section\<open>Invariance of Domain\<close>
theory Invariance_of_Domain
imports Brouwer_Degree "HOL-Analysis.Continuous_Extension" "HOL-Analysis.Homeomorphism"
begin
subsection\<open>Degree invariance mod 2 for map between pairs\<close>
theorem Borsuk_odd_mapping_degree_step:
assumes cmf: "continuous_map (nsphere n) (nsphere n) f"
and f: "\<And>x. x \<in> topspace(nsphere n) \<Longrightarrow> (f \<circ> (\<lambda>x i. -x i)) x = ((\<lambda>x i. -x i) \<circ> f) x"
and fim: "f ` (topspace(nsphere(n - Suc 0))) \<subseteq> topspace(nsphere(n - Suc 0))"
shows "even (Brouwer_degree2 n f - Brouwer_degree2 (n - Suc 0) f)"
proof (cases "n = 0")
case False
define neg where "neg \<equiv> \<lambda>x::nat\<Rightarrow>real. \<lambda>i. -x i"
define upper where "upper \<equiv> \<lambda>n. {x::nat\<Rightarrow>real. x n \<ge> 0}"
define lower where "lower \<equiv> \<lambda>n. {x::nat\<Rightarrow>real. x n \<le> 0}"
define equator where "equator \<equiv> \<lambda>n. {x::nat\<Rightarrow>real. x n = 0}"
define usphere where "usphere \<equiv> \<lambda>n. subtopology (nsphere n) (upper n)"
define lsphere where "lsphere \<equiv> \<lambda>n. subtopology (nsphere n) (lower n)"
have [simp]: "neg x i = -x i" for x i
by (force simp: neg_def)
have equator_upper: "equator n \<subseteq> upper n"
by (force simp: equator_def upper_def)
have upper_usphere: "subtopology (nsphere n) (upper n) = usphere n"
by (simp add: usphere_def)
let ?rhgn = "relative_homology_group n (nsphere n)"
let ?hi_ee = "hom_induced n (nsphere n) (equator n) (nsphere n) (equator n)"
interpret GE: comm_group "?rhgn (equator n)"
by simp
interpret HB: group_hom "?rhgn (equator n)"
"homology_group (int n - 1) (subtopology (nsphere n) (equator n))"
"hom_boundary n (nsphere n) (equator n)"
by (simp add: group_hom_axioms_def group_hom_def hom_boundary_hom)
interpret HIU: group_hom "?rhgn (equator n)"
"?rhgn (upper n)"
"hom_induced n (nsphere n) (equator n) (nsphere n) (upper n) id"
by (simp add: group_hom_axioms_def group_hom_def hom_induced_hom)
have subt_eq: "subtopology (nsphere n) {x. x n = 0} = nsphere (n - Suc 0)"
by (metis False Suc_pred le_zero_eq not_le subtopology_nsphere_equator)
then have equ: "subtopology (nsphere n) (equator n) = nsphere(n - Suc 0)"
"subtopology (lsphere n) (equator n) = nsphere(n - Suc 0)"
"subtopology (usphere n) (equator n) = nsphere(n - Suc 0)"
using False by (auto simp: lsphere_def usphere_def equator_def lower_def upper_def subtopology_subtopology simp flip: Collect_conj_eq cong: rev_conj_cong)
have cmr: "continuous_map (nsphere(n - Suc 0)) (nsphere(n - Suc 0)) f"
by (metis (no_types, lifting) IntE cmf fim continuous_map_from_subtopology continuous_map_in_subtopology equ(1) image_subset_iff topspace_subtopology)
have "f x n = 0" if "x \<in> topspace (nsphere n)" "x n = 0" for x
proof -
have "x \<in> topspace (nsphere (n - Suc 0))"
by (simp add: that topspace_nsphere_minus1)
moreover have "topspace (nsphere n) \<inter> {f. f n = 0} = topspace (nsphere (n - Suc 0))"
by (metis subt_eq topspace_subtopology)
ultimately show ?thesis
using cmr continuous_map_def by fastforce
qed
then have fimeq: "f ` (topspace (nsphere n) \<inter> equator n) \<subseteq> topspace (nsphere n) \<inter> equator n"
using fim cmf by (auto simp: equator_def continuous_map_def image_subset_iff)
have "\<And>k. continuous_map (powertop_real UNIV) euclideanreal (\<lambda>x. - x k)"
by (metis UNIV_I continuous_map_product_projection continuous_map_minus)
then have cm_neg: "continuous_map (nsphere m) (nsphere m) neg" for m
by (force simp: nsphere continuous_map_in_subtopology neg_def continuous_map_componentwise_UNIV intro: continuous_map_from_subtopology)
then have cm_neg_lu: "continuous_map (lsphere n) (usphere n) neg"
by (auto simp: lsphere_def usphere_def lower_def upper_def continuous_map_from_subtopology continuous_map_in_subtopology)
have neg_in_top_iff: "neg x \<in> topspace(nsphere m) \<longleftrightarrow> x \<in> topspace(nsphere m)" for m x
by (simp add: nsphere_def neg_def topspace_Euclidean_space)
obtain z where zcarr: "z \<in> carrier (reduced_homology_group (int n - 1) (nsphere (n - Suc 0)))"
and zeq: "subgroup_generated (reduced_homology_group (int n - 1) (nsphere (n - Suc 0))) {z}
= reduced_homology_group (int n - 1) (nsphere (n - Suc 0))"
using cyclic_reduced_homology_group_nsphere [of "int n - 1" "n - Suc 0"] by (auto simp: cyclic_group_def)
have "hom_boundary n (subtopology (nsphere n) {x. x n \<le> 0}) {x. x n = 0}
\<in> Group.iso (relative_homology_group n
(subtopology (nsphere n) {x. x n \<le> 0}) {x. x n = 0})
(reduced_homology_group (int n - 1) (nsphere (n - Suc 0)))"
using iso_lower_hemisphere_reduced_homology_group [of "int n - 1" "n - Suc 0"] False by simp
then obtain gp where g: "group_isomorphisms
(relative_homology_group n (subtopology (nsphere n) {x. x n \<le> 0}) {x. x n = 0})
(reduced_homology_group (int n - 1) (nsphere (n - Suc 0)))
(hom_boundary n (subtopology (nsphere n) {x. x n \<le> 0}) {x. x n = 0})
gp"
by (auto simp: group.iso_iff_group_isomorphisms)
then interpret gp: group_hom "reduced_homology_group (int n - 1) (nsphere (n - Suc 0))"
"relative_homology_group n (subtopology (nsphere n) {x. x n \<le> 0}) {x. x n = 0}" gp
by (simp add: group_hom_axioms_def group_hom_def group_isomorphisms_def)
obtain zp where zpcarr: "zp \<in> carrier(relative_homology_group n (lsphere n) (equator n))"
and zp_z: "hom_boundary n (lsphere n) (equator n) zp = z"
and zp_sg: "subgroup_generated (relative_homology_group n (lsphere n) (equator n)) {zp}
= relative_homology_group n (lsphere n) (equator n)"
proof
show "gp z \<in> carrier (relative_homology_group n (lsphere n) (equator n))"
"hom_boundary n (lsphere n) (equator n) (gp z) = z"
using g zcarr by (auto simp: lsphere_def equator_def lower_def group_isomorphisms_def)
have giso: "gp \<in> Group.iso (reduced_homology_group (int n - 1) (nsphere (n - Suc 0)))
(relative_homology_group n (subtopology (nsphere n) {x. x n \<le> 0}) {x. x n = 0})"
by (metis (mono_tags, lifting) g group_isomorphisms_imp_iso group_isomorphisms_sym)
show "subgroup_generated (relative_homology_group n (lsphere n) (equator n)) {gp z} =
relative_homology_group n (lsphere n) (equator n)"
apply (rule monoid.equality)
using giso gp.subgroup_generated_by_image [of "{z}"] zcarr
by (auto simp: lsphere_def equator_def lower_def zeq gp.iso_iff)
qed
have hb_iso: "hom_boundary n (subtopology (nsphere n) {x. x n \<ge> 0}) {x. x n = 0}
\<in> iso (relative_homology_group n (subtopology (nsphere n) {x. x n \<ge> 0}) {x. x n = 0})
(reduced_homology_group (int n - 1) (nsphere (n - Suc 0)))"
using iso_upper_hemisphere_reduced_homology_group [of "int n - 1" "n - Suc 0"] False by simp
then obtain gn where g: "group_isomorphisms
(relative_homology_group n (subtopology (nsphere n) {x. x n \<ge> 0}) {x. x n = 0})
(reduced_homology_group (int n - 1) (nsphere (n - Suc 0)))
(hom_boundary n (subtopology (nsphere n) {x. x n \<ge> 0}) {x. x n = 0})
gn"
by (auto simp: group.iso_iff_group_isomorphisms)
then interpret gn: group_hom "reduced_homology_group (int n - 1) (nsphere (n - Suc 0))"
"relative_homology_group n (subtopology (nsphere n) {x. x n \<ge> 0}) {x. x n = 0}" gn
by (simp add: group_hom_axioms_def group_hom_def group_isomorphisms_def)
obtain zn where zncarr: "zn \<in> carrier(relative_homology_group n (usphere n) (equator n))"
and zn_z: "hom_boundary n (usphere n) (equator n) zn = z"
and zn_sg: "subgroup_generated (relative_homology_group n (usphere n) (equator n)) {zn}
= relative_homology_group n (usphere n) (equator n)"
proof
show "gn z \<in> carrier (relative_homology_group n (usphere n) (equator n))"
"hom_boundary n (usphere n) (equator n) (gn z) = z"
using g zcarr by (auto simp: usphere_def equator_def upper_def group_isomorphisms_def)
have giso: "gn \<in> Group.iso (reduced_homology_group (int n - 1) (nsphere (n - Suc 0)))
(relative_homology_group n (subtopology (nsphere n) {x. x n \<ge> 0}) {x. x n = 0})"
by (metis (mono_tags, lifting) g group_isomorphisms_imp_iso group_isomorphisms_sym)
show "subgroup_generated (relative_homology_group n (usphere n) (equator n)) {gn z} =
relative_homology_group n (usphere n) (equator n)"
apply (rule monoid.equality)
using giso gn.subgroup_generated_by_image [of "{z}"] zcarr
by (auto simp: usphere_def equator_def upper_def zeq gn.iso_iff)
qed
let ?hi_lu = "hom_induced n (lsphere n) (equator n) (nsphere n) (upper n) id"
interpret gh_lu: group_hom "relative_homology_group n (lsphere n) (equator n)" "?rhgn (upper n)" ?hi_lu
by (simp add: group_hom_axioms_def group_hom_def hom_induced_hom)
interpret gh_eef: group_hom "?rhgn (equator n)" "?rhgn (equator n)" "?hi_ee f"
by (simp add: group_hom_axioms_def group_hom_def hom_induced_hom)
define wp where "wp \<equiv> ?hi_lu zp"
then have wpcarr: "wp \<in> carrier(?rhgn (upper n))"
by (simp add: hom_induced_carrier)
have "hom_induced n (nsphere n) {} (nsphere n) {x. x n \<ge> 0} id
\<in> iso (reduced_homology_group n (nsphere n))
(?rhgn {x. x n \<ge> 0})"
using iso_reduced_homology_group_upper_hemisphere [of n n n] by auto
then have "carrier(?rhgn {x. x n \<ge> 0})
\<subseteq> (hom_induced n (nsphere n) {} (nsphere n) {x. x n \<ge> 0} id)
` carrier(reduced_homology_group n (nsphere n))"
by (simp add: iso_iff)
then obtain vp where vpcarr: "vp \<in> carrier(reduced_homology_group n (nsphere n))"
and eqwp: "hom_induced n (nsphere n) {} (nsphere n) (upper n) id vp = wp"
using wpcarr by (auto simp: upper_def)
define wn where "wn \<equiv> hom_induced n (usphere n) (equator n) (nsphere n) (lower n) id zn"
then have wncarr: "wn \<in> carrier(?rhgn (lower n))"
by (simp add: hom_induced_carrier)
have "hom_induced n (nsphere n) {} (nsphere n) {x. x n \<le> 0} id
\<in> iso (reduced_homology_group n (nsphere n))
(?rhgn {x. x n \<le> 0})"
using iso_reduced_homology_group_lower_hemisphere [of n n n] by auto
then have "carrier(?rhgn {x. x n \<le> 0})
\<subseteq> (hom_induced n (nsphere n) {} (nsphere n) {x. x n \<le> 0} id)
` carrier(reduced_homology_group n (nsphere n))"
by (simp add: iso_iff)
then obtain vn where vpcarr: "vn \<in> carrier(reduced_homology_group n (nsphere n))"
and eqwp: "hom_induced n (nsphere n) {} (nsphere n) (lower n) id vn = wn"
using wncarr by (auto simp: lower_def)
define up where "up \<equiv> hom_induced n (lsphere n) (equator n) (nsphere n) (equator n) id zp"
then have upcarr: "up \<in> carrier(?rhgn (equator n))"
by (simp add: hom_induced_carrier)
define un where "un \<equiv> hom_induced n (usphere n) (equator n) (nsphere n) (equator n) id zn"
then have uncarr: "un \<in> carrier(?rhgn (equator n))"
by (simp add: hom_induced_carrier)
have *: "(\<lambda>(x, y).
hom_induced n (lsphere n) (equator n) (nsphere n) (equator n) id x
\<otimes>\<^bsub>?rhgn (equator n)\<^esub>
hom_induced n (usphere n) (equator n) (nsphere n) (equator n) id y)
\<in> Group.iso
(relative_homology_group n (lsphere n) (equator n) \<times>\<times>
relative_homology_group n (usphere n) (equator n))
(?rhgn (equator n))"
proof (rule conjunct1 [OF exact_sequence_sum_lemma [OF abelian_relative_homology_group]])
show "hom_induced n (lsphere n) (equator n) (nsphere n) (upper n) id
\<in> Group.iso (relative_homology_group n (lsphere n) (equator n))
(?rhgn (upper n))"
apply (simp add: lsphere_def usphere_def equator_def lower_def upper_def)
using iso_relative_homology_group_lower_hemisphere by blast
show "hom_induced n (usphere n) (equator n) (nsphere n) (lower n) id
\<in> Group.iso (relative_homology_group n (usphere n) (equator n))
(?rhgn (lower n))"
apply (simp_all add: lsphere_def usphere_def equator_def lower_def upper_def)
using iso_relative_homology_group_upper_hemisphere by blast+
show "exact_seq
([?rhgn (lower n),
?rhgn (equator n),
relative_homology_group n (lsphere n) (equator n)],
[hom_induced n (nsphere n) (equator n) (nsphere n) (lower n) id,
hom_induced n (lsphere n) (equator n) (nsphere n) (equator n) id])"
unfolding lsphere_def usphere_def equator_def lower_def upper_def
by (rule homology_exactness_triple_3) force
show "exact_seq
([?rhgn (upper n),
?rhgn (equator n),
relative_homology_group n (usphere n) (equator n)],
[hom_induced n (nsphere n) (equator n) (nsphere n) (upper n) id,
hom_induced n (usphere n) (equator n) (nsphere n) (equator n) id])"
unfolding lsphere_def usphere_def equator_def lower_def upper_def
by (rule homology_exactness_triple_3) force
next
fix x
assume "x \<in> carrier (relative_homology_group n (lsphere n) (equator n))"
show "hom_induced n (nsphere n) (equator n) (nsphere n) (upper n) id
(hom_induced n (lsphere n) (equator n) (nsphere n) (equator n) id x) =
hom_induced n (lsphere n) (equator n) (nsphere n) (upper n) id x"
by (simp add: hom_induced_compose' subset_iff lsphere_def usphere_def equator_def lower_def upper_def)
next
fix x
assume "x \<in> carrier (relative_homology_group n (usphere n) (equator n))"
show "hom_induced n (nsphere n) (equator n) (nsphere n) (lower n) id
(hom_induced n (usphere n) (equator n) (nsphere n) (equator n) id x) =
hom_induced n (usphere n) (equator n) (nsphere n) (lower n) id x"
by (simp add: hom_induced_compose' subset_iff lsphere_def usphere_def equator_def lower_def upper_def)
qed
then have sb: "carrier (?rhgn (equator n))
\<subseteq> (\<lambda>(x, y).
hom_induced n (lsphere n) (equator n) (nsphere n) (equator n) id x
\<otimes>\<^bsub>?rhgn (equator n)\<^esub>
hom_induced n (usphere n) (equator n) (nsphere n) (equator n) id y)
` carrier (relative_homology_group n (lsphere n) (equator n) \<times>\<times>
relative_homology_group n (usphere n) (equator n))"
by (simp add: iso_iff)
obtain a b::int
where up_ab: "?hi_ee f up
= up [^]\<^bsub>?rhgn (equator n)\<^esub> a\<otimes>\<^bsub>?rhgn (equator n)\<^esub> un [^]\<^bsub>?rhgn (equator n)\<^esub> b"
proof -
have hiupcarr: "?hi_ee f up \<in> carrier(?rhgn (equator n))"
by (simp add: hom_induced_carrier)
obtain u v where u: "u \<in> carrier (relative_homology_group n (lsphere n) (equator n))"
and v: "v \<in> carrier (relative_homology_group n (usphere n) (equator n))"
and eq: "?hi_ee f up =
hom_induced n (lsphere n) (equator n) (nsphere n) (equator n) id u
\<otimes>\<^bsub>?rhgn (equator n)\<^esub>
hom_induced n (usphere n) (equator n) (nsphere n) (equator n) id v"
using subsetD [OF sb hiupcarr] by auto
have "u \<in> carrier (subgroup_generated (relative_homology_group n (lsphere n) (equator n)) {zp})"
by (simp_all add: u zp_sg)
then obtain a::int where a: "u = zp [^]\<^bsub>relative_homology_group n (lsphere n) (equator n)\<^esub> a"
by (metis group.carrier_subgroup_generated_by_singleton group_relative_homology_group rangeE zpcarr)
have ae: "hom_induced n (lsphere n) (equator n) (nsphere n) (equator n) id
(pow (relative_homology_group n (lsphere n) (equator n)) zp a)
= pow (?rhgn (equator n)) (hom_induced n (lsphere n) (equator n) (nsphere n) (equator n) id zp) a"
by (meson group_hom.hom_int_pow group_hom_axioms_def group_hom_def group_relative_homology_group hom_induced zpcarr)
have "v \<in> carrier (subgroup_generated (relative_homology_group n (usphere n) (equator n)) {zn})"
by (simp_all add: v zn_sg)
then obtain b::int where b: "v = zn [^]\<^bsub>relative_homology_group n (usphere n) (equator n)\<^esub> b"
by (metis group.carrier_subgroup_generated_by_singleton group_relative_homology_group rangeE zncarr)
have be: "hom_induced n (usphere n) (equator n) (nsphere n) (equator n) id
(zn [^]\<^bsub>relative_homology_group n (usphere n) (equator n)\<^esub> b)
= hom_induced n (usphere n) (equator n) (nsphere n) (equator n) id
zn [^]\<^bsub>relative_homology_group n (nsphere n) (equator n)\<^esub> b"
by (meson group_hom.hom_int_pow group_hom_axioms_def group_hom_def group_relative_homology_group hom_induced zncarr)
show thesis
proof
show "?hi_ee f up
= up [^]\<^bsub>?rhgn (equator n)\<^esub> a \<otimes>\<^bsub>?rhgn (equator n)\<^esub> un [^]\<^bsub>?rhgn (equator n)\<^esub> b"
using a ae b be eq local.up_def un_def by auto
qed
qed
have "(hom_boundary n (nsphere n) (equator n)
\<circ> hom_induced n (lsphere n) (equator n) (nsphere n) (equator n) id) zp = z"
using zp_z equ apply (simp add: lsphere_def naturality_hom_induced)
by (metis hom_boundary_carrier hom_induced_id)
then have up_z: "hom_boundary n (nsphere n) (equator n) up = z"
by (simp add: up_def)
have "(hom_boundary n (nsphere n) (equator n)
\<circ> hom_induced n (usphere n) (equator n) (nsphere n) (equator n) id) zn = z"
using zn_z equ apply (simp add: usphere_def naturality_hom_induced)
by (metis hom_boundary_carrier hom_induced_id)
then have un_z: "hom_boundary n (nsphere n) (equator n) un = z"
by (simp add: un_def)
have Bd_ab: "Brouwer_degree2 (n - Suc 0) f = a + b"
proof (rule Brouwer_degree2_unique_generator; use False int_ops in simp_all)
show "continuous_map (nsphere (n - Suc 0)) (nsphere (n - Suc 0)) f"
using cmr by auto
show "subgroup_generated (reduced_homology_group (int n - 1) (nsphere (n - Suc 0))) {z} =
reduced_homology_group (int n - 1) (nsphere (n - Suc 0))"
using zeq by blast
have "(hom_induced (int n - 1) (nsphere (n - Suc 0)) {} (nsphere (n - Suc 0)) {} f
\<circ> hom_boundary n (nsphere n) (equator n)) up
= (hom_boundary n (nsphere n) (equator n) \<circ>
?hi_ee f) up"
using naturality_hom_induced [OF cmf fimeq, of n, symmetric]
by (simp add: subtopology_restrict equ fun_eq_iff)
also have "\<dots> = hom_boundary n (nsphere n) (equator n)
(up [^]\<^bsub>relative_homology_group n (nsphere n) (equator n)\<^esub>
a \<otimes>\<^bsub>relative_homology_group n (nsphere n) (equator n)\<^esub>
un [^]\<^bsub>relative_homology_group n (nsphere n) (equator n)\<^esub> b)"
by (simp add: o_def up_ab)
also have "\<dots> = z [^]\<^bsub>reduced_homology_group (int n - 1) (nsphere (n - Suc 0))\<^esub> (a + b)"
using zcarr
apply (simp add: HB.hom_int_pow reduced_homology_group_def group.int_pow_subgroup_generated upcarr uncarr)
by (metis equ(1) group.int_pow_mult group_relative_homology_group hom_boundary_carrier un_z up_z)
finally show "hom_induced (int n - 1) (nsphere (n - Suc 0)) {} (nsphere (n - Suc 0)) {} f z =
z [^]\<^bsub>reduced_homology_group (int n - 1) (nsphere (n - Suc 0))\<^esub> (a + b)"
by (simp add: up_z)
qed
define u where "u \<equiv> up \<otimes>\<^bsub>?rhgn (equator n)\<^esub> inv\<^bsub>?rhgn (equator n)\<^esub> un"
have ucarr: "u \<in> carrier (?rhgn (equator n))"
by (simp add: u_def uncarr upcarr)
then have "u [^]\<^bsub>?rhgn (equator n)\<^esub> Brouwer_degree2 n f = u [^]\<^bsub>?rhgn (equator n)\<^esub> (a - b)
\<longleftrightarrow> (GE.ord u) dvd a - b - Brouwer_degree2 n f"
by (simp add: GE.int_pow_eq)
moreover
have "GE.ord u = 0"
proof (clarsimp simp add: GE.ord_eq_0 ucarr)
fix d :: nat
assume "0 < d"
and "u [^]\<^bsub>?rhgn (equator n)\<^esub> d = singular_relboundary_set n (nsphere n) (equator n)"
then have "hom_induced n (nsphere n) (equator n) (nsphere n) (upper n) id u [^]\<^bsub>?rhgn (upper n)\<^esub> d
= \<one>\<^bsub>?rhgn (upper n)\<^esub>"
by (metis HIU.hom_one HIU.hom_nat_pow one_relative_homology_group ucarr)
moreover
have "?hi_lu
= hom_induced n (nsphere n) (equator n) (nsphere n) (upper n) id \<circ>
hom_induced n (lsphere n) (equator n) (nsphere n) (equator n) id"
by (simp add: lsphere_def image_subset_iff equator_upper flip: hom_induced_compose)
then have p: "wp = hom_induced n (nsphere n) (equator n) (nsphere n) (upper n) id up"
by (simp add: local.up_def wp_def)
have n: "hom_induced n (nsphere n) (equator n) (nsphere n) (upper n) id un = \<one>\<^bsub>?rhgn (upper n)\<^esub>"
using homology_exactness_triple_3 [OF equator_upper, of n "nsphere n"]
using un_def zncarr by (auto simp: upper_usphere kernel_def)
have "hom_induced n (nsphere n) (equator n) (nsphere n) (upper n) id u = wp"
unfolding u_def
using p n HIU.inv_one HIU.r_one uncarr upcarr by auto
ultimately have "(wp [^]\<^bsub>?rhgn (upper n)\<^esub> d) = \<one>\<^bsub>?rhgn (upper n)\<^esub>"
by simp
moreover have "infinite (carrier (subgroup_generated (?rhgn (upper n)) {wp}))"
proof -
have "?rhgn (upper n) \<cong> reduced_homology_group n (nsphere n)"
unfolding upper_def
using iso_reduced_homology_group_upper_hemisphere [of n n n]
by (blast intro: group.iso_sym group_reduced_homology_group is_isoI)
also have "\<dots> \<cong> integer_group"
by (simp add: reduced_homology_group_nsphere)
finally have iso: "?rhgn (upper n) \<cong> integer_group" .
