src/HOL/RComplete.thy
author huffman
Fri Sep 02 14:27:55 2011 -0700 (2011-09-02)
changeset 44667 ee5772ca7d16
parent 41550 efa734d9b221
child 44668 31d41a0f6b5d
permissions -rw-r--r--
remove unused, unnecessary lemmas
     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_nn_real: fixes x::real
   117 assumes "0\<le>x" and "x<y" shows "\<exists>r \<in> \<rat>.  x<r \<and> r<y"
   118 proof -
   119   from `x<y` have "0 < y-x" by simp
   120   with reals_Archimedean obtain q::nat 
   121     where q: "inverse (real q) < y-x" and "0 < real q" by auto  
   122   def p \<equiv> "LEAST n.  y \<le> real (Suc n)/real q"  
   123   from reals_Archimedean2 obtain n::nat where "y * real q < real n" by auto
   124   with `0 < real q` have ex: "y \<le> real n/real q" (is "?P n")
   125     by (simp add: pos_less_divide_eq[THEN sym])
   126   also from assms have "\<not> y \<le> real (0::nat) / real q" by simp
   127   ultimately have main: "(LEAST n. y \<le> real n/real q) = Suc p"
   128     by (unfold p_def) (rule Least_Suc)
   129   also from ex have "?P (LEAST x. ?P x)" by (rule LeastI)
   130   ultimately have suc: "y \<le> real (Suc p) / real q" by simp
   131   def r \<equiv> "real p/real q"
   132   have "x = y-(y-x)" by simp
   133   also from suc q have "\<dots> < real (Suc p)/real q - inverse (real q)" by arith
   134   also have "\<dots> = real p / real q"
   135     by (simp only: inverse_eq_divide diff_minus real_of_nat_Suc 
   136     minus_divide_left add_divide_distrib[THEN sym]) simp
   137   finally have "x<r" by (unfold r_def)
   138   have "p<Suc p" .. also note main[THEN sym]
   139   finally have "\<not> ?P p"  by (rule not_less_Least)
   140   hence "r<y" by (simp add: r_def)
   141   from r_def have "r \<in> \<rat>" by simp
   142   with `x<r` `r<y` show ?thesis by fast
   143 qed
   144 
   145 theorem Rats_dense_in_real: fixes x y :: real
   146 assumes "x<y" shows "\<exists>r \<in> \<rat>.  x<r \<and> r<y"
   147 proof -
   148   from reals_Archimedean2 obtain n::nat where "-x < real n" by auto
   149   hence "0 \<le> x + real n" by arith
   150   also from `x<y` have "x + real n < y + real n" by arith
   151   ultimately have "\<exists>r \<in> \<rat>. x + real n < r \<and> r < y + real n"
   152     by(rule Rats_dense_in_nn_real)
   153   then obtain r where "r \<in> \<rat>" and r2: "x + real n < r" 
   154     and r3: "r < y + real n"
   155     by blast
   156   have "r - real n = r + real (int n)/real (-1::int)" by simp
   157   also from `r\<in>\<rat>` have "r + real (int n)/real (-1::int) \<in> \<rat>" by simp
   158   also from r2 have "x < r - real n" by arith
   159   moreover from r3 have "r - real n < y" by arith
   160   ultimately show ?thesis by fast
   161 qed
   162 
   163 
   164 subsection{*Floor and Ceiling Functions from the Reals to the Integers*}
   165 
   166 lemma number_of_less_real_of_int_iff [simp]:
   167      "((number_of n) < real (m::int)) = (number_of n < m)"
   168 apply auto
   169 apply (rule real_of_int_less_iff [THEN iffD1])
   170 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
   171 done
   172 
   173 lemma number_of_less_real_of_int_iff2 [simp]:
   174      "(real (m::int) < (number_of n)) = (m < number_of n)"
   175 apply auto
   176 apply (rule real_of_int_less_iff [THEN iffD1])
   177 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
   178 done
   179 
   180 lemma number_of_le_real_of_int_iff [simp]:
   181      "((number_of n) \<le> real (m::int)) = (number_of n \<le> m)"
