Quelle Characteristic_Functions.thy Sprache: Isabelle

(* Title: HOL/Probability/Characteristic_Functions.thy
 Authors: Jeremy Avigad (CMU), Luke Serafin (CMU), Johannes Hölzl (TUM)
*)

section \<open>Characteristic Functions\<close>

theoryCharacteristic_Functions
 imports Weak_Convergence Independent_Family Distributions
begin

lemma mult_min_right: "a \ 0 \ (a :: real) * min b c = min (a * b) (a * c)"
 by (metis min.absorb_iff2 min_def mult_left_mono)

lemma sequentially_even_odd:
 assumes E: "eventually (\n. P (2 * n)) sequentially" and O: "eventually (\n. P (2 * n + 1)) sequentially"
 shows "eventually P sequentially"
proof -
 from E obtain n_e where [intro]: "\n. n \ n_e \ P (2 * n)"
 auto)
 moreover
 java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
byauto:)
 show
 unfolding
 proof (intro (uto:eventually_sequentiallyfromobtain [intro
fix " 2* (2*n_o 1 \ n" then show "P n"
 by (cases ?thesis eventually_sequentially
 qed
qed

lemma limseq_even_odd:
 assumes "(\n. f (2 * n)) \ (l :: 'a :: topological_space)"
 and " proof( exI allI impIjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
"f\java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

subsection and "

abbreviation iexp
 " \ (\x. exp (\ * complex_of_real x))"

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 by (introiexp

lemma [simp"
 "(iexp has_vector_derivative \ * iexp x) (at x within s)"
 by (auto intro!: derivative_eq_intros continuous_intros [derivative_intros(

lemma interval_integral_iexp:
 fixes a b :: real interval_integral_iexp a b : real
 shows [where F="\x. -\ * iexp x"])
 by subst
 (auto intro!: derivative_eq_intros continuous_intros

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

definition
"\Rightarrow>real\ complex"
java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51

(in) char_zero "
 unfolding char_def by (simp del: space_eq_univ add: prob_space)

lemma (in prob_space) java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 assumesfjava.lang.NullPointerException
 "integrableM(x. exp (\ * (f x)))"
proof (intro integrable_const_bound [of _ 1])
 from "\x. of_real (Re (f x)) = f x"
 by (simp "
bysimp: complex_eq_iff show" x in M cmod(exp (\ * f x)) \ 1"
 norm_exp_i_times Re) forx] simp
qed (insert f, simp)

lemma (in real_distribution) cmod_char_le_1: "norm (char M t) \ 1"
-
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 unfoldingproof java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 also haveunfolding char_def integral_norm_bound
by ( del)
 finallyshow thesis
qed

( real_distributionisCont_char isContM "
 unfolding continuous_at_sequentially
proof safe
 fix X assume X: "X \ t"
 show "(char M \ X) \ char M t"
 unfolding comp_def char_def show .
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
qed

lemma unfolding continuous_at_sequentially safe
 ( intro! borel_measurable_continuous_onI isCont_char

subsection "(har M \ X) \ char M t"

(* the automation can probably be improved *) integral_dominated_convergence w"<>_ 1] autointro: X)
lemmain) char_distr_add
fixes X2:: ' \ real" and t :: real
 assumes "indep_var auto !: borel_measurable_continuous_onI continuous_at_imp_continuous_on isCont_char)
 shows "char (distr M borel (\\. X1 \ + X2 \)) t =
 char (distr M borel X1) t * char (distr
proof -
 from assmshavemeasurable"andom_variable X1 by ( indep_var_rv1)
 from assms have [measurable

 have (in) char_distr_add
 by simp: char_def)
 also (
 by ( char X1* (distr ) "
alsojava.lang.NullPointerException
 from []: " borel X2 java.lang.StringIndexOutOfBoundsException: Range [82, 81) out of bounds for length 82
 (auto intro indep_var_compose comp_defOF])
 also( intro:integrable_iexp)
 simp: char_def)
 finally( add: char_def)
qed

lemma (in prob_space showthesis .
 indep_vars
 char (distr M java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
proofinductrule infinite_finite_induct
 case (insert x F) with indep_vars_subset" (\i. borel) X A \
 by auto add indep_vars_sum
qed (simp_allproofinductrule:infinite_finite_induct)

subsection \<open>Approximations to $e^{ix}$\<close>

text \<open>Proofs from Billingsley, page 343.\<close>

lemma CLBINT_I0c_power_mirror_iexp:
 fixes andn: java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
 defines s m \<equiv> complex_of_real ((x - s) ^ m)"
 "(CLBINT s=0..x. fs iexps =
 x^Suc n / Suc n + (\ / Suc n) * (CLBINT s=0..x. f s (Suc n) * iexp s)"
proof
 have 1:
 "
text <> from Billingsley 34.
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 intro) java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40

