src/HOL/RComplete.thy
author hoelzl
Tue Mar 23 16:17:41 2010 +0100 (2010-03-23)
changeset 35928 d31f55f97663
parent 35578 384ad08a1d1b
child 36795 e05e1283c550
permissions -rw-r--r--
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     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 lemma posreal_complete:
    34   assumes positive_P: "\<forall>x \<in> P. (0::real) < x"
    35     and not_empty_P: "\<exists>x. x \<in> P"
    36     and upper_bound_Ex: "\<exists>y. \<forall>x \<in> P. x<y"
    37   shows "\<exists>S. \<forall>y. (\<exists>x \<in> P. y < x) = (y < S)"
    38 proof (rule exI, rule allI)
    39   fix y
    40   let ?pP = "{w. real_of_preal w \<in> P}"
    41 
    42   show "(\<exists>x\<in>P. y < x) = (y < real_of_preal (psup ?pP))"
    43   proof (cases "0 < y")
    44     assume neg_y: "\<not> 0 < y"
    45     show ?thesis
    46     proof
    47       assume "\<exists>x\<in>P. y < x"
    48       have "\<forall>x. y < real_of_preal x"
    49         using neg_y by (rule real_less_all_real2)
    50       thus "y < real_of_preal (psup ?pP)" ..
    51     next
    52       assume "y < real_of_preal (psup ?pP)"
    53       obtain "x" where x_in_P: "x \<in> P" using not_empty_P ..
    54       hence "0 < x" using positive_P by simp
    55       hence "y < x" using neg_y by simp
    56       thus "\<exists>x \<in> P. y < x" using x_in_P ..
    57     qed
    58   next
    59     assume pos_y: "0 < y"
    60 
    61     then obtain py where y_is_py: "y = real_of_preal py"
    62       by (auto simp add: real_gt_zero_preal_Ex)
    63 
    64     obtain a where "a \<in> P" using not_empty_P ..
    65     with positive_P have a_pos: "0 < a" ..
    66     then obtain pa where "a = real_of_preal pa"
    67       by (auto simp add: real_gt_zero_preal_Ex)
    68     hence "pa \<in> ?pP" using `a \<in> P` by auto
    69     hence pP_not_empty: "?pP \<noteq> {}" by auto
    70 
    71     obtain sup where sup: "\<forall>x \<in> P. x < sup"
    72       using upper_bound_Ex ..
    73     from this and `a \<in> P` have "a < sup" ..
    74     hence "0 < sup" using a_pos by arith
    75     then obtain possup where "sup = real_of_preal possup"
    76       by (auto simp add: real_gt_zero_preal_Ex)
    77     hence "\<forall>X \<in> ?pP. X \<le> possup"
    78       using sup by (auto simp add: real_of_preal_lessI)
    79     with pP_not_empty have psup: "\<And>Z. (\<exists>X \<in> ?pP. Z < X) = (Z < psup ?pP)"
    80       by (rule preal_complete)
    81 
    82     show ?thesis
    83     proof
    84       assume "\<exists>x \<in> P. y < x"
    85       then obtain x where x_in_P: "x \<in> P" and y_less_x: "y < x" ..
    86       hence "0 < x" using pos_y by arith
    87       then obtain px where x_is_px: "x = real_of_preal px"
    88         by (auto simp add: real_gt_zero_preal_Ex)
    89 
    90       have py_less_X: "\<exists>X \<in> ?pP. py < X"
    91       proof
    92         show "py < px" using y_is_py and x_is_px and y_less_x
    93           by (simp add: real_of_preal_lessI)
    94         show "px \<in> ?pP" using x_in_P and x_is_px by simp
    95       qed
    96 
    97       have "(\<exists>X \<in> ?pP. py < X) ==> (py < psup ?pP)"
    98         using psup by simp
    99       hence "py < psup ?pP" using py_less_X by simp
   100       thus "y < real_of_preal (psup {w. real_of_preal w \<in> P})"
   101         using y_is_py and pos_y by (simp add: real_of_preal_lessI)
   102     next
   103       assume y_less_psup: "y < real_of_preal (psup ?pP)"
   104 
   105       hence "py < psup ?pP" using y_is_py
   106         by (simp add: real_of_preal_lessI)
   107       then obtain "X" where py_less_X: "py < X" and X_in_pP: "X \<in> ?pP"
   108         using psup by auto
   109       then obtain x where x_is_X: "x = real_of_preal X"
   110         by (simp add: real_gt_zero_preal_Ex)
   111       hence "y < x" using py_less_X and y_is_py
   112         by (simp add: real_of_preal_lessI)
   113 
   114       moreover have "x \<in> P" using x_is_X and X_in_pP by simp
   115 
   116       ultimately show "\<exists> x \<in> P. y < x" ..
   117     qed
   118   qed
   119 qed
   120 
   121 text {*
   122   \medskip Completeness properties using @{text "isUb"}, @{text "isLub"} etc.
   123 *}
   124 
   125 lemma real_isLub_unique: "[| isLub R S x; isLub R S y |] ==> x = (y::real)"
   126   apply (frule isLub_isUb)
   127   apply (frule_tac x = y in isLub_isUb)
   128   apply (blast intro!: order_antisym dest!: isLub_le_isUb)
   129   done
   130 
   131 
   132 text {*
   133   \medskip Completeness theorem for the positive reals (again).
   134 *}
   135 
   136 lemma posreals_complete:
   137   assumes positive_S: "\<forall>x \<in> S. 0 < x"
   138     and not_empty_S: "\<exists>x. x \<in> S"
   139     and upper_bound_Ex: "\<exists>u. isUb (UNIV::real set) S u"
   140   shows "\<exists>t. isLub (UNIV::real set) S t"
   141 proof
   142   let ?pS = "{w. real_of_preal w \<in> S}"
   143 
   144   obtain u where "isUb UNIV S u" using upper_bound_Ex ..
   145   hence sup: "\<forall>x \<in> S. x \<le> u" by (simp add: isUb_def setle_def)
   146 
   147   obtain x where x_in_S: "x \<in> S" using not_empty_S ..
   148   hence x_gt_zero: "0 < x" using positive_S by simp
   149   have  "x \<le> u" using sup and x_in_S ..
