Theory Homeomorphism

theory Homeomorphism
imports Path_Connected
```(*  Title:      HOL/Analysis/Homeomorphism.thy
Author: LC Paulson (ported from HOL Light)
*)

section ‹Homeomorphism Theorems›

theory Homeomorphism
imports Path_Connected
begin

lemma homeomorphic_spheres':
fixes a ::"'a::euclidean_space" and b ::"'b::euclidean_space"
assumes "0 < δ" and dimeq: "DIM('a) = DIM('b)"
shows "(sphere a δ) homeomorphic (sphere b δ)"
proof -
obtain f :: "'a⇒'b" and g where "linear f" "linear g"
and fg: "⋀x. norm(f x) = norm x" "⋀y. norm(g y) = norm y" "⋀x. g(f x) = x" "⋀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="λx. b + f(x - a)" in exI)
apply(rule_tac x="λx. 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 ‹Homeomorphism of all convex compact sets with nonempty interior›

proposition ray_to_rel_frontier:
fixes a :: "'a::real_inner"
assumes "bounded S"
and a: "a ∈ rel_interior S"
and aff: "(a + l) ∈ affine hull S"
and "l ≠ 0"
obtains d where "0 < d" "(a + d *⇩R l) ∈ rel_frontier S"
"⋀e. ⟦0 ≤ e; e < d⟧ ⟹ (a + e *⇩R l) ∈ rel_interior S"
proof -
have aaff: "a ∈ affine hull S"
by (meson a hull_subset rel_interior_subset rev_subsetD)
let ?D = "{d. 0 < d ∧ a + d *⇩R l ∉ rel_interior S}"
obtain B where "B > 0" and B: "S ⊆ ball a B"
using bounded_subset_ballD [OF ‹bounded S›] by blast
have "a + (B / norm l) *⇩R l ∉ ball a B"
by (simp add: dist_norm ‹l ≠ 0›)
with B have "a + (B / norm l) *⇩R l ∉ rel_interior S"
using rel_interior_subset subsetCE by blast
with ‹B > 0› ‹l ≠ 0› have nonMT: "?D ≠ {}"
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: "⋀x. x ∈ rel_interior S ⟹
∃e>0. ∀x'∈affine hull S. dist x' x < e ⟶ x' ∈ rel_interior S"
using openin_rel_interior [of S] by (simp add: openin_euclidean_subtopology_iff)
define d where "d = Inf ?D"
obtain ε where "0 < ε" and ε: "⋀η. ⟦0 ≤ η; η < ε⟧ ⟹ (a + η *⇩R l) ∈ rel_interior S"
proof -
obtain e where "e>0"
and e: "⋀x'. x' ∈ affine hull S ⟹ dist x' a < e ⟹ x' ∈ rel_interior S"
using relin_Ex a by blast
show thesis
proof (rule_tac ε = "e / norm l" in that)
show "0 < e / norm l" by (simp add: ‹0 < e› ‹l ≠ 0›)
next
show "a + η *⇩R l ∈ rel_interior S" if "0 ≤ η" "η < e / norm l" for η
proof (rule e)
show "a + η *⇩R l ∈ affine hull S"
by (metis (no_types) add_diff_cancel_left' aff affine_affine_hull mem_affine_3_minus aaff)
show "dist (a + η *⇩R l) a < e"
using that by (simp add: ‹l ≠ 0› dist_norm pos_less_divide_eq)
qed
qed
qed
have inint: "⋀e. ⟦0 ≤ e; e < d⟧ ⟹ a + e *⇩R l ∈ 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 "ε ≤ d"
unfolding d_def
apply (rule cInf_greatest [OF nonMT])
using ε dual_order.strict_implies_order le_less_linear by blast
with ‹0 < ε› have "0 < d" by simp
have "a + d *⇩R l ∉ rel_interior S"
proof
assume adl: "a + d *⇩R l ∈ rel_interior S"
obtain e where "e > 0"
and e: "⋀x'. x' ∈ affine hull S ⟹ dist x' (a + d *⇩R l) < e ⟹ x' ∈ rel_interior S"
have "d + e / norm l ≤ Inf {d. 0 < d ∧ a + d *⇩R l ∉ rel_interior S}"
proof (rule cInf_greatest [OF nonMT], clarsimp)
fix x::real
assume "0 < x" and nonrel: "a + x *⇩R l ∉ rel_interior S"
show "d + e / norm l ≤ x"
proof (cases "x < d")
case True with inint nonrel ‹0 < x›
show ?thesis by auto
next
case False
then have dle: "x < d + e / norm l ⟹ dist (a + x *⇩R l) (a + d *⇩R l) < e"
by (simp add: field_simps ‹l ≠ 0›)
have ain: "a + x *⇩R l ∈ 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 ‹0 < e› ‹l ≠ 0› by (simp add: d_def [symmetric] divide_simps)
qed
moreover have "a + d *⇩R l ∈ closure S"
proof (clarsimp simp: closure_approachable)
fix η::real assume "0 < η"
have 1: "a + (d - min d (η / 2 / norm l)) *⇩R l ∈ S"
apply (rule subsetD [OF rel_interior_subset inint])
using ‹l ≠ 0› ‹0 < d› ‹0 < η› by auto
have "norm l * min d (η / (norm l * 2)) ≤ norm l * (η / (norm l * 2))"
by (metis min_def mult_left_mono norm_ge_zero order_refl)
also have "... < η"
using ‹l ≠ 0› ‹0 < η› by (simp add: divide_simps)
finally have 2: "norm l * min d (η / (norm l * 2)) < η" .
show "∃y∈S. dist y (a + d *⇩R l) < η"
apply (rule_tac x="a + (d - min d (η / 2 / norm l)) *⇩R l" in bexI)
using 1 2 ‹0 < d› ‹0 < η› apply (auto simp: algebra_simps)
done
qed
ultimately have infront: "a + d *⇩R l ∈ rel_frontier S"
show ?thesis
by (rule that [OF ‹0 < d› infront inint])
qed

corollary ray_to_frontier:
fixes a :: "'a::euclidean_space"
assumes "bounded S"
and a: "a ∈ interior S"
and "l ≠ 0"
obtains d where "0 < d" "(a + d *⇩R l) ∈ frontier S"
"⋀e. ⟦0 ≤ e; e < d⟧ ⟹ (a + e *⇩R l) ∈ interior S"
proof -
have "interior S = rel_interior S"
using a rel_interior_nonempty_interior by auto
then have "a ∈ rel_interior S"
using a by simp
then show ?thesis
apply (rule ray_to_rel_frontier [OF ‹bounded S› _ _ ‹l ≠ 0›])
using a affine_hull_nonempty_interior apply blast
by (simp add: ‹interior S = rel_interior S› 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 ∈ rel_interior S" and y: "y ∈ S" and xy: "x ≠ y"
obtains z where "z ∈ rel_frontier S" "y ∈ closed_segment x z"
"open_segment x z ⊆ rel_interior S"
proof -
have "x + (y - x) ∈ affine hull S"
using hull_inc [OF y] by auto
then obtain d where "0 < d" and df: "(x + d *⇩R (y-x)) ∈ rel_frontier S"
and di: "⋀e. ⟦0 ≤ e; e < d⟧ ⟹ (x + e *⇩R (y-x)) ∈ rel_interior S"
by (rule ray_to_rel_frontier [OF ‹bounded S› x]) (use xy in auto)
show ?thesis
proof
show "x + d *⇩R (y - x) ∈ rel_frontier S"
next
have "open_segment x y ⊆ rel_interior S"
using rel_interior_closure_convex_segment [OF ‹convex S› x] closure_subset y by blast
moreover have "x + d *⇩R (y - x) ∈ open_segment x y" if "d < 1"
using xy
apply (auto simp: in_segment)
apply (rule_tac x="d" in exI)
using ‹0 < d› that apply (auto simp: divide_simps algebra_simps)
done
ultimately have "1 ≤ d"
using df rel_frontier_def by fastforce
moreover have "x = (1 / d) *⇩R x + ((d - 1) / d) *⇩R x"
ultimately show "y ∈ closed_segment x (x + d *⇩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 *⇩R (y - x)) ⊆ rel_interior S"
apply (rule rel_interior_closure_convex_segment [OF ‹convex S› 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 ∈ rel_interior S"
and y: "y ∈ S" and xy: "~(x = y ∧ S = {x})"
obtains z where "z ∈ rel_frontier S" "y ∈ closed_segment x z"
"open_segment x z ⊆ rel_interior S"
proof (cases "x=y")
case True
with xy have "S ≠ {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 ≠ {a}"
proof (cases "S = {}")
case True  then show ?thesis  by simp
next
case False
then obtain z where "z ∈ 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 ∈ S" "w ≠ z"
using ‹z ∈ S› by blast
show ?thesis
proof
assume "rel_frontier S = {a}"
then consider "w ∉ rel_frontier S" | "z ∉ rel_frontier S"
using ‹w ≠ z› by auto
then show False
proof cases
case 1
then have w: "w ∈ rel_interior S"
using ‹w ∈ S› closure_subset rel_frontier_def by fastforce
have "w + (w - z) ∈ affine hull S"
by (metis ‹w ∈ S› ‹z ∈ S› affine_affine_hull hull_inc mem_affine_3_minus scaleR_one)
then obtain e where "0 < e" "(w + e *⇩R (w - z)) ∈ rel_frontier S"
using ‹w ≠ z›  ‹z ∈ S› by (metis assms ray_to_rel_frontier right_minus_eq w)
moreover obtain d where "0 < d" "(w + d *⇩R (z - w)) ∈ rel_frontier S"
using ray_to_rel_frontier [OF ‹bounded S› w, of "1 *⇩R (z - w)"]  ‹w ≠ z›  ‹z ∈ S›
ultimately have "d *⇩R (z - w) = e *⇩R (w - z)"
using ‹rel_frontier S = {a}› by force
moreover have "e ≠ -d "
using ‹0 < e› ‹0 < d› by force
ultimately show False
by (metis (no_types, lifting) ‹w ≠ z› 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 ∈ rel_interior S"
using ‹z ∈ S› closure_subset rel_frontier_def by fastforce
have "z + (z - w) ∈ affine hull S"
by (metis ‹z ∈ S› ‹w ∈ S› affine_affine_hull hull_inc mem_affine_3_minus scaleR_one)
then obtain e where "0 < e" "(z + e *⇩R (z - w)) ∈ rel_frontier S"
using ‹w ≠ z›  ‹w ∈ S› by (metis assms ray_to_rel_frontier right_minus_eq z)
moreover obtain d where "0 < d" "(z + d *⇩R (w - z)) ∈ rel_frontier S"
using ray_to_rel_frontier [OF ‹bounded S› z, of "1 *⇩R (w - z)"]  ‹w ≠ z›  ‹w ∈ S›
ultimately have "d *⇩R (w - z) = e *⇩R (z - w)"
using ‹rel_frontier S = {a}› by force
moreover have "e ≠ -d "
using ‹0 < e› ‹0 < d› by force
ultimately show False
by (metis (no_types, lifting) ‹w ≠ z› 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 ∈ rel_interior S"
and star: "⋀x. x ∈ S ⟹ open_segment 0 x ⊆ rel_interior S"
shows starlike_compact_projective1_0:
"S - rel_interior S homeomorphic sphere 0 1 ∩ affine hull S"
(is "?SMINUS homeomorphic ?SPHER")
and starlike_compact_projective2_0:
"S homeomorphic cball 0 1 ∩ affine hull S"
(is "S homeomorphic ?CBALL")
proof -
have starI: "(u *⇩R x) ∈ rel_interior S" if "x ∈ S" "0 ≤ u" "u < 1" for x u
proof (cases "x=0 ∨ 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 ∈ S"  using assms rel_interior_subset by auto
define proj where "proj ≡ λx::'a. x /⇩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: "⋀x y. (proj x = proj y ∧ norm x = norm y) ⟷ x = y"
by blast
have projI: "x ∈ affine hull S ⟹ proj x ∈ affine hull S" for x
by (metis ‹0 ∈ S› affine_hull_span_0 hull_inc span_mul proj_def)
have nproj1 [simp]: "x ≠ 0 ⟹ norm(proj x) = 1" for x
have proj0_iff [simp]: "proj x = 0 ⟷ x = 0" for x
have cont_proj: "continuous_on (UNIV - {0}) proj"
unfolding proj_def by (rule continuous_intros | force)+
have proj_spherI: "⋀x. ⟦x ∈ affine hull S; x ≠ 0⟧ ⟹ proj x ∈ ?SPHER"
have "bounded S" "closed S"
using ‹compact S› compact_eq_bounded_closed by blast+
have inj_on_proj: "inj_on proj (S - rel_interior S)"
proof
fix x y
assume x: "x ∈ S - rel_interior S"
and y: "y ∈ S - rel_interior S" and eq: "proj x = proj y"
then have xynot: "x ≠ 0" "y ≠ 0" "x ∈ S" "y ∈ S" "x ∉ rel_interior S" "y ∉ 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 /⇩R norm x = norm x *⇩R (x /⇩R norm x) /⇩R norm (norm x *⇩R (x /⇩R norm x))"
by force
then have "proj ((norm x / norm y) *⇩R y) = proj x"
by (metis (no_types) divide_inverse local.proj_def eq scaleR_scaleR)
