src/HOL/RComplete.thy
 author huffman Fri Sep 02 15:19:59 2011 -0700 (2011-09-02) changeset 44668 31d41a0f6b5d parent 44667 ee5772ca7d16 child 44669 8e6cdb9c00a7 permissions -rw-r--r--
simplify proof of Rats_dense_in_real;
remove lemma Rats_dense_in_nn_real;
     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   assumes positive_P: "\<forall>x \<in> P. (0::real) < x"

    37     and 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     unfolding isLub_def leastP_def setle_def setge_def Ball_def

    80       Collect_def mem_def isUb_def UNIV_def by simp

    81 qed

    82

    83 text{*A version of the same theorem without all those predicates!*}

    84 lemma reals_complete2:

    85   fixes S :: "(real set)"

    86   assumes "\<exists>y. y\<in>S" and "\<exists>(x::real). \<forall>y\<in>S. y \<le> x"

    87   shows "\<exists>x. (\<forall>y\<in>S. y \<le> x) &

    88                (\<forall>z. ((\<forall>y\<in>S. y \<le> z) --> x \<le> z))"

    89 using assms by (rule complete_real)

    90

    91

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

    93

    94 theorem reals_Archimedean:

    95   assumes x_pos: "0 < x"

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

    97   unfolding real_of_nat_def using x_pos

    98   by (rule ex_inverse_of_nat_Suc_less)

    99

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

   101   unfolding real_of_nat_def by (rule ex_less_of_nat)

   102

   103 lemma reals_Archimedean3:

   104   assumes x_greater_zero: "0 < x"

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

   106   unfolding real_of_nat_def using 0 < x

   107   by (auto intro: ex_less_of_nat_mult)

   108

   109

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

   111

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

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

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

   115

   116 lemma Rats_dense_in_real:

   117   fixes x :: real

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

   119 proof -

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

   121   with reals_Archimedean obtain q::nat

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

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

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

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

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

   127     unfolding r_def p_def

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

   129   finally have "x < r" .

   130   moreover have "r < y"

   131     unfolding r_def p_def

   132     by (simp add: divide_less_eq diff_less_eq 0 < q

   133       less_ceiling_iff [symmetric])

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

   135   ultimately show ?thesis by fast

   136 qed

   137

   138

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

   140

   141 lemma number_of_less_real_of_int_iff [simp]:

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

   143 apply auto

   144 apply (rule real_of_int_less_iff [THEN iffD1])

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

   146 done

   147

   148 lemma number_of_less_real_of_int_iff2 [simp]:

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

   150 apply auto

   151 apply (rule real_of_int_less_iff [THEN iffD1])

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

   153 done

   154

   155 lemma number_of_le_real_of_int_iff [simp]:

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

   157 by (simp add: linorder_not_less [symmetric])

   158

   159 lemma number_of_le_real_of_int_iff2 [simp]:

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

   161 by (simp add: linorder_not_less [symmetric])

   162

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

   164 unfolding real_of_nat_def by simp

   165

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

   167 unfolding real_of_nat_def by (simp add: floor_minus)

   168

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

   170 unfolding real_of_int_def by simp

   171

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

   173 unfolding real_of_int_def by (simp add: floor_minus)

   174

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

   176 unfolding real_of_int_def by (rule floor_exists)

   177

   178 lemma lemma_floor:

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

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

   181 proof -

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

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

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

   185   thus ?thesis by arith

   186 qed

   187

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

   189 unfolding real_of_int_def by (rule of_int_floor_le)

   190

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

   192 by (auto intro: lemma_floor)

   193

   194 lemma real_of_int_floor_cancel [simp]:

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

   196   using floor_real_of_int by metis

   197

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

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

   200

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

   202   unfolding real_of_int_def by (rule floor_unique)

   203

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

   205 apply (rule inj_int [THEN injD])

   206 apply (simp add: real_of_nat_Suc)

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

   208 done

   209

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

   211 apply (drule order_le_imp_less_or_eq)

   212 apply (auto intro: floor_eq3)

   213 done

   214

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

   216   unfolding real_of_int_def using floor_correct [of r] by simp

   217

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

   219   unfolding real_of_int_def using floor_correct [of r] by simp

   220

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

   222   unfolding real_of_int_def using floor_correct [of r] by simp

   223

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

   225   unfolding real_of_int_def using floor_correct [of r] by simp

   226

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

   228   unfolding real_of_int_def by (simp add: le_floor_iff)

   229

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

   231   unfolding real_of_int_def by (simp add: le_floor_iff)

   232

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

   234   unfolding real_of_int_def by (rule le_floor_iff)

   235

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

   237   unfolding real_of_int_def by (rule floor_less_iff)

   238

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

   240   unfolding real_of_int_def by (rule less_floor_iff)

