--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Sat Sep 14 22:50:15 2013 +0200
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Sat Sep 14 23:52:36 2013 +0200
@@ -25,7 +25,7 @@
using dist_triangle[of y z x] by (simp add: dist_commute)
(* LEGACY *)
-lemma lim_subseq: "subseq r \<Longrightarrow> s ----> l \<Longrightarrow> (s o r) ----> l"
+lemma lim_subseq: "subseq r \<Longrightarrow> s ----> l \<Longrightarrow> (s \<circ> r) ----> l"
by (rule LIMSEQ_subseq_LIMSEQ)
lemmas real_isGlb_unique = isGlb_unique[where 'a=real]
@@ -85,7 +85,7 @@
show "topological_basis B"
using assms unfolding topological_basis_def
proof safe
- fix O'::"'a set"
+ 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)
@@ -138,14 +138,14 @@
qed
lemma basis_dense:
- fixes B::"'a set set"
- and f::"'a set \<Rightarrow> 'a"
+ fixes B :: "'a set set"
+ and f :: "'a set \<Rightarrow> 'a"
assumes "topological_basis B"
and choosefrom_basis: "\<And>B'. B' \<noteq> {} \<Longrightarrow> f B' \<in> B'"
shows "(\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>B' \<in> B. f B' \<in> X))"
proof (intro allI impI)
- fix X::"'a set"
- assume "open X" "X \<noteq> {}"
+ fix X :: "'a set"
+ assume "open X" and "X \<noteq> {}"
from topological_basisE[OF `topological_basis B` `open X` choosefrom_basis[OF `X \<noteq> {}`]]
guess B' . note B' = this
then show "\<exists>B'\<in>B. f B' \<in> X"
@@ -180,7 +180,7 @@
subsection {* Countable Basis *}
locale countable_basis =
- fixes B::"'a::topological_space set set"
+ fixes B :: "'a::topological_space set set"
assumes is_basis: "topological_basis B"
and countable_basis: "countable B"
begin
@@ -283,8 +283,9 @@
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"
- with A(2, 3)[of a] B(2, 3)[of b] show "x \<in> a \<times> b" "open (a \<times> b)"
- unfolding mem_Times_iff by (auto intro: open_Times)
+ with A(2, 3)[of a] B(2, 3)[of b] show "x \<in> a \<times> b" and "open (a \<times> b)"
+ unfolding mem_Times_iff
+ by (auto intro: open_Times)
next
fix S
assume "open S" "x \<in> S"
@@ -418,7 +419,7 @@
text{* Infer the "universe" from union of all sets in the topology. *}
-definition "topspace T = \<Union>{S. openin T S}"
+definition "topspace T = \<Union>{S. openin T S}"
subsubsection {* Main properties of open sets *}
@@ -1007,7 +1008,7 @@
lemma islimpt_approachable_le:
fixes x :: "'a::metric_space"
- shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"
+ shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x \<le> e)"
unfolding islimpt_approachable
using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",
THEN arg_cong [where f=Not]]
@@ -1043,7 +1044,7 @@
lemma finite_set_avoid:
fixes a :: "'a::metric_space"
assumes fS: "finite S"
- shows "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"
+ shows "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x"
proof (induct rule: finite_induct[OF fS])
case 1
then show ?case by (auto intro: zero_less_one)
@@ -1423,8 +1424,9 @@
apply (drule_tac x=UNIV in spec, simp)
done
-lemma not_trivial_limit_within: "~trivial_limit (at x within S) = (x:closure(S-{x}))"
- using islimpt_in_closure by (metis trivial_limit_within)
+lemma not_trivial_limit_within: "\<not> trivial_limit (at x within S) = (x \<in> closure (S - {x}))"
+ using islimpt_in_closure
+ by (metis trivial_limit_within)
text {* Some property holds "sufficiently close" to the limit point. *}
@@ -1463,19 +1465,19 @@
text{* Show that they yield usual definitions in the various cases. *}
lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
- (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> dist (f x) l < e)"
+ (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a \<le> d \<longrightarrow> dist (f x) l < e)"
by (auto simp add: tendsto_iff eventually_at_le dist_nz)
lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
- (\<forall>e >0. \<exists>d>0. \<forall>x \<in> S. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"
+ (\<forall>e >0. \<exists>d>0. \<forall>x \<in> S. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"
by (auto simp add: tendsto_iff eventually_at dist_nz)
lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
- (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"
+ (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"
by (auto simp add: tendsto_iff eventually_at2)
lemma Lim_at_infinity:
- "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"
+ "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x \<ge> b \<longrightarrow> dist (f x) l < e)"
by (auto simp add: tendsto_iff eventually_at_infinity)
lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
@@ -1489,7 +1491,8 @@
lemma Lim_Un:
assumes "(f ---> l) (at x within S)" "(f ---> l) (at x within T)"
shows "(f ---> l) (at x within (S \<union> T))"
- using assms unfolding tendsto_def eventually_at_filter
+ using assms
+ unfolding tendsto_def eventually_at_filter
apply clarify
apply (drule spec, drule (1) mp, drule (1) mp)
apply (drule spec, drule (1) mp, drule (1) mp)
@@ -1515,10 +1518,10 @@
from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..
