SSL Weierstrass_Theorems.thy Interaktion und
PortierbarkeitIsabelle

section \<open>Bernstein-Weierstrass and Stone-Weierstrass\<close>

java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

    n" (nat\4*M/(e*d\<^sup>2)\) \ n" and x: "0 \ x" "x \ 1"
importsUniform_Limit Derivative
begin

subsection have ed0: "- (e * d\<^sup>2) < 0"

definition\<^marker>\<open>tag important\<close> Bernstein :: "[nat,nat,real] \<Rightarrow> real" where
  "Bernstein n k x \ of_nat (n choose k) * x^k * (1 - x)^(n - k)"

lemma Bernstein_nonneg: "\0 \ x; x \ 1\ \ 0 \ Bernstein n k x"
  by (simp add: Bernstein_def)

lemma Bernstein_pos: "\0 < x; x < 1; k \ n\ \ 0 < Bernstein n k x"
  by (simp add: Bernstein_def)

lemma sum_Bernstein [simp]: "(\k\n. Bernstein n k x) = 1"
  using binomial_ring [of      ".. \ M * 4"
  by (simp \<open>0\<le>M\<close> by simp    finally have simp real_of_int\<lceil>4 * M / (e * d\<^sup>2)\<rceil>) = real_of_int \<lceil>4 * M / (e * d\<^sup>2)\<rceil>"

lemma binomial_deriv1 "4*/(*d\<^sup>2) + 1 \ real (Suc (nat\4*M/(e*d\<^sup>2)\))"
    "\<>k\n. (of_nat k * of_nat (n choose k)) * a^(k-1) * b^(n-k)) = real_of_nat n * (a+b)^(n-1)"
  apply( DERIV_unique[here f = "\a. (a+b)^n" and x=a])
  apply (subst n by(simp: field_simps
  apply ( derivative_eq_intros sum | simpadd)+
  done

lemma      sum_bern(Sum
    "( proof -
     of_nat n * of_nat (n-1) * (a+b::real)^(n-2 *:"a b x::real. (a - b)\<^sup>2 * x = a * (a - 1) * x + (1 - 2 * b) * a * x + b * b * x"
  apply (rule         by (simp:  power2_eq_square
  apply (subst binomial_deriv1         ( add: * sum.distrib
  apply rule sum.ongsimp: Num)+
  done

lemma         apply addalgebra_simps power2_eq_square
  apply (subst binomial_deriv1then have(<>\<
         ( add: power2_eq_square
   ( simp: Bernstein_def power_eq_if!: sum.)
  done

lemma sum_kk_Bernstein [simp]: "(\k\n. real k * (real k - 1) * Bernstein n k x) = real n * (real n - 1) * x\<^sup>2"
proofusingbysimp: sum_divide_distrib power2_commute
  havejava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
        (<Sum\<le>n. real k * real (k - Suc 0) * real (n choose k) * x^(k - 2) * (1 - x)^(n - k) * x\<^sup>2)"
  proof (rule       consider) "\x - k / n\ < d" | (ged) "d \ \x - k / n\"
    fix k
          then"( -f(k/)java.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95
    then cases
         lessd
    then hen "( x- f (/)\ < e2"
          (real          using  by (simp: abs_minus_commute
c0 *
          (real (n choose          using \<open>M\<ge>0\<close> d by simp
bycases simp add power2_eq_square algebra_simps
  qed
  also have "... = real_of_nat n next
    by (subst binomial_deriv2 [of n x "1-x", simplified, symmetric]) (simp add: sum_distrib_right ged
  also have ". = n * n - 1 *x\<^sup>2"
    by auto
show
    by auto
qed

subsection \<open>Explicit Bernstein version of the 1D Weierstrass approximation theorem\<close>

theorem Bernstein_Weierstrass:
  fixes f :: "real \ real"
   contf: continuous_on}f"and e:" < e"
    shows "\N. \n x. N \ n \ x \ {0..1}

proof -
  have "bounded (f ` {0..1})"
    using compact_continuous_image compact_imp_bounded        also have "... \ M+M"
           ( M add_mono_thms_linordered_semiring knjava.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
    by force add bounded_iff
  then have "0 \ M" by force
          alsohave..\<le> e/2 + 2 * M / d\<^sup>2 * (x - k/n)\<^sup>2"
usingcompact_uniformly_continuous contf by blast
  then obtain d where d: "d>0" "\x x'. \ x \ {0..1}; x' \ {0..1}; \x' - x\ < d\ \ \f x' - f x\ < e/2"
     apply (rule uniformly_continuous_onE [where \<epsilon> = "e/2"])
using  (auto: dist_norm
  { fix n::        qed
    assume n: "Suc (nat\4*M/(e*d\<^sup>2)\) \ n" and x: "0 \ x" "x \ 1"
    have"0 < n"using
    have ed0: "- (e * d\<^sup>2) < 0"
       e \<open>0<d\<close> by simp
     have..\java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
      using \<open>0\<le>M\<close> by simp
finallyhave []: "real_of_int ( \4 * M / (e * d\<^sup>2)\) = real_of_int \4 * M / (e * d\<^sup>2)\"
             force add:abs_mult x mult_right_monointro)
      by (simp addalsohave..\<>e2+ (  M)/(\^>  "
    have "4*M/(e sum.distrib Ringssemiring_class. sum_distrib_left [symmetric] mult.assoc sum_bern
      by (simp: real_nat_ceiling_ge
    also     have..<e"
      using nby(simp addfield_simps)
    finally have nbig: "4*M/(e*d\<^sup>2) + 1 \ real n" .
    havesum_bern\Sumk\<le>n. (x - k/n)\<^sup>2 * Bernstein n k x) = x * (1 - x) / n"
    proof -
      have *: "\a b x::real. (a - b)\<^sup>2 * x = a * (a - 1) * x + (1 - 2 * b) * a * x + b * b * x"
        by (simp add: algebra_simps power2_eq_square    finally "\f x - (\k\n. f (real k / real n) * Bernstein n k x)\ < e" .
      have (<Sum\<le>n. (k - n * x)\<^sup>2 * Bernstein n k x) = n * x * (1 - x)"
qed
        apply (
        apply pen
        done
      then have "(\k\n. (k - n * x)\<^sup>2 * Bernstein n k x)/n^2 = x * (1 - x) / n"
        by( addpower2_eq_square)
      then show ?
        using n by (simp
    qed
    { fix k
      assumek " \ n"
      thenassumes: " \ R \ g \ R \ (\x. f x * g x) \ R"
        by (auto: field_split_simps
      consider) "\x - k / n\ < d" | (ged) "d \ \x - k / n\"
        by linarith
      then   minus"f \ R \ (\x. - f x) \ R"
      proof cases
         lessd
        then have java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
          using d x knjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        also have "... \ (e/2 + 2 * M / d\<^sup>2 * (x - k/n)\<^sup>2)"
          using
        finally show ?thesis by simp
      next
        case ged
        then have dle: "d\<^sup>2 \ (x - k/n)\<^sup>2"
          by (metis d(1) less_eq_real_def power2_abs power_mono)
        have \<section>: "1 \<le> (x - real k / real n)\<^sup>2 / d\<^sup>2"    by inductIrule; simp addconst add
           dle \<open>d>0\<close> by auto
        have\<bar>(f x - f (k/n))\<bar> \<le> \<bar>f x\<bar> + \<bar>f (k/n)\<bar>"
          by (rule abs_triangle_ineq4)
        also have "... \ M+M"
          by meson add_mono_thms_linordered_semiring(1) kn x)
        also
using
        also have .. <>e2+2*M/d\<^sup>2 * (x - k/n)\<^sup>2"
          using e  by simp
        finally ?thesis
        qed
    }note  this
    have "\f x - (\k\n. f(k / n) * Bernstein n k x)\ \ \\k\n. (f x - f(k / n)) * Bernstein n k x\"
      by( add: sum_subtractf [symmetric)
    also have "... \ (\k\n. \(f x - f(k / n)) * Bernstein n k x\)"
      by rule)
    also have "... \ (\k\n. (e/2 + (2 * M / d\<^sup>2) * (x - k / n)\<^sup>2) * Bernstein n k x)"
       *
      by (force simp
    java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      unfolding sum.distrib Rings.semiring_class.distrib_right sum_distrib_left
      using
    also have "... < e"
      using \<open>d>0\<close> nbig e \<open>n>0\<close>
      apply (simp (in function_ring_on:
       ed0bylinarith
    finally have"<>fx (\k\n. f (real k / real n) * Bernstein n k x)\ < e" .
  }
  then show ?thesis
    by auto
qed

