--- a/Admin/Release/build_library Tue Oct 18 16:04:44 2016 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,93 +0,0 @@
-#!/usr/bin/env bash
-#
-# build Isabelle HTML library from platform bundle
-
-## diagnostics
-
-PRG=$(basename "$0")
-
-function usage()
-{
- echo
- echo "Usage: $PRG [OPTIONS] ARCHIVE"
- echo
- echo " Options are:"
- echo " -j INT maximum number of parallel jobs (default 1)"
- echo
- echo " Build Isabelle HTML library from platform bundle."
- echo
- exit 1
-}
-
-function fail()
-{
- echo "$1" >&2
- exit 2
-}
-
-
-## process command line
-
-# options
-
-JOBS=""
-
-while getopts "j:" OPT
-do
- case "$OPT" in
- j)
- JOBS="-j $OPTARG"
- ;;
- \?)
- usage
- ;;
- esac
-done
-
-shift $(($OPTIND - 1))
-
-
-# args
-
-[ "$#" -ne 1 ] && usage
-
-ARCHIVE="$1"; shift
-
-[ ! -f "$ARCHIVE" ] && fail "Bad archive file $ARCHIVE"
-ARCHIVE_BASE="$(basename "$ARCHIVE")"
-ARCHIVE_DIR="$(cd "$(dirname "$ARCHIVE")"; echo "$PWD")"
-ARCHIVE_FULL="$ARCHIVE_DIR/$ARCHIVE_BASE"
-
-
-## main
-
-#GNU tar (notably on Mac OS X)
-type -p gnutar >/dev/null && function tar() { gnutar "$@"; }
-
-TMP="/var/tmp/isabelle-makedist$$"
-mkdir "$TMP" || fail "Cannot create directory: \"$TMP\""
-
-cd "$TMP"
-tar -x -z -f "$ARCHIVE_FULL"
-
-cd *
-ISABELLE_NAME="$(basename "$PWD")"
-
-env ISABELLE_IDENTIFIER="${ISABELLE_NAME}-build" \
- ./bin/isabelle build $JOBS -s -c -a -d '~~/src/Benchmarks' -o browser_info \
- -o "document=pdf" -o "document_variants=document:outline=/proof,/ML"
-RC="$?"
-
-cd ..
-
-if [ "$RC" = 0 ]; then
- chmod -R a+r "$ISABELLE_NAME"
- chmod -R g=o "$ISABELLE_NAME"
- tar -c -z -f "$ARCHIVE_DIR/${ISABELLE_NAME}_library.tar.gz" "$ISABELLE_NAME/browser_info"
-fi
-
-# clean up
-cd /tmp
-rm -rf "$TMP"
-
-exit "$RC"
--- a/Admin/Windows/Cygwin/README Tue Oct 18 16:04:44 2016 +0200
+++ b/Admin/Windows/Cygwin/README Thu Oct 20 19:39:27 2016 +0200
@@ -16,3 +16,21 @@
http://isabelle.in.tum.de/cygwin_2016 (Isabelle2016-1)
* Quasi-component: "isabelle makedist_cygwin" (as administrator)
+
+* SSH server:
+
+ - run Cygwin terminal as Administrator
+
+ - run ssh-host-config
+
+ StrictMode: yes
+ privilege separation: yes
+ new local account 'sshd': yes
+ sshd as a service: yes
+ different name (than cyg_server): no
+ new privileged account cyg_server: yes
+
+ - mkpasswd -l > /etc/passwd
+ mkgroup -l > /etc/group
+
+ passwd -R USER
--- a/Admin/components/components.sha1 Tue Oct 18 16:04:44 2016 +0200
+++ b/Admin/components/components.sha1 Thu Oct 20 19:39:27 2016 +0200
@@ -65,6 +65,7 @@
baa6de37bb6f7a104ce5fe6506bca3d2572d601a jdk-7u80.tar.gz
7d5b152ac70f720bb9e783fa45ecadcf95069584 jdk-7u9.tar.gz
baf275a68d3f799a841932e4e9a95a1a604058ae jdk-8u102.tar.gz
+741de6a4a805a0f9fb917d1845409e99346c2747 jdk-8u112.tar.gz
5442f1015a0657259be0590b04572cd933431df7 jdk-8u11.tar.gz
cfecb1383faaf027ffbabfcd77a0b6a6521e0969 jdk-8u20.tar.gz
44ffeeae219782d40ce6822b580e608e72fd4c76 jdk-8u31.tar.gz
--- a/Admin/components/main Tue Oct 18 16:04:44 2016 +0200
+++ b/Admin/components/main Thu Oct 20 19:39:27 2016 +0200
@@ -5,7 +5,7 @@
e-1.8
Haskabelle-2015
isabelle_fonts-20160830
-jdk-8u102
+jdk-8u112
jedit_build-20160330
jfreechart-1.0.14-1
jortho-1.0-2
--- a/Admin/java/build Tue Oct 18 16:04:44 2016 +0200
+++ b/Admin/java/build Thu Oct 20 19:39:27 2016 +0200
@@ -14,8 +14,8 @@
## parameters
-VERSION="8u102"
-FULL_VERSION="1.8.0_102"
+VERSION="8u112"
+FULL_VERSION="1.8.0_112"
ARCHIVE_LINUX32="jdk-${VERSION}-linux-i586.tar.gz"
ARCHIVE_LINUX64="jdk-${VERSION}-linux-x64.tar.gz"
--- a/Admin/profiling_report Tue Oct 18 16:04:44 2016 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,32 +0,0 @@
-#!/usr/bin/env perl
-#
-# Author: Makarius
-#
-# DESCRIPTION: Simple report generator for Poly/ML profiling output.
-
-use strict;
-
-my %log = ();
-my @output = ();
-
-while (<ARGV>) {
- if (m,^([ 0-9]{10}) (\S+$|GARBAGE COLLECTION.*$),) {
- my $count = $1;
- my $fun = $2;
- $fun =~ s,-?\(\d+\).*$,,g;
- $fun =~ s,/\d+$,,g;
- if ($count =~ m,^\s*(\d)+$,) {
- if (defined($log{$fun})) {
- $log{$fun} += $count;
- } else {
- $log{$fun} = $count;
- }
- }
- }
-}
-
-foreach my $fun (keys %log) {
- push @output, (sprintf "%14u %s\n", $log{$fun}, $fun);
-}
-
-print (sort @output);
--- a/Admin/profiling_reports Tue Oct 18 16:04:44 2016 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,19 +0,0 @@
-#!/usr/bin/env bash
-#
-# Author: Makarius
-#
-# DESCRIPTION: Cumulative reports for Poly/ML profiling output.
-
-THIS="$(cd $(dirname "$0"); pwd)"
-
-SRC="$1"
-DST="$2"
-
-mkdir -p "$DST"
-
-for FILE in "$SRC"/*.gz
-do
- echo "$FILE"
- NAME="$(basename "$FILE" .gz)"
- gzip -dc "$FILE" | "$THIS/profiling_report" > "$DST/$NAME"
-done
--- a/NEWS Tue Oct 18 16:04:44 2016 +0200
+++ b/NEWS Thu Oct 20 19:39:27 2016 +0200
@@ -281,7 +281,12 @@
mod_1 ~> mod_by_Suc_0
INCOMPATIBILITY.
-* Renamed "setsum" ~> "sum" and "setprod" ~> "prod".
+* Renamed constants "setsum" ~> "sum" and "setprod" ~> "prod".
+ Corresponding renaming of theorems.
+
+* New type class "idom_abs_sgn" specifies algebraic properties
+of sign and absolute value functions. Type class "sgn_if" has
+disappeared. Slight INCOMPATIBILITY.
* Dedicated syntax LENGTH('a) for length of types.
@@ -1035,6 +1040,9 @@
exhaust the small 32-bit address space of the ML process (which is used
by default).
+* System option "profiling" specifies the mode for global ML profiling
+in "isabelle build". Possible values are "time", "allocations".
+
* System option "ML_process_policy" specifies an optional command prefix
for the underlying ML process, e.g. to control CPU affinity on
multiprocessor systems. The "isabelle jedit" tool allows to override the
--- a/etc/options Tue Oct 18 16:04:44 2016 +0200
+++ b/etc/options Thu Oct 20 19:39:27 2016 +0200
@@ -108,6 +108,9 @@
option checkpoint : bool = false
-- "checkpoint for theories during build process (heap compression)"
+option profiling : string = ""
+ -- "ML profiling (possible values: time, allocations)"
+
section "ML System"
--- a/src/Doc/System/Sessions.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/Doc/System/Sessions.thy Thu Oct 20 19:39:27 2016 +0200
@@ -215,6 +215,11 @@
Isabelle/Scala. Thus it is relatively reliable in canceling processes that
get out of control, even if there is a deadlock without CPU time usage.
+ \<^item> @{system_option_def "profiling"} specifies a mode for global ML
+ profiling. Possible values are the empty string (disabled), \<^verbatim>\<open>time\<close> for
+ @{ML profile_time} and \<^verbatim>\<open>allocations\<close> for @{ML profile_allocations}.
+ Results appear near the bottom of the session log file.
+
The @{tool_def options} tool prints Isabelle system options. Its
command-line usage is:
@{verbatim [display]
--- a/src/HOL/Analysis/Analysis.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Analysis/Analysis.thy Thu Oct 20 19:39:27 2016 +0200
@@ -8,9 +8,10 @@
Determinants
Homeomorphism
Bounded_Continuous_Function
+ Function_Topology
Weierstrass_Theorems
Polytope
- FurtherTopology
+ Further_Topology
Poly_Roots
Conformal_Mappings
Generalised_Binomial_Theorem
--- a/src/HOL/Analysis/Bochner_Integration.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Analysis/Bochner_Integration.thy Thu Oct 20 19:39:27 2016 +0200
@@ -1157,7 +1157,7 @@
let ?s = "\<lambda>n. simple_bochner_integral M (s n)"
have "\<exists>x. ?s \<longlonglongrightarrow> x"
- unfolding convergent_eq_cauchy
+ unfolding convergent_eq_Cauchy
proof (rule metric_CauchyI)
fix e :: real assume "0 < e"
then have "0 < ennreal (e / 2)" by auto
--- a/src/HOL/Analysis/Borel_Space.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Analysis/Borel_Space.thy Thu Oct 20 19:39:27 2016 +0200
@@ -345,6 +345,14 @@
"A \<in> sets borel \<Longrightarrow> insert x A \<in> sets (borel :: 'a::t1_space measure)"
unfolding insert_def by (rule sets.Un) auto
+lemma sets_borel_eq_count_space: "sets (borel :: 'a::{countable, t2_space} measure) = count_space UNIV"
+proof -
+ have "(\<Union>a\<in>A. {a}) \<in> sets borel" for A :: "'a set"
+ by (intro sets.countable_UN') auto
+ then show ?thesis
+ by auto
+qed
+
lemma borel_comp[measurable]: "A \<in> sets borel \<Longrightarrow> - A \<in> sets borel"
unfolding Compl_eq_Diff_UNIV by simp
@@ -1797,13 +1805,13 @@
proof -
define u' where "u' x = lim (\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0)" for x
then have *: "\<And>x. lim (\<lambda>i. f i x) = (if Cauchy (\<lambda>i. f i x) then u' x else (THE x. False))"
- by (auto simp: lim_def convergent_eq_cauchy[symmetric])
+ by (auto simp: lim_def convergent_eq_Cauchy[symmetric])
have "u' \<in> borel_measurable M"
proof (rule borel_measurable_LIMSEQ_metric)
fix x
have "convergent (\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0)"
by (cases "Cauchy (\<lambda>i. f i x)")
- (auto simp add: convergent_eq_cauchy[symmetric] convergent_def)
+ (auto simp add: convergent_eq_Cauchy[symmetric] convergent_def)
then show "(\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0) \<longlonglongrightarrow> u' x"
unfolding u'_def
by (rule convergent_LIMSEQ_iff[THEN iffD1])
--- a/src/HOL/Analysis/Complex_Transcendental.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Analysis/Complex_Transcendental.thy Thu Oct 20 19:39:27 2016 +0200
@@ -236,6 +236,23 @@
finally show ?thesis .
qed
+lemma exp_plus_2pin [simp]: "exp (z + \<i> * (of_int n * (of_real pi * 2))) = exp z"
+ by (simp add: exp_eq)
+
+lemma inj_on_exp_pi:
+ fixes z::complex shows "inj_on exp (ball z pi)"
+proof (clarsimp simp: inj_on_def exp_eq)
+ fix y n
+ assume "dist z (y + 2 * of_int n * of_real pi * \<i>) < pi"
+ "dist z y < pi"
+ then have "dist y (y + 2 * of_int n * of_real pi * \<i>) < pi+pi"
+ using dist_commute_lessI dist_triangle_less_add by blast
+ then have "norm (2 * of_int n * of_real pi * \<i>) < 2*pi"
+ by (simp add: dist_norm)
+ then show "n = 0"
+ by (auto simp: norm_mult)
+qed
+
lemma sin_cos_eq_iff: "sin y = sin x \<and> cos y = cos x \<longleftrightarrow> (\<exists>n::int. y = x + 2 * n * pi)"
proof -
{ assume "sin y = sin x" "cos y = cos x"
--- a/src/HOL/Analysis/Convex_Euclidean_Space.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Analysis/Convex_Euclidean_Space.thy Thu Oct 20 19:39:27 2016 +0200
@@ -4378,6 +4378,15 @@
definition "rel_interior S =
{x. \<exists>T. openin (subtopology euclidean (affine hull S)) T \<and> x \<in> T \<and> T \<subseteq> S}"
+lemma rel_interior_mono:
+ "\<lbrakk>S \<subseteq> T; affine hull S = affine hull T\<rbrakk>
+ \<Longrightarrow> (rel_interior S) \<subseteq> (rel_interior T)"
+ by (auto simp: rel_interior_def)
+
+lemma rel_interior_maximal:
+ "\<lbrakk>T \<subseteq> S; openin(subtopology euclidean (affine hull S)) T\<rbrakk> \<Longrightarrow> T \<subseteq> (rel_interior S)"
+ by (auto simp: rel_interior_def)
+
lemma rel_interior:
"rel_interior S = {x \<in> S. \<exists>T. open T \<and> x \<in> T \<and> T \<inter> affine hull S \<subseteq> S}"
unfolding rel_interior_def[of S] openin_open[of "affine hull S"]
--- a/src/HOL/Analysis/Derivative.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Analysis/Derivative.thy Thu Oct 20 19:39:27 2016 +0200
@@ -1922,7 +1922,7 @@
using assms(1,2,3) by (rule has_derivative_sequence_lipschitz)
have "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially"
apply (rule bchoice)
- unfolding convergent_eq_cauchy
+ unfolding convergent_eq_Cauchy
proof
fix x
assume "x \<in> s"
--- a/src/HOL/Analysis/Extended_Real_Limits.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Analysis/Extended_Real_Limits.thy Thu Oct 20 19:39:27 2016 +0200
@@ -56,6 +56,15 @@
by simp
qed
+instance enat :: second_countable_topology
+proof
+ show "\<exists>B::enat set set. countable B \<and> open = generate_topology B"
+ proof (intro exI conjI)
+ show "countable (range lessThan \<union> range greaterThan::enat set set)"
+ by auto
+ qed (simp add: open_enat_def)
+qed
+
instance ereal :: second_countable_topology
proof (standard, intro exI conjI)
let ?B = "(\<Union>r\<in>\<rat>. {{..< r}, {r <..}} :: ereal set set)"
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Function_Topology.thy Thu Oct 20 19:39:27 2016 +0200
@@ -0,0 +1,1392 @@
+(* Author: Sébastien Gouëzel sebastien.gouezel@univ-rennes1.fr
+ License: BSD
+*)
+
+theory Function_Topology
+imports Topology_Euclidean_Space Bounded_Linear_Function Finite_Product_Measure
+begin
+
+
+section {*Product topology*}
+
+text {*We want to define the product topology.
+
+The product topology on a product of topological spaces is generated by
+the sets which are products of open sets along finitely many coordinates, and the whole
+space along the other coordinates. This is the coarsest topology for which the projection
+to each factor is continuous.
+
+To form a product of objects in Isabelle/HOL, all these objects should be subsets of a common type
+'a. The product is then @{term "PiE I X"}, the set of elements from 'i to 'a such that the $i$-th
+coordinate belongs to $X\;i$ for all $i \in I$.
+
+Hence, to form a product of topological spaces, all these spaces should be subsets of a common type.
+This means that type classes can not be used to define such a product if one wants to take the
+product of different topological spaces (as the type 'a can only be given one structure of
+topological space using type classes). On the other hand, one can define different topologies (as
+introduced in \verb+Topology_Euclidean_Space.thy+) on one type, and these topologies do not need to
+share the same maximal open set. Hence, one can form a product of topologies in this sense, and
+this works well. The big caveat is that it does not interact well with the main body of
+topology in Isabelle/HOL defined in terms of type classes... For instance, continuity of maps
+is not defined in this setting.
+
+As the product of different topological spaces is very important in several areas of
+mathematics (for instance adeles), I introduce below the product topology in terms of topologies,
+and reformulate afterwards the consequences in terms of type classes (which are of course very
+handy for applications).
+
+Given this limitation, it looks to me that it would be very beneficial to revamp the theory
+of topological spaces in Isabelle/HOL in terms of topologies, and keep the statements involving
+type classes as consequences of more general statements in terms of topologies (but I am
+probably too naive here).
+
+Here is an example of a reformulation using topologies. Let
+\begin{verbatim}
+continuous_on_topo T1 T2 f = ((\<forall> U. openin T2 U \<longrightarrow> openin T1 (f-`U \<inter> topspace(T1)))
+ \<and> (f`(topspace T1) \<subseteq> (topspace T2)))
+\end{verbatim}
+be the natural continuity definition of a map from the topology $T1$ to the topology $T2$. Then
+the current \verb+continuous_on+ (with type classes) can be redefined as
+\begin{verbatim}
+continuous_on s f = continuous_on_topo (subtopology euclidean s) (topology euclidean) f
+\end{verbatim}
+
+In fact, I need \verb+continuous_on_topo+ to express the continuity of the projection on subfactors
+for the product topology, in Lemma~\verb+continuous_on_restrict_product_topology+, and I show
+the above equivalence in Lemma~\verb+continuous_on_continuous_on_topo+.
+
+I only develop the basics of the product topology in this theory. The most important missing piece
+is Tychonov theorem, stating that a product of compact spaces is always compact for the product
+topology, even when the product is not finite (or even countable).
+
+I realized afterwards that this theory has a lot in common with \verb+Fin_Map.thy+.
+*}
+
+subsection {*Topology without type classes*}
+
+subsubsection {*The topology generated by some (open) subsets*}
+
+text {* In the definition below of a generated topology, the \<open>Empty\<close> case is not necessary,
+as it follows from \<open>UN\<close> taking for $K$ the empty set. However, it is convenient to have,
+and is never a problem in proofs, so I prefer to write it down explicitly.
+
+We do not require UNIV to be an open set, as this will not be the case in applications. (We are
+thinking of a topology on a subset of UNIV, the remaining part of UNIV being irrelevant.)*}
+
+inductive generate_topology_on for S where
+Empty: "generate_topology_on S {}"
+|Int: "generate_topology_on S a \<Longrightarrow> generate_topology_on S b \<Longrightarrow> generate_topology_on S (a \<inter> b)"
+| UN: "(\<And>k. k \<in> K \<Longrightarrow> generate_topology_on S k) \<Longrightarrow> generate_topology_on S (\<Union>K)"
+| Basis: "s \<in> S \<Longrightarrow> generate_topology_on S s"
+
+lemma istopology_generate_topology_on:
+ "istopology (generate_topology_on S)"
+unfolding istopology_def by (auto intro: generate_topology_on.intros)
+
+text {*The basic property of the topology generated by a set $S$ is that it is the
+smallest topology containing all the elements of $S$:*}
+
+lemma generate_topology_on_coarsest:
+ assumes "istopology T"
+ "\<And>s. s \<in> S \<Longrightarrow> T s"
+ "generate_topology_on S s0"
+ shows "T s0"
+using assms(3) apply (induct rule: generate_topology_on.induct)
+using assms(1) assms(2) unfolding istopology_def by auto
+
+definition topology_generated_by::"('a set set) \<Rightarrow> ('a topology)"
+ where "topology_generated_by S = topology (generate_topology_on S)"
+
+lemma openin_topology_generated_by_iff:
+ "openin (topology_generated_by S) s \<longleftrightarrow> generate_topology_on S s"
+using topology_inverse'[OF istopology_generate_topology_on[of S]]
+unfolding topology_generated_by_def by simp
+
+lemma openin_topology_generated_by:
+ "openin (topology_generated_by S) s \<Longrightarrow> generate_topology_on S s"
+using openin_topology_generated_by_iff by auto
+
+lemma topology_generated_by_topspace:
+ "topspace (topology_generated_by S) = (\<Union>S)"
+proof
+ {
+ fix s assume "openin (topology_generated_by S) s"
+ then have "generate_topology_on S s" by (rule openin_topology_generated_by)
+ then have "s \<subseteq> (\<Union>S)" by (induct, auto)
+ }
+ then show "topspace (topology_generated_by S) \<subseteq> (\<Union>S)"
+ unfolding topspace_def by auto
+next
+ have "generate_topology_on S (\<Union>S)"
+ using generate_topology_on.UN[OF generate_topology_on.Basis, of S S] by simp
+ then show "(\<Union>S) \<subseteq> topspace (topology_generated_by S)"
+ unfolding topspace_def using openin_topology_generated_by_iff by auto
+qed
+
+lemma topology_generated_by_Basis:
+ "s \<in> S \<Longrightarrow> openin (topology_generated_by S) s"
+by (simp only: openin_topology_generated_by_iff, auto simp: generate_topology_on.Basis)
+
+subsubsection {*Continuity*}
+
+text {*We will need to deal with continuous maps in terms of topologies and not in terms
+of type classes, as defined below.*}
+
+definition continuous_on_topo::"'a topology \<Rightarrow> 'b topology \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
+ where "continuous_on_topo T1 T2 f = ((\<forall> U. openin T2 U \<longrightarrow> openin T1 (f-`U \<inter> topspace(T1)))
+ \<and> (f`(topspace T1) \<subseteq> (topspace T2)))"
+
+lemma continuous_on_continuous_on_topo:
+ "continuous_on s f \<longleftrightarrow> continuous_on_topo (subtopology euclidean s) euclidean f"
+unfolding continuous_on_open_invariant openin_open vimage_def continuous_on_topo_def
+topspace_euclidean_subtopology open_openin topspace_euclidean by fast
+
+lemma continuous_on_topo_UNIV:
+ "continuous_on UNIV f \<longleftrightarrow> continuous_on_topo euclidean euclidean f"
+using continuous_on_continuous_on_topo[of UNIV f] subtopology_UNIV[of euclidean] by auto
+
+lemma continuous_on_topo_open [intro]:
+ "continuous_on_topo T1 T2 f \<Longrightarrow> openin T2 U \<Longrightarrow> openin T1 (f-`U \<inter> topspace(T1))"
+unfolding continuous_on_topo_def by auto
+
+lemma continuous_on_topo_topspace [intro]:
+ "continuous_on_topo T1 T2 f \<Longrightarrow> f`(topspace T1) \<subseteq> (topspace T2)"
+unfolding continuous_on_topo_def by auto
+
+lemma continuous_on_generated_topo_iff:
+ "continuous_on_topo T1 (topology_generated_by S) f \<longleftrightarrow>
+ ((\<forall>U. U \<in> S \<longrightarrow> openin T1 (f-`U \<inter> topspace(T1))) \<and> (f`(topspace T1) \<subseteq> (\<Union> S)))"
+unfolding continuous_on_topo_def topology_generated_by_topspace
+proof (auto simp add: topology_generated_by_Basis)
+ assume H: "\<forall>U. U \<in> S \<longrightarrow> openin T1 (f -` U \<inter> topspace T1)"
+ fix U assume "openin (topology_generated_by S) U"
+ then have "generate_topology_on S U" by (rule openin_topology_generated_by)
+ then show "openin T1 (f -` U \<inter> topspace T1)"
+ proof (induct)
+ fix a b
+ assume H: "openin T1 (f -` a \<inter> topspace T1)" "openin T1 (f -` b \<inter> topspace T1)"
+ have "f -` (a \<inter> b) \<inter> topspace T1 = (f-`a \<inter> topspace T1) \<inter> (f-`b \<inter> topspace T1)"
+ by auto
+ then show "openin T1 (f -` (a \<inter> b) \<inter> topspace T1)" using H by auto
+ next
+ fix K
+ assume H: "openin T1 (f -` k \<inter> topspace T1)" if "k\<in> K" for k
+ define L where "L = {f -` k \<inter> topspace T1|k. k \<in> K}"
+ have *: "openin T1 l" if "l \<in>L" for l using that H unfolding L_def by auto
+ have "openin T1 (\<Union>L)" using openin_Union[OF *] by simp
+ moreover have "(\<Union>L) = (f -` \<Union>K \<inter> topspace T1)" unfolding L_def by auto
+ ultimately show "openin T1 (f -` \<Union>K \<inter> topspace T1)" by simp
+ qed (auto simp add: H)
+qed
+
+lemma continuous_on_generated_topo:
+ assumes "\<And>U. U \<in>S \<Longrightarrow> openin T1 (f-`U \<inter> topspace(T1))"
+ "f`(topspace T1) \<subseteq> (\<Union> S)"
+ shows "continuous_on_topo T1 (topology_generated_by S) f"
+using assms continuous_on_generated_topo_iff by blast
+
+lemma continuous_on_topo_compose:
+ assumes "continuous_on_topo T1 T2 f" "continuous_on_topo T2 T3 g"
+ shows "continuous_on_topo T1 T3 (g o f)"
+using assms unfolding continuous_on_topo_def
+proof (auto)
+ fix U :: "'c set"
+ assume H: "openin T3 U"
+ have "openin T1 (f -` (g -` U \<inter> topspace T2) \<inter> topspace T1)"
+ using H assms by blast
+ moreover have "f -` (g -` U \<inter> topspace T2) \<inter> topspace T1 = (g \<circ> f) -` U \<inter> topspace T1"
+ using H assms continuous_on_topo_topspace by fastforce
+ ultimately show "openin T1 ((g \<circ> f) -` U \<inter> topspace T1)"
+ by simp
+qed (blast)
+
+lemma continuous_on_topo_preimage_topspace [intro]:
+ assumes "continuous_on_topo T1 T2 f"
+ shows "f-`(topspace T2) \<inter> topspace T1 = topspace T1"
+using assms unfolding continuous_on_topo_def by auto
+
+
+subsubsection {*Pullback topology*}
+
+text {*Pulling back a topology by map gives again a topology. \<open>subtopology\<close> is
+a special case of this notion, pulling back by the identity. We introduce the general notion as
+we will need it to define the strong operator topology on the space of continuous linear operators,
+by pulling back the product topology on the space of all functions.*}
+
+text {*\verb+pullback_topology A f T+ is the pullback of the topology $T$ by the map $f$ on
+the set $A$.*}
+
+definition pullback_topology::"('a set) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('b topology) \<Rightarrow> ('a topology)"
+ where "pullback_topology A f T = topology (\<lambda>S. \<exists>U. openin T U \<and> S = f-`U \<inter> A)"
+
+lemma istopology_pullback_topology:
+ "istopology (\<lambda>S. \<exists>U. openin T U \<and> S = f-`U \<inter> A)"
+unfolding istopology_def proof (auto)
+ fix K assume "\<forall>S\<in>K. \<exists>U. openin T U \<and> S = f -` U \<inter> A"
+ then have "\<exists>U. \<forall>S\<in>K. openin T (U S) \<and> S = f-`(U S) \<inter> A"
+ by (rule bchoice)
+ then obtain U where U: "\<forall>S\<in>K. openin T (U S) \<and> S = f-`(U S) \<inter> A"
+ by blast
+ define V where "V = (\<Union>S\<in>K. U S)"
+ have "openin T V" "\<Union>K = f -` V \<inter> A" unfolding V_def using U by auto
+ then show "\<exists>V. openin T V \<and> \<Union>K = f -` V \<inter> A" by auto
+qed
+
+lemma openin_pullback_topology:
+ "openin (pullback_topology A f T) S \<longleftrightarrow> (\<exists>U. openin T U \<and> S = f-`U \<inter> A)"
+unfolding pullback_topology_def topology_inverse'[OF istopology_pullback_topology] by auto
+
+lemma topspace_pullback_topology:
+ "topspace (pullback_topology A f T) = f-`(topspace T) \<inter> A"
+by (auto simp add: topspace_def openin_pullback_topology)
+
+lemma continuous_on_topo_pullback [intro]:
+ assumes "continuous_on_topo T1 T2 g"
+ shows "continuous_on_topo (pullback_topology A f T1) T2 (g o f)"
+unfolding continuous_on_topo_def
+proof (auto)
+ fix U::"'b set" assume "openin T2 U"
+ then have "openin T1 (g-`U \<inter> topspace T1)"
+ using assms unfolding continuous_on_topo_def by auto
+ have "(g o f)-`U \<inter> topspace (pullback_topology A f T1) = (g o f)-`U \<inter> A \<inter> f-`(topspace T1)"
+ unfolding topspace_pullback_topology by auto
+ also have "... = f-`(g-`U \<inter> topspace T1) \<inter> A "
+ by auto
+ also have "openin (pullback_topology A f T1) (...)"
+ unfolding openin_pullback_topology using `openin T1 (g-\`U \<inter> topspace T1)` by auto
+ finally show "openin (pullback_topology A f T1) ((g \<circ> f) -` U \<inter> topspace (pullback_topology A f T1))"
+ by auto
+next
+ fix x assume "x \<in> topspace (pullback_topology A f T1)"
+ then have "f x \<in> topspace T1"
+ unfolding topspace_pullback_topology by auto
+ then show "g (f x) \<in> topspace T2"
+ using assms unfolding continuous_on_topo_def by auto
+qed
+
+lemma continuous_on_topo_pullback' [intro]:
+ assumes "continuous_on_topo T1 T2 (f o g)" "topspace T1 \<subseteq> g-`A"
+ shows "continuous_on_topo T1 (pullback_topology A f T2) g"
+unfolding continuous_on_topo_def
+proof (auto)
+ fix U assume "openin (pullback_topology A f T2) U"
+ then have "\<exists>V. openin T2 V \<and> U = f-`V \<inter> A"
+ unfolding openin_pullback_topology by auto
+ then obtain V where "openin T2 V" "U = f-`V \<inter> A"
+ by blast
+ then have "g -` U \<inter> topspace T1 = g-`(f-`V \<inter> A) \<inter> topspace T1"
+ by blast
+ also have "... = (f o g)-`V \<inter> (g-`A \<inter> topspace T1)"
+ by auto
+ also have "... = (f o g)-`V \<inter> topspace T1"
+ using assms(2) by auto
+ also have "openin T1 (...)"
+ using assms(1) `openin T2 V` by auto
+ finally show "openin T1 (g -` U \<inter> topspace T1)" by simp
+next
+ fix x assume "x \<in> topspace T1"
+ have "(f o g) x \<in> topspace T2"
+ using assms(1) `x \<in> topspace T1` unfolding continuous_on_topo_def by auto
+ then have "g x \<in> f-`(topspace T2)"
+ unfolding comp_def by blast
+ moreover have "g x \<in> A" using assms(2) `x \<in> topspace T1` by blast
+ ultimately show "g x \<in> topspace (pullback_topology A f T2)"
+ unfolding topspace_pullback_topology by blast
+qed
+
+subsubsection {*Miscellaneous*}
+
+text {*The following could belong to \verb+Topology_Euclidean_Spaces.thy+, and will be needed
+below.*}
+lemma openin_INT [intro]:
+ assumes "finite I"
+ "\<And>i. i \<in> I \<Longrightarrow> openin T (U i)"
+ shows "openin T ((\<Inter>i \<in> I. U i) \<inter> topspace T)"
+using assms by (induct, auto simp add: inf_sup_aci(2) openin_Int)
+
+lemma openin_INT2 [intro]:
+ assumes "finite I" "I \<noteq> {}"
+ "\<And>i. i \<in> I \<Longrightarrow> openin T (U i)"
+ shows "openin T (\<Inter>i \<in> I. U i)"
+proof -
+ have "(\<Inter>i \<in> I. U i) \<subseteq> topspace T"
+ using `I \<noteq> {}` openin_subset[OF assms(3)] by auto
+ then show ?thesis
+ using openin_INT[of _ _ U, OF assms(1) assms(3)] by (simp add: inf.absorb2 inf_commute)
+qed
+
+
+subsection {*The product topology*}
+
+text {*We can now define the product topology, as generated by
+the sets which are products of open sets along finitely many coordinates, and the whole
+space along the other coordinates. Equivalently, it is generated by sets which are one open
+set along one single coordinate, and the whole space along other coordinates. In fact, this is only
+equivalent for nonempty products, but for the empty product the first formulation is better
+(the second one gives an empty product space, while an empty product should have exactly one
+point, equal to \verb+undefined+ along all coordinates.
+
+So, we use the first formulation, which moreover seems to give rise to more straightforward proofs.
+*}
+
+definition product_topology::"('i \<Rightarrow> ('a topology)) \<Rightarrow> ('i set) \<Rightarrow> (('i \<Rightarrow> 'a) topology)"
+ where "product_topology T I =
+ topology_generated_by {(\<Pi>\<^sub>E i\<in>I. X i) |X. (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)}}"
+
+text {*The total set of the product topology is the product of the total sets
+along each coordinate.*}
+
+lemma product_topology_topspace:
+ "topspace (product_topology T I) = (\<Pi>\<^sub>E i\<in>I. topspace(T i))"
+proof
+ show "topspace (product_topology T I) \<subseteq> (\<Pi>\<^sub>E i\<in>I. topspace (T i))"
+ unfolding product_topology_def apply (simp only: topology_generated_by_topspace)
+ unfolding topspace_def by auto
+ have "(\<Pi>\<^sub>E i\<in>I. topspace (T i)) \<in> {(\<Pi>\<^sub>E i\<in>I. X i) |X. (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)}}"
+ using openin_topspace not_finite_existsD by auto
+ then show "(\<Pi>\<^sub>E i\<in>I. topspace (T i)) \<subseteq> topspace (product_topology T I)"
+ unfolding product_topology_def using PiE_def by (auto simp add: topology_generated_by_topspace)
+qed
+
+lemma product_topology_basis:
+ assumes "\<And>i. openin (T i) (X i)" "finite {i. X i \<noteq> topspace (T i)}"
+ shows "openin (product_topology T I) (\<Pi>\<^sub>E i\<in>I. X i)"
+unfolding product_topology_def apply (rule topology_generated_by_Basis) using assms by auto
+
+lemma product_topology_open_contains_basis:
+ assumes "openin (product_topology T I) U"
+ "x \<in> U"
+ shows "\<exists>X. x \<in> (\<Pi>\<^sub>E i\<in>I. X i) \<and> (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)} \<and> (\<Pi>\<^sub>E i\<in>I. X i) \<subseteq> U"
+proof -
+ have "generate_topology_on {(\<Pi>\<^sub>E i\<in>I. X i) |X. (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)}} U"
+ using assms unfolding product_topology_def by (intro openin_topology_generated_by) auto
+ then have "\<And>x. x\<in>U \<Longrightarrow> \<exists>X. x \<in> (\<Pi>\<^sub>E i\<in>I. X i) \<and> (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)} \<and> (\<Pi>\<^sub>E i\<in>I. X i) \<subseteq> U"
+ proof induction
+ case (Int U V x)
+ then obtain XU XV where H:
+ "x \<in> Pi\<^sub>E I XU" "(\<forall>i. openin (T i) (XU i))" "finite {i. XU i \<noteq> topspace (T i)}" "Pi\<^sub>E I XU \<subseteq> U"
+ "x \<in> Pi\<^sub>E I XV" "(\<forall>i. openin (T i) (XV i))" "finite {i. XV i \<noteq> topspace (T i)}" "Pi\<^sub>E I XV \<subseteq> V"
+ by auto meson
+ define X where "X = (\<lambda>i. XU i \<inter> XV i)"
+ have "Pi\<^sub>E I X \<subseteq> Pi\<^sub>E I XU \<inter> Pi\<^sub>E I XV"
+ unfolding X_def by (auto simp add: PiE_iff)
+ then have "Pi\<^sub>E I X \<subseteq> U \<inter> V" using H by auto
+ moreover have "\<forall>i. openin (T i) (X i)"
+ unfolding X_def using H by auto
+ moreover have "finite {i. X i \<noteq> topspace (T i)}"
+ apply (rule rev_finite_subset[of "{i. XU i \<noteq> topspace (T i)} \<union> {i. XV i \<noteq> topspace (T i)}"])
+ unfolding X_def using H by auto
+ moreover have "x \<in> Pi\<^sub>E I X"
+ unfolding X_def using H by auto
+ ultimately show ?case
+ by auto
+ next
+ case (UN K x)
+ then obtain k where "k \<in> K" "x \<in> k" by auto
+ with UN have "\<exists>X. x \<in> Pi\<^sub>E I X \<and> (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)} \<and> Pi\<^sub>E I X \<subseteq> k"
+ by simp
+ then obtain X where "x \<in> Pi\<^sub>E I X \<and> (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)} \<and> Pi\<^sub>E I X \<subseteq> k"
+ by blast
+ then have "x \<in> Pi\<^sub>E I X \<and> (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)} \<and> Pi\<^sub>E I X \<subseteq> (\<Union>K)"
+ using `k \<in> K` by auto
+ then show ?case
+ by auto
+ qed auto
+ then show ?thesis using `x \<in> U` by auto
+qed
+
+
+text {*The basic property of the product topology is the continuity of projections:*}
+
+lemma continuous_on_topo_product_coordinates [simp]:
+ assumes "i \<in> I"
+ shows "continuous_on_topo (product_topology T I) (T i) (\<lambda>x. x i)"
+proof -
+ {
+ fix U assume "openin (T i) U"
+ define X where "X = (\<lambda>j. if j = i then U else topspace (T j))"
+ then have *: "(\<lambda>x. x i) -` U \<inter> (\<Pi>\<^sub>E i\<in>I. topspace (T i)) = (\<Pi>\<^sub>E j\<in>I. X j)"
+ unfolding X_def using assms openin_subset[OF `openin (T i) U`]
+ by (auto simp add: PiE_iff, auto, metis subsetCE)
+ have **: "(\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)}"
+ unfolding X_def using `openin (T i) U` by auto
+ have "openin (product_topology T I) ((\<lambda>x. x i) -` U \<inter> (\<Pi>\<^sub>E i\<in>I. topspace (T i)))"
+ unfolding product_topology_def
+ apply (rule topology_generated_by_Basis)
+ apply (subst *)
+ using ** by auto
+ }
+ then show ?thesis unfolding continuous_on_topo_def
+ by (auto simp add: assms product_topology_topspace PiE_iff)
+qed
+
+lemma continuous_on_topo_coordinatewise_then_product [intro]:
+ assumes "\<And>i. i \<in> I \<Longrightarrow> continuous_on_topo T1 (T i) (\<lambda>x. f x i)"
+ "\<And>i x. i \<notin> I \<Longrightarrow> x \<in> topspace T1 \<Longrightarrow> f x i = undefined"
+ shows "continuous_on_topo T1 (product_topology T I) f"
+unfolding product_topology_def
+proof (rule continuous_on_generated_topo)
+ fix U assume "U \<in> {Pi\<^sub>E I X |X. (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)}}"
+ then obtain X where H: "U = Pi\<^sub>E I X" "\<And>i. openin (T i) (X i)" "finite {i. X i \<noteq> topspace (T i)}"
+ by blast
+ define J where "J = {i \<in> I. X i \<noteq> topspace (T i)}"
+ have "finite J" "J \<subseteq> I" unfolding J_def using H(3) by auto
+ have "(\<lambda>x. f x i)-`(topspace(T i)) \<inter> topspace T1 = topspace T1" if "i \<in> I" for i
+ using that assms(1) by (simp add: continuous_on_topo_preimage_topspace)
+ then have *: "(\<lambda>x. f x i)-`(X i) \<inter> topspace T1 = topspace T1" if "i \<in> I-J" for i
+ using that unfolding J_def by auto
+ have "f-`U \<inter> topspace T1 = (\<Inter>i\<in>I. (\<lambda>x. f x i)-`(X i) \<inter> topspace T1) \<inter> (topspace T1)"
+ by (subst H(1), auto simp add: PiE_iff assms)
+ also have "... = (\<Inter>i\<in>J. (\<lambda>x. f x i)-`(X i) \<inter> topspace T1) \<inter> (topspace T1)"
+ using * `J \<subseteq> I` by auto
+ also have "openin T1 (...)"
+ apply (rule openin_INT)
+ apply (simp add: `finite J`)
+ using H(2) assms(1) `J \<subseteq> I` by auto
+ ultimately show "openin T1 (f-`U \<inter> topspace T1)" by simp
+next
+ show "f `topspace T1 \<subseteq> \<Union>{Pi\<^sub>E I X |X. (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)}}"
+ apply (subst topology_generated_by_topspace[symmetric])
+ apply (subst product_topology_def[symmetric])
+ apply (simp only: product_topology_topspace)
+ apply (auto simp add: PiE_iff)
+ using assms unfolding continuous_on_topo_def by auto
+qed
+
+lemma continuous_on_topo_product_then_coordinatewise [intro]:
+ assumes "continuous_on_topo T1 (product_topology T I) f"
+ shows "\<And>i. i \<in> I \<Longrightarrow> continuous_on_topo T1 (T i) (\<lambda>x. f x i)"
+ "\<And>i x. i \<notin> I \<Longrightarrow> x \<in> topspace T1 \<Longrightarrow> f x i = undefined"
+proof -
+ fix i assume "i \<in> I"
+ have "(\<lambda>x. f x i) = (\<lambda>y. y i) o f" by auto
+ also have "continuous_on_topo T1 (T i) (...)"
+ apply (rule continuous_on_topo_compose[of _ "product_topology T I"])
+ using assms `i \<in> I` by auto
+ ultimately show "continuous_on_topo T1 (T i) (\<lambda>x. f x i)"
+ by simp
+next
+ fix i x assume "i \<notin> I" "x \<in> topspace T1"
+ have "f x \<in> topspace (product_topology T I)"
+ using assms `x \<in> topspace T1` unfolding continuous_on_topo_def by auto
+ then have "f x \<in> (\<Pi>\<^sub>E i\<in>I. topspace (T i))"
+ using product_topology_topspace by metis
+ then show "f x i = undefined"
+ using `i \<notin> I` by (auto simp add: PiE_iff extensional_def)
+qed
+
+lemma continuous_on_restrict:
+ assumes "J \<subseteq> I"
+ shows "continuous_on_topo (product_topology T I) (product_topology T J) (\<lambda>x. restrict x J)"
+proof (rule continuous_on_topo_coordinatewise_then_product)
+ fix i assume "i \<in> J"
+ then have "(\<lambda>x. restrict x J i) = (\<lambda>x. x i)" unfolding restrict_def by auto
+ then show "continuous_on_topo (product_topology T I) (T i) (\<lambda>x. restrict x J i)"
+ using `i \<in> J` `J \<subseteq> I` by auto
+next
+ fix i assume "i \<notin> J"
+ then show "restrict x J i = undefined" for x::"'a \<Rightarrow> 'b"
+ unfolding restrict_def by auto
+qed
+
+
+subsubsection {*Powers of a single topological space as a topological space, using type classes*}
+
+instantiation "fun" :: (type, topological_space) topological_space
+begin
+
+definition open_fun_def:
+ "open U = openin (product_topology (\<lambda>i. euclidean) UNIV) U"
+
+instance proof
+ have "topspace (product_topology (\<lambda>(i::'a). euclidean::('b topology)) UNIV) = UNIV"
+ unfolding product_topology_topspace topspace_euclidean by auto
+ then show "open (UNIV::('a \<Rightarrow> 'b) set)"
+ unfolding open_fun_def by (metis openin_topspace)
+qed (auto simp add: open_fun_def)
+
+end
+
+lemma euclidean_product_topology:
+ "euclidean = product_topology (\<lambda>i. euclidean::('b::topological_space) topology) UNIV"
+by (metis open_openin topology_eq open_fun_def)
+
+lemma product_topology_basis':
+ fixes x::"'i \<Rightarrow> 'a" and U::"'i \<Rightarrow> ('b::topological_space) set"
+ assumes "finite I" "\<And>i. i \<in> I \<Longrightarrow> open (U i)"
+ shows "open {f. \<forall>i\<in>I. f (x i) \<in> U i}"
+proof -
+ define J where "J = x`I"
+ define V where "V = (\<lambda>y. if y \<in> J then \<Inter>{U i|i. i\<in>I \<and> x i = y} else UNIV)"
+ define X where "X = (\<lambda>y. if y \<in> J then V y else UNIV)"
+ have *: "open (X i)" for i
+ unfolding X_def V_def using assms by auto
+ have **: "finite {i. X i \<noteq> UNIV}"
+ unfolding X_def V_def J_def using assms(1) by auto
+ have "open (Pi\<^sub>E UNIV X)"
+ unfolding open_fun_def apply (rule product_topology_basis)
+ using * ** unfolding open_openin topspace_euclidean by auto
+ moreover have "Pi\<^sub>E UNIV X = {f. \<forall>i\<in>I. f (x i) \<in> U i}"
+ apply (auto simp add: PiE_iff) unfolding X_def V_def J_def
+ proof (auto)
+ fix f :: "'a \<Rightarrow> 'b" and i :: 'i
+ assume a1: "i \<in> I"
+ assume a2: "\<forall>i. f i \<in> (if i \<in> x`I then if i \<in> x`I then \<Inter>{U ia |ia. ia \<in> I \<and> x ia = i} else UNIV else UNIV)"
+ have f3: "x i \<in> x`I"
+ using a1 by blast
+ have "U i \<in> {U ia |ia. ia \<in> I \<and> x ia = x i}"
+ using a1 by blast
+ then show "f (x i) \<in> U i"
+ using f3 a2 by (meson Inter_iff)
+ qed
+ ultimately show ?thesis by simp
+qed
+
+text {*The results proved in the general situation of products of possibly different
+spaces have their counterparts in this simpler setting.*}
+
+lemma continuous_on_product_coordinates [simp]:
+ "continuous_on UNIV (\<lambda>x. x i::('b::topological_space))"
+unfolding continuous_on_topo_UNIV euclidean_product_topology
+by (rule continuous_on_topo_product_coordinates, simp)
+
+lemma continuous_on_coordinatewise_then_product [intro, continuous_intros]:
+ assumes "\<And>i. continuous_on UNIV (\<lambda>x. f x i)"
+ shows "continuous_on UNIV f"
+using assms unfolding continuous_on_topo_UNIV euclidean_product_topology
+by (rule continuous_on_topo_coordinatewise_then_product, simp)
+
+lemma continuous_on_product_then_coordinatewise:
+ assumes "continuous_on UNIV f"
+ shows "continuous_on UNIV (\<lambda>x. f x i)"
+using assms unfolding continuous_on_topo_UNIV euclidean_product_topology
+by (rule continuous_on_topo_product_then_coordinatewise(1), simp)
+
+lemma continuous_on_product_coordinatewise_iff:
+ "continuous_on UNIV f \<longleftrightarrow> (\<forall>i. continuous_on UNIV (\<lambda>x. f x i))"
+by (auto intro: continuous_on_product_then_coordinatewise)
+
+subsubsection {*Topological countability for product spaces*}
+
+text {*The next two lemmas are useful to prove first or second countability
+of product spaces, but they have more to do with countability and could
+be put in the corresponding theory.*}
+
+lemma countable_nat_product_event_const:
+ fixes F::"'a set" and a::'a
+ assumes "a \<in> F" "countable F"
+ shows "countable {x::(nat \<Rightarrow> 'a). (\<forall>i. x i \<in> F) \<and> finite {i. x i \<noteq> a}}"
+proof -
+ have *: "{x::(nat \<Rightarrow> 'a). (\<forall>i. x i \<in> F) \<and> finite {i. x i \<noteq> a}}
+ \<subseteq> (\<Union>N. {x. (\<forall>i. x i \<in> F) \<and> (\<forall>i\<ge>N. x i = a)})"
+ using infinite_nat_iff_unbounded_le by fastforce
+ have "countable {x. (\<forall>i. x i \<in> F) \<and> (\<forall>i\<ge>N. x i = a)}" for N::nat
+ proof (induction N)
+ case 0
+ have "{x. (\<forall>i. x i \<in> F) \<and> (\<forall>i\<ge>(0::nat). x i = a)} = {(\<lambda>i. a)}"
+ using `a \<in> F` by auto
+ then show ?case by auto
+ next
+ case (Suc N)
+ define f::"((nat \<Rightarrow> 'a) \<times> 'a) \<Rightarrow> (nat \<Rightarrow> 'a)"
+ where "f = (\<lambda>(x, b). (\<lambda>i. if i = N then b else x i))"
+ have "{x. (\<forall>i. x i \<in> F) \<and> (\<forall>i\<ge>Suc N. x i = a)} \<subseteq> f`({x. (\<forall>i. x i \<in> F) \<and> (\<forall>i\<ge>N. x i = a)} \<times> F)"
+ proof (auto)
+ fix x assume H: "\<forall>i::nat. x i \<in> F" "\<forall>i\<ge>Suc N. x i = a"
+ define y where "y = (\<lambda>i. if i = N then a else x i)"
+ have "f (y, x N) = x"
+ unfolding f_def y_def by auto
+ moreover have "(y, x N) \<in> {x. (\<forall>i. x i \<in> F) \<and> (\<forall>i\<ge>N. x i = a)} \<times> F"
+ unfolding y_def using H `a \<in> F` by auto
+ ultimately show "x \<in> f`({x. (\<forall>i. x i \<in> F) \<and> (\<forall>i\<ge>N. x i = a)} \<times> F)"
+ by (metis (no_types, lifting) image_eqI)
+ qed
+ moreover have "countable ({x. (\<forall>i. x i \<in> F) \<and> (\<forall>i\<ge>N. x i = a)} \<times> F)"
+ using Suc.IH assms(2) by auto
+ ultimately show ?case
+ by (meson countable_image countable_subset)
+ qed
+ then show ?thesis using countable_subset[OF *] by auto
+qed
+
+lemma countable_product_event_const:
+ fixes F::"('a::countable) \<Rightarrow> 'b set" and b::'b
+ assumes "\<And>i. countable (F i)"
+ shows "countable {f::('a \<Rightarrow> 'b). (\<forall>i. f i \<in> F i) \<and> (finite {i. f i \<noteq> b})}"
+proof -
+ define G where "G = (\<Union>i. F i) \<union> {b}"
+ have "countable G" unfolding G_def using assms by auto
+ have "b \<in> G" unfolding G_def by auto
+ define pi where "pi = (\<lambda>(x::(nat \<Rightarrow> 'b)). (\<lambda> i::'a. x ((to_nat::('a \<Rightarrow> nat)) i)))"
+ have "{f::('a \<Rightarrow> 'b). (\<forall>i. f i \<in> F i) \<and> (finite {i. f i \<noteq> b})}
+ \<subseteq> pi`{g::(nat \<Rightarrow> 'b). (\<forall>j. g j \<in> G) \<and> (finite {j. g j \<noteq> b})}"
+ proof (auto)
+ fix f assume H: "\<forall>i. f i \<in> F i" "finite {i. f i \<noteq> b}"
+ define I where "I = {i. f i \<noteq> b}"
+ define g where "g = (\<lambda>j. if j \<in> to_nat`I then f (from_nat j) else b)"
+ have "{j. g j \<noteq> b} \<subseteq> to_nat`I" unfolding g_def by auto
+ then have "finite {j. g j \<noteq> b}"
+ unfolding I_def using H(2) using finite_surj by blast
+ moreover have "g j \<in> G" for j
+ unfolding g_def G_def using H by auto
+ ultimately have "g \<in> {g::(nat \<Rightarrow> 'b). (\<forall>j. g j \<in> G) \<and> (finite {j. g j \<noteq> b})}"
+ by auto
+ moreover have "f = pi g"
+ unfolding pi_def g_def I_def using H by fastforce
+ ultimately show "f \<in> pi`{g. (\<forall>j. g j \<in> G) \<and> finite {j. g j \<noteq> b}}"
+ by auto
+ qed
+ then show ?thesis
+ using countable_nat_product_event_const[OF `b \<in> G` `countable G`]
+ by (meson countable_image countable_subset)
+qed
+
+instance "fun" :: (countable, first_countable_topology) first_countable_topology
+proof
+ fix x::"'a \<Rightarrow> 'b"
+ have "\<exists>A::('b \<Rightarrow> nat \<Rightarrow> 'b set). \<forall>x. (\<forall>i. x \<in> A x i \<and> open (A x i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A x i \<subseteq> S))"
+ apply (rule choice) using first_countable_basis by auto
+ then obtain A::"('b \<Rightarrow> nat \<Rightarrow> 'b set)" where A: "\<And>x i. x \<in> A x i"
+ "\<And>x i. open (A x i)"
+ "\<And>x S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>i. A x i \<subseteq> S)"
+ by metis
+ text {*$B i$ is a countable basis of neighborhoods of $x_i$.*}
+ define B where "B = (\<lambda>i. (A (x i))`UNIV \<union> {UNIV})"
+ have "countable (B i)" for i unfolding B_def by auto
+
+ define K where "K = {Pi\<^sub>E UNIV X | X. (\<forall>i. X i \<in> B i) \<and> finite {i. X i \<noteq> UNIV}}"
+ have "Pi\<^sub>E UNIV (\<lambda>i. UNIV) \<in> K"
+ unfolding K_def B_def by auto
+ then have "K \<noteq> {}" by auto
+ have "countable {X. (\<forall>i. X i \<in> B i) \<and> finite {i. X i \<noteq> UNIV}}"
+ apply (rule countable_product_event_const) using `\<And>i. countable (B i)` by auto
+ moreover have "K = (\<lambda>X. Pi\<^sub>E UNIV X)`{X. (\<forall>i. X i \<in> B i) \<and> finite {i. X i \<noteq> UNIV}}"
+ unfolding K_def by auto
+ ultimately have "countable K" by auto
+ have "x \<in> k" if "k \<in> K" for k
+ using that unfolding K_def B_def apply auto using A(1) by auto
+ have "open k" if "k \<in> K" for k
+ using that unfolding open_fun_def K_def B_def apply (auto)
+ apply (rule product_topology_basis)
+ unfolding topspace_euclidean by (auto, metis imageE open_UNIV A(2))
+
+ have Inc: "\<exists>k\<in>K. k \<subseteq> U" if "open U \<and> x \<in> U" for U
+ proof -
+ have "openin (product_topology (\<lambda>i. euclidean) UNIV) U" "x \<in> U"
+ using `open U \<and> x \<in> U` unfolding open_fun_def by auto
+ with product_topology_open_contains_basis[OF this]
+ have "\<exists>X. x \<in> (\<Pi>\<^sub>E i\<in>UNIV. X i) \<and> (\<forall>i. open (X i)) \<and> finite {i. X i \<noteq> UNIV} \<and> (\<Pi>\<^sub>E i\<in>UNIV. X i) \<subseteq> U"
+ unfolding topspace_euclidean open_openin by simp
+ then obtain X where H: "x \<in> (\<Pi>\<^sub>E i\<in>UNIV. X i)"
+ "\<And>i. open (X i)"
+ "finite {i. X i \<noteq> UNIV}"
+ "(\<Pi>\<^sub>E i\<in>UNIV. X i) \<subseteq> U"
+ by auto
+ define I where "I = {i. X i \<noteq> UNIV}"
+ define Y where "Y = (\<lambda>i. if i \<in> I then (SOME y. y \<in> B i \<and> y \<subseteq> X i) else UNIV)"
+ have *: "\<exists>y. y \<in> B i \<and> y \<subseteq> X i" for i
+ unfolding B_def using A(3)[OF H(2)] H(1) by (metis PiE_E UNIV_I UnCI image_iff)
+ have **: "Y i \<in> B i \<and> Y i \<subseteq> X i" for i
+ apply (cases "i \<in> I")
+ unfolding Y_def using * that apply (auto)
+ apply (metis (no_types, lifting) someI, metis (no_types, lifting) someI_ex subset_iff)
+ unfolding B_def apply simp
+ unfolding I_def apply auto
+ done
+ have "{i. Y i \<noteq> UNIV} \<subseteq> I"
+ unfolding Y_def by auto
+ then have ***: "finite {i. Y i \<noteq> UNIV}"
+ unfolding I_def using H(3) rev_finite_subset by blast
+ have "(\<forall>i. Y i \<in> B i) \<and> finite {i. Y i \<noteq> UNIV}"
+ using ** *** by auto
+ then have "Pi\<^sub>E UNIV Y \<in> K"
+ unfolding K_def by auto
+
+ have "Y i \<subseteq> X i" for i
+ apply (cases "i \<in> I") using ** apply simp unfolding Y_def I_def by auto
+ then have "Pi\<^sub>E UNIV Y \<subseteq> Pi\<^sub>E UNIV X" by auto
+ then have "Pi\<^sub>E UNIV Y \<subseteq> U" using H(4) by auto
+ then show ?thesis using `Pi\<^sub>E UNIV Y \<in> K` by auto
+ qed
+
+ show "\<exists>L. (\<forall>(i::nat). x \<in> L i \<and> open (L i)) \<and> (\<forall>U. open U \<and> x \<in> U \<longrightarrow> (\<exists>i. L i \<subseteq> U))"
+ apply (rule first_countableI[of K])
+ using `countable K` `\<And>k. k \<in> K \<Longrightarrow> x \<in> k` `\<And>k. k \<in> K \<Longrightarrow> open k` Inc by auto
+qed
+
+lemma product_topology_countable_basis:
+ shows "\<exists>K::(('a::countable \<Rightarrow> 'b::second_countable_topology) set set).
+ topological_basis K \<and> countable K \<and>
+ (\<forall>k\<in>K. \<exists>X. (k = PiE UNIV X) \<and> (\<forall>i. open (X i)) \<and> finite {i. X i \<noteq> UNIV})"
+proof -
+ obtain B::"'b set set" where B: "countable B \<and> topological_basis B"
+ using ex_countable_basis by auto
+ then have "B \<noteq> {}" by (meson UNIV_I empty_iff open_UNIV topological_basisE)
+ define B2 where "B2 = B \<union> {UNIV}"
+ have "countable B2"
+ unfolding B2_def using B by auto
+ have "open U" if "U \<in> B2" for U
+ using that unfolding B2_def using B topological_basis_open by auto
+
+ define K where "K = {Pi\<^sub>E UNIV X | X. (\<forall>i::'a. X i \<in> B2) \<and> finite {i. X i \<noteq> UNIV}}"
+ have i: "\<forall>k\<in>K. \<exists>X. (k = PiE UNIV X) \<and> (\<forall>i. open (X i)) \<and> finite {i. X i \<noteq> UNIV}"
+ unfolding K_def using `\<And>U. U \<in> B2 \<Longrightarrow> open U` by auto
+
+ have "countable {X. (\<forall>(i::'a). X i \<in> B2) \<and> finite {i. X i \<noteq> UNIV}}"
+ apply (rule countable_product_event_const) using `countable B2` by auto
+ moreover have "K = (\<lambda>X. Pi\<^sub>E UNIV X)`{X. (\<forall>i. X i \<in> B2) \<and> finite {i. X i \<noteq> UNIV}}"
+ unfolding K_def by auto
+ ultimately have ii: "countable K" by auto
+
+ have iii: "topological_basis K"
+ proof (rule topological_basisI)
+ fix U and x::"'a\<Rightarrow>'b" assume "open U" "x \<in> U"
+ then have "openin (product_topology (\<lambda>i. euclidean) UNIV) U"
+ unfolding open_fun_def by auto
+ with product_topology_open_contains_basis[OF this `x \<in> U`]
+ have "\<exists>X. x \<in> (\<Pi>\<^sub>E i\<in>UNIV. X i) \<and> (\<forall>i. open (X i)) \<and> finite {i. X i \<noteq> UNIV} \<and> (\<Pi>\<^sub>E i\<in>UNIV. X i) \<subseteq> U"
+ unfolding topspace_euclidean open_openin by simp
+ then obtain X where H: "x \<in> (\<Pi>\<^sub>E i\<in>UNIV. X i)"
+ "\<And>i. open (X i)"
+ "finite {i. X i \<noteq> UNIV}"
+ "(\<Pi>\<^sub>E i\<in>UNIV. X i) \<subseteq> U"
+ by auto
+ then have "x i \<in> X i" for i by auto
+ define I where "I = {i. X i \<noteq> UNIV}"
+ define Y where "Y = (\<lambda>i. if i \<in> I then (SOME y. y \<in> B2 \<and> y \<subseteq> X i \<and> x i \<in> y) else UNIV)"
+ have *: "\<exists>y. y \<in> B2 \<and> y \<subseteq> X i \<and> x i \<in> y" for i
+ unfolding B2_def using B `open (X i)` `x i \<in> X i` by (meson UnCI topological_basisE)
+ have **: "Y i \<in> B2 \<and> Y i \<subseteq> X i \<and> x i \<in> Y i" for i
+ using someI_ex[OF *]
+ apply (cases "i \<in> I")
+ unfolding Y_def using * apply (auto)
+ unfolding B2_def I_def by auto
+ have "{i. Y i \<noteq> UNIV} \<subseteq> I"
+ unfolding Y_def by auto
+ then have ***: "finite {i. Y i \<noteq> UNIV}"
+ unfolding I_def using H(3) rev_finite_subset by blast
+ have "(\<forall>i. Y i \<in> B2) \<and> finite {i. Y i \<noteq> UNIV}"
+ using ** *** by auto
+ then have "Pi\<^sub>E UNIV Y \<in> K"
+ unfolding K_def by auto
+
+ have "Y i \<subseteq> X i" for i
+ apply (cases "i \<in> I") using ** apply simp unfolding Y_def I_def by auto
+ then have "Pi\<^sub>E UNIV Y \<subseteq> Pi\<^sub>E UNIV X" by auto
+ then have "Pi\<^sub>E UNIV Y \<subseteq> U" using H(4) by auto
+
+ have "x \<in> Pi\<^sub>E UNIV Y"
+ using ** by auto
+
+ show "\<exists>V\<in>K. x \<in> V \<and> V \<subseteq> U"
+ using `Pi\<^sub>E UNIV Y \<in> K` `Pi\<^sub>E UNIV Y \<subseteq> U` `x \<in> Pi\<^sub>E UNIV Y` by auto
+ next
+ fix U assume "U \<in> K"
+ show "open U"
+ using `U \<in> K` unfolding open_fun_def K_def apply (auto)
+ apply (rule product_topology_basis)
+ using `\<And>V. V \<in> B2 \<Longrightarrow> open V` open_UNIV unfolding topspace_euclidean open_openin[symmetric]
+ by auto
+ qed
+
+ show ?thesis using i ii iii by auto
+qed
+
+instance "fun" :: (countable, second_countable_topology) second_countable_topology
+ apply standard
+ using product_topology_countable_basis topological_basis_imp_subbasis by auto
+
+
+subsection {*The strong operator topology on continuous linear operators*}
+
+text {*Let 'a and 'b be two normed real vector spaces. Then the space of linear continuous
+operators from 'a to 'b has a canonical norm, and therefore a canonical corresponding topology
+(the type classes instantiation are given in \verb+Bounded_Linear_Function.thy+).
+
+However, there is another topology on this space, the strong operator topology, where $T_n$ tends to
+$T$ iff, for all $x$ in 'a, then $T_n x$ tends to $T x$. This is precisely the product topology
+where the target space is endowed with the norm topology. It is especially useful when 'b is the set
+of real numbers, since then this topology is compact.
+
+We can not implement it using type classes as there is already a topology, but at least we
+can define it as a topology.
+
+Note that there is yet another (common and useful) topology on operator spaces, the weak operator
+topology, defined analogously using the product topology, but where the target space is given the
+weak-* topology, i.e., the pullback of the weak topology on the bidual of the space under the
+canonical embedding of a space into its bidual. We do not define it there, although it could also be
+defined analogously.
+*}
+
+definition strong_operator_topology::"('a::real_normed_vector \<Rightarrow>\<^sub>L'b::real_normed_vector) topology"
+where "strong_operator_topology = pullback_topology UNIV blinfun_apply euclidean"
+
+lemma strong_operator_topology_topspace:
+ "topspace strong_operator_topology = UNIV"
+unfolding strong_operator_topology_def topspace_pullback_topology topspace_euclidean by auto
+
+lemma strong_operator_topology_basis:
+ fixes f::"('a::real_normed_vector \<Rightarrow>\<^sub>L'b::real_normed_vector)" and U::"'i \<Rightarrow> 'b set" and x::"'i \<Rightarrow> 'a"
+ assumes "finite I" "\<And>i. i \<in> I \<Longrightarrow> open (U i)"
+ shows "openin strong_operator_topology {f. \<forall>i\<in>I. blinfun_apply f (x i) \<in> U i}"
+proof -
+ have "open {g::('a\<Rightarrow>'b). \<forall>i\<in>I. g (x i) \<in> U i}"
+ by (rule product_topology_basis'[OF assms])
+ moreover have "{f. \<forall>i\<in>I. blinfun_apply f (x i) \<in> U i}
+ = blinfun_apply-`{g::('a\<Rightarrow>'b). \<forall>i\<in>I. g (x i) \<in> U i} \<inter> UNIV"
+ by auto
+ ultimately show ?thesis
+ unfolding strong_operator_topology_def open_openin apply (subst openin_pullback_topology) by auto
+qed
+
+lemma strong_operator_topology_continuous_evaluation:
+ "continuous_on_topo strong_operator_topology euclidean (\<lambda>f. blinfun_apply f x)"
+proof -
+ have "continuous_on_topo strong_operator_topology euclidean ((\<lambda>f. f x) o blinfun_apply)"
+ unfolding strong_operator_topology_def apply (rule continuous_on_topo_pullback)
+ using continuous_on_topo_UNIV continuous_on_product_coordinates by fastforce
+ then show ?thesis unfolding comp_def by simp
+qed
+
+lemma continuous_on_strong_operator_topo_iff_coordinatewise:
+ "continuous_on_topo T strong_operator_topology f
+ \<longleftrightarrow> (\<forall>x. continuous_on_topo T euclidean (\<lambda>y. blinfun_apply (f y) x))"
+proof (auto)
+ fix x::"'b"
+ assume "continuous_on_topo T strong_operator_topology f"
+ with continuous_on_topo_compose[OF this strong_operator_topology_continuous_evaluation]
+ have "continuous_on_topo T euclidean ((\<lambda>z. blinfun_apply z x) o f)"
+ by simp
+ then show "continuous_on_topo T euclidean (\<lambda>y. blinfun_apply (f y) x)"
+ unfolding comp_def by auto
+next
+ assume *: "\<forall>x. continuous_on_topo T euclidean (\<lambda>y. blinfun_apply (f y) x)"
+ have "continuous_on_topo T euclidean (blinfun_apply o f)"
+ unfolding euclidean_product_topology
+ apply (rule continuous_on_topo_coordinatewise_then_product, auto)
+ using * unfolding comp_def by auto
+ show "continuous_on_topo T strong_operator_topology f"
+ unfolding strong_operator_topology_def
+ apply (rule continuous_on_topo_pullback')
+ by (auto simp add: `continuous_on_topo T euclidean (blinfun_apply o f)`)
+qed
+
+lemma strong_operator_topology_weaker_than_euclidean:
+ "continuous_on_topo euclidean strong_operator_topology (\<lambda>f. f)"
+by (subst continuous_on_strong_operator_topo_iff_coordinatewise,
+ auto simp add: continuous_on_topo_UNIV[symmetric] linear_continuous_on)
+
+
+subsection {*Metrics on product spaces*}
+
+
+text {*In general, the product topology is not metrizable, unless the index set is countable.
+When the index set is countable, essentially any (convergent) combination of the metrics on the
+factors will do. We use below the simplest one, based on $L^1$, but $L^2$ would also work,
+for instance.
+
+What is not completely trivial is that the distance thus defined induces the same topology
+as the product topology. This is what we have to prove to show that we have an instance
+of \verb+metric_space+.
