| import OAI.NumberTheory.Jacobsthal.Partitions.ActualBoxFourSum |
| import OAI.NumberTheory.Jacobsthal.Primes.BoundedEdgeHighPrimes |
| import OAI.NumberTheory.Jacobsthal.Primes.PaperMissingPrimeWindow |
| import OAI.NumberTheory.Jacobsthal.Probability.PaperSubsetTagEvent |
| |
| namespace OAI |
|
|
| namespace Erdos970 |
| open scoped _root_.Erdos970 |
|
|
|
|
| namespace NumberTheoryLean.SubsetTagFailureMass |
| open ErdosCofactorChoices ErdosSubsetWord ErdosTagEvent ActualSourceTags ActualBinOwners |
| open LogarithmicBinScale LogarithmicBinEndpoints LogarithmicBinLabels CanonicalSubsetBox |
| open WrongOwnerBinMass ErdosInverseAlignment |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| noncomputable def badSelections {Y w top Cs eta xi : ℝ} (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (mult : Fin (binCount w top xi) → ℕ) (a : ℕ → ℤ) (q : ℚ) : Finset (Fin (binCount w top xi) → Finset ℕ) := |
| (selections (globalBins w top xi) mult).filter (fun f => |
| (∀ b ∈ occupiedSearch w top xi mult,∀ p ∈ f b,aligns a q p) ∧ |
| ¬∃ t,sourceTag Y w Cs eta (lower w top xi) (width w top xi) |
| (label (zero_lt_one.trans hw) htop hxi) a (descendingWord f)=some t ∧ t.rational=q) |
|
|
| noncomputable def badMass {Y w top Cs eta xi : ℝ} (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (mult : Fin (binCount w top xi) → ℕ) (a : ℕ → ℤ) (q : ℚ) : ℝ := |
| ∑ f ∈ badSelections (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi mult a q,(selectionProduct f:ℝ)⁻¹ |
|
|
| theorem bad_mass_le_alignment {Y w top Cs eta xi : ℝ} (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (mult : Fin (binCount w top xi) → ℕ) (a : ℕ → ℤ) (q : ℚ) : |
| badMass (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi mult a q ≤ |
| ∑ f ∈ (selections (globalBins w top xi) mult).filter |
| (fun f => ∀ b ∈ occupiedSearch w top xi mult,∀ p ∈ f b,aligns a q p),(selectionProduct f:ℝ)⁻¹ := by |
| apply Finset.sum_le_sum_of_subset_of_nonneg |
| · intro f hf |
| obtain ⟨hf,ha,hbad⟩ := Finset.mem_filter.mp hf |
| exact Finset.mem_filter.mpr ⟨hf,ha⟩ |
| · intro f _ _ |
| exact inv_nonneg.mpr (Nat.cast_nonneg _) |
|
|
| theorem bad_mass_le_wrong {Y w top Cs eta xi : ℝ} (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (mult : Fin (binCount w top xi) → ℕ) (a : ℕ → ℤ) (q : ℚ) |
| (howner : ∀ b ∈ occupiedSearch w top xi mult, |
| nearFull eta (lower w top xi b) (width w top xi b) a q → |
| owner Y w Cs eta (lower w top xi b) (width w top xi b) a=some q) |
| (hnear : ∃ b ∈ occupiedSearch w top xi mult, |
| nearFull eta (lower w top xi b) (width w top xi b) a q) : |
| badMass (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi mult a q ≤ |
| ∑ f ∈ (selections (globalBins w top xi) mult).filter |
| (fun f => ∃ b ∈ occupiedSearch w top xi mult,∃ p ∈ f b, |
| p ∈ wrongOwnerPrimes Y w Cs eta (lower w top xi b) (width w top xi b) a q),(selectionProduct f:ℝ)⁻¹ := by |
| apply Finset.sum_le_sum_of_subset_of_nonneg |
| · intro f hf |
| obtain ⟨hf,halign,hbad⟩ := Finset.mem_filter.mp hf |
| have hh := subset_bad_tag_inclusion hw htop hxi mult f a q hf howner halign hbad |
| rcases hh with hnone | hwrong |
| · obtain ⟨b,hb,hn⟩ := hnear |
| exact False.elim (hnone b hb hn) |
| · exact Finset.mem_filter.mpr ⟨hf,hwrong⟩ |
| · intro f _ _ |
| exact inv_nonneg.mpr (Nat.cast_nonneg _) |
| end NumberTheoryLean.SubsetTagFailureMass |
| |
| |
| |
| namespace NumberTheoryLean.HardTagBoxMass |
| open SubsetTagFailureMass ActualSourceTags ActualBinOwners |
| open LogarithmicBinScale LogarithmicBinEndpoints LogarithmicBinLabels PrimeBinRepresentatives |
| open ErdosCofactorChoices ErdosSubsetWord ErdosInverseAlignment |
| open ErdosPrimeInputs.HarmonicPrimeMeasure |
| |
| attribute [local instance] Classical.propDecidable |
| |
| noncomputable def hardBadSelections {Y w top Cs eta xi : ℝ} (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (mult : Fin (binCount w top xi) → ℕ) (a : ℕ → ℤ) (q : ℚ) : Finset (Fin (binCount w top xi) → Finset ℕ) := |
| (selections (globalBins w top xi) mult).filter (fun f => |
| (∀ p ∈ descendingWord f,Cs < primeExponent w p → aligns a q p) ∧ |
| ¬∃ t,sourceTag Y w Cs eta (lower w top xi) (width w top xi) |
| (label (zero_lt_one.trans hw) htop hxi) a (descendingWord f)=some t ∧ t.rational=q) |
| |
| noncomputable def hardBadMass {Y w top Cs eta xi : ℝ} (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (mult : Fin (binCount w top xi) → ℕ) (a : ℕ → ℤ) (q : ℚ) : ℝ := |
