--- a/src/HOL/Analysis/Topology_Euclidean_Space.thy Sat Dec 31 11:09:19 2022 +0000
+++ b/src/HOL/Analysis/Topology_Euclidean_Space.thy Sun Jan 01 00:45:55 2023 +0000
@@ -1307,18 +1307,11 @@
assume i: "i \<in> Basis"
have "dist (x - (e / 2) *\<^sub>R i) x < e"
and "dist (x + (e / 2) *\<^sub>R i) x < e"
- unfolding dist_norm
- apply auto
- unfolding norm_minus_cancel
- using norm_Basis[OF i] \<open>e>0\<close>
- apply auto
- done
+ using norm_Basis[OF i] \<open>e>0\<close> by (auto simp: dist_norm)
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_box
- using i
- by blast+
+ unfolding mem_box using i by blast+
then have "a \<bullet> i < x \<bullet> i" and "x \<bullet> i < b \<bullet> i"
using \<open>e>0\<close> i
by (auto simp: inner_diff_left inner_Basis inner_add_left)
@@ -1349,8 +1342,7 @@
lemma bounded_box [simp]:
fixes a :: "'a::euclidean_space"
shows "bounded (box a b)"
- using bounded_cbox[of a b] box_subset_cbox[of a b] bounded_subset[of "cbox a b" "box a b"]
- by simp
+ by (metis bounded_cbox bounded_interior interior_cbox)
lemma not_interval_UNIV [simp]:
fixes a :: "'a::euclidean_space"
@@ -1385,12 +1377,7 @@
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)"
- proof (rule add_less_le_mono)
- show "e * (a \<bullet> i) < e * (x \<bullet> i)"
- using \<open>0 < e\<close> i mem_box(1) x by auto
- show "(1 - e) * (a \<bullet> i) \<le> (1 - e) * (y \<bullet> i)"
- by (meson diff_ge_0_iff_ge \<open>e \<le> 1\<close> i mem_box(2) mult_left_mono y)
- qed
+ by (smt (verit, best) e i mem_box mult_le_cancel_left_pos mult_left_mono x y)
finally have "a \<bullet> i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i"
unfolding inner_simps by auto
moreover
@@ -1398,12 +1385,7 @@
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)"
- proof (rule add_less_le_mono)
- show "e * (x \<bullet> i) < e * (b \<bullet> i)"
- using \<open>0 < e\<close> i mem_box(1) x by auto
- show "(1 - e) * (y \<bullet> i) \<le> (1 - e) * (b \<bullet> i)"
- by (meson diff_ge_0_iff_ge \<open>e \<le> 1\<close> i mem_box(2) mult_left_mono y)
- qed
+ by (smt (verit, best) e i mem_box mult_le_cancel_left_pos mult_left_mono x y)
finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i < b \<bullet> i"
unfolding inner_simps by auto
}
@@ -1427,7 +1409,7 @@
let ?c = "(1 / 2) *\<^sub>R (a + b)"
{
fix x
- assume as:"x \<in> cbox a b"
+ assume as: "x \<in> cbox a b"
define f where [abs_def]: "f n = x + (inverse (real n + 1)) *\<^sub>R (?c - x)" for n
{
fix n
@@ -1445,22 +1427,16 @@
}
moreover
{
- assume "\<not> (f \<longlongrightarrow> x) sequentially"
- {
- fix e :: real
- assume "e > 0"
- then obtain N :: nat where N: "inverse (real (N + 1)) < e"
- using reals_Archimedean by auto
- have "inverse (real n + 1) < e" if "N \<le> n" for n
- by (auto intro!: that le_less_trans [OF _ N])
- then have "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto
- }
- then have "((\<lambda>n. inverse (real n + 1)) \<longlongrightarrow> 0) sequentially"
+ have "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < \<epsilon>" if "\<epsilon> > 0" for \<epsilon>
+ using reals_Archimedean [of \<epsilon>] that
