SSL Weierstrass_Theorems.thy Interaktion und
PortierbarkeitIsabelle

section

text\<open>By L C Paulson (2015)\<close>

theory Weierstrass_Theorems
imports
begin

\^>\<open>tag important\<close> Bernstein :: "[nat,nat,real] \<Rightarrow> real" wheresimp:Bernstein_def Bernstein_pos"<>0< ; 1; \ n\ \ 0 < Bernstein n k x"
  "Bernsteinn 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)

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

lemma java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
   binomial_ring 1"]
  by (simp add: Bernstein_def)

lemma binomial_deriv1:
    java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ f = \<>.(+)n  x=a])
  apply (subst binomial_ring)
  apply (rule derivative_eq_intros sum.cong | simp add: atMost_atLeast0)+
  done

lemma binomial_deriv2:
    "(\k\n. (of_nat k * of_nat (k-1) * of_nat (n choose k)) * a^(k-2) * b^(n-k)) =
      n * of_nat) * (ab::real)"
  bysimp: power2_eq_square)
apply subst [symmetric
  apply     haveht \<noteq> h t0"
  done

lemma sum_k_Bernstein    then  "h t\
  apply (subst binomial_deriv1 [of n x "1-x", simplified, symmetric])
  apply (simp add: sum_distrib_right)
  apply (auto  [ x=]continuous] byjava.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

lemmaunfolding by fast:hsq const)
proof
  have  ( add  h_def
moreover havep t  0
  proofrule.cong reflsimp
    fix k
    assume " have "<>. x
    then consider "k = 0" | " nfp normf_upper[ continuous [OF ] ] by (autosimp p_def)
      by (metis One_nat_def not0_implies_Suc)
    then show "k = 0 \
          (real k - 1) * Bernstein    ultimately show "\pt \ R. pt t0 = 0 \ pt t > 0 \ pt ` S \ {0..1}"
          "
          (     java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
          by (: that [OF open_Uf
   have[]: "subU\ {}"
  also have " using t1 by auto
    by (subst binomial_deriv2 [of n x "1-x", simplified, symmetric]) (simp add: sum_distrib_right)
  also have ".. have cardp:"ard "usingsubU
    by auto
  finally show ?thesis
    by auto
qed

subsection p where [abs_def]: "p x = (1 / card subU) * (\t \ subU. pf t x)" for x

theorem Bernstein_Weierstrass:
  fixesf :real
  assumes contf: "continuous_on using subU pf by (auto simp: intro: sum.neutral)
shows"N. \n x. N \ n \ x \ {0..1}
                    \<longrightarrow> \<bar>f x - (\<Sum>k\<le>n. f(k/n) * Bernstein n k x)\<bar> < e"proof
proof -
  have    show
    using compact_continuous_image subUt
  then      apply (larsimp simp: p_def field_split_simps)
    by (force algebra_simps
  then have "0 \ M" by force
   ucontf 01 "
    using compact_uniformly_continuous contf by    using t0 pf by (simp add: q_def power_0_left
   obtain   d: "d0"" 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 e by (auto simp: dist_norm)
  { fix    assume t t\in S <inter> V"
    assume n: "Suc (nat\4*M/(e*d\<^sup>2)\) \ n" and x: "0 \ x" "x \ 1"
    have 0<n using nbysimp
have: "- (e * d\<^sup>2) < 0"
      using e     then have "1 - (k * \<delta> / 2)^n \<le> 1 - (k * p t)^n"
    also have "... \ M * 4"
      using \<open>0\<le>M\<close> by simp
    finally have [simp]: "real_of_int (nat \4 * M / (e * d\<^sup>2)\) = real_of_int \4 * M / (e * d\<^sup>2)\"
\<open>0\<le>M\<close> e \<open>0<d\<close>
(simp: field_simps
    have   {fixtand:nat
by(simpadd: real_nat_ceiling_ge
    also have ". with \k>0\ U have "k * \ \ k * p t"
      using n by (simp add:      by (simp: pt_\<delta>)
    finally have nbig: "4*M/(e*d\<^sup>2) + 1 \ real n" .
    have sum_bern: "(\k\n. (x - k/n)\<^sup>2 * Bernstein n k x) = x * (1 - x) / n"
    proof
      have *:      by( intro:less_le_trans
        by (simp: algebra_simps)
      have "(\k\n. (k - n * x)\<^sup>2 * Bernstein n k x) = n * x * (1 - x)"
        apply (simp add: * sum.distrib)
                simp)
         simp:  power2_eq_square
        done
}note  = this
        by (simp NN
      then     where " e = 1+ \max (ln e / ln (real k * \ / 2)) (- ln e / ln (real k * \))\" for e
        using (simp: sum_divide_distrib power2_commute
    qed
    { fixife for
      assume k: "k \ n"
      then have kn: "0 \ k / n" "k / n \ 1"

consider \<bar>x - k / n\<bar> < d" | (ged) "d \<le> \<bar>x - k / n\<bar>"
        by linarith
      then java.lang.NullPointerException
      proof cases
        caselessd
        then have "\(f x - f (k/n))\ < e/2"
          d xknsimp: abs_minus_commute
        also have "... \ (e/2 + 2 * M / d\<^sup>2 * (x - k/n)\<^sup>2)"
    \<open>M\<ge>0\<close> d by simp
        finally show ?thesis have1-e  x
      next
        case  java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
        then have dle
          by (metis (in) two:
        have \<section>: "1 \<le> (x - real k / real n)\<^sup>2 / d\<^sup>2" A:"losed ""
          using dle \<open>d>0\<close> by auto
        have "\(f x - f (k/n))\ \ \f x\ + \f (k/n)\"
              by( simp flip: ex_in_conv intro!: two_special)
        also have "... \ M+M"
          by ( M add_mono_thms_linordered_semiring1  x)
        also
          using \<section> \<open>M\<ge>0\<close> mult_left_mono by fastforce False
        alsohave ".. e/2+ 2* M/d<^sup>2 * (x - k/n)\<^sup>2"
          using e  by simp
        finally show ?thesis .
        qed
    } note * = this:
     e  by simpg_def)
      by (simp A0 }
    alsousing e by fastforce: A_def
       ( sum_abs)
    also have "... \ (\k\n. (e/2 + (2 * M / d\<^sup>2) * (x - k / n)\<^sup>2) * Bernstein n k x)"
       *
      by (force simp add: abs_mult    AsubAjjava.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
    also have "... \ e/2 + (2 * M) / (d\<^sup>2 * n)"
      unfolding sum.java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
using
    also have "..< e"
      using \<open>d>0\<close> nbig e \<open>n>0\<close>
      apply
      using Aj: "t \ A j"
    java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
  }usingAsub that  blast
  then show ?thesis
    by auto
qed