   150   hence "0 < u" using x_gt_zero by arith
   151 
   152   then obtain pu where u_is_pu: "u = real_of_preal pu"
   153     by (auto simp add: real_gt_zero_preal_Ex)
   154 
   155   have pS_less_pu: "\<forall>pa \<in> ?pS. pa \<le> pu"
   156   proof
   157     fix pa
   158     assume "pa \<in> ?pS"
   159     then obtain a where "a \<in> S" and "a = real_of_preal pa"
   160       by simp
   161     moreover hence "a \<le> u" using sup by simp
   162     ultimately show "pa \<le> pu"
   163       using sup and u_is_pu by (simp add: real_of_preal_le_iff)
   164   qed
   165 
   166   have "\<forall>y \<in> S. y \<le> real_of_preal (psup ?pS)"
   167   proof
   168     fix y
   169     assume y_in_S: "y \<in> S"
   170     hence "0 < y" using positive_S by simp
   171     then obtain py where y_is_py: "y = real_of_preal py"
   172       by (auto simp add: real_gt_zero_preal_Ex)
   173     hence py_in_pS: "py \<in> ?pS" using y_in_S by simp
   174     with pS_less_pu have "py \<le> psup ?pS"
   175       by (rule preal_psup_le)
   176     thus "y \<le> real_of_preal (psup ?pS)"
   177       using y_is_py by (simp add: real_of_preal_le_iff)
   178   qed
   179 
   180   moreover {
   181     fix x
   182     assume x_ub_S: "\<forall>y\<in>S. y \<le> x"
   183     have "real_of_preal (psup ?pS) \<le> x"
   184     proof -
   185       obtain "s" where s_in_S: "s \<in> S" using not_empty_S ..
   186       hence s_pos: "0 < s" using positive_S by simp
   187 
   188       hence "\<exists> ps. s = real_of_preal ps" by (simp add: real_gt_zero_preal_Ex)
   189       then obtain "ps" where s_is_ps: "s = real_of_preal ps" ..
   190       hence ps_in_pS: "ps \<in> {w. real_of_preal w \<in> S}" using s_in_S by simp
   191 
   192       from x_ub_S have "s \<le> x" using s_in_S ..
   193       hence "0 < x" using s_pos by simp
   194       hence "\<exists> px. x = real_of_preal px" by (simp add: real_gt_zero_preal_Ex)
   195       then obtain "px" where x_is_px: "x = real_of_preal px" ..
   196 
   197       have "\<forall>pe \<in> ?pS. pe \<le> px"
   198       proof
   199         fix pe
   200         assume "pe \<in> ?pS"
   201         hence "real_of_preal pe \<in> S" by simp
   202         hence "real_of_preal pe \<le> x" using x_ub_S by simp
   203         thus "pe \<le> px" using x_is_px by (simp add: real_of_preal_le_iff)
   204       qed
   205 
   206       moreover have "?pS \<noteq> {}" using ps_in_pS by auto
   207       ultimately have "(psup ?pS) \<le> px" by (simp add: psup_le_ub)
   208       thus "real_of_preal (psup ?pS) \<le> x" using x_is_px by (simp add: real_of_preal_le_iff)
   209     qed
   210   }
   211   ultimately show "isLub UNIV S (real_of_preal (psup ?pS))"
   212     by (simp add: isLub_def leastP_def isUb_def setle_def setge_def)
   213 qed
   214 
   215 text {*
   216   \medskip reals Completeness (again!)
   217 *}
   218 
   219 lemma reals_complete:
   220   assumes notempty_S: "\<exists>X. X \<in> S"
   221     and exists_Ub: "\<exists>Y. isUb (UNIV::real set) S Y"
   222   shows "\<exists>t. isLub (UNIV :: real set) S t"
   223 proof -
   224   obtain X where X_in_S: "X \<in> S" using notempty_S ..
   225   obtain Y where Y_isUb: "isUb (UNIV::real set) S Y"
   226     using exists_Ub ..
   227   let ?SHIFT = "{z. \<exists>x \<in>S. z = x + (-X) + 1} \<inter> {x. 0 < x}"
   228 
   229   {
   230     fix x
   231     assume "isUb (UNIV::real set) S x"
   232     hence S_le_x: "\<forall> y \<in> S. y <= x"
   233       by (simp add: isUb_def setle_def)
   234     {
   235       fix s
   236       assume "s \<in> {z. \<exists>x\<in>S. z = x + - X + 1}"
   237       hence "\<exists> x \<in> S. s = x + -X + 1" ..
   238       then obtain x1 where "x1 \<in> S" and "s = x1 + (-X) + 1" ..
   239       moreover hence "x1 \<le> x" using S_le_x by simp
   240       ultimately have "s \<le> x + - X + 1" by arith
   241     }
   242     then have "isUb (UNIV::real set) ?SHIFT (x + (-X) + 1)"
   243       by (auto simp add: isUb_def setle_def)
   244   } note S_Ub_is_SHIFT_Ub = this
   245 
   246   hence "isUb UNIV ?SHIFT (Y + (-X) + 1)" using Y_isUb by simp
   247   hence "\<exists>Z. isUb UNIV ?SHIFT Z" ..
   248   moreover have "\<forall>y \<in> ?SHIFT. 0 < y" by auto
   249   moreover have shifted_not_empty: "\<exists>u. u \<in> ?SHIFT"
   250     using X_in_S and Y_isUb by auto
   251   ultimately obtain t where t_is_Lub: "isLub UNIV ?SHIFT t"
   252     using posreals_complete [of ?SHIFT] by blast
   253 
   254   show ?thesis
   255   proof
   256     show "isLub UNIV S (t + X + (-1))"
   257     proof (rule isLubI2)
   258       {
   259         fix x
   260         assume "isUb (UNIV::real set) S x"
   261         hence "isUb (UNIV::real set) (?SHIFT) (x + (-X) + 1)"
   262           using S_Ub_is_SHIFT_Ub by simp
   263         hence "t \<le> (x + (-X) + 1)"
   264           using t_is_Lub by (simp add: isLub_le_isUb)
   265         hence "t + X + -1 \<le> x" by arith
   266       }
   267       then show "(t + X + -1) <=* Collect (isUb UNIV S)"
   268         by (simp add: setgeI)
   269     next
   270       show "isUb UNIV S (t + X + -1)"
   271       proof -
   272         {
   273           fix y
   274           assume y_in_S: "y \<in> S"
   275           have "y \<le> t + X + -1"
   276           proof -
   277             obtain "u" where u_in_shift: "u \<in> ?SHIFT" using shifted_not_empty ..
   278             hence "\<exists> x \<in> S. u = x + - X + 1" by simp
   279             then obtain "x" where x_and_u: "u = x + - X + 1" ..
