src/HOL/RComplete.thy
 author wenzelm Mon Mar 16 18:24:30 2009 +0100 (2009-03-16) changeset 30549 d2d7874648bd parent 30242 aea5d7fa7ef5 child 32707 836ec9d0a0c8 permissions -rw-r--r--
simplified method setup;
     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: algebra_simps)

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

   381 qed

   382

   383 instance real :: archimedean_field

   384 proof

   385   fix r :: real

   386   obtain n :: nat where "r < real n"

   387     using reals_Archimedean2 ..

   388   then have "r \<le> of_int (int n)"

   389     unfolding real_eq_of_nat by simp

   390   then show "\<exists>z. r \<le> of_int z" ..

   391 qed

   392

   393 lemma reals_Archimedean3:

   394   assumes x_greater_zero: "0 < x"

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

   396   unfolding real_of_nat_def using 0 < x

   397   by (auto intro: ex_less_of_nat_mult)

   398

   399 lemma reals_Archimedean6:

   400      "0 \<le> r ==> \<exists>(n::nat). real (n - 1) \<le> r & r < real (n)"

   401 unfolding real_of_nat_def

   402 apply (rule exI [where x="nat (floor r + 1)"])

   403 apply (insert floor_correct [of r])

   404 apply (simp add: nat_add_distrib of_nat_nat)

   405 done

   406

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

   408   by (drule reals_Archimedean6) auto

   409

   410 lemma reals_Archimedean_6b_int:

   411      "0 \<le> r ==> \<exists>n::int. real n \<le> r & r < real (n+1)"

   412   unfolding real_of_int_def by (rule floor_exists)

   413

   414 lemma reals_Archimedean_6c_int:

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

   416   unfolding real_of_int_def by (rule floor_exists)

   417

   418

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

   420

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

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

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

   424

   425 lemma Rats_dense_in_nn_real: fixes x::real

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

   427 proof -

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

   429   with reals_Archimedean obtain q::nat

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

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

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

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

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

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

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

   437     by (unfold p_def) (rule Least_Suc)

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

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

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

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

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

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

   444     by (simp only: inverse_eq_divide real_diff_def real_of_nat_Suc

   445     minus_divide_left add_divide_distrib[THEN sym]) simp

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

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

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

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

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

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

   452 qed

   453

   454 theorem Rats_dense_in_real: fixes x y :: real

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

   456 proof -

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

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

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

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

   461     by(rule Rats_dense_in_nn_real)

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

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

   464     by blast

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

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

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

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

   469   ultimately show ?thesis by fast

   470 qed

   471

   472

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

   474

   475 lemma number_of_less_real_of_int_iff [simp]:

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

   477 apply auto

   478 apply (rule real_of_int_less_iff [THEN iffD1])

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

   480 done

   481

   482 lemma number_of_less_real_of_int_iff2 [simp]:

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

   484 apply auto

   485 apply (rule real_of_int_less_iff [THEN iffD1])

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

   487 done

   488

   489 lemma number_of_le_real_of_int_iff [simp]:

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

   491 by (simp add: linorder_not_less [symmetric])

   492

   493 lemma number_of_le_real_of_int_iff2 [simp]:

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

   495 by (simp add: linorder_not_less [symmetric])

   496

   497 lemma floor_real_of_nat_zero: "floor (real (0::nat)) = 0"

   498 by auto (* delete? *)

   499

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

   501 unfolding real_of_nat_def by simp

   502

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

   504 unfolding real_of_nat_def by (simp add: floor_minus)

   505

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

   507 unfolding real_of_int_def by simp

   508

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

   510 unfolding real_of_int_def by (simp add: floor_minus)

   511

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

   513 unfolding real_of_int_def by (rule floor_exists)

   514

   515 lemma lemma_floor:

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

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

   518 proof -

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

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

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

   522   thus ?thesis by arith

   523 qed

   524

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

   526 unfolding real_of_int_def by (rule of_int_floor_le)

   527

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

   529 by (auto intro: lemma_floor)

   530

   531 lemma real_of_int_floor_cancel [simp]:

