(* 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" using relin_Ex adl by blast 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" by (simp add: rel_frontier_def) 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" by (simp add: df) 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" by (metis ‹0 < d› add.commute add_divide_distrib diff_add_cancel divide_self_if less_irrefl scaleR_add_left scaleR_one) 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› by (metis add.commute add.right_neutral diff_add_cancel hull_inc scaleR_one) 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› by (metis add.commute add.right_neutral diff_add_cancel hull_inc scaleR_one) 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 by (simp add: proj_def) have proj0_iff [simp]: "proj x = 0 ⟷ x = 0" for x by (simp add: proj_def) have cont_proj: "continuous_on (UNIV - {0}) proj" unfolding proj_def by (rule continuous_intros | force)+ have proj_spherI: "⋀x. ⟦x ∈ affine hull S; x ≠ 0⟧ ⟹ proj x ∈ ?SPHER" by (simp add: projI) 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" by (simp add: closed_affine_hull compact_Int_closed) show "?SMINUS homeomorphic ?SPHER" apply (subst homeomorphic_sym) apply (rule homeomorphic_compact [OF co01 cont_nosp [unfolded o_def] no_sp_im inj_spher]) done have proj_scaleR: "⋀a x. 0 < a ⟹ proj (a *⇩_{R}x) = proj x" by (simp add: proj_def) have cont_sp0: "continuous_on (affine hull S - {0}) (surf o proj)" apply (rule continuous_on_compose [OF continuous_on_subset [OF cont_proj]], force) apply (rule continuous_on_subset [OF cont_surf]) using homeomorphism_image1 proj_spherI surf by fastforce obtain B where "B>0" and B: "⋀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 (simp add: continuous_within) apply (rule lim_null_scaleR_bounded [where B=B]) apply (simp_all add: tendsto_norm_zero eventually_at) apply (rule_tac x=B in exI) using B surfpS proj_def projI apply (auto simp: ‹B > 0›) done next case False then have "∀⇩_{F}x in at x. (x ∈ affine hull S - {0}) = (x ∈ affine hull S)" apply (simp add: eventually_at) apply (rule_tac x="norm x" in exI) apply (auto simp: False) done with cont_sp0 have *: "continuous (at x within affine hull S) (λx. surf (proj x))" apply (simp add: continuous_on_eq_continuous_within) apply (drule_tac x=x in bspec, force simp: False that) apply (simp add: continuous_within Lim_transform_within_set) done show ?thesis apply (rule continuous_within_subset [where s = "affine hull S", OF _ Int_lower2]) apply (rule continuous_intros *)+ done qed have cont_nosp2: "continuous_on ?CBALL (λx. norm x *⇩_{R}((surf o proj) x))" by (simp add: continuous_on_eq_continuous_within cont_nosp) 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))" by (simp add: proj_def) 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" by (simp add: False proj_def) 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" by (simp add: False local.proj_def) 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" by (simp add: divide_inverse local.proj_def) 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))" by (simp add: le_divide_eq_1) 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" by (simp add: closed_affine_hull compact_Int_closed) show "S homeomorphic ?CBALL" apply (subst homeomorphic_sym) apply (rule homeomorphic_compact [OF co01 cont_nosp2 [unfolded o_def] im_cball inj_cball]) done qed corollary fixes S :: "'a::euclidean_space set" assumes "compact S" and a: "a ∈ 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 ((+) (-a) ` S)" by (meson assms compact_translation) have 2: "0 ∈ rel_interior ((+) (-a) ` S)" by (simp add: a rel_interior_translation) have 3: "open_segment 0 x ⊆ rel_interior ((+) (-a) ` S)" if "x ∈ ((+) (-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 ((+) (-a) ` S) - rel_interior ((+) (-a) ` S)" by (metis rel_interior_translation translation_diff homeomorphic_translation) also have "... homeomorphic sphere 0 1 ∩ affine hull ((+) (-a) ` S)" by (rule starlike_compact_projective1_0 [OF 1 2 3]) also have "... = (+) (-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 ((+) (-a) ` S)" by (metis homeomorphic_translation) also have "... homeomorphic cball 0 1 ∩ affine hull ((+) (-a) ` S)" by (rule