--- a/src/HOL/Analysis/Abstract_Metric_Spaces.thy Wed Jun 28 14:45:29 2023 +0200
+++ b/src/HOL/Analysis/Abstract_Metric_Spaces.thy Wed Jun 28 16:01:34 2023 +0200
@@ -144,21 +144,6 @@
by (auto simp: openin_mtopology)
qed
-lemma mcball_eq_cball [simp]: "Met_TC.mcball = cball"
- by force
-
-lemma mball_eq_ball [simp]: "Met_TC.mball = ball"
- by force
-
-lemma mopen_eq_open [simp]: "Met_TC.mopen = open"
- by (force simp: open_contains_ball Met_TC.mopen_def)
-
-lemma limitin_iff_tendsto [iff]: "limitin Met_TC.mtopology \<sigma> x F = tendsto \<sigma> x F"
- by (simp add: Met_TC.mtopology_def)
-
-lemma mtopology_is_euclideanreal [simp]: "Met_TC.mtopology = euclideanreal"
- by (simp add: Met_TC.mtopology_def)
-
(*
lemma metric_injective_image:
"\<And>f m s.
@@ -419,7 +404,26 @@
qed auto
qed
-end
+
+end (*Metric_space*)
+
+lemma mcball_eq_cball [simp]: "Met_TC.mcball = cball"
+ by force
+
+lemma mball_eq_ball [simp]: "Met_TC.mball = ball"
+ by force
+
+lemma mopen_eq_open [simp]: "Met_TC.mopen = open"
+ by (force simp: open_contains_ball Met_TC.mopen_def)
+
+lemma limitin_iff_tendsto [iff]: "limitin Met_TC.mtopology \<sigma> x F = tendsto \<sigma> x F"
+ by (simp add: Met_TC.mtopology_def)
+
+lemma mtopology_is_euclidean [simp]: "Met_TC.mtopology = euclidean"
+ by (simp add: Met_TC.mtopology_def)
+
+lemma mbounded_iff_bounded [iff]: "Met_TC.mbounded A \<longleftrightarrow> bounded A"
+ by (metis Met_TC.mbounded UNIV_I all_not_in_conv bounded_def)
subsection\<open>Subspace of a metric space\<close>
@@ -472,7 +476,6 @@
subsection \<open>Abstract type of metric spaces\<close>
-
typedef 'a metric = "{(M::'a set,d). Metric_space M d}"
morphisms "dest_metric" "metric"
proof -
@@ -486,7 +489,7 @@
definition mdist where "mdist m \<equiv> snd (dest_metric m)"
-lemma Metric_space_mspace_mdist: "Metric_space (mspace m) (mdist m)"
+lemma Metric_space_mspace_mdist [iff]: "Metric_space (mspace m) (mdist m)"
by (metis Product_Type.Collect_case_prodD dest_metric mdist_def mspace_def)
lemma mdist_nonneg [simp]: "\<And>x y. 0 \<le> mdist m x y"
@@ -524,12 +527,18 @@
definition "mball_of \<equiv> \<lambda>m. Metric_space.mball (mspace m) (mdist m)"
+lemma in_mball_of [simp]: "y \<in> mball_of m x r \<longleftrightarrow> x \<in> mspace m \<and> y \<in> mspace m \<and> mdist m x y < r"
+ by (simp add: Metric_space.in_mball mball_of_def)
+
lemma (in Metric_space) mball_of [simp]:
"mball_of (metric (M,d)) = mball"
by (simp add: mball_of_def)
definition "mcball_of \<equiv> \<lambda>m. Metric_space.mcball (mspace m) (mdist m)"
+lemma in_mcball_of [simp]: "y \<in> mcball_of m x r \<longleftrightarrow> x \<in> mspace m \<and> y \<in> mspace m \<and> mdist m x y \<le> r"
+ by (simp add: Metric_space.in_mcball mcball_of_def)
+
lemma (in Metric_space) mcball_of [simp]:
"mcball_of (metric (M,d)) = mcball"
by (simp add: mcball_of_def)
@@ -545,6 +554,15 @@
lemma mtopology_of_euclidean [simp]: "mtopology_of euclidean_metric = euclidean"
by (simp add: Met_TC.mtopology_def mtopology_of_def)
+text \<open>Allows reference to the current metric space within the locale as a value\<close>
+definition (in Metric_space) "Self \<equiv> metric (M,d)"
+
+lemma (in Metric_space) mspace_Self [simp]: "mspace Self = M"
+ by (simp add: Self_def)
+
+lemma (in Metric_space) mdist_Self [simp]: "mdist Self = d"
+ by (simp add: Self_def)
+
text\<open> Subspace of a metric space\<close>
definition submetric where
@@ -822,12 +840,6 @@
"\<lbrakk>metrizable_space X; openin X S\<rbrakk> \<Longrightarrow> fsigma_in X S"
by (meson closed_imp_gdelta_in fsigma_in_gdelta_in openin_closedin openin_subset)
-lemma mball_eq_ball [simp]: "Met_TC.mball = ball"
- by force
-
-lemma mopen_eq_open [simp]: "Met_TC.mopen = open"
- by (force simp: open_contains_ball Met_TC.mopen_def)
-
lemma metrizable_space_euclidean:
"metrizable_space (euclidean :: 'a::metric_space topology)"
unfolding metrizable_space_def
@@ -1551,6 +1563,23 @@
end
+definition mcomplete_of :: "'a metric \<Rightarrow> bool"
+ where "mcomplete_of \<equiv> \<lambda>m. Metric_space.mcomplete (mspace m) (mdist m)"
+
+lemma (in Metric_space) mcomplete_of [simp]: "mcomplete_of (metric (M,d)) = mcomplete"
+ by (simp add: mcomplete_of_def)
+
+lemma mcomplete_trivial: "Metric_space.mcomplete {} (\<lambda>x y. 0)"
+ using Metric_space.intro Metric_space.mcomplete_empty_mspace by force
+
+lemma mcomplete_trivial_singleton: "Metric_space.mcomplete {\<lambda>x. a} (\<lambda>x y. 0)"
+proof -
+ interpret Metric_space "{\<lambda>x. a}" "\<lambda>x y. 0"
+ by unfold_locales auto
+ show ?thesis
+ unfolding mcomplete_def MCauchy_def image_subset_iff by (metis UNIV_I limit_metric_sequentially)
+qed
+
lemma MCauchy_iff_Cauchy [iff]: "Met_TC.MCauchy = Cauchy"
by (force simp: Cauchy_def Met_TC.MCauchy_def)
@@ -1558,9 +1587,6 @@
"Met_TC.mcomplete (Pure.type ::'a::metric_space itself) \<longleftrightarrow> complete (UNIV::'a set)"
by (auto simp: Met_TC.mcomplete_def complete_def)
-lemma euclidean_metric: "Met_TC.mcomplete (Pure.type ::'a::euclidean_space itself)"
- using complete_UNIV mcomplete_iff_complete by blast
-
context Submetric
begin
@@ -1593,7 +1619,6 @@
end
-
context Metric_space
begin
@@ -2745,9 +2770,9 @@
unfolding completely_metrizable_space_def
by (metis discrete_metric.mcomplete_discrete_metric discrete_metric.mtopology_discrete_metric metric_M_dd)
-lemma completely_metrizable_space_euclideanreal:
- "completely_metrizable_space euclideanreal"
- by (metis Met_TC.mtopology_is_euclideanreal Met_TC.completely_metrizable_space_mtopology euclidean_metric)
+lemma completely_metrizable_space_euclidean:
+ "completely_metrizable_space (euclidean:: 'a::complete_space topology)"
+ using Met_TC.completely_metrizable_space_mtopology complete_UNIV by auto
lemma completely_metrizable_space_closedin:
assumes X: "completely_metrizable_space X" and S: "closedin X S"
@@ -2772,6 +2797,9 @@
qed
qed
+lemma completely_metrizable_space_cbox: "completely_metrizable_space (top_of_set (cbox a b))"
+ using closed_closedin completely_metrizable_space_closedin completely_metrizable_space_euclidean by blast
+
lemma homeomorphic_completely_metrizable_space_aux:
assumes homXY: "X homeomorphic_space Y" and X: "completely_metrizable_space X"
shows "completely_metrizable_space Y"
@@ -3364,6 +3392,14 @@
finally show ?thesis by (simp add: True)
qed (simp add: atin_within_def)
+lemma eventually_within_imp:
+ "eventually P (atin_within X a S) \<longleftrightarrow> eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) (atin X a)"
+ by (auto simp add: eventually_atin_within eventually_atin)
+
+lemma limit_within_subset:
+ "\<lbrakk>limitin X f l (atin_within Y a S); T \<subseteq> S\<rbrakk> \<Longrightarrow> limitin X f l (atin_within Y a T)"
+ by (smt (verit) eventually_atin_within limitin_def subset_eq)
+
lemma atin_subtopology_within:
assumes "a \<in> S"
shows "atin (subtopology X S) a = atin_within X a S"
@@ -4504,7 +4540,7 @@
using assms continuous_map_compose continuous_map_metric by blast
qed
-lemma continuous_map_mdist:
+lemma continuous_map_mdist [continuous_intros]:
assumes f: "continuous_map X (mtopology_of m) f"
and g: "continuous_map X (mtopology_of m) g"
shows "continuous_map X euclidean (\<lambda>x. mdist m (f x) (g x))"
@@ -4560,6 +4596,542 @@
assumes fim: "f ` M1 = M2" and d: "\<And>x y. \<lbrakk>x \<in> M1; y \<in> M1\<rbrakk> \<Longrightarrow> d2 (f x) (f y) = d1 x y"
shows "homeomorphic_map M1.mtopology M2.mtopology f"
by (metis M1.topspace_mtopology M2.subtopology_mspace d embedding_map_def fim isometry_imp_embedding_map subsetI)
+subsection\<open>"Capped" equivalent bounded metrics and general product metrics\<close>
+
+definition (in Metric_space) capped_dist where
+ "capped_dist \<equiv> \<lambda>\<delta> x y. if \<delta> > 0 then min \<delta> (d x y) else d x y"
+
+lemma (in Metric_space) capped_dist: "Metric_space M (capped_dist \<delta>)"
+proof
+ fix x y
+ assume "x \<in> M" "y \<in> M"
+ then show "(capped_dist \<delta> x y = 0) = (x = y)"
+ by (smt (verit, best) capped_dist_def zero)
+ fix z
+ assume "z \<in> M"
+ show "capped_dist \<delta> x z \<le> capped_dist \<delta> x y + capped_dist \<delta> y z"
+ unfolding capped_dist_def using \<open>x \<in> M\<close> \<open>y \<in> M\<close> \<open>z \<in> M\<close>
+ by (smt (verit, del_insts) Metric_space.mdist_pos_eq Metric_space_axioms mdist_reverse_triangle)
+qed (use capped_dist_def commute in auto)
+
+
+definition capped_metric where
+ "capped_metric \<delta> m \<equiv> metric(mspace m, Metric_space.capped_dist (mdist m) \<delta>)"
+
+lemma capped_metric:
+ "capped_metric \<delta> m = (if \<delta> \<le> 0 then m else metric(mspace m, \<lambda>x y. min \<delta> (mdist m x y)))"
+proof -
+ interpret Metric_space "mspace m" "mdist m"
+ by (simp add: Metric_space_mspace_mdist)
+ show ?thesis
+ by (auto simp: capped_metric_def capped_dist_def)
+qed
+
+lemma capped_metric_mspace [simp]:
+ "mspace (capped_metric \<delta> m) = mspace m"
+ apply (simp add: capped_metric not_le)
+ by (smt (verit, ccfv_threshold) Metric_space.mspace_metric Metric_space_def Metric_space_mspace_mdist)
+
+lemma capped_metric_mdist:
+ "mdist (capped_metric \<delta> m) = (\<lambda>x y. if \<delta> \<le> 0 then mdist m x y else min \<delta> (mdist m x y))"
+ apply (simp add: capped_metric not_le)
+ by (smt (verit, del_insts) Metric_space.mdist_metric Metric_space_def Metric_space_mspace_mdist)
+
+lemma mdist_capped_le: "mdist (capped_metric \<delta> m) x y \<le> mdist m x y"
+ by (simp add: capped_metric_mdist)
+
+lemma mdist_capped: "\<delta> > 0 \<Longrightarrow> mdist (capped_metric \<delta> m) x y \<le> \<delta>"
+ by (simp add: capped_metric_mdist)
+
+lemma mball_of_capped_metric [simp]:
+ assumes "x \<in> mspace m" "r > \<delta>" "\<delta> > 0"
+ shows "mball_of (capped_metric \<delta> m) x r = mspace m"
+proof -
+ interpret Metric_space "mspace m" "mdist m"
+ by (simp add: Metric_space_mspace_mdist)
+ have "Metric_space.mball (mspace m) (mdist (capped_metric \<delta> m)) x r \<subseteq> mspace m"
+ by (metis Metric_space.mball_subset_mspace Metric_space_mspace_mdist capped_metric_mspace)
+ moreover have "mspace m \<subseteq> Metric_space.mball (mspace m) (mdist (capped_metric \<delta> m)) x r"
+ by (smt (verit) Metric_space.in_mball Metric_space_mspace_mdist assms capped_metric_mspace mdist_capped subset_eq)
+ ultimately show ?thesis
+ by (simp add: mball_of_def)
+qed
+
+lemma Metric_space_capped_dist[simp]:
+ "Metric_space (mspace m) (Metric_space.capped_dist (mdist m) \<delta>)"
+ using Metric_space.capped_dist Metric_space_mspace_mdist by blast
+
+lemma mtopology_capped_metric:
+ "mtopology_of(capped_metric \<delta> m) = mtopology_of m"
+proof (cases "\<delta> > 0")
+ case True
+ interpret Metric_space "mspace m" "mdist m"
+ by (simp add: Metric_space_mspace_mdist)
+ interpret Cap: Metric_space "mspace m" "mdist (capped_metric \<delta> m)"
+ by (metis Metric_space_mspace_mdist capped_metric_mspace)
+ show ?thesis
+ unfolding topology_eq
+ proof
+ fix S
+ show "openin (mtopology_of (capped_metric \<delta> m)) S = openin (mtopology_of m) S"
+ proof (cases "S \<subseteq> mspace m")
+ case True
+ have "mball x r \<subseteq> Cap.mball x r" for x r
+ by (smt (verit, ccfv_SIG) Cap.in_mball in_mball mdist_capped_le subsetI)
+ moreover have "\<exists>r>0. Cap.mball x r \<subseteq> S" if "r>0" "x \<in> S" and r: "mball x r \<subseteq> S" for r x
+ proof (intro exI conjI)
+ show "min (\<delta>/2) r > 0"
+ using \<open>r>0\<close> \<open>\<delta>>0\<close> by force
+ show "Cap.mball x (min (\<delta>/2) r) \<subseteq> S"
+ using that
+ by clarsimp (smt (verit) capped_metric_mdist field_sum_of_halves in_mball subsetD)
+ qed
+ ultimately have "(\<exists>r>0. Cap.mball x r \<subseteq> S) = (\<exists>r>0. mball x r \<subseteq> S)" if "x \<in> S" for x
+ by (meson subset_trans that)
+ then show ?thesis
+ by (simp add: mtopology_of_def openin_mtopology Cap.openin_mtopology)
+ qed (simp add: openin_closedin_eq)
+ qed
+qed (simp add: capped_metric)
+
+text \<open>Might have been easier to prove this within the locale to start with (using Self)\<close>
+lemma (in Metric_space) mtopology_capped_metric:
+ "Metric_space.mtopology M (capped_dist \<delta>) = mtopology"
+ using mtopology_capped_metric [of \<delta> "metric(M,d)"]
+ by (simp add: Metric_space.mtopology_of capped_dist capped_metric_def)
+
+lemma (in Metric_space) MCauchy_capped_metric:
+ "Metric_space.MCauchy M (capped_dist \<delta>) \<sigma> \<longleftrightarrow> MCauchy \<sigma>"
+proof (cases "\<delta> > 0")
+ case True
+ interpret Cap: Metric_space "M" "capped_dist \<delta>"
+ by (simp add: capped_dist)
+ show ?thesis
+ proof
+ assume \<sigma>: "Cap.MCauchy \<sigma>"
+ show "MCauchy \<sigma>"
+ unfolding MCauchy_def
+ proof (intro conjI strip)
+ show "range \<sigma> \<subseteq> M"
+ using Cap.MCauchy_def \<sigma> by presburger
+ fix \<epsilon> :: real
+ assume "\<epsilon> > 0"
+ with True \<sigma>
+ obtain N where "\<forall>n n'. N \<le> n \<longrightarrow> N \<le> n' \<longrightarrow> capped_dist \<delta> (\<sigma> n) (\<sigma> n') < min \<delta> \<epsilon>"
+ unfolding Cap.MCauchy_def by (metis min_less_iff_conj)
+ with True show "\<exists>N. \<forall>n n'. N \<le> n \<longrightarrow> N \<le> n' \<longrightarrow> d (\<sigma> n) (\<sigma> n') < \<epsilon>"
+ by (force simp: capped_dist_def)
+ qed
+ next
+ assume "MCauchy \<sigma>"
+ then show "Cap.MCauchy \<sigma>"
+ unfolding MCauchy_def Cap.MCauchy_def by (force simp: capped_dist_def)
+ qed
+qed (simp add: capped_dist_def)
+
+
+lemma (in Metric_space) mcomplete_capped_metric:
+ "Metric_space.mcomplete M (capped_dist \<delta>) \<longleftrightarrow> mcomplete"
+ by (simp add: MCauchy_capped_metric Metric_space.mcomplete_def capped_dist mtopology_capped_metric mcomplete_def)
+
+lemma bounded_equivalent_metric:
+ assumes "\<delta> > 0"
+ obtains m' where "mspace m' = mspace m" "mtopology_of m' = mtopology_of m" "\<And>x y. mdist m' x y < \<delta>"
+proof
+ let ?m = "capped_metric (\<delta>/2) m"
+ fix x y
+ show "mdist ?m x y < \<delta>"
+ by (smt (verit, best) assms field_sum_of_halves mdist_capped)
+qed (auto simp: mtopology_capped_metric)
+
+text \<open>A technical lemma needed below\<close>
+lemma Sup_metric_cartesian_product:
+ fixes I m
+ defines "S \<equiv> PiE I (mspace \<circ> m)"
+ defines "D \<equiv> \<lambda>x y. if x \<in> S \<and> y \<in> S then SUP i\<in>I. mdist (m i) (x i) (y i) else 0"
+ defines "m' \<equiv> metric(S,D)"
+ assumes "I \<noteq> {}"
+ and c: "\<And>i x y. \<lbrakk>i \<in> I; x \<in> mspace(m i); y \<in> mspace(m i)\<rbrakk> \<Longrightarrow> mdist (m i) x y \<le> c"
+ shows "Metric_space S D"
+ and "\<forall>x \<in> S. \<forall>y \<in> S. \<forall>b. D x y \<le> b \<longleftrightarrow> (\<forall>i \<in> I. mdist (m i) (x i) (y i) \<le> b)" (is "?the2")
+proof -
+ have bdd: "bdd_above ((\<lambda>i. mdist (m i) (x i) (y i)) ` I)"
+ if "x \<in> S" "y \<in> S" for x y
+ using c that by (force simp: S_def bdd_above_def)
+ have D_iff: "D x y \<le> b \<longleftrightarrow> (\<forall>i \<in> I. mdist (m i) (x i) (y i) \<le> b)"
+ if "x \<in> S" "y \<in> S" for x y b
+ using that \<open>I \<noteq> {}\<close> by (simp add: D_def PiE_iff cSup_le_iff bdd)
+ show "Metric_space S D"
+ proof
+ fix x y
+ show D0: "0 \<le> D x y"
+ using bdd
+ apply (simp add: D_def)
+ by (meson \<open>I \<noteq> {}\<close> cSUP_upper dual_order.trans ex_in_conv mdist_nonneg)
+ show "D x y = D y x"
+ by (simp add: D_def mdist_commute)
+ assume "x \<in> S" and "y \<in> S"
+ then
+ have "D x y = 0 \<longleftrightarrow> (\<forall>i\<in>I. mdist (m i) (x i) (y i) = 0)"
+ using D0 D_iff [of x y 0] nle_le by fastforce
+ also have "... \<longleftrightarrow> x = y"
+ using \<open>x \<in> S\<close> \<open>y \<in> S\<close> by (fastforce simp: S_def PiE_iff extensional_def)
+ finally show "(D x y = 0) \<longleftrightarrow> (x = y)" .
+ fix z
+ assume "z \<in> S"
+ have "mdist (m i) (x i) (z i) \<le> D x y + D y z" if "i \<in> I" for i
+ proof -
+ have "mdist (m i) (x i) (z i) \<le> mdist (m i) (x i) (y i) + mdist (m i) (y i) (z i)"
+ by (metis PiE_E S_def \<open>x \<in> S\<close> \<open>y \<in> S\<close> \<open>z \<in> S\<close> comp_apply mdist_triangle that)
+ also have "... \<le> D x y + D y z"
+ using \<open>x \<in> S\<close> \<open>y \<in> S\<close> \<open>z \<in> S\<close> by (meson D_iff add_mono order_refl that)
+ finally show ?thesis .
+ qed
+ then show "D x z \<le> D x y + D y z"
+ by (simp add: D_iff \<open>x \<in> S\<close> \<open>z \<in> S\<close>)
+ qed
+ then interpret Metric_space S D .
