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>

theory java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
 imports Weak_Convergenceby( simp: eventually_sequentially
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 -
 (auto simp eventually_sequentiallyshow ?thesis eventually_sequentially
 byauto simp )
 moreover
 O n_o where]: "\n. n \ n_o \ P (Suc (2 * n))"
 by (auto n assume"max( n_e)( +1 \ n" then show "P n"
 show
 unfoldingjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
intro)
 fix n assume "max (2 * n_e) (2 * n_o + 1) \ n" then show "P n"
 by (cases "even n") (auto elim!: evenE shows <longlonglongrightarrow> l"
 qed
qed

lemma limseq_even_odd:
 assumes "(\n. f (2 * n)) \ (l :: 'a :: topological_space)"
(\<lambda>n. f (2 * n + 1)) \<longlonglongrightarrow> l"
 shows "f \ l"
 usingiexp

subsection

abbreviation iexp :: "real \ complex" where
 "iexp \ (\x. exp (\ * complex_of_real x))"

lemmaisCont_iexp]: "isCont iexp xjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
 by (intro)

lemmahas_vector_derivative_iexp]:
 "iexphas_vector_derivative\ * iexp x) (at x within s)"
 by (auto intro!:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma:
 fixes:real
 shows "(CLBINT x=a..b. iexp x) = \ * iexp a - \ * iexp b"
 by (substinterval_integral_FTC_finite
 (by( interval_integral_FTC_finite[where F ="\x. -\ * iexp x"])

subsection

definition
 char :: "real measure \ real \ complex"
 where

lemma (in "eal measure CLINT x|M. iexp (t * x)"
 lemma real_distribution: "char M 0=1java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54

lemma (in prob_space) integrable_iexp:
f: "f borel_measurable M" "\x. Im (f x) = 0"
 showsshows java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
proof fhave
 fromfhave"\x. of_real (Re (f x)) = f x"
 ( add)
then AE. exp
 using norm_exp_i_times[ofusing[of" (f x"for ]byjava.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
qed (proof

lemma (in real_distribution) cmod_char_le_1: "norm (char M t) \ 1"
proof-
 have "norm (char M t) \ (\x. norm (iexp (t * x)) \M)"
 unfolding by (intro)
 by (simp: of_real_mult show? .
 lemmain) isCont_char:" (char )tjava.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
 finally ?thesis
qed

lemma (in real_distribution) isCont_char
 unfolding
proof
 fix X assume X byauto:borel_measurable_continuous_onIcontinuous_at_imp_continuous_on)
show(
 unfolding comp_def
 by (rule[where=\lambda.")( !:tendsto_introsX)
qed ( prob_space:

lemma X1 :"a
by(introjava.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95

subsectionfrom []: "borel"byelim

(* the automation can probably be improved *)
lemma prob_space:
 fixesby( add integral_distr
 assumes "indep_var borel X1 borel X2"
 shows "char (distr M borel \\. X1 \ + X2 \)) t =
 char (distrMborel) t *char MborelX2t"
proof -
 from assms have " = (CLINT x|M. iexp (t * (X1 x))) * (CLINT x|M. iexp (t * (X2 x)))"
 assmshave[easurablerandom_variable"by(elimindep_var_rv2)

 have "char (distr M borel (\\. X1 \ + X2 \)) t = (CLINT x|M. iexp (t * (X1 x + X2 x)))"
 by (simp add: char_def integral_distr)
 also have "\ = (CLINT x|M. iexp (t * (X1 x)) * iexp (t * (X2 x))) "
 by (simp add: field_simps exp_add)
 also have "\ = (CLINT x|M. iexp (t * (X1 x))) * (CLINT x|M. iexp (t * (X2 x)))"
 by( indep_var_lebesgue_integral[unfolded, assms
 (uto! integrable_iexp
by( add integral_distr
 by simp integral_distr
 finally ?thesis
qed" (\i. borel) X A \

lemma (in prob_space) ( A :)
 indep_vars
 char by( simp: char_distr_add)
( A rule infinite_finite_inductjava.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
 x ::real ::nat
 by (auto simp add: char_distr_add indep_vars_sum "f s
qedshowsx n* )java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42

