src/HOL/Analysis/Weierstrass_Theorems.thy
author wenzelm
Mon Mar 25 17:21:26 2019 +0100 (3 months ago)
changeset 69981 3dced198b9ec
parent 69737 ec3cc98c38db
child 70136 f03a01a18c6e
permissions -rw-r--r--
more strict AFP properties;
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section \<open>Bernstein-Weierstrass and Stone-Weierstrass\<close>
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text\<open>By L C Paulson (2015)\<close>
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theory Weierstrass_Theorems
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imports Uniform_Limit Path_Connected Derivative
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begin
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subsection \<open>Bernstein polynomials\<close>
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definition%important Bernstein :: "[nat,nat,real] \<Rightarrow> real" where
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  "Bernstein n k x \<equiv> of_nat (n choose k) * x ^ k * (1 - x) ^ (n - k)"
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lemma Bernstein_nonneg: "\<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> 0 \<le> Bernstein n k x"
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  by (simp add: Bernstein_def)
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lemma Bernstein_pos: "\<lbrakk>0 < x; x < 1; k \<le> n\<rbrakk> \<Longrightarrow> 0 < Bernstein n k x"
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  by (simp add: Bernstein_def)
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lemma sum_Bernstein [simp]: "(\<Sum>k\<le>n. Bernstein n k x) = 1"
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  using binomial_ring [of x "1-x" n]
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  by (simp add: Bernstein_def)
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lemma binomial_deriv1:
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    "(\<Sum>k\<le>n. (of_nat k * of_nat (n choose k)) * a^(k-1) * b^(n-k)) = real_of_nat n * (a+b) ^ (n-1)"
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  apply (rule DERIV_unique [where f = "\<lambda>a. (a+b)^n" and x=a])
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  apply (subst binomial_ring)
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  apply (rule derivative_eq_intros sum.cong | simp add: atMost_atLeast0)+
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  done
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lemma binomial_deriv2:
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    "(\<Sum>k\<le>n. (of_nat k * of_nat (k-1) * of_nat (n choose k)) * a^(k-2) * b^(n-k)) =
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     of_nat n * of_nat (n-1) * (a+b::real) ^ (n-2)"
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  apply (rule DERIV_unique [where f = "\<lambda>a. of_nat n * (a+b::real) ^ (n-1)" and x=a])
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  apply (subst binomial_deriv1 [symmetric])
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  apply (rule derivative_eq_intros sum.cong | simp add: Num.numeral_2_eq_2)+
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  done
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lemma sum_k_Bernstein [simp]: "(\<Sum>k\<le>n. real k * Bernstein n k x) = of_nat n * x"
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  apply (subst binomial_deriv1 [of n x "1-x", simplified, symmetric])
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  apply (simp add: sum_distrib_right)
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  apply (auto simp: Bernstein_def algebra_simps power_eq_if intro!: sum.cong)
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  done
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lemma sum_kk_Bernstein [simp]: "(\<Sum>k\<le>n. real k * (real k - 1) * Bernstein n k x) = real n * (real n - 1) * x\<^sup>2"
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proof -
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  have "(\<Sum>k\<le>n. real k * (real k - 1) * Bernstein n k x) =
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        (\<Sum>k\<le>n. real k * real (k - Suc 0) * real (n choose k) * x ^ (k - 2) * (1 - x) ^ (n - k) * x\<^sup>2)"
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  proof (rule sum.cong [OF refl], simp)
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    fix k
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    assume "k \<le> n"
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    then consider "k = 0" | "k = 1" | k' where "k = Suc (Suc k')"
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      by (metis One_nat_def not0_implies_Suc)
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    then show "k = 0 \<or>
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          (real k - 1) * Bernstein n k x =
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          real (k - Suc 0) *
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          (real (n choose k) * (x ^ (k - 2) * ((1 - x) ^ (n - k) * x\<^sup>2)))"
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      by cases (auto simp add: Bernstein_def power2_eq_square algebra_simps)
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  qed
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  also have "... = real_of_nat n * real_of_nat (n - Suc 0) * x\<^sup>2"
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    by (subst binomial_deriv2 [of n x "1-x", simplified, symmetric]) (simp add: sum_distrib_right)
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  also have "... = n * (n - 1) * x\<^sup>2"
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    by auto
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  finally show ?thesis
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    by auto
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qed
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subsection \<open>Explicit Bernstein version of the 1D Weierstrass approximation theorem\<close>
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theorem Bernstein_Weierstrass:
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  fixes f :: "real \<Rightarrow> real"
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  assumes contf: "continuous_on {0..1} f" and e: "0 < e"
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    shows "\<exists>N. \<forall>n x. N \<le> n \<and> x \<in> {0..1}
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                    \<longrightarrow> \<bar>f x - (\<Sum>k\<le>n. f(k/n) * Bernstein n k x)\<bar> < e"
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proof -
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  have "bounded (f ` {0..1})"
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    using compact_continuous_image compact_imp_bounded contf by blast
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  then obtain M where M: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> \<bar>f x\<bar> \<le> M"
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    by (force simp add: bounded_iff)
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  then have Mge0: "0 \<le> M" by force
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  have ucontf: "uniformly_continuous_on {0..1} f"
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    using compact_uniformly_continuous contf by blast
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  then obtain d where d: "d>0" "\<And>x x'. \<lbrakk> x \<in> {0..1}; x' \<in> {0..1}; \<bar>x' - x\<bar> < d\<rbrakk> \<Longrightarrow> \<bar>f x' - f x\<bar> < e/2"
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     apply (rule uniformly_continuous_onE [where e = "e/2"])
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     using e by (auto simp: dist_norm)
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  { fix n::nat and x::real
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    assume n: "Suc (nat\<lceil>4*M/(e*d\<^sup>2)\<rceil>) \<le> n" and x: "0 \<le> x" "x \<le> 1"
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    have "0 < n" using n by simp
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    have ed0: "- (e * d\<^sup>2) < 0"
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      using e \<open>0<d\<close> by simp
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    also have "... \<le> M * 4"
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      using \<open>0\<le>M\<close> by simp
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    finally have [simp]: "real_of_int (nat \<lceil>4 * M / (e * d\<^sup>2)\<rceil>) = real_of_int \<lceil>4 * M / (e * d\<^sup>2)\<rceil>"
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      using \<open>0\<le>M\<close> e \<open>0<d\<close>
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      by (simp add: field_simps)
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    have "4*M/(e*d\<^sup>2) + 1 \<le> real (Suc (nat\<lceil>4*M/(e*d\<^sup>2)\<rceil>))"
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      by (simp add: real_nat_ceiling_ge)
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    also have "... \<le> real n"
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      using n by (simp add: field_simps)
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    finally have nbig: "4*M/(e*d\<^sup>2) + 1 \<le> real n" .
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    have sum_bern: "(\<Sum>k\<le>n. (x - k/n)\<^sup>2 * Bernstein n k x) = x * (1 - x) / n"
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    proof -
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      have *: "\<And>a b x::real. (a - b)\<^sup>2 * x = a * (a - 1) * x + (1 - 2 * b) * a * x + b * b * x"
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        by (simp add: algebra_simps power2_eq_square)
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      have "(\<Sum>k\<le>n. (k - n * x)\<^sup>2 * Bernstein n k x) = n * x * (1 - x)"
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        apply (simp add: * sum.distrib)
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        apply (simp flip: sum_distrib_left add: mult.assoc)
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        apply (simp add: algebra_simps power2_eq_square)
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        done
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      then have "(\<Sum>k\<le>n. (k - n * x)\<^sup>2 * Bernstein n k x)/n^2 = x * (1 - x) / n"
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        by (simp add: power2_eq_square)
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      then show ?thesis
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        using n by (simp add: sum_divide_distrib divide_simps mult.commute power2_commute)
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    qed
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    { fix k
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      assume k: "k \<le> n"
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      then have kn: "0 \<le> k / n" "k / n \<le> 1"
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        by (auto simp: divide_simps)
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      consider (lessd) "\<bar>x - k / n\<bar> < d" | (ged) "d \<le> \<bar>x - k / n\<bar>"
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        by linarith
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      then have "\<bar>(f x - f (k/n))\<bar> \<le> e/2 + 2 * M / d\<^sup>2 * (x - k/n)\<^sup>2"
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      proof cases
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        case lessd
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        then have "\<bar>(f x - f (k/n))\<bar> < e/2"
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          using d x kn by (simp add: abs_minus_commute)
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        also have "... \<le> (e/2 + 2 * M / d\<^sup>2 * (x - k/n)\<^sup>2)"
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          using Mge0 d by simp
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        finally show ?thesis by simp
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      next
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        case ged
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        then have dle: "d\<^sup>2 \<le> (x - k/n)\<^sup>2"
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          by (metis d(1) less_eq_real_def power2_abs power_mono)
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        have "\<bar>(f x - f (k/n))\<bar> \<le> \<bar>f x\<bar> + \<bar>f (k/n)\<bar>"
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          by (rule abs_triangle_ineq4)
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        also have "... \<le> M+M"
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          by (meson M add_mono_thms_linordered_semiring(1) kn x)
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        also have "... \<le> 2 * M * ((x - k/n)\<^sup>2 / d\<^sup>2)"
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          apply simp
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          apply (rule Rings.ordered_semiring_class.mult_left_mono [of 1 "((x - k/n)\<^sup>2 / d\<^sup>2)", simplified])
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          using dle \<open>d>0\<close> \<open>M\<ge>0\<close> by auto
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        also have "... \<le> e/2 + 2 * M / d\<^sup>2 * (x - k/n)\<^sup>2"
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          using e  by simp
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        finally show ?thesis .
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        qed
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    } note * = this
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    have "\<bar>f x - (\<Sum>k\<le>n. f(k / n) * Bernstein n k x)\<bar> \<le> \<bar>\<Sum>k\<le>n. (f x - f(k / n)) * Bernstein n k x\<bar>"
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      by (simp add: sum_subtractf sum_distrib_left [symmetric] algebra_simps)
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    also have "... \<le> (\<Sum>k\<le>n. (e/2 + (2 * M / d\<^sup>2) * (x - k / n)\<^sup>2) * Bernstein n k x)"
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      apply (rule order_trans [OF sum_abs sum_mono])
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      using *
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      apply (simp add: abs_mult Bernstein_nonneg x mult_right_mono)
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      done
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    also have "... \<le> e/2 + (2 * M) / (d\<^sup>2 * n)"
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      apply (simp only: sum.distrib Rings.semiring_class.distrib_right sum_distrib_left [symmetric] mult.assoc sum_bern)
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      using \<open>d>0\<close> x
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      apply (simp add: divide_simps Mge0 mult_le_one mult_left_le)
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      done
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    also have "... < e"
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      apply (simp add: field_simps)
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      using \<open>d>0\<close> nbig e \<open>n>0\<close>
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      apply (simp add: divide_simps algebra_simps)
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      using ed0 by linarith
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    finally have "\<bar>f x - (\<Sum>k\<le>n. f (real k / real n) * Bernstein n k x)\<bar> < e" .
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  }
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  then show ?thesis
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    by auto
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qed
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subsection \<open>General Stone-Weierstrass theorem\<close>
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text\<open>Source:
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Bruno Brosowski and Frank Deutsch.
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An Elementary Proof of the Stone-Weierstrass Theorem.
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Proceedings of the American Mathematical Society
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Volume 81, Number 1, January 1981.
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DOI: 10.2307/2043993  https://www.jstor.org/stable/2043993\<close>
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locale function_ring_on =
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  fixes R :: "('a::t2_space \<Rightarrow> real) set" and S :: "'a set"
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  assumes compact: "compact S"
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  assumes continuous: "f \<in> R \<Longrightarrow> continuous_on S f"
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  assumes add: "f \<in> R \<Longrightarrow> g \<in> R \<Longrightarrow> (\<lambda>x. f x + g x) \<in> R"
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  assumes mult: "f \<in> R \<Longrightarrow> g \<in> R \<Longrightarrow> (\<lambda>x. f x * g x) \<in> R"
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  assumes const: "(\<lambda>_. c) \<in> R"
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  assumes separable: "x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> \<exists>f\<in>R. f x \<noteq> f y"
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begin
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  lemma minus: "f \<in> R \<Longrightarrow> (\<lambda>x. - f x) \<in> R"
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    by (frule mult [OF const [of "-1"]]) simp
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  lemma diff: "f \<in> R \<Longrightarrow> g \<in> R \<Longrightarrow> (\<lambda>x. f x - g x) \<in> R"
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    unfolding diff_conv_add_uminus by (metis add minus)
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  lemma power: "f \<in> R \<Longrightarrow> (\<lambda>x. f x ^ n) \<in> R"
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    by (induct n) (auto simp: const mult)
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  lemma sum: "\<lbrakk>finite I; \<And>i. i \<in> I \<Longrightarrow> f i \<in> R\<rbrakk> \<Longrightarrow> (\<lambda>x. \<Sum>i \<in> I. f i x) \<in> R"
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    by (induct I rule: finite_induct; simp add: const add)
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  lemma prod: "\<lbrakk>finite I; \<And>i. i \<in> I \<Longrightarrow> f i \<in> R\<rbrakk> \<Longrightarrow> (\<lambda>x. \<Prod>i \<in> I. f i x) \<in> R"
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    by (induct I rule: finite_induct; simp add: const mult)
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  definition%important normf :: "('a::t2_space \<Rightarrow> real) \<Rightarrow> real"
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    where "normf f \<equiv> SUP x\<in>S. \<bar>f x\<bar>"
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  lemma normf_upper: "\<lbrakk>continuous_on S f; x \<in> S\<rbrakk> \<Longrightarrow> \<bar>f x\<bar> \<le> normf f"
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    apply (simp add: normf_def)
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    apply (rule cSUP_upper, assumption)
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    by (simp add: bounded_imp_bdd_above compact compact_continuous_image compact_imp_bounded continuous_on_rabs)
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  lemma normf_least: "S \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> \<bar>f x\<bar> \<le> M) \<Longrightarrow> normf f \<le> M"
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    by (simp add: normf_def cSUP_least)
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end
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lemma (in function_ring_on) one:
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  assumes U: "open U" and t0: "t0 \<in> S" "t0 \<in> U" and t1: "t1 \<in> S-U"
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    shows "\<exists>V. open V \<and> t0 \<in> V \<and> S \<inter> V \<subseteq> U \<and>
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               (\<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))"
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proof -
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  have "\<exists>pt \<in> R. pt t0 = 0 \<and> pt t > 0 \<and> pt ` S \<subseteq> {0..1}" if t: "t \<in> S - U" for t
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  proof -
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    have "t \<noteq> t0" using t t0 by auto
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    then obtain g where g: "g \<in> R" "g t \<noteq> g t0"
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      using separable t0  by (metis Diff_subset subset_eq t)
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    define h where [abs_def]: "h x = g x - g t0" for x
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    have "h \<in> R"
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      unfolding h_def by (fast intro: g const diff)
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    then have hsq: "(\<lambda>w. (h w)\<^sup>2) \<in> R"
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      by (simp add: power2_eq_square mult)
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    have "h t \<noteq> h t0"
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      by (simp add: h_def g)
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    then have "h t \<noteq> 0"
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      by (simp add: h_def)
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    then have ht2: "0 < (h t)^2"
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      by simp
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    also have "... \<le> normf (\<lambda>w. (h w)\<^sup>2)"
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      using t normf_upper [where x=t] continuous [OF hsq] by force
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    finally have nfp: "0 < normf (\<lambda>w. (h w)\<^sup>2)" .