have "carrier (subgroup_generated (?rhgn (upper n)) {wp}) = carrier (?rhgn (upper n))"
using gh_lu.subgroup_generated_by_image [of "{zp}"] zpcarr HIU.carrier_subgroup_generated_subset
gh_lu.iso_iff iso_relative_homology_group_lower_hemisphere zp_sg
by (auto simp: lower_def lsphere_def upper_def equator_def wp_def)
then show ?thesis
using infinite_UNIV_int iso_finite [OF iso] by simp
qed
ultimately show False
using HIU.finite_cyclic_subgroup \<open>0 < d\<close> wpcarr by blast
qed
ultimately have iff: "u [^]\<^bsub>?rhgn (equator n)\<^esub> Brouwer_degree2 n f = u [^]\<^bsub>?rhgn (equator n)\<^esub> (a - b)
\<longleftrightarrow> Brouwer_degree2 n f = a - b"
by auto
have "u [^]\<^bsub>?rhgn (equator n)\<^esub> Brouwer_degree2 n f = ?hi_ee f u"
proof -
have ne: "topspace (nsphere n) \<inter> equator n \<noteq> {}"
by (metis equ(1) nonempty_nsphere topspace_subtopology)
have eq1: "hom_boundary n (nsphere n) (equator n) u
= \<one>\<^bsub>reduced_homology_group (int n - 1) (subtopology (nsphere n) (equator n))\<^esub>"
using one_reduced_homology_group u_def un_z uncarr up_z upcarr by force
then have uhom: "u \<in> hom_induced n (nsphere n) {} (nsphere n) (equator n) id `
carrier (reduced_homology_group (int n) (nsphere n))"
using homology_exactness_reduced_1 [OF ne, of n] eq1 ucarr by (auto simp: kernel_def)
then obtain v where vcarr: "v \<in> carrier (reduced_homology_group (int n) (nsphere n))"
and ueq: "u = hom_induced n (nsphere n) {} (nsphere n) (equator n) id v"
by blast
interpret GH_hi: group_hom "homology_group n (nsphere n)"
"?rhgn (equator n)"
"hom_induced n (nsphere n) {} (nsphere n) (equator n) id"
by (simp add: group_hom_axioms_def group_hom_def hom_induced_hom)
have poweq: "pow (homology_group n (nsphere n)) x i = pow (reduced_homology_group n (nsphere n)) x i"
for x and i::int
by (simp add: False un_reduced_homology_group)
have vcarr': "v \<in> carrier (homology_group n (nsphere n))"
using carrier_reduced_homology_group_subset vcarr by blast
have "u [^]\<^bsub>?rhgn (equator n)\<^esub> Brouwer_degree2 n f
= hom_induced n (nsphere n) {} (nsphere n) (equator n) f v"
using vcarr vcarr'
by (simp add: ueq poweq hom_induced_compose' cmf flip: GH_hi.hom_int_pow Brouwer_degree2)
also have "\<dots> = hom_induced n (nsphere n) (topspace(nsphere n) \<inter> equator n) (nsphere n) (equator n) f
(hom_induced n (nsphere n) {} (nsphere n) (topspace(nsphere n) \<inter> equator n) id v)"
using fimeq by (simp add: hom_induced_compose' cmf)
also have "\<dots> = ?hi_ee f u"
by (metis hom_induced inf.left_idem ueq)
finally show ?thesis .
qed
moreover
interpret gh_een: group_hom "?rhgn (equator n)" "?rhgn (equator n)" "?hi_ee neg"
by (simp add: group_hom_axioms_def group_hom_def hom_induced_hom)
have hi_up_eq_un: "?hi_ee neg up = un [^]\<^bsub>?rhgn (equator n)\<^esub> Brouwer_degree2 (n - Suc 0) neg"
proof -
have "?hi_ee neg (hom_induced n (lsphere n) (equator n) (nsphere n) (equator n) id zp)
= hom_induced n (lsphere n) (equator n) (nsphere n) (equator n) (neg \<circ> id) zp"
by (intro hom_induced_compose') (auto simp: lsphere_def equator_def cm_neg)
also have "\<dots> = hom_induced n (usphere n) (equator n) (nsphere n) (equator n) id
(hom_induced n (lsphere n) (equator n) (usphere n) (equator n) neg zp)"
by (subst hom_induced_compose' [OF cm_neg_lu]) (auto simp: usphere_def equator_def)
also have "hom_induced n (lsphere n) (equator n) (usphere n) (equator n) neg zp
= zn [^]\<^bsub>relative_homology_group n (usphere n) (equator n)\<^esub> Brouwer_degree2 (n - Suc 0) neg"
proof -
let ?hb = "hom_boundary n (usphere n) (equator n)"
have eq: "subtopology (nsphere n) {x. x n \<ge> 0} = usphere n \<and> {x. x n = 0} = equator n"
by (auto simp: usphere_def upper_def equator_def)
with hb_iso have inj: "inj_on (?hb) (carrier (relative_homology_group n (usphere n) (equator n)))"
by (simp add: iso_iff)
interpret hb_hom: group_hom "relative_homology_group n (usphere n) (equator n)"
"reduced_homology_group (int n - 1) (nsphere (n - Suc 0))"
"?hb"
using hb_iso iso_iff eq group_hom_axioms_def group_hom_def by fastforce
show ?thesis
proof (rule inj_onD [OF inj])
have *: "hom_induced (int n - 1) (nsphere (n - Suc 0)) {} (nsphere (n - Suc 0)) {} neg z
= z [^]\<^bsub>homology_group (int n - 1) (nsphere (n - Suc 0))\<^esub> Brouwer_degree2 (n - Suc 0) neg"
using Brouwer_degree2 [of z "n - Suc 0" neg] False zcarr
by (simp add: int_ops group.int_pow_subgroup_generated reduced_homology_group_def)
have "?hb \<circ>
hom_induced n (lsphere n) (equator n) (usphere n) (equator n) neg
= hom_induced (int n - 1) (nsphere (n - Suc 0)) {} (nsphere (n - Suc 0)) {} neg \<circ>
hom_boundary n (lsphere n) (equator n)"
apply (subst naturality_hom_induced [OF cm_neg_lu])
apply (force simp: equator_def neg_def)
by (simp add: equ)
then have "?hb
(hom_induced n (lsphere n) (equator n) (usphere n) (equator n) neg zp)
= (z [^]\<^bsub>homology_group (int n - 1) (nsphere (n - Suc 0))\<^esub> Brouwer_degree2 (n - Suc 0) neg)"
by (metis "*" comp_apply zp_z)
also have "\<dots> = ?hb (zn [^]\<^bsub>relative_homology_group n (usphere n) (equator n)\<^esub>
Brouwer_degree2 (n - Suc 0) neg)"
by (metis group.int_pow_subgroup_generated group_relative_homology_group hb_hom.hom_int_pow reduced_homology_group_def zcarr zn_z zncarr)
finally show "?hb (hom_induced n (lsphere n) (equator n) (usphere n) (equator n) neg zp) =
?hb (zn [^]\<^bsub>relative_homology_group n (usphere n) (equator n)\<^esub>
Brouwer_degree2 (n - Suc 0) neg)" by simp
qed (auto simp: hom_induced_carrier group.int_pow_closed zncarr)
qed
finally show ?thesis
by (metis (no_types, lifting) group_hom.hom_int_pow group_hom_axioms_def group_hom_def group_relative_homology_group hom_induced local.up_def un_def zncarr)
qed
have "continuous_map (nsphere (n - Suc 0)) (nsphere (n - Suc 0)) neg"
using cm_neg by blast
then have "homeomorphic_map (nsphere (n - Suc 0)) (nsphere (n - Suc 0)) neg"
apply (auto simp: homeomorphic_map_maps homeomorphic_maps_def)
apply (rule_tac x=neg in exI, auto)
done
then have Brouwer_degree2_21: "Brouwer_degree2 (n - Suc 0) neg ^ 2 = 1"
using Brouwer_degree2_homeomorphic_map power2_eq_1_iff by force
have hi_un_eq_up: "?hi_ee neg un = up [^]\<^bsub>?rhgn (equator n)\<^esub> Brouwer_degree2 (n - Suc 0) neg" (is "?f un = ?y")
proof -
have [simp]: "neg \<circ> neg = id"
by force
have "?f (?f ?y) = ?y"
apply (subst hom_induced_compose' [OF cm_neg _ cm_neg])
apply(force simp: equator_def)
apply (simp add: upcarr hom_induced_id_gen)
done
moreover have "?f ?y = un"
using upcarr apply (simp only: gh_een.hom_int_pow hi_up_eq_un)
by (metis (no_types, lifting) Brouwer_degree2_21 GE.group_l_invI GE.l_inv_ex group.int_pow_1 group.int_pow_pow power2_eq_1_iff uncarr zmult_eq_1_iff)
ultimately show "?f un = ?y"
by simp
qed
have "?hi_ee f un = un [^]\<^bsub>?rhgn (equator n)\<^esub> a \<otimes>\<^bsub>?rhgn (equator n)\<^esub> up [^]\<^bsub>?rhgn (equator n)\<^esub> b"
proof -
let ?TE = "topspace (nsphere n) \<inter> equator n"
have fneg: "(f \<circ> neg) x = (neg \<circ> f) x" if "x \<in> topspace (nsphere n)" for x
using f [OF that] by (force simp: neg_def)
have neg_im: "neg ` (topspace (nsphere n) \<inter> equator n) \<subseteq> topspace (nsphere n) \<inter> equator n"
by (metis cm_neg continuous_map_image_subset_topspace equ(1) topspace_subtopology)
have 1: "hom_induced n (nsphere n) ?TE (nsphere n) ?TE f \<circ> hom_induced n (nsphere n) ?TE (nsphere n) ?TE neg
= hom_induced n (nsphere n) ?TE (nsphere n) ?TE neg \<circ> hom_induced n (nsphere n) ?TE (nsphere n) ?TE f"
using neg_im fimeq cm_neg cmf
apply (simp add: flip: hom_induced_compose del: hom_induced_restrict)
using fneg by (auto intro: hom_induced_eq)
have "(un [^]\<^bsub>?rhgn (equator n)\<^esub> a) \<otimes>\<^bsub>?rhgn (equator n)\<^esub> (up [^]\<^bsub>?rhgn (equator n)\<^esub> b)
= un [^]\<^bsub>?rhgn (equator n)\<^esub> (Brouwer_degree2 (n - 1) neg * a * Brouwer_degree2 (n - 1) neg)
\<otimes>\<^bsub>?rhgn (equator n)\<^esub>
up [^]\<^bsub>?rhgn (equator n)\<^esub> (Brouwer_degree2 (n - 1) neg * b * Brouwer_degree2 (n - 1) neg)"
proof -
have "Brouwer_degree2 (n - Suc 0) neg = 1 \<or> Brouwer_degree2 (n - Suc 0) neg = - 1"
using Brouwer_degree2_21 power2_eq_1_iff by blast
then show ?thesis
by fastforce
qed
also have "\<dots> = ((un [^]\<^bsub>?rhgn (equator n)\<^esub> Brouwer_degree2 (n - 1) neg) [^]\<^bsub>?rhgn (equator n)\<^esub> a \<otimes>\<^bsub>?rhgn (equator n)\<^esub>
(up [^]\<^bsub>?rhgn (equator n)\<^esub> Brouwer_degree2 (n - 1) neg) [^]\<^bsub>?rhgn (equator n)\<^esub> b) [^]\<^bsub>?rhgn (equator n)\<^esub>
Brouwer_degree2 (n - 1) neg"
by (simp add: GE.int_pow_distrib GE.int_pow_pow uncarr upcarr)
also have "\<dots> = ?hi_ee neg (?hi_ee f up) [^]\<^bsub>?rhgn (equator n)\<^esub> Brouwer_degree2 (n - Suc 0) neg"
by (simp add: gh_een.hom_int_pow hi_un_eq_up hi_up_eq_un uncarr up_ab upcarr)
finally have 2: "(un [^]\<^bsub>?rhgn (equator n)\<^esub> a) \<otimes>\<^bsub>?rhgn (equator n)\<^esub> (up [^]\<^bsub>?rhgn (equator n)\<^esub> b)
= ?hi_ee neg (?hi_ee f up) [^]\<^bsub>?rhgn (equator n)\<^esub> Brouwer_degree2 (n - Suc 0) neg" .
have "un = ?hi_ee neg up [^]\<^bsub>?rhgn (equator n)\<^esub> Brouwer_degree2 (n - Suc 0) neg"
by (metis (no_types, hide_lams) Brouwer_degree2_21 GE.int_pow_1 GE.int_pow_pow hi_up_eq_un power2_eq_1_iff uncarr zmult_eq_1_iff)
moreover have "?hi_ee f ((?hi_ee neg up) [^]\<^bsub>?rhgn (equator n)\<^esub> (Brouwer_degree2 (n - Suc 0) neg))
= un [^]\<^bsub>?rhgn (equator n)\<^esub> a \<otimes>\<^bsub>?rhgn (equator n)\<^esub> up [^]\<^bsub>?rhgn (equator n)\<^esub> b"
using 1 2 by (simp add: hom_induced_carrier gh_eef.hom_int_pow fun_eq_iff)
ultimately show ?thesis
by blast
qed
then have "?hi_ee f u = u [^]\<^bsub>?rhgn (equator n)\<^esub> (a - b)"
by (simp add: u_def upcarr uncarr up_ab GE.int_pow_diff GE.m_ac GE.int_pow_distrib GE.int_pow_inv GE.inv_mult_group)
ultimately
have "Brouwer_degree2 n f = a - b"
using iff by blast
with Bd_ab show ?thesis
by simp
qed simp
subsection \<open>General Jordan-Brouwer separation theorem and invariance of dimension\<close>
proposition relative_homology_group_Euclidean_complement_step:
assumes "closedin (Euclidean_space n) S"
shows "relative_homology_group p (Euclidean_space n) (topspace(Euclidean_space n) - S)
\<cong> relative_homology_group (p + k) (Euclidean_space (n+k)) (topspace(Euclidean_space (n+k)) - S)"
proof -
have *: "relative_homology_group p (Euclidean_space n) (topspace(Euclidean_space n) - S)
\<cong> relative_homology_group (p + 1) (Euclidean_space (Suc n)) (topspace(Euclidean_space (Suc n)) - {x \<in> S. x n = 0})"
(is "?lhs \<cong> ?rhs")
if clo: "closedin (Euclidean_space (Suc n)) S" and cong: "\<And>x y. \<lbrakk>x \<in> S; \<And>i. i \<noteq> n \<Longrightarrow> x i = y i\<rbrakk> \<Longrightarrow> y \<in> S"
for p n S
proof -
have Ssub: "S \<subseteq> topspace (Euclidean_space (Suc n))"
by (meson clo closedin_def)
define lo where "lo \<equiv> {x \<in> topspace(Euclidean_space (Suc n)). x n < (if x \<in> S then 0 else 1)}"
define hi where "hi = {x \<in> topspace(Euclidean_space (Suc n)). x n > (if x \<in> S then 0 else -1)}"
have lo_hi_Int: "lo \<inter> hi = {x \<in> topspace(Euclidean_space (Suc n)) - S. x n \<in> {-1<..<1}}"
by (auto simp: hi_def lo_def)
have lo_hi_Un: "lo \<union> hi = topspace(Euclidean_space (Suc n)) - {x \<in> S. x n = 0}"
by (auto simp: hi_def lo_def)
define ret where "ret \<equiv> \<lambda>c::real. \<lambda>x i. if i = n then c else x i"
have cm_ret: "continuous_map (powertop_real UNIV) (powertop_real UNIV) (ret t)" for t
by (auto simp: ret_def continuous_map_componentwise_UNIV intro: continuous_map_product_projection)
let ?ST = "\<lambda>t. subtopology (Euclidean_space (Suc n)) {x. x n = t}"
define squashable where
"squashable \<equiv> \<lambda>t S. \<forall>x t'. x \<in> S \<and> (x n \<le> t' \<and> t' \<le> t \<or> t \<le> t' \<and> t' \<le> x n) \<longrightarrow> ret t' x \<in> S"
have squashable: "squashable t (topspace(Euclidean_space(Suc n)))" for t
by (simp add: squashable_def topspace_Euclidean_space ret_def)
have squashableD: "\<lbrakk>squashable t S; x \<in> S; x n \<le> t' \<and> t' \<le> t \<or> t \<le> t' \<and> t' \<le> x n\<rbrakk> \<Longrightarrow> ret t' x \<in> S" for x t' t S
by (auto simp: squashable_def)
have "squashable 1 hi"
by (force simp: squashable_def hi_def ret_def topspace_Euclidean_space intro: cong)
have "squashable t UNIV" for t
by (force simp: squashable_def hi_def ret_def topspace_Euclidean_space intro: cong)
have squashable_0_lohi: "squashable 0 (lo \<inter> hi)"
using Ssub
by (auto simp: squashable_def hi_def lo_def ret_def topspace_Euclidean_space intro: cong)
have rm_ret: "retraction_maps (subtopology (Euclidean_space (Suc n)) U)
(subtopology (Euclidean_space (Suc n)) {x. x \<in> U \<and> x n = t})
(ret t) id"
if "squashable t U" for t U
unfolding retraction_maps_def
proof (intro conjI ballI)
show "continuous_map (subtopology (Euclidean_space (Suc n)) U)
(subtopology (Euclidean_space (Suc n)) {x \<in> U. x n = t}) (ret t)"
apply (simp add: cm_ret continuous_map_in_subtopology continuous_map_from_subtopology Euclidean_space_def)
using that by (fastforce simp: squashable_def ret_def)
next
show "continuous_map (subtopology (Euclidean_space (Suc n)) {x \<in> U. x n = t})
(subtopology (Euclidean_space (Suc n)) U) id"
using continuous_map_in_subtopology by fastforce
show "ret t (id x) = x"
if "x \<in> topspace (subtopology (Euclidean_space (Suc n)) {x \<in> U. x n = t})" for x
using that by (simp add: topspace_Euclidean_space ret_def fun_eq_iff)
qed
have cm_snd: "continuous_map (prod_topology (top_of_set {0..1}) (subtopology (powertop_real UNIV) S))
euclideanreal (\<lambda>x. snd x k)" for k::nat and S
using continuous_map_componentwise_UNIV continuous_map_into_fulltopology continuous_map_snd by fastforce
have cm_fstsnd: "continuous_map (prod_topology (top_of_set {0..1}) (subtopology (powertop_real UNIV) S))
euclideanreal (\<lambda>x. fst x * snd x k)" for k::nat and S
by (intro continuous_intros continuous_map_into_fulltopology [OF continuous_map_fst] cm_snd)
have hw_sub: "homotopic_with (\<lambda>k. k ` V \<subseteq> V) (subtopology (Euclidean_space (Suc n)) U)
(subtopology (Euclidean_space (Suc n)) U) (ret t) id"
if "squashable t U" "squashable t V" for U V t
unfolding homotopic_with_def
proof (intro exI conjI allI ballI)
let ?h = "\<lambda>(z,x). ret ((1 - z) * t + z * x n) x"
show "(\<lambda>x. ?h (u, x)) ` V \<subseteq> V" if "u \<in> {0..1}" for u
using that
by clarsimp (metis squashableD [OF \<open>squashable t V\<close>] convex_bound_le diff_ge_0_iff_ge eq_diff_eq' le_cases less_eq_real_def segment_bound_lemma)
have 1: "?h ` ({0..1} \<times> ({x. \<forall>i\<ge>Suc n. x i = 0} \<inter> U)) \<subseteq> U"
by clarsimp (metis squashableD [OF \<open>squashable t U\<close>] convex_bound_le diff_ge_0_iff_ge eq_diff_eq' le_cases less_eq_real_def segment_bound_lemma)
show "continuous_map (prod_topology (top_of_set {0..1}) (subtopology (Euclidean_space (Suc n)) U))
(subtopology (Euclidean_space (Suc n)) U) ?h"
apply (simp add: continuous_map_in_subtopology Euclidean_space_def subtopology_subtopology 1)
apply (auto simp: case_prod_unfold ret_def continuous_map_componentwise_UNIV)
apply (intro continuous_map_into_fulltopology [OF continuous_map_fst] cm_snd continuous_intros)
by (auto simp: cm_snd)
qed (auto simp: ret_def)
have cs_hi: "contractible_space(subtopology (Euclidean_space(Suc n)) hi)"
proof -
have "homotopic_with (\<lambda>x. True) (?ST 1) (?ST 1) id (\<lambda>x. (\<lambda>i. if i = n then 1 else 0))"
apply (subst homotopic_with_sym)
apply (simp add: homotopic_with)
apply (rule_tac x="(\<lambda>(z,x) i. if i=n then 1 else z * x i)" in exI)
apply (auto simp: Euclidean_space_def subtopology_subtopology continuous_map_in_subtopology case_prod_unfold continuous_map_componentwise_UNIV cm_fstsnd)
done
then have "contractible_space (?ST 1)"
unfolding contractible_space_def by metis
moreover have "?thesis = contractible_space (?ST 1)"
proof (intro deformation_retract_imp_homotopy_equivalent_space homotopy_equivalent_space_contractibility)
have "{x. \<forall>i\<ge>Suc n. x i = 0} \<inter> {x \<in> hi. x n = 1} = {x. \<forall>i\<ge>Suc n. x i = 0} \<inter> {x. x n = 1}"
by (auto simp: hi_def topspace_Euclidean_space)
then have eq: "subtopology (Euclidean_space (Suc n)) {x. x \<in> hi \<and> x n = 1} = ?ST 1"
by (simp add: Euclidean_space_def subtopology_subtopology)
show "homotopic_with (\<lambda>x. True) (subtopology (Euclidean_space (Suc n)) hi) (subtopology (Euclidean_space (Suc n)) hi) (ret 1) id"
using hw_sub [OF \<open>squashable 1 hi\<close> \<open>squashable 1 UNIV\<close>] eq by simp
show "retraction_maps (subtopology (Euclidean_space (Suc n)) hi) (?ST 1) (ret 1) id"
using rm_ret [OF \<open>squashable 1 hi\<close>] eq by simp
qed
ultimately show ?thesis by metis
qed
have "?lhs \<cong> relative_homology_group p (Euclidean_space (Suc n)) (lo \<inter> hi)"
proof (rule group.iso_sym [OF _ deformation_retract_imp_isomorphic_relative_homology_groups])
have "{x. \<forall>i\<ge>Suc n. x i = 0} \<inter> {x. x n = 0} = {x. \<forall>i\<ge>n. x i = (0::real)}"
by auto (metis le_less_Suc_eq not_le)
then have "?ST 0 = Euclidean_space n"
by (simp add: Euclidean_space_def subtopology_subtopology)
then show "retraction_maps (Euclidean_space (Suc n)) (Euclidean_space n) (ret 0) id"
using rm_ret [OF \<open>squashable 0 UNIV\<close>] by auto
then have "ret 0 x \<in> topspace (Euclidean_space n)"
if "x \<in> topspace (Euclidean_space (Suc n))" "-1 < x n" "x n < 1" for x
using that by (simp add: continuous_map_def retraction_maps_def)
then show "(ret 0) ` (lo \<inter> hi) \<subseteq> topspace (Euclidean_space n) - S"
by (auto simp: local.cong ret_def hi_def lo_def)
show "homotopic_with (\<lambda>h. h ` (lo \<inter> hi) \<subseteq> lo \<inter> hi) (Euclidean_space (Suc n)) (Euclidean_space (Suc n)) (ret 0) id"
using hw_sub [OF squashable squashable_0_lohi] by simp
qed (auto simp: lo_def hi_def Euclidean_space_def)
also have "\<dots> \<cong> relative_homology_group p (subtopology (Euclidean_space (Suc n)) hi) (lo \<inter> hi)"
proof (rule group.iso_sym [OF _ isomorphic_relative_homology_groups_inclusion_contractible])
show "contractible_space (subtopology (Euclidean_space (Suc n)) hi)"
by (simp add: cs_hi)
show "topspace (Euclidean_space (Suc n)) \<inter> hi \<noteq> {}"
apply (simp add: hi_def topspace_Euclidean_space set_eq_iff)
apply (rule_tac x="\<lambda>i. if i = n then 1 else 0" in exI, auto)
done
qed auto
also have "\<dots> \<cong> relative_homology_group p (subtopology (Euclidean_space (Suc n)) (lo \<union> hi)) lo"
proof -
have oo: "openin (Euclidean_space (Suc n)) {x \<in> topspace (Euclidean_space (Suc n)). x n \<in> A}"
if "open A" for A
proof (rule openin_continuous_map_preimage)
