--- a/Admin/components/components.sha1 Tue Apr 22 15:08:14 2025 +0200
+++ b/Admin/components/components.sha1 Thu Apr 24 09:16:33 2025 +0200
@@ -129,6 +129,9 @@
31d6abd58a4c2f7522f14283dfe04e2801a6e828 flatlaf-2.6.tar.gz
95aff7a320715c473ea63294a288ba30a71dec88 flatlaf-3.5.4-1.tar.gz
a93a98c80536a6984db118e129b1b3a778c700ce flatlaf-3.5.4.tar.gz
+7de5b13afd4d9ac09e4d591b63c1979b9276e4f6 flatlaf-3.6-1.tar.gz
+372ac19c1783af818591c379e1646a3fc1fbbfc2 flatlaf-3.6-2.tar.gz
+1363421c87dd24bbff318e37c4788ba0339db7ad flatlaf-3.6.tar.gz
b1c40ce6c087da7e70e221ddd3fcadfa569acb2f foiltex-2.1.4b.tar.gz
f339234ec18369679be0095264e0c0af7762f351 gnu-utils-20210414.tar.gz
71259aa46134e6cf2c6473b4fc408051b3336490 gnu-utils-20211030.tar.gz
@@ -137,6 +140,7 @@
964fb17e87f7a7e8a91895a696f7d059164ae742 hugo-0.142.0.tar.gz
511fa8df8be88eb0500032bbd17742d33bdd4636 hugo-0.88.1.tar.gz
989234b3799fe8750f3c24825d1f717c24fb0214 idea-icons-20210508.tar.gz
+d7f72e7ef6e24ddf3b0d433d35b68658cabd1d88 idea-icons-20250415.tar.gz
20b53cfc3ffc5b15c1eabc91846915b49b4c0367 isabelle_fonts-20151021.tar.gz
736844204b2ef83974cd9f0a215738b767958c41 isabelle_fonts-20151104.tar.gz
9502c1aea938021f154adadff254c5c55da344bd isabelle_fonts-20151106.tar.gz
@@ -266,6 +270,10 @@
12f22ccb48a0a7dc9ad8c1b27552e8fd26d4ae0c jedit-20250215.tar.gz
1aa375149dc5e84cfdd64bf66e7e0f68319b7db7 jedit-20250402.tar.gz
88c9cc14f5618ad21ca17664fdb57b8f5f6f6d2e jedit-20250404.tar.gz
+cab6dca89dc2e0d922e92cc47df07941a3c6b29c jedit-20250415.tar.gz
+23297ab36a853247a17f87b94a29654f3259b341 jedit-20250417.tar.gz
+a291746959e64916e8504b89dca804186d4eb8a1 jedit-20250422.tar.gz
+07360418d691f6bb0e250e8efeb6b935e23eb0cd jedit-20250423.tar.gz
44775a22f42a9d665696bfb49e53c79371c394b0 jedit_build-20111217.tar.gz
a242a688810f2bccf24587b0062ce8027bf77fa2 jedit_build-20120304.tar.gz
4c948dee53f74361c097c08f49a1a5ff9b17bd1d jedit_build-20120307.tar.gz
@@ -322,6 +330,7 @@
df8bb213d39a7eecae97e6af3b11752d6c704c90 jsoup-1.15.4.tar.gz
b1c8e2a289e40cbc139a3c371348cef3b537b1c7 jsoup-1.17.2.tar.gz
342f12a07889ec173b13df9327db684b45118252 jsoup-1.18.3.tar.gz
+a7419f9dab68e50d05533980b6729d1dd7aafb3b jsvg-1.7.1.tar.gz
6c737137cc597fc920943783382e928ea79e3feb kodkodi-1.2.16.tar.gz
afb04f4048a87bb888fe7b05b0139cb060c7925b kodkodi-1.5.2-1.tar.gz
5f95c96bb99927f3a026050f85bd056f37a9189e kodkodi-1.5.2.tar.gz
--- a/Admin/components/main Tue Apr 22 15:08:14 2025 +0200
+++ b/Admin/components/main Thu Apr 24 09:16:33 2025 +0200
@@ -9,17 +9,17 @@
easychair-3.5
eptcs-1.7.0
find_facts_web-20250223
-flatlaf-3.5.4-1
+flatlaf-3.6-2
foiltex-2.1.4b
-idea-icons-20210508
isabelle_fonts-20241227
isabelle_setup-20240327
javamail-20250122
jdk-21.0.6
-jedit-20250404
+jedit-20250423
jfreechart-1.5.3
jortho-1.0-2
jsoup-1.18.3
+jsvg-1.7.1
kodkodi-1.5.7
lipics-3.1.3
llncs-2.25
--- a/NEWS Tue Apr 22 15:08:14 2025 +0200
+++ b/NEWS Thu Apr 24 09:16:33 2025 +0200
@@ -37,6 +37,27 @@
longer included. The old-fashioned toolbar, with its old Navigator
icons, is now disabled by default.
+* Update GUI look-and-feel to current FlatLaf 3.6, with native library
+support on all Isabelle platforms (now including arm64-linux).
+
+* Additional look-and-feels "FlatLaf macOS Light" and "FlatLaf macOS
+Dark" imitate original macOS more closely. This is the default for the
+bundled application on macOS.
+
+* GUI rendering of the gutter is now more accurate, using scaled icons
+to fit precisely into the available space.
+
+* GUI rendering for dark look-and-feels has been slightly improved, e.g.
+menu accelerator font/color.
+
+* Provide scalable icons for old-fashioned "tango" (from early Gnome or
+KDE), and "idea-icons" from current IntelliJ IDEA community edition. All
+icons are available for jEdit and Isabelle/jEdit add-ons. The special
+notation "myicon.svg?scale=0.5" allows to resize icons as specified in
+jEdit properties. The default properties of Isabelle/jEdit usually
+prefer scalable icons over fixed old bitmaps, but some notable icons are
+not available as SVG.
+
*** HOL ***
@@ -191,6 +212,10 @@
* SSH connections allow both bash and zsh as remote shell. This is
particularly important for macOS, where zsh is the default user shell.
+* Isabelle/Scala supports SVG via the "jsvg" library
+(com.github.weisj.jsvg.SVGDocument), as well as the "flatlaf-extras"
+library (com.formdev.flatlaf.extras.FlatSVGIcon).
+
New in Isabelle2025 (March 2025)
--- a/etc/build.props Tue Apr 22 15:08:14 2025 +0200
+++ b/etc/build.props Thu Apr 24 09:16:33 2025 +0200
@@ -36,6 +36,7 @@
src/Pure/Admin/component_jdk.scala \
src/Pure/Admin/component_jedit.scala \
src/Pure/Admin/component_jsoup.scala \
+ src/Pure/Admin/component_jsvg.scala \
src/Pure/Admin/component_lipics.scala \
src/Pure/Admin/component_llncs.scala \
src/Pure/Admin/component_minisat.scala \
--- a/src/HOL/Analysis/Cartesian_Euclidean_Space.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Analysis/Cartesian_Euclidean_Space.thy Thu Apr 24 09:16:33 2025 +0200
@@ -202,8 +202,8 @@
unfolding bounded_def
apply clarify
apply (rule_tac x="x $ i" in exI)
- apply (rule_tac x="e" in exI)
- apply clarify
+ apply (rule_tac x="\<epsilon>" in exI)
+ apply clarify
apply (rule order_trans [OF dist_vec_nth_le], simp)
done
--- a/src/HOL/Analysis/Cartesian_Space.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Analysis/Cartesian_Space.thy Thu Apr 24 09:16:33 2025 +0200
@@ -205,10 +205,13 @@
proof -
have "\<And>B x. A ** B = mat 1 \<Longrightarrow> \<exists>y. x = A *v y"
by (metis matrix_vector_mul_assoc matrix_vector_mul_lid)
- moreover have "\<forall>x. \<exists>xa. x = A *v xa \<Longrightarrow> \<exists>B. A ** B = mat 1"
- by (metis (mono_tags, lifting) matrix_compose_gen matrix_id_mat_1 matrix_of_matrix_vector_mul surj_def vec.linear_axioms vec.linear_surjective_right_inverse)
+ moreover
+ have "\<exists>B. A ** B = mat 1" if "surj ((*v) A)"
+ by (metis (no_types, opaque_lifting) matrix_compose_gen matrix_id_mat_1
+ matrix_of_matrix_vector_mul vec.linear_axioms
+ vec.linear_surjective_right_inverse that)
ultimately show ?thesis
- by (auto simp: image_def set_eq_iff)
+ by (auto simp: image_def set_eq_iff surj_def)
qed
lemma matrix_left_invertible_independent_columns:
@@ -301,24 +304,27 @@
"(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
using matrix_left_invertible_span_rows_gen[of A] by (simp add: span_vec_eq)
+
+lemma matrix_left_right_inverse1:
+ fixes A A' :: "'a::{field}^'n^'n"
+ assumes AA': "A ** A' = mat 1"
+ shows "A' ** A = mat 1"
+proof -
+ have sA: "surj ((*v) A)"
+ using AA' matrix_right_invertible_surjective by auto
+ obtain f' :: "'a ^'n \<Rightarrow> 'a ^'n"
+ where f': "Vector_Spaces.linear (*s) (*s) f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x"
+ using sA vec.linear_surjective_isomorphism by blast
+ have "matrix f' ** A = mat 1"
+ by (metis f' matrix_eq matrix_vector_mul_assoc matrix_vector_mul_lid matrix_works)
+ thus "A' ** A = mat 1"
+ by (metis AA' matrix_mul_assoc matrix_mul_lid)
+qed
+
lemma matrix_left_right_inverse:
fixes A A' :: "'a::{field}^'n^'n"
shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
-proof -
- { fix A A' :: "'a ^'n^'n"
- assume AA': "A ** A' = mat 1"
- have sA: "surj ((*v) A)"
- using AA' matrix_right_invertible_surjective by auto
- obtain f' :: "'a ^'n \<Rightarrow> 'a ^'n"
- where f': "Vector_Spaces.linear (*s) (*s) f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x"
- using sA vec.linear_surjective_isomorphism by blast
- have "matrix f' ** A = mat 1"
- by (metis f' matrix_eq matrix_vector_mul_assoc matrix_vector_mul_lid matrix_works)
- hence "A' ** A = mat 1"
- by (metis AA' matrix_mul_assoc matrix_mul_lid)
- }
- then show ?thesis by blast
-qed
+ using matrix_left_right_inverse1 by blast
lemma invertible_left_inverse:
fixes A :: "'a::{field}^'n^'n"
@@ -445,8 +451,8 @@
then obtain i where "v = row i A"
by (auto simp: rows_def)
with 0 show ?case
- unfolding orthogonal_def inner_vec_def matrix_vector_mult_def row_def
- by (simp add: mult.commute) (metis (no_types) vec_lambda_beta zero_index)
+ by (metis inner_commute matrix_vector_mul_component orthogonal_def row_def vec_lambda_eta
+ zero_index)
qed
lemma nullspace_inter_rowspace:
@@ -490,7 +496,7 @@
finally show ?thesis .
qed
then show ?thesis
- by (simp)
+ by simp
qed
lemma column_rank_def:
@@ -612,15 +618,15 @@
by (metis exhaust_4)
lemma UNIV_1 [simp]: "UNIV = {1::1}"
- by (auto simp add: num1_eq_iff)
+ by auto
-lemma UNIV_2: "UNIV = {1::2, 2::2}"
+lemma UNIV_2: "UNIV = {1, 2::2}"
using exhaust_2 by auto
-lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
+lemma UNIV_3: "UNIV = {1, 2, 3::3}"
using exhaust_3 by auto
-lemma UNIV_4: "UNIV = {1::4, 2::4, 3::4, 4::4}"
+lemma UNIV_4: "UNIV = {1, 2, 3, 4::4}"
using exhaust_4 by auto
lemma sum_1: "sum f (UNIV::1 set) = f 1"
@@ -715,7 +721,7 @@
have "P v" if "\<And>x y. P (vector [x, y])" for v
proof -
have "vector [v$1, v$2] = v"
- by (smt (verit, best) exhaust_2 vec_eq_iff vector_2)
+ unfolding vec_eq_iff by (metis (mono_tags) exhaust_2 vector_2)
then show ?thesis
by (metis that)
qed
@@ -727,7 +733,7 @@
have "P v" if "\<And>x y z. P (vector [x, y, z])" for v
proof -
have "vector [v$1, v$2, v$3] = v"
- by (smt (verit, best) exhaust_3 vec_eq_iff vector_3)
+ unfolding vec_eq_iff by (metis (mono_tags) exhaust_3 vector_3)
then show ?thesis
by (metis that)
qed
@@ -825,8 +831,7 @@
lemma vector_cart:
fixes f :: "real^'n \<Rightarrow> real"
shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"
- unfolding euclidean_eq_iff[where 'a="real^'n"]
- by simp (simp add: Basis_vec_def inner_axis)
+ by (simp add: euclidean_eq_iff[where 'a="real^'n"]) (simp add: Basis_vec_def inner_axis)
lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
by (rule vector_cart)
@@ -1258,10 +1263,9 @@
next
fix A :: "real^'n^'n" and i
assume "row i A = 0"
- show "P ((*v) A)"
- using matrix_vector_mul_linear
- by (rule zeroes[where i=i])
- (metis \<open>row i A = 0\<close> inner_zero_left matrix_vector_mul_component row_def vec_lambda_eta)
+ with matrix_vector_mul_linear show "P ((*v) A)"
+ by (metis matrix_vector_mul_component matrix_vector_mult_0 row_def
+ vec_lambda_eta zero_index zeroes)
next
fix A :: "real^'n^'n"
assume 0: "\<And>i j. i \<noteq> j \<Longrightarrow> A $ i $ j = 0"
--- a/src/HOL/Analysis/Connected.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Analysis/Connected.thy Thu Apr 24 09:16:33 2025 +0200
@@ -2,7 +2,7 @@
Material split off from Topology_Euclidean_Space
*)
-section \<open>Connected Components\<close>
+chapter \<open>Connected Components\<close>
theory Connected
imports
@@ -38,7 +38,7 @@
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)")
- by (simp add: closed_def) metis
+ unfolding closed_def by metis
have th1: "?rhs \<longleftrightarrow> \<not> (\<exists>t' t. closed t'\<and>t = S\<inter>t' \<and> t\<noteq>{} \<and> t\<noteq>S \<and> (\<exists>t'. open t' \<and> t = S \<inter> t'))"
(is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
unfolding connected_def openin_open closedin_closed by auto
@@ -49,8 +49,6 @@
then show ?thesis
by metis
qed
- then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)"
- by blast
then show ?thesis
by (simp add: th0 th1)
qed
@@ -94,7 +92,10 @@
lemma connected_iff_eq_connected_component_set:
"connected S \<longleftrightarrow> (\<forall>x \<in> S. connected_component_set S x = S)"
- by (metis connected_component_def connected_component_in connected_connected_component mem_Collect_eq subsetI subset_antisym)
+proof
+ show "\<forall>x\<in>S. connected_component_set S x = S \<Longrightarrow> connected S"
+ by (metis connectedI_const connected_connected_component)
+qed (auto simp: connected_component_def)
lemma connected_component_subset: "connected_component_set S x \<subseteq> S"
using connected_component_in by blast
@@ -152,8 +153,7 @@
lemma connected_component_disjoint:
"connected_component_set S a \<inter> connected_component_set S b = {} \<longleftrightarrow>
a \<notin> connected_component_set S b"
- by (smt (verit) connected_component_eq connected_component_eq_empty connected_component_refl_eq
- disjoint_iff_not_equal mem_Collect_eq)
+ using connected_component_eq connected_component_sym by fastforce
lemma connected_component_nonoverlap:
"connected_component_set S a \<inter> connected_component_set S b = {} \<longleftrightarrow>
@@ -180,13 +180,18 @@
lemma connected_component_idemp:
"connected_component_set (connected_component_set S x) x = connected_component_set S x"
- by (metis Int_absorb connected_component_disjoint connected_component_eq_empty connected_component_eq_self connected_connected_component)
+proof
+ show "connected_component_set S x \<subseteq> connected_component_set (connected_component_set S x) x"
+ by (metis connected_component_eq_empty connected_component_maximal order.refl
+ connected_component_refl connected_connected_component mem_Collect_eq)
+qed (simp add: connected_component_subset)
lemma connected_component_unique:
"\<lbrakk>x \<in> c; c \<subseteq> S; connected c;
\<And>c'. \<lbrakk>x \<in> c'; c' \<subseteq> S; connected c'\<rbrakk> \<Longrightarrow> c' \<subseteq> c\<rbrakk>
\<Longrightarrow> connected_component_set S x = c"
- by (meson connected_component_maximal connected_component_subset connected_connected_component subsetD subset_antisym)
+ by (simp add: connected_component_maximal connected_component_subset subsetD
+ subset_antisym)
lemma joinable_connected_component_eq:
"\<lbrakk>connected T; T \<subseteq> S;
@@ -237,12 +242,13 @@
lemma connected_component_set_homeomorphism:
assumes "homeomorphism A B f g" "x \<in> A"
- shows "connected_component_set B (f x) = f ` connected_component_set A x" (is "?lhs = ?rhs")
+ shows "connected_component_set B (f x) = f ` connected_component_set A x"
proof -
- have "y \<in> ?lhs \<longleftrightarrow> y \<in> ?rhs" for y
- by (smt (verit, best) assms connected_component_homeomorphism_iff homeomorphism_def image_iff mem_Collect_eq)
- thus ?thesis
- by blast
+ have "\<And>y. connected_component B (f x) y
+ \<Longrightarrow> \<exists>u. u \<in> A \<and> connected_component B (f x) (f u) \<and> y = f u"
+ using assms by (metis connected_component_in homeomorphism_def image_iff)
+ with assms show ?thesis
+ by (auto simp: image_iff connected_component_homeomorphism_iff)
qed
subsection \<open>The set of connected components of a set\<close>
@@ -279,7 +285,7 @@
lemma in_components_maximal:
"C \<in> components S \<longleftrightarrow>
- C \<noteq> {} \<and> C \<subseteq> S \<and> connected C \<and> (\<forall>d. d \<noteq> {} \<and> C \<subseteq> d \<and> d \<subseteq> S \<and> connected d \<longrightarrow> d = C)"
+ C \<noteq> {} \<and> C \<subseteq> S \<and> connected C \<and> (\<forall>D. D \<noteq> {} \<and> C \<subseteq> D \<and> D \<subseteq> S \<and> connected D \<longrightarrow> D = C)"
(is "?lhs \<longleftrightarrow> ?rhs")
proof
assume L: ?lhs
@@ -289,7 +295,8 @@
by (metis (full_types) L components_iff connected_component_maximal connected_component_refl empty_iff mem_Collect_eq subsetD subset_antisym)
next
show "?rhs \<Longrightarrow> ?lhs"
- by (metis bot.extremum_uniqueI components_iff connected_component_eq_empty connected_component_maximal connected_component_subset connected_connected_component subset_emptyI)
+ by (metis bot.extremum componentsI connected_component_maximal connected_component_subset
+ connected_connected_component order_antisym_conv subset_iff)
qed
@@ -335,7 +342,8 @@
by (metis components_empty components_eq_sing_iff empty_iff in_components_connected insertI1)
lemma components_maximal: "\<lbrakk>C \<in> components S; connected T; T \<subseteq> S; C \<inter> T \<noteq> {}\<rbrakk> \<Longrightarrow> T \<subseteq> C"
- by (smt (verit, best) Int_Un_eq(4) Un_upper1 bot_eq_sup_iff connected_Un in_components_maximal inf.orderE sup.mono sup.orderI)
+ by (metis (lifting) ext Int_Un_eq(4) Int_absorb Un_upper1 bot_eq_sup_iff connected_Un
+ in_components_maximal sup.mono sup.orderI)
lemma exists_component_superset: "\<lbrakk>T \<subseteq> S; S \<noteq> {}; connected T\<rbrakk> \<Longrightarrow> \<exists>C. C \<in> components S \<and> T \<subseteq> C"
by (meson componentsI connected_component_maximal equals0I subset_eq)
@@ -583,8 +591,7 @@
assume clo3: "closedin (top_of_set (U - C)) H3"
and clo4: "closedin (top_of_set (U - C)) H4"
and H34: "H3 \<union> H4 = U - C" "H3 \<inter> H4 = {}" and "H3 \<noteq> {}" and "H4 \<noteq> {}"
- and * [rule_format]:
- "\<forall>H1 H2. \<not> closedin (top_of_set S) H1 \<or>
+ and * [rule_format]: "\<forall>H1 H2. \<not> closedin (top_of_set S) H1 \<or>
\<not> closedin (top_of_set S) H2 \<or>
H1 \<union> H2 \<noteq> S \<or> H1 \<inter> H2 \<noteq> {} \<or> \<not> H1 \<noteq> {} \<or> \<not> H2 \<noteq> {}"
then have "H3 \<subseteq> U-C" and ope3: "openin (top_of_set (U - C)) (U - C - H3)"
@@ -641,7 +648,7 @@
lemma finite_range_constant_imp_connected:
assumes "\<And>f::'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1.
\<lbrakk>continuous_on S f; finite(f ` S)\<rbrakk> \<Longrightarrow> f constant_on S"
- shows "connected S"
+ shows "connected S"
proof -
{ fix T U
assume clt: "closedin (top_of_set S) T"
--- a/src/HOL/Analysis/Derivative.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Analysis/Derivative.thy Thu Apr 24 09:16:33 2025 +0200
@@ -2866,7 +2866,7 @@
have "dist (x + (y - f x)) (xh + (y - f xh)) * 2 \<le> dist x xh"
if "norm x \<le> \<delta>" and "norm xh \<le> \<delta>" for x xh
using that 4 [of x "xh-x"] by (auto simp: dist_norm norm_minus_commute algebra_simps)
- then show "\<forall>x\<in>cball 0 \<delta>. \<forall>ya\<in>cball 0 \<delta>. dist (x + (y - f x)) (ya + (y - f ya)) \<le> (1/2) * dist x ya"
+ then show "\<And>x z. \<lbrakk>x\<in>cball 0 \<delta>; z\<in>cball 0 \<delta>\<rbrakk> \<Longrightarrow> dist (x + (y - f x)) (z + (y - f z)) \<le> (1/2) * dist x z"
by auto
qed (auto simp: complete_eq_closed)
then show ?thesis
--- a/src/HOL/Analysis/Elementary_Metric_Spaces.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Analysis/Elementary_Metric_Spaces.thy Thu Apr 24 09:16:33 2025 +0200
@@ -16,21 +16,21 @@
section \<open>Open and closed balls\<close>
definition\<^marker>\<open>tag important\<close> ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
- where "ball x e = {y. dist x y < e}"
+ where "ball x \<epsilon> = {y. dist x y < \<epsilon>}"
definition\<^marker>\<open>tag important\<close> cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
- where "cball x e = {y. dist x y \<le> e}"
+ where "cball x \<epsilon> = {y. dist x y \<le> \<epsilon>}"
definition\<^marker>\<open>tag important\<close> sphere :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
- where "sphere x e = {y. dist x y = e}"
-
-lemma mem_ball [simp, metric_unfold]: "y \<in> ball x e \<longleftrightarrow> dist x y < e"
+ where "sphere x \<epsilon> = {y. dist x y = \<epsilon>}"
+
+lemma mem_ball [simp, metric_unfold]: "y \<in> ball x \<epsilon> \<longleftrightarrow> dist x y < \<epsilon>"
by (simp add: ball_def)
-lemma mem_cball [simp, metric_unfold]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e"
+lemma mem_cball [simp, metric_unfold]: "y \<in> cball x \<epsilon> \<longleftrightarrow> dist x y \<le> \<epsilon>"
by (simp add: cball_def)
-lemma mem_sphere [simp]: "y \<in> sphere x e \<longleftrightarrow> dist x y = e"
+lemma mem_sphere [simp]: "y \<in> sphere x \<epsilon> \<longleftrightarrow> dist x y = \<epsilon>"
by (simp add: sphere_def)
lemma ball_trivial [simp]: "ball x 0 = {}"
@@ -52,16 +52,16 @@
for a :: "'a::metric_space"
by auto
-lemma centre_in_ball [simp]: "x \<in> ball x e \<longleftrightarrow> 0 < e"
+lemma centre_in_ball [simp]: "x \<in> ball x \<epsilon> \<longleftrightarrow> 0 < \<epsilon>"
by simp
-lemma centre_in_cball [simp]: "x \<in> cball x e \<longleftrightarrow> 0 \<le> e"
+lemma centre_in_cball [simp]: "x \<in> cball x \<epsilon> \<longleftrightarrow> 0 \<le> \<epsilon>"
by simp
-lemma ball_subset_cball [simp, intro]: "ball x e \<subseteq> cball x e"
+lemma ball_subset_cball [simp, intro]: "ball x \<epsilon> \<subseteq> cball x \<epsilon>"
by (simp add: subset_eq)
-lemma mem_ball_imp_mem_cball: "x \<in> ball y e \<Longrightarrow> x \<in> cball y e"
+lemma mem_ball_imp_mem_cball: "x \<in> ball y \<epsilon> \<Longrightarrow> x \<in> cball y \<epsilon>"
by auto
lemma sphere_cball [simp,intro]: "sphere z r \<subseteq> cball z r"
@@ -70,16 +70,16 @@
lemma cball_diff_sphere: "cball a r - sphere a r = ball a r"
by auto
-lemma subset_ball[intro]: "d \<le> e \<Longrightarrow> ball x d \<subseteq> ball x e"
+lemma subset_ball[intro]: "\<delta> \<le> \<epsilon> \<Longrightarrow> ball x \<delta> \<subseteq> ball x \<epsilon>"
by auto
-lemma subset_cball[intro]: "d \<le> e \<Longrightarrow> cball x d \<subseteq> cball x e"
+lemma subset_cball[intro]: "\<delta> \<le> \<epsilon> \<Longrightarrow> cball x \<delta> \<subseteq> cball x \<epsilon>"
by auto
-lemma mem_ball_leI: "x \<in> ball y e \<Longrightarrow> e \<le> f \<Longrightarrow> x \<in> ball y f"
+lemma mem_ball_leI: "x \<in> ball y \<epsilon> \<Longrightarrow> \<epsilon> \<le> f \<Longrightarrow> x \<in> ball y f"
by auto
-lemma mem_cball_leI: "x \<in> cball y e \<Longrightarrow> e \<le> f \<Longrightarrow> x \<in> cball y f"
+lemma mem_cball_leI: "x \<in> cball y \<epsilon> \<Longrightarrow> \<epsilon> \<le> f \<Longrightarrow> x \<in> cball y f"
by auto
lemma cball_trans: "y \<in> cball z b \<Longrightarrow> x \<in> cball y a \<Longrightarrow> x \<in> cball z (b + a)"
@@ -100,70 +100,66 @@
lemma cball_diff_eq_sphere: "cball a r - ball a r = sphere a r"
by auto
-lemma open_ball [intro, simp]: "open (ball x e)"
+lemma open_ball [intro, simp]: "open (ball x \<epsilon>)"
proof -
- have "open (dist x -` {..<e})"
+ have "open (dist x -` {..<\<epsilon>})"
by (intro open_vimage open_lessThan continuous_intros)
- also have "dist x -` {..<e} = ball x e"
+ also have "dist x -` {..<\<epsilon>} = ball x \<epsilon>"
by auto
finally show ?thesis .
qed
-lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
+lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>\<epsilon>>0. ball x \<epsilon> \<subseteq> S)"
by (simp add: open_dist subset_eq Ball_def dist_commute)
-lemma openI [intro?]: "(\<And>x. x\<in>S \<Longrightarrow> \<exists>e>0. ball x e \<subseteq> S) \<Longrightarrow> open S"
+lemma openI [intro?]: "(\<And>x. x\<in>S \<Longrightarrow> \<exists>\<epsilon>>0. ball x \<epsilon> \<subseteq> S) \<Longrightarrow> open S"
by (auto simp: open_contains_ball)
lemma openE[elim?]:
assumes "open S" "x\<in>S"
- obtains e where "e>0" "ball x e \<subseteq> S"
+ obtains \<epsilon> where "\<epsilon>>0" "ball x \<epsilon> \<subseteq> S"
using assms unfolding open_contains_ball by auto
-lemma open_contains_ball_eq: "open S \<Longrightarrow> x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
+lemma open_contains_ball_eq: "open S \<Longrightarrow> x\<in>S \<longleftrightarrow> (\<exists>\<epsilon>>0. ball x \<epsilon> \<subseteq> S)"
by (metis open_contains_ball subset_eq centre_in_ball)
-lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
+lemma ball_eq_empty[simp]: "ball x \<epsilon> = {} \<longleftrightarrow> \<epsilon> \<le> 0"
unfolding mem_ball set_eq_iff
by (simp add: not_less) metric
-lemma ball_empty: "e \<le> 0 \<Longrightarrow> ball x e = {}"
+lemma ball_empty: "\<epsilon> \<le> 0 \<Longrightarrow> ball x \<epsilon> = {}"
by simp
-lemma closed_cball [iff]: "closed (cball x e)"
+lemma closed_cball [iff]: "closed (cball x \<epsilon>)"
proof -
- have "closed (dist x -` {..e})"
+ have "closed (dist x -` {..\<epsilon>})"
by (intro closed_vimage closed_atMost continuous_intros)
- also have "dist x -` {..e} = cball x e"
+ also have "dist x -` {..\<epsilon>} = cball x \<epsilon>"
by auto
finally show ?thesis .
qed
-lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. cball x e \<subseteq> S)"
-proof -
- {
- fix x and e::real
- assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
- then have "\<exists>d>0. cball x d \<subseteq> S"
- unfolding subset_eq by (intro exI [where x="e/2"], auto)
- }
- then show ?thesis
- unfolding open_contains_ball by force
-qed
-
-lemma open_contains_cball_eq: "open S \<Longrightarrow> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
+lemma cball_subset_ball:
+ assumes "\<epsilon>>0"
+ shows "\<exists>\<delta>>0. cball x \<delta> \<subseteq> ball x \<epsilon>"
+ using assms unfolding subset_eq by (intro exI [where x="\<epsilon>/2"], auto)
+
+lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>\<epsilon>>0. cball x \<epsilon> \<subseteq> S)"
+ by (meson ball_subset_cball cball_subset_ball open_contains_ball subset_trans)
+
+lemma open_contains_cball_eq: "open S \<Longrightarrow> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>\<epsilon>>0. cball x \<epsilon> \<subseteq> S))"
by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)
-lemma eventually_nhds_ball: "d > 0 \<Longrightarrow> eventually (\<lambda>x. x \<in> ball z d) (nhds z)"
+lemma eventually_nhds_ball: "\<delta> > 0 \<Longrightarrow> eventually (\<lambda>x. x \<in> ball z \<delta>) (nhds z)"
by (rule eventually_nhds_in_open) simp_all
-lemma eventually_at_ball: "d > 0 \<Longrightarrow> eventually (\<lambda>t. t \<in> ball z d \<and> t \<in> A) (at z within A)"
- unfolding eventually_at by (intro exI[of _ d]) (simp_all add: dist_commute)
-
-lemma eventually_at_ball': "d > 0 \<Longrightarrow> eventually (\<lambda>t. t \<in> ball z d \<and> t \<noteq> z \<and> t \<in> A) (at z within A)"
- unfolding eventually_at by (intro exI[of _ d]) (simp_all add: dist_commute)
-
-lemma at_within_ball: "e > 0 \<Longrightarrow> dist x y < e \<Longrightarrow> at y within ball x e = at y"
+lemma eventually_at_ball: "\<delta> > 0 \<Longrightarrow> eventually (\<lambda>t. t \<in> ball z \<delta> \<and> t \<in> A) (at z within A)"
+ unfolding eventually_at by (intro exI[of _ \<delta>]) (simp_all add: dist_commute)
+
+lemma eventually_at_ball': "\<delta> > 0 \<Longrightarrow> eventually (\<lambda>t. t \<in> ball z \<delta> \<and> t \<noteq> z \<and> t \<in> A) (at z within A)"
+ unfolding eventually_at by (intro exI[of _ \<delta>]) (simp_all add: dist_commute)
+
+lemma at_within_ball: "\<epsilon> > 0 \<Longrightarrow> dist x y < \<epsilon> \<Longrightarrow> at y within ball x \<epsilon> = at y"
by (subst at_within_open) auto
lemma atLeastAtMost_eq_cball:
@@ -186,30 +182,42 @@
shows "ball a b = {a - b <..< a + b}"
by (auto simp: dist_real_def)
-lemma interior_ball [simp]: "interior (ball x e) = ball x e"
+lemma interior_ball [simp]: "interior (ball x \<epsilon>) = ball x \<epsilon>"
by (simp add: interior_open)
-lemma cball_eq_empty [simp]: "cball x e = {} \<longleftrightarrow> e < 0"
- by (smt (verit, best) Diff_empty ball_eq_empty cball_diff_sphere centre_in_ball centre_in_cball sphere_empty)
-
-lemma cball_empty [simp]: "e < 0 \<Longrightarrow> cball x e = {}"
+lemma cball_eq_empty [simp]: "cball x \<epsilon> = {} \<longleftrightarrow> \<epsilon> < 0"
+ by (metis centre_in_cball order.trans ex_in_conv linorder_not_le mem_cball
+ zero_le_dist)
+
+lemma cball_empty [simp]: "\<epsilon> < 0 \<Longrightarrow> cball x \<epsilon> = {}"
by simp
lemma cball_sing:
fixes x :: "'a::metric_space"
- shows "e = 0 \<Longrightarrow> cball x e = {x}"
+ shows "\<epsilon> = 0 \<Longrightarrow> cball x \<epsilon> = {x}"
by simp
-lemma ball_divide_subset: "d \<ge> 1 \<Longrightarrow> ball x (e/d) \<subseteq> ball x e"
+lemma ball_divide_subset: "\<delta> \<ge> 1 \<Longrightarrow> ball x (\<epsilon>/\<delta>) \<subseteq> ball x \<epsilon>"
by (metis ball_eq_empty div_by_1 frac_le linear subset_ball zero_less_one)
-lemma ball_divide_subset_numeral: "ball x (e / numeral w) \<subseteq> ball x e"
+lemma ball_divide_subset_numeral: "ball x (\<epsilon> / numeral w) \<subseteq> ball x \<epsilon>"
using ball_divide_subset one_le_numeral by blast
-lemma cball_divide_subset: "d \<ge> 1 \<Longrightarrow> cball x (e/d) \<subseteq> cball x e"
- by (smt (verit, best) cball_empty div_by_1 frac_le subset_cball zero_le_divide_iff)
-
-lemma cball_divide_subset_numeral: "cball x (e / numeral w) \<subseteq> cball x e"
+lemma cball_divide_subset:
+ assumes "\<delta> \<ge> 1"
+ shows "cball x (\<epsilon>/\<delta>) \<subseteq> cball x \<epsilon>"
+proof (cases "\<epsilon>\<ge>0")
+ case True
+ then show ?thesis
+ by (metis assms div_by_1 divide_mono order_le_less subset_cball zero_less_one)
+next
+ case False
+ then have "(\<epsilon>/\<delta>) < 0"
+ using assms divide_less_0_iff by fastforce
+ then show ?thesis by auto
+qed
+
+lemma cball_divide_subset_numeral: "cball x (\<epsilon> / numeral w) \<subseteq> cball x \<epsilon>"
using cball_divide_subset one_le_numeral by blast
lemma cball_scale:
@@ -302,21 +310,21 @@
have "filterlim f (nhds x) sequentially"
unfolding tendsto_iff
proof clarify
- fix e :: real
- assume e: "e > 0"
- then obtain n where n: "Suc n > 1 / e"
+ fix \<epsilon> :: real
+ assume \<epsilon>: "\<epsilon> > 0"
+ then obtain n where n: "Suc n > 1 / \<epsilon>"
by (meson le_nat_floor lessI not_le)
- have "dist (f k) x < e" if "k \<ge> n" for k
+ have "dist (f k) x < \<epsilon>" if "k \<ge> n" for k
proof -
have "dist (f k) x < 1 / real (Suc k)"
using f[of k] by (auto simp: dist_commute)
also have "\<dots> \<le> 1 / real (Suc n)"
using that by (intro divide_left_mono) auto
- also have "\<dots> < e"
- using n e by (simp add: field_simps)
+ also have "\<dots> < \<epsilon>"
+ using n \<epsilon> by (simp add: field_simps)
finally show ?thesis .
