# Theory Abstract_Euclidean_Space

theory Abstract_Euclidean_Space
imports Homotopy Locally
```(*  Author:  LCP, ported from HOL Light
*)

section‹Euclidean space and n-spheres, as subtopologies of n-dimensional space›

theory Abstract_Euclidean_Space
imports Homotopy Locally
begin

subsection ‹Euclidean spaces as abstract topologies›

definition Euclidean_space :: "nat ⇒ (nat ⇒ real) topology"
where "Euclidean_space n ≡ subtopology (powertop_real UNIV) {x. ∀i≥n. x i = 0}"

lemma topspace_Euclidean_space:
"topspace(Euclidean_space n) = {x. ∀i≥n. x i = 0}"

lemma nonempty_Euclidean_space: "topspace(Euclidean_space n) ≠ {}"
by (force simp: topspace_Euclidean_space)

lemma subset_Euclidean_space [simp]:
"topspace(Euclidean_space m) ⊆ topspace(Euclidean_space n) ⟷ m ≤ n"
apply (simp add: topspace_Euclidean_space subset_iff, safe)
apply (drule_tac x="(λi. if i < m then 1 else 0)" in spec)
apply auto
using not_less by fastforce

lemma topspace_Euclidean_space_alt:
"topspace(Euclidean_space n) = (⋂i ∈ {n..}. {x. x ∈ topspace(powertop_real UNIV) ∧ x i ∈ {0}})"
by (auto simp: topspace_Euclidean_space)

lemma closedin_Euclidean_space:
"closedin (powertop_real UNIV) (topspace(Euclidean_space n))"
proof -
have "closedin (powertop_real UNIV) {x. x i = 0}" if "n ≤ i" for i
proof -
have "closedin (powertop_real UNIV) {x ∈ topspace (powertop_real UNIV). x i ∈ {0}}"
proof (rule closedin_continuous_map_preimage)
show "continuous_map (powertop_real UNIV) euclideanreal (λx. x i)"
by (metis UNIV_I continuous_map_product_coordinates)
show "closedin euclideanreal {0}"
by simp
qed
then show ?thesis
by auto
qed
then show ?thesis
unfolding topspace_Euclidean_space_alt
by force
qed

lemma closedin_Euclidean_imp_closed: "closedin (Euclidean_space m) S ⟹ closed S"
by (metis Euclidean_space_def closed_closedin closedin_Euclidean_space closedin_closed_subtopology euclidean_product_topology topspace_Euclidean_space)

lemma closedin_Euclidean_space_iff:
"closedin (Euclidean_space m) S ⟷ closed S ∧ S ⊆ topspace (Euclidean_space m)"
(is "?lhs ⟷ ?rhs")
proof
show "?lhs ⟹ ?rhs"
using closedin_closed_subtopology topspace_Euclidean_space
by (fastforce simp: topspace_Euclidean_space_alt closedin_Euclidean_imp_closed)
show "?rhs ⟹ ?lhs"
by (metis (no_types) closed_closedin euclidean_product_topology inf.orderE)
qed

lemma continuous_map_componentwise_Euclidean_space:
"continuous_map X (Euclidean_space n) (λx i. if i < n then f x i else 0) ⟷
(∀i < n. continuous_map X euclideanreal (λx. f x i))"
proof -
have *: "continuous_map X euclideanreal (λx. if k < n then f x k else 0)"
if "⋀i. i<n ⟹ continuous_map X euclideanreal (λx. f x i)" for k
by (intro continuous_intros that)
show ?thesis
unfolding Euclidean_space_def continuous_map_in_subtopology
by (fastforce simp: continuous_map_componentwise_UNIV * elim: continuous_map_eq)
qed

"⟦continuous_map X (Euclidean_space n) f; continuous_map X (Euclidean_space n) g⟧
⟹ continuous_map X (Euclidean_space n) (λx i. f x i + g x i)"
unfolding Euclidean_space_def continuous_map_in_subtopology

lemma continuous_map_Euclidean_space_diff [continuous_intros]:
"⟦continuous_map X (Euclidean_space n) f; continuous_map X (Euclidean_space n) g⟧
⟹ continuous_map X (Euclidean_space n) (λx i. f x i - g x i)"