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    define p where [abs_def]: "p x = (1 / normf (\<lambda>w. (h w)\<^sup>2)) * (h x)^2" for x
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    have "p \<in> R"
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      unfolding p_def by (fast intro: hsq const mult)
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    moreover have "p t0 = 0"
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      by (simp add: p_def h_def)
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    moreover have "p t > 0"
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      using nfp ht2 by (simp add: p_def)
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    moreover have "\<And>x. x \<in> S \<Longrightarrow> p x \<in> {0..1}"
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      using nfp normf_upper [OF continuous [OF hsq] ] by (auto simp: p_def)
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    ultimately show "\<exists>pt \<in> R. pt t0 = 0 \<and> pt t > 0 \<and> pt ` S \<subseteq> {0..1}"
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      by auto
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  qed
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  then obtain pf where pf: "\<And>t. t \<in> S-U \<Longrightarrow> pf t \<in> R \<and> pf t t0 = 0 \<and> pf t t > 0"
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                   and pf01: "\<And>t. t \<in> S-U \<Longrightarrow> pf t ` S \<subseteq> {0..1}"
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    by metis
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   256
  have com_sU: "compact (S-U)"
lp15@62843
   257
    using compact closed_Int_compact U by (simp add: Diff_eq compact_Int_closed open_closed)
lp15@63938
   258
  have "\<And>t. t \<in> S-U \<Longrightarrow> \<exists>A. open A \<and> A \<inter> S = {x\<in>S. 0 < pf t x}"
lp15@60987
   259
    apply (rule open_Collect_positive)
lp15@60987
   260
    by (metis pf continuous)
lp15@63938
   261
  then obtain Uf where Uf: "\<And>t. t \<in> S-U \<Longrightarrow> open (Uf t) \<and> (Uf t) \<inter> S = {x\<in>S. 0 < pf t x}"
lp15@60987
   262
    by metis
lp15@63938
   263
  then have open_Uf: "\<And>t. t \<in> S-U \<Longrightarrow> open (Uf t)"
lp15@60987
   264
    by blast
lp15@63938
   265
  have tUft: "\<And>t. t \<in> S-U \<Longrightarrow> t \<in> Uf t"
lp15@60987
   266
    using pf Uf by blast
lp15@63938
   267
  then have *: "S-U \<subseteq> (\<Union>x \<in> S-U. Uf x)"
lp15@60987
   268
    by blast
lp15@63938
   269
  obtain subU where subU: "subU \<subseteq> S - U" "finite subU" "S - U \<subseteq> (\<Union>x \<in> subU. Uf x)"
lp15@65585
   270
    by (blast intro: that compactE_image [OF com_sU open_Uf *])
lp15@60987
   271
  then have [simp]: "subU \<noteq> {}"
lp15@60987
   272
    using t1 by auto
lp15@60987
   273
  then have cardp: "card subU > 0" using subU
lp15@60987
   274
    by (simp add: card_gt_0_iff)
wenzelm@63040
   275
  define p where [abs_def]: "p x = (1 / card subU) * (\<Sum>t \<in> subU. pf t x)" for x
lp15@60987
   276
  have pR: "p \<in> R"
nipkow@64267
   277
    unfolding p_def using subU pf by (fast intro: pf const mult sum)
lp15@60987
   278
  have pt0 [simp]: "p t0 = 0"
nipkow@64267
   279
    using subU pf by (auto simp: p_def intro: sum.neutral)
lp15@63938
   280
  have pt_pos: "p t > 0" if t: "t \<in> S-U" for t
lp15@60987
   281
  proof -
lp15@60987
   282
    obtain i where i: "i \<in> subU" "t \<in> Uf i" using subU t by blast
lp15@60987
   283
    show ?thesis
lp15@60987
   284
      using subU i t
lp15@60987
   285
      apply (clarsimp simp: p_def divide_simps)
nipkow@64267
   286
      apply (rule sum_pos2 [OF \<open>finite subU\<close>])
lp15@60987
   287
      using Uf t pf01 apply auto
lp15@60987
   288
      apply (force elim!: subsetCE)
lp15@60987
   289
      done
lp15@60987
   290
  qed
lp15@63938
   291
  have p01: "p x \<in> {0..1}" if t: "x \<in> S" for x
lp15@60987
   292
  proof -
lp15@60987
   293
    have "0 \<le> p x"
lp15@60987
   294
      using subU cardp t
nipkow@64267
   295
      apply (simp add: p_def divide_simps sum_nonneg)
nipkow@64267
   296
      apply (rule sum_nonneg)
lp15@60987
   297
      using pf01 by force
lp15@60987
   298
    moreover have "p x \<le> 1"
lp15@60987
   299
      using subU cardp t
nipkow@64267
   300
      apply (simp add: p_def divide_simps sum_nonneg)
nipkow@64267
   301
      apply (rule sum_bounded_above [where 'a=real and K=1, simplified])
lp15@60987
   302
      using pf01 by force
lp15@60987
   303
    ultimately show ?thesis
lp15@60987
   304
      by auto
lp15@60987
   305
  qed
lp15@63938
   306
  have "compact (p ` (S-U))"
lp15@60987
   307
    by (meson Diff_subset com_sU compact_continuous_image continuous continuous_on_subset pR)
lp15@63938
   308
  then have "open (- (p ` (S-U)))"
lp15@60987
   309
    by (simp add: compact_imp_closed open_Compl)
lp15@63938
   310
  moreover have "0 \<in> - (p ` (S-U))"
lp15@60987
   311
    by (metis (no_types) ComplI image_iff not_less_iff_gr_or_eq pt_pos)
lp15@63938
   312
  ultimately obtain delta0 where delta0: "delta0 > 0" "ball 0 delta0 \<subseteq> - (p ` (S-U))"
lp15@60987
   313
    by (auto simp: elim!: openE)
lp15@63938
   314
  then have pt_delta: "\<And>x. x \<in> S-U \<Longrightarrow> p x \<ge> delta0"
lp15@60987
   315
    by (force simp: ball_def dist_norm dest: p01)
wenzelm@63040
   316
  define \<delta> where "\<delta> = delta0/2"
lp15@60987
   317
  have "delta0 \<le> 1" using delta0 p01 [of t1] t1
lp15@60987
   318
      by (force simp: ball_def dist_norm dest: p01)
lp15@60987
   319
  with delta0 have \<delta>01: "0 < \<delta>" "\<delta> < 1"
lp15@60987
   320
    by (auto simp: \<delta>_def)
lp15@63938
   321
  have pt_\<delta>: "\<And>x. x \<in> S-U \<Longrightarrow> p x \<ge> \<delta>"
lp15@60987
   322
    using pt_delta delta0 by (force simp: \<delta>_def)
lp15@63938
   323
  have "\<exists>A. open A \<and> A \<inter> S = {x\<in>S. p x < \<delta>/2}"
lp15@60987
   324
    by (rule open_Collect_less_Int [OF continuous [OF pR] continuous_on_const])
lp15@63938
   325
  then obtain V where V: "open V" "V \<inter> S = {x\<in>S. p x < \<delta>/2}"
lp15@60987
   326
    by blast
wenzelm@63040
   327
  define k where "k = nat\<lfloor>1/\<delta>\<rfloor> + 1"
lp15@60987
   328
  have "k>0"  by (simp add: k_def)
lp15@60987
   329
  have "k-1 \<le> 1/\<delta>"
lp15@60987
   330
    using \<delta>01 by (simp add: k_def)
lp15@60987
   331
  with \<delta>01 have "k \<le> (1+\<delta>)/\<delta>"
lp15@60987
   332
    by (auto simp: algebra_simps add_divide_distrib)
lp15@60987
   333
  also have "... < 2/\<delta>"
lp15@60987
   334
    using \<delta>01 by (auto simp: divide_simps)
lp15@60987
   335
  finally have k2\<delta>: "k < 2/\<delta>" .
lp15@60987
   336
  have "1/\<delta> < k"
lp15@60987
   337
    using \<delta>01 unfolding k_def by linarith
lp15@60987
   338
  with \<delta>01 k2\<delta> have k\<delta>: "1 < k*\<delta>" "k*\<delta> < 2"
lp15@60987
   339
    by (auto simp: divide_simps)
wenzelm@63040
   340
  define q where [abs_def]: "q n t = (1 - p t ^ n) ^ (k^n)" for n t
lp15@60987
   341
  have qR: "q n \<in> R" for n
lp15@60987
   342
    by (simp add: q_def const diff power pR)
lp15@63938
   343
  have q01: "\<And>n t. t \<in> S \<Longrightarrow> q n t \<in> {0..1}"
lp15@60987
   344
    using p01 by (simp add: q_def power_le_one algebra_simps)
lp15@60987
   345
  have qt0 [simp]: "\<And>n. n>0 \<Longrightarrow> q n t0 = 1"
lp15@60987
   346
    using t0 pf by (simp add: q_def power_0_left)
lp15@60987
   347
  { fix t and n::nat
lp15@63938
   348
    assume t: "t \<in> S \<inter> V"
wenzelm@61222
   349
    with \<open>k>0\<close> V have "k * p t < k * \<delta> / 2"
lp15@60987
   350
       by force
lp15@60987
   351
    then have "1 - (k * \<delta> / 2)^n \<le> 1 - (k * p t)^n"
wenzelm@61222
   352
      using  \<open>k>0\<close> p01 t by (simp add: power_mono)
lp15@60987
   353
    also have "... \<le> q n t"
lp15@60987
   354
      using Bernoulli_inequality [of "- ((p t)^n)" "k^n"]
lp15@60987
   355
      apply (simp add: q_def)
lp15@60987
   356
      by (metis IntE atLeastAtMost_iff p01 power_le_one power_mult_distrib t)
lp15@60987
   357
    finally have "1 - (k * \<delta> / 2) ^ n \<le> q n t" .
lp15@60987
   358
  } note limitV = this
lp15@60987
   359
  { fix t and n::nat
lp15@63938
   360
    assume t: "t \<in> S - U"
wenzelm@61222
   361
    with \<open>k>0\<close> U have "k * \<delta> \<le> k * p t"
lp15@60987
   362
      by (simp add: pt_\<delta>)
lp15@60987
   363
    with k\<delta> have kpt: "1 < k * p t"
lp15@60987
   364
      by (blast intro: less_le_trans)
lp15@60987
   365
    have ptn_pos: "0 < p t ^ n"
lp15@60987
   366
      using pt_pos [OF t] by simp
lp15@60987
   367
    have ptn_le: "p t ^ n \<le> 1"
lp15@60987
   368
      by (meson DiffE atLeastAtMost_iff p01 power_le_one t)
lp15@60987
   369
    have "q n t = (1/(k^n * (p t)^n)) * (1 - p t ^ n) ^ (k^n) * k^n * (p t)^n"
wenzelm@61222
   370
      using pt_pos [OF t] \<open>k>0\<close> by (simp add: q_def)
lp15@60987
   371
    also have "... \<le> (1/(k * (p t))^n) * (1 - p t ^ n) ^ (k^n) * (1 + k^n * (p t)^n)"
wenzelm@61222
   372
      using pt_pos [OF t] \<open>k>0\<close>
lp15@60987
   373
      apply simp
lp15@60987
   374
      apply (simp only: times_divide_eq_right [symmetric])
lp15@60987
   375
      apply (rule mult_left_mono [of "1::real", simplified])
lp15@60987
   376
      apply (simp_all add: power_mult_distrib)
lp15@60987
   377
      apply (rule zero_le_power)
lp15@60987
   378
      using ptn_le by linarith
lp15@60987
   379
    also have "... \<le> (1/(k * (p t))^n) * (1 - p t ^ n) ^ (k^n) * (1 + (p t)^n) ^ (k^n)"
lp15@60987
   380
      apply (rule mult_left_mono [OF Bernoulli_inequality [of "p t ^ n" "k^n"]])
wenzelm@61222
   381
      using \<open>k>0\<close> ptn_pos ptn_le
lp15@60987
   382
      apply (auto simp: power_mult_distrib)
lp15@60987
   383
      done
lp15@60987
   384
    also have "... = (1/(k * (p t))^n) * (1 - p t ^ (2*n)) ^ (k^n)"
wenzelm@61222
   385
      using pt_pos [OF t] \<open>k>0\<close>
nipkow@68403
   386
      by (simp add: algebra_simps power_mult power2_eq_square flip: power_mult_distrib)
lp15@60987
   387
    also have "... \<le> (1/(k * (p t))^n) * 1"
lp15@60987
   388
      apply (rule mult_left_mono [OF power_le_one])
lp15@61762
   389
      using pt_pos \<open>k>0\<close> p01 power_le_one t apply auto
lp15@60987
   390
      done
lp15@60987
   391
    also have "... \<le> (1 / (k*\<delta>))^n"
wenzelm@61222
   392
      using \<open>k>0\<close> \<delta>01  power_mono pt_\<delta> t
lp15@60987
   393
      by (fastforce simp: field_simps)
lp15@60987
   394
    finally have "q n t \<le> (1 / (real k * \<delta>)) ^ n " .