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

   533   using floor_real_of_int by metis

   534

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

   536   unfolding real_of_int_def using floor_unique [of n x] by simp

   537

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

   539   unfolding real_of_int_def by (rule floor_unique)

   540

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

   542 apply (rule inj_int [THEN injD])

   543 apply (simp add: real_of_nat_Suc)

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

   545 done

   546

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

   548 apply (drule order_le_imp_less_or_eq)

   549 apply (auto intro: floor_eq3)

   550 done

   551

   552 lemma floor_number_of_eq:

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

   554   by (rule floor_number_of) (* already declared [simp] *)

   555

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

   557   unfolding real_of_int_def using floor_correct [of r] by simp

   558

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

   560   unfolding real_of_int_def using floor_correct [of r] by simp

   561

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

   563   unfolding real_of_int_def using floor_correct [of r] by simp

   564

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

   566   unfolding real_of_int_def using floor_correct [of r] by simp

   567

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

   569   unfolding real_of_int_def by (simp add: le_floor_iff)

   570

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

   572   unfolding real_of_int_def by (simp add: le_floor_iff)

   573

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

   575   unfolding real_of_int_def by (rule le_floor_iff)

   576

   577 lemma le_floor_eq_number_of:

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

   579   by (rule number_of_le_floor) (* already declared [simp] *)

   580

   581 lemma le_floor_eq_zero: "(0 <= floor x) = (0 <= x)"

   582   by (rule zero_le_floor) (* already declared [simp] *)

   583

   584 lemma le_floor_eq_one: "(1 <= floor x) = (1 <= x)"

   585   by (rule one_le_floor) (* already declared [simp] *)

   586

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

   588   unfolding real_of_int_def by (rule floor_less_iff)

   589

   590 lemma floor_less_eq_number_of:

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

   592   by (rule floor_less_number_of) (* already declared [simp] *)

   593

   594 lemma floor_less_eq_zero: "(floor x < 0) = (x < 0)"

   595   by (rule floor_less_zero) (* already declared [simp] *)

   596

   597 lemma floor_less_eq_one: "(floor x < 1) = (x < 1)"

   598   by (rule floor_less_one) (* already declared [simp] *)

   599

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

   601   unfolding real_of_int_def by (rule less_floor_iff)

   602

   603 lemma less_floor_eq_number_of:

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

   605   by (rule number_of_less_floor) (* already declared [simp] *)

   606

   607 lemma less_floor_eq_zero: "(0 < floor x) = (1 <= x)"

   608   by (rule zero_less_floor) (* already declared [simp] *)

   609

   610 lemma less_floor_eq_one: "(1 < floor x) = (2 <= x)"

   611   by (rule one_less_floor) (* already declared [simp] *)

   612

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

   614   unfolding real_of_int_def by (rule floor_le_iff)

   615

   616 lemma floor_le_eq_number_of:

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

   618   by (rule floor_le_number_of) (* already declared [simp] *)

   619

   620 lemma floor_le_eq_zero: "(floor x <= 0) = (x < 1)"

   621   by (rule floor_le_zero) (* already declared [simp] *)

   622

   623 lemma floor_le_eq_one: "(floor x <= 1) = (x < 2)"

   624   by (rule floor_le_one) (* already declared [simp] *)

   625

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

   627   unfolding real_of_int_def by (rule floor_add_of_int)

   628

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

   630   unfolding real_of_int_def by (rule floor_diff_of_int)

   631

   632 lemma floor_subtract_number_of: "floor (x - number_of n) =

   633     floor x - number_of n"

   634   by (rule floor_diff_number_of) (* already declared [simp] *)

   635

   636 lemma floor_subtract_one: "floor (x - 1) = floor x - 1"

   637   by (rule floor_diff_one) (* already declared [simp] *)

   638

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

   640   unfolding real_of_nat_def by simp

   641

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

   643 by auto (* delete? *)

   644

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

   646   unfolding real_of_int_def by simp

   647

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

   649   unfolding real_of_int_def by simp

   650

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

   652   unfolding real_of_int_def by (rule le_of_int_ceiling)