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

xt\openProofs, page3\<close>

lemma CLBINT_I0c_power_mirror_iexp:
 fixes x :: realby ( derivative_eq_introsauto
 defines "f s m \ complex_of_real ((x - s) ^ m)"
 shows
 let?F= \<>s complex_of_real(-((x - ) ^ ( n / ( n) * iexp
proof
 have 1:
 "(s. complex_of_real(-((x - s) ^ (Suc n) / (Suc n))) * iexp s)
 has_vector_derivative complex_of_real s^)*iexp (\ * iexp s) * complex_of_real(-((x - s) ^ (Suc n) / (Suc n))))
 (atswithin A"fors java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
 by (intro derivative_eq_introsunfolding zero_ereal_def 1

 let ( interval_integral_FTC_finite
havex( n Suc=CLBINT. fsn*iexp (
 proof -
 "?RHS= CLBINT =0x ( s n * s + (\ * iexp s) *
 complex_of_real(-((x - s finallyshow ?thesis
 by cases"0 \ x") (auto intro!: simp: f_def[abs_def])
 also have "... = qed
 unfolding zero_ereal_def using 1
 by (intro interval_integral_FTC_finite)
 (auto ?thesis
 !: continuous_at_imp_continuous_oncontinuous_intros
 finally ?thesis
 by auto
 qed
 show ?thesis
 unfolding \<open>?LHS = ?RHS\<close> f_def interval_lebesgue_integral_mult_right [symmetric]
 by (subst interval_lebesgue_integral_add(2) [symmetric
fixes
qed "fsm\ complex_of_real ((x - s) ^ m)"

 "iexp x =
 fixes x :: real
 defines "f s m \ complex_of_real ((x - s) ^ m)"
 showsx=
 (\<Sum>k \<le> n. (\ * x)^k / (fact k)) + ((\ ^ (Suc n)) / (fact n)) * (CLBINT s=0..x. (f s n) * (iexp s))" (is "?P n")( n)
proof (induction n)
 show "?P 0"
 by ( simpadd:field_simps interval_integral_iexp zero_ereal_def
 ** "a :: real a -b \ a + b = 0"
 b
 *: of_nat fact
 byunfolding[symmetricjava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
 have:" n * of_nat (fact n) - (of_nat (fact n)::complex)"
 unfolding of_nat_mult[symmetric]
 by (simp add: complex_eq_iff ** of_nat_add[symmetric] del: of_nat_mult show"P ih CLBINT_I0c_power_mirror_iexp[ ]
 showP Suc
 unfolding sum.java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 3
bysimpadd_eq_0_iff )
qed

lemma iexp_eq2:
 x :: real
 proof
 shows "iexp x = (\k\Suc n. (\*x)^k/fact k) + \^Suc n/fact n * (CLBINT s=0..x. f s n*(iexp s - 1))"
proof -
 haveisCont_fisCont
 by (auto simp: f_def)
 let? java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
 have : "(CLBINT s=0..x.fsn* (exp s - 1))=
CLBINT * s)( s=0..f s )
 unfolding
 by by ( interval_lebesgue_integral_diff])
 field_simps

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 unfoldingproof interval_integral_FTC_finite a b = f ="<>.fsF = ?)
 proof (subst interval_integral_FTC_finite [where a = 0 and b = x and f = "\s. f s n" and F = ?F])
"Ff n (ywithin{ .0x) y
 unfolding f_def
 intro)
 (auto intro!: derivative_eq_introsbyintro)
qed : continuous_at_imp_continuous_on)

 qed auto: continuous_at_imp_continuous_on)
 ( addfield_simps)