show "continuous_map (Euclidean_space (Suc n)) euclideanreal (\<lambda>x. x n)"
proof -
have "\<forall>n f. continuous_map (product_topology f UNIV) (f (n::nat)) (\<lambda>f. f n::real)"
by (simp add: continuous_map_product_projection)
then show ?thesis
using Euclidean_space_def continuous_map_from_subtopology
by (metis (mono_tags))
qed
qed (auto intro: that)
have "openin (Euclidean_space(Suc n)) lo"
apply (simp add: openin_subopen [of _ lo])
apply (simp add: lo_def, safe)
apply (force intro: oo [of "lessThan 0", simplified] open_Collect_less)
apply (rule_tac x="{x \<in> topspace(Euclidean_space(Suc n)). x n < 1}
\<inter> (topspace(Euclidean_space(Suc n)) - S)" in exI)
using clo apply (force intro: oo [of "lessThan 1", simplified] open_Collect_less)
done
moreover have "openin (Euclidean_space(Suc n)) hi"
apply (simp add: openin_subopen [of _ hi])
apply (simp add: hi_def, safe)
apply (force intro: oo [of "greaterThan 0", simplified] open_Collect_less)
apply (rule_tac x="{x \<in> topspace(Euclidean_space(Suc n)). x n > -1}
\<inter> (topspace(Euclidean_space(Suc n)) - S)" in exI)
using clo apply (force intro: oo [of "greaterThan (-1)", simplified] open_Collect_less)
done
ultimately
have *: "subtopology (Euclidean_space (Suc n)) (lo \<union> hi) closure_of
(topspace (subtopology (Euclidean_space (Suc n)) (lo \<union> hi)) - hi)
\<subseteq> subtopology (Euclidean_space (Suc n)) (lo \<union> hi) interior_of lo"
by (metis (no_types, lifting) Diff_idemp Diff_subset_conv Un_commute Un_upper2 closure_of_interior_of interior_of_closure_of interior_of_complement interior_of_eq lo_hi_Un openin_Un openin_open_subtopology topspace_subtopology_subset)
have eq: "((lo \<union> hi) \<inter> (lo \<union> hi - (topspace (Euclidean_space (Suc n)) \<inter> (lo \<union> hi) - hi))) = hi"
"(lo - (topspace (Euclidean_space (Suc n)) \<inter> (lo \<union> hi) - hi)) = lo \<inter> hi"
by (auto simp: lo_def hi_def Euclidean_space_def)
show ?thesis
using homology_excision_axiom [OF *, of "lo \<union> hi" p]
by (force simp: subtopology_subtopology eq is_iso_def)
qed
also have "\<dots> \<cong> relative_homology_group (p + 1 - 1) (subtopology (Euclidean_space (Suc n)) (lo \<union> hi)) lo"
by simp
also have "\<dots> \<cong> relative_homology_group (p + 1) (Euclidean_space (Suc n)) (lo \<union> hi)"
proof (rule group.iso_sym [OF _ isomorphic_relative_homology_groups_relboundary_contractible])
have proj: "continuous_map (powertop_real UNIV) euclideanreal (\<lambda>f. f n)"
by (metis UNIV_I continuous_map_product_projection)
have hilo: "\<And>x. x \<in> hi \<Longrightarrow> (\<lambda>i. if i = n then - x i else x i) \<in> lo"
"\<And>x. x \<in> lo \<Longrightarrow> (\<lambda>i. if i = n then - x i else x i) \<in> hi"
using local.cong
by (auto simp: hi_def lo_def topspace_Euclidean_space split: if_split_asm)
have "subtopology (Euclidean_space (Suc n)) hi homeomorphic_space subtopology (Euclidean_space (Suc n)) lo"
unfolding homeomorphic_space_def
apply (rule_tac x="\<lambda>x i. if i = n then -(x i) else x i" in exI)+
using proj
apply (auto simp: homeomorphic_maps_def Euclidean_space_def continuous_map_in_subtopology
hilo continuous_map_componentwise_UNIV continuous_map_from_subtopology continuous_map_minus
intro: continuous_map_from_subtopology continuous_map_product_projection)
done
then have "contractible_space(subtopology (Euclidean_space(Suc n)) hi)
\<longleftrightarrow> contractible_space (subtopology (Euclidean_space (Suc n)) lo)"
by (rule homeomorphic_space_contractibility)
then show "contractible_space (subtopology (Euclidean_space (Suc n)) lo)"
using cs_hi by auto
show "topspace (Euclidean_space (Suc n)) \<inter> lo \<noteq> {}"
apply (simp add: lo_def Euclidean_space_def set_eq_iff)
apply (rule_tac x="\<lambda>i. if i = n then -1 else 0" in exI, auto)
done
qed auto
also have "\<dots> \<cong> ?rhs"
by (simp flip: lo_hi_Un)
finally show ?thesis .
qed
show ?thesis
proof (induction k)
case (Suc m)
with assms obtain T where cloT: "closedin (powertop_real UNIV) T"
and SeqT: "S = T \<inter> {x. \<forall>i\<ge>n. x i = 0}"
by (auto simp: Euclidean_space_def closedin_subtopology)
then have "closedin (Euclidean_space (m + n)) S"
apply (simp add: Euclidean_space_def closedin_subtopology)
apply (rule_tac x="T \<inter> topspace(Euclidean_space n)" in exI)
using closedin_Euclidean_space topspace_Euclidean_space by force
moreover have "relative_homology_group p (Euclidean_space n) (topspace (Euclidean_space n) - S)
\<cong> relative_homology_group (p + 1) (Euclidean_space (Suc n)) (topspace (Euclidean_space (Suc n)) - S)"
if "closedin (Euclidean_space n) S" for p n
proof -
define S' where "S' \<equiv> {x \<in> topspace(Euclidean_space(Suc n)). (\<lambda>i. if i < n then x i else 0) \<in> S}"
have Ssub_n: "S \<subseteq> topspace (Euclidean_space n)"
by (meson that closedin_def)
have "relative_homology_group p (Euclidean_space n) (topspace(Euclidean_space n) - S')
\<cong> relative_homology_group (p + 1) (Euclidean_space (Suc n)) (topspace(Euclidean_space (Suc n)) - {x \<in> S'. x n = 0})"
proof (rule *)
have cm: "continuous_map (powertop_real UNIV) euclideanreal (\<lambda>f. f u)" for u
by (metis UNIV_I continuous_map_product_projection)
have "continuous_map (subtopology (powertop_real UNIV) {x. \<forall>i>n. x i = 0}) euclideanreal
(\<lambda>x. if k \<le> n then x k else 0)" for k
by (simp add: continuous_map_from_subtopology [OF cm])
moreover have "\<forall>i\<ge>n. (if i < n then x i else 0) = 0"
if "x \<in> topspace (subtopology (powertop_real UNIV) {x. \<forall>i>n. x i = 0})" for x
using that by simp
ultimately have "continuous_map (Euclidean_space (Suc n)) (Euclidean_space n) (\<lambda>x i. if i < n then x i else 0)"
by (simp add: Euclidean_space_def continuous_map_in_subtopology continuous_map_componentwise_UNIV
continuous_map_from_subtopology [OF cm] image_subset_iff)
then show "closedin (Euclidean_space (Suc n)) S'"
unfolding S'_def using that by (rule closedin_continuous_map_preimage)
next
fix x y
assume xy: "\<And>i. i \<noteq> n \<Longrightarrow> x i = y i" "x \<in> S'"
then have "(\<lambda>i. if i < n then x i else 0) = (\<lambda>i. if i < n then y i else 0)"
by (simp add: S'_def Euclidean_space_def fun_eq_iff)
with xy show "y \<in> S'"
by (simp add: S'_def Euclidean_space_def)
qed
moreover
have abs_eq: "(\<lambda>i. if i < n then x i else 0) = x" if "\<And>i. i \<ge> n \<Longrightarrow> x i = 0" for x :: "nat \<Rightarrow> real" and n
using that by auto
then have "topspace (Euclidean_space n) - S' = topspace (Euclidean_space n) - S"
by (simp add: S'_def Euclidean_space_def set_eq_iff cong: conj_cong)
moreover
have "topspace (Euclidean_space (Suc n)) - {x \<in> S'. x n = 0} = topspace (Euclidean_space (Suc n)) - S"
using Ssub_n
apply (auto simp: S'_def subset_iff Euclidean_space_def set_eq_iff abs_eq cong: conj_cong)
by (metis abs_eq le_antisym not_less_eq_eq)
ultimately show ?thesis
by simp
qed
ultimately have "relative_homology_group (p + m)(Euclidean_space (m + n))(topspace (Euclidean_space (m + n)) - S)
\<cong> relative_homology_group (p + m + 1) (Euclidean_space (Suc (m + n))) (topspace (Euclidean_space (Suc (m + n))) - S)"
by (metis \<open>closedin (Euclidean_space (m + n)) S\<close>)
then show ?case
using Suc.IH iso_trans by (force simp: algebra_simps)
qed (simp add: iso_refl)
qed
lemma iso_Euclidean_complements_lemma1:
assumes S: "closedin (Euclidean_space m) S" and cmf: "continuous_map(subtopology (Euclidean_space m) S) (Euclidean_space n) f"
obtains g where "continuous_map (Euclidean_space m) (Euclidean_space n) g"
"\<And>x. x \<in> S \<Longrightarrow> g x = f x"
proof -
have cont: "continuous_on (topspace (Euclidean_space m) \<inter> S) (\<lambda>x. f x i)" for i
by (metis (no_types) continuous_on_product_then_coordinatewise
cm_Euclidean_space_iff_continuous_on cmf topspace_subtopology)
have "f ` (topspace (Euclidean_space m) \<inter> S) \<subseteq> topspace (Euclidean_space n)"
using cmf continuous_map_image_subset_topspace by fastforce
then
have "\<exists>g. continuous_on (topspace (Euclidean_space m)) g \<and> (\<forall>x \<in> S. g x = f x i)" for i
using S Tietze_unbounded [OF cont [of i]]
by (metis closedin_Euclidean_space_iff closedin_closed_Int topspace_subtopology topspace_subtopology_subset)
then obtain g where cmg: "\<And>i. continuous_map (Euclidean_space m) euclideanreal (g i)"
and gf: "\<And>i x. x \<in> S \<Longrightarrow> g i x = f x i"
unfolding continuous_map_Euclidean_space_iff by metis
let ?GG = "\<lambda>x i. if i < n then g i x else 0"
show thesis
proof
show "continuous_map (Euclidean_space m) (Euclidean_space n) ?GG"
unfolding Euclidean_space_def [of n]
by (auto simp: continuous_map_in_subtopology continuous_map_componentwise cmg)
show "?GG x = f x" if "x \<in> S" for x
proof -
have "S \<subseteq> topspace (Euclidean_space m)"
by (meson S closedin_def)
then have "f x \<in> topspace (Euclidean_space n)"
using cmf that unfolding continuous_map_def topspace_subtopology by blast
then show ?thesis
by (force simp: topspace_Euclidean_space gf that)
qed
qed
qed
lemma iso_Euclidean_complements_lemma2:
assumes S: "closedin (Euclidean_space m) S"
and T: "closedin (Euclidean_space n) T"
and hom: "homeomorphic_map (subtopology (Euclidean_space m) S) (subtopology (Euclidean_space n) T) f"
obtains g where "homeomorphic_map (prod_topology (Euclidean_space m) (Euclidean_space n))
(prod_topology (Euclidean_space n) (Euclidean_space m)) g"
"\<And>x. x \<in> S \<Longrightarrow> g(x,(\<lambda>i. 0)) = (f x,(\<lambda>i. 0))"
proof -
obtain g where cmf: "continuous_map (subtopology (Euclidean_space m) S) (subtopology (Euclidean_space n) T) f"
and cmg: "continuous_map (subtopology (Euclidean_space n) T) (subtopology (Euclidean_space m) S) g"
and gf: "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
and fg: "\<And>y. y \<in> T \<Longrightarrow> f (g y) = y"
using hom S T closedin_subset unfolding homeomorphic_map_maps homeomorphic_maps_def
by fastforce
obtain f' where cmf': "continuous_map (Euclidean_space m) (Euclidean_space n) f'"
and f'f: "\<And>x. x \<in> S \<Longrightarrow> f' x = f x"
using iso_Euclidean_complements_lemma1 S cmf continuous_map_into_fulltopology by metis
obtain g' where cmg': "continuous_map (Euclidean_space n) (Euclidean_space m) g'"
and g'g: "\<And>x. x \<in> T \<Longrightarrow> g' x = g x"
using iso_Euclidean_complements_lemma1 T cmg continuous_map_into_fulltopology by metis
define p where "p \<equiv> \<lambda>(x,y). (x,(\<lambda>i. y i + f' x i))"
define p' where "p' \<equiv> \<lambda>(x,y). (x,(\<lambda>i. y i - f' x i))"
define q where "q \<equiv> \<lambda>(x,y). (x,(\<lambda>i. y i + g' x i))"
define q' where "q' \<equiv> \<lambda>(x,y). (x,(\<lambda>i. y i - g' x i))"
have "homeomorphic_maps (prod_topology (Euclidean_space m) (Euclidean_space n))
(prod_topology (Euclidean_space m) (Euclidean_space n))
p p'"
"homeomorphic_maps (prod_topology (Euclidean_space n) (Euclidean_space m))
(prod_topology (Euclidean_space n) (Euclidean_space m))
q q'"
"homeomorphic_maps (prod_topology (Euclidean_space m) (Euclidean_space n))
(prod_topology (Euclidean_space n) (Euclidean_space m)) (\<lambda>(x,y). (y,x)) (\<lambda>(x,y). (y,x))"
apply (simp_all add: p_def p'_def q_def q'_def homeomorphic_maps_def continuous_map_pairwise)
apply (force simp: case_prod_unfold continuous_map_of_fst [unfolded o_def] cmf' cmg' intro: continuous_intros)+
done
then have "homeomorphic_maps (prod_topology (Euclidean_space m) (Euclidean_space n))
(prod_topology (Euclidean_space n) (Euclidean_space m))
(q' \<circ> (\<lambda>(x,y). (y,x)) \<circ> p) (p' \<circ> ((\<lambda>(x,y). (y,x)) \<circ> q))"
using homeomorphic_maps_compose homeomorphic_maps_sym by (metis (no_types, lifting))
moreover
have "\<And>x. x \<in> S \<Longrightarrow> (q' \<circ> (\<lambda>(x,y). (y,x)) \<circ> p) (x, \<lambda>i. 0) = (f x, \<lambda>i. 0)"
apply (simp add: q'_def p_def f'f)
apply (simp add: fun_eq_iff)
by (metis S T closedin_subset g'g gf hom homeomorphic_imp_surjective_map image_eqI topspace_subtopology_subset)
ultimately
show thesis
using homeomorphic_map_maps that by blast
qed
proposition isomorphic_relative_homology_groups_Euclidean_complements:
assumes S: "closedin (Euclidean_space n) S" and T: "closedin (Euclidean_space n) T"
and hom: "(subtopology (Euclidean_space n) S) homeomorphic_space (subtopology (Euclidean_space n) T)"
shows "relative_homology_group p (Euclidean_space n) (topspace(Euclidean_space n) - S)
\<cong> relative_homology_group p (Euclidean_space n) (topspace(Euclidean_space n) - T)"
proof -
have subST: "S \<subseteq> topspace(Euclidean_space n)" "T \<subseteq> topspace(Euclidean_space n)"
by (meson S T closedin_def)+
have "relative_homology_group p (Euclidean_space n) (topspace (Euclidean_space n) - S)
\<cong> relative_homology_group (p + int n) (Euclidean_space (n + n)) (topspace (Euclidean_space (n + n)) - S)"
using relative_homology_group_Euclidean_complement_step [OF S] by blast
moreover have "relative_homology_group p (Euclidean_space n) (topspace (Euclidean_space n) - T)
\<cong> relative_homology_group (p + int n) (Euclidean_space (n + n)) (topspace (Euclidean_space (n + n)) - T)"
using relative_homology_group_Euclidean_complement_step [OF T] by blast
moreover have "relative_homology_group (p + int n) (Euclidean_space (n + n)) (topspace (Euclidean_space (n + n)) - S)
\<cong> relative_homology_group (p + int n) (Euclidean_space (n + n)) (topspace (Euclidean_space (n + n)) - T)"
proof -
obtain f where f: "homeomorphic_map (subtopology (Euclidean_space n) S)
(subtopology (Euclidean_space n) T) f"
using hom unfolding homeomorphic_space by blast
obtain g where g: "homeomorphic_map (prod_topology (Euclidean_space n) (Euclidean_space n))
(prod_topology (Euclidean_space n) (Euclidean_space n)) g"
and gf: "\<And>x. x \<in> S \<Longrightarrow> g(x,(\<lambda>i. 0)) = (f x,(\<lambda>i. 0))"
using S T f iso_Euclidean_complements_lemma2 by blast
define h where "h \<equiv> \<lambda>x::nat \<Rightarrow>real. ((\<lambda>i. if i < n then x i else 0), (\<lambda>j. if j < n then x(n + j) else 0))"
define k where "k \<equiv> \<lambda>(x,y) i. if i < 2 * n then if i < n then x i else y(i - n) else (0::real)"
have hk: "homeomorphic_maps (Euclidean_space(2 * n)) (prod_topology (Euclidean_space n) (Euclidean_space n)) h k"
unfolding homeomorphic_maps_def
proof safe
show "continuous_map (Euclidean_space (2 * n))
(prod_topology (Euclidean_space n) (Euclidean_space n)) h"
apply (simp add: h_def continuous_map_pairwise o_def continuous_map_componentwise_Euclidean_space)
unfolding Euclidean_space_def
by (metis (mono_tags) UNIV_I continuous_map_from_subtopology continuous_map_product_projection)
have "continuous_map (prod_topology (Euclidean_space n) (Euclidean_space n)) euclideanreal (\<lambda>p. fst p i)" for i
using Euclidean_space_def continuous_map_into_fulltopology continuous_map_fst by fastforce
moreover
have "continuous_map (prod_topology (Euclidean_space n) (Euclidean_space n)) euclideanreal (\<lambda>p. snd p (i - n))" for i
using Euclidean_space_def continuous_map_into_fulltopology continuous_map_snd by fastforce
ultimately
show "continuous_map (prod_topology (Euclidean_space n) (Euclidean_space n))
(Euclidean_space (2 * n)) k"
by (simp add: k_def continuous_map_pairwise o_def continuous_map_componentwise_Euclidean_space case_prod_unfold)
qed (auto simp: k_def h_def fun_eq_iff topspace_Euclidean_space)
define kgh where "kgh \<equiv> k \<circ> g \<circ> h"
let ?i = "hom_induced (p + n) (Euclidean_space(2 * n)) (topspace(Euclidean_space(2 * n)) - S)
(Euclidean_space(2 * n)) (topspace(Euclidean_space(2 * n)) - T) kgh"
have "?i \<in> iso (relative_homology_group (p + int n) (Euclidean_space (2 * n))
(topspace (Euclidean_space (2 * n)) - S))
(relative_homology_group (p + int n) (Euclidean_space (2 * n))
(topspace (Euclidean_space (2 * n)) - T))"
proof (rule homeomorphic_map_relative_homology_iso)
show hm: "homeomorphic_map (Euclidean_space (2 * n)) (Euclidean_space (2 * n)) kgh"
unfolding kgh_def by (meson hk g homeomorphic_map_maps homeomorphic_maps_compose homeomorphic_maps_sym)
have Teq: "T = f ` S"
using f homeomorphic_imp_surjective_map subST(1) subST(2) topspace_subtopology_subset by blast
have khf: "\<And>x. x \<in> S \<Longrightarrow> k(h(f x)) = f x"
by (metis (no_types, lifting) Teq hk homeomorphic_maps_def image_subset_iff le_add1 mult_2 subST(2) subsetD subset_Euclidean_space)
have gh: "g(h x) = h(f x)" if "x \<in> S" for x
proof -
have [simp]: "(\<lambda>i. if i < n then x i else 0) = x"
using subST(1) that topspace_Euclidean_space by (auto simp: fun_eq_iff)
have "f x \<in> topspace(Euclidean_space n)"
using Teq subST(2) that by blast
moreover have "(\<lambda>j. if j < n then x (n + j) else 0) = (\<lambda>j. 0::real)"
using Euclidean_space_def subST(1) that by force
ultimately show ?thesis
by (simp add: topspace_Euclidean_space h_def gf \<open>x \<in> S\<close> fun_eq_iff)
qed
have *: "\<lbrakk>S \<subseteq> U; T \<subseteq> U; kgh ` U = U; inj_on kgh U; kgh ` S = T\<rbrakk> \<Longrightarrow> kgh ` (U - S) = U - T" for U
unfolding inj_on_def set_eq_iff by blast
show "kgh ` (topspace (Euclidean_space (2 * n)) - S) = topspace (Euclidean_space (2 * n)) - T"
proof (rule *)
show "kgh ` topspace (Euclidean_space (2 * n)) = topspace (Euclidean_space (2 * n))"
by (simp add: hm homeomorphic_imp_surjective_map)
show "inj_on kgh (topspace (Euclidean_space (2 * n)))"
using hm homeomorphic_map_def by auto
show "kgh ` S = T"
by (simp add: Teq kgh_def gh khf)
qed (use subST topspace_Euclidean_space in \<open>fastforce+\<close>)
qed auto
then show ?thesis
by (simp add: is_isoI mult_2)
qed
ultimately show ?thesis
by (meson group.iso_sym iso_trans group_relative_homology_group)
qed
lemma lemma_iod:
assumes "S \<subseteq> T" "S \<noteq> {}" and Tsub: "T \<subseteq> topspace(Euclidean_space n)"
and S: "\<And>a b u. \<lbrakk>a \<in> S; b \<in> T; 0 < u; u < 1\<rbrakk> \<Longrightarrow> (\<lambda>i. (1 - u) * a i + u * b i) \<in> S"
shows "path_connectedin (Euclidean_space n) T"
proof -
obtain a where "a \<in> S"
using assms by blast
have "path_component_of (subtopology (Euclidean_space n) T) a b" if "b \<in> T" for b
unfolding path_component_of_def
proof (intro exI conjI)
have [simp]: "\<forall>i\<ge>n. a i = 0"
using Tsub \<open>a \<in> S\<close> assms(1) topspace_Euclidean_space by auto
have [simp]: "\<forall>i\<ge>n. b i = 0"
using Tsub that topspace_Euclidean_space by auto
have inT: "(\<lambda>i. (1 - x) * a i + x * b i) \<in> T" if "0 \<le> x" "x \<le> 1" for x
proof (cases "x = 0 \<or> x = 1")
case True
with \<open>a \<in> S\<close> \<open>b \<in> T\<close> \<open>S \<subseteq> T\<close> show ?thesis
by force
next
case False
then show ?thesis
