|
|
|
|
|
Quellcode-Bibliothek
© Kompilation durch diese Firma
[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]
Datei:
default.txvpck
Sprache: Isabelle
|
|
(*
Material originally from HOL Light, ported by Larry Paulson, moved here by Manuel Eberl
*)
section\<^marker>\<open>tag unimportant\<close> \<open>Smooth paths\<close>
theory Smooth_Paths
imports
Retracts
begin
subsection\<^marker>\<open>tag unimportant\<close> \<open>Homeomorphisms of arc images\<close>
lemma path_connected_arc_complement:
fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space"
assumes "arc \" "2 \ DIM('a)"
shows "path_connected(- path_image \)"
proof -
have "path_image \ homeomorphic {0..1::real}"
by (simp add: assms homeomorphic_arc_image_interval)
then
show ?thesis
apply (rule path_connected_complement_homeomorphic_convex_compact)
apply (auto simp: assms)
done
qed
lemma connected_arc_complement:
fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space"
assumes "arc \" "2 \ DIM('a)"
shows "connected(- path_image \)"
by (simp add: assms path_connected_arc_complement path_connected_imp_connected)
lemma inside_arc_empty:
fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space"
assumes "arc \"
shows "inside(path_image \) = {}"
proof (cases "DIM('a) = 1")
case True
then show ?thesis
using assms connected_arc_image connected_convex_1_gen inside_convex by blast
next
case False
show ?thesis
proof (rule inside_bounded_complement_connected_empty)
show "connected (- path_image \)"
apply (rule connected_arc_complement [OF assms])
using False
by (metis DIM_ge_Suc0 One_nat_def Suc_1 not_less_eq_eq order_class.order.antisym)
show "bounded (path_image \)"
by (simp add: assms bounded_arc_image)
qed
qed
lemma inside_simple_curve_imp_closed:
fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space"
shows "\simple_path \; x \ inside(path_image \)\ \ pathfinish \ = pathstart \"
using arc_simple_path inside_arc_empty by blast
subsection \<open>Piecewise differentiability of paths\<close>
lemma continuous_on_joinpaths_D1:
"continuous_on {0..1} (g1 +++ g2) \ continuous_on {0..1} g1"
apply (rule continuous_on_eq [of _ "(g1 +++ g2) \ ((*)(inverse 2))"])
apply (rule continuous_intros | simp)+
apply (auto elim!: continuous_on_subset simp: joinpaths_def)
done
lemma continuous_on_joinpaths_D2:
"\continuous_on {0..1} (g1 +++ g2); pathfinish g1 = pathstart g2\ \ continuous_on {0..1} g2"
apply (rule continuous_on_eq [of _ "(g1 +++ g2) \ (\x. inverse 2*x + 1/2)"])
apply (rule continuous_intros | simp)+
apply (auto elim!: continuous_on_subset simp add: joinpaths_def pathfinish_def pathstart_def Ball_def)
done
lemma piecewise_differentiable_D1:
assumes "(g1 +++ g2) piecewise_differentiable_on {0..1}"
shows "g1 piecewise_differentiable_on {0..1}"
proof -
obtain S where cont: "continuous_on {0..1} g1" and "finite S"
and S: "\x. x \ {0..1} - S \ g1 +++ g2 differentiable at x within {0..1}"
using assms unfolding piecewise_differentiable_on_def
by (blast dest!: continuous_on_joinpaths_D1)
show ?thesis
unfolding piecewise_differentiable_on_def
proof (intro exI conjI ballI cont)
show "finite (insert 1 (((*)2) ` S))"
by (simp add: \<open>finite S\<close>)
show "g1 differentiable at x within {0..1}" if "x \ {0..1} - insert 1 ((*) 2 ` S)" for x
proof (rule_tac d="dist (x/2) (1/2)" in differentiable_transform_within)
have "g1 +++ g2 differentiable at (x / 2) within {0..1/2}"