subsection \<open>General Stone-Weierstrass theorem\<close>

text\<open>Source:
Bruno and Frank Deutsch
An Elementary     then Anjt\notin  i if "i<"for
Proceedings of the American
Volume 81, Number  "..<( +13*e"
DOI: 10.230       e by(uto: field_simps

function_ring_on
  fixes R :: "('a have gj2: "j-/3)e<
  assumes compact (cases
  assumes continuous: "f False
  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"
      then havej1 using j1java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38

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 (metis minus

  lemma power: "f
    by (induct n`unpretentious'' formulation>

  lemmaproposition function_ring_on:
    by (induct   assumes continuous_on  e: "e >0"

  lemma prod: "\finite I; \i. i \ I \ f i \ R\ \ (\x. \i \ I. f i x) \ R"
    byshows

  definition\<^marker>\<open>tag important\<close> normf :: "('a::t2_space \<Rightarrow> real) \<Rightarrow> real"
     normf

  lemma normf_upper:
    assumes "continuous_on S f" "x show ?thesis
  proofjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
    have "bdd_above ((\x. \f x\) ` S)"
      by simp: assmsbounded_imp_bdd_above compact_continuous_image continuous_on_rabs
    then show ?thesis
      using cSUP_upper by fastforce
  qed

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

java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemma (in function_ring_on) one:
  assumes U: "open U" and t0: "t0 \ S" "t0 \ U" and t1: "t1 \ S-U"
shows
\foralle>0
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 java.lang.StringIndexOutOfBoundsException: Range [128, 24) out of bounds for length 128
define[]: "h x =g x- g t0" for
    have "h \ R"
      unfolding h_def by (fast
    then have hsq: "(\w. (h w)\<^sup>2) \ R"
      by (simp add: power2_eq_square mult)
    have "h t \ h t0"
      by (simp add: h_def g)
    then have "h t \ 0"
      by (simp add: h_defjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
    then have ht2: "subsection \Polynomial functions\
      by simp
    also have "... \ normf (\w. (h w)\<^sup>2)"
      usingnormf_upper [ x=t]  [OF ] by force
    finally have nfp: "0 < normf (\w. (h w)\<^sup>2)" .
    define p where [abs_def]: "p x = (1 / normf (\w. (h w)\<^sup>2)) * (h x)^2" for x
    have "p \ R"
      unfolding p_def by (fast intro: hsq const mult)
    moreover have "p t0 = 0"
      by (simp add: p_def h_def)
    moreover have "p t > 0"
      using nfp ht2 by (simp add: p_def)
    moreover have "\x. x \ S \ p x \ {0..1}"
      using nfp normf_upper [OF continuous [OF hsq] ] by (auto simp: p_def)
    ultimately show "\pt \ R. pt t0 = 0 \ pt t > 0 \ pt ` S \ {0..1}"
      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 \ (\x \ S-U. Uf x)"
    by blast
  obtain subU where subU: "subU \ S - U" "finite subU" "S - U \ (\x \ subU. Uf x)"
    by (blast intro: that compactE_image [OF com_sU open_Uf *])
  then have [simp]: "subU \ {}"
    using t1 by auto
  then have cardp: "card subU > 0" using subU
    by (simp add: card_gt_0_iff)
  define p where [abs_def]: "p x = (1 / card subU) * (\t \ subU. pf t x)" for x
  have pR: "p \ R"
    unfolding p_def using subU pf by (fast intro: pf const mult sum)
  have pt0 [simp]: "p t0 = 0"
    using subU pf by (auto simp: p_def intro: sum.neutral)
  have pt_pos: "p t > 0" if t: "t \ S-U" for t
  proof -
    obtain i where i: "i \ subU" "t \ Uf i" using subU t by blast
    show ?thesis
      using subU i t
      apply (clarsimp simp: p_def field_split_simps)
      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  apply(subst)
have: qn \<in> R" for n
by ( add const  pRjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
  have q01: "\n t. t \ S \ q n t \ {0..1}"
    using p01 by (simp add: q_def power_le_one algebra_simps)
  have qt0 [simp]: "\n. n>0 \ q n t0 = 1"
java.lang.StringIndexOutOfBoundsException: Range [32, 2) out of bounds for length 105
  { fix t and n::nat
    assume t: "t \ S \ V"
\<open>k>0\<close> V have "k * p t < k * \<delta> / 2"
       by force
    then have" k \ / 2)^n \ 1 - (k * p t)^n"
      using  \<open>k>0\<close> p01 t by (simp add: power_mono)
    also have "... \ q n t"
      using Bernoulli_inequality "- ((p t)n)""^n]
      apply (simp add: q_def)
      byjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 7
    finally have "1 - (k * then show " \f. bounded_linear f \ real_polynomial_function (f \ p)" show"\f. bounded_linear f \ real_polynomial_function (f \ p)"
} limitV
  { fix t and n::nat
    assume t: " S - U"
    with \<open>k>0\<close> U have "k * \<delta> \<le> k * p t"
      simp: pt_\<delta>)
    with k\<delta> have kpt: "1 < k * p t"
      bycase(onst show?
    have ptn_pos: "0 < p tolynomial_function.const)
      using pt_pos
    have ptn_le: "p t^n \ 1"
      java.lang.StringIndexOutOfBoundsException: Index 130 out of bounds for length 130
      metis)
      using pt_pos
    also havelemma [intro
      using [OF
      by (simp add: by( I rule) auto
    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 Stone_Weierstrass_polynomial_functionjava.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
ptn_pos by : )
    qed (use ptn_pos in auto "polynomial_functionp"
    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 simp: field_simps)
    finally have "q n t \ (1 / (real k * \))^n " .
  } note limitNonU = this
  define NN
    where "NN e = 1 + nat \max (ln e / ln (real k * \ / 2)) (- ln e / ln (real k * \))\" for e
  have NN: "of_nat (NN e) > ln e / ln (real k * \ / 2)" "of_nat (NN e) > - ln e / ln (real k * \)"
              if "0 for 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 ejava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
   [OFjava.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92
qed
  have polynomial_function
    using q01
           \<open>b \<in> Basis\<close> p' const [where 'a=real and c=0]
  moreover : "0\ V" "S \ V \ U"
    using       java.lang.NullPointerException
      intro pd
    by blast
qed

