|
|
|
|
|
Quellcode-Bibliothek
© Kompilation durch diese Firma
[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]
Datei:
measure_theory.prf
Sprache: Lisp
|
|
(* Title: HOL/Analysis/Path_Connected.thy
Authors: LC Paulson and Robert Himmelmann (TU Muenchen), based on material from HOL Light
*)
section \<open>Path-Connectedness\<close>
theory Path_Connected
imports
Starlike
T1_Spaces
begin
subsection \<open>Paths and Arcs\<close>
definition\<^marker>\<open>tag important\<close> path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
where "path g \ continuous_on {0..1} g"
definition\<^marker>\<open>tag important\<close> pathstart :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a"
where "pathstart g = g 0"
definition\<^marker>\<open>tag important\<close> pathfinish :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a"
where "pathfinish g = g 1"
definition\<^marker>\<open>tag important\<close> path_image :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a set"
where "path_image g = g ` {0 .. 1}"
definition\<^marker>\<open>tag important\<close> reversepath :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> real \<Rightarrow> 'a"
where "reversepath g = (\x. g(1 - x))"
definition\<^marker>\<open>tag important\<close> joinpaths :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> real \<Rightarrow> 'a"
(infixr "+++" 75)
where "g1 +++ g2 = (\x. if x \ 1/2 then g1 (2 * x) else g2 (2 * x - 1))"
definition\<^marker>\<open>tag important\<close> simple_path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
where "simple_path g \
path g \<and> (\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. g x = g y \<longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0)"
definition\<^marker>\<open>tag important\<close> arc :: "(real \<Rightarrow> 'a :: topological_space) \<Rightarrow> bool"
where "arc g \ path g \ inj_on g {0..1}"
subsection\<^marker>\<open>tag unimportant\<close>\<open>Invariance theorems\<close>
lemma path_eq: "path p \ (\t. t \ {0..1} \ p t = q t) \ path q"
using continuous_on_eq path_def by blast
lemma path_continuous_image: "path g \ continuous_on (path_image g) f \ path(f \ g)"
unfolding path_def path_image_def
using continuous_on_compose by blast
lemma continuous_on_translation_eq:
fixes g :: "'a :: real_normed_vector \ 'b :: real_normed_vector"
shows "continuous_on A ((+) a \ g) = continuous_on A g"
proof -
have g: "g = (\x. -a + x) \ ((\x. a + x) \ g)"
by (rule ext) simp
show ?thesis
by (metis (no_types, hide_lams) g continuous_on_compose homeomorphism_def homeomorphism_translation)
qed
lemma path_translation_eq:
fixes g :: "real \ 'a :: real_normed_vector"
shows "path((\x. a + x) \ g) = path g"
using continuous_on_translation_eq path_def by blast
lemma path_linear_image_eq:
fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
assumes "linear f" "inj f"
shows "path(f \ g) = path g"
proof -
from linear_injective_left_inverse [OF assms]
obtain h where h: "linear h" "h \ f = id"
by blast
then have g: "g = h \ (f \ g)"
by (metis comp_assoc id_comp)
show ?thesis
unfolding path_def
using h assms
by (metis g continuous_on_compose linear_continuous_on linear_conv_bounded_linear)
qed
lemma pathstart_translation: "pathstart((\x. a + x) \ g) = a + pathstart g"
by (simp add: pathstart_def)
lemma pathstart_linear_image_eq: "linear f \ pathstart(f \ g) = f(pathstart g)"
by (simp add: pathstart_def)
lemma pathfinish_translation: "pathfinish((\x. a + x) \ g) = a + pathfinish g"
by (simp add: pathfinish_def)
lemma pathfinish_linear_image: "linear f \ pathfinish(f \ g) = f(pathfinish g)"
by (simp add: pathfinish_def)
lemma path_image_translation: "path_image((\x. a + x) \ g) = (\x. a + x) ` (path_image g)"
by (simp add: image_comp path_image_def)
lemma path_image_linear_image: "linear f \ path_image(f \ g) = f ` (path_image g)"
by (simp add: image_comp path_image_def)
lemma reversepath_translation: "reversepath((\x. a + x) \ g) = (\x. a + x) \ reversepath g"
by (rule ext) (simp add: reversepath_def)
lemma reversepath_linear_image: "linear f \ reversepath(f \ g) = f \ reversepath g"
by (rule ext) (simp add: reversepath_def)
lemma joinpaths_translation:
"((\x. a + x) \ g1) +++ ((\x. a + x) \ g2) = (\x. a + x) \ (g1 +++ g2)"
by (rule ext) (simp add: joinpaths_def)
lemma joinpaths_linear_image: "linear f \ (f \ g1) +++ (f \ g2) = f \ (g1 +++ g2)"
by (rule ext) (simp add: joinpaths_def)
lemma simple_path_translation_eq:
fixes g :: "real \ 'a::euclidean_space"
shows "simple_path((\x. a + x) \ g) = simple_path g"
by (simp add: simple_path_def path_translation_eq)
lemma simple_path_linear_image_eq:
fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
assumes "linear f" "inj f"
shows "simple_path(f \ g) = simple_path g"
using assms inj_on_eq_iff [of f]
by (auto simp: path_linear_image_eq simple_path_def path_translation_eq)
lemma arc_translation_eq:
fixes g :: "real \ 'a::euclidean_space"
shows "arc((\x. a + x) \ g) = arc g"
by (auto simp: arc_def inj_on_def path_translation_eq)
lemma arc_linear_image_eq:
fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
assumes "linear f" "inj f"
shows "arc(f \ g) = arc g"
using assms inj_on_eq_iff [of f]
by (auto simp: arc_def inj_on_def path_linear_image_eq)
subsection\<^marker>\<open>tag unimportant\<close>\<open>Basic lemmas about paths\<close>
lemma pathin_iff_path_real [simp]: "pathin euclideanreal g \ path g"
by (simp add: pathin_def path_def)
lemma continuous_on_path: "path f \ t \ {0..1} \ continuous_on t f"
using continuous_on_subset path_def by blast
lemma arc_imp_simple_path: "arc g \ simple_path g"
by (simp add: arc_def inj_on_def simple_path_def)
lemma arc_imp_path: "arc g \ path g"
using arc_def by blast
lemma arc_imp_inj_on: "arc g \ inj_on g {0..1}"
by (auto simp: arc_def)
lemma simple_path_imp_path: "simple_path g \ path g"
using simple_path_def by blast
lemma simple_path_cases: "simple_path g \ arc g \ pathfinish g = pathstart g"
unfolding simple_path_def arc_def inj_on_def pathfinish_def pathstart_def
by force
lemma simple_path_imp_arc: "simple_path g \ pathfinish g \ pathstart g \ arc g"
using simple_path_cases by auto
lemma arc_distinct_ends: "arc g \ pathfinish g \ pathstart g"
unfolding arc_def inj_on_def pathfinish_def pathstart_def
by fastforce
lemma arc_simple_path: "arc g \ simple_path g \ pathfinish g \ pathstart g"
using arc_distinct_ends arc_imp_simple_path simple_path_cases by blast
lemma simple_path_eq_arc: "pathfinish g \ pathstart g \ (simple_path g = arc g)"
by (simp add: arc_simple_path)
lemma path_image_const [simp]: "path_image (\t. a) = {a}"
by (force simp: path_image_def)
lemma path_image_nonempty [simp]: "path_image g \ {}"
unfolding path_image_def image_is_empty box_eq_empty
by auto
lemma pathstart_in_path_image[intro]: "pathstart g \ path_image g"
unfolding pathstart_def path_image_def
by auto
lemma pathfinish_in_path_image[intro]: "pathfinish g \ path_image g"
unfolding pathfinish_def path_image_def
by auto
lemma connected_path_image[intro]: "path g \ connected (path_image g)"
