src/HOL/Real/RComplete.thy
author wenzelm
Thu Jul 14 17:16:52 2005 +0200 (2005-07-14)
changeset 16827 c90a1f450327
parent 16820 5c9d597e4cb9
child 16893 0cc94e6f6ae5
permissions -rw-r--r--
accomodate change of real_of_XXX;
     1 (*  Title       : RComplete.thy
     2     ID          : $Id$
     3     Author      : Jacques D. Fleuriot
     4                   Converted to Isar and polished by lcp
     5                   Most floor and ceiling lemmas by Jeremy Avigad
     6     Copyright   : 1998  University of Cambridge
     7     Copyright   : 2001,2002  University of Edinburgh
     8 *) 
     9 
    10 header{*Completeness of the Reals; Floor and Ceiling Functions*}
    11 
    12 theory RComplete
    13 imports Lubs RealDef
    14 begin
    15 
    16 lemma real_sum_of_halves: "x/2 + x/2 = (x::real)"
    17 by simp
    18 
    19 
    20 subsection{*Completeness of Reals by Supremum Property of type @{typ preal}*} 
    21 
    22  (*a few lemmas*)
    23 lemma real_sup_lemma1:
    24      "\<forall>x \<in> P. 0 < x ==>   
    25       ((\<exists>x \<in> P. y < x) = (\<exists>X. real_of_preal X \<in> P & y < real_of_preal X))"
    26 by (blast dest!: bspec real_gt_zero_preal_Ex [THEN iffD1])
    27 
    28 lemma real_sup_lemma2:
    29      "[| \<forall>x \<in> P. 0 < x;  a \<in> P;   \<forall>x \<in> P. x < y |]  
    30       ==> (\<exists>X. X\<in> {w. real_of_preal w \<in> P}) &  
    31           (\<exists>Y. \<forall>X\<in> {w. real_of_preal w \<in> P}. X < Y)"
    32 apply (rule conjI)
    33 apply (blast dest: bspec real_gt_zero_preal_Ex [THEN iffD1], auto)
    34 apply (drule bspec, assumption)
    35 apply (frule bspec, assumption)
    36 apply (drule order_less_trans, assumption)
    37 apply (drule real_gt_zero_preal_Ex [THEN iffD1], force) 
    38 done
    39 
    40 (*-------------------------------------------------------------
    41             Completeness of Positive Reals
    42  -------------------------------------------------------------*)
    43 
    44 (**
    45  Supremum property for the set of positive reals
    46  FIXME: long proof - should be improved
    47 **)
    48 
    49 (*Let P be a non-empty set of positive reals, with an upper bound y.
    50   Then P has a least upper bound (written S).  
    51 FIXME: Can the premise be weakened to \<forall>x \<in> P. x\<le> y ??*)
    52 lemma posreal_complete: "[| \<forall>x \<in> P. (0::real) < x;  \<exists>x. x \<in> P;  \<exists>y. \<forall>x \<in> P. x<y |]  
    53       ==> (\<exists>S. \<forall>y. (\<exists>x \<in> P. y < x) = (y < S))"
    54 apply (rule_tac x = "real_of_preal (psup ({w. real_of_preal w \<in> P}))" in exI)
    55 apply clarify
    56 apply (case_tac "0 < ya", auto)
    57 apply (frule real_sup_lemma2, assumption+)
    58 apply (drule real_gt_zero_preal_Ex [THEN iffD1])
    59 apply (drule_tac [3] real_less_all_real2, auto)
    60 apply (rule preal_complete [THEN iffD1])
    61 apply (auto intro: order_less_imp_le)
    62 apply (frule real_gt_preal_preal_Ex, force)
    63 (* second part *)
    64 apply (rule real_sup_lemma1 [THEN iffD2], assumption)
    65 apply (auto dest!: real_less_all_real2 real_gt_zero_preal_Ex [THEN iffD1])
    66 apply (frule_tac [2] real_sup_lemma2)
    67 apply (frule real_sup_lemma2, assumption+, clarify) 
    68 apply (rule preal_complete [THEN iffD2, THEN bexE])
    69 prefer 3 apply blast
    70 apply (blast intro!: order_less_imp_le)+
    71 done
    72 
    73 (*--------------------------------------------------------
    74    Completeness properties using isUb, isLub etc.
    75  -------------------------------------------------------*)
    76 
    77 lemma real_isLub_unique: "[| isLub R S x; isLub R S y |] ==> x = (y::real)"
    78 apply (frule isLub_isUb)
    79 apply (frule_tac x = y in isLub_isUb)
    80 apply (blast intro!: order_antisym dest!: isLub_le_isUb)
    81 done
    82 
    83 lemma real_order_restrict: "[| (x::real) <=* S'; S <= S' |] ==> x <=* S"
    84 by (unfold setle_def setge_def, blast)
    85 
    86 (*----------------------------------------------------------------
    87            Completeness theorem for the positive reals(again)
    88  ----------------------------------------------------------------*)
    89 
    90 lemma posreals_complete:
    91      "[| \<forall>x \<in>S. 0 < x;  
    92          \<exists>x. x \<in>S;  
    93          \<exists>u. isUb (UNIV::real set) S u  
    94       |] ==> \<exists>t. isLub (UNIV::real set) S t"
    95 apply (rule_tac x = "real_of_preal (psup ({w. real_of_preal w \<in> S}))" in exI)
    96 apply (auto simp add: isLub_def leastP_def isUb_def)
    97 apply (auto intro!: setleI setgeI dest!: real_gt_zero_preal_Ex [THEN iffD1])
    98 apply (frule_tac x = y in bspec, assumption)
    99 apply (drule real_gt_zero_preal_Ex [THEN iffD1])
   100 apply (auto simp add: real_of_preal_le_iff)
   101 apply (frule_tac y = "real_of_preal ya" in setleD, assumption)
   102 apply (frule real_ge_preal_preal_Ex, safe)
   103 apply (blast intro!: preal_psup_le dest!: setleD intro: real_of_preal_le_iff [THEN iffD1])
   104 apply (frule_tac x = x in bspec, assumption)
   105 apply (frule isUbD2)
   106 apply (drule real_gt_zero_preal_Ex [THEN iffD1])
   107 apply (auto dest!: isUbD real_ge_preal_preal_Ex simp add: real_of_preal_le_iff)
   108 apply (blast dest!: setleD intro!: psup_le_ub intro: real_of_preal_le_iff [THEN iffD1])
   109 done
   110 
   111 
   112 (*-------------------------------
   113     Lemmas
   114  -------------------------------*)
   115 lemma real_sup_lemma3: "\<forall>y \<in> {z. \<exists>x \<in> P. z = x + (-xa) + 1} Int {x. 0 < x}. 0 < y"
   116 by auto
   117  
   118 lemma lemma_le_swap2: "(xa <= S + X + (-Z)) = (xa + (-X) + Z <= (S::real))"
   119 by auto
   120 
   121 lemma lemma_real_complete2b: "[| (x::real) + (-X) + 1 <= S; xa <= x |] ==> xa <= S + X + (- 1)"
   122 by arith
   123 
   124 (*----------------------------------------------------------
   125       reals Completeness (again!)
