src/HOL/Analysis/Homeomorphism.thy
 author paulson Fri, 29 Sep 2017 16:55:08 +0100 changeset 66710 676258a1cf01 parent 66690 6953b1a29e19 child 66827 c94531b5007d permissions -rw-r--r--
eliminated a needless dependence on the theorem homeomorphic_punctured_sphere_affine_gen
```
(*  Title:      HOL/Analysis/Homeomorphism.thy
Author: LC Paulson (ported from HOL Light)
*)

section \<open>Homeomorphism Theorems\<close>

theory Homeomorphism
imports Path_Connected
begin

lemma homeomorphic_spheres':
fixes a ::"'a::euclidean_space" and b ::"'b::euclidean_space"
assumes "0 < \<delta>" and dimeq: "DIM('a) = DIM('b)"
shows "(sphere a \<delta>) homeomorphic (sphere b \<delta>)"
proof -
obtain f :: "'a\<Rightarrow>'b" and g where "linear f" "linear g"
and fg: "\<And>x. norm(f x) = norm x" "\<And>y. norm(g y) = norm y" "\<And>x. g(f x) = x" "\<And>y. f(g y) = y"
by (blast intro: isomorphisms_UNIV_UNIV [OF dimeq])
then have "continuous_on UNIV f" "continuous_on UNIV g"
using linear_continuous_on linear_linear by blast+
then show ?thesis
unfolding homeomorphic_minimal
apply(rule_tac x="\<lambda>x. b + f(x - a)" in exI)
apply(rule_tac x="\<lambda>x. a + g(x - b)" in exI)
using assms
apply (force intro: continuous_intros
continuous_on_compose2 [of _ f] continuous_on_compose2 [of _ g] simp: dist_commute dist_norm fg)
done
qed

lemma homeomorphic_spheres_gen:
fixes a :: "'a::euclidean_space" and b :: "'b::euclidean_space"
assumes "0 < r" "0 < s" "DIM('a::euclidean_space) = DIM('b::euclidean_space)"
shows "(sphere a r homeomorphic sphere b s)"
apply (rule homeomorphic_trans [OF homeomorphic_spheres homeomorphic_spheres'])
using assms  apply auto
done

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"
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"
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

lemma segment_to_rel_frontier_aux:
fixes x :: "'a::euclidean_space"
assumes "convex S" "bounded S" and x: "x \<in> rel_interior S" and y: "y \<in> S" and xy: "x \<noteq> y"
obtains z where "z \<in> rel_frontier S" "y \<in> closed_segment x z"
"open_segment x z \<subseteq> rel_interior S"
proof -
have "x + (y - x) \<in> affine hull S"
using hull_inc [OF y] by auto
then obtain d where "0 < d" and df: "(x + d *\<^sub>R (y-x)) \<in> rel_frontier S"
and di: "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (x + e *\<^sub>R (y-x)) \<in> rel_interior S"
by (rule ray_to_rel_frontier [OF \<open>bounded S\<close> x]) (use xy in auto)
show ?thesis
proof
show "x + d *\<^sub>R (y - x) \<in> rel_frontier S"
next
have "open_segment x y \<subseteq> rel_interior S"
using rel_interior_closure_convex_segment [OF \<open>convex S\<close> x] closure_subset y by blast
moreover have "x + d *\<^sub>R (y - x) \<in> open_segment x y" if "d < 1"
using xy
apply (auto simp: in_segment)
apply (rule_tac x="d" in exI)
using \<open>0 < d\<close> that apply (auto simp: divide_simps algebra_simps)
done
ultimately have "1 \<le> d"
using df rel_frontier_def by fastforce
moreover have "x = (1 / d) *\<^sub>R x + ((d - 1) / d) *\<^sub>R x"
ultimately show "y \<in> closed_segment x (x + d *\<^sub>R (y - x))"
apply (auto simp: in_segment)
apply (rule_tac x="1/d" in exI)
apply (auto simp: divide_simps algebra_simps)
done
next
show "open_segment x (x + d *\<^sub>R (y - x)) \<subseteq> rel_interior S"
apply (rule rel_interior_closure_convex_segment [OF \<open>convex S\<close> x])
using df rel_frontier_def by auto
qed
qed

lemma segment_to_rel_frontier:
fixes x :: "'a::euclidean_space"
assumes S: "convex S" "bounded S" and x: "x \<in> rel_interior S"
and y: "y \<in> S" and xy: "~(x = y \<and> S = {x})"
obtains z where "z \<in> rel_frontier S" "y \<in> closed_segment x z"
"open_segment x z \<subseteq> rel_interior S"
proof (cases "x=y")
case True
with xy have "S \<noteq> {x}"
by blast
with True show ?thesis
by (metis Set.set_insert all_not_in_conv ends_in_segment(1) insert_iff segment_to_rel_frontier_aux[OF S x] that y)
next
case False
then show ?thesis
using segment_to_rel_frontier_aux [OF S x y] that by blast
qed

proposition rel_frontier_not_sing:
fixes a :: "'a::euclidean_space"
assumes "bounded S"
shows "rel_frontier S \<noteq> {a}"
proof (cases "S = {}")
case True  then show ?thesis  by simp
next
case False
then obtain z where "z \<in> S"
by blast
then show ?thesis
proof (cases "S = {z}")
case True then show ?thesis  by simp
next
case False
then obtain w where "w \<in> S" "w \<noteq> z"
using \<open>z \<in> S\<close> by blast
show ?thesis
proof
assume "rel_frontier S = {a}"
then consider "w \<notin> rel_frontier S" | "z \<notin> rel_frontier S"
using \<open>w \<noteq> z\<close> by auto
then show False
proof cases
case 1
then have w: "w \<in> rel_interior S"
using \<open>w \<in> S\<close> closure_subset rel_frontier_def by fastforce
have "w + (w - z) \<in> affine hull S"
by (metis \<open>w \<in> S\<close> \<open>z \<in> S\<close> affine_affine_hull hull_inc mem_affine_3_minus scaleR_one)
then obtain e where "0 < e" "(w + e *\<^sub>R (w - z)) \<in> rel_frontier S"
using \<open>w \<noteq> z\<close>  \<open>z \<in> S\<close> by (metis assms ray_to_rel_frontier right_minus_eq w)
moreover obtain d where "0 < d" "(w + d *\<^sub>R (z - w)) \<in> rel_frontier S"
using ray_to_rel_frontier [OF \<open>bounded S\<close> w, of "1 *\<^sub>R (z - w)"]  \<open>w \<noteq> z\<close>  \<open>z \<in> S\<close>
ultimately have "d *\<^sub>R (z - w) = e *\<^sub>R (w - z)"
using \<open>rel_frontier S = {a}\<close> by force
moreover have "e \<noteq> -d "
using \<open>0 < e\<close> \<open>0 < d\<close> by force
ultimately show False
by (metis (no_types, lifting) \<open>w \<noteq> z\<close> eq_iff_diff_eq_0 minus_diff_eq real_vector.scale_cancel_right real_vector.scale_minus_right scaleR_left.minus)
next
case 2
then have z: "z \<in> rel_interior S"
using \<open>z \<in> S\<close> closure_subset rel_frontier_def by fastforce
have "z + (z - w) \<in> affine hull S"
by (metis \<open>z \<in> S\<close> \<open>w \<in> S\<close> affine_affine_hull hull_inc mem_affine_3_minus scaleR_one)
then obtain e where "0 < e" "(z + e *\<^sub>R (z - w)) \<in> rel_frontier S"
using \<open>w \<noteq> z\<close>  \<open>w \<in> S\<close> by (metis assms ray_to_rel_frontier right_minus_eq z)
moreover obtain d where "0 < d" "(z + d *\<^sub>R (w - z)) \<in> rel_frontier S"
using ray_to_rel_frontier [OF \<open>bounded S\<close> z, of "1 *\<^sub>R (w - z)"]  \<open>w \<noteq> z\<close>  \<open>w \<in> S\<close>
ultimately have "d *\<^sub>R (w - z) = e *\<^sub>R (z - w)"
using \<open>rel_frontier S = {a}\<close> by force
moreover have "e \<noteq> -d "
using \<open>0 < e\<close> \<open>0 < d\<close> by force
ultimately show False
by (metis (no_types, lifting) \<open>w \<noteq> z\<close> eq_iff_diff_eq_0 minus_diff_eq real_vector.scale_cancel_right real_vector.scale_minus_right scaleR_left.minus)
qed
qed
qed
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
have proj0_iff [simp]: "proj x = 0 \<longleftrightarrow> x = 0" for x
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"
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"
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"
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 (rule lim_null_scaleR_bounded [where B=B])
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 (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 (drule_tac x=x in bspec, force simp: False that)
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))"
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))"
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"
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"
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"
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))"
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"
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)"
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"]

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"
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)"
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)
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 (auto simp: algebra_simps)
done
have "subspace T"
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)
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 (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 (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))"
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

corollary 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_affine_sphere_affine [of r b a UNIV T] assms by auto

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

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 -
obtain U :: "'a set" where "affine U" "convex 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 affine_imp_convex)
have "S \<noteq> {}" using assms by auto
then obtain z where "z \<in> U"
by (metis aff_dim_negative_iff equals0I affdS)
then have bne: "ball z 1 \<inter> U \<noteq> {}" by force
then 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]
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 have "rel_frontier S homeomorphic sphere z 1 \<inter> U" .
