src/HOL/RComplete.thy
 author bulwahn Sat Feb 25 09:07:41 2012 +0100 (2012-02-25) changeset 46671 3a40ea076230 parent 45966 03ce2b2a29a2 child 47108 2a1953f0d20d permissions -rw-r--r--
removing unnecessary assumptions in RComplete;
simplifying proof in Probability
     1 (*  Title:      HOL/RComplete.thy

     2     Author:     Jacques D. Fleuriot, University of Edinburgh

     3     Author:     Larry Paulson, University of Cambridge

     4     Author:     Jeremy Avigad, Carnegie Mellon University

     5     Author:     Florian Zuleger, Johannes Hoelzl, and Simon Funke, TU Muenchen

     6 *)

     7

     8 header {* Completeness of the Reals; Floor and Ceiling Functions *}

     9

    10 theory RComplete

    11 imports Lubs RealDef

    12 begin

    13

    14 lemma real_sum_of_halves: "x/2 + x/2 = (x::real)"

    15   by simp

    16

    17 lemma abs_diff_less_iff:

    18   "(\<bar>x - a\<bar> < (r::'a::linordered_idom)) = (a - r < x \<and> x < a + r)"

    19   by auto

    20

    21 subsection {* Completeness of Positive Reals *}

    22

    23 text {*

    24   Supremum property for the set of positive reals

    25

    26   Let @{text "P"} be a non-empty set of positive reals, with an upper

    27   bound @{text "y"}.  Then @{text "P"} has a least upper bound

    28   (written @{text "S"}).

    29

    30   FIXME: Can the premise be weakened to @{text "\<forall>x \<in> P. x\<le> y"}?

    31 *}

    32

    33 text {* Only used in HOL/Import/HOL4Compat.thy; delete? *}

    34

    35 lemma posreal_complete:

    36   fixes P :: "real set"

    37   assumes not_empty_P: "\<exists>x. x \<in> P"

    38     and upper_bound_Ex: "\<exists>y. \<forall>x \<in> P. x<y"

    39   shows "\<exists>S. \<forall>y. (\<exists>x \<in> P. y < x) = (y < S)"

    40 proof -

    41   from upper_bound_Ex have "\<exists>z. \<forall>x\<in>P. x \<le> z"

    42     by (auto intro: less_imp_le)

    43   from complete_real [OF not_empty_P this] obtain S

    44   where S1: "\<And>x. x \<in> P \<Longrightarrow> x \<le> S" and S2: "\<And>z. \<forall>x\<in>P. x \<le> z \<Longrightarrow> S \<le> z" by fast

    45   have "\<forall>y. (\<exists>x \<in> P. y < x) = (y < S)"

    46   proof

    47     fix y show "(\<exists>x\<in>P. y < x) = (y < S)"

    48       apply (cases "\<exists>x\<in>P. y < x", simp_all)

    49       apply (clarify, drule S1, simp)

    50       apply (simp add: not_less S2)

    51       done

    52   qed

    53   thus ?thesis ..

    54 qed

    55

    56 text {*

    57   \medskip Completeness properties using @{text "isUb"}, @{text "isLub"} etc.

    58 *}

    59

    60 lemma real_isLub_unique: "[| isLub R S x; isLub R S y |] ==> x = (y::real)"

    61   apply (frule isLub_isUb)

    62   apply (frule_tac x = y in isLub_isUb)

    63   apply (blast intro!: order_antisym dest!: isLub_le_isUb)

    64   done

    65

    66

    67 text {*

    68   \medskip reals Completeness (again!)

    69 *}

    70

    71 lemma reals_complete:

    72   assumes notempty_S: "\<exists>X. X \<in> S"

    73     and exists_Ub: "\<exists>Y. isUb (UNIV::real set) S Y"

    74   shows "\<exists>t. isLub (UNIV :: real set) S t"

    75 proof -

    76   from assms have "\<exists>X. X \<in> S" and "\<exists>Y. \<forall>x\<in>S. x \<le> Y"

    77     unfolding isUb_def setle_def by simp_all

    78   from complete_real [OF this] show ?thesis

    79     by (simp add: isLub_def leastP_def isUb_def setle_def setge_def)

    80 qed

    81

    82

    83 subsection {* The Archimedean Property of the Reals *}

    84

    85 theorem reals_Archimedean:

    86   assumes x_pos: "0 < x"

