# Theory Sinc_Integral

```(*  Title:     HOL/Probability/Sinc_Integral.thy
Authors:   Jeremy Avigad (CMU), Luke Serafin (CMU)
Authors:   Johannes Hölzl, TU München
*)

section ‹Integral of sinc›

theory Sinc_Integral
imports Distributions
begin

subsection ‹Various preparatory integrals›

text ‹ Naming convention
The theorem name consists of the following parts:
▪ Kind of integral: ‹has_bochner_integral› / ‹integrable› / ‹LBINT›
▪ Interval: Interval (0 / infinity / open / closed) (infinity / open / closed)
▪ Name of the occurring constants: power, exp, m (for minus), scale, sin, \$\ldots\$
›

lemma has_bochner_integral_I0i_power_exp_m':
"has_bochner_integral lborel (λx. x^k * exp (-x) * indicator {0 ..} x::real) (fact k)"
using nn_intergal_power_times_exp_Ici[of k]
by (intro has_bochner_integral_nn_integral)
(auto simp: nn_integral_set_ennreal split: split_indicator)

lemma has_bochner_integral_I0i_power_exp_m:
"has_bochner_integral lborel (λx. x^k * exp (-x) * indicator {0 <..} x::real) (fact k)"
using AE_lborel_singleton[of 0]
by (intro has_bochner_integral_cong_AE[THEN iffD1, OF _ _ _ has_bochner_integral_I0i_power_exp_m'])
(auto split: split_indicator)

lemma integrable_I0i_exp_mscale: "0 < (u::real) ⟹ set_integrable lborel {0 <..} (λx. exp (-(x * u)))"
using lborel_integrable_real_affine_iff[of u "λx. indicator {0 <..} x *⇩R exp (- x)" 0]
has_bochner_integral_I0i_power_exp_m[of 0]
by (simp add: indicator_def zero_less_mult_iff mult_ac integrable.intros set_integrable_def)

lemma LBINT_I0i_exp_mscale: "0 < (u::real) ⟹ LBINT x=0..∞. exp (-(x * u)) = 1 / u"
using lborel_integral_real_affine[of u "λx. indicator {0<..} x *⇩R exp (- x)" 0]
has_bochner_integral_I0i_power_exp_m[of 0]
by (auto simp: indicator_def zero_less_mult_iff interval_lebesgue_integral_0_infty set_lebesgue_integral_def field_simps
dest!: has_bochner_integral_integral_eq)

lemma LBINT_I0c_exp_mscale_sin:
"LBINT x=0..t. exp (-(u * x)) * sin x =
(1 / (1 + u^2)) * (1 - exp (-(u * t)) * (u * sin t + cos t))" (is "_ = ?F t")
unfolding zero_ereal_def
proof (subst interval_integral_FTC_finite)
show "(?F has_vector_derivative exp (- (u * x)) * sin x) (at x within {min 0 t..max 0 t})" for x
by (auto intro!: derivative_eq_intros
simp: has_real_derivative_iff_has_vector_derivative[symmetric] power2_eq_square)
qed (auto intro: continuous_at_imp_continuous_on)

lemma LBINT_I0i_exp_mscale_sin:
assumes "0 < x"
shows "LBINT u=0..∞. ¦exp (-u * x) * sin x¦ = ¦sin x¦ / x"
proof (subst interval_integral_FTC_nonneg)
let ?F = "λu. 1 / x * (1 - exp (- u * x)) * ¦sin x¦"
show "⋀t. (?F has_real_derivative ¦exp (- t * x) * sin x¦) (at t)"
using ‹0 < x› by (auto intro!: derivative_eq_intros simp: abs_mult)
show "((?F ∘ real_of_ereal) ⤏ 0) (at_right 0)"
using ‹0 < x› by (auto simp: zero_ereal_def ereal_tendsto_simps intro!: tendsto_eq_intros)
have *: "((λt. exp (- t * x)) ⤏ 0) at_top"
using ‹0 < x›
by (auto intro!: exp_at_bot[THEN filterlim_compose] filterlim_tendsto_pos_mult_at_top filterlim_ident
simp: filterlim_uminus_at_bot mult.commute[of _ x])
