(* Author: L C Paulson, University of CambridgeAuthorChaieb,java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 53 Author:ChaiebUniversityofCambridge Author:Robertjava.lang.StringIndexOutOfBoundsException: Range [35, 33) out of bounds for length 46 Author:BrianHuffman,PortlandStateUniversity
*)
section‹Elementary Normed Vector Spaces›
theory java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 14
"HOL-Library.FuncSet"
Elementary_Metric_Spaces Cartesian_Spacein e>0" and e: "∧x'. dist x' x < e
Connected begin subsection‹y ∈ diff_add_cancel dist_add_cancel2)
✐e>0. ∀ <\⟶ (∃x∈y∈T. z = x + y)"
oqed
fixes T :: "('b::real_normed_vector) set"
assumes "open S ∨ open T"
shows "open (∪open_d
using assms
assume S: "open S"
show ?thesis
proof (clarsimp simp: open_dist)
fix x y
assume "x ∈ S" "y ∈a "x ∈ T"
with S obtain e where "e > 0" and e: "∧ and e:: "\And>x'. dist x' y < e' n"
by (auto simp: open_dist)
then have "∧aut simp: open_dist)
by (metis ‹y ∈ then have "🚫
then show "∃z. dist z (x + y) < ex∈y∈
using ‹0 < e" "∀⟶ z ∈
qed
assume T: "open T"
?thess
proof (clarsimp simp: open_dist)
fix x y
assume "x ∈x y))"
with T obtain e where "e > 0" and e: "∧x'. dist x' y < e_f: "z= clR (1- k - )"
by (auto simp: open_dist)
then have "∧have "d"dist z y < r"
by (metis ‹x ∈ S› add_diff_cancel_left' add_diff_eq diff_diff_add dist_norm)
then show "∃e>0. ∀z. dist z (x + y) < eunfolding z_def k_def using ‹0 < r›
using ‹0 < e›z n
qed
image_orthogonal_transformation_ball:
fixes f :: "'a::euclidean_space → 'a"
assumes "orthogonal_t x y"
shows "f ` ball x r = ball (f x) r"
(intro equalityI subsetI)
fix y assume "y ∈ f ` ball x r"
with assms show "y \<using
by(force simp: add.commute)
fix y assume y: "y ∈ ball (f x) r"
then obtain z where z: "y = f z"
using assms orthogonal_transformation_surj by blast
with y assms show "y ∈ f ` ball x r"
by (auto simp: orthogonal_transformation_isometry)
image_orthogonal_transformation_cball:
fixes f :: "'a::euclidean_space → 'a"
assumes "orthogonal_transformation f"
shows "f ` cball x r = cball (f x) r"
(intro equalityI subsetI)
fix y assume "y ∈
with assms show "y ∈
by (auto simp: orthogonal_transformation_isometry)
fix y assume ubsection✐‹tag unimportant›
then obtain z wheemma closed_of_nat_image: "closed (of_nat ` A ::: 'a::eal_normed_algeb set)"
using assms orthogonal_transformation_surj by bl by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_nat)
with y assms show "y ∈ f ` cball x r"
by (auto simp: orthogonal_transformation_isometry)
‹Support›
(in monoid_add) support_on :: "'b set → ('b → 'a) →
esupport_on S f = {x∈S. f x ≠ 0}"
support_on_simps[sim
"support_on {} f = {}"
"support_on (insert x S) f =
(if f x = 0 then support_on S f else insert x (support_on S f))"
"support_on (S ∪ T) f = support_on S f ∪clo)
"support_on (S ∩clo[simp]: "close (\< :: 'a :: real_normed_algebra_1 set)"
"support_on (S - T) f = support_on S f - support_on T f"
"support_on (f def by (rule closed_of_in)
unfolding support_on_def by auto
support_on_cong:
"(∧x. x ∈
by (aut fixes A :: "'a :: real_normed_algebra_1 set"
support_on_if: "a ≠ 0 ==>ℤ
by (auto simp: support_on_def)
support_on_if_subset: "support_on A (λx. if P x then a else 0) ⊆ discrete_imp_c[OF zero_les] ballI impI, goal_case)
y (auo simp:: support_on_def)
finite_support[intro]: "finite S ==>"x \<nt" and "y ∈ℤ" by auto unfoldingsupport_on_defbyauto
lemmaimage_affinity_interval: fixesc::"'a::ordered_real_vector" shows"((\<lambda>x.m*\<^sub>Rx+c)`{a..b})= (if{a..b}={}then{} elseif0\<le>mthen{m*\<^sub>Ra+c..m*\<^sub>Rb+c} else{m*\<^sub>Rb+c..m*\<^sub>Ra+c})" (is"?lhs=?rhs") proof(cases"m=0") caseTrue thenshow?thesis byforce next caseFalse show?thesis proof show"?lhs\<subseteq>?rhs" by(autosimp:scaleR_left_monoscaleR_left_mono_neg) show"?rhs\<subseteq>?lhs" proof(clarsimp,introconjIimpIsubsetI) show"\<lbrakk>0\<le>m; a \<le> b; x \<in> {m *\<^sub>R a + c..m *\<^sub>R b + c}\<rbrakk> \<Longrightarrow>x\<in>(\<lambda>x.m*\<^sub>Rx+c)`{a..b}"forx usingFalse by(rule_tacx="inversem*\<^sub>R(x-c)"inimage_eqI) (auto with \<open>dist y <1\<close>show"y=x" show"\<lbrakk>\<not>0\<le>m; a \<le> b; x \<in> {m *\<^sub>R b + c..m *\<^sub>R a + c}\<rbrakk> <
> 0\B<>x x<e> B \longrightarrow f)l\le
(auto simp add: neg_le_divideR_eq neg_divideR_le_eq le_diff_eq java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 49
dense java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
qed
qed
subsection \<fixes : ": l : ':metric_spacejava.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
lemma islimpt_ball:
fixes y: ':r,}java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
ws " ballxe <> 0<e <and <>cball x ejava.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
(is "?lhs \<longleftrightarrow> ?rhs")
proof
show ?rhs if ?lhs
proof
{
assume "e \<le> 0"
then have *: "ball x e = {}"
using ball_eq_empty[of x e] by auto
haveFalse \<open>?lhs\<lose>
unfolding * using shows "(f \<longlongrightarrow> \> (<x x )<java.lang.StringIndexOutOfBoundsException: Range [108, 107) out of bounds for length 116
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
then show "> "by(not_less
show "y \<in> cball x e"
e]islimpt_subsetofy"all x e "ballx e"]
ball_subset_cball[of x e] \<open>?lhs\<close>
unfolding closed_limpt by auto
qed
show ?lhs if ?rhs
proof -
from that have "e > 0" by auto
{
fix d :: real
d"
( java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 66
proof (cases "d \< fixes : ' \<Rightarrow>b::eal_normed_vector
True
then show ?thesis
proof (cases "x = y") case" \lambda>n.norm (f )\le ( n))net"
then have Falseand ( < 0net
using \<open>d \<le> dist x y\<close> \<open>d>0\<close> by auto
java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 48
java.lang.StringIndexOutOfBoundsException: Range [19, 20) out of bounds for length 19
f(<java.lang.StringIndexOutOfBoundsException: Range [39, 37) out of bounds for length 98
java.lang.StringIndexOutOfBoundsException: Range [0, 14) out of bounds for length 7
norm ( )\< x "
unfolding simp add eventually_mono OF]mult_left_mono)
((\<>.norm ( )*B <> 0)F
also ?hesis
( [symmetric)( add )
unfolding scaleR_minus_left java.lang.StringIndexOutOfBoundsException: Range [0, 50) out of bounds for length 3
by (utosimp norm_minus_commute
also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
unfoldingabs_mult_posof n (x -y),OFnorm_ge_zero[of" y]]
unfolding distrib_right using \<open>x\<noteq>y\<close> by auto
alsohave"<><>-d/"<d<> dist y<>and<d0< \open?hs\closejava.lang.StringIndexOutOfBoundsException: Index 128 out of bounds for length 128
by (auto simp: dist_norm)
\^Fin. norm( x java.lang.StringIndexOutOfBoundsException: Index 88 out of bounds for length 88
by auto
moreover
have "(d / ( by simpadd tendsto_norm_zero
using <>x<>\<close[ dist_nz <>>\close>unfoldingscaleR_eq_0_iff
by (auto simp: dist_commute)
moreover
have then ?hesis
using \<open>0 ( tendsto_norm_zero_iff] ( add )
ultimately show ?thesis
by (rule_tac x = "ylemma lim_null_scaleR_bounded
qed
next case False
have d dist x auto
show "\<exists>x' \<in> ball java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
proof (cases "x = y") case True
obtain z where z: "z \<noteq> y""dist z y < min e d"
using perfect_choose_dist[of "min e d" y]
sing \open> > 0\<close> \<open>e0<close> by auto
show ?hesis
by (metis True z dist_commute mem_ball min_less_iff_conj)
next case False
have \<arfx<ar ( )\le\bar xbar> * B"
using \<open>d>0\<close> \<open>d > dist x y s addjava.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 37
qed
by s add )
}
show?