+ by (metis inverse_inverse_eq inverse_less_imp_less nat_le_real_less order_less_trans
+ reals_Archimedean2)
+ then have "(\<lambda>n. inverse (real n + 1)) \<longlonglongrightarrow> 0"
unfolding lim_sequentially by(auto simp: dist_norm)
- then have "(f \<longlongrightarrow> x) sequentially"
+ then have "f \<longlonglongrightarrow> x"
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)"]
+ using tendsto_add[OF tendsto_const, of "\<lambda>n. (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. inverse (real n + 1)" 0 sequentially "((1 / 2) *\<^sub>R (a + b) - x)"]
by auto
}
ultimately have "x \<in> closure (box a b)"
@@ -1491,12 +1467,7 @@
fixes S :: "('a::euclidean_space) set"
assumes "bounded S"
obtains a where "S \<subseteq> cbox (-a) a"
-proof -
- obtain a where "S \<subseteq> box (-a) a"
- using bounded_subset_box_symmetric[OF assms] by auto
- then show ?thesis
- by (meson box_subset_cbox dual_order.trans that)
-qed
+ by (meson assms bounded_subset_box_symmetric box_subset_cbox order.trans)
lemma frontier_cbox:
fixes a b :: "'a::euclidean_space"
@@ -1506,16 +1477,7 @@
lemma frontier_box:
fixes a b :: "'a::euclidean_space"
shows "frontier (box a b) = (if box a b = {} then {} else cbox a b - box a b)"
-proof (cases "box a b = {}")
- case True
- then show ?thesis
- using frontier_empty by auto
-next
- case False
- then show ?thesis
- unfolding frontier_def and closure_box[OF False] and interior_open[OF open_box]
- by auto
-qed
+ by (simp add: frontier_def interior_open open_box)
lemma Int_interval_mixed_eq_empty:
fixes a :: "'a::euclidean_space"
@@ -1549,28 +1511,21 @@
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
moreover
- {
- fix n
+ { fix n
assume n: "\<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e / (real_of_nat DIM('a))"
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
- apply (rule L2_set_le_sum)
- done
+ using L2_set_le_sum [OF zero_le_dist] by (subst euclidean_dist_l2)
also have "\<dots> < (\<Sum>i\<in>(Basis::'a set). e / (real_of_nat DIM('a)))"
- apply (rule sum_strict_mono)
- using n
- apply auto
- done
+ by (meson eucl.finite_Basis n nonempty_Basis sum_strict_mono)
finally have "dist (f (r n)) l < e"
by auto
}
- ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
+ ultimately have "\<forall>\<^sub>F n in sequentially. dist (f (r n)) l < e"
by (rule eventually_mono)
}
- then have *: "((f \<circ> r) \<longlongrightarrow> l) sequentially"
+ then have *: "(f \<circ> r) \<longlonglongrightarrow> l"
unfolding o_def tendsto_iff by simp
- with r show "\<exists>l r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
+ with r show "\<exists>l r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
by auto
qed
@@ -1592,10 +1547,8 @@
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>open A\<close>])
- apply (auto simp: a b B_def)
- done
+ using open_UNION_box[OF \<open>open A\<close>]
+ by (smt (verit, ccfv_threshold) B_def a b image_iff mem_Collect_eq subsetI)
qed
ultimately
have "topological_basis B"
@@ -1658,10 +1611,11 @@
by (simp add: False b cSUP_least)
finally have bi: "x \<bullet> i \<le> b \<bullet> i" .
