| import OAI.NumberTheory.Jacobsthal.Primes.PrimeHistoryCompactCount | |
| import OAI.NumberTheory.Jacobsthal.Renewal.NonnegativeOriginalOccupation | |
| namespace OAI | |
| namespace Erdos970 | |
| open scoped _root_.Erdos970 | |
| section | |
| namespace NumberTheoryLean.FullPathCompactSelection | |
| open _root_.Set _root_.MeasureTheory ProbabilityTheory | |
| open scoped ENNReal | |
| open FinitePathGeometry PrimeHistories PrimeKilledChain ActualProcessCoupling PersistentFailureFlag | |
| open FiniteHistoryTransport ActualCoupledHistories PrimeHistoryCompactCount | |
| open PrimeCompactVisits PrimeCompactWeights SourceCompactOccupation | |
| variable {w ell S B : ℝ} {start : Node} | |
| noncomputable def fullCompactReward (R w B : ℝ) {ell S : ℝ} (start : Node) (N : ℕ) | |
| (h : Hist (FlagState (JointState w ell S start)) N) : ℝ≥0∞ := | |
| ∑ j : Fin N, compactReward R w B start (h ⟨j,Finset.mem_Iic.mpr j.isLt.le⟩).1.1 | |
| theorem fullCompactReward_measurable (R w B : ℝ) {ell S : ℝ} (start : Node) (N : ℕ) : | |
| Measurable (fullCompactReward R w B (ell:=ell) (S:=S) start N) := by | |
| unfold fullCompactReward | |
| apply Finset.measurable_sum | |
| intro j _ | |
| exact (measurable_of_countable (compactReward R w B start)).comp | |
| ((measurable_fst.comp measurable_fst).comp (measurable_pi_apply _)) | |
| theorem fullCompactReward_envelope {R D : ℝ} | |
| (hbound : ∀ h : History w ell S start, h.node.gap ≤ R → scaledWeight w B start (some h) ≤ D) | |
| (N : ℕ) (h : Hist (FlagState (JointState w ell S start)) N) : | |
| fullCompactReward R w B start N h ≤ | |
| ENNReal.ofReal D * ENNReal.ofReal (pathSum (visit R) N (mapHist (fun q => q.1.1) N h)) := by | |
| unfold fullCompactReward pathSum mapHist | |
| rw [ENNReal.ofReal_sum_of_nonneg (fun _ _ => visit_nonneg _ _),Finset.mul_sum] | |
| apply Finset.sum_le_sum | |
| intro j _ | |
| exact compact_reward_envelope hbound _ | |
| variable (hw : normalizationThreshold ≤ w) (hell : 1 ≤ ell) (hS0 : 0 ≤ S) | |
| variable (hS : S ≤ (Real.log w)^3) (hr : 0 < start.gap) | |
| variable (hs : Valid start.side start.ratio) (hsS : start.ratio ≤ S) | |
| theorem selected_compact_history_bound {R D : ℝ} (hR : 0 ≤ R) (hD : 0 ≤ D) | |
| (hbound : ∀ h : History w ell S start, h.node.gap ≤ R → scaledWeight w B start (some h) ≤ D) | |
| (mesh : ℝ) (N : ℕ) {A : Set (Hist (FlagState (JointState w ell S start)) N)} | |
| (hA : MeasurableSet A) : | |
| (∫⁻ h, A.indicator (fullCompactReward R w B start N) h | |
| ∂sourceHistoryLaw hw hell hS0 hS hr hs hsS mesh N) ≤ | |
| ENNReal.ofReal (D*(R+1)) * sourceHistoryLaw hw hell hS0 hS hr hs hsS mesh N A := by | |
| calc | |
| _ ≤ ∫⁻ h, A.indicator (fun _ => ENNReal.ofReal (D*(R+1))) h | |
| ∂sourceHistoryLaw hw hell hS0 hS hr hs hsS mesh N := by | |
| apply lintegral_mono_ae | |
| filter_upwards [sourceHistory_compact_count hw hell hS0 hS hr hs hsS hR mesh N] with h hh | |
| apply Set.indicator_le_indicator | |
| refine (fullCompactReward_envelope hbound N h).trans ?_ | |
| rw [ENNReal.ofReal_mul hD] | |
| exact mul_le_mul_of_nonneg_left (ENNReal.ofReal_le_ofReal hh) zero_le | |
| _ = _ := lintegral_indicator_const hA _ | |
| end NumberTheoryLean.FullPathCompactSelection | |
| end | |
| end Erdos970 | |
| end OAI | |