| import OAI.NumberTheory.Jacobsthal.Estimates.PositiveGapMajorant |
|
|
| namespace OAI |
|
|
| namespace Erdos970 |
| open scoped _root_.Erdos970 |
|
|
| section |
|
|
| namespace NumberTheoryLean.UncappedCompactMass |
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|
| attribute [local instance] Classical.propDecidable |
| open _root_.Set _root_.Finset |
| open FinitePathGeometry PrimeHistories PrimeBinMembership PrimeCompactWeights |
| open ActualPrimeHigh SourceNodeCoordinates LiteralCompactRemoval LiteralPrimeOccupation |
| open PositiveGapMajorant OriginalPrimeOccupationLimit CompactLogOccupationTest NormalizedOccupationLimit |
| open ErdosPrimeInputs.PrimePrefixMass ErdosPrimeInputs.PrimePrefixTail |
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|
| noncomputable def compactPrefixMass (K w ell : ℝ) (start : Node) : ℝ := |
| ∑ ps ∈ uncappedPrefixes w ell start,prefixWeight ps* |
| (if 2 ≤ (terminal w start ps).gap ∧ (terminal w start ps).gap ≤ K then 1 else 0) |
|
|
| theorem uncapped_terminal_ratio_le_gap {w ell : ℝ} {start : Node} {ps : List ℕ} |
| (hell : 1 ≤ ell) (hr : 0 < start.gap) (hs : Valid start.side start.ratio) |
| (hc : Consistent start) (hcut : ell ≤ start.cutoff) (hp : uncappedAllowed w ell start ps) : |
| 0 < (terminal w start ps).ratio ∧ (terminal w start ps).ratio ≤ (terminal w start ps).gap := by |
| obtain ⟨U,_hU,hpU⟩ := (uncapped_iff_exists_ceiling w ell start ps).mp hp |
| exact ⟨valid_pos (terminal_valid hs hpU),history_ratio_le_gap hell hr hs hc hcut ⟨ps,hpU⟩⟩ |
|
|
| theorem uniform_uncapped_compact_mass (K ell d : ℝ) (hK : 3 ≤ K) (hell : 1 ≤ ell) (hd : 0 < d) : |
| ∃ C B₀ w₀ : ℝ,0 < C ∧ 0 < B₀ ∧ 1 < w₀ ∧ ∀ B w : ℝ,B₀ ≤ B → w₀ ≤ w → |
| Real.log B ≤ d*Real.log w → ∀ start : Node, |
| start.side=.even → 199/100 ≤ start.ratio → start.ratio ≤ 23/10 → |
| Consistent start → start.cutoff=B → B^2*compactPrefixMass K w ell start ≤ C := by |
| obtain ⟨H,hcompact,hbounds,hOne,hzero⟩ := exists_positive_gap_majorant K |
| let HH : Side → ℝ×ℝ → ℝ := fun _ => H |
| have hH : ∀ i,Continuous (HH i) := fun _ => H.continuous |
| have hCpt : ∀ i,HasCompactSupport (HH i) := fun _ => hcompact |
| have hsupport : ∀ i : Side,∀ r s : ℝ,r<1 ∨ K+1<r → HH i (r,s)=0 := fun _ => hzero |
| obtain ⟨BL,w₀,hBL,hw₀,hlim⟩ := original_prime_continuous_occupation_limit hH hCpt |
| (by norm_num : 0 < (1:ℝ)) (by linarith : 0 < K+1) hell hsupport d 1 hd (by norm_num) |
| let L : ℝ := |invariantAverage (arrivalTest ell HH)|+1 |
| let W : ℝ := weight .even (199/100) |
| have hL : 0 < L := by dsimp [L]; positivity |
| have hW : 0 < W := weight_pos (by norm_num [Valid]) |
| refine ⟨W*L,max BL ell,w₀,mul_pos hW hL,hBL.trans_le (le_max_left _ _),hw₀,?_⟩ |
| intro B w hB hw hcomp start hi h199 h23 hcons hcut |
| have hBL' : BL ≤ B := (le_max_left _ _).trans hB |
| have hellB : ell ≤ B := (le_max_right _ _).trans hB |
| have hBpos : 0 < B := hBL.trans_le hBL' |
| obtain ⟨hs,hr,hBgap,_hsize⟩ := source_node_bounds hBpos start hi h199 h23 hcons hcut |
| have hlimit := hlim B w hBL' hw hcomp start hi h199 h23 hcons hcut |
| have hnorm : uncappedFamilyValue HH w ell start ≤ L := by |
| have hu := (abs_le.mp hlimit).2 |
| have hab := le_abs_self (invariantAverage (arrivalTest ell HH)) |
| dsimp only [L] |
| linarith |
| let Z : ℝ := ∑ ps ∈ uncappedPrefixes w ell start,prefixWeight ps*nodeReward HH w start ps |
| have hZ : 0 ≤ Z := Finset.sum_nonneg (fun ps _ => mul_nonneg (prefixWeight_nonneg ps) (hbounds _).1) |
| have hdom : compactPrefixMass K w ell start ≤ Z := by |
| apply Finset.sum_le_sum |
| intro ps hp |
| apply mul_le_mul_of_nonneg_left _ (prefixWeight_nonneg ps) |
| change (if 2 ≤ (terminal w start ps).gap ∧ (terminal w start ps).gap ≤ K then 1 else 0) ≤ |
| H ((terminal w start ps).gap,(terminal w start ps).ratio) |
| split_ifs with hgap |
| · have hpA : uncappedAllowed w ell start ps := (Finset.mem_filter.mp hp).2 |
| have hratio := uncapped_terminal_ratio_le_gap hell hr hs hcons (by rwa [hcut]) hpA |
| rw [hOne _ _ hgap.1 hgap.2 hratio.1.le (hratio.2.trans hgap.2)] |
| · exact (hbounds _).1 |
| have hwpos := weight_pos hs |
| have hwupper : weight start.side start.ratio ≤ W := by |
| rw [hi] |
| exact TwoStepDensityBounds.weight_antitone (by norm_num [Valid]) (by rwa [hi] at hs) h199 |
| have hsq : B^2 ≤ start.gap^2 := by nlinarith |
| calc |
| B^2*compactPrefixMass K w ell start ≤ B^2*Z := mul_le_mul_of_nonneg_left hdom (sq_nonneg B) |
| _ ≤ start.gap^2*Z := mul_le_mul_of_nonneg_right hsq hZ |
| _ = weight start.side start.ratio*uncappedFamilyValue HH w ell start := by |
| unfold uncappedFamilyValue |
| dsimp only [Z] |
| field_simp |
| _ ≤ weight start.side start.ratio*L := mul_le_mul_of_nonneg_left hnorm hwpos.le |
| _ ≤ W*L := mul_le_mul_of_nonneg_right hwupper hL.le |
| |
| end NumberTheoryLean.UncappedCompactMass |
| |
| end |
| |
| end Erdos970 |
| |
| end OAI |
| |