show "((?F ∘ real_of_ereal) ⤏ ¦sin x¦ / x) (at_left ∞)"
using ‹0 < x› unfolding ereal_tendsto_simps
by (intro filterlim_compose[OF _ *]) (auto intro!: tendsto_eq_intros filterlim_ident)
qed auto

lemma
shows integrable_inverse_1_plus_square:
"set_integrable lborel (einterval (-∞) ∞) (λx. inverse (1 + x^2))"
and LBINT_inverse_1_plus_square:
"LBINT x=-∞..∞. inverse (1 + x^2) = pi"
proof -
have 1: "- (pi / 2) < x ⟹ x * 2 < pi ⟹ (tan has_real_derivative 1 + (tan x)⇧2) (at x)" for x
using cos_gt_zero_pi[of x] by (subst tan_sec) (auto intro!: DERIV_tan simp: power_inverse)
have 2: "- (pi / 2) < x ⟹ x * 2 < pi ⟹ isCont (λx. 1 + (tan x)⇧2) x" for x
using cos_gt_zero_pi[of x] by auto
have 3: "⟦- (pi / 2) < x; x * 2 < pi⟧  ⟹ isCont (λx. inverse (1 + x⇧2)) (tan x)" for x
show "LBINT x=-∞..∞. inverse (1 + x^2) = pi"
by (subst interval_integral_substitution_nonneg[of "-pi/2" "pi/2" tan "λx. 1 + (tan x)^2"])
(auto intro: derivative_eq_intros 1 2 3 filterlim_tan_at_right
show "set_integrable lborel (einterval (-∞) ∞) (λx. inverse (1 + x^2))"
by (subst interval_integral_substitution_nonneg[of "-pi/2" "pi/2" tan "λx. 1 + (tan x)^2"])
(auto intro: derivative_eq_intros 1 2 3 filterlim_tan_at_right
qed

lemma
shows integrable_I0i_1_div_plus_square:
"interval_lebesgue_integrable lborel 0 ∞ (λx. 1 / (1 + x^2))"
and LBINT_I0i_1_div_plus_square:
"LBINT x=0..∞. 1 / (1 + x^2) = pi / 2"
proof -
have 1: "0 < x ⟹ x * 2 < pi ⟹ (tan has_real_derivative 1 + (tan x)⇧2) (at x)" for x
using cos_gt_zero_pi[of x] by (subst tan_sec) (auto intro!: DERIV_tan simp: power_inverse)
have 2: "0 < x ⟹ x * 2 < pi ⟹ isCont (λx. 1 + (tan x)⇧2) x" for x
using cos_gt_zero_pi[of x] by auto
show "LBINT x=0..∞. 1 / (1 + x^2) = pi / 2"
by (subst interval_integral_substitution_nonneg[of "0" "pi/2" tan "λx. 1 + (tan x)^2"])
(auto intro: derivative_eq_intros 1 2 tendsto_eq_intros
show "interval_lebesgue_integrable lborel 0 ∞ (λx. 1 / (1 + x^2))"
unfolding interval_lebesgue_integrable_def
by (subst interval_integral_substitution_nonneg[of "0" "pi/2" tan "λx. 1 + (tan x)^2"])
(auto intro: derivative_eq_intros 1 2 tendsto_eq_intros
qed

section ‹The sinc function, and the sine integral (Si)›

abbreviation sinc :: "real ⇒ real" where
"sinc ≡ (λx. if x = 0 then 1 else sin x / x)"

lemma sinc_at_0: "((λx. sin x / x::real) ⤏ 1) (at 0)"
using DERIV_sin [of 0] by (auto simp add: has_field_derivative_def field_has_derivative_at)

lemma isCont_sinc: "isCont sinc x"
proof cases
assume "x = 0" then show ?thesis
using LIM_equal [where g = "λx. sin x / x" and a=0 and f=sinc and l=1]
by (auto simp: isCont_def sinc_at_0)
next
assume "x ≠ 0" show ?thesis
by (rule continuous_transform_within [where d = "abs x" and f = "λx. sin x / x"])
(auto simp add: dist_real_def ‹x ≠ 0›)
qed

lemma continuous_on_sinc[continuous_intros]:
"continuous_on S f ⟹ continuous_on S (λx. sinc (f x))"
using continuous_on_compose[of S f sinc, OF _ continuous_at_imp_continuous_on]
by (auto simp: isCont_sinc)

lemma borel_measurable_sinc[measurable]: "sinc ∈ borel_measurable borel"
by (intro borel_measurable_continuous_onI continuous_at_imp_continuous_on ballI isCont_sinc)