+
+The proofs below would work verbatim for general countable products of metric spaces. However,
+since distances are only implemented in terms of type classes, we only develop the theory
+for countable products of the same space.*}
+
+instantiation "fun" :: (countable, metric_space) metric_space
+begin
+
+definition dist_fun_def:
+ "dist x y = (\<Sum>n. (1/2)^n * min (dist (x (from_nat n)) (y (from_nat n))) 1)"
+
+definition uniformity_fun_def:
+ "(uniformity::(('a \<Rightarrow> 'b) \<times> ('a \<Rightarrow> 'b)) filter) = (INF e:{0<..}. principal {(x, y). dist (x::('a\<Rightarrow>'b)) y < e})"
+
+text {*Except for the first one, the auxiliary lemmas below are only useful when proving the
+instance: once it is proved, they become trivial consequences of the general theory of metric
+spaces. It would thus be desirable to hide them once the instance is proved, but I do not know how
+to do this.*}
+
+lemma dist_fun_le_dist_first_terms:
+ "dist x y \<le> 2 * Max {dist (x (from_nat n)) (y (from_nat n))|n. n \<le> N} + (1/2)^N"
+proof -
+ have "(\<Sum>n. (1 / 2) ^ (n+Suc N) * min (dist (x (from_nat (n+Suc N))) (y (from_nat (n+Suc N)))) 1)
+ = (\<Sum>n. (1 / 2) ^ (Suc N) * ((1/2) ^ n * min (dist (x (from_nat (n+Suc N))) (y (from_nat (n+Suc N)))) 1))"
+ by (rule suminf_cong, simp add: power_add)
+ also have "... = (1/2)^(Suc N) * (\<Sum>n. (1 / 2) ^ n * min (dist (x (from_nat (n+Suc N))) (y (from_nat (n+Suc N)))) 1)"
+ apply (rule suminf_mult)
+ by (rule summable_comparison_test'[of "\<lambda>n. (1/2)^n"], auto simp add: summable_geometric_iff)
+ also have "... \<le> (1/2)^(Suc N) * (\<Sum>n. (1 / 2) ^ n)"
+ apply (simp, rule suminf_le, simp)
+ by (rule summable_comparison_test'[of "\<lambda>n. (1/2)^n"], auto simp add: summable_geometric_iff)
+ also have "... = (1/2)^(Suc N) * 2"
+ using suminf_geometric[of "1/2"] by auto
+ also have "... = (1/2)^N"
+ by auto
+ finally have *: "(\<Sum>n. (1 / 2) ^ (n+Suc N) * min (dist (x (from_nat (n+Suc N))) (y (from_nat (n+Suc N)))) 1) \<le> (1/2)^N"
+ by simp
+
+ define M where "M = Max {dist (x (from_nat n)) (y (from_nat n))|n. n \<le> N}"
+ have "dist (x (from_nat 0)) (y (from_nat 0)) \<le> M"
+ unfolding M_def by (rule Max_ge, auto)
+ then have [simp]: "M \<ge> 0" by (meson dual_order.trans zero_le_dist)
+ have "dist (x (from_nat n)) (y (from_nat n)) \<le> M" if "n\<le>N" for n
+ unfolding M_def apply (rule Max_ge) using that by auto
+ then have i: "min (dist (x (from_nat n)) (y (from_nat n))) 1 \<le> M" if "n\<le>N" for n
+ using that by force
+ have "(\<Sum>n< Suc N. (1 / 2) ^ n * min (dist (x (from_nat n)) (y (from_nat n))) 1) \<le>
+ (\<Sum>n< Suc N. M * (1 / 2) ^ n)"
+ by (rule sum_mono, simp add: i)
+ also have "... = M * (\<Sum>n<Suc N. (1 / 2) ^ n)"
+ by (rule sum_distrib_left[symmetric])
+ also have "... \<le> M * (\<Sum>n. (1 / 2) ^ n)"
+ by (rule mult_left_mono, rule sum_le_suminf, auto simp add: summable_geometric_iff)
+ also have "... = M * 2"
+ using suminf_geometric[of "1/2"] by auto
+ finally have **: "(\<Sum>n< Suc N. (1 / 2) ^ n * min (dist (x (from_nat n)) (y (from_nat n))) 1) \<le> 2 * M"
+ by simp
+
+ have "dist x y = (\<Sum>n. (1 / 2) ^ n * min (dist (x (from_nat n)) (y (from_nat n))) 1)"
+ unfolding dist_fun_def by simp
+ also have "... = (\<Sum>n. (1 / 2) ^ (n+Suc N) * min (dist (x (from_nat (n+Suc N))) (y (from_nat (n+Suc N)))) 1)
+ + (\<Sum>n<Suc N. (1 / 2) ^ n * min (dist (x (from_nat n)) (y (from_nat n))) 1)"
+ apply (rule suminf_split_initial_segment)
+ by (rule summable_comparison_test'[of "\<lambda>n. (1/2)^n"], auto simp add: summable_geometric_iff)
+ also have "... \<le> 2 * M + (1/2)^N"
+ using * ** by auto
+ finally show ?thesis unfolding M_def by simp
+qed
+
+lemma open_fun_contains_ball_aux:
+ assumes "open (U::(('a \<Rightarrow> 'b) set))"
+ "x \<in> U"
+ shows "\<exists>e>0. {y. dist x y < e} \<subseteq> U"
+proof -
+ have *: "openin (product_topology (\<lambda>i. euclidean) UNIV) U"
+ using open_fun_def assms by auto
+ obtain X where H: "Pi\<^sub>E UNIV X \<subseteq> U"
+ "\<And>i. openin euclidean (X i)"
+ "finite {i. X i \<noteq> topspace euclidean}"
+ "x \<in> Pi\<^sub>E UNIV X"
+ using product_topology_open_contains_basis[OF * `x \<in> U`] by auto
+ define I where "I = {i. X i \<noteq> topspace euclidean}"
+ have "finite I" unfolding I_def using H(3) by auto
+ {
+ fix i
+ have "x i \<in> X i" using `x \<in> U` H by auto
+ then have "\<exists>e. e>0 \<and> ball (x i) e \<subseteq> X i"
+ using `openin euclidean (X i)` open_openin open_contains_ball by blast
+ then obtain e where "e>0" "ball (x i) e \<subseteq> X i" by blast
+ define f where "f = min e (1/2)"
+ have "f>0" "f<1" unfolding f_def using `e>0` by auto
+ moreover have "ball (x i) f \<subseteq> X i" unfolding f_def using `ball (x i) e \<subseteq> X i` by auto
+ ultimately have "\<exists>f. f > 0 \<and> f < 1 \<and> ball (x i) f \<subseteq> X i" by auto
+ } note * = this
+ have "\<exists>e. \<forall>i. e i > 0 \<and> e i < 1 \<and> ball (x i) (e i) \<subseteq> X i"
+ by (rule choice, auto simp add: *)
+ then obtain e where "\<And>i. e i > 0" "\<And>i. e i < 1" "\<And>i. ball (x i) (e i) \<subseteq> X i"
+ by blast
+ define m where "m = Min {(1/2)^(to_nat i) * e i|i. i \<in> I}"
+ have "m > 0" if "I\<noteq>{}"
+ unfolding m_def apply (rule Min_grI) using `finite I` `I \<noteq> {}` `\<And>i. e i > 0` by auto
+ moreover have "{y. dist x y < m} \<subseteq> U"
+ proof (auto)
+ fix y assume "dist x y < m"
+ have "y i \<in> X i" if "i \<in> I" for i
+ proof -
+ have *: "summable (\<lambda>n. (1/2)^n * min (dist (x (from_nat n)) (y (from_nat n))) 1)"
+ by (rule summable_comparison_test'[of "\<lambda>n. (1/2)^n"], auto simp add: summable_geometric_iff)
+ define n where "n = to_nat i"
+ have "(1/2)^n * min (dist (x (from_nat n)) (y (from_nat n))) 1 < m"
+ using `dist x y < m` unfolding dist_fun_def using sum_le_suminf[OF *, of "{n}"] by auto
+ then have "(1/2)^(to_nat i) * min (dist (x i) (y i)) 1 < m"
+ using `n = to_nat i` by auto
+ also have "... \<le> (1/2)^(to_nat i) * e i"
+ unfolding m_def apply (rule Min_le) using `finite I` `i \<in> I` by auto
+ ultimately have "min (dist (x i) (y i)) 1 < e i"
+ by (auto simp add: divide_simps)
+ then have "dist (x i) (y i) < e i"
+ using `e i < 1` by auto
+ then show "y i \<in> X i" using `ball (x i) (e i) \<subseteq> X i` by auto
+ qed
+ then have "y \<in> Pi\<^sub>E UNIV X" using H(1) unfolding I_def topspace_euclidean by (auto simp add: PiE_iff)
+ then show "y \<in> U" using `Pi\<^sub>E UNIV X \<subseteq> U` by auto
+ qed
+ ultimately have *: "\<exists>m>0. {y. dist x y < m} \<subseteq> U" if "I \<noteq> {}" using that by auto
+
+ have "U = UNIV" if "I = {}"
+ using that H(1) unfolding I_def topspace_euclidean by (auto simp add: PiE_iff)
+ then have "\<exists>m>0. {y. dist x y < m} \<subseteq> U" if "I = {}" using that zero_less_one by blast
+ then show "\<exists>m>0. {y. dist x y < m} \<subseteq> U" using * by blast
+qed
+
+lemma ball_fun_contains_open_aux:
+ fixes x::"('a \<Rightarrow> 'b)" and e::real
+ assumes "e>0"
+ shows "\<exists>U. open U \<and> x \<in> U \<and> U \<subseteq> {y. dist x y < e}"
+proof -
+ have "\<exists>N::nat. 2^N > 8/e"
+ by (simp add: real_arch_pow)
+ then obtain N::nat where "2^N > 8/e" by auto
+ define f where "f = e/4"
+ have [simp]: "e>0" "f > 0" unfolding f_def using assms by auto
+ define X::"('a \<Rightarrow> 'b set)" where "X = (\<lambda>i. if (\<exists>n\<le>N. i = from_nat n) then {z. dist (x i) z < f} else UNIV)"
+ have "{i. X i \<noteq> UNIV} \<subseteq> from_nat`{0..N}"
+ unfolding X_def by auto
+ then have "finite {i. X i \<noteq> topspace euclidean}"
+ unfolding topspace_euclidean using finite_surj by blast
+ have "\<And>i. open (X i)"
+ unfolding X_def using metric_space_class.open_ball open_UNIV by auto
+ then have "\<And>i. openin euclidean (X i)"
+ using open_openin by auto
+ define U where "U = Pi\<^sub>E UNIV X"
+ have "open U"
+ unfolding open_fun_def product_topology_def apply (rule topology_generated_by_Basis)
+ unfolding U_def using `\<And>i. openin euclidean (X i)` `finite {i. X i \<noteq> topspace euclidean}`
+ by auto
+ moreover have "x \<in> U"
+ unfolding U_def X_def by (auto simp add: PiE_iff)
+ moreover have "dist x y < e" if "y \<in> U" for y
+ proof -
+ have *: "dist (x (from_nat n)) (y (from_nat n)) \<le> f" if "n \<le> N" for n
+ using `y \<in> U` unfolding U_def apply (auto simp add: PiE_iff)
+ unfolding X_def using that by (metis less_imp_le mem_Collect_eq)
+ have **: "Max {dist (x (from_nat n)) (y (from_nat n))|n. n \<le> N} \<le> f"
+ apply (rule Max.boundedI) using * by auto
+
+ have "dist x y \<le> 2 * Max {dist (x (from_nat n)) (y (from_nat n))|n. n \<le> N} + (1/2)^N"
+ by (rule dist_fun_le_dist_first_terms)
+ also have "... \<le> 2 * f + e / 8"
+ apply (rule add_mono) using ** `2^N > 8/e` by(auto simp add: algebra_simps divide_simps)
+ also have "... \<le> e/2 + e/8"
+ unfolding f_def by auto
+ also have "... < e"
+ by auto
+ finally show "dist x y < e" by simp
+ qed
+ ultimately show ?thesis by auto
+qed
+
+lemma fun_open_ball_aux:
+ fixes U::"('a \<Rightarrow> 'b) set"
+ shows "open U \<longleftrightarrow> (\<forall>x\<in>U. \<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> y \<in> U)"
+proof (auto)
+ assume "open U"
+ fix x assume "x \<in> U"
+ then show "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> y \<in> U"
+ using open_fun_contains_ball_aux[OF `open U` `x \<in> U`] by auto
+next
+ assume H: "\<forall>x\<in>U. \<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> y \<in> U"
+ define K where "K = {V. open V \<and> V \<subseteq> U}"
+ {
+ fix x assume "x \<in> U"
+ then obtain e where e: "e>0" "{y. dist x y < e} \<subseteq> U" using H by blast
+ then obtain V where V: "open V" "x \<in> V" "V \<subseteq> {y. dist x y < e}"
+ using ball_fun_contains_open_aux[OF `e>0`, of x] by auto
+ have "V \<in> K"
+ unfolding K_def using e(2) V(1) V(3) by auto
+ then have "x \<in> (\<Union>K)" using `x \<in> V` by auto
+ }
+ then have "(\<Union>K) = U"
+ unfolding K_def by auto
+ moreover have "open (\<Union>K)"
+ unfolding K_def by auto
+ ultimately show "open U" by simp
+qed
+
+instance proof
+ fix x y::"'a \<Rightarrow> 'b" show "(dist x y = 0) = (x = y)"
+ proof
+ assume "x = y"
+ then show "dist x y = 0" unfolding dist_fun_def using `x = y` by auto
+ next
+ assume "dist x y = 0"
+ have *: "summable (\<lambda>n. (1/2)^n * min (dist (x (from_nat n)) (y (from_nat n))) 1)"
+ by (rule summable_comparison_test'[of "\<lambda>n. (1/2)^n"], auto simp add: summable_geometric_iff)
+ have "(1/2)^n * min (dist (x (from_nat n)) (y (from_nat n))) 1 = 0" for n
+ using `dist x y = 0` unfolding dist_fun_def by (simp add: "*" suminf_eq_zero_iff)
+ then have "dist (x (from_nat n)) (y (from_nat n)) = 0" for n
+ by (metis div_0 min_def nonzero_mult_div_cancel_left power_eq_0_iff
+ zero_eq_1_divide_iff zero_neq_numeral)
+ then have "x (from_nat n) = y (from_nat n)" for n
+ by auto
+ then have "x i = y i" for i
+ by (metis from_nat_to_nat)
+ then show "x = y"
+ by auto
+ qed
+next
+ text{*The proof of the triangular inequality is trivial, modulo the fact that we are dealing
+ with infinite series, hence we should justify the convergence at each step. In this
+ respect, the following simplification rule is extremely handy.*}
+ have [simp]: "summable (\<lambda>n. (1/2)^n * min (dist (u (from_nat n)) (v (from_nat n))) 1)" for u v::"'a \<Rightarrow> 'b"
+ by (rule summable_comparison_test'[of "\<lambda>n. (1/2)^n"], auto simp add: summable_geometric_iff)
+ fix x y z::"'a \<Rightarrow> 'b"
+ {
+ fix n
+ have *: "dist (x (from_nat n)) (y (from_nat n)) \<le>
+ dist (x (from_nat n)) (z (from_nat n)) + dist (y (from_nat n)) (z (from_nat n))"
+ by (simp add: dist_triangle2)
+ have "0 \<le> dist (y (from_nat n)) (z (from_nat n))"
+ using zero_le_dist by blast
+ moreover
+ {
+ assume "min (dist (y (from_nat n)) (z (from_nat n))) 1 \<noteq> dist (y (from_nat n)) (z (from_nat n))"
+ then have "1 \<le> min (dist (x (from_nat n)) (z (from_nat n))) 1 + min (dist (y (from_nat n)) (z (from_nat n))) 1"
+ by (metis (no_types) diff_le_eq diff_self min_def zero_le_dist zero_le_one)
+ }
+ ultimately have "min (dist (x (from_nat n)) (y (from_nat n))) 1 \<le>
+ min (dist (x (from_nat n)) (z (from_nat n))) 1 + min (dist (y (from_nat n)) (z (from_nat n))) 1"
+ using * by linarith
+ } note ineq = this
+ have "dist x y = (\<Sum>n. (1/2)^n * min (dist (x (from_nat n)) (y (from_nat n))) 1)"
+ unfolding dist_fun_def by simp
+ also have "... \<le> (\<Sum>n. (1/2)^n * min (dist (x (from_nat n)) (z (from_nat n))) 1
+ + (1/2)^n * min (dist (y (from_nat n)) (z (from_nat n))) 1)"
+ apply (rule suminf_le)
+ using ineq apply (metis (no_types, hide_lams) add.right_neutral distrib_left
+ le_divide_eq_numeral1(1) mult_2_right mult_left_mono zero_le_one zero_le_power)
+ by (auto simp add: summable_add)
+ also have "... = (\<Sum>n. (1/2)^n * min (dist (x (from_nat n)) (z (from_nat n))) 1)
+ + (\<Sum>n. (1/2)^n * min (dist (y (from_nat n)) (z (from_nat n))) 1)"
+ by (rule suminf_add[symmetric], auto)
+ also have "... = dist x z + dist y z"
+ unfolding dist_fun_def by simp
+ finally show "dist x y \<le> dist x z + dist y z"
+ by simp
+next
+ text{*Finally, we show that the topology generated by the distance and the product
+ topology coincide. This is essentially contained in Lemma \verb+fun_open_ball_aux+,
+ except that the condition to prove is expressed with filters. To deal with this,
+ we copy some mumbo jumbo from Lemma \verb+eventually_uniformity_metric+ in
+ \verb+Real_Vector_Spaces.thy+*}
+ fix U::"('a \<Rightarrow> 'b) set"
+ have "eventually P uniformity \<longleftrightarrow> (\<exists>e>0. \<forall>x (y::('a \<Rightarrow> 'b)). dist x y < e \<longrightarrow> P (x, y))" for P
+ unfolding uniformity_fun_def apply (subst eventually_INF_base)
+ by (auto simp: eventually_principal subset_eq intro: bexI[of _ "min _ _"])
+ then show "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U)"
+ unfolding fun_open_ball_aux by simp
+qed (simp add: uniformity_fun_def)
+
+end
+
+text {*Nice properties of spaces are preserved under countable products. In addition
+to first countability, second countability and metrizability, as we have seen above,
+completeness is also preserved, and therefore being Polish.*}
+
+instance "fun" :: (countable, complete_space) complete_space
+proof
+ fix u::"nat \<Rightarrow> ('a \<Rightarrow> 'b)" assume "Cauchy u"
+ have "Cauchy (\<lambda>n. u n i)" for i
+ unfolding cauchy_def proof (intro impI allI)
+ fix e assume "e>(0::real)"
+ obtain k where "i = from_nat k"
+ using from_nat_to_nat[of i] by metis
+ have "(1/2)^k * min e 1 > 0" using `e>0` by auto
+ then have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (u m) (u n) < (1/2)^k * min e 1"
+ using `Cauchy u` unfolding cauchy_def by blast
+ then obtain N::nat where C: "\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (u m) (u n) < (1/2)^k * min e 1"
+ by blast
+ have "\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (u m i) (u n i) < e"
+ proof (auto)
+ fix m n::nat assume "m \<ge> N" "n \<ge> N"
+ have "(1/2)^k * min (dist (u m i) (u n i)) 1
+ = sum (\<lambda>p. (1/2)^p * min (dist (u m (from_nat p)) (u n (from_nat p))) 1) {k}"
+ using `i = from_nat k` by auto
+ also have "... \<le> (\<Sum>p. (1/2)^p * min (dist (u m (from_nat p)) (u n (from_nat p))) 1)"
+ apply (rule sum_le_suminf)
+ by (rule summable_comparison_test'[of "\<lambda>n. (1/2)^n"], auto simp add: summable_geometric_iff)
+ also have "... = dist (u m) (u n)"
+ unfolding dist_fun_def by auto
+ also have "... < (1/2)^k * min e 1"
+ using C `m \<ge> N` `n \<ge> N` by auto
+ finally have "min (dist (u m i) (u n i)) 1 < min e 1"
+ by (auto simp add: algebra_simps divide_simps)
+ then show "dist (u m i) (u n i) < e" by auto
+ qed
+ then show "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (u m i) (u n i) < e"
+ by blast
+ qed
+ then have "\<exists>x. (\<lambda>n. u n i) \<longlonglongrightarrow> x" for i
+ using Cauchy_convergent convergent_def by auto
+ then have "\<exists>x. \<forall>i. (\<lambda>n. u n i) \<longlonglongrightarrow> x i"
+ using choice by force
+ then obtain x where *: "\<And>i. (\<lambda>n. u n i) \<longlonglongrightarrow> x i" by blast
+ have "u \<longlonglongrightarrow> x"
+ proof (rule metric_LIMSEQ_I)
+ fix e assume [simp]: "e>(0::real)"
+ have i: "\<exists>K. \<forall>n\<ge>K. dist (u n i) (x i) < e/4" for i
+ by (rule metric_LIMSEQ_D, auto simp add: *)
+ have "\<exists>K. \<forall>i. \<forall>n\<ge>K i. dist (u n i) (x i) < e/4"
+ apply (rule choice) using i by auto
+ then obtain K where K: "\<And>i n. n \<ge> K i \<Longrightarrow> dist (u n i) (x i) < e/4"
+ by blast
+
+ have "\<exists>N::nat. 2^N > 4/e"
+ by (simp add: real_arch_pow)
+ then obtain N::nat where "2^N > 4/e" by auto
+ define L where "L = Max {K (from_nat n)|n. n \<le> N}"
+ have "dist (u k) x < e" if "k \<ge> L" for k
+ proof -
+ have *: "dist (u k (from_nat n)) (x (from_nat n)) \<le> e / 4" if "n \<le> N" for n
+ proof -
+ have "K (from_nat n) \<le> L"
+ unfolding L_def apply (rule Max_ge) using `n \<le> N` by auto
+ then have "k \<ge> K (from_nat n)" using `k \<ge> L` by auto
+ then show ?thesis using K less_imp_le by auto
+ qed
+ have **: "Max {dist (u k (from_nat n)) (x (from_nat n)) |n. n \<le> N} \<le> e/4"
+ apply (rule Max.boundedI) using * by auto
+ have "dist (u k) x \<le> 2 * Max {dist (u k (from_nat n)) (x (from_nat n)) |n. n \<le> N} + (1/2)^N"
+ using dist_fun_le_dist_first_terms by auto
+ also have "... \<le> 2 * e/4 + e/4"
+ apply (rule add_mono)
+ using ** `2^N > 4/e` less_imp_le by (auto simp add: algebra_simps divide_simps)
+ also have "... < e" by auto
+ finally show ?thesis by simp
+ qed
+ then show "\<exists>L. \<forall>k\<ge>L. dist (u k) x < e" by blast
+ qed
+ then show "convergent u" using convergent_def by blast
+qed
+
+instance "fun" :: (countable, polish_space) polish_space
+by standard
+
+
+subsection {*Measurability*}
+
+text {*There are two natural sigma-algebras on a product space: the borel sigma algebra,
+generated by open sets in the product, and the product sigma algebra, countably generated by
+products of measurable sets along finitely many coordinates. The second one is defined and studied
+in \verb+Finite_Product_Measure.thy+.
+
+These sigma-algebra share a lot of natural properties (measurability of coordinates, for instance),
+but there is a fundamental difference: open sets are generated by arbitrary unions, not only
+countable ones, so typically many open sets will not be measurable with respect to the product sigma
+algebra (while all sets in the product sigma algebra are borel). The two sigma algebras coincide
+only when everything is countable (i.e., the product is countable, and the borel sigma algebra in
+the factor is countably generated).
+
+In this paragraph, we develop basic measurability properties for the borel sigma algebra, and
+compare it with the product sigma algebra as explained above.
+*}
+
+lemma measurable_product_coordinates [measurable (raw)]:
+ "(\<lambda>x. x i) \<in> measurable borel borel"
+by (rule borel_measurable_continuous_on1[OF continuous_on_product_coordinates])
+
+lemma measurable_product_then_coordinatewise:
+ fixes f::"'a \<Rightarrow> 'b \<Rightarrow> ('c::topological_space)"
+ assumes [measurable]: "f \<in> borel_measurable M"
+ shows "(\<lambda>x. f x i) \<in> borel_measurable M"
+proof -
+ have "(\<lambda>x. f x i) = (\<lambda>y. y i) o f"
+ unfolding comp_def by auto
+ then show ?thesis by simp
+qed
+
+text {*To compare the Borel sigma algebra with the product sigma algebra, we give a presentation
+of the product sigma algebra that is more similar to the one we used above for the product
+topology.*}
+
+lemma sets_PiM_finite:
+ "sets (Pi\<^sub>M I M) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i))
+ {(\<Pi>\<^sub>E i\<in>I. X i) |X. (\<forall>i. X i \<in> sets (M i)) \<and> finite {i. X i \<noteq> space (M i)}}"
+proof
+ have "{(\<Pi>\<^sub>E i\<in>I. X i) |X. (\<forall>i. X i \<in> sets (M i)) \<and> finite {i. X i \<noteq> space (M i)}} \<subseteq> sets (Pi\<^sub>M I M)"
+ proof (auto)
+ fix X assume H: "\<forall>i. X i \<in> sets (M i)" "finite {i. X i \<noteq> space (M i)}"
+ then have *: "X i \<in> sets (M i)" for i by simp
+ define J where "J = {i \<in> I. X i \<noteq> space (M i)}"
+ have "finite J" "J \<subseteq> I" unfolding J_def using H by auto
+ define Y where "Y = (\<Pi>\<^sub>E j\<in>J. X j)"
+ have "prod_emb I M J Y \<in> sets (Pi\<^sub>M I M)"
+ unfolding Y_def apply (rule sets_PiM_I) using `finite J` `J \<subseteq> I` * by auto
+ moreover have "prod_emb I M J Y = (\<Pi>\<^sub>E i\<in>I. X i)"
+ unfolding prod_emb_def Y_def J_def using H sets.sets_into_space[OF *]
+ by (auto simp add: PiE_iff, blast)
+ ultimately show "Pi\<^sub>E I X \<in> sets (Pi\<^sub>M I M)" by simp
+ qed
+ then show "sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) {(\<Pi>\<^sub>E i\<in>I. X i) |X. (\<forall>i. X i \<in> sets (M i)) \<and> finite {i. X i \<noteq> space (M i)}}
+ \<subseteq> sets (Pi\<^sub>M I M)"
+ by (metis (mono_tags, lifting) sets.sigma_sets_subset' sets.top space_PiM)
+
+ have *: "\<exists>X. {f. (\<forall>i\<in>I. f i \<in> space (M i)) \<and> f \<in> extensional I \<and> f i \<in> A} = Pi\<^sub>E I X \<and>
+ (\<forall>i. X i \<in> sets (M i)) \<and> finite {i. X i \<noteq> space (M i)}"
+ if "i \<in> I" "A \<in> sets (M i)" for i A
+ proof -
+ define X where "X = (\<lambda>j. if j = i then A else space (M j))"
+ have "{f. (\<forall>i\<in>I. f i \<in> space (M i)) \<and> f \<in> extensional I \<and> f i \<in> A} = Pi\<^sub>E I X"
+ unfolding X_def using sets.sets_into_space[OF `A \<in> sets (M i)`] `i \<in> I`
+ by (auto simp add: PiE_iff extensional_def, metis subsetCE, metis)
+ moreover have "X j \<in> sets (M j)" for j
+ unfolding X_def using `A \<in> sets (M i)` by auto
+ moreover have "finite {j. X j \<noteq> space (M j)}"
+ unfolding X_def by simp
+ ultimately show ?thesis by auto
+ qed
+ show "sets (Pi\<^sub>M I M) \<subseteq> sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) {(\<Pi>\<^sub>E i\<in>I. X i) |X. (\<forall>i. X i \<in> sets (M i)) \<and> finite {i. X i \<noteq> space (M i)}}"
+ unfolding sets_PiM_single
+ apply (rule sigma_sets_mono')
+ apply (auto simp add: PiE_iff *)
+ done
+qed
+
+lemma sets_PiM_subset_borel:
+ "sets (Pi\<^sub>M UNIV (\<lambda>_. borel)) \<subseteq> sets borel"
+proof -
+ have *: "Pi\<^sub>E UNIV X \<in> sets borel" if [measurable]: "\<And>i. X i \<in> sets borel" "finite {i. X i \<noteq> UNIV}" for X::"'a \<Rightarrow> 'b set"
+ proof -
+ define I where "I = {i. X i \<noteq> UNIV}"
+ have "finite I" unfolding I_def using that by simp
+ have "Pi\<^sub>E UNIV X = (\<Inter>i\<in>I. (\<lambda>x. x i)-`(X i) \<inter> space borel) \<inter> space borel"
+ unfolding I_def by auto
+ also have "... \<in> sets borel"
+ using that `finite I` by measurable
+ finally show ?thesis by simp
+ qed
+ then have "{(\<Pi>\<^sub>E i\<in>UNIV. X i) |X::('a \<Rightarrow> 'b set). (\<forall>i. X i \<in> sets borel) \<and> finite {i. X i \<noteq> space borel}} \<subseteq> sets borel"
+ by auto
+ then show ?thesis unfolding sets_PiM_finite space_borel
+ by (simp add: * sets.sigma_sets_subset')
+qed
+
+lemma sets_PiM_equal_borel:
+ "sets (Pi\<^sub>M UNIV (\<lambda>i::('a::countable). borel::('b::second_countable_topology measure))) = sets borel"
+proof
+ obtain K::"('a \<Rightarrow> 'b) set set" where K: "topological_basis K" "countable K"
+ "\<And>k. k \<in> K \<Longrightarrow> \<exists>X. (k = PiE UNIV X) \<and> (\<forall>i. open (X i)) \<and> finite {i. X i \<noteq> UNIV}"
+ using product_topology_countable_basis by fast
+ have *: "k \<in> sets (Pi\<^sub>M UNIV (\<lambda>_. borel))" if "k \<in> K" for k
+ proof -
+ obtain X where H: "k = PiE UNIV X" "\<And>i. open (X i)" "finite {i. X i \<noteq> UNIV}"
+ using K(3)[OF `k \<in> K`] by blast
+ show ?thesis unfolding H(1) sets_PiM_finite space_borel using borel_open[OF H(2)] H(3) by auto
+ qed
+ have **: "U \<in> sets (Pi\<^sub>M UNIV (\<lambda>_. borel))" if "open U" for U::"('a \<Rightarrow> 'b) set"
+ proof -
+ obtain B where "B \<subseteq> K" "U = (\<Union>B)"
+ using `open U` `topological_basis K` by (metis topological_basis_def)
+ have "countable B" using `B \<subseteq> K` `countable K` countable_subset by blast
+ moreover have "k \<in> sets (Pi\<^sub>M UNIV (\<lambda>_. borel))" if "k \<in> B" for k
+ using `B \<subseteq> K` * that by auto
+ ultimately show ?thesis unfolding `U = (\<Union>B)` by auto
+ qed
+ have "sigma_sets UNIV (Collect open) \<subseteq> sets (Pi\<^sub>M UNIV (\<lambda>i::'a. (borel::('b measure))))"
+ apply (rule sets.sigma_sets_subset') using ** by auto
+ then show "sets (borel::('a \<Rightarrow> 'b) measure) \<subseteq> sets (Pi\<^sub>M UNIV (\<lambda>_. borel))"
+ unfolding borel_def by auto
+qed (simp add: sets_PiM_subset_borel)
+
+lemma measurable_coordinatewise_then_product:
+ fixes f::"'a \<Rightarrow> ('b::countable) \<Rightarrow> ('c::second_countable_topology)"
+ assumes [measurable]: "\<And>i. (\<lambda>x. f x i) \<in> borel_measurable M"
+ shows "f \<in> borel_measurable M"
+proof -
+ have "f \<in> measurable M (Pi\<^sub>M UNIV (\<lambda>_. borel))"
+ by (rule measurable_PiM_single', auto simp add: assms)
+ then show ?thesis using sets_PiM_equal_borel measurable_cong_sets by blast
+qed
+
+end
--- a/src/HOL/Analysis/FurtherTopology.thy Tue Oct 18 16:04:44 2016 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,2355 +0,0 @@
-section \<open>Extending Continous Maps, Invariance of Domain, etc..\<close>
-
-text\<open>Ported from HOL Light (moretop.ml) by L C Paulson\<close>
-
-theory "FurtherTopology"
- imports Equivalence_Lebesgue_Henstock_Integration Weierstrass_Theorems Polytope
-
-begin
-
-subsection\<open>A map from a sphere to a higher dimensional sphere is nullhomotopic\<close>
-
-lemma spheremap_lemma1:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
- assumes "subspace S" "subspace T" and dimST: "dim S < dim T"
- and "S \<subseteq> T"
- and diff_f: "f differentiable_on sphere 0 1 \<inter> S"
- shows "f ` (sphere 0 1 \<inter> S) \<noteq> sphere 0 1 \<inter> T"
-proof
- assume fim: "f ` (sphere 0 1 \<inter> S) = sphere 0 1 \<inter> T"
- have inS: "\<And>x. \<lbrakk>x \<in> S; x \<noteq> 0\<rbrakk> \<Longrightarrow> (x /\<^sub>R norm x) \<in> S"
- using subspace_mul \<open>subspace S\<close> by blast
- have subS01: "(\<lambda>x. x /\<^sub>R norm x) ` (S - {0}) \<subseteq> sphere 0 1 \<inter> S"
- using \<open>subspace S\<close> subspace_mul by fastforce
- then have diff_f': "f differentiable_on (\<lambda>x. x /\<^sub>R norm x) ` (S - {0})"
- by (rule differentiable_on_subset [OF diff_f])
- define g where "g \<equiv> \<lambda>x. norm x *\<^sub>R f(inverse(norm x) *\<^sub>R x)"
- have gdiff: "g differentiable_on S - {0}"
- unfolding g_def
- by (rule diff_f' derivative_intros differentiable_on_compose [where f=f] | force)+
- have geq: "g ` (S - {0}) = T - {0}"
- proof
- have "g ` (S - {0}) \<subseteq> T"
- apply (auto simp: g_def subspace_mul [OF \<open>subspace T\<close>])
- apply (metis (mono_tags, lifting) DiffI subS01 subspace_mul [OF \<open>subspace T\<close>] fim image_subset_iff inf_le2 singletonD)
- done
- moreover have "g ` (S - {0}) \<subseteq> UNIV - {0}"
- proof (clarsimp simp: g_def)
- fix y
- assume "y \<in> S" and f0: "f (y /\<^sub>R norm y) = 0"
- then have "y \<noteq> 0 \<Longrightarrow> y /\<^sub>R norm y \<in> sphere 0 1 \<inter> S"
- by (auto simp: subspace_mul [OF \<open>subspace S\<close>])
- then show "y = 0"
- by (metis fim f0 Int_iff image_iff mem_sphere_0 norm_eq_zero zero_neq_one)
- qed
- ultimately show "g ` (S - {0}) \<subseteq> T - {0}"
- by auto
- next
- have *: "sphere 0 1 \<inter> T \<subseteq> f ` (sphere 0 1 \<inter> S)"
- using fim by (simp add: image_subset_iff)
- have "x \<in> (\<lambda>x. norm x *\<^sub>R f (x /\<^sub>R norm x)) ` (S - {0})"
- if "x \<in> T" "x \<noteq> 0" for x
- proof -
- have "x /\<^sub>R norm x \<in> T"
- using \<open>subspace T\<close> subspace_mul that by blast
- then show ?thesis
- using * [THEN subsetD, of "x /\<^sub>R norm x"] that apply clarsimp
- apply (rule_tac x="norm x *\<^sub>R xa" in image_eqI, simp)
- apply (metis norm_eq_zero right_inverse scaleR_one scaleR_scaleR)
- using \<open>subspace S\<close> subspace_mul apply force
- done
- qed
- then have "T - {0} \<subseteq> (\<lambda>x. norm x *\<^sub>R f (x /\<^sub>R norm x)) ` (S - {0})"
- by force
- then show "T - {0} \<subseteq> g ` (S - {0})"
- by (simp add: g_def)
- qed
- define T' where "T' \<equiv> {y. \<forall>x \<in> T. orthogonal x y}"
- have "subspace T'"
- by (simp add: subspace_orthogonal_to_vectors T'_def)
- have dim_eq: "dim T' + dim T = DIM('a)"
- using dim_subspace_orthogonal_to_vectors [of T UNIV] \<open>subspace T\<close>
- by (simp add: dim_UNIV T'_def)
- have "\<exists>v1 v2. v1 \<in> span T \<and> (\<forall>w \<in> span T. orthogonal v2 w) \<and> x = v1 + v2" for x
- by (force intro: orthogonal_subspace_decomp_exists [of T x])
- then obtain p1 p2 where p1span: "p1 x \<in> span T"
- and "\<And>w. w \<in> span T \<Longrightarrow> orthogonal (p2 x) w"
- and eq: "p1 x + p2 x = x" for x
- by metis
- then have p1: "\<And>z. p1 z \<in> T" and ortho: "\<And>w. w \<in> T \<Longrightarrow> orthogonal (p2 x) w" for x
- using span_eq \<open>subspace T\<close> by blast+
- then have p2: "\<And>z. p2 z \<in> T'"
- by (simp add: T'_def orthogonal_commute)
- have p12_eq: "\<And>x y. \<lbrakk>x \<in> T; y \<in> T'\<rbrakk> \<Longrightarrow> p1(x + y) = x \<and> p2(x + y) = y"
- proof (rule orthogonal_subspace_decomp_unique [OF eq p1span, where T=T'])
- show "\<And>x y. \<lbrakk>x \<in> T; y \<in> T'\<rbrakk> \<Longrightarrow> p2 (x + y) \<in> span T'"
- using span_eq p2 \<open>subspace T'\<close> by blast
- show "\<And>a b. \<lbrakk>a \<in> T; b \<in> T'\<rbrakk> \<Longrightarrow> orthogonal a b"
- using T'_def by blast
- qed (auto simp: span_superset)
- then have "\<And>c x. p1 (c *\<^sub>R x) = c *\<^sub>R p1 x \<and> p2 (c *\<^sub>R x) = c *\<^sub>R p2 x"
- by (metis eq \<open>subspace T\<close> \<open>subspace T'\<close> p1 p2 scaleR_add_right subspace_mul)
- moreover have lin_add: "\<And>x y. p1 (x + y) = p1 x + p1 y \<and> p2 (x + y) = p2 x + p2 y"
- proof (rule orthogonal_subspace_decomp_unique [OF _ p1span, where T=T'])
- show "\<And>x y. p1 (x + y) + p2 (x + y) = p1 x + p1 y + (p2 x + p2 y)"
- by (simp add: add.assoc add.left_commute eq)
- show "\<And>a b. \<lbrakk>a \<in> T; b \<in> T'\<rbrakk> \<Longrightarrow> orthogonal a b"
- using T'_def by blast
- qed (auto simp: p1span p2 span_superset subspace_add)
- ultimately have "linear p1" "linear p2"
- by unfold_locales auto
- have "(\<lambda>z. g (p1 z)) differentiable_on {x + y |x y. x \<in> S - {0} \<and> y \<in> T'}"
- apply (rule differentiable_on_compose [where f=g])
- apply (rule linear_imp_differentiable_on [OF \<open>linear p1\<close>])
- apply (rule differentiable_on_subset [OF gdiff])
- using p12_eq \<open>S \<subseteq> T\<close> apply auto
- done
- then have diff: "(\<lambda>x. g (p1 x) + p2 x) differentiable_on {x + y |x y. x \<in> S - {0} \<and> y \<in> T'}"
- by (intro derivative_intros linear_imp_differentiable_on [OF \<open>linear p2\<close>])
- have "dim {x + y |x y. x \<in> S - {0} \<and> y \<in> T'} \<le> dim {x + y |x y. x \<in> S \<and> y \<in> T'}"
- by (blast intro: dim_subset)
- also have "... = dim S + dim T' - dim (S \<inter> T')"
- using dim_sums_Int [OF \<open>subspace S\<close> \<open>subspace T'\<close>]
- by (simp add: algebra_simps)
- also have "... < DIM('a)"
- using dimST dim_eq by auto
- finally have neg: "negligible {x + y |x y. x \<in> S - {0} \<and> y \<in> T'}"
- by (rule negligible_lowdim)
- have "negligible ((\<lambda>x. g (p1 x) + p2 x) ` {x + y |x y. x \<in> S - {0} \<and> y \<in> T'})"
- by (rule negligible_differentiable_image_negligible [OF order_refl neg diff])
- then have "negligible {x + y |x y. x \<in> g ` (S - {0}) \<and> y \<in> T'}"
- proof (rule negligible_subset)
- have "\<lbrakk>t' \<in> T'; s \<in> S; s \<noteq> 0\<rbrakk>
- \<Longrightarrow> g s + t' \<in> (\<lambda>x. g (p1 x) + p2 x) `
- {x + t' |x t'. x \<in> S \<and> x \<noteq> 0 \<and> t' \<in> T'}" for t' s
- apply (rule_tac x="s + t'" in image_eqI)
- using \<open>S \<subseteq> T\<close> p12_eq by auto
- then show "{x + y |x y. x \<in> g ` (S - {0}) \<and> y \<in> T'}
- \<subseteq> (\<lambda>x. g (p1 x) + p2 x) ` {x + y |x y. x \<in> S - {0} \<and> y \<in> T'}"
- by auto
- qed
- moreover have "- T' \<subseteq> {x + y |x y. x \<in> g ` (S - {0}) \<and> y \<in> T'}"
- proof clarsimp
- fix z assume "z \<notin> T'"
- show "\<exists>x y. z = x + y \<and> x \<in> g ` (S - {0}) \<and> y \<in> T'"
- apply (rule_tac x="p1 z" in exI)
- apply (rule_tac x="p2 z" in exI)
- apply (simp add: p1 eq p2 geq)
- by (metis \<open>z \<notin> T'\<close> add.left_neutral eq p2)
- qed
- ultimately have "negligible (-T')"
- using negligible_subset by blast
- moreover have "negligible T'"
- using negligible_lowdim
- by (metis add.commute assms(3) diff_add_inverse2 diff_self_eq_0 dim_eq le_add1 le_antisym linordered_semidom_class.add_diff_inverse not_less0)
- ultimately have "negligible (-T' \<union> T')"
- by (metis negligible_Un_eq)
- then show False
- using negligible_Un_eq non_negligible_UNIV by simp
-qed
-
-
-lemma spheremap_lemma2:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
- assumes ST: "subspace S" "subspace T" "dim S < dim T"
- and "S \<subseteq> T"
- and contf: "continuous_on (sphere 0 1 \<inter> S) f"
- and fim: "f ` (sphere 0 1 \<inter> S) \<subseteq> sphere 0 1 \<inter> T"
- shows "\<exists>c. homotopic_with (\<lambda>x. True) (sphere 0 1 \<inter> S) (sphere 0 1 \<inter> T) f (\<lambda>x. c)"
-proof -
- have [simp]: "\<And>x. \<lbrakk>norm x = 1; x \<in> S\<rbrakk> \<Longrightarrow> norm (f x) = 1"
- using fim by (simp add: image_subset_iff)
- have "compact (sphere 0 1 \<inter> S)"
- by (simp add: \<open>subspace S\<close> closed_subspace compact_Int_closed)
- then obtain g where pfg: "polynomial_function g" and gim: "g ` (sphere 0 1 \<inter> S) \<subseteq> T"
- and g12: "\<And>x. x \<in> sphere 0 1 \<inter> S \<Longrightarrow> norm(f x - g x) < 1/2"
- apply (rule Stone_Weierstrass_polynomial_function_subspace [OF _ contf _ \<open>subspace T\<close>, of "1/2"])
- using fim apply auto
- done
- have gnz: "g x \<noteq> 0" if "x \<in> sphere 0 1 \<inter> S" for x
- proof -
- have "norm (f x) = 1"
- using fim that by (simp add: image_subset_iff)
- then show ?thesis
- using g12 [OF that] by auto
- qed
- have diffg: "g differentiable_on sphere 0 1 \<inter> S"
- by (metis pfg differentiable_on_polynomial_function)
- define h where "h \<equiv> \<lambda>x. inverse(norm(g x)) *\<^sub>R g x"
- have h: "x \<in> sphere 0 1 \<inter> S \<Longrightarrow> h x \<in> sphere 0 1 \<inter> T" for x
- unfolding h_def
- using gnz [of x]
- by (auto simp: subspace_mul [OF \<open>subspace T\<close>] subsetD [OF gim])
- have diffh: "h differentiable_on sphere 0 1 \<inter> S"
- unfolding h_def
- apply (intro derivative_intros diffg differentiable_on_compose [OF diffg])
- using gnz apply auto
- done
- have homfg: "homotopic_with (\<lambda>z. True) (sphere 0 1 \<inter> S) (T - {0}) f g"
- proof (rule homotopic_with_linear [OF contf])
- show "continuous_on (sphere 0 1 \<inter> S) g"
- using pfg by (simp add: differentiable_imp_continuous_on diffg)
- next
- have non0fg: "0 \<notin> closed_segment (f x) (g x)" if "norm x = 1" "x \<in> S" for x
- proof -
- have "f x \<in> sphere 0 1"
- using fim that by (simp add: image_subset_iff)
- moreover have "norm(f x - g x) < 1/2"
- apply (rule g12)
- using that by force
- ultimately show ?thesis
- by (auto simp: norm_minus_commute dest: segment_bound)
- qed
- show "\<And>x. x \<in> sphere 0 1 \<inter> S \<Longrightarrow> closed_segment (f x) (g x) \<subseteq> T - {0}"
- apply (simp add: subset_Diff_insert non0fg)
- apply (simp add: segment_convex_hull)
- apply (rule hull_minimal)
- using fim image_eqI gim apply force
- apply (rule subspace_imp_convex [OF \<open>subspace T\<close>])
- done
- qed
- obtain d where d: "d \<in> (sphere 0 1 \<inter> T) - h ` (sphere 0 1 \<inter> S)"
- using h spheremap_lemma1 [OF ST \<open>S \<subseteq> T\<close> diffh] by force
- then have non0hd: "0 \<notin> closed_segment (h x) (- d)" if "norm x = 1" "x \<in> S" for x
- using midpoint_between [of 0 "h x" "-d"] that h [of x]
- by (auto simp: between_mem_segment midpoint_def)
- have conth: "continuous_on (sphere 0 1 \<inter> S) h"
- using differentiable_imp_continuous_on diffh by blast
- have hom_hd: "homotopic_with (\<lambda>z. True) (sphere 0 1 \<inter> S) (T - {0}) h (\<lambda>x. -d)"
- apply (rule homotopic_with_linear [OF conth continuous_on_const])
- apply (simp add: subset_Diff_insert non0hd)
- apply (simp add: segment_convex_hull)
- apply (rule hull_minimal)
- using h d apply (force simp: subspace_neg [OF \<open>subspace T\<close>])
- apply (rule subspace_imp_convex [OF \<open>subspace T\<close>])
- done
- have conT0: "continuous_on (T - {0}) (\<lambda>y. inverse(norm y) *\<^sub>R y)"
- by (intro continuous_intros) auto
- have sub0T: "(\<lambda>y. y /\<^sub>R norm y) ` (T - {0}) \<subseteq> sphere 0 1 \<inter> T"
- by (fastforce simp: assms(2) subspace_mul)
- obtain c where homhc: "homotopic_with (\<lambda>z. True) (sphere 0 1 \<inter> S) (sphere 0 1 \<inter> T) h (\<lambda>x. c)"
- apply (rule_tac c="-d" in that)
- apply (rule homotopic_with_eq)
- apply (rule homotopic_compose_continuous_left [OF hom_hd conT0 sub0T])
- using d apply (auto simp: h_def)
- done
- show ?thesis
- apply (rule_tac x=c in exI)
- apply (rule homotopic_with_trans [OF _ homhc])
- apply (rule homotopic_with_eq)
- apply (rule homotopic_compose_continuous_left [OF homfg conT0 sub0T])
- apply (auto simp: h_def)
- done
-qed
-
-
-lemma spheremap_lemma3:
- assumes "bounded S" "convex S" "subspace U" and affSU: "aff_dim S \<le> dim U"
- obtains T where "subspace T" "T \<subseteq> U" "S \<noteq> {} \<Longrightarrow> aff_dim T = aff_dim S"
- "(rel_frontier S) homeomorphic (sphere 0 1 \<inter> T)"
-proof (cases "S = {}")
- case True
- with \<open>subspace U\<close> subspace_0 show ?thesis
- by (rule_tac T = "{0}" in that) auto
-next
- case False
- then obtain a where "a \<in> S"
- by auto
- then have affS: "aff_dim S = int (dim ((\<lambda>x. -a+x) ` S))"
- by (metis hull_inc aff_dim_eq_dim)
- with affSU have "dim ((\<lambda>x. -a+x) ` S) \<le> dim U"
- by linarith
- with choose_subspace_of_subspace
- obtain T where "subspace T" "T \<subseteq> span U" and dimT: "dim T = dim ((\<lambda>x. -a+x) ` S)" .
- show ?thesis
- proof (rule that [OF \<open>subspace T\<close>])
- show "T \<subseteq> U"
- using span_eq \<open>subspace U\<close> \<open>T \<subseteq> span U\<close> by blast
- show "aff_dim T = aff_dim S"
- using dimT \<open>subspace T\<close> affS aff_dim_subspace by fastforce
- show "rel_frontier S homeomorphic sphere 0 1 \<inter> T"
- proof -
- have "aff_dim (ball 0 1 \<inter> T) = aff_dim (T)"
- by (metis IntI interior_ball \<open>subspace T\<close> aff_dim_convex_Int_nonempty_interior centre_in_ball empty_iff inf_commute subspace_0 subspace_imp_convex zero_less_one)
- then have affS_eq: "aff_dim S = aff_dim (ball 0 1 \<inter> T)"
- using \<open>aff_dim T = aff_dim S\<close> by simp
- have "rel_frontier S homeomorphic rel_frontier(ball 0 1 \<inter> T)"
- apply (rule homeomorphic_rel_frontiers_convex_bounded_sets [OF \<open>convex S\<close> \<open>bounded S\<close>])
- apply (simp add: \<open>subspace T\<close> convex_Int subspace_imp_convex)
- apply (simp add: bounded_Int)
- apply (rule affS_eq)
- done
- also have "... = frontier (ball 0 1) \<inter> T"
- apply (rule convex_affine_rel_frontier_Int [OF convex_ball])
- apply (simp add: \<open>subspace T\<close> subspace_imp_affine)
- using \<open>subspace T\<close> subspace_0 by force
- also have "... = sphere 0 1 \<inter> T"
- by auto
- finally show ?thesis .
- qed
- qed
-qed
-
-
-proposition inessential_spheremap_lowdim_gen:
- fixes f :: "'M::euclidean_space \<Rightarrow> 'a::euclidean_space"
- assumes "convex S" "bounded S" "convex T" "bounded T"
- and affST: "aff_dim S < aff_dim T"
- and contf: "continuous_on (rel_frontier S) f"
- and fim: "f ` (rel_frontier S) \<subseteq> rel_frontier T"
- obtains c where "homotopic_with (\<lambda>z. True) (rel_frontier S) (rel_frontier T) f (\<lambda>x. c)"
-proof (cases "S = {}")
- case True
- then show ?thesis
- by (simp add: that)
-next
- case False
- then show ?thesis
- proof (cases "T = {}")
- case True
- then show ?thesis
- using fim that by auto
- next
- case False
- obtain T':: "'a set"
- where "subspace T'" and affT': "aff_dim T' = aff_dim T"
- and homT: "rel_frontier T homeomorphic sphere 0 1 \<inter> T'"
- apply (rule spheremap_lemma3 [OF \<open>bounded T\<close> \<open>convex T\<close> subspace_UNIV, where 'b='a])
- apply (simp add: dim_UNIV aff_dim_le_DIM)
- using \<open>T \<noteq> {}\<close> by blast
- with homeomorphic_imp_homotopy_eqv
- have relT: "sphere 0 1 \<inter> T' homotopy_eqv rel_frontier T"
- using homotopy_eqv_sym by blast
- have "aff_dim S \<le> int (dim T')"
- using affT' \<open>subspace T'\<close> affST aff_dim_subspace by force
- with spheremap_lemma3 [OF \<open>bounded S\<close> \<open>convex S\<close> \<open>subspace T'\<close>] \<open>S \<noteq> {}\<close>
- obtain S':: "'a set" where "subspace S'" "S' \<subseteq> T'"
- and affS': "aff_dim S' = aff_dim S"
- and homT: "rel_frontier S homeomorphic sphere 0 1 \<inter> S'"
- by metis
- with homeomorphic_imp_homotopy_eqv
- have relS: "sphere 0 1 \<inter> S' homotopy_eqv rel_frontier S"
- using homotopy_eqv_sym by blast
- have dimST': "dim S' < dim T'"
- by (metis \<open>S' \<subseteq> T'\<close> \<open>subspace S'\<close> \<open>subspace T'\<close> affS' affST affT' less_irrefl not_le subspace_dim_equal)
- have "\<exists>c. homotopic_with (\<lambda>z. True) (rel_frontier S) (rel_frontier T) f (\<lambda>x. c)"
- apply (rule homotopy_eqv_homotopic_triviality_null_imp [OF relT contf fim])
- apply (rule homotopy_eqv_cohomotopic_triviality_null[OF relS, THEN iffD1, rule_format])
- apply (metis dimST' \<open>subspace S'\<close> \<open>subspace T'\<close> \<open>S' \<subseteq> T'\<close> spheremap_lemma2, blast)
- done
- with that show ?thesis by blast
- qed
-qed
-
-lemma inessential_spheremap_lowdim:
- fixes f :: "'M::euclidean_space \<Rightarrow> 'a::euclidean_space"
- assumes
- "DIM('M) < DIM('a)" and f: "continuous_on (sphere a r) f" "f ` (sphere a r) \<subseteq> (sphere b s)"
- obtains c where "homotopic_with (\<lambda>z. True) (sphere a r) (sphere b s) f (\<lambda>x. c)"
-proof (cases "s \<le> 0")
- case True then show ?thesis
- by (meson nullhomotopic_into_contractible f contractible_sphere that)
-next
- case False
- show ?thesis
- proof (cases "r \<le> 0")
- case True then show ?thesis
- by (meson f nullhomotopic_from_contractible contractible_sphere that)
- next
- case False
- with \<open>~ s \<le> 0\<close> have "r > 0" "s > 0" by auto
- show ?thesis
- apply (rule inessential_spheremap_lowdim_gen [of "cball a r" "cball b s" f])
- using \<open>0 < r\<close> \<open>0 < s\<close> assms(1)
- apply (simp_all add: f aff_dim_cball)
- using that by blast
- qed
-qed
-
-
-
-subsection\<open> Some technical lemmas about extending maps from cell complexes.\<close>
-
-lemma extending_maps_Union_aux:
- assumes fin: "finite \<F>"
- and "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"
- and "\<And>S T. \<lbrakk>S \<in> \<F>; T \<in> \<F>; S \<noteq> T\<rbrakk> \<Longrightarrow> S \<inter> T \<subseteq> K"
- and "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>g. continuous_on S g \<and> g ` S \<subseteq> T \<and> (\<forall>x \<in> S \<inter> K. g x = h x)"
- shows "\<exists>g. continuous_on (\<Union>\<F>) g \<and> g ` (\<Union>\<F>) \<subseteq> T \<and> (\<forall>x \<in> \<Union>\<F> \<inter> K. g x = h x)"
-using assms
-proof (induction \<F>)
- case empty show ?case by simp
-next
- case (insert S \<F>)
- then obtain f where contf: "continuous_on (S) f" and fim: "f ` S \<subseteq> T" and feq: "\<forall>x \<in> S \<inter> K. f x = h x"
- by (meson insertI1)
- obtain g where contg: "continuous_on (\<Union>\<F>) g" and gim: "g ` \<Union>\<F> \<subseteq> T" and geq: "\<forall>x \<in> \<Union>\<F> \<inter> K. g x = h x"
- using insert by auto
- have fg: "f x = g x" if "x \<in> T" "T \<in> \<F>" "x \<in> S" for x T
- proof -
- have "T \<inter> S \<subseteq> K \<or> S = T"
- using that by (metis (no_types) insert.prems(2) insertCI)
- then show ?thesis
- using UnionI feq geq \<open>S \<notin> \<F>\<close> subsetD that by fastforce
- qed
- show ?case
- apply (rule_tac x="\<lambda>x. if x \<in> S then f x else g x" in exI, simp)
- apply (intro conjI continuous_on_cases)
- apply (simp_all add: insert closed_Union contf contg)
- using fim gim feq geq
- apply (force simp: insert closed_Union contf contg inf_commute intro: fg)+
- done
-qed
-
-lemma extending_maps_Union:
- assumes fin: "finite \<F>"
- and "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>g. continuous_on S g \<and> g ` S \<subseteq> T \<and> (\<forall>x \<in> S \<inter> K. g x = h x)"
- and "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"
- and K: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>; ~ X \<subseteq> Y; ~ Y \<subseteq> X\<rbrakk> \<Longrightarrow> X \<inter> Y \<subseteq> K"
- shows "\<exists>g. continuous_on (\<Union>\<F>) g \<and> g ` (\<Union>\<F>) \<subseteq> T \<and> (\<forall>x \<in> \<Union>\<F> \<inter> K. g x = h x)"
-apply (simp add: Union_maximal_sets [OF fin, symmetric])
-apply (rule extending_maps_Union_aux)
-apply (simp_all add: Union_maximal_sets [OF fin] assms)
-by (metis K psubsetI)
-
-
-lemma extend_map_lemma:
- assumes "finite \<F>" "\<G> \<subseteq> \<F>" "convex T" "bounded T"
- and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
- and aff: "\<And>X. X \<in> \<F> - \<G> \<Longrightarrow> aff_dim X < aff_dim T"
- and face: "\<And>S T. \<lbrakk>S \<in> \<F>; T \<in> \<F>\<rbrakk> \<Longrightarrow> (S \<inter> T) face_of S \<and> (S \<inter> T) face_of T"
- and contf: "continuous_on (\<Union>\<G>) f" and fim: "f ` (\<Union>\<G>) \<subseteq> rel_frontier T"
- obtains g where "continuous_on (\<Union>\<F>) g" "g ` (\<Union>\<F>) \<subseteq> rel_frontier T" "\<And>x. x \<in> \<Union>\<G> \<Longrightarrow> g x = f x"
-proof (cases "\<F> - \<G> = {}")
- case True
- then have "\<Union>\<F> \<subseteq> \<Union>\<G>"
- by (simp add: Union_mono)
- then show ?thesis
- apply (rule_tac g=f in that)
- using contf continuous_on_subset apply blast
- using fim apply blast
- by simp
-next
- case False
- then have "0 \<le> aff_dim T"
- by (metis aff aff_dim_empty aff_dim_geq aff_dim_negative_iff all_not_in_conv not_less)
- then obtain i::nat where i: "int i = aff_dim T"
- by (metis nonneg_eq_int)
- have Union_empty_eq: "\<Union>{D. D = {} \<and> P D} = {}" for P :: "'a set \<Rightarrow> bool"
- by auto
- have extendf: "\<exists>g. continuous_on (\<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < i})) g \<and>
- g ` (\<Union> (\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < i})) \<subseteq> rel_frontier T \<and>
- (\<forall>x \<in> \<Union>\<G>. g x = f x)"
- if "i \<le> aff_dim T" for i::nat
- using that
- proof (induction i)
- case 0 then show ?case
- apply (simp add: Union_empty_eq)
- apply (rule_tac x=f in exI)
- apply (intro conjI)
- using contf continuous_on_subset apply blast
- using fim apply blast
- by simp
- next
- case (Suc p)
- with \<open>bounded T\<close> have "rel_frontier T \<noteq> {}"
- by (auto simp: rel_frontier_eq_empty affine_bounded_eq_lowdim [of T])
- then obtain t where t: "t \<in> rel_frontier T" by auto
- have ple: "int p \<le> aff_dim T" using Suc.prems by force
- obtain h where conth: "continuous_on (\<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p})) h"
- and him: "h ` (\<Union> (\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p}))
- \<subseteq> rel_frontier T"
- and heq: "\<And>x. x \<in> \<Union>\<G> \<Longrightarrow> h x = f x"
- using Suc.IH [OF ple] by auto
- let ?Faces = "{D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D \<le> p}"
- have extendh: "\<exists>g. continuous_on D g \<and>
- g ` D \<subseteq> rel_frontier T \<and>
- (\<forall>x \<in> D \<inter> \<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p}). g x = h x)"
- if D: "D \<in> \<G> \<union> ?Faces" for D
- proof (cases "D \<subseteq> \<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p})")
- case True
- then show ?thesis
- apply (rule_tac x=h in exI)
- apply (intro conjI)
- apply (blast intro: continuous_on_subset [OF conth])
- using him apply blast
- by simp
- next
- case False
- note notDsub = False
- show ?thesis
- proof (cases "\<exists>a. D = {a}")
- case True
- then obtain a where "D = {a}" by auto
- with notDsub t show ?thesis
- by (rule_tac x="\<lambda>x. t" in exI) simp
- next
- case False
- have "D \<noteq> {}" using notDsub by auto
- have Dnotin: "D \<notin> \<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p}"
- using notDsub by auto
- then have "D \<notin> \<G>" by simp
- have "D \<in> ?Faces - {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p}"
- using Dnotin that by auto
- then obtain C where "C \<in> \<F>" "D face_of C" and affD: "aff_dim D = int p"
- by auto
- then have "bounded D"
- using face_of_polytope_polytope poly polytope_imp_bounded by blast
- then have [simp]: "\<not> affine D"
- using affine_bounded_eq_trivial False \<open>D \<noteq> {}\<close> \<open>bounded D\<close> by blast
- have "{F. F facet_of D} \<subseteq> {E. E face_of C \<and> aff_dim E < int p}"
- apply clarify
- apply (metis \<open>D face_of C\<close> affD eq_iff face_of_trans facet_of_def zle_diff1_eq)
- done
- moreover have "polyhedron D"
- using \<open>C \<in> \<F>\<close> \<open>D face_of C\<close> face_of_polytope_polytope poly polytope_imp_polyhedron by auto
- ultimately have relf_sub: "rel_frontier D \<subseteq> \<Union> {E. E face_of C \<and> aff_dim E < p}"
- by (simp add: rel_frontier_of_polyhedron Union_mono)
- then have him_relf: "h ` rel_frontier D \<subseteq> rel_frontier T"
- using \<open>C \<in> \<F>\<close> him by blast
- have "convex D"
- by (simp add: \<open>polyhedron D\<close> polyhedron_imp_convex)
- have affD_lessT: "aff_dim D < aff_dim T"
- using Suc.prems affD by linarith
- have contDh: "continuous_on (rel_frontier D) h"
- using \<open>C \<in> \<F>\<close> relf_sub by (blast intro: continuous_on_subset [OF conth])
- then have *: "(\<exists>c. homotopic_with (\<lambda>x. True) (rel_frontier D) (rel_frontier T) h (\<lambda>x. c)) =
- (\<exists>g. continuous_on UNIV g \<and> range g \<subseteq> rel_frontier T \<and>
- (\<forall>x\<in>rel_frontier D. g x = h x))"
- apply (rule nullhomotopic_into_rel_frontier_extension [OF closed_rel_frontier])
- apply (simp_all add: assms rel_frontier_eq_empty him_relf)
- done
- have "(\<exists>c. homotopic_with (\<lambda>x. True) (rel_frontier D)
- (rel_frontier T) h (\<lambda>x. c))"
- by (metis inessential_spheremap_lowdim_gen
- [OF \<open>convex D\<close> \<open>bounded D\<close> \<open>convex T\<close> \<open>bounded T\<close> affD_lessT contDh him_relf])
- then obtain g where contg: "continuous_on UNIV g"
- and gim: "range g \<subseteq> rel_frontier T"
- and gh: "\<And>x. x \<in> rel_frontier D \<Longrightarrow> g x = h x"
- by (metis *)
- have "D \<inter> E \<subseteq> rel_frontier D"
- if "E \<in> \<G> \<union> {D. Bex \<F> (op face_of D) \<and> aff_dim D < int p}" for E
- proof (rule face_of_subset_rel_frontier)
- show "D \<inter> E face_of D"
- using that \<open>C \<in> \<F>\<close> \<open>D face_of C\<close> face
- apply auto
- apply (meson face_of_Int_subface \<open>\<G> \<subseteq> \<F>\<close> face_of_refl_eq poly polytope_imp_convex subsetD)
- using face_of_Int_subface apply blast
- done
- show "D \<inter> E \<noteq> D"
- using that notDsub by auto
- qed
- then show ?thesis
- apply (rule_tac x=g in exI)
- apply (intro conjI ballI)
- using continuous_on_subset contg apply blast
- using gim apply blast
- using gh by fastforce
- qed
- qed
- have intle: "i < 1 + int j \<longleftrightarrow> i \<le> int j" for i j
- by auto
- have "finite \<G>"
- using \<open>finite \<F>\<close> \<open>\<G> \<subseteq> \<F>\<close> rev_finite_subset by blast
- then have fin: "finite (\<G> \<union> ?Faces)"
- apply simp
- apply (rule_tac B = "\<Union>{{D. D face_of C}| C. C \<in> \<F>}" in finite_subset)
- by (auto simp: \<open>finite \<F>\<close> finite_polytope_faces poly)
- have clo: "closed S" if "S \<in> \<G> \<union> ?Faces" for S
- using that \<open>\<G> \<subseteq> \<F>\<close> face_of_polytope_polytope poly polytope_imp_closed by blast
- have K: "X \<inter> Y \<subseteq> \<Union>(\<G> \<union> {D. \<exists>C\<in>\<F>. D face_of C \<and> aff_dim D < int p})"
- if "X \<in> \<G> \<union> ?Faces" "Y \<in> \<G> \<union> ?Faces" "~ Y \<subseteq> X" for X Y
- proof -
- have ff: "X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
- if XY: "X face_of D" "Y face_of E" and DE: "D \<in> \<F>" "E \<in> \<F>" for D E
- apply (rule face_of_Int_subface [OF _ _ XY])
- apply (auto simp: face DE)
- done
- show ?thesis
- using that
- apply auto
- apply (drule_tac x="X \<inter> Y" in spec, safe)
- using ff face_of_imp_convex [of X] face_of_imp_convex [of Y]
- apply (fastforce dest: face_of_aff_dim_lt)
- by (meson face_of_trans ff)
- qed
- obtain g where "continuous_on (\<Union>(\<G> \<union> ?Faces)) g"
- "g ` \<Union>(\<G> \<union> ?Faces) \<subseteq> rel_frontier T"
- "(\<forall>x \<in> \<Union>(\<G> \<union> ?Faces) \<inter>
- \<Union>(\<G> \<union> {D. \<exists>C\<in>\<F>. D face_of C \<and> aff_dim D < p}). g x = h x)"
- apply (rule exE [OF extending_maps_Union [OF fin extendh clo K]], blast+)
- done
- then show ?case
- apply (simp add: intle local.heq [symmetric], blast)
- done
- qed
- have eq: "\<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < i}) = \<Union>\<F>"
- proof
- show "\<Union>(\<G> \<union> {D. \<exists>C\<in>\<F>. D face_of C \<and> aff_dim D < int i}) \<subseteq> \<Union>\<F>"
- apply (rule Union_subsetI)
- using \<open>\<G> \<subseteq> \<F>\<close> face_of_imp_subset apply force
- done
- show "\<Union>\<F> \<subseteq> \<Union>(\<G> \<union> {D. \<exists>C\<in>\<F>. D face_of C \<and> aff_dim D < i})"
- apply (rule Union_mono)
- using face apply (fastforce simp: aff i)
- done
- qed
- have "int i \<le> aff_dim T" by (simp add: i)
- then show ?thesis
- using extendf [of i] unfolding eq by (metis that)
-qed
-
-lemma extend_map_lemma_cofinite0:
- assumes "finite \<F>"
- and "pairwise (\<lambda>S T. S \<inter> T \<subseteq> K) \<F>"
- and "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>a g. a \<notin> U \<and> continuous_on (S - {a}) g \<and> g ` (S - {a}) \<subseteq> T \<and> (\<forall>x \<in> S \<inter> K. g x = h x)"
- and "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"
- shows "\<exists>C g. finite C \<and> disjnt C U \<and> card C \<le> card \<F> \<and>
- continuous_on (\<Union>\<F> - C) g \<and> g ` (\<Union>\<F> - C) \<subseteq> T
- \<and> (\<forall>x \<in> (\<Union>\<F> - C) \<inter> K. g x = h x)"
- using assms
-proof induction
- case empty then show ?case
- by force
-next
- case (insert X \<F>)
- then have "closed X" and clo: "\<And>X. X \<in> \<F> \<Longrightarrow> closed X"
- and \<F>: "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>a g. a \<notin> U \<and> continuous_on (S - {a}) g \<and> g ` (S - {a}) \<subseteq> T \<and> (\<forall>x \<in> S \<inter> K. g x = h x)"
- and pwX: "\<And>Y. Y \<in> \<F> \<and> Y \<noteq> X \<longrightarrow> X \<inter> Y \<subseteq> K \<and> Y \<inter> X \<subseteq> K"
- and pwF: "pairwise (\<lambda> S T. S \<inter> T \<subseteq> K) \<F>"
- by (simp_all add: pairwise_insert)
- obtain C g where C: "finite C" "disjnt C U" "card C \<le> card \<F>"
- and contg: "continuous_on (\<Union>\<F> - C) g"
- and gim: "g ` (\<Union>\<F> - C) \<subseteq> T"
- and gh: "\<And>x. x \<in> (\<Union>\<F> - C) \<inter> K \<Longrightarrow> g x = h x"
- using insert.IH [OF pwF \<F> clo] by auto
- obtain a f where "a \<notin> U"
- and contf: "continuous_on (X - {a}) f"
- and fim: "f ` (X - {a}) \<subseteq> T"
- and fh: "(\<forall>x \<in> X \<inter> K. f x = h x)"
- using insert.prems by (meson insertI1)
- show ?case
- proof (intro exI conjI)
- show "finite (insert a C)"
- by (simp add: C)
- show "disjnt (insert a C) U"
- using C \<open>a \<notin> U\<close> by simp
- show "card (insert a C) \<le> card (insert X \<F>)"
- by (simp add: C card_insert_if insert.hyps le_SucI)
- have "closed (\<Union>\<F>)"
- using clo insert.hyps by blast
- have "continuous_on (X - insert a C \<union> (\<Union>\<F> - insert a C)) (\<lambda>x. if x \<in> X then f x else g x)"
- apply (rule continuous_on_cases_local)
- apply (simp_all add: closedin_closed)
- using \<open>closed X\<close> apply blast
- using \<open>closed (\<Union>\<F>)\<close> apply blast
- using contf apply (force simp: elim: continuous_on_subset)
- using contg apply (force simp: elim: continuous_on_subset)
- using fh gh insert.hyps pwX by fastforce
- then show "continuous_on (\<Union>insert X \<F> - insert a C) (\<lambda>a. if a \<in> X then f a else g a)"
- by (blast intro: continuous_on_subset)
- show "\<forall>x\<in>(\<Union>insert X \<F> - insert a C) \<inter> K. (if x \<in> X then f x else g x) = h x"
- using gh by (auto simp: fh)
- show "(\<lambda>a. if a \<in> X then f a else g a) ` (\<Union>insert X \<F> - insert a C) \<subseteq> T"
- using fim gim by auto force
- qed
-qed
-
-
-lemma extend_map_lemma_cofinite1:
-assumes "finite \<F>"
- and \<F>: "\<And>X. X \<in> \<F> \<Longrightarrow> \<exists>a g. a \<notin> U \<and> continuous_on (X - {a}) g \<and> g ` (X - {a}) \<subseteq> T \<and> (\<forall>x \<in> X \<inter> K. g x = h x)"
- and clo: "\<And>X. X \<in> \<F> \<Longrightarrow> closed X"
- and K: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>; ~(X \<subseteq> Y); ~(Y \<subseteq> X)\<rbrakk> \<Longrightarrow> X \<inter> Y \<subseteq> K"
- obtains C g where "finite C" "disjnt C U" "card C \<le> card \<F>" "continuous_on (\<Union>\<F> - C) g"
- "g ` (\<Union>\<F> - C) \<subseteq> T"
- "\<And>x. x \<in> (\<Union>\<F> - C) \<inter> K \<Longrightarrow> g x = h x"
-proof -
- let ?\<F> = "{X \<in> \<F>. \<forall>Y\<in>\<F>. \<not> X \<subset> Y}"
- have [simp]: "\<Union>?\<F> = \<Union>\<F>"
- by (simp add: Union_maximal_sets assms)
- have fin: "finite ?\<F>"
- by (force intro: finite_subset [OF _ \<open>finite \<F>\<close>])
- have pw: "pairwise (\<lambda> S T. S \<inter> T \<subseteq> K) ?\<F>"
- by (simp add: pairwise_def) (metis K psubsetI)
- have "card {X \<in> \<F>. \<forall>Y\<in>\<F>. \<not> X \<subset> Y} \<le> card \<F>"
- by (simp add: \<open>finite \<F>\<close> card_mono)
- moreover
- obtain C g where "finite C \<and> disjnt C U \<and> card C \<le> card ?\<F> \<and>
- continuous_on (\<Union>?\<F> - C) g \<and> g ` (\<Union>?\<F> - C) \<subseteq> T
- \<and> (\<forall>x \<in> (\<Union>?\<F> - C) \<inter> K. g x = h x)"
- apply (rule exE [OF extend_map_lemma_cofinite0 [OF fin pw, of U T h]])
- apply (fastforce intro!: clo \<F>)+
- done
- ultimately show ?thesis
- by (rule_tac C=C and g=g in that) auto
-qed
-
-
-lemma extend_map_lemma_cofinite:
- assumes "finite \<F>" "\<G> \<subseteq> \<F>" and T: "convex T" "bounded T"
- and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
- and contf: "continuous_on (\<Union>\<G>) f" and fim: "f ` (\<Union>\<G>) \<subseteq> rel_frontier T"
- and face: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> (X \<inter> Y) face_of X \<and> (X \<inter> Y) face_of Y"
- and aff: "\<And>X. X \<in> \<F> - \<G> \<Longrightarrow> aff_dim X \<le> aff_dim T"
- obtains C g where
- "finite C" "disjnt C (\<Union>\<G>)" "card C \<le> card \<F>" "continuous_on (\<Union>\<F> - C) g"
- "g ` (\<Union> \<F> - C) \<subseteq> rel_frontier T" "\<And>x. x \<in> \<Union>\<G> \<Longrightarrow> g x = f x"
-proof -
- define \<H> where "\<H> \<equiv> \<G> \<union> {D. \<exists>C \<in> \<F> - \<G>. D face_of C \<and> aff_dim D < aff_dim T}"
- have "finite \<G>"
- using assms finite_subset by blast
- moreover have "finite (\<Union>{{D. D face_of C} |C. C \<in> \<F>})"
- apply (rule finite_Union)
- apply (simp add: \<open>finite \<F>\<close>)
- using finite_polytope_faces poly by auto
- ultimately have "finite \<H>"
- apply (simp add: \<H>_def)
- apply (rule finite_subset [of _ "\<Union> {{D. D face_of C} | C. C \<in> \<F>}"], auto)
- done
- have *: "\<And>X Y. \<lbrakk>X \<in> \<H>; Y \<in> \<H>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
- unfolding \<H>_def
- apply (elim UnE bexE CollectE DiffE)
- using subsetD [OF \<open>\<G> \<subseteq> \<F>\<close>] apply (simp_all add: face)
- apply (meson subsetD [OF \<open>\<G> \<subseteq> \<F>\<close>] face face_of_Int_subface face_of_imp_subset face_of_refl poly polytope_imp_convex)+
- done
- obtain h where conth: "continuous_on (\<Union>\<H>) h" and him: "h ` (\<Union>\<H>) \<subseteq> rel_frontier T"
- and hf: "\<And>x. x \<in> \<Union>\<G> \<Longrightarrow> h x = f x"
- using \<open>finite \<H>\<close>
- unfolding \<H>_def
- apply (rule extend_map_lemma [OF _ Un_upper1 T _ _ _ contf fim])
- using \<open>\<G> \<subseteq> \<F>\<close> face_of_polytope_polytope poly apply fastforce
- using * apply (auto simp: \<H>_def)
- done
- have "bounded (\<Union>\<G>)"
- using \<open>finite \<G>\<close> \<open>\<G> \<subseteq> \<F>\<close> poly polytope_imp_bounded by blast
- then have "\<Union>\<G> \<noteq> UNIV"
- by auto
- then obtain a where a: "a \<notin> \<Union>\<G>"
- by blast
- have \<F>: "\<exists>a g. a \<notin> \<Union>\<G> \<and> continuous_on (D - {a}) g \<and>
- g ` (D - {a}) \<subseteq> rel_frontier T \<and> (\<forall>x \<in> D \<inter> \<Union>\<H>. g x = h x)"
- if "D \<in> \<F>" for D
- proof (cases "D \<subseteq> \<Union>\<H>")
- case True
- then show ?thesis
- apply (rule_tac x=a in exI)
- apply (rule_tac x=h in exI)
- using him apply (blast intro!: \<open>a \<notin> \<Union>\<G>\<close> continuous_on_subset [OF conth]) +
- done
- next
- case False
- note D_not_subset = False
- show ?thesis
- proof (cases "D \<in> \<G>")
- case True
- with D_not_subset show ?thesis
- by (auto simp: \<H>_def)
- next
- case False
- then have affD: "aff_dim D \<le> aff_dim T"
- by (simp add: \<open>D \<in> \<F>\<close> aff)
- show ?thesis
- proof (cases "rel_interior D = {}")
- case True
- with \<open>D \<in> \<F>\<close> poly a show ?thesis
- by (force simp: rel_interior_eq_empty polytope_imp_convex)
- next
- case False
- then obtain b where brelD: "b \<in> rel_interior D"
- by blast
- have "polyhedron D"
- by (simp add: poly polytope_imp_polyhedron that)
- have "rel_frontier D retract_of affine hull D - {b}"
- by (simp add: rel_frontier_retract_of_punctured_affine_hull poly polytope_imp_bounded polytope_imp_convex that brelD)
- then obtain r where relfD: "rel_frontier D \<subseteq> affine hull D - {b}"
- and contr: "continuous_on (affine hull D - {b}) r"
- and rim: "r ` (affine hull D - {b}) \<subseteq> rel_frontier D"
- and rid: "\<And>x. x \<in> rel_frontier D \<Longrightarrow> r x = x"
- by (auto simp: retract_of_def retraction_def)
- show ?thesis
- proof (intro exI conjI ballI)
- show "b \<notin> \<Union>\<G>"
- proof clarify
- fix E
- assume "b \<in> E" "E \<in> \<G>"
- then have "E \<inter> D face_of E \<and> E \<inter> D face_of D"
- using \<open>\<G> \<subseteq> \<F>\<close> face that by auto
- with face_of_subset_rel_frontier \<open>E \<in> \<G>\<close> \<open>b \<in> E\<close> brelD rel_interior_subset [of D]
- D_not_subset rel_frontier_def \<H>_def
- show False
- by blast
- qed
- have "r ` (D - {b}) \<subseteq> r ` (affine hull D - {b})"
- by (simp add: Diff_mono hull_subset image_mono)
- also have "... \<subseteq> rel_frontier D"
- by (rule rim)
- also have "... \<subseteq> \<Union>{E. E face_of D \<and> aff_dim E < aff_dim T}"
- using affD
- by (force simp: rel_frontier_of_polyhedron [OF \<open>polyhedron D\<close>] facet_of_def)
- also have "... \<subseteq> \<Union>(\<H>)"
- using D_not_subset \<H>_def that by fastforce
- finally have rsub: "r ` (D - {b}) \<subseteq> \<Union>(\<H>)" .
- show "continuous_on (D - {b}) (h \<circ> r)"
- apply (intro conjI \<open>b \<notin> \<Union>\<G>\<close> continuous_on_compose)
- apply (rule continuous_on_subset [OF contr])
- apply (simp add: Diff_mono hull_subset)
- apply (rule continuous_on_subset [OF conth rsub])
- done
- show "(h \<circ> r) ` (D - {b}) \<subseteq> rel_frontier T"
- using brelD him rsub by fastforce
- show "(h \<circ> r) x = h x" if x: "x \<in> D \<inter> \<Union>\<H>" for x
- proof -
- consider A where "x \<in> D" "A \<in> \<G>" "x \<in> A"
- | A B where "x \<in> D" "A face_of B" "B \<in> \<F>" "B \<notin> \<G>" "aff_dim A < aff_dim T" "x \<in> A"
- using x by (auto simp: \<H>_def)
- then have xrel: "x \<in> rel_frontier D"
- proof cases
- case 1 show ?thesis
- proof (rule face_of_subset_rel_frontier [THEN subsetD])
- show "D \<inter> A face_of D"
- using \<open>A \<in> \<G>\<close> \<open>\<G> \<subseteq> \<F>\<close> face \<open>D \<in> \<F>\<close> by blast
- show "D \<inter> A \<noteq> D"
- using \<open>A \<in> \<G>\<close> D_not_subset \<H>_def by blast
- qed (auto simp: 1)
- next
- case 2 show ?thesis
- proof (rule face_of_subset_rel_frontier [THEN subsetD])
- show "D \<inter> A face_of D"
- apply (rule face_of_Int_subface [of D B _ A, THEN conjunct1])
- apply (simp_all add: 2 \<open>D \<in> \<F>\<close> face)
- apply (simp add: \<open>polyhedron D\<close> polyhedron_imp_convex face_of_refl)
- done
- show "D \<inter> A \<noteq> D"
- using "2" D_not_subset \<H>_def by blast
- qed (auto simp: 2)
- qed
- show ?thesis
- by (simp add: rid xrel)
- qed
- qed
- qed
- qed
- qed
- have clo: "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"
- by (simp add: poly polytope_imp_closed)
- obtain C g where "finite C" "disjnt C (\<Union>\<G>)" "card C \<le> card \<F>" "continuous_on (\<Union>\<F> - C) g"
- "g ` (\<Union>\<F> - C) \<subseteq> rel_frontier T"
- and gh: "\<And>x. x \<in> (\<Union>\<F> - C) \<inter> \<Union>\<H> \<Longrightarrow> g x = h x"
- proof (rule extend_map_lemma_cofinite1 [OF \<open>finite \<F>\<close> \<F> clo])
- show "X \<inter> Y \<subseteq> \<Union>\<H>" if XY: "X \<in> \<F>" "Y \<in> \<F>" and "\<not> X \<subseteq> Y" "\<not> Y \<subseteq> X" for X Y
- proof (cases "X \<in> \<G>")
- case True
- then show ?thesis
- by (auto simp: \<H>_def)
- next
- case False
- have "X \<inter> Y \<noteq> X"
- using \<open>\<not> X \<subseteq> Y\<close> by blast
- with XY
- show ?thesis
- by (clarsimp simp: \<H>_def)
- (metis Diff_iff Int_iff aff antisym_conv face face_of_aff_dim_lt face_of_refl
- not_le poly polytope_imp_convex)
- qed
- qed (blast)+
- with \<open>\<G> \<subseteq> \<F>\<close> show ?thesis
- apply (rule_tac C=C and g=g in that)
- apply (auto simp: disjnt_def hf [symmetric] \<H>_def intro!: gh)
- done
-qed
-
-text\<open>The next two proofs are similar\<close>
-theorem extend_map_cell_complex_to_sphere:
- assumes "finite \<F>" and S: "S \<subseteq> \<Union>\<F>" "closed S" and T: "convex T" "bounded T"
- and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
- and aff: "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X < aff_dim T"
- and face: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> (X \<inter> Y) face_of X \<and> (X \<inter> Y) face_of Y"
- and contf: "continuous_on S f" and fim: "f ` S \<subseteq> rel_frontier T"
- obtains g where "continuous_on (\<Union>\<F>) g"
- "g ` (\<Union>\<F>) \<subseteq> rel_frontier T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-proof -
- obtain V g where "S \<subseteq> V" "open V" "continuous_on V g" and gim: "g ` V \<subseteq> rel_frontier T" and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
- using neighbourhood_extension_into_ANR [OF contf fim _ \<open>closed S\<close>] ANR_rel_frontier_convex T by blast
- have "compact S"
- by (meson assms compact_Union poly polytope_imp_compact seq_compact_closed_subset seq_compact_eq_compact)
- then obtain d where "d > 0" and d: "\<And>x y. \<lbrakk>x \<in> S; y \<in> - V\<rbrakk> \<Longrightarrow> d \<le> dist x y"
- using separate_compact_closed [of S "-V"] \<open>open V\<close> \<open>S \<subseteq> V\<close> by force
- obtain \<G> where "finite \<G>" "\<Union>\<G> = \<Union>\<F>"
- and diaG: "\<And>X. X \<in> \<G> \<Longrightarrow> diameter X < d"
- and polyG: "\<And>X. X \<in> \<G> \<Longrightarrow> polytope X"
- and affG: "\<And>X. X \<in> \<G> \<Longrightarrow> aff_dim X \<le> aff_dim T - 1"
- and faceG: "\<And>X Y. \<lbrakk>X \<in> \<G>; Y \<in> \<G>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
- proof (rule cell_complex_subdivision_exists [OF \<open>d>0\<close> \<open>finite \<F>\<close> poly _ face])
- show "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X \<le> aff_dim T - 1"
- by (simp add: aff)
- qed auto
- obtain h where conth: "continuous_on (\<Union>\<G>) h" and him: "h ` \<Union>\<G> \<subseteq> rel_frontier T" and hg: "\<And>x. x \<in> \<Union>(\<G> \<inter> Pow V) \<Longrightarrow> h x = g x"
- proof (rule extend_map_lemma [of \<G> "\<G> \<inter> Pow V" T g])
- show "continuous_on (\<Union>(\<G> \<inter> Pow V)) g"
- by (metis Union_Int_subset Union_Pow_eq \<open>continuous_on V g\<close> continuous_on_subset le_inf_iff)
- qed (use \<open>finite \<G>\<close> T polyG affG faceG gim in fastforce)+
- show ?thesis
- proof
- show "continuous_on (\<Union>\<F>) h"
- using \<open>\<Union>\<G> = \<Union>\<F>\<close> conth by auto
- show "h ` \<Union>\<F> \<subseteq> rel_frontier T"
- using \<open>\<Union>\<G> = \<Union>\<F>\<close> him by auto
- show "h x = f x" if "x \<in> S" for x
- proof -
- have "x \<in> \<Union>\<G>"
- using \<open>\<Union>\<G> = \<Union>\<F>\<close> \<open>S \<subseteq> \<Union>\<F>\<close> that by auto
- then obtain X where "x \<in> X" "X \<in> \<G>" by blast
- then have "diameter X < d" "bounded X"
- by (auto simp: diaG \<open>X \<in> \<G>\<close> polyG polytope_imp_bounded)
- then have "X \<subseteq> V" using d [OF \<open>x \<in> S\<close>] diameter_bounded_bound [OF \<open>bounded X\<close> \<open>x \<in> X\<close>]
- by fastforce
- have "h x = g x"
- apply (rule hg)
- using \<open>X \<in> \<G>\<close> \<open>X \<subseteq> V\<close> \<open>x \<in> X\<close> by blast
- also have "... = f x"
- by (simp add: gf that)
- finally show "h x = f x" .