by (rule differentiable_subset [OF S [of "x/2"]] | use that in force)+
then show "g1 +++ g2 \ (*) (inverse 2) differentiable at x within {0..1}"
using image_affinity_atLeastAtMost_div [of 2 0 "0::real" 1]
by (auto intro: differentiable_chain_within)
qed (use that in \<open>auto simp: joinpaths_def\<close>)
qed
qed
lemma piecewise_differentiable_D2:
assumes "(g1 +++ g2) piecewise_differentiable_on {0..1}" and eq: "pathfinish g1 = pathstart g2"
shows "g2 piecewise_differentiable_on {0..1}"
proof -
have [simp]: "g1 1 = g2 0"
using eq by (simp add: pathfinish_def pathstart_def)
obtain S where cont: "continuous_on {0..1} g2" and "finite S"
and S: "\x. x \ {0..1} - S \ g1 +++ g2 differentiable at x within {0..1}"
using assms unfolding piecewise_differentiable_on_def
by (blast dest!: continuous_on_joinpaths_D2)
show ?thesis
unfolding piecewise_differentiable_on_def
proof (intro exI conjI ballI cont)
show "finite (insert 0 ((\x. 2*x-1)`S))"
by (simp add: \<open>finite S\<close>)
show "g2 differentiable at x within {0..1}" if "x \ {0..1} - insert 0 ((\x. 2*x-1)`S)" for x
proof (rule_tac d="dist ((x+1)/2) (1/2)" in differentiable_transform_within)
have x2: "(x + 1) / 2 \ S"
using that
apply (clarsimp simp: image_iff)
by (metis add.commute add_diff_cancel_left' mult_2 field_sum_of_halves)
have "g1 +++ g2 \ (\x. (x+1) / 2) differentiable at x within {0..1}"
by (rule differentiable_chain_within differentiable_subset [OF S [of "(x+1)/2"]] | use x2 that in force)+
then show "g1 +++ g2 \ (\x. (x+1) / 2) differentiable at x within {0..1}"
by (auto intro: differentiable_chain_within)
show "(g1 +++ g2 \ (\x. (x + 1) / 2)) x' = g2 x'" if "x' \ {0..1}" "dist x' x < dist ((x + 1) / 2) (1/2)" for x'
proof -
have [simp]: "(2*x'+2)/2 = x'+1"
by (simp add: field_split_simps)
show ?thesis
using that by (auto simp: joinpaths_def)
qed
qed (use that in \<open>auto simp: joinpaths_def\<close>)
qed
qed
lemma piecewise_C1_differentiable_D1:
fixes g1 :: "real \ 'a::real_normed_field"
assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}"
shows "g1 piecewise_C1_differentiable_on {0..1}"
proof -
obtain S where "finite S"
and co12: "continuous_on ({0..1} - S) (\x. vector_derivative (g1 +++ g2) (at x))"
and g12D: "\x\{0..1} - S. g1 +++ g2 differentiable at x"
using assms by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
have g1D: "g1 differentiable at x" if "x \ {0..1} - insert 1 ((*) 2 ` S)" for x
proof (rule differentiable_transform_within)
show "g1 +++ g2 \ (*) (inverse 2) differentiable at x"
using that g12D
apply (simp only: joinpaths_def)
by (rule differentiable_chain_at derivative_intros | force)+
show "\x'. \dist x' x < dist (x/2) (1/2)\
\<Longrightarrow> (g1 +++ g2 \<circ> (*) (inverse 2)) x' = g1 x'"
using that by (auto simp: dist_real_def joinpaths_def)
qed (use that in \<open>auto simp: dist_real_def\<close>)
have [simp]: "vector_derivative (g1 \ (*) 2) (at (x/2)) = 2 *\<^sub>R vector_derivative g1 (at x)"
if "x \ {0..1} - insert 1 ((*) 2 ` S)" for x
apply (subst vector_derivative_chain_at)
using that
apply (rule derivative_eq_intros g1D | simp)+
done
have "continuous_on ({0..1/2} - insert (1/2) S) (\x. vector_derivative (g1 +++ g2) (at x))"
using co12 by (rule continuous_on_subset) force
then have coDhalf: "continuous_on ({0..1/2} - insert (1/2) S) (\x. vector_derivative (g1 \ (*)2) (at x))"
proof (rule continuous_on_eq [OF _ vector_derivative_at])