unfolding mem_cball islimpt_approachable mem_ball by r )
qed
lemma :
fixes x y : * java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 41
shows "y islimpt ball x (dist x y)"
roof (java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
java.lang.StringIndexOutOfBoundsException: Range [7, 8) out of bounds for length 7
assume "y \<in> T""open T"
then obtain r where "0 < r""\<forall>z. dist z y < r \<longrightarrow> z \<in> T"
unfolding open_dist by fast
\<comment>\<open>choose point between @{term x} and @{term y}, within distance @{term r} of @{term y}.\<close>
define k where "k = min 1 (r / (2 * dist x y))"
define z where "z = y + scaleR k (x - y)"
have z_def2: "z = x + scaleR (1 - k) (y - x)"
unfolding z_def by (simp add: algebra_simps)
have "dist z y < r"
unfolding z_def k_def using \<open>0 < r\<close>
by (simp add: dist_norm min_def)
then have "z \<in> T"
using \<open>\<forall>z. dist z y < r \<longrightarrow> z \<in> T\<close> by simp
z "
using \<open>0 < using assms (fast intro tendsto_letendsto_intros)
java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 0
by simp
assumes \>net
unfolding z_def "f\java.lang.StringIndexOutOfBoundsException: Range [34, 32) out of bounds for length 41
:java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 26
\exists\<ball distx.z <>T\and n "
using \<open>z \<in> ball x (dist x y)\<close> \<open>z \<in> T\<close> \<open>z \<noteq> y\<close>
by fast
qed
<Ballsand Normed Spaces\<lose
lemma mem_ball_0 [simp]: "x \<lemma Lim_bilinear
x :"':real_normed_vectorjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
by simp
lemma mem_cball_0 [simp]: "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
for x :: "' shows "((\lambdax.h ( x) g x) \longlongrightarrow>( lm)net"
by simp
lemma closure_ball [simp]:
fixes x :: " \open>h<>\>f\ )\ o(\java.lang.StringIndexOutOfBoundsException: Range [121, 119) out of bounds for length 135
assumes "<"
shows "closure (ball x e) = cball x e"
proof
show "closure (ball x e) \<subseteq> cball x e"
using closed_cball closure_minimal by blast And.x e<> <java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 94
by (metis
then subsection<marker\open unimportant\close><LimitPoint of Filter<java.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 87
by force
qed
lemma mem_sphere_0 [simp]: "x \<in> java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 29
forx: "a:real_normed_vector"
by simp
(* In a trivial vector space, this fails for e = 0. *)
lemma interior_cball [simp] have "<>d 0<d\Longrightarrow>\exists>.x\<> a <> ( -a)<djava.lang.StringIndexOutOfBoundsException: Index 90 out of bounds for length 90
fixes x :: "'a::{real_normed_vector, perfect_space}"
shows "interior (cball x e) = ball x e"
proof (cases "e \<ge> 0") case False note cs = this
from cs have null: "ball x e = {}"
using ball_empty[of e x] by auto
moreover
have "cball x e = {}"
proof (rule equals0I)
fix y "y \<n> cball x e"
then show False
by (metis ball_eq_empty null cs dist_eq_0_iff dist_le_zero_iff empty_subsetI mem_cball
subset_antisym subset_ball)
qed
then have (Lim_ident_at[fa)
using interior_empty by auto
ultimately show ?thesis by blast
next case note cs = this
have "ball x e \<subseteq> cball x e"
using ball_subset_cball by auto
moreover
{
fix S y
assume as: "S \<subseteq> cball x e""open S""y\<in>S"
obtaind ">0"andd \forall>' dist x'y<d \<longrightarrow> x' \<in> S"
unfolding open_dist by blast
then obtain xa where xa_y: "xa assumes continuous_on (closure s)f"
using perfect_choose_dist [of d] by auto
have "xa <in>Sjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
dspecwhere x =]]
using xa by (auto simp: dist_commute)
then have xa_cball: "xa \<in> cball x e"
using as(1) by auto
then have "y \<in> ball x e"
proof (cases "x (2)unfolded ]
True
then have "e > 0" using cs order.order_iff_strict xa_cball xa_y by fastforce
then show " \in> xejava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
using \java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
next case False
have "dist (y + (d / 2 by (meson less_imp_le not_le order_trans zero_less_one)
unfolding dist_norm
using \<open>d>0\<close> norm_ge_zero[of " " +( 2 y )*<^> ( <> cball x e"
using d as(1)[unfolded subset_eq] by blast
by (metis metis less_imp_lelinordered_field_no_ub)
hence **:"java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
nfolding zero_less_norm_iff[ using\openjava.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
have
:
by (auto simp: java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 21
(y - x) *<subRy- x)java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
by (auto simp: algebra_simps)
also have "\<dots> = \<bar>1
* auto
also have "\<dots> = (dist y x) + fixes f :: "nat \<Rightarrow> 'a::real_
* asimp java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
finallyhave " \ge>dist +/"
using *[unfolded mem_cball] by (auto simp: dist_commute)
then show "y \<in> ball x e"
unfolding mem_ball using \<open>d>0\
qed
}
then have "\<forall>S \ "ounded"
by auto
ultimately show ?thesis
shows(`S"
using open_ball[of x e]
by auto
qed
lemma frontier_ball [simp]:
fixes a :: "'a::real_normed_vector"
<> (all )= java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
by (force simp: frontier_def)
lemma frontier_cball [simp]:
fixes a :: "'a::{real_normed_vector, perfect_space}"
shows "frontier (cball a e) = sphere a e"
java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 32
fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
shows "compact (sphere a r)"
[ "balla r" by simp
corollary bounded_sphere [simp]:
fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
shows "lemma bounded_scaling:
by (simp add: compact_imp_bounded)
corollary by (simp add: bounded_linear_i)
fixes a :: "
shows :
by (simp add:compact_imp_closed)
lemma [simp:
fixes a :: "'a::real_normed_vector"
shows"+ b ` ball a r = ball (a+b) r"
proof -
{ fix x :: 'a
assume "dist (a + b) x < r"
moreover
have "b + (x - by (auto simp: image_image)
by simp
\<>()b a r"
by (metis add.fixes S :: "'a::real_normed_vector set"
then show ?thesis
by (autosimp:add.java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
java.lang.StringIndexOutOfBoundsException: Range [9, 10) out of bounds for length 9
lemma image_add_cball [simp]:
fixes a :: "'a::real_normed_vector"
shows "(+) b ` java.lang.StringIndexOutOfBoundsException: Range [0, 22) out of bounds for length 19
proof -
have "\<And>x. dist (a + b) x \<le> r \<Longrightarrow> \<exists>y\<in>cball a r. x = b + y"
()add. diff_add_cancel dist_add_cancel2 mem_cballjava.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
then show ?thesis
by (force simp: add.commute)
qed
lemma closed_of_nat_image: "closed (of_nat ` A :: 'a::real_normed_algebra_1 set)"
by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_nat)
lemmaint_image: "closed (of_int ` A :: 'a::real_normed_algebra_1 set)"
by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_int)
lemma closed_Nats [simp]: "closed (\<nat> :: 'a :: real_normed_algebra_1 setusing [ S"-"] simp
unfolding Nats_def by (rule closed_of_nat_image)
lemma closed_Ints [simp]: "closed (\<int> :: 'java.lang.StringIndexOutOfBoundsException: Range [0, 47) out of bounds for length 41
unfolding Ints_def by (rule closed_of_int_image)
lemma closed_subset_Ints:
fixes A :: "'a :: real_normed_algebra_1 set"
java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
shows "closed A"
proof (intro discrete_imp_closed[OF fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" case (1 x y)
have x\in <> " <> <"
with \<open>dist of f`"java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
elim! Ints_cases : dist_of_int)
qed
subsection \<open>Filters\<close>
definition indirection :: "'ajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
where " indirection =at a within {.\<exists>c\<ge>0. b-a =scaleR c v}"
subsection \<open>Trivial Limits\<close>
: "\<not> trivial_limit "ounded( )
proof -
obtain x::'a where "x \<noteq> 0"
perfect_choose_dist zero_less_one)
then have "b \<le> norm ((roof -
by simp
then ?
