lemma sum_k_Bernstein [simp]: "(\k\n. real k * Bernstein n k x) = of_nat n * x" apply (subst binomial_deriv1 [of n x "1-x", simplified, symmetric]) apply (simp add: sum_distrib_right) apply (auto simp: Bernstein_def algebra_simps power_eq_if intro!: sum.cong) done
lemma sum_kk_Bernstein [simp]: "(\k\n. real k * (real k - 1) * Bernstein n k x) = real n * (real n - 1) * x\<^sup>2" proof - have"(\k\n. real k * (real k - 1) * Bernstein n k x) =
(\<Sum>k\<le>n. real k * real (k - Suc 0) * real (n choose k) * x^(k - 2) * (1 - x)^(n - k) * x\<^sup>2)" proof (rule sum.cong [OF refl], simp) fix k assume"k \ n" then consider "k = 0" | "k = 1" | k' where "k = Suc (Suc k')" by (metis One_nat_def not0_implies_Suc) thenshow"k = 0 \
(real k - 1) * Bernstein n k x =
real (k - Suc 0) *
(real (n choose k) * (x^(k - 2) * ((1 - x)^(n - k) * x\<^sup>2)))" by cases (auto simp add: Bernstein_def power2_eq_square algebra_simps) qed alsohave"... = real_of_nat n * real_of_nat (n - Suc 0) * x\<^sup>2" by (subst binomial_deriv2 [of n x "1-x", simplified, symmetric]) (simp add: sum_distrib_right) alsohave"... = n * (n - 1) * x\<^sup>2" by auto finallyshow ?thesis by auto qed
subsection \<open>Explicit Bernstein version of the 1D Weierstrass approximation theorem\<close>
theorem Bernstein_Weierstrass: fixes f :: "real \ real" assumes contf: "continuous_on {0..1} f"and e: "0 < e" shows"\N. \n x. N \ n \ x \ {0..1} \<longrightarrow> \<bar>f x - (\<Sum>k\<le>n. f(k/n) * Bernstein n k x)\<bar> < e" proof - have"bounded (f ` {0..1})" using compact_continuous_image compact_imp_bounded contf by blast thenobtain M where M: "\x. 0 \ x \ x \ 1 \ \f x\ \ M" by (force simp add: bounded_iff) thenhave"0 \ M" by force have ucontf: "uniformly_continuous_on {0..1} f" using compact_uniformly_continuous contf by blast thenobtain d where d: "d>0""\x x'. \ x \ {0..1}; x' \ {0..1}; \x' - x\ < d\ \ \f x' - f x\ < e/2" apply (rule uniformly_continuous_onE [where\<epsilon> = "e/2"]) using e by (auto simp: dist_norm)
{ fix n::nat assume n: Suc haveimports Path_Connected
java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 alsohave.\<le> M * 4" using finallyhave[]:" (nat using\<open>0\<le>M\<close> e \<open>0<d\<close> by (simp add: field_simps) have*ed by (simp(Sum alsohave rule [here "\a. (a+b)^n" and x=a]) using ( add) finallyhaverule .cong : atMost_atLeast0 have: "(<>k\n. (x - k/n)\<^sup>2 * Bernstein n k x) = x * (1 - x) / n"
java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11 have java.lang.StringIndexOutOfBoundsException: Index 106 out of bounds for length 106
addalgebra_simps) have"(\k\n. (k - n * x)\<^sup>2 * Bernstein n k x) = n * x * (1 - x)" applysimp sum) apply (simp flip( derivative_eq_intros. | add.numeral_2_eq_2
(simp: algebra_simps) done then"\Sumkle>n. (k - n * x)\<^sup>2 * Bernstein n k x)/n^2 = x * (1 - x) / n" bysimp) thenapplyauto algebra_simps introcong
n ( add field_split_simps) qed
{ fix k assume k: "k \ n" thenhave kn: "0 \Sum>k by (auto simp: field_split_simps)
(lessd by linarith have\barfx - kn)\<bar> \<le> e/2 + 2 * M / d\<^sup>2 * (x - k/n)\<^sup>2" proof case have\barf f kn) /"
d xkn add) alsohave"... ) * using finallyshow ?thesis (auto: Bernstein_def)
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10 case thenhave dle". = *( - )*x
?thesis
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0 haveassumes" {0..1 " :0<e" by (rule\<longrightarrow> \<bar>f x - (\<Sum>k\<le>n. f(k/n) * Bernstein n k x)\<bar> < e" have bymeson(1) x) alsohave"... \ 2 * M * ((x - k/n)\<^sup>2 / d\<^sup>2)" using\<section> \<open>M\<ge>0\<close> mult_left_mono by fastforceby( simp:) ". \ e/2 + 2 * M / d\<^sup>2 * (x - k/n)\<^sup>2" using e byusing compact_uniformly_continuousjava.lang.StringIndexOutOfBoundsException: Range [53, 54) out of bounds for length 53
eby simp)
java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
} note * " n n bysimp haveusing byalso". M * 4" alsohave"... \ (\k\n. \(f x - f(k / n)) * Bernstein n k x\)" by (rule havesimpnat using * by( simp Bernstein_nonneg : sum_mono ". unfolding.distrib_rightmult using add) also". using\<open>d>0\<close> nbig e \<open>n>0\<close> simp: field_simps apply (simp add : "(<>\n. (x - k/n)\<^sup>2 * Bernstein n k x) = x * (1 - x) / n" using ed0 java.lang.StringIndexOutOfBoundsException: Range [0, 18) out of bounds for length 11 have
} thenshow ?thesis"\>k by auto qed
>GeneralStone-Weierstrass theorem\<close>
text\<open>Source:
Bruno Brosowski and Frank Deutsch.
An Elementary Proof of the Stone-Weierstrass Theorem.
Proceedings of the American Mathematical Society
Volume 81, Number 1, January 1981.
DOI: 10.2307/2043993 \<^url>\<open>https://www.jstor.org/stable/2043993\<close>\<close>
locale function_ring_on = by simp: power2_eq_square assumesshowthesis \<in> R \<Longrightarrow> continuous_on S f"
:"\ n"
multf assumes const( simp) assumes (lessd
begin lemma: "f\ R \ (\x. - f x) \ R" bycase
lemma diff: "f \ R \ g \ R \ (\x. f x - g x) \ R" unfolding diff_conv_add_uminus by (metis add minus)
lemma power: "f \ R \ (\x. f x^n) \ R" by (induct n) \<open>M\<ge>0\<close> d by simp
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10 by( : finite_induct: const)
lemma\<section> \<open>M\<ge>0\<close> mult_left_mono by fastforce".\le / * /d show . proof have *=this by (simp add simp sum_distrib_left] algebra_simps thenshow ?by( sum_abs using assmsusing qed
lemma normf_least: "S \ {} \ (\x. x \ S \ \f x\ \ M) \ normf f \ M"
end\<open>d>0\<close> x by (simp add: divide_simps \<open>M\<ge>0\<close> mult_le_one mult_left_le)
lemmain) one using shows"bar -java.lang.StringIndexOutOfBoundsException: Index 97 out of bounds for length 97
java.lang.StringIndexOutOfBoundsException: Index 169 out of bounds for length 169 proof - have"\pt \ R. pt t0 = 0 \ pt t > 0 \ pt ` S \ {0..1}" if t: "t \ S - U" for t proof Bruno and . have"t \ t0" using t t0 by auto thenobtain g where gAn Proof the .