qed
- thus "\<forall>\<^sub>F k in sequentially. dist (f k) x < e"
+ thus "\<forall>\<^sub>F k in sequentially. dist (f k) x < \<epsilon>"
unfolding eventually_at_top_linorder by blast
qed
moreover have "f n \<noteq> x" for n
@@ -331,10 +339,10 @@
lemma islimpt_approachable:
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>\<epsilon>>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < \<epsilon>)"
unfolding islimpt_iff_eventually eventually_at by fast
-lemma islimpt_approachable_le: "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x \<le> e)"
+lemma islimpt_approachable_le: "x islimpt S \<longleftrightarrow> (\<forall>\<epsilon>>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x \<le> \<epsilon>)"
for x :: "'a::metric_space"
unfolding islimpt_approachable
using approachable_lt_le2 [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x" and Q="\<lambda>x. True"]
@@ -351,13 +359,13 @@
for x :: "'a::metric_space"
by (auto simp: closure_def islimpt_Un dest: limpt_of_limpts)
-lemma islimpt_eq_infinite_ball: "x islimpt S \<longleftrightarrow> (\<forall>e>0. infinite(S \<inter> ball x e))"
+lemma islimpt_eq_infinite_ball: "x islimpt S \<longleftrightarrow> (\<forall>\<epsilon>>0. infinite(S \<inter> ball x \<epsilon>))"
unfolding islimpt_eq_acc_point
by (metis open_ball Int_commute Int_mono finite_subset open_contains_ball_eq subset_eq)
-lemma islimpt_eq_infinite_cball: "x islimpt S \<longleftrightarrow> (\<forall>e>0. infinite(S \<inter> cball x e))"
+lemma islimpt_eq_infinite_cball: "x islimpt S \<longleftrightarrow> (\<forall>\<epsilon>>0. infinite(S \<inter> cball x \<epsilon>))"
unfolding islimpt_eq_infinite_ball
- by (metis open_ball ball_subset_cball centre_in_ball finite_Int inf.absorb_iff2 inf_assoc open_contains_cball_eq)
+ by (metis ball_subset_cball cball_subset_ball finite_Int inf.absorb_iff2 inf_assoc)
section \<open>Perfect Metric Spaces\<close>
@@ -374,9 +382,10 @@
lemma cball_eq_sing:
fixes x :: "'a::{metric_space,perfect_space}"
- shows "cball x e = {x} \<longleftrightarrow> e = 0"
- by (metis cball_trivial centre_in_cball finite.emptyI finite.insertI finite_Int
- islimpt_UNIV islimpt_eq_infinite_cball less_eq_real_def)
+ shows "cball x \<epsilon> = {x} \<longleftrightarrow> \<epsilon> = 0"
+ using cball_eq_empty [of x \<epsilon>]
+ by (metis open_ball ball_subset_cball cball_trivial
+ centre_in_ball not_less_iff_gr_or_eq not_open_singleton subset_singleton_iff)
section \<open>Finite and discrete\<close>
@@ -384,26 +393,26 @@
lemma finite_ball_include:
fixes a :: "'a::metric_space"
assumes "finite S"
- shows "\<exists>e>0. S \<subseteq> ball a e"
+ shows "\<exists>\<epsilon>>0. S \<subseteq> ball a \<epsilon>"
using assms
proof induction
case (insert x S)
- then obtain e0 where "e0>0" and e0:"S \<subseteq> ball a e0" by auto
- define e where "e = max e0 (2 * dist a x)"
- have "e>0" unfolding e_def using \<open>e0>0\<close> by auto
- moreover have "insert x S \<subseteq> ball a e"
- using e0 \<open>e>0\<close> unfolding e_def by auto
+ then obtain e0 where "e0>0" and e0:"S \<subseteq> ball a e0" by auto
+ define \<epsilon> where "\<epsilon> = max e0 (2 * dist a x)"
+ have "\<epsilon>>0" unfolding \<epsilon>_def using \<open>e0>0\<close> by auto
+ moreover have "insert x S \<subseteq> ball a \<epsilon>"
+ using e0 \<open>\<epsilon>>0\<close> unfolding \<epsilon>_def by auto
ultimately show ?case by auto
qed (auto intro: zero_less_one)
lemma finite_set_avoid:
fixes a :: "'a::metric_space"
assumes "finite S"
- shows "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x"
+ shows "\<exists>\<delta>>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> \<delta> \<le> dist a x"
using assms
proof induction
case (insert x S)
- then obtain d where "d > 0" and d: "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x"
+ then obtain \<delta> where "\<delta> > 0" and \<delta>: "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> \<delta> \<le> dist a x"
by blast
then show ?case
by (metis dist_nz dual_order.strict_implies_order insertE leI order.strict_trans2)
@@ -411,17 +420,17 @@
lemma discrete_imp_closed:
fixes S :: "'a::metric_space set"
- assumes e: "0 < e"
- and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
+ assumes \<epsilon>: "0 < \<epsilon>"
+ and \<delta>: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < \<epsilon> \<longrightarrow> y = x"
shows "closed S"
proof -
- have False if C: "\<And>e. e>0 \<Longrightarrow> \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" for x
+ have False if C: "\<And>\<epsilon>. \<epsilon>>0 \<Longrightarrow> \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < \<epsilon>" for x
proof -
- obtain y where y: "y \<in> S" "y \<noteq> x" "dist y x < e/2"
- by (meson C e half_gt_zero)
- then have mp: "min (e/2) (dist x y) > 0"
+ obtain y where y: "y \<in> S" "y \<noteq> x" "dist y x < \<epsilon>/2"
+ by (meson C \<epsilon> half_gt_zero)
+ then have mp: "min (\<epsilon>/2) (dist x y) > 0"
by (simp add: dist_commute)
- from d y C[OF mp] show ?thesis
+ from \<delta> y C[OF mp] show ?thesis
by metric
qed
then show ?thesis
@@ -429,19 +438,19 @@
qed
lemma discrete_imp_not_islimpt:
- assumes e: "0 < e"
- and d: "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> dist y x < e \<Longrightarrow> y = x"
+ assumes \<epsilon>: "0 < \<epsilon>"
+ and \<delta>: "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> dist y x < \<epsilon> \<Longrightarrow> y = x"
shows "\<not> x islimpt S"
- by (metis closed_limpt d discrete_imp_closed e islimpt_approachable)
+ by (metis closed_limpt \<delta> discrete_imp_closed \<epsilon> islimpt_approachable)
section \<open>Interior\<close>
-lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
+lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>\<epsilon>>0. ball x \<epsilon> \<subseteq> S)"
using open_contains_ball_eq [where S="interior S"]
by (simp add: open_subset_interior)
-lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"
+lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>\<epsilon>>0. cball x \<epsilon> \<subseteq> S)"
by (meson ball_subset_cball interior_subset mem_interior open_contains_cball open_interior
subset_trans)
@@ -453,33 +462,33 @@
lemma frontier_straddle:
fixes a :: "'a::metric_space"
- shows "a \<in> frontier S \<longleftrightarrow> (\<forall>e>0. (\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e))"
+ shows "a \<in> frontier S \<longleftrightarrow> (\<forall>\<epsilon>>0. (\<exists>x\<in>S. dist a x < \<epsilon>) \<and> (\<exists>x. x \<notin> S \<and> dist a x < \<epsilon>))"
unfolding frontier_def closure_interior
by (auto simp: mem_interior subset_eq ball_def)
section \<open>Limits\<close>
-proposition Lim: "(f \<longlongrightarrow> l) net \<longleftrightarrow> trivial_limit net \<or> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
+proposition Lim: "(f \<longlongrightarrow> l) net \<longleftrightarrow> trivial_limit net \<or> (\<forall>\<epsilon>>0. eventually (\<lambda>x. dist (f x) l < \<epsilon>) net)"
by (auto simp: tendsto_iff trivial_limit_eq)
text \<open>Show that they yield usual definitions in the various cases.\<close>
proposition Lim_within_le: "(f \<longlongrightarrow> l)(at a within S) \<longleftrightarrow>
- (\<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)"
+ (\<forall>\<epsilon>>0. \<exists>\<delta>>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a \<le> \<delta> \<longrightarrow> dist (f x) l < \<epsilon>)"
by (auto simp: tendsto_iff eventually_at_le)
proposition Lim_within: "(f \<longlongrightarrow> 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>\<epsilon> >0. \<exists>\<delta>>0. \<forall>x \<in> S. 0 < dist x a \<and> dist x a < \<delta> \<longrightarrow> dist (f x) l < \<epsilon>)"
by (auto simp: tendsto_iff eventually_at)
corollary Lim_withinI [intro?]:
- assumes "\<And>e. e > 0 \<Longrightarrow> \<exists>d>0. \<forall>x \<in> S. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l \<le> e"
+ assumes "\<And>\<epsilon>. \<epsilon> > 0 \<Longrightarrow> \<exists>\<delta>>0. \<forall>x \<in> S. 0 < dist x a \<and> dist x a < \<delta> \<longrightarrow> dist (f x) l \<le> \<epsilon>"
shows "(f \<longlongrightarrow> l) (at a within S)"
- unfolding Lim_within by (smt (verit, ccfv_SIG) assms zero_less_dist_iff)
+ unfolding Lim_within by (smt (verit, best) assms)
proposition Lim_at: "(f \<longlongrightarrow> 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>\<epsilon> >0. \<exists>\<delta>>0. \<forall>x. 0 < dist x a \<and> dist x a < \<delta> \<longrightarrow> dist (f x) l < \<epsilon>)"
by (auto simp: tendsto_iff eventually_at)
lemma Lim_transform_within_set:
@@ -497,9 +506,9 @@
(is "?lhs = ?rhs")
proof
assume ?lhs
- then have "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
+ then have "\<forall>\<epsilon>>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < \<epsilon>"
by (force simp: islimpt_approachable)
- then obtain y where y: "\<And>e. e>0 \<Longrightarrow> y e \<in> S \<and> y e \<noteq> x \<and> dist (y e) x < e"
+ then obtain y where y: "\<And>\<epsilon>. \<epsilon>>0 \<Longrightarrow> y \<epsilon> \<in> S \<and> y \<epsilon> \<noteq> x \<and> dist (y \<epsilon>) x < \<epsilon>"
by metis
define f where "f \<equiv> rec_nat (y 1) (\<lambda>n fn. y (min (inverse(2 ^ (Suc n))) (dist fn x)))"
have [simp]: "f 0 = y 1"
@@ -510,8 +519,13 @@
case 0 show ?case
by (simp add: y)
next
- case (Suc n) then show ?case
- by (smt (verit, best) fSuc dist_pos_lt inverse_positive_iff_positive y zero_less_power)
+ case (Suc n)
+ then have "dist (f (Suc n)) x < inverse (2 ^ Suc n)"
+ unfolding fSuc
+ by (metis dist_nz min_less_iff_conj positive_imp_inverse_positive y zero_less_numeral
+ zero_less_power)
+ with Suc show ?case
+ by (auto simp: y fSuc)
qed
show ?rhs
proof (intro exI conjI allI)
@@ -525,14 +539,15 @@
case (Suc n)
then consider "m < n" | "m = n" using less_Suc_eq by blast
then show ?case
- by (smt (verit, ccfv_threshold) Suc.IH dist_nz f fSuc y)
+ unfolding fSuc
+ by (smt (verit, ccfv_threshold) Suc.IH dist_nz f y)
qed
then show "inj f"
by (metis less_irrefl linorder_injI)
- have "\<And>e n. \<lbrakk>0 < e; nat \<lceil>1 / e::real\<rceil> \<le> n\<rbrakk> \<Longrightarrow> inverse (2 ^ n) < e"
+ have "\<And>\<epsilon> n. \<lbrakk>0 < \<epsilon>; nat \<lceil>1 / \<epsilon>::real\<rceil> \<le> n\<rbrakk> \<Longrightarrow> inverse (2 ^ n) < \<epsilon>"
by (simp add: divide_simps order_le_less_trans)
then show "f \<longlonglongrightarrow> x"
- by (metis dual_order.strict_trans f lim_sequentially)
+ by (meson order.strict_trans f lim_sequentially)
qed
next
assume ?rhs
@@ -543,8 +558,8 @@
lemma Lim_dist_ubound:
assumes "\<not>(trivial_limit net)"
and "(f \<longlongrightarrow> l) net"
- and "eventually (\<lambda>x. dist a (f x) \<le> e) net"
- shows "dist a l \<le> e"
+ and "eventually (\<lambda>x. dist a (f x) \<le> \<epsilon>) net"
+ shows "dist a l \<le> \<epsilon>"
using assms by (fast intro: tendsto_le tendsto_intros)
@@ -553,24 +568,24 @@
text\<open>Derive the epsilon-delta forms, which we often use as "definitions"\<close>
proposition 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)"
+ "continuous (at x within s) f \<longleftrightarrow> (\<forall>\<epsilon>>0. \<exists>\<delta>>0. \<forall>x'\<in> s. dist x' x < \<delta> --> dist (f x') (f x) < \<epsilon>)"
unfolding continuous_within and Lim_within by fastforce
corollary 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)"
+ "continuous (at x) f \<longleftrightarrow> (\<forall>\<epsilon> > 0. \<exists>\<delta> > 0. \<forall>x'. dist x' x < \<delta> \<longrightarrow> dist (f x') (f x) < \<epsilon>)"
using continuous_within_eps_delta [of x UNIV f] by simp
lemma continuous_at_right_real_increasing:
fixes f :: "real \<Rightarrow> real"
assumes nondecF: "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y"
- shows "continuous (at_right a) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f (a + d) - f a < e)"
+ shows "continuous (at_right a) f \<longleftrightarrow> (\<forall>\<epsilon>>0. \<exists>\<delta>>0. f (a + \<delta>) - f a < \<epsilon>)"
apply (simp add: greaterThan_def dist_real_def continuous_within Lim_within_le)
apply (intro all_cong ex_cong)
by (smt (verit, best) nondecF)
lemma continuous_at_left_real_increasing:
assumes nondecF: "\<And> x y. x \<le> y \<Longrightarrow> f x \<le> ((f y) :: real)"
- shows "(continuous (at_left (a :: real)) f) \<longleftrightarrow> (\<forall>e > 0. \<exists>delta > 0. f a - f (a - delta) < e)"
+ shows "(continuous (at_left (a :: real)) f) \<longleftrightarrow> (\<forall>\<epsilon> > 0. \<exists>\<delta> > 0. f a - f (a - \<delta>) < \<epsilon>)"
apply (simp add: lessThan_def dist_real_def continuous_within Lim_within_le)
apply (intro all_cong ex_cong)
by (smt (verit) nondecF)
@@ -579,21 +594,19 @@
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)"
+ (\<forall>\<epsilon> > 0. \<exists>\<delta> > 0. f ` (ball x \<delta> \<inter> S) \<subseteq> ball (f x) \<epsilon>)"
(is "?lhs = ?rhs")
proof
assume ?lhs
{
- fix e :: real
- assume "e > 0"
- then obtain d where "d>0" and d: "\<forall>y\<in>S. 0 < dist y x \<and> dist y x < d \<longrightarrow> dist (f y) (f x) < e"
+ fix \<epsilon> :: real
+ assume "\<epsilon> > 0"
+ then obtain \<delta> where "\<delta>>0" and \<delta>: "\<forall>y\<in>S. 0 < dist y x \<and> dist y x < \<delta> \<longrightarrow> dist (f y) (f x) < \<epsilon>"
using \<open>?lhs\<close>[unfolded continuous_within Lim_within] by auto
- { fix y
- assume "y \<in> f ` (ball x d \<inter> S)" then have "y \<in> ball (f x) e"
- using d \<open>e > 0\<close> by (auto simp: dist_commute)
- }
- then have "\<exists>d>0. f ` (ball x d \<inter> S) \<subseteq> ball (f x) e"
- using \<open>d > 0\<close> by blast
+ have "y \<in> ball (f x) \<epsilon>" if "y \<in> f ` (ball x \<delta> \<inter> S)" for y
+ using that \<delta> \<open>\<epsilon> > 0\<close> by (auto simp: dist_commute)
+ then have "\<exists>\<delta>>0. f ` (ball x \<delta> \<inter> S) \<subseteq> ball (f x) \<epsilon>"
+ using \<open>\<delta> > 0\<close> by blast
}
then show ?rhs by auto
next
@@ -603,38 +616,35 @@
by (metis (mono_tags, lifting) Int_iff dist_commute mem_Collect_eq)
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)"
- apply (simp add: continuous_at Lim_at subset_eq Ball_def Bex_def image_iff)
- by (smt (verit, ccfv_threshold) dist_commute dist_self)
+corollary continuous_at_ball:
+ "continuous (at x) f \<longleftrightarrow> (\<forall>\<epsilon>>0. \<exists>\<delta>>0. f ` (ball x \<delta>) \<subseteq> ball (f x) \<epsilon>)"
+ by (simp add: continuous_within_ball)
text\<open>Define setwise continuity in terms of limits within the set.\<close>
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)"
+ (\<forall>x\<in>s. \<forall>\<epsilon>>0. \<exists>\<delta>>0. \<forall>x'\<in>s. dist x' x < \<delta> \<longrightarrow> dist (f x') (f x) < \<epsilon>)"
unfolding continuous_on_def Lim_within
by (metis dist_pos_lt dist_self)
lemma continuous_within_E:
- assumes "continuous (at x within S) f" "e>0"
- obtains d where "d>0" "\<And>x'. \<lbrakk>x'\<in> S; dist x' x \<le> d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
+ assumes "continuous (at x within S) f" "\<epsilon>>0"
+ obtains \<delta> where "\<delta>>0" "\<And>x'. \<lbrakk>x'\<in> S; dist x' x \<le> \<delta>\<rbrakk> \<Longrightarrow> dist (f x') (f x) < \<epsilon>"
using assms unfolding continuous_within_eps_delta
by (metis dense order_le_less_trans)
lemma continuous_onI [intro?]:
- assumes "\<And>x e. \<lbrakk>e > 0; x \<in> S\<rbrakk> \<Longrightarrow> \<exists>d>0. \<forall>x'\<in>S. dist x' x < d \<longrightarrow> dist (f x') (f x) \<le> e"
+ assumes "\<And>x \<epsilon>. \<lbrakk>\<epsilon> > 0; x \<in> S\<rbrakk> \<Longrightarrow> \<exists>\<delta>>0. \<forall>x'\<in>S. dist x' x < \<delta> \<longrightarrow> dist (f x') (f x) \<le> \<epsilon>"
shows "continuous_on S f"
-apply (simp add: continuous_on_iff, clarify)
-apply (rule ex_forward [OF assms [OF half_gt_zero]], auto)
-done
+ using assms [OF half_gt_zero] by (force simp add: continuous_on_iff)
text\<open>Some simple consequential lemmas.\<close>
lemma continuous_onE:
- assumes "continuous_on s f" "x\<in>s" "e>0"
- obtains d where "d>0" "\<And>x'. \<lbrakk>x' \<in> s; dist x' x \<le> d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
+ assumes "continuous_on s f" "x\<in>s" "\<epsilon>>0"
+ obtains \<delta> where "\<delta>>0" "\<And>x'. \<lbrakk>x' \<in> s; dist x' x \<le> \<delta>\<rbrakk> \<Longrightarrow> dist (f x') (f x) < \<epsilon>"
using assms
unfolding continuous_on_iff by (metis dense order_le_less_trans)
@@ -643,9 +653,9 @@
lemma continuous_transform_within:
fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
assumes "continuous (at x within s) f"
- and "0 < d"
+ and "0 < \<delta>"
and "x \<in> s"
- and "\<And>x'. \<lbrakk>x' \<in> s; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'"
+ and "\<And>x'. \<lbrakk>x' \<in> s; dist x' x < \<delta>\<rbrakk> \<Longrightarrow> f x' = g x'"
shows "continuous (at x within s) g"
using assms
unfolding continuous_within by (force intro: Lim_transform_within)
@@ -655,23 +665,23 @@
lemma closure_approachable:
fixes S :: "'a::metric_space set"
- shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
+ shows "x \<in> closure S \<longleftrightarrow> (\<forall>\<epsilon>>0. \<exists>y\<in>S. dist y x < \<epsilon>)"
using dist_self by (force simp: closure_def islimpt_approachable)
lemma closure_approachable_le:
fixes S :: "'a::metric_space set"
- shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x \<le> e)"
+ shows "x \<in> closure S \<longleftrightarrow> (\<forall>\<epsilon>>0. \<exists>y\<in>S. dist y x \<le> \<epsilon>)"
unfolding closure_approachable
using dense by force
lemma closure_approachableD:
- assumes "x \<in> closure S" "e>0"
- shows "\<exists>y\<in>S. dist x y < e"
+ assumes "x \<in> closure S" "\<epsilon>>0"
+ shows "\<exists>y\<in>S. dist x y < \<epsilon>"
using assms unfolding closure_approachable by (auto simp: dist_commute)
lemma closed_approachable:
fixes S :: "'a::metric_space set"
- shows "closed S \<Longrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
+ shows "closed S \<Longrightarrow> (\<forall>\<epsilon>>0. \<exists>y\<in>S. dist y x < \<epsilon>) \<longleftrightarrow> x \<in> S"
by (metis closure_closed closure_approachable)
lemma closure_contains_Inf:
@@ -681,12 +691,12 @@
proof -
have *: "\<forall>x\<in>S. Inf S \<le> x"
using cInf_lower[of _ S] assms by metis
- { 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"
+ { fix \<epsilon> :: real
+ assume "\<epsilon> > 0"
+ then have "Inf S < Inf S + \<epsilon>" by simp
+ with assms obtain x where "x \<in> S" "x < Inf S + \<epsilon>"
using cInf_lessD by blast
- with * have "\<exists>x\<in>S. dist x (Inf S) < e"
+ with * have "\<exists>x\<in>S. dist x (Inf S) < \<epsilon>"
using dist_real_def by force
}
then show ?thesis unfolding closure_approachable by auto
@@ -700,43 +710,43 @@
have *: "\<forall>x\<in>S. x \<le> Sup S"
using cSup_upper[of _ S] assms by metis
{
- fix e :: real
- assume "e > 0"
- then have "Sup S - e < Sup S" by simp
- with assms obtain x where "x \<in> S" "Sup S - e < x"
+ fix \<epsilon> :: real
+ assume "\<epsilon> > 0"
+ then have "Sup S - \<epsilon> < Sup S" by simp
+ with assms obtain x where "x \<in> S" "Sup S - \<epsilon> < x"
using less_cSupE by blast
- with * have "\<exists>x\<in>S. dist x (Sup S) < e"
+ with * have "\<exists>x\<in>S. dist x (Sup S) < \<epsilon>"
using dist_real_def by force
}
then show ?thesis unfolding closure_approachable by auto
qed
lemma not_trivial_limit_within_ball:
- "\<not> trivial_limit (at x within S) \<longleftrightarrow> (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"
+ "\<not> trivial_limit (at x within S) \<longleftrightarrow> (\<forall>\<epsilon>>0. S \<inter> ball x \<epsilon> - {x} \<noteq> {})"
(is "?lhs \<longleftrightarrow> ?rhs")
proof
show ?rhs if ?lhs
proof -
- { fix e :: real
- assume "e > 0"
- then obtain y where "y \<in> S - {x}" and "dist y x < e"
+ { fix \<epsilon> :: real
+ assume "\<epsilon> > 0"
+ then obtain y where "y \<in> S - {x}" and "dist y x < \<epsilon>"
using \<open>?lhs\<close> not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]
by auto
- then have "y \<in> S \<inter> ball x e - {x}"
+ then have "y \<in> S \<inter> ball x \<epsilon> - {x}"
unfolding ball_def by (simp add: dist_commute)
- then have "S \<inter> ball x e - {x} \<noteq> {}" by blast
+ then have "S \<inter> ball x \<epsilon> - {x} \<noteq> {}" by blast
}
then show ?thesis by auto
qed
show ?lhs if ?rhs
proof -
- { fix e :: real
- assume "e > 0"
- then obtain y where "y \<in> S \<inter> ball x e - {x}"
+ { fix \<epsilon> :: real
+ assume "\<epsilon> > 0"
+ then obtain y where "y \<in> S \<inter> ball x \<epsilon> - {x}"
using \<open>?rhs\<close> by blast
- then have "y \<in> S - {x}" and "dist y x < e"
+ then have "y \<in> S - {x}" and "dist y x < \<epsilon>"
unfolding ball_def by (simp_all add: dist_commute)
- then have "\<exists>y \<in> S - {x}. dist y x < e"
+ then have "\<exists>y \<in> S - {x}. dist y x < \<epsilon>"
by auto
}
then show ?thesis
@@ -750,12 +760,12 @@
(* FIXME: This has to be unified with BSEQ!! *)
definition\<^marker>\<open>tag important\<close> (in metric_space) bounded :: "'a set \<Rightarrow> bool"
- where "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
-
-lemma bounded_subset_cball: "bounded S \<longleftrightarrow> (\<exists>e x. S \<subseteq> cball x e \<and> 0 \<le> e)"
+ where "bounded S \<longleftrightarrow> (\<exists>x \<epsilon>. \<forall>y\<in>S. dist x y \<le> \<epsilon>)"
+
+lemma bounded_subset_cball: "bounded S \<longleftrightarrow> (\<exists>\<epsilon> x. S \<subseteq> cball x \<epsilon> \<and> 0 \<le> \<epsilon>)"
unfolding bounded_def subset_eq by auto (meson order_trans zero_le_dist)
-lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
+lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>\<epsilon>. \<forall>y\<in>S. dist a y \<le> \<epsilon>)"
unfolding bounded_def
by auto (metis add.commute add_le_cancel_right dist_commute dist_triangle_le)
@@ -806,14 +816,15 @@
lemma bounded_closure_image: "bounded (f ` closure S) \<Longrightarrow> bounded (f ` S)"
by (simp add: bounded_subset closure_subset image_mono)
-lemma bounded_cball[simp,intro]: "bounded (cball x e)"
+lemma bounded_cball[simp,intro]: "bounded (cball x \<epsilon>)"
unfolding bounded_def using mem_cball by blast
-lemma bounded_ball[simp,intro]: "bounded (ball x e)"
+lemma bounded_ball[simp,intro]: "bounded (ball x \<epsilon>)"
by (metis ball_subset_cball bounded_cball bounded_subset)
lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
- by (auto simp: bounded_def) (metis Un_iff bounded_any_center le_max_iff_disj)
+ unfolding bounded_def
+ by (metis Un_iff bounded_any_center order.trans linorder_linear)
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
@@ -822,14 +833,7 @@
by auto
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)
-qed
+ by (metis bounded_Un bounded_cball cball_trivial insert_is_Un)
lemma bounded_subset_ballI: "S \<subseteq> ball x r \<Longrightarrow> bounded S"
by (meson bounded_ball bounded_subset)
@@ -837,10 +841,10 @@
lemma bounded_subset_ballD:
assumes "bounded S" shows "\<exists>r. 0 < r \<and> S \<subseteq> ball x r"
proof -
- obtain e::real and y where "S \<subseteq> cball y e" "0 \<le> e"
+ obtain \<epsilon>::real and y where "S \<subseteq> cball y \<epsilon>" "0 \<le> \<epsilon>"
using assms by (auto simp: bounded_subset_cball)
then show ?thesis
- by (intro exI[where x="dist x y + e + 1"]) metric
+ by (intro exI[where x="dist x y + \<epsilon> + 1"]) metric
qed
lemma finite_imp_bounded [intro]: "finite S \<Longrightarrow> bounded S"
@@ -866,12 +870,12 @@
qed
lemma bounded_Times:
- assumes "bounded s" "bounded t"
- shows "bounded (s \<times> t)"
+ 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"
+ 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)"
+ then have "\<forall>z\<in>S \<times> T. dist (x, y) z \<le> sqrt (a\<^sup>2 + b\<^sup>2)"
by (auto simp: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)
then show ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
qed
@@ -931,26 +935,26 @@
\<close>
lemma compact_subset_open_imp_ball_epsilon_subset:
assumes "compact A" "open B" "A \<subseteq> B"
- obtains e where "e > 0" "(\<Union>x\<in>A. ball x e) \<subseteq> B"
+ obtains \<epsilon> where "\<epsilon> > 0" "(\<Union>x\<in>A. ball x \<epsilon>) \<subseteq> B"
proof -
- have "\<forall>x\<in>A. \<exists>e. e > 0 \<and> ball x e \<subseteq> B"
+ have "\<forall>x\<in>A. \<exists>\<epsilon>. \<epsilon> > 0 \<and> ball x \<epsilon> \<subseteq> B"
using assms unfolding open_contains_ball by blast
- then obtain e where e: "\<And>x. x \<in> A \<Longrightarrow> e x > 0" "\<And>x. x \<in> A \<Longrightarrow> ball x (e x) \<subseteq> B"
+ then obtain \<epsilon> where \<epsilon>: "\<And>x. x \<in> A \<Longrightarrow> \<epsilon> x > 0" "\<And>x. x \<in> A \<Longrightarrow> ball x (\<epsilon> x) \<subseteq> B"
by metis
- define C where "C = e ` A"
- obtain X where X: "X \<subseteq> A" "finite X" "A \<subseteq> (\<Union>c\<in>X. ball c (e c / 2))"
- using assms(1)
+ define C where "C = \<epsilon> ` A"
+ obtain X where X: "X \<subseteq> A" "finite X" "A \<subseteq> (\<Union>c\<in>X. ball c (\<epsilon> c / 2))"
+ using \<open>compact A\<close>
proof (rule compactE_image)
- show "open (ball x (e x / 2))" if "x \<in> A" for x
+ show "open (ball x (\<epsilon> x / 2))" if "x \<in> A" for x
by simp
- show "A \<subseteq> (\<Union>c\<in>A. ball c (e c / 2))"
- using e by auto
+ show "A \<subseteq> (\<Union>c\<in>A. ball c (\<epsilon> c / 2))"
+ using \<epsilon> by auto
qed auto
- define e' where "e' = Min (insert 1 ((\<lambda>x. e x / 2) ` X))"
+ define e' where "e' = Min (insert 1 ((\<lambda>x. \<epsilon> x / 2) ` X))"
have "e' > 0"
- unfolding e'_def using e X by (subst Min_gr_iff) auto
- have e': "e' \<le> e x / 2" if "x \<in> X" for x
+ unfolding e'_def using \<epsilon> X by (subst Min_gr_iff) auto
+ have e': "e' \<le> \<epsilon> x / 2" if "x \<in> X" for x
using that X unfolding e'_def by (intro Min.coboundedI) auto
show ?thesis
@@ -961,20 +965,20 @@
show "(\<Union>x\<in>A. ball x e') \<subseteq> B"
proof clarify
fix x y assume xy: "x \<in> A" "y \<in> ball x e'"
- from xy(1) X obtain z where z: "z \<in> X" "x \<in> ball z (e z / 2)"
+ from xy(1) X obtain z where z: "z \<in> X" "x \<in> ball z (\<epsilon> z / 2)"
by auto
have "dist y z \<le> dist x y + dist z x"
by (metis dist_commute dist_triangle)
- also have "dist z x < e z / 2"
+ also have "dist z x < \<epsilon> z / 2"
using xy z by auto
also have "dist x y < e'"
using xy by auto
- also have "\<dots> \<le> e z / 2"
+ also have "\<dots> \<le> \<epsilon> z / 2"
using z by (intro e') auto
- finally have "y \<in> ball z (e z)"
+ finally have "y \<in> ball z (\<epsilon> z)"
by (simp add: dist_commute)
also have "\<dots> \<subseteq> B"
- using z X by (intro e) auto
+ using z X by (intro \<epsilon>) auto
finally show "y \<in> B" .
qed
qed
@@ -982,42 +986,42 @@
lemma compact_subset_open_imp_cball_epsilon_subset:
assumes "compact A" "open B" "A \<subseteq> B"
- obtains e where "e > 0" "(\<Union>x\<in>A. cball x e) \<subseteq> B"
+ obtains \<epsilon> where "\<epsilon> > 0" "(\<Union>x\<in>A. cball x \<epsilon>) \<subseteq> B"
proof -
- obtain e where "e > 0" and e: "(\<Union>x\<in>A. ball x e) \<subseteq> B"
+ obtain \<epsilon> where "\<epsilon> > 0" and \<epsilon>: "(\<Union>x\<in>A. ball x \<epsilon>) \<subseteq> B"
using compact_subset_open_imp_ball_epsilon_subset [OF assms] by blast
- then have "(\<Union>x\<in>A. cball x (e / 2)) \<subseteq> (\<Union>x\<in>A. ball x e)"
+ then have "(\<Union>x\<in>A. cball x (\<epsilon> / 2)) \<subseteq> (\<Union>x\<in>A. ball x \<epsilon>)"
by auto
- with \<open>0 < e\<close> that show ?thesis
- by (metis e half_gt_zero_iff order_trans)
+ with \<open>0 < \<epsilon>\<close> that show ?thesis
+ by (metis \<epsilon> half_gt_zero_iff order_trans)
qed
subsubsection\<open>Totally bounded\<close>
proposition seq_compact_imp_totally_bounded:
assumes "seq_compact S"
- shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> S \<and> S \<subseteq> (\<Union>x\<in>k. ball x e)"
+ shows "\<forall>\<epsilon>>0. \<exists>k. finite k \<and> k \<subseteq> S \<and> S \<subseteq> (\<Union>x\<in>k. ball x \<epsilon>)"
proof -
- { fix e::real assume "e > 0" assume *: "\<And>k. finite k \<Longrightarrow> k \<subseteq> S \<Longrightarrow> \<not> S \<subseteq> (\<Union>x\<in>k. ball x e)"
- let ?Q = "\<lambda>x n r. r \<in> S \<and> (\<forall>m < (n::nat). \<not> (dist (x m) r < e))"
+ { fix \<epsilon>::real assume "\<epsilon> > 0" assume *: "\<And>k. finite k \<Longrightarrow> k \<subseteq> S \<Longrightarrow> \<not> S \<subseteq> (\<Union>x\<in>k. ball x \<epsilon>)"
+ let ?Q = "\<lambda>x n r. r \<in> S \<and> (\<forall>m < (n::nat). \<not> (dist (x m) r < \<epsilon>))"
have "\<exists>x. \<forall>n::nat. ?Q x n (x n)"
proof (rule dependent_wellorder_choice)
fix n x assume "\<And>y. y < n \<Longrightarrow> ?Q x y (x y)"
- then have "\<not> S \<subseteq> (\<Union>x\<in>x ` {0..<n}. ball x e)"
+ then have "\<not> S \<subseteq> (\<Union>x\<in>x ` {0..<n}. ball x \<epsilon>)"
using *[of "x ` {0 ..< n}"] by (auto simp: subset_eq)
- then obtain z where z:"z\<in>S" "z \<notin> (\<Union>x\<in>x ` {0..<n}. ball x e)"
+ then obtain z where z:"z\<in>S" "z \<notin> (\<Union>x\<in>x ` {0..<n}. ball x \<epsilon>)"
unfolding subset_eq by auto
show "\<exists>r. ?Q x n r"
using z by auto
qed simp
- then obtain x where "\<forall>n::nat. x n \<in> S" and x:"\<And>n m. m < n \<Longrightarrow> \<not> (dist (x m) (x n) < e)"
+ then obtain x where "\<forall>n::nat. x n \<in> S" and x:"\<And>n m. m < n \<Longrightarrow> \<not> (dist (x m) (x n) < \<epsilon>)"
by blast
- then obtain l r where "l \<in> S" and r:"strict_mono r" and "((x \<circ> r) \<longlongrightarrow> l) sequentially"
+ then obtain l r where "l \<in> S" and r:"strict_mono r" and "(x \<circ> r) \<longlonglongrightarrow> l"
using assms by (metis seq_compact_def)
then have "Cauchy (x \<circ> r)"
using LIMSEQ_imp_Cauchy by auto
- then obtain N::nat where "\<And>m n. N \<le> m \<Longrightarrow> N \<le> n \<Longrightarrow> dist ((x \<circ> r) m) ((x \<circ> r) n) < e"
- unfolding Cauchy_def using \<open>e > 0\<close> by blast
+ then obtain N::nat where "\<And>m n. N \<le> m \<Longrightarrow> N \<le> n \<Longrightarrow> dist ((x \<circ> r) m) ((x \<circ> r) n) < \<epsilon>"
+ unfolding Cauchy_def using \<open>\<epsilon> > 0\<close> by blast
then have False
using x[of "r N" "r (N+1)"] r by (auto simp: strict_mono_def) }
then show ?thesis
@@ -1031,39 +1035,38 @@
assumes "seq_compact S"
shows "compact S"
proof -
- from seq_compact_imp_totally_bounded[OF \<open>seq_compact S\<close>]
- obtain f where f: "\<forall>e>0. finite (f e) \<and> f e \<subseteq> S \<and> S \<subseteq> (\<Union>x\<in>f e. ball x e)"
- unfolding choice_iff' ..
- define K where "K = (\<lambda>(x, r). ball x r) ` ((\<Union>e \<in> \<rat> \<inter> {0 <..}. f e) \<times> \<rat>)"
+ obtain f where f: "\<forall>\<epsilon>>0. finite (f \<epsilon>) \<and> f \<epsilon> \<subseteq> S \<and> S \<subseteq> (\<Union>x\<in>f \<epsilon>. ball x \<epsilon>)"
+ by (metis assms seq_compact_imp_totally_bounded)
+ define K where "K = (\<lambda>(x, r). ball x r) ` ((\<Union>\<epsilon> \<in> \<rat> \<inter> {0 <..}. f \<epsilon>) \<times> \<rat>)"
have "countably_compact S"
using \<open>seq_compact S\<close> by (rule seq_compact_imp_countably_compact)
then show "compact S"
proof (rule countably_compact_imp_compact)
show "countable K"
unfolding K_def using f
- by (auto intro: countable_finite countable_subset countable_rat
- intro!: countable_image countable_SIGMA countable_UN)
+ by (meson countable_Int1 countable_SIGMA countable_UN countable_finite
+ countable_image countable_rat greaterThan_iff inf_le2 subset_iff)
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"
+ from openE[OF T] obtain \<epsilon> where "0 < \<epsilon>" "ball x \<epsilon> \<subseteq> T"
by auto
- then have "0 < e/2" "ball x (e/2) \<subseteq> T"
+ then have "0 < \<epsilon>/2" "ball x (\<epsilon>/2) \<subseteq> T"
by auto
- from Rats_dense_in_real[OF \<open>0 < e/2\<close>] obtain r where "r \<in> \<rat>" "0 < r" "r < e/2"
+ from Rats_dense_in_real[OF \<open>0 < \<epsilon>/2\<close>] obtain r where "r \<in> \<rat>" "0 < r" "r < \<epsilon>/2"
by auto
from f[rule_format, of r] \<open>0 < r\<close> \<open>x \<in> S\<close> obtain k where "k \<in> f r" "x \<in> ball k r"
by auto
from \<open>r \<in> \<rat>\<close> \<open>0 < r\<close> \<open>k \<in> f r\<close> 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)
+ proof (rule rev_bexI, intro conjI subsetI)
fix y
- assume "y \<in> ball k r"
- with \<open>r < e/2\<close> \<open>x \<in> ball k r\<close> have "dist x y < e"
- by (intro dist_triangle_half_r [of k _ e]) (auto simp: dist_commute)
- with \<open>ball x e \<subseteq> T\<close> show "y \<in> T"
+ assume "y \<in> ball k r \<inter> S"
+ with \<open>r < \<epsilon>/2\<close> \<open>x \<in> ball k r\<close> have "dist x y < \<epsilon>"
+ by (intro dist_triangle_half_r [of k _ \<epsilon>]) (auto simp: dist_commute)
+ with \<open>ball x \<epsilon> \<subseteq> T\<close> show "y \<in> T"
by auto
next
show "x \<in> ball k r" by fact
@@ -1095,37 +1098,34 @@
section \<open>Banach fixed point theorem\<close>
theorem Banach_fix:
- assumes s: "complete s" "s \<noteq> {}"
+ 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"
+ and f: "f ` S \<subseteq> S"
+ and lipschitz: "\<And>x y. \<lbrakk>x\<in>S; y\<in>S\<rbrakk> \<Longrightarrow> dist (f x) (f y) \<le> c * dist x y"
+ shows "\<exists>!x\<in>S. f x = x"
proof -
- from c have "1 - c > 0" by simp
-
- from s(2) obtain z0 where z0: "z0 \<in> s" by blast
+ from S obtain z0 where z0: "z0 \<in> S" by blast
define z where "z n = (f ^^ n) z0" for n
- with f z0 have z_in_s: "z n \<in> s" for n :: nat
+ with f z0 have z_in_s: "z n \<in> S" for n :: nat
by (induct n) auto
- define d where "d = dist (z 0) (z 1)"
+ define \<delta> where "\<delta> = dist (z 0) (z 1)"
have fzn: "f (z n) = z (Suc n)" for n
by (simp add: z_def)
- have cf_z: "dist (z n) (z (Suc n)) \<le> (c ^ n) * d" for n :: nat
+ have cf_z: "dist (z n) (z (Suc n)) \<le> (c ^ n) * \<delta>" for n :: nat
proof (induct n)
case 0
then show ?case
- by (simp add: d_def)
+ by (simp add: \<delta>_def)
next
case (Suc m)
- with \<open>0 \<le> c\<close> have "c * dist (z m) (z (Suc m)) \<le> c ^ Suc m * d"
- using mult_left_mono[of "dist (z m) (z (Suc m))" "c ^ m * d" c] by simp
+ with \<open>0 \<le> c\<close> have "c * dist (z m) (z (Suc m)) \<le> c ^ Suc m * \<delta>"
+ using mult_left_mono[of "dist (z m) (z (Suc m))" "c ^ m * \<delta>" c] by simp
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]
- by (simp add: fzn mult_le_cancel_left)
+ by (metis fzn lipschitz order_trans z_in_s)
qed
- have cf_z2: "(1 - c) * dist (z m) (z (m + n)) \<le> (c ^ m) * d * (1 - c ^ n)" for n m :: nat
+ have cf_z2: "(1 - c) * dist (z m) (z (m + n)) \<le> (c ^ m) * \<delta> * (1 - c ^ n)" for n m :: nat
proof (induct n)
case 0
show ?case by simp
@@ -1134,100 +1134,97 @@
from c have "(1 - c) * dist (z m) (z (m + Suc k)) \<le>
(1 - c) * (dist (z m) (z (m + k)) + dist (z (m + k)) (z (Suc (m + k))))"
by (simp add: dist_triangle)
- also from c cf_z[of "m + k"] have "\<dots> \<le> (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)"
+ also from c cf_z[of "m + k"] have "\<dots> \<le> (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * \<delta>)"
by simp
- also from Suc have "\<dots> \<le> c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d"
+ also from Suc have "\<dots> \<le> c ^ m * \<delta> * (1 - c ^ k) + (1 - c) * c ^ (m + k) * \<delta>"
by (simp add: field_simps)
- also have "\<dots> = (c ^ m) * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)"
+ also have "\<dots> = (c ^ m) * (\<delta> * (1 - c ^ k) + (1 - c) * c ^ k * \<delta>)"
by (simp add: power_add field_simps)
- also from c have "\<dots> \<le> (c ^ m) * d * (1 - c ^ Suc k)"
+ also from c have "\<dots> \<le> (c ^ m) * \<delta> * (1 - c ^ Suc k)"
by (simp add: field_simps)
- finally show ?case by simp
+ finally show ?case .
qed
- have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (z m) (z n) < e" if "e > 0" for e
- proof (cases "d = 0")
+ have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (z m) (z n) < \<epsilon>" if "\<epsilon> > 0" for \<epsilon>
+ proof (cases "\<delta> = 0")
case True
- from \<open>1 - c > 0\<close> have "(1 - c) * x \<le> 0 \<longleftrightarrow> x \<le> 0" for x
- by (simp add: mult_le_0_iff)
+ have "(1 - c) * x \<le> 0 \<longleftrightarrow> x \<le> 0" for x
+ using c mult_le_0_iff nle_le by fastforce
with c cf_z2[of 0] True have "z n = z0" for n
by (simp add: z_def)
- with \<open>e > 0\<close> show ?thesis by simp
+ with \<open>\<epsilon> > 0\<close> show ?thesis by simp
next
case False
- with zero_le_dist[of "z 0" "z 1"] have "d > 0"
- by (metis d_def less_le)
- with \<open>1 - c > 0\<close> \<open>e > 0\<close> have "0 < e * (1 - c) / d"
+ with zero_le_dist[of "z 0" "z 1"] have "\<delta> > 0"
+ by (metis \<delta>_def less_le)
+ with c \<open>\<epsilon> > 0\<close> have "0 < \<epsilon> * (1 - c) / \<delta>"
by simp
- with c obtain N where N: "c ^ N < e * (1 - c) / d"
- using real_arch_pow_inv[of "e * (1 - c) / d" c] by auto
- have *: "dist (z m) (z n) < e" if "m > n" and as: "m \<ge> N" "n \<ge> N" for m n :: nat
+ with c obtain N where N: "c ^ N < \<epsilon> * (1 - c) / \<delta>"
+ using real_arch_pow_inv[of "\<epsilon> * (1 - c) / \<delta>" c] by auto
+ have *: "dist (z m) (z n) < \<epsilon>" if "m > n" and as: "m \<ge> N" "n \<ge> N" for m n :: nat
proof -
- from c \<open>n \<ge> N\<close> have *: "c ^ n \<le> c ^ N"
- using power_decreasing[OF \<open>n\<ge>N\<close>, of c] by simp
- from c \<open>m > n\<close> have "1 - c ^ (m - n) > 0"
- using power_strict_mono[of c 1 "m - n"] by simp
- with \<open>d > 0\<close> \<open>0 < 1 - c\<close> have **: "d * (1 - c ^ (m - n)) / (1 - c) > 0"
+ have *: "c ^ n \<le> c ^ N"
+ using power_decreasing[OF \<open>n\<ge>N\<close>, of c] c by simp
+ have "1 - c ^ (m - n) > 0"
+ using power_strict_mono[of c 1 "m - n"] c \<open>m > n\<close> by simp
+ with \<open>\<delta> > 0\<close> c have **: "\<delta> * (1 - c ^ (m - n)) / (1 - c) > 0"
by simp
from cf_z2[of n "m - n"] \<open>m > n\<close>
- have "dist (z m) (z n) \<le> c ^ n * d * (1 - c ^ (m - n)) / (1 - c)"
- by (simp add: pos_le_divide_eq[OF \<open>1 - c > 0\<close>] mult.commute dist_commute)
- also have "\<dots> \<le> c ^ N * d * (1 - c ^ (m - n)) / (1 - c)"
+ have "dist (z m) (z n) \<le> c ^ n * \<delta> * (1 - c ^ (m - n)) / (1 - c)"
+ by (simp add: pos_le_divide_eq c mult.commute dist_commute)
+ also have "\<dots> \<le> c ^ N * \<delta> * (1 - c ^ (m - n)) / (1 - c)"
using mult_right_mono[OF * order_less_imp_le[OF **]]
by (simp add: mult.assoc)
- also have "\<dots> < (e * (1 - c) / d) * d * (1 - c ^ (m - n)) / (1 - c)"
+ also have "\<dots> < (\<epsilon> * (1 - c) / \<delta>) * \<delta> * (1 - c ^ (m - n)) / (1 - c)"
using mult_strict_right_mono[OF N **] by (auto simp: mult.assoc)
- also from c \<open>1 - c ^ (m - n) > 0\<close> \<open>e > 0\<close> have "\<dots> \<le> e"
- using mult_right_le_one_le[of e "1 - c ^ (m - n)"] by auto
- finally show ?thesis by simp
+ also from c \<open>1 - c ^ (m - n) > 0\<close> \<open>\<epsilon> > 0\<close> have "\<dots> \<le> \<epsilon>"
+ using mult_right_le_one_le[of \<epsilon> "1 - c ^ (m - n)"] by auto
+ finally show ?thesis .
qed
- have "dist (z n) (z m) < e" if "N \<le> m" "N \<le> n" for m n :: nat
- using *[of n m] *[of m n]
- using \<open>0 < e\<close> dist_commute_lessI that by fastforce
+ have "dist (z n) (z m) < \<epsilon>" if "N \<le> m" "N \<le> n" for m n :: nat
+ using *[of n m] *[of m n]
+ using \<open>0 < \<epsilon>\<close> dist_commute_lessI that by fastforce
then show ?thesis by auto
qed
then have "Cauchy z"
by (metis metric_CauchyI)
- then obtain x where "x\<in>s" and x:"(z \<longlongrightarrow> x) sequentially"
- using s(1)[unfolded compact_def complete_def, THEN spec[where x=z]] and z_in_s by auto
-
- define e where "e = dist (f x) x"
- have "e = 0"
+ then obtain x where "x\<in>S" and x:"(z \<longlongrightarrow> x) sequentially"
+ using \<open>complete S\<close> complete_def z_in_s by blast
+
+ define \<epsilon> where "\<epsilon> = dist (f x) x"
+ have "\<epsilon> = 0"
proof (rule ccontr)
- assume "e \<noteq> 0"
- 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 lim_sequentially, THEN spec[where x="e/2"]] by auto
- then have N':"dist (z N) x < e/2" by auto
+ assume "\<epsilon> \<noteq> 0"
+ then have "\<epsilon> > 0"
+ unfolding \<epsilon>_def using zero_le_dist[of "f x" x]
+ by (metis dist_eq_0_iff dist_nz \<epsilon>_def)
+ then obtain N where N:"\<forall>n\<ge>N. dist (z n) x < \<epsilon>/2"
+ using x[unfolded lim_sequentially, THEN spec[where x="\<epsilon>/2"]] by auto
+ then have N':"dist (z N) x < \<epsilon>/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)
have "dist (f (z N)) (f x) \<le> c * dist (z N) x"
- using lipschitz[THEN bspec[where x="z N"], THEN bspec[where x=x]]
- using z_in_s[of N] \<open>x\<in>s\<close> c
- by auto
- also have "\<dots> < e/2"
+ by (simp add: \<open>x \<in> S\<close> lipschitz z_in_s)
+ also have "\<dots> < \<epsilon>/2"
using N' and c using * by auto
finally show False
unfolding fzn
- using N[THEN spec[where x="Suc N"]] and dist_triangle_half_r[of "z (Suc N)" "f x" e x]
- unfolding e_def
- by auto
+ by (metis N \<epsilon>_def dist_commute dist_triangle_half_l le_eq_less_or_eq lessI
+ order_less_irrefl)
qed
- then have "f x = x" by (auto simp: e_def)
- moreover have "y = x" if "f y = y" "y \<in> s" for y
+ then have "f x = x" by (auto simp: \<epsilon>_def)
+ moreover have "y = x" if "f y = y" "y \<in> S" for y
proof -
- from \<open>x \<in> s\<close> \<open>f x = x\<close> that have "dist x y \<le> c * dist x y"
+ from \<open>x \<in> S\<close> \<open>f x = x\<close> that have "dist x y \<le> c * dist x y"
using lipschitz by fastforce
with c and zero_le_dist[of x y] have "dist x y = 0"
by (simp add: mult_le_cancel_right1)
then show ?thesis by simp
qed
ultimately show ?thesis
- using \<open>x\<in>s\<close> by blast
+ using \<open>x\<in>S\<close> by blast
qed
@@ -1237,7 +1234,7 @@
fixes S :: "'a::metric_space set"
assumes S: "compact S" "S \<noteq> {}"
and gs: "(g ` S) \<subseteq> S"
- and dist: "\<forall>x\<in>S. \<forall>y\<in>S. x \<noteq> y \<longrightarrow> dist (g x) (g y) < dist x y"
+ and dist: "\<And>x y. \<lbrakk>x\<in>S; y\<in>S\<rbrakk> \<Longrightarrow> x \<noteq> y \<longrightarrow> dist (g x) (g y) < dist x y"
shows "\<exists>!x\<in>S. g x = x"
proof -
let ?D = "(\<lambda>x. (x, x)) ` S"
@@ -1245,7 +1242,7 @@
by (rule compact_continuous_image)
(auto intro!: S continuous_Pair continuous_ident simp: continuous_on_eq_continuous_within)
- have "\<And>x y e. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> 0 < e \<Longrightarrow> dist y x < e \<Longrightarrow> dist (g y) (g x) < e"
+ have "\<And>x y \<epsilon>. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> 0 < \<epsilon> \<Longrightarrow> dist y x < \<epsilon> \<Longrightarrow> dist (g y) (g x) < \<epsilon>"
using dist by fastforce
then have "continuous_on S g"
by (auto simp: continuous_on_iff)
@@ -1260,7 +1257,7 @@
have "g a = a"
by (metis \<open>a \<in> S\<close> dist gs image_subset_iff le order.strict_iff_not)
moreover have "\<And>x. x \<in> S \<Longrightarrow> g x = x \<Longrightarrow> x = a"
- using dist[THEN bspec[where x=a]] \<open>g a = a\<close> and \<open>a\<in>S\<close> by auto
+ using \<open>a \<in> S\<close> calculation dist by fastforce
ultimately show "\<exists>!x\<in>S. g x = x"
using \<open>a \<in> S\<close> by blast
qed
@@ -1277,9 +1274,9 @@
by (auto simp: diameter_def)
lemma diameter_le:
- assumes "S \<noteq> {} \<or> 0 \<le> d"
- and no: "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> norm(x - y) \<le> d"
- shows "diameter S \<le> d"
+ assumes "S \<noteq> {} \<or> 0 \<le> \<delta>"
+ and no: "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> norm(x - y) \<le> \<delta>"
+ shows "diameter S \<le> \<delta>"
using assms
by (auto simp: dist_norm diameter_def intro: cSUP_least)
@@ -1288,14 +1285,14 @@
assumes S: "bounded S" "x \<in> S" "y \<in> S"
shows "dist x y \<le> diameter S"
proof -
- from S obtain z d where z: "\<And>x. x \<in> S \<Longrightarrow> dist z x \<le> d"
+ from S obtain z \<delta> where z: "\<And>x. x \<in> S \<Longrightarrow> dist z x \<le> \<delta>"
unfolding bounded_def by auto
have "bdd_above (case_prod dist ` (S\<times>S))"
proof (intro bdd_aboveI, safe)
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"
+ show "dist a b \<le> 2 * \<delta>"
by (simp add: dist_commute)
qed
moreover have "(x,y) \<in> S\<times>S" using S by auto
@@ -1308,25 +1305,25 @@
lemma diameter_lower_bounded:
fixes S :: "'a :: metric_space set"
assumes S: "bounded S"
- and d: "0 < d" "d < diameter S"
- shows "\<exists>x\<in>S. \<exists>y\<in>S. d < dist x y"
+ and \<delta>: "0 < \<delta>" "\<delta> < diameter S"
+ shows "\<exists>x\<in>S. \<exists>y\<in>S. \<delta> < dist x y"
proof (rule ccontr)
assume contr: "\<not> ?thesis"
moreover have "S \<noteq> {}"
- using d by (auto simp: diameter_def)
- ultimately have "diameter S \<le> d"
+ using \<delta> by (auto simp: diameter_def)
+ ultimately have "diameter S \<le> \<delta>"
by (auto simp: not_less diameter_def intro!: cSUP_least)
- with \<open>d < diameter S\<close> show False by auto
+ with \<open>\<delta> < diameter S\<close> show False by auto
qed
lemma diameter_bounded:
assumes "bounded S"
shows "\<forall>x\<in>S. \<forall>y\<in>S. dist x y \<le> diameter S"
- and "\<forall>d>0. d < diameter S \<longrightarrow> (\<exists>x\<in>S. \<exists>y\<in>S. dist x y > d)"
+ and "\<forall>\<delta>>0. \<delta> < diameter S \<longrightarrow> (\<exists>x\<in>S. \<exists>y\<in>S. dist x y > \<delta>)"
using diameter_bounded_bound[of S] diameter_lower_bounded[of S] assms
by auto
-lemma bounded_two_points: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>x\<in>S. \<forall>y\<in>S. dist x y \<le> e)"
+lemma bounded_two_points: "bounded S \<longleftrightarrow> (\<exists>\<epsilon>. \<forall>x\<in>S. \<forall>y\<in>S. dist x y \<le> \<epsilon>)"
by (meson bounded_def diameter_bounded(1))
lemma diameter_compact_attained:
@@ -1347,7 +1344,7 @@
lemma diameter_ge_0:
assumes "bounded S" shows "0 \<le> diameter S"
- by (metis all_not_in_conv assms diameter_bounded_bound diameter_empty dist_self order_refl)
+ by (metis assms diameter_bounded(1) diameter_empty dist_self equals0I order.refl)
lemma diameter_subset:
assumes "S \<subseteq> T" "bounded T"
@@ -1361,38 +1358,43 @@
then have "bdd_above ((\<lambda>x. case x of (x, xa) \<Rightarrow> dist x xa) ` (T \<times> T))"
using \<open>bounded T\<close> diameter_bounded_bound by (force simp: bdd_above_def)
with False \<open>S \<subseteq> T\<close> show ?thesis
- apply (simp add: diameter_def)
- apply (rule cSUP_subset_mono, auto)
- done
+ by (simp add: diameter_def cSUP_subset_mono times_subset_iff)
qed
lemma diameter_closure:
assumes "bounded S"
shows "diameter(closure S) = diameter S"
proof (rule order_antisym)
- have "False" if d_less_d: "diameter S < diameter (closure S)"
+ have False if d_less_d: "diameter S < diameter (closure S)"
proof -
- define d where "d = diameter(closure S) - diameter(S)"
- have "d > 0"
- using that by (simp add: d_def)
- then have dle: "diameter(closure(S)) - d / 2 < diameter(closure(S))"
+ define \<delta> where "\<delta> = diameter(closure S) - diameter(S)"
+ have "\<delta> > 0"
+ using that by (simp add: \<delta>_def)
+ then have dle: "diameter(closure(S)) - \<delta> / 2 < diameter(closure(S))"
by simp
- have dd: "diameter (closure S) - d / 2 = (diameter(closure(S)) + diameter(S)) / 2"
- by (simp add: d_def field_split_simps)
+ have dd: "diameter (closure S) - \<delta> / 2 = (diameter(closure(S)) + diameter(S)) / 2"
+ by (simp add: \<delta>_def field_split_simps)
have bocl: "bounded (closure S)"
using assms by blast
- moreover have "0 \<le> diameter S"
- using assms diameter_ge_0 by blast
- ultimately obtain x y where "x \<in> closure S" "y \<in> closure S" and xy: "diameter(closure(S)) - d / 2 < dist x y"
- by (smt (verit) dle d_less_d d_def dd diameter_lower_bounded)
- then obtain x' y' where x'y': "x' \<in> S" "dist x' x < d/4" "y' \<in> S" "dist y' y < d/4"
- by (metis \<open>0 < d\<close> zero_less_divide_iff zero_less_numeral closure_approachable)
+ moreover
+ have "diameter S \<noteq> 0"
+ using diameter_bounded_bound [OF assms]
+ by (metis closure_closed discrete_imp_closed dist_le_zero_iff not_less_iff_gr_or_eq
+ that)
+ then have "0 < diameter S"
+ using assms diameter_ge_0 by fastforce
+ ultimately obtain x y where "x \<in> closure S" "y \<in> closure S" and xy: "diameter(closure(S)) - \<delta> / 2 < dist x y"
+ using diameter_lower_bounded [OF bocl, of "diameter S"]
+ by (metis d_less_d diameter_bounded(2) dist_not_less_zero dist_self dle
+ not_less_iff_gr_or_eq)
+ then obtain x' y' where x'y': "x' \<in> S" "dist x' x < \<delta>/4" "y' \<in> S" "dist y' y < \<delta>/4"
+ by (metis \<open>0 < \<delta>\<close> zero_less_divide_iff zero_less_numeral closure_approachable)
then have "dist x' y' \<le> diameter S"
using assms diameter_bounded_bound by blast
- with x'y' have "dist x y \<le> d / 4 + diameter S + d / 4"
+ with x'y' have "dist x y \<le> \<delta> / 4 + diameter S + \<delta> / 4"
by (meson add_mono dist_triangle dist_triangle3 less_eq_real_def order_trans)
then show ?thesis
- using xy d_def by linarith
+ using xy \<delta>_def by linarith
qed
then show "diameter (closure S) \<le> diameter S"
by fastforce
@@ -1410,14 +1412,9 @@
by (metis \<open>\<C> \<noteq> {}\<close> zero_less_one empty_subsetI equals0I subset_trans that)
next
case False
- { fix x assume "x \<in> S"
- then obtain C where C: "x \<in> C" "C \<in> \<C>"
- using \<open>S \<subseteq> \<Union>\<C>\<close> by blast
- then obtain r where r: "r>0" "ball x (2*r) \<subseteq> C"
- by (metis field_sum_of_halves half_gt_zero mult.commute mult_2_right ope open_contains_ball)
- then have "\<exists>r C. r > 0 \<and> ball x (2*r) \<subseteq> C \<and> C \<in> \<C>"
- using C by blast
- }
+ have "\<exists>r C. r > 0 \<and> ball x (2*r) \<subseteq> C \<and> C \<in> \<C>" if "x \<in> S" for x
+ by (metis \<open>S \<subseteq> \<Union>\<C>\<close> field_sum_of_halves half_gt_zero mult.commute mult_2_right
+ UnionE ope open_contains_ball subset_eq that)
then obtain r where r: "\<And>x. x \<in> S \<Longrightarrow> r x > 0 \<and> (\<exists>C \<in> \<C>. ball x (2*r x) \<subseteq> C)"
by metis
then have "S \<subseteq> (\<Union>x \<in> S. ball x (r x))"
@@ -1437,10 +1434,6 @@
by (simp add: \<open>0 < \<delta>\<close>)
show "\<exists>B \<in> \<C>. T \<subseteq> B" if "T \<subseteq> S" and dia: "diameter T < \<delta>" for T
proof (cases "T = {}")
- case True
- then show ?thesis
- using \<open>\<C> \<noteq> {}\<close> by blast
- next
case False
then obtain y where "y \<in> T" by blast
then have "y \<in> S"
@@ -1449,7 +1442,7 @@
using \<open>S \<subseteq> \<Union>\<T>\<close> S0 that by blast
have "ball y \<delta> \<subseteq> ball y (r x)"
by (metis \<delta>_def \<open>S0 \<noteq> {}\<close> \<open>finite S0\<close> \<open>x \<in> S0\<close> empty_is_image finite_imageI finite_less_Inf_iff imageI less_irrefl not_le subset_ball)
- also have "... \<subseteq> ball x (2*r x)"
+ also have "\<dots> \<subseteq> ball x (2*r x)"
using x by metric
finally obtain C where "C \<in> \<C>" "ball y \<delta> \<subseteq> C"
by (meson r \<open>S0 \<subseteq> S\<close> \<open>x \<in> S0\<close> dual_order.trans subsetCE)
@@ -1459,7 +1452,7 @@
using \<open>y \<in> T\<close> dia diameter_bounded_bound by fastforce
then show ?thesis
by (meson \<open>C \<in> \<C>\<close> \<open>ball y \<delta> \<subseteq> C\<close> subset_eq)
- qed
+ qed (use \<open>\<C> \<noteq> {}\<close> in auto)
qed
qed
@@ -1536,30 +1529,30 @@
fixes unproj:: "('b \<Rightarrow> 'c) \<Rightarrow> 'a"
assumes finite_basis: "finite basis"
assumes bounded_proj: "\<And>k. k \<in> basis \<Longrightarrow> bounded ((\<lambda>x. x proj k) ` range f)"
- assumes proj_unproj: "\<And>e k. k \<in> basis \<Longrightarrow> (unproj e) proj k = e k"
+ assumes proj_unproj: "\<And>\<epsilon> k. k \<in> basis \<Longrightarrow> (unproj \<epsilon>) proj k = \<epsilon> k"
assumes unproj_proj: "\<And>x. unproj (\<lambda>k. x proj k) = x"
- shows "\<forall>d\<subseteq>basis. \<exists>l::'a. \<exists> r::nat\<Rightarrow>nat.
