--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Thu Aug 29 18:44:03 2013 +0200
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Thu Aug 29 19:22:48 2013 +0200
@@ -30,7 +30,7 @@
lemmas real_isGlb_unique = isGlb_unique[where 'a=real]
-lemma countable_PiE:
+lemma countable_PiE:
"finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> countable (F i)) \<Longrightarrow> countable (PiE I F)"
by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq)
@@ -77,45 +77,48 @@
proof safe
fix O' and x::'a
assume H: "topological_basis B" "open O'" "x \<in> O'"
- hence "(\<exists>B'\<subseteq>B. \<Union>B' = O')" by (simp add: topological_basis_def)
+ then have "(\<exists>B'\<subseteq>B. \<Union>B' = O')" by (simp add: topological_basis_def)
then obtain B' where "B' \<subseteq> B" "O' = \<Union>B'" by auto
- thus "\<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'" using H by auto
+ then show "\<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'" using H by auto
next
assume H: ?rhs
- show "topological_basis B" using assms unfolding topological_basis_def
+ show "topological_basis B"
+ using assms unfolding topological_basis_def
proof safe
- fix O'::"'a set" assume "open O'"
+ fix O'::"'a set"
+ assume "open O'"
with H obtain f where "\<forall>x\<in>O'. f x \<in> B \<and> x \<in> f x \<and> f x \<subseteq> O'"
by (force intro: bchoice simp: Bex_def)
- thus "\<exists>B'\<subseteq>B. \<Union>B' = O'"
+ then show "\<exists>B'\<subseteq>B. \<Union>B' = O'"
by (auto intro: exI[where x="{f x |x. x \<in> O'}"])
qed
qed
lemma topological_basisI:
assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"
- assumes "\<And>O' x. open O' \<Longrightarrow> x \<in> O' \<Longrightarrow> \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'"
+ and "\<And>O' x. open O' \<Longrightarrow> x \<in> O' \<Longrightarrow> \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'"
shows "topological_basis B"
using assms by (subst topological_basis_iff) auto
lemma topological_basisE:
fixes O'
assumes "topological_basis B"
- assumes "open O'"
- assumes "x \<in> O'"
+ and "open O'"
+ and "x \<in> O'"
obtains B' where "B' \<in> B" "x \<in> B'" "B' \<subseteq> O'"
proof atomize_elim
- from assms have "\<And>B'. B'\<in>B \<Longrightarrow> open B'" by (simp add: topological_basis_def)
+ from assms have "\<And>B'. B'\<in>B \<Longrightarrow> open B'"
+ by (simp add: topological_basis_def)
with topological_basis_iff assms
- show "\<exists>B'. B' \<in> B \<and> x \<in> B' \<and> B' \<subseteq> O'" using assms by (simp add: Bex_def)
+ show "\<exists>B'. B' \<in> B \<and> x \<in> B' \<and> B' \<subseteq> O'"
+ using assms by (simp add: Bex_def)
qed
lemma topological_basis_open:
assumes "topological_basis B"
- assumes "X \<in> B"
+ and "X \<in> B"
shows "open X"
- using assms
- by (simp add: topological_basis_def)
+ using assms by (simp add: topological_basis_def)
lemma topological_basis_imp_subbasis:
assumes B: "topological_basis B"
@@ -179,7 +182,7 @@
locale countable_basis =
fixes B::"'a::topological_space set set"
assumes is_basis: "topological_basis B"
- assumes countable_basis: "countable B"
+ and countable_basis: "countable B"
begin
lemma open_countable_basis_ex:
@@ -275,7 +278,8 @@
fix x :: "'a \<times> 'b"
from first_countable_basisE[of "fst x"] guess A :: "'a set set" . note A = this
from first_countable_basisE[of "snd x"] guess B :: "'b set set" . note B = this
- show "\<exists>A::nat \<Rightarrow> ('a \<times> 'b) set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
+ show "\<exists>A::nat \<Rightarrow> ('a \<times> 'b) set.
+ (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
proof (rule first_countableI[of "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"], safe)
fix a b
assume x: "a \<in> A" "b \<in> B"
@@ -293,7 +297,8 @@
qed
class second_countable_topology = topological_space +
- assumes ex_countable_subbasis: "\<exists>B::'a::topological_space set set. countable B \<and> open = generate_topology B"
+ assumes ex_countable_subbasis:
+ "\<exists>B::'a::topological_space set set. countable B \<and> open = generate_topology B"
begin
lemma ex_countable_basis: "\<exists>B::'a set set. countable B \<and> topological_basis B"
@@ -337,7 +342,7 @@
unfolding subset_image_iff by blast }
then show "topological_basis ?B"
unfolding topological_space_class.topological_basis_def
- by (safe intro!: topological_space_class.open_Inter)
+ by (safe intro!: topological_space_class.open_Inter)
(simp_all add: B generate_topology.Basis subset_eq)
qed
qed
@@ -368,7 +373,8 @@
then have B: "countable B" "topological_basis B"
using countable_basis is_basis
by (auto simp: countable_basis is_basis)
- then show "\<exists>A::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
+ then show "\<exists>A::nat \<Rightarrow> 'a set.
+ (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
by (intro first_countableI[of "{b\<in>B. x \<in> b}"])
(fastforce simp: topological_space_class.topological_basis_def)+
qed
@@ -418,11 +424,11 @@
lemma openin_clauses:
fixes U :: "'a topology"
- shows "openin U {}"
- "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
- "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
- using openin[of U] unfolding istopology_def mem_Collect_eq
- by fast+
+ shows
+ "openin U {}"
+ "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
+ "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
+ using openin[of U] unfolding istopology_def mem_Collect_eq by fast+
lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
unfolding topspace_def by blast
@@ -502,7 +508,8 @@
proof -
have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S] oS cT
by (auto simp add: topspace_def openin_subset)
- then show ?thesis using oS cT by (auto simp add: closedin_def)
+ then show ?thesis using oS cT
+ by (auto simp add: closedin_def)
qed
lemma closedin_diff[intro]:
@@ -511,8 +518,7 @@
shows "closedin U (S - T)"
proof -
have "S - T = S \<inter> (topspace U - T)"
- using closedin_subset[of U S] oS cT
- by (auto simp add: topspace_def)
+ using closedin_subset[of U S] oS cT by (auto simp add: topspace_def)
then show ?thesis
using oS cT by (auto simp add: openin_closedin_eq)
qed
@@ -537,7 +543,8 @@
}
moreover
{
- fix K assume K: "K \<subseteq> Collect ?L"
+ fix K
+ assume K: "K \<subseteq> Collect ?L"
have th0: "Collect ?L = (\<lambda>S. S \<inter> V) ` Collect (openin U)"
apply (rule set_eqI)
apply (simp add: Ball_def image_iff)
@@ -663,7 +670,8 @@
lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U \<inter> S)"
by (metis closedin_closed)
-lemma closed_closedin_trans: "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"
+lemma closed_closedin_trans:
+ "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"
apply (subgoal_tac "S \<inter> T = T" )
apply auto
apply (frule closedin_closed_Int[of T S])
@@ -680,7 +688,8 @@
(is "?lhs \<longleftrightarrow> ?rhs")
proof
assume ?lhs
- then show ?rhs unfolding openin_open open_dist by blast
+ then show ?rhs
+ unfolding openin_open open_dist by blast
next
def T \<equiv> "{x. \<exists>a\<in>S. \<exists>d>0. (\<forall>y\<in>U. dist y a < d \<longrightarrow> y \<in> S) \<and> dist x a < d}"
have 1: "\<forall>x\<in>T. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> T"
@@ -692,7 +701,7 @@
apply (rule_tac x=d in exI, clarify)
apply (erule le_less_trans [OF dist_triangle])
done
- assume ?rhs hence 2: "S = U \<inter> T"
+ assume ?rhs then have 2: "S = U \<inter> T"
unfolding T_def
apply auto
apply (drule (1) bspec, erule rev_bexI)
@@ -754,10 +763,10 @@
lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e"
by (simp add: subset_eq)
-lemma subset_ball[intro]: "d <= e ==> ball x d \<subseteq> ball x e"
+lemma subset_ball[intro]: "d \<le> e \<Longrightarrow> ball x d \<subseteq> ball x e"
by (simp add: subset_eq)
-lemma subset_cball[intro]: "d <= e ==> cball x d \<subseteq> cball x e"
+lemma subset_cball[intro]: "d \<le> e \<Longrightarrow> cball x d \<subseteq> cball x e"
by (simp add: subset_eq)
lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
@@ -796,7 +805,7 @@
unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..
lemma openE[elim?]:
- assumes "open S" "x\<in>S"
+ assumes "open S" "x\<in>S"
obtains e where "e>0" "ball x e \<subseteq> S"
using assms unfolding open_contains_ball by auto
@@ -876,7 +885,7 @@
lemma open_UNION_box:
fixes M :: "'a\<Colon>euclidean_space set"
- assumes "open M"
+ assumes "open M"
defines "a' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. fst (f i) *\<^sub>R i)"
defines "b' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. snd (f i) *\<^sub>R i)"
defines "I \<equiv> {f\<in>Basis \<rightarrow>\<^sub>E \<rat> \<times> \<rat>. box (a' f) (b' f) \<subseteq> M}"
@@ -886,8 +895,11 @@
fix x assume "x \<in> M"
obtain e where e: "e > 0" "ball x e \<subseteq> M"
using openE[OF `open M` `x \<in> M`] by auto
- moreover then obtain a b where ab: "x \<in> box a b"
- "\<forall>i \<in> Basis. a \<bullet> i \<in> \<rat>" "\<forall>i\<in>Basis. b \<bullet> i \<in> \<rat>" "box a b \<subseteq> ball x e"
+ moreover obtain a b where ab:
+ "x \<in> box a b"
+ "\<forall>i \<in> Basis. a \<bullet> i \<in> \<rat>"
+ "\<forall>i\<in>Basis. b \<bullet> i \<in> \<rat>"
+ "box a b \<subseteq> ball x e"
using rational_boxes[OF e(1)] by metis
ultimately have "x \<in> (\<Union>f\<in>I. box (a' f) (b' f))"
by (intro UN_I[of "\<lambda>i\<in>Basis. (a \<bullet> i, b \<bullet> i)"])
@@ -908,7 +920,10 @@
e1 \<inter> e2 = {} \<and>
e1 \<noteq> {} \<and>
e2 \<noteq> {})"
- unfolding connected_def openin_open by (safe, blast+)
+ unfolding connected_def openin_open
+ apply safe
+ apply blast+
+ done
lemma exists_diff:
fixes P :: "'a set \<Rightarrow> bool"
@@ -938,7 +953,7 @@
apply (subst exists_diff)
apply blast
done
- hence th0: "connected S \<longleftrightarrow>
+ then have th0: "connected S \<longleftrightarrow>
\<not> (\<exists>e2 e1. closed e2 \<and> open e1 \<and> S \<subseteq> e1 \<union> (- e2) \<and> e1 \<inter> (- e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (- e2) \<inter> S \<noteq> {})"
(is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)")
apply (simp add: closed_def)
@@ -951,7 +966,7 @@
fix e2
{
fix e1
- have "?P e2 e1 \<longleftrightarrow> (\<exists>t. closed e2 \<and> t = S\<inter>e2 \<and> open e1 \<and> t = S\<inter>e1 \<and> t\<noteq>{} \<and> t\<noteq>S)"
+ have "?P e2 e1 \<longleftrightarrow> (\<exists>t. closed e2 \<and> t = S\<inter>e2 \<and> open e1 \<and> t = S\<inter>e1 \<and> t\<noteq>{} \<and> t \<noteq> S)"
by auto
}
then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)"
@@ -969,7 +984,7 @@
subsection{* Limit points *}
-definition (in topological_space) islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "islimpt" 60)
+definition (in topological_space) islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "islimpt" 60)
where "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
lemma islimptI:
@@ -1068,12 +1083,12 @@
fix x
assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
from e have e2: "e/2 > 0" by arith
- from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2"
+ from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y \<noteq> x" "dist y x < e/2"
by blast
let ?m = "min (e/2) (dist x y) "
from e2 y(2) have mp: "?m > 0"
by (simp add: dist_nz[THEN sym])
- from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m"
+ from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z \<noteq> x" "dist z x < ?m"
by blast
have th: "dist z y < e" using z y
by (intro dist_triangle_lt [where z=x], simp)
@@ -1170,10 +1185,12 @@
assume "x \<notin> interior S"
with `x \<in> R` `open R` obtain y where "y \<in> R - S"
unfolding interior_def by fast
- from `open R` `closed S` have "open (R - S)" by (rule open_Diff)
- from `R \<subseteq> S \<union> T` have "R - S \<subseteq> T" by fast
- from `y \<in> R - S` `open (R - S)` `R - S \<subseteq> T` `interior T = {}`
- show "False" unfolding interior_def by fast
+ from `open R` `closed S` have "open (R - S)"
+ by (rule open_Diff)
+ from `R \<subseteq> S \<union> T` have "R - S \<subseteq> T"
+ by fast
+ from `y \<in> R - S` `open (R - S)` `R - S \<subseteq> T` `interior T = {}` show False
+ unfolding interior_def by fast
qed
qed
qed
@@ -1187,7 +1204,7 @@
fix T
assume "T \<subseteq> A \<times> B" and "open T"
then show "T \<subseteq> interior A \<times> interior B"
- proof (safe)
+ proof safe
fix x y
assume "(x, y) \<in> T"
then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"
@@ -1271,7 +1288,7 @@
fix x
assume as: "open S" "x \<in> S \<inter> closure T"
{
- assume *:"x islimpt T"
+ assume *: "x islimpt T"
have "x islimpt (S \<inter> T)"
proof (rule islimptI)
fix A
@@ -1302,7 +1319,9 @@
by (intro Sigma_mono closure_subset)
show "closed (closure A \<times> closure B)"
by (intro closed_Times closed_closure)
- fix T assume "A \<times> B \<subseteq> T" and "closed T" thus "closure A \<times> closure B \<subseteq> T"
+ fix T
+ assume "A \<times> B \<subseteq> T" and "closed T"
+ then show "closure A \<times> closure B \<subseteq> T"
apply (simp add: closed_def open_prod_def, clarify)
apply (rule ccontr)
apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)
@@ -1432,7 +1451,7 @@
"eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
using eventually_mono [of P Q] by fast
-lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"
+lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> \<not> trivial_limit net \<Longrightarrow> \<not> eventually (\<lambda>x. P x) net"
by (simp add: eventually_False)
@@ -1481,7 +1500,7 @@
done
lemma Lim_Un_univ:
- "(f ---> l) (at x within S) \<Longrightarrow> (f ---> l) (at x within T) \<Longrightarrow>
+ "(f ---> l) (at x within S) \<Longrightarrow> (f ---> l) (at x within T) \<Longrightarrow>
S \<union> T = UNIV \<Longrightarrow> (f ---> l) (at x)"
by (metis Lim_Un)
@@ -1537,7 +1556,8 @@
assume e: "{x<..} \<inter> I \<noteq> {}"
show ?thesis
proof (rule order_tendstoI)
- fix a assume a: "a < Inf (f ` ({x<..} \<inter> I))"
+ fix a
+ assume a: "a < Inf (f ` ({x<..} \<inter> I))"
{
fix y
assume "y \<in> {x<..} \<inter> I"
@@ -1620,8 +1640,8 @@
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
shows "(f ---> 0) net"
+ using assms(2)
proof (rule metric_tendsto_imp_tendsto)
- show "(g ---> 0) net" by fact
show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"
using assms(1) by (rule eventually_elim1) (simp add: dist_norm)
qed
@@ -1717,7 +1737,9 @@
text{* Limit under bilinear function *}
lemma Lim_bilinear:
- assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"
+ assumes "(f ---> l) net"
+ and "(g ---> m) net"
+ and "bounded_bilinear h"
shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"
using `bounded_bilinear h` `(f ---> l) net` `(g ---> m) net`
by (rule bounded_bilinear.tendsto)
@@ -1733,7 +1755,7 @@
lemma Lim_at_zero:
fixes a :: "'a::real_normed_vector"
fixes l :: "'b::topological_space"
- shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")
+ shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)"
using LIM_offset_zero LIM_offset_zero_cancel ..
text{* It's also sometimes useful to extract the limit point from the filter. *}
@@ -1741,8 +1763,7 @@
abbreviation netlimit :: "'a::t2_space filter \<Rightarrow> 'a"
where "netlimit F \<equiv> Lim F (\<lambda>x. x)"
-lemma netlimit_within:
- "\<not> trivial_limit (at a within S) \<Longrightarrow> netlimit (at a within S) = a"
+lemma netlimit_within: "\<not> trivial_limit (at a within S) \<Longrightarrow> netlimit (at a within S) = a"
by (rule tendsto_Lim) (auto intro: tendsto_intros)
lemma netlimit_at:
@@ -1776,8 +1797,9 @@
done
lemma Lim_transform_within:
- assumes "0 < d" and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
- and "(f ---> l) (at x within S)"
+ assumes "0 < d"
+ and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
+ and "(f ---> l) (at x within S)"
shows "(g ---> l) (at x within S)"
proof (rule Lim_transform_eventually)
show "eventually (\<lambda>x. f x = g x) (at x within S)"
@@ -1786,22 +1808,24 @@
qed
lemma Lim_transform_at:
- assumes "0 < d" and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
- and "(f ---> l) (at x)"
+ assumes "0 < d"
+ and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
+ and "(f ---> l) (at x)"
shows "(g ---> l) (at x)"
+ using _ assms(3)
proof (rule Lim_transform_eventually)
show "eventually (\<lambda>x. f x = g x) (at x)"
unfolding eventually_at2
using assms(1,2) by auto
- show "(f ---> l) (at x)" by fact
qed
text{* Common case assuming being away from some crucial point like 0. *}
lemma Lim_transform_away_within:
fixes a b :: "'a::t1_space"
- assumes "a \<noteq> b" and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
- and "(f ---> l) (at a within S)"
+ assumes "a \<noteq> b"
+ and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
+ and "(f ---> l) (at a within S)"
shows "(g ---> l) (at a within S)"
proof (rule Lim_transform_eventually)
show "(f ---> l) (at a within S)" by fact
@@ -1821,8 +1845,9 @@
text{* Alternatively, within an open set. *}
lemma Lim_transform_within_open:
- assumes "open S" and "a \<in> S" and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"
- and "(f ---> l) (at a)"
+ assumes "open S" and "a \<in> S"
+ and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"
+ and "(f ---> l) (at a)"
shows "(g ---> l) (at a)"
proof (rule Lim_transform_eventually)
show "eventually (\<lambda>x. f x = g x) (at a)"
@@ -1836,15 +1861,17 @@
(* FIXME: Only one congruence rule for tendsto can be used at a time! *)
lemma Lim_cong_within(*[cong add]*):
- assumes "a = b" "x = y" "S = T"
- assumes "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
+ assumes "a = b"
+ and "x = y"
+ and "S = T"
+ and "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"
unfolding tendsto_def eventually_at_topological
using assms by simp
lemma Lim_cong_at(*[cong add]*):
assumes "a = b" "x = y"
- assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
+ and "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"
unfolding tendsto_def eventually_at_topological
using assms by simp
@@ -1855,19 +1882,22 @@
fixes l :: "'a::first_countable_topology"
shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")
proof
- assume "?lhs" moreover
- { assume "l \<in> S"
- hence "?rhs" using tendsto_const[of l sequentially] by auto
+ assume "?lhs"
+ moreover
+ {
+ assume "l \<in> S"
+ then have "?rhs" using tendsto_const[of l sequentially] by auto
}
moreover
- { assume "l islimpt S"
- hence "?rhs" unfolding islimpt_sequential by auto
+ {
+ assume "l islimpt S"
+ then have "?rhs" unfolding islimpt_sequential by auto
}
ultimately show "?rhs"
unfolding closure_def by auto
next
assume "?rhs"
- thus "?lhs" unfolding closure_def islimpt_sequential by auto
+ then show "?lhs" unfolding closure_def islimpt_sequential by auto
qed
lemma closed_sequential_limits:
@@ -1898,7 +1928,8 @@
have *: "\<forall>x\<in>S. Inf S \<le> x"
using cInf_lower_EX[of _ S] assms by metis
{
- fix e :: real assume "0 < e"
+ fix e :: real
+ assume "e > 0"
then have "Inf S < Inf S + e" by simp
with assms obtain x where "x \<in> S" "x < Inf S + e"
by (subst (asm) cInf_less_iff[of _ B]) auto
@@ -1920,9 +1951,11 @@
"(\<not> trivial_limit (at x within S)) = (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"
(is "?lhs = ?rhs")
proof -
- { assume "?lhs"
- { fix e :: real
- assume "e>0"
+ {
+ assume "?lhs"
+ {
+ fix e :: real
+ assume "e > 0"
then obtain y where "y:(S-{x}) & dist y x < e"
using `?lhs` not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]
by auto
@@ -1933,16 +1966,21 @@
then have "?rhs" by auto
}
moreover
- { assume "?rhs"
- { fix e :: real
- assume "e>0"
- then obtain y where "y : (S Int ball x e - {x})" using `?rhs` by blast
+ {
+ assume "?rhs"
+ {
+ fix e :: real
+ assume "e > 0"
+ then obtain y where "y : (S Int ball x e - {x})"
+ using `?rhs` by blast
then have "y:(S-{x}) & dist y x < e"
unfolding ball_def by (simp add: dist_commute)
- then have "EX y:(S-{x}). dist y x < e" by auto
+ then have "EX y:(S-{x}). dist y x < e"
+ by auto
}
then have "?lhs"
- using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"] by auto
+ using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]
+ by auto
}
ultimately show ?thesis by auto
qed
@@ -1975,34 +2013,47 @@
lemma infdist_triangle: "infdist x A \<le> infdist y A + dist x y"
proof cases
assume "A = {}"
- thus ?thesis by (simp add: infdist_def)
+ then show ?thesis by (simp add: infdist_def)
next
assume "A \<noteq> {}"
then obtain a where "a \<in> A" by auto
have "infdist x A \<le> Inf {dist x y + dist y a |a. a \<in> A}"
proof (rule cInf_greatest)
- from `A \<noteq> {}` show "{dist x y + dist y a |a. a \<in> A} \<noteq> {}" by simp
- fix d assume "d \<in> {dist x y + dist y a |a. a \<in> A}"
- then obtain a where d: "d = dist x y + dist y a" "a \<in> A" by auto
+ from `A \<noteq> {}` show "{dist x y + dist y a |a. a \<in> A} \<noteq> {}"
+ by simp
+ fix d
+ assume "d \<in> {dist x y + dist y a |a. a \<in> A}"
+ then obtain a where d: "d = dist x y + dist y a" "a \<in> A"
+ by auto
show "infdist x A \<le> d"
unfolding infdist_notempty[OF `A \<noteq> {}`]
proof (rule cInf_lower2)
- show "dist x a \<in> {dist x a |a. a \<in> A}" using `a \<in> A` by auto
- show "dist x a \<le> d" unfolding d by (rule dist_triangle)
- fix d assume "d \<in> {dist x a |a. a \<in> A}"
- then obtain a where "a \<in> A" "d = dist x a" by auto
- thus "infdist x A \<le> d" by (rule infdist_le)
+ show "dist x a \<in> {dist x a |a. a \<in> A}"
+ using `a \<in> A` by auto
+ show "dist x a \<le> d"
+ unfolding d by (rule dist_triangle)
+ fix d
+ assume "d \<in> {dist x a |a. a \<in> A}"
+ then obtain a where "a \<in> A" "d = dist x a"
+ by auto
+ then show "infdist x A \<le> d"
+ by (rule infdist_le)
qed
qed
also have "\<dots> = dist x y + infdist y A"
proof (rule cInf_eq, safe)
- fix a assume "a \<in> A"
- thus "dist x y + infdist y A \<le> dist x y + dist y a" by (auto intro: infdist_le)
+ fix a
+ assume "a \<in> A"
+ then show "dist x y + infdist y A \<le> dist x y + dist y a"
+ by (auto intro: infdist_le)
next
- fix i assume inf: "\<And>d. d \<in> {dist x y + dist y a |a. a \<in> A} \<Longrightarrow> i \<le> d"
- hence "i - dist x y \<le> infdist y A" unfolding infdist_notempty[OF `A \<noteq> {}`] using `a \<in> A`
+ fix i
+ assume inf: "\<And>d. d \<in> {dist x y + dist y a |a. a \<in> A} \<Longrightarrow> i \<le> d"
+ then have "i - dist x y \<le> infdist y A"
+ unfolding infdist_notempty[OF `A \<noteq> {}`] using `a \<in> A`
by (intro cInf_greatest) (auto simp: field_simps)
- thus "i \<le> dist x y + infdist y A" by simp
+ then show "i \<le> dist x y + infdist y A"
+ by simp
qed
finally show ?thesis by simp
qed
@@ -2015,27 +2066,32 @@
show "infdist x A = 0"
proof (rule ccontr)
assume "infdist x A \<noteq> 0"
- with infdist_nonneg[of x A] have "infdist x A > 0" by auto
- hence "ball x (infdist x A) \<inter> closure A = {}"
+ with infdist_nonneg[of x A] have "infdist x A > 0"
+ by auto
+ then have "ball x (infdist x A) \<inter> closure A = {}"
apply auto
apply (metis `0 < infdist x A` `x \<in> closure A` closure_approachable dist_commute
eucl_less_not_refl euclidean_trans(2) infdist_le)
done
- hence "x \<notin> closure A"
+ then have "x \<notin> closure A"
by (metis `0 < infdist x A` centre_in_ball disjoint_iff_not_equal)
- thus False using `x \<in> closure A` by simp
+ then show False using `x \<in> closure A` by simp
qed
next
assume x: "infdist x A = 0"
- then obtain a where "a \<in> A" by atomize_elim (metis all_not_in_conv assms)
- show "x \<in> closure A" unfolding closure_approachable
- proof (safe, rule ccontr)
- fix e::real assume "0 < e"
+ then obtain a where "a \<in> A"
+ by atomize_elim (metis all_not_in_conv assms)
+ show "x \<in> closure A"
+ unfolding closure_approachable
+ apply safe
+ proof (rule ccontr)
+ fix e :: real
+ assume "e > 0"
assume "\<not> (\<exists>y\<in>A. dist y x < e)"
- hence "infdist x A \<ge> e" using `a \<in> A`
+ then have "infdist x A \<ge> e" using `a \<in> A`
unfolding infdist_def
by (force simp: dist_commute intro: cInf_greatest)
- with x `0 < e` show False by auto
+ with x `e > 0` show False by auto
qed
qed
@@ -2052,7 +2108,8 @@
assumes f: "(f ---> l) F"
shows "((\<lambda>x. infdist (f x) A) ---> infdist l A) F"
proof (rule tendstoI)
- fix e ::real assume "0 < e"
+ fix e ::real
+ assume "e > 0"
from tendstoD[OF f this]
show "eventually (\<lambda>x. dist (infdist (f x) A) (infdist l A) < e) F"
proof (eventually_elim)
@@ -2111,13 +2168,13 @@
{
fix x and e::real
assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
- hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
+ then have "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
}
moreover
{
fix x and e::real
assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
- hence "\<exists>d>0. ball x d \<subseteq> S"
+ then have "\<exists>d>0. ball x d \<subseteq> S"
unfolding subset_eq
apply(rule_tac x="e/2" in exI)
apply auto
@@ -2142,11 +2199,14 @@
shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")
proof
assume "?lhs"
- { assume "e \<le> 0"
- hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto
- have False using `?lhs` unfolding * using islimpt_EMPTY[of y] by auto
+ {
+ assume "e \<le> 0"
+ then have *:"ball x e = {}"
+ using ball_eq_empty[of x e] by auto
+ have False using `?lhs`
+ unfolding * using islimpt_EMPTY[of y] by auto
}
- hence "e > 0" by (metis not_less)
+ then have "e > 0" by (metis not_less)
moreover
have "y \<in> cball x e"
using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"]
@@ -2154,44 +2214,62 @@
unfolding closed_limpt by auto
ultimately show "?rhs" by auto
next
- assume "?rhs" hence "e>0" by auto
- { fix d::real assume "d>0"
+ assume "?rhs"
+ then have "e>0" by auto
+ {
+ fix d :: real
+ assume "d > 0"
have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
- proof(cases "d \<le> dist x y")
- case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
- proof(cases "x=y")
- case True hence False using `d \<le> dist x y` `d>0` by auto
- thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto
+ proof (cases "d \<le> dist x y")
+ case True
+ then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
+ proof (cases "x = y")
+ case True
+ then have False
+ using `d \<le> dist x y` `d>0` by auto
+ then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
+ by auto
next
case False
-
- have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))
- = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
- unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto
+ have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) =
+ norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
+ unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym]
+ by auto
also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]
unfolding scaleR_minus_left scaleR_one
by (auto simp add: norm_minus_commute)
also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
- unfolding distrib_right using `x\<noteq>y`[unfolded dist_nz, unfolded dist_norm] by auto
- also have "\<dots> \<le> e - d/2" using `d \<le> dist x y` and `d>0` and `?rhs` by(auto simp add: dist_norm)
- finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using `d>0` by auto
-
+ unfolding distrib_right using `x\<noteq>y`[unfolded dist_nz, unfolded dist_norm]
+ by auto
+ also have "\<dots> \<le> e - d/2" using `d \<le> dist x y` and `d>0` and `?rhs`
+ by (auto simp add: dist_norm)
+ finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using `d>0`
+ by auto
moreover
-
have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
- using `x\<noteq>y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)
+ using `x\<noteq>y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff
+ by (auto simp add: dist_commute)
moreover
- have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel
- using `d>0` `x\<noteq>y`[unfolded dist_nz] dist_commute[of x y]
- unfolding dist_norm by auto
- ultimately show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by (rule_tac x="y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI) auto
+ have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d"
+ unfolding dist_norm
+ apply simp
+ unfolding norm_minus_cancel
+ using `d > 0` `x\<noteq>y`[unfolded dist_nz] dist_commute[of x y]
+ unfolding dist_norm
+ apply auto
+ done
+ ultimately show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