lemma sinc_AE: "AE x in lborel. sin x / x = sinc x"
by (rule AE_I [where N = "{0}"], auto)

definition Si :: "real ⇒ real" where "Si t ≡ LBINT x=0..t. sin x / x"

lemma sinc_neg [simp]: "sinc (- x) = sinc x"
by auto

(* TODO: Determine whether this can reasonably be eliminated by redefining Si. *)
lemma Si_alt_def : "Si t = LBINT x=0..t. sinc x"
proof cases
assume "0 ≤ t" then show ?thesis
using AE_lborel_singleton[of 0]
by (auto simp: Si_def intro!: interval_lebesgue_integral_cong_AE)
next
assume "¬ 0 ≤ t" then show ?thesis
unfolding Si_def using AE_lborel_singleton[of 0]
by (subst (1 2) interval_integral_endpoints_reverse)
(auto simp: Si_def intro!: interval_lebesgue_integral_cong_AE)
qed

lemma Si_neg:
assumes "T ≥ 0" shows "Si (- T) = - Si T"
proof -
have "LBINT x=ereal 0..T. -1 *⇩R sinc (- x) = LBINT y= ereal (- 0)..ereal (- T). sinc y"
by (rule interval_integral_substitution_finite [OF assms])
(auto intro: derivative_intros continuous_at_imp_continuous_on isCont_sinc)
also have "(LBINT x=ereal 0..T. -1 *⇩R sinc (- x)) = -(LBINT x=ereal 0..T. sinc x)"
by (subst sinc_neg) (simp_all add: interval_lebesgue_integral_uminus)
finally have *: "-(LBINT x=ereal 0..T. sinc x) = LBINT y= ereal 0..ereal (- T). sinc y"
by simp
show ?thesis
using assms unfolding Si_alt_def
by (subst zero_ereal_def)+ (auto simp add: * [symmetric])
qed

lemma integrable_sinc':
"interval_lebesgue_integrable lborel (ereal 0) (ereal T) (λt. sin (t * θ) / t)"
proof -
have *: "interval_lebesgue_integrable lborel (ereal 0) (ereal T) (λt. θ * sinc (t * θ))"
by (intro interval_lebesgue_integrable_mult_right interval_integrable_isCont continuous_within_compose3 [OF isCont_sinc])
auto
show ?thesis
by (rule interval_lebesgue_integrable_cong_AE[THEN iffD1, OF _ _ _ *])
(insert AE_lborel_singleton[of 0], auto)
qed

(* TODO: need a better version of FTC2 *)
lemma DERIV_Si: "(Si has_real_derivative sinc x) (at x)"
proof -
have "(at x within {min 0 (x - 1)..max 0 (x + 1)}) = at x"
by (intro at_within_interior) auto
moreover have "min 0 (x - 1) ≤ x" "x ≤ max 0 (x + 1)"
by auto
ultimately show ?thesis
using interval_integral_FTC2[of "min 0 (x - 1)" 0 "max 0 (x + 1)" sinc x]
by (auto simp: continuous_at_imp_continuous_on isCont_sinc Si_alt_def[abs_def] zero_ereal_def
has_real_derivative_iff_has_vector_derivative
split del: if_split)
qed

lemma isCont_Si: "isCont Si x"
using DERIV_Si by (rule DERIV_isCont)

lemma borel_measurable_Si[measurable]: "Si ∈ borel_measurable borel"
by (auto intro: isCont_Si continuous_at_imp_continuous_on borel_measurable_continuous_onI)

lemma Si_at_top_LBINT:
"((λt. (LBINT x=0..∞. exp (-(x * t)) * (x * sin t + cos t) / (1 + x^2))) ⤏ 0) at_top"
proof -
let ?F = "λx t. exp (- (x * t)) * (x * sin t + cos t) / (1 + x⇧2) :: real"
have int: "set_integrable lborel {0<..} (λx. exp (- x) * (x + 1) :: real)"
unfolding distrib_left
using has_bochner_integral_I0i_power_exp_m[of 0] has_bochner_integral_I0i_power_exp_m[of 1]
by (intro set_integral_add) (auto dest!: integrable.intros simp: ac_simps set_integrable_def)

