| import OAI.NumberTheory.Jacobsthal.Estimates.CandidateExceptionMass |
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| namespace OAI |
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| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| namespace NumberTheoryLean.GeometricEdgeWitness |
| open PrimeHistories GeometricRegularWords SourceCompactSingletons ActualWordSelection |
| open ErdosCofactorChoices ErdosSubsetWord BoundedCandidateCount BoundedEdgeBins FullBoxLengthBounds |
| open BinCutSelections BinCutPrefixGeometry SelectionWordSplit |
| open LogarithmicBinScale LogarithmicBinLabels LogarithmicBinPartition |
| open ErdosPrimeInputs.PrimePrefixMass ErdosPrimeInputs.HarmonicPrimeMeasure |
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| attribute [local instance] Classical.propDecidable |
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| theorem decreasing_edge_prefix_unique (ps pre tail pre' tail' : List ℕ) (u : ℕ) |
| (hd : ps.Pairwise (· > ·)) (he : ps=pre++u::tail) (he' : ps=pre'++u::tail') : pre=pre' := by |
| have hno : u ∉ pre := by |
| intro hu |
| have hh := hd |
| rw [he,List.pairwise_append] at hh |
| exact (lt_irrefl u) (hh.2.2 u hu u (by simp)) |
| have hno' : u ∉ pre' := by |
| intro hu |
| have hh := hd |
| rw [he',List.pairwise_append] at hh |
| exact (lt_irrefl u) (hh.2.2 u hu u (by simp)) |
| have hidx : ps.idxOf u=pre.length := by rw [he,List.idxOf_append_of_notMem hno]; simp |
| have hidx' : ps.idxOf u=pre'.length := by rw [he',List.idxOf_append_of_notMem hno']; simp |
| have hlen : pre.length=pre'.length := hidx.symm.trans hidx' |
| calc |
| pre=ps.take pre.length := by rw [he]; simp |
| _=ps.take pre'.length := by rw [hlen] |
| _=pre' := by rw [he']; simp |
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| theorem geometric_word_bounded_edge {w top xi C B R L alpha beta M X : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (Y : ℕ) (z : Node) (ps : List ℕ) |
| (hp : ps ∈ geometricWords hw htop hxi C B R L alpha beta z) |
| (pre tail : List ℕ) (u : ℕ) (he : ps=pre++u::tail) |
| (hgap : (terminal w z pre).gap ≤ L) |
| (hM : Real.log ((Y:ℝ)/(pre.prod:ℝ))/Real.log w ≤ M) (hX : primeExponent w u ≤ X) : |
| boundedEdgeBin w top xi Y (wordMultiplicity (label (zero_lt_one.trans hw) htop hxi) ps) M X |
| (label (zero_lt_one.trans hw) htop hxi u) := by |
| have hreg := (Finset.mem_filter.mp hp).1 |
| have hd := mem_decreasingPrefixes.mp (Finset.mem_filter.mp hreg).1 |
| have hn : ps.Nodup := hd.1.imp (fun h => ne_of_gt h) |
| let lab := label (zero_lt_one.trans hw) htop hxi |
| let m := wordMultiplicity lab ps |
| let f := wordSelection lab ps |
| let j := lab u |
| have hf : f ∈ selections (globalBins w top xi) m := wordSelection_mem hw htop hxi ps hd.2 |
| have hsingle := GeometricRegularWords.geometric_compact_singleton hw htop hxi z ps hp pre tail u he hgap |
| have hcard : (f j).card=1 := (wordMultiplicity_count lab ps hn j).trans hsingle |
| have hu : u ∈ f j := by |
| change u ∈ ps.toFinset.filter (fun p => lab p=j) |
| simp [he,j] |
| obtain ⟨v,hfv⟩ := Finset.card_eq_one.mp hcard |
| have huv : u=v := by simpa only [hfv,Finset.mem_singleton] using hu |
| have hfu : f j={u} := by rwa [← huv] at hfv |
| have hword := wordSelection_recovers lab ps hd.1 |
| have hsplit := selected_word_split hw htop hxi m f hf j u hfu |
| rw [hword] at hsplit |
| have hpre := decreasing_edge_prefix_unique ps pre tail (descendingWord (aboveSelection f j)) |
| (descendingWord (belowSelection f j)) u hd.1 he hsplit |
| refine ⟨f,hf,u,hfu,?_,hX⟩ |
| unfold parentLength |
| rw [← above_word_product hw htop hxi m f hf j,← hpre] |
| exact hM |
| end NumberTheoryLean.GeometricEdgeWitness |
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| end Erdos970 |
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| end OAI |
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