   126  ----------------------------------------------------------*)
   127 lemma reals_complete: "[| \<exists>X. X \<in>S;  \<exists>Y. isUb (UNIV::real set) S Y |]   
   128       ==> \<exists>t. isLub (UNIV :: real set) S t"
   129 apply safe
   130 apply (subgoal_tac "\<exists>u. u\<in> {z. \<exists>x \<in>S. z = x + (-X) + 1} Int {x. 0 < x}")
   131 apply (subgoal_tac "isUb (UNIV::real set) ({z. \<exists>x \<in>S. z = x + (-X) + 1} Int {x. 0 < x}) (Y + (-X) + 1) ")
   132 apply (cut_tac P = S and xa = X in real_sup_lemma3)
   133 apply (frule posreals_complete [OF _ _ exI], blast, blast, safe)
   134 apply (rule_tac x = "t + X + (- 1) " in exI)
   135 apply (rule isLubI2)
   136 apply (rule_tac [2] setgeI, safe)
   137 apply (subgoal_tac [2] "isUb (UNIV:: real set) ({z. \<exists>x \<in>S. z = x + (-X) + 1} Int {x. 0 < x}) (y + (-X) + 1) ")
   138 apply (drule_tac [2] y = " (y + (- X) + 1) " in isLub_le_isUb)
   139  prefer 2 apply assumption
   140  prefer 2
   141 apply arith
   142 apply (rule setleI [THEN isUbI], safe)
   143 apply (rule_tac x = x and y = y in linorder_cases)
   144 apply (subst lemma_le_swap2)
   145 apply (frule isLubD2)
   146  prefer 2 apply assumption
   147 apply safe
   148 apply blast
   149 apply arith
   150 apply (subst lemma_le_swap2)
   151 apply (frule isLubD2)
   152  prefer 2 apply assumption
   153 apply blast
   154 apply (rule lemma_real_complete2b)
   155 apply (erule_tac [2] order_less_imp_le)
   156 apply (blast intro!: isLubD2, blast) 
   157 apply (simp (no_asm_use) add: add_assoc)
   158 apply (blast dest: isUbD intro!: setleI [THEN isUbI] intro: add_right_mono)
   159 apply (blast dest: isUbD intro!: setleI [THEN isUbI] intro: add_right_mono, auto)
   160 done
   161 
   162 
   163 subsection{*Corollary: the Archimedean Property of the Reals*}
   164 
   165 lemma reals_Archimedean: "0 < x ==> \<exists>n. inverse (real(Suc n)) < x"
   166 apply (rule ccontr)
   167 apply (subgoal_tac "\<forall>n. x * real (Suc n) <= 1")
   168  prefer 2
   169 apply (simp add: linorder_not_less inverse_eq_divide, clarify) 
   170 apply (drule_tac x = n in spec)
   171 apply (drule_tac c = "real (Suc n)"  in mult_right_mono)
   172 apply (rule real_of_nat_ge_zero)
   173 apply (simp add: times_divide_eq_right real_of_nat_Suc_gt_zero [THEN real_not_refl2, THEN not_sym] mult_commute)
   174 apply (subgoal_tac "isUb (UNIV::real set) {z. \<exists>n. z = x* (real (Suc n))} 1")
   175 apply (subgoal_tac "\<exists>X. X \<in> {z. \<exists>n. z = x* (real (Suc n))}")
   176 apply (drule reals_complete)
   177 apply (auto intro: isUbI setleI)
   178 apply (subgoal_tac "\<forall>m. x* (real (Suc m)) <= t")
   179 apply (simp add: real_of_nat_Suc right_distrib)
   180 prefer 2 apply (blast intro: isLubD2)
   181 apply (simp add: le_diff_eq [symmetric] real_diff_def)
   182 apply (subgoal_tac "isUb (UNIV::real set) {z. \<exists>n. z = x* (real (Suc n))} (t + (-x))")
   183 prefer 2 apply (blast intro!: isUbI setleI)
   184 apply (drule_tac y = "t+ (-x) " in isLub_le_isUb)
   185 apply (auto simp add: real_of_nat_Suc right_distrib)
   186 done
   187 
   188 (*There must be other proofs, e.g. Suc of the largest integer in the
   189   cut representing x*)
   190 lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"
   191 apply (rule_tac x = x and y = 0 in linorder_cases)
   192 apply (rule_tac x = 0 in exI)
   193 apply (rule_tac [2] x = 1 in exI)
   194 apply (auto elim: order_less_trans simp add: real_of_nat_one)
   195 apply (frule positive_imp_inverse_positive [THEN reals_Archimedean], safe)
   196 apply (rule_tac x = "Suc n" in exI)
   197 apply (frule_tac b = "inverse x" in mult_strict_right_mono, auto)
   198 done
   199 
   200 lemma reals_Archimedean3: "0 < x ==> \<forall>y. \<exists>(n::nat). y < real n * x"
   201 apply safe
   202 apply (cut_tac x = "y*inverse (x) " in reals_Archimedean2)
   203 apply safe
   204 apply (frule_tac a = "y * inverse x" in mult_strict_right_mono)
   205 apply (auto simp add: mult_assoc real_of_nat_def)
   206 done
   207 
   208 lemma reals_Archimedean6:
   209      "0 \<le> r ==> \<exists>(n::nat). real (n - 1) \<le> r & r < real (n)"
   210 apply (insert reals_Archimedean2 [of r], safe)
   211 apply (frule_tac P = "%k. r < real k" and k = n and m = "%x. x"
   212        in ex_has_least_nat, auto)
   213 apply (rule_tac x = x in exI)
   214 apply (case_tac x, simp)
   215 apply (rename_tac x')
   216 apply (drule_tac x = x' in spec, simp)
   217 done
   218 
   219 lemma reals_Archimedean6a: "0 \<le> r ==> \<exists>n. real (n) \<le> r & r < real (Suc n)"
   220 by (drule reals_Archimedean6, auto)
   221 
   222 lemma reals_Archimedean_6b_int:
   223      "0 \<le> r ==> \<exists>n::int. real n \<le> r & r < real (n+1)"
   224 apply (drule reals_Archimedean6a, auto)
   225 apply (rule_tac x = "int n" in exI)
   226 apply (simp add: real_of_int_real_of_nat real_of_nat_Suc)
   227 done
   228 
   229 lemma reals_Archimedean_6c_int:
   230      "r < 0 ==> \<exists>n::int. real n \<le> r & r < real (n+1)"
   231 apply (rule reals_Archimedean_6b_int [of "-r", THEN exE], simp, auto)
   232 apply (rename_tac n)
   233 apply (drule real_le_imp_less_or_eq, auto)
   234 apply (rule_tac x = "- n - 1" in exI)
   235 apply (rule_tac [2] x = "- n" in exI, auto)
   236 done
   237 
   238 
   239 ML
   240 {*
   241 val real_sum_of_halves = thm "real_sum_of_halves";
   242 val posreal_complete = thm "posreal_complete";
   243 val real_isLub_unique = thm "real_isLub_unique";
   244 val real_order_restrict = thm "real_order_restrict";
   245 val posreals_complete = thm "posreals_complete";
