(* Title: HOL/Real.thy
Author: Jacques D. Fleuriot, University of Edinburgh, 1998
Author: Larry Paulson, University of Cambridge
Author: Jeremy Avigad, Carnegie Mellon University
Author: Florian Zuleger, Johannes Hoelzl, and Simon Funke, TU Muenchen
Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
Construction of Cauchy Reals by Brian Huffman, 2010
*)
section ‹Development of the Reals
using Cauchy Sequences
›
theory Real
imports Rat
begin
text ‹
This
theory contains a formalization of the real numbers as equivalence
classes of Cauchy sequences of rationals. See the AFP entry
@{
text Dedekind_Real}
for an alternative construction
using
Dedekind cuts.
›
subsection ‹Preliminary
lemmas›
text‹Useful
in convergence arguments
›
lemma inverse_of_nat_le:
fixes n::nat
shows "\n \ m; n\0\ \ 1 / of_nat m \ (1::'a::linordered_field) / of_nat n"
by (simp add: frac_le)
lemma add_diff_add:
"(a + c) - (b + d) = (a - b) + (c - d)"
for a b c d ::
"'a::ab_group_add"
by simp
lemma minus_diff_minus:
"- a - - b = - (a - b)"
for a b ::
"'a::ab_group_add"
by simp
lemma mult_diff_mult:
"(x * y - a * b) = x * (y - b) + (x - a) * b"
for x y a b ::
"'a::ring"
by (simp add: algebra_simps)
lemma inverse_diff_inverse:
fixes a b ::
"'a::division_ring"
assumes "a \ 0" and "b \ 0"
shows "inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
using assms
by (simp add: algebra_simps)
lemma obtain_pos_sum:
fixes r :: rat
assumes r:
"0 < r"
obtains s t
where "0 < s" and "0 < t" and "r = s + t"
proof
from r
show "0 < r/2" by simp
from r
show "0 < r/2" by simp
show "r = r/2 + r/2" by simp
qed
subsection ‹Sequences that converge
to zero
›
definition vanishes ::
"(nat \ rat) \ bool"
where "vanishes X \ (\r>0. \k. \n\k. \X n\ < r)"
lemma vanishesI:
"(\r. 0 < r \ \k. \n\k. \X n\ < r) \ vanishes X"
unfolding vanishes_def
by simp
lemma vanishesD:
"vanishes X \ 0 < r \ \k. \n\k. \X n\ < r"
unfolding vanishes_def
by simp
lemma vanishes_const [simp]:
"vanishes (\n. c) \ c = 0"
proof (cases
"c = 0")
case True
then show ?thesis
by (simp add: vanishesI)
next
case False
then show ?thesis
unfolding vanishes_def
using zero_less_abs_iff
by blast
qed
lemma vanishes_minus:
"vanishes X \ vanishes (\n. - X n)"
unfolding vanishes_def
by simp
lemma vanishes_add:
assumes X:
"vanishes X"
and Y:
"vanishes Y"
shows "vanishes (\n. X n + Y n)"
proof (rule vanishesI)
fix r :: rat
assume "0 < r"
then obtain s t
where s:
"0 < s" and t:
"0 < t" and r:
"r = s + t"
by (rule obtain_pos_sum)
obtain i
where i:
"\n\i. \X n\ < s"
using vanishesD [OF X s] ..
obtain j
where j:
"\n\j. \Y n\ < t"
using vanishesD [OF Y t] ..
have "\n\max i j. \X n + Y n\ < r"
proof clarsimp
fix n
assume n:
"i \ n" "j \ n"
have "\X n + Y n\ \ \X n\ + \Y n\"
by (rule abs_triangle_ineq)
also have "\ < s + t"
by (simp add: add_strict_mono i j n)
finally show "\X n + Y n\ < r"
by (simp only: r)
qed
then show "\k. \n\k. \X n + Y n\ < r" ..
qed
lemma vanishes_diff:
assumes "vanishes X" "vanishes Y"
shows "vanishes (\n. X n - Y n)"
unfolding diff_conv_add_uminus
by (intro vanishes_add vanishes_minus assms)
lemma vanishes_mult_bounded:
assumes X:
"\a>0. \n. \X n\ < a"
assumes Y:
"vanishes (\n. Y n)"
shows "vanishes (\n. X n * Y n)"
proof (rule vanishesI)
fix r :: rat
assume r:
"0 < r"
obtain a
where a:
"0 < a" "\n. \X n\ < a"
using X
by blast
obtain b
where b:
"0 < b" "r = a * b"
proof
show "0 < r / a" using r a
by simp
show "r = a * (r / a)" using a
by simp
qed
obtain k
where k:
"\n\k. \Y n\ < b"
using vanishesD [OF Y b(1)] ..
have "\n\k. \X n * Y n\ < r"
by (simp add: b(2) abs_mult mult_strict_mono
' a k)
then show "\k. \n\k. \X n * Y n\ < r" ..
qed
subsection ‹Cauchy sequences
›
definition cauchy ::
"(nat \ rat) \ bool"
where "cauchy X \ (\r>0. \k. \m\k. \n\k. \X m - X n\ < r)"
lemma cauchyI:
"(\r. 0 < r \ \k. \m\k. \n\k. \X m - X n\ < r) \ cauchy X"
unfolding cauchy_def
by simp
lemma cauchyD:
"cauchy X \ 0 < r \ \k. \m\k. \n\k. \X m - X n\ < r"
unfolding cauchy_def
by simp
lemma cauchy_const [simp]:
"cauchy (\n. x)"
unfolding cauchy_def
by simp
lemma cauchy_add [simp]:
assumes X:
"cauchy X" and Y:
"cauchy Y"
shows "cauchy (\n. X n + Y n)"
proof (rule cauchyI)
fix r :: rat
assume "0 < r"
then obtain s t
where s:
"0 < s" and t:
"0 < t" and r:
"r = s + t"
by (rule obtain_pos_sum)
obtain i
where i:
"\m\i. \n\i. \X m - X n\ < s"
using cauchyD [OF X s] ..
obtain j
where j:
"\m\j. \n\j. \Y m - Y n\ < t"
using cauchyD [OF Y t] ..
have "\m\max i j. \n\max i j. \(X m + Y m) - (X n + Y n)\ < r"
proof clarsimp
fix m n
assume *:
"i \ m" "j \ m" "i \ n" "j \ n"
have "\(X m + Y m) - (X n + Y n)\ \ \X m - X n\ + \Y m - Y n\"
unfolding add_diff_add
by (rule abs_triangle_ineq)
also have "\ < s + t"
by (rule add_strict_mono) (simp_all add: i j *)
finally show "\(X m + Y m) - (X n + Y n)\ < r" by (simp only: r)
qed
then show "\k. \m\k. \n\k. \(X m + Y m) - (X n + Y n)\ < r" ..
qed
lemma cauchy_minus [simp]:
assumes X:
"cauchy X"
shows "cauchy (\n. - X n)"
using assms
unfolding cauchy_def
unfolding minus_diff_minus abs_minus_cancel .
lemma cauchy_diff [simp]:
assumes "cauchy X" "cauchy Y"
shows "cauchy (\n. X n - Y n)"
using assms
unfolding diff_conv_add_uminus
by (simp del: add_uminus_conv_diff)
lemma cauchy_imp_bounded:
assumes "cauchy X"
shows "\b>0. \n. \X n\ < b"
proof -
obtain k
where k:
"\m\k. \n\k. \X m - X n\ < 1"
using cauchyD [OF assms zero_less_one] ..
show "\b>0. \n. \X n\ < b"
proof (intro exI conjI allI)
have "0 \ \X 0\" by simp
also have "\X 0\ \ Max (abs ` X ` {..k})" by simp
finally have "0 \ Max (abs ` X ` {..k})" .
then show "0 < Max (abs ` X ` {..k}) + 1" by simp
next
fix n :: nat
show "\X n\ < Max (abs ` X ` {..k}) + 1"
proof (rule linorder_le_cases)
assume "n \ k"
then have "\X n\ \ Max (abs ` X ` {..k})" by simp
then show "\X n\ < Max (abs ` X ` {..k}) + 1" by simp
next
assume "k \ n"
have "\X n\ = \X k + (X n - X k)\" by simp
also have "\X k + (X n - X k)\ \ \X k\ + \X n - X k\"
by (rule abs_triangle_ineq)
also have "\ < Max (abs ` X ` {..k}) + 1"
by (rule add_le_less_mono) (simp_all add: k
‹k
≤ n
›)
finally show "\X n\ < Max (abs ` X ` {..k}) + 1" .
