src/HOL/Real.thy
 author paulson Mon May 23 15:33:24 2016 +0100 (2016-05-23) changeset 63114 27afe7af7379 parent 63040 eb4ddd18d635 child 63331 247eac9758dd permissions -rw-r--r--
Lots of new material for multivariate analysis
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 section \<open>Development of the Reals using Cauchy Sequences\<close>
12 theory Real
13 imports Rat Conditionally_Complete_Lattices
14 begin
16 text \<open>
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 \<close>
23 subsection \<open>Preliminary lemmas\<close>
26   fixes x :: "'a::cancel_semigroup_add" shows "inj (op+ x)"
29 lemma inj_mult_left [simp]: "inj (op* x) \<longleftrightarrow> x \<noteq> (0::'a::idom)"
30   by (metis injI mult_cancel_left the_inv_f_f zero_neq_one)
33   fixes a b c d :: "'a::ab_group_add"
34   shows "(a + c) - (b + d) = (a - b) + (c - d)"
35   by simp
37 lemma minus_diff_minus:
38   fixes a b :: "'a::ab_group_add"
39   shows "- a - - b = - (a - b)"
40   by simp
42 lemma mult_diff_mult:
43   fixes x y a b :: "'a::ring"
44   shows "(x * y - a * b) = x * (y - b) + (x - a) * b"
47 lemma inverse_diff_inverse:
48   fixes a b :: "'a::division_ring"
49   assumes "a \<noteq> 0" and "b \<noteq> 0"
50   shows "inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
51   using assms by (simp add: algebra_simps)
53 lemma obtain_pos_sum:
54   fixes r :: rat assumes r: "0 < r"
55   obtains s t where "0 < s" and "0 < t" and "r = s + t"
56 proof
57     from r show "0 < r/2" by simp
58     from r show "0 < r/2" by simp
59     show "r = r/2 + r/2" by simp
60 qed
62 subsection \<open>Sequences that converge to zero\<close>
64 definition
65   vanishes :: "(nat \<Rightarrow> rat) \<Rightarrow> bool"
66 where
67   "vanishes X = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r)"
69 lemma vanishesI: "(\<And>r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r) \<Longrightarrow> vanishes X"
70   unfolding vanishes_def by simp
72 lemma vanishesD: "\<lbrakk>vanishes X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r"
73   unfolding vanishes_def by simp
75 lemma vanishes_const [simp]: "vanishes (\<lambda>n. c) \<longleftrightarrow> c = 0"
76   unfolding vanishes_def
77   apply (cases "c = 0", auto)
78   apply (rule exI [where x="\<bar>c\<bar>"], auto)
79   done
81 lemma vanishes_minus: "vanishes X \<Longrightarrow> vanishes (\<lambda>n. - X n)"
82   unfolding vanishes_def by simp
85   assumes X: "vanishes X" and Y: "vanishes Y"
86   shows "vanishes (\<lambda>n. X n + Y n)"
87 proof (rule vanishesI)
88   fix r :: rat assume "0 < r"
89   then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
90     by (rule obtain_pos_sum)
91   obtain i where i: "\<forall>n\<ge>i. \<bar>X n\<bar> < s"
92     using vanishesD [OF X s] ..
93   obtain j where j: "\<forall>n\<ge>j. \<bar>Y n\<bar> < t"
94     using vanishesD [OF Y t] ..
95   have "\<forall>n\<ge>max i j. \<bar>X n + Y n\<bar> < r"
96   proof (clarsimp)
97     fix n assume n: "i \<le> n" "j \<le> n"
98     have "\<bar>X n + Y n\<bar> \<le> \<bar>X n\<bar> + \<bar>Y n\<bar>" by (rule abs_triangle_ineq)
99     also have "\<dots> < s + t" by (simp add: add_strict_mono i j n)
100     finally show "\<bar>X n + Y n\<bar> < r" unfolding r .
101   qed
102   thus "\<exists>k. \<forall>n\<ge>k. \<bar>X n + Y n\<bar> < r" ..
103 qed
105 lemma vanishes_diff:
106   assumes X: "vanishes X" and Y: "vanishes Y"
107   shows "vanishes (\<lambda>n. X n - Y n)"
110 lemma vanishes_mult_bounded:
111   assumes X: "\<exists>a>0. \<forall>n. \<bar>X n\<bar> < a"
112   assumes Y: "vanishes (\<lambda>n. Y n)"
113   shows "vanishes (\<lambda>n. X n * Y n)"
114 proof (rule vanishesI)
115   fix r :: rat assume r: "0 < r"
116   obtain a where a: "0 < a" "\<forall>n. \<bar>X n\<bar> < a"
117     using X by blast
118   obtain b where b: "0 < b" "r = a * b"
119   proof
120     show "0 < r / a" using r a by simp
121     show "r = a * (r / a)" using a by simp
122   qed
123   obtain k where k: "\<forall>n\<ge>k. \<bar>Y n\<bar> < b"
124     using vanishesD [OF Y b(1)] ..
125   have "\<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r"
126     by (simp add: b(2) abs_mult mult_strict_mono' a k)
127   thus "\<exists>k. \<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r" ..
128 qed
130 subsection \<open>Cauchy sequences\<close>
132 definition
133   cauchy :: "(nat \<Rightarrow> rat) \<Rightarrow> bool"
134 where
135   "cauchy X \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r)"
137 lemma cauchyI:
138   "(\<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"
139   unfolding cauchy_def by simp
141 lemma cauchyD:
142   "\<lbrakk>cauchy X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r"
143   unfolding cauchy_def by simp
145 lemma cauchy_const [simp]: "cauchy (\<lambda>n. x)"
146   unfolding cauchy_def by simp
149   assumes X: "cauchy X" and Y: "cauchy Y"
150   shows "cauchy (\<lambda>n. X n + Y n)"
151 proof (rule cauchyI)
152   fix r :: rat assume "0 < r"
153   then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
154     by (rule obtain_pos_sum)
155   obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"
156     using cauchyD [OF X s] ..
157   obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>Y m - Y n\<bar> < t"
158     using cauchyD [OF Y t] ..
159   have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>(X m + Y m) - (X n + Y n)\<bar> < r"
160   proof (clarsimp)
161     fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"
162     have "\<bar>(X m + Y m) - (X n + Y n)\<bar> \<le> \<bar>X m - X n\<bar> + \<bar>Y m - Y n\<bar>"
164     also have "\<dots> < s + t"
166     finally show "\<bar>(X m + Y m) - (X n + Y n)\<bar> < r" unfolding r .
167   qed
168   thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>(X m + Y m) - (X n + Y n)\<bar> < r" ..
169 qed
171 lemma cauchy_minus [simp]:
172   assumes X: "cauchy X"
173   shows "cauchy (\<lambda>n. - X n)"
174 using assms unfolding cauchy_def
175 unfolding minus_diff_minus abs_minus_cancel .
177 lemma cauchy_diff [simp]:
178   assumes X: "cauchy X" and Y: "cauchy Y"
179   shows "cauchy (\<lambda>n. X n - Y n)"
182 lemma cauchy_imp_bounded:
183   assumes "cauchy X" shows "\<exists>b>0. \<forall>n. \<bar>X n\<bar> < b"
184 proof -
185   obtain k where k: "\<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < 1"
186     using cauchyD [OF assms zero_less_one] ..
187   show "\<exists>b>0. \<forall>n. \<bar>X n\<bar> < b"
188   proof (intro exI conjI allI)
189     have "0 \<le> \<bar>X 0\<bar>" by simp
190     also have "\<bar>X 0\<bar> \<le> Max (abs  X  {..k})" by simp
191     finally have "0 \<le> Max (abs  X  {..k})" .
192     thus "0 < Max (abs  X  {..k}) + 1" by simp
193   next
194     fix n :: nat
195     show "\<bar>X n\<bar> < Max (abs  X  {..k}) + 1"
196     proof (rule linorder_le_cases)
197       assume "n \<le> k"
198       hence "\<bar>X n\<bar> \<le> Max (abs  X  {..k})" by simp
199       thus "\<bar>X n\<bar> < Max (abs  X  {..k}) + 1" by simp
200     next
201       assume "k \<le> n"
202       have "\<bar>X n\<bar> = \<bar>X k + (X n - X k)\<bar>" by simp
203       also have "\<bar>X k + (X n - X k)\<bar> \<le> \<bar>X k\<bar> + \<bar>X n - X k\<bar>"
204         by (rule abs_triangle_ineq)
205       also have "\<dots> < Max (abs  X  {..k}) + 1"
207       finally show "\<bar>X n\<bar> < Max (abs  X  {..k}) + 1" .
208     qed
209   qed
210 qed
212 lemma cauchy_mult [simp]:
213   assumes X: "cauchy X" and Y: "cauchy Y"
214   shows "cauchy (\<lambda>n. X n * Y n)"
215 proof (rule cauchyI)
216   fix r :: rat assume "0 < r"
217   then obtain u v where u: "0 < u" and v: "0 < v" and "r = u + v"
218     by (rule obtain_pos_sum)
219   obtain a where a: "0 < a" "\<forall>n. \<bar>X n\<bar> < a"
220     using cauchy_imp_bounded [OF X] by blast
221   obtain b where b: "0 < b" "\<forall>n. \<bar>Y n\<bar> < b"
222     using cauchy_imp_bounded [OF Y] by blast
223   obtain s t where s: "0 < s" and t: "0 < t" and r: "r = a * t + s * b"
224   proof
225     show "0 < v/b" using v b(1) by simp
226     show "0 < u/a" using u a(1) by simp
227     show "r = a * (u/a) + (v/b) * b"
228       using a(1) b(1) \<open>r = u + v\<close> by simp
229   qed
230   obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"
231     using cauchyD [OF X s] ..
