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Datei: Wellorder_Constructions.thy Sprache: Isabelle

Original von: Isabelle^©

(*  Author:     L C Paulson, University of Cambridge
    Author:     Amine Chaieb, University of Cambridge
    Author:     Robert Himmelmann, TU Muenchen
    Author:     Brian Huffman, Portland State University
*)

section \<open>Elementary Normed Vector Spaces\<close>

theory Elementary_Normed_Spaces
  imports
  "HOL-Library.FuncSet"
  Elementary_Metric_Spaces Cartesian_Space
  Connected
begin
subsection \<open>Orthogonal Transformation of Balls\<close>

subsection\<^marker>\<open>tag unimportant\<close> \<open>Various Lemmas Combining Imports\<close>

lemma open_sums:
  fixes T :: "('b::real_normed_vector) set"
  assumes "open S \ open T"
  shows "open (\x\ S. \y \ T. {x + y})"
  using assms
proof
  assume S: "open S"
  show ?thesis
  proof (clarsimp simp: open_dist)
    fix x y
    assume "x \ S" "y \ T"
    with S obtain e where "e > 0" and e: "\x'. dist x' x < e \ x' \ S"
      by (auto simp: open_dist)
    then have "\z. dist z (x + y) < e \ \x\S. \y\T. z = x + y"
      by (metis \<open>y \<in> T\<close> diff_add_cancel dist_add_cancel2)
    then show "\e>0. \z. dist z (x + y) < e \ (\x\S. \y\T. z = x + y)"
      using \<open>0 < e\<close> \<open>x \<in> S\<close> by blast
  qed
next
  assume T: "open T"
  show ?thesis
  proof (clarsimp simp: open_dist)
    fix x y
    assume "x \ S" "y \ T"
    with T obtain e where "e > 0" and e: "\x'. dist x' y < e \ x' \ T"
      by (auto simp: open_dist)
    then have "\z. dist z (x + y) < e \ \x\S. \y\T. z = x + y"
      by (metis \<open>x \<in> S\<close> add_diff_cancel_left' add_diff_eq diff_diff_add dist_norm)
    then show "\e>0. \z. dist z (x + y) < e \ (\x\S. \y\T. z = x + y)"
      using \<open>0 < e\<close> \<open>y \<in> T\<close> by blast
  qed
qed

lemma image_orthogonal_transformation_ball:
  fixes f :: "'a::euclidean_space \ 'a"
  assumes "orthogonal_transformation f"
  shows "f ` ball x r = ball (f x) r"
proof (intro equalityI subsetI)
  fix y assume "y \ f ` ball x r"
  with assms show "y \ ball (f x) r"
    by (auto simp: orthogonal_transformation_isometry)
next
  fix y assume y: "y \ ball (f x) r"
  then obtain z where z: "y = f z"
    using assms orthogonal_transformation_surj by blast
  with y assms show "y \ f ` ball x r"
    by (auto simp: orthogonal_transformation_isometry)
qed

lemma image_orthogonal_transformation_cball:
  fixes f :: "'a::euclidean_space \ 'a"
  assumes "orthogonal_transformation f"
  shows "f ` cball x r = cball (f x) r"
proof (intro equalityI subsetI)
  fix y assume "y \ f ` cball x r"
  with assms show "y \ cball (f x) r"
    by (auto simp: orthogonal_transformation_isometry)
next
  fix y assume y: "y \ cball (f x) r"
  then obtain z where z: "y = f z"
    using assms orthogonal_transformation_surj by blast
  with y assms show "y \ f ` cball x r"
    by (auto simp: orthogonal_transformation_isometry)
qed

subsection \<open>Support\<close>

definition (in monoid_add) support_on :: "'b set \ ('b \ 'a) \ 'b set"
  where "support_on S f = {x\S. f x \ 0}"

lemma in_support_on: "x \ support_on S f \ x \ S \ f x \ 0"
  by (simp add: support_on_def)

lemma support_on_simps[simp]:
  "support_on {} f = {}"
  "support_on (insert x S) f =
    (if f x = 0 then support_on S f else insert x (support_on S f))"
  "support_on (S \ T) f = support_on S f \ support_on T f"
  "support_on (S \ T) f = support_on S f \ support_on T f"
  "support_on (S - T) f = support_on S f - support_on T f"
  "support_on (f ` S) g = f ` (support_on S (g \ f))"
  unfolding support_on_def by auto

lemma support_on_cong:
  "(\x. x \ S \ f x = 0 \ g x = 0) \ support_on S f = support_on S g"
  by (auto simp: support_on_def)

lemma support_on_if: "a \ 0 \ support_on A (\x. if P x then a else 0) = {x\A. P x}"
  by (auto simp: support_on_def)

lemma support_on_if_subset: "support_on A (\x. if P x then a else 0) \ {x \ A. P x}"
  by (auto simp: support_on_def)

lemma finite_support[intro]: "finite S \ finite (support_on S f)"
  unfolding support_on_def by auto

(* TODO: is supp_sum really needed? TODO: Generalize to Finite_Set.fold *)
definition (in comm_monoid_add) supp_sum :: "('b \ 'a) \ 'b set \ 'a"
  where "supp_sum f S = (\x\support_on S f. f x)"

lemma supp_sum_empty[simp]: "supp_sum f {} = 0"
  unfolding supp_sum_def by auto

lemma supp_sum_insert[simp]:
  "finite (support_on S f) \
    supp_sum f (insert x S) = (if x \<in> S then supp_sum f S else f x + supp_sum f S)"
  by (simp add: supp_sum_def in_support_on insert_absorb)

lemma supp_sum_divide_distrib: "supp_sum f A / (r::'a::field) = supp_sum (\n. f n / r) A"
  by (cases "r = 0")
     (auto simp: supp_sum_def sum_divide_distrib intro!: sum.cong support_on_cong)

subsection \<open>Intervals\<close>

lemma image_affinity_interval:
  fixes c :: "'a::ordered_real_vector"
  shows "((\x. m *\<^sub>R x + c) ` {a..b}) =
           (if {a..b}={} then {}
            else if 0 \<le> m then {m *\<^sub>R a + c .. m  *\<^sub>R b + c}
            else {m *\<^sub>R b + c .. m *\<^sub>R a + c})"
         (is "?lhs = ?rhs")
proof (cases "m=0")
  case True
  then show ?thesis
    by force
next
  case False
  show ?thesis
  proof
    show "?lhs \ ?rhs"
      by (auto simp: scaleR_left_mono scaleR_left_mono_neg)
    show "?rhs \ ?lhs"
    proof (clarsimp, intro conjI impI subsetI)
      show "\0 \ m; a \ b; x \ {m *\<^sub>R a + c..m *\<^sub>R b + c}\
            \<Longrightarrow> x \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}" for x
        using False
        by (rule_tac x="inverse m *\<^sub>R (x-c)" in image_eqI)
           (auto simp: pos_le_divideR_eq pos_divideR_le_eq le_diff_eq diff_le_eq)
      show "\\ 0 \ m; a \ b; x \ {m *\<^sub>R b + c..m *\<^sub>R a + c}\
            \<Longrightarrow> x \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}" for x
        by (rule_tac x="inverse m *\<^sub>R (x-c)" in image_eqI)
           (auto simp add: neg_le_divideR_eq neg_divideR_le_eq le_diff_eq diff_le_eq)
    qed
  qed
qed

