src/HOL/Real/RComplete.thy
author paulson
Thu Apr 22 10:45:56 2004 +0200 (2004-04-22)
changeset 14641 79b7bd936264
parent 14476 758e7acdea2f
child 15131 c69542757a4d
permissions -rw-r--r--
moved Complex/NSInduct and Hyperreal/IntFloor to more appropriate
places
     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 = Lubs + RealDef:
    12 
    13 lemma real_sum_of_halves: "x/2 + x/2 = (x::real)"
    14 by simp
    15 
    16 
    17 subsection{*Completeness of Reals by Supremum Property of type @{typ preal}*} 
    18 
    19  (*a few lemmas*)
    20 lemma real_sup_lemma1:
    21      "\<forall>x \<in> P. 0 < x ==>   
    22       ((\<exists>x \<in> P. y < x) = (\<exists>X. real_of_preal X \<in> P & y < real_of_preal X))"
    23 by (blast dest!: bspec real_gt_zero_preal_Ex [THEN iffD1])
    24 
    25 lemma real_sup_lemma2:
    26      "[| \<forall>x \<in> P. 0 < x;  a \<in> P;   \<forall>x \<in> P. x < y |]  
    27       ==> (\<exists>X. X\<in> {w. real_of_preal w \<in> P}) &  
    28           (\<exists>Y. \<forall>X\<in> {w. real_of_preal w \<in> P}. X < Y)"
    29 apply (rule conjI)
    30 apply (blast dest: bspec real_gt_zero_preal_Ex [THEN iffD1], auto)
    31 apply (drule bspec, assumption)
    32 apply (frule bspec, assumption)
    33 apply (drule order_less_trans, assumption)
    34 apply (drule real_gt_zero_preal_Ex [THEN iffD1], force) 
    35 done
    36 
    37 (*-------------------------------------------------------------
    38             Completeness of Positive Reals
    39  -------------------------------------------------------------*)
    40 
    41 (**
    42  Supremum property for the set of positive reals
    43  FIXME: long proof - should be improved
    44 **)
    45 
    46 (*Let P be a non-empty set of positive reals, with an upper bound y.
    47   Then P has a least upper bound (written S).  
    48 FIXME: Can the premise be weakened to \<forall>x \<in> P. x\<le> y ??*)
    49 lemma posreal_complete: "[| \<forall>x \<in> P. (0::real) < x;  \<exists>x. x \<in> P;  \<exists>y. \<forall>x \<in> P. x<y |]  
    50       ==> (\<exists>S. \<forall>y. (\<exists>x \<in> P. y < x) = (y < S))"
    51 apply (rule_tac x = "real_of_preal (psup ({w. real_of_preal w \<in> P}))" in exI)
    52 apply clarify
    53 apply (case_tac "0 < ya", auto)
    54 apply (frule real_sup_lemma2, assumption+)
    55 apply (drule real_gt_zero_preal_Ex [THEN iffD1])
    56 apply (drule_tac [3] real_less_all_real2, auto)
    57 apply (rule preal_complete [THEN iffD1])
    58 apply (auto intro: order_less_imp_le)
    59 apply (frule real_gt_preal_preal_Ex, force)
    60 (* second part *)
    61 apply (rule real_sup_lemma1 [THEN iffD2], assumption)
    62 apply (auto dest!: real_less_all_real2 real_gt_zero_preal_Ex [THEN iffD1])
    63 apply (frule_tac [2] real_sup_lemma2)
    64 apply (frule real_sup_lemma2, assumption+, clarify) 
    65 apply (rule preal_complete [THEN iffD2, THEN bexE])