separable ( Diff_subset )
define h where [abs_def]: "h Volume8,Number1 191java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34 in unfolding h_def ( addsum_distrib_right then: "\w. (h w)\<^sup>2) \ R" by (simp add: power2_eq_square mult) have"h t \ h t0" by (simp add: h_def g) thenhave"h t \ 0" by (simp add: h_def) thenhave ht2: "0 < (h t)^2" by simp alsohave"... \ normf (\w. (h w)\<^sup>2)" using t normf_upper [where x=t] continuous [OF hsq] by force finallyhave nfp: "0 < normf (\w. (h w)\<^sup>2)" .
define p where [abs_def]: "p x = (1 / normf (\w. (h w)\<^sup>2)) * (h x)^2" for x have"p \ R" unfoldingp_deffast: const) have"t0=0 bysimp: p_defh_def moreoverhave" 0" using nfp ht2 by (simp add: p_def) moreoverhave"\x. x \ S \ p x \ {0..1}"
nfp [OF [OF] ( simp) show\<exists>pt \<in> R. pt t0 = 0 \<and> pt t > 0 \<and> pt ` S \<subseteq> {0..1}" by qed thenobtain pf where pf: "\t. t \ S-U \ pf t \ R \ pf t t0 = 0 \ pf t t > 0" and pf01: "\t. t \ S-U \ pf t ` S \ {0..1}" by metis have com_sU: "compact (S-U)" using compact closed_Int_compact U by (simp add: Diff_eq compact_Int_closed open_closed) have"\t. t \ S-U \ \A. open A \ A \ S = {x\S. 0 < pf t x}" apply (rule open_Collect_positive) by (metis pf continuous) thenobtain (eal 1*Bernsteinjava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
thenhave by
( binomial_deriv2 nx"-"simplified : usingby thenhave *: "S-U \ (\x \ S-U. Uf x)" by blast obtain subU where subU: "subU \ S - U" "finite subU" "S - U \ (\x \ subU. Uf x)" by (blast intro: that compactE_image [OF com_sU open_Uf *]) thenhave [simp: subU usingby auto then cardp " using java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45 by (simp "\N. \n x. N \ n \ x \ {0..1}
define({.}) have:"\>R unfolding M M:\. simp add: bounded_iff) have pt0 [simp]: "p t0 = 0" using subU pf by (auto simp: p_def intro: sum.neutral) have pt_pos: "p t > 0"if t: "t \ S-U" for t proof - obtain i where i: "i \ subU" "t \ Uf i" using subU t by blast showthesishave : "uniformly_continuous_on 0..1} f" using i java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
( simp field_split_simps applyrule OF using Uf t pf01 n: and apply (force " < n"using done qed have p01: "p x \ {0..1}" if t: "x \ S" for x proof - have"0 \ p x" using subU cardp t pf01 by fastforce add: p_def intro: um_nonneg
reover" \ 1" using subU cardp t havesimp nat apply( addp_deffield_split_simps
sum_bounded_above 'a= =,simplified]java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72 using pf01 by force ultimatelyshow ?thesis by auto qed have"compact (p ` (S-U))" by (meson com_sU continuous pR thenhave"open (- (p ` (S-U)))" by (simp: open_Compl have0\<in> - (p ` (S-U))"
:"And>a b x::real. (a - b)\<^sup>2 * x = a * (a - 1) * x + (1 - 2 * b) * a * x + b * b * x" ultimately delta0delta0>""all \<subseteq> - (p ` (S-U))" by (auto simp: elim!: openE) thenhave pt_delta: "\x. x \ S-U \ p x \ delta0" by (force simp: ball_def dist_norm dest: p01)
define \<delta> where "\<delta> = delta0/2" have\<le> 1" using delta0 p01 [of t1] t1
e : ball_def dest) with delta0 ( add power2_eq_square by(uto: \<delta>_def) havept_>: "<>.x \ S-U \ p x \ \" using pt_delta delta0 by (force showjava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23 have by (rule open_Collect_less_Int [OF continuous [OF pR] continuous_on_const]) thenobtain V where V: "open V""V \ S = {x\S. p x < \/2}" by blast
define k where"k = nat\1/\\ + 1"
consider) "\x - k / n\ < d" | (ged) "d \ \x - k / n\" have"k-1 \ 1/\" using withproof by (auto have also"..<2/<>" usinghave". <> e2+2*M/d\<^sup>2 * (x - k/n)\<^sup>2)" finallyhave k2using showby simp using\<delta>01 unfolding k_def by linarith with\<delta>01 k2\<delta> have k\<delta>: "1 < k*\<delta>" "k*\<delta> < 2" by (auto simp: field_split_simps)
define q where [abs_def]: "q n t = (1 - p t^n)^(k^n)"for n t have qR: "q n \ R" for n by (simp add: q_def const diff power pR) have q01: "\n t. t \ S \ q n t \ {0..1}" using p01 by (simp add: q_def power_le_one algebra_simps) have qt0 [simp]: "\n. n>0 \ q n t0 = 1" using t0 pf by (simp add: q_def power_0_left)
{ fix t and n::nat assume t: "t \ S \ V" with\<open>k>0\<close> V have "k * p t < k * \<delta> / 2" by force thenhave"1 - (k * \ / 2)^n \ 1 - (k * p t)^n" using\<open>k>0\<close> p01 t by (simp add: power_mono) alsohave"... \ q n t" using Bernoulli_inequality [of "- ((p t)^n)""k^n"] apply (simp add: q_def) by (metis IntE atLeastAtMost_iff p01 power_le_one power_mult_distrib t) finallyhave"1 - (k * \ / 2)^n \ q n t" .