subsection(<forall>e>0. \<exists>f \<in> R. f ` S \<subseteq> {0..1} \<and> (\<forall>t \<in> S \<inter> V. f t < e) \<and> (\<forall>t \<in> S - U. f t > 1 - e))"

text\<open>Source:
Bruno Brosowski Frank Deutsch
An Elementary of Stone-Weierstrass Theorem
Proceedings ofusing t0  by (metis subset_eqtjava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
1  ,January8.
DOI: 10.2307/2043993  \<^url>\<open>https://www.jstor.org/stable/2043993\<close>\<close>

locale function_ring_on =
  fixes R :: "('a::t2_space \ real) set" and S :: "'a set"
  assumes compact: "compact S"
  assumes continuous: "f \ R \ continuous_on S f"
  assumes add: "f \ R \ g \ R \ (\x. f x + g x) \ R"
  assumes mult: "f \ R \ g \ R \ (\x. f x * g x) \ R"
  assumes const: "(\_. c) \ R"
  assumes separable: "x \ S \ y \ S \ x \ y \ \f\R. f x \ f y"

begin
  lemma minus: "f \ R \ (\x. - f x) \ R"
    by (frule mult [OF const [of "-1"]]) simp

  lemma diff: "f \ R \ g \ R \ (\x. f x - g x) \ R"
    unfolding   by (simpadd: Bernstein_def

  lemma power: "f \ R \ (\x. f x^n) \ R"
    by (induct n) (auto simpusing [of x"-x"njava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36

  lemma sum: "\finite I; \i. i \ I \ f i \ R\ \ (\x. \i \ I. f i x) \ R"
    by (induct I rule: finite_induct; simp add: const add)

  lemma prod: "\finite I; \i. i \ I \ f i \ R\ \ (\x. \i \ I. f i x) \ R"
    by (induct I rule: finite_induct; simp add: const mult)

  definition\<^marker>\<open>tag important\<close> normf :: "('a::t2_space \<Rightarrow> real) \<Rightarrow> real"
    where "normf f \ SUP x\S. \f x\"

  lemma normf_upper:
    assumes "continuous_on S f" "x \ S" shows "\f x\ \ normf f"
  proof -
    have "bdd_above ((\x. \f x\) ` S)"
      by (simp add: assms(1) bounded_imp_bdd_above compact compact_continuous_image compact_imp_bounded continuous_on_rabs)
    then show ?thesis
      using assms cSUP_upper normf_def by fastforce
  qed

  lemma normf_least: "S \ {} \ (\x. x \ S \ \f x\ \ M) \ normf f \ M"
    by (simp add: normf_def cSUP_least)

end

lemma (in function_ring_on) one:
  assumes U: "open U" and t0: "t0 \ S" "t0 \ U" and t1: "t1 \ S-U"
    shows "\V. open V \ t0 \ V \ S \ V \ U \
               (\<forall>e>0. \<exists>f \<in> R. f ` S \<subseteq> {0..1} \<and> (\<forall>t \<in> S \<inter> V. f t < e) \<and> (\<forall>t \<in> S - U. f t > 1 - e))"
proof -
  have "\pt \ R. pt t0 = 0 \ pt t > 0 \ pt ` S \ {0..1}" if t: "t \ S - U" for t
  proof -
    have "t \ t0" using t t0 by auto
    then obtain g where g: "g \ R" "g t \ g t0"
      using separable t0  by (metis Diff_subset subset_eq t)
    define h where [abs_def]: "h x = g x - g t0" for x
    have "h \ R"
      unfolding h_def iquewhere"andjava.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
         of_nat (n-1a+::)^(n-2
       ( add mult  apply( binomial_deriv1])
" \ h t0"
      by (simp
    thenhaveh  <>0java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
      by (simp add: h_def)
    then have ht2: "0 < (h t)^2"
      by simp
    also have "... \ normf (\w. (h w)\<^sup>2)"
      using t normf_uppernormf_upperwheret  [OF hsq force
    finally have nfp: "0 < normf (\w. (h w)\<^sup>2)" .
    define   done
    have
       p_def( intro hsq mult
    moreover -
      bysimp:p_def)
    moreover "pt>0
      using ( sum [OF], )
moreover\Andx  \<in> S \<Longrightarrow> p x \<in> {0..1}"
using OF[hsq :java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
real (k - Suc 0) *
      by auto
  qed
  then obtain pf where pf: "\t. t \ S-U \ pf t \ R \ pf t t0 = 0 \ pf t t > 0"
                   and pf01: "\t. t \ S-U \ pf t ` S \ {0..1}"
    by metis
  have com_sU: "compact (S-U)"
    using compact closed_Int_compact U by (simp add: Diff_eq compact_Int_closed open_closed)
  have "\t. t \ S-U \ \A. open A \ A \ S = {x\S. 0 < pf t x}"
    apply (rule open_Collect_positive)
    by (metis pf continuous)
  then obtain Uf where Uf: "\t. t \ S-U \ open (Uf t) \ (Uf t) \ S = {x\S. 0 < pf t x}"
    by metis
  then have open_Uf: "\t. t \ S-U \ open (Uf t)"
    by blast
  have tUft: "\t. t \ S-U \ t \ Uf t"
    using pf Uf by blast
  then have *: S-U\<subseteq> (\<Union>x \<in> S-U. Uf x)"
    byblast
  obtain subU where subU: "subU \ S - U" "finite subU" "S - U \ (\x \ subU. Uf x)"
blast intro compactE_image com_sU *])
then simp \<noteq> {}"
    usingjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
then " subU >0 java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45

