rename HOL-Multivariate_Analysis to HOL-Analysis.
(* Title: HOL/Analysis/Homeomorphism.thy
Author: LC Paulson (ported from HOL Light)
*)
section \<open>Homeomorphism Theorems\<close>
theory Homeomorphism
imports Path_Connected
begin
subsection \<open>Homeomorphism of all convex compact sets with nonempty interior\<close>
proposition ray_to_rel_frontier:
fixes a :: "'a::real_inner"
assumes "bounded S"
and a: "a \<in> rel_interior S"
and aff: "(a + l) \<in> affine hull S"
and "l \<noteq> 0"
obtains d where "0 < d" "(a + d *\<^sub>R l) \<in> rel_frontier S"
"\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (a + e *\<^sub>R l) \<in> rel_interior S"
proof -
have aaff: "a \<in> affine hull S"
by (meson a hull_subset rel_interior_subset rev_subsetD)
let ?D = "{d. 0 < d \<and> a + d *\<^sub>R l \<notin> rel_interior S}"
obtain B where "B > 0" and B: "S \<subseteq> ball a B"
using bounded_subset_ballD [OF \<open>bounded S\<close>] by blast
have "a + (B / norm l) *\<^sub>R l \<notin> ball a B"
by (simp add: dist_norm \<open>l \<noteq> 0\<close>)
with B have "a + (B / norm l) *\<^sub>R l \<notin> rel_interior S"
using rel_interior_subset subsetCE by blast
with \<open>B > 0\<close> \<open>l \<noteq> 0\<close> have nonMT: "?D \<noteq> {}"
using divide_pos_pos zero_less_norm_iff by fastforce
have bdd: "bdd_below ?D"
by (metis (no_types, lifting) bdd_belowI le_less mem_Collect_eq)
have relin_Ex: "\<And>x. x \<in> rel_interior S \<Longrightarrow>
\<exists>e>0. \<forall>x'\<in>affine hull S. dist x' x < e \<longrightarrow> x' \<in> rel_interior S"
using openin_rel_interior [of S] by (simp add: openin_euclidean_subtopology_iff)
define d where "d = Inf ?D"
obtain \<epsilon> where "0 < \<epsilon>" and \<epsilon>: "\<And>\<eta>. \<lbrakk>0 \<le> \<eta>; \<eta> < \<epsilon>\<rbrakk> \<Longrightarrow> (a + \<eta> *\<^sub>R l) \<in> rel_interior S"
proof -
obtain e where "e>0"
and e: "\<And>x'. x' \<in> affine hull S \<Longrightarrow> dist x' a < e \<Longrightarrow> x' \<in> rel_interior S"
using relin_Ex a by blast
show thesis
proof (rule_tac \<epsilon> = "e / norm l" in that)
show "0 < e / norm l" by (simp add: \<open>0 < e\<close> \<open>l \<noteq> 0\<close>)
next
show "a + \<eta> *\<^sub>R l \<in> rel_interior S" if "0 \<le> \<eta>" "\<eta> < e / norm l" for \<eta>
proof (rule e)
show "a + \<eta> *\<^sub>R l \<in> affine hull S"
by (metis (no_types) add_diff_cancel_left' aff affine_affine_hull mem_affine_3_minus aaff)
show "dist (a + \<eta> *\<^sub>R l) a < e"
using that by (simp add: \<open>l \<noteq> 0\<close> dist_norm pos_less_divide_eq)
qed
qed
qed
have inint: "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> a + e *\<^sub>R l \<in> rel_interior S"
unfolding d_def using cInf_lower [OF _ bdd]
by (metis (no_types, lifting) a add.right_neutral le_less mem_Collect_eq not_less real_vector.scale_zero_left)
have "\<epsilon> \<le> d"
unfolding d_def
apply (rule cInf_greatest [OF nonMT])
using \<epsilon> dual_order.strict_implies_order le_less_linear by blast
with \<open>0 < \<epsilon>\<close> have "0 < d" by simp
have "a + d *\<^sub>R l \<notin> rel_interior S"
proof
assume adl: "a + d *\<^sub>R l \<in> rel_interior S"
obtain e where "e > 0"
and e: "\<And>x'. x' \<in> affine hull S \<Longrightarrow> dist x' (a + d *\<^sub>R l) < e \<Longrightarrow> x' \<in> rel_interior S"
using relin_Ex adl by blast
have "d + e / norm l \<le> Inf {d. 0 < d \<and> a + d *\<^sub>R l \<notin> rel_interior S}"
proof (rule cInf_greatest [OF nonMT], clarsimp)
fix x::real
assume "0 < x" and nonrel: "a + x *\<^sub>R l \<notin> rel_interior S"
show "d + e / norm l \<le> x"
proof (cases "x < d")
case True with inint nonrel \<open>0 < x\<close>
show ?thesis by auto
next
case False
then have dle: "x < d + e / norm l \<Longrightarrow> dist (a + x *\<^sub>R l) (a + d *\<^sub>R l) < e"
by (simp add: field_simps \<open>l \<noteq> 0\<close>)
have ain: "a + x *\<^sub>R l \<in> affine hull S"
by (metis add_diff_cancel_left' aff affine_affine_hull mem_affine_3_minus aaff)
show ?thesis
using e [OF ain] nonrel dle by force
qed
qed
then show False
using \<open>0 < e\<close> \<open>l \<noteq> 0\<close> by (simp add: d_def [symmetric] divide_simps)
qed
moreover have "a + d *\<^sub>R l \<in> closure S"
proof (clarsimp simp: closure_approachable)
fix \<eta>::real assume "0 < \<eta>"
have 1: "a + (d - min d (\<eta> / 2 / norm l)) *\<^sub>R l \<in> S"
apply (rule subsetD [OF rel_interior_subset inint])
using \<open>l \<noteq> 0\<close> \<open>0 < d\<close> \<open>0 < \<eta>\<close> by auto
have "norm l * min d (\<eta> / (norm l * 2)) \<le> norm l * (\<eta> / (norm l * 2))"
by (metis min_def mult_left_mono norm_ge_zero order_refl)
also have "... < \<eta>"
using \<open>l \<noteq> 0\<close> \<open>0 < \<eta>\<close> by (simp add: divide_simps)
finally have 2: "norm l * min d (\<eta> / (norm l * 2)) < \<eta>" .
show "\<exists>y\<in>S. dist y (a + d *\<^sub>R l) < \<eta>"
apply (rule_tac x="a + (d - min d (\<eta> / 2 / norm l)) *\<^sub>R l" in bexI)
using 1 2 \<open>0 < d\<close> \<open>0 < \<eta>\<close> apply (auto simp: algebra_simps)
done
qed
ultimately have infront: "a + d *\<^sub>R l \<in> rel_frontier S"
by (simp add: rel_frontier_def)
show ?thesis
by (rule that [OF \<open>0 < d\<close> infront inint])
qed
corollary ray_to_frontier:
fixes a :: "'a::euclidean_space"
assumes "bounded S"
and a: "a \<in> interior S"
and "l \<noteq> 0"
obtains d where "0 < d" "(a + d *\<^sub>R l) \<in> frontier S"
"\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (a + e *\<^sub>R l) \<in> interior S"
proof -
have "interior S = rel_interior S"
using a rel_interior_nonempty_interior by auto
then have "a \<in> rel_interior S"
using a by simp
then show ?thesis
apply (rule ray_to_rel_frontier [OF \<open>bounded S\<close> _ _ \<open>l \<noteq> 0\<close>])
using a affine_hull_nonempty_interior apply blast
by (simp add: \<open>interior S = rel_interior S\<close> frontier_def rel_frontier_def that)
qed
proposition
fixes S :: "'a::euclidean_space set"
assumes "compact S" and 0: "0 \<in> rel_interior S"
and star: "\<And>x. x \<in> S \<Longrightarrow> open_segment 0 x \<subseteq> rel_interior S"
shows starlike_compact_projective1_0:
"S - rel_interior S homeomorphic sphere 0 1 \<inter> affine hull S"
(is "?SMINUS homeomorphic ?SPHER")
and starlike_compact_projective2_0:
"S homeomorphic cball 0 1 \<inter> affine hull S"
(is "S homeomorphic ?CBALL")
proof -
have starI: "(u *\<^sub>R x) \<in> rel_interior S" if "x \<in> S" "0 \<le> u" "u < 1" for x u
proof (cases "x=0 \<or> u=0")
case True with 0 show ?thesis by force
next
case False with that show ?thesis
by (auto simp: in_segment intro: star [THEN subsetD])
qed
have "0 \<in> S" using assms rel_interior_subset by auto
define proj where "proj \<equiv> \<lambda>x::'a. x /\<^sub>R norm x"
