src/HOL/Real/RComplete.thy
author nipkow
Wed Aug 18 11:09:40 2004 +0200 (2004-08-18)
changeset 15140 322485b816ac
parent 15131 c69542757a4d
child 15234 ec91a90c604e
permissions -rw-r--r--
import -> imports
     1 (*  Title       : RComplete.thy
     2     ID          : $Id$
     3     Author      : Jacques D. Fleuriot
     4     Copyright   : 1998  University of Cambridge
     5     Copyright   : 2001,2002  University of Edinburgh
     6 Converted to Isar and polished by lcp
     7 *) 
     8 
     9 header{*Completeness of the Reals; Floor and Ceiling Functions*}
    10 
    11 theory RComplete
    12 imports Lubs RealDef
    13 begin
    14 
    15 lemma real_sum_of_halves: "x/2 + x/2 = (x::real)"
    16 by simp
    17 
    18 
    19 subsection{*Completeness of Reals by Supremum Property of type @{typ preal}*} 
    20 
    21  (*a few lemmas*)
    22 lemma real_sup_lemma1:
    23      "\<forall>x \<in> P. 0 < x ==>   
    24       ((\<exists>x \<in> P. y < x) = (\<exists>X. real_of_preal X \<in> P & y < real_of_preal X))"
    25 by (blast dest!: bspec real_gt_zero_preal_Ex [THEN iffD1])
    26 
    27 lemma real_sup_lemma2:
    28      "[| \<forall>x \<in> P. 0 < x;  a \<in> P;   \<forall>x \<in> P. x < y |]  
    29       ==> (\<exists>X. X\<in> {w. real_of_preal w \<in> P}) &  
    30           (\<exists>Y. \<forall>X\<in> {w. real_of_preal w \<in> P}. X < Y)"
    31 apply (rule conjI)
    32 apply (blast dest: bspec real_gt_zero_preal_Ex [THEN iffD1], auto)
    33 apply (drule bspec, assumption)
    34 apply (frule bspec, assumption)
    35 apply (drule order_less_trans, assumption)
    36 apply (drule real_gt_zero_preal_Ex [THEN iffD1], force) 
    37 done
    38 
    39 (*-------------------------------------------------------------
    40             Completeness of Positive Reals
    41  -------------------------------------------------------------*)
    42 
    43 (**
    44  Supremum property for the set of positive reals
    45  FIXME: long proof - should be improved
    46 **)
    47 
    48 (*Let P be a non-empty set of positive reals, with an upper bound y.
    49   Then P has a least upper bound (written S).  
    50 FIXME: Can the premise be weakened to \<forall>x \<in> P. x\<le> y ??*)
    51 lemma posreal_complete: "[| \<forall>x \<in> P. (0::real) < x;  \<exists>x. x \<in> P;  \<exists>y. \<forall>x \<in> P. x<y |]  
    52       ==> (\<exists>S. \<forall>y. (\<exists>x \<in> P. y < x) = (y < S))"
    53 apply (rule_tac x = "real_of_preal (psup ({w. real_of_preal w \<in> P}))" in exI)
    54 apply clarify
    55 apply (case_tac "0 < ya", auto)
    56 apply (frule real_sup_lemma2, assumption+)
    57 apply (drule real_gt_zero_preal_Ex [THEN iffD1])
    58 apply (drule_tac [3] real_less_all_real2, auto)
    59 apply (rule preal_complete [THEN iffD1])
    60 apply (auto intro: order_less_imp_le)
    61 apply (frule real_gt_preal_preal_Ex, force)
    62 (* second part *)
    63 apply (rule real_sup_lemma1 [THEN iffD2], assumption)
    64 apply (auto dest!: real_less_all_real2 real_gt_zero_preal_Ex [THEN iffD1])
    65 apply (frule_tac [2] real_sup_lemma2)
    66 apply (frule real_sup_lemma2, assumption+, clarify) 
    67 apply (rule preal_complete [THEN iffD2, THEN bexE])