   246 val reals_complete = thm "reals_complete";
   247 val reals_Archimedean = thm "reals_Archimedean";
   248 val reals_Archimedean2 = thm "reals_Archimedean2";
   249 val reals_Archimedean3 = thm "reals_Archimedean3";
   250 *}
   251 
   252 
   253 subsection{*Floor and Ceiling Functions from the Reals to the Integers*}
   254 
   255 constdefs
   256 
   257   floor :: "real => int"
   258    "floor r == (LEAST n::int. r < real (n+1))"
   259 
   260   ceiling :: "real => int"
   261     "ceiling r == - floor (- r)"
   262 
   263 syntax (xsymbols)
   264   floor :: "real => int"     ("\<lfloor>_\<rfloor>")
   265   ceiling :: "real => int"   ("\<lceil>_\<rceil>")
   266 
   267 syntax (HTML output)
   268   floor :: "real => int"     ("\<lfloor>_\<rfloor>")
   269   ceiling :: "real => int"   ("\<lceil>_\<rceil>")
   270 
   271 
   272 lemma number_of_less_real_of_int_iff [simp]:
   273      "((number_of n) < real (m::int)) = (number_of n < m)"
   274 apply auto
   275 apply (rule real_of_int_less_iff [THEN iffD1])
   276 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
   277 done
   278 
   279 lemma number_of_less_real_of_int_iff2 [simp]:
   280      "(real (m::int) < (number_of n)) = (m < number_of n)"
   281 apply auto
   282 apply (rule real_of_int_less_iff [THEN iffD1])
   283 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
   284 done
   285 
   286 lemma number_of_le_real_of_int_iff [simp]:
   287      "((number_of n) \<le> real (m::int)) = (number_of n \<le> m)"
   288 by (simp add: linorder_not_less [symmetric])
   289 
   290 lemma number_of_le_real_of_int_iff2 [simp]:
   291      "(real (m::int) \<le> (number_of n)) = (m \<le> number_of n)"
   292 by (simp add: linorder_not_less [symmetric])
   293 
   294 lemma floor_zero [simp]: "floor 0 = 0"
   295 apply (simp add: floor_def del: real_of_int_add)
   296 apply (rule Least_equality)
   297 apply simp_all
   298 done
   299 
   300 lemma floor_real_of_nat_zero [simp]: "floor (real (0::nat)) = 0"
   301 by auto
   302 
   303 lemma floor_real_of_nat [simp]: "floor (real (n::nat)) = int n"
   304 apply (simp only: floor_def)
   305 apply (rule Least_equality)
   306 apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
   307 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1])
   308 apply (simp_all add: real_of_int_real_of_nat)
   309 done
   310 
   311 lemma floor_minus_real_of_nat [simp]: "floor (- real (n::nat)) = - int n"
   312 apply (simp only: floor_def)
   313 apply (rule Least_equality)
   314 apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
   315 apply (drule_tac [2] real_of_int_minus [THEN sym, THEN subst])
   316 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1])
   317 apply (simp_all add: real_of_int_real_of_nat)
   318 done
   319 
   320 lemma floor_real_of_int [simp]: "floor (real (n::int)) = n"
   321 apply (simp only: floor_def)
   322 apply (rule Least_equality)
   323 apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
   324 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1], auto)
   325 done
   326 
   327 lemma floor_minus_real_of_int [simp]: "floor (- real (n::int)) = - n"
   328 apply (simp only: floor_def)
   329 apply (rule Least_equality)
   330 apply (drule_tac [2] real_of_int_minus [THEN sym, THEN subst])
   331 apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
   332 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1], auto)
   333 done
   334 
   335 lemma real_lb_ub_int: " \<exists>n::int. real n \<le> r & r < real (n+1)"
   336 apply (case_tac "r < 0")
   337 apply (blast intro: reals_Archimedean_6c_int)
   338 apply (simp only: linorder_not_less)
   339 apply (blast intro: reals_Archimedean_6b_int reals_Archimedean_6c_int)
   340 done
   341 
   342 lemma lemma_floor:
   343   assumes a1: "real m \<le> r" and a2: "r < real n + 1"
   344   shows "m \<le> (n::int)"
   345 proof -
   346   have "real m < real n + 1" by (rule order_le_less_trans)
   347   also have "... = real(n+1)" by simp
   348   finally have "m < n+1" by (simp only: real_of_int_less_iff)
   349   thus ?thesis by arith
   350 qed
   351 
   352 lemma real_of_int_floor_le [simp]: "real (floor r) \<le> r"
   353 apply (simp add: floor_def Least_def)
   354 apply (insert real_lb_ub_int [of r], safe)
   355 apply (rule theI2)
   356 apply auto
   357 done
   358 
   359 lemma floor_mono: "x < y ==> floor x \<le> floor y"
   360 apply (simp add: floor_def Least_def)
   361 apply (insert real_lb_ub_int [of x])
   362 apply (insert real_lb_ub_int [of y], safe)
   363 apply (rule theI2)
   364 apply (rule_tac [3] theI2)
   365 apply simp
   366 apply (erule conjI)
   367 apply (auto simp add: order_eq_iff int_le_real_less)
   368 done
   369 
   370 lemma floor_mono2: "x \<le> y ==> floor x \<le> floor y"
   371 by (auto dest: real_le_imp_less_or_eq simp add: floor_mono)
   372 
   373 lemma lemma_floor2: "real n < real (x::int) + 1 ==> n \<le> x"
   374 by (auto intro: lemma_floor)
   375 
   376 lemma real_of_int_floor_cancel [simp]:
   377     "(real (floor x) = x) = (\<exists>n::int. x = real n)"
   378 apply (simp add: floor_def Least_def)
   379 apply (insert real_lb_ub_int [of x], erule exE)
   380 apply (rule theI2)
   381 apply (auto intro: lemma_floor) 
   382 done
   383 
   384 lemma floor_eq: "[| real n < x; x < real n + 1 |] ==> floor x = n"
   385 apply (simp add: floor_def)
   386 apply (rule Least_equality)
   387 apply (auto intro: lemma_floor)
   388 done
   389 
   390 lemma floor_eq2: "[| real n \<le> x; x < real n + 1 |] ==> floor x = n"