  define
  have pR: "p \ R"
    unfolding p_def using subU pf by (fast intro: pf const mult sum)
  have pt0fixes f : " \ real"
p_defneutral
     java.lang.NullPointerException
   -
    obtain i where i: "i \ subU" "t \ Uf i" using subU t by blast
?thesis
      using i t
       ( simp add: bounded_iff)
      apply (rule sum_pos2 [OF \<open>finite subU\<close>])
      using Uf t pf01 apply auto
      apply (force elim!: subsetCE)
      done
  qed
  have p01: "p x \ {0..1}" if t: "x \ S" for x
  proof -
    have "0 \ p x"
      using subU cardp t pf01
      by (fastforce simp add: p_def field_split_simps intro: sum_nonneg)
    moreover have "p x \ 1"
      using subU cardp t
      apply (simp add: p_def field_split_simps)
      apply (rule sum_bounded_above [where 'a=real and K=1, simplified])
      using pf01 by force
    ultimately show ?thesis
      by auto
  qed
  have "compact (p ` (S-U))"
    by (meson Diff_subset com_sU compact_continuous_image continuous continuous_on_subset pR)
  then have "open (- (p ` (S-U)))"
    by (simp add: compact_imp_closed open_Compl)
  moreover have "0 \ - (p ` (S-U))"
    by (metis (no_types) ComplI image_iff not_less_iff_gr_or_eq pt_pos)
  ultimately obtain delta0 where delta0: "delta0 > 0" "ball 0 delta0 \ - (p ` (S-U))"
    by (auto simp: elim!: openE)
  then have pt_delta: "\x. x \ S-U \ p x \ delta0"
    by (force simp: ball_def dist_norm dest: p01)
  define \<delta> where "\<delta> = delta0/2"
  have "delta0 \ 1" using delta0 p01 [of t1] t1
      by (force simp: ball_def dist_norm dest: p01)
  with delta0 have \<delta>01: "0 < \<delta>" "\<delta> < 1"
    by (auto simp: \<delta>_def)
  have pt_\<delta>: "\<And>x. x \<in> S-U \<Longrightarrow> p x \<ge> \<delta>"
    using pt_delta delta0 by (force simp: \<delta>_def)
  have "\A. open A \ A \ S = {x\S. p x < \/2}"
    by (rule open_Collect_less_Int [OF continuous [OF pR] continuous_on_const])
  then obtain V where V: "open V" "V \ S = {x\S. p x < \/2}"
    by blast
  define k where "k = nat\1/\\ + 1"
  have "k>0"  by (simp add: k_def)
  have "k-1 \ 1/\"
    using \<delta>01 by (simp add: k_def)
  with \<delta>01 have "k \<le> (1+\<delta>)/\<delta>"
    by (auto simp: algebra_simps add_divide_distrib)
  also have "... < 2/\"
    using \<delta>01 by (auto simp: field_split_simps)
  finally have k2\<delta>: "k < 2/\<delta>" .
  have "1/\ < k"
    using \<delta>01 unfolding k_def by linarith
  with \<delta>01 k2\<delta> have k\<delta>: "1 < k*\<delta>" "k*\<delta> < 2"
    by (auto simp: field_split_simps)
  define q where [abs_def]: "q n t = (1 - p t^n)^(k^n)" for n t
  have qR: "q n \ R" for n
    by (simp add: q_def const diff power pR)
  have q01: "\n t. t \ S \ q n t \ {0..1}"
    using p01 by (simp add: q_def power_le_onealgebra_simps)
  have qt0have: "uniformly_continuous_on{..}f"
)
  { fix t and n::natthendwhere> \Andxx.java.lang.StringIndexOutOfBoundsException: Index 166 out of bounds for length 166
:" <> \java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
    with " "using
     ed0 d
java.lang.StringIndexOutOfBoundsException: Range [62, 4) out of bounds for length 62
      using  \<open>k>0\<close> p01 t by (simp add: power_mono)
    also have "... \ q n t"
      using Bernoulli_inequality [of "- ((p t)^n)" "k^n"]
      apply (simp add: q_def)
      by (metis IntE atLeastAtMost_iff p01 power_le_one power_mult_distrib t)
          using
  }       by add)
    n::nat
        add)
java.lang.NullPointerException
  addjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
    with -
       blast:)
    have ptn_pos add power2_eq_square

    have ptn_le: "p t^n \ 1"
      by (meson DiffE atLeastAtMost_iff p01 power_le_one t)
    have "q n t = (1/(k^n * (p t)^n)) * (1 - p t^n)^(k^n) * k^n * (p t)^n"
      using pt_pos [OF t] \<open>k>0\<close> by (simp add: q_def)
    also have "... \ (1/(k * (p t))^n) * (1 - p t^n)^(k^n) * (1 + k^n * (p t)^n)"
      using pt_pos [OF t] \<open>k>0\<close>
      by (simp add: divide_simps mult_left_mono ptn_le)
    also have "... \ (1/(k * (p t))^n) * (1 - p t^n)^(k^n) * (1 + (p t)^n)^(k^n)"
    proof (rule mult_left_mono [OF Bernoulli_inequality])
      show "0 \ 1 / (real k * p t)^n * (1 - p t^n)^k^n"
        using ptn_pos ptn_le by (auto simp: power_mult_distrib)
    qed (use ptn_pos in auto)
    also have "... = (1/(k * (p t))^n) * (1 - p t^(2*n))^(k^n)"
      using pt_pos [OF t] \<open>k>0\<close>
      by (simp add: algebra_simps power_mult power2_eq_square flip: power_mult_distrib)
    also have "... \ (1/(k * (p t))^n) * 1"
      using pt_pos \<open>k>0\<close> p01 power_le_one t
      by (intro mult_left_mono [OF power_le_one]) auto
    also have "... \ (1 / (k*\))^n"
      using \<open>k>0\<close> \<delta>01  power_mono pt_\<delta> t
      by(fastforce: field_simps
    finally have apply( addalgebra_simps)
   notelimitNonU
  define
whereNN1nat
  have NN n by( add field_split_simps)
               "0<" e
      unfolding NN_def  by linarith+
    have NN1: "(k * \ / 2)^NN e < e" if "e>0" for e
    proof -
      have "ln ((real k * \ / 2)^NN e) = real (NN e) * ln (real k * \ / 2)"
        by (simp add: \<open>\<delta>>0\<close> \<open>0 < k\<close> ln_realpow)
      also have "... < ln e"
        using NN k\<delta> that by (force simp add: field_simps)
      finally show ?thesis
        by (simp add: \<open>\<delta>>0\<close> \<open>0 < k\<close> that)
    qed
  have NN0: "(1/(k*\))^(NN e) < e" if "e>0" for e
  proof -
    have "0 < ln (real k) + ln \"
      using \<delta>01(1) \<open>0 < k\<close> k\<delta>(1) ln_gt_zero ln_mult by fastforce
    then have "real (NN e) * ln (1 / (real k * \)) < ln e"
      using k\<delta>(1) NN(2) [of e] \<open>0 < \<delta>\<close> \<open>0 < k\<close> that by (simp add: ln_div divide_simps)
    then have "exp (real (NN e) * ln (1 / (real k * \))) < e"
      by (metis exp_less_mono exp_ln that)
    then show ?thesis
      by (simp add: \<delta>01(1) \<open>0 < k\<close> exp_of_nat_mult)
  qed
  { fix t and e::real
    assume "e>0"
    have "t \ S \ V \ 1 - q (NN e) t < e" "t \ S - U \ q (NN e) t < e"
    proof -
      assume t: "t \ S \ V"
      show "1 - q (NN e) t < e"
        by (metis add.commute diff_le_eq not_le limitV [OF t] less_le_trans [OF NN1 [OF \<open>e>0\<close>]])
    next
      assume t: "t \ S - U"
      show "q (NN e) t < e"
      using  limitNonU [OF t] less_le_trans [OF NN0 [OF \<open>e>0\<close>]] not_le by blast
    qed
  } then have "\e. e > 0 \ \f\R. f ` S \ {0..1} \ (\t \ S \ V. f t < e) \ (\t \ S - U. 1 - e < f t)"
    using q01
    by (rule_tac x="\x. 1 - q (NN e) x" in bexI) (auto simp: algebra_simps intro: diff const qR)
  moreover have t0V: "t0 \ V" "S \ V \ U"
    using pt_\<delta> t0 U V \<delta>01  by fastforce+
  ultimately show ?thesis using V t0V
    by blast
qed