+ apply (rule_tac x = "y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI)
+ apply auto
+ done
qed
next
- case False hence "d > dist x y" by auto
- show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
- proof(cases "x=y")
+ case False
+ then have "d > dist x y" by auto
+ show "\<exists>x' \<in> ball x e. x' \<noteq> y \<and> dist x' y < d"
+ proof (cases "x = y")
case True
obtain z where **: "z \<noteq> y" "dist z y < min e d"
using perfect_choose_dist[of "min e d" y]
@@ -2199,20 +2277,30 @@
show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
unfolding `x = y`
using `z \<noteq> y` **
- by (rule_tac x=z in bexI, auto simp add: dist_commute)
+ apply (rule_tac x=z in bexI)
+ apply (auto simp add: dist_commute)
+ done
next
- case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
- using `d>0` `d > dist x y` `?rhs` by(rule_tac x=x in bexI, auto)
+ case False
+ then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
+ using `d>0` `d > dist x y` `?rhs`
+ apply (rule_tac x=x in bexI)
+ apply auto
+ done
qed
- qed }
- thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto
+ qed
+ }
+ then show "?lhs"
+ unfolding mem_cball islimpt_approachable mem_ball by auto
qed
lemma closure_ball_lemma:
fixes x y :: "'a::real_normed_vector"
- assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"
+ assumes "x \<noteq> y"
+ shows "y islimpt ball x (dist x y)"
proof (rule islimptI)
- fix T assume "y \<in> T" "open T"
+ fix T
+ 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
(* choose point between x and y, within distance r of y. *)
@@ -2223,7 +2311,8 @@
have "dist z y < r"
unfolding z_def k_def using `0 < r`
by (simp add: dist_norm min_def)
- hence "z \<in> T" using `\<forall>z. dist z y < r \<longrightarrow> z \<in> T` by simp
+ then have "z \<in> T"
+ using `\<forall>z. dist z y < r \<longrightarrow> z \<in> T` by simp
have "dist x z < dist x y"
unfolding z_def2 dist_norm
apply (simp add: norm_minus_commute)
@@ -2233,7 +2322,8 @@
apply (simp add: k_def divide_pos_pos zero_less_dist_iff `0 < r` `x \<noteq> y`)
apply (simp add: zero_less_dist_iff `x \<noteq> y`)
done
- hence "z \<in> ball x (dist x y)" by simp
+ then have "z \<in> ball x (dist x y)"
+ by simp
have "z \<noteq> y"
unfolding z_def k_def using `x \<noteq> y` `0 < r`
by (simp add: min_def)
@@ -2265,51 +2355,72 @@
shows "interior (cball x e) = ball x e"
proof (cases "e\<ge>0")
case False note cs = this
- from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover
- { fix y assume "y \<in> cball x e"
- hence False unfolding mem_cball using dist_nz[of x y] cs by auto }
- hence "cball x e = {}" by auto
- hence "interior (cball x e) = {}" using interior_empty by auto
+ from cs have "ball x e = {}"
+ using ball_empty[of e x] by auto
+ moreover
+ {
+ fix y
+ assume "y \<in> cball x e"
+ then have False
+ unfolding mem_cball using dist_nz[of x y] cs by auto
+ }
+ then have "cball x e = {}" by auto
+ then have "interior (cball x e) = {}"
+ using interior_empty by auto
ultimately show ?thesis by blast
next
case True 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"
- then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast
-
+ 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"
+ then obtain d where "d>0" and d: "\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S"
+ unfolding open_dist by blast
then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"
using perfect_choose_dist [of d] by auto
- have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)
- hence xa_cball:"xa \<in> cball x e" using as(1) by auto
-
- hence "y \<in> ball x e" proof(cases "x = y")
+ have "xa \<in> S"
+ using d[THEN spec[where x = xa]]
+ using xa by (auto simp add: 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 = y")
case True
- hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball]
+ then have "e > 0"
+ using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball]
by (auto simp add: dist_commute)
- thus "y \<in> ball x e" using `x = y ` by simp
+ then show "y \<in> ball x e"
+ using `x = y ` by simp
next
case False
- have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm
+ have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d"
+ unfolding dist_norm
using `d>0` norm_ge_zero[of "y - x"] `x \<noteq> y` by auto
- hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e"
+ then have *: "y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e"
using d as(1)[unfolded subset_eq] by blast
have "y - x \<noteq> 0" using `x \<noteq> y` by auto
- hence **:"d / (2 * norm (y - x)) > 0"
+ then have **:"d / (2 * norm (y - x)) > 0"
unfolding zero_less_norm_iff[THEN sym]
using `d>0` divide_pos_pos[of d "2*norm (y - x)"] by auto
-
- have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x = norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"
+ have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x =
+ norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"
by (auto simp add: dist_norm algebra_simps)
also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
by (auto simp add: algebra_simps)
also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"
using ** by auto
- also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: distrib_right dist_norm)
- finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)
- thus "y \<in> ball x e" unfolding mem_ball using `d>0` by auto
+ also have "\<dots> = (dist y x) + d/2"
+ using ** by (auto simp add: distrib_right dist_norm)
+ finally have "e \<ge> dist x y +d/2"
+ using *[unfolded mem_cball] by (auto simp add: dist_commute)
+ then show "y \<in> ball x e"
+ unfolding mem_ball using `d>0` by auto
qed
}
- hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto
+ then have "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e"
+ by auto
ultimately show ?thesis
using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto
qed
@@ -2335,7 +2446,7 @@
apply (metis zero_le_dist dist_self order_less_le_trans)
done
-lemma cball_empty: "e < 0 ==> cball x e = {}"
+lemma cball_empty: "e < 0 \<Longrightarrow> cball x e = {}"
by (simp add: cball_eq_empty)
lemma cball_eq_sing:
@@ -2345,7 +2456,8 @@
assume e: "0 < e"
obtain a where "a \<noteq> x" "dist a x < e"
using perfect_choose_dist [OF e] by auto
- hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)
+ then have "a \<noteq> x" "dist x a \<le> e"
+ by (auto simp add: dist_commute)
with e show ?thesis by (auto simp add: set_eq_iff)
qed auto
@@ -2377,9 +2489,14 @@
unfolding bounded_any_center [where a=0]
by (simp add: dist_norm)
-lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s"
- unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI)
- using assms by auto
+lemma bounded_realI:
+ assumes "\<forall>x\<in>s. abs (x::real) \<le> B"
+ shows "bounded s"
+ unfolding bounded_def dist_real_def
+ apply (rule_tac x=0 in exI)
+ using assms
+ apply auto
+ done
lemma bounded_empty [simp]: "bounded {}"
by (simp add: bounded_def)
@@ -2402,7 +2519,7 @@
then obtain f where f: "\<forall>n. f n \<in> S" "(f ---> y) sequentially"
unfolding closure_sequential by auto
have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
- hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
+ then have "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
by (rule eventually_mono, simp add: f(1))
have "dist x y \<le> a"
apply (rule Lim_dist_ubound [of sequentially f])
@@ -2411,7 +2528,8 @@
apply fact
done
}
- thus ?thesis unfolding bounded_def by auto
+ then show ?thesis
+ unfolding bounded_def by auto
qed
lemma bounded_cball[simp,intro]: "bounded (cball x e)"
@@ -2445,8 +2563,8 @@
lemma bounded_insert [simp]: "bounded (insert x S) \<longleftrightarrow> bounded S"
proof -
have "\<forall>y\<in>{x}. dist x y \<le> 0" by simp
- hence "bounded {x}" unfolding bounded_def by fast
- thus ?thesis by (metis insert_is_Un bounded_Un)
+ then have "bounded {x}" unfolding bounded_def by fast
+ then show ?thesis by (metis insert_is_Un bounded_Un)
qed
lemma finite_imp_bounded [intro]: "finite S \<Longrightarrow> bounded S"
@@ -2489,14 +2607,16 @@
from assms(2) obtain B where B:"B>0" "\<forall>x. norm (f x) \<le> B * norm x"
using bounded_linear.pos_bounded by (auto simp add: mult_ac)
{
- fix x assume "x\<in>S"
- hence "norm x \<le> b" using b by auto
- hence "norm (f x) \<le> B * b" using B(2)
+ fix x
+ assume "x\<in>S"
+ then have "norm x \<le> b" using b by auto
+ then have "norm (f x) \<le> B * b" using B(2)
apply (erule_tac x=x in allE)
apply (metis B(1) B(2) order_trans mult_le_cancel_left_pos)
done
}
- thus ?thesis unfolding bounded_pos
+ then show ?thesis
+ unfolding bounded_pos
apply (rule_tac x="b*B" in exI)
using b B mult_pos_pos [of b B]
apply (auto simp add: mult_commute)
@@ -2514,16 +2634,16 @@
fixes S :: "'a::real_normed_vector set"
assumes "bounded S"
shows "bounded ((\<lambda>x. a + x) ` S)"
-proof-
+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>S"
- hence "norm (a + x) \<le> b + norm a"
+ then have "norm (a + x) \<le> b + norm a"
using norm_triangle_ineq[of a x] b by auto
}
- thus ?thesis
+ then show ?thesis
unfolding bounded_pos
using norm_ge_zero[of a] b(1) and add_strict_increasing[of b 0 "norm a"]
by (auto intro!: exI[of _ "b + norm a"])
@@ -2540,19 +2660,21 @@
lemma bounded_has_Sup:
fixes S :: "real set"
assumes "bounded S" "S \<noteq> {}"
- shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"
+ shows "\<forall>x\<in>S. x \<le> Sup S"
+ and "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"
proof
- fix x assume "x\<in>S"
- thus "x \<le> Sup S"
+ fix x
+ assume "x\<in>S"
+ then show "x \<le> Sup S"
by (metis cSup_upper abs_le_D1 assms(1) bounded_real)
next
- show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms
- by (metis cSup_least)
+ show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"
+ using assms by (metis cSup_least)
qed
lemma Sup_insert:
fixes S :: "real set"
- shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
+ shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
apply (subst cSup_insert_If)
apply (rule bounded_has_Sup(1)[of S, rule_format])
apply (auto simp: sup_max)
@@ -2569,22 +2691,23 @@
lemma bounded_has_Inf:
fixes S :: "real set"
assumes "bounded S" "S \<noteq> {}"
- shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"
+ shows "\<forall>x\<in>S. x \<ge> Inf S"
+ and "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"
proof
fix x
assume "x\<in>S"
- from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a"
+ from assms(1) obtain a where a: "\<forall>x\<in>S. \<bar>x\<bar> \<le> a"
unfolding bounded_real by auto
- thus "x \<ge> Inf S" using `x\<in>S`
+ then show "x \<ge> Inf S" using `x\<in>S`
by (metis cInf_lower_EX abs_le_D2 minus_le_iff)
next
- show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms
- by (metis cInf_greatest)
+ show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b"
+ using assms by (metis cInf_greatest)
qed
lemma Inf_insert:
fixes S :: "real set"
- shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
+ shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
apply (subst cInf_insert_if)
apply (rule bounded_has_Inf(1)[of S, rule_format])
apply (auto simp: inf_min)
@@ -2592,8 +2715,11 @@
lemma Inf_insert_finite:
fixes S :: "real set"
- shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
- by (rule Inf_insert, rule finite_imp_bounded, simp)
+ shows "finite S \<Longrightarrow> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
+ apply (rule Inf_insert)
+ apply (rule finite_imp_bounded)
+ apply simp
+ done
subsection {* Compactness *}
@@ -2606,22 +2732,23 @@
assume "\<not> (\<exists>x \<in> s. x islimpt t)"
then obtain f where f:"\<forall>x\<in>s. x \<in> f x \<and> open (f x) \<and> (\<forall>y\<in>t. y \<in> f x \<longrightarrow> y = x)"
unfolding islimpt_def
- using bchoice[of s "\<lambda> x T. x \<in> T \<and> open T \<and> (\<forall>y\<in>t. y \<in> T \<longrightarrow> y = x)"] by auto
- obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
+ using bchoice[of s "\<lambda> x T. x \<in> T \<and> open T \<and> (\<forall>y\<in>t. y \<in> T \<longrightarrow> y = x)"]
+ by auto
+ obtain g where g: "g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]]
using f by auto
from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto
{
fix x y
assume "x\<in>t" "y\<in>t" "f x = f y"
- hence "x \<in> f x" "y \<in> f x \<longrightarrow> y = x"
+ then have "x \<in> f x" "y \<in> f x \<longrightarrow> y = x"
using f[THEN bspec[where x=x]] and `t\<subseteq>s` by auto
- hence "x = y"
+ then have "x = y"
using `f x = f y` and f[THEN bspec[where x=y]] and `y\<in>t` and `t\<subseteq>s` by auto
}
- hence "inj_on f t"
+ then have "inj_on f t"
unfolding inj_on_def by simp
- hence "infinite (f ` t)"
+ then have "infinite (f ` t)"
using assms(2) using finite_imageD by auto
moreover
{
@@ -2630,12 +2757,12 @@
from g(3) assms(3) `x\<in>t` obtain h where "h\<in>g" and "x\<in>h" by auto
then obtain y where "y\<in>s" "h = f y"
using g'[THEN bspec[where x=h]] by auto
- hence "y = x"
+ then have "y = x"
using f[THEN bspec[where x=y]] and `x\<in>t` and `x\<in>h`[unfolded `h = f y`] by auto
- hence False
+ then have False
using `f x \<notin> g` `h\<in>g` unfolding `h = f y` by auto
}
- hence "f ` t \<subseteq> g" by auto
+ then have "f ` t \<subseteq> g" by auto
ultimately show False
using g(2) using finite_subset by auto
qed
@@ -2668,12 +2795,14 @@
moreover
have "(\<lambda>n. f (r n)) ----> l"
proof (rule topological_tendstoI)
- fix S assume "open S" "l \<in> S"
+ fix S
+ assume "open S" "l \<in> S"
with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially" by auto
moreover
{
fix i
- assume "Suc 0 \<le> i" then have "f (r i) \<in> A i"
+ assume "Suc 0 \<le> i"
+ then have "f (r i) \<in> A i"
by (cases i) (simp_all add: r_def s)
}
then have "eventually (\<lambda>i. f (r i) \<in> A i) sequentially"
@@ -2687,16 +2816,17 @@
lemma sequence_infinite_lemma:
fixes f :: "nat \<Rightarrow> 'a::t1_space"
- assumes "\<forall>n. f n \<noteq> l" and "(f ---> l) sequentially"
+ assumes "\<forall>n. f n \<noteq> l"
+ and "(f ---> l) sequentially"
shows "infinite (range f)"
proof
assume "finite (range f)"
- hence "closed (range f)" by (rule finite_imp_closed)
- hence "open (- range f)" by (rule open_Compl)
+ then have "closed (range f)" by (rule finite_imp_closed)
+ then have "open (- range f)" by (rule open_Compl)
from assms(1) have "l \<in> - range f" by auto
from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"
using `open (- range f)` `l \<in> - range f` by (rule topological_tendstoD)
- thus False unfolding eventually_sequentially by auto
+ then show False unfolding eventually_sequentially by auto
qed
lemma closure_insert:
@@ -2715,7 +2845,8 @@
assume *: "x islimpt (insert a s)"
show "x islimpt s"
proof (rule islimptI)
- fix t assume t: "x \<in> t" "open t"
+ fix t
+ assume t: "x \<in> t" "open t"
show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"
proof (cases "x = a")
case True
@@ -2728,11 +2859,12 @@
by (simp_all add: open_Diff)
obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"
using * t' by (rule islimptE)
- thus ?thesis by auto
+ then show ?thesis by auto
qed
qed
next
- assume "x islimpt s" thus "x islimpt (insert a s)"
+ assume "x islimpt s"
+ then show "x islimpt (insert a s)"
by (rule islimpt_subset) auto
qed
@@ -2750,14 +2882,17 @@
fixes l :: "'a :: t1_space"
shows "l islimpt S \<longleftrightarrow> (\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"
proof (safe intro!: islimptI)
- fix U assume "l islimpt S" "l \<in> U" "open U" "finite (U \<inter> S)"
+ fix U
+ assume "l islimpt S" "l \<in> U" "open U" "finite (U \<inter> S)"
then have "l islimpt S" "l \<in> (U - (U \<inter> S - {l}))" "open (U - (U \<inter> S - {l}))"
by (auto intro: finite_imp_closed)
then show False
by (rule islimptE) auto
next
- fix T assume *: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S)" "l \<in> T" "open T"
- then have "infinite (T \<inter> S - {l})" by auto
+ fix T
+ assume *: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S)" "l \<in> T" "open T"
+ then have "infinite (T \<inter> S - {l})"
+ by auto
then have "\<exists>x. x \<in> (T \<inter> S - {l})"
unfolding ex_in_conv by (intro notI) simp
then show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> l"
@@ -2773,7 +2908,8 @@
lemma sequence_unique_limpt:
fixes f :: "nat \<Rightarrow> 'a::t2_space"
- assumes "(f ---> l) sequentially" and "l' islimpt (range f)"
+ assumes "(f ---> l) sequentially"
+ and "l' islimpt (range f)"
shows "l' = l"
proof (rule ccontr)
assume "l' \<noteq> l"
@@ -2784,15 +2920,22 @@
then obtain N where "\<forall>n\<ge>N. f n \<in> t"
unfolding eventually_sequentially by auto
- have "UNIV = {..<N} \<union> {N..}" by auto
- hence "l' islimpt (f ` ({..<N} \<union> {N..}))" using assms(2) by simp
- hence "l' islimpt (f ` {..<N} \<union> f ` {N..})" by (simp add: image_Un)
- hence "l' islimpt (f ` {N..})" by (simp add: islimpt_union_finite)
+ have "UNIV = {..<N} \<union> {N..}"
+ by auto
+ then have "l' islimpt (f ` ({..<N} \<union> {N..}))"
+ using assms(2) by simp
+ then have "l' islimpt (f ` {..<N} \<union> f ` {N..})"
+ by (simp add: image_Un)
+ then have "l' islimpt (f ` {N..})"
+ by (simp add: islimpt_union_finite)
then obtain y where "y \<in> f ` {N..}" "y \<in> s" "y \<noteq> l'"
using `l' \<in> s` `open s` by (rule islimptE)
- then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'" by auto
- with `\<forall>n\<ge>N. f n \<in> t` have "f n \<in> s \<inter> t" by simp
- with `s \<inter> t = {}` show False by simp
+ then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'"
+ by auto
+ with `\<forall>n\<ge>N. f n \<in> t` have "f n \<in> s \<inter> t"
+ by simp
+ with `s \<inter> t = {}` show False
+ by simp
qed
lemma bolzano_weierstrass_imp_closed:
@@ -2803,21 +2946,22 @@
{
fix x l
assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"
- hence "l \<in> s"
+ then have "l \<in> s"
proof (cases "\<forall>n. x n \<noteq> l")
case False
- thus "l\<in>s" using as(1) by auto
+ then show "l\<in>s" using as(1) by auto
next
case True note cas = this
with as(2) have "infinite (range x)"
using sequence_infinite_lemma[of x l] by auto
then obtain l' where "l'\<in>s" "l' islimpt (range x)"
using assms[THEN spec[where x="range x"]] as(1) by auto
- thus "l\<in>s" using sequence_unique_limpt[of x l l']
+ then show "l\<in>s" using sequence_unique_limpt[of x l l']
using as cas by auto
qed
}
- thus ?thesis unfolding closed_sequential_limits by fast
+ then show ?thesis
+ unfolding closed_sequential_limits by fast
qed
lemma compact_imp_bounded:
@@ -2830,14 +2974,15 @@
by (rule compactE_image)
from `finite D` have "bounded (\<Union>x\<in>D. ball x 1)"
by (simp add: bounded_UN)
- thus "bounded U" using `U \<subseteq> (\<Union>x\<in>D. ball x 1)`
+ then show "bounded U" using `U \<subseteq> (\<Union>x\<in>D. ball x 1)`
by (rule bounded_subset)
qed
text{* In particular, some common special cases. *}
lemma compact_union [intro]:
- assumes "compact s" "compact t" shows " compact (s \<union> t)"
+ assumes "compact s" "compact t"
+ shows " compact (s \<union> t)"
proof (rule compactI)
fix f
assume *: "Ball f open" "s \<union> t \<subseteq> \<Union>f"
@@ -2858,14 +3003,16 @@
unfolding SUP_def by (rule compact_Union) auto
lemma closed_inter_compact [intro]:
- assumes "closed s" and "compact t"
+ assumes "closed s"
+ and "compact t"
shows "compact (s \<inter> t)"
using compact_inter_closed [of t s] assms
by (simp add: Int_commute)
lemma compact_inter [intro]:
fixes s t :: "'a :: t2_space set"
- assumes "compact s" and "compact t"
+ assumes "compact s"
+ and "compact t"
shows "compact (s \<inter> t)"
using assms by (intro compact_inter_closed compact_imp_closed)
@@ -2873,11 +3020,12 @@
unfolding compact_eq_heine_borel by auto
lemma compact_insert [simp]:
- assumes "compact s" shows "compact (insert x s)"
+ assumes "compact s"
+ shows "compact (insert x s)"
proof -
have "compact ({x} \<union> s)"
using compact_sing assms by (rule compact_union)
- thus ?thesis by simp
+ then show ?thesis by simp
qed
lemma finite_imp_compact: "finite s \<Longrightarrow> compact s"
@@ -2898,7 +3046,9 @@
(\<forall>A. (\<forall>a\<in>A. closed a) \<longrightarrow> (\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}) \<longrightarrow> U \<inter> \<Inter>A \<noteq> {})"
(is "_ \<longleftrightarrow> ?R")
proof (safe intro!: compact_eq_heine_borel[THEN iffD2])
- fix A assume "compact U" and A: "\<forall>a\<in>A. closed a" "U \<inter> \<Inter>A = {}"
+ fix A
+ assume "compact U"
+ and A: "\<forall>a\<in>A. closed a" "U \<inter> \<Inter>A = {}"
and fi: "\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}"
from A have "(\<forall>a\<in>uminus`A. open a) \<and> U \<subseteq> \<Union>(uminus`A)"
by auto
@@ -2907,8 +3057,10 @@
with fi[THEN spec, of B] show False
by (auto dest: finite_imageD intro: inj_setminus)
next
- fix A assume ?R and cover: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"
- from cover have "U \<inter> \<Inter>(uminus`A) = {}" "\<forall>a\<in>uminus`A. closed a"
+ fix A
+ assume ?R
+ assume "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"
+ then have "U \<inter> \<Inter>(uminus`A) = {}" "\<forall>a\<in>uminus`A. closed a"
by auto
with `?R` obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<inter> \<Inter>(uminus`B) = {}"
by (metis subset_image_iff)
@@ -2956,8 +3108,9 @@
"x \<in> closure X \<longleftrightarrow> (\<forall>A. \<forall>S\<subseteq>A. open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {})"
proof safe
assume x: "x \<in> closure X"
- fix S A assume "open S" "x \<in> S" "X \<inter> A = {}" "S \<subseteq> A"
- then have "x \<notin> closure (-S)"
+ fix S A
+ assume "open S" "x \<in> S" "X \<inter> A = {}" "S \<subseteq> A"
+ then have "x \<notin> closure (-S)"
by (auto simp: closure_complement subset_eq[symmetric] intro: interiorI)
with x have "x \<in> closure X - closure (-S)"
by auto
@@ -2975,14 +3128,16 @@
lemma compact_filter:
"compact U \<longleftrightarrow> (\<forall>F. F \<noteq> bot \<longrightarrow> eventually (\<lambda>x. x \<in> U) F \<longrightarrow> (\<exists>x\<in>U. inf (nhds x) F \<noteq> bot))"
proof (intro allI iffI impI compact_fip[THEN iffD2] notI)
- fix F assume "compact U" and F: "F \<noteq> bot" "eventually (\<lambda>x. x \<in> U) F"
- from F have "U \<noteq> {}"
+ fix F
+ assume "compact U"
+ assume F: "F \<noteq> bot" "eventually (\<lambda>x. x \<in> U) F"
+ then have "U \<noteq> {}"
by (auto simp: eventually_False)
def Z \<equiv> "closure ` {A. eventually (\<lambda>x. x \<in> A) F}"
then have "\<forall>z\<in>Z. closed z"
by auto
- moreover
+ moreover
have ev_Z: "\<And>z. z \<in> Z \<Longrightarrow> eventually (\<lambda>x. x \<in> z) F"
unfolding Z_def by (auto elim: eventually_elim1 intro: set_mp[OF closure_subset])
have "(\<forall>B \<subseteq> Z. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {})"
@@ -2997,7 +3152,8 @@
qed
ultimately have "U \<inter> \<Inter>Z \<noteq> {}"
using `compact U` unfolding compact_fip by blast
- then obtain x where "x \<in> U" and x: "\<And>z. z \<in> Z \<Longrightarrow> x \<in> z" by auto
+ then obtain x where "x \<in> U" and x: "\<And>z. z \<in> Z \<Longrightarrow> x \<in> z"
+ by auto
have "\<And>P. eventually P (inf (nhds x) F) \<Longrightarrow> P \<noteq> bot"
unfolding eventually_inf eventually_nhds
@@ -3012,8 +3168,8 @@
with `x \<in> U` show "\<exists>x\<in>U. inf (nhds x) F \<noteq> bot"
by (metis eventually_bot)
next
- fix A assume A: "\<forall>a\<in>A. closed a" "\<forall>B\<subseteq>A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}" "U \<inter> \<Inter>A = {}"
-
+ fix A
+ assume A: "\<forall>a\<in>A. closed a" "\<forall>B\<subseteq>A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}" "U \<inter> \<Inter>A = {}"
def P' \<equiv> "(\<lambda>a (x::'a). x \<in> a)"
then have inj_P': "\<And>A. inj_on P' A"
by (auto intro!: inj_onI simp: fun_eq_iff)
@@ -3021,33 +3177,42 @@
have "F \<noteq> bot"
unfolding F_def
proof (safe intro!: filter_from_subbase_not_bot)
- fix X assume "X \<subseteq> P' ` insert U A" "finite X" "Inf X = bot"
+ fix X
+ assume "X \<subseteq> P' ` insert U A" "finite X" "Inf X = bot"
then obtain B where "B \<subseteq> insert U A" "finite B" and B: "Inf (P' ` B) = bot"
unfolding subset_image_iff by (auto intro: inj_P' finite_imageD)
- with A(2)[THEN spec, of "B - {U}"] have "U \<inter> \<Inter>(B - {U}) \<noteq> {}" by auto
- with B show False by (auto simp: P'_def fun_eq_iff)
+ with A(2)[THEN spec, of "B - {U}"] have "U \<inter> \<Inter>(B - {U}) \<noteq> {}"
+ by auto
+ with B show False
+ by (auto simp: P'_def fun_eq_iff)
qed
moreover have "eventually (\<lambda>x. x \<in> U) F"
unfolding F_def by (rule eventually_filter_from_subbaseI) (auto simp: P'_def)
- moreover assume "\<forall>F. F \<noteq> bot \<longrightarrow> eventually (\<lambda>x. x \<in> U) F \<longrightarrow> (\<exists>x\<in>U. inf (nhds x) F \<noteq> bot)"
+ moreover
+ assume "\<forall>F. F \<noteq> bot \<longrightarrow> eventually (\<lambda>x. x \<in> U) F \<longrightarrow> (\<exists>x\<in>U. inf (nhds x) F \<noteq> bot)"
ultimately obtain x where "x \<in> U" and x: "inf (nhds x) F \<noteq> bot"
by auto
- { fix V assume "V \<in> A"
+ {
+ fix V
+ assume "V \<in> A"
then have V: "eventually (\<lambda>x. x \<in> V) F"
by (auto simp add: F_def image_iff P'_def intro!: eventually_filter_from_subbaseI)
have "x \<in> closure V"
unfolding closure_iff_nhds_not_empty
proof (intro impI allI)
- fix S A assume "open S" "x \<in> S" "S \<subseteq> A"
- then have "eventually (\<lambda>x. x \<in> A) (nhds x)" by (auto simp: eventually_nhds)
+ fix S A
+ assume "open S" "x \<in> S" "S \<subseteq> A"
+ then have "eventually (\<lambda>x. x \<in> A) (nhds x)"
+ by (auto simp: eventually_nhds)
with V have "eventually (\<lambda>x. x \<in> V \<inter> A) (inf (nhds x) F)"
by (auto simp: eventually_inf)
with x show "V \<inter> A \<noteq> {}"
by (auto simp del: Int_iff simp add: trivial_limit_def)
qed
then have "x \<in> V"
- using `V \<in> A` A(1) by simp }
+ using `V \<in> A` A(1) by simp
+ }
with `x\<in>U` have "x \<in> U \<inter> \<Inter>A" by auto
with `U \<inter> \<Inter>A = {}` show False by auto
qed
@@ -3070,12 +3235,14 @@
lemma countably_compact_imp_compact:
assumes "countably_compact U"
- assumes ccover: "countable B" "\<forall>b\<in>B. open b"
- assumes basis: "\<And>T x. open T \<Longrightarrow> x \<in> T \<Longrightarrow> x \<in> U \<Longrightarrow> \<exists>b\<in>B. x \<in> b \<and> b \<inter> U \<subseteq> T"
+ and ccover: "countable B" "\<forall>b\<in>B. open b"
+ and basis: "\<And>T x. open T \<Longrightarrow> x \<in> T \<Longrightarrow> x \<in> U \<Longrightarrow> \<exists>b\<in>B. x \<in> b \<and> b \<inter> U \<subseteq> T"
shows "compact U"
- using `countably_compact U` unfolding compact_eq_heine_borel countably_compact_def
+ using `countably_compact U`
+ unfolding compact_eq_heine_borel countably_compact_def
proof safe
- fix A assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"
+ fix A
+ assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"
assume *: "\<forall>A. countable A \<longrightarrow> (\<forall>a\<in>A. open a) \<longrightarrow> U \<subseteq> \<Union>A \<longrightarrow> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"
moreover def C \<equiv> "{b\<in>B. \<exists>a\<in>A. b \<inter> U \<subseteq> a}"
@@ -3084,10 +3251,12 @@
moreover
have "\<Union>A \<inter> U \<subseteq> \<Union>C"
proof safe
- fix x a assume "x \<in> U" "x \<in> a" "a \<in> A"
- with basis[of a x] A obtain b where "b \<in> B" "x \<in> b" "b \<inter> U \<subseteq> a" by blast
- with `a \<in> A` show "x \<in> \<Union>C" unfolding C_def
- by auto
+ fix x a
+ assume "x \<in> U" "x \<in> a" "a \<in> A"
+ with basis[of a x] A obtain b where "b \<in> B" "x \<in> b" "b \<inter> U \<subseteq> a"
+ by blast
+ with `a \<in> A` show "x \<in> \<Union>C"
+ unfolding C_def by auto
qed
then have "U \<subseteq> \<Union>C" using `U \<subseteq> \<Union>A` by auto
ultimately obtain T where "T\<subseteq>C" "finite T" "U \<subseteq> \<Union>T"
@@ -3102,22 +3271,22 @@
lemma countably_compact_imp_compact_second_countable:
"countably_compact U \<Longrightarrow> compact (U :: 'a :: second_countable_topology set)"
proof (rule countably_compact_imp_compact)
- fix T and x :: 'a assume "open T" "x \<in> T"
+ fix T and x :: 'a
+ assume "open T" "x \<in> T"
from topological_basisE[OF is_basis this] guess b .