   280             have u_le_t: "u \<le> t" using u_in_shift and t_is_Lub by (simp add: isLubD2)
   281 
   282             show ?thesis
   283             proof cases
   284               assume "y \<le> x"
   285               moreover have "x = u + X + - 1" using x_and_u by arith
   286               moreover have "u + X + - 1  \<le> t + X + -1" using u_le_t by arith
   287               ultimately show "y  \<le> t + X + -1" by arith
   288             next
   289               assume "~(y \<le> x)"
   290               hence x_less_y: "x < y" by arith
   291 
   292               have "x + (-X) + 1 \<in> ?SHIFT" using x_and_u and u_in_shift by simp
   293               hence "0 < x + (-X) + 1" by simp
   294               hence "0 < y + (-X) + 1" using x_less_y by arith
   295               hence "y + (-X) + 1 \<in> ?SHIFT" using y_in_S by simp
   296               hence "y + (-X) + 1 \<le> t" using t_is_Lub  by (simp add: isLubD2)
   297               thus ?thesis by simp
   298             qed
   299           qed
   300         }
   301         then show ?thesis by (simp add: isUb_def setle_def)
   302       qed
   303     qed
   304   qed
   305 qed
   306 
   307 text{*A version of the same theorem without all those predicates!*}
   308 lemma reals_complete2:
   309   fixes S :: "(real set)"
   310   assumes "\<exists>y. y\<in>S" and "\<exists>(x::real). \<forall>y\<in>S. y \<le> x"
   311   shows "\<exists>x. (\<forall>y\<in>S. y \<le> x) & 
   312                (\<forall>z. ((\<forall>y\<in>S. y \<le> z) --> x \<le> z))"
   313 proof -
   314   have "\<exists>x. isLub UNIV S x" 
   315     by (rule reals_complete)
   316        (auto simp add: isLub_def isUb_def leastP_def setle_def setge_def prems)
   317   thus ?thesis
   318     by (metis UNIV_I isLub_isUb isLub_le_isUb isUbD isUb_def setleI)
   319 qed
   320 
   321 
   322 subsection {* The Archimedean Property of the Reals *}
   323 
   324 theorem reals_Archimedean:
   325   assumes x_pos: "0 < x"
   326   shows "\<exists>n. inverse (real (Suc n)) < x"
   327 proof (rule ccontr)
   328   assume contr: "\<not> ?thesis"
   329   have "\<forall>n. x * real (Suc n) <= 1"
   330   proof
   331     fix n
   332     from contr have "x \<le> inverse (real (Suc n))"
   333       by (simp add: linorder_not_less)
   334     hence "x \<le> (1 / (real (Suc n)))"
   335       by (simp add: inverse_eq_divide)
   336     moreover have "0 \<le> real (Suc n)"
   337       by (rule real_of_nat_ge_zero)
   338     ultimately have "x * real (Suc n) \<le> (1 / real (Suc n)) * real (Suc n)"
   339       by (rule mult_right_mono)
   340     thus "x * real (Suc n) \<le> 1" by simp
   341   qed
   342   hence "{z. \<exists>n. z = x * (real (Suc n))} *<= 1"
   343     by (simp add: setle_def, safe, rule spec)
   344   hence "isUb (UNIV::real set) {z. \<exists>n. z = x * (real (Suc n))} 1"
   345     by (simp add: isUbI)
   346   hence "\<exists>Y. isUb (UNIV::real set) {z. \<exists>n. z = x* (real (Suc n))} Y" ..
   347   moreover have "\<exists>X. X \<in> {z. \<exists>n. z = x* (real (Suc n))}" by auto
   348   ultimately have "\<exists>t. isLub UNIV {z. \<exists>n. z = x * real (Suc n)} t"
   349     by (simp add: reals_complete)
   350   then obtain "t" where
   351     t_is_Lub: "isLub UNIV {z. \<exists>n. z = x * real (Suc n)} t" ..
   352 
   353   have "\<forall>n::nat. x * real n \<le> t + - x"
   354   proof
   355     fix n
   356     from t_is_Lub have "x * real (Suc n) \<le> t"
   357       by (simp add: isLubD2)
   358     hence  "x * (real n) + x \<le> t"
   359       by (simp add: right_distrib real_of_nat_Suc)
   360     thus  "x * (real n) \<le> t + - x" by arith
   361   qed
   362 
   363   hence "\<forall>m. x * real (Suc m) \<le> t + - x" by simp
   364   hence "{z. \<exists>n. z = x * (real (Suc n))}  *<= (t + - x)"
   365     by (auto simp add: setle_def)
   366   hence "isUb (UNIV::real set) {z. \<exists>n. z = x * (real (Suc n))} (t + (-x))"
   367     by (simp add: isUbI)
   368   hence "t \<le> t + - x"
   369     using t_is_Lub by (simp add: isLub_le_isUb)
   370   thus False using x_pos by arith
   371 qed
   372 
   373 text {*
   374   There must be other proofs, e.g. @{text "Suc"} of the largest
   375   integer in the cut representing @{text "x"}.
   376 *}
   377 
   378 lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"
   379 proof cases
   380   assume "x \<le> 0"
   381   hence "x < real (1::nat)" by simp
   382   thus ?thesis ..
   383 next
   384   assume "\<not> x \<le> 0"
   385   hence x_greater_zero: "0 < x" by simp
   386   hence "0 < inverse x" by simp
   387   then obtain n where "inverse (real (Suc n)) < inverse x"
   388     using reals_Archimedean by blast
   389   hence "inverse (real (Suc n)) * x < inverse x * x"
   390     using x_greater_zero by (rule mult_strict_right_mono)
   391   hence "inverse (real (Suc n)) * x < 1"
   392     using x_greater_zero by simp
   393   hence "real (Suc n) * (inverse (real (Suc n)) * x) < real (Suc n) * 1"
   394     by (rule mult_strict_left_mono) simp
   395   hence "x < real (Suc n)"
   396     by (simp add: algebra_simps)
   397   thus "\<exists>(n::nat). x < real n" ..
   398 qed
   399 
   400 instance real :: archimedean_field
   401 proof
   402   fix r :: real
   403   obtain n :: nat where "r < real n"
   404     using reals_Archimedean2 ..
   405   then have "r \<le> of_int (int n)"
   406     unfolding real_eq_of_nat by simp
   407   then show "\<exists>z. r \<le> of_int z" ..
   408 qed
   409 
   410 lemma reals_Archimedean3:
   411   assumes x_greater_zero: "0 < x"
   412   shows "\<forall>(y::real). \<exists>(n::nat). y < real n * x"
   413   unfolding real_of_nat_def using `0 < x`
   414   by (auto intro: ex_less_of_nat_mult)
   415 
   416 lemma reals_Archimedean6:
   417      "0 \<le> r ==> \<exists>(n::nat). real (n - 1) \<le> r & r < real (n)"
   418 unfolding real_of_nat_def
   419 apply (rule exI [where x="nat (floor r + 1)"])
   420 apply (insert floor_correct [of r])
   421 apply (simp add: nat_add_distrib of_nat_nat)
   422 done
   423 
   424 lemma reals_Archimedean6a: "0 \<le> r ==> \<exists>n. real (n) \<le> r & r < real (Suc n)"
   425   by (drule reals_Archimedean6) auto
   426 
   427 lemma reals_Archimedean_6b_int:
   428      "0 \<le> r ==> \<exists>n::int. real n \<le> r & r < real (n+1)"
   429   unfolding real_of_int_def by (rule floor_exists)
   430 
   431 lemma reals_Archimedean_6c_int:
   432      "r < 0 ==> \<exists>n::int. real n \<le> r & r < real (n+1)"
   433   unfolding real_of_int_def by (rule floor_exists)
   434 
   435 
   436 subsection{*Density of the Rational Reals in the Reals*}
   437 
   438 text{* This density proof is due to Stefan Richter and was ported by TN.  The
   439 original source is \emph{Real Analysis} by H.L. Royden.