unfolding Euclidean_space_def continuous_map_in_subtopology
by (fastforce simp add: continuous_map_componentwise_UNIV continuous_map_diff)

lemma continuous_map_Euclidean_space_iff:
"continuous_map (Euclidean_space m) euclidean g
= continuous_on (topspace (Euclidean_space m)) g"
proof
assume "continuous_map (Euclidean_space m) euclidean g"
then have "continuous_map (top_of_set {f. ∀n≥m. f n = 0}) euclidean g"
then show "continuous_on (topspace (Euclidean_space m)) g"
by (metis continuous_map_subtopology_eu subtopology_topspace topspace_Euclidean_space)
next
assume "continuous_on (topspace (Euclidean_space m)) g"
then have "continuous_map (top_of_set {f. ∀n≥m. f n = 0}) euclidean g"
by (metis (lifting) continuous_map_into_fulltopology continuous_map_subtopology_eu order_refl topspace_Euclidean_space)
then show "continuous_map (Euclidean_space m) euclidean g"
qed

lemma cm_Euclidean_space_iff_continuous_on:
"continuous_map (subtopology (Euclidean_space m) S) (Euclidean_space n) f
⟷ continuous_on (topspace (subtopology (Euclidean_space m) S)) f ∧
f ` (topspace (subtopology (Euclidean_space m) S)) ⊆ topspace (Euclidean_space n)"
(is "?P ⟷ ?Q ∧ ?R")
proof -
have ?Q if ?P
proof -
have "⋀n. Euclidean_space n = top_of_set {f. ∀m≥n. f m = 0}"
with that show ?thesis
qed
moreover
have ?R if ?P
using that by (simp add: image_subset_iff continuous_map_def)
moreover
have ?P if ?Q ?R
proof -
have "continuous_map (top_of_set (topspace (subtopology (subtopology (powertop_real UNIV) {f. ∀n≥m. f n = 0}) S))) (top_of_set (topspace (subtopology (powertop_real UNIV) {f. ∀na≥n. f na = 0}))) f"
using Euclidean_space_def that by auto
then show ?thesis
by (simp add: Euclidean_space_def euclidean_product_topology subtopology_subtopology)
qed
ultimately show ?thesis
by auto
qed

lemma homeomorphic_Euclidean_space_product_topology:
"Euclidean_space n homeomorphic_space product_topology (λi. euclideanreal) {..<n}"
proof -
have cm: "continuous_map (product_topology (λi. euclideanreal) {..<n})
euclideanreal (λx. if k < n then x k else 0)" for k
by (auto intro: continuous_map_if continuous_map_product_projection)
show ?thesis
unfolding homeomorphic_space_def homeomorphic_maps_def
apply (rule_tac x="λf. restrict f {..<n}" in exI)
apply (rule_tac x="λf i. if i < n then f i else 0" in exI)
apply (intro conjI continuous_map_from_subtopology)
apply (force simp: continuous_map_componentwise cm intro: continuous_map_product_projection)+
done
qed

lemma contractible_Euclidean_space [simp]: "contractible_space (Euclidean_space n)"
using homeomorphic_Euclidean_space_product_topology contractible_space_euclideanreal
contractible_space_product_topology homeomorphic_space_contractibility by blast

lemma path_connected_Euclidean_space: "path_connected_space (Euclidean_space n)"

lemma connected_Euclidean_space: "connected_space (Euclidean_space n)"

lemma locally_path_connected_Euclidean_space:
"locally_path_connected_space (Euclidean_space n)"
apply (simp add: homeomorphic_locally_path_connected_space [OF homeomorphic_Euclidean_space_product_topology [of n]]
locally_path_connected_space_product_topology)
using locally_path_connected_space_euclideanreal by auto

lemma compact_Euclidean_space:
"compact_space (Euclidean_space n) ⟷ n = 0"
by (auto simp: homeomorphic_compact_space [OF homeomorphic_Euclidean_space_product_topology] compact_space_product_topology)

subsection‹n-dimensional spheres›

definition nsphere where
"nsphere n ≡ subtopology (Euclidean_space (Suc n)) { x. (∑i≤n. x i ^ 2) = 1 }"

lemma nsphere:
"nsphere n = subtopology (powertop_real UNIV)
{x. (∑i≤n. x i ^ 2) = 1 ∧ (∀i>n. x i = 0)}"
by (simp add: nsphere_def Euclidean_space_def subtopology_subtopology Suc_le_eq Collect_conj_eq Int_commute)

lemma continuous_map_nsphere_projection: "continuous_map (nsphere n) euclideanreal (λx. x k)"
unfolding nsphere
by (blast intro: continuous_map_from_subtopology [OF continuous_map_product_projection])

lemma in_topspace_nsphere: "(λn. if n = 0 then 1 else 0) ∈ topspace (nsphere n)"
by (simp add: nsphere_def topspace_Euclidean_space power2_eq_square if_distrib [where f = "λx. x * _"] cong: if_cong)

lemma nonempty_nsphere [simp]: "~ (topspace(nsphere n) = {})"
using in_topspace_nsphere by auto

lemma subtopology_nsphere_equator:
"subtopology (nsphere (Suc n)) {x. x(Suc n) = 0} = nsphere n"
proof -
have "({x. (∑i≤n. (x i)⇧2) + (x (Suc n))⇧2 = 1 ∧ (∀i>Suc n. x i = 0)} ∩ {x. x (Suc n) = 0})
= {x. (∑i≤n. (x i)⇧2) = 1 ∧ (∀i>n. x i = (0::real))}"
using Suc_lessI [of n] by (fastforce simp: set_eq_iff)
then show ?thesis
qed

lemma topspace_nsphere_minus1:
assumes x: "x ∈ topspace (nsphere n)" and "x n = 0"
shows "x ∈ topspace (nsphere (n - Suc 0))"
proof (cases "n = 0")
case True
then show ?thesis
using x by auto
next
case False
have subt_eq: "nsphere (n - Suc 0) = subtopology (nsphere n) {x. x n = 0}"
by (metis False Suc_pred le_zero_eq not_le subtopology_nsphere_equator)
with x show ?thesis
qed

lemma continuous_map_nsphere_reflection:
"continuous_map (nsphere n) (nsphere n) (λx i. if i = k then -x i else x i)"
proof -
have cm: "continuous_map (powertop_real UNIV) euclideanreal (λx. if j = k then - x j else x j)" for j
proof (cases "j=k")
case True
then show ?thesis
by simp (metis UNIV_I continuous_map_product_projection)
next
case False
then show ?thesis
by (auto intro: continuous_map_product_projection)
qed
have eq: "(if i = k then x k * x k else x i * x i) = x i * x i" for i and x :: "nat ⇒ real"
by simp
show ?thesis
apply (simp add: nsphere continuous_map_in_subtopology continuous_map_componentwise_UNIV
continuous_map_from_subtopology cm)
apply (intro conjI allI impI continuous_intros continuous_map_from_subtopology continuous_map_product_projection)
apply (auto simp: power2_eq_square if_distrib [where f = "λx. x * _"] eq cong: if_cong)
done
qed

proposition contractible_space_upper_hemisphere:
assumes "k ≤ n"
shows "contractible_space(subtopology (nsphere n) {x. x k ≥ 0})"
proof -
define p:: "nat ⇒ real" where "p ≡ λi. if i = k then 1 else 0"
have "p ∈ topspace(nsphere n)"
using assms
by (simp add: nsphere p_def power2_eq_square if_distrib [where f = "λx. x * _"] cong: if_cong)
let ?g = "λx i. x i / sqrt(∑j≤n. x j ^ 2)"
let ?h = "λ(t,q) i. (1 - t) * q i + t * p i"
let ?Y = "subtopology (Euclidean_space (Suc n)) {x. 0 ≤ x k ∧ (∃i≤n. x i ≠ 0)}"
have "continuous_map (prod_topology (top_of_set {0..1}) (subtopology (nsphere n) {x. 0 ≤ x k}))
(subtopology (nsphere n) {x. 0 ≤ x k}) (?g ∘ ?h)"