- strict_mono r \<and> (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) proj i) (l proj i) < e) sequentially)"
+ shows "\<forall>\<delta>\<subseteq>basis. \<exists>l::'a. \<exists> r::nat\<Rightarrow>nat.
+ strict_mono r \<and> (\<forall>\<epsilon>>0. eventually (\<lambda>n. \<forall>i\<in>\<delta>. dist (f (r n) proj i) (l proj i) < \<epsilon>) sequentially)"
proof safe
- fix d :: "'b set"
- assume d: "d \<subseteq> basis"
- with finite_basis have "finite d"
+ fix \<delta> :: "'b set"
+ assume \<delta>: "\<delta> \<subseteq> basis"
+ with finite_basis have "finite \<delta>"
by (blast intro: finite_subset)
- from this d show "\<exists>l::'a. \<exists>r::nat\<Rightarrow>nat. strict_mono r \<and>
- (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) proj i) (l proj i) < e) sequentially)"
- proof (induct d)
+ from this \<delta> show "\<exists>l::'a. \<exists>r::nat\<Rightarrow>nat. strict_mono r \<and>
+ (\<forall>\<epsilon>>0. eventually (\<lambda>n. \<forall>i\<in>\<delta>. dist (f (r n) proj i) (l proj i) < \<epsilon>) sequentially)"
+ proof (induct \<delta>)
case empty
then show ?case
unfolding strict_mono_def by auto
next
- case (insert k d)
+ case (insert k \<delta>)
have k[intro]: "k \<in> basis"
using insert by auto
have s': "bounded ((\<lambda>x. x proj k) ` range f)"
using k
by (rule bounded_proj)
obtain l1::"'a" and r1 where r1: "strict_mono r1"
- and lr1: "\<forall>e > 0. \<forall>\<^sub>F n in sequentially. \<forall>i\<in>d. dist (f (r1 n) proj i) (l1 proj i) < e"
+ and lr1: "\<forall>\<epsilon> > 0. \<forall>\<^sub>F n in sequentially. \<forall>i\<in>\<delta>. dist (f (r1 n) proj i) (l1 proj i) < \<epsilon>"
using insert by auto
have f': "\<forall>n. f (r1 n) proj k \<in> (\<lambda>x. x proj k) ` range f"
by simp
@@ -1573,15 +1566,15 @@
using r1 and r2 unfolding r_def o_def strict_mono_def by auto
moreover
define l where "l = unproj (\<lambda>i. if i = k then l2 else l1 proj i)"
- { fix e::real
- assume "e > 0"
- from lr1 \<open>e > 0\<close> have N1: "\<forall>\<^sub>F n in sequentially. \<forall>i\<in>d. dist (f (r1 n) proj i) (l1 proj i) < e"
+ { fix \<epsilon>::real
+ assume "\<epsilon> > 0"
+ from lr1 \<open>\<epsilon> > 0\<close> have N1: "\<forall>\<^sub>F n in sequentially. \<forall>i\<in>\<delta>. dist (f (r1 n) proj i) (l1 proj i) < \<epsilon>"
by blast
- from lr2 \<open>e > 0\<close> have N2: "\<forall>\<^sub>F n in sequentially. dist (f (r1 (r2 n)) proj k) l2 < e"
+ from lr2 \<open>\<epsilon> > 0\<close> have N2: "\<forall>\<^sub>F n in sequentially. dist (f (r1 (r2 n)) proj k) l2 < \<epsilon>"
by (rule tendstoD)
- from r2 N1 have N1': "\<forall>\<^sub>F n in sequentially. \<forall>i\<in>d. dist (f (r1 (r2 n)) proj i) (l1 proj i) < e"
+ from r2 N1 have N1': "\<forall>\<^sub>F n in sequentially. \<forall>i\<in>\<delta>. dist (f (r1 (r2 n)) proj i) (l1 proj i) < \<epsilon>"
by (rule eventually_subseq)
- have "\<forall>\<^sub>F n in sequentially. \<forall>i\<in>insert k d. dist (f (r n) proj i) (l proj i) < e"
+ have "\<forall>\<^sub>F n in sequentially. \<forall>i\<in>insert k \<delta>. dist (f (r n) proj i) (l proj i) < \<epsilon>"
using N1' N2
by eventually_elim (use insert.prems in \<open>auto simp: l_def r_def o_def proj_unproj\<close>)
}
@@ -1649,37 +1642,37 @@
note lr' = seq_suble [OF lr(2)]
{
- 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 \<open>e > 0\<close> by (meson half_gt_zero)
- then obtain M where M:"\<forall>n\<ge>M. dist ((f \<circ> r) n) l < e/2"
+ fix \<epsilon> :: real
+ assume "\<epsilon> > 0"
+ from as(2) obtain N where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (f m) (f n) < \<epsilon>/2"
+ unfolding Cauchy_def using \<open>\<epsilon> > 0\<close> by (meson half_gt_zero)
+ then obtain M where M:"\<forall>n\<ge>M. dist ((f \<circ> r) n) l < \<epsilon>/2"
by (metis dist_self lim_sequentially lr(3))
{
fix n :: nat
assume n: "n \<ge> max N M"
- have "dist ((f \<circ> r) n) l < e/2"
+ have "dist ((f \<circ> r) n) l < \<epsilon>/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"
+ then have "dist (f n) ((f \<circ> r) n) < \<epsilon>/2"
using N and n by auto
- ultimately have "dist (f n) l < e" using n M
+ ultimately have "dist (f n) l < \<epsilon>" using n M
by metric
}
- then have "\<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 < \<epsilon>" by blast
}
then show "\<exists>l\<in>S. (f \<longlongrightarrow> l) sequentially" using \<open>l\<in>S\<close>
unfolding lim_sequentially by auto
qed
proposition compact_eq_totally_bounded:
- "compact S \<longleftrightarrow> complete S \<and> (\<forall>e>0. \<exists>k. finite k \<and> S \<subseteq> (\<Union>x\<in>k. ball x e))"
+ "compact S \<longleftrightarrow> complete S \<and> (\<forall>\<epsilon>>0. \<exists>k. finite k \<and> S \<subseteq> (\<Union>x\<in>k. ball x \<epsilon>))"
(is "_ \<longleftrightarrow> ?rhs")
proof
assume assms: "?rhs"
- then obtain k where k: "\<And>e. 0 < e \<Longrightarrow> finite (k e)" "\<And>e. 0 < e \<Longrightarrow> S \<subseteq> (\<Union>x\<in>k e. ball x e)"
- by (auto simp: choice_iff')
+ then obtain k where k: "\<And>\<epsilon>. 0 < \<epsilon> \<Longrightarrow> finite (k \<epsilon>)" "\<And>\<epsilon>. 0 < \<epsilon> \<Longrightarrow> S \<subseteq> (\<Union>x\<in>k \<epsilon>. ball x \<epsilon>)"
+ by metis
show "compact S"
proof cases
@@ -1693,21 +1686,21 @@
fix f :: "nat \<Rightarrow> 'a"
assume f: "\<forall>n. f n \<in> S"
- define e where "e n = 1 / (2 * Suc n)" for n
- then have [simp]: "\<And>n. 0 < e n" by auto
- define B where "B n U = (SOME b. infinite {n. f n \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U))" for n U
+ define \<epsilon> where "\<epsilon> n = 1 / (2 * Suc n)" for n
+ then have [simp]: "\<And>n. 0 < \<epsilon> n" by auto
+ define B where "B n U = (SOME b. infinite {n. f n \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (\<epsilon> n) \<inter> U))" for n 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)}"
+ then have "\<exists>b\<in>k (\<epsilon> n). infinite {i\<in>{n. f n \<in> U}. f i \<in> ball b (\<epsilon> n)}"
using k f by (intro pigeonhole_infinite_rel) (auto simp: subset_eq)
then obtain a where
- "a \<in> k (e n)"
- "infinite {i \<in> {n. f n \<in> U}. f i \<in> ball a (e n)}" ..
- then have "\<exists>b. infinite {i. f i \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U)"
- by (intro exI[of _ "ball a (e n) \<inter> U"] exI[of _ a]) (auto simp: ac_simps)
+ "a \<in> k (\<epsilon> n)"
+ "infinite {i \<in> {n. f n \<in> U}. f i \<in> ball a (\<epsilon> n)}" ..
+ then have "\<exists>b. infinite {i. f i \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (\<epsilon> n) \<inter> U)"
+ by (intro exI[of _ "ball a (\<epsilon> 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"
+ have "infinite {i. f i \<in> B n U}" "\<exists>x. B n U \<subseteq> ball x (\<epsilon> n) \<inter> U"
unfolding B_def by auto
}
note B = this
@@ -1718,20 +1711,20 @@
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"
+ then have F: "\<And>n. \<exists>x. F (Suc n) \<subseteq> ball x (\<epsilon> 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"
using decseq_SucI[of F] by (auto simp: decseq_def)
-
- obtain sel where sel: "\<And>k i. i < sel k i" "\<And>k i. f (sel k i) \<in> F k"
- proof (atomize_elim, unfold all_conj_distrib[symmetric], intro choice allI)
- fix k i
+ have "\<exists>x>i. f x \<in> F k" for k i
+ proof -
have "infinite ({n. f n \<in> F k} - {.. i})"
using \<open>infinite {n. f n \<in> F k}\<close> by auto
from infinite_imp_nonempty[OF this]
show "\<exists>x>i. f x \<in> F k"
by (simp add: set_eq_iff not_le conj_commute)
qed
+ then obtain sel where sel: "\<And>k i. i < sel k i" "\<And>k i. f (sel k i) \<in> F k"
+ by meson
define t where "t = rec_nat (sel 0 0) (\<lambda>n i. sel (Suc n) i)"
have "strict_mono t"
@@ -1746,11 +1739,11 @@
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"
- 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)
- with F[of N] obtain x where "dist x ((f \<circ> t) n) < e N" "dist x ((f \<circ> t) m) < e N"
+ then have "(f \<circ> t) n \<in> F (Suc N)" "(f \<circ> t) m \<in> F (Suc N)" "2 * \<epsilon> N < r"
+ using F_dec t by (auto simp: \<epsilon>_def field_simps)
+ with F[of N] obtain x where "dist x ((f \<circ> t) n) < \<epsilon> N" "dist x ((f \<circ> t) m) < \<epsilon> N"
by (auto simp: subset_eq)
- with \<open>2 * e N < r\<close> show "dist ((f \<circ> t) m) ((f \<circ> t) n) < r"
+ with \<open>2 * \<epsilon> N < r\<close> show "dist ((f \<circ> t) m) ((f \<circ> t) n) < r"
by metric
qed
@@ -1758,7 +1751,10 @@
using assms unfolding complete_def by blast
qed
qed
-qed (metis compact_imp_complete compact_imp_seq_compact seq_compact_imp_totally_bounded)
+next
+ show "compact S \<Longrightarrow> ?rhs"
+ by (metis compact_imp_complete compact_imp_seq_compact seq_compact_imp_totally_bounded)
+qed
lemma cauchy_imp_bounded:
assumes "Cauchy S"
@@ -1772,11 +1768,10 @@
using finite_imp_bounded[of "S ` {1..N}"] by auto
then obtain a where a:"\<forall>x\<in>S ` {0..N}. dist (S N) x \<le> a"
unfolding bounded_any_center [where a="S N"] by auto
- ultimately show "?thesis"
- unfolding bounded_any_center [where a="S N"]
- apply (intro exI [where x="max a 1"])
- apply (force simp: le_max_iff_disj less_le_not_le)
- done
+ ultimately have "\<forall>y\<in>range S. dist (S N) y \<le> max a 1"
+ by (force simp: le_max_iff_disj less_le_not_le)
+ then show ?thesis
+ unfolding bounded_any_center [where a="S N"] by blast
qed
instance heine_borel < complete_space
@@ -1805,18 +1800,12 @@
fixes S :: "'a::metric_space set"
assumes "closed S" and "S \<subseteq> t" and "complete t"
shows "complete S"
- using assms complete_Int_closed [of t S] by (simp add: Int_absorb1)
+ by (metis assms complete_Int_closed inf.absorb_iff2)
lemma complete_eq_closed:
fixes S :: "('a::complete_space) set"
shows "complete S \<longleftrightarrow> closed S"
-proof
- assume "closed S" then show "complete S"
- using subset_UNIV complete_UNIV by (rule complete_closed_subset)
-next
- assume "complete S" then show "closed S"
- by (rule complete_imp_closed)
-qed
+ using complete_UNIV complete_closed_subset complete_imp_closed by auto
lemma convergent_eq_Cauchy:
fixes S :: "nat \<Rightarrow> 'a::complete_space"
@@ -1856,7 +1845,8 @@
lemma uniformly_continuous_imp_Cauchy_continuous:
fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
shows "uniformly_continuous_on S f \<Longrightarrow> Cauchy_continuous_on S f"
- by (simp add: uniformly_continuous_on_def Cauchy_continuous_on_def Cauchy_def image_subset_iff) metis
+ by (metis (no_types, lifting) ext Cauchy_continuous_on_def UNIV_I image_subset_iff
+ o_apply uniformly_continuous_on_Cauchy)
lemma Cauchy_continuous_on_imp_continuous:
fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
@@ -1873,14 +1863,15 @@
have "\<sigma> \<longlonglongrightarrow> x"
unfolding tendsto_iff
proof (intro strip)
- fix e :: real
- assume "e>0"
- then obtain N where "inverse (Suc N) < e"
+ fix \<epsilon> :: real
+ assume "\<epsilon>>0"
+ then obtain N where "inverse (Suc N) < \<epsilon>"
using reals_Archimedean by blast
- then have "\<forall>n. N \<le> n \<longrightarrow> dist (\<rho> n) x < e"
- by (smt (verit, ccfv_SIG) dx inverse_Suc inverse_less_iff_less inverse_positive_iff_positive of_nat_Suc of_nat_mono)
- with \<open>e>0\<close> show "\<forall>\<^sub>F n in sequentially. dist (\<sigma> n) x < e"
- by (auto simp add: eventually_sequentially \<sigma>_def)
+ then have "\<forall>n. N \<le> n \<longrightarrow> dist (\<rho> n) x < \<epsilon>"
+ by (metis dx inverse_positive_iff_positive le_imp_inverse_le nless_le not_less_eq_eq
+ of_nat_mono order_le_less_trans zero_le_dist)
+ with \<open>\<epsilon>>0\<close> show "\<forall>\<^sub>F n in sequentially. dist (\<sigma> n) x < \<epsilon>"
+ by (auto simp: eventually_sequentially \<sigma>_def)
qed
then have "Cauchy \<sigma>"
by (intro LIMSEQ_imp_Cauchy)
@@ -1889,15 +1880,15 @@
have "(f \<circ> \<sigma>) \<longlonglongrightarrow> f x"
unfolding tendsto_iff
proof (intro strip)
- fix e :: real
- assume "e>0"
- then obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. dist ((f \<circ> \<sigma>) m) ((f \<circ> \<sigma>) n) < e"
+ fix \<epsilon> :: real
+ assume "\<epsilon>>0"
+ then obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. dist ((f \<circ> \<sigma>) m) ((f \<circ> \<sigma>) n) < \<epsilon>"
using Cf unfolding Cauchy_def by presburger
moreover have "(f \<circ> \<sigma>) (Suc(N+N)) = f x"
by (simp add: \<sigma>_def)
- ultimately have "\<forall>n\<ge>N. dist ((f \<circ> \<sigma>) n) (f x) < e"
+ ultimately have "\<forall>n\<ge>N. dist ((f \<circ> \<sigma>) n) (f x) < \<epsilon>"
by (metis add_Suc le_add2)
- then show "\<forall>\<^sub>F n in sequentially. dist ((f \<circ> \<sigma>) n) (f x) < e"
+ then show "\<forall>\<^sub>F n in sequentially. dist ((f \<circ> \<sigma>) n) (f x) < \<epsilon>"
using eventually_sequentially by blast
qed
moreover have "\<And>n. \<not> dist (f (\<sigma> (2*n))) (f x) < \<epsilon>"
@@ -1919,13 +1910,19 @@
assumes "closed S"
and T: "T \<in> \<F>" "bounded T"
and clof: "\<And>T. T \<in> \<F> \<Longrightarrow> closed T"
- and "\<And>\<F>'. \<lbrakk>finite \<F>'; \<F>' \<subseteq> \<F>\<rbrakk> \<Longrightarrow> S \<inter> \<Inter>\<F>' \<noteq> {}"
+ and \<section>: "\<And>\<F>'. \<lbrakk>finite \<F>'; \<F>' \<subseteq> \<F>\<rbrakk> \<Longrightarrow> S \<inter> \<Inter>\<F>' \<noteq> {}"
shows "S \<inter> \<Inter>\<F> \<noteq> {}"
proof -
- have "compact (S \<inter> T)"
- using \<open>closed S\<close> clof compact_eq_bounded_closed T by blast
- then have "(S \<inter> T) \<inter> \<Inter>\<F> \<noteq> {}"
- by (smt (verit, best) Inf_insert Int_assoc assms compact_imp_fip finite_insert insert_subset)
+ have "(S \<inter> T) \<inter> \<Inter>\<F> \<noteq> {}"
+ proof (rule compact_imp_fip)
+ show "compact (S \<inter> T)"
+ using \<open>closed S\<close> clof compact_eq_bounded_closed T by blast
+ next
+ fix F'
+ assume "finite F'" and "F' \<subseteq> \<F>"
+ then show "S \<inter> T \<inter> \<Inter> F' \<noteq> {}"
+ by (metis Inf_insert Int_assoc \<open>T \<in> \<F>\<close> finite_insert insert_subset \<section>)
+ qed (simp add: clof)
then show ?thesis by blast
qed
@@ -1961,7 +1958,7 @@
lemma compact_cball[simp]:
fixes x :: "'a::heine_borel"
- shows "compact (cball x e)"
+ shows "compact (cball x \<epsilon>)"
using compact_eq_bounded_closed bounded_cball closed_cball by blast
lemma compact_frontier_bounded[intro]:
@@ -2041,7 +2038,7 @@
lemma infdist_le: "a \<in> A \<Longrightarrow> infdist x A \<le> dist x a"
by (auto intro: cINF_lower simp add: infdist_def)
-lemma infdist_le2: "a \<in> A \<Longrightarrow> dist x a \<le> d \<Longrightarrow> infdist x A \<le> d"
+lemma infdist_le2: "a \<in> A \<Longrightarrow> dist x a \<le> \<delta> \<Longrightarrow> infdist x A \<le> \<delta>"
by (auto intro!: cINF_lower2 simp add: infdist_def)
lemma infdist_zero[simp]: "a \<in> A \<Longrightarrow> infdist a A = 0"
@@ -2063,13 +2060,13 @@
proof (rule cInf_greatest)
from \<open>A \<noteq> {}\<close> 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"
+ fix \<delta>
+ assume "\<delta> \<in> {dist x y + dist y a |a. a \<in> A}"
+ then obtain a where \<delta>: "\<delta> = dist x y + dist y a" "a \<in> A"
by auto
- show "infdist x A \<le> d"
+ show "infdist x A \<le> \<delta>"
using infdist_notempty[OF \<open>A \<noteq> {}\<close>]
- by (metis d dist_commute dist_triangle3 infdist_le2)
+ by (metis \<delta> dist_commute dist_triangle3 infdist_le2)
qed
also have "\<dots> = dist x y + infdist y A"
proof (rule cInf_eq, safe)
@@ -2079,7 +2076,7 @@
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"
+ assume inf: "\<And>\<delta>. \<delta> \<in> {dist x y + dist y a |a. a \<in> A} \<Longrightarrow> i \<le> \<delta>"
then have "i - dist x y \<le> infdist y A"
unfolding infdist_notempty[OF \<open>A \<noteq> {}\<close>] using \<open>a \<in> A\<close>
by (intro cINF_greatest) (auto simp: field_simps)
@@ -2112,12 +2109,12 @@
assume x: "infdist x A = 0"
then obtain a where "a \<in> A"
by atomize_elim (metis all_not_in_conv assms)
- have False if "e > 0" "\<not> (\<exists>y\<in>A. dist y x < e)" for e
+ have False if "\<epsilon> > 0" "\<not> (\<exists>y\<in>A. dist y x < \<epsilon>)" for \<epsilon>
proof -
- have "infdist x A \<ge> e" using \<open>a \<in> A\<close>
+ have "infdist x A \<ge> \<epsilon>" using \<open>a \<in> A\<close>
unfolding infdist_def using that
by (force simp: dist_commute intro: cINF_greatest)
- with x \<open>e > 0\<close> show False by auto
+ with x \<open>\<epsilon> > 0\<close> show False by auto
qed
then show "x \<in> closure A"
using closure_approachable by blast
@@ -2179,12 +2176,12 @@
fix z assume H: "dist x z * 2 < infdist x T" "dist y z * 2 < infdist y S"
have "2 * dist x y \<le> 2 * dist x z + 2 * dist y z"
by metric
- also have "... < infdist x T + infdist y S"
+ also have "\<dots> < infdist x T + infdist y S"
using H by auto
finally have "dist x y < infdist x T \<or> dist x y < infdist y S"
by auto
then show False
- using infdist_le[OF \<open>x \<in> S\<close>, of y] infdist_le[OF \<open>y \<in> T\<close>, of x] by (auto simp add: dist_commute)
+ using infdist_le[OF \<open>x \<in> S\<close>, of y] infdist_le[OF \<open>y \<in> T\<close>, of x] by (auto simp: dist_commute)
qed
then have E: "U \<inter> V = {}"
unfolding U_def V_def by auto
@@ -2197,17 +2194,17 @@
assumes f: "(f \<longlongrightarrow> l) F"
shows "((\<lambda>x. infdist (f x) A) \<longlongrightarrow> infdist l A) F"
proof (rule tendstoI)
- fix e ::real
- assume "e > 0"
+ fix \<epsilon> ::real
+ assume "\<epsilon> > 0"
from tendstoD[OF f this]
- show "eventually (\<lambda>x. dist (infdist (f x) A) (infdist l A) < e) F"
+ show "eventually (\<lambda>x. dist (infdist (f x) A) (infdist l A) < \<epsilon>) F"
proof (eventually_elim)
fix x
from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l]
have "dist (infdist (f x) A) (infdist l A) \<le> dist (f x) l"
by (simp add: dist_commute dist_real_def)
- also assume "dist (f x) l < e"
- finally show "dist (infdist (f x) A) (infdist l A) < e" .
+ also assume "dist (f x) l < \<epsilon>"
+ finally show "dist (infdist (f x) A) (infdist l A) < \<epsilon>" .
qed
qed
@@ -2225,27 +2222,27 @@
fixes A::"'a::heine_borel set"
assumes "A \<noteq> {}"
assumes "compact A"
- assumes "e > 0"
- shows "compact {x. infdist x A \<le> e}"
+ assumes "\<epsilon> > 0"
+ shows "compact {x. infdist x A \<le> \<epsilon>}"
proof -
- from continuous_closed_vimage[of "{0..e}" "\<lambda>x. infdist x A"]
+ from continuous_closed_vimage[of "{0..\<epsilon>}" "\<lambda>x. infdist x A"]
continuous_infdist[OF continuous_ident, of _ UNIV A]
- have "closed {x. infdist x A \<le> e}" by (auto simp: vimage_def infdist_nonneg)
+ have "closed {x. infdist x A \<le> \<epsilon>}" by (auto simp: vimage_def infdist_nonneg)
moreover
from assms obtain x0 b where b: "\<And>x. x \<in> A \<Longrightarrow> dist x0 x \<le> b" "closed A"
by (auto simp: compact_eq_bounded_closed bounded_def)
{
fix y
- assume "infdist y A \<le> e"
+ assume "infdist y A \<le> \<epsilon>"
moreover
from infdist_attains_inf[OF \<open>closed A\<close> \<open>A \<noteq> {}\<close>, of y]
obtain z where "z \<in> A" "infdist y A = dist y z" by blast
ultimately
- have "dist x0 y \<le> b + e" using b by metric
+ have "dist x0 y \<le> b + \<epsilon>" using b by metric
} then
- have "bounded {x. infdist x A \<le> e}"
- by (auto simp: bounded_any_center[where a=x0] intro!: exI[where x="b + e"])
- ultimately show "compact {x. infdist x A \<le> e}"
+ have "bounded {x. infdist x A \<le> \<epsilon>}"
+ by (auto simp: bounded_any_center[where a=x0] intro!: exI[where x="b + \<epsilon>"])
+ ultimately show "compact {x. infdist x A \<le> \<epsilon>}"
by (simp add: compact_eq_bounded_closed)
qed
@@ -2255,14 +2252,14 @@
proposition separate_point_closed:
fixes S :: "'a::heine_borel set"
assumes "closed S" and "a \<notin> S"
- shows "\<exists>d>0. \<forall>x\<in>S. d \<le> dist a x"
+ shows "\<exists>\<delta>>0. \<forall>x\<in>S. \<delta> \<le> dist a x"
by (metis assms distance_attains_inf empty_iff gt_ex zero_less_dist_iff)
proposition separate_compact_closed:
fixes S T :: "'a::heine_borel set"
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"
+ shows "\<exists>\<delta>>0. \<forall>x\<in>S. \<forall>y\<in>T. \<delta> \<le> dist x y"
proof cases
assume "S \<noteq> {} \<and> T \<noteq> {}"
then have "S \<noteq> {}" "T \<noteq> {}" by auto
@@ -2283,7 +2280,7 @@
assumes S: "closed S"
and T: "compact T"
and dis: "S \<inter> T = {}"
- shows "\<exists>d>0. \<forall>x\<in>S. \<forall>y\<in>T. d \<le> dist x y"
+ shows "\<exists>\<delta>>0. \<forall>x\<in>S. \<forall>y\<in>T. \<delta> \<le> dist x y"
by (metis separate_compact_closed[OF T S] dis dist_commute inf_commute)
proposition compact_in_open_separated:
@@ -2291,24 +2288,24 @@
assumes A: "A \<noteq> {}" "compact A"
assumes "open B"
assumes "A \<subseteq> B"
- obtains e where "e > 0" "{x. infdist x A \<le> e} \<subseteq> B"
+ obtains \<epsilon> where "\<epsilon> > 0" "{x. infdist x A \<le> \<epsilon>} \<subseteq> B"
proof atomize_elim
have "closed (- B)" "compact A" "- B \<inter> A = {}"
using assms by (auto simp: open_Diff compact_eq_bounded_closed)
from separate_closed_compact[OF this]
obtain d'::real where d': "d'>0" "\<And>x y. x \<notin> B \<Longrightarrow> y \<in> A \<Longrightarrow> d' \<le> dist x y"
by auto
- define d where "d = d' / 2"
- hence "d>0" "d < d'" using d' by auto
- with d' have d: "\<And>x y. x \<notin> B \<Longrightarrow> y \<in> A \<Longrightarrow> d < dist x y"
+ define \<delta> where "\<delta> = d' / 2"
+ hence "\<delta>>0" "\<delta> < d'" using d' by auto
+ with d' have \<delta>: "\<And>x y. x \<notin> B \<Longrightarrow> y \<in> A \<Longrightarrow> \<delta> < dist x y"
by force
- show "\<exists>e>0. {x. infdist x A \<le> e} \<subseteq> B"
+ show "\<exists>\<epsilon>>0. {x. infdist x A \<le> \<epsilon>} \<subseteq> B"
proof (rule ccontr)
- assume "\<nexists>e. 0 < e \<and> {x. infdist x A \<le> e} \<subseteq> B"
- with \<open>d > 0\<close> obtain x where x: "infdist x A \<le> d" "x \<notin> B"
+ assume "\<nexists>\<epsilon>. 0 < \<epsilon> \<and> {x. infdist x A \<le> \<epsilon>} \<subseteq> B"
+ with \<open>\<delta> > 0\<close> obtain x where x: "infdist x A \<le> \<delta>" "x \<notin> B"
by auto
then show False
- by (metis A compact_eq_bounded_closed infdist_attains_inf x d linorder_not_less)
+ by (metis A compact_eq_bounded_closed infdist_attains_inf x \<delta> linorder_not_less)
qed
qed
@@ -2316,8 +2313,8 @@
section \<open>Uniform Continuity\<close>
lemma uniformly_continuous_onE:
- assumes "uniformly_continuous_on s f" "0 < e"
- obtains d where "d>0" "\<And>x x'. \<lbrakk>x\<in>s; x'\<in>s; dist x' x < d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
+ assumes "uniformly_continuous_on s f" "0 < \<epsilon>"
+ obtains \<delta> where "\<delta>>0" "\<And>x x'. \<lbrakk>x\<in>s; x'\<in>s; dist x' x < \<delta>\<rbrakk> \<Longrightarrow> dist (f x') (f x) < \<epsilon>"
using assms
by (auto simp: uniformly_continuous_on_def)
@@ -2332,14 +2329,14 @@
and y: "\<forall>n. y n \<in> s"
and xy: "((\<lambda>n. dist (x n) (y n)) \<longlongrightarrow> 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 \<epsilon> :: real
+ assume "\<epsilon> > 0"
+ then obtain \<delta> where "\<delta> > 0" and \<delta>: "\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < \<delta> \<longrightarrow> dist (f x') (f x) < \<epsilon>"
by (metis \<open>?lhs\<close> uniformly_continuous_onE)
- obtain N where N: "\<forall>n\<ge>N. dist (x n) (y n) < d"
- using xy[unfolded lim_sequentially dist_norm] and \<open>d>0\<close> by auto
- then have "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"
- using d x y by blast
+ obtain N where N: "\<forall>n\<ge>N. dist (x n) (y n) < \<delta>"
+ using xy[unfolded lim_sequentially dist_norm] and \<open>\<delta>>0\<close> by auto
+ then have "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < \<epsilon>"
+ using \<delta> x y by blast
}
then have "((\<lambda>n. dist (f(x n)) (f(y n))) \<longlongrightarrow> 0) sequentially"
unfolding lim_sequentially and dist_real_def by auto
@@ -2349,32 +2346,31 @@
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"
+ then obtain \<epsilon> where "\<epsilon> > 0" "\<forall>\<delta>>0. \<exists>x\<in>s. \<exists>x'\<in>s. dist x' x < \<delta> \<and> \<not> dist (f x') (f x) < \<epsilon>"
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"]
+ "\<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))) < \<epsilon>"
+ using choice[of "\<lambda>\<delta> x. \<delta>>0 \<longrightarrow> fst x \<in> s \<and> snd x \<in> s \<and> dist (snd x) (fst x) < \<delta> \<and> \<not> dist (f (snd x)) (f (fst x)) < \<epsilon>"]
by (auto simp: Bex_def dist_commute)
define x where "x n = fst (fa (inverse (real n + 1)))" for n
define y where "y n = snd (fa (inverse (real n + 1)))" for n
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"
+ and fxy:"\<forall>n. \<not> dist (f (x n)) (f (y n)) < \<epsilon>"
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_inverse[of e] by auto
- with xy0 have "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e"
+ fix \<epsilon> :: real
+ assume "\<epsilon> > 0"
+ then obtain N :: nat where "N \<noteq> 0" and N: "0 < inverse (real N) \<and> inverse (real N) < \<epsilon>"
+ unfolding real_arch_inverse[of \<epsilon>] by auto
+ with xy0 have "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < \<epsilon>"
by (metis order.strict_trans inverse_positive_iff_positive less_imp_inverse_less
nat_le_real_less)
}
- then have "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"
- using \<open>?rhs\<close>[THEN spec[where x=x], THEN spec[where x=y]] and xyn
- unfolding lim_sequentially dist_real_def by auto
- then have False using fxy and \<open>e>0\<close> by auto
+ then have "\<forall>\<epsilon>>0. \<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < \<epsilon>"
+ using \<open>?rhs\<close> xyn by (auto simp: lim_sequentially dist_real_def)
+ then have False using fxy and \<open>\<epsilon>>0\<close> by auto
}
then show ?lhs
unfolding uniformly_continuous_on_def by blast
@@ -2388,9 +2384,9 @@
lemma Heine_Borel_lemma:
assumes "compact S" and Ssub: "S \<subseteq> \<Union>\<G>" and opn: "\<And>G. G \<in> \<G> \<Longrightarrow> open G"
- obtains e where "0 < e" "\<And>x. x \<in> S \<Longrightarrow> \<exists>G \<in> \<G>. ball x e \<subseteq> G"
+ obtains \<epsilon> where "0 < \<epsilon>" "\<And>x. x \<in> S \<Longrightarrow> \<exists>G \<in> \<G>. ball x \<epsilon> \<subseteq> G"
proof -
- have False if neg: "\<And>e. 0 < e \<Longrightarrow> \<exists>x \<in> S. \<forall>G \<in> \<G>. \<not> ball x e \<subseteq> G"
+ have False if neg: "\<And>\<epsilon>. 0 < \<epsilon> \<Longrightarrow> \<exists>x \<in> S. \<forall>G \<in> \<G>. \<not> ball x \<epsilon> \<subseteq> G"
proof -
have "\<exists>x \<in> S. \<forall>G \<in> \<G>. \<not> ball x (1 / Suc n) \<subseteq> G" for n
using neg by simp
@@ -2400,29 +2396,29 @@
using \<open>compact S\<close> compact_def that by metis
then obtain G where "l \<in> G" "G \<in> \<G>"
using Ssub by auto
- then obtain e where "0 < e" and e: "\<And>z. dist z l < e \<Longrightarrow> z \<in> G"
+ then obtain \<epsilon> where "0 < \<epsilon>" and \<epsilon>: "\<And>z. dist z l < \<epsilon> \<Longrightarrow> z \<in> G"
using opn open_dist by blast
- obtain N1 where N1: "\<And>n. n \<ge> N1 \<Longrightarrow> dist (f (r n)) l < e/2"
- by (metis \<open>0 < e\<close> half_gt_zero lim_sequentially o_apply to_l)
- obtain N2 where N2: "of_nat N2 > 2/e"
+ obtain N1 where N1: "\<And>n. n \<ge> N1 \<Longrightarrow> dist (f (r n)) l < \<epsilon>/2"
+ by (metis \<open>0 < \<epsilon>\<close> half_gt_zero lim_sequentially o_apply to_l)
+ obtain N2 where N2: "of_nat N2 > 2/\<epsilon>"
using reals_Archimedean2 by blast
obtain x where "x \<in> ball (f (r (max N1 N2))) (1 / real (Suc (r (max N1 N2))))" and "x \<notin> G"
using fG [OF \<open>G \<in> \<G>\<close>, of "r (max N1 N2)"] by blast
then have "dist (f (r (max N1 N2))) x < 1 / real (Suc (r (max N1 N2)))"
by simp
- also have "... \<le> 1 / real (Suc (max N1 N2))"
+ also have "\<dots> \<le> 1 / real (Suc (max N1 N2))"
by (meson Suc_le_mono \<open>strict_mono r\<close> inverse_of_nat_le nat.discI seq_suble)
- also have "... \<le> 1 / real (Suc N2)"
+ also have "\<dots> \<le> 1 / real (Suc N2)"
by (simp add: field_simps)
- also have "... < e/2"
- using N2 \<open>0 < e\<close> by (simp add: field_simps)
- finally have "dist (f (r (max N1 N2))) x < e/2" .
- moreover have "dist (f (r (max N1 N2))) l < e/2"
+ also have "\<dots> < \<epsilon>/2"
+ using N2 \<open>0 < \<epsilon>\<close> by (simp add: field_simps)
+ finally have "dist (f (r (max N1 N2))) x < \<epsilon>/2" .
+ moreover have "dist (f (r (max N1 N2))) l < \<epsilon>/2"
using N1 max.cobounded1 by blast
- ultimately have "dist x l < e"
+ ultimately have "dist x l < \<epsilon>"
by metric
then show ?thesis
- using e \<open>x \<notin> G\<close> by blast
+ using \<epsilon> \<open>x \<notin> G\<close> by blast
qed
then show ?thesis
by (meson that)
@@ -2430,33 +2426,33 @@
lemma compact_uniformly_equicontinuous:
assumes "compact S"
- and cont: "\<And>x e. \<lbrakk>x \<in> S; 0 < e\<rbrakk>
- \<Longrightarrow> \<exists>d. 0 < d \<and>
- (\<forall>f \<in> \<F>. \<forall>x' \<in> S. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
- and "0 < e"
- obtains d where "0 < d"
- "\<And>f x x'. \<lbrakk>f \<in> \<F>; x \<in> S; x' \<in> S; dist x' x < d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
+ and cont: "\<And>x \<epsilon>. \<lbrakk>x \<in> S; 0 < \<epsilon>\<rbrakk>
+ \<Longrightarrow> \<exists>\<delta>. 0 < \<delta> \<and>
+ (\<forall>f \<in> \<F>. \<forall>x' \<in> S. dist x' x < \<delta> \<longrightarrow> dist (f x') (f x) < \<epsilon>)"
+ and "0 < \<epsilon>"
+ obtains \<delta> where "0 < \<delta>"
+ "\<And>f x x'. \<lbrakk>f \<in> \<F>; x \<in> S; x' \<in> S; dist x' x < \<delta>\<rbrakk> \<Longrightarrow> dist (f x') (f x) < \<epsilon>"
proof -
- obtain d where d_pos: "\<And>x e. \<lbrakk>x \<in> S; 0 < e\<rbrakk> \<Longrightarrow> 0 < d x e"
- and d_dist : "\<And>x x' e f. \<lbrakk>dist x' x < d x e; x \<in> S; x' \<in> S; 0 < e; f \<in> \<F>\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
+ obtain \<delta> where d_pos: "\<And>x \<epsilon>. \<lbrakk>x \<in> S; 0 < \<epsilon>\<rbrakk> \<Longrightarrow> 0 < \<delta> x \<epsilon>"
+ and d_dist : "\<And>x x' \<epsilon> f. \<lbrakk>dist x' x < \<delta> x \<epsilon>; x \<in> S; x' \<in> S; 0 < \<epsilon>; f \<in> \<F>\<rbrakk> \<Longrightarrow> dist (f x') (f x) < \<epsilon>"
using cont by metis
- let ?\<G> = "((\<lambda>x. ball x (d x (e/2))) ` S)"
+ let ?\<G> = "((\<lambda>x. ball x (\<delta> x (\<epsilon>/2))) ` S)"
have Ssub: "S \<subseteq> \<Union> ?\<G>"
- using \<open>0 < e\<close> d_pos by auto
+ using \<open>0 < \<epsilon>\<close> d_pos by auto
then obtain k where "0 < k" and k: "\<And>x. x \<in> S \<Longrightarrow> \<exists>G \<in> ?\<G>. ball x k \<subseteq> G"
by (rule Heine_Borel_lemma [OF \<open>compact S\<close>]) auto
- moreover have "dist (f v) (f u) < e" if "f \<in> \<F>" "u \<in> S" "v \<in> S" "dist v u < k" for f u v
+ moreover have "dist (f v) (f u) < \<epsilon>" if "f \<in> \<F>" "u \<in> S" "v \<in> S" "dist v u < k" for f u v
proof -
obtain G where "G \<in> ?\<G>" "u \<in> G" "v \<in> G"
using k that
by (metis \<open>dist v u < k\<close> \<open>u \<in> S\<close> \<open>0 < k\<close> centre_in_ball subsetD dist_commute mem_ball)
- then obtain w where w: "dist w u < d w (e/2)" "dist w v < d w (e/2)" "w \<in> S"
+ then obtain w where w: "dist w u < \<delta> w (\<epsilon>/2)" "dist w v < \<delta> w (\<epsilon>/2)" "w \<in> S"
by auto
- with that d_dist have "dist (f w) (f v) < e/2"
- by (metis \<open>0 < e\<close> dist_commute half_gt_zero)
+ with that d_dist have "dist (f w) (f v) < \<epsilon>/2"
+ by (metis \<open>0 < \<epsilon>\<close> dist_commute half_gt_zero)
moreover
- have "dist (f w) (f u) < e/2"
- using that d_dist w by (metis \<open>0 < e\<close> dist_commute divide_pos_pos zero_less_numeral)
+ have "dist (f w) (f u) < \<epsilon>/2"
+ using that d_dist w by (metis \<open>0 < \<epsilon>\<close> dist_commute divide_pos_pos zero_less_numeral)
ultimately show ?thesis
using dist_triangle_half_r by blast
qed
@@ -2476,8 +2472,8 @@
lemma continuous_on_closure:
"continuous_on (closure S) f \<longleftrightarrow>
- (\<forall>x e. x \<in> closure S \<and> 0 < e
- \<longrightarrow> (\<exists>d. 0 < d \<and> (\<forall>y. y \<in> S \<and> dist y x < d \<longrightarrow> dist (f y) (f x) < e)))"
+ (\<forall>x \<epsilon>. x \<in> closure S \<and> 0 < \<epsilon>
+ \<longrightarrow> (\<exists>\<delta>. 0 < \<delta> \<and> (\<forall>y. y \<in> S \<and> dist y x < \<delta> \<longrightarrow> dist (f y) (f x) < \<epsilon>)))"
(is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
@@ -2486,27 +2482,27 @@
assume R [rule_format]: ?rhs
show ?lhs
proof
- fix x and e::real
- assume "0 < e" and x: "x \<in> closure S"
+ fix x and \<epsilon>::real
+ assume "0 < \<epsilon>" and x: "x \<in> closure S"
obtain \<delta>::real where "\<delta> > 0"
- and \<delta>: "\<And>y. \<lbrakk>y \<in> S; dist y x < \<delta>\<rbrakk> \<Longrightarrow> dist (f y) (f x) < e/2"
- using R [of x "e/2"] \<open>0 < e\<close> x by auto
- have "dist (f y) (f x) \<le> e" if y: "y \<in> closure S" and dyx: "dist y x < \<delta>/2" for y
+ and \<delta>: "\<And>y. \<lbrakk>y \<in> S; dist y x < \<delta>\<rbrakk> \<Longrightarrow> dist (f y) (f x) < \<epsilon>/2"
+ using R [of x "\<epsilon>/2"] \<open>0 < \<epsilon>\<close> x by auto
+ have "dist (f y) (f x) \<le> \<epsilon>" if y: "y \<in> closure S" and dyx: "dist y x < \<delta>/2" for y
proof -
obtain \<delta>'::real where "\<delta>' > 0"
- and \<delta>': "\<And>z. \<lbrakk>z \<in> S; dist z y < \<delta>'\<rbrakk> \<Longrightarrow> dist (f z) (f y) < e/2"
- using R [of y "e/2"] \<open>0 < e\<close> y by auto
+ and \<delta>': "\<And>z. \<lbrakk>z \<in> S; dist z y < \<delta>'\<rbrakk> \<Longrightarrow> dist (f z) (f y) < \<epsilon>/2"
+ using R [of y "\<epsilon>/2"] \<open>0 < \<epsilon>\<close> y by auto
obtain z where "z \<in> S" and z: "dist z y < min \<delta>' \<delta> / 2"
using closure_approachable y
by (metis \<open>0 < \<delta>'\<close> \<open>0 < \<delta>\<close> divide_pos_pos min_less_iff_conj zero_less_numeral)
- have "dist (f z) (f y) < e/2"
+ have "dist (f z) (f y) < \<epsilon>/2"
using \<delta>' [OF \<open>z \<in> S\<close>] z \<open>0 < \<delta>'\<close> by metric
- moreover have "dist (f z) (f x) < e/2"
+ moreover have "dist (f z) (f x) < \<epsilon>/2"
using \<delta>[OF \<open>z \<in> S\<close>] z dyx by metric
ultimately show ?thesis
by metric
qed
- then show "\<exists>d>0. \<forall>x'\<in>closure S. dist x' x < d \<longrightarrow> dist (f x') (f x) \<le> e"
+ then show "\<exists>\<delta>>0. \<forall>x'\<in>closure S. dist x' x < \<delta> \<longrightarrow> dist (f x') (f x) \<le> \<epsilon>"
by (rule_tac x="\<delta>/2" in exI) (simp add: \<open>\<delta> > 0\<close>)
qed
qed
@@ -2519,9 +2515,9 @@
(is "?lhs = ?rhs")
proof -
have "continuous_on (closure S) f \<longleftrightarrow>
- (\<forall>x \<in> closure S. continuous (at x within S) f)"
+ (\<forall>x \<in> closure S. continuous (at x within S) f)"
by (force simp: continuous_on_closure continuous_within_eps_delta)
- also have "... = ?rhs"
+ also have "\<dots> = ?rhs"
by (force simp: continuous_within_sequentially)
finally show ?thesis .