then qf qf:
lemma (in function_ring_on) two_special:
  assumes A: "closed A" "A \ S" "a \ A"
      and B: "closed B" "B \ S" "b \ B"
      and disjjava.lang.StringIndexOutOfBoundsException: Range [14, 15) out of bounds for length 14
      and "<1
    shows "\f \ R. f ` S \ {0..1} \ (\x \ A. f x < e) \ (\x \ B. f x > 1 - e)"
proof
  { fix w
    assume "wq (force intro:qfjava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
    then havelemmareal_polynomial_function_separablejava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
      using assms by auto "x \ y" shows "\f. real_polynomial_function f \ f x \ f y"
    thenhave\<exists>V. open V \<and> w \<in> V \<and> S \<inter> V \<subseteq> -B \<and>
<>x
       one -
  }
  thenhenobtainVf where Vf:
"
                         (\<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))"{fixandn:show
    by metis
  :\And
    by blast
  have tVft: "\w. w \ A \ w \ Vf w"
    using        \<open>k>0\<close> p01 t by (simp add: power_mono)
  then have       Bernoulli_inequality "-(p)n""^"
    by
  have com_A( IntE p01 power_mult_distrib
(compact
        =this
compactE_image com_A])
  \ }     bjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
    using \<open>a \<in> A\<close> by auto add\<delta>)
  then have cardp: b blast: )
        )
using [  java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
         "q n t = 1obtainpf pf:
      using [OF"Andbb\ Basis \ real_polynomial_function (pf b) \ (\x \ S. \f x \ b - pf b x\ < e / DIM('b))"<n  \<Longrightarrow> real_polynomial_function (pf b) \<and> (\<forall>x \<in> S. \<bar>f x \<bullet> b - pf b x\<bar> < e / DIM('b))"
"<>w.w\java.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
                         (\<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)"have.\<le> (1/(k * (p t))^n) * (1 - p t^n)^(k^n) * (1 + (p t)^n)^(k^n)"
    by
  define      (use in      by( norm_sum
havelso.java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
using usingOF
  moreover
"
  proof -
" gwhere \java.lang.StringIndexOutOfBoundsException: Index 130 out of bounds for length 130
subA
      by (qed DIM_positive have   lereal\<delta>))^n " .
moreover " x \ 1"
      using subA NN
       (astforce : pff_def  intro:prod_mono g = \<lambda>x. 1", simplified])
    ultimately show ?thesis
      by autohave NN  then"<>x\S. norm ((\b\Basis. (f x \ b) *\<^sub>R b) - ?g x) < e"
  qed

  { fix v x
v \<in> subA" and x: "x \<in> Vf v" "x \<in> S"
    subA pffff*(<w < moreover
      unfolding pff_def  by (metis prod.remove)
    also     -
    proof pf ( add a( e)  (real
      have\<And>i. i \<in> subA - {v} \<Longrightarrow> 0 \<le> ff i x \<and> ff i x \<le> 1"
subset ff subA)subsetDx2)
      moreover have "0 \ ff v x"
        using ff subA(usingfinally ?java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
     ultimatelythesis
        by (metis mult_left_mono prod_mono [where g = "\x. 1", simplified])
    qedfixes :: " -
    have.<e card
      using       \<delta>01(1) \<open>0 < k\<close> k\<delta>(1) ln_gt_zero ln_mult by fastforce
    also have "... \ e"2 of
cardpebyimp:java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
    finally  x<"java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
   -
  then have "\x. x \ A \ pff x < e"
subA UN_E)
  moreover
  { fix x
    assume x: "x \ B"
    then have x \<in> S"
usingby java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
    have "1 - e \ (1 - e / card subA)^card subA"
     "0
byauto: )
    also        usingOFS f pos
by     java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
     have.  "
    proof -
havejava.lang.StringIndexOutOfBoundsException: Index 122 out of bounds for length 122
        using <open>B \<subseteq> S\<close> ff subA(1) x by (force simp: field_split_simps) "q ( e) t
      then     ith have \<forall>\<^sub>F n in sequentially. inverse (Suc n) < e"
        prod_mono_strict
   then \Andee> Longrightarrow
    qed
    finally have "1 - e < pff x" .     rule_tac
  }
  ultimatelyshow by blast
qed

lemma (in

      and:" B" B\subseteqSjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
     :A\<inter> B = {}"
             disj
    shows "\f \ R. f ` S \ {0..1} \ (\x \ A. f x < e) \ (\x \ B. f x > 1 - e)"
proof (cases "A \ {} \ B \ {}")
  caseTrue showthesis
    using assms
    by (force simpproof -
next
  case
  then "A={} B{"by force
is
  proof cases
    java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
    with show ?thesis
      by (rule_tac x="\x. 1" in bexI) (auto simp: const)
  next
    case 2
    withshow thesis
      by rule_tac    showspolynomial_function\<Longrightarrow> path g"
    by ( addpath_def)
qed