show "(g1 +++ g2 has_vector_derivative vector_derivative (g1 \ (*) 2) (at x)) (at x)"
if "x \ {0..1/2} - insert (1/2) S" for x
proof (rule has_vector_derivative_transform_within)
show "(g1 \ (*) 2 has_vector_derivative vector_derivative (g1 \ (*) 2) (at x)) (at x)"
using that
by (force intro: g1D differentiable_chain_at simp: vector_derivative_works [symmetric])
show "\x'. \dist x' x < dist x (1/2)\ \ (g1 \ (*) 2) x' = (g1 +++ g2) x'"
using that by (auto simp: dist_norm joinpaths_def)
qed (use that in \<open>auto simp: dist_norm\<close>)
qed
have "continuous_on ({0..1} - insert 1 ((*) 2 ` S))
((\<lambda>x. 1/2 * vector_derivative (g1 \<circ> (*)2) (at x)) \<circ> (*)(1/2))"
apply (rule continuous_intros)+
using coDhalf
apply (simp add: scaleR_conv_of_real image_set_diff image_image)
done
then have con_g1: "continuous_on ({0..1} - insert 1 ((*) 2 ` S)) (\x. vector_derivative g1 (at x))"
by (rule continuous_on_eq) (simp add: scaleR_conv_of_real)
have "continuous_on {0..1} g1"
using continuous_on_joinpaths_D1 assms piecewise_C1_differentiable_on_def by blast
with \<open>finite S\<close> show ?thesis
apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
apply (rule_tac x="insert 1 (((*)2)`S)" in exI)
apply (simp add: g1D con_g1)
done
qed
lemma piecewise_C1_differentiable_D2:
fixes g2 :: "real \ 'a::real_normed_field"
assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}" "pathfinish g1 = pathstart g2"
shows "g2 piecewise_C1_differentiable_on {0..1}"
proof -
obtain S where "finite S"
and co12: "continuous_on ({0..1} - S) (\x. vector_derivative (g1 +++ g2) (at x))"
and g12D: "\x\{0..1} - S. g1 +++ g2 differentiable at x"
using assms by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
have g2D: "g2 differentiable at x" if "x \ {0..1} - insert 0 ((\x. 2*x-1) ` S)" for x
proof (rule differentiable_transform_within)
show "g1 +++ g2 \ (\x. (x + 1) / 2) differentiable at x"
using g12D that
apply (simp only: joinpaths_def)
apply (drule_tac x= "(x+1) / 2" in bspec, force simp: field_split_simps)
apply (rule differentiable_chain_at derivative_intros | force)+
done
show "\x'. dist x' x < dist ((x + 1) / 2) (1/2) \ (g1 +++ g2 \ (\x. (x + 1) / 2)) x' = g2 x'"
using that by (auto simp: dist_real_def joinpaths_def field_simps)
qed (use that in \<open>auto simp: dist_norm\<close>)
have [simp]: "vector_derivative (g2 \ (\x. 2*x-1)) (at ((x+1)/2)) = 2 *\<^sub>R vector_derivative g2 (at x)"
if "x \ {0..1} - insert 0 ((\x. 2*x-1) ` S)" for x
using that by (auto simp: vector_derivative_chain_at field_split_simps g2D)
have "continuous_on ({1/2..1} - insert (1/2) S) (\x. vector_derivative (g1 +++ g2) (at x))"
using co12 by (rule continuous_on_subset) force
then have coDhalf: "continuous_on ({1/2..1} - insert (1/2) S) (\x. vector_derivative (g2 \ (\x. 2*x-1)) (at x))"
proof (rule continuous_on_eq [OF _ vector_derivative_at])
show "(g1 +++ g2 has_vector_derivative vector_derivative (g2 \ (\x. 2 * x - 1)) (at x))
(at x)"
if "x \ {1 / 2..1} - insert (1 / 2) S" for x
proof (rule_tac f="g2 \ (\x. 2*x-1)" and d="dist (3/4) ((x+1)/2)" in has_vector_derivative_transform_within)
show "(g2 \ (\x. 2 * x - 1) has_vector_derivative vector_derivative (g2 \ (\x. 2 * x - 1)) (at x))
(at x)"
using that by (force intro: g2D differentiable_chain_at simp: vector_derivative_works [symmetric])
show "\x'. \dist x' x < dist (3 / 4) ((x + 1) / 2)\ \ (g2 \ (\x. 2 * x - 1)) x' = (g1 +++ g2) x'"