{
assume "?lhs"
- then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
+ then obtain A where "open A" and "x \<in> A" and "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
unfolding eventually_at_topological
by auto
- with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"
+ with T have "open (A \<inter> T)" and "x \<in> A \<inter> T" and "\<forall>y \<in> A \<inter> T. y \<noteq> x \<longrightarrow> P y"
by auto
then show "?rhs"
unfolding eventually_at_topological by auto
@@ -1546,11 +1549,11 @@
assumes mono: "\<And>a b. a \<in> I \<Longrightarrow> b \<in> I \<Longrightarrow> x < a \<Longrightarrow> a \<le> b \<Longrightarrow> f a \<le> f b"
and bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"
shows "(f ---> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
-proof cases
- assume "{x<..} \<inter> I = {}"
+proof (cases "{x<..} \<inter> I = {}")
+ case True
then show ?thesis by (simp add: Lim_within_empty)
next
- assume e: "{x<..} \<inter> I \<noteq> {}"
+ case False
show ?thesis
proof (rule order_tendstoI)
fix a
@@ -1558,7 +1561,7 @@
{
fix y
assume "y \<in> {x<..} \<inter> I"
- with e bnd have "Inf (f ` ({x<..} \<inter> I)) \<le> f y"
+ with False bnd have "Inf (f ` ({x<..} \<inter> I)) \<le> f y"
by (auto intro: cInf_lower)
with a have "a < f y"
by (blast intro: less_le_trans)
@@ -1568,7 +1571,7 @@
next
fix a
assume "Inf (f ` ({x<..} \<inter> I)) < a"
- from cInf_lessD[OF _ this] e obtain y where y: "x < y" "y \<in> I" "f y < a"
+ from cInf_lessD[OF _ this] False obtain y where y: "x < y" "y \<in> I" "f y < a"
by auto
then have "eventually (\<lambda>x. x \<in> I \<longrightarrow> f x < a) (at_right x)"
unfolding eventually_at_right by (metis less_imp_le le_less_trans mono)
@@ -1625,7 +1628,7 @@
and A :: "'a filter"
assumes "(f ---> l) A"
and "l \<noteq> 0"
- shows "((inverse o f) ---> inverse l) A"
+ shows "((inverse \<circ> f) ---> inverse l) A"
unfolding o_def using assms by (rule tendsto_inverse)
lemma Lim_null:
@@ -1646,7 +1649,7 @@
lemma Lim_transform_bound:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
and g :: "'a \<Rightarrow> 'c::real_normed_vector"
- assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net"
+ assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) net"
and "(g ---> 0) net"
shows "(f ---> 0) net"
using assms(1) tendsto_norm_zero [OF assms(2)]
@@ -1657,7 +1660,7 @@
lemma Lim_in_closed_set:
assumes "closed S"
and "eventually (\<lambda>x. f(x) \<in> S) net"
- and "\<not>(trivial_limit net)" "(f ---> l) net"
+ and "\<not> trivial_limit net" "(f ---> l) net"
shows "l \<in> S"
proof (rule ccontr)
assume "l \<notin> S"
@@ -1676,8 +1679,8 @@
lemma Lim_dist_ubound:
assumes "\<not>(trivial_limit net)"
and "(f ---> l) net"
- and "eventually (\<lambda>x. dist a (f x) <= e) net"
- shows "dist a l <= e"
+ and "eventually (\<lambda>x. dist a (f x) \<le> e) net"
+ shows "dist a l \<le> e"
proof -
have "dist a l \<in> {..e}"
proof (rule Lim_in_closed_set)
@@ -1714,7 +1717,9 @@
lemma Lim_norm_lbound:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- assumes "\<not> (trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. e <= norm(f x)) net"
+ assumes "\<not> trivial_limit net"
+ and "(f ---> l) net"