subsection \<open>Limit Points\<close>

lemma islimpt_ball:
  fixes x y :: "'a::{real_normed_vector,perfect_space}"
  shows "y islimpt ball x e \ 0 < e \ y \ cball x e"
  (is "?lhs \ ?rhs")
proof
  show ?rhs if ?lhs
  proof
    {
      assume "e \ 0"
      then have *: "ball x e = {}"
        using ball_eq_empty[of x e] by auto
      have False using \<open>?lhs\<close>
        unfolding * using islimpt_EMPTY[of y] by auto
    }
    then show "e > 0" by (metis not_less)
    show "y \ cball x e"
      using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"]
        ball_subset_cball[of x e] \<open>?lhs\<close>
      unfolding closed_limpt by auto
  qed
  show ?lhs if ?rhs
  proof -
    from that have "e > 0" by auto
    {
      fix d :: real
      assume "d > 0"
      have "\x'\ball x e. x' \ y \ dist x' y < d"
      proof (cases "d \ dist x y")
        case True
        then show ?thesis
        proof (cases "x = y")
          case True
          then have False
            using \<open>d \<le> dist x y\<close> \<open>d>0\<close> by auto
          then show ?thesis
            by auto
        next
          case False
          have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) =
            norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
            unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[symmetric]
            by auto
          also have "\ = \- 1 + d / (2 * norm (x - y))\ * norm (x - y)"
            using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", symmetric, of "y - x"]
            unfolding scaleR_minus_left scaleR_one
            by (auto simp: norm_minus_commute)
          also have "\ = \- norm (x - y) + d / 2\"
            unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
            unfolding distrib_right using \<open>x\<noteq>y\<close>  by auto
          also have "\ \ e - d/2" using \d \ dist x y\ and \d>0\ and \?rhs\
            by (auto simp: dist_norm)
          finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \ ball x e" using \d>0\
            by auto
          moreover
          have "(d / (2*dist y x)) *\<^sub>R (y - x) \ 0"
            using \<open>x\<noteq>y\<close>[unfolded dist_nz] \<open>d>0\<close> unfolding scaleR_eq_0_iff
            by (auto simp: dist_commute)
          moreover
          have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d"
            using \<open>0 < d\<close> by (fastforce simp: dist_norm)
          ultimately show ?thesis
            by (rule_tac x = "y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI) auto
        qed
      next
        case False
        then have "d > dist x y" by auto
        show "\x' \ ball x e. x' \ y \ dist x' y < d"
        proof (cases "x = y")
          case True
          obtain z where z: "z \ y" "dist z y < min e d"
            using perfect_choose_dist[of "min e d" y]
            using \<open>d > 0\<close> \<open>e>0\<close> by auto
          show ?thesis
            by (metis True z dist_commute mem_ball min_less_iff_conj)
        next
          case False
          then show ?thesis
            using \<open>d>0\<close> \<open>d > dist x y\<close> \<open>?rhs\<close> by force
        qed
      qed
    }
    then show ?thesis
      unfolding mem_cball islimpt_approachable mem_ball by auto
  qed
qed

lemma closure_ball_lemma:
  fixes x y :: "'a::real_normed_vector"
  assumes "x \ y"
  shows "y islimpt ball x (dist x y)"
proof (rule islimptI)
  fix T
  assume "y \ T" "open T"
  then obtain r where "0 < r" "\z. dist z y < r \ z \ T"
    unfolding open_dist by fast
  \<comment>\<open>choose point between @{term x} and @{term y}, within distance @{term r} of @{term y}.\<close>
  define k where "k = min 1 (r / (2 * dist x y))"
  define z where "z = y + scaleR k (x - y)"
  have z_def2: "z = x + scaleR (1 - k) (y - x)"
    unfolding z_def by (simp add: algebra_simps)
  have "dist z y < r"
    unfolding z_def k_def using \<open>0 < r\<close>
    by (simp add: dist_norm min_def)
  then have "z \ T"
    using \<open>\<forall>z. dist z y < r \<longrightarrow> z \<in> T\<close> by simp
  have "dist x z < dist x y"
    using \<open>0 < r\<close> assms by (simp add: z_def2 k_def dist_norm norm_minus_commute)
  then have "z \ ball x (dist x y)"
    by simp
  have "z \ y"
    unfolding z_def k_def using \<open>x \<noteq> y\<close> \<open>0 < r\<close>
    by (simp add: min_def)
  show "\z\ball x (dist x y). z \ T \ z \ y"
    using \<open>z \<in> ball x (dist x y)\<close> \<open>z \<in> T\<close> \<open>z \<noteq> y\<close>
    by fast
qed

subsection \<open>Balls and Spheres in Normed Spaces\<close>

lemma mem_ball_0 [simp]: "x \ ball 0 e \ norm x < e"
  for x :: "'a::real_normed_vector"
  by simp

lemma mem_cball_0 [simp]: "x \ cball 0 e \ norm x \ e"
  for x :: "'a::real_normed_vector"
  by simp

lemma closure_ball [simp]:
  fixes x :: "'a::real_normed_vector"
  assumes "0 < e"
  shows "closure (ball x e) = cball x e"
proof
  show "closure (ball x e) \ cball x e"
    using closed_cball closure_minimal by blast
  have "\y. dist x y < e \ dist x y = e \ y \ closure (ball x e)"
    by (metis Un_iff assms closure_ball_lemma closure_def dist_eq_0_iff mem_Collect_eq mem_ball)
  then show "cball x e \ closure (ball x e)"
    by force
qed

lemma mem_sphere_0 [simp]: "x \ sphere 0 e \ norm x = e"
  for x :: "'a::real_normed_vector"
  by simp

(* In a trivial vector space, this fails for e = 0. *)
lemma interior_cball [simp]:
  fixes x :: "'a::{real_normed_vector, perfect_space}"
  shows "interior (cball x e) = ball x e"
proof (cases "e \ 0")
  case False note cs = this
  from cs have null: "ball x e = {}"
    using ball_empty[of e x] by auto
  moreover
  have "cball x e = {}"
  proof (rule equals0I)
    fix y
    assume "y \ cball x e"
    then show False
      by (metis ball_eq_empty null cs dist_eq_0_iff dist_le_zero_iff empty_subsetI mem_cball
          subset_antisym subset_ball)
  qed
  then have "interior (cball x e) = {}"
    using interior_empty by auto
  ultimately show ?thesis by blast
next
  case True note cs = this
  have "ball x e \ cball x e"
    using ball_subset_cball by auto
  moreover
  {
    fix S y
    assume as: "S \ cball x e" "open S" "y\S"
    then obtain d where "d>0" and d: "\x'. dist x' y < d \ x' \ S"
      unfolding open_dist by blast
    then obtain xa where xa_y: "xa \ y" and xa: "dist xa y < d"
      using perfect_choose_dist [of d] by auto
    have "xa \ S"
      using d[THEN spec[where x = xa]]
      using xa by (auto simp: dist_commute)
    then have xa_cball: "xa \ cball x e"
      using as(1) by auto
    then have "y \ ball x e"
    proof (cases "x = y")
      case True
      then have "e > 0" using cs order.order_iff_strict xa_cball xa_y by fastforce
      then show "y \ ball x e"
        using \<open>x = y \<close> by simp
    next
      case False
      have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d"
        unfolding dist_norm
        using \<open>d>0\<close> norm_ge_zero[of "y - x"] \<open>x \<noteq> y\<close> by auto
      then have *: "y + (d / 2 / dist y x) *\<^sub>R (y - x) \ cball x e"
        using d as(1)[unfolded subset_eq] by blast
      have "y - x \ 0" using \x \ y\ by auto
      hence **:"d / (2 * norm (y - x)) > 0"
        unfolding zero_less_norm_iff[symmetric] using \<open>d>0\<close> by auto
      have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x =
        norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"
        by (auto simp: dist_norm algebra_simps)
      also have "\ = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
        by (auto simp: algebra_simps)
      also have "\ = \1 + d / (2 * norm (y - x))\ * norm (y - x)"
        using ** by auto
      also have "\ = (dist y x) + d/2"
        using ** by (auto simp: distrib_right dist_norm)
      finally have "e \ dist x y +d/2"
        using *[unfolded mem_cball] by (auto simp: dist_commute)
      then show "y \ ball x e"
        unfolding mem_ball using \<open>d>0\<close> by auto
    qed
  }
  then have "\S \ cball x e. open S \ S \ ball x e"
    by auto
  ultimately show ?thesis
    using interior_unique[of "ball x e" "cball x e"]
    using open_ball[of x e]
    by auto
qed

lemma frontier_ball [simp]:
  fixes a :: "'a::real_normed_vector"
  shows "0 < e \ frontier (ball a e) = sphere a e"
  by (force simp: frontier_def)

lemma frontier_cball [simp]:
  fixes a :: "'a::{real_normed_vector, perfect_space}"
  shows "frontier (cball a e) = sphere a e"
  by (force simp: frontier_def)

corollary compact_sphere [simp]:
  fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
  shows "compact (sphere a r)"
using compact_frontier [of "cball a r"] by simp

corollary bounded_sphere [simp]:
  fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
  shows "bounded (sphere a r)"
by (simp add: compact_imp_bounded)

corollary closed_sphere  [simp]:
  fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
  shows "closed (sphere a r)"
by (simp add: compact_imp_closed)

lemma image_add_ball [simp]:
  fixes a :: "'a::real_normed_vector"
  shows "(+) b ` ball a r = ball (a+b) r"
proof -
  { fix x :: 'a
    assume "dist (a + b) x < r"
    moreover
    have "b + (x - b) = x"
      by simp
    ultimately have "x \ (+) b ` ball a r"
      by (metis add.commute dist_add_cancel image_eqI mem_ball) }
  then show ?thesis
    by (auto simp: add.commute)
qed

lemma image_add_cball [simp]:
  fixes a :: "'a::real_normed_vector"
  shows "(+) b ` cball a r = cball (a+b) r"
proof -
  have "\x. dist (a + b) x \ r \ \y\cball a r. x = b + y"
    by (metis (no_types) add.commute diff_add_cancel dist_add_cancel2 mem_cball)
  then show ?thesis
    by (force simp: add.commute)
qed