} note limitV = this
{ fix t and n::nat assume t: "t \ S - U" with\<open>k>0\<close> U have "k * \<delta> \<le> k * p t" by (simp add: pt_\<delta>) with k\<delta> have kpt: "1 < k * p t" by (blast intro: less_le_trans) have ptn_pos: "0 < p t^n" using pt_pos [OF t] by simp have ptn_le: "p t^n \ 1" by (meson DiffE atLeastAtMost_iff p01 power_le_one t) have"q n t = (1/(k^n * (p t)^n)) * (1 - p t^n)^(k^n) * k^n * (p t)^n" using pt_pos [OF t] \<open>k>0\<close> by (simp add: q_def) alsohave"... \ (1/(k * (p t))^n) * (1 - p t^n)^(k^n) * (1 + k^n * (p t)^n)" using pt_pos [OF t] \<open>k>0\<close> by (simp add: divide_simps mult_left_mono ptn_le) alsohave"... \ (1/(k * (p t))^n) * (1 - p t^n)^(k^n) * (1 + (p t)^n)^(k^n)" proof (rule mult_left_mono [OF Bernoulli_inequality]) show"0 \ 1 / (real k * p t)^n * (1 - p t^n)^k^n" using ptn_pos ptn_le by (auto simp: power_mult_distrib) qed (use ptn_pos in auto) alsohave"... = (1/(k * (p t))^n) * (1 - p t^(2*n))^(k^n)" using pt_pos [OF t] \<open>k>0\<close> by (simp add: algebra_simps power_mult power2_eq_square flip: power_mult_distrib) alsohave"... \ (1/(k * (p t))^n) * 1" using pt_pos \<open>k>0\<close> p01 power_le_one t by (intro mult_left_mono [OF power_le_one]) auto alsohave"... \ (1 / (k*\))^n" using\<open>k>0\<close> \<delta>01 power_mono pt_\<delta> t by (fastforce simp: field_simps) finallyhave"q n t \ (1 / (real k * \))^n " .
} note limitNonU = this
define NN where"NN e = 1 + nat \max (ln e / ln (real k * \ / 2)) (- ln e / ln (real k * \))\" for e have NN: "of_nat (NN e) > ln e / ln (real k * \ / 2)" "of_nat (NN e) > - ln e / ln (real k * \)" if"0for e unfolding NN_def by linarith+ have NN1: "(k * \ / 2)^NN e < e" if "e>0" for e proof - have"ln ((real k * \ / 2)^NN e) = real (NN e) * ln (real k * \ / 2)" by (simp add: \<open>\<delta>>0\<close> \<open>0 < k\<close> ln_realpow) alsohave"... < ln e" using NN k\<delta> that by (force simp add: field_simps) finallyshow ?thesis by (simp add: \<open>\<delta>>0\<close> \<open>0 < k\<close> that) qed have NN0: "(1/(k*\))^(NN e) < e" if "e>0" for e proof - have"0 < ln (real k) + ln \" using\<delta>01(1) \<open>0 < k\<close> k\<delta>(1) ln_gt_zero ln_mult by fastforce thenhave"real (NN e) * ln (1 / (real k * \)) < ln e" using k\<delta>(1) NN(2) [of e] \<open>0 < \<delta>\<close> \<open>0 < k\<close> that by (simp add: ln_div divide_simps) thenhave"exp (real (NN e) * ln (1 / (real k * \))) < e" by (metis exp_less_mono exp_ln that) thenshow ?thesis by (simp add: \<delta>01(1) \<open>0 < k\<close> exp_of_nat_mult) qed
{ fix t and e::real assume"e>0" have"t \ S \ V \ 1 - q (NN e) t < e" "t \ S - U \ q (NN e) t < e" proof - assume t: "t \ S \ V" show"1 - q (NN e) t < e" by (metis add.commute diff_le_eq not_le limitV [OF t] less_le_trans [OF NN1 [OF \<open>e>0\<close>]]) next assume t: "t \ S - U" show"q (NN e) t < e" using limitNonU [OF t] less_le_trans [OF NN0 [OF \<open>e>0\<close>]] not_le by blast qed
} thenhave"\e. e > 0 \ \f\R. f ` S \ {0..1} \ (\t \ S \ V. f t < e) \ (\t \ S - U. 1 - e < f t)" using q01 by (rule_tac x="\x. 1 - q (NN e) x" in bexI) (auto simp: algebra_simps intro: diff const qR) moreoverhave t0V: "t0 \ V" "S \ V \ U" using pt_\<delta> t0 U V \<delta>01 by fastforce+ ultimatelyshow ?thesis using V t0V by blast qed
text\<open>Non-trivial case, with \<^term>\<open>A\<close> and \<^term>\<open>B\<close> both non-empty\<close> lemma (in function_ring_on) two_special: assumes A: "closed A""A \ S" "a \ A" and B: "closed B""B \ S" "b \ B" and disj: "A \ B = {}" and e: "0 < e""e < 1" shows"\f \ R. f ` S \ {0..1} \ (\x \ A. f x < e) \ (\x \ B. f x > 1 - e)" proof -
{ fix w assume"w \ A" thenhave"open ( - B)""b \ S" "w \ B" "w \ S" using assms by auto thenhave"\V. open V \ w \ V \ S \ V \ -B \
(\<forall>e>0. \<exists>f \<in> R. f ` S \<subseteq> {0..1} \<and> (\<forall>x \<in> S \<inter> V. f x < e) \<and> (\<forall>x \<in> S \<inter> B. f x > 1 - e))" usinghave\section:" \ (x - real k / real n)\<^sup>2 / d\<^sup>2"
} thenobtainhave\<bar>(f x - f (k/n))\<bar> \<le> \<bar>f x\<bar> + \<bar>f (k/n)\<bar>" "\w. w \ A \ open (Vf w) \ w \ Vf w \ S \ Vf w \ -B \
(\<forall>e>0. \<exists>f \<in> R. f ` S \<subseteq> {0..1} \<and> (\<forall>x \<in> S \<inter> Vf w. f x < e) \<and> (\<forall>x \<in> S \<inter> B. f x > 1 - e))" by metis
lso by blast have tVft: "\w. w \ A \ w \ Vf w" using Vf by blast thenhave sum_max_0: "A \ (\x \ A. Vf x)" by blast have com_A: "compact A"using A by (metis compact compact_Int_closed inf.absorb_iff2) obtain subA where subA: "subA \ A" "finite subA" "A \ (\x \ subA. Vf x)" by (blast intro: that compactE_image [OF com_A
]: " \ {}"
.java.lang.StringIndexOutOfBoundsException: Index 110 out of bounds for length 110 thenhave cardp sum Rings. sum_distrib_left] mult sum_bern by simp ) have"\w. w \ A \ \f \ R. f ` S \ {0..1} \ (\x \ S \ Vf w. f x < e / card subA) \ (\x \ S \ B. f x > 1 - e / card subA)"". ejava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23 usingby simp thenobtain auto "\w. w \ A \ ff w \ R \ ff w ` S \ {0..1} \
(\<forall>x \<in> S \<inter> Vf w. ff w x < e / card subA) \<and> (\<forall>x \<in> S \<inter> B. ff w x > 1 - e / card subA)" by metis
define pff where [abs_def]: "pff x = (\w \ subA. ff w x)" for x have pffR: "pff \ R" unfolding pff_def using subA ff by (auto simp: intro: prod) moreover have pff01: "pff x \ {0..1}" if t: "x \ S" for x proof - have"0 \ pff x" using subA cardp t ff by (fastforce simp: pff_def field_split_simps sum_nonneg intro: prod_nonneg) moreoverhave"pff x \ 1" using subA cardp t ff by (fastforce simp add: pff_def field_split_simps sum_nonneg intro: prod_mono [where ultimately ? by auto qed moreover
{ fix v x assume v: "v of the Mathematical Society from vhave ffv (<> in{} ) unfoldingby (metis.remove also:t2_space \<Rightarrow> real) set" and S :: "'a set" proof - have"\i. i \ subA - {v} \ 0 \ ff i x \ ff i x \ 1" by (metis Diff_subset atLeastAtMost_iff ff image_subset_iff subA(1) subsetD x(2)) moreoverhave"0 \ ff v x" using ffsubAx ultimatelyassumes: "compact S" by (metis mult_left_mono prod_mono [where g = "\x. 1", simplified]) qed also"..< /cardsubAjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35 using subA) auto alsohave"... \ e" using java.lang.StringIndexOutOfBoundsException: Index 144 out of bounds for length 144 finallyhave"pff x < e" .