have eqI: "x = y" if "proj x = proj y" "norm x = norm y" for x y
using that by (force simp: proj_def)
then have iff_eq: "\<And>x y. (proj x = proj y \<and> norm x = norm y) \<longleftrightarrow> x = y"
by blast
have projI: "x \<in> affine hull S \<Longrightarrow> proj x \<in> affine hull S" for x
by (metis \<open>0 \<in> S\<close> affine_hull_span_0 hull_inc span_mul proj_def)
have nproj1 [simp]: "x \<noteq> 0 \<Longrightarrow> norm(proj x) = 1" for x
by (simp add: proj_def)
have proj0_iff [simp]: "proj x = 0 \<longleftrightarrow> x = 0" for x
by (simp add: proj_def)
have cont_proj: "continuous_on (UNIV - {0}) proj"
unfolding proj_def by (rule continuous_intros | force)+
have proj_spherI: "\<And>x. \<lbrakk>x \<in> affine hull S; x \<noteq> 0\<rbrakk> \<Longrightarrow> proj x \<in> ?SPHER"
by (simp add: projI)
have "bounded S" "closed S"
using \<open>compact S\<close> compact_eq_bounded_closed by blast+
have inj_on_proj: "inj_on proj (S - rel_interior S)"
proof
fix x y
assume x: "x \<in> S - rel_interior S"
and y: "y \<in> S - rel_interior S" and eq: "proj x = proj y"
then have xynot: "x \<noteq> 0" "y \<noteq> 0" "x \<in> S" "y \<in> S" "x \<notin> rel_interior S" "y \<notin> rel_interior S"
using 0 by auto
consider "norm x = norm y" | "norm x < norm y" | "norm x > norm y" by linarith
then show "x = y"
proof cases
assume "norm x = norm y"
with iff_eq eq show "x = y" by blast
next
assume *: "norm x < norm y"
have "x /\<^sub>R norm x = norm x *\<^sub>R (x /\<^sub>R norm x) /\<^sub>R norm (norm x *\<^sub>R (x /\<^sub>R norm x))"
by force
then have "proj ((norm x / norm y) *\<^sub>R y) = proj x"
by (metis (no_types) divide_inverse local.proj_def eq scaleR_scaleR)
then have [simp]: "(norm x / norm y) *\<^sub>R y = x"
by (rule eqI) (simp add: \<open>y \<noteq> 0\<close>)
have no: "0 \<le> norm x / norm y" "norm x / norm y < 1"
using * by (auto simp: divide_simps)
then show "x = y"
using starI [OF \<open>y \<in> S\<close> no] xynot by auto
next
assume *: "norm x > norm y"
have "y /\<^sub>R norm y = norm y *\<^sub>R (y /\<^sub>R norm y) /\<^sub>R norm (norm y *\<^sub>R (y /\<^sub>R norm y))"
by force
then have "proj ((norm y / norm x) *\<^sub>R x) = proj y"
by (metis (no_types) divide_inverse local.proj_def eq scaleR_scaleR)
then have [simp]: "(norm y / norm x) *\<^sub>R x = y"
by (rule eqI) (simp add: \<open>x \<noteq> 0\<close>)
have no: "0 \<le> norm y / norm x" "norm y / norm x < 1"
using * by (auto simp: divide_simps)
then show "x = y"
using starI [OF \<open>x \<in> S\<close> no] xynot by auto
qed
qed
have "\<exists>surf. homeomorphism (S - rel_interior S) ?SPHER proj surf"
proof (rule homeomorphism_compact)
show "compact (S - rel_interior S)"
using \<open>compact S\<close> compact_rel_boundary by blast
show "continuous_on (S - rel_interior S) proj"
using 0 by (blast intro: continuous_on_subset [OF cont_proj])
show "proj ` (S - rel_interior S) = ?SPHER"
proof
show "proj ` (S - rel_interior S) \<subseteq> ?SPHER"
using 0 by (force simp: hull_inc projI intro: nproj1)
show "?SPHER \<subseteq> proj ` (S - rel_interior S)"
proof (clarsimp simp: proj_def)
fix x
assume "x \<in> affine hull S" and nox: "norm x = 1"
then have "x \<noteq> 0" by auto
obtain d where "0 < d" and dx: "(d *\<^sub>R x) \<in> rel_frontier S"
and ri: "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (e *\<^sub>R x) \<in> rel_interior S"
using ray_to_rel_frontier [OF \<open>bounded S\<close> 0] \<open>x \<in> affine hull S\<close> \<open>x \<noteq> 0\<close> by auto
show "x \<in> (\<lambda>x. x /\<^sub>R norm x) ` (S - rel_interior S)"
apply (rule_tac x="d *\<^sub>R x" in image_eqI)
using \<open>0 < d\<close>
using dx \<open>closed S\<close> apply (auto simp: rel_frontier_def divide_simps nox)
done
qed
qed
qed (rule inj_on_proj)
then obtain surf where surf: "homeomorphism (S - rel_interior S) ?SPHER proj surf"
by blast
then have cont_surf: "continuous_on (proj ` (S - rel_interior S)) surf"
by (auto simp: homeomorphism_def)
have surf_nz: "\<And>x. x \<in> ?SPHER \<Longrightarrow> surf x \<noteq> 0"
by (metis "0" DiffE homeomorphism_def imageI surf)
have cont_nosp: "continuous_on (?SPHER) (\<lambda>x. norm x *\<^sub>R ((surf o proj) x))"
apply (rule continuous_intros)+
apply (rule continuous_on_subset [OF cont_proj], force)
apply (rule continuous_on_subset [OF cont_surf])
apply (force simp: homeomorphism_image1 [OF surf] dest: proj_spherI)
done
have surfpS: "\<And>x. \<lbrakk>norm x = 1; x \<in> affine hull S\<rbrakk> \<Longrightarrow> surf (proj x) \<in> S"
by (metis (full_types) DiffE \<open>0 \<in> S\<close> homeomorphism_def image_eqI norm_zero proj_spherI real_vector.scale_zero_left scaleR_one surf)
have *: "\<exists>y. norm y = 1 \<and> y \<in> affine hull S \<and> x = surf (proj y)"
if "x \<in> S" "x \<notin> rel_interior S" for x
proof -
have "proj x \<in> ?SPHER"
by (metis (full_types) "0" hull_inc proj_spherI that)
moreover have "surf (proj x) = x"
by (metis Diff_iff homeomorphism_def surf that)
ultimately show ?thesis
by (metis \<open>\<And>x. x \<in> ?SPHER \<Longrightarrow> surf x \<noteq> 0\<close> hull_inc inverse_1 local.proj_def norm_sgn projI scaleR_one sgn_div_norm that(1))
qed
have surfp_notin: "\<And>x. \<lbrakk>norm x = 1; x \<in> affine hull S\<rbrakk> \<Longrightarrow> surf (proj x) \<notin> rel_interior S"
by (metis (full_types) DiffE one_neq_zero homeomorphism_def image_eqI norm_zero proj_spherI surf)
have no_sp_im: "(\<lambda>x. norm x *\<^sub>R surf (proj x)) ` (?SPHER) = S - rel_interior S"
by (auto simp: surfpS image_def Bex_def surfp_notin *)
have inj_spher: "inj_on (\<lambda>x. norm x *\<^sub>R surf (proj x)) ?SPHER"
proof
fix x y
assume xy: "x \<in> ?SPHER" "y \<in> ?SPHER"
and eq: " norm x *\<^sub>R surf (proj x) = norm y *\<^sub>R surf (proj y)"
then have "norm x = 1" "norm y = 1" "x \<in> affine hull S" "y \<in> affine hull S"
using 0 by auto
with eq show "x = y"
by (simp add: proj_def) (metis surf xy homeomorphism_def)
qed
have co01: "compact ?SPHER"
by (simp add: closed_affine_hull compact_Int_closed)
show "?SMINUS homeomorphic ?SPHER"
apply (subst homeomorphic_sym)
apply (rule homeomorphic_compact [OF co01 cont_nosp [unfolded o_def] no_sp_im inj_spher])
done
have proj_scaleR: "\<And>a x. 0 < a \<Longrightarrow> proj (a *\<^sub>R x) = proj x"
by (simp add: proj_def)
have cont_sp0: "continuous_on (affine hull S - {0}) (surf o proj)"
apply (rule continuous_on_compose [OF continuous_on_subset [OF cont_proj]], force)
apply (rule continuous_on_subset [OF cont_surf])
using homeomorphism_image1 proj_spherI surf by fastforce
obtain B where "B>0" and B: "\<And>x. x \<in> S \<Longrightarrow> norm x \<le> B"
by (metis compact_imp_bounded \<open>compact S\<close> bounded_pos_less less_eq_real_def)