    66 prefer 3 apply blast
    67 apply (blast intro!: order_less_imp_le)+
    68 done
    69 
    70 (*--------------------------------------------------------
    71    Completeness properties using isUb, isLub etc.
    72  -------------------------------------------------------*)
    73 
    74 lemma real_isLub_unique: "[| isLub R S x; isLub R S y |] ==> x = (y::real)"
    75 apply (frule isLub_isUb)
    76 apply (frule_tac x = y in isLub_isUb)
    77 apply (blast intro!: order_antisym dest!: isLub_le_isUb)
    78 done
    79 
    80 lemma real_order_restrict: "[| (x::real) <=* S'; S <= S' |] ==> x <=* S"
    81 by (unfold setle_def setge_def, blast)
    82 
    83 (*----------------------------------------------------------------
    84            Completeness theorem for the positive reals(again)
    85  ----------------------------------------------------------------*)
    86 
    87 lemma posreals_complete:
    88      "[| \<forall>x \<in>S. 0 < x;  
    89          \<exists>x. x \<in>S;  
    90          \<exists>u. isUb (UNIV::real set) S u  
    91       |] ==> \<exists>t. isLub (UNIV::real set) S t"
    92 apply (rule_tac x = "real_of_preal (psup ({w. real_of_preal w \<in> S}))" in exI)
    93 apply (auto simp add: isLub_def leastP_def isUb_def)
    94 apply (auto intro!: setleI setgeI dest!: real_gt_zero_preal_Ex [THEN iffD1])
    95 apply (frule_tac x = y in bspec, assumption)
    96 apply (drule real_gt_zero_preal_Ex [THEN iffD1])
    97 apply (auto simp add: real_of_preal_le_iff)
    98 apply (frule_tac y = "real_of_preal ya" in setleD, assumption)
    99 apply (frule real_ge_preal_preal_Ex, safe)
   100 apply (blast intro!: preal_psup_le dest!: setleD intro: real_of_preal_le_iff [THEN iffD1])
   101 apply (frule_tac x = x in bspec, assumption)
   102 apply (frule isUbD2)
   103 apply (drule real_gt_zero_preal_Ex [THEN iffD1])
   104 apply (auto dest!: isUbD real_ge_preal_preal_Ex simp add: real_of_preal_le_iff)
   105 apply (blast dest!: setleD intro!: psup_le_ub intro: real_of_preal_le_iff [THEN iffD1])
   106 done
   107 
   108 
   109 (*-------------------------------
   110     Lemmas
   111  -------------------------------*)
   112 lemma real_sup_lemma3: "\<forall>y \<in> {z. \<exists>x \<in> P. z = x + (-xa) + 1} Int {x. 0 < x}. 0 < y"
   113 by auto
   114  
   115 lemma lemma_le_swap2: "(xa <= S + X + (-Z)) = (xa + (-X) + Z <= (S::real))"
   116 by auto
   117 
   118 lemma lemma_real_complete2b: "[| (x::real) + (-X) + 1 <= S; xa <= x |] ==> xa <= S + X + (- 1)"
   119 by arith
   120 
   121 (*----------------------------------------------------------
   122       reals Completeness (again!)
   123  ----------------------------------------------------------*)
   124 lemma reals_complete: "[| \<exists>X. X \<in>S;  \<exists>Y. isUb (UNIV::real set) S Y |]   
   125       ==> \<exists>t. isLub (UNIV :: real set) S t"
   126 apply safe
   127 apply (subgoal_tac "\<exists>u. u\<in> {z. \<exists>x \<in>S. z = x + (-X) + 1} Int {x. 0 < x}")
   128 apply (subgoal_tac "isUb (UNIV::real set) ({z. \<exists>x \<in>S. z = x + (-X) + 1} Int {x. 0 < x}) (Y + (-X) + 1) ")
   129 apply (cut_tac P = S and xa = X in real_sup_lemma3)
   130 apply (frule posreals_complete [OF _ _ exI], blast, blast, safe)
   131 apply (rule_tac x = "t + X + (- 1) " in exI)
   132 apply (rule isLubI2)
