src/HOL/Real/RComplete.thy
 author wenzelm Wed Sep 17 21:27:14 2008 +0200 (2008-09-17) changeset 28263 69eaa97e7e96 parent 28091 50f2d6ba024c child 28562 4e74209f113e permissions -rw-r--r--
moved global ML bindings to global place;
1 (*  Title       : HOL/Real/RComplete.thy
2     ID          : $Id$
3     Author      : Jacques D. Fleuriot, University of Edinburgh
4     Author      : Larry Paulson, University of Cambridge
5     Author      : Jeremy Avigad, Carnegie Mellon University
6     Author      : Florian Zuleger, Johannes Hoelzl, and Simon Funke, TU Muenchen
7 *)
9 header {* Completeness of the Reals; Floor and Ceiling Functions *}
11 theory RComplete
12 imports Lubs RealDef
13 begin
15 lemma real_sum_of_halves: "x/2 + x/2 = (x::real)"
16   by simp
19 subsection {* Completeness of Positive Reals *}
21 text {*
22   Supremum property for the set of positive reals
24   Let @{text "P"} be a non-empty set of positive reals, with an upper
25   bound @{text "y"}.  Then @{text "P"} has a least upper bound
26   (written @{text "S"}).
28   FIXME: Can the premise be weakened to @{text "\<forall>x \<in> P. x\<le> y"}?
29 *}
31 lemma posreal_complete:
32   assumes positive_P: "\<forall>x \<in> P. (0::real) < x"
33     and not_empty_P: "\<exists>x. x \<in> P"
34     and upper_bound_Ex: "\<exists>y. \<forall>x \<in> P. x<y"
35   shows "\<exists>S. \<forall>y. (\<exists>x \<in> P. y < x) = (y < S)"
36 proof (rule exI, rule allI)
37   fix y
38   let ?pP = "{w. real_of_preal w \<in> P}"
40   show "(\<exists>x\<in>P. y < x) = (y < real_of_preal (psup ?pP))"
41   proof (cases "0 < y")
42     assume neg_y: "\<not> 0 < y"
43     show ?thesis
44     proof
45       assume "\<exists>x\<in>P. y < x"
46       have "\<forall>x. y < real_of_preal x"
47         using neg_y by (rule real_less_all_real2)
48       thus "y < real_of_preal (psup ?pP)" ..
49     next
50       assume "y < real_of_preal (psup ?pP)"
51       obtain "x" where x_in_P: "x \<in> P" using not_empty_P ..
52       hence "0 < x" using positive_P by simp
53       hence "y < x" using neg_y by simp
54       thus "\<exists>x \<in> P. y < x" using x_in_P ..
55     qed
56   next
57     assume pos_y: "0 < y"
59     then obtain py where y_is_py: "y = real_of_preal py"
60       by (auto simp add: real_gt_zero_preal_Ex)
62     obtain a where "a \<in> P" using not_empty_P ..
63     with positive_P have a_pos: "0 < a" ..
64     then obtain pa where "a = real_of_preal pa"
65       by (auto simp add: real_gt_zero_preal_Ex)
66     hence "pa \<in> ?pP" using a \<in> P by auto
67     hence pP_not_empty: "?pP \<noteq> {}" by auto
69     obtain sup where sup: "\<forall>x \<in> P. x < sup"
70       using upper_bound_Ex ..
71     from this and a \<in> P have "a < sup" ..
72     hence "0 < sup" using a_pos by arith
73     then obtain possup where "sup = real_of_preal possup"
74       by (auto simp add: real_gt_zero_preal_Ex)
75     hence "\<forall>X \<in> ?pP. X \<le> possup"
76       using sup by (auto simp add: real_of_preal_lessI)
77     with pP_not_empty have psup: "\<And>Z. (\<exists>X \<in> ?pP. Z < X) = (Z < psup ?pP)"
78       by (rule preal_complete)
80     show ?thesis
81     proof
82       assume "\<exists>x \<in> P. y < x"
83       then obtain x where x_in_P: "x \<in> P" and y_less_x: "y < x" ..
84       hence "0 < x" using pos_y by arith
85       then obtain px where x_is_px: "x = real_of_preal px"
86         by (auto simp add: real_gt_zero_preal_Ex)
88       have py_less_X: "\<exists>X \<in> ?pP. py < X"
89       proof
90         show "py < px" using y_is_py and x_is_px and y_less_x
91           by (simp add: real_of_preal_lessI)
92         show "px \<in> ?pP" using x_in_P and x_is_px by simp
93       qed
95       have "(\<exists>X \<in> ?pP. py < X) ==> (py < psup ?pP)"
96         using psup by simp
97       hence "py < psup ?pP" using py_less_X by simp
98       thus "y < real_of_preal (psup {w. real_of_preal w \<in> P})"
99         using y_is_py and pos_y by (simp add: real_of_preal_lessI)
100     next
101       assume y_less_psup: "y < real_of_preal (psup ?pP)"
103       hence "py < psup ?pP" using y_is_py
104         by (simp add: real_of_preal_lessI)
105       then obtain "X" where py_less_X: "py < X" and X_in_pP: "X \<in> ?pP"
106         using psup by auto
107       then obtain x where x_is_X: "x = real_of_preal X"
108         by (simp add: real_gt_zero_preal_Ex)
109       hence "y < x" using py_less_X and y_is_py
110         by (simp add: real_of_preal_lessI)
112       moreover have "x \<in> P" using x_is_X and X_in_pP by simp
114       ultimately show "\<exists> x \<in> P. y < x" ..
115     qed
116   qed
117 qed
119 text {*
120   \medskip Completeness properties using @{text "isUb"}, @{text "isLub"} etc.
121 *}
123 lemma real_isLub_unique: "[| isLub R S x; isLub R S y |] ==> x = (y::real)"
124   apply (frule isLub_isUb)
125   apply (frule_tac x = y in isLub_isUb)
126   apply (blast intro!: order_antisym dest!: isLub_le_isUb)
127   done
130 text {*
131   \medskip Completeness theorem for the positive reals (again).
132 *}
134 lemma posreals_complete:
135   assumes positive_S: "\<forall>x \<in> S. 0 < x"
136     and not_empty_S: "\<exists>x. x \<in> S"
137     and upper_bound_Ex: "\<exists>u. isUb (UNIV::real set) S u"
138   shows "\<exists>t. isLub (UNIV::real set) S t"
139 proof
140   let ?pS = "{w. real_of_preal w \<in> S}"
142   obtain u where "isUb UNIV S u" using upper_bound_Ex ..
143   hence sup: "\<forall>x \<in> S. x \<le> u" by (simp add: isUb_def setle_def)
145   obtain x where x_in_S: "x \<in> S" using not_empty_S ..
146   hence x_gt_zero: "0 < x" using positive_S by simp
147   have  "x \<le> u" using sup and x_in_S ..
