src/HOL/RComplete.thy
 author huffman Sun Mar 25 20:15:39 2012 +0200 (2012-03-25) changeset 47108 2a1953f0d20d parent 46671 3a40ea076230 child 47596 c031e65c8ddc permissions -rw-r--r--
merged fork with new numeral representation (see NEWS)
     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 (* FIXME: theorems for negative numerals *)

   133 lemma numeral_less_real_of_int_iff [simp]:

   134      "((numeral n) < real (m::int)) = (numeral n < m)"

   135 apply auto

   136 apply (rule real_of_int_less_iff [THEN iffD1])

   137 apply (drule_tac  real_of_int_less_iff [THEN iffD2], auto)

   138 done

   139

   140 lemma numeral_less_real_of_int_iff2 [simp]:

   141      "(real (m::int) < (numeral n)) = (m < numeral n)"

   142 apply auto

   143 apply (rule real_of_int_less_iff [THEN iffD1])

   144 apply (drule_tac  real_of_int_less_iff [THEN iffD2], auto)

   145 done

   146

   147 lemma numeral_le_real_of_int_iff [simp]:

   148      "((numeral n) \<le> real (m::int)) = (numeral n \<le> m)"

   149 by (simp add: linorder_not_less [symmetric])

   150

   151 lemma numeral_le_real_of_int_iff2 [simp]:

   152      "(real (m::int) \<le> (numeral n)) = (m \<le> numeral n)"

   153 by (simp add: linorder_not_less [symmetric])

   154

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

   156 unfolding real_of_nat_def by simp

   157

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

   159 unfolding real_of_nat_def by (simp add: floor_minus)

   160

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

   162 unfolding real_of_int_def by simp

   163

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

   165 unfolding real_of_int_def by (simp add: floor_minus)

   166

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

   168 unfolding real_of_int_def by (rule floor_exists)

   169

   170 lemma lemma_floor:

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

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

   173 proof -

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

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

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

   177   thus ?thesis by arith

   178 qed

   179

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

   181 unfolding real_of_int_def by (rule of_int_floor_le)

   182

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

   184 by (auto intro: lemma_floor)

   185

   186 lemma real_of_int_floor_cancel [simp]:

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

   188   using floor_real_of_int by metis

   189

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

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

   192

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

   194   unfolding real_of_int_def by (rule floor_unique)

   195

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

   197 apply (rule inj_int [THEN injD])

   198 apply (simp add: real_of_nat_Suc)

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

   200 done

   201

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

   203 apply (drule order_le_imp_less_or_eq)

   204 apply (auto intro: floor_eq3)

   205 done

   206

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

   208   unfolding real_of_int_def using floor_correct [of r] by simp

   209

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

   211   unfolding real_of_int_def using floor_correct [of r] by simp

   212

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

   214   unfolding real_of_int_def using floor_correct [of r] by simp

   215

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

   217   unfolding real_of_int_def using floor_correct [of r] by simp

   218

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

   220   unfolding real_of_int_def by (simp add: le_floor_iff)

   221

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

   223   unfolding real_of_int_def by (simp add: le_floor_iff)

   224

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

   226   unfolding real_of_int_def by (rule le_floor_iff)

   227

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

   229   unfolding real_of_int_def by (rule floor_less_iff)

   230

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

   232   unfolding real_of_int_def by (rule less_floor_iff)

   233

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

   235   unfolding real_of_int_def by (rule floor_le_iff)

   236

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

   238   unfolding real_of_int_def by (rule floor_add_of_int)

   239

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

   241   unfolding real_of_int_def by (rule floor_diff_of_int)

   242

   243 lemma le_mult_floor:

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

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

   246 proof -

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

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

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

   250     using assms by (auto intro!: mult_mono)

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

   252   finally show ?thesis unfolding real_of_int_less_iff by simp

   253 qed

   254

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

   256   unfolding real_of_nat_def by simp

   257

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

   259   unfolding real_of_int_def by (rule le_of_int_ceiling)

   260

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

   262   unfolding real_of_int_def by simp

   263

   264 lemma real_of_int_ceiling_cancel [simp]:

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

   266   using ceiling_real_of_int by metis

   267

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

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

   270

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

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

   273

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

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

   276

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

   278   unfolding real_of_int_def using ceiling_correct [of r] by simp

   279

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

   281   unfolding real_of_int_def using ceiling_correct [of r] by simp

   282

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

   284   unfolding real_of_int_def by (simp add: ceiling_le_iff)

   285

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

   287   unfolding real_of_int_def by (simp add: ceiling_le_iff)

   288

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

   290   unfolding real_of_int_def by (rule ceiling_le_iff)

   291

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

   293   unfolding real_of_int_def by (rule less_ceiling_iff)

   294

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

   296   unfolding real_of_int_def by (rule ceiling_less_iff)

   297

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

   299   unfolding real_of_int_def by (rule le_ceiling_iff)

   300

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

   302   unfolding real_of_int_def by (rule ceiling_add_of_int)

   303

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

   305   unfolding real_of_int_def by (rule ceiling_diff_of_int)

   306

   307

   308 subsection {* Versions for the natural numbers *}

   309

   310 definition

   311   natfloor :: "real => nat" where

   312   "natfloor x = nat(floor x)"

   313

   314 definition

   315   natceiling :: "real => nat" where

   316   "natceiling x = nat(ceiling x)"

   317

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

   319   by (unfold natfloor_def, simp)

   320

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

   322   by (unfold natfloor_def, simp)

   323

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

   325   by (unfold natfloor_def, simp)

   326

   327 lemma natfloor_numeral_eq [simp]: "natfloor (numeral n) = numeral n"