qed
@@ -2533,38 +2529,38 @@
shows "uniformly_continuous_on (closure S) f"
unfolding uniformly_continuous_on_def
proof (intro allI impI)
- fix e::real
- assume "0 < e"
- then obtain d::real
- where "d>0"
- and d: "\<And>x x'. \<lbrakk>x\<in>S; x'\<in>S; dist x' x < d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e/3"
- using ucont [unfolded uniformly_continuous_on_def, rule_format, of "e/3"] by auto
- show "\<exists>d>0. \<forall>x\<in>closure S. \<forall>x'\<in>closure S. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"
- proof (rule exI [where x="d/3"], clarsimp simp: \<open>d > 0\<close>)
+ fix \<epsilon>::real
+ assume "0 < \<epsilon>"
+ then obtain \<delta>::real
+ where "\<delta>>0"
+ and \<delta>: "\<And>x x'. \<lbrakk>x\<in>S; x'\<in>S; dist x' x < \<delta>\<rbrakk> \<Longrightarrow> dist (f x') (f x) < \<epsilon>/3"
+ using ucont [unfolded uniformly_continuous_on_def, rule_format, of "\<epsilon>/3"] by auto
+ show "\<exists>\<delta>>0. \<forall>x\<in>closure S. \<forall>x'\<in>closure S. dist x' x < \<delta> \<longrightarrow> dist (f x') (f x) < \<epsilon>"
+ proof (rule exI [where x="\<delta>/3"], clarsimp simp: \<open>\<delta> > 0\<close>)
fix x y
- assume x: "x \<in> closure S" and y: "y \<in> closure S" and dyx: "dist y x * 3 < d"
+ assume x: "x \<in> closure S" and y: "y \<in> closure S" and dyx: "dist y x * 3 < \<delta>"
obtain d1::real where "d1 > 0"
- and d1: "\<And>w. \<lbrakk>w \<in> closure S; dist w x < d1\<rbrakk> \<Longrightarrow> dist (f w) (f x) < e/3"
- using cont [unfolded continuous_on_iff, rule_format, of "x" "e/3"] \<open>0 < e\<close> x by auto
- obtain x' where "x' \<in> S" and x': "dist x' x < min d1 (d / 3)"
+ and d1: "\<And>w. \<lbrakk>w \<in> closure S; dist w x < d1\<rbrakk> \<Longrightarrow> dist (f w) (f x) < \<epsilon>/3"
+ using cont [unfolded continuous_on_iff, rule_format, of "x" "\<epsilon>/3"] \<open>0 < \<epsilon>\<close> x by auto
+ obtain x' where "x' \<in> S" and x': "dist x' x < min d1 (\<delta> / 3)"
using closure_approachable [of x S]
- by (metis \<open>0 < d1\<close> \<open>0 < d\<close> divide_pos_pos min_less_iff_conj x zero_less_numeral)
+ by (metis \<open>0 < d1\<close> \<open>0 < \<delta>\<close> divide_pos_pos min_less_iff_conj x zero_less_numeral)
obtain d2::real where "d2 > 0"
- and d2: "\<forall>w \<in> closure S. dist w y < d2 \<longrightarrow> dist (f w) (f y) < e/3"
- using cont [unfolded continuous_on_iff, rule_format, of "y" "e/3"] \<open>0 < e\<close> y by auto
- obtain y' where "y' \<in> S" and y': "dist y' y < min d2 (d / 3)"
+ and d2: "\<forall>w \<in> closure S. dist w y < d2 \<longrightarrow> dist (f w) (f y) < \<epsilon>/3"
+ using cont [unfolded continuous_on_iff, rule_format, of "y" "\<epsilon>/3"] \<open>0 < \<epsilon>\<close> y by auto
+ obtain y' where "y' \<in> S" and y': "dist y' y < min d2 (\<delta> / 3)"
using closure_approachable [of y S]
- by (metis \<open>0 < d2\<close> \<open>0 < d\<close> divide_pos_pos min_less_iff_conj y zero_less_numeral)
- have "dist x' x < d/3" using x' by auto
- then have "dist x' y' < d"
+ by (metis \<open>0 < d2\<close> \<open>0 < \<delta>\<close> divide_pos_pos min_less_iff_conj y zero_less_numeral)
+ have "dist x' x < \<delta>/3" using x' by auto
+ then have "dist x' y' < \<delta>"
using dyx y' by metric
- then have "dist (f x') (f y') < e/3"
- by (rule d [OF \<open>y' \<in> S\<close> \<open>x' \<in> S\<close>])
- moreover have "dist (f x') (f x) < e/3" using \<open>x' \<in> S\<close> closure_subset x' d1
+ then have "dist (f x') (f y') < \<epsilon>/3"
+ by (rule \<delta> [OF \<open>y' \<in> S\<close> \<open>x' \<in> S\<close>])
+ moreover have "dist (f x') (f x) < \<epsilon>/3" using \<open>x' \<in> S\<close> closure_subset x' d1
by (simp add: closure_def)
- moreover have "dist (f y') (f y) < e/3" using \<open>y' \<in> S\<close> closure_subset y' d2
+ moreover have "dist (f y') (f y) < \<epsilon>/3" using \<open>y' \<in> S\<close> closure_subset y' d2
by (simp add: closure_def)
- ultimately show "dist (f y) (f x) < e" by metric
+ ultimately show "dist (f y) (f x) < \<epsilon>" by metric
qed
qed
@@ -2594,28 +2590,27 @@
show "(\<lambda>n. f (xs' n)) \<longlonglongrightarrow> l"
proof (rule tendstoI)
- fix e::real assume "e > 0"
- define e' where "e' \<equiv> e/2"
- have "e' > 0" using \<open>e > 0\<close> by (simp add: e'_def)
+ fix \<epsilon>::real assume "\<epsilon> > 0"
+ define e' where "e' \<equiv> \<epsilon>/2"
+ have "e' > 0" using \<open>\<epsilon> > 0\<close> by (simp add: e'_def)
have "\<forall>\<^sub>F n in sequentially. dist (f (xs n)) l < e'"
by (simp add: \<open>0 < e'\<close> l tendstoD)
moreover
- from uc[unfolded uniformly_continuous_on_def, rule_format, OF \<open>e' > 0\<close>]
- obtain d where d: "d > 0" "\<And>x x'. x \<in> X \<Longrightarrow> x' \<in> X \<Longrightarrow> dist x x' < d \<Longrightarrow> dist (f x) (f x') < e'"
- by auto
- have "\<forall>\<^sub>F n in sequentially. dist (xs n) (xs' n) < d"
- by (auto intro!: \<open>0 < d\<close> order_tendstoD tendsto_eq_intros xs xs')
+ obtain \<delta> where \<delta>: "\<delta> > 0" "\<And>x x'. x \<in> X \<Longrightarrow> x' \<in> X \<Longrightarrow> dist x x' < \<delta> \<Longrightarrow> dist (f x) (f x') < e'"
+ by (metis \<open>0 < e'\<close> uc uniformly_continuous_on_def)
+ have "\<forall>\<^sub>F n in sequentially. dist (xs n) (xs' n) < \<delta>"
+ by (auto intro!: \<open>0 < \<delta>\<close> order_tendstoD tendsto_eq_intros xs xs')
ultimately
- show "\<forall>\<^sub>F n in sequentially. dist (f (xs' n)) l < e"
+ show "\<forall>\<^sub>F n in sequentially. dist (f (xs' n)) l < \<epsilon>"
proof eventually_elim
case (elim n)
have "dist (f (xs' n)) l \<le> dist (f (xs n)) (f (xs' n)) + dist (f (xs n)) l"
by metric
also have "dist (f (xs n)) (f (xs' n)) < e'"
- by (auto intro!: d xs \<open>xs' _ \<in> _\<close> elim)
+ by (auto intro!: \<delta> xs \<open>xs' _ \<in> _\<close> elim)
also note \<open>dist (f (xs n)) l < e'\<close>
- also have "e' + e' = e" by (simp add: e'_def)
+ also have "e' + e' = \<epsilon>" by (simp add: e'_def)
finally show ?case by simp
qed
qed
@@ -2638,15 +2633,14 @@
have "uniformly_continuous_on (closure X) ?g"
unfolding uniformly_continuous_on_def
proof safe
- fix e::real assume "e > 0"
- define e' where "e' \<equiv> e / 3"
- have "e' > 0" using \<open>e > 0\<close> by (simp add: e'_def)
- from uc[unfolded uniformly_continuous_on_def, rule_format, OF \<open>0 < e'\<close>]
- obtain d where "d > 0" and d: "\<And>x x'. x \<in> X \<Longrightarrow> x' \<in> X \<Longrightarrow> dist x' x < d \<Longrightarrow> dist (f x') (f x) < e'"
- by auto
- define d' where "d' = d / 3"
- have "d' > 0" using \<open>d > 0\<close> by (simp add: d'_def)
- show "\<exists>d>0. \<forall>x\<in>closure X. \<forall>x'\<in>closure X. dist x' x < d \<longrightarrow> dist (?g x') (?g x) < e"
+ fix \<epsilon>::real assume "\<epsilon> > 0"
+ define e' where "e' \<equiv> \<epsilon> / 3"
+ have "e' > 0" using \<open>\<epsilon> > 0\<close> by (simp add: e'_def)
+ obtain \<delta> where "\<delta> > 0" and \<delta>: "\<And>x x'. x \<in> X \<Longrightarrow> x' \<in> X \<Longrightarrow> dist x' x < \<delta> \<Longrightarrow> dist (f x') (f x) < e'"
+ using \<open>0 < e'\<close> uc uniformly_continuous_onE by blast
+ define d' where "d' = \<delta> / 3"
+ have "d' > 0" using \<open>\<delta> > 0\<close> by (simp add: d'_def)
+ show "\<exists>\<delta>>0. \<forall>x\<in>closure X. \<forall>x'\<in>closure X. dist x' x < \<delta> \<longrightarrow> dist (?g x') (?g x) < \<epsilon>"
proof (safe intro!: exI[where x=d'] \<open>d' > 0\<close>)
fix x x' assume x: "x \<in> closure X" and x': "x' \<in> closure X" and dist: "dist x' x < d'"
then obtain xs xs' where xs: "xs \<longlonglongrightarrow> x" "\<And>n. xs n \<in> X"
@@ -2656,7 +2650,8 @@
and "\<forall>\<^sub>F n in sequentially. dist (xs n) x < d'"
by (auto intro!: \<open>0 < d'\<close> order_tendstoD tendsto_eq_intros xs xs')
moreover
- have "(\<lambda>x. f (xs x)) \<longlonglongrightarrow> y x" if "x \<in> closure X" "x \<notin> X" "xs \<longlonglongrightarrow> x" "\<And>n. xs n \<in> X" for xs x
+ have "(\<lambda>x. f (xs x)) \<longlonglongrightarrow> y x"
+ if "x \<in> closure X" "x \<notin> X" "xs \<longlonglongrightarrow> x" "\<And>n. xs n \<in> X" for xs x
using that not_eventuallyD
by (force intro!: filterlim_compose[OF y[OF \<open>x \<in> closure X\<close>]] simp: filterlim_at)
then have "(\<lambda>x. f (xs' x)) \<longlonglongrightarrow> ?g x'" "(\<lambda>x. f (xs x)) \<longlonglongrightarrow> ?g x"
@@ -2666,7 +2661,7 @@
"\<forall>\<^sub>F n in sequentially. dist (f (xs n)) (?g x) < e'"
by (auto intro!: \<open>0 < e'\<close> order_tendstoD tendsto_eq_intros)
ultimately
- have "\<forall>\<^sub>F n in sequentially. dist (?g x') (?g x) < e"
+ have "\<forall>\<^sub>F n in sequentially. dist (?g x') (?g x) < \<epsilon>"
proof eventually_elim
case (elim n)
have "dist (?g x') (?g x) \<le>
@@ -2674,17 +2669,18 @@
by (metis add.commute add_le_cancel_left dist_commute dist_triangle dist_triangle_le)
also
from \<open>dist (xs' n) x' < d'\<close> \<open>dist x' x < d'\<close> \<open>dist (xs n) x < d'\<close>
- have "dist (xs' n) (xs n) < d" unfolding d'_def by metric
+ have "dist (xs' n) (xs n) < \<delta>" unfolding d'_def by metric
with \<open>xs _ \<in> X\<close> \<open>xs' _ \<in> X\<close> have "dist (f (xs' n)) (f (xs n)) < e'"
- by (rule d)
+ by (rule \<delta>)
also note \<open>dist (f (xs' n)) (?g x') < e'\<close>
also note \<open>dist (f (xs n)) (?g x) < e'\<close>
finally show ?case by (simp add: e'_def)
qed
- then show "dist (?g x') (?g x) < e" by simp
+ then show "dist (?g x') (?g x) < \<epsilon>" by simp
qed
qed
- moreover have "f x = ?g x" if "x \<in> X" for x using that by simp
+ moreover have "f x = ?g x" if "x \<in> X" for x
+ using that by simp
moreover
{
fix Y h x
@@ -2702,8 +2698,8 @@
using \<open>x \<notin> X\<close> not_eventuallyD xs(2)
by (force intro!: filterlim_compose[OF y[OF \<open>x \<in> closure X\<close>]] simp: filterlim_at xs)
with hx have "h x = y x" by (rule LIMSEQ_unique)
- } then
- have "h x = ?g x"
+ }
+ then have "h x = ?g x"
using extension by auto
}
ultimately show ?thesis ..
@@ -2713,28 +2709,39 @@
fixes f :: "'a :: heine_borel \<Rightarrow> 'b :: heine_borel"
assumes "uniformly_continuous_on S f" "bounded S"
shows "bounded(f ` S)"
- by (metis (no_types) assms bounded_closure_image compact_closure compact_continuous_image compact_eq_bounded_closed image_cong uniformly_continuous_imp_continuous uniformly_continuous_on_extension_on_closure)
+proof -
+ obtain g where "uniformly_continuous_on (closure S) g" and g: "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
+ using uniformly_continuous_on_extension_on_closure assms by metis
+ then have "compact (g ` closure S)"
+ using \<open>bounded S\<close> compact_closure compact_continuous_image
+ uniformly_continuous_imp_continuous by blast
+ then show ?thesis
+ using g bounded_closure_image compact_eq_bounded_closed
+ by auto
+qed
section \<open>With Abstract Topology (TODO: move and remove dependency?)\<close>
lemma openin_contains_ball:
"openin (top_of_set T) S \<longleftrightarrow>
- S \<subseteq> T \<and> (\<forall>x \<in> S. \<exists>e. 0 < e \<and> ball x e \<inter> T \<subseteq> S)"
+ S \<subseteq> T \<and> (\<forall>x \<in> S. \<exists>\<epsilon>. 0 < \<epsilon> \<and> ball x \<epsilon> \<inter> T \<subseteq> S)"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
by (metis IntD2 Int_commute Int_lower1 Int_mono inf.idem openE openin_open)
next
- assume ?rhs
- then show ?lhs
- by (smt (verit) open_ball Int_commute Int_iff centre_in_ball in_mono openin_open_Int openin_subopen)
+ assume R: ?rhs
+ then have "\<forall>x\<in>S. \<exists>R. openin (top_of_set T) R \<and> x \<in> R \<and> R \<subseteq> S"
+ by (metis open_ball Int_iff centre_in_ball inf_sup_aci(1) openin_open_Int subsetD)
+ with R show ?lhs
+ using openin_subopen by auto
qed
lemma openin_contains_cball:
"openin (top_of_set T) S \<longleftrightarrow>
- S \<subseteq> T \<and> (\<forall>x \<in> S. \<exists>e. 0 < e \<and> cball x e \<inter> T \<subseteq> S)"
+ S \<subseteq> T \<and> (\<forall>x \<in> S. \<exists>\<epsilon>. 0 < \<epsilon> \<and> cball x \<epsilon> \<inter> T \<subseteq> S)"
by (fastforce simp: openin_contains_ball intro: exI [where x="_/2"])
@@ -2752,7 +2759,7 @@
obtains a where "\<And>n. a \<in> S n"
proof -
from assms(2) obtain x where x: "\<forall>n. x n \<in> S n"
- using choice[of "\<lambda>n x. x \<in> S n"] by auto
+ by (meson ex_in_conv)
from assms(4,1) have "seq_compact (S 0)"
by (simp add: bounded_closed_imp_seq_compact)
then obtain l r where lr: "l \<in> S 0" "strict_mono r" "(x \<circ> r) \<longlonglongrightarrow> l"
@@ -2766,10 +2773,8 @@
using x and assms(3) and lr(2) [THEN seq_suble] by auto
then have "\<forall>i. (x \<circ> r) (i + n) \<in> S n"
using assms(3) by (fast intro!: le_add2)
- moreover have "(\<lambda>i. (x \<circ> r) (i + n)) \<longlonglongrightarrow> l"
- using lr(3) by (rule LIMSEQ_ignore_initial_segment)
ultimately show "l \<in> S n"
- by (metis closed_sequentially)
+ by (metis LIMSEQ_ignore_initial_segment closed_sequential_limits lr(3))
qed
then show ?thesis
using that by blast
@@ -2782,7 +2787,7 @@
assumes "\<And>n. closed (S n)"
"\<And>n. S n \<noteq> {}"
"\<And>m n. m \<le> n \<Longrightarrow> S n \<subseteq> S m"
- "\<And>e. e>0 \<Longrightarrow> \<exists>n. \<forall>x\<in>S n. \<forall>y\<in>S n. dist x y < e"
+ "\<And>\<epsilon>. \<epsilon>>0 \<Longrightarrow> \<exists>n. \<forall>x\<in>S n. \<forall>y\<in>S n. dist x y < \<epsilon>"
obtains a where "\<And>n. a \<in> S n"
proof -
obtain t where t: "\<forall>n. t n \<in> S n"
@@ -2814,7 +2819,7 @@
then obtain N :: nat where N: "\<forall>n\<ge>N. dist (t n) l < \<epsilon>"
using l[unfolded lim_sequentially] by auto
have "t (max n N) \<in> S n"
- by (meson assms(3) contra_subsetD max.cobounded1 t)
+ by (meson assms(3) subsetD max.cobounded1 t)
then have "\<exists>y\<in>S n. dist y l < \<epsilon>"
using N max.cobounded2 by blast
}
@@ -2835,11 +2840,8 @@
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)"
- then have "dist a b = 0"
- by (meson InterE a \<section> linorder_neq_iff linorder_not_less range_eqI zero_le_dist)
- }
+ then have "dist a b = 0" if "b \<in> \<Inter>(range S)" for b
+ by (meson that InterE \<section> linorder_neq_iff linorder_not_less range_eqI zero_le_dist)
with a have "\<Inter>(range S) = {a}"
unfolding image_def by auto
then show ?thesis ..
@@ -2851,26 +2853,22 @@
fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"
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"
+ shows "\<exists>\<epsilon>>0. \<forall>y \<in> S. dist x y < \<epsilon> \<longrightarrow> f y \<noteq> a"
proof -
obtain U where "open U" and "f x \<in> U" and "a \<notin> U"
using t1_space [OF \<open>f x \<noteq> a\<close>] by fast
- have "(f \<longlongrightarrow> f x) (at x within S)"
- using assms(1) by (simp add: continuous_within)
- then have "eventually (\<lambda>y. f y \<in> U) (at x within S)"
+ have "\<forall>\<^sub>F y in at x within S. f y \<in> U"
using \<open>open U\<close> and \<open>f x \<in> U\<close>
- unfolding tendsto_def by fast
- then have "eventually (\<lambda>y. f y \<noteq> a) (at x within S)"
- using \<open>a \<notin> U\<close> by (fast elim: eventually_mono)
- then show ?thesis
- using \<open>f x \<noteq> a\<close> by (auto simp: dist_commute eventually_at)
+ using assms(1) continuous_within tendsto_def by blast
+ with \<open>f x \<noteq> a\<close> \<open>a \<notin> U\<close> show ?thesis
+ by (metis (no_types, lifting) dist_commute 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"
- shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
+ shows "\<exists>\<epsilon>>0. \<forall>y. dist x y < \<epsilon> \<longrightarrow> f y \<noteq> a"
using assms continuous_within_avoid[of x UNIV f a] by simp
lemma continuous_on_avoid:
@@ -2878,7 +2876,7 @@
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"
+ shows "\<exists>\<epsilon>>0. \<forall>y \<in> S. dist x y < \<epsilon> \<longrightarrow> f y \<noteq> a"
using continuous_within_avoid[of x S f a] assms
by (meson continuous_on_eq_continuous_within)
@@ -2888,7 +2886,7 @@
and "open S"
and "x \<in> S"
and "f x \<noteq> a"
- shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
+ shows "\<exists>\<epsilon>>0. \<forall>y. dist x y < \<epsilon> \<longrightarrow> f y \<noteq> a"
using continuous_at_avoid[of x f a] assms
by (meson continuous_on_eq_continuous_at)
@@ -2907,7 +2905,7 @@
lemma closed_contains_Sup:
fixes S :: "real set"
shows "S \<noteq> {} \<Longrightarrow> bdd_above S \<Longrightarrow> closed S \<Longrightarrow> Sup S \<in> S"
- by (subst closure_closed[symmetric], assumption, rule closure_contains_Sup)
+ by (metis closure_closed closure_contains_Sup)
lemma closed_subset_contains_Sup:
fixes A C :: "real set"
@@ -2917,8 +2915,8 @@
lemma atLeastAtMost_subset_contains_Inf:
fixes A :: "real set" and a b :: real
shows "A \<noteq> {} \<Longrightarrow> a \<le> b \<Longrightarrow> A \<subseteq> {a..b} \<Longrightarrow> Inf A \<in> {a..b}"
- by (rule closed_subset_contains_Inf)
- (auto intro: closed_real_atLeastAtMost intro!: bdd_belowI[of A a])
+ by (meson bdd_below_Icc bdd_below_mono closed_real_atLeastAtMost
+ closed_subset_contains_Inf)
lemma bounded_real: "bounded (S::real set) \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. \<bar>x\<bar> \<le> a)"
by (simp add: bounded_iff)
@@ -2966,49 +2964,49 @@
by (auto simp: bounded_imp_bdd_below inf_min cInf_insert_If)
lemma open_real:
- fixes s :: "real set"
- shows "open s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. \<bar>x' - x\<bar> < e --> x' \<in> s)"
+ fixes S :: "real set"
+ shows "open S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>\<epsilon>>0. \<forall>x'. \<bar>x' - x\<bar> < \<epsilon> --> x' \<in> S)"
unfolding open_dist dist_norm by simp
lemma islimpt_approachable_real:
- fixes s :: "real set"
- shows "x islimpt s \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> s. x' \<noteq> x \<and> \<bar>x' - x\<bar> < e)"
+ fixes S :: "real set"
+ shows "x islimpt S \<longleftrightarrow> (\<forall>\<epsilon>>0. \<exists>x'\<in> S. x' \<noteq> x \<and> \<bar>x' - x\<bar> < \<epsilon>)"
unfolding islimpt_approachable dist_norm by simp
lemma closed_real:
- fixes s :: "real set"
- shows "closed s \<longleftrightarrow> (\<forall>x. (\<forall>e>0. \<exists>x' \<in> s. x' \<noteq> x \<and> \<bar>x' - x\<bar> < e) \<longrightarrow> x \<in> s)"
+ fixes S :: "real set"
+ shows "closed S \<longleftrightarrow> (\<forall>x. (\<forall>\<epsilon>>0. \<exists>x' \<in> S. x' \<noteq> x \<and> \<bar>x' - x\<bar> < \<epsilon>) \<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 --> \<bar>f x' - f x\<bar> < e)"
- by (metis (mono_tags, opaque_lifting) LIM_eq continuous_within norm_eq_zero real_norm_def right_minus_eq)
+ shows "continuous (at x) f \<longleftrightarrow> (\<forall>\<epsilon>>0. \<exists>\<delta>>0. \<forall>x'. norm(x' - x) < \<delta> \<longrightarrow> \<bar>f x' - f x\<bar> < \<epsilon>)"
+ by (simp add: continuous_at_eps_delta dist_norm)
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 \<longrightarrow> \<bar>f x' - f x\<bar> < e))"
+ shows "continuous_on S f \<longleftrightarrow>
+ (\<forall>x \<in> S. \<forall>\<epsilon>>0. \<exists>\<delta>>0. (\<forall>x' \<in> S. norm(x' - x) < \<delta> \<longrightarrow> \<bar>f x' - f x\<bar> < \<epsilon>))"
unfolding continuous_on_iff dist_norm by simp
lemma continuous_on_closed_Collect_le:
fixes f g :: "'a::topological_space \<Rightarrow> real"
- assumes f: "continuous_on s f" and g: "continuous_on s g" and s: "closed s"
- shows "closed {x \<in> s. f x \<le> g x}"
+ assumes f: "continuous_on S f" and g: "continuous_on S g" and S: "closed S"
+ shows "closed {x \<in> S. f x \<le> g x}"
proof -
- have "closed ((\<lambda>x. g x - f x) -` {0..} \<inter> s)"
+ have "closed ((\<lambda>x. g x - f x) -` {0..} \<inter> S)"
using closed_real_atLeast continuous_on_diff [OF g f]
- by (simp add: continuous_on_closed_vimage [OF s])
- also have "((\<lambda>x. g x - f x) -` {0..} \<inter> s) = {x\<in>s. f x \<le> g x}"
+ by (simp add: continuous_on_closed_vimage [OF S])
+ also have "((\<lambda>x. g x - f x) -` {0..} \<inter> S) = {x\<in>S. f x \<le> g x}"
by auto
finally show ?thesis .
qed
lemma continuous_le_on_closure:
fixes a::real
- assumes f: "continuous_on (closure s) f"
- and x: "x \<in> closure(s)"
- and xlo: "\<And>x. x \<in> s ==> f(x) \<le> a"
+ assumes f: "continuous_on (closure S) f"
+ and x: "x \<in> closure(S)"
+ and xlo: "\<And>x. x \<in> S \<Longrightarrow> f(x) \<le> a"
shows "f(x) \<le> a"
using image_closure_subset [OF f, where T=" {x. x \<le> a}" ] assms
continuous_on_closed_Collect_le[of "UNIV" "\<lambda>x. x" "\<lambda>x. a"]
@@ -3016,9 +3014,9 @@
lemma continuous_ge_on_closure:
fixes a::real
- assumes f: "continuous_on (closure s) f"
- and x: "x \<in> closure(s)"
- and xlo: "\<And>x. x \<in> s ==> f(x) \<ge> a"
+ assumes f: "continuous_on (closure S) f"
+ and x: "x \<in> closure S"
+ and xlo: "\<And>x. x \<in> S \<Longrightarrow> f(x) \<ge> a"
shows "f(x) \<ge> a"
using image_closure_subset [OF f, where T=" {x. a \<le> x}"] assms
continuous_on_closed_Collect_le[of "UNIV" "\<lambda>x. a" "\<lambda>x. x"]
@@ -3028,33 +3026,36 @@
section\<open>The infimum of the distance between two sets\<close>
definition\<^marker>\<open>tag important\<close> setdist :: "'a::metric_space set \<Rightarrow> 'a set \<Rightarrow> real" where
- "setdist s t \<equiv>
- (if s = {} \<or> t = {} then 0
- else Inf {dist x y| x y. x \<in> s \<and> y \<in> t})"
-
-lemma setdist_empty1 [simp]: "setdist {} t = 0"
+ "setdist S T \<equiv>
+ (if S = {} \<or> T = {} then 0
+ else Inf {dist x y| x y. x \<in> S \<and> y \<in> T})"
+
+lemma setdist_empty1 [simp]: "setdist {} T = 0"
by (simp add: setdist_def)
-lemma setdist_empty2 [simp]: "setdist t {} = 0"
+lemma setdist_empty2 [simp]: "setdist T {} = 0"
by (simp add: setdist_def)
-lemma setdist_pos_le [simp]: "0 \<le> setdist s t"
+lemma setdist_pos_le [simp]: "0 \<le> setdist S T"
by (auto simp: setdist_def ex_in_conv [symmetric] intro: cInf_greatest)
lemma le_setdistI:
- assumes "s \<noteq> {}" "t \<noteq> {}" "\<And>x y. \<lbrakk>x \<in> s; y \<in> t\<rbrakk> \<Longrightarrow> d \<le> dist x y"
- shows "d \<le> setdist s t"
+ assumes "S \<noteq> {}" "T \<noteq> {}" "\<And>x y. \<lbrakk>x \<in> S; y \<in> T\<rbrakk> \<Longrightarrow> \<delta> \<le> dist x y"
+ shows "\<delta> \<le> setdist S T"
using assms
by (auto simp: setdist_def Set.ex_in_conv [symmetric] intro: cInf_greatest)
-lemma setdist_le_dist: "\<lbrakk>x \<in> s; y \<in> t\<rbrakk> \<Longrightarrow> setdist s t \<le> dist x y"
+lemma setdist_le_dist: "\<lbrakk>x \<in> S; y \<in> T\<rbrakk> \<Longrightarrow> setdist S T \<le> dist x y"
unfolding setdist_def
by (auto intro!: bdd_belowI [where m=0] cInf_lower)
lemma le_setdist_iff:
- "d \<le> setdist S T \<longleftrightarrow>
- (\<forall>x \<in> S. \<forall>y \<in> T. d \<le> dist x y) \<and> (S = {} \<or> T = {} \<longrightarrow> d \<le> 0)"
- by (smt (verit) le_setdistI setdist_def setdist_le_dist)
+ "\<delta> \<le> setdist S T \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y \<in> T. \<delta> \<le> dist x y) \<and> (S = {} \<or> T = {} \<longrightarrow> \<delta> \<le> 0)"
+ (is "?lhs = ?rhs")
+proof
+ show "?rhs \<Longrightarrow> ?lhs"
+ by (meson le_setdistI order_trans setdist_pos_le)
+qed (use setdist_le_dist in fastforce)
lemma setdist_ltE:
assumes "setdist S T < b" "S \<noteq> {}" "T \<noteq> {}"
@@ -3063,7 +3064,10 @@
by (auto simp: not_le [symmetric] le_setdist_iff)
lemma setdist_refl: "setdist S S = 0"
- by (metis dist_eq_0_iff ex_in_conv order_antisym setdist_def setdist_le_dist setdist_pos_le)
+proof (rule antisym)
+ show "setdist S S \<le> 0"
+ by (metis dist_self equals0I order_refl setdist_empty1 setdist_le_dist)
+qed simp
lemma setdist_sym: "setdist S T = setdist T S"
by (force simp: setdist_def dist_commute intro!: arg_cong [where f=Inf])
@@ -3074,8 +3078,10 @@
using setdist_pos_le by fastforce
next
case False
- then have "\<And>x. x \<in> S \<Longrightarrow> setdist S T - dist x a \<le> setdist {a} T"
- by (smt (verit, best) dist_self dist_triangle3 empty_not_insert le_setdist_iff setdist_le_dist singleton_iff)
+ then have "setdist S T - dist x a \<le> setdist {a} T" if "x \<in> S" for x
+ unfolding le_setdist_iff
+ by (metis diff_le_eq dist_commute dist_triangle3 order.trans empty_not_insert
+ setdist_le_dist singleton_iff that)
then have "setdist S T - setdist {a} T \<le> setdist S {a}"
using False by (fastforce intro: le_setdistI)
then show ?thesis
@@ -3127,11 +3133,11 @@
by (metis antisym dist_self setdist_le_dist setdist_pos_le)
lemma setdist_unique:
- "\<lbrakk>a \<in> S; b \<in> T; \<And>x y. x \<in> S \<and> y \<in> T ==> dist a b \<le> dist x y\<rbrakk>
+ "\<lbrakk>a \<in> S; b \<in> T; \<And>x y. x \<in> S \<and> y \<in> T \<Longrightarrow> dist a b \<le> dist x y\<rbrakk>
\<Longrightarrow> setdist S T = dist a b"
by (force simp: setdist_le_dist le_setdist_iff intro: antisym)
-lemma setdist_le_sing: "x \<in> S ==> setdist S T \<le> setdist {x} T"
+lemma setdist_le_sing: "x \<in> S \<Longrightarrow> setdist S T \<le> setdist {x} T"
using setdist_subset_left by auto
lemma infdist_eq_setdist: "infdist x A = setdist {x} A"
@@ -3143,12 +3149,12 @@
if "b \<in> B" "a \<in> A" for a b
proof (rule order_antisym)
have "Inf {dist x y |x y. x \<in> A \<and> y \<in> B} \<le> Inf (dist x ` B)"
- if "b \<in> B" "a \<in> A" "x \<in> A" for x
+ if "b \<in> B" "a \<in> A" "x \<in> A" for x
proof -
have "\<And>b'. b' \<in> B \<Longrightarrow> Inf {dist x y |x y. x \<in> A \<and> y \<in> B} \<le> dist x b'"
- by (metis (mono_tags, lifting) ex_in_conv setdist_def setdist_le_dist that(3))
+ by (metis (mono_tags, lifting) ex_in_conv setdist_def setdist_le_dist \<open>x \<in> A\<close>)
then show ?thesis
- by (smt (verit) cINF_greatest ex_in_conv \<open>b \<in> B\<close>)
+ by (metis (lifting) cINF_greatest emptyE \<open>b \<in> B\<close>)
qed
then show "Inf {dist x y |x y. x \<in> A \<and> y \<in> B} \<le> (INF x\<in>A. Inf (dist x ` B))"
by (metis (mono_tags, lifting) cINF_greatest emptyE that)
@@ -3207,11 +3213,12 @@
lemma diameter_comp_strict_mono:
fixes s :: "nat \<Rightarrow> 'a :: metric_space"
- assumes "strict_mono r" "bounded {s i |i. r n \<le> i}"
+ assumes "strict_mono r" and bnd: "bounded {s i |i. r n \<le> i}"
shows "diameter {s (r i) | i. n \<le> i} \<le> diameter {s i | i. r n \<le> i}"
proof (rule diameter_subset)
- show "{s (r i) | i. n \<le> i} \<subseteq> {s i | i. r n \<le> i}" using assms(1) monotoneD strict_mono_mono by fastforce
-qed (intro assms(2))
+ show "{s (r i) | i. n \<le> i} \<subseteq> {s i | i. r n \<le> i}"
+ using \<open>strict_mono r\<close> monotoneD strict_mono_mono by fastforce
+qed (intro bnd)
lemma diameter_bounded_bound':
fixes S :: "'a :: metric_space set"
@@ -3227,44 +3234,63 @@
lemma bounded_imp_dist_bounded:
assumes "bounded (range s)"
shows "bounded ((\<lambda>(i, j). dist (s i) (s j)) ` ({n..} \<times> {n..}))"
- using bounded_dist_comp[OF bounded_fst bounded_snd, OF bounded_Times(1,1)[OF assms(1,1)]] by (rule bounded_subset, force)
-
-text \<open>A sequence is Cauchy, if and only if it is bounded and its diameter tends to zero. The diameter is well-defined only if the sequence is bounded.\<close>
+ unfolding image_iff case_prod_unfold
+ by (intro bounded_dist_comp; meson assms bounded_dist_comp bounded_dist_comp bounded_subset image_subset_iff rangeI)
+
+text \<open>A sequence is Cauchy, if and only if it is bounded and its diameter tends to zero.
+ The diameter is well-defined only if the sequence is bounded.\<close>
lemma cauchy_iff_diameter_tends_to_zero_and_bounded:
fixes s :: "nat \<Rightarrow> 'a :: metric_space"
shows "Cauchy s \<longleftrightarrow> ((\<lambda>n. diameter {s i | i. i \<ge> n}) \<longlonglongrightarrow> 0 \<and> bounded (range s))"
+ (is "_ = ?rhs")
proof -
have "{s i |i. N \<le> i} \<noteq> {}" for N by blast
- hence diameter_SUP: "diameter {s i |i. N \<le> i} = (SUP (i, j) \<in> {N..} \<times> {N..}. dist (s i) (s j))" for N unfolding diameter_def by (auto intro!: arg_cong[of _ _ Sup])
+ hence diameter_SUP: "diameter {s i |i. N \<le> i} = (SUP (i, j) \<in> {N..} \<times> {N..}. dist (s i) (s j))" for N
+ unfolding diameter_def by (auto intro!: arg_cong[of _ _ Sup])
show ?thesis
proof (intro iffI)
- assume asm: "Cauchy s"
- have "\<exists>N. \<forall>n\<ge>N. norm (diameter {s i |i. n \<le> i}) < e" if e_pos: "e > 0" for e
+ assume "Cauchy s"
+ have "\<exists>N. \<forall>n\<ge>N. norm (diameter {s i |i. n \<le> i}) < \<epsilon>" if e_pos: "\<epsilon> > 0" for \<epsilon>
proof -
- obtain N where dist_less: "dist (s n) (s m) < (1/2) * e" if "n \<ge> N" "m \<ge> N" for n m using asm e_pos by (metis Cauchy_def mult_pos_pos zero_less_divide_iff zero_less_numeral zero_less_one)
- {
- fix r assume "r \<ge> N"
- hence "dist (s n) (s m) < (1/2) * e" if "n \<ge> r" "m \<ge> r" for n m using dist_less that by simp
- hence "(SUP (i, j) \<in> {r..} \<times> {r..}. dist (s i) (s j)) \<le> (1/2) * e" by (intro cSup_least) fastforce+
- also have "... < e" using e_pos by simp
- finally have "diameter {s i |i. r \<le> i} < e" using diameter_SUP by presburger
- }
- moreover have "diameter {s i |i. r \<le> i} \<ge> 0" for r unfolding diameter_SUP using bounded_imp_dist_bounded[OF cauchy_imp_bounded, THEN bounded_imp_bdd_above, OF asm] by (intro cSup_upper2, auto)
- ultimately show ?thesis by auto
+ obtain N where dist_less: "dist (s n) (s m) < (1/2) * \<epsilon>"
+ if "n \<ge> N" "m \<ge> N" for n m
+ using \<open>Cauchy s\<close> e_pos
+ by (meson half_gt_zero less_numeral_extra(1) metric_CauchyD mult_pos_pos)
+ have "diameter {s i |i. r \<le> i} < \<epsilon>"
+ if "r \<ge> N" for r
+ proof -
+ have "dist (s n) (s m) < (1/2) * \<epsilon>" if "n \<ge> r" "m \<ge> r" for n m
+ using \<open>r \<ge> N\<close> dist_less that by simp
+ hence "(SUP (i, j) \<in> {r..} \<times> {r..}. dist (s i) (s j)) \<le> (1/2) * \<epsilon>"
+ by (intro cSup_least) fastforce+
+ also have "\<dots> < \<epsilon>" using e_pos by simp
+ finally show ?thesis
+ using diameter_SUP by presburger
+ qed
+ moreover have "diameter {s i |i. r \<le> i} \<ge> 0" for r
+ unfolding diameter_SUP
+ using bounded_imp_dist_bounded[OF cauchy_imp_bounded, THEN bounded_imp_bdd_above] \<open>Cauchy s\<close>
+ by (force intro: cSup_upper2)
+ ultimately show ?thesis
+ by auto
qed
- thus "(\<lambda>n. diameter {s i |i. n \<le> i}) \<longlonglongrightarrow> 0 \<and> bounded (range s)" using cauchy_imp_bounded[OF asm] by (simp add: LIMSEQ_iff)
+ thus "(\<lambda>n. diameter {s i |i. n \<le> i}) \<longlonglongrightarrow> 0 \<and> bounded (range s)"
+ using cauchy_imp_bounded[OF \<open>Cauchy s\<close>] by (simp add: LIMSEQ_iff)
next
- assume asm: "(\<lambda>n. diameter {s i |i. n \<le> i}) \<longlonglongrightarrow> 0 \<and> bounded (range s)"
- have "\<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. dist (s n) (s m) < e" if e_pos: "e > 0" for e
+ assume R: ?rhs
+ have "\<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. dist (s n) (s m) < \<epsilon>" if e_pos: "\<epsilon> > 0" for \<epsilon>
proof -
- obtain N where diam_less: "diameter {s i |i. r \<le> i} < e" if "r \<ge> N" for r using LIMSEQ_D asm(1) e_pos by fastforce
- {
- fix n m assume "n \<ge> N" "m \<ge> N"
- hence "dist (s n) (s m) < e" using cSUP_lessD[OF bounded_imp_dist_bounded[THEN bounded_imp_bdd_above], OF _ diam_less[unfolded diameter_SUP]] asm by auto
- }
+ obtain N where diam_less: "diameter {s i |i. r \<le> i} < \<epsilon>" if "r \<ge> N" for r
+ using LIMSEQ_D R e_pos by fastforce
+ have "dist (s n) (s m) < \<epsilon>"
+ if "n \<ge> N" "m \<ge> N" for n m
+ using cSUP_lessD[OF bounded_imp_dist_bounded[THEN bounded_imp_bdd_above],
+ OF _ diam_less[unfolded diameter_SUP]]
+ using R that by auto
thus ?thesis by blast
qed
- then show "Cauchy s" by (simp add: Cauchy_def)
+ then show "Cauchy s"
+ by (simp add: Cauchy_def)
qed
qed
--- a/src/HOL/Analysis/FPS_Convergence.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Analysis/FPS_Convergence.thy Thu Apr 24 09:16:33 2025 +0200
@@ -691,6 +691,10 @@
finally show ?thesis .
qed
+lemma eval_fps_has_fps_expansion:
+ "fps_conv_radius F > 0 \<Longrightarrow> eval_fps F has_fps_expansion F"
+ unfolding has_fps_expansion_def by simp
+
lemma has_fps_expansion_imp_continuous:
fixes F :: "'a::{real_normed_field,banach} fps"
assumes "f has_fps_expansion F"
--- a/src/HOL/Analysis/Homotopy.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Analysis/Homotopy.thy Thu Apr 24 09:16:33 2025 +0200
@@ -5222,13 +5222,12 @@
next
case False
show ?thesis
- proof (rule continuous_transform_within [where f=g and d = "norm(x-a)"])
- have "\<exists>d>0. \<forall>x'\<in>cball a r.
- dist x' x < d \<longrightarrow> dist (g x') (g x) < e" if "e>0" for e
+ proof (rule continuous_transform_within [where f=g and \<delta> = "norm(x-a)"])
+ have "\<exists>d>0. \<forall>x'\<in>cball a r. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"
+ if "e>0" for e
proof -
obtain d where "d > 0"
- and d: "\<And>x'. \<lbrakk>dist x' a \<le> r; x' \<noteq> a; dist x' x < d\<rbrakk> \<Longrightarrow>
- dist (g x') (g x) < e"
+ and d: "\<And>y. \<lbrakk>dist y a \<le> r; y \<noteq> a; dist y x < d\<rbrakk> \<Longrightarrow> dist (g y) (g x) < e"
using contg False x \<open>e>0\<close>
unfolding continuous_on_iff by (fastforce simp: dist_commute intro: that)
show ?thesis
--- a/src/HOL/Analysis/Infinite_Products.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Analysis/Infinite_Products.thy Thu Apr 24 09:16:33 2025 +0200
@@ -280,6 +280,7 @@
by linarith
qed
+
subsection\<open>Absolutely convergent products\<close>
definition\<^marker>\<open>tag important\<close> abs_convergent_prod :: "(nat \<Rightarrow> _) \<Rightarrow> bool" where
@@ -1777,6 +1778,26 @@
by (auto simp: prod_defs sums_def_le ln_prod order_tendstoD)
qed
+lemma has_prod_imp_sums_ln_real':
+ fixes P :: real
+ assumes "f has_prod P" "\<And>n. f n > 0"
+ shows "(\<lambda>n. ln (f n)) sums (ln P)"
+proof -
+ have nz: "f n \<noteq> 0" for n
+ using assms(2)[of n] by simp
+ have "P \<noteq> 0"
+ using has_prod_eq_0_iff[OF assms(1)] by (auto simp: nz)
+
+ have "(\<lambda>n. \<Prod>k<n. f k) \<longlonglongrightarrow> P"
+ using has_prod_imp_tendsto'[OF assms(1)] by simp
+ hence "(\<lambda>n. ln (\<Prod>k<n. f k)) \<longlonglongrightarrow> ln P"
+ by (intro tendsto_intros \<open>P \<noteq> 0\<close>)
+ also have "(\<lambda>n. ln (\<Prod>k<n. f k)) = (\<lambda>n. \<Sum>k<n. ln (f k))"
+ by (subst ln_prod) (auto simp: nz)
+ finally show ?thesis
+ by (simp add: sums_def)
+qed
+
lemma summable_ln_real:
fixes f :: "nat \<Rightarrow> real"
assumes f: "convergent_prod f" and 0: "\<And>x. f x > 0"
--- a/src/HOL/Analysis/Infinite_Sum.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Analysis/Infinite_Sum.thy Thu Apr 24 09:16:33 2025 +0200
@@ -2845,6 +2845,31 @@
using has_sum_reindex_bij_witness[of S i j T h g, OF assms]
by (simp add: summable_on_def)
+lemma infsum_reindex_bij_witness:
+ assumes "\<And>a. a \<in> S \<Longrightarrow> i (j a) = a"
+ assumes "\<And>a. a \<in> S \<Longrightarrow> j a \<in> T"
+ assumes "\<And>b. b \<in> T \<Longrightarrow> j (i b) = b"
+ assumes "\<And>b. b \<in> T \<Longrightarrow> i b \<in> S"
+ assumes "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a"
+ shows "infsum g S = infsum h T"
+proof (cases "g summable_on S")
+ case True
+ then obtain s where s: "(g has_sum s) S"
+ by (auto simp: summable_on_def)
+ also have "?this \<longleftrightarrow> (h has_sum s) T"
+ by (rule has_sum_reindex_bij_witness[of _ i j]) (use assms in auto)
+ finally have s': "(h has_sum s) T" .
+ show ?thesis
+ using infsumI[OF s] infsumI[OF s'] by simp
+next
+ case False
+ note \<open>\<not>g summable_on S\<close>
+ also have "g summable_on S \<longleftrightarrow> h summable_on T"
+ by (rule summable_on_reindex_bij_witness[of _ i j]) (use assms in auto)
+ finally show ?thesis
+ using False by (simp add: infsum_not_exists)
+qed
+
lemma has_sum_homomorphism:
assumes "(f has_sum S) A" "h 0 = 0" "\<And>a b. h (a + b) = h a + h b" "continuous_on UNIV h"
shows "((\<lambda>x. h (f x)) has_sum (h S)) A"
--- a/src/HOL/Analysis/Measurable.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Analysis/Measurable.thy Thu Apr 24 09:16:33 2025 +0200
@@ -290,7 +290,7 @@
{x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else
if (\<exists>i. P i x) then P n x \<and> (\<forall>i>n. \<not> P i x)
else Max {} = n}"
- by (intro arg_cong[where f=Collect] ext conj_cong)
+ by (intro arg_cong[where f=Collect] ext)
(auto simp add: 1 2 3 not_le[symmetric] intro!: Max_eqI)
also have "\<dots> \<in> sets M"
by measurable
@@ -317,7 +317,7 @@
{x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else
if (\<exists>i. P i x) then P n x \<and> (\<forall>i<n. \<not> P i x)
else Min {} = n}"
- by (intro arg_cong[where f=Collect] ext conj_cong)
+ by (intro arg_cong[where f=Collect] ext)
(auto simp add: 1 2 3 not_le[symmetric] intro!: Min_eqI)
also have "\<dots> \<in> sets M"
by measurable
@@ -378,7 +378,7 @@
assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> measurable M (count_space UNIV)"
shows "(\<lambda>x. SUP i\<in>I. F i x) \<in> measurable M (count_space UNIV)"
unfolding measurable_count_space_eq2_countable
-proof (safe intro!: UNIV_I)
+proof (intro conjI strip)
fix a
have "(\<lambda>x. SUP i\<in>I. F i x) -` {a} \<inter> space M =
{x\<in>space M. (\<forall>i\<in>I. F i x \<le> a) \<and> (\<forall>b. (\<forall>i\<in>I. F i x \<le> b) \<longrightarrow> a \<le> b)}"
@@ -386,7 +386,7 @@
also have "\<dots> \<in> sets M"
by measurable
finally show "(\<lambda>x. SUP i\<in>I. F i x) -` {a} \<inter> space M \<in> sets M" .
-qed
+qed auto
lemma measurable_INF[measurable]:
fixes F :: "'i \<Rightarrow> 'a \<Rightarrow> 'b::{complete_lattice, countable}"
@@ -394,7 +394,7 @@
assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> measurable M (count_space UNIV)"
shows "(\<lambda>x. INF i\<in>I. F i x) \<in> measurable M (count_space UNIV)"
unfolding measurable_count_space_eq2_countable
-proof (safe intro!: UNIV_I)
+proof (intro conjI strip)
fix a
have "(\<lambda>x. INF i\<in>I. F i x) -` {a} \<inter> space M =
{x\<in>space M. (\<forall>i\<in>I. a \<le> F i x) \<and> (\<forall>b. (\<forall>i\<in>I. b \<le> F i x) \<longrightarrow> b \<le> a)}"
@@ -402,7 +402,7 @@
also have "\<dots> \<in> sets M"
by measurable
finally show "(\<lambda>x. INF i\<in>I. F i x) -` {a} \<inter> space M \<in> sets M" .
-qed
+qed auto
lemma measurable_lfp_coinduct[consumes 1, case_names continuity step]:
fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_lattice, countable})"
@@ -628,10 +628,10 @@
lemma measurable_case_enat[measurable (raw)]:
assumes f: "f \<in> M \<rightarrow>\<^sub>M count_space UNIV" and g: "\<And>i. g i \<in> M \<rightarrow>\<^sub>M N" and h: "h \<in> M \<rightarrow>\<^sub>M N"
shows "(\<lambda>x. case f x of enat i \<Rightarrow> g i x | \<infinity> \<Rightarrow> h x) \<in> M \<rightarrow>\<^sub>M N"
- apply (rule measurable_compose_countable[OF _ f])
- subgoal for i
+proof (rule measurable_compose_countable[OF _ f])
+ show "(\<lambda>x. case i of enat i \<Rightarrow> g i x | \<infinity> \<Rightarrow> h x) \<in> M \<rightarrow>\<^sub>M N" for i
by (cases i) (auto intro: g h)
- done
+qed
hide_const (open) pred
--- a/src/HOL/Analysis/Measure_Space.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Analysis/Measure_Space.thy Thu Apr 24 09:16:33 2025 +0200
@@ -68,13 +68,10 @@
proof
fix n
show "(\<Sum>i < Suc (Suc n). f (binaryset A B i)) = f A + f B"
- by (induct n) (auto simp add: binaryset_def f)
+ by (induct n) (auto simp: binaryset_def f)
qed
- moreover
- have "\<dots> \<longlonglongrightarrow> f A + f B" by (rule tendsto_const)
- ultimately have "(\<lambda>n. \<Sum>i< n+2. f (binaryset A B i)) \<longlonglongrightarrow> f A + f B"
- by simp
- thus ?thesis by (rule LIMSEQ_offset [where k=2])
+ thus ?thesis
+ by (simp add: LIMSEQ_imp_Suc)
qed
lemma binaryset_sums:
@@ -98,7 +95,7 @@
"subadditive M f \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<inter> y = {} \<longrightarrow> f (x \<union> y) \<le> f x + f y)"
lemma subadditiveD: "subadditive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> M \<Longrightarrow> y \<in> M \<Longrightarrow> f (x \<union> y) \<le> f x + f y"
- by (auto simp add: subadditive_def)
+ by (auto simp: subadditive_def)
definition countably_subadditive where
"countably_subadditive M f \<longleftrightarrow>
@@ -108,20 +105,20 @@
fixes f :: "'a set \<Rightarrow> ennreal"
assumes f: "positive M f" and cs: "countably_subadditive M f"
shows "subadditive M f"
-proof (auto simp add: subadditive_def)
+proof (auto simp: subadditive_def)
fix x y
assume x: "x \<in> M" and y: "y \<in> M" and "x \<inter> y = {}"
hence "disjoint_family (binaryset x y)"
- by (auto simp add: disjoint_family_on_def binaryset_def)
+ by (auto simp: disjoint_family_on_def binaryset_def)
hence "range (binaryset x y) \<subseteq> M \<longrightarrow>
(\<Union>i. binaryset x y i) \<in> M \<longrightarrow>
f (\<Union>i. binaryset x y i) \<le> (\<Sum> n. f (binaryset x y n))"
- using cs by (auto simp add: countably_subadditive_def)
+ using cs by (auto simp: countably_subadditive_def)
hence "{x,y,{}} \<subseteq> M \<longrightarrow> x \<union> y \<in> M \<longrightarrow>
f (x \<union> y) \<le> (\<Sum> n. f (binaryset x y n))"
by (simp add: range_binaryset_eq UN_binaryset_eq)
thus "f (x \<union> y) \<le> f x + f y" using f x y
- by (auto simp add: Un o_def suminf_binaryset_eq positive_def)
+ by (auto simp: Un o_def suminf_binaryset_eq positive_def)
qed
definition additive where
@@ -134,15 +131,15 @@
lemma positiveD_empty:
"positive M f \<Longrightarrow> f {} = 0"
- by (auto simp add: positive_def)
+ by (auto simp: positive_def)
lemma additiveD:
"additive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> M \<Longrightarrow> y \<in> M \<Longrightarrow> f (x \<union> y) = f x + f y"
- by (auto simp add: additive_def)
+ by (auto simp: additive_def)
lemma increasingD:
"increasing M f \<Longrightarrow> x \<subseteq> y \<Longrightarrow> x\<in>M \<Longrightarrow> y\<in>M \<Longrightarrow> f x \<le> f y"
- by (auto simp add: increasing_def)
+ by (auto simp: increasing_def)
lemma countably_additiveI[case_names countably]:
"(\<And>A. \<lbrakk>range A \<subseteq> M; disjoint_family A; (\<Union>i. A i) \<in> M\<rbrakk> \<Longrightarrow> (\<Sum>i. f(A i)) = f(\<Union>i. A i))
@@ -175,22 +172,22 @@
next
case (insert s S)
then have "A s \<inter> (\<Union>i\<in>S. A i) = {}"
- by (auto simp add: disjoint_family_on_def neq_iff)
+ by (auto simp: disjoint_family_on_def neq_iff)
moreover
have "A s \<in> M" using insert by blast
moreover have "(\<Union>i\<in>S. A i) \<in> M"
using insert \<open>finite S\<close> by auto
ultimately have "f (A s \<union> (\<Union>i\<in>S. A i)) = f (A s) + f(\<Union>i\<in>S. A i)"
- using ad UNION_in_sets A by (auto simp add: additive_def)
+ using ad UNION_in_sets A by (auto simp: additive_def)
with insert show ?case using ad disjoint_family_on_mono[of S "insert s S" A]
- by (auto simp add: additive_def subset_insertI)
+ by (auto simp: additive_def subset_insertI)
qed
lemma (in ring_of_sets) additive_increasing:
fixes f :: "'a set \<Rightarrow> ennreal"
assumes posf: "positive M f" and addf: "additive M f"
shows "increasing M f"
-proof (auto simp add: increasing_def)
+proof (auto simp: increasing_def)
fix x y
assume xy: "x \<in> M" "y \<in> M" "x \<subseteq> y"
then have "y - x \<in> M" by auto
@@ -231,11 +228,11 @@
fixes f :: "'a set \<Rightarrow> ennreal"
assumes posf: "positive M f" and ca: "countably_additive M f"
shows "additive M f"
-proof (auto simp add: additive_def)
+proof (auto simp: additive_def)
fix x y
assume x: "x \<in> M" and y: "y \<in> M" and "x \<inter> y = {}"
hence "disjoint_family (binaryset x y)"
- by (auto simp add: disjoint_family_on_def binaryset_def)
+ by (auto simp: disjoint_family_on_def binaryset_def)
hence "range (binaryset x y) \<subseteq> M \<longrightarrow>
(\<Union>i. binaryset x y i) \<in> M \<longrightarrow>
f (\<Union>i. binaryset x y i) = (\<Sum> n. f (binaryset x y n))"
@@ -243,7 +240,7 @@
hence "{x,y,{}} \<subseteq> M \<longrightarrow> x \<union> y \<in> M \<longrightarrow> f (x \<union> y) = (\<Sum>n. f (binaryset x y n))"
by (simp add: range_binaryset_eq UN_binaryset_eq)
thus "f (x \<union> y) = f x + f y" using posf x y
- by (auto simp add: Un suminf_binaryset_eq positive_def)
+ by (auto simp: Un suminf_binaryset_eq positive_def)
qed
lemma (in algebra) increasing_additive_bound:
@@ -261,7 +258,7 @@
also have "\<dots> \<le> f \<Omega>" using space_closed A
by (intro increasingD[OF inc] finite_UN) auto
finally show "(\<Sum>i<N. f (A i)) \<le> f \<Omega>" by simp
-qed (insert f A, auto simp: positive_def)
+qed (use f A in \<open>auto simp: positive_def\<close>)
lemma (in ring_of_sets) countably_additiveI_finite:
fixes \<mu> :: "'a set \<Rightarrow> ennreal"
@@ -388,7 +385,8 @@
moreover {
fix n
have "f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i) = f ((A n - (\<Inter>i. A i)) \<union> (\<Inter>i. A i))"
- using A by (subst f(2)[THEN additiveD]) auto
+ using A f(2)
+ by (metis (no_types) Diff Diff_disjoint add.commute additiveD range_subsetD sup_commute)
also have "(A n - (\<Inter>i. A i)) \<union> (\<Inter>i. A i) = A n"
by auto
finally have "f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i) = f (A n)" . }
@@ -493,21 +491,27 @@
by (metis emeasure_mono emeasure_notin_sets sets.sets_into_space sets.top zero_le)
lemma emeasure_Diff:
- assumes "emeasure M B \<noteq> \<infinity>"
+ assumes \<infinity>: "emeasure M B \<noteq> \<infinity>"
and "A \<in> sets M" "B \<in> sets M" and "B \<subseteq> A"
- shows "emeasure M (A - B) = emeasure M A - emeasure M B"
- by (smt (verit, best) add_diff_self_ennreal assms emeasure_Un emeasure_mono
- ennreal_add_left_cancel le_iff_sup)
+shows "emeasure M (A - B) = emeasure M A - emeasure M B"
+proof -
+ have "emeasure M B + emeasure M (A - B) = emeasure M (B \<union> (A-B))"
+ by (simp add: assms emeasure_Un)
+ also have "... = emeasure M A"
+ using Diff_partition \<open>B \<subseteq> A\<close> by fastforce
+ finally show ?thesis
+ by (metis \<infinity> ennreal_add_diff_cancel_left infinity_ennreal_def)
+qed
lemma emeasure_compl:
"s \<in> sets M \<Longrightarrow> emeasure M s \<noteq> \<infinity> \<Longrightarrow> emeasure M (space M - s) = emeasure M (space M) - emeasure M s"
- by (rule emeasure_Diff) (auto dest: sets.sets_into_space)
+ by (simp add: emeasure_Diff sets.sets_into_space)
lemma Lim_emeasure_incseq:
"range A \<subseteq> sets M \<Longrightarrow> incseq A \<Longrightarrow> (\<lambda>i. (emeasure M (A i))) \<longlonglongrightarrow> emeasure M (\<Union>i. A i)"
using emeasure_countably_additive
- by (auto simp add: sets.countably_additive_iff_continuous_from_below emeasure_positive
- emeasure_additive)
+ by (metis emeasure_additive emeasure_positive sets.countable_UN
+ sets.countably_additive_iff_continuous_from_below)
lemma incseq_emeasure:
assumes "range B \<subseteq> sets M" "incseq B"
@@ -543,12 +547,15 @@
show "range (\<lambda>n. A 0 - A n) \<subseteq> sets M"
using A by auto
show "incseq (\<lambda>n. A 0 - A n)"
- using \<open>decseq A\<close> by (auto simp add: incseq_def decseq_def)
+ using \<open>decseq A\<close> by (auto simp: incseq_def decseq_def)
qed
also have "\<dots> = emeasure M (A 0) - emeasure M (\<Inter>i. A i)"
using A finite * by (simp, subst emeasure_Diff) auto
- finally show ?thesis
- by (smt (verit, best) Inf_lower diff_diff_ennreal le_MI finite range_eqI)
+ finally have "emeasure M (A 0) - (INF n. emeasure M (A n)) =
+ emeasure M (A 0) - emeasure M (\<Inter> (range A))" .
