File size: 3,159 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 | 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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