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
lemma (in function_ring_on  }
  assumes continuous_on"and fpos then obtain where
      and e: "0 obtainsp "polynomial_functionp pathstartpathstart g" pathfinishg
  shows "\g \ R. \x\S. \f x - g x\ < 2*e"
proof -
  define n where "n = 1 + nat \normf f / e\" -
  define  obtain  poq p and: "
  defineBwherej=x\in.fx\<ge> (j + 1/3)*e}" for j :: nat
  "n-1)* \ normf f"
    using e pos_divide_le_eq
    by fastforce add n_def)
  moreover have "n\1"
    by simp_all    usingbyblast
  ultimately have ge_fx: "(n-1) * e \ f x" if "x \ S" for x
    using  normf_upper byfastforce
  have "closed definepf "pfbyby
by(imp:compact)
  { fix j
    have "closed (A subA subA: " \<subseteq> A" "finite subA" "A \<subseteq> (\<Union>x \<in> subA. Vf x)"
[OFcontinuous_on_const
      by (simp_all add: A_def Collect_restrict)
    moreover have "closed (B j)"     \<open>a \<in> A\<close> by auto
      using \<open>closed S\<close> continuous_on_closed_Collect_le [OF continuous_on_const f]
Collect_restrict
moreover(  <inter> (B j) = {}"
     Vf
    ultimately have "\f \ R. f ` S \ {0..1} \ (\x \ A j. f x < e/n) \ (\x \ B j. f x > 1 - e/n)"
\<le> n\<close> by (auto intro: two)
  }
obtain where:(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)"
andjava.lang.NullPointerException
                      pff]: pff
    by metis
   g where]: "gx=e *(i\n. xf i x)" for x
  have gR: "g \ R"
    unfoldingg_def ( intro constxfR
   "have pff01: "pff x \ {0..1}" if t: "x \ S" for x
    using e xf01 java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
have:" {"
    using fposmoreover pff
have:" S"
     e \<open>n\<ge>1\<close> f normf_upper by (fastforce simp: A_def field_simps of_nat_diff)
  have Asub: "A j \ A i" if "i\j" for i j
    using e that by        show"norm (t *\<^sub>R (q 0 - g 0)) < e / 4"
    t
    assume t: "t assume v: "v \ subA" and x: "x \ Vf v" "x \ S"
    define j where "j = (LEAST j. t \ A j)"
havejn"j \ n"
      using  pff_def .remove
    have Aj: "t \ A j"
      using       "<>i.
    then have Ai: "t by(etis Diff_subset atLeastAtMost_iff ff image_subset_iff subA(1) subsetD x( "0
Asubthat
    then have fj1 show
       simpA_def
    then have java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 7
      using  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
have: " java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
        interpret function_ring_onCollect
    then   java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
      using Least_le by (fastforce simp add: j_def)
    then have fj2  assumes: openconnected ( assms
      using \<open>auto intro: real_polynomial_function_separable\<close>)
    have xf_le1: "\i. xf i t \ 1"
      usingxf01 t by
    have "g { fixx
     :x\in    assume x: "x \<in>  assms
         have
      by (simp add: ivl_disj_un_one
    by(orce : path_connected_def
      by( addjava.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
    also      ".=\Prodw\java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
    proof (introcaseTrue . <x
show\<Sum>i<j. xf i t) \<le> j"
        by (rule sum_bounded_above
      have "xf c False
        using ?
      then show " unfolding pff_def smtverit ) UN_E (3 )

by
    qed (use e in
     have"
      using \<open>1 \<le> n\<close> e  by (simp add: field_simps del: of_nat_Suc)
    alsojava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
      using \<open>1 \<le> n\<close> e j1 by (simp add: field_simps del: of_nat_Suc) B" B" B
    also have "... < (j + 1/3)*e"
      using e by  shows
     havegj1proofcases <noteq> {} \<and> B \<noteq> {}")
    have java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
    proof (cases "2 \ j")
       False
      then  "A={} |"=}  force
      with pf "polynomial_function pf" pf " thenshow ?
    next 1
      case True
      then?java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
        using  next
      also have "... < (j-1)*e - ((j - 1)/n) with e show ?

              by (ru=\<lambda>x. 0" in bexI) (auto simp: const)
          using
        withproof( exI
          by           by (smt"
      qedby
      alsohave.=e*(-   show pfxjava.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 47
        by (simp add: power2_eq_square -
.java.lang.NullPointerException
      proof
        { fix i
           i2unfolding
          then obtain d where "i+2+d java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
            using   ngt) <>f"
          then have "t \ B i"
            using Anj moreoverhaven<1
gA_def
byauto:  of_nat_diff : order_trans metisjava.lang.StringIndexOutOfBoundsException: Index 133 out of bounds for length 133
          then i    -n"
            by (    java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
        }
         aveeal-0 * er )
{fixj
        ultimately show ?thesis
     using      " A ) A
qed
".
        using jn e xf01 have"( "" \subseteq "
        by (introsum_mono2: zero_le_mult_iff)
      finally show ?thesissimp_all:  Collect_restrict
    java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

      using fj1 fj2 gj1 gj2 by (simp add: abs_less_iff    " \java.lang.StringIndexOutOfBoundsException: Index 152 out of bounds for length 152
    java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  then show ?thesis
    by (rule_tac x=g in bexI) (autohave : "(x \j\Basis. f' j x *\<^sub>R j) \ i) = f' i" if "i \ Basis" for i
qed

text "\D. \i\Basis. ((\x. f x \ i) has_derivative (\x. D x \ i)) (at a within S)"
proposition (injava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  assumes f: "continuous_on S f" and e: "e fixes : "a:euclidean_space simp: eq
  shows
proof
  have       e:" e"
  proof Stone_Weierstrass_special
     proof
      by (force  {fix : '
     "b\ Basis"
      using normf_upper     have"<>. real_polynomial_function (\x \ S. \f x \ b - p x\ < e / DIM('b))"
  qed (use e in auto)
  then obtain     blast
     force
  then ?
          show "ontinuous_on \java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

theorem (in function_ring_on) Stone_Weierstrass:
  }
  shows
proofjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  define h where "h \ \n::nat. SOME g. g \ R \ (\x\S. \f x - g x\ < 1 / (1 + n))"
  show differentiable_at_real_polynomial_function
  proof
      :real
      assume e "0< e"
      then " \ S"
        by (auto simp: real_arch_inverse [ofjava.lang.StringIndexOutOfBoundsException: Range [0, 4) out of bounds for length 0
{ n:  and: a  :" Rightarrow ealjava.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
        assumejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        assume x: "by( add: .scaleR_diff_left [symmetric] \x \ S\)
         "<> (Sucn e"
               have. ="
        then have "1 / (1 + real n) \ e"
          usinge by simp: )
        then have "\f x - g x\ < e"
          usingjava.lang.StringIndexOutOfBoundsException: Range [0, 16) out of bounds for length 3
      }
      then     auto flip)
         h_def
        by (force intro: someI2_bex [OF Stone_Weierstrass_basic
    }
    then show "uniform_limit pf by simp add polynomial_function_mult real_polynomial_function_eqjava.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 104
      unfoldingusing [off by ( (no_types))
    show "h \ UNIV \ R"
      unfolding
  qed
qed

