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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 | import OAI.NumberTheory.Jacobsthal.Analysis.CanonicalBadCompact
import OAI.NumberTheory.Jacobsthal.Analysis.SourceBadCompactOccurrence
namespace OAI
namespace Erdos970
open scoped _root_.Erdos970
section
namespace NumberTheoryLean.MarkedOccurrenceProbability
open _root_.Set _root_.Filter _root_.MeasureTheory ProbabilityTheory
open scoped ENNReal
open FinitePathGeometry FinitePathMeasures PrimeHistories PrimeKilledChain PrimeBinMembership
open ActualCoupledHistories ActualSuccessfulHistories ActualProcessCoupling FlaggedSourceStart
open FiniteHistoryTransport PriorPrimeWindow CompactPrefixOccurrence CompletedMarkEvents
open SourceBadCompactOccurrence CanonicalBadCompact ContinuousKilledBins PersistentFailureFlag
open Erdos970Dependency.MarkedVisits
variable {w ell S R b₀ b₁ : ℝ} {start : Node}
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 missing_occurrence_probability (hc : Consistent start)
(hb₀ : 0 < b₀) (hb₁ : 0 < b₁) (hR : (11/10)*R < 6*b₀)
{mesh : ℝ} (hm : 0 < mesh) (hmesh : mesh ≤ 1/100) (N : ℕ)
(herror : 4*(N:ℝ)*mesh ≤ Real.log (11/10)) (E : Set (List ℕ))
(hE : ∀ p : History w ell S start,p.primes ∈ E →
p.node.gap ≤ R ∧ ¬hasPrimeWindow w b₀ b₁ start p.primes) :
sourceHistoryLaw hw hell hS0 hS hr hs hsS mesh N {h | occurs E N h} ≤
sourceLaw hw hell hS0 hS hr hs hsS mesh N failed +
finitePathMeasure (typedState start.side start.ratio hs) N
(closedNoHitEvent 1 N start.gap b₀ b₁ 1) := by
let μ := sourceHistoryLaw hw hell hS0 hS hr hs hsS mesh N
let F : Set (Hist (FlagState (JointState w ell S start)) N) := {h | (last N h).2 = true}
let C : Set (Hist (FlagState (JointState w ell S start)) N) :=
mapHist (fun q => q.1.2) N ⁻¹' badCompactPath (Real.log start.gap) ((11/10)*R) b₀ b₁ N
have hF : μ F = sourceLaw hw hell hS0 hS hr hs hsS mesh N failed :=
sourceHistory_failure_probability hw hell hS0 hS hr hs hsS mesh N
have hC : μ C ≤ finitePathMeasure (typedState start.side start.ratio hs) N
(closedNoHitEvent 1 N start.gap b₀ b₁ 1) := by
dsimp [μ,C]
rw [← Measure.map_apply
(by exact mapHist_measurable (by exact measurable_snd.comp measurable_fst) N)
(badCompactPath_measurable _ _ _ _ _),sourceHistoryLaw_continuous]
exact completed_bad_compact_mass hr hb₀ hb₁ hR ell S (typedState start.side start.ratio hs) N
calc
_ ≤ μ (F ∪ C) := measure_mono_ae
(missing_occurrence_inclusion hw hell hS0 hS hr hs hsS hc hb₀ hm hmesh N herror E hE)
_ ≤ μ F + μ C := measure_union_le F C
_ ≤ _ := by rw [hF]; exact add_le_add le_rfl hC
end NumberTheoryLean.MarkedOccurrenceProbability
end
end Erdos970
end OAI
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