- qed
- qed
-qed
-
-
-theorem extend_map_cell_complex_to_sphere_cofinite:
- assumes "finite \<F>" and S: "S \<subseteq> \<Union>\<F>" "closed S" and T: "convex T" "bounded T"
- and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
- and aff: "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X \<le> aff_dim T"
- and face: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> (X \<inter> Y) face_of X \<and> (X \<inter> Y) face_of Y"
- and contf: "continuous_on S f" and fim: "f ` S \<subseteq> rel_frontier T"
- obtains C g where "finite C" "disjnt C S" "continuous_on (\<Union>\<F> - C) g"
- "g ` (\<Union>\<F> - C) \<subseteq> rel_frontier T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-proof -
- obtain V g where "S \<subseteq> V" "open V" "continuous_on V g" and gim: "g ` V \<subseteq> rel_frontier T" and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
- using neighbourhood_extension_into_ANR [OF contf fim _ \<open>closed S\<close>] ANR_rel_frontier_convex T by blast
- have "compact S"
- by (meson assms compact_Union poly polytope_imp_compact seq_compact_closed_subset seq_compact_eq_compact)
- then obtain d where "d > 0" and d: "\<And>x y. \<lbrakk>x \<in> S; y \<in> - V\<rbrakk> \<Longrightarrow> d \<le> dist x y"
- using separate_compact_closed [of S "-V"] \<open>open V\<close> \<open>S \<subseteq> V\<close> by force
- obtain \<G> where "finite \<G>" "\<Union>\<G> = \<Union>\<F>"
- and diaG: "\<And>X. X \<in> \<G> \<Longrightarrow> diameter X < d"
- and polyG: "\<And>X. X \<in> \<G> \<Longrightarrow> polytope X"
- and affG: "\<And>X. X \<in> \<G> \<Longrightarrow> aff_dim X \<le> aff_dim T"
- and faceG: "\<And>X Y. \<lbrakk>X \<in> \<G>; Y \<in> \<G>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
- by (rule cell_complex_subdivision_exists [OF \<open>d>0\<close> \<open>finite \<F>\<close> poly aff face]) auto
- obtain C h where "finite C" and dis: "disjnt C (\<Union>(\<G> \<inter> Pow V))"
- and card: "card C \<le> card \<G>" and conth: "continuous_on (\<Union>\<G> - C) h"
- and him: "h ` (\<Union>\<G> - C) \<subseteq> rel_frontier T"
- and hg: "\<And>x. x \<in> \<Union>(\<G> \<inter> Pow V) \<Longrightarrow> h x = g x"
- proof (rule extend_map_lemma_cofinite [of \<G> "\<G> \<inter> Pow V" T g])
- show "continuous_on (\<Union>(\<G> \<inter> Pow V)) g"
- by (metis Union_Int_subset Union_Pow_eq \<open>continuous_on V g\<close> continuous_on_subset le_inf_iff)
- show "g ` \<Union>(\<G> \<inter> Pow V) \<subseteq> rel_frontier T"
- using gim by force
- qed (auto intro: \<open>finite \<G>\<close> T polyG affG dest: faceG)
- have Ssub: "S \<subseteq> \<Union>(\<G> \<inter> Pow V)"
- proof
- fix x
- assume "x \<in> S"
- then have "x \<in> \<Union>\<G>"
- using \<open>\<Union>\<G> = \<Union>\<F>\<close> \<open>S \<subseteq> \<Union>\<F>\<close> by auto
- then obtain X where "x \<in> X" "X \<in> \<G>" by blast
- then have "diameter X < d" "bounded X"
- by (auto simp: diaG \<open>X \<in> \<G>\<close> polyG polytope_imp_bounded)
- then have "X \<subseteq> V" using d [OF \<open>x \<in> S\<close>] diameter_bounded_bound [OF \<open>bounded X\<close> \<open>x \<in> X\<close>]
- by fastforce
- then show "x \<in> \<Union>(\<G> \<inter> Pow V)"
- using \<open>X \<in> \<G>\<close> \<open>x \<in> X\<close> by blast
- qed
- show ?thesis
- proof
- show "continuous_on (\<Union>\<F>-C) h"
- using \<open>\<Union>\<G> = \<Union>\<F>\<close> conth by auto
- show "h ` (\<Union>\<F> - C) \<subseteq> rel_frontier T"
- using \<open>\<Union>\<G> = \<Union>\<F>\<close> him by auto
- show "h x = f x" if "x \<in> S" for x
- proof -
- have "h x = g x"
- apply (rule hg)
- using Ssub that by blast
- also have "... = f x"
- by (simp add: gf that)
- finally show "h x = f x" .
- qed
- show "disjnt C S"
- using dis Ssub by (meson disjnt_iff subset_eq)
- qed (intro \<open>finite C\<close>)
-qed
-
-
-
-subsection\<open> Special cases and corollaries involving spheres.\<close>
-
-lemma disjnt_Diff1: "X \<subseteq> Y' \<Longrightarrow> disjnt (X - Y) (X' - Y')"
- by (auto simp: disjnt_def)
-
-proposition extend_map_affine_to_sphere_cofinite_simple:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "compact S" "convex U" "bounded U"
- and aff: "aff_dim T \<le> aff_dim U"
- and "S \<subseteq> T" and contf: "continuous_on S f"
- and fim: "f ` S \<subseteq> rel_frontier U"
- obtains K g where "finite K" "K \<subseteq> T" "disjnt K S" "continuous_on (T - K) g"
- "g ` (T - K) \<subseteq> rel_frontier U"
- "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-proof -
- have "\<exists>K g. finite K \<and> disjnt K S \<and> continuous_on (T - K) g \<and>
- g ` (T - K) \<subseteq> rel_frontier U \<and> (\<forall>x \<in> S. g x = f x)"
- if "affine T" "S \<subseteq> T" and aff: "aff_dim T \<le> aff_dim U" for T
- proof (cases "S = {}")
- case True
- show ?thesis
- proof (cases "rel_frontier U = {}")
- case True
- with \<open>bounded U\<close> have "aff_dim U \<le> 0"
- using affine_bounded_eq_lowdim rel_frontier_eq_empty by auto
- with aff have "aff_dim T \<le> 0" by auto
- then obtain a where "T \<subseteq> {a}"
- using \<open>affine T\<close> affine_bounded_eq_lowdim affine_bounded_eq_trivial by auto
- then show ?thesis
- using \<open>S = {}\<close> fim
- by (metis Diff_cancel contf disjnt_empty2 finite.emptyI finite_insert finite_subset)
- next
- case False
- then obtain a where "a \<in> rel_frontier U"
- by auto
- then show ?thesis
- using continuous_on_const [of _ a] \<open>S = {}\<close> by force
- qed
- next
- case False
- have "bounded S"
- by (simp add: \<open>compact S\<close> compact_imp_bounded)
- then obtain b where b: "S \<subseteq> cbox (-b) b"
- using bounded_subset_cbox_symmetric by blast
- define bbox where "bbox \<equiv> cbox (-(b+One)) (b+One)"
- have "cbox (-b) b \<subseteq> bbox"
- by (auto simp: bbox_def algebra_simps intro!: subset_box_imp)
- with b \<open>S \<subseteq> T\<close> have "S \<subseteq> bbox \<inter> T"
- by auto
- then have Ssub: "S \<subseteq> \<Union>{bbox \<inter> T}"
- by auto
- then have "aff_dim (bbox \<inter> T) \<le> aff_dim U"
- by (metis aff aff_dim_subset inf_commute inf_le1 order_trans)
- obtain K g where K: "finite K" "disjnt K S"
- and contg: "continuous_on (\<Union>{bbox \<inter> T} - K) g"
- and gim: "g ` (\<Union>{bbox \<inter> T} - K) \<subseteq> rel_frontier U"
- and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
- proof (rule extend_map_cell_complex_to_sphere_cofinite
- [OF _ Ssub _ \<open>convex U\<close> \<open>bounded U\<close> _ _ _ contf fim])
- show "closed S"
- using \<open>compact S\<close> compact_eq_bounded_closed by auto
- show poly: "\<And>X. X \<in> {bbox \<inter> T} \<Longrightarrow> polytope X"
- by (simp add: polytope_Int_polyhedron bbox_def polytope_interval affine_imp_polyhedron \<open>affine T\<close>)
- show "\<And>X Y. \<lbrakk>X \<in> {bbox \<inter> T}; Y \<in> {bbox \<inter> T}\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
- by (simp add:poly face_of_refl polytope_imp_convex)
- show "\<And>X. X \<in> {bbox \<inter> T} \<Longrightarrow> aff_dim X \<le> aff_dim U"
- by (simp add: \<open>aff_dim (bbox \<inter> T) \<le> aff_dim U\<close>)
- qed auto
- define fro where "fro \<equiv> \<lambda>d. frontier(cbox (-(b + d *\<^sub>R One)) (b + d *\<^sub>R One))"
- obtain d where d12: "1/2 \<le> d" "d \<le> 1" and dd: "disjnt K (fro d)"
- proof (rule disjoint_family_elem_disjnt [OF _ \<open>finite K\<close>])
- show "infinite {1/2..1::real}"
- by (simp add: infinite_Icc)
- have dis1: "disjnt (fro x) (fro y)" if "x<y" for x y
- by (auto simp: algebra_simps that subset_box_imp disjnt_Diff1 frontier_def fro_def)
- then show "disjoint_family_on fro {1/2..1}"
- by (auto simp: disjoint_family_on_def disjnt_def neq_iff)
- qed auto
- define c where "c \<equiv> b + d *\<^sub>R One"
- have cbsub: "cbox (-b) b \<subseteq> box (-c) c" "cbox (-b) b \<subseteq> cbox (-c) c" "cbox (-c) c \<subseteq> bbox"
- using d12 by (auto simp: algebra_simps subset_box_imp c_def bbox_def)
- have clo_cbT: "closed (cbox (- c) c \<inter> T)"
- by (simp add: affine_closed closed_Int closed_cbox \<open>affine T\<close>)
- have cpT_ne: "cbox (- c) c \<inter> T \<noteq> {}"
- using \<open>S \<noteq> {}\<close> b cbsub(2) \<open>S \<subseteq> T\<close> by fastforce
- have "closest_point (cbox (- c) c \<inter> T) x \<notin> K" if "x \<in> T" "x \<notin> K" for x
- proof (cases "x \<in> cbox (-c) c")
- case True with that show ?thesis
- by (simp add: closest_point_self)
- next
- case False
- have int_ne: "interior (cbox (-c) c) \<inter> T \<noteq> {}"
- using \<open>S \<noteq> {}\<close> \<open>S \<subseteq> T\<close> b \<open>cbox (- b) b \<subseteq> box (- c) c\<close> by force
- have "convex T"
- by (meson \<open>affine T\<close> affine_imp_convex)
- then have "x \<in> affine hull (cbox (- c) c \<inter> T)"
- by (metis Int_commute Int_iff \<open>S \<noteq> {}\<close> \<open>S \<subseteq> T\<close> cbsub(1) \<open>x \<in> T\<close> affine_hull_convex_Int_nonempty_interior all_not_in_conv b hull_inc inf.orderE interior_cbox)
- then have "x \<in> affine hull (cbox (- c) c \<inter> T) - rel_interior (cbox (- c) c \<inter> T)"
- by (meson DiffI False Int_iff rel_interior_subset subsetCE)
- then have "closest_point (cbox (- c) c \<inter> T) x \<in> rel_frontier (cbox (- c) c \<inter> T)"
- by (rule closest_point_in_rel_frontier [OF clo_cbT cpT_ne])
- moreover have "(rel_frontier (cbox (- c) c \<inter> T)) \<subseteq> fro d"
- apply (subst convex_affine_rel_frontier_Int [OF _ \<open>affine T\<close> int_ne])
- apply (auto simp: fro_def c_def)
- done
- ultimately show ?thesis
- using dd by (force simp: disjnt_def)
- qed
- then have cpt_subset: "closest_point (cbox (- c) c \<inter> T) ` (T - K) \<subseteq> \<Union>{bbox \<inter> T} - K"
- using closest_point_in_set [OF clo_cbT cpT_ne] cbsub(3) by force
- show ?thesis
- proof (intro conjI ballI exI)
- have "continuous_on (T - K) (closest_point (cbox (- c) c \<inter> T))"
- apply (rule continuous_on_closest_point)
- using \<open>S \<noteq> {}\<close> cbsub(2) b that
- by (auto simp: affine_imp_convex convex_Int affine_closed closed_Int closed_cbox \<open>affine T\<close>)
- then show "continuous_on (T - K) (g \<circ> closest_point (cbox (- c) c \<inter> T))"
- by (metis continuous_on_compose continuous_on_subset [OF contg cpt_subset])
- have "(g \<circ> closest_point (cbox (- c) c \<inter> T)) ` (T - K) \<subseteq> g ` (\<Union>{bbox \<inter> T} - K)"
- by (metis image_comp image_mono cpt_subset)
- also have "... \<subseteq> rel_frontier U"
- by (rule gim)
- finally show "(g \<circ> closest_point (cbox (- c) c \<inter> T)) ` (T - K) \<subseteq> rel_frontier U" .
- show "(g \<circ> closest_point (cbox (- c) c \<inter> T)) x = f x" if "x \<in> S" for x
- proof -
- have "(g \<circ> closest_point (cbox (- c) c \<inter> T)) x = g x"
- unfolding o_def
- by (metis IntI \<open>S \<subseteq> T\<close> b cbsub(2) closest_point_self subset_eq that)
- also have "... = f x"
- by (simp add: that gf)
- finally show ?thesis .
- qed
- qed (auto simp: K)
- qed
- then obtain K g where "finite K" "disjnt K S"
- and contg: "continuous_on (affine hull T - K) g"
- and gim: "g ` (affine hull T - K) \<subseteq> rel_frontier U"
- and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
- by (metis aff affine_affine_hull aff_dim_affine_hull
- order_trans [OF \<open>S \<subseteq> T\<close> hull_subset [of T affine]])
- then obtain K g where "finite K" "disjnt K S"
- and contg: "continuous_on (T - K) g"
- and gim: "g ` (T - K) \<subseteq> rel_frontier U"
- and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
- by (rule_tac K=K and g=g in that) (auto simp: hull_inc elim: continuous_on_subset)
- then show ?thesis
- by (rule_tac K="K \<inter> T" and g=g in that) (auto simp: disjnt_iff Diff_Int contg)
-qed
-
-subsection\<open>Extending maps to spheres\<close>
-
-(*Up to extend_map_affine_to_sphere_cofinite_gen*)
-
-lemma closedin_closed_subset:
- "\<lbrakk>closedin (subtopology euclidean U) V; T \<subseteq> U; S = V \<inter> T\<rbrakk>
- \<Longrightarrow> closedin (subtopology euclidean T) S"
- by (metis (no_types, lifting) Int_assoc Int_commute closedin_closed inf.orderE)
-
-lemma extend_map_affine_to_sphere1:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::topological_space"
- assumes "finite K" "affine U" and contf: "continuous_on (U - K) f"
- and fim: "f ` (U - K) \<subseteq> T"
- and comps: "\<And>C. \<lbrakk>C \<in> components(U - S); C \<inter> K \<noteq> {}\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
- and clo: "closedin (subtopology euclidean U) S" and K: "disjnt K S" "K \<subseteq> U"
- obtains g where "continuous_on (U - L) g" "g ` (U - L) \<subseteq> T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-proof (cases "K = {}")
- case True
- then show ?thesis
- by (metis Diff_empty Diff_subset contf fim continuous_on_subset image_subsetI rev_image_eqI subset_iff that)
-next
- case False
- have "S \<subseteq> U"
- using clo closedin_limpt by blast
- then have "(U - S) \<inter> K \<noteq> {}"
- by (metis Diff_triv False Int_Diff K disjnt_def inf.absorb_iff2 inf_commute)
- then have "\<Union>(components (U - S)) \<inter> K \<noteq> {}"
- using Union_components by simp
- then obtain C0 where C0: "C0 \<in> components (U - S)" "C0 \<inter> K \<noteq> {}"
- by blast
- have "convex U"
- by (simp add: affine_imp_convex \<open>affine U\<close>)
- then have "locally connected U"
- by (rule convex_imp_locally_connected)
- have "\<exists>a g. a \<in> C \<and> a \<in> L \<and> continuous_on (S \<union> (C - {a})) g \<and>
- g ` (S \<union> (C - {a})) \<subseteq> T \<and> (\<forall>x \<in> S. g x = f x)"
- if C: "C \<in> components (U - S)" and CK: "C \<inter> K \<noteq> {}" for C
- proof -
- have "C \<subseteq> U-S" "C \<inter> L \<noteq> {}"
- by (simp_all add: in_components_subset comps that)
- then obtain a where a: "a \<in> C" "a \<in> L" by auto
- have opeUC: "openin (subtopology euclidean U) C"
- proof (rule openin_trans)
- show "openin (subtopology euclidean (U-S)) C"
- by (simp add: \<open>locally connected U\<close> clo locally_diff_closed openin_components_locally_connected [OF _ C])
- show "openin (subtopology euclidean U) (U - S)"
- by (simp add: clo openin_diff)
- qed
- then obtain d where "C \<subseteq> U" "0 < d" and d: "cball a d \<inter> U \<subseteq> C"
- using openin_contains_cball by (metis \<open>a \<in> C\<close>)
- then have "ball a d \<inter> U \<subseteq> C"
- by auto
- obtain h k where homhk: "homeomorphism (S \<union> C) (S \<union> C) h k"
- and subC: "{x. (~ (h x = x \<and> k x = x))} \<subseteq> C"
- and bou: "bounded {x. (~ (h x = x \<and> k x = x))}"
- and hin: "\<And>x. x \<in> C \<inter> K \<Longrightarrow> h x \<in> ball a d \<inter> U"
- proof (rule homeomorphism_grouping_points_exists_gen [of C "ball a d \<inter> U" "C \<inter> K" "S \<union> C"])
- show "openin (subtopology euclidean C) (ball a d \<inter> U)"
- by (metis Topology_Euclidean_Space.open_ball \<open>C \<subseteq> U\<close> \<open>ball a d \<inter> U \<subseteq> C\<close> inf.absorb_iff2 inf.orderE inf_assoc open_openin openin_subtopology)
- show "openin (subtopology euclidean (affine hull C)) C"
- by (metis \<open>a \<in> C\<close> \<open>openin (subtopology euclidean U) C\<close> affine_hull_eq affine_hull_openin all_not_in_conv \<open>affine U\<close>)
- show "ball a d \<inter> U \<noteq> {}"
- using \<open>0 < d\<close> \<open>C \<subseteq> U\<close> \<open>a \<in> C\<close> by force
- show "finite (C \<inter> K)"
- by (simp add: \<open>finite K\<close>)
- show "S \<union> C \<subseteq> affine hull C"
- by (metis \<open>C \<subseteq> U\<close> \<open>S \<subseteq> U\<close> \<open>a \<in> C\<close> opeUC affine_hull_eq affine_hull_openin all_not_in_conv assms(2) sup.bounded_iff)
- show "connected C"
- by (metis C in_components_connected)
- qed auto
- have a_BU: "a \<in> ball a d \<inter> U"
- using \<open>0 < d\<close> \<open>C \<subseteq> U\<close> \<open>a \<in> C\<close> by auto
- have "rel_frontier (cball a d \<inter> U) retract_of (affine hull (cball a d \<inter> U) - {a})"
- apply (rule rel_frontier_retract_of_punctured_affine_hull)
- apply (auto simp: \<open>convex U\<close> convex_Int)
- by (metis \<open>affine U\<close> convex_cball empty_iff interior_cball a_BU rel_interior_convex_Int_affine)
- moreover have "rel_frontier (cball a d \<inter> U) = frontier (cball a d) \<inter> U"
- apply (rule convex_affine_rel_frontier_Int)
- using a_BU by (force simp: \<open>affine U\<close>)+
- moreover have "affine hull (cball a d \<inter> U) = U"
- by (metis \<open>convex U\<close> a_BU affine_hull_convex_Int_nonempty_interior affine_hull_eq \<open>affine U\<close> equals0D inf.commute interior_cball)
- ultimately have "frontier (cball a d) \<inter> U retract_of (U - {a})"
- by metis
- then obtain r where contr: "continuous_on (U - {a}) r"
- and rim: "r ` (U - {a}) \<subseteq> sphere a d" "r ` (U - {a}) \<subseteq> U"
- and req: "\<And>x. x \<in> sphere a d \<inter> U \<Longrightarrow> r x = x"
- using \<open>affine U\<close> by (auto simp: retract_of_def retraction_def hull_same)
- define j where "j \<equiv> \<lambda>x. if x \<in> ball a d then r x else x"
- have kj: "\<And>x. x \<in> S \<Longrightarrow> k (j x) = x"
- using \<open>C \<subseteq> U - S\<close> \<open>S \<subseteq> U\<close> \<open>ball a d \<inter> U \<subseteq> C\<close> j_def subC by auto
- have Uaeq: "U - {a} = (cball a d - {a}) \<inter> U \<union> (U - ball a d)"
- using \<open>0 < d\<close> by auto
- have jim: "j ` (S \<union> (C - {a})) \<subseteq> (S \<union> C) - ball a d"
- proof clarify
- fix y assume "y \<in> S \<union> (C - {a})"
- then have "y \<in> U - {a}"
- using \<open>C \<subseteq> U - S\<close> \<open>S \<subseteq> U\<close> \<open>a \<in> C\<close> by auto
- then have "r y \<in> sphere a d"
- using rim by auto
- then show "j y \<in> S \<union> C - ball a d"
- apply (simp add: j_def)
- using \<open>r y \<in> sphere a d\<close> \<open>y \<in> U - {a}\<close> \<open>y \<in> S \<union> (C - {a})\<close> d rim by fastforce
- qed
- have contj: "continuous_on (U - {a}) j"
- unfolding j_def Uaeq
- proof (intro continuous_on_cases_local continuous_on_id, simp_all add: req closedin_closed Uaeq [symmetric])
- show "\<exists>T. closed T \<and> (cball a d - {a}) \<inter> U = (U - {a}) \<inter> T"
- apply (rule_tac x="(cball a d) \<inter> U" in exI)
- using affine_closed \<open>affine U\<close> by blast
- show "\<exists>T. closed T \<and> U - ball a d = (U - {a}) \<inter> T"
- apply (rule_tac x="U - ball a d" in exI)
- using \<open>0 < d\<close> by (force simp: affine_closed \<open>affine U\<close> closed_Diff)
- show "continuous_on ((cball a d - {a}) \<inter> U) r"
- by (force intro: continuous_on_subset [OF contr])
- qed
- have fT: "x \<in> U - K \<Longrightarrow> f x \<in> T" for x
- using fim by blast
- show ?thesis
- proof (intro conjI exI)
- show "continuous_on (S \<union> (C - {a})) (f \<circ> k \<circ> j)"
- proof (intro continuous_on_compose)
- show "continuous_on (S \<union> (C - {a})) j"
- apply (rule continuous_on_subset [OF contj])
- using \<open>C \<subseteq> U - S\<close> \<open>S \<subseteq> U\<close> \<open>a \<in> C\<close> by force
- show "continuous_on (j ` (S \<union> (C - {a}))) k"
- apply (rule continuous_on_subset [OF homeomorphism_cont2 [OF homhk]])
- using jim \<open>C \<subseteq> U - S\<close> \<open>S \<subseteq> U\<close> \<open>ball a d \<inter> U \<subseteq> C\<close> j_def by fastforce
- show "continuous_on (k ` j ` (S \<union> (C - {a}))) f"
- proof (clarify intro!: continuous_on_subset [OF contf])
- fix y assume "y \<in> S \<union> (C - {a})"
- have ky: "k y \<in> S \<union> C"
- using homeomorphism_image2 [OF homhk] \<open>y \<in> S \<union> (C - {a})\<close> by blast
- have jy: "j y \<in> S \<union> C - ball a d"
- using Un_iff \<open>y \<in> S \<union> (C - {a})\<close> jim by auto
- show "k (j y) \<in> U - K"
- apply safe
- using \<open>C \<subseteq> U\<close> \<open>S \<subseteq> U\<close> homeomorphism_image2 [OF homhk] jy apply blast
- by (metis DiffD1 DiffD2 Int_iff Un_iff \<open>disjnt K S\<close> disjnt_def empty_iff hin homeomorphism_apply2 homeomorphism_image2 homhk imageI jy)
- qed
- qed
- have ST: "\<And>x. x \<in> S \<Longrightarrow> (f \<circ> k \<circ> j) x \<in> T"
- apply (simp add: kj)
- apply (metis DiffI \<open>S \<subseteq> U\<close> \<open>disjnt K S\<close> subsetD disjnt_iff fim image_subset_iff)
- done
- moreover have "(f \<circ> k \<circ> j) x \<in> T" if "x \<in> C" "x \<noteq> a" "x \<notin> S" for x
- proof -
- have rx: "r x \<in> sphere a d"
- using \<open>C \<subseteq> U\<close> rim that by fastforce
- have jj: "j x \<in> S \<union> C - ball a d"
- using jim that by blast
- have "k (j x) = j x \<longrightarrow> k (j x) \<in> C \<or> j x \<in> C"
- by (metis Diff_iff Int_iff Un_iff \<open>S \<subseteq> U\<close> subsetD d j_def jj rx sphere_cball that(1))
- then have "k (j x) \<in> C"
- using homeomorphism_apply2 [OF homhk, of "j x"] \<open>C \<subseteq> U\<close> \<open>S \<subseteq> U\<close> a rx
- by (metis (mono_tags, lifting) Diff_iff subsetD jj mem_Collect_eq subC)
- with jj \<open>C \<subseteq> U\<close> show ?thesis
- apply safe
- using ST j_def apply fastforce
- apply (auto simp: not_less intro!: fT)
- by (metis DiffD1 DiffD2 Int_iff hin homeomorphism_apply2 [OF homhk] jj)
- qed
- ultimately show "(f \<circ> k \<circ> j) ` (S \<union> (C - {a})) \<subseteq> T"
- by force
- show "\<forall>x\<in>S. (f \<circ> k \<circ> j) x = f x" using kj by simp
- qed (auto simp: a)
- qed
- then obtain a h where
- ah: "\<And>C. \<lbrakk>C \<in> components (U - S); C \<inter> K \<noteq> {}\<rbrakk>
- \<Longrightarrow> a C \<in> C \<and> a C \<in> L \<and> continuous_on (S \<union> (C - {a C})) (h C) \<and>
- h C ` (S \<union> (C - {a C})) \<subseteq> T \<and> (\<forall>x \<in> S. h C x = f x)"
- using that by metis
- define F where "F \<equiv> {C \<in> components (U - S). C \<inter> K \<noteq> {}}"
- define G where "G \<equiv> {C \<in> components (U - S). C \<inter> K = {}}"
- define UF where "UF \<equiv> (\<Union>C\<in>F. C - {a C})"
- have "C0 \<in> F"
- by (auto simp: F_def C0)
- have "finite F"
- proof (subst finite_image_iff [of "\<lambda>C. C \<inter> K" F, symmetric])
- show "inj_on (\<lambda>C. C \<inter> K) F"
- unfolding F_def inj_on_def
- using components_nonoverlap by blast
- show "finite ((\<lambda>C. C \<inter> K) ` F)"
- unfolding F_def
- by (rule finite_subset [of _ "Pow K"]) (auto simp: \<open>finite K\<close>)
- qed
- obtain g where contg: "continuous_on (S \<union> UF) g"
- and gh: "\<And>x i. \<lbrakk>i \<in> F; x \<in> (S \<union> UF) \<inter> (S \<union> (i - {a i}))\<rbrakk>
- \<Longrightarrow> g x = h i x"
- proof (rule pasting_lemma_exists_closed [OF \<open>finite F\<close>, of "S \<union> UF" "\<lambda>C. S \<union> (C - {a C})" h])
- show "S \<union> UF \<subseteq> (\<Union>C\<in>F. S \<union> (C - {a C}))"
- using \<open>C0 \<in> F\<close> by (force simp: UF_def)
- show "closedin (subtopology euclidean (S \<union> UF)) (S \<union> (C - {a C}))"
- if "C \<in> F" for C
- proof (rule closedin_closed_subset [of U "S \<union> C"])
- show "closedin (subtopology euclidean U) (S \<union> C)"
- apply (rule closedin_Un_complement_component [OF \<open>locally connected U\<close> clo])
- using F_def that by blast
- next
- have "x = a C'" if "C' \<in> F" "x \<in> C'" "x \<notin> U" for x C'
- proof -
- have "\<forall>A. x \<in> \<Union>A \<or> C' \<notin> A"
- using \<open>x \<in> C'\<close> by blast
- with that show "x = a C'"
- by (metis (lifting) DiffD1 F_def Union_components mem_Collect_eq)
- qed
- then show "S \<union> UF \<subseteq> U"
- using \<open>S \<subseteq> U\<close> by (force simp: UF_def)
- next
- show "S \<union> (C - {a C}) = (S \<union> C) \<inter> (S \<union> UF)"
- using F_def UF_def components_nonoverlap that by auto
- qed
- next
- show "continuous_on (S \<union> (C' - {a C'})) (h C')" if "C' \<in> F" for C'
- using ah F_def that by blast
- show "\<And>i j x. \<lbrakk>i \<in> F; j \<in> F;
- x \<in> (S \<union> UF) \<inter> (S \<union> (i - {a i})) \<inter> (S \<union> (j - {a j}))\<rbrakk>
- \<Longrightarrow> h i x = h j x"
- using components_eq by (fastforce simp: components_eq F_def ah)
- qed blast
- have SU': "S \<union> \<Union>G \<union> (S \<union> UF) \<subseteq> U"
- using \<open>S \<subseteq> U\<close> in_components_subset by (auto simp: F_def G_def UF_def)
- have clo1: "closedin (subtopology euclidean (S \<union> \<Union>G \<union> (S \<union> UF))) (S \<union> \<Union>G)"
- proof (rule closedin_closed_subset [OF _ SU'])
- have *: "\<And>C. C \<in> F \<Longrightarrow> openin (subtopology euclidean U) C"
- unfolding F_def
- by clarify (metis (no_types, lifting) \<open>locally connected U\<close> clo closedin_def locally_diff_closed openin_components_locally_connected openin_trans topspace_euclidean_subtopology)
- show "closedin (subtopology euclidean U) (U - UF)"
- unfolding UF_def
- by (force intro: openin_delete *)
- show "S \<union> \<Union>G = (U - UF) \<inter> (S \<union> \<Union>G \<union> (S \<union> UF))"
- using \<open>S \<subseteq> U\<close> apply (auto simp: F_def G_def UF_def)
- apply (metis Diff_iff UnionI Union_components)
- apply (metis DiffD1 UnionI Union_components)
- by (metis (no_types, lifting) IntI components_nonoverlap empty_iff)
- qed
- have clo2: "closedin (subtopology euclidean (S \<union> \<Union>G \<union> (S \<union> UF))) (S \<union> UF)"
- proof (rule closedin_closed_subset [OF _ SU'])
- show "closedin (subtopology euclidean U) (\<Union>C\<in>F. S \<union> C)"
- apply (rule closedin_Union)
- apply (simp add: \<open>finite F\<close>)
- using F_def \<open>locally connected U\<close> clo closedin_Un_complement_component by blast
- show "S \<union> UF = (\<Union>C\<in>F. S \<union> C) \<inter> (S \<union> \<Union>G \<union> (S \<union> UF))"
- using \<open>S \<subseteq> U\<close> apply (auto simp: F_def G_def UF_def)
- using C0 apply blast
- by (metis components_nonoverlap disjnt_def disjnt_iff)
- qed
- have SUG: "S \<union> \<Union>G \<subseteq> U - K"
- using \<open>S \<subseteq> U\<close> K apply (auto simp: G_def disjnt_iff)
- by (meson Diff_iff subsetD in_components_subset)
- then have contf': "continuous_on (S \<union> \<Union>G) f"
- by (rule continuous_on_subset [OF contf])
- have contg': "continuous_on (S \<union> UF) g"
- apply (rule continuous_on_subset [OF contg])
- using \<open>S \<subseteq> U\<close> by (auto simp: F_def G_def)
- have "\<And>x. \<lbrakk>S \<subseteq> U; x \<in> S\<rbrakk> \<Longrightarrow> f x = g x"
- by (subst gh) (auto simp: ah C0 intro: \<open>C0 \<in> F\<close>)
- then have f_eq_g: "\<And>x. x \<in> S \<union> UF \<and> x \<in> S \<union> \<Union>G \<Longrightarrow> f x = g x"
- using \<open>S \<subseteq> U\<close> apply (auto simp: F_def G_def UF_def dest: in_components_subset)
- using components_eq by blast
- have cont: "continuous_on (S \<union> \<Union>G \<union> (S \<union> UF)) (\<lambda>x. if x \<in> S \<union> \<Union>G then f x else g x)"
- by (blast intro: continuous_on_cases_local [OF clo1 clo2 contf' contg' f_eq_g, of "\<lambda>x. x \<in> S \<union> \<Union>G"])
- show ?thesis
- proof
- have UF: "\<Union>F - L \<subseteq> UF"
- unfolding F_def UF_def using ah by blast
- have "U - S - L = \<Union>(components (U - S)) - L"
- by simp
- also have "... = \<Union>F \<union> \<Union>G - L"
- unfolding F_def G_def by blast
- also have "... \<subseteq> UF \<union> \<Union>G"
- using UF by blast
- finally have "U - L \<subseteq> S \<union> \<Union>G \<union> (S \<union> UF)"
- by blast
- then show "continuous_on (U - L) (\<lambda>x. if x \<in> S \<union> \<Union>G then f x else g x)"
- by (rule continuous_on_subset [OF cont])
- have "((U - L) \<inter> {x. x \<notin> S \<and> (\<forall>xa\<in>G. x \<notin> xa)}) \<subseteq> ((U - L) \<inter> (-S \<inter> UF))"
- using \<open>U - L \<subseteq> S \<union> \<Union>G \<union> (S \<union> UF)\<close> by auto
- moreover have "g ` ((U - L) \<inter> (-S \<inter> UF)) \<subseteq> T"
- proof -
- have "g x \<in> T" if "x \<in> U" "x \<notin> L" "x \<notin> S" "C \<in> F" "x \<in> C" "x \<noteq> a C" for x C
- proof (subst gh)
- show "x \<in> (S \<union> UF) \<inter> (S \<union> (C - {a C}))"
- using that by (auto simp: UF_def)
- show "h C x \<in> T"
- using ah that by (fastforce simp add: F_def)
- qed (rule that)
- then show ?thesis
- by (force simp: UF_def)
- qed
- ultimately have "g ` ((U - L) \<inter> {x. x \<notin> S \<and> (\<forall>xa\<in>G. x \<notin> xa)}) \<subseteq> T"
- using image_mono order_trans by blast
- moreover have "f ` ((U - L) \<inter> (S \<union> \<Union>G)) \<subseteq> T"
- using fim SUG by blast
- ultimately show "(\<lambda>x. if x \<in> S \<union> \<Union>G then f x else g x) ` (U - L) \<subseteq> T"
- by force
- show "\<And>x. x \<in> S \<Longrightarrow> (if x \<in> S \<union> \<Union>G then f x else g x) = f x"
- by (simp add: F_def G_def)
- qed
-qed
-
-
-lemma extend_map_affine_to_sphere2:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "compact S" "convex U" "bounded U" "affine T" "S \<subseteq> T"
- and affTU: "aff_dim T \<le> aff_dim U"
- and contf: "continuous_on S f"
- and fim: "f ` S \<subseteq> rel_frontier U"
- and ovlap: "\<And>C. C \<in> components(T - S) \<Longrightarrow> C \<inter> L \<noteq> {}"
- obtains K g where "finite K" "K \<subseteq> L" "K \<subseteq> T" "disjnt K S"
- "continuous_on (T - K) g" "g ` (T - K) \<subseteq> rel_frontier U"
- "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-proof -
- obtain K g where K: "finite K" "K \<subseteq> T" "disjnt K S"
- and contg: "continuous_on (T - K) g"
- and gim: "g ` (T - K) \<subseteq> rel_frontier U"
- and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
- using assms extend_map_affine_to_sphere_cofinite_simple by metis
- have "(\<exists>y C. C \<in> components (T - S) \<and> x \<in> C \<and> y \<in> C \<and> y \<in> L)" if "x \<in> K" for x
- proof -
- have "x \<in> T-S"
- using \<open>K \<subseteq> T\<close> \<open>disjnt K S\<close> disjnt_def that by fastforce
- then obtain C where "C \<in> components(T - S)" "x \<in> C"
- by (metis UnionE Union_components)
- with ovlap [of C] show ?thesis
- by blast
- qed
- then obtain \<xi> where \<xi>: "\<And>x. x \<in> K \<Longrightarrow> \<exists>C. C \<in> components (T - S) \<and> x \<in> C \<and> \<xi> x \<in> C \<and> \<xi> x \<in> L"
- by metis
- obtain h where conth: "continuous_on (T - \<xi> ` K) h"
- and him: "h ` (T - \<xi> ` K) \<subseteq> rel_frontier U"
- and hg: "\<And>x. x \<in> S \<Longrightarrow> h x = g x"
- proof (rule extend_map_affine_to_sphere1 [OF \<open>finite K\<close> \<open>affine T\<close> contg gim, of S "\<xi> ` K"])
- show cloTS: "closedin (subtopology euclidean T) S"
- by (simp add: \<open>compact S\<close> \<open>S \<subseteq> T\<close> closed_subset compact_imp_closed)
- show "\<And>C. \<lbrakk>C \<in> components (T - S); C \<inter> K \<noteq> {}\<rbrakk> \<Longrightarrow> C \<inter> \<xi> ` K \<noteq> {}"
- using \<xi> components_eq by blast
- qed (use K in auto)
- show ?thesis
- proof
- show *: "\<xi> ` K \<subseteq> L"
- using \<xi> by blast
- show "finite (\<xi> ` K)"
- by (simp add: K)
- show "\<xi> ` K \<subseteq> T"
- by clarify (meson \<xi> Diff_iff contra_subsetD in_components_subset)
- show "continuous_on (T - \<xi> ` K) h"
- by (rule conth)
- show "disjnt (\<xi> ` K) S"
- using K
- apply (auto simp: disjnt_def)
- by (metis \<xi> DiffD2 UnionI Union_components)
- qed (simp_all add: him hg gf)
-qed
-
-
-proposition extend_map_affine_to_sphere_cofinite_gen:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes SUT: "compact S" "convex U" "bounded U" "affine T" "S \<subseteq> T"
- and aff: "aff_dim T \<le> aff_dim U"
- and contf: "continuous_on S f"
- and fim: "f ` S \<subseteq> rel_frontier U"
- and dis: "\<And>C. \<lbrakk>C \<in> components(T - S); bounded C\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
- obtains K g where "finite K" "K \<subseteq> L" "K \<subseteq> T" "disjnt K S" "continuous_on (T - K) g"
- "g ` (T - K) \<subseteq> rel_frontier U"
- "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-proof (cases "S = {}")
- case True
- show ?thesis
- proof (cases "rel_frontier U = {}")
- case True
- with aff have "aff_dim T \<le> 0"
- apply (simp add: rel_frontier_eq_empty)
- using affine_bounded_eq_lowdim \<open>bounded U\<close> order_trans by auto
- with aff_dim_geq [of T] consider "aff_dim T = -1" | "aff_dim T = 0"
- by linarith
- then show ?thesis
- proof cases
- assume "aff_dim T = -1"
- then have "T = {}"
- by (simp add: aff_dim_empty)
- then show ?thesis
- by (rule_tac K="{}" in that) auto
- next
- assume "aff_dim T = 0"
- then obtain a where "T = {a}"
- using aff_dim_eq_0 by blast
- then have "a \<in> L"
- using dis [of "{a}"] \<open>S = {}\<close> by (auto simp: in_components_self)
- with \<open>S = {}\<close> \<open>T = {a}\<close> show ?thesis
- by (rule_tac K="{a}" and g=f in that) auto
- qed
- next
- case False
- then obtain y where "y \<in> rel_frontier U"
- by auto
- with \<open>S = {}\<close> show ?thesis
- by (rule_tac K="{}" and g="\<lambda>x. y" in that) (auto simp: continuous_on_const)
- qed
-next
- case False
- have "bounded S"
- by (simp add: assms compact_imp_bounded)
- then obtain b where b: "S \<subseteq> cbox (-b) b"
- using bounded_subset_cbox_symmetric by blast
- define LU where "LU \<equiv> L \<union> (\<Union> {C \<in> components (T - S). ~bounded C} - cbox (-(b+One)) (b+One))"
- obtain K g where "finite K" "K \<subseteq> LU" "K \<subseteq> T" "disjnt K S"
- and contg: "continuous_on (T - K) g"
- and gim: "g ` (T - K) \<subseteq> rel_frontier U"
- and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
- proof (rule extend_map_affine_to_sphere2 [OF SUT aff contf fim])
- show "C \<inter> LU \<noteq> {}" if "C \<in> components (T - S)" for C
- proof (cases "bounded C")
- case True
- with dis that show ?thesis
- unfolding LU_def by fastforce
- next
- case False
- then have "\<not> bounded (\<Union>{C \<in> components (T - S). \<not> bounded C})"
- by (metis (no_types, lifting) Sup_upper bounded_subset mem_Collect_eq that)
- then show ?thesis
- apply (clarsimp simp: LU_def Int_Un_distrib Diff_Int_distrib Int_UN_distrib)
- by (metis (no_types, lifting) False Sup_upper bounded_cbox bounded_subset inf.orderE mem_Collect_eq that)
- qed
- qed blast
- have *: False if "x \<in> cbox (- b - m *\<^sub>R One) (b + m *\<^sub>R One)"
- "x \<notin> box (- b - n *\<^sub>R One) (b + n *\<^sub>R One)"
- "0 \<le> m" "m < n" "n \<le> 1" for m n x
- using that by (auto simp: mem_box algebra_simps)
- have "disjoint_family_on (\<lambda>d. frontier (cbox (- b - d *\<^sub>R One) (b + d *\<^sub>R One))) {1 / 2..1}"
- by (auto simp: disjoint_family_on_def neq_iff frontier_def dest: *)
- then obtain d where d12: "1/2 \<le> d" "d \<le> 1"
- and ddis: "disjnt K (frontier (cbox (-(b + d *\<^sub>R One)) (b + d *\<^sub>R One)))"
- using disjoint_family_elem_disjnt [of "{1/2..1::real}" K "\<lambda>d. frontier (cbox (-(b + d *\<^sub>R One)) (b + d *\<^sub>R One))"]
- by (auto simp: \<open>finite K\<close>)
- define c where "c \<equiv> b + d *\<^sub>R One"
- have cbsub: "cbox (-b) b \<subseteq> box (-c) c"
- "cbox (-b) b \<subseteq> cbox (-c) c"
- "cbox (-c) c \<subseteq> cbox (-(b+One)) (b+One)"
- using d12 by (simp_all add: subset_box c_def inner_diff_left inner_left_distrib)
- have clo_cT: "closed (cbox (- c) c \<inter> T)"
- using affine_closed \<open>affine T\<close> by blast
- have cT_ne: "cbox (- c) c \<inter> T \<noteq> {}"
- using \<open>S \<noteq> {}\<close> \<open>S \<subseteq> T\<close> b cbsub by fastforce
- have S_sub_cc: "S \<subseteq> cbox (- c) c"
- using \<open>cbox (- b) b \<subseteq> cbox (- c) c\<close> b by auto
- show ?thesis
- proof
- show "finite (K \<inter> cbox (-(b+One)) (b+One))"
- using \<open>finite K\<close> by blast
- show "K \<inter> cbox (- (b + One)) (b + One) \<subseteq> L"
- using \<open>K \<subseteq> LU\<close> by (auto simp: LU_def)
- show "K \<inter> cbox (- (b + One)) (b + One) \<subseteq> T"
- using \<open>K \<subseteq> T\<close> by auto
- show "disjnt (K \<inter> cbox (- (b + One)) (b + One)) S"
- using \<open>disjnt K S\<close> by (simp add: disjnt_def disjoint_eq_subset_Compl inf.coboundedI1)
- have cloTK: "closest_point (cbox (- c) c \<inter> T) x \<in> T - K"
- if "x \<in> T" and Knot: "x \<in> K \<longrightarrow> x \<notin> cbox (- b - One) (b + One)" for x
- proof (cases "x \<in> cbox (- c) c")
- case True
- with \<open>x \<in> T\<close> show ?thesis
- using cbsub(3) Knot by (force simp: closest_point_self)
- next
- case False
- have clo_in_rf: "closest_point (cbox (- c) c \<inter> T) x \<in> rel_frontier (cbox (- c) c \<inter> T)"
- proof (intro closest_point_in_rel_frontier [OF clo_cT cT_ne] DiffI notI)
- have "T \<inter> interior (cbox (- c) c) \<noteq> {}"
- using \<open>S \<noteq> {}\<close> \<open>S \<subseteq> T\<close> b cbsub(1) by fastforce
- then show "x \<in> affine hull (cbox (- c) c \<inter> T)"
- by (simp add: Int_commute affine_hull_affine_Int_nonempty_interior \<open>affine T\<close> hull_inc that(1))
- next
- show "False" if "x \<in> rel_interior (cbox (- c) c \<inter> T)"
- proof -
- have "interior (cbox (- c) c) \<inter> T \<noteq> {}"
- using \<open>S \<noteq> {}\<close> \<open>S \<subseteq> T\<close> b cbsub(1) by fastforce
- then have "affine hull (T \<inter> cbox (- c) c) = T"
- using affine_hull_convex_Int_nonempty_interior [of T "cbox (- c) c"]
- by (simp add: affine_imp_convex \<open>affine T\<close> inf_commute)
- then show ?thesis
- by (meson subsetD le_inf_iff rel_interior_subset that False)
- qed
- qed
- have "closest_point (cbox (- c) c \<inter> T) x \<notin> K"
- proof
- assume inK: "closest_point (cbox (- c) c \<inter> T) x \<in> K"
- have "\<And>x. x \<in> K \<Longrightarrow> x \<notin> frontier (cbox (- (b + d *\<^sub>R One)) (b + d *\<^sub>R One))"
- by (metis ddis disjnt_iff)
- then show False
- by (metis DiffI Int_iff \<open>affine T\<close> cT_ne c_def clo_cT clo_in_rf closest_point_in_set
- convex_affine_rel_frontier_Int convex_box(1) empty_iff frontier_cbox inK interior_cbox)
- qed
- then show ?thesis
- using cT_ne clo_cT closest_point_in_set by blast
- qed
- show "continuous_on (T - K \<inter> cbox (- (b + One)) (b + One)) (g \<circ> closest_point (cbox (-c) c \<inter> T))"
- apply (intro continuous_on_compose continuous_on_closest_point continuous_on_subset [OF contg])
- apply (simp_all add: clo_cT affine_imp_convex \<open>affine T\<close> convex_Int cT_ne)
- using cloTK by blast
- have "g (closest_point (cbox (- c) c \<inter> T) x) \<in> rel_frontier U"
- if "x \<in> T" "x \<in> K \<longrightarrow> x \<notin> cbox (- b - One) (b + One)" for x
- apply (rule gim [THEN subsetD])
- using that cloTK by blast
- then show "(g \<circ> closest_point (cbox (- c) c \<inter> T)) ` (T - K \<inter> cbox (- (b + One)) (b + One))
- \<subseteq> rel_frontier U"
- by force
- show "\<And>x. x \<in> S \<Longrightarrow> (g \<circ> closest_point (cbox (- c) c \<inter> T)) x = f x"
- by simp (metis (mono_tags, lifting) IntI \<open>S \<subseteq> T\<close> cT_ne clo_cT closest_point_refl gf subsetD S_sub_cc)
- qed
-qed
-
-
-corollary extend_map_affine_to_sphere_cofinite:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes SUT: "compact S" "affine T" "S \<subseteq> T"
- and aff: "aff_dim T \<le> DIM('b)" and "0 \<le> r"
- and contf: "continuous_on S f"
- and fim: "f ` S \<subseteq> sphere a r"
- and dis: "\<And>C. \<lbrakk>C \<in> components(T - S); bounded C\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
- obtains K g where "finite K" "K \<subseteq> L" "K \<subseteq> T" "disjnt K S" "continuous_on (T - K) g"
- "g ` (T - K) \<subseteq> sphere a r" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-proof (cases "r = 0")
- case True
- with fim show ?thesis
- by (rule_tac K="{}" and g = "\<lambda>x. a" in that) (auto simp: continuous_on_const)
-next
- case False
- with assms have "0 < r" by auto
- then have "aff_dim T \<le> aff_dim (cball a r)"
- by (simp add: aff aff_dim_cball)
- then show ?thesis
- apply (rule extend_map_affine_to_sphere_cofinite_gen
- [OF \<open>compact S\<close> convex_cball bounded_cball \<open>affine T\<close> \<open>S \<subseteq> T\<close> _ contf])
- using fim apply (auto simp: assms False that dest: dis)
- done
-qed
-
-corollary extend_map_UNIV_to_sphere_cofinite:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes aff: "DIM('a) \<le> DIM('b)" and "0 \<le> r"
- and SUT: "compact S"
- and contf: "continuous_on S f"
- and fim: "f ` S \<subseteq> sphere a r"
- and dis: "\<And>C. \<lbrakk>C \<in> components(- S); bounded C\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
- obtains K g where "finite K" "K \<subseteq> L" "disjnt K S" "continuous_on (- K) g"
- "g ` (- K) \<subseteq> sphere a r" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-apply (rule extend_map_affine_to_sphere_cofinite
- [OF \<open>compact S\<close> affine_UNIV subset_UNIV _ \<open>0 \<le> r\<close> contf fim dis])
- apply (auto simp: assms that Compl_eq_Diff_UNIV [symmetric])
-done
-
-corollary extend_map_UNIV_to_sphere_no_bounded_component:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes aff: "DIM('a) \<le> DIM('b)" and "0 \<le> r"
- and SUT: "compact S"
- and contf: "continuous_on S f"
- and fim: "f ` S \<subseteq> sphere a r"
- and dis: "\<And>C. C \<in> components(- S) \<Longrightarrow> \<not> bounded C"
- obtains g where "continuous_on UNIV g" "g ` UNIV \<subseteq> sphere a r" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-apply (rule extend_map_UNIV_to_sphere_cofinite [OF aff \<open>0 \<le> r\<close> \<open>compact S\<close> contf fim, of "{}"])
- apply (auto simp: that dest: dis)
-done
-
-theorem Borsuk_separation_theorem_gen:
- fixes S :: "'a::euclidean_space set"
- assumes "compact S"
- shows "(\<forall>c \<in> components(- S). ~bounded c) \<longleftrightarrow>
- (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> sphere (0::'a) 1
- \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) S (sphere 0 1) f (\<lambda>x. c)))"
- (is "?lhs = ?rhs")
-proof
- assume L [rule_format]: ?lhs
- show ?rhs
- proof clarify
- fix f :: "'a \<Rightarrow> 'a"
- assume contf: "continuous_on S f" and fim: "f ` S \<subseteq> sphere 0 1"
- obtain g where contg: "continuous_on UNIV g" and gim: "range g \<subseteq> sphere 0 1"
- and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
- by (rule extend_map_UNIV_to_sphere_no_bounded_component [OF _ _ \<open>compact S\<close> contf fim L]) auto
- then show "\<exists>c. homotopic_with (\<lambda>x. True) S (sphere 0 1) f (\<lambda>x. c)"
- using nullhomotopic_from_contractible [OF contg gim]
- by (metis assms compact_imp_closed contf empty_iff fim homotopic_with_equal nullhomotopic_into_sphere_extension)
- qed
-next
- assume R [rule_format]: ?rhs
- show ?lhs
- unfolding components_def
- proof clarify
- fix a
- assume "a \<notin> S" and a: "bounded (connected_component_set (- S) a)"
- have cont: "continuous_on S (\<lambda>x. inverse(norm(x - a)) *\<^sub>R (x - a))"
- apply (intro continuous_intros)
- using \<open>a \<notin> S\<close> by auto
- have im: "(\<lambda>x. inverse(norm(x - a)) *\<^sub>R (x - a)) ` S \<subseteq> sphere 0 1"
- by clarsimp (metis \<open>a \<notin> S\<close> eq_iff_diff_eq_0 left_inverse norm_eq_zero)
- show False
- using R cont im Borsuk_map_essential_bounded_component [OF \<open>compact S\<close> \<open>a \<notin> S\<close>] a by blast
- qed
-qed
-
-
-corollary Borsuk_separation_theorem:
- fixes S :: "'a::euclidean_space set"
- assumes "compact S" and 2: "2 \<le> DIM('a)"
- shows "connected(- S) \<longleftrightarrow>
- (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> sphere (0::'a) 1
- \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) S (sphere 0 1) f (\<lambda>x. c)))"
- (is "?lhs = ?rhs")
-proof
- assume L: ?lhs
- show ?rhs
- proof (cases "S = {}")
- case True
- then show ?thesis by auto
- next
- case False
- then have "(\<forall>c\<in>components (- S). \<not> bounded c)"
- by (metis L assms(1) bounded_empty cobounded_imp_unbounded compact_imp_bounded in_components_maximal order_refl)
- then show ?thesis
- by (simp add: Borsuk_separation_theorem_gen [OF \<open>compact S\<close>])
- qed
-next
- assume R: ?rhs
- then show ?lhs
- apply (simp add: Borsuk_separation_theorem_gen [OF \<open>compact S\<close>, symmetric])
- apply (auto simp: components_def connected_iff_eq_connected_component_set)
- using connected_component_in apply fastforce
- using cobounded_unique_unbounded_component [OF _ 2, of "-S"] \<open>compact S\<close> compact_eq_bounded_closed by fastforce
-qed
-
-
-lemma homotopy_eqv_separation:
- fixes S :: "'a::euclidean_space set" and T :: "'a set"
- assumes "S homotopy_eqv T" and "compact S" and "compact T"
- shows "connected(- S) \<longleftrightarrow> connected(- T)"
-proof -
- consider "DIM('a) = 1" | "2 \<le> DIM('a)"
- by (metis DIM_ge_Suc0 One_nat_def Suc_1 dual_order.antisym not_less_eq_eq)
- then show ?thesis
- proof cases
- case 1
- then show ?thesis
- using bounded_connected_Compl_1 compact_imp_bounded homotopy_eqv_empty1 homotopy_eqv_empty2 assms by metis
- next
- case 2
- with assms show ?thesis
- by (simp add: Borsuk_separation_theorem homotopy_eqv_cohomotopic_triviality_null)
- qed
-qed
-
-lemma Jordan_Brouwer_separation:
- fixes S :: "'a::euclidean_space set" and a::'a
- assumes hom: "S homeomorphic sphere a r" and "0 < r"
- shows "\<not> connected(- S)"
-proof -
- have "- sphere a r \<inter> ball a r \<noteq> {}"
- using \<open>0 < r\<close> by (simp add: Int_absorb1 subset_eq)
- moreover
- have eq: "- sphere a r - ball a r = - cball a r"
- by auto
- have "- cball a r \<noteq> {}"
- proof -
- have "frontier (cball a r) \<noteq> {}"
- using \<open>0 < r\<close> by auto
- then show ?thesis
- by (metis frontier_complement frontier_empty)
- qed
- with eq have "- sphere a r - ball a r \<noteq> {}"
- by auto
- moreover
- have "connected (- S) = connected (- sphere a r)"
- proof (rule homotopy_eqv_separation)
- show "S homotopy_eqv sphere a r"
- using hom homeomorphic_imp_homotopy_eqv by blast
- show "compact (sphere a r)"
- by simp
- then show " compact S"
- using hom homeomorphic_compactness by blast
- qed
- ultimately show ?thesis
- using connected_Int_frontier [of "- sphere a r" "ball a r"] by (auto simp: \<open>0 < r\<close>)
-qed
-
-
-lemma Jordan_Brouwer_frontier:
- fixes S :: "'a::euclidean_space set" and a::'a
- assumes S: "S homeomorphic sphere a r" and T: "T \<in> components(- S)" and 2: "2 \<le> DIM('a)"
- shows "frontier T = S"
-proof (cases r rule: linorder_cases)
- assume "r < 0"
- with S T show ?thesis by auto
-next
- assume "r = 0"
- with S T card_eq_SucD obtain b where "S = {b}"
- by (auto simp: homeomorphic_finite [of "{a}" S])
- have "components (- {b}) = { -{b}}"
- using T \<open>S = {b}\<close> by (auto simp: components_eq_sing_iff connected_punctured_universe 2)
- with T show ?thesis
- by (metis \<open>S = {b}\<close> cball_trivial frontier_cball frontier_complement singletonD sphere_trivial)
-next
- assume "r > 0"
- have "compact S"
- using homeomorphic_compactness compact_sphere S by blast
- show ?thesis
- proof (rule frontier_minimal_separating_closed)
- show "closed S"
- using \<open>compact S\<close> compact_eq_bounded_closed by blast
- show "\<not> connected (- S)"
- using Jordan_Brouwer_separation S \<open>0 < r\<close> by blast
- obtain f g where hom: "homeomorphism S (sphere a r) f g"
- using S by (auto simp: homeomorphic_def)
- show "connected (- T)" if "closed T" "T \<subset> S" for T
- proof -
- have "f ` T \<subseteq> sphere a r"
- using \<open>T \<subset> S\<close> hom homeomorphism_image1 by blast
- moreover have "f ` T \<noteq> sphere a r"
- using \<open>T \<subset> S\<close> hom
- by (metis homeomorphism_image2 homeomorphism_of_subsets order_refl psubsetE)
- ultimately have "f ` T \<subset> sphere a r" by blast
- then have "connected (- f ` T)"
- by (rule psubset_sphere_Compl_connected [OF _ \<open>0 < r\<close> 2])
- moreover have "compact T"
- using \<open>compact S\<close> bounded_subset compact_eq_bounded_closed that by blast
- moreover then have "compact (f ` T)"
- by (meson compact_continuous_image continuous_on_subset hom homeomorphism_def psubsetE \<open>T \<subset> S\<close>)
- moreover have "T homotopy_eqv f ` T"
- by (meson \<open>f ` T \<subseteq> sphere a r\<close> dual_order.strict_implies_order hom homeomorphic_def homeomorphic_imp_homotopy_eqv homeomorphism_of_subsets \<open>T \<subset> S\<close>)
- ultimately show ?thesis
- using homotopy_eqv_separation [of T "f`T"] by blast
- qed
- qed (rule T)
-qed
-
-lemma Jordan_Brouwer_nonseparation:
- fixes S :: "'a::euclidean_space set" and a::'a
- assumes S: "S homeomorphic sphere a r" and "T \<subset> S" and 2: "2 \<le> DIM('a)"
- shows "connected(- T)"
-proof -
- have *: "connected(C \<union> (S - T))" if "C \<in> components(- S)" for C
- proof (rule connected_intermediate_closure)
- show "connected C"
- using in_components_connected that by auto
- have "S = frontier C"
- using "2" Jordan_Brouwer_frontier S that by blast
- with closure_subset show "C \<union> (S - T) \<subseteq> closure C"
- by (auto simp: frontier_def)
- qed auto
- have "components(- S) \<noteq> {}"
- by (metis S bounded_empty cobounded_imp_unbounded compact_eq_bounded_closed compact_sphere
- components_eq_empty homeomorphic_compactness)
- then have "- T = (\<Union>C \<in> components(- S). C \<union> (S - T))"
- using Union_components [of "-S"] \<open>T \<subset> S\<close> by auto
- then show ?thesis
- apply (rule ssubst)
- apply (rule connected_Union)
- using \<open>T \<subset> S\<close> apply (auto simp: *)
- done
-qed
-
-subsection\<open> Invariance of domain and corollaries\<close>
-
-lemma invariance_of_domain_ball:
- fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
- assumes contf: "continuous_on (cball a r) f" and "0 < r"
- and inj: "inj_on f (cball a r)"
- shows "open(f ` ball a r)"
-proof (cases "DIM('a) = 1")
- case True
- obtain h::"'a\<Rightarrow>real" and k
- where "linear h" "linear k" "h ` UNIV = UNIV" "k ` UNIV = UNIV"
- "\<And>x. norm(h x) = norm x" "\<And>x. norm(k x) = norm x"
- "\<And>x. k(h x) = x" "\<And>x. h(k x) = x"
- apply (rule isomorphisms_UNIV_UNIV [where 'M='a and 'N=real])
- using True
- apply force
- by (metis UNIV_I UNIV_eq_I imageI)
- have cont: "continuous_on S h" "continuous_on T k" for S T
- by (simp_all add: \<open>linear h\<close> \<open>linear k\<close> linear_continuous_on linear_linear)
- have "continuous_on (h ` cball a r) (h \<circ> f \<circ> k)"
- apply (intro continuous_on_compose cont continuous_on_subset [OF contf])
- apply (auto simp: \<open>\<And>x. k (h x) = x\<close>)
- done
- moreover have "is_interval (h ` cball a r)"
- by (simp add: is_interval_connected_1 \<open>linear h\<close> linear_continuous_on linear_linear connected_continuous_image)
- moreover have "inj_on (h \<circ> f \<circ> k) (h ` cball a r)"
- using inj by (simp add: inj_on_def) (metis \<open>\<And>x. k (h x) = x\<close>)
- ultimately have *: "\<And>T. \<lbrakk>open T; T \<subseteq> h ` cball a r\<rbrakk> \<Longrightarrow> open ((h \<circ> f \<circ> k) ` T)"
- using injective_eq_1d_open_map_UNIV by blast
- have "open ((h \<circ> f \<circ> k) ` (h ` ball a r))"
- by (rule *) (auto simp: \<open>linear h\<close> \<open>range h = UNIV\<close> open_surjective_linear_image)
- then have "open ((h \<circ> f) ` ball a r)"
- by (simp add: image_comp \<open>\<And>x. k (h x) = x\<close> cong: image_cong)
- then show ?thesis
- apply (simp add: image_comp [symmetric])
- apply (metis open_bijective_linear_image_eq \<open>linear h\<close> \<open>\<And>x. k (h x) = x\<close> \<open>range h = UNIV\<close> bijI inj_on_def)
- done
-next
- case False
- then have 2: "DIM('a) \<ge> 2"
- by (metis DIM_ge_Suc0 One_nat_def Suc_1 antisym not_less_eq_eq)
- have fimsub: "f ` ball a r \<subseteq> - f ` sphere a r"
- using inj by clarsimp (metis inj_onD less_eq_real_def mem_cball order_less_irrefl)
- have hom: "f ` sphere a r homeomorphic sphere a r"
- by (meson compact_sphere contf continuous_on_subset homeomorphic_compact homeomorphic_sym inj inj_on_subset sphere_cball)
- then have nconn: "\<not> connected (- f ` sphere a r)"
- by (rule Jordan_Brouwer_separation) (auto simp: \<open>0 < r\<close>)
- obtain C where C: "C \<in> components (- f ` sphere a r)" and "bounded C"
- apply (rule cobounded_has_bounded_component [OF _ nconn])
- apply (simp_all add: 2)
- by (meson compact_imp_bounded compact_continuous_image_eq compact_sphere contf inj sphere_cball)
- moreover have "f ` (ball a r) = C"
- proof
- have "C \<noteq> {}"
- by (rule in_components_nonempty [OF C])
- show "C \<subseteq> f ` ball a r"
- proof (rule ccontr)
- assume nonsub: "\<not> C \<subseteq> f ` ball a r"
- have "- f ` cball a r \<subseteq> C"
- proof (rule components_maximal [OF C])
- have "f ` cball a r homeomorphic cball a r"
- using compact_cball contf homeomorphic_compact homeomorphic_sym inj by blast
- then show "connected (- f ` cball a r)"
- by (auto intro: connected_complement_homeomorphic_convex_compact 2)
- show "- f ` cball a r \<subseteq> - f ` sphere a r"
- by auto
- then show "C \<inter> - f ` cball a r \<noteq> {}"
- using \<open>C \<noteq> {}\<close> in_components_subset [OF C] nonsub
- using image_iff by fastforce
- qed
- then have "bounded (- f ` cball a r)"
- using bounded_subset \<open>bounded C\<close> by auto
- then have "\<not> bounded (f ` cball a r)"
- using cobounded_imp_unbounded by blast
- then show "False"
- using compact_continuous_image [OF contf] compact_cball compact_imp_bounded by blast
- qed
- with \<open>C \<noteq> {}\<close> have "C \<inter> f ` ball a r \<noteq> {}"
- by (simp add: inf.absorb_iff1)
- then show "f ` ball a r \<subseteq> C"
- by (metis components_maximal [OF C _ fimsub] connected_continuous_image ball_subset_cball connected_ball contf continuous_on_subset)
- qed
- moreover have "open (- f ` sphere a r)"
- using hom compact_eq_bounded_closed compact_sphere homeomorphic_compactness by blast
- ultimately show ?thesis
- using open_components by blast
-qed
-
-
-text\<open>Proved by L. E. J. Brouwer (1912)\<close>
-theorem invariance_of_domain:
- fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
- assumes "continuous_on S f" "open S" "inj_on f S"
- shows "open(f ` S)"
- unfolding open_subopen [of "f`S"]
-proof clarify
- fix a
- assume "a \<in> S"
- obtain \<delta> where "\<delta> > 0" and \<delta>: "cball a \<delta> \<subseteq> S"
- using \<open>open S\<close> \<open>a \<in> S\<close> open_contains_cball_eq by blast
- show "\<exists>T. open T \<and> f a \<in> T \<and> T \<subseteq> f ` S"
- proof (intro exI conjI)
- show "open (f ` (ball a \<delta>))"
- by (meson \<delta> \<open>0 < \<delta>\<close> assms continuous_on_subset inj_on_subset invariance_of_domain_ball)
- show "f a \<in> f ` ball a \<delta>"
- by (simp add: \<open>0 < \<delta>\<close>)
- show "f ` ball a \<delta> \<subseteq> f ` S"
- using \<delta> ball_subset_cball by blast
- qed
-qed
-
-lemma inv_of_domain_ss0:
- fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
- assumes contf: "continuous_on U f" and injf: "inj_on f U" and fim: "f ` U \<subseteq> S"
- and "subspace S" and dimS: "dim S = DIM('b::euclidean_space)"
- and ope: "openin (subtopology euclidean S) U"
- shows "openin (subtopology euclidean S) (f ` U)"
-proof -
- have "U \<subseteq> S"
- using ope openin_imp_subset by blast
- have "(UNIV::'b set) homeomorphic S"
- by (simp add: \<open>subspace S\<close> dimS dim_UNIV homeomorphic_subspaces)
- then obtain h k where homhk: "homeomorphism (UNIV::'b set) S h k"
- using homeomorphic_def by blast
- have homkh: "homeomorphism S (k ` S) k h"
- using homhk homeomorphism_image2 homeomorphism_sym by fastforce
- have "open ((k \<circ> f \<circ> h) ` k ` U)"
- proof (rule invariance_of_domain)
- show "continuous_on (k ` U) (k \<circ> f \<circ> h)"
- proof (intro continuous_intros)
- show "continuous_on (k ` U) h"
- by (meson continuous_on_subset [OF homeomorphism_cont1 [OF homhk]] top_greatest)
- show "continuous_on (h ` k ` U) f"
- apply (rule continuous_on_subset [OF contf], clarify)
- apply (metis homhk homeomorphism_def ope openin_imp_subset rev_subsetD)
- done
- show "continuous_on (f ` h ` k ` U) k"
- apply (rule continuous_on_subset [OF homeomorphism_cont2 [OF homhk]])
- using fim homhk homeomorphism_apply2 ope openin_subset by fastforce
- qed
- have ope_iff: "\<And>T. open T \<longleftrightarrow> openin (subtopology euclidean (k ` S)) T"
- using homhk homeomorphism_image2 open_openin by fastforce
- show "open (k ` U)"
- by (simp add: ope_iff homeomorphism_imp_open_map [OF homkh ope])
- show "inj_on (k \<circ> f \<circ> h) (k ` U)"
- apply (clarsimp simp: inj_on_def)
- by (metis subsetD fim homeomorphism_apply2 [OF homhk] image_subset_iff inj_on_eq_iff injf \<open>U \<subseteq> S\<close>)
- qed
- moreover
- have eq: "f ` U = h ` (k \<circ> f \<circ> h \<circ> k) ` U"
- apply (auto simp: image_comp [symmetric])
- apply (metis homkh \<open>U \<subseteq> S\<close> fim homeomorphism_image2 homeomorphism_of_subsets homhk imageI subset_UNIV)
- by (metis \<open>U \<subseteq> S\<close> subsetD fim homeomorphism_def homhk image_eqI)
- ultimately show ?thesis
- by (metis (no_types, hide_lams) homeomorphism_imp_open_map homhk image_comp open_openin subtopology_UNIV)
-qed
-
-lemma inv_of_domain_ss1:
- fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
- assumes contf: "continuous_on U f" and injf: "inj_on f U" and fim: "f ` U \<subseteq> S"
- and "subspace S"
- and ope: "openin (subtopology euclidean S) U"
- shows "openin (subtopology euclidean S) (f ` U)"
-proof -
- define S' where "S' \<equiv> {y. \<forall>x \<in> S. orthogonal x y}"
- have "subspace S'"
- by (simp add: S'_def subspace_orthogonal_to_vectors)
- define g where "g \<equiv> \<lambda>z::'a*'a. ((f \<circ> fst)z, snd z)"
- have "openin (subtopology euclidean (S \<times> S')) (g ` (U \<times> S'))"
- proof (rule inv_of_domain_ss0)
- show "continuous_on (U \<times> S') g"
- apply (simp add: g_def)
- apply (intro continuous_intros continuous_on_compose2 [OF contf continuous_on_fst], auto)
- done
- show "g ` (U \<times> S') \<subseteq> S \<times> S'"
- using fim by (auto simp: g_def)
- show "inj_on g (U \<times> S')"
- using injf by (auto simp: g_def inj_on_def)
- show "subspace (S \<times> S')"
- by (simp add: \<open>subspace S'\<close> \<open>subspace S\<close> subspace_Times)
- show "openin (subtopology euclidean (S \<times> S')) (U \<times> S')"
- by (simp add: openin_Times [OF ope])
- have "dim (S \<times> S') = dim S + dim S'"
- by (simp add: \<open>subspace S'\<close> \<open>subspace S\<close> dim_Times)
- also have "... = DIM('a)"
- using dim_subspace_orthogonal_to_vectors [OF \<open>subspace S\<close> subspace_UNIV]
- by (simp add: add.commute S'_def)
- finally show "dim (S \<times> S') = DIM('a)" .
