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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 | import Mathlib
import OAI.Probability.SKBarriers.Coverage.UniformCoverage
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
open scoped Topology
theorem sqrt_div_positive {a L : ℝ} (hL : 0 < L) :
Real.sqrt a/L = Real.sqrt (a/L^2) := by
rw [Real.sqrt_div' a (sq_nonneg L),Real.sqrt_sq_eq_abs,abs_of_pos hL]
theorem freshCoverageGain_div {n : ℕ} (hn : 0 < n) (b q L : ℝ) (hL : 0 < L) :
freshCoverageGain n b q/L =
(b^2*(n:ℝ)/L)*(1-1/(n:ℝ))/4-Real.log 2/L-
Real.sqrt (Real.pi^2*(b^2*(n:ℝ)/L)*((b^2*(n:ℝ)/L)*q^2/2+Real.log 8/L)/4)-
Real.sqrt (Real.pi^2*(b^2*(n:ℝ)/L)*(Real.log 2/L)/4) := by
have hn' : (n:ℝ) ≠ 0 := by exact_mod_cast hn.ne'
unfold freshCoverageGain
rw [sub_div,sub_div,sub_div,sqrt_div_positive hL,sqrt_div_positive hL]
have hterm : b^2*((n:ℝ)-1)/4/L = (b^2*(n:ℝ)/L)*(1-1/(n:ℝ))/4 := by
field_simp
rw [hterm]
change _-Real.sqrt ((2*(Real.pi^2*b^2*(n:ℝ)/8)*(b^2*(n:ℝ)*q^2/2+Real.log 8))/L^2)-
Real.sqrt ((2*(Real.pi^2*b^2*(n:ℝ)/8)*Real.log 2)/L^2) = _
congr 2 <;> field_simp <;> ring_nf
theorem freshCoverageGain_div_tendsto (L b : ℕ → ℝ) {v q : ℝ}
(hL : Tendsto L atTop atTop)
(hV : Tendsto (fun n => (b n)^2*(n:ℝ)/L n) atTop (𝓝 v)) :
Tendsto (fun n => freshCoverageGain n (b n) q/L n) atTop
(𝓝 (v/4-Real.sqrt (Real.pi^2*v^2*q^2/8))) := by
have hLi : Tendsto (fun n => (L n)⁻¹) atTop (𝓝 (0:ℝ)) :=
tendsto_inv_atTop_zero.comp hL
have hN : Tendsto (fun n : ℕ => (n:ℝ)⁻¹) atTop (𝓝 (0:ℝ)) :=
tendsto_inv_atTop_zero.comp (tendsto_natCast_atTop_atTop (R := ℝ))
have h2 : Tendsto (fun n => Real.log 2/L n) atTop (𝓝 (0:ℝ)) := by
simpa only [mul_zero,div_eq_mul_inv] using hLi.const_mul (Real.log 2)
have h8 : Tendsto (fun n => Real.log 8/L n) atTop (𝓝 (0:ℝ)) := by
simpa only [mul_zero,div_eq_mul_inv] using hLi.const_mul (Real.log 8)
have H := (((hV.mul ((tendsto_const_nhds (x := (1:ℝ))).sub hN)).div_const 4).sub h2).sub
((hV.const_mul (Real.pi^2) |>.mul ((hV.mul_const (q^2) |>.div_const 2).add h8)).div_const 4).sqrt
have H' := H.sub ((hV.const_mul (Real.pi^2) |>.mul h2).div_const 4).sqrt
have HH : Tendsto (fun n =>
((b n)^2*(n:ℝ)/L n)*(1-1/(n:ℝ))/4-Real.log 2/L n-
Real.sqrt (Real.pi^2*((b n)^2*(n:ℝ)/L n)*(((b n)^2*(n:ℝ)/L n)*q^2/2+Real.log 8/L n)/4)-
Real.sqrt (Real.pi^2*((b n)^2*(n:ℝ)/L n)*(Real.log 2/L n)/4)) atTop
(𝓝 (v/4-Real.sqrt (Real.pi^2*v^2*q^2/8))) := by
convert H' using 1 <;> simp only [one_div,sub_zero,mul_one,mul_zero,zero_div,Real.sqrt_zero,add_zero]
congr 2
ring_nf
apply HH.congr'
filter_upwards [hL.eventually (eventually_gt_atTop (0:ℝ)),eventually_gt_atTop (0:ℕ)] with n hLn hn
exact (freshCoverageGain_div hn (b n) q (L n) hLn).symm
end SK.Analytic
end
end
end
end OAI
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