src/HOL/RComplete.thy
 author bulwahn Wed Jan 20 11:56:45 2010 +0100 (2010-01-20) changeset 34948 2d5f2a9f7601 parent 32960 69916a850301 child 35028 108662d50512 permissions -rw-r--r--
     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::ordered_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  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  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 ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"

   657   unfolding real_of_nat_def by simp

   658

   659 lemma ceiling_real_of_nat_zero: "ceiling (real (0::nat)) = 0"

   660 by auto (* delete? *)

   661

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

   663   unfolding real_of_int_def by simp

   664

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

   666   unfolding real_of_int_def by simp

   667

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

   669   unfolding real_of_int_def by (rule le_of_int_ceiling)

   670

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

   672   unfolding real_of_int_def by simp

   673

   674 lemma real_of_int_ceiling_cancel [simp]:

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

   676   using ceiling_real_of_int by metis

   677

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

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

   680

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

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

   683

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

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

   686

   687 lemma ceiling_number_of_eq:

   688      "ceiling (number_of n :: real) = (number_of n)"

   689   by (rule ceiling_number_of) (* already declared [simp] *)

   690

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

   692   unfolding real_of_int_def using ceiling_correct [of r] by simp

   693

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

   695   unfolding real_of_int_def using ceiling_correct [of r] by simp

   696

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

   698   unfolding real_of_int_def by (simp add: ceiling_le_iff)

   699

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

   701   unfolding real_of_int_def by (simp add: ceiling_le_iff)

   702

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

   704   unfolding real_of_int_def by (rule ceiling_le_iff)

   705

   706 lemma ceiling_le_eq_number_of:

   707     "(ceiling x <= number_of n) = (x <= number_of n)"

   708   by (rule ceiling_le_number_of) (* already declared [simp] *)

   709

   710 lemma ceiling_le_zero_eq: "(ceiling x <= 0) = (x <= 0)"

   711   by (rule ceiling_le_zero) (* already declared [simp] *)

   712

   713 lemma ceiling_le_eq_one: "(ceiling x <= 1) = (x <= 1)"

   714   by (rule ceiling_le_one) (* already declared [simp] *)

   715

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

   717   unfolding real_of_int_def by (rule less_ceiling_iff)

   718

   719 lemma less_ceiling_eq_number_of:

   720     "(number_of n < ceiling x) = (number_of n < x)"

   721   by (rule number_of_less_ceiling) (* already declared [simp] *)

   722

   723 lemma less_ceiling_eq_zero: "(0 < ceiling x) = (0 < x)"

   724   by (rule zero_less_ceiling) (* already declared [simp] *)

   725

   726 lemma less_ceiling_eq_one: "(1 < ceiling x) = (1 < x)"

   727   by (rule one_less_ceiling) (* already declared [simp] *)

   728

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

   730   unfolding real_of_int_def by (rule ceiling_less_iff)

   731

   732 lemma ceiling_less_eq_number_of:

   733     "(ceiling x < number_of n) = (x <= number_of n - 1)"

   734   by (rule ceiling_less_number_of) (* already declared [simp] *)

   735

   736 lemma ceiling_less_eq_zero: "(ceiling x < 0) = (x <= -1)"

   737   by (rule ceiling_less_zero) (* already declared [simp] *)

   738

   739 lemma ceiling_less_eq_one: "(ceiling x < 1) = (x <= 0)"

   740   by (rule ceiling_less_one) (* already declared [simp] *)

   741

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

   743   unfolding real_of_int_def by (rule le_ceiling_iff)

   744

   745 lemma le_ceiling_eq_number_of:

   746     "(number_of n <= ceiling x) = (number_of n - 1 < x)"

   747   by (rule number_of_le_ceiling) (* already declared [simp] *)

   748

   749 lemma le_ceiling_eq_zero: "(0 <= ceiling x) = (-1 < x)"

   750   by (rule zero_le_ceiling) (* already declared [simp] *)