- qed
- moreover have "g ` (U \<times> S') = f ` U \<times> S'"
- by (auto simp: g_def image_iff)
- moreover have "0 \<in> S'"
- using \<open>subspace S'\<close> subspace_affine by blast
- ultimately show ?thesis
- by (auto simp: openin_Times_eq)
-qed
-
-
-corollary invariance_of_domain_subspaces:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes ope: "openin (subtopology euclidean U) S"
- and "subspace U" "subspace V" and VU: "dim V \<le> dim U"
- and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
- and injf: "inj_on f S"
- shows "openin (subtopology euclidean V) (f ` S)"
-proof -
- obtain V' where "subspace V'" "V' \<subseteq> U" "dim V' = dim V"
- using choose_subspace_of_subspace [OF VU]
- by (metis span_eq \<open>subspace U\<close>)
- then have "V homeomorphic V'"
- by (simp add: \<open>subspace V\<close> homeomorphic_subspaces)
- then obtain h k where homhk: "homeomorphism V V' h k"
- using homeomorphic_def by blast
- have eq: "f ` S = k ` (h \<circ> f) ` S"
- proof -
- have "k ` h ` f ` S = f ` S"
- by (meson fim homeomorphism_def homeomorphism_of_subsets homhk subset_refl)
- then show ?thesis
- by (simp add: image_comp)
- qed
- show ?thesis
- unfolding eq
- proof (rule homeomorphism_imp_open_map)
- show homkh: "homeomorphism V' V k h"
- by (simp add: homeomorphism_symD homhk)
- have hfV': "(h \<circ> f) ` S \<subseteq> V'"
- using fim homeomorphism_image1 homhk by fastforce
- moreover have "openin (subtopology euclidean U) ((h \<circ> f) ` S)"
- proof (rule inv_of_domain_ss1)
- show "continuous_on S (h \<circ> f)"
- by (meson contf continuous_on_compose continuous_on_subset fim homeomorphism_cont1 homhk)
- show "inj_on (h \<circ> f) S"
- apply (clarsimp simp: inj_on_def)
- by (metis fim homeomorphism_apply2 [OF homkh] image_subset_iff inj_onD injf)
- show "(h \<circ> f) ` S \<subseteq> U"
- using \<open>V' \<subseteq> U\<close> hfV' by auto
- qed (auto simp: assms)
- ultimately show "openin (subtopology euclidean V') ((h \<circ> f) ` S)"
- using openin_subset_trans \<open>V' \<subseteq> U\<close> by force
- qed
-qed
-
-corollary invariance_of_dimension_subspaces:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes ope: "openin (subtopology euclidean U) S"
- and "subspace U" "subspace V"
- and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
- and injf: "inj_on f S" and "S \<noteq> {}"
- shows "dim U \<le> dim V"
-proof -
- have "False" if "dim V < dim U"
- proof -
- obtain T where "subspace T" "T \<subseteq> U" "dim T = dim V"
- using choose_subspace_of_subspace [of "dim V" U]
- by (metis span_eq \<open>subspace U\<close> \<open>dim V < dim U\<close> linear not_le)
- then have "V homeomorphic T"
- by (simp add: \<open>subspace V\<close> homeomorphic_subspaces)
- then obtain h k where homhk: "homeomorphism V T h k"
- using homeomorphic_def by blast
- have "continuous_on S (h \<circ> f)"
- by (meson contf continuous_on_compose continuous_on_subset fim homeomorphism_cont1 homhk)
- moreover have "(h \<circ> f) ` S \<subseteq> U"
- using \<open>T \<subseteq> U\<close> fim homeomorphism_image1 homhk by fastforce
- moreover have "inj_on (h \<circ> f) S"
- apply (clarsimp simp: inj_on_def)
- by (metis fim homeomorphism_apply1 homhk image_subset_iff inj_onD injf)
- ultimately have ope_hf: "openin (subtopology euclidean U) ((h \<circ> f) ` S)"
- using invariance_of_domain_subspaces [OF ope \<open>subspace U\<close> \<open>subspace U\<close>] by auto
- have "(h \<circ> f) ` S \<subseteq> T"
- using fim homeomorphism_image1 homhk by fastforce
- then show ?thesis
- by (metis dim_openin \<open>dim T = dim V\<close> ope_hf \<open>subspace U\<close> \<open>S \<noteq> {}\<close> dim_subset image_is_empty not_le that)
- qed
- then show ?thesis
- using not_less by blast
-qed
-
-corollary invariance_of_domain_affine_sets:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes ope: "openin (subtopology euclidean U) S"
- and aff: "affine U" "affine V" "aff_dim V \<le> aff_dim U"
- and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
- and injf: "inj_on f S"
- shows "openin (subtopology euclidean V) (f ` S)"
-proof (cases "S = {}")
- case True
- then show ?thesis by auto
-next
- case False
- obtain a b where "a \<in> S" "a \<in> U" "b \<in> V"
- using False fim ope openin_contains_cball by fastforce
- have "openin (subtopology euclidean (op + (- b) ` V)) ((op + (- b) \<circ> f \<circ> op + a) ` op + (- a) ` S)"
- proof (rule invariance_of_domain_subspaces)
- show "openin (subtopology euclidean (op + (- a) ` U)) (op + (- a) ` S)"
- by (metis ope homeomorphism_imp_open_map homeomorphism_translation translation_galois)
- show "subspace (op + (- a) ` U)"
- by (simp add: \<open>a \<in> U\<close> affine_diffs_subspace \<open>affine U\<close>)
- show "subspace (op + (- b) ` V)"
- by (simp add: \<open>b \<in> V\<close> affine_diffs_subspace \<open>affine V\<close>)
- show "dim (op + (- b) ` V) \<le> dim (op + (- a) ` U)"
- by (metis \<open>a \<in> U\<close> \<open>b \<in> V\<close> aff_dim_eq_dim affine_hull_eq aff of_nat_le_iff)
- show "continuous_on (op + (- a) ` S) (op + (- b) \<circ> f \<circ> op + a)"
- by (metis contf continuous_on_compose homeomorphism_cont2 homeomorphism_translation translation_galois)
- show "(op + (- b) \<circ> f \<circ> op + a) ` op + (- a) ` S \<subseteq> op + (- b) ` V"
- using fim by auto
- show "inj_on (op + (- b) \<circ> f \<circ> op + a) (op + (- a) ` S)"
- by (auto simp: inj_on_def) (meson inj_onD injf)
- qed
- then show ?thesis
- by (metis (no_types, lifting) homeomorphism_imp_open_map homeomorphism_translation image_comp translation_galois)
-qed
-
-corollary invariance_of_dimension_affine_sets:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes ope: "openin (subtopology euclidean U) S"
- and aff: "affine U" "affine V"
- and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
- and injf: "inj_on f S" and "S \<noteq> {}"
- shows "aff_dim U \<le> aff_dim V"
-proof -
- obtain a b where "a \<in> S" "a \<in> U" "b \<in> V"
- using \<open>S \<noteq> {}\<close> fim ope openin_contains_cball by fastforce
- have "dim (op + (- a) ` U) \<le> dim (op + (- b) ` V)"
- proof (rule invariance_of_dimension_subspaces)
- show "openin (subtopology euclidean (op + (- a) ` U)) (op + (- a) ` S)"
- by (metis ope homeomorphism_imp_open_map homeomorphism_translation translation_galois)
- show "subspace (op + (- a) ` U)"
- by (simp add: \<open>a \<in> U\<close> affine_diffs_subspace \<open>affine U\<close>)
- show "subspace (op + (- b) ` V)"
- by (simp add: \<open>b \<in> V\<close> affine_diffs_subspace \<open>affine V\<close>)
- show "continuous_on (op + (- a) ` S) (op + (- b) \<circ> f \<circ> op + a)"
- by (metis contf continuous_on_compose homeomorphism_cont2 homeomorphism_translation translation_galois)
- show "(op + (- b) \<circ> f \<circ> op + a) ` op + (- a) ` S \<subseteq> op + (- b) ` V"
- using fim by auto
- show "inj_on (op + (- b) \<circ> f \<circ> op + a) (op + (- a) ` S)"
- by (auto simp: inj_on_def) (meson inj_onD injf)
- qed (use \<open>S \<noteq> {}\<close> in auto)
- then show ?thesis
- by (metis \<open>a \<in> U\<close> \<open>b \<in> V\<close> aff_dim_eq_dim affine_hull_eq aff of_nat_le_iff)
-qed
-
-corollary invariance_of_dimension:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes contf: "continuous_on S f" and "open S"
- and injf: "inj_on f S" and "S \<noteq> {}"
- shows "DIM('a) \<le> DIM('b)"
- using invariance_of_dimension_subspaces [of UNIV S UNIV f] assms
- by auto
-
-
-corollary continuous_injective_image_subspace_dim_le:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "subspace S" "subspace T"
- and contf: "continuous_on S f" and fim: "f ` S \<subseteq> T"
- and injf: "inj_on f S"
- shows "dim S \<le> dim T"
- apply (rule invariance_of_dimension_subspaces [of S S _ f])
- using assms by (auto simp: subspace_affine)
-
-lemma invariance_of_dimension_convex_domain:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "convex S"
- and contf: "continuous_on S f" and fim: "f ` S \<subseteq> affine hull T"
- and injf: "inj_on f S"
- shows "aff_dim S \<le> aff_dim T"
-proof (cases "S = {}")
- case True
- then show ?thesis by (simp add: aff_dim_geq)
-next
- case False
- have "aff_dim (affine hull S) \<le> aff_dim (affine hull T)"
- proof (rule invariance_of_dimension_affine_sets)
- show "openin (subtopology euclidean (affine hull S)) (rel_interior S)"
- by (simp add: openin_rel_interior)
- show "continuous_on (rel_interior S) f"
- using contf continuous_on_subset rel_interior_subset by blast
- show "f ` rel_interior S \<subseteq> affine hull T"
- using fim rel_interior_subset by blast
- show "inj_on f (rel_interior S)"
- using inj_on_subset injf rel_interior_subset by blast
- show "rel_interior S \<noteq> {}"
- by (simp add: False \<open>convex S\<close> rel_interior_eq_empty)
- qed auto
- then show ?thesis
- by simp
-qed
-
-
-lemma homeomorphic_convex_sets_le:
- assumes "convex S" "S homeomorphic T"
- shows "aff_dim S \<le> aff_dim T"
-proof -
- obtain h k where homhk: "homeomorphism S T h k"
- using homeomorphic_def assms by blast
- show ?thesis
- proof (rule invariance_of_dimension_convex_domain [OF \<open>convex S\<close>])
- show "continuous_on S h"
- using homeomorphism_def homhk by blast
- show "h ` S \<subseteq> affine hull T"
- by (metis homeomorphism_def homhk hull_subset)
- show "inj_on h S"
- by (meson homeomorphism_apply1 homhk inj_on_inverseI)
- qed
-qed
-
-lemma homeomorphic_convex_sets:
- assumes "convex S" "convex T" "S homeomorphic T"
- shows "aff_dim S = aff_dim T"
- by (meson assms dual_order.antisym homeomorphic_convex_sets_le homeomorphic_sym)
-
-lemma homeomorphic_convex_compact_sets_eq:
- assumes "convex S" "compact S" "convex T" "compact T"
- shows "S homeomorphic T \<longleftrightarrow> aff_dim S = aff_dim T"
- by (meson assms homeomorphic_convex_compact_sets homeomorphic_convex_sets)
-
-lemma invariance_of_domain_gen:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "open S" "continuous_on S f" "inj_on f S" "DIM('b) \<le> DIM('a)"
- shows "open(f ` S)"
- using invariance_of_domain_subspaces [of UNIV S UNIV f] assms by auto
-
-lemma injective_into_1d_imp_open_map_UNIV:
- fixes f :: "'a::euclidean_space \<Rightarrow> real"
- assumes "open T" "continuous_on S f" "inj_on f S" "T \<subseteq> S"
- shows "open (f ` T)"
- apply (rule invariance_of_domain_gen [OF \<open>open T\<close>])
- using assms apply (auto simp: elim: continuous_on_subset subset_inj_on)
- done
-
-lemma continuous_on_inverse_open:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "open S" "continuous_on S f" "DIM('b) \<le> DIM('a)" and gf: "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"
- shows "continuous_on (f ` S) g"
-proof (clarsimp simp add: continuous_openin_preimage_eq)
- fix T :: "'a set"
- assume "open T"
- have eq: "{x. x \<in> f ` S \<and> g x \<in> T} = f ` (S \<inter> T)"
- by (auto simp: gf)
- show "openin (subtopology euclidean (f ` S)) {x \<in> f ` S. g x \<in> T}"
- apply (subst eq)
- apply (rule open_openin_trans)
- apply (rule invariance_of_domain_gen)
- using assms
- apply auto
- using inj_on_inverseI apply auto[1]
- by (metis \<open>open T\<close> continuous_on_subset inj_onI inj_on_subset invariance_of_domain_gen openin_open openin_open_eq)
-qed
-
-lemma invariance_of_domain_homeomorphism:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "open S" "continuous_on S f" "DIM('b) \<le> DIM('a)" "inj_on f S"
- obtains g where "homeomorphism S (f ` S) f g"
-proof
- show "homeomorphism S (f ` S) f (inv_into S f)"
- by (simp add: assms continuous_on_inverse_open homeomorphism_def)
-qed
-
-corollary invariance_of_domain_homeomorphic:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "open S" "continuous_on S f" "DIM('b) \<le> DIM('a)" "inj_on f S"
- shows "S homeomorphic (f ` S)"
- using invariance_of_domain_homeomorphism [OF assms]
- by (meson homeomorphic_def)
-
-end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Further_Topology.thy Thu Oct 20 19:39:27 2016 +0200
@@ -0,0 +1,3097 @@
+section \<open>Extending Continous Maps, Invariance of Domain, etc..\<close>
+
+text\<open>Ported from HOL Light (moretop.ml) by L C Paulson\<close>
+
+theory Further_Topology
+ imports Equivalence_Lebesgue_Henstock_Integration Weierstrass_Theorems Polytope Complex_Transcendental
+begin
+
+subsection\<open>A map from a sphere to a higher dimensional sphere is nullhomotopic\<close>
+
+lemma spheremap_lemma1:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+ assumes "subspace S" "subspace T" and dimST: "dim S < dim T"
+ and "S \<subseteq> T"
+ and diff_f: "f differentiable_on sphere 0 1 \<inter> S"
+ shows "f ` (sphere 0 1 \<inter> S) \<noteq> sphere 0 1 \<inter> T"
+proof
+ assume fim: "f ` (sphere 0 1 \<inter> S) = sphere 0 1 \<inter> T"
+ have inS: "\<And>x. \<lbrakk>x \<in> S; x \<noteq> 0\<rbrakk> \<Longrightarrow> (x /\<^sub>R norm x) \<in> S"
+ using subspace_mul \<open>subspace S\<close> by blast
+ have subS01: "(\<lambda>x. x /\<^sub>R norm x) ` (S - {0}) \<subseteq> sphere 0 1 \<inter> S"
+ using \<open>subspace S\<close> subspace_mul by fastforce
+ then have diff_f': "f differentiable_on (\<lambda>x. x /\<^sub>R norm x) ` (S - {0})"
+ by (rule differentiable_on_subset [OF diff_f])
+ define g where "g \<equiv> \<lambda>x. norm x *\<^sub>R f(inverse(norm x) *\<^sub>R x)"
+ have gdiff: "g differentiable_on S - {0}"
+ unfolding g_def
+ by (rule diff_f' derivative_intros differentiable_on_compose [where f=f] | force)+
+ have geq: "g ` (S - {0}) = T - {0}"
+ proof
+ have "g ` (S - {0}) \<subseteq> T"
+ apply (auto simp: g_def subspace_mul [OF \<open>subspace T\<close>])
+ apply (metis (mono_tags, lifting) DiffI subS01 subspace_mul [OF \<open>subspace T\<close>] fim image_subset_iff inf_le2 singletonD)
+ done
+ moreover have "g ` (S - {0}) \<subseteq> UNIV - {0}"
+ proof (clarsimp simp: g_def)
+ fix y
+ assume "y \<in> S" and f0: "f (y /\<^sub>R norm y) = 0"
+ then have "y \<noteq> 0 \<Longrightarrow> y /\<^sub>R norm y \<in> sphere 0 1 \<inter> S"
+ by (auto simp: subspace_mul [OF \<open>subspace S\<close>])
+ then show "y = 0"
+ by (metis fim f0 Int_iff image_iff mem_sphere_0 norm_eq_zero zero_neq_one)
+ qed
+ ultimately show "g ` (S - {0}) \<subseteq> T - {0}"
+ by auto
+ next
+ have *: "sphere 0 1 \<inter> T \<subseteq> f ` (sphere 0 1 \<inter> S)"
+ using fim by (simp add: image_subset_iff)
+ have "x \<in> (\<lambda>x. norm x *\<^sub>R f (x /\<^sub>R norm x)) ` (S - {0})"
+ if "x \<in> T" "x \<noteq> 0" for x
+ proof -
+ have "x /\<^sub>R norm x \<in> T"
+ using \<open>subspace T\<close> subspace_mul that by blast
+ then show ?thesis
+ using * [THEN subsetD, of "x /\<^sub>R norm x"] that apply clarsimp
+ apply (rule_tac x="norm x *\<^sub>R xa" in image_eqI, simp)
+ apply (metis norm_eq_zero right_inverse scaleR_one scaleR_scaleR)
+ using \<open>subspace S\<close> subspace_mul apply force
+ done
+ qed
+ then have "T - {0} \<subseteq> (\<lambda>x. norm x *\<^sub>R f (x /\<^sub>R norm x)) ` (S - {0})"
+ by force
+ then show "T - {0} \<subseteq> g ` (S - {0})"
+ by (simp add: g_def)
+ qed
+ define T' where "T' \<equiv> {y. \<forall>x \<in> T. orthogonal x y}"
+ have "subspace T'"
+ by (simp add: subspace_orthogonal_to_vectors T'_def)
+ have dim_eq: "dim T' + dim T = DIM('a)"
+ using dim_subspace_orthogonal_to_vectors [of T UNIV] \<open>subspace T\<close>
+ by (simp add: dim_UNIV T'_def)
+ have "\<exists>v1 v2. v1 \<in> span T \<and> (\<forall>w \<in> span T. orthogonal v2 w) \<and> x = v1 + v2" for x
+ by (force intro: orthogonal_subspace_decomp_exists [of T x])
+ then obtain p1 p2 where p1span: "p1 x \<in> span T"
+ and "\<And>w. w \<in> span T \<Longrightarrow> orthogonal (p2 x) w"
+ and eq: "p1 x + p2 x = x" for x
+ by metis
+ then have p1: "\<And>z. p1 z \<in> T" and ortho: "\<And>w. w \<in> T \<Longrightarrow> orthogonal (p2 x) w" for x
+ using span_eq \<open>subspace T\<close> by blast+
+ then have p2: "\<And>z. p2 z \<in> T'"
+ by (simp add: T'_def orthogonal_commute)
+ have p12_eq: "\<And>x y. \<lbrakk>x \<in> T; y \<in> T'\<rbrakk> \<Longrightarrow> p1(x + y) = x \<and> p2(x + y) = y"
+ proof (rule orthogonal_subspace_decomp_unique [OF eq p1span, where T=T'])
+ show "\<And>x y. \<lbrakk>x \<in> T; y \<in> T'\<rbrakk> \<Longrightarrow> p2 (x + y) \<in> span T'"
+ using span_eq p2 \<open>subspace T'\<close> by blast
+ show "\<And>a b. \<lbrakk>a \<in> T; b \<in> T'\<rbrakk> \<Longrightarrow> orthogonal a b"
+ using T'_def by blast
+ qed (auto simp: span_superset)
+ then have "\<And>c x. p1 (c *\<^sub>R x) = c *\<^sub>R p1 x \<and> p2 (c *\<^sub>R x) = c *\<^sub>R p2 x"
+ by (metis eq \<open>subspace T\<close> \<open>subspace T'\<close> p1 p2 scaleR_add_right subspace_mul)
+ moreover have lin_add: "\<And>x y. p1 (x + y) = p1 x + p1 y \<and> p2 (x + y) = p2 x + p2 y"
+ proof (rule orthogonal_subspace_decomp_unique [OF _ p1span, where T=T'])
+ show "\<And>x y. p1 (x + y) + p2 (x + y) = p1 x + p1 y + (p2 x + p2 y)"
+ by (simp add: add.assoc add.left_commute eq)
+ show "\<And>a b. \<lbrakk>a \<in> T; b \<in> T'\<rbrakk> \<Longrightarrow> orthogonal a b"
+ using T'_def by blast
+ qed (auto simp: p1span p2 span_superset subspace_add)
+ ultimately have "linear p1" "linear p2"
+ by unfold_locales auto
+ have "(\<lambda>z. g (p1 z)) differentiable_on {x + y |x y. x \<in> S - {0} \<and> y \<in> T'}"
+ apply (rule differentiable_on_compose [where f=g])
+ apply (rule linear_imp_differentiable_on [OF \<open>linear p1\<close>])
+ apply (rule differentiable_on_subset [OF gdiff])
+ using p12_eq \<open>S \<subseteq> T\<close> apply auto
+ done
+ then have diff: "(\<lambda>x. g (p1 x) + p2 x) differentiable_on {x + y |x y. x \<in> S - {0} \<and> y \<in> T'}"
+ by (intro derivative_intros linear_imp_differentiable_on [OF \<open>linear p2\<close>])
+ have "dim {x + y |x y. x \<in> S - {0} \<and> y \<in> T'} \<le> dim {x + y |x y. x \<in> S \<and> y \<in> T'}"
+ by (blast intro: dim_subset)
+ also have "... = dim S + dim T' - dim (S \<inter> T')"
+ using dim_sums_Int [OF \<open>subspace S\<close> \<open>subspace T'\<close>]
+ by (simp add: algebra_simps)
+ also have "... < DIM('a)"
+ using dimST dim_eq by auto
+ finally have neg: "negligible {x + y |x y. x \<in> S - {0} \<and> y \<in> T'}"
+ by (rule negligible_lowdim)
+ have "negligible ((\<lambda>x. g (p1 x) + p2 x) ` {x + y |x y. x \<in> S - {0} \<and> y \<in> T'})"
+ by (rule negligible_differentiable_image_negligible [OF order_refl neg diff])
+ then have "negligible {x + y |x y. x \<in> g ` (S - {0}) \<and> y \<in> T'}"
+ proof (rule negligible_subset)
+ have "\<lbrakk>t' \<in> T'; s \<in> S; s \<noteq> 0\<rbrakk>
+ \<Longrightarrow> g s + t' \<in> (\<lambda>x. g (p1 x) + p2 x) `
+ {x + t' |x t'. x \<in> S \<and> x \<noteq> 0 \<and> t' \<in> T'}" for t' s
+ apply (rule_tac x="s + t'" in image_eqI)
+ using \<open>S \<subseteq> T\<close> p12_eq by auto
+ then show "{x + y |x y. x \<in> g ` (S - {0}) \<and> y \<in> T'}
+ \<subseteq> (\<lambda>x. g (p1 x) + p2 x) ` {x + y |x y. x \<in> S - {0} \<and> y \<in> T'}"
+ by auto
+ qed
+ moreover have "- T' \<subseteq> {x + y |x y. x \<in> g ` (S - {0}) \<and> y \<in> T'}"
+ proof clarsimp
+ fix z assume "z \<notin> T'"
+ show "\<exists>x y. z = x + y \<and> x \<in> g ` (S - {0}) \<and> y \<in> T'"
+ apply (rule_tac x="p1 z" in exI)
+ apply (rule_tac x="p2 z" in exI)
+ apply (simp add: p1 eq p2 geq)
+ by (metis \<open>z \<notin> T'\<close> add.left_neutral eq p2)
+ qed
+ ultimately have "negligible (-T')"
+ using negligible_subset by blast
+ moreover have "negligible T'"
+ using negligible_lowdim
+ by (metis add.commute assms(3) diff_add_inverse2 diff_self_eq_0 dim_eq le_add1 le_antisym linordered_semidom_class.add_diff_inverse not_less0)
+ ultimately have "negligible (-T' \<union> T')"
+ by (metis negligible_Un_eq)
+ then show False
+ using negligible_Un_eq non_negligible_UNIV by simp
+qed
+
+
+lemma spheremap_lemma2:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+ assumes ST: "subspace S" "subspace T" "dim S < dim T"
+ and "S \<subseteq> T"
+ and contf: "continuous_on (sphere 0 1 \<inter> S) f"
+ and fim: "f ` (sphere 0 1 \<inter> S) \<subseteq> sphere 0 1 \<inter> T"
+ shows "\<exists>c. homotopic_with (\<lambda>x. True) (sphere 0 1 \<inter> S) (sphere 0 1 \<inter> T) f (\<lambda>x. c)"
+proof -
+ have [simp]: "\<And>x. \<lbrakk>norm x = 1; x \<in> S\<rbrakk> \<Longrightarrow> norm (f x) = 1"
+ using fim by (simp add: image_subset_iff)
+ have "compact (sphere 0 1 \<inter> S)"
+ by (simp add: \<open>subspace S\<close> closed_subspace compact_Int_closed)
+ then obtain g where pfg: "polynomial_function g" and gim: "g ` (sphere 0 1 \<inter> S) \<subseteq> T"
+ and g12: "\<And>x. x \<in> sphere 0 1 \<inter> S \<Longrightarrow> norm(f x - g x) < 1/2"
+ apply (rule Stone_Weierstrass_polynomial_function_subspace [OF _ contf _ \<open>subspace T\<close>, of "1/2"])
+ using fim apply auto
+ done
+ have gnz: "g x \<noteq> 0" if "x \<in> sphere 0 1 \<inter> S" for x
+ proof -
+ have "norm (f x) = 1"
+ using fim that by (simp add: image_subset_iff)
+ then show ?thesis
+ using g12 [OF that] by auto
+ qed
+ have diffg: "g differentiable_on sphere 0 1 \<inter> S"
+ by (metis pfg differentiable_on_polynomial_function)
+ define h where "h \<equiv> \<lambda>x. inverse(norm(g x)) *\<^sub>R g x"
+ have h: "x \<in> sphere 0 1 \<inter> S \<Longrightarrow> h x \<in> sphere 0 1 \<inter> T" for x
+ unfolding h_def
+ using gnz [of x]
+ by (auto simp: subspace_mul [OF \<open>subspace T\<close>] subsetD [OF gim])
+ have diffh: "h differentiable_on sphere 0 1 \<inter> S"
+ unfolding h_def
+ apply (intro derivative_intros diffg differentiable_on_compose [OF diffg])
+ using gnz apply auto
+ done
+ have homfg: "homotopic_with (\<lambda>z. True) (sphere 0 1 \<inter> S) (T - {0}) f g"
+ proof (rule homotopic_with_linear [OF contf])
+ show "continuous_on (sphere 0 1 \<inter> S) g"
+ using pfg by (simp add: differentiable_imp_continuous_on diffg)
+ next
+ have non0fg: "0 \<notin> closed_segment (f x) (g x)" if "norm x = 1" "x \<in> S" for x
+ proof -
+ have "f x \<in> sphere 0 1"
+ using fim that by (simp add: image_subset_iff)
+ moreover have "norm(f x - g x) < 1/2"
+ apply (rule g12)
+ using that by force
+ ultimately show ?thesis
+ by (auto simp: norm_minus_commute dest: segment_bound)
+ qed
+ show "\<And>x. x \<in> sphere 0 1 \<inter> S \<Longrightarrow> closed_segment (f x) (g x) \<subseteq> T - {0}"
+ apply (simp add: subset_Diff_insert non0fg)
+ apply (simp add: segment_convex_hull)
+ apply (rule hull_minimal)
+ using fim image_eqI gim apply force
+ apply (rule subspace_imp_convex [OF \<open>subspace T\<close>])
+ done
+ qed
+ obtain d where d: "d \<in> (sphere 0 1 \<inter> T) - h ` (sphere 0 1 \<inter> S)"
+ using h spheremap_lemma1 [OF ST \<open>S \<subseteq> T\<close> diffh] by force
+ then have non0hd: "0 \<notin> closed_segment (h x) (- d)" if "norm x = 1" "x \<in> S" for x
+ using midpoint_between [of 0 "h x" "-d"] that h [of x]
+ by (auto simp: between_mem_segment midpoint_def)
+ have conth: "continuous_on (sphere 0 1 \<inter> S) h"
+ using differentiable_imp_continuous_on diffh by blast
+ have hom_hd: "homotopic_with (\<lambda>z. True) (sphere 0 1 \<inter> S) (T - {0}) h (\<lambda>x. -d)"
+ apply (rule homotopic_with_linear [OF conth continuous_on_const])
+ apply (simp add: subset_Diff_insert non0hd)
+ apply (simp add: segment_convex_hull)
+ apply (rule hull_minimal)
+ using h d apply (force simp: subspace_neg [OF \<open>subspace T\<close>])
+ apply (rule subspace_imp_convex [OF \<open>subspace T\<close>])
+ done
+ have conT0: "continuous_on (T - {0}) (\<lambda>y. inverse(norm y) *\<^sub>R y)"
+ by (intro continuous_intros) auto
+ have sub0T: "(\<lambda>y. y /\<^sub>R norm y) ` (T - {0}) \<subseteq> sphere 0 1 \<inter> T"
+ by (fastforce simp: assms(2) subspace_mul)
+ obtain c where homhc: "homotopic_with (\<lambda>z. True) (sphere 0 1 \<inter> S) (sphere 0 1 \<inter> T) h (\<lambda>x. c)"
+ apply (rule_tac c="-d" in that)
+ apply (rule homotopic_with_eq)
+ apply (rule homotopic_compose_continuous_left [OF hom_hd conT0 sub0T])
+ using d apply (auto simp: h_def)
+ done
+ show ?thesis
+ apply (rule_tac x=c in exI)
+ apply (rule homotopic_with_trans [OF _ homhc])
+ apply (rule homotopic_with_eq)
+ apply (rule homotopic_compose_continuous_left [OF homfg conT0 sub0T])
+ apply (auto simp: h_def)
+ done
+qed
+
+
+lemma spheremap_lemma3:
+ assumes "bounded S" "convex S" "subspace U" and affSU: "aff_dim S \<le> dim U"
+ obtains T where "subspace T" "T \<subseteq> U" "S \<noteq> {} \<Longrightarrow> aff_dim T = aff_dim S"
+ "(rel_frontier S) homeomorphic (sphere 0 1 \<inter> T)"
+proof (cases "S = {}")
+ case True
+ with \<open>subspace U\<close> subspace_0 show ?thesis
+ by (rule_tac T = "{0}" in that) auto
+next
+ case False
+ then obtain a where "a \<in> S"
+ by auto
+ then have affS: "aff_dim S = int (dim ((\<lambda>x. -a+x) ` S))"
+ by (metis hull_inc aff_dim_eq_dim)
+ with affSU have "dim ((\<lambda>x. -a+x) ` S) \<le> dim U"
+ by linarith
+ with choose_subspace_of_subspace
+ obtain T where "subspace T" "T \<subseteq> span U" and dimT: "dim T = dim ((\<lambda>x. -a+x) ` S)" .
+ show ?thesis
+ proof (rule that [OF \<open>subspace T\<close>])
+ show "T \<subseteq> U"
+ using span_eq \<open>subspace U\<close> \<open>T \<subseteq> span U\<close> by blast
+ show "aff_dim T = aff_dim S"
+ using dimT \<open>subspace T\<close> affS aff_dim_subspace by fastforce
+ show "rel_frontier S homeomorphic sphere 0 1 \<inter> T"
+ proof -
+ have "aff_dim (ball 0 1 \<inter> T) = aff_dim (T)"
+ by (metis IntI interior_ball \<open>subspace T\<close> aff_dim_convex_Int_nonempty_interior centre_in_ball empty_iff inf_commute subspace_0 subspace_imp_convex zero_less_one)
+ then have affS_eq: "aff_dim S = aff_dim (ball 0 1 \<inter> T)"
+ using \<open>aff_dim T = aff_dim S\<close> by simp
+ have "rel_frontier S homeomorphic rel_frontier(ball 0 1 \<inter> T)"
+ apply (rule homeomorphic_rel_frontiers_convex_bounded_sets [OF \<open>convex S\<close> \<open>bounded S\<close>])
+ apply (simp add: \<open>subspace T\<close> convex_Int subspace_imp_convex)
+ apply (simp add: bounded_Int)
+ apply (rule affS_eq)
+ done
+ also have "... = frontier (ball 0 1) \<inter> T"
+ apply (rule convex_affine_rel_frontier_Int [OF convex_ball])
+ apply (simp add: \<open>subspace T\<close> subspace_imp_affine)
+ using \<open>subspace T\<close> subspace_0 by force
+ also have "... = sphere 0 1 \<inter> T"
+ by auto
+ finally show ?thesis .
+ qed
+ qed
+qed
+
+
+proposition inessential_spheremap_lowdim_gen:
+ fixes f :: "'M::euclidean_space \<Rightarrow> 'a::euclidean_space"
+ assumes "convex S" "bounded S" "convex T" "bounded T"
+ and affST: "aff_dim S < aff_dim T"
+ and contf: "continuous_on (rel_frontier S) f"
+ and fim: "f ` (rel_frontier S) \<subseteq> rel_frontier T"
+ obtains c where "homotopic_with (\<lambda>z. True) (rel_frontier S) (rel_frontier T) f (\<lambda>x. c)"
+proof (cases "S = {}")
+ case True
+ then show ?thesis
+ by (simp add: that)
+next
+ case False
+ then show ?thesis
+ proof (cases "T = {}")
+ case True
+ then show ?thesis
+ using fim that by auto
+ next
+ case False
+ obtain T':: "'a set"
+ where "subspace T'" and affT': "aff_dim T' = aff_dim T"
+ and homT: "rel_frontier T homeomorphic sphere 0 1 \<inter> T'"
+ apply (rule spheremap_lemma3 [OF \<open>bounded T\<close> \<open>convex T\<close> subspace_UNIV, where 'b='a])
+ apply (simp add: dim_UNIV aff_dim_le_DIM)
+ using \<open>T \<noteq> {}\<close> by blast
+ with homeomorphic_imp_homotopy_eqv
+ have relT: "sphere 0 1 \<inter> T' homotopy_eqv rel_frontier T"
+ using homotopy_eqv_sym by blast
+ have "aff_dim S \<le> int (dim T')"
+ using affT' \<open>subspace T'\<close> affST aff_dim_subspace by force
+ with spheremap_lemma3 [OF \<open>bounded S\<close> \<open>convex S\<close> \<open>subspace T'\<close>] \<open>S \<noteq> {}\<close>
+ obtain S':: "'a set" where "subspace S'" "S' \<subseteq> T'"
+ and affS': "aff_dim S' = aff_dim S"
+ and homT: "rel_frontier S homeomorphic sphere 0 1 \<inter> S'"
+ by metis
+ with homeomorphic_imp_homotopy_eqv
+ have relS: "sphere 0 1 \<inter> S' homotopy_eqv rel_frontier S"
+ using homotopy_eqv_sym by blast
+ have dimST': "dim S' < dim T'"
+ by (metis \<open>S' \<subseteq> T'\<close> \<open>subspace S'\<close> \<open>subspace T'\<close> affS' affST affT' less_irrefl not_le subspace_dim_equal)
+ have "\<exists>c. homotopic_with (\<lambda>z. True) (rel_frontier S) (rel_frontier T) f (\<lambda>x. c)"
+ apply (rule homotopy_eqv_homotopic_triviality_null_imp [OF relT contf fim])
+ apply (rule homotopy_eqv_cohomotopic_triviality_null[OF relS, THEN iffD1, rule_format])
+ apply (metis dimST' \<open>subspace S'\<close> \<open>subspace T'\<close> \<open>S' \<subseteq> T'\<close> spheremap_lemma2, blast)
+ done
+ with that show ?thesis by blast
+ qed
+qed
+
+lemma inessential_spheremap_lowdim:
+ fixes f :: "'M::euclidean_space \<Rightarrow> 'a::euclidean_space"
+ assumes
+ "DIM('M) < DIM('a)" and f: "continuous_on (sphere a r) f" "f ` (sphere a r) \<subseteq> (sphere b s)"
+ obtains c where "homotopic_with (\<lambda>z. True) (sphere a r) (sphere b s) f (\<lambda>x. c)"
+proof (cases "s \<le> 0")
+ case True then show ?thesis
+ by (meson nullhomotopic_into_contractible f contractible_sphere that)
+next
+ case False
+ show ?thesis
+ proof (cases "r \<le> 0")
+ case True then show ?thesis
+ by (meson f nullhomotopic_from_contractible contractible_sphere that)
+ next
+ case False
+ with \<open>~ s \<le> 0\<close> have "r > 0" "s > 0" by auto
+ show ?thesis
+ apply (rule inessential_spheremap_lowdim_gen [of "cball a r" "cball b s" f])
+ using \<open>0 < r\<close> \<open>0 < s\<close> assms(1)
+ apply (simp_all add: f aff_dim_cball)
+ using that by blast
+ qed
+qed
+
+
+
+subsection\<open> Some technical lemmas about extending maps from cell complexes.\<close>
+
+lemma extending_maps_Union_aux:
+ assumes fin: "finite \<F>"
+ and "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"
+ and "\<And>S T. \<lbrakk>S \<in> \<F>; T \<in> \<F>; S \<noteq> T\<rbrakk> \<Longrightarrow> S \<inter> T \<subseteq> K"
+ and "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>g. continuous_on S g \<and> g ` S \<subseteq> T \<and> (\<forall>x \<in> S \<inter> K. g x = h x)"
+ shows "\<exists>g. continuous_on (\<Union>\<F>) g \<and> g ` (\<Union>\<F>) \<subseteq> T \<and> (\<forall>x \<in> \<Union>\<F> \<inter> K. g x = h x)"
+using assms
+proof (induction \<F>)
+ case empty show ?case by simp
+next
+ case (insert S \<F>)
+ then obtain f where contf: "continuous_on (S) f" and fim: "f ` S \<subseteq> T" and feq: "\<forall>x \<in> S \<inter> K. f x = h x"
+ by (meson insertI1)
+ obtain g where contg: "continuous_on (\<Union>\<F>) g" and gim: "g ` \<Union>\<F> \<subseteq> T" and geq: "\<forall>x \<in> \<Union>\<F> \<inter> K. g x = h x"
+ using insert by auto
+ have fg: "f x = g x" if "x \<in> T" "T \<in> \<F>" "x \<in> S" for x T
+ proof -
+ have "T \<inter> S \<subseteq> K \<or> S = T"
+ using that by (metis (no_types) insert.prems(2) insertCI)
+ then show ?thesis
+ using UnionI feq geq \<open>S \<notin> \<F>\<close> subsetD that by fastforce
+ qed
+ show ?case
+ apply (rule_tac x="\<lambda>x. if x \<in> S then f x else g x" in exI, simp)
+ apply (intro conjI continuous_on_cases)
+ apply (simp_all add: insert closed_Union contf contg)
+ using fim gim feq geq
+ apply (force simp: insert closed_Union contf contg inf_commute intro: fg)+
+ done
+qed
+
+lemma extending_maps_Union:
+ assumes fin: "finite \<F>"
+ and "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>g. continuous_on S g \<and> g ` S \<subseteq> T \<and> (\<forall>x \<in> S \<inter> K. g x = h x)"
+ and "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"
+ and K: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>; ~ X \<subseteq> Y; ~ Y \<subseteq> X\<rbrakk> \<Longrightarrow> X \<inter> Y \<subseteq> K"
+ shows "\<exists>g. continuous_on (\<Union>\<F>) g \<and> g ` (\<Union>\<F>) \<subseteq> T \<and> (\<forall>x \<in> \<Union>\<F> \<inter> K. g x = h x)"
+apply (simp add: Union_maximal_sets [OF fin, symmetric])
+apply (rule extending_maps_Union_aux)
+apply (simp_all add: Union_maximal_sets [OF fin] assms)
+by (metis K psubsetI)
+
+
+lemma extend_map_lemma:
+ assumes "finite \<F>" "\<G> \<subseteq> \<F>" "convex T" "bounded T"
+ and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
+ and aff: "\<And>X. X \<in> \<F> - \<G> \<Longrightarrow> aff_dim X < aff_dim T"
+ and face: "\<And>S T. \<lbrakk>S \<in> \<F>; T \<in> \<F>\<rbrakk> \<Longrightarrow> (S \<inter> T) face_of S \<and> (S \<inter> T) face_of T"
+ and contf: "continuous_on (\<Union>\<G>) f" and fim: "f ` (\<Union>\<G>) \<subseteq> rel_frontier T"
+ obtains g where "continuous_on (\<Union>\<F>) g" "g ` (\<Union>\<F>) \<subseteq> rel_frontier T" "\<And>x. x \<in> \<Union>\<G> \<Longrightarrow> g x = f x"
+proof (cases "\<F> - \<G> = {}")
+ case True
+ then have "\<Union>\<F> \<subseteq> \<Union>\<G>"
+ by (simp add: Union_mono)
+ then show ?thesis
+ apply (rule_tac g=f in that)
+ using contf continuous_on_subset apply blast
+ using fim apply blast
+ by simp
+next
+ case False
+ then have "0 \<le> aff_dim T"
+ by (metis aff aff_dim_empty aff_dim_geq aff_dim_negative_iff all_not_in_conv not_less)
+ then obtain i::nat where i: "int i = aff_dim T"
+ by (metis nonneg_eq_int)
+ have Union_empty_eq: "\<Union>{D. D = {} \<and> P D} = {}" for P :: "'a set \<Rightarrow> bool"
+ by auto
+ have extendf: "\<exists>g. continuous_on (\<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < i})) g \<and>
+ g ` (\<Union> (\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < i})) \<subseteq> rel_frontier T \<and>
+ (\<forall>x \<in> \<Union>\<G>. g x = f x)"
+ if "i \<le> aff_dim T" for i::nat
+ using that
+ proof (induction i)
+ case 0 then show ?case
+ apply (simp add: Union_empty_eq)
+ apply (rule_tac x=f in exI)
+ apply (intro conjI)
+ using contf continuous_on_subset apply blast
+ using fim apply blast
+ by simp
+ next
+ case (Suc p)
+ with \<open>bounded T\<close> have "rel_frontier T \<noteq> {}"
+ by (auto simp: rel_frontier_eq_empty affine_bounded_eq_lowdim [of T])
+ then obtain t where t: "t \<in> rel_frontier T" by auto
+ have ple: "int p \<le> aff_dim T" using Suc.prems by force
+ obtain h where conth: "continuous_on (\<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p})) h"
+ and him: "h ` (\<Union> (\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p}))
+ \<subseteq> rel_frontier T"
+ and heq: "\<And>x. x \<in> \<Union>\<G> \<Longrightarrow> h x = f x"
+ using Suc.IH [OF ple] by auto
+ let ?Faces = "{D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D \<le> p}"
+ have extendh: "\<exists>g. continuous_on D g \<and>
+ g ` D \<subseteq> rel_frontier T \<and>
+ (\<forall>x \<in> D \<inter> \<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p}). g x = h x)"
+ if D: "D \<in> \<G> \<union> ?Faces" for D
+ proof (cases "D \<subseteq> \<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p})")
+ case True
+ then show ?thesis
+ apply (rule_tac x=h in exI)
+ apply (intro conjI)
+ apply (blast intro: continuous_on_subset [OF conth])
+ using him apply blast
+ by simp
+ next
+ case False
+ note notDsub = False
+ show ?thesis
+ proof (cases "\<exists>a. D = {a}")
+ case True
+ then obtain a where "D = {a}" by auto
+ with notDsub t show ?thesis
+ by (rule_tac x="\<lambda>x. t" in exI) simp
+ next
+ case False
+ have "D \<noteq> {}" using notDsub by auto
+ have Dnotin: "D \<notin> \<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p}"
+ using notDsub by auto
+ then have "D \<notin> \<G>" by simp
+ have "D \<in> ?Faces - {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p}"
+ using Dnotin that by auto
+ then obtain C where "C \<in> \<F>" "D face_of C" and affD: "aff_dim D = int p"
+ by auto
+ then have "bounded D"
+ using face_of_polytope_polytope poly polytope_imp_bounded by blast
+ then have [simp]: "\<not> affine D"
+ using affine_bounded_eq_trivial False \<open>D \<noteq> {}\<close> \<open>bounded D\<close> by blast
+ have "{F. F facet_of D} \<subseteq> {E. E face_of C \<and> aff_dim E < int p}"
+ apply clarify
+ apply (metis \<open>D face_of C\<close> affD eq_iff face_of_trans facet_of_def zle_diff1_eq)
+ done
+ moreover have "polyhedron D"
+ using \<open>C \<in> \<F>\<close> \<open>D face_of C\<close> face_of_polytope_polytope poly polytope_imp_polyhedron by auto
+ ultimately have relf_sub: "rel_frontier D \<subseteq> \<Union> {E. E face_of C \<and> aff_dim E < p}"
+ by (simp add: rel_frontier_of_polyhedron Union_mono)
+ then have him_relf: "h ` rel_frontier D \<subseteq> rel_frontier T"
+ using \<open>C \<in> \<F>\<close> him by blast
+ have "convex D"
+ by (simp add: \<open>polyhedron D\<close> polyhedron_imp_convex)
+ have affD_lessT: "aff_dim D < aff_dim T"
+ using Suc.prems affD by linarith
+ have contDh: "continuous_on (rel_frontier D) h"
+ using \<open>C \<in> \<F>\<close> relf_sub by (blast intro: continuous_on_subset [OF conth])
+ then have *: "(\<exists>c. homotopic_with (\<lambda>x. True) (rel_frontier D) (rel_frontier T) h (\<lambda>x. c)) =
+ (\<exists>g. continuous_on UNIV g \<and> range g \<subseteq> rel_frontier T \<and>
+ (\<forall>x\<in>rel_frontier D. g x = h x))"
+ apply (rule nullhomotopic_into_rel_frontier_extension [OF closed_rel_frontier])
+ apply (simp_all add: assms rel_frontier_eq_empty him_relf)
+ done
+ have "(\<exists>c. homotopic_with (\<lambda>x. True) (rel_frontier D)
+ (rel_frontier T) h (\<lambda>x. c))"
+ by (metis inessential_spheremap_lowdim_gen
+ [OF \<open>convex D\<close> \<open>bounded D\<close> \<open>convex T\<close> \<open>bounded T\<close> affD_lessT contDh him_relf])
+ then obtain g where contg: "continuous_on UNIV g"
+ and gim: "range g \<subseteq> rel_frontier T"
+ and gh: "\<And>x. x \<in> rel_frontier D \<Longrightarrow> g x = h x"
+ by (metis *)
+ have "D \<inter> E \<subseteq> rel_frontier D"
+ if "E \<in> \<G> \<union> {D. Bex \<F> (op face_of D) \<and> aff_dim D < int p}" for E
+ proof (rule face_of_subset_rel_frontier)
+ show "D \<inter> E face_of D"
+ using that \<open>C \<in> \<F>\<close> \<open>D face_of C\<close> face
+ apply auto
+ apply (meson face_of_Int_subface \<open>\<G> \<subseteq> \<F>\<close> face_of_refl_eq poly polytope_imp_convex subsetD)
+ using face_of_Int_subface apply blast
+ done
+ show "D \<inter> E \<noteq> D"
+ using that notDsub by auto
+ qed
+ then show ?thesis
+ apply (rule_tac x=g in exI)
+ apply (intro conjI ballI)
+ using continuous_on_subset contg apply blast
+ using gim apply blast
+ using gh by fastforce
+ qed
+ qed
+ have intle: "i < 1 + int j \<longleftrightarrow> i \<le> int j" for i j
+ by auto
+ have "finite \<G>"
+ using \<open>finite \<F>\<close> \<open>\<G> \<subseteq> \<F>\<close> rev_finite_subset by blast
+ then have fin: "finite (\<G> \<union> ?Faces)"
+ apply simp
+ apply (rule_tac B = "\<Union>{{D. D face_of C}| C. C \<in> \<F>}" in finite_subset)
+ by (auto simp: \<open>finite \<F>\<close> finite_polytope_faces poly)
+ have clo: "closed S" if "S \<in> \<G> \<union> ?Faces" for S
+ using that \<open>\<G> \<subseteq> \<F>\<close> face_of_polytope_polytope poly polytope_imp_closed by blast
+ have K: "X \<inter> Y \<subseteq> \<Union>(\<G> \<union> {D. \<exists>C\<in>\<F>. D face_of C \<and> aff_dim D < int p})"
+ if "X \<in> \<G> \<union> ?Faces" "Y \<in> \<G> \<union> ?Faces" "~ Y \<subseteq> X" for X Y
+ proof -
+ have ff: "X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+ if XY: "X face_of D" "Y face_of E" and DE: "D \<in> \<F>" "E \<in> \<F>" for D E
+ apply (rule face_of_Int_subface [OF _ _ XY])
+ apply (auto simp: face DE)
+ done
+ show ?thesis
+ using that
+ apply auto
+ apply (drule_tac x="X \<inter> Y" in spec, safe)
+ using ff face_of_imp_convex [of X] face_of_imp_convex [of Y]
+ apply (fastforce dest: face_of_aff_dim_lt)
+ by (meson face_of_trans ff)
+ qed
+ obtain g where "continuous_on (\<Union>(\<G> \<union> ?Faces)) g"
+ "g ` \<Union>(\<G> \<union> ?Faces) \<subseteq> rel_frontier T"
+ "(\<forall>x \<in> \<Union>(\<G> \<union> ?Faces) \<inter>
+ \<Union>(\<G> \<union> {D. \<exists>C\<in>\<F>. D face_of C \<and> aff_dim D < p}). g x = h x)"
+ apply (rule exE [OF extending_maps_Union [OF fin extendh clo K]], blast+)
+ done
+ then show ?case
+ apply (simp add: intle local.heq [symmetric], blast)
+ done
+ qed
+ have eq: "\<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < i}) = \<Union>\<F>"
+ proof
+ show "\<Union>(\<G> \<union> {D. \<exists>C\<in>\<F>. D face_of C \<and> aff_dim D < int i}) \<subseteq> \<Union>\<F>"
+ apply (rule Union_subsetI)
+ using \<open>\<G> \<subseteq> \<F>\<close> face_of_imp_subset apply force
+ done
+ show "\<Union>\<F> \<subseteq> \<Union>(\<G> \<union> {D. \<exists>C\<in>\<F>. D face_of C \<and> aff_dim D < i})"
+ apply (rule Union_mono)
+ using face apply (fastforce simp: aff i)
+ done
+ qed
+ have "int i \<le> aff_dim T" by (simp add: i)
+ then show ?thesis
+ using extendf [of i] unfolding eq by (metis that)
+qed
+
+lemma extend_map_lemma_cofinite0:
+ assumes "finite \<F>"
+ and "pairwise (\<lambda>S T. S \<inter> T \<subseteq> K) \<F>"
+ and "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>a g. a \<notin> U \<and> continuous_on (S - {a}) g \<and> g ` (S - {a}) \<subseteq> T \<and> (\<forall>x \<in> S \<inter> K. g x = h x)"
+ and "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"
+ shows "\<exists>C g. finite C \<and> disjnt C U \<and> card C \<le> card \<F> \<and>
+ continuous_on (\<Union>\<F> - C) g \<and> g ` (\<Union>\<F> - C) \<subseteq> T
+ \<and> (\<forall>x \<in> (\<Union>\<F> - C) \<inter> K. g x = h x)"
+ using assms
+proof induction
+ case empty then show ?case
+ by force
+next
+ case (insert X \<F>)
+ then have "closed X" and clo: "\<And>X. X \<in> \<F> \<Longrightarrow> closed X"
+ and \<F>: "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>a g. a \<notin> U \<and> continuous_on (S - {a}) g \<and> g ` (S - {a}) \<subseteq> T \<and> (\<forall>x \<in> S \<inter> K. g x = h x)"
+ and pwX: "\<And>Y. Y \<in> \<F> \<and> Y \<noteq> X \<longrightarrow> X \<inter> Y \<subseteq> K \<and> Y \<inter> X \<subseteq> K"
+ and pwF: "pairwise (\<lambda> S T. S \<inter> T \<subseteq> K) \<F>"
+ by (simp_all add: pairwise_insert)
+ obtain C g where C: "finite C" "disjnt C U" "card C \<le> card \<F>"
+ and contg: "continuous_on (\<Union>\<F> - C) g"
+ and gim: "g ` (\<Union>\<F> - C) \<subseteq> T"
+ and gh: "\<And>x. x \<in> (\<Union>\<F> - C) \<inter> K \<Longrightarrow> g x = h x"
+ using insert.IH [OF pwF \<F> clo] by auto
+ obtain a f where "a \<notin> U"
+ and contf: "continuous_on (X - {a}) f"
+ and fim: "f ` (X - {a}) \<subseteq> T"
+ and fh: "(\<forall>x \<in> X \<inter> K. f x = h x)"
+ using insert.prems by (meson insertI1)
+ show ?case
+ proof (intro exI conjI)
+ show "finite (insert a C)"
+ by (simp add: C)
+ show "disjnt (insert a C) U"
+ using C \<open>a \<notin> U\<close> by simp
+ show "card (insert a C) \<le> card (insert X \<F>)"
+ by (simp add: C card_insert_if insert.hyps le_SucI)
+ have "closed (\<Union>\<F>)"
+ using clo insert.hyps by blast
+ have "continuous_on (X - insert a C \<union> (\<Union>\<F> - insert a C)) (\<lambda>x. if x \<in> X then f x else g x)"
+ apply (rule continuous_on_cases_local)
+ apply (simp_all add: closedin_closed)
+ using \<open>closed X\<close> apply blast
+ using \<open>closed (\<Union>\<F>)\<close> apply blast
+ using contf apply (force simp: elim: continuous_on_subset)
+ using contg apply (force simp: elim: continuous_on_subset)
+ using fh gh insert.hyps pwX by fastforce
+ then show "continuous_on (\<Union>insert X \<F> - insert a C) (\<lambda>a. if a \<in> X then f a else g a)"
+ by (blast intro: continuous_on_subset)
+ show "\<forall>x\<in>(\<Union>insert X \<F> - insert a C) \<inter> K. (if x \<in> X then f x else g x) = h x"
+ using gh by (auto simp: fh)
+ show "(\<lambda>a. if a \<in> X then f a else g a) ` (\<Union>insert X \<F> - insert a C) \<subseteq> T"
+ using fim gim by auto force
+ qed
+qed
+
+
+lemma extend_map_lemma_cofinite1:
+assumes "finite \<F>"
+ and \<F>: "\<And>X. X \<in> \<F> \<Longrightarrow> \<exists>a g. a \<notin> U \<and> continuous_on (X - {a}) g \<and> g ` (X - {a}) \<subseteq> T \<and> (\<forall>x \<in> X \<inter> K. g x = h x)"
+ and clo: "\<And>X. X \<in> \<F> \<Longrightarrow> closed X"
+ and K: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>; ~(X \<subseteq> Y); ~(Y \<subseteq> X)\<rbrakk> \<Longrightarrow> X \<inter> Y \<subseteq> K"
+ obtains C g where "finite C" "disjnt C U" "card C \<le> card \<F>" "continuous_on (\<Union>\<F> - C) g"
+ "g ` (\<Union>\<F> - C) \<subseteq> T"
+ "\<And>x. x \<in> (\<Union>\<F> - C) \<inter> K \<Longrightarrow> g x = h x"
+proof -
+ let ?\<F> = "{X \<in> \<F>. \<forall>Y\<in>\<F>. \<not> X \<subset> Y}"
+ have [simp]: "\<Union>?\<F> = \<Union>\<F>"
+ by (simp add: Union_maximal_sets assms)
+ have fin: "finite ?\<F>"
+ by (force intro: finite_subset [OF _ \<open>finite \<F>\<close>])
+ have pw: "pairwise (\<lambda> S T. S \<inter> T \<subseteq> K) ?\<F>"
+ by (simp add: pairwise_def) (metis K psubsetI)
+ have "card {X \<in> \<F>. \<forall>Y\<in>\<F>. \<not> X \<subset> Y} \<le> card \<F>"
+ by (simp add: \<open>finite \<F>\<close> card_mono)
+ moreover
+ obtain C g where "finite C \<and> disjnt C U \<and> card C \<le> card ?\<F> \<and>
+ continuous_on (\<Union>?\<F> - C) g \<and> g ` (\<Union>?\<F> - C) \<subseteq> T
+ \<and> (\<forall>x \<in> (\<Union>?\<F> - C) \<inter> K. g x = h x)"
+ apply (rule exE [OF extend_map_lemma_cofinite0 [OF fin pw, of U T h]])
+ apply (fastforce intro!: clo \<F>)+
+ done
+ ultimately show ?thesis
+ by (rule_tac C=C and g=g in that) auto
+qed
+
+
+lemma extend_map_lemma_cofinite:
+ assumes "finite \<F>" "\<G> \<subseteq> \<F>" and T: "convex T" "bounded T"
+ and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
+ and contf: "continuous_on (\<Union>\<G>) f" and fim: "f ` (\<Union>\<G>) \<subseteq> rel_frontier T"
+ and face: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> (X \<inter> Y) face_of X \<and> (X \<inter> Y) face_of Y"
+ and aff: "\<And>X. X \<in> \<F> - \<G> \<Longrightarrow> aff_dim X \<le> aff_dim T"
+ obtains C g where
+ "finite C" "disjnt C (\<Union>\<G>)" "card C \<le> card \<F>" "continuous_on (\<Union>\<F> - C) g"
+ "g ` (\<Union> \<F> - C) \<subseteq> rel_frontier T" "\<And>x. x \<in> \<Union>\<G> \<Longrightarrow> g x = f x"
+proof -
+ define \<H> where "\<H> \<equiv> \<G> \<union> {D. \<exists>C \<in> \<F> - \<G>. D face_of C \<and> aff_dim D < aff_dim T}"
+ have "finite \<G>"
+ using assms finite_subset by blast
+ moreover have "finite (\<Union>{{D. D face_of C} |C. C \<in> \<F>})"
+ apply (rule finite_Union)
+ apply (simp add: \<open>finite \<F>\<close>)
+ using finite_polytope_faces poly by auto
+ ultimately have "finite \<H>"
+ apply (simp add: \<H>_def)
+ apply (rule finite_subset [of _ "\<Union> {{D. D face_of C} | C. C \<in> \<F>}"], auto)
+ done
+ have *: "\<And>X Y. \<lbrakk>X \<in> \<H>; Y \<in> \<H>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+ unfolding \<H>_def
+ apply (elim UnE bexE CollectE DiffE)
+ using subsetD [OF \<open>\<G> \<subseteq> \<F>\<close>] apply (simp_all add: face)
+ apply (meson subsetD [OF \<open>\<G> \<subseteq> \<F>\<close>] face face_of_Int_subface face_of_imp_subset face_of_refl poly polytope_imp_convex)+
+ done
+ obtain h where conth: "continuous_on (\<Union>\<H>) h" and him: "h ` (\<Union>\<H>) \<subseteq> rel_frontier T"
+ and hf: "\<And>x. x \<in> \<Union>\<G> \<Longrightarrow> h x = f x"
+ using \<open>finite \<H>\<close>
+ unfolding \<H>_def
+ apply (rule extend_map_lemma [OF _ Un_upper1 T _ _ _ contf fim])
+ using \<open>\<G> \<subseteq> \<F>\<close> face_of_polytope_polytope poly apply fastforce
+ using * apply (auto simp: \<H>_def)
+ done
+ have "bounded (\<Union>\<G>)"
+ using \<open>finite \<G>\<close> \<open>\<G> \<subseteq> \<F>\<close> poly polytope_imp_bounded by blast
+ then have "\<Union>\<G> \<noteq> UNIV"
+ by auto
+ then obtain a where a: "a \<notin> \<Union>\<G>"
+ by blast
+ have \<F>: "\<exists>a g. a \<notin> \<Union>\<G> \<and> continuous_on (D - {a}) g \<and>
+ g ` (D - {a}) \<subseteq> rel_frontier T \<and> (\<forall>x \<in> D \<inter> \<Union>\<H>. g x = h x)"
+ if "D \<in> \<F>" for D
+ proof (cases "D \<subseteq> \<Union>\<H>")
+ case True
+ then show ?thesis
+ apply (rule_tac x=a in exI)
+ apply (rule_tac x=h in exI)
+ using him apply (blast intro!: \<open>a \<notin> \<Union>\<G>\<close> continuous_on_subset [OF conth]) +
+ done
+ next
+ case False
+ note D_not_subset = False
+ show ?thesis
+ proof (cases "D \<in> \<G>")
+ case True
+ with D_not_subset show ?thesis
+ by (auto simp: \<H>_def)
+ next
+ case False
+ then have affD: "aff_dim D \<le> aff_dim T"
+ by (simp add: \<open>D \<in> \<F>\<close> aff)
+ show ?thesis
+ proof (cases "rel_interior D = {}")
+ case True
+ with \<open>D \<in> \<F>\<close> poly a show ?thesis
+ by (force simp: rel_interior_eq_empty polytope_imp_convex)
+ next
+ case False
+ then obtain b where brelD: "b \<in> rel_interior D"
+ by blast
+ have "polyhedron D"
+ by (simp add: poly polytope_imp_polyhedron that)
+ have "rel_frontier D retract_of affine hull D - {b}"
+ by (simp add: rel_frontier_retract_of_punctured_affine_hull poly polytope_imp_bounded polytope_imp_convex that brelD)
+ then obtain r where relfD: "rel_frontier D \<subseteq> affine hull D - {b}"
+ and contr: "continuous_on (affine hull D - {b}) r"
+ and rim: "r ` (affine hull D - {b}) \<subseteq> rel_frontier D"
+ and rid: "\<And>x. x \<in> rel_frontier D \<Longrightarrow> r x = x"
+ by (auto simp: retract_of_def retraction_def)
+ show ?thesis
+ proof (intro exI conjI ballI)
+ show "b \<notin> \<Union>\<G>"
+ proof clarify
+ fix E
+ assume "b \<in> E" "E \<in> \<G>"
+ then have "E \<inter> D face_of E \<and> E \<inter> D face_of D"
+ using \<open>\<G> \<subseteq> \<F>\<close> face that by auto
+ with face_of_subset_rel_frontier \<open>E \<in> \<G>\<close> \<open>b \<in> E\<close> brelD rel_interior_subset [of D]
+ D_not_subset rel_frontier_def \<H>_def
+ show False
+ by blast
+ qed
+ have "r ` (D - {b}) \<subseteq> r ` (affine hull D - {b})"
+ by (simp add: Diff_mono hull_subset image_mono)
+ also have "... \<subseteq> rel_frontier D"
+ by (rule rim)
+ also have "... \<subseteq> \<Union>{E. E face_of D \<and> aff_dim E < aff_dim T}"
+ using affD
+ by (force simp: rel_frontier_of_polyhedron [OF \<open>polyhedron D\<close>] facet_of_def)
+ also have "... \<subseteq> \<Union>(\<H>)"
+ using D_not_subset \<H>_def that by fastforce
+ finally have rsub: "r ` (D - {b}) \<subseteq> \<Union>(\<H>)" .
+ show "continuous_on (D - {b}) (h \<circ> r)"
+ apply (intro conjI \<open>b \<notin> \<Union>\<G>\<close> continuous_on_compose)
+ apply (rule continuous_on_subset [OF contr])
+ apply (simp add: Diff_mono hull_subset)
+ apply (rule continuous_on_subset [OF conth rsub])
+ done
+ show "(h \<circ> r) ` (D - {b}) \<subseteq> rel_frontier T"
+ using brelD him rsub by fastforce
+ show "(h \<circ> r) x = h x" if x: "x \<in> D \<inter> \<Union>\<H>" for x
+ proof -
+ consider A where "x \<in> D" "A \<in> \<G>" "x \<in> A"
+ | A B where "x \<in> D" "A face_of B" "B \<in> \<F>" "B \<notin> \<G>" "aff_dim A < aff_dim T" "x \<in> A"
+ using x by (auto simp: \<H>_def)
+ then have xrel: "x \<in> rel_frontier D"
+ proof cases
+ case 1 show ?thesis
+ proof (rule face_of_subset_rel_frontier [THEN subsetD])
+ show "D \<inter> A face_of D"
+ using \<open>A \<in> \<G>\<close> \<open>\<G> \<subseteq> \<F>\<close> face \<open>D \<in> \<F>\<close> by blast
+ show "D \<inter> A \<noteq> D"
+ using \<open>A \<in> \<G>\<close> D_not_subset \<H>_def by blast
+ qed (auto simp: 1)
+ next
+ case 2 show ?thesis
+ proof (rule face_of_subset_rel_frontier [THEN subsetD])
+ show "D \<inter> A face_of D"
+ apply (rule face_of_Int_subface [of D B _ A, THEN conjunct1])
+ apply (simp_all add: 2 \<open>D \<in> \<F>\<close> face)
+ apply (simp add: \<open>polyhedron D\<close> polyhedron_imp_convex face_of_refl)
+ done
+ show "D \<inter> A \<noteq> D"
+ using "2" D_not_subset \<H>_def by blast
+ qed (auto simp: 2)
+ qed
+ show ?thesis
+ by (simp add: rid xrel)
+ qed
+ qed
+ qed
+ qed
+ qed
+ have clo: "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"
+ by (simp add: poly polytope_imp_closed)
+ obtain C g where "finite C" "disjnt C (\<Union>\<G>)" "card C \<le> card \<F>" "continuous_on (\<Union>\<F> - C) g"
+ "g ` (\<Union>\<F> - C) \<subseteq> rel_frontier T"
+ and gh: "\<And>x. x \<in> (\<Union>\<F> - C) \<inter> \<Union>\<H> \<Longrightarrow> g x = h x"
+ proof (rule extend_map_lemma_cofinite1 [OF \<open>finite \<F>\<close> \<F> clo])
+ show "X \<inter> Y \<subseteq> \<Union>\<H>" if XY: "X \<in> \<F>" "Y \<in> \<F>" and "\<not> X \<subseteq> Y" "\<not> Y \<subseteq> X" for X Y
+ proof (cases "X \<in> \<G>")
+ case True
+ then show ?thesis
+ by (auto simp: \<H>_def)
+ next
+ case False
+ have "X \<inter> Y \<noteq> X"
+ using \<open>\<not> X \<subseteq> Y\<close> by blast
+ with XY
+ show ?thesis
+ by (clarsimp simp: \<H>_def)
+ (metis Diff_iff Int_iff aff antisym_conv face face_of_aff_dim_lt face_of_refl
+ not_le poly polytope_imp_convex)
+ qed
+ qed (blast)+
+ with \<open>\<G> \<subseteq> \<F>\<close> show ?thesis
+ apply (rule_tac C=C and g=g in that)
+ apply (auto simp: disjnt_def hf [symmetric] \<H>_def intro!: gh)
+ done
+qed
+
+text\<open>The next two proofs are similar\<close>
+theorem extend_map_cell_complex_to_sphere:
+ assumes "finite \<F>" and S: "S \<subseteq> \<Union>\<F>" "closed S" and T: "convex T" "bounded T"
+ and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
+ and aff: "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X < aff_dim T"
+ and face: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> (X \<inter> Y) face_of X \<and> (X \<inter> Y) face_of Y"
+ and contf: "continuous_on S f" and fim: "f ` S \<subseteq> rel_frontier T"
+ obtains g where "continuous_on (\<Union>\<F>) g"
+ "g ` (\<Union>\<F>) \<subseteq> rel_frontier T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof -
+ obtain V g where "S \<subseteq> V" "open V" "continuous_on V g" and gim: "g ` V \<subseteq> rel_frontier T" and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+ using neighbourhood_extension_into_ANR [OF contf fim _ \<open>closed S\<close>] ANR_rel_frontier_convex T by blast
+ have "compact S"
+ by (meson assms compact_Union poly polytope_imp_compact seq_compact_closed_subset seq_compact_eq_compact)
+ then obtain d where "d > 0" and d: "\<And>x y. \<lbrakk>x \<in> S; y \<in> - V\<rbrakk> \<Longrightarrow> d \<le> dist x y"
+ using separate_compact_closed [of S "-V"] \<open>open V\<close> \<open>S \<subseteq> V\<close> by force
+ obtain \<G> where "finite \<G>" "\<Union>\<G> = \<Union>\<F>"
+ and diaG: "\<And>X. X \<in> \<G> \<Longrightarrow> diameter X < d"
+ and polyG: "\<And>X. X \<in> \<G> \<Longrightarrow> polytope X"
+ and affG: "\<And>X. X \<in> \<G> \<Longrightarrow> aff_dim X \<le> aff_dim T - 1"
+ and faceG: "\<And>X Y. \<lbrakk>X \<in> \<G>; Y \<in> \<G>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+ proof (rule cell_complex_subdivision_exists [OF \<open>d>0\<close> \<open>finite \<F>\<close> poly _ face])
+ show "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X \<le> aff_dim T - 1"
+ by (simp add: aff)
+ qed auto
+ obtain h where conth: "continuous_on (\<Union>\<G>) h" and him: "h ` \<Union>\<G> \<subseteq> rel_frontier T" and hg: "\<And>x. x \<in> \<Union>(\<G> \<inter> Pow V) \<Longrightarrow> h x = g x"
+ proof (rule extend_map_lemma [of \<G> "\<G> \<inter> Pow V" T g])
+ show "continuous_on (\<Union>(\<G> \<inter> Pow V)) g"
+ by (metis Union_Int_subset Union_Pow_eq \<open>continuous_on V g\<close> continuous_on_subset le_inf_iff)
+ qed (use \<open>finite \<G>\<close> T polyG affG faceG gim in fastforce)+
+ show ?thesis
+ proof
+ show "continuous_on (\<Union>\<F>) h"
+ using \<open>\<Union>\<G> = \<Union>\<F>\<close> conth by auto
+ show "h ` \<Union>\<F> \<subseteq> rel_frontier T"
+ using \<open>\<Union>\<G> = \<Union>\<F>\<close> him by auto
+ show "h x = f x" if "x \<in> S" for x
+ proof -
+ have "x \<in> \<Union>\<G>"
+ using \<open>\<Union>\<G> = \<Union>\<F>\<close> \<open>S \<subseteq> \<Union>\<F>\<close> that by auto
+ then obtain X where "x \<in> X" "X \<in> \<G>" by blast
+ then have "diameter X < d" "bounded X"
+ by (auto simp: diaG \<open>X \<in> \<G>\<close> polyG polytope_imp_bounded)
+ then have "X \<subseteq> V" using d [OF \<open>x \<in> S\<close>] diameter_bounded_bound [OF \<open>bounded X\<close> \<open>x \<in> X\<close>]
+ by fastforce
+ have "h x = g x"
+ apply (rule hg)
+ using \<open>X \<in> \<G>\<close> \<open>X \<subseteq> V\<close> \<open>x \<in> X\<close> by blast
+ also have "... = f x"
+ by (simp add: gf that)
+ finally show "h x = f x" .
+ qed
+ qed
+qed
+
+
+theorem extend_map_cell_complex_to_sphere_cofinite:
+ assumes "finite \<F>" and S: "S \<subseteq> \<Union>\<F>" "closed S" and T: "convex T" "bounded T"
+ and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
+ and aff: "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X \<le> aff_dim T"
+ and face: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> (X \<inter> Y) face_of X \<and> (X \<inter> Y) face_of Y"
+ and contf: "continuous_on S f" and fim: "f ` S \<subseteq> rel_frontier T"
+ obtains C g where "finite C" "disjnt C S" "continuous_on (\<Union>\<F> - C) g"
+ "g ` (\<Union>\<F> - C) \<subseteq> rel_frontier T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof -
+ obtain V g where "S \<subseteq> V" "open V" "continuous_on V g" and gim: "g ` V \<subseteq> rel_frontier T" and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+ using neighbourhood_extension_into_ANR [OF contf fim _ \<open>closed S\<close>] ANR_rel_frontier_convex T by blast
+ have "compact S"
+ by (meson assms compact_Union poly polytope_imp_compact seq_compact_closed_subset seq_compact_eq_compact)
+ then obtain d where "d > 0" and d: "\<And>x y. \<lbrakk>x \<in> S; y \<in> - V\<rbrakk> \<Longrightarrow> d \<le> dist x y"
+ using separate_compact_closed [of S "-V"] \<open>open V\<close> \<open>S \<subseteq> V\<close> by force
+ obtain \<G> where "finite \<G>" "\<Union>\<G> = \<Union>\<F>"
+ and diaG: "\<And>X. X \<in> \<G> \<Longrightarrow> diameter X < d"
+ and polyG: "\<And>X. X \<in> \<G> \<Longrightarrow> polytope X"
+ and affG: "\<And>X. X \<in> \<G> \<Longrightarrow> aff_dim X \<le> aff_dim T"
+ and faceG: "\<And>X Y. \<lbrakk>X \<in> \<G>; Y \<in> \<G>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+ by (rule cell_complex_subdivision_exists [OF \<open>d>0\<close> \<open>finite \<F>\<close> poly aff face]) auto
+ obtain C h where "finite C" and dis: "disjnt C (\<Union>(\<G> \<inter> Pow V))"
+ and card: "card C \<le> card \<G>" and conth: "continuous_on (\<Union>\<G> - C) h"
+ and him: "h ` (\<Union>\<G> - C) \<subseteq> rel_frontier T"
+ and hg: "\<And>x. x \<in> \<Union>(\<G> \<inter> Pow V) \<Longrightarrow> h x = g x"
+ proof (rule extend_map_lemma_cofinite [of \<G> "\<G> \<inter> Pow V" T g])
+ show "continuous_on (\<Union>(\<G> \<inter> Pow V)) g"
+ by (metis Union_Int_subset Union_Pow_eq \<open>continuous_on V g\<close> continuous_on_subset le_inf_iff)
+ show "g ` \<Union>(\<G> \<inter> Pow V) \<subseteq> rel_frontier T"
+ using gim by force
+ qed (auto intro: \<open>finite \<G>\<close> T polyG affG dest: faceG)
+ have Ssub: "S \<subseteq> \<Union>(\<G> \<inter> Pow V)"
+ proof
+ fix x
+ assume "x \<in> S"
+ then have "x \<in> \<Union>\<G>"
+ using \<open>\<Union>\<G> = \<Union>\<F>\<close> \<open>S \<subseteq> \<Union>\<F>\<close> by auto
+ then obtain X where "x \<in> X" "X \<in> \<G>" by blast
+ then have "diameter X < d" "bounded X"
+ by (auto simp: diaG \<open>X \<in> \<G>\<close> polyG polytope_imp_bounded)
+ then have "X \<subseteq> V" using d [OF \<open>x \<in> S\<close>] diameter_bounded_bound [OF \<open>bounded X\<close> \<open>x \<in> X\<close>]
+ by fastforce
+ then show "x \<in> \<Union>(\<G> \<inter> Pow V)"
+ using \<open>X \<in> \<G>\<close> \<open>x \<in> X\<close> by blast
+ qed
+ show ?thesis
+ proof
+ show "continuous_on (\<Union>\<F>-C) h"
+ using \<open>\<Union>\<G> = \<Union>\<F>\<close> conth by auto
+ show "h ` (\<Union>\<F> - C) \<subseteq> rel_frontier T"
+ using \<open>\<Union>\<G> = \<Union>\<F>\<close> him by auto
+ show "h x = f x" if "x \<in> S" for x
+ proof -
+ have "h x = g x"
+ apply (rule hg)
+ using Ssub that by blast
+ also have "... = f x"
+ by (simp add: gf that)
+ finally show "h x = f x" .