have cont_nosp: "continuous (at x within ?CBALL) (\<lambda>x. norm x *\<^sub>R surf (proj x))"
if "norm x \<le> 1" "x \<in> affine hull S" for x
proof (cases "x=0")
case True
show ?thesis using True
apply (simp add: continuous_within)
apply (rule lim_null_scaleR_bounded [where B=B])
apply (simp_all add: tendsto_norm_zero eventually_at)
apply (rule_tac x=B in exI)
using B surfpS proj_def projI apply (auto simp: \<open>B > 0\<close>)
done
next
case False
then have "\<forall>\<^sub>F x in at x. (x \<in> affine hull S - {0}) = (x \<in> affine hull S)"
apply (simp add: eventually_at)
apply (rule_tac x="norm x" in exI)
apply (auto simp: False)
done
with cont_sp0 have *: "continuous (at x within affine hull S) (\<lambda>x. surf (proj x))"
apply (simp add: continuous_on_eq_continuous_within)
apply (drule_tac x=x in bspec, force simp: False that)
apply (simp add: continuous_within Lim_transform_within_set)
done
show ?thesis
apply (rule continuous_within_subset [where s = "affine hull S", OF _ Int_lower2])
apply (rule continuous_intros *)+
done
qed
have cont_nosp2: "continuous_on ?CBALL (\<lambda>x. norm x *\<^sub>R ((surf o proj) x))"
by (simp add: continuous_on_eq_continuous_within cont_nosp)
have "norm y *\<^sub>R surf (proj y) \<in> S" if "y \<in> cball 0 1" and yaff: "y \<in> affine hull S" for y
proof (cases "y=0")
case True then show ?thesis
by (simp add: \<open>0 \<in> S\<close>)
next
case False
then have "norm y *\<^sub>R surf (proj y) = norm y *\<^sub>R surf (proj (y /\<^sub>R norm y))"
by (simp add: proj_def)
have "norm y \<le> 1" using that by simp
have "surf (proj (y /\<^sub>R norm y)) \<in> S"
apply (rule surfpS)
using proj_def projI yaff
by (auto simp: False)
then have "surf (proj y) \<in> S"
by (simp add: False proj_def)
then show "norm y *\<^sub>R surf (proj y) \<in> S"
by (metis dual_order.antisym le_less_linear norm_ge_zero rel_interior_subset scaleR_one
starI subset_eq \<open>norm y \<le> 1\<close>)
qed
moreover have "x \<in> (\<lambda>x. norm x *\<^sub>R surf (proj x)) ` (?CBALL)" if "x \<in> S" for x
proof (cases "x=0")
case True with that hull_inc show ?thesis by fastforce
next
case False
then have psp: "proj (surf (proj x)) = proj x"
by (metis homeomorphism_def hull_inc proj_spherI surf that)
have nxx: "norm x *\<^sub>R proj x = x"
by (simp add: False local.proj_def)
have affineI: "(1 / norm (surf (proj x))) *\<^sub>R x \<in> affine hull S"
by (metis \<open>0 \<in> S\<close> affine_hull_span_0 hull_inc span_clauses(4) that)
have sproj_nz: "surf (proj x) \<noteq> 0"
by (metis False proj0_iff psp)
then have "proj x = proj (proj x)"
by (metis False nxx proj_scaleR zero_less_norm_iff)
moreover have scaleproj: "\<And>a r. r *\<^sub>R proj a = (r / norm a) *\<^sub>R a"
by (simp add: divide_inverse local.proj_def)
ultimately have "(norm (surf (proj x)) / norm x) *\<^sub>R x \<notin> rel_interior S"
by (metis (no_types) sproj_nz divide_self_if hull_inc norm_eq_zero nproj1 projI psp scaleR_one surfp_notin that)
then have "(norm (surf (proj x)) / norm x) \<ge> 1"
using starI [OF that] by (meson starI [OF that] le_less_linear norm_ge_zero zero_le_divide_iff)
then have nole: "norm x \<le> norm (surf (proj x))"
by (simp add: le_divide_eq_1)
show ?thesis
apply (rule_tac x="inverse(norm(surf (proj x))) *\<^sub>R x" in image_eqI)
apply (metis (no_types, hide_lams) mult.commute scaleproj abs_inverse abs_norm_cancel divide_inverse norm_scaleR nxx positive_imp_inverse_positive proj_scaleR psp sproj_nz zero_less_norm_iff)
apply (auto simp: divide_simps nole affineI)
done
qed
ultimately have im_cball: "(\<lambda>x. norm x *\<^sub>R surf (proj x)) ` ?CBALL = S"
by blast
have inj_cball: "inj_on (\<lambda>x. norm x *\<^sub>R surf (proj x)) ?CBALL"
proof
fix x y
assume "x \<in> ?CBALL" "y \<in> ?CBALL"
and eq: "norm x *\<^sub>R surf (proj x) = norm y *\<^sub>R surf (proj y)"
then have x: "x \<in> affine hull S" and y: "y \<in> affine hull S"
using 0 by auto
show "x = y"
proof (cases "x=0 \<or> y=0")
case True then show "x = y" using eq proj_spherI surf_nz x y by force
next
case False
with x y have speq: "surf (proj x) = surf (proj y)"
by (metis eq homeomorphism_apply2 proj_scaleR proj_spherI surf zero_less_norm_iff)
then have "norm x = norm y"
by (metis \<open>x \<in> affine hull S\<close> \<open>y \<in> affine hull S\<close> eq proj_spherI real_vector.scale_cancel_right surf_nz)
moreover have "proj x = proj y"
by (metis (no_types) False speq homeomorphism_apply2 proj_spherI surf x y)
ultimately show "x = y"
using eq eqI by blast
qed
qed
have co01: "compact ?CBALL"
by (simp add: closed_affine_hull compact_Int_closed)
show "S homeomorphic ?CBALL"
apply (subst homeomorphic_sym)
apply (rule homeomorphic_compact [OF co01 cont_nosp2 [unfolded o_def] im_cball inj_cball])
done
qed
corollary
fixes S :: "'a::euclidean_space set"
assumes "compact S" and a: "a \<in> rel_interior S"
and star: "\<And>x. x \<in> S \<Longrightarrow> open_segment a x \<subseteq> rel_interior S"
shows starlike_compact_projective1:
"S - rel_interior S homeomorphic sphere a 1 \<inter> affine hull S"
and starlike_compact_projective2:
"S homeomorphic cball a 1 \<inter> affine hull S"
proof -
have 1: "compact (op+ (-a) ` S)" by (meson assms compact_translation)
have 2: "0 \<in> rel_interior (op+ (-a) ` S)"
by (simp add: a rel_interior_translation)
have 3: "open_segment 0 x \<subseteq> rel_interior (op+ (-a) ` S)" if "x \<in> (op+ (-a) ` S)" for x
proof -
have "x+a \<in> S" using that by auto
then have "open_segment a (x+a) \<subseteq> rel_interior S" by (metis star)
then show ?thesis using open_segment_translation
using rel_interior_translation by fastforce
qed
have "S - rel_interior S homeomorphic (op+ (-a) ` S) - rel_interior (op+ (-a) ` S)"
by (metis rel_interior_translation translation_diff homeomorphic_translation)
also have "... homeomorphic sphere 0 1 \<inter> affine hull (op+ (-a) ` S)"
by (rule starlike_compact_projective1_0 [OF 1 2 3])
also have "... = op+ (-a) ` (sphere a 1 \<inter> affine hull S)"
by (metis affine_hull_translation left_minus sphere_translation translation_Int)
also have "... homeomorphic sphere a 1 \<inter> affine hull S"
using homeomorphic_translation homeomorphic_sym by blast
finally show "S - rel_interior S homeomorphic sphere a 1 \<inter> affine hull S" .
have "S homeomorphic (op+ (-a) ` S)"
by (metis homeomorphic_translation)
also have "... homeomorphic cball 0 1 \<inter> affine hull (op+ (-a) ` S)"
by (rule starlike_compact_projective2_0 [OF 1 2 3])
also have "... = op+ (-a) ` (cball a 1 \<inter> affine hull S)"
by (metis affine_hull_translation left_minus cball_translation translation_Int)
also have "... homeomorphic cball a 1 \<inter> affine hull S"
using homeomorphic_translation homeomorphic_sym by blast
finally show "S homeomorphic cball a 1 \<inter> affine hull S" .
