src/HOL/RComplete.thy
 author haftmann Mon Apr 27 10:11:44 2009 +0200 (2009-04-27) changeset 31001 7e6ffd8f51a9 parent 30242 aea5d7fa7ef5 child 32707 836ec9d0a0c8 permissions -rw-r--r--
cleaned up theory power further
1 (*  Title:      HOL/RComplete.thy
2     Author:     Jacques D. Fleuriot, University of Edinburgh
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 *)
8 header {* Completeness of the Reals; Floor and Ceiling Functions *}
10 theory RComplete
11 imports Lubs RealDef
12 begin
14 lemma real_sum_of_halves: "x/2 + x/2 = (x::real)"
15   by simp
18 subsection {* Completeness of Positive Reals *}
20 text {*
21   Supremum property for the set of positive reals
23   Let @{text "P"} be a non-empty set of positive reals, with an upper
24   bound @{text "y"}.  Then @{text "P"} has a least upper bound
25   (written @{text "S"}).
27   FIXME: Can the premise be weakened to @{text "\<forall>x \<in> P. x\<le> y"}?
28 *}
30 lemma posreal_complete:
31   assumes positive_P: "\<forall>x \<in> P. (0::real) < x"
32     and not_empty_P: "\<exists>x. x \<in> P"
33     and upper_bound_Ex: "\<exists>y. \<forall>x \<in> P. x<y"
34   shows "\<exists>S. \<forall>y. (\<exists>x \<in> P. y < x) = (y < S)"
35 proof (rule exI, rule allI)
36   fix y
37   let ?pP = "{w. real_of_preal w \<in> P}"
39   show "(\<exists>x\<in>P. y < x) = (y < real_of_preal (psup ?pP))"
40   proof (cases "0 < y")
41     assume neg_y: "\<not> 0 < y"
42     show ?thesis
43     proof
44       assume "\<exists>x\<in>P. y < x"
45       have "\<forall>x. y < real_of_preal x"
46         using neg_y by (rule real_less_all_real2)
47       thus "y < real_of_preal (psup ?pP)" ..
48     next
49       assume "y < real_of_preal (psup ?pP)"
50       obtain "x" where x_in_P: "x \<in> P" using not_empty_P ..
51       hence "0 < x" using positive_P by simp
52       hence "y < x" using neg_y by simp
53       thus "\<exists>x \<in> P. y < x" using x_in_P ..
54     qed
55   next
56     assume pos_y: "0 < y"
58     then obtain py where y_is_py: "y = real_of_preal py"
59       by (auto simp add: real_gt_zero_preal_Ex)
61     obtain a where "a \<in> P" using not_empty_P ..
62     with positive_P have a_pos: "0 < a" ..
63     then obtain pa where "a = real_of_preal pa"
64       by (auto simp add: real_gt_zero_preal_Ex)
65     hence "pa \<in> ?pP" using a \<in> P by auto
66     hence pP_not_empty: "?pP \<noteq> {}" by auto
68     obtain sup where sup: "\<forall>x \<in> P. x < sup"
69       using upper_bound_Ex ..
70     from this and a \<in> P have "a < sup" ..
71     hence "0 < sup" using a_pos by arith
72     then obtain possup where "sup = real_of_preal possup"
73       by (auto simp add: real_gt_zero_preal_Ex)
74     hence "\<forall>X \<in> ?pP. X \<le> possup"
75       using sup by (auto simp add: real_of_preal_lessI)
76     with pP_not_empty have psup: "\<And>Z. (\<exists>X \<in> ?pP. Z < X) = (Z < psup ?pP)"
77       by (rule preal_complete)
79     show ?thesis
80     proof
81       assume "\<exists>x \<in> P. y < x"
82       then obtain x where x_in_P: "x \<in> P" and y_less_x: "y < x" ..
83       hence "0 < x" using pos_y by arith
84       then obtain px where x_is_px: "x = real_of_preal px"
85         by (auto simp add: real_gt_zero_preal_Ex)
87       have py_less_X: "\<exists>X \<in> ?pP. py < X"
88       proof
89         show "py < px" using y_is_py and x_is_px and y_less_x
90           by (simp add: real_of_preal_lessI)
91         show "px \<in> ?pP" using x_in_P and x_is_px by simp
92       qed
94       have "(\<exists>X \<in> ?pP. py < X) ==> (py < psup ?pP)"
95         using psup by simp
96       hence "py < psup ?pP" using py_less_X by simp
97       thus "y < real_of_preal (psup {w. real_of_preal w \<in> P})"
98         using y_is_py and pos_y by (simp add: real_of_preal_lessI)
99     next
100       assume y_less_psup: "y < real_of_preal (psup ?pP)"
102       hence "py < psup ?pP" using y_is_py
103         by (simp add: real_of_preal_lessI)
104       then obtain "X" where py_less_X: "py < X" and X_in_pP: "X \<in> ?pP"
105         using psup by auto
106       then obtain x where x_is_X: "x = real_of_preal X"
107         by (simp add: real_gt_zero_preal_Ex)
108       hence "y < x" using py_less_X and y_is_py
109         by (simp add: real_of_preal_lessI)
111       moreover have "x \<in> P" using x_is_X and X_in_pP by simp
113       ultimately show "\<exists> x \<in> P. y < x" ..
114     qed
115   qed
116 qed
118 text {*
119   \medskip Completeness properties using @{text "isUb"}, @{text "isLub"} etc.
120 *}
122 lemma real_isLub_unique: "[| isLub R S x; isLub R S y |] ==> x = (y::real)"
123   apply (frule isLub_isUb)
124   apply (frule_tac x = y in isLub_isUb)
125   apply (blast intro!: order_antisym dest!: isLub_le_isUb)
126   done
129 text {*
130   \medskip Completeness theorem for the positive reals (again).
131 *}
133 lemma posreals_complete:
134   assumes positive_S: "\<forall>x \<in> S. 0 < x"
135     and not_empty_S: "\<exists>x. x \<in> S"
136     and upper_bound_Ex: "\<exists>u. isUb (UNIV::real set) S u"
137   shows "\<exists>t. isLub (UNIV::real set) S t"
138 proof
139   let ?pS = "{w. real_of_preal w \<in> S}"
141   obtain u where "isUb UNIV S u" using upper_bound_Ex ..
