Quellcode-Bibliothek Weierstrass_Theorems.thy Sprache: Isabelle

section

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

theory Weierstrass_Theorems
imports Uniform_Limit Path_Connected Derivative
begin

subsection \<open>Bernstein polynomials\<close>

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_ringof "-"njava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
  by (simp add: Bernstein_def)

lemma binomial_deriv1:
    "(\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 (rule DERIV_unique [where f java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
   ( binomial_ring
  apply (rule derivative_eq_intros sum.   qR "
  done

lemma    simp: q_def diffpower)
    "(\k\n. (of_nat k * of_nat (k-1) * of_nat (n choose k)) * a^(k-2) * b^(n-k)) =
     of_nat n * of_nat (n-1) * (a+b::real)^(n-2)"
  apply (rule DERIV_unique [where f = "\a. of_nat n * (a+b::real)^(n-1)" and x=a])
  apply (subst binomial_deriv1 [symmetric])
  apply (rule derivative_eq_intros sum.cong    using t0 pf by (simp add: q_def power_0_left)
  done

lemma sum_k_Bernstein [simp]: "(\k\n. real k * Bernstein n k x) = of_nat n * x"
  apply (subst binomial_deriv1 [of n x "1 have "h\<> R"
  apply(imp: sum_distrib_right)
  apply (auto simp:     have hsq(\<lambda>w. (h w)\<^sup>2) \<in> R"
  done

lemma sum_kk_Bernstein [  by ( introhsq mult
moreover " 0
( add h_def)
        (\<Sum>k\<le>n. real k * real (k - Suc 0) * real (n choose k) * x^(k - 2) * (1 - x)^(n - k) * x\<^sup>2)"pt>0java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
  proofusing normf_uppercontinuous hsq]by(auto: p_def
    fix    ultimately "
    assume "k \ n"
    then consider "k = auto
      by (metis One_nat_def not0_implies_Suc)
    then show java.lang.StringIndexOutOfBoundsException: Index 110 out of bounds for length 110
( k-1   n k x =
          real (k - Suc 0) *
          (real (n choose k) * (x^(k - 2) * ((1 - x)^(n - k) * x\<^sup>2)))"
      by cases (auto simp add: Bernstein_def
  qed
  also have ". blast
bysubst [of  1x" simplified, symmetric]) (simpadd sum_distrib_right)
  also      pf Ufblast
    by auto
  finally show ?thesis
    by auto
qed

subsection]" \ {}"

theorem Bernstein_Weierstrass t1 auto
  fixes f have: "card subU>0"subU
  assumes contf: "continuous_on {0..1} f" and e: "0 < e"
shows
                    \<longrightarrow> \<bar>f x - (\<Sum>k\<le>n. f(k/n) * Bernstein n k x)\<bar> < e"
proof -
  have "bounded f ` 0.1}"
    using   pR "
  thenobtain where "x 0 \ x \ x \ 1 \ \f x\ \ M"
    by (force
       ?
  haveucontf{
    using compact_uniformly_continuous contf        subUt
  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(clarsimp: p_def)
     using e by        ( sum_pos2[ \<open>finite subU\<close>])
  { fix:nat x::real
    assume n: "Suc (nat\4*M/(e*d\<^sup>2)\) \ n" and x: "0 \ x" "x \ 1"
    have0n" n by simp
    have ed0: "- (e * d\<^sup>2) < 0"
      using e \<open>0<d\<close> by simp
    also ( simp field_split_simpss)
       have px\<le> 1"
    finally []: "real_of_int(nat \4 * M / (e * d\<^sup>2)\) = real_of_int \4 * M / (e * d\<^sup>2)\"
      using simp: p_def field_split_simps)
      by (simp le [where'a=ealandK=,)
    have "4*M/(e*d\<^sup>2) + 1 \ real (Suc (nat\4*M/(e*d\<^sup>2)\))"
      by (simp add: real_nat_ceiling_ge)
    also have "... \ real n"
      using n by (simp add: field_simps Diff_subset compact_continuous_image continuous_on_subset)
    finally have nbig: add compact_imp_closed)
     moreover "
    proof -
have* \<
        by (simp add: algebra_simps obtain where : "delta0 >0 " 0 delta0
      have "(\k\n. (k - n * x)\<^sup>2 * Bernstein n k x) = n * x * (1 - x)"
        apply (simp add:   "delta0 \ 1" using delta0 p01 [of t1] t1
        apply      by (forcsimp dist_norm: p01
        apply(imp: algebra_simps)
        done
( simp \<delta>_def)
        by (simp add: power2_eq_square \<delta\Andx  \<in> S-U \<Longrightarrow> p x \<ge> \<delta>"
      then ?thesis
        using n by (simp add: sum_divide_distrib field_split_simps power2_commute)
    qed
    { fix k
      assume k: "k \ n"
      then   "\A. open A \ A \ S = {x\S. p x < \/2}"
        by (auto simp: field_split_simps)
(lessd
        by linarith
      then using \<delta>01 by (simp add: k_def)
       cases
        case lessd
        then "\(f x - f (k/n))\ < e/2"
           have.  2/delta
        also have ..\le(/   *   \<^sup>2 * (x - k/n)\<^sup>2)"
           \<open>M\<ge>0\<close> d by simp
        finally ?thesis
      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)
<> " \ (x - real k / real n)\<^sup>2 / d\<^sup>2"
          using dle \<open>d>0\<close> by auto
         "\(f x - f (k/n))\ \ \f x\ + \f (k/n)\"
          by (rule abs_triangle_ineq4)
        also have "... \ M+M"
          by (meson M add_mono_thms_linordered_semiring(1)
        also have "... \ 2 * M * ((x - k/n)\<^sup>2 / d\<^sup>2)"
          using \<section> \<open>M\<ge>0\<close> mult_left_mono by fastforce
a have "... \ e/2 + 2 * M / d\<^sup>2 * (x - k/n)\<^sup>2"
          java.lang.StringIndexOutOfBoundsException: Range [0, 15) out of bounds for length 12
        finally show ?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 (simp add: sum_subtractf sum_distrib_left [symmetric] algebra_simps)
    also have "... \ (\k\n. \(f x - f(k / n)) * Bernstein n k x\)"
        thenhave[simpsubA
alsohave".. (\k\n. (e/2 + (2 * M / d\<^sup>2) * (x - k / n)\<^sup>2) * Bernstein n k x)"
      using *
      by (force simp add: abs_mult Bernstein_nonneg x mult_right_mono intro: sum_mono)
    also have "... \ e/2 + (2 * M) / (d\<^sup>2 * n)"
      unfolding.distrib.semiring_classdistrib_right [symmetric.assoc
      using \<open>d>0\<close> x by (simp add: divide_simps \<open>M\<ge>0\<close> mult_le_one mult_left_le)( add:card_gt_0_iff
    also have..<e"
      using \<open>d>0\<close> nbig e \<open>n>0\<close>
      apply (simp add: field_split_simps)
      using ed0 by linarith
    finally Vf e cardp
  }
  then show ?thesis
    byjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
qed