- then show "\<exists>b\<in>SOME B. countable B \<and> topological_basis B. x \<in> b \<and> b \<inter> U \<subseteq> T" by auto
+ then show "\<exists>b\<in>SOME B. countable B \<and> topological_basis B. x \<in> b \<and> b \<inter> U \<subseteq> T"
+ by auto
qed (insert countable_basis topological_basis_open[OF is_basis], auto)
lemma countably_compact_eq_compact:
"countably_compact U \<longleftrightarrow> compact (U :: 'a :: second_countable_topology set)"
using countably_compact_imp_compact_second_countable compact_imp_countably_compact by blast
-
+
subsubsection{* Sequential compactness *}
-definition
- seq_compact :: "'a::topological_space set \<Rightarrow> bool" where
- "seq_compact S \<longleftrightarrow>
- (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>
- (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"
+definition seq_compact :: "'a::topological_space set \<Rightarrow> bool"
+ where "seq_compact S \<longleftrightarrow>
+ (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"
lemma seq_compact_imp_countably_compact:
fixes U :: "'a :: first_countable_topology set"
@@ -3138,13 +3307,17 @@
show ?thesis
proof (rule ccontr)
assume "\<not> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"
- then have "\<forall>T. \<exists>x. T \<subseteq> A \<and> finite T \<longrightarrow> (x \<in> U - \<Union>T)" by auto
- then obtain X' where T: "\<And>T. T \<subseteq> A \<Longrightarrow> finite T \<Longrightarrow> X' T \<in> U - \<Union>T" by metis
+ then have "\<forall>T. \<exists>x. T \<subseteq> A \<and> finite T \<longrightarrow> (x \<in> U - \<Union>T)"
+ by auto
+ then obtain X' where T: "\<And>T. T \<subseteq> A \<Longrightarrow> finite T \<Longrightarrow> X' T \<in> U - \<Union>T"
+ by metis
def X \<equiv> "\<lambda>n. X' (from_nat_into A ` {.. n})"
have X: "\<And>n. X n \<in> U - (\<Union>i\<le>n. from_nat_into A i)"
using `A \<noteq> {}` unfolding X_def SUP_def by (intro T) (auto intro: from_nat_into)
- then have "range X \<subseteq> U" by auto
- with subseq[of X] obtain r x where "x \<in> U" and r: "subseq r" "(X \<circ> r) ----> x" by auto
+ then have "range X \<subseteq> U"
+ by auto
+ with subseq[of X] obtain r x where "x \<in> U" and r: "subseq r" "(X \<circ> r) ----> x"
+ by auto
from `x\<in>U` `U \<subseteq> \<Union>A` from_nat_into_surj[OF `countable A`]
obtain n where "x \<in> from_nat_into A n" by auto
with r(2) A(1) from_nat_into[OF `A \<noteq> {}`, of n]
@@ -3164,7 +3337,8 @@
lemma compact_imp_seq_compact:
fixes U :: "'a :: first_countable_topology set"
- assumes "compact U" shows "seq_compact U"
+ assumes "compact U"
+ shows "seq_compact U"
unfolding seq_compact_def
proof safe
fix X :: "nat \<Rightarrow> 'a"
@@ -3185,7 +3359,8 @@
have "\<exists>a. i < a \<and> X a \<in> A (Suc n)"
proof (rule ccontr)
assume "\<not> (\<exists>a>i. X a \<in> A (Suc n))"
- then have "\<And>a. Suc i \<le> a \<Longrightarrow> X a \<notin> A (Suc n)" by auto
+ then have "\<And>a. Suc i \<le> a \<Longrightarrow> X a \<notin> A (Suc n)"
+ by auto
then have "eventually (\<lambda>x. x \<notin> A (Suc n)) (filtermap X sequentially)"
by (auto simp: eventually_filtermap eventually_sequentially)
moreover have "eventually (\<lambda>x. x \<in> A (Suc n)) (nhds x)"
@@ -3207,7 +3382,8 @@
proof (rule topological_tendstoI)
fix S
assume "open S" "x \<in> S"
- with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially" by auto
+ with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"
+ by auto
moreover
{
fix i
@@ -3238,9 +3414,9 @@
assumes "countably_compact s" "countable t" "infinite t" "t \<subseteq> s"
shows "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)"
proof (rule ccontr)
- def C \<equiv> "(\<lambda>F. interior (F \<union> (- t))) ` {F. finite F \<and> F \<subseteq> t }"
+ def C \<equiv> "(\<lambda>F. interior (F \<union> (- t))) ` {F. finite F \<and> F \<subseteq> t }"
note `countably_compact s`
- moreover have "\<forall>t\<in>C. open t"
+ moreover have "\<forall>t\<in>C. open t"
by (auto simp: C_def)
moreover
assume "\<not> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"
@@ -3256,19 +3432,20 @@
from `countable t` have "countable C"
unfolding C_def by (auto intro: countable_Collect_finite_subset)
ultimately guess D by (rule countably_compactE)
- then obtain E where E: "E \<subseteq> {F. finite F \<and> F \<subseteq> t }" "finite E" and
- s: "s \<subseteq> (\<Union>F\<in>E. interior (F \<union> (- t)))"
+ then obtain E where E: "E \<subseteq> {F. finite F \<and> F \<subseteq> t }" "finite E"
+ and s: "s \<subseteq> (\<Union>F\<in>E. interior (F \<union> (- t)))"
by (metis (lifting) Union_image_eq finite_subset_image C_def)
from s `t \<subseteq> s` have "t \<subseteq> \<Union>E"
using interior_subset by blast
moreover have "finite (\<Union>E)"
using E by auto
- ultimately show False using `infinite t` by (auto simp: finite_subset)
+ ultimately show False using `infinite t`
+ by (auto simp: finite_subset)
qed
lemma countable_acc_point_imp_seq_compact:
fixes s :: "'a::first_countable_topology set"
- assumes "\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s --> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"
+ assumes "\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"
shows "seq_compact s"
proof -
{
@@ -3281,20 +3458,22 @@
using pigeonhole_infinite[OF _ True] by auto
then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = f l"
using infinite_enumerate by blast
- hence "subseq r \<and> (f \<circ> r) ----> f l"
+ then have "subseq r \<and> (f \<circ> r) ----> f l"
by (simp add: fr tendsto_const o_def)
with f show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"
by auto
next
case False
- with f assms have "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" by auto
+ with f assms have "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)"
+ by auto
then obtain l where "l \<in> s" "\<forall>U. l\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" ..
from this(2) have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
using acc_point_range_imp_convergent_subsequence[of l f] by auto
with `l \<in> s` show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..
qed
}
- thus ?thesis unfolding seq_compact_def by auto
+ then show ?thesis
+ unfolding seq_compact_def by auto
qed
lemma seq_compact_eq_countably_compact:
@@ -3327,11 +3506,11 @@
subsubsection{* Total boundedness *}
-lemma cauchy_def:
- "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"
+lemma cauchy_def: "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"
unfolding Cauchy_def by metis
-fun helper_1 :: "('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where
+fun helper_1 :: "('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a"
+where
"helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"
declare helper_1.simps[simp del]
@@ -3340,7 +3519,8 @@
shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e) ` k))"
proof (rule, rule, rule ccontr)
fix e::real
- assume "e>0" and assm:"\<not> (\<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> \<Union>((\<lambda>x. ball x e) ` k))"
+ assume "e > 0"
+ assume assm: "\<not> (\<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> \<Union>((\<lambda>x. ball x e) ` k))"
def x \<equiv> "helper_1 s e"
{
fix n
@@ -3365,11 +3545,11 @@
using z
apply auto
done
- thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"
+ then show "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"
unfolding Q_def by auto
qed
}
- hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)"
+ then have "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)"
by blast+
then obtain l r where "l\<in>s" and r:"subseq r" and "((x \<circ> r) ---> l) sequentially"
using assms(1)[unfolded seq_compact_def, THEN spec[where x=x]] by auto
@@ -3388,7 +3568,8 @@
lemma seq_compact_imp_heine_borel:
fixes s :: "'a :: metric_space set"
- assumes "seq_compact s" shows "compact s"
+ assumes "seq_compact s"
+ shows "compact s"
proof -
from seq_compact_imp_totally_bounded[OF `seq_compact s`]
guess f unfolding choice_iff' .. note f = this
@@ -3403,20 +3584,29 @@
intro!: countable_image countable_SIGMA countable_UN)
show "\<forall>b\<in>K. open b" by (auto simp: K_def)
next
- fix T x assume T: "open T" "x \<in> T" and x: "x \<in> s"
- from openE[OF T] obtain e where "0 < e" "ball x e \<subseteq> T" by auto
- then have "0 < e / 2" "ball x (e / 2) \<subseteq> T" by auto
- from Rats_dense_in_real[OF `0 < e / 2`] obtain r where "r \<in> \<rat>" "0 < r" "r < e / 2" by auto
+ fix T x
+ assume T: "open T" "x \<in> T" and x: "x \<in> s"
+ from openE[OF T] obtain e where "0 < e" "ball x e \<subseteq> T"
+ by auto
+ then have "0 < e / 2" "ball x (e / 2) \<subseteq> T"
+ by auto
+ from Rats_dense_in_real[OF `0 < e / 2`] obtain r where "r \<in> \<rat>" "0 < r" "r < e / 2"
+ by auto
from f[rule_format, of r] `0 < r` `x \<in> s` obtain k where "k \<in> f r" "x \<in> ball k r"
unfolding Union_image_eq by auto
- from `r \<in> \<rat>` `0 < r` `k \<in> f r` have "ball k r \<in> K" by (auto simp: K_def)
+ from `r \<in> \<rat>` `0 < r` `k \<in> f r` have "ball k r \<in> K"
+ by (auto simp: K_def)
then show "\<exists>b\<in>K. x \<in> b \<and> b \<inter> s \<subseteq> T"
proof (rule bexI[rotated], safe)
- fix y assume "y \<in> ball k r"
+ fix y
+ assume "y \<in> ball k r"
with `r < e / 2` `x \<in> ball k r` have "dist x y < e"
by (intro dist_double[where x = k and d=e]) (auto simp: dist_commute)
- with `ball x e \<subseteq> T` show "y \<in> T" by auto
- qed (rule `x \<in> ball k r`)
+ with `ball x e \<subseteq> T` show "y \<in> T"
+ by auto
+ next
+ show "x \<in> ball k r" by fact
+ qed
qed
qed
@@ -3433,18 +3623,20 @@
lemma compact_eq_bolzano_weierstrass:
fixes s :: "'a::metric_space set"
- shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")
+ shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))"
+ (is "?lhs = ?rhs")
proof
assume ?lhs
- thus ?rhs using heine_borel_imp_bolzano_weierstrass[of s] by auto
+ then show ?rhs
+ using heine_borel_imp_bolzano_weierstrass[of s] by auto
next
assume ?rhs
- thus ?lhs
+ then show ?lhs
unfolding compact_eq_seq_compact_metric by (rule bolzano_weierstrass_imp_seq_compact)
qed
lemma bolzano_weierstrass_imp_bounded:
- "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> bounded s"
+ "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> bounded s"
using compact_imp_bounded unfolding compact_eq_bolzano_weierstrass .
text {*
@@ -3458,13 +3650,18 @@
lemma bounded_closed_imp_seq_compact:
fixes s::"'a::heine_borel set"
- assumes "bounded s" and "closed s" shows "seq_compact s"
+ assumes "bounded s"
+ and "closed s"
+ shows "seq_compact s"
proof (unfold seq_compact_def, clarify)
- fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
- with `bounded s` have "bounded (range f)" by (auto intro: bounded_subset)
+ fix f :: "nat \<Rightarrow> 'a"
+ assume f: "\<forall>n. f n \<in> s"
+ with `bounded s` have "bounded (range f)"
+ by (auto intro: bounded_subset)
obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
using bounded_imp_convergent_subsequence [OF `bounded (range f)`] by auto
- from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp
+ from f have fr: "\<forall>n. (f \<circ> r) n \<in> s"
+ by simp
have "l \<in> s" using `closed s` fr l
unfolding closed_sequential_limits by blast
show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
@@ -3476,12 +3673,12 @@
shows "compact s \<longleftrightarrow> bounded s \<and> closed s" (is "?lhs = ?rhs")
proof
assume ?lhs
- thus ?rhs
+ then show ?rhs
using compact_imp_closed compact_imp_bounded
by blast
next
assume ?rhs
- thus ?lhs
+ then show ?lhs
using bounded_closed_imp_seq_compact[of s]
unfolding compact_eq_seq_compact_metric
by auto
@@ -3514,41 +3711,48 @@
(\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"
proof safe
fix d :: "'a set"
- assume d: "d \<subseteq> Basis"
- with finite_Basis have "finite d" by (blast intro: finite_subset)
+ assume d: "d \<subseteq> Basis"
+ with finite_Basis have "finite d"
+ by (blast intro: finite_subset)
from this d show "\<exists>l::'a. \<exists>r. subseq r \<and>
(\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"
proof (induct d)
case empty
- thus ?case unfolding subseq_def by auto
+ then show ?case
+ unfolding subseq_def by auto
next
case (insert k d)
- have k[intro]:"k\<in>Basis" using insert by auto
- have s': "bounded ((\<lambda>x. x \<bullet> k) ` range f)" using `bounded (range f)`
+ have k[intro]: "k \<in> Basis"
+ using insert by auto
+ have s': "bounded ((\<lambda>x. x \<bullet> k) ` range f)"
+ using `bounded (range f)`
by (auto intro!: bounded_linear_image bounded_linear_inner_left)
- obtain l1::"'a" and r1 where r1:"subseq r1" and
- lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"
+ obtain l1::"'a" and r1 where r1: "subseq r1"
+ and lr1: "\<forall>e > 0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"
using insert(3) using insert(4) by auto
- have f': "\<forall>n. f (r1 n) \<bullet> k \<in> (\<lambda>x. x \<bullet> k) ` range f" by simp
+ have f': "\<forall>n. f (r1 n) \<bullet> k \<in> (\<lambda>x. x \<bullet> k) ` range f"
+ by simp
have "bounded (range (\<lambda>i. f (r1 i) \<bullet> k))"
by (metis (lifting) bounded_subset f' image_subsetI s')
then obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) \<bullet> k) ---> l2) sequentially"
- using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) \<bullet> k"] by (auto simp: o_def)
- def r \<equiv> "r1 \<circ> r2" have r:"subseq r"
+ using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) \<bullet> k"]
+ by (auto simp: o_def)
+ def r \<equiv> "r1 \<circ> r2"
+ have r:"subseq r"
using r1 and r2 unfolding r_def o_def subseq_def by auto
moreover
def l \<equiv> "(\<Sum>i\<in>Basis. (if i = k then l2 else l1\<bullet>i) *\<^sub>R i)::'a"
{
fix e::real
- assume "e>0"
- from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"
+ assume "e > 0"
+ from lr1 `e > 0` have N1: "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"
by blast
- from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) \<bullet> k) l2 < e) sequentially"
+ from lr2 `e > 0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) \<bullet> k) l2 < e) sequentially"
by (rule tendstoD)
from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) \<bullet> i) (l1 \<bullet> i) < e) sequentially"
by (rule eventually_subseq)
have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"
- using N1' N2
+ using N1' N2
by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def)
}
ultimately show ?case by auto
@@ -3564,8 +3768,8 @@
using compact_lemma [OF f] by blast
{
fix e::real
- assume "e>0"
- hence "0 < e / real_of_nat DIM('a)"
+ assume "e > 0"
+ then have "e / real_of_nat DIM('a) > 0"
by (auto intro!: divide_pos_pos DIM_positive)
with l have "eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e / (real_of_nat DIM('a))) sequentially"
by simp
@@ -3576,18 +3780,20 @@
have "dist (f (r n)) l \<le> (\<Sum>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i))"
apply (subst euclidean_dist_l2)
using zero_le_dist
- by (rule setL2_le_setsum)
+ apply (rule setL2_le_setsum)
+ done
also have "\<dots> < (\<Sum>i\<in>(Basis::'a set). e / (real_of_nat DIM('a)))"
apply (rule setsum_strict_mono)
using n
- by auto
- finally have "dist (f (r n)) l < e"
+ apply auto
+ done
+ finally have "dist (f (r n)) l < e"
by auto
}
ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
by (rule eventually_elim1)
}
- hence *: "((f \<circ> r) ---> l) sequentially"
+ then have *: "((f \<circ> r) ---> l) sequentially"
unfolding o_def tendsto_iff by simp
with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
by auto
@@ -3619,13 +3825,13 @@
proof
fix f :: "nat \<Rightarrow> 'a \<times> 'b"
assume f: "bounded (range f)"
- from f have s1: "bounded (range (fst \<circ> f))" unfolding image_comp by (rule bounded_fst)
+ from f have s1: "bounded (range (fst \<circ> f))"
+ unfolding image_comp by (rule bounded_fst)
obtain l1 r1 where r1: "subseq r1" and l1: "(\<lambda>n. fst (f (r1 n))) ----> l1"
using bounded_imp_convergent_subsequence [OF s1] unfolding o_def by fast
from f have s2: "bounded (range (snd \<circ> f \<circ> r1))"
by (auto simp add: image_comp intro: bounded_snd bounded_subset)
- obtain l2 r2 where r2: "subseq r2"
- and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"
+ obtain l2 r2 where r2: "subseq r2" and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"
using bounded_imp_convergent_subsequence [OF s2]
unfolding o_def by fast
have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"
@@ -3656,36 +3862,42 @@
note lr' = seq_suble [OF lr(2)]
{
- fix e::real
- assume "e>0"
+ fix e :: real
+ assume "e > 0"
from as(2) obtain N where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (f m) (f n) < e/2"
unfolding cauchy_def
- using `e>0` apply (erule_tac x="e/2" in allE)
+ using `e > 0`
+ apply (erule_tac x="e/2" in allE)
apply auto
done
from lr(3)[unfolded LIMSEQ_def, THEN spec[where x="e/2"]]
- obtain M where M:"\<forall>n\<ge>M. dist ((f \<circ> r) n) l < e/2" using `e>0` by auto
+ obtain M where M:"\<forall>n\<ge>M. dist ((f \<circ> r) n) l < e/2"
+ using `e > 0` by auto
{
- fix n::nat
- assume n:"n \<ge> max N M"
- have "dist ((f \<circ> r) n) l < e/2" using n M by auto
- moreover have "r n \<ge> N" using lr'[of n] n by auto
- hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto
+ fix n :: nat
+ assume n: "n \<ge> max N M"
+ have "dist ((f \<circ> r) n) l < e/2"
+ using n M by auto
+ moreover have "r n \<ge> N"
+ using lr'[of n] n by auto
+ then have "dist (f n) ((f \<circ> r) n) < e / 2"
+ using N and n by auto
ultimately have "dist (f n) l < e"
- using dist_triangle_half_r[of "f (r n)" "f n" e l] by (auto simp add: dist_commute)
+ using dist_triangle_half_r[of "f (r n)" "f n" e l]
+ by (auto simp add: dist_commute)
}
- hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast
+ then have "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast
}
- hence "\<exists>l\<in>s. (f ---> l) sequentially" using `l\<in>s`
+ then have "\<exists>l\<in>s. (f ---> l) sequentially" using `l\<in>s`
unfolding LIMSEQ_def by auto
}
- thus ?thesis unfolding complete_def by auto
+ then show ?thesis unfolding complete_def by auto
qed
lemma nat_approx_posE:
fixes e::real
assumes "0 < e"
- obtains n::nat where "1 / (Suc n) < e"
+ obtains n :: nat where "1 / (Suc n) < e"
proof atomize_elim
have " 1 / real (Suc (nat (ceiling (1/e)))) < 1 / (ceiling (1/e))"
by (rule divide_strict_left_mono) (auto intro!: mult_pos_pos simp: `0 < e`)
@@ -3705,18 +3917,22 @@
show "compact s"
proof cases
- assume "s = {}" thus "compact s" by (simp add: compact_def)
+ assume "s = {}"
+ then show "compact s" by (simp add: compact_def)
next
assume "s \<noteq> {}"
show ?thesis
unfolding compact_def
proof safe
- fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
-
+ fix f :: "nat \<Rightarrow> 'a"
+ assume f: "\<forall>n. f n \<in> s"
+
def e \<equiv> "\<lambda>n. 1 / (2 * Suc n)"
then have [simp]: "\<And>n. 0 < e n" by auto
def B \<equiv> "\<lambda>n U. SOME b. infinite {n. f n \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U)"
- { fix n U assume "infinite {n. f n \<in> U}"
+ {
+ fix n U
+ assume "infinite {n. f n \<in> U}"
then have "\<exists>b\<in>k (e n). infinite {i\<in>{n. f n \<in> U}. f i \<in> ball b (e n)}"
using k f by (intro pigeonhole_infinite_rel) (auto simp: subset_eq)
then guess a ..
@@ -3724,12 +3940,16 @@
by (intro exI[of _ "ball a (e n) \<inter> U"] exI[of _ a]) (auto simp: ac_simps)
from someI_ex[OF this]
have "infinite {i. f i \<in> B n U}" "\<exists>x. B n U \<subseteq> ball x (e n) \<inter> U"
- unfolding B_def by auto }
+ unfolding B_def by auto
+ }
note B = this
def F \<equiv> "nat_rec (B 0 UNIV) B"
- { fix n have "infinite {i. f i \<in> F n}"
- by (induct n) (auto simp: F_def B) }
+ {
+ fix n
+ have "infinite {i. f i \<in> F n}"
+ by (induct n) (auto simp: F_def B)
+ }
then have F: "\<And>n. \<exists>x. F (Suc n) \<subseteq> ball x (e n) \<inter> F n"
using B by (simp add: F_def)
then have F_dec: "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m"
@@ -3751,13 +3971,17 @@
moreover have "\<forall>i. (f \<circ> t) i \<in> s"
using f by auto
moreover
- { fix n have "(f \<circ> t) n \<in> F n"
- by (cases n) (simp_all add: t_def sel) }
+ {
+ fix n
+ have "(f \<circ> t) n \<in> F n"
+ by (cases n) (simp_all add: t_def sel)
+ }
note t = this
have "Cauchy (f \<circ> t)"
proof (safe intro!: metric_CauchyI exI elim!: nat_approx_posE)
- fix r :: real and N n m assume "1 / Suc N < r" "Suc N \<le> n" "Suc N \<le> m"
+ fix r :: real and N n m
+ assume "1 / Suc N < r" "Suc N \<le> n" "Suc N \<le> m"
then have "(f \<circ> t) n \<in> F (Suc N)" "(f \<circ> t) m \<in> F (Suc N)" "2 * e N < r"
using F_dec t by (auto simp: e_def field_simps real_of_nat_Suc)
with F[of N] obtain x where "dist x ((f \<circ> t) n) < e N" "dist x ((f \<circ> t) m) < e N"
@@ -3774,39 +3998,44 @@
qed (metis compact_imp_complete compact_imp_seq_compact seq_compact_imp_totally_bounded)
lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")
-proof-
- { assume ?rhs
- { fix e::real
+proof -
+ {
+ assume ?rhs
+ {
+ fix e::real
assume "e>0"
with `?rhs` obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"
by (erule_tac x="e/2" in allE) auto
- { fix n m
+ {
+ fix n m
assume nm:"N \<le> m \<and> N \<le> n"
- hence "dist (s m) (s n) < e" using N
+ then have "dist (s m) (s n) < e" using N
using dist_triangle_half_l[of "s m" "s N" "e" "s n"]
by blast
}
- hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"
+ then have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"
by blast
}
- hence ?lhs
+ then have ?lhs
unfolding cauchy_def
by blast
}
- thus ?thesis
+ then show ?thesis
unfolding cauchy_def
using dist_triangle_half_l
by blast
qed
-lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)"
-proof-
- from assms obtain N::nat where "\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < 1"
+lemma cauchy_imp_bounded:
+ assumes "Cauchy s"
+ shows "bounded (range s)"
+proof -
+ from assms obtain N :: nat where "\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < 1"
unfolding cauchy_def
apply (erule_tac x= 1 in allE)
apply auto
done
- hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto
+ then have N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto
moreover
have "bounded (s ` {0..N})"
using finite_imp_bounded[of "s ` {1..N}"] by auto
@@ -3825,11 +4054,11 @@
instance heine_borel < complete_space
proof
fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
- hence "bounded (range f)"
+ then have "bounded (range f)"
by (rule cauchy_imp_bounded)
- hence "compact (closure (range f))"
+ then have "compact (closure (range f))"
unfolding compact_eq_bounded_closed by auto
- hence "complete (closure (range f))"
+ then have "complete (closure (range f))"
by (rule compact_imp_complete)
moreover have "\<forall>n. f n \<in> closure (range f)"
using closure_subset [of "range f"] by auto
@@ -3842,29 +4071,36 @@
instance euclidean_space \<subseteq> banach ..
lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"
-proof(simp add: complete_def, rule, rule)
- fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
- hence "convergent f" by (rule Cauchy_convergent)
- thus "\<exists>l. f ----> l" unfolding convergent_def .
-qed
-
-lemma complete_imp_closed: assumes "complete s" shows "closed s"
+proof (simp add: complete_def, rule, rule)
+ fix f :: "nat \<Rightarrow> 'a"
+ assume "Cauchy f"
+ then have "convergent f" by (rule Cauchy_convergent)
+ then show "\<exists>l. f ----> l" unfolding convergent_def .
+qed
+
+lemma complete_imp_closed:
+ assumes "complete s"
+ shows "closed s"
proof -
- { fix x assume "x islimpt s"
+ {
+ fix x
+ assume "x islimpt s"
then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"
unfolding islimpt_sequential by auto
then obtain l where l: "l\<in>s" "(f ---> l) sequentially"
using `complete s`[unfolded complete_def] using LIMSEQ_imp_Cauchy[of f x] by auto
- hence "x \<in> s" using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto
+ then have "x \<in> s"
+ using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto
}
- thus "closed s" unfolding closed_limpt by auto
+ then show "closed s" unfolding closed_limpt by auto
qed
lemma complete_eq_closed:
fixes s :: "'a::complete_space set"
shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")
proof
- assume ?lhs thus ?rhs by (rule complete_imp_closed)
+ assume ?lhs
+ then show ?rhs by (rule complete_imp_closed)
next
assume ?rhs
{
@@ -3872,11 +4108,11 @@
assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"
then obtain l where "(f ---> l) sequentially"
using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto
- hence "\<exists>l\<in>s. (f ---> l) sequentially"
+ then have "\<exists>l\<in>s. (f ---> l) sequentially"
using `?rhs`[unfolded closed_sequential_limits, THEN spec[where x=f], THEN spec[where x=l]]
using as(1) by auto
}
- thus ?lhs unfolding complete_def by auto
+ then show ?lhs unfolding complete_def by auto
qed
lemma convergent_eq_cauchy:
@@ -3933,56 +4169,94 @@
apply blast
done
- { fix n::nat
- { fix e::real assume "e>0"
+ {
+ fix n :: nat
+ {
+ fix e :: real
+ assume "e>0"
with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e"
unfolding LIMSEQ_def by auto
- hence "dist ((x \<circ> r) (max N n)) l < e" by auto
+ then have "dist ((x \<circ> r) (max N n)) l < e" by auto
moreover
- have "r (max N n) \<ge> n" using lr(2) using seq_suble[of r "max N n"] by auto
- hence "(x \<circ> r) (max N n) \<in> s n"
- using x apply (erule_tac x=n in allE)
- using x apply (erule_tac x="r (max N n)" in allE)
- using assms(3) apply (erule_tac x=n in allE)
+ have "r (max N n) \<ge> n" using lr(2) using seq_suble[of r "max N n"]
+ by auto
+ then have "(x \<circ> r) (max N n) \<in> s n"
+ using x
+ apply (erule_tac x=n in allE)
+ using x
+ apply (erule_tac x="r (max N n)" in allE)
+ using assms(3)
+ apply (erule_tac x=n in allE)
apply (erule_tac x="r (max N n)" in allE)
apply auto
done
- ultimately have "\<exists>y\<in>s n. dist y l < e" by auto
+ ultimately have "\<exists>y\<in>s n. dist y l < e"
+ by auto
}
- hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast
+ then have "l \<in> s n"
+ using closed_approachable[of "s n" l] assms(1) by blast
}
- thus ?thesis by auto
+ then show ?thesis by auto
qed
text {* Decreasing case does not even need compactness, just completeness. *}
lemma decreasing_closed_nest:
- assumes "\<forall>n. closed(s n)"
- "\<forall>n. (s n \<noteq> {})"
- "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
- "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"
+ assumes
+ "\<forall>n. closed(s n)"
+ "\<forall>n. (s n \<noteq> {})"
+ "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
+ "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"
shows "\<exists>a::'a::complete_space. \<forall>n::nat. a \<in> s n"
proof-
- have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto
- hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto
+ have "\<forall>n. \<exists> x. x\<in>s n"
+ using assms(2) by auto
+ then have "\<exists>t. \<forall>n. t n \<in> s n"
+ using choice[of "\<lambda> n x. x \<in> s n"] by auto
then obtain t where t: "\<forall>n. t n \<in> s n" by auto
- { fix e::real assume "e>0"
- then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e" using assms(4) by auto
- { fix m n ::nat assume "N \<le> m \<and> N \<le> n"
- hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding subset_eq t by blast+
- hence "dist (t m) (t n) < e" using N by auto
+ {
+ fix e :: real
+ assume "e > 0"
+ then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e"
+ using assms(4) by auto
+ {
+ fix m n :: nat
+ assume "N \<le> m \<and> N \<le> n"
+ then have "t m \<in> s N" "t n \<in> s N"
+ using assms(3) t unfolding subset_eq t by blast+
+ then have "dist (t m) (t n) < e"
+ using N by auto
}
- hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto
+ then have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e"
+ by auto
}
- hence "Cauchy t" unfolding cauchy_def by auto
- then obtain l where l:"(t ---> l) sequentially" using complete_univ unfolding complete_def by auto
- { fix n::nat
- { fix e::real assume "e>0"
- then obtain N::nat where N:"\<forall>n\<ge>N. dist (t n) l < e" using l[unfolded LIMSEQ_def] by auto
- have "t (max n N) \<in> s n" using assms(3) unfolding subset_eq apply(erule_tac x=n in allE) apply (erule_tac x="max n N" in allE) using t by auto
- hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto
+ then have "Cauchy t"
+ unfolding cauchy_def by auto
+ then obtain l where l:"(t ---> l) sequentially"
+ using complete_univ unfolding complete_def by auto
+ {
+ fix n :: nat
+ {
+ fix e :: real
+ assume "e > 0"
+ then obtain N :: nat where N: "\<forall>n\<ge>N. dist (t n) l < e"
+ using l[unfolded LIMSEQ_def] by auto
+ have "t (max n N) \<in> s n"
+ using assms(3)
+ unfolding subset_eq
+ apply (erule_tac x=n in allE)
+ apply (erule_tac x="max n N" in allE)
+ using t
+ apply auto
+ done
+ then have "\<exists>y\<in>s n. dist y l < e"
+ apply (rule_tac x="t (max n N)" in bexI)
+ using N
+ apply auto
+ done
}
- hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto
+ then have "l \<in> s n"
+ using closed_approachable[of "s n" l] assms(1) by auto
}
then show ?thesis by auto
qed
@@ -3991,56 +4265,94 @@
lemma decreasing_closed_nest_sing:
fixes s :: "nat \<Rightarrow> 'a::complete_space set"
- assumes "\<forall>n. closed(s n)"
- "\<forall>n. s n \<noteq> {}"
- "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
- "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"
+ assumes
+ "\<forall>n. closed(s n)"
+ "\<forall>n. s n \<noteq> {}"
+ "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
+ "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"
shows "\<exists>a. \<Inter>(range s) = {a}"
-proof-
- obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto
- { fix b assume b:"b \<in> \<Inter>(range s)"
- { fix e::real assume "e>0"
- hence "dist a b < e" using assms(4 )using b using a by blast
+proof -
+ obtain a where a: "\<forall>n. a \<in> s n"
+ using decreasing_closed_nest[of s] using assms by auto
+ {
+ fix b
+ assume b: "b \<in> \<Inter>(range s)"
+ {
+ fix e :: real
+ assume "e > 0"
+ then have "dist a b < e"
+ using assms(4) and b and a by blast
}
- hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz less_le)
+ then have "dist a b = 0"
+ by (metis dist_eq_0_iff dist_nz less_le)
}
- with a have "\<Inter>(range s) = {a}" unfolding image_def by auto
- thus ?thesis ..