qed
qed
qed
lemma cauchy_mult [simp]:
assumes X:
"cauchy X" and Y:
"cauchy Y"
shows "cauchy (\n. X n * Y n)"
proof (rule cauchyI)
fix r :: rat
assume "0 < r"
then obtain u v
where u:
"0 < u" and v:
"0 < v" and "r = u + v"
by (rule obtain_pos_sum)
obtain a
where a:
"0 < a" "\n. \X n\ < a"
using cauchy_imp_bounded [OF X]
by blast
obtain b
where b:
"0 < b" "\n. \Y n\ < b"
using cauchy_imp_bounded [OF Y]
by blast
obtain s t
where s:
"0 < s" and t:
"0 < t" and r:
"r = a * t + s * b"
proof
show "0 < v/b" using v b(1)
by simp
show "0 < u/a" using u a(1)
by simp
show "r = a * (u/a) + (v/b) * b"
using a(1) b(1)
‹r = u + v
› by simp
qed
obtain i
where i:
"\m\i. \n\i. \X m - X n\ < s"
using cauchyD [OF X s] ..
obtain j
where j:
"\m\j. \n\j. \Y m - Y n\ < t"
using cauchyD [OF Y t] ..
have "\m\max i j. \n\max i j. \X m * Y m - X n * Y n\ < r"
proof clarsimp
fix m n
assume *:
"i \ m" "j \ m" "i \ n" "j \ n"
have "\X m * Y m - X n * Y n\ = \X m * (Y m - Y n) + (X m - X n) * Y n\"
unfolding mult_diff_mult ..
also have "\ \ \X m * (Y m - Y n)\ + \(X m - X n) * Y n\"
by (rule abs_triangle_ineq)
also have "\ = \X m\ * \Y m - Y n\ + \X m - X n\ * \Y n\"
unfolding abs_mult ..
also have "\ < a * t + s * b"
by (simp_all add: add_strict_mono mult_strict_mono
' a b i j *)
finally show "\X m * Y m - X n * Y n\ < r"
by (simp only: r)
qed
then show "\k. \m\k. \n\k. \X m * Y m - X n * Y n\ < r" ..
qed
lemma cauchy_not_vanishes_cases:
assumes X:
"cauchy X"
assumes nz:
"\ vanishes X"
shows "\b>0. \k. (\n\k. b < - X n) \ (\n\k. b < X n)"
proof -
obtain r
where "0 < r" and r:
"\k. \n\k. r \ \X n\"
using nz
unfolding vanishes_def
by (auto simp add: not_less)
obtain s t
where s:
"0 < s" and t:
"0 < t" and "r = s + t"
using ‹0 < r
› by (rule obtain_pos_sum)
obtain i
where i:
"\m\i. \n\i. \X m - X n\ < s"
using cauchyD [OF X s] ..
obtain k
where "i \ k" and "r \ \X k\"
using r
by blast
have k:
"\n\k. \X n - X k\ < s"
using i
‹i
≤ k
› by auto
have "X k \ - r \ r \ X k"
using ‹r
≤ ∣X k
∣› by auto
then have "(\n\k. t < - X n) \ (\n\k. t < X n)"
unfolding ‹r = s + t
› using k
by auto
then have "\k. (\n\k. t < - X n) \ (\n\k. t < X n)" ..
then show "\t>0. \k. (\n\k. t < - X n) \ (\n\k. t < X n)"
using t
by auto
qed
lemma cauchy_not_vanishes:
assumes X:
"cauchy X"
and nz:
"\ vanishes X"
shows "\b>0. \k. \n\k. b < \X n\"
using cauchy_not_vanishes_cases [OF assms]
by (elim ex_forward conj_forward asm_rl) auto
lemma cauchy_inverse [simp]:
assumes X:
"cauchy X"
and nz:
"\ vanishes X"
shows "cauchy (\n. inverse (X n))"
proof (rule cauchyI)
fix r :: rat
assume "0 < r"
obtain b i
where b:
"0 < b" and i:
"\n\i. b < \X n\"
using cauchy_not_vanishes [OF X nz]
by blast
from b i
have nz:
"\n\i. X n \ 0" by auto
obtain s
where s:
"0 < s" and r:
"r = inverse b * s * inverse b"
proof
show "0 < b * r * b" by (simp add:
‹0 < r
› b)
show "r = inverse b * (b * r * b) * inverse b"
using b
by simp
qed
obtain j
where j:
"\m\j. \n\j. \X m - X n\ < s"
using cauchyD [OF X s] ..
have "\m\max i j. \n\max i j. \inverse (X m) - inverse (X n)\ < r"
proof clarsimp
fix m n
assume *:
"i \ m" "j \ m" "i \ n" "j \ n"
have "\inverse (X m) - inverse (X n)\ = inverse \X m\ * \X m - X n\ * inverse \X n\"
by (simp add: inverse_diff_inverse nz * abs_mult)
also have "\ < inverse b * s * inverse b"
by (simp add: mult_strict_mono less_imp_inverse_less i j b * s)
finally show "\inverse (X m) - inverse (X n)\ < r" by (simp only: r)
qed
then show "\k. \m\k. \n\k. \inverse (X m) - inverse (X n)\ < r" ..
qed
lemma vanishes_diff_inverse:
assumes X:
"cauchy X" "\ vanishes X"
and Y:
"cauchy Y" "\ vanishes Y"
and XY:
"vanishes (\n. X n - Y n)"
shows "vanishes (\n. inverse (X n) - inverse (Y n))"
proof (rule vanishesI)
fix r :: rat
assume r:
"0 < r"
obtain a i
where a:
"0 < a" and i:
"\n\i. a < \X n\"
using cauchy_not_vanishes [OF X]
by blast
obtain b j
where b:
"0 < b" and j:
"\n\j. b < \Y n\"
using cauchy_not_vanishes [OF Y]
by blast
obtain s
where s:
"0 < s" and "inverse a * s * inverse b = r"
proof
show "0 < a * r * b"
using a r b
by simp
show "inverse a * (a * r * b) * inverse b = r"
using a r b
by simp
qed
obtain k
where k:
"\n\k. \X n - Y n\ < s"
using vanishesD [OF XY s] ..
have "\n\max (max i j) k. \inverse (X n) - inverse (Y n)\ < r"
proof clarsimp
fix n
assume n:
"i \ n" "j \ n" "k \ n"
with i j a b
have "X n \ 0" and "Y n \ 0"
by auto
then have "\inverse (X n) - inverse (Y n)\ = inverse \X n\ * \X n - Y n\ * inverse \Y n\"
by (simp add: inverse_diff_inverse abs_mult)
also have "\ < inverse a * s * inverse b"
by (intro mult_strict_mono
' less_imp_inverse_less) (simp_all add: a b i j k n)
also note ‹inverse a * s * inverse b = r
›
finally show "\inverse (X n) - inverse (Y n)\ < r" .
qed
then show "\k. \n\k. \inverse (X n) - inverse (Y n)\ < r" ..