+ qed
+ show "disjnt C S"
+ using dis Ssub by (meson disjnt_iff subset_eq)
+ qed (intro \<open>finite C\<close>)
+qed
+
+
+
+subsection\<open> Special cases and corollaries involving spheres.\<close>
+
+lemma disjnt_Diff1: "X \<subseteq> Y' \<Longrightarrow> disjnt (X - Y) (X' - Y')"
+ by (auto simp: disjnt_def)
+
+proposition extend_map_affine_to_sphere_cofinite_simple:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes "compact S" "convex U" "bounded U"
+ and aff: "aff_dim T \<le> aff_dim U"
+ and "S \<subseteq> T" and contf: "continuous_on S f"
+ and fim: "f ` S \<subseteq> rel_frontier U"
+ obtains K g where "finite K" "K \<subseteq> T" "disjnt K S" "continuous_on (T - K) g"
+ "g ` (T - K) \<subseteq> rel_frontier U"
+ "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof -
+ have "\<exists>K g. finite K \<and> disjnt K S \<and> continuous_on (T - K) g \<and>
+ g ` (T - K) \<subseteq> rel_frontier U \<and> (\<forall>x \<in> S. g x = f x)"
+ if "affine T" "S \<subseteq> T" and aff: "aff_dim T \<le> aff_dim U" for T
+ proof (cases "S = {}")
+ case True
+ show ?thesis
+ proof (cases "rel_frontier U = {}")
+ case True
+ with \<open>bounded U\<close> have "aff_dim U \<le> 0"
+ using affine_bounded_eq_lowdim rel_frontier_eq_empty by auto
+ with aff have "aff_dim T \<le> 0" by auto
+ then obtain a where "T \<subseteq> {a}"
+ using \<open>affine T\<close> affine_bounded_eq_lowdim affine_bounded_eq_trivial by auto
+ then show ?thesis
+ using \<open>S = {}\<close> fim
+ by (metis Diff_cancel contf disjnt_empty2 finite.emptyI finite_insert finite_subset)
+ next
+ case False
+ then obtain a where "a \<in> rel_frontier U"
+ by auto
+ then show ?thesis
+ using continuous_on_const [of _ a] \<open>S = {}\<close> by force
+ qed
+ next
+ case False
+ have "bounded S"
+ by (simp add: \<open>compact S\<close> compact_imp_bounded)
+ then obtain b where b: "S \<subseteq> cbox (-b) b"
+ using bounded_subset_cbox_symmetric by blast
+ define bbox where "bbox \<equiv> cbox (-(b+One)) (b+One)"
+ have "cbox (-b) b \<subseteq> bbox"
+ by (auto simp: bbox_def algebra_simps intro!: subset_box_imp)
+ with b \<open>S \<subseteq> T\<close> have "S \<subseteq> bbox \<inter> T"
+ by auto
+ then have Ssub: "S \<subseteq> \<Union>{bbox \<inter> T}"
+ by auto
+ then have "aff_dim (bbox \<inter> T) \<le> aff_dim U"
+ by (metis aff aff_dim_subset inf_commute inf_le1 order_trans)
+ obtain K g where K: "finite K" "disjnt K S"
+ and contg: "continuous_on (\<Union>{bbox \<inter> T} - K) g"
+ and gim: "g ` (\<Union>{bbox \<inter> T} - K) \<subseteq> rel_frontier U"
+ and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+ proof (rule extend_map_cell_complex_to_sphere_cofinite
+ [OF _ Ssub _ \<open>convex U\<close> \<open>bounded U\<close> _ _ _ contf fim])
+ show "closed S"
+ using \<open>compact S\<close> compact_eq_bounded_closed by auto
+ show poly: "\<And>X. X \<in> {bbox \<inter> T} \<Longrightarrow> polytope X"
+ by (simp add: polytope_Int_polyhedron bbox_def polytope_interval affine_imp_polyhedron \<open>affine T\<close>)
+ show "\<And>X Y. \<lbrakk>X \<in> {bbox \<inter> T}; Y \<in> {bbox \<inter> T}\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+ by (simp add:poly face_of_refl polytope_imp_convex)
+ show "\<And>X. X \<in> {bbox \<inter> T} \<Longrightarrow> aff_dim X \<le> aff_dim U"
+ by (simp add: \<open>aff_dim (bbox \<inter> T) \<le> aff_dim U\<close>)
+ qed auto
+ define fro where "fro \<equiv> \<lambda>d. frontier(cbox (-(b + d *\<^sub>R One)) (b + d *\<^sub>R One))"
+ obtain d where d12: "1/2 \<le> d" "d \<le> 1" and dd: "disjnt K (fro d)"
+ proof (rule disjoint_family_elem_disjnt [OF _ \<open>finite K\<close>])
+ show "infinite {1/2..1::real}"
+ by (simp add: infinite_Icc)
+ have dis1: "disjnt (fro x) (fro y)" if "x<y" for x y
+ by (auto simp: algebra_simps that subset_box_imp disjnt_Diff1 frontier_def fro_def)
+ then show "disjoint_family_on fro {1/2..1}"
+ by (auto simp: disjoint_family_on_def disjnt_def neq_iff)
+ qed auto
+ define c where "c \<equiv> b + d *\<^sub>R One"
+ have cbsub: "cbox (-b) b \<subseteq> box (-c) c" "cbox (-b) b \<subseteq> cbox (-c) c" "cbox (-c) c \<subseteq> bbox"
+ using d12 by (auto simp: algebra_simps subset_box_imp c_def bbox_def)
+ have clo_cbT: "closed (cbox (- c) c \<inter> T)"
+ by (simp add: affine_closed closed_Int closed_cbox \<open>affine T\<close>)
+ have cpT_ne: "cbox (- c) c \<inter> T \<noteq> {}"
+ using \<open>S \<noteq> {}\<close> b cbsub(2) \<open>S \<subseteq> T\<close> by fastforce
+ have "closest_point (cbox (- c) c \<inter> T) x \<notin> K" if "x \<in> T" "x \<notin> K" for x
+ proof (cases "x \<in> cbox (-c) c")
+ case True with that show ?thesis
+ by (simp add: closest_point_self)
+ next
+ case False
+ have int_ne: "interior (cbox (-c) c) \<inter> T \<noteq> {}"
+ using \<open>S \<noteq> {}\<close> \<open>S \<subseteq> T\<close> b \<open>cbox (- b) b \<subseteq> box (- c) c\<close> by force
+ have "convex T"
+ by (meson \<open>affine T\<close> affine_imp_convex)
+ then have "x \<in> affine hull (cbox (- c) c \<inter> T)"
+ by (metis Int_commute Int_iff \<open>S \<noteq> {}\<close> \<open>S \<subseteq> T\<close> cbsub(1) \<open>x \<in> T\<close> affine_hull_convex_Int_nonempty_interior all_not_in_conv b hull_inc inf.orderE interior_cbox)
+ then have "x \<in> affine hull (cbox (- c) c \<inter> T) - rel_interior (cbox (- c) c \<inter> T)"
+ by (meson DiffI False Int_iff rel_interior_subset subsetCE)
+ then have "closest_point (cbox (- c) c \<inter> T) x \<in> rel_frontier (cbox (- c) c \<inter> T)"
+ by (rule closest_point_in_rel_frontier [OF clo_cbT cpT_ne])
+ moreover have "(rel_frontier (cbox (- c) c \<inter> T)) \<subseteq> fro d"
+ apply (subst convex_affine_rel_frontier_Int [OF _ \<open>affine T\<close> int_ne])
+ apply (auto simp: fro_def c_def)
+ done
+ ultimately show ?thesis
+ using dd by (force simp: disjnt_def)
+ qed
+ then have cpt_subset: "closest_point (cbox (- c) c \<inter> T) ` (T - K) \<subseteq> \<Union>{bbox \<inter> T} - K"
+ using closest_point_in_set [OF clo_cbT cpT_ne] cbsub(3) by force
+ show ?thesis
+ proof (intro conjI ballI exI)
+ have "continuous_on (T - K) (closest_point (cbox (- c) c \<inter> T))"
+ apply (rule continuous_on_closest_point)
+ using \<open>S \<noteq> {}\<close> cbsub(2) b that
+ by (auto simp: affine_imp_convex convex_Int affine_closed closed_Int closed_cbox \<open>affine T\<close>)
+ then show "continuous_on (T - K) (g \<circ> closest_point (cbox (- c) c \<inter> T))"
+ by (metis continuous_on_compose continuous_on_subset [OF contg cpt_subset])
+ have "(g \<circ> closest_point (cbox (- c) c \<inter> T)) ` (T - K) \<subseteq> g ` (\<Union>{bbox \<inter> T} - K)"
+ by (metis image_comp image_mono cpt_subset)
+ also have "... \<subseteq> rel_frontier U"
+ by (rule gim)
+ finally show "(g \<circ> closest_point (cbox (- c) c \<inter> T)) ` (T - K) \<subseteq> rel_frontier U" .
+ show "(g \<circ> closest_point (cbox (- c) c \<inter> T)) x = f x" if "x \<in> S" for x
+ proof -
+ have "(g \<circ> closest_point (cbox (- c) c \<inter> T)) x = g x"
+ unfolding o_def
+ by (metis IntI \<open>S \<subseteq> T\<close> b cbsub(2) closest_point_self subset_eq that)
+ also have "... = f x"
+ by (simp add: that gf)
+ finally show ?thesis .
+ qed
+ qed (auto simp: K)
+ qed
+ then obtain K g where "finite K" "disjnt K S"
+ and contg: "continuous_on (affine hull T - K) g"
+ and gim: "g ` (affine hull T - K) \<subseteq> rel_frontier U"
+ and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+ by (metis aff affine_affine_hull aff_dim_affine_hull
+ order_trans [OF \<open>S \<subseteq> T\<close> hull_subset [of T affine]])
+ then obtain K g where "finite K" "disjnt K S"
+ and contg: "continuous_on (T - K) g"
+ and gim: "g ` (T - K) \<subseteq> rel_frontier U"
+ and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+ by (rule_tac K=K and g=g in that) (auto simp: hull_inc elim: continuous_on_subset)
+ then show ?thesis
+ by (rule_tac K="K \<inter> T" and g=g in that) (auto simp: disjnt_iff Diff_Int contg)
+qed
+
+subsection\<open>Extending maps to spheres\<close>
+
+(*Up to extend_map_affine_to_sphere_cofinite_gen*)
+
+lemma closedin_closed_subset:
+ "\<lbrakk>closedin (subtopology euclidean U) V; T \<subseteq> U; S = V \<inter> T\<rbrakk>
+ \<Longrightarrow> closedin (subtopology euclidean T) S"
+ by (metis (no_types, lifting) Int_assoc Int_commute closedin_closed inf.orderE)
+
+lemma extend_map_affine_to_sphere1:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::topological_space"
+ assumes "finite K" "affine U" and contf: "continuous_on (U - K) f"
+ and fim: "f ` (U - K) \<subseteq> T"
+ and comps: "\<And>C. \<lbrakk>C \<in> components(U - S); C \<inter> K \<noteq> {}\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
+ and clo: "closedin (subtopology euclidean U) S" and K: "disjnt K S" "K \<subseteq> U"
+ obtains g where "continuous_on (U - L) g" "g ` (U - L) \<subseteq> T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof (cases "K = {}")
+ case True
+ then show ?thesis
+ by (metis Diff_empty Diff_subset contf fim continuous_on_subset image_subsetI rev_image_eqI subset_iff that)
+next
+ case False
+ have "S \<subseteq> U"
+ using clo closedin_limpt by blast
+ then have "(U - S) \<inter> K \<noteq> {}"
+ by (metis Diff_triv False Int_Diff K disjnt_def inf.absorb_iff2 inf_commute)
+ then have "\<Union>(components (U - S)) \<inter> K \<noteq> {}"
+ using Union_components by simp
+ then obtain C0 where C0: "C0 \<in> components (U - S)" "C0 \<inter> K \<noteq> {}"
+ by blast
+ have "convex U"
+ by (simp add: affine_imp_convex \<open>affine U\<close>)
+ then have "locally connected U"
+ by (rule convex_imp_locally_connected)
+ have "\<exists>a g. a \<in> C \<and> a \<in> L \<and> continuous_on (S \<union> (C - {a})) g \<and>
+ g ` (S \<union> (C - {a})) \<subseteq> T \<and> (\<forall>x \<in> S. g x = f x)"
+ if C: "C \<in> components (U - S)" and CK: "C \<inter> K \<noteq> {}" for C
+ proof -
+ have "C \<subseteq> U-S" "C \<inter> L \<noteq> {}"
+ by (simp_all add: in_components_subset comps that)
+ then obtain a where a: "a \<in> C" "a \<in> L" by auto
+ have opeUC: "openin (subtopology euclidean U) C"
+ proof (rule openin_trans)
+ show "openin (subtopology euclidean (U-S)) C"
+ by (simp add: \<open>locally connected U\<close> clo locally_diff_closed openin_components_locally_connected [OF _ C])
+ show "openin (subtopology euclidean U) (U - S)"
+ by (simp add: clo openin_diff)
+ qed
+ then obtain d where "C \<subseteq> U" "0 < d" and d: "cball a d \<inter> U \<subseteq> C"
+ using openin_contains_cball by (metis \<open>a \<in> C\<close>)
+ then have "ball a d \<inter> U \<subseteq> C"
+ by auto
+ obtain h k where homhk: "homeomorphism (S \<union> C) (S \<union> C) h k"
+ and subC: "{x. (~ (h x = x \<and> k x = x))} \<subseteq> C"
+ and bou: "bounded {x. (~ (h x = x \<and> k x = x))}"
+ and hin: "\<And>x. x \<in> C \<inter> K \<Longrightarrow> h x \<in> ball a d \<inter> U"
+ proof (rule homeomorphism_grouping_points_exists_gen [of C "ball a d \<inter> U" "C \<inter> K" "S \<union> C"])
+ show "openin (subtopology euclidean C) (ball a d \<inter> U)"
+ by (metis Topology_Euclidean_Space.open_ball \<open>C \<subseteq> U\<close> \<open>ball a d \<inter> U \<subseteq> C\<close> inf.absorb_iff2 inf.orderE inf_assoc open_openin openin_subtopology)
+ show "openin (subtopology euclidean (affine hull C)) C"
+ by (metis \<open>a \<in> C\<close> \<open>openin (subtopology euclidean U) C\<close> affine_hull_eq affine_hull_openin all_not_in_conv \<open>affine U\<close>)
+ show "ball a d \<inter> U \<noteq> {}"
+ using \<open>0 < d\<close> \<open>C \<subseteq> U\<close> \<open>a \<in> C\<close> by force
+ show "finite (C \<inter> K)"
+ by (simp add: \<open>finite K\<close>)
+ show "S \<union> C \<subseteq> affine hull C"
+ by (metis \<open>C \<subseteq> U\<close> \<open>S \<subseteq> U\<close> \<open>a \<in> C\<close> opeUC affine_hull_eq affine_hull_openin all_not_in_conv assms(2) sup.bounded_iff)
+ show "connected C"
+ by (metis C in_components_connected)
+ qed auto
+ have a_BU: "a \<in> ball a d \<inter> U"
+ using \<open>0 < d\<close> \<open>C \<subseteq> U\<close> \<open>a \<in> C\<close> by auto
+ have "rel_frontier (cball a d \<inter> U) retract_of (affine hull (cball a d \<inter> U) - {a})"
+ apply (rule rel_frontier_retract_of_punctured_affine_hull)
+ apply (auto simp: \<open>convex U\<close> convex_Int)
+ by (metis \<open>affine U\<close> convex_cball empty_iff interior_cball a_BU rel_interior_convex_Int_affine)
+ moreover have "rel_frontier (cball a d \<inter> U) = frontier (cball a d) \<inter> U"
+ apply (rule convex_affine_rel_frontier_Int)
+ using a_BU by (force simp: \<open>affine U\<close>)+
+ moreover have "affine hull (cball a d \<inter> U) = U"
+ by (metis \<open>convex U\<close> a_BU affine_hull_convex_Int_nonempty_interior affine_hull_eq \<open>affine U\<close> equals0D inf.commute interior_cball)
+ ultimately have "frontier (cball a d) \<inter> U retract_of (U - {a})"
+ by metis
+ then obtain r where contr: "continuous_on (U - {a}) r"
+ and rim: "r ` (U - {a}) \<subseteq> sphere a d" "r ` (U - {a}) \<subseteq> U"
+ and req: "\<And>x. x \<in> sphere a d \<inter> U \<Longrightarrow> r x = x"
+ using \<open>affine U\<close> by (auto simp: retract_of_def retraction_def hull_same)
+ define j where "j \<equiv> \<lambda>x. if x \<in> ball a d then r x else x"
+ have kj: "\<And>x. x \<in> S \<Longrightarrow> k (j x) = x"
+ using \<open>C \<subseteq> U - S\<close> \<open>S \<subseteq> U\<close> \<open>ball a d \<inter> U \<subseteq> C\<close> j_def subC by auto
+ have Uaeq: "U - {a} = (cball a d - {a}) \<inter> U \<union> (U - ball a d)"
+ using \<open>0 < d\<close> by auto
+ have jim: "j ` (S \<union> (C - {a})) \<subseteq> (S \<union> C) - ball a d"
+ proof clarify
+ fix y assume "y \<in> S \<union> (C - {a})"
+ then have "y \<in> U - {a}"
+ using \<open>C \<subseteq> U - S\<close> \<open>S \<subseteq> U\<close> \<open>a \<in> C\<close> by auto
+ then have "r y \<in> sphere a d"
+ using rim by auto
+ then show "j y \<in> S \<union> C - ball a d"
+ apply (simp add: j_def)
+ using \<open>r y \<in> sphere a d\<close> \<open>y \<in> U - {a}\<close> \<open>y \<in> S \<union> (C - {a})\<close> d rim by fastforce
+ qed
+ have contj: "continuous_on (U - {a}) j"
+ unfolding j_def Uaeq
+ proof (intro continuous_on_cases_local continuous_on_id, simp_all add: req closedin_closed Uaeq [symmetric])
+ show "\<exists>T. closed T \<and> (cball a d - {a}) \<inter> U = (U - {a}) \<inter> T"
+ apply (rule_tac x="(cball a d) \<inter> U" in exI)
+ using affine_closed \<open>affine U\<close> by blast
+ show "\<exists>T. closed T \<and> U - ball a d = (U - {a}) \<inter> T"
+ apply (rule_tac x="U - ball a d" in exI)
+ using \<open>0 < d\<close> by (force simp: affine_closed \<open>affine U\<close> closed_Diff)
+ show "continuous_on ((cball a d - {a}) \<inter> U) r"
+ by (force intro: continuous_on_subset [OF contr])
+ qed
+ have fT: "x \<in> U - K \<Longrightarrow> f x \<in> T" for x
+ using fim by blast
+ show ?thesis
+ proof (intro conjI exI)
+ show "continuous_on (S \<union> (C - {a})) (f \<circ> k \<circ> j)"
+ proof (intro continuous_on_compose)
+ show "continuous_on (S \<union> (C - {a})) j"
+ apply (rule continuous_on_subset [OF contj])
+ using \<open>C \<subseteq> U - S\<close> \<open>S \<subseteq> U\<close> \<open>a \<in> C\<close> by force
+ show "continuous_on (j ` (S \<union> (C - {a}))) k"
+ apply (rule continuous_on_subset [OF homeomorphism_cont2 [OF homhk]])
+ using jim \<open>C \<subseteq> U - S\<close> \<open>S \<subseteq> U\<close> \<open>ball a d \<inter> U \<subseteq> C\<close> j_def by fastforce
+ show "continuous_on (k ` j ` (S \<union> (C - {a}))) f"
+ proof (clarify intro!: continuous_on_subset [OF contf])
+ fix y assume "y \<in> S \<union> (C - {a})"
+ have ky: "k y \<in> S \<union> C"
+ using homeomorphism_image2 [OF homhk] \<open>y \<in> S \<union> (C - {a})\<close> by blast
+ have jy: "j y \<in> S \<union> C - ball a d"
+ using Un_iff \<open>y \<in> S \<union> (C - {a})\<close> jim by auto
+ show "k (j y) \<in> U - K"
+ apply safe
+ using \<open>C \<subseteq> U\<close> \<open>S \<subseteq> U\<close> homeomorphism_image2 [OF homhk] jy apply blast
+ by (metis DiffD1 DiffD2 Int_iff Un_iff \<open>disjnt K S\<close> disjnt_def empty_iff hin homeomorphism_apply2 homeomorphism_image2 homhk imageI jy)
+ qed
+ qed
+ have ST: "\<And>x. x \<in> S \<Longrightarrow> (f \<circ> k \<circ> j) x \<in> T"
+ apply (simp add: kj)
+ apply (metis DiffI \<open>S \<subseteq> U\<close> \<open>disjnt K S\<close> subsetD disjnt_iff fim image_subset_iff)
+ done
+ moreover have "(f \<circ> k \<circ> j) x \<in> T" if "x \<in> C" "x \<noteq> a" "x \<notin> S" for x
+ proof -
+ have rx: "r x \<in> sphere a d"
+ using \<open>C \<subseteq> U\<close> rim that by fastforce
+ have jj: "j x \<in> S \<union> C - ball a d"
+ using jim that by blast
+ have "k (j x) = j x \<longrightarrow> k (j x) \<in> C \<or> j x \<in> C"
+ by (metis Diff_iff Int_iff Un_iff \<open>S \<subseteq> U\<close> subsetD d j_def jj rx sphere_cball that(1))
+ then have "k (j x) \<in> C"
+ using homeomorphism_apply2 [OF homhk, of "j x"] \<open>C \<subseteq> U\<close> \<open>S \<subseteq> U\<close> a rx
+ by (metis (mono_tags, lifting) Diff_iff subsetD jj mem_Collect_eq subC)
+ with jj \<open>C \<subseteq> U\<close> show ?thesis
+ apply safe
+ using ST j_def apply fastforce
+ apply (auto simp: not_less intro!: fT)
+ by (metis DiffD1 DiffD2 Int_iff hin homeomorphism_apply2 [OF homhk] jj)
+ qed
+ ultimately show "(f \<circ> k \<circ> j) ` (S \<union> (C - {a})) \<subseteq> T"
+ by force
+ show "\<forall>x\<in>S. (f \<circ> k \<circ> j) x = f x" using kj by simp
+ qed (auto simp: a)
+ qed
+ then obtain a h where
+ ah: "\<And>C. \<lbrakk>C \<in> components (U - S); C \<inter> K \<noteq> {}\<rbrakk>
+ \<Longrightarrow> a C \<in> C \<and> a C \<in> L \<and> continuous_on (S \<union> (C - {a C})) (h C) \<and>
+ h C ` (S \<union> (C - {a C})) \<subseteq> T \<and> (\<forall>x \<in> S. h C x = f x)"
+ using that by metis
+ define F where "F \<equiv> {C \<in> components (U - S). C \<inter> K \<noteq> {}}"
+ define G where "G \<equiv> {C \<in> components (U - S). C \<inter> K = {}}"
+ define UF where "UF \<equiv> (\<Union>C\<in>F. C - {a C})"
+ have "C0 \<in> F"
+ by (auto simp: F_def C0)
+ have "finite F"
+ proof (subst finite_image_iff [of "\<lambda>C. C \<inter> K" F, symmetric])
+ show "inj_on (\<lambda>C. C \<inter> K) F"
+ unfolding F_def inj_on_def
+ using components_nonoverlap by blast
+ show "finite ((\<lambda>C. C \<inter> K) ` F)"
+ unfolding F_def
+ by (rule finite_subset [of _ "Pow K"]) (auto simp: \<open>finite K\<close>)
+ qed
+ obtain g where contg: "continuous_on (S \<union> UF) g"
+ and gh: "\<And>x i. \<lbrakk>i \<in> F; x \<in> (S \<union> UF) \<inter> (S \<union> (i - {a i}))\<rbrakk>
+ \<Longrightarrow> g x = h i x"
+ proof (rule pasting_lemma_exists_closed [OF \<open>finite F\<close>, of "S \<union> UF" "\<lambda>C. S \<union> (C - {a C})" h])
+ show "S \<union> UF \<subseteq> (\<Union>C\<in>F. S \<union> (C - {a C}))"
+ using \<open>C0 \<in> F\<close> by (force simp: UF_def)
+ show "closedin (subtopology euclidean (S \<union> UF)) (S \<union> (C - {a C}))"
+ if "C \<in> F" for C
+ proof (rule closedin_closed_subset [of U "S \<union> C"])
+ show "closedin (subtopology euclidean U) (S \<union> C)"
+ apply (rule closedin_Un_complement_component [OF \<open>locally connected U\<close> clo])
+ using F_def that by blast
+ next
+ have "x = a C'" if "C' \<in> F" "x \<in> C'" "x \<notin> U" for x C'
+ proof -
+ have "\<forall>A. x \<in> \<Union>A \<or> C' \<notin> A"
+ using \<open>x \<in> C'\<close> by blast
+ with that show "x = a C'"
+ by (metis (lifting) DiffD1 F_def Union_components mem_Collect_eq)
+ qed
+ then show "S \<union> UF \<subseteq> U"
+ using \<open>S \<subseteq> U\<close> by (force simp: UF_def)
+ next
+ show "S \<union> (C - {a C}) = (S \<union> C) \<inter> (S \<union> UF)"
+ using F_def UF_def components_nonoverlap that by auto
+ qed
+ next
+ show "continuous_on (S \<union> (C' - {a C'})) (h C')" if "C' \<in> F" for C'
+ using ah F_def that by blast
+ show "\<And>i j x. \<lbrakk>i \<in> F; j \<in> F;
+ x \<in> (S \<union> UF) \<inter> (S \<union> (i - {a i})) \<inter> (S \<union> (j - {a j}))\<rbrakk>
+ \<Longrightarrow> h i x = h j x"
+ using components_eq by (fastforce simp: components_eq F_def ah)
+ qed blast
+ have SU': "S \<union> \<Union>G \<union> (S \<union> UF) \<subseteq> U"
+ using \<open>S \<subseteq> U\<close> in_components_subset by (auto simp: F_def G_def UF_def)
+ have clo1: "closedin (subtopology euclidean (S \<union> \<Union>G \<union> (S \<union> UF))) (S \<union> \<Union>G)"
+ proof (rule closedin_closed_subset [OF _ SU'])
+ have *: "\<And>C. C \<in> F \<Longrightarrow> openin (subtopology euclidean U) C"
+ unfolding F_def
+ by clarify (metis (no_types, lifting) \<open>locally connected U\<close> clo closedin_def locally_diff_closed openin_components_locally_connected openin_trans topspace_euclidean_subtopology)
+ show "closedin (subtopology euclidean U) (U - UF)"
+ unfolding UF_def
+ by (force intro: openin_delete *)
+ show "S \<union> \<Union>G = (U - UF) \<inter> (S \<union> \<Union>G \<union> (S \<union> UF))"
+ using \<open>S \<subseteq> U\<close> apply (auto simp: F_def G_def UF_def)
+ apply (metis Diff_iff UnionI Union_components)
+ apply (metis DiffD1 UnionI Union_components)
+ by (metis (no_types, lifting) IntI components_nonoverlap empty_iff)
+ qed
+ have clo2: "closedin (subtopology euclidean (S \<union> \<Union>G \<union> (S \<union> UF))) (S \<union> UF)"
+ proof (rule closedin_closed_subset [OF _ SU'])
+ show "closedin (subtopology euclidean U) (\<Union>C\<in>F. S \<union> C)"
+ apply (rule closedin_Union)
+ apply (simp add: \<open>finite F\<close>)
+ using F_def \<open>locally connected U\<close> clo closedin_Un_complement_component by blast
+ show "S \<union> UF = (\<Union>C\<in>F. S \<union> C) \<inter> (S \<union> \<Union>G \<union> (S \<union> UF))"
+ using \<open>S \<subseteq> U\<close> apply (auto simp: F_def G_def UF_def)
+ using C0 apply blast
+ by (metis components_nonoverlap disjnt_def disjnt_iff)
+ qed
+ have SUG: "S \<union> \<Union>G \<subseteq> U - K"
+ using \<open>S \<subseteq> U\<close> K apply (auto simp: G_def disjnt_iff)
+ by (meson Diff_iff subsetD in_components_subset)
+ then have contf': "continuous_on (S \<union> \<Union>G) f"
+ by (rule continuous_on_subset [OF contf])
+ have contg': "continuous_on (S \<union> UF) g"
+ apply (rule continuous_on_subset [OF contg])
+ using \<open>S \<subseteq> U\<close> by (auto simp: F_def G_def)
+ have "\<And>x. \<lbrakk>S \<subseteq> U; x \<in> S\<rbrakk> \<Longrightarrow> f x = g x"
+ by (subst gh) (auto simp: ah C0 intro: \<open>C0 \<in> F\<close>)
+ then have f_eq_g: "\<And>x. x \<in> S \<union> UF \<and> x \<in> S \<union> \<Union>G \<Longrightarrow> f x = g x"
+ using \<open>S \<subseteq> U\<close> apply (auto simp: F_def G_def UF_def dest: in_components_subset)
+ using components_eq by blast
+ have cont: "continuous_on (S \<union> \<Union>G \<union> (S \<union> UF)) (\<lambda>x. if x \<in> S \<union> \<Union>G then f x else g x)"
+ by (blast intro: continuous_on_cases_local [OF clo1 clo2 contf' contg' f_eq_g, of "\<lambda>x. x \<in> S \<union> \<Union>G"])
+ show ?thesis
+ proof
+ have UF: "\<Union>F - L \<subseteq> UF"
+ unfolding F_def UF_def using ah by blast
+ have "U - S - L = \<Union>(components (U - S)) - L"
+ by simp
+ also have "... = \<Union>F \<union> \<Union>G - L"
+ unfolding F_def G_def by blast
+ also have "... \<subseteq> UF \<union> \<Union>G"
+ using UF by blast
+ finally have "U - L \<subseteq> S \<union> \<Union>G \<union> (S \<union> UF)"
+ by blast
+ then show "continuous_on (U - L) (\<lambda>x. if x \<in> S \<union> \<Union>G then f x else g x)"
+ by (rule continuous_on_subset [OF cont])
+ have "((U - L) \<inter> {x. x \<notin> S \<and> (\<forall>xa\<in>G. x \<notin> xa)}) \<subseteq> ((U - L) \<inter> (-S \<inter> UF))"
+ using \<open>U - L \<subseteq> S \<union> \<Union>G \<union> (S \<union> UF)\<close> by auto
+ moreover have "g ` ((U - L) \<inter> (-S \<inter> UF)) \<subseteq> T"
+ proof -
+ have "g x \<in> T" if "x \<in> U" "x \<notin> L" "x \<notin> S" "C \<in> F" "x \<in> C" "x \<noteq> a C" for x C
+ proof (subst gh)
+ show "x \<in> (S \<union> UF) \<inter> (S \<union> (C - {a C}))"
+ using that by (auto simp: UF_def)
+ show "h C x \<in> T"
+ using ah that by (fastforce simp add: F_def)
+ qed (rule that)
+ then show ?thesis
+ by (force simp: UF_def)
+ qed
+ ultimately have "g ` ((U - L) \<inter> {x. x \<notin> S \<and> (\<forall>xa\<in>G. x \<notin> xa)}) \<subseteq> T"
+ using image_mono order_trans by blast
+ moreover have "f ` ((U - L) \<inter> (S \<union> \<Union>G)) \<subseteq> T"
+ using fim SUG by blast
+ ultimately show "(\<lambda>x. if x \<in> S \<union> \<Union>G then f x else g x) ` (U - L) \<subseteq> T"
+ by force
+ show "\<And>x. x \<in> S \<Longrightarrow> (if x \<in> S \<union> \<Union>G then f x else g x) = f x"
+ by (simp add: F_def G_def)
+ qed
+qed
+
+
+lemma extend_map_affine_to_sphere2:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes "compact S" "convex U" "bounded U" "affine T" "S \<subseteq> T"
+ and affTU: "aff_dim T \<le> aff_dim U"
+ and contf: "continuous_on S f"
+ and fim: "f ` S \<subseteq> rel_frontier U"
+ and ovlap: "\<And>C. C \<in> components(T - S) \<Longrightarrow> C \<inter> L \<noteq> {}"
+ obtains K g where "finite K" "K \<subseteq> L" "K \<subseteq> T" "disjnt K S"
+ "continuous_on (T - K) g" "g ` (T - K) \<subseteq> rel_frontier U"
+ "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof -
+ obtain K g where K: "finite K" "K \<subseteq> T" "disjnt K S"
+ and contg: "continuous_on (T - K) g"
+ and gim: "g ` (T - K) \<subseteq> rel_frontier U"
+ and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+ using assms extend_map_affine_to_sphere_cofinite_simple by metis
+ have "(\<exists>y C. C \<in> components (T - S) \<and> x \<in> C \<and> y \<in> C \<and> y \<in> L)" if "x \<in> K" for x
+ proof -
+ have "x \<in> T-S"
+ using \<open>K \<subseteq> T\<close> \<open>disjnt K S\<close> disjnt_def that by fastforce
+ then obtain C where "C \<in> components(T - S)" "x \<in> C"
+ by (metis UnionE Union_components)
+ with ovlap [of C] show ?thesis
+ by blast
+ qed
+ then obtain \<xi> where \<xi>: "\<And>x. x \<in> K \<Longrightarrow> \<exists>C. C \<in> components (T - S) \<and> x \<in> C \<and> \<xi> x \<in> C \<and> \<xi> x \<in> L"
+ by metis
+ obtain h where conth: "continuous_on (T - \<xi> ` K) h"
+ and him: "h ` (T - \<xi> ` K) \<subseteq> rel_frontier U"
+ and hg: "\<And>x. x \<in> S \<Longrightarrow> h x = g x"
+ proof (rule extend_map_affine_to_sphere1 [OF \<open>finite K\<close> \<open>affine T\<close> contg gim, of S "\<xi> ` K"])
+ show cloTS: "closedin (subtopology euclidean T) S"
+ by (simp add: \<open>compact S\<close> \<open>S \<subseteq> T\<close> closed_subset compact_imp_closed)
+ show "\<And>C. \<lbrakk>C \<in> components (T - S); C \<inter> K \<noteq> {}\<rbrakk> \<Longrightarrow> C \<inter> \<xi> ` K \<noteq> {}"
+ using \<xi> components_eq by blast
+ qed (use K in auto)
+ show ?thesis
+ proof
+ show *: "\<xi> ` K \<subseteq> L"
+ using \<xi> by blast
+ show "finite (\<xi> ` K)"
+ by (simp add: K)
+ show "\<xi> ` K \<subseteq> T"
+ by clarify (meson \<xi> Diff_iff contra_subsetD in_components_subset)
+ show "continuous_on (T - \<xi> ` K) h"
+ by (rule conth)
+ show "disjnt (\<xi> ` K) S"
+ using K
+ apply (auto simp: disjnt_def)
+ by (metis \<xi> DiffD2 UnionI Union_components)
+ qed (simp_all add: him hg gf)
+qed
+
+
+proposition extend_map_affine_to_sphere_cofinite_gen:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes SUT: "compact S" "convex U" "bounded U" "affine T" "S \<subseteq> T"
+ and aff: "aff_dim T \<le> aff_dim U"
+ and contf: "continuous_on S f"
+ and fim: "f ` S \<subseteq> rel_frontier U"
+ and dis: "\<And>C. \<lbrakk>C \<in> components(T - S); bounded C\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
+ obtains K g where "finite K" "K \<subseteq> L" "K \<subseteq> T" "disjnt K S" "continuous_on (T - K) g"
+ "g ` (T - K) \<subseteq> rel_frontier U"
+ "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof (cases "S = {}")
+ case True
+ show ?thesis
+ proof (cases "rel_frontier U = {}")
+ case True
+ with aff have "aff_dim T \<le> 0"
+ apply (simp add: rel_frontier_eq_empty)
+ using affine_bounded_eq_lowdim \<open>bounded U\<close> order_trans by auto
+ with aff_dim_geq [of T] consider "aff_dim T = -1" | "aff_dim T = 0"
+ by linarith
+ then show ?thesis
+ proof cases
+ assume "aff_dim T = -1"
+ then have "T = {}"
+ by (simp add: aff_dim_empty)
+ then show ?thesis
+ by (rule_tac K="{}" in that) auto
+ next
+ assume "aff_dim T = 0"
+ then obtain a where "T = {a}"
+ using aff_dim_eq_0 by blast
+ then have "a \<in> L"
+ using dis [of "{a}"] \<open>S = {}\<close> by (auto simp: in_components_self)
+ with \<open>S = {}\<close> \<open>T = {a}\<close> show ?thesis
+ by (rule_tac K="{a}" and g=f in that) auto
+ qed
+ next
+ case False
+ then obtain y where "y \<in> rel_frontier U"
+ by auto
+ with \<open>S = {}\<close> show ?thesis
+ by (rule_tac K="{}" and g="\<lambda>x. y" in that) (auto simp: continuous_on_const)
+ qed
+next
+ case False
+ have "bounded S"
+ by (simp add: assms compact_imp_bounded)
+ then obtain b where b: "S \<subseteq> cbox (-b) b"
+ using bounded_subset_cbox_symmetric by blast
+ define LU where "LU \<equiv> L \<union> (\<Union> {C \<in> components (T - S). ~bounded C} - cbox (-(b+One)) (b+One))"
+ obtain K g where "finite K" "K \<subseteq> LU" "K \<subseteq> T" "disjnt K S"
+ and contg: "continuous_on (T - K) g"
+ and gim: "g ` (T - K) \<subseteq> rel_frontier U"
+ and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+ proof (rule extend_map_affine_to_sphere2 [OF SUT aff contf fim])
+ show "C \<inter> LU \<noteq> {}" if "C \<in> components (T - S)" for C
+ proof (cases "bounded C")
+ case True
+ with dis that show ?thesis
+ unfolding LU_def by fastforce
+ next
+ case False
+ then have "\<not> bounded (\<Union>{C \<in> components (T - S). \<not> bounded C})"
+ by (metis (no_types, lifting) Sup_upper bounded_subset mem_Collect_eq that)
+ then show ?thesis
+ apply (clarsimp simp: LU_def Int_Un_distrib Diff_Int_distrib Int_UN_distrib)
+ by (metis (no_types, lifting) False Sup_upper bounded_cbox bounded_subset inf.orderE mem_Collect_eq that)
+ qed
+ qed blast
+ have *: False if "x \<in> cbox (- b - m *\<^sub>R One) (b + m *\<^sub>R One)"
+ "x \<notin> box (- b - n *\<^sub>R One) (b + n *\<^sub>R One)"
+ "0 \<le> m" "m < n" "n \<le> 1" for m n x
+ using that by (auto simp: mem_box algebra_simps)
+ have "disjoint_family_on (\<lambda>d. frontier (cbox (- b - d *\<^sub>R One) (b + d *\<^sub>R One))) {1 / 2..1}"
+ by (auto simp: disjoint_family_on_def neq_iff frontier_def dest: *)
+ then obtain d where d12: "1/2 \<le> d" "d \<le> 1"
+ and ddis: "disjnt K (frontier (cbox (-(b + d *\<^sub>R One)) (b + d *\<^sub>R One)))"
+ using disjoint_family_elem_disjnt [of "{1/2..1::real}" K "\<lambda>d. frontier (cbox (-(b + d *\<^sub>R One)) (b + d *\<^sub>R One))"]
+ by (auto simp: \<open>finite K\<close>)
+ define c where "c \<equiv> b + d *\<^sub>R One"
+ have cbsub: "cbox (-b) b \<subseteq> box (-c) c"
+ "cbox (-b) b \<subseteq> cbox (-c) c"
+ "cbox (-c) c \<subseteq> cbox (-(b+One)) (b+One)"
+ using d12 by (simp_all add: subset_box c_def inner_diff_left inner_left_distrib)
+ have clo_cT: "closed (cbox (- c) c \<inter> T)"
+ using affine_closed \<open>affine T\<close> by blast
+ have cT_ne: "cbox (- c) c \<inter> T \<noteq> {}"
+ using \<open>S \<noteq> {}\<close> \<open>S \<subseteq> T\<close> b cbsub by fastforce
+ have S_sub_cc: "S \<subseteq> cbox (- c) c"
+ using \<open>cbox (- b) b \<subseteq> cbox (- c) c\<close> b by auto
+ show ?thesis
+ proof
+ show "finite (K \<inter> cbox (-(b+One)) (b+One))"
+ using \<open>finite K\<close> by blast
+ show "K \<inter> cbox (- (b + One)) (b + One) \<subseteq> L"
+ using \<open>K \<subseteq> LU\<close> by (auto simp: LU_def)
+ show "K \<inter> cbox (- (b + One)) (b + One) \<subseteq> T"
+ using \<open>K \<subseteq> T\<close> by auto
+ show "disjnt (K \<inter> cbox (- (b + One)) (b + One)) S"
+ using \<open>disjnt K S\<close> by (simp add: disjnt_def disjoint_eq_subset_Compl inf.coboundedI1)
+ have cloTK: "closest_point (cbox (- c) c \<inter> T) x \<in> T - K"
+ if "x \<in> T" and Knot: "x \<in> K \<longrightarrow> x \<notin> cbox (- b - One) (b + One)" for x
+ proof (cases "x \<in> cbox (- c) c")
+ case True
+ with \<open>x \<in> T\<close> show ?thesis
+ using cbsub(3) Knot by (force simp: closest_point_self)
+ next
+ case False
+ have clo_in_rf: "closest_point (cbox (- c) c \<inter> T) x \<in> rel_frontier (cbox (- c) c \<inter> T)"
+ proof (intro closest_point_in_rel_frontier [OF clo_cT cT_ne] DiffI notI)
+ have "T \<inter> interior (cbox (- c) c) \<noteq> {}"
+ using \<open>S \<noteq> {}\<close> \<open>S \<subseteq> T\<close> b cbsub(1) by fastforce
+ then show "x \<in> affine hull (cbox (- c) c \<inter> T)"
+ by (simp add: Int_commute affine_hull_affine_Int_nonempty_interior \<open>affine T\<close> hull_inc that(1))
+ next
+ show "False" if "x \<in> rel_interior (cbox (- c) c \<inter> T)"
+ proof -
+ have "interior (cbox (- c) c) \<inter> T \<noteq> {}"
+ using \<open>S \<noteq> {}\<close> \<open>S \<subseteq> T\<close> b cbsub(1) by fastforce
+ then have "affine hull (T \<inter> cbox (- c) c) = T"
+ using affine_hull_convex_Int_nonempty_interior [of T "cbox (- c) c"]
+ by (simp add: affine_imp_convex \<open>affine T\<close> inf_commute)
+ then show ?thesis
+ by (meson subsetD le_inf_iff rel_interior_subset that False)
+ qed
+ qed
+ have "closest_point (cbox (- c) c \<inter> T) x \<notin> K"
+ proof
+ assume inK: "closest_point (cbox (- c) c \<inter> T) x \<in> K"
+ have "\<And>x. x \<in> K \<Longrightarrow> x \<notin> frontier (cbox (- (b + d *\<^sub>R One)) (b + d *\<^sub>R One))"
+ by (metis ddis disjnt_iff)
+ then show False
+ by (metis DiffI Int_iff \<open>affine T\<close> cT_ne c_def clo_cT clo_in_rf closest_point_in_set
+ convex_affine_rel_frontier_Int convex_box(1) empty_iff frontier_cbox inK interior_cbox)
+ qed
+ then show ?thesis
+ using cT_ne clo_cT closest_point_in_set by blast
+ qed
+ show "continuous_on (T - K \<inter> cbox (- (b + One)) (b + One)) (g \<circ> closest_point (cbox (-c) c \<inter> T))"
+ apply (intro continuous_on_compose continuous_on_closest_point continuous_on_subset [OF contg])
+ apply (simp_all add: clo_cT affine_imp_convex \<open>affine T\<close> convex_Int cT_ne)
+ using cloTK by blast
+ have "g (closest_point (cbox (- c) c \<inter> T) x) \<in> rel_frontier U"
+ if "x \<in> T" "x \<in> K \<longrightarrow> x \<notin> cbox (- b - One) (b + One)" for x
+ apply (rule gim [THEN subsetD])
+ using that cloTK by blast
+ then show "(g \<circ> closest_point (cbox (- c) c \<inter> T)) ` (T - K \<inter> cbox (- (b + One)) (b + One))
+ \<subseteq> rel_frontier U"
+ by force
+ show "\<And>x. x \<in> S \<Longrightarrow> (g \<circ> closest_point (cbox (- c) c \<inter> T)) x = f x"
+ by simp (metis (mono_tags, lifting) IntI \<open>S \<subseteq> T\<close> cT_ne clo_cT closest_point_refl gf subsetD S_sub_cc)
+ qed
+qed
+
+
+corollary extend_map_affine_to_sphere_cofinite:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes SUT: "compact S" "affine T" "S \<subseteq> T"
+ and aff: "aff_dim T \<le> DIM('b)" and "0 \<le> r"
+ and contf: "continuous_on S f"
+ and fim: "f ` S \<subseteq> sphere a r"
+ and dis: "\<And>C. \<lbrakk>C \<in> components(T - S); bounded C\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
+ obtains K g where "finite K" "K \<subseteq> L" "K \<subseteq> T" "disjnt K S" "continuous_on (T - K) g"
+ "g ` (T - K) \<subseteq> sphere a r" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof (cases "r = 0")
+ case True
+ with fim show ?thesis
+ by (rule_tac K="{}" and g = "\<lambda>x. a" in that) (auto simp: continuous_on_const)
+next
+ case False
+ with assms have "0 < r" by auto
+ then have "aff_dim T \<le> aff_dim (cball a r)"
+ by (simp add: aff aff_dim_cball)
+ then show ?thesis
+ apply (rule extend_map_affine_to_sphere_cofinite_gen
+ [OF \<open>compact S\<close> convex_cball bounded_cball \<open>affine T\<close> \<open>S \<subseteq> T\<close> _ contf])
+ using fim apply (auto simp: assms False that dest: dis)
+ done
+qed
+
+corollary extend_map_UNIV_to_sphere_cofinite:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes aff: "DIM('a) \<le> DIM('b)" and "0 \<le> r"
+ and SUT: "compact S"
+ and contf: "continuous_on S f"
+ and fim: "f ` S \<subseteq> sphere a r"
+ and dis: "\<And>C. \<lbrakk>C \<in> components(- S); bounded C\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
+ obtains K g where "finite K" "K \<subseteq> L" "disjnt K S" "continuous_on (- K) g"
+ "g ` (- K) \<subseteq> sphere a r" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+apply (rule extend_map_affine_to_sphere_cofinite
+ [OF \<open>compact S\<close> affine_UNIV subset_UNIV _ \<open>0 \<le> r\<close> contf fim dis])
+ apply (auto simp: assms that Compl_eq_Diff_UNIV [symmetric])
+done
+
+corollary extend_map_UNIV_to_sphere_no_bounded_component:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes aff: "DIM('a) \<le> DIM('b)" and "0 \<le> r"
+ and SUT: "compact S"
+ and contf: "continuous_on S f"
+ and fim: "f ` S \<subseteq> sphere a r"
+ and dis: "\<And>C. C \<in> components(- S) \<Longrightarrow> \<not> bounded C"
+ obtains g where "continuous_on UNIV g" "g ` UNIV \<subseteq> sphere a r" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+apply (rule extend_map_UNIV_to_sphere_cofinite [OF aff \<open>0 \<le> r\<close> \<open>compact S\<close> contf fim, of "{}"])
+ apply (auto simp: that dest: dis)
+done
+
+theorem Borsuk_separation_theorem_gen:
+ fixes S :: "'a::euclidean_space set"
+ assumes "compact S"
+ shows "(\<forall>c \<in> components(- S). ~bounded c) \<longleftrightarrow>
+ (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> sphere (0::'a) 1
+ \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) S (sphere 0 1) f (\<lambda>x. c)))"
+ (is "?lhs = ?rhs")
+proof
+ assume L [rule_format]: ?lhs
+ show ?rhs
+ proof clarify
+ fix f :: "'a \<Rightarrow> 'a"
+ assume contf: "continuous_on S f" and fim: "f ` S \<subseteq> sphere 0 1"
+ obtain g where contg: "continuous_on UNIV g" and gim: "range g \<subseteq> sphere 0 1"
+ and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+ by (rule extend_map_UNIV_to_sphere_no_bounded_component [OF _ _ \<open>compact S\<close> contf fim L]) auto
+ then show "\<exists>c. homotopic_with (\<lambda>x. True) S (sphere 0 1) f (\<lambda>x. c)"
+ using nullhomotopic_from_contractible [OF contg gim]
+ by (metis assms compact_imp_closed contf empty_iff fim homotopic_with_equal nullhomotopic_into_sphere_extension)
+ qed
+next
+ assume R [rule_format]: ?rhs
+ show ?lhs
+ unfolding components_def
+ proof clarify
+ fix a
+ assume "a \<notin> S" and a: "bounded (connected_component_set (- S) a)"
+ have cont: "continuous_on S (\<lambda>x. inverse(norm(x - a)) *\<^sub>R (x - a))"
+ apply (intro continuous_intros)
+ using \<open>a \<notin> S\<close> by auto
+ have im: "(\<lambda>x. inverse(norm(x - a)) *\<^sub>R (x - a)) ` S \<subseteq> sphere 0 1"
+ by clarsimp (metis \<open>a \<notin> S\<close> eq_iff_diff_eq_0 left_inverse norm_eq_zero)
+ show False
+ using R cont im Borsuk_map_essential_bounded_component [OF \<open>compact S\<close> \<open>a \<notin> S\<close>] a by blast
+ qed
+qed
+
+
+corollary Borsuk_separation_theorem:
+ fixes S :: "'a::euclidean_space set"
+ assumes "compact S" and 2: "2 \<le> DIM('a)"
+ shows "connected(- S) \<longleftrightarrow>
+ (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> sphere (0::'a) 1
+ \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) S (sphere 0 1) f (\<lambda>x. c)))"
+ (is "?lhs = ?rhs")
+proof
+ assume L: ?lhs
+ show ?rhs
+ proof (cases "S = {}")
+ case True
+ then show ?thesis by auto
+ next
+ case False
+ then have "(\<forall>c\<in>components (- S). \<not> bounded c)"
+ by (metis L assms(1) bounded_empty cobounded_imp_unbounded compact_imp_bounded in_components_maximal order_refl)
+ then show ?thesis
+ by (simp add: Borsuk_separation_theorem_gen [OF \<open>compact S\<close>])
+ qed
+next
+ assume R: ?rhs
+ then show ?lhs
+ apply (simp add: Borsuk_separation_theorem_gen [OF \<open>compact S\<close>, symmetric])
+ apply (auto simp: components_def connected_iff_eq_connected_component_set)
+ using connected_component_in apply fastforce
+ using cobounded_unique_unbounded_component [OF _ 2, of "-S"] \<open>compact S\<close> compact_eq_bounded_closed by fastforce
+qed
+
+
+lemma homotopy_eqv_separation:
+ fixes S :: "'a::euclidean_space set" and T :: "'a set"
+ assumes "S homotopy_eqv T" and "compact S" and "compact T"
+ shows "connected(- S) \<longleftrightarrow> connected(- T)"
+proof -
+ consider "DIM('a) = 1" | "2 \<le> DIM('a)"
+ by (metis DIM_ge_Suc0 One_nat_def Suc_1 dual_order.antisym not_less_eq_eq)
+ then show ?thesis
+ proof cases
+ case 1
+ then show ?thesis
+ using bounded_connected_Compl_1 compact_imp_bounded homotopy_eqv_empty1 homotopy_eqv_empty2 assms by metis
+ next
+ case 2
+ with assms show ?thesis
+ by (simp add: Borsuk_separation_theorem homotopy_eqv_cohomotopic_triviality_null)
+ qed
+qed
+
+lemma Jordan_Brouwer_separation:
+ fixes S :: "'a::euclidean_space set" and a::'a
+ assumes hom: "S homeomorphic sphere a r" and "0 < r"
+ shows "\<not> connected(- S)"
+proof -
+ have "- sphere a r \<inter> ball a r \<noteq> {}"
+ using \<open>0 < r\<close> by (simp add: Int_absorb1 subset_eq)
+ moreover
+ have eq: "- sphere a r - ball a r = - cball a r"
+ by auto
+ have "- cball a r \<noteq> {}"
+ proof -
+ have "frontier (cball a r) \<noteq> {}"
+ using \<open>0 < r\<close> by auto
+ then show ?thesis
+ by (metis frontier_complement frontier_empty)
+ qed
+ with eq have "- sphere a r - ball a r \<noteq> {}"
+ by auto
+ moreover
+ have "connected (- S) = connected (- sphere a r)"
+ proof (rule homotopy_eqv_separation)
+ show "S homotopy_eqv sphere a r"
+ using hom homeomorphic_imp_homotopy_eqv by blast
+ show "compact (sphere a r)"
+ by simp
+ then show " compact S"
+ using hom homeomorphic_compactness by blast
+ qed
+ ultimately show ?thesis
+ using connected_Int_frontier [of "- sphere a r" "ball a r"] by (auto simp: \<open>0 < r\<close>)
+qed
+
+
+lemma Jordan_Brouwer_frontier:
+ fixes S :: "'a::euclidean_space set" and a::'a
+ assumes S: "S homeomorphic sphere a r" and T: "T \<in> components(- S)" and 2: "2 \<le> DIM('a)"
+ shows "frontier T = S"
+proof (cases r rule: linorder_cases)
+ assume "r < 0"
+ with S T show ?thesis by auto
+next
+ assume "r = 0"
+ with S T card_eq_SucD obtain b where "S = {b}"
+ by (auto simp: homeomorphic_finite [of "{a}" S])
+ have "components (- {b}) = { -{b}}"
+ using T \<open>S = {b}\<close> by (auto simp: components_eq_sing_iff connected_punctured_universe 2)
+ with T show ?thesis
+ by (metis \<open>S = {b}\<close> cball_trivial frontier_cball frontier_complement singletonD sphere_trivial)
+next
+ assume "r > 0"
+ have "compact S"
+ using homeomorphic_compactness compact_sphere S by blast
+ show ?thesis
+ proof (rule frontier_minimal_separating_closed)
+ show "closed S"
+ using \<open>compact S\<close> compact_eq_bounded_closed by blast
+ show "\<not> connected (- S)"
+ using Jordan_Brouwer_separation S \<open>0 < r\<close> by blast
+ obtain f g where hom: "homeomorphism S (sphere a r) f g"
+ using S by (auto simp: homeomorphic_def)
+ show "connected (- T)" if "closed T" "T \<subset> S" for T
+ proof -
+ have "f ` T \<subseteq> sphere a r"
+ using \<open>T \<subset> S\<close> hom homeomorphism_image1 by blast
+ moreover have "f ` T \<noteq> sphere a r"
+ using \<open>T \<subset> S\<close> hom
+ by (metis homeomorphism_image2 homeomorphism_of_subsets order_refl psubsetE)
+ ultimately have "f ` T \<subset> sphere a r" by blast
+ then have "connected (- f ` T)"
+ by (rule psubset_sphere_Compl_connected [OF _ \<open>0 < r\<close> 2])
+ moreover have "compact T"
+ using \<open>compact S\<close> bounded_subset compact_eq_bounded_closed that by blast
+ moreover then have "compact (f ` T)"
+ by (meson compact_continuous_image continuous_on_subset hom homeomorphism_def psubsetE \<open>T \<subset> S\<close>)
+ moreover have "T homotopy_eqv f ` T"
+ by (meson \<open>f ` T \<subseteq> sphere a r\<close> dual_order.strict_implies_order hom homeomorphic_def homeomorphic_imp_homotopy_eqv homeomorphism_of_subsets \<open>T \<subset> S\<close>)
+ ultimately show ?thesis
+ using homotopy_eqv_separation [of T "f`T"] by blast
+ qed
+ qed (rule T)
+qed
+
+lemma Jordan_Brouwer_nonseparation:
+ fixes S :: "'a::euclidean_space set" and a::'a
+ assumes S: "S homeomorphic sphere a r" and "T \<subset> S" and 2: "2 \<le> DIM('a)"
+ shows "connected(- T)"
+proof -
+ have *: "connected(C \<union> (S - T))" if "C \<in> components(- S)" for C
+ proof (rule connected_intermediate_closure)
+ show "connected C"
+ using in_components_connected that by auto
+ have "S = frontier C"
+ using "2" Jordan_Brouwer_frontier S that by blast
+ with closure_subset show "C \<union> (S - T) \<subseteq> closure C"
+ by (auto simp: frontier_def)
+ qed auto
+ have "components(- S) \<noteq> {}"
+ by (metis S bounded_empty cobounded_imp_unbounded compact_eq_bounded_closed compact_sphere
+ components_eq_empty homeomorphic_compactness)
+ then have "- T = (\<Union>C \<in> components(- S). C \<union> (S - T))"
+ using Union_components [of "-S"] \<open>T \<subset> S\<close> by auto
+ then show ?thesis
+ apply (rule ssubst)
+ apply (rule connected_Union)
+ using \<open>T \<subset> S\<close> apply (auto simp: *)
+ done
+qed
+
+subsection\<open> Invariance of domain and corollaries\<close>
+
+lemma invariance_of_domain_ball:
+ fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
+ assumes contf: "continuous_on (cball a r) f" and "0 < r"
+ and inj: "inj_on f (cball a r)"
+ shows "open(f ` ball a r)"
+proof (cases "DIM('a) = 1")
+ case True
+ obtain h::"'a\<Rightarrow>real" and k
+ where "linear h" "linear k" "h ` UNIV = UNIV" "k ` UNIV = UNIV"
+ "\<And>x. norm(h x) = norm x" "\<And>x. norm(k x) = norm x"
+ "\<And>x. k(h x) = x" "\<And>x. h(k x) = x"
+ apply (rule isomorphisms_UNIV_UNIV [where 'M='a and 'N=real])
+ using True
+ apply force
+ by (metis UNIV_I UNIV_eq_I imageI)
+ have cont: "continuous_on S h" "continuous_on T k" for S T
+ by (simp_all add: \<open>linear h\<close> \<open>linear k\<close> linear_continuous_on linear_linear)
+ have "continuous_on (h ` cball a r) (h \<circ> f \<circ> k)"
+ apply (intro continuous_on_compose cont continuous_on_subset [OF contf])
+ apply (auto simp: \<open>\<And>x. k (h x) = x\<close>)
+ done
+ moreover have "is_interval (h ` cball a r)"
+ by (simp add: is_interval_connected_1 \<open>linear h\<close> linear_continuous_on linear_linear connected_continuous_image)
+ moreover have "inj_on (h \<circ> f \<circ> k) (h ` cball a r)"
+ using inj by (simp add: inj_on_def) (metis \<open>\<And>x. k (h x) = x\<close>)
+ ultimately have *: "\<And>T. \<lbrakk>open T; T \<subseteq> h ` cball a r\<rbrakk> \<Longrightarrow> open ((h \<circ> f \<circ> k) ` T)"
+ using injective_eq_1d_open_map_UNIV by blast
+ have "open ((h \<circ> f \<circ> k) ` (h ` ball a r))"
+ by (rule *) (auto simp: \<open>linear h\<close> \<open>range h = UNIV\<close> open_surjective_linear_image)
+ then have "open ((h \<circ> f) ` ball a r)"
+ by (simp add: image_comp \<open>\<And>x. k (h x) = x\<close> cong: image_cong)
+ then show ?thesis
+ apply (simp add: image_comp [symmetric])
+ apply (metis open_bijective_linear_image_eq \<open>linear h\<close> \<open>\<And>x. k (h x) = x\<close> \<open>range h = UNIV\<close> bijI inj_on_def)
+ done
+next
+ case False
+ then have 2: "DIM('a) \<ge> 2"
+ by (metis DIM_ge_Suc0 One_nat_def Suc_1 antisym not_less_eq_eq)
+ have fimsub: "f ` ball a r \<subseteq> - f ` sphere a r"
+ using inj by clarsimp (metis inj_onD less_eq_real_def mem_cball order_less_irrefl)
+ have hom: "f ` sphere a r homeomorphic sphere a r"
+ by (meson compact_sphere contf continuous_on_subset homeomorphic_compact homeomorphic_sym inj inj_on_subset sphere_cball)
+ then have nconn: "\<not> connected (- f ` sphere a r)"
+ by (rule Jordan_Brouwer_separation) (auto simp: \<open>0 < r\<close>)
+ obtain C where C: "C \<in> components (- f ` sphere a r)" and "bounded C"
+ apply (rule cobounded_has_bounded_component [OF _ nconn])
+ apply (simp_all add: 2)
+ by (meson compact_imp_bounded compact_continuous_image_eq compact_sphere contf inj sphere_cball)
+ moreover have "f ` (ball a r) = C"
+ proof
+ have "C \<noteq> {}"
+ by (rule in_components_nonempty [OF C])
+ show "C \<subseteq> f ` ball a r"
+ proof (rule ccontr)
+ assume nonsub: "\<not> C \<subseteq> f ` ball a r"
+ have "- f ` cball a r \<subseteq> C"
+ proof (rule components_maximal [OF C])
+ have "f ` cball a r homeomorphic cball a r"
+ using compact_cball contf homeomorphic_compact homeomorphic_sym inj by blast
+ then show "connected (- f ` cball a r)"
+ by (auto intro: connected_complement_homeomorphic_convex_compact 2)
+ show "- f ` cball a r \<subseteq> - f ` sphere a r"
+ by auto
+ then show "C \<inter> - f ` cball a r \<noteq> {}"
+ using \<open>C \<noteq> {}\<close> in_components_subset [OF C] nonsub
+ using image_iff by fastforce
+ qed
+ then have "bounded (- f ` cball a r)"
+ using bounded_subset \<open>bounded C\<close> by auto
+ then have "\<not> bounded (f ` cball a r)"
+ using cobounded_imp_unbounded by blast
+ then show "False"
+ using compact_continuous_image [OF contf] compact_cball compact_imp_bounded by blast
+ qed
+ with \<open>C \<noteq> {}\<close> have "C \<inter> f ` ball a r \<noteq> {}"
+ by (simp add: inf.absorb_iff1)
+ then show "f ` ball a r \<subseteq> C"
+ by (metis components_maximal [OF C _ fimsub] connected_continuous_image ball_subset_cball connected_ball contf continuous_on_subset)
+ qed
+ moreover have "open (- f ` sphere a r)"
+ using hom compact_eq_bounded_closed compact_sphere homeomorphic_compactness by blast
+ ultimately show ?thesis
+ using open_components by blast
+qed
+
+
+text\<open>Proved by L. E. J. Brouwer (1912)\<close>
+theorem invariance_of_domain:
+ fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
+ assumes "continuous_on S f" "open S" "inj_on f S"
+ shows "open(f ` S)"
+ unfolding open_subopen [of "f`S"]
+proof clarify
+ fix a
+ assume "a \<in> S"
+ obtain \<delta> where "\<delta> > 0" and \<delta>: "cball a \<delta> \<subseteq> S"
+ using \<open>open S\<close> \<open>a \<in> S\<close> open_contains_cball_eq by blast
+ show "\<exists>T. open T \<and> f a \<in> T \<and> T \<subseteq> f ` S"
+ proof (intro exI conjI)
+ show "open (f ` (ball a \<delta>))"
+ by (meson \<delta> \<open>0 < \<delta>\<close> assms continuous_on_subset inj_on_subset invariance_of_domain_ball)
+ show "f a \<in> f ` ball a \<delta>"
+ by (simp add: \<open>0 < \<delta>\<close>)
+ show "f ` ball a \<delta> \<subseteq> f ` S"
+ using \<delta> ball_subset_cball by blast
+ qed
+qed
+
+lemma inv_of_domain_ss0:
+ fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
+ assumes contf: "continuous_on U f" and injf: "inj_on f U" and fim: "f ` U \<subseteq> S"
+ and "subspace S" and dimS: "dim S = DIM('b::euclidean_space)"
+ and ope: "openin (subtopology euclidean S) U"
+ shows "openin (subtopology euclidean S) (f ` U)"
+proof -
+ have "U \<subseteq> S"
+ using ope openin_imp_subset by blast
+ have "(UNIV::'b set) homeomorphic S"
+ by (simp add: \<open>subspace S\<close> dimS dim_UNIV homeomorphic_subspaces)
+ then obtain h k where homhk: "homeomorphism (UNIV::'b set) S h k"
+ using homeomorphic_def by blast
+ have homkh: "homeomorphism S (k ` S) k h"
+ using homhk homeomorphism_image2 homeomorphism_sym by fastforce
+ have "open ((k \<circ> f \<circ> h) ` k ` U)"
+ proof (rule invariance_of_domain)
+ show "continuous_on (k ` U) (k \<circ> f \<circ> h)"
+ proof (intro continuous_intros)
+ show "continuous_on (k ` U) h"
+ by (meson continuous_on_subset [OF homeomorphism_cont1 [OF homhk]] top_greatest)
+ show "continuous_on (h ` k ` U) f"
+ apply (rule continuous_on_subset [OF contf], clarify)
+ apply (metis homhk homeomorphism_def ope openin_imp_subset rev_subsetD)
+ done
+ show "continuous_on (f ` h ` k ` U) k"
+ apply (rule continuous_on_subset [OF homeomorphism_cont2 [OF homhk]])
+ using fim homhk homeomorphism_apply2 ope openin_subset by fastforce
+ qed
+ have ope_iff: "\<And>T. open T \<longleftrightarrow> openin (subtopology euclidean (k ` S)) T"
+ using homhk homeomorphism_image2 open_openin by fastforce
+ show "open (k ` U)"
+ by (simp add: ope_iff homeomorphism_imp_open_map [OF homkh ope])
+ show "inj_on (k \<circ> f \<circ> h) (k ` U)"
+ apply (clarsimp simp: inj_on_def)
+ by (metis subsetD fim homeomorphism_apply2 [OF homhk] image_subset_iff inj_on_eq_iff injf \<open>U \<subseteq> S\<close>)
+ qed
+ moreover
+ have eq: "f ` U = h ` (k \<circ> f \<circ> h \<circ> k) ` U"
+ apply (auto simp: image_comp [symmetric])
+ apply (metis homkh \<open>U \<subseteq> S\<close> fim homeomorphism_image2 homeomorphism_of_subsets homhk imageI subset_UNIV)
+ by (metis \<open>U \<subseteq> S\<close> subsetD fim homeomorphism_def homhk image_eqI)
+ ultimately show ?thesis
+ by (metis (no_types, hide_lams) homeomorphism_imp_open_map homhk image_comp open_openin subtopology_UNIV)
+qed
+
+lemma inv_of_domain_ss1:
+ fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
+ assumes contf: "continuous_on U f" and injf: "inj_on f U" and fim: "f ` U \<subseteq> S"
+ and "subspace S"
+ and ope: "openin (subtopology euclidean S) U"
+ shows "openin (subtopology euclidean S) (f ` U)"
+proof -
+ define S' where "S' \<equiv> {y. \<forall>x \<in> S. orthogonal x y}"
+ have "subspace S'"
+ by (simp add: S'_def subspace_orthogonal_to_vectors)
+ define g where "g \<equiv> \<lambda>z::'a*'a. ((f \<circ> fst)z, snd z)"
+ have "openin (subtopology euclidean (S \<times> S')) (g ` (U \<times> S'))"
+ proof (rule inv_of_domain_ss0)
+ show "continuous_on (U \<times> S') g"
+ apply (simp add: g_def)
+ apply (intro continuous_intros continuous_on_compose2 [OF contf continuous_on_fst], auto)
+ done
+ show "g ` (U \<times> S') \<subseteq> S \<times> S'"
+ using fim by (auto simp: g_def)
+ show "inj_on g (U \<times> S')"
+ using injf by (auto simp: g_def inj_on_def)
+ show "subspace (S \<times> S')"
+ by (simp add: \<open>subspace S'\<close> \<open>subspace S\<close> subspace_Times)
+ show "openin (subtopology euclidean (S \<times> S')) (U \<times> S')"
+ by (simp add: openin_Times [OF ope])
+ have "dim (S \<times> S') = dim S + dim S'"
+ by (simp add: \<open>subspace S'\<close> \<open>subspace S\<close> dim_Times)
+ also have "... = DIM('a)"
+ using dim_subspace_orthogonal_to_vectors [OF \<open>subspace S\<close> subspace_UNIV]
+ by (simp add: add.commute S'_def)
+ finally show "dim (S \<times> S') = DIM('a)" .