then have [simp]: "(norm x / norm y) *⇩R y = x"
by (rule eqI) (simp add: ‹y ≠ 0›)
have no: "0 ≤ norm x / norm y" "norm x / norm y < 1"
using * by (auto simp: divide_simps)
then show "x = y"
using starI [OF ‹y ∈ S› no] xynot by auto
next
assume *: "norm x > norm y"
have "y /⇩R norm y = norm y *⇩R (y /⇩R norm y) /⇩R norm (norm y *⇩R (y /⇩R norm y))"
by force
then have "proj ((norm y / norm x) *⇩R x) = proj y"
by (metis (no_types) divide_inverse local.proj_def eq scaleR_scaleR)
then have [simp]: "(norm y / norm x) *⇩R x = y"
by (rule eqI) (simp add: ‹x ≠ 0›)
have no: "0 ≤ norm y / norm x" "norm y / norm x < 1"
using * by (auto simp: divide_simps)
then show "x = y"
using starI [OF ‹x ∈ S› no] xynot by auto
qed
qed
have "∃surf. homeomorphism (S - rel_interior S) ?SPHER proj surf"
proof (rule homeomorphism_compact)
show "compact (S - rel_interior S)"
using ‹compact S› 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) ⊆ ?SPHER"
using 0 by (force simp: hull_inc projI intro: nproj1)
show "?SPHER ⊆ proj ` (S - rel_interior S)"
proof (clarsimp simp: proj_def)
fix x
assume "x ∈ affine hull S" and nox: "norm x = 1"
then have "x ≠ 0" by auto
obtain d where "0 < d" and dx: "(d *⇩R x) ∈ rel_frontier S"
and ri: "⋀e. ⟦0 ≤ e; e < d⟧ ⟹ (e *⇩R x) ∈ rel_interior S"
using ray_to_rel_frontier [OF ‹bounded S› 0] ‹x ∈ affine hull S› ‹x ≠ 0› by auto
show "x ∈ (λx. x /⇩R norm x) ` (S - rel_interior S)"
apply (rule_tac x="d *⇩R x" in image_eqI)
using ‹0 < d›
using dx ‹closed S› 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: "⋀x. x ∈ ?SPHER ⟹ surf x ≠ 0"
by (metis "0" DiffE homeomorphism_def imageI surf)
have cont_nosp: "continuous_on (?SPHER) (λx. norm x *⇩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: "⋀x. ⟦norm x = 1; x ∈ affine hull S⟧ ⟹ surf (proj x) ∈ S"
by (metis (full_types) DiffE ‹0 ∈ S› homeomorphism_def image_eqI norm_zero proj_spherI real_vector.scale_zero_left scaleR_one surf)
have *: "∃y. norm y = 1 ∧ y ∈ affine hull S ∧ x = surf (proj y)"
if "x ∈ S" "x ∉ rel_interior S" for x
proof -
have "proj x ∈ ?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 ‹⋀x. x ∈ ?SPHER ⟹ surf x ≠ 0› hull_inc inverse_1 local.proj_def norm_sgn projI scaleR_one sgn_div_norm that(1))
qed
have surfp_notin: "⋀x. ⟦norm x = 1; x ∈ affine hull S⟧ ⟹ surf (proj x) ∉ rel_interior S"
by (metis (full_types) DiffE one_neq_zero homeomorphism_def image_eqI norm_zero proj_spherI surf)
have no_sp_im: "(λx. norm x *⇩R surf (proj x)) ` (?SPHER) = S - rel_interior S"
by (auto simp: surfpS image_def Bex_def surfp_notin *)
have inj_spher: "inj_on (λx. norm x *⇩R surf (proj x)) ?SPHER"
proof
fix x y
assume xy: "x ∈ ?SPHER" "y ∈ ?SPHER"
and eq: " norm x *⇩R surf (proj x) = norm y *⇩R surf (proj y)"
then have "norm x = 1" "norm y = 1" "x ∈ affine hull S" "y ∈ 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: "⋀a x. 0 < a ⟹ proj (a *⇩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: "⋀x. x ∈ S ⟹ norm x ≤ B"
by (metis compact_imp_bounded ‹compact S› bounded_pos_less less_eq_real_def)
have cont_nosp: "continuous (at x within ?CBALL) (λx. norm x *⇩R surf (proj x))"
if "norm x ≤ 1" "x ∈ 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: ‹B > 0›)
done
next
case False
then have "∀⇩F x in at x. (x ∈ affine hull S - {0}) = (x ∈ 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) (λx. 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 (λx. norm x *⇩R ((surf o proj) x))"
have "norm y *⇩R surf (proj y) ∈ S"  if "y ∈ cball 0 1" and yaff: "y ∈ affine hull S" for y
proof (cases "y=0")
case True then show ?thesis
by (simp add: ‹0 ∈ S›)
next
case False
then have "norm y *⇩R surf (proj y) = norm y *⇩R surf (proj (y /⇩R norm y))"
have "norm y ≤ 1" using that by simp
have "surf (proj (y /⇩R norm y)) ∈ S"
apply (rule surfpS)
using proj_def projI yaff
by (auto simp: False)
then have "surf (proj y) ∈ S"
then show "norm y *⇩R surf (proj y) ∈ S"
by (metis dual_order.antisym le_less_linear norm_ge_zero rel_interior_subset scaleR_one
starI subset_eq ‹norm y ≤ 1›)
qed
moreover have "x ∈ (λx. norm x *⇩R surf (proj x)) ` (?CBALL)" if "x ∈ 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 *⇩R proj x = x"
have affineI: "(1 / norm (surf (proj x))) *⇩R x ∈ affine hull S"
by (metis ‹0 ∈ S› affine_hull_span_0 hull_inc span_clauses(4) that)
have sproj_nz: "surf (proj x) ≠ 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: "⋀a r. r *⇩R proj a = (r / norm a) *⇩R a"
ultimately have "(norm (surf (proj x)) / norm x) *⇩R x ∉ 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) ≥ 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 ≤ norm (surf (proj x))"
show ?thesis
apply (rule_tac x="inverse(norm(surf (proj x))) *⇩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: "(λx. norm x *⇩R surf (proj x)) ` ?CBALL = S"
by blast
have inj_cball: "inj_on (λx. norm x *⇩R surf (proj x)) ?CBALL"
proof
fix x y
assume "x ∈ ?CBALL" "y ∈ ?CBALL"
and eq: "norm x *⇩R surf (proj x) = norm y *⇩R surf (proj y)"
then have x: "x ∈ affine hull S" and y: "y ∈ affine hull S"
using 0 by auto
show "x = y"
proof (cases "x=0 ∨ 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 ‹x ∈ affine hull S› ‹y ∈ affine hull S› 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 ∈ rel_interior S"
and star: "⋀x. x ∈ S ⟹ open_segment a x ⊆ rel_interior S"
shows starlike_compact_projective1:
"S - rel_interior S homeomorphic sphere a 1 ∩ affine hull S"
and starlike_compact_projective2:
"S homeomorphic cball a 1 ∩ affine hull S"
proof -
have 1: "compact (op+ (-a) ` S)" by (meson assms compact_translation)
have 2: "0 ∈ rel_interior (op+ (-a) ` S)"
have 3: "open_segment 0 x ⊆ rel_interior (op+ (-a) ` S)" if "x ∈ (op+ (-a) ` S)" for x
proof -
have "x+a ∈ S" using that by auto
then have "open_segment a (x+a) ⊆ 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 ∩ affine hull (op+ (-a) ` S)"
by (rule starlike_compact_projective1_0 [OF 1 2 3])
also have "... = op+ (-a) ` (sphere a 1 ∩ affine hull S)"
by (metis affine_hull_translation left_minus sphere_translation translation_Int)
also have "... homeomorphic sphere a 1 ∩ affine hull S"
using homeomorphic_translation homeomorphic_sym by blast
finally show "S - rel_interior S homeomorphic sphere a 1 ∩ affine hull S" .

have "S homeomorphic (op+ (-a) ` S)"
by (metis homeomorphic_translation)
also have "... homeomorphic cball 0 1 ∩ affine hull (op+ (-a) ` S)"
by (rule starlike_compact_projective2_0 [OF 1 2 3])
also have "... = op+ (-a) ` (cball a 1 ∩ affine hull S)"
by (metis affine_hull_translation left_minus cball_translation translation_Int)
also have "... homeomorphic cball a 1 ∩ affine hull S"
using homeomorphic_translation homeomorphic_sym by blast
finally show "S homeomorphic cball a 1 ∩ affine hull S" .
qed

corollary starlike_compact_projective_special:
assumes "compact S"
and cb01: "cball (0::'a::euclidean_space) 1 ⊆ S"
and scale: "⋀x u. ⟦x ∈ S; 0 ≤ u; u < 1⟧ ⟹ u *⇩R x ∈ S - frontier S"
shows "S homeomorphic (cball (0::'a::euclidean_space) 1)"
proof -
have "ball 0 1 ⊆ interior S"
using cb01 interior_cball interior_mono by blast
then have 0: "0 ∈ 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 ‹ball 0 1 ⊆ interior S› by (auto intro!: affine_hull_nonempty_interior)
have star: "open_segment 0 x ⊆ rel_interior S" if "x ∈ S" for x
proof
fix p assume "p ∈ open_segment 0 x"
then obtain u where "x ≠ 0" and u: "0 ≤ u" "u < 1" and p: "u *⇩R x = p"
by (auto simp: in_segment)
then show "p ∈ 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 ‹compact S› 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) ∧
S homeomorphic T"
proof (cases "rel_interior S = {} ∨ rel_interior T = {}")
case True
then show ?thesis
by (metis Diff_empty affeq ‹convex S› ‹convex T› aff_dim_empty homeomorphic_empty rel_interior_eq_empty aff_dim_empty)
next
case False
then obtain a b where a: "a ∈ rel_interior S" and b: "b ∈ rel_interior T" by auto
have starS: "⋀x. x ∈ S ⟹ open_segment a x ⊆ rel_interior S"
using rel_interior_closure_convex_segment
a ‹convex S› closure_subset subsetCE by blast
have starT: "⋀x. x ∈ T ⟹ open_segment b x ⊆ rel_interior T"
using rel_interior_closure_convex_segment
b ‹convex T› closure_subset subsetCE by blast
let ?aS = "op+ (-a) ` S" and ?bT = "op+ (-b) ` T"
have 0: "0 ∈ affine hull ?aS" "0 ∈ 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: "⋀x. x ∈ span ?aS ⟹ norm(f x) = norm x"
and gno: "⋀x. x ∈ span ?bT ⟹ norm(g x) = norm x"
and gf: "⋀x. x ∈ span ?aS ⟹ g(f x) = x"
and fg: "⋀x. x ∈ span ?bT ⟹ f(g x) = x"
by (rule isometries_subspaces) blast
have [simp]: "continuous_on A f" for A
using ‹linear f› linear_conv_bounded_linear linear_continuous_on by blast
have [simp]: "continuous_on B g" for B
using ‹linear g› 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 ∩ affine hull S"
by (rule starlike_compact_projective2 [OF ‹compact S› a starS])
also have "... homeomorphic op+ (-a) ` (cball a 1 ∩ affine hull S)"
by (metis homeomorphic_translation)
also have "... = cball 0 1 ∩ op+ (-a) ` (affine hull S)"
by (auto simp: dist_norm)
also have "... = cball 0 1 ∩ span ?aS"
using eqspanS affine_hull_translation by blast
also have "... homeomorphic cball 0 1 ∩ span ?bT"
proof (rule homeomorphicI [where f=f and g=g])
show fim1: "f ` (cball 0 1 ∩ span ?aS) = cball 0 1 ∩ 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 ∩ span ?bT) = cball 0 1 ∩ 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 ∩ op+ (-b) ` (affine hull T)"
using eqspanT affine_hull_translation by blast
also have "... = op+ (-b) ` (cball b 1 ∩ affine hull T)"
by (auto simp: dist_norm)
also have "... homeomorphic (cball b 1 ∩ affine hull T)"
by (metis homeomorphic_translation homeomorphic_sym)
also have "... homeomorphic T"
by (metis starlike_compact_projective2 [OF ‹compact T› b starT] homeomorphic_sym)
finally have 1: "S homeomorphic T" .