show "x \<bullet> i \<in> (\<lambda>x. x \<bullet> i) ` S"
- apply (rule_tac x="\<Sum>j\<in>Basis. (if j = i then x \<bullet> i else a \<bullet> j) *\<^sub>R j" in image_eqI)
+ apply (rule_tac x="\<Sum>j\<in>Basis. (((\<bullet>)a)(i := x \<bullet> j))j *\<^sub>R j" in image_eqI)
apply (simp add: i)
apply (rule mem_is_intervalI [OF \<open>is_interval S\<close> \<open>a \<in> S\<close> \<open>b \<in> S\<close>])
- using i ai bi apply force
+ using i ai bi
+ apply force
done
qed
have "S = cbox a b"
@@ -1714,10 +1668,7 @@
assume ?lhs
then show ?rhs
unfolding tendsto_def
- apply clarify
- apply (drule_tac x="{s. s \<bullet> i \<in> S}" in spec)
- apply (auto simp: open_preimage_inner)
- done
+ by (smt (verit) eventually_elim2 mem_Collect_eq open_preimage_inner)
next
assume R: ?rhs
then have "\<And>e. e > 0 \<Longrightarrow> \<forall>i\<in>Basis. \<forall>\<^sub>F x in F. dist (f x \<bullet> i) (l \<bullet> i) < e"
@@ -1735,19 +1686,15 @@
have "L2_set (\<lambda>i. dist (f x \<bullet> i) (l \<bullet> i)) Basis \<le> sum (\<lambda>i. dist (f x \<bullet> i) (l \<bullet> i)) Basis"
by (simp add: L2_set_le_sum)
also have "... < DIM('b) * (e / real DIM('b))"
- apply (rule sum_bounded_above_strict)
- using that by auto
+ by (meson DIM_positive sum_bounded_above_strict that)
also have "... = e"
by (simp add: field_simps)
finally show "L2_set (\<lambda>i. dist (f x \<bullet> i) (l \<bullet> i)) Basis < e" .
qed
have "\<forall>\<^sub>F x in F. \<forall>i\<in>Basis. dist (f x \<bullet> i) (l \<bullet> i) < e / DIM('b)"
- apply (rule R')
- using \<open>0 < e\<close> by simp
+ by (simp add: R' \<open>0 < e\<close>)
then show "\<forall>\<^sub>F x in F. dist (f x) l < e"
- apply (rule eventually_mono)
- apply (subst euclidean_dist_l2)
- using * by blast
+ by eventually_elim (metis (full_types) "*" euclidean_dist_l2)
qed
qed
@@ -1759,22 +1706,21 @@
corollary continuous_on_componentwise:
fixes S :: "'a :: t2_space set"
shows "continuous_on S f \<longleftrightarrow> (\<forall>i \<in> Basis. continuous_on S (\<lambda>x. (f x \<bullet> i)))"
- apply (simp add: continuous_on_eq_continuous_within)
- using continuous_componentwise by blast
+ by (metis continuous_componentwise continuous_on_eq_continuous_within)
lemma linear_componentwise_iff:
- "(linear f') \<longleftrightarrow> (\<forall>i\<in>Basis. linear (\<lambda>x. f' x \<bullet> i))"
- apply (auto simp: linear_iff inner_left_distrib)
- apply (metis inner_left_distrib euclidean_eq_iff)
- by (metis euclidean_eqI inner_scaleR_left)
+ "linear f' \<longleftrightarrow> (\<forall>i\<in>Basis. linear (\<lambda>x. f' x \<bullet> i))" (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+ show "?lhs \<Longrightarrow> ?rhs"
+ by (simp add: Real_Vector_Spaces.linear_iff inner_left_distrib)
+ show "?rhs \<Longrightarrow> ?lhs"
+ by (simp add: linear_iff) (metis euclidean_eqI inner_left_distrib inner_scaleR_left)
+qed
lemma bounded_linear_componentwise_iff:
"(bounded_linear f') \<longleftrightarrow> (\<forall>i\<in>Basis. bounded_linear (\<lambda>x. f' x \<bullet> i))"
(is "?lhs = ?rhs")
proof
- assume ?lhs then show ?rhs
- by (simp add: bounded_linear_inner_left_comp)
-next
assume ?rhs
then have "(\<forall>i\<in>Basis. \<exists>K. \<forall>x. \<bar>f' x \<bullet> i\<bar> \<le> norm x * K)" "linear f'"
by (auto simp: bounded_linear_def bounded_linear_axioms_def linear_componentwise_iff [symmetric] ball_conj_distrib)