   241

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

   243   unfolding real_of_int_def by (rule floor_le_iff)

   244

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

   246   unfolding real_of_int_def by (rule floor_add_of_int)

   247

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

   249   unfolding real_of_int_def by (rule floor_diff_of_int)

   250

   251 lemma le_mult_floor:

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

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

   254 proof -

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

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

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

   258     using assms by (auto intro!: mult_mono)

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

   260   finally show ?thesis unfolding real_of_int_less_iff by simp

   261 qed

   262

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

   264   unfolding real_of_nat_def by simp

   265

   266 lemma ceiling_floor [simp]: "ceiling (real (floor r)) = floor r"

   267   unfolding real_of_int_def by simp

   268

   269 lemma floor_ceiling [simp]: "floor (real (ceiling r)) = ceiling r"

   270   unfolding real_of_int_def by simp

   271

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

   273   unfolding real_of_int_def by (rule le_of_int_ceiling)

   274

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

   276   unfolding real_of_int_def by simp

   277

   278 lemma real_of_int_ceiling_cancel [simp]:

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

   280   using ceiling_real_of_int by metis

   281

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

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

   284

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

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

   287

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

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

   290

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

   292   unfolding real_of_int_def using ceiling_correct [of r] by simp

   293

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

   295   unfolding real_of_int_def using ceiling_correct [of r] by simp

   296

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

   298   unfolding real_of_int_def by (simp add: ceiling_le_iff)

   299

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

   301   unfolding real_of_int_def by (simp add: ceiling_le_iff)

   302

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

   304   unfolding real_of_int_def by (rule ceiling_le_iff)

   305

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

   307   unfolding real_of_int_def by (rule less_ceiling_iff)

   308

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

   310   unfolding real_of_int_def by (rule ceiling_less_iff)

   311

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

   313   unfolding real_of_int_def by (rule le_ceiling_iff)

   314

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

   316   unfolding real_of_int_def by (rule ceiling_add_of_int)

   317

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

   319   unfolding real_of_int_def by (rule ceiling_diff_of_int)

   320

   321

   322 subsection {* Versions for the natural numbers *}

   323

   324 definition

   325   natfloor :: "real => nat" where

   326   "natfloor x = nat(floor x)"

   327

   328 definition

   329   natceiling :: "real => nat" where

   330   "natceiling x = nat(ceiling x)"

   331

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

   333   by (unfold natfloor_def, simp)

   334

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

   336   by (unfold natfloor_def, simp)

   337

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

   339   by (unfold natfloor_def, simp)

   340

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

   342   by (unfold natfloor_def, simp)

   343

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

   345   by (unfold natfloor_def, simp)

   346

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

   348   by (unfold natfloor_def, simp)

   349

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

   351   apply (unfold natfloor_def)

   352   apply (subgoal_tac "floor x <= floor 0")

   353   apply simp

   354   apply (erule floor_mono)

   355 done

   356

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

   358   apply (case_tac "0 <= x")

   359   apply (subst natfloor_def)+

   360   apply (subst nat_le_eq_zle)

   361   apply force

   362   apply (erule floor_mono)

   363   apply (subst natfloor_neg)

   364   apply simp

   365   apply simp

   366 done

   367

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

   369   apply (unfold natfloor_def)

   370   apply (subst nat_int [THEN sym])

   371   apply (subst nat_le_eq_zle)

   372   apply simp

   373   apply (rule le_floor)

   374   apply simp

   375 done

   376

   377 lemma less_natfloor:

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

   379   shows "natfloor x < n"

   380 proof (rule ccontr)

   381   assume "\<not> ?thesis" hence *: "n \<le> natfloor x" by simp

   382   note assms(2)

   383   also have "real n \<le> real (natfloor x)"

   384     using * unfolding real_of_nat_le_iff .

   385   finally have "x < real (natfloor x)" .

   386   with real_natfloor_le[OF assms(1)]

   387   show False by auto

   388 qed

   389

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

   391   apply (rule iffI)

   392   apply (rule order_trans)

   393   prefer 2

   394   apply (erule real_natfloor_le)

   395   apply (subst real_of_nat_le_iff)

   396   apply assumption

   397   apply (erule le_natfloor)

   398 done

   399

   400 lemma le_natfloor_eq_number_of [simp]:

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

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

   403   apply (subst le_natfloor_eq, assumption)

   404   apply simp

   405 done

   406

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

   408   apply (case_tac "0 <= x")

   409   apply (subst le_natfloor_eq, assumption, simp)

   410   apply (rule iffI)

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

   412   apply simp

   413   apply (rule natfloor_mono)

   414   apply simp

   415   apply simp

   416 done

   417

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

   419   apply (unfold natfloor_def)