 (imp add)
 unfolding
 (subst [where auto
qed

 by( CLBINT_I0c_power_mirror_iexp n = ]
 java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 "\LBINT s=0..x. $x - s)^n\\ = \LBINT s=0..x. (x - s)^n\"
 "\LBINT s=0..x. \(x - s)^n\\ = \LBINT s=0..x. (x - s)^n\"
cases
 assume" \ x \ even n"
have( s=..\<bar>(x - s)^n\<bar>) = LBINT s=0..x. (x - s)^n"
 by autosimp: power_even_abs min_absorb1
 intro!: simp
 then thesis
 next
 assume then ha "(LBINT s=0.x (x - s)^n$ = LBINT s=0..x. -((x - s)^n)"
 then have "(LBINT s=0..x. $x - s)^n$ = LBINT s=0..x. -((x - s)^n)"
 by (auto simp add: zero_ereal_def power_abs min_absorb1 max_absorb2 (auto simp add: power_abs max_absorb2
 ereal_min[symmetric] ereal_max[symmetric] power_minus_oddsimp: ereal_min ereal_max!: interval_integral_cong
 simp del: ereal_min ereal_max by(subst interval_lebesgue_integral_uminus, rule)
also"
 by (subst interval_lebesgue_integral_uminus, rule refl)
 show by simp
 qed
 also have "LBINT s=0..x. (x - s)^n = x^Suc n / Suc n"
 proof let F ="<>t ((x t)(Suc n) /Suc n)"
 ?="<>t.-(x - )(Suc n) /Suc )"
 have "LBINT s=0..x. (x - s) zero_ereal_def
unfoldingzero_ereal_def
 by (intro interval_integral_FTC_finite continuous_at_imp_continuous_on
 [THEN ])
( simp: power_Suc!:d simp: add_nonneg_eq_0_iff
also "<> =x^(n)/ Suc n) by simp
 finally show showthesisjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
 qed
 finally show ?thesis -
qed

lemma iexp_approx1: "cmod (iexp x - (\k \ n. (\ * x)^k / fact k)) \ \x\^(Suc n) / fact (Suc n)"
proof -
 have"iexp x - (\k \ n. (\ * x)^k / fact k) =
 (<>^Suc) ( n) CLBINTx x-s^ iexp is = ?t2
 by ( then cmod cmod
 then cmod =cmod)java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
 by simp
 also have by (simp: norm_mult norm_divide)
 by (simp add: norm_mult norm_power)
 have"\ \ (1 / of_nat (fact n)) * \LBINT s=0..x. cmod ((x - s)^n * (iexp s))\"
(intro interval_integral_norm2)
 have" \ (1 / of_nat (fact n)) * \LBINT s=0..x. \(x - s)^n\\"
 alsohave" (1 / of_nat (fact n)) * \LBINT s=0..x. \(x - s)^n\\"
 by ((simp: norm_mult delof_real_diffof_real_power
 alsohave" \ (1 / of_nat (fact n)) * \x ^ (Suc n) / (Suc n)\"
by( add: abs_LBINT_I0c_abs_power_diff
 also have "1 / real_of_nat (fact n::nat) * \x ^ Suc n / real (Suc n)\ =
 \<bar>x\<bar> ^ Suc n / fact (Suc n)"
 by (simp add: abs_mult power_abs)
 finallyshowthesis
qed

lemma iexp_approx2: "cmod (iexp x - (\k \ n. (\ * x)^k / fact k)) \ 2 * \x\^n / fact n"
proof ? .

 : java.lang.StringIndexOutOfBoundsException: Index 141 out of bounds for length 141
 \<bar>LBINT s=a..b. f s\<bar> \<le> \<bar>LBINT s=a..b. g s\<bar>"caseSuc
 if f: "\s. 0 \ f s" and g: "\s. f s \ g s" for f g :: "_ \ real"
 using order_trans[OF f g] f g
 nfolding interval_lebesgue_integrable_defset_lebesgue_integral_def
 by (auto order_transOFg] f

 " x - \ Suc n. (\ * x)^k / fact k) =
 ((\ ^ (Suc n)) / (fact n)) * (CLBINT s=0..x. (x - s)^n * (iexp s - 1))" (is "?t1 = ?t2") : integral_nonneg_AE AE_I2!: mult_mono
 unfolding iexp_eq2java.lang.StringIndexOutOfBoundsException: Index 97 out of bounds for length 97
 then have "cmod (?t1 then have "cmod?)=cmod)"
 by
 also have "\ = (1 / (fact n)) * cmod (CLBINT s=0..x. (x - s)^n * (iexp s - 1))"
 by (simp add: norm_mult norm_divide norm_power ( add norm_mult norm_power
 also have "\ \ (1 / (fact n)) * \LBINT s=0..x. cmod ((x - s)^n * (iexp s - 1))\"
 by (intromult_left_mono)
 ( ( intro simp)
 also have "\ = (1 / (fact n)) * \LBINT s=0..x. abs ((x - s)^n) * cmod((iexp s - 1))\"
 by ( add norm_multdel of_real_power
 lso"\ \ (1 / (fact n)) * \LBINT s=0..x. abs ((x - s)^n) * 2\"
byintro *o [OF])
 ( auto: zero_ereal_definterval_integrable_isCont
 add)
 fact
 also power_abs
bysimp: abs_mult)
 finally show ?case .
qedinsert[of" "1,s)

