(* Title: HOL/Multivariate_Analysis/Path_Connected.thy
Author: Robert Himmelmann, TU Muenchen
*)
header {* Continuous paths and path-connected sets *}
theory Path_Connected
imports Convex_Euclidean_Space
begin
lemma continuous_on_cong: (* MOVE to Topological_Spaces *)
"s = t \ (\x. x \ t \ f x = g x) \ continuous_on s f \ continuous_on t g"
unfolding continuous_on_def by (intro ball_cong Lim_cong_within) auto
lemma continuous_on_compose2:
shows "continuous_on t g \ continuous_on s f \ t = f ` s \ continuous_on s (\x. g (f x))"
using continuous_on_compose[of s f g] by (simp add: comp_def)
subsection {* Paths. *}
definition path :: "(real \ 'a::topological_space) \ bool"
where "path g \ continuous_on {0 .. 1} g"
definition pathstart :: "(real \ 'a::topological_space) \ 'a"
where "pathstart g = g 0"
definition pathfinish :: "(real \ 'a::topological_space) \ 'a"
where "pathfinish g = g 1"
definition path_image :: "(real \ 'a::topological_space) \ 'a set"
where "path_image g = g ` {0 .. 1}"
definition reversepath :: "(real \ 'a::topological_space) \ (real \ 'a)"
where "reversepath g = (\x. g(1 - x))"
definition joinpaths :: "(real \ 'a::topological_space) \ (real \ 'a) \ (real \ 'a)"
(infixr "+++" 75)
where "g1 +++ g2 = (\x. if x \ 1/2 then g1 (2 * x) else g2 (2 * x - 1))"
definition simple_path :: "(real \ 'a::topological_space) \ bool"
where "simple_path g \
(\x\{0..1}. \y\{0..1}. g x = g y \ x = y \ x = 0 \ y = 1 \ x = 1 \ y = 0)"
definition injective_path :: "(real \ 'a::topological_space) \ bool"
where "injective_path g \ (\x\{0..1}. \y\{0..1}. g x = g y \ x = y)"
subsection {* Some lemmas about these concepts. *}
lemma injective_imp_simple_path: "injective_path g \ simple_path g"
unfolding injective_path_def simple_path_def by auto
lemma path_image_nonempty: "path_image g \ {}"
unfolding path_image_def image_is_empty interval_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
apply (erule connected_continuous_image)
apply (rule convex_connected, rule convex_real_interval)
done
lemma compact_path_image[intro]: "path g \ compact(path_image g)"
unfolding path_def path_image_def
by (erule compact_continuous_image, rule compact_interval)
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
apply(rule,rule,erule bexE)
apply(rule_tac x="1 - xa" in bexI)
apply auto
done
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 (intro continuous_on_intros)
apply (rule continuous_on_subset[of "{0..1}"], assumption)
apply auto
done
show ?thesis
using *[of "reversepath g"] *[of g]
unfolding reversepath_reversepath
by (rule iffI)
qed
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" "continuous_on {0..1} g2"
unfolding g1 g2 by (auto intro!: continuous_on_intros continuous_on_subset[OF cont])
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" "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" "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
by (intro continuous_on_cases closed_atLeastAtMost g1g2[THEN continuous_on_compose2] continuous_on_intros)
(auto simp: field_simps pathfinish_def pathstart_def intro!: 1 2)
qed
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" "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)"
apply (rule, rule path_image_join_subset, rule)
unfolding Un_iff
proof (erule disjE)
fix x
assume "x \ path_image g1"
then obtain y where y: "y\{0..1}" "x = g1 y"
unfolding path_image_def image_iff by auto
then show "x \ path_image (g1 +++ g2)"
unfolding joinpaths_def path_image_def image_iff
apply (rule_tac x="(1/2) *\<^sub>R y" in bexI)
apply auto
done
next
fix x
assume "x \ path_image g2"
then obtain y where y: "y\{0..1}" "x = g2 y"
unfolding path_image_def image_iff by auto
then show "x \ path_image (g1 +++ g2)"
unfolding joinpaths_def path_image_def image_iff
apply(rule_tac x="(1/2) *\<^sub>R (y + 1)" in bexI)
using assms(1)[unfolded pathfinish_def pathstart_def]
apply (auto simp add: add_divide_distrib)
done
qed
lemma not_in_path_image_join:
assumes "x \ path_image g1" "x \ path_image g2"
