(* Title : Lim.thy
ID : $Id$
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Conversion to Isar and new proofs by Lawrence C Paulson, 2004
*)
header{* Limits and Continuity *}
theory Lim
imports HSEQ
begin
text{*Standard and Nonstandard Definitions*}
definition
LIM :: "['a::real_normed_vector => 'b::real_normed_vector, 'a, 'b] => bool"
("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where
"f -- a --> L =
(\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x - a) < s
--> norm (f x - L) < r)"
definition
NSLIM :: "['a::real_normed_vector => 'b::real_normed_vector, 'a, 'b] => bool"
("((_)/ -- (_)/ --NS> (_))" [60, 0, 60] 60) where
"f -- a --NS> L =
(\<forall>x. (x \<noteq> star_of a & x @= star_of a --> ( *f* f) x @= star_of L))"
definition
isCont :: "['a::real_normed_vector => 'b::real_normed_vector, 'a] => bool" where
"isCont f a = (f -- a --> (f a))"
definition
isNSCont :: "['a::real_normed_vector => 'b::real_normed_vector, 'a] => bool" where
--{*NS definition dispenses with limit notions*}
"isNSCont f a = (\<forall>y. y @= star_of a -->
( *f* f) y @= star_of (f a))"
definition
isUCont :: "['a::real_normed_vector => 'b::real_normed_vector] => bool" where
"isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r)"
definition
isNSUCont :: "['a::real_normed_vector => 'b::real_normed_vector] => bool" where
"isNSUCont f = (\<forall>x y. x @= y --> ( *f* f) x @= ( *f* f) y)"
subsection {* Limits of Functions *}
subsubsection {* Purely standard proofs *}
lemma LIM_eq:
"f -- a --> L =
(\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
by (simp add: LIM_def diff_minus)
lemma LIM_I:
"(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
==> f -- a --> L"
by (simp add: LIM_eq)
lemma LIM_D:
"[| f -- a --> L; 0<r |]
==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
by (simp add: LIM_eq)
lemma LIM_offset: "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
apply (rule LIM_I)
apply (drule_tac r="r" in LIM_D, safe)
apply (rule_tac x="s" in exI, safe)
apply (drule_tac x="x + k" in spec)
apply (simp add: compare_rls)
done
lemma LIM_offset_zero: "f -- a --> L \<Longrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
by (drule_tac k="a" in LIM_offset, simp add: add_commute)
lemma LIM_offset_zero_cancel: "(\<lambda>h. f (a + h)) -- 0 --> L \<Longrightarrow> f -- a --> L"
by (drule_tac k="- a" in LIM_offset, simp)
lemma LIM_const [simp]: "(%x. k) -- x --> k"
by (simp add: LIM_def)
lemma LIM_add:
fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
assumes f: "f -- a --> L" and g: "g -- a --> M"
shows "(%x. f x + g(x)) -- a --> (L + M)"
proof (rule LIM_I)
fix r :: real
assume r: "0 < r"
from LIM_D [OF f half_gt_zero [OF r]]
obtain fs
where fs: "0 < fs"
and fs_lt: "\<forall>x. x \<noteq> a & norm (x-a) < fs --> norm (f x - L) < r/2"
by blast
from LIM_D [OF g half_gt_zero [OF r]]
obtain gs
where gs: "0 < gs"
and gs_lt: "\<forall>x. x \<noteq> a & norm (x-a) < gs --> norm (g x - M) < r/2"
by blast
show "\<exists>s>0.\<forall>x. x \<noteq> a \<and> norm (x-a) < s \<longrightarrow> norm (f x + g x - (L + M)) < r"
proof (intro exI conjI strip)
show "0 < min fs gs" by (simp add: fs gs)
fix x :: 'a
assume "x \<noteq> a \<and> norm (x-a) < min fs gs"
hence "x \<noteq> a \<and> norm (x-a) < fs \<and> norm (x-a) < gs" by simp
with fs_lt gs_lt
have "norm (f x - L) < r/2" and "norm (g x - M) < r/2" by blast+
hence "norm (f x - L) + norm (g x - M) < r" by arith
thus "norm (f x + g x - (L + M)) < r"
by (blast intro: norm_diff_triangle_ineq order_le_less_trans)
qed
qed
lemma LIM_add_zero:
"\<lbrakk>f -- a --> 0; g -- a --> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. f x + g x) -- a --> 0"
by (drule (1) LIM_add, simp)
lemma minus_diff_minus:
fixes a b :: "'a::ab_group_add"
shows "(- a) - (- b) = - (a - b)"
by simp
lemma LIM_minus: "f -- a --> L ==> (%x. -f(x)) -- a --> -L"
by (simp only: LIM_eq minus_diff_minus norm_minus_cancel)
lemma LIM_add_minus:
"[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + -g(x)) -- x --> (l + -m)"
by (intro LIM_add LIM_minus)
lemma LIM_diff:
"[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) - g(x)) -- x --> l-m"