   653

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

   655   unfolding real_of_int_def by simp

   656

   657 lemma real_of_int_ceiling_cancel [simp]:

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

   659   using ceiling_real_of_int by metis

   660

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

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

   663

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

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

   666

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

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

   669

   670 lemma ceiling_number_of_eq:

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

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

   673

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

   675   unfolding real_of_int_def using ceiling_correct [of r] by simp

   676

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

   678   unfolding real_of_int_def using ceiling_correct [of r] by simp

   679

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

   681   unfolding real_of_int_def by (simp add: ceiling_le_iff)

   682

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

   684   unfolding real_of_int_def by (simp add: ceiling_le_iff)

   685

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

   687   unfolding real_of_int_def by (rule ceiling_le_iff)

   688

   689 lemma ceiling_le_eq_number_of:

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

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

   692

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

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

   695

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

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

   698

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

   700   unfolding real_of_int_def by (rule less_ceiling_iff)

   701

   702 lemma less_ceiling_eq_number_of:

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

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

   705

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

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

   708

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

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

   711

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

   713   unfolding real_of_int_def by (rule ceiling_less_iff)

   714

   715 lemma ceiling_less_eq_number_of:

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

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

   718

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

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

   721

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

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

   724

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

   726   unfolding real_of_int_def by (rule le_ceiling_iff)

   727

   728 lemma le_ceiling_eq_number_of:

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

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

   731

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

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

   734

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

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

   737

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

   739   unfolding real_of_int_def by (rule ceiling_add_of_int)

   740

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

   742   unfolding real_of_int_def by (rule ceiling_diff_of_int)

   743

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

   745     ceiling x - number_of n"

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

   747

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

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

   750

   751

   752 subsection {* Versions for the natural numbers *}

   753

   754 definition

   755   natfloor :: "real => nat" where

   756   "natfloor x = nat(floor x)"

   757

   758 definition

   759   natceiling :: "real => nat" where

   760   "natceiling x = nat(ceiling x)"

   761

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

   763   by (unfold natfloor_def, simp)

   764

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

   766   by (unfold natfloor_def, simp)

   767

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

   769   by (unfold natfloor_def, simp)

   770

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

   772   by (unfold natfloor_def, simp)

   773

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

   775   by (unfold natfloor_def, simp)

   776

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

   778   by (unfold natfloor_def, simp)

   779

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

   781   apply (unfold natfloor_def)

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

   783   apply simp

   784   apply (erule floor_mono)

   785 done

   786

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

   788   apply (case_tac "0 <= x")

   789   apply (subst natfloor_def)+

   790   apply (subst nat_le_eq_zle)

   791   apply force

   792   apply (erule floor_mono)

   793   apply (subst natfloor_neg)

   794   apply simp

   795   apply simp

   796 done

   797

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

   799   apply (unfold natfloor_def)

   800   apply (subst nat_int [THEN sym])

   801   apply (subst nat_le_eq_zle)

   802   apply simp

   803   apply (rule le_floor)

   804   apply simp

   805 done

   806

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

   808   apply (rule iffI)

   809   apply (rule order_trans)

   810   prefer 2

   811   apply (erule real_natfloor_le)

   812   apply (subst real_of_nat_le_iff)

   813   apply assumption

   814   apply (erule le_natfloor)

   815 done

   816

   817 lemma le_natfloor_eq_number_of [simp]:

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

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

   820   apply (subst le_natfloor_eq, assumption)

   821   apply simp

   822 done

   823

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

   825   apply (case_tac "0 <= x")

   826   apply (subst le_natfloor_eq, assumption, simp)

   827   apply (rule iffI)

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

   829   apply simp

   830   apply (rule natfloor_mono)

   831   apply simp

   832   apply simp

   833 done

   834

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

   836   apply (unfold natfloor_def)

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

   838   apply (subst eq_nat_nat_iff)

   839   apply simp

   840   apply simp

   841   apply (rule floor_eq2)

   842   apply auto

   843 done

   844

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

   846   apply (case_tac "0 <= x")

   847   apply (unfold natfloor_def)

   848   apply simp

   849   apply simp_all

   850 done

   851

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

   853 using real_natfloor_add_one_gt by (simp add: algebra_simps)

