| import OAI.NumberTheory.Jacobsthal.Estimates.ActualSourceInverse |
| import OAI.NumberTheory.Jacobsthal.Estimates.IsolatedPowerOrdering |
| import OAI.NumberTheory.Jacobsthal.Sieve.BoxWitnessFactorization |
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| namespace OAI |
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| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| namespace NumberTheoryLean.NormalizedBoxException |
| open ErdosCofactorChoices ErdosInverseCells ErdosInverseSampling CanonicalSubsetBox |
| open BoxWitnessFactorization ParentCofactorChoices BinCutSelections |
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| attribute [local instance] Classical.propDecidable |
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| theorem weighted_witness_le_total (P Q U : Finset ℕ) (W : (ℕ × ℕ) → ℕ → Prop) : |
| weightedWitness P Q U W ≤ harmonicMass P*harmonicMass Q*harmonicMass U := by |
| have hs : witnessPairs (P.product Q) U W ⊆ tripleSpace P Q U := Finset.filter_subset _ _ |
| have hh := Finset.sum_le_sum_of_subset_of_nonneg hs (fun x _ _ => tripleWeight_nonneg x) |
| exact hh.trans_eq (triple_weight_sum P Q U) |
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| theorem unnormalize_witness (P Q U : Finset ℕ) (W : (ℕ × ℕ) → ℕ → Prop) (sigma : ℝ) |
| (h : weightedWitness P Q U W/(harmonicMass P*harmonicMass Q*harmonicMass U) ≤ sigma) : |
| weightedWitness P Q U W ≤ sigma*(harmonicMass P*harmonicMass Q*harmonicMass U) := by |
| have hm : 0 ≤ harmonicMass P*harmonicMass Q*harmonicMass U := |
| mul_nonneg (mul_nonneg (harmonicMass_nonneg P) (harmonicMass_nonneg Q)) (harmonicMass_nonneg U) |
| rcases hm.eq_or_lt with hm | hm |
| · rw [← hm,mul_zero] |
| have hh := weighted_witness_le_total P Q U W |
| rwa [← hm] at hh |
| · exact (div_le_iff₀ hm).mp h |
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| theorem selection_exception_bound {n : ℕ} (P : Fin n → Finset ℕ) (m : Fin n → ℕ) |
| (hP : ∀ i,∀ p ∈ P i,p.Prime) (hd : Pairwise (fun i j => Disjoint (P i) (P j))) |
| (i j : Fin n) (hji : j < i) (hi : m i=1) (hj : m j=1) (W : (ℕ × ℕ) → ℕ → Prop) (sigma : ℝ) |
| (h : weightedWitness (P i) (cofactorChoices P (parentCofactorMultiplicity m i j)) (P j) W/ |
| (harmonicMass (P i)*harmonicMass (cofactorChoices P (parentCofactorMultiplicity m i j))*harmonicMass (P j)) ≤ sigma) : |
| selectionWitnessMass P m i j W ≤ sigma*selectionMass P m := by |
| have hw := unnormalize_witness (P i) (cofactorChoices P (parentCofactorMultiplicity m i j)) (P j) W sigma h |
| rw [selection_witness_factor P m hP hd i j hji hi hj,full_box_mass_factor P m hP hd i j hji hi hj] |
| have hh := mul_le_mul_of_nonneg_left hw (harmonicMass_nonneg (cofactorChoices P (belowMultiplicity m j))) |
| nlinarith |
