Theory RComplete

(*  Title       : HOL/Real/RComplete.thy
    ID          : $Id: RComplete.thy,v 1.30 2007/10/23 21:27:24 nipkow Exp $
    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 "∀x ∈ P. x≤ y"}?
*}

lemma posreal_complete:
  assumes positive_P: "∀x ∈ P. (0::real) < x"
    and not_empty_P: "∃x. x ∈ P"
    and upper_bound_Ex: "∃y. ∀x ∈ P. x<y"
  shows "∃S. ∀y. (∃x ∈ P. y < x) = (y < S)"
proof (rule exI, rule allI)
  fix y
  let ?pP = "{w. real_of_preal w ∈ P}"

  show "(∃x∈P. y < x) = (y < real_of_preal (psup ?pP))"
  proof (cases "0 < y")
    assume neg_y: "¬ 0 < y"
    show ?thesis
    proof
      assume "∃x∈P. y < x"
      have "∀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 ∈ P" using not_empty_P ..
      hence "0 < x" using positive_P by simp
      hence "y < x" using neg_y by simp
      thus "∃x ∈ 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 ∈ 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 ∈ ?pP" using `a ∈ P` by auto
    hence pP_not_empty: "?pP ≠ {}" by auto

    obtain sup where sup: "∀x ∈ P. x < sup"
      using upper_bound_Ex ..
    from this and `a ∈ 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 "∀X ∈ ?pP. X ≤ possup"
      using sup by (auto simp add: real_of_preal_lessI)
    with pP_not_empty have psup: "!!Z. (∃X ∈ ?pP. Z < X) = (Z < psup ?pP)"
      by (rule preal_complete)

    show ?thesis
    proof
      assume "∃x ∈ P. y < x"
      then obtain x where x_in_P: "x ∈ 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: "∃X ∈ ?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 ∈ ?pP" using x_in_P and x_is_px by simp
      qed

      have "(∃X ∈ ?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 ∈ 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 ∈ ?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 ∈ P" using x_is_X and X_in_pP by simp

      ultimately show "∃ x ∈ 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: "∀x ∈ S. 0 < x"
    and not_empty_S: "∃x. x ∈ S"
    and upper_bound_Ex: "∃u. isUb (UNIV::real set) S u"
  shows "∃t. isLub (UNIV::real set) S t"
proof
  let ?pS = "{w. real_of_preal w ∈ S}"

  obtain u where "isUb UNIV S u" using upper_bound_Ex ..
  hence sup: "∀x ∈ S. x ≤ u" by (simp add: isUb_def setle_def)

  obtain x where x_in_S: "x ∈ S" using not_empty_S ..
  hence x_gt_zero: "0 < x" using positive_S by simp
  have  "x ≤ 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: "∀pa ∈ ?pS. pa ≤ pu"
  proof
    fix pa
    assume "pa ∈ ?pS"
    then obtain a where "a ∈ S" and "a = real_of_preal pa"
      by simp
    moreover hence "a ≤ u" using sup by simp
    ultimately show "pa ≤ pu"
      using sup and u_is_pu by (simp add: real_of_preal_le_iff)
  qed

  have "∀y ∈ S. y ≤ real_of_preal (psup ?pS)"
  proof
    fix y
    assume y_in_S: "y ∈ 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 ∈ ?pS" using y_in_S by simp
    with pS_less_pu have "py ≤ psup ?pS"
      by (rule preal_psup_le)
    thus "y ≤ 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: "∀y∈S. y ≤ x"
    have "real_of_preal (psup ?pS) ≤ x"
    proof -
      obtain "s" where s_in_S: "s ∈ S" using not_empty_S ..
      hence s_pos: "0 < s" using positive_S by simp

      hence "∃ 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 ∈ {w. real_of_preal w ∈ S}" using s_in_S by simp

      from x_ub_S have "s ≤ x" using s_in_S ..
      hence "0 < x" using s_pos by simp
      hence "∃ 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 "∀pe ∈ ?pS. pe ≤ px"
      proof
        fix pe
        assume "pe ∈ ?pS"
        hence "real_of_preal pe ∈ S" by simp
        hence "real_of_preal pe ≤ x" using x_ub_S by simp
        thus "pe ≤ px" using x_is_px by (simp add: real_of_preal_le_iff)
      qed

      moreover have "?pS ≠ {}" using ps_in_pS by auto
      ultimately have "(psup ?pS) ≤ px" by (simp add: psup_le_ub)
      thus "real_of_preal (psup ?pS) ≤ 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: "∃X. X ∈ S"
    and exists_Ub: "∃Y. isUb (UNIV::real set) S Y"
  shows "∃t. isLub (UNIV :: real set) S t"
proof -
  obtain X where X_in_S: "X ∈ S" using notempty_S ..
  obtain Y where Y_isUb: "isUb (UNIV::real set) S Y"
    using exists_Ub ..
  let ?SHIFT = "{z. ∃x ∈S. z = x + (-X) + 1} ∩ {x. 0 < x}"

  {
    fix x
    assume "isUb (UNIV::real set) S x"
    hence S_le_x: "∀ y ∈ S. y <= x"
      by (simp add: isUb_def setle_def)
    {
      fix s
      assume "s ∈ {z. ∃x∈S. z = x + - X + 1}"
      hence "∃ x ∈ S. s = x + -X + 1" ..
      then obtain x1 where "x1 ∈ S" and "s = x1 + (-X) + 1" ..
      moreover hence "x1 ≤ x" using S_le_x by simp
      ultimately have "s ≤ 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 "∃Z. isUb UNIV ?SHIFT Z" ..
  moreover have "∀y ∈ ?SHIFT. 0 < y" by auto
  moreover have shifted_not_empty: "∃u. u ∈ ?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 ≤ (x + (-X) + 1)"
          using t_is_Lub by (simp add: isLub_le_isUb)
        hence "t + X + -1 ≤ 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 ∈ S"
          have "y ≤ t + X + -1"
          proof -
            obtain "u" where u_in_shift: "u ∈ ?SHIFT" using shifted_not_empty ..
            hence "∃ x ∈ S. u = x + - X + 1" by simp
            then obtain "x" where x_and_u: "u = x + - X + 1" ..
            have u_le_t: "u ≤ t" using u_in_shift and t_is_Lub by (simp add: isLubD2)

            show ?thesis
            proof cases
              assume "y ≤ x"
              moreover have "x = u + X + - 1" using x_and_u by arith
              moreover have "u + X + - 1  ≤ t + X + -1" using u_le_t by arith
              ultimately show "y  ≤ t + X + -1" by arith
            next
              assume "~(y ≤ x)"
              hence x_less_y: "x < y" by arith

              have "x + (-X) + 1 ∈ ?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 ∈ ?SHIFT" using y_in_S by simp
              hence "y + (-X) + 1 ≤ 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 "∃n. inverse (real (Suc n)) < x"
proof (rule ccontr)
  assume contr: "¬ ?thesis"
  have "∀n. x * real (Suc n) <= 1"
  proof
    fix n
    from contr have "x ≤ inverse (real (Suc n))"
      by (simp add: linorder_not_less)
    hence "x ≤ (1 / (real (Suc n)))"
      by (simp add: inverse_eq_divide)
    moreover have "0 ≤ real (Suc n)"
      by (rule real_of_nat_ge_zero)
    ultimately have "x * real (Suc n) ≤ (1 / real (Suc n)) * real (Suc n)"
      by (rule mult_right_mono)
    thus "x * real (Suc n) ≤ 1" by simp
  qed
  hence "{z. ∃n. z = x * (real (Suc n))} *<= 1"
    by (simp add: setle_def, safe, rule spec)
  hence "isUb (UNIV::real set) {z. ∃n. z = x * (real (Suc n))} 1"
    by (simp add: isUbI)
  hence "∃Y. isUb (UNIV::real set) {z. ∃n. z = x* (real (Suc n))} Y" ..
  moreover have "∃X. X ∈ {z. ∃n. z = x* (real (Suc n))}" by auto
  ultimately have "∃t. isLub UNIV {z. ∃n. z = x * real (Suc n)} t"
    by (simp add: reals_complete)
  then obtain "t" where
    t_is_Lub: "isLub UNIV {z. ∃n. z = x * real (Suc n)} t" ..

  have "∀n::nat. x * real n ≤ t + - x"
  proof
    fix n
    from t_is_Lub have "x * real (Suc n) ≤ t"
      by (simp add: isLubD2)
    hence  "x * (real n) + x ≤ t"
      by (simp add: right_distrib real_of_nat_Suc)
    thus  "x * (real n) ≤ t + - x" by arith
  qed

  hence "∀m. x * real (Suc m) ≤ t + - x" by simp
  hence "{z. ∃n. z = x * (real (Suc n))}  *<= (t + - x)"
    by (auto simp add: setle_def)
  hence "isUb (UNIV::real set) {z. ∃n. z = x * (real (Suc n))} (t + (-x))"
    by (simp add: isUbI)
  hence "t ≤ 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: "∃n. (x::real) < real (n::nat)"
proof cases
  assume "x ≤ 0"
  hence "x < real (1::nat)" by simp
  thus ?thesis ..
next
  assume "¬ x ≤ 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 "∃(n::nat). x < real n" ..
qed

lemma reals_Archimedean3:
  assumes x_greater_zero: "0 < x"
  shows "∀(y::real). ∃(n::nat). y < real n * x"
proof
  fix y
  have x_not_zero: "x ≠ 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 "∃(n::nat). y < real n * x" ..
qed

lemma reals_Archimedean6:
     "0 ≤ r ==> ∃(n::nat). real (n - 1) ≤ r & r < real (n)"
apply (insert reals_Archimedean2 [of r], safe)
apply (subgoal_tac "∃x::nat. r < real x ∧ (∀y. r < real y --> x ≤ 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 ≤ r ==> ∃n. real (n) ≤ r & r < real (Suc n)"
  by (drule reals_Archimedean6) auto

lemma reals_Archimedean_6b_int:
     "0 ≤ r ==> ∃n::int. real n ≤ 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 ==> ∃n::int. real n ≤ 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{*Floor and Ceiling Functions from the Reals to the Integers*}

definition
  floor :: "real => int" where
  "floor r = (LEAST n::int. r < real (n+1))"

definition
  ceiling :: "real => int" where
  "ceiling r = - floor (- r)"

notation (xsymbols)
  floor  ("⌊_⌋") and
  ceiling  ("⌈_⌉")

notation (HTML output)
  floor  ("⌊_⌋") and
  ceiling  ("⌈_⌉")


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) ≤ real (m::int)) = (number_of n ≤ m)"
by (simp add: linorder_not_less [symmetric])

lemma number_of_le_real_of_int_iff2 [simp]:
     "(real (m::int) ≤ (number_of n)) = (m ≤ 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: " ∃n::int. real n ≤ 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 ≤ r" and a2: "r < real n + 1"
  shows "m ≤ (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) ≤ 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 ≤ 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 ≤ y ==> floor x ≤ 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 ≤ x"
by (auto intro: lemma_floor)

lemma real_of_int_floor_cancel [simp]:
    "(real (floor x) = x) = (∃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 ≤ 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 ≤ 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 ≤ 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 ≤ 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 ≤ real (ceiling r)"
apply (simp add: ceiling_def)
apply (subst le_minus_iff, simp)
done

lemma ceiling_mono: "x < y ==> ceiling x ≤ ceiling y"
by (simp add: floor_mono ceiling_def)

lemma ceiling_mono2: "x ≤ y ==> ceiling x ≤ ceiling y"
by (simp add: floor_mono2 ceiling_def)

lemma real_of_int_ceiling_cancel [simp]:
     "(real (ceiling x) = x) = (∃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 ≤ 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 ≤ 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 ≤ 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) ≤ 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