+ show ?the2
+ proof (intro strip)
+ show "(D x y \<le> b) = (\<forall>i\<in>I. mdist (m i) (x i) (y i) \<le> b)"
+ if "x \<in> S" and "y \<in> S" for x y b
+ using that by (simp add: D_iff m'_def)
+ qed
+qed
+
+lemma metrizable_topology_A:
+ assumes "metrizable_space (product_topology X I)"
+ shows "topspace (product_topology X I) = {} \<or> (\<forall>i \<in> I. metrizable_space (X i))"
+ by (meson assms metrizable_space_retraction_map_image retraction_map_product_projection)
+
+lemma metrizable_topology_C:
+ assumes "completely_metrizable_space (product_topology X I)"
+ shows "topspace (product_topology X I) = {} \<or> (\<forall>i \<in> I. completely_metrizable_space (X i))"
+ by (meson assms completely_metrizable_space_retraction_map_image retraction_map_product_projection)
+
+lemma metrizable_topology_B:
+ fixes a X I
+ defines "L \<equiv> {i \<in> I. \<nexists>a. topspace (X i) \<subseteq> {a}}"
+ assumes "topspace (product_topology X I) \<noteq> {}"
+ and met: "metrizable_space (product_topology X I)"
+ and "\<And>i. i \<in> I \<Longrightarrow> metrizable_space (X i)"
+ shows "countable L"
+proof -
+ have "\<And>i. \<exists>p q. i \<in> L \<longrightarrow> p \<in> topspace(X i) \<and> q \<in> topspace(X i) \<and> p \<noteq> q"
+ unfolding L_def by blast
+ then obtain \<phi> \<psi> where \<phi>: "\<And>i. i \<in> L \<Longrightarrow> \<phi> i \<in> topspace(X i) \<and> \<psi> i \<in> topspace(X i) \<and> \<phi> i \<noteq> \<psi> i"
+ by metis
+ obtain z where z: "z \<in> (\<Pi>\<^sub>E i\<in>I. topspace (X i))"
+ using assms(2) by fastforce
+ define p where "p \<equiv> \<lambda>i. if i \<in> L then \<phi> i else z i"
+ define q where "q \<equiv> \<lambda>i j. if j = i then \<psi> i else p j"
+ have p: "p \<in> topspace(product_topology X I)"
+ using z \<phi> by (auto simp: p_def L_def)
+ then have q: "\<And>i. i \<in> L \<Longrightarrow> q i \<in> topspace (product_topology X I)"
+ by (auto simp: L_def q_def \<phi>)
+ have fin: "finite {i \<in> L. q i \<notin> U}" if U: "openin (product_topology X I) U" "p \<in> U" for U
+ proof -
+ obtain V where V: "finite {i \<in> I. V i \<noteq> topspace (X i)}" "(\<forall>i\<in>I. openin (X i) (V i))" "p \<in> Pi\<^sub>E I V" "Pi\<^sub>E I V \<subseteq> U"
+ using U by (force simp: openin_product_topology_alt)
+ moreover
+ have "V x \<noteq> topspace (X x)" if "x \<in> L" and "q x \<notin> U" for x
+ using that V q
+ by (smt (verit, del_insts) PiE_iff q_def subset_eq topspace_product_topology)
+ then have "{i \<in> L. q i \<notin> U} \<subseteq> {i \<in> I. V i \<noteq> topspace (X i)}"
+ by (force simp: L_def)
+ ultimately show ?thesis
+ by (meson finite_subset)
+ qed
+ obtain M d where "Metric_space M d" and XI: "product_topology X I = Metric_space.mtopology M d"
+ using met metrizable_space_def by blast
+ then interpret Metric_space M d
+ by blast
+ define C where "C \<equiv> \<Union>n::nat. {i \<in> L. q i \<notin> mball p (inverse (Suc n))}"
+ have "finite {i \<in> L. q i \<notin> mball p (inverse (real (Suc n)))}" for n
+ using XI p by (intro fin; force)
+ then have "countable C"
+ unfolding C_def
+ by (meson countableI_type countable_UN countable_finite)
+ moreover have "L \<subseteq> C"
+ proof (clarsimp simp: C_def)
+ fix i
+ assume "i \<in> L" and "q i \<in> M" and "p \<in> M"
+ then show "\<exists>n. \<not> d p (q i) < inverse (1 + real n)"
+ using reals_Archimedean [of "d p (q i)"]
+ by (metis \<phi> mdist_pos_eq not_less_iff_gr_or_eq of_nat_Suc p_def q_def)
+ qed
+ ultimately show ?thesis
+ using countable_subset by blast
+qed
+
+lemma metrizable_topology_DD:
+ assumes "topspace (product_topology X I) \<noteq> {}"
+ and co: "countable {i \<in> I. \<nexists>a. topspace (X i) \<subseteq> {a}}"
+ and m: "\<And>i. i \<in> I \<Longrightarrow> X i = mtopology_of (m i)"
+ obtains M d where "Metric_space M d" "product_topology X I = Metric_space.mtopology M d"
+ "(\<And>i. i \<in> I \<Longrightarrow> mcomplete_of (m i)) \<Longrightarrow> Metric_space.mcomplete M d"
+proof (cases "I = {}")
+ case True
+ then show ?thesis
+ by (metis discrete_metric.mcomplete_discrete_metric discrete_metric.mtopology_discrete_metric metric_M_dd product_topology_empty_discrete that)
+next
+ case False
+ obtain nk and C:: "nat set" where nk: "{i \<in> I. \<nexists>a. topspace (X i) \<subseteq> {a}} = nk ` C" and "inj_on nk C"
+ using co by (force simp: countable_as_injective_image_subset)
+ then obtain kn where kn: "\<And>w. w \<in> C \<Longrightarrow> kn (nk w) = w"
+ by (metis inv_into_f_f)
+ define cm where "cm \<equiv> \<lambda>i. capped_metric (inverse(Suc(kn i))) (m i)"
+ have mspace_cm: "mspace (cm i) = mspace (m i)" for i
+ by (simp add: cm_def)
+ have c1: "\<And>i x y. mdist (cm i) x y \<le> 1"
+ by (simp add: cm_def capped_metric_mdist min_le_iff_disj divide_simps)
+ then have bdd: "bdd_above ((\<lambda>i. mdist (cm i) (x i) (y i)) ` I)" for x y
+ by (meson bdd_above.I2)
+ define M where "M \<equiv> Pi\<^sub>E I (mspace \<circ> cm)"
+ define d where "d \<equiv> \<lambda>x y. if x \<in> M \<and> y \<in> M then SUP i\<in>I. mdist (cm i) (x i) (y i) else 0"
+
+ have d_le1: "d x y \<le> 1" for x y
+ using \<open>I \<noteq> {}\<close> c1 by (simp add: d_def bdd cSup_le_iff)
+ with \<open>I \<noteq> {}\<close> Sup_metric_cartesian_product [of I cm]
+ have "Metric_space M d"
+ and *: "\<forall>x\<in>M. \<forall>y\<in>M. \<forall>b. (d x y \<le> b) \<longleftrightarrow> (\<forall>i\<in>I. mdist (cm i) (x i) (y i) \<le> b)"
+ by (auto simp: False bdd M_def d_def cSUP_le_iff intro: c1)
+ then interpret Metric_space M d
+ by metis
+ have le_d: "mdist (cm i) (x i) (y i) \<le> d x y" if "i \<in> I" "x \<in> M" "y \<in> M" for i x y
+ using "*" that by blast
+ have product_m: "PiE I (\<lambda>i. mspace (m i)) = topspace(product_topology X I)"
+ using m by force
+
+ define m' where "m' = metric (M,d)"
+ define J where "J \<equiv> \<lambda>U. {i \<in> I. U i \<noteq> topspace (X i)}"
+ have 1: "\<exists>U. finite (J U) \<and> (\<forall>i\<in>I. openin (X i) (U i)) \<and> x \<in> Pi\<^sub>E I U \<and> Pi\<^sub>E I U \<subseteq> mball z r"
+ if "x \<in> M" "z \<in> M" and r: "0 < r" "d z x < r" for x z r
+ proof -
+ have x: "\<And>i. i \<in> I \<Longrightarrow> x i \<in> topspace(X i)"
+ using M_def m mspace_cm that(1) by auto
+ have z: "\<And>i. i \<in> I \<Longrightarrow> z i \<in> topspace(X i)"
+ using M_def m mspace_cm that(2) by auto
+ obtain R where "0 < R" "d z x < R" "R < r"
+ using r dense by (smt (verit, ccfv_threshold))
+ define U where "U \<equiv> \<lambda>i. if R \<le> inverse(Suc(kn i)) then mball_of (m i) (z i) R else topspace(X i)"
+ show ?thesis
+ proof (intro exI conjI)
+ obtain n where n: "real n * R > 1"
+ using \<open>0 < R\<close> ex_less_of_nat_mult by blast
+ have "finite (nk ` (C \<inter> {..n}))"
+ by force
+ moreover
+ have "\<exists>m. m \<in> C \<and> m \<le> n \<and> i = nk m"
+ if R: "R \<le> inverse (1 + real (kn i))" and "i \<in> I"
+ and neq: "mball_of (m i) (z i) R \<noteq> topspace (X i)" for i
+ proof -
+ interpret MI: Metric_space "mspace (m i)" "mdist (m i)"
+ by auto
+ have "MI.mball (z i) R \<noteq> topspace (X i)"
+ by (metis mball_of_def neq)
+ then have "\<nexists>a. topspace (X i) \<subseteq> {a}"
+ using \<open>0 < R\<close> m subset_antisym \<open>i \<in> I\<close> z by fastforce
+ then have "i \<in> nk ` C"
+ using nk \<open>i \<in> I\<close> by auto
+ then show ?thesis
+ by (smt (verit, ccfv_SIG) R \<open>0 < R\<close> image_iff kn lift_Suc_mono_less_iff mult_mono n not_le_imp_less of_nat_0_le_iff of_nat_Suc right_inverse)
+ qed
+ then have "J U \<subseteq> nk ` (C \<inter> {..n})"
+ by (auto simp: image_iff Bex_def J_def U_def split: if_split_asm)
+ ultimately show "finite (J U)"
+ using finite_subset by blast
+ show "\<forall>i\<in>I. openin (X i) (U i)"
+ by (simp add: Metric_space.openin_mball U_def mball_of_def mtopology_of_def m)
+ have xin: "x \<in> Pi\<^sub>E I (topspace \<circ> X)"
+ using M_def \<open>x \<in> M\<close> x by auto
+ moreover
+ have "\<And>i. \<lbrakk>i \<in> I; R \<le> inverse (1 + real (kn i))\<rbrakk> \<Longrightarrow> mdist (m i) (z i) (x i) < R"
+ by (smt (verit, ccfv_SIG) \<open>d z x < R\<close> capped_metric_mdist cm_def le_d of_nat_Suc that)
+ ultimately show "x \<in> Pi\<^sub>E I U"
+ using m z by (auto simp: U_def PiE_iff)
+ show "Pi\<^sub>E I U \<subseteq> mball z r"
+ proof
+ fix y
+ assume y: "y \<in> Pi\<^sub>E I U"
+ then have "y \<in> M"
+ by (force simp: PiE_iff M_def U_def m mspace_cm split: if_split_asm)
+ moreover
+ have "\<forall>i\<in>I. mdist (cm i) (z i) (y i) \<le> R"
+ by (smt (verit) PiE_mem U_def cm_def in_mball_of inverse_Suc mdist_capped mdist_capped_le y)
+ then have "d z y \<le> R"
+ by (simp add: \<open>y \<in> M\<close> \<open>z \<in> M\<close> *)
+ ultimately show "y \<in> mball z r"
+ using \<open>R < r\<close> \<open>z \<in> M\<close> by force
+ qed
+ qed
+ qed
+ have 2: "\<exists>r>0. mball x r \<subseteq> S"
+ if "finite (J U)" and x: "x \<in> Pi\<^sub>E I U" and S: "Pi\<^sub>E I U \<subseteq> S"
+ and U: "\<And>i. i\<in>I \<Longrightarrow> openin (X i) (U i)"
+ and "x \<in> S" for U S x
+ proof -
+ { fix i
+ assume "i \<in> J U"
+ then have "i \<in> I"
+ by (auto simp: J_def)
+ then have "openin (mtopology_of (m i)) (U i)"
+ using U m by force
+ then have "openin (mtopology_of (cm i)) (U i)"
+ by (simp add: Abstract_Metric_Spaces.mtopology_capped_metric cm_def)
+ then have "\<exists>r>0. mball_of (cm i) (x i) r \<subseteq> U i"
+ using x
+ by (simp add: Metric_space.openin_mtopology PiE_mem \<open>i \<in> I\<close> mball_of_def mtopology_of_def)
+ }
+ then obtain rf where rf: "\<And>j. j \<in> J U \<Longrightarrow> rf j >0 \<and> mball_of (cm j) (x j) (rf j) \<subseteq> U j"
+ by metis
+ define r where "r \<equiv> Min (insert 1 (rf ` J U))"
+ show ?thesis
+ proof (intro exI conjI)
+ show "r > 0"
+ by (simp add: \<open>finite (J U)\<close> r_def rf)
+ have r [simp]: "\<And>j. j \<in> J U \<Longrightarrow> r \<le> rf j" "r \<le> 1"
+ by (auto simp: r_def that(1))
+ have *: "mball_of (cm i) (x i) r \<subseteq> U i" if "i \<in> I" for i
+ proof (cases "i \<in> J U")
+ case True
+ with r show ?thesis
+ by (smt (verit) Metric_space.in_mball Metric_space_mspace_mdist mball_of_def rf subset_eq)
+ next
+ case False
+ then show ?thesis
+ by (simp add: J_def cm_def m subset_eq that)
+ qed
+ show "mball x r \<subseteq> S"
+ by (smt (verit) x * in_mball_of M_def Metric_space.in_mball Metric_space_axioms PiE_iff le_d o_apply subset_eq S)
+ qed
+ qed
+ have 3: "x \<in> M"
+ if \<section>: "\<And>x. x\<in>S \<Longrightarrow> \<exists>U. finite (J U) \<and> (\<forall>i\<in>I. openin (X i) (U i)) \<and> x \<in> Pi\<^sub>E I U \<and> Pi\<^sub>E I U \<subseteq> S"
+ and "x \<in> S" for S x
+ using \<section> [OF \<open>x \<in> S\<close>] m openin_subset by (fastforce simp: M_def PiE_iff cm_def)
+ show thesis
+ proof
+ show "Metric_space M d"
+ using Metric_space_axioms by blast
+ show eq: "product_topology X I = Metric_space.mtopology M d"
+ unfolding topology_eq openin_mtopology openin_product_topology_alt
+ using J_def 1 2 3 subset_iff zero by (smt (verit, ccfv_threshold))
+ show "mcomplete" if "\<And>i. i \<in> I \<Longrightarrow> mcomplete_of (m i)"
+ unfolding mcomplete_def
+ proof (intro strip)
+ fix \<sigma>
+ assume \<sigma>: "MCauchy \<sigma>"
+ have "\<exists>y. i \<in> I \<longrightarrow> limitin (X i) (\<lambda>n. \<sigma> n i) y sequentially" for i
+ proof (cases "i \<in> I")
+ case True
+ interpret MI: Metric_space "mspace (m i)" "mdist (m i)"
+ by auto
+ have "\<And>\<sigma>. MI.MCauchy \<sigma> \<longrightarrow> (\<exists>x. limitin MI.mtopology \<sigma> x sequentially)"
+ by (meson MI.mcomplete_def True mcomplete_of_def that)
+ moreover have "MI.MCauchy (\<lambda>n. \<sigma> n i)"
+ unfolding MI.MCauchy_def
+ proof (intro conjI strip)
+ show "range (\<lambda>n. \<sigma> n i) \<subseteq> mspace (m i)"
+ by (smt (verit, ccfv_threshold) MCauchy_def PiE_iff True \<sigma> eq image_subset_iff m topspace_mtopology topspace_mtopology_of topspace_product_topology)
+ fix \<epsilon>::real
+ define r where "r \<equiv> min \<epsilon> (inverse(Suc (kn i)))"
+ assume "\<epsilon> > 0"
+ then have "r > 0"
+ by (simp add: r_def)
+ then obtain N where N: "\<And>n n'. N \<le> n \<and> N \<le> n' \<Longrightarrow> d (\<sigma> n) (\<sigma> n') < r"
+ using \<sigma> unfolding MCauchy_def by meson
+ show "\<exists>N. \<forall>n n'. N \<le> n \<longrightarrow> N \<le> n' \<longrightarrow> mdist (m i) (\<sigma> n i) (\<sigma> n' i) < \<epsilon>"
+ proof (intro strip exI)
+ fix n n'
+ assume "N \<le> n" and "N \<le> n'"
+ then have "mdist (cm i) (\<sigma> n i) (\<sigma> n' i) < r"
+ using *
+ by (smt (verit) Metric_space.MCauchy_def Metric_space_axioms N True \<sigma> rangeI subsetD)
+ then
+ show "mdist (m i) (\<sigma> n i) (\<sigma> n' i) < \<epsilon>"
+ unfolding cm_def r_def
+ by (smt (verit, ccfv_SIG) capped_metric_mdist)
+ qed
+ qed
+ ultimately show ?thesis
+ by (simp add: m mtopology_of_def)
+ qed auto
+ then obtain y where "\<And>i. i \<in> I \<Longrightarrow> limitin (X i) (\<lambda>n. \<sigma> n i) (y i) sequentially"
+ by metis
+ with \<sigma> show "\<exists>x. limitin mtopology \<sigma> x sequentially"
+ apply (rule_tac x="\<lambda>i\<in>I. y i" in exI)
+ apply (simp add: MCauchy_def limitin_componentwise flip: eq)
+ by (metis eq eventually_at_top_linorder range_subsetD topspace_mtopology topspace_product_topology)
+ qed
+ qed
+qed
+
+
+lemma metrizable_topology_D:
+ assumes "topspace (product_topology X I) \<noteq> {}"
+ and co: "countable {i \<in> I. \<nexists>a. topspace (X i) \<subseteq> {a}}"
+ and met: "\<And>i. i \<in> I \<Longrightarrow> metrizable_space (X i)"
+ shows "metrizable_space (product_topology X I)"
+proof -
+ have "\<And>i. i \<in> I \<Longrightarrow> \<exists>m. X i = mtopology_of m"
+ by (metis Metric_space.mtopology_of met metrizable_space_def)
+ then obtain m where m: "\<And>i. i \<in> I \<Longrightarrow> X i = mtopology_of (m i)"
+ by metis
+ then show ?thesis
+ using metrizable_topology_DD [of X I m] assms by (force simp: metrizable_space_def)
+qed
+
+lemma metrizable_topology_E:
+ assumes "topspace (product_topology X I) \<noteq> {}"
+ and "countable {i \<in> I. \<nexists>a. topspace (X i) \<subseteq> {a}}"
+ and met: "\<And>i. i \<in> I \<Longrightarrow> completely_metrizable_space (X i)"
+ shows "completely_metrizable_space (product_topology X I)"
+proof -
+ have "\<And>i. i \<in> I \<Longrightarrow> \<exists>m. mcomplete_of m \<and> X i = mtopology_of m"
+ using met Metric_space.mtopology_of Metric_space.mcomplete_of unfolding completely_metrizable_space_def
+ by metis
+ then obtain m where "\<And>i. i \<in> I \<Longrightarrow> mcomplete_of (m i) \<and> X i = mtopology_of (m i)"
+ by metis
+ then show ?thesis
+ using metrizable_topology_DD [of X I m] assms unfolding metrizable_space_def
+ by (metis (full_types) completely_metrizable_space_def)
+qed
+
+
+lemma metrizable_space_product_topology:
+ "metrizable_space (product_topology X I) \<longleftrightarrow>
+ topspace (product_topology X I) = {} \<or>
+ countable {i \<in> I. \<not> (\<exists>a. topspace(X i) \<subseteq> {a})} \<and>
+ (\<forall>i \<in> I. metrizable_space (X i))"
+ by (metis (mono_tags, lifting) empty_metrizable_space metrizable_topology_A metrizable_topology_B metrizable_topology_D)
+
+lemma completely_metrizable_space_product_topology:
+ "completely_metrizable_space (product_topology X I) \<longleftrightarrow>
+ topspace (product_topology X I) = {} \<or>
+ countable {i \<in> I. \<not> (\<exists>a. topspace(X i) \<subseteq> {a})} \<and>
+ (\<forall>i \<in> I. completely_metrizable_space (X i))"
+ by (metis (mono_tags, lifting) completely_metrizable_imp_metrizable_space empty_completely_metrizable_space metrizable_topology_B metrizable_topology_C metrizable_topology_E)
+
+
+lemma completely_metrizable_Euclidean_space:
+ "completely_metrizable_space(Euclidean_space n)"
+ unfolding Euclidean_space_def
+proof (rule completely_metrizable_space_closedin)
+ show "completely_metrizable_space (powertop_real (UNIV::nat set))"
+ by (simp add: completely_metrizable_space_product_topology completely_metrizable_space_euclidean)
+ show "closedin (powertop_real UNIV) {x. \<forall>i\<ge>n. x i = 0}"
+ using closedin_Euclidean_space topspace_Euclidean_space by auto
+qed
+
+lemma metrizable_Euclidean_space:
+ "metrizable_space(Euclidean_space n)"
+ by (simp add: completely_metrizable_Euclidean_space completely_metrizable_imp_metrizable_space)
+
+lemma locally_connected_Euclidean_space:
+ "locally_connected_space(Euclidean_space n)"
+ by (simp add: locally_path_connected_Euclidean_space locally_path_connected_imp_locally_connected_space)
+
end
--- a/src/HOL/Analysis/Abstract_Topological_Spaces.thy Wed Jun 28 14:45:29 2023 +0200
+++ b/src/HOL/Analysis/Abstract_Topological_Spaces.thy Wed Jun 28 16:01:34 2023 +0200
@@ -3,7 +3,7 @@
section \<open>Various Forms of Topological Spaces\<close>
theory Abstract_Topological_Spaces
- imports Lindelof_Spaces Locally Sum_Topology FSigma
+ imports Lindelof_Spaces Locally Abstract_Euclidean_Space Sum_Topology FSigma
begin
@@ -1122,7 +1122,7 @@
lemma open_map_Kolmogorov_quotient_gen:
- "open_map (subtopology X S) (subtopology X (image (Kolmogorov_quotient X) S)) (Kolmogorov_quotient X)"
+ "open_map (subtopology X S) (subtopology X (Kolmogorov_quotient X ` S)) (Kolmogorov_quotient X)"
proof (clarsimp simp: open_map_def openin_subtopology_alt image_iff)
fix U
assume "openin X U"
@@ -1233,11 +1233,10 @@
lemma Kolmogorov_quotient_lift_exists:
assumes "S \<subseteq> topspace X" "t0_space Y" and f: "continuous_map (subtopology X S) Y f"
- obtains g where "continuous_map (subtopology X (image (Kolmogorov_quotient X) S)) Y g"
+ obtains g where "continuous_map (subtopology X (Kolmogorov_quotient X ` S)) Y g"
"\<And>x. x \<in> S \<Longrightarrow> g(Kolmogorov_quotient X x) = f x"
proof -
- have "\<And>x y. \<lbrakk>x \<in> S; y \<in> S; Kolmogorov_quotient X x = Kolmogorov_quotient X y\<rbrakk>
- \<Longrightarrow> f x = f y"
+ have "\<And>x y. \<lbrakk>x \<in> S; y \<in> S; Kolmogorov_quotient X x = Kolmogorov_quotient X y\<rbrakk> \<Longrightarrow> f x = f y"
using assms
apply (simp add: Kolmogorov_quotient_eq t0_space_def continuous_map_def Int_absorb1 openin_subtopology)
by (smt (verit, del_insts) Int_iff mem_Collect_eq)
@@ -1253,8 +1252,7 @@
subsection\<open>Closed diagonals and graphs\<close>
lemma Hausdorff_space_closedin_diagonal:
- "Hausdorff_space X \<longleftrightarrow>
- closedin (prod_topology X X) ((\<lambda>x. (x,x)) ` topspace X)"
+ "Hausdorff_space X \<longleftrightarrow> closedin (prod_topology X X) ((\<lambda>x. (x,x)) ` topspace X)"
proof -
have \<section>: "((\<lambda>x. (x, x)) ` topspace X) \<subseteq> topspace X \<times> topspace X"
by auto
@@ -1296,6 +1294,26 @@
qed
qed
+lemma forall_in_closure_of:
+ assumes "x \<in> X closure_of S" "\<And>x. x \<in> S \<Longrightarrow> P x"
+ and "closedin X {x \<in> topspace X. P x}"
+ shows "P x"
+ by (smt (verit, ccfv_threshold) Diff_iff assms closedin_def in_closure_of mem_Collect_eq)
+
+lemma forall_in_closure_of_eq:
+ assumes x: "x \<in> X closure_of S"
+ and Y: "Hausdorff_space Y"
+ and f: "continuous_map X Y f" and g: "continuous_map X Y g"
+ and fg: "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
+ shows "f x = g x"
+proof -
+ have "closedin X {x \<in> topspace X. f x = g x}"
+ using Y closedin_continuous_maps_eq f g by blast
+ then show ?thesis
+ using forall_in_closure_of [OF x fg]
+ by fastforce
+qed
+
lemma retract_of_space_imp_closedin:
assumes "Hausdorff_space X" and S: "S retract_of_space X"
shows "closedin X S"
@@ -3067,6 +3085,23 @@
qed
qed
+lemma locally_compact_space_euclidean:
+ "locally_compact_space (euclidean::'a::heine_borel topology)"
+ unfolding locally_compact_space_def
+proof (intro strip)
+ fix x::'a
+ assume "x \<in> topspace euclidean"
+ have "ball x 1 \<subseteq> cball x 1"
+ by auto
+ then show "\<exists>U K. openin euclidean U \<and> compactin euclidean K \<and> x \<in> U \<and> U \<subseteq> K"
+ by (metis Elementary_Metric_Spaces.open_ball centre_in_ball compact_cball compactin_euclidean_iff open_openin zero_less_one)
+qed
+
+lemma locally_compact_Euclidean_space:
+ "locally_compact_space(Euclidean_space n)"
+ using homeomorphic_locally_compact_space [OF homeomorphic_Euclidean_space_product_topology]
+ using locally_compact_space_product_topology locally_compact_space_euclidean by fastforce