148   hence "0 < u" using x_gt_zero by arith
150   then obtain pu where u_is_pu: "u = real_of_preal pu"
151     by (auto simp add: real_gt_zero_preal_Ex)
153   have pS_less_pu: "\<forall>pa \<in> ?pS. pa \<le> pu"
154   proof
155     fix pa
156     assume "pa \<in> ?pS"
157     then obtain a where "a \<in> S" and "a = real_of_preal pa"
158       by simp
159     moreover hence "a \<le> u" using sup by simp
160     ultimately show "pa \<le> pu"
161       using sup and u_is_pu by (simp add: real_of_preal_le_iff)
162   qed
164   have "\<forall>y \<in> S. y \<le> real_of_preal (psup ?pS)"
165   proof
166     fix y
167     assume y_in_S: "y \<in> S"
168     hence "0 < y" using positive_S by simp
169     then obtain py where y_is_py: "y = real_of_preal py"
170       by (auto simp add: real_gt_zero_preal_Ex)
171     hence py_in_pS: "py \<in> ?pS" using y_in_S by simp
172     with pS_less_pu have "py \<le> psup ?pS"
173       by (rule preal_psup_le)
174     thus "y \<le> real_of_preal (psup ?pS)"
175       using y_is_py by (simp add: real_of_preal_le_iff)
176   qed
178   moreover {
179     fix x
180     assume x_ub_S: "\<forall>y\<in>S. y \<le> x"
181     have "real_of_preal (psup ?pS) \<le> x"
182     proof -
183       obtain "s" where s_in_S: "s \<in> S" using not_empty_S ..
184       hence s_pos: "0 < s" using positive_S by simp
186       hence "\<exists> ps. s = real_of_preal ps" by (simp add: real_gt_zero_preal_Ex)
187       then obtain "ps" where s_is_ps: "s = real_of_preal ps" ..
188       hence ps_in_pS: "ps \<in> {w. real_of_preal w \<in> S}" using s_in_S by simp
190       from x_ub_S have "s \<le> x" using s_in_S ..
191       hence "0 < x" using s_pos by simp
192       hence "\<exists> px. x = real_of_preal px" by (simp add: real_gt_zero_preal_Ex)
193       then obtain "px" where x_is_px: "x = real_of_preal px" ..
195       have "\<forall>pe \<in> ?pS. pe \<le> px"
196       proof
197 	fix pe
198 	assume "pe \<in> ?pS"
199 	hence "real_of_preal pe \<in> S" by simp
200 	hence "real_of_preal pe \<le> x" using x_ub_S by simp
201 	thus "pe \<le> px" using x_is_px by (simp add: real_of_preal_le_iff)
202       qed
204       moreover have "?pS \<noteq> {}" using ps_in_pS by auto
205       ultimately have "(psup ?pS) \<le> px" by (simp add: psup_le_ub)
206       thus "real_of_preal (psup ?pS) \<le> x" using x_is_px by (simp add: real_of_preal_le_iff)
207     qed
208   }
209   ultimately show "isLub UNIV S (real_of_preal (psup ?pS))"
210     by (simp add: isLub_def leastP_def isUb_def setle_def setge_def)
211 qed
213 text {*
214   \medskip reals Completeness (again!)
215 *}
217 lemma reals_complete:
218   assumes notempty_S: "\<exists>X. X \<in> S"
219     and exists_Ub: "\<exists>Y. isUb (UNIV::real set) S Y"
220   shows "\<exists>t. isLub (UNIV :: real set) S t"
221 proof -
222   obtain X where X_in_S: "X \<in> S" using notempty_S ..
223   obtain Y where Y_isUb: "isUb (UNIV::real set) S Y"
224     using exists_Ub ..
225   let ?SHIFT = "{z. \<exists>x \<in>S. z = x + (-X) + 1} \<inter> {x. 0 < x}"
227   {
228     fix x
229     assume "isUb (UNIV::real set) S x"
230     hence S_le_x: "\<forall> y \<in> S. y <= x"
231       by (simp add: isUb_def setle_def)
232     {
233       fix s
234       assume "s \<in> {z. \<exists>x\<in>S. z = x + - X + 1}"
235       hence "\<exists> x \<in> S. s = x + -X + 1" ..
236       then obtain x1 where "x1 \<in> S" and "s = x1 + (-X) + 1" ..
237       moreover hence "x1 \<le> x" using S_le_x by simp
238       ultimately have "s \<le> x + - X + 1" by arith
239     }
240     then have "isUb (UNIV::real set) ?SHIFT (x + (-X) + 1)"
241       by (auto simp add: isUb_def setle_def)
242   } note S_Ub_is_SHIFT_Ub = this
244   hence "isUb UNIV ?SHIFT (Y + (-X) + 1)" using Y_isUb by simp
245   hence "\<exists>Z. isUb UNIV ?SHIFT Z" ..
246   moreover have "\<forall>y \<in> ?SHIFT. 0 < y" by auto
247   moreover have shifted_not_empty: "\<exists>u. u \<in> ?SHIFT"
248     using X_in_S and Y_isUb by auto
249   ultimately obtain t where t_is_Lub: "isLub UNIV ?SHIFT t"
250     using posreals_complete [of ?SHIFT] by blast
252   show ?thesis
253   proof
254     show "isLub UNIV S (t + X + (-1))"
255     proof (rule isLubI2)
256       {
257         fix x
258         assume "isUb (UNIV::real set) S x"
259         hence "isUb (UNIV::real set) (?SHIFT) (x + (-X) + 1)"
260 	  using S_Ub_is_SHIFT_Ub by simp
261         hence "t \<le> (x + (-X) + 1)"
262 	  using t_is_Lub by (simp add: isLub_le_isUb)
263         hence "t + X + -1 \<le> x" by arith
264       }
265       then show "(t + X + -1) <=* Collect (isUb UNIV S)"
266 	by (simp add: setgeI)
267     next
268       show "isUb UNIV S (t + X + -1)"
269       proof -
270         {
271           fix y
272           assume y_in_S: "y \<in> S"
273           have "y \<le> t + X + -1"
274           proof -
275             obtain "u" where u_in_shift: "u \<in> ?SHIFT" using shifted_not_empty ..
276             hence "\<exists> x \<in> S. u = x + - X + 1" by simp
277             then obtain "x" where x_and_u: "u = x + - X + 1" ..