subsection g = "\x. 1", simplified])

text\<open>Source:
Bruno Brosowski    ultimately show?hesis
An Elementary Proofjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
ProceedingsAmerican
Volume 81, Number 1, January 1981     subA  "pff x= ff vx*\ subA - {v.ffwx"
DOI: 10.2307      unfolding pff_def prodremove)

locale function_ring_on have "... \ ff v x * 1"
  fixes R :: "('a:ff subA(1) v (2)by fastforce
   compactjava.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
  assumes continuous:
  assumes add:      have. <e/ "
  assumes mult: "f \ R \ g \ R \ (\x. f x * g x) \ R"
  assumes const ff(1)vxbyjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
  assumes separable: "x \ S \ y \ S \ x \ y \ \f\R. f x \ f y"

begin
       B
    by (frule mult [OF const " e (1 - e / card subA)^card subA"le(    card)^cardsubA"

  lemma diff ( simp: )
    unfolding diff_conv_add_uminus by (metis  ".. \<>w \ subA. 1 - e / card subA)"

  lemma power:    also  ".. pff "
    by (induct n) (      have "\<And>i. i > subA

  lemma sum: "\finite I; \i. i \ I \ f i \ R\ \ (\x. \i \ I. f i x) \ R"
    by (inductthesis

      java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
    byinduct: ; simp:const

  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    show "\x. x \ S \ 0 \ f x + normf f"
    then show ?thesis
      using assms cSUP_upper normf_def by fastforce
  qed

  lemma normf_least:  qed (use  auto
    by (simpthen g  "g \ R" "\x\S. \g x - (f x + normf f)\ < e"

end

lemma (in function_ring_on) one:
  assumes U: "open U" and t0:   show ?thesis
    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 proof -
  proof -
    have "t\<
    then obtain g where g: "g \ R" "g t \ g t0"
sing separable  by (metis Diff_subset t)
      ereal
    have "h \ R"
      unfoldingfast
    then have hsq: "(\w. (h w)\<^sup>2) \ R"
       simp  mult
    have