   391 apply (simp add: floor_def)
   392 apply (rule Least_equality)
   393 apply (auto intro: lemma_floor)
   394 done
   395 
   396 lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"
   397 apply (rule inj_int [THEN injD])
   398 apply (simp add: real_of_nat_Suc)
   399 apply (simp add: real_of_nat_Suc floor_eq floor_eq [where n = "int n"])
   400 done
   401 
   402 lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"
   403 apply (drule order_le_imp_less_or_eq)
   404 apply (auto intro: floor_eq3)
   405 done
   406 
   407 lemma floor_number_of_eq [simp]:
   408      "floor(number_of n :: real) = (number_of n :: int)"
   409 apply (subst real_number_of [symmetric])
   410 apply (rule floor_real_of_int)
   411 done
   412 
   413 lemma floor_one [simp]: "floor 1 = 1"
   414   apply (rule trans)
   415   prefer 2
   416   apply (rule floor_real_of_int)
   417   apply simp
   418 done
   419 
   420 lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real(floor r)"
   421 apply (simp add: floor_def Least_def)
   422 apply (insert real_lb_ub_int [of r], safe)
   423 apply (rule theI2)
   424 apply (auto intro: lemma_floor)
   425 done
   426 
   427 lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real(floor r)"
   428 apply (simp add: floor_def Least_def)
   429 apply (insert real_lb_ub_int [of r], safe)
   430 apply (rule theI2)
   431 apply (auto intro: lemma_floor)
   432 done
   433 
   434 lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real(floor r) + 1"
   435 apply (insert real_of_int_floor_ge_diff_one [of r])
   436 apply (auto simp del: real_of_int_floor_ge_diff_one)
   437 done
   438 
   439 lemma real_of_int_floor_add_one_gt [simp]: "r < real(floor r) + 1"
   440 apply (insert real_of_int_floor_gt_diff_one [of r])
   441 apply (auto simp del: real_of_int_floor_gt_diff_one)
   442 done
   443 
   444 lemma le_floor: "real a <= x ==> a <= floor x"
   445   apply (subgoal_tac "a < floor x + 1")
   446   apply arith
   447   apply (subst real_of_int_less_iff [THEN sym])
   448   apply simp
   449   apply (insert real_of_int_floor_add_one_gt [of x]) 
   450   apply arith
   451 done
   452 
   453 lemma real_le_floor: "a <= floor x ==> real a <= x"
   454   apply (rule order_trans)
   455   prefer 2
   456   apply (rule real_of_int_floor_le)
   457   apply (subst real_of_int_le_iff)
   458   apply assumption
   459 done
   460 
   461 lemma le_floor_eq: "(a <= floor x) = (real a <= x)"
   462   apply (rule iffI)
   463   apply (erule real_le_floor)
   464   apply (erule le_floor)
   465 done
   466 
   467 lemma le_floor_eq_number_of [simp]: 
   468     "(number_of n <= floor x) = (number_of n <= x)"
   469 by (simp add: le_floor_eq)
   470 
   471 lemma le_floor_eq_zero [simp]: "(0 <= floor x) = (0 <= x)"
   472 by (simp add: le_floor_eq)
   473 
   474 lemma le_floor_eq_one [simp]: "(1 <= floor x) = (1 <= x)"
   475 by (simp add: le_floor_eq)
   476 
   477 lemma floor_less_eq: "(floor x < a) = (x < real a)"
   478   apply (subst linorder_not_le [THEN sym])+
   479   apply simp
   480   apply (rule le_floor_eq)
   481 done
   482 
   483 lemma floor_less_eq_number_of [simp]: 
   484     "(floor x < number_of n) = (x < number_of n)"
   485 by (simp add: floor_less_eq)
   486 
   487 lemma floor_less_eq_zero [simp]: "(floor x < 0) = (x < 0)"
   488 by (simp add: floor_less_eq)
   489 
   490 lemma floor_less_eq_one [simp]: "(floor x < 1) = (x < 1)"
   491 by (simp add: floor_less_eq)
   492 
   493 lemma less_floor_eq: "(a < floor x) = (real a + 1 <= x)"
   494   apply (insert le_floor_eq [of "a + 1" x])
   495   apply auto
   496 done
   497 
   498 lemma less_floor_eq_number_of [simp]: 
   499     "(number_of n < floor x) = (number_of n + 1 <= x)"
   500 by (simp add: less_floor_eq)
   501 
   502 lemma less_floor_eq_zero [simp]: "(0 < floor x) = (1 <= x)"
   503 by (simp add: less_floor_eq)
   504 
   505 lemma less_floor_eq_one [simp]: "(1 < floor x) = (2 <= x)"
   506 by (simp add: less_floor_eq)
   507 
   508 lemma floor_le_eq: "(floor x <= a) = (x < real a + 1)"
   509   apply (insert floor_less_eq [of x "a + 1"])
   510   apply auto
   511 done
   512 
   513 lemma floor_le_eq_number_of [simp]: 
   514     "(floor x <= number_of n) = (x < number_of n + 1)"
   515 by (simp add: floor_le_eq)
   516 
   517 lemma floor_le_eq_zero [simp]: "(floor x <= 0) = (x < 1)"
   518 by (simp add: floor_le_eq)
   519 
   520 lemma floor_le_eq_one [simp]: "(floor x <= 1) = (x < 2)"
   521 by (simp add: floor_le_eq)
   522 
   523 lemma floor_add [simp]: "floor (x + real a) = floor x + a"
   524   apply (subst order_eq_iff)
   525   apply (rule conjI)
   526   prefer 2
   527   apply (subgoal_tac "floor x + a < floor (x + real a) + 1")
   528   apply arith
   529   apply (subst real_of_int_less_iff [THEN sym])
   530   apply simp
   531   apply (subgoal_tac "x + real a < real(floor(x + real a)) + 1")
   532   apply (subgoal_tac "real (floor x) <= x")
   533   apply arith
   534   apply (rule real_of_int_floor_le)
   535   apply (rule real_of_int_floor_add_one_gt)
   536   apply (subgoal_tac "floor (x + real a) < floor x + a + 1")
   537   apply arith
   538   apply (subst real_of_int_less_iff [THEN sym])  
   539   apply simp
   540   apply (subgoal_tac "real(floor(x + real a)) <= x + real a") 
   541   apply (subgoal_tac "x < real(floor x) + 1")
   542   apply arith
   543   apply (rule real_of_int_floor_add_one_gt)
   544   apply (rule real_of_int_floor_le)
   545 done
   546 
   547 lemma floor_add_number_of [simp]: 
   548     "floor (x + number_of n) = floor x + number_of n"