   133 apply (rule_tac [2] setgeI, safe)
   134 apply (subgoal_tac [2] "isUb (UNIV:: real set) ({z. \<exists>x \<in>S. z = x + (-X) + 1} Int {x. 0 < x}) (y + (-X) + 1) ")
   135 apply (drule_tac [2] y = " (y + (- X) + 1) " in isLub_le_isUb)
   136  prefer 2 apply assumption
   137  prefer 2
   138 apply arith
   139 apply (rule setleI [THEN isUbI], safe)
   140 apply (rule_tac x = x and y = y in linorder_cases)
   141 apply (subst lemma_le_swap2)
   142 apply (frule isLubD2)
   143  prefer 2 apply assumption
   144 apply safe
   145 apply blast
   146 apply arith
   147 apply (subst lemma_le_swap2)
   148 apply (frule isLubD2)
   149  prefer 2 apply assumption
   150 apply blast
   151 apply (rule lemma_real_complete2b)
   152 apply (erule_tac [2] order_less_imp_le)
   153 apply (blast intro!: isLubD2, blast) 
   154 apply (simp (no_asm_use) add: real_add_assoc)
   155 apply (blast dest: isUbD intro!: setleI [THEN isUbI] intro: add_right_mono)
   156 apply (blast dest: isUbD intro!: setleI [THEN isUbI] intro: add_right_mono, auto)
   157 done
   158 
   159 
   160 subsection{*Corollary: the Archimedean Property of the Reals*}
   161 
   162 lemma reals_Archimedean: "0 < x ==> \<exists>n. inverse (real(Suc n)) < x"
   163 apply (rule ccontr)
   164 apply (subgoal_tac "\<forall>n. x * real (Suc n) <= 1")
   165  prefer 2
   166 apply (simp add: linorder_not_less inverse_eq_divide, clarify) 
   167 apply (drule_tac x = n in spec)
   168 apply (drule_tac c = "real (Suc n)"  in mult_right_mono)
   169 apply (rule real_of_nat_ge_zero)
   170 apply (simp add: real_of_nat_Suc_gt_zero [THEN real_not_refl2, THEN not_sym] real_mult_commute)
   171 apply (subgoal_tac "isUb (UNIV::real set) {z. \<exists>n. z = x* (real (Suc n))} 1")
   172 apply (subgoal_tac "\<exists>X. X \<in> {z. \<exists>n. z = x* (real (Suc n))}")
   173 apply (drule reals_complete)
   174 apply (auto intro: isUbI setleI)
   175 apply (subgoal_tac "\<forall>m. x* (real (Suc m)) <= t")
   176 apply (simp add: real_of_nat_Suc right_distrib)
   177 prefer 2 apply (blast intro: isLubD2)
   178 apply (simp add: le_diff_eq [symmetric] real_diff_def)
   179 apply (subgoal_tac "isUb (UNIV::real set) {z. \<exists>n. z = x* (real (Suc n))} (t + (-x))")
   180 prefer 2 apply (blast intro!: isUbI setleI)
   181 apply (drule_tac y = "t+ (-x) " in isLub_le_isUb)
   182 apply (auto simp add: real_of_nat_Suc right_distrib)
   183 done
   184 
   185 (*There must be other proofs, e.g. Suc of the largest integer in the
   186   cut representing x*)
   187 lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"
   188 apply (rule_tac x = x and y = 0 in linorder_cases)
   189 apply (rule_tac x = 0 in exI)
   190 apply (rule_tac [2] x = 1 in exI)
   191 apply (auto elim: order_less_trans simp add: real_of_nat_one)
   192 apply (frule positive_imp_inverse_positive [THEN reals_Archimedean], safe)
   193 apply (rule_tac x = "Suc n" in exI)
   194 apply (frule_tac b = "inverse x" in mult_strict_right_mono, auto)
   195 done
   196 
   197 lemma reals_Archimedean3: "0 < x ==> \<forall>y. \<exists>(n::nat). y < real n * x"
   198 apply safe
   199 apply (cut_tac x = "y*inverse (x) " in reals_Archimedean2)
   200 apply safe
   201 apply (frule_tac a = "y * inverse x" in mult_strict_right_mono)
   202 apply (auto simp add: mult_assoc real_of_nat_def)
   203 done
   204 