    then have "h t \ 0"
      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_upper lemma: "real_polynomial_function p = polynomial_functionp"
    finally have nfp0<normf
    define p where [abs_def]:java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
    have "p \ R"
      unfolding p_def by (fast intro iff](lambda
moreover "="
      by (simp add: p_def h_def)
    moreover have "p t > 0lemma polynomial_function_bounded_linear:
      using nfpbounded_linear
    moreover have "\x. x \ S \ p x \ {0..1}"
      using normf_upper continuous hsq  (auto simpp_def
    ultimately show " ultimately show "\pt \ R. pt t0 = 0 \ pt t > 0 \ pt ` S \ {0..1}"
      by java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
  qed
then pfwhere:"And> S-U \ pf t \ R \ pf t t0 = 0 \ pf t t > 0">S-U
                   and -
by
com_sU (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 assmsby java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
    ( 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
    byjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  have tUft:qed (simp: gbinomial_prod_rev
    usingjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  then :" \ (\x \ S-U. Uf x)"
    by blast
      real_polynomial_function
    by (blast intro  by(nductsimp_all: const)
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    using by auto
  then have cardp f: "polynomial_function f" and :" g"
    by( add: card_gt_0_iff
define [abs_def:" (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 by(rule_tac x=\."exI
using pfby(uto:p_def: .neutral
  have pt_pos: "p t > obtain p1 p2
  proof -
    obtain i where i: "i \ subU" "t \ Uf i" using subU t by blast
    show ?thesis
      using
      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 n t
  have : "q \ R" for n
    by (simp add: q_def const(ule_tac\lambda    p2  exI auto!: derivative_eq_intros)
  have q01: "\n t. t \ S \ q n t \ {0..1}"
    using by ( addq_def  algebra_simps
  have qt0 [simpthen obtainp1 where
defjava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
  {  t  "
    assume t: "t \ S \ V"
    with
       by force    have"\p. real_polynomial_function p \ (\x \ S. \f x \ b - p x\ < e / DIM('b))"
    thenthenopen_Vf>. <in> A \<Longrightarrow> open (Vf w)"
using
    also have "... \ q n t"
using [of ( t^)"kn]
      apply (simpblast
      by metis atLeastAtMost_iff power_le_one t)
    finally "by etis inf.absorb_iff2)
  }notelimitV
   show   ultimately show ?thesisOFopen_Vf
     byjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
   withjava.lang.StringIndexOutOfBoundsException: Range [3, 1) out of bounds for length 3
      by (simp: pt_
    with k\<delta> have kpt: "1 < k * p t"
      y( introless_le_trans
    have ptn_posby (simpaddcard_gt_0_iff
       pt_pos[OF }
    have ptn_le: "p t^n \ 1"
      by (meson DiffE atLeastAtMost_iff p01 power_le_one t)
haveq(/(  where
pt_pos"And>. \ Basis \ real_polynomial_function (pf b) \ (\x \ S. \f x \ b - pf b x\ < e / DIM('b))"
    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          \Andw. <in> A \<Longrightarrow> ff w \<in> R \<and> ff w ` S \<subseteq> {0..1} \<and>
      ".. \ (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])
show0\<le> 1 / (real k * p t)^n * (1 - p t^n)^k^n"
        using ptn_pos metis
    qed ptn_pos auto      by(ule)
    also have "... = (1/(k * pffR:a "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
       pt_posjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
bypff01pff
    also have "... \ (1/(k * (p t))^n) * 1"
          have    "And>i. i \ Basis \ norm ((f x \ i) *\<^sub>R i - pf i x *\<^sub>R i) < e / real DIM('b)"
      by (intro mult_left_mono [OF power_le_one]) auto
    also have "... \ (1 / (k*\))^n"
sing
      by (fastforce simp: field_simps      using cardp t ff
    finally "qnt\<> (1 / (real k *
  } note     have pff
  define
    where by( simpaddpff_deffield_split_simpsjava.lang.StringIndexOutOfBoundsException: Range [27, 28) out of bounds for length 27
  haveNNhave"foralljava.lang.StringIndexOutOfBoundsException: Index 100 out of bounds for length 100
              if  reover
      unfolding
    have NN1:from v have" x = v x * (\in java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
    proofjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
l NN   ( k *<delta> / 2)"
        by        "\i. i \ subA - {v} \ 0 \ ff i x \ ff i x \ 1"
      alsohave. _ atLeastAtMost_iff image_subset_iff(1 subsetD ()
        using NN k\<delta> that by (force simp add: field_simps)
       show
        by (simp add: \<open>\<delta>>0\<close> \<open>0 < k\<close> that)
    qed
"(1/ultimately show ?java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
proof
    have "0 also . < ardsubA"
using
      assumes S:" S"
()[ e] \<open>0 < \<delta>\<close> \<open>0 < k\<close> that by (simp add: ln_div divide_simps):"ontinuous_on using cardp e ( add field_split_simps)
    havepffejava.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
      by (metis pos: f bymetis ontra_subsetD
    then show ?thesis have"
       B byauto
  qed
  {   obtain where g: "\n. polynomial_function (g n)" "\x n. x \ S \ norm(f x - g n x) < inverse (Suc n)"
assume">"
       ( simpfield_simps
[ ]
      assume t:      metis
       " - q (NNe t< e"
    also ".. < pffx"
    next
      assume t: " "<And>i. i \<in> subA \<Longrightarrow> e / real (card subA) \<le> 1 \<and> 1 - e / real (card subA) < ff i x")
      show NN<"
      using  limitNonU [OF java.lang.StringIndexOutOfBoundsException: Range [0, 28) out of bounds for length 23
    qed
} have"<>. 0\<> \f\R. f ` S \ {0..1} \ (\t \ S \ V. f t < e) \ (\t \ S - U. 1 - e < f t)"
    using q01
    by( x="\x. 1 - q (NN e) x" in bexI) (auto simp: algebra_simps intro: diff const qR)
  moreover}
    usingultimately  ?thesis
  ultimately show ?thesis
    by blast
qed

      ndclosed<>"
lemma (in function_ring_on (eventually_elim
  assumesA closed"A\ S" "a \ A"
      andB closed  and"A
and:java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
      and e: "0 < e" "e < 1"
    shows\<open  then ?java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
proof
  { fix w
  " False
consider}|"=} by force
      using assms
    then have "case 1
               (<forall( " g \ path g"
bysimp path_defjava.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
then Vf
         "java.lang.StringIndexOutOfBoundsException: Range [0, 85) out of bounds for length 30
                         (\<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))"\<And>t. t \<in> {0..1} \<Longrightarrow> norm(p t - g t) < e"
    by metis   "B { <> S
  have:() e\<ge> normf f"
    byblast
  have( simp: divide_simpsjava.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
Vf java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
  usingf that java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
    by
  have com_A:  (imp compact
    by (metis
obtainwheresubA
    by (blast intro: that compactE_image [OF s_on_closed_Collect_le f       using \<open>closed S\<close> continuous_on_closed_Collect_le [OF f continuous_on_const
  then have [simp]: "subA \ {}"
    using
  then have       ( simpadd pf_def
    by (simp add: card_gt_0_iff" (pf t- t) Collect_restrict
if     have(Aj)java.lang.NullPointerException
using ejava.lang.StringIndexOutOfBoundsException: Range [15, 11) out of bounds for length 11
  thenwhere
           java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
                        <java.lang.StringIndexOutOfBoundsException: Range [151, 34) out of bounds for length 151
                  xfA And>x j. x \<in> A j \<Longrightarrow> xf j x < e/n"
define where [abs_def" x = (\w \ subA. ff w x)" for x
  have pffR: "pff \ R"
    unfoldingdefine [abs_def    *java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
  moreover    unfolding g_def by (ast: mult sum xfR)
  java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
  proof -
    have "0\ pff x"
      using subA cardp t ff
      by   A0 A = {}"
     have" x \ 1"
   An A n="
      by (fastforceusing ngt
    ultimatelyshow ?thesis
      by auto
  qed
  {fix
  { fix v x
java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
    from    have jn: "j \ n"
      unfolding  by (prod
    also have "... \ ff v x * 1"
    proof -
have\Andi i\<in> subA - {v} \<Longrightarrow> 0 \<le> ff i x \<and> ff i x \<le> 1"
         (Diff_subset atLeastAtMost_iff ff image_subset_iff subA(1) subsetD x(2))
      moreover" \ ff v x"
        using         g AsubOF]] byblast
      ultimately ?thesis
        by (metis mult_left_monoby( add: A_def)
    qed
    also have "... qed (auto simpadd: path_defs pf_def)
      using ff subA
    also have "... \ e"
      using cardp e by (simp add: field_split_simps     j11\<le> j"
    finallyusing  : function_ring_on
}
have\Andx  <  <>   "
   S:" S" "java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 35
  moreover      singby
    java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
assumex:" " usingjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
then "x\ S"
      using B by auto
    have \<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"
      using Bernoulli_inequality [of simphave<exists>e. 0 < e \<and> (\<forall>x \<in> path_image p. ball x e \<subseteq> S)"
      y( simp)
alsohave..  (<> <in> subA. 1 - e / card subA)"
      by (simp add: subA(2))
alsohave".. < "
    proof -
       "(
        using e \<open>B \<subseteq> S\<close> ff subA(1) x by (force simp: field_split_simps)
      then show ?thesis
        using prod_mono_strict[of _ subAusing    show?hesis
        unfoldingby( (,bestassms) subsetD