+ with a have "\<Inter>(range s) = {a}"
+ unfolding image_def by auto
+ then show ?thesis ..
qed
text{* Cauchy-type criteria for uniform convergence. *}
-lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::complete_space" shows
- "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>
- (\<forall>e>0. \<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x --> dist (s m x) (s n x) < e)" (is "?lhs = ?rhs")
-proof(rule)
+lemma uniformly_convergent_eq_cauchy:
+ fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::complete_space"
+ shows
+ "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>
+ (\<forall>e>0. \<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x --> dist (s m x) (s n x) < e)"
+ (is "?lhs = ?rhs")
+proof
assume ?lhs
- then obtain l where l:"\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e" by auto
- { fix e::real assume "e>0"
- then obtain N::nat where N:"\<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e / 2" using l[THEN spec[where x="e/2"]] by auto
- { fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"
- hence "dist (s m x) (s n x) < e"
+ then obtain l where l:"\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e"
+ by auto
+ {
+ fix e :: real
+ assume "e > 0"
+ then obtain N :: nat where N: "\<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e / 2"
+ using l[THEN spec[where x="e/2"]] by auto
+ {
+ fix n m :: nat and x :: "'b"
+ assume "N \<le> m \<and> N \<le> n \<and> P x"
+ then have "dist (s m x) (s n x) < e"
using N[THEN spec[where x=m], THEN spec[where x=x]]
using N[THEN spec[where x=n], THEN spec[where x=x]]
- using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto }
- hence "\<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x --> dist (s m x) (s n x) < e" by auto }
- thus ?rhs by auto
+ using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto
+ }
+ then have "\<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x --> dist (s m x) (s n x) < e" by auto
+ }
+ then show ?rhs by auto
next
assume ?rhs
- hence "\<forall>x. P x \<longrightarrow> Cauchy (\<lambda>n. s n x)" unfolding cauchy_def apply auto by (erule_tac x=e in allE)auto
- then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]
- using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto
- { fix e::real assume "e>0"
+ then have "\<forall>x. P x \<longrightarrow> Cauchy (\<lambda>n. s n x)"
+ unfolding cauchy_def
+ apply auto
+ apply (erule_tac x=e in allE)
+ apply auto
+ done
+ then obtain l where l: "\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially"
+ unfolding convergent_eq_cauchy[THEN sym]
+ using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"]
+ by auto
+ {
+ fix e :: real
+ assume "e > 0"
then obtain N where N:"\<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x \<longrightarrow> dist (s m x) (s n x) < e/2"
using `?rhs`[THEN spec[where x="e/2"]] by auto
- { fix x assume "P x"
+ {
+ fix x
+ assume "P x"
then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"
- using l[THEN spec[where x=x], unfolded LIMSEQ_def] using `e>0` by(auto elim!: allE[where x="e/2"])
- fix n::nat assume "n\<ge>N"
- hence "dist(s n x)(l x) < e" using `P x`and N[THEN spec[where x=n], THEN spec[where x="N+M"], THEN spec[where x=x]]
- using M[THEN spec[where x="N+M"]] and dist_triangle_half_l[of "s n x" "s (N+M) x" e "l x"] by (auto simp add: dist_commute) }
- hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }
- thus ?lhs by auto
+ using l[THEN spec[where x=x], unfolded LIMSEQ_def] and `e > 0`
+ by (auto elim!: allE[where x="e/2"])
+ fix n :: nat
+ assume "n \<ge> N"
+ then have "dist(s n x)(l x) < e"
+ using `P x`and N[THEN spec[where x=n], THEN spec[where x="N+M"], THEN spec[where x=x]]
+ using M[THEN spec[where x="N+M"]] and dist_triangle_half_l[of "s n x" "s (N+M) x" e "l x"]
+ by (auto simp add: dist_commute)
+ }
+ then have "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e"
+ by auto
+ }
+ then show ?lhs by auto
qed
lemma uniformly_cauchy_imp_uniformly_convergent:
@@ -4048,13 +4360,17 @@
assumes "\<forall>e>0.\<exists>N. \<forall>m (n::nat) x. N \<le> m \<and> N \<le> n \<and> P x --> dist(s m x)(s n x) < e"
"\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"
shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"
-proof-
+proof -
obtain l' where l:"\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l' x) < e"
using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto
moreover
- { fix x assume "P x"
- hence "l x = l' x" using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]
- using l and assms(2) unfolding LIMSEQ_def by blast }
+ {
+ fix x
+ assume "P x"
+ then have "l x = l' x"
+ using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]
+ using l and assms(2) unfolding LIMSEQ_def by blast
+ }
ultimately show ?thesis by auto
qed
@@ -4066,42 +4382,86 @@
lemma continuous_within_eps_delta:
"continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. \<forall>x'\<in> s. dist x' x < d --> dist (f x') (f x) < e)"
unfolding continuous_within and Lim_within
- apply auto unfolding dist_nz[THEN sym] apply(auto del: allE elim!:allE) apply(rule_tac x=d in exI) by auto
-
-lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
- \<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"
+ apply auto
+ unfolding dist_nz[THEN sym]
+ apply (auto del: allE elim!:allE)
+ apply(rule_tac x=d in exI)
+ apply auto
+ done
+
+lemma continuous_at_eps_delta:
+ "continuous (at x) f \<longleftrightarrow> (\<forall>e > 0. \<exists>d > 0. \<forall>x'. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
using continuous_within_eps_delta [of x UNIV f] by simp
text{* Versions in terms of open balls. *}
lemma continuous_within_ball:
- "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
- f ` (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
+ "continuous (at x within s) f \<longleftrightarrow>
+ (\<forall>e > 0. \<exists>d > 0. f ` (ball x d \<inter> s) \<subseteq> ball (f x) e)"
+ (is "?lhs = ?rhs")
proof
assume ?lhs
- { fix e::real assume "e>0"
+ {
+ fix e :: real
+ assume "e > 0"
then obtain d where d: "d>0" "\<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (f x) < e"
using `?lhs`[unfolded continuous_within Lim_within] by auto
- { fix y assume "y\<in>f ` (ball x d \<inter> s)"
- hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]
- apply (auto simp add: dist_commute) apply(erule_tac x=xa in ballE) apply auto using `e>0` by auto
+ {
+ fix y
+ assume "y \<in> f ` (ball x d \<inter> s)"
+ then have "y \<in> ball (f x) e"
+ using d(2)
+ unfolding dist_nz[THEN sym]
+ apply (auto simp add: dist_commute)
+ apply (erule_tac x=xa in ballE)
+ apply auto
+ using `e > 0`
+ apply auto
+ done
}
- hence "\<exists>d>0. f ` (ball x d \<inter> s) \<subseteq> ball (f x) e" using `d>0` unfolding subset_eq ball_def by (auto simp add: dist_commute) }
- thus ?rhs by auto
+ then have "\<exists>d>0. f ` (ball x d \<inter> s) \<subseteq> ball (f x) e"
+ using `d > 0`
+ unfolding subset_eq ball_def by (auto simp add: dist_commute)
+ }
+ then show ?rhs by auto
next
- assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq
- apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto
+ assume ?rhs
+ then show ?lhs
+ unfolding continuous_within Lim_within ball_def subset_eq
+ apply (auto simp add: dist_commute)
+ apply (erule_tac x=e in allE)
+ apply auto
+ done
qed
lemma continuous_at_ball:
"continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f ` (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
proof
- assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
- apply auto apply(erule_tac x=e in allE) apply auto apply(rule_tac x=d in exI) apply auto apply(erule_tac x=xa in allE) apply (auto simp add: dist_commute dist_nz)
- unfolding dist_nz[THEN sym] by auto
+ assume ?lhs
+ then show ?rhs
+ unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
+ apply auto
+ apply (erule_tac x=e in allE)
+ apply auto
+ apply (rule_tac x=d in exI)
+ apply auto
+ apply (erule_tac x=xa in allE)
+ apply (auto simp add: dist_commute dist_nz)
+ unfolding dist_nz[THEN sym]
+ apply auto
+ done
next
- assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
- apply auto apply(erule_tac x=e in allE) apply auto apply(rule_tac x=d in exI) apply auto apply(erule_tac x="f xa" in allE) by (auto simp add: dist_commute dist_nz)
+ assume ?rhs
+ then show ?lhs
+ unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
+ apply auto
+ apply (erule_tac x=e in allE)
+ apply auto
+ apply (rule_tac x=d in exI)
+ apply auto
+ apply (erule_tac x="f xa" in allE)
+ apply (auto simp add: dist_commute dist_nz)
+ done
qed
text{* Define setwise continuity in terms of limits within the set. *}
@@ -4109,27 +4469,25 @@
lemma continuous_on_iff:
"continuous_on s f \<longleftrightarrow>
(\<forall>x\<in>s. \<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
-unfolding continuous_on_def Lim_within
-apply (intro ball_cong [OF refl] all_cong ex_cong)
-apply (rename_tac y, case_tac "y = x", simp)
-apply (simp add: dist_nz)
-done
-
-definition
- uniformly_continuous_on ::
- "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"
-where
- "uniformly_continuous_on s f \<longleftrightarrow>
+ unfolding continuous_on_def Lim_within
+ apply (intro ball_cong [OF refl] all_cong ex_cong)
+ apply (rename_tac y, case_tac "y = x")
+ apply simp
+ apply (simp add: dist_nz)
+ done
+
+definition uniformly_continuous_on :: "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"
+ where "uniformly_continuous_on s f \<longleftrightarrow>
(\<forall>e>0. \<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
text{* Some simple consequential lemmas. *}
lemma uniformly_continuous_imp_continuous:
- " uniformly_continuous_on s f ==> continuous_on s f"
+ "uniformly_continuous_on s f \<Longrightarrow> continuous_on s f"
unfolding uniformly_continuous_on_def continuous_on_iff by blast
lemma continuous_at_imp_continuous_within:
- "continuous (at x) f ==> continuous (at x within s) f"
+ "continuous (at x) f \<Longrightarrow> continuous (at x within s) f"
unfolding continuous_within continuous_at using Lim_at_within by auto
lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"
@@ -4138,14 +4496,15 @@
lemmas continuous_on = continuous_on_def -- "legacy theorem name"
lemma continuous_within_subset:
- "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s
- ==> continuous (at x within t) f"
+ "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous (at x within t) f"
unfolding continuous_within by(metis tendsto_within_subset)
lemma continuous_on_interior:
- shows "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"
- by (erule interiorE, drule (1) continuous_on_subset,
- simp add: continuous_on_eq_continuous_at)
+ "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"
+ apply (erule interiorE)
+ apply (drule (1) continuous_on_subset)
+ apply (simp add: continuous_on_eq_continuous_at)
+ done
lemma continuous_on_eq:
"(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"
@@ -4157,25 +4516,32 @@
lemma continuous_within_sequentially:
fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
shows "continuous (at a within s) f \<longleftrightarrow>
- (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially
- --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")
+ (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially
+ \<longrightarrow> ((f o x) ---> f a) sequentially)"
+ (is "?lhs = ?rhs")
proof
assume ?lhs
- { fix x::"nat \<Rightarrow> 'a" assume x:"\<forall>n. x n \<in> s" "\<forall>e>0. eventually (\<lambda>n. dist (x n) a < e) sequentially"
- fix T::"'b set" assume "open T" and "f a \<in> T"
+ {
+ fix x :: "nat \<Rightarrow> 'a"
+ assume x: "\<forall>n. x n \<in> s" "\<forall>e>0. eventually (\<lambda>n. dist (x n) a < e) sequentially"
+ fix T :: "'b set"
+ assume "open T" and "f a \<in> T"
with `?lhs` obtain d where "d>0" and d:"\<forall>x\<in>s. 0 < dist x a \<and> dist x a < d \<longrightarrow> f x \<in> T"
unfolding continuous_within tendsto_def eventually_at by (auto simp: dist_nz)
have "eventually (\<lambda>n. dist (x n) a < d) sequentially"
using x(2) `d>0` by simp
- hence "eventually (\<lambda>n. (f \<circ> x) n \<in> T) sequentially"
+ then have "eventually (\<lambda>n. (f \<circ> x) n \<in> T) sequentially"
proof eventually_elim
- case (elim n) thus ?case
+ case (elim n)
+ then show ?case
using d x(1) `f a \<in> T` unfolding dist_nz[THEN sym] by auto
qed
}
- thus ?rhs unfolding tendsto_iff unfolding tendsto_def by simp
+ then show ?rhs
+ unfolding tendsto_iff tendsto_def by simp
next
- assume ?rhs thus ?lhs
+ assume ?rhs
+ then show ?lhs
unfolding continuous_within tendsto_def [where l="f a"]
by (simp add: sequentially_imp_eventually_within)
qed
@@ -4192,9 +4558,17 @@
(\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially
--> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")
proof
- assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto
+ assume ?rhs
+ then show ?lhs
+ using continuous_within_sequentially[of _ s f]
+ unfolding continuous_on_eq_continuous_within
+ by auto
next
- assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto
+ assume ?lhs
+ then show ?rhs
+ unfolding continuous_on_eq_continuous_within
+ using continuous_within_sequentially[of _ s f]
+ by auto
qed
lemma uniformly_continuous_on_sequentially:
@@ -4203,67 +4577,108 @@
\<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")
proof
assume ?lhs
- { fix x y assume x:"\<forall>n. x n \<in> s" and y:"\<forall>n. y n \<in> s" and xy:"((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially"
- { fix e::real assume "e>0"
- then obtain d where "d>0" and d:"\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"
+ {
+ fix x y
+ assume x: "\<forall>n. x n \<in> s"
+ and y: "\<forall>n. y n \<in> s"
+ and xy: "((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially"
+ {
+ fix e :: real
+ assume "e > 0"
+ then obtain d where "d > 0" and d: "\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"
using `?lhs`[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto
- obtain N where N:"\<forall>n\<ge>N. dist (x n) (y n) < d" using xy[unfolded LIMSEQ_def dist_norm] and `d>0` by auto
- { fix n assume "n\<ge>N"
- hence "dist (f (x n)) (f (y n)) < e"
- using N[THEN spec[where x=n]] using d[THEN bspec[where x="x n"], THEN bspec[where x="y n"]] using x and y
- unfolding dist_commute by simp }
- hence "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e" by auto }
- hence "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially" unfolding LIMSEQ_def and dist_real_def by auto }
- thus ?rhs by auto
+ obtain N where N: "\<forall>n\<ge>N. dist (x n) (y n) < d"
+ using xy[unfolded LIMSEQ_def dist_norm] and `d>0` by auto
+ {
+ fix n
+ assume "n\<ge>N"
+ then have "dist (f (x n)) (f (y n)) < e"
+ using N[THEN spec[where x=n]]
+ using d[THEN bspec[where x="x n"], THEN bspec[where x="y n"]]
+ using x and y
+ unfolding dist_commute
+ by simp
+ }
+ then have "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"
+ by auto
+ }
+ then have "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially"
+ unfolding LIMSEQ_def and dist_real_def by auto
+ }
+ then show ?rhs by auto
next
assume ?rhs
- { assume "\<not> ?lhs"
- then obtain e where "e>0" "\<forall>d>0. \<exists>x\<in>s. \<exists>x'\<in>s. dist x' x < d \<and> \<not> dist (f x') (f x) < e" unfolding uniformly_continuous_on_def by auto
- then obtain fa where fa:"\<forall>x. 0 < x \<longrightarrow> fst (fa x) \<in> s \<and> snd (fa x) \<in> s \<and> dist (fst (fa x)) (snd (fa x)) < x \<and> \<not> dist (f (fst (fa x))) (f (snd (fa x))) < e"
- using choice[of "\<lambda>d x. d>0 \<longrightarrow> fst x \<in> s \<and> snd x \<in> s \<and> dist (snd x) (fst x) < d \<and> \<not> dist (f (snd x)) (f (fst x)) < e"] unfolding Bex_def
+ {
+ assume "\<not> ?lhs"
+ then obtain e where "e > 0" "\<forall>d>0. \<exists>x\<in>s. \<exists>x'\<in>s. dist x' x < d \<and> \<not> dist (f x') (f x) < e"
+ unfolding uniformly_continuous_on_def by auto
+ then obtain fa where fa:
+ "\<forall>x. 0 < x \<longrightarrow> fst (fa x) \<in> s \<and> snd (fa x) \<in> s \<and> dist (fst (fa x)) (snd (fa x)) < x \<and> \<not> dist (f (fst (fa x))) (f (snd (fa x))) < e"
+ using choice[of "\<lambda>d x. d>0 \<longrightarrow> fst x \<in> s \<and> snd x \<in> s \<and> dist (snd x) (fst x) < d \<and> \<not> dist (f (snd x)) (f (fst x)) < e"]
+ unfolding Bex_def
by (auto simp add: dist_commute)
def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"
def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"
- have xyn:"\<forall>n. x n \<in> s \<and> y n \<in> s" and xy0:"\<forall>n. dist (x n) (y n) < inverse (real n + 1)" and fxy:"\<forall>n. \<not> dist (f (x n)) (f (y n)) < e"
- unfolding x_def and y_def using fa by auto
- { fix e::real assume "e>0"
- then obtain N::nat where "N \<noteq> 0" and N:"0 < inverse (real N) \<and> inverse (real N) < e" unfolding real_arch_inv[of e] by auto
- { fix n::nat assume "n\<ge>N"
- hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and `N\<noteq>0` by auto
+ have xyn: "\<forall>n. x n \<in> s \<and> y n \<in> s"
+ and xy0: "\<forall>n. dist (x n) (y n) < inverse (real n + 1)"
+ and fxy:"\<forall>n. \<not> dist (f (x n)) (f (y n)) < e"
+ unfolding x_def and y_def using fa
+ by auto
+ {
+ fix e :: real
+ assume "e > 0"
+ then obtain N :: nat where "N \<noteq> 0" and N: "0 < inverse (real N) \<and> inverse (real N) < e"
+ unfolding real_arch_inv[of e] by auto
+ {
+ fix n :: nat
+ assume "n \<ge> N"
+ then have "inverse (real n + 1) < inverse (real N)"
+ using real_of_nat_ge_zero and `N\<noteq>0` by auto
also have "\<dots> < e" using N by auto
finally have "inverse (real n + 1) < e" by auto
- hence "dist (x n) (y n) < e" using xy0[THEN spec[where x=n]] by auto }
- hence "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto }
- hence "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e" using `?rhs`[THEN spec[where x=x], THEN spec[where x=y]] and xyn unfolding LIMSEQ_def dist_real_def by auto
- hence False using fxy and `e>0` by auto }
- thus ?lhs unfolding uniformly_continuous_on_def by blast
+ then have "dist (x n) (y n) < e"
+ using xy0[THEN spec[where x=n]] by auto
+ }
+ then have "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto
+ }
+ then have "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"
+ using `?rhs`[THEN spec[where x=x], THEN spec[where x=y]] and xyn
+ unfolding LIMSEQ_def dist_real_def by auto
+ then have False using fxy and `e>0` by auto
+ }
+ then show ?lhs
+ unfolding uniformly_continuous_on_def by blast
qed
text{* The usual transformation theorems. *}
lemma continuous_transform_within:
fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
- assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"
- "continuous (at x within s) f"
+ assumes "0 < d"
+ and "x \<in> s"
+ and "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"
+ and "continuous (at x within s) f"
shows "continuous (at x within s) g"
-unfolding continuous_within
+ unfolding continuous_within
proof (rule Lim_transform_within)
show "0 < d" by fact
show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
using assms(3) by auto
have "f x = g x"
using assms(1,2,3) by auto
- thus "(f ---> g x) (at x within s)"
+ then show "(f ---> g x) (at x within s)"
using assms(4) unfolding continuous_within by simp
qed
lemma continuous_transform_at:
fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
- assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"
- "continuous (at x) f"
+ assumes "0 < d"
+ and "\<forall>x'. dist x' x < d --> f x' = g x'"
+ and "continuous (at x) f"
shows "continuous (at x) g"
using continuous_transform_within [of d x UNIV f g] assms by simp
+
subsubsection {* Structural rules for pointwise continuity *}
lemmas continuous_within_id = continuous_ident
@@ -4276,11 +4691,12 @@
using assms unfolding continuous_def by (rule tendsto_infdist)
lemma continuous_infnorm[continuous_intros]:
- shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. infnorm (f x))"
+ "continuous F f \<Longrightarrow> continuous F (\<lambda>x. infnorm (f x))"
unfolding continuous_def by (rule tendsto_infnorm)
lemma continuous_inner[continuous_intros]:
- assumes "continuous F f" and "continuous F g"
+ assumes "continuous F f"
+ and "continuous F g"
shows "continuous F (\<lambda>x. inner (f x) (g x))"
using assms unfolding continuous_def by (rule tendsto_inner)
@@ -4289,12 +4705,13 @@
subsubsection {* Structural rules for setwise continuity *}
lemma continuous_on_infnorm[continuous_on_intros]:
- shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. infnorm (f x))"
+ "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. infnorm (f x))"
unfolding continuous_on by (fast intro: tendsto_infnorm)
lemma continuous_on_inner[continuous_on_intros]:
fixes g :: "'a::topological_space \<Rightarrow> 'b::real_inner"
- assumes "continuous_on s f" and "continuous_on s g"
+ assumes "continuous_on s f"
+ and "continuous_on s g"
shows "continuous_on s (\<lambda>x. inner (f x) (g x))"
using bounded_bilinear_inner assms
by (rule bounded_bilinear.continuous_on)
@@ -4302,32 +4719,36 @@
subsubsection {* Structural rules for uniform continuity *}
lemma uniformly_continuous_on_id[continuous_on_intros]:
- shows "uniformly_continuous_on s (\<lambda>x. x)"
+ "uniformly_continuous_on s (\<lambda>x. x)"
unfolding uniformly_continuous_on_def by auto
lemma uniformly_continuous_on_const[continuous_on_intros]:
- shows "uniformly_continuous_on s (\<lambda>x. c)"
+ "uniformly_continuous_on s (\<lambda>x. c)"
unfolding uniformly_continuous_on_def by simp
lemma uniformly_continuous_on_dist[continuous_on_intros]:
fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"
assumes "uniformly_continuous_on s f"
- assumes "uniformly_continuous_on s g"
+ 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 d"
+ {
+ fix a b c d :: 'b
+ have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"
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 le = this
- { fix x y
+ {
+ fix x y
assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) ----> 0"
assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) ----> 0"
have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) ----> 0"
by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],
simp add: le)
}
- thus ?thesis using assms unfolding uniformly_continuous_on_sequentially
+ then show ?thesis
+ using assms unfolding uniformly_continuous_on_sequentially
unfolding dist_real_def by simp
qed
@@ -4364,15 +4785,17 @@
lemma uniformly_continuous_on_add[continuous_on_intros]:
fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
assumes "uniformly_continuous_on s f"
- assumes "uniformly_continuous_on s g"
+ and "uniformly_continuous_on s g"
shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
- using assms unfolding uniformly_continuous_on_sequentially
+ using assms
+ 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[continuous_on_intros]:
fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
- assumes "uniformly_continuous_on s f" and "uniformly_continuous_on s g"
+ assumes "uniformly_continuous_on s f"
+ and "uniformly_continuous_on s g"
shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"
unfolding ab_diff_minus using assms
by (intro uniformly_continuous_on_add uniformly_continuous_on_minus)
@@ -4385,22 +4808,32 @@
assumes "uniformly_continuous_on s f" "uniformly_continuous_on (f ` s) g"
shows "uniformly_continuous_on s (g o f)"
proof-
- { fix e::real assume "e>0"
- then obtain d where "d>0" and d:"\<forall>x\<in>f ` s. \<forall>x'\<in>f ` s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e" using assms(2) unfolding uniformly_continuous_on_def by auto
- obtain d' where "d'>0" "\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d' \<longrightarrow> dist (f x') (f x) < d" using `d>0` using assms(1) unfolding uniformly_continuous_on_def by auto
- hence "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist ((g \<circ> f) x') ((g \<circ> f) x) < e" using `d>0` using d by auto }
- thus ?thesis using assms unfolding uniformly_continuous_on_def by auto
+ {
+ fix e :: real
+ assume "e > 0"
+ then obtain d where "d > 0"
+ and d: "\<forall>x\<in>f ` s. \<forall>x'\<in>f ` s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"
+ using assms(2) unfolding uniformly_continuous_on_def by auto
+ obtain d' where "d'>0" "\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d' \<longrightarrow> dist (f x') (f x) < d"
+ using `d > 0` using assms(1) unfolding uniformly_continuous_on_def by auto
+ then have "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist ((g \<circ> f) x') ((g \<circ> f) x) < e"
+ using `d>0` using d by auto
+ }
+ then show ?thesis
+ using assms unfolding uniformly_continuous_on_def by auto
qed
text{* Continuity in terms of open preimages. *}
lemma continuous_at_open:
- shows "continuous (at x) f \<longleftrightarrow> (\<forall>t. open t \<and> f x \<in> t --> (\<exists>s. open s \<and> x \<in> s \<and> (\<forall>x' \<in> s. (f x') \<in> t)))"
-unfolding continuous_within_topological [of x UNIV f]
-unfolding imp_conjL by (intro all_cong imp_cong ex_cong conj_cong refl) auto
+ "continuous (at x) f \<longleftrightarrow> (\<forall>t. open t \<and> f x \<in> t --> (\<exists>s. open s \<and> x \<in> s \<and> (\<forall>x' \<in> s. (f x') \<in> t)))"
+ unfolding continuous_within_topological [of x UNIV f]
+ unfolding imp_conjL
+ by (intro all_cong imp_cong ex_cong conj_cong refl) auto
lemma continuous_imp_tendsto:
- assumes "continuous (at x0) f" and "x ----> x0"
+ assumes "continuous (at x0) f"
+ and "x ----> x0"
shows "(f \<circ> x) ----> (f x0)"
proof (rule topological_tendstoI)
fix S
@@ -4415,15 +4848,17 @@
lemma continuous_on_open:
"continuous_on s f \<longleftrightarrow>
- (\<forall>t. openin (subtopology euclidean (f ` s)) t
- --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
+ (\<forall>t. openin (subtopology euclidean (f ` s)) t \<longrightarrow>
+ openin (subtopology euclidean s) {x \<in> s. f x \<in> t})"
unfolding continuous_on_open_invariant openin_open Int_def vimage_def Int_commute
by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)
text {* Similarly in terms of closed sets. *}
lemma continuous_on_closed:
- shows "continuous_on s f \<longleftrightarrow> (\<forall>t. closedin (subtopology euclidean (f ` s)) t --> closedin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
+ "continuous_on s f \<longleftrightarrow>
+ (\<forall>t. closedin (subtopology euclidean (f ` s)) t \<longrightarrow>
+ closedin (subtopology euclidean s) {x \<in> s. f x \<in> t})"
unfolding continuous_on_closed_invariant closedin_closed Int_def vimage_def Int_commute
by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)
@@ -4432,22 +4867,28 @@
lemma continuous_open_in_preimage:
assumes "continuous_on s f" "open t"
shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
-proof-
- have *:"\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f ` s)" by auto
+proof -
+ have *: "\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f ` s)"
+ by auto
have "openin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
using openin_open_Int[of t "f ` s", OF assms(2)] unfolding openin_open by auto
- thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f ` s"]] using * by auto
+ then show ?thesis
+ using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f ` s"]]
+ using * by auto
qed
lemma continuous_closed_in_preimage:
assumes "continuous_on s f" "closed t"
shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
-proof-
- have *:"\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f ` s)" by auto
+proof -
+ have *: "\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f ` s)"
+ by auto
have "closedin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
- using closedin_closed_Int[of t "f ` s", OF assms(2)] unfolding Int_commute by auto
- thus ?thesis
- using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f ` s"]] using * by auto
+ using closedin_closed_Int[of t "f ` s", OF assms(2)] unfolding Int_commute
+ by auto
+ then show ?thesis
+ using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f ` s"]]
+ using * by auto
qed
lemma continuous_open_preimage:
@@ -4456,32 +4897,32 @@
proof-
obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"
using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto
- thus ?thesis using open_Int[of s T, OF assms(2)] by auto
+ then show ?thesis
+ using open_Int[of s T, OF assms(2)] by auto
qed
lemma continuous_closed_preimage:
assumes "continuous_on s f" "closed s" "closed t"
shows "closed {x \<in> s. f x \<in> t}"
proof-
- obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"
- using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto
- thus ?thesis using closed_Int[of s T, OF assms(2)] by auto
+ obtain T where "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"
+ using continuous_closed_in_preimage[OF assms(1,3)]
+ unfolding closedin_closed by auto
+ then show ?thesis using closed_Int[of s T, OF assms(2)] by auto
qed
lemma continuous_open_preimage_univ:
- shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"
+ "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"
using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto
lemma continuous_closed_preimage_univ:
- shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"
+ "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"
using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto
-lemma continuous_open_vimage:
- shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f -` s)"
+lemma continuous_open_vimage: "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f -` s)"
unfolding vimage_def by (rule continuous_open_preimage_univ)
-lemma continuous_closed_vimage:
- shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f -` s)"
+lemma continuous_closed_vimage: "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f -` s)"
unfolding vimage_def by (rule continuous_closed_preimage_univ)
lemma interior_image_subset:
@@ -4490,7 +4931,7 @@
proof
fix x assume "x \<in> interior (f ` s)"
then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f ` s" ..