   182 by (simp add: linorder_not_less [symmetric])
   183 
   184 lemma number_of_le_real_of_int_iff2 [simp]:
   185      "(real (m::int) \<le> (number_of n)) = (m \<le> number_of n)"
   186 by (simp add: linorder_not_less [symmetric])
   187 
   188 lemma floor_real_of_nat [simp]: "floor (real (n::nat)) = int n"
   189 unfolding real_of_nat_def by simp
   190 
   191 lemma floor_minus_real_of_nat [simp]: "floor (- real (n::nat)) = - int n"
   192 unfolding real_of_nat_def by (simp add: floor_minus)
   193 
   194 lemma floor_real_of_int [simp]: "floor (real (n::int)) = n"
   195 unfolding real_of_int_def by simp
   196 
   197 lemma floor_minus_real_of_int [simp]: "floor (- real (n::int)) = - n"
   198 unfolding real_of_int_def by (simp add: floor_minus)
   199 
   200 lemma real_lb_ub_int: " \<exists>n::int. real n \<le> r & r < real (n+1)"
   201 unfolding real_of_int_def by (rule floor_exists)
   202 
   203 lemma lemma_floor:
   204   assumes a1: "real m \<le> r" and a2: "r < real n + 1"
   205   shows "m \<le> (n::int)"
   206 proof -
   207   have "real m < real n + 1" using a1 a2 by (rule order_le_less_trans)
   208   also have "... = real (n + 1)" by simp
   209   finally have "m < n + 1" by (simp only: real_of_int_less_iff)
   210   thus ?thesis by arith
   211 qed
   212 
   213 lemma real_of_int_floor_le [simp]: "real (floor r) \<le> r"
   214 unfolding real_of_int_def by (rule of_int_floor_le)
   215 
   216 lemma lemma_floor2: "real n < real (x::int) + 1 ==> n \<le> x"
   217 by (auto intro: lemma_floor)
   218 
   219 lemma real_of_int_floor_cancel [simp]:
   220     "(real (floor x) = x) = (\<exists>n::int. x = real n)"
   221   using floor_real_of_int by metis
   222 
   223 lemma floor_eq: "[| real n < x; x < real n + 1 |] ==> floor x = n"
   224   unfolding real_of_int_def using floor_unique [of n x] by simp
   225 
   226 lemma floor_eq2: "[| real n \<le> x; x < real n + 1 |] ==> floor x = n"
   227   unfolding real_of_int_def by (rule floor_unique)
   228 
   229 lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"
   230 apply (rule inj_int [THEN injD])
   231 apply (simp add: real_of_nat_Suc)
   232 apply (simp add: real_of_nat_Suc floor_eq floor_eq [where n = "int n"])
   233 done
   234 
   235 lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"
   236 apply (drule order_le_imp_less_or_eq)
   237 apply (auto intro: floor_eq3)
   238 done
   239 
   240 lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real(floor r)"
   241   unfolding real_of_int_def using floor_correct [of r] by simp
   242 
   243 lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real(floor r)"
   244   unfolding real_of_int_def using floor_correct [of r] by simp
   245 
   246 lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real(floor r) + 1"
   247   unfolding real_of_int_def using floor_correct [of r] by simp
   248 
   249 lemma real_of_int_floor_add_one_gt [simp]: "r < real(floor r) + 1"
   250   unfolding real_of_int_def using floor_correct [of r] by simp
   251 
   252 lemma le_floor: "real a <= x ==> a <= floor x"
   253   unfolding real_of_int_def by (simp add: le_floor_iff)
   254 
   255 lemma real_le_floor: "a <= floor x ==> real a <= x"
   256   unfolding real_of_int_def by (simp add: le_floor_iff)
   257 
   258 lemma le_floor_eq: "(a <= floor x) = (real a <= x)"
   259   unfolding real_of_int_def by (rule le_floor_iff)
   260 
   261 lemma floor_less_eq: "(floor x < a) = (x < real a)"
   262   unfolding real_of_int_def by (rule floor_less_iff)
   263 