    87   shows "\<exists>n. inverse (real (Suc n)) < x"

    88   unfolding real_of_nat_def using x_pos

    89   by (rule ex_inverse_of_nat_Suc_less)

    90

    91 lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"

    92   unfolding real_of_nat_def by (rule ex_less_of_nat)

    93

    94 lemma reals_Archimedean3:

    95   assumes x_greater_zero: "0 < x"

    96   shows "\<forall>(y::real). \<exists>(n::nat). y < real n * x"

    97   unfolding real_of_nat_def using 0 < x

    98   by (auto intro: ex_less_of_nat_mult)

    99

   100

   101 subsection{*Density of the Rational Reals in the Reals*}

   102

   103 text{* This density proof is due to Stefan Richter and was ported by TN.  The

   104 original source is \emph{Real Analysis} by H.L. Royden.

   105 It employs the Archimedean property of the reals. *}

   106

   107 lemma Rats_dense_in_real:

   108   fixes x :: real

   109   assumes "x < y" shows "\<exists>r\<in>\<rat>. x < r \<and> r < y"

   110 proof -

   111   from x<y have "0 < y-x" by simp

   112   with reals_Archimedean obtain q::nat

   113     where q: "inverse (real q) < y-x" and "0 < q" by auto

   114   def p \<equiv> "ceiling (y * real q) - 1"

   115   def r \<equiv> "of_int p / real q"

   116   from q have "x < y - inverse (real q)" by simp

   117   also have "y - inverse (real q) \<le> r"

   118     unfolding r_def p_def

   119     by (simp add: le_divide_eq left_diff_distrib le_of_int_ceiling 0 < q)

   120   finally have "x < r" .

   121   moreover have "r < y"

   122     unfolding r_def p_def

   123     by (simp add: divide_less_eq diff_less_eq 0 < q

   124       less_ceiling_iff [symmetric])

   125   moreover from r_def have "r \<in> \<rat>" by simp

   126   ultimately show ?thesis by fast

   127 qed

   128

   129

   130 subsection{*Floor and Ceiling Functions from the Reals to the Integers*}

   131

   132 lemma number_of_less_real_of_int_iff [simp]:

   133      "((number_of n) < real (m::int)) = (number_of n < m)"

   134 apply auto

   135 apply (rule real_of_int_less_iff [THEN iffD1])

   136 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)

   137 done

   138

   139 lemma number_of_less_real_of_int_iff2 [simp]:

   140      "(real (m::int) < (number_of n)) = (m < number_of n)"

   141 apply auto

   142 apply (rule real_of_int_less_iff [THEN iffD1])

   143 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)

   144 done

   145

   146 lemma number_of_le_real_of_int_iff [simp]:

   147      "((number_of n) \<le> real (m::int)) = (number_of n \<le> m)"

   148 by (simp add: linorder_not_less [symmetric])

   149

   150 lemma number_of_le_real_of_int_iff2 [simp]:

   151      "(real (m::int) \<le> (number_of n)) = (m \<le> number_of n)"

   152 by (simp add: linorder_not_less [symmetric])

   153

   154 lemma floor_real_of_nat [simp]: "floor (real (n::nat)) = int n"

   155 unfolding real_of_nat_def by simp

   156

   157 lemma floor_minus_real_of_nat [simp]: "floor (- real (n::nat)) = - int n"

   158 unfolding real_of_nat_def by (simp add: floor_minus)

   159

   160 lemma floor_real_of_int [simp]: "floor (real (n::int)) = n"

   161 unfolding real_of_int_def by simp

   162

   163 lemma floor_minus_real_of_int [simp]: "floor (- real (n::int)) = - n"

   164 unfolding real_of_int_def by (simp add: floor_minus)

   165

   166 lemma real_lb_ub_int: " \<exists>n::int. real n \<le> r & r < real (n+1)"

   167 unfolding real_of_int_def by (rule floor_exists)

   168

   169 lemma lemma_floor:

   170   assumes a1: "real m \<le> r" and a2: "r < real n + 1"

   171   shows "m \<le> (n::int)"

   172 proof -

   173   have "real m < real n + 1" using a1 a2 by (rule order_le_less_trans)

   174   also have "... = real (n + 1)" by simp

   175   finally have "m < n + 1" by (simp only: real_of_int_less_iff)