| end NumberTheoryLean.NormalizedBoxException |
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| namespace NumberTheoryLean.ActualInverseFullBoxMass |
| open ErdosInverseSampleCost ErdosCofactorChoices ErdosInverseCells ErdosInverseSampling ErdosInverseBoxApplication |
| open NormalizedBoxException BoxWitnessFactorization CanonicalSubsetBox ParentCofactorChoices |
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| theorem actual_inverse_full_box_mass {n : ℕ} (P : Fin n → Finset ℕ) (m : Fin n → ℕ) |
| (hP : ∀ i,∀ p ∈ P i,p.Prime) (hd : Pairwise (fun i j => Disjoint (P i) (P j))) |
| (i j : Fin n) (hji : j < i) (hi : m i=1) (hj : m j=1) |
| (Y : ℕ) (small : Finset ℕ) (a : ℕ → ℕ) (delta V0 S R Z cp sigma : ℝ) (F : ℕ) |
| (hInv : SourceInverseConclusion Y small a delta V0 S R Z cp sigma F |
| (P i) (cofactorChoices P (parentCofactorMultiplicity m i j)) (P j)) : |
| selectionWitnessMass P m i j (actualWitness Y small a delta V0 (P i) S R Z cp) ≤ sigma*selectionMass P m := by |
| have hh := hInv.2.2.1 |
| rw [actual_exception_weight] at hh |
| exact selection_exception_bound P m hP hd i j hji hi hj _ sigma hh |
| end NumberTheoryLean.ActualInverseFullBoxMass |
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| namespace NumberTheoryLean.GeometricFullBoxInverse |
| open _root_.Filter FinitePathGeometry PrimeHistories PrimeBinMembership |
| open GeometricBoxImages GeometricInverseGeometry BoundedEdgeBins ParentCofactorChoices |
| open JacobsthalSourceScale IsolatedSourceScales SourceNodeCoordinates |
| open ActualInverseFullBoxMass BoxWitnessFactorization CanonicalSubsetBox IsolatedPowerOrdering ErdosInverseSampleCost |
| open ErdosCofactorChoices ErdosSubsetWord ErdosInverseBoxApplication |
| open LogarithmicBinScale LogarithmicBinEndpoints |
| open scoped Topology |
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| theorem geometric_full_box_inverse (C Kstar aStar alpha delta sigma : ℝ) |
| (hC : 1 ≤ C) (hKstar : 0 ≤ Kstar) (ha : 0 < aStar) (ha1 : aStar ≤ 1) |
| (halpha : 0 < alpha) (hd : 0 < delta) (hsigma : 0 < sigma) (hsigma1 : sigma ≤ 1) : |
| ∃ cp : ℝ,0 < cp ∧ ∃ F : ℕ,∃ xi₀ : ℝ,0 < xi₀ ∧ xi₀ ≤ 1 ∧ |
| ∀ xi : ℝ,∀ hxi : 0 < xi,xi ≤ xi₀ → ∀ᶠ top : ℝ in atTop, |
| ∃ hw : 1 < ErdosInverseBoxHeight.sourceW top,∃ htop : ErdosInverseBoxHeight.sourceW top < top, |
| ∀ R L : ℝ,∀ z : Node,z.side=.even → 199/100 ≤ z.ratio → Consistent z → |
| z.cutoff=ErdosInverseBoxHeight.sourceB top → z.closed=true → |
| z.gap=Real.log (ErdosInverseBoxHeight.sourceY top:ℝ)/Real.log (ErdosInverseBoxHeight.sourceW top)-aStar+2 → |
| ∀ m ∈ geometricBoxes hw htop hxi C (ErdosInverseBoxHeight.sourceB top) R L alpha (1/100) z, |
| ∃ i : Fin (binCount (ErdosInverseBoxHeight.sourceW top) top xi),m i=1 ∧ |
