HOL-Analysis: move Function Topology from AFP/Ergodict_Theory; HOL-Probability: move Essential Supremum from AFP/Lp
--- a/src/HOL/Analysis/Analysis.thy Tue Oct 18 16:05:24 2016 +0100
+++ b/src/HOL/Analysis/Analysis.thy Tue Oct 18 17:29:28 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
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Function_Topology.thy Tue Oct 18 17:29:28 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 16:05:24 2016 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,3098 +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 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
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Further_Topology.thy Tue Oct 18 17:29:28 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
+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/Probability/Probability.thy Tue Oct 18 16:05:24 2016 +0100
+++ b/src/HOL/Probability/Probability.thy Tue Oct 18 17:29:28 2016 +0200
@@ -12,6 +12,7 @@
SPMF
Stream_Space
Conditional_Expectation
+ Essential_Supremum
begin
end