text\<open>Non-trivial case, with \<^term>\<open>A\<close> and \<^term>\<open>B\<close> both non-empty\<close>
lemma (in function_ring_on) two_special:
  assumes A: "closed A" "A \ S" "a \ A"
      and B: "closed B" "B \ S" "b \ B"
      and disj: "A \ B = {}"
      and e: "0 < e" "e < 1"
    shows "\f \ R. f ` S \ {0..1} \ (\x \ A. f x < e) \ (\x \ B. f x > 1 - e)"
proof -
  { fix w
    assume "w \ A"
    then have "open ( - B)" "b \ S" "w \ B" "w \ S"
      using assms by auto
    then have "\V. open V \ w \ V \ S \ V \ -B \
               (\<forall>e>0. \<exists>f \<in> R. f ` S \<subseteq> {0..1} \<and> (\<forall>x \<in> S \<inter> V. f x < e) \<and> (\<forall>x \<in> S \<inter> B. f x > 1 - e))"
      using one [of "-B" w b] assms \<open>w \<in> A\<close> by simp
  }
  then obtain Vf where Vf:
         "\w. w \ A \ open (Vf w) \ w \ Vf w \ S \ Vf w \ -B \
                         (\<forall>e>0. \<exists>f \<in> R. f ` S \<subseteq> {0..1} \<and> (\<forall>x \<in> S \<inter> Vf w. f x < e) \<and> (\<forall>x \<in> S \<inter> B. f x > 1 - e))"
    by metis
  then have open_Vf: "\w. w \ A \ open (Vf w)"
    by blast
  have tVft: "\w. w \ A \ w \ Vf w"
    using Vf by blast
  then have sum_max_0: "A \ (\x \ A. Vf x)"
    by blast
  have com_A: "compact A" using A
    by (metis compact compact_Int_closed inf.absorb_iff2)
  obtain subA where subA: "subA \ A" "finite subA" "A \ (\x \ subA. Vf x)"
    by (blast intro: that compactE_image [OF com_A open_Vf sum_max_0])
  then have [simp]: "subA \ {}"
    using \<open>a \<in> A\<close> by auto
  then have cardp: "card subA > 0" using subA
    by (simp add: card_gt_0_iff)
  have "\w. w \ A \ \f \ R. f ` S \ {0..1} \ (\x \ S \ Vf w. f x < e / card subA) \ (\x \ S \ B. f x > 1 - e / card subA)"
    using Vf e cardp by simp
  then obtain ff where ff:
         "\w. w \ A \ ff w \ R \ ff w ` S \ {0..1} \
                         (\<forall>x \<in> S \<inter> Vf w. ff w x < e / card subA) \<and> (\<forall>x \<in> S \<inter> B. ff w x > 1 - e / card subA)"
    by metis
  define pff where [abs_def]: "pff x = (\w \ subA. ff w x)" for x
  have pffR: "pff \ R"
    unfolding pff_def using subA ff by (auto simp: intro: prod)
  moreover
  have pff01: "pff x \ {0..1}" if t: "x \ S" for x
  proof -
    have "0 \ pff x"
      using subA cardp t ff
      by (fastforce simp: pff_def field_split_simps sum_nonneg intro: prod_nonneg)
    moreover have "pff x \ 1"
      using subA cardp t ff
      by (fastforce simp add: pff_def field_split_simps sum_nonneg intro: prod_mono [where g = "\x. 1", simplified])
    ultimately show ?thesis
      by auto
  qed
  moreover
  { fix v x
    assume v: "v \ subA" and x: "x \ Vf v" "x \ S"
    from subA v have "pff x = ff v x * (\w \ subA - {v}. ff w x)"
      unfolding pff_def  by (metis prod.remove)
    also have "... \ ff v x * 1"
    proof -
      have "\i. i \ subA - {v} \ 0 \ ff i x \ ff i x \ 1"
        by (metis Diff_subset atLeastAtMost_iff ff image_subset_iff subA(1) subsetD x(2))
      moreover have "0 \ ff v x"
        using ff subA(1) v x(2) by fastforce
      ultimately show ?thesis
        by (metis mult_left_mono prod_mono [where g = "\x. 1", simplified])
    qed
    also have "... < e / card subA"
      using ff subA(1) v x by auto
    also have "... \ e"
      using cardp e by (simp add: field_split_simps)
    finally have "pff x < e" .
  }
  then have "\x. x \ A \ pff x < e"
    using A Vf subA by (metis UN_E contra_subsetD)
  moreover
  { fix x
    assume x: "x \ B"
    then have "x \ S"
      using B by auto
    have "1 - e \ (1 - e / card subA)^card subA"
      using Bernoulli_inequality [of "-e / card subA" "card subA"] e cardp
      by (auto simp: field_simps)
    also have "... = (\w \ subA. 1 - e / card subA)"
      by (simp add: subA(2))
    also have "... < pff x"
    proof -
      have "\i. i \ subA \ e / real (card subA) \ 1 \ 1 - e / real (card subA) < ff i x"
        using e         by (auto simp: field_split_simps)       (lessd) "
      then show have"(f x - f (k/n))\ \ e/2 + 2 * M / d\<^sup>2 * (x - k/n)\<^sup>2"
        using lessd
        unfolding pff_def by (smt ( using  by ( addabs_minus_commute)
           using
    finally " -e .
  }
  ultimately show ?thesis by blast
qed

lemma function_ring_on:
  assumes:"losedA A\ S"
      and B: "closed B" "B \ S"
      and disj: "A \ B = {}"
      and e: "0 < e" "e < 1"
    shows "\f \ R. f ` S \ {0..1} \ (\x \ A. f x < e) \ (\x \ B. f x > 1 - e)"
proof (cases "A \ {} \ B \ {}")
  case True then show ?thesis
    using assms
(forcemeson()kn

  case
  then have.\lee      java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
  then show ?thesis
  proof cases
    case 1
    with e show ?thesis
      by (rule_tac x="\x. 1" in bexI) (auto simp: const)
  next
    case 2
    with e show ?thesis
      by (rule_tac x="\x. 0" in bexI) (auto simp: const)
  qed
qed

