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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 51 52 53 54 55 56 | import OAI.NumberTheory.Jacobsthal.Estimates.SourceRootErrorLower
import OAI.NumberTheory.Jacobsthal.Paths.SourceStoppedCorrection
namespace OAI
namespace Erdos970
open scoped _root_.Erdos970
namespace Erdos970Final
open _root_.Filter NumberTheoryLean NumberTheoryLean.PrimeHistories
NumberTheoryLean.SourceRootErrorLower NumberTheoryLean.StoppedCountVertex
NumberTheoryLean.StoppedCountAdapters NumberTheoryLean.LogarithmicBinPartition
NumberTheoryLean.LargePrimeDeletion
ErdosInverseBoxHeight ErdosCorrectionLimit
theorem source_root_survivor_lower : ∃ c : ℝ,0 < c ∧ ∀ᶠ top : ℝ in atTop,
∀ a : ℕ → ℕ,
c*((sourceY top : ℝ)*SmallSieveFinite.smallEuler ⌊sourceW top⌋₊/(sourceB top)^2) ≤
countSurvivors (sourceY top) (cutoffPrimes ⌊sourceW top⌋₊) a
(rootVertex (sourceRootNode (1/100) top) ∅ (sourcePrimeSet (sourceW top) top)
((sourceY top : ℝ)*SmallSieveFinite.smallEuler ⌊sourceW top⌋₊)) := by
obtain ⟨eta0,heta0,heta01,hCorrection⟩ := source_stopped_correction
obtain ⟨c,Clen,Cs,xi0,hc,hClen,hCs,_hCsMin,hxi0,hxi01,hError⟩ :=
source_root_lower_with_stopped_correction 1
have hClenPos : 0 < Clen := by linarith
obtain ⟨K,hK,hCorrection⟩ := hCorrection Cs hCs.le
let eta : ℝ := eta0/2
let xi : ℝ := min (xi0/2) (1/(40*Clen))
have heta : 0 < eta := by dsimp [eta];positivity
have hetaU : eta ≤ eta0 := by dsimp [eta];linarith
have hetaHalf : eta < 1/2 := by dsimp [eta];linarith
have hxi : 0 < xi := lt_min (by positivity) (by positivity)
have hxiU : xi ≤ xi0 := (min_le_left _ _).trans (by linarith)
have hxi1 : xi ≤ 1 := hxiU.trans hxi01
obtain ⟨rho,hrho,hError⟩ := hError eta heta hetaHalf xi hxi hxiU
have hKstop : 0 < 2*K := by positivity
have hCorrection := hCorrection Clen hClenPos.le eta heta.le hetaU xi hxi hxi1 rho (by linarith)
refine ⟨c,hc,?_⟩
filter_upwards [hError (2*K) hKstop,hCorrection] with top hError hCorrection
obtain ⟨_hw,_htop,hError⟩ := hError
obtain ⟨_hwC,_htopC,hCorrection⟩ := hCorrection
intro a
have hLower := hError a
have hSign := hCorrection a
dsimp only at hLower hSign
linarith
end Erdos970Final
end Erdos970
end OAI
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