src/HOL/RComplete.thy
 author huffman Mon Jan 12 12:09:54 2009 -0800 (2009-01-12) changeset 29460 ad87e5d1488b parent 28952 15a4b2cf8c34 child 29667 53103fc8ffa3 permissions -rw-r--r--
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
     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

    18 subsection {* Completeness of Positive Reals *}

    19

    20 text {*

    21   Supremum property for the set of positive reals

    22

    23   Let @{text "P"} be a non-empty set of positive reals, with an upper

    24   bound @{text "y"}.  Then @{text "P"} has a least upper bound

    25   (written @{text "S"}).

    26

    27   FIXME: Can the premise be weakened to @{text "\<forall>x \<in> P. x\<le> y"}?

    28 *}

    29

    30 lemma posreal_complete:

    31   assumes positive_P: "\<forall>x \<in> P. (0::real) < x"

    32     and not_empty_P: "\<exists>x. x \<in> P"

    33     and upper_bound_Ex: "\<exists>y. \<forall>x \<in> P. x<y"

    34   shows "\<exists>S. \<forall>y. (\<exists>x \<in> P. y < x) = (y < S)"

    35 proof (rule exI, rule allI)

    36   fix y

    37   let ?pP = "{w. real_of_preal w \<in> P}"

    38

    39   show "(\<exists>x\<in>P. y < x) = (y < real_of_preal (psup ?pP))"

    40   proof (cases "0 < y")

    41     assume neg_y: "\<not> 0 < y"

    42     show ?thesis

    43     proof

    44       assume "\<exists>x\<in>P. y < x"

    45       have "\<forall>x. y < real_of_preal x"

    46         using neg_y by (rule real_less_all_real2)

    47       thus "y < real_of_preal (psup ?pP)" ..

    48     next

    49       assume "y < real_of_preal (psup ?pP)"

    50       obtain "x" where x_in_P: "x \<in> P" using not_empty_P ..

    51       hence "0 < x" using positive_P by simp

    52       hence "y < x" using neg_y by simp

    53       thus "\<exists>x \<in> P. y < x" using x_in_P ..

    54     qed

    55   next

    56     assume pos_y: "0 < y"

    57

    58     then obtain py where y_is_py: "y = real_of_preal py"

    59       by (auto simp add: real_gt_zero_preal_Ex)

    60

    61     obtain a where "a \<in> P" using not_empty_P ..

    62     with positive_P have a_pos: "0 < a" ..

    63     then obtain pa where "a = real_of_preal pa"

    64       by (auto simp add: real_gt_zero_preal_Ex)

    65     hence "pa \<in> ?pP" using a \<in> P by auto

    66     hence pP_not_empty: "?pP \<noteq> {}" by auto

    67

    68     obtain sup where sup: "\<forall>x \<in> P. x < sup"

    69       using upper_bound_Ex ..

    70     from this and a \<in> P have "a < sup" ..

    71     hence "0 < sup" using a_pos by arith

    72     then obtain possup where "sup = real_of_preal possup"

    73       by (auto simp add: real_gt_zero_preal_Ex)

    74     hence "\<forall>X \<in> ?pP. X \<le> possup"

    75       using sup by (auto simp add: real_of_preal_lessI)

    76     with pP_not_empty have psup: "\<And>Z. (\<exists>X \<in> ?pP. Z < X) = (Z < psup ?pP)"

    77       by (rule preal_complete)

    78

    79     show ?thesis

    80     proof

    81       assume "\<exists>x \<in> P. y < x"

    82       then obtain x where x_in_P: "x \<in> P" and y_less_x: "y < x" ..

    83       hence "0 < x" using pos_y by arith

    84       then obtain px where x_is_px: "x = real_of_preal px"

    85         by (auto simp add: real_gt_zero_preal_Ex)

    86

    87       have py_less_X: "\<exists>X \<in> ?pP. py < X"

    88       proof

    89         show "py < px" using y_is_py and x_is_px and y_less_x

    90           by (simp add: real_of_preal_lessI)

    91         show "px \<in> ?pP" using x_in_P and x_is_px by simp

    92       qed

    93

    94       have "(\<exists>X \<in> ?pP. py < X) ==> (py < psup ?pP)"

    95         using psup by simp

    96       hence "py < psup ?pP" using py_less_X by simp

    97       thus "y < real_of_preal (psup {w. real_of_preal w \<in> P})"

    98         using y_is_py and pos_y by (simp add: real_of_preal_lessI)

    99     next

   100       assume y_less_psup: "y < real_of_preal (psup ?pP)"

   101

   102       hence "py < psup ?pP" using y_is_py

   103         by (simp add: real_of_preal_lessI)

   104       then obtain "X" where py_less_X: "py < X" and X_in_pP: "X \<in> ?pP"

   105         using psup by auto

   106       then obtain x where x_is_X: "x = real_of_preal X"

   107         by (simp add: real_gt_zero_preal_Ex)

   108       hence "y < x" using py_less_X and y_is_py

   109         by (simp add: real_of_preal_lessI)

   110

   111       moreover have "x \<in> P" using x_is_X and X_in_pP by simp

   112

   113       ultimately show "\<exists> x \<in> P. y < x" ..

   114     qed

   115   qed

   116 qed

   117

   118 text {*

   119   \medskip Completeness properties using @{text "isUb"}, @{text "isLub"} etc.

   120 *}

   121

   122 lemma real_isLub_unique: "[| isLub R S x; isLub R S y |] ==> x = (y::real)"

   123   apply (frule isLub_isUb)

   124   apply (frule_tac x = y in isLub_isUb)

   125   apply (blast intro!: order_antisym dest!: isLub_le_isUb)

   126   done

   127

   128

   129 text {*

   130   \medskip Completeness theorem for the positive reals (again).

   131 *}

   132

   133 lemma posreals_complete:

   134   assumes positive_S: "\<forall>x \<in> S. 0 < x"

   135     and not_empty_S: "\<exists>x. x \<in> S"

   136     and upper_bound_Ex: "\<exists>u. isUb (UNIV::real set) S u"

   137   shows "\<exists>t. isLub (UNIV::real set) S t"

   138 proof

   139   let ?pS = "{w. real_of_preal w \<in> S}"

   140

   141   obtain u where "isUb UNIV S u" using upper_bound_Ex ..

   142   hence sup: "\<forall>x \<in> S. x \<le> u" by (simp add: isUb_def setle_def)

   143

   144   obtain x where x_in_S: "x \<in> S" using not_empty_S ..

   145   hence x_gt_zero: "0 < x" using positive_S by simp

   146   have  "x \<le> u" using sup and x_in_S ..