subsection\<^marker>\<open>tag unimportant\<close> \<open>Various Lemmas on Normed Algebras\<close>

lemma closed_of_nat_image: "closed (of_nat ` A :: 'a::real_normed_algebra_1 set)"
  by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_nat)

lemma closed_of_int_image: "closed (of_int ` A :: 'a::real_normed_algebra_1 set)"
  by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_int)

lemma closed_Nats [simp]: "closed (\ :: 'a :: real_normed_algebra_1 set)"
  unfolding Nats_def by (rule closed_of_nat_image)

lemma closed_Ints [simp]: "closed (\ :: 'a :: real_normed_algebra_1 set)"
  unfolding Ints_def by (rule closed_of_int_image)

lemma closed_subset_Ints:
  fixes A :: "'a :: real_normed_algebra_1 set"
  assumes "A \ \"
  shows   "closed A"
proof (intro discrete_imp_closed[OF zero_less_one] ballI impI, goal_cases)
  case (1 x y)
  with assms have "x \ \" and "y \ \" by auto
  with \<open>dist y x < 1\<close> show "y = x"
    by (auto elim!: Ints_cases simp: dist_of_int)
qed

subsection \<open>Filters\<close>

definition indirection :: "'a::real_normed_vector \ 'a \ 'a filter" (infixr "indirection" 70)
  where "a indirection v = at a within {b. \c\0. b - a = scaleR c v}"

subsection \<open>Trivial Limits\<close>

lemma trivial_limit_at_infinity:
  "\ trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
proof -
  obtain x::'a where "x \ 0"
    by (meson perfect_choose_dist zero_less_one)
  then have "b \ norm ((b / norm x) *\<^sub>R x)" for b
    by simp
  then show ?thesis
    unfolding trivial_limit_def eventually_at_infinity
    by blast
qed

lemma at_within_ball_bot_iff:
  fixes x y :: "'a::{real_normed_vector,perfect_space}"
  shows "at x within ball y r = bot \ (r=0 \ x \ cball y r)"
  unfolding trivial_limit_within
  by (metis (no_types) cball_empty equals0D islimpt_ball less_linear)

subsection \<open>Limits\<close>

proposition Lim_at_infinity: "(f \ l) at_infinity \ (\e>0. \b. \x. norm x \ b \ dist (f x) l < e)"
  by (auto simp: tendsto_iff eventually_at_infinity)

corollary Lim_at_infinityI [intro?]:
  assumes "\e. e > 0 \ \B. \x. norm x \ B \ dist (f x) l \ e"
  shows "(f \ l) at_infinity"
proof -
  have "\e. e > 0 \ \B. \x. norm x \ B \ dist (f x) l < e"
    by (meson assms dense le_less_trans)
  then show ?thesis
    using Lim_at_infinity by blast
qed

lemma Lim_transform_within_set_eq:
  fixes a :: "'a::metric_space" and l :: "'b::metric_space"
  shows "eventually (\x. x \ S \ x \ T) (at a)
         \<Longrightarrow> ((f \<longlongrightarrow> l) (at a within S) \<longleftrightarrow> (f \<longlongrightarrow> l) (at a within T))"
  by (force intro: Lim_transform_within_set elim: eventually_mono)

lemma Lim_null:
  fixes f :: "'a \ 'b::real_normed_vector"
  shows "(f \ l) net \ ((\x. f(x) - l) \ 0) net"
  by (simp add: Lim dist_norm)

lemma Lim_null_comparison:
  fixes f :: "'a \ 'b::real_normed_vector"
  assumes "eventually (\x. norm (f x) \ g x) net" "(g \ 0) net"
  shows "(f \ 0) net"
  using assms(2)
proof (rule metric_tendsto_imp_tendsto)
  show "eventually (\x. dist (f x) 0 \ dist (g x) 0) net"
    using assms(1) by (rule eventually_mono) (simp add: dist_norm)
qed

lemma Lim_transform_bound:
  fixes f :: "'a \ 'b::real_normed_vector"
    and g :: "'a \ 'c::real_normed_vector"
  assumes "eventually (\n. norm (f n) \ norm (g n)) net"
    and "(g \ 0) net"
  shows "(f \ 0) net"
  using assms(1) tendsto_norm_zero [OF assms(2)]
  by (rule Lim_null_comparison)

lemma lim_null_mult_right_bounded:
  fixes f :: "'a \ 'b::real_normed_div_algebra"
  assumes f: "(f \ 0) F" and g: "eventually (\x. norm(g x) \ B) F"
    shows "((\z. f z * g z) \ 0) F"
proof -
  have "((\x. norm (f x) * norm (g x)) \ 0) F"
  proof (rule Lim_null_comparison)
    show "\\<^sub>F x in F. norm (norm (f x) * norm (g x)) \ norm (f x) * B"
      by (simp add: eventually_mono [OF g] mult_left_mono)
    show "((\x. norm (f x) * B) \ 0) F"
      by (simp add: f tendsto_mult_left_zero tendsto_norm_zero)
  qed
  then show ?thesis
    by (subst tendsto_norm_zero_iff [symmetric]) (simp add: norm_mult)
qed

lemma lim_null_mult_left_bounded:
  fixes f :: "'a \ 'b::real_normed_div_algebra"
  assumes g: "eventually (\x. norm(g x) \ B) F" and f: "(f \ 0) F"
    shows "((\z. g z * f z) \ 0) F"
proof -
  have "((\x. norm (g x) * norm (f x)) \ 0) F"
  proof (rule Lim_null_comparison)
    show "\\<^sub>F x in F. norm (norm (g x) * norm (f x)) \ B * norm (f x)"
      by (simp add: eventually_mono [OF g] mult_right_mono)
    show "((\x. B * norm (f x)) \ 0) F"
      by (simp add: f tendsto_mult_right_zero tendsto_norm_zero)
  qed
  then show ?thesis
    by (subst tendsto_norm_zero_iff [symmetric]) (simp add: norm_mult)
qed

lemma lim_null_scaleR_bounded:
  assumes f: "(f \ 0) net" and gB: "eventually (\a. f a = 0 \ norm(g a) \ B) net"
    shows "((\n. f n *\<^sub>R g n) \ 0) net"
proof
  fix \<epsilon>::real
  assume "0 < \"
  then have B: "0 < \ / (abs B + 1)" by simp
  have *: "\f x\ * norm (g x) < \" if f: "\f x\ * (\B\ + 1) < \" and g: "norm (g x) \ B" for x
  proof -
    have "\f x\ * norm (g x) \ \f x\ * B"
      by (simp add: mult_left_mono g)
    also have "\ \ \f x\ * (\B\ + 1)"
      by (simp add: mult_left_mono)
    also have "\ < \"
      by (rule f)
    finally show ?thesis .
  qed
  have "\x. \\f x\ < \ / (\B\ + 1); norm (g x) \ B\ \ \f x\ * norm (g x) < \"
    by (simp add: "*" pos_less_divide_eq)
  then show "\\<^sub>F x in net. dist (f x *\<^sub>R g x) 0 < \"
    using \<open>0 < \<epsilon>\<close> by (auto intro: eventually_mono [OF eventually_conj [OF tendstoD [OF f B] gB]])
qed

lemma Lim_norm_ubound:
  fixes f :: "'a \ 'b::real_normed_vector"
  assumes "\(trivial_limit net)" "(f \ l) net" "eventually (\x. norm(f x) \ e) net"
  shows "norm(l) \ e"
  using assms by (fast intro: tendsto_le tendsto_intros)

lemma Lim_norm_lbound:
  fixes f :: "'a \ 'b::real_normed_vector"
  assumes "\ trivial_limit net"
    and "(f \ l) net"
    and "eventually (\x. e \ norm (f x)) net"
  shows "e \ norm l"
  using assms by (fast intro: tendsto_le tendsto_intros)

text\<open>Limit under bilinear function\<close>

lemma Lim_bilinear:
  assumes "(f \ l) net"
    and "(g \ m) net"
    and "bounded_bilinear h"
  shows "((\x. h (f x) (g x)) \ (h l m)) net"
  using \<open>bounded_bilinear h\<close> \<open>(f \<longlongrightarrow> l) net\<close> \<open>(g \<longlongrightarrow> m) net\<close>
  by (rule bounded_bilinear.tendsto)

lemma Lim_at_zero:
  fixes a :: "'a::real_normed_vector"
    and l :: "'b::topological_space"
  shows "(f \ l) (at a) \ ((\x. f(a + x)) \ l) (at 0)"
  using LIM_offset_zero LIM_offset_zero_cancel ..

