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Datei: Ordered_Euclidean_Space.thy Sprache: Isabelle

Original von: Isabelle^©

section \<open>Ordered Euclidean Space\<close>

theory Ordered_Euclidean_Space
imports
  Convex_Euclidean_Space
  "HOL-Library.Product_Order"
begin

text \<open>An ordering on euclidean spaces that will allow us to talk about intervals\<close>

class ordered_euclidean_space = ord + inf + sup + abs + Inf + Sup + euclidean_space +
  assumes eucl_le: "x \ y \ (\i\Basis. x \ i \ y \ i)"
  assumes eucl_less_le_not_le: "x < y \ x \ y \ \ y \ x"
  assumes eucl_inf: "inf x y = (\i\Basis. inf (x \ i) (y \ i) *\<^sub>R i)"
  assumes eucl_sup: "sup x y = (\i\Basis. sup (x \ i) (y \ i) *\<^sub>R i)"
  assumes eucl_Inf: "Inf X = (\i\Basis. (INF x\X. x \ i) *\<^sub>R i)"
  assumes eucl_Sup: "Sup X = (\i\Basis. (SUP x\X. x \ i) *\<^sub>R i)"
  assumes eucl_abs: "\x\ = (\i\Basis. \x \ i\ *\<^sub>R i)"
begin

subclass order
  by standard
    (auto simp: eucl_le eucl_less_le_not_le intro!: euclidean_eqI antisym intro: order.trans)

subclass ordered_ab_group_add_abs
  by standard (auto simp: eucl_le inner_add_left eucl_abs abs_leI)

subclass ordered_real_vector
  by standard (auto simp: eucl_le intro!: mult_left_mono mult_right_mono)

subclass lattice
  by standard (auto simp: eucl_inf eucl_sup eucl_le)

subclass distrib_lattice
  by standard (auto simp: eucl_inf eucl_sup sup_inf_distrib1 intro!: euclidean_eqI)

subclass conditionally_complete_lattice
proof
  fix z::'a and X::"'a set"
  assume "X \ {}"
  hence "\i. (\x. x \ i) ` X \ {}" by simp
  thus "(\x. x \ X \ z \ x) \ z \ Inf X" "(\x. x \ X \ x \ z) \ Sup X \ z"
    by (auto simp: eucl_Inf eucl_Sup eucl_le
      intro!: cInf_greatest cSup_least)
qed (force intro!: cInf_lower cSup_upper
      simp: bdd_below_def bdd_above_def preorder_class.bdd_below_def preorder_class.bdd_above_def
        eucl_Inf eucl_Sup eucl_le)+

lemma inner_Basis_inf_left: "i \ Basis \ inf x y \ i = inf (x \ i) (y \ i)"
  and inner_Basis_sup_left: "i \ Basis \ sup x y \ i = sup (x \ i) (y \ i)"
  by (simp_all add: eucl_inf eucl_sup inner_sum_left inner_Basis if_distrib
      cong: if_cong)

lemma inner_Basis_INF_left: "i \ Basis \ (INF x\X. f x) \ i = (INF x\X. f x \ i)"
  and inner_Basis_SUP_left: "i \ Basis \ (SUP x\X. f x) \ i = (SUP x\X. f x \ i)"
  using eucl_Sup [of "f ` X"] eucl_Inf [of "f ` X"] by (simp_all add: image_comp)

lemma abs_inner: "i \ Basis \ \x\ \ i = \x \ i\"
  by (auto simp: eucl_abs)

lemma
  abs_scaleR: "\a *\<^sub>R b\ = \a\ *\<^sub>R \b\"
  by (auto simp: eucl_abs abs_mult intro!: euclidean_eqI)

lemma interval_inner_leI:
  assumes "x \ {a .. b}" "0 \ i"
  shows "a\i \ x\i" "x\i \ b\i"
  using assms
  unfolding euclidean_inner[of a i] euclidean_inner[of x i] euclidean_inner[of b i]
  by (auto intro!: ordered_comm_monoid_add_class.sum_mono mult_right_mono simp: eucl_le)