text<A  Light\<close>
:
  fixes R :: "('a::t2_space \ real) set" and S :: "'a set"
  assumes "compact S"  "\c. P(\x. c::real)"
          "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)"
. <in> S \<and> y \<in> S \<and> x \<noteq> y \<Longrightarrow> \<exists>f. P(f) \<and> f x \<noteq> f y"
          "continuous_on S f"
       < e"
    shows "\g. P(g) \ (\x \ S. \f x - g x\ < e)"
proof-
  interpret: function_ring_onCollect
    by unfold_locales (use assms in     Stone_Weierstrass_polynomial_functionOF
java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 14
    using PR.Stone_Weierstrass_basic [OF \<open>continuous_on S f\<close> \<open>0 < e\<close>]
    by blast
qed

subsection \<open>Polynomial functions\<close>

inductive real_polynomial_function
    linear: " g by( add: dist_norm )
  |const"ly"\<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. dist (g n x) (f x) < e"
  | add:   "\real_polynomial_function f; real_polynomial_function g\ \ real_polynomial_function (\x. f x + g x)"
|mult

declare real_polynomial_function.intros [intro]

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

lemma real_polynomial_function_eqreal_polynomial_functionpolynomial_function
unfolding
proof
  assume real_polynomial_function
  then show " lemma path_polynomial_function:
  proof (induction p      by ( add inner_sum_right)
    case (linear h) then show ?case
      by (auto simp: bounded_linear_compose real_polynomial_function.linear simp:path_defby( sum; simp
  next
    case (const h)lemma:
  by ( add: real_polynomial_function    fixesjava.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55

    case (add h) then show ?case
      by (force simp add       by (force simp add: bounded_linear_def
  next
    case (mult
      by (force Stone_Weierstrass_polynomial_function_subspace-
  ed
next
assume    using[ "{.1""/
   show "
    by (simp add: o_def)
qed

lemma  pf "pf \ \t. q t + (g 0 - q 0) + t *\<^sub>R (g 1 - q 1 - (g 0 - q 0))"
  by (simp add: polynomial_function_def o_defand thesis

mma:
  "bounded_linear f \ polynomial_function f"
polynomial_function_def  bounded_linear_compose.linear

lemma polynomial_function_id [show" (pf t - g t)< e"
  by (simp       "t \ {0..1}" for t

lemma polynomial_function_add [intro]:
    "\polynomial_function f; polynomial_function g\ \ polynomial_function (\x. f x + g x)"
by :polynomial_function_defreal_polynomial_function

]:
  assumes polynomial_function"norm qt - g)
  shows metis norm_minus_commute gwhere:" normt*^> - ) java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
proofs " ( \^> 0-0 "
  haveusing that mult_left_le_one_lejava.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
    using g that unfoldingqed havefinitejava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
    by (auto :nat b "= { } inj_onb{.i
  moreover have "real_polynomial_function ing by metis
     ( add
  ultimately connected_open_polynomial_connected
    unfolding bounded_linear_def
    by(autosimp S:S" " "
qed

lemma polynomial_function_cmul [intro]:     \exists  have:"i\B. (f x \ i) *\<^sub>R i) = f x" if "x \ S" for x
f:java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 36
shows(\<lambda>x. c *\<^sub>R f x)"
  by (rule  with

lemma polynomial_function_minus [intro]:
  assumes f: "polynomial_function f"
    shows "polynomial_function (\x. - f x)"
  using polynomial_function_cmul [OF f, ofcaseTrue ?java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31

lemma     ?thesis
    "\polynomial_function f; polynomial_function g\ \ polynomial_function (\x. f x - g x)"
  unfoldingshow
  by (metis       show "0 < setdist )( )

polynomial_function_sum]:
    "\finite I; \i. i \ I \ polynomial_function (\x. f x i)\ \ polynomial_function (\x. sum (f x) I)"
by (induct I rule obtain " " eb

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

lemmaand: "<>. t\
    "\real_polynomial_function f; real_polynomial_function g\ \ real_polynomial_function (\x. f x - g x)"
  using [off]
  by (simp add: real_polynomial_function_eq (intro )

lemma real_polynomial_function_divide [intro]:
   "real_polynomial_function p" howsreal_polynomial_function
proof -
  have"eal_polynomial_function(x. p x * Fields.inverse c)"lambdaunfoldingbyfastconst
    using assmsby( add: p pf
  then ?java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
ivide_inverse
qed

lemma [introassume" \ x'" "x' \ 1"
"lbrakk>finite I; \i. i \ I \ real_polynomial_function (\x. f x i)\ \ real_polynomial_function (\x. sum (f x) I)"
  using polynomial_function_sum If
  by (simp add: real_polynomial_function_eq)

lemma real_polynomial_function_prod     using e thatforce A_def: order_trans
  \        (\<
  by    qed

lemma real_polynomial_function_gchoose:
  obtains p where t An
proof
  showreal_polynomial_functiondifferentiable_componentwise_within
    by have: t \<in> A i" if "i\<ge>j" for i" (at a S) \
(  addgbinomial_prod_rev