232   obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>Y m - Y n\<bar> < t"
233     using cauchyD [OF Y t] ..
234   have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>X m * Y m - X n * Y n\<bar> < r"
235   proof (clarsimp)
236     fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"
237     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>"
238       unfolding mult_diff_mult ..
239     also have "\<dots> \<le> \<bar>X m * (Y m - Y n)\<bar> + \<bar>(X m - X n) * Y n\<bar>"
240       by (rule abs_triangle_ineq)
241     also have "\<dots> = \<bar>X m\<bar> * \<bar>Y m - Y n\<bar> + \<bar>X m - X n\<bar> * \<bar>Y n\<bar>"
242       unfolding abs_mult ..
243     also have "\<dots> < a * t + s * b"
245     finally show "\<bar>X m * Y m - X n * Y n\<bar> < r" unfolding r .
246   qed
247   thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m * Y m - X n * Y n\<bar> < r" ..
248 qed
250 lemma cauchy_not_vanishes_cases:
251   assumes X: "cauchy X"
252   assumes nz: "\<not> vanishes X"
253   shows "\<exists>b>0. \<exists>k. (\<forall>n\<ge>k. b < - X n) \<or> (\<forall>n\<ge>k. b < X n)"
254 proof -
255   obtain r where "0 < r" and r: "\<forall>k. \<exists>n\<ge>k. r \<le> \<bar>X n\<bar>"
256     using nz unfolding vanishes_def by (auto simp add: not_less)
257   obtain s t where s: "0 < s" and t: "0 < t" and "r = s + t"
258     using \<open>0 < r\<close> by (rule obtain_pos_sum)
259   obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"
260     using cauchyD [OF X s] ..
261   obtain k where "i \<le> k" and "r \<le> \<bar>X k\<bar>"
262     using r by blast
263   have k: "\<forall>n\<ge>k. \<bar>X n - X k\<bar> < s"
264     using i \<open>i \<le> k\<close> by auto
265   have "X k \<le> - r \<or> r \<le> X k"
266     using \<open>r \<le> \<bar>X k\<bar>\<close> by auto
267   hence "(\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)"
268     unfolding \<open>r = s + t\<close> using k by auto
269   hence "\<exists>k. (\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)" ..
270   thus "\<exists>t>0. \<exists>k. (\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)"
271     using t by auto
272 qed
274 lemma cauchy_not_vanishes:
275   assumes X: "cauchy X"
276   assumes nz: "\<not> vanishes X"
277   shows "\<exists>b>0. \<exists>k. \<forall>n\<ge>k. b < \<bar>X n\<bar>"
278 using cauchy_not_vanishes_cases [OF assms]
279 by clarify (rule exI, erule conjI, rule_tac x=k in exI, auto)
281 lemma cauchy_inverse [simp]:
282   assumes X: "cauchy X"
283   assumes nz: "\<not> vanishes X"
284   shows "cauchy (\<lambda>n. inverse (X n))"
285 proof (rule cauchyI)
286   fix r :: rat assume "0 < r"
287   obtain b i where b: "0 < b" and i: "\<forall>n\<ge>i. b < \<bar>X n\<bar>"
288     using cauchy_not_vanishes [OF X nz] by blast
289   from b i have nz: "\<forall>n\<ge>i. X n \<noteq> 0" by auto
290   obtain s where s: "0 < s" and r: "r = inverse b * s * inverse b"
291   proof
292     show "0 < b * r * b" by (simp add: \<open>0 < r\<close> b)
293     show "r = inverse b * (b * r * b) * inverse b"
294       using b by simp
295   qed
296   obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>X m - X n\<bar> < s"
297     using cauchyD [OF X s] ..
298   have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>inverse (X m) - inverse (X n)\<bar> < r"
299   proof (clarsimp)
300     fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"
301     have "\<bar>inverse (X m) - inverse (X n)\<bar> =
302           inverse \<bar>X m\<bar> * \<bar>X m - X n\<bar> * inverse \<bar>X n\<bar>"
303       by (simp add: inverse_diff_inverse nz * abs_mult)
304     also have "\<dots> < inverse b * s * inverse b"
305       by (simp add: mult_strict_mono less_imp_inverse_less
306                     i j b * s)
307     finally show "\<bar>inverse (X m) - inverse (X n)\<bar> < r" unfolding r .
308   qed
309   thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>inverse (X m) - inverse (X n)\<bar> < r" ..
310 qed
312 lemma vanishes_diff_inverse:
313   assumes X: "cauchy X" "\<not> vanishes X"
314   assumes Y: "cauchy Y" "\<not> vanishes Y"
315   assumes XY: "vanishes (\<lambda>n. X n - Y n)"
316   shows "vanishes (\<lambda>n. inverse (X n) - inverse (Y n))"
317 proof (rule vanishesI)
318   fix r :: rat assume r: "0 < r"
319   obtain a i where a: "0 < a" and i: "\<forall>n\<ge>i. a < \<bar>X n\<bar>"
320     using cauchy_not_vanishes [OF X] by blast
321   obtain b j where b: "0 < b" and j: "\<forall>n\<ge>j. b < \<bar>Y n\<bar>"
322     using cauchy_not_vanishes [OF Y] by blast
323   obtain s where s: "0 < s" and "inverse a * s * inverse b = r"
324   proof
325     show "0 < a * r * b"
326       using a r b by simp
327     show "inverse a * (a * r * b) * inverse b = r"
328       using a r b by simp
329   qed
330   obtain k where k: "\<forall>n\<ge>k. \<bar>X n - Y n\<bar> < s"
331     using vanishesD [OF XY s] ..
332   have "\<forall>n\<ge>max (max i j) k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r"
333   proof (clarsimp)
334     fix n assume n: "i \<le> n" "j \<le> n" "k \<le> n"
335     have "X n \<noteq> 0" and "Y n \<noteq> 0"
336       using i j a b n by auto
337     hence "\<bar>inverse (X n) - inverse (Y n)\<bar> =
338         inverse \<bar>X n\<bar> * \<bar>X n - Y n\<bar> * inverse \<bar>Y n\<bar>"
339       by (simp add: inverse_diff_inverse abs_mult)
340     also have "\<dots> < inverse a * s * inverse b"
341       apply (intro mult_strict_mono' less_imp_inverse_less)
342       apply (simp_all add: a b i j k n)
343       done
344     also note \<open>inverse a * s * inverse b = r\<close>
345     finally show "\<bar>inverse (X n) - inverse (Y n)\<bar> < r" .
346   qed
347   thus "\<exists>k. \<forall>n\<ge>k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r" ..