| ∀ j : Fin (binCount (ErdosInverseBoxHeight.sourceW top) top xi), |
| boundedEdgeBin (ErdosInverseBoxHeight.sourceW top) top xi (ErdosInverseBoxHeight.sourceY top) m |
| (2*Kstar+3) (2*Kstar+3) j → ∀ residue : ℕ → ℕ, |
| j < i ∧ m j=1 ∧ |
| let cm := parentCofactorMultiplicity m i j |
| let w := ErdosInverseBoxHeight.sourceW top |
| SourceInverseConclusion (ErdosInverseBoxHeight.sourceY top) (LargePrimeDeletion.cutoffPrimes ⌊w⌋₊) |
| residue delta (SmallSieveFinite.smallEuler ⌊w⌋₊) (cofactorScale top xi cm) |
| (lower w top xi i) (ErdosInverseBoxHeight.sourceZ top) cp sigma F |
| (actualBins top xi i) (cofactorChoices (actualBins top xi) cm) (actualBins top xi j) ∧ |
| selectionWitnessMass (actualBins top xi) m i j |
| (actualWitness (ErdosInverseBoxHeight.sourceY top) (LargePrimeDeletion.cutoffPrimes ⌊w⌋₊) residue delta |
| (SmallSieveFinite.smallEuler ⌊w⌋₊) (actualBins top xi i) (cofactorScale top xi cm) |
| (lower w top xi i) (ErdosInverseBoxHeight.sourceZ top) cp) ≤ sigma*selectionMass (actualBins top xi) m := by |
| obtain ⟨cp,hcp,F,xiI,hxiI,hxiI1,hInv⟩ := actual_source_inverse C (2*Kstar+4) aStar (2*Kstar+3) alpha delta sigma |
| (by linarith) (by linarith) ha ha1 (by linarith) halpha hd hsigma hsigma1 |
| have hC0 : 0 < C := by linarith |
| refine ⟨cp,hcp,F,min xiI (1/(2*C)),lt_min hxiI (by positivity),(min_le_left _ _).trans hxiI1,?_⟩ |
| intro xi hxi hxi₀ |
| have hxI := hxi₀.trans (min_le_left _ _) |
| have hDelta : 2*C*xi ≤ 1 := by |
| have hh := (le_div_iff₀ (show 0 < 2*C by positivity)).mp (hxi₀.trans (min_le_right _ _)) |
| nlinarith |
| have hWT := sourceW_tendsto.comp Real.tendsto_log_atTop |
| have hBT := sourceB_tendsto.comp Real.tendsto_log_atTop |
| have hlargeT := (hBT.const_mul_atTop halpha).eventually_gt_atTop (2*Kstar+3) |
| filter_upwards [hInv xi hxi hxI,ErdosCofactorChoices.source_span_geometry, |
| Real.tendsto_log_atTop.eventually source_scale_eventually, |
| Real.tendsto_log_atTop.eventually (source_isolated_search_scale halpha), |
| hBT.eventually_ge_atTop 3,hWT.eventually_ge_atTop (Real.exp 1),hlargeT, |
| eventually_gt_atTop (1:ℝ)] with top hInverse hspan hscale hsearch hB3 hWexp hlarge htop1 |
| change Real.exp 1 ≤ ErdosInverseBoxHeight.sourceW top at hWexp |
| change 3 ≤ ErdosInverseBoxHeight.sourceB top at hB3 |
| change 2*Kstar+3 < alpha*ErdosInverseBoxHeight.sourceB top at hlarge |
| have hw : 1 < ErdosInverseBoxHeight.sourceW top := hscale.1 |
| have htop : ErdosInverseBoxHeight.sourceW top < top := hspan.2.1 |
| have hlog : 1 ≤ Real.log (ErdosInverseBoxHeight.sourceW top) := by |
| have hh := Real.log_le_log (Real.exp_pos 1) hWexp |
| simpa only [Real.log_exp] using hh |
| have hpow : (ErdosInverseBoxHeight.sourceW top)^(ErdosInverseBoxHeight.sourceB top)=top := |