+
proposition quotient_map_prod_right:
assumes loc: "locally_compact_space Z"
and reg: "Hausdorff_space Z \<or> regular_space Z"
--- a/src/HOL/Analysis/Abstract_Topology.thy Wed Jun 28 14:45:29 2023 +0200
+++ b/src/HOL/Analysis/Abstract_Topology.thy Wed Jun 28 16:01:34 2023 +0200
@@ -3033,7 +3033,7 @@
definition homeomorphic_space (infixr "homeomorphic'_space" 50)
where "X homeomorphic_space Y \<equiv> \<exists>f g. homeomorphic_maps X Y f g"
-lemma homeomorphic_space_refl: "X homeomorphic_space X"
+lemma homeomorphic_space_refl [iff]: "X homeomorphic_space X"
by (meson homeomorphic_maps_id homeomorphic_space_def)
lemma homeomorphic_space_sym:
--- a/src/HOL/Analysis/Abstract_Topology_2.thy Wed Jun 28 14:45:29 2023 +0200
+++ b/src/HOL/Analysis/Abstract_Topology_2.thy Wed Jun 28 16:01:34 2023 +0200
@@ -1644,6 +1644,11 @@
shows "(x / (1 + \<bar>x\<bar>) = y) \<longleftrightarrow> (\<bar>y\<bar> < 1 \<and> y / (1 - \<bar>y\<bar>) = x)"
using real_grow_shrink by (fastforce simp add: distrib_left)
+lemma real_shrink_eq:
+ fixes x y::real
+ shows "(x / (1 + \<bar>x\<bar>) = y / (1 + \<bar>y\<bar>)) \<longleftrightarrow> x = y"
+ by (metis real_shrink_Galois)
+
lemma real_shrink_lt:
fixes x::real
shows "x / (1 + \<bar>x\<bar>) < y / (1 + \<bar>y\<bar>) \<longleftrightarrow> x < y"
--- a/src/HOL/Analysis/Function_Topology.thy Wed Jun 28 14:45:29 2023 +0200
+++ b/src/HOL/Analysis/Function_Topology.thy Wed Jun 28 16:01:34 2023 +0200
@@ -194,6 +194,10 @@
unfolding product_topology_def using PiE_def by (auto)
qed
+lemma topspace_product_topology_alt:
+ "topspace (product_topology X I) = {x \<in> extensional I. \<forall>i \<in> I. x i \<in> topspace(X i)}"
+ by (fastforce simp: PiE_iff)
+
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)"
@@ -1577,4 +1581,76 @@
then show ?thesis by metis
qed
+subsection \<open>Limits\<close>
+
+text \<open>The original HOL Light proof was a mess, yuk\<close>
+lemma limitin_componentwise:
+ "limitin (product_topology X I) f l F \<longleftrightarrow>
+ l \<in> extensional I \<and>
+ eventually (\<lambda>a. f a \<in> topspace(product_topology X I)) F \<and>
+ (\<forall>i \<in> I. limitin (X i) (\<lambda>c. f c i) (l i) F)"
+ (is "?L \<longleftrightarrow> _ \<and> ?R1 \<and> ?R2")
+proof (cases "l \<in> extensional I")
+ case l: True
+ show ?thesis
+ proof (cases "\<forall>i\<in>I. l i \<in> topspace (X i)")
+ case True
+ have ?R1 if ?L
+ by (metis limitin_subtopology subtopology_topspace that)
+ moreover
+ have ?R2 if ?L
+ unfolding limitin_def
+ proof (intro conjI strip)
+ fix i U
+ assume "i \<in> I" and U: "openin (X i) U \<and> l i \<in> U"
+ then have "openin (product_topology X I) ({y. y i \<in> U} \<inter> topspace (product_topology X I))"
+ unfolding openin_product_topology arbitrary_union_of_relative_to [symmetric]
+ apply (simp add: relative_to_def topspace_product_topology_alt)
+ by (smt (verit, del_insts) Collect_cong arbitrary_union_of_inc finite_intersection_of_inc inf_commute)
+ moreover have "l \<in> {y. y i \<in> U} \<inter> topspace (product_topology X I)"
+ using U True l by (auto simp: extensional_def)
+ ultimately have "eventually (\<lambda>x. f x \<in> {y. y i \<in> U} \<inter> topspace (product_topology X I)) F"
+ by (metis limitin_def that)
+ then show "\<forall>\<^sub>F x in F. f x i \<in> U"
+ by (simp add: eventually_conj_iff)
+ qed (use True in auto)
+ moreover
+ have ?L if R1: ?R1 and R2: ?R2
+ unfolding limitin_def openin_product_topology all_union_of imp_conjL arbitrary_def
+ proof (intro conjI strip)
+ show l: "l \<in> topspace (product_topology X I)"
+ by (simp add: PiE_iff True l)
+ fix \<V>
+ assume "\<V> \<subseteq> Collect (finite intersection_of (\<lambda>F. \<exists>i U. F = {f. f i \<in> U} \<and> i \<in> I \<and> openin (X i) U)
+ relative_to topspace (product_topology X I))"
+ and "l \<in> \<Union> \<V>"
+ then obtain \<W> where "finite \<W>" and \<W>X: "\<forall>X\<in>\<W>. l \<in> X"
+ and \<W>: "\<And>C. C \<in> \<W> \<Longrightarrow> C \<in> {{x. x i \<in> U} |i U. i \<in> I \<and> openin (X i) U}"
+ and \<W>\<V>: "topspace (product_topology X I) \<inter> \<Inter> \<W> \<in> \<V>"
+ by (fastforce simp: intersection_of_def relative_to_def subset_eq)
+ have "\<forall>\<^sub>F x in F. f x \<in> topspace (product_topology X I) \<inter> \<Inter>\<W>"
+ proof -
+ have "\<And>W. W \<in> {{x. x i \<in> U} | i U. i \<in> I \<and> openin (X i) U} \<Longrightarrow> W \<in> \<W> \<Longrightarrow> \<forall>\<^sub>F x in F. f x \<in> W"
+ using \<W>X R2 by (auto simp: limitin_def)
+ with \<W> have "\<forall>\<^sub>F x in F. \<forall>W\<in>\<W>. f x \<in> W"
+ by (simp add: \<open>finite \<W>\<close> eventually_ball_finite)
+ with R1 show ?thesis
+ by (simp add: eventually_conj_iff)
+ qed
+ then show "\<forall>\<^sub>F x in F. f x \<in> \<Union>\<V>"
+ by (smt (verit, ccfv_threshold) \<W>\<V> UnionI eventually_mono)
+ qed
+ ultimately show ?thesis
+ using l by blast
+ next
+ case False
+ then show ?thesis
+ by (metis PiE_iff limitin_def topspace_product_topology)
+ qed
+next
+ case False
+ then show ?thesis
+ by (simp add: limitin_def PiE_iff)
+qed
+
end
--- a/src/HOL/Analysis/Ordered_Euclidean_Space.thy Wed Jun 28 14:45:29 2023 +0200
+++ b/src/HOL/Analysis/Ordered_Euclidean_Space.thy Wed Jun 28 16:01:34 2023 +0200
@@ -157,27 +157,27 @@
shows "((\<lambda>i. inf (X i) (Y i)) \<longlongrightarrow> inf x y) net"
unfolding inf_min eucl_inf by (intro assms tendsto_intros)
-lemma tendsto_Inf:
+lemma tendsto_Inf[tendsto_intros]:
fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::ordered_euclidean_space"
assumes "finite K" "\<And>i. i \<in> K \<Longrightarrow> ((\<lambda>x. f x i) \<longlongrightarrow> l i) F"
shows "((\<lambda>x. Inf (f x ` K)) \<longlongrightarrow> Inf (l ` K)) F"
using assms
by (induction K rule: finite_induct) (auto simp: cInf_insert_If tendsto_inf)
-lemma tendsto_Sup:
+lemma tendsto_Sup[tendsto_intros]:
fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::ordered_euclidean_space"
assumes "finite K" "\<And>i. i \<in> K \<Longrightarrow> ((\<lambda>x. f x i) \<longlongrightarrow> l i) F"
shows "((\<lambda>x. Sup (f x ` K)) \<longlongrightarrow> Sup (l ` K)) F"
using assms
by (induction K rule: finite_induct) (auto simp: cSup_insert_If tendsto_sup)
-lemma continuous_map_Inf:
+lemma continuous_map_Inf [continuous_intros]:
fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::ordered_euclidean_space"
assumes "finite K" "\<And>i. i \<in> K \<Longrightarrow> continuous_map X euclidean (\<lambda>x. f x i)"
shows "continuous_map X euclidean (\<lambda>x. INF i\<in>K. f x i)"
using assms by (simp add: continuous_map_atin tendsto_Inf)
-lemma continuous_map_Sup:
+lemma continuous_map_Sup [continuous_intros]:
fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::ordered_euclidean_space"
assumes "finite K" "\<And>i. i \<in> K \<Longrightarrow> continuous_map X euclidean (\<lambda>x. f x i)"
shows "continuous_map X euclidean (\<lambda>x. SUP i\<in>K. f x i)"
--- a/src/HOL/Analysis/Urysohn.thy Wed Jun 28 14:45:29 2023 +0200
+++ b/src/HOL/Analysis/Urysohn.thy Wed Jun 28 16:01:34 2023 +0200
@@ -1,8 +1,8 @@
-(* Title: HOL/Analysis/Arcwise_Connected.thy
+(* Title: HOL/Analysis/Urysohn.thy
Authors: LC Paulson, based on material from HOL Light
*)
-section \<open>Urysohn lemma and its Consequences\<close>
+section \<open>The Urysohn lemma, its consequences and other advanced material about metric spaces\<close>
theory Urysohn
imports Abstract_Topological_Spaces Abstract_Metric_Spaces Infinite_Sum Arcwise_Connected
@@ -71,7 +71,7 @@
have **: "r \<le> f x" if r: "r \<in> dyadics \<inter> {0..1}" "x \<notin> G r" for r x
proof (rule *)
show "r \<le> s" if "s \<in> dyadics \<inter> {0..1}" and "x \<in> G s" for s :: real
- using that r G [of s r] by (force simp add: dest: closure_of_subset openin_subset)
+ using that r G [of s r] by (force simp: dest: closure_of_subset openin_subset)
qed (use that in force)
have "\<exists>U. openin X U \<and> x \<in> U \<and> (\<forall>y \<in> U. \<bar>f y - f x\<bar> < \<epsilon>)"
@@ -133,7 +133,7 @@
then show "x \<in> G'"
by (simp add: G'_def that)
show "\<forall>y \<in> G'. \<bar>f y - f x\<bar> < \<epsilon>"
- using ** f_le1 in_closure_of r by (fastforce simp add: True G'_def)
+ using ** f_le1 in_closure_of r by (fastforce simp: True G'_def)
qed
next
case False
@@ -161,7 +161,7 @@
qed
qed
then have contf: "continuous_map X (top_of_set {0..1}) f"
- by (force simp add: Met_TC.continuous_map_to_metric dist_real_def continuous_map_in_subtopology fim simp flip: Met_TC.mtopology_is_euclideanreal)
+ by (force simp: Met_TC.continuous_map_to_metric dist_real_def continuous_map_in_subtopology fim simp flip: mtopology_is_euclidean)
define g where "g \<equiv> \<lambda>x. a + (b - a) * f x"
show thesis
proof
@@ -328,7 +328,7 @@
\<Longrightarrow> \<exists>i<j. x \<in> X closure_of k i"
by (metis in_closure_of linorder_not_less opek openin_subset subsetD)
then show "disjnt U V"
- by (force simp add: U_def V_def disjnt_iff)
+ by (force simp: U_def V_def disjnt_iff)
qed
qed
qed
@@ -1117,7 +1117,7 @@
using Urysohn_lemma [of "subtopology X M" K "M-U" 0 1] by auto
then obtain g::"'a\<Rightarrow>real" where contg: "continuous_map (subtopology X M) euclidean g" and gim: "g ` M \<subseteq> {0..1}"
and g0: "\<And>x. x \<in> K \<Longrightarrow> g x = 0" and g1: "\<And>x. \<lbrakk>x \<in> M; x \<notin> U\<rbrakk> \<Longrightarrow> g x = 1"
- using \<open>M \<subseteq> topspace X\<close> by (force simp add: continuous_map_in_subtopology image_subset_iff)
+ using \<open>M \<subseteq> topspace X\<close> by (force simp: continuous_map_in_subtopology image_subset_iff)
show "\<exists>f::'a\<Rightarrow>real. continuous_map X (top_of_set {0..1}) f \<and> f x = 0 \<and> f ` S \<subseteq> {1}"
proof (intro exI conjI)
show "continuous_map X (top_of_set {0..1}) (\<lambda>x. if x \<in> M then g x else 1)"
@@ -1703,7 +1703,7 @@
(is "?lhs = ?rhs")
proof
show "?lhs \<subseteq> ?rhs"
- by (force simp add: topspace_def openin_Alexandroff_compactification)
+ by (force simp: topspace_def openin_Alexandroff_compactification)
have "None \<in> topspace (Alexandroff_compactification X)"
by (meson closedin_empty compactin_empty insert_subset openin_Alexandroff_compactification openin_subset)
moreover have "Some x \<in> topspace (Alexandroff_compactification X)"
@@ -1877,7 +1877,7 @@
then show ?thesis
using t1_space_one_point_compactification [of "Alexandroff_compactification X" None]
using embedding_map_Some embedding_map_imp_homeomorphic_space homeomorphic_t1_space
- by (fastforce simp add: compactin_subtopology closedin_Alexandroff_I closedin_subtopology)
+ by (fastforce simp: compactin_subtopology closedin_Alexandroff_I closedin_subtopology)
qed
@@ -1976,8 +1976,8 @@
regular_space (subtopology ?Y (topspace ?Y - {None})) \<and> locally_compact_space (subtopology ?Y (topspace ?Y - {None}))"
by (rule regular_space_one_point_compactification) (auto simp: compactin_subtopology closedin_subtopology closedin_Alexandroff_I)
also have "... \<longleftrightarrow> regular_space X \<and> locally_compact_space X"
- using embedding_map_Some embedding_map_imp_homeomorphic_space homeomorphic_locally_compact_space homeomorphic_regular_space
- by fastforce
+ by (metis embedding_map_Some embedding_map_imp_homeomorphic_space homeomorphic_locally_compact_space
+ homeomorphic_regular_space topspace_Alexandroff_compactification_delete)
finally show ?thesis .
qed
@@ -2130,7 +2130,7 @@
qed auto
then show ?thesis
unfolding Hausdorff_space_closedin_diagonal closure_of_subset_eq [symmetric]
- by (force simp add: closure_of_def)
+ by (force simp: closure_of_def)
qed
qed
next
@@ -2237,4 +2237,2065 @@
using homeomorphic_space_sym homeomorphic_space homeomorphic_map_def by blast
qed
+subsection\<open>Extending continuous maps "pointwise" in a regular space\<close>
+
+lemma continuous_map_on_intermediate_closure_of:
+ assumes Y: "regular_space Y"
+ and T: "T \<subseteq> X closure_of S"
+ and f: "\<And>t. t \<in> T \<Longrightarrow> limitin Y f (f t) (atin_within X t S)"
+ shows "continuous_map (subtopology X T) Y f"
+proof (clarsimp simp add: continuous_map_atin)
+ fix a
+ assume "a \<in> topspace X" and "a \<in> T"
+ have "f ` T \<subseteq> topspace Y"
+ by (metis f image_subsetI limitin_topspace)
+ have "\<forall>\<^sub>F x in atin_within X a T. f x \<in> W"
+ if W: "openin Y W" "f a \<in> W" for W
+ proof -
+ obtain V C where "openin Y V" "closedin Y C" "f a \<in> V" "V \<subseteq> C" "C \<subseteq> W"
+ by (metis Y W neighbourhood_base_of neighbourhood_base_of_closedin)
+ have "\<forall>\<^sub>F x in atin_within X a S. f x \<in> V"
+ by (metis \<open>a \<in> T\<close> \<open>f a \<in> V\<close> \<open>openin Y V\<close> f limitin_def)
+ then obtain U where "openin X U" "a \<in> U" and U: "\<forall>x \<in> U - {a}. x \<in> S \<longrightarrow> f x \<in> V"
+ by (smt (verit) Diff_iff \<open>a \<in> topspace X\<close> eventually_atin_within insert_iff)
+ moreover have "f z \<in> W" if "z \<in> U" "z \<noteq> a" "z \<in> T" for z
+ proof -
+ have "z \<in> topspace X"
+ using \<open>openin X U\<close> openin_subset \<open>z \<in> U\<close> by blast
+ then have "f z \<in> topspace Y"
+ using \<open>f ` T \<subseteq> topspace Y\<close> \<open>z \<in> T\<close> by blast
+ { assume "f z \<in> topspace Y" "f z \<notin> C"
+ then have "\<forall>\<^sub>F x in atin_within X z S. f x \<in> topspace Y - C"
+ by (metis Diff_iff \<open>closedin Y C\<close> closedin_def f limitinD \<open>z \<in> T\<close>)
+ then obtain U' where U': "openin X U'" "z \<in> U'"
+ "\<And>x. x \<in> U' - {z} \<Longrightarrow> x \<in> S \<Longrightarrow> f x \<notin> C"
+ by (smt (verit) Diff_iff \<open>z \<in> topspace X\<close> eventually_atin_within insertCI)
+ then have *: "\<And>D. z \<in> D \<and> openin X D \<Longrightarrow> \<exists>y. y \<in> S \<and> y \<in> D"
+ by (meson T in_closure_of subsetD \<open>z \<in> T\<close>)
+ have False
+ using * [of "U \<inter> U'"] U' U \<open>V \<subseteq> C\<close> \<open>f a \<in> V\<close> \<open>f z \<notin> C\<close> \<open>openin X U\<close> that
+ by blast
+ }
+ then show ?thesis
+ using \<open>C \<subseteq> W\<close> \<open>f z \<in> topspace Y\<close> by auto
+ qed
+ ultimately have "\<exists>U. openin X U \<and> a \<in> U \<and> (\<forall>x \<in> U - {a}. x \<in> T \<longrightarrow> f x \<in> W)"
+ by blast
+ then show ?thesis
+ using eventually_atin_within by fastforce
+ qed
+ then show "limitin Y f (f a) (atin (subtopology X T) a)"
+ by (metis \<open>a \<in> T\<close> atin_subtopology_within f limitin_def)
+qed
+
+
+lemma continuous_map_on_intermediate_closure_of_eq:
+ assumes "regular_space Y" "S \<subseteq> T" and Tsub: "T \<subseteq> X closure_of S"
+ shows "continuous_map (subtopology X T) Y f \<longleftrightarrow> (\<forall>t \<in> T. limitin Y f (f t) (atin_within X t S))"
+ (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+ assume L: ?lhs
+ show ?rhs
+ proof (clarsimp simp add: continuous_map_atin)
+ fix x
+ assume "x \<in> T"
+ with L Tsub closure_of_subset_topspace
+ have "limitin Y f (f x) (atin (subtopology X T) x)"
+ by (fastforce simp: continuous_map_atin)
+ then show "limitin Y f (f x) (atin_within X x S)"
+ using \<open>x \<in> T\<close> \<open>S \<subseteq> T\<close>
+ by (force simp: limitin_def atin_subtopology_within eventually_atin_within)
+ qed
+next
+ show "?rhs \<Longrightarrow> ?lhs"
+ using assms continuous_map_on_intermediate_closure_of by blast
+qed
+
+lemma continuous_map_extension_pointwise_alt:
+ assumes \<section>: "regular_space Y" "S \<subseteq> T" "T \<subseteq> X closure_of S"
+ and f: "continuous_map (subtopology X S) Y f"
+ and lim: "\<And>t. t \<in> T-S \<Longrightarrow> \<exists>l. limitin Y f l (atin_within X t S)"
+ obtains g where "continuous_map (subtopology X T) Y g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof -
+ obtain g where g: "\<And>t. t \<in> T \<and> t \<notin> S \<Longrightarrow> limitin Y f (g t) (atin_within X t S)"
+ by (metis Diff_iff lim)
+ let ?h = "\<lambda>x. if x \<in> S then f x else g x"
+ show thesis
+ proof
+ have T: "T \<subseteq> topspace X"
+ using \<section> closure_of_subset_topspace by fastforce
+ have "limitin Y ?h (f t) (atin_within X t S)" if "t \<in> T" "t \<in> S" for t
+ proof -
+ have "limitin Y f (f t) (atin_within X t S)"
+ by (meson T f limit_continuous_map_within subset_eq that)
+ then show ?thesis
+ by (simp add: eventually_atin_within limitin_def)
+ qed
+ moreover have "limitin Y ?h (g t) (atin_within X t S)" if "t \<in> T" "t \<notin> S" for t
+ by (smt (verit, del_insts) eventually_atin_within g limitin_def that)
+ ultimately show "continuous_map (subtopology X T) Y ?h"
+ unfolding continuous_map_on_intermediate_closure_of_eq [OF \<section>]
+ by (auto simp: \<section> atin_subtopology_within)
+ qed auto
+qed
+
+
+lemma continuous_map_extension_pointwise:
+ assumes "regular_space Y" "S \<subseteq> T" and Tsub: "T \<subseteq> X closure_of S"
+ and ex: " \<And>x. x \<in> T \<Longrightarrow> \<exists>g. continuous_map (subtopology X (insert x S)) Y g \<and>
+ (\<forall>x \<in> S. g x = f x)"
+ obtains g where "continuous_map (subtopology X T) Y g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof (rule continuous_map_extension_pointwise_alt)
+ show "continuous_map (subtopology X S) Y f"
+ proof (clarsimp simp add: continuous_map_atin)
+ fix t
+ assume "t \<in> topspace X" and "t \<in> S"
+ then obtain g where g: "limitin Y g (g t) (atin (subtopology X (insert t S)) t)" and gf: "\<forall>x \<in> S. g x = f x"
+ by (metis Int_iff \<open>S \<subseteq> T\<close> continuous_map_atin ex inf.orderE insert_absorb topspace_subtopology)
+ with \<open>t \<in> S\<close> show "limitin Y f (f t) (atin (subtopology X S) t)"
+ by (simp add: limitin_def atin_subtopology_within_if eventually_atin_within gf insert_absorb)
+ qed
+ show "\<exists>l. limitin Y f l (atin_within X t S)" if "t \<in> T - S" for t
+ proof -
+ obtain g where g: "continuous_map (subtopology X (insert t S)) Y g" and gf: "\<forall>x \<in> S. g x = f x"
+ using \<open>S \<subseteq> T\<close> ex \<open>t \<in> T - S\<close> by force
+ then have "limitin Y g (g t) (atin_within X t (insert t S))"
+ using Tsub in_closure_of limit_continuous_map_within that by fastforce
+ then show ?thesis
+ unfolding limitin_def
+ by (smt (verit) eventually_atin_within gf subsetD subset_insertI)
+ qed
+qed (use assms in auto)
+
+
+subsection\<open>Extending Cauchy continuous functions to the closure\<close>
+
+lemma Cauchy_continuous_map_extends_to_continuous_closure_of_aux:
+ assumes m2: "mcomplete_of m2" and f: "Cauchy_continuous_map (submetric m1 S) m2 f"
+ and "S \<subseteq> mspace m1"
+ obtains g
+ where "continuous_map (subtopology (mtopology_of m1) (mtopology_of m1 closure_of S))
+ (mtopology_of m2) g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof (rule continuous_map_extension_pointwise_alt)
+ interpret L: Metric_space12 "mspace m1" "mdist m1" "mspace m2" "mdist m2"
+ by (simp add: Metric_space12_mspace_mdist)
+ interpret S: Metric_space "S \<inter> mspace m1" "mdist m1"
+ by (simp add: L.M1.subspace)
+ show "regular_space (mtopology_of m2)"
+ by (simp add: Metric_space.regular_space_mtopology mtopology_of_def)
+ show "S \<subseteq> mtopology_of m1 closure_of S"
+ by (simp add: assms(3) closure_of_subset)
+ show "continuous_map (subtopology (mtopology_of m1) S) (mtopology_of m2) f"
+ by (metis Cauchy_continuous_imp_continuous_map f mtopology_of_submetric)
+ fix a
+ assume a: "a \<in> mtopology_of m1 closure_of S - S"
+ then obtain \<sigma> where ran\<sigma>: "range \<sigma> \<subseteq> S" "range \<sigma> \<subseteq> mspace m1"
+ and lim\<sigma>: "limitin L.M1.mtopology \<sigma> a sequentially"
+ by (force simp: mtopology_of_def L.M1.closure_of_sequentially)
+ then have "L.M1.MCauchy \<sigma>"
+ by (simp add: L.M1.convergent_imp_MCauchy mtopology_of_def)
+ then have "L.M2.MCauchy (f \<circ> \<sigma>)"
+ using f ran\<sigma> by (simp add: Cauchy_continuous_map_def L.M1.subspace Metric_space.MCauchy_def)
+ then obtain l where l: "limitin L.M2.mtopology (f \<circ> \<sigma>) l sequentially"
+ by (meson L.M2.mcomplete_def m2 mcomplete_of_def)
+ have "limitin L.M2.mtopology f l (atin_within L.M1.mtopology a S)"
+ unfolding L.limit_atin_sequentially_within imp_conjL
+ proof (intro conjI strip)
+ show "l \<in> mspace m2"
+ using L.M2.limitin_mspace l by blast
+ fix \<rho>