unfolding path_def path_image_def
using connected_continuous_image connected_Icc by blast
lemma compact_path_image[intro]: "path g \ compact (path_image g)"
unfolding path_def path_image_def
using compact_continuous_image connected_Icc by blast
lemma reversepath_reversepath[simp]: "reversepath (reversepath g) = g"
unfolding reversepath_def
by auto
lemma pathstart_reversepath[simp]: "pathstart (reversepath g) = pathfinish g"
unfolding pathstart_def reversepath_def pathfinish_def
by auto
lemma pathfinish_reversepath[simp]: "pathfinish (reversepath g) = pathstart g"
unfolding pathstart_def reversepath_def pathfinish_def
by auto
lemma pathstart_join[simp]: "pathstart (g1 +++ g2) = pathstart g1"
unfolding pathstart_def joinpaths_def pathfinish_def
by auto
lemma pathfinish_join[simp]: "pathfinish (g1 +++ g2) = pathfinish g2"
unfolding pathstart_def joinpaths_def pathfinish_def
by auto
lemma path_image_reversepath[simp]: "path_image (reversepath g) = path_image g"
proof -
have *: "\g. path_image (reversepath g) \ path_image g"
unfolding path_image_def subset_eq reversepath_def Ball_def image_iff
by force
show ?thesis
using *[of g] *[of "reversepath g"]
unfolding reversepath_reversepath
by auto
qed
lemma path_reversepath [simp]: "path (reversepath g) \ path g"
proof -
have *: "\g. path g \ path (reversepath g)"
unfolding path_def reversepath_def
apply (rule continuous_on_compose[unfolded o_def, of _ "\x. 1 - x"])
apply (auto intro: continuous_intros continuous_on_subset[of "{0..1}"])
done
show ?thesis
using "*" by force
qed
lemma arc_reversepath:
assumes "arc g" shows "arc(reversepath g)"
proof -
have injg: "inj_on g {0..1}"
using assms
by (simp add: arc_def)
have **: "\x y::real. 1-x = 1-y \ x = y"
by simp
show ?thesis
using assms by (clarsimp simp: arc_def intro!: inj_onI) (simp add: inj_onD reversepath_def **)
qed
lemma simple_path_reversepath: "simple_path g \ simple_path (reversepath g)"
apply (simp add: simple_path_def)
apply (force simp: reversepath_def)
done
lemmas reversepath_simps =
path_reversepath path_image_reversepath pathstart_reversepath pathfinish_reversepath
lemma path_join[simp]:
assumes "pathfinish g1 = pathstart g2"
shows "path (g1 +++ g2) \ path g1 \ path g2"
unfolding path_def pathfinish_def pathstart_def
proof safe
assume cont: "continuous_on {0..1} (g1 +++ g2)"
have g1: "continuous_on {0..1} g1 \ continuous_on {0..1} ((g1 +++ g2) \ (\x. x / 2))"
by (intro continuous_on_cong refl) (auto simp: joinpaths_def)
have g2: "continuous_on {0..1} g2 \ continuous_on {0..1} ((g1 +++ g2) \ (\x. x / 2 + 1/2))"
using assms
by (intro continuous_on_cong refl) (auto simp: joinpaths_def pathfinish_def pathstart_def)
show "continuous_on {0..1} g1" and "continuous_on {0..1} g2"
unfolding g1 g2
by (auto intro!: continuous_intros continuous_on_subset[OF cont] simp del: o_apply)
next
assume g1g2: "continuous_on {0..1} g1" "continuous_on {0..1} g2"
have 01: "{0 .. 1} = {0..1/2} \ {1/2 .. 1::real}"
by auto
{
fix x :: real
assume "0 \ x" and "x \ 1"
then have "x \ (\x. x * 2) ` {0..1 / 2}"
by (intro image_eqI[where x="x/2"]) auto
}
note 1 = this
{
fix x :: real
assume "0 \ x" and "x \ 1"
then have "x \ (\x. x * 2 - 1) ` {1 / 2..1}"
by (intro image_eqI[where x="x/2 + 1/2"]) auto
}
note 2 = this
show "continuous_on {0..1} (g1 +++ g2)"
using assms
unfolding joinpaths_def 01
apply (intro continuous_on_cases closed_atLeastAtMost g1g2[THEN continuous_on_compose2] continuous_intros)
apply (auto simp: field_simps pathfinish_def pathstart_def intro!: 1 2)
done
qed
subsection\<^marker>\<open>tag unimportant\<close> \<open>Path Images\<close>
lemma bounded_path_image: "path g \ bounded(path_image g)"
by (simp add: compact_imp_bounded compact_path_image)
lemma closed_path_image:
fixes g :: "real \ 'a::t2_space"
shows "path g \ closed(path_image g)"
by (metis compact_path_image compact_imp_closed)
lemma connected_simple_path_image: "simple_path g \ connected(path_image g)"
by (metis connected_path_image simple_path_imp_path)
lemma compact_simple_path_image: "simple_path g \ compact(path_image g)"
by (metis compact_path_image simple_path_imp_path)
lemma bounded_simple_path_image: "simple_path g \ bounded(path_image g)"
by (metis bounded_path_image simple_path_imp_path)
lemma closed_simple_path_image:
fixes g :: "real \ 'a::t2_space"
shows "simple_path g \ closed(path_image g)"
by (metis closed_path_image simple_path_imp_path)
lemma connected_arc_image: "arc g \ connected(path_image g)"
by (metis connected_path_image arc_imp_path)
lemma compact_arc_image: "arc g \ compact(path_image g)"
by (metis compact_path_image arc_imp_path)
lemma bounded_arc_image: "arc g \ bounded(path_image g)"
by (metis bounded_path_image arc_imp_path)
lemma closed_arc_image:
fixes g :: "real \ 'a::t2_space"
shows "arc g \ closed(path_image g)"
by (metis closed_path_image arc_imp_path)
lemma path_image_join_subset: "path_image (g1 +++ g2) \ path_image g1 \ path_image g2"
unfolding path_image_def joinpaths_def
by auto
lemma subset_path_image_join:
assumes "path_image g1 \ s"
and "path_image g2 \ s"
shows "path_image (g1 +++ g2) \ s"
using path_image_join_subset[of g1 g2] and assms
by auto
lemma path_image_join:
assumes "pathfinish g1 = pathstart g2"
shows "path_image(g1 +++ g2) = path_image g1 \ path_image g2"
proof -
have "path_image g1 \ path_image (g1 +++ g2)"
proof (clarsimp simp: path_image_def joinpaths_def)
fix u::real
assume "0 \ u" "u \ 1"
then show "g1 u \ (\x. g1 (2 * x)) ` ({0..1} \ {x. x * 2 \ 1})"
by (rule_tac x="u/2" in image_eqI) auto
qed
moreover
have \<section>: "g2 u \<in> (\<lambda>x. g2 (2 * x - 1)) ` ({0..1} \<inter> {x. \<not> x * 2 \<le> 1})"
if "0 < u" "u \ 1" for u
using that assms
by (rule_tac x="(u+1)/2" in image_eqI) (auto simp: field_simps pathfinish_def pathstart_def)
have "g2 0 \ (\x. g1 (2 * x)) ` ({0..1} \ {x. x * 2 \ 1})"
using assms
by (rule_tac x="1/2" in image_eqI) (auto simp: pathfinish_def pathstart_def)
then have "path_image g2 \ path_image (g1 +++ g2)"
by (auto simp: path_image_def joinpaths_def intro!: \<section>)
ultimately show ?thesis
using path_image_join_subset by blast
qed
lemma not_in_path_image_join:
assumes "x \ path_image g1"
and "x \ path_image g2"
shows "x \ path_image (g1 +++ g2)"
using assms and path_image_join_subset[of g1 g2]
by auto
lemma pathstart_compose: "pathstart(f \ p) = f(pathstart p)"
by (simp add: pathstart_def)
lemma pathfinish_compose: "pathfinish(f \ p) = f(pathfinish p)"
by (simp add: pathfinish_def)
lemma path_image_compose: "path_image (f \ p) = f ` (path_image p)"
by (simp add: image_comp path_image_def)
lemma path_compose_join: "f \ (p +++ q) = (f \ p) +++ (f \ q)"
by (rule ext) (simp add: joinpaths_def)
lemma path_compose_reversepath: "f \ reversepath p = reversepath(f \ p)"
by (rule ext) (simp add: reversepath_def)
lemma joinpaths_eq:
"(\t. t \ {0..1} \ p t = p' t) \
(\<And>t. t \<in> {0..1} \<Longrightarrow> q t = q' t)
\<Longrightarrow> t \<in> {0..1} \<Longrightarrow> (p +++ q) t = (p' +++ q') t"