@@ -1792,7 +1738,7 @@
qed
then show ?lhs
by (force simp: bounded_linear_def bounded_linear_axioms_def \<open>linear f'\<close>)
-qed
+qed (simp add: bounded_linear_inner_left_comp)
subsection\<^marker>\<open>tag unimportant\<close> \<open>Continuous Extension\<close>
@@ -1853,8 +1799,7 @@
obtain d where d: "0 < d"
"\<And>x'. x' \<in> cbox a b \<Longrightarrow> dist x' (clamp a b x) < d \<Longrightarrow> dist (f x') (f (clamp a b x)) < e"
by force
- show "\<exists>d>0. \<forall>x'. dist x' x < d \<longrightarrow>
- dist (f (clamp a b x')) (f (clamp a b x)) < e"
+ show "\<exists>d>0. \<forall>x'. dist x' x < d \<longrightarrow> dist (f (clamp a b x')) (f (clamp a b x)) < e"
using le
by (auto intro!: d clamp_in_interval dist_clamps_le_dist_args[THEN le_less_trans])
qed
@@ -1876,8 +1821,7 @@
from bounded obtain c where f_bound: "\<forall>x\<in>f ` cbox a b. dist undefined x \<le> c"
by (auto simp: bounded_any_center[where a=undefined])
then show ?thesis
- by (auto intro!: exI[where x=c] clamp_in_interval[OF le[rule_format]]
- simp: bounded_any_center[where a=undefined])
+ by (metis bounded bounded_subset clamp_in_interval image_mono image_subsetI le range_composition)
qed (auto simp: clamp_empty_interval image_def)
@@ -1888,9 +1832,7 @@
fixes x a b :: "'a::euclidean_space"
assumes x: "x \<in> cbox a b"
shows "ext_cont f a b x = f x"
- using assms
- unfolding ext_cont_def
- by (auto simp: clamp_def mem_box intro!: euclidean_eqI[where 'a='a] arg_cong[where f=f])
+ using assms by (simp add: ext_cont_def)
lemma continuous_on_ext_cont[continuous_intros]:
"continuous_on (cbox a b) f \<Longrightarrow> continuous_on S (ext_cont f a b)"
@@ -1909,18 +1851,16 @@
and Un: "\<And>S. open S \<Longrightarrow> \<exists>U. U \<subseteq> \<B> \<and> S = \<Union>U"
using univ_second_countable by blast
have *: "infinite (range (\<lambda>n. ball (0::'a) (inverse(Suc n))))"
- apply (rule Infinite_Set.range_inj_infinite)
- apply (simp add: inj_on_def ball_eq_ball_iff)
- done
+ by (simp add: inj_on_def ball_eq_ball_iff Infinite_Set.range_inj_infinite)
have "infinite \<B>"
proof
assume "finite \<B>"
then have "finite (Union ` (Pow \<B>))"
by simp
- then have "finite (range (\<lambda>n. ball (0::'a) (inverse(Suc n))))"
- apply (rule rev_finite_subset)
+ moreover have "range (\<lambda>n. ball 0 (inverse (real (Suc n)))) \<subseteq> \<Union> ` Pow \<B>"
by (metis (no_types, lifting) PowI image_eqI image_subset_iff Un [OF open_ball])
- with * show False by simp
+ ultimately show False
+ by (metis finite_subset *)
qed
obtain f :: "nat \<Rightarrow> 'a set" where "\<B> = range f" "inj f"
by (blast intro: countable_as_injective_image [OF \<open>countable \<B>\<close> \<open>infinite \<B>\<close>])
@@ -1931,9 +1871,7 @@
using \<open>inj f\<close> \<open>\<B> = range f\<close> apply force
done
show ?thesis
- apply (rule that [OF \<open>inj f\<close> _ *])
- apply (auto simp: \<open>\<B> = range f\<close> opn)
- done
+ using "*" \<open>\<B> = range f\<close> \<open>inj f\<close> opn that by force
qed
proposition separable:
@@ -1969,14 +1907,8 @@
using \<open>0 < e\<close> \<U> \<open>x \<in> S\<close> by auto
next
case False
- then obtain C where "C \<in> \<U>" by blast
- show ?thesis
- proof
- show "dist (f C) x < e"
- by (metis Int_iff Union_iff \<U> \<open>C \<in> \<U>\<close> dist_commute f mem_ball subsetCE)
- show "C \<in> \<B>"