using subsetD [OF \<open>S \<subseteq> T\<close> S] \<open>a \<in> S\<close> \<open>b \<in> T\<close> that by auto
qed
have "continuous_on {0..1} (\<lambda>x. (1 - x) * a k + x * b k)" for k
by (intro continuous_intros)
then show "pathin (subtopology (Euclidean_space n) T) (\<lambda>t i. (1 - t) * a i + t * b i)"
apply (simp add: Euclidean_space_def subtopology_subtopology pathin_subtopology)
apply (simp add: pathin_def continuous_map_componentwise_UNIV inT)
done
qed auto
then have "path_connected_space (subtopology (Euclidean_space n) T)"
by (metis Tsub path_component_of_equiv path_connected_space_iff_path_component topspace_subtopology_subset)
then show ?thesis
by (simp add: Tsub path_connectedin_def)
qed
lemma invariance_of_dimension_closedin_Euclidean_space:
assumes "closedin (Euclidean_space n) S"
shows "subtopology (Euclidean_space n) S homeomorphic_space Euclidean_space n
\<longleftrightarrow> S = topspace(Euclidean_space n)"
(is "?lhs = ?rhs")
proof
assume L: ?lhs
have Ssub: "S \<subseteq> topspace (Euclidean_space n)"
by (meson assms closedin_def)
moreover have False if "a \<notin> S" and "a \<in> topspace (Euclidean_space n)" for a
proof -
have cl_n: "closedin (Euclidean_space (Suc n)) (topspace(Euclidean_space n))"
using Euclidean_space_def closedin_Euclidean_space closedin_subtopology by fastforce
then have sub: "subtopology (Euclidean_space(Suc n)) (topspace(Euclidean_space n)) = Euclidean_space n"
by (metis (no_types, lifting) Euclidean_space_def closedin_subset subtopology_subtopology topspace_Euclidean_space topspace_subtopology topspace_subtopology_subset)
then have cl_S: "closedin (Euclidean_space(Suc n)) S"
using cl_n assms closedin_closed_subtopology by fastforce
have sub_SucS: "subtopology (Euclidean_space (Suc n)) S = subtopology (Euclidean_space n) S"
by (metis Ssub sub subtopology_subtopology topspace_subtopology topspace_subtopology_subset)
have non0: "{y. \<exists>x::nat\<Rightarrow>real. (\<forall>i\<ge>Suc n. x i = 0) \<and> (\<exists>i\<ge>n. x i \<noteq> 0) \<and> y = x n} = -{0}"
proof safe
show "False" if "\<forall>i\<ge>Suc n. f i = 0" "0 = f n" "n \<le> i" "f i \<noteq> 0" for f::"nat\<Rightarrow>real" and i
by (metis that le_antisym not_less_eq_eq)
show "\<exists>f::nat\<Rightarrow>real. (\<forall>i\<ge>Suc n. f i = 0) \<and> (\<exists>i\<ge>n. f i \<noteq> 0) \<and> a = f n" if "a \<noteq> 0" for a
by (rule_tac x="(\<lambda>i. 0)(n:= a)" in exI) (force simp: that)
qed
have "homology_group 0 (subtopology (Euclidean_space (Suc n)) (topspace (Euclidean_space (Suc n)) - S))
\<cong> homology_group 0 (subtopology (Euclidean_space (Suc n)) (topspace (Euclidean_space (Suc n)) - topspace (Euclidean_space n)))"
proof (rule isomorphic_relative_contractible_space_imp_homology_groups)
show "(topspace (Euclidean_space (Suc n)) - S = {}) =
(topspace (Euclidean_space (Suc n)) - topspace (Euclidean_space n) = {})"
using cl_n closedin_subset that by auto
next
fix p
show "relative_homology_group p (Euclidean_space (Suc n))
(topspace (Euclidean_space (Suc n)) - S) \<cong>
relative_homology_group p (Euclidean_space (Suc n))
(topspace (Euclidean_space (Suc n)) - topspace (Euclidean_space n))"
by (simp add: L sub_SucS cl_S cl_n isomorphic_relative_homology_groups_Euclidean_complements sub)
qed (auto simp: L)
moreover
have "continuous_map (powertop_real UNIV) euclideanreal (\<lambda>x. x n)"
by (metis (no_types) UNIV_I continuous_map_product_projection)
then have cm: "continuous_map (subtopology (Euclidean_space (Suc n)) (topspace (Euclidean_space (Suc n)) - topspace (Euclidean_space n)))
euclideanreal (\<lambda>x. x n)"
by (simp add: Euclidean_space_def continuous_map_from_subtopology)
have False if "path_connected_space
(subtopology (Euclidean_space (Suc n))
(topspace (Euclidean_space (Suc n)) - topspace (Euclidean_space n)))"
using path_connectedin_continuous_map_image [OF cm that [unfolded path_connectedin_topspace [symmetric]]]
bounded_path_connected_Compl_real [of "{0}"]
by (simp add: topspace_Euclidean_space image_def Bex_def non0 flip: path_connectedin_topspace)
moreover
have eq: "T = T \<inter> {x. x n \<le> 0} \<union> T \<inter> {x. x n \<ge> 0}" for T :: "(nat \<Rightarrow> real) set"
by auto
have "path_connectedin (Euclidean_space (Suc n)) (topspace (Euclidean_space (Suc n)) - S)"
proof (subst eq, rule path_connectedin_Un)
have "topspace(Euclidean_space(Suc n)) \<inter> {x. x n = 0} = topspace(Euclidean_space n)"
apply (auto simp: topspace_Euclidean_space)
by (metis Suc_leI inf.absorb_iff2 inf.orderE leI)
let ?S = "topspace(Euclidean_space(Suc n)) \<inter> {x. x n < 0}"
show "path_connectedin (Euclidean_space (Suc n))
((topspace (Euclidean_space (Suc n)) - S) \<inter> {x. x n \<le> 0})"
proof (rule lemma_iod)
show "?S \<subseteq> (topspace (Euclidean_space (Suc n)) - S) \<inter> {x. x n \<le> 0}"
using Ssub topspace_Euclidean_space by auto
show "?S \<noteq> {}"
apply (simp add: topspace_Euclidean_space set_eq_iff)
apply (rule_tac x="(\<lambda>i. 0)(n:= -1)" in exI)
apply auto
done
fix a b and u::real
assume
"a \<in> ?S" "0 < u" "u < 1"
"b \<in> (topspace (Euclidean_space (Suc n)) - S) \<inter> {x. x n \<le> 0}"
then show "(\<lambda>i. (1 - u) * a i + u * b i) \<in> ?S"
by (simp add: topspace_Euclidean_space add_neg_nonpos less_eq_real_def mult_less_0_iff)
qed (simp add: topspace_Euclidean_space subset_iff)
let ?T = "topspace(Euclidean_space(Suc n)) \<inter> {x. x n > 0}"
show "path_connectedin (Euclidean_space (Suc n))
((topspace (Euclidean_space (Suc n)) - S) \<inter> {x. 0 \<le> x n})"
proof (rule lemma_iod)
show "?T \<subseteq> (topspace (Euclidean_space (Suc n)) - S) \<inter> {x. 0 \<le> x n}"
using Ssub topspace_Euclidean_space by auto
show "?T \<noteq> {}"
apply (simp add: topspace_Euclidean_space set_eq_iff)
apply (rule_tac x="(\<lambda>i. 0)(n:= 1)" in exI)
apply auto
done
fix a b and u::real
assume "a \<in> ?T" "0 < u" "u < 1" "b \<in> (topspace (Euclidean_space (Suc n)) - S) \<inter> {x. 0 \<le> x n}"
then show "(\<lambda>i. (1 - u) * a i + u * b i) \<in> ?T"
by (simp add: topspace_Euclidean_space add_pos_nonneg)
qed (simp add: topspace_Euclidean_space subset_iff)
show "(topspace (Euclidean_space (Suc n)) - S) \<inter> {x. x n \<le> 0} \<inter>
((topspace (Euclidean_space (Suc n)) - S) \<inter> {x. 0 \<le> x n}) \<noteq> {}"
using that
apply (auto simp: Set.set_eq_iff topspace_Euclidean_space)
by (metis Suc_leD order_refl)
qed
then have "path_connected_space (subtopology (Euclidean_space (Suc n))
(topspace (Euclidean_space (Suc n)) - S))"
apply (simp add: path_connectedin_subtopology flip: path_connectedin_topspace)
by (metis Int_Diff inf_idem)
ultimately
show ?thesis
using isomorphic_homology_imp_path_connectedness by blast
qed
ultimately show ?rhs
by blast
qed (simp add: homeomorphic_space_refl)
lemma isomorphic_homology_groups_Euclidean_complements:
assumes "closedin (Euclidean_space n) S" "closedin (Euclidean_space n) T"
"(subtopology (Euclidean_space n) S) homeomorphic_space (subtopology (Euclidean_space n) T)"
shows "homology_group p (subtopology (Euclidean_space n) (topspace(Euclidean_space n) - S))
\<cong> homology_group p (subtopology (Euclidean_space n) (topspace(Euclidean_space n) - T))"
proof (rule isomorphic_relative_contractible_space_imp_homology_groups)
show "topspace (Euclidean_space n) - S \<subseteq> topspace (Euclidean_space n)"
using assms homeomorphic_space_sym invariance_of_dimension_closedin_Euclidean_space subtopology_superset by fastforce
show "topspace (Euclidean_space n) - T \<subseteq> topspace (Euclidean_space n)"
using assms invariance_of_dimension_closedin_Euclidean_space subtopology_superset by force
show "(topspace (Euclidean_space n) - S = {}) = (topspace (Euclidean_space n) - T = {})"
by (metis Diff_eq_empty_iff assms closedin_subset homeomorphic_space_sym invariance_of_dimension_closedin_Euclidean_space subset_antisym subtopology_topspace)
show "relative_homology_group p (Euclidean_space n) (topspace (Euclidean_space n) - S) \<cong>
relative_homology_group p (Euclidean_space n) (topspace (Euclidean_space n) - T)" for p
using assms isomorphic_relative_homology_groups_Euclidean_complements by blast
qed auto
lemma eqpoll_path_components_Euclidean_complements:
assumes "closedin (Euclidean_space n) S" "closedin (Euclidean_space n) T"
"(subtopology (Euclidean_space n) S) homeomorphic_space (subtopology (Euclidean_space n) T)"
shows "path_components_of
(subtopology (Euclidean_space n)
(topspace(Euclidean_space n) - S))
\<approx> path_components_of
(subtopology (Euclidean_space n)
(topspace(Euclidean_space n) - T))"
by (simp add: assms isomorphic_homology_groups_Euclidean_complements isomorphic_homology_imp_path_components)
lemma path_connectedin_Euclidean_complements:
assumes "closedin (Euclidean_space n) S" "closedin (Euclidean_space n) T"
"(subtopology (Euclidean_space n) S) homeomorphic_space (subtopology (Euclidean_space n) T)"
shows "path_connectedin (Euclidean_space n) (topspace(Euclidean_space n) - S)
\<longleftrightarrow> path_connectedin (Euclidean_space n) (topspace(Euclidean_space n) - T)"
by (meson Diff_subset assms isomorphic_homology_groups_Euclidean_complements isomorphic_homology_imp_path_connectedness path_connectedin_def)
lemma eqpoll_connected_components_Euclidean_complements:
assumes S: "closedin (Euclidean_space n) S" and T: "closedin (Euclidean_space n) T"
and ST: "(subtopology (Euclidean_space n) S) homeomorphic_space (subtopology (Euclidean_space n) T)"
shows "connected_components_of
(subtopology (Euclidean_space n)
(topspace(Euclidean_space n) - S))
\<approx> connected_components_of
(subtopology (Euclidean_space n)
(topspace(Euclidean_space n) - T))"
using eqpoll_path_components_Euclidean_complements [OF assms]
by (metis S T closedin_def locally_path_connected_Euclidean_space locally_path_connected_space_open_subset path_components_eq_connected_components_of)
lemma connected_in_Euclidean_complements:
assumes "closedin (Euclidean_space n) S" "closedin (Euclidean_space n) T"
"(subtopology (Euclidean_space n) S) homeomorphic_space (subtopology (Euclidean_space n) T)"
shows "connectedin (Euclidean_space n) (topspace(Euclidean_space n) - S)
\<longleftrightarrow> connectedin (Euclidean_space n) (topspace(Euclidean_space n) - T)"
apply (simp add: connectedin_def connected_space_iff_components_subset_singleton subset_singleton_iff_lepoll)
using eqpoll_connected_components_Euclidean_complements [OF assms]
by (meson eqpoll_sym lepoll_trans1)
theorem invariance_of_dimension_Euclidean_space:
"Euclidean_space m homeomorphic_space Euclidean_space n \<longleftrightarrow> m = n"
proof (cases m n rule: linorder_cases)
case less
then have *: "topspace (Euclidean_space m) \<subseteq> topspace (Euclidean_space n)"
by (meson le_cases not_le subset_Euclidean_space)
then have "Euclidean_space m = subtopology (Euclidean_space n) (topspace(Euclidean_space m))"
by (simp add: Euclidean_space_def inf.absorb_iff2 subtopology_subtopology)
then show ?thesis
by (metis (no_types, lifting) * Euclidean_space_def closedin_Euclidean_space closedin_closed_subtopology eq_iff invariance_of_dimension_closedin_Euclidean_space subset_Euclidean_space topspace_Euclidean_space)
next
case equal
then show ?thesis
by (simp add: homeomorphic_space_refl)
next
case greater
then have *: "topspace (Euclidean_space n) \<subseteq> topspace (Euclidean_space m)"
by (meson le_cases not_le subset_Euclidean_space)
then have "Euclidean_space n = subtopology (Euclidean_space m) (topspace(Euclidean_space n))"
by (simp add: Euclidean_space_def inf.absorb_iff2 subtopology_subtopology)
then show ?thesis
by (metis (no_types, lifting) "*" Euclidean_space_def closedin_Euclidean_space closedin_closed_subtopology eq_iff homeomorphic_space_sym invariance_of_dimension_closedin_Euclidean_space subset_Euclidean_space topspace_Euclidean_space)
qed
lemma biglemma:
assumes "n \<noteq> 0" and S: "compactin (Euclidean_space n) S"
and cmh: "continuous_map (subtopology (Euclidean_space n) S) (Euclidean_space n) h"
and "inj_on h S"
shows "path_connectedin (Euclidean_space n) (topspace(Euclidean_space n) - h ` S)
\<longleftrightarrow> path_connectedin (Euclidean_space n) (topspace(Euclidean_space n) - S)"
proof (rule path_connectedin_Euclidean_complements)
have hS_sub: "h ` S \<subseteq> topspace(Euclidean_space n)"
by (metis (no_types) S cmh compactin_subspace continuous_map_image_subset_topspace topspace_subtopology_subset)
show clo_S: "closedin (Euclidean_space n) S"
using assms by (simp add: continuous_map_in_subtopology Hausdorff_Euclidean_space compactin_imp_closedin)
show clo_hS: "closedin (Euclidean_space n) (h ` S)"
using Hausdorff_Euclidean_space S cmh compactin_absolute compactin_imp_closedin image_compactin by blast
have "homeomorphic_map (subtopology (Euclidean_space n) S) (subtopology (Euclidean_space n) (h ` S)) h"
proof (rule continuous_imp_homeomorphic_map)
show "compact_space (subtopology (Euclidean_space n) S)"
by (simp add: S compact_space_subtopology)
show "Hausdorff_space (subtopology (Euclidean_space n) (h ` S))"
using hS_sub
by (simp add: Hausdorff_Euclidean_space Hausdorff_space_subtopology)
show "continuous_map (subtopology (Euclidean_space n) S) (subtopology (Euclidean_space n) (h ` S)) h"
using cmh continuous_map_in_subtopology by fastforce
show "h ` topspace (subtopology (Euclidean_space n) S) = topspace (subtopology (Euclidean_space n) (h ` S))"
using clo_hS clo_S closedin_subset by auto
show "inj_on h (topspace (subtopology (Euclidean_space n) S))"
by (metis \<open>inj_on h S\<close> clo_S closedin_def topspace_subtopology_subset)
qed
then show "subtopology (Euclidean_space n) (h ` S) homeomorphic_space subtopology (Euclidean_space n) S"
using homeomorphic_space homeomorphic_space_sym by blast
qed
lemma lemmaIOD:
assumes
"\<exists>T. T \<in> U \<and> c \<subseteq> T" "\<exists>T. T \<in> U \<and> d \<subseteq> T" "\<Union>U = c \<union> d" "\<And>T. T \<in> U \<Longrightarrow> T \<noteq> {}"
"pairwise disjnt U" "~(\<exists>T. U \<subseteq> {T})"
shows "c \<in> U"
using assms
apply safe
subgoal for C' D'
proof (cases "C'=D'")
show "c \<in> U"
if UU: "\<Union> U = c \<union> d"
and U: "\<And>T. T \<in> U \<Longrightarrow> T \<noteq> {}" "disjoint U" and "\<nexists>T. U \<subseteq> {T}" "c \<subseteq> C'" "D' \<in> U" "d \<subseteq> D'" "C' = D'"
proof -
have "c \<union> d = D'"
using Union_upper sup_mono UU that(5) that(6) that(7) that(8) by auto
then have "\<Union>U = D'"
by (simp add: UU)
with U have "U = {D'}"
by (metis (no_types, lifting) disjnt_Union1 disjnt_self_iff_empty insertCI pairwiseD subset_iff that(4) that(6))
then show ?thesis
using that(4) by auto
qed
show "c \<in> U"
if "\<Union> U = c \<union> d""disjoint U" "C' \<in> U" "c \<subseteq> C'""D' \<in> U" "d \<subseteq> D'" "C' \<noteq> D'"
proof -
have "C' \<inter> D' = {}"
using \<open>disjoint U\<close> \<open>C' \<in> U\<close> \<open>D' \<in> U\<close> \<open>C' \<noteq> D'\<close>unfolding disjnt_iff pairwise_def
by blast
then show ?thesis
using subset_antisym that(1) \<open>C' \<in> U\<close> \<open>c \<subseteq> C'\<close> \<open>d \<subseteq> D'\<close> by fastforce
qed
qed
done
theorem invariance_of_domain_Euclidean_space:
assumes U: "openin (Euclidean_space n) U"
and cmf: "continuous_map (subtopology (Euclidean_space n) U) (Euclidean_space n) f"
and "inj_on f U"
shows "openin (Euclidean_space n) (f ` U)" (is "openin ?E (f ` U)")
proof (cases "n = 0")
case True
have [simp]: "Euclidean_space 0 = discrete_topology {\<lambda>i. 0}"
by (auto simp: subtopology_eq_discrete_topology_sing topspace_Euclidean_space)
show ?thesis
using cmf True U by auto
next
case False
define enorm where "enorm \<equiv> \<lambda>x. sqrt(\<Sum>i<n. x i ^ 2)"
have enorm_if [simp]: "enorm (\<lambda>i. if i = k then d else 0) = (if k < n then \<bar>d\<bar> else 0)" for k d
using \<open>n \<noteq> 0\<close> by (auto simp: enorm_def power2_eq_square if_distrib [of "\<lambda>x. x * _"] cong: if_cong)
define zero::"nat\<Rightarrow>real" where "zero \<equiv> \<lambda>i. 0"
have zero_in [simp]: "zero \<in> topspace ?E"
using False by (simp add: zero_def topspace_Euclidean_space)
have enorm_eq_0 [simp]: "enorm x = 0 \<longleftrightarrow> x = zero"
if "x \<in> topspace(Euclidean_space n)" for x
using that unfolding zero_def enorm_def
apply (simp add: sum_nonneg_eq_0_iff fun_eq_iff topspace_Euclidean_space)
using le_less_linear by blast
have [simp]: "enorm zero = 0"
by (simp add: zero_def enorm_def)
have cm_enorm: "continuous_map ?E euclideanreal enorm"
unfolding enorm_def
proof (intro continuous_intros)
show "continuous_map ?E euclideanreal (\<lambda>x. x i)"
if "i \<in> {..<n}" for i
using that by (auto simp: Euclidean_space_def intro: continuous_map_product_projection continuous_map_from_subtopology)
qed auto
have enorm_ge0: "0 \<le> enorm x" for x
by (auto simp: enorm_def sum_nonneg)
have le_enorm: "\<bar>x i\<bar> \<le> enorm x" if "i < n" for i x
proof -
have "\<bar>x i\<bar> \<le> sqrt (\<Sum>k\<in>{i}. (x k)\<^sup>2)"
by auto
also have "\<dots> \<le> sqrt (\<Sum>k<n. (x k)\<^sup>2)"
by (rule real_sqrt_le_mono [OF sum_mono2]) (use that in auto)
finally show ?thesis
by (simp add: enorm_def)
qed
define B where "B \<equiv> \<lambda>r. {x \<in> topspace ?E. enorm x < r}"
define C where "C \<equiv> \<lambda>r. {x \<in> topspace ?E. enorm x \<le> r}"
define S where "S \<equiv> \<lambda>r. {x \<in> topspace ?E. enorm x = r}"
have BC: "B r \<subseteq> C r" and SC: "S r \<subseteq> C r" and disjSB: "disjnt (S r) (B r)" and eqC: "B r \<union> S r = C r" for r
by (auto simp: B_def C_def S_def disjnt_def)
consider "n = 1" | "n \<ge> 2"
using False by linarith
then have **: "openin ?E (h ` (B r))"
if "r > 0" and cmh: "continuous_map(subtopology ?E (C r)) ?E h" and injh: "inj_on h (C r)" for r h
proof cases
case 1
define e :: "[real,nat]\<Rightarrow>real" where "e \<equiv> \<lambda>x i. if i = 0 then x else 0"
define e' :: "(nat\<Rightarrow>real)\<Rightarrow>real" where "e' \<equiv> \<lambda>x. x 0"
have "continuous_map euclidean euclideanreal (\<lambda>f. f (0::nat))"
by auto
then have "continuous_map (subtopology (powertop_real UNIV) {f. \<forall>n\<ge>Suc 0. f n = 0}) euclideanreal (\<lambda>f. f 0)"
by (metis (mono_tags) continuous_map_from_subtopology euclidean_product_topology)
then have hom_ee': "homeomorphic_maps euclideanreal (Euclidean_space 1) e e'"
by (auto simp: homeomorphic_maps_def e_def e'_def continuous_map_in_subtopology Euclidean_space_def)
have eBr: "e ` {-r<..<r} = B r"
unfolding B_def e_def C_def
by(force simp: "1" topspace_Euclidean_space enorm_def power2_eq_square if_distrib [of "\<lambda>x. x * _"] cong: if_cong)
have in_Cr: "\<And>x. \<lbrakk>-r < x; x < r\<rbrakk> \<Longrightarrow> (\<lambda>i. if i = 0 then x else 0) \<in> C r"
using \<open>n \<noteq> 0\<close> by (auto simp: C_def topspace_Euclidean_space)
have inj: "inj_on (e' \<circ> h \<circ> e) {- r<..<r}"