then 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)
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

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)"
also have "... < DIM('n)"
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)
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"
also have Shom: "\<dots> homeomorphic (g \<circ> h) ` op + (- a) ` S"
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 (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)
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)

lemma lowerdim_embeddings:
assumes  "DIM('a) < DIM('b)"
obtains f :: "'a::euclidean_space*real \<Rightarrow> 'b::euclidean_space"
and g :: "'b \<Rightarrow> 'a*real"
and j :: 'b
where "linear f" "linear g" "\<And>z. g (f z) = z" "j \<in> Basis" "\<And>x. f(x,0) \<bullet> j = 0"
proof -
let ?B = "Basis :: ('a*real) set"
have b01: "(0,1) \<in> ?B"
have "DIM('a * real) \<le> DIM('b)"
then obtain basf :: "'a*real \<Rightarrow> 'b" where sbf: "basf ` ?B \<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. if i \<in> basf ` Basis then inv_into Basis basf i else (0,1)"
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)
have sbg: "basg ` Basis \<subseteq> ?B"
by (force simp: basg_def injbf b01)
define f :: "'a*real \<Rightarrow> 'b" where "f \<equiv> \<lambda>u. \<Sum>j\<in>Basis. (u \<bullet> basg j) *\<^sub>R j"
define g :: "'b \<Rightarrow> 'a*real" where "g \<equiv> \<lambda>z. (\<Sum>i\<in>Basis. (z \<bullet> basf i) *\<^sub>R i)"
show ?thesis
proof
show "linear f"
unfolding f_def
by (intro linear_compose_sum linearI ballI) (auto simp: algebra_simps)
show "linear g"
unfolding g_def
by (intro linear_compose_sum linearI ballI) (auto simp: algebra_simps)
have *: "(\<Sum>a \<in> Basis. a \<bullet> basf b * (x \<bullet> basg a)) = x \<bullet> b" if "b \<in> Basis" for x b
using sbf that by auto
show gf: "g (f x) = x" for x
apply (rule euclidean_eqI)
apply (simp add: f_def g_def inner_sum_left scaleR_sum_left algebra_simps)
apply (simp add: Groups_Big.sum_distrib_left [symmetric] *)
done
show "basf(0,1) \<in> Basis"
using b01 sbf by auto
then show "f(x,0) \<bullet> basf(0,1) = 0" for x
using b01 inner_not_same_Basis by fastforce
qed
qed

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
obtain f :: "'a*real \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'a*real"
where "linear f" "linear g" and gf [simp]: "\<And>z. g (f z) = z"
using lowerdim_embeddings [OF dimlt] by metis
then have "inj f"
by (metis injI)
have gfU: "g ` f ` U = U"
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 simp: o_def)
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
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)
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>
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"

lemma covering_space_imp_surjective: "covering_space c p S \<Longrightarrow> p ` c = S"

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 (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>X. openin (subtopology euclidean S) X \<and> y \<in> X \<and> X \<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])
define G12 where "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

lemma covering_space_locally:
fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
assumes loc: "locally \<phi> C" and cov: "covering_space C p S"
and pim: "\<And>T. \<lbrakk>T \<subseteq> C; \<phi> T\<rbrakk> \<Longrightarrow> \<psi>(p ` T)"
shows "locally \<psi> S"
proof -
have "locally \<psi> (p ` C)"
apply (rule locally_open_map_image [OF loc])
using cov covering_space_imp_continuous apply blast
using cov covering_space_imp_surjective covering_space_open_map apply blast
then show ?thesis
using covering_space_imp_surjective [OF cov] by metis
qed

proposition covering_space_locally_eq:
fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
assumes cov: "covering_space C p S"
and pim: "\<And>T. \<lbrakk>T \<subseteq> C; \<phi> T\<rbrakk> \<Longrightarrow> \<psi>(p ` T)"
and qim: "\<And>q U. \<lbrakk>U \<subseteq> S; continuous_on U q; \<psi> U\<rbrakk> \<Longrightarrow> \<phi>(q ` U)"
shows "locally \<psi> S \<longleftrightarrow> locally \<phi> C"
(is "?lhs = ?rhs")
proof
assume L: ?lhs
show ?rhs
proof (rule locallyI)
fix V x
assume V: "openin (subtopology euclidean C) V" and "x \<in> V"
have "p x \<in> p ` C"
by (metis IntE V \<open>x \<in> V\<close> imageI openin_open)
then obtain T \<V> where "p x \<in> T"
and opeT: "openin (subtopology euclidean S) T"
and veq: "\<Union>\<V> = {x \<in> C. p x \<in> T}"
and ope: "\<forall>U\<in>\<V>. openin (subtopology euclidean C) U"
and hom: "\<forall>U\<in>\<V>. \<exists>q. homeomorphism U T p q"
using cov unfolding covering_space_def by (blast intro: that)
have "x \<in> \<Union>\<V>"
using V veq \<open>p x \<in> T\<close> \<open>x \<in> V\<close> openin_imp_subset by fastforce
then obtain U where "x \<in> U" "U \<in> \<V>"
by blast
then obtain q where opeU: "openin (subtopology euclidean C) U" and q: "homeomorphism U T p q"
using ope hom by blast
with V have "openin (subtopology euclidean C) (U \<inter> V)"
by blast
then have UV: "openin (subtopology euclidean S) (p ` (U \<inter> V))"
using cov covering_space_open_map by blast
obtain W W' where opeW: "openin (subtopology euclidean S) W" and "\<psi> W'" "p x \<in> W" "W \<subseteq> W'" and W'sub: "W' \<subseteq> p ` (U \<inter> V)"
using locallyE [OF L UV] \<open>x \<in> U\<close> \<open>x \<in> V\<close> by blast
then have "W \<subseteq> T"
by (metis Int_lower1 q homeomorphism_image1 image_Int_subset order_trans)
show "\<exists>U Z. openin (subtopology euclidean C) U \<and>
\<phi> Z \<and> x \<in> U \<and> U \<subseteq> Z \<and> Z \<subseteq> V"
proof (intro exI conjI)
have "openin (subtopology euclidean T) W"
by (meson opeW opeT openin_imp_subset openin_subset_trans \<open>W \<subseteq> T\<close>)
then have "openin (subtopology euclidean U) (q ` W)"
by (meson homeomorphism_imp_open_map homeomorphism_symD q)
then show "openin (subtopology euclidean C) (q ` W)"
using opeU openin_trans by blast
show "\<phi> (q ` W')"
by (metis (mono_tags, lifting) Int_subset_iff UV W'sub \<open>\<psi> W'\<close> continuous_on_subset dual_order.trans homeomorphism_def image_Int_subset openin_imp_subset q qim)
show "x \<in> q ` W"
by (metis \<open>p x \<in> W\<close> \<open>x \<in> U\<close> homeomorphism_def imageI q)
show "q ` W \<subseteq> q ` W'"
using \<open>W \<subseteq> W'\<close> by blast
have "W' \<subseteq> p ` V"
using W'sub by blast
then show "q ` W' \<subseteq> V"
using W'sub homeomorphism_apply1 [OF q] by auto
qed
qed
next
assume ?rhs
then show ?lhs
using cov covering_space_locally pim by blast
qed

lemma covering_space_locally_compact_eq:
fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
assumes "covering_space C p S"
shows "locally compact S \<longleftrightarrow> locally compact C"
apply (rule covering_space_locally_eq [OF assms])
apply (meson assms compact_continuous_image continuous_on_subset covering_space_imp_continuous)
using compact_continuous_image by blast

lemma covering_space_locally_connected_eq:
fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
assumes "covering_space C p S"
shows "locally connected S \<longleftrightarrow> locally connected C"
apply (rule covering_space_locally_eq [OF assms])
apply (meson connected_continuous_image assms continuous_on_subset covering_space_imp_continuous)
using connected_continuous_image by blast

lemma covering_space_locally_path_connected_eq:
fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
assumes "covering_space C p S"
shows "locally path_connected S \<longleftrightarrow> locally path_connected C"
apply (rule covering_space_locally_eq [OF assms])
apply (meson path_connected_continuous_image assms continuous_on_subset covering_space_imp_continuous)
using path_connected_continuous_image by blast