lp15@60987
   395
  } note limitNonU = this
wenzelm@63040
   396
  define NN
wenzelm@63040
   397
    where "NN e = 1 + nat \<lceil>max (ln e / ln (real k * \<delta> / 2)) (- ln e / ln (real k * \<delta>))\<rceil>" for e
lp15@60987
   398
  have NN: "of_nat (NN e) > ln e / ln (real k * \<delta> / 2)"  "of_nat (NN e) > - ln e / ln (real k * \<delta>)"
lp15@60987
   399
              if "0<e" for e
lp15@60987
   400
      unfolding NN_def  by linarith+
lp15@60987
   401
  have NN1: "\<And>e. e>0 \<Longrightarrow> (k * \<delta> / 2)^NN e < e"
lp15@60987
   402
    apply (subst Transcendental.ln_less_cancel_iff [symmetric])
lp15@60987
   403
      prefer 3 apply (subst ln_realpow)
wenzelm@61222
   404
    using \<open>k>0\<close> \<open>\<delta>>0\<close> NN  k\<delta>
lp15@60987
   405
    apply (force simp add: field_simps)+
lp15@60987
   406
    done
lp15@65578
   407
  have NN0: "(1/(k*\<delta>)) ^ (NN e) < e" if "e>0" for e
lp15@65578
   408
  proof -
lp15@65578
   409
    have "0 < ln (real k) + ln \<delta>"
lp15@65585
   410
      using \<delta>01(1) \<open>0 < k\<close> k\<delta>(1) ln_gt_zero ln_mult by fastforce 
lp15@65578
   411
    then have "real (NN e) * ln (1 / (real k * \<delta>)) < ln e"
lp15@65578
   412
      using k\<delta>(1) NN(2) [of e] that by (simp add: ln_div divide_simps)
lp15@65578
   413
    then have "exp (real (NN e) * ln (1 / (real k * \<delta>))) < e"
lp15@65578
   414
      by (metis exp_less_mono exp_ln that)
lp15@65578
   415
    then show ?thesis
lp15@65583
   416
      by (simp add: \<delta>01(1) \<open>0 < k\<close> exp_of_nat_mult)
lp15@65578
   417
  qed
lp15@60987
   418
  { fix t and e::real
lp15@60987
   419
    assume "e>0"
lp15@63938
   420
    have "t \<in> S \<inter> V \<Longrightarrow> 1 - q (NN e) t < e" "t \<in> S - U \<Longrightarrow> q (NN e) t < e"
lp15@60987
   421
    proof -
lp15@63938
   422
      assume t: "t \<in> S \<inter> V"
lp15@60987
   423
      show "1 - q (NN e) t < e"
wenzelm@61222
   424
        by (metis add.commute diff_le_eq not_le limitV [OF t] less_le_trans [OF NN1 [OF \<open>e>0\<close>]])
lp15@60987
   425
    next
lp15@63938
   426
      assume t: "t \<in> S - U"
lp15@60987
   427
      show "q (NN e) t < e"
wenzelm@61222
   428
      using  limitNonU [OF t] less_le_trans [OF NN0 [OF \<open>e>0\<close>]] not_le by blast
lp15@60987
   429
    qed
lp15@63938
   430
  } then have "\<And>e. e > 0 \<Longrightarrow> \<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. 1 - e < f t)"
lp15@60987
   431
    using q01
lp15@60987
   432
    by (rule_tac x="\<lambda>x. 1 - q (NN e) x" in bexI) (auto simp: algebra_simps intro: diff const qR)
lp15@63938
   433
  moreover have t0V: "t0 \<in> V"  "S \<inter> V \<subseteq> U"
lp15@60987
   434
    using pt_\<delta> t0 U V \<delta>01  by fastforce+
lp15@60987
   435
  ultimately show ?thesis using V t0V
lp15@60987
   436
    by blast
lp15@60987
   437
qed
lp15@60987
   438
wenzelm@69597
   439
text\<open>Non-trivial case, with \<^term>\<open>A\<close> and \<^term>\<open>B\<close> both non-empty\<close>
ak2110@69737
   440
lemma (in function_ring_on) two_special:
lp15@63938
   441
  assumes A: "closed A" "A \<subseteq> S" "a \<in> A"
lp15@63938
   442
      and B: "closed B" "B \<subseteq> S" "b \<in> B"
lp15@60987
   443
      and disj: "A \<inter> B = {}"
lp15@60987
   444
      and e: "0 < e" "e < 1"
lp15@63938
   445
    shows "\<exists>f \<in> R. f ` S \<subseteq> {0..1} \<and> (\<forall>x \<in> A. f x < e) \<and> (\<forall>x \<in> B. f x > 1 - e)"
lp15@60987
   446
proof -
lp15@60987
   447
  { fix w
lp15@60987
   448
    assume "w \<in> A"
lp15@63938
   449
    then have "open ( - B)" "b \<in> S" "w \<notin> B" "w \<in> S"
lp15@60987
   450
      using assms by auto
lp15@63938
   451
    then have "\<exists>V. open V \<and> w \<in> V \<and> S \<inter> V \<subseteq> -B \<and>
lp15@63938
   452
               (\<forall>e>0. \<exists>f \<in> R. f ` S \<subseteq> {0..1} \<and> (\<forall>x \<in> S \<inter> V. f x < e) \<and> (\<forall>x \<in> S \<inter> B. f x > 1 - e))"
wenzelm@61222
   453
      using one [of "-B" w b] assms \<open>w \<in> A\<close> by simp
lp15@60987
   454
  }
lp15@60987
   455
  then obtain Vf where Vf:
lp15@63938
   456
         "\<And>w. w \<in> A \<Longrightarrow> open (Vf w) \<and> w \<in> Vf w \<and> S \<inter> Vf w \<subseteq> -B \<and>
lp15@63938
   457
                         (\<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))"
lp15@60987
   458
    by metis
lp15@60987
   459
  then have open_Vf: "\<And>w. w \<in> A \<Longrightarrow> open (Vf w)"
lp15@60987
   460
    by blast
lp15@60987
   461
  have tVft: "\<And>w. w \<in> A \<Longrightarrow> w \<in> Vf w"
lp15@60987
   462
    using Vf by blast
nipkow@64267
   463
  then have sum_max_0: "A \<subseteq> (\<Union>x \<in> A. Vf x)"
lp15@60987
   464
    by blast
lp15@60987
   465
  have com_A: "compact A" using A
lp15@62843
   466
    by (metis compact compact_Int_closed inf.absorb_iff2)
lp15@60987
   467
  obtain subA where subA: "subA \<subseteq> A" "finite subA" "A \<subseteq> (\<Union>x \<in> subA. Vf x)"
lp15@65585
   468
    by (blast intro: that compactE_image [OF com_A open_Vf sum_max_0])
lp15@60987
   469
  then have [simp]: "subA \<noteq> {}"
wenzelm@61222
   470
    using \<open>a \<in> A\<close> by auto
lp15@60987
   471
  then have cardp: "card subA > 0" using subA
lp15@60987
   472
    by (simp add: card_gt_0_iff)
lp15@63938
   473
  have "\<And>w. w \<in> A \<Longrightarrow> \<exists>f \<in> R. f ` S \<subseteq> {0..1} \<and> (\<forall>x \<in> S \<inter> Vf w. f x < e / card subA) \<and> (\<forall>x \<in> S \<inter> B. f x > 1 - e / card subA)"
lp15@60987
   474
    using Vf e cardp by simp
lp15@60987
   475
  then obtain ff where ff:
lp15@63938
   476
         "\<And>w. w \<in> A \<Longrightarrow> ff w \<in> R \<and> ff w ` S \<subseteq> {0..1} \<and>
lp15@63938
   477
                         (\<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)"
lp15@60987
   478
    by metis
wenzelm@63040
   479
  define pff where [abs_def]: "pff x = (\<Prod>w \<in> subA. ff w x)" for x
lp15@60987
   480
  have pffR: "pff \<in> R"
nipkow@64272
   481
    unfolding pff_def using subA ff by (auto simp: intro: prod)
lp15@60987
   482
  moreover
lp15@63938
   483
  have pff01: "pff x \<in> {0..1}" if t: "x \<in> S" for x
lp15@60987
   484
  proof -
lp15@60987
   485
    have "0 \<le> pff x"
lp15@60987
   486
      using subA cardp t
nipkow@64267
   487
      apply (simp add: pff_def divide_simps sum_nonneg)
nipkow@64272
   488
      apply (rule Groups_Big.linordered_semidom_class.prod_nonneg)
lp15@60987
   489
      using ff by fastforce
lp15@60987
   490
    moreover have "pff x \<le> 1"
lp15@60987
   491
      using subA cardp t
nipkow@64267
   492
      apply (simp add: pff_def divide_simps sum_nonneg)
nipkow@64272
   493
      apply (rule prod_mono [where g = "\<lambda>x. 1", simplified])
lp15@60987
   494
      using ff by fastforce
lp15@60987
   495
    ultimately show ?thesis
lp15@60987
   496
      by auto
lp15@60987
   497
  qed
lp15@60987
   498
  moreover
lp15@60987
   499
  { fix v x
lp15@63938
   500
    assume v: "v \<in> subA" and x: "x \<in> Vf v" "x \<in> S"
lp15@60987
   501
    from subA v have "pff x = ff v x * (\<Prod>w \<in> subA - {v}. ff w x)"
nipkow@64272
   502
      unfolding pff_def  by (metis prod.remove)
lp15@60987
   503
    also have "... \<le> ff v x * 1"
lp15@60987
   504
      apply (rule Rings.ordered_semiring_class.mult_left_mono)
nipkow@64272
   505
      apply (rule prod_mono [where g = "\<lambda>x. 1", simplified])
lp15@60987
   506
      using ff [THEN conjunct2, THEN conjunct1] v subA x
lp15@60987
   507
      apply auto
lp15@60987
   508
      apply (meson atLeastAtMost_iff contra_subsetD imageI)
lp15@60987
   509
      apply (meson atLeastAtMost_iff contra_subsetD image_eqI)
lp15@60987
   510
      using atLeastAtMost_iff by blast
lp15@60987
   511
    also have "... < e / card subA"
lp15@60987
   512
      using ff [THEN conjunct2, THEN conjunct2, THEN conjunct1] v subA x
lp15@60987
   513
      by auto
lp15@60987
   514
    also have "... \<le> e"
lp15@60987
   515
      using cardp e by (simp add: divide_simps)
lp15@60987
   516
    finally have "pff x < e" .
lp15@60987
   517
  }
lp15@60987
   518
  then have "\<And>x. x \<in> A \<Longrightarrow> pff x < e"
lp15@60987
   519
    using A Vf subA by (metis UN_E contra_subsetD)
lp15@60987
   520
  moreover
lp15@60987
   521
  { fix x
lp15@60987
   522
    assume x: "x \<in> B"
lp15@63938
   523
    then have "x \<in> S"
lp15@60987
   524
      using B by auto
lp15@60987
   525
    have "1 - e \<le> (1 - e / card subA) ^ card subA"
lp15@60987
   526
      using Bernoulli_inequality [of "-e / card subA" "card subA"] e cardp
lp15@60987
   527
      by (auto simp: field_simps)
lp15@60987
   528
    also have "... = (\<Prod>w \<in> subA. 1 - e / card subA)"
nipkow@64272
   529
      by (simp add: prod_constant subA(2))
lp15@60987
   530
    also have "... < pff x"
lp15@60987
   531
      apply (simp add: pff_def)
nipkow@64272
   532
      apply (rule prod_mono_strict [where f = "\<lambda>x. 1 - e / card subA", simplified])
lp15@60987
   533
      apply (simp_all add: subA(2))
lp15@60987
   534
      apply (intro ballI conjI)
lp15@60987
   535
      using e apply (force simp: divide_simps)
lp15@60987
   536
      using ff [THEN conjunct2, THEN conjunct2, THEN conjunct2] subA B x
lp15@60987
   537
      apply blast
lp15@60987
   538
      done
lp15@60987
   539
    finally have "1 - e < pff x" .