   264 lemma less_floor_eq: "(a < floor x) = (real a + 1 <= x)"
   265   unfolding real_of_int_def by (rule less_floor_iff)
   266 
   267 lemma floor_le_eq: "(floor x <= a) = (x < real a + 1)"
   268   unfolding real_of_int_def by (rule floor_le_iff)
   269 
   270 lemma floor_add [simp]: "floor (x + real a) = floor x + a"
   271   unfolding real_of_int_def by (rule floor_add_of_int)
   272 
   273 lemma floor_subtract [simp]: "floor (x - real a) = floor x - a"
   274   unfolding real_of_int_def by (rule floor_diff_of_int)
   275 
   276 lemma le_mult_floor:
   277   assumes "0 \<le> (a :: real)" and "0 \<le> b"
   278   shows "floor a * floor b \<le> floor (a * b)"
   279 proof -
   280   have "real (floor a) \<le> a"
   281     and "real (floor b) \<le> b" by auto
   282   hence "real (floor a * floor b) \<le> a * b"
   283     using assms by (auto intro!: mult_mono)
   284   also have "a * b < real (floor (a * b) + 1)" by auto
   285   finally show ?thesis unfolding real_of_int_less_iff by simp
   286 qed
   287 
   288 lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"
   289   unfolding real_of_nat_def by simp
   290 
   291 lemma ceiling_floor [simp]: "ceiling (real (floor r)) = floor r"
   292   unfolding real_of_int_def by simp
   293 
   294 lemma floor_ceiling [simp]: "floor (real (ceiling r)) = ceiling r"
   295   unfolding real_of_int_def by simp
   296 
   297 lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)"
   298   unfolding real_of_int_def by (rule le_of_int_ceiling)
   299 
   300 lemma ceiling_real_of_int [simp]: "ceiling (real (n::int)) = n"
   301   unfolding real_of_int_def by simp
   302 
   303 lemma real_of_int_ceiling_cancel [simp]:
   304      "(real (ceiling x) = x) = (\<exists>n::int. x = real n)"
   305   using ceiling_real_of_int by metis
   306 
   307 lemma ceiling_eq: "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1"
   308   unfolding real_of_int_def using ceiling_unique [of "n + 1" x] by simp
   309 
   310 lemma ceiling_eq2: "[| real n < x; x \<le> real n + 1 |] ==> ceiling x = n + 1"
   311   unfolding real_of_int_def using ceiling_unique [of "n + 1" x] by simp
   312 
   313 lemma ceiling_eq3: "[| real n - 1 < x; x \<le> real n  |] ==> ceiling x = n"
   314   unfolding real_of_int_def using ceiling_unique [of n x] by simp
   315 
   316 lemma real_of_int_ceiling_diff_one_le [simp]: "real (ceiling r) - 1 \<le> r"
   317   unfolding real_of_int_def using ceiling_correct [of r] by simp
   318 
   319 lemma real_of_int_ceiling_le_add_one [simp]: "real (ceiling r) \<le> r + 1"
   320   unfolding real_of_int_def using ceiling_correct [of r] by simp
   321 
   322 lemma ceiling_le: "x <= real a ==> ceiling x <= a"
   323   unfolding real_of_int_def by (simp add: ceiling_le_iff)
   324 
   325 lemma ceiling_le_real: "ceiling x <= a ==> x <= real a"
   326   unfolding real_of_int_def by (simp add: ceiling_le_iff)
   327 
   328 lemma ceiling_le_eq: "(ceiling x <= a) = (x <= real a)"
   329   unfolding real_of_int_def by (rule ceiling_le_iff)
   330 
   331 lemma less_ceiling_eq: "(a < ceiling x) = (real a < x)"
   332   unfolding real_of_int_def by (rule less_ceiling_iff)
   333 
   334 lemma ceiling_less_eq: "(ceiling x < a) = (x <= real a - 1)"
   335   unfolding real_of_int_def by (rule ceiling_less_iff)
   336 
   337 lemma le_ceiling_eq: "(a <= ceiling x) = (real a - 1 < x)"
   338   unfolding real_of_int_def by (rule le_ceiling_iff)
   339 
   340 lemma ceiling_add [simp]: "ceiling (x + real a) = ceiling x + a"