    68 prefer 3 apply blast
    69 apply (blast intro!: order_less_imp_le)+
    70 done
    71 
    72 (*--------------------------------------------------------
    73    Completeness properties using isUb, isLub etc.
    74  -------------------------------------------------------*)
    75 
    76 lemma real_isLub_unique: "[| isLub R S x; isLub R S y |] ==> x = (y::real)"
    77 apply (frule isLub_isUb)
    78 apply (frule_tac x = y in isLub_isUb)
    79 apply (blast intro!: order_antisym dest!: isLub_le_isUb)
    80 done
    81 
    82 lemma real_order_restrict: "[| (x::real) <=* S'; S <= S' |] ==> x <=* S"
    83 by (unfold setle_def setge_def, blast)
    84 
    85 (*----------------------------------------------------------------
    86            Completeness theorem for the positive reals(again)
    87  ----------------------------------------------------------------*)
    88 
    89 lemma posreals_complete:
    90      "[| \<forall>x \<in>S. 0 < x;  
    91          \<exists>x. x \<in>S;  
    92          \<exists>u. isUb (UNIV::real set) S u  
    93       |] ==> \<exists>t. isLub (UNIV::real set) S t"
    94 apply (rule_tac x = "real_of_preal (psup ({w. real_of_preal w \<in> S}))" in exI)
    95 apply (auto simp add: isLub_def leastP_def isUb_def)
    96 apply (auto intro!: setleI setgeI dest!: real_gt_zero_preal_Ex [THEN iffD1])
    97 apply (frule_tac x = y in bspec, assumption)
    98 apply (drule real_gt_zero_preal_Ex [THEN iffD1])
    99 apply (auto simp add: real_of_preal_le_iff)
   100 apply (frule_tac y = "real_of_preal ya" in setleD, assumption)
   101 apply (frule real_ge_preal_preal_Ex, safe)
   102 apply (blast intro!: preal_psup_le dest!: setleD intro: real_of_preal_le_iff [THEN iffD1])
   103 apply (frule_tac x = x in bspec, assumption)
   104 apply (frule isUbD2)
   105 apply (drule real_gt_zero_preal_Ex [THEN iffD1])
   106 apply (auto dest!: isUbD real_ge_preal_preal_Ex simp add: real_of_preal_le_iff)
   107 apply (blast dest!: setleD intro!: psup_le_ub intro: real_of_preal_le_iff [THEN iffD1])
   108 done
   109 
   110 
   111 (*-------------------------------
   112     Lemmas
   113  -------------------------------*)
   114 lemma real_sup_lemma3: "\<forall>y \<in> {z. \<exists>x \<in> P. z = x + (-xa) + 1} Int {x. 0 < x}. 0 < y"
   115 by auto
   116  
   117 lemma lemma_le_swap2: "(xa <= S + X + (-Z)) = (xa + (-X) + Z <= (S::real))"
   118 by auto
   119 
   120 lemma lemma_real_complete2b: "[| (x::real) + (-X) + 1 <= S; xa <= x |] ==> xa <= S + X + (- 1)"
   121 by arith
   122 
   123 (*----------------------------------------------------------
   124       reals Completeness (again!)
