| import OAI.NumberTheory.Jacobsthal.Estimates.SourceRootErrorLower |
| import OAI.NumberTheory.Jacobsthal.Paths.SourceStoppedCorrection |
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| namespace OAI |
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| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| namespace Erdos970Final |
| open _root_.Filter NumberTheoryLean NumberTheoryLean.PrimeHistories |
| NumberTheoryLean.SourceRootErrorLower NumberTheoryLean.StoppedCountVertex |
| NumberTheoryLean.StoppedCountAdapters NumberTheoryLean.LogarithmicBinPartition |
| NumberTheoryLean.LargePrimeDeletion |
| ErdosInverseBoxHeight ErdosCorrectionLimit |
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| 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 |
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| end Erdos970Final |
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| end Erdos970 |
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| end OAI |
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