lemma (in real_distribution) char_approx1:
 assumes integrable_moments: "\k. k \ n \ integrable M (\x. x^k)"
 shows
 (*<bar>t\<bar>^n / fact n) * expectation (\<lambda>x. \<bar>x\<bar>^n)" (is "cmod (char M t - ?t1) \<le> _")
proof -
 have "cmod (char M -(\k \ n. ((\ * t)^k / fact k) * expectation (\x. x^k))) \
 by (intro integrable_const_boundauto

 define c where [abs_def]: "c k x = (\ * t)^k / fact k * complex_of_real (x^k)" for k x
 have integ_c: "\k. k \ n \ integrable M (\x. c k x)"
 unfolding c_def( integrable_mult_right integrable_of_real)

 have "k integrable_const_bound auto
 unfolding c where abs_def"c x (\ * t)^k / fact k * complex_of_real (x^k)" for k x
 then c_def( integrable_mult_right integrable_moments
 by (simp " \ n \ expectation (c k) = (\*t) ^ k * (expectation (\x. x ^ k)) / fact k" for k
 also have "\ = norm ((CLINT x | M. iexp (t * x) - (\k \ n. c k x)))"
 unfolding char_def - (M-|.
 also"dots ((CLINTx|M (t *x - \
 byintrojava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
 also have "\ \ expectation (\x. 2 * \t\ ^ n / fact n * \x\ ^ n)"
 proof "
 show "integrable M (\x. cmod (iexp (t * x) - (\k\n. c k x)))"
byintro Bochner_Integration Bochner_Integration integ_c
 show "integrable \\java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
power_abs]
 by (intro integrable_mult_right integrable_absshow \<lambda>x. 2 * \<bar>t\<bar> ^ n / fact n * \<bar>x\<bar> ^ n)" power_abs]
 c ( t*x) \<Sum>k\<le>n. c k x)) \<le> 2 * \<bar>t\<bar> ^ n / fact n * \<bar>x\<bar> ^ n" for x
 iexp_approx2 "] auto simp dd java.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 85
 qed
 finallyshow
 unfolding
qed

lemmaassumes: \And. <
 assumes : "\k. k \ n \ integrable M (\x. x ^ k)"
 shows java.lang.StringIndexOutOfBoundsException: Index 137 out of bounds for length 137
 (\<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))" t1
 (isbyintro) auto
proof -
 have integ_iexp: "integrable M (\x. iexp (t * x))"
 by (intro integrable_const_bound

define []: "c x (<> t)^ k * complex_of_real(^)"for
 integ_c
 unfolding c_def byjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

 have
 \<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
 ( mult_min_right
 apply simp
apply( arg_cong2 f=min
 apply (simp_all add: field_simps abs_mult ( add)
 apply (simp_all