   549   apply (subst floor_add [THEN sym])
   550   apply simp
   551 done
   552 
   553 lemma floor_add_one [simp]: "floor (x + 1) = floor x + 1"
   554   apply (subst floor_add [THEN sym])
   555   apply simp
   556 done
   557 
   558 lemma floor_subtract [simp]: "floor (x - real a) = floor x - a"
   559   apply (subst diff_minus)+
   560   apply (subst real_of_int_minus [THEN sym])
   561   apply (rule floor_add)
   562 done
   563 
   564 lemma floor_subtract_number_of [simp]: "floor (x - number_of n) = 
   565     floor x - number_of n"
   566   apply (subst floor_subtract [THEN sym])
   567   apply simp
   568 done
   569 
   570 lemma floor_subtract_one [simp]: "floor (x - 1) = floor x - 1"
   571   apply (subst floor_subtract [THEN sym])
   572   apply simp
   573 done
   574 
   575 lemma ceiling_zero [simp]: "ceiling 0 = 0"
   576 by (simp add: ceiling_def)
   577 
   578 lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"
   579 by (simp add: ceiling_def)
   580 
   581 lemma ceiling_real_of_nat_zero [simp]: "ceiling (real (0::nat)) = 0"
   582 by auto
   583 
   584 lemma ceiling_floor [simp]: "ceiling (real (floor r)) = floor r"
   585 by (simp add: ceiling_def)
   586 
   587 lemma floor_ceiling [simp]: "floor (real (ceiling r)) = ceiling r"
   588 by (simp add: ceiling_def)
   589 
   590 lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)"
   591 apply (simp add: ceiling_def)
   592 apply (subst le_minus_iff, simp)
   593 done
   594 
   595 lemma ceiling_mono: "x < y ==> ceiling x \<le> ceiling y"
   596 by (simp add: floor_mono ceiling_def)
   597 
   598 lemma ceiling_mono2: "x \<le> y ==> ceiling x \<le> ceiling y"
   599 by (simp add: floor_mono2 ceiling_def)
   600 
   601 lemma real_of_int_ceiling_cancel [simp]:
   602      "(real (ceiling x) = x) = (\<exists>n::int. x = real n)"
   603 apply (auto simp add: ceiling_def)
   604 apply (drule arg_cong [where f = uminus], auto)
   605 apply (rule_tac x = "-n" in exI, auto)
   606 done
   607 
   608 lemma ceiling_eq: "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1"
   609 apply (simp add: ceiling_def)
   610 apply (rule minus_equation_iff [THEN iffD1])
   611 apply (simp add: floor_eq [where n = "-(n+1)"])
   612 done
   613 
   614 lemma ceiling_eq2: "[| real n < x; x \<le> real n + 1 |] ==> ceiling x = n + 1"
   615 by (simp add: ceiling_def floor_eq2 [where n = "-(n+1)"])
   616 
   617 lemma ceiling_eq3: "[| real n - 1 < x; x \<le> real n  |] ==> ceiling x = n"
   618 by (simp add: ceiling_def floor_eq2 [where n = "-n"])
   619 
   620 lemma ceiling_real_of_int [simp]: "ceiling (real (n::int)) = n"
   621 by (simp add: ceiling_def)
   622 
   623 lemma ceiling_number_of_eq [simp]:
   624      "ceiling (number_of n :: real) = (number_of n)"
   625 apply (subst real_number_of [symmetric])
   626 apply (rule ceiling_real_of_int)
   627 done
   628 
   629 lemma ceiling_one [simp]: "ceiling 1 = 1"
   630   by (unfold ceiling_def, simp)
   631 
   632 lemma real_of_int_ceiling_diff_one_le [simp]: "real (ceiling r) - 1 \<le> r"
   633 apply (rule neg_le_iff_le [THEN iffD1])
   634 apply (simp add: ceiling_def diff_minus)
   635 done
   636 
   637 lemma real_of_int_ceiling_le_add_one [simp]: "real (ceiling r) \<le> r + 1"
   638 apply (insert real_of_int_ceiling_diff_one_le [of r])
   639 apply (simp del: real_of_int_ceiling_diff_one_le)
   640 done
   641 
   642 lemma ceiling_le: "x <= real a ==> ceiling x <= a"
   643   apply (unfold ceiling_def)
   644   apply (subgoal_tac "-a <= floor(- x)")
   645   apply simp
   646   apply (rule le_floor)
   647   apply simp
   648 done
   649 
   650 lemma ceiling_le_real: "ceiling x <= a ==> x <= real a"
   651   apply (unfold ceiling_def)
   652   apply (subgoal_tac "real(- a) <= - x")
   653   apply simp
   654   apply (rule real_le_floor)
   655   apply simp
   656 done
   657 
   658 lemma ceiling_le_eq: "(ceiling x <= a) = (x <= real a)"
   659   apply (rule iffI)
   660   apply (erule ceiling_le_real)
   661   apply (erule ceiling_le)
   662 done
   663 
   664 lemma ceiling_le_eq_number_of [simp]: 
   665     "(ceiling x <= number_of n) = (x <= number_of n)"
   666 by (simp add: ceiling_le_eq)
   667 
   668 lemma ceiling_le_zero_eq [simp]: "(ceiling x <= 0) = (x <= 0)" 
   669 by (simp add: ceiling_le_eq)
   670 
   671 lemma ceiling_le_eq_one [simp]: "(ceiling x <= 1) = (x <= 1)" 
   672 by (simp add: ceiling_le_eq)
   673 
   674 lemma less_ceiling_eq: "(a < ceiling x) = (real a < x)"
   675   apply (subst linorder_not_le [THEN sym])+
   676   apply simp
   677   apply (rule ceiling_le_eq)
   678 done
   679 
   680 lemma less_ceiling_eq_number_of [simp]: 
   681     "(number_of n < ceiling x) = (number_of n < x)"
   682 by (simp add: less_ceiling_eq)
   683 
   684 lemma less_ceiling_eq_zero [simp]: "(0 < ceiling x) = (0 < x)"
   685 by (simp add: less_ceiling_eq)
   686 
   687 lemma less_ceiling_eq_one [simp]: "(1 < ceiling x) = (1 < x)"
   688 by (simp add: less_ceiling_eq)
   689 
   690 lemma ceiling_less_eq: "(ceiling x < a) = (x <= real a - 1)"
   691   apply (insert ceiling_le_eq [of x "a - 1"])
   692   apply auto
   693 done
   694 
   695 lemma ceiling_less_eq_number_of [simp]: 
   696     "(ceiling x < number_of n) = (x <= number_of n - 1)"
   697 by (simp add: ceiling_less_eq)
   698 
   699 lemma ceiling_less_eq_zero [simp]: "(ceiling x < 0) = (x <= -1)"
   700 by (simp add: ceiling_less_eq)
   701 
   702 lemma ceiling_less_eq_one [simp]: "(ceiling x < 1) = (x <= 0)"
   703 by (simp add: ceiling_less_eq)
   704 
   705 lemma le_ceiling_eq: "(a <= ceiling x) = (real a - 1 < x)"