+ assume "range \<rho> \<subseteq> S \<inter> mspace m1 - {a}" and lim\<rho>: "limitin L.M1.mtopology \<rho> a sequentially"
+ then have ran\<rho>: "range \<rho> \<subseteq> S" "range \<rho> \<subseteq> mspace m1" "\<And>n. \<rho> n \<noteq> a"
+ by auto
+ have "a \<in> mspace m1"
+ using L.M1.limitin_mspace lim\<rho> by auto
+ have "S.MCauchy \<sigma>" "S.MCauchy \<rho>"
+ using L.M1.convergent_imp_MCauchy L.M1.MCauchy_def S.MCauchy_def lim\<sigma> ran\<sigma> lim\<rho> ran\<rho> by force+
+ then have "L.M2.MCauchy (f \<circ> \<rho>)" "L.M2.MCauchy (f \<circ> \<sigma>)"
+ using f by (auto simp: Cauchy_continuous_map_def)
+ then have ran_f: "range (\<lambda>x. f (\<rho> x)) \<subseteq> mspace m2" "range (\<lambda>x. f (\<sigma> x)) \<subseteq> mspace m2"
+ by (auto simp: L.M2.MCauchy_def)
+ have "(\<lambda>n. mdist m2 (f (\<rho> n)) l) \<longlonglongrightarrow> 0"
+ proof (rule Lim_null_comparison)
+ have "mdist m2 (f (\<rho> n)) l \<le> mdist m2 (f (\<sigma> n)) l + mdist m2 (f (\<sigma> n)) (f (\<rho> n))" for n
+ using \<open>l \<in> mspace m2\<close> ran_f L.M2.triangle'' by (smt (verit, best) range_subsetD)
+ then show "\<forall>\<^sub>F n in sequentially. norm (mdist m2 (f (\<rho> n)) l) \<le> mdist m2 (f (\<sigma> n)) l + mdist m2 (f (\<sigma> n)) (f (\<rho> n))"
+ by force
+ define \<psi> where "\<psi> \<equiv> \<lambda>n. if even n then \<sigma> (n div 2) else \<rho> (n div 2)"
+ have "(\<lambda>n. mdist m1 (\<sigma> n) (\<rho> n)) \<longlonglongrightarrow> 0"
+ proof (rule Lim_null_comparison)
+ show "\<forall>\<^sub>F n in sequentially. norm (mdist m1 (\<sigma> n) (\<rho> n)) \<le> mdist m1 (\<sigma> n) a + mdist m1 (\<rho> n) a"
+ using L.M1.triangle' [of _ a] ran\<sigma> ran\<rho> \<open>a \<in> mspace m1\<close> by (simp add: range_subsetD)
+ have "(\<lambda>n. mdist m1 (\<sigma> n) a) \<longlonglongrightarrow> 0"
+ using L.M1.limitin_metric_dist_null lim\<sigma> by blast
+ moreover have "(\<lambda>n. mdist m1 (\<rho> n) a) \<longlonglongrightarrow> 0"
+ using L.M1.limitin_metric_dist_null lim\<rho> by blast
+ ultimately show "(\<lambda>n. mdist m1 (\<sigma> n) a + mdist m1 (\<rho> n) a) \<longlonglongrightarrow> 0"
+ by (simp add: tendsto_add_zero)
+ qed
+ with \<open>S.MCauchy \<sigma>\<close> \<open>S.MCauchy \<rho>\<close> have "S.MCauchy \<psi>"
+ by (simp add: S.MCauchy_interleaving_gen \<psi>_def)
+ then have "L.M2.MCauchy (f \<circ> \<psi>)"
+ by (metis Cauchy_continuous_map_def f mdist_submetric mspace_submetric)
+ then have "(\<lambda>n. mdist m2 (f (\<sigma> n)) (f (\<rho> n))) \<longlonglongrightarrow> 0"
+ using L.M2.MCauchy_interleaving_gen [of "f \<circ> \<sigma>" "f \<circ> \<rho>"]
+ by (simp add: if_distrib \<psi>_def o_def cong: if_cong)
+ moreover have "\<forall>\<^sub>F n in sequentially. f (\<sigma> n) \<in> mspace m2 \<and> (\<lambda>x. mdist m2 (f (\<sigma> x)) l) \<longlonglongrightarrow> 0"
+ using l by (auto simp: L.M2.limitin_metric_dist_null \<open>l \<in> mspace m2\<close>)
+ ultimately show "(\<lambda>n. mdist m2 (f (\<sigma> n)) l + mdist m2 (f (\<sigma> n)) (f (\<rho> n))) \<longlonglongrightarrow> 0"
+ by (metis (mono_tags) tendsto_add_zero eventually_sequentially order_refl)
+ qed
+ with ran_f show "limitin L.M2.mtopology (f \<circ> \<rho>) l sequentially"
+ by (auto simp: L.M2.limitin_metric_dist_null eventually_sequentially \<open>l \<in> mspace m2\<close>)
+ qed
+ then show "\<exists>l. limitin (mtopology_of m2) f l (atin_within (mtopology_of m1) a S)"
+ by (force simp: mtopology_of_def)
+qed auto
+
+
+lemma Cauchy_continuous_map_extends_to_continuous_closure_of:
+ assumes "mcomplete_of m2"
+ and f: "Cauchy_continuous_map (submetric m1 S) m2 f"
+ obtains g
+ where "continuous_map (subtopology (mtopology_of m1) ((mtopology_of m1) closure_of S))
+ (mtopology_of m2) g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof -
+ obtain g where cmg:
+ "continuous_map (subtopology (mtopology_of m1) ((mtopology_of m1) closure_of (mspace m1 \<inter> S)))
+ (mtopology_of m2) g"
+ and gf: "(\<forall>x \<in> mspace m1 \<inter> S. g x = f x)"
+ using Cauchy_continuous_map_extends_to_continuous_closure_of_aux assms
+ by (metis inf_commute inf_le2 submetric_restrict)
+ define h where "h \<equiv> \<lambda>x. if x \<in> topspace(mtopology_of m1) then g x else f x"
+ show thesis
+ proof
+ show "continuous_map (subtopology (mtopology_of m1) ((mtopology_of m1) closure_of S))
+ (mtopology_of m2) h"
+ unfolding h_def
+ proof (rule continuous_map_eq)
+ show "continuous_map (subtopology (mtopology_of m1) (mtopology_of m1 closure_of S)) (mtopology_of m2) g"
+ by (metis closure_of_restrict cmg topspace_mtopology_of)
+ qed auto
+ qed (auto simp: gf h_def)
+qed
+
+
+lemma Cauchy_continuous_map_extends_to_continuous_intermediate_closure_of:
+ assumes "mcomplete_of m2"
+ and f: "Cauchy_continuous_map (submetric m1 S) m2 f"
+ and T: "T \<subseteq> mtopology_of m1 closure_of S"
+ obtains g
+ where "continuous_map (subtopology (mtopology_of m1) T) (mtopology_of m2) g"
+ "(\<forall>x \<in> S. g x = f x)"
+ by (metis Cauchy_continuous_map_extends_to_continuous_closure_of T assms(1) continuous_map_from_subtopology_mono f)
+
+text \<open>Technical lemma helpful for porting particularly ugly HOL Light proofs\<close>
+lemma all_in_closure_of:
+ assumes P: "\<forall>x \<in> S. P x" and clo: "closedin X {x \<in> topspace X. P x}"
+ shows "\<forall>x \<in> X closure_of S. P x"
+proof -
+ have *: "topspace X \<inter> S \<subseteq> {x \<in> topspace X. P x}"
+ using P by auto
+ show ?thesis
+ using closure_of_minimal [OF * clo] closure_of_restrict by fastforce
+qed
+
+lemma Lipschitz_continuous_map_on_intermediate_closure_aux:
+ assumes lcf: "Lipschitz_continuous_map (submetric m1 S) m2 f"
+ and "S \<subseteq> T" and Tsub: "T \<subseteq> (mtopology_of m1) closure_of S"
+ and cmf: "continuous_map (subtopology (mtopology_of m1) T) (mtopology_of m2) f"
+ and "S \<subseteq> mspace m1"
+ shows "Lipschitz_continuous_map (submetric m1 T) m2 f"
+proof -
+ interpret L: Metric_space12 "mspace m1" "mdist m1" "mspace m2" "mdist m2"
+ by (simp add: Metric_space12_mspace_mdist)
+ interpret S: Metric_space "S \<inter> mspace m1" "mdist m1"
+ by (simp add: L.M1.subspace)
+ have "T \<subseteq> mspace m1"
+ using Tsub by (auto simp: mtopology_of_def closure_of_def)
+ show ?thesis
+ unfolding Lipschitz_continuous_map_pos
+ proof
+ show "f ` mspace (submetric m1 T) \<subseteq> mspace m2"
+ by (metis cmf Metric_space.metric_continuous_map Metric_space_mspace_mdist mtopology_of_def mtopology_of_submetric)
+ define X where "X \<equiv> prod_topology (subtopology L.M1.mtopology T) (subtopology L.M1.mtopology T)"
+ obtain B::real where "B > 0" and B: "\<forall>(x,y) \<in> S\<times>S. mdist m2 (f x) (f y) \<le> B * mdist m1 x y"
+ using lcf \<open>S \<subseteq> mspace m1\<close> by (force simp: Lipschitz_continuous_map_pos)
+ have eq: "{z \<in> A. case z of (x,y) \<Rightarrow> p x y \<le> B * q x y} = {z \<in> A. ((\<lambda>(x,y). B * q x y - p x y)z) \<in> {0..}}"
+ for p q and A::"('a*'a)set"
+ by auto
+ have clo: "closedin X {z \<in> topspace X. case z of (x, y) \<Rightarrow> mdist m2 (f x) (f y) \<le> B * mdist m1 x y}"
+ unfolding eq
+ proof (rule closedin_continuous_map_preimage)
+ have *: "continuous_map X L.M2.mtopology (f \<circ> fst)" "continuous_map X L.M2.mtopology (f \<circ> snd)"
+ using cmf by (auto simp: mtopology_of_def X_def intro: continuous_map_compose continuous_map_fst continuous_map_snd)
+ then show "continuous_map X euclidean (\<lambda>x. case x of (x, y) \<Rightarrow> B * mdist m1 x y - mdist m2 (f x) (f y))"
+ unfolding case_prod_unfold
+ proof (intro continuous_intros; simp add: mtopology_of_def o_def)
+ show "continuous_map X L.M1.mtopology fst" "continuous_map X L.M1.mtopology snd"
+ by (simp_all add: X_def continuous_map_subtopology_fst continuous_map_subtopology_snd flip: subtopology_Times)
+ qed
+ qed auto
+ have "mdist m2 (f x) (f y) \<le> B * mdist m1 x y" if "x \<in> T" "y \<in> T" for x y
+ using all_in_closure_of [OF B clo] \<open>S \<subseteq> T\<close> Tsub
+ by (fastforce simp: X_def subset_iff closure_of_Times closure_of_subtopology inf.absorb2
+ mtopology_of_def that)
+ then show "\<exists>B>0. \<forall>x\<in>mspace (submetric m1 T).
+ \<forall>y\<in>mspace (submetric m1 T).
+ mdist m2 (f x) (f y) \<le> B * mdist (submetric m1 T) x y"
+ using \<open>0 < B\<close> by auto
+ qed
+qed
+
+
+lemma Lipschitz_continuous_map_on_intermediate_closure:
+ assumes "Lipschitz_continuous_map (submetric m1 S) m2 f"
+ and "S \<subseteq> T" "T \<subseteq> (mtopology_of m1) closure_of S"
+ and "continuous_map (subtopology (mtopology_of m1) T) (mtopology_of m2) f"
+ shows "Lipschitz_continuous_map (submetric m1 T) m2 f"
+ by (metis Lipschitz_continuous_map_on_intermediate_closure_aux assms closure_of_subset_topspace subset_trans topspace_mtopology_of)
+
+
+lemma Lipschitz_continuous_map_extends_to_closure_of:
+ assumes m2: "mcomplete_of m2"
+ and f: "Lipschitz_continuous_map (submetric m1 S) m2 f"
+ obtains g
+ where "Lipschitz_continuous_map (submetric m1 (mtopology_of m1 closure_of S)) m2 g"
+ "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof -
+ obtain g
+ where g: "continuous_map (subtopology (mtopology_of m1) ((mtopology_of m1) closure_of S))
+ (mtopology_of m2) g" "(\<forall>x \<in> S. g x = f x)"
+ by (metis Cauchy_continuous_map_extends_to_continuous_closure_of Lipschitz_imp_Cauchy_continuous_map f m2)
+ have "Lipschitz_continuous_map (submetric m1 (mtopology_of m1 closure_of S)) m2 g"
+ proof (rule Lipschitz_continuous_map_on_intermediate_closure)
+ show "Lipschitz_continuous_map (submetric m1 (mspace m1 \<inter> S)) m2 g"
+ by (smt (verit, best) IntD2 Lipschitz_continuous_map_eq f g(2) inf_commute mspace_submetric submetric_restrict)
+ show "mspace m1 \<inter> S \<subseteq> mtopology_of m1 closure_of S"
+ using closure_of_subset_Int by force
+ show "mtopology_of m1 closure_of S \<subseteq> mtopology_of m1 closure_of (mspace m1 \<inter> S)"
+ by (metis closure_of_restrict subset_refl topspace_mtopology_of)
+ show "continuous_map (subtopology (mtopology_of m1) (mtopology_of m1 closure_of S)) (mtopology_of m2) g"
+ by (simp add: g)
+ qed
+ with g that show thesis
+ by metis
+qed
+
+
+lemma Lipschitz_continuous_map_extends_to_intermediate_closure_of:
+ assumes "mcomplete_of m2"
+ and "Lipschitz_continuous_map (submetric m1 S) m2 f"
+ and "T \<subseteq> mtopology_of m1 closure_of S"
+ obtains g
+ where "Lipschitz_continuous_map (submetric m1 T) m2 g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+ by (metis Lipschitz_continuous_map_extends_to_closure_of Lipschitz_continuous_map_from_submetric_mono assms)
+
+text \<open>This proof uses the same trick to extend the function's domain to its closure\<close>
+lemma uniformly_continuous_map_on_intermediate_closure_aux:
+ assumes ucf: "uniformly_continuous_map (submetric m1 S) m2 f"
+ and "S \<subseteq> T" and Tsub: "T \<subseteq> (mtopology_of m1) closure_of S"
+ and cmf: "continuous_map (subtopology (mtopology_of m1) T) (mtopology_of m2) f"
+ and "S \<subseteq> mspace m1"
+ shows "uniformly_continuous_map (submetric m1 T) m2 f"
+proof -
+ interpret L: Metric_space12 "mspace m1" "mdist m1" "mspace m2" "mdist m2"
+ by (simp add: Metric_space12_mspace_mdist)
+ interpret S: Metric_space "S \<inter> mspace m1" "mdist m1"
+ by (simp add: L.M1.subspace)
+ have "T \<subseteq> mspace m1"
+ using Tsub by (auto simp: mtopology_of_def closure_of_def)
+ show ?thesis
+ unfolding uniformly_continuous_map_def
+ proof (intro conjI strip)
+ show "f ` mspace (submetric m1 T) \<subseteq> mspace m2"
+ by (metis cmf Metric_space.metric_continuous_map Metric_space_mspace_mdist mtopology_of_def mtopology_of_submetric)
+ fix \<epsilon>::real
+ assume "\<epsilon> > 0"
+ then obtain \<delta> where "\<delta>>0" and \<delta>: "\<forall>(x,y) \<in> S\<times>S. mdist m1 x y < \<delta> \<longrightarrow> mdist m2 (f x) (f y) \<le> \<epsilon>/2"
+ using ucf \<open>S \<subseteq> mspace m1\<close> unfolding uniformly_continuous_map_def mspace_submetric
+ apply (simp add: Ball_def del: divide_const_simps)
+ by (metis IntD2 half_gt_zero inf.orderE less_eq_real_def)
+ define X where "X \<equiv> prod_topology (subtopology L.M1.mtopology T) (subtopology L.M1.mtopology T)"
+ text \<open>A clever construction involving the union of two closed sets\<close>
+ have eq: "{z \<in> A. case z of (x,y) \<Rightarrow> p x y < d \<longrightarrow> q x y \<le> e}
+ = {z \<in> A. ((\<lambda>(x,y). p x y - d)z) \<in> {0..}} \<union> {z \<in> A. ((\<lambda>(x,y). e - q x y)z) \<in> {0..}}"
+ for p q and d e::real and A::"('a*'a)set"
+ by auto
+ have clo: "closedin X {z \<in> topspace X. case z of (x, y) \<Rightarrow> mdist m1 x y < \<delta> \<longrightarrow> mdist m2 (f x) (f y) \<le> \<epsilon>/2}"
+ unfolding eq
+ proof (intro closedin_Un closedin_continuous_map_preimage)
+ have *: "continuous_map X L.M1.mtopology fst" "continuous_map X L.M1.mtopology snd"
+ by (metis X_def continuous_map_subtopology_fst subtopology_Times continuous_map_subtopology_snd)+
+ show "continuous_map X euclidean (\<lambda>x. case x of (x, y) \<Rightarrow> mdist m1 x y - \<delta>)"
+ unfolding case_prod_unfold
+ by (intro continuous_intros; simp add: mtopology_of_def *)
+ have *: "continuous_map X L.M2.mtopology (f \<circ> fst)" "continuous_map X L.M2.mtopology (f \<circ> snd)"
+ using cmf by (auto simp: mtopology_of_def X_def intro: continuous_map_compose continuous_map_fst continuous_map_snd)
+ then show "continuous_map X euclidean (\<lambda>x. case x of (x, y) \<Rightarrow> \<epsilon> / 2 - mdist m2 (f x) (f y))"
+ unfolding case_prod_unfold
+ by (intro continuous_intros; simp add: mtopology_of_def o_def)
+ qed auto
+ have "mdist m2 (f x) (f y) \<le> \<epsilon>/2" if "x \<in> T" "y \<in> T" "mdist m1 x y < \<delta>" for x y
+ using all_in_closure_of [OF \<delta> clo] \<open>S \<subseteq> T\<close> Tsub
+ by (fastforce simp: X_def subset_iff closure_of_Times closure_of_subtopology inf.absorb2
+ mtopology_of_def that)
+ then show "\<exists>\<delta>>0. \<forall>x\<in>mspace (submetric m1 T). \<forall>y\<in>mspace (submetric m1 T). mdist (submetric m1 T) y x < \<delta> \<longrightarrow> mdist m2 (f y) (f x) < \<epsilon>"
+ using \<open>0 < \<delta>\<close> \<open>0 < \<epsilon>\<close> by fastforce
+ qed
+qed
+
+lemma uniformly_continuous_map_on_intermediate_closure:
+ assumes "uniformly_continuous_map (submetric m1 S) m2 f"
+ and "S \<subseteq> T" and"T \<subseteq> (mtopology_of m1) closure_of S"
+ and "continuous_map (subtopology (mtopology_of m1) T) (mtopology_of m2) f"
+ shows "uniformly_continuous_map (submetric m1 T) m2 f"
+ by (metis assms closure_of_subset_topspace subset_trans topspace_mtopology_of
+ uniformly_continuous_map_on_intermediate_closure_aux)
+
+lemma uniformly_continuous_map_extends_to_closure_of:
+ assumes m2: "mcomplete_of m2"
+ and f: "uniformly_continuous_map (submetric m1 S) m2 f"
+ obtains g
+ where "uniformly_continuous_map (submetric m1 (mtopology_of m1 closure_of S)) m2 g"
+ "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof -
+ obtain g
+ where g: "continuous_map (subtopology (mtopology_of m1) ((mtopology_of m1) closure_of S))
+ (mtopology_of m2) g" "(\<forall>x \<in> S. g x = f x)"
+ by (metis Cauchy_continuous_map_extends_to_continuous_closure_of uniformly_imp_Cauchy_continuous_map f m2)
+ have "uniformly_continuous_map (submetric m1 (mtopology_of m1 closure_of S)) m2 g"
+ proof (rule uniformly_continuous_map_on_intermediate_closure)
+ show "uniformly_continuous_map (submetric m1 (mspace m1 \<inter> S)) m2 g"
+ by (smt (verit, best) IntD2 uniformly_continuous_map_eq f g(2) inf_commute mspace_submetric submetric_restrict)
+ show "mspace m1 \<inter> S \<subseteq> mtopology_of m1 closure_of S"
+ using closure_of_subset_Int by force
+ show "mtopology_of m1 closure_of S \<subseteq> mtopology_of m1 closure_of (mspace m1 \<inter> S)"
+ by (metis closure_of_restrict subset_refl topspace_mtopology_of)
+ show "continuous_map (subtopology (mtopology_of m1) (mtopology_of m1 closure_of S)) (mtopology_of m2) g"
+ by (simp add: g)
+ qed
+ with g that show thesis
+ by metis
+qed
+
+
+lemma uniformly_continuous_map_extends_to_intermediate_closure_of:
+ assumes "mcomplete_of m2"
+ and "uniformly_continuous_map (submetric m1 S) m2 f"
+ and "T \<subseteq> mtopology_of m1 closure_of S"
+ obtains g
+ where "uniformly_continuous_map (submetric m1 T) m2 g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+ by (metis uniformly_continuous_map_extends_to_closure_of uniformly_continuous_map_from_submetric_mono assms)
+
+
+lemma Cauchy_continuous_map_on_intermediate_closure_aux:
+ assumes ucf: "Cauchy_continuous_map (submetric m1 S) m2 f"
+ and "S \<subseteq> T" and Tsub: "T \<subseteq> (mtopology_of m1) closure_of S"
+ and cmf: "continuous_map (subtopology (mtopology_of m1) T) (mtopology_of m2) f"
+ and "S \<subseteq> mspace m1"
+ shows "Cauchy_continuous_map (submetric m1 T) m2 f"
+proof -
+ interpret L: Metric_space12 "mspace m1" "mdist m1" "mspace m2" "mdist m2"
+ by (simp add: Metric_space12_mspace_mdist)
+ interpret S: Metric_space "S \<inter> mspace m1" "mdist m1"
+ by (simp add: L.M1.subspace)
+ interpret T: Metric_space T "mdist m1"
+ by (metis L.M1.subspace Tsub closure_of_subset_topspace dual_order.trans topspace_mtopology_of)
+ have "T \<subseteq> mspace m1"
+ using Tsub by (auto simp: mtopology_of_def closure_of_def)
+ then show ?thesis
+ proof (clarsimp simp: Cauchy_continuous_map_def Int_absorb2)
+ fix \<sigma>
+ assume \<sigma>: "T.MCauchy \<sigma>"
+ have "\<exists>y\<in>S. mdist m1 (\<sigma> n) y < inverse (Suc n) \<and> mdist m2 (f (\<sigma> n)) (f y) < inverse (Suc n)" for n
+ proof -
+ have "\<sigma> n \<in> T"
+ using \<sigma> by (force simp: T.MCauchy_def)
+ moreover have "continuous_map (mtopology_of (submetric m1 T)) L.M2.mtopology f"
+ by (metis cmf mtopology_of_def mtopology_of_submetric)
+ ultimately obtain \<delta> where "\<delta>>0" and \<delta>: "\<forall>x \<in> T. mdist m1 (\<sigma> n) x < \<delta> \<longrightarrow> mdist m2 (f(\<sigma> n)) (f x) < inverse (Suc n)"
+ using \<open>T \<subseteq> mspace m1\<close>
+ apply (simp add: mtopology_of_def Metric_space.metric_continuous_map L.M1.subspace Int_absorb2)
+ by (metis inverse_Suc of_nat_Suc)
+ have "\<exists>y \<in> S. mdist m1 (\<sigma> n) y < min \<delta> (inverse (Suc n))"
+ using \<open>\<sigma> n \<in> T\<close> Tsub \<open>\<delta>>0\<close>
+ unfolding mtopology_of_def L.M1.metric_closure_of subset_iff mem_Collect_eq L.M1.in_mball
+ by (smt (verit, del_insts) inverse_Suc )
+ with \<delta> \<open>S \<subseteq> T\<close> show ?thesis
+ by auto
+ qed
+ then obtain \<rho> where \<rho>S: "\<And>n. \<rho> n \<in> S" and \<rho>1: "\<And>n. mdist m1 (\<sigma> n) (\<rho> n) < inverse (Suc n)"
+ and \<rho>2: "\<And>n. mdist m2 (f (\<sigma> n)) (f (\<rho> n)) < inverse (Suc n)"
+ by metis
+ have "S.MCauchy \<rho>"
+ unfolding S.MCauchy_def
+ proof (intro conjI strip)
+ show "range \<rho> \<subseteq> S \<inter> mspace m1"
+ using \<open>S \<subseteq> mspace m1\<close> by (auto simp: \<rho>S)
+ fix \<epsilon> :: real
+ assume "\<epsilon>>0"
+ then obtain M where M: "\<And>n n'. M \<le> n \<Longrightarrow> M \<le> n' \<Longrightarrow> mdist m1 (\<sigma> n) (\<sigma> n') < \<epsilon>/2"
+ using \<sigma> unfolding T.MCauchy_def by (meson half_gt_zero)
+ have "\<forall>\<^sub>F n in sequentially. inverse (Suc n) < \<epsilon>/4"
+ using Archimedean_eventually_inverse \<open>0 < \<epsilon>\<close> divide_pos_pos zero_less_numeral by blast
+ then obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> inverse (Suc n) < \<epsilon>/4"
+ by (meson eventually_sequentially)
+ have "mdist m1 (\<rho> n) (\<rho> n') < \<epsilon>" if "n \<ge> max M N" "n' \<ge> max M N" for n n'
+ proof -
+ have "mdist m1 (\<rho> n) (\<rho> n') \<le> mdist m1 (\<rho> n) (\<sigma> n) + mdist m1 (\<sigma> n) (\<rho> n')"
+ by (meson T.MCauchy_def T.triangle \<rho>S \<sigma> \<open>S \<subseteq> T\<close> rangeI subset_iff)
+ also have "\<dots> \<le> mdist m1 (\<rho> n) (\<sigma> n) + mdist m1 (\<sigma> n) (\<sigma> n') + mdist m1 (\<sigma> n') (\<rho> n')"
+ by (smt (verit, best) T.MCauchy_def T.triangle \<rho>S \<sigma> \<open>S \<subseteq> T\<close> in_mono rangeI)
+ also have "\<dots> < \<epsilon>/4 + \<epsilon>/2 + \<epsilon>/4"
+ using \<rho>1[of n] \<rho>1[of n'] N[of n] N[of n'] that M[of n n'] by (simp add: T.commute)
+ also have "... \<le> \<epsilon>"
+ by simp
+ finally show ?thesis .