proof (rule continuous_map_compose)
have *: "⟦0 ≤ b k; (∑i≤n. (b i)⇧2) = 1; ∀i>n. b i = 0; 0 ≤ a; a ≤ 1⟧
⟹ ∃i. (i = k ⟶ (1 - a) * b k + a ≠ 0) ∧
(i ≠ k ⟶ i ≤ n ∧ a ≠ 1 ∧ b i ≠ 0)" for a::real and b
apply (cases "a ≠ 1 ∧ b k = 0"; simp)
apply (metis (no_types, lifting) atMost_iff sum.neutral zero_power2)
by (metis add.commute add_le_same_cancel2 diff_ge_0_iff_ge diff_zero less_eq_real_def mult_eq_0_iff mult_nonneg_nonneg not_le numeral_One zero_neq_numeral)
show "continuous_map (prod_topology (top_of_set {0..1}) (subtopology (nsphere n) {x. 0 ≤ x k})) ?Y ?h"
using assms
apply (auto simp: * nsphere continuous_map_componentwise_UNIV
prod_topology_subtopology subtopology_subtopology case_prod_unfold
continuous_map_in_subtopology Euclidean_space_def p_def if_distrib [where f = "λx. _ * x"] cong: if_cong)
apply (intro continuous_map_prod_snd continuous_intros continuous_map_from_subtopology)
apply auto
done
next
have 1: "⋀x i. ⟦ i ≤ n; x i ≠ 0⟧ ⟹ (∑i≤n. (x i / sqrt (∑j≤n. (x j)⇧2))⇧2) = 1"
by (force simp: sum_nonneg sum_nonneg_eq_0_iff field_split_simps simp flip: sum_divide_distrib)
have cm: "continuous_map ?Y (nsphere n) (λx i. x i / sqrt (∑j≤n. (x j)⇧2))"
unfolding Euclidean_space_def nsphere subtopology_subtopology continuous_map_in_subtopology
proof (intro continuous_intros conjI)
show "continuous_map
(subtopology (powertop_real UNIV) ({x. ∀i≥Suc n. x i = 0} ∩ {x. 0 ≤ x k ∧ (∃i≤n. x i ≠ 0)}))
(powertop_real UNIV) (λx i. x i / sqrt (∑j≤n. (x j)⇧2))"
unfolding continuous_map_componentwise
by (intro continuous_intros conjI ballI) (auto simp: sum_nonneg_eq_0_iff)
qed (auto simp: 1)
show "continuous_map ?Y (subtopology (nsphere n) {x. 0 ≤ x k}) (λx i. x i / sqrt (∑j≤n. (x j)⇧2))"
by (force simp: cm sum_nonneg continuous_map_in_subtopology if_distrib [where f = "λx. _ * x"] cong: if_cong)
qed
moreover have "(?g ∘ ?h) (0, x) = x"
if "x ∈ topspace (subtopology (nsphere n) {x. 0 ≤ x k})" for x
using that
moreover
have "(?g ∘ ?h) (1, x) = p"
if "x ∈ topspace (subtopology (nsphere n) {x. 0 ≤ x k})" for x
by (force simp: assms p_def power2_eq_square if_distrib [where f = "λx. x * _"] cong: if_cong)
ultimately
show ?thesis
apply (rule_tac x=p in exI)
apply (rule_tac x="?g ∘ ?h" in exI, force)
done
qed

corollary contractible_space_lower_hemisphere:
assumes "k ≤ n"
shows "contractible_space(subtopology (nsphere n) {x. x k ≤ 0})"
proof -
have "contractible_space (subtopology (nsphere n) {x. 0 ≤ x k}) = ?thesis"
proof (rule homeomorphic_space_contractibility)
show "subtopology (nsphere n) {x. 0 ≤ x k} homeomorphic_space subtopology (nsphere n) {x. x k ≤ 0}"
unfolding homeomorphic_space_def homeomorphic_maps_def
apply (rule_tac x="λx i. if i = k then -(x i) else x i" in exI)+
apply (auto simp: continuous_map_in_subtopology continuous_map_from_subtopology
continuous_map_nsphere_reflection)
done
qed
then show ?thesis
using contractible_space_upper_hemisphere [OF assms] by metis
qed

proposition nullhomotopic_nonsurjective_sphere_map:
assumes f: "continuous_map (nsphere p) (nsphere p) f"
and fim: "f ` (topspace(nsphere p)) ≠ topspace(nsphere p)"
obtains a where "homotopic_with (λx. True) (nsphere p) (nsphere p) f (λx. a)"
proof -