   420   apply (subst (2) nat_int [THEN sym])

   421   apply (subst eq_nat_nat_iff)

   422   apply simp

   423   apply simp

   424   apply (rule floor_eq2)

   425   apply auto

   426 done

   427

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

   429   apply (case_tac "0 <= x")

   430   apply (unfold natfloor_def)

   431   apply simp

   432   apply simp_all

   433 done

   434

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

   436 using real_natfloor_add_one_gt by (simp add: algebra_simps)

   437

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

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

   440   apply arith

   441   apply (rule real_natfloor_add_one_gt)

   442 done

   443

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

   445   apply (unfold natfloor_def)

   446   apply (subgoal_tac "real a = real (int a)")

   447   apply (erule ssubst)

   448   apply (simp add: nat_add_distrib del: real_of_int_of_nat_eq)

   449   apply simp

   450 done

   451

   452 lemma natfloor_add_number_of [simp]:

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

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

   455   apply (subst natfloor_add [THEN sym])

   456   apply simp_all

   457 done

   458

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

   460   apply (subst natfloor_add [THEN sym])

   461   apply assumption

   462   apply simp

   463 done

   464

   465 lemma natfloor_subtract [simp]: "real a <= x ==>

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

   467   apply (unfold natfloor_def)

   468   apply (subgoal_tac "real a = real (int a)")

   469   apply (erule ssubst)

   470   apply (simp del: real_of_int_of_nat_eq)

   471   apply simp

   472 done

   473

   474 lemma natfloor_div_nat:

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

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

   477 proof -

   478   have "natfloor x = (natfloor x) div y * y + (natfloor x) mod y"

   479     by simp

   480   then have a: "real(natfloor x) = real ((natfloor x) div y) * real y +

   481     real((natfloor x) mod y)"

   482     by (simp only: real_of_nat_add [THEN sym] real_of_nat_mult [THEN sym])

   483   have "x = real(natfloor x) + (x - real(natfloor x))"

   484     by simp

   485   then have "x = real ((natfloor x) div y) * real y +

   486       real((natfloor x) mod y) + (x - real(natfloor x))"

   487     by (simp add: a)

   488   then have "x / real y = ... / real y"

   489     by simp

   490   also have "... = real((natfloor x) div y) + real((natfloor x) mod y) /

   491     real y + (x - real(natfloor x)) / real y"

   492     by (auto simp add: algebra_simps add_divide_distrib diff_divide_distrib)

   493   finally have "natfloor (x / real y) = natfloor(...)" by simp

   494   also have "... = natfloor(real((natfloor x) mod y) /

   495     real y + (x - real(natfloor x)) / real y + real((natfloor x) div y))"

   496     by (simp add: add_ac)

   497   also have "... = natfloor(real((natfloor x) mod y) /

   498     real y + (x - real(natfloor x)) / real y) + (natfloor x) div y"

   499     apply (rule natfloor_add)

   500     apply (rule add_nonneg_nonneg)

   501     apply (rule divide_nonneg_pos)

   502     apply simp

   503     apply (simp add: assms)

   504     apply (rule divide_nonneg_pos)

   505     apply (simp add: algebra_simps)

   506     apply (rule real_natfloor_le)

   507     using assms apply auto

   508     done

   509   also have "natfloor(real((natfloor x) mod y) /

   510     real y + (x - real(natfloor x)) / real y) = 0"

   511     apply (rule natfloor_eq)

   512     apply simp

   513     apply (rule add_nonneg_nonneg)

   514     apply (rule divide_nonneg_pos)

   515     apply force

   516     apply (force simp add: assms)

   517     apply (rule divide_nonneg_pos)

   518     apply (simp add: algebra_simps)

   519     apply (rule real_natfloor_le)

   520     apply (auto simp add: assms)

   521     using assms apply arith

   522     using assms apply (simp add: add_divide_distrib [THEN sym])

   523     apply (subgoal_tac "real y = real y - 1 + 1")

   524     apply (erule ssubst)

   525     apply (rule add_le_less_mono)

   526     apply (simp add: algebra_simps)

   527     apply (subgoal_tac "1 + real(natfloor x mod y) =

   528       real(natfloor x mod y + 1)")

   529     apply (erule ssubst)

   530     apply (subst real_of_nat_le_iff)

   531     apply (subgoal_tac "natfloor x mod y < y")

   532     apply arith

   533     apply (rule mod_less_divisor)

   534     apply auto

   535     using real_natfloor_add_one_gt

   536     apply (simp add: algebra_simps)

   537     done

   538   finally show ?thesis by simp

   539 qed

   540

   541 lemma le_mult_natfloor:

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

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

   544   unfolding natfloor_def

   545   apply (subst nat_mult_distrib[symmetric])

