_ge real_of_nat_def)
1983 lemma Rats_no_bot_less: "\<exists> q \<in> \<rat>. q < (x :: real)"
1984   apply (auto intro!: bexI[of _ "of_int (floor x - 1)"])
1985   apply (rule less_le_trans[OF _ of_int_floor_le])
1986   apply simp
1987   done
1989 subsection {* Exponentiation with floor *}
1991 lemma floor_power:
1992   assumes "x = real (floor x)"
1993   shows "floor (x ^ n) = floor x ^ n"
1994 proof -
1995   have *: "x ^ n = real (floor x ^ n)"
1996     using assms by (induct n arbitrary: x) simp_all
1997   show ?thesis unfolding real_of_int_inject[symmetric]
1998     unfolding * floor_real_of_int ..
1999 qed
2001 lemma natfloor_power:
2002   assumes "x = real (natfloor x)"
2003   shows "natfloor (x ^ n) = natfloor x ^ n"
2004 proof -
2005   from assms have "0 \<le> floor x" by auto
2006   note assms[unfolded natfloor_def real_nat_eq_real[OF 0 \<le> floor x]]
2007   from floor_power[OF this]
2008   show ?thesis unfolding natfloor_def nat_power_eq[OF 0 \<le> floor x`, symmetric]
2009     by simp
2010 qed
2013 subsection {* Implementation of rational real numbers *}
2015 text {* Formal constructor *}
2017 definition Ratreal :: "rat \<Rightarrow> real" where
2018   [code_abbrev, simp]: "Ratreal = of_rat"
2020 code_datatype Ratreal
2023 text {* Numerals *}
2025 lemma [code_abbrev]:
2026   "(of_rat (of_int a) :: real) = of_int a"
2027   by simp
2029 lemma [code_abbrev]:
2030   "(of_rat 0 :: real) = 0"
2031   by simp
2033 lemma [code_abbrev]:
2034   "(of_rat 1 :: real) = 1"
2035   by simp
2037 lemma [code_abbrev]:
2038   "(of_rat (- 1) :: real) = - 1"
2039   by simp
2041 lemma [code_abbrev]:
2042   "(of_rat (numeral k) :: real) = numeral k"
2043   by simp
2045 lemma [code_abbrev]:
2046   "(of_rat (- numeral k) :: real) = - numeral k"
2047   by simp
2049 lemma [code_post]:
2050   "(of_rat (1 / numeral k) :: real) = 1 / numeral k"
2051   "(of_rat (numeral k / numeral l) :: real) = numeral k / numeral l"
2052   "(of_rat (- (1 / numeral k)) :: real) = - (1 / numeral k)"
2053   "(of_rat (- (numeral k / numeral l)) :: real) = - (numeral k / numeral l)"
2054   by (simp_all add: of_rat_divide of_rat_minus)
2057 text {* Operations *}
2059 lemma zero_real_code [code]:
2060   "0 = Ratreal 0"
2061 by simp
2063 lemma one_real_code [code]:
2064   "1 = Ratreal 1"
2065 by simp
2067 instantiation real :: equal
2068 begin
2070 definition "HOL.equal (x\<Colon>real) y \<longleftrightarrow> x - y = 0"
2072 instance proof
2075 lemma real_equal_code [code]:
2076   "HOL.equal (Ratreal x) (Ratreal y) \<longleftrightarrow> HOL.equal x y"
2077   by (simp add: equal_real_def equal)
2079 lemma [code nbe]:
2080   "HOL.equal (x::real) x \<longleftrightarrow> True"
2081   by (rule equal_refl)
2083 end
2085 lemma real_less_eq_code [code]: "Ratreal x \<le> Ratreal y \<longleftrightarrow> x \<le> y"
2088 lemma real_less_code [code]: "Ratreal x < Ratreal y \<longleftrightarrow> x < y"
2091 lemma real_plus_code [code]: "Ratreal x + Ratreal y = Ratreal (x + y)"
2094 lemma real_times_code [code]: "Ratreal x * Ratreal y = Ratreal (x * y)"
2097 lemma real_uminus_code [code]: "- Ratreal x = Ratreal (- x)"
2100 lemma real_minus_code [code]: "Ratreal x - Ratreal y = Ratreal (x - y)"
2103 lemma real_inverse_code [code]: "inverse (Ratreal x) = Ratreal (inverse x)"
2106 lemma real_divide_code [code]: "Ratreal x / Ratreal y = Ratreal (x / y)"
2109 lemma real_floor_code [code]: "floor (Ratreal x) = floor x"
2110   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)
2113 text {* Quickcheck *}
2115 definition (in term_syntax)
2116   valterm_ratreal :: "rat \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> real \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
2117   [code_unfold]: "valterm_ratreal k = Code_Evaluation.valtermify Ratreal {\<cdot>} k"
2119 notation fcomp (infixl "\<circ>>" 60)
2120 notation scomp (infixl "\<circ>\<rightarrow>" 60)
2122 instantiation real :: random
2123 begin
2125 definition
2126   "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>r. Pair (valterm_ratreal r))"
2128 instance ..
2130 end
2132 no_notation fcomp (infixl "\<circ>>" 60)
2133 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
2135 instantiation real :: exhaustive
2136 begin
2138 definition
2139   "exhaustive_real f d = Quickcheck_Exhaustive.exhaustive (%r. f (Ratreal r)) d"
2141 instance ..
2143 end
2145 instantiation real :: full_exhaustive
2146 begin
2148 definition
2149   "full_exhaustive_real f d = Quickcheck_Exhaustive.full_exhaustive (%r. f (valterm_ratreal r)) d"
2151 instance ..
2153 end
2155 instantiation real :: narrowing
2156 begin
2158 definition
2159   "narrowing = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.cons Ratreal) narrowing"
2161 instance ..
2163 end
2166 subsection {* Setup for Nitpick *}
2168 declaration {*
2169   Nitpick_HOL.register_frac_type @{type_name real}
2170    [(@{const_name zero_real_inst.zero_real}, @{const_name Nitpick.zero_frac}),
2171     (@{const_name one_real_inst.one_real}, @{const_name Nitpick.one_frac}),
2172     (@{const_name plus_real_inst.plus_real}, @{const_name Nitpick.plus_frac}),
2173     (@{const_name times_real_inst.times_real}, @{const_name Nitpick.times_frac}),
2174     (@{const_name uminus_real_inst.uminus_real}, @{const_name Nitpick.uminus_frac}),
2175     (@{const_name inverse_real_inst.inverse_real}, @{const_name Nitpick.inverse_frac}),
2176     (@{const_name ord_real_inst.less_real}, @{const_name Nitpick.less_frac}),
2177     (@{const_name ord_real_inst.less_eq_real}, @{const_name Nitpick.less_eq_frac})]
2178 *}
2180 lemmas [nitpick_unfold] = inverse_real_inst.inverse_real one_real_inst.one_real
2181     ord_real_inst.less_real ord_real_inst.less_eq_real plus_real_inst.plus_real
2182     times_real_inst.times_real uminus_real_inst.uminus_real
2183     zero_real_inst.zero_real
2186 subsection {* Setup for SMT *}
2188 ML_file "Tools/SMT/smt_real.ML"
2189 ML_file "Tools/SMT/z3_real.ML"
2191 lemma [z3_rule]:
2192   "0 + (x::real) = x"
2193   "x + 0 = x"
2194   "0 * x = 0"
2195   "1 * x = x"
2196   "x + y = y + x"
2197   by auto
2199 end