lemma inner_nonneg_nonneg:
  shows "0 \ a \ 0 \ b \ 0 \ a \ b"
  using interval_inner_leI[of a 0 a b]
  by auto

lemma inner_Basis_mono:
  shows "a \ b \ c \ Basis \ a \ c \ b \ c"
  by (simp add: eucl_le)

lemma Basis_nonneg[intro, simp]: "i \ Basis \ 0 \ i"
  by (auto simp: eucl_le inner_Basis)

lemma Sup_eq_maximum_componentwise:
  fixes s::"'a set"
  assumes i: "\b. b \ Basis \ X \ b = i b \ b"
  assumes sup: "\b x. b \ Basis \ x \ s \ x \ b \ X \ b"
  assumes i_s: "\b. b \ Basis \ (i b \ b) \ (\x. x \ b) ` s"
  shows "Sup s = X"
  using assms
  unfolding eucl_Sup euclidean_representation_sum
  by (auto intro!: conditionally_complete_lattice_class.cSup_eq_maximum)

lemma Inf_eq_minimum_componentwise:
  assumes i: "\b. b \ Basis \ X \ b = i b \ b"
  assumes sup: "\b x. b \ Basis \ x \ s \ X \ b \ x \ b"
  assumes i_s: "\b. b \ Basis \ (i b \ b) \ (\x. x \ b) ` s"
  shows "Inf s = X"
  using assms
  unfolding eucl_Inf euclidean_representation_sum
  by (auto intro!: conditionally_complete_lattice_class.cInf_eq_minimum)

end

proposition  compact_attains_Inf_componentwise:
  fixes b::"'a::ordered_euclidean_space"
  assumes "b \ Basis" assumes "X \ {}" "compact X"
  obtains x where "x \ X" "x \ b = Inf X \ b" "\y. y \ X \ x \ b \ y \ b"
proof atomize_elim
  let ?proj = "(\x. x \ b) ` X"
  from assms have "compact ?proj" "?proj \ {}"
    by (auto intro!: compact_continuous_image continuous_intros)
  from compact_attains_inf[OF this]
  obtain s x
    where s: "s\(\x. x \ b) ` X" "\t. t\(\x. x \ b) ` X \ s \ t"
      and x: "x \ X" "s = x \ b" "\y. y \ X \ x \ b \ y \ b"
    by auto
  hence "Inf ?proj = x \ b"
    by (auto intro!: conditionally_complete_lattice_class.cInf_eq_minimum)
  hence "x \ b = Inf X \ b"
    by (auto simp: eucl_Inf inner_sum_left inner_Basis if_distrib \<open>b \<in> Basis\<close>
      cong: if_cong)
  with x show "\x. x \ X \ x \ b = Inf X \ b \ (\y. y \ X \ x \ b \ y \ b)" by blast
qed

proposition
  compact_attains_Sup_componentwise:
  fixes b::"'a::ordered_euclidean_space"
  assumes "b \ Basis" assumes "X \ {}" "compact X"
  obtains x where "x \ X" "x \ b = Sup X \ b" "\y. y \ X \ y \ b \ x \ b"
proof atomize_elim
  let ?proj = "(\x. x \ b) ` X"
  from assms have "compact ?proj" "?proj \ {}"
    by (auto intro!: compact_continuous_image continuous_intros)
  from compact_attains_sup[OF this]
  obtain s x
    where s: "s\(\x. x \ b) ` X" "\t. t\(\x. x \ b) ` X \ t \ s"
      and x: "x \ X" "s = x \ b" "\y. y \ X \ y \ b \ x \ b"
    by auto
  hence "Sup ?proj = x \ b"
    by (auto intro!: cSup_eq_maximum)
  hence "x \ b = Sup X \ b"
    by (auto simp: eucl_Sup[where 'a='a] inner_sum_left inner_Basis if_distrib \<open>b \<in> Basis\<close>
      cong: if_cong)
  with x show "\x. x \ X \ x \ b = Sup X \ b \ (\y. y \ X \ y \ b \ x \ b)" by blast
qed