| ∑ f ∈ hardBadSelections (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi mult a q,(selectionProduct f:ℝ)⁻¹ |
| |
| theorem hardBad_subset_bad {Y w top Cs eta xi : ℝ} (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (hsearch : Cs ≤ w^((1/4:ℝ))) (mult : Fin (binCount w top xi) → ℕ) (a : ℕ → ℤ) (q : ℚ) : |
| hardBadSelections (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi mult a q ⊆ |
| badSelections (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi mult a q := by |
| intro f hf |
| obtain ⟨hf,halign,hbad⟩ := Finset.mem_filter.mp hf |
| refine Finset.mem_filter.mpr ⟨hf,?_,hbad⟩ |
| intro b hb p hp |
| have hbin := ((mem_selections _ mult f).mp hf b).1 hp |
| have hx := (bin_exponent_bounds hw (endpoint_pos (zero_lt_one.trans hw) _) |
| (effectiveWidth_pos (zero_lt_one.trans hw) htop hxi).le hbin).1 |
| have hs := (Finset.mem_filter.mp hb).2.1 |
| change w^((1/4:ℝ)) ≤ leftExponent w (lower w top xi b) at hs |
| exact halign p ((mem_descendingWord f p).mpr ⟨b,hp⟩) (hsearch.trans hs |>.trans_lt hx) |
|
|
| theorem hardBadMass_le_badMass {Y w top Cs eta xi : ℝ} (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (hsearch : Cs ≤ w^((1/4:ℝ))) (mult : Fin (binCount w top xi) → ℕ) (a : ℕ → ℤ) (q : ℚ) : |
| hardBadMass (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi mult a q ≤ |
| badMass (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi mult a q := by |
| exact Finset.sum_le_sum_of_subset_of_nonneg (hardBad_subset_bad hw htop hxi hsearch mult a q) |
| (fun f _ _ => inv_nonneg.mpr (Nat.cast_nonneg _)) |
| end NumberTheoryLean.HardTagBoxMass |
|
|
|
|
|
|
| namespace NumberTheoryLean.SafeAnchorSearchCount |
| open _root_.Filter _root_.Erdos970.Filter FinitePathGeometry PrimeHistories PrimeBinMembership ActualPrimeHigh |
| open StrongReferenceTransport StrongSourceFamilies SafeSubsetBoxGeometry |
| open LogarithmicBinScale LogarithmicBinEndpoints LogarithmicBinLabels |
| open ErdosCofactorChoices ErdosSubsetWord ErdosSearchCount |
|
|
|
|
| theorem safe_anchor_search_count (K Clen : ℝ) (hC : 0 ≤ Clen) : |
| ∀ᶠ w : ℝ in atTop,∀ (top B xi : ℝ) (hw : 1 < w) (htop : w < top) (hxi : 0 < xi), |
| xi ≤ 1 → Real.log B ≤ 2*Real.log w → ∀ z : Node, |
| Valid z.side z.ratio → Consistent z → StrongState z → z.closed=true → w^z.cutoff=top → w ≤ z.gap → |
| ∀ mult : Fin (binCount w top xi) → ℕ,SafeAnchor hw htop hxi Clen B K z mult → |
| (∀ b ∈ occupiedSearch w top xi mult,mult b=1) → |
| searchConstant*Real.log w ≤ ((occupiedSearch w top xi mult).card:ℝ) ∧ |
| (∑ b,(mult b:ℝ)) ≤ Clen*Real.log B := by |
| filter_upwards [actual_search_count_uniform K Clen hC] with w hcount |
| intro top B xi hw htop hxi hxi1 hcomp z hs hz hstrong hclosed hcap hstart mult ha hsingle |
| obtain ⟨f,hf,hlen,hsafe,hK⟩ := ha |
| have ha' : SafeAnchor hw htop hxi Clen B K z mult := ⟨f,hf,hlen,hsafe,hK⟩ |
| have href := (anchored_choice_geometry hw htop hxi hC hcomp z mult ha' f hf).1 |
| have htransport := source_reference_transport hw htop z hs hz hstrong hclosed hcap (descendingWord f) href |
| obtain ⟨U,_hU,hallowed⟩ := (uncapped_iff_exists_ceiling w 1 z (descendingWord f)).mp |
| ((mem_uncappedPrefixes hw z (descendingWord f)).mp htransport.1) |
| have hsource := word_source_membership hw htop hxi mult f hf |
| have hc := hcount top B xi 1 U hw htop hxi hxi1 hcomp (by norm_num) z (descendingWord f) |
| hs hstart hallowed hK hlen hsource |
| rw [word_search_count_eq_occupied hw htop hxi mult f hf hsingle] at hc |
| refine ⟨hc,?_⟩ |
| have hl : ((descendingWord f).length:ℝ)=∑ b,(mult b:ℝ) := by |
| exact_mod_cast global_word_length hw htop hxi mult f hf |
| rwa [← hl] |
| end NumberTheoryLean.SafeAnchorSearchCount |
|
|
|
|
|
|
| namespace NumberTheoryLean.HardTagWordFamily |
| open HardTagBoxMass FiniteBoxWordMass ErdosCofactorChoices ErdosSubsetWord |
| open LogarithmicBinScale LogarithmicBinPartition |
| open ErdosPrimeInputs.PrimePrefixMass ErdosPrimeInputs.PrimePrefixTail |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| private theorem finite_union_mass_le {α β : Type*} [DecidableEq α] [DecidableEq β] |
| (M : Finset β) (T : β → Finset α) (W : α → ℝ) (hW : ∀ x,0 ≤ W x) : |
| (∑ x ∈ M.biUnion T,W x) ≤ ∑ b ∈ M,∑ x ∈ T b,W x := by |
| induction M using Finset.induction_on with |