subsection\<^marker>\<open>tag unimportant\<close> \<open>Limit Point of Filter\<close>

lemma netlimit_at_vector:
  fixes a :: "'a::real_normed_vector"
  shows "netlimit (at a) = a"
proof (cases "\x. x \ a")
  case True then obtain x where x: "x \ a" ..
  have "\d. 0 < d \ \x. x \ a \ norm (x - a) < d"
    by (rule_tac x="a + scaleR (d / 2) (sgn (x - a))" in exI) (simp add: norm_sgn sgn_zero_iff x)
  then have "\ trivial_limit (at a)"
    by (auto simp: trivial_limit_def eventually_at dist_norm)
  then show ?thesis
    by (rule Lim_ident_at [of a UNIV])
qed simp

subsection \<open>Boundedness\<close>

lemma continuous_on_closure_norm_le:
  fixes f :: "'a::metric_space \ 'b::real_normed_vector"
  assumes "continuous_on (closure s) f"
    and "\y \ s. norm(f y) \ b"
    and "x \ (closure s)"
  shows "norm (f x) \ b"
proof -
  have *: "f ` s \ cball 0 b"
    using assms(2)[unfolded mem_cball_0[symmetric]] by auto
  show ?thesis
    by (meson "*" assms(1) assms(3) closed_cball image_closure_subset image_subset_iff mem_cball_0)
qed

lemma bounded_pos: "bounded S \ (\b>0. \x\ S. norm x \ b)"
  unfolding bounded_iff
  by (meson less_imp_le not_le order_trans zero_less_one)

lemma bounded_pos_less: "bounded S \ (\b>0. \x\ S. norm x < b)"
  by (metis bounded_pos le_less_trans less_imp_le linordered_field_no_ub)

lemma Bseq_eq_bounded:
  fixes f :: "nat \ 'a::real_normed_vector"
  shows "Bseq f \ bounded (range f)"
  unfolding Bseq_def bounded_pos by auto

lemma bounded_linear_image:
  assumes "bounded S"
    and "bounded_linear f"
  shows "bounded (f ` S)"
proof -
  from assms(1) obtain b where "b > 0" and b: "\x\S. norm x \ b"
    unfolding bounded_pos by auto
  from assms(2) obtain B where B: "B > 0" "\x. norm (f x) \ B * norm x"
    using bounded_linear.pos_bounded by (auto simp: ac_simps)
  show ?thesis
    unfolding bounded_pos
  proof (intro exI, safe)
    show "norm (f x) \ B * b" if "x \ S" for x
      by (meson B b less_imp_le mult_left_mono order_trans that)
  qed (use \<open>b > 0\<close> \<open>B > 0\<close> in auto)
qed

lemma bounded_scaling:
  fixes S :: "'a::real_normed_vector set"
  shows "bounded S \ bounded ((\x. c *\<^sub>R x) ` S)"
  by (simp add: bounded_linear_image bounded_linear_scaleR_right)

lemma bounded_scaleR_comp:
  fixes f :: "'a \ 'b::real_normed_vector"
  assumes "bounded (f ` S)"
  shows "bounded ((\x. r *\<^sub>R f x) ` S)"
  using bounded_scaling[of "f ` S" r] assms
  by (auto simp: image_image)

lemma bounded_translation:
  fixes S :: "'a::real_normed_vector set"
  assumes "bounded S"
  shows "bounded ((\x. a + x) ` S)"
proof -
  from assms obtain b where b: "b > 0" "\x\S. norm x \ b"
    unfolding bounded_pos by auto
  {
    fix x
    assume "x \ S"
    then have "norm (a + x) \ b + norm a"
      using norm_triangle_ineq[of a x] b by auto
  }
  then show ?thesis
    unfolding bounded_pos
    using norm_ge_zero[of a] b(1) and add_strict_increasing[of b 0 "norm a"]
    by (auto intro!: exI[of _ "b + norm a"])
qed

lemma bounded_translation_minus:
  fixes S :: "'a::real_normed_vector set"
  shows "bounded S \ bounded ((\x. x - a) ` S)"
using bounded_translation [of S "-a"] by simp

lemma bounded_uminus [simp]:
  fixes X :: "'a::real_normed_vector set"
  shows "bounded (uminus ` X) \ bounded X"
by (auto simp: bounded_def dist_norm; rule_tac x="-x" in exI; force simp: add.commute norm_minus_commute)

lemma uminus_bounded_comp [simp]:
  fixes f :: "'a \ 'b::real_normed_vector"
  shows "bounded ((\x. - f x) ` S) \ bounded (f ` S)"
  using bounded_uminus[of "f ` S"]
  by (auto simp: image_image)

lemma bounded_plus_comp:
  fixes f g::"'a \ 'b::real_normed_vector"
  assumes "bounded (f ` S)"
  assumes "bounded (g ` S)"
  shows "bounded ((\x. f x + g x) ` S)"
proof -
  {
    fix B C
    assume "\x. x\S \ norm (f x) \ B" "\x. x\S \ norm (g x) \ C"
    then have "\x. x \ S \ norm (f x + g x) \ B + C"
      by (auto intro!: norm_triangle_le add_mono)
  } then show ?thesis
    using assms by (fastforce simp: bounded_iff)
qed

lemma bounded_plus:
  fixes S ::"'a::real_normed_vector set"
  assumes "bounded S" "bounded T"
  shows "bounded ((\(x,y). x + y) ` (S \ T))"
  using bounded_plus_comp [of fst "S \ T" snd] assms
  by (auto simp: split_def split: if_split_asm)

lemma bounded_minus_comp:
  "bounded (f ` S) \ bounded (g ` S) \ bounded ((\x. f x - g x) ` S)"
  for f g::"'a \ 'b::real_normed_vector"
  using bounded_plus_comp[of "f" S "\x. - g x"]
  by auto

lemma bounded_minus:
  fixes S ::"'a::real_normed_vector set"
  assumes "bounded S" "bounded T"
  shows "bounded ((\(x,y). x - y) ` (S \ T))"
  using bounded_minus_comp [of fst "S \ T" snd] assms
  by (auto simp: split_def split: if_split_asm)

lemma not_bounded_UNIV[simp]:
  "\ bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
proof (auto simp: bounded_pos not_le)
  obtain x :: 'a where "x \ 0"
    using perfect_choose_dist [OF zero_less_one] by fast
  fix b :: real
  assume b: "b >0"
  have b1: "b +1 \ 0"
    using b by simp
  with \<open>x \<noteq> 0\<close> have "b < norm (scaleR (b + 1) (sgn x))"
    by (simp add: norm_sgn)
  then show "\x::'a. b < norm x" ..
qed

corollary cobounded_imp_unbounded:
    fixes S :: "'a::{real_normed_vector, perfect_space} set"
    shows "bounded (- S) \ \ bounded S"
  using bounded_Un [of S "-S"]  by (simp)

subsection\<^marker>\<open>tag unimportant\<close>\<open>Relations among convergence and absolute convergence for power series\<close>

lemma summable_imp_bounded:
  fixes f :: "nat \ 'a::real_normed_vector"
  shows "summable f \ bounded (range f)"
by (frule summable_LIMSEQ_zero) (simp add: convergent_imp_bounded)

lemma summable_imp_sums_bounded:
   "summable f \ bounded (range (\n. sum f {..
by (auto simp: summable_def sums_def dest: convergent_imp_bounded)

lemma power_series_conv_imp_absconv_weak:
  fixes a:: "nat \ 'a::{real_normed_div_algebra,banach}" and w :: 'a
  assumes sum: "summable (\n. a n * z ^ n)" and no: "norm w < norm z"
    shows "summable (\n. of_real(norm(a n)) * w ^ n)"
proof -
  obtain M where M: "\x. norm (a x * z ^ x) \ M"
    using summable_imp_bounded [OF sum] by (force simp: bounded_iff)
  show ?thesis
  proof (rule series_comparison_complex)
    have "\n. norm (a n) * norm z ^ n \ M"
      by (metis (no_types) M norm_mult norm_power)
    then show "summable (\n. complex_of_real (norm (a n) * norm w ^ n))"
      using Abel_lemma no norm_ge_zero summable_of_real by blast
  qed (auto simp: norm_mult norm_power)
qed

subsection \<open>Normed spaces with the Heine-Borel property\<close>

lemma not_compact_UNIV[simp]:
  fixes s :: "'a::{real_normed_vector,perfect_space,heine_borel} set"
  shows "\ compact (UNIV::'a set)"
    by (simp add: compact_eq_bounded_closed)

lemma not_compact_space_euclideanreal [simp]: "\ compact_space euclideanreal"
  by (simp add: compact_space_def)