   125  ----------------------------------------------------------*)
   126 lemma reals_complete: "[| \<exists>X. X \<in>S;  \<exists>Y. isUb (UNIV::real set) S Y |]   
   127       ==> \<exists>t. isLub (UNIV :: real set) S t"
   128 apply safe
   129 apply (subgoal_tac "\<exists>u. u\<in> {z. \<exists>x \<in>S. z = x + (-X) + 1} Int {x. 0 < x}")
   130 apply (subgoal_tac "isUb (UNIV::real set) ({z. \<exists>x \<in>S. z = x + (-X) + 1} Int {x. 0 < x}) (Y + (-X) + 1) ")
   131 apply (cut_tac P = S and xa = X in real_sup_lemma3)
   132 apply (frule posreals_complete [OF _ _ exI], blast, blast, safe)
   133 apply (rule_tac x = "t + X + (- 1) " in exI)
   134 apply (rule isLubI2)
   135 apply (rule_tac [2] setgeI, safe)
   136 apply (subgoal_tac [2] "isUb (UNIV:: real set) ({z. \<exists>x \<in>S. z = x + (-X) + 1} Int {x. 0 < x}) (y + (-X) + 1) ")
   137 apply (drule_tac [2] y = " (y + (- X) + 1) " in isLub_le_isUb)
   138  prefer 2 apply assumption
   139  prefer 2
   140 apply arith
   141 apply (rule setleI [THEN isUbI], safe)
   142 apply (rule_tac x = x and y = y in linorder_cases)
   143 apply (subst lemma_le_swap2)
   144 apply (frule isLubD2)
   145  prefer 2 apply assumption
   146 apply safe
   147 apply blast
   148 apply arith
   149 apply (subst lemma_le_swap2)
   150 apply (frule isLubD2)
   151  prefer 2 apply assumption
   152 apply blast
   153 apply (rule lemma_real_complete2b)
   154 apply (erule_tac [2] order_less_imp_le)
   155 apply (blast intro!: isLubD2, blast) 
   156 apply (simp (no_asm_use) add: real_add_assoc)
   157 apply (blast dest: isUbD intro!: setleI [THEN isUbI] intro: add_right_mono)
   158 apply (blast dest: isUbD intro!: setleI [THEN isUbI] intro: add_right_mono, auto)
   159 done
   160 
   161 
   162 subsection{*Corollary: the Archimedean Property of the Reals*}
   163 
   164 lemma reals_Archimedean: "0 < x ==> \<exists>n. inverse (real(Suc n)) < x"
   165 apply (rule ccontr)
   166 apply (subgoal_tac "\<forall>n. x * real (Suc n) <= 1")
   167  prefer 2
   168 apply (simp add: linorder_not_less inverse_eq_divide, clarify) 
   169 apply (drule_tac x = n in spec)
   170 apply (drule_tac c = "real (Suc n)"  in mult_right_mono)
   171 apply (rule real_of_nat_ge_zero)
   172 apply (simp add: real_of_nat_Suc_gt_zero [THEN real_not_refl2, THEN not_sym] real_mult_commute)
   173 apply (subgoal_tac "isUb (UNIV::real set) {z. \<exists>n. z = x* (real (Suc n))} 1")
   174 apply (subgoal_tac "\<exists>X. X \<in> {z. \<exists>n. z = x* (real (Suc n))}")
   175 apply (drule reals_complete)
   176 apply (auto intro: isUbI setleI)
   177 apply (subgoal_tac "\<forall>m. x* (real (Suc m)) <= t")
   178 apply (simp add: real_of_nat_Suc right_distrib)
   179 prefer 2 apply (blast intro: isLubD2)
   180 apply (simp add: le_diff_eq [symmetric] real_diff_def)
   181 apply (subgoal_tac "isUb (UNIV::real set) {z. \<exists>n. z = x* (real (Suc n))} (t + (-x))")
   182 prefer 2 apply (blast intro!: isUbI setleI)
   183 apply (drule_tac y = "t+ (-x) " in isLub_le_isUb)
   184 apply (auto simp add: real_of_nat_Suc right_distrib)
   185 done
   186 
   187 (*There must be other proofs, e.g. Suc of the largest integer in the
   188   cut representing x*)
   189 lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"
   190 apply (rule_tac x = x and y = 0 in linorder_cases)
   191 apply (rule_tac x = 0 in exI)
   192 apply (rule_tac [2] x = 1 in exI)
   193 apply (auto elim: order_less_trans simp add: real_of_nat_one)
