(*  Title       : HOL/RComplete.thy

     Author      : Jacques D. Fleuriot, University of Edinburgh

     Author      : Larry Paulson, University of Cambridge

     Author      : Jeremy Avigad, Carnegie Mellon University

     Author      : Florian Zuleger, Johannes Hoelzl, and Simon Funke, TU Muenchen

 *)

 header {* Completeness of the Reals; Floor and Ceiling Functions *}

 theory RComplete

 imports Lubs RealDef

 begin

 lemma real_sum_of_halves: "x/2 + x/2 = (x::real)"

   by simp

 subsection {* Completeness of Positive Reals *}

 text {*

   Supremum property for the set of positive reals

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

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

   (written @{text "S"}).

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

 *}

 lemma posreal_complete:

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

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

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

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

 proof (rule exI, rule allI)

   fix y

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

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

   proof (cases "0 < y")

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

     show ?thesis

     proof

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

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

         using neg_y by (rule real_less_all_real2)

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

     next

       assume "y < real_of_preal (psup ?pP)"

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

       hence "0 < x" using positive_P by simp

       hence "y < x" using neg_y by simp

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

     qed

   next

     assume pos_y: "0 < y"

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

       by (auto simp add: real_gt_zero_preal_Ex)

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

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

     then obtain pa where "a = real_of_preal pa"

       by (auto simp add: real_gt_zero_preal_Ex)

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

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

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

       using upper_bound_Ex ..

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

     hence "0 < sup" using a_pos by arith

     then obtain possup where "sup = real_of_preal possup"

       by (auto simp add: real_gt_zero_preal_Ex)

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

       using sup by (auto simp add: real_of_preal_lessI)

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

       by (rule preal_complete)

     show ?thesis

     proof

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

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

       hence "0 < x" using pos_y by arith

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

         by (auto simp add: real_gt_zero_preal_Ex)

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

       proof

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

           by (simp add: real_of_preal_lessI)

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

       qed

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

         using psup by simp

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

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

         using y_is_py and pos_y by (simp add: real_of_preal_lessI)

     next

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

       hence "py < psup ?pP" using y_is_py

         by (simp add: real_of_preal_lessI)

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

         using psup by auto

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

         by (simp add: real_gt_zero_preal_Ex)

       hence "y < x" using py_less_X and y_is_py

         by (simp add: real_of_preal_lessI)

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

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

     qed

   qed

 qed

 text {*

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

 *}

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

   apply (frule isLub_isUb)

   apply (frule_tac x = y in isLub_isUb)

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

   done

 text {*

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

 *}

 lemma posreals_complete:

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

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

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

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

 proof

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

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

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

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

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

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

   hence "0 < u" using x_gt_zero by arith

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

     by (auto simp add: real_gt_zero_preal_Ex)

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

   proof

     fix pa

     assume "pa \<in> ?pS"

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

       by simp

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

     ultimately show "pa \<le> pu"

       using sup and u_is_pu by (simp add: real_of_preal_le_iff)

   qed

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

   proof

     fix y

     assume y_in_S: "y \<in> S"

     hence "0 < y" using positive_S by simp

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

       by (auto simp add: real_gt_zero_preal_Ex)

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

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

       by (rule preal_psup_le)

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

       using y_is_py by (simp add: real_of_preal_le_iff)

   qed

   moreover {

     fix x

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

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

     proof -

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

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

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

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

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

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

       hence "0 < x" using s_pos by simp

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

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

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

       proof

 	fix pe

 	assume "pe \<in> ?pS"

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

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

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

       qed

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

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

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

     qed

   }

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

     by (simp add: isLub_def leastP_def isUb_def setle_def setge_def)

 qed

 text {*

   \medskip reals Completeness (again!)

 *}

 lemma reals_complete:

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

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

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

 proof -

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

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

     using exists_Ub ..