- using \<open>\<U> \<subseteq> \<B>\<close> \<open>C \<in> \<U>\<close> by blast
- qed
+ then show ?thesis
+ by (metis IntI Union_iff \<U> \<open>0 < e\<close> \<open>x \<in> S\<close> dist_commute dist_self f inf_le2 mem_ball subset_eq)
qed
qed
qed
@@ -2001,8 +1933,7 @@
then show "norm (x - y) \<le> 2*r" by simp
qed (simp add: that)
have "2*r = dist (a + r *\<^sub>R (SOME i. i \<in> Basis)) (a - r *\<^sub>R (SOME i. i \<in> Basis))"
- apply (simp add: dist_norm)
- by (metis abs_of_nonneg mult.right_neutral norm_numeral norm_scaleR norm_some_Basis real_norm_def scaleR_2 that)
+ using \<open>0 \<le> r\<close> that by (simp add: dist_norm flip: scaleR_2)
also have "... \<le> diameter (cball a r)"
apply (rule diameter_bounded_bound)
using that by (auto simp: dist_norm)
@@ -2023,8 +1954,8 @@
lemma diameter_closed_interval [simp]: "diameter {a..b} = (if b < a then 0 else b-a)"
proof -
- have "{a .. b} = cball ((a+b)/2) ((b-a)/2)"
- by (auto simp: dist_norm abs_if field_split_simps split: if_split_asm)
+ have "{a..b} = cball ((a+b)/2) ((b-a)/2)"
+ using atLeastAtMost_eq_cball by blast
then show ?thesis
by simp
qed
@@ -2032,7 +1963,7 @@
lemma diameter_open_interval [simp]: "diameter {a<..<b} = (if b < a then 0 else b-a)"
proof -
have "{a <..< b} = ball ((a+b)/2) ((b-a)/2)"
- by (auto simp: dist_norm abs_if field_split_simps split: if_split_asm)
+ using greaterThanLessThan_eq_ball by blast
then show ?thesis
by simp
qed
@@ -2101,10 +2032,8 @@
shows "open(f ` A) \<longleftrightarrow> open A"
proof
assume "open(f ` A)"
- then have "open(f -` (f ` A))"
- using assms by (force simp: linear_continuous_at linear_conv_bounded_linear continuous_open_vimage)
then show "open A"
- by (simp add: assms bij_is_inj inj_vimage_image_eq)
+ by (metis assms bij_is_inj continuous_open_vimage inj_vimage_image_eq linear_continuous_at linear_linear)
next
assume "open A"
then show "open(f ` A)"
@@ -2114,20 +2043,9 @@
corollary interior_bijective_linear_image:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "linear f" "bij f"
- shows "interior (f ` S) = f ` interior S" (is "?lhs = ?rhs")
-proof safe
- fix x
- assume x: "x \<in> ?lhs"
- then obtain T where "open T" and "x \<in> T" and "T \<subseteq> f ` S"
- by (metis interiorE)
- then show "x \<in> ?rhs"
- by (metis (no_types, opaque_lifting) assms subsetD interior_maximal open_bijective_linear_image_eq subset_image_iff)
-next
- fix x
- assume x: "x \<in> interior S"
- then show "f x \<in> interior (f ` S)"
- by (meson assms imageI image_mono interiorI interior_subset open_bijective_linear_image_eq open_interior)
-qed
+ shows "interior (f ` S) = f ` interior S"
+ by (smt (verit) assms bij_is_inj inj_image_subset_iff interior_maximal interior_subset
+ open_bijective_linear_image_eq open_interior subset_antisym subset_imageE)
lemma interior_injective_linear_image:
fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
@@ -2234,10 +2152,9 @@
shows "closed(f ` s)"
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 \<open>e>0\<close> assms(1,2) e] and assms(4)
- unfolding complete_eq_closed[symmetric] by auto
+ using assms injective_imp_isometric by blast
+ with assms show ?thesis
+ by (meson complete_eq_closed complete_isometric_image)
qed
@@ -2274,9 +2191,8 @@
show "closedin (top_of_set (range f)) (f ` S)"
using continuous_closedin_preimage [OF confg cgf] by simp
show "closed (range f)"
