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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 | import OAI.NumberTheory.Jacobsthal.Partitions.FullBoxTotalLength
namespace OAI
namespace Erdos970
open scoped _root_.Erdos970
section
namespace ErdosVarianceWeighted
open NumberTheoryLean FinitePathGeometry PrimeHistories PrimeBinMembership StrongReferenceTransport SafeSubsetBoxGeometry
ErdosCofactorChoices ErdosSubsetWord SingletonBinSelection LogarithmicBinScale
theorem product_le_Y_of_length (w a : ℝ) (hw : 1 < w) (ha : 0 ≤ a) (Y d : ℕ)
(hY : 0 < Y) (hd : 0 < d) (hlen : a ≤ Real.log ((Y : ℝ)/(d : ℝ))/Real.log w) : d ≤ Y := by
have hdR : (0 : ℝ) < d := by exact_mod_cast hd
have hYR : (0 : ℝ) < Y := by exact_mod_cast hY
have hh : w^a ≤ (Y : ℝ)/(d : ℝ) := (Real.le_logb_iff_rpow_le hw (div_pos hYR hdR)).mp hlen
have hratio : 1 ≤ (Y : ℝ)/(d : ℝ) := (Real.one_le_rpow hw.le ha).trans hh
have hreal := (le_div_iff₀ hdR).mp hratio
have hdle : (d : ℝ) ≤ Y := by simpa only [one_mul] using hreal
exact_mod_cast hdle
theorem full_box_variance_fibres {w top xi B C K a : ℝ}
(hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (hC : 0 ≤ C) (ha0 : 0 ≤ a)
(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) (hanchor : SafeAnchor hw htop hxi C B K z m)
(i : Fin (binCount w top xi)) (hi : m i = 1) :
∀ f ∈ selections (globalBins w top xi) (eraseMultiplicity m i),Squarefree (selectionProduct f) ∧
(∀ t ∈ (selectionProduct f).primeFactors,w < (t : ℝ)) ∧
(∀ p ∈ globalBins w top xi i,(selectionProduct f).Coprime p ∧
3*a/4 ≤ Real.log ((Y : ℝ)/((p : ℝ)*selectionProduct f))/Real.log w) := by
intro f hf
have harith := erased_cofactor_arithmetic hw htop hxi m i f hf
have hlen := erased_cofactor_all_lengths hw htop hxi hC hcomp Y hY z hroot hs hz hg hclosed hcap m hanchor i hi f hf
refine ⟨harith.1,harith.2.1,?_⟩
intro p hp
exact ⟨harith.2.2 p hp,(by linarith : 3*a/4 ≤ a).trans (hlen p hp)⟩
end ErdosVarianceWeighted
end
end Erdos970
end OAI
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