   176   thus ?thesis by arith

   177 qed

   178

   179 lemma real_of_int_floor_le [simp]: "real (floor r) \<le> r"

   180 unfolding real_of_int_def by (rule of_int_floor_le)

   181

   182 lemma lemma_floor2: "real n < real (x::int) + 1 ==> n \<le> x"

   183 by (auto intro: lemma_floor)

   184

   185 lemma real_of_int_floor_cancel [simp]:

   186     "(real (floor x) = x) = (\<exists>n::int. x = real n)"

   187   using floor_real_of_int by metis

   188

   189 lemma floor_eq: "[| real n < x; x < real n + 1 |] ==> floor x = n"

   190   unfolding real_of_int_def using floor_unique [of n x] by simp

   191

   192 lemma floor_eq2: "[| real n \<le> x; x < real n + 1 |] ==> floor x = n"

   193   unfolding real_of_int_def by (rule floor_unique)

   194

   195 lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"

   196 apply (rule inj_int [THEN injD])

   197 apply (simp add: real_of_nat_Suc)

   198 apply (simp add: real_of_nat_Suc floor_eq floor_eq [where n = "int n"])

   199 done

   200

   201 lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"

   202 apply (drule order_le_imp_less_or_eq)

   203 apply (auto intro: floor_eq3)

   204 done

   205

   206 lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real(floor r)"

   207   unfolding real_of_int_def using floor_correct [of r] by simp

   208

   209 lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real(floor r)"

   210   unfolding real_of_int_def using floor_correct [of r] by simp

   211

   212 lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real(floor r) + 1"

   213   unfolding real_of_int_def using floor_correct [of r] by simp

   214

   215 lemma real_of_int_floor_add_one_gt [simp]: "r < real(floor r) + 1"

   216   unfolding real_of_int_def using floor_correct [of r] by simp

   217

   218 lemma le_floor: "real a <= x ==> a <= floor x"

   219   unfolding real_of_int_def by (simp add: le_floor_iff)

   220

   221 lemma real_le_floor: "a <= floor x ==> real a <= x"

   222   unfolding real_of_int_def by (simp add: le_floor_iff)

   223

   224 lemma le_floor_eq: "(a <= floor x) = (real a <= x)"

   225   unfolding real_of_int_def by (rule le_floor_iff)

   226

   227 lemma floor_less_eq: "(floor x < a) = (x < real a)"

   228   unfolding real_of_int_def by (rule floor_less_iff)

   229

   230 lemma less_floor_eq: "(a < floor x) = (real a + 1 <= x)"

   231   unfolding real_of_int_def by (rule less_floor_iff)

   232

   233 lemma floor_le_eq: "(floor x <= a) = (x < real a + 1)"

   234   unfolding real_of_int_def by (rule floor_le_iff)

   235

   236 lemma floor_add [simp]: "floor (x + real a) = floor x + a"

   237   unfolding real_of_int_def by (rule floor_add_of_int)

   238

   239 lemma floor_subtract [simp]: "floor (x - real a) = floor x - a"

   240   unfolding real_of_int_def by (rule floor_diff_of_int)

   241

   242 lemma le_mult_floor:

   243   assumes "0 \<le> (a :: real)" and "0 \<le> b"

   244   shows "floor a * floor b \<le> floor (a * b)"

   245 proof -

   246   have "real (floor a) \<le> a"

   247     and "real (floor b) \<le> b" by auto

   248   hence "real (floor a * floor b) \<le> a * b"

   249     using assms by (auto intro!: mult_mono)

   250   also have "a * b < real (floor (a * b) + 1)" by auto

   251   finally show ?thesis unfolding real_of_int_less_iff by simp

   252 qed

   253

   254 lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"

   255   unfolding real_of_nat_def by simp

   256

   257 lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)"

   258   unfolding real_of_int_def by (rule le_of_int_ceiling)

   259

   260 lemma ceiling_real_of_int [simp]: "ceiling (real (n::int)) = n"

   261   unfolding real_of_int_def by simp

   262

   263 lemma real_of_int_ceiling_cancel [simp]:

   264      "(real (ceiling x) = x) = (\<exists>n::int. x = real n)"

   265   using ceiling_real_of_int by metis

   266

   267 lemma ceiling_eq: "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1"

   268   unfolding real_of_int_def using ceiling_unique [of "n + 1" x] by simp

   269

   270 lemma ceiling_eq2: "[| real n < x; x \<le> real n + 1 |] ==> ceiling x = n + 1"