text\<open>The special case where \<^term>\<open>f\<close> is non-negative and \<^term>\<open>e<1/3\<close>\<close>
lemma (in function_ring_on) Stone_Weierstrass_special:
  assumes f: "continuous_on S f" and fpos: "\x. x \ S \ f x \ 0"
      and e: "0 < e" "e < 1/3"
  shows "\g \ R. \x\S. \f x - g x\ < 2*e"
proof -
  define n where "n = 1 + nat \normf f / e\"
  define A where "A j = {x \ S. f x \ (j - 1/3)*e}" for j :: nat
  define B where "B j = {x \ S. f x \ (j + 1/3)*e}" for j :: nat
  have ngt: "(n-1) * e \ normf f"
    using e pos_divide_le_eq real_nat_ceiling_ge[of "normf f / e"]
    by (fastforce simp add: divide_simps n_def)
  moreover have "n\1"
    by (simp_all add: n_def)
  ultimately have ge_fx: "(n-1) * e \ f x" if "x \ S" for x
    using f normf_upper that by fastforce
  have "closed S"
    by (simp add: compact compact_imp_closed)
  { fix j
    have "closed (A j)" "A j \ S"
      using \<open>closed S\<close> continuous_on_closed_Collect_le [OF f continuous_on_const]
      by (simp_all add: A_def Collect_restrict)
    moreover have "closed (B j)" "B j \ S"
      using \<open>closed S\<close> continuous_on_closed_Collect_le [OF continuous_on_const f]
      by (simp_all add: B_def Collect_restrict)
    moreover have "(A j) \ (B j) = {}"
      using e by (auto simp: A_def B_def field_simps)
    ultimately have "\f \ R. f ` S \ {0..1} \ (\x \ A j. f x < e/n) \ (\x \ B j. f x > 1 - e/n)"
      using e \<open>1 \<le> n\<close> by (auto intro: two)
  }
  then obtain xf where xfR: "\j. xf j \ R" and xf01: "\j. xf j ` S \ {0..1}"
                   and xfA: "\x j. x \ A j \ xf j x < e/n"
                   and xfB: "\x j. x \ B j \ xf j x > 1 - e/n"
    by metis
  define g where [abs_def]: "g x = e * (\i\n. xf i x)" for x
  have gR: "g \ R"
    unfolding g_def by (fast intro: mult const sum xfR)
  have gge0 "\x. x \ S \ g x \ 0"
    usingxf01 ( add: g_def zero_le_mult_iff image_subset_iff sum_nonneg
  have: "A 0 ={}
     fpos( simp)
  havebyrule)
    using e ngtusing
have: " A i" if "i\j" for i j
    using e that by (force simp: A_def intro: order_transdistrib Rings.semiring_class.distrib_right sum_distrib_left [symmetric] mult.assoc sum_bern
  { fix t
    assume t: "t \ S"
    define j where       \<open>d>0\<close> x by (simp add: divide_simps \<open>M\<ge>0\<close> mult_le_one mult_left_le). <e
    have jn: "j \ n"
      using t An by (simp add: Least_le j_def)
    havefinally have "\f x - (\k\n. f (real k / real n) * Bernstein n k x)\ < e" .
      using t An by (fastforce simp add: j_def intro: LeastI)
    then have Ai: "t \ A i" if "i\j" for i
      using  [OF]byjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
    then BrosowskiFrank.
      by (simp add: A_def)
have: " <>A"ifj  i
      using  Aj \<open>i<j\<close> not_less_Least by (fastforce simp add: j_def)
    have j1: "1 \ j"
      using A0 Aj j_def not_less_eq_eq by (fastforce simp add: j_def)
    then have Anj: "t \ A (j-1)"
      using Least_le by (fastforce simp add: j_def)
    then have fj2: "(j - 4/3)*e < f t"
      using j1 t  by (simp add: A_def of_nat_diff)
    have xf_le1: "\i. xf i t \ 1"
      using xf01 t by force
    have "g t = e * (\i\n. xf i t)"
      by (simp add: g_def flip: distrib_left)
    also have "... = e * (\i \ {.. {j..n}. xf i t)"
      by (simp add: ivl_disj_un_one(4) jn)
    also have "... = e * (\ii=j..n. xf i t)"
      by (simp add: distrib_left ivl_disj_int sum.union_disjoint)
    also have "... \ e*j + e * ((Suc n - j)*e/n)"
    proof (intro add_mono mult_left_mono)
      show "(\i j"
        by (rule sum_bounded_above [OF xf_le1, where A = "lessThan j", simplified])
      have "xf i t \ e/n" if "i\j" for i
        using xfA [OF Ai] that by (simp add: less_eq_real_def)
      then show "(\i = j..n. xf i t) \ real (Suc n - j) * e / real n"
        using sum_bounded_above [of "{j..n}" "\i. xf i t"]
        by fastforce
    qed (use e in auto)
    also have "... \ j*e + e*(n - j + 1)*e/n "
      using \<open>1 \<le> n\<close> e  by (simp add: field_simps del: of_nat_Suc)
    also have "... \ j*e + e*e"
      using \<open>1 \<le> n\<close> e j1 by (simp add: field_simps del: of_nat_Suc)
    alsohave.  j+/)e
      using ( simp)
    finally locale =
    have ( - 43*  g t"
    proof "2 \ j")
casejava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
      then "="using by simp
      with t gge0 e show ?thesis by force
    next
      case True
      then have "(j - 4/3)*e < (j-1)*e - e^2"
        using e by (auto simp: of_nat_diff algebra_simps power2_eq_square)
      also have "... < (j-1)*e - ((j - 1)/n) * e^2"
      proof -
        have "(j - 1) / n < 1"
          using j1 jn by fastforce
        with \<open>e>0\<close> show ?thesis
          by (smt (verit, best) mult_less_cancel_right2 zero_less_power)
      qed
      also have "... = e * (j-1) * (1 - e/n)"
        by (simp add: power2_eq_square field_simps)
      also have "... \ e * (\i\j-2. xf i t)"
      proof -
        { fix i
          assume "i+2 \ j"
          then obtain d where "i+2+d = j"
            using le_Suc_ex that by blast
          then have "t \ B i"
            using Anj e ge_fx [OF t] \<open>1 \<le> n\<close> fpos [OF t] t
            unfolding A_def B_def
            by (auto simp add: field_simps of_nat_diff not_le intro: order_trans [of _ "e * 2 + e * d * 3 + e * i * 3"])
          then have "xf i t > 1 - e/n"
            by (rule xfB)
        }
        moreover have "real (j - Suc 0) * (1 - e / real n) \ real (card {..j - 2}) * (1 - e / real n)"
          using Suc_diff_le True by fastforce
        ultimately show ?thesis
          using e True by (auto intro: order_trans [OF _ sum_bounded_below [OF less_imp_le]])
      qed
      also have "... \ g t"
        using jn e xf01 t
        by (auto intro!: Groups_Big.sum_mono2 simp add: g_def zero_le_mult_iff image_subset_iff sum_nonneg)
      finally show ?thesis .
    qed
    have "\f t - g t\ < 2 * e"
      using fj1 fj2 gj1 gj2 by (simp add: abs_less_iff field_simps)
  }
  then show ?thesis
    by (rule_tac x=g java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 0
qed    unfoldingdiff_conv_add_uminus add)

unpretentious\<close
(in) Stone_Weierstrass_basic
   f:" S f"and
  shows "\g \ R. \x\S. \f x - g x\ < e"
proof -
  have "\g \ R. \x\S. \(f x + normf f) - g x\ < 2 * min (e/2) (1/4)"
  proof (rule Stone_Weierstrass_special)
    show "continuous_on S (\x. f x + normf f)"
      by (force intro: Limits.continuous_on_add [OF f Topological_Spaces.continuous_on_const])
    show "\x. x \ S \ 0 \ f x + normf f"
      using normf_upper [OF f] by force
  qed (use e in auto)
  then obtain g where "g \ R" "\x\S. \g x - (f x + normf f)\ < e"
    by force
  then show ?thesis
    by (rule_tac x="\x. g x - normf f" in bexI) (auto simp: algebra_simps intro: diff const)
qed

theorem (in function_ring_on) Stone_Weierstrass:
  assumes f: "continuous_on S f"
   "\F\UNIV \ R. LIM n sequentially. F n :> uniformly_on S f"
proof -
  define h where "h where" f \<equiv> SUP x\<in>S. \<bar>f x\<bar>"
  show
  proof
    { fix e::real
      assume e: "0 < e"
            ( add(1)  compact compact_imp_bounded)
        by (auto simp assms normf_def
      { fix n :: nat and x :: 'a and g :: "'a \<Rightarrow> real"
        assume n: "N \ n" "\x\S. \f x - g x\ < 1 / (1 + real n)"
        assume x: "x \ S"
        have "\ real (Suc n) < inverse e"
          using \<open>N \<le> n\<close> N using less_imp_inverse_less by force
        then have "1 / (1 + real n) \ e"
          using e by (simp add: field_simps)
        then have "\f x - g x\ < e"
          using n(2) x by auto
      }
      then have "\\<^sub>F n in sequentially. \x\S. \f x - h n x\ < e"
        unfolding h_def
        by (force intro: someI2_bex [OF Stone_Weierstrass_basic [OF f]] eventually_sequentiallyI [of N])
    }
    then show "uniform_limit S h f sequentially"
      unfolding uniform_limit_iff by (auto simp: dist_norm abs_minus_commute)
    show "h \ UNIV \ R"
      unfolding h_def by (force intro: someI2_bex [OF Stone_Weierstrass_basic [OF f]])
  qed
qed

textend
corollary Stone_Weierstrass_HOL:
  fixes R :: "('a::t2_space java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  assumes "compact "\<exists>V. open V \<and> t0 \<in> V \<and> S \<inter> V \<subseteq> U \<and>               (<>e>0.\<exists>f \<in> R. f ` S \<subseteq> {0..1} \<and> (\<forall>t \<in> S \<inter> V. f t < e) \<and> (\<forall>t \<in> S - U. f t > 1 - e))"
          "\f. P f \ continuous_on S f"
          "\f g. P(f) \ P(g) \ P(\x. f x + g x)" "\f g. P(f) \ P(g) \ P(\x. f x * g x)"
          "\x y. x \ S \ y \ S \ x \ y \ \f. P(f) \ f x \ f y"
          "continuous_on S f"
       "0 < e"
    shows     h where abs_def   t0 x
java.lang.StringIndexOutOfBoundsException: Range [31, 7) out of bounds for length 7
  interpret PR: function_ring_on "Collect P"
    by unfold_locales (use assms in auto)
  show ?thesis
    using PR.Stone_Weierstrass_basic [OF \<open>continuous_on S f\<close> \<open>0 < e\<close>]
    by blast
qed

subsection

inductive real_polynomial_function :: t normf_upper[wherecontinuoushsqjava.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
    linear: "bounded_linear f \ real_polynomial_function f"
  | const: "real_polynomial_function (\x. c)"
  | add:   "\real_polynomial_function f; real_polynomial_function g\ \ real_polynomial_function (\x. f x + g x)"
  | mult:  "\real_polynomial_function f; real_polynomial_function g\ \ real_polynomial_function (\x. f x * g x)"    with