348 qed
350 subsection \<open>Equivalence relation on Cauchy sequences\<close>
352 definition realrel :: "(nat \<Rightarrow> rat) \<Rightarrow> (nat \<Rightarrow> rat) \<Rightarrow> bool"
353   where "realrel = (\<lambda>X Y. cauchy X \<and> cauchy Y \<and> vanishes (\<lambda>n. X n - Y n))"
355 lemma realrelI [intro?]:
356   assumes "cauchy X" and "cauchy Y" and "vanishes (\<lambda>n. X n - Y n)"
357   shows "realrel X Y"
358   using assms unfolding realrel_def by simp
360 lemma realrel_refl: "cauchy X \<Longrightarrow> realrel X X"
361   unfolding realrel_def by simp
363 lemma symp_realrel: "symp realrel"
364   unfolding realrel_def
365   by (rule sympI, clarify, drule vanishes_minus, simp)
367 lemma transp_realrel: "transp realrel"
368   unfolding realrel_def
369   apply (rule transpI, clarify)
372   done
374 lemma part_equivp_realrel: "part_equivp realrel"
375   by (blast intro: part_equivpI symp_realrel transp_realrel
376     realrel_refl cauchy_const)
378 subsection \<open>The field of real numbers\<close>
380 quotient_type real = "nat \<Rightarrow> rat" / partial: realrel
381   morphisms rep_real Real
382   by (rule part_equivp_realrel)
384 lemma cr_real_eq: "pcr_real = (\<lambda>x y. cauchy x \<and> Real x = y)"
385   unfolding real.pcr_cr_eq cr_real_def realrel_def by auto
387 lemma Real_induct [induct type: real]: (* TODO: generate automatically *)
388   assumes "\<And>X. cauchy X \<Longrightarrow> P (Real X)" shows "P x"
389 proof (induct x)
390   case (1 X)
391   hence "cauchy X" by (simp add: realrel_def)
392   thus "P (Real X)" by (rule assms)
393 qed
395 lemma eq_Real:
396   "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> Real X = Real Y \<longleftrightarrow> vanishes (\<lambda>n. X n - Y n)"
397   using real.rel_eq_transfer
398   unfolding real.pcr_cr_eq cr_real_def rel_fun_def realrel_def by simp
400 lemma Domainp_pcr_real [transfer_domain_rule]: "Domainp pcr_real = cauchy"
401 by (simp add: real.domain_eq realrel_def)
403 instantiation real :: field
404 begin
406 lift_definition zero_real :: "real" is "\<lambda>n. 0"
409 lift_definition one_real :: "real" is "\<lambda>n. 1"
412 lift_definition plus_real :: "real \<Rightarrow> real \<Rightarrow> real" is "\<lambda>X Y n. X n + Y n"
416 lift_definition uminus_real :: "real \<Rightarrow> real" is "\<lambda>X n. - X n"
417   unfolding realrel_def minus_diff_minus
418   by (simp only: cauchy_minus vanishes_minus simp_thms)
420 lift_definition times_real :: "real \<Rightarrow> real \<Rightarrow> real" is "\<lambda>X Y n. X n * Y n"
421   unfolding realrel_def mult_diff_mult
422   by (subst (4) mult.commute, simp only: cauchy_mult vanishes_add
423     vanishes_mult_bounded cauchy_imp_bounded simp_thms)
425 lift_definition inverse_real :: "real \<Rightarrow> real"
426   is "\<lambda>X. if vanishes X then (\<lambda>n. 0) else (\<lambda>n. inverse (X n))"
427 proof -
428   fix X Y assume "realrel X Y"
429   hence X: "cauchy X" and Y: "cauchy Y" and XY: "vanishes (\<lambda>n. X n - Y n)"
430     unfolding realrel_def by simp_all
431   have "vanishes X \<longleftrightarrow> vanishes Y"
432   proof
433     assume "vanishes X"
434     from vanishes_diff [OF this XY] show "vanishes Y" by simp
435   next
436     assume "vanishes Y"
437     from vanishes_add [OF this XY] show "vanishes X" by simp
438   qed
439   thus "?thesis X Y"
440     unfolding realrel_def
441     by (simp add: vanishes_diff_inverse X Y XY)
442 qed
444 definition
445   "x - y = (x::real) + - y"
447 definition
448   "x div y = (x::real) * inverse y"
451   assumes X: "cauchy X" and Y: "cauchy Y"
452   shows "Real X + Real Y = Real (\<lambda>n. X n + Y n)"
453   using assms plus_real.transfer
454   unfolding cr_real_eq rel_fun_def by simp
456 lemma minus_Real:
457   assumes X: "cauchy X"
458   shows "- Real X = Real (\<lambda>n. - X n)"
459   using assms uminus_real.transfer
460   unfolding cr_real_eq rel_fun_def by simp
462 lemma diff_Real:
463   assumes X: "cauchy X" and Y: "cauchy Y"
464   shows "Real X - Real Y = Real (\<lambda>n. X n - Y n)"
465   unfolding minus_real_def
468 lemma mult_Real:
469   assumes X: "cauchy X" and Y: "cauchy Y"
470   shows "Real X * Real Y = Real (\<lambda>n. X n * Y n)"
471   using assms times_real.transfer
472   unfolding cr_real_eq rel_fun_def by simp
474 lemma inverse_Real:
475   assumes X: "cauchy X"
476   shows "inverse (Real X) =
477     (if vanishes X then 0 else Real (\<lambda>n. inverse (X n)))"
478   using assms inverse_real.transfer zero_real.transfer
479   unfolding cr_real_eq rel_fun_def by (simp split: if_split_asm, metis)
481 instance proof
482   fix a b c :: real
483   show "a + b = b + a"
484     by transfer (simp add: ac_simps realrel_def)
485   show "(a + b) + c = a + (b + c)"
486     by transfer (simp add: ac_simps realrel_def)
487   show "0 + a = a"
488     by transfer (simp add: realrel_def)
489   show "- a + a = 0"
490     by transfer (simp add: realrel_def)
491   show "a - b = a + - b"
492     by (rule minus_real_def)
493   show "(a * b) * c = a * (b * c)"
494     by transfer (simp add: ac_simps realrel_def)
495   show "a * b = b * a"
496     by transfer (simp add: ac_simps realrel_def)
497   show "1 * a = a"
498     by transfer (simp add: ac_simps realrel_def)
499   show "(a + b) * c = a * c + b * c"
500     by transfer (simp add: distrib_right realrel_def)
501   show "(0::real) \<noteq> (1::real)"
502     by transfer (simp add: realrel_def)
503   show "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
504     apply transfer
506     apply (rule vanishesI)
507     apply (frule (1) cauchy_not_vanishes, clarify)
508     apply (rule_tac x=k in exI, clarify)
509     apply (drule_tac x=n in spec, simp)
510     done
511   show "a div b = a * inverse b"
512     by (rule divide_real_def)
513   show "inverse (0::real) = 0"
514     by transfer (simp add: realrel_def)
515 qed
517 end
519 subsection \<open>Positive reals\<close>
521 lift_definition positive :: "real \<Rightarrow> bool"
522   is "\<lambda>X. \<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n"
523 proof -
524   { fix X Y
525     assume "realrel X Y"
526     hence XY: "vanishes (\<lambda>n. X n - Y n)"
527       unfolding realrel_def by simp_all
528     assume "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n"
529     then obtain r i where "0 < r" and i: "\<forall>n\<ge>i. r < X n"
530       by blast
531     obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
532       using \<open>0 < r\<close> by (rule obtain_pos_sum)
533     obtain j where j: "\<forall>n\<ge>j. \<bar>X n - Y n\<bar> < s"
534       using vanishesD [OF XY s] ..
535     have "\<forall>n\<ge>max i j. t < Y n"
536     proof (clarsimp)
537       fix n assume n: "i \<le> n" "j \<le> n"
538       have "\<bar>X n - Y n\<bar> < s" and "r < X n"
539         using i j n by simp_all
540       thus "t < Y n" unfolding r by simp
541     qed
542     hence "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < Y n" using t by blast
543   } note 1 = this
544   fix X Y assume "realrel X Y"
545   hence "realrel X Y" and "realrel Y X"
546     using symp_realrel unfolding symp_def by auto
547   thus "?thesis X Y"
548     by (safe elim!: 1)
549 qed
551 lemma positive_Real:
552   assumes X: "cauchy X"
553   shows "positive (Real X) \<longleftrightarrow> (\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n)"
554   using assms positive.transfer
555   unfolding cr_real_eq rel_fun_def by simp
557 lemma positive_zero: "\<not> positive 0"
558   by transfer auto
561   "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x + y)"
562 apply transfer
563 apply (clarify, rename_tac a b i j)
564 apply (rule_tac x="a + b" in exI, simp)
565 apply (rule_tac x="max i j" in exI, clarsimp)
567 done
569 lemma positive_mult:
570   "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x * y)"
571 apply transfer
572 apply (clarify, rename_tac a b i j)
573 apply (rule_tac x="a * b" in exI, simp)
574 apply (rule_tac x="max i j" in exI, clarsimp)
575 apply (rule mult_strict_mono, auto)
576 done
578 lemma positive_minus:
579   "\<not> positive x \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> positive (- x)"
580 apply transfer
582 apply (drule (1) cauchy_not_vanishes_cases, safe)
583 apply blast+
584 done
586 instantiation real :: linordered_field
587 begin
589 definition
590   "x < y \<longleftrightarrow> positive (y - x)"
592 definition
593   "x \<le> (y::real) \<longleftrightarrow> x < y \<or> x = y"
595 definition
596   "\<bar>a::real\<bar> = (if a < 0 then - a else a)"
598 definition
599   "sgn (a::real) = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
601 instance proof
602   fix a b c :: real
603   show "\<bar>a\<bar> = (if a < 0 then - a else a)"
604     by (rule abs_real_def)
605   show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a"
606     unfolding less_eq_real_def less_real_def
608   show "a \<le> a"
609     unfolding less_eq_real_def by simp
610   show "a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
611     unfolding less_eq_real_def less_real_def
613   show "a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> a = b"
614     unfolding less_eq_real_def less_real_def
616   show "a \<le> b \<Longrightarrow> c + a \<le> c + b"
617     unfolding less_eq_real_def less_real_def by auto
618     (* FIXME: Procedure int_combine_numerals: c + b - (c + a) \<equiv> b + - a *)
619     (* Should produce c + b - (c + a) \<equiv> b - a *)
620   show "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
621     by (rule sgn_real_def)
622   show "a \<le> b \<or> b \<le> a"
623     unfolding less_eq_real_def less_real_def
624     by (auto dest!: positive_minus)
625   show "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
626     unfolding less_real_def
627     by (drule (1) positive_mult, simp add: algebra_simps)
628 qed
630 end
632 instantiation real :: distrib_lattice
633 begin
635 definition
636   "(inf :: real \<Rightarrow> real \<Rightarrow> real) = min"
638 definition
639   "(sup :: real \<Rightarrow> real \<Rightarrow> real) = max"
641 instance proof
642 qed (auto simp add: inf_real_def sup_real_def max_min_distrib2)
644 end
646 lemma of_nat_Real: "of_nat x = Real (\<lambda>n. of_nat x)"
647 apply (induct x)
650 done
652 lemma of_int_Real: "of_int x = Real (\<lambda>n. of_int x)"
653 apply (cases x rule: int_diff_cases)
654 apply (simp add: of_nat_Real diff_Real)
655 done
657 lemma of_rat_Real: "of_rat x = Real (\<lambda>n. x)"
658 apply (induct x)
659 apply (simp add: Fract_of_int_quotient of_rat_divide)
660 apply (simp add: of_int_Real divide_inverse)
661 apply (simp add: inverse_Real mult_Real)
662 done
664 instance real :: archimedean_field
665 proof
666   fix x :: real
667   show "\<exists>z. x \<le> of_int z"
668     apply (induct x)
669     apply (frule cauchy_imp_bounded, clarify)
670     apply (rule_tac x="\<lceil>b\<rceil> + 1" in exI)
671     apply (rule less_imp_le)
672     apply (simp add: of_int_Real less_real_def diff_Real positive_Real)
673     apply (rule_tac x=1 in exI, simp add: algebra_simps)
674     apply (rule_tac x=0 in exI, clarsimp)
675     apply (rule le_less_trans [OF abs_ge_self])
676     apply (rule less_le_trans [OF _ le_of_int_ceiling])
677     apply simp
678     done
679 qed
681 instantiation real :: floor_ceiling
682 begin
684 definition [code del]:
685   "\<lfloor>x::real\<rfloor> = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))"
687 instance
688 proof
689   fix x :: real
690   show "of_int \<lfloor>x\<rfloor> \<le> x \<and> x < of_int (\<lfloor>x\<rfloor> + 1)"
691     unfolding floor_real_def using floor_exists1 by (rule theI')
692 qed
694 end
696 subsection \<open>Completeness\<close>
698 lemma not_positive_Real:
699   assumes X: "cauchy X"
700   shows "\<not> positive (Real X) \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> r)"
701 unfolding positive_Real [OF X]
702 apply (auto, unfold not_less)
703 apply (erule obtain_pos_sum)
704 apply (drule_tac x=s in spec, simp)
705 apply (drule_tac r=t in cauchyD [OF X], clarify)
706 apply (drule_tac x=k in spec, clarsimp)
707 apply (rule_tac x=n in exI, clarify, rename_tac m)
708 apply (drule_tac x=m in spec, simp)
709 apply (drule_tac x=n in spec, simp)
710 apply (drule spec, drule (1) mp, clarify, rename_tac i)
711 apply (rule_tac x="max i k" in exI, simp)
712 done
714 lemma le_Real:
715   assumes X: "cauchy X" and Y: "cauchy Y"
716   shows "Real X \<le> Real Y = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r)"
717 unfolding not_less [symmetric, where 'a=real] less_real_def
718 apply (simp add: diff_Real not_positive_Real X Y)
719 apply (simp add: diff_le_eq ac_simps)
720 done
722 lemma le_RealI:
723   assumes Y: "cauchy Y"
724   shows "\<forall>n. x \<le> of_rat (Y n) \<Longrightarrow> x \<le> Real Y"
725 proof (induct x)
726   fix X assume X: "cauchy X" and "\<forall>n. Real X \<le> of_rat (Y n)"
727   hence le: "\<And>m r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. X n \<le> Y m + r"
728     by (simp add: of_rat_Real le_Real)
729   {
730     fix r :: rat assume "0 < r"
731     then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
732       by (rule obtain_pos_sum)
733     obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>Y m - Y n\<bar> < s"
734       using cauchyD [OF Y s] ..