   854

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

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

   857   apply arith

   858   apply (rule real_natfloor_add_one_gt)

   859 done

   860

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

   862   apply (unfold natfloor_def)

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

   864   apply (erule ssubst)

   865   apply (simp add: nat_add_distrib del: real_of_int_of_nat_eq)

   866   apply simp

   867 done

   868

   869 lemma natfloor_add_number_of [simp]:

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

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

   872   apply (subst natfloor_add [THEN sym])

   873   apply simp_all

   874 done

   875

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

   877   apply (subst natfloor_add [THEN sym])

   878   apply assumption

   879   apply simp

   880 done

   881

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

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

   884   apply (unfold natfloor_def)

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

   886   apply (erule ssubst)

   887   apply (simp del: real_of_int_of_nat_eq)

   888   apply simp

   889 done

   890

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

   892   by (unfold natceiling_def, simp)

   893

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

   895   by (unfold natceiling_def, simp)

   896

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

   898   by (unfold natceiling_def, simp)

   899

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

   901   by (unfold natceiling_def, simp)

   902

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

   904   by (unfold natceiling_def, simp)

   905

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

   907   apply (unfold natceiling_def)

   908   apply (case_tac "x < 0")

   909   apply simp

   910   apply (subst real_nat_eq_real)

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

   912   apply simp

   913   apply (rule ceiling_mono)

   914   apply simp

   915   apply simp

   916 done

   917

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

   919   apply (unfold natceiling_def)

   920   apply simp

   921 done

   922

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

   924   apply (case_tac "0 <= x")

   925   apply (subst natceiling_def)+

   926   apply (subst nat_le_eq_zle)

   927   apply (rule disjI2)

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

   929   apply simp

   930   apply (rule order_trans)

   931   apply simp

   932   apply (erule order_trans)

   933   apply simp

   934   apply (erule ceiling_mono)

   935   apply (subst natceiling_neg)

   936   apply simp_all

   937 done

   938

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

   940   apply (unfold natceiling_def)

   941   apply (case_tac "x < 0")

   942   apply simp

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

   944   apply (subst nat_le_eq_zle)

   945   apply simp

   946   apply (rule ceiling_le)

   947   apply simp

   948 done

   949

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

   951   apply (rule iffI)

   952   apply (rule order_trans)

   953   apply (rule real_natceiling_ge)

   954   apply (subst real_of_nat_le_iff)

   955   apply assumption

   956   apply (erule natceiling_le)

   957 done

   958

   959 lemma natceiling_le_eq_number_of [simp]:

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

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

   962   apply (subst natceiling_le_eq, assumption)

   963   apply simp

   964 done

   965

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

   967   apply (case_tac "0 <= x")

   968   apply (subst natceiling_le_eq)

   969   apply assumption

   970   apply simp

   971   apply (subst natceiling_neg)

   972   apply simp

   973   apply simp

   974 done

   975

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

   977   apply (unfold natceiling_def)

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

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

   980   apply (erule ssubst)

   981   apply (subst eq_nat_nat_iff)

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

   983   apply simp

   984   apply (rule ceiling_mono)

   985   apply force

   986   apply force

   987   apply (rule ceiling_eq2)

   988   apply (simp, simp)

   989   apply (subst nat_add_distrib)

   990   apply auto

   991 done

   992

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

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

   995   apply (unfold natceiling_def)

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

   997   apply (erule ssubst)

   998   apply (simp del: real_of_int_of_nat_eq)

   999   apply (subst nat_add_distrib)

  1000   apply (subgoal_tac "0 = ceiling 0")

  1001   apply (erule ssubst)

  1002   apply (erule ceiling_mono)

  1003   apply simp_all

  1004 done

  1005

  1006 lemma natceiling_add_number_of [simp]:

  1007     "~ neg ((number_of n)::int) ==> 0 <= x ==>

  1008       natceiling (x + number_of n) = natceiling x + number_of n"

  1009   apply (subst natceiling_add [THEN sym])

  1010   apply simp_all

  1011 done

  1012

  1013 lemma natceiling_add_one: "0 <= x ==> natceiling(x + 1) = natceiling x + 1"