   440 It employs the Archimedean property of the reals. *}
   441 
   442 lemma Rats_dense_in_nn_real: fixes x::real
   443 assumes "0\<le>x" and "x<y" shows "\<exists>r \<in> \<rat>.  x<r \<and> r<y"
   444 proof -
   445   from `x<y` have "0 < y-x" by simp
   446   with reals_Archimedean obtain q::nat 
   447     where q: "inverse (real q) < y-x" and "0 < real q" by auto  
   448   def p \<equiv> "LEAST n.  y \<le> real (Suc n)/real q"  
   449   from reals_Archimedean2 obtain n::nat where "y * real q < real n" by auto
   450   with `0 < real q` have ex: "y \<le> real n/real q" (is "?P n")
   451     by (simp add: pos_less_divide_eq[THEN sym])
   452   also from assms have "\<not> y \<le> real (0::nat) / real q" by simp
   453   ultimately have main: "(LEAST n. y \<le> real n/real q) = Suc p"
   454     by (unfold p_def) (rule Least_Suc)
   455   also from ex have "?P (LEAST x. ?P x)" by (rule LeastI)
   456   ultimately have suc: "y \<le> real (Suc p) / real q" by simp
   457   def r \<equiv> "real p/real q"
   458   have "x = y-(y-x)" by simp
   459   also from suc q have "\<dots> < real (Suc p)/real q - inverse (real q)" by arith
   460   also have "\<dots> = real p / real q"
   461     by (simp only: inverse_eq_divide real_diff_def real_of_nat_Suc 
   462     minus_divide_left add_divide_distrib[THEN sym]) simp
   463   finally have "x<r" by (unfold r_def)
   464   have "p<Suc p" .. also note main[THEN sym]
   465   finally have "\<not> ?P p"  by (rule not_less_Least)
   466   hence "r<y" by (simp add: r_def)
   467   from r_def have "r \<in> \<rat>" by simp
   468   with `x<r` `r<y` show ?thesis by fast
   469 qed
   470 
   471 theorem Rats_dense_in_real: fixes x y :: real
   472 assumes "x<y" shows "\<exists>r \<in> \<rat>.  x<r \<and> r<y"
   473 proof -
   474   from reals_Archimedean2 obtain n::nat where "-x < real n" by auto
   475   hence "0 \<le> x + real n" by arith
   476   also from `x<y` have "x + real n < y + real n" by arith
   477   ultimately have "\<exists>r \<in> \<rat>. x + real n < r \<and> r < y + real n"
   478     by(rule Rats_dense_in_nn_real)
   479   then obtain r where "r \<in> \<rat>" and r2: "x + real n < r" 
   480     and r3: "r < y + real n"
   481     by blast
   482   have "r - real n = r + real (int n)/real (-1::int)" by simp
   483   also from `r\<in>\<rat>` have "r + real (int n)/real (-1::int) \<in> \<rat>" by simp
   484   also from r2 have "x < r - real n" by arith
   485   moreover from r3 have "r - real n < y" by arith
   486   ultimately show ?thesis by fast
   487 qed
   488 
   489 
   490 subsection{*Floor and Ceiling Functions from the Reals to the Integers*}
   491 
   492 lemma number_of_less_real_of_int_iff [simp]:
   493      "((number_of n) < real (m::int)) = (number_of n < m)"
   494 apply auto
   495 apply (rule real_of_int_less_iff [THEN iffD1])
   496 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
   497 done
   498 
   499 lemma number_of_less_real_of_int_iff2 [simp]:
   500      "(real (m::int) < (number_of n)) = (m < number_of n)"
   501 apply auto
   502 apply (rule real_of_int_less_iff [THEN iffD1])
   503 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
   504 done
   505 
   506 lemma number_of_le_real_of_int_iff [simp]:
   507      "((number_of n) \<le> real (m::int)) = (number_of n \<le> m)"
   508 by (simp add: linorder_not_less [symmetric])
   509 
   510 lemma number_of_le_real_of_int_iff2 [simp]:
   511      "(real (m::int) \<le> (number_of n)) = (m \<le> number_of n)"
   512 by (simp add: linorder_not_less [symmetric])
   513 
   514 lemma floor_real_of_nat_zero: "floor (real (0::nat)) = 0"
   515 by auto (* delete? *)
   516 
   517 lemma floor_real_of_nat [simp]: "floor (real (n::nat)) = int n"
   518 unfolding real_of_nat_def by simp
   519 
   520 lemma floor_minus_real_of_nat [simp]: "floor (- real (n::nat)) = - int n"
   521 unfolding real_of_nat_def by (simp add: floor_minus)
   522 
   523 lemma floor_real_of_int [simp]: "floor (real (n::int)) = n"
   524 unfolding real_of_int_def by simp
   525 
   526 lemma floor_minus_real_of_int [simp]: "floor (- real (n::int)) = - n"
   527 unfolding real_of_int_def by (simp add: floor_minus)
   528 
   529 lemma real_lb_ub_int: " \<exists>n::int. real n \<le> r & r < real (n+1)"
   530 unfolding real_of_int_def by (rule floor_exists)
   531 
   532 lemma lemma_floor:
   533   assumes a1: "real m \<le> r" and a2: "r < real n + 1"
   534   shows "m \<le> (n::int)"
   535 proof -
   536   have "real m < real n + 1" using a1 a2 by (rule order_le_less_trans)
   537   also have "... = real (n + 1)" by simp
   538   finally have "m < n + 1" by (simp only: real_of_int_less_iff)
   539   thus ?thesis by arith
   540 qed
   541 
   542 lemma real_of_int_floor_le [simp]: "real (floor r) \<le> r"
   543 unfolding real_of_int_def by (rule of_int_floor_le)
   544 
   545 lemma lemma_floor2: "real n < real (x::int) + 1 ==> n \<le> x"
   546 by (auto intro: lemma_floor)
   547 
   548 lemma real_of_int_floor_cancel [simp]:
   549     "(real (floor x) = x) = (\<exists>n::int. x = real n)"
   550   using floor_real_of_int by metis
   551 
   552 lemma floor_eq: "[| real n < x; x < real n + 1 |] ==> floor x = n"
   553   unfolding real_of_int_def using floor_unique [of n x] by simp
   554 
   555 lemma floor_eq2: "[| real n \<le> x; x < real n + 1 |] ==> floor x = n"
   556   unfolding real_of_int_def by (rule floor_unique)
   557 
   558 lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"
   559 apply (rule inj_int [THEN injD])
   560 apply (simp add: real_of_nat_Suc)
   561 apply (simp add: real_of_nat_Suc floor_eq floor_eq [where n = "int n"])
   562 done
   563 
   564 lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"
   565 apply (drule order_le_imp_less_or_eq)
   566 apply (auto intro: floor_eq3)
