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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 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | import OAI.Probability.FirstPassage.BKBitLaw
import OAI.Probability.FirstPassage.BKLimit
namespace OAI
namespace PlanarFPP
open MeasureTheory ProbabilityTheory Set Finset Filter
open scoped ENNReal
def StrictPathEvent (S : Finset Edge) (F : Set StrictPathWitness) (w : Edge → ℝ) : Prop :=
∃ p ∈ F, p.edges ⊆ S ∧ p.Good w
lemma dyadicPathFamily_subset (S : Finset Edge) (t : ℕ) (F : Set StrictPathWitness) :
dyadicPathFamily S t F ⊆ (S ×ˢ Finset.univ).powerset := by
intro x hx
exact mem_powerset.mpr ((mem_dyadicPathFamily ..).mp hx).1
lemma dyadicDisjointPathFamily_subset (S : Finset Edge) (t : ℕ) (Fs : List (Set StrictPathWitness)) :
dyadicDisjointPathFamily S t Fs ⊆ (S ×ˢ Finset.univ).powerset := by
classical
exact Finset.filter_subset _ _
variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {τ : Edge → Ω → ℝ}
omit [MeasurableSpace Ω] in
lemma dyadic_single_cell_original {S : Finset Edge} {t : ℕ} {F : Set StrictPathWitness}
{x : Finset (Edge × Fin t)} {ω : Ω}
(hx : x ∈ dyadicPathFamily S t F) (hcell : ω ∈ dyadicRankCell τ S t x) :
StrictPathEvent S F (fun e => τ e ω) := by
obtain ⟨_,p,hp,he,hgood⟩ := (mem_dyadicPathFamily ..).mp hx
exact ⟨p,hp,he,((dyadicRankCell_path_iff hcell p he).mpr hgood).original⟩
lemma ExponentialEnvironment.dyadic_single_mass_le (h : ExponentialEnvironment P τ)
(S : Finset Edge) (t : ℕ) (F : Set StrictPathWitness) :
fairFamilyMass (S ×ˢ Finset.univ) (dyadicPathFamily S t F) ≤
P.real {ω | StrictPathEvent S F (fun e => τ e ω)} := by
let _ := h.probability
rw [← h.dyadicRankCells_real S t _ (dyadicPathFamily_subset S t F)]
apply measureReal_mono ?_ (measure_ne_top _ _)
intro ω hω
obtain ⟨x,hx⟩ := Set.mem_iUnion.mp hω
obtain ⟨hx,hcell⟩ := Set.mem_iUnion.mp hx
exact dyadic_single_cell_original hx hcell
omit [MeasurableSpace Ω] in
lemma rounded_disjoint_cell_mem {S : Finset Edge} {t : ℕ} {Fs : List (Set StrictPathWitness)}
{x : Finset (Edge × Fin t)} {ω : Ω} (hx : x ⊆ S ×ˢ Finset.univ)
(hcell : ω ∈ dyadicRankCell τ S t x)
(hd : RoundedDisjointPaths S Fs t (fun e => τ e ω)) :
x ∈ dyadicDisjointPathFamily S t Fs := by
classical
obtain ⟨ps,hps,hdis⟩ := hd
apply Finset.mem_filter.mpr
refine ⟨mem_powerset.mpr hx,ps,?_,hdis⟩
exact hps.imp (fun p _ hp => ⟨hp.1,hp.2.1,(dyadicRankCell_path_iff hcell p hp.2.1).mp hp.2.2⟩)
lemma ExponentialEnvironment.rounded_disjoint_mass_le (h : ExponentialEnvironment P τ)
(S : Finset Edge) (t : ℕ) (Fs : List (Set StrictPathWitness)) :
P.real {ω | RoundedDisjointPaths S Fs t (fun e => τ e ω)} ≤
fairFamilyMass (S ×ˢ Finset.univ) (dyadicDisjointPathFamily S t Fs) := by
let _ := h.probability
rw [← h.dyadicRankCells_real S t _ (dyadicDisjointPathFamily_subset S t Fs)]
apply ENNReal.toReal_mono (measure_ne_top _ _)
apply measure_mono_ae
filter_upwards [h.ae_dyadicRankCell S t] with ω hω
intro hd
obtain ⟨x,hx,hcell⟩ := hω
exact Set.mem_iUnion.mpr ⟨x,Set.mem_iUnion.mpr
⟨rounded_disjoint_cell_mem (mem_powerset.mp hx) hcell hd,hcell⟩⟩
lemma strict_path_probabilities_nonneg (P : Measure Ω) (τ : Edge → Ω → ℝ)
(S : Finset Edge) (Fs : List (Set StrictPathWitness)) :
0 ≤ (Fs.map (fun F => P.real {ω | StrictPathEvent S F (fun e => τ e ω)})).prod := by
induction Fs with
| nil => simp
| cons F Fs ih => exact mul_nonneg measureReal_nonneg ih
lemma ExponentialEnvironment.dyadic_factors_le (h : ExponentialEnvironment P τ)
(S : Finset Edge) (t : ℕ) (Fs : List (Set StrictPathWitness)) :
(Fs.map (fun F => fairFamilyMass (S ×ˢ Finset.univ) (dyadicPathFamily S t F))).prod ≤
(Fs.map (fun F => P.real {ω | StrictPathEvent S F (fun e => τ e ω)})).prod := by
induction Fs with
| nil => simp
| cons F Fs ih =>
simp only [List.map_cons,List.prod_cons]
exact (mul_le_mul_of_nonneg_left ih (fairFamilyMass_nonneg _ _)).trans
(mul_le_mul_of_nonneg_right (h.dyadic_single_mass_le S t F)
(strict_path_probabilities_nonneg P τ S Fs))
theorem ExponentialEnvironment.rounded_strict_path_BK (h : ExponentialEnvironment P τ)
(S : Finset Edge) (t : ℕ) (Fs : List (Set StrictPathWitness)) :
P.real {ω | RoundedDisjointPaths S Fs t (fun e => τ e ω)} ≤
(Fs.map (fun F => P.real {ω | StrictPathEvent S F (fun e => τ e ω)})).prod :=
(h.rounded_disjoint_mass_le S t Fs).trans
((dyadic_strict_path_BK S t Fs).trans (h.dyadic_factors_le S t Fs))
theorem exponential_strict_path_BK (h : ExponentialEnvironment P τ)
(S : Finset Edge) (Fs : List (Set StrictPathWitness)) :
P.real {ω | StrictDisjointPaths S Fs (fun e => τ e ω)} ≤
(Fs.map (fun F => P.real {ω | StrictPathEvent S F (fun e => τ e ω)})).prod := by
let _ := h.probability
let C : ℝ := (Fs.map (fun F => P.real {ω | StrictPathEvent S F (fun e => τ e ω)})).prod
have hC : 0 ≤ C := strict_path_probabilities_nonneg P τ S Fs
have hh := strict_disjoint_measure_le_of_rounded P τ S Fs (ENNReal.ofReal C) (fun t => by
calc
P {ω | RoundedDisjointPaths S Fs t (fun e => τ e ω)} =
ENNReal.ofReal (P.real {ω | RoundedDisjointPaths S Fs t (fun e => τ e ω)}) :=
(ENNReal.ofReal_toReal (measure_ne_top _ _)).symm
_ ≤ ENNReal.ofReal C := ENNReal.ofReal_le_ofReal (h.rounded_strict_path_BK S t Fs))
exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top hh).trans_eq (ENNReal.toReal_ofReal hC)
end PlanarFPP
end OAI
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