+ then show ?thesis
+ by (metis Inf_lower ennreal_minus_cancel infinity_ennreal_def le_MI local.finite
+ range_eqI)
qed
lemma INF_emeasure_decseq':
@@ -606,17 +613,17 @@
proof (intro INF_emeasure_decseq[symmetric])
show "decseq (\<lambda>i. F (L i))"
using L by (intro antimonoI F L_mono) auto
- qed (insert L fin, auto)
+ qed (use L fin in auto)
also have "\<dots> = (INF i\<in>I. emeasure M (F i))"
proof (intro antisym INF_greatest)
show "i \<in> I \<Longrightarrow> (INF i. emeasure M (F (L i))) \<le> emeasure M (F i)" for i
by (intro INF_lower2[of i]) auto
- qed (insert L, auto intro: INF_lower)
+ qed (use L in \<open>auto intro: INF_lower\<close>)
also have "(\<Inter>i. F (L i)) = (\<Inter>i\<in>I. F i)"
proof (intro antisym INF_greatest)
show "i \<in> I \<Longrightarrow> (\<Inter>i. F (L i)) \<subseteq> F i" for i
- by (intro INF_lower2[of i]) auto
- qed (insert L, auto)
+ by (metis Inf_lower L_eq rangeI)
+ qed (use L in auto)
finally show ?thesis .
qed
@@ -633,9 +640,9 @@
shows "emeasure M {x\<in>space M. lfp F x} = (SUP i. emeasure M {x\<in>space M. (F ^^ i) (\<lambda>x. False) x})"
proof -
have "emeasure M {x\<in>space M. lfp F x} = emeasure M (\<Union>i. {x\<in>space M. (F ^^ i) (\<lambda>x. False) x})"
- using sup_continuous_lfp[OF cont] by (auto simp add: bot_fun_def intro!: arg_cong2[where f=emeasure])
+ using sup_continuous_lfp[OF cont] by (auto simp: bot_fun_def intro!: arg_cong2[where f=emeasure])
moreover { fix i from \<open>P M\<close> have "{x\<in>space M. (F ^^ i) (\<lambda>x. False) x} \<in> sets M"
- by (induct i arbitrary: M) (auto simp add: pred_def[symmetric] intro: *) }
+ by (induct i arbitrary: M) (auto simp: pred_def[symmetric] intro: *) }
moreover have "incseq (\<lambda>i. {x\<in>space M. (F ^^ i) (\<lambda>x. False) x})"
proof (rule incseq_SucI)
fix i
@@ -663,7 +670,7 @@
then show "(\<lambda>s. emeasure (M s) {x \<in> space N. (SUP i. C i) x}) = (SUP i. (\<lambda>s. emeasure (M s) {x \<in> space N. C i x}))"
unfolding SUP_apply
by (subst SUP_emeasure_incseq) (auto simp: mono_def fun_eq_iff intro!: arg_cong2[where f=emeasure])
-qed (auto simp add: iter le_fun_def SUP_apply intro!: meas cont)
+qed (auto simp: iter le_fun_def SUP_apply intro!: meas cont)
lemma emeasure_subadditive_finite:
"finite I \<Longrightarrow> A ` I \<subseteq> sets M \<Longrightarrow> emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))"
@@ -717,7 +724,7 @@
show "disjoint_family_on (\<lambda>i. A \<inter> B i) S"
using \<open>disjoint_family_on B S\<close>
unfolding disjoint_family_on_def by auto
- qed (insert assms, auto)
+ qed (use assms in auto)
also have "(\<Union>i\<in>S. A \<inter> (B i)) = A"
using A by auto
finally show ?thesis by simp
@@ -768,11 +775,13 @@
using sets.top[of M] sets.space_closed[of M] S.top S.space_closed \<open>sets M = sigma_sets \<Omega> E\<close>
by blast
- { fix F D assume "F \<in> E" and "?\<mu> F \<noteq> \<infinity>"
- then have [intro]: "F \<in> sigma_sets \<Omega> E" by auto
+ have *: "emeasure M (F \<inter> D) = emeasure N (F \<inter> D)"
+ if "F \<in> E" and "?\<mu> F \<noteq> \<infinity>" and D: "D \<in> sets M" for F D
+ proof -
+ have [intro]: "F \<in> sigma_sets \<Omega> E"
+ using that by auto
have "?\<nu> F \<noteq> \<infinity>" using \<open>?\<mu> F \<noteq> \<infinity>\<close> \<open>F \<in> E\<close> eq by simp
- assume "D \<in> sets M"
- with \<open>Int_stable E\<close> \<open>E \<subseteq> Pow \<Omega>\<close> have "emeasure M (F \<inter> D) = emeasure N (F \<inter> D)"
+ from \<open>Int_stable E\<close> \<open>E \<subseteq> Pow \<Omega>\<close> D show ?thesis
unfolding M
proof (induct rule: sigma_sets_induct_disjoint)
case (basic A)
@@ -803,8 +812,8 @@
by (subst (1 2) suminf_emeasure[symmetric]) (auto simp: disjoint_family_on_def subset_eq M N)
with A show ?case
by auto
- qed }
- note * = this
+ qed
+ qed
show "M = N"
proof (rule measure_eqI)
show "sets M = sets N"
@@ -815,7 +824,7 @@
let ?D = "disjointed (\<lambda>i. F \<inter> A i)"
from \<open>space M = \<Omega>\<close> have F_eq: "F = (\<Union>i. ?D i)"
using \<open>F \<in> sets M\<close>[THEN sets.sets_into_space] A(2)[symmetric] by (auto simp: UN_disjointed_eq)
- have [simp, intro]: "\<And>i. ?D i \<in> sets M"
+ have DinM[simp]: "\<And>i. ?D i \<in> sets M"
using sets.range_disjointed_sets[of "\<lambda>i. F \<inter> A i" M] \<open>F \<in> sets M\<close>
by (auto simp: subset_eq)
have "disjoint_family ?D"
@@ -826,11 +835,11 @@
fix i
have "A i \<inter> ?D i = ?D i"
by (auto simp: disjointed_def)
- then show "emeasure M (?D i) = emeasure N (?D i)"
- using *[of "A i" "?D i", OF _ A(3)] A(1) by auto
+ with A show "emeasure M (?D i) = emeasure N (?D i)"
+ by (metis "*" DinM range_subsetD)
qed
ultimately show "emeasure M F = emeasure N F"
- by (simp add: image_subset_iff \<open>sets M = sets N\<close>[symmetric] F_eq[symmetric] suminf_emeasure)
+ by (metis DinM F_eq \<open>sets M = sets N\<close> image_subset_iff suminf_emeasure)
qed
qed
@@ -847,8 +856,8 @@
assume "\<Omega> = {}"
have *: "sigma_sets \<Omega> E = sets (sigma \<Omega> E)"
using E(2) by simp
- have "space M = \<Omega>" "space N = \<Omega>"
- using sets E(2) unfolding * by (auto dest: sets_eq_imp_space_eq simp del: sets_measure_of)
+ obtain "space M = \<Omega>" "space N = \<Omega>"
+ by (simp add: "*" sets sets_eq_imp_space_eq space_measure_of_conv)
then show "M = N"
unfolding \<open>\<Omega> = {}\<close> by (auto dest: space_empty)
next
@@ -910,13 +919,7 @@
lemma UN_from_nat_into:
assumes I: "countable I" "I \<noteq> {}"
shows "(\<Union>i\<in>I. N i) = (\<Union>i. N (from_nat_into I i))"
-proof -
- have "(\<Union>i\<in>I. N i) = \<Union>(N ` range (from_nat_into I))"
- using I by simp
- also have "\<dots> = (\<Union>i. (N \<circ> from_nat_into I) i)"
- by simp
- finally show ?thesis by simp
-qed
+ using assms by (simp add: UN_extend_simps)
lemma null_sets_UN':
assumes "countable I"
@@ -949,7 +952,7 @@
proof (intro CollectI conjI null_setsI)
show "emeasure M (A \<inter> B) = 0" using assms
by (intro emeasure_eq_0[of B _ "A \<inter> B"]) auto
-qed (insert assms, auto)
+qed (use assms in auto)
lemma null_set_Int2:
assumes "B \<in> null_sets M" "A \<in> sets M" shows "B \<inter> A \<in> null_sets M"
@@ -969,15 +972,17 @@
lemma null_set_Diff:
assumes "B \<in> null_sets M" "A \<in> sets M" shows "B - A \<in> null_sets M"
proof (intro CollectI conjI null_setsI)
- show "emeasure M (B - A) = 0" using assms by (intro emeasure_eq_0[of B _ "B - A"]) auto
-qed (insert assms, auto)
+ show "emeasure M (B - A) = 0"
+ using assms by (intro emeasure_eq_0[of B _ "B - A"]) auto
+qed (use assms in auto)
lemma emeasure_Un_null_set:
assumes "A \<in> sets M" "B \<in> null_sets M"
shows "emeasure M (A \<union> B) = emeasure M A"
proof -
have *: "A \<union> B = A \<union> (B - A)" by auto
- have "B - A \<in> null_sets M" using assms(2,1) by (rule null_set_Diff)
+ have "B - A \<in> null_sets M" using assms
+ using null_set_Diff by blast
then show ?thesis
unfolding * using assms
by (subst plus_emeasure[symmetric]) auto
@@ -1035,8 +1040,8 @@
proof
assume "AE x in M. P x" then obtain N where N: "N \<in> sets M" "?P \<subseteq> N" "emeasure M N = 0"
unfolding eventually_ae_filter by auto
- have "emeasure M ?P \<le> emeasure M N"
- using assms N(1,2) by (auto intro: emeasure_mono)
+ then have "emeasure M ?P \<le> emeasure M N"
+ using emeasure_mono by blast
then have "emeasure M ?P = 0"
unfolding \<open>emeasure M N = 0\<close> by auto
then show "?P \<in> null_sets M" using assms by auto
@@ -1132,7 +1137,7 @@
qed
lemma AE_space: "AE x in M. x \<in> space M"
- by (rule AE_I[where N="{}"]) auto
+ by (auto intro: AE_I[where N="{}"])
lemma AE_I2[simp, intro]:
"(\<And>x. x \<in> space M \<Longrightarrow> P x) \<Longrightarrow> AE x in M. P x"
@@ -1153,8 +1158,8 @@
"(AE x in M. \<forall>i. P i x) \<longleftrightarrow> (\<forall>i::'i::countable. AE x in M. P i x)"
proof
assume "\<forall>i. AE x in M. P i x"
- from this[unfolded eventually_ae_filter Bex_def, THEN choice]
- obtain N where N: "\<And>i. N i \<in> null_sets M" "\<And>i. {x\<in>space M. \<not> P i x} \<subseteq> N i" by auto
+ then obtain N where N: "\<And>i. N i \<in> null_sets M" "\<And>i. {x\<in>space M. \<not> P i x} \<subseteq> N i"
+ unfolding eventually_ae_filter by metis
have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. {x\<in>space M. \<not> P i x})" by auto
also have "\<dots> \<subseteq> (\<Union>i. N i)" using N by auto
finally have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. N i)" .
@@ -1169,9 +1174,8 @@
shows "(AE x in M. \<forall>y\<in>X. P x y) \<longleftrightarrow> (\<forall>y\<in>X. AE x in M. P x y)"
proof
assume "\<forall>y\<in>X. AE x in M. P x y"
- from this[unfolded eventually_ae_filter Bex_def, THEN bchoice]
- obtain N where N: "\<And>y. y \<in> X \<Longrightarrow> N y \<in> null_sets M" "\<And>y. y \<in> X \<Longrightarrow> {x\<in>space M. \<not> P x y} \<subseteq> N y"
- by auto
+ then obtain N where N: "\<And>y. y \<in> X \<Longrightarrow> N y \<in> null_sets M" "\<And>y. y \<in> X \<Longrightarrow> {x\<in>space M. \<not> P x y} \<subseteq> N y"
+ unfolding eventually_ae_filter by metis
have "{x\<in>space M. \<not> (\<forall>y\<in>X. P x y)} \<subseteq> (\<Union>y\<in>X. {x\<in>space M. \<not> P x y})"
by auto
also have "\<dots> \<subseteq> (\<Union>y\<in>X. N y)"
@@ -1202,14 +1206,12 @@
by auto
qed
-lemma AE_finite_all:
- assumes f: "finite S" shows "(AE x in M. \<forall>i\<in>S. P i x) \<longleftrightarrow> (\<forall>i\<in>S. AE x in M. P i x)"
- using f by induct auto
+lemmas AE_finite_all = eventually_ball_finite_distrib
lemma AE_finite_allI:
assumes "finite S"
shows "(\<And>s. s \<in> S \<Longrightarrow> AE x in M. Q s x) \<Longrightarrow> AE x in M. \<forall>s\<in>S. Q s x"
- using AE_finite_all[OF \<open>finite S\<close>] by auto
+ by (simp add: AE_ball_countable' assms countable_finite)
lemma emeasure_mono_AE:
assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B"
@@ -1229,10 +1231,9 @@
qed (simp add: emeasure_notin_sets)
lemma emeasure_eq_AE:
- assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"
- assumes A: "A \<in> sets M" and B: "B \<in> sets M"
+ assumes "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B" "A \<in> sets M" "B \<in> sets M"
shows "emeasure M A = emeasure M B"
- using assms by (safe intro!: antisym emeasure_mono_AE) auto
+ using assms by (force intro!: antisym emeasure_mono_AE)
lemma emeasure_Collect_eq_AE:
"AE x in M. P x \<longleftrightarrow> Q x \<Longrightarrow> Measurable.pred M Q \<Longrightarrow> Measurable.pred M P \<Longrightarrow>
@@ -1255,11 +1256,11 @@
shows "emeasure M C = emeasure M A + emeasure M B"
proof -
have "emeasure M C = emeasure M (A \<union> B)"
- by (rule emeasure_eq_AE) (insert 1, auto)
+ by (rule emeasure_eq_AE) (use 1 in auto)
also have "\<dots> = emeasure M A + emeasure M (B - A)"
by (subst plus_emeasure) auto
also have "emeasure M (B - A) = emeasure M B"
- by (rule emeasure_eq_AE) (insert 2, auto)
+ by (rule emeasure_eq_AE) (use 2 in auto)
finally show ?thesis .
qed
@@ -1309,16 +1310,12 @@
proof (rule that[of "disjointed A"])
show "range (disjointed A) \<subseteq> sets M"
by (rule sets.range_disjointed_sets[OF range])
- show "(\<Union>i. disjointed A i) = space M"
- and "disjoint_family (disjointed A)"
+ show "(\<Union>i. disjointed A i) = space M" and "disjoint_family (disjointed A)"
using disjoint_family_disjointed UN_disjointed_eq[of A] space range
by auto
show "emeasure M (disjointed A i) \<noteq> \<infinity>" for i
- proof -
- have "emeasure M (disjointed A i) \<le> emeasure M (A i)"
- using range disjointed_subset[of A i] by (auto intro!: emeasure_mono)
- then show ?thesis using measure[of i] by (auto simp: top_unique)
- qed
+ using range disjointed_subset[of A i] measure[of i]
+ by (simp add: emeasure_mono neq_top_trans)
qed
qed
@@ -1334,10 +1331,7 @@
show "range (\<lambda>n. \<Union>i\<le>n. F i) \<subseteq> sets M"
using F by (force simp: incseq_def)
show "(\<Union>n. \<Union>i\<le>n. F i) = space M"
- proof -
- from F have "\<And>x. x \<in> space M \<Longrightarrow> \<exists>i. x \<in> F i" by auto
- with F show ?thesis by fastforce
- qed
+ using F(2) by fastforce
show "emeasure M (\<Union>i\<le>n. F i) \<noteq> \<infinity>" for n
proof -
have "emeasure M (\<Union>i\<le>n. F i) \<le> (\<Sum>i\<le>n. emeasure M (F i))"
@@ -1361,16 +1355,19 @@
where A: "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "incseq A"
using sigma_finite_incseq by blast
define B where "B = (\<lambda>i. W \<inter> A i)"
- have B_meas: "\<And>i. B i \<in> sets M" using W_meas \<open>range A \<subseteq> sets M\<close> B_def by blast
- have b: "\<And>i. B i \<subseteq> W" using B_def by blast
-
- { fix i
+ have B_meas: "\<And>i. B i \<in> sets M"
+ using W_meas \<open>range A \<subseteq> sets M\<close> B_def by blast
+ have BsubW: "\<And>i. B i \<subseteq> W"
+ using B_def by blast
+
+ have Bfinite: "emeasure M (B i) < \<infinity>" for i
+ proof -
have "emeasure M (B i) \<le> emeasure M (A i)"
using A by (intro emeasure_mono) (auto simp: B_def)
also have "emeasure M (A i) < \<infinity>"
using \<open>\<And>i. emeasure M (A i) \<noteq> \<infinity>\<close> by (simp add: less_top)
- finally have "emeasure M (B i) < \<infinity>" . }
- note c = this
+ finally show ?thesis .
+ qed
have "W = (\<Union>i. B i)" using B_def \<open>(\<Union>i. A i) = space M\<close> W_meas by auto
moreover have "incseq B" using B_def \<open>incseq A\<close> by (simp add: incseq_def subset_eq)
@@ -1378,10 +1375,10 @@
by (simp add: B_meas Lim_emeasure_incseq image_subset_iff)
then have "(\<lambda>i. emeasure M (B i)) \<longlonglongrightarrow> \<infinity>" using W_inf by simp
from order_tendstoD(1)[OF this, of C]
- obtain i where d: "emeasure M (B i) > C"
+ obtain i where "emeasure M (B i) > C"
by (auto simp: eventually_sequentially)
- have "B i \<in> sets M" "B i \<subseteq> W" "emeasure M (B i) < \<infinity>" "emeasure M (B i) > C"
- using B_meas b c d by auto
+ then have "B i \<in> sets M" "B i \<subseteq> W" "emeasure M (B i) < \<infinity>" "emeasure M (B i) > C"
+ using B_meas BsubW Bfinite by auto
then show ?thesis using that by blast
qed
@@ -1446,8 +1443,8 @@
proof (subst (1 2) emeasure_Collect_distr[symmetric, where X=f])
show "f \<in> measurable M' M" "f \<in> measurable M' M"
using f[OF \<open>P M\<close>] by auto
- { fix i show "Measurable.pred M ((F ^^ i) (\<lambda>x. False))"
- using \<open>P M\<close> by (induction i arbitrary: M) (auto intro!: *) }
+ show "Measurable.pred M ((F ^^ i) (\<lambda>x. False))" for i
+ using \<open>P M\<close> by (induction i arbitrary: M) (auto intro!: *)
show "Measurable.pred M (lfp F)"
using \<open>P M\<close> cont * by (rule measurable_lfp_coinduct[of P])
@@ -1484,8 +1481,8 @@
proof (rule measure_eqI)
fix A assume "A \<in> sets (distr M N f)"
with assms show "emeasure (distr M N f) A = emeasure (distr K L g) A"
- by (auto simp add: emeasure_distr intro!: emeasure_eq_AE measurable_sets)
-qed (insert 1, simp)
+ by (auto simp: emeasure_distr intro!: emeasure_eq_AE measurable_sets)
+qed (use 1 in simp)
lemma AE_distrD:
assumes f: "f \<in> measurable M M'"
@@ -1509,31 +1506,34 @@
have "f -` {x\<in>space N. \<not> P x} \<inter> space M = {x \<in> space M. \<not> P (f x)}"
using f[THEN measurable_space] by auto
then show "(emeasure (distr M N f) {x \<in> space (distr M N f). \<not> P x} = 0) =
- (emeasure M {x \<in> space M. \<not> P (f x)} = 0)"
+ (emeasure M {x \<in> space M. \<not> P (f x)} = 0)"
by (simp add: emeasure_distr)
qed auto
lemma null_sets_distr_iff:
"f \<in> measurable M N \<Longrightarrow> A \<in> null_sets (distr M N f) \<longleftrightarrow> f -` A \<inter> space M \<in> null_sets M \<and> A \<in> sets N"
- by (auto simp add: null_sets_def emeasure_distr)
+ by (auto simp: null_sets_def emeasure_distr)
proposition distr_distr:
"g \<in> measurable N L \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> distr (distr M N f) L g = distr M L (g \<circ> f)"
- by (auto simp add: emeasure_distr measurable_space
+ by (auto simp: emeasure_distr measurable_space
intro!: arg_cong[where f="emeasure M"] measure_eqI)
subsection\<^marker>\<open>tag unimportant\<close> \<open>Real measure values\<close>
lemma ring_of_finite_sets: "ring_of_sets (space M) {A\<in>sets M. emeasure M A \<noteq> top}"
-proof (rule ring_of_setsI)
- show "a \<in> {A \<in> sets M. emeasure M A \<noteq> top} \<Longrightarrow> b \<in> {A \<in> sets M. emeasure M A \<noteq> top} \<Longrightarrow>
- a \<union> b \<in> {A \<in> sets M. emeasure M A \<noteq> top}" for a b
- using emeasure_subadditive[of a M b] by (auto simp: top_unique)
-
- show "a \<in> {A \<in> sets M. emeasure M A \<noteq> top} \<Longrightarrow> b \<in> {A \<in> sets M. emeasure M A \<noteq> top} \<Longrightarrow>
- a - b \<in> {A \<in> sets M. emeasure M A \<noteq> top}" for a b
- using emeasure_mono[of "a - b" a M] by (auto simp: top_unique)
-qed (auto dest: sets.sets_into_space)
+proof -
+ have False
+ if "a \<in> sets M" and "emeasure M a \<noteq> top"
+ and "b \<in> sets M" and "emeasure M b \<noteq> top"
+ and "emeasure M (a - b) = top"
+ for a b
+ using that
+ by (metis emeasure_Un emeasure_Un_Int ennreal_add_eq_top)
+ then show ?thesis
+ using emeasure_Un_Int
+ by (fastforce intro!: sets.sets_into_space ring_of_setsI)
+qed
lemma measure_nonneg[simp]: "0 \<le> measure M A"
unfolding measure_def by auto
@@ -1542,7 +1542,7 @@
using measure_nonneg not_le by blast
lemma zero_less_measure_iff: "0 < measure M A \<longleftrightarrow> measure M A \<noteq> 0"
- using measure_nonneg[of M A] by (auto simp add: le_less)
+ using measure_nonneg[of M A] by (auto simp: le_less)
lemma measure_le_0_iff: "measure M X \<le> 0 \<longleftrightarrow> measure M X = 0"
using measure_nonneg[of M X] by linarith
@@ -1602,16 +1602,12 @@
assumes finite: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"
shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"
proof -
- have "(\<lambda>i. emeasure M (A i)) sums (emeasure M (\<Union>i. A i))"
+ have \<section>: "(\<lambda>i. emeasure M (A i)) sums (emeasure M (\<Union>i. A i))"
unfolding suminf_emeasure[OF measurable, symmetric] by (simp add: summable_sums)
- moreover
- { fix i
- have "emeasure M (A i) \<le> emeasure M (\<Union>i. A i)"
- using measurable by (auto intro!: emeasure_mono)
- then have "emeasure M (A i) = ennreal ((measure M (A i)))"
- using finite by (intro emeasure_eq_ennreal_measure) (auto simp: top_unique) }
- ultimately show ?thesis using finite
- by (subst (asm) (2) emeasure_eq_ennreal_measure) simp_all
+ then have "emeasure M (A i) = ennreal ((measure M (A i)))" for i
+ by (metis assms(3) emeasure_eq_ennreal_measure ennreal_suminf_lessD
+ infinity_ennreal_def less_top sums_unique)
+ with \<section> show ?thesis using finite emeasure_eq_ennreal_measure by fastforce
qed
lemma measure_subadditive:
@@ -1623,7 +1619,7 @@
using emeasure_subadditive[OF measurable] fin by (auto simp: top_unique)
then show "(measure M (A \<union> B)) \<le> (measure M A) + (measure M B)"
unfolding measure_def
- by (metis emeasure_subadditive[OF measurable] fin enn2real_mono enn2real_plus
+ by (metis emeasure_subadditive[OF measurable] fin enn2real_mono enn2real_plus
ennreal_add_less_top infinity_ennreal_def less_top)
qed
@@ -1631,12 +1627,9 @@
assumes A: "finite I" "A`I \<subseteq> sets M" and fin: "\<And>i. i \<in> I \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>"
shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))"
proof -
- { have "emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))"
- using emeasure_subadditive_finite[OF A] .
- also have "\<dots> < \<infinity>"
- using fin by (simp add: less_top A)
- finally have "emeasure M (\<Union>i\<in>I. A i) \<noteq> top" by simp }
- note * = this
+ have *: "emeasure M (\<Union>i\<in>I. A i) \<noteq> top"
+ using emeasure_subadditive_finite[OF A] fin
+ by (metis \<open>finite I\<close> ennreal_sum_eq_top infinity_ennreal_def neq_top_trans)
show ?thesis
using emeasure_subadditive_finite[OF A] fin
unfolding emeasure_eq_ennreal_measure[OF *]
@@ -1660,7 +1653,7 @@
using emeasure_subadditive_countably[OF A] .
also have "\<dots> = ennreal (\<Sum>i. measure M (A i))"
using fin unfolding emeasure_eq_ennreal_measure[OF **]
- by (subst suminf_ennreal) (auto simp: **)
+ by (metis infinity_ennreal_def measure_nonneg suminf_ennreal)
finally show ?thesis
using ge0 ennreal_le_iff by blast
qed
@@ -1781,7 +1774,7 @@
lemma measurable_Un_null_set:
assumes "B \<in> null_sets M"
shows "(A \<union> B \<in> fmeasurable M \<and> A \<in> sets M) \<longleftrightarrow> A \<in> fmeasurable M"
- using assms by (fastforce simp add: fmeasurable.Un fmeasurableI_null_sets intro: fmeasurableI2)
+ using assms by (fastforce simp: fmeasurable.Un fmeasurableI_null_sets intro: fmeasurableI2)
lemma measurable_Diff_null_set:
assumes "B \<in> null_sets M"
@@ -1800,7 +1793,7 @@
qed
lemma measure_Un2:
- "A \<in> fmeasurable M \<Longrightarrow> B \<in> fmeasurable M \<Longrightarrow> measure M (A \<union> B) = measure M A + measure M (B - A)"
+ "\<lbrakk>A \<in> fmeasurable M; B \<in> fmeasurable M\<rbrakk> \<Longrightarrow> measure M (A\<union>B) = measure M A + measure M (B - A)"
using measure_Union[of M A "B - A"] by (auto simp: fmeasurableD2 fmeasurable.Diff)
lemma measure_Un3:
@@ -1991,7 +1984,7 @@
fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
assumes "summable f"
shows "(\<lambda>n. (\<Sum>k. f(k+n))) \<longlonglongrightarrow> 0"
-by (subst lim_sequentially, auto simp add: dist_norm suminf_exist_split[OF _ assms])
+by (subst lim_sequentially, auto simp: dist_norm suminf_exist_split[OF _ assms])
lemma emeasure_union_summable:
assumes [measurable]: "\<And>n. A n \<in> sets M"
@@ -2000,17 +1993,18 @@
proof -
define B where "B = (\<lambda>N. (\<Union>n\<in>{..<N}. A n))"
have [measurable]: "B N \<in> sets M" for N unfolding B_def by auto
- have "(\<lambda>N. emeasure M (B N)) \<longlonglongrightarrow> emeasure M (\<Union>N. B N)"
- apply (rule Lim_emeasure_incseq) unfolding B_def by (auto simp add: SUP_subset_mono incseq_def)
+ have "incseq B"
+ by (auto simp: SUP_subset_mono B_def incseq_def)
+ then have "(\<lambda>N. emeasure M (B N)) \<longlonglongrightarrow> emeasure M (\<Union>N. B N)"
+ by (simp add: Lim_emeasure_incseq image_subset_iff)
moreover have "emeasure M (B N) \<le> ennreal (\<Sum>n. measure M (A n))" for N
proof -
have *: "(\<Sum>n<N. measure M (A n)) \<le> (\<Sum>n. measure M (A n))"
- using assms(3) measure_nonneg sum_le_suminf by blast
-
+ using \<open>summable _\<close> measure_nonneg sum_le_suminf by blast
have "emeasure M (B N) \<le> (\<Sum>n<N. emeasure M (A n))"
unfolding B_def by (rule emeasure_subadditive_finite, auto)
also have "\<dots> = (\<Sum>n<N. ennreal(measure M (A n)))"
- using assms(2) by (simp add: emeasure_eq_ennreal_measure less_top)
+ using assms by (simp add: emeasure_eq_ennreal_measure less_top)
also have "\<dots> = ennreal (\<Sum>n<N. measure M (A n))"
by auto
also have "\<dots> \<le> ennreal (\<Sum>n. measure M (A n))"
@@ -2027,24 +2021,24 @@
lemma borel_cantelli_limsup1:
assumes [measurable]: "\<And>n. A n \<in> sets M"
- and "\<And>n. emeasure M (A n) < \<infinity>" "summable (\<lambda>n. measure M (A n))"
+ and "\<And>n. emeasure M (A n) < \<infinity>" and sum: "summable (\<lambda>n. measure M (A n))"
shows "limsup A \<in> null_sets M"
proof -
have "emeasure M (limsup A) \<le> 0"
proof (rule LIMSEQ_le_const)
- have "(\<lambda>n. (\<Sum>k. measure M (A (k+n)))) \<longlonglongrightarrow> 0" by (rule suminf_exist_split2[OF assms(3)])
+ have "(\<lambda>n. (\<Sum>k. measure M (A (k+n)))) \<longlonglongrightarrow> 0" by (rule suminf_exist_split2[OF sum])
then show "(\<lambda>n. ennreal (\<Sum>k. measure M (A (k+n)))) \<longlonglongrightarrow> 0"
unfolding ennreal_0[symmetric] by (intro tendsto_ennrealI)
have "emeasure M (limsup A) \<le> (\<Sum>k. measure M (A (k+n)))" for n
proof -
have I: "(\<Union>k\<in>{n..}. A k) = (\<Union>k. A (k+n))" by (auto, metis le_add_diff_inverse2, fastforce)
have "emeasure M (limsup A) \<le> emeasure M (\<Union>k\<in>{n..}. A k)"
- by (rule emeasure_mono, auto simp add: limsup_INF_SUP)
+ by (rule emeasure_mono, auto simp: limsup_INF_SUP)
also have "\<dots> = emeasure M (\<Union>k. A (k+n))"
using I by auto
also have "\<dots> \<le> (\<Sum>k. measure M (A (k+n)))"
apply (rule emeasure_union_summable)
- using assms summable_ignore_initial_segment[OF assms(3), of n] by auto
+ using assms summable_ignore_initial_segment[OF sum, of n] by auto
finally show ?thesis by simp
qed
then show "\<exists>N. \<forall>n\<ge>N. emeasure M (limsup A) \<le> (\<Sum>k. measure M (A (k+n)))"
@@ -2113,11 +2107,6 @@
assumes "A \<subseteq> B" "B \<in> sets M" shows "measure M A \<le> measure M B"
using emeasure_mono[OF assms] emeasure_real[of A] emeasure_real[of B] by (auto simp: measure_def)
-lemma (in finite_measure) finite_measure_subadditive:
- assumes m: "A \<in> sets M" "B \<in> sets M"
- shows "measure M (A \<union> B) \<le> measure M A + measure M B"
- using measure_subadditive[OF m] by simp
-
lemma (in finite_measure) finite_measure_subadditive_finite:
assumes "finite I" "A`I \<subseteq> sets M" shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))"
using measure_subadditive_finite[OF assms] by simp
@@ -2163,21 +2152,21 @@
by (auto intro!: finite_measure_mono simp: increasing_def)
lemma (in finite_measure) measure_zero_union:
- assumes "s \<in> sets M" "t \<in> sets M" "measure M t = 0"
- shows "measure M (s \<union> t) = measure M s"
+ assumes "S \<in> sets M" "T \<in> sets M" "measure M T = 0"
+ shows "measure M (S \<union> T) = measure M S"
using assms
proof -
- have "measure M (s \<union> t) \<le> measure M s"
- using finite_measure_subadditive[of s t] assms by auto
- moreover have "measure M (s \<union> t) \<ge> measure M s"
+ have "measure M (S \<union> T) \<le> measure M S"
+ by (metis add.right_neutral assms measure_Un_le)
+ moreover have "measure M (S \<union> T) \<ge> measure M S"
using assms by (blast intro: finite_measure_mono)
ultimately show ?thesis by simp
qed
lemma (in finite_measure) measure_eq_compl:
- assumes "s \<in> sets M" "t \<in> sets M"
- assumes "measure M (space M - s) = measure M (space M - t)"
- shows "measure M s = measure M t"
+ assumes "S \<in> sets M" "T \<in> sets M"
+ assumes "measure M (space M - S) = measure M (space M - T)"
+ shows "measure M S = measure M T"
using assms finite_measure_compl by auto
lemma (in finite_measure) measure_eq_bigunion_image:
@@ -2204,53 +2193,53 @@
qed simp
lemma (in finite_measure) measure_space_inter:
- assumes events:"s \<in> sets M" "t \<in> sets M"
- assumes "measure M t = measure M (space M)"
- shows "measure M (s \<inter> t) = measure M s"
+ assumes events:"S \<in> sets M" "T \<in> sets M"
+ assumes "measure M T = measure M (space M)"
+ shows "measure M (S \<inter> T) = measure M S"
proof -
- have "measure M ((space M - s) \<union> (space M - t)) = measure M (space M - s)"
- using events assms finite_measure_compl[of "t"] by (auto intro!: measure_zero_union)
- also have "(space M - s) \<union> (space M - t) = space M - (s \<inter> t)"
+ have "measure M ((space M - S) \<union> (space M - T)) = measure M (space M - S)"
+ using events assms finite_measure_compl[of "T"] by (auto intro!: measure_zero_union)
+ also have "(space M - S) \<union> (space M - T) = space M - (S \<inter> T)"
by blast
- finally show "measure M (s \<inter> t) = measure M s"
- using events by (auto intro!: measure_eq_compl[of "s \<inter> t" s])
+ finally show "measure M (S \<inter> T) = measure M S"
+ using events by (auto intro!: measure_eq_compl[of "S \<inter> T" S])
qed
lemma (in finite_measure) measure_equiprobable_finite_unions:
- assumes s: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> sets M"
- assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> measure M {x} = measure M {y}"
- shows "measure M s = real (card s) * measure M {SOME x. x \<in> s}"
+ assumes S: "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
+ assumes "\<And> x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> measure M {x} = measure M {y}"
+ shows "measure M S = real (card S) * measure M {SOME x. x \<in> S}"
proof cases
- assume "s \<noteq> {}"
- then have "\<exists> x. x \<in> s" by blast
+ assume "S \<noteq> {}"
+ then have "\<exists> x. x \<in> S" by blast
from someI_ex[OF this] assms
- have prob_some: "\<And> x. x \<in> s \<Longrightarrow> measure M {x} = measure M {SOME y. y \<in> s}" by blast
- have "measure M s = (\<Sum> x \<in> s. measure M {x})"
- using finite_measure_eq_sum_singleton[OF s] by simp
- also have "\<dots> = (\<Sum> x \<in> s. measure M {SOME y. y \<in> s})" using prob_some by auto
- also have "\<dots> = real (card s) * measure M {(SOME x. x \<in> s)}"
+ have prob_some: "\<And> x. x \<in> S \<Longrightarrow> measure M {x} = measure M {SOME y. y \<in> S}" by blast
+ have "measure M S = (\<Sum> x \<in> S. measure M {x})"
+ using finite_measure_eq_sum_singleton[OF S] by simp
+ also have "\<dots> = (\<Sum> x \<in> S. measure M {SOME y. y \<in> S})" using prob_some by auto
+ also have "\<dots> = real (card S) * measure M {(SOME x. x \<in> S)}"
using sum_constant assms by simp
finally show ?thesis by simp
qed simp
lemma (in finite_measure) measure_real_sum_image_fn:
assumes "e \<in> sets M"
- assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> sets M"
- assumes "finite s"
- assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}"
- assumes upper: "space M \<subseteq> (\<Union>i \<in> s. f i)"
- shows "measure M e = (\<Sum> x \<in> s. measure M (e \<inter> f x))"
+ assumes "\<And> x. x \<in> S \<Longrightarrow> e \<inter> f x \<in> sets M"
+ assumes "finite S"
+ assumes disjoint: "\<And> x y. \<lbrakk>x \<in> S ; y \<in> S ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}"
+ assumes upper: "space M \<subseteq> (\<Union>i \<in> S. f i)"
+ shows "measure M e = (\<Sum> x \<in> S. measure M (e \<inter> f x))"
proof -
- have "e \<subseteq> (\<Union>i\<in>s. f i)"
+ have "e \<subseteq> (\<Union>i\<in>S. f i)"
using \<open>e \<in> sets M\<close> sets.sets_into_space upper by blast
- then have e: "e = (\<Union>i \<in> s. e \<inter> f i)"
+ then have e: "e = (\<Union>i \<in> S. e \<inter> f i)"
by auto
- hence "measure M e = measure M (\<Union>i \<in> s. e \<inter> f i)" by simp
- also have "\<dots> = (\<Sum> x \<in> s. measure M (e \<inter> f x))"
+ hence "measure M e = measure M (\<Union>i \<in> S. e \<inter> f i)" by simp
+ also have "\<dots> = (\<Sum> x \<in> S. measure M (e \<inter> f x))"
proof (rule finite_measure_finite_Union)
- show "finite s" by fact
- show "(\<lambda>i. e \<inter> f i)`s \<subseteq> sets M" using assms(2) by auto
- show "disjoint_family_on (\<lambda>i. e \<inter> f i) s"
+ show "finite S" by fact
+ show "(\<lambda>i. e \<inter> f i)`S \<subseteq> sets M" using assms(2) by auto
+ show "disjoint_family_on (\<lambda>i. e \<inter> f i) S"
using disjoint by (auto simp: disjoint_family_on_def)
qed
finally show ?thesis .
@@ -2286,7 +2275,7 @@
next
show "f x \<le> (\<lambda>s. emeasure (M s) {x \<in> space N. F top x})" for x
using bound[of x] sets_eq_imp_space_eq[OF sets] by (simp add: iter)
-qed (auto simp add: iter le_fun_def INF_apply[abs_def] intro!: meas cont)
+qed (auto simp: iter le_fun_def INF_apply[abs_def] intro!: meas cont)
subsection\<^marker>\<open>tag unimportant\<close> \<open>Counting space\<close>
@@ -2357,8 +2346,8 @@
by (rule infinite_super[OF _ 1]) auto
then have "infinite (\<Union>i. F i)"
by auto
- ultimately show ?thesis by (simp only:) simp
-
+ ultimately show ?thesis
+ by (metis (mono_tags, lifting) infinity_ennreal_def)
qed
qed
qed
@@ -2391,8 +2380,9 @@
using emeasure_count_space[of X A] by simp
lemma measure_count_space: "measure (count_space A) X = (if X \<subseteq> A then of_nat (card X) else 0)"
- by (cases "finite X") (auto simp: measure_notin_sets ennreal_of_nat_eq_real_of_nat
- measure_zero_top measure_eq_emeasure_eq_ennreal)
+ by (cases "finite X")
+ (auto simp: measure_notin_sets ennreal_of_nat_eq_real_of_nat
+ measure_zero_top measure_eq_emeasure_eq_ennreal)
lemma emeasure_count_space_eq_0:
"emeasure (count_space A) X = 0 \<longleftrightarrow> (X \<subseteq> A \<longrightarrow> X = {})"
@@ -2407,19 +2397,19 @@
qed (simp add: emeasure_notin_sets)
lemma null_sets_count_space: "null_sets (count_space A) = { {} }"
- unfolding null_sets_def by (auto simp add: emeasure_count_space_eq_0)
+ unfolding null_sets_def by (auto simp: emeasure_count_space_eq_0)
lemma AE_count_space: "(AE x in count_space A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
- unfolding eventually_ae_filter by (auto simp add: null_sets_count_space)
+ unfolding eventually_ae_filter by (auto simp: null_sets_count_space)
lemma sigma_finite_measure_count_space_countable:
assumes A: "countable A"
shows "sigma_finite_measure (count_space A)"
- proof qed (insert A, auto intro!: exI[of _ "(\<lambda>a. {a}) ` A"])
+ proof qed (use A in \<open>auto intro!: exI[of _ "(\<lambda>a. {a}) ` A"]\<close>)
lemma sigma_finite_measure_count_space:
fixes A :: "'a::countable set" shows "sigma_finite_measure (count_space A)"
- by (rule sigma_finite_measure_count_space_countable) auto
+ using countableI_type sigma_finite_measure_count_space_countable by blast
lemma finite_measure_count_space:
assumes [simp]: "finite A"
@@ -2428,10 +2418,7 @@
lemma sigma_finite_measure_count_space_finite:
assumes A: "finite A" shows "sigma_finite_measure (count_space A)"
-proof -
- interpret finite_measure "count_space A" using A by (rule finite_measure_count_space)
- show "sigma_finite_measure (count_space A)" ..
