| import OAI.NumberTheory.Jacobsthal.Paths.FullBoxPrefixLower |
| import OAI.NumberTheory.Jacobsthal.Sieve.ErasedCofactorGeometry |
|
|
| namespace OAI |
|
|
| namespace Erdos970 |
| open scoped _root_.Erdos970 |
|
|
| section |
|
|
| namespace ErdosVarianceWeighted |
| open NumberTheoryLean FinitePathGeometry PrimeHistories PrimeBinMembership StrongReferenceTransport |
| StrongSourceFamilies SafeSubsetBoxGeometry ActualReferencePrefixes PrimeProductLogCoordinates |
| ErdosCofactorChoices ErdosSubsetWord SingletonBinSelection SingletonBinProduct |
| LogarithmicBinScale LogarithmicBinLabels LogarithmicBinPartition LogarithmicBinEndpoints |
| attribute [local instance] Classical.propDecidable |
|
|
| theorem full_box_total_length_lower {w top xi B C K a : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (hC : 0 ≤ C) |
| (hcomp : Real.log B ≤ 2*Real.log w) (Y : ℕ) (hY : 0 < Y) (z : Node) |
| (hroot : z.gap = Real.log (Y : ℝ)/Real.log w-a+2) |
| (hs : Valid z.side z.ratio) (hz : Consistent z) (hg : StrongState z) |
| (hclosed : z.closed = true) (hcap : w^z.cutoff = top) |
| (m : Fin (binCount w top xi) → ℕ) (ha : SafeAnchor hw htop hxi C B K z m) |
| (f : Fin (binCount w top xi) → Finset ℕ) (hf : f ∈ selections (globalBins w top xi) m) : |
| a ≤ Real.log ((Y : ℝ)/(selectionProduct f : ℝ))/Real.log w := by |
| have hfull := (anchored_choice_geometry hw htop hxi hC hcomp z m ha f hf).1 |
| have hprefix : descendingWord f ∈ (descendingWord f).inits := |
| (List.mem_inits _ _).mpr ⟨[],by simp⟩ |
| have hgap := reference_prefix_gap_floor hw htop z hs hz hg hclosed hcap _ _ hfull hprefix |
| have hsource := word_source_membership hw htop hxi m f hf |
| have hlog := terminal_gap_log_quotient Y hY z hroot (descendingWord f) |
| (fun p hp => ((mem_sourcePrimeSet (zero_lt_one.trans hw) htop p).mp (hsource p hp)).1.pos) |
| have hprod := descendingWord_product (globalBins w top xi) |
| (fun j k hjk => bins_pairwise_disjoint (zero_lt_one.trans hw) htop hxi j k hjk) |
| f (fun j => ((mem_selections _ m f).mp hf j).1) |
| rw [hprod] at hlog |
| linarith |
|
|
| theorem erased_cofactor_all_lengths {w top xi B C K a : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (hC : 0 ≤ C) |
| (hcomp : Real.log B ≤ 2*Real.log w) (Y : ℕ) (hY : 0 < Y) (z : Node) |
| (hroot : z.gap = Real.log (Y : ℝ)/Real.log w-a+2) |
| (hs : Valid z.side z.ratio) (hz : Consistent z) (hg : StrongState z) |
| (hclosed : z.closed = true) (hcap : w^z.cutoff = top) |
| (m : Fin (binCount w top xi) → ℕ) (ha : SafeAnchor hw htop hxi C B K z m) |
| (i : Fin (binCount w top xi)) (hi : m i = 1) |
| (f : Fin (binCount w top xi) → Finset ℕ) |
| (hf : f ∈ selections (globalBins w top xi) (eraseMultiplicity m i)) : |
| ∀ p ∈ globalBins w top xi i,a ≤ Real.log ((Y : ℝ)/((p : ℝ)*selectionProduct f))/Real.log w := by |
| intro p hp |
| have hfill := fill_selection_mem (globalBins w top xi) m i hi f hf p hp |
| have hh := full_box_total_length_lower hw htop hxi hC hcomp Y hY z hroot hs hz hg hclosed hcap m ha |
| (fillSelection f i p) hfill |
| rw [filled_product f i p (erased_coordinate_empty _ m i f hf),Nat.cast_mul] at hh |
| exact hh |
|
|
| end ErdosVarianceWeighted |
|
|
| end |
|
|
| end Erdos970 |
|
|
| end OAI |
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|