- hence "x \<in> f ` s" by auto
+ then have "x \<in> f ` s" by auto
then obtain y where y: "y \<in> s" "x = f y" by auto
have "open (vimage f T)"
using assms(1) `open T` by (rule continuous_open_vimage)
@@ -4517,71 +4958,89 @@
lemma continuous_constant_on_closure:
fixes f :: "_ \<Rightarrow> 'b::t1_space"
assumes "continuous_on (closure s) f"
- "\<forall>x \<in> s. f x = a"
+ and "\<forall>x \<in> s. f x = a"
shows "\<forall>x \<in> (closure s). f x = a"
using continuous_closed_preimage_constant[of "closure s" f a]
- assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto
+ assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset
+ unfolding subset_eq
+ by auto
lemma image_closure_subset:
assumes "continuous_on (closure s) f" "closed t" "(f ` s) \<subseteq> t"
shows "f ` (closure s) \<subseteq> t"
-proof-
- have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto
+proof -
+ have "s \<subseteq> {x \<in> closure s. f x \<in> t}"
+ using assms(3) closure_subset by auto
moreover have "closed {x \<in> closure s. f x \<in> t}"
using continuous_closed_preimage[OF assms(1)] and assms(2) by auto
ultimately have "closure s = {x \<in> closure s . f x \<in> t}"
using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto
- thus ?thesis by auto
+ then show ?thesis by auto
qed
lemma continuous_on_closure_norm_le:
fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
- assumes "continuous_on (closure s) f" "\<forall>y \<in> s. norm(f y) \<le> b" "x \<in> (closure s)"
+ assumes "continuous_on (closure s) f"
+ and "\<forall>y \<in> s. norm(f y) \<le> b"
+ and "x \<in> (closure s)"
shows "norm(f x) \<le> b"
-proof-
- have *:"f ` s \<subseteq> cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] by auto
+proof -
+ have *: "f ` s \<subseteq> cball 0 b"
+ using assms(2)[unfolded mem_cball_0[THEN sym]] by auto
show ?thesis
using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)
- unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)
+ unfolding subset_eq
+ apply (erule_tac x="f x" in ballE)
+ apply (auto simp add: dist_norm)
+ done
qed
text {* Making a continuous function avoid some value in a neighbourhood. *}
lemma continuous_within_avoid:
fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"
- assumes "continuous (at x within s) f" and "f x \<noteq> a"
+ assumes "continuous (at x within s) f"
+ and "f x \<noteq> a"
shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"
proof-
obtain U where "open U" and "f x \<in> U" and "a \<notin> U"
using t1_space [OF `f x \<noteq> a`] by fast
have "(f ---> f x) (at x within s)"
using assms(1) by (simp add: continuous_within)
- hence "eventually (\<lambda>y. f y \<in> U) (at x within s)"
+ then have "eventually (\<lambda>y. f y \<in> U) (at x within s)"
using `open U` and `f x \<in> U`
unfolding tendsto_def by fast
- hence "eventually (\<lambda>y. f y \<noteq> a) (at x within s)"
+ then have "eventually (\<lambda>y. f y \<noteq> a) (at x within s)"
using `a \<notin> U` by (fast elim: eventually_mono [rotated])
- thus ?thesis
+ then show ?thesis
using `f x \<noteq> a` by (auto simp: dist_commute zero_less_dist_iff eventually_at)
qed
lemma continuous_at_avoid:
fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"
- assumes "continuous (at x) f" and "f x \<noteq> a"
+ assumes "continuous (at x) f"
+ and "f x \<noteq> a"
shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
using assms continuous_within_avoid[of x UNIV f a] by simp
lemma continuous_on_avoid:
fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"
- assumes "continuous_on s f" "x \<in> s" "f x \<noteq> a"
+ assumes "continuous_on s f"
+ and "x \<in> s"
+ and "f x \<noteq> a"
shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"
-using assms(1)[unfolded continuous_on_eq_continuous_within, THEN bspec[where x=x], OF assms(2)] continuous_within_avoid[of x s f a] assms(3) by auto
+ using assms(1)[unfolded continuous_on_eq_continuous_within, THEN bspec[where x=x],
+ OF assms(2)] continuous_within_avoid[of x s f a]
+ using assms(3)
+ by auto
lemma continuous_on_open_avoid:
fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"
assumes "continuous_on s f" "open s" "x \<in> s" "f x \<noteq> a"
shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
-using assms(1)[unfolded continuous_on_eq_continuous_at[OF assms(2)], THEN bspec[where x=x], OF assms(3)] continuous_at_avoid[of x f a] assms(4) by auto
+ using assms(1)[unfolded continuous_on_eq_continuous_at[OF assms(2)], THEN bspec[where x=x], OF assms(3)]
+ using continuous_at_avoid[of x f a] assms(4)
+ by auto
text {* Proving a function is constant by proving open-ness of level set. *}
@@ -4589,22 +5048,28 @@
fixes f :: "_ \<Rightarrow> 'b::t1_space"
shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
openin (subtopology euclidean s) {x \<in> s. f x = a}
- ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"
-unfolding connected_clopen using continuous_closed_in_preimage_constant by auto
+ \<Longrightarrow> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"
+ unfolding connected_clopen
+ using continuous_closed_in_preimage_constant by auto
lemma continuous_levelset_open_in:
fixes f :: "_ \<Rightarrow> 'b::t1_space"
shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>
(\<exists>x \<in> s. f x = a) ==> (\<forall>x \<in> s. f x = a)"
-using continuous_levelset_open_in_cases[of s f ]
-by meson
+ using continuous_levelset_open_in_cases[of s f ]
+ by meson
lemma continuous_levelset_open:
fixes f :: "_ \<Rightarrow> 'b::t1_space"
- assumes "connected s" "continuous_on s f" "open {x \<in> s. f x = a}" "\<exists>x \<in> s. f x = a"
+ assumes "connected s"
+ and "continuous_on s f"
+ and "open {x \<in> s. f x = a}"
+ and "\<exists>x \<in> s. f x = a"
shows "\<forall>x \<in> s. f x = a"
-using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast
+ using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open]
+ using assms (3,4)
+ by fast
text {* Some arithmetical combinations (more to prove). *}
@@ -4612,18 +5077,38 @@
fixes s :: "'a::real_normed_vector set"
assumes "c \<noteq> 0" "open s"
shows "open((\<lambda>x. c *\<^sub>R x) ` s)"
-proof-
- { fix x assume "x \<in> s"
- then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]] by auto
- have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using mult_pos_pos[OF `e>0`] by auto
+proof -
+ {
+ fix x
+ assume "x \<in> s"
+ then obtain e where "e>0"
+ and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]]
+ by auto
+ have "e * abs c > 0"
+ using assms(1)[unfolded zero_less_abs_iff[THEN sym]]
+ using mult_pos_pos[OF `e>0`]
+ by auto
moreover
- { fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"
- hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm
+ {
+ fix y
+ assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"
+ then have "norm ((1 / c) *\<^sub>R y - x) < e"
+ unfolding dist_norm
using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)
- hence "y \<in> op *\<^sub>R c ` s" using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "op *\<^sub>R c"] e[THEN spec[where x="(1 / c) *\<^sub>R y"]] assms(1) unfolding dist_norm scaleR_scaleR by auto }
- ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> op *\<^sub>R c ` s" apply(rule_tac x="e * abs c" in exI) by auto }
- thus ?thesis unfolding open_dist by auto
+ then have "y \<in> op *\<^sub>R c ` s"
+ using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "op *\<^sub>R c"]
+ using e[THEN spec[where x="(1 / c) *\<^sub>R y"]]
+ using assms(1)
+ unfolding dist_norm scaleR_scaleR
+ by auto
+ }
+ ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> op *\<^sub>R c ` s"
+ apply (rule_tac x="e * abs c" in exI)
+ apply auto
+ done
+ }
+ then show ?thesis unfolding open_dist by auto
qed
lemma minus_image_eq_vimage:
@@ -4640,83 +5125,124 @@
lemma open_translation:
fixes s :: "'a::real_normed_vector set"
assumes "open s" shows "open((\<lambda>x. a + x) ` s)"
-proof-
- { fix x have "continuous (at x) (\<lambda>x. x - a)"
- by (intro continuous_diff continuous_at_id continuous_const) }
- moreover have "{x. x - a \<in> s} = op + a ` s" by force
- ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto
+proof -
+ {
+ fix x
+ have "continuous (at x) (\<lambda>x. x - a)"
+ by (intro continuous_diff continuous_at_id continuous_const)
+ }
+ moreover have "{x. x - a \<in> s} = op + a ` s"
+ by force
+ ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s]
+ using assms by auto
qed
lemma open_affinity:
fixes s :: "'a::real_normed_vector set"
assumes "open s" "c \<noteq> 0"
shows "open ((\<lambda>x. a + c *\<^sub>R x) ` s)"
-proof-
- have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..
- have "op + a ` op *\<^sub>R c ` s = (op + a \<circ> op *\<^sub>R c) ` s" by auto
- thus ?thesis using assms open_translation[of "op *\<^sub>R c ` s" a] unfolding * by auto
+proof -
+ have *: "(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)"
+ unfolding o_def ..
+ have "op + a ` op *\<^sub>R c ` s = (op + a \<circ> op *\<^sub>R c) ` s"
+ by auto
+ then show ?thesis
+ using assms open_translation[of "op *\<^sub>R c ` s" a]
+ unfolding *
+ by auto
qed
lemma interior_translation:
fixes s :: "'a::real_normed_vector set"
shows "interior ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (interior s)"
proof (rule set_eqI, rule)
- fix x assume "x \<in> interior (op + a ` s)"
- then obtain e where "e>0" and e:"ball x e \<subseteq> op + a ` s" unfolding mem_interior by auto
- hence "ball (x - a) e \<subseteq> s" unfolding subset_eq Ball_def mem_ball dist_norm apply auto apply(erule_tac x="a + xa" in allE) unfolding ab_group_add_class.diff_diff_eq[THEN sym] by auto
- thus "x \<in> op + a ` interior s" unfolding image_iff apply(rule_tac x="x - a" in bexI) unfolding mem_interior using `e > 0` by auto
+ fix x
+ assume "x \<in> interior (op + a ` s)"
+ then obtain e where "e > 0" and e: "ball x e \<subseteq> op + a ` s"
+ unfolding mem_interior by auto
+ then have "ball (x - a) e \<subseteq> s"
+ unfolding subset_eq Ball_def mem_ball dist_norm
+ apply auto
+ apply (erule_tac x="a + xa" in allE)
+ unfolding ab_group_add_class.diff_diff_eq[THEN sym]
+ apply auto
+ done
+ then show "x \<in> op + a ` interior s"
+ unfolding image_iff
+ apply (rule_tac x="x - a" in bexI)
+ unfolding mem_interior
+ using `e > 0`
+ apply auto
+ done
next
- fix x assume "x \<in> op + a ` interior s"
- then obtain y e where "e>0" and e:"ball y e \<subseteq> s" and y:"x = a + y" unfolding image_iff Bex_def mem_interior by auto
- { fix z have *:"a + y - z = y + a - z" by auto
- assume "z\<in>ball x e"
- hence "z - a \<in> s" using e[unfolded subset_eq, THEN bspec[where x="z - a"]] unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 * by auto
- hence "z \<in> op + a ` s" unfolding image_iff by(auto intro!: bexI[where x="z - a"]) }
- hence "ball x e \<subseteq> op + a ` s" unfolding subset_eq by auto
- thus "x \<in> interior (op + a ` s)" unfolding mem_interior using `e>0` by auto
+ fix x
+ assume "x \<in> op + a ` interior s"
+ then obtain y e where "e > 0" and e: "ball y e \<subseteq> s" and y: "x = a + y"
+ unfolding image_iff Bex_def mem_interior by auto
+ {
+ fix z
+ have *: "a + y - z = y + a - z" by auto
+ assume "z \<in> ball x e"
+ then have "z - a \<in> s"
+ using e[unfolded subset_eq, THEN bspec[where x="z - a"]]
+ unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 *
+ by auto
+ then have "z \<in> op + a ` s"
+ unfolding image_iff by (auto intro!: bexI[where x="z - a"])
+ }
+ then have "ball x e \<subseteq> op + a ` s"
+ unfolding subset_eq by auto
+ then show "x \<in> interior (op + a ` s)"
+ unfolding mem_interior using `e > 0` by auto
qed
text {* Topological properties of linear functions. *}
lemma linear_lim_0:
- assumes "bounded_linear f" shows "(f ---> 0) (at (0))"
-proof-
+ assumes "bounded_linear f"
+ shows "(f ---> 0) (at (0))"
+proof -
interpret f: bounded_linear f by fact
have "(f ---> f 0) (at 0)"
using tendsto_ident_at by (rule f.tendsto)
- thus ?thesis unfolding f.zero .
+ then show ?thesis unfolding f.zero .
qed
lemma linear_continuous_at:
- assumes "bounded_linear f" shows "continuous (at a) f"
+ assumes "bounded_linear f"
+ shows "continuous (at a) f"
unfolding continuous_at using assms
apply (rule bounded_linear.tendsto)
apply (rule tendsto_ident_at)
done
lemma linear_continuous_within:
- shows "bounded_linear f ==> continuous (at x within s) f"
+ "bounded_linear f ==> continuous (at x within s) f"
using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto
lemma linear_continuous_on:
- shows "bounded_linear f ==> continuous_on s f"
+ "bounded_linear f ==> continuous_on s f"
using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto
text {* Also bilinear functions, in composition form. *}
lemma bilinear_continuous_at_compose:
- shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h
- ==> continuous (at x) (\<lambda>x. h (f x) (g x))"
- unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto
+ "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h \<Longrightarrow>
+ continuous (at x) (\<lambda>x. h (f x) (g x))"
+ unfolding continuous_at
+ using Lim_bilinear[of f "f x" "(at x)" g "g x" h]
+ by auto
lemma bilinear_continuous_within_compose:
- shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h
- ==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"
- unfolding continuous_within using Lim_bilinear[of f "f x"] by auto
+ "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h \<Longrightarrow>
+ continuous (at x within s) (\<lambda>x. h (f x) (g x))"
+ unfolding continuous_within
+ using Lim_bilinear[of f "f x"]
+ by auto
lemma bilinear_continuous_on_compose:
- shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h
- ==> continuous_on s (\<lambda>x. h (f x) (g x))"
+ "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h \<Longrightarrow>
+ continuous_on s (\<lambda>x. h (f x) (g x))"
unfolding continuous_on_def
by (fast elim: bounded_bilinear.tendsto)
@@ -4729,13 +5255,13 @@
proof safe
fix C
assume "compact S" and "\<forall>c\<in>C. openin (subtopology euclidean S) c" and "S \<subseteq> \<Union>C"
- hence "\<forall>c\<in>{T. open T \<and> S \<inter> T \<in> C}. open c" and "S \<subseteq> \<Union>{T. open T \<and> S \<inter> T \<in> C}"
+ then have "\<forall>c\<in>{T. open T \<and> S \<inter> T \<in> C}. open c" and "S \<subseteq> \<Union>{T. open T \<and> S \<inter> T \<in> C}"
unfolding openin_open by force+
with `compact S` obtain D where "D \<subseteq> {T. open T \<and> S \<inter> T \<in> C}" and "finite D" and "S \<subseteq> \<Union>D"
by (rule compactE)
- hence "image (\<lambda>T. S \<inter> T) D \<subseteq> C \<and> finite (image (\<lambda>T. S \<inter> T) D) \<and> S \<subseteq> \<Union>(image (\<lambda>T. S \<inter> T) D)"
+ then have "image (\<lambda>T. S \<inter> T) D \<subseteq> C \<and> finite (image (\<lambda>T. S \<inter> T) D) \<and> S \<subseteq> \<Union>(image (\<lambda>T. S \<inter> T) D)"
by auto
- thus "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..
+ then show "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..
next
assume 1: "\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>
(\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D)"
@@ -4744,7 +5270,7 @@
fix C
let ?C = "image (\<lambda>T. S \<inter> T) C"
assume "\<forall>t\<in>C. open t" and "S \<subseteq> \<Union>C"
- hence "(\<forall>c\<in>?C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>?C"
+ then have "(\<forall>c\<in>?C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>?C"
unfolding openin_open by auto
with 1 obtain D where "D \<subseteq> ?C" and "finite D" and "S \<subseteq> \<Union>D"
by metis
@@ -4762,37 +5288,43 @@
apply (erule rev_bexI, fast)
done
qed
- thus "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..
+ then show "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..
qed
qed
lemma connected_continuous_image:
assumes "continuous_on s f" "connected s"
shows "connected(f ` s)"
-proof-
- { fix T assume as: "T \<noteq> {}" "T \<noteq> f ` s" "openin (subtopology euclidean (f ` s)) T" "closedin (subtopology euclidean (f ` s)) T"
+proof -
+ {
+ fix T
+ assume as: "T \<noteq> {}" "T \<noteq> f ` s" "openin (subtopology euclidean (f ` s)) T" "closedin (subtopology euclidean (f ` s)) T"
have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"
using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]
using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]
using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto
- hence False using as(1,2)
- using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }
- thus ?thesis unfolding connected_clopen by auto
+ then have False using as(1,2)
+ using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto
+ }
+ then show ?thesis
+ unfolding connected_clopen by auto
qed
text {* Continuity implies uniform continuity on a compact domain. *}
-
+
lemma compact_uniformly_continuous:
assumes f: "continuous_on s f" and s: "compact s"
shows "uniformly_continuous_on s f"
unfolding uniformly_continuous_on_def
proof (cases, safe)
- fix e :: real assume "0 < e" "s \<noteq> {}"
+ fix e :: real
+ assume "0 < e" "s \<noteq> {}"
def [simp]: R \<equiv> "{(y, d). y \<in> s \<and> 0 < d \<and> ball y d \<inter> s \<subseteq> {x \<in> s. f x \<in> ball (f y) (e/2) } }"
let ?b = "(\<lambda>(y, d). ball y (d/2))"
have "(\<forall>r\<in>R. open (?b r))" "s \<subseteq> (\<Union>r\<in>R. ?b r)"
proof safe
- fix y assume "y \<in> s"
+ fix y
+ assume "y \<in> s"
from continuous_open_in_preimage[OF f open_ball]
obtain T where "open T" and T: "{x \<in> s. f x \<in> ball (f y) (e/2)} = T \<inter> s"
unfolding openin_subtopology open_openin by metis
@@ -4807,7 +5339,8 @@
by (subst Min_gr_iff) auto
show "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"
proof (rule, safe)
- fix x x' assume in_s: "x' \<in> s" "x \<in> s"
+ fix x x'
+ assume in_s: "x' \<in> s" "x \<in> s"
with D obtain y d where x: "x \<in> ball y (d/2)" "(y, d) \<in> D"
by blast
moreover assume "dist x x' < Min (snd`D) / 2"
@@ -4823,11 +5356,13 @@
lemma continuous_uniform_limit:
fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::metric_space"
assumes "\<not> trivial_limit F"
- assumes "eventually (\<lambda>n. continuous_on s (f n)) F"
- assumes "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"
+ and "eventually (\<lambda>n. continuous_on s (f n)) F"
+ and "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"
shows "continuous_on s g"
-proof-
- { fix x and e::real assume "x\<in>s" "e>0"
+proof -
+ {
+ fix x and e :: real
+ assume "x\<in>s" "e>0"
have "eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e / 3) F"
using `e>0` assms(3)[THEN spec[where x="e/3"]] by auto
from eventually_happens [OF eventually_conj [OF this assms(2)]]
@@ -4836,29 +5371,33 @@
have "e / 3 > 0" using `e>0` by auto
then obtain d where "d>0" and d:"\<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f n x') (f n x) < e / 3"
using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF `x\<in>s`, THEN spec[where x="e/3"]] by blast
- { fix y assume "y \<in> s" and "dist y x < d"
- hence "dist (f n y) (f n x) < e / 3"
+ {
+ fix y
+ assume "y \<in> s" and "dist y x < d"
+ then have "dist (f n y) (f n x) < e / 3"
by (rule d [rule_format])
- hence "dist (f n y) (g x) < 2 * e / 3"
+ then have "dist (f n y) (g x) < 2 * e / 3"
using dist_triangle [of "f n y" "g x" "f n x"]
using n(1)[THEN bspec[where x=x], OF `x\<in>s`]
by auto
- hence "dist (g y) (g x) < e"
+ then have "dist (g y) (g x) < e"
using n(1)[THEN bspec[where x=y], OF `y\<in>s`]
using dist_triangle3 [of "g y" "g x" "f n y"]
- by auto }
- hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"
- using `d>0` by auto }
- thus ?thesis unfolding continuous_on_iff by auto
+ by auto
+ }
+ then have "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"
+ using `d>0` by auto
+ }
+ then show ?thesis
+ unfolding continuous_on_iff by auto
qed
subsection {* Topological stuff lifted from and dropped to R *}
lemma open_real:
- fixes s :: "real set" shows
- "open s \<longleftrightarrow>
- (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")
+ fixes s :: "real set"
+ shows "open s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)"
unfolding open_dist dist_norm by simp
lemma islimpt_approachable_real:
@@ -4868,23 +5407,31 @@
lemma closed_real:
fixes s :: "real set"
- shows "closed s \<longleftrightarrow>
- (\<forall>x. (\<forall>e>0. \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)
- --> x \<in> s)"
+ shows "closed s \<longleftrightarrow> (\<forall>x. (\<forall>e>0. \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e) \<longrightarrow> x \<in> s)"
unfolding closed_limpt islimpt_approachable dist_norm by simp
lemma continuous_at_real_range:
fixes f :: "'a::real_normed_vector \<Rightarrow> real"
- shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
- \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"
- unfolding continuous_at unfolding Lim_at
- unfolding dist_nz[THEN sym] unfolding dist_norm apply auto
- apply(erule_tac x=e in allE) apply auto apply (rule_tac x=d in exI) apply auto apply (erule_tac x=x' in allE) apply auto
- apply(erule_tac x=e in allE) by auto
+ shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"
+ unfolding continuous_at
+ unfolding Lim_at
+ unfolding dist_nz[THEN sym]
+ unfolding dist_norm
+ apply auto
+ apply (erule_tac x=e in allE)
+ apply auto
+ apply (rule_tac x=d in exI)
+ apply auto
+ apply (erule_tac x=x' in allE)
+ apply auto
+ apply (erule_tac x=e in allE)
+ apply auto
+ done
lemma continuous_on_real_range:
fixes f :: "'a::real_normed_vector \<Rightarrow> real"
- shows "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. \<forall>e>0. \<exists>d>0. (\<forall>x' \<in> s. norm(x' - x) < d --> abs(f x' - f x) < e))"
+ shows "continuous_on s f \<longleftrightarrow>
+ (\<forall>x \<in> s. \<forall>e>0. \<exists>d>0. (\<forall>x' \<in> s. norm(x' - x) < d \<longrightarrow> abs(f x' - f x) < e))"
unfolding continuous_on_iff dist_norm by simp
text {* Hence some handy theorems on distance, diameter etc. of/from a set. *}
@@ -4893,11 +5440,13 @@
assumes "compact s" "s \<noteq> {}"
shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a y \<le> dist a x"
proof (rule continuous_attains_sup [OF assms])
- { fix x assume "x\<in>s"
+ {
+ fix x
+ assume "x\<in>s"
have "(dist a ---> dist a x) (at x within s)"
by (intro tendsto_dist tendsto_const tendsto_ident_at)
}
- thus "continuous_on s (dist a)"
+ then show "continuous_on s (dist a)"
unfolding continuous_on ..
qed
@@ -4907,63 +5456,68 @@
fixes a :: "'a::heine_borel"
assumes "closed s" "s \<noteq> {}"
shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a x \<le> dist a y"
-proof-
+proof -
from assms(2) obtain b where "b \<in> s" by auto
let ?B = "s \<inter> cball a (dist b a)"
- have "?B \<noteq> {}" using `b \<in> s` by (auto simp add: dist_commute)
+ have "?B \<noteq> {}" using `b \<in> s`
+ by (auto simp add: dist_commute)
moreover have "continuous_on ?B (dist a)"
by (auto intro!: continuous_at_imp_continuous_on continuous_dist continuous_at_id continuous_const)
moreover have "compact ?B"
by (intro closed_inter_compact `closed s` compact_cball)
ultimately obtain x where "x \<in> ?B" "\<forall>y\<in>?B. dist a x \<le> dist a y"
by (metis continuous_attains_inf)
- thus ?thesis by fastforce
+ then show ?thesis by fastforce
qed
subsection {* Pasted sets *}
lemma bounded_Times:
- assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"
-proof-
+ assumes "bounded s" "bounded t"
+ shows "bounded (s \<times> t)"
+proof -
obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"
using assms [unfolded bounded_def] by auto
then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<^sup>2 + b\<^sup>2)"
by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)
- thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
+ then show ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
qed
lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
-by (induct x) simp
+ by (induct x) simp
lemma seq_compact_Times: "seq_compact s \<Longrightarrow> seq_compact t \<Longrightarrow> seq_compact (s \<times> t)"
-unfolding seq_compact_def
-apply clarify
-apply (drule_tac x="fst \<circ> f" in spec)
-apply (drule mp, simp add: mem_Times_iff)
-apply (clarify, rename_tac l1 r1)
-apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
-apply (drule mp, simp add: mem_Times_iff)
-apply (clarify, rename_tac l2 r2)
-apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
-apply (rule_tac x="r1 \<circ> r2" in exI)
-apply (rule conjI, simp add: subseq_def)
-apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption)
-apply (drule (1) tendsto_Pair) back
-apply (simp add: o_def)
-done
-
-lemma compact_Times:
+ unfolding seq_compact_def
+ apply clarify
+ apply (drule_tac x="fst \<circ> f" in spec)
+ apply (drule mp, simp add: mem_Times_iff)
+ apply (clarify, rename_tac l1 r1)
+ apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
+ apply (drule mp, simp add: mem_Times_iff)
+ apply (clarify, rename_tac l2 r2)
+ apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
+ apply (rule_tac x="r1 \<circ> r2" in exI)
+ apply (rule conjI, simp add: subseq_def)
+ apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption)
+ apply (drule (1) tendsto_Pair) back
+ apply (simp add: o_def)
+ done
+
+lemma compact_Times:
assumes "compact s" "compact t"
shows "compact (s \<times> t)"
proof (rule compactI)
- fix C assume C: "\<forall>t\<in>C. open t" "s \<times> t \<subseteq> \<Union>C"
+ fix C
+ assume C: "\<forall>t\<in>C. open t" "s \<times> t \<subseteq> \<Union>C"
have "\<forall>x\<in>s. \<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"
proof
- fix x assume "x \<in> s"
+ fix x
+ assume "x \<in> s"
have "\<forall>y\<in>t. \<exists>a b c. c \<in> C \<and> open a \<and> open b \<and> x \<in> a \<and> y \<in> b \<and> a \<times> b \<subseteq> c" (is "\<forall>y\<in>t. ?P y")
- proof
- fix y assume "y \<in> t"
+ proof
+ fix y
+ assume "y \<in> t"
with `x \<in> s` C obtain c where "c \<in> C" "(x, y) \<in> c" "open c" by auto
then show "?P y" by (auto elim!: open_prod_elim)
qed
@@ -4981,12 +5535,18 @@
then obtain a d where a: "\<forall>x\<in>s. open (a x)" "s \<subseteq> (\<Union>x\<in>s. a x)"
and d: "\<And>x. x \<in> s \<Longrightarrow> d x \<subseteq> C \<and> finite (d x) \<and> a x \<times> t \<subseteq> \<Union>d x"
unfolding subset_eq UN_iff by metis
- moreover from compactE_image[OF `compact s` a] obtain e where e: "e \<subseteq> s" "finite e"
- and s: "s \<subseteq> (\<Union>x\<in>e. a x)" by auto
+ moreover
+ from compactE_image[OF `compact s` a]
+ obtain e where e: "e \<subseteq> s" "finite e" and s: "s \<subseteq> (\<Union>x\<in>e. a x)"
+ by auto
moreover
- { from s have "s \<times> t \<subseteq> (\<Union>x\<in>e. a x \<times> t)" by auto
- also have "\<dots> \<subseteq> (\<Union>x\<in>e. \<Union>d x)" using d `e \<subseteq> s` by (intro UN_mono) auto
- finally have "s \<times> t \<subseteq> (\<Union>x\<in>e. \<Union>d x)" . }
+ {
+ from s have "s \<times> t \<subseteq> (\<Union>x\<in>e. a x \<times> t)"
+ by auto
+ also have "\<dots> \<subseteq> (\<Union>x\<in>e. \<Union>d x)"
+ using d `e \<subseteq> s` by (intro UN_mono) auto
+ finally have "s \<times> t \<subseteq> (\<Union>x\<in>e. \<Union>d x)" .