unfolding "A>.<>\ l ""Ax \S<java.lang.StringIndexOutOfBoundsException: Range [103, 101) out of bounds for length 122
java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 12
qedby (intro )
lemma at_within_ball_bot_iff:
fixes x y :: "'a::{real_normed_vector,perfect_space}"
unfolding trivial_limit_within
((
subsection \<open>Limits\<close>
proposition Lim_at_infinity: "(f \<longlongrightarrow> l "((lambdax) +y S\>T"
by (auto simp: tendsto_iff eventually_at_infinity)
corollary Lim_at_infinityI [intro?]:
assumes "\<And by (auto simp: split_def split: if_split_asm)
shows "(f \<longlongrightarrow> l) at_infinity"
proof -
have "\<And>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
show ?hesis
using Lim_at_infinity by blast
qed
lemma Lim_transform_within_set_eq:
fixes a :: "'a::metric_space"and l :: "'b::java.lang.StringIndexOutOfBoundsException: Range [0, 58) out of bounds for length 9
shows "eventually (\ :"a:java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 40
\<Longrightarrow> ((f "java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 60
by (force intro: Lim_transform_within_set elim: eventually_mono)
lemma Lim_null:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
shows (f\ ) \ (< ()-l ljava.lang.StringIndexOutOfBoundsException: Range [109, 107) out of bounds for length 116
by (simp add:
lemma Lim_null_comparison:
fixes f :: "'a \<Rightarrow> 'b::java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 37
assumes "eventually (\<lambda bounded\\< . <y\in>T{ -}"
shows "(f \<longlongrightarrow> 0) net
singassms(
proof (rule metric_tendsto_imp_tendsto"<not>bounded (NIV :'::real_normed_vector, "
show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"
using assms(java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 37
qed
lemma Lim_transform_bound:
fixes f :: andjava.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 34
assumes "using bounded_Un -] java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 41 and"(g \<longlongrightarrow> 0) net"
shows "(f \<longlongrightarrow> 0) net"
using assmsl java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 27
by (rule s :java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 66
lemma lim_null_mult_right_bounded:
java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 59
java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 98
shows "((\<lambda>z. f java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
-
have "((\<lambda>x. norm (f x) * norm (g x)) \<longlongrightarrow> 0) F"
Lim_null_comparison)
show "\<forall>\<^sub>F shows "summable \lambdan.of_realnormjava.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 61
java.lang.StringIndexOutOfBoundsException: Range [58, 57) out of bounds for length 58
show"(<x java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 67
by (simp add: f tendsto_mult_left_zerojava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 40
qed
then show ?thesis
by (java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 70
qed
lemma :
fixes f :: "'a \<Rightarrow> 'b::java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 39
assumesqed
shows "((\<lambda>z
proof -
) * (fx)\ 0"
proof (rule Lim_null_comparison)
show "\<forall>\<^sub>F x in F. java.lang.StringIndexOutOfBoundsException: Range [0, 40) out of bounds for length 29
by (simp add: eventually_mono [OF g] mult_right_mono)
show "((\<lambda>x. B * norm (f x))osed)
by (simp add: f lemma not_compact_space_euclideanreal [simp]: "\<not> compact_space euclideanreal"
qed
then (simpadd
by (subst java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 0
qed
lemma :
assumes f: " fixes ::"':{heine_borel,real_normed_vector} set"
shows ((<lambdan \^Rg n < )net"
proof
fix \<epsilon>::real
assume "0 < \<epsilon>"
then have B: "0 < \< obtains F where "\<And>n compact(F n)""<ndn F \subseteq> S""<>n.F \<subseteq> FSuc n)java.lang.StringIndexOutOfBoundsException: Index 106 out of bounds for length 106
* \bar> \bar ( x <>" "barfx<bar> * (\<bar>B\<bar> + 1) < \<epsilon>" and g: "norm (g x) \<le> B" for x
proof -
have "\<bar>f x\<bar> * of_nat n)"forn
by (simp add: mult_left_mono g)
also have "\<dots> \<le> \<bar>f x\<bar> * (\<bar>B\<bar> + 1)"
by (simp add: mult_left_monojava.lang.StringIndexOutOfBoundsException: Range [4, 5) out of bounds for length 4
also have "\<dots> < \<epsilon>"
by (rule f)
finally show ?thesis .
qed
have "\<And>x. \<java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 19
by (simp add: "*" pos_less_divide_eq)
thenshow "<>\^ub> x in net. dist (f *<>R x)0 < \<>"
using \<open>0 < \<epsilon>\<close> by (auto intro: eventually_mono [OF eventually_conj [OF tendstoD [OF f B] java.lang.StringIndexOutOfBoundsException: Index 115 out of bounds for length 66
qeded
lemma Lim_norm_ubound:
fixes f : "a \<Rightarrow b:real_normed_vector
assumes "\<not>(trivial_limit net)""(f \qed auto
shows "norm(l) \<le \open>ntersecting chains sets the property\<close>
using assms by (fast intro: tendsto_le tendsto_intros)
lemmaLim_norm_lbound:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 36
nd( <java.lang.StringIndexOutOfBoundsException: Range [35, 32) out of bounds for length 41 and"eventually (\<lambda>x. e \<le> java.lang.StringIndexOutOfBoundsException: Range [7, 5) out of bounds for length 7
shows "e \<le> norm l"
using assms by (fast intro: tendsto_le tendsto_intros)
text\<open>Limit under bilinear function\<close>
lemma Lim_bilinear:
assumes "(f \<longlongrightarrow> l) net" and"(g \<longlongrightarrow> m) net" and"bounded_bilinear java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 8
shows"\x fx)(x) < ( lm)net
using \<open>bounded_bilinear h\<close> \<open>(f \<longlongrightarrow consider"B\<java.lang.StringIndexOutOfBoundsException: Range [35, 33) out of bounds for length 57
by (rule bounded_bilinear.tendsto)
lemma Lim_at_zero:
fixes a :: "'a::real_normed_vector" and l :: "'b::topological_space"
showsarrow ) (at a)\<ongleftrightarrow>((<lambda>x. f(a + x)) \<longlongrightarrow> l) (at 0)f(a+ ) \<longlongrightarrow> ) (at0)
using LIM_offset_zero LIM_offset_zero_cancel ..
subsection\<^marker>\<open>tag unimportant\<close> \<open>Limit Point of Filter\<close>
lemma netlimit_at_vector:
fixes a :: "'a::real_normed_vector"
hows"etlimit(at a) a"
proof (cases "\<exists>x. x \<noteq> a") case True then obtain x where x: "x \ have False if "<x x\in> <>\existsY\in\G>.\ "
have "\<And>d. 0have"\x \in \java.lang.StringIndexOutOfBoundsException: Range [55, 53) out of bounds for length 92 2) (( -) in exI ( add:norm_sgn java.lang.StringIndexOutOfBoundsException: Index 97 out of bounds for length 97
henhave "not>trivial_limit a ajava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
by (auto simp: trivial_limit_def eventually_at dist_norm)
then show obtain x where " <java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 58
by (rule Lim_ident_at [ ultimately show ?
qed simp
subsection \<open>Boundednessjava.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 29
lemma continuous_on_closure_norm_le:
fixes f ::thenshow?thesisblast
assumes "continuous_on (closure s) f"
:
x\in(java.lang.StringIndexOutOfBoundsException: Range [28, 25) out of bounds for length 29
shows "norm (f x) \<le> b"
proof-
have *: "f ` s \<subseteq> cball 0 java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 37
java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 59
show ?thesis
by (meson "*
qed
lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x \<le> b)"
java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 24
by (meson less_imp_le not_le order_transtrans zero_less_one)
lemma bounded_pos_less: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x < b)"
java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 73
lemma bounded_normE:
assumes thesis
using closure_subset by auto
by (meson assms bounded_pos)
lemma bounded_normE_less:
assumes "bounded A"
obtains B where "B > 0""\<java.lang.StringIndexOutOfBoundsException: Range [0, 32) out of bounds for length 12
by (meson assms bounded_pos_less)
lemma Bseq_eq_bounded:
fixes f :: "nat \<Rightarrow>x< ""0 java.lang.StringIndexOutOfBoundsException: Range [30, 31) out of bounds for length 30
shows"Bseq\<java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 56
unfolding Bseq_def bounded_pos by auto
lemma bounded_linear_image:
assumes "bounded S"
java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 26
b ( )
proof - "b>0 b \forall>\<>S.norm xjava.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
byauto 2)java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 83
using bounded_linear.pos_bounded by (auto simp: ac_simps)
show ?thesis
ng
proof (intro exI whereT:" "" T\<java.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 53
show "norm (f x) \<le> B * b"if"x \<in> S" for x
mono java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 64
qed (use \< using \open T<>byblast
java.lang.StringIndexOutOfBoundsException: Range [21, 3) out of bounds for length 3
java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 22
( add:\<>closed\<closure_minimal)
shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
by (simp add: bounded_linear_image bounded_linear_scaleR_right)
..\ "
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
assumes "bounded (f ` S)"
shows "bounded ((\<lambda>x. r *\<^sub>R f x) ` using closure_subset d blast
using bounded_scaling[of "f ` S" r] assms
by (auto simp: image_image)
lemma bounded_translation:
fixes S :: "'a::real_normed_vector set"
assumes "bounded S"
shows "bounded ((\<lambda>x. a + x) ` S)"
proof -
from assms obtain b where b: "b > 0""\<forall>x\<in>S. norm x \<le> b"
unfolding bounded_pos by auto
{
fix x
assume "x \<in show S\inter d/)\subseteq>Ujava.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
then have "norm (a +qed
using norm_triangle_ineq[of a x] b by auto
qed
then show ?thesis
unfoldinglet\Phi \<lambdan.openintop_of_set ) < X\noteq>}\andjava.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
usingnorm_ge_zero[of a] b(1) andadd_strict_increasing[ b0""
by (auto intro!: exI[of _ "b + norm a"])
qed
lemma bounded_translation_minus:
: 'a: setjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. x - a) ` S)"
using bounded_translation [of S "-a"] by simp
lemma finally ( <> xe2)\subseteq x ".