 "lambda.-xs^Suc)/Suc ))* s"
 have " -
 proof(\<
 has_vector_derivative((x-)n s + (
 complex_of_real(-((x - s) ^ (Sucat within ) sA
 by (cases "0 \ x") (auto intro!: simp: f_def[abs_def])

 unfolding using
 byintro)
 (auto simp "^Suc n)/( n) =( s=0.x.(f s + (\ * iexp s) * -(f s (Suc n) / (Suc n))))" (is "?LHS = ?RHS")
 have (s0..(sniexp
show
 by auto(
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
 show
intro )
 by show
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
qed

lemma iexp_eq1:
 x :: real
 defines \<equiv> complex_of_real ((x - s) ^ m)"
 showsjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
 "iexp java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
proof induction
 show "?P 0"
 by (auto simp add: java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 19
next
 fix n assume ihauto add field_simps f_def)
have:\ab: .=b
y linarith
 have" n * of_nat( n) \ - (of_nat (fact n)::complex)"
 of_nat_mult]
 by (simp add: complex_eq_iff ** of_nat_add[symmetric] del: of_nat_mult of_nat_fact of_nat_add * of_nat)java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
" (Suc n)"
 unfolding sum.atMost_Suc f_def[of _ n
 by (simp add: divide_simps "?( n)"
qed

lemma iexp_eq2:
 fixes x :: real
 defines "f (simp add add: divide_simps *) (simp add:field_simps)
fixes
-
 have isCont_f: "isCont have isCont_f: " (\<lambda>s. f s n) x" for n x
 by (auto F= "s. complex_of_real (-((x - s) ^ (Suc n) / real (Suc n)))"
 let ?F = "\s. complex_of_real (-((x - s) ^ (Suc n) / real (Suc n)))"
 havecalc1CLBINT (exp)
 (CLBINT s=0. ( s=0..x. f sn *iexp)- CLBINT.x. f s n"
 unfolding zero_ereal_def
by (ubst(2) [symmetric
 (simp_all add: interval_integrable_isCont(simp_all add: interval_integrable_isCont isCont_f field_simps)

 have calc2: "(CLBINT s=0..x. f s n) = x^Suc n / Suc n"
 unfolding zero_ereal_def
 (subst [where=0and xand=" and F = ?F]java.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 107
 show "(?F has_vector_derivative f y n) (at y within {min 0 x..max 0 x})" show"(? has_vector_derivative yn) (t min0 x..max x}"for
 unfoldingby( has_vector_derivative_of_realjava.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
 ( has_vector_derivative_of_real)
 (auto intro!: derivative_eq_intros (autointro isCont_f
 qed( intro isCont_f

 haveby(imp: field_simps)
 by(imp : field_simps

 showby( CLBINT_I0c_power_mirror_iexp n = n])auto
 unfolding
 by subst [where= n)auto
qed

lemma abs_LBINT_I0c_abs_power_diff:
 "\LBINT s=0..x. \(x - s)^n\\ = \x ^ (Suc n) / (Suc n)\"
proof -
have
 proof caseshave
 oof
 then 0
 by (auto simp then "LBINTs0.x
 intro!: interval_integral_cong(auto addzero_ereal_def power_abs max_absorb2
ysimp
 next
 show? by simp
ve(LBINT..java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
byautosimpadd zero_ereal_def min_absorb1
 ereal_min[symmetric] ereal_max[symmetric] power_minus_odd[symmetric]
 del intro)
 also have "\ = - (LBINT s=0..x. (x - s)^n)"
subst refl
 finally show ? have "\ = - (LBINT s=0..x. (x - s)^n)"
 qed
 also havefinally ?thesis
 java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
let? \lambda.-(-t^Suc/Suc
 haveletF \lambdat ( - t)Suc/n"
unfolding
 by ( unfolding java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
 has_real_derivative_iff_has_vector_derivativeiffD1
 auto del intro erivative_eq_intros add)
 have"dots Suc )/(n"bysimp
 finally ?thesis .
 qed
 finally show ?thesis .
qed