qed
subsection ‹Equivalence relation on Cauchy sequences
›
definition realrel ::
"(nat \ rat) \ (nat \ rat) \ bool"
where "realrel = (\X Y. cauchy X \ cauchy Y \ vanishes (\n. X n - Y n))"
lemma realrelI [intro?]:
"cauchy X \ cauchy Y \ vanishes (\n. X n - Y n) \ realrel X Y"
by (simp add: realrel_def)
lemma realrel_refl:
"cauchy X \ realrel X X"
by (simp add: realrel_def)
lemma symp_realrel:
"symp realrel"
by (simp add: abs_minus_commute realrel_def symp_def vanishes_def)
lemma transp_realrel:
"transp realrel"
unfolding realrel_def
by (rule transpI) (force simp add: dest: vanishes_add)
lemma part_equivp_realrel:
"part_equivp realrel"
by (blast intro: part_equivpI symp_realrel transp_realrel realrel_refl cauchy_const)
subsection ‹The field of real numbers
›
quotient_type real =
"nat \ rat" / partial: realrel
morphisms rep_real Real
by (rule part_equivp_realrel)
lemma cr_real_eq:
"pcr_real = (\x y. cauchy x \ Real x = y)"
unfolding real.pcr_cr_eq cr_real_def realrel_def
by auto
lemma Real_induct [induct type: real]:
(* TODO: generate automatically *)
assumes "\X. cauchy X \ P (Real X)"
shows "P x"
proof (induct x)
case (1 X)
then have "cauchy X" by (simp add: realrel_def)
then show "P (Real X)" by (rule assms)
qed
lemma eq_Real:
"cauchy X \ cauchy Y \ Real X = Real Y \ vanishes (\n. X n - Y n)"
using real.rel_eq_transfer
unfolding real.pcr_cr_eq cr_real_def rel_fun_def realrel_def
by simp
lemma Domainp_pcr_real [transfer_domain_rule]:
"Domainp pcr_real = cauchy"
by (simp add: real.domain_eq realrel_def)
instantiation real :: field
begin
lift_definition zero_real ::
"real" is "\n. 0"
by (simp add: realrel_refl)
lift_definition one_real ::
"real" is "\n. 1"
by (simp add: realrel_refl)
lift_definition plus_real ::
"real \ real \ real" is "\X Y n. X n + Y n"
unfolding realrel_def add_diff_add
by (simp only: cauchy_add vanishes_add simp_thms)
lift_definition uminus_real ::
"real \ real" is "\X n. - X n"
unfolding realrel_def minus_diff_minus
by (simp only: cauchy_minus vanishes_minus simp_thms)
lift_definition times_real ::
"real \ real \ real" is "\X Y n. X n * Y n"
proof -
fix f1 f2 f3 f4
have "\cauchy f1; cauchy f4; vanishes (\n. f1 n - f2 n); vanishes (\n. f3 n - f4 n)\
==> vanishes (λn. f1 n * (f3 n - f4 n) + f4 n * (f1 n - f2 n))
"
by (simp add: vanishes_add vanishes_mult_bounded cauchy_imp_bounded)
then show "\realrel f1 f2; realrel f3 f4\ \ realrel (\n. f1 n * f3 n) (\n. f2 n * f4 n)"
by (simp add: mult.commute realrel_def mult_diff_mult)
qed
lift_definition inverse_real ::
"real \ real"
is "\X. if vanishes X then (\n. 0) else (\n. inverse (X n))"
proof -
fix X Y
assume "realrel X Y"
then have X:
"cauchy X" and Y:
"cauchy Y" and XY:
"vanishes (\n. X n - Y n)"
by (simp_all add: realrel_def)
have "vanishes X \ vanishes Y"
proof
assume "vanishes X"
from vanishes_diff [OF this XY]
show "vanishes Y"
by simp
next
assume "vanishes Y"
from vanishes_add [OF this XY]
show "vanishes X"
by simp
qed
then show "?thesis X Y"
by (simp add: vanishes_diff_inverse X Y XY realrel_def)
qed
definition "x - y = x + - y" for x y :: real
definition "x div y = x * inverse y" for x y :: real
lemma add_Real:
"cauchy X \ cauchy Y \ Real X + Real Y = Real (\n. X n + Y n)"
using plus_real.transfer
by (simp add: cr_real_eq rel_fun_def)
lemma minus_Real:
"cauchy X \ - Real X = Real (\n. - X n)"
using uminus_real.transfer
by (simp add: cr_real_eq rel_fun_def)
lemma diff_Real:
"cauchy X \ cauchy Y \ Real X - Real Y = Real (\n. X n - Y n)"
by (simp add: minus_Real add_Real minus_real_def)
lemma mult_Real:
"cauchy X \ cauchy Y \ Real X * Real Y = Real (\n. X n * Y n)"
using times_real.transfer
by (simp add: cr_real_eq rel_fun_def)
lemma inverse_Real:
"cauchy X \ inverse (Real X) = (if vanishes X then 0 else Real (\n. inverse (X n)))"
using inverse_real.transfer zero_real.transfer
unfolding cr_real_eq rel_fun_def
by (simp split: if_split_asm, metis)
instance
proof
fix a b c :: real
show "a + b = b + a"
by transfer (simp add: ac_simps realrel_def)
show "(a + b) + c = a + (b + c)"
by transfer (simp add: ac_simps realrel_def)
show "0 + a = a"
by transfer (simp add: realrel_def)
show "- a + a = 0"
by transfer (simp add: realrel_def)
show "a - b = a + - b"
by (rule minus_real_def)
show "(a * b) * c = a * (b * c)"
by transfer (simp add: ac_simps realrel_def)
show "a * b = b * a"
by transfer (simp add: ac_simps realrel_def)
show "1 * a = a"
by transfer (simp add: ac_simps realrel_def)
show "(a + b) * c = a * c + b * c"
by transfer (simp add: distrib_right realrel_def)
show "(0::real) \ (1::real)"
by transfer (simp add: realrel_def)
have "vanishes (\n. inverse (X n) * X n - 1)" if X:
"cauchy X" "\ vanishes X" for X
proof (rule vanishesI)
fix r::rat
assume "0 < r"
obtain b k
where "b>0" "\n\k. b < \X n\"
using X cauchy_not_vanishes
by blast
then show "\k. \n\k. \inverse (X n) * X n - 1\ < r"
using ‹0 < r
› by force
qed
then show "a \ 0 \ inverse a * a = 1"
by transfer (simp add: realrel_def)
show "a div b = a * inverse b"
by (rule divide_real_def)
show "inverse (0::real) = 0"
by transfer (simp add: realrel_def)
qed
end
subsection ‹Positive reals
›
lift_definition positive ::
"real \ bool"
is "\X. \r>0. \k. \n\k. r < X n"
proof -
have 1:
"\r>0. \k. \n\k. r < Y n"
if *:
"realrel X Y" and **:
"\r>0. \k. \n\k. r < X n" for X Y
proof -
from *
have XY:
"vanishes (\n. X n - Y n)"
by (simp_all add: realrel_def)
from **
obtain r i
where "0 < r" and i:
"\n\i. r < X n"
by blast
obtain s t
where s:
"0 < s" and t:
"0 < t" and r:
"r = s + t"
using ‹0 < r
› by (rule obtain_pos_sum)
obtain j
where j:
"\n\j. \X n - Y n\ < s"
using vanishesD [OF XY s] ..
have "\n\max i j. t < Y n"
proof clarsimp
fix n
assume n:
"i \ n" "j \ n"
have "\X n - Y n\ < s" and "r < X n"
using i j n
by simp_all
then show "t < Y n" by (simp add: r)
qed
then show ?thesis
using t
by blast
qed
fix X Y
assume "realrel X Y"
then have "realrel X Y" and "realrel Y X"
using symp_realrel
by (auto simp: symp_def)
then show "?thesis X Y"
by (safe elim!: 1)
qed
lemma positive_Real:
"cauchy X \ positive (Real X) \ (\r>0. \k. \n\k. r < X n)"
using positive.transfer
by (simp add: cr_real_eq rel_fun_def)
lemma positive_zero:
"\ positive 0"
by transfer auto
lemma positive_add:
assumes "positive x" "positive y" shows "positive (x + y)"
proof -
have *:
"\\n\i. a < x n; \n\j. b < y n; 0 < a; 0 < b; n \ max i j\
==> a+b < x n + y n
" for x y and a b::rat and i j n::nat
by (simp add: add_strict_mono)
show ?thesis
using assms
by transfer (blast intro: * pos_add_strict)
qed
lemma positive_mult:
assumes "positive x" "positive y" shows "positive (x * y)"
proof -
have *:
"\\n\i. a < x n; \n\j. b < y n; 0 < a; 0 < b; n \ max i j\
==> a*b < x n * y n
" for x y and a b::rat and i j n::nat
by (simp add: mult_strict_mono
')
show ?thesis
using assms
by transfer (blast intro: * mult_pos_pos)
qed
lemma positive_minus:
"\ positive x \ x \ 0 \ positive (- x)"
apply transfer
apply (simp add: realrel_def)
apply (blast dest: cauchy_not_vanishes_cases)
done
instantiation real :: linordered_field
begin
definition "x < y \ positive (y - x)"
definition "x \ y \ x < y \ x = y" for x y :: real
definition "\a\ = (if a < 0 then - a else a)" for a :: real
definition "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)" for a :: real
instance
proof
fix a b c :: real
show "\a\ = (if a < 0 then - a else a)"
by (rule abs_real_def)
show "a < b \ a \ b \ \ b \ a"
"a \ b \ b \ c \ a \ c" "a \ a"
"a \ b \ b \ a \ a = b"
"a \ b \ c + a \ c + b"
unfolding less_eq_real_def less_real_def
by (force simp add: positive_zero dest: positive_add)+
show "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
by (rule sgn_real_def)
show "a \ b \ b \ a"
by (auto dest!: positive_minus simp: less_eq_real_def less_real_def)
show "a < b \ 0 < c \ c * a < c * b"
unfolding less_real_def
by (force simp add: algebra_simps dest: positive_mult)
qed
end
instantiation real :: distrib_lattice
begin
definition "(inf :: real \ real \ real) = min"
definition "(sup :: real \ real \ real) = max"
instance
by standard (auto simp add: inf_real_def sup_real_def max_min_distrib2)
end
lemma of_nat_Real:
"of_nat x = Real (\n. of_nat x)"
by (induct x) (simp_all add: zero_real_def one_real_def add_Real)
lemma of_int_Real:
"of_int x = Real (\n. of_int x)"
by (cases x rule: int_diff_cases) (simp add: of_nat_Real diff_Real)
lemma of_rat_Real:
"of_rat x = Real (\n. x)"
proof (induct x)
case (Fract a b)
then show ?