278             have u_le_t: "u \<le> t" using u_in_shift and t_is_Lub by (simp add: isLubD2)
280             show ?thesis
281             proof cases
282               assume "y \<le> x"
283               moreover have "x = u + X + - 1" using x_and_u by arith
284               moreover have "u + X + - 1  \<le> t + X + -1" using u_le_t by arith
285               ultimately show "y  \<le> t + X + -1" by arith
286             next
287               assume "~(y \<le> x)"
288               hence x_less_y: "x < y" by arith
290               have "x + (-X) + 1 \<in> ?SHIFT" using x_and_u and u_in_shift by simp
291               hence "0 < x + (-X) + 1" by simp
292               hence "0 < y + (-X) + 1" using x_less_y by arith
293               hence "y + (-X) + 1 \<in> ?SHIFT" using y_in_S by simp
294               hence "y + (-X) + 1 \<le> t" using t_is_Lub  by (simp add: isLubD2)
295               thus ?thesis by simp
296             qed
297           qed
298         }
299         then show ?thesis by (simp add: isUb_def setle_def)
300       qed
301     qed
302   qed
303 qed
306 subsection {* The Archimedean Property of the Reals *}
308 theorem reals_Archimedean:
309   assumes x_pos: "0 < x"
310   shows "\<exists>n. inverse (real (Suc n)) < x"
311 proof (rule ccontr)
312   assume contr: "\<not> ?thesis"
313   have "\<forall>n. x * real (Suc n) <= 1"
314   proof
315     fix n
316     from contr have "x \<le> inverse (real (Suc n))"
317       by (simp add: linorder_not_less)
318     hence "x \<le> (1 / (real (Suc n)))"
319       by (simp add: inverse_eq_divide)
320     moreover have "0 \<le> real (Suc n)"
321       by (rule real_of_nat_ge_zero)
322     ultimately have "x * real (Suc n) \<le> (1 / real (Suc n)) * real (Suc n)"
323       by (rule mult_right_mono)
324     thus "x * real (Suc n) \<le> 1" by simp
325   qed
326   hence "{z. \<exists>n. z = x * (real (Suc n))} *<= 1"
327     by (simp add: setle_def, safe, rule spec)
328   hence "isUb (UNIV::real set) {z. \<exists>n. z = x * (real (Suc n))} 1"
329     by (simp add: isUbI)
330   hence "\<exists>Y. isUb (UNIV::real set) {z. \<exists>n. z = x* (real (Suc n))} Y" ..
331   moreover have "\<exists>X. X \<in> {z. \<exists>n. z = x* (real (Suc n))}" by auto
332   ultimately have "\<exists>t. isLub UNIV {z. \<exists>n. z = x * real (Suc n)} t"
333     by (simp add: reals_complete)
334   then obtain "t" where
335     t_is_Lub: "isLub UNIV {z. \<exists>n. z = x * real (Suc n)} t" ..
337   have "\<forall>n::nat. x * real n \<le> t + - x"
338   proof
339     fix n
340     from t_is_Lub have "x * real (Suc n) \<le> t"
341       by (simp add: isLubD2)
342     hence  "x * (real n) + x \<le> t"
343       by (simp add: right_distrib real_of_nat_Suc)
344     thus  "x * (real n) \<le> t + - x" by arith
345   qed
347   hence "\<forall>m. x * real (Suc m) \<le> t + - x" by simp
348   hence "{z. \<exists>n. z = x * (real (Suc n))}  *<= (t + - x)"
349     by (auto simp add: setle_def)
350   hence "isUb (UNIV::real set) {z. \<exists>n. z = x * (real (Suc n))} (t + (-x))"
351     by (simp add: isUbI)
352   hence "t \<le> t + - x"
353     using t_is_Lub by (simp add: isLub_le_isUb)
354   thus False using x_pos by arith
355 qed
357 text {*
358   There must be other proofs, e.g. @{text "Suc"} of the largest
359   integer in the cut representing @{text "x"}.
360 *}
362 lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"
363 proof cases
364   assume "x \<le> 0"
365   hence "x < real (1::nat)" by simp
366   thus ?thesis ..
367 next
368   assume "\<not> x \<le> 0"
369   hence x_greater_zero: "0 < x" by simp
370   hence "0 < inverse x" by simp
371   then obtain n where "inverse (real (Suc n)) < inverse x"
372     using reals_Archimedean by blast
373   hence "inverse (real (Suc n)) * x < inverse x * x"
374     using x_greater_zero by (rule mult_strict_right_mono)
375   hence "inverse (real (Suc n)) * x < 1"
376     using x_greater_zero by simp
377   hence "real (Suc n) * (inverse (real (Suc n)) * x) < real (Suc n) * 1"
378     by (rule mult_strict_left_mono) simp
379   hence "x < real (Suc n)"
380     by (simp add: ring_simps)
381   thus "\<exists>(n::nat). x < real n" ..
382 qed
384 lemma reals_Archimedean3:
385   assumes x_greater_zero: "0 < x"
386   shows "\<forall>(y::real). \<exists>(n::nat). y < real n * x"
387 proof
388   fix y
389   have x_not_zero: "x \<noteq> 0" using x_greater_zero by simp
390   obtain n where "y * inverse x < real (n::nat)"
391     using reals_Archimedean2 ..
392   hence "y * inverse x * x < real n * x"
393     using x_greater_zero by (simp add: mult_strict_right_mono)
394   hence "x * inverse x * y < x * real n"
395     by (simp add: ring_simps)
396   hence "y < real (n::nat) * x"
397     using x_not_zero by (simp add: ring_simps)
398   thus "\<exists>(n::nat). y < real n * x" ..
399 qed
401 lemma reals_Archimedean6:
402      "0 \<le> r ==> \<exists>(n::nat). real (n - 1) \<le> r & r < real (n)"
403 apply (insert reals_Archimedean2 [of r], safe)
404 apply (subgoal_tac "\<exists>x::nat. r < real x \<and> (\<forall>y. r < real y \<longrightarrow> x \<le> y)", auto)
405 apply (rule_tac x = x in exI)
406 apply (case_tac x, simp)
407 apply (rename_tac x')
408 apply (drule_tac x = x' in spec, simp)
409 apply (rule_tac x="LEAST n. r < real n" in exI, safe)
410 apply (erule LeastI, erule Least_le)
411 done
413 lemma reals_Archimedean6a: "0 \<le> r ==> \<exists>n. real (n) \<le> r & r < real (Suc n)"
414   by (drule reals_Archimedean6) auto
416 lemma reals_Archimedean_6b_int:
417      "0 \<le> r ==> \<exists>n::int. real n \<le> r & r < real (n+1)"
418 apply (drule reals_Archimedean6a, auto)
419 apply (rule_tac x = "int n" in exI)
420 apply (simp add: real_of_int_real_of_nat real_of_nat_Suc)
421 done
423 lemma reals_Archimedean_6c_int:
424      "r < 0 ==> \<exists>n::int. real n \<le> r & r < real (n+1)"
425 apply (rule reals_Archimedean_6b_int [of "-r", THEN exE], simp, auto)
426 apply (rename_tac n)
427 apply (drule order_le_imp_less_or_eq, auto)
428 apply (rule_tac x = "- n - 1" in exI)
429 apply (rule_tac  x = "- n" in exI, auto)
430 done
433 subsection{*Density of the Rational Reals in the Reals*}
435 text{* This density proof is due to Stefan Richter and was ported by TN.  The
436 original source is \emph{Real Analysis} by H.L. Royden.