| | empty => simp |
| | @insert b M hb ih => |
| rw [Finset.biUnion_insert,Finset.sum_insert hb] |
| have he : (∑ x ∈ T b ∪ M.biUnion T,W x)+(∑ x ∈ T b ∩ M.biUnion T,W x)= |
| (∑ x ∈ T b,W x)+(∑ x ∈ M.biUnion T,W x) := Finset.sum_union_inter |
| have hn : 0 ≤ ∑ x ∈ T b ∩ M.biUnion T,W x := Finset.sum_nonneg (fun x _ => hW x) |
| linarith |
|
|
| noncomputable def hardBadWords {Y w top Cs eta xi : ℝ} (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (mult : Fin (binCount w top xi) → ℕ) (a : ℕ → ℤ) (q : ℚ) : Finset (List ℕ) := |
| (hardBadSelections (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi mult a q).image descendingWord |
|
|
| theorem hardBadWords_mass {Y w top Cs eta xi : ℝ} (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (mult : Fin (binCount w top xi) → ℕ) (a : ℕ → ℤ) (q : ℚ) : |
| (∑ ps ∈ hardBadWords (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi mult a q,prefixWeight ps)= |
| hardBadMass (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi mult a q := by |
| have hd : Pairwise (fun i j => Disjoint (globalBins w top xi i) (globalBins w top xi j)) := |
| fun i j hij => bins_pairwise_disjoint (zero_lt_one.trans hw) htop hxi i j hij |
| rw [hardBadWords,Finset.sum_image] |
| · apply Finset.sum_congr rfl |
| intro f hf |
| have hs := (Finset.mem_filter.mp hf).1 |
| rw [StoppedVertexHistory.prefixWeight_product,descendingWord_product (globalBins w top xi) hd f |
| (fun i => ((mem_selections _ mult f).mp hs i).1)] |
| · intro f hf g hg he |
| exact word_injective (globalBins w top xi) hd f g |
| (fun i => ((mem_selections _ mult f).mp (Finset.mem_filter.mp hf).1 i).1) |
| (fun i => ((mem_selections _ mult g).mp (Finset.mem_filter.mp hg).1 i).1) he |
|
|
| noncomputable def hardWordFamily {Y w top Cs eta xi : ℝ} (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (M : Finset (Fin (binCount w top xi) → ℕ)) (Q : (Fin (binCount w top xi) → ℕ) → Finset ℚ) |
| (a : ℕ → ℤ) : Finset (List ℕ) := |
| M.biUnion (fun mult => (Q mult).biUnion (fun q => hardBadWords (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi mult a q)) |
|
|
| theorem hardWordFamily_mass_le {Y w top Cs eta xi : ℝ} (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (M : Finset (Fin (binCount w top xi) → ℕ)) (Q : (Fin (binCount w top xi) → ℕ) → Finset ℚ) |
| (a : ℕ → ℤ) : |
| (∑ ps ∈ hardWordFamily (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi M Q a,prefixWeight ps) ≤ |
| ∑ mult ∈ M,∑ q ∈ Q mult,hardBadMass (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi mult a q := by |
| apply (finite_union_mass_le M _ prefixWeight prefixWeight_nonneg).trans |
| apply Finset.sum_le_sum |
| intro mult _hm |
| have hh := finite_union_mass_le (Q mult) |
| (fun q => hardBadWords (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi mult a q) prefixWeight prefixWeight_nonneg |
| simpa only [hardBadWords_mass] using hh |
| end NumberTheoryLean.HardTagWordFamily |
|
|
|
|
|
|
| namespace NumberTheoryLean.SafeBoxTagRate |
| open _root_.Filter _root_.Erdos970.Filter FinitePathGeometry PrimeHistories PrimeBinMembership StrongReferenceTransport |
| open SafeSubsetBoxGeometry SafeAnchorSearchCount SubsetTagFailureMass CanonicalSubsetBox |
| open LogarithmicBinScale LogarithmicBinEndpoints LogarithmicBinLabels LogarithmicBinMaps |
| open CorrectSearchTags UniformSearchOwner ActualBinOwners ActualSourceTags |
| open ErdosCofactorChoices ErdosSubsetWord ErdosTagEvent ErdosSearchCount |
| open ErdosInversePrimeBin ErdosInverseAlignment |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| theorem safe_box_tag_rate {K Clen Cs eta xi : ℝ} (hC : 0 ≤ Clen) (hCs : 0 ≤ Cs) |
| (heta0 : 0 < eta) (heta : eta < 1/2) (hxi : 0 < xi) (hxi1 : xi ≤ 1) : |
| ∀ᶠ w : ℝ in atTop,∀ (B top : ℝ) (hw : 1 < w) (htop : w < top),3 ≤ B → |
| Real.log B ≤ 2*Real.log w → ∀ Y : ℝ,0 ≤ Y → Y ≤ Real.exp (w^2) → ∀ z : Node, |
| z.side=.even → 199/100 ≤ z.ratio → Consistent z → z.cutoff=B → z.closed=true → w^B=top → w ≤ z.gap → |
| ∀ mult : Fin (binCount w top xi) → ℕ,SafeAnchor hw htop hxi Clen B K z mult → |
| (∀ b ∈ occupiedSearch w top xi mult,mult b=1) → ∀ (a : ℕ → ℤ) (q : ℚ),eligible Y w Cs q → |
| badMass (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi mult a q ≤ |
| (w^(-(searchConstant*eta/2))+w^(-1:ℝ))*selectionMass (globalBins w top xi) mult := by |
| have htheta : 0 < xi/4 := by positivity |