   194 apply (frule positive_imp_inverse_positive [THEN reals_Archimedean], safe)
   195 apply (rule_tac x = "Suc n" in exI)
   196 apply (frule_tac b = "inverse x" in mult_strict_right_mono, auto)
   197 done
   198 
   199 lemma reals_Archimedean3: "0 < x ==> \<forall>y. \<exists>(n::nat). y < real n * x"
   200 apply safe
   201 apply (cut_tac x = "y*inverse (x) " in reals_Archimedean2)
   202 apply safe
   203 apply (frule_tac a = "y * inverse x" in mult_strict_right_mono)
   204 apply (auto simp add: mult_assoc real_of_nat_def)
   205 done
   206 
   207 ML
   208 {*
   209 val real_sum_of_halves = thm "real_sum_of_halves";
   210 val posreal_complete = thm "posreal_complete";
   211 val real_isLub_unique = thm "real_isLub_unique";
   212 val real_order_restrict = thm "real_order_restrict";
   213 val posreals_complete = thm "posreals_complete";
   214 val reals_complete = thm "reals_complete";
   215 val reals_Archimedean = thm "reals_Archimedean";
   216 val reals_Archimedean2 = thm "reals_Archimedean2";
   217 val reals_Archimedean3 = thm "reals_Archimedean3";
   218 *}
   219 
   220 
   221 subsection{*Floor and Ceiling Functions from the Reals to the Integers*}
   222 
   223 constdefs
   224 
   225   floor :: "real => int"
   226    "floor r == (LEAST n::int. r < real (n+1))"
   227 
   228   ceiling :: "real => int"
   229     "ceiling r == - floor (- r)"
   230 
   231 syntax (xsymbols)
   232   floor :: "real => int"     ("\<lfloor>_\<rfloor>")
   233   ceiling :: "real => int"   ("\<lceil>_\<rceil>")
   234 
   235 syntax (HTML output)
   236   floor :: "real => int"     ("\<lfloor>_\<rfloor>")
   237   ceiling :: "real => int"   ("\<lceil>_\<rceil>")
   238 
   239 
   240 lemma number_of_less_real_of_int_iff [simp]:
   241      "((number_of n) < real (m::int)) = (number_of n < m)"
   242 apply auto
   243 apply (rule real_of_int_less_iff [THEN iffD1])
   244 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
   245 done
   246 
   247 lemma number_of_less_real_of_int_iff2 [simp]:
   248      "(real (m::int) < (number_of n)) = (m < number_of n)"
   249 apply auto
   250 apply (rule real_of_int_less_iff [THEN iffD1])
   251 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
   252 done
   253 
   254 lemma number_of_le_real_of_int_iff [simp]:
   255      "((number_of n) \<le> real (m::int)) = (number_of n \<le> m)"
   256 by (simp add: linorder_not_less [symmetric])
   257 
   258 lemma number_of_le_real_of_int_iff2 [simp]:
   259      "(real (m::int) \<le> (number_of n)) = (m \<le> number_of n)"
   260 by (simp add: linorder_not_less [symmetric])
   261 
   262 lemma floor_zero [simp]: "floor 0 = 0"
   263 apply (simp add: floor_def)
   264 apply (rule Least_equality, auto)
   265 done
   266 
   267 lemma floor_real_of_nat_zero [simp]: "floor (real (0::nat)) = 0"
   268 by auto
   269 
   270 lemma floor_real_of_nat [simp]: "floor (real (n::nat)) = int n"
   271 apply (simp only: floor_def)
   272 apply (rule Least_equality)
   273 apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
   274 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1])
   275 apply (simp_all add: real_of_int_real_of_nat)
   276 done
   277 
   278 lemma floor_minus_real_of_nat [simp]: "floor (- real (n::nat)) = - int n"
   279 apply (simp only: floor_def)
   280 apply (rule Least_equality)
   281 apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
   282 apply (drule_tac [2] real_of_int_minus [THEN subst])