text\<open>Representing sets as the union of a chain of compact sets.\<close>
lemma closed_Union_compact_subsets:
  fixes S :: "'a::{heine_borel,real_normed_vector} set"
  assumes "closed S"
  obtains F where "\n. compact(F n)" "\n. F n \ S" "\n. F n \ F(Suc n)"
                  "(\n. F n) = S" "\K. \compact K; K \ S\ \ \N. \n \ N. K \ F n"
proof
  show "compact (S \ cball 0 (of_nat n))" for n
    using assms compact_eq_bounded_closed by auto
next
  show "(\n. S \ cball 0 (real n)) = S"
    by (auto simp: real_arch_simple)
next
  fix K :: "'a set"
  assume "compact K" "K \ S"
  then obtain N where "K \ cball 0 N"
    by (meson bounded_pos mem_cball_0 compact_imp_bounded subsetI)
  then show "\N. \n\N. K \ S \ cball 0 (real n)"
    by (metis of_nat_le_iff Int_subset_iff \<open>K \<subseteq> S\<close> real_arch_simple subset_cball subset_trans)
qed auto

subsection \<open>Intersecting chains of compact sets and the Baire property\<close>

proposition bounded_closed_chain:
  fixes \<F> :: "'a::heine_borel set set"
  assumes "B \ \" "bounded B" and \: "\S. S \ \ \ closed S" and "{} \ \"
      and chain: "\S T. S \ \ \ T \ \ \ S \ T \ T \ S"
    shows "\\ \ {}"
proof -
  have "B \ \\ \ {}"
  proof (rule compact_imp_fip)
    show "compact B" "\T. T \ \ \ closed T"
      by (simp_all add: assms compact_eq_bounded_closed)
    show "\finite \; \ \ \\ \ B \ \\ \ {}" for \
    proof (induction \<G> rule: finite_induct)
      case empty
      with assms show ?case by force
    next
      case (insert U \<G>)
      then have "U \ \" and ne: "B \ \\ \ {}" by auto
      then consider "B \ U" | "U \ B"
          using \<open>B \<in> \<F>\<close> chain by blast
        then show ?case
        proof cases
          case 1
          then show ?thesis
            using Int_left_commute ne by auto
        next
          case 2
          have "U \ {}"
            using \<open>U \<in> \<F>\<close> \<open>{} \<notin> \<F>\<close> by blast
          moreover
          have False if "\x. x \ U \ \Y\\. x \ Y"
          proof -
            have "\x. x \ U \ \Y\\. Y \ U"
              by (metis chain contra_subsetD insert.prems insert_subset that)
            then obtain Y where "Y \ \" "Y \ U"
              by (metis all_not_in_conv \<open>U \<noteq> {}\<close>)
            moreover obtain x where "x \ \\"
              by (metis Int_emptyI ne)
            ultimately show ?thesis
              by (metis Inf_lower subset_eq that)
          qed
          with 2 show ?thesis
            by blast
        qed
      qed
  qed
  then show ?thesis by blast
qed

corollary compact_chain:
  fixes \<F> :: "'a::heine_borel set set"
  assumes "\S. S \ \ \ compact S" "{} \ \"
          "\S T. S \ \ \ T \ \ \ S \ T \ T \ S"
    shows "\ \ \ {}"
proof (cases "\ = {}")
  case True
  then show ?thesis by auto
next
  case False
  show ?thesis
    by (metis False all_not_in_conv assms compact_imp_bounded compact_imp_closed bounded_closed_chain)
qed

lemma compact_nest:
  fixes F :: "'a::linorder \ 'b::heine_borel set"
  assumes F: "\n. compact(F n)" "\n. F n \ {}" and mono: "\m n. m \ n \ F n \ F m"
  shows "\(range F) \ {}"
proof -
  have *: "\S T. S \ range F \ T \ range F \ S \ T \ T \ S"
    by (metis mono image_iff le_cases)
  show ?thesis
    using F by (intro compact_chain [OF _ _ *]; blast dest: *)
qed

text\<open>The Baire property of dense sets\<close>
theorem Baire:
  fixes S::"'a::{real_normed_vector,heine_borel} set"
  assumes "closed S" "countable \"
      and ope: "\T. T \ \ \ openin (top_of_set S) T \ S \ closure T"
shows "S \ closure(\\)"
proof (cases "\ = {}")
  case True
  then show ?thesis
    using closure_subset by auto
next
  let ?g = "from_nat_into \"
  case False
  then have gin: "?g n \ \" for n
    by (simp add: from_nat_into)
  show ?thesis
  proof (clarsimp simp: closure_approachable)
    fix x and e::real
    assume "x \ S" "0 < e"
    obtain TF where opeF: "\n. openin (top_of_set S) (TF n)"
               and ne: "\n. TF n \ {}"
               and subg: "\n. S \ closure(TF n) \ ?g n"
               and subball: "\n. closure(TF n) \ ball x e"
               and decr: "\n. TF(Suc n) \ TF n"
    proof -
      have *: "\Y. (openin (top_of_set S) Y \ Y \ {} \
                   S \<inter> closure Y \<subseteq> ?g n \<and> closure Y \<subseteq> ball x e) \<and> Y \<subseteq> U"
        if opeU: "openin (top_of_set S) U" and "U \ {}" and cloU: "closure U \ ball x e" for U n
      proof -
        obtain T where T: "open T" "U = T \ S"
          using \<open>openin (top_of_set S) U\<close> by (auto simp: openin_subtopology)
        with \<open>U \<noteq> {}\<close> have "T \<inter> closure (?g n) \<noteq> {}"
          using gin ope by fastforce
        then have "T \ ?g n \ {}"
          using \<open>open T\<close> open_Int_closure_eq_empty by blast
        then obtain y where "y \ U" "y \ ?g n"
          using T ope [of "?g n", OF gin] by (blast dest:  openin_imp_subset)
        moreover have "openin (top_of_set S) (U \ ?g n)"
          using gin ope opeU by blast
        ultimately obtain d where U: "U \ ?g n \ S" and "d > 0" and d: "ball y d \ S \ U \ ?g n"
          by (force simp: openin_contains_ball)
        show ?thesis
        proof (intro exI conjI)
          show "openin (top_of_set S) (S \ ball y (d/2))"
            by (simp add: openin_open_Int)
          show "S \ ball y (d/2) \ {}"
            using \<open>0 < d\<close> \<open>y \<in> U\<close> opeU openin_imp_subset by fastforce
          have "S \ closure (S \ ball y (d/2)) \ S \ closure (ball y (d/2))"
            using closure_mono by blast
          also have "... \ ?g n"
            using \<open>d > 0\<close> d by force
          finally show "S \ closure (S \ ball y (d/2)) \ ?g n" .
          have "closure (S \ ball y (d/2)) \ S \ ball y d"
          proof -
            have "closure (ball y (d/2)) \ ball y d"
              using \<open>d > 0\<close> by auto
            then have "closure (S \ ball y (d/2)) \ ball y d"
              by (meson closure_mono inf.cobounded2 subset_trans)
            then show ?thesis
              by (simp add: \<open>closed S\<close> closure_minimal)
          qed
          also have "... \ ball x e"
            using cloU closure_subset d by blast
          finally show "closure (S \ ball y (d/2)) \ ball x e" .
          show "S \ ball y (d/2) \ U"
            using ball_divide_subset_numeral d by blast
        qed
      qed
      let ?\<Phi> = "\<lambda>n X. openin (top_of_set S) X \<and> X \<noteq> {} \<and>
                      S \<inter> closure X \<subseteq> ?g n \<and> closure X \<subseteq> ball x e"
      have "closure (S \ ball x (e/2)) \ closure(ball x (e/2))"
        by (simp add: closure_mono)
      also have "... \ ball x e"
        using \<open>e > 0\<close> by auto
      finally have "closure (S \ ball x (e/2)) \ ball x e" .
      moreover have"openin (top_of_set S) (S \ ball x (e/2))" "S \ ball x (e/2) \ {}"
        using \<open>0 < e\<close> \<open>x \<in> S\<close> by auto
      ultimately obtain Y where Y: "?\ 0 Y \ Y \ S \ ball x (e/2)"
            using * [of "S \ ball x (e/2)" 0] by metis
      show thesis
      proof (rule exE [OF dependent_nat_choice])
        show "\x. ?\ 0 x"
          using Y by auto
        show "\Y. ?\ (Suc n) Y \ Y \ X" if "?\ n X" for X n
          using that by (blast intro: *)
      qed (use that in metis)
    qed
    have "(\n. S \ closure (TF n)) \ {}"
    proof (rule compact_nest)
      show "\n. compact (S \ closure (TF n))"
        by (metis closed_closure subball bounded_subset_ballI compact_eq_bounded_closed closed_Int_compact [OF \<open>closed S\<close>])
      show "\n. S \ closure (TF n) \ {}"
        by (metis Int_absorb1 opeF \<open>closed S\<close> closure_eq_empty closure_minimal ne openin_imp_subset)
      show "\m n. m \ n \ S \ closure (TF n) \ S \ closure (TF m)"
        by (meson closure_mono decr dual_order.refl inf_mono lift_Suc_antimono_le)
    qed
    moreover have "(\n. S \ closure (TF n)) \ {y \ \\. dist y x < e}"
    proof (clarsimp, intro conjI)
      fix y
      assume "y \ S" and y: "\n. y \ closure (TF n)"
      then show "\T\\. y \ T"
        by (metis Int_iff from_nat_into_surj [OF \<open>countable \<G>\<close>] subsetD subg)
      show "dist y x < e"
        by (metis y dist_commute mem_ball subball subsetCE)
    qed
    ultimately show "\y \ \\. dist y x < e"
      by auto
  qed
qed

