| import OAI.Probability.FirstPassage.WitnessBorelCantelli |
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| namespace OAI |
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| namespace PlanarFPP |
| open MeasureTheory Filter |
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| 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)] |
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| variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {τ : Edge → Ω → ℝ} |
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| 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 |
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| end PlanarFPP |
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| end OAI |
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