  1014   apply (subst natceiling_add [THEN sym])

  1015   apply assumption

  1016   apply simp

  1017 done

  1018

  1019 lemma natceiling_subtract [simp]: "real a <= x ==>

  1020     natceiling(x - real a) = natceiling x - a"

  1021   apply (unfold natceiling_def)

  1022   apply (subgoal_tac "real a = real (int a)")

  1023   apply (erule ssubst)

  1024   apply (simp del: real_of_int_of_nat_eq)

  1025   apply simp

  1026 done

  1027

  1028 lemma natfloor_div_nat: "1 <= x ==> y > 0 ==>

  1029   natfloor (x / real y) = natfloor x div y"

  1030 proof -

  1031   assume "1 <= (x::real)" and "(y::nat) > 0"

  1032   have "natfloor x = (natfloor x) div y * y + (natfloor x) mod y"

  1033     by simp

  1034   then have a: "real(natfloor x) = real ((natfloor x) div y) * real y +

  1035     real((natfloor x) mod y)"

  1036     by (simp only: real_of_nat_add [THEN sym] real_of_nat_mult [THEN sym])

  1037   have "x = real(natfloor x) + (x - real(natfloor x))"

  1038     by simp

  1039   then have "x = real ((natfloor x) div y) * real y +

  1040       real((natfloor x) mod y) + (x - real(natfloor x))"

  1041     by (simp add: a)

  1042   then have "x / real y = ... / real y"

  1043     by simp

  1044   also have "... = real((natfloor x) div y) + real((natfloor x) mod y) /

  1045     real y + (x - real(natfloor x)) / real y"

  1046     by (auto simp add: algebra_simps add_divide_distrib

  1047       diff_divide_distrib prems)

  1048   finally have "natfloor (x / real y) = natfloor(...)" by simp

  1049   also have "... = natfloor(real((natfloor x) mod y) /

  1050     real y + (x - real(natfloor x)) / real y + real((natfloor x) div y))"

  1051     by (simp add: add_ac)

  1052   also have "... = natfloor(real((natfloor x) mod y) /

  1053     real y + (x - real(natfloor x)) / real y) + (natfloor x) div y"

  1054     apply (rule natfloor_add)

  1055     apply (rule add_nonneg_nonneg)

  1056     apply (rule divide_nonneg_pos)

  1057     apply simp

  1058     apply (simp add: prems)

  1059     apply (rule divide_nonneg_pos)

  1060     apply (simp add: algebra_simps)

  1061     apply (rule real_natfloor_le)

  1062     apply (insert prems, auto)

  1063     done

  1064   also have "natfloor(real((natfloor x) mod y) /

  1065     real y + (x - real(natfloor x)) / real y) = 0"

  1066     apply (rule natfloor_eq)

  1067     apply simp

  1068     apply (rule add_nonneg_nonneg)

  1069     apply (rule divide_nonneg_pos)

  1070     apply force

  1071     apply (force simp add: prems)

  1072     apply (rule divide_nonneg_pos)

  1073     apply (simp add: algebra_simps)

  1074     apply (rule real_natfloor_le)

  1075     apply (auto simp add: prems)

  1076     apply (insert prems, arith)

  1077     apply (simp add: add_divide_distrib [THEN sym])

  1078     apply (subgoal_tac "real y = real y - 1 + 1")

  1079     apply (erule ssubst)

  1080     apply (rule add_le_less_mono)

  1081     apply (simp add: algebra_simps)

  1082     apply (subgoal_tac "1 + real(natfloor x mod y) =

  1083       real(natfloor x mod y + 1)")

  1084     apply (erule ssubst)

  1085     apply (subst real_of_nat_le_iff)

  1086     apply (subgoal_tac "natfloor x mod y < y")

  1087     apply arith

  1088     apply (rule mod_less_divisor)

  1089     apply auto

  1090     using real_natfloor_add_one_gt

  1091     apply (simp add: algebra_simps)

  1092     done

  1093   finally show ?thesis by simp

  1094 qed

  1095

  1096 end