qed
corollary starlike_compact_projective_special:
assumes "compact S"
and cb01: "cball (0::'a::euclidean_space) 1 \<subseteq> S"
and scale: "\<And>x u. \<lbrakk>x \<in> S; 0 \<le> u; u < 1\<rbrakk> \<Longrightarrow> u *\<^sub>R x \<in> S - frontier S"
shows "S homeomorphic (cball (0::'a::euclidean_space) 1)"
proof -
have "ball 0 1 \<subseteq> interior S"
using cb01 interior_cball interior_mono by blast
then have 0: "0 \<in> rel_interior S"
by (meson centre_in_ball subsetD interior_subset_rel_interior le_numeral_extra(2) not_le)
have [simp]: "affine hull S = UNIV"
using \<open>ball 0 1 \<subseteq> interior S\<close> by (auto intro!: affine_hull_nonempty_interior)
have star: "open_segment 0 x \<subseteq> rel_interior S" if "x \<in> S" for x
proof
fix p assume "p \<in> open_segment 0 x"
then obtain u where "x \<noteq> 0" and u: "0 \<le> u" "u < 1" and p: "u *\<^sub>R x = p"
by (auto simp: in_segment)
then show "p \<in> rel_interior S"
using scale [OF that u] closure_subset frontier_def interior_subset_rel_interior by fastforce
qed
show ?thesis
using starlike_compact_projective2_0 [OF \<open>compact S\<close> 0 star] by simp
qed
lemma homeomorphic_convex_lemma:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes "convex S" "compact S" "convex T" "compact T"
and affeq: "aff_dim S = aff_dim T"
shows "(S - rel_interior S) homeomorphic (T - rel_interior T) \<and>
S homeomorphic T"
proof (cases "rel_interior S = {} \<or> rel_interior T = {}")
case True
then show ?thesis
by (metis Diff_empty affeq \<open>convex S\<close> \<open>convex T\<close> aff_dim_empty homeomorphic_empty rel_interior_eq_empty aff_dim_empty)
next
case False
then obtain a b where a: "a \<in> rel_interior S" and b: "b \<in> rel_interior T" by auto
have starS: "\<And>x. x \<in> S \<Longrightarrow> open_segment a x \<subseteq> rel_interior S"
using rel_interior_closure_convex_segment
a \<open>convex S\<close> closure_subset subsetCE by blast
have starT: "\<And>x. x \<in> T \<Longrightarrow> open_segment b x \<subseteq> rel_interior T"
using rel_interior_closure_convex_segment
b \<open>convex T\<close> closure_subset subsetCE by blast
let ?aS = "op+ (-a) ` S" and ?bT = "op+ (-b) ` T"
have 0: "0 \<in> affine hull ?aS" "0 \<in> affine hull ?bT"
by (metis a b subsetD hull_inc image_eqI left_minus rel_interior_subset)+
have subs: "subspace (span ?aS)" "subspace (span ?bT)"
by (rule subspace_span)+
moreover
have "dim (span (op + (- a) ` S)) = dim (span (op + (- b) ` T))"
by (metis 0 aff_dim_translation_eq aff_dim_zero affeq dim_span nat_int)
ultimately obtain f g where "linear f" "linear g"
and fim: "f ` span ?aS = span ?bT"
and gim: "g ` span ?bT = span ?aS"
and fno: "\<And>x. x \<in> span ?aS \<Longrightarrow> norm(f x) = norm x"
and gno: "\<And>x. x \<in> span ?bT \<Longrightarrow> norm(g x) = norm x"
and gf: "\<And>x. x \<in> span ?aS \<Longrightarrow> g(f x) = x"
and fg: "\<And>x. x \<in> span ?bT \<Longrightarrow> f(g x) = x"
by (rule isometries_subspaces) blast
have [simp]: "continuous_on A f" for A
using \<open>linear f\<close> linear_conv_bounded_linear linear_continuous_on by blast
have [simp]: "continuous_on B g" for B
using \<open>linear g\<close> linear_conv_bounded_linear linear_continuous_on by blast
have eqspanS: "affine hull ?aS = span ?aS"
by (metis a affine_hull_span_0 subsetD hull_inc image_eqI left_minus rel_interior_subset)
have eqspanT: "affine hull ?bT = span ?bT"
by (metis b affine_hull_span_0 subsetD hull_inc image_eqI left_minus rel_interior_subset)
have "S homeomorphic cball a 1 \<inter> affine hull S"
by (rule starlike_compact_projective2 [OF \<open>compact S\<close> a starS])
also have "... homeomorphic op+ (-a) ` (cball a 1 \<inter> affine hull S)"
by (metis homeomorphic_translation)
also have "... = cball 0 1 \<inter> op+ (-a) ` (affine hull S)"
by (auto simp: dist_norm)
also have "... = cball 0 1 \<inter> span ?aS"
using eqspanS affine_hull_translation by blast
also have "... homeomorphic cball 0 1 \<inter> span ?bT"
proof (rule homeomorphicI [where f=f and g=g])
show fim1: "f ` (cball 0 1 \<inter> span ?aS) = cball 0 1 \<inter> span ?bT"
apply (rule subset_antisym)
using fim fno apply (force simp:, clarify)
by (metis IntI fg gim gno image_eqI mem_cball_0)
show "g ` (cball 0 1 \<inter> span ?bT) = cball 0 1 \<inter> span ?aS"
apply (rule subset_antisym)
using gim gno apply (force simp:, clarify)
by (metis IntI fim1 gf image_eqI)
qed (auto simp: fg gf)
also have "... = cball 0 1 \<inter> op+ (-b) ` (affine hull T)"
using eqspanT affine_hull_translation by blast
also have "... = op+ (-b) ` (cball b 1 \<inter> affine hull T)"
by (auto simp: dist_norm)
also have "... homeomorphic (cball b 1 \<inter> affine hull T)"
by (metis homeomorphic_translation homeomorphic_sym)
also have "... homeomorphic T"
by (metis starlike_compact_projective2 [OF \<open>compact T\<close> b starT] homeomorphic_sym)
finally have 1: "S homeomorphic T" .
have "S - rel_interior S homeomorphic sphere a 1 \<inter> affine hull S"
by (rule starlike_compact_projective1 [OF \<open>compact S\<close> a starS])
also have "... homeomorphic op+ (-a) ` (sphere a 1 \<inter> affine hull S)"
by (metis homeomorphic_translation)
also have "... = sphere 0 1 \<inter> op+ (-a) ` (affine hull S)"
by (auto simp: dist_norm)
also have "... = sphere 0 1 \<inter> span ?aS"
using eqspanS affine_hull_translation by blast
also have "... homeomorphic sphere 0 1 \<inter> span ?bT"
proof (rule homeomorphicI [where f=f and g=g])
show fim1: "f ` (sphere 0 1 \<inter> span ?aS) = sphere 0 1 \<inter> span ?bT"
apply (rule subset_antisym)
using fim fno apply (force simp:, clarify)
by (metis IntI fg gim gno image_eqI mem_sphere_0)
show "g ` (sphere 0 1 \<inter> span ?bT) = sphere 0 1 \<inter> span ?aS"
apply (rule subset_antisym)
using gim gno apply (force simp:, clarify)
by (metis IntI fim1 gf image_eqI)
qed (auto simp: fg gf)
also have "... = sphere 0 1 \<inter> op+ (-b) ` (affine hull T)"
using eqspanT affine_hull_translation by blast
also have "... = op+ (-b) ` (sphere b 1 \<inter> affine hull T)"
by (auto simp: dist_norm)
also have "... homeomorphic (sphere b 1 \<inter> affine hull T)"
by (metis homeomorphic_translation homeomorphic_sym)
also have "... homeomorphic T - rel_interior T"
by (metis starlike_compact_projective1 [OF \<open>compact T\<close> b starT] homeomorphic_sym)
finally have 2: "S - rel_interior S homeomorphic T - rel_interior T" .
show ?thesis
using 1 2 by blast
qed
lemma homeomorphic_convex_compact_sets:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes "convex S" "compact S" "convex T" "compact T"
and affeq: "aff_dim S = aff_dim T"
shows "S homeomorphic T"
using homeomorphic_convex_lemma [OF assms] assms
by (auto simp: rel_frontier_def)
lemma homeomorphic_rel_frontiers_convex_bounded_sets:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes "convex S" "bounded S" "convex T" "bounded T"
and affeq: "aff_dim S = aff_dim T"
shows "rel_frontier S homeomorphic rel_frontier T"
using assms homeomorphic_convex_lemma [of "closure S" "closure T"]
by (simp add: rel_frontier_def convex_rel_interior_closure)
subsection\<open>Homeomorphisms between punctured spheres and affine sets\<close>
text\<open>Including the famous stereoscopic projection of the 3-D sphere to the complex plane\<close>
text\<open>The special case with centre 0 and radius 1\<close>
lemma homeomorphic_punctured_affine_sphere_affine_01:
assumes "b \<in> sphere 0 1" "affine T" "0 \<in> T" "b \<in> T" "affine p"
and affT: "aff_dim T = aff_dim p + 1"
shows "(sphere 0 1 \<inter> T) - {b} homeomorphic p"
proof -
have [simp]: "norm b = 1" "b\<bullet>b = 1"
using assms by (auto simp: norm_eq_1)
have [simp]: "T \<inter> {v. b\<bullet>v = 0} \<noteq> {}"
using \<open>0 \<in> T\<close> by auto
have [simp]: "\<not> T \<subseteq> {v. b\<bullet>v = 0}"
using \<open>norm b = 1\<close> \<open>b \<in> T\<close> by auto
define f where "f \<equiv> \<lambda>x. 2 *\<^sub>R b + (2 / (1 - b\<bullet>x)) *\<^sub>R (x - b)"
define g where "g \<equiv> \<lambda>y. b + (4 / (norm y ^ 2 + 4)) *\<^sub>R (y - 2 *\<^sub>R b)"
have [simp]: "\<And>x. \<lbrakk>x \<in> T; b\<bullet>x = 0\<rbrakk> \<Longrightarrow> f (g x) = x"