   147   hence "0 < u" using x_gt_zero by arith

   148

   149   then obtain pu where u_is_pu: "u = real_of_preal pu"

   150     by (auto simp add: real_gt_zero_preal_Ex)

   151

   152   have pS_less_pu: "\<forall>pa \<in> ?pS. pa \<le> pu"

   153   proof

   154     fix pa

   155     assume "pa \<in> ?pS"

   156     then obtain a where "a \<in> S" and "a = real_of_preal pa"

   157       by simp

   158     moreover hence "a \<le> u" using sup by simp

   159     ultimately show "pa \<le> pu"

   160       using sup and u_is_pu by (simp add: real_of_preal_le_iff)

   161   qed

   162

   163   have "\<forall>y \<in> S. y \<le> real_of_preal (psup ?pS)"

   164   proof

   165     fix y

   166     assume y_in_S: "y \<in> S"

   167     hence "0 < y" using positive_S by simp

   168     then obtain py where y_is_py: "y = real_of_preal py"

   169       by (auto simp add: real_gt_zero_preal_Ex)

   170     hence py_in_pS: "py \<in> ?pS" using y_in_S by simp

   171     with pS_less_pu have "py \<le> psup ?pS"

   172       by (rule preal_psup_le)

   173     thus "y \<le> real_of_preal (psup ?pS)"

   174       using y_is_py by (simp add: real_of_preal_le_iff)

   175   qed

   176

   177   moreover {

   178     fix x

   179     assume x_ub_S: "\<forall>y\<in>S. y \<le> x"

   180     have "real_of_preal (psup ?pS) \<le> x"

   181     proof -

   182       obtain "s" where s_in_S: "s \<in> S" using not_empty_S ..

   183       hence s_pos: "0 < s" using positive_S by simp

   184

   185       hence "\<exists> ps. s = real_of_preal ps" by (simp add: real_gt_zero_preal_Ex)

   186       then obtain "ps" where s_is_ps: "s = real_of_preal ps" ..

   187       hence ps_in_pS: "ps \<in> {w. real_of_preal w \<in> S}" using s_in_S by simp

   188

   189       from x_ub_S have "s \<le> x" using s_in_S ..

   190       hence "0 < x" using s_pos by simp

   191       hence "\<exists> px. x = real_of_preal px" by (simp add: real_gt_zero_preal_Ex)

   192       then obtain "px" where x_is_px: "x = real_of_preal px" ..

   193

   194       have "\<forall>pe \<in> ?pS. pe \<le> px"

   195       proof

   196 	fix pe

   197 	assume "pe \<in> ?pS"

   198 	hence "real_of_preal pe \<in> S" by simp

   199 	hence "real_of_preal pe \<le> x" using x_ub_S by simp

   200 	thus "pe \<le> px" using x_is_px by (simp add: real_of_preal_le_iff)

   201       qed

   202

   203       moreover have "?pS \<noteq> {}" using ps_in_pS by auto

   204       ultimately have "(psup ?pS) \<le> px" by (simp add: psup_le_ub)

   205       thus "real_of_preal (psup ?pS) \<le> x" using x_is_px by (simp add: real_of_preal_le_iff)

   206     qed

   207   }

   208   ultimately show "isLub UNIV S (real_of_preal (psup ?pS))"

   209     by (simp add: isLub_def leastP_def isUb_def setle_def setge_def)

   210 qed

   211

   212 text {*

   213   \medskip reals Completeness (again!)

   214 *}

   215

   216 lemma reals_complete:

   217   assumes notempty_S: "\<exists>X. X \<in> S"

   218     and exists_Ub: "\<exists>Y. isUb (UNIV::real set) S Y"

   219   shows "\<exists>t. isLub (UNIV :: real set) S t"

   220 proof -

   221   obtain X where X_in_S: "X \<in> S" using notempty_S ..

   222   obtain Y where Y_isUb: "isUb (UNIV::real set) S Y"

   223     using exists_Ub ..

   224   let ?SHIFT = "{z. \<exists>x \<in>S. z = x + (-X) + 1} \<inter> {x. 0 < x}"

   225

   226   {

   227     fix x

   228     assume "isUb (UNIV::real set) S x"

   229     hence S_le_x: "\<forall> y \<in> S. y <= x"

   230       by (simp add: isUb_def setle_def)

   231     {

   232       fix s

   233       assume "s \<in> {z. \<exists>x\<in>S. z = x + - X + 1}"

   234       hence "\<exists> x \<in> S. s = x + -X + 1" ..

   235       then obtain x1 where "x1 \<in> S" and "s = x1 + (-X) + 1" ..

   236       moreover hence "x1 \<le> x" using S_le_x by simp

   237       ultimately have "s \<le> x + - X + 1" by arith

   238     }

   239     then have "isUb (UNIV::real set) ?SHIFT (x + (-X) + 1)"

   240       by (auto simp add: isUb_def setle_def)

   241   } note S_Ub_is_SHIFT_Ub = this

   242

   243   hence "isUb UNIV ?SHIFT (Y + (-X) + 1)" using Y_isUb by simp

   244   hence "\<exists>Z. isUb UNIV ?SHIFT Z" ..

   245   moreover have "\<forall>y \<in> ?SHIFT. 0 < y" by auto

   246   moreover have shifted_not_empty: "\<exists>u. u \<in> ?SHIFT"

   247     using X_in_S and Y_isUb by auto

   248   ultimately obtain t where t_is_Lub: "isLub UNIV ?SHIFT t"

   249     using posreals_complete [of ?SHIFT] by blast

   250

   251   show ?thesis

   252   proof

   253     show "isLub UNIV S (t + X + (-1))"

   254     proof (rule isLubI2)

   255       {

   256         fix x

   257         assume "isUb (UNIV::real set) S x"

   258         hence "isUb (UNIV::real set) (?SHIFT) (x + (-X) + 1)"

   259 	  using S_Ub_is_SHIFT_Ub by simp

   260         hence "t \<le> (x + (-X) + 1)"

   261 	  using t_is_Lub by (simp add: isLub_le_isUb)

   262         hence "t + X + -1 \<le> x" by arith

   263       }

   264       then show "(t + X + -1) <=* Collect (isUb UNIV S)"

   265 	by (simp add: setgeI)

   266     next

   267       show "isUb UNIV S (t + X + -1)"

   268       proof -

   269         {

   270           fix y

   271           assume y_in_S: "y \<in> S"

   272           have "y \<le> t + X + -1"

   273           proof -

   274             obtain "u" where u_in_shift: "u \<in> ?SHIFT" using shifted_not_empty ..

   275             hence "\<exists> x \<in> S. u = x + - X + 1" by simp

   276             then obtain "x" where x_and_u: "u = x + - X + 1" ..