 have"
 c_def integral_complex_of_real
 then have "norm (char M t - ?t1) = have " ( t ) Mt-CLINT
 by also "dots =norm(CLINT .( * )-(k \ n. c k x)))"
 also have "\ = norm ((CLINT x | M. iexp (t * x) - (\k \ n. c k x)))"
 unfolding char_def by (subst Bochner_Integration.integral_diff[OF integ_iexp]) unfolding by (subst.integral_diff integ_iexp( introinteg_c
 also have"\ \ expectation (\x. min (2 * \t * x\^n / fact n) (\t * x\^(Suc n) / fact (Suc n)))"
 by (rule integral_norm_bound (rule)
alsojava.lang.StringIndexOutOfBoundsException: Index 135 out of bounds for length 135
 ( "_ \ expectation ?f")
 proof (ruleshowintegrablejava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
showM(<lambda>x. cmod (iexp (t * x) - (\<Sum>k\<le>n. c k x)))"
 by (intro integrable_norm Bochner_Integration.integrable_diff integ_iexp Bochner_Integration.integrable_sum(auto: integrable_moments power_abssymmetric)
 "integrableM?f"
 by (ruleusing[of "t * x n]iexp_approx2[of "t*x njava.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
 (autosimp: integrable_momentspower_abssymmetricpower_mult_distrib)
 show "cmod (iexp qed
 using iexp_approx1[of "t * x" n] iexp_approx2[of "t * x" n]
 by "\ = (\t\^n / fact (Suc n)) * expectation (\x. min (2 * \x\^n * Suc n) (\t\ * \x\^Suc n))"
 proof (rule Bochner_Integration.integral_mult_right

 *
 proof (rule Bochner_Integration.integral_mult_right)
 show "integrable M (\x. min (2 * \x\ ^ n * real (Suc n)) (\t\ * \x\ ^ Suc n))"
byrule.integrable_boundwhere\<lambda>x. 2 * \<bar>x\<bar> ^ n * real (Suc n)"])
 auto:i power_abs] power_mult_distrib
 qed
 finally show ?thesis
 unfolding .
qed

lemma (in
 fixes
 assumes
 integrable_1 integrable(
 integral_1: "java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 9
 integrable_2: : "variance \2"
 integral_2 (\<lambda>x. x) = \<sigma>2"
M t 1-^ \<sigma>2 / 2)) \<le>
 ( real_distribution [of ,simplified]
proof-
 notef integral_2 [simp] expectation
havesimpprob"by metisprob_spacespace_eq_univ)
 from integral_2 have [simp]: "expectation (\x. x * x) = \2"
 simp java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
 havejava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
 usingassms auto: eval_nat_numeral)
 note char_approx1
 note 2 = char_approx1using [oft 1] byjava.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
also havefact"bysimpadd eval_nat_numeral)
also have t\<^sup>2 * expectation (\<lambda>x. min (6 * x\<^sup>2) (\<bar>t\<bar> * \<bar>x\<bar> ^ 3)) / 6 =
 using char_approx2 [of 2 t OF by simp
 also have "(\k\2. (\ * t) ^ k * expectation (\x. x ^ k) / (fact k)) = 1 - t^2 * \2 / 2"
 by (simp show ?thesis
 alsojava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 also have "\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)\<close>
 finally ?thesis
qed

text
 This [simp" M "" \java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
 we and\mu_: \mu distrX"
\<close>

lemmain) char_approx3
 fixes \<mu> :: "real measure" and X
 assumes [simprandom_variable
 and [applyintro.char_approx3
 var_XX <sigma>2"
 and \<mu>_def: "\<mu> = distr M borel X"
 shows "cmod (char \ t - (1 - t^2 * \2 / 2)) \
 (t^2 / 6) * expectation (\<lambda>x. min (6 * (X x)^2) (\<bar>t\<bar> * \<bar>X x\<bar>^3))"
 usingvar_X \<mu>_def
applysubst [symmetric rv_X)
 applygoback forth them
 lemmain) char_approx1
 done

n>
 this the inthe --in of variable** the,
rather distribution . I don hich,in java.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95
 go back and forth shows" (char t - (\k \ n. ((\ * t)^k / fact k) * expectation (\x. (X x)^k))) \
\<close>

lemma (in prob_space) char_approx1':
 fixes \<mu> :: "real measure" and X
 assumes : "k. k \ n \ integrable M (\x. X x ^ k)"
 and rv_X[measurable]: "random_variable borel X"
 and \<mu>_distr : "distr M borel X = \<mu>"
 shows ( <> \Sumk\<le> n. ((\ * t)^k / fact k) * expectation (\<lambda>x. (X x)^k))) \<le>
 (done
 subsection
 apply (
 real_distributionborel] rv_X
(* TODO: should this be an instance statement? *)
d

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

abbreviation
 "std_normal_distribution \ density lborel std_normal_density"