unfolding f_def g_def by (simp add: algebra_simps divide_simps add_nonneg_eq_0_iff)
have no: "\<And>x. \<lbrakk>norm x = 1; b\<bullet>x \<noteq> 1\<rbrakk> \<Longrightarrow> (norm (f x))\<^sup>2 = 4 * (1 + b\<bullet>x) / (1 - b\<bullet>x)"
apply (simp add: dot_square_norm [symmetric])
apply (simp add: f_def vector_add_divide_simps divide_simps norm_eq_1)
apply (simp add: algebra_simps inner_commute)
done
have [simp]: "\<And>u::real. 8 + u * (u * 8) = u * 16 \<longleftrightarrow> u=1"
by algebra
have [simp]: "\<And>x. \<lbrakk>norm x = 1; b \<bullet> x \<noteq> 1\<rbrakk> \<Longrightarrow> g (f x) = x"
unfolding g_def no by (auto simp: f_def divide_simps)
have [simp]: "\<And>x. \<lbrakk>x \<in> T; b \<bullet> x = 0\<rbrakk> \<Longrightarrow> norm (g x) = 1"
unfolding g_def
apply (rule power2_eq_imp_eq)
apply (simp_all add: dot_square_norm [symmetric] divide_simps vector_add_divide_simps)
apply (simp add: algebra_simps inner_commute)
done
have [simp]: "\<And>x. \<lbrakk>x \<in> T; b \<bullet> x = 0\<rbrakk> \<Longrightarrow> b \<bullet> g x \<noteq> 1"
unfolding g_def
apply (simp_all add: dot_square_norm [symmetric] divide_simps vector_add_divide_simps add_nonneg_eq_0_iff)
apply (auto simp: algebra_simps)
done
have "subspace T"
by (simp add: assms subspace_affine)
have [simp]: "\<And>x. \<lbrakk>x \<in> T; b \<bullet> x = 0\<rbrakk> \<Longrightarrow> g x \<in> T"
unfolding g_def
by (blast intro: \<open>subspace T\<close> \<open>b \<in> T\<close> subspace_add subspace_mul subspace_diff)
have "f ` {x. norm x = 1 \<and> b\<bullet>x \<noteq> 1} \<subseteq> {x. b\<bullet>x = 0}"
unfolding f_def using \<open>norm b = 1\<close> norm_eq_1
by (force simp: field_simps inner_add_right inner_diff_right)
moreover have "f ` T \<subseteq> T"
unfolding f_def using assms
apply (auto simp: field_simps inner_add_right inner_diff_right)
by (metis add_0 diff_zero mem_affine_3_minus)
moreover have "{x. b\<bullet>x = 0} \<inter> T \<subseteq> f ` ({x. norm x = 1 \<and> b\<bullet>x \<noteq> 1} \<inter> T)"
apply clarify
apply (rule_tac x = "g x" in image_eqI, auto)
done
ultimately have imf: "f ` ({x. norm x = 1 \<and> b\<bullet>x \<noteq> 1} \<inter> T) = {x. b\<bullet>x = 0} \<inter> T"
by blast
have no4: "\<And>y. b\<bullet>y = 0 \<Longrightarrow> norm ((y\<bullet>y + 4) *\<^sub>R b + 4 *\<^sub>R (y - 2 *\<^sub>R b)) = y\<bullet>y + 4"
apply (rule power2_eq_imp_eq)
apply (simp_all add: dot_square_norm [symmetric])
apply (auto simp: power2_eq_square algebra_simps inner_commute)
done
have [simp]: "\<And>x. \<lbrakk>norm x = 1; b \<bullet> x \<noteq> 1\<rbrakk> \<Longrightarrow> b \<bullet> f x = 0"
by (simp add: f_def algebra_simps divide_simps)
have [simp]: "\<And>x. \<lbrakk>x \<in> T; norm x = 1; b \<bullet> x \<noteq> 1\<rbrakk> \<Longrightarrow> f x \<in> T"
unfolding f_def
by (blast intro: \<open>subspace T\<close> \<open>b \<in> T\<close> subspace_add subspace_mul subspace_diff)
have "g ` {x. b\<bullet>x = 0} \<subseteq> {x. norm x = 1 \<and> b\<bullet>x \<noteq> 1}"
unfolding g_def
apply (clarsimp simp: no4 vector_add_divide_simps divide_simps add_nonneg_eq_0_iff dot_square_norm [symmetric])
apply (auto simp: algebra_simps)
done
moreover have "g ` T \<subseteq> T"
unfolding g_def
by (blast intro: \<open>subspace T\<close> \<open>b \<in> T\<close> subspace_add subspace_mul subspace_diff)
moreover have "{x. norm x = 1 \<and> b\<bullet>x \<noteq> 1} \<inter> T \<subseteq> g ` ({x. b\<bullet>x = 0} \<inter> T)"
apply clarify
apply (rule_tac x = "f x" in image_eqI, auto)
done
ultimately have img: "g ` ({x. b\<bullet>x = 0} \<inter> T) = {x. norm x = 1 \<and> b\<bullet>x \<noteq> 1} \<inter> T"
by blast
have aff: "affine ({x. b\<bullet>x = 0} \<inter> T)"
by (blast intro: affine_hyperplane assms)
have contf: "continuous_on ({x. norm x = 1 \<and> b\<bullet>x \<noteq> 1} \<inter> T) f"
unfolding f_def by (rule continuous_intros | force)+
have contg: "continuous_on ({x. b\<bullet>x = 0} \<inter> T) g"
unfolding g_def by (rule continuous_intros | force simp: add_nonneg_eq_0_iff)+
have "(sphere 0 1 \<inter> T) - {b} = {x. norm x = 1 \<and> (b\<bullet>x \<noteq> 1)} \<inter> T"
using \<open>norm b = 1\<close> by (auto simp: norm_eq_1) (metis vector_eq \<open>b\<bullet>b = 1\<close>)
also have "... homeomorphic {x. b\<bullet>x = 0} \<inter> T"
by (rule homeomorphicI [OF imf img contf contg]) auto
also have "... homeomorphic p"
apply (rule homeomorphic_affine_sets [OF aff \<open>affine p\<close>])
apply (simp add: Int_commute aff_dim_affine_Int_hyperplane [OF \<open>affine T\<close>] affT)
done
finally show ?thesis .
qed
theorem homeomorphic_punctured_affine_sphere_affine:
fixes a :: "'a :: euclidean_space"
assumes "0 < r" "b \<in> sphere a r" "affine T" "a \<in> T" "b \<in> T" "affine p"
and aff: "aff_dim T = aff_dim p + 1"
shows "((sphere a r \<inter> T) - {b}) homeomorphic p"
proof -
have "a \<noteq> b" using assms by auto
then have inj: "inj (\<lambda>x::'a. x /\<^sub>R norm (a - b))"
by (simp add: inj_on_def)
have "((sphere a r \<inter> T) - {b}) homeomorphic
op+ (-a) ` ((sphere a r \<inter> T) - {b})"
by (rule homeomorphic_translation)
also have "... homeomorphic op *\<^sub>R (inverse r) ` op + (- a) ` (sphere a r \<inter> T - {b})"
by (metis \<open>0 < r\<close> homeomorphic_scaling inverse_inverse_eq inverse_zero less_irrefl)
also have "... = sphere 0 1 \<inter> (op *\<^sub>R (inverse r) ` op + (- a) ` T) - {(b - a) /\<^sub>R r}"
using assms by (auto simp: dist_norm norm_minus_commute divide_simps)
also have "... homeomorphic p"
apply (rule homeomorphic_punctured_affine_sphere_affine_01)
using assms
apply (auto simp: dist_norm norm_minus_commute affine_scaling affine_translation [symmetric] aff_dim_translation_eq inj)
done
finally show ?thesis .
qed
proposition homeomorphic_punctured_sphere_affine_gen:
fixes a :: "'a :: euclidean_space"
assumes "convex S" "bounded S" and a: "a \<in> rel_frontier S"
and "affine T" and affS: "aff_dim S = aff_dim T + 1"
shows "rel_frontier S - {a} homeomorphic T"
proof -
have "S \<noteq> {}" using assms by auto
obtain U :: "'a set" where "affine U" and affdS: "aff_dim U = aff_dim S"
using choose_affine_subset [OF affine_UNIV aff_dim_geq]
by (meson aff_dim_affine_hull affine_affine_hull)
have "convex U"
by (simp add: affine_imp_convex \<open>affine U\<close>)
have "U \<noteq> {}"
by (metis \<open>S \<noteq> {}\<close> \<open>aff_dim U = aff_dim S\<close> aff_dim_empty)
then obtain z where "z \<in> U"
by auto
then have bne: "ball z 1 \<inter> U \<noteq> {}" by force
have [simp]: "aff_dim(ball z 1 \<inter> U) = aff_dim U"
using aff_dim_convex_Int_open [OF \<open>convex U\<close> open_ball] bne
by (fastforce simp add: Int_commute)
have "rel_frontier S homeomorphic rel_frontier (ball z 1 \<inter> U)"
apply (rule homeomorphic_rel_frontiers_convex_bounded_sets)
apply (auto simp: \<open>affine U\<close> affine_imp_convex convex_Int affdS assms)
done
also have "... = sphere z 1 \<inter> U"
using convex_affine_rel_frontier_Int [of "ball z 1" U]
by (simp add: \<open>affine U\<close> bne)
finally obtain h k where him: "h ` rel_frontier S = sphere z 1 \<inter> U"
and kim: "k ` (sphere z 1 \<inter> U) = rel_frontier S"
and hcon: "continuous_on (rel_frontier S) h"
and kcon: "continuous_on (sphere z 1 \<inter> U) k"
and kh: "\<And>x. x \<in> rel_frontier S \<Longrightarrow> k(h(x)) = x"
and hk: "\<And>y. y \<in> sphere z 1 \<inter> U \<Longrightarrow> h(k(y)) = y"
unfolding homeomorphic_def homeomorphism_def by auto
have "rel_frontier S - {a} homeomorphic (sphere z 1 \<inter> U) - {h a}"
proof (rule homeomorphicI [where f=h and g=k])
show h: "h ` (rel_frontier S - {a}) = sphere z 1 \<inter> U - {h a}"
using him a kh by auto metis
show "k ` (sphere z 1 \<inter> U - {h a}) = rel_frontier S - {a}"
by (force simp: h [symmetric] image_comp o_def kh)
qed (auto intro: continuous_on_subset hcon kcon simp: kh hk)
also have "... homeomorphic T"
apply (rule homeomorphic_punctured_affine_sphere_affine)
using a him
by (auto simp: affS affdS \<open>affine T\<close> \<open>affine U\<close> \<open>z \<in> U\<close>)
finally show ?thesis .