- apply (rule closed_injective_image_subspace)
- using f apply (auto simp: linear_linear linear_injective_0)
- done
+ using closed_injective_image_subspace f linear_conv_bounded_linear
+ linear_injective_0 subspace_UNIV by blast
qed
qed
@@ -2289,25 +2205,28 @@
lemma closure_injective_linear_image:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows "\<lbrakk>linear f; inj f\<rbrakk> \<Longrightarrow> f ` (closure S) = closure (f ` S)"
- apply (rule subset_antisym)
- apply (simp add: closure_linear_image_subset)
- by (simp add: closure_minimal closed_injective_linear_image closure_subset image_mono)
+ by (simp add: closed_injective_linear_image closure_linear_image_subset
+ closure_minimal closure_subset image_mono subset_antisym)
lemma closure_bounded_linear_image:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- shows "\<lbrakk>linear f; bounded S\<rbrakk> \<Longrightarrow> f ` (closure S) = closure (f ` S)"
- apply (rule subset_antisym, simp add: closure_linear_image_subset)
- apply (rule closure_minimal, simp add: closure_subset image_mono)
- by (meson bounded_closure closed_closure compact_continuous_image compact_eq_bounded_closed linear_continuous_on linear_conv_bounded_linear)
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes "linear f" "bounded S"
+ shows "f ` (closure S) = closure (f ` S)" (is "?lhs = ?rhs")
+proof
+ show "?lhs \<subseteq> ?rhs"
+ using assms closure_linear_image_subset by blast
+ show "?rhs \<subseteq> ?lhs"
+ using assms by (meson closure_minimal closure_subset compact_closure compact_eq_bounded_closed
+ compact_continuous_image image_mono linear_continuous_on linear_linear)
+qed
lemma closure_scaleR:
fixes S :: "'a::real_normed_vector set"
- shows "((*\<^sub>R) c) ` (closure S) = closure (((*\<^sub>R) c) ` S)"
+ shows "((*\<^sub>R) c) ` (closure S) = closure (((*\<^sub>R) c) ` S)" (is "?lhs = ?rhs")
proof
- show "((*\<^sub>R) c) ` (closure S) \<subseteq> closure (((*\<^sub>R) c) ` S)"
- using bounded_linear_scaleR_right
- by (rule closure_bounded_linear_image_subset)
- show "closure (((*\<^sub>R) c) ` S) \<subseteq> ((*\<^sub>R) c) ` (closure S)"
+ show "?lhs \<subseteq> ?rhs"
+ using bounded_linear_scaleR_right by (rule closure_bounded_linear_image_subset)
+ show "?rhs \<subseteq> ?lhs"
by (intro closure_minimal image_mono closure_subset closed_scaling closed_closure)
qed
@@ -2326,73 +2245,55 @@
qed
lemma closed_subspace:
- fixes s :: "'a::euclidean_space set"
- assumes "subspace s"
- shows "closed s"
+ fixes S :: "'a::euclidean_space set"
+ assumes "subspace S"
+ shows "closed S"
proof -
- have "dim s \<le> card (Basis :: 'a set)"
+ have "dim S \<le> card (Basis :: 'a set)"
using dim_subset_UNIV by auto
- with ex_card[OF this] obtain d :: "'a set" where t: "card d = dim s" and d: "d \<subseteq> Basis"
- by auto
+ with obtain_subset_with_card_n
+ obtain d :: "'a set" where cd: "card d = dim S" and d: "d \<subseteq> Basis"
+ by metis
let ?t = "{x::'a. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
- have "\<exists>f. linear f \<and> f ` {x::'a. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} = s \<and>
+ have "\<exists>f. linear f \<and> f ` {x::'a. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} = S \<and>
inj_on f {x::'a. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"
- using dim_substandard[of d] t d assms
+ using dim_substandard[of d] cd d assms