shows "x \ path_image(g1 +++ g2)"
using assms and path_image_join_subset[of g1 g2] by auto
lemma simple_path_reversepath:
assumes "simple_path g"
shows "simple_path (reversepath g)"
using assms
unfolding simple_path_def reversepath_def
apply -
apply (rule ballI)+
apply (erule_tac x="1-x" in ballE, erule_tac x="1-y" in ballE)
apply auto
done
lemma simple_path_join_loop:
assumes "injective_path g1" "injective_path g2" "pathfinish g2 = pathstart g1"
"(path_image g1 \ path_image g2) \ {pathstart g1,pathstart g2}"
shows "simple_path(g1 +++ g2)"
unfolding simple_path_def
proof ((rule ballI)+, rule impI)
let ?g = "g1 +++ g2"
note inj = assms(1,2)[unfolded injective_path_def, rule_format]
fix x y :: real
assume xy: "x \ {0..1}" "y \ {0..1}" "?g x = ?g y"
show "x = y \ x = 0 \ y = 1 \ x = 1 \ y = 0"
proof (case_tac "x \ 1/2", case_tac[!] "y \ 1/2", unfold not_le)
assume as: "x \ 1 / 2" "y \ 1 / 2"
then have "g1 (2 *\<^sub>R x) = g1 (2 *\<^sub>R y)"
using xy(3) unfolding joinpaths_def by auto
moreover
have "2 *\<^sub>R x \ {0..1}" "2 *\<^sub>R y \ {0..1}" using xy(1,2) as
by auto
ultimately
show ?thesis using inj(1)[of "2*\<^sub>R x" "2*\<^sub>R y"] by auto
next
assume as:"x > 1 / 2" "y > 1 / 2"
then have "g2 (2 *\<^sub>R x - 1) = g2 (2 *\<^sub>R y - 1)"
using xy(3) unfolding joinpaths_def by auto
moreover
have "2 *\<^sub>R x - 1 \ {0..1}" "2 *\<^sub>R y - 1 \ {0..1}"
using xy(1,2) as by auto
ultimately
show ?thesis using inj(2)[of "2*\<^sub>R x - 1" "2*\<^sub>R y - 1"] by auto
next
assume as:"x \ 1 / 2" "y > 1 / 2"
then have "?g x \ path_image g1" "?g y \ path_image g2"
unfolding path_image_def joinpaths_def
using xy(1,2) by auto
moreover
have "?g y \ pathstart g2" using as(2) unfolding pathstart_def joinpaths_def
using inj(2)[of "2 *\<^sub>R y - 1" 0] and xy(2)
by (auto simp add: field_simps)
ultimately
have *:"?g x = pathstart g1" using assms(4) unfolding xy(3) by auto
then have "x = 0" unfolding pathstart_def joinpaths_def using as(1) and xy(1)
using inj(1)[of "2 *\<^sub>R x" 0] by auto
moreover
have "y = 1" using * unfolding xy(3) assms(3)[THEN sym]
unfolding joinpaths_def pathfinish_def using as(2) and xy(2)
using inj(2)[of "2 *\<^sub>R y - 1" 1] by auto
ultimately show ?thesis by auto
next
assume as: "x > 1 / 2" "y \ 1 / 2"
then have "?g x \ path_image g2" "?g y \ path_image g1"
unfolding path_image_def joinpaths_def
using xy(1,2) by auto
moreover
have "?g x \ pathstart g2" using as(1) unfolding pathstart_def joinpaths_def
using inj(2)[of "2 *\<^sub>R x - 1" 0] and xy(1)
by (auto simp add: field_simps)
ultimately
have *: "?g y = pathstart g1" using assms(4) unfolding xy(3) by auto
then have "y = 0" unfolding pathstart_def joinpaths_def using as(2) and xy(2)
using inj(1)[of "2 *\<^sub>R y" 0] by auto
moreover
have "x = 1" using * unfolding xy(3)[THEN sym] assms(3)[THEN sym]
unfolding joinpaths_def pathfinish_def using as(1) and xy(1)
using inj(2)[of "2 *\<^sub>R x - 1" 1] by auto
ultimately show ?thesis by auto
qed
qed
lemma injective_path_join:
assumes "injective_path g1" "injective_path g2" "pathfinish g1 = pathstart g2"
"(path_image g1 \ path_image g2) \ {pathstart g2}"
shows "injective_path(g1 +++ g2)"
unfolding injective_path_def
proof (rule, rule, rule)
let ?g = "g1 +++ g2"
note inj = assms(1,2)[unfolded injective_path_def, rule_format]
fix x y
assume xy: "x \ {0..1}" "y \ {0..1}" "(g1 +++ g2) x = (g1 +++ g2) y"
show "x = y"
proof (cases "x \ 1/2", case_tac[!] "y \ 1/2", unfold not_le)
assume "x \ 1 / 2" "y \ 1 / 2"
then show ?thesis using inj(1)[of "2*\<^sub>R x" "2*\<^sub>R y"] and xy
unfolding joinpaths_def by auto
next
assume "x > 1 / 2" "y > 1 / 2"
then show ?thesis using inj(2)[of "2*\<^sub>R x - 1" "2*\<^sub>R y - 1"] and xy
unfolding joinpaths_def by auto
next
assume as: "x \ 1 / 2" "y > 1 / 2"
then have "?g x \ path_image g1" "?g y \ path_image g2"
unfolding path_image_def joinpaths_def
using xy(1,2) by auto
then have "?g x = pathfinish g1" "?g y = pathstart g2"
using assms(4) unfolding assms(3) xy(3) by auto