have "S - rel_interior S homeomorphic sphere a 1 ∩ affine hull S"
by (rule starlike_compact_projective1 [OF ‹compact S› a starS])
also have "... homeomorphic op+ (-a) ` (sphere a 1 ∩ affine hull S)"
by (metis homeomorphic_translation)
also have "... = sphere 0 1 ∩ op+ (-a) ` (affine hull S)"
by (auto simp: dist_norm)
also have "... = sphere 0 1 ∩ span ?aS"
using eqspanS affine_hull_translation by blast
also have "... homeomorphic sphere 0 1 ∩ span ?bT"
proof (rule homeomorphicI [where f=f and g=g])
show fim1: "f ` (sphere 0 1 ∩ span ?aS) = sphere 0 1 ∩ 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 ∩ span ?bT) = sphere 0 1 ∩ 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 ∩ op+ (-b) ` (affine hull T)"
using eqspanT affine_hull_translation by blast
also have "... = op+ (-b) ` (sphere b 1 ∩ affine hull T)"
by (auto simp: dist_norm)
also have "... homeomorphic (sphere b 1 ∩ affine hull T)"
by (metis homeomorphic_translation homeomorphic_sym)
also have "... homeomorphic T - rel_interior T"
by (metis starlike_compact_projective1 [OF ‹compact T› 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‹Homeomorphisms between punctured spheres and affine sets›
text‹Including the famous stereoscopic projection of the 3-D sphere to the complex plane›

text‹The special case with centre 0 and radius 1›
lemma homeomorphic_punctured_affine_sphere_affine_01:
assumes "b ∈ sphere 0 1" "affine T" "0 ∈ T" "b ∈ T" "affine p"
and affT: "aff_dim T = aff_dim p + 1"
shows "(sphere 0 1 ∩ T) - {b} homeomorphic p"
proof -
have [simp]: "norm b = 1" "b∙b = 1"
using assms by (auto simp: norm_eq_1)
have [simp]: "T ∩ {v. b∙v = 0} ≠ {}"
using ‹0 ∈ T› by auto
have [simp]: "¬ T ⊆ {v. b∙v = 0}"
using ‹norm b = 1› ‹b ∈ T› by auto
define f where "f ≡ λx. 2 *⇩R b + (2 / (1 - b∙x)) *⇩R (x - b)"
define g where "g ≡ λy. b + (4 / (norm y ^ 2 + 4)) *⇩R (y - 2 *⇩R b)"
have [simp]: "⋀x. ⟦x ∈ T; b∙x = 0⟧ ⟹ f (g x) = x"
have no: "⋀x. ⟦norm x = 1; b∙x ≠ 1⟧ ⟹ (norm (f x))⇧2 = 4 * (1 + b∙x) / (1 - b∙x)"
done
have [simp]: "⋀u::real. 8 + u * (u * 8) = u * 16 ⟷ u=1"
by algebra
have [simp]: "⋀x. ⟦norm x = 1; b ∙ x ≠ 1⟧ ⟹ g (f x) = x"
unfolding g_def no by (auto simp: f_def divide_simps)
have [simp]: "⋀x. ⟦x ∈ T; b ∙ x = 0⟧ ⟹ norm (g x) = 1"
unfolding g_def
apply (rule power2_eq_imp_eq)
done
have [simp]: "⋀x. ⟦x ∈ T; b ∙ x = 0⟧ ⟹ b ∙ g x ≠ 1"
unfolding g_def
apply (auto simp: algebra_simps)
done
have "subspace T"
have [simp]: "⋀x. ⟦x ∈ T; b ∙ x = 0⟧ ⟹ g x ∈ T"
unfolding g_def
by (blast intro: ‹subspace T› ‹b ∈ T› subspace_add subspace_mul subspace_diff)
have "f ` {x. norm x = 1 ∧ b∙x ≠ 1} ⊆ {x. b∙x = 0}"
unfolding f_def using ‹norm b = 1› norm_eq_1
by (force simp: field_simps inner_add_right inner_diff_right)
moreover have "f ` T ⊆ T"
unfolding f_def using assms
apply (auto simp: field_simps inner_add_right inner_diff_right)
moreover have "{x. b∙x = 0} ∩ T ⊆ f ` ({x. norm x = 1 ∧ b∙x ≠ 1} ∩ T)"
apply clarify
apply (rule_tac x = "g x" in image_eqI, auto)
done
ultimately have imf: "f ` ({x. norm x = 1 ∧ b∙x ≠ 1} ∩ T) = {x. b∙x = 0} ∩ T"
by blast
have no4: "⋀y. b∙y = 0 ⟹ norm ((y∙y + 4) *⇩R b + 4 *⇩R (y - 2 *⇩R b)) = y∙y + 4"
apply (rule power2_eq_imp_eq)
apply (auto simp: power2_eq_square algebra_simps inner_commute)
done
have [simp]: "⋀x. ⟦norm x = 1; b ∙ x ≠ 1⟧ ⟹ b ∙ f x = 0"
by (simp add: f_def algebra_simps divide_simps)
have [simp]: "⋀x. ⟦x ∈ T; norm x = 1; b ∙ x ≠ 1⟧ ⟹ f x ∈ T"
unfolding f_def
by (blast intro: ‹subspace T› ‹b ∈ T› subspace_add subspace_mul subspace_diff)
have "g ` {x. b∙x = 0} ⊆ {x. norm x = 1 ∧ b∙x ≠ 1}"
unfolding g_def
apply (auto simp: algebra_simps)
done
moreover have "g ` T ⊆ T"
unfolding g_def
by (blast intro: ‹subspace T› ‹b ∈ T› subspace_add subspace_mul subspace_diff)
moreover have "{x. norm x = 1 ∧ b∙x ≠ 1} ∩ T ⊆ g ` ({x. b∙x = 0} ∩ T)"
apply clarify
apply (rule_tac x = "f x" in image_eqI, auto)
done
ultimately have img: "g ` ({x. b∙x = 0} ∩ T) = {x. norm x = 1 ∧ b∙x ≠ 1} ∩ T"
by blast
have aff: "affine ({x. b∙x = 0} ∩ T)"
by (blast intro: affine_hyperplane assms)
have contf: "continuous_on ({x. norm x = 1 ∧ b∙x ≠ 1} ∩ T) f"
unfolding f_def by (rule continuous_intros | force)+
have contg: "continuous_on ({x. b∙x = 0} ∩ T) g"
unfolding g_def by (rule continuous_intros | force simp: add_nonneg_eq_0_iff)+
have "(sphere 0 1 ∩ T) - {b} = {x. norm x = 1 ∧ (b∙x ≠ 1)} ∩ T"
using  ‹norm b = 1› by (auto simp: norm_eq_1) (metis vector_eq  ‹b∙b = 1›)
also have "... homeomorphic {x. b∙x = 0} ∩ T"
by (rule homeomorphicI [OF imf img contf contg]) auto
also have "... homeomorphic p"
apply (rule homeomorphic_affine_sets [OF aff ‹affine p›])
apply (simp add: Int_commute aff_dim_affine_Int_hyperplane [OF ‹affine T›] affT)
done
finally show ?thesis .
qed

theorem homeomorphic_punctured_affine_sphere_affine:
fixes a :: "'a :: euclidean_space"
assumes "0 < r" "b ∈ sphere a r" "affine T" "a ∈ T" "b ∈ T" "affine p"
and aff: "aff_dim T = aff_dim p + 1"
shows "((sphere a r ∩ T) - {b}) homeomorphic p"
proof -
have "a ≠ b" using assms by auto
then have inj: "inj (λx::'a. x /⇩R norm (a - b))"
have "((sphere a r ∩ T) - {b}) homeomorphic
op+ (-a) ` ((sphere a r ∩ T) - {b})"
by (rule homeomorphic_translation)
also have "... homeomorphic op *⇩R (inverse r) ` op + (- a) ` (sphere a r ∩ T - {b})"
by (metis ‹0 < r› homeomorphic_scaling inverse_inverse_eq inverse_zero less_irrefl)
also have "... = sphere 0 1 ∩ (op *⇩R (inverse r) ` op + (- a) ` T) - {(b - a) /⇩R r}"
using assms by (auto simp: dist_norm norm_minus_commute divide_simps)
also have "... homeomorphic p"
apply (rule homeomorphic_punctured_affine_sphere_affine_01)
using assms
apply (auto simp: dist_norm norm_minus_commute affine_scaling affine_translation [symmetric] aff_dim_translation_eq inj)
done
finally show ?thesis .
qed

proposition homeomorphic_punctured_sphere_affine_gen:
fixes a :: "'a :: euclidean_space"
assumes "convex S" "bounded S" and a: "a ∈ rel_frontier S"
and "affine T" and affS: "aff_dim S = aff_dim T + 1"
shows "rel_frontier S - {a} homeomorphic T"
proof -
have "S ≠ {}" using assms by auto
obtain U :: "'a set" where "affine U" and affdS: "aff_dim U = aff_dim S"
using choose_affine_subset [OF affine_UNIV aff_dim_geq]
by (meson aff_dim_affine_hull affine_affine_hull)
have "convex U"
by (simp add: affine_imp_convex ‹affine U›)
have "U ≠ {}"
by (metis ‹S ≠ {}› ‹aff_dim U = aff_dim S› aff_dim_empty)
then obtain z where "z ∈ U"
by auto
then have bne: "ball z 1 ∩ U ≠ {}" by force
have [simp]: "aff_dim(ball z 1 ∩ U) = aff_dim U"
using aff_dim_convex_Int_open [OF ‹convex U› open_ball] bne
have "rel_frontier S homeomorphic rel_frontier (ball z 1 ∩ U)"
apply (rule homeomorphic_rel_frontiers_convex_bounded_sets)
apply (auto simp: ‹affine U› affine_imp_convex convex_Int affdS assms)
done
also have "... = sphere z 1 ∩ U"
using convex_affine_rel_frontier_Int [of "ball z 1" U]
by (simp add: ‹affine U› bne)
finally obtain h k where him: "h ` rel_frontier S = sphere z 1 ∩ U"
and kim: "k ` (sphere z 1 ∩ U) = rel_frontier S"
and hcon: "continuous_on (rel_frontier S) h"
and kcon: "continuous_on (sphere z 1 ∩ U) k"
and kh:  "⋀x. x ∈ rel_frontier S ⟹ k(h(x)) = x"
and hk:  "⋀y. y ∈ sphere z 1 ∩ U ⟹ h(k(y)) = y"
unfolding homeomorphic_def homeomorphism_def by auto
have "rel_frontier S - {a} homeomorphic (sphere z 1 ∩ U) - {h a}"
proof (rule homeomorphicI [where f=h and g=k])
show h: "h ` (rel_frontier S - {a}) = sphere z 1 ∩ U - {h a}"
using him a kh by auto metis
show "k ` (sphere z 1 ∩ 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 ‹affine T›  ‹affine U› ‹z ∈ U›)
finally show ?thesis .
qed

lemma homeomorphic_punctured_sphere_affine:
fixes a :: "'a :: euclidean_space"
assumes "0 < r" and b: "b ∈ sphere a r"
and "affine T" and affS: "aff_dim T + 1 = DIM('a)"
shows "(sphere a r - {b}) homeomorphic T"
using homeomorphic_punctured_sphere_affine_gen [of "cball a r" b T]
assms aff_dim_cball by force

corollary homeomorphic_punctured_sphere_hyperplane:
fixes a :: "'a :: euclidean_space"
assumes "0 < r" and b: "b ∈ sphere a r"
and "c ≠ 0"
shows "(sphere a r - {b}) homeomorphic {x::'a. c ∙ x = d}"
apply (rule homeomorphic_punctured_sphere_affine)
using assms
apply (auto simp: affine_hyperplane of_nat_diff)
done

text‹ 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.›

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 ≠ {}" "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 ∈ S" by auto
obtain i::'n where i: "i ∈ Basis" "i ≠ 0"
using SOME_Basis Basis_zero by force
have "0 ∈ affine hull (op + (- a) ` S)"
by (simp add: ‹a ∈ S› 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 ⊆ {x. i ∙ x = 0}"
and dimT: "dim T = dim (op + (- a) ` S)"
apply (rule choose_subspace_of_subspace [of "dim (op + (- a) ` S)" "{x::'n. i ∙ x = 0}"])
apply (simp add: dim_hyperplane [OF ‹i ≠ 0›])
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]:  "⋀x. x ∈ span (op + (- a) ` S) ⟹ k(h x) = x"
and kinv [simp]:  "⋀x. x ∈ T ⟹ h(k x) = x"
apply (rule isometries_subspaces [OF _ ‹subspace T›])
apply (auto simp: dimT)
done
have hcont: "continuous_on A h" and kcont: "continuous_on B k" for A B
using ‹linear h› ‹linear k› linear_continuous_on linear_conv_bounded_linear by blast+
have ihhhh[simp]: "⋀x. x ∈ S ⟹ i ∙ h (x - a) = 0"
using Tsub [THEN subsetD] heq span_inc by fastforce
have "sphere 0 1 - {i} homeomorphic {x. i ∙ 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 ∙ x = 0} f g"
by (force simp: homeomorphic_def)
have "h ` op + (- a) ` S ⊆ T"
using heq span_clauses(1) span_linear_image by blast
then have "g ` h ` op + (- a) ` S ⊆ g ` {x. i ∙ x = 0}"
using Tsub by (simp add: image_mono)
also have "... ⊆ sphere 0 1 - {i}"
by (simp add: fg [unfolded homeomorphism_def])
finally have gh_sub_sph: "(g ∘ h) ` op + (- a) ` S ⊆ sphere 0 1 - {i}"
by (metis image_comp)
then have gh_sub_cb: "(g ∘ h) ` op + (- a) ` S ⊆ cball 0 1"
by (metis Diff_subset order_trans sphere_cball)
have [simp]: "⋀u. u ∈ S ⟹ norm (g (h (u - a))) = 1"
using gh_sub_sph [THEN subsetD] by (auto simp: o_def)
have ghcont: "continuous_on (op + (- a) ` S) (λx. g (h x))"
apply (rule continuous_on_compose2 [OF homeomorphism_cont2 [OF fg] hcont], force)
done
have kfcont: "continuous_on ((g ∘ h ∘ op + (- a)) ` S) (λx. 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: "… homeomorphic (g ∘ h) ` op + (- a) ` S"
apply (rule_tac x="g ∘ h" in exI)
apply (rule_tac x="k ∘ 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 ∘ h) ` op + (- a) ` S" .