142   hence sup: "\<forall>x \<in> S. x \<le> u" by (simp add: isUb_def setle_def)
144   obtain x where x_in_S: "x \<in> S" using not_empty_S ..
145   hence x_gt_zero: "0 < x" using positive_S by simp
146   have  "x \<le> u" using sup and x_in_S ..
147   hence "0 < u" using x_gt_zero by arith
149   then obtain pu where u_is_pu: "u = real_of_preal pu"
150     by (auto simp add: real_gt_zero_preal_Ex)
152   have pS_less_pu: "\<forall>pa \<in> ?pS. pa \<le> pu"
153   proof
154     fix pa
155     assume "pa \<in> ?pS"
156     then obtain a where "a \<in> S" and "a = real_of_preal pa"
157       by simp
158     moreover hence "a \<le> u" using sup by simp
159     ultimately show "pa \<le> pu"
160       using sup and u_is_pu by (simp add: real_of_preal_le_iff)
161   qed
163   have "\<forall>y \<in> S. y \<le> real_of_preal (psup ?pS)"
164   proof
165     fix y
166     assume y_in_S: "y \<in> S"
167     hence "0 < y" using positive_S by simp
168     then obtain py where y_is_py: "y = real_of_preal py"
169       by (auto simp add: real_gt_zero_preal_Ex)
170     hence py_in_pS: "py \<in> ?pS" using y_in_S by simp
171     with pS_less_pu have "py \<le> psup ?pS"
172       by (rule preal_psup_le)
173     thus "y \<le> real_of_preal (psup ?pS)"
174       using y_is_py by (simp add: real_of_preal_le_iff)
175   qed
177   moreover {
178     fix x
179     assume x_ub_S: "\<forall>y\<in>S. y \<le> x"
180     have "real_of_preal (psup ?pS) \<le> x"
181     proof -
182       obtain "s" where s_in_S: "s \<in> S" using not_empty_S ..
183       hence s_pos: "0 < s" using positive_S by simp
185       hence "\<exists> ps. s = real_of_preal ps" by (simp add: real_gt_zero_preal_Ex)
186       then obtain "ps" where s_is_ps: "s = real_of_preal ps" ..
187       hence ps_in_pS: "ps \<in> {w. real_of_preal w \<in> S}" using s_in_S by simp
189       from x_ub_S have "s \<le> x" using s_in_S ..
190       hence "0 < x" using s_pos by simp
191       hence "\<exists> px. x = real_of_preal px" by (simp add: real_gt_zero_preal_Ex)
192       then obtain "px" where x_is_px: "x = real_of_preal px" ..
194       have "\<forall>pe \<in> ?pS. pe \<le> px"
195       proof
196 	fix pe
197 	assume "pe \<in> ?pS"
198 	hence "real_of_preal pe \<in> S" by simp
199 	hence "real_of_preal pe \<le> x" using x_ub_S by simp
200 	thus "pe \<le> px" using x_is_px by (simp add: real_of_preal_le_iff)
201       qed
203       moreover have "?pS \<noteq> {}" using ps_in_pS by auto
204       ultimately have "(psup ?pS) \<le> px" by (simp add: psup_le_ub)
205       thus "real_of_preal (psup ?pS) \<le> x" using x_is_px by (simp add: real_of_preal_le_iff)
206     qed
207   }
208   ultimately show "isLub UNIV S (real_of_preal (psup ?pS))"
209     by (simp add: isLub_def leastP_def isUb_def setle_def setge_def)
210 qed
212 text {*
213   \medskip reals Completeness (again!)
214 *}
216 lemma reals_complete:
217   assumes notempty_S: "\<exists>X. X \<in> S"
218     and exists_Ub: "\<exists>Y. isUb (UNIV::real set) S Y"
219   shows "\<exists>t. isLub (UNIV :: real set) S t"
220 proof -
221   obtain X where X_in_S: "X \<in> S" using notempty_S ..
222   obtain Y where Y_isUb: "isUb (UNIV::real set) S Y"
223     using exists_Ub ..
224   let ?SHIFT = "{z. \<exists>x \<in>S. z = x + (-X) + 1} \<inter> {x. 0 < x}"
226   {
227     fix x
228     assume "isUb (UNIV::real set) S x"
229     hence S_le_x: "\<forall> y \<in> S. y <= x"
230       by (simp add: isUb_def setle_def)
231     {
232       fix s
233       assume "s \<in> {z. \<exists>x\<in>S. z = x + - X + 1}"
234       hence "\<exists> x \<in> S. s = x + -X + 1" ..
235       then obtain x1 where "x1 \<in> S" and "s = x1 + (-X) + 1" ..
236       moreover hence "x1 \<le> x" using S_le_x by simp
237       ultimately have "s \<le> x + - X + 1" by arith
238     }
239     then have "isUb (UNIV::real set) ?SHIFT (x + (-X) + 1)"
240       by (auto simp add: isUb_def setle_def)
241   } note S_Ub_is_SHIFT_Ub = this
243   hence "isUb UNIV ?SHIFT (Y + (-X) + 1)" using Y_isUb by simp
244   hence "\<exists>Z. isUb UNIV ?SHIFT Z" ..
245   moreover have "\<forall>y \<in> ?SHIFT. 0 < y" by auto
246   moreover have shifted_not_empty: "\<exists>u. u \<in> ?SHIFT"
247     using X_in_S and Y_isUb by auto
248   ultimately obtain t where t_is_Lub: "isLub UNIV ?SHIFT t"
249     using posreals_complete [of ?SHIFT] by blast
251   show ?thesis
252   proof
253     show "isLub UNIV S (t + X + (-1))"
254     proof (rule isLubI2)
255       {
256         fix x
257         assume "isUb (UNIV::real set) S x"
258         hence "isUb (UNIV::real set) (?SHIFT) (x + (-X) + 1)"
259 	  using S_Ub_is_SHIFT_Ub by simp
260         hence "t \<le> (x + (-X) + 1)"
261 	  using t_is_Lub by (simp add: isLub_le_isUb)
262         hence "t + X + -1 \<le> x" by arith
263       }
264       then show "(t + X + -1) <=* Collect (isUb UNIV S)"
265 	by (simp add: setgeI)
266     next
267       show "isUb UNIV S (t + X + -1)"
268       proof -
269         {
270           fix y
271           assume y_in_S: "y \<in> S"
272           have "y \<le> t + X + -1"
273           proof -
274             obtain "u" where u_in_shift: "u \<in> ?SHIFT" using shifted_not_empty ..