case
apply (simp add: Fract_of_int_quotient of_rat_divide)
apply (simp add: of_int_Real divide_inverse inverse_Real mult_Real)
done
qed
instance real :: archimedean_field
proof
show "\z. x \ of_int z" for x :: real
proof (induct x)
case (1 X)
then obtain b
where "0 < b" and b:
"\n. \X n\ < b"
by (blast dest: cauchy_imp_bounded)
then have "Real X < of_int (\b\ + 1)"
using 1
apply (simp add: of_int_Real less_real_def diff_Real positive_Real)
apply (rule_tac x=1
in exI)
apply (simp add: algebra_simps)
by (metis abs_ge_self le_less_trans le_of_int_ceiling less_le)
then show ?
case
using less_eq_real_def
by blast
qed
qed
instantiation real :: floor_ceiling
begin
definition [code del]:
"\x::real\ = (THE z. of_int z \ x \ x < of_int (z + 1))"
instance
proof
show "of_int \x\ \ x \ x < of_int (\x\ + 1)" for x :: real
unfolding floor_real_def
using floor_exists1
by (rule theI
')
qed
end
subsection ‹Completeness
›
lemma not_positive_Real:
assumes "cauchy X" shows "\ positive (Real X) \ (\r>0. \k. \n\k. X n \ r)" (
is "?lhs = ?rhs")
unfolding positive_Real [OF assms]
proof (intro iffI allI notI impI)
show "\k. \n\k. X n \ r" if r:
"\ (\r>0. \k. \n\k. r < X n)" and "0 < r" for r
proof -
obtain s t
where "s > 0" "t > 0" "r = s+t"
using ‹r > 0
› obtain_pos_sum
by blast
obtain k
where k:
"\m n. \m\k; n\k\ \ \X m - X n\ < t"
using cauchyD [OF assms
‹t > 0
›]
by blast
obtain n
where "n \ k" "X n \ s"
by (meson r
‹0 < s
› not_less)
then have "X l \ r" if "l \ n" for l
using k [OF
‹n
≥ k
›, of l] that
‹r = s+t
› by linarith
then show ?thesis
by blast
qed
qed (meson le_cases not_le)
lemma le_Real:
assumes "cauchy X" "cauchy Y"
shows "Real X \ Real Y = (\r>0. \k. \n\k. X n \ Y n + r)"
unfolding not_less [symmetric,
where 'a=real] less_real_def
apply (simp add: diff_Real not_positive_Real assms)
apply (simp add: diff_le_eq ac_simps)
done
lemma le_RealI:
assumes Y:
"cauchy Y"
shows "\n. x \ of_rat (Y n) \ x \ Real Y"
proof (induct x)
fix X
assume X:
"cauchy X" and "\n. Real X \ of_rat (Y n)"
then have le:
"\m r. 0 < r \ \k. \n\k. X n \ Y m + r"
by (simp add: of_rat_Real le_Real)
then have "\k. \n\k. X n \ Y n + r" if "0 < r" for r :: rat
proof -
from that
obtain s t
where s:
"0 < s" and t:
"0 < t" and r:
"r = s + t"
by (rule obtain_pos_sum)
obtain i
where i:
"\m\i. \n\i. \Y m - Y n\ < s"
using cauchyD [OF Y s] ..
obtain j
where j:
"\n\j. X n \ Y i + t"
using le [OF t] ..
have "\n\max i j. X n \ Y n + r"
proof clarsimp
fix n
assume n:
"i \ n" "j \ n"
have "X n \ Y i + t"
using n j
by simp
moreover have "\Y i - Y n\ < s"
using n i
by simp
ultimately show "X n \ Y n + r"
unfolding r
by simp
qed
then show ?thesis ..
qed
then show "Real X \ Real Y"
by (simp add: of_rat_Real le_Real X Y)
qed
lemma Real_leI:
assumes X:
"cauchy X"
assumes le:
"\n. of_rat (X n) \ y"
shows "Real X \ y"
proof -
have "- y \ - Real X"
by (simp add: minus_Real X le_RealI of_rat_minus le)
then show ?thesis
by simp
qed
lemma less_RealD:
assumes "cauchy Y"
shows "x < Real Y \ \n. x < of_rat (Y n)"
by (meson Real_leI assms leD leI)
lemma of_nat_less_two_power [simp]:
"of_nat n < (2::'a::linordered_idom) ^ n"
by auto
lemma complete_real:
fixes S ::
"real set"
assumes "\x. x \ S" and "\z. \x\S. x \ z"
shows "\y. (\x\S. x \ y) \ (\z. (\x\S. x \ z) \ y \ z)"
proof -
obtain x
where x:
"x \ S" using assms(1) ..
obtain z
where z:
"\x\S. x \ z" using assms(2) ..
define P
where "P x \ (\y\S. y \ of_rat x)" for x
obtain a
where a:
"\ P a"
proof
have "of_int \x - 1\ \ x - 1" by (rule of_int_floor_le)
also have "x - 1 < x" by simp
finally have "of_int \x - 1\ < x" .
then have "\ x \ of_int \x - 1\" by (simp only: not_le)
then show "\ P (of_int \x - 1\)"
unfolding P_def of_rat_of_int_eq
using x
by blast
qed
obtain b
where b:
"P b"
proof
show "P (of_int \z\)"
unfolding P_def of_rat_of_int_eq
proof
fix y
assume "y \ S"
then have "y \ z" using z
by simp
also have "z \ of_int \z\" by (rule le_of_int_ceiling)
finally show "y \ of_int \z\" .
qed
qed
define avg
where "avg x y = x/2 + y/2" for x y :: rat
define bisect
where "bisect = (\(x, y). if P (avg x y) then (x, avg x y) else (avg x y, y))"
define A
where "A n = fst ((bisect ^^ n) (a, b))" for n
define B
where "B n = snd ((bisect ^^ n) (a, b))" for n
define C
where "C n = avg (A n) (B n)" for n
have A_0 [simp]:
"A 0 = a" unfolding A_def
by simp
have B_0 [simp]:
"B 0 = b" unfolding B_def
by simp
have A_Suc [simp]:
"\n. A (Suc n) = (if P (C n) then A n else C n)"
unfolding A_def B_def C_def bisect_def split_def
by simp
have B_Suc [simp]:
"\n. B (Suc n) = (if P (C n) then C n else B n)"
unfolding A_def B_def C_def bisect_def split_def
by simp
have width:
"B n - A n = (b - a) / 2^n" for n
proof (induct n)
case (Suc n)
then show ?
case
by (simp add: C_def eq_divide_eq avg_def algebra_simps)
qed simp
have twos:
"\n. y / 2 ^ n < r" if "0 < r" for y r :: rat
proof -
obtain n
where "y / r < rat_of_nat n"
using ‹0 < r
› reals_Archimedean2
by blast
then have "\n. y < r * 2 ^ n"
by (metis divide_less_eq less_trans mult.commute of_nat_less_two_power that)
then show ?thesis
by (simp add: field_split_simps)
qed
have PA:
"\ P (A n)" for n
by (induct n) (simp_all add: a)
have PB:
"P (B n)" for n
by (induct n) (simp_all add: b)
have ab:
"a < b"
using a b
unfolding P_def
by (meson leI less_le_trans of_rat_less)
have AB:
"A n < B n" for n
by (induct n) (simp_all add: ab C_def avg_def)
have "A i \ A j \ B j \ B i" if "i < j" for i j
using that
proof (
induction rule: less_Suc_induct)
case (1 i)
then show ?
case
apply (clarsimp simp add: C_def avg_def add_divide_distrib [symmetric])
apply (rule AB [
THEN less_imp_le])
done
qed simp
then have A_mono:
"A i \ A j" and B_mono:
"B j \ B i" if "i \ j" for i j
by (metis eq_refl le_neq_implies_less that)+
have cauchy_lemma:
"cauchy X" if *:
"\n i. i\n \ A n \ X i \ X i \ B n" for X
proof (rule cauchyI)
fix r::rat
assume "0 < r"
then obtain k
where k:
"(b - a) / 2 ^ k < r"
using twos
by blast
have "\X m - X n\ < r" if "m\k" "n\k" for m n
proof -
have "\X m - X n\ \ B k - A k"
by (simp add: * abs_rat_def diff_mono that)
also have "... < r"
by (simp add: k width)
finally show ?thesis .