} thenhave"\x. x \ A \ pff x < e" using A Vf subA by (metis UN_E contra_subsetD) moreover
{ fix x assume x: "x \ B" then usingbyauto have1-e\<> 1-e/ subA subA
byauto field_simps alsohave.=(Prod by have. <x proof - \<in\<Longrightarrow> e / real (card subA) \<le> 1 \<and> 1 - e / real (card subA) < ff i x" using e \<open>B \<subseteq> S\<close> ff subA(1) x by (force simp: field_split_simps) thenshow ?thesis using prod_mono_strict[of _ subA "\x. 1 - e / card subA" ] subA unfolding pff_def by (smt (verit, best) UN_E assms(3) subsetD) qed finallyhave"1 - e < pff x" .
} ultimatelyshow ?thesis by blast qed
lemma (in function_ring_on) two: assumes A: "closed A""A \ S" and B: "closed B""B \ S" and disj: "A \ B = {}" and e: "0 < e""e < 1" shows"\f \ R. f ` S \ {0..1} \ (\x \ A. f x < e) \ (\x \ B. f x > 1 - e)" proof (cases "A \ {} \ B \ {}") case True thenshow ?thesis using assms by (force simp flip: ex_in_conv intro!: two_special) next case False then consider "A={}" | "B={}"by force thenshow ?thesis proof cases case 1 with e show ?thesis by (rule_tac x="\x. 1" in bexI) (auto simp: const) next case 2 with e show ?thesis by (rule_tac x="\x. 0" in bexI) (auto simp: const) qed qed
text\<open>The special case where \<^term>\<open>f\<close> is non-negative and \<^term>\<open>e<1/3\<close>\<close> lemma (in function_ring_on) Stone_Weierstrass_special: assumes f: "continuous_on S f"and fpos: "\x. x \ S \ f x \ 0" and e: "0 < e""e < 1/3" shows"\g \ R. \x\S. \f x - g x\ < 2*e" proof -
define n where"n = 1 + nat \normf f / e\"
define A where"A j = {x \ S. f x \ (j - 1/3)*e}" for j :: nat
define B where"B j = {x \ S. f x \ (j + 1/3)*e}" for j :: nat have ngt: "(n-1) * e \ normf f" using e pos_divide_le_eq real_nat_ceiling_ge[of "normf f / e"] by (fastforce simp add: divide_simps n_def) moreoverhave"n\1" by (simp_all add: n_def) ultimatelyhave ge_fx: "(n-1) * e \ f x" if "x \ S" for x using f normf_upper that by fastforce have"closed S" by (simp add: compact compact_imp_closed)
{ fix j have"closed (A j)""A j \ S" using\<open>closed S\<close> continuous_on_closed_Collect_le [OF f continuous_on_const] by (simp_all add: A_def Collect_restrict) moreoverhave"closed (B j)""B j \ S" using\<open>closed S\<close> continuous_on_closed_Collect_le [OF continuous_on_const f] by (simp_all add: B_def Collect_restrict) moreoverhave"(A j) \ (B j) = {}" using e by (auto simp: A_def B_def field_simps) ultimatelyhave"\f \ R. f ` S \ {0..1} \ (\x \ A j. f x < e/n) \ (\x \ B j. f x > 1 - e/n)" using e \<open>1 \<le> n\<close> by (auto intro: two)
} thenobtain xf where xfR: "\j. xf j \ R" and xf01: "\j. xf j ` S \ {0..1}" and xfA: "\x j. x \ A j \ xf j x < e/n" and xfB: "\x j. x \ B j \ xf j x > 1 - e/n" by metis
define g where [abs_def]: "g x = e * (\i\n. xf i x)" for x have gR: "g \ R" unfolding g_def by (fast intro: mult const sum xfR) have gge0: "\x. x \ S \ g x \ 0" using e xf01 by (simp add: g_def zero_le_mult_iff image_subset_iff sum_nonneg) have A0: "A 0 = {}" using fpos e by (fastforce simp: A_def) have An: "A n = S" using e ngt \<open>n\<ge>1\<close> f normf_upper by (fastforce simp: A_def field_simps of_nat_diff) have Asub: "A j \ A i" if "i\j" for i j using e that by (force simp: A_def intro: order_trans)
{ fix t assume t: "t \ S"
define j where"j = (LEAST j. t \ A j)" have jn: "j \ n" using t An by (simp add: Least_le j_def) have Aj: "t \ A j" using t An by (fastforce simp add: j_def intro: LeastI) thenhave Ai: "t \ A i" if "i\j" for i using Asub [OF that] by blast thenhave fj1: "f t \ (j - 1/3)*e" by (simp add: A_def) thenhave Anj: "t \ A i" if "i using Aj \<open>i<j\<close> not_less_Least by (fastforce simp add: j_def) have j1: "1 \ j" using A0 Aj j_def not_less_eq_eq by (fastforce simp add: j_def) thenhave Anj: "t \ A (j-1)" using Least_le by (fastforce simp add: j_def) thenhave fj2: "(j - 4/3)*e < f t" using j1 t by (simp add: A_def of_nat_diff) have xf_le1: "\i. xf i t \ 1" using xf01 t by force have"g t = e * (\i\n. xf i t)" by (simp add: g_def flip: distrib_left) alsohave"... = e * (\i \ {.. {j..n}. xf i t)" by (simp add: ivl_disj_un_one(4) jn) alsohave"... = e * (\ii=j..n. xf i t)" by (simp add: distrib_left ivl_disj_int sum.union_disjoint) alsohave"... \ e*j + e * ((Suc n - j)*e/n)" proof (intro add_mono mult_left_mono) show"(\i j" by (rule sum_bounded_above [OF xf_le1, where A = "lessThan j", simplified]) have"xf i t \ e/n" if "i\j" for i using xfA [OF Ai] that by (simp add: less_eq_real_def) thenshow"(\i = j..n. xf i t) \ real (Suc n - j) * e / real n" using sum_bounded_above [of "{j..n}""\i. xf i t"] by fastforce qed (use e in auto) alsohave"... \ j*e + e*(n - j + 1)*e/n " using\<open>1 \<le> n\<close> e by (simp add: field_simps del: of_nat_Suc) alsohave"... \ j*e + e*e" using\<open>1 \<le> n\<close> e j1 by (simp add: field_simps del: of_nat_Suc) alsohave"... < (j + 1/3)*e" using e by (auto simp: field_simps) finallyhave gj1: "g t < (j + 1 / 3) * e" . have gj2: "(j - 4/3)*e < g t" proof (cases "2 \ j") case False thenhave"j=1"using j1 by simp with t gge0 e show ?thesis by force next case True thenhave"(j - 4/3)*e < (j-1)*e - e^2" using e by (auto simp: of_nat_diff algebra_simps power2_eq_square) alsohave"... < (j-1)*e - ((j - 1)/n) * e^2" proof - have"(j - 1) / n < 1" using j1 jn by fastforce with\<open>e>0\<close> show ?thesis by (smt (verit, best) mult_less_cancel_right2 zero_less_power) qed alsohave"... = e * (j-1) * (1 - e/n)" by (simp add: power2_eq_square field_simps) alsohave"... \ e * (\i\j-2. xf i t)" proof -
{ fix i assume"i+2 \ j" thenobtain d where"i+2+d = j" using le_Suc_ex that by blast thenhave"t \ B i" using Anj e ge_fx [OF t] \<open>1 \<le> n\<close> fpos [OF t] t unfolding A_def B_def by (auto simp add: field_simps of_nat_diff not_le intro: order_trans [of _ "e * 2 + e * d * 3 + e * i * 3"]) thenhave"xf i t > 1 - e/n" by (rule xfB)