using that by (auto simp: dist_norm joinpaths_def add_divide_distrib)
qed (use that in \<open>auto simp: dist_norm\<close>)
qed
have [simp]: "((\x. (x+1) / 2) ` ({0..1} - insert 0 ((\x. 2 * x - 1) ` S))) = ({1/2..1} - insert (1/2) S)"
apply (simp add: image_set_diff inj_on_def image_image)
apply (auto simp: image_affinity_atLeastAtMost_div add_divide_distrib)
done
have "continuous_on ({0..1} - insert 0 ((\x. 2*x-1) ` S))
((\<lambda>x. 1/2 * vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at x)) \<circ> (\<lambda>x. (x+1)/2))"
by (rule continuous_intros | simp add: coDhalf)+
then have con_g2: "continuous_on ({0..1} - insert 0 ((\x. 2*x-1) ` S)) (\x. vector_derivative g2 (at x))"
by (rule continuous_on_eq) (simp add: scaleR_conv_of_real)
have "continuous_on {0..1} g2"
using continuous_on_joinpaths_D2 assms piecewise_C1_differentiable_on_def by blast
with \<open>finite S\<close> show ?thesis
apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
apply (rule_tac x="insert 0 ((\x. 2 * x - 1) ` S)" in exI)
apply (simp add: g2D con_g2)
done
qed
subsection \<open>Valid paths, and their start and finish\<close>
definition\<^marker>\<open>tag important\<close> valid_path :: "(real \<Rightarrow> 'a :: real_normed_vector) \<Rightarrow> bool"
where "valid_path f \ f piecewise_C1_differentiable_on {0..1::real}"
definition closed_path :: "(real \ 'a :: real_normed_vector) \ bool"
where "closed_path g \ g 0 = g 1"
text\<open>In particular, all results for paths apply\<close>
lemma valid_path_imp_path: "valid_path g \ path g"
by (simp add: path_def piecewise_C1_differentiable_on_def valid_path_def)
lemma connected_valid_path_image: "valid_path g \ connected(path_image g)"
by (metis connected_path_image valid_path_imp_path)
lemma compact_valid_path_image: "valid_path g \ compact(path_image g)"
by (metis compact_path_image valid_path_imp_path)
lemma bounded_valid_path_image: "valid_path g \ bounded(path_image g)"
by (metis bounded_path_image valid_path_imp_path)
lemma closed_valid_path_image: "valid_path g \ closed(path_image g)"
by (metis closed_path_image valid_path_imp_path)
lemma valid_path_compose:
assumes "valid_path g"
and der: "\x. x \ path_image g \ f field_differentiable (at x)"
and con: "continuous_on (path_image g) (deriv f)"
shows "valid_path (f \ g)"
proof -
obtain S where "finite S" and g_diff: "g C1_differentiable_on {0..1} - S"
using \<open>valid_path g\<close> unfolding valid_path_def piecewise_C1_differentiable_on_def by auto
have "f \ g differentiable at t" when "t\{0..1} - S" for t
proof (rule differentiable_chain_at)
show "g differentiable at t" using \<open>valid_path g\<close>
by (meson C1_differentiable_on_eq \<open>g C1_differentiable_on {0..1} - S\<close> that)
next
have "g t\path_image g" using that DiffD1 image_eqI path_image_def by metis
then show "f differentiable at (g t)"
using der[THEN field_differentiable_imp_differentiable] by auto
qed
moreover have "continuous_on ({0..1} - S) (\x. vector_derivative (f \ g) (at x))"
proof (rule continuous_on_eq [where f = "\x. vector_derivative g (at x) * deriv f (g x)"],
rule continuous_intros)
show "continuous_on ({0..1} - S) (\x. vector_derivative g (at x))"
using g_diff C1_differentiable_on_eq by auto
next
have "continuous_on {0..1} (\x. deriv f (g x))"
using continuous_on_compose[OF _ con[unfolded path_image_def],unfolded comp_def]