lemma covering_space_locally_compact:
fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
assumes "locally compact C" "covering_space C p S"
shows "locally compact S"
using assms covering_space_locally_compact_eq by blast

lemma covering_space_locally_connected:
fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
assumes "locally connected C" "covering_space C p S"
shows "locally connected S"
using assms covering_space_locally_connected_eq by blast

lemma covering_space_locally_path_connected:
fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
assumes "locally path_connected C" "covering_space C p S"
shows "locally path_connected S"
using assms covering_space_locally_path_connected_eq by blast

proposition covering_space_lift_homotopy:
fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
and h :: "real \<times> 'c::real_normed_vector \<Rightarrow> 'b"
assumes cov: "covering_space C p S"
and conth: "continuous_on ({0..1} \<times> U) h"
and him: "h ` ({0..1} \<times> U) \<subseteq> S"
and heq: "\<And>y. y \<in> U \<Longrightarrow> h (0,y) = p(f y)"
and contf: "continuous_on U f" and fim: "f ` U \<subseteq> C"
obtains k where "continuous_on ({0..1} \<times> U) k"
"k ` ({0..1} \<times> U) \<subseteq> C"
"\<And>y. y \<in> U \<Longrightarrow> k(0, y) = f y"
"\<And>z. z \<in> {0..1} \<times> U \<Longrightarrow> h z = p(k z)"
proof -
have "\<exists>V k. openin (subtopology euclidean U) V \<and> y \<in> V \<and>
continuous_on ({0..1} \<times> V) k \<and> k ` ({0..1} \<times> V) \<subseteq> C \<and>
(\<forall>z \<in> V. k(0, z) = f z) \<and> (\<forall>z \<in> {0..1} \<times> V. h z = p(k z))"
if "y \<in> U" for y
proof -
obtain UU where UU: "\<And>s. s \<in> S \<Longrightarrow> s \<in> (UU s) \<and> openin (subtopology euclidean S) (UU s) \<and>
(\<exists>\<V>. \<Union>\<V> = {x. x \<in> C \<and> p x \<in> (UU s)} \<and>
(\<forall>U \<in> \<V>. openin (subtopology euclidean C) U) \<and>
pairwise disjnt \<V> \<and>
(\<forall>U \<in> \<V>. \<exists>q. homeomorphism U (UU s) p q))"
using cov unfolding covering_space_def by (metis (mono_tags))
then have ope: "\<And>s. s \<in> S \<Longrightarrow> s \<in> (UU s) \<and> openin (subtopology euclidean S) (UU s)"
by blast
have "\<exists>k n i. open k \<and> open n \<and>
t \<in> k \<and> y \<in> n \<and> i \<in> S \<and> h ` (({0..1} \<inter> k) \<times> (U \<inter> n)) \<subseteq> UU i" if "t \<in> {0..1}" for t
proof -
have hinS: "h (t, y) \<in> S"
using \<open>y \<in> U\<close> him that by blast
then have "(t,y) \<in> {z \<in> {0..1} \<times> U. h z \<in> UU (h (t, y))}"
using \<open>y \<in> U\<close> \<open>t \<in> {0..1}\<close>  by (auto simp: ope)
moreover have ope_01U: "openin (subtopology euclidean ({0..1} \<times> U)) {z. z \<in> ({0..1} \<times> U) \<and> h z \<in> UU(h(t, y))}"
using hinS ope continuous_on_open_gen [OF him] conth by blast
ultimately obtain V W where opeV: "open V" and "t \<in> {0..1} \<inter> V" "t \<in> {0..1} \<inter> V"
and opeW: "open W" and "y \<in> U" "y \<in> W"
and VW: "({0..1} \<inter> V) \<times> (U \<inter> W)  \<subseteq> {z. z \<in> ({0..1} \<times> U) \<and> h z \<in> UU(h(t, y))}"
by (rule Times_in_interior_subtopology) (auto simp: openin_open)
then show ?thesis
using hinS by blast
qed
then obtain K NN X where
K: "\<And>t. t \<in> {0..1} \<Longrightarrow> open (K t)"
and NN: "\<And>t. t \<in> {0..1} \<Longrightarrow> open (NN t)"
and inUS: "\<And>t. t \<in> {0..1} \<Longrightarrow> t \<in> K t \<and> y \<in> NN t \<and> X t \<in> S"
and him: "\<And>t. t \<in> {0..1} \<Longrightarrow> h ` (({0..1} \<inter> K t) \<times> (U \<inter> NN t)) \<subseteq> UU (X t)"
by (metis (mono_tags))
obtain \<T> where "\<T> \<subseteq> ((\<lambda>i. K i \<times> NN i)) ` {0..1}" "finite \<T>" "{0::real..1} \<times> {y} \<subseteq> \<Union>\<T>"
proof (rule compactE)
show "compact ({0::real..1} \<times> {y})"
show "{0..1} \<times> {y} \<subseteq> (\<Union>i\<in>{0..1}. K i \<times> NN i)"
using K inUS by auto
show "\<And>B. B \<in> (\<lambda>i. K i \<times> NN i) ` {0..1} \<Longrightarrow> open B"
using K NN by (auto simp: open_Times)
qed blast
then obtain tk where "tk \<subseteq> {0..1}" "finite tk"
and tk: "{0::real..1} \<times> {y} \<subseteq> (\<Union>i \<in> tk. K i \<times> NN i)"
by (metis (no_types, lifting) finite_subset_image)
then have "tk \<noteq> {}"
by auto
define n where "n = INTER tk NN"
have "y \<in> n" "open n"
using inUS NN \<open>tk \<subseteq> {0..1}\<close> \<open>finite tk\<close>
by (auto simp: n_def open_INT subset_iff)
obtain \<delta> where "0 < \<delta>" and \<delta>: "\<And>T. \<lbrakk>T \<subseteq> {0..1}; diameter T < \<delta>\<rbrakk> \<Longrightarrow> \<exists>B\<in>K ` tk. T \<subseteq> B"
proof (rule Lebesgue_number_lemma [of "{0..1}" "K ` tk"])
show "K ` tk \<noteq> {}"
using \<open>tk \<noteq> {}\<close> by auto
show "{0..1} \<subseteq> UNION tk K"
using tk by auto
show "\<And>B. B \<in> K ` tk \<Longrightarrow> open B"
using \<open>tk \<subseteq> {0..1}\<close> K by auto
qed auto
obtain N::nat where N: "N > 1 / \<delta>"
using reals_Archimedean2 by blast
then have "N > 0"
using \<open>0 < \<delta>\<close> order.asym by force
have *: "\<exists>V k. openin (subtopology euclidean U) V \<and> y \<in> V \<and>
continuous_on ({0..of_nat n / N} \<times> V) k \<and>
k ` ({0..of_nat n / N} \<times> V) \<subseteq> C \<and>
(\<forall>z\<in>V. k (0, z) = f z) \<and>
(\<forall>z\<in>{0..of_nat n / N} \<times> V. h z = p (k z))" if "n \<le> N" for n
using that
proof (induction n)
case 0
show ?case
apply (rule_tac x=U in exI)
apply (rule_tac x="f \<circ> snd" in exI)
apply (intro conjI \<open>y \<in> U\<close> continuous_intros continuous_on_subset [OF contf])
using fim  apply (auto simp: heq)
done
next
case (Suc n)
then obtain V k where opeUV: "openin (subtopology euclidean U) V"
and "y \<in> V"
and contk: "continuous_on ({0..real n / real N} \<times> V) k"
and kim: "k ` ({0..real n / real N} \<times> V) \<subseteq> C"
and keq: "\<And>z. z \<in> V \<Longrightarrow> k (0, z) = f z"
and heq: "\<And>z. z \<in> {0..real n / real N} \<times> V \<Longrightarrow> h z = p (k z)"
using Suc_leD by auto
have "n \<le> N"
using Suc.prems by auto
obtain t where "t \<in> tk" and t: "{real n / real N .. (1 + real n) / real N} \<subseteq> K t"
proof (rule bexE [OF \<delta>])
show "{real n / real N .. (1 + real n) / real N} \<subseteq> {0..1}"
using Suc.prems by (auto simp: divide_simps algebra_simps)
show diameter_less: "diameter {real n / real N .. (1 + real n) / real N} < \<delta>"
using \<open>0 < \<delta>\<close> N by (auto simp: divide_simps algebra_simps)
qed blast
have t01: "t \<in> {0..1}"
using \<open>t \<in> tk\<close> \<open>tk \<subseteq> {0..1}\<close> by blast
obtain \<V> where "\<Union>\<V> = {x. x \<in> C \<and> p x \<in> (UU (X t))}"
and opeC: "\<And>U. U \<in> \<V> \<Longrightarrow> openin (subtopology euclidean C) U"
and "pairwise disjnt \<V>"
and homuu: "\<And>U. U \<in> \<V> \<Longrightarrow> \<exists>q. homeomorphism U (UU (X t)) p q"
using inUS [OF t01] UU by meson
have n_div_N_in: "real n / real N \<in> {real n / real N .. (1 + real n) / real N}"
using N by (auto simp: divide_simps algebra_simps)
with t have nN_in_kkt: "real n / real N \<in> K t"
by blast
have "k (real n / real N, y) \<in> {x. x \<in> C \<and> p x \<in> (UU (X t))}"
proof (simp, rule conjI)
show "k (real n / real N, y) \<in> C"
using \<open>y \<in> V\<close> kim keq by force
have "p (k (real n / real N, y)) = h (real n / real N, y)"
by (simp add: \<open>y \<in> V\<close> heq)
also have "... \<in> h ` (({0..1} \<inter> K t) \<times> (U \<inter> NN t))"
apply (rule imageI)
using \<open>y \<in> V\<close> t01 \<open>n \<le> N\<close>
apply (simp add: nN_in_kkt \<open>y \<in> U\<close> inUS divide_simps)
done
also have "... \<subseteq> UU (X t)"
using him t01 by blast
finally show "p (k (real n / real N, y)) \<in> UU (X t)" .