have "((λt::real. LBINT x:{0<..}. ?F x t) ⤏ (LBINT x::real:{0<..}. 0)) at_top"
unfolding set_lebesgue_integral_def
proof (rule integral_dominated_convergence_at_top[OF _ _ int [unfolded set_integrable_def]],
simp_all del: abs_divide split: split_indicator)
have *: "0 < x ⟹ ¦x * sin t + cos t¦ / (1 + x⇧2) ≤ (x * 1 + 1) / 1" for x t :: real
by (intro frac_le abs_triangle_ineq[THEN order_trans] add_mono)
(auto simp add: abs_mult simp del: mult_1_right intro!: mult_mono)
then have "1 ≤ t ⟹ 0 < x ⟹ ¦?F x t¦ ≤ exp (- x) * (x + 1)" for x t :: real
by (auto simp add: abs_mult times_divide_eq_right[symmetric] simp del: times_divide_eq_right
intro!:  mult_mono)
then show "∀⇩F i in at_top. AE x in lborel. 0 < x ⟶ ¦?F x i¦ ≤ exp (- x) * (x + 1)"
by (auto intro: eventually_mono eventually_ge_at_top[of "1::real"])
show "AE x in lborel. 0 < x ⟶ (?F x ⤏ 0) at_top"
proof (intro always_eventually impI allI)
fix x :: real assume "0 < x"
show "((λt. exp (- (x * t)) * (x * sin t + cos t) / (1 + x⇧2)) ⤏ 0) at_top"
proof (rule Lim_null_comparison)
show "∀⇩F t in at_top. norm (?F x t) ≤ exp (- (x * t)) * (x + 1)"
using eventually_ge_at_top[of "1::real"] * ‹0 < x›
by (auto simp add: abs_mult times_divide_eq_right[symmetric] simp del: times_divide_eq_right
intro!: mult_mono elim: eventually_mono)
show "((λt. exp (- (x * t)) * (x + 1)) ⤏ 0) at_top"
by (auto simp: filterlim_uminus_at_top [symmetric]
intro!: filterlim_tendsto_pos_mult_at_top[OF tendsto_const] ‹0<x› filterlim_ident
tendsto_mult_left_zero filterlim_compose[OF exp_at_bot])
qed
qed
qed
then show "((λt. (LBINT x=0..∞. exp (-(x * t)) * (x * sin t + cos t) / (1 + x^2))) ⤏ 0) at_top"
qed

lemma Si_at_top_integrable:
assumes "t ≥ 0"
shows "interval_lebesgue_integrable lborel 0 ∞ (λx. exp (- (x * t)) * (x * sin t + cos t) / (1 + x⇧2))"
using ‹0 ≤ t› unfolding le_less
proof
assume "0 = t" then show ?thesis
using integrable_I0i_1_div_plus_square by simp
next
assume [arith]: "0 < t"
have *: "0 ≤ a ⟹ 0 < x ⟹ a / (1 + x⇧2) ≤ a" for a x :: real
using zero_le_power2[of x, arith] using divide_left_mono[of 1 "1+x⇧2" a] by auto
have "set_integrable lborel {0<..} (λx. (exp (- x) * x) * (sin t/t) + exp (- x) * cos t)"
using has_bochner_integral_I0i_power_exp_m[of 0] has_bochner_integral_I0i_power_exp_m[of 1]
(auto dest!: integrable.intros simp: ac_simps set_integrable_def)
from lborel_integrable_real_affine[OF this [unfolded set_integrable_def], of t 0]
show ?thesis
unfolding interval_lebesgue_integral_0_infty set_integrable_def
by (rule Bochner_Integration.integrable_bound) (auto simp: field_simps * split: split_indicator)
qed

lemma Si_at_top: "(Si ⤏ pi / 2) at_top"
proof -
have †: "∀⇩F t in at_top. pi / 2 - (LBINT u=0..∞. exp (-(u * t)) * (u * sin t + cos t) / (1 + u^2)) = Si t"
using eventually_ge_at_top[of 0]
proof eventually_elim
fix t :: real assume "t ≥ 0"
have "Si t = LBINT x=0..t. sin x * (LBINT u=0..∞. exp (-(u * x)))"
unfolding Si_def using ‹0 ≤ t›
by (intro interval_integral_discrete_difference[where X="{0}"]) (auto simp: LBINT_I0i_exp_mscale)