+ }
ultimately show "\<exists>C'\<subseteq>C. finite C' \<and> s \<times> t \<subseteq> \<Union>C'"
by (intro exI[of _ "(\<Union>x\<in>e. d x)"]) (auto simp add: subset_eq)
qed
@@ -4995,28 +5555,38 @@
lemma compact_scaling:
fixes s :: "'a::real_normed_vector set"
- assumes "compact s" shows "compact ((\<lambda>x. c *\<^sub>R x) ` s)"
-proof-
+ assumes "compact s"
+ shows "compact ((\<lambda>x. c *\<^sub>R x) ` s)"
+proof -
let ?f = "\<lambda>x. scaleR c x"
- have *:"bounded_linear ?f" by (rule bounded_linear_scaleR_right)
- show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
- using linear_continuous_at[OF *] assms by auto
+ have *: "bounded_linear ?f" by (rule bounded_linear_scaleR_right)
+ show ?thesis
+ using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
+ using linear_continuous_at[OF *] assms
+ by auto
qed
lemma compact_negations:
fixes s :: "'a::real_normed_vector set"
- assumes "compact s" shows "compact ((\<lambda>x. -x) ` s)"
+ assumes "compact s"
+ shows "compact ((\<lambda>x. -x) ` s)"
using compact_scaling [OF assms, of "- 1"] by auto
lemma compact_sums:
fixes s t :: "'a::real_normed_vector set"
- assumes "compact s" "compact t" shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"
-proof-
- have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z) ` (s \<times> t)"
- apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto
+ assumes "compact s" and "compact t"
+ shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"
+proof -
+ have *: "{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z) ` (s \<times> t)"
+ apply auto
+ unfolding image_iff
+ apply (rule_tac x="(xa, y)" in bexI)
+ apply auto
+ done
have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"
unfolding continuous_on by (rule ballI) (intro tendsto_intros)
- thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto
+ then show ?thesis
+ unfolding * using compact_continuous_image compact_Times [OF assms] by auto
qed
lemma compact_differences:
@@ -5024,24 +5594,36 @@
assumes "compact s" "compact t" shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"
proof-
have "{x - y | x y. x\<in>s \<and> y \<in> t} = {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
- apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
- thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
+ apply auto
+ apply (rule_tac x= xa in exI)
+ apply auto
+ apply (rule_tac x=xa in exI)
+ apply auto
+ done
+ then show ?thesis
+ using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
qed
lemma compact_translation:
fixes s :: "'a::real_normed_vector set"
- assumes "compact s" shows "compact ((\<lambda>x. a + x) ` s)"
-proof-
- have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x) ` s" by auto
- thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto
+ assumes "compact s"
+ shows "compact ((\<lambda>x. a + x) ` s)"
+proof -
+ have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x) ` s"
+ by auto
+ then show ?thesis
+ using compact_sums[OF assms compact_sing[of a]] by auto
qed
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 "op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
- thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto
+ assumes "compact s"
+ shows "compact ((\<lambda>x. a + c *\<^sub>R x) ` s)"
+proof -
+ have "op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s"
+ by auto
+ then show ?thesis
+ using compact_translation[OF compact_scaling[OF assms], of a c] by auto
qed
text {* Hence we get the following. *}
@@ -5050,9 +5632,11 @@
fixes s :: "'a::metric_space set"
assumes "compact s" "s \<noteq> {}"
shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"
-proof-
- have "compact (s \<times> s)" using `compact s` by (intro compact_Times)
- moreover have "s \<times> s \<noteq> {}" using `s \<noteq> {}` by auto
+proof -
+ have "compact (s \<times> s)"
+ using `compact s` by (intro compact_Times)
+ moreover have "s \<times> s \<noteq> {}"
+ using `s \<noteq> {}` by auto
moreover have "continuous_on (s \<times> s) (\<lambda>x. dist (fst x) (snd x))"
by (intro continuous_at_imp_continuous_on ballI continuous_intros)
ultimately show ?thesis
@@ -5073,7 +5657,8 @@
unfolding bounded_def by auto
have "dist x y \<le> Sup ?D"
proof (rule cSup_upper, safe)
- fix a b assume "a \<in> s" "b \<in> s"
+ fix a b
+ assume "a \<in> s" "b \<in> s"
with z[of a] z[of b] dist_triangle[of a b z]
show "dist a b \<le> 2 * d"
by (simp add: dist_commute)
@@ -5084,7 +5669,8 @@
lemma diameter_lower_bounded:
fixes s :: "'a :: metric_space set"
- assumes s: "bounded s" and d: "0 < d" "d < diameter s"
+ assumes s: "bounded s"
+ and d: "0 < d" "d < diameter s"
shows "\<exists>x\<in>s. \<exists>y\<in>s. d < dist x y"
proof (rule ccontr)
let ?D = "{dist x y |x y. x \<in> s \<and> y \<in> s}"
@@ -5110,12 +5696,17 @@
assumes "compact s" "s \<noteq> {}"
shows "\<exists>x\<in>s. \<exists>y\<in>s. dist x y = diameter s"
proof -
- have b:"bounded s" using assms(1) by (rule compact_imp_bounded)
- then obtain x y where xys:"x\<in>s" "y\<in>s" and xy:"\<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"
+ have b: "bounded s" using assms(1)
+ by (rule compact_imp_bounded)
+ then obtain x y where xys: "x\<in>s" "y\<in>s" and xy: "\<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"
using compact_sup_maxdistance[OF assms] by auto
- hence "diameter s \<le> dist x y"
- unfolding diameter_def by clarsimp (rule cSup_least, fast+)
- thus ?thesis
+ then have "diameter s \<le> dist x y"
+ unfolding diameter_def
+ apply clarsimp
+ apply (rule cSup_least)
+ apply fast+
+ done
+ then show ?thesis
by (metis b diameter_bounded_bound order_antisym xys)
qed
@@ -5123,109 +5714,175 @@
lemma closed_scaling:
fixes s :: "'a::real_normed_vector set"
- assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x) ` s)"
-proof(cases "s={}")
- case True thus ?thesis by auto
+ assumes "closed s"
+ shows "closed ((\<lambda>x. c *\<^sub>R x) ` s)"
+proof (cases "s = {}")
+ case True
+ then show ?thesis by auto
next
case False
show ?thesis
- proof(cases "c=0")
- have *:"(\<lambda>x. 0) ` s = {0}" using `s\<noteq>{}` by auto
- case True thus ?thesis apply auto unfolding * by auto
+ proof (cases "c = 0")
+ have *: "(\<lambda>x. 0) ` s = {0}" using `s\<noteq>{}` by auto
+ case True
+ then show ?thesis
+ apply auto
+ unfolding *
+ apply auto
+ done
next
case False
- { fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c ` s" "(x ---> l) sequentially"
- { fix n::nat have "scaleR (1 / c) (x n) \<in> s"
+ {
+ fix x l
+ assume as: "\<forall>n::nat. x n \<in> scaleR c ` s" "(x ---> l) sequentially"
+ {
+ fix n :: nat
+ have "scaleR (1 / c) (x n) \<in> s"
using as(1)[THEN spec[where x=n]]
- using `c\<noteq>0` by auto
+ using `c\<noteq>0`
+ by auto
}
moreover
- { fix e::real assume "e>0"
- hence "0 < e *\<bar>c\<bar>" using `c\<noteq>0` mult_pos_pos[of e "abs c"] by auto
+ {
+ fix e :: real
+ assume "e > 0"
+ then have "0 < e *\<bar>c\<bar>"
+ using `c\<noteq>0` mult_pos_pos[of e "abs c"] by auto
then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"
using as(2)[unfolded LIMSEQ_def, THEN spec[where x="e * abs c"]] by auto
- hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"
- unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]
- using mult_imp_div_pos_less[of "abs c" _ e] `c\<noteq>0` by auto }
- hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding LIMSEQ_def by auto
+ then have "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"
+ unfolding dist_norm
+ unfolding scaleR_right_diff_distrib[THEN sym]
+ using mult_imp_div_pos_less[of "abs c" _ e] `c\<noteq>0` by auto
+ }
+ then have "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially"
+ unfolding LIMSEQ_def by auto
ultimately have "l \<in> scaleR c ` s"
- using assms[unfolded closed_sequential_limits, THEN spec[where x="\<lambda>n. scaleR (1/c) (x n)"], THEN spec[where x="scaleR (1/c) l"]]
- unfolding image_iff using `c\<noteq>0` apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto }
- thus ?thesis unfolding closed_sequential_limits by fast
+ using assms[unfolded closed_sequential_limits,
+ THEN spec[where x="\<lambda>n. scaleR (1/c) (x n)"],
+ THEN spec[where x="scaleR (1/c) l"]]
+ unfolding image_iff using `c\<noteq>0`
+ apply (rule_tac x="scaleR (1 / c) l" in bexI)
+ apply auto
+ done
+ }
+ then show ?thesis
+ unfolding closed_sequential_limits by fast
qed
qed
lemma closed_negations:
fixes s :: "'a::real_normed_vector set"
- assumes "closed s" shows "closed ((\<lambda>x. -x) ` s)"
+ assumes "closed s"
+ shows "closed ((\<lambda>x. -x) ` s)"
using closed_scaling[OF assms, of "- 1"] by simp
lemma compact_closed_sums:
fixes s :: "'a::real_normed_vector set"
- assumes "compact s" "closed t" shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
-proof-
+ assumes "compact s" and "closed t"
+ shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
+proof -
let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"
- { fix x l assume as:"\<forall>n. x n \<in> ?S" "(x ---> l) sequentially"
- from as(1) obtain f where f:"\<forall>n. x n = fst (f n) + snd (f n)" "\<forall>n. fst (f n) \<in> s" "\<forall>n. snd (f n) \<in> t"
+ {
+ fix x l
+ assume as: "\<forall>n. x n \<in> ?S" "(x ---> l) sequentially"
+ from as(1) obtain f where f: "\<forall>n. x n = fst (f n) + snd (f n)" "\<forall>n. fst (f n) \<in> s" "\<forall>n. snd (f n) \<in> t"
using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto
- obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"
+ obtain l' r where "l'\<in>s" and r: "subseq r" and lr: "(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"
using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"
- using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1) unfolding o_def by auto
- hence "l - l' \<in> t"
+ using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1)
+ 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
- hence "l \<in> ?S" using `l' \<in> s` apply auto apply(rule_tac x=l' in exI) apply(rule_tac x="l - l'" in exI) by auto
+ using f(3)
+ by auto
+ then have "l \<in> ?S"
+ using `l' \<in> s`
+ apply auto
+ apply (rule_tac x=l' in exI)
+ apply (rule_tac x="l - l'" in exI)
+ apply auto
+ done
}
- thus ?thesis unfolding closed_sequential_limits by fast
+ then show ?thesis
+ unfolding closed_sequential_limits by fast
qed
lemma closed_compact_sums:
fixes s t :: "'a::real_normed_vector set"
assumes "closed s" "compact t"
shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
-proof-
- have "{x + y |x y. x \<in> t \<and> y \<in> s} = {x + y |x y. x \<in> s \<and> y \<in> t}" apply auto
- apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto
- thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp
+proof -
+ have "{x + y |x y. x \<in> t \<and> y \<in> s} = {x + y |x y. x \<in> s \<and> y \<in> t}"
+ apply auto
+ apply (rule_tac x=y in exI)
+ apply auto
+ apply (rule_tac x=y in exI)
+ apply auto
+ done
+ then show ?thesis
+ using compact_closed_sums[OF assms(2,1)] by simp
qed
lemma compact_closed_differences:
fixes s t :: "'a::real_normed_vector set"
- assumes "compact s" "closed t"
+ assumes "compact s" and "closed t"
shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
-proof-
+proof -
have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
- apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
- thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto
+ apply auto
+ apply (rule_tac x=xa in exI)
+ apply auto
+ apply (rule_tac x=xa in exI)
+ apply auto
+ done
+ then show ?thesis
+ using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto
qed
lemma closed_compact_differences:
fixes s t :: "'a::real_normed_vector set"
assumes "closed s" "compact t"
shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
-proof-
+proof -
have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
- apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
- thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
+ apply auto
+ apply (rule_tac x=xa in exI)
+ apply auto
+ apply (rule_tac x=xa in exI)
+ apply auto
+ done
+ then show ?thesis
+ using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
qed
lemma closed_translation:
fixes a :: "'a::real_normed_vector"
- assumes "closed s" shows "closed ((\<lambda>x. a + x) ` s)"
-proof-
+ assumes "closed s"
+ shows "closed ((\<lambda>x. a + x) ` s)"
+proof -
have "{a + y |y. y \<in> s} = (op + a ` s)" by auto
- thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto
+ then show ?thesis
+ using compact_closed_sums[OF compact_sing[of a] assms] by auto
qed
lemma translation_Compl:
fixes a :: "'a::ab_group_add"
shows "(\<lambda>x. a + x) ` (- t) = - ((\<lambda>x. a + x) ` t)"
- apply (auto simp add: image_iff) apply(rule_tac x="x - a" in bexI) by auto
+ apply (auto simp add: image_iff)
+ apply (rule_tac x="x - a" in bexI)
+ apply auto
+ done
lemma translation_UNIV:
- fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"
- apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto
+ fixes a :: "'a::ab_group_add"
+ shows "range (\<lambda>x. a + x) = UNIV"
+ apply (auto simp add: image_iff)
+ apply (rule_tac x="x - a" in exI)
+ apply auto
+ done
lemma translation_diff:
fixes a :: "'a::ab_group_add"
@@ -5235,37 +5892,48 @@
lemma closure_translation:
fixes a :: "'a::real_normed_vector"
shows "closure ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (closure s)"
-proof-
- have *:"op + a ` (- s) = - op + a ` s"
- apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto
- show ?thesis unfolding closure_interior translation_Compl
- using interior_translation[of a "- s"] unfolding * by auto
+proof -
+ have *: "op + a ` (- s) = - op + a ` s"
+ apply auto
+ unfolding image_iff
+ apply (rule_tac x="x - a" in bexI)
+ apply auto
+ done
+ show ?thesis
+ unfolding closure_interior translation_Compl
+ using interior_translation[of a "- s"]
+ unfolding *
+ by auto
qed
lemma frontier_translation:
fixes a :: "'a::real_normed_vector"
shows "frontier((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (frontier s)"
- unfolding frontier_def translation_diff interior_translation closure_translation by auto
+ unfolding frontier_def translation_diff interior_translation closure_translation
+ by auto
subsection {* Separation between points and sets *}
lemma separate_point_closed:
fixes s :: "'a::heine_borel set"
- shows "closed s \<Longrightarrow> a \<notin> s ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"
-proof(cases "s = {}")
+ assumes "closed s" and "a \<notin> s"
+ shows "\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x"
+proof (cases "s = {}")
case True
- thus ?thesis by(auto intro!: exI[where x=1])
+ then show ?thesis by(auto intro!: exI[where x=1])
next
case False
- assume "closed s" "a \<notin> s"
- then obtain x where "x\<in>s" "\<forall>y\<in>s. dist a x \<le> dist a y" using `s \<noteq> {}` distance_attains_inf [of s a] by blast
- with `x\<in>s` show ?thesis using dist_pos_lt[of a x] and`a \<notin> s` by blast
+ from assms obtain x where "x\<in>s" "\<forall>y\<in>s. dist a x \<le> dist a y"
+ using `s \<noteq> {}` distance_attains_inf [of s a] by blast
+ with `x\<in>s` show ?thesis using dist_pos_lt[of a x] and`a \<notin> s`
+ by blast
qed
lemma separate_compact_closed:
fixes s t :: "'a::heine_borel set"
- assumes "compact s" and t: "closed t" "s \<inter> t = {}"
+ assumes "compact s"
+ and t: "closed t" "s \<inter> t = {}"
shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
proof cases
assume "s \<noteq> {} \<and> t \<noteq> {}"
@@ -5279,70 +5947,104 @@
using t `t \<noteq> {}` in_closed_iff_infdist_zero by (auto simp: less_le infdist_nonneg)
moreover have "\<forall>x'\<in>s. \<forall>y\<in>t. ?inf x \<le> dist x' y"
using x by (auto intro: order_trans infdist_le)
- ultimately show ?thesis
- by auto
+ ultimately show ?thesis by auto
qed (auto intro!: exI[of _ 1])
lemma separate_closed_compact:
fixes s t :: "'a::heine_borel set"
- assumes "closed s" and "compact t" and "s \<inter> t = {}"
+ assumes "closed s"
+ and "compact t"
+ and "s \<inter> t = {}"
shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
-proof-
- have *:"t \<inter> s = {}" using assms(3) by auto
- show ?thesis using separate_compact_closed[OF assms(2,1) *]
- apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)
- by (auto simp add: dist_commute)
+proof -
+ have *: "t \<inter> s = {}"
+ using assms(3) by auto
+ show ?thesis
+ using separate_compact_closed[OF assms(2,1) *]
+ apply auto
+ apply (rule_tac x=d in exI)
+ apply auto
+ apply (erule_tac x=y in ballE)
+ apply (auto simp add: dist_commute)
+ done
qed
subsection {* Intervals *}
-
-lemma interval: fixes a :: "'a::ordered_euclidean_space" shows
- "{a <..< b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i}" and
- "{a .. b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i}"
- by(auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
-
-lemma mem_interval: fixes a :: "'a::ordered_euclidean_space" shows
- "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i)"
- "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i)"
- using interval[of a b] by(auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
-
-lemma interval_eq_empty: fixes a :: "'a::ordered_euclidean_space" shows
- "({a <..< b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i))" (is ?th1) and
- "({a .. b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i < a\<bullet>i))" (is ?th2)
-proof-
- { fix i x assume i:"i\<in>Basis" and as:"b\<bullet>i \<le> a\<bullet>i" and x:"x\<in>{a <..< b}"
- hence "a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i" unfolding mem_interval by auto
- hence "a\<bullet>i < b\<bullet>i" by auto
- hence False using as by auto }
+
+lemma interval:
+ fixes a :: "'a::ordered_euclidean_space"
+ shows "{a <..< b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i}"
+ and "{a .. b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i}"
+ by (auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
+
+lemma mem_interval:
+ fixes a :: "'a::ordered_euclidean_space"
+ shows "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i)"
+ and "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i)"
+ using interval[of a b]
+ by (auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
+
+lemma interval_eq_empty:
+ fixes a :: "'a::ordered_euclidean_space"
+ shows "({a <..< b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i))" (is ?th1)
+ and "({a .. b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i < a\<bullet>i))" (is ?th2)
+proof -
+ {
+ fix i x
+ assume i: "i\<in>Basis" and as:"b\<bullet>i \<le> a\<bullet>i" and x:"x\<in>{a <..< b}"
+ then have "a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i"
+ unfolding mem_interval by auto
+ then have "a\<bullet>i < b\<bullet>i" by auto
+ then have False using as by auto
+ }
moreover
- { assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i \<le> a\<bullet>i)"
+ {
+ assume as: "\<forall>i\<in>Basis. \<not> (b\<bullet>i \<le> a\<bullet>i)"
let ?x = "(1/2) *\<^sub>R (a + b)"
- { fix i :: 'a assume i:"i\<in>Basis"
- have "a\<bullet>i < b\<bullet>i" using as[THEN bspec[where x=i]] i by auto
- hence "a\<bullet>i < ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i < b\<bullet>i"
- by (auto simp: inner_add_left) }
- hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto }
+ {
+ fix i :: 'a
+ assume i: "i \<in> Basis"
+ have "a\<bullet>i < b\<bullet>i"
+ using as[THEN bspec[where x=i]] i by auto
+ then have "a\<bullet>i < ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i < b\<bullet>i"
+ by (auto simp: inner_add_left)
+ }
+ then have "{a <..< b} \<noteq> {}"
+ using mem_interval(1)[of "?x" a b] by auto
+ }
ultimately show ?th1 by blast
- { fix i x assume i:"i\<in>Basis" and as:"b\<bullet>i < a\<bullet>i" and x:"x\<in>{a .. b}"
- hence "a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i" unfolding mem_interval by auto
- hence "a\<bullet>i \<le> b\<bullet>i" by auto
- hence False using as by auto }
+ {
+ fix i x
+ assume i: "i \<in> Basis" and as:"b\<bullet>i < a\<bullet>i" and x:"x\<in>{a .. b}"
+ then have "a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i"
+ unfolding mem_interval by auto
+ then have "a\<bullet>i \<le> b\<bullet>i" by auto
+ then have False using as by auto
+ }
moreover
- { assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i < a\<bullet>i)"
+ {
+ assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i < a\<bullet>i)"
let ?x = "(1/2) *\<^sub>R (a + b)"
- { fix i :: 'a assume i:"i\<in>Basis"
- have "a\<bullet>i \<le> b\<bullet>i" using as[THEN bspec[where x=i]] i by auto
- hence "a\<bullet>i \<le> ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i \<le> b\<bullet>i"
- by (auto simp: inner_add_left) }
- hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto }
+ {
+ fix i :: 'a
+ assume i:"i \<in> Basis"
+ have "a\<bullet>i \<le> b\<bullet>i"
+ using as[THEN bspec[where x=i]] i by auto
+ then have "a\<bullet>i \<le> ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i \<le> b\<bullet>i"
+ by (auto simp: inner_add_left)
+ }
+ then have "{a .. b} \<noteq> {}"
+ using mem_interval(2)[of "?x" a b] by auto
+ }
ultimately show ?th2 by blast
qed
-lemma interval_ne_empty: fixes a :: "'a::ordered_euclidean_space" shows
- "{a .. b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i)" and
- "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
+lemma interval_ne_empty:
+ fixes a :: "'a::ordered_euclidean_space"
+ shows "{a .. b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i)"
+ and "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
unfolding interval_eq_empty[of a b] by fastforce+
lemma interval_sing:
@@ -5351,11 +6053,12 @@
unfolding set_eq_iff mem_interval eq_iff [symmetric]
by (auto intro: euclidean_eqI simp: ex_in_conv)
-lemma subset_interval_imp: fixes a :: "'a::ordered_euclidean_space" shows
- "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and
- "(\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and
- "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and
- "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
+lemma subset_interval_imp:
+ fixes a :: "'a::ordered_euclidean_space"
+ shows "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}"
+ and "(\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}"
+ and "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}"
+ and "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
unfolding subset_eq[unfolded Ball_def] unfolding mem_interval
by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+
@@ -5365,80 +6068,125 @@
unfolding subset_eq [unfolded Ball_def] mem_interval
by (fast intro: less_imp_le)
-lemma subset_interval: fixes a :: "'a::ordered_euclidean_space" shows
- "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th1) and
- "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i)" (is ?th2) and
- "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th3) and
- "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th4)
-proof-
- show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)
- show ?th2 unfolding subset_eq and Ball_def and mem_interval by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)
- { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"
- hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by auto
- fix i :: 'a assume i:"i\<in>Basis"
+lemma subset_interval:
+ fixes a :: "'a::ordered_euclidean_space"
+ shows "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th1)
+ and "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i)" (is ?th2)
+ and "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th3)
+ and "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th4)
+proof -
+ show ?th1
+ unfolding subset_eq and Ball_def and mem_interval
+ by (auto intro: order_trans)
+ show ?th2
+ unfolding subset_eq and Ball_def and mem_interval
+ by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)
+ {
+ assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"
+ then have "{c<..<d} \<noteq> {}"
+ unfolding interval_eq_empty by auto
+ fix i :: 'a
+ assume i: "i \<in> Basis"
(** TODO combine the following two parts as done in the HOL_light version. **)
- { let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((min (a\<bullet>j) (d\<bullet>j))+c\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"
+ {
+ let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((min (a\<bullet>j) (d\<bullet>j))+c\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"
assume as2: "a\<bullet>i > c\<bullet>i"
- { fix j :: 'a assume j:"j\<in>Basis"
- hence "c \<bullet> j < ?x \<bullet> j \<and> ?x \<bullet> j < d \<bullet> j"
- apply(cases "j=i") using as(2)[THEN bspec[where x=j]] i
- by (auto simp add: as2) }
- hence "?x\<in>{c<..<d}" using i unfolding mem_interval by auto
+ {
+ fix j :: 'a
+ assume j: "j \<in> Basis"
+ then have "c \<bullet> j < ?x \<bullet> j \<and> ?x \<bullet> j < d \<bullet> j"
+ apply (cases "j = i")
+ using as(2)[THEN bspec[where x=j]] i
+ apply (auto simp add: as2)
+ done
+ }
+ then have "?x\<in>{c<..<d}"
+ using i unfolding mem_interval by auto
+ moreover
+ have "?x \<notin> {a .. b}"
+ unfolding mem_interval
+ apply auto
+ apply (rule_tac x=i in bexI)
+ using as(2)[THEN bspec[where x=i]] and as2 i
+ apply auto
+ done
+ ultimately have False using as by auto
+ }
+ then have "a\<bullet>i \<le> c\<bullet>i" by (rule ccontr) auto
+ moreover
+ {
+ let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((max (b\<bullet>j) (c\<bullet>j))+d\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"
+ assume as2: "b\<bullet>i < d\<bullet>i"
+ {
+ fix j :: 'a
+ assume "j\<in>Basis"
+ then have "d \<bullet> j > ?x \<bullet> j \<and> ?x \<bullet> j > c \<bullet> j"
+ apply (cases "j = i")
+ using as(2)[THEN bspec[where x=j]]
+ apply (auto simp add: as2)
+ done
+ }
+ then have "?x\<in>{c<..<d}"
+ unfolding mem_interval by auto
moreover
have "?x\<notin>{a .. b}"
- unfolding mem_interval apply auto apply(rule_tac x=i in bexI)
- using as(2)[THEN bspec[where x=i]] and as2 i
- by auto
- ultimately have False using as by auto }
- hence "a\<bullet>i \<le> c\<bullet>i" by(rule ccontr)auto
- moreover
- { let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((max (b\<bullet>j) (c\<bullet>j))+d\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"
- assume as2: "b\<bullet>i < d\<bullet>i"
- { fix j :: 'a assume "j\<in>Basis"
- hence "d \<bullet> j > ?x \<bullet> j \<and> ?x \<bullet> j > c \<bullet> j"
- apply(cases "j=i") using as(2)[THEN bspec[where x=j]]
- by (auto simp add: as2) }
- hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
- moreover
- have "?x\<notin>{a .. b}"
- unfolding mem_interval apply auto apply(rule_tac x=i in bexI)
+ unfolding mem_interval
+ apply auto
+ apply (rule_tac x=i in bexI)
using as(2)[THEN bspec[where x=i]] and as2 using i
- by auto
- ultimately have False using as by auto }
- hence "b\<bullet>i \<ge> d\<bullet>i" by(rule ccontr)auto
+ apply auto
+ done
+ ultimately have False using as by auto
+ }
+ then have "b\<bullet>i \<ge> d\<bullet>i" by (rule ccontr) auto
ultimately
have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i" by auto
} note part1 = this
show ?th3
- unfolding subset_eq and Ball_def and mem_interval
- apply(rule,rule,rule,rule)
- apply(rule part1)
+ unfolding subset_eq and Ball_def and mem_interval
+ apply (rule, rule, rule, rule)
+ apply (rule part1)
unfolding subset_eq and Ball_def and mem_interval
prefer 4
- apply auto
- by(erule_tac x=xa in allE,erule_tac x=xa in allE,fastforce)+
- { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"
- fix i :: 'a assume i:"i\<in>Basis"
- from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto
- hence "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i" using part1 and as(2) using i by auto } note * = this
- show ?th4 unfolding subset_eq and Ball_def and mem_interval
- apply(rule,rule,rule,rule) apply(rule *) unfolding subset_eq and Ball_def and mem_interval prefer 4
- apply auto by(erule_tac x=xa in allE, simp)+
-qed
-
-lemma inter_interval: fixes a :: "'a::ordered_euclidean_space" shows
- "{a .. b} \<inter> {c .. d} = {(\<Sum>i\<in>Basis. max (a\<bullet>i) (c\<bullet>i) *\<^sub>R i) .. (\<Sum>i\<in>Basis. min (b\<bullet>i) (d\<bullet>i) *\<^sub>R i)}"
- unfolding set_eq_iff and Int_iff and mem_interval by auto
-
-lemma disjoint_interval: fixes a::"'a::ordered_euclidean_space" shows
- "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i < c\<bullet>i \<or> d\<bullet>i < a\<bullet>i))" (is ?th1) and
- "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th2) and
- "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th3) and
- "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th4)
-proof-
+ apply auto
+ apply (erule_tac x=xa in allE, erule_tac x=xa in allE, fastforce)+
+ done
+ {
+ assume as: "{c<..<d} \<subseteq> {a<..<b}" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"
+ fix i :: 'a
+ assume i:"i\<in>Basis"
+ from as(1) have "{c<..<d} \<subseteq> {a..b}"
+ using interval_open_subset_closed[of a b] by auto
+ then have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i"
+ using part1 and as(2) using i by auto
+ } note * = this
+ show ?th4
+ unfolding subset_eq and Ball_def and mem_interval
+ apply (rule, rule, rule, rule)
+ apply (rule *)
+ unfolding subset_eq and Ball_def and mem_interval
+ prefer 4
+ apply auto
+ apply (erule_tac x=xa in allE, simp)+
+ done
+qed
+
+lemma inter_interval:
+ fixes a :: "'a::ordered_euclidean_space"
+ shows "{a .. b} \<inter> {c .. d} = {(\<Sum>i\<in>Basis. max (a\<bullet>i) (c\<bullet>i) *\<^sub>R i) .. (\<Sum>i\<in>Basis. min (b\<bullet>i) (d\<bullet>i) *\<^sub>R i)}"
+ unfolding set_eq_iff and Int_iff and mem_interval
+ by auto
+
+lemma disjoint_interval:
+ fixes a::"'a::ordered_euclidean_space"
+ shows "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i < c\<bullet>i \<or> d\<bullet>i < a\<bullet>i))" (is ?th1)
+ and "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th2)
+ and "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th3)
+ and "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th4)
+proof -
let ?z = "(\<Sum>i\<in>Basis. (((max (a\<bullet>i) (c\<bullet>i)) + (min (b\<bullet>i) (d\<bullet>i))) / 2) *\<^sub>R i)::'a"
have **: "\<And>P Q. (\<And>i :: 'a. i \<in> Basis \<Longrightarrow> Q ?z i \<Longrightarrow> P i) \<Longrightarrow>
- (\<And>i x :: 'a. i \<in> Basis \<Longrightarrow> P i \<Longrightarrow> Q x i) \<Longrightarrow> (\<forall>x. \<exists>i\<in>Basis. Q x i) \<longleftrightarrow> (\<exists>i\<in>Basis. P i)"
+ (\<And>i x :: 'a. i \<in> Basis \<Longrightarrow> P i \<Longrightarrow> Q x i) \<Longrightarrow> (\<forall>x. \<exists>i\<in>Basis. Q x i) \<longleftrightarrow> (\<exists>i\<in>Basis. P i)"
by blast
note * = set_eq_iff Int_iff empty_iff mem_interval ball_conj_distrib[symmetric] eq_False ball_simps(10)
show ?th1 unfolding * by (intro **) auto
@@ -5450,8 +6198,9 @@
(* Moved interval_open_subset_closed a bit upwards *)
lemma open_interval[intro]:
- fixes a b :: "'a::ordered_euclidean_space" shows "open {a<..<b}"
-proof-
+ fixes a b :: "'a::ordered_euclidean_space"
+ shows "open {a<..<b}"
+proof -
have "open (\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) -` {a\<bullet>i<..<b\<bullet>i})"
by (intro open_INT finite_lessThan ballI continuous_open_vimage allI
linear_continuous_at open_real_greaterThanLessThan finite_Basis bounded_linear_inner_left)
@@ -5461,8 +6210,9 @@
qed
lemma closed_interval[intro]:
- fixes a b :: "'a::ordered_euclidean_space" shows "closed {a .. b}"
-proof-
+ fixes a b :: "'a::ordered_euclidean_space"
+ shows "closed {a .. b}"
+proof -
have "closed (\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) -` {a\<bullet>i .. b\<bullet>i})"
by (intro closed_INT ballI continuous_closed_vimage allI
linear_continuous_at closed_real_atLeastAtMost finite_Basis bounded_linear_inner_left)
@@ -5477,127 +6227,237 @@
proof(rule subset_antisym)
show "?R \<subseteq> ?L" using interval_open_subset_closed open_interval
by (rule interior_maximal)
-next
- { fix x assume "x \<in> interior {a..b}"
- then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" ..