qed
then show "\k. \m\k. \n\k. \X m - X n\ < r"
by blast
qed
have "cauchy A"
by (rule cauchy_lemma) (meson AB A_mono B_mono dual_order.strict_implies_order less_le_t
rans)
have "cauchy B"
by (rule cauchy_lemma) (meson AB A_mono B_mono dual_order.strict_implies_order le_less_trans)
have "\x\S. x \ Real B"
proof
fix x
assume "x \ S"
then show "x \ Real B"
using PB [unfolded P_def] ‹cauchy B›
by (simp add: le_RealI)
qed
moreover have "\z. (\x\S. x \ z) \ Real A \ z"
by (meson PA Real_leI P_def ‹cauchy A› le_cases order.trans)
moreover have "vanishes (\n. (b - a) / 2 ^ n)"
proof (rule vanishesI)
fix r :: rat
assume "0 < r"
then obtain k where k: "\b - a\ / 2 ^ k < r"
using twos by blast
have "\n\k. \(b - a) / 2 ^ n\ < r"
proof clarify
fix n
assume n: "k \ n"
have "\(b - a) / 2 ^ n\ = \b - a\ / 2 ^ n"
by simp
also have "\ \ \b - a\ / 2 ^ k"
using n by (simp add: divide_left_mono)
also note k
finally show "\(b - a) / 2 ^ n\ < r" .
qed
then show "\k. \n\k. \(b - a) / 2 ^ n\ < r" ..
qed
then have "Real B = Real A"
by (simp add: eq_Real ‹cauchy A› ‹cauchy B› width)
ultimately show "\y. (\x\S. x \ y) \ (\z. (\x\S. x \ z) \ y \ z)"
by force
qed
instantiation real :: linear_continuum
begin
subsection ‹Supremum of a set of reals›
definition "Sup X = (LEAST z::real. \x\X. x \ z)"
definition "Inf X = - Sup (uminus ` X)" for X :: "real set"
instance
proof
show Sup_upper: "x \ Sup X"
if "x \ X" "bdd_above X"
for x :: real and X :: "real set"
proof -
from that obtain s where s: "\y\X. y \ s" "\z. \y\X. y \ z \ s \ z"
using complete_real[of X] unfolding bdd_above_def by blast
then show ?thesis
unfolding Sup_real_def by (rule LeastI2_order) (auto simp: that)
qed
show Sup_least: "Sup X \ z"
if "X \ {}" and z: "\x. x \ X \ x \ z"
for z :: real and X :: "real set"
proof -
from that obtain s where s: "\y\X. y \ s" "\z. \y\X. y \ z \ s \ z"
using complete_real [of X] by blast
then have "Sup X = s"
unfolding Sup_real_def by (best intro: Least_equality)
also from s z have "\ \ z"
by blast
finally show ?thesis .
qed
show "Inf X \ x" if "x \ X" "bdd_below X"
for x :: real and X :: "real set"
using Sup_upper [of "-x" "uminus ` X"] by (auto simp: Inf_real_def that)
show "z \ Inf X" if "X \ {}" "\x. x \ X \ z \ x"
for z :: real and X :: "real set"
using Sup_least [of "uminus ` X" "- z"] by (force simp: Inf_real_def that)
show "\a b::real. a \ b"
using zero_neq_one by blast
qed
end
subsection ‹Hiding implementation details›
hide_const (open) vanishes cauchy positive Real
declare Real_induct [induct del]
declare Abs_real_induct [induct del]
declare Abs_real_cases [cases del]
lifting_update real.lifting
lifting_forget real.lifting
subsection ‹Embedding numbers into the Reals›
abbreviation real_of_nat :: "nat \ real"
where "real_of_nat \ of_nat"
abbreviation real :: "nat \ real"
where "real \ of_nat"
abbreviation real_of_int :: "int \ real"
where "real_of_int \ of_int"
abbreviation real_of_rat :: "rat \ real"
where "real_of_rat \ of_rat"
declare [[coercion_enabled]]
declare [[coercion "of_nat :: nat \ int"]]
declare [[coercion "of_nat :: nat \ real"]]
declare [[coercion "of_int :: int \ real"]]
(* We do not add rat to the coerced types, this has often unpleasant side effects when writing
inverse (Suc n) which sometimes gets two coercions: of_rat (inverse (of_nat (Suc n))) *)
declare [[coercion_map map]]
declare [[coercion_map "\f g h x. g (h (f x))"]]
declare [[coercion_map "\f g (x,y). (f x, g y)"]]
declare of_int_eq_0_iff [algebra, presburger]
declare of_int_eq_1_iff [algebra, presburger]
declare of_int_eq_iff [algebra, presburger]
declare of_int_less_0_iff [algebra, presburger]
declare of_int_less_1_iff [algebra, presburger]
declare of_int_less_iff [algebra, presburger]
declare of_int_le_0_iff [algebra, presburger]
declare of_int_le_1_iff [algebra, presburger]
declare of_int_le_iff [algebra, presburger]
declare of_int_0_less_iff [algebra, presburger]
declare of_int_0_le_iff [algebra, presburger]
declare of_int_1_less_iff [algebra, presburger]
declare of_int_1_le_iff [algebra, presburger]
lemma int_less_real_le: "n < m \ real_of_int n + 1 \ real_of_int m"
proof -
have "(0::real) \ 1"
by (metis less_eq_real_def zero_less_one)
then show ?thesis
by (metis floor_of_int less_floor_iff)
qed
lemma int_le_real_less: "n \ m \ real_of_int n < real_of_int m + 1"
by (meson int_less_real_le not_le)
lemma (in field_char_0) of_int_div_aux:
"(of_int x) / (of_int d) =
of_int (x div d) + (of_int (x mod d)) / (of_int d)"
proof -
have "x = (x div d) * d + x mod d"
by auto
then have "of_int x = of_int (x div d) * of_int d + of_int(x mod d)"
by (metis local.of_int_add local.of_int_mult)
then show ?thesis
by (simp add: divide_simps)
qed
lemma real_of_int_div:
"d dvd n \ real_of_int (n div d) = real_of_int n / real_of_int d" for d n :: int
by auto
lemma real_of_int_div2: "0 \ real_of_int n / real_of_int x - real_of_int (n div x)"
proof (cases "x = 0")
case False
then show ?thesis
by (metis diff_ge_0_iff_ge floor_divide_of_int_eq of_int_floor_le)
qed simp
lemma real_of_int_div3: "real_of_int n / real_of_int x - real_of_int (n div x) \ 1"
apply (simp add: algebra_simps)
by (metis add.commute floor_correct floor_divide_of_int_eq less_eq_real_def of_int_1 of_int_add)
lemma real_of_int_div4: "real_of_int (n div x) \ real_of_int n / real_of_int x"
using real_of_int_div2 [of n x] by simp
subsection ‹Embedding the Naturals into the Reals›
lemma (in field_char_0) of_nat_of_nat_div_aux:
"of_nat x / of_nat d = of_nat (x div d) + of_nat (x mod d) / of_nat d"
by (metis add_divide_distrib diff_add_cancel of_nat_div)
lemma(in field_char_0) of_nat_of_nat_div: "d dvd n \ of_nat(n div d) = of_nat n / of_nat d"
by auto
lemma (in linordered_field) of_nat_div_le_of_nat: "of_nat (n div x) \ of_nat n / of_nat x"
by (metis le_add_same_cancel1 of_nat_0_le_iff of_nat_of_nat_div_aux zero_le_divide_iff)
lemma real_of_card: "real (card A) = sum (\x. 1) A"
by simp
lemma nat_less_real_le: "n < m \ real n + 1 \ real m"
by (metis less_iff_succ_less_eq of_nat_1 of_nat_add of_nat_le_iff)
lemma nat_le_real_less: "n \ m \ real n < real m + 1"
by (meson nat_less_real_le not_le)
lemma real_of_nat_div: "d dvd n \ real(n div d) = real n / real d"
by auto
lemma real_binomial_eq_mult_binomial_Suc:
assumes "k \ n"
shows "real(n choose k) = (n + 1 - k) / (n + 1) * (Suc n choose k)"
using assms
by (simp add: of_nat_binomial_eq_mult_binomial_Suc [of k n] add.commute)
subsection ‹The Archimedean Property of the Reals›
text ‹Not actually the reals any more!›
lemma real_arch_inverse:
fixes e::"'a::archimedean_field"
shows "0 < e \ (\n::nat. n \ 0 \ 0 < inverse (real n) \ inverse (of_nat n) < e)"
using reals_Archimedean[of e] less_trans[of 0 "1 / of_nat n" e for n::nat]
by (auto simp add: field_simps cong: conj_cong simp del: of_nat_Suc)
lemma reals_Archimedean3:
fixes x::"'a::archimedean_field"
shows "0 < x \ \y. \n. y < of_nat n * x"
by (auto intro: ex_less_of_nat_mult)
lemma real_archimedian_rdiv_eq_0:
fixes x::"'a::archimedean_field"
assumes "x \ 0" and "\m::nat. m > 0 \ of_nat m * x \ c"
shows "x = 0"