qed
then have "k (real n / real N, y) \<in> \<Union>\<V>"
using \<open>\<Union>\<V> = {x \<in> C. p x \<in> UU (X t)}\<close> by blast
then obtain W where W: "k (real n / real N, y) \<in> W" and "W \<in> \<V>"
by blast
then obtain p' where opeC': "openin (subtopology euclidean C) W"
and hom': "homeomorphism W (UU (X t)) p p'"
using homuu opeC by blast
then have "W \<subseteq> C"
using openin_imp_subset by blast
define W' where "W' = UU(X t)"
have opeVW: "openin (subtopology euclidean V) {z \<in> V. (k \<circ> Pair (real n / real N)) z \<in> W}"
apply (rule continuous_openin_preimage [OF _ _ opeC'])
apply (intro continuous_intros continuous_on_subset [OF contk])
using kim apply (auto simp: \<open>y \<in> V\<close> W)
done
obtain N' where opeUN': "openin (subtopology euclidean U) N'"
and "y \<in> N'" and kimw: "k ` ({(real n / real N)} \<times> N') \<subseteq> W"
apply (rule_tac N' = "{z \<in> V. (k \<circ> Pair ((real n / real N))) z \<in> W}" in that)
apply (fastforce simp:  \<open>y \<in> V\<close> W intro!: openin_trans [OF opeVW opeUV])+
done
obtain Q Q' where opeUQ: "openin (subtopology euclidean U) Q"
and cloUQ': "closedin (subtopology euclidean U) Q'"
and "y \<in> Q" "Q \<subseteq> Q'"
and Q': "Q' \<subseteq> (U \<inter> NN(t)) \<inter> N' \<inter> V"
proof -
obtain VO VX where "open VO" "open VX" and VO: "V = U \<inter> VO" and VX: "N' = U \<inter> VX"
using opeUV opeUN' by (auto simp: openin_open)
then have "open (NN(t) \<inter> VO \<inter> VX)"
using NN t01 by blast
then obtain e where "e > 0" and e: "cball y e \<subseteq> NN(t) \<inter> VO \<inter> VX"
by (metis Int_iff \<open>N' = U \<inter> VX\<close> \<open>V = U \<inter> VO\<close> \<open>y \<in> N'\<close> \<open>y \<in> V\<close> inUS open_contains_cball t01)
show ?thesis
proof
show "openin (subtopology euclidean U) (U \<inter> ball y e)"
by blast
show "closedin (subtopology euclidean U) (U \<inter> cball y e)"
using e by (auto simp: closedin_closed)
qed (use \<open>y \<in> U\<close> \<open>e > 0\<close> VO VX e in auto)
qed
then have "y \<in> Q'" "Q \<subseteq> (U \<inter> NN(t)) \<inter> N' \<inter> V"
by blast+
have neq: "{0..real n / real N} \<union> {real n / real N..(1 + real n) / real N} = {0..(1 + real n) / real N}"
apply (auto simp: divide_simps)
by (metis mult_zero_left of_nat_0_le_iff of_nat_0_less_iff order_trans real_mult_le_cancel_iff1)
then have neqQ': "{0..real n / real N} \<times> Q' \<union> {real n / real N..(1 + real n) / real N} \<times> Q' = {0..(1 + real n) / real N} \<times> Q'"
by blast
have cont: "continuous_on ({0..(1 + real n) / real N} \<times> Q')
(\<lambda>x. if x \<in> {0..real n / real N} \<times> Q' then k x else (p' \<circ> h) x)"
unfolding neqQ' [symmetric]
proof (rule continuous_on_cases_local, simp_all add: neqQ' del: comp_apply)
show "closedin (subtopology euclidean ({0..(1 + real n) / real N} \<times> Q'))
({0..real n / real N} \<times> Q')"
apply (rule_tac x="{0 .. real n / real N} \<times> UNIV" in exI)
using n_div_N_in apply (auto simp: closed_Times)
done
show "closedin (subtopology euclidean ({0..(1 + real n) / real N} \<times> Q'))
({real n / real N..(1 + real n) / real N} \<times> Q')"
apply (rule_tac x="{real n / real N .. (1 + real n) / real N} \<times> UNIV" in exI)
apply (auto simp: closed_Times)
by (meson divide_nonneg_nonneg of_nat_0_le_iff order_trans)
show "continuous_on ({0..real n / real N} \<times> Q') k"
apply (rule continuous_on_subset [OF contk])
using Q' by auto
have "continuous_on ({real n / real N..(1 + real n) / real N} \<times> Q') h"
proof (rule continuous_on_subset [OF conth])
show "{real n / real N..(1 + real n) / real N} \<times> Q' \<subseteq> {0..1} \<times> U"
using \<open>N > 0\<close>
apply auto
apply (meson divide_nonneg_nonneg of_nat_0_le_iff order_trans)
using Suc.prems order_trans apply fastforce
apply (metis IntE  cloUQ' closedin_closed)
done
qed
moreover have "continuous_on (h ` ({real n / real N..(1 + real n) / real N} \<times> Q')) p'"
proof (rule continuous_on_subset [OF homeomorphism_cont2 [OF hom']])
have "h ` ({real n / real N..(1 + real n) / real N} \<times> Q') \<subseteq> h ` (({0..1} \<inter> K t) \<times> (U \<inter> NN t))"
apply (rule image_mono)
using \<open>0 < \<delta>\<close> \<open>N > 0\<close> Suc.prems apply auto
apply (meson divide_nonneg_nonneg of_nat_0_le_iff order_trans)
using Suc.prems order_trans apply fastforce
using t Q' apply auto
done
with him show "h ` ({real n / real N..(1 + real n) / real N} \<times> Q') \<subseteq> UU (X t)"
using t01 by blast
qed
ultimately show "continuous_on ({real n / real N..(1 + real n) / real N} \<times> Q') (p' \<circ> h)"
by (rule continuous_on_compose)
have "k (real n / real N, b) = p' (h (real n / real N, b))" if "b \<in> Q'" for b
proof -
have "k (real n / real N, b) \<in> W"
using that Q' kimw  by force
then have "k (real n / real N, b) = p' (p (k (real n / real N, b)))"
by (simp add:  homeomorphism_apply1 [OF hom'])
then show ?thesis
using Q' that by (force simp: heq)
qed
then show "\<And>x. x \<in> {real n / real N..(1 + real n) / real N} \<times> Q' \<and>
x \<in> {0..real n / real N} \<times> Q' \<Longrightarrow> k x = (p' \<circ> h) x"
by auto
qed
have h_in_UU: "h (x, y) \<in> UU (X t)" if "y \<in> Q" "\<not> x \<le> real n / real N" "0 \<le> x" "x \<le> (1 + real n) / real N" for x y
proof -
have "x \<le> 1"
using Suc.prems that order_trans by force
moreover have "x \<in> K t"
by (meson atLeastAtMost_iff le_less not_le subset_eq t that)
moreover have "y \<in> U"
using \<open>y \<in> Q\<close> opeUQ openin_imp_subset by blast
moreover have "y \<in> NN t"
using Q' \<open>Q \<subseteq> Q'\<close> \<open>y \<in> Q\<close> by auto
ultimately have "(x, y) \<in> (({0..1} \<inter> K t) \<times> (U \<inter> NN t))"
using that by auto
then have "h (x, y) \<in> h ` (({0..1} \<inter> K t) \<times> (U \<inter> NN t))"
by blast
also have "... \<subseteq> UU (X t)"
by (metis him t01)
finally show ?thesis .
qed
let ?k = "(\<lambda>x. if x \<in> {0..real n / real N} \<times> Q' then k x else (p' \<circ> h) x)"
show ?case
proof (intro exI conjI)
show "continuous_on ({0..real (Suc n) / real N} \<times> Q) ?k"
apply (rule continuous_on_subset [OF cont])
using \<open>Q \<subseteq> Q'\<close> by auto
have "\<And>a b. \<lbrakk>a \<le> real n / real N; b \<in> Q'; 0 \<le> a\<rbrakk> \<Longrightarrow> k (a, b) \<in> C"
using kim Q' by force
moreover have "\<And>a b. \<lbrakk>b \<in> Q; 0 \<le> a; a \<le> (1 + real n) / real N; \<not> a \<le> real n / real N\<rbrakk> \<Longrightarrow> p' (h (a, b)) \<in> C"
apply (rule \<open>W \<subseteq> C\<close> [THEN subsetD])
using homeomorphism_image2 [OF hom', symmetric]  h_in_UU  Q' \<open>Q \<subseteq> Q'\<close> \<open>W \<subseteq> C\<close>
apply auto
done
ultimately show "?k ` ({0..real (Suc n) / real N} \<times> Q) \<subseteq> C"
using Q' \<open>Q \<subseteq> Q'\<close> by force
show "\<forall>z\<in>Q. ?k (0, z) = f z"
using Q' keq  \<open>Q \<subseteq> Q'\<close> by auto
show "\<forall>z \<in> {0..real (Suc n) / real N} \<times> Q. h z = p(?k z)"
using \<open>Q \<subseteq> U \<inter> NN t \<inter> N' \<inter> V\<close> heq apply clarsimp
using h_in_UU Q' \<open>Q \<subseteq> Q'\<close> apply (auto simp: homeomorphism_apply2 [OF hom', symmetric])
done
qed (auto simp: \<open>y \<in> Q\<close> opeUQ)
qed
show ?thesis
using*[OF order_refl] N \<open>0 < \<delta>\<close> by (simp add: split: if_split_asm)
qed
then obtain V fs where opeV: "\<And>y. y \<in> U \<Longrightarrow> openin (subtopology euclidean U) (V y)"
and V: "\<And>y. y \<in> U \<Longrightarrow> y \<in> V y"
and contfs: "\<And>y. y \<in> U \<Longrightarrow> continuous_on ({0..1} \<times> V y) (fs y)"
and *: "\<And>y. y \<in> U \<Longrightarrow> (fs y) ` ({0..1} \<times> V y) \<subseteq> C \<and>
(\<forall>z \<in> V y. fs y (0, z) = f z) \<and>
(\<forall>z \<in> {0..1} \<times> V y. h z = p(fs y z))"
by (metis (mono_tags))
then have VU: "\<And>y. y \<in> U \<Longrightarrow> V y \<subseteq> U"
by (meson openin_imp_subset)
obtain k where contk: "continuous_on ({0..1} \<times> U) k"
