| import OAI.NumberTheory.Jacobsthal.Paths.FiniteBoxWordMass |
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| namespace OAI |
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| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| namespace NumberTheoryLean.CanonicalKeyReference |
| open _root_.Filter PrimeHistories CanonicalStopKey CanonicalKeyWitness CanonicalKeyReplacement CanonicalKeyBands |
| open ActualStopGroupFiber StoppedCountVertex StoppedVertexHistory StoppedCountAdapters StoppedTraceSets |
| open LogarithmicBinScale LogarithmicBinEndpoints LogarithmicBinLabels LogarithmicBinPartition |
| open ErdosInversePrimeBin ErdosInverseAlignment ErdosPrimeInputs.MertensStrong |
| open scoped Topology |
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| theorem moving_key_reference {Clen xi Kstop rho : ℝ} (hC : 0 ≤ Clen) |
| (hxi : 0 < xi) (hxi1 : xi ≤ 1) (hK : 0 < Kstop) (hrho : 0 < rho) : |
| ∀ᶠ w : ℝ in atTop,∃ hw : 1 < w,∀ top : ℝ,∀ htop : w < top, |
| ∀ B : ℝ,Real.log B ≤ 2*Real.log w → ∀ Y : ℕ,∀ Cs eta : ℝ, |
| ∀ (a : ℕ → ℕ) (z : Node) (k : Key (binCount w top xi)), |
| ∀ _d : Witness hw htop hxi Y Cs eta Clen B (Kstop*(Real.log w)^2) |
| (rho*Kstop*(Real.log w)^2) ((Y:ℝ)*SmallSieveFinite.smallEuler ⌊w⌋₊) a z k, |
| ∀ p∈primeBin (lower w top xi k.tag.bin) (width w top xi k.tag.bin), |
| aligns (SourceStopPredicate.sourceClass a) k.tag.rational p → |
| referenceValue w (after w (rootVertex z ∅ (sourcePrimeSet w top) |
| ((Y:ℝ)*SmallSieveFinite.smallEuler ⌊w⌋₊)) (k.pre++p::k.tail)) |
| ≤ (33/100)*primeProduct (k.tail.getLastD 0:ℝ)* |
| ((Y:ℝ)/((p:ℝ)*(k.pre.prod*k.tail.prod:ℕ))) := by |
| filter_upwards [ErdosStoppedReferenceUpper.actual_stopped_reference_upper hC hxi hxi1 hK hrho] |
| with w hRef |
| obtain ⟨hw,hRef⟩ := hRef |
| refine ⟨hw,?_⟩ |
| intro top htop B hcomp Y Cs eta a z k d p hp ha |
| have hm := aligning_word_mem hw htop hxi d hC p hp ha |
| rw [family,ErdosInverseEuler.smallEuler_eq_primeProduct] at hm |
| have hh := (hRef top htop B hcomp Y Cs eta a z (k.pre++p::k.tail) hm).2 |
| rw [last_fixed_tail k.pre k.tail p d.tail_nonempty, |
| TagBelowStopWindow.replacement_product,Nat.cast_mul] at hh |
| rw [ErdosInverseEuler.smallEuler_eq_primeProduct] |
| convert hh using 1; ring |
| end NumberTheoryLean.CanonicalKeyReference |
| |
| |
| end Erdos970 |
| |
| end OAI |
| |