    finally have         fastforce
  }
  ultimately show ?thesis by blastalsohave      show\<And>x. x \<in> path_image p \<Longrightarrow> ball x (setdist (path_image p) (-S)) \<subseteq> S"
qed

lemma (in function_ring_on) two:
  assumes A: "closed A" "A \ S"
      and: closed" \ S"
      and disj: "A \ B = {}"
      and e: "0 < e" "e < 1"
    "\f \ R. f ` S \ {0..1} \ (\x \ A. f x < e) \ (\x \ B. f x > 1 - e)"
proof Ajava.lang.NullPointerException
  case True then show ?thesis
    using assms
    by (force simp flip
next
  case      aseFalse
  thenconsider" B={}"byjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 11
  show?thesis
  proof cases
    casejava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
show thesis
      by (rule_tac x="\x. 1" in bexI) (auto simp: const)
  next
    case 2
ethesisusing [OF
le_tac x"\x. 0" in bexI) (auto simp: const)

qed

proof intro conjI
lemma (in function_ring_on) Stone_Weierstrass_special(ial_function
       fact
      and e: " also have "..  e  (-    "pathstart pf x" "pathfinish pf = y"
g \<in> R. \<forall>x\<in>S. \<bar>f x - g x\<bar> < 2*e"
proof
d n . <le> e * (\<Sum>i\<le>j-2. xf i t)"e* \Sum>        -
  define A where "A assume"+ \<le> j" java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
  define B where "B j = {assume "0
  have: "(n-1)* e\ge normf java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
    usingby metis dist_commute pf_e)
    by (fastforce simp add: divide_simps n_def)
  moreover have "\1
    by( dd  B_def
             ( simp add field_simps not_leintro [        by(metis <open>0 \<le> x'\<close> \<open>x' \<le> 1\<close> atLeastAtMost_iff eb imageI mem_ball path_image_def subset_iff)
    using f normf_upper have "xf i t>1 -e/"
  have  qed
    by        h"(j Suc) * 1 e / eal n \ real (card {..j - 2}) * (1 - e / real n)"
   fix java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
haveclosed ) \<subseteq> S"

      by      also . \<le> g t"
    moreover "closed Bj) "Bj <>S
      using\<open>closed S\<close> continuous_on_closed_Collect_le [OF continuous_on_const f] auto:Groups_Big  addg_def image_subset_iff)
      by (simp_all addB_def)
    moreover have "(A j qed
      using e by( simp A_def B_defjava.lang.StringIndexOutOfBoundsException: Range [52, 40) out of bounds for length 40
ultimately"exists>f \ R. f ` S \ {0..1} \ (\x \ A j. f x < e/n) \ (\x \ B j. f x > 1 - e/n)"
      using e java.lang.StringIndexOutOfBoundsException: Range [4, 34) out of bounds for length 28
}
  then obtain"\i. i \ Basis \ ((\x. f x \ i) has_derivative f' i) (at a within S)"
                   and xfA: "java.lang.StringIndexOutOfBoundsException: Range [0, 36) out of bounds for length 14
                   and xfB: "\x j. x \ B j \ xf j x > 1 - e/n"
    by metis
  define] g x=  \<Sum>i\<le>n. xf i x)" for x
  have
java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 55
  have gge0 path_image_def
    using clarsimp
  have A0
    using
  have An real_polynomial_function_sum]java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 38
    using e ngt \<open>n\<ge>1\<close> f normf_upper by (fastforce simp: A_def field_simps of_nat_diff) [of ]
  have         byatLeastAtMost_iff
  ( simp intro)
  { fix t
    assume"< 0 \ x'\ \x' \ 1\ atLeastAtMost_iff eb imageI mem_ball path_image_def subset_iff)
    define j where "j = (LEAST j. t \ A j)"
    have jn:
      usingAn
    have Aj: "t \ A j"
      using t An " lemma :
n  Ai" \ A i" if "i\j" for i
      using Asub [(<forall>isimp )
    then have fj1: "f t java.lang.StringIndexOutOfBoundsException: Range [0, 26) out of bounds for length 7
by(simp: A_def
    then have Anjjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
       Ajhave : "(j\Basis. f' j x *\<^sub>R j) \ i) = f' i" if "i \ Basis" for i
           that(u g
      using A0 Aj j_def not_less_eq_eqhave\exists>.\<forall>i\<in>Basis. ((\<lambda>x. f x \<bullet> i) has_derivative (\<lambda>x. D x \<bullet> i)) (at a within S)"
    then have Anj: "t \ A (j-1)"
      using Least_le
    then havenext
      using j1t  simp:differentiable_def
    have xf_le1: "\i. xf i t \ 1"
          usinghas_derivative_componentwise_within showcase
    have" t =e* \i\n. xf i t)"
      by  casemult
    also have "... = e * using fynomial_function_inner []:
      by (simp add
    also have".. =e *(
      by (simp add: distrib_left ivl_disj_intshowsassumesshowspolynomial_function
    also  ".. \ e*j + e * ((Suc n - j)*e/n)"
    proof (intro add_mono mult_left_mono)
show\<Sum>i<j. xf i t) \<le> j"
        by  pplyjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
      have  "
        using xfA [OF Ai] that -
then
        using sum_bounded_above differentiable_at_real_polynomial_function:
        real_polynomial_function
    qeduse   (java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 56
    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)java.lang.StringIndexOutOfBoundsException: Range [48, 46) out of bounds for length 49
also "..<>j* ee"
      sing
    also have "... < (j + 1/3)*e"
      using e by (auto simp:
    finally have gj1: "g t eal_polynomial_function
     j2( - <>n:  =java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
     (cases \<le> j")
      case False
      then have "j=1" using j1 by simp obtain
      tgge0thesis
    next
      case True
      then have "(j - 4/3)*e < (j-1)*e - e^2"
        using e by (auto simp f show
      also have "... < (j-1)*e - ((j - java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
      proof -