by (auto simp: joinpaths_def)
lemma simple_path_inj_on: "simple_path g \ inj_on g {0<..<1}"
by (auto simp: simple_path_def path_image_def inj_on_def less_eq_real_def Ball_def)
subsection\<^marker>\<open>tag unimportant\<close>\<open>Simple paths with the endpoints removed\<close>
lemma simple_path_endless:
assumes "simple_path c"
shows "path_image c - {pathstart c,pathfinish c} = c ` {0<..<1}" (is "?lhs = ?rhs")
proof
show "?lhs \ ?rhs"
using less_eq_real_def by (auto simp: path_image_def pathstart_def pathfinish_def)
show "?rhs \ ?lhs"
using assms
apply (auto simp: simple_path_def path_image_def pathstart_def pathfinish_def Ball_def)
using less_eq_real_def zero_le_one by blast+
qed
lemma connected_simple_path_endless:
assumes "simple_path c"
shows "connected(path_image c - {pathstart c,pathfinish c})"
proof -
have "continuous_on {0<..<1} c"
using assms by (simp add: simple_path_def continuous_on_path path_def subset_iff)
then have "connected (c ` {0<..<1})"
using connected_Ioo connected_continuous_image by blast
then show ?thesis
using assms by (simp add: simple_path_endless)
qed
lemma nonempty_simple_path_endless:
"simple_path c \ path_image c - {pathstart c,pathfinish c} \ {}"
by (simp add: simple_path_endless)
subsection\<^marker>\<open>tag unimportant\<close>\<open>The operations on paths\<close>
lemma path_image_subset_reversepath: "path_image(reversepath g) \ path_image g"
by simp
lemma path_imp_reversepath: "path g \ path(reversepath g)"
by simp
lemma half_bounded_equal: "1 \ x * 2 \ x * 2 \ 1 \ x = (1/2::real)"
by simp
lemma continuous_on_joinpaths:
assumes "continuous_on {0..1} g1" "continuous_on {0..1} g2" "pathfinish g1 = pathstart g2"
shows "continuous_on {0..1} (g1 +++ g2)"
proof -
have "{0..1::real} = {0..1/2} \ {1/2..1}"
by auto
then show ?thesis
using assms by (metis path_def path_join)
qed
lemma path_join_imp: "\path g1; path g2; pathfinish g1 = pathstart g2\ \ path(g1 +++ g2)"
by simp
lemma simple_path_join_loop:
assumes "arc g1" "arc g2"
"pathfinish g1 = pathstart g2" "pathfinish g2 = pathstart g1"
"path_image g1 \ path_image g2 \ {pathstart g1, pathstart g2}"
shows "simple_path(g1 +++ g2)"
proof -
have injg1: "inj_on g1 {0..1}"
using assms
by (simp add: arc_def)
have injg2: "inj_on g2 {0..1}"
using assms
by (simp add: arc_def)
have g12: "g1 1 = g2 0"
and g21: "g2 1 = g1 0"
and sb: "g1 ` {0..1} \ g2 ` {0..1} \ {g1 0, g2 0}"
using assms
by (simp_all add: arc_def pathfinish_def pathstart_def path_image_def)
{ fix x and y::real
assume g2_eq: "g2 (2 * x - 1) = g1 (2 * y)"
and xyI: "x \ 1 \ y \ 0"
and xy: "x \ 1" "0 \ y" " y * 2 \ 1" "\ x * 2 \ 1"
then consider "g1 (2 * y) = g1 0" | "g1 (2 * y) = g2 0"
using sb by force
then have False
proof cases
case 1
then have "y = 0"
using xy g2_eq by (auto dest!: inj_onD [OF injg1])
then show ?thesis
using xy g2_eq xyI by (auto dest: inj_onD [OF injg2] simp flip: g21)
next
case 2
then have "2*x = 1"
using g2_eq g12 inj_onD [OF injg2] atLeastAtMost_iff xy(1) xy(4) by fastforce
with xy show False by auto
qed
} note * = this
{ fix x and y::real
assume xy: "g1 (2 * x) = g2 (2 * y - 1)" "y \ 1" "0 \ x" "\ y * 2 \ 1" "x * 2 \ 1"
then have "x = 0 \ y = 1"
using * xy by force
} note ** = this
show ?thesis
using assms
apply (simp add: arc_def simple_path_def)
apply (auto simp: joinpaths_def split: if_split_asm
dest!: * ** dest: inj_onD [OF injg1] inj_onD [OF injg2])
done
qed
lemma arc_join:
assumes "arc g1" "arc g2"
"pathfinish g1 = pathstart g2"
"path_image g1 \ path_image g2 \ {pathstart g2}"
shows "arc(g1 +++ g2)"
proof -
have injg1: "inj_on g1 {0..1}"
using assms
by (simp add: arc_def)
have injg2: "inj_on g2 {0..1}"
using assms
by (simp add: arc_def)
have g11: "g1 1 = g2 0"
and sb: "g1 ` {0..1} \ g2 ` {0..1} \ {g2 0}"
using assms
by (simp_all add: arc_def pathfinish_def pathstart_def path_image_def)
{ fix x and y::real
assume xy: "g2 (2 * x - 1) = g1 (2 * y)" "x \ 1" "0 \ y" " y * 2 \ 1" "\ x * 2 \ 1"
then have "g1 (2 * y) = g2 0"
using sb by force
then have False
using xy inj_onD injg2 by fastforce
} note * = this
show ?thesis
using assms
apply (simp add: arc_def inj_on_def)
apply (auto simp: joinpaths_def arc_imp_path split: if_split_asm
dest: * *[OF sym] inj_onD [OF injg1] inj_onD [OF injg2])
done
qed
lemma reversepath_joinpaths:
"pathfinish g1 = pathstart g2 \ reversepath(g1 +++ g2) = reversepath g2 +++ reversepath g1"
unfolding reversepath_def pathfinish_def pathstart_def joinpaths_def
by (rule ext) (auto simp: mult.commute)
subsection\<^marker>\<open>tag unimportant\<close>\<open>Some reversed and "if and only if" versions of joining theorems\<close>
lemma path_join_path_ends:
fixes g1 :: "real \ 'a::metric_space"
assumes "path(g1 +++ g2)" "path g2"
shows "pathfinish g1 = pathstart g2"
proof (rule ccontr)
define e where "e = dist (g1 1) (g2 0)"
assume Neg: "pathfinish g1 \ pathstart g2"
then have "0 < dist (pathfinish g1) (pathstart g2)"
by auto
then have "e > 0"
by (metis e_def pathfinish_def pathstart_def)
then have "\e>0. \d>0. \x'\{0..1}. dist x' 0 < d \ dist (g2 x') (g2 0) < e"
using \<open>path g2\<close> atLeastAtMost_iff zero_le_one unfolding path_def continuous_on_iff
by blast
then obtain d1 where "d1 > 0"
and d1: "\x'. \x'\{0..1}; norm x' < d1\ \ dist (g2 x') (g2 0) < e/2"
by (metis \<open>0 < e\<close> half_gt_zero_iff norm_conv_dist)
obtain d2 where "d2 > 0"
and d2: "\x'. \x'\{0..1}; dist x' (1/2) < d2\
\<Longrightarrow> dist ((g1 +++ g2) x') (g1 1) < e/2"
using assms(1) \<open>e > 0\<close> unfolding path_def continuous_on_iff
apply (drule_tac x="1/2" in bspec, simp)
apply (drule_tac x="e/2" in spec, force simp: joinpaths_def)
done
have int01_1: "min (1/2) (min d1 d2) / 2 \ {0..1}"
using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def)
have dist1: "norm (min (1 / 2) (min d1 d2) / 2) < d1"
using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def dist_norm)
have int01_2: "1/2 + min (1/2) (min d1 d2) / 4 \ {0..1}"
using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def)
have dist2: "dist (1 / 2 + min (1 / 2) (min d1 d2) / 4) (1 / 2) < d2"
using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def dist_norm)
have [simp]: "\ min (1 / 2) (min d1 d2) \ 0"
using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def)
have "dist (g2 (min (1 / 2) (min d1 d2) / 2)) (g1 1) < e/2"
"dist (g2 (min (1 / 2) (min d1 d2) / 2)) (g2 0) < e/2"
using d1 [OF int01_1 dist1] d2 [OF int01_2 dist2] by (simp_all add: joinpaths_def)
then have "dist (g1 1) (g2 0) < e/2 + e/2"
using dist_triangle_half_r e_def by blast
then show False
by (simp add: e_def [symmetric])
qed
lemma path_join_eq [simp]:
fixes g1 :: "real \ 'a::metric_space"
assumes "path g1" "path g2"
shows "path(g1 +++ g2) \ pathfinish g1 = pathstart g2"
using assms by (metis path_join_path_ends path_join_imp)
lemma simple_path_joinE:
assumes "simple_path(g1 +++ g2)" and "pathfinish g1 = pathstart g2"
obtains "arc g1" "arc g2"
"path_image g1 \ path_image g2 \ {pathstart g1, pathstart g2}"
proof -
have *: "\x y. \0 \ x; x \ 1; 0 \ y; y \ 1; (g1 +++ g2) x = (g1 +++ g2) y\