   328   by (unfold natfloor_def, simp)

   329

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

   331   by (unfold natfloor_def, simp)

   332

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

   334   by (unfold natfloor_def, simp)

   335

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

   337   unfolding natfloor_def by simp

   338

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

   340   unfolding natfloor_def by (intro nat_mono floor_mono)

   341

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

   343   apply (unfold natfloor_def)

   344   apply (subst nat_int [THEN sym])

   345   apply (rule nat_mono)

   346   apply (rule le_floor)

   347   apply simp

   348 done

   349

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

   351   unfolding natfloor_def real_of_nat_def

   352   by (simp add: nat_less_iff floor_less_iff)

   353

   354 lemma less_natfloor:

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

   356   shows "natfloor x < n"

   357   using assms by (simp add: natfloor_less_iff)

   358

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

   360   apply (rule iffI)

   361   apply (rule order_trans)

   362   prefer 2

   363   apply (erule real_natfloor_le)

   364   apply (subst real_of_nat_le_iff)

   365   apply assumption

   366   apply (erule le_natfloor)

   367 done

   368

   369 lemma le_natfloor_eq_numeral [simp]:

   370     "~ neg((numeral n)::int) ==> 0 <= x ==>

   371       (numeral n <= natfloor x) = (numeral n <= x)"

   372   apply (subst le_natfloor_eq, assumption)

   373   apply simp

   374 done

   375

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

   377   apply (case_tac "0 <= x")

   378   apply (subst le_natfloor_eq, assumption, simp)

   379   apply (rule iffI)

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

   381   apply simp

   382   apply (rule natfloor_mono)

   383   apply simp

   384   apply simp

   385 done

   386

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

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

   389

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

   391   apply (case_tac "0 <= x")

   392   apply (unfold natfloor_def)

   393   apply simp

   394   apply simp_all

   395 done

   396

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

   398 using real_natfloor_add_one_gt by (simp add: algebra_simps)

   399

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

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

   402   apply arith

   403   apply (rule real_natfloor_add_one_gt)

   404 done

   405

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

   407   unfolding natfloor_def

   408   unfolding real_of_int_of_nat_eq [symmetric] floor_add

   409   by (simp add: nat_add_distrib)

   410

   411 lemma natfloor_add_numeral [simp]:

   412     "~neg ((numeral n)::int) ==> 0 <= x ==>

   413       natfloor (x + numeral n) = natfloor x + numeral n"

   414   by (simp add: natfloor_add [symmetric])

   415

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

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

   418

   419 lemma natfloor_subtract [simp]:

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

   421   unfolding natfloor_def real_of_int_of_nat_eq [symmetric] floor_subtract

   422   by simp

   423

   424 lemma natfloor_div_nat:

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

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

   427 proof (rule natfloor_eq)

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

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

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

   431     using assms by (simp add: le_divide_eq le_natfloor_eq)

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

   433     apply (subst mod_div_equality [symmetric])

   434     apply (rule add_strict_left_mono)

   435     apply (rule mod_less_divisor)

   436     apply fact

   437     done

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

   439     using assms

   440     by (simp add: divide_less_eq natfloor_less_iff left_distrib)

   441 qed

   442

   443 lemma le_mult_natfloor:

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

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

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

   447

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

   449   by (unfold natceiling_def, simp)

   450

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

   452   by (unfold natceiling_def, simp)

   453

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

   455   by (unfold natceiling_def, simp)

   456

   457 lemma natceiling_numeral_eq [simp]: "natceiling (numeral n) = numeral n"

   458   by (unfold natceiling_def, simp)

   459

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

   461   by (unfold natceiling_def, simp)

   462

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

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

   465

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

   467   unfolding natceiling_def by simp

   468

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

   470   unfolding natceiling_def by (intro nat_mono ceiling_mono)

   471

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

   473   unfolding natceiling_def real_of_nat_def

   474   by (simp add: nat_le_iff ceiling_le_iff)

   475

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

   477   unfolding natceiling_def real_of_nat_def

   478   by (simp add: nat_le_iff ceiling_le_iff)

   479

   480 lemma natceiling_le_eq_numeral [simp]:

   481     "~ neg((numeral n)::int) ==>

   482       (natceiling x <= numeral n) = (x <= numeral n)"

   483   by (simp add: natceiling_le_eq)

   484

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

   486   unfolding natceiling_def

   487   by (simp add: nat_le_iff ceiling_le_iff)

   488

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

   490   unfolding natceiling_def

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

   492

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

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

   495   unfolding natceiling_def

   496   unfolding real_of_int_of_nat_eq [symmetric] ceiling_add

   497   by (simp add: nat_add_distrib)

   498

   499 lemma natceiling_add_numeral [simp]:

   500     "~ neg ((numeral n)::int) ==> 0 <= x ==>

   501       natceiling (x + numeral n) = natceiling x + numeral n"

   502   by (simp add: natceiling_add [symmetric])

   503

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

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

   506

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

   508   unfolding natceiling_def real_of_int_of_nat_eq [symmetric] ceiling_subtract

   509   by simp

   510

   511 subsection {* Exponentiation with floor *}

   512

   513 lemma floor_power:

   514   assumes "x = real (floor x)"

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

   516 proof -

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

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

   519   show ?thesis unfolding real_of_int_inject[symmetric]

   520     unfolding * floor_real_of_int ..

   521 qed

   522

   523 lemma natfloor_power:

   524   assumes "x = real (natfloor x)"

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

   526 proof -

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

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

   529   from floor_power[OF this]

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

   531     by simp

   532 qed

   533

   534 end