File size: 3,142 Bytes
f89a930 | 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 | 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
|