+ qed
+ then show "\<exists>N. \<forall>n n'. N \<le> n \<longrightarrow> N \<le> n' \<longrightarrow> mdist m1 (\<rho> n) (\<rho> n') < \<epsilon>"
+ by blast
+ qed
+ then have f\<rho>: "L.M2.MCauchy (f \<circ> \<rho>)"
+ using ucf by (simp add: Cauchy_continuous_map_def)
+ show "L.M2.MCauchy (f \<circ> \<sigma>)"
+ unfolding L.M2.MCauchy_def
+ proof (intro conjI strip)
+ show "range (f \<circ> \<sigma>) \<subseteq> mspace m2"
+ using \<open>T \<subseteq> mspace m1\<close> \<sigma> cmf
+ apply (auto simp: )
+ by (metis Metric_space.metric_continuous_map Metric_space_mspace_mdist T.MCauchy_def image_eqI inf.absorb1 mspace_submetric mtopology_of_def mtopology_of_submetric range_subsetD subset_iff)
+ fix \<epsilon> :: real
+ assume "\<epsilon>>0"
+ then obtain M where M: "\<And>n n'. M \<le> n \<Longrightarrow> M \<le> n' \<Longrightarrow> mdist m2 ((f \<circ> \<rho>) n) ((f \<circ> \<rho>) n') < \<epsilon>/2"
+ using f\<rho> unfolding L.M2.MCauchy_def by (meson half_gt_zero)
+ have "\<forall>\<^sub>F n in sequentially. inverse (Suc n) < \<epsilon>/4"
+ using Archimedean_eventually_inverse \<open>0 < \<epsilon>\<close> divide_pos_pos zero_less_numeral by blast
+ then obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> inverse (Suc n) < \<epsilon>/4"
+ by (meson eventually_sequentially)
+ have "mdist m2 ((f \<circ> \<sigma>) n) ((f \<circ> \<sigma>) n') < \<epsilon>" if "n \<ge> max M N" "n' \<ge> max M N" for n n'
+ proof -
+ have "mdist m2 ((f \<circ> \<sigma>) n) ((f \<circ> \<sigma>) n') \<le> mdist m2 ((f \<circ> \<sigma>) n) ((f \<circ> \<rho>) n) + mdist m2 ((f \<circ> \<rho>) n) ((f \<circ> \<sigma>) n')"
+ by (meson L.M2.MCauchy_def \<open>range (f \<circ> \<sigma>) \<subseteq> mspace m2\<close> f\<rho> mdist_triangle rangeI subset_eq)
+ also have "\<dots> \<le> mdist m2 ((f \<circ> \<sigma>) n) ((f \<circ> \<rho>) n) + mdist m2 ((f \<circ> \<rho>) n) ((f \<circ> \<rho>) n') + mdist m2 ((f \<circ> \<rho>) n') ((f \<circ> \<sigma>) n')"
+ by (smt (verit) L.M2.MCauchy_def L.M2.triangle \<open>range (f \<circ> \<sigma>) \<subseteq> mspace m2\<close> f\<rho> range_subsetD)
+ also have "\<dots> < \<epsilon>/4 + \<epsilon>/2 + \<epsilon>/4"
+ using \<rho>2[of n] \<rho>2[of n'] N[of n] N[of n'] that M[of n n'] by (simp add: L.M2.commute)
+ also have "... \<le> \<epsilon>"
+ by simp
+ finally show ?thesis .
+ qed
+ then show "\<exists>N. \<forall>n n'. N \<le> n \<longrightarrow> N \<le> n' \<longrightarrow> mdist m2 ((f \<circ> \<sigma>) n) ((f \<circ> \<sigma>) n') < \<epsilon>"
+ by blast
+ qed
+ qed
+qed
+
+lemma Cauchy_continuous_map_on_intermediate_closure:
+ assumes "Cauchy_continuous_map (submetric m1 S) m2 f"
+ and "S \<subseteq> T" and "T \<subseteq> (mtopology_of m1) closure_of S"
+ and "continuous_map (subtopology (mtopology_of m1) T) (mtopology_of m2) f"
+ shows "Cauchy_continuous_map (submetric m1 T) m2 f"
+ by (metis Cauchy_continuous_map_on_intermediate_closure_aux assms closure_of_subset_topspace order.trans topspace_mtopology_of)
+
+
+lemma Cauchy_continuous_map_extends_to_closure_of:
+ assumes m2: "mcomplete_of m2"
+ and f: "Cauchy_continuous_map (submetric m1 S) m2 f"
+ obtains g
+ where "Cauchy_continuous_map (submetric m1 (mtopology_of m1 closure_of S)) m2 g"
+ "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof -
+ obtain g
+ where g: "continuous_map (subtopology (mtopology_of m1) ((mtopology_of m1) closure_of S))
+ (mtopology_of m2) g" "(\<forall>x \<in> S. g x = f x)"
+ by (metis Cauchy_continuous_map_extends_to_continuous_closure_of f m2)
+ have "Cauchy_continuous_map (submetric m1 (mtopology_of m1 closure_of S)) m2 g"
+ proof (rule Cauchy_continuous_map_on_intermediate_closure)
+ show "Cauchy_continuous_map (submetric m1 (mspace m1 \<inter> S)) m2 g"
+ by (smt (verit, best) IntD2 Cauchy_continuous_map_eq f g(2) inf_commute mspace_submetric submetric_restrict)
+ show "mspace m1 \<inter> S \<subseteq> mtopology_of m1 closure_of S"
+ using closure_of_subset_Int by force
+ show "mtopology_of m1 closure_of S \<subseteq> mtopology_of m1 closure_of (mspace m1 \<inter> S)"
+ by (metis closure_of_restrict subset_refl topspace_mtopology_of)
+ show "continuous_map (subtopology (mtopology_of m1) (mtopology_of m1 closure_of S)) (mtopology_of m2) g"
+ by (simp add: g)
+ qed
+ with g that show thesis
+ by metis
+qed
+
+
+lemma Cauchy_continuous_map_extends_to_intermediate_closure_of:
+ assumes "mcomplete_of m2"
+ and "Cauchy_continuous_map (submetric m1 S) m2 f"
+ and "T \<subseteq> mtopology_of m1 closure_of S"
+ obtains g
+ where "Cauchy_continuous_map (submetric m1 T) m2 g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+ by (metis Cauchy_continuous_map_extends_to_closure_of Cauchy_continuous_map_from_submetric_mono assms)
+
+
+subsection\<open>Metric space of bounded functions\<close>
+
+context Metric_space
+begin
+
+definition fspace :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) set" where
+ "fspace \<equiv> \<lambda>S. {f. f`S \<subseteq> M \<and> f \<in> extensional S \<and> mbounded (f`S)}"
+
+definition fdist :: "['b set, 'b \<Rightarrow> 'a, 'b \<Rightarrow> 'a] \<Rightarrow> real" where
+ "fdist \<equiv> \<lambda>S f g. if f \<in> fspace S \<and> g \<in> fspace S \<and> S \<noteq> {}
+ then Sup ((\<lambda>x. d (f x) (g x)) ` S) else 0"
+
+lemma fspace_empty [simp]: "fspace {} = {\<lambda>x. undefined}"
+ by (auto simp: fspace_def)
+
+lemma fdist_empty [simp]: "fdist {} = (\<lambda>x y. 0)"
+ by (auto simp: fdist_def)
+
+lemma fspace_in_M: "\<lbrakk>f \<in> fspace S; x \<in> S\<rbrakk> \<Longrightarrow> f x \<in> M"
+ by (auto simp: fspace_def)
+
+lemma bdd_above_dist:
+ assumes f: "f \<in> fspace S" and g: "g \<in> fspace S" and "S \<noteq> {}"
+ shows "bdd_above ((\<lambda>u. d (f u) (g u)) ` S)"
+proof -
+ obtain a where "a \<in> S"
+ using \<open>S \<noteq> {}\<close> by blast
+ obtain B x C y where "B>0" and B: "f`S \<subseteq> mcball x B"
+ and "C>0" and C: "g`S \<subseteq> mcball y C"
+ using f g mbounded_pos by (auto simp: fspace_def)
+ have "d (f u) (g u) \<le> B + d x y + C" if "u\<in>S" for u
+ proof -
+ have "f u \<in> M"
+ by (meson B image_eqI mbounded_mcball mbounded_subset_mspace subsetD that)
+ have "g u \<in> M"
+ by (meson C image_eqI mbounded_mcball mbounded_subset_mspace subsetD that)
+ have "x \<in> M" "y \<in> M"
+ using B C that by auto
+ have "d (f u) (g u) \<le> d (f u) x + d x (g u)"
+ by (simp add: \<open>f u \<in> M\<close> \<open>g u \<in> M\<close> \<open>x \<in> M\<close> triangle)
+ also have "\<dots> \<le> d (f u) x + d x y + d y (g u)"
+ by (simp add: \<open>f u \<in> M\<close> \<open>g u \<in> M\<close> \<open>x \<in> M\<close> \<open>y \<in> M\<close> triangle)
+ also have "\<dots> \<le> B + d x y + C"
+ using B C commute that by fastforce
+ finally show ?thesis .
+ qed
+ then show ?thesis
+ by (meson bdd_above.I2)
+qed
+
+
+lemma Metric_space_funspace: "Metric_space (fspace S) (fdist S)"
+proof
+ show *: "0 \<le> fdist S f g" for f g
+ by (auto simp: fdist_def intro: cSUP_upper2 [OF bdd_above_dist])
+ show "fdist S f g = fdist S g f" for f g
+ by (auto simp: fdist_def commute)
+ show "(fdist S f g = 0) = (f = g)"
+ if fg: "f \<in> fspace S" "g \<in> fspace S" for f g
+ proof
+ assume 0: "fdist S f g = 0"
+ show "f = g"
+ proof (cases "S={}")
+ case True
+ with 0 that show ?thesis
+ by (simp add: fdist_def fspace_def)
+ next
+ case False
+ with 0 fg have Sup0: "(SUP x\<in>S. d (f x) (g x)) = 0"
+ by (simp add: fdist_def)
+ have "d (f x) (g x) = 0" if "x\<in>S" for x
+ by (smt (verit) False Sup0 \<open>x\<in>S\<close> bdd_above_dist [OF fg] less_cSUP_iff nonneg)
+ with fg show "f=g"
+ by (simp add: fspace_def extensionalityI image_subset_iff)
+ qed
+ next
+ show "f = g \<Longrightarrow> fdist S f g = 0"
+ using fspace_in_M [OF \<open>g \<in> fspace S\<close>] by (auto simp: fdist_def)
+ qed
+ show "fdist S f h \<le> fdist S f g + fdist S g h"
+ if fgh: "f \<in> fspace S" "g \<in> fspace S" "h \<in> fspace S" for f g h
+ proof (clarsimp simp add: fdist_def that)
+ assume "S \<noteq> {}"
+ have dfh: "d (f x) (h x) \<le> d (f x) (g x) + d (g x) (h x)" if "x\<in>S" for x
+ by (meson fgh fspace_in_M that triangle)
+ have bdd_fgh: "bdd_above ((\<lambda>x. d (f x) (g x)) ` S)" "bdd_above ((\<lambda>x. d (g x) (h x)) ` S)"
+ by (simp_all add: \<open>S \<noteq> {}\<close> bdd_above_dist that)
+ then obtain B C where B: "\<And>x. x\<in>S \<Longrightarrow> d (f x) (g x) \<le> B" and C: "\<And>x. x\<in>S \<Longrightarrow> d (g x) (h x) \<le> C"
+ by (auto simp: bdd_above_def)
+ then have "\<And>x. x\<in>S \<Longrightarrow> d (f x) (g x) + d (g x) (h x) \<le> B+C"
+ by force
+ then have bdd: "bdd_above ((\<lambda>x. d (f x) (g x) + d (g x) (h x)) ` S)"
+ by (auto simp: bdd_above_def)
+ then have "(SUP x\<in>S. d (f x) (h x)) \<le> (SUP x\<in>S. d (f x) (g x) + d (g x) (h x))"
+ by (metis (mono_tags, lifting) cSUP_mono \<open>S \<noteq> {}\<close> dfh)
+ also have "\<dots> \<le> (SUP x\<in>S. d (f x) (g x)) + (SUP x\<in>S. d (g x) (h x))"
+ by (simp add: \<open>S \<noteq> {}\<close> bdd cSUP_le_iff bdd_fgh add_mono cSup_upper)
+ finally show "(SUP x\<in>S. d (f x) (h x)) \<le> (SUP x\<in>S. d (f x) (g x)) + (SUP x\<in>S. d (g x) (h x))" .
+ qed
+qed
+
end
+
+
+definition funspace where
+ "funspace S m \<equiv> metric (Metric_space.fspace (mspace m) (mdist m) S,
+ Metric_space.fdist (mspace m) (mdist m) S)"
+
+lemma mspace_funspace [simp]:
+ "mspace (funspace S m) = Metric_space.fspace (mspace m) (mdist m) S"
+ by (simp add: Metric_space.Metric_space_funspace Metric_space.mspace_metric funspace_def)
+
+lemma mdist_funspace [simp]:
+ "mdist (funspace S m) = Metric_space.fdist (mspace m) (mdist m) S"
+ by (simp add: Metric_space.Metric_space_funspace Metric_space.mdist_metric funspace_def)
+
+lemma funspace_imp_welldefined:
+ "\<lbrakk>f \<in> mspace (funspace S m); x \<in> S\<rbrakk> \<Longrightarrow> f x \<in> mspace m"
+ by (simp add: Metric_space.fspace_def subset_iff)
+
+lemma funspace_imp_extensional:
+ "f \<in> mspace (funspace S m) \<Longrightarrow> f \<in> extensional S"
+ by (simp add: Metric_space.fspace_def)
+
+lemma funspace_imp_bounded_image:
+ "f \<in> mspace (funspace S m) \<Longrightarrow> Metric_space.mbounded (mspace m) (mdist m) (f ` S)"
+ by (simp add: Metric_space.fspace_def)
+
+lemma funspace_imp_bounded:
+ "f \<in> mspace (funspace S m) \<Longrightarrow> S = {} \<or> (\<exists>c B. \<forall>x \<in> S. mdist m c (f x) \<le> B)"
+ by (auto simp: Metric_space.fspace_def Metric_space.mbounded)
+
+
+lemma (in Metric_space) funspace_imp_bounded2:
+ assumes "f \<in> fspace S" "g \<in> fspace S"
+ obtains B where "\<And>x. x \<in> S \<Longrightarrow> d (f x) (g x) \<le> B"
+proof -
+ have "mbounded (f ` S \<union> g ` S)"
+ using mbounded_Un assms by (force simp: fspace_def)
+ then show thesis
+ by (metis UnCI imageI mbounded_alt that)
+qed
+
+lemma funspace_imp_bounded2:
+ assumes "f \<in> mspace (funspace S m)" "g \<in> mspace (funspace S m)"
+ obtains B where "\<And>x. x \<in> S \<Longrightarrow> mdist m (f x) (g x) \<le> B"
+ by (metis Metric_space_mspace_mdist assms mspace_funspace Metric_space.funspace_imp_bounded2)
+
+lemma (in Metric_space) funspace_mdist_le:
+ assumes fg: "f \<in> fspace S" "g \<in> fspace S" and "S \<noteq> {}"
+ shows "fdist S f g \<le> a \<longleftrightarrow> (\<forall>x \<in> S. d (f x) (g x) \<le> a)"
+ using assms bdd_above_dist [OF fg] by (simp add: fdist_def cSUP_le_iff)
+
+lemma funspace_mdist_le:
+ assumes "f \<in> mspace (funspace S m)" "g \<in> mspace (funspace S m)" and "S \<noteq> {}"
+ shows "mdist (funspace S m) f g \<le> a \<longleftrightarrow> (\<forall>x \<in> S. mdist m (f x) (g x) \<le> a)"
+ using assms by (simp add: Metric_space.funspace_mdist_le)
+
+
+lemma (in Metric_space) mcomplete_funspace:
+ assumes "mcomplete"
+ shows "mcomplete_of (funspace S Self)"
+proof -
+ interpret F: Metric_space "fspace S" "fdist S"
+ by (simp add: Metric_space_funspace)
+ show ?thesis
+ proof (cases "S={}")
+ case True
+ then show ?thesis
+ by (simp add: mcomplete_of_def mcomplete_trivial_singleton)
+ next
+ case False
+ show ?thesis
+ proof (clarsimp simp: mcomplete_of_def Metric_space.mcomplete_def)
+ fix \<sigma>
+ assume \<sigma>: "F.MCauchy \<sigma>"
+ then have \<sigma>M: "\<And>n x. x \<in> S \<Longrightarrow> \<sigma> n x \<in> M"
+ by (auto simp: F.MCauchy_def intro: fspace_in_M)
+ have fdist_less: "\<exists>N. \<forall>n n'. N \<le> n \<longrightarrow> N \<le> n' \<longrightarrow> fdist S (\<sigma> n) (\<sigma> n') < \<epsilon>" if "\<epsilon>>0" for \<epsilon>
+ using \<sigma> that by (auto simp: F.MCauchy_def)
+ have \<sigma>ext: "\<And>n. \<sigma> n \<in> extensional S"
+ using \<sigma> unfolding F.MCauchy_def by (auto simp: fspace_def)
+ have \<sigma>bd: "\<And>n. mbounded (\<sigma> n ` S)"
+ using \<sigma> unfolding F.MCauchy_def by (simp add: fspace_def image_subset_iff)
+ have \<sigma>in[simp]: "\<sigma> n \<in> fspace S" for n
+ using F.MCauchy_def \<sigma> by blast
+ have bd2: "\<And>n n'. \<exists>B. \<forall>x \<in> S. d (\<sigma> n x) (\<sigma> n' x) \<le> B"
+ using \<sigma> unfolding F.MCauchy_def by (metis range_subsetD funspace_imp_bounded2)
+ have sup: "\<And>n n' x0. x0 \<in> S \<Longrightarrow> d (\<sigma> n x0) (\<sigma> n' x0) \<le> Sup ((\<lambda>x. d (\<sigma> n x) (\<sigma> n' x)) ` S)"
+ proof (rule cSup_upper)
+ show "bdd_above ((\<lambda>x. d (\<sigma> n x) (\<sigma> n' x)) ` S)" if "x0 \<in> S" for n n' x0
+ using that bd2 by (meson bdd_above.I2)
+ qed auto
+ have pcy: "MCauchy (\<lambda>n. \<sigma> n x)" if "x \<in> S" for x
+ unfolding MCauchy_def
+ proof (intro conjI strip)
+ show "range (\<lambda>n. \<sigma> n x) \<subseteq> M"
+ using \<sigma>M that by blast
+ fix \<epsilon> :: real
+ assume "\<epsilon> > 0"
+ then obtain N where N: "\<And>n n'. N \<le> n \<longrightarrow> N \<le> n' \<longrightarrow> fdist S (\<sigma> n) (\<sigma> n') < \<epsilon>"
+ using \<sigma> by (force simp: F.MCauchy_def)
+ { fix n n'
+ assume n: "N \<le> n" "N \<le> n'"
+ have "d (\<sigma> n x) (\<sigma> n' x) \<le> (SUP x\<in>S. d (\<sigma> n x) (\<sigma> n' x))"
+ using that sup by presburger
+ then have "d (\<sigma> n x) (\<sigma> n' x) \<le> fdist S (\<sigma> n) (\<sigma> n')"
+ by (simp add: fdist_def \<open>S \<noteq> {}\<close>)
+ with N n have "d (\<sigma> n x) (\<sigma> n' x) < \<epsilon>"
+ by fastforce
+ } then show "\<exists>N. \<forall>n n'. N \<le> n \<longrightarrow> N \<le> n' \<longrightarrow> d (\<sigma> n x) (\<sigma> n' x) < \<epsilon>"
+ by blast
+ qed
+ have "\<exists>l. limitin mtopology (\<lambda>n. \<sigma> n x) l sequentially" if "x \<in> S" for x
+ using assms mcomplete_def pcy \<open>x \<in> S\<close> by presburger
+ then obtain g0 where g0: "\<And>x. x \<in> S \<Longrightarrow> limitin mtopology (\<lambda>n. \<sigma> n x) (g0 x) sequentially"
+ by metis
+ define g where "g \<equiv> restrict g0 S"
+ have gext: "g \<in> extensional S"
+ and glim: "\<And>x. x \<in> S \<Longrightarrow> limitin mtopology (\<lambda>n. \<sigma> n x) (g x) sequentially"
+ by (auto simp: g_def g0)
+ have gwd: "g x \<in> M" if "x \<in> S" for x
+ using glim limitin_metric that by blast
+ have unif: "\<exists>N. \<forall>x n. x \<in> S \<longrightarrow> N \<le> n \<longrightarrow> d (\<sigma> n x) (g x) < \<epsilon>" if "\<epsilon>>0" for \<epsilon>
+ proof -
+ obtain N where N: "\<And>n n'. N \<le> n \<and> N \<le> n' \<Longrightarrow> Sup ((\<lambda>x. d (\<sigma> n x) (\<sigma> n' x)) ` S) < \<epsilon>/2"
+ using \<open>S\<noteq>{}\<close> \<open>\<epsilon>>0\<close> fdist_less [of "\<epsilon>/2"]
+ by (metis (mono_tags) \<sigma>in fdist_def half_gt_zero)
+ show ?thesis
+ proof (intro exI strip)
+ fix x n
+ assume "x \<in> S" and "N \<le> n"
+ obtain N' where N': "\<And>n. N' \<le> n \<Longrightarrow> \<sigma> n x \<in> M \<and> d (\<sigma> n x) (g x) < \<epsilon>/2"
+ by (metis \<open>0 < \<epsilon>\<close> \<open>x \<in> S\<close> glim half_gt_zero limit_metric_sequentially)
+ have "d (\<sigma> n x) (g x) \<le> d (\<sigma> n x) (\<sigma> (max N N') x) + d (\<sigma> (max N N') x) (g x)"
+ using \<open>x \<in> S\<close> \<sigma>M gwd triangle by presburger
+ also have "\<dots> < \<epsilon>/2 + \<epsilon>/2"
+ by (smt (verit) N N' \<open>N \<le> n\<close> \<open>x \<in> S\<close> max.cobounded1 max.cobounded2 sup)
+ finally show "d (\<sigma> n x) (g x) < \<epsilon>" by simp
+ qed
+ qed
+ have "limitin F.mtopology \<sigma> g sequentially"
+ unfolding F.limit_metric_sequentially
+ proof (intro conjI strip)
+ obtain N where N: "\<And>n n'. N \<le> n \<and> N \<le> n' \<Longrightarrow> Sup ((\<lambda>x. d (\<sigma> n x) (\<sigma> n' x)) ` S) < 1"
+ using fdist_less [of 1] \<open>S\<noteq>{}\<close> by (auto simp: fdist_def)
+ have "\<And>x. x \<in> \<sigma> N ` S \<Longrightarrow> x \<in> M"
+ using \<sigma>M by blast
+ obtain a B where "a \<in> M" and B: "\<And>x. x \<in> (\<sigma> N) ` S \<Longrightarrow> d a x \<le> B"
+ by (metis False \<sigma>M \<sigma>bd ex_in_conv imageI mbounded_alt_pos)
+ have "d a (g x) \<le> B+1" if "x\<in>S" for x
+ proof -
+ have "d a (g x) \<le> d a (\<sigma> N x) + d (\<sigma> N x) (g x)"
+ by (simp add: \<open>a \<in> M\<close> \<sigma>M gwd that triangle)
+ also have "\<dots> \<le> B+1"
+ proof -
+ have "d a (\<sigma> N x) \<le> B"
+ by (simp add: B that)
+ moreover
+ have False if 1: "d (\<sigma> N x) (g x) > 1"
+ proof -
+ obtain r where "1 < r" and r: "r < d (\<sigma> N x) (g x)"
+ using 1 dense by blast
+ then obtain N' where N': "\<And>n. N' \<le> n \<Longrightarrow> \<sigma> n x \<in> M \<and> d (\<sigma> n x) (g x) < r-1"
+ using glim [OF \<open>x\<in>S\<close>] by (fastforce simp: limit_metric_sequentially)
+ have "d (\<sigma> N x) (g x) \<le> d (\<sigma> N x) (\<sigma> (max N N') x) + d (\<sigma> (max N N') x) (g x)"
+ by (metis \<open>x \<in> S\<close> \<sigma>M commute gwd triangle')
+ also have "\<dots> < 1 + (r-1)"
+ by (smt (verit) N N' \<open>x \<in> S\<close> max.cobounded1 max.cobounded2 max.idem sup)
+ finally have "d (\<sigma> N x) (g x) < r"
+ by simp
+ with r show False
+ by linarith
+ qed
+ ultimately show ?thesis
+ by force
+ qed
+ finally show ?thesis .