275             hence "\<exists> x \<in> S. u = x + - X + 1" by simp
276             then obtain "x" where x_and_u: "u = x + - X + 1" ..
277             have u_le_t: "u \<le> t" using u_in_shift and t_is_Lub by (simp add: isLubD2)
279             show ?thesis
280             proof cases
281               assume "y \<le> x"
282               moreover have "x = u + X + - 1" using x_and_u by arith
283               moreover have "u + X + - 1  \<le> t + X + -1" using u_le_t by arith
284               ultimately show "y  \<le> t + X + -1" by arith
285             next
286               assume "~(y \<le> x)"
287               hence x_less_y: "x < y" by arith
289               have "x + (-X) + 1 \<in> ?SHIFT" using x_and_u and u_in_shift by simp
290               hence "0 < x + (-X) + 1" by simp
291               hence "0 < y + (-X) + 1" using x_less_y by arith
292               hence "y + (-X) + 1 \<in> ?SHIFT" using y_in_S by simp
293               hence "y + (-X) + 1 \<le> t" using t_is_Lub  by (simp add: isLubD2)
294               thus ?thesis by simp
295             qed
296           qed
297         }
298         then show ?thesis by (simp add: isUb_def setle_def)
299       qed
300     qed
301   qed
302 qed
305 subsection {* The Archimedean Property of the Reals *}
307 theorem reals_Archimedean:
308   assumes x_pos: "0 < x"
309   shows "\<exists>n. inverse (real (Suc n)) < x"
310 proof (rule ccontr)
311   assume contr: "\<not> ?thesis"
312   have "\<forall>n. x * real (Suc n) <= 1"
313   proof
314     fix n
315     from contr have "x \<le> inverse (real (Suc n))"
316       by (simp add: linorder_not_less)
317     hence "x \<le> (1 / (real (Suc n)))"
318       by (simp add: inverse_eq_divide)
319     moreover have "0 \<le> real (Suc n)"
320       by (rule real_of_nat_ge_zero)
321     ultimately have "x * real (Suc n) \<le> (1 / real (Suc n)) * real (Suc n)"
322       by (rule mult_right_mono)
323     thus "x * real (Suc n) \<le> 1" by simp
324   qed
325   hence "{z. \<exists>n. z = x * (real (Suc n))} *<= 1"
326     by (simp add: setle_def, safe, rule spec)
327   hence "isUb (UNIV::real set) {z. \<exists>n. z = x * (real (Suc n))} 1"
328     by (simp add: isUbI)
329   hence "\<exists>Y. isUb (UNIV::real set) {z. \<exists>n. z = x* (real (Suc n))} Y" ..
330   moreover have "\<exists>X. X \<in> {z. \<exists>n. z = x* (real (Suc n))}" by auto
331   ultimately have "\<exists>t. isLub UNIV {z. \<exists>n. z = x * real (Suc n)} t"
332     by (simp add: reals_complete)
333   then obtain "t" where
334     t_is_Lub: "isLub UNIV {z. \<exists>n. z = x * real (Suc n)} t" ..
336   have "\<forall>n::nat. x * real n \<le> t + - x"
337   proof
338     fix n
339     from t_is_Lub have "x * real (Suc n) \<le> t"
340       by (simp add: isLubD2)
341     hence  "x * (real n) + x \<le> t"
342       by (simp add: right_distrib real_of_nat_Suc)
343     thus  "x * (real n) \<le> t + - x" by arith
344   qed
346   hence "\<forall>m. x * real (Suc m) \<le> t + - x" by simp
347   hence "{z. \<exists>n. z = x * (real (Suc n))}  *<= (t + - x)"
348     by (auto simp add: setle_def)
349   hence "isUb (UNIV::real set) {z. \<exists>n. z = x * (real (Suc n))} (t + (-x))"
350     by (simp add: isUbI)
351   hence "t \<le> t + - x"
352     using t_is_Lub by (simp add: isLub_le_isUb)
353   thus False using x_pos by arith
354 qed
356 text {*
357   There must be other proofs, e.g. @{text "Suc"} of the largest
358   integer in the cut representing @{text "x"}.
359 *}
361 lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"
362 proof cases
363   assume "x \<le> 0"
364   hence "x < real (1::nat)" by simp
365   thus ?thesis ..
366 next
367   assume "\<not> x \<le> 0"
368   hence x_greater_zero: "0 < x" by simp
369   hence "0 < inverse x" by simp
370   then obtain n where "inverse (real (Suc n)) < inverse x"
371     using reals_Archimedean by blast
372   hence "inverse (real (Suc n)) * x < inverse x * x"
373     using x_greater_zero by (rule mult_strict_right_mono)
374   hence "inverse (real (Suc n)) * x < 1"
375     using x_greater_zero by simp
376   hence "real (Suc n) * (inverse (real (Suc n)) * x) < real (Suc n) * 1"
377     by (rule mult_strict_left_mono) simp
378   hence "x < real (Suc n)"
379     by (simp add: algebra_simps)
380   thus "\<exists>(n::nat). x < real n" ..
381 qed
383 instance real :: archimedean_field
384 proof
385   fix r :: real
386   obtain n :: nat where "r < real n"
387     using reals_Archimedean2 ..
388   then have "r \<le> of_int (int n)"
389     unfolding real_eq_of_nat by simp
390   then show "\<exists>z. r \<le> of_int z" ..