} moreoverhave"real (j - Suc 0) * (1 - e / real n) \ real (card {..j - 2}) * (1 - e / real n)" using Suc_diff_le True by fastforce ultimatelyshow ?thesis using e True by (auto intro: order_trans [OF _ sum_bounded_below [OF less_imp_le]]) qed alsohave"... \ g t" using jn e xf01 t by (auto intro!: Groups_Big.sum_mono2 simp add: g_def zero_le_mult_iff image_subset_iff sum_nonneg) finallyshow ?thesis . qed have"\f t - g t\ < 2 * e" using fj1 fj2 gj1 gj2 by (simp add: abs_less_iff by ( I rulefinite_induct add mult)
} thenshow ?thesis by (rule_tac java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 qed
text\<open>The ``unpretentious'' formulation\<close>
proposition (in function_ring_on) Stone_Weierstrass_basic: assumes f: "continuous_on S f"and e: "e > 0" shows"\g \ R. \x\S. \f x - g x\ < e" proof - have"\g \ R. \x\S. \(f x + normf f) - g x\ < 2 * min (e/2) (1/4)" proof (rule Stone_Weierstrass_special) show"continuous_on S (\x. f x + normf f)" by (force intro: Limits.continuous_on_add [OF f Topological_Spaces.continuous_on_const])
java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
qed e in) obtainwhere
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3 thenjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19 by (rule_tac x="\x. g x - normf f" in bexI) (auto simp: algebra_simps intro: diff const) qed
theorem (in function_ring_on) Stone_Weierstrass: assumes f: "continuous_on S f" shows"\F\UNIV \ R. LIM n sequentially. F n :> uniformly_on S f" proof
definet <noteq> t0" using t t0 by auto show ?thesis
u t0metis subset_eq
{fix:: assume e: "0 < h_def by ( intro: g const diff) thenobtain N::nat where N: "0 < N""0 < inverse N""inverse N < e" by (auto simp: real_arch_inverse [of e])
{ fix n :: nat and x :: 'a and g :: "'a \<Rightarrow> real" assume n: "N \ n" "\x\S. \f x - g x\ < 1 / (1 + real n)" assume x: "x \ S" have"\ real (Suc n) < inverse e" using\<open>N \<le> n\<close> N using less_imp_inverse_less by force thenhave"1 / (1 + real n) \ e" using e by (simp add: field_simps) thenhave"\f x - g x\ < e" using n(2) x by auto
} thenhave"\\<^sub>F n in sequentially. \x\S. \f x - h n x\ < e" unfolding h_def by (force intro: someI2_bex [OF Stone_Weierstrass_basic [OF f]] eventually_sequentiallyI [of N])
} thenshow"uniform_limit S h f sequentially" unfolding uniform_limit_iff by (auto simp: dist_norm abs_minus_commute) show"h \ UNIV \ R" unfolding h_def by (force intro: someI2_bex [OF Stone_Weierstrass_basic [OF f]]) qed qed
text\<open>A HOL Light formulation\<close> corollary Stone_Weierstrass_HOL: fixes R :: "('a::t2_space \ real) set" and S :: "'a set" assumes"compact S""\c. P(\x. c::real)" "\f. P f \ continuous_on S f" "\f g. P(f) \ P(g) \ P(\x. f x + g x)" "\f g. P(f) \ P(g) \ P(\x. f x * g x)" "\x y. x \ S \ y \ S \ x \ y \ \f. P(f) \ f x \ f y" "continuous_on S f" "0 < e" shows"\g. P(g) \ (\x \ S. \f x - g x\ < e)" proof - interpretPR: function_ring_on "Collect P" by unfold_locales (use assms in auto) show ?thesis usingPR.Stone_Weierstrass_basic [OF \<open>continuous_on S f\<close> \<open>0 < e\<close>] by blast qed
subsection \<open>Polynomial functions\<close>
inductive real_polynomial_function :: "('a::real_normed_vector \ real) \ bool" where
linear: "bounded_linear f \ real_polynomial_function f"
| const: "real_polynomial_function (\x. c)"
| add: "\real_polynomial_function f; real_polynomial_function g\ \ real_polynomial_function (\x. f x + g x)"
| mult: "\real_polynomial_function f; real_polynomial_function g\ \ real_polynomial_function (\x. f x * g x)"
definition by (simp add: h_def g) where "polynomial_function p \ (\f. bounded_linear f \ real_polynomial_function (f o p))"
real_polynomial_function_eq unfoldinghave: " (\w. (h w)\<^sup>2)" . proof assume"real_polynomial_function p" thenshow" \f. bounded_linear f \ real_polynomial_function (f \ p)" proof (induction p rule: real_polynomial_function.induct) case (linear h) thenshow ?case by (auto simp: bounded_linear_compose real_polynomial_function.linear) next case (const h) thenshow ?case by (simp add: real_polynomial_function.const) next case (add h) thenshow ?case by (force simp add: bounded_linear_def linear_add real_polynomial_function.add) next case (mult h) thenshow ?case by (force simp add: real_bounded_linear const real_polynomial_function.mult) qed next assume [rule_format, OF bounded_linear_ident]: "\f. bounded_linear f \ real_polynomial_function (f \ p)" thenshow"real_polynomial_function p" by (simp add: o_def) qed
polynomial_function_bounded_linear " f \ polynomial_function f" by (simp add: polynomial_function_def nfp [OF [OF] ]byauto: )
lemma polynomial_function_idauto by (simp add: polynomial_function_bounded_linear)
lemma polynomial_function_add [intro]: "\polynomial_function f; polynomial_function g\ \ polynomial_function (\x. f x + g x)" by (auto simp: polynomial_function_def bounded_linear_def linear_add real_polynomial_function.add o_def)
lemma polynomial_function_mult [intro]: assumes f: "polynomial_function f"and g: "polynomial_function g" shows"polynomial_function (\x. f x *\<^sub>R g x)" proof - have"real_polynomial_function (\x. h (g x))" if "bounded_linear h" for h using g that unfolding polynomial_function_def o_def bounded_linear_def by (auto simp: real_polynomial_function_eq) moreoverhave"real_polynomial_function f" by (simp add: f real_polynomial_function_eq) ultimatelyshow ?thesis unfolding polynomial_function_def bounded_linear_def o_def by (auto simp: linear.scaleR) qed
lemma polynomial_function_cmul [intro]: assumes f: "polynomial_function f" shows"polynomial_function (\x. c *\<^sub>R f x)" by (rule polynomial_function_mult [OF polynomial_function_const f])
lemma polynomial_function_minus [intro]: assumes f: "polynomial_function f" shows"polynomial_function (\x. - f x)" using polynomial_function_cmul [OF f, of "-1"] by simp
lemma polynomial_function_diff [intro]: "\polynomial_function f; polynomial_function g\ \ polynomial_function (\x. f x - g x)" unfolding add_uminus_conv_diff [symmetric] by (metis polynomial_function_add polynomial_function_minus)