          using j1 jn by java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 16
         \<open>e>0\<close> show ?thesis
          by (smt (verit, best   have " -java.lang.StringIndexOutOfBoundsException: Range [20, 11) out of bounds for length 11
      qed
have java.lang.StringIndexOutOfBoundsException: Range [9, 7) out of bounds for length 11
        u  where =2sum_max_0and
       have".orthogonal_self java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
      proof -
         show by simpmult f1 f2)
          assume "i+2 \ j"
          then d where2+d=jjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
            le_Suc_ex
t   b2
            using \<lambda>x. \<Sum>i\<le>n1. b1 i * x^i)" "f2 = (\<lambda>x. \<Sum>i\<le>n2. b2 i * x^i)"
            unfolding A_def B_def
d:field_simpsof_nat_diffnot_le: order_trans _ eshow<Sum>i\<in>B. (x \<bullet> i) *\<^sub>R i) \<in> span B"
          then showcase
            by (rule xfB)
        }
        moreoverlemma"\java.lang.StringIndexOutOfBoundsException: Index 102 out of bounds for length 102
          using Suc_diff_le Trueassume? thenshow?
        ultimately show ?thesis
assume :norm_eq_1pairwise_def
      qed
      also ".\ g t"
        using jn e xf01 t
        by (autolemma :
w? .
    qed
   have"f t - g t\ < 2 * e"
      using fj1 fj2 gj1 gj2  have".=java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 35
  }
      finally  (introimpI
    by (rule_tac"
qed

text\<open>The ``unpretentious'' formulation\<close> simpjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
proposition Stone_Weierstrass_polynomial_function_subspace
   : "
  shows "\g \ R. \x\S. \f x - g x\ < e"
proof -
  have u rp
  proofandsimp
              intro!  [OF "e
      byintro.continuous_on_add [OF fTopological_Spaces  real_polynomial_function
s "\x. x \ S \ 0 \ f x + normf f"
      normf_upper f]java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 3
  qed (use e in \<open>First, we need to show that they are continuous, differentiable and separable.\<close>
  then obtain "real_polynomial_function"
    by forceshows"
  thenand"x. x \ B \ norm x = 1"
    by (rule_tac               "independent "and)
qedjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

theorem  \<open>subspace T\<close> by metis
then "
  shows "\F\UNIV \ R. LIM n sequentially. F n :> uniformly_on S f"
proof -
define  "h \ \n::nat. SOME g. g \ R \ (\x\S. \f x - g x\ < 1 / (1 + n))"
  showthesis     finite_imp_nat_seg_image_inj_on bymetis

    { fix e::real
      assume e: "0 < e"
      thenN"have: (
        by (auto simp:  fixes f:  fixes f :: o_typesB1java.lang.StringIndexOutOfBoundsException: Index 138 out of bounds for length 138
      { fix n :: nat and x :: 'a using continuous_polymonial_function [OFassms continuous_at_imp_continuous_on
        assume n: "N
         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)
         have"\f x - g x\ < e"
          using linear
      }java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
      thenhave "\\<^sub>F n in sequentially. \x\S. \f x - h n x\ < e"
        unfolding
        by    const)
    }
    then show "uniform_limit S h f sequentially"
       uniform_limit_iffauto: dist_norm abs_minus_commute)
    show "h \ UNIV \ R"
      unfolding by (force: someI2_bex[OF Stone_Weierstrass_basic[ f]])
  qed
qed

text\<open>A HOL Light formulation\<close>
corollary Stone_Weierstrass_HOL:then ?case
    by(rule_tac ="< x+p2x exI (auto intro:derivative_eq_intros)
  assumes " case mult f1 f2)
          \AndPfjava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
          "\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"
       "0 < e"
    shows "\g. P(g) \ (\x \ S. \f x - g x\ < e)"
proof -
  interpret: function_ring_on "
    by unfold_locales (usebyrule_tac"\x. f1 x * p2 x + f2 x * p1 x" in exI) (auto intro!: derivative_eq_intros)
  showthesis
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    by blast
qed

subsection \<open>Polynomial functions\<close>

inductive real_polynomial_function :: "('a::real_normed_vector \ real) \ bool" where
    linear: "bounded_linear f \ real_polynomial_function f"
  | const "eal_polynomial_function(x. c)"
  | add:   "\real_polynomial_function f; real_polynomial_function g\ \ real_polynomial_function (\x. f x + g x)"
  |mult"

declare real_polynomial_function.intros [intro" \ Basis"