\<open>valid_path g\<close> piecewise_C1_differentiable_on_def valid_path_def
by blast
then show "continuous_on ({0..1} - S) (\x. deriv f (g x))"
using continuous_on_subset by blast
next
show "vector_derivative g (at t) * deriv f (g t) = vector_derivative (f \ g) (at t)"
when "t \ {0..1} - S" for t
proof (rule vector_derivative_chain_at_general[symmetric])
show "g differentiable at t" by (meson C1_differentiable_on_eq g_diff that)
next
have "g t\path_image g" using that DiffD1 image_eqI path_image_def by metis
then show "f field_differentiable at (g t)" using der by auto
qed
qed
ultimately have "f \ g C1_differentiable_on {0..1} - S"
using C1_differentiable_on_eq by blast
moreover have "path (f \ g)"
apply (rule path_continuous_image[OF valid_path_imp_path[OF \<open>valid_path g\<close>]])
using der
by (simp add: continuous_at_imp_continuous_on field_differentiable_imp_continuous_at)
ultimately show ?thesis unfolding valid_path_def piecewise_C1_differentiable_on_def path_def
using \<open>finite S\<close> by auto
qed
lemma valid_path_uminus_comp[simp]:
fixes g::"real \ 'a ::real_normed_field"
shows "valid_path (uminus \ g) \ valid_path g"
proof
show "valid_path g \ valid_path (uminus \ g)" for g::"real \ 'a"
by (auto intro!: valid_path_compose derivative_intros)
then show "valid_path g" when "valid_path (uminus \ g)"
by (metis fun.map_comp group_add_class.minus_comp_minus id_comp that)
qed
lemma valid_path_offset[simp]:
shows "valid_path (\t. g t - z) \ valid_path g"
proof
show *: "valid_path (g::real\'a) \ valid_path (\t. g t - z)" for g z
unfolding valid_path_def
by (fastforce intro:derivative_intros C1_differentiable_imp_piecewise piecewise_C1_differentiable_diff)
show "valid_path (\t. g t - z) \ valid_path g"
using *[of "\t. g t - z" "-z",simplified] .
qed
lemma valid_path_imp_reverse:
assumes "valid_path g"
shows "valid_path(reversepath g)"
proof -
obtain S where "finite S" and S: "g C1_differentiable_on ({0..1} - S)"
using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
then have "finite ((-) 1 ` S)"
by auto
moreover have "(reversepath g C1_differentiable_on ({0..1} - (-) 1 ` S))"
unfolding reversepath_def
apply (rule C1_differentiable_compose [of "\x::real. 1-x" _ g, unfolded o_def])
using S
by (force simp: finite_vimageI inj_on_def C1_differentiable_on_eq elim!: continuous_on_subset)+
ultimately show ?thesis using assms
by (auto simp: valid_path_def piecewise_C1_differentiable_on_def path_def [symmetric])
qed
lemma valid_path_reversepath [simp]: "valid_path(reversepath g) \ valid_path g"
using valid_path_imp_reverse by force
lemma valid_path_join:
assumes "valid_path g1" "valid_path g2" "pathfinish g1 = pathstart g2"
shows "valid_path(g1 +++ g2)"
proof -
have "g1 1 = g2 0"
using assms by (auto simp: pathfinish_def pathstart_def)
moreover have "(g1 \ (\x. 2*x)) piecewise_C1_differentiable_on {0..1/2}"
apply (rule piecewise_C1_differentiable_compose)
using assms
apply (auto simp: valid_path_def piecewise_C1_differentiable_on_def continuous_on_joinpaths)
apply (force intro: finite_vimageI [where h = "(*)2"] inj_onI)
done
moreover have "(g2 \ (\x. 2*x-1)) piecewise_C1_differentiable_on {1/2..1}"
apply (rule piecewise_C1_differentiable_compose)
using assms unfolding valid_path_def piecewise_C1_differentiable_on_def
by (auto intro!: continuous_intros finite_vimageI [where h = "(\x. 2*x - 1)"] inj_onI