   341   unfolding real_of_int_def by (rule ceiling_add_of_int)
   342 
   343 lemma ceiling_subtract [simp]: "ceiling (x - real a) = ceiling x - a"
   344   unfolding real_of_int_def by (rule ceiling_diff_of_int)
   345 
   346 
   347 subsection {* Versions for the natural numbers *}
   348 
   349 definition
   350   natfloor :: "real => nat" where
   351   "natfloor x = nat(floor x)"
   352 
   353 definition
   354   natceiling :: "real => nat" where
   355   "natceiling x = nat(ceiling x)"
   356 
   357 lemma natfloor_zero [simp]: "natfloor 0 = 0"
   358   by (unfold natfloor_def, simp)
   359 
   360 lemma natfloor_one [simp]: "natfloor 1 = 1"
   361   by (unfold natfloor_def, simp)
   362 
   363 lemma zero_le_natfloor [simp]: "0 <= natfloor x"
   364   by (unfold natfloor_def, simp)
   365 
   366 lemma natfloor_number_of_eq [simp]: "natfloor (number_of n) = number_of n"
   367   by (unfold natfloor_def, simp)
   368 
   369 lemma natfloor_real_of_nat [simp]: "natfloor(real n) = n"
   370   by (unfold natfloor_def, simp)
   371 
   372 lemma real_natfloor_le: "0 <= x ==> real(natfloor x) <= x"
   373   by (unfold natfloor_def, simp)
   374 
   375 lemma natfloor_neg: "x <= 0 ==> natfloor x = 0"
   376   apply (unfold natfloor_def)
   377   apply (subgoal_tac "floor x <= floor 0")
   378   apply simp
   379   apply (erule floor_mono)
   380 done
   381 
   382 lemma natfloor_mono: "x <= y ==> natfloor x <= natfloor y"
   383   apply (case_tac "0 <= x")
   384   apply (subst natfloor_def)+
   385   apply (subst nat_le_eq_zle)
   386   apply force
   387   apply (erule floor_mono)
   388   apply (subst natfloor_neg)
   389   apply simp
   390   apply simp
   391 done
   392 
   393 lemma le_natfloor: "real x <= a ==> x <= natfloor a"
   394   apply (unfold natfloor_def)
   395   apply (subst nat_int [THEN sym])
   396   apply (subst nat_le_eq_zle)
   397   apply simp
   398   apply (rule le_floor)
   399   apply simp
   400 done
   401 
   402 lemma less_natfloor:
   403   assumes "0 \<le> x" and "x < real (n :: nat)"
   404   shows "natfloor x < n"
   405 proof (rule ccontr)
   406   assume "\<not> ?thesis" hence *: "n \<le> natfloor x" by simp
   407   note assms(2)
   408   also have "real n \<le> real (natfloor x)"
   409     using * unfolding real_of_nat_le_iff .
   410   finally have "x < real (natfloor x)" .
   411   with real_natfloor_le[OF assms(1)]
   412   show False by auto
   413 qed
   414 
   415 lemma le_natfloor_eq: "0 <= x ==> (a <= natfloor x) = (real a <= x)"
   416   apply (rule iffI)
   417   apply (rule order_trans)
   418   prefer 2
   419   apply (erule real_natfloor_le)
   420   apply (subst real_of_nat_le_iff)
   421   apply assumption
   422   apply (erule le_natfloor)
   423 done
   424 
   425 lemma le_natfloor_eq_number_of [simp]:
   426     "~ neg((number_of n)::int) ==> 0 <= x ==>
   427       (number_of n <= natfloor x) = (number_of n <= x)"
   428   apply (subst le_natfloor_eq, assumption)
   429   apply simp
   430 done
   431 
   432 lemma le_natfloor_eq_one [simp]: "(1 <= natfloor x) = (1 <= x)"
   433   apply (case_tac "0 <= x")
   434   apply (subst le_natfloor_eq, assumption, simp)
   435   apply (rule iffI)
   436   apply (subgoal_tac "natfloor x <= natfloor 0")
   437   apply simp
   438   apply (rule natfloor_mono)
   439   apply simp
   440   apply simp
   441 done
   442 
   443 lemma natfloor_eq: "real n <= x ==> x < real n + 1 ==> natfloor x = n"
   444   apply (unfold natfloor_def)
   445   apply (subst (2) nat_int [THEN sym])