qed
lemma homeomorphic_punctured_sphere_affine:
fixes a :: "'a :: euclidean_space"
assumes "0 < r" and b: "b \<in> sphere a r"
and "affine T" and affS: "aff_dim T + 1 = DIM('a)"
shows "(sphere a r - {b}) homeomorphic T"
using homeomorphic_punctured_sphere_affine_gen [of "cball a r" b T]
assms aff_dim_cball by force
corollary homeomorphic_punctured_sphere_hyperplane:
fixes a :: "'a :: euclidean_space"
assumes "0 < r" and b: "b \<in> sphere a r"
and "c \<noteq> 0"
shows "(sphere a r - {b}) homeomorphic {x::'a. c \<bullet> x = d}"
apply (rule homeomorphic_punctured_sphere_affine)
using assms
apply (auto simp: affine_hyperplane of_nat_diff)
done
text\<open> When dealing with AR, ANR and ANR later, it's useful to know that every set
is homeomorphic to a closed subset of a convex set, and
if the set is locally compact we can take the convex set to be the universe.\<close>
proposition homeomorphic_closedin_convex:
fixes S :: "'m::euclidean_space set"
assumes "aff_dim S < DIM('n)"
obtains U and T :: "'n::euclidean_space set"
where "convex U" "U \<noteq> {}" "closedin (subtopology euclidean U) T"
"S homeomorphic T"
proof (cases "S = {}")
case True then show ?thesis
by (rule_tac U=UNIV and T="{}" in that) auto
next
case False
then obtain a where "a \<in> S" by auto
obtain i::'n where i: "i \<in> Basis" "i \<noteq> 0"
using SOME_Basis Basis_zero by force
have "0 \<in> affine hull (op + (- a) ` S)"
by (simp add: \<open>a \<in> S\<close> hull_inc)
then have "dim (op + (- a) ` S) = aff_dim (op + (- a) ` S)"
by (simp add: aff_dim_zero)
also have "... < DIM('n)"
by (simp add: aff_dim_translation_eq assms)
finally have dd: "dim (op + (- a) ` S) < DIM('n)"
by linarith
obtain T where "subspace T" and Tsub: "T \<subseteq> {x. i \<bullet> x = 0}"
and dimT: "dim T = dim (op + (- a) ` S)"
apply (rule choose_subspace_of_subspace [of "dim (op + (- a) ` S)" "{x::'n. i \<bullet> x = 0}"])
apply (simp add: dim_hyperplane [OF \<open>i \<noteq> 0\<close>])
apply (metis DIM_positive Suc_pred dd not_le not_less_eq_eq)
apply (metis span_eq subspace_hyperplane)
done
have "subspace (span (op + (- a) ` S))"
using subspace_span by blast
then obtain h k where "linear h" "linear k"
and heq: "h ` span (op + (- a) ` S) = T"
and keq:"k ` T = span (op + (- a) ` S)"
and hinv [simp]: "\<And>x. x \<in> span (op + (- a) ` S) \<Longrightarrow> k(h x) = x"
and kinv [simp]: "\<And>x. x \<in> T \<Longrightarrow> h(k x) = x"
apply (rule isometries_subspaces [OF _ \<open>subspace T\<close>])
apply (auto simp: dimT)
done
have hcont: "continuous_on A h" and kcont: "continuous_on B k" for A B
using \<open>linear h\<close> \<open>linear k\<close> linear_continuous_on linear_conv_bounded_linear by blast+
have ihhhh[simp]: "\<And>x. x \<in> S \<Longrightarrow> i \<bullet> h (x - a) = 0"
using Tsub [THEN subsetD] heq span_inc by fastforce
have "sphere 0 1 - {i} homeomorphic {x. i \<bullet> x = 0}"
apply (rule homeomorphic_punctured_sphere_affine)
using i
apply (auto simp: affine_hyperplane)
by (metis DIM_positive Suc_eq_plus1 add.left_neutral diff_add_cancel not_le not_less_eq_eq of_nat_1 of_nat_diff)
then obtain f g where fg: "homeomorphism (sphere 0 1 - {i}) {x. i \<bullet> x = 0} f g"
by (force simp: homeomorphic_def)
have "h ` op + (- a) ` S \<subseteq> T"
using heq span_clauses(1) span_linear_image by blast
then have "g ` h ` op + (- a) ` S \<subseteq> g ` {x. i \<bullet> x = 0}"
using Tsub by (simp add: image_mono)
also have "... \<subseteq> sphere 0 1 - {i}"
by (simp add: fg [unfolded homeomorphism_def])
finally have gh_sub_sph: "(g \<circ> h) ` op + (- a) ` S \<subseteq> sphere 0 1 - {i}"
by (metis image_comp)
then have gh_sub_cb: "(g \<circ> h) ` op + (- a) ` S \<subseteq> cball 0 1"
by (metis Diff_subset order_trans sphere_cball)
have [simp]: "\<And>u. u \<in> S \<Longrightarrow> norm (g (h (u - a))) = 1"
using gh_sub_sph [THEN subsetD] by (auto simp: o_def)
have ghcont: "continuous_on (op + (- a) ` S) (\<lambda>x. g (h x))"
apply (rule continuous_on_compose2 [OF homeomorphism_cont2 [OF fg] hcont], force)
done
have kfcont: "continuous_on ((g \<circ> h \<circ> op + (- a)) ` S) (\<lambda>x. k (f x))"
apply (rule continuous_on_compose2 [OF kcont])
using homeomorphism_cont1 [OF fg] gh_sub_sph apply (force intro: continuous_on_subset, blast)
done
have "S homeomorphic op + (- a) ` S"
by (simp add: homeomorphic_translation)
also have Shom: "\<dots> homeomorphic (g \<circ> h) ` op + (- a) ` S"
apply (simp add: homeomorphic_def homeomorphism_def)
apply (rule_tac x="g \<circ> h" in exI)
apply (rule_tac x="k \<circ> f" in exI)
apply (auto simp: ghcont kfcont span_clauses(1) homeomorphism_apply2 [OF fg] image_comp)
apply (force simp: o_def homeomorphism_apply2 [OF fg] span_clauses(1))
done
finally have Shom: "S homeomorphic (g \<circ> h) ` op + (- a) ` S" .
show ?thesis
apply (rule_tac U = "ball 0 1 \<union> image (g o h) (op + (- a) ` S)"
and T = "image (g o h) (op + (- a) ` S)"
in that)
apply (rule convex_intermediate_ball [of 0 1], force)
using gh_sub_cb apply force
apply force
apply (simp add: closedin_closed)
apply (rule_tac x="sphere 0 1" in exI)
apply (auto simp: Shom)
done
qed
subsection\<open>Locally compact sets in an open set\<close>
text\<open> Locally compact sets are closed in an open set and are homeomorphic
to an absolutely closed set if we have one more dimension to play with.\<close>
lemma locally_compact_open_Int_closure:
fixes S :: "'a :: metric_space set"
assumes "locally compact S"
obtains T where "open T" "S = T \<inter> closure S"
proof -
have "\<forall>x\<in>S. \<exists>T v u. u = S \<inter> T \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> S \<and> open T \<and> compact v"
by (metis assms locally_compact openin_open)
then obtain t v where
tv: "\<And>x. x \<in> S
\<Longrightarrow> v x \<subseteq> S \<and> open (t x) \<and> compact (v x) \<and> (\<exists>u. x \<in> u \<and> u \<subseteq> v x \<and> u = S \<inter> t x)"
by metis
then have o: "open (UNION S t)"
by blast
have "S = \<Union> (v ` S)"
using tv by blast
also have "... = UNION S t \<inter> closure S"
proof
show "UNION S v \<subseteq> UNION S t \<inter> closure S"
apply safe
apply (metis Int_iff subsetD UN_iff tv)
apply (simp add: closure_def rev_subsetD tv)
done
have "t x \<inter> closure S \<subseteq> v x" if "x \<in> S" for x
proof -
have "t x \<inter> closure S \<subseteq> closure (t x \<inter> S)"
by (simp add: open_Int_closure_subset that tv)
also have "... \<subseteq> v x"
by (metis Int_commute closure_minimal compact_imp_closed that tv)
finally show ?thesis .
qed
then show "UNION S t \<inter> closure S \<subseteq> UNION S v"
by blast
qed
finally have e: "S = UNION S t \<inter> closure S" .