437 It employs the Archimedean property of the reals. *}
439 lemma Rats_dense_in_nn_real: fixes x::real
440 assumes "0\<le>x" and "x<y" shows "\<exists>r \<in> \<rat>.  x<r \<and> r<y"
441 proof -
442   from x<y have "0 < y-x" by simp
443   with reals_Archimedean obtain q::nat
444     where q: "inverse (real q) < y-x" and "0 < real q" by auto
445   def p \<equiv> "LEAST n.  y \<le> real (Suc n)/real q"
446   from reals_Archimedean2 obtain n::nat where "y * real q < real n" by auto
447   with 0 < real q have ex: "y \<le> real n/real q" (is "?P n")
448     by (simp add: pos_less_divide_eq[THEN sym])
449   also from assms have "\<not> y \<le> real (0::nat) / real q" by simp
450   ultimately have main: "(LEAST n. y \<le> real n/real q) = Suc p"
451     by (unfold p_def) (rule Least_Suc)
452   also from ex have "?P (LEAST x. ?P x)" by (rule LeastI)
453   ultimately have suc: "y \<le> real (Suc p) / real q" by simp
454   def r \<equiv> "real p/real q"
455   have "x = y-(y-x)" by simp
456   also from suc q have "\<dots> < real (Suc p)/real q - inverse (real q)" by arith
457   also have "\<dots> = real p / real q"
458     by (simp only: inverse_eq_divide real_diff_def real_of_nat_Suc
459     minus_divide_left add_divide_distrib[THEN sym]) simp
460   finally have "x<r" by (unfold r_def)
461   have "p<Suc p" .. also note main[THEN sym]
462   finally have "\<not> ?P p"  by (rule not_less_Least)
463   hence "r<y" by (simp add: r_def)
464   from r_def have "r \<in> \<rat>" by simp
465   with x<r r<y show ?thesis by fast
466 qed
468 theorem Rats_dense_in_real: fixes x y :: real
469 assumes "x<y" shows "\<exists>r \<in> \<rat>.  x<r \<and> r<y"
470 proof -
471   from reals_Archimedean2 obtain n::nat where "-x < real n" by auto
472   hence "0 \<le> x + real n" by arith
473   also from x<y have "x + real n < y + real n" by arith
474   ultimately have "\<exists>r \<in> \<rat>. x + real n < r \<and> r < y + real n"
475     by(rule Rats_dense_in_nn_real)
476   then obtain r where "r \<in> \<rat>" and r2: "x + real n < r"
477     and r3: "r < y + real n"
478     by blast
479   have "r - real n = r + real (int n)/real (-1::int)" by simp
480   also from r\<in>\<rat> have "r + real (int n)/real (-1::int) \<in> \<rat>" by simp
481   also from r2 have "x < r - real n" by arith
482   moreover from r3 have "r - real n < y" by arith
483   ultimately show ?thesis by fast
484 qed
487 subsection{*Floor and Ceiling Functions from the Reals to the Integers*}
489 definition
490   floor :: "real => int" where
491   [code func del]: "floor r = (LEAST n::int. r < real (n+1))"
493 definition
494   ceiling :: "real => int" where
495   "ceiling r = - floor (- r)"
497 notation (xsymbols)
498   floor  ("\<lfloor>_\<rfloor>") and
499   ceiling  ("\<lceil>_\<rceil>")
501 notation (HTML output)
502   floor  ("\<lfloor>_\<rfloor>") and
503   ceiling  ("\<lceil>_\<rceil>")
506 lemma number_of_less_real_of_int_iff [simp]:
507      "((number_of n) < real (m::int)) = (number_of n < m)"
508 apply auto
509 apply (rule real_of_int_less_iff [THEN iffD1])
510 apply (drule_tac  real_of_int_less_iff [THEN iffD2], auto)
511 done
513 lemma number_of_less_real_of_int_iff2 [simp]:
514      "(real (m::int) < (number_of n)) = (m < number_of n)"
515 apply auto
516 apply (rule real_of_int_less_iff [THEN iffD1])
517 apply (drule_tac  real_of_int_less_iff [THEN iffD2], auto)
518 done
520 lemma number_of_le_real_of_int_iff [simp]:
521      "((number_of n) \<le> real (m::int)) = (number_of n \<le> m)"
522 by (simp add: linorder_not_less [symmetric])
524 lemma number_of_le_real_of_int_iff2 [simp]:
525      "(real (m::int) \<le> (number_of n)) = (m \<le> number_of n)"
526 by (simp add: linorder_not_less [symmetric])
528 lemma floor_zero [simp]: "floor 0 = 0"
529 apply (simp add: floor_def del: real_of_int_add)
530 apply (rule Least_equality)
531 apply simp_all
532 done
534 lemma floor_real_of_nat_zero [simp]: "floor (real (0::nat)) = 0"
535 by auto
537 lemma floor_real_of_nat [simp]: "floor (real (n::nat)) = int n"
538 apply (simp only: floor_def)
539 apply (rule Least_equality)
540 apply (drule_tac  real_of_int_of_nat_eq [THEN ssubst])
541 apply (drule_tac  real_of_int_less_iff [THEN iffD1])
542 apply simp_all
543 done
545 lemma floor_minus_real_of_nat [simp]: "floor (- real (n::nat)) = - int n"
546 apply (simp only: floor_def)
547 apply (rule Least_equality)
548 apply (drule_tac  real_of_int_of_nat_eq [THEN ssubst])
549 apply (drule_tac  real_of_int_minus [THEN sym, THEN subst])
550 apply (drule_tac  real_of_int_less_iff [THEN iffD1])
551 apply simp_all
552 done
554 lemma floor_real_of_int [simp]: "floor (real (n::int)) = n"
555 apply (simp only: floor_def)
556 apply (rule Least_equality)
557 apply auto
558 done
560 lemma floor_minus_real_of_int [simp]: "floor (- real (n::int)) = - n"
561 apply (simp only: floor_def)
562 apply (rule Least_equality)
563 apply (drule_tac  real_of_int_minus [THEN sym, THEN subst])
564 apply auto
565 done
567 lemma real_lb_ub_int: " \<exists>n::int. real n \<le> r & r < real (n+1)"
568 apply (case_tac "r < 0")
569 apply (blast intro: reals_Archimedean_6c_int)
570 apply (simp only: linorder_not_less)
571 apply (blast intro: reals_Archimedean_6b_int reals_Archimedean_6c_int)
572 done
574 lemma lemma_floor:
575   assumes a1: "real m \<le> r" and a2: "r < real n + 1"
576   shows "m \<le> (n::int)"
577 proof -
578   have "real m < real n + 1" using a1 a2 by (rule order_le_less_trans)
579   also have "... = real (n + 1)" by simp
580   finally have "m < n + 1" by (simp only: real_of_int_less_iff)
581   thus ?thesis by arith
582 qed
584 lemma real_of_int_floor_le [simp]: "real (floor r) \<le> r"
585 apply (simp add: floor_def Least_def)
586 apply (insert real_lb_ub_int [of r], safe)
587 apply (rule theI2)
588 apply auto
589 done
591 lemma floor_mono: "x < y ==> floor x \<le> floor y"
592 apply (simp add: floor_def Least_def)
593 apply (insert real_lb_ub_int [of x])