391 qed
393 lemma reals_Archimedean3:
394   assumes x_greater_zero: "0 < x"
395   shows "\<forall>(y::real). \<exists>(n::nat). y < real n * x"
396   unfolding real_of_nat_def using 0 < x
397   by (auto intro: ex_less_of_nat_mult)
399 lemma reals_Archimedean6:
400      "0 \<le> r ==> \<exists>(n::nat). real (n - 1) \<le> r & r < real (n)"
401 unfolding real_of_nat_def
402 apply (rule exI [where x="nat (floor r + 1)"])
403 apply (insert floor_correct [of r])
405 done
407 lemma reals_Archimedean6a: "0 \<le> r ==> \<exists>n. real (n) \<le> r & r < real (Suc n)"
408   by (drule reals_Archimedean6) auto
410 lemma reals_Archimedean_6b_int:
411      "0 \<le> r ==> \<exists>n::int. real n \<le> r & r < real (n+1)"
412   unfolding real_of_int_def by (rule floor_exists)
414 lemma reals_Archimedean_6c_int:
415      "r < 0 ==> \<exists>n::int. real n \<le> r & r < real (n+1)"
416   unfolding real_of_int_def by (rule floor_exists)
419 subsection{*Density of the Rational Reals in the Reals*}
421 text{* This density proof is due to Stefan Richter and was ported by TN.  The
422 original source is \emph{Real Analysis} by H.L. Royden.
423 It employs the Archimedean property of the reals. *}
425 lemma Rats_dense_in_nn_real: fixes x::real
426 assumes "0\<le>x" and "x<y" shows "\<exists>r \<in> \<rat>.  x<r \<and> r<y"
427 proof -
428   from x<y have "0 < y-x" by simp
429   with reals_Archimedean obtain q::nat
430     where q: "inverse (real q) < y-x" and "0 < real q" by auto
431   def p \<equiv> "LEAST n.  y \<le> real (Suc n)/real q"
432   from reals_Archimedean2 obtain n::nat where "y * real q < real n" by auto
433   with 0 < real q have ex: "y \<le> real n/real q" (is "?P n")
434     by (simp add: pos_less_divide_eq[THEN sym])
435   also from assms have "\<not> y \<le> real (0::nat) / real q" by simp
436   ultimately have main: "(LEAST n. y \<le> real n/real q) = Suc p"
437     by (unfold p_def) (rule Least_Suc)
438   also from ex have "?P (LEAST x. ?P x)" by (rule LeastI)
439   ultimately have suc: "y \<le> real (Suc p) / real q" by simp
440   def r \<equiv> "real p/real q"
441   have "x = y-(y-x)" by simp
442   also from suc q have "\<dots> < real (Suc p)/real q - inverse (real q)" by arith
443   also have "\<dots> = real p / real q"
444     by (simp only: inverse_eq_divide real_diff_def real_of_nat_Suc
445     minus_divide_left add_divide_distrib[THEN sym]) simp
446   finally have "x<r" by (unfold r_def)
447   have "p<Suc p" .. also note main[THEN sym]
448   finally have "\<not> ?P p"  by (rule not_less_Least)
449   hence "r<y" by (simp add: r_def)
450   from r_def have "r \<in> \<rat>" by simp
451   with x<r r<y show ?thesis by fast
452 qed
454 theorem Rats_dense_in_real: fixes x y :: real
455 assumes "x<y" shows "\<exists>r \<in> \<rat>.  x<r \<and> r<y"
456 proof -
457   from reals_Archimedean2 obtain n::nat where "-x < real n" by auto
458   hence "0 \<le> x + real n" by arith
459   also from x<y have "x + real n < y + real n" by arith
460   ultimately have "\<exists>r \<in> \<rat>. x + real n < r \<and> r < y + real n"
461     by(rule Rats_dense_in_nn_real)
462   then obtain r where "r \<in> \<rat>" and r2: "x + real n < r"
463     and r3: "r < y + real n"
464     by blast
465   have "r - real n = r + real (int n)/real (-1::int)" by simp
466   also from r\<in>\<rat> have "r + real (int n)/real (-1::int) \<in> \<rat>" by simp
467   also from r2 have "x < r - real n" by arith
468   moreover from r3 have "r - real n < y" by arith
469   ultimately show ?thesis by fast
470 qed
473 subsection{*Floor and Ceiling Functions from the Reals to the Integers*}
475 lemma number_of_less_real_of_int_iff [simp]:
476      "((number_of n) < real (m::int)) = (number_of n < m)"
477 apply auto
478 apply (rule real_of_int_less_iff [THEN iffD1])
479 apply (drule_tac  real_of_int_less_iff [THEN iffD2], auto)
480 done
482 lemma number_of_less_real_of_int_iff2 [simp]:
483      "(real (m::int) < (number_of n)) = (m < number_of n)"
484 apply auto
485 apply (rule real_of_int_less_iff [THEN iffD1])
486 apply (drule_tac  real_of_int_less_iff [THEN iffD2], auto)
487 done
489 lemma number_of_le_real_of_int_iff [simp]:
490      "((number_of n) \<le> real (m::int)) = (number_of n \<le> m)"
491 by (simp add: linorder_not_less [symmetric])
493 lemma number_of_le_real_of_int_iff2 [simp]:
494      "(real (m::int) \<le> (number_of n)) = (m \<le> number_of n)"
495 by (simp add: linorder_not_less [symmetric])
497 lemma floor_real_of_nat_zero: "floor (real (0::nat)) = 0"
498 by auto (* delete? *)
500 lemma floor_real_of_nat [simp]: "floor (real (n::nat)) = int n"
501 unfolding real_of_nat_def by simp
503 lemma floor_minus_real_of_nat [simp]: "floor (- real (n::nat)) = - int n"
504 unfolding real_of_nat_def by (simp add: floor_minus)
506 lemma floor_real_of_int [simp]: "floor (real (n::int)) = n"
507 unfolding real_of_int_def by simp
509 lemma floor_minus_real_of_int [simp]: "floor (- real (n::int)) = - n"
510 unfolding real_of_int_def by (simp add: floor_minus)
512 lemma real_lb_ub_int: " \<exists>n::int. real n \<le> r & r < real (n+1)"