   277             have u_le_t: "u \<le> t" using u_in_shift and t_is_Lub by (simp add: isLubD2)

   278

   279             show ?thesis

   280             proof cases

   281               assume "y \<le> x"

   282               moreover have "x = u + X + - 1" using x_and_u by arith

   283               moreover have "u + X + - 1  \<le> t + X + -1" using u_le_t by arith

   284               ultimately show "y  \<le> t + X + -1" by arith

   285             next

   286               assume "~(y \<le> x)"

   287               hence x_less_y: "x < y" by arith

   288

   289               have "x + (-X) + 1 \<in> ?SHIFT" using x_and_u and u_in_shift by simp

   290               hence "0 < x + (-X) + 1" by simp

   291               hence "0 < y + (-X) + 1" using x_less_y by arith

   292               hence "y + (-X) + 1 \<in> ?SHIFT" using y_in_S by simp

   293               hence "y + (-X) + 1 \<le> t" using t_is_Lub  by (simp add: isLubD2)

   294               thus ?thesis by simp

   295             qed

   296           qed

   297         }

   298         then show ?thesis by (simp add: isUb_def setle_def)

   299       qed

   300     qed

   301   qed

   302 qed

   303

   304

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

   306

   307 theorem reals_Archimedean:

   308   assumes x_pos: "0 < x"

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

   310 proof (rule ccontr)

   311   assume contr: "\<not> ?thesis"

   312   have "\<forall>n. x * real (Suc n) <= 1"

   313   proof

   314     fix n

   315     from contr have "x \<le> inverse (real (Suc n))"

   316       by (simp add: linorder_not_less)

   317     hence "x \<le> (1 / (real (Suc n)))"

   318       by (simp add: inverse_eq_divide)

   319     moreover have "0 \<le> real (Suc n)"

   320       by (rule real_of_nat_ge_zero)

   321     ultimately have "x * real (Suc n) \<le> (1 / real (Suc n)) * real (Suc n)"

   322       by (rule mult_right_mono)

   323     thus "x * real (Suc n) \<le> 1" by simp

   324   qed

   325   hence "{z. \<exists>n. z = x * (real (Suc n))} *<= 1"

   326     by (simp add: setle_def, safe, rule spec)

   327   hence "isUb (UNIV::real set) {z. \<exists>n. z = x * (real (Suc n))} 1"

   328     by (simp add: isUbI)

   329   hence "\<exists>Y. isUb (UNIV::real set) {z. \<exists>n. z = x* (real (Suc n))} Y" ..

   330   moreover have "\<exists>X. X \<in> {z. \<exists>n. z = x* (real (Suc n))}" by auto

   331   ultimately have "\<exists>t. isLub UNIV {z. \<exists>n. z = x * real (Suc n)} t"

   332     by (simp add: reals_complete)

   333   then obtain "t" where

   334     t_is_Lub: "isLub UNIV {z. \<exists>n. z = x * real (Suc n)} t" ..

   335

   336   have "\<forall>n::nat. x * real n \<le> t + - x"

   337   proof

   338     fix n

   339     from t_is_Lub have "x * real (Suc n) \<le> t"

   340       by (simp add: isLubD2)

   341     hence  "x * (real n) + x \<le> t"

   342       by (simp add: right_distrib real_of_nat_Suc)

   343     thus  "x * (real n) \<le> t + - x" by arith

   344   qed

   345

   346   hence "\<forall>m. x * real (Suc m) \<le> t + - x" by simp

   347   hence "{z. \<exists>n. z = x * (real (Suc n))}  *<= (t + - x)"

   348     by (auto simp add: setle_def)

   349   hence "isUb (UNIV::real set) {z. \<exists>n. z = x * (real (Suc n))} (t + (-x))"

   350     by (simp add: isUbI)

   351   hence "t \<le> t + - x"

   352     using t_is_Lub by (simp add: isLub_le_isUb)

   353   thus False using x_pos by arith

   354 qed

   355

   356 text {*

   357   There must be other proofs, e.g. @{text "Suc"} of the largest

   358   integer in the cut representing @{text "x"}.

   359 *}

   360

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

   362 proof cases

   363   assume "x \<le> 0"

   364   hence "x < real (1::nat)" by simp

   365   thus ?thesis ..

   366 next

   367   assume "\<not> x \<le> 0"

   368   hence x_greater_zero: "0 < x" by simp

   369   hence "0 < inverse x" by simp

   370   then obtain n where "inverse (real (Suc n)) < inverse x"

   371     using reals_Archimedean by blast

   372   hence "inverse (real (Suc n)) * x < inverse x * x"

   373     using x_greater_zero by (rule mult_strict_right_mono)

   374   hence "inverse (real (Suc n)) * x < 1"

   375     using x_greater_zero by simp

   376   hence "real (Suc n) * (inverse (real (Suc n)) * x) < real (Suc n) * 1"

   377     by (rule mult_strict_left_mono) simp

   378   hence "x < real (Suc n)"

   379     by (simp add: ring_simps)

   380   thus "\<exists>(n::nat). x < real n" ..

   381 qed

   382

   383 lemma reals_Archimedean3:

   384   assumes x_greater_zero: "0 < x"

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

   386 proof

   387   fix y

   388   have x_not_zero: "x \<noteq> 0" using x_greater_zero by simp

   389   obtain n where "y * inverse x < real (n::nat)"

   390     using reals_Archimedean2 ..

   391   hence "y * inverse x * x < real n * x"

   392     using x_greater_zero by (simp add: mult_strict_right_mono)

   393   hence "x * inverse x * y < x * real n"

   394     by (simp add: ring_simps)

   395   hence "y < real (n::nat) * x"

   396     using x_not_zero by (simp add: ring_simps)

   397   thus "\<exists>(n::nat). y < real n * x" ..

   398 qed

   399

   400 lemma reals_Archimedean6:

   401      "0 \<le> r ==> \<exists>(n::nat). real (n - 1) \<le> r & r < real (n)"

   402 apply (insert reals_Archimedean2 [of r], safe)

   403 apply (subgoal_tac "\<exists>x::nat. r < real x \<and> (\<forall>y. r < real y \<longrightarrow> x \<le> y)", auto)

   404 apply (rule_tac x = x in exI)

   405 apply (case_tac x, simp)

   406 apply (rename_tac x')

   407 apply (drule_tac x = x' in spec, simp)

   408 apply (rule_tac x="LEAST n. r < real n" in exI, safe)

   409 apply (erule LeastI, erule Least_le)

   410 done

   411

   412 lemma reals_Archimedean6a: "0 \<le> r ==> \<exists>n. real (n) \<le> r & r < real (Suc n)"

   413   by (drule reals_Archimedean6) auto

   414

   415 lemma reals_Archimedean_6b_int:

   416      "0 \<le> r ==> \<exists>n::int. real n \<le> r & r < real (n+1)"

   417 apply (drule reals_Archimedean6a, auto)

   418 apply (rule_tac x = "int n" in exI)

   419 apply (simp add: real_of_int_real_of_nat real_of_nat_Suc)

   420 done

   421

   422 lemma reals_Archimedean_6c_int:

   423      "r < 0 ==> \<exists>n::int. real n \<le> r & r < real (n+1)"

   424 apply (rule reals_Archimedean_6b_int [of "-r", THEN exE], simp, auto)

   425 apply (rename_tac n)

   426 apply (drule order_le_imp_less_or_eq, auto)

   427 apply (rule_tac x = "- n - 1" in exI)

   428 apply (rule_tac  x = "- n" in exI, auto)