735     obtain j where j: "\<forall>n\<ge>j. X n \<le> Y i + t"
736       using le [OF t] ..
737     have "\<forall>n\<ge>max i j. X n \<le> Y n + r"
738     proof (clarsimp)
739       fix n assume n: "i \<le> n" "j \<le> n"
740       have "X n \<le> Y i + t" using n j by simp
741       moreover have "\<bar>Y i - Y n\<bar> < s" using n i by simp
742       ultimately show "X n \<le> Y n + r" unfolding r by simp
743     qed
744     hence "\<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r" ..
745   }
746   thus "Real X \<le> Real Y"
747     by (simp add: of_rat_Real le_Real X Y)
748 qed
750 lemma Real_leI:
751   assumes X: "cauchy X"
752   assumes le: "\<forall>n. of_rat (X n) \<le> y"
753   shows "Real X \<le> y"
754 proof -
755   have "- y \<le> - Real X"
756     by (simp add: minus_Real X le_RealI of_rat_minus le)
757   thus ?thesis by simp
758 qed
760 lemma less_RealD:
761   assumes Y: "cauchy Y"
762   shows "x < Real Y \<Longrightarrow> \<exists>n. x < of_rat (Y n)"
763 by (erule contrapos_pp, simp add: not_less, erule Real_leI [OF Y])
765 lemma of_nat_less_two_power [simp]:
766   "of_nat n < (2::'a::linordered_idom) ^ n"
767 apply (induct n, simp)
768 by (metis add_le_less_mono mult_2 of_nat_Suc one_le_numeral one_le_power power_Suc)
770 lemma complete_real:
771   fixes S :: "real set"
772   assumes "\<exists>x. x \<in> S" and "\<exists>z. \<forall>x\<in>S. x \<le> z"
773   shows "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)"
774 proof -
775   obtain x where x: "x \<in> S" using assms(1) ..
776   obtain z where z: "\<forall>x\<in>S. x \<le> z" using assms(2) ..
778   define P where "P x \<longleftrightarrow> (\<forall>y\<in>S. y \<le> of_rat x)" for x
779   obtain a where a: "\<not> P a"
780   proof
781     have "of_int \<lfloor>x - 1\<rfloor> \<le> x - 1" by (rule of_int_floor_le)
782     also have "x - 1 < x" by simp
783     finally have "of_int \<lfloor>x - 1\<rfloor> < x" .
784     hence "\<not> x \<le> of_int \<lfloor>x - 1\<rfloor>" by (simp only: not_le)
785     then show "\<not> P (of_int \<lfloor>x - 1\<rfloor>)"
786       unfolding P_def of_rat_of_int_eq using x by blast
787   qed
788   obtain b where b: "P b"
789   proof
790     show "P (of_int \<lceil>z\<rceil>)"
791     unfolding P_def of_rat_of_int_eq
792     proof
793       fix y assume "y \<in> S"
794       hence "y \<le> z" using z by simp
795       also have "z \<le> of_int \<lceil>z\<rceil>" by (rule le_of_int_ceiling)
796       finally show "y \<le> of_int \<lceil>z\<rceil>" .
797     qed
798   qed
800   define avg where "avg x y = x/2 + y/2" for x y :: rat
801   define bisect where "bisect = (\<lambda>(x, y). if P (avg x y) then (x, avg x y) else (avg x y, y))"
802   define A where "A n = fst ((bisect ^^ n) (a, b))" for n
803   define B where "B n = snd ((bisect ^^ n) (a, b))" for n
804   define C where "C n = avg (A n) (B n)" for n
805   have A_0 [simp]: "A 0 = a" unfolding A_def by simp
806   have B_0 [simp]: "B 0 = b" unfolding B_def by simp
807   have A_Suc [simp]: "\<And>n. A (Suc n) = (if P (C n) then A n else C n)"
808     unfolding A_def B_def C_def bisect_def split_def by simp
809   have B_Suc [simp]: "\<And>n. B (Suc n) = (if P (C n) then C n else B n)"
810     unfolding A_def B_def C_def bisect_def split_def by simp
812   have width: "\<And>n. B n - A n = (b - a) / 2^n"
814     apply (induct_tac n, simp)
815     apply (simp add: C_def avg_def algebra_simps)
816     done
818   have twos: "\<And>y r :: rat. 0 < r \<Longrightarrow> \<exists>n. y / 2 ^ n < r"
820     apply (subst mult.commute)
821     apply (frule_tac y=y in ex_less_of_nat_mult)
822     apply clarify
823     apply (rule_tac x=n in exI)
824     apply (erule less_trans)
825     apply (rule mult_strict_right_mono)
826     apply (rule le_less_trans [OF _ of_nat_less_two_power])
827     apply simp
828     apply assumption
829     done
831   have PA: "\<And>n. \<not> P (A n)"
832     by (induct_tac n, simp_all add: a)
833   have PB: "\<And>n. P (B n)"
834     by (induct_tac n, simp_all add: b)
835   have ab: "a < b"
836     using a b unfolding P_def
837     apply (clarsimp simp add: not_le)
838     apply (drule (1) bspec)
839     apply (drule (1) less_le_trans)
841     done
842   have AB: "\<And>n. A n < B n"
844   have A_mono: "\<And>i j. i \<le> j \<Longrightarrow> A i \<le> A j"
845     apply (auto simp add: le_less [where 'a=nat])
846     apply (erule less_Suc_induct)
847     apply (clarsimp simp add: C_def avg_def)
849     apply (rule AB [THEN less_imp_le])
850     apply simp
851     done
852   have B_mono: "\<And>i j. i \<le> j \<Longrightarrow> B j \<le> B i"
853     apply (auto simp add: le_less [where 'a=nat])
854     apply (erule less_Suc_induct)
855     apply (clarsimp simp add: C_def avg_def)
857     apply (rule AB [THEN less_imp_le])
858     apply simp
859     done
860   have cauchy_lemma:
861     "\<And>X. \<forall>n. \<forall>i\<ge>n. A n \<le> X i \<and> X i \<le> B n \<Longrightarrow> cauchy X"
862     apply (rule cauchyI)
863     apply (drule twos [where y="b - a"])
864     apply (erule exE)
865     apply (rule_tac x=n in exI, clarify, rename_tac i j)
866     apply (rule_tac y="B n - A n" in le_less_trans) defer
868     apply (drule_tac x=n in spec)
869     apply (frule_tac x=i in spec, drule (1) mp)
870     apply (frule_tac x=j in spec, drule (1) mp)
871     apply (frule A_mono, drule B_mono)
872     apply (frule A_mono, drule B_mono)
873     apply arith
874     done
875   have "cauchy A"
876     apply (rule cauchy_lemma [rule_format])
878     apply (erule order_trans [OF less_imp_le [OF AB] B_mono])
879     done
880   have "cauchy B"
881     apply (rule cauchy_lemma [rule_format])
883     apply (erule order_trans [OF A_mono less_imp_le [OF AB]])
884     done
885   have 1: "\<forall>x\<in>S. x \<le> Real B"
886   proof
887     fix x assume "x \<in> S"
888     then show "x \<le> Real B"
889       using PB [unfolded P_def] \<open>cauchy B\<close>
891   qed
892   have 2: "\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> Real A \<le> z"
893     apply clarify
894     apply (erule contrapos_pp)
896     apply (drule less_RealD [OF \<open>cauchy A\<close>], clarify)
897     apply (subgoal_tac "\<not> P (A n)")
898     apply (simp add: P_def not_le, clarify)
899     apply (erule rev_bexI)
900     apply (erule (1) less_trans)
902     done
903   have "vanishes (\<lambda>n. (b - a) / 2 ^ n)"
904   proof (rule vanishesI)
905     fix r :: rat assume "0 < r"
906     then obtain k where k: "\<bar>b - a\<bar> / 2 ^ k < r"
907       using twos by blast
908     have "\<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r"
909     proof (clarify)
910       fix n assume n: "k \<le> n"
911       have "\<bar>(b - a) / 2 ^ n\<bar> = \<bar>b - a\<bar> / 2 ^ n"
912         by simp
913       also have "\<dots> \<le> \<bar>b - a\<bar> / 2 ^ k"
914         using n by (simp add: divide_left_mono)
915       also note k
916       finally show "\<bar>(b - a) / 2 ^ n\<bar> < r" .