subsection \<open>Continuity\<close>

subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Structural rules for uniform continuity\<close>

lemma (in bounded_linear) uniformly_continuous_on[continuous_intros]:
  fixes g :: "_::metric_space \ _"
  assumes "uniformly_continuous_on s g"
  shows "uniformly_continuous_on s (\x. f (g x))"
  using assms unfolding uniformly_continuous_on_sequentially
  unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]
  by (auto intro: tendsto_zero)

lemma uniformly_continuous_on_dist[continuous_intros]:
  fixes f g :: "'a::metric_space \ 'b::metric_space"
  assumes "uniformly_continuous_on s f"
    and "uniformly_continuous_on s g"
  shows "uniformly_continuous_on s (\x. dist (f x) (g x))"
proof -
  {
    fix a b c d :: 'b
    have "\dist a b - dist c d\ \ dist a c + dist b d"
      using dist_triangle2 [of a b c] dist_triangle2 [of b c d]
      using dist_triangle3 [of c d a] dist_triangle [of a d b]
      by arith
  } note le = this
  {
    fix x y
    assume f: "(\n. dist (f (x n)) (f (y n))) \ 0"
    assume g: "(\n. dist (g (x n)) (g (y n))) \ 0"
    have "(\n. \dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\) \ 0"
      by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],
        simp add: le)
  }
  then show ?thesis
    using assms unfolding uniformly_continuous_on_sequentially
    unfolding dist_real_def by simp
qed

lemma uniformly_continuous_on_cmul_right [continuous_intros]:
  fixes f :: "'a::real_normed_vector \ 'b::real_normed_algebra"
  shows "uniformly_continuous_on s f \ uniformly_continuous_on s (\x. f x * c)"
  using bounded_linear.uniformly_continuous_on[OF bounded_linear_mult_left] .

lemma uniformly_continuous_on_cmul_left[continuous_intros]:
  fixes f :: "'a::real_normed_vector \ 'b::real_normed_algebra"
  assumes "uniformly_continuous_on s f"
    shows "uniformly_continuous_on s (\x. c * f x)"
by (metis assms bounded_linear.uniformly_continuous_on bounded_linear_mult_right)

lemma uniformly_continuous_on_norm[continuous_intros]:
  fixes f :: "'a :: metric_space \ 'b :: real_normed_vector"
  assumes "uniformly_continuous_on s f"
  shows "uniformly_continuous_on s (\x. norm (f x))"
  unfolding norm_conv_dist using assms
  by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)

lemma uniformly_continuous_on_cmul[continuous_intros]:
  fixes f :: "'a::metric_space \ 'b::real_normed_vector"
  assumes "uniformly_continuous_on s f"
  shows "uniformly_continuous_on s (\x. c *\<^sub>R f(x))"
  using bounded_linear_scaleR_right assms
  by (rule bounded_linear.uniformly_continuous_on)

lemma dist_minus:
  fixes x y :: "'a::real_normed_vector"
  shows "dist (- x) (- y) = dist x y"
  unfolding dist_norm minus_diff_minus norm_minus_cancel ..

lemma uniformly_continuous_on_minus[continuous_intros]:
  fixes f :: "'a::metric_space \ 'b::real_normed_vector"
  shows "uniformly_continuous_on s f \ uniformly_continuous_on s (\x. - f x)"
  unfolding uniformly_continuous_on_def dist_minus .

lemma uniformly_continuous_on_add[continuous_intros]:
  fixes f g :: "'a::metric_space \ 'b::real_normed_vector"
  assumes "uniformly_continuous_on s f"
    and "uniformly_continuous_on s g"
  shows "uniformly_continuous_on s (\x. f x + g x)"
  using assms
  unfolding uniformly_continuous_on_sequentially
  unfolding dist_norm tendsto_norm_zero_iff add_diff_add
  by (auto intro: tendsto_add_zero)

lemma uniformly_continuous_on_diff[continuous_intros]:
  fixes f :: "'a::metric_space \ 'b::real_normed_vector"
  assumes "uniformly_continuous_on s f"
    and "uniformly_continuous_on s g"
  shows "uniformly_continuous_on s (\x. f x - g x)"
  using assms uniformly_continuous_on_add [of s f "- g"]
    by (simp add: fun_Compl_def uniformly_continuous_on_minus)

subsection\<^marker>\<open>tag unimportant\<close> \<open>Arithmetic Preserves Topological Properties\<close>

lemma open_scaling[intro]:
  fixes s :: "'a::real_normed_vector set"
  assumes "c \ 0"
    and "open s"
  shows "open((\x. c *\<^sub>R x) ` s)"
proof -
  {
    fix x
    assume "x \ s"
    then obtain e where "e>0"
      and e:"\x'. dist x' x < e \ x' \ s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]]
      by auto
    have "e * \c\ > 0"
      using assms(1)[unfolded zero_less_abs_iff[symmetric]] \<open>e>0\<close> by auto
    moreover
    {
      fix y
      assume "dist y (c *\<^sub>R x) < e * \c\"
      then have "norm (c *\<^sub>R ((1 / c) *\<^sub>R y - x)) < e * norm c"
        by (simp add: \<open>c \<noteq> 0\<close> dist_norm scale_right_diff_distrib)
      then have "norm ((1 / c) *\<^sub>R y - x) < e"
        by (simp add: \<open>c \<noteq> 0\<close>)
      then have "y \ (*\<^sub>R) c ` s"
        using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "(*\<^sub>R) c"]
        by (simp add: \<open>c \<noteq> 0\<close> dist_norm e)
    }
    ultimately have "\e>0. \x'. dist x' (c *\<^sub>R x) < e \ x' \ (*\<^sub>R) c ` s"
      by (rule_tac x="e * \c\" in exI, auto)
  }
  then show ?thesis unfolding open_dist by auto
qed

lemma minus_image_eq_vimage:
  fixes A :: "'a::ab_group_add set"
  shows "(\x. - x) ` A = (\x. - x) -` A"
  by (auto intro!: image_eqI [where f="\x. - x"])

lemma open_negations:
  fixes S :: "'a::real_normed_vector set"
  shows "open S \ open ((\x. - x) ` S)"
  using open_scaling [of "- 1" S] by simp

lemma open_translation:
  fixes S :: "'a::real_normed_vector set"
  assumes "open S"
  shows "open((\x. a + x) ` S)"
proof -
  {
    fix x
    have "continuous (at x) (\x. x - a)"
      by (intro continuous_diff continuous_ident continuous_const)
  }
  moreover have "{x. x - a \ S} = (+) a ` S"
    by force
  ultimately show ?thesis
    by (metis assms continuous_open_vimage vimage_def)
qed

lemma open_translation_subtract:
  fixes S :: "'a::real_normed_vector set"
  assumes "open S"
  shows "open ((\x. x - a) ` S)"
  using assms open_translation [of S "- a"] by (simp cong: image_cong_simp)

lemma open_neg_translation:
  fixes S :: "'a::real_normed_vector set"
  assumes "open S"
  shows "open((\x. a - x) ` S)"
  using open_translation[OF open_negations[OF assms], of a]
  by (auto simp: image_image)

lemma open_affinity:
  fixes S :: "'a::real_normed_vector set"
  assumes "open S"  "c \ 0"
  shows "open ((\x. a + c *\<^sub>R x) ` S)"
proof -
  have *: "(\x. a + c *\<^sub>R x) = (\x. a + x) \ (\x. c *\<^sub>R x)"
    unfolding o_def ..
  have "(+) a ` (*\<^sub>R) c ` S = ((+) a \ (*\<^sub>R) c) ` S"
    by auto
  then show ?thesis
    using assms open_translation[of "(*\<^sub>R) c ` S" a]
    unfolding *
    by auto
qed