lp15@60987
   540
  }
lp15@60987
   541
  ultimately
lp15@60987
   542
  show ?thesis by blast
lp15@60987
   543
qed
lp15@60987
   544
ak2110@69737
   545
lemma (in function_ring_on) two:
lp15@63938
   546
  assumes A: "closed A" "A \<subseteq> S"
lp15@63938
   547
      and B: "closed B" "B \<subseteq> S"
lp15@60987
   548
      and disj: "A \<inter> B = {}"
lp15@60987
   549
      and e: "0 < e" "e < 1"
lp15@63938
   550
    shows "\<exists>f \<in> R. f ` S \<subseteq> {0..1} \<and> (\<forall>x \<in> A. f x < e) \<and> (\<forall>x \<in> B. f x > 1 - e)"
lp15@60987
   551
proof (cases "A \<noteq> {} \<and> B \<noteq> {}")
lp15@60987
   552
  case True then show ?thesis
nipkow@68403
   553
    apply (simp flip: ex_in_conv)
lp15@60987
   554
    using assms
lp15@60987
   555
    apply safe
lp15@60987
   556
    apply (force simp add: intro!: two_special)
lp15@60987
   557
    done
lp15@60987
   558
next
lp15@60987
   559
  case False with e show ?thesis
lp15@60987
   560
    apply simp
lp15@60987
   561
    apply (erule disjE)
lp15@60987
   562
    apply (rule_tac [2] x="\<lambda>x. 0" in bexI)
lp15@60987
   563
    apply (rule_tac x="\<lambda>x. 1" in bexI)
lp15@60987
   564
    apply (auto simp: const)
lp15@60987
   565
    done
lp15@60987
   566
qed
lp15@60987
   567
wenzelm@69597
   568
text\<open>The special case where \<^term>\<open>f\<close> is non-negative and \<^term>\<open>e<1/3\<close>\<close>
ak2110@69737
   569
lemma (in function_ring_on) Stone_Weierstrass_special:
lp15@63938
   570
  assumes f: "continuous_on S f" and fpos: "\<And>x. x \<in> S \<Longrightarrow> f x \<ge> 0"
lp15@60987
   571
      and e: "0 < e" "e < 1/3"
lp15@63938
   572
  shows "\<exists>g \<in> R. \<forall>x\<in>S. \<bar>f x - g x\<bar> < 2*e"
ak2110@69737
   573
proof -
wenzelm@63040
   574
  define n where "n = 1 + nat \<lceil>normf f / e\<rceil>"
lp15@63938
   575
  define A where "A j = {x \<in> S. f x \<le> (j - 1/3)*e}" for j :: nat
lp15@63938
   576
  define B where "B j = {x \<in> S. f x \<ge> (j + 1/3)*e}" for j :: nat
lp15@60987
   577
  have ngt: "(n-1) * e \<ge> normf f" "n\<ge>1"
lp15@60987
   578
    using e
lp15@61609
   579
    apply (simp_all add: n_def field_simps of_nat_Suc)
lp15@60987
   580
    by (metis real_nat_ceiling_ge mult.commute not_less pos_less_divide_eq)
lp15@63938
   581
  then have ge_fx: "(n-1) * e \<ge> f x" if "x \<in> S" for x
lp15@60987
   582
    using f normf_upper that by fastforce
lp15@60987
   583
  { fix j
lp15@63938
   584
    have A: "closed (A j)" "A j \<subseteq> S"
lp15@60987
   585
      apply (simp_all add: A_def Collect_restrict)
lp15@60987
   586
      apply (rule continuous_on_closed_Collect_le [OF f continuous_on_const])
lp15@60987
   587
      apply (simp add: compact compact_imp_closed)
lp15@60987
   588
      done
lp15@63938
   589
    have B: "closed (B j)" "B j \<subseteq> S"
lp15@60987
   590
      apply (simp_all add: B_def Collect_restrict)
lp15@60987
   591
      apply (rule continuous_on_closed_Collect_le [OF continuous_on_const f])
lp15@60987
   592
      apply (simp add: compact compact_imp_closed)
lp15@60987
   593
      done
lp15@60987
   594
    have disj: "(A j) \<inter> (B j) = {}"
lp15@60987
   595
      using e by (auto simp: A_def B_def field_simps)
lp15@63938
   596
    have "\<exists>f \<in> R. f ` S \<subseteq> {0..1} \<and> (\<forall>x \<in> A j. f x < e/n) \<and> (\<forall>x \<in> B j. f x > 1 - e/n)"
lp15@60987
   597
      apply (rule two)
lp15@60987
   598
      using e A B disj ngt
lp15@60987
   599
      apply simp_all
lp15@60987
   600
      done
lp15@60987
   601
  }
lp15@63938
   602
  then obtain xf where xfR: "\<And>j. xf j \<in> R" and xf01: "\<And>j. xf j ` S \<subseteq> {0..1}"
lp15@60987
   603
                   and xfA: "\<And>x j. x \<in> A j \<Longrightarrow> xf j x < e/n"
lp15@60987
   604
                   and xfB: "\<And>x j. x \<in> B j \<Longrightarrow> xf j x > 1 - e/n"
lp15@60987
   605
    by metis
wenzelm@63040
   606
  define g where [abs_def]: "g x = e * (\<Sum>i\<le>n. xf i x)" for x
lp15@60987
   607
  have gR: "g \<in> R"
nipkow@64267
   608
    unfolding g_def by (fast intro: mult const sum xfR)
lp15@63938
   609
  have gge0: "\<And>x. x \<in> S \<Longrightarrow> g x \<ge> 0"
nipkow@64267
   610
    using e xf01 by (simp add: g_def zero_le_mult_iff image_subset_iff sum_nonneg)
lp15@60987
   611
  have A0: "A 0 = {}"
lp15@60987
   612
    using fpos e by (fastforce simp: A_def)
lp15@63938
   613
  have An: "A n = S"
lp15@61609
   614
    using e ngt f normf_upper by (fastforce simp: A_def field_simps of_nat_diff)
lp15@60987
   615
  have Asub: "A j \<subseteq> A i" if "i\<ge>j" for i j
lp15@60987
   616
    using e that apply (clarsimp simp: A_def)
lp15@60987
   617
    apply (erule order_trans, simp)
lp15@60987
   618
    done
lp15@60987
   619
  { fix t
lp15@63938
   620
    assume t: "t \<in> S"
wenzelm@63040
   621
    define j where "j = (LEAST j. t \<in> A j)"
lp15@60987
   622
    have jn: "j \<le> n"
lp15@60987
   623
      using t An by (simp add: Least_le j_def)
lp15@60987
   624
    have Aj: "t \<in> A j"
lp15@60987
   625
      using t An by (fastforce simp add: j_def intro: LeastI)
lp15@60987
   626
    then have Ai: "t \<in> A i" if "i\<ge>j" for i
lp15@60987
   627
      using Asub [OF that] by blast
lp15@60987
   628
    then have fj1: "f t \<le> (j - 1/3)*e"
lp15@60987
   629
      by (simp add: A_def)
lp15@60987
   630
    then have Anj: "t \<notin> A i" if "i<j" for i
wenzelm@61222
   631
      using  Aj  \<open>i<j\<close>
lp15@60987
   632
      apply (simp add: j_def)
lp15@60987
   633
      using not_less_Least by blast
lp15@60987
   634
    have j1: "1 \<le> j"
lp15@60987
   635
      using A0 Aj j_def not_less_eq_eq by (fastforce simp add: j_def)
lp15@60987
   636
    then have Anj: "t \<notin> A (j-1)"
lp15@60987
   637
      using Least_le by (fastforce simp add: j_def)
lp15@60987
   638
    then have fj2: "(j - 4/3)*e < f t"
lp15@61609
   639
      using j1 t  by (simp add: A_def of_nat_diff)
lp15@60987
   640
    have ***: "xf i t \<le> e/n" if "i\<ge>j" for i
lp15@60987
   641
      using xfA [OF Ai] that by (simp add: less_eq_real_def)
lp15@60987
   642
    { fix i
lp15@60987
   643
      assume "i+2 \<le> j"
lp15@60987
   644
      then obtain d where "i+2+d = j"
lp15@60987
   645
        using le_Suc_ex that by blast
lp15@60987
   646
      then have "t \<in> B i"
wenzelm@61222
   647
        using Anj e ge_fx [OF t] \<open>1 \<le> n\<close> fpos [OF t] t
lp15@60987
   648
        apply (simp add: A_def B_def)
lp15@61609
   649
        apply (clarsimp simp add: field_simps of_nat_diff not_le of_nat_Suc)
lp15@60987
   650
        apply (rule order_trans [of _ "e * 2 + (e * (real d * 3) + e * (real i * 3))"])
lp15@60987
   651
        apply auto
lp15@60987
   652
        done
lp15@60987
   653
      then have "xf i t > 1 - e/n"
lp15@60987
   654
        by (rule xfB)
lp15@60987
   655
    } note **** = this
lp15@60987
   656
    have xf_le1: "\<And>i. xf i t \<le> 1"
lp15@60987
   657
      using xf01 t by force
lp15@60987
   658
    have "g t = e * (\<Sum>i<j. xf i t) + e * (\<Sum>i=j..n. xf i t)"
lp15@60987
   659
      using j1 jn e
nipkow@68403
   660
      apply (simp add: g_def flip: distrib_left)
nipkow@64267
   661
      apply (subst sum.union_disjoint [symmetric])
lp15@60987
   662
      apply (auto simp: ivl_disj_un)
lp15@60987
   663
      done
lp15@60987
   664
    also have "... \<le> e*j + e * ((Suc n - j)*e/n)"
lp15@60987
   665
      apply (rule add_mono)
lp15@61609
   666
      apply (simp_all only: mult_le_cancel_left_pos e)
nipkow@64267
   667
      apply (rule sum_bounded_above [OF xf_le1, where A = "lessThan j", simplified])
nipkow@64267
   668
      using sum_bounded_above [of "{j..n}" "\<lambda>i. xf i t", OF ***]
lp15@60987
   669
      apply simp
lp15@60987
   670
      done
lp15@60987
   671
    also have "... \<le> j*e + e*(n - j + 1)*e/n "
lp15@61609
   672
      using \<open>1 \<le> n\<close> e  by (simp add: field_simps del: of_nat_Suc)
lp15@60987
   673
    also have "... \<le> j*e + e*e"
lp15@61609
   674
      using \<open>1 \<le> n\<close> e j1 by (simp add: field_simps del: of_nat_Suc)
lp15@60987
   675
    also have "... < (j + 1/3)*e"
lp15@60987
   676
      using e by (auto simp: field_simps)
lp15@60987
   677
    finally have gj1: "g t < (j + 1 / 3) * e" .
lp15@60987
   678
    have gj2: "(j - 4/3)*e < g t"
lp15@60987
   679
    proof (cases "2 \<le> j")
lp15@60987
   680
      case False
lp15@60987
   681
      then have "j=1" using j1 by simp
lp15@60987
   682
      with t gge0 e show ?thesis by force
lp15@60987
   683
    next
lp15@60987
   684
      case True
lp15@60987
   685
      then have "(j - 4/3)*e < (j-1)*e - e^2"
lp15@61609
   686
        using e by (auto simp: of_nat_diff algebra_simps power2_eq_square)
lp15@60987
   687
      also have "... < (j-1)*e - ((j - 1)/n) * e^2"
lp15@60987
   688
        using e True jn by (simp add: power2_eq_square field_simps)
lp15@60987
   689
      also have "... = e * (j-1) * (1 - e/n)"
lp15@60987
   690
        by (simp add: power2_eq_square field_simps)
lp15@60987
   691
      also have "... \<le> e * (\<Sum>i\<le>j-2. xf i t)"
lp15@60987
   692
        using e
lp15@60987
   693
        apply simp
nipkow@64267
   694
        apply (rule order_trans [OF _ sum_bounded_below [OF less_imp_le [OF ****]]])
lp15@60987
   695
        using True
lp15@61609
   696
        apply (simp_all add: of_nat_Suc of_nat_diff)
lp15@60987
   697
        done
lp15@60987
   698
      also have "... \<le> g t"
lp15@60987
   699
        using jn e
lp15@60987
   700
        using e xf01 t
nipkow@64267
   701
        apply (simp add: g_def zero_le_mult_iff image_subset_iff sum_nonneg)
nipkow@64267
   702
        apply (rule Groups_Big.sum_mono2, auto)
lp15@60987
   703
        done
lp15@60987
   704
      finally show ?thesis .