   706   apply (insert less_ceiling_eq [of "a - 1" x])
   707   apply auto
   708 done
   709 
   710 lemma le_ceiling_eq_number_of [simp]: 
   711     "(number_of n <= ceiling x) = (number_of n - 1 < x)"
   712 by (simp add: le_ceiling_eq)
   713 
   714 lemma le_ceiling_eq_zero [simp]: "(0 <= ceiling x) = (-1 < x)"
   715 by (simp add: le_ceiling_eq)
   716 
   717 lemma le_ceiling_eq_one [simp]: "(1 <= ceiling x) = (0 < x)"
   718 by (simp add: le_ceiling_eq)
   719 
   720 lemma ceiling_add [simp]: "ceiling (x + real a) = ceiling x + a"
   721   apply (unfold ceiling_def, simp)
   722   apply (subst real_of_int_minus [THEN sym])
   723   apply (subst floor_add)
   724   apply simp
   725 done
   726 
   727 lemma ceiling_add_number_of [simp]: "ceiling (x + number_of n) = 
   728     ceiling x + number_of n"
   729   apply (subst ceiling_add [THEN sym])
   730   apply simp
   731 done
   732 
   733 lemma ceiling_add_one [simp]: "ceiling (x + 1) = ceiling x + 1"
   734   apply (subst ceiling_add [THEN sym])
   735   apply simp
   736 done
   737 
   738 lemma ceiling_subtract [simp]: "ceiling (x - real a) = ceiling x - a"
   739   apply (subst diff_minus)+
   740   apply (subst real_of_int_minus [THEN sym])
   741   apply (rule ceiling_add)
   742 done
   743 
   744 lemma ceiling_subtract_number_of [simp]: "ceiling (x - number_of n) = 
   745     ceiling x - number_of n"
   746   apply (subst ceiling_subtract [THEN sym])
   747   apply simp
   748 done
   749 
   750 lemma ceiling_subtract_one [simp]: "ceiling (x - 1) = ceiling x - 1"
   751   apply (subst ceiling_subtract [THEN sym])
   752   apply simp
   753 done
   754 
   755 subsection {* Versions for the natural numbers *}
   756 
   757 constdefs
   758   natfloor :: "real => nat"
   759   "natfloor x == nat(floor x)"
   760   natceiling :: "real => nat"
   761   "natceiling x == nat(ceiling x)"
   762 
   763 lemma natfloor_zero [simp]: "natfloor 0 = 0"
   764   by (unfold natfloor_def, simp)
   765 
   766 lemma natfloor_one [simp]: "natfloor 1 = 1"
   767   by (unfold natfloor_def, simp)
   768 
   769 lemma zero_le_natfloor [simp]: "0 <= natfloor x"
   770   by (unfold natfloor_def, simp)
   771 
   772 lemma natfloor_number_of_eq [simp]: "natfloor (number_of n) = number_of n"
   773   by (unfold natfloor_def, simp)
   774 
   775 lemma natfloor_real_of_nat [simp]: "natfloor(real n) = n"
   776   by (unfold natfloor_def, simp)
   777 
   778 lemma real_natfloor_le: "0 <= x ==> real(natfloor x) <= x"
   779   by (unfold natfloor_def, simp)
   780 
   781 lemma natfloor_neg: "x <= 0 ==> natfloor x = 0"
   782   apply (unfold natfloor_def)
   783   apply (subgoal_tac "floor x <= floor 0")
   784   apply simp
   785   apply (erule floor_mono2)
   786 done
   787 
   788 lemma natfloor_mono: "x <= y ==> natfloor x <= natfloor y"
   789   apply (case_tac "0 <= x")
   790   apply (subst natfloor_def)+
   791   apply (subst nat_le_eq_zle)
   792   apply force
   793   apply (erule floor_mono2) 
   794   apply (subst natfloor_neg)
   795   apply simp
   796   apply simp
   797 done
   798 
   799 lemma le_natfloor: "real x <= a ==> x <= natfloor a"
   800   apply (unfold natfloor_def)
   801   apply (subst nat_int [THEN sym])
   802   apply (subst nat_le_eq_zle)
   803   apply simp
   804   apply (rule le_floor)
   805   apply simp
   806 done
   807 
   808 lemma le_natfloor_eq: "0 <= x ==> (a <= natfloor x) = (real a <= x)"
   809   apply (rule iffI)
   810   apply (rule order_trans)
   811   prefer 2
   812   apply (erule real_natfloor_le)
   813   apply (subst real_of_nat_le_iff)
   814   apply assumption
   815   apply (erule le_natfloor)
   816 done
   817 
   818 lemma le_natfloor_eq_number_of [simp]: 
   819     "~ neg((number_of n)::int) ==> 0 <= x ==>
   820       (number_of n <= natfloor x) = (number_of n <= x)"
   821   apply (subst le_natfloor_eq, assumption)
   822   apply simp
   823 done
   824 
   825 lemma le_natfloor_eq_one [simp]: "(1 <= natfloor x) = (1 <= x)"
   826   apply (case_tac "0 <= x")
   827   apply (subst le_natfloor_eq, assumption, simp)
   828   apply (rule iffI)
   829   apply (subgoal_tac "natfloor x <= natfloor 0") 
   830   apply simp
   831   apply (rule natfloor_mono)
   832   apply simp
   833   apply simp
   834 done
   835 
   836 lemma natfloor_eq: "real n <= x ==> x < real n + 1 ==> natfloor x = n"
   837   apply (unfold natfloor_def)
   838   apply (subst nat_int [THEN sym]);back;
   839   apply (subst eq_nat_nat_iff)
   840   apply simp
   841   apply simp
   842   apply (rule floor_eq2)
   843   apply auto
   844 done
   845 
   846 lemma real_natfloor_add_one_gt: "x < real(natfloor x) + 1"
   847   apply (case_tac "0 <= x")
   848   apply (unfold natfloor_def)
   849   apply simp
   850   apply simp_all
   851 done
   852 
   853 lemma real_natfloor_gt_diff_one: "x - 1 < real(natfloor x)"
   854   apply (simp add: compare_rls)
   855   apply (rule real_natfloor_add_one_gt)
   856 done
   857 
   858 lemma ge_natfloor_plus_one_imp_gt: "natfloor z + 1 <= n ==> z < real n"
   859   apply (subgoal_tac "z < real(natfloor z) + 1")
   860   apply arith
   861   apply (rule real_natfloor_add_one_gt)
   862 done
   863 
   864 lemma natfloor_add [simp]: "0 <= x ==> natfloor (x + real a) = natfloor x + a"
   865   apply (unfold natfloor_def)
   866   apply (subgoal_tac "real a = real (int a)")
   867   apply (erule ssubst)
   868   apply (simp add: nat_add_distrib)
   869   apply simp
   870 done
   871 
   872 lemma natfloor_add_number_of [simp]: 
   873     "~neg ((number_of n)::int) ==> 0 <= x ==> 