fixes X :: "'a::real_normed_vector set"
shows "bounded (uminus ` X) \<longleftrightarrow> bounded X"
by (auto simp: bounded_def dist_norm; rule_tac x="-x" in exI; force simp: add.commute norm_minus_commute)
lemma uminus_bounded_comp [simp]:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
shows "bounded ((\ using \open0<\close>\<>S\c>by java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
using[ f`"java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
by (auto simp: image_image)
emmabounded_plus_compjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
fg::"'a \<Rightarrow> 'b::real_normed_vector"
assumes "bounded (f ` S)"
assumes "bounded (g ` S)"
shows "bounded ((\<lambda>x. f x + g x) ` S)"
proof -
{
fix B C
assume "\<And>x. x\<in>S \<Longrightarrow> norm (f x) \<le> B""\<And>x. x\< proof ( [ dependent_nat_choice])
then"<x.x\in <>norm( )\le "
by (auto intro!: norm_triangle_le add_mono)
thesis
show \Y \Phi( n <>Y<>X "?Phi>nX" for X n
qed
lemma bounded_plus:
fixes S ::"'a::real_normed_vector set"
assumes "bounded S""bounded T"
shows "bounded ((\<lambda>(x,y). x + yjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
using bounded_plus_comp [of rulecompact_nestjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
by (auto metis subball [OF\open \close]java.lang.StringIndexOutOfBoundsException: Index 136 out of bounds for length 136
lemma by( opeF\openclosed S\<close> closure_eq_empty closure_minimal neopenin_imp_subsetjava.lang.StringIndexOutOfBoundsException: Index 113 out of bounds for length 113
bounded( )\>bounded( )<>(\<lambdax )` S)"
for f g::"'a \<Rightarrow> 'b::real_normed_vector"
usingof f \lambda>x ]
by auto
lemma bounded_sumsjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
fixes S :: "'a::real_normed_vector set"
assumes "bounded S"and"bounded T"
shows "bounded (\<Union>x\<in> S. \<Union>y \<in> java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
using assms by (simp add: bounded_iff) (meson norm_triangle_mono)
lemma
fixes S :: "'a::real_normed_vector set"
assumes "bounded S"and"bounded T"
shows "bounded (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
using assms by (simp add: bounded_iff) (meson add_mono norm_triangle_le_diff)
lemma not_bounded_UNIV[simp]: "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
proof (auto simp: bounded_pos not_le)
obtain x :: 'a where "x \<noteq> 0"
using perfect_choose_dist [OF zero_less_one] by fast
fix b :: real
assume b: "b >0"
have b1: "b +1 \<ge> 0"
using b by simp
with \<open>x \<noteq> 0\<close> have "b < norm (scaleR (b + 1) (sgn x))"
by (simp add: norm_sgn)
then show "\<exists>x::'a. b < norm x" ..
qed
corollary cobounded_imp_unbounded:
fixes S :: "'a::{real_normed_vector, perfect_space} set"
shows "bounded (- S) \<Longrightarrow> \<not> bounded S"
using bounded_Un [of S "-S"] by (simp)
subsection\<^marker>\<open>tag unimportant\<close>\<open>Relations among convergence and absolute convergence for power series\<close>
lemma summable_imp_bounded:
fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
shows "summable f \<Longrightarrow> bounded (range f)"
by (frule summable_LIMSEQ_zero) (simp add: convergent_imp_bounded)
lemma summable_imp_sums_bounded: "summable f \<Longrightarrow> bounded (range (\<lambda>n. sum f {..<n}))"
by (auto simp: summable_def sums_def dest: convergent_imp_bounded)
lemma power_series_conv_imp_absconv_weak:
fixes a:: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}"and w :: 'a
assumes sum: "summable (\<lambda>n. a n * z ^ n)"and no: "norm w < norm z"
shows "summable (\<lambda>n. of_real(norm(a n)) * w ^ n)"
proof -
obtain M where M: "\<And>x. norm (a x * z ^ x) \<le> M"
using summable_imp_bounded [OF sum] by (force simp: bounded_iff)
show ?thesis
proof (rule series_comparison_complex)
have "\<And>n. norm (a n) * norm z ^ n \<le> M"
by (metis (no_types) M norm_mult norm_power)
then show "summable (\<lambda>n. complex_of_real (norm (a n) * norm w ^ n))"
java.lang.StringIndexOutOfBoundsException: Range [0, 11) out of bounds for length 0
qed( :norm_mult
qed
subsection \<open>Normed spaces with " s "
[:
fixes s :: "'a::{real_normed_vector,perfect_space
shows "\<not> compact (UNIV::'a set)"
by (simp add: java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 0
lemma not_compact_space_euclideanreal [ assumes "uniformly_continuous_on s f"
by (simp add: compact_space_def)
>Representingsets as unionofa .<>
lemma:
fixes S :: "'a::{heine_borel,real_normed_vector} set" " "
obtains fix :'java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21 "(\<Union>nusing [ bc java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 63
proof
show "compact (S \<inter> cball 0 (of_nat n))" } le =this
using assms compact_eq_bounded_closed by auto
next
show "(\<Union>n. S \<inter> cball 0 (real fix java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
auto java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
next
fix: ' "
assume "compact K""K \<subseteq> S(x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) \<longlonglongrightarrow> 0"
Nwhere" < Njava.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
by( bounded_poscompact_imp_boundedsubsetI)
then java.lang.StringIndexOutOfBoundsException: Range [0, 11) out of bounds for length 3
by ( using assms unfolding
qed auto
subsection \<open>Intersecting chains of compact sets and the Baire property\<close>
proposition bounded_closed_chain:
fixes \<F> :: "'a::heine_borel set set"
assumes "B \<in> \<F>""bounded B" ::':real_normed_vector <>':java.lang.StringIndexOutOfBoundsException: Range [75, 74) out of bounds for length 75 and chain: "\<And>S T.using [Fbounded_linear_mult_left java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
shows "\<lemma uniformly_cojava.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 59
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
have" \inter>\Inter>\>\noteq }java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
proofrule java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
show compactB \And.T<>\F <Longrightarrow T
by (simp_all add: assms compact_eq_bounded_closed)
show "\<lbrakk>lemma java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 54
proof induction\>rule java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46 caseempty
with assms show ?case by force
next case (insert U \<G>)
then have "U \<in> \<Fby( java.lang.StringIndexOutOfBoundsException: Range [71, 70) out of bounds for length 71
> U" U\>Bjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
<>B\in>\F\close byblast
then show ?case
proof cases case1
shows"s(<lambdax.c \^> x)java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
using Int_left_commute ne by auto
next case2
have "U \<noteq> {}"
using \<open>U \<in> \<F>\<close> \<open>{} \<notin> \<F>\<close> by blast
moreover
have False if"\<And>x. x \<in> U \<Longrightarrow> \<exists>Y\<in>\<G>. x \<notin> Y"
proof -
have "\<And>x. x \<in> U \<Longrightarrow> \<exists>Y\<in>\<G>. Y \<subseteq>unfoldingdist_norm norm_minus_cancel ..