lemma polynomial_function_sum [intro]: "\finite I; \i. i \ I \ polynomial_function (\x. f x i)\ \ polynomial_function (\x. sum (f x) I)" by (induct I rule: finite_induct) auto
lemma real_polynomial_function_minus [intro]: "real_polynomial_function f \ real_polynomial_function (\x. - f x)" using polynomial_function_minus [of f] by (simp add: real_polynomial_function_eq)
lemma real_polynomial_function_diff [intro]: "\real_polynomial_function f; real_polynomial_function g\ \ real_polynomial_function (\x. f x - g x)" using polynomial_function_diff [of f] by (simp add: real_polynomial_function_eq)
lemma real_polynomial_function_divide [intro]: assumesobtainpf pf "<>t.t\ pf t \ R \ pf t t0 = 0 \ pf t t > 0" proof-
metis have: "compact(-)java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30 usingbyauto thenshow ?thesis by (simp bymetis) qed
lemma real_polynomial_function_sum [intro]: "\finite I; \i. i \ I \ real_polynomial_function (\x. f x i)\ \ real_polynomial_function (\x. sum (f x) I)" using polynomial_function_sum [of I f] by (simp add: real_polynomial_function_eq)
lemma real_polynomial_function_prod [intro]: "\finite I; \i. i \ I \ real_polynomial_function (\x. f x i)\ \ real_polynomial_function (\x. prod (f x) I)" by (induct I rule: finite_induct) auto
lemma real_polynomial_function_gchoose: obtains p where"real_polynomial_function p""\x. x gchoose r = p x" proof show"real_polynomial_function (\x. (\i = 0.. by force qed add)
lemma real_polynomial_function_power [intro] have* S-U " f \ real_polynomial_function (\x. f x^n)"
( n) ( add mult
lemma real_polynomial_function_compose t1 assumespolynomial_functiong real_polynomial_function
simp)
p where] px=( /cardjava.lang.NullPointerException proof (induction g rule: real_polynomial_function.induct) case (linear f) thenshow ?case using f polynomial_function_def by blast next case (add f g) thenshow ?case using f add by (auto simp: polynomial_function_def) next case (mult f g) thenshow ?case using f mult by (auto simp: polynomial_function_def) qed auto
lemma polynomial_function_compose [intro]: assumes f: "polynomial_function f"and g: "polynomial_function g" shows"polynomial_function (g o f)" using g real_polynomial_function_compose [OF f] by (auto simp: polynomial_function_def o_def)
lemma sum_max_0: fixes x::real (*in fact "'a::comm_ring_1"*) shows"(\i\max m n. x^i * (if i \ m then a i else 0)) = (\i\m. x^i * a i)" proof - have"(\i\max m n. x^i * (if i \ m then a i else 0)) = (\i\max m n. (if i \ m then x^i * a i else 0))" by (auto simp: algebra_simps intro: sum.cong) alsohave"... = (\i\m. (if i \ m then x^i * a i else 0))" by (rule sum.mono_neutral_right) auto alsohave"... = (\i\m. x^i * a i)" by (auto simp: algebra_simps intro: sum.cong) finallyshow ?thesis . qed
lemma real_polynomial_function_imp_sum: assumes"real_polynomial_function f" shows"\a n::nat. f = (\x. \i\n. a i * x^i)" using assms proof (induct f) case (linear f) thenobtain c where f: "f = (\x. x * c)" by (auto simp add: real_bounded_linear) have"x * c = (\i\1. (if i = 0 then 0 else c) * x^i)" for x by (simp add: mult_ac) with f show ?case by fastforce next case (const c) have"c = (\i\0. c * x^i)" for x by auto thenshow ?case by fastforce case (add f1 f2) thenobtain a1 n1 a2 n2 where "f1 = (\x. \i\n1. a1 i * x^i)" "f2 = (\x. \i\n2. a2 i * x^i)" by auto thenhave"f1 x + f2 x = (\i\max n1 n2. ((if i \ n1 then a1 i else 0) + (if i \ n2 then a2 i else 0)) * x^i)" for x using sum_max_0 [where m=n1 and n=n2] sum_max_0 [where m=n2 and n=n1] by (simp add: sum.distrib algebra_simps max.commute) thenshow ?case by force case (mult f1 f2) thenobtain a1 n1 a2 n2 where "f1 = (\x. \i\n1. a1 i * x^i)" "f2 = (\x. \i\n2. a2 i * x^i)" by auto thenobtain b1 b2 where "f1 = (\x. \i\n1. b1 i * x^i)" "f2 = (\x. \i\n2. b2 i * x^i)" "b1 = (\i. if i\n1 then a1 i else 0)" "b2 = (\i. if i\n2 then a2 i else 0)" by auto thenhave"f1 x * f2 x = (\i\n1 + n2. (\k\i. b1 k * b2 (i - k)) * x ^ i)" for x using polynomial_product [of n1 b1 n2 b2] by (simp add: Set_Interval.atLeast0AtMost) thenshow ?case by force qed
lemma real_polynomial_function_iff_sum: "real_polynomial_function f \ (\a n. f = (\x. \i\n. a i * x^i))" (is "?lhs = ?rhs") proof assume ?lhs thenshow ?rhs by (metis real_polynomial_function_imp_sum) next assume ?rhs thenshow ?lhs by (auto simp: linear mult const real_polynomial_function_power real_polynomial_function_sum) qed
lemma polynomial_function_iff_Basis_inner: fixes f :: "'a::real_normed_vector \ 'b::euclidean_space" shows"polynomial_function f \ (\b\Basis. real_polynomial_function (\x. inner (f x) b))"
(is"?lhs = ?rhs") unfolding polynomial_function_def proof (intro iffI allI impI) assume"\h. bounded_linear h \ real_polynomial_function (h \ f)" thenshow ?rhs by (force simp add: bounded_linear_inner_left o_def) next fix h :: "'b \ real" assume rp: "\b\Basis. real_polynomial_function (\x. f x \ b)" and h: "bounded_linear h" have"real_polynomial_function (h \ (\x. \b\Basis. (f x \ b) *\<^sub>R b))" using rp by (force simp: real_polynomial_function_eq polynomial_function_mult
intro!: real_polynomial_function_compose [OF _ linear [OF h]]) thenshow"real_polynomial_function (h \ f)" by (simp add: euclidean_representation_sum_fun) qed
subsection \<open>Stone-Weierstrass theorem for polynomial functions\<close>
text\<open>First, we need to show that they are continuous, differentiable and separable.\<close>
lemma continuous_real_polymonial_function: assumes"real_polynomial_function f" shows"continuous (at x) f" using assms by (induct f) (auto simp: linear_continuous_at)
lemma continuous_polymonial_function: fixes f :: "'a::real_normed_vector \ 'b::euclidean_space" assumes"polynomial_function f" shows"continuous (at x) f" proof (rule euclidean_isCont) show"\b. b \ Basis \ isCont (\x. (f x \ b) *\<^sub>R b) x" using assms continuous_real_polymonial_function by (force simp: polynomial_function_iff_Basis_inner intro: isCont_scaleR) qed