by (intro subspace_isomorphism[OF subspace_substandard[of "\<lambda>i. i \<notin> d"]]) (auto simp: inner_Basis)
then obtain f where f:
"linear f"
- "f ` {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} = s"
+ "f ` {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} = S"
"inj_on f {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"
by blast
interpret f: bounded_linear f
using f by (simp add: linear_conv_bounded_linear)
have "x \<in> ?t \<Longrightarrow> f x = 0 \<Longrightarrow> x = 0" for x
using f.zero d f(3)[THEN inj_onD, of x 0] by auto
- moreover have "closed ?t" by (rule closed_substandard)
- moreover have "subspace ?t" by (rule subspace_substandard)
- ultimately show ?thesis
- using closed_injective_image_subspace[of ?t f]
- unfolding f(2) using f(1) unfolding linear_conv_bounded_linear by auto
+ then show ?thesis
+ using closed_injective_image_subspace[of ?t f] closed_substandard subspace_substandard
+ using f(2) f.bounded_linear_axioms by force
qed
-lemma complete_subspace: "subspace s \<Longrightarrow> complete s"
- for s :: "'a::euclidean_space set"
+lemma complete_subspace: "subspace S \<Longrightarrow> complete S"
+ for S :: "'a::euclidean_space set"
using complete_eq_closed closed_subspace by auto
-lemma closed_span [iff]: "closed (span s)"
- for s :: "'a::euclidean_space set"
+lemma closed_span [iff]: "closed (span S)"
+ for S :: "'a::euclidean_space set"
by (simp add: closed_subspace)
-lemma dim_closure [simp]: "dim (closure s) = dim s" (is "?dc = ?d")
- for s :: "'a::euclidean_space set"
-proof -
- have "?dc \<le> ?d"
- using closure_minimal[OF span_superset, of s]
- using closed_subspace[OF subspace_span, of s]
- using dim_subset[of "closure s" "span s"]
- by simp
- then show ?thesis
- using dim_subset[OF closure_subset, of s]
- by simp
-qed
+lemma dim_closure [simp]: "dim (closure S) = dim S" (is "?dc = ?d")
+ for S :: "'a::euclidean_space set"
+ by (metis closed_span closure_minimal closure_subset dim_eq_span span_eq_dim span_superset subset_le_dim)
subsection \<open>Set Distance\<close>
lemma setdist_compact_closed:
fixes A :: "'a::heine_borel set"
- assumes A: "compact A" and B: "closed B"
+ assumes "compact A" "closed B"
and "A \<noteq> {}" "B \<noteq> {}"
shows "\<exists>x \<in> A. \<exists>y \<in> B. dist x y = setdist A B"
-proof -
- obtain x where "x \<in> A" "setdist A B = infdist x B"
- by (metis A assms(3) setdist_attains_inf setdist_sym)
- moreover
- obtain y where"y \<in> B" "infdist x B = dist x y"
- using B \<open>B \<noteq> {}\<close> infdist_attains_inf by blast
- ultimately show ?thesis
- using \<open>x \<in> A\<close> \<open>y \<in> B\<close> by auto
-qed
+ by (metis assms infdist_attains_inf setdist_attains_inf setdist_sym)
lemma setdist_closed_compact:
fixes S :: "'a::heine_borel set"
@@ -2406,14 +2307,10 @@
assumes S: "compact S" and T: "closed T"
shows "setdist S T = 0 \<longleftrightarrow> S = {} \<or> T = {} \<or> S \<inter> T \<noteq> {}"
proof (cases "S = {} \<or> T = {}")
- case True
- then show ?thesis
- by force
-next
case False
then show ?thesis
- by (metis S T disjoint_iff_not_equal in_closed_iff_infdist_zero setdist_attains_inf setdist_eq_0I setdist_sym)
-qed
+ by (metis S T disjoint_iff in_closed_iff_infdist_zero setdist_attains_inf setdist_eq_0I setdist_sym)
+qed auto
corollary setdist_gt_0_compact_closed:
assumes S: "compact S" and T: "closed T"