File size: 3,899 Bytes
f61e547 | 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 | import OAI.Probability.FirstPassage.WitnessBorelCantelli
namespace OAI
namespace PlanarFPP
open MeasureTheory Filter
lemma witness_scale_inequalities (b m₀ am ap η s L : ℝ) (hm : 0 < m₀)
(hη : 0 < η) (hη1 : η ≤ 1) (hs : 2 ≤ s)
(hS : max |m₀-((am+ap)/2)*b| |(am+ap)/2| / η ≤ s)
(hD : ((η+2*|(am+ap)/2|+1)/m₀)*s ≤ L) :
max |m₀-((am+ap)/2)*b| |(am+ap)/2| ≤ η*s ∧
η*s < 2*m₀*L-|(am+ap)/2| *(s+1) ∧
3*η*s+2*|(am+ap)/2| *(s+1) < 100*(1+|am|+|ap|)*s := by
have hstep := (div_le_iff₀ hη).mp hS
have hD' := mul_le_mul_of_nonneg_left hD hm.le
have he : m₀*((η+2*|(am+ap)/2|+1)/m₀*s)=(η+2*|(am+ap)/2|+1)*s := by field_simp
rw [he] at hD'
have hA : 2*|(am+ap)/2| ≤ |am|+|ap| := by
rw [abs_div,abs_of_pos (by norm_num : (0 : ℝ)<2)]
linarith [abs_add_le am ap]
have hAs := mul_le_mul_of_nonneg_right hA (by linarith : 0 ≤ s+1)
have hηs := mul_le_mul_of_nonneg_right hη1 (by linarith : 0 ≤ s)
have haS := mul_nonneg (abs_nonneg am) (by linarith : 0 ≤ s-1)
have hapS := mul_nonneg (abs_nonneg ap) (by linarith : 0 ≤ s-1)
have hmidS := mul_nonneg (abs_nonneg ((am+ap)/2)) (by linarith : 0 ≤ s-1)
refine ⟨by nlinarith,?_,?_⟩ <;> nlinarith [abs_nonneg am,abs_nonneg ap,abs_nonneg ((am+ap)/2),mul_pos hη (by linarith : 0<s)]
variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {τ : Edge → Ω → ℝ}
theorem actual_witness_scale (henv : ExponentialEnvironment P τ) (b η : ℝ)
(hb : |b| ≤ 1) (hη : 0 < η) (hη1 : η ≤ 1) :
∃ k m : ℕ, 2 ≤ (2 : ℝ)^k ∧
max |actualSourceTimeConstant (direction b)-
((extremeSlope actualSourceTimeConstant b false+extremeSlope actualSourceTimeConstant b true)/2)*b|
|(extremeSlope actualSourceTimeConstant b false+extremeSlope actualSourceTimeConstant b true)/2| ≤ η*(2 : ℝ)^k ∧
η*(2 : ℝ)^k < 2*actualSourceTimeConstant (direction b)*(2 : ℝ)^m-
|(extremeSlope actualSourceTimeConstant b false+extremeSlope actualSourceTimeConstant b true)/2| *((2 : ℝ)^k+1) ∧
3*η*(2 : ℝ)^k+2*|(extremeSlope actualSourceTimeConstant b false+extremeSlope actualSourceTimeConstant b true)/2| *((2 : ℝ)^k+1)
< sourceCrossingK actualSourceTimeConstant b*(2 : ℝ)^k ∧
∀ is : List (ℤ×ℤ), P.real {ω | StrictDisjointPaths (spatialListSupport b k m is)
(is.map (spatialWitnessFamily b (actualSourceTimeConstant (direction b))
(extremeSlope actualSourceTimeConstant b false) (extremeSlope actualSourceTimeConstant b true)
(sourceCrossingK actualSourceTimeConstant b) η k m)) (fun e => τ e ω)} ≤
(((1 : ℝ)/1024)^16)^is.length := by
let μ := actualSourceTimeConstant
let am := extremeSlope μ b false
let ap := extremeSlope μ b true
let a := (am+ap)/2
let m₀ := μ (direction b)
let q₀ : ℝ := ((1/1024)^16)/2
have hq : 0 < q₀ := by dsimp [q₀]; positivity
have hq1 : q₀ < 1 := by dsimp [q₀]; norm_num
let S := max 2 (max |m₀-a*b| |a| / η)
let D := (η+2*|a|+1)/m₀
obtain ⟨k,hk,hs,hd,hBK⟩ := actual_spatial_witness_BK_scales henv b q₀ η hb hq hq1 hη 0 S D
let m := characteristicExponent P τ μ b q₀ k
have hpow : (characteristicLength P τ μ b q₀ k : ℝ)=(2 : ℝ)^m := by
simp [characteristicLength,m]
rw [hpow] at hd
have hs2 : 2 ≤ (2 : ℝ)^k := (le_max_left _ _).trans hs
have hsC : max |m₀-a*b| |a| / η ≤ (2 : ℝ)^k := (le_max_right _ _).trans hs
obtain ⟨hstep,hscale₁,hscale₂⟩ := witness_scale_inequalities b m₀ am ap η ((2 : ℝ)^k) ((2 : ℝ)^m)
(norm_direction_pos μ actualSourceTimeConstant_norm b) hη hη1 hs2 hsC hd
refine ⟨k,m,hs2,hstep,hscale₁,hscale₂,?_⟩
intro is
have hh := hBK is
have he : 2*q₀=((1 : ℝ)/1024)^16 := by dsimp [q₀]; ring
rw [he] at hh
exact hh
end PlanarFPP
end OAI
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