\<Longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0"
using assms by (simp add: simple_path_def)
have "path g1"
using assms path_join simple_path_imp_path by blast
moreover have "inj_on g1 {0..1}"
proof (clarsimp simp: inj_on_def)
fix x y
assume "g1 x = g1 y" "0 \ x" "x \ 1" "0 \ y" "y \ 1"
then show "x = y"
using * [of "x/2" "y/2"] by (simp add: joinpaths_def split_ifs)
qed
ultimately have "arc g1"
using assms by (simp add: arc_def)
have [simp]: "g2 0 = g1 1"
using assms by (metis pathfinish_def pathstart_def)
have "path g2"
using assms path_join simple_path_imp_path by blast
moreover have "inj_on g2 {0..1}"
proof (clarsimp simp: inj_on_def)
fix x y
assume "g2 x = g2 y" "0 \ x" "x \ 1" "0 \ y" "y \ 1"
then show "x = y"
using * [of "(x + 1) / 2" "(y + 1) / 2"]
by (force simp: joinpaths_def split_ifs field_split_simps)
qed
ultimately have "arc g2"
using assms by (simp add: arc_def)
have "g2 y = g1 0 \ g2 y = g1 1"
if "g1 x = g2 y" "0 \ x" "x \ 1" "0 \ y" "y \ 1" for x y
using * [of "x / 2" "(y + 1) / 2"] that
by (auto simp: joinpaths_def split_ifs field_split_simps)
then have "path_image g1 \ path_image g2 \ {pathstart g1, pathstart g2}"
by (fastforce simp: pathstart_def pathfinish_def path_image_def)
with \<open>arc g1\<close> \<open>arc g2\<close> show ?thesis using that by blast
qed
lemma simple_path_join_loop_eq:
assumes "pathfinish g2 = pathstart g1" "pathfinish g1 = pathstart g2"
shows "simple_path(g1 +++ g2) \
arc g1 \<and> arc g2 \<and> path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}"
by (metis assms simple_path_joinE simple_path_join_loop)
lemma arc_join_eq:
assumes "pathfinish g1 = pathstart g2"
shows "arc(g1 +++ g2) \
arc g1 \<and> arc g2 \<and> path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g2}"
(is "?lhs = ?rhs")
proof
assume ?lhs
then have "simple_path(g1 +++ g2)" by (rule arc_imp_simple_path)
then have *: "\x y. \0 \ x; x \ 1; 0 \ y; y \ 1; (g1 +++ g2) x = (g1 +++ g2) y\
\<Longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0"
using assms by (simp add: simple_path_def)
have False if "g1 0 = g2 u" "0 \ u" "u \ 1" for u
using * [of 0 "(u + 1) / 2"] that assms arc_distinct_ends [OF \<open>?lhs\<close>]
by (auto simp: joinpaths_def pathstart_def pathfinish_def split_ifs field_split_simps)
then have n1: "pathstart g1 \ path_image g2"
unfolding pathstart_def path_image_def
using atLeastAtMost_iff by blast
show ?rhs using \<open>?lhs\<close>
using \<open>simple_path (g1 +++ g2)\<close> assms n1 simple_path_joinE by auto
next
assume ?rhs then show ?lhs
using assms
by (fastforce simp: pathfinish_def pathstart_def intro!: arc_join)
qed
lemma arc_join_eq_alt:
"pathfinish g1 = pathstart g2
\<Longrightarrow> (arc(g1 +++ g2) \<longleftrightarrow>
arc g1 \<and> arc g2 \<and>
path_image g1 \<inter> path_image g2 = {pathstart g2})"
using pathfinish_in_path_image by (fastforce simp: arc_join_eq)
subsection\<^marker>\<open>tag unimportant\<close>\<open>The joining of paths is associative\<close>
lemma path_assoc:
"\pathfinish p = pathstart q; pathfinish q = pathstart r\
\<Longrightarrow> path(p +++ (q +++ r)) \<longleftrightarrow> path((p +++ q) +++ r)"
by simp
lemma simple_path_assoc:
assumes "pathfinish p = pathstart q" "pathfinish q = pathstart r"
shows "simple_path (p +++ (q +++ r)) \ simple_path ((p +++ q) +++ r)"
proof (cases "pathstart p = pathfinish r")
case True show ?thesis
proof
assume "simple_path (p +++ q +++ r)"
with assms True show "simple_path ((p +++ q) +++ r)"
by (fastforce simp add: simple_path_join_loop_eq arc_join_eq path_image_join
dest: arc_distinct_ends [of r])
next
assume 0: "simple_path ((p +++ q) +++ r)"
with assms True have q: "pathfinish r \ path_image q"
using arc_distinct_ends
by (fastforce simp add: simple_path_join_loop_eq arc_join_eq path_image_join)
have "pathstart r \ path_image p"
using assms
by (metis 0 IntI arc_distinct_ends arc_join_eq_alt empty_iff insert_iff
pathfinish_in_path_image pathfinish_join simple_path_joinE)
with assms 0 q True show "simple_path (p +++ q +++ r)"
by (auto simp: simple_path_join_loop_eq arc_join_eq path_image_join
dest!: subsetD [OF _ IntI])
qed
next
case False
{ fix x :: 'a
assume a: "path_image p \ path_image q \ {pathstart q}"
"(path_image p \ path_image q) \ path_image r \ {pathstart r}"
"x \ path_image p" "x \ path_image r"
have "pathstart r \ path_image q"
by (metis assms(2) pathfinish_in_path_image)
with a have "x = pathstart q"
by blast
}
with False assms show ?thesis
by (auto simp: simple_path_eq_arc simple_path_join_loop_eq arc_join_eq path_image_join)
qed
lemma arc_assoc:
"\pathfinish p = pathstart q; pathfinish q = pathstart r\
\<Longrightarrow> arc(p +++ (q +++ r)) \<longleftrightarrow> arc((p +++ q) +++ r)"
by (simp add: arc_simple_path simple_path_assoc)
subsubsection\<^marker>\<open>tag unimportant\<close>\<open>Symmetry and loops\<close>
lemma path_sym:
"\pathfinish p = pathstart q; pathfinish q = pathstart p\ \ path(p +++ q) \ path(q +++ p)"
by auto
lemma simple_path_sym:
"\pathfinish p = pathstart q; pathfinish q = pathstart p\
\<Longrightarrow> simple_path(p +++ q) \<longleftrightarrow> simple_path(q +++ p)"
by (metis (full_types) inf_commute insert_commute simple_path_joinE simple_path_join_loop)
lemma path_image_sym:
"\pathfinish p = pathstart q; pathfinish q = pathstart p\
\<Longrightarrow> path_image(p +++ q) = path_image(q +++ p)"
by (simp add: path_image_join sup_commute)
subsection\<open>Subpath\<close>
definition\<^marker>\<open>tag important\<close> subpath :: "real \<Rightarrow> real \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> real \<Rightarrow> 'a::real_normed_vector"
where "subpath a b g \ \x. g((b - a) * x + a)"
lemma path_image_subpath_gen:
fixes g :: "_ \ 'a::real_normed_vector"
shows "path_image(subpath u v g) = g ` (closed_segment u v)"
by (auto simp add: closed_segment_real_eq path_image_def subpath_def)
lemma path_image_subpath:
fixes g :: "real \ 'a::real_normed_vector"
shows "path_image(subpath u v g) = (if u \ v then g ` {u..v} else g ` {v..u})"
by (simp add: path_image_subpath_gen closed_segment_eq_real_ivl)
lemma path_image_subpath_commute:
fixes g :: "real \ 'a::real_normed_vector"
shows "path_image(subpath u v g) = path_image(subpath v u g)"
by (simp add: path_image_subpath_gen closed_segment_eq_real_ivl)
lemma path_subpath [simp]:
fixes g :: "real \ 'a::real_normed_vector"
assumes "path g" "u \ {0..1}" "v \ {0..1}"
shows "path(subpath u v g)"
proof -
have "continuous_on {0..1} (g \ (\x. ((v-u) * x+ u)))"
using assms
apply (intro continuous_intros; simp add: image_affinity_atLeastAtMost [where c=u])
apply (auto simp: path_def continuous_on_subset)
done
then show ?thesis
by (simp add: path_def subpath_def)
qed
lemma pathstart_subpath [simp]: "pathstart(subpath u v g) = g(u)"
by (simp add: pathstart_def subpath_def)
lemma pathfinish_subpath [simp]: "pathfinish(subpath u v g) = g(v)"
by (simp add: pathfinish_def subpath_def)
lemma subpath_trivial [simp]: "subpath 0 1 g = g"
by (simp add: subpath_def)
