| import OAI.NumberTheory.Jacobsthal.Probability.SourceCompactCoupling |
| import OAI.NumberTheory.Jacobsthal.Renewal.ContinuousCompletedOccupation |
|
|
| namespace OAI |
|
|
| namespace Erdos970 |
| open scoped _root_.Erdos970 |
|
|
| section |
|
|
| namespace NumberTheoryLean.ContinuousArrivalSupport |
|
|
| open _root_.Set _root_.MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
| open FinitePathGeometry FinitePathMeasures PrimeHistories PrimeKilledChain |
| open ContinuousKilledBins CemeteryKernel LowStateHorizon ArrivalKernelGeometry |
|
|
| def arrivalSupported (v ell : ℝ) : Space CostState → Prop |
| | .inl z => z ∈ arrivalSet v ell |
| | .inr _ => True |
|
|
| theorem arrivalSupported_measurable (v ell : ℝ) : MeasurableSet {z | arrivalSupported v ell z} := by |
| apply measurableSet_sum_iff.mpr |
| exact ⟨arrivalSet_measurable v ell,MeasurableSet.univ⟩ |
|
|
| theorem unsupported_eq (v ell : ℝ) : |
| {z : Space CostState | ¬arrivalSupported v ell z} = Sum.inl '' (arrivalSet v ell)ᶜ := by |
| ext z |
| cases z <;> simp [arrivalSupported] |
|
|
| theorem continuous_row_arrival (v ell S : ℝ) (q : Space CostState) : |
| ∀ᵐ y ∂continuousChain v ell S q,arrivalSupported v ell y := by |
| apply ae_iff.mpr |
| rw [unsupported_eq] |
| cases q with |
| | inl z => |
| rw [continuousChain,complete_live_mass _ _ (arrivalSet_measurable v ell).compl, |
| lowKernel,Kernel.restrict_apply,Measure.restrict_apply (arrivalSet_measurable v ell).compl] |
| have he : (arrivalSet v ell)ᶜ ∩ lowDomain v ell S = ∅ := by |
| apply Set.eq_empty_iff_forall_notMem.mpr |
| intro y hy |
| exact hy.1 hy.2.1 |
| rw [he,measure_empty] |
| | inr u => |
| cases u |
| change Measure.dirac (dead : Space CostState) (Sum.inl '' (arrivalSet v ell)ᶜ) = 0 |
| rw [Measure.dirac_apply' _ (arrivalSet_measurable v ell).compl.inl_image, |
| indicator_of_notMem (by simp [dead])] |
|
|
| theorem continuous_power_arrival (v ell S : ℝ) (q : Space CostState) (n : ℕ) (hn : 1 ≤ n) : |
| ∀ᵐ y ∂((continuousChain v ell S)^n) q,arrivalSupported v ell y := by |
| cases n with |
| | zero => omega |
| | succ n => |
| have hp : (continuousChain v ell S)^(n+1) = continuousChain v ell S ∘ₖ ((continuousChain v ell S)^n) := pow_succ' _ _ |
| rw [hp] |
| apply Kernel.ae_comp_of_ae_ae (arrivalSupported_measurable v ell) |
| exact Filter.Eventually.of_forall (continuous_row_arrival v ell S) |
|
|
| variable {w ell S : ℝ} {start : Node} |
| variable (hw : normalizationThreshold ≤ w) (hell : 1 ≤ ell) (hS0 : 0 ≤ S) (hS : S ≤ (Real.log w)^3) |
| (hr : 0 < start.gap) (hs : Valid start.side start.ratio) (hsS : start.ratio ≤ S) |
|
|
| include hw hell hS0 hS hr hsS in |
|
|
| theorem source_arrival (mesh : ℝ) (n : ℕ) (hn : 1 ≤ n) : |
| ∀ᵐ q ∂CouplingData.sourceLaw w ell S start hs mesh n, |
| arrivalSupported (Real.log start.gap) ell q.1.2 := by |
| have he := FlaggedSourceStart.sourceLaw_continuous hw hell hS0 hS hr hs hsS mesh n |
| rw [CouplingData.sourceLaw_eq hw hell hS0 hS hr hs hsS] at he |
| have h : ∀ᵐ y ∂(CouplingData.sourceLaw w ell S start hs mesh n).map (fun q => q.1.2), |
| arrivalSupported (Real.log start.gap) ell y := by |
| rw [he] |
| exact continuous_power_arrival _ _ _ _ n hn |
| exact (ae_map_iff (measurable_snd.comp measurable_fst).aemeasurable |
| (arrivalSupported_measurable _ _)).mp h |
|
|
| end NumberTheoryLean.ContinuousArrivalSupport |
|
|
| end |
|
|
| end Erdos970 |
|
|
| end OAI |
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|