   446   apply (subst eq_nat_nat_iff)
   447   apply simp
   448   apply simp
   449   apply (rule floor_eq2)
   450   apply auto
   451 done
   452 
   453 lemma real_natfloor_add_one_gt: "x < real(natfloor x) + 1"
   454   apply (case_tac "0 <= x")
   455   apply (unfold natfloor_def)
   456   apply simp
   457   apply simp_all
   458 done
   459 
   460 lemma real_natfloor_gt_diff_one: "x - 1 < real(natfloor x)"
   461 using real_natfloor_add_one_gt by (simp add: algebra_simps)
   462 
   463 lemma ge_natfloor_plus_one_imp_gt: "natfloor z + 1 <= n ==> z < real n"
   464   apply (subgoal_tac "z < real(natfloor z) + 1")
   465   apply arith
   466   apply (rule real_natfloor_add_one_gt)
   467 done
   468 
   469 lemma natfloor_add [simp]: "0 <= x ==> natfloor (x + real a) = natfloor x + a"
   470   apply (unfold natfloor_def)
   471   apply (subgoal_tac "real a = real (int a)")
   472   apply (erule ssubst)
   473   apply (simp add: nat_add_distrib del: real_of_int_of_nat_eq)
   474   apply simp
   475 done
   476 
   477 lemma natfloor_add_number_of [simp]:
   478     "~neg ((number_of n)::int) ==> 0 <= x ==>
   479       natfloor (x + number_of n) = natfloor x + number_of n"
   480   apply (subst natfloor_add [THEN sym])
   481   apply simp_all
   482 done
   483 
   484 lemma natfloor_add_one: "0 <= x ==> natfloor(x + 1) = natfloor x + 1"
   485   apply (subst natfloor_add [THEN sym])
   486   apply assumption
   487   apply simp
   488 done
   489 
   490 lemma natfloor_subtract [simp]: "real a <= x ==>
   491     natfloor(x - real a) = natfloor x - a"
   492   apply (unfold natfloor_def)
   493   apply (subgoal_tac "real a = real (int a)")
   494   apply (erule ssubst)
   495   apply (simp del: real_of_int_of_nat_eq)
   496   apply simp
   497 done
   498 
   499 lemma natfloor_div_nat:
   500   assumes "1 <= x" and "y > 0"
   501   shows "natfloor (x / real y) = natfloor x div y"
   502 proof -
   503   have "natfloor x = (natfloor x) div y * y + (natfloor x) mod y"
   504     by simp
   505   then have a: "real(natfloor x) = real ((natfloor x) div y) * real y +
   506     real((natfloor x) mod y)"
   507     by (simp only: real_of_nat_add [THEN sym] real_of_nat_mult [THEN sym])
   508   have "x = real(natfloor x) + (x - real(natfloor x))"
   509     by simp
   510   then have "x = real ((natfloor x) div y) * real y +
   511       real((natfloor x) mod y) + (x - real(natfloor x))"
   512     by (simp add: a)
   513   then have "x / real y = ... / real y"
   514     by simp
   515   also have "... = real((natfloor x) div y) + real((natfloor x) mod y) /
   516     real y + (x - real(natfloor x)) / real y"
   517     by (auto simp add: algebra_simps add_divide_distrib diff_divide_distrib)
   518   finally have "natfloor (x / real y) = natfloor(...)" by simp
   519   also have "... = natfloor(real((natfloor x) mod y) /
   520     real y + (x - real(natfloor x)) / real y + real((natfloor x) div y))"
   521     by (simp add: add_ac)
   522   also have "... = natfloor(real((natfloor x) mod y) /
   523     real y + (x - real(natfloor x)) / real y) + (natfloor x) div y"
   524     apply (rule natfloor_add)
   525     apply (rule add_nonneg_nonneg)
   526     apply (rule divide_nonneg_pos)
   527     apply simp
   528     apply (simp add: assms)
   529     apply (rule divide_nonneg_pos)
   530     apply (simp add: algebra_simps)
   531     apply (rule real_natfloor_le)
   532     using assms apply auto
   533     done
   534   also have "natfloor(real((natfloor x) mod y) /