lemma subpath_reversepath: "subpath 1 0 g = reversepath g"
by (simp add: reversepath_def subpath_def)
lemma reversepath_subpath: "reversepath(subpath u v g) = subpath v u g"
by (simp add: reversepath_def subpath_def algebra_simps)
lemma subpath_translation: "subpath u v ((\x. a + x) \ g) = (\x. a + x) \ subpath u v g"
by (rule ext) (simp add: subpath_def)
lemma subpath_image: "subpath u v (f \ g) = f \ subpath u v g"
by (rule ext) (simp add: subpath_def)
lemma affine_ineq:
fixes x :: "'a::linordered_idom"
assumes "x \ 1" "v \ u"
shows "v + x * u \ u + x * v"
proof -
have "(1-x)*(u-v) \ 0"
using assms by auto
then show ?thesis
by (simp add: algebra_simps)
qed
lemma sum_le_prod1:
fixes a::real shows "\a \ 1; b \ 1\ \ a + b \ 1 + a * b"
by (metis add.commute affine_ineq mult.right_neutral)
lemma simple_path_subpath_eq:
"simple_path(subpath u v g) \
path(subpath u v g) \<and> u\<noteq>v \<and>
(\<forall>x y. x \<in> closed_segment u v \<and> y \<in> closed_segment u v \<and> g x = g y
\<longrightarrow> x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u)"
(is "?lhs = ?rhs")
proof
assume ?lhs
then have p: "path (\x. g ((v - u) * x + u))"
and sim: "(\x y. \x\{0..1}; y\{0..1}; g ((v - u) * x + u) = g ((v - u) * y + u)\
\<Longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0)"
by (auto simp: simple_path_def subpath_def)
{ fix x y
assume "x \ closed_segment u v" "y \ closed_segment u v" "g x = g y"
then have "x = y \ x = u \ y = v \ x = v \ y = u"
using sim [of "(x-u)/(v-u)" "(y-u)/(v-u)"] p
by (auto split: if_split_asm simp add: closed_segment_real_eq image_affinity_atLeastAtMost)
(simp_all add: field_split_simps)
} moreover
have "path(subpath u v g) \ u\v"
using sim [of "1/3" "2/3"] p
by (auto simp: subpath_def)
ultimately show ?rhs
by metis
next
assume ?rhs
then
have d1: "\x y. \g x = g y; u \ x; x \ v; u \ y; y \ v\ \ x = y \ x = u \ y = v \ x = v \ y = u"
and d2: "\x y. \g x = g y; v \ x; x \ u; v \ y; y \ u\ \ x = y \ x = u \ y = v \ x = v \ y = u"
and ne: "u < v \ v < u"
and psp: "path (subpath u v g)"
by (auto simp: closed_segment_real_eq image_affinity_atLeastAtMost)
have [simp]: "\x. u + x * v = v + x * u \ u=v \ x=1"
by algebra
show ?lhs using psp ne
unfolding simple_path_def subpath_def
by (fastforce simp add: algebra_simps affine_ineq mult_left_mono crossproduct_eq dest: d1 d2)
qed
lemma arc_subpath_eq:
"arc(subpath u v g) \ path(subpath u v g) \ u\v \ inj_on g (closed_segment u v)"
(is "?lhs = ?rhs")
proof
assume ?lhs
then have p: "path (\x. g ((v - u) * x + u))"
and sim: "(\x y. \x\{0..1}; y\{0..1}; g ((v - u) * x + u) = g ((v - u) * y + u)\
\<Longrightarrow> x = y)"
by (auto simp: arc_def inj_on_def subpath_def)
{ fix x y
assume "x \ closed_segment u v" "y \ closed_segment u v" "g x = g y"
then have "x = y"
using sim [of "(x-u)/(v-u)" "(y-u)/(v-u)"] p
by (cases "v = u")
(simp_all split: if_split_asm add: inj_on_def closed_segment_real_eq image_affinity_atLeastAtMost,
simp add: field_simps)
} moreover
have "path(subpath u v g) \ u\v"
using sim [of "1/3" "2/3"] p
by (auto simp: subpath_def)
ultimately show ?rhs
unfolding inj_on_def
by metis
next
assume ?rhs
then
have d1: "\x y. \g x = g y; u \ x; x \ v; u \ y; y \ v\ \ x = y"
and d2: "\x y. \g x = g y; v \ x; x \ u; v \ y; y \ u\ \ x = y"
and ne: "u < v \ v < u"
and psp: "path (subpath u v g)"
by (auto simp: inj_on_def closed_segment_real_eq image_affinity_atLeastAtMost)
show ?lhs using psp ne
unfolding arc_def subpath_def inj_on_def
by (auto simp: algebra_simps affine_ineq mult_left_mono crossproduct_eq dest: d1 d2)
qed
lemma simple_path_subpath:
assumes "simple_path g" "u \ {0..1}" "v \ {0..1}" "u \ v"
shows "simple_path(subpath u v g)"
using assms
apply (simp add: simple_path_subpath_eq simple_path_imp_path)
apply (simp add: simple_path_def closed_segment_real_eq image_affinity_atLeastAtMost, fastforce)
done
lemma arc_simple_path_subpath:
"\simple_path g; u \ {0..1}; v \ {0..1}; g u \ g v\ \ arc(subpath u v g)"
by (force intro: simple_path_subpath simple_path_imp_arc)
lemma arc_subpath_arc:
"\arc g; u \ {0..1}; v \ {0..1}; u \ v\ \ arc(subpath u v g)"
by (meson arc_def arc_imp_simple_path arc_simple_path_subpath inj_onD)
lemma arc_simple_path_subpath_interior:
"\simple_path g; u \ {0..1}; v \ {0..1}; u \ v; \u-v\ < 1\ \ arc(subpath u v g)"
by (force simp: simple_path_def intro: arc_simple_path_subpath)
lemma path_image_subpath_subset:
"\u \ {0..1}; v \ {0..1}\ \ path_image(subpath u v g) \ path_image g"
by (metis atLeastAtMost_iff atLeastatMost_subset_iff path_image_def path_image_subpath subset_image_iff)
lemma join_subpaths_middle: "subpath (0) ((1 / 2)) p +++ subpath ((1 / 2)) 1 p = p"
by (rule ext) (simp add: joinpaths_def subpath_def field_split_simps)
subsection\<^marker>\<open>tag unimportant\<close>\<open>There is a subpath to the frontier\<close>
lemma subpath_to_frontier_explicit:
fixes S :: "'a::metric_space set"
assumes g: "path g" and "pathfinish g \ S"
obtains u where "0 \ u" "u \ 1"
"\x. 0 \ x \ x < u \ g x \ interior S"
"(g u \ interior S)" "(u = 0 \ g u \ closure S)"
proof -
have gcon: "continuous_on {0..1} g"
using g by (simp add: path_def)
moreover have "bounded ({u. g u \ closure (- S)} \ {0..1})"
using compact_eq_bounded_closed by fastforce
ultimately have com: "compact ({0..1} \ {u. g u \ closure (- S)})"
using closed_vimage_Int
by (metis (full_types) Int_commute closed_atLeastAtMost closed_closure compact_eq_bounded_closed vimage_def)
have "1 \ {u. g u \ closure (- S)}"
using assms by (simp add: pathfinish_def closure_def)
then have dis: "{0..1} \ {u. g u \ closure (- S)} \ {}"
using atLeastAtMost_iff zero_le_one by blast
then obtain u where "0 \ u" "u \ 1" and gu: "g u \ closure (- S)"
and umin: "\t. \0 \ t; t \ 1; g t \ closure (- S)\ \ u \ t"
using compact_attains_inf [OF com dis] by fastforce
then have umin': "\t. \0 \ t; t \ 1; t < u\ \ g t \ S"
using closure_def by fastforce
have \<section>: "g u \<in> closure S" if "u \<noteq> 0"
proof -
have "u > 0" using that \<open>0 \<le> u\<close> by auto
{ fix e::real assume "e > 0"
obtain d where "d>0" and d: "\x'. \x' \ {0..1}; dist x' u \ d\ \ dist (g x') (g u) < e"
using continuous_onE [OF gcon _ \<open>e > 0\<close>] \<open>0 \<le> _\<close> \<open>_ \<le> 1\<close> atLeastAtMost_iff by auto
have *: "dist (max 0 (u - d / 2)) u \ d"
using \<open>0 \<le> u\<close> \<open>u \<le> 1\<close> \<open>d > 0\<close> by (simp add: dist_real_def)
have "\y\S. dist y (g u) < e"
using \<open>0 < u\<close> \<open>u \<le> 1\<close> \<open>d > 0\<close>
by (force intro: d [OF _ *] umin')
}
then show ?thesis
by (simp add: frontier_def closure_approachable)
qed
show ?thesis
proof
show "\x. 0 \ x \ x < u \ g x \ interior S"
using \<open>u \<le> 1\<close> interior_closure umin by fastforce
show "g u \ interior S"
by (simp add: gu interior_closure)
qed (use \<open>0 \<le> u\<close> \<open>u \<le> 1\<close> \<section> in auto)
qed
lemma subpath_to_frontier_strong:
assumes g: "path g" and "pathfinish g \ S"