   429 done

   430

   431

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

   433

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

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

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

   437

   438 lemma Rats_dense_in_nn_real: fixes x::real

   439 assumes "0\<le>x" and "x<y" shows "\<exists>r \<in> \<rat>.  x<r \<and> r<y"

   440 proof -

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

   442   with reals_Archimedean obtain q::nat

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

   444   def p \<equiv> "LEAST n.  y \<le> real (Suc n)/real q"

   445   from reals_Archimedean2 obtain n::nat where "y * real q < real n" by auto

   446   with 0 < real q have ex: "y \<le> real n/real q" (is "?P n")

   447     by (simp add: pos_less_divide_eq[THEN sym])

   448   also from assms have "\<not> y \<le> real (0::nat) / real q" by simp

   449   ultimately have main: "(LEAST n. y \<le> real n/real q) = Suc p"

   450     by (unfold p_def) (rule Least_Suc)

   451   also from ex have "?P (LEAST x. ?P x)" by (rule LeastI)

   452   ultimately have suc: "y \<le> real (Suc p) / real q" by simp

   453   def r \<equiv> "real p/real q"

   454   have "x = y-(y-x)" by simp

   455   also from suc q have "\<dots> < real (Suc p)/real q - inverse (real q)" by arith

   456   also have "\<dots> = real p / real q"

   457     by (simp only: inverse_eq_divide real_diff_def real_of_nat_Suc

   458     minus_divide_left add_divide_distrib[THEN sym]) simp

   459   finally have "x<r" by (unfold r_def)

   460   have "p<Suc p" .. also note main[THEN sym]

   461   finally have "\<not> ?P p"  by (rule not_less_Least)

   462   hence "r<y" by (simp add: r_def)

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

   464   with x<r r<y show ?thesis by fast

   465 qed

   466

   467 theorem Rats_dense_in_real: fixes x y :: real

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

   469 proof -

   470   from reals_Archimedean2 obtain n::nat where "-x < real n" by auto

   471   hence "0 \<le> x + real n" by arith

   472   also from x<y have "x + real n < y + real n" by arith

   473   ultimately have "\<exists>r \<in> \<rat>. x + real n < r \<and> r < y + real n"

   474     by(rule Rats_dense_in_nn_real)

   475   then obtain r where "r \<in> \<rat>" and r2: "x + real n < r"

   476     and r3: "r < y + real n"

   477     by blast

   478   have "r - real n = r + real (int n)/real (-1::int)" by simp

   479   also from r\<in>\<rat> have "r + real (int n)/real (-1::int) \<in> \<rat>" by simp

   480   also from r2 have "x < r - real n" by arith

   481   moreover from r3 have "r - real n < y" by arith

   482   ultimately show ?thesis by fast

   483 qed

   484

   485

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

   487

   488 definition

   489   floor :: "real => int" where

   490   [code del]: "floor r = (LEAST n::int. r < real (n+1))"

   491

   492 definition

   493   ceiling :: "real => int" where

   494   "ceiling r = - floor (- r)"

   495

   496 notation (xsymbols)

   497   floor  ("\<lfloor>_\<rfloor>") and

   498   ceiling  ("\<lceil>_\<rceil>")

   499

   500 notation (HTML output)

   501   floor  ("\<lfloor>_\<rfloor>") and

   502   ceiling  ("\<lceil>_\<rceil>")

   503

   504

   505 lemma number_of_less_real_of_int_iff [simp]:

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

   507 apply auto

   508 apply (rule real_of_int_less_iff [THEN iffD1])

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

   510 done

   511

   512 lemma number_of_less_real_of_int_iff2 [simp]:

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

   514 apply auto

   515 apply (rule real_of_int_less_iff [THEN iffD1])

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

   517 done

   518

   519 lemma number_of_le_real_of_int_iff [simp]:

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

   521 by (simp add: linorder_not_less [symmetric])

   522

   523 lemma number_of_le_real_of_int_iff2 [simp]:

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

   525 by (simp add: linorder_not_less [symmetric])

   526

   527 lemma floor_zero [simp]: "floor 0 = 0"

   528 apply (simp add: floor_def del: real_of_int_add)

   529 apply (rule Least_equality)

   530 apply simp_all

   531 done

   532

   533 lemma floor_real_of_nat_zero [simp]: "floor (real (0::nat)) = 0"

   534 by auto

   535

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

   537 apply (simp only: floor_def)

   538 apply (rule Least_equality)

   539 apply (drule_tac  real_of_int_of_nat_eq [THEN ssubst])

   540 apply (drule_tac  real_of_int_less_iff [THEN iffD1])

   541 apply simp_all

   542 done

   543

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

   545 apply (simp only: floor_def)

   546 apply (rule Least_equality)

   547 apply (drule_tac  real_of_int_of_nat_eq [THEN ssubst])

   548 apply (drule_tac  real_of_int_minus [THEN sym, THEN subst])

   549 apply (drule_tac  real_of_int_less_iff [THEN iffD1])

   550 apply simp_all

   551 done

   552

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

   554 apply (simp only: floor_def)

   555 apply (rule Least_equality)

   556 apply auto

   557 done

   558

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

   560 apply (simp only: floor_def)

   561 apply (rule Least_equality)

   562 apply (drule_tac  real_of_int_minus [THEN sym, THEN subst])

   563 apply auto

   564 done

   565

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

   567 apply (case_tac "r < 0")

   568 apply (blast intro: reals_Archimedean_6c_int)

   569 apply (simp only: linorder_not_less)

   570 apply (blast intro: reals_Archimedean_6b_int reals_Archimedean_6c_int)

   571 done

   572

   573 lemma lemma_floor:

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

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

   576 proof -

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

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

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

   580   thus ?thesis by arith

   581 qed

   582

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

   584 apply (simp add: floor_def Least_def)

   585 apply (insert real_lb_ub_int [of r], safe)

   586 apply (rule theI2)

   587 apply auto

   588 done

   589

   590 lemma floor_mono: "x < y ==> floor x \<le> floor y"

   591 apply (simp add: floor_def Least_def)

   592 apply (insert real_lb_ub_int [of x])

   593 apply (insert real_lb_ub_int [of y], safe)

   594 apply (rule theI2)

   595 apply (rule_tac  theI2)

   596 apply simp

   597 apply (erule conjI)

   598 apply (auto simp add: order_eq_iff int_le_real_less)

   599 done

   600

   601 lemma floor_mono2: "x \<le> y ==> floor x \<le> floor y"

   602 by (auto dest: order_le_imp_less_or_eq simp add: floor_mono)

   603

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

   605 by (auto intro: lemma_floor)

   606

   607 lemma real_of_int_floor_cancel [simp]:

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

   609 apply (simp add: floor_def Least_def)

   610 apply (insert real_lb_ub_int [of x], erule exE)

   611 apply (rule theI2)

   612 apply (auto intro: lemma_floor)

   613 done

   614

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

   616 apply (simp add: floor_def)

   617 apply (rule Least_equality)