   535     real y + (x - real(natfloor x)) / real y) = 0"
   536     apply (rule natfloor_eq)
   537     apply simp
   538     apply (rule add_nonneg_nonneg)
   539     apply (rule divide_nonneg_pos)
   540     apply force
   541     apply (force simp add: assms)
   542     apply (rule divide_nonneg_pos)
   543     apply (simp add: algebra_simps)
   544     apply (rule real_natfloor_le)
   545     apply (auto simp add: assms)
   546     using assms apply arith
   547     using assms apply (simp add: add_divide_distrib [THEN sym])
   548     apply (subgoal_tac "real y = real y - 1 + 1")
   549     apply (erule ssubst)
   550     apply (rule add_le_less_mono)
   551     apply (simp add: algebra_simps)
   552     apply (subgoal_tac "1 + real(natfloor x mod y) =
   553       real(natfloor x mod y + 1)")
   554     apply (erule ssubst)
   555     apply (subst real_of_nat_le_iff)
   556     apply (subgoal_tac "natfloor x mod y < y")
   557     apply arith
   558     apply (rule mod_less_divisor)
   559     apply auto
   560     using real_natfloor_add_one_gt
   561     apply (simp add: algebra_simps)
   562     done
   563   finally show ?thesis by simp
   564 qed
   565 
   566 lemma le_mult_natfloor:
   567   assumes "0 \<le> (a :: real)" and "0 \<le> b"
   568   shows "natfloor a * natfloor b \<le> natfloor (a * b)"
   569   unfolding natfloor_def
   570   apply (subst nat_mult_distrib[symmetric])
   571   using assms apply simp
   572   apply (subst nat_le_eq_zle)
   573   using assms le_mult_floor by (auto intro!: mult_nonneg_nonneg)
   574 
   575 lemma natceiling_zero [simp]: "natceiling 0 = 0"
   576   by (unfold natceiling_def, simp)
   577 
   578 lemma natceiling_one [simp]: "natceiling 1 = 1"
   579   by (unfold natceiling_def, simp)
   580 
   581 lemma zero_le_natceiling [simp]: "0 <= natceiling x"
   582   by (unfold natceiling_def, simp)
   583 
   584 lemma natceiling_number_of_eq [simp]: "natceiling (number_of n) = number_of n"
   585   by (unfold natceiling_def, simp)
   586 
   587 lemma natceiling_real_of_nat [simp]: "natceiling(real n) = n"
   588   by (unfold natceiling_def, simp)
   589 
   590 lemma real_natceiling_ge: "x <= real(natceiling x)"
   591   apply (unfold natceiling_def)
   592   apply (case_tac "x < 0")
   593   apply simp
   594   apply (subst real_nat_eq_real)
   595   apply (subgoal_tac "ceiling 0 <= ceiling x")
   596   apply simp
   597   apply (rule ceiling_mono)
   598   apply simp
   599   apply simp
   600 done
   601 
   602 lemma natceiling_neg: "x <= 0 ==> natceiling x = 0"
   603   apply (unfold natceiling_def)
   604   apply simp
   605 done
   606 
   607 lemma natceiling_mono: "x <= y ==> natceiling x <= natceiling y"
   608   apply (case_tac "0 <= x")
   609   apply (subst natceiling_def)+
   610   apply (subst nat_le_eq_zle)
   611   apply (rule disjI2)
   612   apply (subgoal_tac "real (0::int) <= real(ceiling y)")
   613   apply simp
   614   apply (rule order_trans)
   615   apply simp
   616   apply (erule order_trans)
   617   apply simp
   618   apply (erule ceiling_mono)
   619   apply (subst natceiling_neg)
   620   apply simp_all
   621 done
   622 
   623 lemma natceiling_le: "x <= real a ==> natceiling x <= a"
   624   apply (unfold natceiling_def)
   625   apply (case_tac "x < 0")
   626   apply simp
   627   apply (subst (2) nat_int [THEN sym])
   628   apply (subst nat_le_eq_zle)
   629   apply simp
   630   apply (rule ceiling_le)
   631   apply simp
   632 done
   633 