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

   {

     fix x

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

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

       by (simp add: isUb_def setle_def)

     {

       fix s

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

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

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

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

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

     }

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

       by (auto simp add: isUb_def setle_def)

   } note S_Ub_is_SHIFT_Ub = this

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

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

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

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

     using X_in_S and Y_isUb by auto

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

     using posreals_complete [of ?SHIFT] by blast

   show ?thesis

   proof

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

     proof (rule isLubI2)

       {

         fix x

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

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

 	  using S_Ub_is_SHIFT_Ub by simp

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

 	  using t_is_Lub by (simp add: isLub_le_isUb)

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

       }

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

 	by (simp add: setgeI)

     next

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

       proof -

         {

           fix y

           assume y_in_S: "y \<in> S"

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

           proof -

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

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

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

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

             show ?thesis

             proof cases

               assume "y \<le> x"

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

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

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

             next

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

               hence x_less_y: "x < y" by arith

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

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

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

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

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

               thus ?thesis by simp

             qed

           qed

         }

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

       qed

     qed

   qed

 qed

 subsection {* The Archimedean Property of the Reals *}

 theorem reals_Archimedean:

   assumes x_pos: "0 < x"

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

 proof (rule ccontr)

   assume contr: "\<not> ?thesis"

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

   proof

     fix n

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

       by (simp add: linorder_not_less)

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

       by (simp add: inverse_eq_divide)

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

       by (rule real_of_nat_ge_zero)

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

       by (rule mult_right_mono)

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

   qed

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

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

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

     by (simp add: isUbI)

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

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

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

     by (simp add: reals_complete)

   then obtain "t" where

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

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

   proof

     fix n

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

       by (simp add: isLubD2)

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

       by (simp add: right_distrib real_of_nat_Suc)

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

   qed

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

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

     by (auto simp add: setle_def)

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

     by (simp add: isUbI)

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

     using t_is_Lub by (simp add: isLub_le_isUb)

   thus False using x_pos by arith

 qed

 text {*

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

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

 *}

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

 proof cases

   assume "x \<le> 0"

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

   thus ?thesis ..

 next

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

   hence x_greater_zero: "0 < x" by simp

   hence "0 < inverse x" by simp

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

     using reals_Archimedean by blast

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

     using x_greater_zero by (rule mult_strict_right_mono)

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

     using x_greater_zero by simp

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

     by (rule mult_strict_left_mono) simp

   hence "x < real (Suc n)"

     by (simp add: ring_simps)

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

 qed

 lemma reals_Archimedean3:

   assumes x_greater_zero: "0 < x"

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

 proof

   fix y

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

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

     using reals_Archimedean2 ..

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

     using x_greater_zero by (simp add: mult_strict_right_mono)

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

     by (simp add: ring_simps)

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

     using x_not_zero by (simp add: ring_simps)

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

 qed

 lemma reals_Archimedean6:

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

 apply (insert reals_Archimedean2 [of r], safe)

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

 apply (rule_tac x = x in exI)

 apply (case_tac x, simp)

 apply (rename_tac x')

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

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

 apply (erule LeastI, erule Least_le)

 done

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

   by (drule reals_Archimedean6) auto

 lemma reals_Archimedean_6b_int:

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

 apply (drule reals_Archimedean6a, auto)

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

 apply (simp add: real_of_int_real_of_nat real_of_nat_Suc)

 done

 lemma reals_Archimedean_6c_int:

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

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

 apply (rename_tac n)

 apply (drule order_le_imp_less_or_eq, auto)

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

 apply (rule_tac [2] x = "- n" in exI, auto)

 done

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

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

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

 It employs the Archimedean property of the reals. *}

 lemma Rats_dense_in_nn_real: fixes x::real

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

 proof -

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

   with reals_Archimedean obtain q::nat 

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

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

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

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

     by (simp add: pos_less_divide_eq[THEN sym])

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

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

     by (unfold p_def) (rule Least_Suc)

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

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

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

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

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

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

     by (simp only: inverse_eq_divide real_diff_def real_of_nat_Suc 

     minus_divide_left add_divide_distrib[THEN sym]) simp

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

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

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

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

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

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

 qed

 theorem Rats_dense_in_real: fixes x y :: real

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

 proof -

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

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

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

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

     by(rule Rats_dense_in_nn_real)

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

     and r3: "r < y + real n"

     by blast

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

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

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

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

   ultimately show ?thesis by fast

 qed

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

 definition

   floor :: "real => int" where

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

 definition

   ceiling :: "real => int" where

   "ceiling r = - floor (- r)"

 notation (xsymbols)

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

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

 notation (HTML output)