lp15@60987
   705
    qed
lp15@60987
   706
    have "\<bar>f t - g t\<bar> < 2 * e"
lp15@60987
   707
      using fj1 fj2 gj1 gj2 by (simp add: abs_less_iff field_simps)
lp15@60987
   708
  }
lp15@60987
   709
  then show ?thesis
lp15@60987
   710
    by (rule_tac x=g in bexI) (auto intro: gR)
lp15@60987
   711
qed
lp15@60987
   712
lp15@60987
   713
text\<open>The ``unpretentious'' formulation\<close>
ak2110@69737
   714
proposition (in function_ring_on) Stone_Weierstrass_basic:
lp15@63938
   715
  assumes f: "continuous_on S f" and e: "e > 0"
lp15@63938
   716
  shows "\<exists>g \<in> R. \<forall>x\<in>S. \<bar>f x - g x\<bar> < e"
ak2110@69737
   717
proof -
lp15@63938
   718
  have "\<exists>g \<in> R. \<forall>x\<in>S. \<bar>(f x + normf f) - g x\<bar> < 2 * min (e/2) (1/4)"
lp15@60987
   719
    apply (rule Stone_Weierstrass_special)
lp15@60987
   720
    apply (rule Limits.continuous_on_add [OF f Topological_Spaces.continuous_on_const])
lp15@60987
   721
    using normf_upper [OF f] apply force
lp15@60987
   722
    apply (simp add: e, linarith)
lp15@60987
   723
    done
lp15@63938
   724
  then obtain g where "g \<in> R" "\<forall>x\<in>S. \<bar>g x - (f x + normf f)\<bar> < e"
lp15@60987
   725
    by force
lp15@60987
   726
  then show ?thesis
lp15@60987
   727
    apply (rule_tac x="\<lambda>x. g x - normf f" in bexI)
lp15@60987
   728
    apply (auto simp: algebra_simps intro: diff const)
lp15@60987
   729
    done
lp15@60987
   730
qed
lp15@60987
   731
lp15@60987
   732
ak2110@69737
   733
theorem (in function_ring_on) Stone_Weierstrass:
lp15@63938
   734
  assumes f: "continuous_on S f"
lp15@63938
   735
  shows "\<exists>F\<in>UNIV \<rightarrow> R. LIM n sequentially. F n :> uniformly_on S f"
ak2110@69737
   736
proof -
lp15@60987
   737
  { fix e::real
lp15@60987
   738
    assume e: "0 < e"
lp15@60987
   739
    then obtain N::nat where N: "0 < N" "0 < inverse N" "inverse N < e"
lp15@62623
   740
      by (auto simp: real_arch_inverse [of e])
lp15@60987
   741
    { fix n :: nat and x :: 'a and g :: "'a \<Rightarrow> real"
lp15@63938
   742
      assume n: "N \<le> n"  "\<forall>x\<in>S. \<bar>f x - g x\<bar> < 1 / (1 + real n)"
lp15@63938
   743
      assume x: "x \<in> S"
lp15@60987
   744
      have "\<not> real (Suc n) < inverse e"
wenzelm@61222
   745
        using \<open>N \<le> n\<close> N using less_imp_inverse_less by force
lp15@60987
   746
      then have "1 / (1 + real n) \<le> e"
lp15@61609
   747
        using e by (simp add: field_simps of_nat_Suc)
lp15@60987
   748
      then have "\<bar>f x - g x\<bar> < e"
lp15@60987
   749
        using n(2) x by auto
lp15@60987
   750
    } note * = this
lp15@63938
   751
    have "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. \<bar>f x - (SOME g. g \<in> R \<and> (\<forall>x\<in>S. \<bar>f x - g x\<bar> < 1 / (1 + real n))) x\<bar> < e"
lp15@60987
   752
      apply (rule eventually_sequentiallyI [of N])
lp15@60987
   753
      apply (auto intro: someI2_bex [OF Stone_Weierstrass_basic [OF f]] *)
lp15@60987
   754
      done
lp15@60987
   755
  } then
lp15@60987
   756
  show ?thesis
lp15@63938
   757
    apply (rule_tac x="\<lambda>n::nat. SOME g. g \<in> R \<and> (\<forall>x\<in>S. \<bar>f x - g x\<bar> < 1 / (1 + n))" in bexI)
lp15@60987
   758
    prefer 2  apply (force intro: someI2_bex [OF Stone_Weierstrass_basic [OF f]])
lp15@60987
   759
    unfolding uniform_limit_iff
lp15@60987
   760
    apply (auto simp: dist_norm abs_minus_commute)
lp15@60987
   761
    done
lp15@60987
   762
qed
lp15@60987
   763
wenzelm@61222
   764
text\<open>A HOL Light formulation\<close>
ak2110@69737
   765
corollary Stone_Weierstrass_HOL:
lp15@63938
   766
  fixes R :: "('a::t2_space \<Rightarrow> real) set" and S :: "'a set"
lp15@63938
   767
  assumes "compact S"  "\<And>c. P(\<lambda>x. c::real)"
lp15@63938
   768
          "\<And>f. P f \<Longrightarrow> continuous_on S f"
lp15@60987
   769
          "\<And>f g. P(f) \<and> P(g) \<Longrightarrow> P(\<lambda>x. f x + g x)"  "\<And>f g. P(f) \<and> P(g) \<Longrightarrow> P(\<lambda>x. f x * g x)"
nipkow@69508
   770
          "\<And>x y. x \<in> S \<and> y \<in> S \<and> x \<noteq> y \<Longrightarrow> \<exists>f. P(f) \<and> f x \<noteq> f y"
lp15@63938
   771
          "continuous_on S f"
lp15@60987
   772
       "0 < e"
lp15@63938
   773
    shows "\<exists>g. P(g) \<and> (\<forall>x \<in> S. \<bar>f x - g x\<bar> < e)"
ak2110@69737
   774
proof -
lp15@60987
   775
  interpret PR: function_ring_on "Collect P"
lp15@60987
   776
    apply unfold_locales
lp15@60987
   777
    using assms
lp15@60987
   778
    by auto
lp15@60987
   779
  show ?thesis
lp15@63938
   780
    using PR.Stone_Weierstrass_basic [OF \<open>continuous_on S f\<close> \<open>0 < e\<close>]
lp15@60987
   781
    by blast
lp15@60987
   782
qed
lp15@60987
   783
lp15@60987
   784
immler@69683
   785
subsection \<open>Polynomial functions\<close>
lp15@60987
   786
lp15@60987
   787
inductive real_polynomial_function :: "('a::real_normed_vector \<Rightarrow> real) \<Rightarrow> bool" where
lp15@60987
   788
    linear: "bounded_linear f \<Longrightarrow> real_polynomial_function f"
lp15@60987
   789
  | const: "real_polynomial_function (\<lambda>x. c)"
lp15@60987
   790
  | add:   "\<lbrakk>real_polynomial_function f; real_polynomial_function g\<rbrakk> \<Longrightarrow> real_polynomial_function (\<lambda>x. f x + g x)"
lp15@60987
   791
  | mult:  "\<lbrakk>real_polynomial_function f; real_polynomial_function g\<rbrakk> \<Longrightarrow> real_polynomial_function (\<lambda>x. f x * g x)"
lp15@60987
   792
lp15@60987
   793
declare real_polynomial_function.intros [intro]
lp15@60987
   794
ak2110@68833
   795
definition%important polynomial_function :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> bool"
lp15@60987
   796
  where
lp15@60987
   797
   "polynomial_function p \<equiv> (\<forall>f. bounded_linear f \<longrightarrow> real_polynomial_function (f o p))"
lp15@60987
   798
ak2110@69737
   799
lemma real_polynomial_function_eq: "real_polynomial_function p = polynomial_function p"
lp15@60987
   800
unfolding polynomial_function_def
lp15@60987
   801
proof
lp15@60987
   802
  assume "real_polynomial_function p"
lp15@60987
   803
  then show " \<forall>f. bounded_linear f \<longrightarrow> real_polynomial_function (f \<circ> p)"
lp15@60987
   804
  proof (induction p rule: real_polynomial_function.induct)
lp15@60987
   805
    case (linear h) then show ?case
lp15@60987
   806
      by (auto simp: bounded_linear_compose real_polynomial_function.linear)
lp15@60987
   807
  next
lp15@60987
   808
    case (const h) then show ?case
lp15@60987
   809
      by (simp add: real_polynomial_function.const)
lp15@60987
   810
  next
lp15@60987
   811
    case (add h) then show ?case
lp15@60987
   812
      by (force simp add: bounded_linear_def linear_add real_polynomial_function.add)
lp15@60987
   813
  next
lp15@60987
   814
    case (mult h) then show ?case
lp15@60987
   815
      by (force simp add: real_bounded_linear const real_polynomial_function.mult)
lp15@60987
   816
  qed
lp15@60987
   817
next
lp15@60987
   818
  assume [rule_format, OF bounded_linear_ident]: "\<forall>f. bounded_linear f \<longrightarrow> real_polynomial_function (f \<circ> p)"
lp15@60987
   819
  then show "real_polynomial_function p"
lp15@60987
   820
    by (simp add: o_def)
lp15@60987
   821
qed
lp15@60987
   822
ak2110@69737
   823
lemma polynomial_function_const [iff]: "polynomial_function (\<lambda>x. c)"
lp15@60987
   824
  by (simp add: polynomial_function_def o_def const)
lp15@60987
   825
ak2110@69737
   826
lemma polynomial_function_bounded_linear:
lp15@60987
   827
  "bounded_linear f \<Longrightarrow> polynomial_function f"
lp15@60987
   828
  by (simp add: polynomial_function_def o_def bounded_linear_compose real_polynomial_function.linear)
lp15@60987
   829
ak2110@69737
   830
lemma polynomial_function_id [iff]: "polynomial_function(\<lambda>x. x)"
lp15@60987
   831
  by (simp add: polynomial_function_bounded_linear)
lp15@60987
   832
ak2110@69737
   833
lemma polynomial_function_add [intro]:
lp15@60987
   834
    "\<lbrakk>polynomial_function f; polynomial_function g\<rbrakk> \<Longrightarrow> polynomial_function (\<lambda>x. f x + g x)"
lp15@60987
   835
  by (auto simp: polynomial_function_def bounded_linear_def linear_add real_polynomial_function.add o_def)
lp15@60987
   836
ak2110@69737
   837
lemma polynomial_function_mult [intro]:
lp15@60987
   838
  assumes f: "polynomial_function f" and g: "polynomial_function g"
lp15@60987
   839
    shows "polynomial_function (\<lambda>x. f x *\<^sub>R g x)"
lp15@60987
   840
  using g
lp15@60987
   841
  apply (auto simp: polynomial_function_def bounded_linear_def Real_Vector_Spaces.linear.scaleR  const real_polynomial_function.mult o_def)
lp15@60987
   842
  apply (rule mult)
lp15@60987
   843
  using f
lp15@60987
   844
  apply (auto simp: real_polynomial_function_eq)
lp15@60987
   845
  done
lp15@60987
   846
ak2110@69737
   847
lemma polynomial_function_cmul [intro]:
lp15@60987
   848
  assumes f: "polynomial_function f"
lp15@60987
   849
    shows "polynomial_function (\<lambda>x. c *\<^sub>R f x)"
lp15@60987
   850
  by (rule polynomial_function_mult [OF polynomial_function_const f])
lp15@60987
   851
ak2110@69737
   852
lemma polynomial_function_minus [intro]:
lp15@60987
   853
  assumes f: "polynomial_function f"
lp15@60987
   854
    shows "polynomial_function (\<lambda>x. - f x)"
lp15@60987
   855
  using polynomial_function_cmul [OF f, of "-1"] by simp
lp15@60987
   856
ak2110@69737
   857
lemma polynomial_function_diff [intro]:
lp15@60987
   858
    "\<lbrakk>polynomial_function f; polynomial_function g\<rbrakk> \<Longrightarrow> polynomial_function (\<lambda>x. f x - g x)"
lp15@60987
   859
  unfolding add_uminus_conv_diff [symmetric]
lp15@60987
   860
  by (metis polynomial_function_add polynomial_function_minus)
lp15@60987
   861
ak2110@69737
   862
lemma polynomial_function_sum [intro]:
nipkow@64267
   863
    "\<lbrakk>finite I; \<And>i. i \<in> I \<Longrightarrow> polynomial_function (\<lambda>x. f x i)\<rbrakk> \<Longrightarrow> polynomial_function (\<lambda>x. sum (f x) I)"
lp15@60987
   864
by (induct I rule: finite_induct) auto
lp15@60987
   865
ak2110@69737
   866
lemma real_polynomial_function_minus [intro]:
lp15@60987
   867
    "real_polynomial_function f \<Longrightarrow> real_polynomial_function (\<lambda>x. - f x)"
lp15@60987
   868
  using polynomial_function_minus [of f]
lp15@60987
   869
  by (simp add: real_polynomial_function_eq)
lp15@60987
   870
ak2110@69737
   871
lemma real_polynomial_function_diff [intro]:
lp15@60987
   872
    "\<lbrakk>real_polynomial_function f; real_polynomial_function g\<rbrakk> \<Longrightarrow> real_polynomial_function (\<lambda>x. f x - g x)"
lp15@60987
   873
  using polynomial_function_diff [of f]
lp15@60987
   874
  by (simp add: real_polynomial_function_eq)
lp15@60987
   875
ak2110@69737
   876
lemma real_polynomial_function_sum [intro]:
nipkow@64267
   877
    "\<lbrakk>finite I; \<And>i. i \<in> I \<Longrightarrow> real_polynomial_function (\<lambda>x. f x i)\<rbrakk> \<Longrightarrow> real_polynomial_function (\<lambda>x. sum (f x) I)"
nipkow@64267
   878
  using polynomial_function_sum [of I f]
lp15@60987
   879
  by (simp add: real_polynomial_function_eq)
lp15@60987
   880
ak2110@69737
   881
lemma real_polynomial_function_power [intro]:
lp15@60987
   882
    "real_polynomial_function f \<Longrightarrow> real_polynomial_function (\<lambda>x. f x ^ n)"
lp15@60987
   883
  by (induct n) (simp_all add: const mult)
lp15@60987
   884
ak2110@69737
   885
lemma real_polynomial_function_compose [intro]:
lp15@60987
   886
  assumes f: "polynomial_function f" and g: "real_polynomial_function g"
lp15@60987
   887
    shows "real_polynomial_function (g o f)"
lp15@60987
   888
  using g
lp15@60987
   889
  apply (induction g rule: real_polynomial_function.induct)
lp15@60987
   890
  using f
lp15@60987
   891
  apply (simp_all add: polynomial_function_def o_def const add mult)
lp15@60987
   892
  done
lp15@60987
   893
ak2110@69737
   894
lemma polynomial_function_compose [intro]:
lp15@60987
   895
  assumes f: "polynomial_function f" and g: "polynomial_function g"
lp15@60987
   896
    shows "polynomial_function (g o f)"
lp15@60987
   897
  using g real_polynomial_function_compose [OF f]
lp15@60987
   898
  by (auto simp: polynomial_function_def o_def)
lp15@60987
   899
ak2110@69737
   900
lemma sum_max_0:
lp15@60987
   901
  fixes x::real (*in fact "'a::comm_ring_1"*)
lp15@68077
   902
  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)"
lp15@60987
   903
proof -
lp15@68077
   904
  have "(\<Sum>i\<le>max m n. x^i * (if i \<le> m then a i else 0)) = (\<Sum>i\<le>max m n. (if i \<le> m then x^i * a i else 0))"
nipkow@64267
   905
    by (auto simp: algebra_simps intro: sum.cong)
lp15@68077
   906
  also have "... = (\<Sum>i\<le>m. (if i \<le> m then x^i * a i else 0))"
nipkow@64267
   907
    by (rule sum.mono_neutral_right) auto
lp15@68077
   908
  also have "... = (\<Sum>i\<le>m. x^i * a i)"
nipkow@64267
   909
    by (auto simp: algebra_simps intro: sum.cong)
lp15@60987
   910
  finally show ?thesis .