proof (clarsimp simp: inj_on_def e_def e'_def)
show "(x::real) = y"
if f: "h (\<lambda>i. if i = 0 then x else 0) 0 = h (\<lambda>i. if i = 0 then y else 0) 0"
and "-r < x" "x < r" "-r < y" "y < r"
for x y :: real
proof -
have x: "(\<lambda>i. if i = 0 then x else 0) \<in> C r" and y: "(\<lambda>i. if i = 0 then y else 0) \<in> C r"
by (blast intro: inj_onD [OF \<open>inj_on h (C r)\<close>] that in_Cr)+
have "continuous_map (subtopology (Euclidean_space (Suc 0)) (C r)) (Euclidean_space (Suc 0)) h"
using cmh by (simp add: 1)
then have "h ` ({x. \<forall>i\<ge>Suc 0. x i = 0} \<inter> C r) \<subseteq> {x. \<forall>i\<ge>Suc 0. x i = 0}"
by (force simp: Euclidean_space_def subtopology_subtopology continuous_map_def)
have "h (\<lambda>i. if i = 0 then x else 0) j = h (\<lambda>i. if i = 0 then y else 0) j" for j
proof (cases j)
case (Suc j')
have "h ` ({x. \<forall>i\<ge>Suc 0. x i = 0} \<inter> C r) \<subseteq> {x. \<forall>i\<ge>Suc 0. x i = 0}"
using continuous_map_image_subset_topspace [OF cmh]
by (simp add: 1 Euclidean_space_def subtopology_subtopology)
with Suc f x y show ?thesis
by (simp add: "1" image_subset_iff)
qed (use f in blast)
then have "(\<lambda>i. if i = 0 then x else 0) = (\<lambda>i::nat. if i = 0 then y else 0)"
by (blast intro: inj_onD [OF \<open>inj_on h (C r)\<close>] that in_Cr)
then show ?thesis
by (simp add: fun_eq_iff) presburger
qed
qed
have hom_e': "homeomorphic_map (Euclidean_space 1) euclideanreal e'"
using hom_ee' homeomorphic_maps_map by blast
have "openin (Euclidean_space n) (h ` e ` {- r<..<r})"
unfolding 1
proof (subst homeomorphic_map_openness [OF hom_e', symmetric])
show "h ` e ` {- r<..<r} \<subseteq> topspace (Euclidean_space 1)"
using "1" C_def \<open>\<And>r. B r \<subseteq> C r\<close> cmh continuous_map_image_subset_topspace eBr by fastforce
have cont: "continuous_on {- r<..<r} (e' \<circ> h \<circ> e)"
proof (intro continuous_on_compose)
have "\<And>i. continuous_on {- r<..<r} (\<lambda>x. if i = 0 then x else 0)"
by (auto simp: continuous_on_topological)
then show "continuous_on {- r<..<r} e"
by (force simp: e_def intro: continuous_on_coordinatewise_then_product)
have subCr: "e ` {- r<..<r} \<subseteq> topspace (subtopology ?E (C r))"
by (auto simp: eBr \<open>\<And>r. B r \<subseteq> C r\<close>) (auto simp: B_def)
with cmh show "continuous_on (e ` {- r<..<r}) h"
by (meson cm_Euclidean_space_iff_continuous_on continuous_on_subset)
have "h ` (e ` {- r<..<r}) \<subseteq> topspace ?E"
using subCr cmh by (simp add: continuous_map_def image_subset_iff)
moreover have "continuous_on (topspace ?E) e'"
by (metis "1" continuous_map_Euclidean_space_iff hom_ee' homeomorphic_maps_def)
ultimately show "continuous_on (h ` e ` {- r<..<r}) e'"
by (simp add: e'_def continuous_on_subset)
qed
show "openin euclideanreal (e' ` h ` e ` {- r<..<r})"
using injective_eq_1d_open_map_UNIV [OF cont] inj by (simp add: image_image is_interval_1)
qed
then show ?thesis
by (simp flip: eBr)
next
case 2
have cloC: "\<And>r. closedin (Euclidean_space n) (C r)"
unfolding C_def
by (rule closedin_continuous_map_preimage [OF cm_enorm, of concl: "{.._}", simplified])
have cloS: "\<And>r. closedin (Euclidean_space n) (S r)"
unfolding S_def
by (rule closedin_continuous_map_preimage [OF cm_enorm, of concl: "{_}", simplified])
have C_subset: "C r \<subseteq> UNIV \<rightarrow>\<^sub>E {- \<bar>r\<bar>..\<bar>r\<bar>}"
using le_enorm \<open>r > 0\<close>
apply (auto simp: C_def topspace_Euclidean_space abs_le_iff)
apply (metis add.inverse_neutral le_cases less_minus_iff not_le order_trans)
by (metis enorm_ge0 not_le order.trans)
have compactinC: "compactin (Euclidean_space n) (C r)"
unfolding Euclidean_space_def compactin_subtopology
proof
show "compactin (powertop_real UNIV) (C r)"
proof (rule closed_compactin [OF _ C_subset])
show "closedin (powertop_real UNIV) (C r)"
by (metis Euclidean_space_def cloC closedin_Euclidean_space closedin_closed_subtopology topspace_Euclidean_space)
qed (simp add: compactin_PiE)
qed (auto simp: C_def topspace_Euclidean_space)
have compactinS: "compactin (Euclidean_space n) (S r)"
unfolding Euclidean_space_def compactin_subtopology
proof
show "compactin (powertop_real UNIV) (S r)"
proof (rule closed_compactin)
show "S r \<subseteq> UNIV \<rightarrow>\<^sub>E {- \<bar>r\<bar>..\<bar>r\<bar>}"
using C_subset \<open>\<And>r. S r \<subseteq> C r\<close> by blast
show "closedin (powertop_real UNIV) (S r)"
by (metis Euclidean_space_def cloS closedin_Euclidean_space closedin_closed_subtopology topspace_Euclidean_space)
qed (simp add: compactin_PiE)
qed (auto simp: S_def topspace_Euclidean_space)
have h_if_B: "\<And>y. y \<in> B r \<Longrightarrow> h y \<in> topspace ?E"
using B_def \<open>\<And>r. B r \<union> S r = C r\<close> cmh continuous_map_image_subset_topspace by fastforce
have com_hSr: "compactin (Euclidean_space n) (h ` S r)"
by (meson \<open>\<And>r. S r \<subseteq> C r\<close> cmh compactinS compactin_subtopology image_compactin)
have ope_comp_hSr: "openin (Euclidean_space n) (topspace (Euclidean_space n) - h ` S r)"
proof (rule openin_diff)
show "closedin (Euclidean_space n) (h ` S r)"
using Hausdorff_Euclidean_space com_hSr compactin_imp_closedin by blast
qed auto
have h_pcs: "h ` (B r) \<in> path_components_of (subtopology ?E (topspace ?E - h ` (S r)))"
proof (rule lemmaIOD)
have pc_interval: "path_connectedin (Euclidean_space n) {x \<in> topspace(Euclidean_space n). enorm x \<in> T}"
if T: "is_interval T" for T
proof -
define mul :: "[real, nat \<Rightarrow> real, nat] \<Rightarrow> real" where "mul \<equiv> \<lambda>a x i. a * x i"
let ?neg = "mul (-1)"
have neg_neg [simp]: "?neg (?neg x) = x" for x
by (simp add: mul_def)
have enorm_mul [simp]: "enorm(mul a x) = abs a * enorm x" for a x
by (simp add: enorm_def mul_def power_mult_distrib) (metis real_sqrt_abs real_sqrt_mult sum_distrib_left)
have mul_in_top: "mul a x \<in> topspace ?E"
if "x \<in> topspace ?E" for a x
using mul_def that topspace_Euclidean_space by auto
have neg_in_S: "?neg x \<in> S r"
if "x \<in> S r" for x r
using that topspace_Euclidean_space S_def by simp (simp add: mul_def)
have *: "path_connectedin ?E (S d)"
if "d \<ge> 0" for d
proof (cases "d = 0")
let ?ES = "subtopology ?E (S d)"
case False
then have "d > 0"
using that by linarith
moreover have "path_connected_space ?ES"
unfolding path_connected_space_iff_path_component
proof clarify
have **: "path_component_of ?ES x y"
if x: "x \<in> topspace ?ES" and y: "y \<in> topspace ?ES" "x \<noteq> ?neg y" for x y
proof -
show ?thesis
unfolding path_component_of_def pathin_def S_def
proof (intro exI conjI)
let ?g = "(\<lambda>x. mul (d / enorm x) x) \<circ> (\<lambda>t i. (1 - t) * x i + t * y i)"
show "continuous_map (top_of_set {0::real..1}) (subtopology ?E {x \<in> topspace ?E. enorm x = d}) ?g"
proof (rule continuous_map_compose)
let ?Y = "subtopology ?E (- {zero})"
have **: False
if eq0: "\<And>j. (1 - r) * x j + r * y j = 0"
and ne: "x i \<noteq> - y i"
and d: "enorm x = d" "enorm y = d"
and r: "0 \<le> r" "r \<le> 1"
for i r
proof -
have "mul (1-r) x = ?neg (mul r y)"
using eq0 by (simp add: mul_def fun_eq_iff algebra_simps)
then have "enorm (mul (1-r) x) = enorm (?neg (mul r y))"
by metis
with r have "(1-r) * enorm x = r * enorm y"
by simp
then have r12: "r = 1/2"
using \<open>d \<noteq> 0\<close> d by auto
show ?thesis
using ne eq0 [of i] unfolding r12 by (simp add: algebra_simps)
qed
show "continuous_map (top_of_set {0..1}) ?Y (\<lambda>t i. (1 - t) * x i + t * y i)"
using x y
unfolding continuous_map_componentwise_UNIV Euclidean_space_def continuous_map_in_subtopology
apply (intro conjI allI continuous_intros)
apply (auto simp: zero_def mul_def S_def Euclidean_space_def fun_eq_iff)
using ** by blast
have cm_enorm': "continuous_map (subtopology (powertop_real UNIV) A) euclideanreal enorm" for A
unfolding enorm_def by (intro continuous_intros) auto
have "continuous_map ?Y (subtopology ?E {x. enorm x = d}) (\<lambda>x. mul (d / enorm x) x)"
unfolding continuous_map_in_subtopology
proof (intro conjI)
show "continuous_map ?Y (Euclidean_space n) (\<lambda>x. mul (d / enorm x) x)"
unfolding continuous_map_in_subtopology Euclidean_space_def mul_def zero_def subtopology_subtopology continuous_map_componentwise_UNIV
proof (intro conjI allI cm_enorm' continuous_intros)
show "enorm x \<noteq> 0"
if "x \<in> topspace (subtopology (powertop_real UNIV) ({x. \<forall>i\<ge>n. x i = 0} \<inter> - {\<lambda>i. 0}))" for x
using that by simp (metis abs_le_zero_iff le_enorm not_less)
qed auto
qed (use \<open>d > 0\<close> enorm_ge0 in auto)
moreover have "subtopology ?E {x \<in> topspace ?E. enorm x = d} = subtopology ?E {x. enorm x = d}"
by (simp add: subtopology_restrict Collect_conj_eq)
ultimately show "continuous_map ?Y (subtopology (Euclidean_space n) {x \<in> topspace (Euclidean_space n). enorm x = d}) (\<lambda>x. mul (d / enorm x) x)"
by metis
qed
show "?g (0::real) = x" "?g (1::real) = y"
using that by (auto simp: S_def zero_def mul_def fun_eq_iff)
qed
qed
obtain a b where a: "a \<in> topspace ?ES" and b: "b \<in> topspace ?ES"
and "a \<noteq> b" and negab: "?neg a \<noteq> b"
proof
let ?v = "\<lambda>j i::nat. if i = j then d else 0"
show "?v 0 \<in> topspace (subtopology ?E (S d))" "?v 1 \<in> topspace (subtopology ?E (S d))"
using \<open>n \<ge> 2\<close> \<open>d \<ge> 0\<close> by (auto simp: S_def topspace_Euclidean_space)
show "?v 0 \<noteq> ?v 1" "?neg (?v 0) \<noteq> (?v 1)"
using \<open>d > 0\<close> by (auto simp: mul_def fun_eq_iff)
qed
show "path_component_of ?ES x y"
if x: "x \<in> topspace ?ES" and y: "y \<in> topspace ?ES"
for x y
proof -
have "path_component_of ?ES x (?neg x)"
proof -
have "path_component_of ?ES x a"
by (metis (no_types, hide_lams) ** a b \<open>a \<noteq> b\<close> negab path_component_of_trans path_component_of_sym x)
moreover
have pa_ab: "path_component_of ?ES a b" using "**" a b negab neg_neg by blast
then have "path_component_of ?ES a (?neg x)"
by (metis "**" \<open>a \<noteq> b\<close> cloS closedin_def neg_in_S path_component_of_equiv topspace_subtopology_subset x)
ultimately show ?thesis
by (meson path_component_of_trans)
qed
then show ?thesis
using "**" x y by force
qed
qed
ultimately show ?thesis
by (simp add: cloS closedin_subset path_connectedin_def)
qed (simp add: S_def cong: conj_cong)
have "path_component_of (subtopology ?E {x \<in> topspace ?E. enorm x \<in> T}) x y"
if "enorm x = a" "x \<in> topspace ?E" "enorm x \<in> T" "enorm y = b" "y \<in> topspace ?E" "enorm y \<in> T"
for x y a b
using that
proof (induction a b arbitrary: x y rule: linorder_less_wlog)
case (less a b)
then have "a \<ge> 0"
using enorm_ge0 by blast
with less.hyps have "b > 0"
by linarith
show ?case
proof (rule path_component_of_trans)
have y'_ts: "mul (a / b) y \<in> topspace ?E"
using \<open>y \<in> topspace ?E\<close> mul_in_top by blast
moreover have "enorm (mul (a / b) y) = a"
unfolding enorm_mul using \<open>0 < b\<close> \<open>0 \<le> a\<close> less.prems by simp
ultimately have y'_S: "mul (a / b) y \<in> S a"
using S_def by blast
have "x \<in> S a"
using S_def less.prems by blast
with \<open>x \<in> topspace ?E\<close> y'_ts y'_S
have "path_component_of (subtopology ?E (S a)) x (mul (a / b) y)"
by (metis * [OF \<open>a \<ge> 0\<close>] path_connected_space_iff_path_component path_connectedin_def topspace_subtopology_subset)
moreover
have "{f \<in> topspace ?E. enorm f = a} \<subseteq> {f \<in> topspace ?E. enorm f \<in> T}"
using \<open>enorm x = a\<close> \<open>enorm x \<in> T\<close> by force
ultimately
show "path_component_of (subtopology ?E {x. x \<in> topspace ?E \<and> enorm x \<in> T}) x (mul (a / b) y)"
by (simp add: S_def path_component_of_mono)
have "pathin ?E (\<lambda>t. mul (((1 - t) * b + t * a) / b) y)"
using \<open>b > 0\<close> \<open>y \<in> topspace ?E\<close>
unfolding pathin_def Euclidean_space_def mul_def continuous_map_in_subtopology continuous_map_componentwise_UNIV
by (intro allI conjI continuous_intros) auto
moreover have "mul (((1 - t) * b + t * a) / b) y \<in> topspace ?E"
if "t \<in> {0..1}" for t
using \<open>y \<in> topspace ?E\<close> mul_in_top by blast
moreover have "enorm (mul (((1 - t) * b + t * a) / b) y) \<in> T"
if "t \<in> {0..1}" for t
proof -
have "a \<in> T" "b \<in> T"
using less.prems by auto
then have "\<bar>(1 - t) * b + t * a\<bar> \<in> T"
proof (rule mem_is_interval_1_I [OF T])
show "a \<le> \<bar>(1 - t) * b + t * a\<bar>"
using that \<open>a \<ge> 0\<close> less.hyps segment_bound_lemma by auto
show "\<bar>(1 - t) * b + t * a\<bar> \<le> b"
using that \<open>a \<ge> 0\<close> less.hyps by (auto intro: convex_bound_le)
qed
then show ?thesis
unfolding enorm_mul \<open>enorm y = b\<close> using that \<open>b > 0\<close> by simp
qed
ultimately have pa: "pathin (subtopology ?E {x \<in> topspace ?E. enorm x \<in> T})
(\<lambda>t. mul (((1 - t) * b + t * a) / b) y)"
by (auto simp: pathin_subtopology)
have ex_pathin: "\<exists>g. pathin (subtopology ?E {x \<in> topspace ?E. enorm x \<in> T}) g \<and>
g 0 = y \<and> g 1 = mul (a / b) y"
apply (rule_tac x="\<lambda>t. mul (((1 - t) * b + t * a) / b) y" in exI)
using \<open>b > 0\<close> pa by (auto simp: mul_def)
show "path_component_of (subtopology ?E {x. x \<in> topspace ?E \<and> enorm x \<in> T}) (mul (a / b) y) y"
by (rule path_component_of_sym) (simp add: path_component_of_def ex_pathin)
qed
next
case (refl a)
then have pc: "path_component_of (subtopology ?E (S (enorm u))) u v"
if "u \<in> topspace ?E \<inter> S (enorm x)" "v \<in> topspace ?E \<inter> S (enorm u)" for u v
using * [of a] enorm_ge0 that
by (auto simp: path_connectedin_def path_connected_space_iff_path_component S_def)
have sub: "{u \<in> topspace ?E. enorm u = enorm x} \<subseteq> {u \<in> topspace ?E. enorm u \<in> T}"
using \<open>enorm x \<in> T\<close> by auto
show ?case
using pc [of x y] refl by (auto simp: S_def path_component_of_mono [OF _ sub])
next
case (sym a b)
then show ?case
by (blast intro: path_component_of_sym)
qed
then show ?thesis
by (simp add: path_connectedin_def path_connected_space_iff_path_component)
qed
have "h ` S r \<subseteq> topspace ?E"
by (meson SC cmh compact_imp_compactin_subtopology compactinS compactin_subset_topspace image_compactin)
moreover
have "\<not> compact_space ?E "
by (metis compact_Euclidean_space \<open>n \<noteq> 0\<close>)
then have "\<not> compactin ?E (topspace ?E)"
by (simp add: compact_space_def topspace_Euclidean_space)
then have "h ` S r \<noteq> topspace ?E"
using com_hSr by auto
ultimately have top_hSr_ne: "topspace (subtopology ?E (topspace ?E - h ` S r)) \<noteq> {}"
by auto
show pc1: "\<exists>T. T \<in> path_components_of (subtopology ?E (topspace ?E - h ` S r)) \<and> h ` B r \<subseteq> T"
proof (rule exists_path_component_of_superset [OF _ top_hSr_ne])
have "path_connectedin ?E (h ` B r)"
proof (rule path_connectedin_continuous_map_image)
show "continuous_map (subtopology ?E (C r)) ?E h"
by (simp add: cmh)
have "path_connectedin ?E (B r)"
using pc_interval[of "{..<r}"] is_interval_convex_1 unfolding B_def by auto
then show "path_connectedin (subtopology ?E (C r)) (B r)"
by (simp add: path_connectedin_subtopology BC)
qed
moreover have "h ` B r \<subseteq> topspace ?E - h ` S r"
apply (auto simp: h_if_B)
by (metis BC SC disjSB disjnt_iff inj_onD [OF injh] subsetD)
ultimately show "path_connectedin (subtopology ?E (topspace ?E - h ` S r)) (h ` B r)"
by (simp add: path_connectedin_subtopology)
qed metis
show "\<exists>T. T \<in> path_components_of (subtopology ?E (topspace ?E - h ` S r)) \<and> topspace ?E - h ` (C r) \<subseteq> T"
proof (rule exists_path_component_of_superset [OF _ top_hSr_ne])
have eq: "topspace ?E - {x \<in> topspace ?E. enorm x \<le> r} = {x \<in> topspace ?E. r < enorm x}"
by auto
have "path_connectedin ?E (topspace ?E - C r)"
using pc_interval[of "{r<..}"] is_interval_convex_1 unfolding C_def eq by auto
then have "path_connectedin ?E (topspace ?E - h ` C r)"
by (metis biglemma [OF \<open>n \<noteq> 0\<close> compactinC cmh injh])
then show "path_connectedin (subtopology ?E (topspace ?E - h ` S r)) (topspace ?E - h ` C r)"
by (simp add: Diff_mono SC image_mono path_connectedin_subtopology)
qed metis
have "topspace ?E \<inter> (topspace ?E - h ` S r) = h ` B r \<union> (topspace ?E - h ` C r)" (is "?lhs = ?rhs")
proof
show "?lhs \<subseteq> ?rhs"
using \<open>\<And>r. B r \<union> S r = C r\<close> by auto
have "h ` B r \<inter> h ` S r = {}"
by (metis Diff_triv \<open>\<And>r. B r \<union> S r = C r\<close> \<open>\<And>r. disjnt (S r) (B r)\<close> disjnt_def inf_commute inj_on_Un injh)
then show "?rhs \<subseteq> ?lhs"
using path_components_of_subset pc1 \<open>\<And>r. B r \<union> S r = C r\<close>
by (fastforce simp add: h_if_B)
qed
then show "\<Union> (path_components_of (subtopology ?E (topspace ?E - h ` S r))) = h ` B r \<union> (topspace ?E - h ` (C r))"
by (simp add: Union_path_components_of)
show "T \<noteq> {}"
if "T \<in> path_components_of (subtopology ?E (topspace ?E - h ` S r))" for T
using that by (simp add: nonempty_path_components_of)
show "disjoint (path_components_of (subtopology ?E (topspace ?E - h ` S r)))"
by (simp add: pairwise_disjoint_path_components_of)
have "\<not> path_connectedin ?E (topspace ?E - h ` S r)"
proof (subst biglemma [OF \<open>n \<noteq> 0\<close> compactinS])
show "continuous_map (subtopology ?E (S r)) ?E h"
by (metis Un_commute Un_upper1 cmh continuous_map_from_subtopology_mono eqC)
show "inj_on h (S r)"
using SC inj_on_subset injh by blast
show "\<not> path_connectedin ?E (topspace ?E - S r)"
proof
have "topspace ?E - S r = {x \<in> topspace ?E. enorm x \<noteq> r}"
by (auto simp: S_def)
moreover have "enorm ` {x \<in> topspace ?E. enorm x \<noteq> r} = {0..} - {r}"
proof
have "\<exists>x. x \<in> topspace ?E \<and> enorm x \<noteq> r \<and> d = enorm x"
if "d \<noteq> r" "d \<ge> 0" for d
proof (intro exI conjI)
show "(\<lambda>i. if i = 0 then d else 0) \<in> topspace ?E"
using \<open>n \<noteq> 0\<close> by (auto simp: Euclidean_space_def)
show "enorm (\<lambda>i. if i = 0 then d else 0) \<noteq> r" "d = enorm (\<lambda>i. if i = 0 then d else 0)"
using \<open>n \<noteq> 0\<close> that by simp_all
qed
then show "{0..} - {r} \<subseteq> enorm ` {x \<in> topspace ?E. enorm x \<noteq> r}"
by (auto simp: image_def)
qed (auto simp: enorm_ge0)
ultimately have non_r: "enorm ` (topspace ?E - S r) = {0..} - {r}"
by simp
have "\<exists>x\<ge>0. x \<noteq> r \<and> r \<le> x"
by (metis gt_ex le_cases not_le order_trans)
then have "\<not> is_interval ({0..} - {r})"
unfolding is_interval_1
using \<open>r > 0\<close> by (auto simp: Bex_def)
then show False
if "path_connectedin ?E (topspace ?E - S r)"
using path_connectedin_continuous_map_image [OF cm_enorm that] by (simp add: is_interval_path_connected_1 non_r)