starlike_compact_projective2_0 [OF 1 2 3]) also have "... = (+) (-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 = "(+) (-a) ` S" and ?bT = "(+) (-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 ((+) (- a) ` S)) = dim (span ((+) (- 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 (+) (-a) ` (cball a 1 ∩ affine hull S)" by (metis homeomorphic_translation) also have "... = cball 0 1 ∩ (+) (-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 ∩ (+) (-b) ` (affine hull T)" using eqspanT affine_hull_translation by blast also have "... = (+) (-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 (+) (-a) ` (sphere a 1 ∩ affine hull S)" by (metis homeomorphic_translation) also have "... = sphere 0 1 ∩ (+) (-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 ∩ (+) (-b) ` (affine hull T)" using eqspanT affine_hull_translation by blast also have "... = (+) (-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"] by (simp add: rel_frontier_def convex_rel_interior_closure) 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" unfolding f_def g_def by (simp add: algebra_simps divide_simps add_nonneg_eq_0_iff) have no: "⋀x. ⟦norm x = 1; b∙x ≠ 1⟧ ⟹ (norm (f x))⇧^{2}= 4 * (1 + b∙x) / (1 - b∙x)" apply (simp add: dot_square_norm [symmetric]) apply (simp add: f_def vector_add_divide_simps divide_simps norm_eq_1) apply (simp add: algebra_simps inner_commute) done have [simp]: "⋀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) apply (simp_all add: dot_square_norm [symmetric] divide_simps vector_add_divide_simps) apply (simp add: algebra_simps inner_commute) done have [simp]: "⋀x. ⟦x ∈ T; b ∙ x = 0⟧ ⟹ b ∙ g x ≠ 1" unfolding g_def apply (simp_all add: dot_square_norm [symmetric] divide_simps vector_add_divide_simps add_nonneg_eq_0_iff) apply (auto simp: algebra_simps) done have "subspace T" by (simp add: assms subspace_affine) have [simp]: "⋀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) by (metis add_0 diff_zero mem_affine_3_minus) 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 (simp_all add: dot_square_norm [symmetric]) 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 (clarsimp simp: no4 vector_add_divide_simps divide_simps add_nonneg_eq_0_iff dot_square_norm [symmetric]) apply (auto simp: algebra_simps) done moreover have "g ` T ⊆ 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))" by (simp add: inj_on_def) have "((sphere a r ∩ T) - {b}) homeomorphic (+) (-a) ` ((sphere a r ∩ T) - {b})" by (rule homeomorphic_translation) also have "... homeomorphic ( *⇩_{R}) (inverse r) ` (+) (- a) ` (sphere a r ∩ T - {b})" by (metis ‹0 < r› homeomorphic_scaling inverse_inverse_eq inverse_zero less_irrefl) also have "... = sphere 0 1 ∩ (( *⇩_{R}) (inverse r) ` (+) (- 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 corollary 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_affine_sphere_affine [of r b a UNIV T] assms by auto corollary homeomorphic_punctured_sphere_hyperplane: fixes a :: "'a :: euclidean_space" assumes "0 < r" and b: "b ∈ 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 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 - obtain U :: "'a set" where "affine U" "convex U" and affdS: "aff_dim U = aff_dim S" using choose_affine_subset [OF affine_UNIV aff_dim_geq] by (meson aff_dim_affine_hull affine_affine_hull affine_imp_convex) have "S ≠ {}" using assms by auto then obtain z where "z ∈ U" by (metis aff_dim_negative_iff equals0I affdS) then have bne: "ball z 1 ∩ U ≠ {}" by force then have [simp]: "aff_dim(ball z 1 ∩ U) = aff_dim U" using aff_dim_convex_Int_open [OF ‹convex U› open_ball] by (fastforce simp add: Int_commute) have "rel_frontier S homeomorphic rel_frontier (ball z 1 ∩ U)" by (rule homeomorphic_rel_frontiers_convex_bounded_sets) (auto simp: ‹affine U› affine_imp_convex convex_Int affdS assms) also have "... = sphere z 1 ∩ U" using convex_affine_rel_frontier_Int [of "ball z 1" U] by (simp add: ‹affine U› bne) finally have "rel_frontier S homeomorphic sphere z 1 ∩ U" . then 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) 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" by (rule homeomorphic_punctured_affine_sphere_affine) (use a him in ‹auto simp: affS affdS ‹affine T› ‹affine