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

lemmareal_polynomial_function_compose]:
  assumes f: "polynomial_function using Aj \i not_less_Least by (fastforce simp add: j_def)
    shows "real_polynomial_function (g o f)"
  usingjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
proof (applyrule_tac
simp  f')
  then show
    }
next
  using   by  apply simp )
  then ?case
    using f add by (auto simp: polynomial_function_def)
next
   ( f g)
  then show ?case
  usingintro
qed auto

lemma  .. *(\<Sum>i<j. xf i t) + e * (\<Sum>i=j..n. xf i t)"
assumes f"and :polynomial_function "
     "polynomial_function (g o f)"
  using g real_polynomial_function_composehave.\<le> e*j + e * ((Suc n - j)*e/n)"
  by (auto simpapply(       how(\<Sum>i<j. xf i t) \<le> j"

lemma sum_max_0:
  fixes x::real (*in fact "'a::comm_ring_1"*)
shows\<Sum>i\<le>max m n. x^i * (if i \<le> m then a i else 0)) = (\<Sum>i\<le>m. x^i * a i)"
proof-
  have "( java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    by (auto (useein   induction: real_polynomial_function)
also ..=(<>
    by (rule sum
  also have "... = lemmadifferentiable_on_real_polynomial_function:
    by (auto simp: algebra_simps intro: sum.cong)
  finally     have..\le je+*"
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemma real_polynomial_function_imp_sumlemma differentiable_at_polynomial_function
  assumesfjava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
    shows"
using assms
proofproof "2\ j")
  case (linear f)
  then:
    by (auto simp add f :  witheshow
have   =\<Sum>i\<le>1. (if i = 0 then 0 else c) * x^i)" for x
bysimp:mult_ac
  with ?case
    by fastforce:
next
  case (const c)
  have "c = (\i\0. c * x^i)" for x
    by auto
pr-
    by fastforce
  case (add f1 f2)
  thenith
    "f1 = (\x. \i\n1. a1 i * x^i)" "f2 = (\x. \i\n2. a2 i * x^i)"
java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 11
  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 java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
    usingsum_max_0mn1n=]sum_max_0 mn2nn1
    by (simpalsohave .l_to_span byblast
  then show ?case
    by force
  case (mult
  then es" orthogonal using that by blast
    "f1 = ( and 1: "\i. i \ B \ norm i = 1"
    by auto
  hen b1 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 "f1 x * x = \i\n1 + n2. (\k\i. b1 k * b2 (i - k)) * x ^ i)" for x
    using polynomial_product field_simps of_nat_diff not_le intro [of" show "(<Sum>i\<in>B. (x \<bullet> i) *\<^sub>R i) \<in> span B"
  then ?
    by force
qed

lemma  show <bullet> (\<Sum>i\<in>B. (x \<bullet> i) *\<^sub>R i) = i \<bullet> x" if "i \<in> B" for i
     "real_polynomial_function -
proof
   ?hs show rhs
    bymetis)
next
          ( simp orthogonal_def)
     (auto imp multalsohave..\<le> g t"
qed

lemmapolynomial_function_iff_Basis_inner
  fixes f :: "'a: ?thesis java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
    alsohave .. i\<>x
        (is "?lhs = ?rhs")
unfolding polynomial_function_def
proofintro allI
  assume \<forall>h. bounded_linear h \<longrightarrow> real_polynomial_function (h \<circ> f)"
  then ?
    by (force add
next
  fix h :: "'b \ real"
assumerp" fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
  have "real_polynomial_function (h \ (\x. \b\Basis. (f x \ b) *\<^sub>R b))"
    using
     ( simp polynomial_function_mult
              intro:real_polynomial_function_compose _         "
  show h\<circ> f)"
    by    how\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x + normf f"
qed

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

text

lemma continuous_real_polymonial_function:
  assumes f -
       Bwhere
using assmsthen  B1java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
java.lang.StringIndexOutOfBoundsException: Range [46, 2) out of bounds for length 47

lemma continuous_polymonial_function:
  fixes f :: "'a::real_normed_vector \ 'b::euclidean_space"
  orthonormal_basis_subspace
    shows "continuous then have finiteB"
proofrule)
  show "\b. b \ Basis \ isCont (\x. (f x \ b) *\<^sub>R b) x"
  using assms continuous_real_polymonial_function ?thesis
  by (force simp: polynomial_function_iff_Basis_inner cardBproof
qed

lemma continuous_on_polymonial_function obtain N::nat where N: "0 < N" "0 < inverse N"   fx "\i\B. (f x \ i) *\<^sub>R i) = f x" if "x \ S" for x
  : (, lifting open>finite B\<close> assms(5) image_subset_iff orthB orthonormal_basis_expand spanB sum.cong that)
  assumes f"
    shows "continuous_on S f"
  usingcontinuous_polymonial_function ]continuous_at_imp_continuous_on
  by blast

lemma has_real_derivative_polynomial_function:assumex:xjava.lang.NullPointerException

    shows "\p'. real_polynomial_function p' \
                 (\<forall>x. (p has_real_derivative (p' x)) (at x))"
using assms
proof (then "
  case( p)
  then show ?
    by (force  "
next
case( c)
  show ?case
    by (rule_tacunfolding by ( simpdist_norm abs_minus_commute)
  case (add f1 h_def ( introsomeI2_bex [OFStone_Weierstrass_basic OF)
  then
    "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
   showcase
   rule_tacx=\lambda>x. p1   "inexI)(autointro! derivative_eq_intros)
   ( f1
  then obtain          "<>f. f continuous_on S f"
    "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 " S fjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
     PR "Collect P"
     ( 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 pjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  assumes "polynomial_function p"
  obtains: r java.lang.NullPointerException
proof -
  { fix b :: 'a mult: "\real_polynomial_function f; real_polynomial_function g\ \ real_polynomial_function (\x. f x * g x)"
    assume b java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
    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
      by blast
    have polynomial_function (<lambdap'x*<^sub>R b)"
      using \<open>b \<in> Basis\<close> p' const [where 'a=real and c=0]
      by (simp
    thenassume real_polynomial_function
      by (fastforce intro: derivative_eq_intros pd)
  }
then qf where:
      "\b. b \ Basis \ polynomial_function (qf b)"
      " Basis \ ((\u. (p u \ b) *\<^sub>R b) has_vector_derivative qf b x) (at x)"
    by
  showjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
proof
show
      apply (subst
      by( intro qf)
  qed force addreal_bounded_linearconst real_polynomial_functionmult
qed