   751

   752 lemma le_ceiling_eq_one: "(1 <= ceiling x) = (0 < x)"

   753   by (rule one_le_ceiling) (* already declared [simp] *)

   754

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

   756   unfolding real_of_int_def by (rule ceiling_add_of_int)

   757

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

   759   unfolding real_of_int_def by (rule ceiling_diff_of_int)

   760

   761 lemma ceiling_subtract_number_of: "ceiling (x - number_of n) =

   762     ceiling x - number_of n"

   763   by (rule ceiling_diff_number_of) (* already declared [simp] *)

   764

   765 lemma ceiling_subtract_one: "ceiling (x - 1) = ceiling x - 1"

   766   by (rule ceiling_diff_one) (* already declared [simp] *)

   767

   768

   769 subsection {* Versions for the natural numbers *}

   770

   771 definition

   772   natfloor :: "real => nat" where

   773   "natfloor x = nat(floor x)"

   774

   775 definition

   776   natceiling :: "real => nat" where

   777   "natceiling x = nat(ceiling x)"

   778

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

   780   by (unfold natfloor_def, simp)

   781

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

   783   by (unfold natfloor_def, simp)

   784

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

   786   by (unfold natfloor_def, simp)

   787

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

   789   by (unfold natfloor_def, simp)

   790

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

   792   by (unfold natfloor_def, simp)

   793

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

   795   by (unfold natfloor_def, simp)

   796

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

   798   apply (unfold natfloor_def)

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

   800   apply simp

   801   apply (erule floor_mono)

   802 done

   803

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

   805   apply (case_tac "0 <= x")

   806   apply (subst natfloor_def)+

   807   apply (subst nat_le_eq_zle)

   808   apply force

   809   apply (erule floor_mono)

   810   apply (subst natfloor_neg)

   811   apply simp

   812   apply simp

   813 done

   814

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

   816   apply (unfold natfloor_def)

   817   apply (subst nat_int [THEN sym])

   818   apply (subst nat_le_eq_zle)

   819   apply simp

   820   apply (rule le_floor)

   821   apply simp

   822 done

   823

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

   825   apply (rule iffI)

   826   apply (rule order_trans)

   827   prefer 2

   828   apply (erule real_natfloor_le)

   829   apply (subst real_of_nat_le_iff)

   830   apply assumption

   831   apply (erule le_natfloor)

   832 done

   833

   834 lemma le_natfloor_eq_number_of [simp]:

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

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

   837   apply (subst le_natfloor_eq, assumption)

   838   apply simp

   839 done

   840

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

   842   apply (case_tac "0 <= x")

   843   apply (subst le_natfloor_eq, assumption, simp)

   844   apply (rule iffI)

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

   846   apply simp

   847   apply (rule natfloor_mono)

   848   apply simp

   849   apply simp

   850 done

   851

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

   853   apply (unfold natfloor_def)

   854   apply (subst nat_int [THEN sym]);back;

   855   apply (subst eq_nat_nat_iff)

   856   apply simp

   857   apply simp

   858   apply (rule floor_eq2)

   859   apply auto

   860 done

   861

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

   863   apply (case_tac "0 <= x")

   864   apply (unfold natfloor_def)

   865   apply simp

   866   apply simp_all

   867 done

   868

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

   870 using real_natfloor_add_one_gt by (simp add: algebra_simps)

   871

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

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

   874   apply arith

   875   apply (rule real_natfloor_add_one_gt)

   876 done

   877

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

   879   apply (unfold natfloor_def)

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

   881   apply (erule ssubst)

   882   apply (simp add: nat_add_distrib del: real_of_int_of_nat_eq)

   883   apply simp

   884 done

   885

   886 lemma natfloor_add_number_of [simp]:

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

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

   889   apply (subst natfloor_add [THEN sym])

   890   apply simp_all

   891 done

   892

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

   894   apply (subst natfloor_add [THEN sym])