   205 ML
   206 {*
   207 val real_sum_of_halves = thm "real_sum_of_halves";
   208 val posreal_complete = thm "posreal_complete";
   209 val real_isLub_unique = thm "real_isLub_unique";
   210 val real_order_restrict = thm "real_order_restrict";
   211 val posreals_complete = thm "posreals_complete";
   212 val reals_complete = thm "reals_complete";
   213 val reals_Archimedean = thm "reals_Archimedean";
   214 val reals_Archimedean2 = thm "reals_Archimedean2";
   215 val reals_Archimedean3 = thm "reals_Archimedean3";
   216 *}
   217 
   218 
   219 subsection{*Floor and Ceiling Functions from the Reals to the Integers*}
   220 
   221 constdefs
   222 
   223   floor :: "real => int"
   224    "floor r == (LEAST n::int. r < real (n+1))"
   225 
   226   ceiling :: "real => int"
   227     "ceiling r == - floor (- r)"
   228 
   229 syntax (xsymbols)
   230   floor :: "real => int"     ("\<lfloor>_\<rfloor>")
   231   ceiling :: "real => int"   ("\<lceil>_\<rceil>")
   232 
   233 syntax (HTML output)
   234   floor :: "real => int"     ("\<lfloor>_\<rfloor>")
   235   ceiling :: "real => int"   ("\<lceil>_\<rceil>")
   236 
   237 
   238 lemma number_of_less_real_of_int_iff [simp]:
   239      "((number_of n) < real (m::int)) = (number_of n < m)"
   240 apply auto
   241 apply (rule real_of_int_less_iff [THEN iffD1])
   242 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
   243 done
   244 
   245 lemma number_of_less_real_of_int_iff2 [simp]:
   246      "(real (m::int) < (number_of n)) = (m < number_of n)"
   247 apply auto
   248 apply (rule real_of_int_less_iff [THEN iffD1])
   249 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
   250 done
   251 
   252 lemma number_of_le_real_of_int_iff [simp]:
   253      "((number_of n) \<le> real (m::int)) = (number_of n \<le> m)"
   254 by (simp add: linorder_not_less [symmetric])
   255 
   256 lemma number_of_le_real_of_int_iff2 [simp]:
   257      "(real (m::int) \<le> (number_of n)) = (m \<le> number_of n)"
   258 by (simp add: linorder_not_less [symmetric])
   259 
   260 lemma floor_zero [simp]: "floor 0 = 0"
   261 apply (simp add: floor_def)
   262 apply (rule Least_equality, auto)
   263 done
   264 
   265 lemma floor_real_of_nat_zero [simp]: "floor (real (0::nat)) = 0"
   266 by auto
   267 
   268 lemma floor_real_of_nat [simp]: "floor (real (n::nat)) = int n"
   269 apply (simp only: floor_def)
   270 apply (rule Least_equality)
   271 apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
   272 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1])
   273 apply (simp_all add: real_of_int_real_of_nat)
   274 done
   275 
   276 lemma floor_minus_real_of_nat [simp]: "floor (- real (n::nat)) = - int n"
   277 apply (simp only: floor_def)
   278 apply (rule Least_equality)
   279 apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
   280 apply (drule_tac [2] real_of_int_minus [THEN subst])
   281 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1])
   282 apply (simp_all add: real_of_int_real_of_nat)
   283 done
   284 
   285 lemma floor_real_of_int [simp]: "floor (real (n::int)) = n"
   286 apply (simp only: floor_def)
   287 apply (rule Least_equality)
   288 apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
   289 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1], auto)
   290 done
   291 