by (metis (no_types, opaque_lifting) reals_Archimedean3 order.order_iff_strict le0 le_less_trans not_le assms)
lemma inverse_Suc: "inverse (of_nat (Suc n)) > (0::'a::archimedean_field)"
using of_nat_0_less_iff positive_imp_inverse_positive zero_less_Suc by blast
lemma Archimedean_eventually_inverse:
fixes ε::"'a::archimedean_field" shows "(\\<^sub>F n in sequentially. inverse (of_nat (Suc n)) < \) \ 0 < \"
(is "?lhs=?rhs")
proof
assume ?lhs
then show ?rhs
unfolding eventually_at_top_dense
by (metis (no_types, lifting) gt_ex inverse_Suc nat.distinct(1) real_arch_inverse)
next
assume ?rhs
then obtain N where "inverse (of_nat (Suc N)) < \"
using reals_Archimedean by blast
then have "inverse (of_nat (Suc n)) < \" if "n \ N" for n
using that Suc_le_mono inverse_Suc inverse_less_imp_less
by (meson inverse_positive_iff_positive linorder_not_less of_nat_less_iff order_le_less_trans)
then show ?lhs
unfolding eventually_sequentially by blast
qed
(*HOL Light's FORALL_POS_MONO_1_EQ*)
text ‹On the relationship between two different ways of converting to 0›
lemma Inter_eq_Inter_inverse_Suc:
assumes "\r' r. r' < r \ A r' \ A r"
shows "\ (A ` {0<..}) = (\n. A(inverse(Suc n)))"
proof
have "x \ A \"
if x: "\n. x \ A (inverse (Suc n))" and "\>0" for x and ε :: real
proof -
obtain n where "inverse (Suc n) < \"
using ‹ε>0› reals_Archimedean by blast
with assms x show ?thesis
by blast
qed
then show "(\n. A(inverse(Suc n))) \ (\\\{0<..}. A \)"
by auto
qed (use inverse_Suc in fastforce)
subsection ‹Rationals›
lemma Rats_abs_iff[simp]:
"\(x::real)\ \ \ \ x \ \"
by(simp add: abs_real_def split: if_splits)
lemma Rats_eq_int_div_int: "\ = {real_of_int i / real_of_int j | i j. j \ 0}" (is "_ = ?S")
proof
show "\ \ ?S"
proof
fix x :: real
assume "x \ \"
then obtain r where "x = of_rat r"
unfolding Rats_def ..
have "of_rat r \ ?S"
by (cases r) (auto simp add: of_rat_rat)
then show "x \ ?S"
using ‹x = of_rat r› by simp
qed
next
show "?S \ \"
proof (auto simp: Rats_def)
fix i j :: int
assume "j \ 0"
then have "real_of_int i / real_of_int j = of_rat (Fract i j)"
by (simp add: of_rat_rat)
then show "real_of_int i / real_of_int j \ range of_rat"
by blast
qed
qed
lemma Rats_eq_int_div_nat: "\ = { real_of_int i / real n | i n. n \ 0}"
proof (auto simp: Rats_eq_int_div_int)
fix i j :: int
assume "j \ 0"
show "\(i'::int) (n::nat). real_of_int i / real_of_int j = real_of_int i' / real n \ 0 < n"
proof (cases "j > 0")
case True
then have "real_of_int i / real_of_int j = real_of_int i / real (nat j) \ 0 < nat j"
by simp
then show ?thesis by blast
next
case False
with ‹j ≠ 0›
have "real_of_int i / real_of_int j = real_of_int (- i) / real (nat (- j)) \ 0 < nat (- j)"
by simp
then show ?thesis by blast
qed
next
fix i :: int and n :: nat
assume "0 < n"
then have "real_of_int i / real n = real_of_int i / real_of_int(int n) \ int n \ 0"
by simp
then show "\i' j. real_of_int i / real n = real_of_int i' / real_of_int j \ j \ 0"
by blast
qed
lemma Rats_abs_nat_div_natE:
assumes "x \ \"
obtains m n :: nat where "n \ 0" and "\x\ = real m / real n" and "coprime m n"
proof -
from ‹x ∈ ℚ› obtain i :: int and n :: nat where "n \ 0" and "x = real_of_int i / real n"
by (auto simp add: Rats_eq_int_div_nat)
then have "\x\ = real (nat \i\) / real n" by simp
then obtain m :: nat where x_rat: "\x\ = real m / real n" by blast
let ?gcd = "gcd m n"
from ‹n ≠ 0› have gcd: "?gcd \ 0" by simp
let ?k = "m div ?gcd"
let ?l = "n div ?gcd"
let ?gcd' = "gcd ?k ?l"
have "?gcd dvd m" ..
then have gcd_k: "?gcd * ?k = m"
by (rule dvd_mult_div_cancel)
have "?gcd dvd n" ..
then have gcd_l: "?gcd * ?l = n"
by (rule dvd_mult_div_cancel)
from ‹n ≠ 0› and gcd_l have "?gcd * ?l \ 0" by simp
then have "?l \ 0" by (blast dest!: mult_not_zero)
moreover
have "\x\ = real ?k / real ?l"
proof -
from gcd have "real ?k / real ?l = real (?gcd * ?k) / real (?gcd * ?l)"
by (simp add: real_of_nat_div)
also from gcd_k and gcd_l have "\ = real m / real n" by simp
also from x_rat have "\ = \x\" ..
finally show ?thesis ..
qed
moreover
have "?gcd' = 1"
proof -
have "?gcd * ?gcd' = gcd (?gcd * ?k) (?gcd * ?l)"
by (rule gcd_mult_distrib_nat)
with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp
with gcd show ?thesis by auto
qed
then have "coprime ?k ?l"
by (simp only: coprime_iff_gcd_eq_1)
ultimately show ?thesis ..
qed
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 🚫‹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 "\r\\. x < r \ 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 blast
define p where "p = \y * real q\ - 1"
define r where "r = of_int p / real q"
from q have "x < y - inverse (real q)"
by simp
also from ‹0 < q› have "y - inverse (real q) \ r"
by (simp add: r_def p_def le_divide_eq left_diff_distrib)
finally have "x < r" .
moreover from ‹0 < q› have "r < y"
by (simp add: r_def p_def divide_less_eq diff_less_eq less_ceiling_iff [symmetric])
moreover have "r \ \"
by (simp add: r_def)
ultimately show ?thesis by blast
qed
lemma of_rat_dense:
fixes x y :: real
assumes "x < y"
shows "\q :: rat. x < of_rat q \ of_rat q < y"
using Rats_dense_in_real [OF ‹x < y›]
by (auto elim: Rats_cases)
subsection ‹Numerals and Arithmetic›
declaration ‹
K (Lin_Arith.add_inj_const (🍋‹of_nat›, 🍋‹nat ==> real›)
#> Lin_Arith.add_inj_const (🍋‹of_int›, 🍋‹int ==> real›))
›
subsection ‹Simprules combining ‹x + y› and ‹0›› (* FIXME ARE THEY NEEDED? *)
lemma real_add_minus_iff [simp]: "x + - a = 0 \ x = a"
for x a :: real
by arith
lemma real_add_less_0_iff: "x + y < 0 \ y < - x"
for x y :: real
by auto
lemma real_0_less_add_iff: "0 < x + y \ - x < y"
for x y :: real
by auto
lemma real_add_le_0_iff: "x + y \ 0 \ y \ - x"
for x y :: real
by auto
lemma real_0_le_add_iff: "0 \ x + y \ - x \ y"
for x y :: real
by auto
lemma mult_ge1_I: "\x\1; y\1\ \ x*y \ (1::real)"
using mult_mono by fastforce
subsection ‹Lemmas about powers›
lemma two_realpow_ge_one: "(1::real) \ 2 ^ n"
by simp
declare sum_squares_eq_zero_iff [simp] sum_power2_eq_zero_iff [simp]
lemma real_minus_mult_self_le [simp]: "- (u * u) \ x * x"
for u x :: real
by (rule order_trans [where y = 0]) auto
lemma realpow_square_minus_le [simp]: "- u\<^sup>2 \ x\<^sup>2"
for u x :: real
by (auto simp add: power2_eq_square)
subsection ‹Density of the Reals›
lemma field_lbound_gt_zero: "0 < d1 \ 0 < d2 \ \e. 0 < e \ e < d1 \ e < d2"
for d1 d2 :: "'a::linordered_field"
by (rule exI [where x = "min d1 d2 / 2"]) (simp add: min_def)
lemma field_less_half_sum: "x < y \ x < (x + y) / 2"
for x y :: "'a::linordered_field"
by auto
lemma field_sum_of_halves: "x / 2 + x / 2 = x"
for x :: "'a::linordered_field"
by simp
subsection ‹Archimedean properties and useful consequences›
text‹Bernoulli's inequality\
proposition Bernoulli_inequality:
fixes x :: "'a :: linordered_field"
assumes "-1 \ x"
shows "1 + of_nat n * x \ (1 + x) ^ n"
proof (induct n)
case 0
then show ?case by simp
next
case (Suc n)
have "1 + of_nat (Suc n) * x \ 1 + of_nat(Suc n) * x + of_nat n * x^2"
by simp
also have "... = (1 + x) * (1 + of_nat n * x)"
by (auto simp: power2_eq_square algebra_simps)
also have "\ \ (1 + x) ^ Suc n"
using Suc.hyps assms mult_left_mono by fastforce
finally show ?case .