then show ?thesis
using as and inj(1)[of "2 *\<^sub>R x" 1] inj(2)[of "2 *\<^sub>R y - 1" 0] and xy(1,2)
unfolding pathstart_def pathfinish_def joinpaths_def
by auto
next
assume as:"x > 1 / 2" "y \ 1 / 2"
then have "?g x \ path_image g2" "?g y \ path_image g1"
unfolding path_image_def joinpaths_def
using xy(1,2) by auto
then have "?g x = pathstart g2" "?g y = pathfinish g1"
using assms(4) unfolding assms(3) xy(3) by auto
then show ?thesis using as and inj(2)[of "2 *\<^sub>R x - 1" 0] inj(1)[of "2 *\<^sub>R y" 1] and xy(1,2)
unfolding pathstart_def pathfinish_def joinpaths_def
by auto
qed
qed
lemmas join_paths_simps = path_join path_image_join pathstart_join pathfinish_join
subsection {* Reparametrizing a closed curve to start at some chosen point. *}
definition "shiftpath a (f::real \ 'a::topological_space) =
(\x. if (a + x) \ 1 then f(a + x) else f(a + x - 1))"
lemma pathstart_shiftpath: "a \ 1 \ pathstart(shiftpath a g) = g a"
unfolding pathstart_def shiftpath_def by auto
lemma pathfinish_shiftpath:
assumes "0 \ a" "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" "a \ {0 .. 1}"
shows "pathfinish(shiftpath a g) = g a" "pathstart(shiftpath a g) = g a"
using assms by(auto intro!:pathfinish_shiftpath pathstart_shiftpath)
lemma closed_shiftpath:
assumes "pathfinish g = pathstart g" "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" "pathfinish g = pathstart g" "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 *
apply (rule continuous_on_union)
apply (rule closed_real_atLeastAtMost)+
apply (rule continuous_on_eq[of _ "g \ (\x. a + x)"]) prefer 3
apply (rule continuous_on_eq[of _ "g \ (\x. a - 1 + x)"]) defer prefer 3
apply (rule continuous_on_intros)+ prefer 2
apply (rule continuous_on_intros)+
apply (rule_tac[1-2] continuous_on_subset[OF assms(1)[unfolded path_def]])
using assms(3) and **
apply (auto, auto simp add: field_simps)
done
qed
lemma shiftpath_shiftpath:
assumes "pathfinish g = pathstart g" "a \ {0..1}" "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 \ {0..1}" "pathfinish g = pathstart g"
shows "path_image(shiftpath a g) = path_image g"
proof -
{ fix x
assume as:"g 1 = g 0" "x \ {0..1::real}" " \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 as(1,2) and as(3)[THEN bspec[where x="1 + x - a"]] and assms(1)
apply (auto simp add: field_simps atomize_not)
done
next
case True
then show ?thesis using as(1-2) and assms(1) apply(rule_tac x="x - a" in bexI)
by(auto simp add: field_simps)
qed
}
then show ?thesis
using assms unfolding shiftpath_def path_image_def pathfinish_def pathstart_def
by(auto simp add: image_iff)
qed
subsection {* Special case of straight-line paths. *}
definition linepath :: "'a::real_normed_vector \ 'a \ real \ '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 continuous_linepath_at[intro]: "continuous (at x) (linepath a b)"
unfolding linepath_def by (intro continuous_intros)
lemma continuous_on_linepath[intro]: "continuous_on s (linepath a b)"
using continuous_linepath_at by(auto intro!: continuous_at_imp_continuous_on)
lemma path_linepath[intro]: "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
apply (rule set_eqI, rule) defer
unfolding mem_Collect_eq image_iff
apply(erule exE)
apply(rule_tac x="u *\<^sub>R 1" in bexI)
apply auto
done
lemma reversepath_linepath[simp]: "reversepath(linepath a b) = linepath b a"
unfolding reversepath_def linepath_def
by auto
lemma injective_path_linepath:
assumes "a \ b"
shows "injective_path (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 injective_path_def linepath_def
by (auto simp add: algebra_simps)
qed
lemma simple_path_linepath[intro]: "a \ b \ simple_path(linepath a b)"
by(auto intro!: injective_imp_simple_path injective_path_linepath)
subsection {* Bounding a point away from a path. *}
lemma not_on_path_ball:
fixes g :: "real \ 'a::heine_borel"
assumes "path g" "z \ path_image g"
shows "\e > 0. ball z e \ (path_image g) = {}"
proof -
obtain a where "a \ path_image g" "\y \ path_image g. dist z a \ dist z y"