and k: "\<And>x i. \<lbrakk>i \<in> U; x \<in> {0..1} \<times> U \<inter> {0..1} \<times> V i\<rbrakk> \<Longrightarrow> k x = fs i x"
proof (rule pasting_lemma_exists)
show "{0..1} \<times> U \<subseteq> (\<Union>i\<in>U. {0..1} \<times> V i)"
apply auto
using V by blast
show "\<And>i. i \<in> U \<Longrightarrow> openin (subtopology euclidean ({0..1} \<times> U)) ({0..1} \<times> V i)"
show "\<And>i. i \<in> U \<Longrightarrow> continuous_on ({0..1} \<times> V i) (fs i)"
using contfs by blast
show "fs i x = fs j x"  if "i \<in> U" "j \<in> U" and x: "x \<in> {0..1} \<times> U \<inter> {0..1} \<times> V i \<inter> {0..1} \<times> V j"
for i j x
proof -
obtain u y where "x = (u, y)" "y \<in> V i" "y \<in> V j" "0 \<le> u" "u \<le> 1"
using x by auto
show ?thesis
proof (rule covering_space_lift_unique [OF cov, of _ "(0,y)" _ "{0..1} \<times> {y}" h])
show "fs i (0, y) = fs j (0, y)"
using*V by (simp add: \<open>y \<in> V i\<close> \<open>y \<in> V j\<close> that)
show conth_y: "continuous_on ({0..1} \<times> {y}) h"
apply (rule continuous_on_subset [OF conth])
using VU \<open>y \<in> V j\<close> that by auto
show "h ` ({0..1} \<times> {y}) \<subseteq> S"
using \<open>y \<in> V i\<close> assms(3) VU that by fastforce
show "continuous_on ({0..1} \<times> {y}) (fs i)"
using continuous_on_subset [OF contfs] \<open>i \<in> U\<close>
by (simp add: \<open>y \<in> V i\<close> subset_iff)
show "fs i ` ({0..1} \<times> {y}) \<subseteq> C"
using "*" \<open>y \<in> V i\<close> \<open>i \<in> U\<close> by fastforce
show "\<And>x. x \<in> {0..1} \<times> {y} \<Longrightarrow> h x = p (fs i x)"
using "*" \<open>y \<in> V i\<close> \<open>i \<in> U\<close> by blast
show "continuous_on ({0..1} \<times> {y}) (fs j)"
using continuous_on_subset [OF contfs] \<open>j \<in> U\<close>
by (simp add: \<open>y \<in> V j\<close> subset_iff)
show "fs j ` ({0..1} \<times> {y}) \<subseteq> C"
using "*" \<open>y \<in> V j\<close> \<open>j \<in> U\<close> by fastforce
show "\<And>x. x \<in> {0..1} \<times> {y} \<Longrightarrow> h x = p (fs j x)"
using "*" \<open>y \<in> V j\<close> \<open>j \<in> U\<close> by blast
show "connected ({0..1::real} \<times> {y})"
using connected_Icc connected_Times connected_sing by blast
show "(0, y) \<in> {0..1::real} \<times> {y}"
by force
show "x \<in> {0..1} \<times> {y}"
using \<open>x = (u, y)\<close> x by blast
qed
qed
qed blast
show ?thesis
proof
show "k ` ({0..1} \<times> U) \<subseteq> C"
using V*k VU by fastforce
show "\<And>y. y \<in> U \<Longrightarrow> k (0, y) = f y"
show "\<And>z. z \<in> {0..1} \<times> U \<Longrightarrow> h z = p (k z)"
using V*k by auto
qed (auto simp: contk)
qed

corollary covering_space_lift_homotopy_alt:
fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
and h :: "'c::real_normed_vector \<times> real \<Rightarrow> 'b"
assumes cov: "covering_space C p S"
and conth: "continuous_on (U \<times> {0..1}) h"
and him: "h ` (U \<times> {0..1}) \<subseteq> S"
and heq: "\<And>y. y \<in> U \<Longrightarrow> h (y,0) = p(f y)"
and contf: "continuous_on U f" and fim: "f ` U \<subseteq> C"
obtains k where "continuous_on (U \<times> {0..1}) k"
"k ` (U \<times> {0..1}) \<subseteq> C"
"\<And>y. y \<in> U \<Longrightarrow> k(y, 0) = f y"
"\<And>z. z \<in> U \<times> {0..1} \<Longrightarrow> h z = p(k z)"
proof -
have "continuous_on ({0..1} \<times> U) (h \<circ> (\<lambda>z. (snd z, fst z)))"
by (intro continuous_intros continuous_on_subset [OF conth]) auto
then obtain k where contk: "continuous_on ({0..1} \<times> U) k"
and kim:  "k ` ({0..1} \<times> U) \<subseteq> C"
and k0: "\<And>y. y \<in> U \<Longrightarrow> k(0, y) = f y"
and heqp: "\<And>z. z \<in> {0..1} \<times> U \<Longrightarrow> (h \<circ> (\<lambda>z. Pair (snd z) (fst z))) z = p(k z)"
apply (rule covering_space_lift_homotopy [OF cov _ _ _ contf fim])
using him  by (auto simp: contf heq)
show ?thesis
apply (rule_tac k="k \<circ> (\<lambda>z. Pair (snd z) (fst z))" in that)
apply (intro continuous_intros continuous_on_subset [OF contk])
using kim heqp apply (auto simp: k0)
done
qed

corollary covering_space_lift_homotopic_function:
fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" and g:: "'c::real_normed_vector \<Rightarrow> 'a"
assumes cov: "covering_space C p S"
and contg: "continuous_on U g"
and gim: "g ` U \<subseteq> C"
and pgeq: "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
and hom: "homotopic_with (\<lambda>x. True) U S f f'"
obtains g' where "continuous_on U g'" "image g' U \<subseteq> C" "\<And>y. y \<in> U \<Longrightarrow> p(g' y) = f' y"
proof -
obtain h where conth: "continuous_on ({0..1::real} \<times> U) h"
and him: "h ` ({0..1} \<times> U) \<subseteq> S"
and h0:  "\<And>x. h(0, x) = f x"
and h1: "\<And>x. h(1, x) = f' x"
using hom by (auto simp: homotopic_with_def)
have "\<And>y. y \<in> U \<Longrightarrow> h (0, y) = p (g y)"
then obtain k where contk: "continuous_on ({0..1} \<times> U) k"
and kim: "k ` ({0..1} \<times> U) \<subseteq> C"
and k0: "\<And>y. y \<in> U \<Longrightarrow> k(0, y) = g y"
and heq: "\<And>z. z \<in> {0..1} \<times> U \<Longrightarrow> h z = p(k z)"
using covering_space_lift_homotopy [OF cov conth him _ contg gim] by metis
show ?thesis
proof
show "continuous_on U (k \<circ> Pair 1)"
by (meson contk atLeastAtMost_iff continuous_on_o_Pair order_refl zero_le_one)
show "(k \<circ> Pair 1) ` U \<subseteq> C"
using kim by auto
show "\<And>y. y \<in> U \<Longrightarrow> p ((k \<circ> Pair 1) y) = f' y"
by (auto simp: h1 heq [symmetric])
qed
qed

corollary covering_space_lift_inessential_function:
fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" and U :: "'c::real_normed_vector set"
assumes cov: "covering_space C p S"
and hom: "homotopic_with (\<lambda>x. True) U S f (\<lambda>x. a)"
obtains g where "continuous_on U g" "g ` U \<subseteq> C" "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
proof (cases "U = {}")
case True
then show ?thesis
using that continuous_on_empty by blast
next
case False
then obtain b where b: "b \<in> C" "p b = a"
using covering_space_imp_surjective [OF cov] homotopic_with_imp_subset2 [OF hom]
by auto
then have gim: "(\<lambda>y. b) ` U \<subseteq> C"
by blast
show ?thesis
apply (rule covering_space_lift_homotopic_function
[OF cov continuous_on_const gim _ homotopic_with_symD [OF hom]])
using b that apply auto
done
qed

subsection\<open> Lifting of general functions to covering space\<close>

proposition covering_space_lift_path_strong:
fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
and f :: "'c::real_normed_vector \<Rightarrow> 'b"
assumes cov: "covering_space C p S" and "a \<in> C"
and "path g" and pag: "path_image g \<subseteq> S" and pas: "pathstart g = p a"
obtains h where "path h" "path_image h \<subseteq> C" "pathstart h = a"
and "\<And>t. t \<in> {0..1} \<Longrightarrow> p(h t) = g t"
proof -
obtain k:: "real \<times> 'c \<Rightarrow> 'a"
where contk: "continuous_on ({0..1} \<times> {undefined}) k"
and kim: "k ` ({0..1} \<times> {undefined}) \<subseteq> C"
and k0:  "k (0, undefined) = a"
and pk: "\<And>z. z \<in> {0..1} \<times> {undefined} \<Longrightarrow> p(k z) = (g \<circ> fst) z"
proof (rule covering_space_lift_homotopy [OF cov, of "{undefined}" "g \<circ> fst"])
show "continuous_on ({0..1::real} \<times> {undefined::'c}) (g \<circ> fst)"
apply (intro continuous_intros)
using \<open>path g\<close> by (simp add: path_def)
show "(g \<circ> fst) ` ({0..1} \<times> {undefined}) \<subseteq> S"
using pag by (auto simp: path_image_def)
show "(g \<circ> fst) (0, y) = p a" if "y \<in> {undefined}" for y::'c
by (metis comp_def fst_conv pas pathstart_def)
qed (use assms in auto)
show ?thesis
proof
show "path (k \<circ> (\<lambda>t. Pair t undefined))"
unfolding path_def
by (intro continuous_on_compose continuous_intros continuous_on_subset [OF contk]) auto
show "path_image (k \<circ> (\<lambda>t. (t, undefined))) \<subseteq> C"
using kim by (auto simp: path_image_def)
show "pathstart (k \<circ> (\<lambda>t. (t, undefined))) = a"
by (auto simp: pathstart_def k0)
show "\<And>t. t \<in> {0..1} \<Longrightarrow> p ((k \<circ> (\<lambda>t. (t, undefined))) t) = g t"