- then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a..b}" unfolding open_dist and subset_eq by auto
- { fix i :: 'a assume i:"i\<in>Basis"
+ {
+ fix x
+ assume "x \<in> interior {a..b}"
+ then obtain s where s: "open s" "x \<in> s" "s \<subseteq> {a..b}" ..
+ then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a..b}"
+ unfolding open_dist and subset_eq by auto
+ {
+ fix i :: 'a
+ assume i: "i \<in> Basis"
have "dist (x - (e / 2) *\<^sub>R i) x < e"
- "dist (x + (e / 2) *\<^sub>R i) x < e"
- unfolding dist_norm apply auto
- unfolding norm_minus_cancel using norm_Basis[OF i] `e>0` by auto
- hence "a \<bullet> i \<le> (x - (e / 2) *\<^sub>R i) \<bullet> i"
- "(x + (e / 2) *\<^sub>R i) \<bullet> i \<le> b \<bullet> i"
+ and "dist (x + (e / 2) *\<^sub>R i) x < e"
+ unfolding dist_norm
+ apply auto
+ unfolding norm_minus_cancel
+ using norm_Basis[OF i] `e>0`
+ apply auto
+ done
+ then have "a \<bullet> i \<le> (x - (e / 2) *\<^sub>R i) \<bullet> i" and "(x + (e / 2) *\<^sub>R i) \<bullet> i \<le> b \<bullet> i"
using e[THEN spec[where x="x - (e/2) *\<^sub>R i"]]
- and e[THEN spec[where x="x + (e/2) *\<^sub>R i"]]
- unfolding mem_interval using i by blast+
- hence "a \<bullet> i < x \<bullet> i" and "x \<bullet> i < b \<bullet> i"
- using `e>0` i by (auto simp: inner_diff_left inner_Basis inner_add_left) }
- hence "x \<in> {a<..<b}" unfolding mem_interval by auto }
- thus "?L \<subseteq> ?R" ..
-qed
-
-lemma bounded_closed_interval: fixes a :: "'a::ordered_euclidean_space" shows "bounded {a .. b}"
-proof-
+ and e[THEN spec[where x="x + (e/2) *\<^sub>R i"]]
+ unfolding mem_interval
+ using i
+ by blast+
+ then have "a \<bullet> i < x \<bullet> i" and "x \<bullet> i < b \<bullet> i"
+ using `e>0` i
+ by (auto simp: inner_diff_left inner_Basis inner_add_left)
+ }
+ then have "x \<in> {a<..<b}"
+ unfolding mem_interval by auto
+ }
+ then show "?L \<subseteq> ?R" ..
+qed
+
+lemma bounded_closed_interval:
+ fixes a :: "'a::ordered_euclidean_space"
+ shows "bounded {a .. b}"
+proof -
let ?b = "\<Sum>i\<in>Basis. \<bar>a\<bullet>i\<bar> + \<bar>b\<bullet>i\<bar>"
- { fix x::"'a" assume x:"\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i"
- { fix i :: 'a assume "i\<in>Basis"
- hence "\<bar>x\<bullet>i\<bar> \<le> \<bar>a\<bullet>i\<bar> + \<bar>b\<bullet>i\<bar>" using x[THEN bspec[where x=i]] by auto }
- hence "(\<Sum>i\<in>Basis. \<bar>x \<bullet> i\<bar>) \<le> ?b" apply-apply(rule setsum_mono) by auto
- hence "norm x \<le> ?b" using norm_le_l1[of x] by auto }
- thus ?thesis unfolding interval and bounded_iff by auto
-qed
-
-lemma bounded_interval: fixes a :: "'a::ordered_euclidean_space" shows
- "bounded {a .. b} \<and> bounded {a<..<b}"
+ {
+ fix x :: "'a"
+ assume x: "\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i"
+ {
+ fix i :: 'a
+ assume "i \<in> Basis"
+ then have "\<bar>x\<bullet>i\<bar> \<le> \<bar>a\<bullet>i\<bar> + \<bar>b\<bullet>i\<bar>"
+ using x[THEN bspec[where x=i]] by auto
+ }
+ then have "(\<Sum>i\<in>Basis. \<bar>x \<bullet> i\<bar>) \<le> ?b"
+ apply -
+ apply (rule setsum_mono)
+ apply auto
+ done
+ then have "norm x \<le> ?b"
+ using norm_le_l1[of x] by auto
+ }
+ then show ?thesis
+ unfolding interval and bounded_iff by auto
+qed
+
+lemma bounded_interval:
+ fixes a :: "'a::ordered_euclidean_space"
+ shows "bounded {a .. b} \<and> bounded {a<..<b}"
using bounded_closed_interval[of a b]
using interval_open_subset_closed[of a b]
using bounded_subset[of "{a..b}" "{a<..<b}"]
by simp
-lemma not_interval_univ: fixes a :: "'a::ordered_euclidean_space" shows
- "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"
+lemma not_interval_univ:
+ fixes a :: "'a::ordered_euclidean_space"
+ shows "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"
using bounded_interval[of a b] by auto
-lemma compact_interval: fixes a :: "'a::ordered_euclidean_space" shows "compact {a .. b}"
+lemma compact_interval:
+ fixes a :: "'a::ordered_euclidean_space"
+ shows "compact {a .. b}"
using bounded_closed_imp_seq_compact[of "{a..b}"] using bounded_interval[of a b]
by (auto simp: compact_eq_seq_compact_metric)
-lemma open_interval_midpoint: fixes a :: "'a::ordered_euclidean_space"
- assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"
-proof-
- { fix i :: 'a assume "i\<in>Basis"
- hence "a \<bullet> i < ((1 / 2) *\<^sub>R (a + b)) \<bullet> i \<and> ((1 / 2) *\<^sub>R (a + b)) \<bullet> i < b \<bullet> i"
- using assms[unfolded interval_ne_empty, THEN bspec[where x=i]] by (auto simp: inner_add_left) }
- thus ?thesis unfolding mem_interval by auto
-qed
-
-lemma open_closed_interval_convex: fixes x :: "'a::ordered_euclidean_space"
- assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"
+lemma open_interval_midpoint:
+ fixes a :: "'a::ordered_euclidean_space"
+ assumes "{a<..<b} \<noteq> {}"
+ shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"
+proof -
+ {
+ fix i :: 'a
+ assume "i \<in> Basis"
+ then have "a \<bullet> i < ((1 / 2) *\<^sub>R (a + b)) \<bullet> i \<and> ((1 / 2) *\<^sub>R (a + b)) \<bullet> i < b \<bullet> i"
+ using assms[unfolded interval_ne_empty, THEN bspec[where x=i]] by (auto simp: inner_add_left)
+ }
+ then show ?thesis unfolding mem_interval by auto
+qed
+
+lemma open_closed_interval_convex:
+ fixes x :: "'a::ordered_euclidean_space"
+ assumes x: "x \<in> {a<..<b}"
+ and y: "y \<in> {a .. b}"
+ and e: "0 < e" "e \<le> 1"
shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"
-proof-
- { fix i :: 'a assume i:"i\<in>Basis"
- have "a \<bullet> i = e * (a \<bullet> i) + (1 - e) * (a \<bullet> i)" unfolding left_diff_distrib by simp
- also have "\<dots> < e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)" apply(rule add_less_le_mono)
- using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
- using x unfolding mem_interval using i apply simp
- using y unfolding mem_interval using i apply simp
+proof -
+ {
+ fix i :: 'a
+ assume i: "i \<in> Basis"
+ have "a \<bullet> i = e * (a \<bullet> i) + (1 - e) * (a \<bullet> i)"
+ unfolding left_diff_distrib by simp
+ also have "\<dots> < e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)"
+ apply (rule add_less_le_mono)
+ using e unfolding mult_less_cancel_left and mult_le_cancel_left
+ apply simp_all
+ using x unfolding mem_interval using i
+ apply simp
+ using y unfolding mem_interval using i
+ apply simp
done
- finally have "a \<bullet> i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i" unfolding inner_simps by auto
- moreover {
- have "b \<bullet> i = e * (b\<bullet>i) + (1 - e) * (b\<bullet>i)" unfolding left_diff_distrib by simp
- also have "\<dots> > e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)" apply(rule add_less_le_mono)
- using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
- using x unfolding mem_interval using i apply simp
- using y unfolding mem_interval using i apply simp
- done
- finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i < b \<bullet> i" unfolding inner_simps by auto
- } ultimately have "a \<bullet> i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i \<and> (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i < b \<bullet> i" by auto }
- thus ?thesis unfolding mem_interval by auto
-qed
-
-lemma closure_open_interval: fixes a :: "'a::ordered_euclidean_space"
+ finally have "a \<bullet> i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i"
+ unfolding inner_simps by auto
+ moreover
+ {
+ have "b \<bullet> i = e * (b\<bullet>i) + (1 - e) * (b\<bullet>i)"
+ unfolding left_diff_distrib by simp
+ also have "\<dots> > e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)"
+ apply (rule add_less_le_mono)
+ using e unfolding mult_less_cancel_left and mult_le_cancel_left
+ apply simp_all
+ using x
+ unfolding mem_interval
+ using i
+ apply simp
+ using y
+ unfolding mem_interval
+ using i
+ apply simp
+ done
+ finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i < b \<bullet> i"
+ unfolding inner_simps by auto
+ }
+ ultimately have "a \<bullet> i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i \<and> (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i < b \<bullet> i"
+ by auto
+ }
+ then show ?thesis
+ unfolding mem_interval by auto
+qed
+
+lemma closure_open_interval:
+ fixes a :: "'a::ordered_euclidean_space"
assumes "{a<..<b} \<noteq> {}"
shows "closure {a<..<b} = {a .. b}"
-proof-
- have ab:"a < b" using assms[unfolded interval_ne_empty] apply(subst eucl_less) by auto
+proof -
+ have ab: "a < b"
+ using assms[unfolded interval_ne_empty]
+ apply (subst eucl_less)
+ apply auto
+ done
let ?c = "(1 / 2) *\<^sub>R (a + b)"
- { fix x assume as:"x \<in> {a .. b}"
- def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"
- { fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"
- have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto
+ {
+ fix x
+ assume as:"x \<in> {a .. b}"
+ def f \<equiv> "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"
+ {
+ fix n
+ assume fn: "f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc: "x \<noteq> ?c"
+ have *: "0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1"
+ unfolding inverse_le_1_iff by auto
have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =
x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"
by (auto simp add: algebra_simps)
- hence "f n < b" and "a < f n" using open_closed_interval_convex[OF open_interval_midpoint[OF assms] as *] unfolding f_def by auto
- hence False using fn unfolding f_def using xc by auto }
+ then have "f n < b" and "a < f n"
+ using open_closed_interval_convex[OF open_interval_midpoint[OF assms] as *]
+ unfolding f_def by auto
+ then have False
+ using fn unfolding f_def using xc by auto
+ }
moreover
- { assume "\<not> (f ---> x) sequentially"
- { fix e::real assume "e>0"
- hence "\<exists>N::nat. inverse (real (N + 1)) < e" using real_arch_inv[of e] apply (auto simp add: Suc_pred') apply(rule_tac x="n - 1" in exI) by auto
- then obtain N::nat where "inverse (real (N + 1)) < e" by auto
- hence "\<forall>n\<ge>N. inverse (real n + 1) < e" by (auto, metis Suc_le_mono le_SucE less_imp_inverse_less nat_le_real_less order_less_trans real_of_nat_Suc real_of_nat_Suc_gt_zero)
- hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto }
- hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"
+ {
+ assume "\<not> (f ---> x) sequentially"
+ {
+ fix e :: real
+ assume "e > 0"
+ then have "\<exists>N::nat. inverse (real (N + 1)) < e"
+ using real_arch_inv[of e]
+ apply (auto simp add: Suc_pred')
+ apply (rule_tac x="n - 1" in exI)
+ apply auto
+ done
+ then obtain N :: nat where "inverse (real (N + 1)) < e"
+ by auto
+ then have "\<forall>n\<ge>N. inverse (real n + 1) < e"
+ apply auto
+ apply (metis Suc_le_mono le_SucE less_imp_inverse_less nat_le_real_less order_less_trans
+ real_of_nat_Suc real_of_nat_Suc_gt_zero)
+ done
+ then have "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto
+ }
+ then have "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"
unfolding LIMSEQ_def by(auto simp add: dist_norm)
- hence "(f ---> x) sequentially" unfolding f_def
+ then have "(f ---> x) sequentially"
+ unfolding f_def
using tendsto_add[OF tendsto_const, of "\<lambda>n::nat. (inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b) - x)" 0 sequentially x]
- using tendsto_scaleR [OF _ tendsto_const, of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *\<^sub>R (a + b) - x)"] by auto }
+ using tendsto_scaleR [OF _ tendsto_const, of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *\<^sub>R (a + b) - x)"]
+ by auto
+ }
ultimately have "x \<in> closure {a<..<b}"
- using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto }
- thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast
-qed
-
-lemma bounded_subset_open_interval_symmetric: fixes s::"('a::ordered_euclidean_space) set"
- assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a<..<a}"
-proof-
- obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto
+ using as and open_interval_midpoint[OF assms]
+ unfolding closure_def
+ unfolding islimpt_sequential
+ by (cases "x=?c") auto
+ }
+ then show ?thesis
+ using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast
+qed
+
+lemma bounded_subset_open_interval_symmetric:
+ fixes s::"('a::ordered_euclidean_space) set"
+ assumes "bounded s"
+ shows "\<exists>a. s \<subseteq> {-a<..<a}"
+proof -
+ obtain b where "b>0" and b: "\<forall>x\<in>s. norm x \<le> b"
+ using assms[unfolded bounded_pos] by auto
def a \<equiv> "(\<Sum>i\<in>Basis. (b + 1) *\<^sub>R i)::'a"
- { fix x assume "x\<in>s"
- fix i :: 'a assume i:"i\<in>Basis"
- hence "(-a)\<bullet>i < x\<bullet>i" and "x\<bullet>i < a\<bullet>i" using b[THEN bspec[where x=x], OF `x\<in>s`]
- and Basis_le_norm[OF i, of x] unfolding inner_simps and a_def by auto }
- thus ?thesis by(auto intro: exI[where x=a] simp add: eucl_less[where 'a='a])
+ {
+ fix x
+ assume "x \<in> s"
+ fix i :: 'a
+ assume i: "i \<in> Basis"
+ then have "(-a)\<bullet>i < x\<bullet>i" and "x\<bullet>i < a\<bullet>i"
+ using b[THEN bspec[where x=x], OF `x\<in>s`]
+ using Basis_le_norm[OF i, of x]
+ unfolding inner_simps and a_def
+ by auto
+ }
+ then show ?thesis
+ by (auto intro: exI[where x=a] simp add: eucl_less[where 'a='a])
qed
lemma bounded_subset_open_interval:
@@ -5608,14 +6468,16 @@
lemma bounded_subset_closed_interval_symmetric:
fixes s :: "('a::ordered_euclidean_space) set"
assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a .. a}"
-proof-
- obtain a where "s \<subseteq> {- a<..<a}" using bounded_subset_open_interval_symmetric[OF assms] by auto
- thus ?thesis using interval_open_subset_closed[of "-a" a] by auto
+proof -
+ obtain a where "s \<subseteq> {- a<..<a}"
+ using bounded_subset_open_interval_symmetric[OF assms] by auto
+ then show ?thesis
+ using interval_open_subset_closed[of "-a" a] by auto
qed
lemma bounded_subset_closed_interval:
fixes s :: "('a::ordered_euclidean_space) set"
- shows "bounded s ==> (\<exists>a b. s \<subseteq> {a .. b})"
+ shows "bounded s \<Longrightarrow> (\<exists>a b. s \<subseteq> {a .. b})"
using bounded_subset_closed_interval_symmetric[of s] by auto
lemma frontier_closed_interval:
@@ -5626,44 +6488,74 @@
lemma frontier_open_interval:
fixes a b :: "'a::ordered_euclidean_space"
shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"
-proof(cases "{a<..<b} = {}")
- case True thus ?thesis using frontier_empty by auto
+proof (cases "{a<..<b} = {}")
+ case True
+ then show ?thesis
+ using frontier_empty by auto
next
- case False thus ?thesis unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval] by auto
-qed
-
-lemma inter_interval_mixed_eq_empty: fixes a :: "'a::ordered_euclidean_space"
- assumes "{c<..<d} \<noteq> {}" shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"
- unfolding closure_open_interval[OF assms, THEN sym] unfolding open_inter_closure_eq_empty[OF open_interval] ..
+ case False
+ then show ?thesis
+ unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval]
+ by auto
+qed
+
+lemma inter_interval_mixed_eq_empty:
+ fixes a :: "'a::ordered_euclidean_space"
+ assumes "{c<..<d} \<noteq> {}"
+ shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"
+ unfolding closure_open_interval[OF assms, THEN sym]
+ unfolding open_inter_closure_eq_empty[OF open_interval] ..
(* Some stuff for half-infinite intervals too; FIXME: notation? *)
-lemma closed_interval_left: fixes b::"'a::euclidean_space"
+lemma closed_interval_left:
+ fixes b :: "'a::euclidean_space"
shows "closed {x::'a. \<forall>i\<in>Basis. x\<bullet>i \<le> b\<bullet>i}"
-proof-
- { fix i :: 'a assume i:"i\<in>Basis"
- fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i\<in>Basis. x \<bullet> i \<le> b \<bullet> i}. x' \<noteq> x \<and> dist x' x < e"
- { assume "x\<bullet>i > b\<bullet>i"
+proof -
+ {
+ fix i :: 'a
+ assume i: "i \<in> Basis"
+ fix x :: "'a"
+ assume x: "\<forall>e>0. \<exists>x'\<in>{x. \<forall>i\<in>Basis. x \<bullet> i \<le> b \<bullet> i}. x' \<noteq> x \<and> dist x' x < e"
+ {
+ assume "x\<bullet>i > b\<bullet>i"
then obtain y where "y \<bullet> i \<le> b \<bullet> i" "y \<noteq> x" "dist y x < x\<bullet>i - b\<bullet>i"
using x[THEN spec[where x="x\<bullet>i - b\<bullet>i"]] using i by auto
- hence False using Basis_le_norm[OF i, of "y - x"] unfolding dist_norm inner_simps using i
- by auto }
- hence "x\<bullet>i \<le> b\<bullet>i" by(rule ccontr)auto }
- thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
-qed
-
-lemma closed_interval_right: fixes a::"'a::euclidean_space"
+ then have False
+ using Basis_le_norm[OF i, of "y - x"]
+ unfolding dist_norm inner_simps
+ using i
+ by auto
+ }
+ then have "x\<bullet>i \<le> b\<bullet>i" by (rule ccontr)auto
+ }
+ then show ?thesis
+ unfolding closed_limpt unfolding islimpt_approachable by blast
+qed
+
+lemma closed_interval_right:
+ fixes a :: "'a::euclidean_space"
shows "closed {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i}"
-proof-
- { fix i :: 'a assume i:"i\<in>Basis"
- fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i}. x' \<noteq> x \<and> dist x' x < e"
- { assume "a\<bullet>i > x\<bullet>i"
+proof -
+ {
+ fix i :: 'a
+ assume i: "i \<in> Basis"
+ fix x :: "'a"
+ assume x: "\<forall>e>0. \<exists>x'\<in>{x. \<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i}. x' \<noteq> x \<and> dist x' x < e"
+ {
+ assume "a\<bullet>i > x\<bullet>i"
then obtain y where "a \<bullet> i \<le> y \<bullet> i" "y \<noteq> x" "dist y x < a\<bullet>i - x\<bullet>i"
using x[THEN spec[where x="a\<bullet>i - x\<bullet>i"]] i by auto
- hence False using Basis_le_norm[OF i, of "y - x"] unfolding dist_norm inner_simps by auto }
- hence "a\<bullet>i \<le> x\<bullet>i" by(rule ccontr)auto }
- thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
+ then have False
+ using Basis_le_norm[OF i, of "y - x"]
+ unfolding dist_norm inner_simps
+ by auto
+ }
+ then have "a\<bullet>i \<le> x\<bullet>i" by (rule ccontr) auto
+ }
+ then show ?thesis
+ unfolding closed_limpt unfolding islimpt_approachable by blast
qed
lemma open_box: "open (box a b)"
@@ -5678,15 +6570,18 @@
instance euclidean_space \<subseteq> second_countable_topology
proof
def a \<equiv> "\<lambda>f :: 'a \<Rightarrow> (real \<times> real). \<Sum>i\<in>Basis. fst (f i) *\<^sub>R i"
- then have a: "\<And>f. (\<Sum>i\<in>Basis. fst (f i) *\<^sub>R i) = a f" by simp
+ then have a: "\<And>f. (\<Sum>i\<in>Basis. fst (f i) *\<^sub>R i) = a f"
+ by simp
def b \<equiv> "\<lambda>f :: 'a \<Rightarrow> (real \<times> real). \<Sum>i\<in>Basis. snd (f i) *\<^sub>R i"
- then have b: "\<And>f. (\<Sum>i\<in>Basis. snd (f i) *\<^sub>R i) = b f" by simp
+ then have b: "\<And>f. (\<Sum>i\<in>Basis. snd (f i) *\<^sub>R i) = b f"
+ by simp
def B \<equiv> "(\<lambda>f. box (a f) (b f)) ` (Basis \<rightarrow>\<^sub>E (\<rat> \<times> \<rat>))"
have "Ball B open" by (simp add: B_def open_box)
moreover have "(\<forall>A. open A \<longrightarrow> (\<exists>B'\<subseteq>B. \<Union>B' = A))"
proof safe
- fix A::"'a set" assume "open A"
+ fix A::"'a set"
+ assume "open A"
show "\<exists>B'\<subseteq>B. \<Union>B' = A"
apply (rule exI[of _ "{b\<in>B. b \<subseteq> A}"])
apply (subst (3) open_UNION_box[OF `open A`])
@@ -5694,9 +6589,11 @@
done
qed
ultimately
- have "topological_basis B" unfolding topological_basis_def by blast
+ have "topological_basis B"
+ unfolding topological_basis_def by blast
moreover
- have "countable B" unfolding B_def
+ have "countable B"
+ unfolding B_def
by (intro countable_image countable_PiE finite_Basis countable_SIGMA countable_rat)
ultimately show "\<exists>B::'a set set. countable B \<and> open = generate_topology B"
by (blast intro: topological_basis_imp_subbasis)
@@ -5728,25 +6625,29 @@
subsection {* Closure of halfspaces and hyperplanes *}
lemma isCont_open_vimage:
- assumes "\<And>x. isCont f x" and "open s" shows "open (f -` s)"
+ assumes "\<And>x. isCont f x"
+ and "open s"
+ shows "open (f -` s)"
proof -
from assms(1) have "continuous_on UNIV f"
unfolding isCont_def continuous_on_def by simp
- hence "open {x \<in> UNIV. f x \<in> s}"
+ then have "open {x \<in> UNIV. f x \<in> s}"
using open_UNIV `open s` by (rule continuous_open_preimage)
- thus "open (f -` s)"
+ then show "open (f -` s)"
by (simp add: vimage_def)
qed
lemma isCont_closed_vimage:
- assumes "\<And>x. isCont f x" and "closed s" shows "closed (f -` s)"
+ assumes "\<And>x. isCont f x"
+ and "closed s"
+ shows "closed (f -` s)"
using assms unfolding closed_def vimage_Compl [symmetric]
by (rule isCont_open_vimage)
lemma open_Collect_less:
fixes f g :: "'a::t2_space \<Rightarrow> real"
assumes f: "\<And>x. isCont f x"
- assumes g: "\<And>x. isCont g x"
+ and g: "\<And>x. isCont g x"
shows "open {x. f x < g x}"
proof -
have "open ((\<lambda>x. g x - f x) -` {0<..})"
@@ -5760,7 +6661,7 @@
lemma closed_Collect_le:
fixes f g :: "'a::t2_space \<Rightarrow> real"
assumes f: "\<And>x. isCont f x"
- assumes g: "\<And>x. isCont g x"
+ and g: "\<And>x. isCont g x"
shows "closed {x. f x \<le> g x}"
proof -
have "closed ((\<lambda>x. g x - f x) -` {0..})"
@@ -5774,12 +6675,12 @@
lemma closed_Collect_eq:
fixes f g :: "'a::t2_space \<Rightarrow> 'b::t2_space"
assumes f: "\<And>x. isCont f x"
- assumes g: "\<And>x. isCont g x"
+ and g: "\<And>x. isCont g x"
shows "closed {x. f x = g x}"
proof -
have "open {(x::'b, y::'b). x \<noteq> y}"
unfolding open_prod_def by (auto dest!: hausdorff)
- hence "closed {(x::'b, y::'b). x = y}"
+ then have "closed {(x::'b, y::'b). x = y}"
unfolding closed_def split_def Collect_neg_eq .