declare real_polynomial_function.then 1-(*\<delta> / 2)^n \<le> 1 - (k * p t)^n"

definition\<^marker>\<open>tag important\<close> polynomial_function :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> bool" [of)) k"
  where
   "polynomial_function p \ (\f. bounded_linear f \ real_polynomial_function (f o p))"

lemma real_polynomial_function_eq: "real_polynomial_function p = polynomial_function p"
unfolding polynomial_function_def
proof
  assume "real_polynomial_function p"
  then
  proof (induction p    note = this
    case (linear tjava.lang.NullPointerException
      by (by( addpt_
  next
     ( h) then ?case
olynomial_function
  next
    case (add h) then show ?case
      by (force simp add: bounded_linear_def linear_add real_polynomial_function.add)
  next
    case (mult h) then show ?case
      by (force simp add: real_bounded_linear const real_polynomial_function.mult)
  qed
next
  assume [rule_format, OF bounded_linear_ident]: "\f. bounded_linear f \ real_polynomial_function (f \ p)"
  then show "real_polynomial_function p"
    by (simp add: o_def)
qed

lemma polynomial_function_const [iff]: "polynomial_function (\x. c)"
  by (simp add: polynomial_function_def o_def const)

lemma polynomial_function_bounded_linear:
  "bounded_linear f \ polynomial_function f"
  by (simp add: polynomial_function_def o_def bounded_linear_compose real_polynomial_function.linear)

lemma polynomial_function_id [iff]: "polynomial_function(\x. x)"
  by (simp add: polynomial_function_bounded_linear)

lemma polynomial_function_add [intro]:
    "\polynomial_function f; polynomial_function g\ \ polynomial_function (\x. f x + g x)"
  by (auto simp: polynomial_function_def bounded_linear_def linear_add real_polynomial_function.add o_def)

lemma polynomial_function_mult [intro]:
  assumes f: "polynomial_function f" and g: "polynomial_function g"
  shows "polynomial_function (\x. f x *\<^sub>R g x)"
proof -
  have "real_polynomial_function (\x. h (g x))" if "bounded_linear h" for h
    using g that unfolding polynomial_function_def o_def bounded_linear_def
    by (auto simp: real_polynomial_function_eq)
  moreover have "real_polynomial_function f"
    by (simp add: f real_polynomial_function_eq)
  ultimately show ?thesis
    unfolding polynomial_function_def bounded_linear_def o_def
    by (auto simp: linear.scaleR)
qed

lemma polynomial_function_cmul [intro]:
  assumes f: "polynomial_function f"
    shows "polynomial_function (\x. c *\<^sub>R f x)"
  by (rule polynomial_function_mult [OF polynomial_function_const f])

lemma polynomial_function_minus [intro]:
  assumes f: "polynomial_function f"
    shows "polynomial_function (\x. - f x)"
  using polynomial_function_cmul [OF f, of "-1"] by simp

lemma polynomial_function_diff [intro]:
    "\polynomial_function f; polynomial_function g\ \ polynomial_function (\x. f x - g x)"
  unfolding add_uminus_conv_diff [symmetric]
by( polynomial_function_add polynomial_function_minus

polynomial_function_sum]:
pt_pos t]\<open>k>0\<close>
(nduct: finite_induct

lemma real_polynomial_function_minus [intro]:
    "real_polynomial_function f \ real_polynomial_function (\x. - f x)"
  using polynomial_function_minus [of f]
  by (simp add: real_polynomial_function_eq)

lemma real_polynomial_function_diff [intro]:
    "\real_polynomial_function f; real_polynomial_function g\ \ real_polynomial_function (\x. f x - g x)"
  using polynomial_function_diff [of f]
  by (simp add: real_polynomial_function_eq)

lemma real_polynomial_function_divide [intro]:
  assumes "real_polynomial_function p" shows "real_polynomial_function (\x. p x / c)"
proof -
  have "real_polynomial_function (\x. p x * Fields.inverse c)"
    using assms by auto
  then show ?thesis
    by (simp add: divide_inverse)
qed

lemma real_polynomial_function_sum [intro]:
    "\finite I; \i. i \ I \ real_polynomial_function (\x. f x i)\ \ real_polynomial_function (\x. sum (f x) I)"
  using polynomial_function_sum [of I f]
  by (simp add: real_polynomial_function_eq)

lemma real_polynomial_function_prod [intro]:
  "\finite I; \i. i \ I \ real_polynomial_function (\x. f x i)\ \ real_polynomial_function (\x. prod (f x) I)"
  by (induct I rule: finite_induct) auto

lemma real_polynomial_function_gchoose:
  obtains p where "real_polynomial_function p" "\x. x gchoose r = p x"
proof
  show "real_polynomial_function (\x. (\i = 0..
    by force
qed (simp add: gbinomial_prod_rev)

lemma real_polynomial_function_power [intro]:
    "real_polynomial_function f \ real_polynomial_function (\x. f x^n)"
  by (induct n) (simp_all add: const mult)

lemma real_polynomial_function_compose [intro]:
  assumes f: "polynomial_function f" and g: "real_polynomial_function g"
    shows "real_polynomial_function (g o f)"
  using g
proof (induction g rule: real_polynomial_function.induct)
  case (linear f)
  then show ?case
    using f polynomial_function_def by blast
next
  case (add f g)
  then show ?case
    using f add by (auto simp: polynomial_function_def)
next
  case (mult f g)
  then show ?case
  using f mult by (auto simp: polynomial_function_def)
qed auto

lemma polynomial_function_compose [intro]:
  assumes f: "polynomial_function f" and g: "polynomial_function g"
    shows "polynomial_function (g o f)"
  using g real_polynomial_function_compose [OF f]
  by (auto simp: polynomial_function_def o_def)

lemma sum_max_0:
  fixes x::real (*in fact "'a::comm_ring_1"*)
  shows "(\i\max m n. x^i * (if i \ m then a i else 0)) = (\i\m. x^i * a i)"
proof -
  have "(\i\max m n. x^i * (if i \ m then a i else 0)) = (\i\max m n. (if i \ m then x^i * a i else 0))"
    by (auto simp: algebra_simps intro: sum.cong)
  also have "... = (\i\m. (if i \ m then x^i * a i else 0))"
    by (rule sum.mono_neutral_right) auto
  also have "... = (\i\m. x^i * a i)"
    by (auto simp: algebra_simps intro: sum.cong)
  finally show ?thesis .
qed

lemma real_polynomial_function_imp_sum:
  assumes "real_polynomial_function f"
    shows "\a n::nat. f = (\x. \i\n. a i * x^i)"
using assms
proof (induct f)
  case (linear f)
  then obtain c where f: "f = (\x. x * c)"
    by (auto simp add: real_bounded_linear)
  have "x * c = (\i\1. (if i = 0 then 0 else c) * x^i)" for x
    by (simp add: mult_ac)
  with f show ?case
    by fastforce
next
  case (const c)
  have "c = (\i\0. c * x^i)" for x
    by auto
  then show ?case
    by fastforce
  case (add f1 f2)
  then obtain a1 n1 a2 n2 where
    "f1 = (\x. \i\n1. a1 i * x^i)" "f2 = (\x. \i\n2. a2 i * x^i)"
    by auto
  then have "f1 x + f2 x = (\i\max n1 n2. ((if i \ n1 then a1 i else 0) + (if i \ n2 then a2 i else 0)) * x^i)"
      for x
    using sum_max_0 [where m=n1 and n=n2] sum_max_0 [where m=n2 and n=n1]
    by (simp add: sum.distrib algebra_simps max.commute)
  then show ?case
    by force
  case (mult f1 f2)
  then obtain a1 n1 a2 n2 where
    "f1 = (\x. \i\n1. a1 i * x^i)" "f2 = (\x. \i\n2. a2 i * x^i)"
    by auto
  then obtain b1 b2 where
    "f1 = (\x. \i\n1. b1 i * x^i)" "f2 = (\x. \i\n2. b2 i * x^i)"
    "b1 = (\i. if i\n1 then a1 i else 0)" "b2 = (\i. if i\n2 then a2 i else 0)"
    by auto
  then have "f1 x * f2 x = (\i\n1 + n2. (\k\i. b1 k * b2 (i - k)) * x ^ i)" for x
    using polynomial_product [of n1 b1 n2 b2] by (simp add: Set_Interval.atLeast0AtMost)
  then show ?case
    by force
qed