by (metis chain contra_subsetD insert.prems insert_subset that)
then obtain Y where "Y \<in> \<G>""Y \<subseteq> U"
by (java.lang.StringIndexOutOfBoundsException: Range [0, 23) out of bounds for length 0
moreoverobtainx where"x <in>\<>\G"
by (metis Int_emptyI ne)
ultimately show ?thesis
by (metis Inf_lower subset_eq that)
qed
with 2 show ?thesis
qed
qed
qed
then show ?thesis by blast
qed
corollary compact_chain:
fixesf :"a:\Rightarrow b:java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
assumes "\<And>S. S \<in> \<F> and " "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \< shows uniformly_continuous_on \lambdax )java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
\Inter <>\>{"
proof (cases "\<F> = {}") case True
then show ?thesis by auto
next
False
show ?thesis
:"a:metric_space\Rightarrow b:java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
qed
lemma compact_nestshows \lambdax )java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
fixes F :: "'a::linorder \<Rightarrow> 'b::heine_borel set"
assumes F: "\<And>n. compact(Fjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
shows "\<Inter>(range F) \<noteq> {}"
proof -
have *: "\<And>S T. S \<in> range F \<and> T \<in> range F \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
by (metis mono image_iff le_cases)
show ?thesis
using F by (intro compact_chain [OF _ _ *]; blast dest: *)
qed
text\<open>The Baire property of dense sets\<close>
theorem Baire:
fixes S::"'a::{real_normed_vector,heine_borel} set"
assumes "closed S""countable \<G>" and ope: "\<And>T. T \<in> \<G> \<Longrightarrow> openin (top_of_set S) T \<and> S \<subseteq> closure T"
shows "S \<subseteq> closure(\<Inter>\<G>)"
proof (cases "\<G> = {}") case True
then show ?thesis
using closure_subset by auto
next let ?g = "from_nat_into \<G>" case False
then have gin: "?g n \<in> \<G>" for n
by (simp add: from_nat_into)
show ?thesis
proof (clarsimp simp: closure_approachable)
fix x and e::real
assume "x \<in> S""0 < e"
obtain TF where opeF: "\<And>n. openin (top_of_set S) (TF n)" and ne: "\<And>n. TF n \<noteq> {}" and subg: "\<And>n. S \<inter> closure(TF n) \<subseteq> ?g n" and subball: "\<And>n. closure(TF n) \<subseteq> ball x e" and decr: "\<And>n. TF(Suc n) \<subseteq> TF n"
proof -
have *: "\<exists>Y. (openin (top_of_set S) Y \<and> Y \<noteq> {} \<and>
S \<inter> closure Y \<subseteq> ?g n \<and> closure Y \<subseteq> ball x e) \<and> Y \<subseteq> U" if opeU: "openin (top_of_set S) U"and"U \<noteq> {}"and cloU: "closure U \<subseteq> ball x e" for U n
proof -
obtain T where T: "open T""U = T \<inter> S"
using \<open>openin (top_of_set S) U\<close> by (auto simp: openin_subtopology)
with \<open>U \<noteq> {}\<close> have "T \<inter> closure (?g n) \<noteq> {}"
using gin ope by fastforce
then have "T \<inter> ?g n \<noteq> {}"
using \<open>open T\<close> open_Int_closure_eq_empty by blast
then obtain y where "y \<in> U""y \<in> ?g n"
using T ope [of "?g n", OF gin] by (blast dest: openin_imp_subset)
moreover have "openin (top_of_set S) (U \<inter> ?g n)"
using gin ope opeU by blast
ultimately obtain djava.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 121
:)
show ?thesis
proof (intro exI conjI)
show "openin (op_of_set S (S\<inter> ball y(d/2)java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
simp java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 42
show "S \<inter> ball y (d/2) \<noteq> {}"
<> \close> \<open>y \<in> U\<close> opeU openin_imp_subset by fastforce
have "S \<inter> closure (S \<inter> ball y (d/2)) \<subseteq> S \<inter> closure (ball y (d/2))"
using closure_mono by blast
also have "... \<subseteq> ? byforce
using \<open>d > 0\<close> d by force
finally show "S \<inter> closure (S \<inter> ball y (d/2)) \<subseteq> ?g n" .
have "closure ( ultimately show ?thesis
proof -
have "closure (ball y (d/2)) \<subseteq> ball y d"
using \<open>d > 0\<close> by auto
then have "closure (S \<inter> ball y (d/2)) \<subseteq> ball y d"
shows "open (<>x ) )
then show ?thesis
simpadd:<> \close>
qed
also have "... \<subseteq> java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
using cloU closure_subset d by java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 18
finally show "closure (S \<inter> ballusing open_translation[OF [ assms,of ]
show "S \<inter> ball y (d/2 ( : java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
lemma open_affinity:
qed
qed let ?\<Phi> = "\< fixesS:: "'::real_normed_vector set"
S \<inter> closure X \<subseteq> ?g n \<and> closure X \<subseteq> ball x e"
have"closure ( \ (2)\subseteq>closure(x e2)
by (simp add: closure_mono)
also have "... \<subseteq> ball x java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
using \<open>e > 0\<close> java.lang.StringIndexOutOfBoundsException: Range [0, 37) out of bounds for length 22
finally have "closure (S \<by auto
moreover have"openin (top_of_set S) (S \<inter> ball x (e/2))""S \<inter> ball x (e/2) \<noteq> {}"
using \< open_translationof"*<sub>)c`S ]
java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
using * [of "" ()a`S + S)" :'a: setjava.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 85
show thesis
rule OFjava.lang.StringIndexOutOfBoundsException: Range [48, 46) out of bounds for length 48
show "\<exists>x. have ball( )e <Sjava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
by( java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
show "\<exists>Y. ?\<Phi> (Suc n) Y \<and> Y \<subseteq> X"if"?\<Phi> n X" for X n
using that by (blast intro: *)
qed (use that in metis)
qed
fixx
proof (rule compact_nest) "<>.compact( \inter>closure (TF n))"
by (metis closed_closure subball bounded_subset_ballI compact_eq_bounded_closed closed_Int_compact [OF \<open>closed S\<close>])
show "\<And>n. S \<inter> closure (TF n) \<noteq> {}"
by (metis Int_absorb1 opeF \<open>closed S\<close> closure_eq_empty closure_minimal ne openin_imp_subset) "\<And> .m <> <>S\inter T n) <>S\inter closure(TFm)java.lang.StringIndexOutOfBoundsException: Index 115 out of bounds for length 115
by (meson closure_mono decr dual_order.refl inf_mono lift_Suc_antimono_le)
qed
moreover have "(\<Inter>n. S \<inter> closure (TF n)) \<subseteq> {y \<in> \<Inter>\<G>assume z< x e"
proof (clarsimp, intro conjI)
java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
assume "y \<in>by java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
\>\in\G <>T
by (metis Int_iff from_nat_into_surj [OF \<open>countable \<G>\ }
show "dist y x < e"
by (metis y dist_commute mem_ball subball subsetCE)
qed
ultimately show "\<exists>y \<in> \<Inter>\<G>. dist y x < e"
by auto
qed
qed
subsection \<open>Continuity\<close>
interior(<>.x-a )=(<>.x- a) ` S :"a:java.lang.StringIndexOutOfBoundsException: Range [107, 106) out of bounds for length 111
lemma (inbounded_linear)]:
fixes g :: "_::metric_space \<Rightarrow> _"
assumes "uniformly_continuous_on s g"
shows "uniformly_continuous_on s (\<lambda>x. f (g x))"
using assms unfolding uniformly_continuous_on_sequentially
unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]
by (auto intro: tendsto_zero)
uniformly_continuous_on_dist[continuous_intros]:
fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space" " sfjava.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39 and"uniformly_continuous_on s g"
shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"
proof -
{
fix a b c d :: 'b
have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b compact(\lambda>x -x `s"
using dist_triangle2 [of a b c] dist_triangle2 [of b c d]
using dist_triangle3 [of c d a] dist_triangle [of a d b]
by arith
}note java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
{
fix x y
assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) \<longlonglongrightarrow> 0"
assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) \<longlonglongrightarrow> 0"
have " and"compact t"
by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],
simpadd le)
}
then show ?thesis
using assms unfolding uniformly_continuous_on_sequentially
bysimp
qed
uniformly_continuous_on_cmul_right []:
fixes f :: "'a::real_normed_vector continuous_on( <>)\lambda>z. fst z + snd z)"
shows "uniformly_continuous_on s f \<Longrightarrow java.lang.StringIndexOutOfBoundsException: Range [80, 78) out of bounds for length 103
using uniformly_continuous_on[Fbounded_linear_mult_left java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
lemma uniformly_continuous_on_cmul_left[continuous_intros]:
fixes f * compact_continuous_image compact_Times[ assms]byauto
assumes "uniformly_continuous_on s f"
shows "uniformly_continuous_on s (\<lambda>x. c * f x)"
by (metis assms bounded_linear
lemma compact_differences:
fixes f :: "'a :: metric_space \<Rightarrow> 'b :: real_normed_vector"
assumes "uniformly_continuous_on s f"
shows assumes compact java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
unfolding norm_conv_dist using assms
by( uniformly_continuous_on_dist )
uniformly_continuous_on_cmul[]:
fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
assumes "uniformly_continuous_on s f"
shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"
using bounded_linear_scaleR_right assms
rmly_continuous_on)
then show?thesis
fixes x y :: "'a::real_normed_vector"
shows "dist (- x) (- y) = dist x y"
using [ assms()compact_negations[ 2)]byauto
lemma uniformly_continuous_on_minus[continuous_intros]:
fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
shows "uniformly_continuous_on s f \<Longrightarrow> java.lang.StringIndexOutOfBoundsException: Range [0, 78) out of bounds for length 0
unfolding uniformly_continuous_on_def dist_minus .