+ and "eventually (\<lambda>x. e \<le> norm (f x)) net"
shows "e \<le> norm l"
proof -
have "norm l \<in> {e..}"
@@ -1946,7 +1951,7 @@
lemma not_trivial_limit_within_ball:
- "(\<not> trivial_limit (at x within S)) = (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"
+ "\<not> trivial_limit (at x within S) \<longleftrightarrow> (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"
(is "?lhs = ?rhs")
proof -
{
@@ -1954,12 +1959,12 @@
{
fix e :: real
assume "e > 0"
- then obtain y where "y:(S-{x}) & dist y x < e"
+ then obtain y where "y \<in> S - {x}" and "dist y x < e"
using `?lhs` not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]
by auto
- then have "y : (S Int ball x e - {x})"
+ then have "y \<in> S \<inter> ball x e - {x}"
unfolding ball_def by (simp add: dist_commute)
- then have "S Int ball x e - {x} ~= {}" by blast
+ then have "S \<inter> ball x e - {x} \<noteq> {}" by blast
}
then have "?rhs" by auto
}
@@ -1969,11 +1974,11 @@
{
fix e :: real
assume "e > 0"
- then obtain y where "y : (S Int ball x e - {x})"
+ then obtain y where "y \<in> S \<inter> 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"
+ then have "y \<in> S - {x}" and "dist y x < e"
+ unfolding ball_def by (simp_all add: dist_commute)
+ then have "\<exists>y \<in> S - {x}. dist y x < e"
by auto
}
then have "?lhs"
@@ -2004,16 +2009,18 @@
assumes "a \<in> A"
shows "infdist a A = 0"
proof -
- from infdist_le[OF assms, of "dist a a"] have "infdist a A \<le> 0" by auto
- with infdist_nonneg[of a A] assms show "infdist a A = 0" by auto
+ from infdist_le[OF assms, of "dist a a"] have "infdist a A \<le> 0"
+ by auto
+ with infdist_nonneg[of a A] assms show "infdist a A = 0"
+ by auto
qed
lemma infdist_triangle: "infdist x A \<le> infdist y A + dist x y"
-proof cases
- assume "A = {}"
+proof (cases "A = {}")
+ case True
then show ?thesis by (simp add: infdist_def)
next
- assume "A \<noteq> {}"
+ case False
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)
@@ -2202,7 +2209,7 @@
ultimately show "?rhs" by auto
next
assume "?rhs"
- then have "e>0" by auto
+ then have "e > 0" by auto
{
fix d :: real
assume "d > 0"
@@ -2340,7 +2347,7 @@
lemma interior_cball:
fixes x :: "'a::{real_normed_vector, perfect_space}"
shows "interior (cball x e) = ball x e"
-proof (cases "e\<ge>0")
+proof (cases "e \<ge> 0")
case False note cs = this
from cs have "ball x e = {}"
using ball_empty[of e x] by auto
@@ -2409,7 +2416,9 @@
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
+ using interior_unique[of "ball x e" "cball x e"]
+ using open_ball[of x e]
+ by auto
qed
lemma frontier_ball:
@@ -2422,13 +2431,13 @@
lemma frontier_cball:
fixes a :: "'a::{real_normed_vector, perfect_space}"
- shows "frontier(cball a e) = {x. dist a x = e}"
+ shows "frontier (cball a e) = {x. dist a x = e}"
apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)
apply (simp add: set_eq_iff)
apply arith
done
-lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"
+lemma cball_eq_empty: "cball x e = {} \<longleftrightarrow> e < 0"
apply (simp add: set_eq_iff not_le)
apply (metis zero_le_dist dist_self order_less_le_trans)
done
@@ -2438,7 +2447,7 @@
lemma cball_eq_sing:
fixes x :: "'a::{metric_space,perfect_space}"