simp: image_affinity_atLeastAtMost_diff continuous_on_joinpaths)
ultimately show ?thesis
apply (simp only: valid_path_def continuous_on_joinpaths joinpaths_def)
apply (rule piecewise_C1_differentiable_cases)
apply (auto simp: o_def)
done
qed
lemma valid_path_join_D1:
fixes g1 :: "real \ 'a::real_normed_field"
shows "valid_path (g1 +++ g2) \ valid_path g1"
unfolding valid_path_def
by (rule piecewise_C1_differentiable_D1)
lemma valid_path_join_D2:
fixes g2 :: "real \ 'a::real_normed_field"
shows "\valid_path (g1 +++ g2); pathfinish g1 = pathstart g2\ \ valid_path g2"
unfolding valid_path_def
by (rule piecewise_C1_differentiable_D2)
lemma valid_path_join_eq [simp]:
fixes g2 :: "real \ 'a::real_normed_field"
shows "pathfinish g1 = pathstart g2 \ (valid_path(g1 +++ g2) \ valid_path g1 \ valid_path g2)"
using valid_path_join_D1 valid_path_join_D2 valid_path_join by blast
lemma valid_path_shiftpath [intro]:
assumes "valid_path g" "pathfinish g = pathstart g" "a \ {0..1}"
shows "valid_path(shiftpath a g)"
using assms
apply (auto simp: valid_path_def shiftpath_alt_def)
apply (rule piecewise_C1_differentiable_cases)
apply (auto simp: algebra_simps)
apply (rule piecewise_C1_differentiable_affine [of g 1 a, simplified o_def scaleR_one])
apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset)
apply (rule piecewise_C1_differentiable_affine [of g 1 "a-1", simplified o_def scaleR_one algebra_simps])
apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset)
done
lemma vector_derivative_linepath_within:
"x \ {0..1} \ vector_derivative (linepath a b) (at x within {0..1}) = b - a"
apply (rule vector_derivative_within_cbox [of 0 "1::real", simplified])
apply (auto simp: has_vector_derivative_linepath_within)
done
lemma vector_derivative_linepath_at [simp]: "vector_derivative (linepath a b) (at x) = b - a"
by (simp add: has_vector_derivative_linepath_within vector_derivative_at)
lemma valid_path_linepath [iff]: "valid_path (linepath a b)"
apply (simp add: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_on_linepath)
apply (rule_tac x="{}" in exI)
apply (simp add: differentiable_on_def differentiable_def)
using has_vector_derivative_def has_vector_derivative_linepath_within
apply (fastforce simp add: continuous_on_eq_continuous_within)
done
lemma valid_path_subpath:
fixes g :: "real \ 'a :: real_normed_vector"
assumes "valid_path g" "u \ {0..1}" "v \ {0..1}"
shows "valid_path(subpath u v g)"
proof (cases "v=u")
case True
then show ?thesis
unfolding valid_path_def subpath_def
by (force intro: C1_differentiable_on_const C1_differentiable_imp_piecewise)
next
case False
have "(g \ (\x. ((v-u) * x + u))) piecewise_C1_differentiable_on {0..1}"
apply (rule piecewise_C1_differentiable_compose)
apply (simp add: C1_differentiable_imp_piecewise)
apply (simp add: image_affinity_atLeastAtMost)
using assms False
apply (auto simp: algebra_simps valid_path_def piecewise_C1_differentiable_on_subset)
apply (subst Int_commute)
apply (auto simp: inj_on_def algebra_simps crossproduct_eq finite_vimage_IntI)
done
then show ?thesis
by (auto simp: o_def valid_path_def subpath_def)
qed
lemma valid_path_rectpath [simp, intro]: "valid_path (rectpath a b)"
by (simp add: Let_def rectpath_def)
end
¤ Dauer der Verarbeitung: 0.25 Sekunden
(vorverarbeitet)
¤
|
Haftungshinweis
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung ist noch experimentell.
|
|
|
|
|
|
