src/HOL/RComplete.thy
author huffman
Fri Sep 02 20:58:31 2011 -0700 (2011-09-02)
changeset 44678 21eb31192850
parent 44669 8e6cdb9c00a7
child 44679 a89d0ad8ed20
permissions -rw-r--r--
remove redundant simp rules ceiling_floor and floor_ceiling
     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 
    84 subsection {* The Archimedean Property of the Reals *}
    85 
    86 theorem reals_Archimedean:
    87   assumes x_pos: "0 < x"
    88   shows "\<exists>n. inverse (real (Suc n)) < x"
    89   unfolding real_of_nat_def using x_pos
    90   by (rule ex_inverse_of_nat_Suc_less)
    91 
    92 lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"
    93   unfolding real_of_nat_def by (rule ex_less_of_nat)
    94 
    95 lemma reals_Archimedean3:
    96   assumes x_greater_zero: "0 < x"
    97   shows "\<forall>(y::real). \<exists>(n::nat). y < real n * x"
    98   unfolding real_of_nat_def using `0 < x`
    99   by (auto intro: ex_less_of_nat_mult)
   100 
   101 
   102 subsection{*Density of the Rational Reals in the Reals*}
   103 
   104 text{* This density proof is due to Stefan Richter and was ported by TN.  The
   105 original source is \emph{Real Analysis} by H.L. Royden.
   106 It employs the Archimedean property of the reals. *}
   107 
   108 lemma Rats_dense_in_real:
   109   fixes x :: real
   110   assumes "x < y" shows "\<exists>r\<in>\<rat>. x < r \<and> r < y"
   111 proof -
   112   from `x<y` have "0 < y-x" by simp
   113   with reals_Archimedean obtain q::nat 
   114     where q: "inverse (real q) < y-x" and "0 < q" by auto
   115   def p \<equiv> "ceiling (y * real q) - 1"
   116   def r \<equiv> "of_int p / real q"
   117   from q have "x < y - inverse (real q)" by simp
   118   also have "y - inverse (real q) \<le> r"
   119     unfolding r_def p_def
   120     by (simp add: le_divide_eq left_diff_distrib le_of_int_ceiling `0 < q`)
   121   finally have "x < r" .
   122   moreover have "r < y"
   123     unfolding r_def p_def
   124     by (simp add: divide_less_eq diff_less_eq `0 < q`
   125       less_ceiling_iff [symmetric])
   126   moreover from r_def have "r \<in> \<rat>" by simp
   127   ultimately show ?thesis by fast
   128 qed
   129 
   130 
   131 subsection{*Floor and Ceiling Functions from the Reals to the Integers*}
   132 
   133 lemma number_of_less_real_of_int_iff [simp]:
   134      "((number_of n) < real (m::int)) = (number_of n < m)"
   135 apply auto
   136 apply (rule real_of_int_less_iff [THEN iffD1])
   137 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
   138 done
   139 
   140 lemma number_of_less_real_of_int_iff2 [simp]:
   141      "(real (m::int) < (number_of n)) = (m < number_of n)"
   142 apply auto
   143 apply (rule real_of_int_less_iff [THEN iffD1])
   144 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
   145 done
   146 
   147 lemma number_of_le_real_of_int_iff [simp]:
   148      "((number_of n) \<le> real (m::int)) = (number_of n \<le> m)"
   149 by (simp add: linorder_not_less [symmetric])
   150 
   151 lemma number_of_le_real_of_int_iff2 [simp]:
   152      "(real (m::int) \<le> (number_of n)) = (m \<le> number_of 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_number_of_eq [simp]: "natfloor (number_of n) = number_of 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   apply (unfold natfloor_def)
   338   apply (subgoal_tac "floor x <= floor 0")
   339   apply simp
   340   apply (erule floor_mono)
   341 done
   342 
   343 lemma natfloor_mono: "x <= y ==> natfloor x <= natfloor y"
   344   apply (case_tac "0 <= x")
   345   apply (subst natfloor_def)+
   346   apply (subst nat_le_eq_zle)
   347   apply force
   348   apply (erule floor_mono)
   349   apply (subst natfloor_neg)
   350   apply simp
   351   apply simp
   352 done
   353 
   354 lemma le_natfloor: "real x <= a ==> x <= natfloor a"
   355   apply (unfold natfloor_def)
   356   apply (subst nat_int [THEN sym])
   357   apply (subst nat_le_eq_zle)
   358   apply simp
   359   apply (rule le_floor)
   360   apply simp
   361 done
   362 
   363 lemma less_natfloor:
   364   assumes "0 \<le> x" and "x < real (n :: nat)"
   365   shows "natfloor x < n"
   366 proof (rule ccontr)
   367   assume "\<not> ?thesis" hence *: "n \<le> natfloor x" by simp
   368   note assms(2)
   369   also have "real n \<le> real (natfloor x)"
   370     using * unfolding real_of_nat_le_iff .