594 apply (insert real_lb_ub_int [of y], safe)
595 apply (rule theI2)
596 apply (rule_tac  theI2)
597 apply simp
598 apply (erule conjI)
599 apply (auto simp add: order_eq_iff int_le_real_less)
600 done
602 lemma floor_mono2: "x \<le> y ==> floor x \<le> floor y"
603 by (auto dest: order_le_imp_less_or_eq simp add: floor_mono)
605 lemma lemma_floor2: "real n < real (x::int) + 1 ==> n \<le> x"
606 by (auto intro: lemma_floor)
608 lemma real_of_int_floor_cancel [simp]:
609     "(real (floor x) = x) = (\<exists>n::int. x = real n)"
610 apply (simp add: floor_def Least_def)
611 apply (insert real_lb_ub_int [of x], erule exE)
612 apply (rule theI2)
613 apply (auto intro: lemma_floor)
614 done
616 lemma floor_eq: "[| real n < x; x < real n + 1 |] ==> floor x = n"
617 apply (simp add: floor_def)
618 apply (rule Least_equality)
619 apply (auto intro: lemma_floor)
620 done
622 lemma floor_eq2: "[| real n \<le> x; x < real n + 1 |] ==> floor x = n"
623 apply (simp add: floor_def)
624 apply (rule Least_equality)
625 apply (auto intro: lemma_floor)
626 done
628 lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"
629 apply (rule inj_int [THEN injD])
630 apply (simp add: real_of_nat_Suc)
631 apply (simp add: real_of_nat_Suc floor_eq floor_eq [where n = "int n"])
632 done
634 lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"
635 apply (drule order_le_imp_less_or_eq)
636 apply (auto intro: floor_eq3)
637 done
639 lemma floor_number_of_eq [simp]:
640      "floor(number_of n :: real) = (number_of n :: int)"
641 apply (subst real_number_of [symmetric])
642 apply (rule floor_real_of_int)
643 done
645 lemma floor_one [simp]: "floor 1 = 1"
646   apply (rule trans)
647   prefer 2
648   apply (rule floor_real_of_int)
649   apply simp
650 done
652 lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real(floor r)"
653 apply (simp add: floor_def Least_def)
654 apply (insert real_lb_ub_int [of r], safe)
655 apply (rule theI2)
656 apply (auto intro: lemma_floor)
657 done
659 lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real(floor r)"
660 apply (simp add: floor_def Least_def)
661 apply (insert real_lb_ub_int [of r], safe)
662 apply (rule theI2)
663 apply (auto intro: lemma_floor)
664 done
666 lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real(floor r) + 1"
667 apply (insert real_of_int_floor_ge_diff_one [of r])
668 apply (auto simp del: real_of_int_floor_ge_diff_one)
669 done
671 lemma real_of_int_floor_add_one_gt [simp]: "r < real(floor r) + 1"
672 apply (insert real_of_int_floor_gt_diff_one [of r])
673 apply (auto simp del: real_of_int_floor_gt_diff_one)
674 done
676 lemma le_floor: "real a <= x ==> a <= floor x"
677   apply (subgoal_tac "a < floor x + 1")
678   apply arith
679   apply (subst real_of_int_less_iff [THEN sym])
680   apply simp
681   apply (insert real_of_int_floor_add_one_gt [of x])
682   apply arith
683 done
685 lemma real_le_floor: "a <= floor x ==> real a <= x"
686   apply (rule order_trans)
687   prefer 2
688   apply (rule real_of_int_floor_le)
689   apply (subst real_of_int_le_iff)
690   apply assumption
691 done
693 lemma le_floor_eq: "(a <= floor x) = (real a <= x)"
694   apply (rule iffI)
695   apply (erule real_le_floor)
696   apply (erule le_floor)
697 done
699 lemma le_floor_eq_number_of [simp]:
700     "(number_of n <= floor x) = (number_of n <= x)"
701 by (simp add: le_floor_eq)
703 lemma le_floor_eq_zero [simp]: "(0 <= floor x) = (0 <= x)"
704 by (simp add: le_floor_eq)
706 lemma le_floor_eq_one [simp]: "(1 <= floor x) = (1 <= x)"
707 by (simp add: le_floor_eq)
709 lemma floor_less_eq: "(floor x < a) = (x < real a)"
710   apply (subst linorder_not_le [THEN sym])+
711   apply simp
712   apply (rule le_floor_eq)
713 done
715 lemma floor_less_eq_number_of [simp]:
716     "(floor x < number_of n) = (x < number_of n)"
717 by (simp add: floor_less_eq)
719 lemma floor_less_eq_zero [simp]: "(floor x < 0) = (x < 0)"
720 by (simp add: floor_less_eq)
722 lemma floor_less_eq_one [simp]: "(floor x < 1) = (x < 1)"
723 by (simp add: floor_less_eq)
725 lemma less_floor_eq: "(a < floor x) = (real a + 1 <= x)"
726   apply (insert le_floor_eq [of "a + 1" x])
727   apply auto
728 done
730 lemma less_floor_eq_number_of [simp]:
731     "(number_of n < floor x) = (number_of n + 1 <= x)"
732 by (simp add: less_floor_eq)
734 lemma less_floor_eq_zero [simp]: "(0 < floor x) = (1 <= x)"
735 by (simp add: less_floor_eq)
737 lemma less_floor_eq_one [simp]: "(1 < floor x) = (2 <= x)"
738 by (simp add: less_floor_eq)
740 lemma floor_le_eq: "(floor x <= a) = (x < real a + 1)"
741   apply (insert floor_less_eq [of x "a + 1"])
742   apply auto
743 done
745 lemma floor_le_eq_number_of [simp]:
746     "(floor x <= number_of n) = (x < number_of n + 1)"
747 by (simp add: floor_le_eq)
749 lemma floor_le_eq_zero [simp]: "(floor x <= 0) = (x < 1)"
750 by (simp add: floor_le_eq)
752 lemma floor_le_eq_one [simp]: "(floor x <= 1) = (x < 2)"
753 by (simp add: floor_le_eq)
755 lemma floor_add [simp]: "floor (x + real a) = floor x + a"
756   apply (subst order_eq_iff)
757   apply (rule conjI)
758   prefer 2
759   apply (subgoal_tac "floor x + a < floor (x + real a) + 1")
760   apply arith
761   apply (subst real_of_int_less_iff [THEN sym])
762   apply simp
763   apply (subgoal_tac "x + real a < real(floor(x + real a)) + 1")
764   apply (subgoal_tac "real (floor x) <= x")
765   apply arith
766   apply (rule real_of_int_floor_le)
767   apply (rule real_of_int_floor_add_one_gt)
768   apply (subgoal_tac "floor (x + real a) < floor x + a + 1")
769   apply arith
770   apply (subst real_of_int_less_iff [THEN sym])
771   apply simp
772   apply (subgoal_tac "real(floor(x + real a)) <= x + real a")
773   apply (subgoal_tac "x < real(floor x) + 1")
774   apply arith
775   apply (rule real_of_int_floor_add_one_gt)
776   apply (rule real_of_int_floor_le)
777 done
779 lemma floor_add_number_of [simp]:
780     "floor (x + number_of n) = floor x + number_of n"