513 unfolding real_of_int_def by (rule floor_exists)
515 lemma lemma_floor:
516   assumes a1: "real m \<le> r" and a2: "r < real n + 1"
517   shows "m \<le> (n::int)"
518 proof -
519   have "real m < real n + 1" using a1 a2 by (rule order_le_less_trans)
520   also have "... = real (n + 1)" by simp
521   finally have "m < n + 1" by (simp only: real_of_int_less_iff)
522   thus ?thesis by arith
523 qed
525 lemma real_of_int_floor_le [simp]: "real (floor r) \<le> r"
526 unfolding real_of_int_def by (rule of_int_floor_le)
528 lemma lemma_floor2: "real n < real (x::int) + 1 ==> n \<le> x"
529 by (auto intro: lemma_floor)
531 lemma real_of_int_floor_cancel [simp]:
532     "(real (floor x) = x) = (\<exists>n::int. x = real n)"
533   using floor_real_of_int by metis
535 lemma floor_eq: "[| real n < x; x < real n + 1 |] ==> floor x = n"
536   unfolding real_of_int_def using floor_unique [of n x] by simp
538 lemma floor_eq2: "[| real n \<le> x; x < real n + 1 |] ==> floor x = n"
539   unfolding real_of_int_def by (rule floor_unique)
541 lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"
542 apply (rule inj_int [THEN injD])
543 apply (simp add: real_of_nat_Suc)
544 apply (simp add: real_of_nat_Suc floor_eq floor_eq [where n = "int n"])
545 done
547 lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"
548 apply (drule order_le_imp_less_or_eq)
549 apply (auto intro: floor_eq3)
550 done
552 lemma floor_number_of_eq:
553      "floor(number_of n :: real) = (number_of n :: int)"
554   by (rule floor_number_of) (* already declared [simp] *)
556 lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real(floor r)"
557   unfolding real_of_int_def using floor_correct [of r] by simp
559 lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real(floor r)"
560   unfolding real_of_int_def using floor_correct [of r] by simp
562 lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real(floor r) + 1"
563   unfolding real_of_int_def using floor_correct [of r] by simp
565 lemma real_of_int_floor_add_one_gt [simp]: "r < real(floor r) + 1"
566   unfolding real_of_int_def using floor_correct [of r] by simp
568 lemma le_floor: "real a <= x ==> a <= floor x"
569   unfolding real_of_int_def by (simp add: le_floor_iff)
571 lemma real_le_floor: "a <= floor x ==> real a <= x"
572   unfolding real_of_int_def by (simp add: le_floor_iff)
574 lemma le_floor_eq: "(a <= floor x) = (real a <= x)"
575   unfolding real_of_int_def by (rule le_floor_iff)
577 lemma le_floor_eq_number_of:
578     "(number_of n <= floor x) = (number_of n <= x)"
579   by (rule number_of_le_floor) (* already declared [simp] *)
581 lemma le_floor_eq_zero: "(0 <= floor x) = (0 <= x)"
582   by (rule zero_le_floor) (* already declared [simp] *)
584 lemma le_floor_eq_one: "(1 <= floor x) = (1 <= x)"
585   by (rule one_le_floor) (* already declared [simp] *)
587 lemma floor_less_eq: "(floor x < a) = (x < real a)"
588   unfolding real_of_int_def by (rule floor_less_iff)
590 lemma floor_less_eq_number_of:
591     "(floor x < number_of n) = (x < number_of n)"
592   by (rule floor_less_number_of) (* already declared [simp] *)
594 lemma floor_less_eq_zero: "(floor x < 0) = (x < 0)"
595   by (rule floor_less_zero) (* already declared [simp] *)
597 lemma floor_less_eq_one: "(floor x < 1) = (x < 1)"
598   by (rule floor_less_one) (* already declared [simp] *)
600 lemma less_floor_eq: "(a < floor x) = (real a + 1 <= x)"
601   unfolding real_of_int_def by (rule less_floor_iff)
603 lemma less_floor_eq_number_of:
604     "(number_of n < floor x) = (number_of n + 1 <= x)"
605   by (rule number_of_less_floor) (* already declared [simp] *)
607 lemma less_floor_eq_zero: "(0 < floor x) = (1 <= x)"
608   by (rule zero_less_floor) (* already declared [simp] *)
610 lemma less_floor_eq_one: "(1 < floor x) = (2 <= x)"
611   by (rule one_less_floor) (* already declared [simp] *)
613 lemma floor_le_eq: "(floor x <= a) = (x < real a + 1)"
614   unfolding real_of_int_def by (rule floor_le_iff)
616 lemma floor_le_eq_number_of:
617     "(floor x <= number_of n) = (x < number_of n + 1)"
618   by (rule floor_le_number_of) (* already declared [simp] *)
620 lemma floor_le_eq_zero: "(floor x <= 0) = (x < 1)"
621   by (rule floor_le_zero) (* already declared [simp] *)
623 lemma floor_le_eq_one: "(floor x <= 1) = (x < 2)"
624   by (rule floor_le_one) (* already declared [simp] *)
626 lemma floor_add [simp]: "floor (x + real a) = floor x + a"
627   unfolding real_of_int_def by (rule floor_add_of_int)
629 lemma floor_subtract [simp]: "floor (x - real a) = floor x - a"
630   unfolding real_of_int_def by (rule floor_diff_of_int)
632 lemma floor_subtract_number_of: "floor (x - number_of n) =
633     floor x - number_of n"
634   by (rule floor_diff_number_of) (* already declared [simp] *)
636 lemma floor_subtract_one: "floor (x - 1) = floor x - 1"
637   by (rule floor_diff_one) (* already declared [simp] *)
639 lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"
640   unfolding real_of_nat_def by simp
642 lemma ceiling_real_of_nat_zero: "ceiling (real (0::nat)) = 0"
643 by auto (* delete? *)
645 lemma ceiling_floor [simp]: "ceiling (real (floor r)) = floor r"