lemma iexp_approx1: "cmod (iexp x - (\k \ n. (\ * x)^k / fact k)) \ \x\^(Suc n) / fact (Suc n)"
proof
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
iexp
 by (subst(\i ( n) /(act)*( s=0..x.( -)n*( s))"( "?t1")
have" (?t1) = (?t2)"
 have" (?t1) = (?t2"
 also have "\ = (1 / of_nat (fact n)) * cmod (CLBINT s=0..x. (x - s)^n * (iexp s))"
add norm_power
 bysimp norm_divide
 byalso
 mult_left_mono
also java.lang.NullPointerException
 by ">
 add: )
 java.lang.NullPointerException
 simp)
 ? .
 byjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 finallyshowthesis
d

lemmahave*"\a b. interval_lebesgue_integrable lborel a b f \ interval_lebesgue_integrable lborel a b g \
proof (induction n) \<comment> \<open>really cases\<close>
 case ( n)
 have *: "\a b. interval_lebesgue_integrable lborel a b f \ interval_lebesgue_integrable lborel a b g \
 \<bar>LBINT s=a..b. f s\<bar> \<le> \<bar>LBINT s=a..b. g s\<bar>"
 ifu interval_lebesgue_integral_def set_integrable_def
 using[ f ] fg
 unfoldinghaveiexp(<>
 by (autosimp[OF] introintegral_mono)

 have "iexp x - (\k \ Suc n. (\ * x)^k / fact k) =
 ((\ ^ (Suc n)) / (fact n)) * (CLBINT s=0..x. (x - s)^n * (iexp s - 1))" (is "?t1 = ?t2")
 unfolding iexp_eq2 [of x n] by (simp add: field_simps)
then (?1 (?t2
 by simpbysimp
 also have "\ = (1 / (fact n)) * cmod (CLBINT s=0..x. (x - s)^n * (iexp s - 1))"
 by (simp: norm_divide)
 also have "\ \ (1 / (fact n)) * \LBINT s=0..x. cmod ((x - s)^n * (iexp s - 1))\"
 by (intro mult_left_mono interval_integral_norm2(intro interval_integral_norm2
auto!:interval_integrable_isCont: zero_ereal_def
 also have "\ = (1 / (fact n)) * \LBINT s=0..x. abs ((x - s)^n) * cmod((iexp s - 1))\"
 by (simp add: norm_multsimp: norm_mult : of_real_diff)
a have "
 by ( ( mult_left_mono rder_trans norm_triangle_ineq4
(auto simp mult_ac intro!: interval_integrable_isCont)
 also have "\ = (2 / (fact n)) * \x ^ (Suc n) / (Suc n)\"
 by (simp: abs_LBINT_I0c_abs_power_diff abs_mult
also have"2/ n * \x ^ Suc n / real (Suc n)\ = 2 * \x\ ^ Suc n / (fact (Suc n))"
 by (simp add: abs_mult)
 finally ( add power_abs
qed (insert norm_triangle_ineq4 ( norm_triangle_ineq4 "expx ] impjava.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53

 2java.lang.StringIndexOutOfBoundsException: Index 114 out of bounds for length 114
 assumes -
showsMt
 (2*\<bar>t\<bar>^n / fact n) * expectation (\<lambda>x. \<bar>x\<bar>^n)" (is "cmod (char M t - ?t1) \<le> _")) java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
proof -
 haveunfolding by intro integrable_moments
 by (intro)auto

 define[]: "kx=\
 have integ_c: "\k. k \ n \ integrable M (\x. c k x)"
 unfolding by (intro integrable_of_real)