   292 lemma floor_minus_real_of_int [simp]: "floor (- real (n::int)) = - n"
   293 apply (simp only: floor_def)
   294 apply (rule Least_equality)
   295 apply (drule_tac [2] real_of_int_minus [THEN subst])
   296 apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
   297 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1], auto)
   298 done
   299 
   300 lemma reals_Archimedean6:
   301      "0 \<le> r ==> \<exists>(n::nat). real (n - 1) \<le> r & r < real (n)"
   302 apply (insert reals_Archimedean2 [of r], safe)
   303 apply (frule_tac P = "%k. r < real k" and k = n and m = "%x. x"
   304        in ex_has_least_nat, auto)
   305 apply (rule_tac x = x in exI)
   306 apply (case_tac x, simp)
   307 apply (rename_tac x')
   308 apply (drule_tac x = x' in spec, simp)
   309 done
   310 
   311 lemma reals_Archimedean6a: "0 \<le> r ==> \<exists>n. real (n) \<le> r & r < real (Suc n)"
   312 by (drule reals_Archimedean6, auto)
   313 
   314 lemma reals_Archimedean_6b_int:
   315      "0 \<le> r ==> \<exists>n::int. real n \<le> r & r < real (n+1)"
   316 apply (drule reals_Archimedean6a, auto)
   317 apply (rule_tac x = "int n" in exI)
   318 apply (simp add: real_of_int_real_of_nat real_of_nat_Suc)
   319 done
   320 
   321 lemma reals_Archimedean_6c_int:
   322      "r < 0 ==> \<exists>n::int. real n \<le> r & r < real (n+1)"
   323 apply (rule reals_Archimedean_6b_int [of "-r", THEN exE], simp, auto)
   324 apply (rename_tac n)
   325 apply (drule real_le_imp_less_or_eq, auto)
   326 apply (rule_tac x = "- n - 1" in exI)
   327 apply (rule_tac [2] x = "- n" in exI, auto)
   328 done
   329 
   330 lemma real_lb_ub_int: " \<exists>n::int. real n \<le> r & r < real (n+1)"
   331 apply (case_tac "r < 0")
   332 apply (blast intro: reals_Archimedean_6c_int)
   333 apply (simp only: linorder_not_less)
   334 apply (blast intro: reals_Archimedean_6b_int reals_Archimedean_6c_int)
   335 done
   336 
   337 lemma lemma_floor:
   338   assumes a1: "real m \<le> r" and a2: "r < real n + 1"
   339   shows "m \<le> (n::int)"
   340 proof -
   341   have "real m < real n + 1" by (rule order_le_less_trans)
   342   also have "... = real(n+1)" by simp
   343   finally have "m < n+1" by (simp only: real_of_int_less_iff)
   344   thus ?thesis by arith
   345 qed
   346 
   347 lemma real_of_int_floor_le [simp]: "real (floor r) \<le> r"
   348 apply (simp add: floor_def Least_def)
   349 apply (insert real_lb_ub_int [of r], safe)
   350 apply (rule theI2, auto)
   351 done
   352 
   353 lemma floor_le: "x < y ==> floor x \<le> floor y"
   354 apply (simp add: floor_def Least_def)
   355 apply (insert real_lb_ub_int [of x])
   356 apply (insert real_lb_ub_int [of y], safe)
   357 apply (rule theI2)
   358 apply (rule_tac [3] theI2, auto)
   359 done
   360 
   361 lemma floor_le2: "x \<le> y ==> floor x \<le> floor y"
   362 by (auto dest: real_le_imp_less_or_eq simp add: floor_le)
   363 
   364 lemma lemma_floor2: "real n < real (x::int) + 1 ==> n \<le> x"
   365 by (auto intro: lemma_floor)
   366 
   367 lemma real_of_int_floor_cancel [simp]:
   368     "(real (floor x) = x) = (\<exists>n::int. x = real n)"
   369 apply (simp add: floor_def Least_def)
   370 apply (insert real_lb_ub_int [of x], erule exE)
   371 apply (rule theI2)
   372 apply (auto intro: lemma_floor)
   373 done
   374 