781   apply (subst floor_add [THEN sym])
782   apply simp
783 done
785 lemma floor_add_one [simp]: "floor (x + 1) = floor x + 1"
786   apply (subst floor_add [THEN sym])
787   apply simp
788 done
790 lemma floor_subtract [simp]: "floor (x - real a) = floor x - a"
791   apply (subst diff_minus)+
792   apply (subst real_of_int_minus [THEN sym])
793   apply (rule floor_add)
794 done
796 lemma floor_subtract_number_of [simp]: "floor (x - number_of n) =
797     floor x - number_of n"
798   apply (subst floor_subtract [THEN sym])
799   apply simp
800 done
802 lemma floor_subtract_one [simp]: "floor (x - 1) = floor x - 1"
803   apply (subst floor_subtract [THEN sym])
804   apply simp
805 done
807 lemma ceiling_zero [simp]: "ceiling 0 = 0"
808 by (simp add: ceiling_def)
810 lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"
811 by (simp add: ceiling_def)
813 lemma ceiling_real_of_nat_zero [simp]: "ceiling (real (0::nat)) = 0"
814 by auto
816 lemma ceiling_floor [simp]: "ceiling (real (floor r)) = floor r"
817 by (simp add: ceiling_def)
819 lemma floor_ceiling [simp]: "floor (real (ceiling r)) = ceiling r"
820 by (simp add: ceiling_def)
822 lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)"
823 apply (simp add: ceiling_def)
824 apply (subst le_minus_iff, simp)
825 done
827 lemma ceiling_mono: "x < y ==> ceiling x \<le> ceiling y"
828 by (simp add: floor_mono ceiling_def)
830 lemma ceiling_mono2: "x \<le> y ==> ceiling x \<le> ceiling y"
831 by (simp add: floor_mono2 ceiling_def)
833 lemma real_of_int_ceiling_cancel [simp]:
834      "(real (ceiling x) = x) = (\<exists>n::int. x = real n)"
835 apply (auto simp add: ceiling_def)
836 apply (drule arg_cong [where f = uminus], auto)
837 apply (rule_tac x = "-n" in exI, auto)
838 done
840 lemma ceiling_eq: "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1"
841 apply (simp add: ceiling_def)
842 apply (rule minus_equation_iff [THEN iffD1])
843 apply (simp add: floor_eq [where n = "-(n+1)"])
844 done
846 lemma ceiling_eq2: "[| real n < x; x \<le> real n + 1 |] ==> ceiling x = n + 1"
847 by (simp add: ceiling_def floor_eq2 [where n = "-(n+1)"])
849 lemma ceiling_eq3: "[| real n - 1 < x; x \<le> real n  |] ==> ceiling x = n"
850 by (simp add: ceiling_def floor_eq2 [where n = "-n"])
852 lemma ceiling_real_of_int [simp]: "ceiling (real (n::int)) = n"
853 by (simp add: ceiling_def)
855 lemma ceiling_number_of_eq [simp]:
856      "ceiling (number_of n :: real) = (number_of n)"
857 apply (subst real_number_of [symmetric])
858 apply (rule ceiling_real_of_int)
859 done
861 lemma ceiling_one [simp]: "ceiling 1 = 1"
862   by (unfold ceiling_def, simp)
864 lemma real_of_int_ceiling_diff_one_le [simp]: "real (ceiling r) - 1 \<le> r"
865 apply (rule neg_le_iff_le [THEN iffD1])
866 apply (simp add: ceiling_def diff_minus)
867 done
869 lemma real_of_int_ceiling_le_add_one [simp]: "real (ceiling r) \<le> r + 1"
870 apply (insert real_of_int_ceiling_diff_one_le [of r])
871 apply (simp del: real_of_int_ceiling_diff_one_le)
872 done
874 lemma ceiling_le: "x <= real a ==> ceiling x <= a"
875   apply (unfold ceiling_def)
876   apply (subgoal_tac "-a <= floor(- x)")
877   apply simp
878   apply (rule le_floor)
879   apply simp
880 done
882 lemma ceiling_le_real: "ceiling x <= a ==> x <= real a"
883   apply (unfold ceiling_def)
884   apply (subgoal_tac "real(- a) <= - x")
885   apply simp
886   apply (rule real_le_floor)
887   apply simp
888 done
890 lemma ceiling_le_eq: "(ceiling x <= a) = (x <= real a)"
891   apply (rule iffI)
892   apply (erule ceiling_le_real)
893   apply (erule ceiling_le)
894 done
896 lemma ceiling_le_eq_number_of [simp]:
897     "(ceiling x <= number_of n) = (x <= number_of n)"
898 by (simp add: ceiling_le_eq)
900 lemma ceiling_le_zero_eq [simp]: "(ceiling x <= 0) = (x <= 0)"
901 by (simp add: ceiling_le_eq)
903 lemma ceiling_le_eq_one [simp]: "(ceiling x <= 1) = (x <= 1)"
904 by (simp add: ceiling_le_eq)
906 lemma less_ceiling_eq: "(a < ceiling x) = (real a < x)"
907   apply (subst linorder_not_le [THEN sym])+
908   apply simp
909   apply (rule ceiling_le_eq)
910 done
912 lemma less_ceiling_eq_number_of [simp]:
913     "(number_of n < ceiling x) = (number_of n < x)"
914 by (simp add: less_ceiling_eq)
916 lemma less_ceiling_eq_zero [simp]: "(0 < ceiling x) = (0 < x)"
917 by (simp add: less_ceiling_eq)
919 lemma less_ceiling_eq_one [simp]: "(1 < ceiling x) = (1 < x)"
920 by (simp add: less_ceiling_eq)
922 lemma ceiling_less_eq: "(ceiling x < a) = (x <= real a - 1)"
923   apply (insert ceiling_le_eq [of x "a - 1"])
924   apply auto
925 done
927 lemma ceiling_less_eq_number_of [simp]:
928     "(ceiling x < number_of n) = (x <= number_of n - 1)"
929 by (simp add: ceiling_less_eq)
931 lemma ceiling_less_eq_zero [simp]: "(ceiling x < 0) = (x <= -1)"
932 by (simp add: ceiling_less_eq)
934 lemma ceiling_less_eq_one [simp]: "(ceiling x < 1) = (x <= 0)"
935 by (simp add: ceiling_less_eq)
937 lemma le_ceiling_eq: "(a <= ceiling x) = (real a - 1 < x)"
938   apply (insert less_ceiling_eq [of "a - 1" x])
939   apply auto
940 done
942 lemma le_ceiling_eq_number_of [simp]:
943     "(number_of n <= ceiling x) = (number_of n - 1 < x)"
944 by (simp add: le_ceiling_eq)
946 lemma le_ceiling_eq_zero [simp]: "(0 <= ceiling x) = (-1 < x)"
947 by (simp add: le_ceiling_eq)
949 lemma le_ceiling_eq_one [simp]: "(1 <= ceiling x) = (0 < x)"
950 by (simp add: le_ceiling_eq)
952 lemma ceiling_add [simp]: "ceiling (x + real a) = ceiling x + a"
953   apply (unfold ceiling_def, simp)
954   apply (subst real_of_int_minus [THEN sym])
955   apply (subst floor_add)
956   apply simp
957 done
959 lemma ceiling_add_number_of [simp]: "ceiling (x + number_of n) =
960     ceiling x + number_of n"
961   apply (subst ceiling_add [THEN sym])
962   apply simp
963 done
965 lemma ceiling_add_one [simp]: "ceiling (x + 1) = ceiling x + 1"
966   apply (subst ceiling_add [THEN sym])
967   apply simp
968 done