qed
corollary Bernoulli_inequality_even:
fixes x :: "'a :: linordered_field"
assumes "even n"
shows "1 + of_nat n * x \ (1 + x) ^ n"
proof (cases "-1 \ x \ n=0")
case True
then show ?thesis
by (auto simp: Bernoulli_inequality)
next
case False
then have "of_nat n \ (1::'a)"
by simp
with False have "of_nat n * x \ -1"
by (metis linear minus_zero mult.commute mult.left_neutral mult_left_mono_neg neg_le_iff_le order_trans zero_le_one)
then have "1 + of_nat n * x \ 0"
by auto
also have "... \ (1 + x) ^ n"
using assms zero_le_even_power by blast
finally show ?thesis .
qed
corollary real_arch_pow:
fixes x :: real
assumes x: "1 < x"
shows "\n. y < x^n"
proof -
from x have x0: "x - 1 > 0"
by arith
from reals_Archimedean3[OF x0, rule_format, of y]
obtain n :: nat where n: "y < real n * (x - 1)" by metis
from x0 have x00: "x- 1 \ -1" by arith
from Bernoulli_inequality[OF x00, of n] n
have "y < x^n" by auto
then show ?thesis by metis
qed
corollary real_arch_pow_inv:
fixes x y :: real
assumes y: "y > 0"
and x1: "x < 1"
shows "\n. x^n < y"
proof (cases "x > 0")
case True
with x1 have ix: "1 < 1/x" by (simp add: field_simps)
from real_arch_pow[OF ix, of "1/y"]
obtain n where n: "1/y < (1/x)^n" by blast
then show ?thesis using y ‹x > 0›
by (auto simp add: field_simps)
next
case False
with y x1 show ?thesis
by (metis less_le_trans not_less power_one_right)
qed
lemma forall_pos_mono:
"(\d e::real. d < e \ P d \ P e) \
(∧n::nat. n ≠ 0 ==> P (inverse (real n))) ==> (∧e. 0 < e ==> P e)"
by (metis real_arch_inverse)
lemma forall_pos_mono_1:
"(\d e::real. d < e \ P d \ P e) \
(∧n. P (inverse (real (Suc n)))) ==> 0 < e ==> P e"
using reals_Archimedean by blast
lemma Archimedean_eventually_pow:
fixes x::real
assumes "1 < x"
shows "\\<^sub>F n in sequentially. b < x ^ n"
proof -
obtain N where "\n. n\N \ b < x ^ n"
by (metis assms le_less order_less_trans power_strict_increasing_iff real_arch_pow)
then show ?thesis
using eventually_sequentially by blast
qed
lemma Archimedean_eventually_pow_inverse:
fixes x::real
assumes "\x\ < 1" "\ > 0"
shows "\\<^sub>F n in sequentially. \x^n\ < \"
proof (cases "x = 0")
case True
then show ?thesis
by (simp add: assms eventually_at_top_dense zero_power)
next
case False
then have "\\<^sub>F n in sequentially. inverse \ < inverse \x\ ^ n"
by (simp add: Archimedean_eventually_pow assms(1) one_less_inverse)
then show ?thesis
by eventually_elim (metis ‹ε > 0› inverse_less_imp_less power_abs power_inverse)
qed
subsection ‹Floor and Ceiling Functions from the Reals to the Integers›
(* FIXME: theorems for negative numerals. Many duplicates, e.g. from Archimedean_Field.thy. *)
lemma real_of_nat_less_numeral_iff [simp]: "real n < numeral w \ n < numeral w"
for n :: nat
by (metis of_nat_less_iff of_nat_numeral)
lemma numeral_less_real_of_nat_iff [simp]: "numeral w < real n \ numeral w < n"
for n :: nat
by (metis of_nat_less_iff of_nat_numeral)
lemma numeral_le_real_of_nat_iff [simp]: "numeral n \ real m \ numeral n \ m"
for m :: nat
by (metis not_le real_of_nat_less_numeral_iff)
lemma of_int_floor_cancel [simp]: "of_int \x\ = x \ (\n::int. x = of_int n)"
by (metis floor_of_int)
lemma of_int_floor [simp]: "a \ \ \ of_int (floor a) = a"
by (metis Ints_cases of_int_floor_cancel)
lemma floor_frac [simp]: "\frac r\ = 0"
by (simp add: frac_def)
lemma frac_1 [simp]: "frac 1 = 0"
by (simp add: frac_def)
lemma frac_in_Rats_iff [simp]:
fixes r::"'a::{floor_ceiling,field_char_0}"
shows "frac r \ \ \ r \ \"
by (metis Rats_add Rats_diff Rats_of_int diff_add_cancel frac_def)
lemma floor_eq: "real_of_int n < x \ x < real_of_int n + 1 \ \x\ = n"
by linarith
lemma floor_eq2: "real_of_int n \ x \ x < real_of_int n + 1 \ \x\ = n"
by (fact floor_unique)
lemma floor_eq3: "real n < x \ x < real (Suc n) \ nat \x\ = n"
by linarith
lemma floor_eq4: "real n \ x \ x < real (Suc n) \ nat \x\ = n"
by linarith
lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \ real_of_int \r\"
by linarith
lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real_of_int \r\"
by linarith
lemma real_of_int_floor_add_one_ge [simp]: "r \ real_of_int \r\ + 1"
by linarith
lemma real_of_int_floor_add_one_gt [simp]: "r < real_of_int \r\ + 1"
by linarith
lemma floor_divide_real_eq_div:
assumes "0 \ b"
shows "\a / real_of_int b\ = \a\ div b"
proof (cases "b = 0")
case True
then show ?thesis by simp
next
case False
with assms have b: "b > 0" by simp
have "j = i div b"
if "real_of_int i \ a" "a < 1 + real_of_int i"
"real_of_int j * real_of_int b \ a" "a < real_of_int b + real_of_int j * real_of_int b"
for i j :: int
proof -
from that have "i < b + j * b"
by (metis le_less_trans of_int_add of_int_less_iff of_int_mult)
moreover have "j * b < 1 + i"
proof -
have "real_of_int (j * b) < real_of_int i + 1"
using ‹a < 1 + real_of_int i› ‹real_of_int j * real_of_int b ≤ a› by force
then show "j * b < 1 + i" by linarith
qed
ultimately have "(j - i div b) * b \ i mod b" "i mod b < ((j - i div b) + 1) * b"
by (auto simp: field_simps)
then have "(j - i div b) * b < 1 * b" "0 * b < ((j - i div b) + 1) * b"
using pos_mod_bound [OF b, of i] pos_mod_sign [OF b, of i]
by linarith+
then show ?thesis using b unfolding mult_less_cancel_right by auto
qed
with b show ?thesis by (auto split: floor_split simp: field_simps)
qed
lemma floor_one_divide_eq_div_numeral [simp]:
"\1 / numeral b::real\ = 1 div numeral b"
by (metis floor_divide_of_int_eq of_int_1 of_int_numeral)
lemma floor_minus_one_divide_eq_div_numeral [simp]:
"\- (1 / numeral b)::real\ = - 1 div numeral b"
by (metis (mono_tags, opaque_lifting) div_minus_right minus_divide_right
floor_divide_of_int_eq of_int_neg_numeral of_int_1)
lemma floor_divide_eq_div_numeral [simp]:
"\numeral a / numeral b::real\ = numeral a div numeral b"
by (metis floor_divide_of_int_eq of_int_numeral)
lemma floor_minus_divide_eq_div_numeral [simp]:
"\- (numeral a / numeral b)::real\ = - numeral a div numeral b"
by (metis divide_minus_left floor_divide_of_int_eq of_int_neg_numeral of_int_numeral)
lemma of_int_ceiling_cancel [simp]: "of_int \x\ = x \ (\n::int. x = of_int n)"
using ceiling_of_int by metis
lemma of_int_ceiling [simp]: "a \ \ \ of_int (ceiling a) = a"
by (metis Ints_cases of_int_ceiling_cancel)
lemma ceiling_eq: "of_int n < x \ x \ of_int n + 1 \ \x\ = n + 1"
by (simp add: ceiling_unique)
lemma of_int_ceiling_diff_one_le [simp]: "of_int \r\ - 1 \ r"
by linarith
lemma of_int_ceiling_le_add_one [simp]: "of_int \r\ \ r + 1"
by linarith
lemma ceiling_le: "x \ of_int a \ \x\ \ a"
by (simp add: ceiling_le_iff)
lemma ceiling_divide_eq_div: "\of_int a / of_int b\ = - (- a div b)"
by (metis ceiling_def floor_divide_of_int_eq minus_divide_left of_int_minus)
lemma ceiling_divide_eq_div_numeral [simp]:
"\numeral a / numeral b :: real\ = - (- numeral a div numeral b)"
using ceiling_divide_eq_div[of "numeral a" "numeral b"] by simp
lemma ceiling_minus_divide_eq_div_numeral [simp]:
"\- (numeral a / numeral b :: real)\ = - (numeral a div numeral b)"
using ceiling_divide_eq_div[of "- numeral a" "numeral b"] by simp
text ‹
The following lemmas are remnants of the erstwhile functions natfloor
and natceiling.