 havek\<le> n \<Longrightarrow> expectation (c k) = (\*t) ^ k * (expectation (\<lambda>x. x ^ k)) / fact k" for k
 unfolding c_def integral_mult_right_zero integral_complex_of_real by simp
 then have "norm (char M t -?t1) = norm (char t (CLINT x M (\k \ n. c k x)))"
 by (simp add: integ_c)
have \<> =norm |.iexp )-(<Sum>k \<le> n. c k x)))"
 unfolding char_def by (subst Bochner_Integration.integral_diff[OF integ_iexp]) (auto intro ( integral_norm_bound)
 also have "\ \ expectation (\x. cmod (iexp (t * x) - (\k \ n. c k x)))"
 by (intro integral_norm_bound)
 alsohave\<dots> \<le> expectation (\<lambda>x. 2 * \<bar>t\<bar> ^ n / fact n * \<bar>x\<bar> ^ n)"
 proof (rule ( integrable_norm.integrable_diffinteg_iexp.integrable_sum) simp
showM (<>.cmod( )-\Sumk<le>n. c k x)))"
 by (intro integrable_norm Bochner_Integration.integrable_diff unfolding[symmetric
 "integrableM(
 unfolding[symmetric
 by (intro integrable_mult_right integrable_abs integrable_moments) simp
show"modiexp( *x)-(\k\n. c k x)) \ 2 * \t\ ^ n / fact n * \x\ ^ n" for x
 using iexp_approx2[of "t * x" n] by (auto simp add: abs_mult field_simps using[of "t *x n]by( a:abs_multfield_simpsc_def)
 qed
 finally show ?thesis
 unfoldingfinally ?thesis
qed

lemma (in
 integrable_moments "<>k k\le>n\ integrable M (\x. x ^ k)"
 shows assumesintegrable_moments
(<bar>t\<bar>^n / fact (Suc n)) * expectation (\<lambda>x. min (2 * \<bar>x\<bar>^n * Suc n) (\<bar>t\<bar> * \<bar>x\<bar>^Suc n))"
 (is "cmod (char M t-?t1) \ _")
proof -
 have integ_iexp: "integrable M (\x. iexp (t * x))"
 ( integrable_const_boundjava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42

 define c where
 c whereabs_defk=(i>*t)k/fact xk" k x
 unfolding c_defhave: "\k. k \ n \ integrable M (\x. c k x)"

 have *: "min (2 * \t * x\^n / fact n) (\t * x\^Suc n / fact (Suc n)) =
 \<bar>t\<bar>^n / fact (Suc n) * min (2 * \<bar>x\<bar>^n * real (Suc n)) (\<bar>t\<bar> * \<bar>x\<bar>^(Suc n))" for x
 apply (substhave *: "min(2* \t * x\^n / fact n) (\t * x\^Suc n / fact (Suc n)) =
 apply simp
 apply (apply subst)
 apply apply rule[where])
 apply(simp_all: field_simps
 done

 done
 unfolding k\<le> n \<Longrightarrow> expectation (c k) = (\*t) ^ k * (expectation (\<lambda>x. x ^ k)) / fact k" for kunfolding integral_mult_right_zero by simp
thennormcharM -?t1 = norm(charM t (CLINT x | M. (\<Sum>k \<le> n. c k x)))"
 by (simp add: integ_c)
 have"<> (CLINTx|M iexp t * x \
char_def Bochner_Integration[OF]) auto!: )
 also have "\ \ expectation (\x. cmod (iexp (t * x) - (\k \ n. c k x)))"
 by (rule integral_norm_bound)
also "\ \ expectation (\x. min (2 * \t * x\^n / fact n) (\t * x\^(Suc n) / fact (Suc n)))"
 (is "_ \ expectation ?f")
 proof integral_mono
 show " have "\<dots> \<le> expectation (\<lambda>x. min (2 * \<bar>t * x\<bar>^n / fact n) (\<bar>t * x\<bar>^(Suc n) / fact (Suc n)))"
 by (intro integrable_norm Bochner_Integrationis
 " M ?f"
 by (rule Bochner_Integration.integrable_bound[where "integrable \java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
 simp[] power_mult_distrib
 showshow f
 iexp_approx1" [" "]
 by (auto simp [] power_mult_distrib
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
ave
 unfolding *
)
 show "integrable also have"\<dots> = (\<bar>t\<bar>^n / fact (Suc n)) * expectation (\<lambda>x. min (2 * \<bar>x\<bar>^n * Suc n) (\<bar>t\<bar> * \<bar>x\<bar>^Suc n))"unfoldingjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 ( Bochner_Integration[ f="
 (auto simp: integrable_moments power_abs( simp ntegrable_moments[symmetric)
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
 finally integral_mult_right_zero
 unfolding
qed