   375 lemma floor_eq: "[| real n < x; x < real n + 1 |] ==> floor x = n"
   376 apply (simp add: floor_def)
   377 apply (rule Least_equality)
   378 apply (auto intro: lemma_floor)
   379 done
   380 
   381 lemma floor_eq2: "[| real n \<le> x; x < real n + 1 |] ==> floor x = n"
   382 apply (simp add: floor_def)
   383 apply (rule Least_equality)
   384 apply (auto intro: lemma_floor)
   385 done
   386 
   387 lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"
   388 apply (rule inj_int [THEN injD])
   389 apply (simp add: real_of_nat_Suc)
   390 apply (simp add: real_of_nat_Suc floor_eq floor_eq [where n = "of_nat n"])
   391 done
   392 
   393 lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"
   394 apply (drule order_le_imp_less_or_eq)
   395 apply (auto intro: floor_eq3)
   396 done
   397 
   398 lemma floor_number_of_eq [simp]:
   399      "floor(number_of n :: real) = (number_of n :: int)"
   400 apply (subst real_number_of [symmetric])
   401 apply (rule floor_real_of_int)
   402 done
   403 
   404 lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real(floor r)"
   405 apply (simp add: floor_def Least_def)
   406 apply (insert real_lb_ub_int [of r], safe)
   407 apply (rule theI2)
   408 apply (auto intro: lemma_floor)
   409 done
   410 
   411 lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real(floor r) + 1"
   412 apply (insert real_of_int_floor_ge_diff_one [of r])
   413 apply (auto simp del: real_of_int_floor_ge_diff_one)
   414 done
   415 
   416 
   417 subsection{*Ceiling Function for Positive Reals*}
   418 
   419 lemma ceiling_zero [simp]: "ceiling 0 = 0"
   420 by (simp add: ceiling_def)
   421 
   422 lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"
   423 by (simp add: ceiling_def)
   424 
   425 lemma ceiling_real_of_nat_zero [simp]: "ceiling (real (0::nat)) = 0"
   426 by auto
   427 
   428 lemma ceiling_floor [simp]: "ceiling (real (floor r)) = floor r"
   429 by (simp add: ceiling_def)
   430 
   431 lemma floor_ceiling [simp]: "floor (real (ceiling r)) = ceiling r"
   432 by (simp add: ceiling_def)
   433 
   434 lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)"
   435 apply (simp add: ceiling_def)
   436 apply (subst le_minus_iff, simp)
   437 done
   438 
   439 lemma ceiling_le: "x < y ==> ceiling x \<le> ceiling y"
   440 by (simp add: floor_le ceiling_def)
   441 
   442 lemma ceiling_le2: "x \<le> y ==> ceiling x \<le> ceiling y"
   443 by (simp add: floor_le2 ceiling_def)
   444 
   445 lemma real_of_int_ceiling_cancel [simp]:
   446      "(real (ceiling x) = x) = (\<exists>n::int. x = real n)"
   447 apply (auto simp add: ceiling_def)
   448 apply (drule arg_cong [where f = uminus], auto)
   449 apply (rule_tac x = "-n" in exI, auto)
   450 done
   451 
   452 lemma ceiling_eq: "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1"
   453 apply (simp add: ceiling_def)
   454 apply (rule minus_equation_iff [THEN iffD1])
   455 apply (simp add: floor_eq [where n = "-(n+1)"])
   456 done
   457 
   458 lemma ceiling_eq2: "[| real n < x; x \<le> real n + 1 |] ==> ceiling x = n + 1"
   459 by (simp add: ceiling_def floor_eq2 [where n = "-(n+1)"])
   460 
   461 lemma ceiling_eq3: "[| real n - 1 < x; x \<le> real n  |] ==> ceiling x = n"
   462 by (simp add: ceiling_def floor_eq2 [where n = "-n"])
   463 
   464 lemma ceiling_real_of_int [simp]: "ceiling (real (n::int)) = n"