by (simp only: diff_minus LIM_add LIM_minus)
lemma LIM_zero: "f -- a --> l \<Longrightarrow> (\<lambda>x. f x - l) -- a --> 0"
by (simp add: LIM_def)
lemma LIM_zero_cancel: "(\<lambda>x. f x - l) -- a --> 0 \<Longrightarrow> f -- a --> l"
by (simp add: LIM_def)
lemma LIM_zero_iff: "(\<lambda>x. f x - l) -- a --> 0 = f -- a --> l"
by (simp add: LIM_def)
lemma LIM_imp_LIM:
assumes f: "f -- a --> l"
assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
shows "g -- a --> m"
apply (rule LIM_I, drule LIM_D [OF f], safe)
apply (rule_tac x="s" in exI, safe)
apply (drule_tac x="x" in spec, safe)
apply (erule (1) order_le_less_trans [OF le])
done
lemma LIM_norm: "f -- a --> l \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> norm l"
by (erule LIM_imp_LIM, simp add: norm_triangle_ineq3)
lemma LIM_norm_zero: "f -- a --> 0 \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> 0"
by (drule LIM_norm, simp)
lemma LIM_norm_zero_cancel: "(\<lambda>x. norm (f x)) -- a --> 0 \<Longrightarrow> f -- a --> 0"
by (erule LIM_imp_LIM, simp)
lemma LIM_norm_zero_iff: "(\<lambda>x. norm (f x)) -- a --> 0 = f -- a --> 0"
by (rule iffI [OF LIM_norm_zero_cancel LIM_norm_zero])
lemma LIM_rabs: "f -- a --> (l::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> \<bar>l\<bar>"
by (fold real_norm_def, rule LIM_norm)
lemma LIM_rabs_zero: "f -- a --> (0::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> 0"
by (fold real_norm_def, rule LIM_norm_zero)
lemma LIM_rabs_zero_cancel: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) \<Longrightarrow> f -- a --> 0"
by (fold real_norm_def, rule LIM_norm_zero_cancel)
lemma LIM_rabs_zero_iff: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) = f -- a --> 0"
by (fold real_norm_def, rule LIM_norm_zero_iff)
lemma LIM_const_not_eq:
fixes a :: "'a::real_normed_div_algebra"
shows "k \<noteq> L ==> ~ ((%x. k) -- a --> L)"
apply (simp add: LIM_eq)
apply (rule_tac x="norm (k - L)" in exI, simp, safe)
apply (rule_tac x="a + of_real (s/2)" in exI, simp add: norm_of_real)
done
lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
lemma LIM_const_eq:
fixes a :: "'a::real_normed_div_algebra"
shows "(%x. k) -- a --> L ==> k = L"
apply (rule ccontr)
apply (blast dest: LIM_const_not_eq)
done
lemma LIM_unique:
fixes a :: "'a::real_normed_div_algebra"
shows "[| f -- a --> L; f -- a --> M |] ==> L = M"
apply (drule LIM_diff, assumption)
apply (auto dest!: LIM_const_eq)
done
lemma LIM_self: "(%x. x) -- a --> a"
by (auto simp add: LIM_def)
text{*Limits are equal for functions equal except at limit point*}
lemma LIM_equal:
"[| \<forall>x. x \<noteq> a --> (f x = g x) |] ==> (f -- a --> l) = (g -- a --> l)"
by (simp add: LIM_def)
lemma LIM_cong:
"\<lbrakk>a = b; \<And>x. x \<noteq> b \<Longrightarrow> f x = g x; l = m\<rbrakk>
\<Longrightarrow> ((\<lambda>x. f x) -- a --> l) = ((\<lambda>x. g x) -- b --> m)"
by (simp add: LIM_def)
lemma LIM_equal2:
assumes 1: "0 < R"
assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
shows "g -- a --> l \<Longrightarrow> f -- a --> l"
apply (unfold LIM_def, safe)
apply (drule_tac x="r" in spec, safe)
apply (rule_tac x="min s R" in exI, safe)
apply (simp add: 1)
apply (simp add: 2)
done
text{*Two uses in Hyperreal/Transcendental.ML*}
lemma LIM_trans:
"[| (%x. f(x) + -g(x)) -- a --> 0; g -- a --> l |] ==> f -- a --> l"
apply (drule LIM_add, assumption)
apply (auto simp add: add_assoc)
done
lemma LIM_compose:
assumes g: "g -- l --> g l"
assumes f: "f -- a --> l"
shows "(\<lambda>x. g (f x)) -- a --> g l"
proof (rule LIM_I)
fix r::real assume r: "0 < r"
obtain s where s: "0 < s"
and less_r: "\<And>y. \<lbrakk>y \<noteq> l; norm (y - l) < s\<rbrakk> \<Longrightarrow> norm (g y - g l) < r"
using LIM_D [OF g r] by fast
obtain t where t: "0 < t"
and less_s: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < t\<rbrakk> \<Longrightarrow> norm (f x - l) < s"
using LIM_D [OF f s] by fast
show "\<exists>t>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < t \<longrightarrow> norm (g (f x) - g l) < r"
proof (rule exI, safe)
show "0 < t" using t .