show ?thesis
by (rule that [OF o e])
qed
lemma locally_compact_closedin_open:
fixes S :: "'a :: metric_space set"
assumes "locally compact S"
obtains T where "open T" "closedin (subtopology euclidean T) S"
by (metis locally_compact_open_Int_closure [OF assms] closed_closure closedin_closed_Int)
lemma locally_compact_homeomorphism_projection_closed:
assumes "locally compact S"
obtains T and f :: "'a \<Rightarrow> 'a :: euclidean_space \<times> 'b :: euclidean_space"
where "closed T" "homeomorphism S T f fst"
proof (cases "closed S")
case True
then show ?thesis
apply (rule_tac T = "S \<times> {0}" and f = "\<lambda>x. (x, 0)" in that)
apply (auto simp: closed_Times homeomorphism_def continuous_intros)
done
next
case False
obtain U where "open U" and US: "U \<inter> closure S = S"
by (metis locally_compact_open_Int_closure [OF assms])
with False have Ucomp: "-U \<noteq> {}"
using closure_eq by auto
have [simp]: "closure (- U) = -U"
by (simp add: \<open>open U\<close> closed_Compl)
define f :: "'a \<Rightarrow> 'a \<times> 'b" where "f \<equiv> \<lambda>x. (x, One /\<^sub>R setdist {x} (- U))"
have "continuous_on U (\<lambda>x. (x, One /\<^sub>R setdist {x} (- U)))"
apply (intro continuous_intros continuous_on_setdist)
by (simp add: Ucomp setdist_eq_0_sing_1)
then have homU: "homeomorphism U (f`U) f fst"
by (auto simp: f_def homeomorphism_def image_iff continuous_intros)
have cloS: "closedin (subtopology euclidean U) S"
by (metis US closed_closure closedin_closed_Int)
have cont: "isCont ((\<lambda>x. setdist {x} (- U)) o fst) z" for z :: "'a \<times> 'b"
by (rule isCont_o continuous_intros continuous_at_setdist)+
have setdist1D: "setdist {a} (- U) *\<^sub>R b = One \<Longrightarrow> setdist {a} (- U) \<noteq> 0" for a::'a and b::'b
by force
have *: "r *\<^sub>R b = One \<Longrightarrow> b = (1 / r) *\<^sub>R One" for r and b::'b
by (metis One_non_0 nonzero_divide_eq_eq real_vector.scale_eq_0_iff real_vector.scale_scale scaleR_one)
have "f ` U = {z. (setdist {fst z} (- U) *\<^sub>R snd z) \<in> {One}}"
apply (auto simp: f_def setdist_eq_0_sing_1 field_simps Ucomp)
apply (rule_tac x=a in image_eqI)
apply (auto simp: * setdist_eq_0_sing_1 dest: setdist1D)
done
then have clfU: "closed (f ` U)"
apply (rule ssubst)
apply (rule continuous_closed_preimage_univ)
apply (auto intro: continuous_intros cont [unfolded o_def])
done
have "closed (f ` S)"
apply (rule closedin_closed_trans [OF _ clfU])
apply (rule homeomorphism_imp_closed_map [OF homU cloS])
done
then show ?thesis
apply (rule that)
apply (rule homeomorphism_of_subsets [OF homU])
using US apply auto
done
qed
lemma locally_compact_closed_Int_open:
fixes S :: "'a :: euclidean_space set"
shows
"locally compact S \<longleftrightarrow> (\<exists>U u. closed U \<and> open u \<and> S = U \<inter> u)"
by (metis closed_closure closed_imp_locally_compact inf_commute locally_compact_Int locally_compact_open_Int_closure open_imp_locally_compact)
proposition locally_compact_homeomorphic_closed:
fixes S :: "'a::euclidean_space set"
assumes "locally compact S" and dimlt: "DIM('a) < DIM('b)"
obtains T :: "'b::euclidean_space set" where "closed T" "S homeomorphic T"
proof -
obtain U:: "('a*real)set" and h
where "closed U" and homU: "homeomorphism S U h fst"
using locally_compact_homeomorphism_projection_closed assms by metis
let ?BP = "Basis :: ('a*real) set"
have "DIM('a * real) \<le> DIM('b)"
by (simp add: Suc_leI dimlt)
then obtain basf :: "'a*real \<Rightarrow> 'b" where surbf: "basf ` ?BP \<subseteq> Basis" and injbf: "inj_on basf Basis"
by (metis finite_Basis card_le_inj)
define basg:: "'b \<Rightarrow> 'a * real" where
"basg \<equiv> \<lambda>i. inv_into Basis basf i"
have bgf[simp]: "basg (basf i) = i" if "i \<in> Basis" for i
using inv_into_f_f injbf that by (force simp: basg_def)
define f :: "'a*real \<Rightarrow> 'b" where "f \<equiv> \<lambda>u. \<Sum>j\<in>Basis. (u \<bullet> basg j) *\<^sub>R j"
have "linear f"
unfolding f_def
apply (intro linear_compose_setsum linearI ballI)
apply (auto simp: algebra_simps)
done
define g :: "'b \<Rightarrow> 'a*real" where "g \<equiv> \<lambda>z. (\<Sum>i\<in>Basis. (z \<bullet> basf i) *\<^sub>R i)"
have "linear g"
unfolding g_def
apply (intro linear_compose_setsum linearI ballI)
apply (auto simp: algebra_simps)
done
have *: "(\<Sum>a \<in> Basis. a \<bullet> basf b * (x \<bullet> basg a)) = x \<bullet> b" if "b \<in> Basis" for x b
using surbf that by auto
have gf[simp]: "g (f x) = x" for x
apply (rule euclidean_eqI)
apply (simp add: f_def g_def inner_setsum_left scaleR_setsum_left algebra_simps)
apply (simp add: Groups_Big.setsum_right_distrib [symmetric] *)
done
then have "inj f" by (metis injI)
have gfU: "g ` f ` U = U"
by (rule set_eqI) (auto simp: image_def)
have "S homeomorphic U"
using homU homeomorphic_def by blast
also have "... homeomorphic f ` U"
apply (rule homeomorphicI [OF refl gfU])
apply (meson \<open>inj f\<close> \<open>linear f\<close> homeomorphism_cont2 linear_homeomorphism_image)
using \<open>linear g\<close> linear_continuous_on linear_conv_bounded_linear apply blast
apply auto
done
finally show ?thesis
apply (rule_tac T = "f ` U" in that)
apply (rule closed_injective_linear_image [OF \<open>closed U\<close> \<open>linear f\<close> \<open>inj f\<close>], assumption)
done
qed
lemma homeomorphic_convex_compact_lemma:
fixes s :: "'a::euclidean_space set"
assumes "convex s"
and "compact s"
and "cball 0 1 \<subseteq> s"
shows "s homeomorphic (cball (0::'a) 1)"
proof (rule starlike_compact_projective_special[OF assms(2-3)])
fix x u
assume "x \<in> s" and "0 \<le> u" and "u < (1::real)"
have "open (ball (u *\<^sub>R x) (1 - u))"
by (rule open_ball)
moreover have "u *\<^sub>R x \<in> ball (u *\<^sub>R x) (1 - u)"
unfolding centre_in_ball using \<open>u < 1\<close> by simp
moreover have "ball (u *\<^sub>R x) (1 - u) \<subseteq> s"
proof
fix y
assume "y \<in> ball (u *\<^sub>R x) (1 - u)"
then have "dist (u *\<^sub>R x) y < 1 - u"
unfolding mem_ball .
with \<open>u < 1\<close> have "inverse (1 - u) *\<^sub>R (y - u *\<^sub>R x) \<in> cball 0 1"
by (simp add: dist_norm inverse_eq_divide norm_minus_commute)
with assms(3) have "inverse (1 - u) *\<^sub>R (y - u *\<^sub>R x) \<in> s" ..
with assms(1) have "(1 - u) *\<^sub>R ((y - u *\<^sub>R x) /\<^sub>R (1 - u)) + u *\<^sub>R x \<in> s"
using \<open>x \<in> s\<close> \<open>0 \<le> u\<close> \<open>u < 1\<close> [THEN less_imp_le] by (rule convexD_alt)
then show "y \<in> s" using \<open>u < 1\<close>
by simp
qed
ultimately have "u *\<^sub>R x \<in> interior s" ..
then show "u *\<^sub>R x \<in> s - frontier s"
using frontier_def and interior_subset by auto
qed
proposition homeomorphic_convex_compact_cball:
fixes e :: real
and s :: "'a::euclidean_space set"
assumes "convex s"
and "compact s"
and "interior s \<noteq> {}"
and "e > 0"
shows "s homeomorphic (cball (b::'a) e)"
proof -
obtain a where "a \<in> interior s"
using assms(3) by auto
then obtain d where "d > 0" and d: "cball a d \<subseteq> s"
unfolding mem_interior_cball by auto
let ?d = "inverse d" and ?n = "0::'a"
have "cball ?n 1 \<subseteq> (\<lambda>x. inverse d *\<^sub>R (x - a)) ` s"
apply rule
apply (rule_tac x="d *\<^sub>R x + a" in image_eqI)
defer
apply (rule d[unfolded subset_eq, rule_format])
using \<open>d > 0\<close>
unfolding mem_cball dist_norm
apply (auto simp add: mult_right_le_one_le)
done
then have "(\<lambda>x. inverse d *\<^sub>R (x - a)) ` s homeomorphic cball ?n 1"
using homeomorphic_convex_compact_lemma[of "(\<lambda>x. ?d *\<^sub>R -a + ?d *\<^sub>R x) ` s",
OF convex_affinity compact_affinity]
using assms(1,2)
by (auto simp add: scaleR_right_diff_distrib)
then show ?thesis
apply (rule_tac homeomorphic_trans[OF _ homeomorphic_balls(2)[of 1 _ ?n]])
apply (rule homeomorphic_trans[OF homeomorphic_affinity[of "?d" s "?d *\<^sub>R -a"]])
using \<open>d>0\<close> \<open>e>0\<close>
apply (auto simp add: scaleR_right_diff_distrib)
done
qed
corollary homeomorphic_convex_compact:
fixes s :: "'a::euclidean_space set"
and t :: "'a set"
assumes "convex s" "compact s" "interior s \<noteq> {}"
and "convex t" "compact t" "interior t \<noteq> {}"
shows "s homeomorphic t"
using assms