   618 apply (auto intro: lemma_floor)

   619 done

   620

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

   622 apply (simp add: floor_def)

   623 apply (rule Least_equality)

   624 apply (auto intro: lemma_floor)

   625 done

   626

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

   628 apply (rule inj_int [THEN injD])

   629 apply (simp add: real_of_nat_Suc)

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

   631 done

   632

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

   634 apply (drule order_le_imp_less_or_eq)

   635 apply (auto intro: floor_eq3)

   636 done

   637

   638 lemma floor_number_of_eq [simp]:

   639      "floor(number_of n :: real) = (number_of n :: int)"

   640 apply (subst real_number_of [symmetric])

   641 apply (rule floor_real_of_int)

   642 done

   643

   644 lemma floor_one [simp]: "floor 1 = 1"

   645   apply (rule trans)

   646   prefer 2

   647   apply (rule floor_real_of_int)

   648   apply simp

   649 done

   650

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

   652 apply (simp add: floor_def Least_def)

   653 apply (insert real_lb_ub_int [of r], safe)

   654 apply (rule theI2)

   655 apply (auto intro: lemma_floor)

   656 done

   657

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

   659 apply (simp add: floor_def Least_def)

   660 apply (insert real_lb_ub_int [of r], safe)

   661 apply (rule theI2)

   662 apply (auto intro: lemma_floor)

   663 done

   664

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

   666 apply (insert real_of_int_floor_ge_diff_one [of r])

   667 apply (auto simp del: real_of_int_floor_ge_diff_one)

   668 done

   669

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

   671 apply (insert real_of_int_floor_gt_diff_one [of r])

   672 apply (auto simp del: real_of_int_floor_gt_diff_one)

   673 done

   674

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

   676   apply (subgoal_tac "a < floor x + 1")

   677   apply arith

   678   apply (subst real_of_int_less_iff [THEN sym])

   679   apply simp

   680   apply (insert real_of_int_floor_add_one_gt [of x])

   681   apply arith

   682 done

   683

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

   685   apply (rule order_trans)

   686   prefer 2

   687   apply (rule real_of_int_floor_le)

   688   apply (subst real_of_int_le_iff)

   689   apply assumption

   690 done

   691

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

   693   apply (rule iffI)

   694   apply (erule real_le_floor)

   695   apply (erule le_floor)

   696 done

   697

   698 lemma le_floor_eq_number_of [simp]:

   699     "(number_of n <= floor x) = (number_of n <= x)"

   700 by (simp add: le_floor_eq)

   701

   702 lemma le_floor_eq_zero [simp]: "(0 <= floor x) = (0 <= x)"

   703 by (simp add: le_floor_eq)

   704

   705 lemma le_floor_eq_one [simp]: "(1 <= floor x) = (1 <= x)"

   706 by (simp add: le_floor_eq)

   707

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

   709   apply (subst linorder_not_le [THEN sym])+

   710   apply simp

   711   apply (rule le_floor_eq)

   712 done

   713

   714 lemma floor_less_eq_number_of [simp]:

   715     "(floor x < number_of n) = (x < number_of n)"

   716 by (simp add: floor_less_eq)

   717

   718 lemma floor_less_eq_zero [simp]: "(floor x < 0) = (x < 0)"

   719 by (simp add: floor_less_eq)

   720

   721 lemma floor_less_eq_one [simp]: "(floor x < 1) = (x < 1)"

   722 by (simp add: floor_less_eq)

   723

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

   725   apply (insert le_floor_eq [of "a + 1" x])

   726   apply auto

   727 done

   728

   729 lemma less_floor_eq_number_of [simp]:

   730     "(number_of n < floor x) = (number_of n + 1 <= x)"

   731 by (simp add: less_floor_eq)

   732

   733 lemma less_floor_eq_zero [simp]: "(0 < floor x) = (1 <= x)"

   734 by (simp add: less_floor_eq)

   735

   736 lemma less_floor_eq_one [simp]: "(1 < floor x) = (2 <= x)"

   737 by (simp add: less_floor_eq)

   738

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

   740   apply (insert floor_less_eq [of x "a + 1"])

   741   apply auto

   742 done

   743

   744 lemma floor_le_eq_number_of [simp]:

   745     "(floor x <= number_of n) = (x < number_of n + 1)"

   746 by (simp add: floor_le_eq)

   747

   748 lemma floor_le_eq_zero [simp]: "(floor x <= 0) = (x < 1)"

   749 by (simp add: floor_le_eq)

   750

   751 lemma floor_le_eq_one [simp]: "(floor x <= 1) = (x < 2)"

   752 by (simp add: floor_le_eq)

   753

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

   755   apply (subst order_eq_iff)

   756   apply (rule conjI)

   757   prefer 2

   758   apply (subgoal_tac "floor x + a < floor (x + real a) + 1")

   759   apply arith

   760   apply (subst real_of_int_less_iff [THEN sym])

   761   apply simp

   762   apply (subgoal_tac "x + real a < real(floor(x + real a)) + 1")

   763   apply (subgoal_tac "real (floor x) <= x")

   764   apply arith

   765   apply (rule real_of_int_floor_le)

   766   apply (rule real_of_int_floor_add_one_gt)

   767   apply (subgoal_tac "floor (x + real a) < floor x + a + 1")

   768   apply arith

   769   apply (subst real_of_int_less_iff [THEN sym])

   770   apply simp

   771   apply (subgoal_tac "real(floor(x + real a)) <= x + real a")

   772   apply (subgoal_tac "x < real(floor x) + 1")

   773   apply arith

   774   apply (rule real_of_int_floor_add_one_gt)

   775   apply (rule real_of_int_floor_le)

   776 done

   777

   778 lemma floor_add_number_of [simp]:

   779     "floor (x + number_of n) = floor x + number_of n"

   780   apply (subst floor_add [THEN sym])

   781   apply simp

   782 done

   783

   784 lemma floor_add_one [simp]: "floor (x + 1) = floor x + 1"

   785   apply (subst floor_add [THEN sym])

   786   apply simp

   787 done

   788

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

   790   apply (subst diff_minus)+

   791   apply (subst real_of_int_minus [THEN sym])

   792   apply (rule floor_add)

   793 done

   794

   795 lemma floor_subtract_number_of [simp]: "floor (x - number_of n) =

   796     floor x - number_of n"

   797   apply (subst floor_subtract [THEN sym])

   798   apply simp

   799 done

   800

   801 lemma floor_subtract_one [simp]: "floor (x - 1) = floor x - 1"

   802   apply (subst floor_subtract [THEN sym])

   803   apply simp

   804 done

   805

   806 lemma ceiling_zero [simp]: "ceiling 0 = 0"

   807 by (simp add: ceiling_def)

   808

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

   810 by (simp add: ceiling_def)

   811

   812 lemma ceiling_real_of_nat_zero [simp]: "ceiling (real (0::nat)) = 0"

   813 by auto

   814

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

   816 by (simp add: ceiling_def)

   817

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

   819 by (simp add: ceiling_def)