show ?thesis
apply (rule_tac U = "ball 0 1 ∪ 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‹Locally compact sets in an open set›

text‹ 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.›

lemma locally_compact_open_Int_closure:
fixes S :: "'a :: metric_space set"
assumes "locally compact S"
obtains T where "open T" "S = T ∩ closure S"
proof -
have "∀x∈S. ∃T v u. u = S ∩ T ∧ x ∈ u ∧ u ⊆ v ∧ v ⊆ S ∧ open T ∧ compact v"
by (metis assms locally_compact openin_open)
then obtain t v where
tv: "⋀x. x ∈ S
⟹ v x ⊆ S ∧ open (t x) ∧ compact (v x) ∧ (∃u. x ∈ u ∧ u ⊆ v x ∧ u = S ∩ t x)"
by metis
then have o: "open (UNION S t)"
by blast
have "S = ⋃ (v ` S)"
using tv by blast
also have "... = UNION S t ∩ closure S"
proof
show "UNION S v ⊆ UNION S t ∩ closure S"
apply safe
apply (metis Int_iff subsetD UN_iff tv)
apply (simp add: closure_def rev_subsetD tv)
done
have "t x ∩ closure S ⊆ v x" if "x ∈ S" for x
proof -
have "t x ∩ closure S ⊆ closure (t x ∩ S)"
by (simp add: open_Int_closure_subset that tv)
also have "... ⊆ v x"
by (metis Int_commute closure_minimal compact_imp_closed that tv)
finally show ?thesis .
qed
then show "UNION S t ∩ closure S ⊆ UNION S v"
by blast
qed
finally have e: "S = UNION S t ∩ 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 ⇒ 'a :: euclidean_space × '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 × {0}" and f = "λx. (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 ∩ closure S = S"
by (metis locally_compact_open_Int_closure [OF assms])
with False have Ucomp: "-U ≠ {}"
using closure_eq by auto
have [simp]: "closure (- U) = -U"
by (simp add: ‹open U› closed_Compl)
define f :: "'a ⇒ 'a × 'b" where "f ≡ λx. (x, One /⇩R setdist {x} (- U))"
have "continuous_on U (λx. (x, One /⇩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 ((λx. setdist {x} (- U)) o fst) z" for z :: "'a × 'b"
by (rule isCont_o continuous_intros continuous_at_setdist)+
have setdist1D: "setdist {a} (- U) *⇩R b = One ⟹ setdist {a} (- U) ≠ 0" for a::'a and b::'b
by force
have *: "r *⇩R b = One ⟹ b = (1 / r) *⇩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) *⇩R snd z) ∈ {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 ⟷ (∃U u. closed U ∧ open u ∧ S = U ∩ 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 ⇒ 'b::euclidean_space"
and g :: "'b ⇒ 'a*real"
and j :: 'b
where "linear f" "linear g" "⋀z. g (f z) = z" "j ∈ Basis" "⋀x. f(x,0) ∙ j = 0"
proof -
let ?B = "Basis :: ('a*real) set"
have b01: "(0,1) ∈ ?B"
have "DIM('a * real) ≤ DIM('b)"
then obtain basf :: "'a*real ⇒ 'b" where sbf: "basf ` ?B ⊆ Basis" and injbf: "inj_on basf Basis"
by (metis finite_Basis card_le_inj)
define basg:: "'b ⇒ 'a * real" where
"basg ≡ λi. if i ∈ basf ` Basis then inv_into Basis basf i else (0,1)"
have bgf[simp]: "basg (basf i) = i" if "i ∈ Basis" for i
using inv_into_f_f injbf that by (force simp: basg_def)
have sbg: "basg ` Basis ⊆ ?B"
by (force simp: basg_def injbf b01)
define f :: "'a*real ⇒ 'b" where "f ≡ λu. ∑j∈Basis. (u ∙ basg j) *⇩R j"
define g :: "'b ⇒ 'a*real" where "g ≡ λz. (∑i∈Basis. (z ∙ basf i) *⇩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 *: "(∑a ∈ Basis. a ∙ basf b * (x ∙ basg a)) = x ∙ b" if "b ∈ 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) ∈ Basis"
using b01 sbf by auto
then show "f(x,0) ∙ 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 ⇒ 'b" and g :: "'b ⇒ 'a*real"
where "linear f" "linear g" and gf [simp]: "⋀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 ‹inj f› ‹linear f› homeomorphism_cont2 linear_homeomorphism_image)
using ‹linear g› 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 ‹closed U› ‹linear f› ‹inj f›], assumption)
done
qed

lemma homeomorphic_convex_compact_lemma:
fixes S :: "'a::euclidean_space set"
assumes "convex S"
and "compact S"
and "cball 0 1 ⊆ S"
shows "S homeomorphic (cball (0::'a) 1)"
proof (rule starlike_compact_projective_special[OF assms(2-3)])
fix x u
assume "x ∈ S" and "0 ≤ u" and "u < (1::real)"
have "open (ball (u *⇩R x) (1 - u))"
by (rule open_ball)
moreover have "u *⇩R x ∈ ball (u *⇩R x) (1 - u)"
unfolding centre_in_ball using ‹u < 1› by simp
moreover have "ball (u *⇩R x) (1 - u) ⊆ S"
proof
fix y
assume "y ∈ ball (u *⇩R x) (1 - u)"
then have "dist (u *⇩R x) y < 1 - u"
unfolding mem_ball .
with ‹u < 1› have "inverse (1 - u) *⇩R (y - u *⇩R x) ∈ cball 0 1"
by (simp add: dist_norm inverse_eq_divide norm_minus_commute)
with assms(3) have "inverse (1 - u) *⇩R (y - u *⇩R x) ∈ S" ..
with assms(1) have "(1 - u) *⇩R ((y - u *⇩R x) /⇩R (1 - u)) + u *⇩R x ∈ S"
using ‹x ∈ S› ‹0 ≤ u› ‹u < 1› [THEN less_imp_le] by (rule convexD_alt)
then show "y ∈ S" using ‹u < 1›
by simp
qed
ultimately have "u *⇩R x ∈ interior S" ..
then show "u *⇩R x ∈ 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 ≠ {}"
and "e > 0"
shows "S homeomorphic (cball (b::'a) e)"
proof -
obtain a where "a ∈ interior S"
using assms(3) by auto
then obtain d where "d > 0" and d: "cball a d ⊆ S"
unfolding mem_interior_cball by auto
let ?d = "inverse d" and ?n = "0::'a"
have "cball ?n 1 ⊆ (λx. inverse d *⇩R (x - a)) ` S"
apply rule
apply (rule_tac x="d *⇩R x + a" in image_eqI)
defer
apply (rule d[unfolded subset_eq, rule_format])
using ‹d > 0›
unfolding mem_cball dist_norm
done
then have "(λx. inverse d *⇩R (x - a)) ` S homeomorphic cball ?n 1"
using homeomorphic_convex_compact_lemma[of "(λx. ?d *⇩R -a + ?d *⇩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 *⇩R -a"]])
using ‹d>0› ‹e>0›
done
qed

corollary homeomorphic_convex_compact:
fixes S :: "'a::euclidean_space set"
and T :: "'a set"
assumes "convex S" "compact S" "interior S ≠ {}"
and "convex T" "compact T" "interior T ≠ {}"
shows "S homeomorphic T"
using assms
by (meson zero_less_one homeomorphic_trans homeomorphic_convex_compact_cball homeomorphic_sym)

subsection‹Covering spaces and lifting results for them›

definition covering_space
:: "'a::topological_space set ⇒ ('a ⇒ 'b) ⇒ 'b::topological_space set ⇒ bool"
where
"covering_space c p S ≡
continuous_on c p ∧ p ` c = S ∧
(∀x ∈ S. ∃T. x ∈ T ∧ openin (subtopology euclidean S) T ∧
(∃v. ⋃v = {x. x ∈ c ∧ p x ∈ T} ∧
(∀u ∈ v. openin (subtopology euclidean c) u) ∧
pairwise disjnt v ∧
(∀u ∈ v. ∃q. homeomorphism u T p q)))"

lemma covering_space_imp_continuous: "covering_space c p S ⟹ continuous_on c p"

lemma covering_space_imp_surjective: "covering_space c p S ⟹ p ` c = S"

lemma homeomorphism_imp_covering_space: "homeomorphism S T f g ⟹ 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 ∈ c"
obtains T u q where "x ∈ T" "openin (subtopology euclidean c) T"
"p x ∈ 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 ∈ S"
obtains x T u q where "p x = y"
"x ∈ T" "openin (subtopology euclidean c) T"
"y ∈ u" "openin (subtopology euclidean S) u"
"homeomorphism T u p q"
proof -
obtain x where "p x = y" "x ∈ c"
using assms covering_space_imp_surjective by blast
show ?thesis
apply (rule covering_space_local_homeomorphism [OF p ‹x ∈ c›])
using that ‹p x = y› 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:
"⋀x. x ∈ S ⟹
∃X VS. x ∈ X ∧ openin (subtopology euclidean S) X ∧
⋃VS = {x. x ∈ c ∧ p x ∈ X} ∧
(∀u ∈ VS. openin (subtopology euclidean c) u) ∧
pairwise disjnt VS ∧
(∀u ∈ VS. ∃q. homeomorphism u X p q)"
using p by (auto simp: covering_space_def)
have "T ⊆ c"  by (metis openin_euclidean_subtopology_iff T)
have "∃X. openin (subtopology euclidean S) X ∧ y ∈ X ∧ X ⊆ p ` T"
if "y ∈ p ` T" for y
proof -
have "y ∈ S" using ‹T ⊆ c› pce that by blast
obtain U VS where "y ∈ U" and U: "openin (subtopology euclidean S) U"
and VS: "⋃VS = {x. x ∈ c ∧ p x ∈ U}"
and openVS: "∀V ∈ VS. openin (subtopology euclidean c) V"
and homVS: "⋀V. V ∈ VS ⟹ ∃q. homeomorphism V U p q"
using covs [OF ‹y ∈ S›] by auto
obtain x where "x ∈ c" "p x ∈ U" "x ∈ T" "p x = y"
apply simp
using T [unfolded openin_euclidean_subtopology_iff] ‹y ∈ U› ‹y ∈ p ` T› by blast
with VS obtain V where "x ∈ V" "V ∈ VS" by auto
then obtain q where q: "homeomorphism V U p q" using homVS by blast
then have ptV: "p ` (T ∩ V) = U ∩ {z. q z ∈ (T ∩ V)}"
using VS ‹V ∈ VS› by (auto simp: homeomorphism_def)
have ocv: "openin (subtopology euclidean c) V"
by (simp add: ‹V ∈ VS› openVS)
have "openin (subtopology euclidean U) {z ∈ U. q z ∈ T ∩ 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 ∩ {z. q z ∈ T ∩ V})"
using openin_trans [of U] by (simp add: Collect_conj_eq U)
show ?thesis
apply (rule_tac x = "p ` (T ∩ V)" in exI)
apply (rule conjI)
apply (simp only: ptV os)
using ‹p x = y› ‹x ∈ V› ‹x ∈ T› apply blast
done
qed
with openin_subopen show ?thesis by blast
qed

lemma covering_space_lift_unique_gen:
fixes f :: "'a::topological_space ⇒ 'b::topological_space"
fixes g1 :: "'a ⇒ '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 ⊆ S"
and g1: "continuous_on T g1"  "g1 ` T ⊆ c"
and fg1: "⋀x. x ∈ T ⟹ f x = p(g1 x)"
and g2: "continuous_on T g2"  "g2 ` T ⊆ c"
and fg2: "⋀x. x ∈ T ⟹ f x = p(g2 x)"
and u_compt: "U ∈ components T" and "a ∈ U" "x ∈ U"
shows "g1 x = g2 x"
proof -
have "U ⊆ T" by (rule in_components_subset [OF u_compt])
define G12 where "G12 ≡ {x ∈ 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 ‹U ⊆ T› 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 ∈ U" "g1 z - g2 z = 0"