646   unfolding real_of_int_def by simp
648 lemma floor_ceiling [simp]: "floor (real (ceiling r)) = ceiling r"
649   unfolding real_of_int_def by simp
651 lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)"
652   unfolding real_of_int_def by (rule le_of_int_ceiling)
654 lemma ceiling_real_of_int [simp]: "ceiling (real (n::int)) = n"
655   unfolding real_of_int_def by simp
657 lemma real_of_int_ceiling_cancel [simp]:
658      "(real (ceiling x) = x) = (\<exists>n::int. x = real n)"
659   using ceiling_real_of_int by metis
661 lemma ceiling_eq: "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1"
662   unfolding real_of_int_def using ceiling_unique [of "n + 1" x] by simp
664 lemma ceiling_eq2: "[| real n < x; x \<le> real n + 1 |] ==> ceiling x = n + 1"
665   unfolding real_of_int_def using ceiling_unique [of "n + 1" x] by simp
667 lemma ceiling_eq3: "[| real n - 1 < x; x \<le> real n  |] ==> ceiling x = n"
668   unfolding real_of_int_def using ceiling_unique [of n x] by simp
670 lemma ceiling_number_of_eq:
671      "ceiling (number_of n :: real) = (number_of n)"
672   by (rule ceiling_number_of) (* already declared [simp] *)
674 lemma real_of_int_ceiling_diff_one_le [simp]: "real (ceiling r) - 1 \<le> r"
675   unfolding real_of_int_def using ceiling_correct [of r] by simp
677 lemma real_of_int_ceiling_le_add_one [simp]: "real (ceiling r) \<le> r + 1"
678   unfolding real_of_int_def using ceiling_correct [of r] by simp
680 lemma ceiling_le: "x <= real a ==> ceiling x <= a"
681   unfolding real_of_int_def by (simp add: ceiling_le_iff)
683 lemma ceiling_le_real: "ceiling x <= a ==> x <= real a"
684   unfolding real_of_int_def by (simp add: ceiling_le_iff)
686 lemma ceiling_le_eq: "(ceiling x <= a) = (x <= real a)"
687   unfolding real_of_int_def by (rule ceiling_le_iff)
689 lemma ceiling_le_eq_number_of:
690     "(ceiling x <= number_of n) = (x <= number_of n)"
691   by (rule ceiling_le_number_of) (* already declared [simp] *)
693 lemma ceiling_le_zero_eq: "(ceiling x <= 0) = (x <= 0)"
694   by (rule ceiling_le_zero) (* already declared [simp] *)
696 lemma ceiling_le_eq_one: "(ceiling x <= 1) = (x <= 1)"
697   by (rule ceiling_le_one) (* already declared [simp] *)
699 lemma less_ceiling_eq: "(a < ceiling x) = (real a < x)"
700   unfolding real_of_int_def by (rule less_ceiling_iff)
702 lemma less_ceiling_eq_number_of:
703     "(number_of n < ceiling x) = (number_of n < x)"
704   by (rule number_of_less_ceiling) (* already declared [simp] *)
706 lemma less_ceiling_eq_zero: "(0 < ceiling x) = (0 < x)"
707   by (rule zero_less_ceiling) (* already declared [simp] *)
709 lemma less_ceiling_eq_one: "(1 < ceiling x) = (1 < x)"
710   by (rule one_less_ceiling) (* already declared [simp] *)
712 lemma ceiling_less_eq: "(ceiling x < a) = (x <= real a - 1)"
713   unfolding real_of_int_def by (rule ceiling_less_iff)
715 lemma ceiling_less_eq_number_of:
716     "(ceiling x < number_of n) = (x <= number_of n - 1)"
717   by (rule ceiling_less_number_of) (* already declared [simp] *)
719 lemma ceiling_less_eq_zero: "(ceiling x < 0) = (x <= -1)"
720   by (rule ceiling_less_zero) (* already declared [simp] *)
722 lemma ceiling_less_eq_one: "(ceiling x < 1) = (x <= 0)"
723   by (rule ceiling_less_one) (* already declared [simp] *)
725 lemma le_ceiling_eq: "(a <= ceiling x) = (real a - 1 < x)"
726   unfolding real_of_int_def by (rule le_ceiling_iff)
728 lemma le_ceiling_eq_number_of:
729     "(number_of n <= ceiling x) = (number_of n - 1 < x)"
730   by (rule number_of_le_ceiling) (* already declared [simp] *)
732 lemma le_ceiling_eq_zero: "(0 <= ceiling x) = (-1 < x)"
733   by (rule zero_le_ceiling) (* already declared [simp] *)
735 lemma le_ceiling_eq_one: "(1 <= ceiling x) = (0 < x)"
736   by (rule one_le_ceiling) (* already declared [simp] *)
738 lemma ceiling_add [simp]: "ceiling (x + real a) = ceiling x + a"
739   unfolding real_of_int_def by (rule ceiling_add_of_int)
741 lemma ceiling_subtract [simp]: "ceiling (x - real a) = ceiling x - a"
742   unfolding real_of_int_def by (rule ceiling_diff_of_int)
744 lemma ceiling_subtract_number_of: "ceiling (x - number_of n) =
745     ceiling x - number_of n"
746   by (rule ceiling_diff_number_of) (* already declared [simp] *)
748 lemma ceiling_subtract_one: "ceiling (x - 1) = ceiling x - 1"
749   by (rule ceiling_diff_one) (* already declared [simp] *)
752 subsection {* Versions for the natural numbers *}
754 definition
755   natfloor :: "real => nat" where
756   "natfloor x = nat(floor x)"
758 definition
759   natceiling :: "real => nat" where
760   "natceiling x = nat(ceiling x)"
762 lemma natfloor_zero [simp]: "natfloor 0 = 0"
763   by (unfold natfloor_def, simp)
765 lemma natfloor_one [simp]: "natfloor 1 = 1"
766   by (unfold natfloor_def, simp)
768 lemma zero_le_natfloor [simp]: "0 <= natfloor x"
769   by (unfold natfloor_def, simp)
771 lemma natfloor_number_of_eq [simp]: "natfloor (number_of n) = number_of n"
772   by (unfold natfloor_def, simp)
774 lemma natfloor_real_of_nat [simp]: "natfloor(real n) = n"
775   by (unfold natfloor_def, simp)