   371   finally have "x < real (natfloor x)" .
   372   with real_natfloor_le[OF assms(1)]
   373   show False by auto
   374 qed
   375 
   376 lemma le_natfloor_eq: "0 <= x ==> (a <= natfloor x) = (real a <= x)"
   377   apply (rule iffI)
   378   apply (rule order_trans)
   379   prefer 2
   380   apply (erule real_natfloor_le)
   381   apply (subst real_of_nat_le_iff)
   382   apply assumption
   383   apply (erule le_natfloor)
   384 done
   385 
   386 lemma le_natfloor_eq_number_of [simp]:
   387     "~ neg((number_of n)::int) ==> 0 <= x ==>
   388       (number_of n <= natfloor x) = (number_of n <= x)"
   389   apply (subst le_natfloor_eq, assumption)
   390   apply simp
   391 done
   392 
   393 lemma le_natfloor_eq_one [simp]: "(1 <= natfloor x) = (1 <= x)"
   394   apply (case_tac "0 <= x")
   395   apply (subst le_natfloor_eq, assumption, simp)
   396   apply (rule iffI)
   397   apply (subgoal_tac "natfloor x <= natfloor 0")
   398   apply simp
   399   apply (rule natfloor_mono)
   400   apply simp
   401   apply simp
   402 done
   403 
   404 lemma natfloor_eq: "real n <= x ==> x < real n + 1 ==> natfloor x = n"
   405   apply (unfold natfloor_def)
   406   apply (subst (2) nat_int [THEN sym])
   407   apply (subst eq_nat_nat_iff)
   408   apply simp
   409   apply simp
   410   apply (rule floor_eq2)
   411   apply auto
   412 done
   413 
   414 lemma real_natfloor_add_one_gt: "x < real(natfloor x) + 1"
   415   apply (case_tac "0 <= x")
   416   apply (unfold natfloor_def)
   417   apply simp
   418   apply simp_all
   419 done
   420 
   421 lemma real_natfloor_gt_diff_one: "x - 1 < real(natfloor x)"
   422 using real_natfloor_add_one_gt by (simp add: algebra_simps)
   423 
   424 lemma ge_natfloor_plus_one_imp_gt: "natfloor z + 1 <= n ==> z < real n"
   425   apply (subgoal_tac "z < real(natfloor z) + 1")
   426   apply arith
   427   apply (rule real_natfloor_add_one_gt)
   428 done
   429 
   430 lemma natfloor_add [simp]: "0 <= x ==> natfloor (x + real a) = natfloor x + a"
   431   apply (unfold natfloor_def)
   432   apply (subgoal_tac "real a = real (int a)")
   433   apply (erule ssubst)
   434   apply (simp add: nat_add_distrib del: real_of_int_of_nat_eq)
   435   apply simp
   436 done
   437 
   438 lemma natfloor_add_number_of [simp]:
   439     "~neg ((number_of n)::int) ==> 0 <= x ==>
   440       natfloor (x + number_of n) = natfloor x + number_of n"
   441   apply (subst natfloor_add [THEN sym])
   442   apply simp_all
   443 done
   444 
   445 lemma natfloor_add_one: "0 <= x ==> natfloor(x + 1) = natfloor x + 1"
   446   apply (subst natfloor_add [THEN sym])
   447   apply assumption
   448   apply simp
   449 done
   450 
   451 lemma natfloor_subtract [simp]: "real a <= x ==>
   452     natfloor(x - real a) = natfloor x - a"
   453   apply (unfold natfloor_def)
   454   apply (subgoal_tac "real a = real (int a)")
   455   apply (erule ssubst)
   456   apply (simp del: real_of_int_of_nat_eq)
   457   apply simp
   458 done
   459 
   460 lemma natfloor_div_nat:
   461   assumes "1 <= x" and "y > 0"
   462   shows "natfloor (x / real y) = natfloor x div y"
   463 proof -
   464   have "natfloor x = (natfloor x) div y * y + (natfloor x) mod y"
   465     by simp
   466   then have a: "real(natfloor x) = real ((natfloor x) div y) * real y +
   467     real((natfloor x) mod y)"
   468     by (simp only: real_of_nat_add [THEN sym] real_of_nat_mult [THEN sym])
   469   have "x = real(natfloor x) + (x - real(natfloor x))"
   470     by simp
   471   then have "x = real ((natfloor x) div y) * real y +
   472       real((natfloor x) mod y) + (x - real(natfloor x))"
   473     by (simp add: a)
   474   then have "x / real y = ... / real y"
   475     by simp
   476   also have "... = real((natfloor x) div y) + real((natfloor x) mod y) /
   477     real y + (x - real(natfloor x)) / real y"
   478     by (auto simp add: algebra_simps add_divide_distrib diff_divide_distrib)