lp15@60987
   911
qed
lp15@60987
   912
ak2110@69737
   913
lemma real_polynomial_function_imp_sum:
lp15@60987
   914
  assumes "real_polynomial_function f"
lp15@68077
   915
    shows "\<exists>a n::nat. f = (\<lambda>x. \<Sum>i\<le>n. a i * x ^ i)"
lp15@60987
   916
using assms
lp15@60987
   917
proof (induct f)
lp15@60987
   918
  case (linear f)
lp15@60987
   919
  then show ?case
lp15@60987
   920
    apply (clarsimp simp add: real_bounded_linear)
lp15@60987
   921
    apply (rule_tac x="\<lambda>i. if i=0 then 0 else c" in exI)
lp15@60987
   922
    apply (rule_tac x=1 in exI)
lp15@60987
   923
    apply (simp add: mult_ac)
lp15@60987
   924
    done
lp15@60987
   925
next
lp15@60987
   926
  case (const c)
lp15@60987
   927
  show ?case
lp15@60987
   928
    apply (rule_tac x="\<lambda>i. c" in exI)
lp15@60987
   929
    apply (rule_tac x=0 in exI)
lp15@61609
   930
    apply (auto simp: mult_ac of_nat_Suc)
lp15@60987
   931
    done
lp15@60987
   932
  case (add f1 f2)
lp15@60987
   933
  then obtain a1 n1 a2 n2 where
lp15@68077
   934
    "f1 = (\<lambda>x. \<Sum>i\<le>n1. a1 i * x ^ i)" "f2 = (\<lambda>x. \<Sum>i\<le>n2. a2 i * x ^ i)"
lp15@60987
   935
    by auto
lp15@60987
   936
  then show ?case
lp15@60987
   937
    apply (rule_tac x="\<lambda>i. (if i \<le> n1 then a1 i else 0) + (if i \<le> n2 then a2 i else 0)" in exI)
lp15@60987
   938
    apply (rule_tac x="max n1 n2" in exI)
nipkow@64267
   939
    using sum_max_0 [where m=n1 and n=n2] sum_max_0 [where m=n2 and n=n1]
nipkow@64267
   940
    apply (simp add: sum.distrib algebra_simps max.commute)
lp15@60987
   941
    done
lp15@60987
   942
  case (mult f1 f2)
lp15@60987
   943
  then obtain a1 n1 a2 n2 where
lp15@68077
   944
    "f1 = (\<lambda>x. \<Sum>i\<le>n1. a1 i * x ^ i)" "f2 = (\<lambda>x. \<Sum>i\<le>n2. a2 i * x ^ i)"
lp15@60987
   945
    by auto
lp15@60987
   946
  then obtain b1 b2 where
lp15@68077
   947
    "f1 = (\<lambda>x. \<Sum>i\<le>n1. b1 i * x ^ i)" "f2 = (\<lambda>x. \<Sum>i\<le>n2. b2 i * x ^ i)"
lp15@60987
   948
    "b1 = (\<lambda>i. if i\<le>n1 then a1 i else 0)" "b2 = (\<lambda>i. if i\<le>n2 then a2 i else 0)"
lp15@60987
   949
    by auto
lp15@60987
   950
  then show ?case
lp15@60987
   951
    apply (rule_tac x="\<lambda>i. \<Sum>k\<le>i. b1 k * b2 (i - k)" in exI)
lp15@60987
   952
    apply (rule_tac x="n1+n2" in exI)
lp15@60987
   953
    using polynomial_product [of n1 b1 n2 b2]
lp15@60987
   954
    apply (simp add: Set_Interval.atLeast0AtMost)
lp15@60987
   955
    done
lp15@60987
   956
qed
lp15@60987
   957
ak2110@69737
   958
lemma real_polynomial_function_iff_sum:
lp15@68077
   959
     "real_polynomial_function f \<longleftrightarrow> (\<exists>a n::nat. f = (\<lambda>x. \<Sum>i\<le>n. a i * x ^ i))"
lp15@60987
   960
  apply (rule iffI)
nipkow@64267
   961
  apply (erule real_polynomial_function_imp_sum)
nipkow@64267
   962
  apply (auto simp: linear mult const real_polynomial_function_power real_polynomial_function_sum)
lp15@60987
   963
  done
lp15@60987
   964
ak2110@69737
   965
lemma polynomial_function_iff_Basis_inner:
lp15@60987
   966
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
lp15@60987
   967
  shows "polynomial_function f \<longleftrightarrow> (\<forall>b\<in>Basis. real_polynomial_function (\<lambda>x. inner (f x) b))"
lp15@60987
   968
        (is "?lhs = ?rhs")
lp15@60987
   969
unfolding polynomial_function_def
ak2110@69737
   970
proof (intro iffI allI impI)
lp15@60987
   971
  assume "\<forall>h. bounded_linear h \<longrightarrow> real_polynomial_function (h \<circ> f)"
lp15@60987
   972
  then show ?rhs
lp15@60987
   973
    by (force simp add: bounded_linear_inner_left o_def)
lp15@60987
   974
next
lp15@60987
   975
  fix h :: "'b \<Rightarrow> real"
lp15@60987
   976
  assume rp: "\<forall>b\<in>Basis. real_polynomial_function (\<lambda>x. f x \<bullet> b)" and h: "bounded_linear h"
lp15@60987
   977
  have "real_polynomial_function (h \<circ> (\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b))"
lp15@60987
   978
    apply (rule real_polynomial_function_compose [OF _  linear [OF h]])
lp15@60987
   979
    using rp
lp15@60987
   980
    apply (auto simp: real_polynomial_function_eq polynomial_function_mult)
lp15@60987
   981
    done
lp15@60987
   982
  then show "real_polynomial_function (h \<circ> f)"
nipkow@64267
   983
    by (simp add: euclidean_representation_sum_fun)
lp15@60987
   984
qed
lp15@60987
   985
immler@69683
   986
subsection \<open>Stone-Weierstrass theorem for polynomial functions\<close>
lp15@60987
   987
lp15@60987
   988
text\<open>First, we need to show that they are continous, differentiable and separable.\<close>
lp15@60987
   989
ak2110@69737
   990
lemma continuous_real_polymonial_function:
lp15@60987
   991
  assumes "real_polynomial_function f"
lp15@60987
   992
    shows "continuous (at x) f"
lp15@60987
   993
using assms
lp15@60987
   994
by (induct f) (auto simp: linear_continuous_at)
lp15@60987
   995
ak2110@69737
   996
lemma continuous_polymonial_function:
lp15@60987
   997
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
lp15@60987
   998
  assumes "polynomial_function f"
lp15@60987
   999
    shows "continuous (at x) f"
lp15@60987
  1000
  apply (rule euclidean_isCont)
lp15@60987
  1001
  using assms apply (simp add: polynomial_function_iff_Basis_inner)
lp15@60987
  1002
  apply (force dest: continuous_real_polymonial_function intro: isCont_scaleR)
lp15@60987
  1003
  done
lp15@60987
  1004
ak2110@69737
  1005
lemma continuous_on_polymonial_function:
lp15@60987
  1006
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
lp15@60987
  1007
  assumes "polynomial_function f"
lp15@63938
  1008
    shows "continuous_on S f"
lp15@60987
  1009
  using continuous_polymonial_function [OF assms] continuous_at_imp_continuous_on
lp15@60987
  1010
  by blast
lp15@60987
  1011
ak2110@69737
  1012
lemma has_real_derivative_polynomial_function:
lp15@60987
  1013
  assumes "real_polynomial_function p"
lp15@60987
  1014
    shows "\<exists>p'. real_polynomial_function p' \<and>
lp15@60987
  1015
                 (\<forall>x. (p has_real_derivative (p' x)) (at x))"
lp15@60987
  1016
using assms
lp15@60987
  1017
proof (induct p)
lp15@60987
  1018
  case (linear p)
lp15@60987
  1019
  then show ?case
lp15@60987
  1020
    by (force simp: real_bounded_linear const intro!: derivative_eq_intros)
lp15@60987
  1021
next
lp15@60987
  1022
  case (const c)
lp15@60987
  1023
  show ?case
lp15@60987
  1024
    by (rule_tac x="\<lambda>x. 0" in exI) auto
lp15@60987
  1025
  case (add f1 f2)
lp15@60987
  1026
  then obtain p1 p2 where
lp15@60987
  1027
    "real_polynomial_function p1" "\<And>x. (f1 has_real_derivative p1 x) (at x)"
lp15@60987
  1028
    "real_polynomial_function p2" "\<And>x. (f2 has_real_derivative p2 x) (at x)"
lp15@60987
  1029
    by auto
lp15@60987
  1030
  then show ?case
lp15@60987
  1031
    apply (rule_tac x="\<lambda>x. p1 x + p2 x" in exI)
lp15@60987
  1032
    apply (auto intro!: derivative_eq_intros)
lp15@60987
  1033
    done
lp15@60987
  1034
  case (mult f1 f2)
lp15@60987
  1035
  then obtain p1 p2 where
lp15@60987
  1036
    "real_polynomial_function p1" "\<And>x. (f1 has_real_derivative p1 x) (at x)"
lp15@60987
  1037
    "real_polynomial_function p2" "\<And>x. (f2 has_real_derivative p2 x) (at x)"
lp15@60987
  1038
    by auto
lp15@60987
  1039
  then show ?case
lp15@60987
  1040
    using mult
lp15@60987
  1041
    apply (rule_tac x="\<lambda>x. f1 x * p2 x + f2 x * p1 x" in exI)
lp15@60987
  1042
    apply (auto intro!: derivative_eq_intros)
lp15@60987
  1043
    done
lp15@60987
  1044
qed
lp15@60987
  1045
ak2110@69737
  1046
lemma has_vector_derivative_polynomial_function:
lp15@60987
  1047
  fixes p :: "real \<Rightarrow> 'a::euclidean_space"
lp15@60987
  1048
  assumes "polynomial_function p"
lp15@63938
  1049
  obtains p' where "polynomial_function p'" "\<And>x. (p has_vector_derivative (p' x)) (at x)"
lp15@60987
  1050
proof -
lp15@60987
  1051
  { fix b :: 'a
lp15@60987
  1052
    assume "b \<in> Basis"
lp15@60987
  1053
    then
lp15@60987
  1054
    obtain p' where p': "real_polynomial_function p'" and pd: "\<And>x. ((\<lambda>x. p x \<bullet> b) has_real_derivative p' x) (at x)"
wenzelm@61222
  1055
      using assms [unfolded polynomial_function_iff_Basis_inner, rule_format]  \<open>b \<in> Basis\<close>
lp15@60987
  1056
      has_real_derivative_polynomial_function
lp15@60987
  1057
      by blast
lp15@60987
  1058
    have "\<exists>q. polynomial_function q \<and> (\<forall>x. ((\<lambda>u. (p u \<bullet> b) *\<^sub>R b) has_vector_derivative q x) (at x))"
lp15@60987
  1059
      apply (rule_tac x="\<lambda>x. p' x *\<^sub>R b" in exI)
wenzelm@61222
  1060
      using \<open>b \<in> Basis\<close> p'
lp15@60987
  1061
      apply (simp add: polynomial_function_iff_Basis_inner inner_Basis)
lp15@60987
  1062
      apply (auto intro: derivative_eq_intros pd)
lp15@60987
  1063
      done
lp15@60987
  1064
  }
lp15@60987
  1065
  then obtain qf where qf:
lp15@60987
  1066
      "\<And>b. b \<in> Basis \<Longrightarrow> polynomial_function (qf b)"
lp15@60987
  1067
      "\<And>b x. b \<in> Basis \<Longrightarrow> ((\<lambda>u. (p u \<bullet> b) *\<^sub>R b) has_vector_derivative qf b x) (at x)"
lp15@60987
  1068
    by metis
lp15@60987
  1069
  show ?thesis
lp15@63938
  1070
    apply (rule_tac p'="\<lambda>x. \<Sum>b\<in>Basis. qf b x" in that)
lp15@63938
  1071
     apply (force intro: qf)
nipkow@64267
  1072
    apply (subst euclidean_representation_sum_fun [of p, symmetric])
nipkow@64267
  1073
     apply (auto intro: has_vector_derivative_sum qf)
lp15@60987
  1074
    done
lp15@60987
  1075
qed
lp15@60987
  1076
ak2110@69737
  1077
lemma real_polynomial_function_separable:
lp15@60987
  1078
  fixes x :: "'a::euclidean_space"
lp15@60987
  1079
  assumes "x \<noteq> y" shows "\<exists>f. real_polynomial_function f \<and> f x \<noteq> f y"
lp15@60987
  1080
proof -
lp15@60987
  1081
  have "real_polynomial_function (\<lambda>u. \<Sum>b\<in>Basis. (inner (x-u) b)^2)"
nipkow@64267
  1082
    apply (rule real_polynomial_function_sum)
lp15@60987
  1083
    apply (auto simp: algebra_simps real_polynomial_function_power real_polynomial_function_diff
lp15@60987
  1084
                 const linear bounded_linear_inner_left)
lp15@60987
  1085
    done
lp15@60987
  1086
  then show ?thesis
lp15@60987
  1087
    apply (intro exI conjI, assumption)
lp15@60987
  1088
    using assms
nipkow@64267
  1089
    apply (force simp add: euclidean_eq_iff [of x y] sum_nonneg_eq_0_iff algebra_simps)
lp15@60987
  1090
    done
lp15@60987
  1091
qed
lp15@60987
  1092
ak2110@69737
  1093
lemma Stone_Weierstrass_real_polynomial_function:
lp15@60987
  1094
  fixes f :: "'a::euclidean_space \<Rightarrow> real"