   271   unfolding real_of_int_def using ceiling_unique [of "n + 1" x] by simp

   272

   273 lemma ceiling_eq3: "[| real n - 1 < x; x \<le> real n  |] ==> ceiling x = n"

   274   unfolding real_of_int_def using ceiling_unique [of n x] by simp

   275

   276 lemma real_of_int_ceiling_diff_one_le [simp]: "real (ceiling r) - 1 \<le> r"

   277   unfolding real_of_int_def using ceiling_correct [of r] by simp

   278

   279 lemma real_of_int_ceiling_le_add_one [simp]: "real (ceiling r) \<le> r + 1"

   280   unfolding real_of_int_def using ceiling_correct [of r] by simp

   281

   282 lemma ceiling_le: "x <= real a ==> ceiling x <= a"

   283   unfolding real_of_int_def by (simp add: ceiling_le_iff)

   284

   285 lemma ceiling_le_real: "ceiling x <= a ==> x <= real a"

   286   unfolding real_of_int_def by (simp add: ceiling_le_iff)

   287

   288 lemma ceiling_le_eq: "(ceiling x <= a) = (x <= real a)"

   289   unfolding real_of_int_def by (rule ceiling_le_iff)

   290

   291 lemma less_ceiling_eq: "(a < ceiling x) = (real a < x)"

   292   unfolding real_of_int_def by (rule less_ceiling_iff)

   293

   294 lemma ceiling_less_eq: "(ceiling x < a) = (x <= real a - 1)"

   295   unfolding real_of_int_def by (rule ceiling_less_iff)

   296

   297 lemma le_ceiling_eq: "(a <= ceiling x) = (real a - 1 < x)"

   298   unfolding real_of_int_def by (rule le_ceiling_iff)

   299

   300 lemma ceiling_add [simp]: "ceiling (x + real a) = ceiling x + a"

   301   unfolding real_of_int_def by (rule ceiling_add_of_int)

   302

   303 lemma ceiling_subtract [simp]: "ceiling (x - real a) = ceiling x - a"

   304   unfolding real_of_int_def by (rule ceiling_diff_of_int)

   305

   306

   307 subsection {* Versions for the natural numbers *}

   308

   309 definition

   310   natfloor :: "real => nat" where

   311   "natfloor x = nat(floor x)"

   312

   313 definition

   314   natceiling :: "real => nat" where

   315   "natceiling x = nat(ceiling x)"

   316

   317 lemma natfloor_zero [simp]: "natfloor 0 = 0"

   318   by (unfold natfloor_def, simp)

   319

   320 lemma natfloor_one [simp]: "natfloor 1 = 1"

   321   by (unfold natfloor_def, simp)

   322

   323 lemma zero_le_natfloor [simp]: "0 <= natfloor x"

   324   by (unfold natfloor_def, simp)

   325

   326 lemma natfloor_number_of_eq [simp]: "natfloor (number_of n) = number_of n"

   327   by (unfold natfloor_def, simp)

   328

   329 lemma natfloor_real_of_nat [simp]: "natfloor(real n) = n"

   330   by (unfold natfloor_def, simp)

   331

   332 lemma real_natfloor_le: "0 <= x ==> real(natfloor x) <= x"

   333   by (unfold natfloor_def, simp)

   334

   335 lemma natfloor_neg: "x <= 0 ==> natfloor x = 0"

   336   unfolding natfloor_def by simp

   337

   338 lemma natfloor_mono: "x <= y ==> natfloor x <= natfloor y"

   339   unfolding natfloor_def by (intro nat_mono floor_mono)

   340

   341 lemma le_natfloor: "real x <= a ==> x <= natfloor a"

   342   apply (unfold natfloor_def)

   343   apply (subst nat_int [THEN sym])

   344   apply (rule nat_mono)

   345   apply (rule le_floor)

   346   apply simp

   347 done

   348

   349 lemma natfloor_less_iff: "0 \<le> x \<Longrightarrow> natfloor x < n \<longleftrightarrow> x < real n"

   350   unfolding natfloor_def real_of_nat_def

   351   by (simp add: nat_less_iff floor_less_iff)

   352

   353 lemma less_natfloor:

   354   assumes "0 \<le> x" and "x < real (n :: nat)"