lemma uniformly_continuous_on_add[continuous_intros]:
fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
assumes "uniformly_continuous_on s f" and"uniformly_continuous_on s g"
shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
usingassms
unfolding uniformly_continuous_on_sequentially
unfolding dist_norm tendsto_norm_zero_iff add_diff_add
by (auto intro: tendsto_add_zero)
lemma uniformly_continuous_on_diff[]java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
assumes "uniformly_continuous_on s f" and"uniformly_continuous_on s g"
shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"
using assms uniformly_continuous_on_add [of OFassms] by java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
by (simp add: java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
lemma fixes S ::': setjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::real_normed_vector"
shows assumes " " compactTjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
by (induction I rule: infinite_finite_induct)
(auto simp><>.<y<>.{ )={-y| . <>S\and> <>T"
subsection\<^marker java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
lemma open_scaling[intro]:
fixes S :: "'a::real_normed_vector set"
assumes "c \<noteq> 0" and"open S"
shows "open((\<lambda>x. c *\<^sub>R x) ` S)"
proof -
{
fix x
assume "x \<in> S"
thenobtain \<epsilon>where"<>" and \<epsilon>: "\<java.lang.StringIndexOutOfBoundsException: Range [0, 31) out of bounds for length 19
by auto
have "\<epsilon> * \<bar>c\<bar> > 0"
using assms(1)[unfolded zero_less_abs_iff[symmetric]] \<open>\<epsilon>>0\<close> by auto
moreover
{
fix y "dist y( *\<^sub>R x)<\epsilon \barc<bar>java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
then have "norm (c *\<^sub>R ((1 / c) *\<^sub>R y - x)) < \<epsilon :"a:setjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
by (shows compact(\lambdax.a+c*<sub> "
then have "norm ((1 / c) *\<^sub>R y - x) < \<epsilon>"
:<penc\>\close)
java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
using rev_image_eqI[of compact_translation[ compact_scaling[ ,ofac auto
byjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
}
ultimately have "\<exists>e>0. \<forall>x'. dist x' (c
by (rule_tac x="\<epsilon> * \<bar>c\<bar>" in exI assumesclosed S"
}
then show ?thesis unfolding open_dist by auto
qed
lemma open_times_image:
fixes S::"'a::real_normed_field set"
assumes "c\<noteq>0""open S"
shows" (* ) )
proof - let ?f = "\<lambda>x. x/c"and ?g="((*) c)"
have "continuous_on UNIV ?f" using \<open>c\<noteq>0\<close> by (auto intro:continuous_intros)
then have "open (?f -` S)" java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 73
moreover have "?g ` S = ?f -` S" using \<open>c\<noteq>0\<close>
using image_iff by fastforce
ultimately show ?thesis by auto
qed
lemma minus_image_eq_vimage:
fixes A :: "'a::ab_group_add set"
shows "(\<lambda>x. - x) ` A = (\<lambda>x. - x) -` A"
( !image_eqI where=\lambda.-x]java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
lemma open_negations:
fixes S :: "'a::real_normed_vector set"
shows "open S \<Longrightarrow> open ((\<lambda>x. - x) ` S)"
using open_scaling [of "- 1" S] by simp
lemma open_translation:
fixes S :: "'a::real_normed_vector set"
assumes "open S"
shows "open((\<lambda>x. a + x) ` java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 3
proof -
{
fix x
have "continuous ( from as(1) obtain f (1)obtainf where : \forall>.xn=fstf (n""<n fst(fn \ \forall>.snd(n < java.lang.StringIndexOutOfBoundsException: Index 140 out of bounds for length 140
by (intro continuous_diff continuous_ident continuous_const)
}
moreover have "{x. x - a \<in> S} = (+) a ` S"
by force
ultimately show ?thesis
(metis continuous_open_vimagevimage_def)
qed
lemma :
fixes S :: "'a::real_normed_vector set"
assumes "open S"
shows "open ((\<lambda>x. x - a) ` S)"
using assms open_translation [of S "- a"] by (simp cong: image_cong_simp)
lemma open_neg_translation:
fixes S :: "'a::real_normed_vector set"
assumes "open S"
shows "open((\<lambda>x. a - THEN where x=l-l"]
using open_translation[OF open_negations[OF assms], of a]
autosimp:image_image)
lemma open_affinity:
fixes S :: "'a::real_normed_vector set"
assumes "open S""c \<noteq> 0"
shows }
proof -
)\circ>(<>.c\^Rx"
unfolding o_def ..
have "(+) a ` (*\<^sub>R) show thesis
by auto
then show ?thesis
using assmsopen_translation[ (\^sub>R "a]
unfolding *
by auto
qed
lemma interior_translation:
assumes closedS" "java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 32
proof (rule set_eqI, rule)
fix x
java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 39
then obtain e where "e > 0"and thenshow?
mem_interior auto
then have "ball (x - a) e \<subseteq> S"
unfolding subset_eq Ball_def mem_balljava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
by (auto simp: diff_diff_eq)
then show"x <in> ()a` S"
unfolding image_iff
by (metis \<open> (<>\ .<> \in T x+y) \Unionx<>. <y\in x-}"
next
fix x
sume" <>()a S"
thenjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
unfolding image_iff Bex_def mem_interior by auto
{
fix z
have *: "a + y - z = y + a - z" by auto
z\in>ballx "
then have " java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
using e[unfolded subset_eq, THEN bspec[where x="z - a"]]
java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 25
by auto
then have "z \<in> (+) a ` S"
unfolding image_iff by (auto intro!: bexI[where x="z - a"])
}
then have using java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 67
unfolding subset_eq by auto " <>interior)java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
mem_interior \opene>0close> by auto
qed
lemma interior_translation_subtract:
interior(<> a )=(<>. ) " :"a:java.lang.StringIndexOutOfBoundsException: Range [107, 106) out of bounds for length 111
using interior_translation [of "- a"] by (simp cong: image_cong_simp)
lemma compact_scaling:
fixes s :: "'a::real_normed_vector set"
assumes "compact s"
shows "compact ((\<lambda> simpadd java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 55
proof let ?f = "\<lambda>x. scaleR c x"
have *: "bounded_linear java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 72
show ?thesis
compact_continuous_imageof f] [sf
using linear_continuous_at[OF *] assms
by auto
qed
lemma compact_negations:
fixess:"a:
assumes "compact s"
shows "compact ((\<lambda>x. - x) ` s)"
using compact_scaling [OF assms, of java.lang.StringIndexOutOfBoundsException: Range [38, 25) out of bounds for length 25
compact_sumsjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
fixes s t :: "'a::real_normed_vector set"
assumes "compact s"
compact"
+y|xy.x <in> s \> y <>t"
proof -
havelemma java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 24
( :intro:[x=x-" ])
have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"
by( )(introjava.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 66
then show ?thesis
unfolding * using compact_continuous_image compact_Times [OF assms] " a )r +a r" for a :: "'n::real_normed_vector"
qed
lemma compact_differences:
fixes s t :: "'a::real_normed_vector set" "compact " and"compact t"
shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"
proof-
>s \<> y <>t x+y|x y.x\<n s \and <>(uminus t}"
using java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0
then show ?thesis
using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by " (\lambda>.c \^>R x) ` S)"
qed
lemma compact_sums':
fixes S :: "'a::real_normed_vector apply (rule_tac x="\<lambda>x. c *\<^sR x"inexI
assumes "compact S"and" apply (rule_tac x="\<lambda>x(/c *<subRx exIjava.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
shows "compact (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
proof -
have "(\<Union>x\<in>S. \<Union>y\<in>T. {x + y}) = {x + y S ::"a:set"
by blast
then show ?thesis
using [ ]by
qed
lemma compact_differences':
fixes S :: "'a::real_normed_vector set"
assumes "compact S"and"compact T" " \Union><> .<>\in x-y}"
proof -
have "(\<Union>x\<in>S. \<Union>y\<in>T. {x - y}) = {x - y |x y. x \<in> S \<and> y \<in> T}"
by blast
then show ?thesis
using compact_differences [OF assms] by simp
qed
lemma compact_translation:
shows "S (\<>x.a+c*<^ub>R x `S"
proof -
have "{x + y |x y. x \<in> s \<and> y \<proof-
by
then show ?thesis
act_sumsOF [ ] java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
qed
lemma
compact(\>.x- ) compacts :"':java.lang.StringIndexOutOfBoundsException: Index 90 out of bounds for length 90
using that compact_translation [of s "- a"] by (simp cong: image_cong_simp)
lemma compact_affinity:
fixes s :: "'a::real_normed_vector set"
assumes "compact s"
shows "compact ((\<lambda>x. a + c *\<^sub>R x) ` s)"
proof -
have"(`(*\^>)c` \lambda>. \^ubR )`"
by auto
then show ?thesis
using compact_translationshow cthjava.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 42
qed
lemma closed_scaling:
fixes S :: "'a::real_normed_vector set"
assumes "closed S"
shows "closed ((\<lambda>x. c *\<^sub>Rjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
roofcases" "java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21 casethen ?java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 39
next case False
assms closed(\lambda.inverse \^Rx `)java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
by (simp add: continuous_closed_vimage)
also have "(\<lambda>x. inverse c *\<^sub>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
using <>\ \>by auto:[]java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
finally show ?thesis .