obtain v w q where "g1 z ∈ v" and ocv: "openin (subtopology euclidean c) v"
and "p (g1 z) ∈ 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 ‹U ⊆ T› ‹z ∈ U› g1(2) apply blast+
done
have "g2 z ∈ v" using ‹g1 z ∈ v› z by auto
have gg: "{x ∈ U. g x ∈ v} = {x ∈ U. g x ∈ (v ∩ g ` U)}" for g
by auto
have "openin (subtopology euclidean (g1 ` U)) (v ∩ g1 ` U)"
using ocv ‹U ⊆ T› g1 by (fastforce simp add: openin_open)
then have 1: "openin (subtopology euclidean U) {x ∈ U. g1 x ∈ v}"
unfolding gg by (blast intro: contu continuous_on_open [THEN iffD1, rule_format])
have "openin (subtopology euclidean (g2 ` U)) (v ∩ g2 ` U)"
using ocv ‹U ⊆ T› g2 by (fastforce simp add: openin_open)
then have 2: "openin (subtopology euclidean U) {x ∈ U. g2 x ∈ v}"
unfolding gg by (blast intro: contu continuous_on_open [THEN iffD1, rule_format])
show "∃T. openin (subtopology euclidean U) T ∧
z ∈ T ∧ T ⊆ {z ∈ U. g1 z - g2 z = 0}"
using z
apply (rule_tac x = "{x. x ∈ U ∧ g1 x ∈ v} ∩ {x. x ∈ U ∧ g2 x ∈ v}" in exI)
apply (intro conjI)
apply (rule openin_Int [OF 1 2])
using ‹g1 z ∈ v›  ‹g2 z ∈ v›  apply (force simp:, clarify)
apply (metis ‹U ⊆ T› 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 = {} ∨ G12 = U"
by (intro connected_clopen [THEN iffD1, rule_format] ‹connected U› conjI o12 c12)
with eq ‹a ∈ U› have "⋀x. x ∈ U ⟹ g1 x - g2 x = 0" by (auto simp: G12_def)
then show ?thesis
using ‹x ∈ U› by force
qed

proposition covering_space_lift_unique:
fixes f :: "'a::topological_space ⇒ 'b::topological_space"
fixes g1 :: "'a ⇒ 'c::real_normed_vector"
assumes "covering_space c p S"
"g1 a = g2 a"
"continuous_on T f"  "f ` T ⊆ S"
"continuous_on T g1"  "g1 ` T ⊆ c"  "⋀x. x ∈ T ⟹ f x = p(g1 x)"
"continuous_on T g2"  "g2 ` T ⊆ c"  "⋀x. x ∈ T ⟹ f x = p(g2 x)"
"connected T"  "a ∈ T"   "x ∈ 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 ⇒ 'b::real_normed_vector"
assumes loc: "locally φ C" and cov: "covering_space C p S"
and pim: "⋀T. ⟦T ⊆ C; φ T⟧ ⟹ ψ(p ` T)"
shows "locally ψ S"
proof -
have "locally ψ (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 ⇒ 'b::real_normed_vector"
assumes cov: "covering_space C p S"
and pim: "⋀T. ⟦T ⊆ C; φ T⟧ ⟹ ψ(p ` T)"
and qim: "⋀q U. ⟦U ⊆ S; continuous_on U q; ψ U⟧ ⟹ φ(q ` U)"
shows "locally ψ S ⟷ locally φ 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 ∈ V"
have "p x ∈ p ` C"
by (metis IntE V ‹x ∈ V› imageI openin_open)
then obtain T 𝒱 where "p x ∈ T"
and opeT: "openin (subtopology euclidean S) T"
and veq: "⋃𝒱 = {x ∈ C. p x ∈ T}"
and ope: "∀U∈𝒱. openin (subtopology euclidean C) U"
and hom: "∀U∈𝒱. ∃q. homeomorphism U T p q"
using cov unfolding covering_space_def by (blast intro: that)
have "x ∈ ⋃𝒱"
using V veq ‹p x ∈ T› ‹x ∈ V› openin_imp_subset by fastforce
then obtain U where "x ∈ U" "U ∈ 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 ∩ V)"
by blast
then have UV: "openin (subtopology euclidean S) (p ` (U ∩ V))"
using cov covering_space_open_map by blast
obtain W W' where opeW: "openin (subtopology euclidean S) W" and "ψ W'" "p x ∈ W" "W ⊆ W'" and W'sub: "W' ⊆ p ` (U ∩ V)"
using locallyE [OF L UV] ‹x ∈ U› ‹x ∈ V› by blast
then have "W ⊆ T"
by (metis Int_lower1 q homeomorphism_image1 image_Int_subset order_trans)
show "∃U Z. openin (subtopology euclidean C) U ∧
φ Z ∧ x ∈ U ∧ U ⊆ Z ∧ Z ⊆ V"
proof (intro exI conjI)
have "openin (subtopology euclidean T) W"
by (meson opeW opeT openin_imp_subset openin_subset_trans ‹W ⊆ T›)
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 "φ (q ` W')"
by (metis (mono_tags, lifting) Int_subset_iff UV W'sub ‹ψ W'› continuous_on_subset dual_order.trans homeomorphism_def image_Int_subset openin_imp_subset q qim)
show "x ∈ q ` W"
by (metis ‹p x ∈ W› ‹x ∈ U› homeomorphism_def imageI q)
show "q ` W ⊆ q ` W'"
using ‹W ⊆ W'› by blast
have "W' ⊆ p ` V"
using W'sub by blast
then show "q ` W' ⊆ 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 ⇒ 'b::real_normed_vector"
assumes "covering_space C p S"
shows "locally compact S ⟷ 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 ⇒ 'b::real_normed_vector"
assumes "covering_space C p S"
shows "locally connected S ⟷ 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 ⇒ 'b::real_normed_vector"
assumes "covering_space C p S"
shows "locally path_connected S ⟷ 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 ⇒ '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 ⇒ '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 ⇒ '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 ⇒ 'b::real_normed_vector"
and h :: "real × 'c::real_normed_vector ⇒ 'b"
assumes cov: "covering_space C p S"
and conth: "continuous_on ({0..1} × U) h"
and him: "h ` ({0..1} × U) ⊆ S"
and heq: "⋀y. y ∈ U ⟹ h (0,y) = p(f y)"
and contf: "continuous_on U f" and fim: "f ` U ⊆ C"
obtains k where "continuous_on ({0..1} × U) k"
"k ` ({0..1} × U) ⊆ C"
"⋀y. y ∈ U ⟹ k(0, y) = f y"
"⋀z. z ∈ {0..1} × U ⟹ h z = p(k z)"
proof -
have "∃V k. openin (subtopology euclidean U) V ∧ y ∈ V ∧
continuous_on ({0..1} × V) k ∧ k ` ({0..1} × V) ⊆ C ∧
(∀z ∈ V. k(0, z) = f z) ∧ (∀z ∈ {0..1} × V. h z = p(k z))"
if "y ∈ U" for y
proof -
obtain UU where UU: "⋀s. s ∈ S ⟹ s ∈ (UU s) ∧ openin (subtopology euclidean S) (UU s) ∧
(∃𝒱. ⋃𝒱 = {x. x ∈ C ∧ p x ∈ (UU s)} ∧
(∀U ∈ 𝒱. openin (subtopology euclidean C) U) ∧
pairwise disjnt 𝒱 ∧
(∀U ∈ 𝒱. ∃q. homeomorphism U (UU s) p q))"
using cov unfolding covering_space_def by (metis (mono_tags))
then have ope: "⋀s. s ∈ S ⟹ s ∈ (UU s) ∧ openin (subtopology euclidean S) (UU s)"
by blast
have "∃k n i. open k ∧ open n ∧
t ∈ k ∧ y ∈ n ∧ i ∈ S ∧ h ` (({0..1} ∩ k) × (U ∩ n)) ⊆ UU i" if "t ∈ {0..1}" for t
proof -
have hinS: "h (t, y) ∈ S"
using ‹y ∈ U› him that by blast
then have "(t,y) ∈ {z ∈ {0..1} × U. h z ∈ UU (h (t, y))}"
using ‹y ∈ U› ‹t ∈ {0..1}›  by (auto simp: ope)
moreover have ope_01U: "openin (subtopology euclidean ({0..1} × U)) {z. z ∈ ({0..1} × U) ∧ h z ∈ 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 ∈ {0..1} ∩ V" "t ∈ {0..1} ∩ V"
and opeW: "open W" and "y ∈ U" "y ∈ W"
and VW: "({0..1} ∩ V) × (U ∩ W)  ⊆ {z. z ∈ ({0..1} × U) ∧ h z ∈ 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: "⋀t. t ∈ {0..1} ⟹ open (K t)"
and NN: "⋀t. t ∈ {0..1} ⟹ open (NN t)"
and inUS: "⋀t. t ∈ {0..1} ⟹ t ∈ K t ∧ y ∈ NN t ∧ X t ∈ S"
and him: "⋀t. t ∈ {0..1} ⟹ h ` (({0..1} ∩ K t) × (U ∩ NN t)) ⊆ UU (X t)"
by (metis (mono_tags))
obtain 𝒯 where "𝒯 ⊆ ((λi. K i × NN i)) ` {0..1}" "finite 𝒯" "{0::real..1} × {y} ⊆ ⋃𝒯"
proof (rule compactE)
show "compact ({0::real..1} × {y})"
show "{0..1} × {y} ⊆ (⋃i∈{0..1}. K i × NN i)"
using K inUS by auto
show "⋀B. B ∈ (λi. K i × NN i) ` {0..1} ⟹ open B"
using K NN by (auto simp: open_Times)
qed blast
then obtain tk where "tk ⊆ {0..1}" "finite tk"
and tk: "{0::real..1} × {y} ⊆ (⋃i ∈ tk. K i × NN i)"
by (metis (no_types, lifting) finite_subset_image)
then have "tk ≠ {}"
by auto
define n where "n = INTER tk NN"
have "y ∈ n" "open n"
using inUS NN ‹tk ⊆ {0..1}› ‹finite tk›
by (auto simp: n_def open_INT subset_iff)
obtain δ where "0 < δ" and δ: "⋀T. ⟦T ⊆ {0..1}; diameter T < δ⟧ ⟹ ∃B∈K ` tk. T ⊆ B"
proof (rule Lebesgue_number_lemma [of "{0..1}" "K ` tk"])
show "K ` tk ≠ {}"
using ‹tk ≠ {}› by auto
show "{0..1} ⊆ UNION tk K"
using tk by auto
show "⋀B. B ∈ K ` tk ⟹ open B"
using ‹tk ⊆ {0..1}› K by auto
qed auto
obtain N::nat where N: "N > 1 / δ"
using reals_Archimedean2 by blast
then have "N > 0"
using ‹0 < δ› order.asym by force
have *: "∃V k. openin (subtopology euclidean U) V ∧ y ∈ V ∧
continuous_on ({0..of_nat n / N} × V) k ∧
k ` ({0..of_nat n / N} × V) ⊆ C ∧
(∀z∈V. k (0, z) = f z) ∧
(∀z∈{0..of_nat n / N} × V. h z = p (k z))" if "n ≤ N" for n
using that
proof (induction n)
case 0
show ?case
apply (rule_tac x=U in exI)
apply (rule_tac x="f ∘ snd" in exI)
apply (intro conjI ‹y ∈ U› 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 ∈ V"
and contk: "continuous_on ({0..real n / real N} × V) k"
and kim: "k ` ({0..real n / real N} × V) ⊆ C"
and keq: "⋀z. z ∈ V ⟹ k (0, z) = f z"
and heq: "⋀z. z ∈ {0..real n / real N} × V ⟹ h z = p (k z)"
using Suc_leD by auto
have "n ≤ N"
using Suc.prems by auto
obtain t where "t ∈ tk" and t: "{real n / real N .. (1 + real n) / real N} ⊆ K t"
proof (rule bexE [OF δ])
show "{real n / real N .. (1 + real n) / real N} ⊆ {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} < δ"
using ‹0 < δ› N by (auto simp: divide_simps algebra_simps)
qed blast
have t01: "t ∈ {0..1}"
using ‹t ∈ tk› ‹tk ⊆ {0..1}› by blast
obtain 𝒱 where "⋃𝒱 = {x. x ∈ C ∧ p x ∈ (UU (X t))}"
and opeC: "⋀U. U ∈ 𝒱 ⟹ openin (subtopology euclidean C) U"
and "pairwise disjnt 𝒱"
and homuu: "⋀U. U ∈ 𝒱 ⟹ ∃q. homeomorphism U (UU (X t)) p q"
using inUS [OF t01] UU by meson
have n_div_N_in: "real n / real N ∈ {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 ∈ K t"
by blast
have "k (real n / real N, y) ∈ {x. x ∈ C ∧ p x ∈ (UU (X t))}"
proof (simp, rule conjI)
show "k (real n / real N, y) ∈ C"
using ‹y ∈ V› kim keq by force
have "p (k (real n / real N, y)) = h (real n / real N, y)"
by (simp add: ‹y ∈ V› heq)
also have "... ∈ h ` (({0..1} ∩ K t) × (U ∩ NN t))"
apply (rule imageI)
using ‹y ∈ V› t01 ‹n ≤ N›
apply (simp add: nN_in_kkt ‹y ∈ U› inUS divide_simps)
done
also have "... ⊆ UU (X t)"
using him t01 by blast
finally show "p (k (real n / real N, y)) ∈ UU (X t)" .
