src/HOL/Real/RComplete.thy
 author nipkow Tue, 02 Sep 2008 21:31:28 +0200 changeset 28091 50f2d6ba024c parent 27435 b3f8e9bdf9a7 child 28562 4e74209f113e permissions -rw-r--r--
Streamlined parts of Complex/ex/DenumRat and AFP/Integration/Rats and distributed them over Real/ (to do with bijections and density). Library/NatPair became Nat_Int_Bij and made that part of Main.

(*  Title       : HOL/Real/RComplete.thy
ID          : $Id$
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"

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"
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"
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"

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
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
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"
hence "y < x" using py_less_X and y_is_py

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"

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"
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"
{
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)"
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))"
hence "x \<le> (1 / (real (Suc n)))"
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"
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"
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"
hence  "x * (real n) + x \<le> t"
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)"
hence "isUb (UNIV::real set) {z. \<exists>n. z = x * (real (Suc n))} (t + (-x))"
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)"
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"
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)
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")
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
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 func 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)"

lemma number_of_le_real_of_int_iff2 [simp]:
"(real (m::int) \<le> (number_of n)) = (m \<le> number_of n)"

lemma floor_zero [simp]: "floor 0 = 0"
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 (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 (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 (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 (rule Least_equality)
apply (auto intro: lemma_floor)
done

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

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

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

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)"

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

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

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)"

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

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

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)"

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

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

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 (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_le)
done

"floor (x + number_of n) = floor x + number_of n"
apply simp
done

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

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

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

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

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

lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)"
apply (subst le_minus_iff, simp)
done

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

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

lemma real_of_int_ceiling_cancel [simp]:
"(real (ceiling x) = x) = (\<exists>n::int. x = real n)"
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 (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"

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])
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)"

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

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

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)"

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

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

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)"

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

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

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)"

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

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

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 simp
done

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

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

"~neg ((number_of n)::int) ==> 0 <= x ==>
natfloor (x + number_of n) = natfloor x + number_of n"
apply simp_all
done

lemma natfloor_add_one: "0 <= x ==> natfloor(x + 1) = natfloor x + 1"
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 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 (subgoal_tac "0 = ceiling 0")
apply (erule ssubst)
apply (erule ceiling_mono2)
apply simp_all
done

"~ neg ((number_of n)::int) ==> 0 <= x ==>
natceiling (x + number_of n) = natceiling x + number_of n"
apply simp_all
done

lemma natceiling_add_one: "0 <= x ==> natceiling(x + 1) = natceiling x + 1"
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))"
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"
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))"
also have "... = natfloor(real((natfloor x) mod y) /
real y + (x - real(natfloor x)) / real y) + (natfloor x) div y"
apply (rule divide_nonneg_pos)
apply simp
apply (rule divide_nonneg_pos)
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 divide_nonneg_pos)
apply force
apply (rule divide_nonneg_pos)
apply (rule real_natfloor_le)
apply (insert prems, arith)
apply (subgoal_tac "real y = real y - 1 + 1")
apply (erule ssubst)
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