lemma interior_translation:
  "interior ((+) a ` S) = (+) a ` (interior S)" for S :: "'a::real_normed_vector set"
proof (rule set_eqI, rule)
  fix x
  assume "x \ interior ((+) a ` S)"
  then obtain e where "e > 0" and e: "ball x e \ (+) a ` S"
    unfolding mem_interior by auto
  then have "ball (x - a) e \ S"
    unfolding subset_eq Ball_def mem_ball dist_norm
    by (auto simp: diff_diff_eq)
  then show "x \ (+) a ` interior S"
    unfolding image_iff
    by (metis \<open>0 < e\<close> add.commute diff_add_cancel mem_interior)
next
  fix x
  assume "x \ (+) a ` interior S"
  then obtain y e where "e > 0" and e: "ball y e \ S" and y: "x = a + y"
    unfolding image_iff Bex_def mem_interior by auto
  {
    fix z
    have *: "a + y - z = y + a - z" by auto
    assume "z \ ball x e"
    then have "z - a \ S"
      using e[unfolded subset_eq, THEN bspec[where x="z - a"]]
      unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 *
      by auto
    then have "z \ (+) a ` S"
      unfolding image_iff by (auto intro!: bexI[where x="z - a"])
  }
  then have "ball x e \ (+) a ` S"
    unfolding subset_eq by auto
  then show "x \ interior ((+) a ` S)"
    unfolding mem_interior using \<open>e > 0\<close> by auto
qed

lemma interior_translation_subtract:
  "interior ((\x. x - a) ` S) = (\x. x - a) ` interior S" for S :: "'a::real_normed_vector set"
  using interior_translation [of "- a"] by (simp cong: image_cong_simp)

lemma compact_scaling:
  fixes s :: "'a::real_normed_vector set"
  assumes "compact s"
  shows "compact ((\x. c *\<^sub>R x) ` s)"
proof -
  let ?f = "\x. scaleR c x"
  have *: "bounded_linear ?f" by (rule bounded_linear_scaleR_right)
  show ?thesis
    using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
    using linear_continuous_at[OF *] assms
    by auto
qed

lemma compact_negations:
  fixes s :: "'a::real_normed_vector set"
  assumes "compact s"
  shows "compact ((\x. - x) ` s)"
  using compact_scaling [OF assms, of "- 1"] by auto

lemma compact_sums:
  fixes s t :: "'a::real_normed_vector set"
  assumes "compact s"
    and "compact t"
  shows "compact {x + y | x y. x \ s \ y \ t}"
proof -
  have *: "{x + y | x y. x \ s \ y \ t} = (\z. fst z + snd z) ` (s \ t)"
    by (fastforce simp: image_iff)
  have "continuous_on (s \ t) (\z. fst z + snd z)"
    unfolding continuous_on by (rule ballI) (intro tendsto_intros)
  then show ?thesis
    unfolding * using compact_continuous_image compact_Times [OF assms] by auto
qed

lemma compact_differences:
  fixes s t :: "'a::real_normed_vector set"
  assumes "compact s"
    and "compact t"
  shows "compact {x - y | x y. x \ s \ y \ t}"
proof-
  have "{x - y | x y. x\s \ y \ t} = {x + y | x y. x \ s \ y \ (uminus ` t)}"
    using diff_conv_add_uminus by force
  then show ?thesis
    using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
qed

lemma compact_translation:
  "compact ((+) a ` s)" if "compact s" for s :: "'a::real_normed_vector set"
proof -
  have "{x + y |x y. x \ s \ y \ {a}} = (\x. a + x) ` s"
    by auto
  then show ?thesis
    using compact_sums [OF that compact_sing [of a]] by auto
qed

lemma compact_translation_subtract:
  "compact ((\x. x - a) ` s)" if "compact s" for s :: "'a::real_normed_vector set"
  using that compact_translation [of s "- a"] by (simp cong: image_cong_simp)

lemma compact_affinity:
  fixes s :: "'a::real_normed_vector set"
  assumes "compact s"
  shows "compact ((\x. a + c *\<^sub>R x) ` s)"
proof -
  have "(+) a ` (*\<^sub>R) c ` s = (\x. a + c *\<^sub>R x) ` s"
    by auto
  then show ?thesis
    using compact_translation[OF compact_scaling[OF assms], of a c] by auto
qed

lemma closed_scaling:
  fixes S :: "'a::real_normed_vector set"
  assumes "closed S"
  shows "closed ((\x. c *\<^sub>R x) ` S)"
proof (cases "c = 0")
  case True then show ?thesis
    by (auto simp: image_constant_conv)
next
  case False
  from assms have "closed ((\x. inverse c *\<^sub>R x) -` S)"
    by (simp add: continuous_closed_vimage)
  also have "(\x. inverse c *\<^sub>R x) -` S = (\x. c *\<^sub>R x) ` S"
    using \<open>c \<noteq> 0\<close> by (auto elim: image_eqI [rotated])
  finally show ?thesis .
qed

lemma closed_negations:
  fixes S :: "'a::real_normed_vector set"
  assumes "closed S"
  shows "closed ((\x. -x) ` S)"
  using closed_scaling[OF assms, of "- 1"] by simp

lemma compact_closed_sums:
  fixes S :: "'a::real_normed_vector set"
  assumes "compact S" and "closed T"
  shows "closed (\x\ S. \y \ T. {x + y})"
proof -
  let ?S = "{x + y |x y. x \ S \ y \ T}"
  {
    fix x l
    assume as: "\n. x n \ ?S" "(x \ l) sequentially"
    from as(1) obtain f where f: "\n. x n = fst (f n) + snd (f n)" "\n. fst (f n) \ S" "\n. snd (f n) \ T"
      using choice[of "\n y. x n = (fst y) + (snd y) \ fst y \ S \ snd y \ T"] by auto
    obtain l' r where "l'\<in>S" and r: "strict_mono r" and lr: "(((\<lambda>n. fst (f n)) \<circ> r) \<longlongrightarrow> l') sequentially"
      using assms(1)[unfolded compact_def, THEN spec[where x="\ n. fst (f n)"]] using f(2) by auto
    have "((\n. snd (f (r n))) \ l - l') sequentially"
      using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1)
      unfolding o_def
      by auto
    then have "l - l' \ T"
      using assms(2)[unfolded closed_sequential_limits,
        THEN spec[where x="\ n. snd (f (r n))"],
        THEN spec[where x="l - l'"]]
      using f(3)
      by auto
    then have "l \ ?S"
      using \<open>l' \<in> S\<close> by force
  }
  moreover have "?S = (\x\ S. \y \ T. {x + y})"
    by force
  ultimately show ?thesis
    unfolding closed_sequential_limits
    by (metis (no_types, lifting))
qed

lemma closed_compact_sums:
  fixes S T :: "'a::real_normed_vector set"
  assumes "closed S" "compact T"
  shows "closed (\x\ S. \y \ T. {x + y})"
proof -
  have "(\x\ T. \y \ S. {x + y}) = (\x\ S. \y \ T. {x + y})"
    by auto
  then show ?thesis
    using compact_closed_sums[OF assms(2,1)] by simp
qed

lemma compact_closed_differences:
  fixes S T :: "'a::real_normed_vector set"
  assumes "compact S" "closed T"
  shows "closed (\x\ S. \y \ T. {x - y})"
proof -
  have "(\x\ S. \y \ uminus ` T. {x + y}) = (\x\ S. \y \ T. {x - y})"
    by force
  then show ?thesis
    by (metis assms closed_negations compact_closed_sums)
qed

lemma closed_compact_differences:
  fixes S T :: "'a::real_normed_vector set"
  assumes "closed S" "compact T"
  shows "closed (\x\ S. \y \ T. {x - y})"
proof -
  have "(\x\ S. \y \ uminus ` T. {x + y}) = {x - y |x y. x \ S \ y \ T}"
    by auto
then show ?thesis
  using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
qed

lemma closed_translation:
  "closed ((+) a ` S)" if "closed S" for a :: "'a::real_normed_vector"
proof -
  have "(\x\ {a}. \y \ S. {x + y}) = ((+) a ` S)" by auto
  then show ?thesis
    using compact_closed_sums [OF compact_sing [of a] that] by auto
qed

lemma closed_translation_subtract:
  "closed ((\x. x - a) ` S)" if "closed S" for a :: "'a::real_normed_vector"
  using that closed_translation [of S "- a"] by (simp cong: image_cong_simp)

lemma closure_translation:
  "closure ((+) a ` s) = (+) a ` closure s" for a :: "'a::real_normed_vector"
proof -
  have *: "(+) a ` (- s) = - (+) a ` s"
    by (auto intro!: image_eqI [where x = "x - a" for x])
  show ?thesis
    using interior_translation [of a "- s", symmetric]
    by (simp add: closure_interior translation_Compl *)
qed