| have hwrong := canonical_wrong_mass (Cs := Cs) (theta := xi/4) (A := 1) (Clen := Clen) |
| hCs htheta (by norm_num) hC |
| filter_upwards [safe_anchor_search_count K Clen hC,uniform_correct_owner hCs heta htheta,hwrong] with w hcount howner hwrong |
| intro B top hw htop hB hcomp Y hY hYup z hi h199 hz hcut hclosed hcap hstart mult hanchor hsingle a q hq |
| have hw0 : 0 < w := zero_lt_one.trans hw |
| have hspan : (1+xi)*w ≤ top := by |
| have hp : w^3 ≤ top := by |
| have hh := Real.rpow_le_rpow_of_exponent_le hw.le hB |
| rw [hcap] at hh |
| simpa only [Real.rpow_ofNat] using hh |
| have hw2 : 2 ≤ w := howner.1 |
| have hm := mul_nonneg hw0.le (show 0 ≤ w^2-2 by nlinarith) |
| nlinarith |
| have hwidth := common_width_bounds hw0 htop hxi hxi1 hspan |
| have hvalid : Valid z.side z.ratio := by rw [hi] |
| have hstrong := source_strong_state hB z hi h199 hz hcut |
| have hcounts := hcount top B xi hw htop hxi hxi1 hcomp z hvalid hz hstrong hclosed |
| (by rwa [hcut]) hstart mult hanchor hsingle |
| let I := occupiedSearch w top xi mult |
| let R := lower w top xi |
| have hidentify : ∀ b ∈ I,nearFull eta (R b) (width w top xi b) a q → |
| owner Y w Cs eta (R b) (width w top xi b) a=some q := by |
| intro b hb hnear |
| exact howner.2 Y (R b) (width w top xi b) hY hYup (endpoint_pos hw0 _) |
| (Finset.mem_filter.mp hb).2.1 hwidth.2.1 (hwidth.2.2.1.trans hxi1) a q hq hnear |
| have hwm := hwrong Y (effectiveWidth w top xi) eta B hY hYup hwidth.2.1 |
| (hwidth.2.2.1.trans hxi1) hcomp (binCount w top xi) R mult hcounts.2 I hsingle |
| (fun b hb => searchBin_lower hw (endpoint_pos hw0 _) (Finset.mem_filter.mp hb).2.1) a q hq |
| have hmass : 0 ≤ selectionMass (globalBins w top xi) mult := |
| Finset.sum_nonneg (fun f _ => inv_nonneg.mpr (Nat.cast_nonneg _)) |
| by_cases hnone : ∀ b ∈ I,¬nearFull eta (R b) (width w top xi b) a q |
| · have hno := canonical_no_nearfull_mass R w searchConstant (effectiveWidth w top xi) eta mult |
| (fun b => endpoint_pos hw0 _) hw searchConstant_pos hwidth.1.le (hwidth.2.2.1.trans hxi1) |
| heta0 (by linarith) I hsingle hcounts.1 a q (fun b hb => (nonnearfull_card_lt _ _ _ _ _ (hnone b hb)).le) |
| have hh := (bad_mass_le_alignment (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi mult a q).trans hno |
| exact hh.trans (mul_le_mul_of_nonneg_right |
| (le_add_of_nonneg_right (Real.rpow_nonneg hw0.le (-1))) hmass) |
| · push Not at hnone |
| have hh := (bad_mass_le_wrong (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi mult a q hidentify hnone).trans hwm |
| exact hh.trans (mul_le_mul_of_nonneg_right |
| (le_add_of_nonneg_left (Real.rpow_nonneg hw0.le (-(searchConstant*eta/2)))) hmass) |
| end NumberTheoryLean.SafeBoxTagRate |
|
|
|
|
|
|
| namespace NumberTheoryLean.RegularBadTagWords |
| open FinitePathGeometry PrimeHistories SourceStopPredicate ActualWordSelection ActualRegularBoxes |
| open LogarithmicBinScale LogarithmicBinEndpoints LogarithmicBinLabels LogarithmicBinPartition |
| open ErdosSubsetWord ErdosCofactorChoices HardTagWordFamily HardTagBoxMass ActualSourceTags |
| open ErdosPrimeInputs.PrimePrefixMass ErdosPrimeInputs.PrimePrefixTail |
| open ErdosPrimeInputs.HarmonicPrimeMeasure ErdosInverseAlignment |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| noncomputable def badWordTest {Y w top xi Cs eta : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (Q : (Fin (binCount w top xi) → ℕ) → Finset ℚ) (a : ℕ → ℤ) (ps : List ℕ) : Prop := |
| ∃ q ∈ Q (wordMultiplicity (label (zero_lt_one.trans hw) htop hxi) ps), |
| (∀ p ∈ ps,Cs < primeExponent w p → aligns a q p) ∧ |
| ¬∃ t,sourceTag Y w Cs eta (lower w top xi) (width w top xi) |
| (label (zero_lt_one.trans hw) htop hxi) a ps=some t ∧ t.rational=q |
|
|
| noncomputable def regularBadWords {Y w top xi Cs eta : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (Clen B K : ℝ) (z : Node) |
| (Q : (Fin (binCount w top xi) → ℕ) → Finset ℚ) (a : ℕ → ℤ) : Finset (List ℕ) := |
| (regularWords hw htop hxi Clen B K z).filter (badWordTest (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi Q a) |
|
|
| theorem regularBadWords_subset {Y w top xi Cs eta : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (Clen B K : ℝ) (z : Node) |
| (Q : (Fin (binCount w top xi) → ℕ) → Finset ℚ) (a : ℕ → ℤ) : |
| regularBadWords (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi Clen B K z Q a ⊆ |