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

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

 lemma number_of_less_real_of_int_iff [simp]:

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

 apply auto

 apply (rule real_of_int_less_iff [THEN iffD1])

 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)

 done

 lemma number_of_less_real_of_int_iff2 [simp]:

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

 apply auto

 apply (rule real_of_int_less_iff [THEN iffD1])

 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)

 done

 lemma number_of_le_real_of_int_iff [simp]:

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

 by (simp add: linorder_not_less [symmetric])

 lemma number_of_le_real_of_int_iff2 [simp]:

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

 by (simp add: linorder_not_less [symmetric])

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

 apply (simp add: floor_def del: real_of_int_add)

 apply (rule Least_equality)

 apply simp_all

 done

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

 by auto

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

 apply (simp only: floor_def)

 apply (rule Least_equality)

 apply (drule_tac [2] real_of_int_of_nat_eq [THEN ssubst])

 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1])

 apply simp_all

 done

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

 apply (simp only: floor_def)

 apply (rule Least_equality)

 apply (drule_tac [2] real_of_int_of_nat_eq [THEN ssubst])

 apply (drule_tac [2] real_of_int_minus [THEN sym, THEN subst])

 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1])

 apply simp_all

 done

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

 apply (simp only: floor_def)

 apply (rule Least_equality)

 apply auto

 done

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

 apply (simp only: floor_def)

 apply (rule Least_equality)

 apply (drule_tac [2] real_of_int_minus [THEN sym, THEN subst])

 apply auto

 done

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

 apply (case_tac "r < 0")

 apply (blast intro: reals_Archimedean_6c_int)

 apply (simp only: linorder_not_less)

 apply (blast intro: reals_Archimedean_6b_int reals_Archimedean_6c_int)

 done

 lemma lemma_floor:

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

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

 proof -

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

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

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

   thus ?thesis by arith

 qed

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

 apply (simp add: floor_def Least_def)

 apply (insert real_lb_ub_int [of r], safe)

 apply (rule theI2)

 apply auto

 done

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

 apply (simp add: floor_def Least_def)

 apply (insert real_lb_ub_int [of x])

 apply (insert real_lb_ub_int [of y], safe)

 apply (rule theI2)

 apply (rule_tac [3] theI2)

 apply simp

 apply (erule conjI)

 apply (auto simp add: order_eq_iff int_le_real_less)

 done

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

 by (auto dest: order_le_imp_less_or_eq simp add: floor_mono)

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

 by (auto intro: lemma_floor)

 lemma real_of_int_floor_cancel [simp]:

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

 apply (simp add: floor_def Least_def)

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

 apply (rule theI2)

 apply (auto intro: lemma_floor)

 done

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

 apply (simp add: floor_def)

 apply (rule Least_equality)

 apply (auto intro: lemma_floor)

 done

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

 apply (simp add: floor_def)

 apply (rule Least_equality)

 apply (auto intro: lemma_floor)

 done

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

 apply (rule inj_int [THEN injD])

 apply (simp add: real_of_nat_Suc)

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

 done

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

 apply (drule order_le_imp_less_or_eq)

 apply (auto intro: floor_eq3)

 done

 lemma floor_number_of_eq [simp]:

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

 apply (subst real_number_of [symmetric])

 apply (rule floor_real_of_int)

 done

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

   apply (rule trans)

   prefer 2

   apply (rule floor_real_of_int)

   apply simp

 done

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

 apply (simp add: floor_def Least_def)

 apply (insert real_lb_ub_int [of r], safe)

 apply (rule theI2)

 apply (auto intro: lemma_floor)

 done

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

 apply (simp add: floor_def Least_def)

 apply (insert real_lb_ub_int [of r], safe)

 apply (rule theI2)

 apply (auto intro: lemma_floor)

 done

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

 apply (insert real_of_int_floor_ge_diff_one [of r])

 apply (auto simp del: real_of_int_floor_ge_diff_one)

 done

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

 apply (insert real_of_int_floor_gt_diff_one [of r])

 apply (auto simp del: real_of_int_floor_gt_diff_one)

 done

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

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

   apply arith

   apply (subst real_of_int_less_iff [THEN sym])

   apply simp

   apply (insert real_of_int_floor_add_one_gt [of x])

   apply arith

 done

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

   apply (rule order_trans)

   prefer 2

   apply (rule real_of_int_floor_le)

   apply (subst real_of_int_le_iff)

   apply assumption

 done

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

   apply (rule iffI)

   apply (erule real_le_floor)

   apply (erule le_floor)

 done

 lemma le_floor_eq_number_of [simp]:

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

 by (simp add: le_floor_eq)

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

 by (simp add: le_floor_eq)

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

 by (simp add: le_floor_eq)

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

   apply (subst linorder_not_le [THEN sym])+

   apply simp

   apply (rule le_floor_eq)

 done

 lemma floor_less_eq_number_of [simp]:

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

 by (simp add: floor_less_eq)

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

 by (simp add: floor_less_eq)

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

 by (simp add: floor_less_eq)

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

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

   apply auto

 done

 lemma less_floor_eq_number_of [simp]:

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

 by (simp add: less_floor_eq)

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

 by (simp add: less_floor_eq)

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

 by (simp add: less_floor_eq)

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

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

   apply auto

 done

 lemma floor_le_eq_number_of [simp]:

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

 by (simp add: floor_le_eq)

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

 by (simp add: floor_le_eq)

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

 by (simp add: floor_le_eq)

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

   apply (subst order_eq_iff)

   apply (rule conjI)

   prefer 2

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

   apply arith

   apply (subst real_of_int_less_iff [THEN sym])

   apply simp

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

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

   apply arith

   apply (rule real_of_int_floor_le)

   apply (rule real_of_int_floor_add_one_gt)

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

   apply arith

   apply (subst real_of_int_less_iff [THEN sym])

   apply simp

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

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

   apply arith

   apply (rule real_of_int_floor_add_one_gt)

   apply (rule real_of_int_floor_le)

 done

 lemma floor_add_number_of [simp]:

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

   apply (subst floor_add [THEN sym])

   apply simp

 done

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

   apply (subst floor_add [THEN sym])

   apply simp

 done

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

   apply (subst diff_minus)+

   apply (subst real_of_int_minus [THEN sym])

   apply (rule floor_add)

 done

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

     floor x - number_of n"

   apply (subst floor_subtract [THEN sym])

   apply simp

 done

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

   apply (subst floor_subtract [THEN sym])

   apply simp

 done

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

 by (simp add: ceiling_def)

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

 by (simp add: ceiling_def)

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

 by auto

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

 by (simp add: ceiling_def)

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

 by (simp add: ceiling_def)

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

 apply (simp add: ceiling_def)

 apply (subst le_minus_iff, simp)

 done

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

 by (simp add: floor_mono ceiling_def)

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

 by (simp add: floor_mono2 ceiling_def)

 lemma real_of_int_ceiling_cancel [simp]:

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

 apply (auto simp add: ceiling_def)

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

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

 done

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

 apply (simp add: ceiling_def)

 apply (rule minus_equation_iff [THEN iffD1])

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

 done

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

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

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

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

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

 by (simp add: ceiling_def)

 lemma ceiling_number_of_eq [simp]:

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

 apply (subst real_number_of [symmetric])

 apply (rule ceiling_real_of_int)

 done

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

   by (unfold ceiling_def, simp)

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

 apply (rule neg_le_iff_le [THEN iffD1])

 apply (simp add: ceiling_def diff_minus)

 done

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

 apply (insert real_of_int_ceiling_diff_one_le [of r])

 apply (simp del: real_of_int_ceiling_diff_one_le)

 done

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

   apply (unfold ceiling_def)

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

   apply simp

   apply (rule le_floor)

   apply simp

 done

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

   apply (unfold ceiling_def)

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

   apply simp

   apply (rule real_le_floor)

   apply simp

 done

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

   apply (rule iffI)

   apply (erule ceiling_le_real)

   apply (erule ceiling_le)

 done

 lemma ceiling_le_eq_number_of [simp]:

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

 by (simp add: ceiling_le_eq)

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

 by (simp add: ceiling_le_eq)

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

 by (simp add: ceiling_le_eq)

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

   apply (subst linorder_not_le [THEN sym])+

   apply simp

   apply (rule ceiling_le_eq)

 done

 lemma less_ceiling_eq_number_of [simp]:

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

 by (simp add: less_ceiling_eq)