lemma real_polynomial_function_iff_sum:
     "real_polynomial_function f \ (\a n. f = (\x. \i\n. a i * x^i))" (is "?lhs = ?rhs")
proof
  assume ?lhs then show ?rhs
    by (metis real_polynomial_function_imp_sum)
next
  assume ?rhs then show ?lhs
    by (auto simp: linear mult const real_polynomial_function_power real_polynomial_function_sum)
qed

lemma polynomial_function_iff_Basis_inner:
  fixes f :: "'a::real_normed_vector \ 'b::euclidean_space"
  shows "polynomial_function f \ (\b\Basis. real_polynomial_function (\x. inner (f x) b))"
        (is "?lhs = ?rhs")
unfolding polynomial_function_def
proof (intro iffI allI impI)
  assume "\h. bounded_linear h \ real_polynomial_function (h \ f)"
  then show ?rhs
    by (force simp add: bounded_linear_inner_left o_def)
next
  fix h :: "'b \ real"
  assume rp: "\b\Basis. real_polynomial_function (\x. f x \ b)" and h: "bounded_linear h"
  have "real_polynomial_function (h \ (\x. \b\Basis. (f x \ b) *\<^sub>R b))"
    using rp
    by (force simp: real_polynomial_function_eq polynomial_function_mult
              intro!: real_polynomial_function_compose [OF _  linear [OF h]])
  then show "real_polynomial_function (h \ f)"
    by (simp add: euclidean_representation_sum_fun)
qed

subsection \<open>Stone-Weierstrass theorem for polynomial functions\<close>

text\<open>First, we need to show that they are continuous, differentiable and separable.\<close>

lemma continuous_real_polymonial_function:
  assumes "real_polynomial_function f"
    shows "continuous (at x) f"
using assms
by (induct f) (auto simp: linear_continuous_at)

lemma continuous_polymonial_function:
  fixes f :: "'a::real_normed_vector \ 'b::euclidean_space"
  assumes "polynomial_function f"
    shows "continuous (at x) f"
proof (rule euclidean_isCont)
  show "\b. b \ Basis \ isCont (\x. (f x \ b) *\<^sub>R b) x"
  using assms continuous_real_polymonial_function
  by (force simp: polynomial_function_iff_Basis_inner intro: isCont_scaleR)
qed

lemma continuous_on_polymonial_function:
  fixes f :: "'a::real_normed_vector \ 'b::euclidean_space"
  assumes "polynomial_function f"
    shows "continuous_on S f"
  using continuous_polymonial_function [OF assms] continuous_at_imp_continuous_on
  by blast

lemma has_real_derivative_polynomial_function:
  assumes "real_polynomial_function p"
    shows "\p'. real_polynomial_function p' \
                 (\<forall>x. (p has_real_derivative (p' x)) (at x))"
using assms
proof (induct p)
  case (linear p)
  then show ?case
    by (force simp: real_bounded_linear const intro!: derivative_eq_intros)
next
  case (const c)
  show ?case
    by (rule_tac x="\x. 0" in exI) auto
  case (add f1 f2)
  then obtain p1 p2 where
    "real_polynomial_function p1" "\x. (f1 has_real_derivative p1 x) (at x)"
    "real_polynomial_function p2" "\x. (f2 has_real_derivative p2 x) (at x)"
    by auto
  then show ?case
    by (rule_tac x="\x. p1 x + p2 x" in exI) (auto intro!: derivative_eq_intros)
  case (mult f1 f2)
  then obtain p1 p2 where
    "real_polynomial_function p1" "\x. (f1 has_real_derivative p1 x) (at x)"
    "real_polynomial_function p2" "\x. (f2 has_real_derivative p2 x) (at x)"
    by auto
  then show ?case
    using mult
    by (rule_tac x="\x. f1 x * p2 x + f2 x * p1 x" in exI) (auto intro!: derivative_eq_intros)
qed

lemma has_vector_derivative_polynomial_function:
  fixes p :: "real \ 'a::euclidean_space"
  assumes "polynomial_function p"
  obtains p' where "polynomial_function p'" "\<And>x. (p has_vector_derivative (p' x)) (at x)"
proof -
  { fix b :: 'a
    assume "b \ Basis"
    then
    obtain p' where p': "real_polynomial_function p'" and pd: "\x. ((\x. p x \ b) has_real_derivative p' x) (at x)"
      using assms [unfolded polynomial_function_iff_Basis_inner] has_real_derivative_polynomial_function
      by blast
    have "polynomial_function (\x. p' x *\<^sub>R b)"
      using \<open>b \<in> Basis\<close> p' const [where 'a=real and c=0]
      by (simp add: polynomial_function_iff_Basis_inner inner_Basis)
    then have "\q. polynomial_function q \ (\x. ((\u. (p u \ b) *\<^sub>R b) has_vector_derivative q x) (at x))"
      by (fastforce intro: derivative_eq_intros pd)
  }
  then obtain qf where qf:
      "\b. b \ Basis \ polynomial_function (qf b)"
      "\b x. b \ Basis \ ((\u. (p u \ b) *\<^sub>R b) has_vector_derivative qf b x) (at x)"
    by metis
  show ?thesis
  proof
    show "\x. (p has_vector_derivative (\b\Basis. qf b x)) (at x)"
      apply (subst euclidean_representation_sum_fun [of p, symmetric])
      by (auto intro: has_vector_derivative_sum qf)
  qed (force intro: qf)
qed

lemma real_polynomial_function_separable:
  fixes x :: "'a::euclidean_space"
  assumes "x \ y" shows "\f. real_polynomial_function f \ f x \ f y"
proof -
  have "real_polynomial_function (\u. \b\Basis. (inner (x-u) b)^2)"
  proof (rule real_polynomial_function_sum)
    show "\i. i \ Basis \ real_polynomial_function (\u. ((x - u) \ i)\<^sup>2)"
      by (auto simp: algebra_simps real_polynomial_function_diff const linear bounded_linear_inner_left)
  qed auto
  moreover have "(\b\Basis. ((x - y) \ b)\<^sup>2) \ 0"
    using assms by (force simp add: euclidean_eq_iff [of x y] sum_nonneg_eq_0_iff algebra_simps)
  ultimately show ?thesis
    by auto
qed

lemma Stone_Weierstrass_real_polynomial_function:
  fixes f :: "'a::euclidean_space \ real"
  assumes "compact S" "continuous_on S f" "0 < e"
  obtains g where "real_polynomial_function g" "\x. x \ S \ \f x - g x\ < e"
proof -
  interpret PR: function_ring_on "Collect real_polynomial_function"
  proof unfold_locales
  qed (use assms continuous_on_polymonial_function real_polynomial_function_eq
       in \<open>auto intro: real_polynomial_function_separable\<close>)
  show ?thesis
    using PR.Stone_Weierstrass_basic [OF \<open>continuous_on S f\<close> \<open>0 < e\<close>] that by blast
qed

theorem :
  fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
  assumes S: "compact S"
      and f: "continuous_on f"
      and e: "0 < e"
    shows "java.lang.StringIndexOutOfBoundsException: Range [0, 53) out of bounds for length 17