›
lemma nat_floor_neg: "x \ 0 \ nat \x\ = 0"
for x :: real
by linarith
lemma le_nat_floor: "real x \ a \ x \ nat \a\"
by linarith
lemma le_mult_nat_floor: "nat \a\ * nat \b\ \ nat \a * b\"
by (cases "0 \ a \ 0 \ b")
(auto simp add: nat_mult_distrib[symmetric] nat_mono le_mult_floor)
lemma nat_ceiling_le_eq [simp]: "nat \x\ \ a \ x \ real a"
by linarith
lemma real_nat_ceiling_ge: "x \ real (nat \x\)"
by linarith
lemma Rats_no_top_le: "\q \ \. x \ q"
for x :: real
by (auto intro!: bexI[of _ "of_nat (nat \x\)"]) linarith
lemma Rats_no_bot_less: "\q \ \. q < x" for x :: real
by (auto intro!: bexI[of _ "of_int (\x\ - 1)"]) linarith
lemma floor_ceiling_diff_le: "0 \ r \ nat\real k - r\ \ k - nat\r\"
by linarith
lemma floor_ceiling_diff_le': "nat\r - real k\ \ nat\r\ - k"
by linarith
lemma ceiling_floor_diff_ge: "nat\r - real k\ \ nat\r\ - k"
by linarith
lemma ceiling_floor_diff_ge': "r \ k \ nat\r - real k\ \ k - nat\r\"
by linarith
subsection ‹Exponentiation with floor›
lemma floor_power:
assumes "x = of_int \x\"
shows "\x ^ n\ = \x\ ^ n"
proof -
have "x ^ n = of_int (\x\ ^ n)"
using assms by (induct n arbitrary: x) simp_all
then show ?thesis by (metis floor_of_int)
qed
lemma floor_numeral_power [simp]: "\numeral x ^ n\ = numeral x ^ n"
by (metis floor_of_int of_int_numeral of_int_power)
lemma ceiling_numeral_power [simp]: "\numeral x ^ n\ = numeral x ^ n"
by (metis ceiling_of_int of_int_numeral of_int_power)
subsection ‹Implementation of rational real numbers›
text ‹Formal constructor›
definition Ratreal :: "rat \ real"
where [code_abbrev, simp]: "Ratreal = real_of_rat"
code_datatype Ratreal
text ‹Quasi-Numerals›
lemma [code_abbrev]:
"real_of_rat (numeral k) = numeral k"
"real_of_rat (- numeral k) = - numeral k"
"real_of_rat (rat_of_int a) = real_of_int a"
by simp_all
lemma [code_post]:
"real_of_rat 0 = 0"
"real_of_rat 1 = 1"
"real_of_rat (- 1) = - 1"
"real_of_rat (1 / numeral k) = 1 / numeral k"
"real_of_rat (numeral k / numeral l) = numeral k / numeral l"
"real_of_rat (- (1 / numeral k)) = - (1 / numeral k)"
"real_of_rat (- (numeral k / numeral l)) = - (numeral k / numeral l)"
by (simp_all add: of_rat_divide of_rat_minus)
text ‹Operations›
lemma zero_real_code [code]: "0 = Ratreal 0"
by simp
lemma one_real_code [code]: "1 = Ratreal 1"
by simp
instantiation real :: equal
begin
definition "HOL.equal x y \ x - y = 0" for x :: real
instance by standard (simp add: equal_real_def)
lemma real_equal_code [code]: "HOL.equal (Ratreal x) (Ratreal y) \ HOL.equal x y"
by (simp add: equal_real_def equal)
lemma [code nbe]: "HOL.equal x x \ True"
for x :: real
by (rule equal_refl)
end
lemma real_less_eq_code [code]: "Ratreal x \ Ratreal y \ x \ y"
by (simp add: of_rat_less_eq)
lemma real_less_code [code]: "Ratreal x < Ratreal y \ x < y"
by (simp add: of_rat_less)
lemma real_plus_code [code]: "Ratreal x + Ratreal y = Ratreal (x + y)"
by (simp add: of_rat_add)
lemma real_times_code [code]: "Ratreal x * Ratreal y = Ratreal (x * y)"
by (simp add: of_rat_mult)
lemma real_uminus_code [code]: "- Ratreal x = Ratreal (- x)"
by (simp add: of_rat_minus)
lemma real_minus_code [code]: "Ratreal x - Ratreal y = Ratreal (x - y)"
by (simp add: of_rat_diff)
lemma real_inverse_code [code]: "inverse (Ratreal x) = Ratreal (inverse x)"
by (simp add: of_rat_inverse)
lemma real_divide_code [code]: "Ratreal x / Ratreal y = Ratreal (x / y)"
by (simp add: of_rat_divide)
lemma real_floor_code [code]: "\Ratreal x\ = \x\"
by (metis Ratreal_def floor_le_iff floor_unique le_floor_iff
of_int_floor_le of_rat_of_int_eq real_less_eq_code)
text ‹Quickcheck›
context
includes term_syntax
begin
definition
valterm_ratreal :: "rat \ (unit \ Code_Evaluation.term) \ real \ (unit \ Code_Evaluation.term)"
where [code_unfold]: "valterm_ratreal k = Code_Evaluation.valtermify Ratreal {\} k"
end
instantiation real :: random
begin
context
includes state_combinator_syntax
begin
definition
"Quickcheck_Random.random i = Quickcheck_Random.random i \\ (\r. Pair (valterm_ratreal r))"
instance ..
end
end
instantiation real :: exhaustive
begin
definition
"exhaustive_real f d = Quickcheck_Exhaustive.exhaustive (\r. f (Ratreal r)) d"
instance ..
end
instantiation real :: full_exhaustive
begin
definition
"full_exhaustive_real f d = Quickcheck_Exhaustive.full_exhaustive (\r. f (valterm_ratreal r)) d"
instance ..
end
instantiation real :: narrowing
begin
definition
"narrowing_real = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.cons Ratreal) narrowing"
instance ..
end
subsection ‹Setup for Nitpick›
declaration ‹
Nitpick_HOL.register_frac_type 🍋‹real›
[(🍋‹zero_real_inst.zero_real›, 🍋‹Nitpick.zero_frac›),
(🍋‹one_real_inst.one_real›, 🍋‹Nitpick.one_frac›),
(🍋‹plus_real_inst.plus_real›, 🍋‹Nitpick.plus_frac›),
(🍋‹times_real_inst.times_real›, 🍋‹Nitpick.times_frac›),
(🍋‹uminus_real_inst.uminus_real›, 🍋‹Nitpick.uminus_frac›),
--> --------------------
--> maximum size reached
--> --------------------