   820

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

   822 apply (simp add: ceiling_def)

   823 apply (subst le_minus_iff, simp)

   824 done

   825

   826 lemma ceiling_mono: "x < y ==> ceiling x \<le> ceiling y"

   827 by (simp add: floor_mono ceiling_def)

   828

   829 lemma ceiling_mono2: "x \<le> y ==> ceiling x \<le> ceiling y"

   830 by (simp add: floor_mono2 ceiling_def)

   831

   832 lemma real_of_int_ceiling_cancel [simp]:

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

   834 apply (auto simp add: ceiling_def)

   835 apply (drule arg_cong [where f = uminus], auto)

   836 apply (rule_tac x = "-n" in exI, auto)

   837 done

   838

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

   840 apply (simp add: ceiling_def)

   841 apply (rule minus_equation_iff [THEN iffD1])

   842 apply (simp add: floor_eq [where n = "-(n+1)"])

   843 done

   844

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

   846 by (simp add: ceiling_def floor_eq2 [where n = "-(n+1)"])

   847

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

   849 by (simp add: ceiling_def floor_eq2 [where n = "-n"])

   850

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

   852 by (simp add: ceiling_def)

   853

   854 lemma ceiling_number_of_eq [simp]:

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

   856 apply (subst real_number_of [symmetric])

   857 apply (rule ceiling_real_of_int)

   858 done

   859

   860 lemma ceiling_one [simp]: "ceiling 1 = 1"

   861   by (unfold ceiling_def, simp)

   862

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

   864 apply (rule neg_le_iff_le [THEN iffD1])

   865 apply (simp add: ceiling_def diff_minus)

   866 done

   867

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

   869 apply (insert real_of_int_ceiling_diff_one_le [of r])

   870 apply (simp del: real_of_int_ceiling_diff_one_le)

   871 done

   872

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

   874   apply (unfold ceiling_def)

   875   apply (subgoal_tac "-a <= floor(- x)")

   876   apply simp

   877   apply (rule le_floor)

   878   apply simp

   879 done

   880

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

   882   apply (unfold ceiling_def)

   883   apply (subgoal_tac "real(- a) <= - x")

   884   apply simp

   885   apply (rule real_le_floor)

   886   apply simp

   887 done

   888

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

   890   apply (rule iffI)

   891   apply (erule ceiling_le_real)

   892   apply (erule ceiling_le)

   893 done

   894

   895 lemma ceiling_le_eq_number_of [simp]:

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

   897 by (simp add: ceiling_le_eq)

   898

   899 lemma ceiling_le_zero_eq [simp]: "(ceiling x <= 0) = (x <= 0)"

   900 by (simp add: ceiling_le_eq)

   901

   902 lemma ceiling_le_eq_one [simp]: "(ceiling x <= 1) = (x <= 1)"

   903 by (simp add: ceiling_le_eq)

   904

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

   906   apply (subst linorder_not_le [THEN sym])+

   907   apply simp

   908   apply (rule ceiling_le_eq)

   909 done

   910

   911 lemma less_ceiling_eq_number_of [simp]:

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

   913 by (simp add: less_ceiling_eq)

   914

   915 lemma less_ceiling_eq_zero [simp]: "(0 < ceiling x) = (0 < x)"

   916 by (simp add: less_ceiling_eq)

   917

   918 lemma less_ceiling_eq_one [simp]: "(1 < ceiling x) = (1 < x)"

   919 by (simp add: less_ceiling_eq)

   920

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

   922   apply (insert ceiling_le_eq [of x "a - 1"])

   923   apply auto

   924 done

   925

   926 lemma ceiling_less_eq_number_of [simp]:

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

   928 by (simp add: ceiling_less_eq)

   929

   930 lemma ceiling_less_eq_zero [simp]: "(ceiling x < 0) = (x <= -1)"

   931 by (simp add: ceiling_less_eq)

   932

   933 lemma ceiling_less_eq_one [simp]: "(ceiling x < 1) = (x <= 0)"

   934 by (simp add: ceiling_less_eq)

   935

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

   937   apply (insert less_ceiling_eq [of "a - 1" x])

   938   apply auto

   939 done

   940

   941 lemma le_ceiling_eq_number_of [simp]:

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

   943 by (simp add: le_ceiling_eq)

   944

   945 lemma le_ceiling_eq_zero [simp]: "(0 <= ceiling x) = (-1 < x)"

   946 by (simp add: le_ceiling_eq)

   947

   948 lemma le_ceiling_eq_one [simp]: "(1 <= ceiling x) = (0 < x)"

   949 by (simp add: le_ceiling_eq)

   950

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

   952   apply (unfold ceiling_def, simp)

   953   apply (subst real_of_int_minus [THEN sym])

   954   apply (subst floor_add)

   955   apply simp

   956 done

   957

   958 lemma ceiling_add_number_of [simp]: "ceiling (x + number_of n) =

   959     ceiling x + number_of n"

   960   apply (subst ceiling_add [THEN sym])

   961   apply simp

   962 done

   963

   964 lemma ceiling_add_one [simp]: "ceiling (x + 1) = ceiling x + 1"

   965   apply (subst ceiling_add [THEN sym])

   966   apply simp

   967 done

   968

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

   970   apply (subst diff_minus)+

   971   apply (subst real_of_int_minus [THEN sym])

   972   apply (rule ceiling_add)

   973 done

   974

   975 lemma ceiling_subtract_number_of [simp]: "ceiling (x - number_of n) =

   976     ceiling x - number_of n"

   977   apply (subst ceiling_subtract [THEN sym])

   978   apply simp

   979 done

   980

   981 lemma ceiling_subtract_one [simp]: "ceiling (x - 1) = ceiling x - 1"

   982   apply (subst ceiling_subtract [THEN sym])

   983   apply simp

   984 done

   985

   986 subsection {* Versions for the natural numbers *}

   987

   988 definition

   989   natfloor :: "real => nat" where

   990   "natfloor x = nat(floor x)"

   991

   992 definition

   993   natceiling :: "real => nat" where

   994   "natceiling x = nat(ceiling x)"

   995

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

   997   by (unfold natfloor_def, simp)

   998

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

  1000   by (unfold natfloor_def, simp)

  1001

  1002 lemma zero_le_natfloor [simp]: "0 <= natfloor x"

  1003   by (unfold natfloor_def, simp)

  1004

  1005 lemma natfloor_number_of_eq [simp]: "natfloor (number_of n) = number_of n"

  1006   by (unfold natfloor_def, simp)

  1007

  1008 lemma natfloor_real_of_nat [simp]: "natfloor(real n) = n"

  1009   by (unfold natfloor_def, simp)

  1010

  1011 lemma real_natfloor_le: "0 <= x ==> real(natfloor x) <= x"

  1012   by (unfold natfloor_def, simp)

  1013

  1014 lemma natfloor_neg: "x <= 0 ==> natfloor x = 0"

  1015   apply (unfold natfloor_def)