- shows "(cball x e = {x}) \<longleftrightarrow> e = 0"
+ shows "cball x e = {x} \<longleftrightarrow> e = 0"
proof (rule linorder_cases)
assume e: "0 < e"
obtain a where "a \<noteq> x" "dist a x < e"
@@ -2466,7 +2475,8 @@
lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
unfolding bounded_def
apply safe
- apply (rule_tac x="dist a x + e" in exI, clarify)
+ apply (rule_tac x="dist a x + e" in exI)
+ apply clarify
apply (drule (1) bspec)
apply (erule order_trans [OF dist_triangle add_left_mono])
apply auto
@@ -2526,7 +2536,7 @@
apply auto
done
-lemma bounded_ball[simp,intro]: "bounded(ball x e)"
+lemma bounded_ball[simp,intro]: "bounded (ball x e)"
by (metis ball_subset_cball bounded_cball bounded_subset)
lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
@@ -2534,14 +2544,16 @@
apply (rename_tac x y r s)
apply (rule_tac x=x in exI)
apply (rule_tac x="max r (dist x y + s)" in exI)
- apply (rule ballI, rename_tac z, safe)
- apply (drule (1) bspec, simp)
+ apply (rule ballI)
+ apply safe
+ apply (drule (1) bspec)
+ apply simp
apply (drule (1) bspec)
apply (rule min_max.le_supI2)
apply (erule order_trans [OF dist_triangle add_left_mono])
done
-lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"
+lemma bounded_Union[intro]: "finite F \<Longrightarrow> \<forall>S\<in>F. bounded S \<Longrightarrow> bounded (\<Union>F)"
by (induct rule: finite_induct[of F]) auto
lemma bounded_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. bounded (B x) \<Longrightarrow> bounded (\<Union>x\<in>A. B x)"
@@ -2549,22 +2561,27 @@
lemma bounded_insert [simp]: "bounded (insert x S) \<longleftrightarrow> bounded S"
proof -
- have "\<forall>y\<in>{x}. dist x y \<le> 0" by simp
- then have "bounded {x}" unfolding bounded_def by fast
- then show ?thesis by (metis insert_is_Un bounded_Un)
+ have "\<forall>y\<in>{x}. dist x y \<le> 0"
+ by simp
+ 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"
by (induct set: finite) simp_all
-lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"
+lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x \<le> b)"
apply (simp add: bounded_iff)
- apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")
+ apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x \<le> y \<longrightarrow> x \<le> 1 + abs y)")
apply metis
apply arith
done
-lemma Bseq_eq_bounded: "Bseq f \<longleftrightarrow> bounded (range f::_::real_normed_vector set)"
+lemma Bseq_eq_bounded:
+ fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
+ shows "Bseq f \<longleftrightarrow> bounded (range f)"
unfolding Bseq_def bounded_pos by auto
lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
@@ -2575,11 +2592,13 @@
lemma not_bounded_UNIV[simp, intro]:
"\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
-proof(auto simp add: bounded_pos not_le)
+proof (auto simp add: bounded_pos not_le)
obtain x :: 'a where "x \<noteq> 0"
using perfect_choose_dist [OF zero_less_one] by fast
- fix b::real assume b: "b >0"
- have b1: "b +1 \<ge> 0" using b by simp
+ fix b :: real
+ assume b: "b >0"
+ have b1: "b +1 \<ge> 0"
+ using b by simp
with `x \<noteq> 0` have "b < norm (scaleR (b + 1) (sgn x))"
by (simp add: norm_sgn)
then show "\<exists>x::'a. b < norm x" ..
