| import OAI.NumberTheory.Jacobsthal.Partitions.SortedLabelRepetition |
| import OAI.NumberTheory.Jacobsthal.Paths.RepeatedWordData |
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| namespace OAI |
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| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| namespace NumberTheoryLean.RepeatedWordEvents |
| open _root_.Set _root_.MeasureTheory ProbabilityTheory |
| open FinitePathGeometry PrimeHistories PrimeKilledChain ActualPrimeHigh ActualProcessCoupling |
| open FiniteHistoryTransport FiniteHistoryOccurrence FiniteHistoryPairLaw FinitePairOccurrence |
| open ActualCoupledHistories RepeatedPairProbability RepeatedPairEvents RepeatedWordData |
| open SortedLabelRepetition AdjacentPrimeRecovery ActualRepeatedBinStep ActualSourceTags |
| open LogarithmicBinScale LogarithmicBinEndpoints LogarithmicBinLabels |
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| noncomputable def searchRepeatedWord {w top xi : ℝ} (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (ps : List ℕ) : Prop := ∃ b : Fin (binCount w top xi),searchBin w (lower w top xi b) ∧ |
| 2 ≤ (ps.map (label (zero_lt_one.trans hw) htop hxi)).count b |
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| def compactAdjacentWord {n : ℕ} (w : ℝ) (lab : ℕ → Fin n) (L : ℝ) (start : Node) (ps : List ℕ) : Prop := |
| ∃ pre p q tail,ps=pre++p::q::tail ∧ lab p=lab q ∧ (terminal w start (pre++[p])).gap ≤ L |
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| variable {w ell S : ℝ} {start : Node} |
| variable (hwn : 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) |
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| theorem search_word_pair_occurs {top xi : ℝ} (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (hcap : w^start.cutoff=top) (mesh : ℝ) (N : ℕ) : |
| ∀ᵐ h ∂sourceHistoryLaw hwn hell hS0 hS hr hs hsS mesh N, |
| ∀ j : Finset.Iic N,∀ p : History w ell S start,(h j).1.1=some p → searchRepeatedWord hw htop hxi p.primes → |
| h ∈ pairOccurrence (rawRepeatPair (label (zero_lt_one.trans hw) htop hxi) (w^((1/4:ℝ))) S Set.univ) N := by |
| filter_upwards [sourceHistory_recovers_adjacent hwn hell hS0 hS hr hs hsS mesh N] with h hh |
| intro j p hp hrep |
| obtain ⟨b,hsearch,hcount⟩ := hrep |
| have hd := allowed_decreasing hw p.admissible |
| have hsrc := history_source_primes hw htop hell hcap p |
| obtain ⟨pre,a,c,tail,hword,ha,hc⟩ := source_repeated_label_adjacent hw htop hxi p.primes hd hsrc b hcount |
| obtain ⟨k,q,r,hq,hr,hqw,hrw⟩ := hh j p hp pre a c tail hword |
| apply Set.mem_iUnion.mpr |
| refine ⟨k,?_⟩ |
| change primeRepeatPair _ _ _ _ ((atIndex N h k.castSucc).1.1,(atIndex N h k.succ).1.1) |
| rw [hq,hr] |
| apply adjacent_search_pair_data hw htop hell hxi hsS hcap pre a c q r hqw hrw (ha.trans hc.symm) |
| rwa [hc] |
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| theorem compact_word_pair_occurs {n : ℕ} (lab : ℕ → Fin n) (hw : 1 < w) (L mesh : ℝ) (N : ℕ) : |
| ∀ᵐ h ∂sourceHistoryLaw hwn hell hS0 hS hr hs hsS mesh N, |
| ∀ j : Finset.Iic N,∀ p : History w ell S start,(h j).1.1=some p → compactAdjacentWord w lab L start p.primes → |
| h ∈ pairOccurrence (rawRepeatPair lab 1 L (compactParentGate L)) N := by |
| filter_upwards [sourceHistory_recovers_adjacent hwn hell hS0 hS hr hs hsS mesh N] with h hh |
| intro j p hp hrep |
| obtain ⟨pre,a,c,tail,hword,hlab,hgap⟩ := hrep |
| obtain ⟨k,q,r,hq,hr',hqw,hrw⟩ := hh j p hp pre a c tail hword |
| apply Set.mem_iUnion.mpr |
| refine ⟨k,?_⟩ |
| change primeRepeatPair _ _ _ _ ((atIndex N h k.castSucc).1.1,(atIndex N h k.succ).1.1) |
| rw [hq,hr'] |
| exact adjacent_compact_pair_data lab hw hell hr hs L pre a c q r hqw hrw hlab hgap |
| end NumberTheoryLean.RepeatedWordEvents |
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| end Erdos970 |
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| end OAI |
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