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

 by (simp add: less_ceiling_eq)

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

 by (simp add: less_ceiling_eq)

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

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

   apply auto

 done

 lemma ceiling_less_eq_number_of [simp]:

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

 by (simp add: ceiling_less_eq)

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

 by (simp add: ceiling_less_eq)

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

 by (simp add: ceiling_less_eq)

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

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

   apply auto

 done

 lemma le_ceiling_eq_number_of [simp]:

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

 by (simp add: le_ceiling_eq)

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

 by (simp add: le_ceiling_eq)

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

 by (simp add: le_ceiling_eq)

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

   apply (unfold ceiling_def, simp)

   apply (subst real_of_int_minus [THEN sym])

   apply (subst floor_add)

   apply simp

 done

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

     ceiling x + number_of n"

   apply (subst ceiling_add [THEN sym])

   apply simp

 done

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

   apply (subst ceiling_add [THEN sym])

   apply simp

 done

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

   apply (subst diff_minus)+

   apply (subst real_of_int_minus [THEN sym])

   apply (rule ceiling_add)

 done

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

     ceiling x - number_of n"

   apply (subst ceiling_subtract [THEN sym])

   apply simp

 done

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

   apply (subst ceiling_subtract [THEN sym])

   apply simp

 done

 subsection {* Versions for the natural numbers *}

 definition

   natfloor :: "real => nat" where

   "natfloor x = nat(floor x)"

 definition

   natceiling :: "real => nat" where

   "natceiling x = nat(ceiling x)"

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

   by (unfold natfloor_def, simp)

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

   by (unfold natfloor_def, simp)

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

   by (unfold natfloor_def, simp)

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

   by (unfold natfloor_def, simp)

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

   by (unfold natfloor_def, simp)

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

   by (unfold natfloor_def, simp)

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

   apply (unfold natfloor_def)

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

   apply simp

   apply (erule floor_mono2)

 done

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

   apply (case_tac "0 <= x")

   apply (subst natfloor_def)+

   apply (subst nat_le_eq_zle)

   apply force

   apply (erule floor_mono2)

   apply (subst natfloor_neg)

   apply simp

   apply simp

 done

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

   apply (unfold natfloor_def)

   apply (subst nat_int [THEN sym])

   apply (subst nat_le_eq_zle)

   apply simp

   apply (rule le_floor)

   apply simp

 done

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

   apply (rule iffI)

   apply (rule order_trans)

   prefer 2

   apply (erule real_natfloor_le)

   apply (subst real_of_nat_le_iff)

   apply assumption

   apply (erule le_natfloor)

 done

 lemma le_natfloor_eq_number_of [simp]:

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

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

   apply (subst le_natfloor_eq, assumption)

   apply simp

 done

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

   apply (case_tac "0 <= x")

   apply (subst le_natfloor_eq, assumption, simp)

   apply (rule iffI)

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

   apply simp

   apply (rule natfloor_mono)

   apply simp

   apply simp

 done

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

   apply (unfold natfloor_def)

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

   apply (subst eq_nat_nat_iff)

   apply simp

   apply simp

   apply (rule floor_eq2)

   apply auto

 done

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

   apply (case_tac "0 <= x")

   apply (unfold natfloor_def)

   apply simp

   apply simp_all

 done

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

   apply (simp add: compare_rls)

   apply (rule real_natfloor_add_one_gt)

 done

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

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

   apply arith

   apply (rule real_natfloor_add_one_gt)

 done

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

   apply (unfold natfloor_def)

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

   apply (erule ssubst)

   apply (simp add: nat_add_distrib del: real_of_int_of_nat_eq)

   apply simp

 done

 lemma natfloor_add_number_of [simp]:

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

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

   apply (subst natfloor_add [THEN sym])

   apply simp_all

 done

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

   apply (subst natfloor_add [THEN sym])

   apply assumption

   apply simp

 done

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

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

   apply (unfold natfloor_def)

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

   apply (erule ssubst)

   apply (simp del: real_of_int_of_nat_eq)

   apply simp

 done

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

   by (unfold natceiling_def, simp)

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

   by (unfold natceiling_def, simp)

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

   by (unfold natceiling_def, simp)

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

   by (unfold natceiling_def, simp)