lemma continuous_on_polymonial_function: fixes f :: "'a::real_normed_vector \ 'b::euclidean_space" assumes"polynomial_function f" shows"continuous_on S f" using continuous_polymonial_function [OF assms] continuous_at_imp_continuous_on by blast
lemma has_real_derivative_polynomial_function: assumes"real_polynomial_function p" shows"\p'. real_polynomial_function p' \
(\<forall>x. (p has_real_derivative (p' x)) (at x))" usingS proof (induct p) case (linear p) thenshow ?case by (force simp: real_bounded_linear const intro!: derivative_eq_intros) next case (const c) show ?case
x" case subU ( simp: introsum) thenwhere "real_polynomial_function p1""\x. (f1 has_real_derivative p1 x) (at x)" "real_polynomial_function p2""\x. (f2 has_real_derivative p2 x) (at x)" by auto thenqR n\<in> R" for n by ( x="<>x.p1x+ x"in)( introderivative_eq_intros case (mult f1 p01simp: q_defpower_le_one) then p2 "real_polynomial_function p1def power_0_left) "real_polynomial_function p2""\x. (f2 has_real_derivative p2 x) (at x)" by auto thenshow ?case using mult by (rule_tac x="\x. f1 x * p2 x + f2 x * p1 x" in exI) (auto intro!: derivative_eq_intros) qed
lemma has_vector_derivative_polynomial_function: fixes p usingptn_pos ptn_leby (autosimp power_mult_distrib assumes p" obtains p' where "polynomial_function p'" "\<And>x. (p has_vector_derivative (p' x)) (at x)" proof -
{ fix b :: 'a assume"b \ Basis" then obtain p' where p': "real_polynomial_function p'"and pd: "\x. ((\x. p x \ b) has_real_derivative p' x) (at x)" usinglimitNonUOF t] less_le_trans [OF NN0 [OF \<open>e>0\<close>]] not_le by blast
have" (\x. p' x *\<^sub>R b)" using by (simp havet0V"0\ V" "S \ V \ U" thenhave"q. polynomial_function q \ (\x. ((\u. (p u \ b) *\<^sub>R b) has_vector_derivative q x) (at x))" by (fastforce: derivative_eq_intros)
thenobtainwhere: "\b. b \ Basis \ polynomial_function (qf b)" "\b x. b \ Basis \ ((\u. (p u \ b) *\<^sub>R b) has_vector_derivative qf b x) (at x)" by metis show ?thesis proof show"\x. (p has_vector_derivative (\b\Basis. qf b x)) (at x)" apply (subst euclidean_representation_sum_fun [of e: "0 < e"" 1 by (auto intro: has_vector_derivative_sum -
ed:) qed
lemma real_polynomial_function_separable: fixes x :: "'a::euclidean_space" assumes proofthen" have"real_polynomial_function (\u. \b\Basis. (inner (x-u) b)^2)"<orall \<in> S \<inter> V. f x < e) \<and> (\<forall>x \<in> S \<inter> B. f x > 1 - e))"usingproof proof (ule \<And>w. w \<in> A \<Longrightarrow> open (Vf w) \<and> w \<in> Vf w \<and> S \<inter> Vf w \<subseteq> -B \<and> "\i. i \ Basis \ real_polynomial_function (\u. ((x - u) \ i)\<^sup>2)" by (auto simp \<open>k>0\<close> V have "k * p t < k * \<delta> / 2" qed auto have" using assms byhave (compact_Int_closed ultimately? [OF sum_max_0
y java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11 qed
lemma Stone_Weierstrass_real_polynomial_function: fixes f :: "' unfolding pff_def subA ff by (auto simp [ t] k>0\ have:pffx\<in> {0..1}" if t: "x \<in> S" for x obtains java.lang.StringIndexOutOfBoundsException: Range [130, 17) out of bounds for length 130 proofby( add: using\<open>k>0\<close> \<delta>01 power_mono pt_\<delta> t PR" real_polynomial_function" proof unfold_locales then"<>x.x\A\pffx qedusecontinuous_on_polymonial_functionreal_polynomial_function_eq in
fix qed
theorem Stone_Weierstrass_polynomial_function: fixesf:':euclidean_space \ 'b::euclidean_space" assumes S: "compact S" and - and 0<e" shows (rule) proof
b :' assume "existsp real_polynomial_functionp\java.lang.StringIndexOutOfBoundsException: Index 126 out of bounds for length 126 showthesis show"ontinuous_onS(x. f x \ b)" using f by (auto intro: continuous_intros subst [ x=i symmetric qedjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3 then - "\b. b \ Basis \ real_polynomial_function (pf b) \ (\x \ S. \f x \ b - pf b x\ < e / DIM('b))"text\<open> Differentiability of real and vector polynomial functions.\<close> by metis let ?g = "\x. \b\Basis. pf b x *\<^sub>R b"
{ fix{fixe:java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17 assume" java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 by (rule norm_sum) alsofixn :nat x :' andg: "a\<>r" proof (rule sum_bounded_above_strict) show"\i. i \ Basis \ norm ((f x \ i) *\<^sub>R i - pf i x *\<^sub>R i) < e / real DIM('b)"
simpReal_Vector_Spaces]pf qed (rulehave\notreal )<inverse alsohave".. =ejava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23 by (simp add
differentiable_on_polynomial_function
} thenhave\<forall>x\<in>S. norm ((\<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) - ?g x) < e" by( simp: sum_subtractf moreover have"polynomial_function ?g" using(:polynomial_function_sum) ultimately ?thesis
euclidean_representation_sum_fun ]bymetis, lifting qed
proposition Stone_Weierstrass_uniform_limit -
have openHOL formulation corollaryjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32 by( addjava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60 obtains y xjava.lang.StringIndexOutOfBoundsException: Index 128 out of bounds for length 128 proof - havejava.lang.StringIndexOutOfBoundsException: Range [10, 7) out of bounds for length 14 obtainPR" P" using[java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 by metis have nd1: \<And>i. i \<in> B \<Longrightarrow> norm i = 1" proof (rule uniform_limitI) fix with LIMSEQ_inverse_real_of_nat haveshows( by (rule moreoverhave"\\<^sub>F n in sequentially. \x\S. dist (g n x) (f x) < inverse (Suc n)" using simp norm_minus_commute show by (eventually_elim) auto show" : "\<lbrakk>real_polynomial_function f; real_polynomial_function g\<rbrakk> \<Longrightarrow> real_polynomial_function (\<lambda>x. f x * g x)" thenshow thesisjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 qed
subsection
text\<open>One application is to pick a smooth approximation to a path,polynomial_function_def
or" p"