+ qed
+ with gwd \<open>a \<in> M\<close> have "mbounded (g ` S)"
+ unfolding mbounded by blast
+ with gwd gext show "g \<in> fspace S"
+ by (auto simp: fspace_def)
+ fix \<epsilon>::real
+ assume "\<epsilon>>0"
+ then obtain N where "\<And>x n. x \<in> S \<Longrightarrow> N \<le> n \<Longrightarrow> d (\<sigma> n x) (g x) < \<epsilon>/2"
+ by (meson unif half_gt_zero)
+ then have "fdist S (\<sigma> n) g \<le> \<epsilon>/2" if "N \<le> n" for n
+ using \<open>g \<in> fspace S\<close> False that
+ by (force simp: funspace_mdist_le simp del: divide_const_simps)
+ then show "\<exists>N. \<forall>n\<ge>N. \<sigma> n \<in> fspace S \<and> fdist S (\<sigma> n) g < \<epsilon>"
+ by (metis \<open>0 < \<epsilon>\<close> \<sigma>in add_strict_increasing field_sum_of_halves half_gt_zero)
+ qed
+ then show "\<exists>x. limitin F.mtopology \<sigma> x sequentially"
+ by blast
+ qed
+ qed
+qed
+
+
+
+subsection\<open>Metric space of continuous bounded functions\<close>
+
+definition cfunspace where
+ "cfunspace X m \<equiv> submetric (funspace (topspace X) m) {f. continuous_map X (mtopology_of m) f}"
+
+lemma mspace_cfunspace [simp]:
+ "mspace (cfunspace X m) =
+ {f. f ` topspace X \<subseteq> mspace m \<and> f \<in> extensional (topspace X) \<and>
+ Metric_space.mbounded (mspace m) (mdist m) (f ` (topspace X)) \<and>
+ continuous_map X (mtopology_of m) f}"
+ by (auto simp: cfunspace_def Metric_space.fspace_def)
+
+lemma mdist_cfunspace_eq_mdist_funspace:
+ "mdist (cfunspace X m) = mdist (funspace (topspace X) m)"
+ by (auto simp: cfunspace_def)
+
+lemma cfunspace_subset_funspace:
+ "mspace (cfunspace X m) \<subseteq> mspace (funspace (topspace X) m)"
+ by (simp add: cfunspace_def)
+
+lemma cfunspace_mdist_le:
+ "\<lbrakk>f \<in> mspace (cfunspace X m); g \<in> mspace (cfunspace X m); topspace X \<noteq> {}\<rbrakk>
+ \<Longrightarrow> mdist (cfunspace X m) f g \<le> a \<longleftrightarrow> (\<forall>x \<in> topspace X. mdist m (f x) (g x) \<le> a)"
+ by (simp add: cfunspace_def Metric_space.funspace_mdist_le)
+
+lemma cfunspace_imp_bounded2:
+ assumes "f \<in> mspace (cfunspace X m)" "g \<in> mspace (cfunspace X m)"
+ obtains B where "\<And>x. x \<in> topspace X \<Longrightarrow> mdist m (f x) (g x) \<le> B"
+ by (metis assms all_not_in_conv cfunspace_mdist_le nle_le)
+
+lemma cfunspace_mdist_lt:
+ "\<lbrakk>compactin X (topspace X); f \<in> mspace (cfunspace X m);
+ g \<in> mspace (cfunspace X m); mdist (cfunspace X m) f g < a;
+ x \<in> topspace X\<rbrakk>
+ \<Longrightarrow> mdist m (f x) (g x) < a"
+ by (metis (full_types) cfunspace_mdist_le empty_iff less_eq_real_def less_le_not_le)
+
+lemma mdist_cfunspace_le:
+ assumes "0 \<le> B" and B: "\<And>x. x \<in> topspace X \<Longrightarrow> mdist m (f x) (g x) \<le> B"
+ shows "mdist (cfunspace X m) f g \<le> B"
+proof (cases "topspace X = {}")
+ case True
+ then show ?thesis
+ by (simp add: Metric_space.fdist_empty \<open>B\<ge>0\<close> cfunspace_def)
+next
+ case False
+ have bdd: "bdd_above ((\<lambda>u. mdist m (f u) (g u)) ` topspace X)"
+ by (meson B bdd_above.I2)
+ with assms bdd show ?thesis
+ by (simp add: mdist_cfunspace_eq_mdist_funspace Metric_space.fdist_def cSUP_le_iff)
+qed
+
+
+lemma mdist_cfunspace_imp_mdist_le:
+ "\<lbrakk>f \<in> mspace (cfunspace X m); g \<in> mspace (cfunspace X m);
+ mdist (cfunspace X m) f g \<le> a; x \<in> topspace X\<rbrakk> \<Longrightarrow> mdist m (f x) (g x) \<le> a"
+ using cfunspace_mdist_le by blast
+
+lemma compactin_mspace_cfunspace:
+ "compactin X (topspace X)
+ \<Longrightarrow> mspace (cfunspace X m) =
+ {f. (\<forall>x \<in> topspace X. f x \<in> mspace m) \<and>
+ f \<in> extensional (topspace X) \<and>
+ continuous_map X (mtopology_of m) f}"
+ by (auto simp: Metric_space.compactin_imp_mbounded image_compactin mtopology_of_def)
+
+lemma (in Metric_space) mcomplete_cfunspace:
+ assumes "mcomplete"
+ shows "mcomplete_of (cfunspace X Self)"
+proof -
+ interpret F: Metric_space "fspace (topspace X)" "fdist (topspace X)"
+ by (simp add: Metric_space_funspace)
+ interpret S: Submetric "fspace (topspace X)" "fdist (topspace X)" "mspace (cfunspace X Self)"
+ proof
+ show "mspace (cfunspace X Self) \<subseteq> fspace (topspace X)"
+ by (metis cfunspace_subset_funspace mdist_Self mspace_Self mspace_funspace)
+ qed
+ show ?thesis
+ proof (cases "topspace X = {}")
+ case True
+ then show ?thesis
+ by (simp add: mcomplete_of_def mcomplete_trivial_singleton mdist_cfunspace_eq_mdist_funspace cong: conj_cong)
+ next
+ case False
+ have *: "continuous_map X mtopology g"
+ if "range \<sigma> \<subseteq> mspace (cfunspace X Self)"
+ and g: "limitin F.mtopology \<sigma> g sequentially" for \<sigma> g
+ unfolding continuous_map_to_metric
+ proof (intro strip)
+ have \<sigma>: "\<And>n. continuous_map X mtopology (\<sigma> n)"
+ using that by (auto simp: mtopology_of_def)
+ fix x and \<epsilon>::real
+ assume "x \<in> topspace X" and "0 < \<epsilon>"
+ then obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> \<sigma> n \<in> fspace (topspace X) \<and> fdist (topspace X) (\<sigma> n) g < \<epsilon>/3"
+ unfolding mtopology_of_def F.limitin_metric
+ by (metis F.limit_metric_sequentially divide_pos_pos g zero_less_numeral)
+ then obtain U where "openin X U" "x \<in> U"
+ and U: "\<And>y. y \<in> U \<Longrightarrow> \<sigma> N y \<in> mball (\<sigma> N x) (\<epsilon>/3)"
+ by (metis Metric_space.continuous_map_to_metric Metric_space_axioms \<open>0 < \<epsilon>\<close> \<open>x \<in> topspace X\<close> \<sigma> divide_pos_pos zero_less_numeral)
+ moreover
+ have "g y \<in> mball (g x) \<epsilon>" if "y\<in>U" for y
+ proof -
+ have "U \<subseteq> topspace X"
+ using \<open>openin X U\<close> by (simp add: openin_subset)
+ have gx: "g x \<in> M"
+ by (meson F.limitin_mspace \<open>x \<in> topspace X\<close> fspace_in_M g)
+ have "y \<in> topspace X"
+ using \<open>U \<subseteq> topspace X\<close> that by auto
+ have gy: "g y \<in> M"
+ by (meson F.limitin_mspace[OF g] \<open>U \<subseteq> topspace X\<close> fspace_in_M subsetD that)
+ have "d (g x) (g y) < \<epsilon>"
+ proof -
+ have *: "d (\<sigma> N x0) (g x0) \<le> \<epsilon>/3" if "x0 \<in> topspace X" for x0
+ proof -
+ have "bdd_above ((\<lambda>x. d (\<sigma> N x) (g x)) ` topspace X)"
+ by (metis F.limit_metric_sequentially False N bdd_above_dist g order_refl)
+ then have "d (\<sigma> N x0) (g x0) \<le> Sup ((\<lambda>x. d (\<sigma> N x) (g x)) ` topspace X)"
+ by (simp add: cSup_upper that)
+ also have "\<dots> \<le> \<epsilon>/3"
+ by (smt (verit) F.limit_metric_sequentially False N Sup.SUP_cong fdist_def g order_refl)
+ finally show ?thesis .
+ qed
+ have "d (g x) (g y) \<le> d (g x) (\<sigma> N x) + d (\<sigma> N x) (g y)"
+ using U gx gy that triangle by force
+ also have "\<dots> < \<epsilon>/3 + \<epsilon>/3 + \<epsilon>/3"
+ by (smt (verit) "*" U gy \<open>x \<in> topspace X\<close> \<open>y \<in> topspace X\<close> commute in_mball that triangle)
+ finally show ?thesis by simp
+ qed
+ with gx gy show ?thesis by simp
+ qed
+ ultimately show "\<exists>U. openin X U \<and> x \<in> U \<and> (\<forall>y\<in>U. g y \<in> mball (g x) \<epsilon>)"
+ by blast
+ qed
+
+ have "S.sub.mcomplete"
+ proof (rule S.sequentially_closedin_mcomplete_imp_mcomplete)
+ show "F.mcomplete"
+ by (metis assms mcomplete_funspace mcomplete_of_def mdist_Self mdist_funspace mspace_Self mspace_funspace)
+ fix \<sigma> g
+ assume g: "range \<sigma> \<subseteq> mspace (cfunspace X Self) \<and> limitin F.mtopology \<sigma> g sequentially"
+ show "g \<in> mspace (cfunspace X Self)"
+ proof (simp add: mtopology_of_def, intro conjI)
+ show "g ` topspace X \<subseteq> M" "g \<in> extensional (topspace X)" "mbounded (g ` topspace X)"
+ using g F.limitin_mspace by (force simp: fspace_def)+
+ show "continuous_map X mtopology g"
+ using * g by blast
+ qed
+ qed
+ then show ?thesis
+ by (simp add: mcomplete_of_def mdist_cfunspace_eq_mdist_funspace)
+ qed
+qed
+
+
+subsection\<open>Existence of completion for any metric space M as a subspace of M=>R\<close>
+
+lemma (in Metric_space) metric_completion_explicit:
+ obtains f :: "['a,'a] \<Rightarrow> real" and S where
+ "S \<subseteq> mspace(funspace M euclidean_metric)"
+ "mcomplete_of (submetric (funspace M euclidean_metric) S)"
+ "f ` M \<subseteq> S"
+ "mtopology_of(funspace M euclidean_metric) closure_of f ` M = S"
+ "\<And>x y. \<lbrakk>x \<in> M; y \<in> M\<rbrakk>
+ \<Longrightarrow> mdist (funspace M euclidean_metric) (f x) (f y) = d x y"
+proof -
+ define m':: "('a\<Rightarrow>real) metric" where "m' \<equiv> funspace M euclidean_metric"
+ show thesis
+ proof (cases "M = {}")
+ case True
+ then show ?thesis
+ using that by (simp add: mcomplete_of_def mcomplete_trivial)
+ next
+ case False
+ then obtain a where "a \<in> M"
+ by auto
+ define f where "f \<equiv> \<lambda>x. (\<lambda>u \<in> M. d x u - d a u)"
+ define S where "S \<equiv> mtopology_of(funspace M euclidean_metric) closure_of (f ` M)"
+ interpret S: Submetric "Met_TC.fspace M" "Met_TC.fdist M" "S \<inter> Met_TC.fspace M"
+ by (simp add: Met_TC.Metric_space_funspace Submetric.intro Submetric_axioms_def)
+
+ have fim: "f ` M \<subseteq> mspace m'"
+ proof (clarsimp simp: m'_def Met_TC.fspace_def)
+ fix b
+ assume "b \<in> M"
+ then have "\<And>c. \<lbrakk>c \<in> M\<rbrakk> \<Longrightarrow> \<bar>d b c - d a c\<bar> \<le> d a b"
+ by (smt (verit, best) \<open>a \<in> M\<close> commute triangle'')
+ then have "(\<lambda>x. d b x - d a x) ` M \<subseteq> cball 0 (d a b)"
+ by force
+ then show "f b \<in> extensional M \<and> bounded (f b ` M)"
+ by (metis bounded_cball bounded_subset f_def image_restrict_eq restrict_extensional_sub set_eq_subset)
+ qed
+ show thesis
+ proof
+ show "S \<subseteq> mspace (funspace M euclidean_metric)"
+ by (simp add: S_def in_closure_of subset_iff)
+ have "closedin S.mtopology (S \<inter> Met_TC.fspace M)"
+ by (simp add: S_def closedin_Int funspace_def)
+ moreover have "S.mcomplete"
+ using Metric_space.mcomplete_funspace Met_TC.Metric_space_axioms by (fastforce simp: mcomplete_of_def)
+ ultimately show "mcomplete_of (submetric (funspace M euclidean_metric) S)"
+ by (simp add: S.closedin_eq_mcomplete mcomplete_of_def)
+ show "f ` M \<subseteq> S"
+ using S_def fim in_closure_of m'_def by fastforce
+ show "mtopology_of (funspace M euclidean_metric) closure_of f ` M = S"
+ by (auto simp: f_def S_def mtopology_of_def)
+ show "mdist (funspace M euclidean_metric) (f x) (f y) = d x y"
+ if "x \<in> M" "y \<in> M" for x y
+ proof -
+ have "\<forall>c\<in>M. dist (f x c) (f y c) \<le> r \<Longrightarrow> d x y \<le> r" for r
+ using that by (auto simp: f_def dist_real_def)
+ moreover have "dist (f x z) (f y z) \<le> r" if "d x y \<le> r" and "z \<in> M" for r z
+ using that \<open>x \<in> M\<close> \<open>y \<in> M\<close>
+ apply (simp add: f_def Met_TC.fdist_def dist_real_def)
+ by (smt (verit, best) commute triangle')
+ ultimately have "(SUP c \<in> M. dist (f x c) (f y c)) = d x y"
+ by (intro cSup_unique) auto
+ with that fim show ?thesis
+ using that fim by (simp add: Met_TC.fdist_def False m'_def image_subset_iff)
+ qed
+ qed
+ qed
+qed
+
+
+lemma (in Metric_space) metric_completion:
+ obtains f :: "['a,'a] \<Rightarrow> real" and m' where
+ "mcomplete_of m'"
+ "f ` M \<subseteq> mspace m' "
+ "mtopology_of m' closure_of f ` M = mspace m'"
+ "\<And>x y. \<lbrakk>x \<in> M; y \<in> M\<rbrakk> \<Longrightarrow> mdist m' (f x) (f y) = d x y"
+proof -
+ obtain f :: "['a,'a] \<Rightarrow> real" and S where
+ Ssub: "S \<subseteq> mspace(funspace M euclidean_metric)"
+ and mcom: "mcomplete_of (submetric (funspace M euclidean_metric) S)"
+ and fim: "f ` M \<subseteq> S"
+ and eqS: "mtopology_of(funspace M euclidean_metric) closure_of f ` M = S"
+ and eqd: "\<And>x y. \<lbrakk>x \<in> M; y \<in> M\<rbrakk> \<Longrightarrow> mdist (funspace M euclidean_metric) (f x) (f y) = d x y"
+ using metric_completion_explicit by metis
+ define m' where "m' \<equiv> submetric (funspace M euclidean_metric) S"
+ show thesis
+ proof
+ show "mcomplete_of m'"
+ by (simp add: mcom m'_def)
+ show "f ` M \<subseteq> mspace m'"
+ using Ssub fim m'_def by auto
+ show "mtopology_of m' closure_of f ` M = mspace m'"
+ using eqS fim Ssub
+ by (force simp: m'_def mtopology_of_submetric closure_of_subtopology Int_absorb1)
+ show "mdist m' (f x) (f y) = d x y" if "x \<in> M" and "y \<in> M" for x y
+ using that eqd m'_def by force
+ qed
+qed
+
+lemma metrizable_space_completion:
+ assumes "metrizable_space X"
+ obtains f :: "['a,'a] \<Rightarrow> real" and Y where
+ "completely_metrizable_space Y" "embedding_map X Y f"
+ "Y closure_of (f ` (topspace X)) = topspace Y"
+proof -
+ obtain M d where "Metric_space M d" and Xeq: "X = Metric_space.mtopology M d"
+ using assms metrizable_space_def by blast
+ then interpret Metric_space M d by simp
+ obtain f :: "['a,'a] \<Rightarrow> real" and m' where
+ "mcomplete_of m'"
+ and fim: "f ` M \<subseteq> mspace m' "
+ and m': "mtopology_of m' closure_of f ` M = mspace m'"
+ and eqd: "\<And>x y. \<lbrakk>x \<in> M; y \<in> M\<rbrakk> \<Longrightarrow> mdist m' (f x) (f y) = d x y"
+ by (meson metric_completion)
+ show thesis
+ proof
+ show "completely_metrizable_space (mtopology_of m')"
+ using \<open>mcomplete_of m'\<close>
+ unfolding completely_metrizable_space_def mcomplete_of_def mtopology_of_def
+ by (metis Metric_space_mspace_mdist)
+ show "embedding_map X (mtopology_of m') f"
+ using Metric_space12.isometry_imp_embedding_map
+ by (metis Metric_space12_def Metric_space_axioms Metric_space_mspace_mdist Xeq eqd fim mtopology_of_def)
+ show "(mtopology_of m') closure_of f ` topspace X = topspace (mtopology_of m')"
+ by (simp add: Xeq m')
+ qed
+qed
+
+
+
+subsection\<open>Contractions\<close>
+
+lemma (in Metric_space) contraction_imp_unique_fixpoint:
+ assumes "f x = x" "f y = y"
+ and "f ` M \<subseteq> M"
+ and "k < 1"
+ and "\<And>x y. \<lbrakk>x \<in> M; y \<in> M\<rbrakk> \<Longrightarrow> d (f x) (f y) \<le> k * d x y"
+ and "x \<in> M" "y \<in> M"
+ shows "x = y"
+ by (smt (verit, ccfv_SIG) Metric_space.mdist_pos_less Metric_space_axioms assms mult_le_cancel_right1)
+
+text \<open>Banach Fixed-Point Theorem (aka, Contraction Mapping Principle)\<close>
+lemma (in Metric_space) Banach_fixedpoint_thm:
+ assumes mcomplete and "M \<noteq> {}" and fim: "f ` M \<subseteq> M"
+ and "k < 1"
+ and con: "\<And>x y. \<lbrakk>x \<in> M; y \<in> M\<rbrakk> \<Longrightarrow> d (f x) (f y) \<le> k * d x y"
+ obtains x where "x \<in> M" "f x = x"
+proof -
+ obtain a where "a \<in> M"
+ using \<open>M \<noteq> {}\<close> by blast
+ show thesis
+ proof (cases "\<forall>x \<in> M. f x = f a")
+ case True
+ then show ?thesis
+ by (metis \<open>a \<in> M\<close> fim image_subset_iff that)
+ next
+ case False
+ then obtain b where "b \<in> M" and b: "f b \<noteq> f a"
+ by blast
+ have "k>0"
+ using Lipschitz_coefficient_pos [where f=f]
+ by (metis False \<open>a \<in> M\<close> con fim mdist_Self mspace_Self)
+ define \<sigma> where "\<sigma> \<equiv> \<lambda>n. (f^^n) a"
+ have f_iter: "\<sigma> n \<in> M" for n
+ unfolding \<sigma>_def by (induction n) (use \<open>a \<in> M\<close> fim in auto)
+ show ?thesis
+ proof (cases "f a = a")
+ case True
+ then show ?thesis
+ using \<open>a \<in> M\<close> that by blast
+ next
+ case False
+ have "MCauchy \<sigma>"
+ proof -
+ show ?thesis
+ unfolding MCauchy_def
+ proof (intro conjI strip)
+ show "range \<sigma> \<subseteq> M"
+ using f_iter by blast
+ fix \<epsilon>::real
+ assume "\<epsilon>>0"
+ with \<open>k < 1\<close> \<open>f a \<noteq> a\<close> \<open>a \<in> M\<close> fim have gt0: "((1 - k) * \<epsilon>) / d a (f a) > 0"
+ by (fastforce simp: divide_simps)
+ obtain N where "k^N < ((1-k) * \<epsilon>) / d a (f a)"
+ using real_arch_pow_inv [OF gt0 \<open>k < 1\<close>] by blast
+ then have N: "\<And>n. n \<ge> N \<Longrightarrow> k^n < ((1-k) * \<epsilon>) / d a (f a)"
+ by (smt (verit) \<open>0 < k\<close> assms(4) power_decreasing)
+ have "\<forall>n n'. n<n' \<longrightarrow> N \<le> n \<longrightarrow> N \<le> n' \<longrightarrow> d (\<sigma> n) (\<sigma> n') < \<epsilon>"
+ proof (intro exI strip)
+ fix n n'
+ assume "n<n'" "N \<le> n" "N \<le> n'"
+ have "d (\<sigma> n) (\<sigma> n') \<le> (\<Sum>i=n..<n'. d (\<sigma> i) (\<sigma> (Suc i)))"
+ proof -
+ have "n < m \<Longrightarrow> d (\<sigma> n) (\<sigma> m) \<le> (\<Sum>i=n..<m. d (\<sigma> i) (\<sigma> (Suc i)))" for m
+ proof (induction m)
+ case 0
+ then show ?case
+ by simp
+ next
+ case (Suc m)
+ then consider "n<m" | "m=n"
+ by linarith
+ then show ?case
+ proof cases
+ case 1
+ have "d (\<sigma> n) (\<sigma> (Suc m)) \<le> d (\<sigma> n) (\<sigma> m) + d (\<sigma> m) (\<sigma> (Suc m))"
+ by (simp add: f_iter triangle)
+ also have "\<dots> \<le> (\<Sum>i=n..<m. d (\<sigma> i) (\<sigma> (Suc i))) + d (\<sigma> m) (\<sigma> (Suc m))"
+ using Suc 1 by linarith
+ also have "\<dots> = (\<Sum>i = n..<Suc m. d (\<sigma> i) (\<sigma> (Suc i)))"
+ using "1" by force
+ finally show ?thesis .
+ qed auto
+ qed
+ with \<open>n < n'\<close> show ?thesis by blast
+ qed
+ also have "\<dots> \<le> (\<Sum>i=n..<n'. d a (f a) * k^i)"
+ proof (rule sum_mono)
+ fix i
+ assume "i \<in> {n..<n'}"
+ show "d (\<sigma> i) (\<sigma> (Suc i)) \<le> d a (f a) * k ^ i"
+ proof (induction i)
+ case 0
+ then show ?case
+ by (auto simp: \<sigma>_def)
+ next
+ case (Suc i)
+ have "d (\<sigma> (Suc i)) (\<sigma> (Suc (Suc i))) \<le> k * d (\<sigma> i) (\<sigma> (Suc i))"
+ using con \<sigma>_def f_iter fim by fastforce
+ also have "\<dots> \<le> d a (f a) * k ^ Suc i"
+ using Suc \<open>0 < k\<close> by auto
+ finally show ?case .
+ qed
+ qed
+ also have "\<dots> = d a (f a) * (\<Sum>i=n..<n'. k^i)"
+ by (simp add: sum_distrib_left)
+ also have "\<dots> = d a (f a) * (\<Sum>i=0..<n'-n. k^(i+n))"
+ using sum.shift_bounds_nat_ivl [of "power k" 0 n "n'-n"] \<open>n < n'\<close> by simp
+ also have "\<dots> = d a (f a) * k^n * (\<Sum>i<n'-n. k^i)"
+ by (simp add: power_add lessThan_atLeast0 flip: sum_distrib_right)
+ also have "\<dots> = d a (f a) * (k ^ n - k ^ n') / (1 - k)"
+ using \<open>k < 1\<close> \<open>n < n'\<close> apply (simp add: sum_gp_strict)
+ by (simp add: algebra_simps flip: power_add)
+ also have "\<dots> < \<epsilon>"
+ using N \<open>k < 1\<close> \<open>0 < \<epsilon>\<close> \<open>0 < k\<close> \<open>N \<le> n\<close>
+ apply (simp add: field_simps)
+ by (smt (verit) nonneg pos_less_divide_eq zero_less_divide_iff zero_less_power)
+ finally show "d (\<sigma> n) (\<sigma> n') < \<epsilon>" .