+ qed
+ moreover have "g ` (U \<times> S') = f ` U \<times> S'"
+ by (auto simp: g_def image_iff)
+ moreover have "0 \<in> S'"
+ using \<open>subspace S'\<close> subspace_affine by blast
+ ultimately show ?thesis
+ by (auto simp: openin_Times_eq)
+qed
+
+
+corollary invariance_of_domain_subspaces:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes ope: "openin (subtopology euclidean U) S"
+ and "subspace U" "subspace V" and VU: "dim V \<le> dim U"
+ and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
+ and injf: "inj_on f S"
+ shows "openin (subtopology euclidean V) (f ` S)"
+proof -
+ obtain V' where "subspace V'" "V' \<subseteq> U" "dim V' = dim V"
+ using choose_subspace_of_subspace [OF VU]
+ by (metis span_eq \<open>subspace U\<close>)
+ then have "V homeomorphic V'"
+ by (simp add: \<open>subspace V\<close> homeomorphic_subspaces)
+ then obtain h k where homhk: "homeomorphism V V' h k"
+ using homeomorphic_def by blast
+ have eq: "f ` S = k ` (h \<circ> f) ` S"
+ proof -
+ have "k ` h ` f ` S = f ` S"
+ by (meson fim homeomorphism_def homeomorphism_of_subsets homhk subset_refl)
+ then show ?thesis
+ by (simp add: image_comp)
+ qed
+ show ?thesis
+ unfolding eq
+ proof (rule homeomorphism_imp_open_map)
+ show homkh: "homeomorphism V' V k h"
+ by (simp add: homeomorphism_symD homhk)
+ have hfV': "(h \<circ> f) ` S \<subseteq> V'"
+ using fim homeomorphism_image1 homhk by fastforce
+ moreover have "openin (subtopology euclidean U) ((h \<circ> f) ` S)"
+ proof (rule inv_of_domain_ss1)
+ show "continuous_on S (h \<circ> f)"
+ by (meson contf continuous_on_compose continuous_on_subset fim homeomorphism_cont1 homhk)
+ show "inj_on (h \<circ> f) S"
+ apply (clarsimp simp: inj_on_def)
+ by (metis fim homeomorphism_apply2 [OF homkh] image_subset_iff inj_onD injf)
+ show "(h \<circ> f) ` S \<subseteq> U"
+ using \<open>V' \<subseteq> U\<close> hfV' by auto
+ qed (auto simp: assms)
+ ultimately show "openin (subtopology euclidean V') ((h \<circ> f) ` S)"
+ using openin_subset_trans \<open>V' \<subseteq> U\<close> by force
+ qed
+qed
+
+corollary invariance_of_dimension_subspaces:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes ope: "openin (subtopology euclidean U) S"
+ and "subspace U" "subspace V"
+ and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
+ and injf: "inj_on f S" and "S \<noteq> {}"
+ shows "dim U \<le> dim V"
+proof -
+ have "False" if "dim V < dim U"
+ proof -
+ obtain T where "subspace T" "T \<subseteq> U" "dim T = dim V"
+ using choose_subspace_of_subspace [of "dim V" U]
+ by (metis span_eq \<open>subspace U\<close> \<open>dim V < dim U\<close> linear not_le)
+ then have "V homeomorphic T"
+ by (simp add: \<open>subspace V\<close> homeomorphic_subspaces)
+ then obtain h k where homhk: "homeomorphism V T h k"
+ using homeomorphic_def by blast
+ have "continuous_on S (h \<circ> f)"
+ by (meson contf continuous_on_compose continuous_on_subset fim homeomorphism_cont1 homhk)
+ moreover have "(h \<circ> f) ` S \<subseteq> U"
+ using \<open>T \<subseteq> U\<close> fim homeomorphism_image1 homhk by fastforce
+ moreover have "inj_on (h \<circ> f) S"
+ apply (clarsimp simp: inj_on_def)
+ by (metis fim homeomorphism_apply1 homhk image_subset_iff inj_onD injf)
+ ultimately have ope_hf: "openin (subtopology euclidean U) ((h \<circ> f) ` S)"
+ using invariance_of_domain_subspaces [OF ope \<open>subspace U\<close> \<open>subspace U\<close>] by auto
+ have "(h \<circ> f) ` S \<subseteq> T"
+ using fim homeomorphism_image1 homhk by fastforce
+ then show ?thesis
+ by (metis dim_openin \<open>dim T = dim V\<close> ope_hf \<open>subspace U\<close> \<open>S \<noteq> {}\<close> dim_subset image_is_empty not_le that)
+ qed
+ then show ?thesis
+ using not_less by blast
+qed
+
+corollary invariance_of_domain_affine_sets:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes ope: "openin (subtopology euclidean U) S"
+ and aff: "affine U" "affine V" "aff_dim V \<le> aff_dim U"
+ and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
+ and injf: "inj_on f S"
+ shows "openin (subtopology euclidean V) (f ` S)"
+proof (cases "S = {}")
+ case True
+ then show ?thesis by auto
+next
+ case False
+ obtain a b where "a \<in> S" "a \<in> U" "b \<in> V"
+ using False fim ope openin_contains_cball by fastforce
+ have "openin (subtopology euclidean (op + (- b) ` V)) ((op + (- b) \<circ> f \<circ> op + a) ` op + (- a) ` S)"
+ proof (rule invariance_of_domain_subspaces)
+ show "openin (subtopology euclidean (op + (- a) ` U)) (op + (- a) ` S)"
+ by (metis ope homeomorphism_imp_open_map homeomorphism_translation translation_galois)
+ show "subspace (op + (- a) ` U)"
+ by (simp add: \<open>a \<in> U\<close> affine_diffs_subspace \<open>affine U\<close>)
+ show "subspace (op + (- b) ` V)"
+ by (simp add: \<open>b \<in> V\<close> affine_diffs_subspace \<open>affine V\<close>)
+ show "dim (op + (- b) ` V) \<le> dim (op + (- a) ` U)"
+ by (metis \<open>a \<in> U\<close> \<open>b \<in> V\<close> aff_dim_eq_dim affine_hull_eq aff of_nat_le_iff)
+ show "continuous_on (op + (- a) ` S) (op + (- b) \<circ> f \<circ> op + a)"
+ by (metis contf continuous_on_compose homeomorphism_cont2 homeomorphism_translation translation_galois)
+ show "(op + (- b) \<circ> f \<circ> op + a) ` op + (- a) ` S \<subseteq> op + (- b) ` V"
+ using fim by auto
+ show "inj_on (op + (- b) \<circ> f \<circ> op + a) (op + (- a) ` S)"
+ by (auto simp: inj_on_def) (meson inj_onD injf)
+ qed
+ then show ?thesis
+ by (metis (no_types, lifting) homeomorphism_imp_open_map homeomorphism_translation image_comp translation_galois)
+qed
+
+corollary invariance_of_dimension_affine_sets:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes ope: "openin (subtopology euclidean U) S"
+ and aff: "affine U" "affine V"
+ and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
+ and injf: "inj_on f S" and "S \<noteq> {}"
+ shows "aff_dim U \<le> aff_dim V"
+proof -
+ obtain a b where "a \<in> S" "a \<in> U" "b \<in> V"
+ using \<open>S \<noteq> {}\<close> fim ope openin_contains_cball by fastforce
+ have "dim (op + (- a) ` U) \<le> dim (op + (- b) ` V)"
+ proof (rule invariance_of_dimension_subspaces)
+ show "openin (subtopology euclidean (op + (- a) ` U)) (op + (- a) ` S)"
+ by (metis ope homeomorphism_imp_open_map homeomorphism_translation translation_galois)
+ show "subspace (op + (- a) ` U)"
+ by (simp add: \<open>a \<in> U\<close> affine_diffs_subspace \<open>affine U\<close>)
+ show "subspace (op + (- b) ` V)"
+ by (simp add: \<open>b \<in> V\<close> affine_diffs_subspace \<open>affine V\<close>)
+ show "continuous_on (op + (- a) ` S) (op + (- b) \<circ> f \<circ> op + a)"
+ by (metis contf continuous_on_compose homeomorphism_cont2 homeomorphism_translation translation_galois)
+ show "(op + (- b) \<circ> f \<circ> op + a) ` op + (- a) ` S \<subseteq> op + (- b) ` V"
+ using fim by auto
+ show "inj_on (op + (- b) \<circ> f \<circ> op + a) (op + (- a) ` S)"
+ by (auto simp: inj_on_def) (meson inj_onD injf)
+ qed (use \<open>S \<noteq> {}\<close> in auto)
+ then show ?thesis
+ by (metis \<open>a \<in> U\<close> \<open>b \<in> V\<close> aff_dim_eq_dim affine_hull_eq aff of_nat_le_iff)
+qed
+
+corollary invariance_of_dimension:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes contf: "continuous_on S f" and "open S"
+ and injf: "inj_on f S" and "S \<noteq> {}"
+ shows "DIM('a) \<le> DIM('b)"
+ using invariance_of_dimension_subspaces [of UNIV S UNIV f] assms
+ by auto
+
+
+corollary continuous_injective_image_subspace_dim_le:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes "subspace S" "subspace T"
+ and contf: "continuous_on S f" and fim: "f ` S \<subseteq> T"
+ and injf: "inj_on f S"
+ shows "dim S \<le> dim T"
+ apply (rule invariance_of_dimension_subspaces [of S S _ f])
+ using assms by (auto simp: subspace_affine)
+
+lemma invariance_of_dimension_convex_domain:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes "convex S"
+ and contf: "continuous_on S f" and fim: "f ` S \<subseteq> affine hull T"
+ and injf: "inj_on f S"
+ shows "aff_dim S \<le> aff_dim T"
+proof (cases "S = {}")
+ case True
+ then show ?thesis by (simp add: aff_dim_geq)
+next
+ case False
+ have "aff_dim (affine hull S) \<le> aff_dim (affine hull T)"
+ proof (rule invariance_of_dimension_affine_sets)
+ show "openin (subtopology euclidean (affine hull S)) (rel_interior S)"
+ by (simp add: openin_rel_interior)
+ show "continuous_on (rel_interior S) f"
+ using contf continuous_on_subset rel_interior_subset by blast
+ show "f ` rel_interior S \<subseteq> affine hull T"
+ using fim rel_interior_subset by blast
+ show "inj_on f (rel_interior S)"
+ using inj_on_subset injf rel_interior_subset by blast
+ show "rel_interior S \<noteq> {}"
+ by (simp add: False \<open>convex S\<close> rel_interior_eq_empty)
+ qed auto
+ then show ?thesis
+ by simp
+qed
+
+
+lemma homeomorphic_convex_sets_le:
+ assumes "convex S" "S homeomorphic T"
+ shows "aff_dim S \<le> aff_dim T"
+proof -
+ obtain h k where homhk: "homeomorphism S T h k"
+ using homeomorphic_def assms by blast
+ show ?thesis
+ proof (rule invariance_of_dimension_convex_domain [OF \<open>convex S\<close>])
+ show "continuous_on S h"
+ using homeomorphism_def homhk by blast
+ show "h ` S \<subseteq> affine hull T"
+ by (metis homeomorphism_def homhk hull_subset)
+ show "inj_on h S"
+ by (meson homeomorphism_apply1 homhk inj_on_inverseI)
+ qed
+qed
+
+lemma homeomorphic_convex_sets:
+ assumes "convex S" "convex T" "S homeomorphic T"
+ shows "aff_dim S = aff_dim T"
+ by (meson assms dual_order.antisym homeomorphic_convex_sets_le homeomorphic_sym)
+
+lemma homeomorphic_convex_compact_sets_eq:
+ assumes "convex S" "compact S" "convex T" "compact T"
+ shows "S homeomorphic T \<longleftrightarrow> aff_dim S = aff_dim T"
+ by (meson assms homeomorphic_convex_compact_sets homeomorphic_convex_sets)
+
+lemma invariance_of_domain_gen:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes "open S" "continuous_on S f" "inj_on f S" "DIM('b) \<le> DIM('a)"
+ shows "open(f ` S)"
+ using invariance_of_domain_subspaces [of UNIV S UNIV f] assms by auto
+
+lemma injective_into_1d_imp_open_map_UNIV:
+ fixes f :: "'a::euclidean_space \<Rightarrow> real"
+ assumes "open T" "continuous_on S f" "inj_on f S" "T \<subseteq> S"
+ shows "open (f ` T)"
+ apply (rule invariance_of_domain_gen [OF \<open>open T\<close>])
+ using assms apply (auto simp: elim: continuous_on_subset subset_inj_on)
+ done
+
+lemma continuous_on_inverse_open:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes "open S" "continuous_on S f" "DIM('b) \<le> DIM('a)" and gf: "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"
+ shows "continuous_on (f ` S) g"
+proof (clarsimp simp add: continuous_openin_preimage_eq)
+ fix T :: "'a set"
+ assume "open T"
+ have eq: "{x. x \<in> f ` S \<and> g x \<in> T} = f ` (S \<inter> T)"
+ by (auto simp: gf)
+ show "openin (subtopology euclidean (f ` S)) {x \<in> f ` S. g x \<in> T}"
+ apply (subst eq)
+ apply (rule open_openin_trans)
+ apply (rule invariance_of_domain_gen)
+ using assms
+ apply auto
+ using inj_on_inverseI apply auto[1]
+ by (metis \<open>open T\<close> continuous_on_subset inj_onI inj_on_subset invariance_of_domain_gen openin_open openin_open_eq)
+qed
+
+lemma invariance_of_domain_homeomorphism:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes "open S" "continuous_on S f" "DIM('b) \<le> DIM('a)" "inj_on f S"
+ obtains g where "homeomorphism S (f ` S) f g"
+proof
+ show "homeomorphism S (f ` S) f (inv_into S f)"
+ by (simp add: assms continuous_on_inverse_open homeomorphism_def)
+qed
+
+corollary invariance_of_domain_homeomorphic:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes "open S" "continuous_on S f" "DIM('b) \<le> DIM('a)" "inj_on f S"
+ shows "S homeomorphic (f ` S)"
+ using invariance_of_domain_homeomorphism [OF assms]
+ by (meson homeomorphic_def)
+
+lemma continuous_image_subset_interior:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes "continuous_on S f" "inj_on f S" "DIM('b) \<le> DIM('a)"
+ shows "f ` (interior S) \<subseteq> interior(f ` S)"
+ apply (rule interior_maximal)
+ apply (simp add: image_mono interior_subset)
+ apply (rule invariance_of_domain_gen)
+ using assms
+ apply (auto simp: subset_inj_on interior_subset continuous_on_subset)
+ done
+
+lemma homeomorphic_interiors_same_dimension:
+ fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
+ assumes "S homeomorphic T" and dimeq: "DIM('a) = DIM('b)"
+ shows "(interior S) homeomorphic (interior T)"
+ using assms [unfolded homeomorphic_minimal]
+ unfolding homeomorphic_def
+proof (clarify elim!: ex_forward)
+ fix f g
+ assume S: "\<forall>x\<in>S. f x \<in> T \<and> g (f x) = x" and T: "\<forall>y\<in>T. g y \<in> S \<and> f (g y) = y"
+ and contf: "continuous_on S f" and contg: "continuous_on T g"
+ then have fST: "f ` S = T" and gTS: "g ` T = S" and "inj_on f S" "inj_on g T"
+ by (auto simp: inj_on_def intro: rev_image_eqI) metis+
+ have fim: "f ` interior S \<subseteq> interior T"
+ using continuous_image_subset_interior [OF contf \<open>inj_on f S\<close>] dimeq fST by simp
+ have gim: "g ` interior T \<subseteq> interior S"
+ using continuous_image_subset_interior [OF contg \<open>inj_on g T\<close>] dimeq gTS by simp
+ show "homeomorphism (interior S) (interior T) f g"
+ unfolding homeomorphism_def
+ proof (intro conjI ballI)
+ show "\<And>x. x \<in> interior S \<Longrightarrow> g (f x) = x"
+ by (meson \<open>\<forall>x\<in>S. f x \<in> T \<and> g (f x) = x\<close> subsetD interior_subset)
+ have "interior T \<subseteq> f ` interior S"
+ proof
+ fix x assume "x \<in> interior T"
+ then have "g x \<in> interior S"
+ using gim by blast
+ then show "x \<in> f ` interior S"
+ by (metis T \<open>x \<in> interior T\<close> image_iff interior_subset subsetCE)
+ qed
+ then show "f ` interior S = interior T"
+ using fim by blast
+ show "continuous_on (interior S) f"
+ by (metis interior_subset continuous_on_subset contf)
+ show "\<And>y. y \<in> interior T \<Longrightarrow> f (g y) = y"
+ by (meson T subsetD interior_subset)
+ have "interior S \<subseteq> g ` interior T"
+ proof
+ fix x assume "x \<in> interior S"
+ then have "f x \<in> interior T"
+ using fim by blast
+ then show "x \<in> g ` interior T"
+ by (metis S \<open>x \<in> interior S\<close> image_iff interior_subset subsetCE)
+ qed
+ then show "g ` interior T = interior S"
+ using gim by blast
+ show "continuous_on (interior T) g"
+ by (metis interior_subset continuous_on_subset contg)
+ qed
+qed
+
+lemma homeomorphic_open_imp_same_dimension:
+ fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
+ assumes "S homeomorphic T" "open S" "S \<noteq> {}" "open T" "T \<noteq> {}"
+ shows "DIM('a) = DIM('b)"
+ using assms
+ apply (simp add: homeomorphic_minimal)
+ apply (rule order_antisym; metis inj_onI invariance_of_dimension)
+ done
+
+lemma homeomorphic_interiors:
+ fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
+ assumes "S homeomorphic T" "interior S = {} \<longleftrightarrow> interior T = {}"
+ shows "(interior S) homeomorphic (interior T)"
+proof (cases "interior T = {}")
+ case True
+ with assms show ?thesis by auto
+next
+ case False
+ then have "DIM('a) = DIM('b)"
+ using assms
+ apply (simp add: homeomorphic_minimal)
+ apply (rule order_antisym; metis continuous_on_subset inj_onI inj_on_subset interior_subset invariance_of_dimension open_interior)
+ done
+ then show ?thesis
+ by (rule homeomorphic_interiors_same_dimension [OF \<open>S homeomorphic T\<close>])
+qed
+
+lemma homeomorphic_frontiers_same_dimension:
+ fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
+ assumes "S homeomorphic T" "closed S" "closed T" and dimeq: "DIM('a) = DIM('b)"
+ shows "(frontier S) homeomorphic (frontier T)"
+ using assms [unfolded homeomorphic_minimal]
+ unfolding homeomorphic_def
+proof (clarify elim!: ex_forward)
+ fix f g
+ assume S: "\<forall>x\<in>S. f x \<in> T \<and> g (f x) = x" and T: "\<forall>y\<in>T. g y \<in> S \<and> f (g y) = y"
+ and contf: "continuous_on S f" and contg: "continuous_on T g"
+ then have fST: "f ` S = T" and gTS: "g ` T = S" and "inj_on f S" "inj_on g T"
+ by (auto simp: inj_on_def intro: rev_image_eqI) metis+
+ have "g ` interior T \<subseteq> interior S"
+ using continuous_image_subset_interior [OF contg \<open>inj_on g T\<close>] dimeq gTS by simp
+ then have fim: "f ` frontier S \<subseteq> frontier T"
+ apply (simp add: frontier_def)
+ using continuous_image_subset_interior assms(2) assms(3) S by auto
+ have "f ` interior S \<subseteq> interior T"
+ using continuous_image_subset_interior [OF contf \<open>inj_on f S\<close>] dimeq fST by simp
+ then have gim: "g ` frontier T \<subseteq> frontier S"
+ apply (simp add: frontier_def)
+ using continuous_image_subset_interior T assms(2) assms(3) by auto
+ show "homeomorphism (frontier S) (frontier T) f g"
+ unfolding homeomorphism_def
+ proof (intro conjI ballI)
+ show gf: "\<And>x. x \<in> frontier S \<Longrightarrow> g (f x) = x"
+ by (simp add: S assms(2) frontier_def)
+ show fg: "\<And>y. y \<in> frontier T \<Longrightarrow> f (g y) = y"
+ by (simp add: T assms(3) frontier_def)
+ have "frontier T \<subseteq> f ` frontier S"
+ proof
+ fix x assume "x \<in> frontier T"
+ then have "g x \<in> frontier S"
+ using gim by blast
+ then show "x \<in> f ` frontier S"
+ by (metis fg \<open>x \<in> frontier T\<close> imageI)
+ qed
+ then show "f ` frontier S = frontier T"
+ using fim by blast
+ show "continuous_on (frontier S) f"
+ by (metis Diff_subset assms(2) closure_eq contf continuous_on_subset frontier_def)
+ have "frontier S \<subseteq> g ` frontier T"
+ proof
+ fix x assume "x \<in> frontier S"
+ then have "f x \<in> frontier T"
+ using fim by blast
+ then show "x \<in> g ` frontier T"
+ by (metis gf \<open>x \<in> frontier S\<close> imageI)
+ qed
+ then show "g ` frontier T = frontier S"
+ using gim by blast
+ show "continuous_on (frontier T) g"
+ by (metis Diff_subset assms(3) closure_closed contg continuous_on_subset frontier_def)
+ qed
+qed
+
+lemma homeomorphic_frontiers:
+ fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
+ assumes "S homeomorphic T" "closed S" "closed T"
+ "interior S = {} \<longleftrightarrow> interior T = {}"
+ shows "(frontier S) homeomorphic (frontier T)"
+proof (cases "interior T = {}")
+ case True
+ then show ?thesis
+ by (metis Diff_empty assms closure_eq frontier_def)
+next
+ case False
+ show ?thesis
+ apply (rule homeomorphic_frontiers_same_dimension)
+ apply (simp_all add: assms)
+ using False assms homeomorphic_interiors homeomorphic_open_imp_same_dimension by blast
+qed
+
+lemma continuous_image_subset_rel_interior:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes contf: "continuous_on S f" and injf: "inj_on f S" and fim: "f ` S \<subseteq> T"
+ and TS: "aff_dim T \<le> aff_dim S"
+ shows "f ` (rel_interior S) \<subseteq> rel_interior(f ` S)"
+proof (rule rel_interior_maximal)
+ show "f ` rel_interior S \<subseteq> f ` S"
+ by(simp add: image_mono rel_interior_subset)
+ show "openin (subtopology euclidean (affine hull f ` S)) (f ` rel_interior S)"
+ proof (rule invariance_of_domain_affine_sets)
+ show "openin (subtopology euclidean (affine hull S)) (rel_interior S)"
+ by (simp add: openin_rel_interior)
+ show "aff_dim (affine hull f ` S) \<le> aff_dim (affine hull S)"
+ by (metis aff_dim_affine_hull aff_dim_subset fim TS order_trans)
+ show "f ` rel_interior S \<subseteq> affine hull f ` S"
+ by (meson \<open>f ` rel_interior S \<subseteq> f ` S\<close> hull_subset order_trans)
+ show "continuous_on (rel_interior S) f"
+ using contf continuous_on_subset rel_interior_subset by blast
+ show "inj_on f (rel_interior S)"
+ using inj_on_subset injf rel_interior_subset by blast
+ qed auto
+qed
+
+lemma homeomorphic_rel_interiors_same_dimension:
+ fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
+ assumes "S homeomorphic T" and aff: "aff_dim S = aff_dim T"
+ shows "(rel_interior S) homeomorphic (rel_interior T)"
+ using assms [unfolded homeomorphic_minimal]
+ unfolding homeomorphic_def
+proof (clarify elim!: ex_forward)
+ fix f g
+ assume S: "\<forall>x\<in>S. f x \<in> T \<and> g (f x) = x" and T: "\<forall>y\<in>T. g y \<in> S \<and> f (g y) = y"
+ and contf: "continuous_on S f" and contg: "continuous_on T g"
+ then have fST: "f ` S = T" and gTS: "g ` T = S" and "inj_on f S" "inj_on g T"
+ by (auto simp: inj_on_def intro: rev_image_eqI) metis+
+ have fim: "f ` rel_interior S \<subseteq> rel_interior T"
+ by (metis \<open>inj_on f S\<close> aff contf continuous_image_subset_rel_interior fST order_refl)
+ have gim: "g ` rel_interior T \<subseteq> rel_interior S"
+ by (metis \<open>inj_on g T\<close> aff contg continuous_image_subset_rel_interior gTS order_refl)
+ show "homeomorphism (rel_interior S) (rel_interior T) f g"
+ unfolding homeomorphism_def
+ proof (intro conjI ballI)
+ show gf: "\<And>x. x \<in> rel_interior S \<Longrightarrow> g (f x) = x"
+ using S rel_interior_subset by blast
+ show fg: "\<And>y. y \<in> rel_interior T \<Longrightarrow> f (g y) = y"
+ using T mem_rel_interior_ball by blast
+ have "rel_interior T \<subseteq> f ` rel_interior S"
+ proof
+ fix x assume "x \<in> rel_interior T"
+ then have "g x \<in> rel_interior S"
+ using gim by blast
+ then show "x \<in> f ` rel_interior S"
+ by (metis fg \<open>x \<in> rel_interior T\<close> imageI)
+ qed
+ moreover have "f ` rel_interior S \<subseteq> rel_interior T"
+ by (metis \<open>inj_on f S\<close> aff contf continuous_image_subset_rel_interior fST order_refl)
+ ultimately show "f ` rel_interior S = rel_interior T"
+ by blast
+ show "continuous_on (rel_interior S) f"
+ using contf continuous_on_subset rel_interior_subset by blast
+ have "rel_interior S \<subseteq> g ` rel_interior T"
+ proof
+ fix x assume "x \<in> rel_interior S"
+ then have "f x \<in> rel_interior T"
+ using fim by blast
+ then show "x \<in> g ` rel_interior T"
+ by (metis gf \<open>x \<in> rel_interior S\<close> imageI)
+ qed
+ then show "g ` rel_interior T = rel_interior S"
+ using gim by blast
+ show "continuous_on (rel_interior T) g"
+ using contg continuous_on_subset rel_interior_subset by blast
+ qed
+qed
+
+lemma homeomorphic_rel_interiors:
+ fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
+ assumes "S homeomorphic T" "rel_interior S = {} \<longleftrightarrow> rel_interior T = {}"
+ shows "(rel_interior S) homeomorphic (rel_interior T)"
+proof (cases "rel_interior T = {}")
+ case True
+ with assms show ?thesis by auto
+next
+ case False
+ obtain f g
+ where S: "\<forall>x\<in>S. f x \<in> T \<and> g (f x) = x" and T: "\<forall>y\<in>T. g y \<in> S \<and> f (g y) = y"
+ and contf: "continuous_on S f" and contg: "continuous_on T g"
+ using assms [unfolded homeomorphic_minimal] by auto
+ have "aff_dim (affine hull S) \<le> aff_dim (affine hull T)"
+ apply (rule invariance_of_dimension_affine_sets [of _ "rel_interior S" _ f])
+ apply (simp_all add: openin_rel_interior False assms)
+ using contf continuous_on_subset rel_interior_subset apply blast
+ apply (meson S hull_subset image_subsetI rel_interior_subset rev_subsetD)
+ apply (metis S inj_on_inverseI inj_on_subset rel_interior_subset)
+ done
+ moreover have "aff_dim (affine hull T) \<le> aff_dim (affine hull S)"
+ apply (rule invariance_of_dimension_affine_sets [of _ "rel_interior T" _ g])
+ apply (simp_all add: openin_rel_interior False assms)
+ using contg continuous_on_subset rel_interior_subset apply blast
+ apply (meson T hull_subset image_subsetI rel_interior_subset rev_subsetD)
+ apply (metis T inj_on_inverseI inj_on_subset rel_interior_subset)
+ done
+ ultimately have "aff_dim S = aff_dim T" by force
+ then show ?thesis
+ by (rule homeomorphic_rel_interiors_same_dimension [OF \<open>S homeomorphic T\<close>])
+qed
+
+
+lemma homeomorphic_rel_boundaries_same_dimension:
+ fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
+ assumes "S homeomorphic T" and aff: "aff_dim S = aff_dim T"
+ shows "(S - rel_interior S) homeomorphic (T - rel_interior T)"
+ using assms [unfolded homeomorphic_minimal]
+ unfolding homeomorphic_def
+proof (clarify elim!: ex_forward)
+ fix f g
+ assume S: "\<forall>x\<in>S. f x \<in> T \<and> g (f x) = x" and T: "\<forall>y\<in>T. g y \<in> S \<and> f (g y) = y"
+ and contf: "continuous_on S f" and contg: "continuous_on T g"
+ then have fST: "f ` S = T" and gTS: "g ` T = S" and "inj_on f S" "inj_on g T"
+ by (auto simp: inj_on_def intro: rev_image_eqI) metis+
+ have fim: "f ` rel_interior S \<subseteq> rel_interior T"
+ by (metis \<open>inj_on f S\<close> aff contf continuous_image_subset_rel_interior fST order_refl)
+ have gim: "g ` rel_interior T \<subseteq> rel_interior S"
+ by (metis \<open>inj_on g T\<close> aff contg continuous_image_subset_rel_interior gTS order_refl)
+ show "homeomorphism (S - rel_interior S) (T - rel_interior T) f g"
+ unfolding homeomorphism_def
+ proof (intro conjI ballI)
+ show gf: "\<And>x. x \<in> S - rel_interior S \<Longrightarrow> g (f x) = x"
+ using S rel_interior_subset by blast
+ show fg: "\<And>y. y \<in> T - rel_interior T \<Longrightarrow> f (g y) = y"
+ using T mem_rel_interior_ball by blast
+ show "f ` (S - rel_interior S) = T - rel_interior T"
+ using S fST fim gim by auto
+ show "continuous_on (S - rel_interior S) f"
+ using contf continuous_on_subset rel_interior_subset by blast
+ show "g ` (T - rel_interior T) = S - rel_interior S"
+ using T gTS gim fim by auto
+ show "continuous_on (T - rel_interior T) g"
+ using contg continuous_on_subset rel_interior_subset by blast
+ qed
+qed
+
+lemma homeomorphic_rel_boundaries:
+ fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
+ assumes "S homeomorphic T" "rel_interior S = {} \<longleftrightarrow> rel_interior T = {}"
+ shows "(S - rel_interior S) homeomorphic (T - rel_interior T)"
+proof (cases "rel_interior T = {}")
+ case True
+ with assms show ?thesis by auto
+next
+ case False
+ obtain f g
+ where S: "\<forall>x\<in>S. f x \<in> T \<and> g (f x) = x" and T: "\<forall>y\<in>T. g y \<in> S \<and> f (g y) = y"
+ and contf: "continuous_on S f" and contg: "continuous_on T g"
+ using assms [unfolded homeomorphic_minimal] by auto
+ have "aff_dim (affine hull S) \<le> aff_dim (affine hull T)"
+ apply (rule invariance_of_dimension_affine_sets [of _ "rel_interior S" _ f])
+ apply (simp_all add: openin_rel_interior False assms)
+ using contf continuous_on_subset rel_interior_subset apply blast
+ apply (meson S hull_subset image_subsetI rel_interior_subset rev_subsetD)
+ apply (metis S inj_on_inverseI inj_on_subset rel_interior_subset)
+ done
+ moreover have "aff_dim (affine hull T) \<le> aff_dim (affine hull S)"
+ apply (rule invariance_of_dimension_affine_sets [of _ "rel_interior T" _ g])
+ apply (simp_all add: openin_rel_interior False assms)
+ using contg continuous_on_subset rel_interior_subset apply blast
+ apply (meson T hull_subset image_subsetI rel_interior_subset rev_subsetD)
+ apply (metis T inj_on_inverseI inj_on_subset rel_interior_subset)
+ done
+ ultimately have "aff_dim S = aff_dim T" by force
+ then show ?thesis
+ by (rule homeomorphic_rel_boundaries_same_dimension [OF \<open>S homeomorphic T\<close>])
+qed
+
+proposition uniformly_continuous_homeomorphism_UNIV_trivial:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
+ assumes contf: "uniformly_continuous_on S f" and hom: "homeomorphism S UNIV f g"
+ shows "S = UNIV"
+proof (cases "S = {}")
+ case True
+ then show ?thesis
+ by (metis UNIV_I hom empty_iff homeomorphism_def image_eqI)
+next
+ case False
+ have "inj g"
+ by (metis UNIV_I hom homeomorphism_apply2 injI)
+ then have "open (g ` UNIV)"
+ by (blast intro: invariance_of_domain hom homeomorphism_cont2)
+ then have "open S"
+ using hom homeomorphism_image2 by blast
+ moreover have "complete S"
+ unfolding complete_def
+ proof clarify
+ fix \<sigma>
+ assume \<sigma>: "\<forall>n. \<sigma> n \<in> S" and "Cauchy \<sigma>"
+ have "Cauchy (f o \<sigma>)"
+ using uniformly_continuous_imp_Cauchy_continuous \<open>Cauchy \<sigma>\<close> \<sigma> contf by blast
+ then obtain l where "(f \<circ> \<sigma>) \<longlonglongrightarrow> l"
+ by (auto simp: convergent_eq_Cauchy [symmetric])
+ show "\<exists>l\<in>S. \<sigma> \<longlonglongrightarrow> l"
+ proof
+ show "g l \<in> S"
+ using hom homeomorphism_image2 by blast
+ have "(g \<circ> (f \<circ> \<sigma>)) \<longlonglongrightarrow> g l"
+ by (meson UNIV_I \<open>(f \<circ> \<sigma>) \<longlonglongrightarrow> l\<close> continuous_on_sequentially hom homeomorphism_cont2)
+ then show "\<sigma> \<longlonglongrightarrow> g l"
+ proof -
+ have "\<forall>n. \<sigma> n = (g \<circ> (f \<circ> \<sigma>)) n"
+ by (metis (no_types) \<sigma> comp_eq_dest_lhs hom homeomorphism_apply1)
+ then show ?thesis
+ by (metis (no_types) LIMSEQ_iff \<open>(g \<circ> (f \<circ> \<sigma>)) \<longlonglongrightarrow> g l\<close>)
+ qed
+ qed
+ qed
+ then have "closed S"
+ by (simp add: complete_eq_closed)
+ ultimately show ?thesis
+ using clopen [of S] False by simp
+qed
+
+subsection\<open>The power, squaring and exponential functions as covering maps\<close>
+
+proposition covering_space_power_punctured_plane:
+ assumes "0 < n"
+ shows "covering_space (- {0}) (\<lambda>z::complex. z^n) (- {0})"
+proof -
+ consider "n = 1" | "2 \<le> n" using assms by linarith
+ then obtain e where "0 < e"
+ and e: "\<And>w z. cmod(w - z) < e * cmod z \<Longrightarrow> (w^n = z^n \<longleftrightarrow> w = z)"
+ proof cases
+ assume "n = 1" then show ?thesis
+ by (rule_tac e=1 in that) auto
+ next
+ assume "2 \<le> n"
+ have eq_if_pow_eq:
+ "w = z" if lt: "cmod (w - z) < 2 * sin (pi / real n) * cmod z"
+ and eq: "w^n = z^n" for w z
+ proof (cases "z = 0")
+ case True with eq assms show ?thesis by (auto simp: power_0_left)
+ next
+ case False
+ then have "z \<noteq> 0" by auto
+ have "(w/z)^n = 1"
+ by (metis False divide_self_if eq power_divide power_one)
+ then obtain j where j: "w / z = exp (2 * of_real pi * \<i> * j / n)" and "j < n"
+ using Suc_leI assms \<open>2 \<le> n\<close> complex_roots_unity [THEN eqset_imp_iff, of n "w/z"]
+ by force
+ have "cmod (w/z - 1) < 2 * sin (pi / real n)"
+ using lt assms \<open>z \<noteq> 0\<close> by (simp add: divide_simps norm_divide)
+ then have "cmod (exp (\<i> * of_real (2 * pi * j / n)) - 1) < 2 * sin (pi / real n)"
+ by (simp add: j field_simps)
+ then have "2 * \<bar>sin((2 * pi * j / n) / 2)\<bar> < 2 * sin (pi / real n)"
+ by (simp only: dist_exp_ii_1)
+ then have sin_less: "sin((pi * j / n)) < sin (pi / real n)"
+ by (simp add: field_simps)
+ then have "w / z = 1"
+ proof (cases "j = 0")
+ case True then show ?thesis by (auto simp: j)
+ next
+ case False
+ then have "sin (pi / real n) \<le> sin((pi * j / n))"
+ proof (cases "j / n \<le> 1/2")
+ case True
+ show ?thesis
+ apply (rule sin_monotone_2pi_le)
+ using \<open>j \<noteq> 0 \<close> \<open>j < n\<close> True
+ apply (auto simp: field_simps intro: order_trans [of _ 0])
+ done
+ next
+ case False
+ then have seq: "sin(pi * j / n) = sin(pi * (n - j) / n)"
+ using \<open>j < n\<close> by (simp add: algebra_simps diff_divide_distrib of_nat_diff)
+ show ?thesis
+ apply (simp only: seq)
+ apply (rule sin_monotone_2pi_le)
+ using \<open>j < n\<close> False
+ apply (auto simp: field_simps intro: order_trans [of _ 0])
+ done
+ qed
+ with sin_less show ?thesis by force
+ qed
+ then show ?thesis by simp
+ qed
+ show ?thesis
+ apply (rule_tac e = "2 * sin(pi / n)" in that)
+ apply (force simp: \<open>2 \<le> n\<close> sin_pi_divide_n_gt_0)
+ apply (meson eq_if_pow_eq)
+ done
+ qed
+ have zn1: "continuous_on (- {0}) (\<lambda>z::complex. z^n)"
+ by (rule continuous_intros)+
+ have zn2: "(\<lambda>z::complex. z^n) ` (- {0}) = - {0}"
+ using assms by (auto simp: image_def elim: exists_complex_root_nonzero [where n = n])
+ have zn3: "\<exists>T. z^n \<in> T \<and> open T \<and> 0 \<notin> T \<and>
+ (\<exists>v. \<Union>v = {x. x \<noteq> 0 \<and> x^n \<in> T} \<and>
+ (\<forall>u\<in>v. open u \<and> 0 \<notin> u) \<and>
+ pairwise disjnt v \<and>
+ (\<forall>u\<in>v. Ex (homeomorphism u T (\<lambda>z. z^n))))"
+ if "z \<noteq> 0" for z::complex
+ proof -
+ def d \<equiv> "min (1/2) (e/4) * norm z"
+ have "0 < d"
+ by (simp add: d_def \<open>0 < e\<close> \<open>z \<noteq> 0\<close>)
+ have iff_x_eq_y: "x^n = y^n \<longleftrightarrow> x = y"
+ if eq: "w^n = z^n" and x: "x \<in> ball w d" and y: "y \<in> ball w d" for w x y
+ proof -
+ have [simp]: "norm z = norm w" using that
+ by (simp add: assms power_eq_imp_eq_norm)
+ show ?thesis
+ proof (cases "w = 0")
+ case True with \<open>z \<noteq> 0\<close> assms eq
+ show ?thesis by (auto simp: power_0_left)
+ next
+ case False
+ have "cmod (x - y) < 2*d"
+ using x y
+ by (simp add: dist_norm [symmetric]) (metis dist_commute mult_2 dist_triangle_less_add)
+ also have "... \<le> 2 * e / 4 * norm w"
+ using \<open>e > 0\<close> by (simp add: d_def min_mult_distrib_right)
+ also have "... = e * (cmod w / 2)"
+ by simp
+ also have "... \<le> e * cmod y"
+ apply (rule mult_left_mono)
+ using \<open>e > 0\<close> y
+ apply (simp_all add: dist_norm d_def min_mult_distrib_right del: divide_const_simps)
+ apply (metis dist_0_norm dist_complex_def dist_triangle_half_l linorder_not_less order_less_irrefl)
+ done
+ finally have "cmod (x - y) < e * cmod y" .
+ then show ?thesis by (rule e)
+ qed
+ qed
+ then have inj: "inj_on (\<lambda>w. w^n) (ball z d)"
+ by (simp add: inj_on_def)
+ have cont: "continuous_on (ball z d) (\<lambda>w. w ^ n)"
+ by (intro continuous_intros)
+ have noncon: "\<not> (\<lambda>w::complex. w^n) constant_on UNIV"
+ by (metis UNIV_I assms constant_on_def power_one zero_neq_one zero_power)
+ have open_imball: "open ((\<lambda>w. w^n) ` ball z d)"
+ by (rule invariance_of_domain [OF cont open_ball inj])
+ have im_eq: "(\<lambda>w. w^n) ` ball z' d = (\<lambda>w. w^n) ` ball z d"
+ if z': "z'^n = z^n" for z'
+ proof -
+ have nz': "norm z' = norm z" using that assms power_eq_imp_eq_norm by blast
+ have "(w \<in> (\<lambda>w. w^n) ` ball z' d) = (w \<in> (\<lambda>w. w^n) ` ball z d)" for w
+ proof (cases "w=0")
+ case True with assms show ?thesis
+ by (simp add: image_def ball_def nz')
+ next
+ case False
+ have "z' \<noteq> 0" using \<open>z \<noteq> 0\<close> nz' by force
+ have [simp]: "(z*x / z')^n = x^n" if "x \<noteq> 0" for x
+ using z' that by (simp add: field_simps \<open>z \<noteq> 0\<close>)
+ have [simp]: "cmod (z - z * x / z') = cmod (z' - x)" if "x \<noteq> 0" for x
+ proof -
+ have "cmod (z - z * x / z') = cmod z * cmod (1 - x / z')"
+ by (metis (no_types) ab_semigroup_mult_class.mult_ac(1) complex_divide_def mult.right_neutral norm_mult right_diff_distrib')
+ also have "... = cmod z' * cmod (1 - x / z')"
+ by (simp add: nz')
+ also have "... = cmod (z' - x)"
+ by (simp add: \<open>z' \<noteq> 0\<close> diff_divide_eq_iff norm_divide)
+ finally show ?thesis .
+ qed
+ have [simp]: "(z'*x / z)^n = x^n" if "x \<noteq> 0" for x
+ using z' that by (simp add: field_simps \<open>z \<noteq> 0\<close>)
+ have [simp]: "cmod (z' - z' * x / z) = cmod (z - x)" if "x \<noteq> 0" for x
+ proof -
+ have "cmod (z * (1 - x * inverse z)) = cmod (z - x)"
+ by (metis \<open>z \<noteq> 0\<close> diff_divide_distrib divide_complex_def divide_self_if nonzero_eq_divide_eq semiring_normalization_rules(7))
+ then show ?thesis
+ by (metis (no_types) mult.assoc complex_divide_def mult.right_neutral norm_mult nz' right_diff_distrib')
+ qed
+ show ?thesis
+ unfolding image_def ball_def
+ apply safe
+ apply simp_all
+ apply (rule_tac x="z/z' * x" in exI)
+ using assms False apply (simp add: dist_norm)
+ apply (rule_tac x="z'/z * x" in exI)
+ using assms False apply (simp add: dist_norm)
+ done
+ qed
+ then show ?thesis by blast
+ qed
+ have ex_ball: "\<exists>B. (\<exists>z'. B = ball z' d \<and> z'^n = z^n) \<and> x \<in> B"
+ if "x \<noteq> 0" and eq: "x^n = w^n" and dzw: "dist z w < d" for x w
+ proof -
+ have "w \<noteq> 0" by (metis assms power_eq_0_iff that(1) that(2))
+ have [simp]: "cmod x = cmod w"
+ using assms power_eq_imp_eq_norm eq by blast
+ have [simp]: "cmod (x * z / w - x) = cmod (z - w)"
+ proof -
+ have "cmod (x * z / w - x) = cmod x * cmod (z / w - 1)"
+ by (metis (no_types) mult.right_neutral norm_mult right_diff_distrib' times_divide_eq_right)
+ also have "... = cmod w * cmod (z / w - 1)"
+ by simp
+ also have "... = cmod (z - w)"
+ by (simp add: \<open>w \<noteq> 0\<close> divide_diff_eq_iff nonzero_norm_divide)
+ finally show ?thesis .
+ qed
+ show ?thesis
+ apply (rule_tac x="ball (z / w * x) d" in exI)
+ using \<open>d > 0\<close> that
+ apply (simp add: ball_eq_ball_iff)
+ apply (simp add: \<open>z \<noteq> 0\<close> \<open>w \<noteq> 0\<close> field_simps)
+ apply (simp add: dist_norm)
+ done
+ qed
+ have ball1: "\<Union>{ball z' d |z'. z'^n = z^n} = {x. x \<noteq> 0 \<and> x^n \<in> (\<lambda>w. w^n) ` ball z d}"
+ apply (rule equalityI)
+ prefer 2 apply (force simp: ex_ball, clarsimp)
+ apply (subst im_eq [symmetric], assumption)
+ using assms
+ apply (force simp: dist_norm d_def min_mult_distrib_right dest: power_eq_imp_eq_norm)
+ done
+ have ball2: "pairwise disjnt {ball z' d |z'. z'^n = z^n}"
+ proof (clarsimp simp add: pairwise_def disjnt_iff)
+ fix \<xi> \<zeta> x
+ assume "\<xi>^n = z^n" "\<zeta>^n = z^n" "ball \<xi> d \<noteq> ball \<zeta> d"
+ and "dist \<xi> x < d" "dist \<zeta> x < d"
+ then have "dist \<xi> \<zeta> < d+d"
+ using dist_triangle_less_add by blast
+ then have "cmod (\<xi> - \<zeta>) < 2*d"
+ by (simp add: dist_norm)
+ also have "... \<le> e * cmod z"
+ using mult_right_mono \<open>0 < e\<close> that by (auto simp: d_def)
+ finally have "cmod (\<xi> - \<zeta>) < e * cmod z" .
+ with e have "\<xi> = \<zeta>"
+ by (metis \<open>\<xi>^n = z^n\<close> \<open>\<zeta>^n = z^n\<close> assms power_eq_imp_eq_norm)
+ then show "False"
+ using \<open>ball \<xi> d \<noteq> ball \<zeta> d\<close> by blast
+ qed
+ have ball3: "Ex (homeomorphism (ball z' d) ((\<lambda>w. w^n) ` ball z d) (\<lambda>z. z^n))"
+ if zeq: "z'^n = z^n" for z'
+ proof -
+ have inj: "inj_on (\<lambda>z. z ^ n) (ball z' d)"
+ by (meson iff_x_eq_y inj_onI zeq)
+ show ?thesis
+ apply (rule invariance_of_domain_homeomorphism [of "ball z' d" "\<lambda>z. z^n"])
+ apply (rule open_ball continuous_intros order_refl inj)+
+ apply (force simp: im_eq [OF zeq])
+ done
+ qed
+ show ?thesis
+ apply (rule_tac x = "(\<lambda>w. w^n) ` (ball z d)" in exI)
+ apply (intro conjI open_imball)
+ using \<open>d > 0\<close> apply simp
+ using \<open>z \<noteq> 0\<close> assms apply (force simp: d_def)
+ apply (rule_tac x="{ ball z' d |z'. z'^n = z^n}" in exI)
+ apply (intro conjI ball1 ball2)
+ apply (force simp: assms d_def power_eq_imp_eq_norm that, clarify)
+ by (metis ball3)
+ qed
+ show ?thesis
+ using assms
+ apply (simp add: covering_space_def zn1 zn2)
+ apply (subst zn2 [symmetric])
+ apply (simp add: openin_open_eq open_Compl)
+ apply (blast intro: zn3)
+ done
+qed
+
+corollary covering_space_square_punctured_plane:
+ "covering_space (- {0}) (\<lambda>z::complex. z^2) (- {0})"
+ by (simp add: covering_space_power_punctured_plane)
+
+
+
+proposition covering_space_exp_punctured_plane:
+ "covering_space UNIV (\<lambda>z::complex. exp z) (- {0})"
+proof (simp add: covering_space_def, intro conjI ballI)
+ show "continuous_on UNIV (\<lambda>z::complex. exp z)"
+ by (rule continuous_on_exp [OF continuous_on_id])
+ show "range exp = - {0::complex}"
+ by auto (metis exp_Ln range_eqI)
+ show "\<exists>T. z \<in> T \<and> openin (subtopology euclidean (- {0})) T \<and>
+ (\<exists>v. \<Union>v = {z. exp z \<in> T} \<and> (\<forall>u\<in>v. open u) \<and> disjoint v \<and>
+ (\<forall>u\<in>v. \<exists>q. homeomorphism u T exp q))"
+ if "z \<in> - {0::complex}" for z
+ proof -
+ have "z \<noteq> 0"
+ using that by auto
+ have inj_exp: "inj_on exp (ball (Ln z) 1)"
+ apply (rule inj_on_subset [OF inj_on_exp_pi [of "Ln z"]])
+ using pi_ge_two by (simp add: ball_subset_ball_iff)
+ define \<V> where "\<V> \<equiv> range (\<lambda>n. (\<lambda>x. x + of_real (2 * of_int n * pi) * ii) ` (ball(Ln z) 1))"
+ show ?thesis
+ proof (intro exI conjI)
+ show "z \<in> exp ` (ball(Ln z) 1)"
+ by (metis \<open>z \<noteq> 0\<close> centre_in_ball exp_Ln rev_image_eqI zero_less_one)
+ have "open (- {0::complex})"
+ by blast
+ moreover have "inj_on exp (ball (Ln z) 1)"
+ apply (rule inj_on_subset [OF inj_on_exp_pi [of "Ln z"]])
+ using pi_ge_two by (simp add: ball_subset_ball_iff)
+ ultimately show "openin (subtopology euclidean (- {0})) (exp ` ball (Ln z) 1)"
+ by (auto simp: openin_open_eq invariance_of_domain continuous_on_exp [OF continuous_on_id])
+ show "\<Union>\<V> = {w. exp w \<in> exp ` ball (Ln z) 1}"
+ by (auto simp: \<V>_def Complex_Transcendental.exp_eq image_iff)
+ show "\<forall>V\<in>\<V>. open V"
+ by (auto simp: \<V>_def inj_on_def continuous_intros invariance_of_domain)
+ have xy: "2 \<le> cmod (2 * of_int x * of_real pi * \<i> - 2 * of_int y * of_real pi * \<i>)"
+ if "x < y" for x y
+ proof -
+ have "1 \<le> abs (x - y)"
+ using that by linarith
+ then have "1 \<le> cmod (of_int x - of_int y) * 1"
+ by (metis mult.right_neutral norm_of_int of_int_1_le_iff of_int_abs of_int_diff)
+ also have "... \<le> cmod (of_int x - of_int y) * of_real pi"
+ apply (rule mult_left_mono)
+ using pi_ge_two by auto
+ also have "... \<le> cmod ((of_int x - of_int y) * of_real pi * \<i>)"
+ by (simp add: norm_mult)
+ also have "... \<le> cmod (of_int x * of_real pi * \<i> - of_int y * of_real pi * \<i>)"
+ by (simp add: algebra_simps)
+ finally have "1 \<le> cmod (of_int x * of_real pi * \<i> - of_int y * of_real pi * \<i>)" .
+ then have "2 * 1 \<le> cmod (2 * (of_int x * of_real pi * \<i> - of_int y * of_real pi * \<i>))"
+ by (metis mult_le_cancel_left_pos norm_mult_numeral1 zero_less_numeral)
+ then show ?thesis
+ by (simp add: algebra_simps)
+ qed
+ show "disjoint \<V>"
+ apply (clarsimp simp add: \<V>_def pairwise_def disjnt_def add.commute [of _ "x*y" for x y]
+ image_add_ball ball_eq_ball_iff)
+ apply (rule disjoint_ballI)
+ apply (auto simp: dist_norm neq_iff)
+ by (metis norm_minus_commute xy)+
+ show "\<forall>u\<in>\<V>. \<exists>q. homeomorphism u (exp ` ball (Ln z) 1) exp q"
+ proof
+ fix u
+ assume "u \<in> \<V>"
+ then obtain n where n: "u = (\<lambda>x. x + of_real (2 * of_int n * pi) * ii) ` (ball(Ln z) 1)"
+ by (auto simp: \<V>_def)
+ have "compact (cball (Ln z) 1)"
+ by simp
+ moreover have "continuous_on (cball (Ln z) 1) exp"
+ by (rule continuous_on_exp [OF continuous_on_id])
+ moreover have "inj_on exp (cball (Ln z) 1)"
+ apply (rule inj_on_subset [OF inj_on_exp_pi [of "Ln z"]])
+ using pi_ge_two by (simp add: cball_subset_ball_iff)
+ ultimately obtain \<gamma> where hom: "homeomorphism (cball (Ln z) 1) (exp ` cball (Ln z) 1) exp \<gamma>"
+ using homeomorphism_compact by blast
+ have eq1: "exp ` u = exp ` ball (Ln z) 1"
+ unfolding n
+ apply (auto simp: algebra_simps)
+ apply (rename_tac w)
+ apply (rule_tac x = "w + \<i> * (of_int n * (of_real pi * 2))" in image_eqI)
+ apply (auto simp: image_iff)
+ done
+ have \<gamma>exp: "\<gamma> (exp x) + 2 * of_int n * of_real pi * \<i> = x" if "x \<in> u" for x
+ proof -
+ have "exp x = exp (x - 2 * of_int n * of_real pi * \<i>)"
+ by (simp add: exp_eq)
+ then have "\<gamma> (exp x) = \<gamma> (exp (x - 2 * of_int n * of_real pi * \<i>))"
+ by simp
+ also have "... = x - 2 * of_int n * of_real pi * \<i>"
+ apply (rule homeomorphism_apply1 [OF hom])
+ using \<open>x \<in> u\<close> by (auto simp: n)
+ finally show ?thesis
+ by simp
+ qed
+ have exp2n: "exp (\<gamma> (exp x) + 2 * of_int n * complex_of_real pi * \<i>) = exp x"
+ if "dist (Ln z) x < 1" for x
+ using that by (auto simp: exp_eq homeomorphism_apply1 [OF hom])
+ have cont: "continuous_on (exp ` ball (Ln z) 1) (\<lambda>x. \<gamma> x + 2 * of_int n * complex_of_real pi * \<i>)"
+ apply (intro continuous_intros)
+ apply (rule continuous_on_subset [OF homeomorphism_cont2 [OF hom]])
+ apply (force simp:)
+ done
+ show "\<exists>q. homeomorphism u (exp ` ball (Ln z) 1) exp q"
+ apply (rule_tac x="(\<lambda>x. x + of_real(2 * n * pi) * ii) \<circ> \<gamma>" in exI)
+ unfolding homeomorphism_def
+ apply (intro conjI ballI eq1 continuous_on_exp [OF continuous_on_id])
+ apply (auto simp: \<gamma>exp exp2n cont n)
+ apply (simp add: homeomorphism_apply1 [OF hom])
+ apply (simp add: image_comp [symmetric])
+ using hom homeomorphism_apply1 apply (force simp: image_iff)
+ done
+ qed
+ qed
+ qed
+qed
+
+end
--- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy Thu Oct 20 19:39:27 2016 +0200
@@ -960,7 +960,7 @@
using dp p(1) mn d(2) by auto
qed
qed
- then guess y unfolding convergent_eq_cauchy[symmetric] .. note y=this[THEN LIMSEQ_D]
+ then guess y unfolding convergent_eq_Cauchy[symmetric] .. note y=this[THEN LIMSEQ_D]
show ?l
unfolding integrable_on_def has_integral
proof (rule_tac x=y in exI, clarify)
@@ -1798,7 +1798,7 @@
qed
qed
then obtain s where s: "i \<longlonglongrightarrow> s"
- using convergent_eq_cauchy[symmetric] by blast
+ using convergent_eq_Cauchy[symmetric] by blast
show ?thesis
unfolding integrable_on_def has_integral
proof (rule_tac x=s in exI, clarify)
@@ -5437,7 +5437,7 @@
apply auto
done
qed
- from this[unfolded convergent_eq_cauchy[symmetric]] guess i ..
+ from this[unfolded convergent_eq_Cauchy[symmetric]] guess i ..
note i = this[THEN LIMSEQ_D]
show ?l unfolding integrable_on_def has_integral_alt'[of f]
--- a/src/HOL/Analysis/Measurable.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Analysis/Measurable.thy Thu Oct 20 19:39:27 2016 +0200
@@ -646,6 +646,14 @@
shows "liminf A \<in> sets M"
by (subst liminf_SUP_INF, auto)
+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
+ by (cases i) (auto intro: g h)
+ done
+
hide_const (open) pred
end
--- a/src/HOL/Analysis/Measure_Space.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Analysis/Measure_Space.thy Thu Oct 20 19:39:27 2016 +0200
@@ -1469,6 +1469,9 @@
lemma distr_id[simp]: "distr N N (\<lambda>x. x) = N"
by (rule measure_eqI) (auto simp: emeasure_distr)
+lemma distr_id2: "sets M = sets N \<Longrightarrow> distr N M (\<lambda>x. x) = N"
+ by (rule measure_eqI) (auto simp: emeasure_distr)
+
lemma measure_distr:
"f \<in> measurable M N \<Longrightarrow> S \<in> sets N \<Longrightarrow> measure (distr M N f) S = measure M (f -` S \<inter> space M)"
by (simp add: emeasure_distr measure_def)
@@ -3516,6 +3519,11 @@
finally show ?thesis .
qed
+lemma measurable_SUP2:
+ "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f \<in> measurable N (M i)) \<Longrightarrow>
+ (\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> space (M i) = space (M j)) \<Longrightarrow> f \<in> measurable N (SUP i:I. M i)"
+ by (auto intro!: measurable_Sup2)
+
lemma sets_Sup_sigma:
assumes [simp]: "M \<noteq> {}" and M: "\<And>m. m \<in> M \<Longrightarrow> m \<subseteq> Pow \<Omega>"
shows "sets (SUP m:M. sigma \<Omega> m) = sets (sigma \<Omega> (\<Union>M))"
--- a/src/HOL/Analysis/Topology_Euclidean_Space.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Analysis/Topology_Euclidean_Space.thy Thu Oct 20 19:39:27 2016 +0200
@@ -458,6 +458,11 @@
(fastforce simp: topological_space_class.topological_basis_def)+
qed
+instance nat :: second_countable_topology
+proof
+ show "\<exists>B::nat set set. countable B \<and> open = generate_topology B"
+ by (intro exI[of _ "range lessThan \<union> range greaterThan"]) (auto simp: open_nat_def)
+qed
lemma countable_separating_set_linorder1:
shows "\<exists>B::('a::{linorder_topology, second_countable_topology} set). countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b \<in> B. x < b \<and> b \<le> y))"
@@ -1034,6 +1039,14 @@
shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
by (simp add: dist_norm)
+lemma disjoint_ballI:
+ shows "dist x y \<ge> r+s \<Longrightarrow> ball x r \<inter> ball y s = {}"
+ using dist_triangle_less_add not_le by fastforce
+
+lemma disjoint_cballI:
+ shows "dist x y > r+s \<Longrightarrow> cball x r \<inter> cball y s = {}"
+ by (metis add_mono disjoint_iff_not_equal dist_triangle2 dual_order.trans leD mem_cball)
+
lemma mem_sphere_0 [simp]:
fixes x :: "'a::real_normed_vector"
shows "x \<in> sphere 0 e \<longleftrightarrow> norm x = e"
@@ -5435,62 +5448,62 @@
qed
lemma complete_imp_closed:
- fixes s :: "'a::metric_space set"
- assumes "complete s"
- shows "closed s"
+ fixes S :: "'a::metric_space set"
+ assumes "complete S"
+ shows "closed S"
proof (unfold closed_sequential_limits, clarify)
- fix f x assume "\<forall>n. f n \<in> s" and "f \<longlonglongrightarrow> x"
+ fix f x assume "\<forall>n. f n \<in> S" and "f \<longlonglongrightarrow> x"
from \<open>f \<longlonglongrightarrow> x\<close> have "Cauchy f"
by (rule LIMSEQ_imp_Cauchy)
- with \<open>complete s\<close> and \<open>\<forall>n. f n \<in> s\<close> obtain l where "l \<in> s" and "f \<longlonglongrightarrow> l"
+ with \<open>complete S\<close> and \<open>\<forall>n. f n \<in> S\<close> obtain l where "l \<in> S" and "f \<longlonglongrightarrow> l"
by (rule completeE)
from \<open>f \<longlonglongrightarrow> x\<close> and \<open>f \<longlonglongrightarrow> l\<close> have "x = l"
by (rule LIMSEQ_unique)
- with \<open>l \<in> s\<close> show "x \<in> s"
+ with \<open>l \<in> S\<close> show "x \<in> S"
by simp
qed
lemma complete_Int_closed:
- fixes s :: "'a::metric_space set"
- assumes "complete s" and "closed t"
- shows "complete (s \<inter> t)"
+ fixes S :: "'a::metric_space set"
+ assumes "complete S" and "closed t"
+ shows "complete (S \<inter> t)"
proof (rule completeI)
- fix f assume "\<forall>n. f n \<in> s \<inter> t" and "Cauchy f"
- then have "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t"
+ fix f assume "\<forall>n. f n \<in> S \<inter> t" and "Cauchy f"
+ then have "\<forall>n. f n \<in> S" and "\<forall>n. f n \<in> t"
by simp_all
- from \<open>complete s\<close> obtain l where "l \<in> s" and "f \<longlonglongrightarrow> l"
- using \<open>\<forall>n. f n \<in> s\<close> and \<open>Cauchy f\<close> by (rule completeE)
+ from \<open>complete S\<close> obtain l where "l \<in> S" and "f \<longlonglongrightarrow> l"
+ using \<open>\<forall>n. f n \<in> S\<close> and \<open>Cauchy f\<close> by (rule completeE)
from \<open>closed t\<close> and \<open>\<forall>n. f n \<in> t\<close> and \<open>f \<longlonglongrightarrow> l\<close> have "l \<in> t"
by (rule closed_sequentially)
- with \<open>l \<in> s\<close> and \<open>f \<longlonglongrightarrow> l\<close> show "\<exists>l\<in>s \<inter> t. f \<longlonglongrightarrow> l"
+ with \<open>l \<in> S\<close> and \<open>f \<longlonglongrightarrow> l\<close> show "\<exists>l\<in>S \<inter> t. f \<longlonglongrightarrow> l"
by fast
qed
lemma complete_closed_subset:
- 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)
+ 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)
lemma complete_eq_closed:
- fixes s :: "('a::complete_space) set"
- shows "complete s \<longleftrightarrow> closed s"
+ fixes S :: "('a::complete_space) set"
+ shows "complete S \<longleftrightarrow> closed S"
proof
- assume "closed s" then show "complete s"
+ 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"
+ assume "complete S" then show "closed S"
by (rule complete_imp_closed)
qed
-lemma convergent_eq_cauchy:
- fixes s :: "nat \<Rightarrow> 'a::complete_space"
- shows "(\<exists>l. (s \<longlongrightarrow> l) sequentially) \<longleftrightarrow> Cauchy s"
+lemma convergent_eq_Cauchy:
+ fixes S :: "nat \<Rightarrow> 'a::complete_space"
+ shows "(\<exists>l. (S \<longlongrightarrow> l) sequentially) \<longleftrightarrow> Cauchy S"
unfolding Cauchy_convergent_iff convergent_def ..
lemma convergent_imp_bounded:
- fixes s :: "nat \<Rightarrow> 'a::metric_space"
- shows "(s \<longlongrightarrow> l) sequentially \<Longrightarrow> bounded (range s)"
+ fixes S :: "nat \<Rightarrow> 'a::metric_space"
+ shows "(S \<longlongrightarrow> l) sequentially \<Longrightarrow> bounded (range S)"
by (intro cauchy_imp_bounded LIMSEQ_imp_Cauchy)
lemma compact_cball[simp]:
@@ -5500,15 +5513,15 @@
by blast
lemma compact_frontier_bounded[intro]:
- fixes s :: "'a::heine_borel set"
- shows "bounded s \<Longrightarrow> compact (frontier s)"
+ fixes S :: "'a::heine_borel set"
+ shows "bounded S \<Longrightarrow> compact (frontier S)"
unfolding frontier_def
using compact_eq_bounded_closed
by blast
lemma compact_frontier[intro]:
- fixes s :: "'a::heine_borel set"
- shows "compact s \<Longrightarrow> compact (frontier s)"
+ fixes S :: "'a::heine_borel set"
+ shows "compact S \<Longrightarrow> compact (frontier S)"
using compact_eq_bounded_closed compact_frontier_bounded
by blast
@@ -5528,8 +5541,8 @@
by (simp add: compact_imp_closed)
lemma frontier_subset_compact:
- fixes s :: "'a::heine_borel set"
- shows "compact s \<Longrightarrow> frontier s \<subseteq> s"
+ fixes S :: "'a::heine_borel set"
+ shows "compact S \<Longrightarrow> frontier S \<subseteq> S"
using frontier_subset_closed compact_eq_bounded_closed
by blast
@@ -5723,7 +5736,7 @@
apply auto
done
then obtain l where l: "\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) \<longlongrightarrow> l x) sequentially"
- unfolding convergent_eq_cauchy[symmetric]
+ unfolding convergent_eq_Cauchy[symmetric]
using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) \<longlongrightarrow> l) sequentially"]
by auto
{
@@ -6081,6 +6094,11 @@
unfolding uniformly_continuous_on_def by blast
qed
+lemma continuous_closed_imp_Cauchy_continuous:
+ fixes S :: "('a::complete_space) set"
+ shows "\<lbrakk>continuous_on S f; closed S; Cauchy \<sigma>; \<And>n. (\<sigma> n) \<in> S\<rbrakk> \<Longrightarrow> Cauchy(f o \<sigma>)"
+ apply (simp add: complete_eq_closed [symmetric] continuous_on_sequentially)
+ by (meson LIMSEQ_imp_Cauchy complete_def)
text\<open>The usual transformation theorems.\<close>
@@ -6630,7 +6648,7 @@
from uniformly_continuous_on_Cauchy[OF uc LIMSEQ_imp_Cauchy, OF xs]
obtain l where l: "(\<lambda>n. f (xs n)) \<longlonglongrightarrow> l"
- by atomize_elim (simp only: convergent_eq_cauchy)
+ by atomize_elim (simp only: convergent_eq_Cauchy)
have "(f \<longlongrightarrow> l) (at x within X)"
proof (safe intro!: Lim_within_LIMSEQ)
@@ -6641,7 +6659,7 @@
from uniformly_continuous_on_Cauchy[OF uc LIMSEQ_imp_Cauchy, OF \<open>xs' \<longlonglongrightarrow> x\<close> \<open>xs' _ \<in> X\<close>]
obtain l' where l': "(\<lambda>n. f (xs' n)) \<longlonglongrightarrow> l'"
- by atomize_elim (simp only: convergent_eq_cauchy)
+ by atomize_elim (simp only: convergent_eq_Cauchy)
show "(\<lambda>n. f (xs' n)) \<longlonglongrightarrow> l"
proof (rule tendstoI)
--- a/src/HOL/Archimedean_Field.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Archimedean_Field.thy Thu Oct 20 19:39:27 2016 +0200
@@ -736,6 +736,9 @@
by simp
qed
+lemma round_unique': "\<bar>x - of_int n\<bar> < 1/2 \<Longrightarrow> round x = n"
+ by (subst (asm) abs_less_iff, rule round_unique) (simp_all add: field_simps)
+
lemma round_altdef: "round x = (if frac x \<ge> 1/2 then \<lceil>x\<rceil> else \<lfloor>x\<rfloor>)"
by (cases "frac x \<ge> 1/2")
(rule round_unique, ((simp add: frac_def field_simps ceiling_altdef; linarith)+)[2])+
--- a/src/HOL/Complex.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Complex.thy Thu Oct 20 19:39:27 2016 +0200
@@ -381,6 +381,23 @@
by (simp add: complex_sgn_def divide_inverse)
+subsection \<open>Absolute value\<close>
+
+instantiation complex :: field_abs_sgn
+begin
+
+definition abs_complex :: "complex \<Rightarrow> complex"
+ where "abs_complex = of_real \<circ> norm"
+
+instance
+ apply standard
+ apply (auto simp add: abs_complex_def complex_sgn_def norm_mult)
+ apply (auto simp add: scaleR_conv_of_real field_simps)
+ done
+
+end
+
+
subsection \<open>Completeness of the Complexes\<close>
lemma bounded_linear_Re: "bounded_linear Re"
--- a/src/HOL/Data_Structures/document/root.tex Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Data_Structures/document/root.tex Thu Oct 20 19:39:27 2016 +0200
@@ -1,6 +1,8 @@
\documentclass[11pt,a4paper]{article}
\usepackage{isabelle,isabellesym}
\usepackage{latexsym}
+\usepackage{amssymb}
+\usepackage{amsmath}
% this should be the last package used
\usepackage{pdfsetup}
--- a/src/HOL/Enum.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Enum.thy Thu Oct 20 19:39:27 2016 +0200
@@ -580,7 +580,7 @@
instantiation finite_1 ::
"{linordered_ring_strict, linordered_comm_semiring_strict, ordered_comm_ring,
ordered_cancel_comm_monoid_diff, comm_monoid_mult, ordered_ring_abs,
- one, modulo, sgn_if, inverse}"
+ one, modulo, sgn, inverse}"
begin
definition [simp]: "Groups.zero = a\<^sub>1"
definition [simp]: "Groups.one = a\<^sub>1"
@@ -683,7 +683,7 @@
instance finite_2 :: complete_linorder ..
-instantiation finite_2 :: "{field, abs_if, ring_div, sgn_if, semiring_div}" begin
+instantiation finite_2 :: "{field, ring_div, idom_abs_sgn}" begin
definition [simp]: "0 = a\<^sub>1"
definition [simp]: "1 = a\<^sub>2"
definition "x + y = (case (x, y) of (a\<^sub>1, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>1 | _ \<Rightarrow> a\<^sub>2)"
@@ -806,7 +806,7 @@
instance finite_3 :: complete_linorder ..
-instantiation finite_3 :: "{field, abs_if, ring_div, semiring_div, sgn_if}" begin
+instantiation finite_3 :: "{field, ring_div, idom_abs_sgn}" begin
definition [simp]: "0 = a\<^sub>1"
definition [simp]: "1 = a\<^sub>2"
definition
@@ -819,9 +819,9 @@
definition "x * y = (case (x, y) of (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>2 | (a\<^sub>3, a\<^sub>3) \<Rightarrow> a\<^sub>2 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>3 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>3 | _ \<Rightarrow> a\<^sub>1)"
definition "inverse = (\<lambda>x :: finite_3. x)"
definition "x div y = x * inverse (y :: finite_3)"
-definition "abs = (\<lambda>x :: finite_3. x)"
+definition "abs = (\<lambda>x. case x of a\<^sub>3 \<Rightarrow> a\<^sub>2 | _ \<Rightarrow> x)"
definition "x mod y = (case (x, y) of (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>2 | (a\<^sub>3, a\<^sub>1) \<Rightarrow> a\<^sub>3 | _ \<Rightarrow> a\<^sub>1)"
-definition "sgn = (\<lambda>x. case x of a\<^sub>1 \<Rightarrow> a\<^sub>1 | _ \<Rightarrow> a\<^sub>2)"
+definition "sgn = (\<lambda>x :: finite_3. x)"
instance
by intro_classes
(simp_all add: plus_finite_3_def uminus_finite_3_def minus_finite_3_def times_finite_3_def
--- a/src/HOL/Fields.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Fields.thy Thu Oct 20 19:39:27 2016 +0200
@@ -513,6 +513,48 @@
subsection \<open>Ordered fields\<close>
+class field_abs_sgn = field + idom_abs_sgn
+begin
+
+lemma sgn_inverse [simp]:
+ "sgn (inverse a) = inverse (sgn a)"
+proof (cases "a = 0")
+ case True then show ?thesis by simp
+next
+ case False
+ then have "a * inverse a = 1"
+ by simp
+ then have "sgn (a * inverse a) = sgn 1"
+ by simp
+ then have "sgn a * sgn (inverse a) = 1"
+ by (simp add: sgn_mult)
+ then have "inverse (sgn a) * (sgn a * sgn (inverse a)) = inverse (sgn a) * 1"
+ by simp
+ then have "(inverse (sgn a) * sgn a) * sgn (inverse a) = inverse (sgn a)"
+ by (simp add: ac_simps)
+ with False show ?thesis
+ by (simp add: sgn_eq_0_iff)
+qed
+
+lemma abs_inverse [simp]:
+ "\<bar>inverse a\<bar> = inverse \<bar>a\<bar>"
+proof -
+ from sgn_mult_abs [of "inverse a"] sgn_mult_abs [of a]
+ have "inverse (sgn a) * \<bar>inverse a\<bar> = inverse (sgn a * \<bar>a\<bar>)"
+ by simp
+ then show ?thesis by (auto simp add: sgn_eq_0_iff)
+qed
+
+lemma sgn_divide [simp]:
+ "sgn (a / b) = sgn a / sgn b"
+ unfolding divide_inverse sgn_mult by simp
+
+lemma abs_divide [simp]:
+ "\<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
+ unfolding divide_inverse abs_mult by simp
+
+end
+
class linordered_field = field + linordered_idom
begin
@@ -932,16 +974,15 @@
show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)
qed
+subclass field_abs_sgn ..
+
lemma nonzero_abs_inverse:
- "a \<noteq> 0 ==> \<bar>inverse a\<bar> = inverse \<bar>a\<bar>"
-apply (auto simp add: neq_iff abs_if nonzero_inverse_minus_eq
- negative_imp_inverse_negative)
-apply (blast intro: positive_imp_inverse_positive elim: less_asym)
-done
+ "a \<noteq> 0 ==> \<bar>inverse a\<bar> = inverse \<bar>a\<bar>"
+ by (rule abs_inverse)
lemma nonzero_abs_divide:
- "b \<noteq> 0 ==> \<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
- by (simp add: divide_inverse abs_mult nonzero_abs_inverse)
+ "b \<noteq> 0 ==> \<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
+ by (rule abs_divide)
lemma field_le_epsilon:
assumes e: "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e"
@@ -1147,19 +1188,6 @@
"(b/a = 1) = ((a \<noteq> 0 & a = b))"
by (auto simp add: divide_eq_eq)
-lemma abs_inverse [simp]:
- "\<bar>inverse a\<bar> =
- inverse \<bar>a\<bar>"
-apply (cases "a=0", simp)
-apply (simp add: nonzero_abs_inverse)
-done
-
-lemma abs_divide [simp]:
- "\<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
-apply (cases "b=0", simp)
-apply (simp add: nonzero_abs_divide)
-done
-
lemma abs_div_pos: "0 < y ==>
\<bar>x\<bar> / y = \<bar>x / y\<bar>"
apply (subst abs_divide)
@@ -1174,7 +1202,7 @@
lemma inverse_sgn:
"sgn (inverse a) = inverse (sgn a)"
- by (simp add: sgn_if)
+ by (fact sgn_inverse)
lemma field_le_mult_one_interval:
assumes *: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
--- a/src/HOL/Groups.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Groups.thy Thu Oct 20 19:39:27 2016 +0200
@@ -1148,18 +1148,6 @@
class sgn =
fixes sgn :: "'a \<Rightarrow> 'a"
-class abs_if = minus + uminus + ord + zero + abs +
- assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)"
-
-class sgn_if = minus + uminus + zero + one + ord + sgn +
- assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
-begin
-
-lemma sgn0 [simp]: "sgn 0 = 0"
- by (simp add:sgn_if)
-
-end
-
class ordered_ab_group_add_abs = ordered_ab_group_add + abs +
assumes abs_ge_zero [simp]: "\<bar>a\<bar> \<ge> 0"
and abs_ge_self: "a \<le> \<bar>a\<bar>"
--- a/src/HOL/Library/Extended_Nonnegative_Real.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Library/Extended_Nonnegative_Real.thy Thu Oct 20 19:39:27 2016 +0200
@@ -1131,6 +1131,9 @@
lemma enn2real_positive_iff: "0 < enn2real x \<longleftrightarrow> (0 < x \<and> x < top)"
by (cases x rule: ennreal_cases) auto
+lemma enn2real_eq_1_iff[simp]: "enn2real x = 1 \<longleftrightarrow> x = 1"
+ by (cases x) auto
+
subsection \<open>Coercion from @{typ enat} to @{typ ennreal}\<close>
--- a/src/HOL/Library/Fraction_Field.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Library/Fraction_Field.thy Thu Oct 20 19:39:27 2016 +0200
@@ -70,7 +70,7 @@
and "\<And>a c. Fract 0 a = Fract 0 c"
by(transfer; simp)+
-instantiation fract :: (idom) "{comm_ring_1,power}"
+instantiation fract :: (idom) comm_ring_1
begin
lift_definition zero_fract :: "'a fract" is "(0, 1)" by simp
@@ -353,31 +353,20 @@
end
-instantiation fract :: (linordered_idom) "{distrib_lattice,abs_if,sgn_if}"
+instantiation fract :: (linordered_idom) linordered_field
begin
-definition abs_fract_def2: "\<bar>q\<bar> = (if q < 0 then -q else (q::'a fract))"
+definition abs_fract_def2:
+ "\<bar>q\<bar> = (if q < 0 then -q else (q::'a fract))"
definition sgn_fract_def:
"sgn (q::'a fract) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
theorem abs_fract [simp]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
-unfolding abs_fract_def2 not_le[symmetric]
-by transfer(auto simp add: zero_less_mult_iff le_less)
-
-definition inf_fract_def:
- "(inf :: 'a fract \<Rightarrow> 'a fract \<Rightarrow> 'a fract) = min"
+ unfolding abs_fract_def2 not_le [symmetric]
+ by transfer (auto simp add: zero_less_mult_iff le_less)
-definition sup_fract_def:
- "(sup :: 'a fract \<Rightarrow> 'a fract \<Rightarrow> 'a fract) = max"
-
-instance
-by intro_classes (simp_all add: abs_fract_def2 sgn_fract_def inf_fract_def sup_fract_def max_min_distrib2)
-
-end
-
-instance fract :: (linordered_idom) linordered_field
-proof
+instance proof
fix q r s :: "'a fract"
assume "q \<le> r"
then show "s + q \<le> s + r"
@@ -420,7 +409,23 @@
by (simp add: ac_simps)
qed
qed
-qed
+qed (fact sgn_fract_def abs_fract_def2)+
+
+end
+
+instantiation fract :: (linordered_idom) distrib_lattice
+begin
+
+definition inf_fract_def:
+ "(inf :: 'a fract \<Rightarrow> 'a fract \<Rightarrow> 'a fract) = min"
+
+definition sup_fract_def:
+ "(sup :: 'a fract \<Rightarrow> 'a fract \<Rightarrow> 'a fract) = max"
+
+instance
+ by standard (simp_all add: inf_fract_def sup_fract_def max_min_distrib2)
+
+end
lemma fract_induct_pos [case_names Fract]:
fixes P :: "'a::linordered_idom fract \<Rightarrow> bool"
--- a/src/HOL/Library/Linear_Temporal_Logic_on_Streams.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Library/Linear_Temporal_Logic_on_Streams.thy Thu Oct 20 19:39:27 2016 +0200
@@ -662,7 +662,7 @@
lemma sfilter_not_P[simp]: "\<not> P (shd s) \<Longrightarrow> sfilter P s = sfilter P (stl s)"
using sfilter_Stream[of P "shd s" "stl s"] by simp
-lemma sfilter_eq:
+lemma sfilter_eq:
assumes "ev (holds P) s"
shows "sfilter P s = x ## s' \<longleftrightarrow>
P x \<and> (not (holds P) suntil (HLD {x} aand nxt (\<lambda>s. sfilter P s = s'))) s"
@@ -685,7 +685,7 @@
proof
assume "alw Q (sfilter P s)" with * show "alw (\<lambda>x. Q (sfilter P x)) s"
proof (coinduction arbitrary: s rule: alw_coinduct)
- case (stl s)
+ case (stl s)
then have "ev (holds P) s"
by blast
from this stl show ?case
@@ -694,7 +694,7 @@
next
assume "alw (\<lambda>x. Q (sfilter P x)) s" with * show "alw Q (sfilter P s)"
proof (coinduction arbitrary: s rule: alw_coinduct)
- case (stl s)
+ case (stl s)
then have "ev (holds P) s"
by blast
from this stl show ?case
@@ -767,4 +767,22 @@
lemma hld_smap': "HLD x (smap f s) = HLD (f -` x) s"
by (simp add: HLD_def)
+lemma pigeonhole_stream:
+ assumes "alw (HLD s) \<omega>"
+ assumes "finite s"
+ shows "\<exists>x\<in>s. alw (ev (HLD {x})) \<omega>"
+proof -
+ have "\<forall>i\<in>UNIV. \<exists>x\<in>s. \<omega> !! i = x"
+ using `alw (HLD s) \<omega>` by (simp add: alw_iff_sdrop HLD_iff)
+ from pigeonhole_infinite_rel[OF infinite_UNIV_nat `finite s` this]
+ show ?thesis
+ by (simp add: HLD_iff infinite_iff_alw_ev[symmetric])
+qed
+
+lemma ev_eq_suntil: "ev P \<omega> \<longleftrightarrow> (not P suntil P) \<omega>"
+proof
+ assume "ev P \<omega>" then show "((\<lambda>xs. \<not> P xs) suntil P) \<omega>"
+ by (induction rule: ev_induct_strong) (auto intro: suntil.intros)
+qed (auto simp: ev_suntil)
+
end
--- a/src/HOL/Library/Old_SMT/old_smt_solver.ML Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Library/Old_SMT/old_smt_solver.ML Thu Oct 20 19:39:27 2016 +0200
@@ -53,7 +53,7 @@
fun make_cmd command options problem_path proof_path =
space_implode " "
- ("(exec 2>&1;" :: map File.bash_string (command () @ options) @
+ ("(exec 2>&1;" :: map Bash.string (command () @ options) @
[File.bash_path problem_path, ")", ">", File.bash_path proof_path])
fun trace_and ctxt msg f x =
--- a/src/HOL/Library/Stream.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Library/Stream.thy Thu Oct 20 19:39:27 2016 +0200
@@ -242,7 +242,7 @@
lemma sdrop_snth: "sdrop n s !! m = s !! (n + m)"
by (induct n arbitrary: m s) auto
-partial_function (tailrec) sdrop_while :: "('a \<Rightarrow> bool) \<Rightarrow> 'a stream \<Rightarrow> 'a stream" where
+partial_function (tailrec) sdrop_while :: "('a \<Rightarrow> bool) \<Rightarrow> 'a stream \<Rightarrow> 'a stream" where
"sdrop_while P s = (if P (shd s) then sdrop_while P (stl s) else s)"
lemma sdrop_while_SCons[code]:
@@ -342,7 +342,7 @@
by (induct n arbitrary: u) (auto simp: rotate1_rotate_swap rotate1_hd_tl rotate_conv_mod[symmetric])
lemma sset_cycle[simp]:
- assumes "xs \<noteq> []"
+ assumes "xs \<noteq> []"
shows "sset (cycle xs) = set xs"
proof (intro set_eqI iffI)
fix x
@@ -408,6 +408,14 @@
lemma sconst_streams: "x \<in> A \<Longrightarrow> sconst x \<in> streams A"
by (simp add: streams_iff_sset)
+lemma streams_empty_iff: "streams S = {} \<longleftrightarrow> S = {}"
+proof safe
+ fix x assume "x \<in> S" "streams S = {}"
+ then have "sconst x \<in> streams S"
+ by (intro sconst_streams)
+ then show "x \<in> {}"
+ unfolding \<open>streams S = {}\<close> by simp
+qed (auto simp: streams_empty)
subsection \<open>stream of natural numbers\<close>
@@ -442,11 +450,11 @@
lemma flat_unfold: "shd ws \<noteq> [] \<Longrightarrow> flat ws = shd ws @- flat (stl ws)"
by (cases ws) auto
-lemma flat_snth: "\<forall>xs \<in> sset s. xs \<noteq> [] \<Longrightarrow> flat s !! n = (if n < length (shd s) then
+lemma flat_snth: "\<forall>xs \<in> sset s. xs \<noteq> [] \<Longrightarrow> flat s !! n = (if n < length (shd s) then
shd s ! n else flat (stl s) !! (n - length (shd s)))"
by (metis flat_unfold not_less shd_sset shift_snth_ge shift_snth_less)
-lemma sset_flat[simp]: "\<forall>xs \<in> sset s. xs \<noteq> [] \<Longrightarrow>
+lemma sset_flat[simp]: "\<forall>xs \<in> sset s. xs \<noteq> [] \<Longrightarrow>
sset (flat s) = (\<Union>xs \<in> sset s. set xs)" (is "?P \<Longrightarrow> ?L = ?R")
proof safe
fix x assume ?P "x : ?L"
--- a/src/HOL/Limits.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Limits.thy Thu Oct 20 19:39:27 2016 +0200
@@ -2340,6 +2340,11 @@
lemma isUCont_Cauchy: "isUCont f \<Longrightarrow> Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
by (rule uniformly_continuous_on_Cauchy[where S=UNIV and f=f]) simp_all
+
+lemma uniformly_continuous_imp_Cauchy_continuous:
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
+ shows "\<lbrakk>uniformly_continuous_on S f; Cauchy \<sigma>; \<And>n. (\<sigma> n) \<in> S\<rbrakk> \<Longrightarrow> Cauchy(f o \<sigma>)"
+ by (simp add: uniformly_continuous_on_def Cauchy_def) meson
lemma (in bounded_linear) isUCont: "isUCont f"
unfolding isUCont_def dist_norm
--- a/src/HOL/Nonstandard_Analysis/StarDef.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Nonstandard_Analysis/StarDef.thy Thu Oct 20 19:39:27 2016 +0200
@@ -898,19 +898,15 @@
instance star :: (abs_if) abs_if
by (intro_classes; transfer) (fact abs_if)
-instance star :: (sgn_if) sgn_if
- by (intro_classes; transfer) (fact sgn_if)
-
instance star :: (linordered_ring_strict) linordered_ring_strict ..
instance star :: (ordered_comm_ring) ordered_comm_ring ..
instance star :: (linordered_semidom) linordered_semidom
- apply intro_classes
- apply(transfer, fact zero_less_one)
- apply(transfer, fact le_add_diff_inverse2)
- done
+ by (intro_classes; transfer) (fact zero_less_one le_add_diff_inverse2)+
-instance star :: (linordered_idom) linordered_idom ..
+instance star :: (linordered_idom) linordered_idom
+ by (intro_classes; transfer) (fact sgn_if)
+
instance star :: (linordered_field) linordered_field ..
subsection \<open>Power\<close>
--- a/src/HOL/Number_Theory/Fib.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Number_Theory/Fib.thy Thu Oct 20 19:39:27 2016 +0200
@@ -1,12 +1,13 @@
(* Title: HOL/Number_Theory/Fib.thy
Author: Lawrence C. Paulson
Author: Jeremy Avigad
+ Author: Manuel Eberl
*)
section \<open>The fibonacci function\<close>
theory Fib
-imports Main GCD Binomial
+ imports Complex_Main
begin
@@ -38,6 +39,34 @@
by (induct n rule: fib.induct) (auto simp add: )
+subsection \<open>More efficient code\<close>
+
+text \<open>
+ The naive approach is very inefficient since the branching recursion leads to many
+ values of @{term fib} being computed multiple times. We can avoid this by ``remembering''
+ the last two values in the sequence, yielding a tail-recursive version.
+ This is far from optimal (it takes roughly $O(n\cdot M(n))$ time where $M(n)$ is the
+ time required to multiply two $n$-bit integers), but much better than the naive version,
+ which is exponential.