-qed
+ by (simp add: assms finite_measure.axioms(1) finite_measure_count_space)
subsection\<^marker>\<open>tag unimportant\<close> \<open>Measure restricted to space\<close>
@@ -2453,15 +2440,13 @@
by (fastforce simp: sets_restrict_space_iff[OF assms(1)] image_subset_iff
dest: sets.sets_into_space)+
then show "(\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)"
- by (subst suminf_emeasure) (auto simp: disjoint_family_subset)
+ by (simp add: image_subset_iff suminf_emeasure)
qed
qed
next
case False
- with assms have "A \<notin> sets (restrict_space M \<Omega>)"
- by (simp add: sets_restrict_space_iff)
- with False show ?thesis
- by (simp add: emeasure_notin_sets)
+ with assms show ?thesis
+ by (metis emeasure_notin_sets sets_restrict_space_iff)
qed
lemma measure_restrict_space:
@@ -2475,14 +2460,12 @@
proof -
have ex_cong: "\<And>P Q f. (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> (\<And>x. Q x \<Longrightarrow> P (f x)) \<Longrightarrow> (\<exists>x. P x) \<longleftrightarrow> (\<exists>x. Q x)"
by auto
- { fix X assume X: "X \<in> sets M" "emeasure M X = 0"
- then have "emeasure M (\<Omega> \<inter> space M \<inter> X) \<le> emeasure M X"
- by (intro emeasure_mono) auto
- then have "emeasure M (\<Omega> \<inter> space M \<inter> X) = 0"
- using X by (auto intro!: antisym) }
+ have "emeasure M (\<Omega> \<inter> space M \<inter> X) = 0"
+ if "X \<in> sets M" "emeasure M X = 0" for X
+ by (meson emeasure_eq_0 inf_le2 that)
with assms show ?thesis
unfolding eventually_ae_filter
- by (auto simp add: space_restrict_space null_sets_def sets_restrict_space_iff
+ by (auto simp: space_restrict_space null_sets_def sets_restrict_space_iff
emeasure_restrict_space cong: conj_cong
intro!: ex_cong[where f="\<lambda>X. (\<Omega> \<inter> space M) \<inter> X"])
qed
@@ -2509,7 +2492,7 @@
moreover fix X assume "X \<in> sets (restrict_space (count_space B) A)"
ultimately have "X \<subseteq> A \<inter> B" by auto
then show "emeasure (restrict_space (count_space B) A) X = emeasure (count_space (A \<inter> B)) X"
- by (cases "finite X") (auto simp add: emeasure_restrict_space)
+ by (cases "finite X") (auto simp: emeasure_restrict_space)
qed
lemma sigma_finite_measure_restrict_space:
@@ -2523,28 +2506,24 @@
by blast
let ?C = "(\<inter>) A ` C"
from C have "countable ?C" "?C \<subseteq> sets (restrict_space M A)" "(\<Union>?C) = space (restrict_space M A)"
- by(auto simp add: sets_restrict_space space_restrict_space)
+ by(auto simp: sets_restrict_space space_restrict_space)
moreover {
fix a
assume "a \<in> ?C"
then obtain a' where "a = A \<inter> a'" "a' \<in> C" ..
then have "emeasure (restrict_space M A) a \<le> emeasure M a'"
- using A C by(auto simp add: emeasure_restrict_space intro: emeasure_mono)
- also have "\<dots> < \<infinity>" using C(4)[rule_format, of a'] \<open>a' \<in> C\<close> by (simp add: less_top)
+ using A C by(auto simp: emeasure_restrict_space intro: emeasure_mono)
+ also have "\<dots> < \<infinity>" using C(4) \<open>a' \<in> C\<close> top.not_eq_extremum by auto
finally have "emeasure (restrict_space M A) a \<noteq> \<infinity>" by simp }
ultimately show ?thesis
- by unfold_locales (rule exI conjI|assumption|blast)+
+ by (meson sigma_finite_measure_def)
qed
lemma finite_measure_restrict_space:
assumes "finite_measure M"
and A: "A \<in> sets M"
shows "finite_measure (restrict_space M A)"
-proof -
- interpret finite_measure M by fact
- show ?thesis
- by(rule finite_measureI)(simp add: emeasure_restrict_space A space_restrict_space)
-qed
+ by (simp add: assms emeasure_restrict_space finite_measure.emeasure_finite finite_measureI)
lemma restrict_distr:
assumes [measurable]: "f \<in> measurable M N"
@@ -2554,9 +2533,8 @@
proof (rule measure_eqI)
fix A assume "A \<in> sets (restrict_space (distr M N f) \<Omega>)"
with restrict show "emeasure ?l A = emeasure ?r A"
- by (subst emeasure_distr)
- (auto simp: sets_restrict_space_iff emeasure_restrict_space emeasure_distr
- intro!: measurable_restrict_space2)
+ by (simp add: emeasure_distr emeasure_restrict_space measurable_restrict_space2
+ sets_restrict_space_iff)
qed (simp add: sets_restrict_space)
lemma measure_eqI_restrict_generator:
@@ -2570,12 +2548,13 @@
proof (rule measure_eqI)
fix X assume X: "X \<in> sets M"
then have "emeasure M X = emeasure (restrict_space M \<Omega>) (X \<inter> \<Omega>)"
- using ae \<Omega> by (auto simp add: emeasure_restrict_space intro!: emeasure_eq_AE)
+ using ae \<Omega> by (auto simp: emeasure_restrict_space intro!: emeasure_eq_AE)
also have "restrict_space M \<Omega> = restrict_space N \<Omega>"
proof (rule measure_eqI_generator_eq)
fix X assume "X \<in> E"
then show "emeasure (restrict_space M \<Omega>) X = emeasure (restrict_space N \<Omega>) X"
- using E \<Omega> by (subst (1 2) emeasure_restrict_space) (auto simp: sets_eq sets_eq[THEN sets_eq_imp_space_eq])
+ using \<Omega> E
+ by (metis Pow_iff emeasure_restrict_space inf.orderE sets.sets_into_space sets_eq subsetD)
next
show "range (from_nat_into A) \<subseteq> E" "(\<Union>i. from_nat_into A i) = \<Omega>"
using A by (auto cong del: SUP_cong_simp)
@@ -2588,7 +2567,7 @@
by (auto intro: from_nat_into)
qed fact+
also have "emeasure (restrict_space N \<Omega>) (X \<inter> \<Omega>) = emeasure N X"
- using X ae \<Omega> by (auto simp add: emeasure_restrict_space sets_eq intro!: emeasure_eq_AE)
+ using X ae \<Omega> by (auto simp: emeasure_restrict_space sets_eq intro!: emeasure_eq_AE)
finally show "emeasure M X = emeasure N X" .
qed fact
@@ -2695,14 +2674,10 @@
have le_\<gamma>[intro]: "X \<in> sets M \<Longrightarrow> d X \<le> \<gamma>" for X
by (auto simp: \<gamma>_def intro!: cSUP_upper)
- have "\<exists>f. \<forall>n. f n \<in> sets M \<and> d (f n) > \<gamma> - 1 / 2^n"
- proof (intro choice_iff[THEN iffD1] allI)
- fix n
- have "\<exists>X\<in>sets M. \<gamma> - 1 / 2^n < d X"
- unfolding \<gamma>_def by (intro less_cSUP_iff[THEN iffD1]) auto
- then show "\<exists>y. y \<in> sets M \<and> \<gamma> - 1 / 2 ^ n < d y"
- by auto
- qed
+ have "\<exists>X\<in>sets M. \<gamma> - 1 / 2^n < d X" for n
+ unfolding \<gamma>_def by (intro less_cSUP_iff[THEN iffD1]) auto
+ then have "\<exists>f. \<forall>n. f n \<in> sets M \<and> d (f n) > \<gamma> - 1 / 2^n"
+ by metis
then obtain E where [measurable]: "E n \<in> sets M" and E: "d (E n) > \<gamma> - 1 / 2^n" for n
by auto
@@ -2987,9 +2962,8 @@
using sets.sigma_sets_subset[of \<A> x] by auto
lemma sigma_le_iff: "\<A> \<subseteq> Pow \<Omega> \<Longrightarrow> sigma \<Omega> \<A> \<le> x \<longleftrightarrow> (\<Omega> \<subseteq> space x \<and> (space x = \<Omega> \<longrightarrow> \<A> \<subseteq> sets x))"
- by (cases "\<Omega> = space x")
- (simp_all add: eq_commute[of _ "sets x"] le_measure_iff emeasure_sigma le_fun_def
- sigma_sets_superset_generator sigma_sets_le_sets_iff)
+ apply (simp add: le_measure_iff le_fun_def emeasure_sigma)
+ by (metis order_refl sets_measure_of sigma_sets_le_sets_iff)
instantiation measure :: (type) semilattice_sup
begin
@@ -3057,7 +3031,7 @@
by (intro less_eq_measure.intros(2)) simp_all
next
case 3 with \<open>z \<le> y\<close> \<open>x \<le> y\<close> show ?thesis
- by (auto simp add: le_measure intro!: emeasure_sup_measure'_le2)
+ by (auto simp: le_measure intro!: emeasure_sup_measure'_le2)
qed
next
assume **: "x \<le> y" "z \<le> y" "space x = space z" "\<not> sets z \<subseteq> sets x" "\<not> sets x \<subseteq> sets z"
@@ -3083,7 +3057,7 @@
by (auto dest: sets_eq_imp_space_eq simp add: le_measure_iff split: if_split_asm)
lemma sets_sup: "sets A = sets M \<Longrightarrow> sets B = sets M \<Longrightarrow> sets (sup A B) = sets M"
- by (auto simp add: sup_measure_def sup_lexord_def dest: sets_eq_imp_space_eq)
+ by (auto simp: sup_measure_def sup_lexord_def dest: sets_eq_imp_space_eq)
lemma le_measureD1: "A \<le> B \<Longrightarrow> space A \<le> space B"
by (auto simp: le_measure_iff split: if_split_asm)
@@ -3110,18 +3084,13 @@
by (auto simp: Sup_lexord_def Let_def)
lemma Sup_lexord1:
- assumes A: "A \<noteq> {}" "(\<And>a. a \<in> A \<Longrightarrow> k a = (\<Union>a\<in>A. k a))" "P (c A)"
+ assumes A: "A \<noteq> {}" and eq: "(\<And>a. a \<in> A \<Longrightarrow> k a = (\<Union>a\<in>A. k a))" and P: "P (c A)"
shows "P (Sup_lexord k c s A)"
- unfolding Sup_lexord_def Let_def
-proof (clarsimp, safe)
- show "\<forall>a\<in>A. k a \<noteq> (\<Union>x\<in>A. k x) \<Longrightarrow> P (s A)"
- by (metis assms(1,2) ex_in_conv)
-next
- fix a assume "a \<in> A" "k a = (\<Union>x\<in>A. k x)"
- then have "{a \<in> A. k a = (\<Union>x\<in>A. k x)} = {a \<in> A. k a = k a}"
- by (metis A(2)[symmetric])
- then show "P (c {a \<in> A. k a = (\<Union>x\<in>A. k x)})"
- by (simp add: A(3))
+proof -
+ have "{a \<in> A. k a = \<Union>(k ` A)} = A" for a :: 'a
+ by (metis (mono_tags, lifting) Collect_cong Collect_mem_eq eq)
+ then show ?thesis
+ using A P by (auto simp: Sup_lexord_def Let_def)
qed
instantiation measure :: (type) complete_lattice
@@ -3137,19 +3106,8 @@
by simp
next
case (insert i J)
- show ?case
- proof cases
- assume "i \<in> I" with insert show ?thesis
- by (auto simp: insert_absorb)
- next
- assume "i \<notin> I"
- have "sup_measure.F id I \<le> sup_measure.F id (I \<union> J)"
- by (intro insert)
- also have "\<dots> \<le> sup_measure.F id (insert i (I \<union> J))"
- using insert \<open>i \<notin> I\<close> by (subst sup_measure.insert) auto
- finally show ?thesis
- by auto
- qed
+ then show ?case
+ by (metis finite.insertI sup.orderE sup_ge1 sup_ge2 sup_measure.union_diff2 sup_measure.union_inter)
qed
lemma sup_measure_F_mono: "finite I \<Longrightarrow> J \<subseteq> I \<Longrightarrow> sup_measure.F id J \<le> sup_measure.F id I"
@@ -3214,9 +3172,10 @@
proof (intro arg_cong [of _ _ Sup] image_cong refl)
fix i assume i: "i \<in> {P. finite P \<and> P \<subseteq> M}"
show "(\<Sum>n. ?\<mu> i (F n)) = ?\<mu> i (\<Union>(F ` UNIV))"
- proof cases
- assume "i \<noteq> {}" with i ** show ?thesis
- by (smt (verit, best) "1"(2) Measure_Space.sets_sup_measure_F assms(1) mem_Collect_eq subset_eq suminf_cong suminf_emeasure)
+ proof (cases "i = {}")
+ case False
+ with i ** sets_eq show ?thesis
+ by (smt (verit, best) "1"(2) Measure_Space.sets_sup_measure_F mem_Collect_eq subset_eq suminf_cong suminf_emeasure)
qed simp
qed
qed
@@ -3300,7 +3259,7 @@
proof (subst emeasure_Sup_measure'[OF _ \<open>X \<in> sets x\<close>])
show "emeasure x X \<le> (SUP P \<in> {P. finite P \<and> P \<subseteq> S'}. emeasure (sup_measure.F id P) X)"
using \<open>x\<in>S'\<close> by (intro SUP_upper2[where i="{x}"]) auto
- qed (insert \<open>x\<in>S'\<close> S', auto)
+ qed (use \<open>x\<in>S'\<close> S' in auto)
qed
next
assume "x \<notin> S'"
@@ -3309,15 +3268,12 @@
moreover have "sets x \<le> sets b"
using \<open>x\<in>S\<close> unfolding b by auto
ultimately show ?thesis
- using * \<open>x \<in> S\<close>
- by (intro less_eq_measure.intros(2))
- (simp_all add: * \<open>space x = space b\<close> less_le)
+ using * \<open>x \<in> S\<close> by (simp add: le_measure_iff sets_le_imp_space_le)
qed
next
assume "x \<notin> S"
with \<open>x\<in>A\<close> \<open>x \<notin> S\<close> \<open>space b = space a\<close> show ?thesis
- by (intro less_eq_measure.intros)
- (simp_all add: * less_le a SUP_upper S)
+ by (simp add: "*" S SUP_upper2 a le_measure_iff)
qed
qed
show least: "Sup A \<le> x" if x: "\<And>z. z \<in> A \<Longrightarrow> z \<le> x" for x :: "'a measure" and A
@@ -3397,7 +3353,7 @@
next
assume "space x \<noteq> space a"
then have "space a < space x"
- using le_measureD1[OF x[OF \<open>a\<in>A\<close>]] by auto
+ by (simp add: \<open>a \<in> A\<close> le_measureD1 psubsetI x)
then show "Sup_measure' S' \<le> x"
by (intro less_eq_measure.intros) (simp add: * \<open>space b = space a\<close>)
qed
@@ -3447,15 +3403,15 @@
show "\<exists>j\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. sup_measure.F id J X \<le> (SUP i\<in>j. M i) X"
proof cases
assume "J' = {}" with \<open>i \<in> I\<close> show ?thesis
- by (auto simp add: J)
+ by (auto simp: J)
next
assume "J' \<noteq> {}" with J J' show ?thesis
- by (intro bexI[of _ "J'"]) (auto simp add: eq simp del: id_apply)
+ using eq by auto
qed
next
fix J assume J: "J \<in> {P. P \<noteq> {} \<and> finite P \<and> P \<subseteq> I}"
show "\<exists>J'\<in>{J. finite J \<and> J \<subseteq> M`I}. (SUP i\<in>J. M i) X \<le> sup_measure.F id J' X"
- using J by (intro bexI[of _ "M`J"]) (auto simp add: eq simp del: id_apply)
+ using J by (intro bexI[of _ "M`J"]) (auto simp: eq simp del: id_apply)
qed
finally show ?thesis .
qed
@@ -3485,9 +3441,15 @@
lemma space_Sup_eq_UN: "space (Sup M) = (\<Union>x\<in>M. space x)" (is "?L=?R")
proof
show "?L \<subseteq> ?R"
- using Sup_lexord[where P="\<lambda>x. space x = _"]
- apply (clarsimp simp: Sup_measure_def)
- by (smt (verit) Sup_lexord_def UN_E mem_Collect_eq space_Sup_measure'2 space_measure_of_conv)
+ proof -
+ define A where "A \<equiv> {a \<in> M. space a = \<Union> (space ` M)}"
+ have "\<exists>x\<in>M. a \<in> space x"
+ if "a \<in> space (Sup_measure' {a \<in> A. sets a = \<Union> (sets ` A)})"
+ for a
+ by (metis (no_types, lifting) A_def UN_E mem_Collect_eq space_Sup_measure'2 that)
+ then show ?thesis
+ by (auto simp: A_def space_measure_of_conv Sup_measure_def Sup_lexord_def Let_def split: if_splits)
+ qed
qed (use Sup_upper le_measureD1 in fastforce)
@@ -3546,7 +3508,7 @@
shows "f \<in> measurable (Sup M) N"
proof -
have "space (Sup M) = space m"
- using m by (auto simp add: space_Sup_eq_UN dest: const_space)
+ using m by (auto simp: space_Sup_eq_UN dest: const_space)
then show ?thesis
using m f unfolding measurable_def by (auto intro: in_sets_Sup[OF const_space])
qed
@@ -3581,13 +3543,13 @@
assumes [simp]: "M \<noteq> {}" and M: "\<And>m. m \<in> M \<Longrightarrow> m \<subseteq> Pow \<Omega>"
shows "sets (SUP m\<in>M. sigma \<Omega> m) = sets (sigma \<Omega> (\<Union>M))"
proof -
- { fix a m assume "a \<in> sigma_sets \<Omega> m" "m \<in> M"
- then have "a \<in> sigma_sets \<Omega> (\<Union>M)"
- by induction (auto intro: sigma_sets.intros(2-)) }
+ have "a \<in> sigma_sets \<Omega> (\<Union>M)"
+ if "a \<in> sigma_sets \<Omega> m" "m \<in> M" for a m
+ using that by induction (auto intro: sigma_sets.intros)
then have "sigma_sets \<Omega> (\<Union> (sigma_sets \<Omega> ` M)) = sigma_sets \<Omega> (\<Union> M)"
by (smt (verit, best) UN_iff Union_iff sigma_sets.Basic sigma_sets_eqI)
then show "sets (SUP m\<in>M. sigma \<Omega> m) = sets (sigma \<Omega> (\<Union>M))"
- by (subst sets_Sup_eq) (fastforce simp add: M Union_least)+
+ by (subst sets_Sup_eq) (fastforce simp: M Union_least)+
qed
lemma Sup_sigma:
@@ -3600,10 +3562,10 @@
proof (intro less_eq_measure.intros(3))
show "space (sigma \<Omega> (\<Union>M)) = space (SUP m\<in>M. sigma \<Omega> m)"
"sets (sigma \<Omega> (\<Union>M)) = sets (SUP m\<in>M. sigma \<Omega> m)"
- by (auto simp add: M sets_Sup_sigma sets_eq_imp_space_eq space_measure_of_conv)
+ by (auto simp: M sets_Sup_sigma sets_eq_imp_space_eq space_measure_of_conv)
qed (simp add: emeasure_sigma le_fun_def)
fix m assume "m \<in> M" then show "sigma \<Omega> m \<le> sigma \<Omega> (\<Union>M)"
- by (subst sigma_le_iff) (auto simp add: M *)
+ by (subst sigma_le_iff) (auto simp: M *)
qed
lemma SUP_sigma_sigma:
@@ -3615,29 +3577,25 @@
shows "sets (vimage_algebra X f (Sup M)) = sets (SUP m \<in> M. vimage_algebra X f m)"
(is "?L = ?R")
proof
- have "\<And>m. m \<in> M \<Longrightarrow> f \<in> Sup (vimage_algebra X f ` M) \<rightarrow>\<^sub>M m"
- using assms
- by (smt (verit, del_insts) Pi_iff imageE image_eqI measurable_Sup1
- measurable_vimage_algebra1 space_vimage_algebra)
+ { fix m
+ assume "m \<in> M"
+ then have "f \<in> vimage_algebra X f m \<rightarrow>\<^sub>M m"
+ by (simp add: assms measurable_vimage_algebra1)
+ then have "f \<in> Sup (vimage_algebra X f ` M) \<rightarrow>\<^sub>M m"
+ using \<open>m \<in> M\<close> by (force simp: intro: measurable_Sup1)
+ }
then show "?L \<subseteq> ?R"
- by (intro sets_image_in_sets measurable_Sup2) (simp_all add: space_Sup_eq_UN *)
+ by (intro sets_image_in_sets measurable_Sup2) (simp_all add: space_Sup_eq_UN *)
show "?R \<subseteq> ?L"
apply (intro sets_Sup_in_sets)
- apply (force simp add: * space_Sup_eq_UN sets_vimage_algebra2 intro: in_sets_Sup)+
+ apply (force simp: * space_Sup_eq_UN sets_vimage_algebra2 intro: in_sets_Sup)+
done
qed
lemma restrict_space_eq_vimage_algebra':
"sets (restrict_space M \<Omega>) = sets (vimage_algebra (\<Omega> \<inter> space M) (\<lambda>x. x) M)"
-proof -
- have *: "{A \<inter> (\<Omega> \<inter> space M) |A. A \<in> sets M} = {A \<inter> \<Omega> |A. A \<in> sets M}"
- using sets.sets_into_space[of _ M] by blast
-
- show ?thesis
- unfolding restrict_space_def
- by (subst sets_measure_of)
- (auto simp add: image_subset_iff sets_vimage_algebra * dest: sets.sets_into_space intro!: arg_cong2[where f=sigma_sets])
-qed
+ by (metis Int_assoc image_cong inf_le2 restrict_space_eq_vimage_algebra
+ sets.Int_space_eq1 sets_restrict_space)
lemma sigma_le_sets:
assumes [simp]: "A \<subseteq> Pow X" shows "sets (sigma X A) \<subseteq> sets N \<longleftrightarrow> X \<in> sets N \<and> A \<subseteq> sets N"
@@ -3657,8 +3615,13 @@
lemma measurable_iff_sets:
"f \<in> measurable M N \<longleftrightarrow> (f \<in> space M \<rightarrow> space N \<and> sets (vimage_algebra (space M) f N) \<subseteq> sets M)"
- unfolding measurable_def
- by (smt (verit, ccfv_threshold) mem_Collect_eq sets_vimage_algebra sigma_sets_le_sets_iff subset_eq)
+ (is "?L = ?R")
+proof
+ show "?L \<Longrightarrow> ?R"
+ by (simp add: measurable_space sets_image_in_sets)
+ show "?R \<Longrightarrow> ?L"
+ by (simp add: in_vimage_algebra measurable_def subset_eq)
+qed
lemma sets_vimage_algebra_space: "X \<in> sets (vimage_algebra X f M)"
using sets.top[of "vimage_algebra X f M"] by simp
@@ -3669,19 +3632,21 @@
shows "measurable M N \<subseteq> measurable M' N'"
unfolding measurable_def
proof safe
- fix f A assume "f \<in> space M \<rightarrow> space N" "A \<in> sets N'"
- moreover assume "\<forall>y\<in>sets N. f -` y \<inter> space M \<in> sets M" note this[THEN bspec, of A]
- ultimately show "f -` A \<inter> space M' \<in> sets M'"
- using assms by auto
+ fix f A
+ assume "f \<in> space M \<rightarrow> space N" "A \<in> sets N'"
+ "\<forall>y\<in>sets N. f -` y \<inter> space M \<in> sets M"
+ then show "f -` A \<inter> space M' \<in> sets M'"
+ using assms by (metis subset_eq)
qed (use N M in auto)
lemma measurable_Sup_measurable:
assumes f: "f \<in> space N \<rightarrow> A"
shows "f \<in> measurable N (Sup {M. space M = A \<and> f \<in> measurable N M})"
proof (rule measurable_Sup2)
- show "{M. space M = A \<and> f \<in> measurable N M} \<noteq> {}"
- using f unfolding ex_in_conv[symmetric]
- by (intro exI[of _ "sigma A {}"]) (auto intro!: measurable_measure_of)
+ have "f \<in> N \<rightarrow>\<^sub>M sigma A {}"
+ by (meson empty_subsetI equals0D f measurable_measure_of)
+ then show "{M. space M = A \<and> f \<in> measurable N M} \<noteq> {}"
+ by fastforce
qed auto
lemma (in sigma_algebra) sigma_sets_subset':
@@ -3709,9 +3674,7 @@
lemma mono_vimage_algebra:
"sets M \<le> sets N \<Longrightarrow> sets (vimage_algebra X f M) \<subseteq> sets (vimage_algebra X f N)"
- using sets.top[of "sigma X {f -` A \<inter> X |A. A \<in> sets N}"]
- unfolding vimage_algebra_def
- by (smt (verit, del_insts) space_measure_of sigma_le_sets Pow_iff inf_le2 mem_Collect_eq subset_eq)
+ by (simp add: in_vimage_algebra sets_image_in_sets' sets_vimage_algebra_space subsetD)
lemma mono_restrict_space: "sets M \<le> sets N \<Longrightarrow> sets (restrict_space M X) \<subseteq> sets (restrict_space N X)"
unfolding sets_restrict_space by (rule image_mono)
--- a/src/HOL/Analysis/Summation_Tests.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Analysis/Summation_Tests.thy Thu Apr 24 09:16:33 2025 +0200
@@ -287,6 +287,20 @@
finally show ?thesis .
qed
+lemma summable_power_int_real_iff:
+ "summable (\<lambda>n. real n powi c) \<longleftrightarrow> c < -1"
+proof -
+ have "summable (\<lambda>n. real n powi c) \<longleftrightarrow> summable (\<lambda>n. real (Suc n) powi c)"
+ by (subst summable_Suc_iff) auto
+ also have "(\<lambda>n. real (Suc n) powi c) = (\<lambda>n. real (Suc n) powr of_int c)"
+ by (subst powr_real_of_int') auto
+ also have "summable \<dots> \<longleftrightarrow> summable (\<lambda>n. real n powr of_int c)"
+ by (subst summable_Suc_iff) auto
+ also have "\<dots> \<longleftrightarrow> c < -1"
+ by (subst summable_real_powr_iff) auto
+ finally show ?thesis .
+qed
+
lemma inverse_power_summable:
assumes s: "s \<ge> 2"
shows "summable (\<lambda>n. inverse (of_nat n ^ s :: 'a :: {real_normed_div_algebra,banach}))"
--- a/src/HOL/Analysis/Weierstrass_Theorems.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Analysis/Weierstrass_Theorems.thy Thu Apr 24 09:16:33 2025 +0200
@@ -81,7 +81,7 @@
have ucontf: "uniformly_continuous_on {0..1} f"
using compact_uniformly_continuous contf by blast
then obtain d where d: "d>0" "\<And>x x'. \<lbrakk> x \<in> {0..1}; x' \<in> {0..1}; \<bar>x' - x\<bar> < d\<rbrakk> \<Longrightarrow> \<bar>f x' - f x\<bar> < e/2"
- apply (rule uniformly_continuous_onE [where e = "e/2"])
+ apply (rule uniformly_continuous_onE [where \<epsilon> = "e/2"])
using e by (auto simp: dist_norm)
{ fix n::nat and x::real
assume n: "Suc (nat\<lceil>4*M/(e*d\<^sup>2)\<rceil>) \<le> n" and x: "0 \<le> x" "x \<le> 1"
--- a/src/HOL/Archimedean_Field.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Archimedean_Field.thy Thu Apr 24 09:16:33 2025 +0200
@@ -705,6 +705,9 @@
lemma frac_eq: "frac x = x \<longleftrightarrow> 0 \<le> x \<and> x < 1"
by (simp add: frac_unique_iff)
+lemma frac_eq_id [simp]: "x \<in> {0..<1} \<Longrightarrow> frac x = x"
+ by (simp add: frac_eq)
+
lemma frac_neg: "frac (- x) = (if x \<in> \<int> then 0 else 1 - frac x)"
for x :: "'a::floor_ceiling"
apply (auto simp add: frac_unique_iff)
--- a/src/HOL/Codegenerator_Test/Generate_Efficient_Datastructures.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Codegenerator_Test/Generate_Efficient_Datastructures.thy Thu Apr 24 09:16:33 2025 +0200
@@ -30,6 +30,7 @@
"Gcd :: _ poly set \<Rightarrow> _"
"Lcm :: _ poly set \<Rightarrow> _"
nlists
+ Multiset.multisets_of_size
]]
text \<open>
--- a/src/HOL/Complex_Analysis/Cauchy_Integral_Formula.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Complex_Analysis/Cauchy_Integral_Formula.thy Thu Apr 24 09:16:33 2025 +0200
@@ -1887,7 +1887,7 @@
and \<eta>: "\<And>x x'. \<lbrakk>x\<in>?TZ; x'\<in>?TZ; dist x' x < \<eta>\<rbrakk> \<Longrightarrow>
dist ((\<lambda>(x,y). F x y) x') ((\<lambda>(x,y). F x y) x) < \<epsilon>/norm(b-a)"
using \<open>0 < \<epsilon>\<close> \<open>a \<noteq> b\<close>
- by (auto elim: uniformly_continuous_onE [where e = "\<epsilon>/norm(b-a)"])
+ by (auto elim: uniformly_continuous_onE [where \<epsilon> = "\<epsilon>/norm(b-a)"])
have \<eta>: "\<lbrakk>norm (w - x1) \<le> \<delta>; x2 \<in> closed_segment a b;
norm (w - x1') \<le> \<delta>; x2' \<in> closed_segment a b; norm ((x1', x2') - (x1, x2)) < \<eta>\<rbrakk>
\<Longrightarrow> norm (F x1' x2' - F x1 x2) \<le> \<epsilon> / cmod (b-a)"
@@ -2239,7 +2239,7 @@
then obtain kk where "kk>0"
and kk: "\<And>x x'. \<lbrakk>x \<in> ?ddpa; x' \<in> ?ddpa; dist x' x < kk\<rbrakk> \<Longrightarrow>
dist ((\<lambda>(x,y). d x y) x') ((\<lambda>(x,y). d x y) x) < ee"
- by (rule uniformly_continuous_onE [where e = ee]) (use \<open>0 < ee\<close> in auto)
+ by (rule uniformly_continuous_onE [where \<epsilon> = ee]) (use \<open>0 < ee\<close> in auto)
have kk: "\<lbrakk>norm (w-x) \<le> dd; z \<in> path_image \<gamma>; norm ((w, z) - (x, z)) < kk\<rbrakk> \<Longrightarrow> norm (d w z - d x z) < ee"
for w z
using \<open>dd>0\<close> kk [of "(x,z)" "(w,z)"] by (force simp: norm_minus_commute dist_norm)
--- a/src/HOL/Complex_Analysis/Conformal_Mappings.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Complex_Analysis/Conformal_Mappings.thy Thu Apr 24 09:16:33 2025 +0200
@@ -1703,7 +1703,7 @@
then have [simp]: "closure {r'<..<r} = {r'..r}" by simp
show ?thesis
apply (rule continuous_ge_on_closure
- [where f = "\<lambda>r. norm z / (r - norm z) * C" and s = "{r'<..<r}",
+ [where f = "\<lambda>r. norm z / (r - norm z) * C" and S = "{r'<..<r}",
OF _ _ T1])
using that r'
by (auto simp: not_le intro!: continuous_intros)
--- a/src/HOL/Complex_Analysis/Laurent_Convergence.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Complex_Analysis/Laurent_Convergence.thy Thu Apr 24 09:16:33 2025 +0200
@@ -1555,6 +1555,19 @@
using assms unfolding fps_eq_iff
by (auto simp: fps_eq_iff fps_nth_fps_expansion fps_expansion_def)
+lemma holomorphic_on_imp_fps_conv_radius_ge:
+ assumes "f has_fps_expansion F" "f holomorphic_on eball 0 r"
+ shows "fps_conv_radius F \<ge> r"
+proof -
+ define n where "n = subdegree F"
+ have "fps_conv_radius (fps_expansion f 0) \<ge> r"
+ by (intro conv_radius_fps_expansion assms)
+ also have "fps_expansion f 0 = F"
+ using assms by (intro fps_expansion_eqI)
+ finally show ?thesis
+ by simp
+qed
+
lemma has_fps_expansion_imp_eval_fps_eq:
assumes "f has_fps_expansion F" "norm z < r"
assumes "f holomorphic_on ball 0 r"
@@ -1572,6 +1585,31 @@
by simp
qed
+lemma has_fps_expansion_imp_sums_complex:
+ fixes F :: "complex fps"
+ assumes "f has_fps_expansion F" "f holomorphic_on eball 0 r" "ereal (norm z) < r"
+ shows "(\<lambda>n. fps_nth F n * z ^ n) sums f z"
+proof -
+ have r: "fps_conv_radius F \<ge> r"
+ using assms(1,2) by (rule holomorphic_on_imp_fps_conv_radius_ge)
+ from assms obtain R where R: "norm z < R" "ereal R < r"
+ using ereal_dense2 less_ereal.simps(1) by blast
+ have z: "norm z < fps_conv_radius F"
+ using r R assms(3) by order
+
+ have "summable (\<lambda>n. fps_nth F n * z ^ n)"
+ by (rule summable_fps) (use z in auto)
+ moreover have "eval_fps F z = f z"
+ proof (rule has_fps_expansion_imp_eval_fps_eq[where r = R])
+ have *: "ereal (norm z) < r" if "norm z < R" for z :: complex
+ using that R ereal_le_less less_imp_le by blast
+ show "f holomorphic_on ball 0 R"
+ using assms(2) by (rule holomorphic_on_subset) (use * in auto)
+ qed (use R assms(1) in auto)
+ ultimately show ?thesis
+ unfolding eval_fps_def sums_iff by simp
+qed
+
lemma fls_conv_radius_ge:
assumes "f has_laurent_expansion F"
assumes "f holomorphic_on eball 0 r - {0}"
--- a/src/HOL/Complex_Analysis/Winding_Numbers.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Complex_Analysis/Winding_Numbers.thy Thu Apr 24 09:16:33 2025 +0200
@@ -855,7 +855,7 @@
by (rule_tac x="min (pe/4) (e/2*pe^2/L/4)" in exI) (simp add: dist_norm cmod_wn_diff)
qed
then show ?thesis
- apply (rule continuous_transform_within [where d = "min d e/2"])
+ apply (rule continuous_transform_within [where \<delta> = "min d e/2"])
apply (auto simp: \<open>d>0\<close> \<open>e>0\<close> dist_norm wnwn)
done
qed
--- a/src/HOL/Computational_Algebra/Formal_Power_Series.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Computational_Algebra/Formal_Power_Series.thy Thu Apr 24 09:16:33 2025 +0200
@@ -12,6 +12,7 @@
Euclidean_Algorithm
Primes
"HOL-Library.FuncSet"
+ "HOL-Library.Multiset"
begin
@@ -3832,214 +3833,44 @@
subsection \<open>Finite and infinite products\<close>
-text \<open>
- The following operator gives the set of all functions $A \to \mathbb{N}$ with
- $\sum_{x\in A} f(x) = n$, i.e.\ all the different ways to put $n$ unlabelled balls into
- the labelled bins $A$.
-\<close>
-definition nat_partitions :: "'a set \<Rightarrow> nat \<Rightarrow> ('a \<Rightarrow> nat) set" where
- "nat_partitions A n =
- {f. finite {x. f x > 0} \<and> {x. f x > 0} \<subseteq> A \<and> (\<Sum>x\<in>{x\<in>A. f x > 0}. f x) = n}"
-
-lemma
- assumes "h \<in> nat_partitions A n"
- shows nat_partitions_outside: "x \<notin> A \<Longrightarrow> h x = 0"
- and nat_partitions_sum: "(\<Sum>x\<in>{x\<in>A. h x > 0}. h x) = n"
- and nat_partitions_finite_support: "finite {x. h x > 0}"
- using assms by (auto simp: nat_partitions_def)
-
-lemma nat_partitions_finite_def:
- assumes "finite A"
- shows "nat_partitions A n = {f. {x. f x > 0} \<subseteq> A \<and> (\<Sum>x\<in>A. f x) = n}"
-proof (intro equalityI subsetI)
- fix f assume f: "f \<in> nat_partitions A n"
- have "(\<Sum>x\<in>A. f x) = (\<Sum>x | x \<in> A \<and> f x > 0. f x)"
- by (rule sum.mono_neutral_right) (use assms in auto)
- thus "f \<in> {f. {x. f x > 0} \<subseteq> A \<and> (\<Sum>x\<in>A. f x) = n}"
- using f nat_partitions_def by auto
-next
- fix f assume "f \<in> {f. {x. f x > 0} \<subseteq> A \<and> (\<Sum>x\<in>A. f x) = n}"
- hence f: "{x. f x > 0} \<subseteq> A" "(\<Sum>x\<in>A. f x) = n"
- by blast+
- have "(\<Sum>x\<in>A. f x) = (\<Sum>x | x \<in> A \<and> f x > 0. f x)"
- by (rule sum.mono_neutral_right) (use assms in auto)
- moreover have "finite {x. f x > 0}"
- using f(1) assms by (rule finite_subset)
- ultimately show "f \<in> nat_partitions A n"
- using f by (simp add: nat_partitions_def)
-qed
-
-lemma nat_partitions_subset:
- "nat_partitions A n \<subseteq> A \<rightarrow> {0..n}"
-proof
- fix f assume f: "f \<in> nat_partitions A n"
- have "f x \<le> n" if x: "x \<in> A" for x
- proof (cases "f x = 0")
- case False
- have "f x \<le> (\<Sum>x\<in>{x\<in>A. f x \<noteq> 0}. f x)"
- by (rule member_le_sum) (use x False f in \<open>auto simp: nat_partitions_def\<close>)
- also have "\<dots> = n"
- using f by (auto simp: nat_partitions_def)
- finally show "f x \<le> n" .
- qed auto
- thus "f \<in> A \<rightarrow> {0..n}"
- using f by (auto simp: nat_partitions_def)
-qed
-
-lemma conj_mono_strong: "(P1 \<longrightarrow> Q1) \<Longrightarrow> (P1 \<longrightarrow> P2 \<longrightarrow> Q2) \<Longrightarrow> (P1 \<and> P2) \<longrightarrow> (Q1 \<and> Q2)"
- by iprover
-
-lemma nat_partitions_mono:
- assumes "A \<subseteq> B"
- shows "nat_partitions A n \<subseteq> nat_partitions B n"
- unfolding nat_partitions_def
-proof (intro Collect_mono conj_mono_strong impI)
- fix f :: "'a \<Rightarrow> nat"
- assume 1: "finite {x. f x > 0}"
- assume 2: "{x. f x > 0} \<subseteq> A"
- thus "{x. f x > 0} \<subseteq> B"
- using assms by blast
- assume 3: "(\<Sum>x | x \<in> A \<and> f x > 0. f x) = n"
- have *: "{x\<in>B. f x > 0} = {x\<in>A. f x > 0} \<union> {x\<in>B-A. f x > 0}"
- using assms by auto
- have "(\<Sum>x | x \<in> B \<and> f x > 0. f x) = (\<Sum>x | x \<in> A \<and> f x > 0. f x)"
- by (intro sum.mono_neutral_right) (use assms 2 in \<open>auto intro: finite_subset[OF _ 1]\<close>)
- with 3 show "(\<Sum>x | x \<in> B \<and> f x > 0. f x) = n"
- by simp
-qed
-
-lemma in_nat_partitions_imp_le:
- assumes "h \<in> nat_partitions A n" "x \<in> A"
- shows "h x \<le> n"
- using nat_partitions_subset[of A n] assms by (auto simp: PiE_def Pi_def)
-
-lemma nat_partitions_0 [simp]: "nat_partitions A 0 = {\<lambda>_. 0}"
-proof (intro equalityI subsetI)
- fix h :: "'a \<Rightarrow> nat" assume "h \<in> {\<lambda>_. 0}"
- thus "h \<in> nat_partitions A 0"
- by (auto simp: nat_partitions_def)
-qed (auto simp: nat_partitions_def fun_eq_iff)
-
-lemma nat_partitions_empty [simp]: "n > 0 \<Longrightarrow> nat_partitions {} n = {}"
- by (auto simp: nat_partitions_def fun_eq_iff)
-
-lemma nat_partitions_insert:
- assumes [simp]: "a \<notin> A"
- shows "bij_betw (\<lambda>(m,f). f(a := m))
- (SIGMA m:{0..n}. nat_partitions A (n - m)) (nat_partitions (insert a A) n)"
-proof (rule bij_betwI[of _ _ _ "\<lambda>f. (f a, f(a := 0))"], goal_cases)
- case 1
- show ?case
- proof safe
- fix m f assume m: "m \<in> {0..n}" and f: "f \<in> nat_partitions A (n - m)"
- define f' where "f' = f(a := m)"
- have "(\<Sum>x\<in>{x\<in>insert a A. f' x > 0}. f' x) = (\<Sum>x\<in>insert a {x\<in>A. f x > 0}. f' x)"
- by (rule sum.mono_neutral_cong_left)
- (use f in \<open>auto simp: f'_def nat_partitions_def\<close>)
- also have "\<dots> = m + (\<Sum>x\<in>{x\<in>A. f x > 0}. f' x)"
- by (subst sum.insert) (use f in \<open>auto simp: nat_partitions_def f'_def\<close>)
- also have "(\<Sum>x\<in>{x\<in>A. f x > 0}. f' x) = (\<Sum>x\<in>{x\<in>A. f x > 0}. f x)"
- by (intro sum.cong) (auto simp: f'_def)
- also have "\<dots> = n - m"
- using f by (auto simp: nat_partitions_def)
- finally have "(\<Sum>x\<in>{x\<in>insert a A. f' x > 0}. f' x) = n"
- using m by simp
- moreover have "finite {x. f' x > 0}"
- by (rule finite_subset[of _ "insert a {x\<in>A. f x > 0}"])
- (use f in \<open>auto simp: nat_partitions_def f'_def\<close>)
- moreover have "{x. f' x > 0} \<subseteq> insert a A"
- using f by (auto simp: f'_def nat_partitions_def)
- ultimately show "f' \<in> nat_partitions (insert a A) n"
- by (simp add: nat_partitions_def)
- qed
-next
- case 2
- show ?case
- proof safe
- fix f assume "f \<in> nat_partitions (insert a A) n"
- define f' where "f' = f(a := 0)"
- have fin: "finite {x. f x > 0}"
- and subset: "{x. f x > 0} \<subseteq> insert a A"
- and sum: "(\<Sum>x\<in>{x\<in>insert a A. f x > 0}. f x) = n"
- using \<open>f \<in> _\<close> by (auto simp: nat_partitions_def)
-
- show le: "f a \<in> {0..n}"
- proof (cases "f a = 0")
- case False
- hence "f a \<le> (\<Sum>x\<in>{x\<in>insert a A. f x > 0}. f x)"
- by (intro member_le_sum) (use fin in auto)
- with sum show ?thesis
- by simp
- qed auto
-
- have "n = (\<Sum>x\<in>{x\<in>insert a A. f x > 0}. f x)"
- using sum by simp
- also have "(\<Sum>x\<in>{x\<in>insert a A. f x > 0}. f x) = (\<Sum>x\<in>insert a {x\<in>A. f x > 0}. f x)"
- by (rule sum.mono_neutral_left) (auto intro: finite_subset[OF _ fin])
- also have "\<dots> = f a + (\<Sum>x\<in>{x\<in>A. f x > 0}. f x)"
- by (subst sum.insert) (auto intro: finite_subset[OF _ fin])
- also have "(\<Sum>x\<in>{x\<in>A. f x > 0}. f x) = (\<Sum>x\<in>{x\<in>A. f' x > 0}. f' x)"
- by (intro sum.cong) (auto simp: f'_def)
- finally have "(\<Sum>x\<in>{x\<in>A. f' x > 0}. f' x) = n - f a"
- using le by simp
-
- moreover have "finite {x. 0 < f' x}"
- by (rule finite_subset[OF _ fin]) (auto simp: f'_def)
- moreover have "{x. 0 < f' x} \<subseteq> A"
- using subset by (auto simp: f'_def)
- ultimately show "f' \<in> nat_partitions A (n - f a)"
- by (auto simp: nat_partitions_def)
- qed
-next
- case (3 f)
- thus ?case
- by (fastforce simp: nat_partitions_def fun_eq_iff)
-qed (auto simp: nat_partitions_def)
-
-lemma finite_nat_partitions [intro]:
- assumes "finite A"
- shows "finite (nat_partitions A n)"
- using assms
-proof (induction arbitrary: n rule: finite_induct)
- case empty
- thus ?case
- by (cases "n = 0") auto
-next
- case (insert x A n)
- have "finite (SIGMA m:{0..n}. nat_partitions A (n - m))"
- by (auto intro: insert.IH)
- also have "?this \<longleftrightarrow> finite (nat_partitions (insert x A) n)"
- by (rule bij_betw_finite, rule nat_partitions_insert) fact
- finally show ?case .
-qed
-
lemma fps_prod_nth':
assumes "finite A"
- shows "fps_nth (\<Prod>x\<in>A. f x) n = (\<Sum>h\<in>nat_partitions A n. \<Prod>x\<in>A. fps_nth (f x) (h x))"
+ shows "fps_nth (\<Prod>x\<in>A. f x) n = (\<Sum>X\<in>multisets_of_size A n. \<Prod>x\<in>A. fps_nth (f x) (count X x))"
using assms
proof (induction A arbitrary: n rule: finite_induct)
case (insert a A n)
note [simp] = \<open>a \<notin> A\<close>
note [intro, simp] = \<open>finite A\<close>
- have "(\<Sum>h\<in>nat_partitions (insert a A) n. \<Prod>x\<in>insert a A. fps_nth (f x) (h x)) =
- (\<Sum>(m,h)\<in>(SIGMA m:{0..n}. nat_partitions A (n-m)). \<Prod>x\<in>insert a A. fps_nth (f x) ((h(a := m)) x))"
- by (subst sum.reindex_bij_betw[OF nat_partitions_insert, symmetric])
+ have "(\<Sum>X\<in>multisets_of_size (insert a A) n. \<Prod>x\<in>insert a A. fps_nth (f x) (count X x)) =
+ (\<Sum>(m,X)\<in>(SIGMA m:{0..n}. multisets_of_size A (n-m)).
+ \<Prod>x\<in>insert a A. fps_nth (f x) (count (X + replicate_mset m a) x))"
+ by (subst sum.reindex_bij_betw[OF bij_betw_multisets_of_size_insert, symmetric])
(simp_all add: case_prod_unfold)
- also have "\<dots> = (\<Sum>m=0..n. \<Sum>h\<in>nat_partitions A (n-m). \<Prod>x\<in>insert a A. fps_nth (f x) ((h(a := m)) x))"
+ also have "\<dots> = (\<Sum>m=0..n. \<Sum>X\<in>multisets_of_size A (n-m).
+ \<Prod>x\<in>insert a A. fps_nth (f x) (count (X + replicate_mset m a) x))"
by (rule sum.Sigma [symmetric]) auto
also have "\<dots> = (\<Sum>m=0..n. fps_nth (f a) m * fps_nth (\<Prod>x\<in>A. f x) (n - m))"
proof (rule sum.cong)
fix m
assume m: "m \<in> {0..n}"
- have "(\<Sum>h\<in>nat_partitions A (n-m). \<Prod>x\<in>insert a A. fps_nth (f x) ((h(a := m)) x)) =
- fps_nth (f a) m * (\<Sum>h\<in>nat_partitions A (n-m). \<Prod>x\<in>A. fps_nth (f x) ((h(a := m)) x))"
+ have "(\<Sum>X\<in>multisets_of_size A (n-m).
+ \<Prod>x\<in>insert a A. fps_nth (f x) (count (X + replicate_mset m a) x)) =
+ (\<Sum>X\<in>multisets_of_size A (n-m). fps_nth (f a) (count X a + m) *
+ (\<Prod>x\<in>A. fps_nth (f x) (count (X + replicate_mset m a) x)))"
+ by simp
+ also have "\<dots> = (\<Sum>X\<in>multisets_of_size A (n-m). fps_nth (f a) m *
+ (\<Prod>x\<in>A. fps_nth (f x) (count (X + replicate_mset m a) x)))"
+ by (intro sum.cong arg_cong2[of _ _ _ _ "(*)"] arg_cong2[of _ _ _ _ fps_nth] refl)
+ (auto simp: multisets_of_size_def simp flip: not_in_iff)
+ also have "\<dots> = fps_nth (f a) m * (\<Sum>X\<in>multisets_of_size A (n-m).
+ (\<Prod>x\<in>A. fps_nth (f x) (count (X + replicate_mset m a) x)))"
by (simp add: sum_distrib_left)
- also have "(\<Sum>h\<in>nat_partitions A (n-m). \<Prod>x\<in>A. fps_nth (f x) ((h(a := m)) x)) =
- (\<Sum>h\<in>nat_partitions A (n-m). \<Prod>x\<in>A. fps_nth (f x) (h x))"
+ also have "(\<Sum>X\<in>multisets_of_size A (n-m). \<Prod>x\<in>A. fps_nth (f x) (count (X + replicate_mset m a) x)) =
+ (\<Sum>X\<in>multisets_of_size A (n-m). \<Prod>x\<in>A. fps_nth (f x) (count X x))"
by (intro sum.cong prod.cong) auto
also have "\<dots> = fps_nth (\<Prod>x\<in>A. f x) (n - m)"
by (rule insert.IH [symmetric])
- finally show "(\<Sum>h\<in>nat_partitions A (n-m). \<Prod>x\<in>insert a A. fps_nth (f x) ((h(a := m)) x)) =
+ finally show "(\<Sum>X\<in>multisets_of_size A (n-m). \<Prod>x\<in>insert a A. fps_nth (f x) (count (X + replicate_mset m a) x)) =
fps_nth (f a) m * fps_nth (\<Prod>x\<in>A. f x) (n - m)" .
qed auto
also have "\<dots> = fps_nth (\<Prod>x\<in>insert a A. f x) n"
@@ -4051,7 +3882,7 @@
fixes f :: "nat \<Rightarrow> 'a :: {idom, t2_space} fps"
assumes [simp]: "\<And>k. f k \<noteq> 0"
assumes g: "\<And>n k. k > g n \<Longrightarrow> subdegree (f k - 1) > n"
- defines "P \<equiv> Abs_fps (\<lambda>n. (\<Sum>h\<in>nat_partitions {..g n} n. \<Prod>i\<le>g n. fps_nth (f i) (h i)))"
+ defines "P \<equiv> Abs_fps (\<lambda>n. (\<Sum>X\<in>multisets_of_size {..g n} n. \<Prod>i\<le>g n. fps_nth (f i) (count X i)))"
shows "(\<lambda>n. \<Prod>k\<le>n. f k) \<longlonglongrightarrow> P"
proof (rule tendsto_fpsI)
fix n :: nat
@@ -4059,93 +3890,95 @@
using eventually_ge_at_top[of "g n"]
proof eventually_elim
case (elim N)
- have "fps_nth (prod f {..N}) n = (\<Sum>h\<in>nat_partitions {..N} n. \<Prod>x\<le>N. fps_nth (f x) (h x))"
+ have "fps_nth (prod f {..N}) n = (\<Sum>X\<in>multisets_of_size {..N} n. \<Prod>x\<le>N. fps_nth (f x) (count X x))"
by (subst fps_prod_nth') auto
- also have "\<dots> = (\<Sum>h | h \<in> nat_partitions {..N} n \<and> (\<forall>x\<le>N. fps_nth (f x) (h x) \<noteq> 0).
- \<Prod>x\<le>N. fps_nth (f x) (h x))"
+ also have "\<dots> = (\<Sum>X | X \<in> multisets_of_size {..N} n \<and> (\<forall>x\<le>N. fps_nth (f x) (count X x) \<noteq> 0).