obtains u where "0 \ u" "u \ 1" "g u \ interior S"
"u = 0 \ (\x. 0 \ x \ x < 1 \ subpath 0 u g x \ interior S) \ g u \ closure S"
proof -
obtain u where "0 \ u" "u \ 1"
and gxin: "\x. 0 \ x \ x < u \ g x \ interior S"
and gunot: "(g u \ interior S)" and u0: "(u = 0 \ g u \ closure S)"
using subpath_to_frontier_explicit [OF assms] by blast
show ?thesis
proof
show "g u \ interior S"
using gunot by blast
qed (use \<open>0 \<le> u\<close> \<open>u \<le> 1\<close> u0 in \<open>(force simp: subpath_def gxin)+\<close>)
qed
lemma subpath_to_frontier:
assumes g: "path g" and g0: "pathstart g \ closure S" and g1: "pathfinish g \ S"
obtains u where "0 \ u" "u \ 1" "g u \ frontier S" "path_image(subpath 0 u g) - {g u} \ interior S"
proof -
obtain u where "0 \ u" "u \ 1"
and notin: "g u \ interior S"
and disj: "u = 0 \
(\<forall>x. 0 \<le> x \<and> x < 1 \<longrightarrow> subpath 0 u g x \<in> interior S) \<and> g u \<in> closure S"
(is "_ \ ?P")
using subpath_to_frontier_strong [OF g g1] by blast
show ?thesis
proof
show "g u \ frontier S"
by (metis DiffI disj frontier_def g0 notin pathstart_def)
show "path_image (subpath 0 u g) - {g u} \ interior S"
using disj
proof
assume "u = 0"
then show ?thesis
by (simp add: path_image_subpath)
next
assume P: ?P
show ?thesis
proof (clarsimp simp add: path_image_subpath_gen)
fix y
assume y: "y \ closed_segment 0 u" "g y \ interior S"
with \<open>0 \<le> u\<close> have "0 \<le> y" "y \<le> u"
by (auto simp: closed_segment_eq_real_ivl split: if_split_asm)
then have "y=u \ subpath 0 u g (y/u) \ interior S"
using P less_eq_real_def by force
then show "g y = g u"
using y by (auto simp: subpath_def split: if_split_asm)
qed
qed
qed (use \<open>0 \<le> u\<close> \<open>u \<le> 1\<close> in auto)
qed
lemma exists_path_subpath_to_frontier:
fixes S :: "'a::real_normed_vector set"
assumes "path g" "pathstart g \ closure S" "pathfinish g \ S"
obtains h where "path h" "pathstart h = pathstart g" "path_image h \ path_image g"
"path_image h - {pathfinish h} \ interior S"
"pathfinish h \ frontier S"
proof -
obtain u where u: "0 \ u" "u \ 1" "g u \ frontier S" "(path_image(subpath 0 u g) - {g u}) \ interior S"
using subpath_to_frontier [OF assms] by blast
show ?thesis
proof
show "path_image (subpath 0 u g) \ path_image g"
by (simp add: path_image_subpath_subset u)
show "pathstart (subpath 0 u g) = pathstart g"
by (metis pathstart_def pathstart_subpath)
qed (use assms u in \<open>auto simp: path_image_subpath\<close>)
qed
lemma exists_path_subpath_to_frontier_closed:
fixes S :: "'a::real_normed_vector set"
assumes S: "closed S" and g: "path g" and g0: "pathstart g \ S" and g1: "pathfinish g \ S"
obtains h where "path h" "pathstart h = pathstart g" "path_image h \ path_image g \ S"
"pathfinish h \ frontier S"
proof -
obtain h where h: "path h" "pathstart h = pathstart g" "path_image h \ path_image g"
"path_image h - {pathfinish h} \ interior S"
"pathfinish h \ frontier S"
using exists_path_subpath_to_frontier [OF g _ g1] closure_closed [OF S] g0 by auto
show ?thesis
proof
show "path_image h \ path_image g \ S"
using assms h interior_subset [of S] by (auto simp: frontier_def)
qed (use h in auto)
qed
subsection \<open>Shift Path to Start at Some Given Point\<close>
definition\<^marker>\<open>tag important\<close> shiftpath :: "real \<Rightarrow> (real \<Rightarrow> 'a::topological_space) \<Rightarrow> real \<Rightarrow> 'a"
where "shiftpath a f = (\x. if (a + x) \ 1 then f (a + x) else f (a + x - 1))"
lemma shiftpath_alt_def: "shiftpath a f = (\x. if x \ 1-a then f (a + x) else f (a + x - 1))"
by (auto simp: shiftpath_def)
lemma pathstart_shiftpath: "a \ 1 \ pathstart (shiftpath a g) = g a"
unfolding pathstart_def shiftpath_def by auto
lemma pathfinish_shiftpath:
assumes "0 \ a"
and "pathfinish g = pathstart g"
shows "pathfinish (shiftpath a g) = g a"
using assms
unfolding pathstart_def pathfinish_def shiftpath_def
by auto
lemma endpoints_shiftpath:
assumes "pathfinish g = pathstart g"
and "a \ {0 .. 1}"
shows "pathfinish (shiftpath a g) = g a"
and "pathstart (shiftpath a g) = g a"
using assms
by (auto intro!: pathfinish_shiftpath pathstart_shiftpath)
lemma closed_shiftpath:
assumes "pathfinish g = pathstart g"
and "a \ {0..1}"
shows "pathfinish (shiftpath a g) = pathstart (shiftpath a g)"
using endpoints_shiftpath[OF assms]
by auto
lemma path_shiftpath:
assumes "path g"
and "pathfinish g = pathstart g"
and "a \ {0..1}"
shows "path (shiftpath a g)"
proof -
have *: "{0 .. 1} = {0 .. 1-a} \ {1-a .. 1}"
using assms(3) by auto
have **: "\x. x + a = 1 \ g (x + a - 1) = g (x + a)"
using assms(2)[unfolded pathfinish_def pathstart_def]
by auto
show ?thesis
unfolding path_def shiftpath_def *
proof (rule continuous_on_closed_Un)
have contg: "continuous_on {0..1} g"
using \<open>path g\<close> path_def by blast
show "continuous_on {0..1-a} (\x. if a + x \ 1 then g (a + x) else g (a + x - 1))"
proof (rule continuous_on_eq)
show "continuous_on {0..1-a} (g \ (+) a)"
by (intro continuous_intros continuous_on_subset [OF contg]) (use \<open>a \<in> {0..1}\<close> in auto)
qed auto
show "continuous_on {1-a..1} (\x. if a + x \ 1 then g (a + x) else g (a + x - 1))"
proof (rule continuous_on_eq)
show "continuous_on {1-a..1} (g \ (+) (a - 1))"
by (intro continuous_intros continuous_on_subset [OF contg]) (use \<open>a \<in> {0..1}\<close> in auto)
qed (auto simp: "**" add.commute add_diff_eq)
qed auto
qed
lemma shiftpath_shiftpath:
assumes "pathfinish g = pathstart g"
and "a \ {0..1}"
and "x \ {0..1}"
shows "shiftpath (1 - a) (shiftpath a g) x = g x"
using assms
unfolding pathfinish_def pathstart_def shiftpath_def
by auto
lemma path_image_shiftpath:
assumes a: "a \ {0..1}"
and "pathfinish g = pathstart g"
shows "path_image (shiftpath a g) = path_image g"
proof -
{ fix x
assume g: "g 1 = g 0" "x \ {0..1::real}" and gne: "\y. y\{0..1} \ {x. \ a + x \ 1} \ g x \ g (a + y - 1)"
then have "\y\{0..1} \ {x. a + x \ 1}. g x = g (a + y)"
proof (cases "a \ x")
case False
then show ?thesis
apply (rule_tac x="1 + x - a" in bexI)
using g gne[of "1 + x - a"] a by (force simp: field_simps)+
next
case True
then show ?thesis
using g a by (rule_tac x="x - a" in bexI) (auto simp: field_simps)
qed
}
then show ?thesis
using assms
unfolding shiftpath_def path_image_def pathfinish_def pathstart_def
by (auto simp: image_iff)
qed
lemma simple_path_shiftpath:
assumes "simple_path g" "pathfinish g = pathstart g" and a: "0 \ a" "a \ 1"
shows "simple_path (shiftpath a g)"
unfolding simple_path_def
proof (intro conjI impI ballI)
show "path (shiftpath a g)"
by (simp add: assms path_shiftpath simple_path_imp_path)
have *: "\x y. \g x = g y; x \ {0..1}; y \ {0..1}\ \ x = y \ x = 0 \ y = 1 \ x = 1 \ y = 0"
using assms by (simp add: simple_path_def)
show "x = y \ x = 0 \ y = 1 \ x = 1 \ y = 0"
if "x \ {0..1}" "y \ {0..1}" "shiftpath a g x = shiftpath a g y" for x y
using that a unfolding shiftpath_def
by (force split: if_split_asm dest!: *)
qed
subsection \<open>Straight-Line Paths\<close>