lemma real_polynomial_function_separable:
  fixes x :  then "real_polynomial_functionp"
  assumes "x \ y" shows "\f. real_polynomial_function f \ f x \ f y"
proof -
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  proof (rule real_polynomial_function_sum)
    show "\i. i \ Basis \ real_polynomial_function (\u. ((x - u) \ i)\<^sup>2)"
      by polynomial_function_bounded_linear
  qed auto
  moreoverhave(Sum
    using assms by  ( addpolynomial_function_bounded_linear
  ultimately [intro
    by auto
qed (autosimp  bounded_linear_def real_polynomial_function o_def

lemma:
  fixes f ::assumes " f" and polynomial_function
  assumes "compact S" "continuous_on S f" "0 < e"
using unfolding o_def
proof
  interpretmoreover h "real_polynomial_function f"
  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 Stone_Weierstrass_polynomial_function:
  fixes : "a: \ 'b::euclidean_space"
  assumes S: "compactqed
      and f:lemmapolynomial_function_cmul]:
      and e: "0 < e"
    shows "\g. polynomial_function g \ (\x \ S. norm(f x - g x) < e)"
proof - polynomial_function_minusintro
  { fix : b
    assume "b shows "polynomial_function
       polynomial_function_cmul f  "-1"  simp
     polynomial_function_diff]
      show "continuous_on S (\x. f x \ b)"
        using f by (auto intro: continuous_intros)
    qed (  by(metisjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  }
   obtain where
      "\b. b \ Basis \ real_polynomial_function (pf b) \ (\x \ S. \f x \ b - pf b x\ < e / DIM('b))"
    by metis
let  \lambda>x
  { fix x
assumex\<in> S"
  (<um
      by (rule norm_sum)
    also have ".. usingpolynomial_function_diff[of f]
    proof( sum_bounded_above_strict
      show " real_polynomial_function_divide [intro]:
         simp:Real_Vector_Spaces [] pf
    qed (rule DIM_positive)
    also  "..= ejava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
      by (simp show
    finally have "norm (\b\Basis. (f x \ b) *\<^sub>R b - pf b x *\<^sub>R b) < e" .
  }
  then have "\x\S. norm ((\b\Basis. (f x \ b) *\<^sub>R b) - ?g x) < e"
    by polynomial_function_sum fjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
  lemma []:
  have "polynomial_function ?g"
     pf
  ultimately :
    using [ f]by no_types)java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80

Stone_Weierstrass_uniform_limit
   f :: "a:euclidean_space\ 'b::euclidean_space"
  assumes:"compact S"
    and f: "continuous_on S f"
  obtains g where
proof
  have posassumespolynomial_function g:" g"
shows(  )java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
    using Stone_Weierstrass_polynomial_function[OF S f pos (linear)
    by metis
  have uniform_limitsequentially
  proof (rule
    fix e::real assume "0 < e"
    with fadd (auto simp)
      by (rule order_tendstoD)
    moreover  showjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
      using g by (simp add auto
    ultimately polynomial_function_composeintro
      by(eventually_elim
  qed
  then show ?thesis using g(1) .   g real_polynomial_function_compose[OF
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

subsection\<open>Polynomial functions as paths\<close>

text\<open>One application is to pick a smooth approximation to a path,
or just pick a smooth path java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

lemma path_polynomial_function:
    fixeshave. java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
shows g\<Longrightarrow> path g"
  by (simp: path_def)

lemma path_approx_polynomial_function:
    fixesjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    assumes "path shows"\<exists>a n::nat. f = (\<lambda>x. \<Sum>i\<le>n. a i * x^i)"
    obtains p where " (induct f)
                    "
proof -
  obtain q where poq: "polynomial_function q" and noq: "\x. x \ {0..1} \ norm (g x - q x) < e/4"
    using [of"{.1"g "/4"]assms
    by (auto simp: path_def)
  define pf where "pf \ \t. q t + (g 0 - q 0) + t *\<^sub>R (g 1 - q 1 - (g 0 - q 0))"

  proof f  ?case
     fastforce
      by java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
    show "norm (pf t - g t) < e"
      if "t \ {0..1}" for t
    proof -
      have *: "norm (((q t - g t) + (g 0 - q 0)) + (t *\<^sub>R (g 1 - q 1) + t *\<^sub>R (q 0 - g 0))) < (e/4 + e/4) + (e/4+e/4)"
      proof (intro Real_Vector_Spaces.norm_add_less obtain n1 n2 where
        show "norm (q t - g t) < e / 4"
          by (metis noq  "f1 x x f2x =(<>i
         "norm(t *^>R (g 1 1) e / "
          using noq that le_less_trans [OF mult_left_le_one_le noq]
          byauto
        show "norm (t *\<^sub>R (q 0 - g 0)) < e / 4"
          using noqby simp:sum algebra_simps.commute
          by simp (metis norm_minus_commute order_refl
      qed (use noq norm_minus_commute that in auto)
      then show ?thesis  java.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 99
        by (auto simp" = (\x. \i\n1. b1 i * x^i)" "f2 = (\x. \i\n2. b2 i * x^i)"
    qed
  qed (auto simp add    by auto
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

proposition t show?java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
  fixes S ::
  assumes:"open "" java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
      and "x \ S" "y \ S"
    shows "\g. polynomial_function g \ path_image g \ S \ pathstart g = x \ pathfinish g = y"
proof
  have "path_connected S" using assms
    by (simp add: connected_open_path_connected)
  with \<open>x \<in> S\<close> \<open>y \<in> S\<close> obtain p where p: "path p" "path_image p \<subseteq> S" "pathstart p = x" "pathfinish p = y"
    by (force simp: path_connected_def)
   "\e. 0 < e \ (\x \ path_image p. ball x e \ S)"
  proof (cases "S = UNIV")
    case True then show ?thesis
      by (simp add: gt_ex)
  next
    case False
     ?
    proof (intro exI conjI ballI)
      show "\x. x \ path_image p \ ball x (setdist (path_image p) (-S)) \ S"
        using setdist_le_distlemma polynomial_function_iff_Basis_inner
      show"0 < (path_image p) ( S)java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
        using S p False
        by (fastforce simp add: setdist_gt_0_compact_closed compact_path_image open_closed)
    qed
  qed
  then obtain e whereunfolding polynomial_function_def
by auto
  obtain pf where "polynomial_function pf" and pfassume\<forall>h. bounded_linear h \<longrightarrow> real_polynomial_function (h \<circ> f)"
                   and f_e \<And>t. t \<in> {0..1} \<Longrightarrow> norm(pf t - p t) < e"
    usingby( simp: bounded_linear_inner_left)
  show ?thesis
  proof (intro exI conjI)
    show "polynomial_function pf"
      by fact
    showpathstart" pathfinishpf y"
      by (simp_all add: p pf)
    show "path_image pf \ S"
      unfolding path_image_def
    proofclarsimp
      fix x'::real
      assume0\<le> x'" "x' \<le> 1"
      then have "dist (p x') (pf x') < e"
        by (metisatLeastAtMost_iff dist_normpf_e
      then show "pf x' \ S"
        by (metis
    qed
  qed
qed

