| import OAI.NumberTheory.Jacobsthal.Estimates.ActualRegularAdmission |
| import OAI.NumberTheory.Jacobsthal.Paths.FullWordHighAlignment |
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| namespace OAI |
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| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| namespace NumberTheoryLean.RegularStoppedEvent |
| open FinitePathGeometry PrimeHistories PrimeBinMembership SourceStopPredicate ActualWordSelection ActualRegularBoxes |
| open LogarithmicBinScale LogarithmicBinEndpoints LogarithmicBinLabels LogarithmicBinPartition |
| open ActualRegularAdmission ActualPrefixClearance RegularBadTagWords SourceMarkedStopping |
| open StoppedTraceSets StoppedCountAdapters PriorPrimeWindow MissingWindowPrimeMass |
| open StrongReferenceTransport StrongSourceFamilies ActualPrimeHigh ReferenceAdmission |
| open ErdosPrimeInputs.PrimePrefixMass ErdosPrimeInputs.PrimePrefixTail |
| open ErdosPrimeInputs.HarmonicPrimeMeasure ErdosInverseAlignment |
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| attribute [local instance] Classical.propDecidable |
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| noncomputable def hasSourceStop {w top xi : ℝ} (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (Y : ℕ) (Cs eta Clen B b₀ b₁ : ℝ) (a : ℕ → ℕ) (z : Node) (ps : List ℕ) : Prop := |
| ∃ pre tail : List ℕ,ps=pre++tail ∧ pre ∈ stopped w |
| (stopCandidate Y w Cs eta Clen B xi b₀ b₁ (lower w top xi) (width w top xi) |
| (label (zero_lt_one.trans hw) htop hxi) a z) |
| (sourcePrimeSet w top).card (rootVertex z ∅ (sourcePrimeSet w top) 1) |
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| noncomputable def uncapturedTest {w top xi : ℝ} (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (Y : ℕ) (Cs eta Clen B b₀ b₁ : ℝ) (a : ℕ → ℕ) (z : Node) |
| (Q : (Fin (binCount w top xi) → ℕ) → Finset ℚ) (ps : List ℕ) : Prop := |
| (∃ q ∈ Q (wordMultiplicity (label (zero_lt_one.trans hw) htop hxi) ps), |
| ∀ p ∈ ps,Cs < primeExponent w p → aligns (sourceClass a) q p) ∧ |
| ¬hasSourceStop hw htop hxi Y Cs eta Clen B b₀ b₁ a z ps |
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| noncomputable def uncapturedRegular {w top xi : ℝ} (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (Y : ℕ) (Cs eta Clen B K b₀ b₁ : ℝ) (a : ℕ → ℕ) (z : Node) |
| (Q : (Fin (binCount w top xi) → ℕ) → Finset ℚ) : Finset (List ℕ) := |
| (regularWords hw htop hxi Clen B K z).filter (uncapturedTest hw htop hxi Y Cs eta Clen B b₀ b₁ a z Q) |
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| theorem uncaptured_subset_exceptions {w top xi Cs eta Clen B K b₀ b₁ : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (hC : 0 ≤ Clen) |
| (hcomp : Real.log B ≤ 2*Real.log w) (hmesh : 2*(xi/Real.log w) ≤ 6*(2*Clen*xi)) |
| (hb₀ : 0 < b₀) (hCs : Cs < 2*b₀) (hsearch : b₁ < w^((1/4:ℝ))) |
| (hh : xi/Real.log w ≤ b₀) (hsmall : 2*Clen*xi+(206/100)*(xi/Real.log w) ≤ (4/100)*b₀) |
| (Y : ℕ) (a : ℕ → ℕ) (z : Node) (hs : Valid z.side z.ratio) (hz : Consistent z) |
| (hg : StrongState z) (hclosed : z.closed=true) (hcap : w^z.cutoff=top) (hroot : b₁/2 < z.cutoff) |
| (Q : (Fin (binCount w top xi) → ℕ) → Finset ℚ) : |
| uncapturedRegular hw htop hxi Y Cs eta Clen B K b₀ b₁ a z Q ⊆ |
| regularBadWords (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi Clen B K z Q (sourceClass a) ∪ |
| missingWindowPrefixes w 1 K b₀ b₁ z := by |
| intro ps hp |
| obtain ⟨hr,⟨q,hq,ha⟩,hnotstop⟩ := Finset.mem_filter.mp hp |
| have href := regular_word_reference hw htop hxi hC hcomp hmesh z ps hr |
| by_cases hwin : hasPrimeWindow w b₀ b₁ z ps |
| · by_cases hcorrect : ∃ t,ActualSourceTags.sourceTag Y w Cs eta (lower w top xi) (width w top xi) |
| (label (zero_lt_one.trans hw) htop hxi) (sourceClass a) ps=some t ∧ t.rational=q |
| · obtain ⟨t,ht,htq⟩ := hcorrect |
| obtain ⟨pre,tail,heq,hwin⟩ := hwin |
| have hd := mem_decreasingPrefixes.mp (Finset.mem_filter.mp hr).1 |
| have hreg := (Finset.mem_filter.mp hr).2 |
| have hsafe := clearance_supplies_safety hw htop hxi hC hcomp hmesh z ps hd.2 hreg.1 hreg.2.1 |
| have hcount := regular_tag_singleton hw htop hxi z ps hr (sourceClass a) t ht |
| have halign : ∀ p ∈ ps,Cs < primeExponent w p → aligns (sourceClass a) t.rational p := by |
| simpa only [htq] using ha |
| exact False.elim (hnotstop (marked_prefix_forces_actual_stop Y hw htop hxi hC hcomp hb₀ hCs hsearch hh hsmall |
| a ∅ 1 z ps pre tail heq hs hz hg hclosed hcap hroot href hreg.1 hsafe t ht hcount halign hwin)) |
| · exact Finset.mem_union_left _ (Finset.mem_filter.mpr ⟨hr,q,hq,ha,hcorrect⟩) |
| · have hu := (source_reference_transport hw htop z hs hz hg hclosed hcap ps href).1 |
| exact Finset.mem_union_right _ (Finset.mem_filter.mpr ⟨hu,(Finset.mem_filter.mp hr).2.2.2.1,hwin⟩) |
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| end NumberTheoryLean.RegularStoppedEvent |
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| end Erdos970 |
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| end OAI |
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