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

   by (unfold natceiling_def, simp)

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

   apply (unfold natceiling_def)

   apply (case_tac "x < 0")

   apply simp

   apply (subst real_nat_eq_real)

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

   apply simp

   apply (rule ceiling_mono2)

   apply simp

   apply simp

 done

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

   apply (unfold natceiling_def)

   apply simp

 done

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

   apply (case_tac "0 <= x")

   apply (subst natceiling_def)+

   apply (subst nat_le_eq_zle)

   apply (rule disjI2)

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

   apply simp

   apply (rule order_trans)

   apply simp

   apply (erule order_trans)

   apply simp

   apply (erule ceiling_mono2)

   apply (subst natceiling_neg)

   apply simp_all

 done

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

   apply (unfold natceiling_def)

   apply (case_tac "x < 0")

   apply simp

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

   apply (subst nat_le_eq_zle)

   apply simp

   apply (rule ceiling_le)

   apply simp

 done

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

   apply (rule iffI)

   apply (rule order_trans)

   apply (rule real_natceiling_ge)

   apply (subst real_of_nat_le_iff)

   apply assumption

   apply (erule natceiling_le)

 done

 lemma natceiling_le_eq_number_of [simp]:

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

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

   apply (subst natceiling_le_eq, assumption)

   apply simp

 done

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

   apply (case_tac "0 <= x")

   apply (subst natceiling_le_eq)

   apply assumption

   apply simp

   apply (subst natceiling_neg)

   apply simp

   apply simp

 done

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

   apply (unfold natceiling_def)

   apply (simplesubst nat_int [THEN sym]) back back

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

   apply (erule ssubst)

   apply (subst eq_nat_nat_iff)

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

   apply simp

   apply (rule ceiling_mono2)

   apply force

   apply force

   apply (rule ceiling_eq2)

   apply (simp, simp)

   apply (subst nat_add_distrib)

   apply auto

 done

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

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

   apply (unfold natceiling_def)

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

   apply (erule ssubst)

   apply (simp del: real_of_int_of_nat_eq)

   apply (subst nat_add_distrib)

   apply (subgoal_tac "0 = ceiling 0")

   apply (erule ssubst)

   apply (erule ceiling_mono2)

   apply simp_all

 done

 lemma natceiling_add_number_of [simp]:

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

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

   apply (subst natceiling_add [THEN sym])

   apply simp_all

 done

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

   apply (subst natceiling_add [THEN sym])

   apply assumption

   apply simp

 done

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

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

   apply (unfold natceiling_def)

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

   apply (erule ssubst)

   apply (simp del: real_of_int_of_nat_eq)

   apply simp

 done

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

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

 proof -

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

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

     by simp

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

     real((natfloor x) mod y)"

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

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

     by simp

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

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

     by (simp add: a)

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

     by simp

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

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

     by (auto simp add: ring_simps add_divide_distrib

       diff_divide_distrib prems)

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

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

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

     by (simp add: add_ac)

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

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

     apply (rule natfloor_add)

     apply (rule add_nonneg_nonneg)

     apply (rule divide_nonneg_pos)

     apply simp

     apply (simp add: prems)

     apply (rule divide_nonneg_pos)

     apply (simp add: compare_rls)

     apply (rule real_natfloor_le)

     apply (insert prems, auto)

     done

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

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

     apply (rule natfloor_eq)

     apply simp

     apply (rule add_nonneg_nonneg)

     apply (rule divide_nonneg_pos)

     apply force

     apply (force simp add: prems)

     apply (rule divide_nonneg_pos)

     apply (simp add: compare_rls)

     apply (rule real_natfloor_le)

     apply (auto simp add: prems)

     apply (insert prems, arith)

     apply (simp add: add_divide_distrib [THEN sym])

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

     apply (erule ssubst)

     apply (rule add_le_less_mono)

     apply (simp add: compare_rls)

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

       real(natfloor x mod y + 1)")

     apply (erule ssubst)

     apply (subst real_of_nat_le_iff)

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

     apply arith

     apply (rule mod_less_divisor)

     apply auto

     apply (simp add: compare_rls)

     apply (subst add_commute)

     apply (rule real_natfloor_add_one_gt)

     done

   finally show ?thesis by simp

 qed

 end