by (meson zero_less_one homeomorphic_trans homeomorphic_convex_compact_cball homeomorphic_sym)
subsection\<open>Covering spaces and lifting results for them\<close>
definition covering_space
:: "'a::topological_space set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"
where
"covering_space c p s \<equiv>
continuous_on c p \<and> p ` c = s \<and>
(\<forall>x \<in> s. \<exists>t. x \<in> t \<and> openin (subtopology euclidean s) t \<and>
(\<exists>v. \<Union>v = {x. x \<in> c \<and> p x \<in> t} \<and>
(\<forall>u \<in> v. openin (subtopology euclidean c) u) \<and>
pairwise disjnt v \<and>
(\<forall>u \<in> v. \<exists>q. homeomorphism u t p q)))"
lemma covering_space_imp_continuous: "covering_space c p s \<Longrightarrow> continuous_on c p"
by (simp add: covering_space_def)
lemma covering_space_imp_surjective: "covering_space c p s \<Longrightarrow> p ` c = s"
by (simp add: covering_space_def)
lemma homeomorphism_imp_covering_space: "homeomorphism s t f g \<Longrightarrow> covering_space s f t"
apply (simp add: homeomorphism_def covering_space_def, clarify)
apply (rule_tac x=t in exI, simp)
apply (rule_tac x="{s}" in exI, auto)
done
lemma covering_space_local_homeomorphism:
assumes "covering_space c p s" "x \<in> c"
obtains t u q where "x \<in> t" "openin (subtopology euclidean c) t"
"p x \<in> u" "openin (subtopology euclidean s) u"
"homeomorphism t u p q"
using assms
apply (simp add: covering_space_def, clarify)
apply (drule_tac x="p x" in bspec, force)
by (metis (no_types, lifting) Union_iff mem_Collect_eq)
lemma covering_space_local_homeomorphism_alt:
assumes p: "covering_space c p s" and "y \<in> s"
obtains x t u q where "p x = y"
"x \<in> t" "openin (subtopology euclidean c) t"
"y \<in> u" "openin (subtopology euclidean s) u"
"homeomorphism t u p q"
proof -
obtain x where "p x = y" "x \<in> c"
using assms covering_space_imp_surjective by blast
show ?thesis
apply (rule covering_space_local_homeomorphism [OF p \<open>x \<in> c\<close>])
using that \<open>p x = y\<close> by blast
qed
proposition covering_space_open_map:
fixes s :: "'a :: metric_space set" and t :: "'b :: metric_space set"
assumes p: "covering_space c p s" and t: "openin (subtopology euclidean c) t"
shows "openin (subtopology euclidean s) (p ` t)"
proof -
have pce: "p ` c = s"
and covs:
"\<And>x. x \<in> s \<Longrightarrow>
\<exists>X VS. x \<in> X \<and> openin (subtopology euclidean s) X \<and>
\<Union>VS = {x. x \<in> c \<and> p x \<in> X} \<and>
(\<forall>u \<in> VS. openin (subtopology euclidean c) u) \<and>
pairwise disjnt VS \<and>
(\<forall>u \<in> VS. \<exists>q. homeomorphism u X p q)"
using p by (auto simp: covering_space_def)
have "t \<subseteq> c" by (metis openin_euclidean_subtopology_iff t)
have "\<exists>T. openin (subtopology euclidean s) T \<and> y \<in> T \<and> T \<subseteq> p ` t"
if "y \<in> p ` t" for y
proof -
have "y \<in> s" using \<open>t \<subseteq> c\<close> pce that by blast
obtain U VS where "y \<in> U" and U: "openin (subtopology euclidean s) U"
and VS: "\<Union>VS = {x. x \<in> c \<and> p x \<in> U}"
and openVS: "\<forall>V \<in> VS. openin (subtopology euclidean c) V"
and homVS: "\<And>V. V \<in> VS \<Longrightarrow> \<exists>q. homeomorphism V U p q"
using covs [OF \<open>y \<in> s\<close>] by auto
obtain x where "x \<in> c" "p x \<in> U" "x \<in> t" "p x = y"
apply simp
using t [unfolded openin_euclidean_subtopology_iff] \<open>y \<in> U\<close> \<open>y \<in> p ` t\<close> by blast
with VS obtain V where "x \<in> V" "V \<in> VS" by auto
then obtain q where q: "homeomorphism V U p q" using homVS by blast
then have ptV: "p ` (t \<inter> V) = U \<inter> {z. q z \<in> (t \<inter> V)}"
using VS \<open>V \<in> VS\<close> by (auto simp: homeomorphism_def)
have ocv: "openin (subtopology euclidean c) V"
by (simp add: \<open>V \<in> VS\<close> openVS)
have "openin (subtopology euclidean U) {z \<in> U. q z \<in> t \<inter> V}"
apply (rule continuous_on_open [THEN iffD1, rule_format])
using homeomorphism_def q apply blast
using openin_subtopology_Int_subset [of c] q t unfolding homeomorphism_def
by (metis inf.absorb_iff2 Int_commute ocv openin_euclidean_subtopology_iff)
then have os: "openin (subtopology euclidean s) (U \<inter> {z. q z \<in> t \<inter> V})"
using openin_trans [of U] by (simp add: Collect_conj_eq U)
show ?thesis
apply (rule_tac x = "p ` (t \<inter> V)" in exI)
apply (rule conjI)
apply (simp only: ptV os)
using \<open>p x = y\<close> \<open>x \<in> V\<close> \<open>x \<in> t\<close> apply blast
done
qed
with openin_subopen show ?thesis by blast
qed
lemma covering_space_lift_unique_gen:
fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
fixes g1 :: "'a \<Rightarrow> 'c::real_normed_vector"
assumes cov: "covering_space c p s"
and eq: "g1 a = g2 a"
and f: "continuous_on t f" "f ` t \<subseteq> s"
and g1: "continuous_on t g1" "g1 ` t \<subseteq> c"
and fg1: "\<And>x. x \<in> t \<Longrightarrow> f x = p(g1 x)"
and g2: "continuous_on t g2" "g2 ` t \<subseteq> c"
and fg2: "\<And>x. x \<in> t \<Longrightarrow> f x = p(g2 x)"
and u_compt: "u \<in> components t" and "a \<in> u" "x \<in> u"
shows "g1 x = g2 x"
proof -
have "u \<subseteq> t" by (rule in_components_subset [OF u_compt])
def G12 \<equiv> "{x \<in> u. g1 x - g2 x = 0}"
have "connected u" by (rule in_components_connected [OF u_compt])
have contu: "continuous_on u g1" "continuous_on u g2"
using \<open>u \<subseteq> t\<close> continuous_on_subset g1 g2 by blast+
have o12: "openin (subtopology euclidean u) G12"
unfolding G12_def
proof (subst openin_subopen, clarify)
fix z
assume z: "z \<in> u" "g1 z - g2 z = 0"
obtain v w q where "g1 z \<in> v" and ocv: "openin (subtopology euclidean c) v"
and "p (g1 z) \<in> w" and osw: "openin (subtopology euclidean s) w"
and hom: "homeomorphism v w p q"
apply (rule_tac x = "g1 z" in covering_space_local_homeomorphism [OF cov])
using \<open>u \<subseteq> t\<close> \<open>z \<in> u\<close> g1(2) apply blast+
done
have "g2 z \<in> v" using \<open>g1 z \<in> v\<close> z by auto
have gg: "{x \<in> u. g x \<in> v} = {x \<in> u. g x \<in> (v \<inter> g ` u)}" for g
by auto
have "openin (subtopology euclidean (g1 ` u)) (v \<inter> g1 ` u)"
using ocv \<open>u \<subseteq> t\<close> g1 by (fastforce simp add: openin_open)
then have 1: "openin (subtopology euclidean u) {x \<in> u. g1 x \<in> v}"
unfolding gg by (blast intro: contu continuous_on_open [THEN iffD1, rule_format])
have "openin (subtopology euclidean (g2 ` u)) (v \<inter> g2 ` u)"
using ocv \<open>u \<subseteq> t\<close> g2 by (fastforce simp add: openin_open)
then have 2: "openin (subtopology euclidean u) {x \<in> u. g2 x \<in> v}"
unfolding gg by (blast intro: contu continuous_on_open [THEN iffD1, rule_format])
show "\<exists>T. openin (subtopology euclidean u) T \<and>
z \<in> T \<and> T \<subseteq> {z \<in> u. g1 z - g2 z = 0}"
using z
apply (rule_tac x = "{x. x \<in> u \<and> g1 x \<in> v} \<inter> {x. x \<in> u \<and> g2 x \<in> v}" in exI)
apply (intro conjI)
apply (rule openin_Int [OF 1 2])
using \<open>g1 z \<in> v\<close> \<open>g2 z \<in> v\<close> apply (force simp:, clarify)
apply (metis \<open>u \<subseteq> t\<close> subsetD eq_iff_diff_eq_0 fg1 fg2 hom homeomorphism_def)
done
qed
have c12: "closedin (subtopology euclidean u) G12"
unfolding G12_def
by (intro continuous_intros continuous_closedin_preimage_constant contu)
have "G12 = {} \<or> G12 = u"
by (intro connected_clopen [THEN iffD1, rule_format] \<open>connected u\<close> conjI o12 c12)
with eq \<open>a \<in> u\<close> have "\<And>x. x \<in> u \<Longrightarrow> g1 x - g2 x = 0" by (auto simp: G12_def)
then show ?thesis
using \<open>x \<in> u\<close> by force
qed
proposition covering_space_lift_unique:
fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
fixes g1 :: "'a \<Rightarrow> 'c::real_normed_vector"
assumes "covering_space c p s"
"g1 a = g2 a"
"continuous_on t f" "f ` t \<subseteq> s"
"continuous_on t g1" "g1 ` t \<subseteq> c" "\<And>x. x \<in> t \<Longrightarrow> f x = p(g1 x)"
"continuous_on t g2" "g2 ` t \<subseteq> c" "\<And>x. x \<in> t \<Longrightarrow> f x = p(g2 x)"
"connected t" "a \<in> t" "x \<in> t"
shows "g1 x = g2 x"
using covering_space_lift_unique_gen [of c p s] in_components_self assms ex_in_conv by blast
end