+ qed
+ then show "\<exists>N. \<forall>n n'. N \<le> n \<longrightarrow> N \<le> n' \<longrightarrow> d (\<sigma> n) (\<sigma> n') < \<epsilon>"
+ by (metis \<open>0 < \<epsilon>\<close> commute f_iter linorder_not_le local.mdist_zero nat_less_le)
+ qed
+ qed
+ then obtain l where l: "limitin mtopology \<sigma> l sequentially"
+ using \<open>mcomplete\<close> mcomplete_def by blast
+ show ?thesis
+ proof
+ show "l \<in> M"
+ using l limitin_mspace by blast
+ show "f l = l"
+ proof (rule limitin_metric_unique)
+ have "limitin mtopology (f \<circ> \<sigma>) (f l) sequentially"
+ proof (rule continuous_map_limit)
+ have "Lipschitz_continuous_map Self Self f"
+ using con by (auto simp: Lipschitz_continuous_map_def fim)
+ then show "continuous_map mtopology mtopology f"
+ using Lipschitz_continuous_imp_continuous_map Self_def by force
+ qed (use l in auto)
+ moreover have "(f \<circ> \<sigma>) = (\<lambda>i. \<sigma>(i+1))"
+ by (auto simp: \<sigma>_def)
+ ultimately show "limitin mtopology (\<lambda>n. (f^^n)a) (f l) sequentially"
+ using limitin_sequentially_offset_rev [of mtopology \<sigma> 1]
+ by (simp add: \<sigma>_def)
+ qed (use l in \<open>auto simp: \<sigma>_def\<close>)
+ qed
+ qed
+ qed
+qed
+
+
+subsection\<open> The Baire Category Theorem\<close>
+
+text \<open>Possibly relevant to the theorem "Baire" in Elementary Normed Spaces\<close>
+
+lemma (in Metric_space) metric_Baire_category:
+ assumes "mcomplete" "countable \<G>"
+ and "\<And>T. T \<in> \<G> \<Longrightarrow> openin mtopology T \<and> mtopology closure_of T = M"
+shows "mtopology closure_of \<Inter>\<G> = M"
+proof (cases "\<G>={}")
+ case False
+ then obtain U :: "nat \<Rightarrow> 'a set" where U: "range U = \<G>"
+ by (metis \<open>countable \<G>\<close> uncountable_def)
+ with assms have u_open: "\<And>n. openin mtopology (U n)" and u_dense: "\<And>n. mtopology closure_of (U n) = M"
+ by auto
+ have "\<Inter>(range U) \<inter> W \<noteq> {}" if W: "openin mtopology W" "W \<noteq> {}" for W
+ proof -
+ have "W \<subseteq> M"
+ using openin_mtopology W by blast
+ have "\<exists>r' x'. 0 < r' \<and> r' < r/2 \<and> x' \<in> M \<and> mcball x' r' \<subseteq> mball x r \<inter> U n"
+ if "r>0" "x \<in> M" for x r n
+ proof -
+ obtain z where z: "z \<in> U n \<inter> mball x r"
+ using u_dense [of n] \<open>r>0\<close> \<open>x \<in> M\<close>
+ by (metis dense_intersects_open centre_in_mball_iff empty_iff openin_mball topspace_mtopology equals0I)
+ then have "z \<in> M" by auto
+ have "openin mtopology (U n \<inter> mball x r)"
+ by (simp add: openin_Int u_open)
+ with \<open>z \<in> M\<close> z obtain e where "e>0" and e: "mcball z e \<subseteq> U n \<inter> mball x r"
+ by (meson openin_mtopology_mcball)
+ define r' where "r' \<equiv> min e (r/4)"
+ show ?thesis
+ proof (intro exI conjI)
+ show "0 < r'" "r' < r / 2" "z \<in> M"
+ using \<open>e>0\<close> \<open>r>0\<close> \<open>z \<in> M\<close> by (auto simp: r'_def)
+ show "mcball z r' \<subseteq> mball x r \<inter> U n"
+ using Metric_space.mcball_subset_concentric e r'_def by auto
+ qed
+ qed
+ then obtain nextr nextx
+ where nextr: "\<And>r x n. \<lbrakk>r>0; x\<in>M\<rbrakk> \<Longrightarrow> 0 < nextr r x n \<and> nextr r x n < r/2"
+ and nextx: "\<And>r x n. \<lbrakk>r>0; x\<in>M\<rbrakk> \<Longrightarrow> nextx r x n \<in> M"
+ and nextsub: "\<And>r x n. \<lbrakk>r>0; x\<in>M\<rbrakk> \<Longrightarrow> mcball (nextx r x n) (nextr r x n) \<subseteq> mball x r \<inter> U n"
+ by metis
+ obtain x0 where x0: "x0 \<in> U 0 \<inter> W"
+ by (metis W dense_intersects_open topspace_mtopology all_not_in_conv u_dense)
+ then have "x0 \<in> M"
+ using \<open>W \<subseteq> M\<close> by fastforce
+ obtain r0 where "0 < r0" "r0 < 1" and sub: "mcball x0 r0 \<subseteq> U 0 \<inter> W"
+ proof -
+ have "openin mtopology (U 0 \<inter> W)"
+ using W u_open by blast
+ then obtain r where "r>0" and r: "mball x0 r \<subseteq> U 0" "mball x0 r \<subseteq> W"
+ by (meson Int_subset_iff openin_mtopology x0)
+ define r0 where "r0 \<equiv> (min r 1) / 2"
+ show thesis
+ proof
+ show "0 < r0" "r0 < 1"
+ using \<open>r>0\<close> by (auto simp: r0_def)
+ show "mcball x0 r0 \<subseteq> U 0 \<inter> W"
+ using r \<open>0 < r0\<close> r0_def by auto
+ qed
+ qed
+ define b where "b \<equiv> rec_nat (x0,r0) (\<lambda>n (x,r). (nextx r x n, nextr r x n))"
+ have b0[simp]: "b 0 = (x0,r0)"
+ by (simp add: b_def)
+ have bSuc[simp]: "b (Suc n) = (let (x,r) = b n in (nextx r x n, nextr r x n))" for n
+ by (simp add: b_def)
+ define xf where "xf \<equiv> fst \<circ> b"
+ define rf where "rf \<equiv> snd \<circ> b"
+ have rfxf: "0 < rf n \<and> xf n \<in> M" for n
+ proof (induction n)
+ case 0
+ with \<open>0 < r0\<close> \<open>x0 \<in> M\<close> show ?case
+ by (auto simp: rf_def xf_def)
+ next
+ case (Suc n)
+ then show ?case
+ by (auto simp: rf_def xf_def case_prod_unfold nextr nextx Let_def)
+ qed
+ have mcball_sub: "mcball (xf (Suc n)) (rf (Suc n)) \<subseteq> mball (xf n) (rf n) \<inter> U n" for n
+ using rfxf nextsub by (auto simp: xf_def rf_def case_prod_unfold Let_def)
+ have half: "rf (Suc n) < rf n / 2" for n
+ using rfxf nextr by (auto simp: xf_def rf_def case_prod_unfold Let_def)
+ then have "decseq rf"
+ using rfxf by (smt (verit, ccfv_threshold) decseq_SucI field_sum_of_halves)
+ have nested: "mball (xf n) (rf n) \<subseteq> mball (xf m) (rf m)" if "m \<le> n" for m n
+ using that
+ proof (induction n)
+ case (Suc n)
+ then show ?case
+ by (metis mcball_sub order.trans inf.boundedE le_Suc_eq mball_subset_mcball order.refl)
+ qed auto
+ have "MCauchy xf"
+ unfolding MCauchy_def
+ proof (intro conjI strip)
+ show "range xf \<subseteq> M"
+ using rfxf by blast
+ fix \<epsilon> :: real
+ assume "\<epsilon>>0"
+ then obtain N where N: "inverse (2^N) < \<epsilon>"
+ using real_arch_pow_inv by (force simp flip: power_inverse)
+ have "d (xf n) (xf n') < \<epsilon>" if "n \<le> n'" "N \<le> n" "N \<le> n'" for n n'
+ proof -
+ have *: "rf n < inverse (2 ^ n)" for n
+ proof (induction n)
+ case 0
+ then show ?case
+ by (simp add: \<open>r0 < 1\<close> rf_def)
+ next
+ case (Suc n)
+ with half show ?case
+ by simp (smt (verit))
+ qed
+ have "rf n \<le> rf N"
+ using \<open>decseq rf\<close> \<open>N \<le> n\<close> by (simp add: decseqD)
+ moreover
+ have "xf n' \<in> mball (xf n) (rf n)"
+ using nested rfxf \<open>n \<le> n'\<close> by blast
+ ultimately have "d (xf n) (xf n') < rf N"
+ by auto
+ also have "\<dots> < \<epsilon>"
+ using "*" N order.strict_trans by blast
+ finally show ?thesis .
+ qed
+ then show "\<exists>N. \<forall>n n'. N \<le> n \<longrightarrow> N \<le> n' \<longrightarrow> d (xf n) (xf n') < \<epsilon>"
+ by (metis commute linorder_le_cases)
+ qed
+ then obtain l where l: "limitin mtopology xf l sequentially"
+ using \<open>mcomplete\<close> mcomplete_alt by blast
+ have l_in: "l \<in> mcball (xf n) (rf n)" for n
+ proof -
+ have "\<forall>\<^sub>F m in sequentially. xf m \<in> mcball (xf n) (rf n)"
+ unfolding eventually_sequentially
+ by (meson nested rfxf centre_in_mball_iff mball_subset_mcball subset_iff)
+ with l limitin_closedin show ?thesis
+ by (metis closedin_mcball trivial_limit_sequentially)
+ qed
+ then have "\<And>n. l \<in> U n"
+ using mcball_sub by blast
+ moreover have "l \<in> W"
+ using l_in[of 0] sub by (auto simp: xf_def rf_def)
+ ultimately show ?thesis by auto
+ qed
+ with U show ?thesis
+ by (metis dense_intersects_open topspace_mtopology)
+qed auto
+
+
+
+lemma (in Metric_space) metric_Baire_category_alt:
+ assumes "mcomplete" "countable \<G>"
+ and empty: "\<And>T. T \<in> \<G> \<Longrightarrow> closedin mtopology T \<and> mtopology interior_of T = {}"
+ shows "mtopology interior_of \<Union>\<G> = {}"
+proof -
+ have *: "mtopology closure_of \<Inter>((-)M ` \<G>) = M"
+ proof (intro metric_Baire_category conjI \<open>mcomplete\<close>)
+ show "countable ((-) M ` \<G>)"
+ using \<open>countable \<G>\<close> by blast
+ fix T
+ assume "T \<in> (-) M ` \<G>"
+ then obtain U where U: "U \<in> \<G>" "T = M-U" "U \<subseteq> M"
+ using empty metric_closedin_iff_sequentially_closed by force
+ with empty show "openin mtopology T" by blast
+ show "mtopology closure_of T = M"
+ using U by (simp add: closure_of_interior_of double_diff empty)
+ qed
+ with closure_of_eq show ?thesis
+ by (fastforce simp: interior_of_closure_of split: if_split_asm)
+qed
+
+
+text \<open>Since all locally compact Hausdorff spaces are regular, the disjunction in the HOL Light version is redundant.\<close>
+lemma Baire_category_aux:
+ assumes "locally_compact_space X" "regular_space X"
+ and "countable \<G>"
+ and empty: "\<And>G. G \<in> \<G> \<Longrightarrow> closedin X G \<and> X interior_of G = {}"
+ shows "X interior_of \<Union>\<G> = {}"
+proof (cases "\<G> = {}")
+ case True
+ then show ?thesis
+ by simp
+next
+ case False
+ then obtain T :: "nat \<Rightarrow> 'a set" where T: "\<G> = range T"
+ by (metis \<open>countable \<G>\<close> uncountable_def)
+ with empty have Tempty: "\<And>n. X interior_of (T n) = {}"
+ by auto
+ show ?thesis
+ proof (clarsimp simp: T interior_of_def)
+ fix z U
+ assume "z \<in> U" and opeA: "openin X U" and Asub: "U \<subseteq> \<Union> (range T)"
+ with openin_subset have "z \<in> topspace X"
+ by blast
+ have "neighbourhood_base_of (\<lambda>C. compactin X C \<and> closedin X C) X"
+ using assms locally_compact_regular_space_neighbourhood_base by auto
+ then obtain V K where "openin X V" "compactin X K" "closedin X K" "z \<in> V" "V \<subseteq> K" "K \<subseteq> U"
+ by (metis (no_types, lifting) \<open>z \<in> U\<close> neighbourhood_base_of opeA)
+ have nb_closedin: "neighbourhood_base_of (closedin X) X"
+ using \<open>regular_space X\<close> neighbourhood_base_of_closedin by auto
+ have "\<exists>\<Phi>. \<forall>n. (\<Phi> n \<subseteq> K \<and> closedin X (\<Phi> n) \<and> X interior_of \<Phi> n \<noteq> {} \<and> disjnt (\<Phi> n) (T n)) \<and>
+ \<Phi> (Suc n) \<subseteq> \<Phi> n"
+ proof (rule dependent_nat_choice)
+ show "\<exists>x\<subseteq>K. closedin X x \<and> X interior_of x \<noteq> {} \<and> disjnt x (T 0)"
+ proof -
+ have False if "V \<subseteq> T 0"
+ using Tempty \<open>openin X V\<close> \<open>z \<in> V\<close> interior_of_maximal that by fastforce
+ then obtain x where "openin X (V - T 0) \<and> x \<in> V - T 0"
+ using T \<open>openin X V\<close> empty by blast
+ with nb_closedin
+ obtain N C where "openin X N" "closedin X C" "x \<in> N" "N \<subseteq> C" "C \<subseteq> V - T 0"
+ unfolding neighbourhood_base_of by metis
+ show ?thesis
+ proof (intro exI conjI)
+ show "C \<subseteq> K"
+ using \<open>C \<subseteq> V - T 0\<close> \<open>V \<subseteq> K\<close> by auto
+ show "X interior_of C \<noteq> {}"
+ by (metis \<open>N \<subseteq> C\<close> \<open>openin X N\<close> \<open>x \<in> N\<close> empty_iff interior_of_eq_empty)
+ show "disjnt C (T 0)"
+ using \<open>C \<subseteq> V - T 0\<close> disjnt_iff by fastforce
+ qed (use \<open>closedin X C\<close> in auto)
+ qed
+ show "\<exists>L. (L \<subseteq> K \<and> closedin X L \<and> X interior_of L \<noteq> {} \<and> disjnt L (T (Suc n))) \<and> L \<subseteq> C"
+ if \<section>: "C \<subseteq> K \<and> closedin X C \<and> X interior_of C \<noteq> {} \<and> disjnt C (T n)"
+ for C n
+ proof -
+ have False if "X interior_of C \<subseteq> T (Suc n)"
+ by (metis Tempty interior_of_eq_empty \<section> openin_interior_of that)
+ then obtain x where "openin X (X interior_of C - T (Suc n)) \<and> x \<in> X interior_of C - T (Suc n)"
+ using T empty by fastforce
+ with nb_closedin
+ obtain N D where "openin X N" "closedin X D" "x \<in> N" "N \<subseteq> D" and D: "D \<subseteq> X interior_of C - T(Suc n)"
+ unfolding neighbourhood_base_of by metis
+ show ?thesis
+ proof (intro conjI exI)
+ show "D \<subseteq> K"
+ using D interior_of_subset \<section> by fastforce
+ show "X interior_of D \<noteq> {}"
+ by (metis \<open>N \<subseteq> D\<close> \<open>openin X N\<close> \<open>x \<in> N\<close> empty_iff interior_of_eq_empty)
+ show "disjnt D (T (Suc n))"
+ using D disjnt_iff by fastforce
+ show "D \<subseteq> C"
+ using interior_of_subset [of X C] D by blast
+ qed (use \<open>closedin X D\<close> in auto)
+ qed
+ qed
+ then obtain \<Phi> where \<Phi>: "\<And>n. \<Phi> n \<subseteq> K \<and> closedin X (\<Phi> n) \<and> X interior_of \<Phi> n \<noteq> {} \<and> disjnt (\<Phi> n) (T n)"
+ and "\<And>n. \<Phi> (Suc n) \<subseteq> \<Phi> n" by metis
+ then have "decseq \<Phi>"
+ by (simp add: decseq_SucI)
+ moreover have "\<And>n. \<Phi> n \<noteq> {}"
+ by (metis \<Phi> bot.extremum_uniqueI interior_of_subset)
+ ultimately have "\<Inter>(range \<Phi>) \<noteq> {}"
+ by (metis \<Phi> compact_space_imp_nest \<open>compactin X K\<close> compactin_subspace closedin_subset_topspace)
+ moreover have "U \<subseteq> {y. \<exists>x. y \<in> T x}"
+ using Asub by auto
+ with T have "{a. \<forall>n. a \<in> \<Phi> n} \<subseteq> {}"
+ by (smt (verit) Asub \<Phi> Collect_empty_eq UN_iff \<open>K \<subseteq> U\<close> disjnt_iff subset_iff)
+ ultimately show False
+ by blast
+ qed
+qed
+
+
+lemma Baire_category_alt:
+ assumes "completely_metrizable_space X \<or> locally_compact_space X \<and> regular_space X"
+ and "countable \<G>"
+ and "\<And>T. T \<in> \<G> \<Longrightarrow> closedin X T \<and> X interior_of T = {}"
+ shows "X interior_of \<Union>\<G> = {}"
+ using Baire_category_aux [of X \<G>] Metric_space.metric_Baire_category_alt
+ by (metis assms completely_metrizable_space_def)
+
+
+lemma Baire_category:
+ assumes "completely_metrizable_space X \<or> locally_compact_space X \<and> regular_space X"
+ and "countable \<G>"
+ and top: "\<And>T. T \<in> \<G> \<Longrightarrow> openin X T \<and> X closure_of T = topspace X"
+ shows "X closure_of \<Inter>\<G> = topspace X"
+proof (cases "\<G>={}")
+ case False
+ have *: "X interior_of \<Union>((-)(topspace X) ` \<G>) = {}"
+ proof (intro Baire_category_alt conjI assms)
+ show "countable ((-) (topspace X) ` \<G>)"
+ using assms by blast
+ fix T
+ assume "T \<in> (-) (topspace X) ` \<G>"
+ then obtain U where U: "U \<in> \<G>" "T = (topspace X) - U" "U \<subseteq> (topspace X)"
+ by (meson top image_iff openin_subset)
+ then show "closedin X T"
+ by (simp add: closedin_diff top)
+ show "X interior_of T = {}"
+ using U top by (simp add: interior_of_closure_of double_diff)
+ qed
+ then show ?thesis
+ by (simp add: closure_of_eq_topspace interior_of_complement)
+qed auto
+
+
+subsection\<open>Sierpinski-Hausdorff type results about countable closed unions\<close>
+
+lemma locally_connected_not_countable_closed_union:
+ assumes "topspace X \<noteq> {}" and csX: "connected_space X"
+ and lcX: "locally_connected_space X"
+ and X: "completely_metrizable_space X \<or> locally_compact_space X \<and> Hausdorff_space X"
+ and "countable \<U>" and pwU: "pairwise disjnt \<U>"
+ and clo: "\<And>C. C \<in> \<U> \<Longrightarrow> closedin X C \<and> C \<noteq> {}"
+ and UU_eq: "\<Union>\<U> = topspace X"
+ shows "\<U> = {topspace X}"
+proof -
+ define \<V> where "\<V> \<equiv> (frontier_of) X ` \<U>"
+ define B where "B \<equiv> \<Union>\<V>"
+ then have Bsub: "B \<subseteq> topspace X"
+ by (simp add: Sup_le_iff \<V>_def closedin_frontier_of closedin_subset)
+ have allsub: "A \<subseteq> topspace X" if "A \<in> \<U>" for A
+ by (meson clo closedin_def that)
+ show ?thesis
+ proof (rule ccontr)
+ assume "\<U> \<noteq> {topspace X}"
+ with assms have "\<exists>A\<in>\<U>. \<not> (closedin X A \<and> openin X A)"
+ by (metis Union_empty connected_space_clopen_in singletonI subsetI subset_singleton_iff)
+ then have "B \<noteq> {}"
+ by (auto simp: B_def \<V>_def frontier_of_eq_empty allsub)
+ moreover
+ have "subtopology X B interior_of B = B"
+ by (simp add: Bsub interior_of_openin openin_subtopology_refl)
+ ultimately have int_B_nonempty: "subtopology X B interior_of B \<noteq> {}"
+ by auto
+ have "subtopology X B interior_of \<Union>\<V> = {}"
+ proof (intro Baire_category_alt conjI)
+ have "\<Union>\<U> \<subseteq> B \<union> \<Union>((interior_of) X ` \<U>)"
+ using clo closure_of_closedin by (fastforce simp: B_def \<V>_def frontier_of_def)
+ moreover have "B \<union> \<Union>((interior_of) X ` \<U>) \<subseteq> \<Union>\<U>"
+ using allsub clo frontier_of_subset_eq interior_of_subset by (fastforce simp: B_def \<V>_def )
+ moreover have "disjnt B (\<Union>((interior_of) X ` \<U>))"
+ using pwU
+ apply (clarsimp simp: B_def \<V>_def frontier_of_def pairwise_def disjnt_iff)
+ by (metis clo closure_of_eq interior_of_subset subsetD)
+ ultimately have "B = topspace X - \<Union> ((interior_of) X ` \<U>)"
+ by (auto simp: UU_eq disjnt_iff)
+ then have "closedin X B"
+ by fastforce
+ with X show "completely_metrizable_space (subtopology X B) \<or> locally_compact_space (subtopology X B) \<and> regular_space (subtopology X B)"
+ by (metis completely_metrizable_space_closedin locally_compact_Hausdorff_or_regular
+ locally_compact_space_closed_subset regular_space_subtopology)
+ show "countable \<V>"
+ by (simp add: \<V>_def \<open>countable \<U>\<close>)
+ fix V
+ assume "V \<in> \<V>"
+ then obtain S where S: "S \<in> \<U>" "V = X frontier_of S"
+ by (auto simp: \<V>_def)
+ show "closedin (subtopology X B) V"
+ by (metis B_def Sup_upper \<V>_def \<open>V \<in> \<V>\<close> closedin_frontier_of closedin_subset_topspace image_iff)
+ have "subtopology X B interior_of (X frontier_of S) = {}"
+ proof (clarsimp simp: interior_of_def openin_subtopology_alt)
+ fix a U
+ assume "a \<in> B" "a \<in> U" and opeU: "openin X U" and BUsub: "B \<inter> U \<subseteq> X frontier_of S"
+ then have "a \<in> S"
+ by (meson IntI \<open>S \<in> \<U>\<close> clo frontier_of_subset_closedin subsetD)
+ then obtain W C where "openin X W" "connectedin X C" "a \<in> W" "W \<subseteq> C" "C \<subseteq> U"
+ by (metis \<open>a \<in> U\<close> lcX locally_connected_space opeU)
+ have "W \<inter> X frontier_of S \<noteq> {}"
+ using \<open>B \<inter> U \<subseteq> X frontier_of S\<close> \<open>a \<in> B\<close> \<open>a \<in> U\<close> \<open>a \<in> W\<close> by auto
+ with frontier_of_openin_straddle_Int
+ obtain "W \<inter> S \<noteq> {}" "W - S \<noteq> {}" "W \<subseteq> topspace X"
+ using \<open>openin X W\<close> by (metis openin_subset)
+ then obtain b where "b \<in> topspace X" "b \<in> W-S"
+ by blast
+ with UU_eq obtain T where "T \<in> \<U>" "T \<noteq> S" "W \<inter> T \<noteq> {}"
+ by auto
+ then have "disjnt S T"
+ by (metis \<open>S \<in> \<U>\<close> pairwise_def pwU)
+ then have "C - T \<noteq> {}"
+ by (meson Diff_eq_empty_iff \<open>W \<subseteq> C\<close> \<open>a \<in> S\<close> \<open>a \<in> W\<close> disjnt_iff subsetD)
+ then have "C \<inter> X frontier_of T \<noteq> {}"
+ using \<open>W \<inter> T \<noteq> {}\<close> \<open>W \<subseteq> C\<close> \<open>connectedin X C\<close> connectedin_Int_frontier_of by blast
+ moreover have "C \<inter> X frontier_of T = {}"
+ proof -
+ have "X frontier_of S \<subseteq> S" "X frontier_of T \<subseteq> T"
+ using frontier_of_subset_closedin \<open>S \<in> \<U>\<close> \<open>T \<in> \<U>\<close> clo by blast+
+ moreover have "X frontier_of T \<union> B = B"
+ using B_def \<V>_def \<open>T \<in> \<U>\<close> by blast
+ ultimately show ?thesis
+ using BUsub \<open>C \<subseteq> U\<close> \<open>disjnt S T\<close> unfolding disjnt_def by blast
+ qed
+ ultimately show False
+ by simp
+ qed
+ with S show "subtopology X B interior_of V = {}"
+ by meson
+ qed
+ then show False
+ using B_def int_B_nonempty by blast
+ qed
+qed
+
+lemma real_Sierpinski_lemma:
+ fixes a b::real
+ assumes "a \<le> b"
+ and "countable \<U>" and pwU: "pairwise disjnt \<U>"
+ and clo: "\<And>C. C \<in> \<U> \<Longrightarrow> closed C \<and> C \<noteq> {}"
+ and "\<Union>\<U> = {a..b}"
+ shows "\<U> = {{a..b}}"
+proof -
+ have "locally_connected_space (top_of_set {a..b})"
+ by (simp add: locally_connected_real_interval)
+ moreover
+ have "completely_metrizable_space (top_of_set {a..b})"
+ by (metis box_real(2) completely_metrizable_space_cbox)
+ ultimately
+ show ?thesis
+ using locally_connected_not_countable_closed_union [of "subtopology euclidean {a..b}"] assms
+ apply (simp add: closedin_subtopology)
+ by (metis Union_upper inf.orderE)
+qed
+
+
+subsection\<open>The Tychonoff embedding\<close>
+
+lemma completely_regular_space_cube_embedding_explicit:
+ assumes "completely_regular_space X" "Hausdorff_space X"
+ shows "embedding_map X
+ (product_topology (\<lambda>f. top_of_set {0..1::real})
+ (mspace (submetric (cfunspace X euclidean_metric)
+ {f. f ` topspace X \<subseteq> {0..1}})) )
+ (\<lambda>x. \<lambda>f \<in> mspace (submetric (cfunspace X euclidean_metric) {f. f ` topspace X \<subseteq> {0..1}}).