qed
then have "k (real n / real N, y) ∈ ⋃𝒱"
using ‹⋃𝒱 = {x ∈ C. p x ∈ UU (X t)}› by blast
then obtain W where W: "k (real n / real N, y) ∈ W" and "W ∈ 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 ⊆ C"
using openin_imp_subset by blast
define W' where "W' = UU(X t)"
have opeVW: "openin (subtopology euclidean V) {z ∈ V. (k ∘ Pair (real n / real N)) z ∈ W}"
apply (rule continuous_openin_preimage [OF _ _ opeC'])
apply (intro continuous_intros continuous_on_subset [OF contk])
using kim apply (auto simp: ‹y ∈ V› W)
done
obtain N' where opeUN': "openin (subtopology euclidean U) N'"
and "y ∈ N'" and kimw: "k ` ({(real n / real N)} × N') ⊆ W"
apply (rule_tac N' = "{z ∈ V. (k ∘ Pair ((real n / real N))) z ∈ W}" in that)
apply (fastforce simp:  ‹y ∈ V› 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 ∈ Q" "Q ⊆ Q'"
and Q': "Q' ⊆ (U ∩ NN(t)) ∩ N' ∩ V"
proof -
obtain VO VX where "open VO" "open VX" and VO: "V = U ∩ VO" and VX: "N' = U ∩ VX"
using opeUV opeUN' by (auto simp: openin_open)
then have "open (NN(t) ∩ VO ∩ VX)"
using NN t01 by blast
then obtain e where "e > 0" and e: "cball y e ⊆ NN(t) ∩ VO ∩ VX"
by (metis Int_iff ‹N' = U ∩ VX› ‹V = U ∩ VO› ‹y ∈ N'› ‹y ∈ V› inUS open_contains_cball t01)
show ?thesis
proof
show "openin (subtopology euclidean U) (U ∩ ball y e)"
by blast
show "closedin (subtopology euclidean U) (U ∩ cball y e)"
using e by (auto simp: closedin_closed)
qed (use ‹y ∈ U› ‹e > 0› VO VX e in auto)
qed
then have "y ∈ Q'" "Q ⊆ (U ∩ NN(t)) ∩ N' ∩ V"
by blast+
have neq: "{0..real n / real N} ∪ {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} × Q' ∪ {real n / real N..(1 + real n) / real N} × Q' = {0..(1 + real n) / real N} × Q'"
by blast
have cont: "continuous_on ({0..(1 + real n) / real N} × Q')
(λx. if x ∈ {0..real n / real N} × Q' then k x else (p' ∘ 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} × Q'))
({0..real n / real N} × Q')"
apply (rule_tac x="{0 .. real n / real N} × UNIV" in exI)
using n_div_N_in apply (auto simp: closed_Times)
done
show "closedin (subtopology euclidean ({0..(1 + real n) / real N} × Q'))
({real n / real N..(1 + real n) / real N} × Q')"
apply (rule_tac x="{real n / real N .. (1 + real n) / real N} × 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} × 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} × Q') h"
proof (rule continuous_on_subset [OF conth])
show "{real n / real N..(1 + real n) / real N} × Q' ⊆ {0..1} × U"
using ‹N > 0›
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} × Q')) p'"
proof (rule continuous_on_subset [OF homeomorphism_cont2 [OF hom']])
have "h ` ({real n / real N..(1 + real n) / real N} × Q') ⊆ h ` (({0..1} ∩ K t) × (U ∩ NN t))"
apply (rule image_mono)
using ‹0 < δ› ‹N > 0› 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} × Q') ⊆ UU (X t)"
using t01 by blast
qed
ultimately show "continuous_on ({real n / real N..(1 + real n) / real N} × Q') (p' ∘ h)"
by (rule continuous_on_compose)
have "k (real n / real N, b) = p' (h (real n / real N, b))" if "b ∈ Q'" for b
proof -
have "k (real n / real N, b) ∈ 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 "⋀x. x ∈ {real n / real N..(1 + real n) / real N} × Q' ∧
x ∈ {0..real n / real N} × Q' ⟹ k x = (p' ∘ h) x"
by auto
qed
have h_in_UU: "h (x, y) ∈ UU (X t)" if "y ∈ Q" "¬ x ≤ real n / real N" "0 ≤ x" "x ≤ (1 + real n) / real N" for x y
proof -
have "x ≤ 1"
using Suc.prems that order_trans by force
moreover have "x ∈ K t"
by (meson atLeastAtMost_iff le_less not_le subset_eq t that)
moreover have "y ∈ U"
using ‹y ∈ Q› opeUQ openin_imp_subset by blast
moreover have "y ∈ NN t"
using Q' ‹Q ⊆ Q'› ‹y ∈ Q› by auto
ultimately have "(x, y) ∈ (({0..1} ∩ K t) × (U ∩ NN t))"
using that by auto
then have "h (x, y) ∈ h ` (({0..1} ∩ K t) × (U ∩ NN t))"
by blast
also have "... ⊆ UU (X t)"
by (metis him t01)
finally show ?thesis .
qed
let ?k = "(λx. if x ∈ {0..real n / real N} × Q' then k x else (p' ∘ h) x)"
show ?case
proof (intro exI conjI)
show "continuous_on ({0..real (Suc n) / real N} × Q) ?k"
apply (rule continuous_on_subset [OF cont])
using ‹Q ⊆ Q'› by auto
have "⋀a b. ⟦a ≤ real n / real N; b ∈ Q'; 0 ≤ a⟧ ⟹ k (a, b) ∈ C"
using kim Q' by force
moreover have "⋀a b. ⟦b ∈ Q; 0 ≤ a; a ≤ (1 + real n) / real N; ¬ a ≤ real n / real N⟧ ⟹ p' (h (a, b)) ∈ C"
apply (rule ‹W ⊆ C› [THEN subsetD])
using homeomorphism_image2 [OF hom', symmetric]  h_in_UU  Q' ‹Q ⊆ Q'› ‹W ⊆ C›
apply auto
done
ultimately show "?k ` ({0..real (Suc n) / real N} × Q) ⊆ C"
using Q' ‹Q ⊆ Q'› by force
show "∀z∈Q. ?k (0, z) = f z"
using Q' keq  ‹Q ⊆ Q'› by auto
show "∀z ∈ {0..real (Suc n) / real N} × Q. h z = p(?k z)"
using ‹Q ⊆ U ∩ NN t ∩ N' ∩ V› heq apply clarsimp
using h_in_UU Q' ‹Q ⊆ Q'› apply (auto simp: homeomorphism_apply2 [OF hom', symmetric])
done
qed (auto simp: ‹y ∈ Q› opeUQ)
qed
show ?thesis
using*[OF order_refl] N ‹0 < δ› by (simp add: split: if_split_asm)
qed
then obtain V fs where opeV: "⋀y. y ∈ U ⟹ openin (subtopology euclidean U) (V y)"
and V: "⋀y. y ∈ U ⟹ y ∈ V y"
and contfs: "⋀y. y ∈ U ⟹ continuous_on ({0..1} × V y) (fs y)"
and *: "⋀y. y ∈ U ⟹ (fs y) ` ({0..1} × V y) ⊆ C ∧
(∀z ∈ V y. fs y (0, z) = f z) ∧
(∀z ∈ {0..1} × V y. h z = p(fs y z))"
by (metis (mono_tags))
then have VU: "⋀y. y ∈ U ⟹ V y ⊆ U"
by (meson openin_imp_subset)
obtain k where contk: "continuous_on ({0..1} × U) k"
and k: "⋀x i. ⟦i ∈ U; x ∈ {0..1} × U ∩ {0..1} × V i⟧ ⟹ k x = fs i x"
proof (rule pasting_lemma_exists)
show "{0..1} × U ⊆ (⋃i∈U. {0..1} × V i)"
apply auto
using V by blast
show "⋀i. i ∈ U ⟹ openin (subtopology euclidean ({0..1} × U)) ({0..1} × V i)"
show "⋀i. i ∈ U ⟹ continuous_on ({0..1} × V i) (fs i)"
using contfs by blast
show "fs i x = fs j x"  if "i ∈ U" "j ∈ U" and x: "x ∈ {0..1} × U ∩ {0..1} × V i ∩ {0..1} × V j"
for i j x
proof -
obtain u y where "x = (u, y)" "y ∈ V i" "y ∈ V j" "0 ≤ u" "u ≤ 1"
using x by auto
show ?thesis
proof (rule covering_space_lift_unique [OF cov, of _ "(0,y)" _ "{0..1} × {y}" h])
show "fs i (0, y) = fs j (0, y)"
using*V by (simp add: ‹y ∈ V i› ‹y ∈ V j› that)
show conth_y: "continuous_on ({0..1} × {y}) h"
apply (rule continuous_on_subset [OF conth])
using VU ‹y ∈ V j› that by auto
show "h ` ({0..1} × {y}) ⊆ S"
using ‹y ∈ V i› assms(3) VU that by fastforce
show "continuous_on ({0..1} × {y}) (fs i)"
using continuous_on_subset [OF contfs] ‹i ∈ U›
by (simp add: ‹y ∈ V i› subset_iff)
show "fs i ` ({0..1} × {y}) ⊆ C"
using "*" ‹y ∈ V i› ‹i ∈ U› by fastforce
show "⋀x. x ∈ {0..1} × {y} ⟹ h x = p (fs i x)"
using "*" ‹y ∈ V i› ‹i ∈ U› by blast
show "continuous_on ({0..1} × {y}) (fs j)"
using continuous_on_subset [OF contfs] ‹j ∈ U›
by (simp add: ‹y ∈ V j› subset_iff)
show "fs j ` ({0..1} × {y}) ⊆ C"
using "*" ‹y ∈ V j› ‹j ∈ U› by fastforce
show "⋀x. x ∈ {0..1} × {y} ⟹ h x = p (fs j x)"
using "*" ‹y ∈ V j› ‹j ∈ U› by blast
show "connected ({0..1::real} × {y})"
using connected_Icc connected_Times connected_sing by blast
show "(0, y) ∈ {0..1::real} × {y}"
by force
show "x ∈ {0..1} × {y}"
using ‹x = (u, y)› x by blast
qed
qed
qed blast
show ?thesis
proof
show "k ` ({0..1} × U) ⊆ C"
using V*k VU by fastforce
show "⋀y. y ∈ U ⟹ k (0, y) = f y"
show "⋀z. z ∈ {0..1} × U ⟹ 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 ⇒ 'b::real_normed_vector"
and h :: "'c::real_normed_vector × real ⇒ 'b"
assumes cov: "covering_space C p S"
and conth: "continuous_on (U × {0..1}) h"
and him: "h ` (U × {0..1}) ⊆ S"
and heq: "⋀y. y ∈ U ⟹ h (y,0) = p(f y)"
and contf: "continuous_on U f" and fim: "f ` U ⊆ C"
obtains k where "continuous_on (U × {0..1}) k"
"k ` (U × {0..1}) ⊆ C"
"⋀y. y ∈ U ⟹ k(y, 0) = f y"
"⋀z. z ∈ U × {0..1} ⟹ h z = p(k z)"
proof -
have "continuous_on ({0..1} × U) (h ∘ (λz. (snd z, fst z)))"
by (intro continuous_intros continuous_on_subset [OF conth]) auto
then obtain k where contk: "continuous_on ({0..1} × U) k"
and kim:  "k ` ({0..1} × U) ⊆ C"
and k0: "⋀y. y ∈ U ⟹ k(0, y) = f y"
and heqp: "⋀z. z ∈ {0..1} × U ⟹ (h ∘ (λz. 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 ∘ (λz. 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 ⇒ 'b::real_normed_vector" and g:: "'c::real_normed_vector ⇒ 'a"
assumes cov: "covering_space C p S"
and contg: "continuous_on U g"
and gim: "g ` U ⊆ C"
and pgeq: "⋀y. y ∈ U ⟹ p(g y) = f y"
and hom: "homotopic_with (λx. True) U S f f'"
obtains g' where "continuous_on U g'" "image g' U ⊆ C" "⋀y. y ∈ U ⟹ p(g' y) = f' y"
proof -
obtain h where conth: "continuous_on ({0..1::real} × U) h"
and him: "h ` ({0..1} × U) ⊆ S"
and h0:  "⋀x. h(0, x) = f x"
and h1: "⋀x. h(1, x) = f' x"
using hom by (auto simp: homotopic_with_def)
have "⋀y. y ∈ U ⟹ h (0, y) = p (g y)"
then obtain k where contk: "continuous_on ({0..1} × U) k"
and kim: "k ` ({0..1} × U) ⊆ C"
and k0: "⋀y. y ∈ U ⟹ k(0, y) = g y"
and heq: "⋀z. z ∈ {0..1} × U ⟹ 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 ∘ Pair 1)"
by (meson contk atLeastAtMost_iff continuous_on_o_Pair order_refl zero_le_one)
show "(k ∘ Pair 1) ` U ⊆ C"
using kim by auto
show "⋀y. y ∈ U ⟹ p ((k ∘ 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 ⇒ 'b::real_normed_vector" and U :: "'c::real_normed_vector set"
assumes cov: "covering_space C p S"
and hom: "homotopic_with (λx. True) U S f (λx. a)"
obtains g where "continuous_on U g" "g ` U ⊆ C" "⋀y. y ∈ U ⟹ 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 ∈ C" "p b = a"