970 lemma ceiling_subtract [simp]: "ceiling (x - real a) = ceiling x - a"
971   apply (subst diff_minus)+
972   apply (subst real_of_int_minus [THEN sym])
973   apply (rule ceiling_add)
974 done
976 lemma ceiling_subtract_number_of [simp]: "ceiling (x - number_of n) =
977     ceiling x - number_of n"
978   apply (subst ceiling_subtract [THEN sym])
979   apply simp
980 done
982 lemma ceiling_subtract_one [simp]: "ceiling (x - 1) = ceiling x - 1"
983   apply (subst ceiling_subtract [THEN sym])
984   apply simp
985 done
987 subsection {* Versions for the natural numbers *}
989 definition
990   natfloor :: "real => nat" where
991   "natfloor x = nat(floor x)"
993 definition
994   natceiling :: "real => nat" where
995   "natceiling x = nat(ceiling x)"
997 lemma natfloor_zero [simp]: "natfloor 0 = 0"
998   by (unfold natfloor_def, simp)
1000 lemma natfloor_one [simp]: "natfloor 1 = 1"
1001   by (unfold natfloor_def, simp)
1003 lemma zero_le_natfloor [simp]: "0 <= natfloor x"
1004   by (unfold natfloor_def, simp)
1006 lemma natfloor_number_of_eq [simp]: "natfloor (number_of n) = number_of n"
1007   by (unfold natfloor_def, simp)
1009 lemma natfloor_real_of_nat [simp]: "natfloor(real n) = n"
1010   by (unfold natfloor_def, simp)
1012 lemma real_natfloor_le: "0 <= x ==> real(natfloor x) <= x"
1013   by (unfold natfloor_def, simp)
1015 lemma natfloor_neg: "x <= 0 ==> natfloor x = 0"
1016   apply (unfold natfloor_def)
1017   apply (subgoal_tac "floor x <= floor 0")
1018   apply simp
1019   apply (erule floor_mono2)
1020 done
1022 lemma natfloor_mono: "x <= y ==> natfloor x <= natfloor y"
1023   apply (case_tac "0 <= x")
1024   apply (subst natfloor_def)+
1025   apply (subst nat_le_eq_zle)
1026   apply force
1027   apply (erule floor_mono2)
1028   apply (subst natfloor_neg)
1029   apply simp
1030   apply simp
1031 done
1033 lemma le_natfloor: "real x <= a ==> x <= natfloor a"
1034   apply (unfold natfloor_def)
1035   apply (subst nat_int [THEN sym])
1036   apply (subst nat_le_eq_zle)
1037   apply simp
1038   apply (rule le_floor)
1039   apply simp
1040 done
1042 lemma le_natfloor_eq: "0 <= x ==> (a <= natfloor x) = (real a <= x)"
1043   apply (rule iffI)
1044   apply (rule order_trans)
1045   prefer 2
1046   apply (erule real_natfloor_le)
1047   apply (subst real_of_nat_le_iff)
1048   apply assumption
1049   apply (erule le_natfloor)
1050 done
1052 lemma le_natfloor_eq_number_of [simp]:
1053     "~ neg((number_of n)::int) ==> 0 <= x ==>
1054       (number_of n <= natfloor x) = (number_of n <= x)"
1055   apply (subst le_natfloor_eq, assumption)
1056   apply simp
1057 done
1059 lemma le_natfloor_eq_one [simp]: "(1 <= natfloor x) = (1 <= x)"
1060   apply (case_tac "0 <= x")
1061   apply (subst le_natfloor_eq, assumption, simp)
1062   apply (rule iffI)
1063   apply (subgoal_tac "natfloor x <= natfloor 0")
1064   apply simp
1065   apply (rule natfloor_mono)
1066   apply simp
1067   apply simp
1068 done
1070 lemma natfloor_eq: "real n <= x ==> x < real n + 1 ==> natfloor x = n"
1071   apply (unfold natfloor_def)
1072   apply (subst nat_int [THEN sym]);back;
1073   apply (subst eq_nat_nat_iff)
1074   apply simp
1075   apply simp
1076   apply (rule floor_eq2)
1077   apply auto
1078 done
1080 lemma real_natfloor_add_one_gt: "x < real(natfloor x) + 1"
1081   apply (case_tac "0 <= x")
1082   apply (unfold natfloor_def)
1083   apply simp
1084   apply simp_all
1085 done
1087 lemma real_natfloor_gt_diff_one: "x - 1 < real(natfloor x)"
1088   apply (simp add: compare_rls)
1089   apply (rule real_natfloor_add_one_gt)
1090 done
1092 lemma ge_natfloor_plus_one_imp_gt: "natfloor z + 1 <= n ==> z < real n"
1093   apply (subgoal_tac "z < real(natfloor z) + 1")
1094   apply arith
1095   apply (rule real_natfloor_add_one_gt)
1096 done
1098 lemma natfloor_add [simp]: "0 <= x ==> natfloor (x + real a) = natfloor x + a"
1099   apply (unfold natfloor_def)
1100   apply (subgoal_tac "real a = real (int a)")
1101   apply (erule ssubst)
1102   apply (simp add: nat_add_distrib del: real_of_int_of_nat_eq)
1103   apply simp
1104 done
1106 lemma natfloor_add_number_of [simp]:
1107     "~neg ((number_of n)::int) ==> 0 <= x ==>
1108       natfloor (x + number_of n) = natfloor x + number_of n"
1109   apply (subst natfloor_add [THEN sym])
1110   apply simp_all
1111 done
1113 lemma natfloor_add_one: "0 <= x ==> natfloor(x + 1) = natfloor x + 1"
1114   apply (subst natfloor_add [THEN sym])
1115   apply assumption
1116   apply simp
1117 done
1119 lemma natfloor_subtract [simp]: "real a <= x ==>
1120     natfloor(x - real a) = natfloor x - a"
1121   apply (unfold natfloor_def)
1122   apply (subgoal_tac "real a = real (int a)")
1123   apply (erule ssubst)
1124   apply (simp del: real_of_int_of_nat_eq)
1125   apply simp
1126 done
1128 lemma natceiling_zero [simp]: "natceiling 0 = 0"
1129   by (unfold natceiling_def, simp)
1131 lemma natceiling_one [simp]: "natceiling 1 = 1"
1132   by (unfold natceiling_def, simp)
1134 lemma zero_le_natceiling [simp]: "0 <= natceiling x"
1135   by (unfold natceiling_def, simp)
1137 lemma natceiling_number_of_eq [simp]: "natceiling (number_of n) = number_of n"
1138   by (unfold natceiling_def, simp)
1140 lemma natceiling_real_of_nat [simp]: "natceiling(real n) = n"
1141   by (unfold natceiling_def, simp)
1143 lemma real_natceiling_ge: "x <= real(natceiling x)"
1144   apply (unfold natceiling_def)
1145   apply (case_tac "x < 0")
1146   apply simp
1147   apply (subst real_nat_eq_real)
1148   apply (subgoal_tac "ceiling 0 <= ceiling x")
1149   apply simp
1150   apply (rule ceiling_mono2)
1151   apply simp
1152   apply simp
1153 done
1155 lemma natceiling_neg: "x <= 0 ==> natceiling x = 0"
1156   apply (unfold natceiling_def)
1157   apply simp
1158 done
1160 lemma natceiling_mono: "x <= y ==> natceiling x <= natceiling y"
1161   apply (case_tac "0 <= x")
1162   apply (subst natceiling_def)+
1163   apply (subst nat_le_eq_zle)
1164   apply (rule disjI2)