917     qed
918     thus "\<exists>k. \<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r" ..
919   qed
920   hence 3: "Real B = Real A"
921     by (simp add: eq_Real \<open>cauchy A\<close> \<open>cauchy B\<close> width)
922   show "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)"
923     using 1 2 3 by (rule_tac x="Real B" in exI, simp)
924 qed
926 instantiation real :: linear_continuum
927 begin
929 subsection\<open>Supremum of a set of reals\<close>
931 definition "Sup X = (LEAST z::real. \<forall>x\<in>X. x \<le> z)"
932 definition "Inf (X::real set) = - Sup (uminus  X)"
934 instance
935 proof
936   { fix x :: real and X :: "real set"
937     assume x: "x \<in> X" "bdd_above X"
938     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"
939       using complete_real[of X] unfolding bdd_above_def by blast
940     then show "x \<le> Sup X"
941       unfolding Sup_real_def by (rule LeastI2_order) (auto simp: x) }
942   note Sup_upper = this
944   { fix z :: real and X :: "real set"
945     assume x: "X \<noteq> {}" and z: "\<And>x. x \<in> X \<Longrightarrow> x \<le> z"
946     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"
947       using complete_real[of X] by blast
948     then have "Sup X = s"
949       unfolding Sup_real_def by (best intro: Least_equality)
950     also from s z have "... \<le> z"
951       by blast
952     finally show "Sup X \<le> z" . }
953   note Sup_least = this
955   { fix x :: real and X :: "real set" assume x: "x \<in> X" "bdd_below X" then show "Inf X \<le> x"
956       using Sup_upper[of "-x" "uminus  X"] by (auto simp: Inf_real_def) }
957   { 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"
958       using Sup_least[of "uminus ` X" "- z"] by (force simp: Inf_real_def) }
959   show "\<exists>a b::real. a \<noteq> b"
960     using zero_neq_one by blast
961 qed
962 end
965 subsection \<open>Hiding implementation details\<close>
967 hide_const (open) vanishes cauchy positive Real
969 declare Real_induct [induct del]
970 declare Abs_real_induct [induct del]
971 declare Abs_real_cases [cases del]
973 lifting_update real.lifting
974 lifting_forget real.lifting
976 subsection\<open>More Lemmas\<close>
978 text \<open>BH: These lemmas should not be necessary; they should be
979 covered by existing simp rules and simplification procedures.\<close>
981 lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
982 by simp (* solved by linordered_ring_less_cancel_factor simproc *)
984 lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)"
985 by simp (* solved by linordered_ring_le_cancel_factor simproc *)
987 lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
988 by simp (* solved by linordered_ring_le_cancel_factor simproc *)
991 subsection \<open>Embedding numbers into the Reals\<close>
993 abbreviation
994   real_of_nat :: "nat \<Rightarrow> real"
995 where
996   "real_of_nat \<equiv> of_nat"
998 abbreviation
999   real :: "nat \<Rightarrow> real"
1000 where
1001   "real \<equiv> of_nat"
1003 abbreviation
1004   real_of_int :: "int \<Rightarrow> real"
1005 where
1006   "real_of_int \<equiv> of_int"
1008 abbreviation
1009   real_of_rat :: "rat \<Rightarrow> real"
1010 where
1011   "real_of_rat \<equiv> of_rat"
1013 declare [[coercion_enabled]]
1015 declare [[coercion "of_nat :: nat \<Rightarrow> int"]]
1016 declare [[coercion "of_nat :: nat \<Rightarrow> real"]]
1017 declare [[coercion "of_int :: int \<Rightarrow> real"]]
1019 (* We do not add rat to the coerced types, this has often unpleasant side effects when writing
1020 inverse (Suc n) which sometimes gets two coercions: of_rat (inverse (of_nat (Suc n))) *)
1022 declare [[coercion_map map]]
1023 declare [[coercion_map "\<lambda>f g h x. g (h (f x))"]]
1024 declare [[coercion_map "\<lambda>f g (x,y). (f x, g y)"]]
1026 declare of_int_eq_0_iff [algebra, presburger]
1027 declare of_int_eq_1_iff [algebra, presburger]
1028 declare of_int_eq_iff [algebra, presburger]
1029 declare of_int_less_0_iff [algebra, presburger]
1030 declare of_int_less_1_iff [algebra, presburger]
1031 declare of_int_less_iff [algebra, presburger]
1032 declare of_int_le_0_iff [algebra, presburger]
1033 declare of_int_le_1_iff [algebra, presburger]
1034 declare of_int_le_iff [algebra, presburger]
1035 declare of_int_0_less_iff [algebra, presburger]
1036 declare of_int_0_le_iff [algebra, presburger]
1037 declare of_int_1_less_iff [algebra, presburger]
1038 declare of_int_1_le_iff [algebra, presburger]
1040 lemma int_less_real_le: "(n < m) = (real_of_int n + 1 <= real_of_int m)"
1041 proof -
1042   have "(0::real) \<le> 1"
1043     by (metis less_eq_real_def zero_less_one)
1044   thus ?thesis
1045     by (metis floor_of_int less_floor_iff)
1046 qed
1048 lemma int_le_real_less: "(n \<le> m) = (real_of_int n < real_of_int m + 1)"
1049   by (meson int_less_real_le not_le)
1052 lemma real_of_int_div_aux: "(real_of_int x) / (real_of_int d) =
1053     real_of_int (x div d) + (real_of_int (x mod d)) / (real_of_int d)"
1054 proof -
1055   have "x = (x div d) * d + x mod d"
1056     by auto
1057   then have "real_of_int x = real_of_int (x div d) * real_of_int d + real_of_int(x mod d)"
1059   then have "real_of_int x / real_of_int d = ... / real_of_int d"
1060     by simp
1061   then show ?thesis
1063 qed
1065 lemma real_of_int_div:
1066   fixes d n :: int
1067   shows "d dvd n \<Longrightarrow> real_of_int (n div d) = real_of_int n / real_of_int d"
1070 lemma real_of_int_div2:
1071   "0 <= real_of_int n / real_of_int x - real_of_int (n div x)"
1072   apply (case_tac "x = 0", simp)
1073   apply (case_tac "0 < x")
1074    apply (metis add.left_neutral floor_correct floor_divide_of_int_eq le_diff_eq)
1075   apply (metis add.left_neutral floor_correct floor_divide_of_int_eq le_diff_eq)
1076   done
1078 lemma real_of_int_div3:
1079   "real_of_int (n::int) / real_of_int (x) - real_of_int (n div x) <= 1"
1081   apply (subst real_of_int_div_aux)
1082   apply (auto simp add: divide_le_eq intro: order_less_imp_le)
1083 done
1085 lemma real_of_int_div4: "real_of_int (n div x) <= real_of_int (n::int) / real_of_int x"
1086 by (insert real_of_int_div2 [of n x], simp)
1089 subsection\<open>Embedding the Naturals into the Reals\<close>
1091 lemma real_of_card: "real (card A) = setsum (\<lambda>x.1) A"
1092   by simp
1094 lemma nat_less_real_le: "(n < m) = (real n + 1 \<le> real m)"
1095   by (metis discrete of_nat_1 of_nat_add of_nat_le_iff)
1097 lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"
1098   by (meson nat_less_real_le not_le)
1100 lemma real_of_nat_div_aux: "(real x) / (real d) =
1101     real (x div d) + (real (x mod d)) / (real d)"
1102 proof -
1103   have "x = (x div d) * d + x mod d"
1104     by auto
1105   then have "real x = real (x div d) * real d + real(x mod d)"
1107   then have "real x / real d = \<dots> / real d"
1108     by simp
1109   then show ?thesis
1111 qed
1113 lemma real_of_nat_div: "d dvd n \<Longrightarrow> real(n div d) = real n / real d"
1114   by (subst real_of_nat_div_aux)
1115     (auto simp add: dvd_eq_mod_eq_0 [symmetric])
1117 lemma real_of_nat_div2:
1118   "0 <= real (n::nat) / real (x) - real (n div x)"
1120 apply (subst real_of_nat_div_aux)
1121 apply simp
1122 done
1124 lemma real_of_nat_div3:
1125   "real (n::nat) / real (x) - real (n div x) <= 1"
1126 apply(case_tac "x = 0")
1127 apply (simp)
1129 apply (subst real_of_nat_div_aux)
1130 apply simp
1131 done
1133 lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x"
1134 by (insert real_of_nat_div2 [of n x], simp)
1136 subsection \<open>The Archimedean Property of the Reals\<close>
1138 lemma real_arch_inverse: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
1139   using reals_Archimedean[of e] less_trans[of 0 "1 / real n" e for n::nat]
1140   by (auto simp add: field_simps cong: conj_cong simp del: of_nat_Suc)
1142 lemma reals_Archimedean3:
1143   assumes x_greater_zero: "0 < x"
1144   shows "\<forall>y. \<exists>n. y < real n * x"
1145   using \<open>0 < x\<close> by (auto intro: ex_less_of_nat_mult)
1147 lemma real_archimedian_rdiv_eq_0:
1148   assumes x0: "x \<ge> 0"
1149       and c: "c \<ge> 0"
1150       and xc: "\<And>m::nat. m > 0 \<Longrightarrow> real m * x \<le> c"
1151     shows "x = 0"
1152 by (metis reals_Archimedean3 dual_order.order_iff_strict le0 le_less_trans not_le x0 xc)
1155 subsection\<open>Rationals\<close>
1157 lemma Rats_eq_int_div_int:
1158   "\<rat> = { real_of_int i / real_of_int j |i j. j \<noteq> 0}" (is "_ = ?S")
1159 proof
1160   show "\<rat> \<subseteq> ?S"
1161   proof
1162     fix x::real assume "x : \<rat>"
1163     then obtain r where "x = of_rat r" unfolding Rats_def ..