using covering_space_imp_surjective [OF cov] homotopic_with_imp_subset2 [OF hom]
by auto
then have gim: "(λy. b) ` U ⊆ 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‹ Lifting of general functions to covering space›

proposition covering_space_lift_path_strong:
fixes p :: "'a::real_normed_vector ⇒ 'b::real_normed_vector"
and f :: "'c::real_normed_vector ⇒ 'b"
assumes cov: "covering_space C p S" and "a ∈ C"
and "path g" and pag: "path_image g ⊆ S" and pas: "pathstart g = p a"
obtains h where "path h" "path_image h ⊆ C" "pathstart h = a"
and "⋀t. t ∈ {0..1} ⟹ p(h t) = g t"
proof -
obtain k:: "real × 'c ⇒ 'a"
where contk: "continuous_on ({0..1} × {undefined}) k"
and kim: "k ` ({0..1} × {undefined}) ⊆ C"
and k0:  "k (0, undefined) = a"
and pk: "⋀z. z ∈ {0..1} × {undefined} ⟹ p(k z) = (g ∘ fst) z"
proof (rule covering_space_lift_homotopy [OF cov, of "{undefined}" "g ∘ fst"])
show "continuous_on ({0..1::real} × {undefined::'c}) (g ∘ fst)"
apply (intro continuous_intros)
using ‹path g› by (simp add: path_def)
show "(g ∘ fst) ` ({0..1} × {undefined}) ⊆ S"
using pag by (auto simp: path_image_def)
show "(g ∘ fst) (0, y) = p a" if "y ∈ {undefined}" for y::'c
by (metis comp_def fst_conv pas pathstart_def)
qed (use assms in auto)
show ?thesis
proof
show "path (k ∘ (λt. Pair t undefined))"
unfolding path_def
by (intro continuous_on_compose continuous_intros continuous_on_subset [OF contk]) auto
show "path_image (k ∘ (λt. (t, undefined))) ⊆ C"
using kim by (auto simp: path_image_def)
show "pathstart (k ∘ (λt. (t, undefined))) = a"
by (auto simp: pathstart_def k0)
show "⋀t. t ∈ {0..1} ⟹ p ((k ∘ (λt. (t, undefined))) t) = g t"
by (auto simp: pk)
qed
qed

corollary covering_space_lift_path:
fixes p :: "'a::real_normed_vector ⇒ 'b::real_normed_vector"
assumes cov: "covering_space C p S" and "path g" and pig: "path_image g ⊆ S"
obtains h where "path h" "path_image h ⊆ C" "⋀t. t ∈ {0..1} ⟹ p(h t) = g t"
proof -
obtain a where "a ∈ 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 ‹a ∈ C› ‹path g› pig]
by (metis ‹pathstart g = p a› that)
qed

proposition covering_space_lift_homotopic_paths:
fixes p :: "'a::real_normed_vector ⇒ 'b::real_normed_vector"
assumes cov: "covering_space C p S"
and "path g1" and pig1: "path_image g1 ⊆ S"
and "path g2" and pig2: "path_image g2 ⊆ S"
and hom: "homotopic_paths S g1 g2"
and "path h1" and pih1: "path_image h1 ⊆ C" and ph1: "⋀t. t ∈ {0..1} ⟹ p(h1 t) = g1 t"
and "path h2" and pih2: "path_image h2 ⊆ C" and ph2: "⋀t. t ∈ {0..1} ⟹ p(h2 t) = g2 t"
and h1h2: "pathstart h1 = pathstart h2"
shows "homotopic_paths C h1 h2"
proof -
obtain h :: "real × real ⇒ 'b"
where conth: "continuous_on ({0..1} × {0..1}) h"
and him: "h ` ({0..1} × {0..1}) ⊆ S"
and h0: "⋀x. h (0, x) = g1 x" and h1: "⋀x. h (1, x) = g2 x"
and heq0: "⋀t. t ∈ {0..1} ⟹ h (t, 0) = g1 0"
and heq1: "⋀t. t ∈ {0..1} ⟹ 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} × {0..1}) k"
and kim: "k ` ({0..1} × {0..1}) ⊆ C"
and kh2: "⋀y. y ∈ {0..1} ⟹ k (y, 0) = h2 0"
and hpk: "⋀z. z ∈ {0..1} × {0..1} ⟹ h z = p (k z)"
apply (rule covering_space_lift_homotopy_alt [OF cov conth him, of "λx. 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 ‹path g1› ‹path g2› path_def by blast+
have g1im: "g1 ` {0..1} ⊆ S" and g2im: "g2 ` {0..1} ⊆ S"
using path_image_def pig1 pig2 by auto
have conth1: "continuous_on {0..1} h1" and conth2: "continuous_on {0..1} h2"
using ‹path h1› ‹path h2› path_def by blast+
have h1im: "h1 ` {0..1} ⊆ C" and h2im: "h2 ` {0..1} ⊆ 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 ∈ {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} (λa. k (0, a))"
by (intro continuous_intros continuous_on_compose2 [OF contk]) auto
show "⋀x. x ∈ {0..1} ⟹ 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 ‹auto simp: ph1›)
show "k(1, x) = h2 x" if "x ∈ {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} (λa. k (1, a))"
by (intro continuous_intros continuous_on_compose2 [OF contk]) auto
show "⋀x. x ∈ {0..1} ⟹ 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 ‹auto simp: ph2›)
show "⋀t. t ∈ {0..1} ⟹ (k ∘ Pair t) 0 = h1 0"
by (metis comp_apply h1h2 kh2 pathstart_def)
show "(k ∘ Pair t) 1 = h1 1" if "t ∈ {0..1}" for t
proof (rule covering_space_lift_unique
[OF cov, of "λa. (k ∘ Pair a) 1" 0 "λa. h1 1" "{0..1}"  "λx. g1 1"])
show "(k ∘ Pair 0) 1 = h1 1"
using keqh1 by auto
show "continuous_on {0..1} (λa. (k ∘ Pair a) 1)"
apply simp
by (intro continuous_intros continuous_on_compose2 [OF contk]) auto
show "⋀x. x ∈ {0..1} ⟹ g1 1 = p ((k ∘ Pair x) 1)"
using heq1 hpk by auto
qed (use contk kim g1im h1im that in ‹auto simp: ph1 continuous_on_const›)
qed (use contk kim in auto)
qed

corollary covering_space_monodromy:
fixes p :: "'a::real_normed_vector ⇒ 'b::real_normed_vector"
assumes cov: "covering_space C p S"
and "path g1" and pig1: "path_image g1 ⊆ S"
and "path g2" and pig2: "path_image g2 ⊆ S"
and hom: "homotopic_paths S g1 g2"
and "path h1" and pih1: "path_image h1 ⊆ C" and ph1: "⋀t. t ∈ {0..1} ⟹ p(h1 t) = g1 t"
and "path h2" and pih2: "path_image h2 ⊆ C" and ph2: "⋀t. t ∈ {0..1} ⟹ 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 ⇒ '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 ⊆ C"
and a: "pathstart g = a" and b: "pathfinish g = b"
and pgeq: "⋀t. t ∈ {0..1} ⟹ p(g t) = f t"
obtains g' where "path g'" "path_image g' ⊆ C"
"pathstart g' = a" "pathfinish g' = b" "⋀t. t ∈ {0..1} ⟹ p(g' t) = f' t"
proof (rule covering_space_lift_path_strong [OF cov, of a f'])
show "a ∈ C"
using a pig by auto
show "path f'" "path_image f' ⊆ 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 ⇒ 'b::real_normed_vector"
and f :: "'c::real_normed_vector ⇒ 'b"
assumes cov: "covering_space C p S" and "a ∈ C" "z ∈ U"
and U: "path_connected U" "locally path_connected U"
and contf: "continuous_on U f" and fim: "f ` U ⊆ S"
and feq: "f z = p a"
and hom: "⋀r. ⟦path r; path_image r ⊆ U; pathstart r = z; pathfinish r = z⟧
⟹ ∃q. path q ∧ path_image q ⊆ C ∧
pathstart q = a ∧ pathfinish q = a ∧
homotopic_paths S (f ∘ r) (p ∘ q)"
obtains g where "continuous_on U g" "g ` U ⊆ C" "g z = a" "⋀y. y ∈ U ⟹ p(g y) = f y"
proof -
have *: "∃g h. path g ∧ path_image g ⊆ U ∧
pathstart g = z ∧ pathfinish g = y ∧
path h ∧ path_image h ⊆ C ∧ pathstart h = a ∧
(∀t ∈ {0..1}. p(h t) = f(g t))"
if "y ∈ U" for y
proof -
obtain g where "path g" "path_image g ⊆ U" and pastg: "pathstart g = z"
and pafig: "pathfinish g = y"
using U ‹z ∈ U› ‹y ∈ U› by (force simp: path_connected_def)
obtain h where "path h" "path_image h ⊆ C" "pathstart h = a"
and "⋀t. t ∈ {0..1} ⟹ p(h t) = (f ∘ g) t"
proof (rule covering_space_lift_path_strong [OF cov ‹a ∈ C›])
show "path (f ∘ g)"
using ‹path g› ‹path_image g ⊆ U› contf continuous_on_subset path_continuous_image by blast
show "path_image (f ∘ g) ⊆ S"
by (metis ‹path_image g ⊆ U› fim image_mono path_image_compose subset_trans)
show "pathstart (f ∘ g) = p a"
by (simp add: feq pastg pathstart_compose)
qed auto
then show ?thesis
by (metis ‹path g› ‹path_image g ⊆ U› comp_apply pafig pastg)
qed
have "∃l. ∀g h. path g ∧ path_image g ⊆ U ∧ pathstart g = z ∧ pathfinish g = y ∧
path h ∧ path_image h ⊆ C ∧ pathstart h = a ∧
(∀t ∈ {0..1}. p(h t) = f(g t))  ⟶ pathfinish h = l" for y
proof -
have "pathfinish h = pathfinish h'"
if g: "path g" "path_image g ⊆ U" "pathstart g = z" "pathfinish g = y"
and h: "path h" "path_image h ⊆ C" "pathstart h = a"
and phg: "⋀t. t ∈ {0..1} ⟹ p(h t) = f(g t)"
and g': "path g'" "path_image g' ⊆ U" "pathstart g' = z" "pathfinish g' = y"
and h': "path h'" "path_image h' ⊆ C" "pathstart h' = a"
and phg': "⋀t. t ∈ {0..1} ⟹ p(h' t) = f(g' t)"
for g h g' h'
proof -
obtain q where "path q" and piq: "path_image q ⊆ C" and pastq: "pathstart q = a" and pafiq: "pathfinish q = a"
and homS: "homotopic_paths S (f ∘ g +++ reversepath g') (p ∘ q)"
using g g' hom [of "g +++ reversepath g'"] by (auto simp:  subset_path_image_join)
have papq: "path (p ∘ q)"
using homS homotopic_paths_imp_path by blast
have pipq: "path_image (p ∘ q) ⊆ S"
using homS homotopic_paths_imp_subset by blast
obtain q' where "path q'" "path_image q' ⊆ C"
and "pathstart q' = pathstart q" "pathfinish q' = pathfinish q"
and pq'_eq: "⋀t. t ∈ {0..1} ⟹ p (q' t) = (f ∘ g +++ reversepath g') t"
using covering_space_lift_homotopic_path [OF cov homotopic_paths_sym [OF homS] ‹path q› piq refl refl]
by auto
have "q' t = (h ∘ op *⇩R 2) t" if "0 ≤ t" "t ≤ 1/2" for t
proof (rule covering_space_lift_unique [OF cov, of q' 0 "h ∘ op *⇩R 2" "{0..1/2}" "f ∘ g ∘ op *⇩R 2" t])
show "q' 0 = (h ∘ op *⇩R 2) 0"
by (metis ‹pathstart q' = pathstart q› comp_def g h pastq pathstart_def pth_4(2))
show "continuous_on {0..1/2} (f ∘ g ∘ op *⇩R 2)"
apply (intro continuous_intros continuous_on_compose continuous_on_path [OF ‹path g›] continuous_on_subset [OF contf])
using g(2) path_image_def by fastforce+
show "(f ∘ g ∘ op *⇩R 2) ` {0..1/2} ⊆ S"
using g(2) path_image_def fim by fastforce
show "(h ∘ op *⇩R 2) ` {0..1/2} ⊆ C"
using h path_image_def by fastforce
show "q' ` {0..1/2} ⊆ C"
using ‹path_image q' ⊆ C› path_image_def by fastforce
show "⋀x. x ∈ {0..1/2} ⟹ (f ∘ g ∘ op *⇩R 2) x = p (q' x)"
by (auto simp: joinpaths_def pq'_eq)
show "⋀x. x ∈ {0..1/2} ⟹ (f ∘ g ∘ op *⇩R 2) x = p ((h ∘ op *⇩R 2) x)"