+ f x)"
+proof -
+ define K where "K \<equiv> mspace(submetric (cfunspace X euclidean_metric) {f. f ` topspace X \<subseteq> {0..1::real}})"
+ define e where "e \<equiv> \<lambda>x. \<lambda>f\<in>K. f x"
+ have "e x \<noteq> e y" if xy: "x \<noteq> y" "x \<in> topspace X" "y \<in> topspace X" for x y
+ proof -
+ have "closedin X {x}"
+ by (simp add: \<open>Hausdorff_space X\<close> closedin_Hausdorff_singleton \<open>x \<in> topspace X\<close>)
+ then obtain f where contf: "continuous_map X euclideanreal f"
+ and f01: "\<And>x. x \<in> topspace X \<Longrightarrow> f x \<in> {0..1}" and fxy: "f y = 0" "f x = 1"
+ using \<open>completely_regular_space X\<close> xy unfolding completely_regular_space_def
+ by (smt (verit, ccfv_threshold) Diff_iff continuous_map_in_subtopology image_subset_iff singleton_iff)
+ then have "bounded (f ` topspace X)"
+ by (meson bounded_closed_interval bounded_subset image_subset_iff)
+ with contf f01 have "restrict f (topspace X) \<in> K"
+ by (auto simp: K_def)
+ with fxy xy show ?thesis
+ unfolding e_def by (metis restrict_apply' zero_neq_one)
+ qed
+ then have "inj_on e (topspace X)"
+ by (meson inj_onI)
+ then obtain e' where e': "\<And>x. x \<in> topspace X \<Longrightarrow> e' (e x) = x"
+ by (metis inv_into_f_f)
+ have "continuous_map (subtopology (product_topology (\<lambda>f. top_of_set {0..1}) K) (e ` topspace X)) X e'"
+ proof (clarsimp simp add: continuous_map_atin limitin_atin openin_subtopology_alt e')
+ fix x U
+ assume "e x \<in> K \<rightarrow>\<^sub>E {0..1}" and "x \<in> topspace X" and "openin X U" and "x \<in> U"
+ then obtain g where contg: "continuous_map X (top_of_set {0..1}) g" and "g x = 0"
+ and gim: "g ` (topspace X - U) \<subseteq> {1::real}"
+ using \<open>completely_regular_space X\<close> unfolding completely_regular_space_def
+ by (metis Diff_iff openin_closedin_eq)
+ then have "bounded (g ` topspace X)"
+ by (meson bounded_closed_interval bounded_subset continuous_map_in_subtopology)
+ moreover have "g ` topspace X \<subseteq> {0..1}"
+ using contg by (simp add: continuous_map_in_subtopology)
+ ultimately have g_in_K: "restrict g (topspace X) \<in> K"
+ using contg by (simp add: K_def continuous_map_in_subtopology)
+ have "openin (top_of_set {0..1}) {0..<1::real}"
+ using open_real_greaterThanLessThan[of "-1" 1] by (force simp: openin_open)
+ moreover have "e x \<in> (\<Pi>\<^sub>E f\<in>K. if f = restrict g (topspace X) then {0..<1} else {0..1})"
+ using \<open>e x \<in> K \<rightarrow>\<^sub>E {0..1}\<close> by (simp add: e_def \<open>g x = 0\<close> \<open>x \<in> topspace X\<close> PiE_iff)
+ moreover have "e y = e x"
+ if "y \<notin> U" and ey: "e y \<in> (\<Pi>\<^sub>E f\<in>K. if f = restrict g (topspace X) then {0..<1} else {0..1})"
+ and y: "y \<in> topspace X" for y
+ proof -
+ have "e y (restrict g (topspace X)) \<in> {0..<1}"
+ using ey by (smt (verit, ccfv_SIG) PiE_mem g_in_K)
+ with gim g_in_K y \<open>y \<notin> U\<close> show ?thesis
+ by (fastforce simp: e_def)
+ qed
+ ultimately
+ show "\<exists>W. openin (product_topology (\<lambda>f. top_of_set {0..1}) K) W \<and> e x \<in> W \<and> e' ` (e ` topspace X \<inter> W - {e x}) \<subseteq> U"
+ apply (rule_tac x="PiE K (\<lambda>f. if f = restrict g (topspace X) then {0..<1} else {0..1})" in exI)
+ by (auto simp: openin_PiE_gen e')
+ qed
+ with e' have "embedding_map X (product_topology (\<lambda>f. top_of_set {0..1}) K) e"
+ unfolding embedding_map_def homeomorphic_map_maps homeomorphic_maps_def
+ by (fastforce simp: e_def K_def continuous_map_in_subtopology continuous_map_componentwise)
+ then show ?thesis
+ by (simp add: K_def e_def)
+qed
+
+
+lemma completely_regular_space_cube_embedding:
+ fixes X :: "'a topology"
+ assumes "completely_regular_space X" "Hausdorff_space X"
+ obtains K:: "('a\<Rightarrow>real)set" and e
+ where "embedding_map X (product_topology (\<lambda>f. top_of_set {0..1::real}) K) e"
+ using completely_regular_space_cube_embedding_explicit [OF assms] by metis
+
+
+subsection \<open>Urysohn and Tietze analogs for completely regular spaces\<close>
+
+text\<open>"Urysohn and Tietze analogs for completely regular spaces if (()) set is
+assumed compact instead of closed. Note that Hausdorffness is *not* required: inside () proof
+we factor through the Kolmogorov quotient." -- John Harrison\<close>
+
+lemma Urysohn_completely_regular_closed_compact:
+ fixes a b::real
+ assumes "a \<le> b" "completely_regular_space X" "closedin X S" "compactin X T" "disjnt S T"
+ obtains f where "continuous_map X (subtopology euclidean {a..b}) f" "f ` T \<subseteq> {a}" "f ` S \<subseteq> {b}"
+proof -
+ obtain f where contf: "continuous_map X (subtopology euclideanreal {0..1}) f"
+ and f0: "f ` T \<subseteq> {0}" and f1: "f ` S \<subseteq> {1}"
+ proof (cases "T={}")
+ case True
+ show thesis
+ proof
+ show "continuous_map X (top_of_set {0..1}) (\<lambda>x. 1::real)" "(\<lambda>x. 1::real) ` T \<subseteq> {0}" "(\<lambda>x. 1::real) ` S \<subseteq> {1}"
+ using True by auto
+ qed
+ next
+ case False
+ have "\<And>t. t \<in> T \<Longrightarrow> \<exists>f. continuous_map X (subtopology euclideanreal ({0..1})) f \<and> f t = 0 \<and> f ` S \<subseteq> {1}"
+ using assms unfolding completely_regular_space_def
+ by (meson DiffI compactin_subset_topspace disjnt_iff subset_eq)
+ then obtain g where contg: "\<And>t. t \<in> T \<Longrightarrow> continuous_map X (subtopology euclideanreal {0..1}) (g t)"
+ and g0: "\<And>t. t \<in> T \<Longrightarrow> g t t = 0"
+ and g1: "\<And>t. t \<in> T \<Longrightarrow> g t ` S \<subseteq> {1}"
+ by metis
+ then have g01: "\<And>t. t \<in> T \<Longrightarrow> g t ` topspace X \<subseteq> {0..1}"
+ by (meson continuous_map_in_subtopology)
+ define G where "G \<equiv> \<lambda>t. {x \<in> topspace X. g t x \<in> {..<1/2}}"
+ have "Ball (G`T) (openin X)"
+ using contg unfolding G_def continuous_map_in_subtopology
+ by (smt (verit, best) Collect_cong openin_continuous_map_preimage image_iff open_lessThan open_openin)
+ moreover have "T \<subseteq> \<Union>(G`T)"
+ using \<open>compactin X T\<close> g0 compactin_subset_topspace by (force simp: G_def)
+ ultimately have "\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> G`T \<and> T \<subseteq> \<Union> \<F>"
+ using \<open>compactin X T\<close> unfolding compactin_def by blast
+ then obtain K where K: "finite K" "K \<subseteq> T" "T \<subseteq> \<Union>(G`K)"
+ by (metis finite_subset_image)
+ with False have "K \<noteq> {}"
+ by fastforce
+ define f where "f \<equiv> \<lambda>x. 2 * max 0 (Inf ((\<lambda>t. g t x) ` K) - 1/2)"
+ have [simp]: "max 0 (x - 1/2) = 0 \<longleftrightarrow> x \<le> 1/2" for x::real
+ by force
+ have [simp]: "2 * max 0 (x - 1/2) = 1 \<longleftrightarrow> x = 1" for x::real
+ by (simp add: max_def_raw)
+ show thesis
+ proof
+ have "g t s = 1" if "s \<in> S" "t \<in> K" for s t
+ using \<open>K \<subseteq> T\<close> g1 that by auto
+ then show "f ` S \<subseteq> {1}"
+ using \<open>K \<noteq> {}\<close> by (simp add: f_def image_subset_iff)
+ have "(INF t\<in>K. g t x) \<le> 1/2" if "x \<in> T" for x
+ proof -
+ obtain k where "k \<in> K" "g k x < 1/2"
+ using K \<open>x \<in> T\<close> by (auto simp: G_def)
+ then show ?thesis
+ by (meson \<open>finite K\<close> cInf_le_finite dual_order.trans finite_imageI imageI less_le_not_le)
+ qed
+ then show "f ` T \<subseteq> {0}"
+ by (force simp: f_def)
+ have "\<And>t. t \<in> K \<Longrightarrow> continuous_map X euclideanreal (g t)"
+ using \<open>K \<subseteq> T\<close> contg continuous_map_in_subtopology by blast
+ moreover have "2 * max 0 ((INF t\<in>K. g t x) - 1/2) \<le> 1" if "x \<in> topspace X" for x
+ proof -
+ obtain k where "k \<in> K" "g k x \<le> 1"
+ using K \<open>x \<in> topspace X\<close> \<open>K \<noteq> {}\<close> g01 by (fastforce simp: G_def)
+ then have "(INF t\<in>K. g t x) \<le> 1"
+ by (meson \<open>finite K\<close> cInf_le_finite dual_order.trans finite_imageI imageI)
+ then show ?thesis
+ by (simp add: max_def_raw)
+ qed
+ ultimately show "continuous_map X (top_of_set {0..1}) f"
+ by (force simp: f_def continuous_map_in_subtopology intro!: \<open>finite K\<close> continuous_intros)
+ qed
+ qed
+ define g where "g \<equiv> \<lambda>x. a + (b - a) * f x"
+ show thesis
+ proof
+ have "\<forall>x\<in>topspace X. a + (b - a) * f x \<le> b"
+ using contf \<open>a \<le> b\<close> apply (simp add: continuous_map_in_subtopology image_subset_iff)
+ by (smt (verit, best) mult_right_le_one_le)
+ then show "continuous_map X (top_of_set {a..b}) g"
+ using contf \<open>a \<le> b\<close> unfolding g_def continuous_map_in_subtopology image_subset_iff
+ by (intro conjI continuous_intros; simp)
+ show "g ` T \<subseteq> {a}" "g ` S \<subseteq> {b}"
+ using f0 f1 by (auto simp: g_def)
+ qed
+qed
+
+
+lemma Urysohn_completely_regular_compact_closed:
+ fixes a b::real
+ assumes "a \<le> b" "completely_regular_space X" "compactin X S" "closedin X T" "disjnt S T"
+ obtains f where "continuous_map X (subtopology euclidean {a..b}) f" "f ` T \<subseteq> {a}" "f ` S \<subseteq> {b}"
+proof -
+ obtain f where contf: "continuous_map X (subtopology euclidean {-b..-a}) f" and fim: "f ` T \<subseteq> {-a}" "f ` S \<subseteq> {-b}"
+ by (meson Urysohn_completely_regular_closed_compact assms disjnt_sym neg_le_iff_le)
+ show thesis
+ proof
+ show "continuous_map X (top_of_set {a..b}) (uminus \<circ> f)"
+ using contf by (auto simp: continuous_map_in_subtopology o_def)
+ show "(uminus o f) ` T \<subseteq> {a}" "(uminus o f) ` S \<subseteq> {b}"
+ using fim by fastforce+
+ qed
+qed
+
+lemma Urysohn_completely_regular_compact_closed_alt:
+ fixes a b::real
+ assumes "completely_regular_space X" "compactin X S" "closedin X T" "disjnt S T"
+ obtains f where "continuous_map X euclideanreal f" "f ` T \<subseteq> {a}" "f ` S \<subseteq> {b}"
+proof (cases a b rule: le_cases)
+ case le
+ then show ?thesis
+ by (meson Urysohn_completely_regular_compact_closed assms continuous_map_into_fulltopology that)
+next
+ case ge
+ then show ?thesis
+ using Urysohn_completely_regular_closed_compact assms
+ by (metis Urysohn_completely_regular_closed_compact assms continuous_map_into_fulltopology disjnt_sym that)
+qed
+
+
+lemma Tietze_extension_comp_reg_aux:
+ fixes T :: "real set"
+ assumes "completely_regular_space X" "Hausdorff_space X" "compactin X S"
+ and T: "is_interval T" "T\<noteq>{}"
+ and contf: "continuous_map (subtopology X S) euclidean f" and fim: "f`S \<subseteq> T"
+ obtains g where "continuous_map X euclidean g" "g ` topspace X \<subseteq> T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof -
+ obtain K:: "('a\<Rightarrow>real)set" and e
+ where e0: "embedding_map X (product_topology (\<lambda>f. top_of_set {0..1::real}) K) e"
+ using assms completely_regular_space_cube_embedding by blast
+ define cube where "cube \<equiv> product_topology (\<lambda>f. top_of_set {0..1::real}) K"
+ have e: "embedding_map X cube e"
+ using e0 by (simp add: cube_def)
+ obtain e' where e': "homeomorphic_maps X (subtopology cube (e ` topspace X)) e e'"
+ using e by (force simp: cube_def embedding_map_def homeomorphic_map_maps)
+ then have conte: "continuous_map X (subtopology cube (e ` topspace X)) e"
+ and conte': "continuous_map (subtopology cube (e ` topspace X)) X e'"
+ and e'e: "\<forall>x\<in>topspace X. e' (e x) = x"
+ by (auto simp: homeomorphic_maps_def)
+ have "Hausdorff_space cube"
+ unfolding cube_def
+ using Hausdorff_space_euclidean Hausdorff_space_product_topology Hausdorff_space_subtopology by blast
+ have "normal_space cube"
+ proof (rule compact_Hausdorff_or_regular_imp_normal_space)
+ show "compact_space cube"
+ unfolding cube_def
+ using compact_space_product_topology compact_space_subtopology compactin_euclidean_iff by blast
+ qed (use \<open>Hausdorff_space cube\<close> in auto)
+ moreover
+ have comp: "compactin cube (e ` S)"
+ by (meson \<open>compactin X S\<close> conte continuous_map_in_subtopology image_compactin)
+ then have "closedin cube (e ` S)"
+ by (intro compactin_imp_closedin \<open>Hausdorff_space cube\<close>)
+ moreover
+ have "continuous_map (subtopology cube (e ` S)) euclideanreal (f \<circ> e')"
+ proof (intro continuous_map_compose)
+ show "continuous_map (subtopology cube (e ` S)) (subtopology X S) e'"
+ unfolding continuous_map_in_subtopology
+ proof
+ show "continuous_map (subtopology cube (e ` S)) X e'"
+ by (meson \<open>compactin X S\<close> compactin_subset_topspace conte' continuous_map_from_subtopology_mono image_mono)
+ show "e' ` topspace (subtopology cube (e ` S)) \<subseteq> S"
+ using \<open>compactin X S\<close> compactin_subset_topspace e'e by fastforce
+ qed
+ qed (simp add: contf)
+ moreover
+ have "(f \<circ> e') ` e ` S \<subseteq> T"
+ using \<open>compactin X S\<close> compactin_subset_topspace e'e fim by fastforce
+ ultimately
+ obtain g where contg: "continuous_map cube euclidean g" and gsub: "g ` topspace cube \<subseteq> T"
+ and gf: "\<And>x. x \<in> e`S \<Longrightarrow> g x = (f \<circ> e') x"
+ using Tietze_extension_realinterval T by metis
+ show thesis
+ proof
+ show "continuous_map X euclideanreal (g \<circ> e)"
+ by (meson contg conte continuous_map_compose continuous_map_in_subtopology)
+ show "(g \<circ> e) ` topspace X \<subseteq> T"
+ using gsub conte continuous_map_image_subset_topspace by fastforce
+ fix x
+ assume "x \<in> S"
+ then show "(g \<circ> e) x = f x"
+ using gf \<open>compactin X S\<close> compactin_subset_topspace e'e by fastforce
+ qed
+qed
+
+
+lemma Tietze_extension_completely_regular:
+ assumes "completely_regular_space X" "compactin X S" "is_interval T" "T \<noteq> {}"
+ and contf: "continuous_map (subtopology X S) euclidean f" and fim: "f`S \<subseteq> T"
+ obtains g where "continuous_map X euclideanreal g" "g ` topspace X \<subseteq> T"
+ "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof -
+ define Q where "Q \<equiv> Kolmogorov_quotient X ` (topspace X)"
+ obtain g where contg: "continuous_map (subtopology X (Kolmogorov_quotient X ` S)) euclidean g"
+ and gf: "\<And>x. x \<in> S \<Longrightarrow> g(Kolmogorov_quotient X x) = f x"
+ using Kolmogorov_quotient_lift_exists
+ by (metis \<open>compactin X S\<close> contf compactin_subset_topspace open_openin t0_space_def t1_space)
+ have "S \<subseteq> topspace X"
+ by (simp add: \<open>compactin X S\<close> compactin_subset_topspace)
+ then have [simp]: "Q \<inter> Kolmogorov_quotient X ` S = Kolmogorov_quotient X ` S"
+ using Q_def by blast
+ have creg: "completely_regular_space (subtopology X Q)"
+ by (simp add: \<open>completely_regular_space X\<close> completely_regular_space_subtopology)
+ then have "regular_space (subtopology X Q)"
+ by (simp add: completely_regular_imp_regular_space)
+ then have "Hausdorff_space (subtopology X Q)"
+ using Q_def regular_t0_eq_Hausdorff_space t0_space_Kolmogorov_quotient by blast
+ moreover
+ have "compactin (subtopology X Q) (Kolmogorov_quotient X ` S)"
+ by (metis Q_def \<open>compactin X S\<close> image_compactin quotient_imp_continuous_map quotient_map_Kolmogorov_quotient)
+ ultimately obtain h where conth: "continuous_map (subtopology X Q) euclidean h"
+ and him: "h ` topspace (subtopology X Q) \<subseteq> T"
+ and hg: "\<And>x. x \<in> Kolmogorov_quotient X ` S \<Longrightarrow> h x = g x"
+ using Tietze_extension_comp_reg_aux [of "subtopology X Q" "Kolmogorov_quotient X ` S" T g]
+ apply (simp add: subtopology_subtopology creg contg assms)
+ using fim gf by blast
+ show thesis
+ proof
+ show "continuous_map X euclideanreal (h \<circ> Kolmogorov_quotient X)"
+ by (metis Q_def conth continuous_map_compose quotient_imp_continuous_map quotient_map_Kolmogorov_quotient)
+ show "(h \<circ> Kolmogorov_quotient X) ` topspace X \<subseteq> T"
+ using Q_def continuous_map_Kolmogorov_quotient continuous_map_image_subset_topspace him by fastforce
+ fix x
+ assume "x \<in> S" then show "(h \<circ> Kolmogorov_quotient X) x = f x"
+ by (simp add: gf hg)
+ qed
+qed
+
+end
--- a/src/HOL/Data_Structures/Tree23_Set.thy Wed Jun 28 14:45:29 2023 +0200
+++ b/src/HOL/Data_Structures/Tree23_Set.thy Wed Jun 28 16:01:34 2023 +0200
@@ -103,7 +103,7 @@
"node32 t1 a (UF t2) b (Node3 t3 c t4 d t5) = TD(Node3 t1 a (Node2 t2 b t3) c (Node2 t4 d t5))"
fun node33 :: "'a tree23 \<Rightarrow> 'a \<Rightarrow> 'a tree23 \<Rightarrow> 'a \<Rightarrow> 'a upD \<Rightarrow> 'a upD" where
-"node33 l a m b (TD r) = TD(Node3 l a m b r)" |
+"node33 t1 a t2 b (TD t3) = TD(Node3 t1 a t2 b t3)" |
"node33 t1 a (Node2 t2 b t3) c (UF t4) = TD(Node2 t1 a (Node3 t2 b t3 c t4))" |
"node33 t1 a (Node3 t2 b t3 c t4) d (UF t5) = TD(Node3 t1 a (Node2 t2 b t3) c (Node2 t4 d t5))"
--- a/src/HOL/Library/Countable_Set.thy Wed Jun 28 14:45:29 2023 +0200
+++ b/src/HOL/Library/Countable_Set.thy Wed Jun 28 16:01:34 2023 +0200
@@ -14,6 +14,9 @@
definition countable :: "'a set \<Rightarrow> bool" where
"countable S \<longleftrightarrow> (\<exists>f::'a \<Rightarrow> nat. inj_on f S)"
+lemma countable_as_injective_image_subset: "countable S \<longleftrightarrow> (\<exists>f. \<exists>K::nat set. S = f ` K \<and> inj_on f K)"
+ by (metis countable_def inj_on_the_inv_into the_inv_into_onto)
+
lemma countableE:
assumes S: "countable S" obtains f :: "'a \<Rightarrow> nat" where "inj_on f S"
using S by (auto simp: countable_def)
--- a/src/HOL/Library/Equipollence.thy Wed Jun 28 14:45:29 2023 +0200
+++ b/src/HOL/Library/Equipollence.thy Wed Jun 28 16:01:34 2023 +0200
@@ -1,7 +1,7 @@
section \<open>Equipollence and Other Relations Connected with Cardinality\<close>
theory "Equipollence"
- imports FuncSet
+ imports FuncSet Countable_Set
begin
subsection\<open>Eqpoll\<close>
@@ -47,11 +47,7 @@
by (meson card_of_ordLeq eqpoll_iff_card_of_ordIso lepoll_def lepoll_trans ordIso_iff_ordLeq)
lemma lepoll_trans2 [trans]: "\<lbrakk>A \<lesssim> B; B \<approx> C\<rbrakk> \<Longrightarrow> A \<lesssim> C"
- apply (clarsimp simp: eqpoll_def lepoll_def bij_betw_def)
- apply (rename_tac f g)
- apply (rule_tac x="g \<circ> f" in exI)
- apply (auto simp: image_subset_iff inj_on_def)
- done
+ by (metis bij_betw_def eqpoll_def lepoll_def lepoll_trans order_refl)
lemma eqpoll_sym: "A \<approx> B \<Longrightarrow> B \<approx> A"
unfolding eqpoll_def
@@ -87,9 +83,7 @@
by (auto simp: lepoll_iff)
lemma infinite_le_lepoll: "infinite A \<longleftrightarrow> (UNIV::nat set) \<lesssim> A"
-apply (auto simp: lepoll_def)
- apply (simp add: infinite_countable_subset)
- using infinite_iff_countable_subset by blast
+ by (simp add: infinite_iff_countable_subset lepoll_def)
lemma lepoll_Pow_self: "A \<lesssim> Pow A"
unfolding lepoll_def inj_def
@@ -121,12 +115,7 @@
by (blast intro: lepoll_antisym singleton_lepoll)
lemma subset_singleton_iff_lepoll: "(\<exists>x. S \<subseteq> {x}) \<longleftrightarrow> S \<lesssim> {()}"
-proof safe
- show "S \<lesssim> {()}" if "S \<subseteq> {x}" for x
- using subset_imp_lepoll [OF that] by (simp add: singleton_eqpoll lepoll_trans2)
- show "\<exists>x. S \<subseteq> {x}" if "S \<lesssim> {()}"
- by (metis (no_types, opaque_lifting) image_empty image_insert lepoll_iff that)
-qed
+ using lepoll_iff by fastforce
lemma infinite_insert_lepoll:
assumes "infinite A" shows "insert a A \<lesssim> A"
@@ -158,6 +147,12 @@
using \<open>infinite A\<close> infinite_le_lepoll lepoll_trans by auto
qed
+lemma countable_lepoll: "\<lbrakk>countable A; B \<lesssim> A\<rbrakk> \<Longrightarrow> countable B"
+ by (meson countable_image countable_subset lepoll_iff)
+
+lemma countable_eqpoll: "\<lbrakk>countable A; B \<approx> A\<rbrakk> \<Longrightarrow> countable B"
+ using countable_lepoll eqpoll_imp_lepoll by blast
+
subsection\<open>The strict relation\<close>
lemma lesspoll_not_refl [iff]: "~ (i \<prec> i)"
@@ -192,6 +187,9 @@
assumes "infinite A" "finite B" shows "B \<prec> A"
by (meson assms eqpoll_finite_iff finite_lepoll_infinite lesspoll_def)
+lemma countable_lesspoll: "\<lbrakk>countable A; B \<prec> A\<rbrakk> \<Longrightarrow> countable B"
+ using countable_lepoll lesspoll_def by blast
+
subsection\<open>Mapping by an injection\<close>
lemma inj_on_image_eqpoll_self: "inj_on f A \<Longrightarrow> f ` A \<approx> A"