definition\<^marker>\<open>tag important\<close> linepath :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"
where "linepath a b = (\x. (1 - x) *\<^sub>R a + x *\<^sub>R b)"
lemma pathstart_linepath[simp]: "pathstart (linepath a b) = a"
unfolding pathstart_def linepath_def
by auto
lemma pathfinish_linepath[simp]: "pathfinish (linepath a b) = b"
unfolding pathfinish_def linepath_def
by auto
lemma linepath_inner: "linepath a b x \ v = linepath (a \ v) (b \ v) x"
by (simp add: linepath_def algebra_simps)
lemma Re_linepath': "Re (linepath a b x) = linepath (Re a) (Re b) x"
by (simp add: linepath_def)
lemma Im_linepath': "Im (linepath a b x) = linepath (Im a) (Im b) x"
by (simp add: linepath_def)
lemma linepath_0': "linepath a b 0 = a"
by (simp add: linepath_def)
lemma linepath_1': "linepath a b 1 = b"
by (simp add: linepath_def)
lemma continuous_linepath_at[intro]: "continuous (at x) (linepath a b)"
unfolding linepath_def
by (intro continuous_intros)
lemma continuous_on_linepath [intro,continuous_intros]: "continuous_on s (linepath a b)"
using continuous_linepath_at
by (auto intro!: continuous_at_imp_continuous_on)
lemma path_linepath[iff]: "path (linepath a b)"
unfolding path_def
by (rule continuous_on_linepath)
lemma path_image_linepath[simp]: "path_image (linepath a b) = closed_segment a b"
unfolding path_image_def segment linepath_def
by auto
lemma reversepath_linepath[simp]: "reversepath (linepath a b) = linepath b a"
unfolding reversepath_def linepath_def
by auto
lemma linepath_0 [simp]: "linepath 0 b x = x *\<^sub>R b"
by (simp add: linepath_def)
lemma linepath_cnj: "cnj (linepath a b x) = linepath (cnj a) (cnj b) x"
by (simp add: linepath_def)
lemma arc_linepath:
assumes "a \ b" shows [simp]: "arc (linepath a b)"
proof -
{
fix x y :: "real"
assume "x *\<^sub>R b + y *\<^sub>R a = x *\<^sub>R a + y *\<^sub>R b"
then have "(x - y) *\<^sub>R a = (x - y) *\<^sub>R b"
by (simp add: algebra_simps)
with assms have "x = y"
by simp
}
then show ?thesis
unfolding arc_def inj_on_def
by (fastforce simp: algebra_simps linepath_def)
qed
lemma simple_path_linepath[intro]: "a \ b \ simple_path (linepath a b)"
by (simp add: arc_imp_simple_path)
lemma linepath_trivial [simp]: "linepath a a x = a"
by (simp add: linepath_def real_vector.scale_left_diff_distrib)
lemma linepath_refl: "linepath a a = (\x. a)"
by auto
lemma subpath_refl: "subpath a a g = linepath (g a) (g a)"
by (simp add: subpath_def linepath_def algebra_simps)
lemma linepath_of_real: "(linepath (of_real a) (of_real b) x) = of_real ((1 - x)*a + x*b)"
by (simp add: scaleR_conv_of_real linepath_def)
lemma of_real_linepath: "of_real (linepath a b x) = linepath (of_real a) (of_real b) x"
by (metis linepath_of_real mult.right_neutral of_real_def real_scaleR_def)
lemma inj_on_linepath:
assumes "a \ b" shows "inj_on (linepath a b) {0..1}"
proof (clarsimp simp: inj_on_def linepath_def)
fix x y
assume "(1 - x) *\<^sub>R a + x *\<^sub>R b = (1 - y) *\<^sub>R a + y *\<^sub>R b" "0 \ x" "x \ 1" "0 \ y" "y \ 1"
then have "x *\<^sub>R (a - b) = y *\<^sub>R (a - b)"
by (auto simp: algebra_simps)
then show "x=y"
using assms by auto
qed
lemma linepath_le_1:
fixes a::"'a::linordered_idom" shows "\a \ 1; b \ 1; 0 \ u; u \ 1\ \ (1 - u) * a + u * b \ 1"
using mult_left_le [of a "1-u"] mult_left_le [of b u] by auto
lemma linepath_in_path:
shows "x \ {0..1} \ linepath a b x \ closed_segment a b"
by (auto simp: segment linepath_def)
lemma linepath_image_01: "linepath a b ` {0..1} = closed_segment a b"
by (auto simp: segment linepath_def)
lemma linepath_in_convex_hull:
fixes x::real
assumes a: "a \ convex hull S"
and b: "b \ convex hull S"
and x: "0\x" "x\1"
shows "linepath a b x \ convex hull S"
proof -
have "linepath a b x \ closed_segment a b"
using x by (auto simp flip: linepath_image_01)
then show ?thesis
using a b convex_contains_segment by blast
qed
lemma Re_linepath: "Re(linepath (of_real a) (of_real b) x) = (1 - x)*a + x*b"
by (simp add: linepath_def)
lemma Im_linepath: "Im(linepath (of_real a) (of_real b) x) = 0"
by (simp add: linepath_def)
lemma bounded_linear_linepath:
assumes "bounded_linear f"
shows "f (linepath a b x) = linepath (f a) (f b) x"
proof -
interpret f: bounded_linear f by fact
show ?thesis by (simp add: linepath_def f.add f.scale)
qed
lemma bounded_linear_linepath':
assumes "bounded_linear f"
shows "f \ linepath a b = linepath (f a) (f b)"
using bounded_linear_linepath[OF assms] by (simp add: fun_eq_iff)
lemma linepath_cnj': "cnj \ linepath a b = linepath (cnj a) (cnj b)"
by (simp add: linepath_def fun_eq_iff)
lemma differentiable_linepath [intro]: "linepath a b differentiable at x within A"
by (auto simp: linepath_def)
lemma has_vector_derivative_linepath_within:
"(linepath a b has_vector_derivative (b - a)) (at x within S)"
by (force intro: derivative_eq_intros simp add: linepath_def has_vector_derivative_def algebra_simps)
subsection\<^marker>\<open>tag unimportant\<close>\<open>Segments via convex hulls\<close>
lemma segments_subset_convex_hull:
"closed_segment a b \ (convex hull {a,b,c})"
"closed_segment a c \ (convex hull {a,b,c})"
"closed_segment b c \ (convex hull {a,b,c})"
"closed_segment b a \ (convex hull {a,b,c})"
"closed_segment c a \ (convex hull {a,b,c})"
"closed_segment c b \ (convex hull {a,b,c})"
by (auto simp: segment_convex_hull linepath_of_real elim!: rev_subsetD [OF _ hull_mono])
lemma midpoints_in_convex_hull:
assumes "x \ convex hull s" "y \ convex hull s"
shows "midpoint x y \ convex hull s"
proof -
have "(1 - inverse(2)) *\<^sub>R x + inverse(2) *\<^sub>R y \ convex hull s"
by (rule convexD_alt) (use assms in auto)
then show ?thesis
by (simp add: midpoint_def algebra_simps)
qed
lemma not_in_interior_convex_hull_3:
fixes a :: "complex"
shows "a \ interior(convex hull {a,b,c})"
"b \ interior(convex hull {a,b,c})"
"c \ interior(convex hull {a,b,c})"
by (auto simp: card_insert_le_m1 not_in_interior_convex_hull)
lemma midpoint_in_closed_segment [simp]: "midpoint a b \ closed_segment a b"
using midpoints_in_convex_hull segment_convex_hull by blast
lemma midpoint_in_open_segment [simp]: "midpoint a b \ open_segment a b \ a \ b"
by (simp add: open_segment_def)
lemma continuous_IVT_local_extremum:
fixes f :: "'a::euclidean_space \ real"
assumes contf: "continuous_on (closed_segment a b) f"
and "a \ b" "f a = f b"
obtains z where "z \ open_segment a b"
"(\w \ closed_segment a b. (f w) \ (f z)) \
(\<forall>w \<in> closed_segment a b. (f z) \<le> (f w))"
proof -
obtain c where "c \ closed_segment a b" and c: "\y. y \ closed_segment a b \ f y \ f c"
using continuous_attains_sup [of "closed_segment a b" f] contf by auto
obtain d where "d \ closed_segment a b" and d: "\y. y \ closed_segment a b \ f d \ f y"
using continuous_attains_inf [of "closed_segment a b" f] contf by auto
show ?thesis
proof (cases "c \ open_segment a b \ d \ open_segment a b")
case True
then show ?thesis
using c d that by blast
next
case False
then have "(c = a \ c = b) \ (d = a \ d = b)"
by (simp add: \<open>c \<in> closed_segment a b\<close> \<open>d \<in> closed_segment a b\<close> open_segment_def)