with isCont_Pair [OF f g]
have "closed ((\<lambda>x. (f x, g x)) -` {(x, y). x = y})"
@@ -5800,12 +6701,10 @@
lemma closed_hyperplane: "closed {x. inner a x = b}"
by (simp add: closed_Collect_eq)
-lemma closed_halfspace_component_le:
- shows "closed {x::'a::euclidean_space. x\<bullet>i \<le> a}"
+lemma closed_halfspace_component_le: "closed {x::'a::euclidean_space. x\<bullet>i \<le> a}"
by (simp add: closed_Collect_le)
-lemma closed_halfspace_component_ge:
- shows "closed {x::'a::euclidean_space. x\<bullet>i \<ge> a}"
+lemma closed_halfspace_component_ge: "closed {x::'a::euclidean_space. x\<bullet>i \<ge> a}"
by (simp add: closed_Collect_le)
text {* Openness of halfspaces. *}
@@ -5816,12 +6715,10 @@
lemma open_halfspace_gt: "open {x. inner a x > b}"
by (simp add: open_Collect_less)
-lemma open_halfspace_component_lt:
- shows "open {x::'a::euclidean_space. x\<bullet>i < a}"
+lemma open_halfspace_component_lt: "open {x::'a::euclidean_space. x\<bullet>i < a}"
by (simp add: open_Collect_less)
-lemma open_halfspace_component_gt:
- shows "open {x::'a::euclidean_space. x\<bullet>i > a}"
+lemma open_halfspace_component_gt: "open {x::'a::euclidean_space. x\<bullet>i > a}"
by (simp add: open_Collect_less)
text{* Instantiation for intervals on @{text ordered_euclidean_space} *}
@@ -5829,22 +6726,22 @@
lemma eucl_lessThan_eq_halfspaces:
fixes a :: "'a\<Colon>ordered_euclidean_space"
shows "{..<a} = (\<Inter>i\<in>Basis. {x. x \<bullet> i < a \<bullet> i})"
- by (auto simp: eucl_less[where 'a='a])
+ by (auto simp: eucl_less[where 'a='a])
lemma eucl_greaterThan_eq_halfspaces:
fixes a :: "'a\<Colon>ordered_euclidean_space"
shows "{a<..} = (\<Inter>i\<in>Basis. {x. a \<bullet> i < x \<bullet> i})"
- by (auto simp: eucl_less[where 'a='a])
+ by (auto simp: eucl_less[where 'a='a])
lemma eucl_atMost_eq_halfspaces:
fixes a :: "'a\<Colon>ordered_euclidean_space"
shows "{.. a} = (\<Inter>i\<in>Basis. {x. x \<bullet> i \<le> a \<bullet> i})"
- by (auto simp: eucl_le[where 'a='a])
+ by (auto simp: eucl_le[where 'a='a])
lemma eucl_atLeast_eq_halfspaces:
fixes a :: "'a\<Colon>ordered_euclidean_space"
shows "{a ..} = (\<Inter>i\<in>Basis. {x. a \<bullet> i \<le> x \<bullet> i})"
- by (auto simp: eucl_le[where 'a='a])
+ by (auto simp: eucl_le[where 'a='a])
lemma open_eucl_lessThan[simp, intro]:
fixes a :: "'a\<Colon>ordered_euclidean_space"
@@ -5870,26 +6767,37 @@
text {* This gives a simple derivation of limit component bounds. *}
-lemma Lim_component_le: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
- assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. f(x)\<bullet>i \<le> b) net"
+lemma Lim_component_le:
+ fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
+ assumes "(f ---> l) net"
+ and "\<not> (trivial_limit net)"
+ and "eventually (\<lambda>x. f(x)\<bullet>i \<le> b) net"
shows "l\<bullet>i \<le> b"
by (rule tendsto_le[OF assms(2) tendsto_const tendsto_inner[OF assms(1) tendsto_const] assms(3)])
-lemma Lim_component_ge: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
- assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. b \<le> (f x)\<bullet>i) net"
+lemma Lim_component_ge:
+ fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
+ assumes "(f ---> l) net"
+ and "\<not> (trivial_limit net)"
+ and "eventually (\<lambda>x. b \<le> (f x)\<bullet>i) net"
shows "b \<le> l\<bullet>i"
by (rule tendsto_le[OF assms(2) tendsto_inner[OF assms(1) tendsto_const] tendsto_const assms(3)])
-lemma Lim_component_eq: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
- assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)\<bullet>i = b) net"
+lemma Lim_component_eq:
+ fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
+ assumes net: "(f ---> l) net" "~(trivial_limit net)"
+ and ev:"eventually (\<lambda>x. f(x)\<bullet>i = b) net"
shows "l\<bullet>i = b"
using ev[unfolded order_eq_iff eventually_conj_iff]
- using Lim_component_ge[OF net, of b i] and Lim_component_le[OF net, of i b] by auto
-
-text{* Limits relative to a union. *}
+ using Lim_component_ge[OF net, of b i]
+ using Lim_component_le[OF net, of i b]
+ by auto
+
+text {* Limits relative to a union. *}
lemma eventually_within_Un:
- "eventually P (at x within (s \<union> t)) \<longleftrightarrow> eventually P (at x within s) \<and> eventually P (at x within t)"
+ "eventually P (at x within (s \<union> t)) \<longleftrightarrow>
+ eventually P (at x within s) \<and> eventually P (at x within t)"
unfolding eventually_at_filter
by (auto elim!: eventually_rev_mp)
@@ -5900,25 +6808,27 @@
by (auto simp add: eventually_within_Un)
lemma Lim_topological:
- "(f ---> l) net \<longleftrightarrow>
- trivial_limit net \<or>
- (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
+ "(f ---> l) net \<longleftrightarrow>
+ trivial_limit net \<or> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
unfolding tendsto_def trivial_limit_eq by auto
-text{* Some more convenient intermediate-value theorem formulations. *}
+text{* Some more convenient intermediate-value theorem formulations. *}
lemma connected_ivt_hyperplane:
assumes "connected s" "x \<in> s" "y \<in> s" "inner a x \<le> b" "b \<le> inner a y"
shows "\<exists>z \<in> s. inner a z = b"
-proof(rule ccontr)
+proof (rule ccontr)
assume as:"\<not> (\<exists>z\<in>s. inner a z = b)"
let ?A = "{x. inner a x < b}"
let ?B = "{x. inner a x > b}"
- have "open ?A" "open ?B" using open_halfspace_lt and open_halfspace_gt by auto
+ have "open ?A" "open ?B"
+ using open_halfspace_lt and open_halfspace_gt by auto
moreover have "?A \<inter> ?B = {}" by auto
moreover have "s \<subseteq> ?A \<union> ?B" using as by auto
ultimately show False
- using assms(1)[unfolded connected_def not_ex, THEN spec[where x="?A"], THEN spec[where x="?B"]] and assms(2-5)
+ using assms(1)[unfolded connected_def not_ex,
+ THEN spec[where x="?A"], THEN spec[where x="?B"]]
+ using assms(2-5)
by auto
qed
@@ -5939,38 +6849,38 @@
definition
homeomorphic :: "'a::topological_space set \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"
- (infixr "homeomorphic" 60) where
- "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"
+ (infixr "homeomorphic" 60)
+ where "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"
lemma homeomorphic_refl: "s homeomorphic s"
unfolding homeomorphic_def
unfolding homeomorphism_def
using continuous_on_id
- apply(rule_tac x = "(\<lambda>x. x)" in exI)
- apply(rule_tac x = "(\<lambda>x. x)" in exI)
+ apply (rule_tac x = "(\<lambda>x. x)" in exI)
+ apply (rule_tac x = "(\<lambda>x. x)" in exI)
apply blast
done
lemma homeomorphic_sym: "s homeomorphic t \<longleftrightarrow> t homeomorphic s"
unfolding homeomorphic_def
unfolding homeomorphism_def
- by blast
+ by blast
lemma homeomorphic_trans:
- assumes "s homeomorphic t" "t homeomorphic u"
+ assumes "s homeomorphic t"
+ and "t homeomorphic u"
shows "s homeomorphic u"
-proof-
- obtain f1 g1 where fg1:"\<forall>x\<in>s. g1 (f1 x) = x" "f1 ` s = t"
- "continuous_on s f1" "\<forall>y\<in>t. f1 (g1 y) = y" "g1 ` t = s" "continuous_on t g1"
+proof -
+ obtain f1 g1 where fg1: "\<forall>x\<in>s. g1 (f1 x) = x" "f1 ` s = t"
+ "continuous_on s f1" "\<forall>y\<in>t. f1 (g1 y) = y" "g1 ` t = s" "continuous_on t g1"
using assms(1) unfolding homeomorphic_def homeomorphism_def by auto
- obtain f2 g2 where fg2:"\<forall>x\<in>t. g2 (f2 x) = x" "f2 ` t = u" "continuous_on t f2"
- "\<forall>y\<in>u. f2 (g2 y) = y" "g2 ` u = t" "continuous_on u g2"
+ obtain f2 g2 where fg2: "\<forall>x\<in>t. g2 (f2 x) = x" "f2 ` t = u" "continuous_on t f2"
+ "\<forall>y\<in>u. f2 (g2 y) = y" "g2 ` u = t" "continuous_on u g2"
using assms(2) unfolding homeomorphic_def homeomorphism_def by auto
-
{
fix x
assume "x\<in>s"
- hence "(g1 \<circ> g2) ((f2 \<circ> f1) x) = x"
+ then have "(g1 \<circ> g2) ((f2 \<circ> f1) x) = x"
using fg1(1)[THEN bspec[where x=x]] and fg2(1)[THEN bspec[where x="f1 x"]] and fg1(2)
by auto
}
@@ -5978,10 +6888,11 @@
using fg1(2) fg2(2) by auto
moreover have "continuous_on s (f2 \<circ> f1)"
using continuous_on_compose[OF fg1(3)] and fg2(3) unfolding fg1(2) by auto
- moreover {
+ moreover
+ {
fix y
assume "y\<in>u"
- hence "(f2 \<circ> f1) ((g1 \<circ> g2) y) = y"
+ then have "(f2 \<circ> f1) ((g1 \<circ> g2) y) = y"
using fg2(4)[THEN bspec[where x=y]] and fg1(4)[THEN bspec[where x="g2 y"]] and fg2(5)
by auto
}
@@ -6030,17 +6941,19 @@
fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
assumes "compact s" "continuous_on s f" "f ` s = t" "inj_on f s"
shows "\<exists>g. homeomorphism s t f g"
-proof-
+proof -
def g \<equiv> "\<lambda>x. SOME y. y\<in>s \<and> f y = x"
have g: "\<forall>x\<in>s. g (f x) = x"
using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto
{
- fix y assume "y\<in>t"
- then obtain x where x:"f x = y" "x\<in>s" using assms(3) by auto
- hence "g (f x) = x" using g by auto
- hence "f (g y) = y" unfolding x(1)[THEN sym] by auto
+ fix y
+ assume "y \<in> t"
+ then obtain x where x:"f x = y" "x\<in>s"
+ using assms(3) by auto
+ then have "g (f x) = x" using g by auto
+ then have "f (g y) = y" unfolding x(1)[THEN sym] by auto
}
- hence g':"\<forall>x\<in>t. f (g x) = x" by auto
+ then have g':"\<forall>x\<in>t. f (g x) = x" by auto
moreover
{
fix x
@@ -6055,7 +6968,7 @@
then obtain y where y:"y\<in>t" "g y = x" by auto
then obtain x' where x':"x'\<in>s" "f x' = y"
using assms(3) by auto
- hence "x \<in> s"
+ then have "x \<in> s"
unfolding g_def
using someI2[of "\<lambda>b. b\<in>s \<and> f b = y" x' "\<lambda>x. x\<in>s"]
unfolding y(2)[THEN sym] and g_def
@@ -6063,7 +6976,7 @@
}
ultimately have "x\<in>s \<longleftrightarrow> x \<in> g ` t" ..
}
- hence "g ` t = s" by auto
+ then have "g ` t = s" by auto
ultimately show ?thesis
unfolding homeomorphism_def homeomorphic_def
apply (rule_tac x=g in exI)
@@ -6074,21 +6987,21 @@
lemma homeomorphic_compact:
fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
- shows "compact s \<Longrightarrow> continuous_on s f \<Longrightarrow> (f ` s = t) \<Longrightarrow> inj_on f s
- \<Longrightarrow> s homeomorphic t"
+ shows "compact s \<Longrightarrow> continuous_on s f \<Longrightarrow> (f ` s = t) \<Longrightarrow> inj_on f s \<Longrightarrow> s homeomorphic t"
unfolding homeomorphic_def by (metis homeomorphism_compact)
-text{* Preservation of topological properties. *}
+text{* Preservation of topological properties. *}
lemma homeomorphic_compactness: "s homeomorphic t \<Longrightarrow> (compact s \<longleftrightarrow> compact t)"
unfolding homeomorphic_def homeomorphism_def
by (metis compact_continuous_image)
-text{* Results on translation, scaling etc. *}
+text{* Results on translation, scaling etc. *}
lemma homeomorphic_scaling:
fixes s :: "'a::real_normed_vector set"
- assumes "c \<noteq> 0" shows "s homeomorphic ((\<lambda>x. c *\<^sub>R x) ` s)"
+ assumes "c \<noteq> 0"
+ shows "s homeomorphic ((\<lambda>x. c *\<^sub>R x) ` s)"
unfolding homeomorphic_minimal
apply (rule_tac x="\<lambda>x. c *\<^sub>R x" in exI)
apply (rule_tac x="\<lambda>x. (1 / c) *\<^sub>R x" in exI)
@@ -6110,7 +7023,7 @@
fixes s :: "'a::real_normed_vector set"
assumes "c \<noteq> 0"
shows "s homeomorphic ((\<lambda>x. a + c *\<^sub>R x) ` s)"
-proof-
+proof -
have *: "op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
show ?thesis
using homeomorphic_trans
@@ -6124,8 +7037,8 @@
fixes a b ::"'a::real_normed_vector"
assumes "0 < d" "0 < e"
shows "(ball a d) homeomorphic (ball b e)" (is ?th)
- "(cball a d) homeomorphic (cball b e)" (is ?cth)
-proof-
+ and "(cball a d) homeomorphic (cball b e)" (is ?cth)
+proof -
show ?th unfolding homeomorphic_minimal
apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
@@ -6133,7 +7046,6 @@
apply (auto intro!: continuous_on_intros
simp: dist_commute dist_norm pos_divide_less_eq mult_strict_left_mono)
done
-next
show ?cth unfolding homeomorphic_minimal
apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
@@ -6173,9 +7085,9 @@
using `N \<le> n` N unfolding f.diff[THEN sym] by auto
finally have "norm (x n - x N) < d" using `e>0` by simp
}
- hence "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d" by auto
+ then have "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d" by auto
}
- thus ?thesis unfolding cauchy and dist_norm by auto
+ then show ?thesis unfolding cauchy and dist_norm by auto
qed
lemma complete_isometric_image:
@@ -6190,18 +7102,18 @@
{
fix g
assume as:"\<forall>n::nat. g n \<in> f ` s" and cfg:"Cauchy g"
- then obtain x where "\<forall>n. x n \<in> s \<and> g n = f (x n)"
+ then 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
- hence x:"\<forall>n. x n \<in> s" "\<forall>n. g n = f (x n)" by auto
- hence "f \<circ> x = g" unfolding fun_eq_iff 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" unfolding fun_eq_iff by auto
then obtain l where "l\<in>s" and l:"(x ---> l) sequentially"
using cs[unfolded complete_def, THEN spec[where x="x"]]
using cauchy_isometric[OF `0<e` s f normf] and cfg and x(1) by auto
- hence "\<exists>l\<in>f ` s. (g ---> l) sequentially"
+ then have "\<exists>l\<in>f ` s. (g ---> l) sequentially"
using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
unfolding `f \<circ> x = g` by auto
}
- thus ?thesis unfolding complete_def by auto
+ then show ?thesis unfolding complete_def by auto
qed
lemma injective_imp_isometric:
@@ -6214,10 +7126,10 @@
{
fix x
assume "x \<in> s"
- hence "x = 0" using True by auto
- hence "norm x \<le> norm (f x)" by auto
+ then have "x = 0" using True by auto
+ then have "norm x \<le> norm (f x)" by auto
}
- thus ?thesis by (auto intro!: exI[where x=1])
+ then show ?thesis by (auto intro!: exI[where x=1])
next
interpret f: bounded_linear f by fact
case False
@@ -6229,7 +7141,7 @@
have "?S'' = frontier(cball 0 (norm a))"
unfolding frontier_cball and dist_norm by auto
- hence "compact ?S''"
+ then have "compact ?S''"
using compact_frontier[OF compact_cball, of 0 "norm a"] by auto
moreover have "?S' = s \<inter> ?S''" by auto
ultimately have "compact ?S'"
@@ -6237,12 +7149,13 @@
moreover have *:"f ` ?S' = ?S" by auto
ultimately have "compact ?S"
using compact_continuous_image[OF linear_continuous_on[OF f(1)], of ?S'] by auto
- hence "closed ?S" using compact_imp_closed by auto
+ then have "closed ?S" using compact_imp_closed by auto
moreover have "?S \<noteq> {}" using a by auto
ultimately obtain b' where "b'\<in>?S" "\<forall>y\<in>?S. norm b' \<le> norm y"
using distance_attains_inf[of ?S 0] unfolding dist_0_norm by auto
- then obtain b where "b\<in>s" and ba:"norm b = norm a"
- and b:"\<forall>x\<in>{x \<in> s. norm x = norm a}. norm (f b) \<le> norm (f x)"
+ then obtain b where "b\<in>s"
+ and ba: "norm b = norm a"
+ and b: "\<forall>x\<in>{x \<in> s. norm x = norm a}. norm (f b) \<le> norm (f x)"
unfolding *[THEN sym] unfolding image_iff by auto
let ?e = "norm (f b) / norm b"
@@ -6258,21 +7171,21 @@
{
fix x
assume "x\<in>s"
- hence "norm (f b) / norm b * norm x \<le> norm (f x)"
+ then have "norm (f b) / norm b * norm x \<le> norm (f x)"
proof (cases "x=0")
case True
- thus "norm (f b) / norm b * norm x \<le> norm (f x)" by auto
+ then show "norm (f b) / norm b * norm x \<le> norm (f x)" by auto
next
case False
- hence *:"0 < norm a / norm x"
+ then have *: "0 < norm a / norm x"
using `a\<noteq>0`
unfolding zero_less_norm_iff[THEN sym]
by (simp only: divide_pos_pos)
have "\<forall>c. \<forall>x\<in>s. c *\<^sub>R x \<in> s"
using s[unfolded subspace_def] by auto
- hence "(norm a / norm x) *\<^sub>R x \<in> {x \<in> s. norm x = norm a}"
+ then have "(norm a / norm x) *\<^sub>R x \<in> {x \<in> s. norm x = norm a}"
using `x\<in>s` and `x\<noteq>0` by auto
- thus "norm (f b) / norm b * norm x \<le> norm (f x)"
+ then show "norm (f b) / norm b * norm x \<le> norm (f x)"
using b[THEN bspec[where x="(norm a / norm x) *\<^sub>R x"]]
unfolding f.scaleR and ba using `x\<noteq>0` `a\<noteq>0`
by (auto simp add: mult_commute pos_le_divide_eq pos_divide_le_eq)
@@ -6283,10 +7196,10 @@
lemma closed_injective_image_subspace:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "subspace s" "bounded_linear f" "\<forall>x\<in>s. f x = 0 --> x = 0" "closed s"
+ assumes "subspace s" "bounded_linear f" "\<forall>x\<in>s. f x = 0 \<longrightarrow> x = 0" "closed s"
shows "closed(f ` s)"
-proof-
- obtain e where "e>0" and e:"\<forall>x\<in>s. e * norm x \<le> norm (f x)"
+proof -
+ obtain e where "e > 0" and e: "\<forall>x\<in>s. e * norm x \<le> norm (f x)"
using injective_imp_isometric[OF assms(4,1,2,3)] by auto
show ?thesis
using complete_isometric_image[OF `e>0` assms(1,2) e] and assms(4)
@@ -6320,55 +7233,58 @@
moreover
{
fix x::"'a" assume "x \<in> ?A"
- hence "finite d" "x \<in> ?A" using assms by(auto intro: finite_subset[OF _ finite_Basis])
+ then have "finite d" "x \<in> ?A"
+ using assms by (auto intro: finite_subset[OF _ finite_Basis])
from this d have "x \<in> span d"
proof (induct d arbitrary: x)
case empty
- hence "x=0"
+ then have "x = 0"
apply (rule_tac euclidean_eqI)
apply auto
done
- thus ?case
+ then show ?case
using subspace_0[OF subspace_span[of "{}"]] by auto
next
case (insert k F)
- hence *:"\<forall>i\<in>Basis. i \<notin> insert k F \<longrightarrow> x \<bullet> i = 0" by auto
- have **:"F \<subseteq> insert k F" by auto
+ then have *: "\<forall>i\<in>Basis. i \<notin> insert k F \<longrightarrow> x \<bullet> i = 0" by auto
+ have **: "F \<subseteq> insert k F" by auto
def y \<equiv> "x - (x\<bullet>k) *\<^sub>R k"
- have y:"x = y + (x\<bullet>k) *\<^sub>R k" unfolding y_def by auto
+ have y: "x = y + (x\<bullet>k) *\<^sub>R k" unfolding y_def by auto
{ fix i assume i': "i \<notin> F" "i \<in> Basis"
- hence "y \<bullet> i = 0" unfolding y_def
+ then have "y \<bullet> i = 0" unfolding y_def
using *[THEN bspec[where x=i]] insert by (auto simp: inner_simps inner_Basis) }
- hence "y \<in> span F" using insert by auto
- hence "y \<in> span (insert k F)"
+ then have "y \<in> span F" using insert by auto
+ then have "y \<in> span (insert k F)"
using span_mono[of F "insert k F"] using assms by auto
moreover
have "k \<in> span (insert k F)" by(rule span_superset, auto)
- hence "(x\<bullet>k) *\<^sub>R k \<in> span (insert k F)"
+ then have "(x\<bullet>k) *\<^sub>R k \<in> span (insert k F)"
using span_mul by auto
ultimately
have "y + (x\<bullet>k) *\<^sub>R k \<in> span (insert k F)"
using span_add by auto
- thus ?case using y by auto
+ then show ?case using y by auto
qed
}
- hence "?A \<subseteq> span d" by auto
+ then have "?A \<subseteq> span d" by auto
moreover
{
fix x
assume "x \<in> d"
- hence "x \<in> ?D" using assms by auto
+ then have "x \<in> ?D" using assms by auto
}
- hence "independent d"
+ then have "independent d"
using independent_mono[OF independent_Basis, of d] and assms by auto
moreover
have "d \<subseteq> ?D" unfolding subset_eq using assms by auto
ultimately show ?thesis using dim_unique[of d ?A] by auto
qed
-text{* Hence closure and completeness of all subspaces. *}
-
-lemma ex_card: assumes "n \<le> card A" shows "\<exists>S\<subseteq>A. card S = n"
+text{* Hence closure and completeness of all subspaces. *}
+
+lemma ex_card:
+ assumes "n \<le> card A"
+ shows "\<exists>S\<subseteq>A. card S = n"
proof cases
assume "finite A"
from ex_bij_betw_nat_finite[OF this] guess f ..
@@ -6425,7 +7341,7 @@
using dim_subset[of "closure s" "span s"]
unfolding dim_span
by auto
- thus ?thesis using dim_subset[OF closure_subset, of s]
+ then show ?thesis using dim_subset[OF closure_subset, of s]
by auto
qed
@@ -6462,34 +7378,54 @@
(if {a .. b} = {} then {}
else (if 0 \<le> m then {m *\<^sub>R a + c .. m *\<^sub>R b + c}
else {m *\<^sub>R b + c .. m *\<^sub>R a + c}))"
-proof(cases "m=0")
- { fix x assume "x \<le> c" "c \<le> x"
- hence "x=c" unfolding eucl_le[where 'a='a] apply-
- apply(subst euclidean_eq_iff) by (auto intro: order_antisym) }
+proof (cases "m = 0")
+ {
+ fix x
+ assume "x \<le> c" "c \<le> x"
+ then have "x = c"
+ unfolding eucl_le[where 'a='a]
+ apply -
+ apply (subst euclidean_eq_iff)
+ apply (auto intro: order_antisym)
+ done
+ }
moreover case True
moreover have "c \<in> {m *\<^sub>R a + c..m *\<^sub>R b + c}" unfolding True by(auto simp add: eucl_le[where 'a='a])
ultimately show ?thesis by auto
next
case False
- { fix y assume "a \<le> y" "y \<le> b" "m > 0"
- hence "m *\<^sub>R a + c \<le> m *\<^sub>R y + c" "m *\<^sub>R y + c \<le> m *\<^sub>R b + c"
+ {
+ fix y
+ assume "a \<le> y" "y \<le> b" "m > 0"
+ then have "m *\<^sub>R a + c \<le> m *\<^sub>R y + c" and "m *\<^sub>R y + c \<le> m *\<^sub>R b + c"
unfolding eucl_le[where 'a='a] by (auto simp: inner_simps)
- } moreover
- { fix y assume "a \<le> y" "y \<le> b" "m < 0"
- hence "m *\<^sub>R b + c \<le> m *\<^sub>R y + c" "m *\<^sub>R y + c \<le> m *\<^sub>R a + c"
- unfolding eucl_le[where 'a='a] by(auto simp add: mult_left_mono_neg inner_simps)
- } moreover
- { fix y assume "m > 0" "m *\<^sub>R a + c \<le> y" "y \<le> m *\<^sub>R b + c"
- hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
+ }
+ moreover
+ {
+ fix y
+ assume "a \<le> y" "y \<le> b" "m < 0"
+ then have "m *\<^sub>R b + c \<le> m *\<^sub>R y + c" and "m *\<^sub>R y + c \<le> m *\<^sub>R a + c"
+ unfolding eucl_le[where 'a='a] by (auto simp add: mult_left_mono_neg inner_simps)
+ }
+ moreover
+ {
+ fix y
+ assume "m > 0" and "m *\<^sub>R a + c \<le> y" and "y \<le> m *\<^sub>R b + c"
+ then have "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]
apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
- by(auto simp add: pos_le_divide_eq pos_divide_le_eq mult_commute diff_le_iff inner_simps)
- } moreover
- { fix y assume "m *\<^sub>R b + c \<le> y" "y \<le> m *\<^sub>R a + c" "m < 0"
- hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
+ apply (auto simp add: pos_le_divide_eq pos_divide_le_eq mult_commute diff_le_iff inner_simps)
+ done
+ }
+ moreover
+ {
+ fix y
+ assume "m *\<^sub>R b + c \<le> y" "y \<le> m *\<^sub>R a + c" "m < 0"
+ then have "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]
apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
- by(auto simp add: neg_le_divide_eq neg_divide_le_eq mult_commute diff_le_iff inner_simps)
+ apply (auto simp add: neg_le_divide_eq neg_divide_le_eq mult_commute diff_le_iff inner_simps)
+ done
}
ultimately show ?thesis using False by auto
qed
@@ -6502,40 +7438,44 @@
subsection {* Banach fixed point theorem (not really topological...) *}
lemma banach_fix:
- assumes s:"complete s" "s \<noteq> {}" and c:"0 \<le> c" "c < 1" and f:"(f ` s) \<subseteq> s" and
- lipschitz:"\<forall>x\<in>s. \<forall>y\<in>s. dist (f x) (f y) \<le> c * dist x y"
+ assumes s: "complete s" "s \<noteq> {}"
+ and c: "0 \<le> c" "c < 1"
+ and f: "(f ` s) \<subseteq> s"
+ and lipschitz:"\<forall>x\<in>s. \<forall>y\<in>s. dist (f x) (f y) \<le> c * dist x y"
shows "\<exists>! x\<in>s. (f x = x)"
-proof-
+proof -
have "1 - c > 0" using c by auto
from s(2) obtain z0 where "z0 \<in> s" by auto
def z \<equiv> "\<lambda>n. (f ^^ n) z0"
- { fix n::nat
+ {
+ fix n :: nat
have "z n \<in> s" unfolding z_def
proof (induct n)
case 0
- thus ?case using `z0 \<in>s` by auto
+ then show ?case using `z0 \<in> s` by auto
next
case Suc
- thus ?case using f by auto qed
+ then show ?case using f by auto qed
} note z_in_s = this
def d \<equiv> "dist (z 0) (z 1)"
have fzn:"\<And>n. f (z n) = z (Suc n)" unfolding z_def by auto
{
- fix n::nat
+ fix n :: nat
have "dist (z n) (z (Suc n)) \<le> (c ^ n) * d"
proof (induct n)
- case 0 thus ?case
+ case 0
+ then show ?case
unfolding d_def by auto
next
case (Suc m)
- hence "c * dist (z m) (z (Suc m)) \<le> c ^ Suc m * d"
+ then have "c * dist (z m) (z (Suc m)) \<le> c ^ Suc m * d"
using `0 \<le> c`
using mult_left_mono[of "dist (z m) (z (Suc m))" "c ^ m * d" c]
by auto
- thus ?case
+ then show ?case
using lipschitz[THEN bspec[where x="z m"], OF z_in_s, THEN bspec[where x="z (Suc m)"], OF z_in_s]
unfolding fzn and mult_le_cancel_left
by auto
@@ -6543,10 +7483,11 @@
} note cf_z = this
{
- fix n m::nat
+ fix n m :: nat
have "(1 - c) * dist (z m) (z (m+n)) \<le> (c ^ m) * d * (1 - c ^ n)"
proof (induct n)
- case 0 show ?case by auto
+ case 0
+ show ?case by auto
next
case (Suc k)
have "(1 - c) * dist (z m) (z (m + Suc k)) \<le>
@@ -6564,21 +7505,21 @@
qed
} note cf_z2 = this
{
- fix e::real
- assume "e>0"
- hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (z m) (z n) < e"
+ fix e :: real
+ assume "e > 0"
+ then have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (z m) (z n) < e"
proof (cases "d = 0")
case True
have *: "\<And>x. ((1 - c) * x \<le> 0) = (x \<le> 0)" using `1 - c > 0`
by (metis mult_zero_left mult_commute real_mult_le_cancel_iff1)
from True have "\<And>n. z n = z0" using cf_z2[of 0] and c unfolding z_def
by (simp add: *)
- thus ?thesis using `e>0` by auto
+ then show ?thesis using `e>0` by auto
next
case False
- hence "d>0" unfolding d_def using zero_le_dist[of "z 0" "z 1"]
+ then have "d>0" unfolding d_def using zero_le_dist[of "z 0" "z 1"]
by (metis False d_def less_le)
- hence "0 < e * (1 - c) / d"
+ then have "0 < e * (1 - c) / d"
using `e>0` and `1-c>0`
using divide_pos_pos[of "e * (1 - c)" d] and mult_pos_pos[of e "1 - c"]
by auto
@@ -6591,7 +7532,7 @@
using power_decreasing[OF `n\<ge>N`, of c] by auto
have "1 - c ^ (m - n) > 0"
using c and power_strict_mono[of c 1 "m - n"] using `m>n` by auto
- hence **:"d * (1 - c ^ (m - n)) / (1 - c) > 0"
+ then have **: "d * (1 - c ^ (m - n)) / (1 - c) > 0"
using mult_pos_pos[OF `d>0`, of "1 - c ^ (m - n)"]
using divide_pos_pos[of "d * (1 - c ^ (m - n))" "1 - c"]
using `0 < 1 - c`
@@ -6606,27 +7547,30 @@
unfolding mult_assoc by auto
also have "\<dots> < (e * (1 - c) / d) * d * (1 - c ^ (m - n)) / (1 - c)"
using mult_strict_right_mono[OF N **] unfolding mult_assoc by auto
- also have "\<dots> = e * (1 - c ^ (m - n))" using c and `d>0` and `1 - c > 0` by auto
- also have "\<dots> \<le> e" using c and `1 - c ^ (m - n) > 0` and `e>0` using mult_right_le_one_le[of e "1 - c ^ (m - n)"] by auto
+ also have "\<dots> = e * (1 - c ^ (m - n))"
+ using c and `d>0` and `1 - c > 0` by auto
+ also have "\<dots> \<le> e" using c and `1 - c ^ (m - n) > 0` and `e>0`
+ using mult_right_le_one_le[of e "1 - c ^ (m - n)"] by auto
finally have "dist (z m) (z n) < e" by auto
} note * = this
{
- fix m n::nat
- assume as:"N\<le>m" "N\<le>n"
- hence "dist (z n) (z m) < e"
+ fix m n :: nat
+ assume as: "N \<le> m" "N \<le> n"
+ then have "dist (z n) (z m) < e"
proof (cases "n = m")
case True
- thus ?thesis using `e>0` by auto
+ then show ?thesis using `e>0` by auto
next
case False
- thus ?thesis using as and *[of n m] *[of m n]
+ then show ?thesis using as and *[of n m] *[of m n]
unfolding nat_neq_iff by (auto simp add: dist_commute)
qed
}
- thus ?thesis by auto
+ then show ?thesis by auto
qed
}
- hence "Cauchy z" unfolding cauchy_def by auto
+ then have "Cauchy z"
+ unfolding cauchy_def by auto
then obtain x where "x\<in>s" and x:"(z ---> x) sequentially"
using s(1)[unfolded compact_def complete_def, THEN spec[where x=z]] and z_in_s by auto
@@ -6634,13 +7578,14 @@
have "e = 0"
proof (rule ccontr)
assume "e \<noteq> 0"
- hence "e>0" unfolding e_def using zero_le_dist[of "f x" x]
+ then have "e > 0"
+ unfolding e_def using zero_le_dist[of "f x" x]
by (metis dist_eq_0_iff dist_nz e_def)
then obtain N where N:"\<forall>n\<ge>N. dist (z n) x < e / 2"
using x[unfolded LIMSEQ_def, THEN spec[where x="e/2"]] by auto
- hence N':"dist (z N) x < e / 2" by auto
-
- have *:"c * dist (z N) x \<le> dist (z N) x"
+ then have N':"dist (z N) x < e / 2" by auto
+
+ have *: "c * dist (z N) x \<le> dist (z N) x"
unfolding mult_le_cancel_right2
using zero_le_dist[of "z N" x] and c
by (metis dist_eq_0_iff dist_nz order_less_asym less_le)
@@ -6657,24 +7602,25 @@
unfolding e_def
by auto
qed
- hence "f x = x" unfolding e_def by auto
+ then have "f x = x" unfolding e_def by auto
moreover
{
fix y
assume "f y = y" "y\<in>s"
- hence "dist x y \<le> c * dist x y"
+ then have "dist x y \<le> c * dist x y"
using lipschitz[THEN bspec[where x=x], THEN bspec[where x=y]]
using `x\<in>s` and `f x = x`
by auto
- hence "dist x y = 0"
+ then have "dist x y = 0"
unfolding mult_le_cancel_right1
using c and zero_le_dist[of x y]
by auto
- hence "y = x" by auto
+ then have "y = x" by auto
}
ultimately show ?thesis using `x\<in>s` by blast+
qed
+
subsection {* Edelstein fixed point theorem *}
lemma edelstein_fix: