(*  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: algebra_simps)

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

qed

instance real :: archimedean_field

proof

  fix r :: real

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

    using reals_Archimedean2 ..

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

    unfolding real_eq_of_nat by simp

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

qed

lemma reals_Archimedean3:

  assumes x_greater_zero: "0 < x"

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

  unfolding real_of_nat_def using `0 < x`

  by (auto intro: ex_less_of_nat_mult)

lemma reals_Archimedean6:

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

unfolding real_of_nat_def

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

apply (insert floor_correct [of r])

apply (simp add: nat_add_distrib of_nat_nat)

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)"

  unfolding real_of_int_def by (rule floor_exists)

lemma reals_Archimedean_6c_int:

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

  unfolding real_of_int_def by (rule floor_exists)

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*}

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_real_of_nat_zero: "floor (real (0::nat)) = 0"

by auto (* delete? *)

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

unfolding real_of_nat_def by simp

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

unfolding real_of_nat_def by (simp add: floor_minus)

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

unfolding real_of_int_def by simp

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

unfolding real_of_int_def by (simp add: floor_minus)

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

unfolding real_of_int_def by (rule floor_exists)

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"

unfolding real_of_int_def by (rule of_int_floor_le)

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)"

  using floor_real_of_int by metis

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

  unfolding real_of_int_def using floor_unique [of n x] by simp

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

  unfolding real_of_int_def by (rule floor_unique)

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:

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

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

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

  unfolding real_of_int_def using floor_correct [of r] by simp

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

  unfolding real_of_int_def using floor_correct [of r] by simp

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

  unfolding real_of_int_def using floor_correct [of r] by simp

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

  unfolding real_of_int_def using floor_correct [of r] by simp

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

  unfolding real_of_int_def by (simp add: le_floor_iff)

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

  unfolding real_of_int_def by (simp add: le_floor_iff)

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

  unfolding real_of_int_def by (rule le_floor_iff)

lemma le_floor_eq_number_of:

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

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

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

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

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

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

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

  unfolding real_of_int_def by (rule floor_less_iff)

lemma floor_less_eq_number_of:

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

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

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

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

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

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

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

  unfolding real_of_int_def by (rule less_floor_iff)

lemma less_floor_eq_number_of:

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

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

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

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

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

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

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

  unfolding real_of_int_def by (rule floor_le_iff)

lemma floor_le_eq_number_of:

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

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

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

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

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

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

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

  unfolding real_of_int_def by (rule floor_add_of_int)

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

  unfolding real_of_int_def by (rule floor_diff_of_int)

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

    floor x - number_of n"

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

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

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

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

  unfolding real_of_nat_def by simp

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

by auto (* delete? *)

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

  unfolding real_of_int_def by simp

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

  unfolding real_of_int_def by simp

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

  unfolding real_of_int_def by (rule le_of_int_ceiling)

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

  unfolding real_of_int_def by simp

lemma real_of_int_ceiling_cancel [simp]:

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

  using ceiling_real_of_int by metis

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

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

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

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

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

  unfolding real_of_int_def using ceiling_unique [of n x] by simp

lemma ceiling_number_of_eq:

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

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

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

  unfolding real_of_int_def using ceiling_correct [of r] by simp

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

  unfolding real_of_int_def using ceiling_correct [of r] by simp

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

  unfolding real_of_int_def by (simp add: ceiling_le_iff)

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

  unfolding real_of_int_def by (simp add: ceiling_le_iff)

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

  unfolding real_of_int_def by (rule ceiling_le_iff)

lemma ceiling_le_eq_number_of:

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

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

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

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

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

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

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

  unfolding real_of_int_def by (rule less_ceiling_iff)

lemma less_ceiling_eq_number_of:

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

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

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

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

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

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

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

  unfolding real_of_int_def by (rule ceiling_less_iff)

lemma ceiling_less_eq_number_of:

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

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

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

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

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

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

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

  unfolding real_of_int_def by (rule le_ceiling_iff)

lemma le_ceiling_eq_number_of:

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

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

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

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

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

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

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

  unfolding real_of_int_def by (rule ceiling_add_of_int)

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

  unfolding real_of_int_def by (rule ceiling_diff_of_int)

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

    ceiling x - number_of n"

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

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

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

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_mono)

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_mono)

  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)"

using real_natfloor_add_one_gt by (simp add: algebra_simps)

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_mono)

  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_mono)

  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_mono)

  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"