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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 | import Mathlib
import OAI.Probability.SKBarriers.Coverage.CoverageGainRate
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
open scoped Topology
theorem coverage_eventually_parameters (L : ℕ → ℝ) {β e q cE D : ℝ}
(hβ : 0 < β) (hcE : 0 < cE) (hD : 0 < D)
(hK : 0 < coverageGainRate β e q)
(hL : Tendsto L atTop atTop)
(hsub : Tendsto (fun n => L n/(n:ℝ)) atTop (𝓝 0)) :
∃ M D₂ : ℝ, 0 < M ∧ 0 < D₂ ∧ ∀ᶠ n : ℕ in atTop,
0 < n ∧ 0 < L n ∧ β*Real.sin (coverageAngle L M n) ≠ 0 ∧
Real.exp (-cE*(n:ℝ)) ≤ Real.exp (-D*L n)/2 ∧
4*Real.exp (-D₂*L n)*Real.exp ((β*Real.sin (coverageAngle L M n))^2*(n:ℝ)/2) ≤
Real.exp ((β*Real.sin (coverageAngle L M n))^2*(n:ℝ)*q^2/2) ∧
L 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 := by
let M := (D+2)/coverageGainRate β e q
have hM : 0 < M := by dsimp only [M]; positivity
let D₂ := β^2*M+2
have hD₂ : 0 < D₂ := by dsimp only [D₂]; positivity
have hMG : M*coverageGainRate β e q-D = 2 := by dsimp only [M]; field_simp; ring
have hgain := coverage_totalGain_div_tendsto (β := β) (e := e) (D := D) (q := q)
L hM hL hsub
rw [hMG] at hgain
have hsmall : Tendsto (fun n : ℕ => D*(L n/(n:ℝ))+Real.log 2/(n:ℝ))
atTop (𝓝 0) := by
have H := (hsub.const_mul D).add
((tendsto_natCast_atTop_atTop (R := ℝ)).const_div_atTop (Real.log 2))
simpa using H
refine ⟨M,D₂,hM,hD₂,?_⟩
filter_upwards [eventually_gt_atTop (0:ℕ),hL.eventually (eventually_gt_atTop (0:ℝ)),
hL.eventually (eventually_ge_atTop (Real.log 4)),
(coverageAngle_tendsto L M hsub).eventually (eventually_lt_nhds Real.pi_pos),
hsmall.eventually (eventually_lt_nhds hcE),
(coverageAngle_variance_tendsto (β := β) L hM hL hsub).eventually
(eventually_lt_nhds (show β^2*M < β^2*M+1 by linarith)),
hgain.eventually (eventually_gt_nhds (by norm_num : (1:ℝ) < 2))]
with n hn hLn hLlog hθ htail hV hG
have hn' : (0:ℝ) < n := by exact_mod_cast hn
have hθpos := coverageAngle_pos hM hn hLn
have hb : β*Real.sin (coverageAngle L M n) ≠ 0 :=
ne_of_gt (mul_pos hβ (Real.sin_pos_of_pos_of_lt_pi hθpos hθ))
refine ⟨hn,hLn,hb,?_,?_,?_⟩
· have H : D*L n+Real.log 2 < cE*(n:ℝ) := by
have HH := (div_lt_iff₀ hn').mp (show (D*L n+Real.log 2)/(n:ℝ) < cE by
simpa only [add_div,mul_div_assoc] using htail)
exact HH
rw [show Real.exp (-D*L n)/2 = Real.exp (-D*L n-Real.log 2) by
rw [Real.exp_sub,Real.exp_log (by norm_num)]]
apply Real.exp_le_exp.mpr
linarith
· have HV : (β*Real.sin (coverageAngle L M n))^2*(n:ℝ) < (β^2*M+1)*L n :=
(div_lt_iff₀ hLn).mp hV
have hML : 0 ≤ β^2*M*L n := by positivity
have HE : Real.log 4-D₂*L n+(β*Real.sin (coverageAngle L M n))^2*(n:ℝ)/2 ≤ 0 := by
dsimp only [D₂]
nlinarith
calc
_ = Real.exp (Real.log 4-D₂*L n+(β*Real.sin (coverageAngle L M n))^2*(n:ℝ)/2) := by
rw [Real.exp_add,Real.exp_sub,Real.exp_log (by norm_num),neg_mul,Real.exp_neg]
ring
_ ≤ 1 := (Real.exp_le_one_iff).mpr HE
_ ≤ _ := Real.one_le_exp (by positivity)
· simpa only [one_mul] using ((le_div_iff₀ hLn).mp hG.le)
end SK.Analytic
end
end
end
end OAI
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