| hardWordFamily (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi |
| (regularBoxes hw htop hxi Clen B K z) Q a := by |
| intro ps hp |
| obtain ⟨hr,q,hq,ha,hbad⟩ := Finset.mem_filter.mp hp |
| have hd := mem_decreasingPrefixes.mp (Finset.mem_filter.mp hr).1 |
| let lab := label (zero_lt_one.trans hw) htop hxi |
| have hrec := wordSelection_recovers lab ps hd.1 |
| apply Finset.mem_biUnion.mpr |
| refine ⟨wordMultiplicity lab ps,Finset.mem_image.mpr ⟨ps,hr,rfl⟩,?_⟩ |
| apply Finset.mem_biUnion.mpr |
| refine ⟨q,hq,Finset.mem_image.mpr ⟨wordSelection lab ps,?_,hrec⟩⟩ |
| apply Finset.mem_filter.mpr |
| refine ⟨wordSelection_mem hw htop hxi ps hd.2,?_⟩ |
| simpa only [hrec] using (show (∀ p ∈ ps,Cs < primeExponent w p → aligns a q p) ∧ |
| ¬∃ t,sourceTag Y w Cs eta (lower w top xi) (width w top xi) lab a ps=some t ∧ t.rational=q from ⟨ha,hbad⟩) |
|
|
| theorem regularBadWords_mass_le {Y w top xi Cs eta : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) (Clen B K : ℝ) (z : Node) |
| (Q : (Fin (binCount w top xi) → ℕ) → Finset ℚ) (a : ℕ → ℤ) : |
| (∑ ps ∈ regularBadWords (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi Clen B K z Q a,prefixWeight ps) ≤ |
| ∑ ps ∈ hardWordFamily (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi |
| (regularBoxes hw htop hxi Clen B K z) Q a,prefixWeight ps := |
| Finset.sum_le_sum_of_subset_of_nonneg (regularBadWords_subset hw htop hxi Clen B K z Q a) |
| (fun ps _ _ => prefixWeight_nonneg ps) |
| end NumberTheoryLean.RegularBadTagWords |
|
|
|
|
|
|
| namespace NumberTheoryLean.GlobalSafeTagCapture |
| open _root_.Filter _root_.Erdos970.Filter FinitePathGeometry PrimeHistories PrimeBinMembership |
| open SafeSubsetBoxGeometry SafeBoxTagRate ActualSafeBoxMass SubsetTagFailureMass CanonicalSubsetBox |
| open LogarithmicBinScale ActualBinOwners ErdosSubsetWord ErdosSearchCount |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| private theorem finite_candidate_sum {β : Type*} (M : Finset β) (Q : β → Finset ℚ) |
| (H : ℕ) (r : ℝ) (hr : 0 ≤ r) (F : β → ℝ) (hF : ∀ b ∈ M,0 ≤ F b) |
| (G : β → ℚ → ℝ) (hG : ∀ b ∈ M,∀ q ∈ Q b,G b q ≤ r*F b) |
| (hQ : ∀ b ∈ M,(Q b).card ≤ H) : |
| (∑ b ∈ M,∑ q ∈ Q b,G b q) ≤ (H:ℝ)*r*(∑ b ∈ M,F b) := by |
| calc |
| _ ≤ ∑ b ∈ M,∑ _q ∈ Q b,r*F b := Finset.sum_le_sum (fun b hb => Finset.sum_le_sum (fun q hq => hG b hb q hq)) |
| _ = ∑ b ∈ M,((Q b).card:ℝ)*(r*F b) := by simp |
| _ ≤ ∑ b ∈ M,(H:ℝ)*(r*F b) := Finset.sum_le_sum (fun b hb => |
| mul_le_mul_of_nonneg_right (by exact_mod_cast hQ b hb) (mul_nonneg hr (hF b hb))) |
| _ = _ := by rw [Finset.mul_sum]; apply Finset.sum_congr rfl |
|
|
| theorem global_bad_tag_mass_small {K Clen Cs eta xi : ℝ} (hK : 3 ≤ K) (hC : 0 ≤ Clen) (hCs : 0 ≤ Cs) |
| (heta0 : 0 < eta) (heta : eta < 1/2) (hxi : 0 < xi) (hxi1 : xi ≤ 1) |
| (hDelta : 2*Clen*xi ≤ 1) (H : ℕ) (epsilon : ℝ) (hepsilon : 0 < epsilon) : |
| ∃ B₀ w₀ : ℝ,3 ≤ B₀ ∧ 1 < w₀ ∧ ∀ (B w top : ℝ) (hw : 1 < w) (htop : w < top), |
| B₀ ≤ B → w₀ ≤ w → Real.log B ≤ 2*Real.log w → ∀ Y : ℝ,0 ≤ Y → Y ≤ Real.exp (w^2) → |
| ∀ (z : Node) (a : ℕ → ℤ),z.side=.even → 199/100 ≤ z.ratio → z.ratio ≤ 23/10 → |
| Consistent z → z.cutoff=B → z.closed=true → w^B=top → w ≤ z.gap → |
| ∀ (M : Finset (Fin (binCount w top xi) → ℕ)) (Q : (Fin (binCount w top xi) → ℕ) → Finset ℚ), |
| (∀ mult ∈ M,SafeAnchor hw htop hxi Clen B K z mult) → |
| (∀ mult ∈ M,∀ b ∈ occupiedSearch w top xi mult,mult b=1) → |
| (∀ mult ∈ M,(Q mult).card ≤ H) → (∀ mult ∈ M,∀ q ∈ Q mult,eligible Y w Cs q) → |
| B^2*(∑ mult ∈ M,∑ q ∈ Q mult,badMass (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi mult a q) ≤ epsilon := by |
| obtain ⟨C,B₀,wbox,hCpos,hB₀,hwbox,hboxes⟩ := uniform_safe_box_mass K hK |
| have hd : 0 < searchConstant*eta/2 := by have hc := searchConstant_pos |
| have hrate : Tendsto (fun w : ℝ => (H:ℝ)*C*(w^(-(searchConstant*eta/2))+w^(-1:ℝ))) atTop (nhds 0) := by |
| have hh := ((tendsto_rpow_neg_atTop hd).add (tendsto_rpow_neg_atTop (by norm_num : (0:ℝ)<1))).const_mul ((H:ℝ)*C) |
| simpa only [zero_add,mul_zero] using hh |
| have hsmall : ∀ᶠ w : ℝ in atTop,(H:ℝ)*C*(w^(-(searchConstant*eta/2))+w^(-1:ℝ)) ≤ epsilon := |
| ((tendsto_order.mp hrate).2 epsilon hepsilon).mono (fun _ h => h.le) |
| have htag := safe_box_tag_rate (K := K) (Clen := Clen) (Cs := Cs) (eta := eta) (xi := xi) hC hCs heta0 heta hxi hxi1 |