  1016   apply (subgoal_tac "floor x <= floor 0")

  1017   apply simp

  1018   apply (erule floor_mono2)

  1019 done

  1020

  1021 lemma natfloor_mono: "x <= y ==> natfloor x <= natfloor y"

  1022   apply (case_tac "0 <= x")

  1023   apply (subst natfloor_def)+

  1024   apply (subst nat_le_eq_zle)

  1025   apply force

  1026   apply (erule floor_mono2)

  1027   apply (subst natfloor_neg)

  1028   apply simp

  1029   apply simp

  1030 done

  1031

  1032 lemma le_natfloor: "real x <= a ==> x <= natfloor a"

  1033   apply (unfold natfloor_def)

  1034   apply (subst nat_int [THEN sym])

  1035   apply (subst nat_le_eq_zle)

  1036   apply simp

  1037   apply (rule le_floor)

  1038   apply simp

  1039 done

  1040

  1041 lemma le_natfloor_eq: "0 <= x ==> (a <= natfloor x) = (real a <= x)"

  1042   apply (rule iffI)

  1043   apply (rule order_trans)

  1044   prefer 2

  1045   apply (erule real_natfloor_le)

  1046   apply (subst real_of_nat_le_iff)

  1047   apply assumption

  1048   apply (erule le_natfloor)

  1049 done

  1050

  1051 lemma le_natfloor_eq_number_of [simp]:

  1052     "~ neg((number_of n)::int) ==> 0 <= x ==>

  1053       (number_of n <= natfloor x) = (number_of n <= x)"

  1054   apply (subst le_natfloor_eq, assumption)

  1055   apply simp

  1056 done

  1057

  1058 lemma le_natfloor_eq_one [simp]: "(1 <= natfloor x) = (1 <= x)"

  1059   apply (case_tac "0 <= x")

  1060   apply (subst le_natfloor_eq, assumption, simp)

  1061   apply (rule iffI)

  1062   apply (subgoal_tac "natfloor x <= natfloor 0")

  1063   apply simp

  1064   apply (rule natfloor_mono)

  1065   apply simp

  1066   apply simp

  1067 done

  1068

  1069 lemma natfloor_eq: "real n <= x ==> x < real n + 1 ==> natfloor x = n"

  1070   apply (unfold natfloor_def)

  1071   apply (subst nat_int [THEN sym]);back;

  1072   apply (subst eq_nat_nat_iff)

  1073   apply simp

  1074   apply simp

  1075   apply (rule floor_eq2)

  1076   apply auto

  1077 done

  1078

  1079 lemma real_natfloor_add_one_gt: "x < real(natfloor x) + 1"

  1080   apply (case_tac "0 <= x")

  1081   apply (unfold natfloor_def)

  1082   apply simp

  1083   apply simp_all

  1084 done

  1085

  1086 lemma real_natfloor_gt_diff_one: "x - 1 < real(natfloor x)"

  1087   apply (simp add: compare_rls)

  1088   apply (rule real_natfloor_add_one_gt)

  1089 done

  1090

  1091 lemma ge_natfloor_plus_one_imp_gt: "natfloor z + 1 <= n ==> z < real n"

  1092   apply (subgoal_tac "z < real(natfloor z) + 1")

  1093   apply arith

  1094   apply (rule real_natfloor_add_one_gt)

  1095 done

  1096

  1097 lemma natfloor_add [simp]: "0 <= x ==> natfloor (x + real a) = natfloor x + a"

  1098   apply (unfold natfloor_def)

  1099   apply (subgoal_tac "real a = real (int a)")

  1100   apply (erule ssubst)

  1101   apply (simp add: nat_add_distrib del: real_of_int_of_nat_eq)

  1102   apply simp

  1103 done

  1104

  1105 lemma natfloor_add_number_of [simp]:

  1106     "~neg ((number_of n)::int) ==> 0 <= x ==>

  1107       natfloor (x + number_of n) = natfloor x + number_of n"

  1108   apply (subst natfloor_add [THEN sym])

  1109   apply simp_all

  1110 done

  1111

  1112 lemma natfloor_add_one: "0 <= x ==> natfloor(x + 1) = natfloor x + 1"

  1113   apply (subst natfloor_add [THEN sym])

  1114   apply assumption

  1115   apply simp

  1116 done

  1117

  1118 lemma natfloor_subtract [simp]: "real a <= x ==>

  1119     natfloor(x - real a) = natfloor x - a"

  1120   apply (unfold natfloor_def)

  1121   apply (subgoal_tac "real a = real (int a)")

  1122   apply (erule ssubst)

  1123   apply (simp del: real_of_int_of_nat_eq)

  1124   apply simp

  1125 done

  1126

  1127 lemma natceiling_zero [simp]: "natceiling 0 = 0"

  1128   by (unfold natceiling_def, simp)

  1129

  1130 lemma natceiling_one [simp]: "natceiling 1 = 1"

  1131   by (unfold natceiling_def, simp)

  1132

  1133 lemma zero_le_natceiling [simp]: "0 <= natceiling x"

  1134   by (unfold natceiling_def, simp)

  1135

  1136 lemma natceiling_number_of_eq [simp]: "natceiling (number_of n) = number_of n"

  1137   by (unfold natceiling_def, simp)

  1138

  1139 lemma natceiling_real_of_nat [simp]: "natceiling(real n) = n"

  1140   by (unfold natceiling_def, simp)

  1141

  1142 lemma real_natceiling_ge: "x <= real(natceiling x)"

  1143   apply (unfold natceiling_def)

  1144   apply (case_tac "x < 0")

  1145   apply simp

  1146   apply (subst real_nat_eq_real)

  1147   apply (subgoal_tac "ceiling 0 <= ceiling x")

  1148   apply simp

  1149   apply (rule ceiling_mono2)

  1150   apply simp

  1151   apply simp

  1152 done

  1153

  1154 lemma natceiling_neg: "x <= 0 ==> natceiling x = 0"

  1155   apply (unfold natceiling_def)

  1156   apply simp

  1157 done

  1158

  1159 lemma natceiling_mono: "x <= y ==> natceiling x <= natceiling y"

  1160   apply (case_tac "0 <= x")

  1161   apply (subst natceiling_def)+

  1162   apply (subst nat_le_eq_zle)

  1163   apply (rule disjI2)

  1164   apply (subgoal_tac "real (0::int) <= real(ceiling y)")

  1165   apply simp

  1166   apply (rule order_trans)

  1167   apply simp

  1168   apply (erule order_trans)

  1169   apply simp

  1170   apply (erule ceiling_mono2)

  1171   apply (subst natceiling_neg)

  1172   apply simp_all

  1173 done

  1174

  1175 lemma natceiling_le: "x <= real a ==> natceiling x <= a"

  1176   apply (unfold natceiling_def)

  1177   apply (case_tac "x < 0")

  1178   apply simp

  1179   apply (subst nat_int [THEN sym]);back;

  1180   apply (subst nat_le_eq_zle)

