File size: 2,586 Bytes
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 | import Mathlib
import OAI.Probability.SKBarriers.Coverage.EnergyTailGap
import OAI.Probability.SKBarriers.Coverage.CoverageParameters
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
open scoped Topology
theorem uniform_coverage_pair {β : ℝ} (hβ : 1 < β) (L : ℕ → ℝ)
(hL : Tendsto L atTop atTop)
(hsub : Tendsto (fun n => L n/(n:ℝ)) atTop (𝓝 0)) :
∃ q : ℝ, 0 < q ∧ q < 1 ∧ ∀ D : ℝ, 0 < D → ∃ D₂ : ℝ, 0 < D₂ ∧
Tendsto (fun n => (disorderLaw n).real
(coveragePairBad n β (Real.exp (-D*L n)) (Real.exp (-D₂*L n)) q))
atTop (𝓝 0) := by
obtain ⟨e,he,cE,hcE,N₀,hN₀⟩ := coverage_energy_tail hβ
have hβpos : 0 < β := lt_trans zero_lt_one hβ
obtain ⟨q,hq,hq1,hK⟩ := exists_positive_coverageGainRate hβpos he.2
refine ⟨q,hq,hq1,?_⟩
intro D hD
obtain ⟨M,D₂,hM,hD₂,hpars⟩ := coverage_eventually_parameters (e := e) (q := q) L hβpos hcE hD hK hL hsub
refine ⟨D₂,hD₂,?_⟩
have hbound : ∀ᶠ n : ℕ in atTop,
(disorderLaw n).real (coveragePairBad n β (Real.exp (-D*L n)) (Real.exp (-D₂*L n)) q) ≤
Real.exp (-cE*(n:ℝ))+2*Real.exp (-L n/(β^2*M)) := by
filter_upwards [hpars,eventually_ge_atTop N₀] with n hn hlarge
obtain ⟨hnpos,hLn,hb,htrim,hsecond,hgain⟩ := hn
have HH := coveragePairBad_probability_le hnpos hβpos.le
(Real.exp_pos (-D*L n)) (Real.exp_nonneg (-D₂*L n)) hq.le hb htrim hLn.le hsecond hgain
have HE := hN₀ n hlarge
have hn' : (0:ℝ) < n := by exact_mod_cast hnpos
have hθ : coverageAngle L M n^2 = M*L n/(n:ℝ) := Real.sq_sqrt (by positivity)
have hquot : -(L n)^2/((coverageAngle L M n)^2*β^2*(n:ℝ)) = -L n/(β^2*M) := by
rw [hθ]
field_simp
rw [hquot] at HH
exact HH.trans (add_le_add HE le_rfl)
have hE : Tendsto (fun n : ℕ => Real.exp (-cE*(n:ℝ))) atTop (𝓝 0) :=
Real.tendsto_exp_atBot.comp ((tendsto_natCast_atTop_atTop (R := ℝ)).const_mul_atTop_of_neg (neg_neg_of_pos hcE))
have hR : Tendsto (fun n => Real.exp (-L n/(β^2*M))) atTop (𝓝 0) := by
have H := Real.tendsto_exp_atBot.comp
(hL.const_mul_atTop_of_neg (neg_lt_zero.mpr (show 0 < 1/(β^2*M) by positivity)))
convert H using 1
funext n
congr 1
ring
have H := hE.add (hR.const_mul 2)
simp only [mul_zero,add_zero] at H
exact squeeze_zero' (Eventually.of_forall (fun n => measureReal_nonneg)) hbound H
end SK.Analytic
end
end
end
end OAI
|