| obtain ⟨w₁,hw₁⟩ := eventually_atTop.mp (htag.and (hsmall.and (eventually_ge_atTop wbox))) |
| refine ⟨B₀,max 2 w₁,hB₀,by have hh := le_max_left (2:ℝ) w₁ |
| intro B w top hw htop hB hW hcomp Y hY hYup z a hi h199 h23 hz hcut hclosed hcap hstart M Q hanchor hsingle hcard helig |
| have hgood := hw₁ w ((le_max_right _ _).trans hW) |
| have hB3 := hB₀.trans hB |
| let r := w^(-(searchConstant*eta/2))+w^(-1:ℝ) |
| have hw0 : 0 < w := zero_lt_one.trans hw |
| have hr : 0 ≤ r := add_nonneg (Real.rpow_nonneg hw0.le _) (Real.rpow_nonneg hw0.le _) |
| have hm := finite_candidate_sum M Q H r hr (selectionMass (globalBins w top xi)) |
| (fun mult _ => Finset.sum_nonneg (fun f _ => inv_nonneg.mpr (Nat.cast_nonneg _))) |
| (fun mult q => badMass (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi mult a q) |
| (fun mult hmult q hq => hgood.1 B top hw htop hB3 hcomp Y hY hYup z hi h199 hz hcut hclosed hcap hstart |
| mult (hanchor mult hmult) (hsingle mult hmult) a q (helig mult hmult q hq)) hcard |
| have hbox := hboxes B w top hB hgood.2.2 hw htop xi Clen hxi hC hcomp hDelta z hi h199 h23 hz hcut hclosed hcap M hanchor |
| calc |
| _ ≤ B^2*((H:ℝ)*r*(∑ mult ∈ M,selectionMass (globalBins w top xi) mult)) := mul_le_mul_of_nonneg_left hm (sq_nonneg B) |
| _ = (H:ℝ)*r*(B^2*(∑ mult ∈ M,selectionMass (globalBins w top xi) mult)) := by ring |
| _ ≤ (H:ℝ)*r*C := mul_le_mul_of_nonneg_left hbox (mul_nonneg (Nat.cast_nonneg _) hr) |
| _ = (H:ℝ)*C*r := by ring |
| _ ≤ epsilon := hgood.2.1 |
| end NumberTheoryLean.GlobalSafeTagCapture |
|
|
|
|
|
|
| namespace NumberTheoryLean.PaperSafeTagCapture |
| open _root_.Filter _root_.Erdos970.Filter FinitePathGeometry PrimeHistories PrimeBinMembership SourceNodeCoordinates |
| open JacobsthalSourceScale PaperMissingPrimeWindow PaperOwnerHeight PaperBinOwners |
| open SafeSubsetBoxGeometry GlobalSafeTagCapture HardTagBoxMass ActualBinOwners |
| open LogarithmicBinScale ErdosSubsetWord |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| theorem paper_safe_hard_tag_capture {K Clen Cs eta xi : ℝ} (hK : 3 ≤ K) (hC : 0 ≤ Clen) (hCs : 0 ≤ Cs) |
| (heta0 : 0 < eta) (heta : eta < 1/2) (hxi : 0 < xi) (hxi1 : xi ≤ 1) |
| (hDelta : 2*Clen*xi ≤ 1) (H : ℕ) (epsilon : ℝ) (hepsilon : 0 < epsilon) : |
| ∀ᶠ top : ℝ in atTop,∃ hw : 1 < sourceW (Real.log top),∃ htop : sourceW (Real.log top) < top, |
| ∀ Y : ℕ,Y ≤ ⌊top^2/(Real.log top)^2⌋₊ → ∀ (z : Node) (a : ℕ → ℤ), |
| z.side=.even → 199/100 ≤ z.ratio → z.ratio ≤ 23/10 → Consistent z → z.cutoff=sourceB (Real.log top) → z.closed=true → |
| ∀ (M : Finset (Fin (binCount (sourceW (Real.log top)) top xi) → ℕ)) |
| (Q : (Fin (binCount (sourceW (Real.log top)) top xi) → ℕ) → Finset ℚ), |
| (∀ mult ∈ M,SafeAnchor hw htop hxi Clen (sourceB (Real.log top)) K z mult) → |
| (∀ mult ∈ M,∀ b ∈ occupiedSearch (sourceW (Real.log top)) top xi mult,mult b=1) → |
| (∀ mult ∈ M,(Q mult).card ≤ H) → |
| (∀ mult ∈ M,∀ q ∈ Q mult,eligible Y (sourceW (Real.log top)) Cs q) → |
| (sourceB (Real.log top))^2 * |
| (∑ mult ∈ M,∑ q ∈ Q mult,hardBadMass (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi mult a q) ≤ epsilon := by |
| obtain ⟨B₀,w₀,_hB₀,_hw₀,hbound⟩ := global_bad_tag_mass_small hK hC hCs heta0 heta hxi hxi1 hDelta H epsilon hepsilon |
| have hwT := sourceW_tendsto.comp Real.tendsto_log_atTop |
| have hBT := sourceB_tendsto.comp Real.tendsto_log_atTop |
| have hsearch : ∀ᶠ top : ℝ in atTop,Cs ≤ (sourceW (Real.log top))^((1/4:ℝ)) := |
| ((tendsto_rpow_atTop (by norm_num : (0:ℝ)<1/4)).comp hwT).eventually_ge_atTop Cs |
| filter_upwards [paper_bin_span hxi hxi1,Real.tendsto_log_atTop.eventually source_scale_eventually, |
| Real.tendsto_log_atTop.eventually sourceW_le_sourceB_eventually,literal_source_floor_height, |
| hwT.eventually_ge_atTop w₀,hBT.eventually_ge_atTop B₀,eventually_gt_atTop (1:ℝ),hsearch] |
| with top hspan hscale hWB hheight hwBound hBBound htop1 hsearch |
| have hw : 1 < sourceW (Real.log top) := by linarith [hspan.1] |
| refine ⟨hw,hspan.2.1,?_⟩ |
| intro Y hY z a hi h199 h23 hz hcut hclosed M Q hanchors hsingle hcard helig |
| have hYup : (Y:ℝ) ≤ Real.exp ((sourceW (Real.log top))^2) := |
| (show (Y:ℝ) ≤ (⌊top^2/(Real.log top)^2⌋₊:ℕ) by exact_mod_cast hY).trans hheight |
| have hcap : (sourceW (Real.log top))^(sourceB (Real.log top))=top := |
| (sourceW_pow_sourceB hw).trans (Real.exp_log (zero_lt_one.trans htop1)) |