   283 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1])
   284 apply (simp_all add: real_of_int_real_of_nat)
   285 done
   286 
   287 lemma floor_real_of_int [simp]: "floor (real (n::int)) = n"
   288 apply (simp only: floor_def)
   289 apply (rule Least_equality)
   290 apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
   291 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1], auto)
   292 done
   293 
   294 lemma floor_minus_real_of_int [simp]: "floor (- real (n::int)) = - n"
   295 apply (simp only: floor_def)
   296 apply (rule Least_equality)
   297 apply (drule_tac [2] real_of_int_minus [THEN subst])
   298 apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
   299 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1], auto)
   300 done
   301 
   302 lemma reals_Archimedean6:
   303      "0 \<le> r ==> \<exists>(n::nat). real (n - 1) \<le> r & r < real (n)"
   304 apply (insert reals_Archimedean2 [of r], safe)
   305 apply (frule_tac P = "%k. r < real k" and k = n and m = "%x. x"
   306        in ex_has_least_nat, auto)
   307 apply (rule_tac x = x in exI)
   308 apply (case_tac x, simp)
   309 apply (rename_tac x')
   310 apply (drule_tac x = x' in spec, simp)
   311 done
   312 
   313 lemma reals_Archimedean6a: "0 \<le> r ==> \<exists>n. real (n) \<le> r & r < real (Suc n)"
   314 by (drule reals_Archimedean6, auto)
   315 
   316 lemma reals_Archimedean_6b_int:
   317      "0 \<le> r ==> \<exists>n::int. real n \<le> r & r < real (n+1)"
   318 apply (drule reals_Archimedean6a, auto)
   319 apply (rule_tac x = "int n" in exI)
   320 apply (simp add: real_of_int_real_of_nat real_of_nat_Suc)
   321 done
   322 
   323 lemma reals_Archimedean_6c_int:
   324      "r < 0 ==> \<exists>n::int. real n \<le> r & r < real (n+1)"
   325 apply (rule reals_Archimedean_6b_int [of "-r", THEN exE], simp, auto)
   326 apply (rename_tac n)
   327 apply (drule real_le_imp_less_or_eq, auto)
   328 apply (rule_tac x = "- n - 1" in exI)
   329 apply (rule_tac [2] x = "- n" in exI, auto)
   330 done
   331 
   332 lemma real_lb_ub_int: " \<exists>n::int. real n \<le> r & r < real (n+1)"
   333 apply (case_tac "r < 0")
   334 apply (blast intro: reals_Archimedean_6c_int)
   335 apply (simp only: linorder_not_less)
   336 apply (blast intro: reals_Archimedean_6b_int reals_Archimedean_6c_int)
   337 done
   338 
   339 lemma lemma_floor:
   340   assumes a1: "real m \<le> r" and a2: "r < real n + 1"
   341   shows "m \<le> (n::int)"
   342 proof -
   343   have "real m < real n + 1" by (rule order_le_less_trans)
   344   also have "... = real(n+1)" by simp
   345   finally have "m < n+1" by (simp only: real_of_int_less_iff)
   346   thus ?thesis by arith
   347 qed
   348 
   349 lemma real_of_int_floor_le [simp]: "real (floor r) \<le> r"
   350 apply (simp add: floor_def Least_def)
   351 apply (insert real_lb_ub_int [of r], safe)
   352 apply (rule theI2, auto)
   353 done
   354 
   355 lemma floor_le: "x < y ==> floor x \<le> floor y"
   356 apply (simp add: floor_def Least_def)
   357 apply (insert real_lb_ub_int [of x])
   358 apply (insert real_lb_ub_int [of y], safe)
   359 apply (rule theI2)
   360 apply (rule_tac [3] theI2, auto)
   361 done
   362 
   363 lemma floor_le2: "x \<le> y ==> floor x \<le> floor y"
   364 by (auto dest: real_le_imp_less_or_eq simp add: floor_le)
   365 
   366 lemma lemma_floor2: "real n < real (x::int) + 1 ==> n \<le> x"
   367 by (auto intro: lemma_floor)
   368 
   369 lemma real_of_int_floor_cancel [simp]:
