| import OAI.NumberTheory.Jacobsthal.Estimates.RepeatedPositionNeighbors |
| import OAI.NumberTheory.Jacobsthal.Probability.RepeatedWordEvents |
|
|
| namespace OAI |
|
|
| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| namespace NumberTheoryLean.CompactPositionRepetition |
| open FinitePathGeometry PrimeHistories RepeatedWordEvents RepeatedPositionNeighbors |
| open LogarithmicBinScale LogarithmicBinLabels LogarithmicBinPartition |
| open ErdosPrimeInputs.HarmonicPrimeMeasure |
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| theorem compact_position_repetition {n : ℕ} (lab : ℕ → Fin n) (w L : ℝ) (z : Node) |
| (pre tail : List ℕ) (p : ℕ) |
| (hord : ((pre++p::tail).map lab).Pairwise (· ≥ ·)) |
| (hc : 2 ≤ ((pre++p::tail).map lab).count (lab p)) |
| (hgap : (terminal w z pre).gap ≤ L) (hp : 0 ≤ primeExponent w p) : |
| compactAdjacentWord w lab L z (pre++p::tail) := by |
| rcases repeated_position_neighbor lab pre tail p hord hc with ⟨pre',q,hpre,hq⟩ | ⟨q,tail',htail,hq⟩ |
| · refine ⟨pre',q,p,tail,?_,hq,?_⟩ |
| · rw [hpre] |
| simp |
| · rwa [← hpre] |
| · refine ⟨pre,p,q,tail',by rw [htail],hq.symm,?_⟩ |
| have hg : (terminal w z (pre++[p])).gap=(terminal w z pre).gap-primeExponent w p := by |
| simp [terminal,step] |
| rw [hg] |
| linarith |
|
|
| theorem compact_position_singleton {n : ℕ} (lab : ℕ → Fin n) (w L : ℝ) (z : Node) |
| (ps : List ℕ) (hord : (ps.map lab).Pairwise (· ≥ ·)) |
| (hno : ¬compactAdjacentWord w lab L z ps) |
| (pre tail : List ℕ) (p : ℕ) (he : ps=pre++p::tail) |
| (hgap : (terminal w z pre).gap ≤ L) (hp : 0 ≤ primeExponent w p) : |
| (ps.map lab).count (lab p) ≤ 1 := by |
| by_contra hc |
| have hc' : 2 ≤ (ps.map lab).count (lab p) := by omega |
| apply hno |
| rw [he] at hord hc' ⊢ |
| exact compact_position_repetition lab w L z pre tail p hord hc' hgap hp |
| end NumberTheoryLean.CompactPositionRepetition |
| |
| |
| |
| namespace NumberTheoryLean.SourceCompactSingletons |
| open FinitePathGeometry PrimeHistories RepeatedWordEvents CompactPositionRepetition SortedLabelRepetition |
| open LogarithmicBinScale LogarithmicBinLabels LogarithmicBinPartition |
| open ErdosPrimeInputs.HarmonicPrimeMeasure |
| |
| |
| theorem source_compact_singleton {w top xi : ℝ} (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (L : ℝ) (z : Node) (ps : List ℕ) (hd : ps.Pairwise (· > ·)) |
| (hsrc : ∀ p ∈ ps,p ∈ sourcePrimeSet w top) |
| (hno : ¬compactAdjacentWord w (label (zero_lt_one.trans hw) htop hxi) L z ps) |
| (pre tail : List ℕ) (p : ℕ) (he : ps=pre++p::tail) (hgap : (terminal w z pre).gap ≤ L) : |
| (ps.map (label (zero_lt_one.trans hw) htop hxi)).count (label (zero_lt_one.trans hw) htop hxi p)=1 := by |
| have hp : p ∈ ps := by rw [he]; simp |
| have hp' := (mem_sourcePrimeSet (zero_lt_one.trans hw) htop p).mp (hsrc p hp) |
| have hxp : 0 ≤ primeExponent w p := by |
| unfold primeExponent |
| exact div_nonneg (Real.log_nonneg (le_trans hw.le hp'.2.1.le)) (Real.log_nonneg hw.le) |
| have hle := compact_position_singleton _ w L z ps |
| (source_word_labels_ordered hw htop hxi ps hd hsrc) hno pre tail p he hgap hxp |
| have hpos : 0 < (ps.map (label (zero_lt_one.trans hw) htop hxi)).count (label (zero_lt_one.trans hw) htop hxi p) := |
| List.count_pos_iff.mpr (List.mem_map.mpr ⟨p,hp,rfl⟩) |
| omega |
| |
| theorem source_search_singleton {w top xi : ℝ} (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (ps : List ℕ) (hno : ¬searchRepeatedWord hw htop hxi ps) |
| (b : Fin (binCount w top xi)) (hs : ActualSourceTags.searchBin w (LogarithmicBinEndpoints.lower w top xi b)) |
| (hm : b ∈ ps.map (label (zero_lt_one.trans hw) htop hxi)) : |
| (ps.map (label (zero_lt_one.trans hw) htop hxi)).count b=1 := by |
| have hc := List.count_pos_iff.mpr hm |
| have hn : ¬2 ≤ (ps.map (label (zero_lt_one.trans hw) htop hxi)).count b := fun h => hno ⟨b,hs,h⟩ |
| omega |
| end NumberTheoryLean.SourceCompactSingletons |
| |
| |
| end Erdos970 |
| |
| end OAI |
| |