| have hgap := (source_node_bounds hscale.2.1 z hi h199 h23 hz hcut).2.2.1 |
| have hstart := hWB.trans hgap |
| have hh := hbound (sourceB (Real.log top)) (sourceW (Real.log top)) top hw hspan.2.1 hBBound hwBound |
| hscale.2.2.2.2 Y (Nat.cast_nonneg Y) hYup z a hi h199 h23 hz hcut hclosed hcap hstart M Q hanchors hsingle hcard helig |
| apply le_trans _ hh |
| apply mul_le_mul_of_nonneg_left _ (sq_nonneg _) |
| apply Finset.sum_le_sum |
| intro mult _hm |
| apply Finset.sum_le_sum |
| intro q _hq |
| exact hardBadMass_le_badMass hw hspan.2.1 hxi hsearch mult a q |
| end NumberTheoryLean.PaperSafeTagCapture |
|
|
|
|
|
|
| namespace NumberTheoryLean.PaperSafeWordCapture |
| open _root_.Filter _root_.Erdos970.Filter FinitePathGeometry PrimeHistories PrimeBinMembership |
| open JacobsthalSourceScale SafeSubsetBoxGeometry HardTagWordFamily ActualBinOwners |
| open LogarithmicBinScale ErdosSubsetWord PaperSafeTagCapture |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| theorem paper_safe_hard_word_capture {K Clen Cs eta xi : ℝ} (hK : 3 ≤ K) (hC : 0 ≤ Clen) (hCs : 0 ≤ Cs) |
| (heta0 : 0 < eta) (heta : eta < 1/2) (hxi : 0 < xi) (hxi1 : xi ≤ 1) |
| (hDelta : 2*Clen*xi ≤ 1) (H : ℕ) (epsilon : ℝ) (hepsilon : 0 < epsilon) : |
| ∀ᶠ top : ℝ in atTop,∃ hw : 1 < sourceW (Real.log top),∃ htop : sourceW (Real.log top) < top, |
| ∀ Y : ℕ,Y ≤ ⌊top^2/(Real.log top)^2⌋₊ → ∀ (z : Node) (a : ℕ → ℤ), |
| z.side=.even → 199/100 ≤ z.ratio → z.ratio ≤ 23/10 → Consistent z → z.cutoff=sourceB (Real.log top) → z.closed=true → |
| ∀ (M : Finset (Fin (binCount (sourceW (Real.log top)) top xi) → ℕ)) |
| (Q : (Fin (binCount (sourceW (Real.log top)) top xi) → ℕ) → Finset ℚ), |
| (∀ mult ∈ M,SafeAnchor hw htop hxi Clen (sourceB (Real.log top)) K z mult) → |
| (∀ mult ∈ M,∀ b ∈ occupiedSearch (sourceW (Real.log top)) top xi mult,mult b=1) → |
| (∀ mult ∈ M,(Q mult).card ≤ H) → |
| (∀ mult ∈ M,∀ q ∈ Q mult,eligible Y (sourceW (Real.log top)) Cs q) → |
| (sourceB (Real.log top))^2 * |
| (∑ ps ∈ hardWordFamily (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi M Q a,ErdosPrimeInputs.PrimePrefixMass.prefixWeight ps) ≤ epsilon := by |
| filter_upwards [paper_safe_hard_tag_capture hK hC hCs heta0 heta hxi hxi1 hDelta H epsilon hepsilon] with top htopData |
| obtain ⟨hw,htop,hbound⟩ := htopData |
| refine ⟨hw,htop,?_⟩ |
| intro Y hY z a hi h199 h23 hz hcut hclosed M Q hanchors hsingle hcard helig |
| have hh := hbound Y hY z a hi h199 h23 hz hcut hclosed M Q hanchors hsingle hcard helig |
| exact (mul_le_mul_of_nonneg_left (hardWordFamily_mass_le (Y := Y) (Cs := Cs) (eta := eta) |
| hw htop hxi M Q a) (sq_nonneg (sourceB (Real.log top)))).trans hh |
| end NumberTheoryLean.PaperSafeWordCapture |
|
|
|
|
|
|
| namespace NumberTheoryLean.PaperRegularTagCapture |
| open _root_.Filter _root_.Erdos970.Filter FinitePathGeometry PrimeHistories PrimeBinMembership |
| open JacobsthalSourceScale ActualRegularBoxes ActualPrefixClearance RegularBadTagWords |
| open PaperSafeWordCapture ActualBinOwners LogarithmicBinScale |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| theorem paper_regular_bad_tag_mass {K Clen Cs eta xi : ℝ} (hK : 3 ≤ K) (hC : 1 ≤ Clen) (hCs : 0 ≤ Cs) |
| (heta0 : 0 < eta) (heta : eta < 1/2) (hxi : 0 < xi) (hxi1 : xi ≤ 1) |
| (hDelta : 2*Clen*xi ≤ 1) (H : ℕ) (epsilon : ℝ) (hepsilon : 0 < epsilon) : |
| ∀ᶠ top : ℝ in atTop,∃ hw : 1 < sourceW (Real.log top),∃ htop : sourceW (Real.log top) < top, |
| ∀ Y : ℕ,Y ≤ ⌊top^2/(Real.log top)^2⌋₊ → ∀ (z : Node) (a : ℕ → ℤ), |
| z.side=.even → 199/100 ≤ z.ratio → z.ratio ≤ 23/10 → Consistent z → z.cutoff=sourceB (Real.log top) → z.closed=true → |
| ∀ Q : (Fin (binCount (sourceW (Real.log top)) top xi) → ℕ) → Finset ℚ, |
| (∀ mult ∈ regularBoxes hw htop hxi Clen (sourceB (Real.log top)) K z,(Q mult).card ≤ H) → |
| (∀ mult ∈ regularBoxes hw htop hxi Clen (sourceB (Real.log top)) K z, |
| ∀ q ∈ Q mult,eligible Y (sourceW (Real.log top)) Cs q) → |
| (sourceB (Real.log top))^2 * |
| (∑ ps ∈ regularBadWords (Y := Y) (Cs := Cs) (eta := eta) hw htop hxi Clen |
| (sourceB (Real.log top)) K z Q a,ErdosPrimeInputs.PrimePrefixMass.prefixWeight ps) ≤ epsilon := by |
| have hC0 : 0 ≤ Clen := by linarith |
| have hwT := sourceW_tendsto.comp Real.tendsto_log_atTop |
| filter_upwards [paper_safe_hard_word_capture hK hC0 hCs heta0 heta hxi hxi1 hDelta H epsilon hepsilon, |