   567 done
   568 
   569 lemma floor_number_of_eq:
   570      "floor(number_of n :: real) = (number_of n :: int)"
   571   by (rule floor_number_of) (* already declared [simp] *)
   572 
   573 lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real(floor r)"
   574   unfolding real_of_int_def using floor_correct [of r] by simp
   575 
   576 lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real(floor r)"
   577   unfolding real_of_int_def using floor_correct [of r] by simp
   578 
   579 lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real(floor r) + 1"
   580   unfolding real_of_int_def using floor_correct [of r] by simp
   581 
   582 lemma real_of_int_floor_add_one_gt [simp]: "r < real(floor r) + 1"
   583   unfolding real_of_int_def using floor_correct [of r] by simp
   584 
   585 lemma le_floor: "real a <= x ==> a <= floor x"
   586   unfolding real_of_int_def by (simp add: le_floor_iff)
   587 
   588 lemma real_le_floor: "a <= floor x ==> real a <= x"
   589   unfolding real_of_int_def by (simp add: le_floor_iff)
   590 
   591 lemma le_floor_eq: "(a <= floor x) = (real a <= x)"
   592   unfolding real_of_int_def by (rule le_floor_iff)
   593 
   594 lemma le_floor_eq_number_of:
   595     "(number_of n <= floor x) = (number_of n <= x)"
   596   by (rule number_of_le_floor) (* already declared [simp] *)
   597 
   598 lemma le_floor_eq_zero: "(0 <= floor x) = (0 <= x)"
   599   by (rule zero_le_floor) (* already declared [simp] *)
   600 
   601 lemma le_floor_eq_one: "(1 <= floor x) = (1 <= x)"
   602   by (rule one_le_floor) (* already declared [simp] *)
   603 
   604 lemma floor_less_eq: "(floor x < a) = (x < real a)"
   605   unfolding real_of_int_def by (rule floor_less_iff)
   606 
   607 lemma floor_less_eq_number_of:
   608     "(floor x < number_of n) = (x < number_of n)"
   609   by (rule floor_less_number_of) (* already declared [simp] *)
   610 
   611 lemma floor_less_eq_zero: "(floor x < 0) = (x < 0)"
   612   by (rule floor_less_zero) (* already declared [simp] *)
   613 
   614 lemma floor_less_eq_one: "(floor x < 1) = (x < 1)"
   615   by (rule floor_less_one) (* already declared [simp] *)
   616 
   617 lemma less_floor_eq: "(a < floor x) = (real a + 1 <= x)"
   618   unfolding real_of_int_def by (rule less_floor_iff)
   619 
   620 lemma less_floor_eq_number_of:
   621     "(number_of n < floor x) = (number_of n + 1 <= x)"
   622   by (rule number_of_less_floor) (* already declared [simp] *)
   623 
   624 lemma less_floor_eq_zero: "(0 < floor x) = (1 <= x)"
   625   by (rule zero_less_floor) (* already declared [simp] *)
   626 
   627 lemma less_floor_eq_one: "(1 < floor x) = (2 <= x)"
   628   by (rule one_less_floor) (* already declared [simp] *)
   629 
   630 lemma floor_le_eq: "(floor x <= a) = (x < real a + 1)"
   631   unfolding real_of_int_def by (rule floor_le_iff)
   632 
   633 lemma floor_le_eq_number_of:
   634     "(floor x <= number_of n) = (x < number_of n + 1)"
   635   by (rule floor_le_number_of) (* already declared [simp] *)
   636 
   637 lemma floor_le_eq_zero: "(floor x <= 0) = (x < 1)"
   638   by (rule floor_le_zero) (* already declared [simp] *)
   639 
   640 lemma floor_le_eq_one: "(floor x <= 1) = (x < 2)"
   641   by (rule floor_le_one) (* already declared [simp] *)
   642 
   643 lemma floor_add [simp]: "floor (x + real a) = floor x + a"
   644   unfolding real_of_int_def by (rule floor_add_of_int)
   645 
   646 lemma floor_subtract [simp]: "floor (x - real a) = floor x - a"
   647   unfolding real_of_int_def by (rule floor_diff_of_int)
   648 
   649 lemma floor_subtract_number_of: "floor (x - number_of n) =
   650     floor x - number_of n"
   651   by (rule floor_diff_number_of) (* already declared [simp] *)
   652 
   653 lemma floor_subtract_one: "floor (x - 1) = floor x - 1"
   654   by (rule floor_diff_one) (* already declared [simp] *)
   655 
   656 lemma le_mult_floor:
   657   assumes "0 \<le> (a :: real)" and "0 \<le> b"
   658   shows "floor a * floor b \<le> floor (a * b)"
   659 proof -
   660   have "real (floor a) \<le> a"
   661     and "real (floor b) \<le> b" by auto
   662   hence "real (floor a * floor b) \<le> a * b"
   663     using assms by (auto intro!: mult_mono)
   664   also have "a * b < real (floor (a * b) + 1)" by auto
   665   finally show ?thesis unfolding real_of_int_less_iff by simp
   666 qed
   667 
   668 lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"
   669   unfolding real_of_nat_def by simp
   670 
   671 lemma ceiling_real_of_nat_zero: "ceiling (real (0::nat)) = 0"
   672 by auto (* delete? *)
   673 
   674 lemma ceiling_floor [simp]: "ceiling (real (floor r)) = floor r"
   675   unfolding real_of_int_def by simp
   676 
   677 lemma floor_ceiling [simp]: "floor (real (ceiling r)) = ceiling r"
   678   unfolding real_of_int_def by simp
   679 
   680 lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)"
   681   unfolding real_of_int_def by (rule le_of_int_ceiling)
   682 
   683 lemma ceiling_real_of_int [simp]: "ceiling (real (n::int)) = n"
   684   unfolding real_of_int_def by simp
   685 
   686 lemma real_of_int_ceiling_cancel [simp]:
   687      "(real (ceiling x) = x) = (\<exists>n::int. x = real n)"
   688   using ceiling_real_of_int by metis
   689 
   690 lemma ceiling_eq: "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1"
   691   unfolding real_of_int_def using ceiling_unique [of "n + 1" x] by simp
   692 
   693 lemma ceiling_eq2: "[| real n < x; x \<le> real n + 1 |] ==> ceiling x = n + 1"
   694   unfolding real_of_int_def using ceiling_unique [of "n + 1" x] by simp
   695 
   696 lemma ceiling_eq3: "[| real n - 1 < x; x \<le> real n  |] ==> ceiling x = n"
   697   unfolding real_of_int_def using ceiling_unique [of n x] by simp
   698 