t  Vf(
    assume "b \ Basis"
     java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
    proofnote=
      show "then have [simp]: "subA {}assume t: y auto
        using f by (auto intro:with java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    :
  java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  then:
      < \<>Basis
    by metis
  let ?g = "\x. \b\Basis. pf b x *\<^sub>R b"
  { fix x
    assumealsohave .
    have "norm show " \<le> 1 / (real k * p t)^n * (1 - p t^n)^k^n"
       ule
    havejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    proof (rule   \<in> {0..1}" if t: "x \<in> S" for x
showjava.lang.StringIndexOutOfBoundsException: Index 130 out of bounds for length 130
         simp:u java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
     (rule)
    also have "... = e"
      using sum_nonneg  [where"
    finally have "norm (\b\Basis. (f x \ b) *\<^sub>R b - pf b x *\<^sub>R b) < e" .
  }
  <\<in>S. norm ((\<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) - ?g x) < e"
      mojava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
  have
    using by(imp:polynomial_function_sum)*ln *java.lang.NullPointerException
  ultimately  "..< java.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 89
      ?thesis
qed

proposition(1/ultimatelyjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
   f "0< ". e/ "
assumes: compact
    and f "ontinuous_onSf"  e(add
  obtains     "   " .
oof
  havepos "by ( contra_subsetDjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
   gwherejava.lang.StringIndexOutOfBoundsException: Index 133 out of bounds for length 133
Stone_Weierstrass_polynomial_function ]
by
  have "show"1-q )  ejava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
  proof (rule uniform_limitI
    fix e::real ejava.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94
w LIMSEQ_inverse_real_of_nat"
      by (usingjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
      java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
      using
a B:closed"" <subseteq> S" "
      by) auto
   A:" A" "
  then :" B"" disj A
qed

subsectionthenthesis

text
or assume"

lemma\>e>0. \<exists>f \<in> R. f ` S \<subseteq> {0..1} \<and> (\<forall>x \<in> S \<inter> V. f x < e) \<and> (\<forall>x \<in> S \<inter> B. f x > 1 - e))" eshow?thesis
    fixes e show?
shows\<Longrightarrow> path g"
  bysimp: continuous_on_polymonial_function

lemma path_approx_polynomial_function:

    assumes f:" S f"  fposVf:
     where "" p = pathstart" pathfinishp= "
                    "\t. t \ {0..1} \ norm(p t - g t) < e"
proof
qwhere: "olynomial_functionq"  noq\<And>x. x \<in> {0..1} \<Longrightarrow> norm (g x - q x) < e/4"
    using Stone_Weierstrass_polynomial_function [ngtjava.lang.StringIndexOutOfBoundsException: Range [19, 17) out of bounds for length 37
    by (auto simp( add byjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
   where blast
    ( add compact_imp_closed
  proofjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
java.lang.StringIndexOutOfBoundsException: Range [93, 33) out of bounds for length 33
      byforce : poq)
    show norm -) <)
       " have "(A
java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 11
      have *: "norm obtain ff where \ n\ by (auto intro: two)
  then xf xfRjava.lang.StringIndexOutOfBoundsException: Index 151 out of bounds for length 151
        show    by: "
            have gge0:
        showhave"
          using noq that  A0"0 java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
          by show
        show norm
          using noqjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
          by simp tunfoldingmetis)
      qedproofjava.lang.StringIndexOutOfBoundsException: Range [11, 12) out of bounds for length 11
have
by [ java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 35
    qed
  qed :)
qed

proposition1
  fixes S ::  A0java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 67
assumes   Sqed   real_polynomial_function_eq
      and "x \ S" "y \ S"
    u xf01java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 14
proof -
S assms
    by (simphen"\ S"
with
     (orce simp)
   "
  proof (cases "S =b auto: field_simpsjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
    case       .pff
      by (      "\i j"
  next
    ase
    showjava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
    proof (intro exI java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 7
       \<And>x. x \<in> path_image p \<Longrightarrow> ball x (setdist (path_image p) (-S)) \<subseteq> S"
        usingjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
      show "0 < setdist (path_image p) (- S)"
        using S p False
        byfinally gj1 ("A \ {} \ B \ {}")
    qed
  qed
c False

  obtain where and:"athstartt ?java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
                   and pf_e eby
     path_approx_polynomial_functionOF
  show ?thesis  qed
  proofintro)
pf

   show=pf
      by (simp_all
    show "path_image efinen where. e (<> proof-
      path_image_def
    proof clarsimp
      fix x'::real
      assume0\<le> x'" "x' \<le> 1"
      then have "dist (p x') (pf x') < e"
        ( atLeastAtMost_iff dist_normjava.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
      then show "pfby simp_alla B_def
        bymetisjava.lang.StringIndexOutOfBoundsException: Index 133 out of bounds for length 133
qed
  qed
moreover " **- /ealn

     e "(Aj)"j
   "f differentiable (have".
    (\<forall>i \<in> Basis. (\<lambda>x. f x \<bullet> i) differentiable at a within S)" \<open>closed S\<close> continuous_on_closed_Collect_le [OF continuous_on_const f] ! . simp   sum_nonneg
proofusing  auto:A_def       using e by (auto simp: A_def B_def field_simps
{assume\have<
    then obtain f' where f':
           "\i. i \ Basis \ ((\x. f x \ i) has_derivative f' i) (at a within S)"
      by metis
    haveeq"\.(\j\Basis. f' j x *\<^sub>R j) \ i) = f' i" if "i \ Basis" for i
      using that
    have"\D. \i\Basis. ((\x. f x \ i) has_derivative (\x. D x \ i)) (at a within S)"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      apply( add f')
      done
  }
  then show ?thesis
    show -
    using has_derivative_componentwise_within
    by
qed

lemma polynomial_function_inner [    byforce
  fixes i :: "'a::euclidean_space"
  shows "polynomial_function g \ polynomial_function (\x. g x \ i)"
  apply( euclidean_representationwhere, ])qed
  apply (force simp: inner_sum_right java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  done

text

lemma:
   "real_polynomial_function f \ f differentiable (at a within S)"
  assumee:"0< java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
     (simp_all add: bounded_linear_imp_differentiable)

lemma differentiable_on_real_polynomial_function:
   "real_polynomial_function p \ p differentiable_on S"
by (simp add: differentiable_at_imp_differentiable_on differentiable_at_real_polynomial_function)

lemma differentiable_at_polynomial_function:
  fixes f :: "_ \ 'a::euclidean_space"
  shows "polynomial_function f \ f differentiable (at a within S)"
eby( add field_simps

lemma:
  fixes f then"
  shows "polynomial_function f \ f differentiable_on S"
by  unfolding

lemma vector_eq_dot_span:
  assumes   showjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
  shows "x = y"
proof
  \<> Light
    Stone_Weierstrass_HOL
  moreover have "x - y \ span B"
    by simp
  ultimately have "x - y = g whereyx
    using orthogonal_to_span orthogonal_self pos
    then show ?thesis by simp
qed

lemma orthonormal_basis_expand:
  assumes
and 1:"
      and "x \ span B"
      java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
     "(i\B. (x \ i) *\<^sub>R i) = x"
proof
  show "(\i\B. (x \ i) *\<^sub>R i) \ span B"
: " \\<^sub>F n in sequentially. \x\S. dist (g n x) (f x) < e"
showi  java.lang.NullPointerException
  proof ? using
    have [simp
      using\<open>Polynomial functions as paths\<close>: " p = java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      by (
     "
      bysimp:inner_sum_right
     "polynomial_function g \ path g"
       java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 30
    bysimp:real_polynomial_functiong::" \ 'b::euclidean_space"
    asassumes"path g" "0 < "
      next
  qed

theoremjava.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 7
  fixes       Stone_Weierstrass_polynomial_function0}"g]
  assumes
      and contf  where
      and
      and "subspace T" "f ` S \ T"
    btains o_def real_polynomial_function)
                    "\x. x \ S \ norm(f x - g x) < e"
proof -
  obtain B wherejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
             and B1 f:" norm(t- java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 39
             and "independent B" and cardB"norm( <>1-1)
             andproof  " t*R( g0)e/java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
    by ( norm_minus_commute )
  thenB
    byt show
   obtainnat   java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 75
ing metis
bysimpjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    by (auto      auto simpS" " "Sjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
havejava.lang.NullPointerException
   af:" f"
  have cont path_connected assms
--> --------------------

--> maximum size reached

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

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