obtain a where a: "a ∈ topspace(nsphere p)" "a ∉ f ` (topspace(nsphere p))"
using fim continuous_map_image_subset_topspace f by blast
then have a1: "(∑i≤p. (a i)⇧2) = 1" and a0: "⋀i. i > p ⟹ a i = 0"
have f1: "(∑j≤p. (f x j)⇧2) = 1" if "x ∈ topspace (nsphere p)" for x
proof -
have "f x ∈ topspace (nsphere p)"
using continuous_map_image_subset_topspace f that by blast
then show ?thesis
qed
show thesis
proof
let ?g = "λx i. x i / sqrt(∑j≤p. x j ^ 2)"
let ?h = "λ(t,x) i. (1 - t) * f x i - t * a i"
let ?Y = "subtopology (Euclidean_space(Suc p)) (- {λi. 0})"
let ?T01 = "top_of_set {0..1::real}"
have 1: "continuous_map (prod_topology ?T01 (nsphere p)) (nsphere p) (?g ∘ ?h)"
proof (rule continuous_map_compose)
have "continuous_map (prod_topology ?T01 (nsphere p)) euclideanreal ((λx. f x k) ∘ snd)" for k
unfolding nsphere
apply (rule continuous_map_compose [of _ "nsphere p" f, unfolded o_def])
using f apply (simp add: nsphere)
then have "continuous_map (prod_topology ?T01 (nsphere p)) euclideanreal (λr. ?h r k)"
for k
unfolding case_prod_unfold o_def
by (intro continuous_map_into_fulltopology [OF continuous_map_fst] continuous_intros) auto
moreover have "?h ` ({0..1} × topspace (nsphere p)) ⊆ {x. ∀i≥Suc p. x i = 0}"
using continuous_map_image_subset_topspace [OF f]
by (auto simp: nsphere image_subset_iff a0)
moreover have "(λi. 0) ∉ ?h ` ({0..1} × topspace (nsphere p))"
proof clarify
fix t b
assume eq: "(λi. 0) = (λi. (1 - t) * f b i - t * a i)" and "t ∈ {0..1}" and b: "b ∈ topspace (nsphere p)"
have "(1 - t)⇧2 = (∑i≤p. ((1 - t) * f b i)^2)"
using f1 [OF b] by (simp add: power_mult_distrib flip: sum_distrib_left)
also have "… = (∑i≤p. (t * a i)^2)"
using eq by (simp add: fun_eq_iff)
also have "… = t⇧2"
using a1 by (simp add: power_mult_distrib flip: sum_distrib_left)
finally have "1 - t = t"
then have *: "t = 1/2"
by simp
have fba: "f b ≠ a"
using a(2) b by blast
then show False
using eq unfolding * by (simp add: fun_eq_iff)
qed
ultimately show "continuous_map (prod_topology ?T01 (nsphere p)) ?Y ?h"
by (simp add: Euclidean_space_def continuous_map_in_subtopology continuous_map_componentwise_UNIV)
next
have *: "⟦∀i≥Suc p. x i = 0; x ≠ (λi. 0)⟧ ⟹ (∑j≤p. (x j)⇧2) ≠ 0" for x :: "nat ⇒ real"
by (force simp: fun_eq_iff not_less_eq_eq sum_nonneg_eq_0_iff)
show "continuous_map ?Y (nsphere p) ?g"
apply (simp add: Euclidean_space_def continuous_map_in_subtopology continuous_map_componentwise_UNIV
nsphere continuous_map_componentwise subtopology_subtopology)
apply (intro conjI allI continuous_intros continuous_map_from_subtopology [OF continuous_map_product_projection])
apply (force simp: sum_nonneg fun_eq_iff not_less_eq_eq sum_nonneg_eq_0_iff power_divide simp flip: sum_divide_distrib)
done
qed
have 2: "(?g ∘ ?h) (0, x) = f x" if "x ∈ topspace (nsphere p)" for x
using that f1 by simp
have 3: "(?g ∘ ?h) (1, x) = (λi. - a i)" for x
using a by (force simp: field_split_simps nsphere)
then show "homotopic_with (λx. True) (nsphere p) (nsphere p) f (λx. (λi. - a i))"
by (force simp: homotopic_with intro: 1 2 3)
qed
qed

lemma Hausdorff_Euclidean_space:
"Hausdorff_space (Euclidean_space n)"
unfolding Euclidean_space_def
by (rule Hausdorff_space_subtopology) (metis Hausdorff_space_euclidean Hausdorff_space_product_topology)

end

```