| (sourceW_pow_sourceB hw).trans (Real.exp_log (zero_lt_one.trans htop1)) |
| refine ⟨hw,htop,?_⟩ |
| intro R L z hi h199 hz hcut hclosed hroot m hm |
| have hs : Valid z.side z.ratio := by rw [hi]; change 198/100 ≤ z.ratio; linarith |
| have hY : 0 < ErdosInverseBoxHeight.sourceY top := by |
| by_contra! hn |
| have hzero : ErdosInverseBoxHeight.sourceY top=0 := by omega |
| rw [hzero,Nat.cast_zero,Real.log_zero,zero_div] at hroot |
| have hg := node_gap_eq hs hz hcut |
| nlinarith |
| have hgeo := geometric_inverse_geometry (M:=2*Kstar+3) (X:=2*Kstar+3) hw htop hxi hC hlog hscale.2.2.2.2 hDelta hB3 hpow hsearch hlarge |
| (ErdosInverseBoxHeight.sourceY top) hY z hroot hi h199 hz hcut hclosed m hm |
| obtain ⟨i,hmi,hilo,hihi,hgeo⟩ := hgeo |
| refine ⟨i,hmi,?_⟩ |
| intro j hj residue |
| obtain ⟨hij,_hmj,hzi,hzj,hlen,_hUlo,hUhi,hparent,hchild⟩ := hgeo j hj |
| have hji := source_power_before_edge hw htop hxi hpow m i j hilo hj hlarge |
| have hmj := bounded_edge_singleton m j hj |
| refine ⟨hji,hmj,?_⟩ |
| dsimp only |
| have hresult := hInverse (parentCofactorMultiplicity m i j) i j hij hzi hzj hlen residue hilo hihi hUhi |
| (by |
| intro q hq p hp |
| change Real.log ((ErdosInverseBoxHeight.sourceY top:ℝ)/((p:ℝ)*(q:ℝ)))/Real.log (ErdosInverseBoxHeight.sourceW top) ≤ _ |
| have hh := hparent q hq p hp |
| nlinarith) |
| (by |
| intro q hq p hp u hu |
| exact (by linarith : 3*aStar/4 ≤ aStar).trans (hchild q hq p hp u hu)) |
| refine ⟨hresult,?_⟩ |
| exact actual_inverse_full_box_mass (actualBins top xi) m |
| (fun k _p hp => (bin_prime_in_source (zero_lt_one.trans hw) htop hxi k hp).1) |
| (fun k l hkl => LogarithmicBinPartition.bins_pairwise_disjoint (zero_lt_one.trans hw) htop hxi k l hkl) |
| i j hji hmi hmj _ _ residue _ _ _ _ _ _ _ F hresult |
| end NumberTheoryLean.GeometricFullBoxInverse |
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| namespace NumberTheoryLean.GeometricRangeInverse |
| open _root_.Filter FinitePathGeometry PrimeHistories PrimeBinMembership |
| open GeometricBoxImages GeometricInverseGeometry BoundedEdgeBins ParentCofactorChoices |
| open JacobsthalSourceScale IsolatedSourceScales SourceNodeCoordinates |
| open ActualInverseFullBoxMass BoxWitnessFactorization CanonicalSubsetBox IsolatedPowerOrdering ErdosInverseSampleCost |
| open ErdosCofactorChoices ErdosSubsetWord ErdosInverseBoxApplication |
| open LogarithmicBinScale LogarithmicBinEndpoints |
| open scoped Topology |
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| theorem geometric_range_inverse (C Kstar aStar alpha delta sigma : ℝ) |
| (hC : 1 ≤ C) (hKstar : 0 ≤ Kstar) (ha : 0 < aStar) (ha1 : aStar ≤ 1) |
| (halpha : 0 < alpha) (hd : 0 < delta) (hsigma : 0 < sigma) (hsigma1 : sigma ≤ 1) : |
| ∃ cp : ℝ,0 < cp ∧ ∃ F : ℕ,∃ xi₀ : ℝ,0 < xi₀ ∧ xi₀ ≤ 1 ∧ |