+\<close>
+
+fun gen_fib :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat" where
+ "gen_fib a b 0 = a"
+| "gen_fib a b (Suc 0) = b"
+| "gen_fib a b (Suc (Suc n)) = gen_fib b (a + b) (Suc n)"
+
+lemma gen_fib_recurrence: "gen_fib a b (Suc (Suc n)) = gen_fib a b n + gen_fib a b (Suc n)"
+ by (induction a b n rule: gen_fib.induct) simp_all
+
+lemma gen_fib_fib: "gen_fib (fib n) (fib (Suc n)) m = fib (n + m)"
+ by (induction m rule: fib.induct) (simp_all del: gen_fib.simps(3) add: gen_fib_recurrence)
+
+lemma fib_conv_gen_fib: "fib n = gen_fib 0 1 n"
+ using gen_fib_fib[of 0 n] by simp
+
+declare fib_conv_gen_fib [code]
+
+
subsection \<open>A Few Elementary Results\<close>
text \<open>
@@ -104,6 +133,114 @@
by (induct n rule: nat.induct) (auto simp add: field_simps)
+subsection \<open>Closed form\<close>
+
+lemma fib_closed_form:
+ defines "\<phi> \<equiv> (1 + sqrt 5) / (2::real)" and "\<psi> \<equiv> (1 - sqrt 5) / (2::real)"
+ shows "of_nat (fib n) = (\<phi> ^ n - \<psi> ^ n) / sqrt 5"
+proof (induction n rule: fib.induct)
+ fix n :: nat
+ assume IH1: "of_nat (fib n) = (\<phi> ^ n - \<psi> ^ n) / sqrt 5"
+ assume IH2: "of_nat (fib (Suc n)) = (\<phi> ^ Suc n - \<psi> ^ Suc n) / sqrt 5"
+ have "of_nat (fib (Suc (Suc n))) = of_nat (fib (Suc n)) + of_nat (fib n)" by simp
+ also have "... = (\<phi>^n*(\<phi> + 1) - \<psi>^n*(\<psi> + 1)) / sqrt 5"
+ by (simp add: IH1 IH2 field_simps)
+ also have "\<phi> + 1 = \<phi>\<^sup>2" by (simp add: \<phi>_def field_simps power2_eq_square)
+ also have "\<psi> + 1 = \<psi>\<^sup>2" by (simp add: \<psi>_def field_simps power2_eq_square)
+ also have "\<phi>^n * \<phi>\<^sup>2 - \<psi>^n * \<psi>\<^sup>2 = \<phi> ^ Suc (Suc n) - \<psi> ^ Suc (Suc n)"
+ by (simp add: power2_eq_square)
+ finally show "of_nat (fib (Suc (Suc n))) = (\<phi> ^ Suc (Suc n) - \<psi> ^ Suc (Suc n)) / sqrt 5" .
+qed (simp_all add: \<phi>_def \<psi>_def field_simps)
+
+lemma fib_closed_form':
+ defines "\<phi> \<equiv> (1 + sqrt 5) / (2 :: real)" and "\<psi> \<equiv> (1 - sqrt 5) / (2 :: real)"
+ assumes "n > 0"
+ shows "fib n = round (\<phi> ^ n / sqrt 5)"
+proof (rule sym, rule round_unique')
+ have "\<bar>\<phi> ^ n / sqrt 5 - of_int (int (fib n))\<bar> = \<bar>\<psi>\<bar> ^ n / sqrt 5"
+ by (simp add: fib_closed_form[folded \<phi>_def \<psi>_def] field_simps power_abs)
+ also {
+ from assms have "\<bar>\<psi>\<bar>^n \<le> \<bar>\<psi>\<bar>^1"
+ by (intro power_decreasing) (simp_all add: algebra_simps real_le_lsqrt)
+ also have "... < sqrt 5 / 2" by (simp add: \<psi>_def field_simps)
+ finally have "\<bar>\<psi>\<bar>^n / sqrt 5 < 1/2" by (simp add: field_simps)
+ }
+ finally show "\<bar>\<phi> ^ n / sqrt 5 - of_int (int (fib n))\<bar> < 1/2" .
+qed
+
+lemma fib_asymptotics:
+ defines "\<phi> \<equiv> (1 + sqrt 5) / (2 :: real)"
+ shows "(\<lambda>n. real (fib n) / (\<phi> ^ n / sqrt 5)) \<longlonglongrightarrow> 1"
+proof -
+ define \<psi> where "\<psi> \<equiv> (1 - sqrt 5) / (2 :: real)"
+ have "\<phi> > 1" by (simp add: \<phi>_def)
+ hence A: "\<phi> \<noteq> 0" by auto
+ have "(\<lambda>n. (\<psi> / \<phi>) ^ n) \<longlonglongrightarrow> 0"
+ by (rule LIMSEQ_power_zero) (simp_all add: \<phi>_def \<psi>_def field_simps add_pos_pos)
+ hence "(\<lambda>n. 1 - (\<psi> / \<phi>) ^ n) \<longlonglongrightarrow> 1 - 0" by (intro tendsto_diff tendsto_const)
+ with A show ?thesis
+ by (simp add: divide_simps fib_closed_form [folded \<phi>_def \<psi>_def])
+qed
+
+
+subsection \<open>Divide-and-Conquer recurrence\<close>
+
+text \<open>
+ The following divide-and-conquer recurrence allows for a more efficient computation
+ of Fibonacci numbers; however, it requires memoisation of values to be reasonably
+ efficient, cutting the number of values to be computed to logarithmically many instead of
+ linearly many. The vast majority of the computation time is then actually spent on the
+ multiplication, since the output number is exponential in the input number.
+\<close>
+
+lemma fib_rec_odd:
+ defines "\<phi> \<equiv> (1 + sqrt 5) / (2 :: real)" and "\<psi> \<equiv> (1 - sqrt 5) / (2 :: real)"
+ shows "fib (Suc (2*n)) = fib n^2 + fib (Suc n)^2"
+proof -
+ have "of_nat (fib n^2 + fib (Suc n)^2) = ((\<phi> ^ n - \<psi> ^ n)\<^sup>2 + (\<phi> * \<phi> ^ n - \<psi> * \<psi> ^ n)\<^sup>2)/5"
+ by (simp add: fib_closed_form[folded \<phi>_def \<psi>_def] field_simps power2_eq_square)
+ also have "(\<phi> ^ n - \<psi> ^ n)\<^sup>2 + (\<phi> * \<phi> ^ n - \<psi> * \<psi> ^ n)\<^sup>2 =
+ \<phi>^(2*n) + \<psi>^(2*n) - 2*(\<phi>*\<psi>)^n + \<phi>^(2*n+2) + \<psi>^(2*n+2) - 2*(\<phi>*\<psi>)^(n+1)" (is "_ = ?A")
+ by (simp add: power2_eq_square algebra_simps power_mult power_mult_distrib)
+ also have "\<phi> * \<psi> = -1" by (simp add: \<phi>_def \<psi>_def field_simps)
+ hence "?A = \<phi>^(2*n+1) * (\<phi> + inverse \<phi>) + \<psi>^(2*n+1) * (\<psi> + inverse \<psi>)"
+ by (auto simp: field_simps power2_eq_square)
+ also have "1 + sqrt 5 > 0" by (auto intro: add_pos_pos)
+ hence "\<phi> + inverse \<phi> = sqrt 5" by (simp add: \<phi>_def field_simps)
+ also have "\<psi> + inverse \<psi> = -sqrt 5" by (simp add: \<psi>_def field_simps)
+ also have "(\<phi> ^ (2*n+1) * sqrt 5 + \<psi> ^ (2*n+1)* - sqrt 5) / 5 =
+ (\<phi> ^ (2*n+1) - \<psi> ^ (2*n+1)) * (sqrt 5 / 5)" by (simp add: field_simps)
+ also have "sqrt 5 / 5 = inverse (sqrt 5)" by (simp add: field_simps)
+ also have "(\<phi> ^ (2*n+1) - \<psi> ^ (2*n+1)) * ... = of_nat (fib (Suc (2*n)))"
+ by (simp add: fib_closed_form[folded \<phi>_def \<psi>_def] divide_inverse)
+ finally show ?thesis by (simp only: of_nat_eq_iff)
+qed
+
+lemma fib_rec_even: "fib (2*n) = (fib (n - 1) + fib (n + 1)) * fib n"
+proof (induction n)
+ case (Suc n)
+ let ?rfib = "\<lambda>x. real (fib x)"
+ have "2 * (Suc n) = Suc (Suc (2*n))" by simp
+ also have "real (fib ...) = ?rfib n^2 + ?rfib (Suc n)^2 + (?rfib (n - 1) + ?rfib (n + 1)) * ?rfib n"
+ by (simp add: fib_rec_odd Suc)
+ also have "(?rfib (n - 1) + ?rfib (n + 1)) * ?rfib n = (2 * ?rfib (n + 1) - ?rfib n) * ?rfib n"
+ by (cases n) simp_all
+ also have "?rfib n^2 + ?rfib (Suc n)^2 + ... = (?rfib (Suc n) + 2 * ?rfib n) * ?rfib (Suc n)"
+ by (simp add: algebra_simps power2_eq_square)
+ also have "... = real ((fib (Suc n - 1) + fib (Suc n + 1)) * fib (Suc n))" by simp
+ finally show ?case by (simp only: of_nat_eq_iff)
+qed simp
+
+lemma fib_rec_even': "fib (2*n) = (2*fib (n - 1) + fib n) * fib n"
+ by (subst fib_rec_even, cases n) simp_all
+
+lemma fib_rec:
+ "fib n = (if n = 0 then 0 else if n = 1 then 1 else
+ if even n then let n' = n div 2; fn = fib n' in (2 * fib (n' - 1) + fn) * fn
+ else let n' = n div 2 in fib n' ^ 2 + fib (Suc n') ^ 2)"
+ by (auto elim: evenE oddE simp: fib_rec_odd fib_rec_even' Let_def)
+
+
subsection \<open>Fibonacci and Binomial Coefficients\<close>
lemma sum_drop_zero: "(\<Sum>k = 0..Suc n. if 0<k then (f (k - 1)) else 0) = (\<Sum>j = 0..n. f j)"
--- a/src/HOL/Number_Theory/Number_Theory.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Number_Theory/Number_Theory.thy Thu Oct 20 19:39:27 2016 +0200
@@ -2,7 +2,7 @@
section \<open>Comprehensive number theory\<close>
theory Number_Theory
-imports Fib Residues Eratosthenes QuadraticReciprocity Pocklington
+imports Fib Residues Eratosthenes Quadratic_Reciprocity Pocklington
begin
end
--- a/src/HOL/Number_Theory/QuadraticReciprocity.thy Tue Oct 18 16:04:44 2016 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,387 +0,0 @@
-(* Author: Jaime Mendizabal Roche *)
-
-theory QuadraticReciprocity
-imports Gauss
-begin
-
-text {* The proof is based on Gauss's fifth proof, which can be found at http://www.lehigh.edu/~shw2/q-recip/gauss5.pdf *}
-
-locale QR =
- fixes p :: "nat"
- fixes q :: "nat"
-
- assumes p_prime: "prime p"
- assumes p_ge_2: "2 < p"
- assumes q_prime: "prime q"
- assumes q_ge_2: "2 < q"
- assumes pq_neq: "p \<noteq> q"
-begin
-
-lemma odd_p: "odd p" using p_ge_2 p_prime prime_odd_nat by blast
-
-lemma p_ge_0: "0 < int p"
- using p_prime not_prime_0[where 'a = nat] by fastforce+
-
-lemma p_eq2: "int p = (2 * ((int p - 1) div 2)) + 1" using odd_p by simp
-
-lemma odd_q: "odd q" using q_ge_2 q_prime prime_odd_nat by blast
-
-lemma q_ge_0: "0 < int q" using q_prime not_prime_0[where 'a = nat] by fastforce+
-
-lemma q_eq2: "int q = (2 * ((int q - 1) div 2)) + 1" using odd_q by simp
-
-lemma pq_eq2: "int p * int q = (2 * ((int p * int q - 1) div 2)) + 1" using odd_p odd_q by simp
-
-lemma pq_coprime: "coprime p q"
- using pq_neq p_prime primes_coprime_nat q_prime by blast
-
-lemma pq_coprime_int: "coprime (int p) (int q)"
- using pq_coprime transfer_int_nat_gcd(1) by presburger
-
-lemma qp_ineq: "(int p * k \<le> (int p * int q - 1) div 2) = (k \<le> (int q - 1) div 2)"
-proof -
- have "(2 * int p * k \<le> int p * int q - 1) = (2 * k \<le> int q - 1)" using p_ge_0 by auto
- thus ?thesis by auto
-qed
-
-lemma QRqp: "QR q p" using QR_def QR_axioms by simp
-
-lemma pq_commute: "int p * int q = int q * int p" by simp
-
-lemma pq_ge_0: "int p * int q > 0" using p_ge_0 q_ge_0 mult_pos_pos by blast
-
-definition "r = ((p - 1) div 2)*((q - 1) div 2)"
-definition "m = card (GAUSS.E p q)"
-definition "n = card (GAUSS.E q p)"
-
-abbreviation "Res (k::int) \<equiv> {0 .. k - 1}"
-abbreviation "Res_ge_0 (k::int) \<equiv> {0 <.. k - 1}"
-abbreviation "Res_0 (k::int) \<equiv> {0::int}"
-abbreviation "Res_l (k::int) \<equiv> {0 <.. (k - 1) div 2}"
-abbreviation "Res_h (k::int) \<equiv> {(k - 1) div 2 <.. k - 1}"
-
-abbreviation "Sets_pq r0 r1 r2 \<equiv>
- {(x::int). x \<in> r0 (int p * int q) \<and> x mod p \<in> r1 (int p) \<and> x mod q \<in> r2 (int q)}"
-
-definition "A = Sets_pq Res_l Res_l Res_h"
-definition "B = Sets_pq Res_l Res_h Res_l"
-definition "C = Sets_pq Res_h Res_h Res_l"
-definition "D = Sets_pq Res_l Res_h Res_h"
-definition "E = Sets_pq Res_l Res_0 Res_h"
-definition "F = Sets_pq Res_l Res_h Res_0"
-
-definition "a = card A"
-definition "b = card B"
-definition "c = card C"
-definition "d = card D"
-definition "e = card E"
-definition "f = card F"
-
-lemma Gpq: "GAUSS p q" unfolding GAUSS_def
- using p_prime pq_neq p_ge_2 q_prime
- by (auto simp: cong_altdef_int zdvd_int [symmetric] dest: primes_dvd_imp_eq)
-
-lemma Gqp: "GAUSS q p" using QRqp QR.Gpq by simp
-
-lemma QR_lemma_01: "(\<lambda>x. x mod q) ` E = GAUSS.E q p"
-proof
- {
- fix x
- assume a1: "x \<in> E"
- then obtain k where k: "x = int p * k" unfolding E_def by blast
- have "x \<in> Res_l (int p * int q)" using a1 E_def by blast
- hence "k \<in> GAUSS.A q" using Gqp GAUSS.A_def k qp_ineq by (simp add: zero_less_mult_iff)
- hence "x mod q \<in> GAUSS.E q p"
- using GAUSS.C_def[of q p] Gqp k GAUSS.B_def[of q p] a1 GAUSS.E_def[of q p]
- unfolding E_def by force
- hence "x \<in> E \<longrightarrow> x mod int q \<in> GAUSS.E q p" by auto
- }
- thus "(\<lambda>x. x mod int q) ` E \<subseteq> GAUSS.E q p" by auto
-next
- show "GAUSS.E q p \<subseteq> (\<lambda>x. x mod q) ` E"
- proof
- fix x
- assume a1: "x \<in> GAUSS.E q p"
- then obtain ka where ka: "ka \<in> GAUSS.A q" "x = (ka * p) mod q"
- using Gqp GAUSS.B_def GAUSS.C_def GAUSS.E_def by auto
- hence "ka * p \<in> Res_l (int p * int q)"
- using GAUSS.A_def Gqp p_ge_0 qp_ineq by (simp add: Groups.mult_ac(2))
- thus "x \<in> (\<lambda>x. x mod q) ` E" unfolding E_def using ka a1 Gqp GAUSS.E_def q_ge_0 by force
- qed
-qed
-
-lemma QR_lemma_02: "e= n"
-proof -
- {
- fix x y
- assume a: "x \<in> E" "y \<in> E" "x mod q = y mod q"
- obtain p_inv where p_inv: "[int p * p_inv = 1] (mod int q)"
- using pq_coprime_int cong_solve_coprime_int by blast
- obtain kx ky where k: "x = int p * kx" "y = int p * ky" using a E_def dvd_def[of p x] by blast
- hence "0 < x" "int p * kx \<le> (int p * int q - 1) div 2"
- "0 < y" "int p * ky \<le> (int p * int q - 1) div 2"
- using E_def a greaterThanAtMost_iff mem_Collect_eq by blast+
- hence "0 \<le> kx" "kx < q" "0 \<le> ky" "ky < q" using qp_ineq k by (simp add: zero_less_mult_iff)+
- moreover have "(p_inv * (p * kx)) mod q = (p_inv * (p * ky)) mod q"
- using a(3) mod_mult_cong k by blast
- hence "(p * p_inv * kx) mod q = (p * p_inv * ky) mod q" by (simp add:algebra_simps)
- hence "kx mod q = ky mod q"
- using p_inv mod_mult_cong[of "p * p_inv" "q" "1"] cong_int_def by auto
- hence "[kx = ky] (mod q)" using cong_int_def by blast
- ultimately have "x = y" using cong_less_imp_eq_int k by blast
- }
- hence "inj_on (\<lambda>x. x mod q) E" unfolding inj_on_def by auto
- thus ?thesis using QR_lemma_01 card_image e_def n_def by fastforce
-qed
-
-lemma QR_lemma_03: "f = m"
-proof -
- have "F = QR.E q p" unfolding F_def pq_commute using QRqp QR.E_def[of q p] by fastforce
- hence "f = QR.e q p" unfolding f_def using QRqp QR.e_def[of q p] by presburger
- thus ?thesis using QRqp QR.QR_lemma_02 m_def QRqp QR.n_def by presburger
-qed
-
-definition f_1 :: "int \<Rightarrow> int \<times> int" where
- "f_1 x = ((x mod p), (x mod q))"
-
-definition P_1 :: "int \<times> int \<Rightarrow> int \<Rightarrow> bool" where
- "P_1 res x \<longleftrightarrow> x mod p = fst res & x mod q = snd res & x \<in> Res (int p * int q)"
-
-definition g_1 :: "int \<times> int \<Rightarrow> int" where
- "g_1 res = (THE x. P_1 res x)"
-
-lemma P_1_lemma: assumes "0 \<le> fst res" "fst res < p" "0 \<le> snd res" "snd res < q"
- shows "\<exists>! x. P_1 res x"
-proof -
- obtain y k1 k2 where yk: "y = nat (fst res) + k1 * p" "y = nat (snd res) + k2 * q"
- using chinese_remainder[of p q] pq_coprime p_ge_0 q_ge_0 by fastforce
- have h1: "[y = fst res] (mod p)" "[y = snd res] (mod q)"
- using yk(1) assms(1) cong_iff_lin_int[of "fst res"] cong_sym_int apply simp
- using yk(2) assms(3) cong_iff_lin_int[of "snd res"] cong_sym_int by simp
- have "(y mod (int p * int q)) mod int p = fst res" "(y mod (int p * int q)) mod int q = snd res"
- using h1(1) mod_mod_cancel[of "int p"] assms(1) assms(2) cong_int_def apply simp
- using h1(2) mod_mod_cancel[of "int q"] assms(3) assms(4) cong_int_def by simp
- then obtain x where "P_1 res x" unfolding P_1_def
- using Divides.pos_mod_bound Divides.pos_mod_sign pq_ge_0 by fastforce
- moreover {
- fix a b
- assume a: "P_1 res a" "P_1 res b"
- hence "int p * int q dvd a - b"
- using divides_mult[of "int p" "a - b" "int q"] pq_coprime_int zmod_eq_dvd_iff[of a _ b]
- unfolding P_1_def by force
- hence "a = b" using dvd_imp_le_int[of "a - b"] a unfolding P_1_def by fastforce
- }
- ultimately show ?thesis by auto
-qed
-
-lemma g_1_lemma: assumes "0 \<le> fst res" "fst res < p" "0 \<le> snd res" "snd res < q"
- shows "P_1 res (g_1 res)" using assms P_1_lemma theI'[of "P_1 res"] g_1_def by presburger
-
-definition "BuC = Sets_pq Res_ge_0 Res_h Res_l"
-
-lemma QR_lemma_04: "card BuC = card ((Res_h p) \<times> (Res_l q))"
- using card_bij_eq[of f_1 "BuC" "(Res_h p) \<times> (Res_l q)" g_1]
-proof
- {
- fix x y
- assume a: "x \<in> BuC" "y \<in> BuC" "f_1 x = f_1 y"
- hence "int p * int q dvd x - y"
- using f_1_def pq_coprime_int divides_mult[of "int p" "x - y" "int q"]
- zmod_eq_dvd_iff[of x _ y] by auto
- hence "x = y"
- using dvd_imp_le_int[of "x - y" "int p * int q"] a unfolding BuC_def by force
- }
- thus "inj_on f_1 BuC" unfolding inj_on_def by auto
-next
- {
- fix x y
- assume a: "x \<in> (Res_h p) \<times> (Res_l q)" "y \<in> (Res_h p) \<times> (Res_l q)" "g_1 x = g_1 y"
- hence "0 \<le> fst x" "fst x < p" "0 \<le> snd x" "snd x < q"
- "0 \<le> fst y" "fst y < p" "0 \<le> snd y" "snd y < q"
- using mem_Sigma_iff prod.collapse by fastforce+
- hence "x = y" using g_1_lemma[of x] g_1_lemma[of y] a P_1_def by fastforce
- }
- thus "inj_on g_1 ((Res_h p) \<times> (Res_l q))" unfolding inj_on_def by auto
-next
- show "g_1 ` ((Res_h p) \<times> (Res_l q)) \<subseteq> BuC"
- proof
- fix y
- assume "y \<in> g_1 ` ((Res_h p) \<times> (Res_l q))"
- then obtain x where x: "y = g_1 x" "x \<in> ((Res_h p) \<times> (Res_l q))" by blast
- hence "P_1 x y" using g_1_lemma by fastforce
- thus "y \<in> BuC" unfolding P_1_def BuC_def mem_Collect_eq using x SigmaE prod.sel by fastforce
- qed
-qed (auto simp: BuC_def finite_subset f_1_def)
-
-lemma QR_lemma_05: "card ((Res_h p) \<times> (Res_l q)) = r"
-proof -
- have "card (Res_l q) = (q - 1) div 2" "card (Res_h p) = (p - 1) div 2" using p_eq2 by force+
- thus ?thesis unfolding r_def using card_cartesian_product[of "Res_h p" "Res_l q"] by presburger
-qed
-
-lemma QR_lemma_06: "b + c = r"
-proof -
- have "B \<inter> C = {}" "finite B" "finite C" "B \<union> C = BuC" unfolding B_def C_def BuC_def by fastforce+
- thus ?thesis
- unfolding b_def c_def using card_empty card_Un_Int QR_lemma_04 QR_lemma_05 by fastforce
-qed
-
-definition f_2:: "int \<Rightarrow> int" where
- "f_2 x = (int p * int q) - x"
-
-lemma f_2_lemma_1: "\<And>x. f_2 (f_2 x) = x" unfolding f_2_def by simp
-
-lemma f_2_lemma_2: "[f_2 x = int p - x] (mod p)" unfolding f_2_def using cong_altdef_int by simp
-
-lemma f_2_lemma_3: "f_2 x \<in> S \<Longrightarrow> x \<in> f_2 ` S"
- using f_2_lemma_1[of x] image_eqI[of x f_2 "f_2 x" S] by presburger
-
-lemma QR_lemma_07: "f_2 ` Res_l (int p * int q) = Res_h (int p * int q)"
- "f_2 ` Res_h (int p * int q) = Res_l (int p * int q)"
-proof -
- have h1: "f_2 ` Res_l (int p * int q) \<subseteq> Res_h (int p * int q)" using f_2_def by force
- have h2: "f_2 ` Res_h (int p * int q) \<subseteq> Res_l (int p * int q)" using f_2_def pq_eq2 by fastforce
- have h3: "Res_h (int p * int q) \<subseteq> f_2 ` Res_l (int p * int q)" using h2 f_2_lemma_3 by blast
- have h4: "Res_l (int p * int q) \<subseteq> f_2 ` Res_h (int p * int q)" using h1 f_2_lemma_3 by blast
- show "f_2 ` Res_l (int p * int q) = Res_h (int p * int q)" using h1 h3 by blast
- show "f_2 ` Res_h (int p * int q) = Res_l (int p * int q)" using h2 h4 by blast
-qed
-
-lemma QR_lemma_08: "(f_2 x mod p \<in> Res_l p) = (x mod p \<in> Res_h p)"
- "(f_2 x mod p \<in> Res_h p) = (x mod p \<in> Res_l p)"
- using f_2_lemma_2[of x] cong_int_def[of "f_2 x" "p - x" p] minus_mod_self2[of x p]
- zmod_zminus1_eq_if[of x p] p_eq2 by auto
-
-lemma QR_lemma_09: "(f_2 x mod q \<in> Res_l q) = (x mod q \<in> Res_h q)"
- "(f_2 x mod q \<in> Res_h q) = (x mod q \<in> Res_l q)"
- using QRqp QR.QR_lemma_08 f_2_def QR.f_2_def pq_commute by auto+
-
-lemma QR_lemma_10: "a = c" unfolding a_def c_def apply (rule card_bij_eq[of f_2 A C f_2])
- unfolding A_def C_def
- using QR_lemma_07 QR_lemma_08 QR_lemma_09 apply ((simp add: inj_on_def f_2_def),blast)+
- by fastforce+
-
-definition "BuD = Sets_pq Res_l Res_h Res_ge_0"
-definition "BuDuF = Sets_pq Res_l Res_h Res"
-
-definition f_3 :: "int \<Rightarrow> int \<times> int" where
- "f_3 x = (x mod p, x div p + 1)"
-
-definition g_3 :: "int \<times> int \<Rightarrow> int" where
- "g_3 x = fst x + (snd x - 1) * p"
-
-lemma QR_lemma_11: "card BuDuF = card ((Res_h p) \<times> (Res_l q))"
- using card_bij_eq[of f_3 BuDuF "(Res_h p) \<times> (Res_l q)" g_3]
-proof
- show "f_3 ` BuDuF \<subseteq> (Res_h p) \<times> (Res_l q)"
- proof
- fix y
- assume "y \<in> f_3 ` BuDuF"
- then obtain x where x: "y = f_3 x" "x \<in> BuDuF" by blast
- hence "x \<le> int p * (int q - 1) div 2 + (int p - 1) div 2"
- unfolding BuDuF_def using p_eq2 int_distrib(4) by auto
- moreover have "(int p - 1) div 2 \<le> - 1 + x mod p" using x BuDuF_def by auto
- moreover have "int p * (int q - 1) div 2 = int p * ((int q - 1) div 2)"
- using zdiv_zmult1_eq odd_q by auto
- hence "p * (int q - 1) div 2 = p * ((int q + 1) div 2 - 1)" by fastforce
- ultimately have "x \<le> p * ((int q + 1) div 2 - 1) - 1 + x mod p" by linarith
- hence "x div p < (int q + 1) div 2 - 1"
- using mult.commute[of "int p" "x div p"] p_ge_0 div_mult_mod_eq[of x p]
- mult_less_cancel_left_pos[of p "x div p" "(int q + 1) div 2 - 1"] by linarith
- moreover have "0 < x div p + 1"
- using pos_imp_zdiv_neg_iff[of p x] p_ge_0 x mem_Collect_eq BuDuF_def by auto
- ultimately show "y \<in> (Res_h p) \<times> (Res_l q)" using x BuDuF_def f_3_def by auto
- qed
-next
- have h1: "\<And>x. x \<in> ((Res_h p) \<times> (Res_l q)) \<Longrightarrow> f_3 (g_3 x) = x"
- proof -
- fix x
- assume a: "x \<in> ((Res_h p) \<times> (Res_l q))"
- moreover have h: "(fst x + (snd x - 1) * int p) mod int p = fst x" using a by force
- ultimately have "(fst x + (snd x - 1) * int p) div int p + 1 = snd x"
- by (auto simp: semiring_numeral_div_class.div_less)
- with h show "f_3 (g_3 x) = x" unfolding f_3_def g_3_def by simp
- qed
- show "inj_on g_3 ((Res_h p) \<times> (Res_l q))" apply (rule inj_onI[of "(Res_h p) \<times> (Res_l q)" g_3])
- proof -
- fix x y
- assume "x \<in> ((Res_h p) \<times> (Res_l q))" "y \<in> ((Res_h p) \<times> (Res_l q))" "g_3 x = g_3 y"
- thus "x = y" using h1[of x] h1[of y] by presburger
- qed
-next
- show "g_3 ` ((Res_h p) \<times> (Res_l q)) \<subseteq> BuDuF"
- proof
- fix y
- assume "y \<in> g_3 ` ((Res_h p) \<times> (Res_l q))"
- then obtain x where x: "y = g_3 x" "x \<in> (Res_h p) \<times> (Res_l q)" by blast
- hence "snd x \<le> (int q - 1) div 2" by force
- moreover have "int p * ((int q - 1) div 2) = (int p * int q - int p) div 2"
- using int_distrib(4) zdiv_zmult1_eq[of "int p" "int q - 1" 2] odd_q by fastforce
- ultimately have "(snd x) * int p \<le> (int q * int p - int p) div 2"
- using mult_right_mono[of "snd x" "(int q - 1) div 2" p] mult.commute[of "(int q - 1) div 2" p]
- pq_commute by presburger
- hence "(snd x - 1) * int p \<le> (int q * int p - 1) div 2 - int p"
- using p_ge_0 int_distrib(3) by auto
- moreover have "fst x \<le> int p - 1" using x by force
- ultimately have "fst x + (snd x - 1) * int p \<le> (int p * int q - 1) div 2"
- using pq_commute by linarith
- moreover have "0 < fst x" "0 \<le> (snd x - 1) * p" using x(2) by fastforce+
- ultimately show "y \<in> BuDuF" unfolding BuDuF_def using q_ge_0 x g_3_def x(1) by auto
- qed
-next
- show "finite BuDuF" unfolding BuDuF_def by fastforce
-qed (simp add: inj_on_inverseI[of BuDuF g_3] f_3_def g_3_def QR_lemma_05)+
-
-lemma QR_lemma_12: "b + d + m = r"
-proof -
- have "B \<inter> D = {}" "finite B" "finite D" "B \<union> D = BuD" unfolding B_def D_def BuD_def by fastforce+
- hence "b + d = card BuD" unfolding b_def d_def using card_Un_Int by fastforce
- moreover have "BuD \<inter> F = {}" "finite BuD" "finite F" unfolding BuD_def F_def by fastforce+
- moreover have "BuD \<union> F = BuDuF" unfolding BuD_def F_def BuDuF_def
- using q_ge_0 ivl_disj_un_singleton(5)[of 0 "int q - 1"] by auto
- ultimately show ?thesis using QR_lemma_03 QR_lemma_05 QR_lemma_11 card_Un_disjoint[of BuD F]
- unfolding b_def d_def f_def by presburger
-qed
-
-lemma QR_lemma_13: "a + d + n = r"
-proof -
- have "A = QR.B q p" unfolding A_def pq_commute using QRqp QR.B_def[of q p] by blast
- hence "a = QR.b q p" using a_def QRqp QR.b_def[of q p] by presburger
- moreover have "D = QR.D q p" unfolding D_def pq_commute using QRqp QR.D_def[of q p] by blast
- hence "d = QR.d q p" using d_def QRqp QR.d_def[of q p] by presburger
- moreover have "n = QR.m q p" using n_def QRqp QR.m_def[of q p] by presburger
- moreover have "r = QR.r q p" unfolding r_def using QRqp QR.r_def[of q p] by auto
- ultimately show ?thesis using QRqp QR.QR_lemma_12 by presburger
-qed
-
-lemma QR_lemma_14: "(-1::int) ^ (m + n) = (-1) ^ r"
-proof -
- have "m + n + 2 * d = r" using QR_lemma_06 QR_lemma_10 QR_lemma_12 QR_lemma_13 by auto
- thus ?thesis using power_add[of "-1::int" "m + n" "2 * d"] by fastforce
-qed
-
-lemma Quadratic_Reciprocity:
- "(Legendre p q) * (Legendre q p) = (-1::int) ^ ((p - 1) div 2 * ((q - 1) div 2))"
- using Gpq Gqp GAUSS.gauss_lemma power_add[of "-1::int" m n] QR_lemma_14
- unfolding r_def m_def n_def by auto
-
-end
-
-theorem Quadratic_Reciprocity: assumes "prime p" "2 < p" "prime q" "2 < q" "p \<noteq> q"
- shows "(Legendre p q) * (Legendre q p) = (-1::int) ^ ((p - 1) div 2 * ((q - 1) div 2))"
- using QR.Quadratic_Reciprocity QR_def assms by blast
-
-theorem Quadratic_Reciprocity_int: assumes "prime (nat p)" "2 < p" "prime (nat q)" "2 < q" "p \<noteq> q"
- shows "(Legendre p q) * (Legendre q p) = (-1::int) ^ (nat ((p - 1) div 2 * ((q - 1) div 2)))"
-proof -
- have "0 \<le> (p - 1) div 2" using assms by simp
- moreover have "(nat p - 1) div 2 = nat ((p - 1) div 2)" "(nat q - 1) div 2 = nat ((q - 1) div 2)"
- by fastforce+
- ultimately have "(nat p - 1) div 2 * ((nat q - 1) div 2) = nat ((p - 1) div 2 * ((q - 1) div 2))"
- using nat_mult_distrib by presburger
- moreover have "2 < nat p" "2 < nat q" "nat p \<noteq> nat q" "int (nat p) = p" "int (nat q) = q"
- using assms by linarith+
- ultimately show ?thesis using Quadratic_Reciprocity[of "nat p" "nat q"] assms by presburger
-qed
-
-end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Number_Theory/Quadratic_Reciprocity.thy Thu Oct 20 19:39:27 2016 +0200
@@ -0,0 +1,387 @@
+(* Author: Jaime Mendizabal Roche *)
+
+theory Quadratic_Reciprocity
+imports Gauss
+begin
+
+text {* The proof is based on Gauss's fifth proof, which can be found at http://www.lehigh.edu/~shw2/q-recip/gauss5.pdf *}
+
+locale QR =
+ fixes p :: "nat"
+ fixes q :: "nat"
+
+ assumes p_prime: "prime p"
+ assumes p_ge_2: "2 < p"
+ assumes q_prime: "prime q"
+ assumes q_ge_2: "2 < q"
+ assumes pq_neq: "p \<noteq> q"
+begin
+
+lemma odd_p: "odd p" using p_ge_2 p_prime prime_odd_nat by blast
+
+lemma p_ge_0: "0 < int p"
+ using p_prime not_prime_0[where 'a = nat] by fastforce+
+
+lemma p_eq2: "int p = (2 * ((int p - 1) div 2)) + 1" using odd_p by simp
+
+lemma odd_q: "odd q" using q_ge_2 q_prime prime_odd_nat by blast
+
+lemma q_ge_0: "0 < int q" using q_prime not_prime_0[where 'a = nat] by fastforce+
+
+lemma q_eq2: "int q = (2 * ((int q - 1) div 2)) + 1" using odd_q by simp
+
+lemma pq_eq2: "int p * int q = (2 * ((int p * int q - 1) div 2)) + 1" using odd_p odd_q by simp
+
+lemma pq_coprime: "coprime p q"
+ using pq_neq p_prime primes_coprime_nat q_prime by blast
+
+lemma pq_coprime_int: "coprime (int p) (int q)"
+ using pq_coprime transfer_int_nat_gcd(1) by presburger
+
+lemma qp_ineq: "(int p * k \<le> (int p * int q - 1) div 2) = (k \<le> (int q - 1) div 2)"
+proof -
+ have "(2 * int p * k \<le> int p * int q - 1) = (2 * k \<le> int q - 1)" using p_ge_0 by auto
+ thus ?thesis by auto
+qed
+
+lemma QRqp: "QR q p" using QR_def QR_axioms by simp
+
+lemma pq_commute: "int p * int q = int q * int p" by simp
+
+lemma pq_ge_0: "int p * int q > 0" using p_ge_0 q_ge_0 mult_pos_pos by blast
+
+definition "r = ((p - 1) div 2)*((q - 1) div 2)"
+definition "m = card (GAUSS.E p q)"
+definition "n = card (GAUSS.E q p)"
+
+abbreviation "Res (k::int) \<equiv> {0 .. k - 1}"
+abbreviation "Res_ge_0 (k::int) \<equiv> {0 <.. k - 1}"
+abbreviation "Res_0 (k::int) \<equiv> {0::int}"
+abbreviation "Res_l (k::int) \<equiv> {0 <.. (k - 1) div 2}"
+abbreviation "Res_h (k::int) \<equiv> {(k - 1) div 2 <.. k - 1}"
+
+abbreviation "Sets_pq r0 r1 r2 \<equiv>
+ {(x::int). x \<in> r0 (int p * int q) \<and> x mod p \<in> r1 (int p) \<and> x mod q \<in> r2 (int q)}"
+
+definition "A = Sets_pq Res_l Res_l Res_h"
+definition "B = Sets_pq Res_l Res_h Res_l"
+definition "C = Sets_pq Res_h Res_h Res_l"
+definition "D = Sets_pq Res_l Res_h Res_h"
+definition "E = Sets_pq Res_l Res_0 Res_h"
+definition "F = Sets_pq Res_l Res_h Res_0"
+
+definition "a = card A"
+definition "b = card B"
+definition "c = card C"
+definition "d = card D"
+definition "e = card E"
+definition "f = card F"
+
+lemma Gpq: "GAUSS p q" unfolding GAUSS_def
+ using p_prime pq_neq p_ge_2 q_prime
+ by (auto simp: cong_altdef_int zdvd_int [symmetric] dest: primes_dvd_imp_eq)
+
+lemma Gqp: "GAUSS q p" using QRqp QR.Gpq by simp
+
+lemma QR_lemma_01: "(\<lambda>x. x mod q) ` E = GAUSS.E q p"
+proof
+ {
+ fix x
+ assume a1: "x \<in> E"
+ then obtain k where k: "x = int p * k" unfolding E_def by blast
+ have "x \<in> Res_l (int p * int q)" using a1 E_def by blast
+ hence "k \<in> GAUSS.A q" using Gqp GAUSS.A_def k qp_ineq by (simp add: zero_less_mult_iff)
+ hence "x mod q \<in> GAUSS.E q p"
+ using GAUSS.C_def[of q p] Gqp k GAUSS.B_def[of q p] a1 GAUSS.E_def[of q p]
+ unfolding E_def by force
+ hence "x \<in> E \<longrightarrow> x mod int q \<in> GAUSS.E q p" by auto
+ }
+ thus "(\<lambda>x. x mod int q) ` E \<subseteq> GAUSS.E q p" by auto
+next
+ show "GAUSS.E q p \<subseteq> (\<lambda>x. x mod q) ` E"
+ proof
+ fix x
+ assume a1: "x \<in> GAUSS.E q p"
+ then obtain ka where ka: "ka \<in> GAUSS.A q" "x = (ka * p) mod q"
+ using Gqp GAUSS.B_def GAUSS.C_def GAUSS.E_def by auto
+ hence "ka * p \<in> Res_l (int p * int q)"
+ using GAUSS.A_def Gqp p_ge_0 qp_ineq by (simp add: Groups.mult_ac(2))
+ thus "x \<in> (\<lambda>x. x mod q) ` E" unfolding E_def using ka a1 Gqp GAUSS.E_def q_ge_0 by force
+ qed
+qed
+
+lemma QR_lemma_02: "e= n"
+proof -
+ {
+ fix x y
+ assume a: "x \<in> E" "y \<in> E" "x mod q = y mod q"
+ obtain p_inv where p_inv: "[int p * p_inv = 1] (mod int q)"
+ using pq_coprime_int cong_solve_coprime_int by blast
+ obtain kx ky where k: "x = int p * kx" "y = int p * ky" using a E_def dvd_def[of p x] by blast
+ hence "0 < x" "int p * kx \<le> (int p * int q - 1) div 2"
+ "0 < y" "int p * ky \<le> (int p * int q - 1) div 2"
+ using E_def a greaterThanAtMost_iff mem_Collect_eq by blast+
+ hence "0 \<le> kx" "kx < q" "0 \<le> ky" "ky < q" using qp_ineq k by (simp add: zero_less_mult_iff)+
+ moreover have "(p_inv * (p * kx)) mod q = (p_inv * (p * ky)) mod q"
+ using a(3) mod_mult_cong k by blast
+ hence "(p * p_inv * kx) mod q = (p * p_inv * ky) mod q" by (simp add:algebra_simps)
+ hence "kx mod q = ky mod q"
+ using p_inv mod_mult_cong[of "p * p_inv" "q" "1"] cong_int_def by auto
+ hence "[kx = ky] (mod q)" using cong_int_def by blast
+ ultimately have "x = y" using cong_less_imp_eq_int k by blast
+ }
+ hence "inj_on (\<lambda>x. x mod q) E" unfolding inj_on_def by auto
+ thus ?thesis using QR_lemma_01 card_image e_def n_def by fastforce
+qed
+
+lemma QR_lemma_03: "f = m"
+proof -
+ have "F = QR.E q p" unfolding F_def pq_commute using QRqp QR.E_def[of q p] by fastforce
+ hence "f = QR.e q p" unfolding f_def using QRqp QR.e_def[of q p] by presburger
+ thus ?thesis using QRqp QR.QR_lemma_02 m_def QRqp QR.n_def by presburger
+qed
+
+definition f_1 :: "int \<Rightarrow> int \<times> int" where
+ "f_1 x = ((x mod p), (x mod q))"
+
+definition P_1 :: "int \<times> int \<Rightarrow> int \<Rightarrow> bool" where
+ "P_1 res x \<longleftrightarrow> x mod p = fst res & x mod q = snd res & x \<in> Res (int p * int q)"
+
+definition g_1 :: "int \<times> int \<Rightarrow> int" where
+ "g_1 res = (THE x. P_1 res x)"
+
+lemma P_1_lemma: assumes "0 \<le> fst res" "fst res < p" "0 \<le> snd res" "snd res < q"
+ shows "\<exists>! x. P_1 res x"
+proof -
+ obtain y k1 k2 where yk: "y = nat (fst res) + k1 * p" "y = nat (snd res) + k2 * q"
+ using chinese_remainder[of p q] pq_coprime p_ge_0 q_ge_0 by fastforce
+ have h1: "[y = fst res] (mod p)" "[y = snd res] (mod q)"
+ using yk(1) assms(1) cong_iff_lin_int[of "fst res"] cong_sym_int apply simp
+ using yk(2) assms(3) cong_iff_lin_int[of "snd res"] cong_sym_int by simp
+ have "(y mod (int p * int q)) mod int p = fst res" "(y mod (int p * int q)) mod int q = snd res"
+ using h1(1) mod_mod_cancel[of "int p"] assms(1) assms(2) cong_int_def apply simp
+ using h1(2) mod_mod_cancel[of "int q"] assms(3) assms(4) cong_int_def by simp
+ then obtain x where "P_1 res x" unfolding P_1_def
+ using Divides.pos_mod_bound Divides.pos_mod_sign pq_ge_0 by fastforce
+ moreover {
+ fix a b
+ assume a: "P_1 res a" "P_1 res b"
+ hence "int p * int q dvd a - b"
+ using divides_mult[of "int p" "a - b" "int q"] pq_coprime_int zmod_eq_dvd_iff[of a _ b]
+ unfolding P_1_def by force
+ hence "a = b" using dvd_imp_le_int[of "a - b"] a unfolding P_1_def by fastforce
+ }
+ ultimately show ?thesis by auto
+qed
+
+lemma g_1_lemma: assumes "0 \<le> fst res" "fst res < p" "0 \<le> snd res" "snd res < q"
+ shows "P_1 res (g_1 res)" using assms P_1_lemma theI'[of "P_1 res"] g_1_def by presburger
+
+definition "BuC = Sets_pq Res_ge_0 Res_h Res_l"
+
+lemma QR_lemma_04: "card BuC = card ((Res_h p) \<times> (Res_l q))"
+ using card_bij_eq[of f_1 "BuC" "(Res_h p) \<times> (Res_l q)" g_1]
+proof
+ {
+ fix x y
+ assume a: "x \<in> BuC" "y \<in> BuC" "f_1 x = f_1 y"
+ hence "int p * int q dvd x - y"
+ using f_1_def pq_coprime_int divides_mult[of "int p" "x - y" "int q"]
+ zmod_eq_dvd_iff[of x _ y] by auto
+ hence "x = y"
+ using dvd_imp_le_int[of "x - y" "int p * int q"] a unfolding BuC_def by force
+ }
+ thus "inj_on f_1 BuC" unfolding inj_on_def by auto
+next
+ {
+ fix x y
+ assume a: "x \<in> (Res_h p) \<times> (Res_l q)" "y \<in> (Res_h p) \<times> (Res_l q)" "g_1 x = g_1 y"
+ hence "0 \<le> fst x" "fst x < p" "0 \<le> snd x" "snd x < q"
+ "0 \<le> fst y" "fst y < p" "0 \<le> snd y" "snd y < q"
+ using mem_Sigma_iff prod.collapse by fastforce+
+ hence "x = y" using g_1_lemma[of x] g_1_lemma[of y] a P_1_def by fastforce
+ }
+ thus "inj_on g_1 ((Res_h p) \<times> (Res_l q))" unfolding inj_on_def by auto
+next
+ show "g_1 ` ((Res_h p) \<times> (Res_l q)) \<subseteq> BuC"
+ proof
+ fix y
+ assume "y \<in> g_1 ` ((Res_h p) \<times> (Res_l q))"
+ then obtain x where x: "y = g_1 x" "x \<in> ((Res_h p) \<times> (Res_l q))" by blast
+ hence "P_1 x y" using g_1_lemma by fastforce
+ thus "y \<in> BuC" unfolding P_1_def BuC_def mem_Collect_eq using x SigmaE prod.sel by fastforce
+ qed
+qed (auto simp: BuC_def finite_subset f_1_def)
+
+lemma QR_lemma_05: "card ((Res_h p) \<times> (Res_l q)) = r"
+proof -
+ have "card (Res_l q) = (q - 1) div 2" "card (Res_h p) = (p - 1) div 2" using p_eq2 by force+
+ thus ?thesis unfolding r_def using card_cartesian_product[of "Res_h p" "Res_l q"] by presburger
+qed
+
+lemma QR_lemma_06: "b + c = r"
+proof -
+ have "B \<inter> C = {}" "finite B" "finite C" "B \<union> C = BuC" unfolding B_def C_def BuC_def by fastforce+
+ thus ?thesis
+ unfolding b_def c_def using card_empty card_Un_Int QR_lemma_04 QR_lemma_05 by fastforce
+qed
+
+definition f_2:: "int \<Rightarrow> int" where
+ "f_2 x = (int p * int q) - x"
+
+lemma f_2_lemma_1: "\<And>x. f_2 (f_2 x) = x" unfolding f_2_def by simp
+
+lemma f_2_lemma_2: "[f_2 x = int p - x] (mod p)" unfolding f_2_def using cong_altdef_int by simp
+
+lemma f_2_lemma_3: "f_2 x \<in> S \<Longrightarrow> x \<in> f_2 ` S"
+ using f_2_lemma_1[of x] image_eqI[of x f_2 "f_2 x" S] by presburger
+
+lemma QR_lemma_07: "f_2 ` Res_l (int p * int q) = Res_h (int p * int q)"
+ "f_2 ` Res_h (int p * int q) = Res_l (int p * int q)"
+proof -
+ have h1: "f_2 ` Res_l (int p * int q) \<subseteq> Res_h (int p * int q)" using f_2_def by force
+ have h2: "f_2 ` Res_h (int p * int q) \<subseteq> Res_l (int p * int q)" using f_2_def pq_eq2 by fastforce
+ have h3: "Res_h (int p * int q) \<subseteq> f_2 ` Res_l (int p * int q)" using h2 f_2_lemma_3 by blast
+ have h4: "Res_l (int p * int q) \<subseteq> f_2 ` Res_h (int p * int q)" using h1 f_2_lemma_3 by blast
+ show "f_2 ` Res_l (int p * int q) = Res_h (int p * int q)" using h1 h3 by blast
+ show "f_2 ` Res_h (int p * int q) = Res_l (int p * int q)" using h2 h4 by blast
+qed
+
+lemma QR_lemma_08: "(f_2 x mod p \<in> Res_l p) = (x mod p \<in> Res_h p)"
+ "(f_2 x mod p \<in> Res_h p) = (x mod p \<in> Res_l p)"
+ using f_2_lemma_2[of x] cong_int_def[of "f_2 x" "p - x" p] minus_mod_self2[of x p]
+ zmod_zminus1_eq_if[of x p] p_eq2 by auto
+
+lemma QR_lemma_09: "(f_2 x mod q \<in> Res_l q) = (x mod q \<in> Res_h q)"
+ "(f_2 x mod q \<in> Res_h q) = (x mod q \<in> Res_l q)"
+ using QRqp QR.QR_lemma_08 f_2_def QR.f_2_def pq_commute by auto+
+
+lemma QR_lemma_10: "a = c" unfolding a_def c_def apply (rule card_bij_eq[of f_2 A C f_2])
+ unfolding A_def C_def
+ using QR_lemma_07 QR_lemma_08 QR_lemma_09 apply ((simp add: inj_on_def f_2_def),blast)+
+ by fastforce+
+
+definition "BuD = Sets_pq Res_l Res_h Res_ge_0"
+definition "BuDuF = Sets_pq Res_l Res_h Res"
+
+definition f_3 :: "int \<Rightarrow> int \<times> int" where
+ "f_3 x = (x mod p, x div p + 1)"
+
+definition g_3 :: "int \<times> int \<Rightarrow> int" where
+ "g_3 x = fst x + (snd x - 1) * p"
+
+lemma QR_lemma_11: "card BuDuF = card ((Res_h p) \<times> (Res_l q))"
+ using card_bij_eq[of f_3 BuDuF "(Res_h p) \<times> (Res_l q)" g_3]
+proof
+ show "f_3 ` BuDuF \<subseteq> (Res_h p) \<times> (Res_l q)"
+ proof
+ fix y
+ assume "y \<in> f_3 ` BuDuF"
+ then obtain x where x: "y = f_3 x" "x \<in> BuDuF" by blast
+ hence "x \<le> int p * (int q - 1) div 2 + (int p - 1) div 2"
+ unfolding BuDuF_def using p_eq2 int_distrib(4) by auto
+ moreover have "(int p - 1) div 2 \<le> - 1 + x mod p" using x BuDuF_def by auto
+ moreover have "int p * (int q - 1) div 2 = int p * ((int q - 1) div 2)"
+ using zdiv_zmult1_eq odd_q by auto
+ hence "p * (int q - 1) div 2 = p * ((int q + 1) div 2 - 1)" by fastforce
+ ultimately have "x \<le> p * ((int q + 1) div 2 - 1) - 1 + x mod p" by linarith
+ hence "x div p < (int q + 1) div 2 - 1"
+ using mult.commute[of "int p" "x div p"] p_ge_0 div_mult_mod_eq[of x p]
+ mult_less_cancel_left_pos[of p "x div p" "(int q + 1) div 2 - 1"] by linarith
+ moreover have "0 < x div p + 1"
+ using pos_imp_zdiv_neg_iff[of p x] p_ge_0 x mem_Collect_eq BuDuF_def by auto
+ ultimately show "y \<in> (Res_h p) \<times> (Res_l q)" using x BuDuF_def f_3_def by auto
+ qed
+next
+ have h1: "\<And>x. x \<in> ((Res_h p) \<times> (Res_l q)) \<Longrightarrow> f_3 (g_3 x) = x"
+ proof -
+ fix x
+ assume a: "x \<in> ((Res_h p) \<times> (Res_l q))"
+ moreover have h: "(fst x + (snd x - 1) * int p) mod int p = fst x" using a by force
+ ultimately have "(fst x + (snd x - 1) * int p) div int p + 1 = snd x"
+ by (auto simp: semiring_numeral_div_class.div_less)
+ with h show "f_3 (g_3 x) = x" unfolding f_3_def g_3_def by simp
+ qed
+ show "inj_on g_3 ((Res_h p) \<times> (Res_l q))" apply (rule inj_onI[of "(Res_h p) \<times> (Res_l q)" g_3])
+ proof -
+ fix x y
+ assume "x \<in> ((Res_h p) \<times> (Res_l q))" "y \<in> ((Res_h p) \<times> (Res_l q))" "g_3 x = g_3 y"
+ thus "x = y" using h1[of x] h1[of y] by presburger
+ qed
+next
+ show "g_3 ` ((Res_h p) \<times> (Res_l q)) \<subseteq> BuDuF"
+ proof
+ fix y
+ assume "y \<in> g_3 ` ((Res_h p) \<times> (Res_l q))"
+ then obtain x where x: "y = g_3 x" "x \<in> (Res_h p) \<times> (Res_l q)" by blast
+ hence "snd x \<le> (int q - 1) div 2" by force
+ moreover have "int p * ((int q - 1) div 2) = (int p * int q - int p) div 2"
+ using int_distrib(4) zdiv_zmult1_eq[of "int p" "int q - 1" 2] odd_q by fastforce
+ ultimately have "(snd x) * int p \<le> (int q * int p - int p) div 2"
+ using mult_right_mono[of "snd x" "(int q - 1) div 2" p] mult.commute[of "(int q - 1) div 2" p]
+ pq_commute by presburger
+ hence "(snd x - 1) * int p \<le> (int q * int p - 1) div 2 - int p"
+ using p_ge_0 int_distrib(3) by auto
+ moreover have "fst x \<le> int p - 1" using x by force
+ ultimately have "fst x + (snd x - 1) * int p \<le> (int p * int q - 1) div 2"
+ using pq_commute by linarith
+ moreover have "0 < fst x" "0 \<le> (snd x - 1) * p" using x(2) by fastforce+
+ ultimately show "y \<in> BuDuF" unfolding BuDuF_def using q_ge_0 x g_3_def x(1) by auto
+ qed
+next
+ show "finite BuDuF" unfolding BuDuF_def by fastforce
+qed (simp add: inj_on_inverseI[of BuDuF g_3] f_3_def g_3_def QR_lemma_05)+
+
+lemma QR_lemma_12: "b + d + m = r"
+proof -
+ have "B \<inter> D = {}" "finite B" "finite D" "B \<union> D = BuD" unfolding B_def D_def BuD_def by fastforce+
+ hence "b + d = card BuD" unfolding b_def d_def using card_Un_Int by fastforce
+ moreover have "BuD \<inter> F = {}" "finite BuD" "finite F" unfolding BuD_def F_def by fastforce+
+ moreover have "BuD \<union> F = BuDuF" unfolding BuD_def F_def BuDuF_def
+ using q_ge_0 ivl_disj_un_singleton(5)[of 0 "int q - 1"] by auto
+ ultimately show ?thesis using QR_lemma_03 QR_lemma_05 QR_lemma_11 card_Un_disjoint[of BuD F]
+ unfolding b_def d_def f_def by presburger
+qed
+
+lemma QR_lemma_13: "a + d + n = r"
+proof -
+ have "A = QR.B q p" unfolding A_def pq_commute using QRqp QR.B_def[of q p] by blast
+ hence "a = QR.b q p" using a_def QRqp QR.b_def[of q p] by presburger
+ moreover have "D = QR.D q p" unfolding D_def pq_commute using QRqp QR.D_def[of q p] by blast
+ hence "d = QR.d q p" using d_def QRqp QR.d_def[of q p] by presburger
+ moreover have "n = QR.m q p" using n_def QRqp QR.m_def[of q p] by presburger
+ moreover have "r = QR.r q p" unfolding r_def using QRqp QR.r_def[of q p] by auto
+ ultimately show ?thesis using QRqp QR.QR_lemma_12 by presburger
+qed
+
+lemma QR_lemma_14: "(-1::int) ^ (m + n) = (-1) ^ r"
+proof -
+ have "m + n + 2 * d = r" using QR_lemma_06 QR_lemma_10 QR_lemma_12 QR_lemma_13 by auto
+ thus ?thesis using power_add[of "-1::int" "m + n" "2 * d"] by fastforce
+qed
+
+lemma Quadratic_Reciprocity:
+ "(Legendre p q) * (Legendre q p) = (-1::int) ^ ((p - 1) div 2 * ((q - 1) div 2))"
+ using Gpq Gqp GAUSS.gauss_lemma power_add[of "-1::int" m n] QR_lemma_14
+ unfolding r_def m_def n_def by auto
+
+end
+
+theorem Quadratic_Reciprocity: assumes "prime p" "2 < p" "prime q" "2 < q" "p \<noteq> q"
+ shows "(Legendre p q) * (Legendre q p) = (-1::int) ^ ((p - 1) div 2 * ((q - 1) div 2))"
+ using QR.Quadratic_Reciprocity QR_def assms by blast
+
+theorem Quadratic_Reciprocity_int: assumes "prime (nat p)" "2 < p" "prime (nat q)" "2 < q" "p \<noteq> q"
+ shows "(Legendre p q) * (Legendre q p) = (-1::int) ^ (nat ((p - 1) div 2 * ((q - 1) div 2)))"
+proof -
+ have "0 \<le> (p - 1) div 2" using assms by simp
+ moreover have "(nat p - 1) div 2 = nat ((p - 1) div 2)" "(nat q - 1) div 2 = nat ((q - 1) div 2)"
+ by fastforce+
+ ultimately have "(nat p - 1) div 2 * ((nat q - 1) div 2) = nat ((p - 1) div 2 * ((q - 1) div 2))"
+ using nat_mult_distrib by presburger
+ moreover have "2 < nat p" "2 < nat q" "nat p \<noteq> nat q" "int (nat p) = p" "int (nat q) = q"
+ using assms by linarith+
+ ultimately show ?thesis using Quadratic_Reciprocity[of "nat p" "nat q"] assms by presburger
+qed
+
+end
--- a/src/HOL/Number_Theory/document/root.tex Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Number_Theory/document/root.tex Thu Oct 20 19:39:27 2016 +0200
@@ -1,6 +1,8 @@
\documentclass[11pt,a4paper]{article}
\usepackage{graphicx}
\usepackage{isabelle,isabellesym}
+\usepackage{amssymb}
+\usepackage{amsmath}
\usepackage{pdfsetup}
\urlstyle{rm}
--- a/src/HOL/Orderings.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Orderings.thy Thu Oct 20 19:39:27 2016 +0200
@@ -1437,6 +1437,17 @@
apply (erule Least_le)
done
+lemma exists_least_iff: "(\<exists>n. P n) \<longleftrightarrow> (\<exists>n. P n \<and> (\<forall>m < n. \<not> P m))" (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+ assume ?rhs thus ?lhs by blast
+next
+ assume H: ?lhs then obtain n where n: "P n" by blast
+ let ?x = "Least P"
+ { fix m assume m: "m < ?x"
+ from not_less_Least[OF m] have "\<not> P m" . }
+ with LeastI_ex[OF H] show ?rhs by blast
+qed
+
end
--- a/src/HOL/Probability/Distribution_Functions.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Probability/Distribution_Functions.thy Thu Oct 20 19:39:27 2016 +0200
@@ -36,11 +36,11 @@
by (simp add: cdf_def)
locale finite_borel_measure = finite_measure M for M :: "real measure" +
- assumes M_super_borel: "sets borel \<subseteq> sets M"
+ assumes M_is_borel: "sets M = sets borel"
begin
lemma sets_M[intro]: "a \<in> sets borel \<Longrightarrow> a \<in> sets M"
- using M_super_borel by auto
+ using M_is_borel by auto
lemma cdf_diff_eq:
assumes "x < y"
@@ -57,7 +57,7 @@
unfolding cdf_def by (auto intro!: finite_measure_mono)
lemma borel_UNIV: "space M = UNIV"
- by (metis in_mono sets.sets_into_space space_in_borel top_le M_super_borel)
+ by (metis in_mono sets.sets_into_space space_in_borel top_le M_is_borel)
lemma cdf_nonneg: "cdf M x \<ge> 0"
unfolding cdf_def by (rule measure_nonneg)
@@ -142,11 +142,17 @@
end
locale real_distribution = prob_space M for M :: "real measure" +
- assumes events_eq_borel [simp, measurable_cong]: "sets M = sets borel" and space_eq_univ [simp]: "space M = UNIV"
+ assumes events_eq_borel [simp, measurable_cong]: "sets M = sets borel"
begin
+lemma finite_borel_measure_M: "finite_borel_measure M"
+ by standard auto
+
sublocale finite_borel_measure M
- by standard auto
+ by (rule finite_borel_measure_M)
+
+lemma space_eq_univ [simp]: "space M = UNIV"
+ using events_eq_borel[THEN sets_eq_imp_space_eq] by simp
lemma cdf_bounded_prob: "\<And>x. cdf M x \<le> 1"
by (subst prob_space [symmetric], rule cdf_bounded)
@@ -167,20 +173,23 @@
"random_variable borel X \<Longrightarrow> real_distribution (distr M borel X)"
unfolding real_distribution_def real_distribution_axioms_def by (auto intro!: prob_space_distr)
-subsection \<open>uniqueness\<close>
+subsection \<open>Uniqueness\<close>
-lemma (in real_distribution) emeasure_Ioc:
+lemma (in finite_borel_measure) emeasure_Ioc:
assumes "a \<le> b" shows "emeasure M {a <.. b} = cdf M b - cdf M a"
proof -
have "{a <.. b} = {..b} - {..a}"
by auto
- with \<open>a \<le> b\<close> show ?thesis
+ moreover have "{..x} \<in> sets M" for x
+ using atMost_borel[of x] M_is_borel by auto
+ moreover note \<open>a \<le> b\<close>
+ ultimately show ?thesis
by (simp add: emeasure_eq_measure finite_measure_Diff cdf_def)
qed
-lemma cdf_unique:
+lemma cdf_unique':
fixes M1 M2
- assumes "real_distribution M1" and "real_distribution M2"
+ assumes "finite_borel_measure M1" and "finite_borel_measure M2"
assumes "cdf M1 = cdf M2"
shows "M1 = M2"
proof (rule measure_eqI_generator_eq[where \<Omega>=UNIV])
@@ -188,14 +197,56 @@
then obtain a b where Xeq: "X = {a<..b}" by auto
then show "emeasure M1 X = emeasure M2 X"
by (cases "a \<le> b")
- (simp_all add: assms(1,2)[THEN real_distribution.emeasure_Ioc] assms(3))
+ (simp_all add: assms(1,2)[THEN finite_borel_measure.emeasure_Ioc] assms(3))
next
show "(\<Union>i. {- real (i::nat)<..real i}) = UNIV"
by (rule UN_Ioc_eq_UNIV)
-qed (auto simp: real_distribution.emeasure_Ioc[OF assms(1)]
- assms(1,2)[THEN real_distribution.events_eq_borel] borel_sigma_sets_Ioc
+qed (auto simp: finite_borel_measure.emeasure_Ioc[OF assms(1)]
+ assms(1,2)[THEN finite_borel_measure.M_is_borel] borel_sigma_sets_Ioc
Int_stable_def)
+lemma cdf_unique:
+ "real_distribution M1 \<Longrightarrow> real_distribution M2 \<Longrightarrow> cdf M1 = cdf M2 \<Longrightarrow> M1 = M2"
+ using cdf_unique'[of M1 M2] by (simp add: real_distribution.finite_borel_measure_M)
+
+lemma
+ fixes F :: "real \<Rightarrow> real"
+ assumes nondecF : "\<And> x y. x \<le> y \<Longrightarrow> F x \<le> F y"
+ and right_cont_F : "\<And>a. continuous (at_right a) F"
+ and lim_F_at_bot : "(F \<longlongrightarrow> 0) at_bot"
+ and lim_F_at_top : "(F \<longlongrightarrow> m) at_top"
+ and m: "0 \<le> m"
+ shows interval_measure_UNIV: "emeasure (interval_measure F) UNIV = m"
+ and finite_borel_measure_interval_measure: "finite_borel_measure (interval_measure F)"
+proof -
+ let ?F = "interval_measure F"
+ { have "ennreal (m - 0) = (SUP i::nat. ennreal (F (real i) - F (- real i)))"
+ by (intro LIMSEQ_unique[OF _ LIMSEQ_SUP] tendsto_ennrealI tendsto_intros
+ lim_F_at_bot[THEN filterlim_compose] lim_F_at_top[THEN filterlim_compose]
+ lim_F_at_bot[THEN filterlim_compose] filterlim_real_sequentially
+ filterlim_uminus_at_top[THEN iffD1])
+ (auto simp: incseq_def nondecF intro!: diff_mono)
+ also have "\<dots> = (SUP i::nat. emeasure ?F {- real i<..real i})"
+ by (subst emeasure_interval_measure_Ioc) (simp_all add: nondecF right_cont_F)
+ also have "\<dots> = emeasure ?F (\<Union>i::nat. {- real i<..real i})"
+ by (rule SUP_emeasure_incseq) (auto simp: incseq_def)
+ also have "(\<Union>i. {- real (i::nat)<..real i}) = space ?F"
+ by (simp add: UN_Ioc_eq_UNIV)
+ finally have "emeasure ?F (space ?F) = m"
+ by simp }
+ note * = this
+ then show "emeasure (interval_measure F) UNIV = m"
+ by simp
+
+ interpret finite_measure ?F
+ proof
+ show "emeasure ?F (space ?F) \<noteq> \<infinity>"
+ using * by simp
+ qed
+ show "finite_borel_measure (interval_measure F)"
+ proof qed simp_all
+qed
+
lemma real_distribution_interval_measure:
fixes F :: "real \<Rightarrow> real"
assumes nondecF : "\<And> x y. x \<le> y \<Longrightarrow> F x \<le> F y" and
@@ -206,53 +257,47 @@
proof -
let ?F = "interval_measure F"
interpret prob_space ?F
- proof
- have "ennreal (1 - 0) = (SUP i::nat. ennreal (F (real i) - F (- real i)))"
- by (intro LIMSEQ_unique[OF _ LIMSEQ_SUP] tendsto_ennrealI tendsto_intros
- lim_F_at_bot[THEN filterlim_compose] lim_F_at_top[THEN filterlim_compose]
- lim_F_at_bot[THEN filterlim_compose] filterlim_real_sequentially
- filterlim_uminus_at_top[THEN iffD1])
- (auto simp: incseq_def nondecF intro!: diff_mono)
- also have "\<dots> = (SUP i::nat. emeasure ?F {- real i<..real i})"
- by (subst emeasure_interval_measure_Ioc) (simp_all add: nondecF right_cont_F)
- also have "\<dots> = emeasure ?F (\<Union>i::nat. {- real i<..real i})"
- by (rule SUP_emeasure_incseq) (auto simp: incseq_def)
- also have "(\<Union>i. {- real (i::nat)<..real i}) = space ?F"
- by (simp add: UN_Ioc_eq_UNIV)
- finally show "emeasure ?F (space ?F) = 1"
- by (simp add: one_ereal_def)
- qed
+ proof qed (use interval_measure_UNIV[OF assms] in simp)
show ?thesis
proof qed simp_all
qed
-lemma cdf_interval_measure:
+lemma
fixes F :: "real \<Rightarrow> real"
assumes nondecF : "\<And> x y. x \<le> y \<Longrightarrow> F x \<le> F y" and
right_cont_F : "\<And>a. continuous (at_right a) F" and
- lim_F_at_bot : "(F \<longlongrightarrow> 0) at_bot" and
- lim_F_at_top : "(F \<longlongrightarrow> 1) at_top"
- shows "cdf (interval_measure F) = F"
+ lim_F_at_bot : "(F \<longlongrightarrow> 0) at_bot"
+ shows emeasure_interval_measure_Iic: "emeasure (interval_measure F) {.. x} = F x"
+ and measure_interval_measure_Iic: "measure (interval_measure F) {.. x} = F x"
unfolding cdf_def
-proof (intro ext)
- interpret real_distribution "interval_measure F"
- by (rule real_distribution_interval_measure) fact+
- fix x
- have "F x - 0 = measure (interval_measure F) (\<Union>i::nat. {-real i <.. x})"
- proof (intro LIMSEQ_unique[OF _ finite_Lim_measure_incseq])
+proof -
+ have F_nonneg[simp]: "0 \<le> F y" for y
+ using lim_F_at_bot by (rule tendsto_upperbound) (auto simp: eventually_at_bot_linorder nondecF intro!: exI[of _ y])
+
+ have "emeasure (interval_measure F) (\<Union>i::nat. {-real i <.. x}) = F x - ennreal 0"
+ proof (intro LIMSEQ_unique[OF Lim_emeasure_incseq])
have "(\<lambda>i. F x - F (- real i)) \<longlonglongrightarrow> F x - 0"
by (intro tendsto_intros lim_F_at_bot[THEN filterlim_compose] filterlim_real_sequentially
filterlim_uminus_at_top[THEN iffD1])
- then show "(\<lambda>i. measure (interval_measure F) {- real i<..x}) \<longlonglongrightarrow> F x - 0"
- apply (rule filterlim_cong[OF refl refl, THEN iffD1, rotated])
+ from tendsto_ennrealI[OF this]
+ show "(\<lambda>i. emeasure (interval_measure F) {- real i<..x}) \<longlonglongrightarrow> F x - ennreal 0"
+ apply (rule filterlim_cong[THEN iffD1, rotated 3])
+ apply simp
+ apply simp
apply (rule eventually_sequentiallyI[where c="nat (ceiling (- x))"])
- apply (simp add: measure_interval_measure_Ioc right_cont_F nondecF)
+ apply (simp add: emeasure_interval_measure_Ioc right_cont_F nondecF)
done
qed (auto simp: incseq_def)
also have "(\<Union>i::nat. {-real i <.. x}) = {..x}"
by auto (metis minus_minus neg_less_iff_less reals_Archimedean2)
- finally show "measure (interval_measure F) {..x} = F x"
+ finally show "emeasure (interval_measure F) {..x} = F x"
by simp
+ then show "measure (interval_measure F) {..x} = F x"
+ by (simp add: measure_def)
qed
+lemma cdf_interval_measure:
+ "(\<And> x y. x \<le> y \<Longrightarrow> F x \<le> F y) \<Longrightarrow> (\<And>a. continuous (at_right a) F) \<Longrightarrow> (F \<longlongrightarrow> 0) at_bot \<Longrightarrow> cdf (interval_measure F) = F"
+ by (simp add: cdf_def fun_eq_iff measure_interval_measure_Iic)
+
end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Probability/Essential_Supremum.thy Thu Oct 20 19:39:27 2016 +0200
@@ -0,0 +1,199 @@
+(* Author: Sébastien Gouëzel sebastien.gouezel@univ-rennes1.fr
+ License: BSD
+*)
+
+theory Essential_Supremum
+imports "../Analysis/Analysis"
+begin
+
+section {*The essential supremum*}
+
+text {*In this paragraph, we define the essential supremum and give its basic properties. The
+essential supremum of a function is its maximum value if one is allowed to throw away a set
+of measure $0$. It is convenient to define it to be infinity for non-measurable functions, as
+it allows for neater statements in general. This is a prerequisiste to define the space $L^\infty$.*}
+
+definition esssup::"'a measure \<Rightarrow> ('a \<Rightarrow> 'b::{second_countable_topology, dense_linorder, linorder_topology, complete_linorder}) \<Rightarrow> 'b"
+ where "esssup M f = (if f \<in> borel_measurable M then Inf {z. emeasure M {x \<in> space M. f x > z} = 0} else top)"
+
+lemma esssup_zero_measure:
+ "emeasure M {x \<in> space M. f x > esssup M f} = 0"
+proof (cases "esssup M f = top")
+ case True
+ then show ?thesis by auto
+next
+ case False
+ then have [measurable]: "f \<in> borel_measurable M" unfolding esssup_def by meson
+ have "esssup M f < top" using False by (auto simp: less_top)
+ have *: "{x \<in> space M. f x > z} \<in> null_sets M" if "z > esssup M f" for z
+ proof -
+ have "\<exists>w. w < z \<and> emeasure M {x \<in> space M. f x > w} = 0"
+ using `z > esssup M f` unfolding esssup_def apply auto
+ by (metis (mono_tags, lifting) Inf_less_iff mem_Collect_eq)
+ then obtain w where "w < z" "emeasure M {x \<in> space M. f x > w} = 0" by auto
+ then have a: "{x \<in> space M. f x > w} \<in> null_sets M" by auto
+ have b: "{x \<in> space M. f x > z} \<subseteq> {x \<in> space M. f x > w}" using `w < z` by auto
+ show ?thesis using null_sets_subset[OF a _ b] by simp
+ qed
+ obtain u::"nat \<Rightarrow> 'b" where u: "\<And>n. u n > esssup M f" "u \<longlonglongrightarrow> esssup M f"
+ using approx_from_above_dense_linorder[OF `esssup M f < top`] by auto
+ have "{x \<in> space M. f x > esssup M f} = (\<Union>n. {x \<in> space M. f x > u n})"
+ using u apply auto
+ apply (metis (mono_tags, lifting) order_tendsto_iff eventually_mono LIMSEQ_unique)
+ using less_imp_le less_le_trans by blast
+ also have "... \<in> null_sets M"
+ using *[OF u(1)] by auto
+ finally show ?thesis by auto
+qed
+
+lemma esssup_AE:
+ "AE x in M. f x \<le> esssup M f"
+proof (cases "f \<in> borel_measurable M")
+ case True
+ show ?thesis
+ apply (rule AE_I[OF _ esssup_zero_measure[of _ f]]) using True by auto
+next
+ case False
+ then have "esssup M f = top" unfolding esssup_def by auto
+ then show ?thesis by auto
+qed
+
+lemma esssup_pos_measure:
+ assumes "f \<in> borel_measurable M" "z < esssup M f"
+ shows "emeasure M {x \<in> space M. f x > z} > 0"
+using assms Inf_less_iff mem_Collect_eq not_gr_zero unfolding esssup_def by force
+
+lemma esssup_non_measurable:
+ assumes "f \<notin> borel_measurable M"
+ shows "esssup M f = top"
+using assms unfolding esssup_def by auto
+
+lemma esssup_I [intro]:
+ assumes "f \<in> borel_measurable M" "AE x in M. f x \<le> c"
+ shows "esssup M f \<le> c"
+proof -
+ have "emeasure M {x \<in> space M. \<not> f x \<le> c} = 0"
+ apply (rule AE_E2[OF assms(2)]) using assms(1) by simp
+ then have *: "emeasure M {x \<in> space M. f x > c} = 0"
+ by (metis (mono_tags, lifting) Collect_cong not_less)
+ show ?thesis unfolding esssup_def using assms apply simp by (rule Inf_lower, simp add: *)
+qed
+
+lemma esssup_AE_mono:
+ assumes "f \<in> borel_measurable M" "AE x in M. f x \<le> g x"
+ shows "esssup M f \<le> esssup M g"
+proof (cases "g \<in> borel_measurable M")
+ case False
+ then show ?thesis unfolding esssup_def by auto
+next
+ case True
+ have "AE x in M. f x \<le> esssup M g"
+ using assms(2) esssup_AE[of g M] by auto
+ then show ?thesis using esssup_I assms(1) by auto
+qed
+
+lemma esssup_mono:
+ assumes "f \<in> borel_measurable M" "\<And>x. f x \<le> g x"
+ shows "esssup M f \<le> esssup M g"
+apply (rule esssup_AE_mono) using assms by auto
+
+lemma esssup_AE_cong:
+ assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
+ and "AE x in M. f x = g x"
+ shows "esssup M f = esssup M g"
+proof -
+ have "esssup M f \<le> esssup M g"
+ using esssup_AE_mono[OF assms(1), of g] assms(3) by (simp add: eq_iff)
+ moreover have "esssup M g \<le> esssup M f"
+ using esssup_AE_mono[OF assms(2), of f] assms(3) by (simp add: eq_iff)
+ ultimately show ?thesis by simp
+qed
+
+lemma esssup_const:
+ assumes "emeasure M (space M) \<noteq> 0"
+ shows "esssup M (\<lambda>x. c) = c"
+proof -
+ have "emeasure M {x \<in> space M. (\<lambda>x. c) x > z} = (if c > z then emeasure M (space M) else 0)" for z
+ by auto
+ then have "{z. emeasure M {x \<in> space M. (\<lambda>x. c) x > z} = 0} = {c..}" using assms by auto
+ then have "esssup M (\<lambda>x. c) = Inf {c..}" unfolding esssup_def by auto
+ then show ?thesis by auto
+qed
+
+lemma esssup_cmult:
+ assumes "c > (0::real)"
+ shows "esssup M (\<lambda>x. c * f x::ereal) = c * esssup M f"
+proof (cases "f \<in> borel_measurable M")
+ case True
+ then have a [measurable]: "f \<in> borel_measurable M" by simp
+ then have b [measurable]: "(\<lambda>x. c * f x) \<in> borel_measurable M" by simp
+ have a: "{x \<in> space M. c * z < c * f x} = {x \<in> space M. z < f x}" for z::ereal
+ by (meson assms ereal_less(2) ereal_mult_left_mono ereal_mult_strict_left_mono less_ereal.simps(4) less_imp_le not_less)
+ have *: "{z::ereal. emeasure M {x \<in> space M. ereal c * f x > z} = 0} = {c * z| z::ereal. emeasure M {x \<in> space M. f x > z} = 0}"
+ proof (auto)
+ fix y assume *: "emeasure M {x \<in> space M. y < c * f x} = 0"
+ define z where "z = y / c"
+ have **: "y = c * z" unfolding z_def using assms by (simp add: ereal_mult_divide)
+ then have "y = c * z \<and> emeasure M {x \<in> space M. z < f x} = 0"
+ using * unfolding ** unfolding a by auto
+ then show "\<exists>z. y = ereal c * z \<and> emeasure M {x \<in> space M. z < f x} = 0"
+ by auto
+ next
+ fix z assume *: "emeasure M {x \<in> space M. z < f x} = 0"
+ then show "emeasure M {x \<in> space M. c * z < c * f x} = 0"
+ using a by auto
+ qed
+ have "esssup M (\<lambda>x. c * f x) = Inf {z::ereal. emeasure M {x \<in> space M. c * f x > z} = 0}"
+ unfolding esssup_def using b by auto
+ also have "... = Inf {c * z| z::ereal. emeasure M {x \<in> space M. f x > z} = 0}"
+ using * by auto
+ also have "... = ereal c * Inf {z. emeasure M {x \<in> space M. f x > z} = 0}"
+ apply (rule ereal_Inf_cmult) using assms by auto
+ also have "... = c * esssup M f"
+ unfolding esssup_def by auto
+ finally show ?thesis by simp
+next
+ case False
+ have "esssup M f = top" using False unfolding esssup_def by auto
+ then have *: "c * esssup M f = \<infinity>" using assms by (simp add: ennreal_mult_eq_top_iff top_ereal_def)
+ have "(\<lambda>x. c * f x) \<notin> borel_measurable M"
+ proof (rule ccontr)
+ assume "\<not> (\<lambda>x. c * f x) \<notin> borel_measurable M"
+ then have [measurable]: "(\<lambda>x. c * f x) \<in> borel_measurable M" by simp
+ then have "(\<lambda>x. (1/c) * (c * f x)) \<in> borel_measurable M" by measurable
+ moreover have "(1/c) * (c * f x) = f x" for x
+ by (metis "*" PInfty_neq_ereal(1) divide_inverse divide_self_if ereal_zero_mult mult.assoc mult.commute mult.left_neutral one_ereal_def times_ereal.simps(1) zero_ereal_def)
+ ultimately show False using False by auto
+ qed
+ then have "esssup M (\<lambda>x. c * f x) = \<infinity>" unfolding esssup_def by (simp add: top_ereal_def)
+ then show ?thesis using * by auto
+qed
+
+lemma esssup_add:
+ "esssup M (\<lambda>x. f x + g x::ereal) \<le> esssup M f + esssup M g"
+proof (cases "f \<in> borel_measurable M \<and> g \<in> borel_measurable M")
+ case True
+ then have [measurable]: "(\<lambda>x. f x + g x) \<in> borel_measurable M" by auto
+ have "f x + g x \<le> esssup M f + esssup M g" if "f x \<le> esssup M f" "g x \<le> esssup M g" for x
+ using that ereal_add_mono by auto
+ then have "AE x in M. f x + g x \<le> esssup M f + esssup M g"
+ using esssup_AE[of f M] esssup_AE[of g M] by auto
+ then show ?thesis using esssup_I by auto
+next
+ case False
+ then have "esssup M f + esssup M g = \<infinity>" unfolding esssup_def top_ereal_def by auto
+ then show ?thesis by auto
+qed
+
+lemma esssup_zero_space:
+ assumes "emeasure M (space M) = 0"
+ "f \<in> borel_measurable M"
+ shows "esssup M f = (- \<infinity>::ereal)"
+proof -
+ have "emeasure M {x \<in> space M. f x > - \<infinity>} = 0"
+ using assms(1) emeasure_mono emeasure_eq_0 by fastforce
+ then show ?thesis unfolding esssup_def using assms(2) Inf_eq_MInfty by auto
+qed
+
+end
+
--- a/src/HOL/Probability/Giry_Monad.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Probability/Giry_Monad.thy Thu Oct 20 19:39:27 2016 +0200
@@ -1778,4 +1778,10 @@
shows "space (M \<bind> f) = space B"
using M by (intro space_bind[OF sets_kernel[OF f]]) auto
+lemma bind_distr_return:
+ "f \<in> M \<rightarrow>\<^sub>M N \<Longrightarrow> g \<in> N \<rightarrow>\<^sub>M L \<Longrightarrow> space M \<noteq> {} \<Longrightarrow>
+ distr M N f \<bind> (\<lambda>x. return L (g x)) = distr M L (\<lambda>x. g (f x))"
+ by (subst bind_distr[OF _ measurable_compose[OF _ return_measurable]])
+ (auto intro!: bind_return_distr')
+
end
--- a/src/HOL/Probability/Probability.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Probability/Probability.thy Thu Oct 20 19:39:27 2016 +0200
@@ -12,6 +12,8 @@
SPMF
Stream_Space
Conditional_Expectation
+ Essential_Supremum
+ Stopping_Time
begin
end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Probability/Stopping_Time.thy Thu Oct 20 19:39:27 2016 +0200
@@ -0,0 +1,262 @@
+(* Author: Johannes Hölzl <hoelzl@in.tum.de> *)
+
+section {* Stopping times *}
+
+theory Stopping_Time
+ imports "../Analysis/Analysis"
+begin
+
+subsection \<open>Stopping Time\<close>
+
+text \<open>This is also called strong stopping time. Then stopping time is T with alternative is
+ \<open>T x < t\<close> measurable.\<close>
+
+definition stopping_time :: "('t::linorder \<Rightarrow> 'a measure) \<Rightarrow> ('a \<Rightarrow> 't) \<Rightarrow> bool"
+where
+ "stopping_time F T = (\<forall>t. Measurable.pred (F t) (\<lambda>x. T x \<le> t))"
+
+lemma stopping_time_cong: "(\<And>t x. x \<in> space (F t) \<Longrightarrow> T x = S x) \<Longrightarrow> stopping_time F T = stopping_time F S"
+ unfolding stopping_time_def by (intro arg_cong[where f=All] ext measurable_cong) simp
+
+lemma stopping_timeD: "stopping_time F T \<Longrightarrow> Measurable.pred (F t) (\<lambda>x. T x \<le> t)"
+ by (auto simp: stopping_time_def)
+
+lemma stopping_timeD2: "stopping_time F T \<Longrightarrow> Measurable.pred (F t) (\<lambda>x. t < T x)"
+ unfolding not_le[symmetric] by (auto intro: stopping_timeD Measurable.pred_intros_logic)
+
+lemma stopping_timeI[intro?]: "(\<And>t. Measurable.pred (F t) (\<lambda>x. T x \<le> t)) \<Longrightarrow> stopping_time F T"
+ by (auto simp: stopping_time_def)
+
+lemma measurable_stopping_time:
+ fixes T :: "'a \<Rightarrow> 't::{linorder_topology, second_countable_topology}"
+ assumes T: "stopping_time F T"
+ and M: "\<And>t. sets (F t) \<subseteq> sets M" "\<And>t. space (F t) = space M"
+ shows "T \<in> M \<rightarrow>\<^sub>M borel"
+proof (rule borel_measurableI_le)
+ show "{x \<in> space M. T x \<le> t} \<in> sets M" for t
+ using stopping_timeD[OF T] M by (auto simp: Measurable.pred_def)
+qed
+
+lemma stopping_time_const: "stopping_time F (\<lambda>x. c)"
+ by (auto simp: stopping_time_def)
+
+lemma stopping_time_min:
+ "stopping_time F T \<Longrightarrow> stopping_time F S \<Longrightarrow> stopping_time F (\<lambda>x. min (T x) (S x))"
+ by (auto simp: stopping_time_def min_le_iff_disj intro!: pred_intros_logic)
+
+lemma stopping_time_max:
+ "stopping_time F T \<Longrightarrow> stopping_time F S \<Longrightarrow> stopping_time F (\<lambda>x. max (T x) (S x))"
+ by (auto simp: stopping_time_def intro!: pred_intros_logic)
+
+section \<open>Filtration\<close>
+
+locale filtration =
+ fixes \<Omega> :: "'a set" and F :: "'t::{linorder_topology, second_countable_topology} \<Rightarrow> 'a measure"
+ assumes space_F: "\<And>i. space (F i) = \<Omega>"
+ assumes sets_F_mono: "\<And>i j. i \<le> j \<Longrightarrow> sets (F i) \<le> sets (F j)"
+begin
+
+subsection \<open>$\sigma$-algebra of a Stopping Time\<close>
+
+definition pre_sigma :: "('a \<Rightarrow> 't) \<Rightarrow> 'a measure"
+where
+ "pre_sigma T = sigma \<Omega> {A. \<forall>t. {\<omega>\<in>A. T \<omega> \<le> t} \<in> sets (F t)}"
+
+lemma space_pre_sigma: "space (pre_sigma T) = \<Omega>"
+ unfolding pre_sigma_def using sets.space_closed[of "F _"]
+ by (intro space_measure_of) (auto simp: space_F)
+
+lemma measure_pre_sigma[simp]: "emeasure (pre_sigma T) = (\<lambda>_. 0)"
+ by (simp add: pre_sigma_def emeasure_sigma)
+
+lemma sigma_algebra_pre_sigma:
+ assumes T: "stopping_time F T"
+ shows "sigma_algebra \<Omega> {A. \<forall>t. {\<omega>\<in>A. T \<omega> \<le> t} \<in> sets (F t)}"
+ unfolding sigma_algebra_iff2
+proof (intro sigma_algebra_iff2[THEN iffD2] conjI ballI allI impI CollectI)
+ show "{A. \<forall>t. {\<omega> \<in> A. T \<omega> \<le> t} \<in> sets (F t)} \<subseteq> Pow \<Omega>"
+ using sets.space_closed[of "F _"] by (auto simp: space_F)
+next
+ fix A t assume "A \<in> {A. \<forall>t. {\<omega> \<in> A. T \<omega> \<le> t} \<in> sets (F t)}"
+ then have "{\<omega> \<in> space (F t). T \<omega> \<le> t} - {\<omega> \<in> A. T \<omega> \<le> t} \<in> sets (F t)"
+ using T stopping_timeD[measurable] by auto
+ also have "{\<omega> \<in> space (F t). T \<omega> \<le> t} - {\<omega> \<in> A. T \<omega> \<le> t} = {\<omega> \<in> \<Omega> - A. T \<omega> \<le> t}"
+ by (auto simp: space_F)
+ finally show "{\<omega> \<in> \<Omega> - A. T \<omega> \<le> t} \<in> sets (F t)" .
