openAI-math-dataset / lean /OAI /NumberTheory /Jacobsthal /Probability /MarkedOccurrenceProbability.lean
| 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 | |