1164     have "of_rat r : ?S"
1165       by (cases r) (auto simp add:of_rat_rat)
1166     thus "x : ?S" using \<open>x = of_rat r\<close> by simp
1167   qed
1168 next
1169   show "?S \<subseteq> \<rat>"
1170   proof(auto simp:Rats_def)
1171     fix i j :: int assume "j \<noteq> 0"
1172     hence "real_of_int i / real_of_int j = of_rat(Fract i j)"
1174     thus "real_of_int i / real_of_int j \<in> range of_rat" by blast
1175   qed
1176 qed
1178 lemma Rats_eq_int_div_nat:
1179   "\<rat> = { real_of_int i / real n |i n. n \<noteq> 0}"
1180 proof(auto simp:Rats_eq_int_div_int)
1181   fix i j::int assume "j \<noteq> 0"
1182   show "EX (i'::int) (n::nat). real_of_int i / real_of_int j = real_of_int i'/real n \<and> 0<n"
1183   proof cases
1184     assume "j>0"
1185     hence "real_of_int i / real_of_int j = real_of_int i/real(nat j) \<and> 0<nat j"
1187     thus ?thesis by blast
1188   next
1189     assume "~ j>0"
1190     hence "real_of_int i / real_of_int j = real_of_int(-i) / real(nat(-j)) \<and> 0<nat(-j)" using \<open>j\<noteq>0\<close>
1192     thus ?thesis by blast
1193   qed
1194 next
1195   fix i::int and n::nat assume "0 < n"
1196   hence "real_of_int i / real n = real_of_int i / real_of_int(int n) \<and> int n \<noteq> 0" by simp
1197   thus "\<exists>i' j. real_of_int i / real n = real_of_int i' / real_of_int j \<and> j \<noteq> 0" by blast
1198 qed
1200 lemma Rats_abs_nat_div_natE:
1201   assumes "x \<in> \<rat>"
1202   obtains m n :: nat
1203   where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "gcd m n = 1"
1204 proof -
1205   from \<open>x \<in> \<rat>\<close> obtain i::int and n::nat where "n \<noteq> 0" and "x = real_of_int i / real n"
1207   hence "\<bar>x\<bar> = real (nat \<bar>i\<bar>) / real n" by (simp add: of_nat_nat)
1208   then obtain m :: nat where x_rat: "\<bar>x\<bar> = real m / real n" by blast
1209   let ?gcd = "gcd m n"
1210   from \<open>n\<noteq>0\<close> have gcd: "?gcd \<noteq> 0" by simp
1211   let ?k = "m div ?gcd"
1212   let ?l = "n div ?gcd"
1213   let ?gcd' = "gcd ?k ?l"
1214   have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m"
1215     by (rule dvd_mult_div_cancel)
1216   have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n"
1217     by (rule dvd_mult_div_cancel)
1218   from \<open>n \<noteq> 0\<close> and gcd_l
1219   have "?gcd * ?l \<noteq> 0" by simp
1220   then have "?l \<noteq> 0" by (blast dest!: mult_not_zero)
1221   moreover
1222   have "\<bar>x\<bar> = real ?k / real ?l"
1223   proof -
1224     from gcd have "real ?k / real ?l = real (?gcd * ?k) / real (?gcd * ?l)"
1226     also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp
1227     also from x_rat have "\<dots> = \<bar>x\<bar>" ..
1228     finally show ?thesis ..
1229   qed
1230   moreover
1231   have "?gcd' = 1"
1232   proof -
1233     have "?gcd * ?gcd' = gcd (?gcd * ?k) (?gcd * ?l)"
1234       by (rule gcd_mult_distrib_nat)
1235     with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp
1236     with gcd show ?thesis by auto
1237   qed
1238   ultimately show ?thesis ..
1239 qed
1241 subsection\<open>Density of the Rational Reals in the Reals\<close>
1243 text\<open>This density proof is due to Stefan Richter and was ported by TN.  The
1244 original source is \emph{Real Analysis} by H.L. Royden.
1245 It employs the Archimedean property of the reals.\<close>
1247 lemma Rats_dense_in_real:
1248   fixes x :: real
1249   assumes "x < y" shows "\<exists>r\<in>\<rat>. x < r \<and> r < y"
1250 proof -
1251   from \<open>x<y\<close> have "0 < y-x" by simp
1252   with reals_Archimedean obtain q::nat
1253     where q: "inverse (real q) < y-x" and "0 < q" by blast
1254   define p where "p = \<lceil>y * real q\<rceil> - 1"
1255   define r where "r = of_int p / real q"
1256   from q have "x < y - inverse (real q)" by simp
1257   also have "y - inverse (real q) \<le> r"
1258     unfolding r_def p_def
1259     by (simp add: le_divide_eq left_diff_distrib le_of_int_ceiling \<open>0 < q\<close>)
1260   finally have "x < r" .