   699 lemma ceiling_number_of_eq:
   700      "ceiling (number_of n :: real) = (number_of n)"
   701   by (rule ceiling_number_of) (* already declared [simp] *)
   702 
   703 lemma real_of_int_ceiling_diff_one_le [simp]: "real (ceiling r) - 1 \<le> r"
   704   unfolding real_of_int_def using ceiling_correct [of r] by simp
   705 
   706 lemma real_of_int_ceiling_le_add_one [simp]: "real (ceiling r) \<le> r + 1"
   707   unfolding real_of_int_def using ceiling_correct [of r] by simp
   708 
   709 lemma ceiling_le: "x <= real a ==> ceiling x <= a"
   710   unfolding real_of_int_def by (simp add: ceiling_le_iff)
   711 
   712 lemma ceiling_le_real: "ceiling x <= a ==> x <= real a"
   713   unfolding real_of_int_def by (simp add: ceiling_le_iff)
   714 
   715 lemma ceiling_le_eq: "(ceiling x <= a) = (x <= real a)"
   716   unfolding real_of_int_def by (rule ceiling_le_iff)
   717 
   718 lemma ceiling_le_eq_number_of:
   719     "(ceiling x <= number_of n) = (x <= number_of n)"
   720   by (rule ceiling_le_number_of) (* already declared [simp] *)
   721 
   722 lemma ceiling_le_zero_eq: "(ceiling x <= 0) = (x <= 0)"
   723   by (rule ceiling_le_zero) (* already declared [simp] *)
   724 
   725 lemma ceiling_le_eq_one: "(ceiling x <= 1) = (x <= 1)"
   726   by (rule ceiling_le_one) (* already declared [simp] *)
   727 
   728 lemma less_ceiling_eq: "(a < ceiling x) = (real a < x)"
   729   unfolding real_of_int_def by (rule less_ceiling_iff)
   730 
   731 lemma less_ceiling_eq_number_of:
   732     "(number_of n < ceiling x) = (number_of n < x)"
   733   by (rule number_of_less_ceiling) (* already declared [simp] *)
   734 
   735 lemma less_ceiling_eq_zero: "(0 < ceiling x) = (0 < x)"
   736   by (rule zero_less_ceiling) (* already declared [simp] *)
   737 
   738 lemma less_ceiling_eq_one: "(1 < ceiling x) = (1 < x)"
   739   by (rule one_less_ceiling) (* already declared [simp] *)
   740 
   741 lemma ceiling_less_eq: "(ceiling x < a) = (x <= real a - 1)"
   742   unfolding real_of_int_def by (rule ceiling_less_iff)
   743 
   744 lemma ceiling_less_eq_number_of:
   745     "(ceiling x < number_of n) = (x <= number_of n - 1)"
   746   by (rule ceiling_less_number_of) (* already declared [simp] *)
   747 
   748 lemma ceiling_less_eq_zero: "(ceiling x < 0) = (x <= -1)"
   749   by (rule ceiling_less_zero) (* already declared [simp] *)
   750 
   751 lemma ceiling_less_eq_one: "(ceiling x < 1) = (x <= 0)"
   752   by (rule ceiling_less_one) (* already declared [simp] *)
   753 
   754 lemma le_ceiling_eq: "(a <= ceiling x) = (real a - 1 < x)"
   755   unfolding real_of_int_def by (rule le_ceiling_iff)
   756 
   757 lemma le_ceiling_eq_number_of:
   758     "(number_of n <= ceiling x) = (number_of n - 1 < x)"
   759   by (rule number_of_le_ceiling) (* already declared [simp] *)
   760 
   761 lemma le_ceiling_eq_zero: "(0 <= ceiling x) = (-1 < x)"
   762   by (rule zero_le_ceiling) (* already declared [simp] *)
   763 
   764 lemma le_ceiling_eq_one: "(1 <= ceiling x) = (0 < x)"
   765   by (rule one_le_ceiling) (* already declared [simp] *)
   766 
   767 lemma ceiling_add [simp]: "ceiling (x + real a) = ceiling x + a"
   768   unfolding real_of_int_def by (rule ceiling_add_of_int)
   769 
   770 lemma ceiling_subtract [simp]: "ceiling (x - real a) = ceiling x - a"
   771   unfolding real_of_int_def by (rule ceiling_diff_of_int)
   772 
   773 lemma ceiling_subtract_number_of: "ceiling (x - number_of n) =
   774     ceiling x - number_of n"
   775   by (rule ceiling_diff_number_of) (* already declared [simp] *)
   776 
   777 lemma ceiling_subtract_one: "ceiling (x - 1) = ceiling x - 1"
   778   by (rule ceiling_diff_one) (* already declared [simp] *)
   779 
   780 
   781 subsection {* Versions for the natural numbers *}
   782 
   783 definition
   784   natfloor :: "real => nat" where
   785   "natfloor x = nat(floor x)"
   786 
   787 definition
   788   natceiling :: "real => nat" where
   789   "natceiling x = nat(ceiling x)"
   790 
   791 lemma natfloor_zero [simp]: "natfloor 0 = 0"
   792   by (unfold natfloor_def, simp)
   793 
   794 lemma natfloor_one [simp]: "natfloor 1 = 1"
   795   by (unfold natfloor_def, simp)
   796 
   797 lemma zero_le_natfloor [simp]: "0 <= natfloor x"
   798   by (unfold natfloor_def, simp)
   799 
   800 lemma natfloor_number_of_eq [simp]: "natfloor (number_of n) = number_of n"
   801   by (unfold natfloor_def, simp)
   802 
   803 lemma natfloor_real_of_nat [simp]: "natfloor(real n) = n"
   804   by (unfold natfloor_def, simp)
   805 
   806 lemma real_natfloor_le: "0 <= x ==> real(natfloor x) <= x"
   807   by (unfold natfloor_def, simp)
   808 
   809 lemma natfloor_neg: "x <= 0 ==> natfloor x = 0"
   810   apply (unfold natfloor_def)
   811   apply (subgoal_tac "floor x <= floor 0")
   812   apply simp
   813   apply (erule floor_mono)
   814 done
   815 
   816 lemma natfloor_mono: "x <= y ==> natfloor x <= natfloor y"
   817   apply (case_tac "0 <= x")
   818   apply (subst natfloor_def)+
   819   apply (subst nat_le_eq_zle)
   820   apply force
   821   apply (erule floor_mono)
   822   apply (subst natfloor_neg)
   823   apply simp
   824   apply simp
   825 done
   826 
   827 lemma le_natfloor: "real x <= a ==> x <= natfloor a"
   828   apply (unfold natfloor_def)
   829   apply (subst nat_int [THEN sym])
   830   apply (subst nat_le_eq_zle)
   831   apply simp
   832   apply (rule le_floor)
   833   apply simp
   834 done
   835 
   836 lemma less_natfloor:
   837   assumes "0 \<le> x" and "x < real (n :: nat)"
   838   shows "natfloor x < n"
   839 proof (rule ccontr)
   840   assume "\<not> ?thesis" hence *: "n \<le> natfloor x" by simp
   841   note assms(2)
   842   also have "real n \<le> real (natfloor x)"
   843     using * unfolding real_of_nat_le_iff .