by (auto simp: pk)
qed
qed

corollary covering_space_lift_path:
fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
assumes cov: "covering_space C p S" and "path g" and pig: "path_image g \<subseteq> S"
obtains h where "path h" "path_image h \<subseteq> C" "\<And>t. t \<in> {0..1} \<Longrightarrow> p(h t) = g t"
proof -
obtain a where "a \<in> C" "pathstart g = p a"
by (metis pig cov covering_space_imp_surjective imageE pathstart_in_path_image subsetCE)
show ?thesis
using covering_space_lift_path_strong [OF cov \<open>a \<in> C\<close> \<open>path g\<close> pig]
by (metis \<open>pathstart g = p a\<close> that)
qed

proposition covering_space_lift_homotopic_paths:
fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
assumes cov: "covering_space C p S"
and "path g1" and pig1: "path_image g1 \<subseteq> S"
and "path g2" and pig2: "path_image g2 \<subseteq> S"
and hom: "homotopic_paths S g1 g2"
and "path h1" and pih1: "path_image h1 \<subseteq> C" and ph1: "\<And>t. t \<in> {0..1} \<Longrightarrow> p(h1 t) = g1 t"
and "path h2" and pih2: "path_image h2 \<subseteq> C" and ph2: "\<And>t. t \<in> {0..1} \<Longrightarrow> p(h2 t) = g2 t"
and h1h2: "pathstart h1 = pathstart h2"
shows "homotopic_paths C h1 h2"
proof -
obtain h :: "real \<times> real \<Rightarrow> 'b"
where conth: "continuous_on ({0..1} \<times> {0..1}) h"
and him: "h ` ({0..1} \<times> {0..1}) \<subseteq> S"
and h0: "\<And>x. h (0, x) = g1 x" and h1: "\<And>x. h (1, x) = g2 x"
and heq0: "\<And>t. t \<in> {0..1} \<Longrightarrow> h (t, 0) = g1 0"
and heq1: "\<And>t. t \<in> {0..1} \<Longrightarrow> h (t, 1) = g1 1"
using hom by (auto simp: homotopic_paths_def homotopic_with_def pathstart_def pathfinish_def)
obtain k where contk: "continuous_on ({0..1} \<times> {0..1}) k"
and kim: "k ` ({0..1} \<times> {0..1}) \<subseteq> C"
and kh2: "\<And>y. y \<in> {0..1} \<Longrightarrow> k (y, 0) = h2 0"
and hpk: "\<And>z. z \<in> {0..1} \<times> {0..1} \<Longrightarrow> h z = p (k z)"
apply (rule covering_space_lift_homotopy_alt [OF cov conth him, of "\<lambda>x. h2 0"])
using h1h2 ph1 ph2 apply (force simp: heq0 pathstart_def pathfinish_def)
using path_image_def pih2 continuous_on_const by fastforce+
have contg1: "continuous_on {0..1} g1" and contg2: "continuous_on {0..1} g2"
using \<open>path g1\<close> \<open>path g2\<close> path_def by blast+
have g1im: "g1 ` {0..1} \<subseteq> S" and g2im: "g2 ` {0..1} \<subseteq> S"
using path_image_def pig1 pig2 by auto
have conth1: "continuous_on {0..1} h1" and conth2: "continuous_on {0..1} h2"
using \<open>path h1\<close> \<open>path h2\<close> path_def by blast+
have h1im: "h1 ` {0..1} \<subseteq> C" and h2im: "h2 ` {0..1} \<subseteq> C"
using path_image_def pih1 pih2 by auto
show ?thesis
unfolding homotopic_paths pathstart_def pathfinish_def
proof (intro exI conjI ballI)
show keqh1: "k(0, x) = h1 x" if "x \<in> {0..1}" for x
proof (rule covering_space_lift_unique [OF cov _ contg1 g1im])
show "k (0,0) = h1 0"
by (metis atLeastAtMost_iff h1h2 kh2 order_refl pathstart_def zero_le_one)
show "continuous_on {0..1} (\<lambda>a. k (0, a))"
by (intro continuous_intros continuous_on_compose2 [OF contk]) auto
show "\<And>x. x \<in> {0..1} \<Longrightarrow> g1 x = p (k (0, x))"
by (metis atLeastAtMost_iff h0 hpk zero_le_one mem_Sigma_iff order_refl)
qed (use conth1 h1im kim that in \<open>auto simp: ph1\<close>)
show "k(1, x) = h2 x" if "x \<in> {0..1}" for x
proof (rule covering_space_lift_unique [OF cov _ contg2 g2im])
show "k (1,0) = h2 0"
by (metis atLeastAtMost_iff kh2 order_refl zero_le_one)
show "continuous_on {0..1} (\<lambda>a. k (1, a))"
by (intro continuous_intros continuous_on_compose2 [OF contk]) auto
show "\<And>x. x \<in> {0..1} \<Longrightarrow> g2 x = p (k (1, x))"
by (metis atLeastAtMost_iff h1 hpk mem_Sigma_iff order_refl zero_le_one)
qed (use conth2 h2im kim that in \<open>auto simp: ph2\<close>)
show "\<And>t. t \<in> {0..1} \<Longrightarrow> (k \<circ> Pair t) 0 = h1 0"
by (metis comp_apply h1h2 kh2 pathstart_def)
show "(k \<circ> Pair t) 1 = h1 1" if "t \<in> {0..1}" for t
proof (rule covering_space_lift_unique
[OF cov, of "\<lambda>a. (k \<circ> Pair a) 1" 0 "\<lambda>a. h1 1" "{0..1}"  "\<lambda>x. g1 1"])
show "(k \<circ> Pair 0) 1 = h1 1"
using keqh1 by auto
show "continuous_on {0..1} (\<lambda>a. (k \<circ> Pair a) 1)"
apply simp
by (intro continuous_intros continuous_on_compose2 [OF contk]) auto
show "\<And>x. x \<in> {0..1} \<Longrightarrow> g1 1 = p ((k \<circ> Pair x) 1)"
using heq1 hpk by auto
qed (use contk kim g1im h1im that in \<open>auto simp: ph1 continuous_on_const\<close>)
qed (use contk kim in auto)
qed

corollary covering_space_monodromy:
fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
assumes cov: "covering_space C p S"
and "path g1" and pig1: "path_image g1 \<subseteq> S"
and "path g2" and pig2: "path_image g2 \<subseteq> S"
and hom: "homotopic_paths S g1 g2"
and "path h1" and pih1: "path_image h1 \<subseteq> C" and ph1: "\<And>t. t \<in> {0..1} \<Longrightarrow> p(h1 t) = g1 t"
and "path h2" and pih2: "path_image h2 \<subseteq> C" and ph2: "\<And>t. t \<in> {0..1} \<Longrightarrow> p(h2 t) = g2 t"
and h1h2: "pathstart h1 = pathstart h2"
shows "pathfinish h1 = pathfinish h2"
using covering_space_lift_homotopic_paths [OF assms] homotopic_paths_imp_pathfinish by blast

corollary covering_space_lift_homotopic_path:
fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
assumes cov: "covering_space C p S"
and hom: "homotopic_paths S f f'"
and "path g" and pig: "path_image g \<subseteq> C"
and a: "pathstart g = a" and b: "pathfinish g = b"
and pgeq: "\<And>t. t \<in> {0..1} \<Longrightarrow> p(g t) = f t"
obtains g' where "path g'" "path_image g' \<subseteq> C"
"pathstart g' = a" "pathfinish g' = b" "\<And>t. t \<in> {0..1} \<Longrightarrow> p(g' t) = f' t"
proof (rule covering_space_lift_path_strong [OF cov, of a f'])
show "a \<in> C"
using a pig by auto
show "path f'" "path_image f' \<subseteq> S"
using hom homotopic_paths_imp_path homotopic_paths_imp_subset by blast+
show "pathstart f' = p a"
by (metis a atLeastAtMost_iff hom homotopic_paths_imp_pathstart order_refl pathstart_def pgeq zero_le_one)
qed (metis (mono_tags, lifting) assms cov covering_space_monodromy hom homotopic_paths_imp_path homotopic_paths_imp_subset pgeq pig)

proposition covering_space_lift_general:
fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
and f :: "'c::real_normed_vector \<Rightarrow> 'b"
assumes cov: "covering_space C p S" and "a \<in> C" "z \<in> U"
and U: "path_connected U" "locally path_connected U"
and contf: "continuous_on U f" and fim: "f ` U \<subseteq> S"
and feq: "f z = p a"
and hom: "\<And>r. \<lbrakk>path r; path_image r \<subseteq> U; pathstart r = z; pathfinish r = z\<rbrakk>
\<Longrightarrow> \<exists>q. path q \<and> path_image q \<subseteq> C \<and>
pathstart q = a \<and> pathfinish q = a \<and>
homotopic_paths S (f \<circ> r) (p \<circ> q)"
obtains g where "continuous_on U g" "g ` U \<subseteq> C" "g z = a" "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
proof -
have *: "\<exists>g h. path g \<and> path_image g \<subseteq> U \<and>
pathstart g = z \<and> pathfinish g = y \<and>
path h \<and> path_image h \<subseteq> C \<and> pathstart h = a \<and>
(\<forall>t \<in> {0..1}. p(h t) = f(g t))"
if "y \<in> U" for y
proof -
obtain g where "path g" "path_image g \<subseteq> U" and pastg: "pathstart g = z"
and pafig: "pathfinish g = y"
using U \<open>z \<in> U\<close> \<open>y \<in> U\<close> by (force simp: path_connected_def)
obtain h where "path h" "path_image h \<subseteq> C" "pathstart h = a"
and "\<And>t. t \<in> {0..1} \<Longrightarrow> p(h t) = (f \<circ> g) t"
proof (rule covering_space_lift_path_strong [OF cov \<open>a \<in> C\<close>])
show "path (f \<circ> g)"
using \<open>path g\<close> \<open>path_image g \<subseteq> U\<close> contf continuous_on_subset path_continuous_image by blast
show "path_image (f \<circ> g) \<subseteq> S"
by (metis \<open>path_image g \<subseteq> U\<close> fim image_mono path_image_compose subset_trans)