1165   apply (subgoal_tac "real (0::int) <= real(ceiling y)")
1166   apply simp
1167   apply (rule order_trans)
1168   apply simp
1169   apply (erule order_trans)
1170   apply simp
1171   apply (erule ceiling_mono2)
1172   apply (subst natceiling_neg)
1173   apply simp_all
1174 done
1176 lemma natceiling_le: "x <= real a ==> natceiling x <= a"
1177   apply (unfold natceiling_def)
1178   apply (case_tac "x < 0")
1179   apply simp
1180   apply (subst nat_int [THEN sym]);back;
1181   apply (subst nat_le_eq_zle)
1182   apply simp
1183   apply (rule ceiling_le)
1184   apply simp
1185 done
1187 lemma natceiling_le_eq: "0 <= x ==> (natceiling x <= a) = (x <= real a)"
1188   apply (rule iffI)
1189   apply (rule order_trans)
1190   apply (rule real_natceiling_ge)
1191   apply (subst real_of_nat_le_iff)
1192   apply assumption
1193   apply (erule natceiling_le)
1194 done
1196 lemma natceiling_le_eq_number_of [simp]:
1197     "~ neg((number_of n)::int) ==> 0 <= x ==>
1198       (natceiling x <= number_of n) = (x <= number_of n)"
1199   apply (subst natceiling_le_eq, assumption)
1200   apply simp
1201 done
1203 lemma natceiling_le_eq_one: "(natceiling x <= 1) = (x <= 1)"
1204   apply (case_tac "0 <= x")
1205   apply (subst natceiling_le_eq)
1206   apply assumption
1207   apply simp
1208   apply (subst natceiling_neg)
1209   apply simp
1210   apply simp
1211 done
1213 lemma natceiling_eq: "real n < x ==> x <= real n + 1 ==> natceiling x = n + 1"
1214   apply (unfold natceiling_def)
1215   apply (simplesubst nat_int [THEN sym]) back back
1216   apply (subgoal_tac "nat(int n) + 1 = nat(int n + 1)")
1217   apply (erule ssubst)
1218   apply (subst eq_nat_nat_iff)
1219   apply (subgoal_tac "ceiling 0 <= ceiling x")
1220   apply simp
1221   apply (rule ceiling_mono2)
1222   apply force
1223   apply force
1224   apply (rule ceiling_eq2)
1225   apply (simp, simp)
1226   apply (subst nat_add_distrib)
1227   apply auto
1228 done
1230 lemma natceiling_add [simp]: "0 <= x ==>
1231     natceiling (x + real a) = natceiling x + a"
1232   apply (unfold natceiling_def)
1233   apply (subgoal_tac "real a = real (int a)")
1234   apply (erule ssubst)
1235   apply (simp del: real_of_int_of_nat_eq)
1236   apply (subst nat_add_distrib)
1237   apply (subgoal_tac "0 = ceiling 0")
1238   apply (erule ssubst)
1239   apply (erule ceiling_mono2)
1240   apply simp_all
1241 done
1243 lemma natceiling_add_number_of [simp]:
1244     "~ neg ((number_of n)::int) ==> 0 <= x ==>
1245       natceiling (x + number_of n) = natceiling x + number_of n"
1246   apply (subst natceiling_add [THEN sym])
1247   apply simp_all
1248 done
1250 lemma natceiling_add_one: "0 <= x ==> natceiling(x + 1) = natceiling x + 1"
1251   apply (subst natceiling_add [THEN sym])
1252   apply assumption
1253   apply simp
1254 done
1256 lemma natceiling_subtract [simp]: "real a <= x ==>
1257     natceiling(x - real a) = natceiling x - a"
1258   apply (unfold natceiling_def)
1259   apply (subgoal_tac "real a = real (int a)")
1260   apply (erule ssubst)
1261   apply (simp del: real_of_int_of_nat_eq)
1262   apply simp
1263 done
1265 lemma natfloor_div_nat: "1 <= x ==> y > 0 ==>
1266   natfloor (x / real y) = natfloor x div y"
1267 proof -
1268   assume "1 <= (x::real)" and "(y::nat) > 0"
1269   have "natfloor x = (natfloor x) div y * y + (natfloor x) mod y"
1270     by simp
1271   then have a: "real(natfloor x) = real ((natfloor x) div y) * real y +
1272     real((natfloor x) mod y)"
1273     by (simp only: real_of_nat_add [THEN sym] real_of_nat_mult [THEN sym])
1274   have "x = real(natfloor x) + (x - real(natfloor x))"
1275     by simp
1276   then have "x = real ((natfloor x) div y) * real y +
1277       real((natfloor x) mod y) + (x - real(natfloor x))"
1278     by (simp add: a)
1279   then have "x / real y = ... / real y"
1280     by simp
1281   also have "... = real((natfloor x) div y) + real((natfloor x) mod y) /
1282     real y + (x - real(natfloor x)) / real y"
1283     by (auto simp add: ring_simps add_divide_distrib
1284       diff_divide_distrib prems)
1285   finally have "natfloor (x / real y) = natfloor(...)" by simp
1286   also have "... = natfloor(real((natfloor x) mod y) /
1287     real y + (x - real(natfloor x)) / real y + real((natfloor x) div y))"
1289   also have "... = natfloor(real((natfloor x) mod y) /
1290     real y + (x - real(natfloor x)) / real y) + (natfloor x) div y"
1291     apply (rule natfloor_add)
1292     apply (rule add_nonneg_nonneg)
1293     apply (rule divide_nonneg_pos)
1294     apply simp
1295     apply (simp add: prems)
1296     apply (rule divide_nonneg_pos)
1297     apply (simp add: compare_rls)
1298     apply (rule real_natfloor_le)
1299     apply (insert prems, auto)
1300     done
1301   also have "natfloor(real((natfloor x) mod y) /
1302     real y + (x - real(natfloor x)) / real y) = 0"
1303     apply (rule natfloor_eq)
1304     apply simp
1305     apply (rule add_nonneg_nonneg)
1306     apply (rule divide_nonneg_pos)
1307     apply force
1308     apply (force simp add: prems)
1309     apply (rule divide_nonneg_pos)
1310     apply (simp add: compare_rls)
1311     apply (rule real_natfloor_le)
1312     apply (auto simp add: prems)
1313     apply (insert prems, arith)
1314     apply (simp add: add_divide_distrib [THEN sym])
1315     apply (subgoal_tac "real y = real y - 1 + 1")
1316     apply (erule ssubst)
1317     apply (rule add_le_less_mono)
1318     apply (simp add: compare_rls)
1319     apply (subgoal_tac "real(natfloor x mod y) + 1 =
1320       real(natfloor x mod y + 1)")
1321     apply (erule ssubst)
1322     apply (subst real_of_nat_le_iff)
1323     apply (subgoal_tac "natfloor x mod y < y")
1324     apply arith
1325     apply (rule mod_less_divisor)
1326     apply auto
1327     apply (simp add: compare_rls)
1328     apply (subst add_commute)
1329     apply (rule real_natfloor_add_one_gt)
1330     done
1331   finally show ?thesis by simp
1332 qed
1334 end