   479   finally have "natfloor (x / real y) = natfloor(...)" by simp
   480   also have "... = natfloor(real((natfloor x) mod y) /
   481     real y + (x - real(natfloor x)) / real y + real((natfloor x) div y))"
   482     by (simp add: add_ac)
   483   also have "... = natfloor(real((natfloor x) mod y) /
   484     real y + (x - real(natfloor x)) / real y) + (natfloor x) div y"
   485     apply (rule natfloor_add)
   486     apply (rule add_nonneg_nonneg)
   487     apply (rule divide_nonneg_pos)
   488     apply simp
   489     apply (simp add: assms)
   490     apply (rule divide_nonneg_pos)
   491     apply (simp add: algebra_simps)
   492     apply (rule real_natfloor_le)
   493     using assms apply auto
   494     done
   495   also have "natfloor(real((natfloor x) mod y) /
   496     real y + (x - real(natfloor x)) / real y) = 0"
   497     apply (rule natfloor_eq)
   498     apply simp
   499     apply (rule add_nonneg_nonneg)
   500     apply (rule divide_nonneg_pos)
   501     apply force
   502     apply (force simp add: assms)
   503     apply (rule divide_nonneg_pos)
   504     apply (simp add: algebra_simps)
   505     apply (rule real_natfloor_le)
   506     apply (auto simp add: assms)
   507     using assms apply arith
   508     using assms apply (simp add: add_divide_distrib [THEN sym])
   509     apply (subgoal_tac "real y = real y - 1 + 1")
   510     apply (erule ssubst)
   511     apply (rule add_le_less_mono)
   512     apply (simp add: algebra_simps)
   513     apply (subgoal_tac "1 + real(natfloor x mod y) =
   514       real(natfloor x mod y + 1)")
   515     apply (erule ssubst)
   516     apply (subst real_of_nat_le_iff)
   517     apply (subgoal_tac "natfloor x mod y < y")
   518     apply arith
   519     apply (rule mod_less_divisor)
   520     apply auto
   521     using real_natfloor_add_one_gt
   522     apply (simp add: algebra_simps)
   523     done
   524   finally show ?thesis by simp
   525 qed
   526 
   527 lemma le_mult_natfloor:
   528   assumes "0 \<le> (a :: real)" and "0 \<le> b"
   529   shows "natfloor a * natfloor b \<le> natfloor (a * b)"
   530   unfolding natfloor_def
   531   apply (subst nat_mult_distrib[symmetric])
   532   using assms apply simp
   533   apply (subst nat_le_eq_zle)
   534   using assms le_mult_floor by (auto intro!: mult_nonneg_nonneg)
   535 
   536 lemma natceiling_zero [simp]: "natceiling 0 = 0"
   537   by (unfold natceiling_def, simp)
   538 
   539 lemma natceiling_one [simp]: "natceiling 1 = 1"
   540   by (unfold natceiling_def, simp)
   541 
   542 lemma zero_le_natceiling [simp]: "0 <= natceiling x"
   543   by (unfold natceiling_def, simp)
   544 
   545 lemma natceiling_number_of_eq [simp]: "natceiling (number_of n) = number_of n"
   546   by (unfold natceiling_def, simp)
   547 
   548 lemma natceiling_real_of_nat [simp]: "natceiling(real n) = n"
   549   by (unfold natceiling_def, simp)
   550 
   551 lemma real_natceiling_ge: "x <= real(natceiling x)"
   552   apply (unfold natceiling_def)
   553   apply (case_tac "x < 0")
   554   apply simp
   555   apply (subst real_nat_eq_real)
   556   apply (subgoal_tac "ceiling 0 <= ceiling x")
   557   apply simp
   558   apply (rule ceiling_mono)
   559   apply simp
   560   apply simp
   561 done
   562 
   563 lemma natceiling_neg: "x <= 0 ==> natceiling x = 0"
   564   apply (unfold natceiling_def)
   565   apply simp
   566 done
   567 
   568 lemma natceiling_mono: "x <= y ==> natceiling x <= natceiling y"
   569   apply (case_tac "0 <= x")
   570   apply (subst natceiling_def)+
   571   apply (subst nat_le_eq_zle)
   572   apply (rule disjI2)
   573   apply (subgoal_tac "real (0::int) <= real(ceiling y)")
   574   apply simp
   575   apply (rule order_trans)
   576   apply simp
   577   apply (erule order_trans)
   578   apply simp
   579   apply (erule ceiling_mono)
   580   apply (subst natceiling_neg)