qed
qed
then have "\<not> path_connected_space (subtopology ?E (topspace ?E - h ` S r))"
by (simp add: path_connectedin_def)
then show "\<nexists>T. path_components_of (subtopology ?E (topspace ?E - h ` S r)) \<subseteq> {T}"
by (simp add: path_components_of_subset_singleton)
qed
moreover have "openin ?E A"
if "A \<in> path_components_of (subtopology ?E (topspace ?E - h ` (S r)))" for A
using locally_path_connected_Euclidean_space [of n] that ope_comp_hSr
by (simp add: locally_path_connected_space_open_path_components)
ultimately show ?thesis by metis
qed
have "\<exists>T. openin ?E T \<and> f x \<in> T \<and> T \<subseteq> f ` U"
if "x \<in> U" for x
proof -
have x: "x \<in> topspace ?E"
by (meson U in_mono openin_subset that)
obtain V where V: "openin (powertop_real UNIV) V" and Ueq: "U = V \<inter> {x. \<forall>i\<ge>n. x i = 0}"
using U by (auto simp: openin_subtopology Euclidean_space_def)
with \<open>x \<in> U\<close> have "x \<in> V" by blast
then obtain T where Tfin: "finite {i. T i \<noteq> UNIV}" and Topen: "\<And>i. open (T i)"
and Tx: "x \<in> Pi\<^sub>E UNIV T" and TV: "Pi\<^sub>E UNIV T \<subseteq> V"
using V by (force simp: openin_product_topology_alt)
have "\<exists>e>0. \<forall>x'. \<bar>x' - x i\<bar> < e \<longrightarrow> x' \<in> T i" for i
using Topen [of i] Tx by (auto simp: open_real)
then obtain \<beta> where B0: "\<And>i. \<beta> i > 0" and BT: "\<And>i x'. \<bar>x' - x i\<bar> < \<beta> i \<Longrightarrow> x' \<in> T i"
by metis
define r where "r \<equiv> Min (insert 1 (\<beta> ` {i. T i \<noteq> UNIV}))"
have "r > 0"
by (simp add: B0 Tfin r_def)
have inU: "y \<in> U"
if y: "y \<in> topspace ?E" and yxr: "\<And>i. i<n \<Longrightarrow> \<bar>y i - x i\<bar> < r" for y
proof -
have "y i \<in> T i" for i
proof (cases "T i = UNIV")
show "y i \<in> T i" if "T i \<noteq> UNIV"
proof (cases "i < n")
case True
then show ?thesis
using yxr [OF True] that by (simp add: r_def BT Tfin)
next
case False
then show ?thesis
using B0 Ueq \<open>x \<in> U\<close> topspace_Euclidean_space y by (force intro: BT)
qed
qed auto
with TV have "y \<in> V" by auto
then show ?thesis
using that by (auto simp: Ueq topspace_Euclidean_space)
qed
have xinU: "(\<lambda>i. x i + y i) \<in> U" if "y \<in> C(r/2)" for y
proof (rule inU)
have y: "y \<in> topspace ?E"
using C_def that by blast
show "(\<lambda>i. x i + y i) \<in> topspace ?E"
using x y by (simp add: topspace_Euclidean_space)
have "enorm y \<le> r/2"
using that by (simp add: C_def)
then show "\<bar>x i + y i - x i\<bar> < r" if "i < n" for i
using le_enorm enorm_ge0 that \<open>0 < r\<close> leI order_trans by fastforce
qed
show ?thesis
proof (intro exI conjI)
show "openin ?E ((f \<circ> (\<lambda>y i. x i + y i)) ` B (r/2))"
proof (rule **)
have "continuous_map (subtopology ?E (C(r/2))) (subtopology ?E U) (\<lambda>y i. x i + y i)"
by (auto simp: xinU continuous_map_in_subtopology
intro!: continuous_intros continuous_map_Euclidean_space_add x)
then show "continuous_map (subtopology ?E (C(r/2))) ?E (f \<circ> (\<lambda>y i. x i + y i))"
by (rule continuous_map_compose) (simp add: cmf)
show "inj_on (f \<circ> (\<lambda>y i. x i + y i)) (C(r/2))"
proof (clarsimp simp add: inj_on_def C_def topspace_Euclidean_space simp del: divide_const_simps)
show "y' = y"
if ey: "enorm y \<le> r / 2" and ey': "enorm y' \<le> r / 2"
and y0: "\<forall>i\<ge>n. y i = 0" and y'0: "\<forall>i\<ge>n. y' i = 0"
and feq: "f (\<lambda>i. x i + y' i) = f (\<lambda>i. x i + y i)"
for y' y :: "nat \<Rightarrow> real"
proof -
have "(\<lambda>i. x i + y i) \<in> U"
proof (rule inU)
show "(\<lambda>i. x i + y i) \<in> topspace ?E"
using topspace_Euclidean_space x y0 by auto
show "\<bar>x i + y i - x i\<bar> < r" if "i < n" for i
using ey le_enorm [of _ y] \<open>r > 0\<close> that by fastforce
qed
moreover have "(\<lambda>i. x i + y' i) \<in> U"
proof (rule inU)
show "(\<lambda>i. x i + y' i) \<in> topspace ?E"
using topspace_Euclidean_space x y'0 by auto
show "\<bar>x i + y' i - x i\<bar> < r" if "i < n" for i
using ey' le_enorm [of _ y'] \<open>r > 0\<close> that by fastforce
qed
ultimately have "(\<lambda>i. x i + y' i) = (\<lambda>i. x i + y i)"
using feq by (meson \<open>inj_on f U\<close> inj_on_def)
then show ?thesis
by (auto simp: fun_eq_iff)
qed
qed
qed (simp add: \<open>0 < r\<close>)
have "x \<in> (\<lambda>y i. x i + y i) ` B (r / 2)"
proof
show "x = (\<lambda>i. x i + zero i)"
by (simp add: zero_def)
qed (auto simp: B_def \<open>r > 0\<close>)
then show "f x \<in> (f \<circ> (\<lambda>y i. x i + y i)) ` B (r/2)"
by (metis image_comp image_eqI)
show "(f \<circ> (\<lambda>y i. x i + y i)) ` B (r/2) \<subseteq> f ` U"
using \<open>\<And>r. B r \<subseteq> C r\<close> xinU by fastforce
qed
qed
then show ?thesis
using openin_subopen by force
qed
corollary invariance_of_domain_Euclidean_space_embedding_map:
assumes "openin (Euclidean_space n) U"
and cmf: "continuous_map(subtopology (Euclidean_space n) U) (Euclidean_space n) f"
and "inj_on f U"
shows "embedding_map(subtopology (Euclidean_space n) U) (Euclidean_space n) f"
proof (rule injective_open_imp_embedding_map [OF cmf])
show "open_map (subtopology (Euclidean_space n) U) (Euclidean_space n) f"
unfolding open_map_def
by (meson assms continuous_map_from_subtopology_mono inj_on_subset invariance_of_domain_Euclidean_space openin_imp_subset openin_trans_full)
show "inj_on f (topspace (subtopology (Euclidean_space n) U))"
using assms openin_subset topspace_subtopology_subset by fastforce
qed
corollary invariance_of_domain_Euclidean_space_gen:
assumes "n \<le> m" and U: "openin (Euclidean_space m) U"
and cmf: "continuous_map(subtopology (Euclidean_space m) U) (Euclidean_space n) f"
and "inj_on f U"
shows "openin (Euclidean_space n) (f ` U)"
proof -
have *: "Euclidean_space n = subtopology (Euclidean_space m) (topspace(Euclidean_space n))"
by (metis Euclidean_space_def \<open>n \<le> m\<close> inf.absorb_iff2 subset_Euclidean_space subtopology_subtopology topspace_Euclidean_space)
moreover have "U \<subseteq> topspace (subtopology (Euclidean_space m) U)"
by (metis U inf.absorb_iff2 openin_subset openin_subtopology openin_topspace)
ultimately show ?thesis
by (metis (no_types) U \<open>inj_on f U\<close> cmf continuous_map_in_subtopology inf.absorb_iff2
inf.orderE invariance_of_domain_Euclidean_space openin_imp_subset openin_subtopology openin_topspace)
qed
corollary invariance_of_domain_Euclidean_space_embedding_map_gen:
assumes "n \<le> m" and U: "openin (Euclidean_space m) U"
and cmf: "continuous_map(subtopology (Euclidean_space m) U) (Euclidean_space n) f"
and "inj_on f U"
shows "embedding_map(subtopology (Euclidean_space m) U) (Euclidean_space n) f"
proof (rule injective_open_imp_embedding_map [OF cmf])
show "open_map (subtopology (Euclidean_space m) U) (Euclidean_space n) f"
by (meson U \<open>n \<le> m\<close> \<open>inj_on f U\<close> cmf continuous_map_from_subtopology_mono invariance_of_domain_Euclidean_space_gen open_map_def openin_open_subtopology subset_inj_on)
show "inj_on f (topspace (subtopology (Euclidean_space m) U))"
using assms openin_subset topspace_subtopology_subset by fastforce
qed
subsection\<open>Relating two variants of Euclidean space, one within product topology. \<close>
proposition homeomorphic_maps_Euclidean_space_euclidean_gen_OLD:
fixes B :: "'n::euclidean_space set"
assumes "finite B" "independent B" and orth: "pairwise orthogonal B" and n: "card B = n"
obtains f g where "homeomorphic_maps (Euclidean_space n) (top_of_set (span B)) f g"
proof -
note representation_basis [OF \<open>independent B\<close>, simp]
obtain b where injb: "inj_on b {..<n}" and beq: "b ` {..<n} = B"
using finite_imp_nat_seg_image_inj_on [OF \<open>finite B\<close>]
by (metis n card_Collect_less_nat card_image lessThan_def)
then have biB: "\<And>i. i < n \<Longrightarrow> b i \<in> B"
by force
have repr: "\<And>v. v \<in> span B \<Longrightarrow> (\<Sum>i<n. representation B v (b i) *\<^sub>R b i) = v"
using real_vector.sum_representation_eq [OF \<open>independent B\<close> _ \<open>finite B\<close>]
by (metis (no_types, lifting) injb beq order_refl sum.reindex_cong)
let ?f = "\<lambda>x. \<Sum>i<n. x i *\<^sub>R b i"
let ?g = "\<lambda>v i. if i < n then representation B v (b i) else 0"
show thesis
proof
show "homeomorphic_maps (Euclidean_space n) (top_of_set (span B)) ?f ?g"
unfolding homeomorphic_maps_def
proof (intro conjI)
have *: "continuous_map euclidean (top_of_set (span B)) ?f"
by (metis (mono_tags) biB continuous_map_span_sum lessThan_iff)
show "continuous_map (Euclidean_space n) (top_of_set (span B)) ?f"
unfolding Euclidean_space_def
by (rule continuous_map_from_subtopology) (simp add: euclidean_product_topology *)
show "continuous_map (top_of_set (span B)) (Euclidean_space n) ?g"
unfolding Euclidean_space_def
by (auto simp: continuous_map_in_subtopology continuous_map_componentwise_UNIV continuous_on_representation \<open>independent B\<close> biB orth pairwise_orthogonal_imp_finite)
have [simp]: "\<And>x i. i<n \<Longrightarrow> x i *\<^sub>R b i \<in> span B"
by (simp add: biB span_base span_scale)
have "representation B (?f x) (b j) = x j"
if 0: "\<forall>i\<ge>n. x i = (0::real)" and "j < n" for x j
proof -
have "representation B (?f x) (b j) = (\<Sum>i<n. representation B (x i *\<^sub>R b i) (b j))"
by (subst real_vector.representation_sum) (auto simp add: \<open>independent B\<close>)
also have "... = (\<Sum>i<n. x i * representation B (b i) (b j))"
by (simp add: assms(2) biB representation_scale span_base)
also have "... = (\<Sum>i<n. if b j = b i then x i else 0)"
by (simp add: biB if_distrib cong: if_cong)
also have "... = x j"
using that inj_on_eq_iff [OF injb] by auto
finally show ?thesis .
qed
then show "\<forall>x\<in>topspace (Euclidean_space n). ?g (?f x) = x"
by (auto simp: Euclidean_space_def)
show "\<forall>y\<in>topspace (top_of_set (span B)). ?f (?g y) = y"
using repr by (auto simp: Euclidean_space_def)
qed
qed
qed
proposition homeomorphic_maps_Euclidean_space_euclidean_gen:
fixes B :: "'n::euclidean_space set"
assumes "independent B" and orth: "pairwise orthogonal B" and n: "card B = n"
and 1: "\<And>u. u \<in> B \<Longrightarrow> norm u = 1"
obtains f g where "homeomorphic_maps (Euclidean_space n) (top_of_set (span B)) f g"
and "\<And>x. x \<in> topspace (Euclidean_space n) \<Longrightarrow> (norm (f x))\<^sup>2 = (\<Sum>i<n. (x i)\<^sup>2)"
proof -
note representation_basis [OF \<open>independent B\<close>, simp]
have "finite B"
using \<open>independent B\<close> finiteI_independent by metis
obtain b where injb: "inj_on b {..<n}" and beq: "b ` {..<n} = B"
using finite_imp_nat_seg_image_inj_on [OF \<open>finite B\<close>]
by (metis n card_Collect_less_nat card_image lessThan_def)
then have biB: "\<And>i. i < n \<Longrightarrow> b i \<in> B"
by force
have "0 \<notin> B"
using \<open>independent B\<close> dependent_zero by blast
have [simp]: "b i \<bullet> b j = (if j = i then 1 else 0)"
if "i < n" "j < n" for i j
proof (cases "i = j")
case True
with 1 that show ?thesis
by (auto simp: norm_eq_sqrt_inner biB)
next
case False
then have "b i \<noteq> b j"
by (meson inj_onD injb lessThan_iff that)
then show ?thesis
using orth by (auto simp: orthogonal_def pairwise_def norm_eq_sqrt_inner that biB)
qed
have [simp]: "\<And>x i. i<n \<Longrightarrow> x i *\<^sub>R b i \<in> span B"
by (simp add: biB span_base span_scale)
have repr: "\<And>v. v \<in> span B \<Longrightarrow> (\<Sum>i<n. representation B v (b i) *\<^sub>R b i) = v"
using real_vector.sum_representation_eq [OF \<open>independent B\<close> _ \<open>finite B\<close>]
by (metis (no_types, lifting) injb beq order_refl sum.reindex_cong)
define f where "f \<equiv> \<lambda>x. \<Sum>i<n. x i *\<^sub>R b i"
define g where "g \<equiv> \<lambda>v i. if i < n then representation B v (b i) else 0"
show thesis
proof
show "homeomorphic_maps (Euclidean_space n) (top_of_set (span B)) f g"
unfolding homeomorphic_maps_def
proof (intro conjI)
have *: "continuous_map euclidean (top_of_set (span B)) f"
unfolding f_def
by (rule continuous_map_span_sum) (use biB \<open>0 \<notin> B\<close> in auto)
show "continuous_map (Euclidean_space n) (top_of_set (span B)) f"
unfolding Euclidean_space_def
by (rule continuous_map_from_subtopology) (simp add: euclidean_product_topology *)
show "continuous_map (top_of_set (span B)) (Euclidean_space n) g"
unfolding Euclidean_space_def g_def
by (auto simp: continuous_map_in_subtopology continuous_map_componentwise_UNIV continuous_on_representation \<open>independent B\<close> biB orth pairwise_orthogonal_imp_finite)
have "representation B (f x) (b j) = x j"
if 0: "\<forall>i\<ge>n. x i = (0::real)" and "j < n" for x j
proof -
have "representation B (f x) (b j) = (\<Sum>i<n. representation B (x i *\<^sub>R b i) (b j))"
unfolding f_def
by (subst real_vector.representation_sum) (auto simp add: \<open>independent B\<close>)
also have "... = (\<Sum>i<n. x i * representation B (b i) (b j))"
by (simp add: \<open>independent B\<close> biB representation_scale span_base)
also have "... = (\<Sum>i<n. if b j = b i then x i else 0)"
by (simp add: biB if_distrib cong: if_cong)
also have "... = x j"
using that inj_on_eq_iff [OF injb] by auto
finally show ?thesis .
qed
then show "\<forall>x\<in>topspace (Euclidean_space n). g (f x) = x"
by (auto simp: Euclidean_space_def f_def g_def)
show "\<forall>y\<in>topspace (top_of_set (span B)). f (g y) = y"
using repr by (auto simp: Euclidean_space_def f_def g_def)
qed
show normeq: "(norm (f x))\<^sup>2 = (\<Sum>i<n. (x i)\<^sup>2)" if "x \<in> topspace (Euclidean_space n)" for x
unfolding f_def dot_square_norm [symmetric]
by (simp add: power2_eq_square inner_sum_left inner_sum_right if_distrib biB cong: if_cong)
qed
qed
corollary homeomorphic_maps_Euclidean_space_euclidean:
obtains f :: "(nat \<Rightarrow> real) \<Rightarrow> 'n::euclidean_space" and g
where "homeomorphic_maps (Euclidean_space (DIM('n))) euclidean f g"
by (force intro: homeomorphic_maps_Euclidean_space_euclidean_gen [OF independent_Basis orthogonal_Basis refl norm_Basis])
lemma homeomorphic_maps_nsphere_euclidean_sphere:
fixes B :: "'n::euclidean_space set"
assumes B: "independent B" and orth: "pairwise orthogonal B" and n: "card B = n" and "n \<noteq> 0"
and 1: "\<And>u. u \<in> B \<Longrightarrow> norm u = 1"
obtains f :: "(nat \<Rightarrow> real) \<Rightarrow> 'n::euclidean_space" and g
where "homeomorphic_maps (nsphere(n - 1)) (top_of_set (sphere 0 1 \<inter> span B)) f g"
proof -
have "finite B"
using \<open>independent B\<close> finiteI_independent by metis
obtain f g where fg: "homeomorphic_maps (Euclidean_space n) (top_of_set (span B)) f g"
and normf: "\<And>x. x \<in> topspace (Euclidean_space n) \<Longrightarrow> (norm (f x))\<^sup>2 = (\<Sum>i<n. (x i)\<^sup>2)"
using homeomorphic_maps_Euclidean_space_euclidean_gen [OF B orth n 1]
by blast
obtain b where injb: "inj_on b {..<n}" and beq: "b ` {..<n} = B"
using finite_imp_nat_seg_image_inj_on [OF \<open>finite B\<close>]
by (metis n card_Collect_less_nat card_image lessThan_def)
then have biB: "\<And>i. i < n \<Longrightarrow> b i \<in> B"
by force
have [simp]: "\<And>i. i < n \<Longrightarrow> b i \<noteq> 0"
using \<open>independent B\<close> biB dependent_zero by fastforce
have [simp]: "b i \<bullet> b j = (if j = i then (norm (b i))\<^sup>2 else 0)"
if "i < n" "j < n" for i j
proof (cases "i = j")
case False
then have "b i \<noteq> b j"
by (meson inj_onD injb lessThan_iff that)
then show ?thesis
using orth by (auto simp: orthogonal_def pairwise_def norm_eq_sqrt_inner that biB)
qed (auto simp: norm_eq_sqrt_inner)
have [simp]: "Suc (n - Suc 0) = n"
using Suc_pred \<open>n \<noteq> 0\<close> by blast
then have [simp]: "{..card B - Suc 0} = {..<card B}"
using n by fastforce
show thesis
proof
have 1: "norm (f x) = 1"
if "(\<Sum>i<card B. (x i)\<^sup>2) = (1::real)" "x \<in> topspace (Euclidean_space n)" for x
proof -
have "norm (f x)^2 = 1"
using normf that by (simp add: n)
with that show ?thesis
by (simp add: power2_eq_imp_eq)
qed
have "homeomorphic_maps (nsphere (n - 1)) (top_of_set (span B \<inter> sphere 0 1)) f g"
unfolding nsphere_def subtopology_subtopology [symmetric]
proof (rule homeomorphic_maps_subtopologies_alt)
show "homeomorphic_maps (Euclidean_space (Suc (n - 1))) (top_of_set (span B)) f g"
using fg by (force simp add: )
show "f ` (topspace (Euclidean_space (Suc (n - 1))) \<inter> {x. (\<Sum>i\<le>n - 1. (x i)\<^sup>2) = 1}) \<subseteq> sphere 0 1"
using n by (auto simp: image_subset_iff Euclidean_space_def 1)
have "(\<Sum>i\<le>n - Suc 0. (g u i)\<^sup>2) = 1"
if "u \<in> span B" and "norm (u::'n) = 1" for u
proof -
obtain v where [simp]: "u = f v" "v \<in> topspace (Euclidean_space n)"
using fg unfolding homeomorphic_maps_map subset_iff
by (metis \<open>u \<in> span B\<close> homeomorphic_imp_surjective_map image_eqI topspace_euclidean_subtopology)
then have [simp]: "g (f v) = v"
by (meson fg homeomorphic_maps_map)
have fv21: "norm (f v) ^ 2 = 1"
using that by simp
show ?thesis
using that normf fv21 \<open>v \<in> topspace (Euclidean_space n)\<close> n by force
qed
then show "g ` (topspace (top_of_set (span B)) \<inter> sphere 0 1) \<subseteq> {x. (\<Sum>i\<le>n - 1. (x i)\<^sup>2) = 1}"
by auto
qed
then show "homeomorphic_maps (nsphere(n - 1)) (top_of_set (sphere 0 1 \<inter> span B)) f g"
by (simp add: inf_commute)
qed
qed
subsection\<open> Invariance of dimension and domain\<close>
lemma homeomorphic_maps_iff_homeomorphism [simp]:
"homeomorphic_maps (top_of_set S) (top_of_set T) f g \<longleftrightarrow> homeomorphism S T f g"
unfolding homeomorphic_maps_def homeomorphism_def by force
lemma homeomorphic_space_iff_homeomorphic [simp]:
"(top_of_set S) homeomorphic_space (top_of_set T) \<longleftrightarrow> S homeomorphic T"
by (simp add: homeomorphic_def homeomorphic_space_def)
lemma homeomorphic_subspace_Euclidean_space:
fixes S :: "'a::euclidean_space set"
assumes "subspace S"
shows "top_of_set S homeomorphic_space Euclidean_space n \<longleftrightarrow> dim S = n"
proof -
obtain B where B: "B \<subseteq> S" "independent B" "span B = S" "card B = dim S"
and orth: "pairwise orthogonal B" and 1: "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
by (metis assms orthonormal_basis_subspace)
then have "finite B"
by (simp add: pairwise_orthogonal_imp_finite)
have "top_of_set S homeomorphic_space top_of_set (span B)"
unfolding homeomorphic_space_iff_homeomorphic
by (auto simp: assms B intro: homeomorphic_subspaces)
also have "\<dots> homeomorphic_space Euclidean_space (dim S)"
unfolding homeomorphic_space_def
using homeomorphic_maps_Euclidean_space_euclidean_gen [OF \<open>independent B\<close> orth] homeomorphic_maps_sym 1 B
by metis
finally have "top_of_set S homeomorphic_space Euclidean_space (dim S)" .