   874       natfloor (x + number_of n) = natfloor x + number_of n"
   875   apply (subst natfloor_add [THEN sym])
   876   apply simp_all
   877 done
   878 
   879 lemma natfloor_add_one: "0 <= x ==> natfloor(x + 1) = natfloor x + 1"
   880   apply (subst natfloor_add [THEN sym])
   881   apply assumption
   882   apply simp
   883 done
   884 
   885 lemma natfloor_subtract [simp]: "real a <= x ==> 
   886     natfloor(x - real a) = natfloor x - a"
   887   apply (unfold natfloor_def)
   888   apply (subgoal_tac "real a = real (int a)")
   889   apply (erule ssubst)
   890   apply simp
   891   apply (subst nat_diff_distrib)
   892   apply simp
   893   apply (rule le_floor)
   894   apply simp_all
   895 done
   896 
   897 lemma natceiling_zero [simp]: "natceiling 0 = 0"
   898   by (unfold natceiling_def, simp)
   899 
   900 lemma natceiling_one [simp]: "natceiling 1 = 1"
   901   by (unfold natceiling_def, simp)
   902 
   903 lemma zero_le_natceiling [simp]: "0 <= natceiling x"
   904   by (unfold natceiling_def, simp)
   905 
   906 lemma natceiling_number_of_eq [simp]: "natceiling (number_of n) = number_of n"
   907   by (unfold natceiling_def, simp)
   908 
   909 lemma natceiling_real_of_nat [simp]: "natceiling(real n) = n"
   910   by (unfold natceiling_def, simp)
   911 
   912 lemma real_natceiling_ge: "x <= real(natceiling x)"
   913   apply (unfold natceiling_def)
   914   apply (case_tac "x < 0")
   915   apply simp
   916   apply (subst real_nat_eq_real)
   917   apply (subgoal_tac "ceiling 0 <= ceiling x")
   918   apply simp
   919   apply (rule ceiling_mono2)
   920   apply simp
   921   apply simp
   922 done
   923 
   924 lemma natceiling_neg: "x <= 0 ==> natceiling x = 0"
   925   apply (unfold natceiling_def)
   926   apply simp
   927 done
   928 
   929 lemma natceiling_mono: "x <= y ==> natceiling x <= natceiling y"
   930   apply (case_tac "0 <= x")
   931   apply (subst natceiling_def)+
   932   apply (subst nat_le_eq_zle)
   933   apply (rule disjI2)
   934   apply (subgoal_tac "real (0::int) <= real(ceiling y)")
   935   apply simp
   936   apply (rule order_trans)
   937   apply simp
   938   apply (erule order_trans)
   939   apply simp
   940   apply (erule ceiling_mono2)
   941   apply (subst natceiling_neg)
   942   apply simp_all
   943 done
   944 
   945 lemma natceiling_le: "x <= real a ==> natceiling x <= a"
   946   apply (unfold natceiling_def)
   947   apply (case_tac "x < 0")
   948   apply simp
   949   apply (subst nat_int [THEN sym]);back;
   950   apply (subst nat_le_eq_zle)
   951   apply simp
   952   apply (rule ceiling_le)
   953   apply simp
   954 done
   955 
   956 lemma natceiling_le_eq: "0 <= x ==> (natceiling x <= a) = (x <= real a)"
   957   apply (rule iffI)
   958   apply (rule order_trans)
   959   apply (rule real_natceiling_ge)
   960   apply (subst real_of_nat_le_iff)
   961   apply assumption
   962   apply (erule natceiling_le)
   963 done
   964 
   965 lemma natceiling_le_eq_number_of [simp]: 
   966     "~ neg((number_of n)::int) ==> 0 <= x ==>
   967       (natceiling x <= number_of n) = (x <= number_of n)"
   968   apply (subst natceiling_le_eq, assumption)
   969   apply simp
   970 done
   971 
   972 lemma natceiling_le_eq_one: "(natceiling x <= 1) = (x <= 1)"
   973   apply (case_tac "0 <= x")
   974   apply (subst natceiling_le_eq)
   975   apply assumption
   976   apply simp
   977   apply (subst natceiling_neg)
   978   apply simp
   979   apply simp
   980 done
   981 
   982 lemma natceiling_eq: "real n < x ==> x <= real n + 1 ==> natceiling x = n + 1"
   983   apply (unfold natceiling_def)
   984   apply (subst nat_int [THEN sym]);back;
   985   apply (subgoal_tac "nat(int n) + 1 = nat(int n + 1)")
   986   apply (erule ssubst)
   987   apply (subst eq_nat_nat_iff)
   988   apply (subgoal_tac "ceiling 0 <= ceiling x")
   989   apply simp
   990   apply (rule ceiling_mono2)
   991   apply force
   992   apply force
   993   apply (rule ceiling_eq2)
   994   apply (simp, simp)
   995   apply (subst nat_add_distrib)
   996   apply auto
   997 done
   998 
   999 lemma natceiling_add [simp]: "0 <= x ==> 
  1000     natceiling (x + real a) = natceiling x + a"
  1001   apply (unfold natceiling_def)
  1002   apply (subgoal_tac "real a = real (int a)")
  1003   apply (erule ssubst)
  1004   apply simp
  1005   apply (subst nat_add_distrib)
  1006   apply (subgoal_tac "0 = ceiling 0")
  1007   apply (erule ssubst)
  1008   apply (erule ceiling_mono2)
  1009   apply simp_all
  1010 done
  1011 
  1012 lemma natceiling_add_number_of [simp]: 
  1013     "~ neg ((number_of n)::int) ==> 0 <= x ==> 
  1014       natceiling (x + number_of n) = natceiling x + number_of n"
  1015   apply (subst natceiling_add [THEN sym])
  1016   apply simp_all
  1017 done
  1018 
  1019 lemma natceiling_add_one: "0 <= x ==> natceiling(x + 1) = natceiling x + 1"
  1020   apply (subst natceiling_add [THEN sym])
  1021   apply assumption
  1022   apply simp
  1023 done
  1024 
  1025 lemma natceiling_subtract [simp]: "real a <= x ==> 
  1026     natceiling(x - real a) = natceiling x - a"
  1027   apply (unfold natceiling_def)
  1028   apply (subgoal_tac "real a = real (int a)")
  1029   apply (erule ssubst)
  1030   apply simp
  1031   apply (subst nat_diff_distrib)
  1032   apply simp
  1033   apply (rule order_trans)
  1034   prefer 2
  1035   apply (rule ceiling_mono2)
  1036   apply assumption
  1037   apply simp_all
  1038 done
  1039 
  1040 lemma natfloor_div_nat: "1 <= x ==> 0 < y ==> 
  1041   natfloor (x / real y) = natfloor x div y"
  1042 proof -