+ \<Prod>x\<le>N. fps_nth (f x) (count X x))"
by (intro sum.mono_neutral_right) auto
- also have "{h. h \<in> nat_partitions {..N} n \<and> (\<forall>x\<le>N. fps_nth (f x) (h x) \<noteq> 0)} =
- {h. h \<in> nat_partitions {..g n} n \<and> (\<forall>x\<le>N. fps_nth (f x) (h x) \<noteq> 0)}" (is "?lhs = ?rhs")
+ also have "{X. X \<in> multisets_of_size {..N} n \<and> (\<forall>x\<le>N. fps_nth (f x) (count X x) \<noteq> 0)} =
+ {X. X \<in> multisets_of_size {..g n} n \<and> (\<forall>x\<le>N. fps_nth (f x) (count X x) \<noteq> 0)}"
+ (is "?lhs = ?rhs")
proof (intro equalityI subsetI)
- fix h assume "h \<in> ?rhs"
- thus "h \<in> ?lhs" using elim nat_partitions_mono[of "{..g n}" "{..N}"] by auto
+ fix X assume "X \<in> ?rhs"
+ thus "X \<in> ?lhs" using elim multisets_of_size_mono[of "{..g n}" "{..N}"] by auto
next
- fix h assume "h \<in> ?lhs"
- hence h: "{x. 0 < h x} \<subseteq> {..N}" "(\<Sum>x\<le>N. h x) = n" "\<And>x. x \<le> N \<Longrightarrow> fps_nth (f x) (h x) \<noteq> 0"
- by (auto simp: nat_partitions_finite_def)
- have "{x. 0 < h x} \<subseteq> {..g n}"
+ fix X assume "X \<in> ?lhs"
+ hence X: "set_mset X \<subseteq> {..N}" "size X = n" "\<And>x. x \<le> N \<Longrightarrow> fps_nth (f x) (count X x) \<noteq> 0"
+ by (auto simp: multisets_of_size_def)
+ have "set_mset X \<subseteq> {..g n}"
proof
- fix x assume *: "x \<in> {x. h x > 0}"
+ fix x assume *: "x \<in> set_mset X"
show "x \<in> {..g n}"
proof (rule ccontr)
assume "x \<notin> {..g n}"
hence x: "x > g n" "x \<le> N"
- using h(1) * by auto
- have "h x \<le> n"
- using \<open>h \<in> ?lhs\<close> nat_partitions_subset[of "{..N}" n] x by (auto simp: Pi_def)
+ using X(1) * by auto
+ have "count X x \<le> n"
+ using X x count_le_size[of X x] by (auto simp: Pi_def)
also have "n < subdegree (f x - 1)"
by (rule g) (use x in auto)
- finally have "fps_nth (f x - 1) (h x) = 0"
+ finally have "fps_nth (f x - 1) (count X x) = 0"
by blast
- hence "fps_nth (f x) (h x) = 0"
+ hence "fps_nth (f x) (count X x) = 0"
using * by simp
- moreover have "fps_nth (f x) (h x) \<noteq> 0"
- by (intro h(3)) (use x in auto)
+ moreover have "fps_nth (f x) (count X x) \<noteq> 0"
+ by (intro X(3)) (use x in auto)
ultimately show False by contradiction
qed
qed
- moreover from this have "(\<Sum>x\<le>N. h x) = (\<Sum>x\<le>g n. h x)"
- by (intro sum.mono_neutral_right) (use elim in auto)
- ultimately show "h \<in> ?rhs" using elim h
- by (auto simp: nat_partitions_finite_def)
+ thus "X \<in> ?rhs" using X
+ by (auto simp: multisets_of_size_def)
qed
- also have "(\<Sum>h | h \<in> nat_partitions {..g n} n \<and> (\<forall>x\<le>N. fps_nth (f x) (h x) \<noteq> 0).
- \<Prod>x\<le>N. fps_nth (f x) (h x)) =
- (\<Sum>h | h \<in> nat_partitions {..g n} n \<and> (\<forall>x\<le>N. fps_nth (f x) (h x) \<noteq> 0).
- \<Prod>i\<le>g n. fps_nth (f i) (h i))"
+ also have "(\<Sum>X | X \<in> multisets_of_size {..g n} n \<and> (\<forall>x\<le>N. fps_nth (f x) (count X x) \<noteq> 0).
+ \<Prod>x\<le>N. fps_nth (f x) (count X x)) =
+ (\<Sum>X | X \<in> multisets_of_size {..g n} n \<and> (\<forall>x\<le>N. fps_nth (f x) (count X x) \<noteq> 0).
+ \<Prod>i\<le>g n. fps_nth (f i) (count X i))"
proof (intro sum.cong prod.mono_neutral_right ballI)
- fix h i
+ fix X i
assume i: "i \<in> {..N} - {..g n}"
- assume "h \<in> {h. h \<in> nat_partitions {..g n} n \<and> (\<forall>x\<le>N. fps_nth (f x) (h x) \<noteq> 0)}"
- hence h: "h \<in> nat_partitions {..g n} n" "\<And>x. x \<le> N \<Longrightarrow> fps_nth (f x) (h x) \<noteq> 0"
+ assume "X \<in> {X. X \<in> multisets_of_size {..g n} n \<and> (\<forall>x\<le>N. fps_nth (f x) (count X x) \<noteq> 0)}"
+ hence h: "X \<in> multisets_of_size {..g n} n" "\<And>x. x \<le> N \<Longrightarrow> fps_nth (f x) (count X x) \<noteq> 0"
by blast+
- have [simp]: "h i = 0"
- using i h unfolding nat_partitions_def by auto
+ have "i \<notin># X"
+ using i h unfolding multisets_of_size_def by auto
have "n < subdegree (f i - 1)"
by (intro g) (use i in auto)
- moreover have "h i \<le> n"
- by auto
- ultimately have "fps_nth (f i - 1) (h i) = 0"
+ moreover have "count X i \<le> n"
+ using \<open>i \<notin># X\<close> by (simp add: not_in_iff)
+ ultimately have "fps_nth (f i - 1) (count X i) = 0"
by (intro nth_less_subdegree_zero) auto
- thus "fps_nth (f i) (h i) = 1"
- using h(2) i by (auto split: if_splits)
+ thus "fps_nth (f i) (count X i) = 1"
+ using h(2) i \<open>i \<notin># X\<close> by (auto split: if_splits)
qed (use elim in auto)
- also have "(\<Sum>h | h \<in> nat_partitions {..g n} n \<and> (\<forall>x\<le>N. fps_nth (f x) (h x) \<noteq> 0).
- \<Prod>i\<le>g n. fps_nth (f i) (h i)) =
- (\<Sum>h \<in> nat_partitions {..g n} n. \<Prod>i\<le>g n. fps_nth (f i) (h i))"
+ also have "(\<Sum>X | X \<in> multisets_of_size {..g n} n \<and> (\<forall>x\<le>N. fps_nth (f x) (count X x) \<noteq> 0).
+ \<Prod>i\<le>g n. fps_nth (f i) (count X i)) =
+ (\<Sum>X \<in> multisets_of_size {..g n} n. \<Prod>i\<le>g n. fps_nth (f i) (count X i))"
proof (intro sum.mono_neutral_left ballI)
- fix h assume "h \<in> nat_partitions {..g n} n - {h\<in>nat_partitions {..g n} n. \<forall>x\<le>N. fps_nth (f x) (h x) \<noteq> 0}"
- then obtain i where h: "h \<in> nat_partitions {..g n} n" and i: "i \<le> N" "fps_nth (f i) (h i) = 0"
+ fix X assume "X \<in> multisets_of_size {..g n} n -
+ {X\<in>multisets_of_size {..g n} n. \<forall>x\<le>N. fps_nth (f x) (count X x) \<noteq> 0}"
+ then obtain i
+ where h: "X \<in> multisets_of_size {..g n} n"
+ and i: "i \<le> N" "fps_nth (f i) (count X i) = 0"
by blast
have "\<not>(i > g n)"
proof
assume i': "i > g n"
- hence "h i = 0"
- using h by (auto simp: nat_partitions_def)
+ hence "count X i = 0"
+ using h by (auto simp: multisets_of_size_def simp flip: not_in_iff)
have "subdegree (f i - 1) > n"
by (intro g) (use i' in auto)
hence "subdegree (f i - 1) > 0"
by simp
hence "fps_nth (f i - 1) 0 = 0"
by blast
- hence "fps_nth (f i) (h i) = 1"
- using \<open>h i = 0\<close> by simp
+ hence "fps_nth (f i) (count X i) = 1"
+ using \<open>count X i = 0\<close> by simp
thus False using i by simp
qed
- thus " (\<Prod>x\<le>g n. fps_nth (f x) (h x)) = 0"
+ thus " (\<Prod>x\<le>g n. fps_nth (f x) (count X x)) = 0"
using i by auto
qed auto
--- a/src/HOL/Library/Multiset.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Library/Multiset.thy Thu Apr 24 09:16:33 2025 +0200
@@ -4834,4 +4834,185 @@
hide_const (open) wcount
+
+subsection \<open>The set of multisets of a given size\<close>
+
+(* contributed by Manuel Eberl *)
+
+text \<open>
+ The following operator gives the set of all multisets consisting of $n$ elements drawn from
+ the set $A$. In other words: all the different ways to put $n$ unlabelled balls into
+ the labelled bins $A$.
+\<close>
+definition multisets_of_size :: "'a set \<Rightarrow> nat \<Rightarrow> 'a multiset set" where
+ "multisets_of_size A n = {X. set_mset X \<subseteq> A \<and> size X = n}"
+
+lemma
+ assumes "X \<in> multisets_of_size A n"
+ shows multisets_of_size_subset: "set_mset X \<subseteq> A"
+ and multisets_of_size_size: "size X = n"
+ using assms by (auto simp: multisets_of_size_def)
+
+lemma multisets_of_size_mono:
+ assumes "A \<subseteq> B"
+ shows "multisets_of_size A n \<subseteq> multisets_of_size B n"
+ unfolding multisets_of_size_def
+ by (intro Collect_mono) (use assms in auto)
+
+lemma multisets_of_size_0 [simp]: "multisets_of_size A 0 = {{#}}"
+proof (intro equalityI subsetI)
+ fix h :: "'a multiset" assume "h \<in> {{#}}"
+ thus "h \<in> multisets_of_size A 0"
+ by (auto simp: multisets_of_size_def)
+qed (auto simp: multisets_of_size_def fun_eq_iff)
+
+lemma multisets_of_size_empty [simp]: "n > 0 \<Longrightarrow> multisets_of_size {} n = {}"
+ by (auto simp: multisets_of_size_def fun_eq_iff)
+
+lemma count_le_size: "count X x \<le> size X"
+ by (induction X) auto
+
+lemma bij_betw_multisets_of_size_insert:
+ assumes "a \<notin> A"
+ shows "bij_betw (\<lambda>(m,X). X + replicate_mset m a)
+ (SIGMA m:{0..n}. multisets_of_size A (n - m)) (multisets_of_size (insert a A) n)"
+proof -
+ define B where "B = (SIGMA m:{0..n}. multisets_of_size A (n - m))"
+ define C where "C = (multisets_of_size (insert a A) n)"
+ define f where "f = (\<lambda>(m,X). X + replicate_mset m a)"
+ define g where "g = (\<lambda>X. (count X a, filter_mset (\<lambda>x. x \<noteq> a) X))"
+ note defs = B_def C_def f_def g_def
+
+ have *: "size (filter_mset (\<lambda>x. x \<noteq> a) X) = size X - count X a" for X
+ proof -
+ have "size X = size (filter_mset (\<lambda>x. x \<noteq> a) X) + count X a"
+ by (induction X) auto
+ thus ?thesis
+ by linarith
+ qed
+
+ have 1: "f mX \<in> C" if "mX \<in> B" for mX
+ using that by (auto simp: multisets_of_size_def defs split: if_splits)
+
+ have 2: "g X \<in> B" if "X \<in> C" for X
+ using that by (auto simp: multisets_of_size_def count_le_size * defs)
+
+ have 3: "g (f mX) = mX" if "mX \<in> B" for mX
+ using that assms
+ by (auto simp: multisets_of_size_def multiset_eq_iff defs simp flip: not_in_iff)
+
+ have 4: "f (g X) = X" if "X \<in> C" for X
+ using that
+ by (auto simp: multisets_of_size_def multiset_eq_iff defs simp flip: not_in_iff)
+
+ have "f ` B = C"
+ using 1 2 4 unfolding set_eq_iff image_iff by metis
+ moreover have "inj_on f B"
+ using 3 unfolding inj_on_def by metis
+ ultimately show ?thesis
+ unfolding bij_betw_def defs by metis
+qed
+
+lemma multisets_of_size_insert:
+ assumes "a \<notin> A"
+ shows "multisets_of_size (insert a A) n =
+ (\<Union>m\<le>n. (\<lambda>X. X + replicate_mset m a) ` multisets_of_size A (n - m))"
+proof -
+ have "multisets_of_size (insert a A) n =
+ (\<lambda>(m,X). X + replicate_mset m a) ` (SIGMA m:{0..n}. multisets_of_size A (n - m))"
+ using bij_betw_multisets_of_size_insert[OF assms, of n] unfolding bij_betw_def by simp
+ also have "\<dots> = (\<Union>m\<le>n. (\<lambda>X. X + replicate_mset m a) ` multisets_of_size A (n - m))"
+ unfolding Sigma_def image_UN atLeast0AtMost image_insert image_empty prod.case
+ UNION_singleton_eq_range image_image by (rule refl)
+ finally show ?thesis .
+qed
+
+primrec multisets_of_size_list :: "'a list \<Rightarrow> nat \<Rightarrow> 'a list list" where
+ "multisets_of_size_list [] n = (if n = 0 then [[]] else [])"
+| "multisets_of_size_list (x # xs) n =
+ [replicate m x @ ys . m \<leftarrow> [0..<n+1], ys \<leftarrow> multisets_of_size_list xs (n - m)]"
+
+lemma multisets_of_size_list_correct:
+ assumes "distinct xs"
+ shows "mset ` set (multisets_of_size_list xs n) = multisets_of_size (set xs) n"
+ using assms
+proof (induction xs arbitrary: n)
+ case Nil
+ thus ?case
+ by (cases "n = 0") auto
+next
+ case (Cons x xs n)
+ have IH: "multisets_of_size (set xs) n = mset ` set (multisets_of_size_list xs n)" for n
+ by (rule Cons.IH [symmetric]) (use Cons.prems in auto)
+ from Cons.prems have "x \<notin> set xs"
+ by auto
+ thus ?case
+ by (simp add: multisets_of_size_insert image_UN atLeastLessThanSuc_atLeastAtMost image_image
+ add_ac atLeast0AtMost IH del: upt_Suc)
+qed
+
+lemma multisets_of_size_code [code]:
+ "multisets_of_size (set xs) n = set (map mset (multisets_of_size_list (remdups xs) n))"
+ using multisets_of_size_list_correct[of "remdups xs"] by simp
+
+lemma finite_multisets_of_size [intro]:
+ assumes "finite A"
+ shows "finite (multisets_of_size A n)"
+ using assms
+proof (induction arbitrary: n rule: finite_induct)
+ case empty
+ thus ?case
+ by (cases "n = 0") auto
+next
+ case (insert x A n)
+ have "finite (SIGMA m:{0..n}. multisets_of_size A (n - m))"
+ by (auto intro: insert.IH)
+ also have "?this \<longleftrightarrow> finite (multisets_of_size (insert x A) n)"
+ by (rule bij_betw_finite, rule bij_betw_multisets_of_size_insert) fact
+ finally show ?case .
+qed
+
+lemma card_multisets_of_size:
+ assumes "finite A"
+ shows "card (multisets_of_size A n) = (card A + n - 1) choose n"
+ using assms
+proof (induction A arbitrary: n rule: finite_induct)
+ case empty
+ thus ?case
+ by (cases "n = 0") auto
+next
+ case (insert a A n)
+ have "card (multisets_of_size (insert a A) n) = card (SIGMA m:{0..n}. multisets_of_size A (n - m))"
+ using bij_betw_same_card[OF bij_betw_multisets_of_size_insert[of a A], of n] insert.hyps
+ by simp
+ also have "\<dots> = (\<Sum>m=0..n. card (multisets_of_size A (n - m)))"
+ by (intro card_SigmaI) (use insert.hyps in auto)
+ also have "\<dots> = (\<Sum>m=0..n. (card A + (n - m) - 1) choose (n - m))"
+ by (intro sum.cong insert.IH refl)
+ also have "\<dots> = (\<Sum>m=0..n. (card A + m - 1) choose m)"
+ by (intro sum.reindex_bij_witness[of _ "\<lambda>m. n - m" "\<lambda>m. n - m"]) auto
+ also have "\<dots> = (card A + n) choose n"
+ proof (cases "card A = 0")
+ case True
+ have "(\<Sum>m=0..n. (card A + m - 1) choose m) = (\<Sum>m\<in>{0}. (m-1) choose m)"
+ by (intro sum.mono_neutral_cong_right) (use True in auto)
+ also have "\<dots> = 1"
+ by simp
+ also have "\<dots> = (card A + n) choose n"
+ using True by simp
+ finally show ?thesis .
+ next
+ case False
+ have "(\<Sum>m=0..n. (card A + m - 1) choose m) = (\<Sum>m\<le>n. ((card A - 1) + m) choose m)"
+ by (intro sum.cong) (use False in \<open>auto simp: algebra_simps\<close>)
+ also have "\<dots> = (\<Sum>m\<le>n. ((card A - 1) + m) choose (card A - 1))"
+ by (subst binomial_symmetric) auto
+ also have "\<dots> = (card A + n) choose n"
+ using choose_rising_sum(2)[of "card A - 1" n] False by simp
+ finally show ?thesis .
+ qed
+ finally show ?case
+ using insert.hyps by simp
+qed
+
end
--- a/src/HOL/Library/Real_Mod.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Library/Real_Mod.thy Thu Apr 24 09:16:33 2025 +0200
@@ -238,4 +238,90 @@
shows "[a / c = b / c] (rmod m)"
using rcong_mult_modulus[of a b m "1 / c"] assms by (auto simp: field_simps)
+lemma sin_rmod [simp]: "sin (x rmod (2*pi)) = sin x"
+ and cos_rmod [simp]: "cos (x rmod (2*pi)) = cos x"
+ by (simp_all add: rmod_def sin_diff cos_diff)
+
+lemma tan_rmod [simp]: "tan (x rmod (2*pi)) = tan x"
+ and cot_rmod [simp]: "cot (x rmod (2*pi)) = cot x"
+ and cis_rmod [simp]: "cis (x rmod (2*pi)) = cis x"
+ and rcis_rmod [simp]: "rcis r (x rmod (2*pi)) = rcis r x"
+ by (simp_all add: tan_def cot_def complex_eq_iff)
+
+lemma cis_eq_iff: "cis a = cis b \<longleftrightarrow> [a = b] (rmod (2 * pi))"
+proof -
+ have "cis a = cis b \<longleftrightarrow> sin a = sin b \<and> cos a = cos b"
+ by (auto simp: complex_eq_iff)
+ also have "\<dots> \<longleftrightarrow> (\<exists>x. a = b + 2 * pi * real_of_int x)"
+ by (rule sin_cos_eq_iff)
+ also have "\<dots> \<longleftrightarrow> [b = a] (rmod (2 * pi))"
+ by (simp add: rcong_altdef mult_ac)
+ finally show ?thesis
+ by (simp add: rcong_sym_iff)
+qed
+
+lemma cis_eq_1_iff: "cis a = 1 \<longleftrightarrow> (\<exists>n. a = of_int n * (2 * pi))"
+proof -
+ have "cis 0 = cis a \<longleftrightarrow> [0 = a] (rmod (2 * pi))"
+ by (rule cis_eq_iff)
+ also have "\<dots> \<longleftrightarrow> (\<exists>n. a = of_int n * (2 * pi))"
+ unfolding rcong_altdef by simp
+ finally show ?thesis
+ by simp
+qed
+
+lemma cis_cong:
+ assumes "[a = b] (rmod 2 * pi)"
+ shows "cis a = cis b"
+ using cis_eq_iff assms by blast
+
+lemma rcis_cong:
+ assumes "[a = b] (rmod 2 * pi)"
+ shows "rcis r a = rcis r b"
+ using assms by (auto simp: rcis_def intro!: cis_cong)
+
+lemma sin_rcong: "[x = y] (rmod (2 * pi)) \<Longrightarrow> sin x = sin y"
+ and cos_rcong: "[x = y] (rmod (2 * pi)) \<Longrightarrow> cos x = cos y"
+ using cis_cong[of x y] by (simp_all add: complex_eq_iff)
+
+lemma sin_eq_sin_conv_rmod:
+ assumes "sin x = sin y"
+ shows "[x = y] (rmod 2 * pi) \<or> [x = pi - y] (rmod 2 * pi)"
+proof -
+ have "0 = sin y - sin x"
+ using assms by simp
+ hence "sin ((y - x) / 2) = 0 \<or> cos ((y + x) / 2) = 0"
+ unfolding sin_diff_sin by simp
+ hence "(\<exists>i. y = x + real_of_int i * (2 * pi)) \<or>
+ (\<exists>i. pi - y = x + real_of_int (-i) * (2 * pi))"
+ unfolding sin_zero_iff_int2 cos_zero_iff_int2
+ by (auto simp: algebra_simps)
+ thus ?thesis
+ unfolding rcong_altdef by blast
+qed
+
+lemma cos_eq_cos_conv_rmod:
+ assumes "cos x = cos y"
+ shows "[x = y] (rmod 2 * pi) \<or> [x = -y] (rmod 2 * pi)"
+proof -
+ have "0 = cos y - cos x"
+ using assms by simp
+ hence "sin ((y + x) / 2) = 0 \<or> sin ((x - y) / 2) = 0"
+ unfolding cos_diff_cos by simp
+ hence "(\<exists>i. -y = x + real_of_int (-i) * (2 * pi)) \<or>
+ (\<exists>i. y = x + real_of_int (-i) * (2 * pi))"
+ unfolding sin_zero_iff_int2
+ by (auto simp: algebra_simps)
+ thus ?thesis
+ unfolding rcong_altdef by blast
+qed
+
+lemma sin_eq_sin_conv_rmod_iff:
+ "sin x = sin y \<longleftrightarrow> [x = y] (rmod 2 * pi) \<or> [x = pi - y] (rmod 2 * pi)"
+ by (metis sin_eq_sin_conv_rmod sin_pi_minus sin_rcong)
+
+lemma cos_eq_cos_conv_rmod_iff:
+ "cos x = cos y \<longleftrightarrow> [x = y] (rmod 2 * pi) \<or> [x = -y] (rmod 2 * pi)"
+ by (metis cos_eq_cos_conv_rmod cos_minus cos_rcong)
+
end
\ No newline at end of file
--- a/src/HOL/Probability/Sinc_Integral.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Probability/Sinc_Integral.thy Thu Apr 24 09:16:33 2025 +0200
@@ -128,7 +128,7 @@
by (auto simp: isCont_def sinc_at_0)
next
assume "x \<noteq> 0" show ?thesis
- by (rule continuous_transform_within [where d = "abs x" and f = "\<lambda>x. sin x / x"])
+ by (rule continuous_transform_within [where \<delta> = "abs x" and f = "\<lambda>x. sin x / x"])
(auto simp add: dist_real_def \<open>x \<noteq> 0\<close>)
qed
--- a/src/HOL/Product_Type.thy Tue Apr 22 15:08:14 2025 +0200
+++ b/src/HOL/Product_Type.thy Thu Apr 24 09:16:33 2025 +0200
@@ -1316,6 +1316,11 @@
by auto
qed
+lemma bij_betw_map_prod:
+ assumes "bij_betw f A C" "bij_betw g B D"
+ shows "bij_betw (map_prod f g) (A \<times> B) (C \<times> D)"
+ using assms unfolding bij_betw_def inj_on_def by auto
+
subsection \<open>Simproc for rewriting a set comprehension into a pointfree expression\<close>
--- a/src/Pure/Admin/build_release.scala Tue Apr 22 15:08:14 2025 +0200
+++ b/src/Pure/Admin/build_release.scala Thu Apr 24 09:16:33 2025 +0200
@@ -691,7 +691,8 @@
case Platform.Family.macos =>
File.change(isabelle_target + jedit_props) {
- _.replaceAll("delete-line.shortcut=.*", "delete-line.shortcut=C+d")
+ _.replaceAll("lookAndFeel=.*", "lookAndFeel=com.formdev.flatlaf.themes.FlatMacLightLaf")
+ .replaceAll("delete-line.shortcut=.*", "delete-line.shortcut=C+d")
.replaceAll("delete.shortcut2=.*", "delete.shortcut2=A+d")
}
--- a/src/Pure/Admin/component_flatlaf.scala Tue Apr 22 15:08:14 2025 +0200
+++ b/src/Pure/Admin/component_flatlaf.scala Thu Apr 24 09:16:33 2025 +0200
@@ -8,22 +8,31 @@
object Component_FlatLaf {
- /* platform information */
+ /* jars and native libraries */
- sealed case class Platform_Info(name: String, exe: Boolean = false)
+ sealed case class Lib(template: String, exe: Boolean = false) {
+ def path(version: String): Path =
+ Path.explode(template.replace("{V}", version))
+
+ def jar_name(version: String): Option[String] =
+ if (File.is_jar(template)) Some(path(version).file_name) else None
+ }
- private val platforms =
+ private val libs =
List(
- Platform_Info("flatlaf-{V}-macos-arm64.dylib"),
- Platform_Info("flatlaf-{V}-macos-x86_64.dylib"),
- Platform_Info("flatlaf-{V}-linux-x86_64.so"),
- Platform_Info("flatlaf-{V}-windows-x86_64.dll", exe = true))
+ Lib("flatlaf/{V}/flatlaf-{V}-no-natives.jar"),
+ Lib("flatlaf/{V}/flatlaf-{V}-macos-arm64.dylib"),
+ Lib("flatlaf/{V}/flatlaf-{V}-macos-x86_64.dylib"),
+ Lib("flatlaf/{V}/flatlaf-{V}-linux-arm64.so"),
+ Lib("flatlaf/{V}/flatlaf-{V}-linux-x86_64.so"),
+ Lib("flatlaf/{V}/flatlaf-{V}-windows-x86_64.dll", exe = true),
+ Lib("flatlaf-extras/{V}/flatlaf-extras-{V}.jar"))
/* build flatlaf */
- val default_download_url = "https://repo1.maven.org/maven2/com/formdev/flatlaf"
- val default_version = "3.5.4"
+ val default_download_url = "https://repo1.maven.org/maven2/com/formdev"
+ val default_version = "3.6"
def build_flatlaf(
target_dir: Path = Path.current,
@@ -42,28 +51,32 @@
Isabelle_System.make_directory(component_dir.lib)
- def download(name: String, exe: Boolean = false): Unit = {
- val download_name = name.replace("{V}", version)
- val target = component_dir.lib + Path.basic(download_name)
+ for (lib <- libs) {
+ val lib_path = lib.path(version)
+ val target = component_dir.lib + Path.basic(lib_path.file_name)
Isabelle_System.download_file(
- download_url + "/" + version + "/" + download_name, target, progress = progress)
- if (exe) File.set_executable(target)
+ download_url + "/" + lib_path.implode, target, progress = progress)
+ if (lib.exe) File.set_executable(target)
}
- download("flatlaf-{V}-no-natives.jar")
-
- for (platform <- platforms) download(platform.name, exe = platform.exe)
+ val jar_names = libs.flatMap(_.jar_name(version))
/* settings */
+ val classpath =
+ libs.flatMap(_.jar_name(version)).map(a => "$ISABELLE_FLATLAF_HOME/lib/" + a)
+ .mkString(":")
+
component_dir.write_settings("""
ISABELLE_FLATLAF_HOME="$COMPONENT"
-classpath "$ISABELLE_FLATLAF_HOME/lib/flatlaf-""" + version + """-no-natives.jar"
+classpath """ + quote(classpath) + """
isabelle_scala_service "isabelle.FlatLightLaf"
isabelle_scala_service "isabelle.FlatDarkLaf"
+isabelle_scala_service "isabelle.FlatMacLightLaf"
+isabelle_scala_service "isabelle.FlatMacDarkLaf"
""")
@@ -71,11 +84,14 @@
File.write(component_dir.README,
"""This is the FlatLaf Java/Swing look-and-feel from
-https://www.formdev.com/flatlaf and
-https://mvnrepository.com/artifact/com.formdev/flatlaf
+https://mvnrepository.com/artifact/com.formdev
It is covered by the Apache License 2.0 license.
+See also https://www.formdev.com/flatlaf and especially the demo
+application https://download.formdev.com/flatlaf/flatlaf-demo-latest.jar
+(which may be run via "java -jar ...").
+
Makarius
""" + Date.Format.date(Date.now()) + "\n")
--- a/src/Pure/Admin/component_jedit.scala Tue Apr 22 15:08:14 2025 +0200
+++ b/src/Pure/Admin/component_jedit.scala Thu Apr 24 09:16:33 2025 +0200
@@ -144,6 +144,24 @@
Isabelle_System.extract(source_path, jedit_dir)
+ /* tango icons (SVG) */
+
+ val tango_path = Isabelle_System.make_directory(tmp_dir + Path.explode("tango"))
+ Isabelle_System.download_file(
+ "https://github.com/stephenc/tango-icon-theme/archive/41b8f6abd7eb.zip",
+ tango_path.zip, progress = progress)
+ Isabelle_System.extract(tango_path.zip, tango_path, strip = true)
+
+
+ /* IntelliJ IDEA icons (SVG) */
+
+ val idea_path = Isabelle_System.make_directory(tmp_dir + Path.explode("idea"))
+ Isabelle_System.download_file(
+ "https://isabelle.sketis.net/components/idea-icons-20250415.tar.gz",
+ idea_path.tar.gz, progress = progress)
+ Isabelle_System.extract(idea_path.tar.gz, idea_path, strip = true)
+
+
/* patched version */
Isabelle_System.copy_dir(jedit_dir, jedit_patched_dir)
@@ -161,12 +179,30 @@
cwd = source_dir, echo = true).check
}
- for { theme <- List("classic", "tango") } {
- val path = Path.explode("org/gjt/sp/jedit/icons/themes/" + theme + "/32x32/apps/isabelle.gif")
+ progress.echo("Augmenting icons ...")
+
+ val jedit_icons_path = source_dir + Path.explode("org/gjt/sp/jedit/icons/themes")
+ val jedit_classic_path = jedit_icons_path + Path.basic("classic")
+ val jedit_tango_path = jedit_icons_path + Path.basic("tango")
+ val jedit_idea_path = jedit_tango_path + Path.basic("idea-icons")
+
+ for (theme <- List(jedit_classic_path, jedit_tango_path)) {
Isabelle_System.copy_file(Path.explode("~~/lib/logo/isabelle_transparent-32.gif"),
- source_dir + path)
+ theme + Path.explode("32x32/apps/isabelle.gif"))
}
+ for {
+ svg_file <- File.find_files(tango_path.file, pred = file => File.is_svg(file.getName))
+ rel_path <- File.relative_path(tango_path, File.path(svg_file))
+ } {
+ val dir = Isabelle_System.make_directory(jedit_tango_path + rel_path.dir)
+ Isabelle_System.copy_file(File.path(svg_file), dir + rel_path.base)
+ }
+
+ Isabelle_System.extract(idea_path + Path.explode("jar/idea-icons.jar"), jedit_tango_path)
+ Isabelle_System.rm_tree(jedit_tango_path + Path.explode("META-INF"))
+ Isabelle_System.copy_file(idea_path + Path.explode("README"), jedit_idea_path)
+
progress.echo("Building jEdit ...")
Isabelle_System.copy_dir(source_dir, tmp_source_dir)
progress.bash("ant", cwd = tmp_source_dir, echo = true).check
@@ -340,10 +376,8 @@
metal.secondary.fontsize=12
navigate-backwards.label=Navigate backwards
navigate-backwards.shortcut=AS+LEFT
-navigate-backwards.icon=ArrowL.png
navigate-forwards.label=Navigate forwards
navigate-forwards.shortcut=AS+RIGHT
-navigate-forwards.icon=ArrowR.png
navigate-toolbar=navigate-backwards navigate-forwards
navigator.showOnToolbar=true
new-file-in-mode.shortcut=
@@ -363,7 +397,6 @@
recent-buffer.shortcut2=C+CIRCUMFLEX
restore.remote=false
restore=false
-search.find=Search:
search.subdirs.toggle=true
select-block.shortcut2=C+8
sidekick-tree.dock-position=right
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Pure/Admin/component_jsvg.scala Thu Apr 24 09:16:33 2025 +0200
@@ -0,0 +1,92 @@
+/* Title: Pure/Admin/component_jsvg.scala
+ Author: Makarius
+
+Build Isabelle jsvg component from official download.
+*/
+
+package isabelle
+
+
+object Component_JSVG {
+ /* build jsvg */
+
+ val default_download_url =
+ "https://repo1.maven.org/maven2/com/github/weisj/jsvg/1.7.1/jsvg-1.7.1.jar"
+
+ def build_jsvg(
+ download_url: String = default_download_url,
+ progress: Progress = new Progress,
+ target_dir: Path = Path.current
+ ): Unit = {
+ val Download_Name = """^.*/([^/]+)\.jar""".r
+ val download_name =
+ download_url match {
+ case Download_Name(download_name) => download_name
+ case _ => error("Malformed jar download URL: " + quote(download_url))
+ }
+
+
+ /* component */
+
+ val component_dir =
+ Components.Directory(target_dir + Path.basic(download_name)).create(progress = progress)
+
+ File.write(component_dir.LICENSE,
+ Url.read("https://raw.githubusercontent.com/weisJ/jsvg/refs/heads/master/LICENSE"))
+
+
+ /* README */
+
+ File.write(component_dir.README,
+ "This is a Java SVG implementation (JSVG) from\n" + download_url +
+ "\n\nSee also https://github.com/weisJ/jsvg" +
+ "\n\n Makarius\n " + Date.Format.date(Date.now()) + "\n")
+
+
+ /* settings */
+
+ component_dir.write_settings("""
+ISABELLE_JSVG_HOME="$COMPONENT"
+
+classpath "$ISABELLE_JSVG_HOME/lib/""" + download_name + """.jar"
+""")
+
+
+ /* jar */
+
+ val jar = component_dir.lib + Path.basic(download_name).jar
+ Isabelle_System.make_directory(jar.dir)
+ Isabelle_System.download_file(download_url, jar, progress = progress)
+ }
+
+
+ /* Isabelle tool wrapper */
+
+ val isabelle_tool =
+ Isabelle_Tool("component_jsvg", "build Isabelle jsvg component from official download",
+ Scala_Project.here,
+ { args =>
+ var target_dir = Path.current
+ var download_url = default_download_url
+
+ val getopts = Getopts("""
+Usage: isabelle component_jsvg [OPTIONS] DOWNLOAD
+
+ Options are:
+ -D DIR target directory (default ".")
+ -U URL download URL
+ (default: """" + default_download_url + """")
+
+ Build jsvg component from the specified download URL (JAR).
+""",
+ "D:" -> (arg => target_dir = Path.explode(arg)),
+ "U:" -> (arg => download_url = arg))
+
+ val more_args = getopts(args)
+ if (more_args.nonEmpty) getopts.usage()
+
+ val progress = new Console_Progress()
+
+ build_jsvg(download_url = download_url, progress = progress, target_dir = target_dir)
+ })
+}
--- a/src/Pure/GUI/gui.scala Tue Apr 22 15:08:14 2025 +0200
+++ b/src/Pure/GUI/gui.scala Thu Apr 24 09:16:33 2025 +0200
@@ -18,21 +18,20 @@
import scala.swing.{CheckBox, ComboBox, ScrollPane, TextArea, ListView, Separator}
import scala.swing.event.{ButtonClicked, SelectionChanged}
+import com.formdev.flatlaf
+import com.formdev.flatlaf.FlatLaf
+
object GUI {
/* Swing look-and-feel */
- def init_laf(): Unit = {
- val prop = com.formdev.flatlaf.FlatSystemProperties.USE_NATIVE_LIBRARY
- System.setProperty(prop, System.getProperty(prop, "false"))
- com.formdev.flatlaf.FlatLightLaf.setup()
- }
-
def current_laf(): String = UIManager.getLookAndFeel.getClass.getName()
- def is_macos_laf: Boolean =
+ def is_macos_laf(): Boolean =
Platform.is_macos && UIManager.getSystemLookAndFeelClassName() == current_laf()
+ def is_dark_laf(): Boolean = FlatLaf.isLafDark()
+
class Look_And_Feel(laf: LookAndFeel) extends Isabelle_System.Service {
def info: UIManager.LookAndFeelInfo =
new UIManager.LookAndFeelInfo(laf.getName, laf.getClass.getName)
@@ -56,6 +55,11 @@
UIManager.put("Component.arrowType", "triangle")
}
+ def init_laf(): Unit = {
+ init_lafs()
+ flatlaf.FlatLightLaf.setup()
+ }
+
/* additional look-and-feels */
@@ -541,5 +545,7 @@
}
}
-class FlatLightLaf extends GUI.Look_And_Feel(new com.formdev.flatlaf.FlatLightLaf)
-class FlatDarkLaf extends GUI.Look_And_Feel(new com.formdev.flatlaf.FlatDarkLaf)
+class FlatLightLaf extends GUI.Look_And_Feel(new flatlaf.FlatLightLaf)
+class FlatDarkLaf extends GUI.Look_And_Feel(new flatlaf.FlatDarkLaf)
+class FlatMacLightLaf extends GUI.Look_And_Feel(new flatlaf.themes.FlatMacLightLaf)
+class FlatMacDarkLaf extends GUI.Look_And_Feel(new flatlaf.themes.FlatMacDarkLaf)
--- a/src/Pure/GUI/tree_view.scala Tue Apr 22 15:08:14 2025 +0200
+++ b/src/Pure/GUI/tree_view.scala Thu Apr 24 09:16:33 2025 +0200
@@ -80,7 +80,7 @@
}
// follow jEdit
- if (!GUI.is_macos_laf) {
+ if (!GUI.is_macos_laf()) {
putClientProperty("JTree.lineStyle", "Angled")
}
}
--- a/src/Pure/General/file.scala Tue Apr 22 15:08:14 2025 +0200
+++ b/src/Pure/General/file.scala Thu Apr 24 09:16:33 2025 +0200
@@ -102,6 +102,7 @@
def is_pdf(s: String): Boolean = s.endsWith(".pdf")
def is_png(s: String): Boolean = s.endsWith(".png")
def is_scala(s: String): Boolean = s.endsWith(".scala")
+ def is_svg(s: String): Boolean = s.endsWith(".svg")
def is_tar_bz2(s: String): Boolean = s.endsWith(".tar.bz2")
def is_tar_gz(s: String): Boolean = s.endsWith(".tar.gz")
def is_tar_xz(s: String): Boolean = s.endsWith(".tar.xz")
--- a/src/Pure/General/path.scala Tue Apr 22 15:08:14 2025 +0200
+++ b/src/Pure/General/path.scala Thu Apr 24 09:16:33 2025 +0200
@@ -266,6 +266,7 @@
def thy: Path = ext("thy")
def xml: Path = ext("xml")
def xz: Path = ext("xz")
+ def zip: Path = ext("zip")
def zst: Path = ext("zst")
def backup: Path = {
--- a/src/Pure/System/isabelle_tool.scala Tue Apr 22 15:08:14 2025 +0200
+++ b/src/Pure/System/isabelle_tool.scala Thu Apr 24 09:16:33 2025 +0200
@@ -189,6 +189,7 @@
Component_JDK.isabelle_tool,
Component_JEdit.isabelle_tool,
Component_Jsoup.isabelle_tool,
+ Component_JSVG.isabelle_tool,
Component_LIPIcs.isabelle_tool,
Component_LLNCS.isabelle_tool,
Component_Minisat.isabelle_tool,
--- a/src/Tools/jEdit/etc/options Tue Apr 22 15:08:14 2025 +0200
+++ b/src/Tools/jEdit/etc/options Thu Apr 24 09:16:33 2025 +0200
@@ -159,11 +159,11 @@
section "Icons"
-option tooltip_close_icon : string = "idea-icons/actions/closeHovered.png"
-option tooltip_detach_icon : string = "idea-icons/actions/nextfile.png"
-option gutter_information_icon : string = "idea-icons/general/balloonInformation.png"
-option gutter_warning_icon : string = "idea-icons/runConfigurations/testFailed.png"
-option gutter_legacy_icon : string = "idea-icons/general/balloonWarning.png"
-option gutter_error_icon : string = "idea-icons/runConfigurations/testError.png"
-option process_passive_icon : string = "idea-icons/process/step_passive.png"
-option process_active_icons : string = "idea-icons/process/step_1.png:idea-icons/process/step_2.png:idea-icons/process/step_3.png:idea-icons/process/step_4.png:idea-icons/process/step_5.png:idea-icons/process/step_6.png:idea-icons/process/step_7.png:idea-icons/process/step_8.png:idea-icons/process/step_9.png:idea-icons/process/step_10.png:idea-icons/process/step_11.png:idea-icons/process/step_12.png"
+option tooltip_close_icon : string = "idea-icons/actions/closeDarkGrey.svg"
+option tooltip_detach_icon : string = "idea-icons/actions/copy.svg"
+option gutter_information_icon : string = "idea-icons/general/information.svg"
+option gutter_warning_icon : string = "idea-icons/general/warning.svg"
+option gutter_legacy_icon : string = "idea-icons/runConfigurations/testFailed.svg"
+option gutter_error_icon : string = "idea-icons/general/error.svg"
+option process_passive_icon : string = "idea-icons/process/big/step_passive.svg?scale=0.5"
+option process_active_icons : string = "idea-icons/process/big/step_1.svg?scale=0.5:idea-icons/process/big/step_2.svg?scale=0.5:idea-icons/process/big/step_3.svg?scale=0.5:idea-icons/process/big/step_4.svg?scale=0.5:idea-icons/process/big/step_5.svg?scale=0.5:idea-icons/process/big/step_6.svg?scale=0.5:idea-icons/process/big/step_7.svg?scale=0.5:idea-icons/process/big/step_8.svg?scale=0.5"
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Tools/jEdit/patches/gui_utilities Thu Apr 24 09:16:33 2025 +0200
@@ -0,0 +1,137 @@
+diff -ru jedit5.7.0/jEdit/org/gjt/sp/jedit/GUIUtilities.java jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/GUIUtilities.java
+--- jedit5.7.0/jEdit/org/gjt/sp/jedit/GUIUtilities.java 2024-08-03 19:53:15.000000000 +0200
++++ jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/GUIUtilities.java 2025-04-23 14:28:53.149349418 +0200
+@@ -42,6 +42,8 @@
+ import java.net.URL;
+ import java.util.*;
+ import java.util.List;
++import java.util.regex.Pattern;
++import java.util.regex.Matcher;
+ import java.lang.ref.SoftReference;
+
+ import javax.annotation.Nonnull;
+@@ -72,6 +74,8 @@
+ import java.util.concurrent.ScheduledExecutorService;
+ import java.util.concurrent.TimeUnit;
+ import java.util.concurrent.atomic.AtomicLong;
++
++import com.formdev.flatlaf.extras.FlatSVGIcon;
+ //}}}
+
+ /** Various GUI utility functions related to icons, menus, toolbars, keyboard shortcuts, etc.
+@@ -115,14 +119,14 @@
+ * @return the icon
+ * @since jEdit 2.6pre7
+ */
+- public static Icon loadIcon(String iconName)
++ public static Icon loadIcon(String iconSpec)
+ {
+- if(iconName == null)
++ if(iconSpec == null)
+ return null;
+
+ // * Enable old icon naming scheme support
+- if(deprecatedIcons.containsKey(iconName))
+- iconName = deprecatedIcons.get(iconName);
++ if(deprecatedIcons.containsKey(iconSpec))
++ iconSpec = deprecatedIcons.get(iconSpec);
+
+ // check if there is a cached version first
+ Map<String, Icon> cache = null;
+@@ -135,12 +139,25 @@
+ cache = new HashMap<>();
+ iconCache = new SoftReference<>(cache);
+ }
+- Icon icon = cache.get(iconName);
++ Icon icon = cache.get(iconSpec);
+ if(icon != null)
+ return icon;
+
+ URL url;
+
++ float iconScale = 1.0f;
++ String iconName = iconSpec;
++ {
++ Matcher matcher = Pattern.compile("^([^?]+)\\?scale=(.+)$").matcher(iconSpec);
++ if (matcher.matches()) {
++ try {
++ iconScale = Float.valueOf(matcher.group(2));
++ iconName = matcher.group(1);
++ }
++ catch (NumberFormatException e) { }
++ }
++ }
++
+ try
+ {
+ // get the icon
+@@ -164,9 +181,11 @@
+ }
+ }
+
+- icon = new ImageIcon(url);
++ icon =
++ url.toString().endsWith(".svg") ?
++ new FlatSVGIcon(url).derive(iconScale) : new ImageIcon(url);
+
+- cache.put(iconName,icon);
++ cache.put(iconSpec,icon);
+ return icon;
+ } //}}}
+
+@@ -1094,9 +1113,7 @@
+ return new Font("Monospaced", Font.PLAIN, 12);
+ }
+ else {
+- Font font2 =
+- new Font(OperatingSystem.isWindows() ? "Lucida Console" : "Monospaced",
+- Font.PLAIN, font1.getSize());
++ Font font2 = new Font("Isabelle DejaVu Sans Mono", Font.PLAIN, font1.getSize());
+ FontRenderContext frc = new FontRenderContext(null, true, false);
+ float scale =
+ font1.getLineMetrics("", frc).getHeight() / font2.getLineMetrics("", frc).getHeight();
+@@ -1107,6 +1124,22 @@
+
+ //{{{ Colors and styles
+
++ public static Color menuAcceleratorForeground(boolean selection) {
++ Color color =
++ UIManager.getColor(selection ?
++ "MenuItem.acceleratorSelectionForeground" :
++ "MenuItem.acceleratorForeground");
++
++ if (color == null) color = Color.black;
++
++ return color;
++ }
++
++ public static boolean isDarkLaf()
++ {
++ return com.formdev.flatlaf.FlatLaf.isLafDark();
++ }
++
+ //{{{ getStyleString() method
+ /**
+ * Converts a style into it's string representation.
+@@ -1619,6 +1652,21 @@
+ }
+ //}}}
+
++ //{{{ isPopupTrigger() method
++ /**
++ * Returns if the specified event is the popup trigger event.
++ * This implements precisely defined behavior, as opposed to
++ * MouseEvent.isPopupTrigger().