   465 by (simp add: ceiling_def)
   466 
   467 lemma ceiling_number_of_eq [simp]:
   468      "ceiling (number_of n :: real) = (number_of n)"
   469 apply (subst real_number_of [symmetric])
   470 apply (rule ceiling_real_of_int)
   471 done
   472 
   473 lemma real_of_int_ceiling_diff_one_le [simp]: "real (ceiling r) - 1 \<le> r"
   474 apply (rule neg_le_iff_le [THEN iffD1])
   475 apply (simp add: ceiling_def diff_minus)
   476 done
   477 
   478 lemma real_of_int_ceiling_le_add_one [simp]: "real (ceiling r) \<le> r + 1"
   479 apply (insert real_of_int_ceiling_diff_one_le [of r])
   480 apply (simp del: real_of_int_ceiling_diff_one_le)
   481 done
   482 
   483 ML
   484 {*
   485 val number_of_less_real_of_int_iff = thm "number_of_less_real_of_int_iff";
   486 val number_of_less_real_of_int_iff2 = thm "number_of_less_real_of_int_iff2";
   487 val number_of_le_real_of_int_iff = thm "number_of_le_real_of_int_iff";
   488 val number_of_le_real_of_int_iff2 = thm "number_of_le_real_of_int_iff2";
   489 val floor_zero = thm "floor_zero";
   490 val floor_real_of_nat_zero = thm "floor_real_of_nat_zero";
   491 val floor_real_of_nat = thm "floor_real_of_nat";
   492 val floor_minus_real_of_nat = thm "floor_minus_real_of_nat";
   493 val floor_real_of_int = thm "floor_real_of_int";
   494 val floor_minus_real_of_int = thm "floor_minus_real_of_int";
   495 val reals_Archimedean6 = thm "reals_Archimedean6";
   496 val reals_Archimedean6a = thm "reals_Archimedean6a";
   497 val reals_Archimedean_6b_int = thm "reals_Archimedean_6b_int";
   498 val reals_Archimedean_6c_int = thm "reals_Archimedean_6c_int";
   499 val real_lb_ub_int = thm "real_lb_ub_int";
   500 val lemma_floor = thm "lemma_floor";
   501 val real_of_int_floor_le = thm "real_of_int_floor_le";
   502 val floor_le = thm "floor_le";
   503 val floor_le2 = thm "floor_le2";
   504 val lemma_floor2 = thm "lemma_floor2";
   505 val real_of_int_floor_cancel = thm "real_of_int_floor_cancel";
   506 val floor_eq = thm "floor_eq";
   507 val floor_eq2 = thm "floor_eq2";
   508 val floor_eq3 = thm "floor_eq3";
   509 val floor_eq4 = thm "floor_eq4";
   510 val floor_number_of_eq = thm "floor_number_of_eq";
   511 val real_of_int_floor_ge_diff_one = thm "real_of_int_floor_ge_diff_one";
   512 val real_of_int_floor_add_one_ge = thm "real_of_int_floor_add_one_ge";
   513 val ceiling_zero = thm "ceiling_zero";
   514 val ceiling_real_of_nat = thm "ceiling_real_of_nat";
   515 val ceiling_real_of_nat_zero = thm "ceiling_real_of_nat_zero";
   516 val ceiling_floor = thm "ceiling_floor";
   517 val floor_ceiling = thm "floor_ceiling";
   518 val real_of_int_ceiling_ge = thm "real_of_int_ceiling_ge";
   519 val ceiling_le = thm "ceiling_le";
   520 val ceiling_le2 = thm "ceiling_le2";
   521 val real_of_int_ceiling_cancel = thm "real_of_int_ceiling_cancel";
   522 val ceiling_eq = thm "ceiling_eq";
   523 val ceiling_eq2 = thm "ceiling_eq2";
   524 val ceiling_eq3 = thm "ceiling_eq3";
   525 val ceiling_real_of_int = thm "ceiling_real_of_int";
   526 val ceiling_number_of_eq = thm "ceiling_number_of_eq";
   527 val real_of_int_ceiling_diff_one_le = thm "real_of_int_ceiling_diff_one_le";
   528 val real_of_int_ceiling_le_add_one = thm "real_of_int_ceiling_le_add_one";
   529 *}
   530 
   531 
   532 end
   533 
   534 
   535 
   536