then show ?thesis
using homeomorphic_space_sym homeomorphic_space_trans invariance_of_dimension_Euclidean_space by blast
qed
lemma homeomorphic_subspace_Euclidean_space_dim:
fixes S :: "'a::euclidean_space set"
assumes "subspace S"
shows "top_of_set S homeomorphic_space Euclidean_space (dim S)"
by (simp add: homeomorphic_subspace_Euclidean_space assms)
lemma homeomorphic_subspaces_eq:
fixes S T:: "'a::euclidean_space set"
assumes "subspace S" "subspace T"
shows "S homeomorphic T \<longleftrightarrow> dim S = dim T"
proof
show "dim S = dim T"
if "S homeomorphic T"
proof -
have "Euclidean_space (dim S) homeomorphic_space top_of_set S"
using \<open>subspace S\<close> homeomorphic_space_sym homeomorphic_subspace_Euclidean_space_dim by blast
also have "\<dots> homeomorphic_space top_of_set T"
by (simp add: that)
also have "\<dots> homeomorphic_space Euclidean_space (dim T)"
by (simp add: homeomorphic_subspace_Euclidean_space assms)
finally have "Euclidean_space (dim S) homeomorphic_space Euclidean_space (dim T)" .
then show ?thesis
by (simp add: invariance_of_dimension_Euclidean_space)
qed
next
show "S homeomorphic T"
if "dim S = dim T"
by (metis that assms homeomorphic_subspaces)
qed
lemma homeomorphic_affine_Euclidean_space:
assumes "affine S"
shows "top_of_set S homeomorphic_space Euclidean_space n \<longleftrightarrow> aff_dim S = n"
(is "?X homeomorphic_space ?E \<longleftrightarrow> aff_dim S = n")
proof (cases "S = {}")
case True
with assms show ?thesis
using homeomorphic_empty_space nonempty_Euclidean_space by fastforce
next
case False
then obtain a where "a \<in> S"
by force
have "(?X homeomorphic_space ?E)
= (top_of_set (image (\<lambda>x. -a + x) S) homeomorphic_space ?E)"
proof
show "top_of_set ((+) (- a) ` S) homeomorphic_space ?E"
if "?X homeomorphic_space ?E"
using that
by (meson homeomorphic_space_iff_homeomorphic homeomorphic_space_sym homeomorphic_space_trans homeomorphic_translation)
show "?X homeomorphic_space ?E"
if "top_of_set ((+) (- a) ` S) homeomorphic_space ?E"
using that
by (meson homeomorphic_space_iff_homeomorphic homeomorphic_space_trans homeomorphic_translation)
qed
also have "\<dots> \<longleftrightarrow> aff_dim S = n"
by (metis \<open>a \<in> S\<close> aff_dim_eq_dim affine_diffs_subspace affine_hull_eq assms homeomorphic_subspace_Euclidean_space of_nat_eq_iff)
finally show ?thesis .
qed
corollary invariance_of_domain_subspaces:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes ope: "openin (top_of_set U) S"
and "subspace U" "subspace V" and VU: "dim V \<le> dim U"
and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
and injf: "inj_on f S"
shows "openin (top_of_set V) (f ` S)"
proof -
have "S \<subseteq> U"
using openin_imp_subset [OF ope] .
have Uhom: "top_of_set U homeomorphic_space Euclidean_space (dim U)"
and Vhom: "top_of_set V homeomorphic_space Euclidean_space (dim V)"
by (simp_all add: assms homeomorphic_subspace_Euclidean_space_dim)
then obtain \<phi> \<phi>' where hom: "homeomorphic_maps (top_of_set U) (Euclidean_space (dim U)) \<phi> \<phi>'"
by (auto simp: homeomorphic_space_def)
obtain \<psi> \<psi>' where \<psi>: "homeomorphic_map (top_of_set V) (Euclidean_space (dim V)) \<psi>"
and \<psi>'\<psi>: "\<forall>x\<in>V. \<psi>' (\<psi> x) = x"
using Vhom by (auto simp: homeomorphic_space_def homeomorphic_maps_map)
have "((\<psi> \<circ> f \<circ> \<phi>') o \<phi>) ` S = (\<psi> o f) ` S"
proof (rule image_cong [OF refl])
show "(\<psi> \<circ> f \<circ> \<phi>' \<circ> \<phi>) x = (\<psi> \<circ> f) x" if "x \<in> S" for x
using that unfolding o_def
by (metis \<open>S \<subseteq> U\<close> hom homeomorphic_maps_map in_mono topspace_euclidean_subtopology)
qed
moreover
have "openin (Euclidean_space (dim V)) ((\<psi> \<circ> f \<circ> \<phi>') ` \<phi> ` S)"
proof (rule invariance_of_domain_Euclidean_space_gen [OF VU])
show "openin (Euclidean_space (dim U)) (\<phi> ` S)"
using homeomorphic_map_openness_eq hom homeomorphic_maps_map ope by blast
show "continuous_map (subtopology (Euclidean_space (dim U)) (\<phi> ` S)) (Euclidean_space (dim V)) (\<psi> \<circ> f \<circ> \<phi>')"
proof (intro continuous_map_compose)
have "continuous_on ({x. \<forall>i\<ge>dim U. x i = 0} \<inter> \<phi> ` S) \<phi>'"
if "continuous_on {x. \<forall>i\<ge>dim U. x i = 0} \<phi>'"
using that by (force elim: continuous_on_subset)
moreover have "\<phi>' ` ({x. \<forall>i\<ge>dim U. x i = 0} \<inter> \<phi> ` S) \<subseteq> S"
if "\<forall>x\<in>U. \<phi>' (\<phi> x) = x"
using that \<open>S \<subseteq> U\<close> by fastforce
ultimately show "continuous_map (subtopology (Euclidean_space (dim U)) (\<phi> ` S)) (top_of_set S) \<phi>'"
using hom unfolding homeomorphic_maps_def
by (simp add: Euclidean_space_def subtopology_subtopology euclidean_product_topology)
show "continuous_map (top_of_set S) (top_of_set V) f"
by (simp add: contf fim)
show "continuous_map (top_of_set V) (Euclidean_space (dim V)) \<psi>"
by (simp add: \<psi> homeomorphic_imp_continuous_map)
qed
show "inj_on (\<psi> \<circ> f \<circ> \<phi>') (\<phi> ` S)"
using injf hom
unfolding inj_on_def homeomorphic_maps_map
by simp (metis \<open>S \<subseteq> U\<close> \<psi>'\<psi> fim imageI subsetD)
qed
ultimately have "openin (Euclidean_space (dim V)) (\<psi> ` f ` S)"
by (simp add: image_comp)
then show ?thesis
by (simp add: fim homeomorphic_map_openness_eq [OF \<psi>])
qed
lemma invariance_of_domain:
fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
assumes "continuous_on S f" "open S" "inj_on f S" shows "open(f ` S)"
using invariance_of_domain_subspaces [of UNIV S UNIV] assms by (force simp add: )
corollary invariance_of_dimension_subspaces:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes ope: "openin (top_of_set U) S"
and "subspace U" "subspace V"
and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
and injf: "inj_on f S" and "S \<noteq> {}"
shows "dim U \<le> dim V"
proof -
have "False" if "dim V < dim U"
proof -
obtain T where "subspace T" "T \<subseteq> U" "dim T = dim V"
using choose_subspace_of_subspace [of "dim V" U]
by (metis \<open>dim V < dim U\<close> assms(2) order.strict_implies_order span_eq_iff)
then have "V homeomorphic T"
by (simp add: \<open>subspace V\<close> homeomorphic_subspaces)
then obtain h k where homhk: "homeomorphism V T h k"
using homeomorphic_def by blast
have "continuous_on S (h \<circ> f)"
by (meson contf continuous_on_compose continuous_on_subset fim homeomorphism_cont1 homhk)
moreover have "(h \<circ> f) ` S \<subseteq> U"
using \<open>T \<subseteq> U\<close> fim homeomorphism_image1 homhk by fastforce
moreover have "inj_on (h \<circ> f) S"
apply (clarsimp simp: inj_on_def)
by (metis fim homeomorphism_apply1 homhk image_subset_iff inj_onD injf)
ultimately have ope_hf: "openin (top_of_set U) ((h \<circ> f) ` S)"
using invariance_of_domain_subspaces [OF ope \<open>subspace U\<close> \<open>subspace U\<close>] by blast
have "(h \<circ> f) ` S \<subseteq> T"
using fim homeomorphism_image1 homhk by fastforce
then have "dim ((h \<circ> f) ` S) \<le> dim T"
by (rule dim_subset)
also have "dim ((h \<circ> f) ` S) = dim U"
using \<open>S \<noteq> {}\<close> \<open>subspace U\<close>
by (blast intro: dim_openin ope_hf)
finally show False
using \<open>dim V < dim U\<close> \<open>dim T = dim V\<close> by simp
qed
then show ?thesis
using not_less by blast
qed
corollary invariance_of_domain_affine_sets:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes ope: "openin (top_of_set U) S"
and aff: "affine U" "affine V" "aff_dim V \<le> aff_dim U"
and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
and injf: "inj_on f S"
shows "openin (top_of_set V) (f ` S)"
proof (cases "S = {}")
case False
obtain a b where "a \<in> S" "a \<in> U" "b \<in> V"
using False fim ope openin_contains_cball by fastforce
have "openin (top_of_set ((+) (- b) ` V)) (((+) (- b) \<circ> f \<circ> (+) a) ` (+) (- a) ` S)"
proof (rule invariance_of_domain_subspaces)
show "openin (top_of_set ((+) (- a) ` U)) ((+) (- a) ` S)"
by (metis ope homeomorphism_imp_open_map homeomorphism_translation translation_galois)
show "subspace ((+) (- a) ` U)"
by (simp add: \<open>a \<in> U\<close> affine_diffs_subspace_subtract \<open>affine U\<close> cong: image_cong_simp)
show "subspace ((+) (- b) ` V)"
by (simp add: \<open>b \<in> V\<close> affine_diffs_subspace_subtract \<open>affine V\<close> cong: image_cong_simp)
show "dim ((+) (- b) ` V) \<le> dim ((+) (- a) ` U)"
by (metis \<open>a \<in> U\<close> \<open>b \<in> V\<close> aff_dim_eq_dim affine_hull_eq aff of_nat_le_iff)
show "continuous_on ((+) (- a) ` S) ((+) (- b) \<circ> f \<circ> (+) a)"
by (metis contf continuous_on_compose homeomorphism_cont2 homeomorphism_translation translation_galois)
show "((+) (- b) \<circ> f \<circ> (+) a) ` (+) (- a) ` S \<subseteq> (+) (- b) ` V"
using fim by auto
show "inj_on ((+) (- b) \<circ> f \<circ> (+) a) ((+) (- a) ` S)"
by (auto simp: inj_on_def) (meson inj_onD injf)
qed
then show ?thesis
by (metis (no_types, lifting) homeomorphism_imp_open_map homeomorphism_translation image_comp translation_galois)
qed auto
corollary invariance_of_dimension_affine_sets:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes ope: "openin (top_of_set U) S"
and aff: "affine U" "affine V"
and contf: "continuous_on S f" and fim: "f ` S \<subseteq> V"
and injf: "inj_on f S" and "S \<noteq> {}"
shows "aff_dim U \<le> aff_dim V"
proof -
obtain a b where "a \<in> S" "a \<in> U" "b \<in> V"
using \<open>S \<noteq> {}\<close> fim ope openin_contains_cball by fastforce
have "dim ((+) (- a) ` U) \<le> dim ((+) (- b) ` V)"
proof (rule invariance_of_dimension_subspaces)
show "openin (top_of_set ((+) (- a) ` U)) ((+) (- a) ` S)"
by (metis ope homeomorphism_imp_open_map homeomorphism_translation translation_galois)
show "subspace ((+) (- a) ` U)"
by (simp add: \<open>a \<in> U\<close> affine_diffs_subspace_subtract \<open>affine U\<close> cong: image_cong_simp)
show "subspace ((+) (- b) ` V)"
by (simp add: \<open>b \<in> V\<close> affine_diffs_subspace_subtract \<open>affine V\<close> cong: image_cong_simp)
show "continuous_on ((+) (- a) ` S) ((+) (- b) \<circ> f \<circ> (+) a)"
by (metis contf continuous_on_compose homeomorphism_cont2 homeomorphism_translation translation_galois)
show "((+) (- b) \<circ> f \<circ> (+) a) ` (+) (- a) ` S \<subseteq> (+) (- b) ` V"
using fim by auto
show "inj_on ((+) (- b) \<circ> f \<circ> (+) a) ((+) (- a) ` S)"
by (auto simp: inj_on_def) (meson inj_onD injf)
qed (use \<open>S \<noteq> {}\<close> in auto)
then show ?thesis
by (metis \<open>a \<in> U\<close> \<open>b \<in> V\<close> aff_dim_eq_dim affine_hull_eq aff of_nat_le_iff)
qed
corollary invariance_of_dimension:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes contf: "continuous_on S f" and "open S"
and injf: "inj_on f S" and "S \<noteq> {}"
shows "DIM('a) \<le> DIM('b)"
using invariance_of_dimension_subspaces [of UNIV S UNIV f] assms
by auto
corollary continuous_injective_image_subspace_dim_le:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "subspace S" "subspace T"
and contf: "continuous_on S f" and fim: "f ` S \<subseteq> T"
and injf: "inj_on f S"
shows "dim S \<le> dim T"
apply (rule invariance_of_dimension_subspaces [of S S _ f])
using assms by (auto simp: subspace_affine)
lemma invariance_of_dimension_convex_domain:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "convex S"
and contf: "continuous_on S f" and fim: "f ` S \<subseteq> affine hull T"
and injf: "inj_on f S"
shows "aff_dim S \<le> aff_dim T"
proof (cases "S = {}")
case True
then show ?thesis by (simp add: aff_dim_geq)
next
case False
have "aff_dim (affine hull S) \<le> aff_dim (affine hull T)"
proof (rule invariance_of_dimension_affine_sets)
show "openin (top_of_set (affine hull S)) (rel_interior S)"
by (simp add: openin_rel_interior)
show "continuous_on (rel_interior S) f"
using contf continuous_on_subset rel_interior_subset by blast
show "f ` rel_interior S \<subseteq> affine hull T"
using fim rel_interior_subset by blast
show "inj_on f (rel_interior S)"
using inj_on_subset injf rel_interior_subset by blast
show "rel_interior S \<noteq> {}"
by (simp add: False \<open>convex S\<close> rel_interior_eq_empty)
qed auto
then show ?thesis
by simp
qed
lemma homeomorphic_convex_sets_le:
assumes "convex S" "S homeomorphic T"
shows "aff_dim S \<le> aff_dim T"
proof -
obtain h k where homhk: "homeomorphism S T h k"
using homeomorphic_def assms by blast
show ?thesis
proof (rule invariance_of_dimension_convex_domain [OF \<open>convex S\<close>])
show "continuous_on S h"
using homeomorphism_def homhk by blast
show "h ` S \<subseteq> affine hull T"
by (metis homeomorphism_def homhk hull_subset)
show "inj_on h S"
by (meson homeomorphism_apply1 homhk inj_on_inverseI)
qed
qed
lemma homeomorphic_convex_sets:
assumes "convex S" "convex T" "S homeomorphic T"
shows "aff_dim S = aff_dim T"
by (meson assms dual_order.antisym homeomorphic_convex_sets_le homeomorphic_sym)
lemma homeomorphic_convex_compact_sets_eq:
assumes "convex S" "compact S" "convex T" "compact T"
shows "S homeomorphic T \<longleftrightarrow> aff_dim S = aff_dim T"
by (meson assms homeomorphic_convex_compact_sets homeomorphic_convex_sets)
lemma invariance_of_domain_gen:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "open S" "continuous_on S f" "inj_on f S" "DIM('b) \<le> DIM('a)"
shows "open(f ` S)"
using invariance_of_domain_subspaces [of UNIV S UNIV f] assms by auto
lemma injective_into_1d_imp_open_map_UNIV:
fixes f :: "'a::euclidean_space \<Rightarrow> real"
assumes "open T" "continuous_on S f" "inj_on f S" "T \<subseteq> S"
shows "open (f ` T)"
apply (rule invariance_of_domain_gen [OF \<open>open T\<close>])
using assms apply (auto simp: elim: continuous_on_subset subset_inj_on)
done
lemma continuous_on_inverse_open:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "open S" "continuous_on S f" "DIM('b) \<le> DIM('a)" and gf: "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"
shows "continuous_on (f ` S) g"
proof (clarsimp simp add: continuous_openin_preimage_eq)
fix T :: "'a set"
assume "open T"
have eq: "f ` S \<inter> g -` T = f ` (S \<inter> T)"
by (auto simp: gf)
have "openin (top_of_set (f ` S)) (f ` (S \<inter> T))"
proof (rule open_openin_trans [OF invariance_of_domain_gen])
show "inj_on f S"
using inj_on_inverseI gf by auto
show "open (f ` (S \<inter> T))"
by (meson \<open>inj_on f S\<close> \<open>open T\<close> assms(1-3) continuous_on_subset inf_le1 inj_on_subset invariance_of_domain_gen open_Int)
qed (use assms in auto)
then show "openin (top_of_set (f ` S)) (f ` S \<inter> g -` T)"
by (simp add: eq)
qed
lemma invariance_of_domain_homeomorphism:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "open S" "continuous_on S f" "DIM('b) \<le> DIM('a)" "inj_on f S"
obtains g where "homeomorphism S (f ` S) f g"
proof
show "homeomorphism S (f ` S) f (inv_into S f)"
by (simp add: assms continuous_on_inverse_open homeomorphism_def)
qed
corollary invariance_of_domain_homeomorphic:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "open S" "continuous_on S f" "DIM('b) \<le> DIM('a)" "inj_on f S"
shows "S homeomorphic (f ` S)"
using invariance_of_domain_homeomorphism [OF assms]
by (meson homeomorphic_def)
lemma continuous_image_subset_interior:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "continuous_on S f" "inj_on f S" "DIM('b) \<le> DIM('a)"
shows "f ` (interior S) \<subseteq> interior(f ` S)"
proof (rule interior_maximal)
show "f ` interior S \<subseteq> f ` S"
by (simp add: image_mono interior_subset)
show "open (f ` interior S)"
using assms
by (auto simp: subset_inj_on interior_subset continuous_on_subset invariance_of_domain_gen)
qed
lemma homeomorphic_interiors_same_dimension:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes "S homeomorphic T" and dimeq: "DIM('a) = DIM('b)"
shows "(interior S) homeomorphic (interior T)"
using assms [unfolded homeomorphic_minimal]
unfolding homeomorphic_def
proof (clarify elim!: ex_forward)
fix f g
assume S: "\<forall>x\<in>S. f x \<in> T \<and> g (f x) = x" and T: "\<forall>y\<in>T. g y \<in> S \<and> f (g y) = y"
and contf: "continuous_on S f" and contg: "continuous_on T g"
then have fST: "f ` S = T" and gTS: "g ` T = S" and "inj_on f S" "inj_on g T"
by (auto simp: inj_on_def intro: rev_image_eqI) metis+
have fim: "f ` interior S \<subseteq> interior T"
using continuous_image_subset_interior [OF contf \<open>inj_on f S\<close>] dimeq fST by simp
have gim: "g ` interior T \<subseteq> interior S"
using continuous_image_subset_interior [OF contg \<open>inj_on g T\<close>] dimeq gTS by simp
show "homeomorphism (interior S) (interior T) f g"
unfolding homeomorphism_def
proof (intro conjI ballI)
show "\<And>x. x \<in> interior S \<Longrightarrow> g (f x) = x"
by (meson \<open>\<forall>x\<in>S. f x \<in> T \<and> g (f x) = x\<close> subsetD interior_subset)
have "interior T \<subseteq> f ` interior S"
proof
fix x assume "x \<in> interior T"
then have "g x \<in> interior S"
using gim by blast
then show "x \<in> f ` interior S"
by (metis T \<open>x \<in> interior T\<close> image_iff interior_subset subsetCE)
qed
then show "f ` interior S = interior T"
using fim by blast
show "continuous_on (interior S) f"
by (metis interior_subset continuous_on_subset contf)
show "\<And>y. y \<in> interior T \<Longrightarrow> f (g y) = y"
by (meson T subsetD interior_subset)
have "interior S \<subseteq> g ` interior T"
proof
fix x assume "x \<in> interior S"
then have "f x \<in> interior T"
using fim by blast
then show "x \<in> g ` interior T"
by (metis S \<open>x \<in> interior S\<close> image_iff interior_subset subsetCE)
qed
then show "g ` interior T = interior S"
using gim by blast
show "continuous_on (interior T) g"
by (metis interior_subset continuous_on_subset contg)
qed
qed
lemma homeomorphic_open_imp_same_dimension:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes "S homeomorphic T" "open S" "S \<noteq> {}" "open T" "T \<noteq> {}"
shows "DIM('a) = DIM('b)"
using assms
apply (simp add: homeomorphic_minimal)
apply (rule order_antisym; metis inj_onI invariance_of_dimension)
done
proposition homeomorphic_interiors:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes "S homeomorphic T" "interior S = {} \<longleftrightarrow> interior T = {}"
shows "(interior S) homeomorphic (interior T)"
proof (cases "interior T = {}")
case True
with assms show ?thesis by auto
next
case False
then have "DIM('a) = DIM('b)"
using assms
apply (simp add: homeomorphic_minimal)
apply (rule order_antisym; metis continuous_on_subset inj_onI inj_on_subset interior_subset invariance_of_dimension open_interior)
done
then show ?thesis
by (rule homeomorphic_interiors_same_dimension [OF \<open>S homeomorphic T\<close>])
qed
lemma homeomorphic_frontiers_same_dimension:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes "S homeomorphic T" "closed S" "closed T" and dimeq: "DIM('a) = DIM('b)"
shows "(frontier S) homeomorphic (frontier T)"
using assms [unfolded homeomorphic_minimal]
unfolding homeomorphic_def
proof (clarify elim!: ex_forward)
fix f g
assume S: "\<forall>x\<in>S. f x \<in> T \<and> g (f x) = x" and T: "\<forall>y\<in>T. g y \<in> S \<and> f (g y) = y"
and contf: "continuous_on S f" and contg: "continuous_on T g"
then have fST: "f ` S = T" and gTS: "g ` T = S" and "inj_on f S" "inj_on g T"
by (auto simp: inj_on_def intro: rev_image_eqI) metis+
have "g ` interior T \<subseteq> interior S"
using continuous_image_subset_interior [OF contg \<open>inj_on g T\<close>] dimeq gTS by simp
then have fim: "f ` frontier S \<subseteq> frontier T"
apply (simp add: frontier_def)
using continuous_image_subset_interior assms(2) assms(3) S by auto
have "f ` interior S \<subseteq> interior T"
using continuous_image_subset_interior [OF contf \<open>inj_on f S\<close>] dimeq fST by simp
then have gim: "g ` frontier T \<subseteq> frontier S"
apply (simp add: frontier_def)
using continuous_image_subset_interior T assms(2) assms(3) by auto
show "homeomorphism (frontier S) (frontier T) f g"
unfolding homeomorphism_def
proof (intro conjI ballI)