   895   apply assumption

   896   apply simp

   897 done

   898

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

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

   901   apply (unfold natfloor_def)

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

   903   apply (erule ssubst)

   904   apply (simp del: real_of_int_of_nat_eq)

   905   apply simp

   906 done

   907

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

   909   by (unfold natceiling_def, simp)

   910

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

   912   by (unfold natceiling_def, simp)

   913

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

   915   by (unfold natceiling_def, simp)

   916

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

   918   by (unfold natceiling_def, simp)

   919

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

   921   by (unfold natceiling_def, simp)

   922

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

   924   apply (unfold natceiling_def)

   925   apply (case_tac "x < 0")

   926   apply simp

   927   apply (subst real_nat_eq_real)

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

   929   apply simp

   930   apply (rule ceiling_mono)

   931   apply simp

   932   apply simp

   933 done

   934

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

   936   apply (unfold natceiling_def)

   937   apply simp

   938 done

   939

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

   941   apply (case_tac "0 <= x")

   942   apply (subst natceiling_def)+

   943   apply (subst nat_le_eq_zle)

   944   apply (rule disjI2)

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

   946   apply simp

   947   apply (rule order_trans)

   948   apply simp

   949   apply (erule order_trans)

   950   apply simp

   951   apply (erule ceiling_mono)

   952   apply (subst natceiling_neg)

   953   apply simp_all

   954 done

   955

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

   957   apply (unfold natceiling_def)

   958   apply (case_tac "x < 0")

   959   apply simp

   960   apply (subst nat_int [THEN sym]);back;

   961   apply (subst nat_le_eq_zle)

   962   apply simp

   963   apply (rule ceiling_le)

   964   apply simp

   965 done

   966

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

   968   apply (rule iffI)

   969   apply (rule order_trans)

   970   apply (rule real_natceiling_ge)

   971   apply (subst real_of_nat_le_iff)

   972   apply assumption

   973   apply (erule natceiling_le)

   974 done

   975

   976 lemma natceiling_le_eq_number_of [simp]:

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

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

   979   apply (subst natceiling_le_eq, assumption)

   980   apply simp

   981 done

   982

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

   984   apply (case_tac "0 <= x")

   985   apply (subst natceiling_le_eq)

   986   apply assumption

   987   apply simp

   988   apply (subst natceiling_neg)

   989   apply simp

   990   apply simp

   991 done

   992

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

   994   apply (unfold natceiling_def)

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

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

   997   apply (erule ssubst)

   998   apply (subst eq_nat_nat_iff)

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

  1000   apply simp

  1001   apply (rule ceiling_mono)

  1002   apply force

  1003   apply force

  1004   apply (rule ceiling_eq2)

  1005   apply (simp, simp)

  1006   apply (subst nat_add_distrib)

  1007   apply auto

  1008 done

  1009

  1010 lemma natceiling_add [simp]: "0 <= x ==>

  1011     natceiling (x + real a) = natceiling x + a"

  1012   apply (unfold natceiling_def)

  1013   apply (subgoal_tac "real a = real (int a)")

  1014   apply (erule ssubst)

  1015   apply (simp del: real_of_int_of_nat_eq)

  1016   apply (subst nat_add_distrib)

  1017   apply (subgoal_tac "0 = ceiling 0")

  1018   apply (erule ssubst)

  1019   apply (erule ceiling_mono)

  1020   apply simp_all

  1021 done

  1022

  1023 lemma natceiling_add_number_of [simp]:

  1024     "~ neg ((number_of n)::int) ==> 0 <= x ==>

  1025       natceiling (x + number_of n) = natceiling x + number_of n"

  1026   apply (subst natceiling_add [THEN sym])

  1027   apply simp_all

  1028 done

  1029

  1030 lemma natceiling_add_one: "0 <= x ==> natceiling(x + 1) = natceiling x + 1"

  1031   apply (subst natceiling_add [THEN sym])