1261   moreover have "r < y"
1262     unfolding r_def p_def
1263     by (simp add: divide_less_eq diff_less_eq \<open>0 < q\<close>
1264       less_ceiling_iff [symmetric])
1265   moreover from r_def have "r \<in> \<rat>" by simp
1266   ultimately show ?thesis by blast
1267 qed
1269 lemma of_rat_dense:
1270   fixes x y :: real
1271   assumes "x < y"
1272   shows "\<exists>q :: rat. x < of_rat q \<and> of_rat q < y"
1273 using Rats_dense_in_real [OF \<open>x < y\<close>]
1274 by (auto elim: Rats_cases)
1277 subsection\<open>Numerals and Arithmetic\<close>
1279 lemma [code_abbrev]:   (*FIXME*)
1280   "real_of_int (numeral k) = numeral k"
1281   "real_of_int (- numeral k) = - numeral k"
1282   by simp_all
1284 declaration \<open>
1285   K (Lin_Arith.add_inj_thms [@{thm of_nat_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
1286     (* not needed because x < (y::nat) can be rewritten as Suc x <= y: of_nat_less_iff RS iffD2 *)
1287   #> Lin_Arith.add_inj_thms [@{thm of_int_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
1288     (* not needed because x < (y::int) can be rewritten as x + 1 <= y: of_int_less_iff RS iffD2 *)
1290       @{thm of_nat_mult}, @{thm of_int_0}, @{thm of_int_1},
1291       @{thm of_int_add}, @{thm of_int_minus}, @{thm of_int_diff},
1292       @{thm of_int_mult}, @{thm of_int_of_nat_eq},
1293       @{thm of_nat_numeral}, @{thm of_nat_numeral}, @{thm of_int_neg_numeral}]
1294   #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat \<Rightarrow> real"})
1295   #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int \<Rightarrow> real"}))
1296 \<close>
1298 subsection\<open>Simprules combining x+y and 0: ARE THEY NEEDED?\<close>
1300 lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)"
1301 by arith
1303 lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"
1304 by auto
1306 lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)"
1307 by auto
1309 lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> -x)"
1310 by auto
1312 lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (-x \<le> y)"
1313 by auto
1317 lemma two_realpow_ge_one: "(1::real) \<le> 2 ^ n"
1318   by simp
1320 text \<open>FIXME: declare this [simp] for all types, or not at all\<close>
1321 declare sum_squares_eq_zero_iff [simp] sum_power2_eq_zero_iff [simp]
1323 lemma real_minus_mult_self_le [simp]: "-(u * u) \<le> (x * (x::real))"
1324 by (rule_tac y = 0 in order_trans, auto)
1326 lemma realpow_square_minus_le [simp]: "- u\<^sup>2 \<le> (x::real)\<^sup>2"
1327   by (auto simp add: power2_eq_square)
1329 lemma numeral_power_eq_real_of_int_cancel_iff[simp]:
1330      "numeral x ^ n = real_of_int (y::int) \<longleftrightarrow> numeral x ^ n = y"
1331   by (metis of_int_eq_iff of_int_numeral of_int_power)
1333 lemma real_of_int_eq_numeral_power_cancel_iff[simp]:
1334      "real_of_int (y::int) = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"
1335   using numeral_power_eq_real_of_int_cancel_iff[of x n y]
1336   by metis
1338 lemma numeral_power_eq_real_of_nat_cancel_iff[simp]:
1339      "numeral x ^ n = real (y::nat) \<longleftrightarrow> numeral x ^ n = y"
1340   using of_nat_eq_iff by fastforce
1342 lemma real_of_nat_eq_numeral_power_cancel_iff[simp]:
1343   "real (y::nat) = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"
1344   using numeral_power_eq_real_of_nat_cancel_iff[of x n y]
1345   by metis
1347 lemma numeral_power_le_real_of_nat_cancel_iff[simp]:
1348   "(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::nat) ^ n \<le> a"
1349 by (metis of_nat_le_iff of_nat_numeral of_nat_power)
1351 lemma real_of_nat_le_numeral_power_cancel_iff[simp]:
1352   "real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::nat) ^ n"
1353 by (metis of_nat_le_iff of_nat_numeral of_nat_power)
1355 lemma numeral_power_le_real_of_int_cancel_iff[simp]:
1356     "(numeral x::real) ^ n \<le> real_of_int a \<longleftrightarrow> (numeral x::int) ^ n \<le> a"
1357   by (metis ceiling_le_iff ceiling_of_int of_int_numeral of_int_power)
1359 lemma real_of_int_le_numeral_power_cancel_iff[simp]:
1360     "real_of_int a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::int) ^ n"
1361   by (metis floor_of_int le_floor_iff of_int_numeral of_int_power)
1363 lemma numeral_power_less_real_of_nat_cancel_iff[simp]:
1364     "(numeral x::real) ^ n < real a \<longleftrightarrow> (numeral x::nat) ^ n < a"
1365   by (metis of_nat_less_iff of_nat_numeral of_nat_power)
1367 lemma real_of_nat_less_numeral_power_cancel_iff[simp]:
1368   "real a < (numeral x::real) ^ n \<longleftrightarrow> a < (numeral x::nat) ^ n"
1369 by (metis of_nat_less_iff of_nat_numeral of_nat_power)
1371 lemma numeral_power_less_real_of_int_cancel_iff[simp]:
1372     "(numeral x::real) ^ n < real_of_int a \<longleftrightarrow> (numeral x::int) ^ n < a"
1373   by (meson not_less real_of_int_le_numeral_power_cancel_iff)
1375 lemma real_of_int_less_numeral_power_cancel_iff[simp]:
1376      "real_of_int a < (numeral x::real) ^ n \<longleftrightarrow> a < (numeral x::int) ^ n"
1377   by (meson not_less numeral_power_le_real_of_int_cancel_iff)
1379 lemma neg_numeral_power_le_real_of_int_cancel_iff[simp]:
1380     "(- numeral x::real) ^ n \<le> real_of_int a \<longleftrightarrow> (- numeral x::int) ^ n \<le> a"
1381   by (metis of_int_le_iff of_int_neg_numeral of_int_power)
1383 lemma real_of_int_le_neg_numeral_power_cancel_iff[simp]:
1384      "real_of_int a \<le> (- numeral x::real) ^ n \<longleftrightarrow> a \<le> (- numeral x::int) ^ n"
1385   by (metis of_int_le_iff of_int_neg_numeral of_int_power)
1388 subsection\<open>Density of the Reals\<close>
1390 lemma real_lbound_gt_zero:
1391      "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"
1392 apply (rule_tac x = " (min d1 d2) /2" in exI)
1394 done
1397 text\<open>Similar results are proved in \<open>Fields\<close>\<close>
1398 lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"
1399   by auto
1401 lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"
1402   by auto
1404 lemma real_sum_of_halves: "x/2 + x/2 = (x::real)"
1405   by simp
1407 subsection\<open>Floor and Ceiling Functions from the Reals to the Integers\<close>
1409 (* FIXME: theorems for negative numerals. Many duplicates, e.g. from Archimedean_Field.thy. *)
1411 lemma real_of_nat_less_numeral_iff [simp]:
1412      "real (n::nat) < numeral w \<longleftrightarrow> n < numeral w"
1413   by (metis of_nat_less_iff of_nat_numeral)
1415 lemma numeral_less_real_of_nat_iff [simp]:
1416      "numeral w < real (n::nat) \<longleftrightarrow> numeral w < n"
1417   by (metis of_nat_less_iff of_nat_numeral)
1419 lemma numeral_le_real_of_nat_iff[simp]:
1420   "(numeral n \<le> real(m::nat)) = (numeral n \<le> m)"
1421 by (metis not_le real_of_nat_less_numeral_iff)
1423 declare of_int_floor_le [simp] (* FIXME*)
1425 lemma of_int_floor_cancel [simp]:
1426     "(of_int \<lfloor>x\<rfloor> = x) = (\<exists>n::int. x = of_int n)"
1427   by (metis floor_of_int)
1429 lemma floor_eq: "[| real_of_int n < x; x < real_of_int n + 1 |] ==> \<lfloor>x\<rfloor> = n"
1430   by linarith
1432 lemma floor_eq2: "[| real_of_int n \<le> x; x < real_of_int n + 1 |] ==> \<lfloor>x\<rfloor> = n"
1433   by linarith
1435 lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat \<lfloor>x\<rfloor> = n"
1436   by linarith
1438 lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat \<lfloor>x\<rfloor> = n"
1439   by linarith
1441 lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real_of_int \<lfloor>r\<rfloor>"
1442   by linarith
1444 lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real_of_int \<lfloor>r\<rfloor>"
1445   by linarith
1447 lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real_of_int \<lfloor>r\<rfloor> + 1"
1448   by linarith
1450 lemma real_of_int_floor_add_one_gt [simp]: "r < real_of_int \<lfloor>r\<rfloor> + 1"
1451   by linarith
1453 lemma floor_eq_iff: "\<lfloor>x\<rfloor> = b \<longleftrightarrow> of_int b \<le> x \<and> x < of_int (b + 1)"
1456 lemma floor_add2[simp]: "\<lfloor>of_int a + x\<rfloor> = a + \<lfloor>x\<rfloor>"
1459 lemma floor_divide_real_eq_div: "0 \<le> b \<Longrightarrow> \<lfloor>a / real_of_int b\<rfloor> = \<lfloor>a\<rfloor> div b"
1460 proof cases
1461   assume "0 < b"
1462   { fix i j :: int assume "real_of_int i \<le> a" "a < 1 + real_of_int i"
1463       "real_of_int j * real_of_int b \<le> a" "a < real_of_int b + real_of_int j * real_of_int b"
1464     then have "i < b + j * b"
1465       by (metis linorder_not_less of_int_add of_int_le_iff of_int_mult order_trans_rules(21))
1466     moreover have "j * b < 1 + i"
1467     proof -
1468       have "real_of_int (j * b) < real_of_int i + 1"
1469         using \<open>a < 1 + real_of_int i\<close> \<open>real_of_int j * real_of_int b \<le> a\<close> by force
1470       thus "j * b < 1 + i"
1471         by linarith
1472     qed
1473     ultimately have "(j - i div b) * b \<le> i mod b" "i mod b < ((j - i div b) + 1) * b"