   844   finally have "x < real (natfloor x)" .
   845   with real_natfloor_le[OF assms(1)]
   846   show False by auto
   847 qed
   848 
   849 lemma le_natfloor_eq: "0 <= x ==> (a <= natfloor x) = (real a <= x)"
   850   apply (rule iffI)
   851   apply (rule order_trans)
   852   prefer 2
   853   apply (erule real_natfloor_le)
   854   apply (subst real_of_nat_le_iff)
   855   apply assumption
   856   apply (erule le_natfloor)
   857 done
   858 
   859 lemma le_natfloor_eq_number_of [simp]:
   860     "~ neg((number_of n)::int) ==> 0 <= x ==>
   861       (number_of n <= natfloor x) = (number_of n <= x)"
   862   apply (subst le_natfloor_eq, assumption)
   863   apply simp
   864 done
   865 
   866 lemma le_natfloor_eq_one [simp]: "(1 <= natfloor x) = (1 <= x)"
   867   apply (case_tac "0 <= x")
   868   apply (subst le_natfloor_eq, assumption, simp)
   869   apply (rule iffI)
   870   apply (subgoal_tac "natfloor x <= natfloor 0")
   871   apply simp
   872   apply (rule natfloor_mono)
   873   apply simp
   874   apply simp
   875 done
   876 
   877 lemma natfloor_eq: "real n <= x ==> x < real n + 1 ==> natfloor x = n"
   878   apply (unfold natfloor_def)
   879   apply (subst (2) nat_int [THEN sym])
   880   apply (subst eq_nat_nat_iff)
   881   apply simp
   882   apply simp
   883   apply (rule floor_eq2)
   884   apply auto
   885 done
   886 
   887 lemma real_natfloor_add_one_gt: "x < real(natfloor x) + 1"
   888   apply (case_tac "0 <= x")
   889   apply (unfold natfloor_def)
   890   apply simp
   891   apply simp_all
   892 done
   893 
   894 lemma real_natfloor_gt_diff_one: "x - 1 < real(natfloor x)"
   895 using real_natfloor_add_one_gt by (simp add: algebra_simps)
   896 
   897 lemma ge_natfloor_plus_one_imp_gt: "natfloor z + 1 <= n ==> z < real n"
   898   apply (subgoal_tac "z < real(natfloor z) + 1")
   899   apply arith
   900   apply (rule real_natfloor_add_one_gt)
   901 done
   902 
   903 lemma natfloor_add [simp]: "0 <= x ==> natfloor (x + real a) = natfloor x + a"
   904   apply (unfold natfloor_def)
   905   apply (subgoal_tac "real a = real (int a)")
   906   apply (erule ssubst)
   907   apply (simp add: nat_add_distrib del: real_of_int_of_nat_eq)
   908   apply simp
   909 done
   910 
   911 lemma natfloor_add_number_of [simp]:
   912     "~neg ((number_of n)::int) ==> 0 <= x ==>
   913       natfloor (x + number_of n) = natfloor x + number_of n"
   914   apply (subst natfloor_add [THEN sym])
   915   apply simp_all
   916 done
   917 
   918 lemma natfloor_add_one: "0 <= x ==> natfloor(x + 1) = natfloor x + 1"
   919   apply (subst natfloor_add [THEN sym])
   920   apply assumption
   921   apply simp
   922 done
   923 
   924 lemma natfloor_subtract [simp]: "real a <= x ==>
   925     natfloor(x - real a) = natfloor x - a"
   926   apply (unfold natfloor_def)
   927   apply (subgoal_tac "real a = real (int a)")
   928   apply (erule ssubst)
   929   apply (simp del: real_of_int_of_nat_eq)
   930   apply simp
   931 done
   932 
   933 lemma natfloor_div_nat: "1 <= x ==> y > 0 ==>
   934   natfloor (x / real y) = natfloor x div y"
   935 proof -
   936   assume "1 <= (x::real)" and "(y::nat) > 0"
   937   have "natfloor x = (natfloor x) div y * y + (natfloor x) mod y"
   938     by simp
   939   then have a: "real(natfloor x) = real ((natfloor x) div y) * real y +
   940     real((natfloor x) mod y)"
   941     by (simp only: real_of_nat_add [THEN sym] real_of_nat_mult [THEN sym])
   942   have "x = real(natfloor x) + (x - real(natfloor x))"
   943     by simp
   944   then have "x = real ((natfloor x) div y) * real y +
   945       real((natfloor x) mod y) + (x - real(natfloor x))"
   946     by (simp add: a)
   947   then have "x / real y = ... / real y"
   948     by simp
   949   also have "... = real((natfloor x) div y) + real((natfloor x) mod y) /
   950     real y + (x - real(natfloor x)) / real y"
   951     by (auto simp add: algebra_simps add_divide_distrib
   952       diff_divide_distrib prems)
   953   finally have "natfloor (x / real y) = natfloor(...)" by simp
   954   also have "... = natfloor(real((natfloor x) mod y) /
   955     real y + (x - real(natfloor x)) / real y + real((natfloor x) div y))"
   956     by (simp add: add_ac)
   957   also have "... = natfloor(real((natfloor x) mod y) /
   958     real y + (x - real(natfloor x)) / real y) + (natfloor x) div y"
   959     apply (rule natfloor_add)
   960     apply (rule add_nonneg_nonneg)
   961     apply (rule divide_nonneg_pos)
   962     apply simp
   963     apply (simp add: prems)
   964     apply (rule divide_nonneg_pos)
   965     apply (simp add: algebra_simps)
   966     apply (rule real_natfloor_le)
   967     apply (insert prems, auto)
   968     done
   969   also have "natfloor(real((natfloor x) mod y) /
   970     real y + (x - real(natfloor x)) / real y) = 0"
   971     apply (rule natfloor_eq)
   972     apply simp
   973     apply (rule add_nonneg_nonneg)
   974     apply (rule divide_nonneg_pos)
   975     apply force
   976     apply (force simp add: prems)
   977     apply (rule divide_nonneg_pos)
   978     apply (simp add: algebra_simps)
   979     apply (rule real_natfloor_le)
   980     apply (auto simp add: prems)
   981     apply (insert prems, arith)
   982     apply (simp add: add_divide_distrib [THEN sym])
   983     apply (subgoal_tac "real y = real y - 1 + 1")
   984     apply (erule ssubst)
   985     apply (rule add_le_less_mono)
   986     apply (simp add: algebra_simps)