  1043   assume "1 <= (x::real)" and "0 < (y::nat)"
  1044   have "natfloor x = (natfloor x) div y * y + (natfloor x) mod y"
  1045     by simp
  1046   then have a: "real(natfloor x) = real ((natfloor x) div y) * real y + 
  1047     real((natfloor x) mod y)"
  1048     by (simp only: real_of_nat_add [THEN sym] real_of_nat_mult [THEN sym])
  1049   have "x = real(natfloor x) + (x - real(natfloor x))"
  1050     by simp
  1051   then have "x = real ((natfloor x) div y) * real y + 
  1052       real((natfloor x) mod y) + (x - real(natfloor x))"
  1053     by (simp add: a)
  1054   then have "x / real y = ... / real y"
  1055     by simp
  1056   also have "... = real((natfloor x) div y) + real((natfloor x) mod y) / 
  1057     real y + (x - real(natfloor x)) / real y"
  1058     by (auto simp add: ring_distrib ring_eq_simps add_divide_distrib
  1059       diff_divide_distrib prems)
  1060   finally have "natfloor (x / real y) = natfloor(...)" by simp
  1061   also have "... = natfloor(real((natfloor x) mod y) / 
  1062     real y + (x - real(natfloor x)) / real y + real((natfloor x) div y))"
  1063     by (simp add: add_ac)
  1064   also have "... = natfloor(real((natfloor x) mod y) / 
  1065     real y + (x - real(natfloor x)) / real y) + (natfloor x) div y"
  1066     apply (rule natfloor_add)
  1067     apply (rule add_nonneg_nonneg)
  1068     apply (rule divide_nonneg_pos)
  1069     apply simp
  1070     apply (simp add: prems)
  1071     apply (rule divide_nonneg_pos)
  1072     apply (simp add: compare_rls)
  1073     apply (rule real_natfloor_le)
  1074     apply (insert prems, auto)
  1075     done
  1076   also have "natfloor(real((natfloor x) mod y) / 
  1077     real y + (x - real(natfloor x)) / real y) = 0"
  1078     apply (rule natfloor_eq)
  1079     apply simp
  1080     apply (rule add_nonneg_nonneg)
  1081     apply (rule divide_nonneg_pos)
  1082     apply force
  1083     apply (force simp add: prems)
  1084     apply (rule divide_nonneg_pos)
  1085     apply (simp add: compare_rls)
  1086     apply (rule real_natfloor_le)
  1087     apply (auto simp add: prems)
  1088     apply (insert prems, arith)
  1089     apply (simp add: add_divide_distrib [THEN sym])
  1090     apply (subgoal_tac "real y = real y - 1 + 1")
  1091     apply (erule ssubst)
  1092     apply (rule add_le_less_mono)
  1093     apply (simp add: compare_rls)
  1094     apply (subgoal_tac "real(natfloor x mod y) + 1 = 
  1095       real(natfloor x mod y + 1)")
  1096     apply (erule ssubst)
  1097     apply (subst real_of_nat_le_iff)
  1098     apply (subgoal_tac "natfloor x mod y < y")
  1099     apply arith
  1100     apply (rule mod_less_divisor)
  1101     apply assumption
  1102     apply auto
  1103     apply (simp add: compare_rls)
  1104     apply (subst add_commute)
  1105     apply (rule real_natfloor_add_one_gt)
  1106     done
  1107   finally show ?thesis
  1108     by simp
  1109 qed
  1110 
  1111 ML
  1112 {*
  1113 val number_of_less_real_of_int_iff = thm "number_of_less_real_of_int_iff";
  1114 val number_of_less_real_of_int_iff2 = thm "number_of_less_real_of_int_iff2";
  1115 val number_of_le_real_of_int_iff = thm "number_of_le_real_of_int_iff";
  1116 val number_of_le_real_of_int_iff2 = thm "number_of_le_real_of_int_iff2";
  1117 val floor_zero = thm "floor_zero";
  1118 val floor_real_of_nat_zero = thm "floor_real_of_nat_zero";
  1119 val floor_real_of_nat = thm "floor_real_of_nat";
  1120 val floor_minus_real_of_nat = thm "floor_minus_real_of_nat";
  1121 val floor_real_of_int = thm "floor_real_of_int";
  1122 val floor_minus_real_of_int = thm "floor_minus_real_of_int";
  1123 val reals_Archimedean6 = thm "reals_Archimedean6";
  1124 val reals_Archimedean6a = thm "reals_Archimedean6a";
  1125 val reals_Archimedean_6b_int = thm "reals_Archimedean_6b_int";
  1126 val reals_Archimedean_6c_int = thm "reals_Archimedean_6c_int";
  1127 val real_lb_ub_int = thm "real_lb_ub_int";
  1128 val lemma_floor = thm "lemma_floor";
  1129 val real_of_int_floor_le = thm "real_of_int_floor_le";
  1130 (*val floor_le = thm "floor_le";
  1131 val floor_le2 = thm "floor_le2";
  1132 *)
  1133 val lemma_floor2 = thm "lemma_floor2";
  1134 val real_of_int_floor_cancel = thm "real_of_int_floor_cancel";
  1135 val floor_eq = thm "floor_eq";
  1136 val floor_eq2 = thm "floor_eq2";
  1137 val floor_eq3 = thm "floor_eq3";
  1138 val floor_eq4 = thm "floor_eq4";
  1139 val floor_number_of_eq = thm "floor_number_of_eq";
  1140 val real_of_int_floor_ge_diff_one = thm "real_of_int_floor_ge_diff_one";
  1141 val real_of_int_floor_add_one_ge = thm "real_of_int_floor_add_one_ge";
  1142 val ceiling_zero = thm "ceiling_zero";
  1143 val ceiling_real_of_nat = thm "ceiling_real_of_nat";
  1144 val ceiling_real_of_nat_zero = thm "ceiling_real_of_nat_zero";
  1145 val ceiling_floor = thm "ceiling_floor";
  1146 val floor_ceiling = thm "floor_ceiling";
  1147 val real_of_int_ceiling_ge = thm "real_of_int_ceiling_ge";
  1148 (*
  1149 val ceiling_le = thm "ceiling_le";
  1150 val ceiling_le2 = thm "ceiling_le2";
  1151 *)
  1152 val real_of_int_ceiling_cancel = thm "real_of_int_ceiling_cancel";
  1153 val ceiling_eq = thm "ceiling_eq";
  1154 val ceiling_eq2 = thm "ceiling_eq2";
  1155 val ceiling_eq3 = thm "ceiling_eq3";
  1156 val ceiling_real_of_int = thm "ceiling_real_of_int";
  1157 val ceiling_number_of_eq = thm "ceiling_number_of_eq";
  1158 val real_of_int_ceiling_diff_one_le = thm "real_of_int_ceiling_diff_one_le";
  1159 val real_of_int_ceiling_le_add_one = thm "real_of_int_ceiling_le_add_one";
  1160 *}
  1161 
  1162 
  1163 end