show "pathstart (f \<circ> g) = p a"
by (simp add: feq pastg pathstart_compose)
qed auto
then show ?thesis
by (metis \<open>path g\<close> \<open>path_image g \<subseteq> U\<close> comp_apply pafig pastg)
qed
have "\<exists>l. \<forall>g h. path g \<and> path_image g \<subseteq> U \<and> pathstart g = z \<and> pathfinish g = y \<and>
path h \<and> path_image h \<subseteq> C \<and> pathstart h = a \<and>
(\<forall>t \<in> {0..1}. p(h t) = f(g t))  \<longrightarrow> pathfinish h = l" for y
proof -
have "pathfinish h = pathfinish h'"
if g: "path g" "path_image g \<subseteq> U" "pathstart g = z" "pathfinish g = y"
and h: "path h" "path_image h \<subseteq> C" "pathstart h = a"
and phg: "\<And>t. t \<in> {0..1} \<Longrightarrow> p(h t) = f(g t)"
and g': "path g'" "path_image g' \<subseteq> U" "pathstart g' = z" "pathfinish g' = y"
and h': "path h'" "path_image h' \<subseteq> C" "pathstart h' = a"
and phg': "\<And>t. t \<in> {0..1} \<Longrightarrow> p(h' t) = f(g' t)"
for g h g' h'
proof -
obtain q where "path q" and piq: "path_image q \<subseteq> C" and pastq: "pathstart q = a" and pafiq: "pathfinish q = a"
and homS: "homotopic_paths S (f \<circ> g +++ reversepath g') (p \<circ> q)"
using g g' hom [of "g +++ reversepath g'"] by (auto simp:  subset_path_image_join)
have papq: "path (p \<circ> q)"
using homS homotopic_paths_imp_path by blast
have pipq: "path_image (p \<circ> q) \<subseteq> S"
using homS homotopic_paths_imp_subset by blast
obtain q' where "path q'" "path_image q' \<subseteq> C"
and "pathstart q' = pathstart q" "pathfinish q' = pathfinish q"
and pq'_eq: "\<And>t. t \<in> {0..1} \<Longrightarrow> p (q' t) = (f \<circ> g +++ reversepath g') t"
using covering_space_lift_homotopic_path [OF cov homotopic_paths_sym [OF homS] \<open>path q\<close> piq refl refl]
by auto
have "q' t = (h \<circ> op *\<^sub>R 2) t" if "0 \<le> t" "t \<le> 1/2" for t
proof (rule covering_space_lift_unique [OF cov, of q' 0 "h \<circ> op *\<^sub>R 2" "{0..1/2}" "f \<circ> g \<circ> op *\<^sub>R 2" t])
show "q' 0 = (h \<circ> op *\<^sub>R 2) 0"
by (metis \<open>pathstart q' = pathstart q\<close> comp_def g h pastq pathstart_def pth_4(2))
show "continuous_on {0..1/2} (f \<circ> g \<circ> op *\<^sub>R 2)"
apply (intro continuous_intros continuous_on_compose continuous_on_path [OF \<open>path g\<close>] continuous_on_subset [OF contf])
using g(2) path_image_def by fastforce+
show "(f \<circ> g \<circ> op *\<^sub>R 2) ` {0..1/2} \<subseteq> S"
using g(2) path_image_def fim by fastforce
show "(h \<circ> op *\<^sub>R 2) ` {0..1/2} \<subseteq> C"
using h path_image_def by fastforce
show "q' ` {0..1/2} \<subseteq> C"
using \<open>path_image q' \<subseteq> C\<close> path_image_def by fastforce
show "\<And>x. x \<in> {0..1/2} \<Longrightarrow> (f \<circ> g \<circ> op *\<^sub>R 2) x = p (q' x)"
by (auto simp: joinpaths_def pq'_eq)
show "\<And>x. x \<in> {0..1/2} \<Longrightarrow> (f \<circ> g \<circ> op *\<^sub>R 2) x = p ((h \<circ> op *\<^sub>R 2) x)"
show "continuous_on {0..1/2} q'"
by (simp add: continuous_on_path \<open>path q'\<close>)
show "continuous_on {0..1/2} (h \<circ> op *\<^sub>R 2)"
apply (intro continuous_intros continuous_on_compose continuous_on_path [OF \<open>path h\<close>], force)
done
qed (use that in auto)
moreover have "q' t = (reversepath h' \<circ> (\<lambda>t. 2 *\<^sub>R t - 1)) t" if "1/2 < t" "t \<le> 1" for t
proof (rule covering_space_lift_unique [OF cov, of q' 1 "reversepath h' \<circ> (\<lambda>t. 2 *\<^sub>R t - 1)" "{1/2<..1}" "f \<circ> reversepath g' \<circ> (\<lambda>t. 2 *\<^sub>R t - 1)" t])
show "q' 1 = (reversepath h' \<circ> (\<lambda>t. 2 *\<^sub>R t - 1)) 1"
using h' \<open>pathfinish q' = pathfinish q\<close> pafiq
by (simp add: pathstart_def pathfinish_def reversepath_def)
show "continuous_on {1/2<..1} (f \<circ> reversepath g' \<circ> (\<lambda>t. 2 *\<^sub>R t - 1))"
apply (intro continuous_intros continuous_on_compose continuous_on_path \<open>path g'\<close> continuous_on_subset [OF contf])
using g' apply simp_all
by (auto simp: path_image_def reversepath_def)
show "(f \<circ> reversepath g' \<circ> (\<lambda>t. 2 *\<^sub>R t - 1)) ` {1/2<..1} \<subseteq> S"
using g'(2) path_image_def fim by (auto simp: image_subset_iff path_image_def reversepath_def)
show "q' ` {1/2<..1} \<subseteq> C"
using \<open>path_image q' \<subseteq> C\<close> path_image_def by fastforce
show "(reversepath h' \<circ> (\<lambda>t. 2 *\<^sub>R t - 1)) ` {1/2<..1} \<subseteq> C"
using h' by (simp add: path_image_def reversepath_def subset_eq)
show "\<And>x. x \<in> {1/2<..1} \<Longrightarrow> (f \<circ> reversepath g' \<circ> (\<lambda>t. 2 *\<^sub>R t - 1)) x = p (q' x)"
by (auto simp: joinpaths_def pq'_eq)
show "\<And>x. x \<in> {1/2<..1} \<Longrightarrow>
(f \<circ> reversepath g' \<circ> (\<lambda>t. 2 *\<^sub>R t - 1)) x = p ((reversepath h' \<circ> (\<lambda>t. 2 *\<^sub>R t - 1)) x)"
show "continuous_on {1/2<..1} q'"
by (auto intro: continuous_on_path [OF \<open>path q'\<close>])
show "continuous_on {1/2<..1} (reversepath h' \<circ> (\<lambda>t. 2 *\<^sub>R t - 1))"
apply (intro continuous_intros continuous_on_compose continuous_on_path \<open>path h'\<close>)
using h' apply auto
done
qed (use that in auto)
ultimately have "q' t = (h +++ reversepath h') t" if "0 \<le> t" "t \<le> 1" for t
using that by (simp add: joinpaths_def)
then have "path(h +++ reversepath h')"
by (auto intro: path_eq [OF \<open>path q'\<close>])
then show ?thesis
by (auto simp: \<open>path h\<close> \<open>path h'\<close>)
qed
then show ?thesis by metis
qed
then obtain l :: "'c \<Rightarrow> 'a"
where l: "\<And>y g h. \<lbrakk>path g; path_image g \<subseteq> U; pathstart g = z; pathfinish g = y;
path h; path_image h \<subseteq> C; pathstart h = a;
\<And>t. t \<in> {0..1} \<Longrightarrow> p(h t) = f(g t)\<rbrakk> \<Longrightarrow> pathfinish h = l y"
by metis
show ?thesis
proof
show pleq: "p (l y) = f y" if "y \<in> U" for y
using*[OF \<open>y \<in> U\<close>]  by (metis l atLeastAtMost_iff order_refl pathfinish_def zero_le_one)
show "l z = a"
using l [of "linepath z z" z "linepath a a"] by (auto simp: assms)
show LC: "l ` U \<subseteq> C"
by (clarify dest!: *) (metis (full_types) l pathfinish_in_path_image subsetCE)
have "\<exists>T. openin (subtopology euclidean U) T \<and> y \<in> T \<and> T \<subseteq> {x \<in> U. l x \<in> X}"
if X: "openin (subtopology euclidean C) X" and "y \<in> U" "l y \<in> X" for X y
proof -
have "X \<subseteq> C"
using X openin_euclidean_subtopology_iff by blast
have "f y \<in> S"
using fim \<open>y \<in> U\<close> by blast
then obtain W \<V>
where WV: "f y \<in> W \<and> openin (subtopology euclidean S) W \<and>
(\<Union>\<V> = {x. x \<in> C \<and> p x \<in> W} \<and>
(\<forall>U \<in> \<V>. openin (subtopology euclidean C) U) \<and>
pairwise disjnt \<V> \<and>
(\<forall>U \<in> \<V>. \<exists>q. homeomorphism U W p q))"
using cov by (force simp: covering_space_def)
then have "l y \<in> \<Union>\<V>"
using \<open>X \<subseteq> C\<close> pleq that by auto
then obtain W' where "l y \<in> W'" and "W' \<in> \<V>"
by blast
with WV obtain p' where opeCW': "openin (subtopology euclidean C) W'"
and homUW': "homeomorphism W' W p p'"
by blast
then have contp': "continuous_on W p'" and p'im: "p' ` W \<subseteq> W'"
using homUW' homeomorphism_image2 homeomorphism_cont2 by fastforce+
obtain V where "y \<in> V" "y \<in> U" and fimW: "f ` V \<subseteq> W" "V \<subseteq> U"
and "path_connected V" and opeUV: "openin (subtopology euclidean U) V"
proof -
have "openin (subtopology euclidean U) {c \<in> U. f c \<in> W}"
using WV contf continuous_on_open_gen fim by auto
then show ?thesis
using U WV
apply (auto simp: locally_path_connected)
apply (drule_tac x="{x. x \<in> U \<and> f x \<in> W}" in spec)
apply (drule_tac x=y in spec)
apply (auto simp: \<open>y \<in> U\<close> intro: that)
done
qed
have "W' \<subseteq> C" "W \<subseteq> S"
using opeCW' WV openin_imp_subset by auto
have p'im: "p' ` W \<subseteq> W'"
using homUW' homeomorphism_image2 by fastforce
show ?thesis
proof (intro exI conjI)
have "openin (subtopology euclidean S) {x \<in> W. p' x \<in> W' \<inter> X}"