++ * @param evt The event
++ * @since jEdit 3.2pre8
++ * @deprecated use {@link GenericGUIUtilities#requestFocus(Window, Component)}
++ */
++ @Deprecated
++ public static boolean isPopupTrigger(MouseEvent evt)
++ {
++ return GenericGUIUtilities.isPopupTrigger(evt);
++ } //}}}
++
+ //{{{ init() method
+ static void init()
+ {
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Tools/jEdit/patches/gutter Thu Apr 24 09:16:33 2025 +0200
@@ -0,0 +1,12 @@
+diff -ru jedit5.7.0/jEdit/org/gjt/sp/jedit/textarea/Gutter.java jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/textarea/Gutter.java
+--- jedit5.7.0/jEdit/org/gjt/sp/jedit/textarea/Gutter.java 2024-08-03 19:53:18.000000000 +0200
++++ jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/textarea/Gutter.java 2025-04-22 22:08:05.106604316 +0200
+@@ -682,7 +682,7 @@
+ //{{{ Private members
+
+ //{{{ Instance variables
+- private static final int FOLD_MARKER_SIZE = 12;
++ public static final int FOLD_MARKER_SIZE = 12;
+ private static final int SELECTION_GUTTER_WIDTH = 12;
+ // The selection gutter exists only if the gutter is not expanded
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Tools/jEdit/patches/icons Thu Apr 24 09:16:33 2025 +0200
@@ -0,0 +1,580 @@
+diff -ru jedit5.7.0/jEdit/build.xml jedit5.7.0-patched/jEdit/build.xml
+--- jedit5.7.0/jEdit/build.xml 2024-08-03 19:53:28.000000000 +0200
++++ jedit5.7.0-patched/jEdit/build.xml 2025-04-16 17:20:57.401732024 +0200
+@@ -488,6 +488,7 @@
+ <include name="org/gjt/sp/jedit/icons/**/*.gif"/>
+ <include name="org/gjt/sp/jedit/icons/**/*.jpg"/>
+ <include name="org/gjt/sp/jedit/icons/**/*.png"/>
++ <include name="org/gjt/sp/jedit/icons/**/*.svg"/>
+ <include name="org/jedit/localization/*.props"/>
+ </fileset>
+ </jar>
+@@ -783,6 +784,7 @@
+ <include name="*.txt"/>
+ <include name="*.html"/>
+ <include name="*.png"/>
++ <include name="*.svg"/>
+ <include name="tips/**"/>
+ </fileset>
+ </copy>
+diff -ru jedit5.7.0/jEdit/ivy.xml jedit5.7.0-patched/jEdit/ivy.xml
+--- jedit5.7.0/jEdit/ivy.xml 2024-08-03 19:53:28.000000000 +0200
++++ jedit5.7.0-patched/jEdit/ivy.xml 2025-04-16 12:22:57.782535840 +0200
+@@ -94,5 +94,8 @@
+ <dependency org="com.google.code.findbugs" name="jsr305" rev="3.0.2"/>
+
+ <dependency org="com.evolvedbinary.appbundler" name="appbundler" rev="1.3.0" conf="appbundler"/>
++
++ <dependency org="com.formdev" name="flatlaf" rev="3.6"/>
++ <dependency org="com.formdev" name="flatlaf-extras" rev="3.6"/>
+ </dependencies>
+ </ivy-module>
+diff -ru jedit5.7.0/jEdit/org/gjt/sp/jedit/gui/MarkerViewer.java jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/gui/MarkerViewer.java
+--- jedit5.7.0/jEdit/org/gjt/sp/jedit/gui/MarkerViewer.java 2024-08-03 19:53:18.000000000 +0200
++++ jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/gui/MarkerViewer.java 2025-04-16 21:35:23.519418287 +0200
+@@ -50,28 +50,28 @@
+ toolBar.add(Box.createGlue());
+
+ RolloverButton addMarker = new RolloverButton(
+- GUIUtilities.loadIcon("Plus.png"));
++ GUIUtilities.loadIcon(jEdit.getProperty("add-marker.icon.small")));
+ addMarker.setToolTipText(GenericGUIUtilities.prettifyMenuLabel(
+ jEdit.getProperty("add-marker.label")));
+ addMarker.addActionListener(this);
+ addMarker.setActionCommand("add-marker");
+ toolBar.add(addMarker);
+
+- previous = new RolloverButton(GUIUtilities.loadIcon("ArrowL.png"));
++ previous = new RolloverButton(GUIUtilities.loadIcon(jEdit.getProperty("prev-marker.icon.small")));
+ previous.setToolTipText(GenericGUIUtilities.prettifyMenuLabel(
+ jEdit.getProperty("prev-marker.label")));
+ previous.addActionListener(this);
+ previous.setActionCommand("prev-marker");
+ toolBar.add(previous);
+
+- next = new RolloverButton(GUIUtilities.loadIcon("ArrowR.png"));
++ next = new RolloverButton(GUIUtilities.loadIcon(jEdit.getProperty("next-marker.icon.small")));
+ next.setToolTipText(GenericGUIUtilities.prettifyMenuLabel(
+ jEdit.getProperty("next-marker.label")));
+ next.addActionListener(this);
+ next.setActionCommand("next-marker");
+ toolBar.add(next);
+
+- clear = new RolloverButton(GUIUtilities.loadIcon("Clear.png"));
++ clear = new RolloverButton(GUIUtilities.loadIcon(jEdit.getProperty("remove-all-markers.icon.small")));
+ clear.setToolTipText(GenericGUIUtilities.prettifyMenuLabel(
+ jEdit.getProperty("remove-all-markers.label")));
+ clear.addActionListener(this);
+diff -ru jedit5.7.0/jEdit/org/gjt/sp/jedit/jedit_gui.props jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/jedit_gui.props
+--- jedit5.7.0/jEdit/org/gjt/sp/jedit/jedit_gui.props 2024-08-03 19:53:20.000000000 +0200
++++ jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/jedit_gui.props 2025-04-17 00:11:31.583536114 +0200
+@@ -8,13 +8,15 @@
+ ###
+
+ #{{{ Common icons
+-common.add.icon=22x22/actions/list-add.png
+-common.remove.icon=22x22/actions/list-remove.png
+-common.moveUp.icon=22x22/actions/go-up.png
+-common.moveDown.icon=22x22/actions/go-down.png
+-common.clearAll.icon=22x22/actions/edit-clear.png
++common.add.icon=32x32/actions/list-add.svg?scale=0.7
++common.remove.icon=32x32/actions/list-remove.svg?scale=0.7
++common.moveUp.icon=32x32/actions/go-up.svg?scale=0.7
++common.moveDown.icon=32x32/actions/go-down.svg?scale=0.7
++common.clearAll.icon=32x32/actions/edit-clear.svg?scale=0.7
+ logo.icon.small=16x16/apps/jedit.png
+ logo.icon.medium=32x32/apps/jedit.png
++navigate-backwards.icon=idea-icons/expui/general/chevronLeft.svg?scale=1.2
++navigate-forwards.icon=idea-icons/expui/general/chevronRight.svg?scale=1.2
+
+ #}}}
+
+@@ -28,7 +30,7 @@
+ defer=false
+ startup=true
+
+-broken-image.icon=22x22/status/image-missing.png
++broken-image.icon=32x32/status/image-missing.svg?scale=0.7
+ dropdown-arrow.icon=ToolbarMenu.gif
+ #}}}
+
+@@ -39,68 +41,69 @@
+ buffer-options combined-options - \
+ plugin-manager - help
+
+-new-file.icon=22x22/actions/document-new.png
+-open-file.icon=22x22/actions/document-open.png
+-save.icon=22x22/actions/document-save.png
+-close-buffer.icon=22x22/actions/document-close.png
+-global-close-buffer.icon=22x22/actions/document-close.png
+-print.icon=22x22/actions/document-print.png
++new-file.icon=32x32/actions/document-new.svg?scale=0.7
++open-file.icon=32x32/actions/document-open.svg?scale=0.7
++save.icon=32x32/actions/document-save.svg?scale=0.7
++close-buffer.icon=32x32/actions/process-stop.svg?scale=0.7
++global-close-buffer.icon=32x32/actions/process-stop.svg?scale=0.7
++print.icon=32x32/actions/document-print.svg?scale=0.7
+ page-setup.icon=22x22/actions/printer-setup.png
+-undo.icon=22x22/actions/edit-undo.png
+-redo.icon=22x22/actions/edit-redo.png
+-cut.icon=22x22/actions/edit-cut.png
+-copy.icon=22x22/actions/edit-copy.png
+-paste.icon=22x22/actions/edit-paste.png
+-find.icon=22x22/actions/edit-find.png
+-find-next.icon=22x22/actions/edit-find-next.png
+-new-view.icon=22x22/actions/window-new.png
++undo.icon=32x32/actions/edit-undo.svg?scale=0.7
++redo.icon=32x32/actions/edit-redo.svg?scale=0.7
++cut.icon=32x32/actions/edit-cut.svg?scale=0.7
++copy.icon=32x32/actions/edit-copy.svg?scale=0.7
++paste.icon=32x32/actions/edit-paste.svg?scale=0.7
++find.icon=32x32/actions/edit-find.svg?scale=0.7
++find-prev.icon=32x32/actions/go-previous.svg?scale=0.7
++find-next.icon=32x32/actions/go-next.svg?scale=0.7
++new-view.icon=32x32/actions/window-new.svg?scale=0.7
+ unsplit.icon=22x22/actions/window-unsplit.png
+ split-horizontal.icon=22x22/actions/window-split-horizontal.png
+ split-vertical.icon=22x22/actions/window-split-vertical.png
+-buffer-options.icon=22x22/actions/document-properties.png
+-global-options.icon=22x22/categories/preferences-system.png
+-combined-options.icon=22x22/categories/preferences-system.png
++buffer-options.icon=32x32/actions/document-properties.svg?scale=0.7
++global-options.icon=32x32/categories/preferences-system.svg?scale=0.7
++combined-options.icon=32x32/categories/preferences-system.svg?scale=0.7
+ plugin-manager.icon=22x22/places/plugins.png
+-help.icon=22x22/apps/help-browser.png
++help.icon=22x22/apps/help-browser.svg
+
+ #{{{ Icon list for tool bar editor
+ icons=22x22/actions/resize-horisontal.png \
+- 22x22/actions/go-down.png \
+- 22x22/actions/go-previous.png \
+- 22x22/actions/go-next.png \
+- 22x22/actions/go-home.png \
+- 22x22/actions/go-up.png \
+- 22x22/actions/go-first.png \
+- 22x22/actions/go-last.png \
+- 22x22/actions/go-parent.png \
+- 22x22/actions/document-close.png \
+- 22x22/actions/edit-undo.png \
+- 22x22/actions/edit-redo.png \
+- 22x22/actions/edit-cut.png \
+- 22x22/actions/edit-paste.png \
+- 22x22/actions/edit-delete.png \
+- 22x22/actions/edit-clear.png \
+- 22x22/actions/edit-find-next.png \
+- 22x22/actions/edit-find-in-folder.png \
+- 22x22/actions/edit-find.png \
+- 22x22/actions/edit-copy.png \
++ 22x22/actions/go-down.svg \
++ 22x22/actions/go-previous.svg \
++ 22x22/actions/go-next.svg \
++ 32x32/actions/go-home.svg?scale=0.7 \
++ 22x22/actions/go-up.svg \
++ 22x22/actions/go-first.svg \
++ 22x22/actions/go-last.svg \
++ 22x22/actions/go-up.svg \
++ 32x32/actions/process-stop.svg?scale=0.7 \
++ 32x32/actions/edit-undo.svg?scale=0.7 \
++ 32x32/actions/edit-redo.svg?scale=0.7 \
++ 32x32/actions/edit-cut.svg?scale=0.7 \
++ 32x32/actions/edit-paste.svg?scale=0.7 \
++ scalable/actions/edit-delete.svg \
++ 22x22/actions/edit-clear.svg \
++ 22x22/actions/go-next.svg \
++ 32x32/actions/system-search.svg?scale=0.7 \
++ 32x32/actions/edit-find.svg?scale=0.7 \
++ 32x32/actions/edit-copy.svg?scale=0.7 \
+ 22x22/actions/copy-to-buffer.png \
+- 22x22/actions/list-remove.png \
+- 22x22/actions/list-add.png \
+- 22x22/actions/folder-new.png \
+- 22x22/actions/window-new.png \
+- 22x22/actions/document-new.png \
+- 22x22/actions/document-open.png \
++ 32x32/actions/list-remove.svg?scale=0.7 \
++ 32x32/actions/list-add.svg?scale=0.7 \
++ 32x32/actions/folder-new.svg?scale=0.7 \
++ 32x32/actions/document-new.svg?scale=0.7 \
++ 32x32/actions/document-new.svg?scale=0.7 \
++ 32x32/actions/document-open.svg?scale=0.7 \
+ 22x22/actions/document-reload2.png \
+- 22x22/actions/document-properties.png \
+- 22x22/actions/document-save.png \
+- 22x22/actions/document-save-all.png \
+- 22x22/actions/document-save-as.png \
++ 32x32/actions/document-properties.svg?scale=0.7 \
++ 32x32/actions/document-save.svg?scale=0.7 \
++ 32x32/actions/document-save-all.svg?scale=0.5 \
++ 32x32/actions/document-save-as.svg?scale=0.7 \
+ 22x22/actions/printer-setup.png \
+- 22x22/actions/process-stop.png \
+- 22x22/actions/media-playback-pause.png \
+- 22x22/actions/media-playback-start.png \
+- 22x22/actions/view-refresh.png \
++ 22x22/actions/system-log-out.svg \
++ 22x22/actions/media-playback-pause.svg \
++ 22x22/actions/media-playback-start.svg \
++ 22x22/actions/view-refresh.svg \
+ 22x22/actions/application-run.png \
+ 22x22/actions/edit-find-multiple.png \
+ 22x22/actions/edit-find-single.png \
+@@ -109,18 +112,18 @@
+ 22x22/actions/window-unsplit.png \
+ 22x22/actions/zoom-in.png \
+ 22x22/actions/zoom-out.png \
+- 22x22/apps/utilities-terminal.png \
+- 22x22/apps/system-file-manager.png \
+- 22x22/apps/internet-web-browser.png \
+- 22x22/apps/help-browser.png \
+- 22x22/apps/system-installer.png \
+- 22x22/status/image-missing.png \
+- 22x22/status/folder-visiting.png \
+- 22x22/devices/drive-harddisk.png \
+- 22x22/devices/media-floppy.png \
+- 22x22/devices/printer.png \
++ 22x22/apps/utilities-terminal.svg \
++ 32x32/apps/system-file-manager.svg?scale=0.7 \
++ 32x32/apps/internet-web-browser.svg?scale=0.7 \
++ 22x22/apps/help-browser.svg \
++ 32x32/apps/system-installer.svg?scale=0.7 \
++ 32x32/status/image-missing.svg?scale=0.7 \
++ 32x32/status/folder-visiting.svg?scale=0.7 \
++ 32x32/devices/drive-harddisk.svg?scale=0.7 \
++ 22x22/devices/media-floppy.svg \
++ 32x32/devices/printer.svg?scale=0.7 \
+ 22x22/places/plugins.png \
+- 22x22/categories/preferences-system.png \
++ 32x32/categories/preferences-system.svg?scale=0.7 \
+ Blank24.gif
+ #}}}
+
+@@ -163,31 +166,31 @@
+ print \
+ - \
+ exit
+-new-file.icon.small=16x16/actions/document-new.png
+-new-file-in-mode.icon.small=16x16/actions/document-new.png
+-open-file.icon.small=16x16/actions/document-open.png
+-reload.icon.small=16x16/actions/view-refresh.png
+-reload-all.icon.small=16x16/actions/view-refresh.png
+-close-buffer.icon.small=16x16/actions/document-close.png
+-closeall-bufferset.icon.small=16x16/actions/document-close.png
+-closeall-except-active.icon.small=16x16/actions/document-close.png
+-global-close-buffer.icon.small=16x16/actions/document-close.png
+-close-all.icon.small=16x16/actions/document-close.png
+-save.icon.small=16x16/actions/document-save.png
+-save-as.icon.small=16x16/actions/document-save-as.png
+-save-a-copy-as.icon.small=16x16/actions/document-save-as.png
+-save-all.icon.small=16x16/actions/document-save-all.png
+-print.icon.small=16x16/actions/document-print.png
+-page-setup.icon.small=16x16/actions/document-properties.png
+-exit.icon.small=16x16/actions/process-stop.png
+-exit.icon.medium=22x22/actions/process-stop.png
++new-file.icon.small=32x32/actions/document-new.svg?scale=0.5
++new-file-in-mode.icon.small=32x32/actions/document-new.svg?scale=0.5
++open-file.icon.small=32x32/actions/document-open.svg?scale=0.5
++reload.icon.small=16x16/actions/view-refresh.svg
++reload-all.icon.small=16x16/actions/view-refresh.svg
++close-buffer.icon.small=32x32/actions/process-stop.svg?scale=0.5
++closeall-bufferset.icon.small=32x32/actions/process-stop.svg?scale=0.5
++closeall-except-active.icon.small=32x32/actions/process-stop.svg?scale=0.5
++global-close-buffer.icon.small=32x32/actions/process-stop.svg?scale=0.5
++close-all.icon.small=32x32/actions/process-stop.svg?scale=0.5
++save.icon.small=32x32/actions/document-save.svg?scale=0.5
++save-as.icon.small=32x32/actions/document-save-as.svg?scale=0.5
++save-a-copy-as.icon.small=32x32/actions/document-save-as.svg?scale=0.5
++save-all.icon.small=32x32/actions/document-save.svg?scale=0.5
++print.icon.small=32x32/actions/document-print.svg?scale=0.5
++page-setup.icon.small=32x32/actions/document-properties.svg?scale=0.5
++exit.icon.small=16x16/actions/system-log-out.svg
++exit.icon.medium=22x22/actions/system-log-out.svg
+
+ #{{{ Recent Files menu
+ recent-files.code=new RecentFilesProvider();
+ #}}}
+
+ reload-encoding.code=new ReloadWithEncodingProvider();
+-reload-encoding.icon.small=16x16/actions/view-refresh.png
++reload-encoding.icon.small=16x16/actions/view-refresh.svg
+ #}}}
+
+ #{{{ Edit menu
+@@ -211,12 +214,12 @@
+ %text \
+ %indent \
+ %source
+-undo.icon.small=16x16/actions/edit-undo.png
+-redo.icon.small=16x16/actions/edit-redo.png
+-cut.icon.small=16x16/actions/edit-cut.png
+-copy.icon.small=16x16/actions/edit-copy.png
+-paste.icon.small=16x16/actions/edit-paste.png
+-select-all.icon.small=16x16/actions/edit-select-all.png
++undo.icon.small=32x32/actions/edit-undo.svg?scale=0.5
++redo.icon.small=32x32/actions/edit-redo.svg?scale=0.5
++cut.icon.small=32x32/actions/edit-cut.svg?scale=0.5
++copy.icon.small=32x32/actions/edit-copy.svg?scale=0.5
++paste.icon.small=32x32/actions/edit-paste.svg?scale=0.5
++select-all.icon.small=16x16/actions/edit-select-all.svg
+
+ #{{{ More Clipboard menu
+ clipboard=cut-append \
+@@ -308,16 +311,18 @@
+ regexp \
+ - \
+ hypersearch-results
+-find.icon.small=22x22/actions/edit-find.png
+-find-next.icon.small=22x22/actions/edit-find-next.png
+-search-in-directory.icon.small=22x22/actions/edit-find-in-folder.png
+-replace-in-selection.icon.small=22x22/actions/edit-find-replace.png
+-replace-and-find-next.icon.small=22x22/actions/edit-find-replace.png
+-replace-all.icon.small=22x22/actions/edit-find-replace.png
+-quick-search.icon.small=22x22/actions/edit-find.png
+-hypersearch.icon.small=22x22/actions/edit-find-multiple.png
+-quick-search-word.icon.small=22x22/actions/edit-find.png
+-hypersearch-word.icon.small=22x22/actions/edit-find.png
++find.icon.small=32x32/actions/edit-find.svg?scale=0.7
++find-prev.icon.small=32x32/actions/go-previous.svg?scale=0.7
++find-next.icon.small=32x32/actions/go-next.svg?scale=0.7
++search-in-open-buffers.icon.small=32x32/actions/system-search.svg?scale=0.7
++search-in-directory.icon.small=32x32/actions/system-search.svg?scale=0.7
++replace-in-selection.icon.small=32x32/actions/edit-find-replace.svg?scale=0.7
++replace-and-find-next.icon.small=32x32/actions/edit-find-replace.svg?scale=0.7
++replace-all.icon.small=32x32/actions/edit-find-replace.svg?scale=0.7
++quick-search.icon.small=32x32/actions/edit-find.svg?scale=0.7
++hypersearch.icon.small=32x32/actions/edit-find.svg?scale=0.7
++quick-search-word.icon.small=32x32/actions/edit-find.svg?scale=0.7
++hypersearch-word.icon.small=32x32/actions/edit-find.svg?scale=0.7
+ #}}}
+
+ #{{{ Markers menu
+@@ -336,12 +341,12 @@
+ view-markers \
+ -
+ markers.code=new MarkersProvider();
+-add-marker.icon.small=22x22/actions/bookmark-new.png
+-add-marker-shortcut.icon.small=22x22/actions/bookmark-new.png
+-remove-all-markers.icon.small=22x22/actions/edit-clear.png
+-goto-marker.icon.small=22x22/actions/go-jump.png
+-prev-marker.icon.small=22x22/actions/go-previous.png
+-next-marker.icon.small=22x22/actions/go-next.png
++add-marker.icon.small=32x32/actions/bookmark-new.svg?scale=0.7
++add-marker-shortcut.icon.small=32x32/actions/bookmark-new.svg?scale=0.7
++remove-all-markers.icon.small=32x32/actions/edit-clear.svg?scale=0.7
++goto-marker.icon.small=32x32/actions/go-jump.svg?scale=0.7
++prev-marker.icon.small=32x32/actions/go-previous.svg?scale=0.7
++next-marker.icon.small=32x32/actions/go-next.svg?scale=0.7
+ #}}}
+
+ #{{{ Folding menu
+@@ -388,9 +393,12 @@
+ - \
+ set-view-title \
+ toggle-full-screen
+-new-view.icon.small=16x16/actions/window-new.png
+-new-plain-view.icon.small=16x16/actions/window-new.png
+-close-view.icon.small=16x16/actions/document-close.png
++new-view.icon.small=32x32/actions/window-new.svg?scale=0.5
++new-plain-view.icon.small=32x32/actions/window-new.svg?scale=0.5
++close-view.icon.small=32x32/actions/process-stop.svg?scale=0.5
++prev-buffer.icon.small=32x32/actions/go-previous.svg?scale=0.5
++next-buffer.icon.small=32x32/actions/go-next.svg?scale=0.5
++recent-buffer.icon.small=32x32/actions/go-up.svg?scale=0.5
+
+ #{{{ Scrolling menu
+ scrolling=scroll-to-current-line \
+@@ -454,9 +462,9 @@
+ - \
+ %quick-options
+
+-buffer-options.icon.small=16x16/actions/document-properties.png
+-global-options.icon.small=16x16/categories/preferences-system.png
+-combined-options.icon.small=16x16/categories/preferences-system.png
++buffer-options.icon.small=32x32/actions/document-properties.svg?scale=0.5
++global-options.icon.small=32x32/categories/preferences-system.svg?scale=0.5
++combined-options.icon.small=32x32/categories/preferences-system.svg?scale=0.5
+
+ #{{{ Recent Directories menu
+ recent-directories.code=new RecentDirectoriesProvider();
+@@ -518,9 +526,9 @@
+ rescan-macros \
+ -
+ macros.code=new MacrosProvider();
+-new-macro.icon.small=16x16/actions/document-new.png
+-record-macro.icon.small=16x16/actions/media-record.png
+-stop-recording.icon.small=16x16/actions/media-playback-stop.png
++new-macro.icon.small=32x32/actions/document-new.svg?scale=0.5
++record-macro.icon.small=16x16/actions/media-record.svg
++stop-recording.icon.small=32x32/actions/media-playback-stop.svg?scale=0.5
+ #}}}
+
+ #{{{ Plugins menu
+@@ -771,7 +779,7 @@
+
+ #{{{ HyperSearch results dialog
+ hypersearch-results.clear.icon=22x22/actions/edit-clear.png
+-hypersearch-results.stop.icon=22x22/actions/process-stop.png
++hypersearch-results.stop.icon=22x22/actions/system-log-out.png
+ hypersearch-results.multi.multiple.icon=22x22/actions/edit-find-multiple.png
+ hypersearch-results.multi.single.icon=22x22/actions/edit-find-single.png
+ hypersearch-results.match.highlight.icon=22x22/actions/edit-find-highlight-match.png
+@@ -784,8 +792,8 @@
+ #}}}
+
+ #{{{ Help Viewer
+-helpviewer.back.icon=22x22/actions/go-previous.png
+-helpviewer.forward.icon=22x22/actions/go-next.png
++helpviewer.back.icon=32x32/actions/go-previous.svg?scale=0.7
++helpviewer.forward.icon=32x32/actions/go-next.svg?scale=0.7
+ #}}}
+
+ #}}}
+@@ -809,9 +817,9 @@
+
+ #{{{ Abbreviations pane
+ options.abbrevs.code=new AbbrevsOptionPane();
+-options.abbrevs.add.icon=22x22/actions/list-add.png
+-options.abbrevs.edit.icon=22x22/actions/document-properties.png
+-options.abbrevs.remove.icon=22x22/actions/list-remove.png
++options.abbrevs.add.icon=32x32/actions/list-add.svg?scale=0.7
++options.abbrevs.edit.icon=32x32/actions/document-properties.svg?scale=0.7
++options.abbrevs.remove.icon=32x32/actions/list-remove.svg?scale=0.7
+ #}}}
+
+ #{{{ Appearance pane
+@@ -840,11 +848,11 @@
+
+ #{{{ Context Menu pane
+ options.context.code=new ContextOptionPane();
+-options.context.add.icon=22x22/actions/list-add.png
+-options.context.remove.icon=22x22/actions/list-remove.png
+-options.context.moveUp.icon=22x22/actions/go-up.png
+-options.context.moveDown.icon=22x22/actions/go-down.png
+-options.context.reset.icon=22x22/actions/edit-clear.png
++options.context.add.icon=32x32/actions/list-add.svg?scale=0.7
++options.context.remove.icon=32x32/actions/list-remove.svg?scale=0.7
++options.context.moveUp.icon=32x32/actions/go-up.svg?scale=0.7
++options.context.moveDown.icon=32x32/actions/go-down.svg?scale=0.7
++options.context.reset.icon=32x32/actions/edit-clear.svg?scale=0.7
+ options.context.includeOptionsLink=true
+ #}}}
+
+@@ -906,12 +914,12 @@
+
+ #{{{ Tool Bar pane
+ options.toolbar.code=new ToolBarOptionPane();
+-options.toolbar.add.icon=22x22/actions/list-add.png
+-options.toolbar.remove.icon=22x22/actions/list-remove.png
+-options.toolbar.moveUp.icon=22x22/actions/go-up.png
+-options.toolbar.moveDown.icon=22x22/actions/go-down.png
+-options.toolbar.reset.icon=22x22/actions/edit-clear.png
+-options.toolbar.edit.icon=22x22/actions/document-properties.png
++options.toolbar.add.icon=32x32/actions/list-add.svg?scale=0.7
++options.toolbar.remove.icon=32x32/actions/list-remove.svg?scale=0.7
++options.toolbar.moveUp.icon=22x22/actions/go-up.svg
++options.toolbar.moveDown.icon=22x22/actions/go-down.svg
++options.toolbar.reset.icon=22x22/actions/edit-clear.svg
++options.toolbar.edit.icon=32x32/actions/document-properties.svg?scale=0.7
+ #}}}
+
+ #{{{ View pane
+@@ -949,7 +957,8 @@
+ vfs.browser.default-filter=*[^~#]
+ vfs.browser.filter-enabled=true
+ vfs.browser.file.icon=16x16/mimetypes/text-x-generic.png
+-vfs.browser.icon.small=16x16/apps/system-file-manager.png
++vfs.browser.icon=32x32/apps/system-file-manager.svg?scale=0.7
++vfs.browser.icon.small=32x32/apps/system-file-manager.svg?scale=0.5
+ vfs.browser.open-file.icon=16x16/actions/edit-select-all.png
+ vfs.browser.dir.icon=16x16/places/folder.png
+ vfs.browser.open-dir.icon=16x16/status/folder-open.png
+@@ -1007,13 +1016,13 @@
+ plugin-manager.mirror-url=http://plugins.jedit.org/export/mirror_list.php
+
+ #{{{ Plugin management
+-manage-plugins.restore.icon=22x22/actions/document-open.png
+-manage-plugins.save.icon=22x22/actions/document-save.png
++manage-plugins.restore.icon=32x32/actions/document-open.svg?scale=0.7
++manage-plugins.save.icon=32x32/actions/document-save.svg?scale=0.7
+ #}}}
+
+ #{{{ Plugin installation
+-install-plugins.choose-plugin-set.icon=22x22/actions/document-open.png
+-install-plugins.clear-plugin-set.icon=22x22/actions/edit-clear.png
++install-plugins.choose-plugin-set.icon=32x32/actions/document-open.svg?scale=0.7
++install-plugins.clear-plugin-set.icon=22x22/actions/edit-clear.svg
+ #}}}
+
+ #}}}
+diff -ru jedit5.7.0/jEdit/org/gjt/sp/jedit/options/BrowserColorsOptionPane.java jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/options/BrowserColorsOptionPane.java
+--- jedit5.7.0/jEdit/org/gjt/sp/jedit/options/BrowserColorsOptionPane.java 2024-08-03 19:53:15.000000000 +0200
++++ jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/options/BrowserColorsOptionPane.java 2025-04-16 16:12:29.542089148 +0200
+@@ -78,12 +78,12 @@
+ buttons.setBorder(new EmptyBorder(3,0,0,0));
+ buttons.setLayout(new BoxLayout(buttons,BoxLayout.X_AXIS));
+ ActionListener actionHandler = new ActionHandler();
+- JButton add = new RolloverButton(GUIUtilities.loadIcon("Plus.png"));
++ JButton add = new RolloverButton(GUIUtilities.loadIcon(jEdit.getProperty("common.add.icon")));
+ add.setToolTipText(jEdit.getProperty("common.add"));
+ add.addActionListener(e -> colorsModel.add());
+ buttons.add(add);
+ buttons.add(Box.createHorizontalStrut(6));
+- remove = new RolloverButton(GUIUtilities.loadIcon("Minus.png"));
++ remove = new RolloverButton(GUIUtilities.loadIcon(jEdit.getProperty("common.remove.icon")));
+ remove.setToolTipText(jEdit.getProperty("common.remove"));
+ remove.addActionListener(e ->
+ {
+@@ -93,12 +93,12 @@
+ });
+ buttons.add(remove);
+ buttons.add(Box.createHorizontalStrut(6));
+- moveUp = new RolloverButton(GUIUtilities.loadIcon("ArrowU.png"));
++ moveUp = new RolloverButton(GUIUtilities.loadIcon(jEdit.getProperty("common.moveUp.icon")));
+ moveUp.setToolTipText(jEdit.getProperty("common.moveUp"));
+ moveUp.addActionListener(actionHandler);
+ buttons.add(moveUp);
+ buttons.add(Box.createHorizontalStrut(6));
+- moveDown = new RolloverButton(GUIUtilities.loadIcon("ArrowD.png"));
++ moveDown = new RolloverButton(GUIUtilities.loadIcon(jEdit.getProperty("common.moveDown.icon")));
+ moveDown.setToolTipText(jEdit.getProperty("common.moveDown"));
+ moveDown.addActionListener(actionHandler);
+ buttons.add(moveDown);
+diff -ru jedit5.7.0/jEdit/org/gjt/sp/jedit/options/StatusBarOptionPane.java jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/options/StatusBarOptionPane.java
+--- jedit5.7.0/jEdit/org/gjt/sp/jedit/options/StatusBarOptionPane.java 2024-08-03 19:53:15.000000000 +0200
++++ jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/options/StatusBarOptionPane.java 2025-04-16 16:12:37.730958557 +0200
+@@ -160,12 +160,12 @@
+ buttons.setBorder(new EmptyBorder(3,0,0,0));
+ buttons.setLayout(new BoxLayout(buttons,BoxLayout.X_AXIS));
+ buttons.add(Box.createHorizontalStrut(6));
+- moveUp = new RolloverButton(GUIUtilities.loadIcon("ArrowU.png"));
++ moveUp = new RolloverButton(GUIUtilities.loadIcon(jEdit.getProperty("common.moveUp.icon")));
+ moveUp.setToolTipText(jEdit.getProperty("options.status.moveUp"));
+ moveUp.addActionListener(e -> moveUp());
+ buttons.add(moveUp);
+ buttons.add(Box.createHorizontalStrut(6));
+- moveDown = new RolloverButton(GUIUtilities.loadIcon("ArrowD.png"));
++ moveDown = new RolloverButton(GUIUtilities.loadIcon(jEdit.getProperty("common.moveDown.icon")));
+ moveDown.setToolTipText(jEdit.getProperty("options.status.moveDown"));
+ moveDown.addActionListener(e -> moveDown());
+ buttons.add(moveDown);
+diff -ru jedit5.7.0/jEdit/org/gjt/sp/jedit/gui/RegisterViewer.java jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/gui/RegisterViewer.java
+--- jedit5.7.0/jEdit/org/gjt/sp/jedit/gui/RegisterViewer.java 2024-08-03 19:53:18.000000000 +0200
++++ jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/gui/RegisterViewer.java 2025-04-16 21:45:44.861713409 +0200
+@@ -54,7 +54,7 @@
+ toolBar.add(Box.createGlue());
+
+ RolloverButton pasteRegister = new RolloverButton(
+- GUIUtilities.loadIcon("Paste.png"));
++ GUIUtilities.loadIcon(jEdit.getProperty("paste.icon")));
+ pasteRegister.setToolTipText(GenericGUIUtilities.prettifyMenuLabel(
+ jEdit.getProperty("paste-string-register.label")));
+ pasteRegister.addActionListener(e -> insertRegister());
+@@ -62,7 +62,7 @@
+ toolBar.add(pasteRegister);
+
+ RolloverButton clearRegister = new RolloverButton(
+- GUIUtilities.loadIcon("Clear.png"));
++ GUIUtilities.loadIcon(jEdit.getProperty("common.clearAll.icon")));
+ clearRegister.setToolTipText(GenericGUIUtilities.prettifyMenuLabel(
+ jEdit.getProperty("clear-string-register.label")));
+ clearRegister.addActionListener(e -> clearSelectedIndex());
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Tools/jEdit/patches/menu_accelerator Thu Apr 24 09:16:33 2025 +0200
@@ -0,0 +1,144 @@
+diff -ru jedit5.7.0/jEdit/org/gjt/sp/jedit/menu/EnhancedCheckBoxMenuItem.java jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/menu/EnhancedCheckBoxMenuItem.java
+--- jedit5.7.0/jEdit/org/gjt/sp/jedit/menu/EnhancedCheckBoxMenuItem.java 2024-08-03 19:53:18.000000000 +0200
++++ jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/menu/EnhancedCheckBoxMenuItem.java 2025-04-23 14:40:22.714447724 +0200
+@@ -99,7 +99,7 @@
+
+ if(shortcut != null)
+ {
+- FontMetrics fm = getFontMetrics(EnhancedMenuItem.acceleratorFont);
++ FontMetrics fm = getFontMetrics(GUIUtilities.menuAcceleratorFont());
+ d.width += (fm.stringWidth(shortcut) + fm.stringWidth("AAAA"));
+ }
+ return d;
+@@ -114,11 +114,9 @@
+ if(shortcut != null)
+ {
+ Graphics2D g2 = (Graphics2D)g;
+- g.setFont(EnhancedMenuItem.acceleratorFont);
++ g.setFont(GUIUtilities.menuAcceleratorFont());
+ g2.setRenderingHint(RenderingHints.KEY_ANTIALIASING, RenderingHints.VALUE_ANTIALIAS_ON);
+- g.setColor(getModel().isArmed() ?
+- EnhancedMenuItem.acceleratorSelectionForeground :
+- EnhancedMenuItem.acceleratorForeground);
++ g.setColor(GUIUtilities.menuAcceleratorForeground(getModel().isArmed()));
+ FontMetrics fm = g.getFontMetrics();
+ Insets insets = getInsets();
+ g.drawString(shortcut,getWidth() - (fm.stringWidth(
+diff -ru jedit5.7.0/jEdit/org/gjt/sp/jedit/menu/EnhancedMenuItem.java jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/menu/EnhancedMenuItem.java
+--- jedit5.7.0/jEdit/org/gjt/sp/jedit/menu/EnhancedMenuItem.java 2024-08-03 19:53:18.000000000 +0200
++++ jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/menu/EnhancedMenuItem.java 2025-04-23 14:26:16.757848751 +0200
+@@ -94,7 +94,7 @@
+
+ if(shortcut != null)
+ {
+- FontMetrics fm = getFontMetrics(acceleratorFont);
++ FontMetrics fm = getFontMetrics(GUIUtilities.menuAcceleratorFont());
+ d.width += (fm.stringWidth(shortcut) + fm.stringWidth("AAAA"));
+ }
+ return d;
+@@ -109,11 +109,9 @@
+ if(shortcut != null)
+ {
+ Graphics2D g2 = (Graphics2D)g;
+- g.setFont(acceleratorFont);
++ g.setFont(GUIUtilities.menuAcceleratorFont());
+ g2.setRenderingHint(RenderingHints.KEY_ANTIALIASING, RenderingHints.VALUE_ANTIALIAS_ON);
+- g.setColor(getModel().isArmed() ?
+- acceleratorSelectionForeground :
+- acceleratorForeground);
++ g.setColor(GUIUtilities.menuAcceleratorForeground(getModel().isArmed()));
+ FontMetrics fm = g.getFontMetrics();
+ Insets insets = getInsets();
+ g.drawString(shortcut,getWidth() - (fm.stringWidth(
+@@ -122,12 +120,6 @@
+ }
+ } //}}}
+
+- //{{{ Package-private members
+- static Font acceleratorFont;
+- static Color acceleratorForeground;
+- static Color acceleratorSelectionForeground;
+- //}}}
+-
+ //{{{ Private members
+
+ //{{{ Instance variables
+@@ -135,25 +127,5 @@
+ private String shortcut;
+ //}}}
+
+- //{{{ Class initializer
+- static
+- {
+- acceleratorFont = GUIUtilities.menuAcceleratorFont();
+-
+- acceleratorForeground = UIManager
+- .getColor("MenuItem.acceleratorForeground");
+- if(acceleratorForeground == null)
+- {
+- acceleratorForeground = Color.black;
+- }
+-
+- acceleratorSelectionForeground = UIManager
+- .getColor("MenuItem.acceleratorSelectionForeground");
+- if(acceleratorSelectionForeground == null)
+- {
+- acceleratorSelectionForeground = Color.black;
+- }
+- } //}}}
+-
+ //}}}
+ }
+diff -ru jedit5.7.0/jEdit/org/gjt/sp/jedit/menu/MarkersProvider.java jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/menu/MarkersProvider.java
+--- jedit5.7.0/jEdit/org/gjt/sp/jedit/menu/MarkersProvider.java 2024-08-03 19:53:18.000000000 +0200
++++ jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/menu/MarkersProvider.java 2025-04-23 14:27:48.829375470 +0200
+@@ -107,7 +107,7 @@
+
+ if(shortcut != null)
+ {
+- FontMetrics fm = getFontMetrics(acceleratorFont);
++ FontMetrics fm = getFontMetrics(GUIUtilities.menuAcceleratorFont());
+ d.width += (fm.stringWidth(shortcut) + fm.stringWidth("AAAA"));
+ }
+ return d;
+@@ -124,11 +124,9 @@
+ if(shortcut != null)
+ {
+ Graphics2D g2 = (Graphics2D)g;
+- g.setFont(acceleratorFont);
++ g.setFont(GUIUtilities.menuAcceleratorFont());
+ g2.setRenderingHint(RenderingHints.KEY_ANTIALIASING, RenderingHints.VALUE_ANTIALIAS_ON);
+- g.setColor(getModel().isArmed() ?
+- acceleratorSelectionForeground :
+- acceleratorForeground);
++ g.setColor(GUIUtilities.menuAcceleratorForeground(getModel().isArmed()));
+ FontMetrics fm = g.getFontMetrics();
+ Insets insets = getInsets();
+ g.drawString(shortcut,getWidth() - (fm.stringWidth(
+@@ -140,9 +138,6 @@
+ //{{{ Private members
+ private final String shortcutProp;
+ private final char shortcut;
+- private static final Font acceleratorFont;
+- private static final Color acceleratorForeground;
+- private static final Color acceleratorSelectionForeground;
+
+ //{{{ getShortcut() method
+ private String getShortcut()
+@@ -162,16 +157,6 @@
+ }
+ } //}}}
+
+- //{{{ Class initializer
+- static
+- {
+- acceleratorFont = GUIUtilities.menuAcceleratorFont();
+- acceleratorForeground = UIManager
+- .getColor("MenuItem.acceleratorForeground");
+- acceleratorSelectionForeground = UIManager
+- .getColor("MenuItem.acceleratorSelectionForeground");
+- } //}}}
+-
+ //}}}
+ } //}}}
+ }
--- a/src/Tools/jEdit/patches/menus_and_sidekick Tue Apr 22 15:08:14 2025 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,56 +0,0 @@
-diff -ru jedit5.7.0/jEdit/org/gjt/sp/jedit/GUIUtilities.java jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/GUIUtilities.java
---- jedit5.7.0/jEdit/org/gjt/sp/jedit/GUIUtilities.java 2024-08-03 19:53:15.000000000 +0200
-+++ jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/GUIUtilities.java 2024-10-29 11:50:54.062016616 +0100
-@@ -1094,9 +1094,7 @@
- return new Font("Monospaced", Font.PLAIN, 12);
- }
- else {
-- Font font2 =
-- new Font(OperatingSystem.isWindows() ? "Lucida Console" : "Monospaced",
-- Font.PLAIN, font1.getSize());
-+ Font font2 = new Font("Isabelle DejaVu Sans Mono", Font.PLAIN, font1.getSize());
- FontRenderContext frc = new FontRenderContext(null, true, false);
- float scale =
- font1.getLineMetrics("", frc).getHeight() / font2.getLineMetrics("", frc).getHeight();
-diff -ru jedit5.7.0/jEdit/org/gjt/sp/jedit/gui/StatusBar.java jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/gui/StatusBar.java
---- jedit5.7.0/jEdit/org/gjt/sp/jedit/gui/StatusBar.java 2024-08-03 19:53:16.000000000 +0200
-+++ jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/gui/StatusBar.java 2024-11-15 20:22:26.451538237 +0100
-@@ -225,8 +225,11 @@
- else
- this.message.setText(" ");
- }
-- else
-- this.message.setText(message);
-+ else {
-+ Exception exn = new Exception();
-+ if (!exn.getStackTrace()[1].getClassName().startsWith("sidekick."))
-+ this.message.setText(message);
-+ }
- } //}}}
-
- //{{{ setMessageComponent() method
-diff -ru jedit5.7.0/jEdit/org/gjt/sp/jedit/GUIUtilities.java jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/GUIUtilities.java
---- jedit5.7.0/jEdit/org/gjt/sp/jedit/GUIUtilities.java 2024-11-15 18:42:41.560326356 +0100
-+++ jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/GUIUtilities.java 2024-11-15 20:33:52.458587638 +0100
-@@ -1617,6 +1617,21 @@
- }
- //}}}
-
-+ //{{{ isPopupTrigger() method
-+ /**
-+ * Returns if the specified event is the popup trigger event.
-+ * This implements precisely defined behavior, as opposed to
-+ * MouseEvent.isPopupTrigger().
-+ * @param evt The event
-+ * @since jEdit 3.2pre8
-+ * @deprecated use {@link GenericGUIUtilities#requestFocus(Window, Component)}
-+ */
-+ @Deprecated
-+ public static boolean isPopupTrigger(MouseEvent evt)
-+ {
-+ return GenericGUIUtilities.isPopupTrigger(evt);
-+ } //}}}
-+
- //{{{ init() method
- static void init()
- {
--- a/src/Tools/jEdit/patches/props Tue Apr 22 15:08:14 2025 +0200
+++ b/src/Tools/jEdit/patches/props Thu Apr 24 09:16:33 2025 +0200
@@ -1,6 +1,15 @@
diff -ru jedit5.7.0/jEdit/org/jedit/localization/jedit_en.props jedit5.7.0-patched/jEdit/org/jedit/localization/jedit_en.props
--- jedit5.7.0/jEdit/org/jedit/localization/jedit_en.props 2024-08-03 19:53:22.000000000 +0200
-+++ jedit5.7.0-patched/jEdit/org/jedit/localization/jedit_en.props 2024-10-29 11:50:54.066016546 +0100
++++ jedit5.7.0-patched/jEdit/org/jedit/localization/jedit_en.props 2025-04-22 22:08:19.298392180 +0200
+@@ -70,7 +70,7 @@
+ #}}}
+
+ #{{{ Tool bar
+-view.search.find=Search for:
++view.search.find=Search:
+ view.search.close-tooltip=Hide search bar (ESCAPE)
+
+ view.action.prompt=Action:
@@ -1282,8 +1282,7 @@
The most likely reason is that the JAR file is corrupt; try\n\
reinstalling it. See Utilities->Troubleshooting->Activity Log\n\
@@ -11,3 +20,12 @@
plugin-error.already-loaded=Two copies installed. Please remove one of the \
two copies.
plugin-error.dep-jdk=Requires Java {0} or later, but you only have version {1}.
+@@ -1610,7 +1609,7 @@
+ options.gutter.optionalComponents=Optional gutter components
+ options.gutter.lineNumbers=Line numbers
+ options.gutter.minLineNumberDigits=Minimal number of digits to reserve for line numbers:
+-options.gutter.selectionAreaEnabled=Line selection area when line numbers are not shown
++options.gutter.selectionAreaEnabled=Line selection area (with icons) when line numbers are not shown
+ options.gutter.selectionAreaBgColor=Selection area background color:
+ options.gutter.selectionAreaWidth=Selection area width (in pixels):
+ options.gutter.font=Gutter font:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Tools/jEdit/patches/status_bar Thu Apr 24 09:16:33 2025 +0200
@@ -0,0 +1,17 @@
+diff -ru jedit5.7.0/jEdit/org/gjt/sp/jedit/gui/StatusBar.java jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/gui/StatusBar.java
+--- jedit5.7.0/jEdit/org/gjt/sp/jedit/gui/StatusBar.java 2024-08-03 19:53:16.000000000 +0200
++++ jedit5.7.0-patched/jEdit/org/gjt/sp/jedit/gui/StatusBar.java 2024-11-15 20:22:26.451538237 +0100
+@@ -225,8 +225,11 @@
+ else
+ this.message.setText(" ");
+ }
+- else
+- this.message.setText(message);
++ else {
++ Exception exn = new Exception();
++ if (!exn.getStackTrace()[1].getClassName().startsWith("sidekick."))
++ this.message.setText(message);
++ }
+ } //}}}
+
+ //{{{ setMessageComponent() method
--- a/src/Tools/jEdit/src/completion_popup.scala Tue Apr 22 15:08:14 2025 +0200
+++ b/src/Tools/jEdit/src/completion_popup.scala Thu Apr 24 09:16:33 2025 +0200
@@ -403,7 +403,7 @@
text_field =>
// see https://forums.oracle.com/thread/1361677
- if (GUI.is_macos_laf) text_field.setCaret(new DefaultCaret)
+ if (GUI.is_macos_laf()) text_field.setCaret(new DefaultCaret)
// owned by GUI thread
private var completion_popup: Option[Completion_Popup] = None
--- a/src/Tools/jEdit/src/document_view.scala Tue Apr 22 15:08:14 2025 +0200
+++ b/src/Tools/jEdit/src/document_view.scala Thu Apr 24 09:16:33 2025 +0200
@@ -12,11 +12,13 @@
import java.awt.Graphics2D
import java.awt.event.KeyEvent
+import java.awt.geom.AffineTransform
import javax.swing.event.{CaretListener, CaretEvent}
import org.gjt.sp.jedit.jEdit
import org.gjt.sp.jedit.options.GutterOptionPane
-import org.gjt.sp.jedit.textarea.{JEditTextArea, TextArea, TextAreaExtension, TextAreaPainter}
+import org.gjt.sp.jedit.textarea.{JEditTextArea, TextArea, TextAreaExtension, TextAreaPainter,
+ Gutter}
object Document_View {
@@ -140,9 +142,19 @@
GUI_Thread.assert {}
val gutter = text_area.getGutter
- val sel_width = GutterOptionPane.getSelectionAreaWidth
- val border_width = jEdit.getIntegerProperty("view.gutter.borderWidth", 3)
- val FOLD_MARKER_SIZE = 12
+ val gutter_width = gutter.getWidth
+ val gutter_insets = gutter.getBorder.getBorderInsets(gutter)
+
+ val skip_left = gutter_insets.left + Gutter.FOLD_MARKER_SIZE
+ val skip_right = gutter_insets.right
+ val icon_width = gutter_width - skip_left - skip_right
+ val icon_height = line_height
+
+ def scale(a: Int, b: Int): Double = 0.95 * a.toDouble / b.toDouble
+
+ val gutter_icons =
+ !gutter.isExpanded &&
+ gutter.isSelectionAreaEnabled && icon_width >= 12 && icon_height >= 12
val buffer = model.buffer
JEdit_Lib.buffer_lock(buffer) {
@@ -155,19 +167,27 @@
rendering.gutter_content(line_range) match {
case Some((icon, color)) =>
// icons within selection area
- if (!gutter.isExpanded &&
- gutter.isSelectionAreaEnabled && sel_width >= 12 && line_height >= 12) {
- val x0 =
- (FOLD_MARKER_SIZE + sel_width - border_width - icon.getIconWidth) max 10
- val y0 =
- y + i * line_height + (((line_height - icon.getIconHeight) / 2) max 0)
- icon.paintIcon(gutter, gfx, x0, y0)
+ if (gutter_icons && icon.getIconWidth > 0 && icon.getIconHeight > 0) {
+ val w0 = icon.getIconWidth
+ val h0 = icon.getIconHeight
+ val s = Math.min(scale(icon_width, w0), scale(icon_height, h0))
+
+ val w = (s * w0).ceil
+ val h = (s * h0).ceil
+ val x0 = skip_left + (((icon_width - w) / 2) max 0)
+ val y0 = y + i * line_height + (((icon_height - h) / 2) max 0)
+
+ val tr0 = gfx.getTransform
+ val tr = new AffineTransform(tr0); tr.translate(x0, y0); tr.scale(s, s)
+ gfx.setTransform(tr)
+ icon.paintIcon(gutter, gfx, 0, 0)
+ gfx.setTransform(tr0)
}
- // background
+ // background only
else {
val y0 = y + i * line_height
gfx.setColor(color)
- gfx.fillRect(0, y0, gutter.getWidth, line_height)
+ gfx.fillRect(0, y0, gutter_width, line_height)
}
case None =>
}
--- a/src/Tools/jEdit/src/jedit_lib.scala Tue Apr 22 15:08:14 2025 +0200
+++ b/src/Tools/jEdit/src/jedit_lib.scala Thu Apr 24 09:16:33 2025 +0200
@@ -13,7 +13,7 @@
import java.awt.{Component, Container, Toolkit}
import java.awt.event.{InputEvent, KeyEvent, KeyListener}
import java.awt.font.FontRenderContext
-import javax.swing.{Icon, ImageIcon, JScrollBar, JWindow}
+import javax.swing.{ImageIcon, JScrollBar, JWindow}
import scala.util.parsing.input.CharSequenceReader
import scala.util.matching.Regex
@@ -26,6 +26,8 @@
import org.gjt.sp.jedit.buffer.{BufferListener, BufferAdapter, JEditBuffer, LineManager}
import org.gjt.sp.jedit.textarea.{JEditTextArea, TextArea, TextAreaPainter, Selection, AntiAlias}
+import com.formdev.flatlaf.extras.FlatSVGIcon
+
object JEdit_Lib {
/* jEdit directories */
@@ -384,23 +386,8 @@
/* icons */
- def load_icon(name: String): Icon = {
- val name1 =
- if (name.startsWith("idea-icons/")) {
- val file = File.uri(Path.explode("$ISABELLE_IDEA_ICONS")).toASCIIString
- "jar:" + file + "!/" + name
- }
- else name
- val icon = GUIUtilities.loadIcon(name1)
- if (icon.getIconWidth < 0 || icon.getIconHeight < 0) error("Bad icon: " + name)
- else icon
- }
-
- def load_image_icon(name: String): ImageIcon =
- load_icon(name) match {
- case icon: ImageIcon => icon
- case _ => error("Bad image icon: " + name)
- }
+ def load_icon(spec: String): ImageIcon =
+ GUIUtilities.loadIcon(spec).asInstanceOf[ImageIcon]
/* buffer event handling */
--- a/src/Tools/jEdit/src/pretty_text_area.scala Tue Apr 22 15:08:14 2025 +0200
+++ b/src/Tools/jEdit/src/pretty_text_area.scala Thu Apr 24 09:16:33 2025 +0200
@@ -266,7 +266,7 @@
/* search */
- private val search_label: Component = new Label(jEdit.getProperty("search.find")) {
+ private val search_label: Component = new Label(jEdit.getProperty("view.search.find")) {
tooltip = "Search and highlight output via regular expression"
}
--- a/src/Tools/jEdit/src/process_indicator.scala Tue Apr 22 15:08:14 2025 +0200
+++ b/src/Tools/jEdit/src/process_indicator.scala Thu Apr 24 09:16:33 2025 +0200
@@ -18,10 +18,10 @@
private val label = new Label
private val passive_icon =
- JEdit_Lib.load_image_icon(PIDE.options.string("process_passive_icon")).getImage
+ JEdit_Lib.load_icon(PIDE.options.string("process_passive_icon")).getImage
private val active_icons =
space_explode(':', PIDE.options.string("process_active_icons")).map(name =>
- JEdit_Lib.load_image_icon(name).getImage)
+ JEdit_Lib.load_icon(name).getImage)
private class Animation extends ImageIcon(passive_icon) {
private var current_frame = 0