   370     "(real (floor x) = x) = (\<exists>n::int. x = real n)"
   371 apply (simp add: floor_def Least_def)
   372 apply (insert real_lb_ub_int [of x], erule exE)
   373 apply (rule theI2)
   374 apply (auto intro: lemma_floor)
   375 done
   376 
   377 lemma floor_eq: "[| real n < x; x < real n + 1 |] ==> floor x = n"
   378 apply (simp add: floor_def)
   379 apply (rule Least_equality)
   380 apply (auto intro: lemma_floor)
   381 done
   382 
   383 lemma floor_eq2: "[| real n \<le> x; x < real n + 1 |] ==> floor x = n"
   384 apply (simp add: floor_def)
   385 apply (rule Least_equality)
   386 apply (auto intro: lemma_floor)
   387 done
   388 
   389 lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"
   390 apply (rule inj_int [THEN injD])
   391 apply (simp add: real_of_nat_Suc)
   392 apply (simp add: real_of_nat_Suc floor_eq floor_eq [where n = "of_nat n"])
   393 done
   394 
   395 lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"
   396 apply (drule order_le_imp_less_or_eq)
   397 apply (auto intro: floor_eq3)
   398 done
   399 
   400 lemma floor_number_of_eq [simp]:
   401      "floor(number_of n :: real) = (number_of n :: int)"
   402 apply (subst real_number_of [symmetric])
   403 apply (rule floor_real_of_int)
   404 done
   405 
   406 lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real(floor r)"
   407 apply (simp add: floor_def Least_def)
   408 apply (insert real_lb_ub_int [of r], safe)
   409 apply (rule theI2)
   410 apply (auto intro: lemma_floor)
   411 done
   412 
   413 lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real(floor r) + 1"
   414 apply (insert real_of_int_floor_ge_diff_one [of r])
   415 apply (auto simp del: real_of_int_floor_ge_diff_one)
   416 done
   417 
   418 
   419 subsection{*Ceiling Function for Positive Reals*}
   420 
   421 lemma ceiling_zero [simp]: "ceiling 0 = 0"
   422 by (simp add: ceiling_def)
   423 
   424 lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"
   425 by (simp add: ceiling_def)
   426 
   427 lemma ceiling_real_of_nat_zero [simp]: "ceiling (real (0::nat)) = 0"
   428 by auto
   429 
   430 lemma ceiling_floor [simp]: "ceiling (real (floor r)) = floor r"
   431 by (simp add: ceiling_def)
   432 
   433 lemma floor_ceiling [simp]: "floor (real (ceiling r)) = ceiling r"
   434 by (simp add: ceiling_def)
   435 
   436 lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)"
   437 apply (simp add: ceiling_def)
   438 apply (subst le_minus_iff, simp)
   439 done
   440 
   441 lemma ceiling_le: "x < y ==> ceiling x \<le> ceiling y"
   442 by (simp add: floor_le ceiling_def)
   443 
   444 lemma ceiling_le2: "x \<le> y ==> ceiling x \<le> ceiling y"
   445 by (simp add: floor_le2 ceiling_def)
   446 
   447 lemma real_of_int_ceiling_cancel [simp]:
   448      "(real (ceiling x) = x) = (\<exists>n::int. x = real n)"
   449 apply (auto simp add: ceiling_def)
   450 apply (drule arg_cong [where f = uminus], auto)
   451 apply (rule_tac x = "-n" in exI, auto)
   452 done
   453 
   454 lemma ceiling_eq: "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1"
   455 apply (simp add: ceiling_def)
   456 apply (rule minus_equation_iff [THEN iffD1])
   457 apply (simp add: floor_eq [where n = "-(n+1)"])
   458 done
   459 
   460 lemma ceiling_eq2: "[| real n < x; x \<le> real n + 1 |] ==> ceiling x = n + 1"
   461 by (simp add: ceiling_def floor_eq2 [where n = "-(n+1)"])
   462 
   463 lemma ceiling_eq3: "[| real n - 1 < x; x \<le> real n  |] ==> ceiling x = n"