@@ -2590,15 +2609,17 @@
and "bounded_linear f"
shows "bounded (f ` S)"
proof -
- from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b"
+ from assms(1) obtain b where b: "b > 0" "\<forall>x\<in>S. norm x \<le> b"
unfolding bounded_pos by auto
- from assms(2) obtain B where B:"B>0" "\<forall>x. norm (f x) \<le> B * norm x"
+ 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"
- then have "norm x \<le> b" using b by auto
- then have "norm (f x) \<le> B * b" using B(2)
+ 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
@@ -2624,11 +2645,11 @@
assumes "bounded S"
shows "bounded ((\<lambda>x. a + x) ` S)"
proof -
- from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b"
+ 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"
+ assume "x \<in> S"
then have "norm (a + x) \<le> b + norm a"
using norm_triangle_ineq[of a x] b by auto
}
@@ -2648,7 +2669,8 @@
lemma bounded_has_Sup:
fixes S :: "real set"
- assumes "bounded S" "S \<noteq> {}"
+ assumes "bounded S"
+ and "S \<noteq> {}"
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
@@ -2679,18 +2701,19 @@
lemma bounded_has_Inf:
fixes S :: "real set"
- assumes "bounded S" "S \<noteq> {}"
+ assumes "bounded S"
+ and "S \<noteq> {}"
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"
+ assume "x \<in> S"
from assms(1) obtain a where a: "\<forall>x\<in>S. \<bar>x\<bar> \<le> a"
unfolding bounded_real by auto
- then show "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"
+ show "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"
using assms by (metis cInf_greatest)
qed
@@ -2715,25 +2738,29 @@
subsubsection {* Bolzano-Weierstrass property *}
lemma heine_borel_imp_bolzano_weierstrass:
- assumes "compact s" and "infinite t" and "t \<subseteq> s"
+ assumes "compact s"
+ and "infinite t"
+ and "t \<subseteq> s"
shows "\<exists>x \<in> s. x islimpt t"
proof (rule ccontr)
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)"
+ 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"
+ 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
+ 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"
+ assume "x \<in> t" "y \<in> t" "f x = f y"
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
+ using f[THEN bspec[where x=x]] and `t \<subseteq> s` by auto
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
+ using `f x = f y` and f[THEN bspec[where x=y]] and `y \<in> t` and `t \<subseteq> s`
+ by auto
}
then have "inj_on f t"
unfolding inj_on_def by simp
@@ -2742,14 +2769,17 @@
moreover
{
fix x
- assume "x\<in>t" "f x \<notin> g"
- 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"
+ assume "x \<in> t" "f x \<notin> g"
+ 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
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
+ using f[THEN bspec[where x=y]] and `x\<in>t` and `x\<in>h`[unfolded `h = f y`]
+ by auto
then have False
- using `f x \<notin> g` `h\<in>g` unfolding `h = f y` by auto
+ using `f x \<notin> g` `h \<in> g` unfolding `h = f y`
+ by auto
}
then have "f ` t \<subseteq> g" by auto
ultimately show False
@@ -2786,7 +2816,8 @@
proof (rule topological_tendstoI)
fix S
assume "open S" "l \<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
@@ -2810,12 +2841,18 @@
shows "infinite (range f)"
proof
assume "finite (range f)"
- 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
+ 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)
- then show False unfolding eventually_sequentially by auto
+ using `open (- range f)` `l \<in> - range f`
+ by (rule topological_tendstoD)
+ then show False
+ unfolding eventually_sequentially
+ by auto
qed
lemma closure_insert:
@@ -2928,7 +2965,7 @@
qed
lemma bolzano_weierstrass_imp_closed:
- fixes s :: "'a::{first_countable_topology, t2_space} set"
+ fixes s :: "'a::{first_countable_topology,t2_space} set"
assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
shows "closed s"
proof -
@@ -3276,7 +3313,7 @@
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))"
+ (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially))"
lemma seq_compact_imp_countably_compact:
fixes U :: "'a :: first_countable_topology set"
@@ -3391,7 +3428,7 @@
qed
lemma seq_compactI:
- assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"
+ assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
shows "seq_compact S"
unfolding seq_compact_def using assms by fast
@@ -3611,7 +3648,7 @@
lemma compact_def:
