(* 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
lemma abs_diff_less_iff:
"(\<bar>x - a\<bar> < (r::'a::linordered_idom)) = (a - r < x \<and> x < a + r)"
by auto
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"}?
*}
text {* Only used in HOL/Import/HOL4Compat.thy; delete? *}
lemma posreal_complete:
fixes P :: "real set"
assumes 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 -
from upper_bound_Ex have "\<exists>z. \<forall>x\<in>P. x \<le> z"
by (auto intro: less_imp_le)
from complete_real [OF not_empty_P this] obtain S
where S1: "\<And>x. x \<in> P \<Longrightarrow> x \<le> S" and S2: "\<And>z. \<forall>x\<in>P. x \<le> z \<Longrightarrow> S \<le> z" by fast
have "\<forall>y. (\<exists>x \<in> P. y < x) = (y < S)"
proof
fix y show "(\<exists>x\<in>P. y < x) = (y < S)"
apply (cases "\<exists>x\<in>P. y < x", simp_all)
apply (clarify, drule S1, simp)
apply (simp add: not_less S2)
done
qed
thus ?thesis ..
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 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 -
from assms have "\<exists>X. X \<in> S" and "\<exists>Y. \<forall>x\<in>S. x \<le> Y"
unfolding isUb_def setle_def by simp_all
from complete_real [OF this] show ?thesis
by (simp add: isLub_def leastP_def isUb_def setle_def setge_def)
qed
subsection {* The Archimedean Property of the Reals *}
theorem reals_Archimedean:
assumes x_pos: "0 < x"
shows "\<exists>n. inverse (real (Suc n)) < x"
unfolding real_of_nat_def using x_pos
by (rule ex_inverse_of_nat_Suc_less)
lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"
unfolding real_of_nat_def by (rule ex_less_of_nat)
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)
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_real:
fixes x :: real
assumes "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 < q" by auto
def p \<equiv> "ceiling (y * real q) - 1"
def r \<equiv> "of_int p / real q"
from q have "x < y - inverse (real q)" by simp
also have "y - inverse (real q) \<le> r"
unfolding r_def p_def
by (simp add: le_divide_eq left_diff_distrib le_of_int_ceiling `0 < q`)
finally have "x < r" .
moreover have "r < y"
unfolding r_def p_def
by (simp add: divide_less_eq diff_less_eq `0 < q`
less_ceiling_iff [symmetric])
moreover from r_def have "r \<in> \<rat>" by simp
ultimately show ?thesis by fast
qed
subsection{*Floor and Ceiling Functions from the Reals to the Integers*}
(* FIXME: theorems for negative numerals *)
lemma numeral_less_real_of_int_iff [simp]:
"((numeral n) < real (m::int)) = (numeral 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 numeral_less_real_of_int_iff2 [simp]:
"(real (m::int) < (numeral n)) = (m < numeral 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 numeral_le_real_of_int_iff [simp]:
"((numeral n) \<le> real (m::int)) = (numeral n \<le> m)"
by (simp add: linorder_not_less [symmetric])
lemma numeral_le_real_of_int_iff2 [simp]:
"(real (m::int) \<le> (numeral n)) = (m \<le> numeral n)"
by (simp add: linorder_not_less [symmetric])
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 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 floor_less_eq: "(floor x < a) = (x < real a)"
unfolding real_of_int_def by (rule floor_less_iff)
lemma less_floor_eq: "(a < floor x) = (real a + 1 <= x)"
unfolding real_of_int_def by (rule less_floor_iff)
lemma floor_le_eq: "(floor x <= a) = (x < real a + 1)"
unfolding real_of_int_def by (rule floor_le_iff)
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 le_mult_floor:
assumes "0 \<le> (a :: real)" and "0 \<le> b"
shows "floor a * floor b \<le> floor (a * b)"
proof -
have "real (floor a) \<le> a"
and "real (floor b) \<le> b" by auto
hence "real (floor a * floor b) \<le> a * b"
using assms by (auto intro!: mult_mono)
also have "a * b < real (floor (a * b) + 1)" by auto
finally show ?thesis unfolding real_of_int_less_iff by simp
qed
lemma floor_divide_eq_div:
"floor (real a / real b) = a div b"
proof cases
assume "b \<noteq> 0 \<or> b dvd a"
with real_of_int_div3[of a b] show ?thesis
by (auto simp: real_of_int_div[symmetric] intro!: floor_eq2 real_of_int_div4 neq_le_trans)
(metis add_left_cancel zero_neq_one real_of_int_div_aux real_of_int_inject
real_of_int_zero_cancel right_inverse_eq div_self mod_div_trivial)
qed (auto simp: real_of_int_div)
lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"
unfolding real_of_nat_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 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 less_ceiling_eq: "(a < ceiling x) = (real a < x)"
unfolding real_of_int_def by (rule less_ceiling_iff)
lemma ceiling_less_eq: "(ceiling x < a) = (x <= real a - 1)"
unfolding real_of_int_def by (rule ceiling_less_iff)
lemma le_ceiling_eq: "(a <= ceiling x) = (real a - 1 < x)"
unfolding real_of_int_def by (rule le_ceiling_iff)
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)
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_numeral_eq [simp]: "natfloor (numeral n) = numeral 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"
unfolding natfloor_def by simp
lemma natfloor_mono: "x <= y ==> natfloor x <= natfloor y"
unfolding natfloor_def by (intro nat_mono floor_mono)
lemma le_natfloor: "real x <= a ==> x <= natfloor a"
apply (unfold natfloor_def)
apply (subst nat_int [THEN sym])
apply (rule nat_mono)
apply (rule le_floor)
apply simp
done
lemma natfloor_less_iff: "0 \<le> x \<Longrightarrow> natfloor x < n \<longleftrightarrow> x < real n"
unfolding natfloor_def real_of_nat_def
by (simp add: nat_less_iff floor_less_iff)
lemma less_natfloor:
assumes "0 \<le> x" and "x < real (n :: nat)"
shows "natfloor x < n"
using assms by (simp add: natfloor_less_iff)
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_numeral [simp]:
"~ neg((numeral n)::int) ==> 0 <= x ==>
(numeral n <= natfloor x) = (numeral 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"
unfolding natfloor_def by (simp add: floor_eq2 [where n="int n"])
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"
unfolding natfloor_def
unfolding real_of_int_of_nat_eq [symmetric] floor_add
by (simp add: nat_add_distrib)
lemma natfloor_add_numeral [simp]:
"~neg ((numeral n)::int) ==> 0 <= x ==>
natfloor (x + numeral n) = natfloor x + numeral n"
by (simp add: natfloor_add [symmetric])
lemma natfloor_add_one: "0 <= x ==> natfloor(x + 1) = natfloor x + 1"
by (simp add: natfloor_add [symmetric] del: One_nat_def)
lemma natfloor_subtract [simp]:
"natfloor(x - real a) = natfloor x - a"
unfolding natfloor_def real_of_int_of_nat_eq [symmetric] floor_subtract
by simp
lemma natfloor_div_nat:
assumes "1 <= x" and "y > 0"
shows "natfloor (x / real y) = natfloor x div y"
proof (rule natfloor_eq)
have "(natfloor x) div y * y \<le> natfloor x"
by (rule add_leD1 [where k="natfloor x mod y"], simp)
thus "real (natfloor x div y) \<le> x / real y"
using assms by (simp add: le_divide_eq le_natfloor_eq)
have "natfloor x < (natfloor x) div y * y + y"
apply (subst mod_div_equality [symmetric])
apply (rule add_strict_left_mono)
apply (rule mod_less_divisor)
apply fact
done
thus "x / real y < real (natfloor x div y) + 1"
using assms
by (simp add: divide_less_eq natfloor_less_iff left_distrib)
qed
lemma le_mult_natfloor:
shows "natfloor a * natfloor b \<le> natfloor (a * b)"
by (cases "0 <= a & 0 <= b")
(auto simp add: le_natfloor_eq mult_nonneg_nonneg mult_mono' real_natfloor_le natfloor_neg)
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_numeral_eq [simp]: "natceiling (numeral n) = numeral 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)"
unfolding natceiling_def by (cases "x < 0", simp_all)
lemma natceiling_neg: "x <= 0 ==> natceiling x = 0"
unfolding natceiling_def by simp
lemma natceiling_mono: "x <= y ==> natceiling x <= natceiling y"
unfolding natceiling_def by (intro nat_mono ceiling_mono)
lemma natceiling_le: "x <= real a ==> natceiling x <= a"
unfolding natceiling_def real_of_nat_def
by (simp add: nat_le_iff ceiling_le_iff)
lemma natceiling_le_eq: "(natceiling x <= a) = (x <= real a)"
unfolding natceiling_def real_of_nat_def
by (simp add: nat_le_iff ceiling_le_iff)
lemma natceiling_le_eq_numeral [simp]:
"~ neg((numeral n)::int) ==>
(natceiling x <= numeral n) = (x <= numeral n)"
by (simp add: natceiling_le_eq)
lemma natceiling_le_eq_one: "(natceiling x <= 1) = (x <= 1)"
unfolding natceiling_def
by (simp add: nat_le_iff ceiling_le_iff)
lemma natceiling_eq: "real n < x ==> x <= real n + 1 ==> natceiling x = n + 1"
unfolding natceiling_def
by (simp add: ceiling_eq2 [where n="int n"])
lemma natceiling_add [simp]: "0 <= x ==>
natceiling (x + real a) = natceiling x + a"
unfolding natceiling_def
unfolding real_of_int_of_nat_eq [symmetric] ceiling_add
by (simp add: nat_add_distrib)
lemma natceiling_add_numeral [simp]:
"~ neg ((numeral n)::int) ==> 0 <= x ==>
natceiling (x + numeral n) = natceiling x + numeral n"
by (simp add: natceiling_add [symmetric])
lemma natceiling_add_one: "0 <= x ==> natceiling(x + 1) = natceiling x + 1"
by (simp add: natceiling_add [symmetric] del: One_nat_def)
lemma natceiling_subtract [simp]: "natceiling(x - real a) = natceiling x - a"
unfolding natceiling_def real_of_int_of_nat_eq [symmetric] ceiling_subtract
by simp
subsection {* Exponentiation with floor *}
lemma floor_power:
assumes "x = real (floor x)"
shows "floor (x ^ n) = floor x ^ n"
proof -
have *: "x ^ n = real (floor x ^ n)"
using assms by (induct n arbitrary: x) simp_all
show ?thesis unfolding real_of_int_inject[symmetric]
unfolding * floor_real_of_int ..
qed
lemma natfloor_power:
assumes "x = real (natfloor x)"
shows "natfloor (x ^ n) = natfloor x ^ n"
proof -
from assms have "0 \<le> floor x" by auto
note assms[unfolded natfloor_def real_nat_eq_real[OF `0 \<le> floor x`]]
from floor_power[OF this]
show ?thesis unfolding natfloor_def nat_power_eq[OF `0 \<le> floor x`, symmetric]
by simp
qed
end