  1032   apply assumption

  1033   apply simp

  1034 done

  1035

  1036 lemma natceiling_subtract [simp]: "real a <= x ==>

  1037     natceiling(x - real a) = natceiling x - a"

  1038   apply (unfold natceiling_def)

  1039   apply (subgoal_tac "real a = real (int a)")

  1040   apply (erule ssubst)

  1041   apply (simp del: real_of_int_of_nat_eq)

  1042   apply simp

  1043 done

  1044

  1045 lemma natfloor_div_nat: "1 <= x ==> y > 0 ==>

  1046   natfloor (x / real y) = natfloor x div y"

  1047 proof -

  1048   assume "1 <= (x::real)" and "(y::nat) > 0"

  1049   have "natfloor x = (natfloor x) div y * y + (natfloor x) mod y"

  1050     by simp

  1051   then have a: "real(natfloor x) = real ((natfloor x) div y) * real y +

  1052     real((natfloor x) mod y)"

  1053     by (simp only: real_of_nat_add [THEN sym] real_of_nat_mult [THEN sym])

  1054   have "x = real(natfloor x) + (x - real(natfloor x))"

  1055     by simp

  1056   then have "x = real ((natfloor x) div y) * real y +

  1057       real((natfloor x) mod y) + (x - real(natfloor x))"

  1058     by (simp add: a)

  1059   then have "x / real y = ... / real y"

  1060     by simp

  1061   also have "... = real((natfloor x) div y) + real((natfloor x) mod y) /

  1062     real y + (x - real(natfloor x)) / real y"

  1063     by (auto simp add: algebra_simps add_divide_distrib

  1064       diff_divide_distrib prems)

  1065   finally have "natfloor (x / real y) = natfloor(...)" by simp

  1066   also have "... = natfloor(real((natfloor x) mod y) /

  1067     real y + (x - real(natfloor x)) / real y + real((natfloor x) div y))"

  1068     by (simp add: add_ac)

  1069   also have "... = natfloor(real((natfloor x) mod y) /

  1070     real y + (x - real(natfloor x)) / real y) + (natfloor x) div y"

  1071     apply (rule natfloor_add)

  1072     apply (rule add_nonneg_nonneg)

  1073     apply (rule divide_nonneg_pos)

  1074     apply simp

  1075     apply (simp add: prems)

  1076     apply (rule divide_nonneg_pos)

  1077     apply (simp add: algebra_simps)

  1078     apply (rule real_natfloor_le)

  1079     apply (insert prems, auto)

  1080     done

  1081   also have "natfloor(real((natfloor x) mod y) /

  1082     real y + (x - real(natfloor x)) / real y) = 0"

  1083     apply (rule natfloor_eq)

  1084     apply simp

  1085     apply (rule add_nonneg_nonneg)

  1086     apply (rule divide_nonneg_pos)

  1087     apply force

  1088     apply (force simp add: prems)

  1089     apply (rule divide_nonneg_pos)

  1090     apply (simp add: algebra_simps)

  1091     apply (rule real_natfloor_le)

  1092     apply (auto simp add: prems)

  1093     apply (insert prems, arith)

  1094     apply (simp add: add_divide_distrib [THEN sym])

  1095     apply (subgoal_tac "real y = real y - 1 + 1")

  1096     apply (erule ssubst)

  1097     apply (rule add_le_less_mono)

  1098     apply (simp add: algebra_simps)

  1099     apply (subgoal_tac "1 + real(natfloor x mod y) =

  1100       real(natfloor x mod y + 1)")

  1101     apply (erule ssubst)

  1102     apply (subst real_of_nat_le_iff)

  1103     apply (subgoal_tac "natfloor x mod y < y")

  1104     apply arith

  1105     apply (rule mod_less_divisor)

  1106     apply auto

  1107     using real_natfloor_add_one_gt

  1108     apply (simp add: algebra_simps)

  1109     done

  1110   finally show ?thesis by simp

  1111 qed

  1112

  1113 end