with \<open>a \<noteq> b\<close> \<open>f a = f b\<close> c d show ?thesis
by (rule_tac z = "midpoint a b" in that) (fastforce+)
qed
qed
text\<open>An injective map into R is also an open map w.r.T. the universe, and conversely. \<close>
proposition injective_eq_1d_open_map_UNIV:
fixes f :: "real \ real"
assumes contf: "continuous_on S f" and S: "is_interval S"
shows "inj_on f S \ (\T. open T \ T \ S \ open(f ` T))"
(is "?lhs = ?rhs")
proof safe
fix T
assume injf: ?lhs and "open T" and "T \ S"
have "\U. open U \ f x \ U \ U \ f ` T" if "x \ T" for x
proof -
obtain \<delta> where "\<delta> > 0" and \<delta>: "cball x \<delta> \<subseteq> T"
using \<open>open T\<close> \<open>x \<in> T\<close> open_contains_cball_eq by blast
show ?thesis
proof (intro exI conjI)
have "closed_segment (x-\) (x+\) = {x-\..x+\}"
using \<open>0 < \<delta>\<close> by (auto simp: closed_segment_eq_real_ivl)
also have "\ \ S"
using \<delta> \<open>T \<subseteq> S\<close> by (auto simp: dist_norm subset_eq)
finally have "f ` (open_segment (x-\) (x+\)) = open_segment (f (x-\)) (f (x+\))"
using continuous_injective_image_open_segment_1
by (metis continuous_on_subset [OF contf] inj_on_subset [OF injf])
then show "open (f ` {x-\<..})"
using \<open>0 < \<delta>\<close> by (simp add: open_segment_eq_real_ivl)
show "f x \ f ` {x - \<..}"
by (auto simp: \<open>\<delta> > 0\<close>)
show "f ` {x - \<..} \ f ` T"
using \<delta> by (auto simp: dist_norm subset_iff)
qed
qed
with open_subopen show "open (f ` T)"
by blast
next
assume R: ?rhs
have False if xy: "x \ S" "y \ S" and "f x = f y" "x \ y" for x y
proof -
have "open (f ` open_segment x y)"
using R
by (metis S convex_contains_open_segment is_interval_convex open_greaterThanLessThan open_segment_eq_real_ivl xy)
moreover
have "continuous_on (closed_segment x y) f"
by (meson S closed_segment_subset contf continuous_on_subset is_interval_convex that)
then obtain \<xi> where "\<xi> \<in> open_segment x y"
and \<xi>: "(\<forall>w \<in> closed_segment x y. (f w) \<le> (f \<xi>)) \<or>
(\<forall>w \<in> closed_segment x y. (f \<xi>) \<le> (f w))"
using continuous_IVT_local_extremum [of x y f] \<open>f x = f y\<close> \<open>x \<noteq> y\<close> by blast
ultimately obtain e where "e>0" and e: "\u. dist u (f \) < e \ u \ f ` open_segment x y"
using open_dist by (metis image_eqI)
have fin: "f \ + (e/2) \ f ` open_segment x y" "f \ - (e/2) \ f ` open_segment x y"
using e [of "f \ + (e/2)"] e [of "f \ - (e/2)"] \e > 0\ by (auto simp: dist_norm)
show ?thesis
using \<xi> \<open>0 < e\<close> fin open_closed_segment by fastforce
qed
then show ?lhs
by (force simp: inj_on_def)
qed
subsection\<^marker>\<open>tag unimportant\<close> \<open>Bounding a point away from a path\<close>
lemma not_on_path_ball:
fixes g :: "real \ 'a::heine_borel"
assumes "path g"
and z: "z \ path_image g"
shows "\e > 0. ball z e \ path_image g = {}"
proof -
have "closed (path_image g)"
by (simp add: \<open>path g\<close> closed_path_image)
then obtain a where "a \ path_image g" "\y \ path_image g. dist z a \ dist z y"
by (auto intro: distance_attains_inf[OF _ path_image_nonempty, of g z])
then show ?thesis
by (rule_tac x="dist z a" in exI) (use dist_commute z in auto)
qed
lemma not_on_path_cball:
fixes g :: "real \ 'a::heine_borel"
assumes "path g"
and "z \ path_image g"
shows "\e>0. cball z e \ (path_image g) = {}"
proof -
obtain e where "ball z e \ path_image g = {}" "e > 0"
using not_on_path_ball[OF assms] by auto
moreover have "cball z (e/2) \ ball z e"
using \<open>e > 0\<close> by auto
ultimately show ?thesis
by (rule_tac x="e/2" in exI) auto
qed
subsection \<open>Path component\<close>
text \<open>Original formalization by Tom Hales\<close>
definition\<^marker>\<open>tag important\<close> "path_component S x y \<equiv>
(\<exists>g. path g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y)"
abbreviation\<^marker>\<open>tag important\<close>
"path_component_set S x \ Collect (path_component S x)"
lemmas path_defs = path_def pathstart_def pathfinish_def path_image_def path_component_def
lemma path_component_mem:
assumes "path_component S x y"
shows "x \ S" and "y \ S"
using assms
unfolding path_defs
by auto
lemma path_component_refl:
assumes "x \ S"
shows "path_component S x x"
using assms
unfolding path_defs
by (metis (full_types) assms continuous_on_const image_subset_iff path_image_def)
lemma path_component_refl_eq: "path_component S x x \ x \ S"
by (auto intro!: path_component_mem path_component_refl)
lemma path_component_sym: "path_component S x y \ path_component S y x"
unfolding path_component_def
by (metis (no_types) path_image_reversepath path_reversepath pathfinish_reversepath pathstart_reversepath)
lemma path_component_trans:
assumes "path_component S x y" and "path_component S y z"
shows "path_component S x z"
using assms
unfolding path_component_def
by (metis path_join pathfinish_join pathstart_join subset_path_image_join)
lemma path_component_of_subset: "S \ T \ path_component S x y \ path_component T x y"
unfolding path_component_def by auto
lemma path_component_linepath:
fixes S :: "'a::real_normed_vector set"
shows "closed_segment a b \ S \ path_component S a b"
unfolding path_component_def
by (rule_tac x="linepath a b" in exI, auto)
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Path components as sets\<close>
lemma path_component_set:
"path_component_set S x =
{y. (\<exists>g. path g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y)}"
by (auto simp: path_component_def)
lemma path_component_subset: "path_component_set S x \ S"
by (auto simp: path_component_mem(2))
lemma path_component_eq_empty: "path_component_set S x = {} \ x \ S"
using path_component_mem path_component_refl_eq
by fastforce
lemma path_component_mono:
"S \ T \ (path_component_set S x) \ (path_component_set T x)"
by (simp add: Collect_mono path_component_of_subset)
lemma path_component_eq:
"y \ path_component_set S x \ path_component_set S y = path_component_set S x"
by (metis (no_types, lifting) Collect_cong mem_Collect_eq path_component_sym path_component_trans)
subsection \<open>Path connectedness of a space\<close>
definition\<^marker>\<open>tag important\<close> "path_connected S \<longleftrightarrow>
(\<forall>x\<in>S. \<forall>y\<in>S. \<exists>g. path g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y)"
lemma path_connectedin_iff_path_connected_real [simp]:
"path_connectedin euclideanreal S \ path_connected S"
by (simp add: path_connectedin path_connected_def path_defs)
lemma path_connected_component: "path_connected S \ (\x\S. \y\S. path_component S x y)"
--> --------------------
--> maximum size reached
--> --------------------
¤ Diese beiden folgenden Angebotsgruppen bietet das Unternehmen0.62Angebot
Wie Sie bei der Firma Beratungs- und Dienstleistungen beauftragen können
¤
|
Lebenszyklus
Die hierunter aufgelisteten Ziele sind für diese Firma wichtig
Ziele
Entwicklung einer Software für die statische Quellcodeanalyse
|
|
|
|
|
|