   581   apply simp_all
   582 done
   583 
   584 lemma natceiling_le: "x <= real a ==> natceiling x <= a"
   585   apply (unfold natceiling_def)
   586   apply (case_tac "x < 0")
   587   apply simp
   588   apply (subst (2) nat_int [THEN sym])
   589   apply (subst nat_le_eq_zle)
   590   apply simp
   591   apply (rule ceiling_le)
   592   apply simp
   593 done
   594 
   595 lemma natceiling_le_eq: "0 <= x ==> (natceiling x <= a) = (x <= real a)"
   596   apply (rule iffI)
   597   apply (rule order_trans)
   598   apply (rule real_natceiling_ge)
   599   apply (subst real_of_nat_le_iff)
   600   apply assumption
   601   apply (erule natceiling_le)
   602 done
   603 
   604 lemma natceiling_le_eq_number_of [simp]:
   605     "~ neg((number_of n)::int) ==> 0 <= x ==>
   606       (natceiling x <= number_of n) = (x <= number_of n)"
   607   apply (subst natceiling_le_eq, assumption)
   608   apply simp
   609 done
   610 
   611 lemma natceiling_le_eq_one: "(natceiling x <= 1) = (x <= 1)"
   612   apply (case_tac "0 <= x")
   613   apply (subst natceiling_le_eq)
   614   apply assumption
   615   apply simp
   616   apply (subst natceiling_neg)
   617   apply simp
   618   apply simp
   619 done
   620 
   621 lemma natceiling_eq: "real n < x ==> x <= real n + 1 ==> natceiling x = n + 1"
   622   apply (unfold natceiling_def)
   623   apply (simplesubst nat_int [THEN sym]) back back
   624   apply (subgoal_tac "nat(int n) + 1 = nat(int n + 1)")
   625   apply (erule ssubst)
   626   apply (subst eq_nat_nat_iff)
   627   apply (subgoal_tac "ceiling 0 <= ceiling x")
   628   apply simp
   629   apply (rule ceiling_mono)
   630   apply force
   631   apply force
   632   apply (rule ceiling_eq2)
   633   apply (simp, simp)
   634   apply (subst nat_add_distrib)
   635   apply auto
   636 done
   637 
   638 lemma natceiling_add [simp]: "0 <= x ==>
   639     natceiling (x + real a) = natceiling x + a"
   640   apply (unfold natceiling_def)
   641   apply (subgoal_tac "real a = real (int a)")
   642   apply (erule ssubst)
   643   apply (simp del: real_of_int_of_nat_eq)
   644   apply (subst nat_add_distrib)
   645   apply (subgoal_tac "0 = ceiling 0")
   646   apply (erule ssubst)
   647   apply (erule ceiling_mono)
   648   apply simp_all
   649 done
   650 
   651 lemma natceiling_add_number_of [simp]:
   652     "~ neg ((number_of n)::int) ==> 0 <= x ==>
   653       natceiling (x + number_of n) = natceiling x + number_of n"
   654   apply (subst natceiling_add [THEN sym])
   655   apply simp_all
   656 done
   657 
   658 lemma natceiling_add_one: "0 <= x ==> natceiling(x + 1) = natceiling x + 1"
   659   apply (subst natceiling_add [THEN sym])
   660   apply assumption
   661   apply simp
   662 done
   663 
   664 lemma natceiling_subtract [simp]: "real a <= x ==>
   665     natceiling(x - real a) = natceiling x - a"
   666   apply (unfold natceiling_def)
   667   apply (subgoal_tac "real a = real (int a)")
   668   apply (erule ssubst)
   669   apply (simp del: real_of_int_of_nat_eq)
   670   apply simp
   671 done
   672 
   673 subsection {* Exponentiation with floor *}
   674 
   675 lemma floor_power:
   676   assumes "x = real (floor x)"
   677   shows "floor (x ^ n) = floor x ^ n"
   678 proof -
   679   have *: "x ^ n = real (floor x ^ n)"
   680     using assms by (induct n arbitrary: x) simp_all
   681   show ?thesis unfolding real_of_int_inject[symmetric]
   682     unfolding * floor_real_of_int ..
   683 qed
   684 
   685 lemma natfloor_power:
   686   assumes "x = real (natfloor x)"
   687   shows "natfloor (x ^ n) = natfloor x ^ n"
   688 proof -
   689   from assms have "0 \<le> floor x" by auto
   690   note assms[unfolded natfloor_def real_nat_eq_real[OF `0 \<le> floor x`]]
   691   from floor_power[OF this]
   692   show ?thesis unfolding natfloor_def nat_power_eq[OF `0 \<le> floor x`, symmetric]
   693     by simp
   694 qed
   695 
   696 end