  1181   apply simp

  1182   apply (rule ceiling_le)

  1183   apply simp

  1184 done

  1185

  1186 lemma natceiling_le_eq: "0 <= x ==> (natceiling x <= a) = (x <= real a)"

  1187   apply (rule iffI)

  1188   apply (rule order_trans)

  1189   apply (rule real_natceiling_ge)

  1190   apply (subst real_of_nat_le_iff)

  1191   apply assumption

  1192   apply (erule natceiling_le)

  1193 done

  1194

  1195 lemma natceiling_le_eq_number_of [simp]:

  1196     "~ neg((number_of n)::int) ==> 0 <= x ==>

  1197       (natceiling x <= number_of n) = (x <= number_of n)"

  1198   apply (subst natceiling_le_eq, assumption)

  1199   apply simp

  1200 done

  1201

  1202 lemma natceiling_le_eq_one: "(natceiling x <= 1) = (x <= 1)"

  1203   apply (case_tac "0 <= x")

  1204   apply (subst natceiling_le_eq)

  1205   apply assumption

  1206   apply simp

  1207   apply (subst natceiling_neg)

  1208   apply simp

  1209   apply simp

  1210 done

  1211

  1212 lemma natceiling_eq: "real n < x ==> x <= real n + 1 ==> natceiling x = n + 1"

  1213   apply (unfold natceiling_def)

  1214   apply (simplesubst nat_int [THEN sym]) back back

  1215   apply (subgoal_tac "nat(int n) + 1 = nat(int n + 1)")

  1216   apply (erule ssubst)

  1217   apply (subst eq_nat_nat_iff)

  1218   apply (subgoal_tac "ceiling 0 <= ceiling x")

  1219   apply simp

  1220   apply (rule ceiling_mono2)

  1221   apply force

  1222   apply force

  1223   apply (rule ceiling_eq2)

  1224   apply (simp, simp)

  1225   apply (subst nat_add_distrib)

  1226   apply auto

  1227 done

  1228

  1229 lemma natceiling_add [simp]: "0 <= x ==>

  1230     natceiling (x + real a) = natceiling x + a"

  1231   apply (unfold natceiling_def)

  1232   apply (subgoal_tac "real a = real (int a)")

  1233   apply (erule ssubst)

  1234   apply (simp del: real_of_int_of_nat_eq)

  1235   apply (subst nat_add_distrib)

  1236   apply (subgoal_tac "0 = ceiling 0")

  1237   apply (erule ssubst)

  1238   apply (erule ceiling_mono2)

  1239   apply simp_all

  1240 done

  1241

  1242 lemma natceiling_add_number_of [simp]:

  1243     "~ neg ((number_of n)::int) ==> 0 <= x ==>

  1244       natceiling (x + number_of n) = natceiling x + number_of n"

  1245   apply (subst natceiling_add [THEN sym])

  1246   apply simp_all

  1247 done

  1248

  1249 lemma natceiling_add_one: "0 <= x ==> natceiling(x + 1) = natceiling x + 1"

  1250   apply (subst natceiling_add [THEN sym])

  1251   apply assumption

  1252   apply simp

  1253 done

  1254

  1255 lemma natceiling_subtract [simp]: "real a <= x ==>

  1256     natceiling(x - real a) = natceiling x - a"

  1257   apply (unfold natceiling_def)

  1258   apply (subgoal_tac "real a = real (int a)")

  1259   apply (erule ssubst)

  1260   apply (simp del: real_of_int_of_nat_eq)

  1261   apply simp

  1262 done

  1263

  1264 lemma natfloor_div_nat: "1 <= x ==> y > 0 ==>

  1265   natfloor (x / real y) = natfloor x div y"

  1266 proof -

  1267   assume "1 <= (x::real)" and "(y::nat) > 0"

  1268   have "natfloor x = (natfloor x) div y * y + (natfloor x) mod y"

  1269     by simp

  1270   then have a: "real(natfloor x) = real ((natfloor x) div y) * real y +

  1271     real((natfloor x) mod y)"

  1272     by (simp only: real_of_nat_add [THEN sym] real_of_nat_mult [THEN sym])

  1273   have "x = real(natfloor x) + (x - real(natfloor x))"

  1274     by simp

  1275   then have "x = real ((natfloor x) div y) * real y +

  1276       real((natfloor x) mod y) + (x - real(natfloor x))"

  1277     by (simp add: a)

  1278   then have "x / real y = ... / real y"

  1279     by simp

  1280   also have "... = real((natfloor x) div y) + real((natfloor x) mod y) /

  1281     real y + (x - real(natfloor x)) / real y"

  1282     by (auto simp add: ring_simps add_divide_distrib

  1283       diff_divide_distrib prems)

  1284   finally have "natfloor (x / real y) = natfloor(...)" by simp

  1285   also have "... = natfloor(real((natfloor x) mod y) /

  1286     real y + (x - real(natfloor x)) / real y + real((natfloor x) div y))"

  1287     by (simp add: add_ac)

  1288   also have "... = natfloor(real((natfloor x) mod y) /

  1289     real y + (x - real(natfloor x)) / real y) + (natfloor x) div y"

  1290     apply (rule natfloor_add)

  1291     apply (rule add_nonneg_nonneg)

  1292     apply (rule divide_nonneg_pos)

  1293     apply simp

  1294     apply (simp add: prems)

  1295     apply (rule divide_nonneg_pos)

  1296     apply (simp add: compare_rls)

  1297     apply (rule real_natfloor_le)

  1298     apply (insert prems, auto)

  1299     done

  1300   also have "natfloor(real((natfloor x) mod y) /

  1301     real y + (x - real(natfloor x)) / real y) = 0"

  1302     apply (rule natfloor_eq)

  1303     apply simp

  1304     apply (rule add_nonneg_nonneg)

  1305     apply (rule divide_nonneg_pos)

  1306     apply force

  1307     apply (force simp add: prems)

  1308     apply (rule divide_nonneg_pos)

  1309     apply (simp add: compare_rls)

  1310     apply (rule real_natfloor_le)

  1311     apply (auto simp add: prems)

  1312     apply (insert prems, arith)

  1313     apply (simp add: add_divide_distrib [THEN sym])

  1314     apply (subgoal_tac "real y = real y - 1 + 1")

  1315     apply (erule ssubst)

  1316     apply (rule add_le_less_mono)

  1317     apply (simp add: compare_rls)

  1318     apply (subgoal_tac "real(natfloor x mod y) + 1 =

  1319       real(natfloor x mod y + 1)")

  1320     apply (erule ssubst)

  1321     apply (subst real_of_nat_le_iff)

  1322     apply (subgoal_tac "natfloor x mod y < y")

  1323     apply arith

  1324     apply (rule mod_less_divisor)

  1325     apply auto

  1326     apply (simp add: compare_rls)

  1327     apply (subst add_commute)

  1328     apply (rule real_natfloor_add_one_gt)

  1329     done

  1330   finally show ?thesis by simp

  1331 qed

  1332

  1333 end