proof (rule openin_trans)
show "openin (subtopology euclidean W) {x \<in> W. p' x \<in> W' \<inter> X}"
apply (rule continuous_openin_preimage [OF contp' p'im])
using X \<open>W' \<subseteq> C\<close> apply (auto simp: openin_open)
done
show "openin (subtopology euclidean S) W"
using WV by blast
qed
then show "openin (subtopology euclidean U) (V \<inter> {x. x \<in> U \<and> f x \<in> {x. x \<in> W \<and> p' x \<in> W' \<inter> X}})"
by (intro openin_Int opeUV continuous_openin_preimage [OF contf fim])
have "p' (f y) \<in> X"
using \<open>l y \<in> W'\<close> homeomorphism_apply1 [OF homUW'] pleq \<open>y \<in> U\<close> \<open>l y \<in> X\<close> by fastforce
then show "y \<in> V \<inter> {x \<in> U. f x \<in> {x \<in> W. p' x \<in> W' \<inter> X}}"
using \<open>y \<in> U\<close> \<open>y \<in> V\<close> WV p'im by auto
show "V \<inter> {x \<in> U. f x \<in> {x \<in> W. p' x \<in> W' \<inter> X}} \<subseteq> {x \<in> U. l x \<in> X}"
proof clarsimp
fix y'
assume y': "y' \<in> V" "y' \<in> U" "f y' \<in> W" "p' (f y') \<in> W'" "p' (f y') \<in> X"
then obtain \<gamma> where "path \<gamma>" "path_image \<gamma> \<subseteq> V" "pathstart \<gamma> = y" "pathfinish \<gamma> = y'"
by (meson \<open>path_connected V\<close> \<open>y \<in> V\<close> path_connected_def)
obtain pp qq where "path pp" "path_image pp \<subseteq> U"
"pathstart pp = z" "pathfinish pp = y"
"path qq" "path_image qq \<subseteq> C" "pathstart qq = a"
and pqqeq: "\<And>t. t \<in> {0..1} \<Longrightarrow> p(qq t) = f(pp t)"
using*[OF \<open>y \<in> U\<close>] by blast
have finW: "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> f (\<gamma> x) \<in> W"
using \<open>path_image \<gamma> \<subseteq> V\<close> by (auto simp: image_subset_iff path_image_def fimW [THEN subsetD])
have "pathfinish (qq +++ (p' \<circ> f \<circ> \<gamma>)) = l y'"
proof (rule l [of "pp +++ \<gamma>" y' "qq +++ (p' \<circ> f \<circ> \<gamma>)" ])
show "path (pp +++ \<gamma>)"
by (simp add: \<open>path \<gamma>\<close> \<open>path pp\<close> \<open>pathfinish pp = y\<close> \<open>pathstart \<gamma> = y\<close>)
show "path_image (pp +++ \<gamma>) \<subseteq> U"
using \<open>V \<subseteq> U\<close> \<open>path_image \<gamma> \<subseteq> V\<close> \<open>path_image pp \<subseteq> U\<close> not_in_path_image_join by blast
show "pathstart (pp +++ \<gamma>) = z"
by (simp add: \<open>pathstart pp = z\<close>)
show "pathfinish (pp +++ \<gamma>) = y'"
by (simp add: \<open>pathfinish \<gamma> = y'\<close>)
have paqq: "pathfinish qq = pathstart (p' \<circ> f \<circ> \<gamma>)"
apply (simp add: \<open>pathstart \<gamma> = y\<close> pathstart_compose)
apply (metis (mono_tags, lifting) \<open>l y \<in> W'\<close> \<open>path pp\<close> \<open>path qq\<close> \<open>path_image pp \<subseteq> U\<close> \<open>path_image qq \<subseteq> C\<close>
\<open>pathfinish pp = y\<close> \<open>pathstart pp = z\<close> \<open>pathstart qq = a\<close>
homeomorphism_apply1 [OF homUW'] l pleq pqqeq \<open>y \<in> U\<close>)
done
have "continuous_on (path_image \<gamma>) (p' \<circ> f)"
proof (rule continuous_on_compose)
show "continuous_on (path_image \<gamma>) f"
using \<open>path_image \<gamma> \<subseteq> V\<close> \<open>V \<subseteq> U\<close> contf continuous_on_subset by blast
show "continuous_on (f ` path_image \<gamma>) p'"
apply (rule continuous_on_subset [OF contp'])
apply (auto simp: path_image_def pathfinish_def pathstart_def finW)
done
qed
then show "path (qq +++ (p' \<circ> f \<circ> \<gamma>))"
using \<open>path \<gamma>\<close> \<open>path qq\<close> paqq path_continuous_image path_join_imp by blast
show "path_image (qq +++ (p' \<circ> f \<circ> \<gamma>)) \<subseteq> C"
apply (rule subset_path_image_join)
apply (simp add: \<open>path_image qq \<subseteq> C\<close>)
by (metis \<open>W' \<subseteq> C\<close> \<open>path_image \<gamma> \<subseteq> V\<close> dual_order.trans fimW(1) image_comp image_mono p'im path_image_compose)
show "pathstart (qq +++ (p' \<circ> f \<circ> \<gamma>)) = a"
by (simp add: \<open>pathstart qq = a\<close>)
show "p ((qq +++ (p' \<circ> f \<circ> \<gamma>)) \<xi>) = f ((pp +++ \<gamma>) \<xi>)" if \<xi>: "\<xi> \<in> {0..1}" for \<xi>
show "p (qq (2*\<xi>)) = f (pp (2*\<xi>))" if "\<xi>*2 \<le> 1"
using \<open>\<xi> \<in> {0..1}\<close> pqqeq that by auto
show "p (p' (f (\<gamma> (2*\<xi> - 1)))) = f (\<gamma> (2*\<xi> - 1))" if "\<not> \<xi>*2 \<le> 1"
apply (rule homeomorphism_apply2 [OF homUW' finW])
using that \<xi> by auto
qed
qed
with \<open>pathfinish \<gamma> = y'\<close>  \<open>p' (f y') \<in> X\<close> show "l y' \<in> X"
unfolding pathfinish_join by (simp add: pathfinish_def)
qed
qed
qed
then show "continuous_on U l"
using openin_subopen continuous_on_open_gen [OF LC]
by (metis (no_types, lifting) mem_Collect_eq)
qed
qed

corollary covering_space_lift_stronger:
fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
and f :: "'c::real_normed_vector \<Rightarrow> 'b"
assumes cov: "covering_space C p S" "a \<in> C" "z \<in> U"
and U: "path_connected U" "locally path_connected U"
and contf: "continuous_on U f" and fim: "f ` U \<subseteq> S"
and feq: "f z = p a"
and hom: "\<And>r. \<lbrakk>path r; path_image r \<subseteq> U; pathstart r = z; pathfinish r = z\<rbrakk>
\<Longrightarrow> \<exists>b. homotopic_paths S (f \<circ> r) (linepath b b)"
obtains g where "continuous_on U g" "g ` U \<subseteq> C" "g z = a" "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
proof (rule covering_space_lift_general [OF cov U contf fim feq])
fix r
assume "path r" "path_image r \<subseteq> U" "pathstart r = z" "pathfinish r = z"
then obtain b where b: "homotopic_paths S (f \<circ> r) (linepath b b)"
using hom by blast
then have "f (pathstart r) = b"
by (metis homotopic_paths_imp_pathstart pathstart_compose pathstart_linepath)
then have "homotopic_paths S (f \<circ> r) (linepath (f z) (f z))"
by (simp add: b \<open>pathstart r = z\<close>)
then have "homotopic_paths S (f \<circ> r) (p \<circ> linepath a a)"
by (simp add: o_def feq linepath_def)
then show "\<exists>q. path q \<and>
path_image q \<subseteq> C \<and>
pathstart q = a \<and> pathfinish q = a \<and> homotopic_paths S (f \<circ> r) (p \<circ> q)"
by (force simp: \<open>a \<in> C\<close>)
qed auto

corollary covering_space_lift_strong:
fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
and f :: "'c::real_normed_vector \<Rightarrow> 'b"
assumes cov: "covering_space C p S" "a \<in> C" "z \<in> U"
and scU: "simply_connected U" and lpcU: "locally path_connected U"
and contf: "continuous_on U f" and fim: "f ` U \<subseteq> S"
and feq: "f z = p a"
obtains g where "continuous_on U g" "g ` U \<subseteq> C" "g z = a" "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
proof (rule covering_space_lift_stronger [OF cov _ lpcU contf fim feq])
show "path_connected U"
using scU simply_connected_eq_contractible_loop_some by blast
fix r
assume r: "path r" "path_image r \<subseteq> U" "pathstart r = z" "pathfinish r = z"
have "linepath (f z) (f z) = f \<circ> linepath z z"
then have "homotopic_paths S (f \<circ> r) (linepath (f z) (f z))"
by (metis r contf fim homotopic_paths_continuous_image scU simply_connected_eq_contractible_path)
then show "\<exists>b. homotopic_paths S (f \<circ> r) (linepath b b)"
by blast
qed blast

corollary covering_space_lift:
fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
and f :: "'c::real_normed_vector \<Rightarrow> 'b"
assumes cov: "covering_space C p S"
and U: "simply_connected U"  "locally path_connected U"
and contf: "continuous_on U f" and fim: "f ` U \<subseteq> S"
obtains g where "continuous_on U g" "g ` U \<subseteq> C" "\<And>y. y \<in> U \<Longrightarrow> p(g y) = f y"
proof (cases "U = {}")
case True
with that show ?thesis by auto
next
case False
then obtain z where "z \<in> U" by blast
then obtain a where "a \<in> C" "f z = p a"
by (metis cov covering_space_imp_surjective fim image_iff image_subset_iff)
then show ?thesis
by (metis that covering_space_lift_strong [OF cov _ \<open>z \<in> U\<close> U contf fim])
qed

end
```