also have "… = (LBINT x. (LBINT u=ereal 0..∞. indicator {0 <..< t} x *⇩R sin x * exp (-(u * x))))"
using ‹0≤t› by (simp add: zero_ereal_def interval_lebesgue_integral_le_eq mult_ac set_lebesgue_integral_def)
also have "… = (LBINT x. (LBINT u. indicator ({0<..} × {0 <..< t}) (u, x) *⇩R (sin x * exp (-(u * x)))))"
apply (subst interval_integral_Ioi)
unfolding set_lebesgue_integral_def  by(simp_all add: indicator_times ac_simps )
also have "… = (LBINT u. (LBINT x. indicator ({0<..} × {0 <..< t}) (u, x) *⇩R (sin x * exp (-(u * x)))))"
proof (intro lborel_pair.Fubini_integral[symmetric] lborel_pair.Fubini_integrable)
show "(λ(x, y). indicator ({0<..} × {0<..<t}) (y, x) *⇩R (sin x * exp (- (y * x))))
∈ borel_measurable (lborel ⨂⇩M lborel)" (is "?f ∈ borel_measurable _")
by measurable

have "AE x in lborel. indicator {0..t} x *⇩R abs (sinc x) = (LBINT y. norm (?f (x, y)))"
using AE_lborel_singleton[of 0] AE_lborel_singleton[of t]
proof eventually_elim
fix x :: real assume x: "x ≠ 0" "x ≠ t"
have "(LBINT y. ¦indicator ({0<..} × {0<..<t}) (y, x) *⇩R (sin x * exp (- (y * x)))¦) =
(LBINT y. ¦sin x¦ * exp (- (y * x)) * indicator {0<..} y * indicator {0<..<t} x)"
by (intro Bochner_Integration.integral_cong) (auto split: split_indicator simp: abs_mult)
also have "… = ¦sin x¦ * indicator {0<..<t} x * (LBINT y=0..∞.  exp (- (y * x)))"
by (cases "x > 0")
(auto simp: field_simps interval_lebesgue_integral_0_infty set_lebesgue_integral_def split: split_indicator)
also have "… = ¦sin x¦ * indicator {0<..<t} x * (1 / x)"
by (cases "x > 0") (auto simp add: LBINT_I0i_exp_mscale)
also have "… = indicator {0..t} x *⇩R ¦sinc x¦"
using x by (simp add: field_simps split: split_indicator)
finally show "indicator {0..t} x *⇩R abs (sinc x) = (LBINT y. norm (?f (x, y)))"
by simp
qed
moreover have "integrable lborel (λx. indicat_real {0..t} x *⇩R ¦sinc x¦)"
by (auto intro!: borel_integrable_compact continuous_intros simp del: real_scaleR_def)
ultimately show "integrable lborel (λx. LBINT y. norm (?f (x, y)))"
by (rule integrable_cong_AE_imp[rotated 2]) simp

have "0 < x ⟹ set_integrable lborel {0<..} (λy. sin x * exp (- (y * x)))" for x :: real
by (intro set_integrable_mult_right integrable_I0i_exp_mscale)
then show "AE x in lborel. integrable lborel (λy. ?f (x, y))"
by (intro AE_I2) (auto simp: indicator_times set_integrable_def split: split_indicator)
qed
also have "... = (LBINT u=0..∞. (LBINT x=0..t. exp (-(u * x)) * sin x))"
using ‹0≤t›
by (auto simp: interval_lebesgue_integral_def set_lebesgue_integral_def zero_ereal_def ac_simps
split: split_indicator intro!: Bochner_Integration.integral_cong)
also have "… = (LBINT u=0..∞. 1 / (1 + u⇧2) - 1 / (1 + u⇧2) * (exp (- (u * t)) * (u * sin t + cos t)))"
by (auto simp: divide_simps LBINT_I0c_exp_mscale_sin intro!: interval_integral_cong)
also have "... = pi / 2 - (LBINT u=0..∞. exp (- (u * t)) * (u * sin t + cos t) / (1 + u^2))"
using Si_at_top_integrable[OF ‹0≤t›] by (simp add: integrable_I0i_1_div_plus_square LBINT_I0i_1_div_plus_square)
finally show "pi / 2 - (LBINT u=0..∞. exp (-(u * t)) * (u * sin t + cos t) / (1 + u^2)) = Si t" ..