lemma
 fixes:" M \x. x)" and
 assumes
 integrable_1: "integrable M (\x. x)" and
 integral_1: "expectation (\x. x) = 0" and
 integrable_2: "integrable M (\x. x^2)" and
integral_2(<>.
 shows "cmod integral_2: "variance
 (t^2 / 6) * har-( t2*\<sigma>2 / 2)) \<le>
proof -
note.char_approx2 M 2t simplified]
 have [simp -
 rom havesimp]" (\x. x * x) = \2"
 by []: " UNIV = 1 ( java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
 have 1: "k \ 2 \ integrable M (\x. x^k)" for k by( add: integral_1numeral_eq_Suc)
 using assms by (auto simp: eval_nat_numeral le_Suc_eq)
 note char_approx1
 note 2 = char_approx1 [of 2 t, OF 1, simplified]
 have "cmod (char M t - (\k\2. (\ * t) ^ k * (expectation (\x. x ^ k)) / (fact k))) \
 t\<^sup>2 * expectation (\<lambda>x. min (6 * x\<^sup>2) (\<bar>t\<bar> * \<bar>x\<bar> ^ 3)) / fact (3::nat)" by ( simp le_Suc_eq
 char_approx2 2 ,OF simp
 also have "(\k\2. (\ * t) ^ k * expectation (\x. x ^ k) / (fact k)) = 1 - t^2 * \2 / 2"
 by (simp add: complex_eq_iff numeral_eq_Suc integral_1 Re_divide Im_divide)
 also " 3 = 6 ( :eval_nat_numeral
 have"\<^sup>2 * expectation (\x. min (6 * x\<^sup>2) (\t\ * \x\ ^ 3)) / 6 =
 t\<^sup>2 / 6 * expectation (\<lambda>x. min (6 * x\<^sup>2) (\<bar>t\<bar> * \<bar>x\<bar> ^ 3))" by (simp add: field_simps) , 1]by
finally .
qed

text \<open>
 This ishavet<> expectation
 we will
\<close>

lemma (in prob_space) char_approx3':
 fixes \<mu> :: "real measure" and Xshow .
 assumes rv_X \<open>
 and]: "ntegrableMX integrableM(x. (X x)^2)" "expectation X = 0"
 and var_X: "variance X = \2"
and <>def"<>= distr M borel X"
 shows "cmod (char \ t - (1 - t^2 * \2 / 2)) \
( prob_space':
 using var_X unfolding \<mu>_def
 apply (subst integral_distr rv_X]: "random_variable borel X"
 ( real_distribution)
 apply (auto simp addand: "variance =\java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
 done

text \<open>
 using unfolding\<mu>_def
 ( integral_distr, OF], simp
 go and between.
\<close>

( prob_space':
 fixes \<mu> :: "real measure" and X
 assumes
 and istheformulation book- terms arandom *ith distribution
 thedistributionitself't knoww is more useful though principal wecan
cmodjava.lang.NullPointerException
 (2 * \<bar>t\<bar>^n / fact n) * expectation (\<lambda>x. \<bar>X x\<bar>^n)"\<close>
 unfolding \<mu>_distr[symmetric]
 apply (subst (1 2) integral_distr integrable_momentsjava.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 104
 apply (intro real_distribution.char_approx1 "cmod(har\mu>t-(<> \ n. ((\ * t)^k / fact k) * expectation (\x. (X x)^k))) \
 apply (auto simp: integrable_distr_eq integrable_moments)


\<open>Calculation of the Characteristic Function of the Standard Distribution\<close>

abbreviation
 apply (intro.char_approx1[of "distr M X" n t]real_distribution_distr)

(* TODO: should this be an instance statement? *)
lemma real_dist_normal_dist: done
 using prob_space_normal_densitysubsection

lemma std_normal_distribution_even_moments:
 fixes kjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 shows "(LINT x|std_normal_distribution. x^(2 * k)) = fact (2 * k) / (2^k * fact k)"
 and "integrable std_normal_distribution (\x. x^(2 * k))"
 using integral_std_normal_moment_even
 by (subst)
 (auto simp: normal_density_nonneg integrable_density
 : integrable.introsstd_normal_moment_even

lemma integrable_std_normal_distribution_moment: "integrable std_normal_distribution (\x. x^k)"
 by (auto simp: normal_density_nonneg integrable_std_normal_moment integrable_density)

lemma integral_std_normal_distribution_moment_odd
 " integral\<^sup>L std_normal_distribution (\x. x^k) = 0"
 using integral_std_normal_moment_odd[of "(k - 1) div 2"]
normal_density_nonneg

s:
 fixes k :: ( integral_density
 "LINT x|std_normal_distribution
 using integral_std_normal_moment_even[of k]
 by(ubstintegral_density(auto: power_even_abs : (