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_eq: "real_polynomial_function p = polynomial_function p"
unfoldinghave "\<>x. p' \java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
proof
   " p"
  then show " \f. bounded_linear f \ real_polynomial_function (f \ p)"
  proof (induction p rule: real_polynomial_function.induct)
    case (linear h) then show ?  then obtain  qf
      by (auto simp: bounded_linear_compose\>
  next
    case (const ?thesis
      by (simp add: real_polynomial_function.const)
  next
    case (add h  proof
      by (force     "\x. (p has_vector_derivative (\b\Basis. qf b x)) (at x)"
  next
    case (mult h) thenby auto: has_vector_derivative_sum
      by (force simp: real_bounded_linear constreal_polynomial_function.)
  qed
next
  assumejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
show java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
    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:
  "bounded_linear f \ polynomial_function f"
  by (simp  "\<>b\Basis. ((x - y) \ b)\<^sup>2) \ 0"

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

lemma polynomial_function_addintro]:
    "\polynomial_function f; polynomial_function g\ \ polynomial_function (\x. f x + g x)"
  by (auto simp:polynomial_function_def linear_add.add)

lemma polynomial_function_mult [intro Stone_Weierstrass_real_polynomial_function
  f: polynomial_function g:" 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
     g that polynomial_function_def bounded_linear_def
    by (auto simp -
  avereal_polynomial_function
    by (simp add: f real_polynomial_function_eq)
  ultimatelyjava.lang.StringIndexOutOfBoundsException: Range [72, 13) out of bounds for length 72
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    by (auto simp: linear f:"a:euclidean_space \ 'b::euclidean_space"
qed

  [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])

lemmapolynomial_function_minus [intro]:
  assumes f:  {  b : 'b
    shows (\<lambda>x. - f x)"
using [OF ,of1]by

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

lemma polynomial_function_sum [intro]:
    "\finite I; \i. i \ I \ polynomial_function (\x. f x i)\ \ polynomial_function (\x. sum (f x) I)"
by (induct I rule: finite_induct) auto

lemma real_polynomial_function_minus  then pf pf:
    "real_polynomial_function f \ real_polynomial_function (\x. - f x)"
  using polynomial_function_minus [of f]
  by (simp add: real_polynomial_function_eq   ?g="<lambda.\<Sum>b\<in>Basis. pf b x *\<^sub>R b"

lemma real_polynomial_function_diff     assume " \ S"
    " have"norm\<>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b - pf b x *\<^sub>R b) \<le> (\<Sum>b\<in>Basis. norm ((f x \<bullet> b) *\<^sub>R b - pf b x *\<^sub>R b))"
    of
  by (simp add: real_polynomial_function_eq (rule)

lemma
  assumes "real_polynomial_function p" showsby( add Real_Vector_Spaces.scaleR_diff_left[symmetric \<open>x \<in> S\<close>)
proof -     have..  "
  have "real_polynomial_function (\x. p x * Fields.inverse c)"
    using assms by auto
  then ?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 [of I f]
  by (simp add: real_polynomial_function_eq)

lemma real_polynomial_function_prodintro
  "\finite I; \i. i \ I \ real_polynomial_function (\x. f x i)\ \ real_polynomial_function (\x. prod (f x) I)"
  by (induct I rule using by (simp add: polynomial_function_sum polynomial_function_mult real_polynomial_function_eq)

lemma real_polynomial_function_gchoose
  obtains p    using euclidean_representation_sum_fun[of]  (metis(no_types, lifting)
proof
  show "real_polynomial_function (\x. (\i = 0..
    by force
qed

lemma real_polynomial_function_power [introproposition:
fixes:':euclidean_space 'b::euclidean_space"
  by (inductassumes S

lemma real_polynomial_function_compose [intro]: -
   f: "polynomial_function f" and real_polynomial_function
    shows "real_polynomial_function g of"
  using g
proof (induction g rule: real_polynomial_function.induct)
  case(linear f)
  then show ?case
    using f polynomial_function_def"uniform_limit S g f "
next
  case (add f g)
  then show ?case
    using  by (auto : polynomial_function_def
next
  case (mult f g)
  then ?case
  using f mult by (auto simp: polynomial_function_def)
qedauto

lemmapolynomial_function_compose [intro]:
  assumes f: "polynomial_function f" and (eventually_elim) auto
    shows "polynomial_function (g o f)"
  usingreal_polynomial_function_compose [ f]
  by (auto simp: polynomial_function_def o_def)qed

lemma sum_max_0:
  fixesjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 0
  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 (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  also  ".. = (i\m. (if i \ m then x^i * a i else 0))"
    by (rule sum.mono_neutral_right) auto
     "polynomial_function g \ path g"
by(simp add continuous_on_polymonial_function)
  finally show ?thesis .
qed

lemma real_polynomial_function_imp_sum:
  assumes "real_polynomial_function f"
    \<exists>a n::nat. f = (\<lambda>x. \<Sum>i\<le>n. a i * x^i)"
using assms
proofinduct
  case (linear"\t. t \ {0..1} \ norm(p t - g t) < e"
  then obtain java.lang.StringIndexOutOfBoundsException: Range [0, 15) out of bounds for length 7
    by (auto simp add     Stone_Weierstrass_polynomial_function "0.} "" java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
  have "x * c = (\i\1. (if i = 0 then 0 else c) * x^i)" for x
    by (simp  showthesis
  withf showcase
    byfastforce
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 a1 a2n2 java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
    "f1 = (\x. \i\n1. a1 i * x^i)" "f2 = (\x. \i\n2. a2 i * x^i)"
    by auto
  thenhave"f1x+ x \Sumi\max n1 n2. ((if i \ n1 then a1 i else 0) + (if i \ n2 then a2 i else 0)) * x^i)"
      for xshow t*<subg 1 -q))<e /4java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
    using sum_max_0 [where m=n1 and n=n2 java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
    by( add: .distrib max)
  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 (\<lambda>x. \<Sum>i\<le>n1. b1 i * x^i)" "f2 = (\<lambda>x. \<Sum>i\<le>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)"
    byauto
  then have "f1 x * f2 x = (\i\n1 + n2. (\k\i. b1 k * b2 (i - k)) * x ^ i)" for xqed
    using polynomial_product [of n1
  hen ?case
    by force
qed