(* TODO: should this be an instance statement? *) integral_density
intro )
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma:
 fixes k :: nat
 shows "(LINT x|std_normal_distribution. x^(2 * k)) = fact (2 *oddk\java.lang.StringIndexOutOfBoundsException: Index 90 out of bounds for length 90
 and "integrable by (auto simp: integral_density normal_density_nonneg elim: oddE)
 using lemma std_normal_distribution_even_moments_ab
 bysubst)
 (auto simp: normal_density_nonnegshows(LINT. \<bar>x\<bar>^(2 * k)) = fact (2 * k) / (2^k * fact k)"
 intro (ubst ) simpnormal_density_nonneg)

lemmaintegrable_std_normal_distribution_moment"integrablestd_normal_distribution (\x. x^k)"
 by (auto simp std_normal_distribution_odd_moments_abs

lemma integral_std_normal_distribution_moment_odd:
 " k \ integral\<^sup>L std_normal_distribution (\x. x^k) = 0"
 using[of( -1 2]
 by (auto simp: integral_density normal_density_nonneg elim: oddE)

lemma std_normal_distribution_even_moments_abs
fixes k : nat
 shows "( char std_normal_distribution = \lambda>t.complex_of_real exp (-(t^) / ))"
 using integral_std_normal_moment_even[of k]
 by (subst) (auto simp: normal_density_nonneg)

lemma std_normal_distribution_odd_moments_abs:
 k :: java.lang.StringIndexOutOfBoundsException: Range [16, 17) out of bounds for length 16
showsLINT.\<
 using integral_std_normal_moment_abs_odd *"f 2*n (\k < Suc n. (1 / fact k) * (- (t^2) / 2)^k)" for n :: nat
 by( integral_density(autosimp)

theorem(intro.reindex_bij_witness_not_neutral, where
char(lambda( -(^) /2)"
proof (intro ext LIMSEQ_unique)
 fix t ( simp: integral_std_normal_distribution_moment_odd
let'java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
f=\<lambda>n. (\<Sum>k \<le> n. ?f' k)"
 showproof
 ( java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
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) (simp add by(subst) simp_all
then* ? 2n java.lang.NullPointerException
 unfolding
 by "(n. ?f (2 * n + 1)) \ exp (-(t^2) / 2)"
 i="\n. n div 2" and j="\n. 2 * n" and T'="{i. i \ 2 * n \ odd i}" and S'="{}"])
 (auto simp: integral_std_normal_distribution_moment_odd unfolding*byjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
showjava.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
 unfolding * using let \lambdan <bar>t\<bar> ^ n / fact n * (LINT x|std_normal_distribution. \<bar>x\<bar> ^ n)"
( tendsto_of_real) (autos: inverse_eq_divide [symmetric
 *:"f( )=? 2*n"for
 proof -
 have "?f (2 * n + 1) = ?f (2 * n) + ?f' (2 * n + 1)"
 by simp
 "?'(2 +1 = "
 by qed tendsto_mult_right_zero
 finallyhave:? 2*n+1 2 <bar>t\<bar> * sqrt (2 / pi)) * ((2 * t^2)^n * fact n / fact (2 * n + 1))" for n
 by simp (simp: power_mult] power_even_abs
 qed
 show "(\n. ?f (2 * n + 1)) \ exp (-(t^2) / 2)"
 unfolding ** by fact
 qed

 have **: "(\n. x ^ n / fact n) \ 0" for x :: real
 using summable_LIMSEQ_zero [OF summable_exp] by (auto simp by(autosimp add: divide_simpsintro mult_left_mono

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

 show "?f java.lang.StringIndexOutOfBoundsException: Range [0, 63) out of bounds for length 0
 proofrule[OF])
 show "(\n. ?F (2 * n)) \ 0"
proof Lim_transform_eventually
 show
 unfolding std_normal_distribution_even_moments_abs
 qed (intro

 have *: "?F (2 * n + 1) = (2 * \t\ * sqrt (2 / pi)) * ((2 * t^2)^n * fact n / fact (2 * n + 1))" for n
 unfolding std_normal_distribution_odd_moments_abs
 by (simp add: field_simps power_mult[symmetric] power_even_abs)
 have "norm ((2 * t\<^sup>2) ^ n * fact n / fact (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

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