lemma tendsto_sup[tendsto_intros]:
  fixes X :: "'a \ 'b::ordered_euclidean_space"
  assumes "(X \ x) net" "(Y \ y) net"
  shows "((\i. sup (X i) (Y i)) \ sup x y) net"
   unfolding sup_max eucl_sup by (intro assms tendsto_intros)

lemma tendsto_inf[tendsto_intros]:
  fixes X :: "'a \ 'b::ordered_euclidean_space"
  assumes "(X \ x) net" "(Y \ y) net"
  shows "((\i. inf (X i) (Y i)) \ inf x y) net"
   unfolding inf_min eucl_inf by (intro assms tendsto_intros)

lemma tendsto_componentwise_max:
  assumes f: "(f \ l) F" and g: "(g \ m) F"
  shows "((\x. (\i\Basis. max (f x \ i) (g x \ i) *\<^sub>R i)) \ (\i\Basis. max (l \ i) (m \ i) *\<^sub>R i)) F"
  by (intro tendsto_intros assms)

lemma tendsto_componentwise_min:
  assumes f: "(f \ l) F" and g: "(g \ m) F"
  shows "((\x. (\i\Basis. min (f x \ i) (g x \ i) *\<^sub>R i)) \ (\i\Basis. min (l \ i) (m \ i) *\<^sub>R i)) F"
  by (intro tendsto_intros assms)

lemma (in order) atLeastatMost_empty'[simp]:
  "(\ a \ b) \ {a..b} = {}"
  by (auto)

instance real :: ordered_euclidean_space
  by standard auto

lemma in_Basis_prod_iff:
  fixes i::"'a::euclidean_space*'b::euclidean_space"
  shows "i \ Basis \ fst i = 0 \ snd i \ Basis \ snd i = 0 \ fst i \ Basis"
  by (cases i) (auto simp: Basis_prod_def)

instantiation\<^marker>\<open>tag unimportant\<close> prod :: (abs, abs) abs
begin

definition "\x\ = (\fst x\, \snd x\)"

instance ..

end

instance prod :: (ordered_euclidean_space, ordered_euclidean_space) ordered_euclidean_space
  by standard
    (auto intro!: add_mono simp add: euclidean_representation_sum' Ball_def inner_prod_def
      in_Basis_prod_iff inner_Basis_inf_left inner_Basis_sup_left inner_Basis_INF_left Inf_prod_def
      inner_Basis_SUP_left Sup_prod_def less_prod_def less_eq_prod_def eucl_le[where 'a='a]
      eucl_le[where 'a='b] abs_prod_def abs_inner)

text\<open>Instantiation for intervals on \<open>ordered_euclidean_space\<close>\<close>

proposition
  fixes a :: "'a::ordered_euclidean_space"
  shows cbox_interval: "cbox a b = {a..b}"
    and interval_cbox: "{a..b} = cbox a b"
    and eucl_le_atMost: "{x. \i\Basis. x \ i <= a \ i} = {..a}"
    and eucl_le_atLeast: "{x. \i\Basis. a \ i <= x \ i} = {a..}"
  by (auto simp: eucl_le[where 'a='a] eucl_less_def box_def cbox_def)

lemma sums_vec_nth :
  assumes "f sums a"
  shows "(\x. f x $ i) sums a $ i"
  using assms unfolding sums_def
  by (auto dest: tendsto_vec_nth [where i=i])

lemma summable_vec_nth :
  assumes "summable f"
  shows "summable (\x. f x $ i)"
  using assms unfolding summable_def
  by (blast intro: sums_vec_nth)

lemma closed_eucl_atLeastAtMost[simp, intro]:
  fixes a :: "'a::ordered_euclidean_space"
  shows "closed {a..b}"
  by (simp add: cbox_interval[symmetric] closed_cbox)