+next
+ fix AA :: "nat \<Rightarrow> 'a set" and t assume "range AA \<subseteq> {A. \<forall>t. {\<omega> \<in> A. T \<omega> \<le> t} \<in> sets (F t)}"
+ then have "(\<Union>i. {\<omega> \<in> AA i. T \<omega> \<le> t}) \<in> sets (F t)" for t
+ by auto
+ also have "(\<Union>i. {\<omega> \<in> AA i. T \<omega> \<le> t}) = {\<omega> \<in> UNION UNIV AA. T \<omega> \<le> t}"
+ by auto
+ finally show "{\<omega> \<in> UNION UNIV AA. T \<omega> \<le> t} \<in> sets (F t)" .
+qed auto
+
+lemma sets_pre_sigma: "stopping_time F T \<Longrightarrow> sets (pre_sigma T) = {A. \<forall>t. {\<omega>\<in>A. T \<omega> \<le> t} \<in> sets (F t)}"
+ unfolding pre_sigma_def by (rule sigma_algebra.sets_measure_of_eq[OF sigma_algebra_pre_sigma])
+
+lemma sets_pre_sigmaI: "stopping_time F T \<Longrightarrow> (\<And>t. {\<omega>\<in>A. T \<omega> \<le> t} \<in> sets (F t)) \<Longrightarrow> A \<in> sets (pre_sigma T)"
+ unfolding sets_pre_sigma by auto
+
+lemma pred_pre_sigmaI:
+ assumes T: "stopping_time F T"
+ shows "(\<And>t. Measurable.pred (F t) (\<lambda>\<omega>. P \<omega> \<and> T \<omega> \<le> t)) \<Longrightarrow> Measurable.pred (pre_sigma T) P"
+ unfolding pred_def space_F space_pre_sigma by (intro sets_pre_sigmaI[OF T]) simp
+
+lemma sets_pre_sigmaD: "stopping_time F T \<Longrightarrow> A \<in> sets (pre_sigma T) \<Longrightarrow> {\<omega>\<in>A. T \<omega> \<le> t} \<in> sets (F t)"
+ unfolding sets_pre_sigma by auto
+
+lemma stopping_time_le_const: "stopping_time F T \<Longrightarrow> s \<le> t \<Longrightarrow> Measurable.pred (F t) (\<lambda>\<omega>. T \<omega> \<le> s)"
+ using stopping_timeD[of F T] sets_F_mono[of _ t] by (auto simp: pred_def space_F)
+
+lemma measurable_stopping_time_pre_sigma:
+ assumes T: "stopping_time F T" shows "T \<in> pre_sigma T \<rightarrow>\<^sub>M borel"
+proof (intro borel_measurableI_le sets_pre_sigmaI[OF T])
+ fix t t'
+ have "{\<omega>\<in>space (F (min t' t)). T \<omega> \<le> min t' t} \<in> sets (F (min t' t))"
+ using T unfolding pred_def[symmetric] by (rule stopping_timeD)
+ also have "\<dots> \<subseteq> sets (F t)"
+ by (rule sets_F_mono) simp
+ finally show "{\<omega> \<in> {x \<in> space (pre_sigma T). T x \<le> t'}. T \<omega> \<le> t} \<in> sets (F t)"
+ by (simp add: space_pre_sigma space_F)
+qed
+
+lemma mono_pre_sigma:
+ assumes T: "stopping_time F T" and S: "stopping_time F S"
+ and le: "\<And>\<omega>. \<omega> \<in> \<Omega> \<Longrightarrow> T \<omega> \<le> S \<omega>"
+ shows "sets (pre_sigma T) \<subseteq> sets (pre_sigma S)"
+ unfolding sets_pre_sigma[OF S] sets_pre_sigma[OF T]
+proof safe
+ interpret sigma_algebra \<Omega> "{A. \<forall>t. {\<omega>\<in>A. T \<omega> \<le> t} \<in> sets (F t)}"
+ using T by (rule sigma_algebra_pre_sigma)
+ fix A t assume A: "\<forall>t. {\<omega>\<in>A. T \<omega> \<le> t} \<in> sets (F t)"
+ then have "A \<subseteq> \<Omega>"
+ using sets_into_space by auto
+ from A have "{\<omega>\<in>A. T \<omega> \<le> t} \<inter> {\<omega>\<in>space (F t). S \<omega> \<le> t} \<in> sets (F t)"
+ using stopping_timeD[OF S] by (auto simp: pred_def)
+ also have "{\<omega>\<in>A. T \<omega> \<le> t} \<inter> {\<omega>\<in>space (F t). S \<omega> \<le> t} = {\<omega>\<in>A. S \<omega> \<le> t}"
+ using \<open>A \<subseteq> \<Omega>\<close> sets_into_space[of A] le by (auto simp: space_F intro: order_trans)
+ finally show "{\<omega>\<in>A. S \<omega> \<le> t} \<in> sets (F t)"
+ by auto
+qed
+
+lemma stopping_time_less_const:
+ assumes T: "stopping_time F T" shows "Measurable.pred (F t) (\<lambda>\<omega>. T \<omega> < t)"
+proof -
+ guess D :: "'t set" by (rule countable_dense_setE)
+ note D = this
+ show ?thesis
+ proof cases
+ assume *: "\<forall>t'<t. \<exists>t''. t' < t'' \<and> t'' < t"
+ { fix t' assume "t' < t"
+ with * have "{t' <..< t} \<noteq> {}"
+ by fastforce
+ with D(2)[OF _ this]
+ have "\<exists>d\<in>D. t'< d \<and> d < t"
+ by auto }
+ note ** = this
+
+ show ?thesis
+ proof (rule measurable_cong[THEN iffD2])
+ show "T \<omega> < t \<longleftrightarrow> (\<exists>r\<in>{r\<in>D. r < t}. T \<omega> \<le> r)" for \<omega>
+ by (auto dest: ** intro: less_imp_le)
+ show "Measurable.pred (F t) (\<lambda>w. \<exists>r\<in>{r \<in> D. r < t}. T w \<le> r)"
+ by (intro measurable_pred_countable stopping_time_le_const[OF T] countable_Collect D) auto
+ qed
+ next
+ assume "\<not> (\<forall>t'<t. \<exists>t''. t' < t'' \<and> t'' < t)"
+ then obtain t' where t': "t' < t" "\<And>t''. t'' < t \<Longrightarrow> t'' \<le> t'"
+ by (auto simp: not_less[symmetric])
+ show ?thesis
+ proof (rule measurable_cong[THEN iffD2])
+ show "T \<omega> < t \<longleftrightarrow> T \<omega> \<le> t'" for \<omega>
+ using t' by auto
+ show "Measurable.pred (F t) (\<lambda>w. T w \<le> t')"
+ using \<open>t'<t\<close> by (intro stopping_time_le_const[OF T]) auto
+ qed
+ qed
+qed
+
+lemma stopping_time_eq_const: "stopping_time F T \<Longrightarrow> Measurable.pred (F t) (\<lambda>\<omega>. T \<omega> = t)"
+ unfolding eq_iff using stopping_time_less_const[of T t]
+ by (intro pred_intros_logic stopping_time_le_const) (auto simp: not_less[symmetric] )
+
+lemma stopping_time_less:
+ assumes T: "stopping_time F T" and S: "stopping_time F S"
+ shows "Measurable.pred (pre_sigma T) (\<lambda>\<omega>. T \<omega> < S \<omega>)"
+proof (rule pred_pre_sigmaI[OF T])
+ fix t
+ obtain D :: "'t set"
+ where [simp]: "countable D" and semidense_D: "\<And>x y. x < y \<Longrightarrow> (\<exists>b\<in>D. x \<le> b \<and> b < y)"
+ using countable_separating_set_linorder2 by auto
+ show "Measurable.pred (F t) (\<lambda>\<omega>. T \<omega> < S \<omega> \<and> T \<omega> \<le> t)"
+ proof (rule measurable_cong[THEN iffD2])
+ let ?f = "\<lambda>\<omega>. if T \<omega> = t then \<not> S \<omega> \<le> t else \<exists>s\<in>{s\<in>D. s \<le> t}. T \<omega> \<le> s \<and> \<not> (S \<omega> \<le> s)"
+ { fix \<omega> assume "T \<omega> \<le> t" "T \<omega> \<noteq> t" "T \<omega> < S \<omega>"
+ then have "T \<omega> < min t (S \<omega>)"
+ by auto
+ then obtain r where "r \<in> D" "T \<omega> \<le> r" "r < min t (S \<omega>)"
+ by (metis semidense_D)
+ then have "\<exists>s\<in>{s\<in>D. s \<le> t}. T \<omega> \<le> s \<and> s < S \<omega>"
+ by auto }
+ then show "(T \<omega> < S \<omega> \<and> T \<omega> \<le> t) = ?f \<omega>" for \<omega>
+ by (auto simp: not_le)
+ show "Measurable.pred (F t) ?f"
+ by (intro pred_intros_logic measurable_If measurable_pred_countable countable_Collect
+ stopping_time_le_const predE stopping_time_eq_const T S)
+ auto
+ qed
+qed
+
+end
+
+lemma stopping_time_SUP_enat:
+ fixes T :: "nat \<Rightarrow> ('a \<Rightarrow> enat)"
+ shows "(\<And>i. stopping_time F (T i)) \<Longrightarrow> stopping_time F (SUP i. T i)"
+ unfolding stopping_time_def SUP_apply SUP_le_iff by (auto intro!: pred_intros_countable)
+
+lemma less_eSuc_iff: "a < eSuc b \<longleftrightarrow> (a \<le> b \<and> a \<noteq> \<infinity>)"
+ by (cases a) auto
+
+lemma stopping_time_Inf_enat:
+ fixes F :: "enat \<Rightarrow> 'a measure"
+ assumes F: "filtration \<Omega> F"
+ assumes P: "\<And>i. Measurable.pred (F i) (P i)"
+ shows "stopping_time F (\<lambda>\<omega>. Inf {i. P i \<omega>})"
+proof (rule stopping_timeI, cases)
+ fix t :: enat assume "t = \<infinity>" then show "Measurable.pred (F t) (\<lambda>\<omega>. Inf {i. P i \<omega>} \<le> t)"
+ by auto
+next
+ fix t :: enat assume "t \<noteq> \<infinity>"
+ moreover
+ { fix i \<omega> assume "Inf {i. P i \<omega>} \<le> t"
+ with \<open>t \<noteq> \<infinity>\<close> have "(\<exists>i\<le>t. P i \<omega>)"
+ unfolding Inf_le_iff by (cases t) (auto elim!: allE[of _ "eSuc t"] simp: less_eSuc_iff) }
+ ultimately have *: "\<And>\<omega>. Inf {i. P i \<omega>} \<le> t \<longleftrightarrow> (\<exists>i\<in>{..t}. P i \<omega>)"
+ by (auto intro!: Inf_lower2)
+ show "Measurable.pred (F t) (\<lambda>\<omega>. Inf {i. P i \<omega>} \<le> t)"
+ unfolding * using filtration.sets_F_mono[OF F, of _ t] P
+ by (intro pred_intros_countable_bounded) (auto simp: pred_def filtration.space_F[OF F])
+qed
+
+lemma stopping_time_Inf_nat:
+ fixes F :: "nat \<Rightarrow> 'a measure"
+ assumes F: "filtration \<Omega> F"
+ assumes P: "\<And>i. Measurable.pred (F i) (P i)" and wf: "\<And>i \<omega>. \<omega> \<in> \<Omega> \<Longrightarrow> \<exists>n. P n \<omega>"
+ shows "stopping_time F (\<lambda>\<omega>. Inf {i. P i \<omega>})"
+ unfolding stopping_time_def
+proof (intro allI, subst measurable_cong)
+ fix t \<omega> assume "\<omega> \<in> space (F t)"
+ then have "\<omega> \<in> \<Omega>"
+ using filtration.space_F[OF F] by auto
+ from wf[OF this] have "((LEAST n. P n \<omega>) \<le> t) = (\<exists>i\<le>t. P i \<omega>)"
+ by (rule LeastI2_wellorder_ex) auto
+ then show "(Inf {i. P i \<omega>} \<le> t) = (\<exists>i\<in>{..t}. P i \<omega>)"
+ by (simp add: Inf_nat_def Bex_def)
+next
+ fix t from P show "Measurable.pred (F t) (\<lambda>w. \<exists>i\<in>{..t}. P i w)"
+ using filtration.sets_F_mono[OF F, of _ t]
+ by (intro pred_intros_countable_bounded) (auto simp: pred_def filtration.space_F[OF F])
+qed
+
+end
--- a/src/HOL/Probability/Stream_Space.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Probability/Stream_Space.thy Thu Oct 20 19:39:27 2016 +0200
@@ -446,6 +446,17 @@
by (cases "xs = []") (auto simp: * space_stream_space del: in_listsD)
qed (auto simp: * ae sets_M del: in_listsD intro!: streams_sets)
+lemma sets_sstart[measurable]: "sstart \<Omega> xs \<in> sets (stream_space (count_space UNIV))"
+proof (induction xs)
+ case (Cons x xs)
+ note this[measurable]
+ have "sstart \<Omega> (x # xs) = {\<omega>\<in>space (stream_space (count_space UNIV)). \<omega> \<in> sstart \<Omega> (x # xs)}"
+ by (auto simp: space_stream_space)
+ also have "\<dots> \<in> sets (stream_space (count_space UNIV))"
+ unfolding in_sstart by measurable
+ finally show ?case .
+qed (auto intro!: streams_sets)
+
primrec scylinder :: "'a set \<Rightarrow> 'a set list \<Rightarrow> 'a stream set"
where
"scylinder S [] = streams S"
--- a/src/HOL/Quotient_Examples/Quotient_Rat.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Quotient_Examples/Quotient_Rat.thy Thu Oct 20 19:39:27 2016 +0200
@@ -28,7 +28,7 @@
quotient_type rat = "int \<times> int" / partial: ratrel
using ratrel_equivp .
-instantiation rat :: "{zero, one, plus, uminus, minus, times, ord, abs_if, sgn_if}"
+instantiation rat :: "{zero, one, plus, uminus, minus, times, ord, abs, sgn}"
begin
quotient_definition
@@ -100,8 +100,7 @@
definition
sgn_rat_def: "sgn (i::rat) = (if i = 0 then 0 else if 0 < i then 1 else - 1)"
-instance by intro_classes
- (auto simp add: rabs_rat_def sgn_rat_def)
+instance ..
end
--- a/src/HOL/ROOT Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/ROOT Thu Oct 20 19:39:27 2016 +0200
@@ -181,7 +181,7 @@
theories [document = false]
"Less_False"
"~~/src/HOL/Library/Multiset"
- "~~/src/HOL/Library/Float"
+ "~~/src/HOL/Number_Theory/Fib"
theories
Balance
Tree_Map
--- a/src/HOL/Rings.thy Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Rings.thy Thu Oct 20 19:39:27 2016 +0200
@@ -532,6 +532,100 @@
end
+class idom_abs_sgn = idom + abs + sgn +
+ assumes sgn_mult_abs: "sgn a * \<bar>a\<bar> = a"
+ and sgn_sgn [simp]: "sgn (sgn a) = sgn a"
+ and abs_abs [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
+ and abs_0 [simp]: "\<bar>0\<bar> = 0"
+ and sgn_0 [simp]: "sgn 0 = 0"
+ and sgn_1 [simp]: "sgn 1 = 1"
+ and sgn_minus_1: "sgn (- 1) = - 1"
+ and sgn_mult: "sgn (a * b) = sgn a * sgn b"
+begin
+
+lemma sgn_eq_0_iff:
+ "sgn a = 0 \<longleftrightarrow> a = 0"
+proof -
+ { assume "sgn a = 0"
+ then have "sgn a * \<bar>a\<bar> = 0"
+ by simp
+ then have "a = 0"
+ by (simp add: sgn_mult_abs)
+ } then show ?thesis
+ by auto
+qed
+
+lemma abs_eq_0_iff:
+ "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0"
+proof -
+ { assume "\<bar>a\<bar> = 0"
+ then have "sgn a * \<bar>a\<bar> = 0"
+ by simp
+ then have "a = 0"
+ by (simp add: sgn_mult_abs)
+ } then show ?thesis
+ by auto
+qed
+
+lemma abs_mult_sgn:
+ "\<bar>a\<bar> * sgn a = a"
+ using sgn_mult_abs [of a] by (simp add: ac_simps)
+
+lemma abs_1 [simp]:
+ "\<bar>1\<bar> = 1"
+ using sgn_mult_abs [of 1] by simp
+
+lemma sgn_abs [simp]:
+ "\<bar>sgn a\<bar> = of_bool (a \<noteq> 0)"
+ using sgn_mult_abs [of "sgn a"] mult_cancel_left [of "sgn a" "\<bar>sgn a\<bar>" 1]
+ by (auto simp add: sgn_eq_0_iff)
+
+lemma abs_sgn [simp]:
+ "sgn \<bar>a\<bar> = of_bool (a \<noteq> 0)"
+ using sgn_mult_abs [of "\<bar>a\<bar>"] mult_cancel_right [of "sgn \<bar>a\<bar>" "\<bar>a\<bar>" 1]
+ by (auto simp add: abs_eq_0_iff)
+
+lemma abs_mult:
+ "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
+proof (cases "a = 0 \<or> b = 0")
+ case True
+ then show ?thesis
+ by auto
+next
+ case False
+ then have *: "sgn (a * b) \<noteq> 0"
+ by (simp add: sgn_eq_0_iff)
+ from abs_mult_sgn [of "a * b"] abs_mult_sgn [of a] abs_mult_sgn [of b]
+ have "\<bar>a * b\<bar> * sgn (a * b) = \<bar>a\<bar> * sgn a * \<bar>b\<bar> * sgn b"
+ by (simp add: ac_simps)
+ then have "\<bar>a * b\<bar> * sgn (a * b) = \<bar>a\<bar> * \<bar>b\<bar> * sgn (a * b)"
+ by (simp add: sgn_mult ac_simps)
+ with * show ?thesis
+ by simp
+qed
+
+lemma sgn_minus [simp]:
+ "sgn (- a) = - sgn a"
+proof -
+ from sgn_minus_1 have "sgn (- 1 * a) = - 1 * sgn a"
+ by (simp only: sgn_mult)
+ then show ?thesis
+ by simp
+qed
+
+lemma abs_minus [simp]:
+ "\<bar>- a\<bar> = \<bar>a\<bar>"
+proof -
+ have [simp]: "\<bar>- 1\<bar> = 1"
+ using sgn_mult_abs [of "- 1"] by simp
+ then have "\<bar>- 1 * a\<bar> = 1 * \<bar>a\<bar>"
+ by (simp only: abs_mult)
+ then show ?thesis
+ by simp
+qed
+
+end
+
text \<open>
The theory of partially ordered rings is taken from the books:
\<^item> \<^emph>\<open>Lattice Theory\<close> by Garret Birkhoff, American Mathematical Society, 1979
@@ -1599,6 +1693,9 @@
end
+class abs_if = minus + uminus + ord + zero + abs +
+ assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)"
+
class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if
begin
@@ -1842,7 +1939,8 @@
end
class linordered_idom =
- comm_ring_1 + linordered_comm_semiring_strict + ordered_ab_group_add + abs_if + sgn_if
+ comm_ring_1 + linordered_comm_semiring_strict + ordered_ab_group_add + abs_if + sgn +
+ assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
begin
subclass linordered_semiring_1_strict ..
@@ -1857,6 +1955,10 @@
show "b \<le> a \<Longrightarrow> a - b + b = a" for a b by simp
qed
+subclass idom_abs_sgn
+ by standard
+ (auto simp add: sgn_if abs_if zero_less_mult_iff)
+
lemma linorder_neqE_linordered_idom:
assumes "x \<noteq> y"
obtains "x < y" | "y < x"
@@ -1888,11 +1990,8 @@
lemma mult_less_cancel_left2: "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
using mult_less_cancel_left [of c a 1] by simp
-lemma sgn_sgn [simp]: "sgn (sgn a) = sgn a"
- unfolding sgn_if by simp
-
lemma sgn_0_0: "sgn a = 0 \<longleftrightarrow> a = 0"
- unfolding sgn_if by simp
+ by (fact sgn_eq_0_iff)
lemma sgn_1_pos: "sgn a = 1 \<longleftrightarrow> a > 0"
unfolding sgn_if by simp
@@ -1906,9 +2005,6 @@
lemma sgn_neg [simp]: "a < 0 \<Longrightarrow> sgn a = - 1"
by (simp only: sgn_1_neg)
-lemma sgn_mult: "sgn (a * b) = sgn a * sgn b"
- by (auto simp add: sgn_if zero_less_mult_iff)
-
lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"
unfolding sgn_if abs_if by auto
@@ -1920,7 +2016,7 @@
lemma abs_sgn_eq_1 [simp]:
"a \<noteq> 0 \<Longrightarrow> \<bar>sgn a\<bar> = 1"
- by (simp add: abs_if)
+ by simp
lemma abs_sgn_eq: "\<bar>sgn a\<bar> = (if a = 0 then 0 else 1)"
by (simp add: sgn_if)
@@ -2005,10 +2101,10 @@
begin
lemma mult_sgn_abs: "sgn x * \<bar>x\<bar> = x"
- unfolding abs_if sgn_if by auto
+ by (fact sgn_mult_abs)
-lemma abs_one [simp]: "\<bar>1\<bar> = 1"
- by (simp add: abs_if)
+lemma abs_one: "\<bar>1\<bar> = 1"
+ by (fact abs_1)
end
@@ -2022,9 +2118,6 @@
subclass ordered_ring_abs
by standard (auto simp: abs_if not_less mult_less_0_iff)
-lemma abs_mult: "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
- by (rule abs_eq_mult) auto
-
lemma abs_mult_self [simp]: "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
by (simp add: abs_if)
--- a/src/HOL/Tools/Nitpick/kodkod.ML Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Tools/Nitpick/kodkod.ML Thu Oct 20 19:39:27 2016 +0200
@@ -1028,7 +1028,7 @@
val outcome =
let
val code =
- Isabelle_System.bash ("cd " ^ File.bash_string temp_dir ^ ";\n\
+ Isabelle_System.bash ("cd " ^ Bash.string temp_dir ^ ";\n\
\\"$KODKODI/bin/kodkodi\"" ^
(if ms >= 0 then " -max-msecs " ^ string_of_int ms
else "") ^
--- a/src/HOL/Tools/SMT/smt_solver.ML Tue Oct 18 16:04:44 2016 +0200
+++ b/src/HOL/Tools/SMT/smt_solver.ML Thu Oct 20 19:39:27 2016 +0200
@@ -49,7 +49,7 @@
local
fun make_command command options problem_path proof_path =
- "(exec 2>&1;" :: map File.bash_string (command () @ options) @
+ "(exec 2>&1;" :: map Bash.string (command () @ options) @
[File.bash_path problem_path, ")", ">", File.bash_path proof_path]
|> space_implode " "
--- a/src/Pure/Admin/build_doc.scala Tue Oct 18 16:04:44 2016 +0200
+++ b/src/Pure/Admin/build_doc.scala Thu Oct 20 19:39:27 2016 +0200
@@ -78,7 +78,7 @@
val getopts =
Getopts("""
-Usage: isabelle build_doc [OPTIONS] [DOCS ...]"
+Usage: isabelle build_doc [OPTIONS] [DOCS ...]
Options are:
-a select all documentation sessions
@@ -88,9 +88,9 @@
Build Isabelle documentation from documentation sessions with
suitable document_variants entry.
""",
- "a" -> (_ => all_docs = true),
- "j:" -> (arg => max_jobs = Value.Int.parse(arg)),
- "s" -> (_ => system_mode = true))
+ "a" -> (_ => all_docs = true),
+ "j:" -> (arg => max_jobs = Value.Int.parse(arg)),
+ "s" -> (_ => system_mode = true))
val docs = getopts(args)
--- a/src/Pure/Admin/build_history.scala Tue Oct 18 16:04:44 2016 +0200
+++ b/src/Pure/Admin/build_history.scala Thu Oct 20 19:39:27 2016 +0200
@@ -96,7 +96,7 @@
/** build_history **/
private val default_rev = "tip"
- private val default_threads = 1
+ private val default_multicore = (1, 1)
private val default_heap = 1000
private val default_isabelle_identifier = "build_history"
@@ -109,12 +109,13 @@
fresh: Boolean = false,
nonfree: Boolean = false,
multicore_base: Boolean = false,
- threads_list: List[Int] = List(default_threads),
+ multicore_list: List[(Int, Int)] = List(default_multicore),
arch_64: Boolean = false,
heap: Int = default_heap,
max_heap: Option[Int] = None,
more_settings: List[String] = Nil,
verbose: Boolean = false,
+ build_tags: List[String] = Nil,
build_args: List[String] = Nil): List[(Process_Result, Path)] =
{
/* sanity checks */
@@ -122,7 +123,10 @@
if (File.eq(Path.explode("~~"), hg.root))
error("Repository coincides with ISABELLE_HOME=" + Path.explode("~~").expand)
- for (threads <- threads_list if threads < 1) error("Bad threads value < 1: " + threads)
+ for ((threads, _) <- multicore_list if threads < 1)
+ error("Bad threads value < 1: " + threads)
+ for ((_, processes) <- multicore_list if processes < 1)
+ error("Bad processes value < 1: " + processes)
if (heap < 100) error("Bad heap value < 100: " + heap)
@@ -146,11 +150,12 @@
/* main */
+ val build_host = Isabelle_System.hostname()
val build_history_date = Date.now()
- val build_host = Isabelle_System.hostname()
+ val build_group_id = build_host + ":" + build_history_date.time.ms
var first_build = true
- for (threads <- threads_list) yield
+ for ((threads, processes) <- multicore_list) yield
{
/* init settings */
@@ -183,27 +188,34 @@
Isabelle_System.copy_dir(isabelle_base_log, isabelle_output_log)
val build_start = Date.now()
- val res =
- other_isabelle("build -v " + File.bash_args(build_args), redirect = true, echo = verbose)
+ val build_args1 = List("-v", "-j" + processes) ::: build_args
+ val build_result =
+ other_isabelle("build " + Bash.strings(build_args1), redirect = true, echo = verbose)
val build_end = Date.now()
-
- /* output log */
-
val log_path =
other_isabelle.isabelle_home_user +
Build_Log.log_subdir(build_history_date) +
- Build_Log.log_filename(
- BUILD_HISTORY, build_history_date, build_host, ml_platform, "M" + threads)
+ Build_Log.log_filename(BUILD_HISTORY, build_history_date,
+ List(build_host, ml_platform, "M" + threads) ::: build_tags)
- val build_info = Build_Log.Log_File(log_path.base.implode, res.out_lines).parse_build_info()
+ val build_info =
+ Build_Log.Log_File(log_path.base.implode, build_result.out_lines).parse_build_info()
+
+
+ /* output log */
val meta_info =
- List(Build_Log.Field.build_engine -> BUILD_HISTORY,
- Build_Log.Field.build_host -> build_host,
- Build_Log.Field.build_start -> Build_Log.print_date(build_start),
- Build_Log.Field.build_end -> Build_Log.print_date(build_end),
- Build_Log.Field.isabelle_version -> isabelle_version)
+ Build_Log.Prop.multiple(Build_Log.Prop.build_tags, build_tags) :::
+ Build_Log.Prop.multiple(Build_Log.Prop.build_args, build_args1) :::
+ List(
+ Build_Log.Prop.build_group_id -> build_group_id,
+ Build_Log.Prop.build_id -> (build_host + ":" + build_start.time.ms),
+ Build_Log.Prop.build_engine -> BUILD_HISTORY,
+ Build_Log.Prop.build_host -> build_host,
+ Build_Log.Prop.build_start -> Build_Log.print_date(build_start),
+ Build_Log.Prop.build_end -> Build_Log.print_date(build_end),
+ Build_Log.Prop.isabelle_version -> isabelle_version)
val ml_statistics =
build_info.finished_sessions.flatMap(session_name =>
@@ -228,7 +240,7 @@
Isabelle_System.mkdirs(log_path.dir)
File.write_xz(log_path.ext("xz"),
terminate_lines(
- Build_Log.Log_File.print_props(META_INFO_MARKER, meta_info) :: res.out_lines :::
+ Build_Log.Log_File.print_props(META_INFO_MARKER, meta_info) :: build_result.out_lines :::
ml_statistics.map(Build_Log.Log_File.print_props(Build_Log.ML_STATISTICS_MARKER, _)) :::
heap_sizes), XZ.options(6))
@@ -242,13 +254,26 @@
first_build = false
- (res, log_path.ext("xz"))
+ (build_result, log_path.ext("xz"))
}
}
/* command line entry point */
+ private object Multicore
+ {
+ private val Pat1 = """^(\d+)$""".r
+ private val Pat2 = """^(\d+)x(\d+)$""".r
+
+ def parse(s: String): (Int, Int) =
+ s match {
+ case Pat1(Value.Int(x)) => (x, 1)
+ case Pat2(Value.Int(x), Value.Int(y)) => (x, y)
+ case _ => error("Bad multicore configuration: " + quote(s))
+ }
+ }
+
def main(args: Array[String])
{
Command_Line.tool0 {
@@ -256,13 +281,14 @@
var components_base = ""
var heap: Option[Int] = None
var max_heap: Option[Int] = None
- var threads_list = List(default_threads)
+ var multicore_list = List(default_multicore)
var isabelle_identifier = default_isabelle_identifier
var more_settings: List[String] = Nil
var fresh = false
var arch_64 = false
var nonfree = false
var rev = default_rev
+ var build_tags = List.empty[String]
var verbose = false
val getopts = Getopts("""
@@ -272,7 +298,7 @@
-B first multicore build serves as base for scheduling information
-C DIR base directory for Isabelle components (default: $ISABELLE_HOME_USER/../contrib)
-H SIZE minimal ML heap in MB (default: """ + default_heap + """ for x86, """ + default_heap * 2 + """ for x86_64)
- -M THREADS multicore configurations (comma-separated list, default: """ + default_threads + """)
+ -M MULTICORE multicore configurations (see below)
-N NAME alternative ISABELLE_IDENTIFIER (default: """ + default_isabelle_identifier + """)
-U SIZE maximal ML heap in MB (default: unbounded)
-e TEXT additional text for generated etc/settings
@@ -280,15 +306,19 @@
-m ARCH processor architecture (32=x86, 64=x86_64, default: x86)
-n include nonfree components
-r REV update to revision (default: """ + default_rev + """)
+ -t TAG free-form build tag (multiple occurrences possible)
-v verbose
Build Isabelle sessions from the history of another REPOSITORY clone,
passing ARGS directly to its isabelle build tool.
+
+ Each MULTICORE configuration consists of one or two numbers (default 1):
+ THREADS or THREADSxPROCESSES, e.g. -M 1,2,4 or -M 1x4,2x2,4.
""",
"B" -> (_ => multicore_base = true),
"C:" -> (arg => components_base = arg),
"H:" -> (arg => heap = Some(Value.Int.parse(arg))),
- "M:" -> (arg => threads_list = space_explode(',', arg).map(Value.Int.parse(_))),
+ "M:" -> (arg => multicore_list = space_explode(',', arg).map(Multicore.parse(_))),
"N:" -> (arg => isabelle_identifier = arg),
"U:" -> (arg => max_heap = Some(Value.Int.parse(arg))),
"e:" -> (arg => more_settings = more_settings ::: List(arg)),
@@ -301,6 +331,7 @@
},
"n" -> (_ => nonfree = true),
"r:" -> (arg => rev = arg),
+ "t:" -> (arg => build_tags = build_tags ::: List(arg)),
"v" -> (_ => verbose = true))
val more_args = getopts(args)
@@ -315,10 +346,10 @@
val results =
build_history(hg, progress = progress, rev = rev, isabelle_identifier = isabelle_identifier,
components_base = components_base, fresh = fresh, nonfree = nonfree,
- multicore_base = multicore_base, threads_list = threads_list, arch_64 = arch_64,
+ multicore_base = multicore_base, multicore_list = multicore_list, arch_64 = arch_64,
heap = heap.getOrElse(if (arch_64) default_heap * 2 else default_heap),
max_heap = max_heap, more_settings = more_settings, verbose = verbose,
- build_args = build_args)
+ build_tags = build_tags, build_args = build_args)
for ((_, log_path) <- results)
Output.writeln(log_path.implode, stdout = true)
@@ -342,7 +373,7 @@
options: String = "",
args: String = ""): List[(String, Bytes)] =
{
- val isabelle_admin = ssh.remote_path(isabelle_repos_self + Path.explode("Admin"))
+ val isabelle_admin = isabelle_repos_self + Path.explode("Admin")
/* prepare repository clones */
@@ -353,19 +384,19 @@
if (self_update) {
isabelle_hg.pull()
isabelle_hg.update(clean = true)
- ssh.execute(File.bash_string(isabelle_admin + "/build") + " jars_fresh").check
+ ssh.execute(ssh.bash_path(isabelle_admin + Path.explode("build")) + " jars_fresh").check
}
Mercurial.setup_repository(
- ssh.remote_path(isabelle_repos_self), isabelle_repos_other, ssh = Some(ssh))
+ ssh.bash_path(isabelle_repos_self), isabelle_repos_other, ssh = Some(ssh))
/* Admin/build_history */
val result =
ssh.execute(
- File.bash_string(isabelle_admin + "/build_history") + " " + options + " " +
- File.bash_string(ssh.remote_path(isabelle_repos_other)) + " " + args,
+ ssh.bash_path(isabelle_admin + Path.explode("build_history")) + " " + options + " " +
+ ssh.bash_path(isabelle_repos_other) + " " + args,
progress_stderr = progress.echo(_)).check
for (line <- result.out_lines; log = Path.explode(line))
--- a/src/Pure/Admin/build_log.scala Tue Oct 18 16:04:44 2016 +0200
+++ b/src/Pure/Admin/build_log.scala Thu Oct 20 19:39:27 2016 +0200
@@ -18,36 +18,32 @@
object Build_Log
{
- /** directory content **/
+ /** content **/
+
+ /* properties */
- /* file names */
+ object Prop
+ {
+ val separator = '\u000b'
+
+ def multiple(name: String, args: List[String]): Properties.T =
+ if (args.isEmpty) Nil
+ else List(name -> args.mkString(separator.toString))
- def log_date(date: Date): String =
- String.format(Locale.ROOT, "%s.%05d",
- DateTimeFormatter.ofPattern("yyyy-MM-dd").format(date.rep),
- new java.lang.Long((date.time - date.midnight.time).ms / 1000))
-
- def log_subdir(date: Date): Path =
- Path.explode("log") + Path.explode(date.rep.getYear.toString)
-
- def log_filename(engine: String, date: Date, more: String*): Path =
- Path.explode((engine :: log_date(date) :: more.toList).mkString("", "_", ".log"))
+ val build_tags = "build_tags" // multiple
+ val build_args = "build_args" // multiple
+ val build_group_id = "build_group_id"
+ val build_id = "build_id"
+ val build_engine = "build_engine"
+ val build_host = "build_host"
+ val build_start = "build_start"
+ val build_end = "build_end"
+ val isabelle_version = "isabelle_version"
+ val afp_version = "afp_version"
+ }
- /* log file collections */
-
- def is_log(file: JFile): Boolean =
- List(".log", ".log.gz", ".log.xz").exists(ext => file.getName.endsWith(ext))
-
- def isatest_files(dir: Path): List[JFile] =
- File.find_files(dir.file, file => is_log(file) && file.getName.startsWith("isatest-makeall-"))
-
- def afp_test_files(dir: Path): List[JFile] =
- File.find_files(dir.file, file => is_log(file) && file.getName.startsWith("afp-test-devel-"))
-
-
-
- /** settings **/
+ /* settings */
object Settings
{
@@ -78,6 +74,32 @@
}
+ /* file names */
+
+ def log_date(date: Date): String =
+ String.format(Locale.ROOT, "%s.%05d",
+ DateTimeFormatter.ofPattern("yyyy-MM-dd").format(date.rep),
+ new java.lang.Long((date.time - date.midnight.time).ms / 1000))
+
+ def log_subdir(date: Date): Path =
+ Path.explode("log") + Path.explode(date.rep.getYear.toString)
+
+ def log_filename(engine: String, date: Date, more: List[String] = Nil): Path =
+ Path.explode((engine :: log_date(date) :: more).mkString("", "_", ".log"))
+
+
+ /* log file collections */
+
+ def is_log(file: JFile): Boolean =
+ List(".log", ".log.gz", ".log.xz").exists(ext => file.getName.endsWith(ext))
+
+ def isatest_files(dir: Path): List[JFile] =
+ File.find_files(dir.file, file => is_log(file) && file.getName.startsWith("isatest-makeall-"))
+
+ def afp_test_files(dir: Path): List[JFile] =
+ File.find_files(dir.file, file => is_log(file) && file.getName.startsWith("afp-test-devel-"))
+
+
/** log file **/
@@ -245,16 +267,6 @@
/** meta info **/
- object Field
- {
- val build_engine = "build_engine"
- val build_host = "build_host"
- val build_start = "build_start"
- val build_end = "build_end"
- val isabelle_version = "isabelle_version"
- val afp_version = "afp_version"
- }
-
object Meta_Info
{
val empty: Meta_Info = Meta_Info(Nil, Nil)
@@ -303,23 +315,28 @@
def parse(engine: String, host: String, start: Date,
End: Regex, Isabelle_Version: Regex, AFP_Version: Regex): Meta_Info =
{
- val build_engine = if (engine == "") Nil else List(Field.build_engine -> engine)
- val build_host = if (host == "") Nil else List(Field.build_host -> host)
+ val build_id =
+ {
+ val prefix = if (host != "") host else if (engine != "") engine else ""
+ (if (prefix == "") "build" else prefix) + ":" + start.time.ms
+ }
+ val build_engine = if (engine == "") Nil else List(Prop.build_engine -> engine)
+ val build_host = if (host == "") Nil else List(Prop.build_host -> host)
- val start_date = List(Field.build_start -> start.toString)
+ val start_date = List(Prop.build_start -> start.toString)
val end_date =
log_file.lines.last match {
case End(log_file.Strict_Date(end_date)) =>
- List(Field.build_end -> end_date.toString)
+ List(Prop.build_end -> end_date.toString)
case _ => Nil
}
val isabelle_version =
- log_file.find_match(Isabelle_Version).map(Field.isabelle_version -> _)
+ log_file.find_match(Isabelle_Version).map(Prop.isabelle_version -> _)
val afp_version =
- log_file.find_match(AFP_Version).map(Field.afp_version -> _)
+ log_file.find_match(AFP_Version).map(Prop.afp_version -> _)
- Meta_Info(build_engine ::: build_host :::
+ Meta_Info((Prop.build_id -> build_id) :: build_engine ::: build_host :::
start_date ::: end_date ::: isabelle_version.toList ::: afp_version.toList,
log_file.get_settings(Settings.all_settings))
}
--- a/src/Pure/Admin/build_release.scala Tue Oct 18 16:04:44 2016 +0200
+++ b/src/Pure/Admin/build_release.scala Thu Oct 20 19:39:27 2016 +0200
@@ -76,8 +76,8 @@
progress.bash(
"isabelle makedist -d " + File.bash_path(base_dir) + jobs_option +
(if (official_release) " -O" else "") +
- (if (release_name != "") " -r " + File.bash_string(release_name) else "") +
- (if (rev != "") " " + File.bash_string(rev) else ""),
+ (if (release_name != "") " -r " + Bash.string(release_name) else "") +
+ (if (rev != "") " " + Bash.string(rev) else ""),
echo = true).check
}
Library.trim_line(File.read(isabelle_ident_file))
@@ -98,8 +98,8 @@
progress.echo("\nApplication bundle for " + platform_family + ": " + bundle_archive.implode)
progress.bash(
"isabelle makedist_bundle " + File.bash_path(release_info.dist_archive) +
- " " + File.bash_string(platform_family) +
- (if (remote_mac == "") "" else " " + File.bash_string(remote_mac)),
+ " " + Bash.string(platform_family) +
+ (if (remote_mac == "") "" else " " + Bash.string(remote_mac)),
echo = true).check
}
}
@@ -150,10 +150,29 @@
progress.echo("### Library archive already exists: " +
release_info.dist_library_archive.implode)
else {
- progress.bash("\"$ISABELLE_HOME/Admin/Release/build_library\"" + jobs_option + " " +
- File.bash_path(release_info.dist_dir +
- Path.explode(release_info.name + "_" +
- Isabelle_System.getenv_strict("ISABELLE_PLATFORM_FAMILY") + ".tar.gz"))).check
+ Isabelle_System.with_tmp_dir("build_release")(tmp_dir =>
+ {
+ def execute(script: String): Unit =
+ Isabelle_System.bash(script, cwd = tmp_dir.file).check
+
+ val name = release_info.name
+ val platform = Isabelle_System.getenv_strict("ISABELLE_PLATFORM_FAMILY")
+ val bundle = release_info.dist_dir + Path.explode(name + "_" + platform + ".tar.gz")
+ execute("tar xzf " + File.bash_path(bundle))
+
+ val other_isabelle =
+ new Other_Isabelle(progress, tmp_dir + Path.explode(name), name + "-build")
+
+ other_isabelle.bash("bin/isabelle build" + jobs_option +
+ " -o browser_info -o document=pdf -o document_variants=document:outline=/proof,/ML" +
+ " -s -c -a -d '~~/src/Benchmarks'", echo = true).check
+ other_isabelle.isabelle_home_user.file.delete
+
+ execute("chmod -R a+r " + Bash.string(name))
+ execute("chmod -R g=o " + Bash.string(name))
+ execute("tar czf " + File.bash_path(release_info.dist_library_archive) +
+ " " + Bash.string(name + "/browser_info"))
+ })
}
}
@@ -198,7 +217,7 @@
"R:" -> (arg => release_name = arg),
"W:" -> (arg => website = Some(Path.explode(arg))),
"j:" -> (arg => parallel_jobs = Value.Int.parse(arg)),
- "l" -> (_ => build_library),
+ "l" -> (_ => build_library = true),
"p:" -> (arg => platform_families = Library.space_explode(',', arg)),
"r:" -> (arg => rev = arg))
--- a/src/Pure/Admin/ci_profile.scala Tue Oct 18 16:04:44 2016 +0200
+++ b/src/Pure/Admin/ci_profile.scala Thu Oct 20 19:39:27 2016 +0200
@@ -27,7 +27,7 @@
max_jobs = jobs,
dirs = include,
select_dirs = select,
- system_mode = false,
+ system_mode = true,
selection = select_sessions _)
}
val end_time = Time.now()
--- a/src/Pure/Admin/isabelle_cronjob.scala Tue Oct 18 16:04:44 2016 +0200
+++ b/src/Pure/Admin/isabelle_cronjob.scala Thu Oct 20 19:39:27 2016 +0200
@@ -97,19 +97,21 @@
host: String,
user: String = "",
port: Int = SSH.default_port,
- shared_home: Boolean = false,
+ shared_home: Boolean = true,
options: String = "",
args: String = "-o timeout=10800 -a")
private val remote_builds =
List(
- Remote_Build("lxbroy10", options = "-m32 -M4 -N", shared_home = true),
+ Remote_Build("lxbroy10", options = "-m32 -M4 -N"),
Remote_Build("macbroy2", options = "-m32 -M4"),
Remote_Build("macbroy30", options = "-m32 -M2"),
Remote_Build("macbroy31", options = "-m32 -M2"))
private def remote_build_history(rev: String, r: Remote_Build): Logger_Task =
- Logger_Task("build_history-" + r.host, logger =>
+ {
+ val task_name = "build_history-" + r.host
+ Logger_Task(task_name, logger =>
{
using(logger.ssh_context.open_session(host = r.host, user = r.user, port = r.port))(
ssh =>
@@ -120,12 +122,14 @@
isabelle_repos.ext(r.host),
isabelle_repos_source = isabelle_dev_source,
self_update = !r.shared_home,
- options = r.options + " -f -r " + File.bash_string(rev),
+ options =
+ r.options + " -f -r " + Bash.string(rev) + " -N " + Bash.string(task_name),
args = r.args)
for ((log, bytes) <- results)
Bytes.write(logger.log_dir + Path.explode(log), bytes)
})
})
+ }
@@ -167,7 +171,9 @@
val err =
res match {
case Exn.Res(_) => None
- case Exn.Exn(exn) => Some(Exn.message(exn))
+ case Exn.Exn(exn) =>
+ val first_line = Library.split_lines(Exn.message(exn)).headOption getOrElse "exception"
+ Some(first_line)
}
logger.log_end(end_date, err)
}
--- a/src/Pure/Admin/other_isabelle.scala Tue Oct 18 16:04:44 2016 +0200
+++ b/src/Pure/Admin/other_isabelle.scala Thu Oct 20 19:39:27 2016 +0200
@@ -7,7 +7,7 @@
package isabelle
-private class Other_Isabelle(progress: Progress, isabelle_home: Path, isabelle_identifier: String)
+class Other_Isabelle(progress: Progress, val isabelle_home: Path, val isabelle_identifier: String)
{
other_isabelle =>
@@ -16,8 +16,8 @@
def bash(script: String, redirect: Boolean = false, echo: Boolean = false): Process_Result =
progress.bash(
- "export ISABELLE_IDENTIFIER=" + File.bash_string(isabelle_identifier) + "\n" + script,
- env = null, cwd = isabelle_home.file, redirect = redirect)
+ "export ISABELLE_IDENTIFIER=" + Bash.string(isabelle_identifier) + "\n" + script,
+ env = null, cwd = isabelle_home.file, redirect = redirect, echo = echo)
def apply(cmdline: String, redirect: Boolean = false, echo: Boolean = false): Process_Result =
bash("bin/isabelle " + cmdline, redirect, echo)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Pure/Admin/profiling_report.scala Thu Oct 20 19:39:27 2016 +0200
@@ -0,0 +1,55 @@
+/* Title: Pure/Admin/profiling_report.scala
+ Author: Makarius
+
+Report Poly/ML profiling information from log files.
+*/
+
+package isabelle
+
+
+import java.util.Locale
+
+
+object Profiling_Report
+{
+ def profiling_report(log_file: Build_Log.Log_File): List[(Long, String)] =
+ {
+ val Line = """^(?:### )?([ 0-9]{10}) (\S+|GARBAGE COLLECTION.*)$""".r
+ val Count = """ *(\d+)""".r
+ val clean = """-?\(\d+\).*$""".r
+
+ var results = Map.empty[String, Long]
+ for (Line(Count(Value.Long(count)), raw_fun) <- log_file.lines) {
+ val fun = clean.replaceAllIn(raw_fun, "")
+ results += (fun -> (results.getOrElse(fun, 0L) + count))
+ }
+ for ((fun, count) <- results.toList.sortBy(_._2)) yield (count, fun)
+ }
+
+
+ /* Isabelle tool wrapper */
+
+ val isabelle_tool =
+ Isabelle_Tool("profiling_report", "report Poly/ML profiling information from log files", args =>
+ {
+ Command_Line.tool0 {
+ val getopts =
+ Getopts("""
+Usage: isabelle profiling_report [LOGS ...]
+
+ Report Poly/ML profiling output from log files (potentially compressed).
+""")
+ val log_names = getopts(args)
+ for (name <- log_names) {
+ val log_file = Build_Log.Log_File(Path.explode(name))
+ val results =
+ for ((count, fun) <- profiling_report(log_file))
+ yield
+ String.format(Locale.ROOT, "%14d %s",
+ count.asInstanceOf[AnyRef], fun.asInstanceOf[AnyRef])
+ if (results.nonEmpty)
+ Output.writeln(cat_lines((log_file.name + ":") :: results))
+ }
+ }
+ })
+}
--- a/src/Pure/Admin/remote_dmg.scala Tue Oct 18 16:04:44 2016 +0200
+++ b/src/Pure/Admin/remote_dmg.scala Thu Oct 20 19:39:27 2016 +0200
@@ -13,13 +13,13 @@
{
ssh.with_tmp_dir(remote_dir =>
{
- val cd = "cd " + File.bash_string(ssh.remote_path(remote_dir)) + "; "
+ val cd = "cd " + ssh.bash_path(remote_dir) + "; "
ssh.write_file(remote_dir + Path.explode("dmg.tar.gz"), tar_gz_file)
ssh.execute(cd + "mkdir root && tar -C root -xzf dmg.tar.gz").check
ssh.execute(
cd + "hdiutil create -srcfolder root" +
- (if (volume_name == "") "" else " -volname " + File.bash_string(volume_name)) +
+ (if (volume_name == "") "" else " -volname " + Bash.string(volume_name)) +
" dmg.dmg").check
ssh.read_file(remote_dir + Path.explode("dmg.dmg"), dmg_file)
})
--- a/src/Pure/General/file.ML Tue Oct 18 16:04:44 2016 +0200
+++ b/src/Pure/General/file.ML Thu Oct 20 19:39:27 2016 +0200
@@ -8,8 +8,6 @@
sig
val standard_path: Path.T -> string
val platform_path: Path.T -> string
- val bash_string: string -> string
- val bash_args: string list -> string
val bash_path: Path.T -> string
val full_path: Path.T -> Path.T -> Path.T
val tmp_path: Path.T -> Path.T
@@ -46,26 +44,7 @@
val standard_path = Path.implode o Path.expand;
val platform_path = ML_System.platform_path o standard_path;
-fun bash_string "" = "\"\""
- | bash_string str =
- str |> translate_string (fn ch =>
- let val c = ord ch in
- (case ch of
- "\t" => "$'\\t'"
- | "\n" => "$'\\n'"
- | "\f" => "$'\\f'"
- | "\r" => "$'\\r'"
- | _ =>
- if Symbol.is_ascii_letter ch orelse Symbol.is_ascii_digit ch orelse
- exists_string (fn c => c = ch) "-./:_" then ch
- else if c < 16 then "$'\\x0" ^ Int.fmt StringCvt.HEX c ^ "'"
- else if c < 32 orelse c >= 127 then "$'\\x" ^ Int.fmt StringCvt.HEX c ^ "'"
- else "\\" ^ ch)
- end);
-
-val bash_args = space_implode " " o map bash_string;
-
-val bash_path = bash_string o standard_path;
+val bash_path = Bash_Syntax.string o standard_path;
(* full_path *)
--- a/src/Pure/General/file.scala Tue Oct 18 16:04:44 2016 +0200
+++ b/src/Pure/General/file.scala Thu Oct 20 19:39:27 2016 +0200
@@ -108,33 +108,8 @@
/* bash path */
- private def bash_chr(c: Byte): String =
- {
- val ch = c.toChar
- ch match {
- case '\t' => "$'\\t'"
- case '\n' => "$'\\n'"
- case '\f' => "$'\\f'"
- case '\r' => "$'\\r'"
- case _ =>
- if (Symbol.is_ascii_letter(ch) || Symbol.is_ascii_digit(ch) || "-./:_".contains(ch))
- Symbol.ascii(ch)
- else if (c < 0) "$'\\x" + Integer.toHexString(256 + c) + "'"
- else if (c < 16) "$'\\x0" + Integer.toHexString(c) + "'"
- else if (c < 32 || c >= 127) "$'\\x" + Integer.toHexString(c) + "'"
- else "\\" + ch
- }
- }
-
- def bash_string(s: String): String =
- if (s == "") "\"\""
- else UTF8.bytes(s).iterator.map(bash_chr(_)).mkString
-
- def bash_args(args: List[String]): String =
- args.iterator.map(bash_string(_)).mkString(" ")
-
- def bash_path(path: Path): String = bash_string(standard_path(path))
- def bash_path(file: JFile): String = bash_string(standard_path(file))
+ def bash_path(path: Path): String = Bash.string(standard_path(path))
+ def bash_path(file: JFile): String = Bash.string(standard_path(file))
/* directory entries */
--- a/src/Pure/General/mercurial.scala Tue Oct 18 16:04:44 2016 +0200
+++ b/src/Pure/General/mercurial.scala Thu Oct 20 19:39:27 2016 +0200
@@ -16,7 +16,7 @@
/* command-line syntax */
def optional(s: String, prefix: String = ""): String =
- if (s == "") "" else " " + prefix + " " + File.bash_string(s)
+ if (s == "") "" else " " + prefix + " " + Bash.string(s)
def opt_flag(flag: String, b: Boolean): String = if (b) " " + flag else ""
def opt_rev(s: String): String = optional(s, "--rev")
@@ -40,7 +40,7 @@
case None => Isabelle_System.mkdirs(hg.root.dir)
case Some(ssh) => ssh.mkdirs(hg.root.dir)
}
- hg.command("clone", File.bash_string(source) + " " + File.bash_path(hg.root), options).check
+ hg.command("clone", Bash.string(source) + " " + File.bash_path(hg.root), options).check
hg
}
--- a/src/Pure/General/name_space.ML Tue Oct 18 16:04:44 2016 +0200
+++ b/src/Pure/General/name_space.ML Thu Oct 20 19:39:27 2016 +0200
@@ -120,7 +120,7 @@
(* internal names *)
-type internals = (string list * string list) Change_Table.T;
+type internals = (string list * string list) Change_Table.T; (*xname -> visible, hidden*)
fun map_internals f xname : internals -> internals =
Change_Table.map_default (xname, ([], [])) f;
@@ -132,13 +132,15 @@
fun hide_name name = map_internals (apsnd (update (op =) name)) name;
+(* external accesses *)
+
+type accesses = (xstring list * xstring list); (*input / output fragments*)
+type entries = (accesses * entry) Change_Table.T; (*name -> accesses, entry*)
+
+
(* datatype T *)
-datatype T =
- Name_Space of
- {kind: string,
- internals: internals, (*xname -> visible, hidden*)
- entries: (xstring list * entry) Change_Table.T}; (*name -> externals, entry*)
+datatype T = Name_Space of {kind: string, internals: internals, entries: entries};
fun make_name_space (kind, internals, entries) =
Name_Space {kind = kind, internals = internals, entries = entries};
@@ -200,12 +202,13 @@
fun get_accesses (Name_Space {entries, ...}) name =
(case Change_Table.lookup entries name of
- NONE => []
- | SOME (externals, _) => externals);
+ NONE => ([], [])
+ | SOME (accesses, _) => accesses);
-fun valid_accesses (Name_Space {internals, ...}) name =
- Change_Table.fold (fn (xname, (names, _)) =>
- if not (null names) andalso hd names = name then cons xname else I) internals [];
+fun is_valid_access (Name_Space {internals, ...}) name xname =
+ (case Change_Table.lookup internals xname of
+ SOME (name' :: _, _) => name = name'
+ | _ => false);
(* extern *)
@@ -234,7 +237,7 @@
in
if names_long then name
else if names_short then Long_Name.base_name name
- else ext (get_accesses space name)
+ else ext (#2 (get_accesses space name))
end;
fun extern_ord ctxt space = string_ord o apply2 (extern ctxt space);
@@ -426,7 +429,7 @@
fun mandatory_suffixes xs = map rev (mandatory_prefixes (rev xs));
-fun accesses naming binding =
+fun make_accesses naming binding =
(case name_spec naming binding of
{restriction = SOME true, ...} => ([], [])
| {restriction, spec, ...} =>
@@ -443,12 +446,13 @@
space |> map_name_space (fn (kind, internals, entries) =>
let
val _ = the_entry space name;
- val names = valid_accesses space name;
+ val (accs, accs') = get_accesses space name;
+ val xnames = filter (is_valid_access space name) accs;
val internals' = internals
|> hide_name name
|> fold (del_name name)
- (if fully then names else inter (op =) [Long_Name.base_name name] names)
- |> fold (del_name_extra name) (get_accesses space name);
+ (if fully then xnames else inter (op =) [Long_Name.base_name name] xnames)
+ |> fold (del_name_extra name) accs';
in (kind, internals', entries) end);
@@ -458,10 +462,12 @@
space |> map_name_space (fn (kind, internals, entries) =>
let
val _ = the_entry space name;
- val (accs, accs') = accesses naming binding;
- val internals' = internals |> fold (add_name name) accs;
+ val (more_accs, more_accs') = make_accesses naming binding;
+ val internals' = internals |> fold (add_name name) more_accs;
val entries' = entries
- |> Change_Table.map_entry name (apfst (fold_rev (update op =) accs'));
+ |> Change_Table.map_entry name (apfst (fn (accs, accs') =>
+ (fold_rev (update op =) more_accs accs,
+ fold_rev (update op =) more_accs' accs')))
in (kind, internals', entries') end);
@@ -497,7 +503,7 @@
val naming = naming_of context;
val Naming {group, theory_name, ...} = naming;
val {concealed, spec, ...} = name_spec naming binding;
- val (accs, accs') = accesses naming binding;
+ val accesses = make_accesses naming binding;
val name = Long_Name.implode (map fst spec);
val _ = name = "" andalso error (Binding.bad binding);
@@ -512,10 +518,10 @@
val space' =
space |> map_name_space (fn (kind, internals, entries) =>
let
- val internals' = internals |> fold (add_name name) accs;
+ val internals' = internals |> fold (add_name name) (#1 accesses);
val entries' =
(if strict then Change_Table.update_new else Change_Table.update)
- (name, (accs', entry)) entries
+ (name, (accesses, entry)) entries
handle Change_Table.DUP dup =>
err_dup kind (dup, #2 (the (Change_Table.lookup entries dup)))
(name, entry) (#pos entry);
--- a/src/Pure/General/ssh.scala Tue Oct 18 16:04:44 2016 +0200
+++ b/src/Pure/General/ssh.scala Thu Oct 20 19:39:27 2016 +0200
@@ -243,6 +243,7 @@
}
def expand_path(path: Path): Path = path.expand_env(settings)
def remote_path(path: Path): String = expand_path(path).implode
+ def bash_path(path: Path): String = Bash.string(remote_path(path))
def chmod(permissions: Int, path: Path): Unit = sftp.chmod(permissions, remote_path(path))
def mv(path1: Path, path2: Path): Unit = sftp.rename(remote_path(path1), remote_path(path2))
@@ -323,8 +324,10 @@
/* tmp dirs */
+ def rm_tree(dir: Path): Unit = rm_tree(remote_path(dir))
+
def rm_tree(remote_dir: String): Unit =
- execute("rm -r -f " + File.bash_string(remote_dir)).check
+ execute("rm -r -f " + Bash.string(remote_dir)).check
def tmp_dir(): String =
execute("mktemp -d -t tmp.XXXXXXXXXX").check.out
--- a/src/Pure/ROOT Tue Oct 18 16:04:44 2016 +0200
+++ b/src/Pure/ROOT Thu Oct 20 19:39:27 2016 +0200
@@ -4,6 +4,7 @@
description {*
The Pure logical framework
*}
+ options [threads = 1]
global_theories
Pure
theories
--- a/src/Pure/ROOT.ML Tue Oct 18 16:04:44 2016 +0200
+++ b/src/Pure/ROOT.ML Thu Oct 20 19:39:27 2016 +0200
@@ -67,6 +67,7 @@
ML_file "PIDE/xml.ML";
ML_file "General/path.ML";
ML_file "General/url.ML";
+ML_file "System/bash_syntax.ML";
ML_file "General/file.ML";
ML_file "General/long_name.ML";
ML_file "General/binding.ML";
--- a/src/Pure/System/bash.ML Tue Oct 18 16:04:44 2016 +0200
+++ b/src/Pure/System/bash.ML Thu Oct 20 19:39:27 2016 +0200
@@ -6,6 +6,8 @@
signature BASH =
sig
+ val string: string -> string
+ val strings: string list -> string
val process: string -> {out: string, err: string, rc: int, terminate: unit -> unit}
end;
@@ -14,6 +16,9 @@
structure Bash: BASH =
struct
+val string = Bash_Syntax.string;
+val strings = Bash_Syntax.strings;
+
val process = Thread_Attributes.uninterruptible (fn restore_attributes => fn script =>
let
datatype result = Wait | Signal | Result of int;
@@ -105,6 +110,9 @@
structure Bash: BASH =
struct
+val string = Bash_Syntax.string;
+val strings = Bash_Syntax.strings;
+
val process = Thread_Attributes.uninterruptible (fn restore_attributes => fn script =>
let
datatype result = Wait | Signal | Result of int;
--- a/src/Pure/System/bash.scala Tue Oct 18 16:04:44 2016 +0200
+++ b/src/Pure/System/bash.scala Thu Oct 20 19:39:27 2016 +0200
@@ -13,6 +13,36 @@
object Bash
{
+ /* concrete syntax */
+
+ private def bash_chr(c: Byte): String =
+ {
+ val ch = c.toChar
+ ch match {
+ case '\t' => "$'\\t'"
+ case '\n' => "$'\\n'"
+ case '\f' => "$'\\f'"
+ case '\r' => "$'\\r'"
+ case _ =>
+ if (Symbol.is_ascii_letter(ch) || Symbol.is_ascii_digit(ch) || "-./:_".contains(ch))
+ Symbol.ascii(ch)
+ else if (c < 0) "$'\\x" + Integer.toHexString(256 + c) + "'"
+ else if (c < 16) "$'\\x0" + Integer.toHexString(c) + "'"
+ else if (c < 32 || c >= 127) "$'\\x" + Integer.toHexString(c) + "'"
+ else "\\" + ch
+ }
+ }
+
+ def string(s: String): String =
+ if (s == "") "\"\""
+ else UTF8.bytes(s).iterator.map(bash_chr(_)).mkString
+
+ def strings(ss: List[String]): String =
+ ss.iterator.map(Bash.string(_)).mkString(" ")
+
+
+ /* process and result */
+
private class Limited_Progress(proc: Process, progress_limit: Option[Long])
{
private var count = 0L
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Pure/System/bash_syntax.ML Thu Oct 20 19:39:27 2016 +0200
@@ -0,0 +1,35 @@
+(* Title: Pure/System/bash_syntax.ML
+ Author: Makarius
+
+Syntax for GNU bash (see also Pure/System/bash.ML).
+*)
+
+signature BASH_SYNTAX =
+sig
+ val string: string -> string
+ val strings: string list -> string
+end;
+
+structure Bash_Syntax: BASH_SYNTAX =
+struct
+
+fun string "" = "\"\""
+ | string str =
+ str |> translate_string (fn ch =>
+ let val c = ord ch in
+ (case ch of
+ "\t" => "$'\\t'"
+ | "\n" => "$'\\n'"
+ | "\f" => "$'\\f'"
+ | "\r" => "$'\\r'"
+ | _ =>
+ if Symbol.is_ascii_letter ch orelse Symbol.is_ascii_digit ch orelse
+ exists_string (fn c => c = ch) "-./:_" then ch
+ else if c < 16 then "$'\\x0" ^ Int.fmt StringCvt.HEX c ^ "'"
+ else if c < 32 orelse c >= 127 then "$'\\x" ^ Int.fmt StringCvt.HEX c ^ "'"
+ else "\\" ^ ch)
+ end);
+
+val strings = space_implode " " o map string;
+
+end;
--- a/src/Pure/System/isabelle_system.scala Tue Oct 18 16:04:44 2016 +0200
+++ b/src/Pure/System/isabelle_system.scala Thu Oct 20 19:39:27 2016 +0200
@@ -318,7 +318,7 @@
def hostname(): String = bash("hostname -s").check.out
def open(arg: String): Unit =
- bash("exec \"$ISABELLE_OPEN\" " + File.bash_string(arg) + " >/dev/null 2>/dev/null &")
+ bash("exec \"$ISABELLE_OPEN\" " + Bash.string(arg) + " >/dev/null 2>/dev/null &")
def pdf_viewer(arg: Path): Unit =
bash("exec \"$PDF_VIEWER\" " + File.bash_path(arg) + " >/dev/null 2>/dev/null &")
--- a/src/Pure/System/isabelle_tool.scala Tue Oct 18 16:04:44 2016 +0200
+++ b/src/Pure/System/isabelle_tool.scala Thu Oct 20 19:39:27 2016 +0200
@@ -89,7 +89,7 @@
(args: List[String]) =>
{
val tool = dir + Path.basic(name)
- val result = Isabelle_System.bash(File.bash_path(tool) + " " + File.bash_args(args))
+ val result = Isabelle_System.bash(File.bash_path(tool) + " " + Bash.strings(args))
sys.exit(result.print_stdout.rc)
}
})
@@ -106,6 +106,7 @@
Doc.isabelle_tool,
ML_Process.isabelle_tool,
Options.isabelle_tool,
+ Profiling_Report.isabelle_tool,
Remote_DMG.isabelle_tool,
Update_Cartouches.isabelle_tool,
Update_Header.isabelle_tool,
--- a/src/Pure/Tools/build.ML Tue Oct 18 16:04:44 2016 +0200
+++ b/src/Pure/Tools/build.ML Thu Oct 20 19:39:27 2016 +0200
@@ -114,6 +114,12 @@
symbols = symbols,
last_timing = last_timing,
master_dir = master_dir}
+ |>
+ (case Options.string options "profiling" of
+ "" => I
+ | "time" => profile_time
+ | "allocations" => profile_allocations
+ | bad => error ("Bad profiling option: " ^ quote bad))
|> Unsynchronized.setmp print_mode
(space_explode "," (Options.string options "print_mode") @ print_mode_value ())) thys)
else
--- a/src/Pure/Tools/ml_process.scala Tue Oct 18 16:04:44 2016 +0200
+++ b/src/Pure/Tools/ml_process.scala Thu Oct 20 19:39:27 2016 +0200
@@ -107,7 +107,7 @@
Bash.process(
"exec " + options.string("ML_process_policy") + """ "$ML_HOME/poly" -q """ +
- File.bash_args(bash_args),
+ Bash.strings(bash_args),
cwd = cwd,
env =
Isabelle_System.library_path(env ++ env_options ++ env_tmp,
--- a/src/Pure/build-jars Tue Oct 18 16:04:44 2016 +0200
+++ b/src/Pure/build-jars Thu Oct 20 19:39:27 2016 +0200
@@ -19,6 +19,7 @@
Admin/ci_profile.scala
Admin/isabelle_cronjob.scala
Admin/other_isabelle.scala
+ Admin/profiling_report.scala
Admin/remote_dmg.scala
Concurrent/consumer_thread.scala
Concurrent/counter.scala