using distance_attains_inf[OF _ path_image_nonempty, of g z]
using compact_path_image[THEN compact_imp_closed, OF assms(1)] by auto
then show ?thesis
apply (rule_tac x="dist z a" in exI)
using assms(2)
apply (auto intro!: dist_pos_lt)
done
qed
lemma not_on_path_cball:
fixes g :: "real \ 'a::heine_borel"
assumes "path g" "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 `e>0` by auto
ultimately show ?thesis apply(rule_tac x="e/2" in exI) by auto
qed
subsection {* Path component, considered as a "joinability" relation (from Tom Hales). *}
definition "path_component s x y \
(\g. path g \ path_image g \ s \ pathstart g = x \ pathfinish g = y)"
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" "y \ s"
using assms unfolding path_defs by auto
lemma path_component_refl:
assumes "x \ s"
shows "path_component s x x"
unfolding path_defs
apply (rule_tac x="\u. x" in exI)
using assms apply (auto intro!:continuous_on_intros) done
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"
using assms
unfolding path_component_def
apply (erule exE)
apply (rule_tac x="reversepath g" in exI)
apply auto
done
lemma path_component_trans:
assumes "path_component s x y" "path_component s y z"
shows "path_component s x z"
using assms
unfolding path_component_def
apply -
apply (erule exE)+
apply (rule_tac x="g +++ ga" in exI)
apply (auto simp add: path_image_join)
done
lemma path_component_of_subset: "s \ t \ path_component s x y \ path_component t x y"
unfolding path_component_def by auto
subsection {* Can also consider it as a set, as the name suggests. *}
lemma path_component_set:
"{y. path_component s x y} =
{y. (\g. path g \ path_image g \ s \ pathstart g = x \ pathfinish g = y)}"
apply (rule set_eqI)
unfolding mem_Collect_eq
unfolding path_component_def
apply auto
done
lemma path_component_subset: "{y. path_component s x y} \ s"
apply (rule, rule path_component_mem(2))
apply auto
done
lemma path_component_eq_empty: "{y. path_component s x y} = {} \ x \ s"
apply rule
apply (drule equals0D[of _ x]) defer
apply (rule equals0I)
unfolding mem_Collect_eq
apply (drule path_component_mem(1))
using path_component_refl
apply auto
done
subsection {* Path connectedness of a space. *}
definition "path_connected s \
(\x\s. \y\s. \g. path g \ (path_image g) \ s \ pathstart g = x \ pathfinish g = y)"
lemma path_connected_component: "path_connected s \ (\x\s. \y\s. path_component s x y)"
unfolding path_connected_def path_component_def by auto
lemma path_connected_component_set: "path_connected s \ (\x\s. {y. path_component s x y} = s)"
unfolding path_connected_component
apply (rule, rule, rule, rule path_component_subset)
unfolding subset_eq mem_Collect_eq Ball_def
apply auto
done
subsection {* Some useful lemmas about path-connectedness. *}
lemma convex_imp_path_connected:
fixes s :: "'a::real_normed_vector set"
assumes "convex s" shows "path_connected s"
unfolding path_connected_def
apply (rule, rule, rule_tac x = "linepath x y" in exI)
unfolding path_image_linepath
using assms [unfolded convex_contains_segment]
apply auto
done
lemma path_connected_imp_connected:
assumes "path_connected s"
shows "connected s"
unfolding connected_def not_ex
apply (rule, rule, rule ccontr)
unfolding not_not
apply (erule conjE)+
proof -
fix e1 e2
assume as: "open e1" "open e2" "s \ e1 \ e2" "e1 \ e2 \ s = {}" "e1 \ s \ {}" "e2 \ s \ {}"
then obtain x1 x2 where obt:"x1\e1\s" "x2\e2\s" by auto
then obtain g where g:"path g" "path_image g \ s" "pathstart g = x1" "pathfinish g = x2"
using assms[unfolded path_connected_def,rule_format,of x1 x2] by auto
have *: "connected {0..1::real}"
by (auto intro!: convex_connected convex_real_interval)
have "{0..1} \ {x \ {0..1}. g x \ e1} \ {x \ {0..1}. g x \ e2}"
using as(3) g(2)[unfolded path_defs] by blast
moreover have "{x \ {0..1}. g x \ e1} \ {x \ {0..1}. g x \ e2} = {}"
using as(4) g(2)[unfolded path_defs] unfolding subset_eq by auto
moreover have "{x \ {0..1}. g x \ e1} \ {} \ {x \ {0..1}. g x \