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135052a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 | import Mathlib
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
open scoped Topology
def coverageAngle (L : ℕ → ℝ) (M : ℝ) (n : ℕ) : ℝ := Real.sqrt (M*L n/(n:ℝ))
theorem coverageAngle_tendsto (L : ℕ → ℝ) (M : ℝ)
(hsub : Tendsto (fun n => L n/(n:ℝ)) atTop (𝓝 0)) :
Tendsto (coverageAngle L M) atTop (𝓝 0) := by
have H := (hsub.const_mul M).sqrt
change Tendsto (fun n => Real.sqrt (M*L n/(n:ℝ))) atTop (𝓝 0)
simpa only [← mul_div_assoc,mul_zero,Real.sqrt_zero] using H
theorem coverageAngle_pos {L : ℕ → ℝ} {M : ℝ} (hM : 0 < M) {n : ℕ}
(hn : 0 < n) (hL : 0 < L n) : 0 < coverageAngle L M n := by
unfold coverageAngle
have hn' : (0:ℝ) < n := by exact_mod_cast hn
positivity
theorem sin_eq_sinc_mul (x : ℝ) : Real.sin x = Real.sinc x*x := by
by_cases hx : x = 0
· simp [hx]
· rw [Real.sinc_of_ne_zero hx,div_mul_cancel₀ _ hx]
theorem coverageAngle_variance_tendsto (L : ℕ → ℝ) {M β : ℝ} (hM : 0 < M)
(hL : Tendsto L atTop atTop)
(hsub : Tendsto (fun n => L n/(n:ℝ)) atTop (𝓝 0)) :
Tendsto (fun n => (β*Real.sin (coverageAngle L M n))^2*(n:ℝ)/L n)
atTop (𝓝 (β^2*M)) := by
have hsinc : Tendsto (fun n => Real.sinc (coverageAngle L M n)) atTop (𝓝 (1:ℝ)) := by
simpa only [Function.comp_def,Real.sinc_zero] using
Real.continuous_sinc.continuousAt.tendsto.comp (coverageAngle_tendsto L M hsub)
have H := (hsinc.pow 2).const_mul (β^2*M)
simp only [one_pow,mul_one] at H
apply H.congr'
filter_upwards [hL.eventually (eventually_gt_atTop (0:ℝ)),eventually_gt_atTop (0:ℕ)] with n hLn hn
have hn' : (0:ℝ) < n := by exact_mod_cast hn
have hθ : coverageAngle L M n^2 = M*L n/(n:ℝ) :=
Real.sq_sqrt (by positivity)
rw [sin_eq_sinc_mul,mul_pow,mul_pow,hθ]
field_simp
theorem coverageAngle_loss_tendsto (L : ℕ → ℝ) {M β e : ℝ} (hM : 0 < M)
(hL : Tendsto L atTop atTop)
(hsub : Tendsto (fun n => L n/(n:ℝ)) atTop (𝓝 0)) :
Tendsto (fun n => β*(Real.cos (coverageAngle L M n)-1)*(n:ℝ)*e/L n)
atTop (𝓝 (-β*M*e/2)) := by
have hθ := coverageAngle_tendsto L M hsub
have hsinc : Tendsto (fun n => Real.sinc (coverageAngle L M n)) atTop (𝓝 (1:ℝ)) := by
simpa only [Function.comp_def,Real.sinc_zero] using Real.continuous_sinc.continuousAt.tendsto.comp hθ
have hcos : Tendsto (fun n => Real.cos (coverageAngle L M n)) atTop (𝓝 (1:ℝ)) := by
simpa only [Function.comp_def,Real.cos_zero] using Real.continuous_cos.continuousAt.tendsto.comp hθ
have H := ((hsinc.pow 2).const_mul (-β*M*e)).div (tendsto_const_nhds.add hcos)
(by norm_num : (1:ℝ)+1 ≠ 0)
norm_num only [one_pow,mul_one,one_add_one_eq_two] at H
apply H.congr'
filter_upwards [hL.eventually (eventually_gt_atTop (0:ℝ)),eventually_gt_atTop (0:ℕ),
hcos.eventually (eventually_gt_nhds (by norm_num : (0:ℝ)<1))] with n hLn hn hc
have hn' : (0:ℝ) < n := by exact_mod_cast hn
have hθsq : coverageAngle L M n^2 = M*L n/(n:ℝ) := Real.sq_sqrt (by positivity)
have htrig := Real.sin_sq_add_cos_sq (coverageAngle L M n)
rw [sin_eq_sinc_mul,mul_pow,hθsq] at htrig
have hden : 1+Real.cos (coverageAngle L M n) ≠ 0 := by positivity
have hcosEq : Real.cos (coverageAngle L M n)-1 =
-(Real.sinc (coverageAngle L M n)^2*(M*L n/(n:ℝ)))/(1+Real.cos (coverageAngle L M n)) := by
apply (eq_div_iff hden).mpr
nlinarith
change -β*M*e*Real.sinc (coverageAngle L M n)^2/(1+Real.cos (coverageAngle L M n)) =
β*(Real.cos (coverageAngle L M n)-1)*(n:ℝ)*e/L n
rw [hcosEq]
field_simp
end SK.Analytic
end
end
end
end OAI
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