   634 lemma natceiling_le_eq: "0 <= x ==> (natceiling x <= a) = (x <= real a)"
   635   apply (rule iffI)
   636   apply (rule order_trans)
   637   apply (rule real_natceiling_ge)
   638   apply (subst real_of_nat_le_iff)
   639   apply assumption
   640   apply (erule natceiling_le)
   641 done
   642 
   643 lemma natceiling_le_eq_number_of [simp]:
   644     "~ neg((number_of n)::int) ==> 0 <= x ==>
   645       (natceiling x <= number_of n) = (x <= number_of n)"
   646   apply (subst natceiling_le_eq, assumption)
   647   apply simp
   648 done
   649 
   650 lemma natceiling_le_eq_one: "(natceiling x <= 1) = (x <= 1)"
   651   apply (case_tac "0 <= x")
   652   apply (subst natceiling_le_eq)
   653   apply assumption
   654   apply simp
   655   apply (subst natceiling_neg)
   656   apply simp
   657   apply simp
   658 done
   659 
   660 lemma natceiling_eq: "real n < x ==> x <= real n + 1 ==> natceiling x = n + 1"
   661   apply (unfold natceiling_def)
   662   apply (simplesubst nat_int [THEN sym]) back back
   663   apply (subgoal_tac "nat(int n) + 1 = nat(int n + 1)")
   664   apply (erule ssubst)
   665   apply (subst eq_nat_nat_iff)
   666   apply (subgoal_tac "ceiling 0 <= ceiling x")
   667   apply simp
   668   apply (rule ceiling_mono)
   669   apply force
   670   apply force
   671   apply (rule ceiling_eq2)
   672   apply (simp, simp)
   673   apply (subst nat_add_distrib)
   674   apply auto
   675 done
   676 
   677 lemma natceiling_add [simp]: "0 <= x ==>
   678     natceiling (x + real a) = natceiling x + a"
   679   apply (unfold natceiling_def)
   680   apply (subgoal_tac "real a = real (int a)")
   681   apply (erule ssubst)
   682   apply (simp del: real_of_int_of_nat_eq)
   683   apply (subst nat_add_distrib)
   684   apply (subgoal_tac "0 = ceiling 0")
   685   apply (erule ssubst)
   686   apply (erule ceiling_mono)
   687   apply simp_all
   688 done
   689 
   690 lemma natceiling_add_number_of [simp]:
   691     "~ neg ((number_of n)::int) ==> 0 <= x ==>
   692       natceiling (x + number_of n) = natceiling x + number_of n"
   693   apply (subst natceiling_add [THEN sym])
   694   apply simp_all
   695 done
   696 
   697 lemma natceiling_add_one: "0 <= x ==> natceiling(x + 1) = natceiling x + 1"
   698   apply (subst natceiling_add [THEN sym])
   699   apply assumption
   700   apply simp
   701 done
   702 
   703 lemma natceiling_subtract [simp]: "real a <= x ==>
   704     natceiling(x - real a) = natceiling x - a"
   705   apply (unfold natceiling_def)
   706   apply (subgoal_tac "real a = real (int a)")
   707   apply (erule ssubst)
   708   apply (simp del: real_of_int_of_nat_eq)
   709   apply simp
   710 done
   711 
   712 subsection {* Exponentiation with floor *}
   713 
   714 lemma floor_power:
   715   assumes "x = real (floor x)"
   716   shows "floor (x ^ n) = floor x ^ n"
   717 proof -
   718   have *: "x ^ n = real (floor x ^ n)"
   719     using assms by (induct n arbitrary: x) simp_all
   720   show ?thesis unfolding real_of_int_inject[symmetric]
   721     unfolding * floor_real_of_int ..
   722 qed
   723 
   724 lemma natfloor_power:
   725   assumes "x = real (natfloor x)"
   726   shows "natfloor (x ^ n) = natfloor x ^ n"
   727 proof -
   728   from assms have "0 \<le> floor x" by auto
   729   note assms[unfolded natfloor_def real_nat_eq_real[OF `0 \<le> floor x`]]
   730   from floor_power[OF this]
   731   show ?thesis unfolding natfloor_def nat_power_eq[OF `0 \<le> floor x`, symmetric]
   732     by simp
   733 qed
   734 
   735 end