File size: 3,852 Bytes
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 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 | import OAI.NumberTheory.Jacobsthal.Estimates.RepeatedPositionNeighbors
import OAI.NumberTheory.Jacobsthal.Probability.RepeatedWordEvents
namespace OAI
namespace Erdos970
open scoped _root_.Erdos970
namespace NumberTheoryLean.CompactPositionRepetition
open FinitePathGeometry PrimeHistories RepeatedWordEvents RepeatedPositionNeighbors
open LogarithmicBinScale LogarithmicBinLabels LogarithmicBinPartition
open ErdosPrimeInputs.HarmonicPrimeMeasure
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
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