   464 by (simp add: ceiling_def floor_eq2 [where n = "-n"])
   465 
   466 lemma ceiling_real_of_int [simp]: "ceiling (real (n::int)) = n"
   467 by (simp add: ceiling_def)
   468 
   469 lemma ceiling_number_of_eq [simp]:
   470      "ceiling (number_of n :: real) = (number_of n)"
   471 apply (subst real_number_of [symmetric])
   472 apply (rule ceiling_real_of_int)
   473 done
   474 
   475 lemma real_of_int_ceiling_diff_one_le [simp]: "real (ceiling r) - 1 \<le> r"
   476 apply (rule neg_le_iff_le [THEN iffD1])
   477 apply (simp add: ceiling_def diff_minus)
   478 done
   479 
   480 lemma real_of_int_ceiling_le_add_one [simp]: "real (ceiling r) \<le> r + 1"
   481 apply (insert real_of_int_ceiling_diff_one_le [of r])
   482 apply (simp del: real_of_int_ceiling_diff_one_le)
   483 done
   484 
   485 ML
   486 {*
   487 val number_of_less_real_of_int_iff = thm "number_of_less_real_of_int_iff";
   488 val number_of_less_real_of_int_iff2 = thm "number_of_less_real_of_int_iff2";
   489 val number_of_le_real_of_int_iff = thm "number_of_le_real_of_int_iff";
   490 val number_of_le_real_of_int_iff2 = thm "number_of_le_real_of_int_iff2";
   491 val floor_zero = thm "floor_zero";
   492 val floor_real_of_nat_zero = thm "floor_real_of_nat_zero";
   493 val floor_real_of_nat = thm "floor_real_of_nat";
   494 val floor_minus_real_of_nat = thm "floor_minus_real_of_nat";
   495 val floor_real_of_int = thm "floor_real_of_int";
   496 val floor_minus_real_of_int = thm "floor_minus_real_of_int";
   497 val reals_Archimedean6 = thm "reals_Archimedean6";
   498 val reals_Archimedean6a = thm "reals_Archimedean6a";
   499 val reals_Archimedean_6b_int = thm "reals_Archimedean_6b_int";
   500 val reals_Archimedean_6c_int = thm "reals_Archimedean_6c_int";
   501 val real_lb_ub_int = thm "real_lb_ub_int";
   502 val lemma_floor = thm "lemma_floor";
   503 val real_of_int_floor_le = thm "real_of_int_floor_le";
   504 val floor_le = thm "floor_le";
   505 val floor_le2 = thm "floor_le2";
   506 val lemma_floor2 = thm "lemma_floor2";
   507 val real_of_int_floor_cancel = thm "real_of_int_floor_cancel";
   508 val floor_eq = thm "floor_eq";
   509 val floor_eq2 = thm "floor_eq2";
   510 val floor_eq3 = thm "floor_eq3";
   511 val floor_eq4 = thm "floor_eq4";
   512 val floor_number_of_eq = thm "floor_number_of_eq";
   513 val real_of_int_floor_ge_diff_one = thm "real_of_int_floor_ge_diff_one";
   514 val real_of_int_floor_add_one_ge = thm "real_of_int_floor_add_one_ge";
   515 val ceiling_zero = thm "ceiling_zero";
   516 val ceiling_real_of_nat = thm "ceiling_real_of_nat";
   517 val ceiling_real_of_nat_zero = thm "ceiling_real_of_nat_zero";
   518 val ceiling_floor = thm "ceiling_floor";
   519 val floor_ceiling = thm "floor_ceiling";
   520 val real_of_int_ceiling_ge = thm "real_of_int_ceiling_ge";
   521 val ceiling_le = thm "ceiling_le";
   522 val ceiling_le2 = thm "ceiling_le2";
   523 val real_of_int_ceiling_cancel = thm "real_of_int_ceiling_cancel";
   524 val ceiling_eq = thm "ceiling_eq";
   525 val ceiling_eq2 = thm "ceiling_eq2";
   526 val ceiling_eq3 = thm "ceiling_eq3";
   527 val ceiling_real_of_int = thm "ceiling_real_of_int";
   528 val ceiling_number_of_eq = thm "ceiling_number_of_eq";
   529 val real_of_int_ceiling_diff_one_le = thm "real_of_int_ceiling_diff_one_le";
   530 val real_of_int_ceiling_le_add_one = thm "real_of_int_ceiling_le_add_one";
   531 *}
   532 
   533 
   534 end
   535 
   536 
   537 
   538