| ∀ xi : ℝ,∀ hxi : 0 < xi,xi ≤ xi₀ → ∀ᶠ top : ℝ in atTop, |
| ∃ hw : 1 < ErdosInverseBoxHeight.sourceW top,∃ htop : ErdosInverseBoxHeight.sourceW top < top, |
| ∀ R L : ℝ,∀ z : Node,z.side=.even → 199/100 ≤ z.ratio → Consistent z → |
| z.cutoff=ErdosInverseBoxHeight.sourceB top → z.closed=true → |
| z.gap=Real.log (ErdosInverseBoxHeight.sourceY top:ℝ)/Real.log (ErdosInverseBoxHeight.sourceW top)-aStar+2 → |
| ∀ m ∈ geometricBoxes hw htop hxi C (ErdosInverseBoxHeight.sourceB top) R L alpha (1/100) z, |
| ∃ i : Fin (binCount (ErdosInverseBoxHeight.sourceW top) top xi),m i=1 ∧ |
| top^alpha ≤ lower (ErdosInverseBoxHeight.sourceW top) top xi i ∧ |
| lower (ErdosInverseBoxHeight.sourceW top) top xi i ≤ top^((1:ℝ)/100) ∧ |
| ∀ j : Fin (binCount (ErdosInverseBoxHeight.sourceW top) top xi), |
| boundedEdgeBin (ErdosInverseBoxHeight.sourceW top) top xi (ErdosInverseBoxHeight.sourceY top) m |
| (2*Kstar+3) (2*Kstar+3) j → ∀ residue : ℕ → ℕ, |
| j < i ∧ m j=1 ∧ |
| let cm := parentCofactorMultiplicity m i j |
| let w := ErdosInverseBoxHeight.sourceW top |
| SourceInverseConclusion (ErdosInverseBoxHeight.sourceY top) (LargePrimeDeletion.cutoffPrimes ⌊w⌋₊) |
| residue delta (SmallSieveFinite.smallEuler ⌊w⌋₊) (cofactorScale top xi cm) |
| (lower w top xi i) (ErdosInverseBoxHeight.sourceZ top) cp sigma F |
| (actualBins top xi i) (cofactorChoices (actualBins top xi) cm) (actualBins top xi j) ∧ |
| selectionWitnessMass (actualBins top xi) m i j |
| (actualWitness (ErdosInverseBoxHeight.sourceY top) (LargePrimeDeletion.cutoffPrimes ⌊w⌋₊) residue delta |
| (SmallSieveFinite.smallEuler ⌊w⌋₊) (actualBins top xi i) (cofactorScale top xi cm) |
| (lower w top xi i) (ErdosInverseBoxHeight.sourceZ top) cp) ≤ sigma*selectionMass (actualBins top xi) m := by |
| obtain ⟨cp,hcp,F,xiI,hxiI,hxiI1,hInv⟩ := actual_source_inverse C (2*Kstar+4) aStar (2*Kstar+3) alpha delta sigma |
| (by linarith) (by linarith) ha ha1 (by linarith) halpha hd hsigma hsigma1 |
| have hC0 : 0 < C := by linarith |
| refine ⟨cp,hcp,F,min xiI (1/(2*C)),lt_min hxiI (by positivity),(min_le_left _ _).trans hxiI1,?_⟩ |
| intro xi hxi hxi₀ |
| have hxI := hxi₀.trans (min_le_left _ _) |
| have hDelta : 2*C*xi ≤ 1 := by |
| have hh := (le_div_iff₀ (show 0 < 2*C by positivity)).mp (hxi₀.trans (min_le_right _ _)) |
| nlinarith |
| have hWT := sourceW_tendsto.comp Real.tendsto_log_atTop |
| have hBT := sourceB_tendsto.comp Real.tendsto_log_atTop |
| have hlargeT := (hBT.const_mul_atTop halpha).eventually_gt_atTop (2*Kstar+3) |
| filter_upwards [hInv xi hxi hxI,ErdosCofactorChoices.source_span_geometry, |
| Real.tendsto_log_atTop.eventually source_scale_eventually, |
| Real.tendsto_log_atTop.eventually (source_isolated_search_scale halpha), |