next
fix x assume "x \<noteq> a" and "norm (x - a) < t"
hence "norm (f x - l) < s" by (rule less_s)
thus "norm (g (f x) - g l) < r"
using r less_r by (case_tac "f x = l", simp_all)
qed
qed
lemma LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l"
unfolding o_def by (rule LIM_compose)
lemma real_LIM_sandwich_zero:
fixes f g :: "'a::real_normed_vector \<Rightarrow> real"
assumes f: "f -- a --> 0"
assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
shows "g -- a --> 0"
proof (rule LIM_imp_LIM [OF f])
fix x assume x: "x \<noteq> a"
have "norm (g x - 0) = g x" by (simp add: 1 x)
also have "g x \<le> f x" by (rule 2 [OF x])
also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
finally show "norm (g x - 0) \<le> norm (f x - 0)" .
qed
text {* Bounded Linear Operators *}
lemma (in bounded_linear) cont: "f -- a --> f a"
proof (rule LIM_I)
fix r::real assume r: "0 < r"
obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
using pos_bounded by fast
show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - f a) < r"
proof (rule exI, safe)
from r K show "0 < r / K" by (rule divide_pos_pos)
next
fix x assume x: "norm (x - a) < r / K"
have "norm (f x - f a) = norm (f (x - a))" by (simp only: diff)
also have "\<dots> \<le> norm (x - a) * K" by (rule norm_le)
also from K x have "\<dots> < r" by (simp only: pos_less_divide_eq)
finally show "norm (f x - f a) < r" .
qed
qed
lemma (in bounded_linear) LIM:
"g -- a --> l \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> f l"
by (rule LIM_compose [OF cont])
lemma (in bounded_linear) LIM_zero:
"g -- a --> 0 \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> 0"
by (drule LIM, simp only: zero)
text {* Bounded Bilinear Operators *}
lemma (in bounded_bilinear) LIM_prod_zero:
assumes f: "f -- a --> 0"
assumes g: "g -- a --> 0"
shows "(\<lambda>x. f x ** g x) -- a --> 0"
proof (rule LIM_I)
fix r::real assume r: "0 < r"
obtain K where K: "0 < K"
and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
using pos_bounded by fast
from K have K': "0 < inverse K"
by (rule positive_imp_inverse_positive)
obtain s where s: "0 < s"
and norm_f: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < s\<rbrakk> \<Longrightarrow> norm (f x) < r"
using LIM_D [OF f r] by auto
obtain t where t: "0 < t"
and norm_g: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < t\<rbrakk> \<Longrightarrow> norm (g x) < inverse K"
using LIM_D [OF g K'] by auto
show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x ** g x - 0) < r"
proof (rule exI, safe)
from s t show "0 < min s t" by simp
next
fix x assume x: "x \<noteq> a"
assume "norm (x - a) < min s t"
hence xs: "norm (x - a) < s" and xt: "norm (x - a) < t" by simp_all
from x xs have 1: "norm (f x) < r" by (rule norm_f)
from x xt have 2: "norm (g x) < inverse K" by (rule norm_g)
have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K" by (rule norm_le)
also from 1 2 K have "\<dots> < r * inverse K * K"
by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero)
also from K have "r * inverse K * K = r" by simp
finally show "norm (f x ** g x - 0) < r" by simp
qed
qed
lemma (in bounded_bilinear) LIM_left_zero:
"f -- a --> 0 \<Longrightarrow> (\<lambda>x. f x ** c) -- a --> 0"
by (rule bounded_linear.LIM_zero [OF bounded_linear_left])
lemma (in bounded_bilinear) LIM_right_zero:
"f -- a --> 0 \<Longrightarrow> (\<lambda>x. c ** f x) -- a --> 0"
by (rule bounded_linear.LIM_zero [OF bounded_linear_right])
lemma (in bounded_bilinear) LIM:
"\<lbrakk>f -- a --> L; g -- a --> M\<rbrakk> \<Longrightarrow> (\<lambda>x. f x ** g x) -- a --> L ** M"
apply (drule LIM_zero)
apply (drule LIM_zero)
apply (rule LIM_zero_cancel)
apply (subst prod_diff_prod)
apply (rule LIM_add_zero)
apply (rule LIM_add_zero)
apply (erule (1) LIM_prod_zero)
apply (erule LIM_left_zero)
apply (erule LIM_right_zero)
done
lemmas LIM_mult = bounded_bilinear_mult.LIM
lemmas LIM_mult_zero = bounded_bilinear_mult.LIM_prod_zero
lemmas LIM_mult_left_zero = bounded_bilinear_mult.LIM_left_zero
lemmas LIM_mult_right_zero = bounded_bilinear_mult.LIM_right_zero
lemmas LIM_scaleR = bounded_bilinear_scaleR.LIM
lemmas LIM_of_real = bounded_linear_of_real.LIM
lemma LIM_power:
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::{recpower,real_normed_algebra}"
assumes f: "f -- a --> l"
shows "(\<lambda>x. f x ^ n) -- a --> l ^ n"
by (induct n, simp, simp add: power_Suc LIM_mult f)
lemma LIM_inverse_lemma:
fixes x :: "'a::real_normed_div_algebra"
assumes r: "0 < r"
assumes x: "norm (x - 1) < min (1/2) (r/2)"
shows "norm (inverse x - 1) < r"
proof -
from r have r2: "0 < r/2" by simp
from x have 0: "x \<noteq> 0" by clarsimp
from x have x': "norm (1 - x) < min (1/2) (r/2)"
by (simp only: norm_minus_commute)
hence less1: "norm (1 - x) < r/2" by simp
have "norm (1::'a) - norm x \<le> norm (1 - x)" by (rule norm_triangle_ineq2)
also from x' have "norm (1 - x) < 1/2" by simp
finally have "1/2 < norm x" by simp
hence "inverse (norm x) < inverse (1/2)"
by (rule less_imp_inverse_less, simp)
hence less2: "norm (inverse x) < 2"
by (simp add: nonzero_norm_inverse 0)
from less1 less2 r2 norm_ge_zero
have "norm (1 - x) * norm (inverse x) < (r/2) * 2"
by (rule mult_strict_mono)
thus "norm (inverse x - 1) < r"
by (simp only: norm_mult [symmetric] left_diff_distrib, simp add: 0)
qed
lemma LIM_inverse_fun:
assumes a: "a \<noteq> (0::'a::real_normed_div_algebra)"
shows "inverse -- a --> inverse a"
proof (rule LIM_equal2)
from a show "0 < norm a" by simp
next
fix x assume "norm (x - a) < norm a"
hence "x \<noteq> 0" by auto
with a show "inverse x = inverse (inverse a * x) * inverse a"
by (simp add: nonzero_inverse_mult_distrib
nonzero_imp_inverse_nonzero
nonzero_inverse_inverse_eq mult_assoc)
next
have 1: "inverse -- 1 --> inverse (1::'a)"
apply (rule LIM_I)
apply (rule_tac x="min (1/2) (r/2)" in exI)
apply (simp add: LIM_inverse_lemma)
done
have "(\<lambda>x. inverse a * x) -- a --> inverse a * a"
by (intro LIM_mult LIM_self LIM_const)
hence "(\<lambda>x. inverse a * x) -- a --> 1"
by (simp add: a)
with 1 have "(\<lambda>x. inverse (inverse a * x)) -- a --> inverse 1"
by (rule LIM_compose)
hence "(\<lambda>x. inverse (inverse a * x)) -- a --> 1"
by simp
hence "(\<lambda>x. inverse (inverse a * x) * inverse a) -- a --> 1 * inverse a"
by (intro LIM_mult LIM_const)
thus "(\<lambda>x. inverse (inverse a * x) * inverse a) -- a --> inverse a"
by simp
qed
lemma LIM_inverse:
fixes L :: "'a::real_normed_div_algebra"
shows "\<lbrakk>f -- a --> L; L \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. inverse (f x)) -- a --> inverse L"
by (rule LIM_inverse_fun [THEN LIM_compose])
subsubsection {* Purely nonstandard proofs *}
lemma NSLIM_I:
"(\<And>x. \<lbrakk>x \<noteq> star_of a; x \<approx> star_of a\<rbrakk> \<Longrightarrow> starfun f x \<approx> star_of L)
\<Longrightarrow> f -- a --NS> L"
by (simp add: NSLIM_def)
lemma NSLIM_D:
"\<lbrakk>f -- a --NS> L; x \<noteq> star_of a; x \<approx> star_of a\<rbrakk>
\<Longrightarrow> starfun f x \<approx> star_of L"
by (simp add: NSLIM_def)
text{*Proving properties of limits using nonstandard definition.
The properties hold for standard limits as well!*}
lemma NSLIM_mult:
fixes l m :: "'a::real_normed_algebra"
shows "[| f -- x --NS> l; g -- x --NS> m |]
==> (%x. f(x) * g(x)) -- x --NS> (l * m)"
by (auto simp add: NSLIM_def intro!: approx_mult_HFinite)
lemma starfun_scaleR [simp]:
"starfun (\<lambda>x. f x *# g x) = (\<lambda>x. scaleHR (starfun f x) (starfun g x))"
by transfer (rule refl)
lemma NSLIM_scaleR:
"[| f -- x --NS> l; g -- x --NS> m |]
==> (%x. f(x) *# g(x)) -- x --NS> (l *# m)"
by (auto simp add: NSLIM_def intro!: approx_scaleR_HFinite)
lemma NSLIM_add:
"[| f -- x --NS> l; g -- x --NS> m |]
==> (%x. f(x) + g(x)) -- x --NS> (l + m)"
by (auto simp add: NSLIM_def intro!: approx_add)
lemma NSLIM_const [simp]: "(%x. k) -- x --NS> k"
by (simp add: NSLIM_def)
lemma NSLIM_minus: "f -- a --NS> L ==> (%x. -f(x)) -- a --NS> -L"
by (simp add: NSLIM_def)
lemma NSLIM_diff:
"\<lbrakk>f -- x --NS> l; g -- x --NS> m\<rbrakk> \<Longrightarrow> (\<lambda>x. f x - g x) -- x --NS> (l - m)"
by (simp only: diff_def NSLIM_add NSLIM_minus)
lemma NSLIM_add_minus: "[| f -- x --NS> l; g -- x --NS> m |] ==> (%x. f(x) + -g(x)) -- x --NS> (l + -m)"
by (simp only: NSLIM_add NSLIM_minus)
lemma NSLIM_inverse:
fixes L :: "'a::real_normed_div_algebra"
shows "[| f -- a --NS> L; L \<noteq> 0 |]
==> (%x. inverse(f(x))) -- a --NS> (inverse L)"
apply (simp add: NSLIM_def, clarify)
apply (drule spec)
apply (auto simp add: star_of_approx_inverse)
done
lemma NSLIM_zero:
assumes f: "f -- a --NS> l" shows "(%x. f(x) - l) -- a --NS> 0"
proof -
have "(\<lambda>x. f x - l) -- a --NS> l - l"
by (rule NSLIM_diff [OF f NSLIM_const])
thus ?thesis by simp
qed
lemma NSLIM_zero_cancel: "(%x. f(x) - l) -- x --NS> 0 ==> f -- x --NS> l"
apply (drule_tac g = "%x. l" and m = l in NSLIM_add)
apply (auto simp add: diff_minus add_assoc)
done
lemma NSLIM_const_not_eq:
fixes a :: real (* TODO: generalize to real_normed_div_algebra *)
shows "k \<noteq> L ==> ~ ((%x. k) -- a --NS> L)"
apply (simp add: NSLIM_def)
apply (rule_tac x="star_of a + epsilon" in exI)
apply (auto intro: Infinitesimal_add_approx_self [THEN approx_sym]
simp add: hypreal_epsilon_not_zero)
done
lemma NSLIM_not_zero:
fixes a :: real
shows "k \<noteq> 0 ==> ~ ((%x. k) -- a --NS> 0)"
by (rule NSLIM_const_not_eq)
lemma NSLIM_const_eq:
fixes a :: real
shows "(%x. k) -- a --NS> L ==> k = L"
apply (rule ccontr)
apply (blast dest: NSLIM_const_not_eq)
done
text{* can actually be proved more easily by unfolding the definition!*}
lemma NSLIM_unique:
fixes a :: real
shows "[| f -- a --NS> L; f -- a --NS> M |] ==> L = M"
apply (drule NSLIM_minus)
apply (drule NSLIM_add, assumption)
apply (auto dest!: NSLIM_const_eq [symmetric])
apply (simp add: diff_def [symmetric])
done
lemma NSLIM_mult_zero:
fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
shows "[| f -- x --NS> 0; g -- x --NS> 0 |] ==> (%x. f(x)*g(x)) -- x --NS> 0"
by (drule NSLIM_mult, auto)
lemma NSLIM_self: "(%x. x) -- a --NS> a"
by (simp add: NSLIM_def)
subsubsection {* Equivalence of @{term LIM} and @{term NSLIM} *}
lemma LIM_NSLIM:
assumes f: "f -- a --> L" shows "f -- a --NS> L"
proof (rule NSLIM_I)
fix x
assume neq: "x \<noteq> star_of a"
assume approx: "x \<approx> star_of a"
have "starfun f x - star_of L \<in> Infinitesimal"
proof (rule InfinitesimalI2)
fix r::real assume r: "0 < r"
from LIM_D [OF f r]
obtain s where s: "0 < s" and
less_r: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < s\<rbrakk> \<Longrightarrow> norm (f x - L) < r"
by fast
from less_r have less_r':
"\<And>x. \<lbrakk>x \<noteq> star_of a; hnorm (x - star_of a) < star_of s\<rbrakk>
\<Longrightarrow> hnorm (starfun f x - star_of L) < star_of r"
by transfer
from approx have "x - star_of a \<in> Infinitesimal"
by (unfold approx_def)
hence "hnorm (x - star_of a) < star_of s"
using s by (rule InfinitesimalD2)
with neq show "hnorm (starfun f x - star_of L) < star_of r"
by (rule less_r')
qed
thus "starfun f x \<approx> star_of L"
by (unfold approx_def)
qed
lemma NSLIM_LIM:
assumes f: "f -- a --NS> L" shows "f -- a --> L"
proof (rule LIM_I)
fix r::real assume r: "0 < r"
have "\<exists>s>0. \<forall>x. x \<noteq> star_of a \<and> hnorm (x - star_of a) < s
\<longrightarrow> hnorm (starfun f x - star_of L) < star_of r"
proof (rule exI, safe)
show "0 < epsilon" by (rule hypreal_epsilon_gt_zero)
next
fix x assume neq: "x \<noteq> star_of a"
assume "hnorm (x - star_of a) < epsilon"
with Infinitesimal_epsilon
have "x - star_of a \<in> Infinitesimal"
by (rule hnorm_less_Infinitesimal)
hence "x \<approx> star_of a"
by (unfold approx_def)
with f neq have "starfun f x \<approx> star_of L"
by (rule NSLIM_D)
hence "starfun f x - star_of L \<in> Infinitesimal"
by (unfold approx_def)
thus "hnorm (starfun f x - star_of L) < star_of r"
using r by (rule InfinitesimalD2)
qed
thus "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r"
by transfer
qed
theorem LIM_NSLIM_iff: "(f -- x --> L) = (f -- x --NS> L)"
by (blast intro: LIM_NSLIM NSLIM_LIM)
subsection {* Continuity *}
subsubsection {* Purely standard proofs *}
lemma LIM_isCont_iff: "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
lemma isCont_iff: "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
by (simp add: isCont_def LIM_isCont_iff)
lemma isCont_Id: "isCont (\<lambda>x. x) a"
unfolding isCont_def by (rule LIM_self)
lemma isCont_const [simp]: "isCont (\<lambda>x. k) a"
unfolding isCont_def by (rule LIM_const)
lemma isCont_norm: "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
unfolding isCont_def by (rule LIM_norm)
lemma isCont_rabs: "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x :: real\<bar>) a"
unfolding isCont_def by (rule LIM_rabs)
lemma isCont_add: "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
unfolding isCont_def by (rule LIM_add)
lemma isCont_minus: "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
unfolding isCont_def by (rule LIM_minus)
lemma isCont_diff: "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
unfolding isCont_def by (rule LIM_diff)
lemma isCont_mult:
fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
unfolding isCont_def by (rule LIM_mult)
lemma isCont_inverse:
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_div_algebra"
shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. inverse (f x)) a"
unfolding isCont_def by (rule LIM_inverse)
lemma isCont_LIM_compose:
"\<lbrakk>isCont g l; f -- a --> l\<rbrakk> \<Longrightarrow> (\<lambda>x. g (f x)) -- a --> g l"
unfolding isCont_def by (rule LIM_compose)
lemma isCont_o2: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
unfolding isCont_def by (rule LIM_compose)
lemma isCont_o: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (g o f) a"
unfolding o_def by (rule isCont_o2)
lemma (in bounded_linear) isCont: "isCont f a"
unfolding isCont_def by (rule cont)
lemma (in bounded_bilinear) isCont:
"\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
unfolding isCont_def by (rule LIM)
lemmas isCont_scaleR = bounded_bilinear_scaleR.isCont
lemma isCont_of_real:
"isCont f a \<Longrightarrow> isCont (\<lambda>x. of_real (f x)) a"
unfolding isCont_def by (rule LIM_of_real)
lemma isCont_power:
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::{recpower,real_normed_algebra}"
shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
unfolding isCont_def by (rule LIM_power)
subsubsection {* Nonstandard proofs *}
lemma isNSContD:
"\<lbrakk>isNSCont f a; y \<approx> star_of a\<rbrakk> \<Longrightarrow> ( *f* f) y \<approx> star_of (f a)"
by (simp add: isNSCont_def)
lemma isNSCont_NSLIM: "isNSCont f a ==> f -- a --NS> (f a) "
by (simp add: isNSCont_def NSLIM_def)
lemma NSLIM_isNSCont: "f -- a --NS> (f a) ==> isNSCont f a"
apply (simp add: isNSCont_def NSLIM_def, auto)
apply (case_tac "y = star_of a", auto)
done
text{*NS continuity can be defined using NS Limit in
similar fashion to standard def of continuity*}
lemma isNSCont_NSLIM_iff: "(isNSCont f a) = (f -- a --NS> (f a))"
by (blast intro: isNSCont_NSLIM NSLIM_isNSCont)
text{*Hence, NS continuity can be given
in terms of standard limit*}
lemma isNSCont_LIM_iff: "(isNSCont f a) = (f -- a --> (f a))"
by (simp add: LIM_NSLIM_iff isNSCont_NSLIM_iff)
text{*Moreover, it's trivial now that NS continuity
is equivalent to standard continuity*}
lemma isNSCont_isCont_iff: "(isNSCont f a) = (isCont f a)"
apply (simp add: isCont_def)
apply (rule isNSCont_LIM_iff)
done
text{*Standard continuity ==> NS continuity*}
lemma isCont_isNSCont: "isCont f a ==> isNSCont f a"
by (erule isNSCont_isCont_iff [THEN iffD2])
text{*NS continuity ==> Standard continuity*}
lemma isNSCont_isCont: "isNSCont f a ==> isCont f a"
by (erule isNSCont_isCont_iff [THEN iffD1])
text{*Alternative definition of continuity*}
(* Prove equivalence between NS limits - *)
(* seems easier than using standard def *)
lemma NSLIM_h_iff: "(f -- a --NS> L) = ((%h. f(a + h)) -- 0 --NS> L)"
apply (simp add: NSLIM_def, auto)
apply (drule_tac x = "star_of a + x" in spec)
apply (drule_tac [2] x = "- star_of a + x" in spec, safe, simp)
apply (erule mem_infmal_iff [THEN iffD2, THEN Infinitesimal_add_approx_self [THEN approx_sym]])
apply (erule_tac [3] approx_minus_iff2 [THEN iffD1])
prefer 2 apply (simp add: add_commute diff_def [symmetric])
apply (rule_tac x = x in star_cases)
apply (rule_tac [2] x = x in star_cases)
apply (auto simp add: starfun star_of_def star_n_minus star_n_add add_assoc approx_refl star_n_zero_num)
done
lemma NSLIM_isCont_iff: "(f -- a --NS> f a) = ((%h. f(a + h)) -- 0 --NS> f a)"
by (rule NSLIM_h_iff)
lemma isNSCont_minus: "isNSCont f a ==> isNSCont (%x. - f x) a"
by (simp add: isNSCont_def)
lemma isNSCont_inverse:
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_div_algebra"
shows "[| isNSCont f x; f x \<noteq> 0 |] ==> isNSCont (%x. inverse (f x)) x"
by (auto intro: isCont_inverse simp add: isNSCont_isCont_iff)
lemma isNSCont_const [simp]: "isNSCont (%x. k) a"
by (simp add: isNSCont_def)
lemma isNSCont_abs [simp]: "isNSCont abs (a::real)"
apply (simp add: isNSCont_def)
apply (auto intro: approx_hrabs simp add: starfun_rabs_hrabs)
done
lemma isCont_abs [simp]: "isCont abs (a::real)"
by (auto simp add: isNSCont_isCont_iff [symmetric])
(****************************************************************
(%* Leave as commented until I add topology theory or remove? *%)
(%*------------------------------------------------------------
Elementary topology proof for a characterisation of
continuity now: a function f is continuous if and only
if the inverse image, {x. f(x) \<in> A}, of any open set A
is always an open set
------------------------------------------------------------*%)
Goal "[| isNSopen A; \<forall>x. isNSCont f x |]
==> isNSopen {x. f x \<in> A}"
by (auto_tac (claset(),simpset() addsimps [isNSopen_iff1]));
by (dtac (mem_monad_approx RS approx_sym);
by (dres_inst_tac [("x","a")] spec 1);
by (dtac isNSContD 1 THEN assume_tac 1)
by (dtac bspec 1 THEN assume_tac 1)
by (dres_inst_tac [("x","( *f* f) x")] approx_mem_monad2 1);
by (blast_tac (claset() addIs [starfun_mem_starset]);
qed "isNSCont_isNSopen";
Goalw [isNSCont_def]
"\<forall>A. isNSopen A --> isNSopen {x. f x \<in> A} \
\ ==> isNSCont f x";
by (auto_tac (claset() addSIs [(mem_infmal_iff RS iffD1) RS
(approx_minus_iff RS iffD2)],simpset() addsimps
[Infinitesimal_def,SReal_iff]));
by (dres_inst_tac [("x","{z. abs(z + -f(x)) < ya}")] spec 1);
by (etac (isNSopen_open_interval RSN (2,impE));
by (auto_tac (claset(),simpset() addsimps [isNSopen_def,isNSnbhd_def]));
by (dres_inst_tac [("x","x")] spec 1);
by (auto_tac (claset() addDs [approx_sym RS approx_mem_monad],
simpset() addsimps [hypreal_of_real_zero RS sym,STAR_starfun_rabs_add_minus]));
qed "isNSopen_isNSCont";
Goal "(\<forall>x. isNSCont f x) = \
\ (\<forall>A. isNSopen A --> isNSopen {x. f(x) \<in> A})";
by (blast_tac (claset() addIs [isNSCont_isNSopen,
isNSopen_isNSCont]);
qed "isNSCont_isNSopen_iff";
(%*------- Standard version of same theorem --------*%)
Goal "(\<forall>x. isCont f x) = \
\ (\<forall>A. isopen A --> isopen {x. f(x) \<in> A})";
by (auto_tac (claset() addSIs [isNSCont_isNSopen_iff],
simpset() addsimps [isNSopen_isopen_iff RS sym,
isNSCont_isCont_iff RS sym]));
qed "isCont_isopen_iff";
*******************************************************************)
subsection {* Uniform Continuity *}
lemma isNSUContD: "[| isNSUCont f; x \<approx> y|] ==> ( *f* f) x \<approx> ( *f* f) y"
by (simp add: isNSUCont_def)
lemma isUCont_isCont: "isUCont f ==> isCont f x"
by (simp add: isUCont_def isCont_def LIM_def, meson)
lemma isUCont_isNSUCont:
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
assumes f: "isUCont f" shows "isNSUCont f"
proof (unfold isNSUCont_def, safe)
fix x y :: "'a star"
assume approx: "x \<approx> y"
have "starfun f x - starfun f y \<in> Infinitesimal"
proof (rule InfinitesimalI2)
fix r::real assume r: "0 < r"
with f obtain s where s: "0 < s" and
less_r: "\<And>x y. norm (x - y) < s \<Longrightarrow> norm (f x - f y) < r"
by (auto simp add: isUCont_def)
from less_r have less_r':
"\<And>x y. hnorm (x - y) < star_of s
\<Longrightarrow> hnorm (starfun f x - starfun f y) < star_of r"
by transfer
from approx have "x - y \<in> Infinitesimal"
by (unfold approx_def)
hence "hnorm (x - y) < star_of s"
using s by (rule InfinitesimalD2)
thus "hnorm (starfun f x - starfun f y) < star_of r"
by (rule less_r')
qed
thus "starfun f x \<approx> starfun f y"
by (unfold approx_def)
qed
lemma isNSUCont_isUCont:
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
assumes f: "isNSUCont f" shows "isUCont f"
proof (unfold isUCont_def, safe)
fix r::real assume r: "0 < r"
have "\<exists>s>0. \<forall>x y. hnorm (x - y) < s
\<longrightarrow> hnorm (starfun f x - starfun f y) < star_of r"
proof (rule exI, safe)
show "0 < epsilon" by (rule hypreal_epsilon_gt_zero)
next
fix x y :: "'a star"
assume "hnorm (x - y) < epsilon"
with Infinitesimal_epsilon
have "x - y \<in> Infinitesimal"
by (rule hnorm_less_Infinitesimal)
hence "x \<approx> y"
by (unfold approx_def)
with f have "starfun f x \<approx> starfun f y"
by (simp add: isNSUCont_def)
hence "starfun f x - starfun f y \<in> Infinitesimal"
by (unfold approx_def)
thus "hnorm (starfun f x - starfun f y) < star_of r"
using r by (rule InfinitesimalD2)
qed
thus "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
by transfer
qed
subsection {* Relation of LIM and LIMSEQ *}
lemma LIMSEQ_SEQ_conv1:
fixes a :: "'a::real_normed_vector"
assumes X: "X -- a --> L"
shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
proof (safe intro!: LIMSEQ_I)
fix S :: "nat \<Rightarrow> 'a"
fix r :: real
assume rgz: "0 < r"
assume as: "\<forall>n. S n \<noteq> a"
assume S: "S ----> a"
from LIM_D [OF X rgz] obtain s
where sgz: "0 < s"
and aux: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < s\<rbrakk> \<Longrightarrow> norm (X x - L) < r"
by fast
from LIMSEQ_D [OF S sgz]
obtain no where "\<forall>n\<ge>no. norm (S n - a) < s" by blast
hence "\<forall>n\<ge>no. norm (X (S n) - L) < r" by (simp add: aux as)
thus "\<exists>no. \<forall>n\<ge>no. norm (X (S n) - L) < r" ..
qed
lemma LIMSEQ_SEQ_conv2:
fixes a :: real
assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
shows "X -- a --> L"
proof (rule ccontr)
assume "\<not> (X -- a --> L)"
hence "\<not> (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x - a) < s --> norm (X x - L) < r)" by (unfold LIM_def)
hence "\<exists>r > 0. \<forall>s > 0. \<exists>x. \<not>(x \<noteq> a \<and> \<bar>x - a\<bar> < s --> norm (X x - L) < r)" by simp
hence "\<exists>r > 0. \<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> norm (X x - L) \<ge> r)" by (simp add: linorder_not_less)
then obtain r where rdef: "r > 0 \<and> (\<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> norm (X x - L) \<ge> r))" by auto
let ?F = "\<lambda>n::nat. SOME x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> norm (X x - L) \<ge> r"
have "\<And>n. \<exists>x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> norm (X x - L) \<ge> r"
using rdef by simp
hence F: "\<And>n. ?F n \<noteq> a \<and> \<bar>?F n - a\<bar> < inverse (real (Suc n)) \<and> norm (X (?F n) - L) \<ge> r"
by (rule someI_ex)
hence F1: "\<And>n. ?F n \<noteq> a"
and F2: "\<And>n. \<bar>?F n - a\<bar> < inverse (real (Suc n))"
and F3: "\<And>n. norm (X (?F n) - L) \<ge> r"
by fast+
have "?F ----> a"
proof (rule LIMSEQ_I, unfold real_norm_def)
fix e::real
assume "0 < e"
(* choose no such that inverse (real (Suc n)) < e *)
have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)
then obtain m where nodef: "inverse (real (Suc m)) < e" by auto
show "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e"
proof (intro exI allI impI)
fix n
assume mlen: "m \<le> n"
have "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
by (rule F2)
also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
by auto
also from nodef have
"inverse (real (Suc m)) < e" .
finally show "\<bar>?F n - a\<bar> < e" .
qed
qed
moreover have "\<forall>n. ?F n \<noteq> a"
by (rule allI) (rule F1)
moreover from prems have "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L" by simp
ultimately have "(\<lambda>n. X (?F n)) ----> L" by simp
moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"
proof -
{
fix no::nat
obtain n where "n = no + 1" by simp
then have nolen: "no \<le> n" by simp
(* We prove this by showing that for any m there is an n\<ge>m such that |X (?F n) - L| \<ge> r *)
have "norm (X (?F n) - L) \<ge> r"
by (rule F3)
with nolen have "\<exists>n. no \<le> n \<and> norm (X (?F n) - L) \<ge> r" by fast
}
then have "(\<forall>no. \<exists>n. no \<le> n \<and> norm (X (?F n) - L) \<ge> r)" by simp
with rdef have "\<exists>e>0. (\<forall>no. \<exists>n. no \<le> n \<and> norm (X (?F n) - L) \<ge> e)" by auto
thus ?thesis by (unfold LIMSEQ_def, auto simp add: linorder_not_less)
qed
ultimately show False by simp
qed
lemma LIMSEQ_SEQ_conv:
"(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::real) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
(X -- a --> L)"
proof
assume "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
show "X -- a --> L" by (rule LIMSEQ_SEQ_conv2)
next
assume "(X -- a --> L)"
show "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L" by (rule LIMSEQ_SEQ_conv1)
qed
end