| Real.tendsto_log_atTop.eventually source_scale_eventually, |
| (Real.tendsto_log_atTop.comp hwT).eventually_ge_atTop 1] with top hbound hscale hlog |
| obtain ⟨hw,htop,hbound⟩ := hbound |
| refine ⟨hw,htop,?_⟩ |
| intro Y hY z a hi h199 h23 hz hcut hclosed Q hcard helig |
| have hsmall := source_mesh_clearance_condition hlog hxi.le hC |
| have hh := hbound Y hY z a hi h199 h23 hz hcut hclosed |
| (regularBoxes hw htop hxi Clen (sourceB (Real.log top)) K z) Q |
| (regular_boxes_anchors hw htop hxi hC0 hscale.2.2.2.2 hsmall z) |
| (regular_boxes_search_singletons hw htop hxi z) hcard helig |
| exact (mul_le_mul_of_nonneg_left (regularBadWords_mass_le (Y := Y) (Cs := Cs) (eta := eta) |
| hw htop hxi Clen (sourceB (Real.log top)) K z Q a) (sq_nonneg _)).trans hh |
| end NumberTheoryLean.PaperRegularTagCapture |
|
|
|
|
|
|
| namespace NumberTheoryLean.FullWordHighAlignment |
| open ErdosSubsetWord ErdosCofactorChoices ErdosInverseAlignment |
| open LogarithmicBinScale LogarithmicBinLabels LogarithmicBinPartition LogarithmicBinEndpoints |
| open ParentTailPartition ParentCofactorChoices BinCutSelections SingletonBinSelection |
| open TwoPrimeObservableSum BoundedEdgeBins BoundedEdgeHighPrimes |
| open ErdosPrimeInputs.HarmonicPrimeMeasure |
|
|
|
|
| theorem parent_prime_factors {w top xi : ℝ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (m : Fin (binCount w top xi) → ℕ) (f : Fin (binCount w top xi) → Finset ℕ) |
| (hf : f ∈ selections (globalBins w top xi) m) (i j : Fin (binCount w top xi)) : |
| (selectionProduct (parentCofactorSelection f i j)).primeFactors= |
| selectionPrimes (parentCofactorSelection f i j) := by |
| have hc := erase_selection_mem _ _ _ (above_selection_mem _ m f hf j) i |
| exact selectionProduct_primeFactors (globalBins w top xi) |
| (fun k _p hp => (bin_prime_in_source (zero_lt_one.trans hw) htop hxi k hp).1) |
| (fun k l hkl => bins_pairwise_disjoint (zero_lt_one.trans hw) htop hxi k l hkl) |
| (parentCofactorSelection f i j) (fun k => ((mem_selections _ _ _).mp hc k).1) |
|
|
| theorem full_word_high_alignment {w top xi Cs M X : ℝ} {Y : ℕ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (hCs : X+xi/Real.log w ≤ Cs) |
| (m : Fin (binCount w top xi) → ℕ) (f : Fin (binCount w top xi) → Finset ℕ) |
| (hf : f ∈ selections (globalBins w top xi) m) (i j : Fin (binCount w top xi)) |
| (hi : m i=1) (hj : boundedEdgeBin w top xi Y m M X j) |
| (a : ℕ → ℤ) (q : ℚ) (hiso : aligns a q (pickedPrime f i)) |
| (hparent : ∀ p∈(selectionProduct (parentCofactorSelection f i j)).primeFactors, |
| Cs < primeExponent w p → aligns a q p) : |
| ∀ p∈descendingWord f,Cs < primeExponent w p → aligns a q p := by |
| intro p hp hhigh |
| obtain ⟨k,hpk⟩ := (mem_descendingWord f p).mp hp |
| by_cases hki : k=i |
| · subst k |
| obtain ⟨p0,_hp0,hfi,_hfill⟩ := singleton_selection_recovery _ m i hi f hf |
| have he : p=p0 := by simpa only [hfi,Finset.mem_singleton] using hpk |
| simpa [pickedPrime,hfi,he] using hiso |
| · have hjk : j < k := by |
| by_contra! hkj |
| have hpbin := ((mem_selections _ _ _).mp hf k).1 hpk |
| exact (not_le_of_gt hhigh) ((bounded_edge_descendant_exponent hw htop hxi m j k hj hkj hpbin).trans hCs) |
| apply hparent p _ hhigh |
| rw [parent_prime_factors hw htop hxi m f hf i j] |
| apply Finset.mem_biUnion.mpr |
| exact ⟨k,Finset.mem_univ _,by simpa [parentCofactorSelection,eraseSelection,aboveSelection,hki,hjk] using hpk⟩ |
|
|
| theorem full_word_high_alignment_fixed_cutoff {w top xi Cs M X : ℝ} {Y : ℕ} |
| (hw : 1 < w) (htop : w < top) (hxi : 0 < xi) |
| (hlog : 1 ≤ Real.log w) (hxi1 : xi ≤ 1) (hCs : X+1 ≤ Cs) |
| (m : Fin (binCount w top xi) → ℕ) (f : Fin (binCount w top xi) → Finset ℕ) |
| (hf : f ∈ selections (globalBins w top xi) m) (i j : Fin (binCount w top xi)) |
| (hi : m i=1) (hj : boundedEdgeBin w top xi Y m M X j) |
| (a : ℕ → ℤ) (q : ℚ) (hiso : aligns a q (pickedPrime f i)) |
| (hparent : ∀ p∈(selectionProduct (parentCofactorSelection f i j)).primeFactors, |
| Cs < primeExponent w p → aligns a q p) : |
| ∀ p∈descendingWord f,Cs < primeExponent w p → aligns a q p := by |
| apply full_word_high_alignment hw htop hxi _ m f hf i j hi hj a q hiso hparent |
| have hh : xi/Real.log w ≤ 1 := (div_le_one (Real.log_pos hw)).mpr (hxi1.trans hlog) |
| linarith |
| end NumberTheoryLean.FullWordHighAlignment |
|
|
|
|
| end Erdos970 |
|
|
| end OAI |
|
|