U› ‹z ∈ U››) finally show ?thesis . qed 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 ((+) (- a) ` S)" by (simp add: ‹a ∈ S› hull_inc) then have "dim ((+) (- a) ` S) = aff_dim ((+) (- a) ` S)" by (simp add: aff_dim_zero) also have "... < DIM('n)" by (simp add: aff_dim_translation_eq assms) finally have dd: "dim ((+) (- a) ` S) < DIM('n)" by linarith obtain T where "subspace T" and Tsub: "T ⊆ {x. i ∙ x = 0}" and dimT: "dim T = dim ((+) (- a) ` S)" apply (rule choose_subspace_of_subspace [of "dim ((+) (- 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_iff subspace_hyperplane) done have "subspace (span ((+) (- a) ` S))" using subspace_span by blast then obtain h k where "linear h" "linear k" and heq: "h ` span ((+) (- a) ` S) = T" and keq:"k ` T = span ((+) (- a) ` S)" and hinv [simp]: "⋀x. x ∈ span ((+) (- 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_superset 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) by (metis DIM_positive Suc_eq_plus1 add.left_neutral diff_add_cancel not_le not_less_eq_eq of_nat_1 of_nat_diff) then obtain f g where fg: "homeomorphism (sphere 0 1 - {i}) {x. i ∙ x = 0} f g" by (force simp: homeomorphic_def) have "h ` (+) (- a) ` S ⊆ T" using heq span_superset span_linear_image by blast then have "g ` h ` (+) (- 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) ` (+) (- a) ` S ⊆ sphere 0 1 - {i}" by (metis image_comp) then have gh_sub_cb: "(g ∘ h) ` (+) (- 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 ((+) (- 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 ∘ (+) (- 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 (+) (- a) ` S" by (simp add: homeomorphic_translation) also have Shom: "… homeomorphic (g ∘ h) ` (+) (- a) ` S" apply (simp add: homeomorphic_def homeomorphism_def) apply (rule_tac x="g ∘ h" in exI) apply (rule_tac x="k ∘ f" in exI) apply (auto simp: ghcont kfcont span_base homeomorphism_apply2 [OF fg] image_comp) apply (force simp: o_def homeomorphism_apply2 [OF fg] span_base) done finally have Shom: "S homeomorphic (g ∘ h) ` (+) (- a) ` S" . show ?thesis apply (rule_tac U = "ball 0 1 ∪ image (g o h) ((+) (- a) ` S)" and T = "image (g o h) ((+) (- a) ` S)" in that) apply (rule convex_intermediate_ball [of 0 1], force) using gh_sub_cb apply force apply force apply (simp add: closedin_closed) apply (rule_tac x="sphere 0 1" in exI) apply (auto simp: Shom) done qed subsection‹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) by (simp add: Ucomp setdist_eq_0_sing_1) then have homU: "homeomorphism U (f`U) f fst" by (auto simp: f_def homeomorphism_def image_iff continuous_intros) have cloS: "closedin (subtopology euclidean U) S" by (metis US closed_closure closedin_closed_Int) have cont: "isCont ((λx. setdist {x} (- U)) o fst) z" for z :: "'a × 'b" by (rule continuous_at_compose 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 = (λz. (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_vimage) 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" by (simp add: Basis_prod_def) have "DIM('a * real) ≤ DIM('b)" by (simp add: Suc_leI assms) 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 apply (simp add: f_def inner_sum_left) apply (rule comm_monoid_add_class.sum.neutral) 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" by (simp add: image_comp o_def) 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 apply (auto simp add: mult_right_le_one_le) 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) by (auto simp add: scaleR_right_diff_distrib) then show ?thesis apply (rule_tac homeomorphic_trans[OF _ homeomorphic_balls(2)[of 1 _ ?n]]) apply (rule homeomorphic_trans[OF homeomorphic_affinity[of "?d" S "?d *⇩_{R}-a"]]) using ‹d>0› ‹e>0› apply (auto simp add: scaleR_right_diff_distrib) done qed corollary homeomorphic_convex_compact: fixes S :: "'a::euclidean_space set" and T :: "'a set" assumes "convex S" "compact S" "interior S ≠ {}" 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 = c ∩ p -` 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" by (simp add: covering_space_def) lemma covering_space_imp_surjective: "covering_space c p S ⟹ p ` c = S" by (simp add: covering_space_def) 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 (simp add: covering_space_def, clarify) apply (drule_tac x="p x" in bspec, force) by (metis IntI UnionE vimage_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 = c ∩ p -` 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 = c ∩ p -` 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 ∩ q -` (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) (U ∩ q -` (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 ∩ q -` (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: "U ∩ g -` v = U ∩ g -` (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) (U ∩ g1 -` 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) (U ∩ g2 -` 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 = "(U ∩ g1 -` v) ∩ (U ∩ g2 -` 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 by (simp add: pim) 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 = C ∩ p -` 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 = C ∩ p -` 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) ∈ ({0..1} × U) ∩ h -` UU(h(t, y))" using ‹y ∈ U› ‹t ∈ {0..1}› by (auto simp: ope) moreover have ope_01U: "openin (subtopology euclidean ({0..1} × U)) (({0..1} × U) ∩ h -` 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) ⊆ (({0..1} × U) ∩ h -` 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})" by (simp add: compact_Times) 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: "⋃𝒱 = C ∩ p -` 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) ∈ C ∩ p -` 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 with 𝒱 have "k (real n / real N, y) ∈ ⋃𝒱" 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) (V ∩ (k ∘ Pair (n / N)) -` 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' = "(V ∩ (k ∘ Pair (n / N)) -` 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 (simp add: closedin_closed) 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 (simp add: closedin_closed) 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)" by (simp add: opeV openin_Times) 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" by (simp add: V*k) 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)" by (simp add: h0 pgeq) 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 ∘ ( *⇩_{R}) 2) t" if "0 ≤ t" "t ≤ 1/2" for t proof (rule covering_space_lift_unique [OF cov, of q' 0 "h ∘ ( *⇩_{R}) 2" "{0..1/2}" "f ∘ g ∘ ( *⇩_{R}) 2" t]) show "q' 0 = (h ∘ ( *⇩_{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 ∘ ( *⇩_{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 ∘ ( *⇩_{R}) 2) ` {0..1/2} ⊆ S" using g(2) path_image_def fim by fastforce show "(h ∘ ( *⇩_{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 ∘ ( *⇩_{R}) 2) x = p (q' x)" by (auto simp: joinpaths_def pq'_eq) show "⋀x. x ∈ {0..1/2} ⟹ (f ∘ g ∘ ( *⇩_{R}) 2) x = p ((h ∘ ( *⇩_{R}) 2) x)" by (simp add: phg) show "continuous_on {0..1/2} q'" by (simp add: continuous_on_path ‹path q'›) show "continuous_on {0..1/2} (h ∘ ( *⇩_{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)" by (simp add: phg' reversepath_def) 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 ⊆ U ∩ l -` 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 = C ∩ p -` 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) (U ∩ f -` 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="U ∩ f -` 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) (W ∩ p' -` (W' ∩ X))" proof (rule openin_trans) show "openin (subtopology euclidean W) (W ∩ p' -` (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 ∩ (U ∩ (f -` (W ∩ (p' -` (W' ∩ X))))))" by (blast intro: opeUV openin_subtopology_self 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 ∩ (U ∩ f -` (W ∩ p' -` (W' ∩ X)))" using ‹y ∈ U› ‹y ∈ V› WV p'im by auto show "V ∩ (U ∩ f -` (W ∩ p' -` (W' ∩ X))) ⊆ U ∩ l -` X" proof (intro subsetI IntI; clarify) 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 proof (simp add: joinpaths_def, safe) 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 "y' ∈ l -` X" unfolding pathfinish_join by (simp add: pathfinish_def) qed qed qed then show "continuous_on U l" by (metis IntD1 IntD2 vimage_eq openin_subopen continuous_on_open_gen [OF LC]) 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" by (simp add: o_def linepath_def) 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