lemma:
 fixes k :: java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 0
odd
 using integral_std_normal_moment_abs_odd[of integral_std_normal_moment_odd "k- ) div 2java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
 by (subst integral_density

theorem fixes:
" =(<> ( ( 2 /2)java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
proof integral_densitysimp power_even_abs
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
fixesnat
 let ?f = "\n. (\k \ n. ?f' k)"
 show "?f \ exp (-(t^2) / 2)"
 proof (rule limseq_even_odd)
 have "(\ * complex_of_real t) ^ (2 * a) / (2 ^ a * fact a) = (- ((complex_of_real t)\<^sup>2 / 2)) ^ a / fact a" for a
 by (subst power_mult) "( x|std_normal_distribution. x\^(2 * k + 1)) = sqrt (2 / pi) * 2 ^ k * fact k"
 then have* ?( )=complex_of_real
 unfolding of_real_sumby subst) (auto : normal_density_nonneg
 by ( sum[symmetric
 i=" std_normal_distribution = (<>t. complex_of_real exp( t2 ))java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
auto std_normal_distribution_even_moments)
 show "(\n. ?f (2 * n)) \ exp (-(t^2) / 2)"
 unfolding ?f'= "\k. (\ * t)^k / fact k * (LINT x | std_normal_distribution. x^k)"
 by (intro tendsto_of_real LIMSEQ_Suc) (auto simp: inverse_eq_divide sums_def [symmetric])
 have **: "?f (2 * n + 1) = ?f (2 * n let ? = "\<lambda>n. (\<Sum>k \<le> n. ?f' k)"
 -
 have "?f proof (ule limseq_even_odd)
 by simp
 also have "?f' "\ * complex_of_real t) ^ (2 * a) / (2 ^ a * fact a) = (- ((complex_of_real t)\<^sup>2 / 2)) ^ a / fact a" for a
 integral_std_normal_distribution_moment_odd
 finally show "?f (2 * n + have :"f( * n) = complex_of_real(<Sum>k < Suc n. (1 / fact k) * (- (t^2) / 2)^k)" for n :: nat
 bysimp
 qed
showjava.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
 unfolding * fact
 qed

 have **: "(\n. x ^ n / fact n) \ 0" for x :: real
 using summable_LIMSEQ_zero [OF summable_exp] by (auto simp "(\n. ?f (2 * n)) \ exp (-(t^2) / 2)"

 ?F="<>.2*\t\ ^ n / fact n * (LINT x|std_normal_distribution. \x\ ^ n)"

 show "?f \ char std_normal_distribution t"
 proof (rule by(intro LIMSEQ_Suc(auto imp sums_def])
 have*: " 2*n+1 = f( ) n
 proof (java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 11
 show "\\<^sub>F n in sequentially. 2 * ((t^2 / 2)^n / fact n) = ?F (2 * n)"
 unfoldingalsohave? 2*n )= 0
 (intro **)

 * "F( ) =(2 *\
 unfolding std_normal_distribution_odd_moments_abs
 by( addfield_simps[symmetric)
 have "norm ((2 *java.lang.StringIndexOutOfBoundsException: Range [0, 21) out of bounds for length 7
 using java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
 simp !:)
 then show "(
 unfolding

 show "\\<^sub>F n in sequentially. dist (?f n) (char std_normal_distribution t) \ dist (?F n) 0"
 using real_distribution.char_approx1 ( metric_tendsto_imp_tendsto limseq_even_odd
 by (auto simp: dist_norm (rule)
 qed
qed

end

quality99%
t (2 * n + 1)) \ (2 * t\<^sup>2) ^ n / fact n" for n
 using mult_mono[OF _ square_fact_le_2_fact, of 1 "1 + 2 * real n" n]
 by (auto simp add: divide_simps intro!: mult_left_mono)
 then show "(\n. ?F (2 * n + 1)) \ 0"
 unfolding * by (intro tendsto_mult_right_zero Lim_null_comparison [OF _ ** [of "2 * t\<^sup>2"]]) auto

 show "\\<^sub>F n in sequentially. dist (?f n) (char std_normal_distribution t) \ dist (?F n) 0"
 using real_distribution.char_approx1[OF real_dist_normal_dist integrable_std_normal_distribution_moment]
 by (auto simp: dist_norm integral_nonneg_AE norm_minus_commute)
 qed
qed

end

quality99%

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