| import OAI.Probability.LatticePassage.NaturalMoment |
|
|
| namespace OAI |
|
|
| namespace CompanionFPP |
| open MeasureTheory Real |
|
|
| lemma mean_passageTime_nonneg (x y : Vertex) : |
| 0 ≤ ∫ ω : Environment, passageTime ω x y ∂environmentLaw := by |
| apply integral_nonneg_of_ae |
| filter_upwards [passageTime_properties] with ω hω |
| exact hω.1 x y |
|
|
| theorem passageTime_centered_large (x y : Vertex) (hd : 1 ≤ latticeDistance x y) |
| (u : ℝ) (hu : 4*(latticeDistance x y : ℝ) ≤ u) : |
| environmentLaw.real {ω : Environment | |
| u ≤ |passageTime ω x y - ∫ η : Environment, passageTime η x y ∂environmentLaw|} ≤ |
| exp (-u/4) := by |
| have hm := mean_passageTime_le_distance x y |
| have hm0 := mean_passageTime_nonneg x y |
| have hd' : (1 : ℝ) ≤ latticeDistance x y := by exact_mod_cast hd |
| have hsub : environmentLaw {ω : Environment | |
| u ≤ |passageTime ω x y - ∫ η : Environment, passageTime η x y ∂environmentLaw|} ≤ |
| environmentLaw {ω : Environment | u ≤ passageTime ω x y} := by |
| apply measure_mono_ae |
| filter_upwards [passageTime_properties] with ω hω |
| intro he |
| have hn := hω.1 x y |
| rcases le_abs.mp he with he | he <;> linarith |
| apply (ENNReal.toReal_mono (measure_ne_top _ _) hsub).trans |
| apply (passageTime_upper_tail x y u).trans |
| rw [show (2 : ℝ)^latticeDistance x y = exp ((latticeDistance x y : ℝ)*log 2) by |
| rw [exp_nat_mul, exp_log (by norm_num : (0 : ℝ) < 2)], ← exp_add, exp_le_exp] |
| have hlog := log_le_sub_one_of_pos (by norm_num : (0 : ℝ) < 2) |
| nlinarith |
|
|
| theorem passageTime_centered_large_sqrt (x y : Vertex) (hd : 1 ≤ latticeDistance x y) |
| (t : ℝ) (ht : 4*sqrt (latticeDistance x y) ≤ t) : |
| environmentLaw.real {ω : Environment | |
| t*sqrt (latticeDistance x y) ≤ |
| |passageTime ω x y - ∫ η : Environment, passageTime η x y ∂environmentLaw|} ≤ |
| exp (-t/4) := by |
| have hd' : (1 : ℝ) ≤ latticeDistance x y := by exact_mod_cast hd |
| have hs := sqrt_nonneg (latticeDistance x y : ℝ) |
| have hsq := sq_sqrt (show 0 ≤ (latticeDistance x y : ℝ) by positivity) |
| have hs1 : 1 ≤ sqrt (latticeDistance x y : ℝ) := by nlinarith |
| have ht0 : 0 ≤ t := by linarith |
| apply (passageTime_centered_large x y hd (t*sqrt (latticeDistance x y)) |
| (by nlinarith)).trans |
| rw [exp_le_exp] |
| nlinarith |
|
|
| theorem localPassageTime_center_shift (x y : Vertex) (R : ℕ) (hR : 1 ≤ R) |
| (hscale : 40*latticeDistance x y ≤ R) : |
| |(∫ ω : Environment, localPassageTime ω x y R ∂environmentLaw) - |
| (∫ ω : Environment, passageTime ω x y ∂environmentLaw)| ≤ |
| ((11/3 : ℝ)*R+4)*exp (-(log 2/40)*R) := by |
| have hd : latticeDistance x y < R := by omega |
| have hn : 0 ≤ (∫ ω : Environment, localPassageTime ω x y R ∂environmentLaw) - |
| (∫ ω : Environment, passageTime ω x y ∂environmentLaw) := by |
| apply sub_nonneg.mpr |
| apply integral_mono_ae (integrable_passageTime x y) (integrable_localPassageTime x y R hd) |
| filter_upwards [all_edge_weights_positive] with ω hω |
| exact passageTime_le_localPassageTime ω (fun e => (hω e).le) x y R hd |
| rw [abs_of_nonneg hn, ← integral_sub (integrable_localPassageTime x y R hd) |
| (integrable_passageTime x y)] |
| exact localPassageTime_mean_error_exponential x y R hR hscale |
|
|
| theorem passageTime_centered_localization (x y : Vertex) (R : ℕ) (hR : 1 ≤ R) |
| (hscale : 40*latticeDistance x y ≤ R) (u : ℝ) : |
| environmentLaw.real {ω : Environment | |
| u ≤ |passageTime ω x y - ∫ η : Environment, passageTime η x y ∂environmentLaw|} ≤ |
| (11/3 : ℝ)*exp (-(log 2/40)*R) + |
| environmentLaw.real {ω : Environment | |
| u - ((11/3 : ℝ)*R+4)*exp (-(log 2/40)*R) ≤ |
| |localPassageTime ω x y R - |
| ∫ η : Environment, localPassageTime η x y R ∂environmentLaw|} := by |
| let F := {ω : Environment | localPassageTime ω x y R ≠ passageTime ω x y} |
| let G := {ω : Environment | |
| u - ((11/3 : ℝ)*R+4)*exp (-(log 2/40)*R) ≤ |
| |localPassageTime ω x y R - |
| ∫ η : Environment, localPassageTime η x y R ∂environmentLaw|} |
| have hs : {ω : Environment | |
| u ≤ |passageTime ω x y - ∫ η : Environment, passageTime η x y ∂environmentLaw|} ⊆ |
| F ∪ G := by |
| intro ω hω |
| by_cases hf : ω ∈ F |
| · exact Or.inl hf |
| · right |
| have he : localPassageTime ω x y R = passageTime ω x y := not_not.mp hf |
| have hb := abs_add_le (localPassageTime ω x y R - |
| ∫ η : Environment, localPassageTime η x y R ∂environmentLaw) |
| ((∫ η : Environment, localPassageTime η x y R ∂environmentLaw) - |
| ∫ η : Environment, passageTime η x y ∂environmentLaw) |
| rw [sub_add_sub_cancel, he] at hb |
| have hc := localPassageTime_center_shift x y R hR hscale |
| change u ≤ |passageTime ω x y - ∫ η : Environment, passageTime η x y ∂environmentLaw| at hω |
| change u - ((11/3 : ℝ)*R+4)*exp (-(log 2/40)*R) ≤ |
| |localPassageTime ω x y R - ∫ η : Environment, localPassageTime η x y R ∂environmentLaw| |
| rw [he] |
| linarith |
| apply (measureReal_mono hs).trans |
| apply (measureReal_union_le F G).trans |
| exact add_le_add (localPassageTime_failure_probability x y R hR hscale) le_rfl |
|
|
| end CompanionFPP |
|
|
| end OAI |
|
|