show gf: "\<And>x. x \<in> frontier S \<Longrightarrow> g (f x) = x"
by (simp add: S assms(2) frontier_def)
show fg: "\<And>y. y \<in> frontier T \<Longrightarrow> f (g y) = y"
by (simp add: T assms(3) frontier_def)
have "frontier T \<subseteq> f ` frontier S"
proof
fix x assume "x \<in> frontier T"
then have "g x \<in> frontier S"
using gim by blast
then show "x \<in> f ` frontier S"
by (metis fg \<open>x \<in> frontier T\<close> imageI)
qed
then show "f ` frontier S = frontier T"
using fim by blast
show "continuous_on (frontier S) f"
by (metis Diff_subset assms(2) closure_eq contf continuous_on_subset frontier_def)
have "frontier S \<subseteq> g ` frontier T"
proof
fix x assume "x \<in> frontier S"
then have "f x \<in> frontier T"
using fim by blast
then show "x \<in> g ` frontier T"
by (metis gf \<open>x \<in> frontier S\<close> imageI)
qed
then show "g ` frontier T = frontier S"
using gim by blast
show "continuous_on (frontier T) g"
by (metis Diff_subset assms(3) closure_closed contg continuous_on_subset frontier_def)
qed
qed
lemma homeomorphic_frontiers:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes "S homeomorphic T" "closed S" "closed T"
"interior S = {} \<longleftrightarrow> interior T = {}"
shows "(frontier S) homeomorphic (frontier T)"
proof (cases "interior T = {}")
case True
then show ?thesis
by (metis Diff_empty assms closure_eq frontier_def)
next
case False
show ?thesis
apply (rule homeomorphic_frontiers_same_dimension)
apply (simp_all add: assms)
using False assms homeomorphic_interiors homeomorphic_open_imp_same_dimension by blast
qed
lemma continuous_image_subset_rel_interior:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes contf: "continuous_on S f" and injf: "inj_on f S" and fim: "f ` S \<subseteq> T"
and TS: "aff_dim T \<le> aff_dim S"
shows "f ` (rel_interior S) \<subseteq> rel_interior(f ` S)"
proof (rule rel_interior_maximal)
show "f ` rel_interior S \<subseteq> f ` S"
by(simp add: image_mono rel_interior_subset)
show "openin (top_of_set (affine hull f ` S)) (f ` rel_interior S)"
proof (rule invariance_of_domain_affine_sets)
show "openin (top_of_set (affine hull S)) (rel_interior S)"
by (simp add: openin_rel_interior)
show "aff_dim (affine hull f ` S) \<le> aff_dim (affine hull S)"
by (metis aff_dim_affine_hull aff_dim_subset fim TS order_trans)
show "f ` rel_interior S \<subseteq> affine hull f ` S"
by (meson \<open>f ` rel_interior S \<subseteq> f ` S\<close> hull_subset order_trans)
show "continuous_on (rel_interior S) f"
using contf continuous_on_subset rel_interior_subset by blast
show "inj_on f (rel_interior S)"
using inj_on_subset injf rel_interior_subset by blast
qed auto
qed
lemma homeomorphic_rel_interiors_same_dimension:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes "S homeomorphic T" and aff: "aff_dim S = aff_dim T"
shows "(rel_interior S) homeomorphic (rel_interior T)"
using assms [unfolded homeomorphic_minimal]
unfolding homeomorphic_def
proof (clarify elim!: ex_forward)
fix f g
assume S: "\<forall>x\<in>S. f x \<in> T \<and> g (f x) = x" and T: "\<forall>y\<in>T. g y \<in> S \<and> f (g y) = y"
and contf: "continuous_on S f" and contg: "continuous_on T g"
then have fST: "f ` S = T" and gTS: "g ` T = S" and "inj_on f S" "inj_on g T"
by (auto simp: inj_on_def intro: rev_image_eqI) metis+
have fim: "f ` rel_interior S \<subseteq> rel_interior T"
by (metis \<open>inj_on f S\<close> aff contf continuous_image_subset_rel_interior fST order_refl)
have gim: "g ` rel_interior T \<subseteq> rel_interior S"
by (metis \<open>inj_on g T\<close> aff contg continuous_image_subset_rel_interior gTS order_refl)
show "homeomorphism (rel_interior S) (rel_interior T) f g"
unfolding homeomorphism_def
proof (intro conjI ballI)
show gf: "\<And>x. x \<in> rel_interior S \<Longrightarrow> g (f x) = x"
using S rel_interior_subset by blast
show fg: "\<And>y. y \<in> rel_interior T \<Longrightarrow> f (g y) = y"
using T mem_rel_interior_ball by blast
have "rel_interior T \<subseteq> f ` rel_interior S"
proof
fix x assume "x \<in> rel_interior T"
then have "g x \<in> rel_interior S"
using gim by blast
then show "x \<in> f ` rel_interior S"
by (metis fg \<open>x \<in> rel_interior T\<close> imageI)
qed
moreover have "f ` rel_interior S \<subseteq> rel_interior T"
by (metis \<open>inj_on f S\<close> aff contf continuous_image_subset_rel_interior fST order_refl)
ultimately show "f ` rel_interior S = rel_interior T"
by blast
show "continuous_on (rel_interior S) f"
using contf continuous_on_subset rel_interior_subset by blast
have "rel_interior S \<subseteq> g ` rel_interior T"
proof
fix x assume "x \<in> rel_interior S"
then have "f x \<in> rel_interior T"
using fim by blast
then show "x \<in> g ` rel_interior T"
by (metis gf \<open>x \<in> rel_interior S\<close> imageI)
qed
then show "g ` rel_interior T = rel_interior S"
using gim by blast
show "continuous_on (rel_interior T) g"
using contg continuous_on_subset rel_interior_subset by blast
qed
qed
lemma homeomorphic_rel_interiors:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes "S homeomorphic T" "rel_interior S = {} \<longleftrightarrow> rel_interior T = {}"
shows "(rel_interior S) homeomorphic (rel_interior T)"
proof (cases "rel_interior T = {}")
case True
with assms show ?thesis by auto
next
case False
obtain f g
where S: "\<forall>x\<in>S. f x \<in> T \<and> g (f x) = x" and T: "\<forall>y\<in>T. g y \<in> S \<and> f (g y) = y"
and contf: "continuous_on S f" and contg: "continuous_on T g"
using assms [unfolded homeomorphic_minimal] by auto
have "aff_dim (affine hull S) \<le> aff_dim (affine hull T)"
apply (rule invariance_of_dimension_affine_sets [of _ "rel_interior S" _ f])
apply (simp_all add: openin_rel_interior False assms)
using contf continuous_on_subset rel_interior_subset apply blast
apply (meson S hull_subset image_subsetI rel_interior_subset rev_subsetD)
apply (metis S inj_on_inverseI inj_on_subset rel_interior_subset)
done
moreover have "aff_dim (affine hull T) \<le> aff_dim (affine hull S)"
apply (rule invariance_of_dimension_affine_sets [of _ "rel_interior T" _ g])
apply (simp_all add: openin_rel_interior False assms)
using contg continuous_on_subset rel_interior_subset apply blast
apply (meson T hull_subset image_subsetI rel_interior_subset rev_subsetD)
apply (metis T inj_on_inverseI inj_on_subset rel_interior_subset)
done
ultimately have "aff_dim S = aff_dim T" by force
then show ?thesis
by (rule homeomorphic_rel_interiors_same_dimension [OF \<open>S homeomorphic T\<close>])
qed
lemma homeomorphic_rel_boundaries_same_dimension:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes "S homeomorphic T" and aff: "aff_dim S = aff_dim T"
shows "(S - rel_interior S) homeomorphic (T - rel_interior T)"
using assms [unfolded homeomorphic_minimal]
unfolding homeomorphic_def
proof (clarify elim!: ex_forward)
fix f g
assume S: "\<forall>x\<in>S. f x \<in> T \<and> g (f x) = x" and T: "\<forall>y\<in>T. g y \<in> S \<and> f (g y) = y"
and contf: "continuous_on S f" and contg: "continuous_on T g"
then have fST: "f ` S = T" and gTS: "g ` T = S" and "inj_on f S" "inj_on g T"
by (auto simp: inj_on_def intro: rev_image_eqI) metis+
have fim: "f ` rel_interior S \<subseteq> rel_interior T"
by (metis \<open>inj_on f S\<close> aff contf continuous_image_subset_rel_interior fST order_refl)
have gim: "g ` rel_interior T \<subseteq> rel_interior S"
by (metis \<open>inj_on g T\<close> aff contg continuous_image_subset_rel_interior gTS order_refl)
show "homeomorphism (S - rel_interior S) (T - rel_interior T) f g"
unfolding homeomorphism_def
proof (intro conjI ballI)
show gf: "\<And>x. x \<in> S - rel_interior S \<Longrightarrow> g (f x) = x"
using S rel_interior_subset by blast
show fg: "\<And>y. y \<in> T - rel_interior T \<Longrightarrow> f (g y) = y"
using T mem_rel_interior_ball by blast
show "f ` (S - rel_interior S) = T - rel_interior T"
using S fST fim gim by auto
show "continuous_on (S - rel_interior S) f"
using contf continuous_on_subset rel_interior_subset by blast
show "g ` (T - rel_interior T) = S - rel_interior S"
using T gTS gim fim by auto
show "continuous_on (T - rel_interior T) g"
using contg continuous_on_subset rel_interior_subset by blast
qed
qed
lemma homeomorphic_rel_boundaries:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes "S homeomorphic T" "rel_interior S = {} \<longleftrightarrow> rel_interior T = {}"
shows "(S - rel_interior S) homeomorphic (T - rel_interior T)"
proof (cases "rel_interior T = {}")
case True
with assms show ?thesis by auto
next
case False
obtain f g
where S: "\<forall>x\<in>S. f x \<in> T \<and> g (f x) = x" and T: "\<forall>y\<in>T. g y \<in> S \<and> f (g y) = y"
and contf: "continuous_on S f" and contg: "continuous_on T g"
using assms [unfolded homeomorphic_minimal] by auto
have "aff_dim (affine hull S) \<le> aff_dim (affine hull T)"
apply (rule invariance_of_dimension_affine_sets [of _ "rel_interior S" _ f])
apply (simp_all add: openin_rel_interior False assms)
using contf continuous_on_subset rel_interior_subset apply blast
apply (meson S hull_subset image_subsetI rel_interior_subset rev_subsetD)
apply (metis S inj_on_inverseI inj_on_subset rel_interior_subset)
done
moreover have "aff_dim (affine hull T) \<le> aff_dim (affine hull S)"
apply (rule invariance_of_dimension_affine_sets [of _ "rel_interior T" _ g])
apply (simp_all add: openin_rel_interior False assms)
using contg continuous_on_subset rel_interior_subset apply blast
apply (meson T hull_subset image_subsetI rel_interior_subset rev_subsetD)
apply (metis T inj_on_inverseI inj_on_subset rel_interior_subset)
done
ultimately have "aff_dim S = aff_dim T" by force
then show ?thesis
by (rule homeomorphic_rel_boundaries_same_dimension [OF \<open>S homeomorphic T\<close>])
qed
proposition uniformly_continuous_homeomorphism_UNIV_trivial:
fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
assumes contf: "uniformly_continuous_on S f" and hom: "homeomorphism S UNIV f g"
shows "S = UNIV"
proof (cases "S = {}")
case True
then show ?thesis
by (metis UNIV_I hom empty_iff homeomorphism_def image_eqI)
next
case False
have "inj g"
by (metis UNIV_I hom homeomorphism_apply2 injI)
then have "open (g ` UNIV)"
by (blast intro: invariance_of_domain hom homeomorphism_cont2)
then have "open S"
using hom homeomorphism_image2 by blast
moreover have "complete S"
unfolding complete_def
proof clarify
fix \<sigma>
assume \<sigma>: "\<forall>n. \<sigma> n \<in> S" and "Cauchy \<sigma>"
have "Cauchy (f o \<sigma>)"
using uniformly_continuous_imp_Cauchy_continuous \<open>Cauchy \<sigma>\<close> \<sigma> contf by blast
then obtain l where "(f \<circ> \<sigma>) \<longlonglongrightarrow> l"
by (auto simp: convergent_eq_Cauchy [symmetric])
show "\<exists>l\<in>S. \<sigma> \<longlonglongrightarrow> l"
proof
show "g l \<in> S"
using hom homeomorphism_image2 by blast
have "(g \<circ> (f \<circ> \<sigma>)) \<longlonglongrightarrow> g l"
by (meson UNIV_I \<open>(f \<circ> \<sigma>) \<longlonglongrightarrow> l\<close> continuous_on_sequentially hom homeomorphism_cont2)
then show "\<sigma> \<longlonglongrightarrow> g l"
proof -
have "\<forall>n. \<sigma> n = (g \<circ> (f \<circ> \<sigma>)) n"
by (metis (no_types) \<sigma> comp_eq_dest_lhs hom homeomorphism_apply1)
then show ?thesis
by (metis (no_types) LIMSEQ_iff \<open>(g \<circ> (f \<circ> \<sigma>)) \<longlonglongrightarrow> g l\<close>)
qed
qed
qed
then have "closed S"
by (simp add: complete_eq_closed)
ultimately show ?thesis
using clopen [of S] False by simp
qed
proposition invariance_of_domain_sphere_affine_set_gen:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes contf: "continuous_on S f" and injf: "inj_on f S" and fim: "f ` S \<subseteq> T"
and U: "bounded U" "convex U"
and "affine T" and affTU: "aff_dim T < aff_dim U"
and ope: "openin (top_of_set (rel_frontier U)) S"
shows "openin (top_of_set T) (f ` S)"
proof (cases "rel_frontier U = {}")
case True
then show ?thesis
using ope openin_subset by force
next
case False
obtain b c where b: "b \<in> rel_frontier U" and c: "c \<in> rel_frontier U" and "b \<noteq> c"
using \<open>bounded U\<close> rel_frontier_not_sing [of U] subset_singletonD False by fastforce
obtain V :: "'a set" where "affine V" and affV: "aff_dim V = aff_dim U - 1"
proof (rule choose_affine_subset [OF affine_UNIV])
show "- 1 \<le> aff_dim U - 1"
by (metis aff_dim_empty aff_dim_geq aff_dim_negative_iff affTU diff_0 diff_right_mono not_le)
show "aff_dim U - 1 \<le> aff_dim (UNIV::'a set)"
by (metis aff_dim_UNIV aff_dim_le_DIM le_cases not_le zle_diff1_eq)
qed auto
have SU: "S \<subseteq> rel_frontier U"
using ope openin_imp_subset by auto
have homb: "rel_frontier U - {b} homeomorphic V"
and homc: "rel_frontier U - {c} homeomorphic V"
using homeomorphic_punctured_sphere_affine_gen [of U _ V]
by (simp_all add: \<open>affine V\<close> affV U b c)
then obtain g h j k
where gh: "homeomorphism (rel_frontier U - {b}) V g h"
and jk: "homeomorphism (rel_frontier U - {c}) V j k"
by (auto simp: homeomorphic_def)
with SU have hgsub: "(h ` g ` (S - {b})) \<subseteq> S" and kjsub: "(k ` j ` (S - {c})) \<subseteq> S"
by (simp_all add: homeomorphism_def subset_eq)
have [simp]: "aff_dim T \<le> aff_dim V"
by (simp add: affTU affV)
have "openin (top_of_set T) ((f \<circ> h) ` g ` (S - {b}))"
proof (rule invariance_of_domain_affine_sets [OF _ \<open>affine V\<close>])
show "openin (top_of_set V) (g ` (S - {b}))"
apply (rule homeomorphism_imp_open_map [OF gh])
by (meson Diff_mono Diff_subset SU ope openin_delete openin_subset_trans order_refl)
show "continuous_on (g ` (S - {b})) (f \<circ> h)"
apply (rule continuous_on_compose)
apply (meson Diff_mono SU homeomorphism_def homeomorphism_of_subsets gh set_eq_subset)
using contf continuous_on_subset hgsub by blast
show "inj_on (f \<circ> h) (g ` (S - {b}))"
using kjsub
apply (clarsimp simp add: inj_on_def)
by (metis SU b homeomorphism_def inj_onD injf insert_Diff insert_iff gh rev_subsetD)
show "(f \<circ> h) ` g ` (S - {b}) \<subseteq> T"
by (metis fim image_comp image_mono hgsub subset_trans)
qed (auto simp: assms)
moreover
have "openin (top_of_set T) ((f \<circ> k) ` j ` (S - {c}))"
proof (rule invariance_of_domain_affine_sets [OF _ \<open>affine V\<close>])
show "openin (top_of_set V) (j ` (S - {c}))"
apply (rule homeomorphism_imp_open_map [OF jk])
by (meson Diff_mono Diff_subset SU ope openin_delete openin_subset_trans order_refl)
show "continuous_on (j ` (S - {c})) (f \<circ> k)"
apply (rule continuous_on_compose)
apply (meson Diff_mono SU homeomorphism_def homeomorphism_of_subsets jk set_eq_subset)
using contf continuous_on_subset kjsub by blast
show "inj_on (f \<circ> k) (j ` (S - {c}))"
using kjsub
apply (clarsimp simp add: inj_on_def)
by (metis SU c homeomorphism_def inj_onD injf insert_Diff insert_iff jk rev_subsetD)
show "(f \<circ> k) ` j ` (S - {c}) \<subseteq> T"
by (metis fim image_comp image_mono kjsub subset_trans)
qed (auto simp: assms)
ultimately have "openin (top_of_set T) ((f \<circ> h) ` g ` (S - {b}) \<union> ((f \<circ> k) ` j ` (S - {c})))"
by (rule openin_Un)
moreover have "(f \<circ> h) ` g ` (S - {b}) = f ` (S - {b})"
proof -
have "h ` g ` (S - {b}) = (S - {b})"
proof
show "h ` g ` (S - {b}) \<subseteq> S - {b}"
using homeomorphism_apply1 [OF gh] SU
by (fastforce simp add: image_iff image_subset_iff)
show "S - {b} \<subseteq> h ` g ` (S - {b})"
apply clarify
by (metis SU subsetD homeomorphism_apply1 [OF gh] image_iff member_remove remove_def)
qed
then show ?thesis
by (metis image_comp)
qed
moreover have "(f \<circ> k) ` j ` (S - {c}) = f ` (S - {c})"
proof -
have "k ` j ` (S - {c}) = (S - {c})"
proof
show "k ` j ` (S - {c}) \<subseteq> S - {c}"
using homeomorphism_apply1 [OF jk] SU
by (fastforce simp add: image_iff image_subset_iff)
show "S - {c} \<subseteq> k ` j ` (S - {c})"
apply clarify
by (metis SU subsetD homeomorphism_apply1 [OF jk] image_iff member_remove remove_def)
qed
then show ?thesis
by (metis image_comp)
qed
moreover have "f ` (S - {b}) \<union> f ` (S - {c}) = f ` (S)"
using \<open>b \<noteq> c\<close> by blast
ultimately show ?thesis
by simp
qed
lemma invariance_of_domain_sphere_affine_set:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes contf: "continuous_on S f" and injf: "inj_on f S" and fim: "f ` S \<subseteq> T"
and "r \<noteq> 0" "affine T" and affTU: "aff_dim T < DIM('a)"
and ope: "openin (top_of_set (sphere a r)) S"
shows "openin (top_of_set T) (f ` S)"
proof (cases "sphere a r = {}")
case True
then show ?thesis
using ope openin_subset by force
next
case False
show ?thesis
proof (rule invariance_of_domain_sphere_affine_set_gen [OF contf injf fim bounded_cball convex_cball \<open>affine T\<close>])
show "aff_dim T < aff_dim (cball a r)"
by (metis False affTU aff_dim_cball assms(4) linorder_cases sphere_empty)
show "openin (top_of_set (rel_frontier (cball a r))) S"
by (simp add: \<open>r \<noteq> 0\<close> ope)
qed
qed
lemma no_embedding_sphere_lowdim:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes contf: "continuous_on (sphere a r) f" and injf: "inj_on f (sphere a r)" and "r > 0"
shows "DIM('a) \<le> DIM('b)"
proof -
have "False" if "DIM('a) > DIM('b)"
proof -
have "compact (f ` sphere a r)"
using compact_continuous_image
by (simp add: compact_continuous_image contf)
then have "\<not> open (f ` sphere a r)"
using compact_open
by (metis assms(3) image_is_empty not_less_iff_gr_or_eq sphere_eq_empty)
then show False
using invariance_of_domain_sphere_affine_set [OF contf injf subset_UNIV] \<open>r > 0\<close>
by (metis aff_dim_UNIV affine_UNIV less_irrefl of_nat_less_iff open_openin openin_subtopology_self subtopology_UNIV that)
qed
then show ?thesis
using not_less by blast
qed
lemma empty_interior_lowdim_gen:
fixes S :: "'N::euclidean_space set" and T :: "'M::euclidean_space set"
assumes dim: "DIM('M) < DIM('N)" and ST: "S homeomorphic T"
shows "interior S = {}"
proof -
obtain h :: "'M \<Rightarrow> 'N" where "linear h" "\<And>x. norm(h x) = norm x"
by (rule isometry_subset_subspace [OF subspace_UNIV subspace_UNIV, where ?'a = 'M and ?'b = 'N])
(use dim in auto)
then have "inj h"
by (metis linear_inj_iff_eq_0 norm_eq_zero)
then have "h ` T homeomorphic T"
using \<open>linear h\<close> homeomorphic_sym linear_homeomorphic_image by blast
then have "interior (h ` T) homeomorphic interior S"
using homeomorphic_interiors_same_dimension
by (metis ST homeomorphic_sym homeomorphic_trans)
moreover
have "interior (range h) = {}"
by (simp add: \<open>inj h\<close> \<open>linear h\<close> dim dim_image_eq empty_interior_lowdim)
then have "interior (h ` T) = {}"
by (metis image_mono interior_mono subset_empty top_greatest)
ultimately show ?thesis
by simp
qed
lemma empty_interior_lowdim_gen_le:
fixes S :: "'N::euclidean_space set" and T :: "'M::euclidean_space set"
assumes "DIM('M) \<le> DIM('N)" "interior T = {}" "S homeomorphic T"
shows "interior S = {}"
by (metis assms empty_interior_lowdim_gen homeomorphic_empty(1) homeomorphic_interiors_same_dimension less_le)
lemma homeomorphic_affine_sets_eq:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes "affine S" "affine T"
shows "S homeomorphic T \<longleftrightarrow> aff_dim S = aff_dim T"
proof (cases "S = {} \<or> T = {}")
case True
then show ?thesis
using assms homeomorphic_affine_sets by force
next
case False
then obtain a b where "a \<in> S" "b \<in> T"
by blast
then have "subspace ((+) (- a) ` S)" "subspace ((+) (- b) ` T)"
using affine_diffs_subspace assms by blast+
then show ?thesis
by (metis affine_imp_convex assms homeomorphic_affine_sets homeomorphic_convex_sets)
qed
lemma homeomorphic_hyperplanes_eq:
fixes a :: "'M::euclidean_space" and c :: "'N::euclidean_space"
assumes "a \<noteq> 0" "c \<noteq> 0"
shows "({x. a \<bullet> x = b} homeomorphic {x. c \<bullet> x = d} \<longleftrightarrow> DIM('M) = DIM('N))" (is "?lhs = ?rhs")
proof -
have "(DIM('M) - Suc 0 = DIM('N) - Suc 0) \<longleftrightarrow> (DIM('M) = DIM('N))"
by auto (metis DIM_positive Suc_pred)
then show ?thesis
using assms by (simp add: homeomorphic_affine_sets_eq affine_hyperplane)
qed
end