lp15@63938
  1095
  assumes "compact S" "continuous_on S f" "0 < e"
lp15@63938
  1096
  obtains g where "real_polynomial_function g" "\<And>x. x \<in> S \<Longrightarrow> \<bar>f x - g x\<bar> < e"
ak2110@69737
  1097
proof -
lp15@60987
  1098
  interpret PR: function_ring_on "Collect real_polynomial_function"
lp15@60987
  1099
    apply unfold_locales
lp15@60987
  1100
    using assms continuous_on_polymonial_function real_polynomial_function_eq
lp15@60987
  1101
    apply (auto intro: real_polynomial_function_separable)
lp15@60987
  1102
    done
lp15@60987
  1103
  show ?thesis
lp15@63938
  1104
    using PR.Stone_Weierstrass_basic [OF \<open>continuous_on S f\<close> \<open>0 < e\<close>] that
lp15@60987
  1105
    by blast
lp15@60987
  1106
qed
lp15@60987
  1107
ak2110@69737
  1108
theorem Stone_Weierstrass_polynomial_function:
lp15@60987
  1109
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
lp15@63938
  1110
  assumes S: "compact S"
lp15@63938
  1111
      and f: "continuous_on S f"
lp15@60987
  1112
      and e: "0 < e"
lp15@63938
  1113
    shows "\<exists>g. polynomial_function g \<and> (\<forall>x \<in> S. norm(f x - g x) < e)"
ak2110@69737
  1114
proof -
lp15@60987
  1115
  { fix b :: 'b
lp15@60987
  1116
    assume "b \<in> Basis"
lp15@63938
  1117
    have "\<exists>p. real_polynomial_function p \<and> (\<forall>x \<in> S. \<bar>f x \<bullet> b - p x\<bar> < e / DIM('b))"
lp15@63938
  1118
      apply (rule exE [OF Stone_Weierstrass_real_polynomial_function [OF S _, of "\<lambda>x. f x \<bullet> b" "e / card Basis"]])
lp15@60987
  1119
      using e f
lp15@60987
  1120
      apply (auto simp: Euclidean_Space.DIM_positive intro: continuous_intros)
lp15@60987
  1121
      done
lp15@60987
  1122
  }
lp15@60987
  1123
  then obtain pf where pf:
lp15@63938
  1124
      "\<And>b. b \<in> Basis \<Longrightarrow> real_polynomial_function (pf b) \<and> (\<forall>x \<in> S. \<bar>f x \<bullet> b - pf b x\<bar> < e / DIM('b))"
lp15@60987
  1125
      apply (rule bchoice [rule_format, THEN exE])
lp15@60987
  1126
      apply assumption
lp15@60987
  1127
      apply (force simp add: intro: that)
lp15@60987
  1128
      done
lp15@60987
  1129
  have "polynomial_function (\<lambda>x. \<Sum>b\<in>Basis. pf b x *\<^sub>R b)"
lp15@60987
  1130
    using pf
nipkow@64267
  1131
    by (simp add: polynomial_function_sum polynomial_function_mult real_polynomial_function_eq)
lp15@60987
  1132
  moreover
lp15@60987
  1133
  { fix x
lp15@63938
  1134
    assume "x \<in> S"
lp15@60987
  1135
    have "norm (\<Sum>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))"
nipkow@64267
  1136
      by (rule norm_sum)
lp15@60987
  1137
    also have "... < of_nat DIM('b) * (e / DIM('b))"
nipkow@64267
  1138
      apply (rule sum_bounded_above_strict)
lp15@63938
  1139
      apply (simp add: Real_Vector_Spaces.scaleR_diff_left [symmetric] pf \<open>x \<in> S\<close>)
lp15@60987
  1140
      apply (rule DIM_positive)
lp15@60987
  1141
      done
lp15@60987
  1142
    also have "... = e"
lp15@60987
  1143
      using DIM_positive by (simp add: field_simps)
lp15@60987
  1144
    finally have "norm (\<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b - pf b x *\<^sub>R b) < e" .
lp15@60987
  1145
  }
lp15@60987
  1146
  ultimately
lp15@60987
  1147
  show ?thesis
nipkow@64267
  1148
    apply (subst euclidean_representation_sum_fun [of f, symmetric])
lp15@60987
  1149
    apply (rule_tac x="\<lambda>x. \<Sum>b\<in>Basis. pf b x *\<^sub>R b" in exI)
nipkow@68403
  1150
    apply (auto simp flip: sum_subtractf)
lp15@60987
  1151
    done
lp15@60987
  1152
qed
lp15@60987
  1153
ak2110@69737
  1154
proposition Stone_Weierstrass_uniform_limit:
immler@65204
  1155
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
immler@65204
  1156
  assumes S: "compact S"
immler@65204
  1157
    and f: "continuous_on S f"
immler@65204
  1158
  obtains g where "uniform_limit S g f sequentially" "\<And>n. polynomial_function (g n)"
ak2110@69737
  1159
proof -
immler@65204
  1160
  have pos: "inverse (Suc n) > 0" for n by auto
immler@65204
  1161
  obtain g where g: "\<And>n. polynomial_function (g n)" "\<And>x n. x \<in> S \<Longrightarrow> norm(f x - g n x) < inverse (Suc n)"
immler@65204
  1162
    using Stone_Weierstrass_polynomial_function[OF S f pos]
immler@65204
  1163
    by metis
immler@65204
  1164
  have "uniform_limit S g f sequentially"
immler@65204
  1165
  proof (rule uniform_limitI)
immler@65204
  1166
    fix e::real assume "0 < e"
immler@65204
  1167
    with LIMSEQ_inverse_real_of_nat have "\<forall>\<^sub>F n in sequentially. inverse (Suc n) < e"
immler@65204
  1168
      by (rule order_tendstoD)
immler@65204
  1169
    moreover have "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. dist (g n x) (f x) < inverse (Suc n)"
immler@65204
  1170
      using g by (simp add: dist_norm norm_minus_commute)
immler@65204
  1171
    ultimately show "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. dist (g n x) (f x) < e"
immler@65204
  1172
      by (eventually_elim) auto
immler@65204
  1173
  qed
immler@65204
  1174
  then show ?thesis using g(1) ..
immler@65204
  1175
qed
immler@65204
  1176
lp15@60987
  1177
immler@69683
  1178
subsection\<open>Polynomial functions as paths\<close>
lp15@60987
  1179
wenzelm@61222
  1180
text\<open>One application is to pick a smooth approximation to a path,
wenzelm@61222
  1181
or just pick a smooth path anyway in an open connected set\<close>
lp15@60987
  1182
ak2110@69737
  1183
lemma path_polynomial_function:
lp15@60987
  1184
    fixes g  :: "real \<Rightarrow> 'b::euclidean_space"
lp15@60987
  1185
    shows "polynomial_function g \<Longrightarrow> path g"
lp15@60987
  1186
  by (simp add: path_def continuous_on_polymonial_function)
lp15@60987
  1187
ak2110@69737
  1188
lemma path_approx_polynomial_function:
lp15@60987
  1189
    fixes g :: "real \<Rightarrow> 'b::euclidean_space"
lp15@60987
  1190
    assumes "path g" "0 < e"
lp15@60987
  1191
    shows "\<exists>p. polynomial_function p \<and>
lp15@60987
  1192
                pathstart p = pathstart g \<and>
lp15@60987
  1193
                pathfinish p = pathfinish g \<and>
lp15@60987
  1194
                (\<forall>t \<in> {0..1}. norm(p t - g t) < e)"
lp15@60987
  1195
proof -
lp15@60987
  1196
  obtain q where poq: "polynomial_function q" and noq: "\<And>x. x \<in> {0..1} \<Longrightarrow> norm (g x - q x) < e/4"
lp15@60987
  1197
    using Stone_Weierstrass_polynomial_function [of "{0..1}" g "e/4"] assms
lp15@60987
  1198
    by (auto simp: path_def)
lp15@60987
  1199
  have pf: "polynomial_function (\<lambda>t. q t + (g 0 - q 0) + t *\<^sub>R (g 1 - q 1 - (g 0 - q 0)))"
lp15@60987
  1200
    by (force simp add: poq)
lp15@60987
  1201
  have *: "\<And>t. t \<in> {0..1} \<Longrightarrow> 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)"
lp15@60987
  1202
    apply (intro Real_Vector_Spaces.norm_add_less)
lp15@60987
  1203
    using noq
lp15@60987
  1204
    apply (auto simp: norm_minus_commute intro: le_less_trans [OF mult_left_le_one_le noq] simp del: less_divide_eq_numeral1)
lp15@60987
  1205
    done
lp15@60987
  1206
  show ?thesis
lp15@60987
  1207
    apply (intro exI conjI)
lp15@60987
  1208
    apply (rule pf)
lp15@60987
  1209
    using *
lp15@60987
  1210
    apply (auto simp add: pathstart_def pathfinish_def algebra_simps)
lp15@60987
  1211
    done
lp15@60987
  1212
qed
lp15@60987
  1213
ak2110@69737
  1214
proposition connected_open_polynomial_connected:
lp15@63938
  1215
  fixes S :: "'a::euclidean_space set"
lp15@63938
  1216
  assumes S: "open S" "connected S"
lp15@63938
  1217
      and "x \<in> S" "y \<in> S"
lp15@63938
  1218
    shows "\<exists>g. polynomial_function g \<and> path_image g \<subseteq> S \<and>
lp15@60987
  1219
               pathstart g = x \<and> pathfinish g = y"
ak2110@69737
  1220
proof -
lp15@63938
  1221
  have "path_connected S" using assms
lp15@60987
  1222
    by (simp add: connected_open_path_connected)
lp15@63938
  1223
  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"
lp15@60987
  1224
    by (force simp: path_connected_def)
lp15@63938
  1225
  have "\<exists>e. 0 < e \<and> (\<forall>x \<in> path_image p. ball x e \<subseteq> S)"
lp15@63938
  1226
  proof (cases "S = UNIV")
lp15@60987
  1227
    case True then show ?thesis
lp15@60987
  1228
      by (simp add: gt_ex)
lp15@60987
  1229
  next
lp15@60987
  1230
    case False
lp15@63938
  1231
    then have "- S \<noteq> {}" by blast
lp15@60987
  1232
    then show ?thesis
lp15@63938
  1233
      apply (rule_tac x="setdist (path_image p) (-S)" in exI)
lp15@63938
  1234
      using S p
lp15@60987
  1235
      apply (simp add: setdist_gt_0_compact_closed compact_path_image open_closed)
lp15@63938
  1236
      using setdist_le_dist [of _ "path_image p" _ "-S"]
lp15@60987
  1237
      by fastforce
lp15@60987
  1238
  qed
lp15@63938
  1239
  then obtain e where "0 < e"and eb: "\<And>x. x \<in> path_image p \<Longrightarrow> ball x e \<subseteq> S"
lp15@60987
  1240
    by auto
lp15@60987
  1241
  show ?thesis
wenzelm@61222
  1242
    using path_approx_polynomial_function [OF \<open>path p\<close> \<open>0 < e\<close>]
lp15@60987
  1243
    apply clarify
lp15@60987
  1244
    apply (intro exI conjI, assumption)
lp15@60987
  1245
    using p
lp15@60987
  1246
    apply (fastforce simp add: dist_norm path_image_def norm_minus_commute intro: eb [THEN subsetD])+
lp15@60987
  1247
    done
lp15@60987
  1248
qed
lp15@60987
  1249
ak2110@69737
  1250
lemma has_derivative_componentwise_within:
lp15@63938
  1251
   "(f has_derivative f') (at a within S) \<longleftrightarrow>
lp15@63938
  1252
    (\<forall>i \<in> Basis. ((\<lambda>x. f x \<bullet> i) has_derivative (\<lambda>x. f' x \<bullet> i)) (at a within S))"
lp15@63938
  1253
  apply (simp add: has_derivative_within)
lp15@63938
  1254
  apply (subst tendsto_componentwise_iff)
lp15@63938
  1255
  apply (simp add: bounded_linear_componentwise_iff [symmetric] ball_conj_distrib)
lp15@63938
  1256
  apply (simp add: algebra_simps)
lp15@63938
  1257
  done
lp15@63938
  1258
ak2110@69737
  1259
lemma differentiable_componentwise_within:
lp15@63938
  1260
   "f differentiable (at a within S) \<longleftrightarrow>
lp15@63938
  1261
    (\<forall>i \<in> Basis. (\<lambda>x. f x \<bullet> i) differentiable at a within S)"
lp15@63938
  1262
proof -
lp15@63938
  1263
  { assume "\<forall>i\<in>Basis. \<exists>D. ((\<lambda>x. f x \<bullet> i) has_derivative D) (at a within S)"
lp15@63938
  1264
    then obtain f' where f':
lp15@63938
  1265
           "\<And>i. i \<in> Basis \<Longrightarrow> ((\<lambda>x. f x \<bullet> i) has_derivative f' i) (at a within S)"
lp15@63938
  1266
      by metis
lp15@63938
  1267