1474       by (auto simp: field_simps)
1475     then have "(j - i div b) * b < 1 * b" "0 * b < ((j - i div b) + 1) * b"
1476       using pos_mod_bound[OF \<open>0<b\<close>, of i] pos_mod_sign[OF \<open>0<b\<close>, of i] by linarith+
1477     then have "j = i div b"
1478       using \<open>0 < b\<close> unfolding mult_less_cancel_right by auto
1479   }
1480   with \<open>0 < b\<close> show ?thesis
1481     by (auto split: floor_split simp: field_simps)
1482 qed auto
1484 lemma floor_divide_eq_div_numeral[simp]: "\<lfloor>numeral a / numeral b::real\<rfloor> = numeral a div numeral b"
1485   by (metis floor_divide_of_int_eq of_int_numeral)
1487 lemma floor_minus_divide_eq_div_numeral[simp]: "\<lfloor>- (numeral a / numeral b)::real\<rfloor> = - numeral a div numeral b"
1488   by (metis divide_minus_left floor_divide_of_int_eq of_int_neg_numeral of_int_numeral)
1490 lemma of_int_ceiling_cancel [simp]: "(of_int \<lceil>x\<rceil> = x) = (\<exists>n::int. x = of_int n)"
1491   using ceiling_of_int by metis
1493 lemma ceiling_eq: "[| of_int n < x; x \<le> of_int n + 1 |] ==> \<lceil>x\<rceil> = n + 1"
1496 lemma of_int_ceiling_diff_one_le [simp]: "of_int \<lceil>r\<rceil> - 1 \<le> r"
1497   by linarith
1499 lemma of_int_ceiling_le_add_one [simp]: "of_int \<lceil>r\<rceil> \<le> r + 1"
1500   by linarith
1502 lemma ceiling_le: "x <= of_int a ==> \<lceil>x\<rceil> <= a"
1505 lemma ceiling_divide_eq_div: "\<lceil>of_int a / of_int b\<rceil> = - (- a div b)"
1506   by (metis ceiling_def floor_divide_of_int_eq minus_divide_left of_int_minus)
1508 lemma ceiling_divide_eq_div_numeral [simp]:
1509   "\<lceil>numeral a / numeral b :: real\<rceil> = - (- numeral a div numeral b)"
1510   using ceiling_divide_eq_div[of "numeral a" "numeral b"] by simp
1512 lemma ceiling_minus_divide_eq_div_numeral [simp]:
1513   "\<lceil>- (numeral a / numeral b :: real)\<rceil> = - (numeral a div numeral b)"
1514   using ceiling_divide_eq_div[of "- numeral a" "numeral b"] by simp
1516 text\<open>The following lemmas are remnants of the erstwhile functions natfloor
1517 and natceiling.\<close>
1519 lemma nat_floor_neg: "(x::real) <= 0 ==> nat \<lfloor>x\<rfloor> = 0"
1520   by linarith
1522 lemma le_nat_floor: "real x <= a ==> x <= nat \<lfloor>a\<rfloor>"
1523   by linarith
1525 lemma le_mult_nat_floor: "nat \<lfloor>a\<rfloor> * nat \<lfloor>b\<rfloor> \<le> nat \<lfloor>a * b\<rfloor>"
1526   by (cases "0 <= a & 0 <= b")
1527      (auto simp add: nat_mult_distrib[symmetric] nat_mono le_mult_floor)
1529 lemma nat_ceiling_le_eq [simp]: "(nat \<lceil>x\<rceil> <= a) = (x <= real a)"
1530   by linarith
1532 lemma real_nat_ceiling_ge: "x <= real (nat \<lceil>x\<rceil>)"
1533   by linarith
1535 lemma Rats_no_top_le: "\<exists> q \<in> \<rat>. (x :: real) \<le> q"
1536   by (auto intro!: bexI[of _ "of_nat (nat \<lceil>x\<rceil>)"]) linarith
1538 lemma Rats_no_bot_less: "\<exists> q \<in> \<rat>. q < (x :: real)"
1539   apply (auto intro!: bexI[of _ "of_int (\<lfloor>x\<rfloor> - 1)"])
1540   apply (rule less_le_trans[OF _ of_int_floor_le])
1541   apply simp
1542   done
1544 subsection \<open>Exponentiation with floor\<close>
1546 lemma floor_power:
1547   assumes "x = of_int \<lfloor>x\<rfloor>"
1548   shows "\<lfloor>x ^ n\<rfloor> = \<lfloor>x\<rfloor> ^ n"
1549 proof -
1550   have "x ^ n = of_int (\<lfloor>x\<rfloor> ^ n)"
1551     using assms by (induct n arbitrary: x) simp_all
1552   then show ?thesis by (metis floor_of_int)
1553 qed
1555 lemma floor_numeral_power[simp]:
1556   "\<lfloor>numeral x ^ n\<rfloor> = numeral x ^ n"
1557   by (metis floor_of_int of_int_numeral of_int_power)
1559 lemma ceiling_numeral_power[simp]:
1560   "\<lceil>numeral x ^ n\<rceil> = numeral x ^ n"
1561   by (metis ceiling_of_int of_int_numeral of_int_power)
1563 subsection \<open>Implementation of rational real numbers\<close>
1565 text \<open>Formal constructor\<close>
1567 definition Ratreal :: "rat \<Rightarrow> real" where
1568   [code_abbrev, simp]: "Ratreal = of_rat"
1570 code_datatype Ratreal
1573 text \<open>Numerals\<close>
1575 lemma [code_abbrev]:
1576   "(of_rat (of_int a) :: real) = of_int a"
1577   by simp
1579 lemma [code_abbrev]:
1580   "(of_rat 0 :: real) = 0"
1581   by simp
1583 lemma [code_abbrev]:
1584   "(of_rat 1 :: real) = 1"
1585   by simp
1587 lemma [code_abbrev]:
1588   "(of_rat (- 1) :: real) = - 1"
1589   by simp
1591 lemma [code_abbrev]:
1592   "(of_rat (numeral k) :: real) = numeral k"
1593   by simp
1595 lemma [code_abbrev]:
1596   "(of_rat (- numeral k) :: real) = - numeral k"
1597   by simp
1599 lemma [code_post]:
1600   "(of_rat (1 / numeral k) :: real) = 1 / numeral k"
1601   "(of_rat (numeral k / numeral l) :: real) = numeral k / numeral l"
1602   "(of_rat (- (1 / numeral k)) :: real) = - (1 / numeral k)"
1603   "(of_rat (- (numeral k / numeral l)) :: real) = - (numeral k / numeral l)"
1604   by (simp_all add: of_rat_divide of_rat_minus)
1607 text \<open>Operations\<close>
1609 lemma zero_real_code [code]:
1610   "0 = Ratreal 0"
1611 by simp
1613 lemma one_real_code [code]:
1614   "1 = Ratreal 1"
1615 by simp
1617 instantiation real :: equal
1618 begin
1620 definition "HOL.equal (x::real) y \<longleftrightarrow> x - y = 0"
1622 instance proof
1625 lemma real_equal_code [code]:
1626   "HOL.equal (Ratreal x) (Ratreal y) \<longleftrightarrow> HOL.equal x y"
1627   by (simp add: equal_real_def equal)
1629 lemma [code nbe]:
1630   "HOL.equal (x::real) x \<longleftrightarrow> True"
1631   by (rule equal_refl)
1633 end
1635 lemma real_less_eq_code [code]: "Ratreal x \<le> Ratreal y \<longleftrightarrow> x \<le> y"
1638 lemma real_less_code [code]: "Ratreal x < Ratreal y \<longleftrightarrow> x < y"
1641 lemma real_plus_code [code]: "Ratreal x + Ratreal y = Ratreal (x + y)"
1644 lemma real_times_code [code]: "Ratreal x * Ratreal y = Ratreal (x * y)"
1647 lemma real_uminus_code [code]: "- Ratreal x = Ratreal (- x)"
1650 lemma real_minus_code [code]: "Ratreal x - Ratreal y = Ratreal (x - y)"
1653 lemma real_inverse_code [code]: "inverse (Ratreal x) = Ratreal (inverse x)"
1656 lemma real_divide_code [code]: "Ratreal x / Ratreal y = Ratreal (x / y)"
1659 lemma real_floor_code [code]: "\<lfloor>Ratreal x\<rfloor> = \<lfloor>x\<rfloor>"
1660   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)
1663 text \<open>Quickcheck\<close>
1665 definition (in term_syntax)
1666   valterm_ratreal :: "rat \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> real \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
1667   [code_unfold]: "valterm_ratreal k = Code_Evaluation.valtermify Ratreal {\<cdot>} k"
1669 notation fcomp (infixl "\<circ>>" 60)
1670 notation scomp (infixl "\<circ>\<rightarrow>" 60)
1672 instantiation real :: random
1673 begin
1675 definition
1676   "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>r. Pair (valterm_ratreal r))"
1678 instance ..
1680 end
1682 no_notation fcomp (infixl "\<circ>>" 60)
1683 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
1685 instantiation real :: exhaustive
1686 begin
1688 definition
1689   "exhaustive_real f d = Quickcheck_Exhaustive.exhaustive (%r. f (Ratreal r)) d"
1691 instance ..
1693 end
1695 instantiation real :: full_exhaustive
1696 begin
1698 definition
1699   "full_exhaustive_real f d = Quickcheck_Exhaustive.full_exhaustive (%r. f (valterm_ratreal r)) d"
1701 instance ..
1703 end
1705 instantiation real :: narrowing
1706 begin
1708 definition
1709   "narrowing = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.cons Ratreal) narrowing"
1711 instance ..
1713 end
1716 subsection \<open>Setup for Nitpick\<close>
1718 declaration \<open>
1719   Nitpick_HOL.register_frac_type @{type_name real}
1720     [(@{const_name zero_real_inst.zero_real}, @{const_name Nitpick.zero_frac}),
1721      (@{const_name one_real_inst.one_real}, @{const_name Nitpick.one_frac}),
1722      (@{const_name plus_real_inst.plus_real}, @{const_name Nitpick.plus_frac}),
1723      (@{const_name times_real_inst.times_real}, @{const_name Nitpick.times_frac}),
1724      (@{const_name uminus_real_inst.uminus_real}, @{const_name Nitpick.uminus_frac}),
1725      (@{const_name inverse_real_inst.inverse_real}, @{const_name Nitpick.inverse_frac}),
1726      (@{const_name ord_real_inst.less_real}, @{const_name Nitpick.less_frac}),
1727      (@{const_name ord_real_inst.less_eq_real}, @{const_name Nitpick.less_eq_frac})]
1728 \<close>
1730 lemmas [nitpick_unfold] = inverse_real_inst.inverse_real one_real_inst.one_real
1731     ord_real_inst.less_real ord_real_inst.less_eq_real plus_real_inst.plus_real
1732     times_real_inst.times_real uminus_real_inst.uminus_real
1733     zero_real_inst.zero_real
1736 subsection \<open>Setup for SMT\<close>
1738 ML_file "Tools/SMT/smt_real.ML"
1739 ML_file "Tools/SMT/z3_real.ML"
1741 lemma [z3_rule]:
1742   "0 + (x::real) = x"
1743   "x + 0 = x"
1744   "0 * x = 0"
1745   "1 * x = x"
1746   "x + y = y + x"
1747   by auto
1749 end