777 lemma real_natfloor_le: "0 <= x ==> real(natfloor x) <= x"
778   by (unfold natfloor_def, simp)
780 lemma natfloor_neg: "x <= 0 ==> natfloor x = 0"
781   apply (unfold natfloor_def)
782   apply (subgoal_tac "floor x <= floor 0")
783   apply simp
784   apply (erule floor_mono)
785 done
787 lemma natfloor_mono: "x <= y ==> natfloor x <= natfloor y"
788   apply (case_tac "0 <= x")
789   apply (subst natfloor_def)+
790   apply (subst nat_le_eq_zle)
791   apply force
792   apply (erule floor_mono)
793   apply (subst natfloor_neg)
794   apply simp
795   apply simp
796 done
798 lemma le_natfloor: "real x <= a ==> x <= natfloor a"
799   apply (unfold natfloor_def)
800   apply (subst nat_int [THEN sym])
801   apply (subst nat_le_eq_zle)
802   apply simp
803   apply (rule le_floor)
804   apply simp
805 done
807 lemma le_natfloor_eq: "0 <= x ==> (a <= natfloor x) = (real a <= x)"
808   apply (rule iffI)
809   apply (rule order_trans)
810   prefer 2
811   apply (erule real_natfloor_le)
812   apply (subst real_of_nat_le_iff)
813   apply assumption
814   apply (erule le_natfloor)
815 done
817 lemma le_natfloor_eq_number_of [simp]:
818     "~ neg((number_of n)::int) ==> 0 <= x ==>
819       (number_of n <= natfloor x) = (number_of n <= x)"
820   apply (subst le_natfloor_eq, assumption)
821   apply simp
822 done
824 lemma le_natfloor_eq_one [simp]: "(1 <= natfloor x) = (1 <= x)"
825   apply (case_tac "0 <= x")
826   apply (subst le_natfloor_eq, assumption, simp)
827   apply (rule iffI)
828   apply (subgoal_tac "natfloor x <= natfloor 0")
829   apply simp
830   apply (rule natfloor_mono)
831   apply simp
832   apply simp
833 done
835 lemma natfloor_eq: "real n <= x ==> x < real n + 1 ==> natfloor x = n"
836   apply (unfold natfloor_def)
837   apply (subst nat_int [THEN sym]);back;
838   apply (subst eq_nat_nat_iff)
839   apply simp
840   apply simp
841   apply (rule floor_eq2)
842   apply auto
843 done
845 lemma real_natfloor_add_one_gt: "x < real(natfloor x) + 1"
846   apply (case_tac "0 <= x")
847   apply (unfold natfloor_def)
848   apply simp
849   apply simp_all
850 done
852 lemma real_natfloor_gt_diff_one: "x - 1 < real(natfloor x)"
853 using real_natfloor_add_one_gt by (simp add: algebra_simps)
855 lemma ge_natfloor_plus_one_imp_gt: "natfloor z + 1 <= n ==> z < real n"
856   apply (subgoal_tac "z < real(natfloor z) + 1")
857   apply arith
858   apply (rule real_natfloor_add_one_gt)
859 done
861 lemma natfloor_add [simp]: "0 <= x ==> natfloor (x + real a) = natfloor x + a"
862   apply (unfold natfloor_def)
863   apply (subgoal_tac "real a = real (int a)")
864   apply (erule ssubst)
865   apply (simp add: nat_add_distrib del: real_of_int_of_nat_eq)
866   apply simp
867 done
869 lemma natfloor_add_number_of [simp]:
870     "~neg ((number_of n)::int) ==> 0 <= x ==>
871       natfloor (x + number_of n) = natfloor x + number_of n"
872   apply (subst natfloor_add [THEN sym])
873   apply simp_all
874 done
876 lemma natfloor_add_one: "0 <= x ==> natfloor(x + 1) = natfloor x + 1"
877   apply (subst natfloor_add [THEN sym])
878   apply assumption
879   apply simp
880 done
882 lemma natfloor_subtract [simp]: "real a <= x ==>
883     natfloor(x - real a) = natfloor x - a"
884   apply (unfold natfloor_def)
885   apply (subgoal_tac "real a = real (int a)")
886   apply (erule ssubst)
887   apply (simp del: real_of_int_of_nat_eq)
888   apply simp
889 done
891 lemma natceiling_zero [simp]: "natceiling 0 = 0"
892   by (unfold natceiling_def, simp)
894 lemma natceiling_one [simp]: "natceiling 1 = 1"
895   by (unfold natceiling_def, simp)
897 lemma zero_le_natceiling [simp]: "0 <= natceiling x"
898   by (unfold natceiling_def, simp)
900 lemma natceiling_number_of_eq [simp]: "natceiling (number_of n) = number_of n"
901   by (unfold natceiling_def, simp)
903 lemma natceiling_real_of_nat [simp]: "natceiling(real n) = n"
904   by (unfold natceiling_def, simp)
906 lemma real_natceiling_ge: "x <= real(natceiling x)"
907   apply (unfold natceiling_def)
908   apply (case_tac "x < 0")
909   apply simp
910   apply (subst real_nat_eq_real)
911   apply (subgoal_tac "ceiling 0 <= ceiling x")
912   apply simp
913   apply (rule ceiling_mono)
914   apply simp
915   apply simp
916 done
918 lemma natceiling_neg: "x <= 0 ==> natceiling x = 0"
919   apply (unfold natceiling_def)
920   apply simp
921 done
923 lemma natceiling_mono: "x <= y ==> natceiling x <= natceiling y"
924   apply (case_tac "0 <= x")
925   apply (subst natceiling_def)+
926   apply (subst nat_le_eq_zle)
927   apply (rule disjI2)
928   apply (subgoal_tac "real (0::int) <= real(ceiling y)")
929   apply simp
930   apply (rule order_trans)
931   apply simp
932   apply (erule order_trans)
933   apply simp
934   apply (erule ceiling_mono)
935   apply (subst natceiling_neg)
936   apply simp_all
937 done
939 lemma natceiling_le: "x <= real a ==> natceiling x <= a"
940   apply (unfold natceiling_def)
941   apply (case_tac "x < 0")
942   apply simp
943   apply (subst nat_int [THEN sym]);back;
944   apply (subst nat_le_eq_zle)
945   apply simp
946   apply (rule ceiling_le)
947   apply simp
948 done