    have eq: "(\<lambda>x. (\<Sum>j\<in>Basis. f' j x *\<^sub>R j) \<bullet> i) = f' i" if "i \<in> Basis" for i
lp15@63938
  1268
      using that by (simp add: inner_add_left inner_add_right)
lp15@63938
  1269
    have "\<exists>D. \<forall>i\<in>Basis. ((\<lambda>x. f x \<bullet> i) has_derivative (\<lambda>x. D x \<bullet> i)) (at a within S)"
lp15@63938
  1270
      apply (rule_tac x="\<lambda>x::'a. (\<Sum>j\<in>Basis. f' j x *\<^sub>R j) :: 'b" in exI)
lp15@63938
  1271
      apply (simp add: eq f')
lp15@63938
  1272
      done
lp15@63938
  1273
  }
lp15@63938
  1274
  then show ?thesis
lp15@63938
  1275
    apply (simp add: differentiable_def)
lp15@63938
  1276
    using has_derivative_componentwise_within
lp15@63938
  1277
    by blast
lp15@63938
  1278
qed
lp15@63938
  1279
ak2110@69737
  1280
lemma polynomial_function_inner [intro]:
lp15@63938
  1281
  fixes i :: "'a::euclidean_space"
lp15@63938
  1282
  shows "polynomial_function g \<Longrightarrow> polynomial_function (\<lambda>x. g x \<bullet> i)"
lp15@63938
  1283
  apply (subst euclidean_representation [where x=i, symmetric])
nipkow@64267
  1284
  apply (force simp: inner_sum_right polynomial_function_iff_Basis_inner polynomial_function_sum)
lp15@63938
  1285
  done
lp15@63938
  1286
lp15@63938
  1287
text\<open> Differentiability of real and vector polynomial functions.\<close>
lp15@63938
  1288
ak2110@69737
  1289
lemma differentiable_at_real_polynomial_function:
lp15@63938
  1290
   "real_polynomial_function f \<Longrightarrow> f differentiable (at a within S)"
lp15@63938
  1291
  by (induction f rule: real_polynomial_function.induct)
lp15@63938
  1292
     (simp_all add: bounded_linear_imp_differentiable)
lp15@63938
  1293
ak2110@69737
  1294
lemma differentiable_on_real_polynomial_function:
lp15@63938
  1295
   "real_polynomial_function p \<Longrightarrow> p differentiable_on S"
lp15@63938
  1296
by (simp add: differentiable_at_imp_differentiable_on differentiable_at_real_polynomial_function)
lp15@63938
  1297
ak2110@69737
  1298
lemma differentiable_at_polynomial_function:
lp15@63938
  1299
  fixes f :: "_ \<Rightarrow> 'a::euclidean_space"
lp15@63938
  1300
  shows "polynomial_function f \<Longrightarrow> f differentiable (at a within S)"
lp15@63938
  1301
  by (metis differentiable_at_real_polynomial_function polynomial_function_iff_Basis_inner differentiable_componentwise_within)
lp15@63938
  1302
ak2110@69737
  1303
lemma differentiable_on_polynomial_function:
lp15@63938
  1304
  fixes f :: "_ \<Rightarrow> 'a::euclidean_space"
lp15@63938
  1305
  shows "polynomial_function f \<Longrightarrow> f differentiable_on S"
lp15@63938
  1306
by (simp add: differentiable_at_polynomial_function differentiable_on_def)
lp15@63938
  1307
ak2110@69737
  1308
lemma vector_eq_dot_span:
lp15@63938
  1309
  assumes "x \<in> span B" "y \<in> span B" and i: "\<And>i. i \<in> B \<Longrightarrow> i \<bullet> x = i \<bullet> y"
lp15@63938
  1310
  shows "x = y"
lp15@63938
  1311
proof -
lp15@63938
  1312
  have "\<And>i. i \<in> B \<Longrightarrow> orthogonal (x - y) i"
lp15@63938
  1313
    by (simp add: i inner_commute inner_diff_right orthogonal_def)
lp15@63938
  1314
  moreover have "x - y \<in> span B"
lp15@63938
  1315
    by (simp add: assms span_diff)
lp15@63938
  1316
  ultimately have "x - y = 0"
lp15@63938
  1317
    using orthogonal_to_span orthogonal_self by blast
lp15@63938
  1318
    then show ?thesis by simp
lp15@63938
  1319
qed
lp15@63938
  1320
ak2110@69737
  1321
lemma orthonormal_basis_expand:
lp15@63938
  1322
  assumes B: "pairwise orthogonal B"
lp15@63938
  1323
      and 1: "\<And>i. i \<in> B \<Longrightarrow> norm i = 1"
lp15@63938
  1324
      and "x \<in> span B"
lp15@63938
  1325
      and "finite B"
lp15@63938
  1326
    shows "(\<Sum>i\<in>B. (x \<bullet> i) *\<^sub>R i) = x"
lp15@63938
  1327
proof (rule vector_eq_dot_span [OF _ \<open>x \<in> span B\<close>])
lp15@63938
  1328
  show "(\<Sum>i\<in>B. (x \<bullet> i) *\<^sub>R i) \<in> span B"
nipkow@64267
  1329
    by (simp add: span_clauses span_sum)
lp15@63938
  1330
  show "i \<bullet> (\<Sum>i\<in>B. (x \<bullet> i) *\<^sub>R i) = i \<bullet> x" if "i \<in> B" for i
lp15@63938
  1331
  proof -
lp15@63938
  1332
    have [simp]: "i \<bullet> j = (if j = i then 1 else 0)" if "j \<in> B" for j
lp15@63938
  1333
      using B 1 that \<open>i \<in> B\<close>
lp15@63938
  1334
      by (force simp: norm_eq_1 orthogonal_def pairwise_def)
lp15@63938
  1335
    have "i \<bullet> (\<Sum>i\<in>B. (x \<bullet> i) *\<^sub>R i) = (\<Sum>j\<in>B. x \<bullet> j * (i \<bullet> j))"
nipkow@64267
  1336
      by (simp add: inner_sum_right)
lp15@63938
  1337
    also have "... = (\<Sum>j\<in>B. if j = i then x \<bullet> i else 0)"
nipkow@64267
  1338
      by (rule sum.cong; simp)
lp15@63938
  1339
    also have "... = i \<bullet> x"
nipkow@64267
  1340
      by (simp add: \<open>finite B\<close> that inner_commute sum.delta)
lp15@63938
  1341
    finally show ?thesis .
lp15@63938
  1342
  qed
lp15@63938
  1343
qed
lp15@63938
  1344
lp15@63938
  1345
ak2110@69737
  1346
theorem Stone_Weierstrass_polynomial_function_subspace:
lp15@63938
  1347
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
lp15@63938
  1348
  assumes "compact S"
lp15@63938
  1349
      and contf: "continuous_on S f"
lp15@63938
  1350
      and "0 < e"
lp15@63938
  1351
      and "subspace T" "f ` S \<subseteq> T"
lp15@63938
  1352
    obtains g where "polynomial_function g" "g ` S \<subseteq> T"
lp15@63938
  1353
                    "\<And>x. x \<in> S \<Longrightarrow> norm(f x - g x) < e"
ak2110@69737
  1354
proof -
lp15@63938
  1355
  obtain B where "B \<subseteq> T" and orthB: "pairwise orthogonal B"
lp15@63938
  1356
             and B1: "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
lp15@63938
  1357
             and "independent B" and cardB: "card B = dim T"
lp15@63938
  1358
             and spanB: "span B = T"
lp15@63938
  1359
    using orthonormal_basis_subspace \<open>subspace T\<close> by metis
lp15@63938
  1360
  then have "finite B"
lp15@63938
  1361
    by (simp add: independent_imp_finite)
lp15@63938
  1362
  then obtain n::nat and b where "B = b ` {i. i < n}" "inj_on b {i. i < n}"
lp15@63938
  1363
    using finite_imp_nat_seg_image_inj_on by metis
lp15@63938
  1364
  with cardB have "n = card B" "dim T = n"
lp15@63938
  1365
    by (auto simp: card_image)
lp15@63938
  1366
  have fx: "(\<Sum>i\<in>B. (f x \<bullet> i) *\<^sub>R i) = f x" if "x \<in> S" for x
lp15@63938
  1367
    apply (rule orthonormal_basis_expand [OF orthB B1 _ \<open>finite B\<close>])
lp15@63938
  1368
    using \<open>f ` S \<subseteq> T\<close> spanB that by auto
lp15@63938
  1369
  have cont: "continuous_on S (\<lambda>x. \<Sum>i\<in>B. (f x \<bullet> i) *\<^sub>R i)"
lp15@63938
  1370
    by (intro continuous_intros contf)
lp15@63938
  1371
  obtain g where "polynomial_function g"
lp15@63938
  1372
             and g: "\<And>x. x \<in> S \<Longrightarrow> norm ((\<Sum>i\<in>B. (f x \<bullet> i) *\<^sub>R i) - g x) < e / (n+2)"
lp15@63938
  1373
    using Stone_Weierstrass_polynomial_function [OF \<open>compact S\<close> cont, of "e / real (n + 2)"] \<open>0 < e\<close>
lp15@63938
  1374
    by auto
lp15@63938
  1375
  with fx have g: "\<And>x. x \<in> S \<Longrightarrow> norm (f x - g x) < e / (n+2)"
lp15@63938
  1376
    by auto
lp15@63938
  1377
  show ?thesis
lp15@63938
  1378
  proof
lp15@63938
  1379
    show "polynomial_function (\<lambda>x. \<Sum>i\<in>B. (g x \<bullet> i) *\<^sub>R i)"
nipkow@64267
  1380
      apply (rule polynomial_function_sum)
lp15@63938
  1381
       apply (simp add: \<open>finite B\<close>)
lp15@63938
  1382
      using \<open>polynomial_function g\<close>  by auto
lp15@63938
  1383
    show "(\<lambda>x. \<Sum>i\<in>B. (g x \<bullet> i) *\<^sub>R i) ` S \<subseteq> T"
lp15@67986
  1384
      using \<open>B \<subseteq> T\<close>
immler@68072
  1385
      by (blast intro: subspace_sum subspace_mul \<open>subspace T\<close>)
lp15@63938
  1386
    show "norm (f x - (\<Sum>i\<in>B. (g x \<bullet> i) *\<^sub>R i)) < e" if "x \<in> S" for x
lp15@63938
  1387
    proof -
lp15@63938
  1388
      have orth': "pairwise (\<lambda>i j. orthogonal ((f x \<bullet> i) *\<^sub>R i - (g x \<bullet> i) *\<^sub>R i)
lp15@63938
  1389
                                              ((f x \<bullet> j) *\<^sub>R j - (g x \<bullet> j) *\<^sub>R j)) B"
lp15@63938
  1390
        apply (rule pairwise_mono [OF orthB])
lp15@63938
  1391
        apply (auto simp: orthogonal_def inner_diff_right inner_diff_left)
lp15@63938
  1392
        done
lp15@63938
  1393
      then have "(norm (\<Sum>i\<in>B. (f x \<bullet> i) *\<^sub>R i - (g x \<bullet> i) *\<^sub>R i))\<^sup>2 =
lp15@63938
  1394
                 (\<Sum>i\<in>B. (norm ((f x \<bullet> i) *\<^sub>R i - (g x \<bullet> i) *\<^sub>R i))\<^sup>2)"
nipkow@64267
  1395
        by (simp add:  norm_sum_Pythagorean [OF \<open>finite B\<close> orth'])
lp15@63938
  1396
      also have "... = (\<Sum>i\<in>B. (norm (((f x - g x) \<bullet> i) *\<^sub>R i))\<^sup>2)"
lp15@63938
  1397
        by (simp add: algebra_simps)
lp15@63938
  1398
      also have "... \<le> (\<Sum>i\<in>B. (norm (f x - g x))\<^sup>2)"
nipkow@64267
  1399
        apply (rule sum_mono)
lp15@63938
  1400
        apply (simp add: B1)
lp15@63938
  1401
        apply (rule order_trans [OF Cauchy_Schwarz_ineq])
lp15@63938
  1402
        by (simp add: B1 dot_square_norm)
lp15@63938
  1403
      also have "... = n * norm (f x - g x)^2"
lp15@63938
  1404
        by (simp add: \<open>n = card B\<close>)
lp15@63938
  1405
      also have "... \<le> n * (e / (n+2))^2"
lp15@63938
  1406
        apply (rule mult_left_mono)
lp15@63938
  1407
         apply (meson dual_order.order_iff_strict g norm_ge_zero power_mono that, simp)
lp15@63938
  1408
        done
lp15@63938
  1409
      also have "... \<le> e^2 / (n+2)"
lp15@63938
  1410
        using \<open>0 < e\<close> by (simp add: divide_simps power2_eq_square)
lp15@63938
  1411
      also have "... < e^2"
lp15@63938
  1412
        using \<open>0 < e\<close> by (simp add: divide_simps)
lp15@63938
  1413
      finally have "(norm (\<Sum>i\<in>B. (f x \<bullet> i) *\<^sub>R i - (g x \<bullet> i) *\<^sub>R i))\<^sup>2 < e^2" .
lp15@63938
  1414
      then have "(norm (\<Sum>i\<in>B. (f x \<bullet> i) *\<^sub>R i - (g x \<bullet> i) *\<^sub>R i)) < e"
lp15@63938
  1415
        apply (rule power2_less_imp_less)
lp15@63938
  1416
        using  \<open>0 < e\<close> by auto
lp15@63938
  1417
      then show ?thesis
nipkow@64267
  1418
        using fx that by (simp add: sum_subtractf)
lp15@63938
  1419
    qed
lp15@63938
  1420
  qed
lp15@63938
  1421
qed
lp15@63938
  1422
lp15@63938
  1423
lp15@60987
  1424
hide_fact linear add mult const
lp15@60987
  1425
lp15@60987
  1426
end