lemma S " connected S"
     "real_polynomial_function f \ (\a n. f = (\x. \i\n. a i * x^i))" (is "?lhs = ?rhs")
proof
  assume ?lhs  -
    by (have
next
  assume ?rhs then shownext
    by (autoshowthesis
qed

lemmapolynomial_function_iff_Basis_inner:
         setdist -S"
  shows "polynomial_function f \ (\b\Basis. real_polynomial_function (\x. inner (f x) b))"
        (is "?lhs = ?rhs")
unfolding

   "
  then showp:"
     (force add o_def
next
  fix h :: "'b \ real"
" pf = x"" ="
  have "real_polynomial_function (h \ (\x. \b\Basis. (f x \ b) *\<^sub>R b))"
     clarsimp
    by (force simp " \ x'" "x' \ 1"
              intro!: real_polynomial_function_compose [  dist_commute )
  then show \<open>0 \<le> x'\<close> \<open>x' \<le> 1\<close> atLeastAtMost_iff eb imageI mem_ball path_image_def subset_iff)
    by (simp :
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>

lemmausing  bysimp: inner_add_left inner_add_right
  assumes "real_polynomial_function f"
    shows rule_tacjava.lang.NullPointerException
using assms
by (induct f)      apply(simp: eq f')

lemma continuous_polymonial_function:
  fixes: ':real_normed_vector \ 'b::euclidean_space"
  assumes " (simpadd: differentiable_def)
    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 i ::"a:"
qedshowsg\<Longrightarrow> polynomial_function (\<lambda>x. g x \<bullet> i)"

lemma continuous_on_polymonial_function:
  fixes:
  assumes "polynomial_function f"
    showscontinuous_on
  using continuous_polymonial_function [
  by blast

lemma has_real_derivative_polynomial_function:
  assumes "real_polynomial_function p"
     "\p'. real_polynomial_function p' \
                 (\<forall>x. (p has_real_derivative (p' x)) (at x))"
using assms
proof (induct p)
  case linear
  then show
     (force simp:  const!: derivative_eq_intros
next
  case (const c)
  showcase
    by (rule_tac(metis polynomial_function_iff_Basis_inner)
  case (add
  thenobtain p1where
    "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
byrule_tac
  case simp:assms span_diff
  then obtain p1 p2 where
    "real_polynomial_function p1" t show by simp
    "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 proof (rule vector_eq_dot_span [OF _ \<open>x \<in> span B\<close>])
  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)" ( add span_clauses)
proof -
  { fix b :    []:"\bullet if j ithen else 0)" ifjava.lang.NullPointerException
    assume "b \ Basis"
java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 8
    obtain
f:"a:: \ 'b::euclidean_space"
      by blast
    have "polynomial_function (\x. p' x *\<^sub>R b)"
      \<open>b \<in> Basis\<close> p' const [where 'a=real and c=0]
      by (simp add: polynomial_function_iff_Basis_innersubspace`S\<subseteq> T"
        obtains g where"polynomial_function g" "g ` S T"
      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)"
    bymetis
  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)
   (intro
qed

lemma real_polynomial_function_separable:
  fixes ::':java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
  assumes "x : card_image)
proof -
havejava.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
  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 Stone_Weierstrass_polynomial_function:
  fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
  assumes S: "compact S"
      and f: "continuous_on S f"
      and e: "0 < e"
    shows "\g. polynomial_function g \ (\x \ S. norm(f x - g x) < e)"
proof -
  { fix b :: 'b
    assume "b \ Basis"
    have "\p. real_polynomial_function p \ (\x \ S. \f x \ b - p x\ < e / DIM('b))"
    proof (rule Stone_Weierstrass_real_polynomial_function [OF S _, of "\x. f x \ b" "e / card Basis"])
      show "continuous_on S (\x. f x \ b)"
        using f by (auto intro: continuous_intros)
    qed (use e in auto)
  }
  then obtain pf where pf:
      "\b. b \ Basis \ real_polynomial_function (pf b) \ (\x \ S. \f x \ b - pf b x\ < e / DIM('b))"
    by metis
  let ?g = "\x. \b\Basis. pf b x *\<^sub>R b"
  { fix x
    assume "x \ S"
    have "norm (\b\Basis. (f x \ b) *\<^sub>R b - pf b x *\<^sub>R b) \ (\b\Basis. norm ((f x \ b) *\<^sub>R b - pf b x *\<^sub>R b))"
      by (rule norm_sum)
    also have "... < of_nat DIM('b) * (e / DIM('b))"
    proof (rule sum_bounded_above_strict)
      show "\i. i \ Basis \ norm ((f x \ i) *\<^sub>R i - pf i x *\<^sub>R i) < e / real DIM('b)"
        by (simp add: Real_Vector_Spaces.scaleR_diff_left [symmetric] pf \<open>x \<in> S\<close>)
    qed (rule DIM_positive)
    also have "... = e"
      by (simp add: field_simps)
    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 (auto simp flip: sum_subtractf)
  moreover
  have "polynomial_function ?g"
    using pf by (simp add: polynomial_function_sum polynomial_function_mult real_polynomial_function_eq)
  ultimately show ?thesis
    using euclidean_representation_sum_fun [of f] by (metis (no_types, lifting))
qed