show "continuous_on {0..1/2} q'"
by (simp add: continuous_on_path ‹path q'›)
show "continuous_on {0..1/2} (h ∘ op *⇩R 2)"
apply (intro continuous_intros continuous_on_compose continuous_on_path [OF ‹path h›], force)
done
qed (use that in auto)
moreover have "q' t = (reversepath h' ∘ (λt. 2 *⇩R t - 1)) t" if "1/2 < t" "t ≤ 1" for t
proof (rule covering_space_lift_unique [OF cov, of q' 1 "reversepath h' ∘ (λt. 2 *⇩R t - 1)" "{1/2<..1}" "f ∘ reversepath g' ∘ (λt. 2 *⇩R t - 1)" t])
show "q' 1 = (reversepath h' ∘ (λt. 2 *⇩R t - 1)) 1"
using h' ‹pathfinish q' = pathfinish q› pafiq
by (simp add: pathstart_def pathfinish_def reversepath_def)
show "continuous_on {1/2<..1} (f ∘ reversepath g' ∘ (λt. 2 *⇩R t - 1))"
apply (intro continuous_intros continuous_on_compose continuous_on_path ‹path g'› continuous_on_subset [OF contf])
using g' apply simp_all
by (auto simp: path_image_def reversepath_def)
show "(f ∘ reversepath g' ∘ (λt. 2 *⇩R t - 1)) ` {1/2<..1} ⊆ S"
using g'(2) path_image_def fim by (auto simp: image_subset_iff path_image_def reversepath_def)
show "q' ` {1/2<..1} ⊆ C"
using ‹path_image q' ⊆ C› path_image_def by fastforce
show "(reversepath h' ∘ (λt. 2 *⇩R t - 1)) ` {1/2<..1} ⊆ C"
using h' by (simp add: path_image_def reversepath_def subset_eq)
show "⋀x. x ∈ {1/2<..1} ⟹ (f ∘ reversepath g' ∘ (λt. 2 *⇩R t - 1)) x = p (q' x)"
by (auto simp: joinpaths_def pq'_eq)
show "⋀x. x ∈ {1/2<..1} ⟹
(f ∘ reversepath g' ∘ (λt. 2 *⇩R t - 1)) x = p ((reversepath h' ∘ (λt. 2 *⇩R t - 1)) x)"
show "continuous_on {1/2<..1} q'"
by (auto intro: continuous_on_path [OF ‹path q'›])
show "continuous_on {1/2<..1} (reversepath h' ∘ (λt. 2 *⇩R t - 1))"
apply (intro continuous_intros continuous_on_compose continuous_on_path ‹path h'›)
using h' apply auto
done
qed (use that in auto)
ultimately have "q' t = (h +++ reversepath h') t" if "0 ≤ t" "t ≤ 1" for t
using that by (simp add: joinpaths_def)
then have "path(h +++ reversepath h')"
by (auto intro: path_eq [OF ‹path q'›])
then show ?thesis
by (auto simp: ‹path h› ‹path h'›)
qed
then show ?thesis by metis
qed
then obtain l :: "'c ⇒ 'a"
where l: "⋀y g h. ⟦path g; path_image g ⊆ U; pathstart g = z; pathfinish g = y;
path h; path_image h ⊆ C; pathstart h = a;
⋀t. t ∈ {0..1} ⟹ p(h t) = f(g t)⟧ ⟹ pathfinish h = l y"
by metis
show ?thesis
proof
show pleq: "p (l y) = f y" if "y ∈ U" for y
using*[OF ‹y ∈ U›]  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 ⊆ C"
by (clarify dest!: *) (metis (full_types) l pathfinish_in_path_image subsetCE)
have "∃T. openin (subtopology euclidean U) T ∧ y ∈ T ∧ T ⊆ {x ∈ U. l x ∈ X}"
if X: "openin (subtopology euclidean C) X" and "y ∈ U" "l y ∈ X" for X y
proof -
have "X ⊆ C"
using X openin_euclidean_subtopology_iff by blast
have "f y ∈ S"
using fim ‹y ∈ U› by blast
then obtain W 𝒱
where WV: "f y ∈ W ∧ openin (subtopology euclidean S) W ∧
(⋃𝒱 = {x. x ∈ C ∧ p x ∈ W} ∧
(∀U ∈ 𝒱. openin (subtopology euclidean C) U) ∧
pairwise disjnt 𝒱 ∧
(∀U ∈ 𝒱. ∃q. homeomorphism U W p q))"
using cov by (force simp: covering_space_def)
then have "l y ∈ ⋃𝒱"
using ‹X ⊆ C› pleq that by auto
then obtain W' where "l y ∈ W'" and "W' ∈ 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 ⊆ W'"
using homUW' homeomorphism_image2 homeomorphism_cont2 by fastforce+
obtain V where "y ∈ V" "y ∈ U" and fimW: "f ` V ⊆ W" "V ⊆ U"
and "path_connected V" and opeUV: "openin (subtopology euclidean U) V"
proof -
have "openin (subtopology euclidean U) {c ∈ U. f c ∈ 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 ∈ U ∧ f x ∈ W}" in spec)
apply (drule_tac x=y in spec)
apply (auto simp: ‹y ∈ U› intro: that)
done
qed
have "W' ⊆ C" "W ⊆ S"
using opeCW' WV openin_imp_subset by auto
have p'im: "p' ` W ⊆ W'"
using homUW' homeomorphism_image2 by fastforce
show ?thesis
proof (intro exI conjI)
have "openin (subtopology euclidean S) {x ∈ W. p' x ∈ W' ∩ X}"
proof (rule openin_trans)
show "openin (subtopology euclidean W) {x ∈ W. p' x ∈ W' ∩ X}"
apply (rule continuous_openin_preimage [OF contp' p'im])
using X ‹W' ⊆ C› apply (auto simp: openin_open)
done
show "openin (subtopology euclidean S) W"
using WV by blast
qed
then show "openin (subtopology euclidean U) (V ∩ {x. x ∈ U ∧ f x ∈ {x. x ∈ W ∧ p' x ∈ W' ∩ X}})"
by (intro openin_Int opeUV continuous_openin_preimage [OF contf fim])
have "p' (f y) ∈ X"
using ‹l y ∈ W'› homeomorphism_apply1 [OF homUW'] pleq ‹y ∈ U› ‹l y ∈ X› by fastforce
then show "y ∈ V ∩ {x ∈ U. f x ∈ {x ∈ W. p' x ∈ W' ∩ X}}"
using ‹y ∈ U› ‹y ∈ V› WV p'im by auto
show "V ∩ {x ∈ U. f x ∈ {x ∈ W. p' x ∈ W' ∩ X}} ⊆ {x ∈ U. l x ∈ X}"
proof clarsimp
fix y'
assume y': "y' ∈ V" "y' ∈ U" "f y' ∈ W" "p' (f y') ∈ W'" "p' (f y') ∈ X"
then obtain γ where "path γ" "path_image γ ⊆ V" "pathstart γ = y" "pathfinish γ = y'"
by (meson ‹path_connected V› ‹y ∈ V› path_connected_def)
obtain pp qq where "path pp" "path_image pp ⊆ U"
"pathstart pp = z" "pathfinish pp = y"
"path qq" "path_image qq ⊆ C" "pathstart qq = a"
and pqqeq: "⋀t. t ∈ {0..1} ⟹ p(qq t) = f(pp t)"
using*[OF ‹y ∈ U›] by blast
have finW: "⋀x. ⟦0 ≤ x; x ≤ 1⟧ ⟹ f (γ x) ∈ W"
using ‹path_image γ ⊆ V› by (auto simp: image_subset_iff path_image_def fimW [THEN subsetD])
have "pathfinish (qq +++ (p' ∘ f ∘ γ)) = l y'"
proof (rule l [of "pp +++ γ" y' "qq +++ (p' ∘ f ∘ γ)" ])
show "path (pp +++ γ)"
by (simp add: ‹path γ› ‹path pp› ‹pathfinish pp = y› ‹pathstart γ = y›)
show "path_image (pp +++ γ) ⊆ U"
using ‹V ⊆ U› ‹path_image γ ⊆ V› ‹path_image pp ⊆ U› not_in_path_image_join by blast
show "pathstart (pp +++ γ) = z"
by (simp add: ‹pathstart pp = z›)
show "pathfinish (pp +++ γ) = y'"
by (simp add: ‹pathfinish γ = y'›)
have paqq: "pathfinish qq = pathstart (p' ∘ f ∘ γ)"
apply (simp add: ‹pathstart γ = y› pathstart_compose)
apply (metis (mono_tags, lifting) ‹l y ∈ W'› ‹path pp› ‹path qq› ‹path_image pp ⊆ U› ‹path_image qq ⊆ C›
‹pathfinish pp = y› ‹pathstart pp = z› ‹pathstart qq = a›
homeomorphism_apply1 [OF homUW'] l pleq pqqeq ‹y ∈ U›)
done
have "continuous_on (path_image γ) (p' ∘ f)"
proof (rule continuous_on_compose)
show "continuous_on (path_image γ) f"
using ‹path_image γ ⊆ V› ‹V ⊆ U› contf continuous_on_subset by blast
show "continuous_on (f ` path_image γ) 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' ∘ f ∘ γ))"
using ‹path γ› ‹path qq› paqq path_continuous_image path_join_imp by blast
show "path_image (qq +++ (p' ∘ f ∘ γ)) ⊆ C"
apply (rule subset_path_image_join)
apply (simp add: ‹path_image qq ⊆ C›)
by (metis ‹W' ⊆ C› ‹path_image γ ⊆ V› dual_order.trans fimW(1) image_comp image_mono p'im path_image_compose)
show "pathstart (qq +++ (p' ∘ f ∘ γ)) = a"
by (simp add: ‹pathstart qq = a›)
show "p ((qq +++ (p' ∘ f ∘ γ)) ξ) = f ((pp +++ γ) ξ)" if ξ: "ξ ∈ {0..1}" for ξ
show "p (qq (2*ξ)) = f (pp (2*ξ))" if "ξ*2 ≤ 1"
using ‹ξ ∈ {0..1}› pqqeq that by auto
show "p (p' (f (γ (2*ξ - 1)))) = f (γ (2*ξ - 1))" if "¬ ξ*2 ≤ 1"
apply (rule homeomorphism_apply2 [OF homUW' finW])
using that ξ by auto
qed
qed
with ‹pathfinish γ = y'›  ‹p' (f y') ∈ X› show "l y' ∈ 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 ⇒ 'b::real_normed_vector"
and f :: "'c::real_normed_vector ⇒ 'b"
assumes cov: "covering_space C p S" "a ∈ C" "z ∈ U"
and U: "path_connected U" "locally path_connected U"
and contf: "continuous_on U f" and fim: "f ` U ⊆ S"
and feq: "f z = p a"
and hom: "⋀r. ⟦path r; path_image r ⊆ U; pathstart r = z; pathfinish r = z⟧
⟹ ∃b. homotopic_paths S (f ∘ r) (linepath b b)"
obtains g where "continuous_on U g" "g ` U ⊆ C" "g z = a" "⋀y. y ∈ U ⟹ 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 ⊆ U" "pathstart r = z" "pathfinish r = z"
then obtain b where b: "homotopic_paths S (f ∘ 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 ∘ r) (linepath (f z) (f z))"
by (simp add: b ‹pathstart r = z›)
then have "homotopic_paths S (f ∘ r) (p ∘ linepath a a)"
by (simp add: o_def feq linepath_def)
then show "∃q. path q ∧
path_image q ⊆ C ∧
pathstart q = a ∧ pathfinish q = a ∧ homotopic_paths S (f ∘ r) (p ∘ q)"
by (force simp: ‹a ∈ C›)
qed auto

corollary covering_space_lift_strong:
fixes p :: "'a::real_normed_vector ⇒ 'b::real_normed_vector"
and f :: "'c::real_normed_vector ⇒ 'b"
assumes cov: "covering_space C p S" "a ∈ C" "z ∈ U"
and scU: "simply_connected U" and lpcU: "locally path_connected U"
and contf: "continuous_on U f" and fim: "f ` U ⊆ S"
and feq: "f z = p a"
obtains g where "continuous_on U g" "g ` U ⊆ C" "g z = a" "⋀y. y ∈ U ⟹ 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 ⊆ U" "pathstart r = z" "pathfinish r = z"
have "linepath (f z) (f z) = f ∘ linepath z z"
then have "homotopic_paths S (f ∘ r) (linepath (f z) (f z))"
by (metis r contf fim homotopic_paths_continuous_image scU simply_connected_eq_contractible_path)
then show "∃b. homotopic_paths S (f ∘ r) (linepath b b)"
by blast
qed blast

corollary covering_space_lift:
fixes p :: "'a::real_normed_vector ⇒ 'b::real_normed_vector"
and f :: "'c::real_normed_vector ⇒ '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 ⊆ S"
obtains g where "continuous_on U g" "g ` U ⊆ C" "⋀y. y ∈ U ⟹ p(g y) = f y"
proof (cases "U = {}")
case True
with that show ?thesis by auto
next
case False
then obtain z where "z ∈ U" by blast
then obtain a where "a ∈ 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 _ ‹z ∈ U› U contf fim])
qed

end
```