| hBT.eventually_ge_atTop 3,hWT.eventually_ge_atTop (Real.exp 1),hlargeT, |
| eventually_gt_atTop (1:ℝ)] with top hInverse hspan hscale hsearch hB3 hWexp hlarge htop1 |
| change Real.exp 1 ≤ ErdosInverseBoxHeight.sourceW top at hWexp |
| change 3 ≤ ErdosInverseBoxHeight.sourceB top at hB3 |
| change 2*Kstar+3 < alpha*ErdosInverseBoxHeight.sourceB top at hlarge |
| have hw : 1 < ErdosInverseBoxHeight.sourceW top := hscale.1 |
| have htop : ErdosInverseBoxHeight.sourceW top < top := hspan.2.1 |
| have hlog : 1 ≤ Real.log (ErdosInverseBoxHeight.sourceW top) := by |
| have hh := Real.log_le_log (Real.exp_pos 1) hWexp |
| simpa only [Real.log_exp] using hh |
| have hpow : (ErdosInverseBoxHeight.sourceW top)^(ErdosInverseBoxHeight.sourceB top)=top := |
| (sourceW_pow_sourceB hw).trans (Real.exp_log (zero_lt_one.trans htop1)) |
| refine ⟨hw,htop,?_⟩ |
| intro R L z hi h199 hz hcut hclosed hroot m hm |
| have hs : Valid z.side z.ratio := by rw [hi]; change 198/100 ≤ z.ratio; linarith |
| have hY : 0 < ErdosInverseBoxHeight.sourceY top := by |
| by_contra! hn |
| have hzero : ErdosInverseBoxHeight.sourceY top=0 := by omega |
| rw [hzero,Nat.cast_zero,Real.log_zero,zero_div] at hroot |
| have hg := node_gap_eq hs hz hcut |
| nlinarith |
| have hgeo := geometric_inverse_geometry (M:=2*Kstar+3) (X:=2*Kstar+3) hw htop hxi hC hlog hscale.2.2.2.2 hDelta hB3 hpow hsearch hlarge |
| (ErdosInverseBoxHeight.sourceY top) hY z hroot hi h199 hz hcut hclosed m hm |
| obtain ⟨i,hmi,hilo,hihi,hgeo⟩ := hgeo |
| refine ⟨i,hmi,hilo,hihi,?_⟩ |
| intro j hj residue |
| obtain ⟨hij,_hmj,hzi,hzj,hlen,_hUlo,hUhi,hparent,hchild⟩ := hgeo j hj |
| have hji := source_power_before_edge hw htop hxi hpow m i j hilo hj hlarge |
| have hmj := bounded_edge_singleton m j hj |
| refine ⟨hji,hmj,?_⟩ |
| dsimp only |
| have hresult := hInverse (parentCofactorMultiplicity m i j) i j hij hzi hzj hlen residue hilo hihi hUhi |
| (by |
| intro q hq p hp |
| change Real.log ((ErdosInverseBoxHeight.sourceY top:ℝ)/((p:ℝ)*(q:ℝ)))/Real.log (ErdosInverseBoxHeight.sourceW top) ≤ _ |
| have hh := hparent q hq p hp |
| nlinarith) |
| (by |
| intro q hq p hp u hu |
| exact (by linarith : 3*aStar/4 ≤ aStar).trans (hchild q hq p hp u hu)) |
| refine ⟨hresult,?_⟩ |
| exact actual_inverse_full_box_mass (actualBins top xi) m |
| (fun k _p hp => (bin_prime_in_source (zero_lt_one.trans hw) htop hxi k hp).1) |
| (fun k l hkl => LogarithmicBinPartition.bins_pairwise_disjoint (zero_lt_one.trans hw) htop hxi k l hkl) |
| i j hji hmi hmj _ _ residue _ _ _ _ _ _ _ F hresult |
| end NumberTheoryLean.GeometricRangeInverse |
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| end Erdos970 |
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| end OAI |
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