proposition Stone_Weierstrass_uniform_limit:
  fixes f :: "'a::euclidean_space \ 'b::euclidean_space"
  assumes S: "compact S"
    and f: "continuous_on S f"
  obtains g where "uniform_limit S g f sequentially" "\n. polynomial_function (g n)"
proof -
  have pos: "inverse (Suc n) > 0" for n by auto
  obtain g where g: "\n. polynomial_function (g n)" "\x n. x \ S \ norm(f x - g n x) < inverse (Suc n)"
    using Stone_Weierstrass_polynomial_function[OF S f pos]
    by metis
  have "uniform_limit S g f sequentially"
  proof (rule uniform_limitI)
    fix e::real assume "0 < e"
    with LIMSEQ_inverse_real_of_nat have "\\<^sub>F n in sequentially. inverse (Suc n) < e"
      by (rule order_tendstoD)
    moreover have "\\<^sub>F n in sequentially. \x\S. dist (g n x) (f x) < inverse (Suc n)"
      using g by (simp add: dist_norm norm_minus_commute)
    ultimately show "\\<^sub>F n in sequentially. \x\S. dist (g n x) (f x) < e"
      by (eventually_elim) auto
  qed
  then show ?thesis using g(1) ..
qed

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 anyway in an open connected set\<close>

lemma path_polynomial_function:
    fixes g  :: "real \ 'b::euclidean_space"
    shows "polynomial_function g \ path g"
  by (simp add: path_def continuous_on_polymonial_function)

lemma path_approx_polynomial_function:
    fixes g :: "real \ 'b::euclidean_space"
    assumes "path g" "0 < e"
    obtains p where "polynomial_function p" "pathstart p = pathstart g" "pathfinish p = pathfinish g"
                    "\t. t \ {0..1} \ norm(p t - g t) < e"
proof -
  obtain q where poq: "polynomial_function q" and noq: "\x. x \ {0..1} \ norm (g x - q x) < e/4"
    using Stone_Weierstrass_polynomial_function [of "{0..1}" g "e/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))"
  show thesis
  proof
    show "polynomial_function pf"
      by (force simp add: poq pf_def)
    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)
        show "norm (q t - g t) < e / 4"
          by (metis noq norm_minus_commute that)
        show "norm (t *\<^sub>R (g 1 - q 1)) < e / 4"
          using noq that le_less_trans [OF mult_left_le_one_le noq]
          by auto
        show "norm (t *\<^sub>R (q 0 - g 0)) < e / 4"
          using noq that le_less_trans [OF mult_left_le_one_le noq]
          by simp (metis norm_minus_commute order_refl zero_le_one)
      qed (use noq norm_minus_commute that in auto)
      then show ?thesis
        by (auto simp add: algebra_simps pf_def)
    qed
  qed (auto simp add: path_defs pf_def)
qed

proposition connected_open_polynomial_connected:
  fixes S :: "'a::euclidean_space set"
  assumes S: "open S" "connected S"
      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)
  have "\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
    show ?thesis
    proof (intro exI conjI ballI)
      show "\x. x \ path_image p \ ball x (setdist (path_image p) (-S)) \ S"
        using setdist_le_dist [of _ "path_image p" _ "-S"] by fastforce
      show "0 < setdist (path_image p) (- S)"
        using S p False
        by (fastforce simp add: setdist_gt_0_compact_closed compact_path_image open_closed)
    qed
  qed
  then obtain e where "0 < e"and eb: "\x. x \ path_image p \ ball x e \ S"
    by auto
  obtain pf where "polynomial_function pf" and pf: "pathstart pf = pathstart p" "pathfinish pf = pathfinish p"
                   and pf_e: "\t. t \ {0..1} \ norm(pf t - p t) < e"
    using path_approx_polynomial_function [OF \<open>path p\<close> \<open>0 < e\<close>] by blast
  show ?thesis
  proof (intro exI conjI)
    show "polynomial_function pf"
      by fact
    show "pathstart pf = x" "pathfinish pf = y"
      by (simp_all add: p pf)
    show "path_image pf \ S"
      unfolding path_image_def
    proof clarsimp
      fix x'::real
      assume "0 \ x'" "x' \ 1"
      then have "dist (p x') (pf x') < e"
        by (metis atLeastAtMost_iff dist_commute dist_norm pf_e)
      then show "pf x' \ S"
        by (metis \<open>0 \<le> x'\<close> \<open>x' \<le> 1\<close> atLeastAtMost_iff eb imageI mem_ball path_image_def subset_iff)
    qed
  qed
qed

lemma differentiable_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: "(\x. (\j\Basis. f' j x *\<^sub>R j) \ i) = f' i" if "i \ Basis" for i
      using that by (simp add: inner_add_left inner_add_right)
    have "\D. \i\Basis. ((\x. f x \ i) has_derivative (\x. D x \ i)) (at a within S)"
      apply (rule_tac x="\x::'a. (\j\Basis. f' j x *\<^sub>R j) :: 'b" in exI)
      apply (simp add: eq f')
      done
  }
  then show ?thesis
    apply (simp add: differentiable_def)
    using has_derivative_componentwise_within
    by blast
qed

lemma polynomial_function_inner [intro]:
  fixes i :: "'a::euclidean_space"
  shows "polynomial_function g \ polynomial_function (\x. g x \ i)"
  apply (subst euclidean_representation [where x=i, symmetric])
  apply (force simp: inner_sum_right polynomial_function_iff_Basis_inner polynomial_function_sum)
  done

text\<open> Differentiability of real and vector polynomial functions.\<close>

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

lemma differentiable_on_polynomial_function:
  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:
style='color:red'>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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