   987     apply (subgoal_tac "1 + real(natfloor x mod y) =
   988       real(natfloor x mod y + 1)")
   989     apply (erule ssubst)
   990     apply (subst real_of_nat_le_iff)
   991     apply (subgoal_tac "natfloor x mod y < y")
   992     apply arith
   993     apply (rule mod_less_divisor)
   994     apply auto
   995     using real_natfloor_add_one_gt
   996     apply (simp add: algebra_simps)
   997     done
   998   finally show ?thesis by simp
   999 qed
  1000 
  1001 lemma le_mult_natfloor:
  1002   assumes "0 \<le> (a :: real)" and "0 \<le> b"
  1003   shows "natfloor a * natfloor b \<le> natfloor (a * b)"
  1004   unfolding natfloor_def
  1005   apply (subst nat_mult_distrib[symmetric])
  1006   using assms apply simp
  1007   apply (subst nat_le_eq_zle)
  1008   using assms le_mult_floor by (auto intro!: mult_nonneg_nonneg)
  1009 
  1010 lemma natceiling_zero [simp]: "natceiling 0 = 0"
  1011   by (unfold natceiling_def, simp)
  1012 
  1013 lemma natceiling_one [simp]: "natceiling 1 = 1"
  1014   by (unfold natceiling_def, simp)
  1015 
  1016 lemma zero_le_natceiling [simp]: "0 <= natceiling x"
  1017   by (unfold natceiling_def, simp)
  1018 
  1019 lemma natceiling_number_of_eq [simp]: "natceiling (number_of n) = number_of n"
  1020   by (unfold natceiling_def, simp)
  1021 
  1022 lemma natceiling_real_of_nat [simp]: "natceiling(real n) = n"
  1023   by (unfold natceiling_def, simp)
  1024 
  1025 lemma real_natceiling_ge: "x <= real(natceiling x)"
  1026   apply (unfold natceiling_def)
  1027   apply (case_tac "x < 0")
  1028   apply simp
  1029   apply (subst real_nat_eq_real)
  1030   apply (subgoal_tac "ceiling 0 <= ceiling x")
  1031   apply simp
  1032   apply (rule ceiling_mono)
  1033   apply simp
  1034   apply simp
  1035 done
  1036 
  1037 lemma natceiling_neg: "x <= 0 ==> natceiling x = 0"
  1038   apply (unfold natceiling_def)
  1039   apply simp
  1040 done
  1041 
  1042 lemma natceiling_mono: "x <= y ==> natceiling x <= natceiling y"
  1043   apply (case_tac "0 <= x")
  1044   apply (subst natceiling_def)+
  1045   apply (subst nat_le_eq_zle)
  1046   apply (rule disjI2)
  1047   apply (subgoal_tac "real (0::int) <= real(ceiling y)")
  1048   apply simp
  1049   apply (rule order_trans)
  1050   apply simp
  1051   apply (erule order_trans)
  1052   apply simp
  1053   apply (erule ceiling_mono)
  1054   apply (subst natceiling_neg)
  1055   apply simp_all
  1056 done
  1057 
  1058 lemma natceiling_le: "x <= real a ==> natceiling x <= a"
  1059   apply (unfold natceiling_def)
  1060   apply (case_tac "x < 0")
  1061   apply simp
  1062   apply (subst (2) nat_int [THEN sym])
  1063   apply (subst nat_le_eq_zle)
  1064   apply simp
  1065   apply (rule ceiling_le)
  1066   apply simp
  1067 done
  1068 
  1069 lemma natceiling_le_eq: "0 <= x ==> (natceiling x <= a) = (x <= real a)"
  1070   apply (rule iffI)
  1071   apply (rule order_trans)
  1072   apply (rule real_natceiling_ge)
  1073   apply (subst real_of_nat_le_iff)
  1074   apply assumption
  1075   apply (erule natceiling_le)
  1076 done
  1077 
  1078 lemma natceiling_le_eq_number_of [simp]:
  1079     "~ neg((number_of n)::int) ==> 0 <= x ==>
  1080       (natceiling x <= number_of n) = (x <= number_of n)"
  1081   apply (subst natceiling_le_eq, assumption)
  1082   apply simp
  1083 done
  1084 
  1085 lemma natceiling_le_eq_one: "(natceiling x <= 1) = (x <= 1)"
  1086   apply (case_tac "0 <= x")
  1087   apply (subst natceiling_le_eq)
  1088   apply assumption
  1089   apply simp
  1090   apply (subst natceiling_neg)
  1091   apply simp
  1092   apply simp
  1093 done
  1094 
  1095 lemma natceiling_eq: "real n < x ==> x <= real n + 1 ==> natceiling x = n + 1"
  1096   apply (unfold natceiling_def)
  1097   apply (simplesubst nat_int [THEN sym]) back back
  1098   apply (subgoal_tac "nat(int n) + 1 = nat(int n + 1)")
  1099   apply (erule ssubst)
  1100   apply (subst eq_nat_nat_iff)
  1101   apply (subgoal_tac "ceiling 0 <= ceiling x")
  1102   apply simp
  1103   apply (rule ceiling_mono)
  1104   apply force
  1105   apply force
  1106   apply (rule ceiling_eq2)
  1107   apply (simp, simp)
  1108   apply (subst nat_add_distrib)
  1109   apply auto
  1110 done
  1111 
  1112 lemma natceiling_add [simp]: "0 <= x ==>
  1113     natceiling (x + real a) = natceiling x + a"
  1114   apply (unfold natceiling_def)
  1115   apply (subgoal_tac "real a = real (int a)")
  1116   apply (erule ssubst)
  1117   apply (simp del: real_of_int_of_nat_eq)
  1118   apply (subst nat_add_distrib)
  1119   apply (subgoal_tac "0 = ceiling 0")
  1120   apply (erule ssubst)
  1121   apply (erule ceiling_mono)
  1122   apply simp_all
  1123 done
  1124 
  1125 lemma natceiling_add_number_of [simp]:
  1126     "~ neg ((number_of n)::int) ==> 0 <= x ==>
  1127       natceiling (x + number_of n) = natceiling x + number_of n"
  1128   apply (subst natceiling_add [THEN sym])
  1129   apply simp_all
  1130 done
  1131 
  1132 lemma natceiling_add_one: "0 <= x ==> natceiling(x + 1) = natceiling x + 1"
  1133   apply (subst natceiling_add [THEN sym])
  1134   apply assumption
  1135   apply simp
  1136 done
  1137 
  1138 lemma natceiling_subtract [simp]: "real a <= x ==>
  1139     natceiling(x - real a) = natceiling x - a"
  1140   apply (unfold natceiling_def)
  1141   apply (subgoal_tac "real a = real (int a)")
  1142   apply (erule ssubst)
  1143   apply (simp del: real_of_int_of_nat_eq)
  1144   apply simp
  1145 done
  1146 
  1147 
  1148 end