   355   shows "natfloor x < n"

   356   using assms by (simp add: natfloor_less_iff)

   357

   358 lemma le_natfloor_eq: "0 <= x ==> (a <= natfloor x) = (real a <= x)"

   359   apply (rule iffI)

   360   apply (rule order_trans)

   361   prefer 2

   362   apply (erule real_natfloor_le)

   363   apply (subst real_of_nat_le_iff)

   364   apply assumption

   365   apply (erule le_natfloor)

   366 done

   367

   368 lemma le_natfloor_eq_number_of [simp]:

   369     "~ neg((number_of n)::int) ==> 0 <= x ==>

   370       (number_of n <= natfloor x) = (number_of n <= x)"

   371   apply (subst le_natfloor_eq, assumption)

   372   apply simp

   373 done

   374

   375 lemma le_natfloor_eq_one [simp]: "(1 <= natfloor x) = (1 <= x)"

   376   apply (case_tac "0 <= x")

   377   apply (subst le_natfloor_eq, assumption, simp)

   378   apply (rule iffI)

   379   apply (subgoal_tac "natfloor x <= natfloor 0")

   380   apply simp

   381   apply (rule natfloor_mono)

   382   apply simp

   383   apply simp

   384 done

   385

   386 lemma natfloor_eq: "real n <= x ==> x < real n + 1 ==> natfloor x = n"

   387   unfolding natfloor_def by (simp add: floor_eq2 [where n="int n"])

   388

   389 lemma real_natfloor_add_one_gt: "x < real(natfloor x) + 1"

   390   apply (case_tac "0 <= x")

   391   apply (unfold natfloor_def)

   392   apply simp

   393   apply simp_all

   394 done

   395

   396 lemma real_natfloor_gt_diff_one: "x - 1 < real(natfloor x)"

   397 using real_natfloor_add_one_gt by (simp add: algebra_simps)

   398

   399 lemma ge_natfloor_plus_one_imp_gt: "natfloor z + 1 <= n ==> z < real n"

   400   apply (subgoal_tac "z < real(natfloor z) + 1")

   401   apply arith

   402   apply (rule real_natfloor_add_one_gt)

   403 done

   404

   405 lemma natfloor_add [simp]: "0 <= x ==> natfloor (x + real a) = natfloor x + a"

   406   unfolding natfloor_def

   407   unfolding real_of_int_of_nat_eq [symmetric] floor_add

   408   by (simp add: nat_add_distrib)

   409

   410 lemma natfloor_add_number_of [simp]:

   411     "~neg ((number_of n)::int) ==> 0 <= x ==>

   412       natfloor (x + number_of n) = natfloor x + number_of n"

   413   by (simp add: natfloor_add [symmetric])

   414

   415 lemma natfloor_add_one: "0 <= x ==> natfloor(x + 1) = natfloor x + 1"

   416   by (simp add: natfloor_add [symmetric] del: One_nat_def)

   417

   418 lemma natfloor_subtract [simp]:

   419     "natfloor(x - real a) = natfloor x - a"

   420   unfolding natfloor_def real_of_int_of_nat_eq [symmetric] floor_subtract

   421   by simp

   422

   423 lemma natfloor_div_nat:

   424   assumes "1 <= x" and "y > 0"

   425   shows "natfloor (x / real y) = natfloor x div y"

   426 proof (rule natfloor_eq)

   427   have "(natfloor x) div y * y \<le> natfloor x"

   428     by (rule add_leD1 [where k="natfloor x mod y"], simp)

   429   thus "real (natfloor x div y) \<le> x / real y"

   430     using assms by (simp add: le_divide_eq le_natfloor_eq)

   431   have "natfloor x < (natfloor x) div y * y + y"

   432     apply (subst mod_div_equality [symmetric])

   433     apply (rule add_strict_left_mono)

   434     apply (rule mod_less_divisor)

   435     apply fact

   436     done

   437   thus "x / real y < real (natfloor x div y) + 1"

   438     using assms

   439     by (simp add: divide_less_eq natfloor_less_iff left_distrib)

   440 qed

   441

   442 lemma le_mult_natfloor:

   443   shows "natfloor a * natfloor b \<le> natfloor (a * b)"

   444   by (cases "0 <= a & 0 <= b")

   445     (auto simp add: le_natfloor_eq mult_nonneg_nonneg mult_mono' real_natfloor_le natfloor_neg)