qed
show ?thesis
by (rule Lim_transform_eventually [OF _ †])
(auto intro!: tendsto_eq_intros Si_at_top_LBINT)
qed

subsection ‹The final theorems: boundedness and scalability›

lemma bounded_Si: "∃B. ∀T. ¦Si T¦ ≤ B"
proof -
have *: "0 ≤ y ⟹ dist x y < z ⟹ abs x ≤ y + z" for x y z :: real

have "eventually (λT. dist (Si T) (pi / 2) < 1) at_top"
using Si_at_top by (elim tendstoD) simp
then have "eventually (λT. 0 ≤ T ∧ ¦Si T¦ ≤ pi / 2 + 1) at_top"
using eventually_ge_at_top[of 0] by eventually_elim (insert *[of "pi/2" "Si _" 1], auto)
then have "∃T. 0 ≤ T ∧ (∀t ≥ T. ¦Si t¦ ≤ pi / 2 + 1)"
then obtain T where T: "0 ≤ T" "⋀t. t ≥ T ⟹ ¦Si t¦ ≤ pi / 2 + 1"
by auto
moreover have "t ≤ - T ⟹ ¦Si t¦ ≤ pi / 2 + 1" for t
using T(1) T(2)[of "-t"] Si_neg[of "- t"] by simp
moreover have "∃M. ∀t. -T ≤ t ∧ t ≤ T ⟶ ¦Si t¦ ≤ M"
by (rule isCont_bounded) (auto intro!: isCont_Si continuous_intros ‹0 ≤ T›)
then obtain M where M: "⋀t. -T ≤ t ∧ t ≤ T ⟹ ¦Si t¦ ≤ M"
by auto
ultimately show ?thesis
by (intro exI[of _ "max M (pi/2 + 1)"]) (meson le_max_iff_disj linorder_not_le order_le_less)
qed

lemma LBINT_I0c_sin_scale_divide:
assumes "T ≥ 0"
shows "LBINT t=0..T. sin (t * θ) / t = sgn θ * Si (T * ¦θ¦)"
proof -
{ assume "0 < θ"
have "(LBINT t=ereal 0..T. sin (t * θ) / t) = (LBINT t=ereal 0..T. θ *⇩R sinc (t * θ))"
by (rule interval_integral_discrete_difference[of "{0}"]) auto
also have "… = (LBINT t=ereal (0 * θ)..T * θ. sinc t)"
apply (rule interval_integral_substitution_finite [OF assms])
apply (subst mult.commute, rule DERIV_subset)
by (auto intro!: derivative_intros continuous_at_imp_continuous_on isCont_sinc)
also have "… = (LBINT t=ereal (0 * θ)..T * θ. sin t / t)"
by (rule interval_integral_discrete_difference[of "{0}"]) auto
finally have "(LBINT t=ereal 0..T. sin (t * θ) / t) = (LBINT t=ereal 0..T * θ. sin t / t)"
by simp
hence "(LBINT x. indicator {0<..<T} x * sin (x * θ) / x) = (LBINT x. indicator {0<..<T * θ} x * sin x / x)"
using assms ‹0 < θ› unfolding interval_lebesgue_integral_def einterval_eq zero_ereal_def
by (auto simp: ac_simps set_lebesgue_integral_def)
} note aux1 = this
{ assume "0 > θ"
have [simp]: "integrable lborel (λx. sin (x * θ) * indicator {0<..<T} x / x)"
using integrable_sinc' [of T θ] assms
by (simp add: interval_lebesgue_integrable_def set_integrable_def ac_simps)
have "(LBINT t=ereal 0..T. sin (t * -θ) / t) = (LBINT t=ereal 0..T. -θ *⇩R sinc (t * -θ))"
by (rule interval_integral_discrete_difference[of "{0}"]) auto
also have "… = (LBINT t=ereal (0 * -θ)..T * -θ. sinc t)"
apply (rule interval_integral_substitution_finite [OF assms])
apply (subst mult.commute, rule DERIV_subset)
by (auto intro!: derivative_intros continuous_at_imp_continuous_on isCont_sinc)
also have "… = (LBINT t=ereal (0 * -θ)..T * -θ. sin t / t)"
by (rule interval_integral_discrete_difference[of "{0}"]) auto
finally have "(LBINT t=ereal 0..T. sin (t * -θ) / t) = (LBINT t=ereal 0..T * -θ. sin t / t)"
by simp
hence "(LBINT x. indicator {0<..<T} x * sin (x * θ) / x) =
- (LBINT x. indicator {0<..<- (T * θ)} x * sin x / x)"
using assms ‹0 > θ› unfolding interval_lebesgue_integral_def einterval_eq zero_ereal_def
by (auto simp add: field_simps mult_le_0_iff set_lebesgue_integral_def split: if_split_asm)
} note aux2 = this
show ?thesis
using assms unfolding Si_def interval_lebesgue_integral_def set_lebesgue_integral_def sgn_real_def einterval_eq zero_ereal_def
by (auto simp: aux1 aux2)
qed

end
```