"compact (S :: 'a::metric_space set) \<longleftrightarrow>
- (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f o r) ----> l))"
+ (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f \<circ> r) ----> l))"
unfolding compact_eq_seq_compact_metric seq_compact_def by auto
subsubsection {* Complete the chain of compactness variants *}
@@ -4514,7 +4551,7 @@
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
- \<longrightarrow> ((f o x) ---> f a) sequentially)"
+ \<longrightarrow> ((f \<circ> x) ---> f a) sequentially)"
(is "?lhs = ?rhs")
proof
assume ?lhs
@@ -4546,14 +4583,14 @@
lemma continuous_at_sequentially:
fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
shows "continuous (at a) f \<longleftrightarrow>
- (\<forall>x. (x ---> a) sequentially --> ((f o x) ---> f a) sequentially)"
+ (\<forall>x. (x ---> a) sequentially --> ((f \<circ> x) ---> f a) sequentially)"
using continuous_within_sequentially[of a UNIV f] by simp
lemma continuous_on_sequentially:
fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
shows "continuous_on s f \<longleftrightarrow>
(\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially
- --> ((f o x) ---> f(a)) sequentially)"
+ --> ((f \<circ> x) ---> f a) sequentially)"
(is "?lhs = ?rhs")
proof
assume ?rhs
@@ -4804,8 +4841,8 @@
lemma uniformly_continuous_on_compose[continuous_on_intros]:
assumes "uniformly_continuous_on s f" "uniformly_continuous_on (f ` s) g"
- shows "uniformly_continuous_on s (g o f)"
-proof-
+ shows "uniformly_continuous_on s (g \<circ> f)"
+proof -
{
fix e :: real
assume "e > 0"
@@ -6818,7 +6855,7 @@
lemma Lim_component_eq:
fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
- assumes net: "(f ---> l) net" "~(trivial_limit net)"
+ assumes net: "(f ---> l) net" "\<not> 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]
@@ -6887,8 +6924,7 @@
(\<forall>x\<in>s. (g(f x) = x)) \<and> (f ` s = t) \<and> continuous_on s f \<and>
(\<forall>y\<in>t. (f(g y) = y)) \<and> (g ` t = s) \<and> continuous_on t g"
-definition
- homeomorphic :: "'a::topological_space set \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"
+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)"
@@ -7099,12 +7135,12 @@
lemma cauchy_isometric:
fixes x :: "nat \<Rightarrow> 'a::euclidean_space"
- assumes e: "0 < e"
+ assumes e: "e > 0"
and s: "subspace s"
and f: "bounded_linear f"
- and normf: "\<forall>x\<in>s. norm(f x) \<ge> e * norm(x)"
- and xs: "\<forall>n::nat. x n \<in> s"
- and cf: "Cauchy(f o x)"
+ and normf: "\<forall>x\<in>s. norm (f x) \<ge> e * norm x"
+ and xs: "\<forall>n. x n \<in> s"
+ and cf: "Cauchy (f \<circ> x)"
shows "Cauchy x"
proof -
interpret f: bounded_linear f by fact
@@ -7145,24 +7181,31 @@
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)"
- using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"] by auto
- then have x:"\<forall>n. x n \<in> s" "\<forall>n. g n = f (x n)" by auto
- then have "f \<circ> x = g" unfolding fun_eq_iff by auto
+ using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"]
+ by auto
+ then have x:"\<forall>n. x n \<in> s" "\<forall>n. g n = f (x n)"
+ by auto
+ then have "f \<circ> x = g"
+ 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
+ using cauchy_isometric[OF `0<e` s f normf] and cfg and x(1)
+ by auto
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
+ unfolding `f \<circ> x = g`
+ by auto
}
- then show ?thesis unfolding complete_def by auto
+ then show ?thesis
+ unfolding complete_def by auto
qed
lemma injective_imp_isometric:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes s: "closed s" "subspace s"
- and f: "bounded_linear f" "\<forall>x\<in>s. (f x = 0) \<longrightarrow> (x = 0)"
- shows "\<exists>e>0. \<forall>x\<in>s. norm (f x) \<ge> e * norm(x)"
+ and f: "bounded_linear f" "\<forall>x\<in>s. f x = 0 \<longrightarrow> x = 0"
+ shows "\<exists>e>0. \<forall>x\<in>s. norm (f x) \<ge> e * norm x"
proof (cases "s \<subseteq> {0::'a}")
case True
{
@@ -7175,8 +7218,10 @@
next
interpret f: bounded_linear f by fact
case False
- then obtain a where a:"a\<noteq>0" "a\<in>s" by auto
- from False have "s \<noteq> {}" by auto
+ then obtain a where a: "a \<noteq> 0" "a \<in> s"
+ by auto
+ from False have "s \<noteq> {}"
+ by auto
let ?S = "{f x| x. (x \<in> s \<and> norm x = norm a)}"
let ?S' = "{x::'a. x\<in>s \<and> norm x = norm a}"
let ?S'' = "{x::'a. norm x = norm a}"