qed
lemma closed_negations:
fixes S :: "'a::real_normed_vector set"
assumes "closed S"
shows "closed ((\<lambda>x java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
using closed_scaling[OF assms, of "- 1"] by simp
lemma compact_closed_sums:
fixes :"':java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 41
assumes "compact S"and"closed T"
shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
proof - let ?S = "{x + y |x y. x \<in> S \<and> y \< have 0 "<z z\noteq "
{
fix x l
as \n n\in S ( <longlongrightarrow )
from as(1) obtain f where f: " thenshow"continuous_on (\lambdaz \<subR(+ )"
using choice[of "\<lambda>n y. x n = (fst y) + (snd y) by autointro! java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 39
obtain ' rwhere l'<>"andr: r lr ((\<>. fn)<>)\java.lang.StringIndexOutOfBoundsException: Range [123, 122) out of bounds for length 141
using assms(1)[unfolded compact_def, THEN auto java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 37
java.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 82
OFas2 ]]and1
unfolding o_def
by auto
then have "l - l' \<in> T"
using assms(2)[unfolded closed_sequential_limits,
THEN spec[where x="\<lambda> n. snd (f (r n))"],
THEN spec[where x="l - l'"]]
using f(3)
by auto
then have "l \<in> ?S"
using \<open>l' \<in> S\<close> by force
}
moreover have "?S = (\<Union>x\<in> S. \<Union>y \<in> T. {x + yjava.lang.StringIndexOutOfBoundsException: Range [26, 9) out of bounds for length 26
by force
ultimately show ?
metis(,liftingjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
qed
lemma closed_compact_sums:
fixes T: ':real_normed_vector set"
assumes "closed S""compact T"
shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
proof -
have "(\< byblast
by auto
then show ?thesis
using compact_closed_sums[OF assms(2,1)] by using simp
qed
:
show thesis "compactS closed "
shows "closed (\<Union>x\<in> S. \<Union>yjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
proof -
have "(\<Union>x\<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by subsection\^\open> <lose <> Isometry up )<
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
by (assms compact_closed_sums
qed
lemma closed_compact_differences:
fixes S T :: "'a::real_normed_vector set"
assumes "closed S""compact T"
shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 7
have "(\<Union>x\<in> java.lang.StringIndexOutOfBoundsException: Range [39, 40) out of bounds for length 39
by auto
then show ?thesis
using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
qed
lemma closed_translation: "closed ((+) a `olded Cauchy_defo_def, spec[ "*d] java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
have "(\<Union>x\<in> {a}. \<Union>y \<in> xn )"
then show ?thesis
using compact_closed_sums [OF compact_sing [of a] that] by auto
qed
lemma closed_translation_subtract: "closed ((\<lambda>x. x - also have "norm (f (x nxN) e*d"
using that closed_translation [of S "using \<> <>n\<>N .[ java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
lemma closure_translation: "closure ((+) a ` s) = (+) a ` closure s" forqed
show thesis
have *: "(+) a ` (- s) = - (+) a ` s"
by (auto intro!: image_eqI [where x = "x - a" for x])
show ?thesis
using interior_translation [of a "- s", symmetric]
by (simp add: closure_interior translation_Compl :"<orall>\in>.normf )\ge>e x)
qed
lemma closure_translation_subtract:- "closurehave"<>\inf`s g <longlongrightarrow> l) sequentially"
using closure_translation [of "- a" s] by (simp cong: image_cong_simp)
lemma lemmafas:\<oralln:.gn\in f `s cfg:" g java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69 "frontier ((+) a ` s) = (+) a ` frontier s" for a :: "'a::real_normed_vector"
by (auto simp add: frontier_def translation_diff interior_translation closure_translation)
lemma frontier_translation_subtract:
frontier "fora: ':real_normed_vectorjava.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
by (auto simp add: frontier_def translation_diff interior_translation closure_translation)
lemma sphere_translation: "sphere (a + c) r = (+) a ` sphere c r" for a :: "'n::real_normed_vector"
by (auto simp: dist_norm algebra_simps intro!: image_eqI [where x = "x - a" for x])
lemma : "sphere (c - a) r = (\<lambda>x. x - athen thesis
using sphere_translation [of "- a" c] by (simp cong: image_cong_simp)
lemma cball_translation: "cball (a + c) r = (+) a ` cball c r" for a :: "'n::real_normed_vector"
by (auto simp: dist_norm algebra_simps intro!: image_eqI [where x = "x - a" for x])
lemma cball_translation_subtract: "cball (c - a) r = (\<lambda>x. x - a) ` cball c r" for a :: "'n::real_normed_vector"
o -a ] simpcong java.lang.StringIndexOutOfBoundsException: Range [70, 69) out of bounds for length 70
lemma ball_translation: "ball (a + c) r = (+) a ` ball c r" for a :: "'n::real_normed_vector"
to :dist_normalgebra_simpsintro:image_eqI where "forx
lemma ball_translation_subtract: "ball (c - a) r = (\<lambda>x. x - a) `java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
usingball_translation[ "-a ]by simpcong )
subsection\<^ assumes conf: "co :"java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 35
lemma homeomorphic_scaling:
fixes S :: "'a::real_normed_vector set"
c<>0
shows "S homeomorphic ((\<lambda>x. c *\<^sub>R x) ` S)"
unfolding homeomorphic_minimal
( x"<x c*<sub> " exI apply (rule_tac x="\<lambda>x. (1 / c) *\<^sub>R x" then obtaine">" :"<>.\lbrakk> <>S; f y \<noteq> f x\<rbrakk> \<Longrightarrow> e \<le> norm (f y - f x)"
using assms by (auto simp: continuous_intros)
lemma homeomorphic_translation:
fixes S :: defineFwhere"F\equiv> f )f )java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
shows "S homeomorphic ((\<lambda>x. a + x) ` S)"
unfolding homeomorphic_minimal apply (rule_tac x="\< C < cball ( )(/2" apply (rule_tac x="\<lambda>x. -a + x" in exI)
by (auto simp: continuous_intros)
lemma homeomorphic_affinity:
fixes S :: "'a::real_normed_vector set"
assumes" \noteq> 0"
shows "S homeomorphic ((\<lambda>x. a + c *\< proof -
proof -
have *: "(+) a ` (*\<^sub>R) c ` S = (\<lambda>x. a + c *\<^sub>R x) ` S" by auto
show ?thesis
metis * homeomorphic_scaling homeomorphic_trans java.lang.StringIndexOutOfBoundsException: Range [89, 88) out of bounds for length 89
qed
lemma:
fixes a b ::"'a::real_normed_vector"
assumes" d 0<"
shows "(ball a d) homeomorphic (ball b e)" (is ?th) and"(cball a d) homeomorphic (cball b e)" (is ?cth)
proof -
show ?th show? applyjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17 apply(rule_tac x="\<lambda>x. a +ultimatelyhave "C <inter { <> <> =}java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
using assms
by (auto intro!: continuous_intros simpunfolding java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
show ?cth unfolding homeomorphic_minimal applyx"<x ed \^ub> x- )"in java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70 apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
using assms
by (auto intro!: continuous_intros simp: dist_commute dist_norm pos_divide_le_eq)
qed
lemma by metisComplID_defccsdist_norm notnot_lethat()java.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 96
fixes a b ::"'a::real_normed_vector"
assumes "0 < d""0 < e"
shows "(sphere a d) homeomorphic (sphere b e)"
homeomorphic_minimal apply(rule_tac x="\<lambda>x. b + (e/dultimately "( )(fx fx" apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
using assms
by (auto intro!: continuous_intros simp: dist_commute dist_norm pos_divide_less_eq)
lemma homeomorphic_ball01_UNIV: "ball (0::'a::real_normed_vector) 1 homeomorphic (UNIV:: 'a set)"
(is "?B homeomorphic ?U")
proof
\lambdaz. \^> 1norm) " :a apply (rule_tac x="x /\<^sub>R (1 + norm x)" in image_eqI) apply (auto simp: field_split_simps)
using norm_ge_zero [of x] apply linarith+
done
then show "(\<lambda>z::'a. z /\<^sub>R (1 - <>fconstant_on )"( thesis1
by blast
x\in>range(<>.( 1+normz))\<^R) if"normx "x:'
using that
by (rule_tac x="x /\<^sub>R (1 - norm x)" in image_eqI) (auto simp: field_split_simps)
then show "(\<lambda>z::'a. z /\<^sub>R (1 + norm z)) ` ?U = ?B"
by (force simp: field_split_simps dest: add_less_zeroD)
inuous_on ball )(<>.z \^> 1-norm )"
by (rule continuous_intros | force)+
have 0: "\<And>z. 1 + norm z \<noteq \<longrightarrow fconstant_on S)) is?hesis2
by (metis (no_types) le_add_same_cancel1 norm_ge_zero not_one_le_zero)
then show "continuous_on UNIV (\<lambda>z. z /\<^sub>R (1 + norm z))"
by (auto intro!: continuous_intros)
show "\<And>x. x \<in> ball 0 1 \<Longrightarrow>
ormx \^R( ( \^sub> 1 ))=xjava.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
auto:field_split_simps)
show "\<And>y. y /\<^sub>R (1 + norm y) /\<^sub>R (1 - norm \Longrightarrow>( <> )\<nd> ( \longleftrightarrow>u) <> s\longleftrightarrow>v)
using 0 by (auto simp: field_split_simps)
qed
proposition homeomorphic_ball_UNIV:
fixes a ::"'a::real_normed_vector"
assumes "0 < r" shows "ball a r homeomorphic (UNIV:: ' (blast dest java.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 47
using assms homeomorphic_ball01_UNIV homeomorphic_balls(1) homeomorphic_trans zero_less_one by blast
subsection\<^marker>\<openjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
lemma finite_implies_discrete:
fixes S :: "'a::topological_space set"
assumes "finite (f ` S)"
(<x\<>S existse0. \<forall>y \<in>S \and y\<>fx\l>e\le (y-fx)java.lang.StringIndexOutOfBoundsException: Index 134 out of bounds for length 134
proof -
have using continuous_discrete_range_constant_eq[ iffD1, OF ]assmsbyblast
proof (cases "f ` S - {f x} = {}") case True
with zero_less_numeral show ?thesis
by(fastforce add:Set conj_congjava.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
next and" f )java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
then obtain z where "zusingassms byblast
by blast
moreover have finn: "finite {norm (z - f x) |z. z \<in> f ` S - {f x}}"
using assms by simp
ultimately have *: "0 < Inf{norm(z - f x) | z. z \<in> f ` S - {f x}}"
by (force intro: finite_imp_less_Inf)
show ?thesis
by (force intro!: * cInf_le_finite [OF finn])
qed
with assms show ?thesis
by blast
qed
subsection\<^marker>\<open>tag unimportant\<close> \<open>Completeness of "Isometry" (up to constant bounds)\<close>
lemma cauchy_isometric:\<comment> \<open>TODO: rename lemma to \<open>Cauchy_isometric\<close>\<close>
assumes e: "e > 0" and s: "subspace s" and f: "bounded_linear f" and normf: "\<forall>x\<in>s. norm (f x) \<ge> e * norm x" and xs: "\<forall>n. x n \<in> s" and cf: "Cauchy (f \<circ> x)"
shows "Cauchy x"
proof -
interpret f: bounded_linear f by fact
have "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d"if"d > 0" for d :: real
proof -
from that obtain N where N: "\<forall>n\<ge>N. norm (f (x n) - f (x N)) < e * d"
using cf[unfolded Cauchy_def o_def dist_norm, THEN spec[where x="e*d"]] e
by auto
have "norm (x n - x N) < d"if"n \<ge> N" for n
proof -
have "e * norm (x n - x N) \<le> norm (f (x n - x N))"
using subspace_diff[OF s, of "x n""x N"]
using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]
using normf[THEN bspec[where x="x n - x N"]]
by auto
also have "norm (f (x n - x N)) < e * d"
using \<open>N \<le> n\<close> N unfolding f.diff[symmetric] by auto
finally show ?thesis
using \<open>e>0\<close> by simp
qed
then show ?thesis by auto
qed
then show ?thesis
by (simp add: Cauchy_altdef2 dist_norm)
qed
lemma complete_isometric_image:
assumes "0 < e" and s: "subspace s" and f: "bounded_linear f" and normf: "\<forall>x\<in>s. norm(f x) \<ge> e * norm(x)" and cs: "complete s"
shows "complete (f ` s)"
proof -
have "\<exists>l\<in>f ` s. (g \<longlongrightarrow> l) sequentially" if as:"\<forall>n::nat. g n \<in> f ` s"and cfg:"Cauchy g" for g
proof -
from that obtain x where "\<forall>n. x n \<in> s \<and> g n = f (x n)"
using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"] by auto
then have x: "\<forall>n. x n \<in> s""\<forall>n. g n = f (x n)" by auto
then have "f \<circ> x = g" by (simp add: fun_eq_iff)
then obtain l where "l\<in>s"and l:"(x \<longlongrightarrow> l) sequentially"
using cs[unfolded complete_def, THEN spec[where x=x]]
using cauchy_isometric[OF \<open>0 < e\<close> s f normf] and cfg and x(1)
by auto
then show ?thesis
using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
by (auto simp: \<open>f \<circ> x = g\<close>)
qed
then show ?thesis
unfolding complete_def by auto
qed
subsection \<open>Connected Normed Spaces\<close>
lemma compact_components:
fixes s :: "'a::heine_borel set"
shows "\<lbrakk>compact s; c \<in> components s\<rbrakk> \<Longrightarrow> compact c"
by (meson bounded_subset closed_components in_components_subset compact_eq_bounded_closed)
lemma discrete_subset_disconnected:
fixes S :: "'a::topological_space set"
fixes t :: "'b::real_normed_vector set"
assumes conf: "continuous_on S f" and no: "\<And>x. x \<in> S \<Longrightarrow> \<exists>e>0. \<forall>y. y \<in> S \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x)"
shows "f ` S \<subseteq> {y. connected_component_set (f ` S) y = {y}}"
proof -
{ fix x assume x: "x \<in> S"
then obtain e where "e>0"and ele: "\<And>y. \<lbrakk>y \<in> S; f y \<noteq> f x\<rbrakk> \<Longrightarrow> e \<le> norm (f y - f x)"
using conf no [OF x] by auto
then have e2: "0 \<le> e/2"
by simp
define F where "F \<equiv> connected_component_set (f ` S) (f x)"
have False if"y \<in> S"and ccs: "f y \<in> F"andnot: "f y \<noteq> f x" for y
proof -
define C where "C \<equiv> cball (f x) (e/2)"
define D where "D \<equiv> - ball (f x) e"
have disj: "C \<inter> D = {}"
unfolding C_def D_def using \<open>0 < e\<close> by fastforce
moreover have FCD: "F \<subseteq> C \<union> D"
proof -
have "t \<in> C \<or> t \<in> D"if"t \<in> F" for t
proof -
obtain y where "y \<in> S""t = f y"
using F_def \<open>t \<in> F\<close> connected_component_in by blast
then show ?thesis
by (metis C_def ComplI D_def centre_in_cball dist_norm e2 ele mem_ball norm_minus_commute not_le)
qed
then show ?thesis
by auto
qed
ultimately have "C \<inter> F = {} \<or> D \<inter> F = {}"
using connected_closed [of "F"] \<open>e>0\<close> not
unfolding C_def D_def
by (metis Elementary_Metric_Spaces.open_ball F_def closed_cball connected_connected_component inf_bot_left open_closed)
moreover have "C \<inter> F \<noteq> {}"
unfolding disjoint_iff
by (metis FCD ComplD image_eqI mem_Collect_eq subsetD x D_def F_def Un_iff \<open>0 < e\<close> centre_in_ball connected_component_refl_eq)
moreover have "D \<inter> F \<noteq> {}"
unfolding disjoint_iff
by (metis ComplI D_def ccs dist_norm ele mem_ball norm_minus_commute not not_le that(1))
ultimately show ?thesis by metis
qed
moreover have "connected_component_set (f ` S) (f x) \<subseteq> f ` S"
by (auto simp: connected_component_in)
ultimately have "connected_component_set (f ` S) (f x) = {f x}"
by (auto simp: x F_def)
}
with assms show ?thesis
by blast
qed
lemma continuous_disconnected_range_constant_eq: "(connected S \<longleftrightarrow>
(\<forall>f::'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1.
\<forall>t. continuous_on S f \<and> f ` S \<subseteq> t \<and> (\<forall>y \<in> t. connected_component_set t y = {y})
\<longrightarrow> f constant_on S))" (is ?thesis1) and continuous_discrete_range_constant_eq: "(connected S \<longleftrightarrow>
(\<forall>f::'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1.
continuous_on S f \<and>
(\<forall>x \<in> S. \<exists>e. 0 < e \<and> (\<forall>y. y \<in> S \<and> (f y \<noteq> f x) \<longrightarrow> e \<le> norm(f y - f x)))
\<longrightarrow> f constant_on S))" (is ?thesis2) and continuous_finite_range_constant_eq: "(connected S \<longleftrightarrow>
(\<forall>f::'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1.
continuous_on S f \<and> finite (f ` S)
\<longrightarrow> f constant_on S))" (is ?thesis3)
proof -
have *: "\<And>s t u v. \<lbrakk>s \<Longrightarrow> t; t \<Longrightarrow> u; u \<Longrightarrow> v; v \<Longrightarrow> s\<rbrakk>
\<Longrightarrow> (s \<longleftrightarrow> t) \<and> (s \<longleftrightarrow> u) \<and> (s \<longleftrightarrow> v)"
by blast
have "?thesis1 \<and> ?thesis2 \<and> ?thesis3" apply (rule *)
using continuous_disconnected_range_constant apply (metis image_subset_iff_funcset) apply (smt (verit, best) discrete_subset_disconnected mem_Collect_eq subsetD subsetI) apply (blast dest: finite_implies_discrete) apply (blast intro!: finite_range_constant_imp_connected)
done
then show ?thesis1 ?thesis2 ?thesis3
by blast+
qed
lemma continuous_discrete_range_constant:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1"
assumes S: "connected S" and"continuous_on S f" and"\<And>x. x \<in> S \<Longrightarrow> \<exists>e>0. \<forall>y. y \<in> S \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x)"
shows "f constant_on S"
using continuous_discrete_range_constant_eq [THEN iffD1, OF S] assms by blast
lemma continuous_finite_range_constant:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1"
assumes "connected S" and"continuous_on S f" and"finite (f ` S)"
shows "f constant_on S"
using assms continuous_finite_range_constant_eq by blast
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung und die Messung sind noch experimentell.