lemma closed_eucl_atMost[simp, intro]:
  fixes a :: "'a::ordered_euclidean_space"
  shows "closed {..a}"
  by (simp add: closed_interval_left eucl_le_atMost[symmetric])

lemma closed_eucl_atLeast[simp, intro]:
  fixes a :: "'a::ordered_euclidean_space"
  shows "closed {a..}"
  by (simp add: closed_interval_right eucl_le_atLeast[symmetric])

lemma bounded_closed_interval [simp]:
  fixes a :: "'a::ordered_euclidean_space"
  shows "bounded {a .. b}"
  using bounded_cbox[of a b]
  by (metis interval_cbox)

lemma convex_closed_interval [simp]:
  fixes a :: "'a::ordered_euclidean_space"
  shows "convex {a .. b}"
  using convex_box[of a b]
  by (metis interval_cbox)

lemma image_smult_interval:"(\x. m *\<^sub>R (x::_::ordered_euclidean_space)) ` {a .. b} =
  (if {a .. b} = {} then {} else if 0 \<le> m then {m *\<^sub>R a .. m *\<^sub>R b} else {m *\<^sub>R b .. m *\<^sub>R a})"
  using image_smult_cbox[of m a b]
  by (simp add: cbox_interval)

lemma [simp]:
  fixes a b::"'a::ordered_euclidean_space"
  shows is_interval_ic: "is_interval {..a}"
    and is_interval_ci: "is_interval {a..}"
    and is_interval_cc: "is_interval {b..a}"
  by (force simp: is_interval_def eucl_le[where 'a='a])+

lemma connected_interval [simp]:
  fixes a b::"'a::ordered_euclidean_space"
  shows "connected {a..b}"
  using is_interval_cc is_interval_connected by blast

lemma compact_interval [simp]:
  fixes a b::"'a::ordered_euclidean_space"
  shows "compact {a .. b}"
  by (metis compact_cbox interval_cbox)

no_notation
  eucl_less (infix " 50)

lemma One_nonneg: "0 \ (\Basis::'a::ordered_euclidean_space)"
  by (auto intro: sum_nonneg)

lemma
  fixes a b::"'a::ordered_euclidean_space"
  shows bdd_above_cbox[intro, simp]: "bdd_above (cbox a b)"
    and bdd_below_cbox[intro, simp]: "bdd_below (cbox a b)"
    and bdd_above_box[intro, simp]: "bdd_above (box a b)"
    and bdd_below_box[intro, simp]: "bdd_below (box a b)"
  unfolding atomize_conj
  by (metis bdd_above_Icc bdd_above_mono bdd_below_Icc bdd_below_mono bounded_box
            bounded_subset_cbox_symmetric interval_cbox)

instantiation vec :: (ordered_euclidean_space, finite) ordered_euclidean_space
begin

definition\<^marker>\<open>tag important\<close> "inf x y = (\<chi> i. inf (x $ i) (y $ i))"
definition\<^marker>\<open>tag important\<close> "sup x y = (\<chi> i. sup (x $ i) (y $ i))"
definition\<^marker>\<open>tag important\<close> "Inf X = (\<chi> i. (INF x\<in>X. x $ i))"
definition\<^marker>\<open>tag important\<close> "Sup X = (\<chi> i. (SUP x\<in>X. x $ i))"
definition\<^marker>\<open>tag important\<close> "\<bar>x\<bar> = (\<chi> i. \<bar>x $ i\<bar>)"

instance
  apply standard
  unfolding euclidean_representation_sum'
  apply (auto simp: less_eq_vec_def inf_vec_def sup_vec_def Inf_vec_def Sup_vec_def inner_axis
    Basis_vec_def inner_Basis_inf_left inner_Basis_sup_left inner_Basis_INF_left
    inner_Basis_SUP_left eucl_le[where 'a='a] less_le_not_le abs_vec_def abs_inner)
  done

end

end

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