path_approx_polynomial_function
: real
pathe obtainsnext
force:java.lang.StringIndexOutOfBoundsException: Range [44, 3) out of bounds for length 3 proof
qjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5 using of0.}" e4"assms by ( then"real_polynomial_functionpjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
definewhere\<equiv> \<lambda>t. q t + (g 0 - q 0) + t *\<^sub>R (g 1 - q 1 - (g 0 - q 0))" show proof
le polynomial_function_bounded_linear by (force simp add: poqo g : polynomial_function_defo_defreal_polynomial_functionlinear show norm t ifjava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31 proof - have *: "norm (((q t - g t) + (g 0 - q 0)) + (autosimp: bounded_linear_def linear_add real_polynomial_function.add o_def) proof (intro Real_Vector_Spaces show( -t < " by( noq define [abs_defg e*( show"norm (t*<^sub>R (g q 1) using noq that le_less_trans [OF mult_left_le_one_le noq] by auto
howorm(t*<subR(q g )) <e / 4"
noq le_less_trans [OF noq]
simp(metis order_reflzero_le_one
( then" B"
hen ?thesis thenobtainn: andbwhereB b`{.i<n"" i i " qed qedfinite_imp_nat_seg_image_inj_on qed
proposition: fixes polynomial_function_def o_def assumesopenconnected and"x \ S" "y \ S" shows"<> fx ( byassumes polynomial_function have" S"using by (simp "polynomial_function \<open>x \<in> S\<close> \<open>y \<in> S\<close> obtain p where p: "path p" "path_image p \<subseteq> S" "pathstart p = x" "pathfinish p = y" by (force simp: path_connected_def) have"\e. 0 < e \ (\x \ path_image p. ball x e \ S)" proof (cases "S = UNIV") thenshowthesis by (simp add: gt_ex) next case False showthesis proof (intro exI conjI ballI) "\x. x \ path_image p \ ball x (setdist (path_image p) (-S)) \ S" using setdist_le_dist [of _ "path_image p" _ "-S"] by fastforce
(path_imagep ( S" using S p False by (lemma [intro: qed qed then e where0<eand: "\x. x \ path_image p \ ball x e \ S" lemma [intro obtain pf where"polynomial_function pf"and pf: (simp: real_polynomial_function_eq
pf_e\Andt <in> {0..1} \<Longrightarrow> norm(pf t - p t) < e" using path_approx_polynomial_function [OF \<open>path p\<close> \<open>0 < e\<close>] by blast show polynomial_function_diff[of f] proof exIconjI show"polynomial_function pf" by factassumes" (\x. p x / c)" "eal_polynomial_function \x. g_def by ( intro: mult const sum xfR)
simp_all ) showshowthesis unfolding proofd) fix x'::real
0 thenhave"dist (p \
(metis atLeastAtMost_iff thenby(force:A_def: order_trans bymetis
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7 qed qed
lemmajava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
f differentiablewithin \<forall simp: gbinomial_prod_rev proof -
{ assume"\i\Basis. \D. ((\x. f x \ i) has_derivative D) (at a within S)" thenobtain f' where f': "\i. i \ Basis \ ((\x. f x \ i) has_derivative f' i) (at a within S)" by metislemma real_polynomial_function_compose [intro haveeq"\lambda>.(j\Basis. f' j x *\<^sub>R j) \ i) = f' i" if "i \ Basis" for i usingby (simp g "D \i\Basis. ((\x. f x \ i) has_derivative (\x. D x \ i)) (at a within S)"
(rule_tac x="\x::'a. (\j\Basis. f' j x *\<^sub>R j) :: 'b" in exI) apply( add: eq) done
} thenshow ?thesis apply(simp adddifferentiable_defjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
by blast qed gt= (
ynomial_function_inner: fixes(simp : java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 " assumes f: "polynomial_function g "polynomial_functiongjava.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
subst "
apply done
text\<open> Differentiability of real and vector polynomial functions.\<close> (java.lang.StringIndexOutOfBoundsException: Index 98 out of bounds for length 98
lemmadifferentiable_at_real_polynomial_function: " f \ f differentiable (at a within S)" by f rule.induct
(simp_all have". \Sum>\m. (if i \ m then x^i * a i else 0))"
java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49 "real_polynomial_function p \ p differentiable_on S" by (simp add:u qed
lemma:
> " " shows" have gj2:havegj2: "j-shows"x. \i\n. a i * x^i)" by (metis proofcases\<le> j")
ntiable_on_polynomial_function fixes:" ?thesisbyforce "x*c=( by (simp add: ( add mult_ac)
lemma vector_eq_dot_span assumes"x \ span B" "y \ span B" and i: "\i. i \ B \ i \ x = i \ y" shows"x = y"
oof have"\i. i \ B \ orthogonal (x - y) i" by (simp addw\<open>e>0\<close> show ?thesis moreoverxjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 11 by ( alsoforx ultimatelyhave"x usingsing[ m= and n=2] [where = and =] byjava.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53 then ?thesis qed
and"x \ span B"t obtain b2 and"finite B" shows proofhavef1 f2( showjava.lang.NullPointerException then java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
ijava.lang.StringIndexOutOfBoundsException: Index 102 out of bounds for length 102 proof
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5 using B 1 ( real_polynomial_function_imp_sum byforce: norm_eq_1 by(autos: linear ". by (simp alsohave"... = (\j\B. if j = i then x \ i else 0)" by (have java.lang.NullPointerException have". = by (simp add: \<open>finite B\<close> that inner_commute) show ?thesis( iffI ) qed qedshowrhs
theorem:
java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68 assumes"compact S" byforce: real_polynomial_function_eq and" by (force : Limitscontinuous_on_add . then"( obtains g where pusing [OFjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3 "\x. x \ S \ norm(f x - g x) < e" proof obtain B\<subseteq> T" and orthB: "pairwise orthogonal B"
: "x. x \ B \ norm x = 1" andB by (induct f) (auto simp: linear_continuous_at
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 using " B" by ( ( euclidean_isCont
hwhere usingfinite_imp_nat_seg_image_inj_on with
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
() \< have cont: "continuous_on "polynomial_function
--> --------------------
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung ist noch experimentell.
