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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 | import Mathlib
import OAI.Probability.SKBarriers.Coverage.CoverageAsymptotic
import OAI.Probability.SKBarriers.Coverage.CoverageAngle
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
open scoped Topology
def coverageGainRate (β e q : ℝ) : ℝ :=
β^2/4-β*e/2-Real.sqrt (Real.pi^2*(β^2)^2*q^2/8)
theorem exists_positive_coverageGainRate {β e : ℝ} (hβ : 0 < β) (he : e < β/2) :
∃ q : ℝ, 0 < q ∧ q < 1 ∧ 0 < coverageGainRate β e q := by
let g := β^2/4-β*e/2
have hg : 0 < g := by dsimp only [g]; nlinarith
let q := min (1/2:ℝ) (g/(4*Real.pi*β^2))
have hq : 0 < q := lt_min (by norm_num) (by positivity)
have hq1 : q < 1 := (min_le_left _ _).trans_lt (by norm_num)
have hqbound : Real.pi*β^2*q ≤ g/4 := by
have H := (le_div_iff₀ (show 0 < 4*Real.pi*β^2 by positivity)).mp (min_le_right (1/2:ℝ) (g/(4*Real.pi*β^2)))
change q*(4*Real.pi*β^2) ≤ g at H
nlinarith
have hroot : Real.sqrt (Real.pi^2*(β^2)^2*q^2/8) ≤ Real.pi*β^2*q := by
apply Real.sqrt_le_iff.mpr
constructor
· positivity
· have H : 0 ≤ (Real.pi*β^2*q)^2 := sq_nonneg _
nlinarith
refine ⟨q,hq,hq1,?_⟩
change 0 < g-Real.sqrt (Real.pi^2*(β^2)^2*q^2/8)
linarith
theorem coverage_totalGain_div_tendsto (L : ℕ → ℝ) {M β e D q : ℝ} (hM : 0 < M)
(hL : Tendsto L atTop atTop)
(hsub : Tendsto (fun n => L n/(n:ℝ)) atTop (𝓝 0)) :
Tendsto (fun n => (Real.log (Real.exp (-D*L n)/2)+
β*(Real.cos (coverageAngle L M n)-1)*(n:ℝ)*e+
freshCoverageGain n (β*Real.sin (coverageAngle L M n)) q)/L n)
atTop (𝓝 (M*coverageGainRate β e q-D)) := by
have hlog : Tendsto (fun n => Real.log (Real.exp (-D*L n)/2)/L n) atTop (𝓝 (-D)) := by
have H := (tendsto_const_nhds (x := -D)).sub (hL.const_div_atTop (Real.log 2))
simp only [sub_zero] at H
apply H.congr'
filter_upwards [hL.eventually (eventually_gt_atTop (0:ℝ))] with n hLn
rw [Real.log_div (Real.exp_ne_zero _) (by norm_num),Real.log_exp]
field_simp
have hV := coverageAngle_variance_tendsto (β := β) L hM hL hsub
have hf := freshCoverageGain_div_tendsto (q := q) L
(fun n => β*Real.sin (coverageAngle L M n)) hL hV
have he' := coverageAngle_loss_tendsto (β := β) (e := e) L hM hL hsub
have H := (hlog.add he').add hf
have hroot : Real.sqrt (Real.pi^2*(β^2*M)^2*q^2/8) =
M*Real.sqrt (Real.pi^2*(β^2)^2*q^2/8) := by
rw [show Real.pi^2*(β^2*M)^2*q^2/8 = M^2*(Real.pi^2*(β^2)^2*q^2/8) by ring,
Real.sqrt_mul (sq_nonneg M),Real.sqrt_sq_eq_abs,abs_of_pos hM]
convert H using 1
· funext n
rw [add_div,add_div]
· rw [hroot]
unfold coverageGainRate
congr 1
ring
end SK.Analytic
end
end
end
end OAI
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