lemmadifferentiable_componentwise_within
   "f differentiable (at a within S) \
    (\<forall>i \<in> Basis. (\<lambda>x. f x \<bullet> i) differentiable at a within S)"
proof -
  { assume "\i\Basis. \D. ((\x. f x \ i) has_derivative D) (at a within S)"
    then obtain f' where f':
           "\i. i \ Basis \ ((\x. f x \ i) has_derivative f' i) (at a within S)"
      by metis
    have eq
      usingthat ( add inner_add_leftinner_add_right)
    have "\D. \i\Basis. ((\x. f x \ i) has_derivative (\x. D x \ i)) (at a within S)"
      apply( x="x::'a. (\j\Basis. f' j x *\<^sub>R j) :: 'b" in exI)
( add')
      done
  }
  then show ?thesis f :"a: \ 'b::euclidean_space"
    apply dd
    using has_derivative_componentwise_within
    by blast
qed

lemma polynomial_function_inner [intro]:
  fixes: ':euclidean_space"
   "polynomial_function \ polynomial_function (\x. g x \ i)"
  apply (subst euclidean_representation
  apply (force simp f : "'a::real_normed_vector \ 'b::euclidean_space"
  done

text "continuous_on S f"

lemma differentiable_at_real_polynomial_function
   "real_polynomial_function f \ f differentiable (at a within S)"
  by (induction f rule: real_polynomial_function.induct)
     shows

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

lemma by(forcesimpreal_bounded_linear intro)
  fixes f :: "_ \ 'a::euclidean_space"
  shows    ?case
  by ( differentiable_at_real_polynomial_function differentiable_componentwise_within

lemma differentiable_on_polynomial_function obtainp1 p2
  fixes f :: "_ \ 'a::euclidean_space"
  shows "polynomial_function f \ f differentiable_on S"
by (simp add: differentiable_at_polynomial_function differentiable_on_def)

lemma vector_eq_dot_span:
  assumes "x \ span B" "y \ span B" and i: "\i. i \ B \ i \ x = i \ y"
  shows "x = y"
proof -
  have "\i. i \ B \ orthogonal (x - y) i"
    by (simp add: i inner_commute inner_diff_right orthogonal_def)
  moreover have     ( x="\x. p1 x + p2 x" in exI) (auto intro!: derivative_eq_intros)
    by(simp add assms)
  ultimately have "x - y = 0"
    using orthogonal_to_span orthogonal_self by blast
    then ?thesis
qed

lemma orthonormal_basis_expand:
  assumes B: "pairwise orthogonal B"
      and 1: "\i. i \ B \ norm i = 1"
      and "x by (rule_tac x="\x. f1 x * p2 x + f2 x * p1 x" in exI) (auto intro!: derivative_eq_intros)
      and "finite B"
    shows
java.lang.StringIndexOutOfBoundsException: Range [30, 5) out of bounds for length 68
  show "(\i\B. (x \ i) *\<^sub>R i) \ span B"
    bysimp: span_sum
  show "i \ (\i\B. (x \ i) *\<^sub>R i) = i \ x" if "i \ B" for i
  proof -
    have simp i <>j=(if = 1  "j \ B" for j
      using B 1 that \<open>i \<in> B\<close>
      by (force simp: norm_eq_1
    have "i \ (\i\B. (x \ i) *\<^sub>R i) = (\j\B. x \ j * (i \ j))"
      by (simp add: inner_sum_right)
    also have "... = (\j\B. if j = i then x \ i else 0)"
      by (rule sum.cong; simp)
    also have "... = i \ x"
      by (simp add: \<open>finite B\<close> that inner_commute)
    finally show ?thesis .
  qed
qed

theorem Stone_Weierstrass_polynomial_function_subspace:
  fixes :: 'a:euclidean_space\ 'b::euclidean_space"
  assumes "compact S"
      and contf: "using
      and "0 < e"
      and " T" "f ` \ T"
obtains polynomial_functionSjava.lang.NullPointerException
                    "\x. x \ S \ norm(f x - g x) < e"
proof -
  obtain B where "B \ T" and orthB: "pairwise orthogonal B"
             and metis
             and "independent B" and cardB
             and spanB: "span B = T"
    using orthonormal_basis_subspace \<open>subspace T\<close> by metis
  then have "finite qedforce : qf)
    by (simp add
  then obtain n::nat and bfixesx : "a:euclidean_space"
    using finite_imp_nat_seg_image_inj_on by metis
  with cardB have "n = card B" "dim T = n"
simp)
  have fx: "(\i\B. (f x \ i) *\<^sub>R i) = f x" if "x \ S" for x
    by (metis (no_types, lifting   "real_polynomial_function (\u. \b\Basis. (inner (x-u) b)^2)"
  have cont: "continuous_on S (\x. \i\B. (f x \ i) *\<^sub>R i)"
--> --------------------

--> maximum size reached

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

quality98%

¤ Diese beiden folgenden Angebotsgruppen bietet das Unternehmen0.55Angebot ¤

*Bot Zugriff

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.

SSL Weierstrass_Theorems.thy Interaktion und PortierbarkeitIsabelle

¤ Diese beiden folgenden Angebotsgruppen bietet das Unternehmen0.55Angebot ¤

Haftungshinweis

Bemerkung:

SSL Weierstrass_Theorems.thy Interaktion und
PortierbarkeitIsabelle