lemma closure_translation_subtract:
  "closure ((\x. x - a) ` s) = (\x. x - a) ` closure s" for a :: "'a::real_normed_vector"
  using closure_translation [of "- a" s] by (simp cong: image_cong_simp)

lemma frontier_translation:
  "frontier ((+) a ` s) = (+) a ` frontier s" for a :: "'a::real_normed_vector"
  by (auto simp add: frontier_def translation_diff interior_translation closure_translation)

lemma frontier_translation_subtract:
  "frontier ((+) a ` s) = (+) a ` frontier s" for a :: "'a::real_normed_vector"
  by (auto simp add: frontier_def translation_diff interior_translation closure_translation)

lemma sphere_translation:
  "sphere (a + c) r = (+) a ` sphere c r" for a :: "'n::real_normed_vector"
  by (auto simp: dist_norm algebra_simps intro!: image_eqI [where x = "x - a" for x])

lemma sphere_translation_subtract:
  "sphere (c - a) r = (\x. x - a) ` sphere c r" for a :: "'n::real_normed_vector"
  using sphere_translation [of "- a" c] by (simp cong: image_cong_simp)

lemma cball_translation:
  "cball (a + c) r = (+) a ` cball c r" for a :: "'n::real_normed_vector"
  by (auto simp: dist_norm algebra_simps intro!: image_eqI [where x = "x - a" for x])

lemma cball_translation_subtract:
  "cball (c - a) r = (\x. x - a) ` cball c r" for a :: "'n::real_normed_vector"
  using cball_translation [of "- a" c] by (simp cong: image_cong_simp)

lemma ball_translation:
  "ball (a + c) r = (+) a ` ball c r" for a :: "'n::real_normed_vector"
  by (auto simp: dist_norm algebra_simps intro!: image_eqI [where x = "x - a" for x])

lemma ball_translation_subtract:
  "ball (c - a) r = (\x. x - a) ` ball c r" for a :: "'n::real_normed_vector"
  using ball_translation [of "- a" c] by (simp cong: image_cong_simp)

subsection\<^marker>\<open>tag unimportant\<close>\<open>Homeomorphisms\<close>

lemma homeomorphic_scaling:
  fixes S :: "'a::real_normed_vector set"
  assumes "c \ 0"
  shows "S homeomorphic ((\x. c *\<^sub>R x) ` S)"
  unfolding homeomorphic_minimal
  apply (rule_tac x="\x. c *\<^sub>R x" in exI)
  apply (rule_tac x="\x. (1 / c) *\<^sub>R x" in exI)
  using assms by (auto simp: continuous_intros)

lemma homeomorphic_translation:
  fixes S :: "'a::real_normed_vector set"
  shows "S homeomorphic ((\x. a + x) ` S)"
  unfolding homeomorphic_minimal
  apply (rule_tac x="\x. a + x" in exI)
  apply (rule_tac x="\x. -a + x" in exI)
  by (auto simp: continuous_intros)

lemma homeomorphic_affinity:
  fixes S :: "'a::real_normed_vector set"
  assumes "c \ 0"
  shows "S homeomorphic ((\x. a + c *\<^sub>R x) ` S)"
proof -
  have *: "(+) a ` (*\<^sub>R) c ` S = (\x. a + c *\<^sub>R x) ` S" by auto
  show ?thesis
    by (metis "*" assms homeomorphic_scaling homeomorphic_trans homeomorphic_translation)
qed

lemma homeomorphic_balls:
  fixes a b ::"'a::real_normed_vector"
  assumes "0 < d"  "0 < e"
  shows "(ball a d) homeomorphic (ball b e)" (is ?th)
    and "(cball a d) homeomorphic (cball b e)" (is ?cth)
proof -
  show ?th unfolding homeomorphic_minimal
    apply(rule_tac x="\x. b + (e/d) *\<^sub>R (x - a)" in exI)
    apply(rule_tac x="\x. a + (d/e) *\<^sub>R (x - b)" in exI)
    using assms
    by (auto intro!: continuous_intros simp: dist_commute dist_norm pos_divide_less_eq)
  show ?cth unfolding homeomorphic_minimal
    apply(rule_tac x="\x. b + (e/d) *\<^sub>R (x - a)" in exI)
    apply(rule_tac x="\x. a + (d/e) *\<^sub>R (x - b)" in exI)
    using assms
    by (auto intro!: continuous_intros simp: dist_commute dist_norm pos_divide_le_eq)
qed

lemma homeomorphic_spheres:
  fixes a b ::"'a::real_normed_vector"
  assumes "0 < d"  "0 < e"
  shows "(sphere a d) homeomorphic (sphere b e)"
unfolding homeomorphic_minimal
    apply(rule_tac x="\x. b + (e/d) *\<^sub>R (x - a)" in exI)
    apply(rule_tac x="\x. a + (d/e) *\<^sub>R (x - b)" in exI)
    using assms
    by (auto intro!: continuous_intros simp: dist_commute dist_norm pos_divide_less_eq)

lemma homeomorphic_ball01_UNIV:
  "ball (0::'a::real_normed_vector) 1 homeomorphic (UNIV:: 'a set)"
  (is "?B homeomorphic ?U")
proof
  have "x \ (\z. z /\<^sub>R (1 - norm z)) ` ball 0 1" for x::'a
    apply (rule_tac x="x /\<^sub>R (1 + norm x)" in image_eqI)
     apply (auto simp: field_split_simps)
    using norm_ge_zero [of x] apply linarith+
    done
  then show "(\z::'a. z /\<^sub>R (1 - norm z)) ` ?B = ?U"
    by blast
  have "x \ range (\z. (1 / (1 + norm z)) *\<^sub>R z)" if "norm x < 1" for x::'a
    using that
    by (rule_tac x="x /\<^sub>R (1 - norm x)" in image_eqI) (auto simp: field_split_simps)
  then show "(\z::'a. z /\<^sub>R (1 + norm z)) ` ?U = ?B"
    by (force simp: field_split_simps dest: add_less_zeroD)
  show "continuous_on (ball 0 1) (\z. z /\<^sub>R (1 - norm z))"
    by (rule continuous_intros | force)+
  have 0: "\z. 1 + norm z \ 0"
    by (metis (no_types) le_add_same_cancel1 norm_ge_zero not_one_le_zero)
  then show "continuous_on UNIV (\z. z /\<^sub>R (1 + norm z))"
    by (auto intro!: continuous_intros)
  show "\x. x \ ball 0 1 \
         x /\<^sub>R (1 - norm x) /\<^sub>R (1 + norm (x /\<^sub>R (1 - norm x))) = x"
    by (auto simp: field_split_simps)
  show "\y. y /\<^sub>R (1 + norm y) /\<^sub>R (1 - norm (y /\<^sub>R (1 + norm y))) = y"
    using 0 by (auto simp: field_split_simps)
qed

proposition homeomorphic_ball_UNIV:
  fixes a ::"'a::real_normed_vector"
  assumes "0 < r" shows "ball a r homeomorphic (UNIV:: 'a set)"
  using assms homeomorphic_ball01_UNIV homeomorphic_balls(1) homeomorphic_trans zero_less_one by blast

subsection\<^marker>\<open>tag unimportant\<close> \<open>Discrete\<close>

lemma finite_implies_discrete:
  fixes S :: "'a::topological_space set"
  assumes "finite (f ` S)"
  shows "(\x \ S. \e>0. \y. y \ S \ f y \ f x \ e \ norm (f y - f x))"
proof -
  have "\e>0. \y. y \ S \ f y \ f x \ e \ norm (f y - f x)" if "x \ S" for x
  proof (cases "f ` S - {f x} = {}")
    case True
    with zero_less_numeral show ?thesis
      by (fastforce simp add: Set.image_subset_iff cong: conj_cong)
  next
    case False
    then obtain z where "z \ S" "f z \ f x"
      by blast
    moreover have finn: "finite {norm (z - f x) |z. z \ f ` S - {f x}}"
      using assms by simp
    ultimately have *: "0 < Inf{norm(z - f x) | z. z \ f ` S - {f x}}"
      by (force intro: finite_imp_less_Inf)
    show ?thesis
      by (force intro!: * cInf_le_finite [OF finn])
  qed
  with assms show ?thesis
    by blast
qed

subsection\<^marker>\<open>tag unimportant\<close> \<open>Completeness of "Isometry" (up to constant bounds)\<close>

lemma cauchy_isometric:\<comment> \<open>TODO: rename lemma to \<open>Cauchy_isometric\<close>\<close>
  assumes e: "e > 0"
    and s: "subspace s"
    and f: "bounded_linear f"
    and normf: "\x\s. norm (f x) \ e * norm x"
--> --------------------

--> maximum size reached

--> --------------------

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