950 lemma natceiling_le_eq: "0 <= x ==> (natceiling x <= a) = (x <= real a)"
951   apply (rule iffI)
952   apply (rule order_trans)
953   apply (rule real_natceiling_ge)
954   apply (subst real_of_nat_le_iff)
955   apply assumption
956   apply (erule natceiling_le)
957 done
959 lemma natceiling_le_eq_number_of [simp]:
960     "~ neg((number_of n)::int) ==> 0 <= x ==>
961       (natceiling x <= number_of n) = (x <= number_of n)"
962   apply (subst natceiling_le_eq, assumption)
963   apply simp
964 done
966 lemma natceiling_le_eq_one: "(natceiling x <= 1) = (x <= 1)"
967   apply (case_tac "0 <= x")
968   apply (subst natceiling_le_eq)
969   apply assumption
970   apply simp
971   apply (subst natceiling_neg)
972   apply simp
973   apply simp
974 done
976 lemma natceiling_eq: "real n < x ==> x <= real n + 1 ==> natceiling x = n + 1"
977   apply (unfold natceiling_def)
978   apply (simplesubst nat_int [THEN sym]) back back
979   apply (subgoal_tac "nat(int n) + 1 = nat(int n + 1)")
980   apply (erule ssubst)
981   apply (subst eq_nat_nat_iff)
982   apply (subgoal_tac "ceiling 0 <= ceiling x")
983   apply simp
984   apply (rule ceiling_mono)
985   apply force
986   apply force
987   apply (rule ceiling_eq2)
988   apply (simp, simp)
989   apply (subst nat_add_distrib)
990   apply auto
991 done
993 lemma natceiling_add [simp]: "0 <= x ==>
994     natceiling (x + real a) = natceiling x + a"
995   apply (unfold natceiling_def)
996   apply (subgoal_tac "real a = real (int a)")
997   apply (erule ssubst)
998   apply (simp del: real_of_int_of_nat_eq)
999   apply (subst nat_add_distrib)
1000   apply (subgoal_tac "0 = ceiling 0")
1001   apply (erule ssubst)
1002   apply (erule ceiling_mono)
1003   apply simp_all
1004 done
1006 lemma natceiling_add_number_of [simp]:
1007     "~ neg ((number_of n)::int) ==> 0 <= x ==>
1008       natceiling (x + number_of n) = natceiling x + number_of n"
1009   apply (subst natceiling_add [THEN sym])
1010   apply simp_all
1011 done
1013 lemma natceiling_add_one: "0 <= x ==> natceiling(x + 1) = natceiling x + 1"
1014   apply (subst natceiling_add [THEN sym])
1015   apply assumption
1016   apply simp
1017 done
1019 lemma natceiling_subtract [simp]: "real a <= x ==>
1020     natceiling(x - real a) = natceiling x - a"
1021   apply (unfold natceiling_def)
1022   apply (subgoal_tac "real a = real (int a)")
1023   apply (erule ssubst)
1024   apply (simp del: real_of_int_of_nat_eq)
1025   apply simp
1026 done
1028 lemma natfloor_div_nat: "1 <= x ==> y > 0 ==>
1029   natfloor (x / real y) = natfloor x div y"
1030 proof -
1031   assume "1 <= (x::real)" and "(y::nat) > 0"
1032   have "natfloor x = (natfloor x) div y * y + (natfloor x) mod y"
1033     by simp
1034   then have a: "real(natfloor x) = real ((natfloor x) div y) * real y +
1035     real((natfloor x) mod y)"
1036     by (simp only: real_of_nat_add [THEN sym] real_of_nat_mult [THEN sym])
1037   have "x = real(natfloor x) + (x - real(natfloor x))"
1038     by simp
1039   then have "x = real ((natfloor x) div y) * real y +
1040       real((natfloor x) mod y) + (x - real(natfloor x))"
1041     by (simp add: a)
1042   then have "x / real y = ... / real y"
1043     by simp
1044   also have "... = real((natfloor x) div y) + real((natfloor x) mod y) /
1045     real y + (x - real(natfloor x)) / real y"
1046     by (auto simp add: algebra_simps add_divide_distrib
1047       diff_divide_distrib prems)
1048   finally have "natfloor (x / real y) = natfloor(...)" by simp
1049   also have "... = natfloor(real((natfloor x) mod y) /
1050     real y + (x - real(natfloor x)) / real y + real((natfloor x) div y))"
1052   also have "... = natfloor(real((natfloor x) mod y) /
1053     real y + (x - real(natfloor x)) / real y) + (natfloor x) div y"
1054     apply (rule natfloor_add)
1055     apply (rule add_nonneg_nonneg)
1056     apply (rule divide_nonneg_pos)
1057     apply simp
1058     apply (simp add: prems)
1059     apply (rule divide_nonneg_pos)
1060     apply (simp add: algebra_simps)
1061     apply (rule real_natfloor_le)
1062     apply (insert prems, auto)
1063     done
1064   also have "natfloor(real((natfloor x) mod y) /
1065     real y + (x - real(natfloor x)) / real y) = 0"
1066     apply (rule natfloor_eq)
1067     apply simp
1068     apply (rule add_nonneg_nonneg)
1069     apply (rule divide_nonneg_pos)
1070     apply force
1071     apply (force simp add: prems)
1072     apply (rule divide_nonneg_pos)
1073     apply (simp add: algebra_simps)
1074     apply (rule real_natfloor_le)
1075     apply (auto simp add: prems)
1076     apply (insert prems, arith)
1077     apply (simp add: add_divide_distrib [THEN sym])
1078     apply (subgoal_tac "real y = real y - 1 + 1")
1079     apply (erule ssubst)
1080     apply (rule add_le_less_mono)
1081     apply (simp add: algebra_simps)
1082     apply (subgoal_tac "1 + real(natfloor x mod y) =
1083       real(natfloor x mod y + 1)")
1084     apply (erule ssubst)
1085     apply (subst real_of_nat_le_iff)
1086     apply (subgoal_tac "natfloor x mod y < y")
1087     apply arith
1088     apply (rule mod_less_divisor)
1089     apply auto