   446

   447 lemma natceiling_zero [simp]: "natceiling 0 = 0"

   448   by (unfold natceiling_def, simp)

   449

   450 lemma natceiling_one [simp]: "natceiling 1 = 1"

   451   by (unfold natceiling_def, simp)

   452

   453 lemma zero_le_natceiling [simp]: "0 <= natceiling x"

   454   by (unfold natceiling_def, simp)

   455

   456 lemma natceiling_number_of_eq [simp]: "natceiling (number_of n) = number_of n"

   457   by (unfold natceiling_def, simp)

   458

   459 lemma natceiling_real_of_nat [simp]: "natceiling(real n) = n"

   460   by (unfold natceiling_def, simp)

   461

   462 lemma real_natceiling_ge: "x <= real(natceiling x)"

   463   unfolding natceiling_def by (cases "x < 0", simp_all)

   464

   465 lemma natceiling_neg: "x <= 0 ==> natceiling x = 0"

   466   unfolding natceiling_def by simp

   467

   468 lemma natceiling_mono: "x <= y ==> natceiling x <= natceiling y"

   469   unfolding natceiling_def by (intro nat_mono ceiling_mono)

   470

   471 lemma natceiling_le: "x <= real a ==> natceiling x <= a"

   472   unfolding natceiling_def real_of_nat_def

   473   by (simp add: nat_le_iff ceiling_le_iff)

   474

   475 lemma natceiling_le_eq: "(natceiling x <= a) = (x <= real a)"

   476   unfolding natceiling_def real_of_nat_def

   477   by (simp add: nat_le_iff ceiling_le_iff)

   478

   479 lemma natceiling_le_eq_number_of [simp]:

   480     "~ neg((number_of n)::int) ==>

   481       (natceiling x <= number_of n) = (x <= number_of n)"

   482   by (simp add: natceiling_le_eq)

   483

   484 lemma natceiling_le_eq_one: "(natceiling x <= 1) = (x <= 1)"

   485   unfolding natceiling_def

   486   by (simp add: nat_le_iff ceiling_le_iff)

   487

   488 lemma natceiling_eq: "real n < x ==> x <= real n + 1 ==> natceiling x = n + 1"

   489   unfolding natceiling_def

   490   by (simp add: ceiling_eq2 [where n="int n"])

   491

   492 lemma natceiling_add [simp]: "0 <= x ==>

   493     natceiling (x + real a) = natceiling x + a"

   494   unfolding natceiling_def

   495   unfolding real_of_int_of_nat_eq [symmetric] ceiling_add

   496   by (simp add: nat_add_distrib)

   497

   498 lemma natceiling_add_number_of [simp]:

   499     "~ neg ((number_of n)::int) ==> 0 <= x ==>

   500       natceiling (x + number_of n) = natceiling x + number_of n"

   501   by (simp add: natceiling_add [symmetric])

   502

   503 lemma natceiling_add_one: "0 <= x ==> natceiling(x + 1) = natceiling x + 1"

   504   by (simp add: natceiling_add [symmetric] del: One_nat_def)

   505

   506 lemma natceiling_subtract [simp]: "natceiling(x - real a) = natceiling x - a"

   507   unfolding natceiling_def real_of_int_of_nat_eq [symmetric] ceiling_subtract

   508   by simp

   509

   510 subsection {* Exponentiation with floor *}

   511

   512 lemma floor_power:

   513   assumes "x = real (floor x)"

   514   shows "floor (x ^ n) = floor x ^ n"

   515 proof -

   516   have *: "x ^ n = real (floor x ^ n)"

   517     using assms by (induct n arbitrary: x) simp_all

   518   show ?thesis unfolding real_of_int_inject[symmetric]

   519     unfolding * floor_real_of_int ..

   520 qed

   521

   522 lemma natfloor_power:

   523   assumes "x = real (natfloor x)"

   524   shows "natfloor (x ^ n) = natfloor x ^ n"

   525 proof -

   526   from assms have "0 \<le> floor x" by auto

   527   note assms[unfolded natfloor_def real_nat_eq_real[OF 0 \<le> floor x]]

   528   from floor_power[OF this]

   529   show ?thesis unfolding natfloor_def nat_power_eq[OF 0 \<le> floor x, symmetric]

   530     by simp

   531 qed

   532

   533 end