   546   using assms apply simp

   547   apply (subst nat_le_eq_zle)

   548   using assms le_mult_floor by (auto intro!: mult_nonneg_nonneg)

   549

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

   551   by (unfold natceiling_def, simp)

   552

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

   554   by (unfold natceiling_def, simp)

   555

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

   557   by (unfold natceiling_def, simp)

   558

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

   560   by (unfold natceiling_def, simp)

   561

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

   563   by (unfold natceiling_def, simp)

   564

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

   566   apply (unfold natceiling_def)

   567   apply (case_tac "x < 0")

   568   apply simp

   569   apply (subst real_nat_eq_real)

   570   apply (subgoal_tac "ceiling 0 <= ceiling x")

   571   apply simp

   572   apply (rule ceiling_mono)

   573   apply simp

   574   apply simp

   575 done

   576

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

   578   apply (unfold natceiling_def)

   579   apply simp

   580 done

   581

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

   583   apply (case_tac "0 <= x")

   584   apply (subst natceiling_def)+

   585   apply (subst nat_le_eq_zle)

   586   apply (rule disjI2)

   587   apply (subgoal_tac "real (0::int) <= real(ceiling y)")

   588   apply simp

   589   apply (rule order_trans)

   590   apply simp

   591   apply (erule order_trans)

   592   apply simp

   593   apply (erule ceiling_mono)

   594   apply (subst natceiling_neg)

   595   apply simp_all

   596 done

   597

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

   599   apply (unfold natceiling_def)

   600   apply (case_tac "x < 0")

   601   apply simp

   602   apply (subst (2) nat_int [THEN sym])

   603   apply (subst nat_le_eq_zle)

   604   apply simp

   605   apply (rule ceiling_le)

   606   apply simp

   607 done

   608

   609 lemma natceiling_le_eq: "0 <= x ==> (natceiling x <= a) = (x <= real a)"

   610   apply (rule iffI)

   611   apply (rule order_trans)

   612   apply (rule real_natceiling_ge)

   613   apply (subst real_of_nat_le_iff)

   614   apply assumption

   615   apply (erule natceiling_le)

   616 done

   617

   618 lemma natceiling_le_eq_number_of [simp]:

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

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

   621   apply (subst natceiling_le_eq, assumption)

   622   apply simp

   623 done

   624

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

   626   apply (case_tac "0 <= x")

   627   apply (subst natceiling_le_eq)

   628   apply assumption

   629   apply simp

   630   apply (subst natceiling_neg)

   631   apply simp

   632   apply simp

   633 done

   634

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

   636   apply (unfold natceiling_def)

   637   apply (simplesubst nat_int [THEN sym]) back back

   638   apply (subgoal_tac "nat(int n) + 1 = nat(int n + 1)")

   639   apply (erule ssubst)

   640   apply (subst eq_nat_nat_iff)

   641   apply (subgoal_tac "ceiling 0 <= ceiling x")

   642   apply simp

   643   apply (rule ceiling_mono)

   644   apply force

   645   apply force

   646   apply (rule ceiling_eq2)

   647   apply (simp, simp)

   648   apply (subst nat_add_distrib)

   649   apply auto

   650 done

   651

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

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

   654   apply (unfold natceiling_def)

   655   apply (subgoal_tac "real a = real (int a)")

   656   apply (erule ssubst)

   657   apply (simp del: real_of_int_of_nat_eq)

   658   apply (subst nat_add_distrib)

   659   apply (subgoal_tac "0 = ceiling 0")

   660   apply (erule ssubst)

   661   apply (erule ceiling_mono)

   662   apply simp_all

   663 done

   664

   665 lemma natceiling_add_number_of [simp]:

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

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

   668   apply (subst natceiling_add [THEN sym])

   669   apply simp_all

   670 done

   671

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

   673   apply (subst natceiling_add [THEN sym])

   674   apply assumption

   675   apply simp

   676 done

   677

   678 lemma natceiling_subtract [simp]: "real a <= x ==>

   679     natceiling(x - real a) = natceiling x - a"

   680   apply (unfold natceiling_def)

   681   apply (subgoal_tac "real a = real (int a)")

   682   apply (erule ssubst)

   683   apply (simp del: real_of_int_of_nat_eq)

   684   apply simp

   685 done

   686

   687 subsection {* Exponentiation with floor *}

   688

   689 lemma floor_power:

   690   assumes "x = real (floor x)"

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

   692 proof -

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

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

   695   show ?thesis unfolding real_of_int_inject[symmetric]

   696     unfolding * floor_real_of_int ..

   697 qed

   698

   699 lemma natfloor_power:

   700   assumes "x = real (natfloor x)"

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

   702 proof -

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

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

   705   from floor_power[OF this]

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

   707     by simp

   708 qed

   709

   710 end