| import OAI.NumberTheory.Jacobsthal.Partitions.SourceNodeCoordinates |
|
|
| namespace OAI |
|
|
| namespace Erdos970 |
| open scoped _root_.Erdos970 |
|
|
| section |
|
|
| open _root_.MeasureTheory _root_.Set _root_.Finset |
| namespace ErdosRetainedRewardTransfer |
| open NumberTheoryLean.FinitePathGeometry NumberTheoryLean.PrimeHistories |
| open NumberTheoryLean.PrimeKilledChain NumberTheoryLean.PrimeCompactWeights |
| open NumberTheoryLean.PrimeCorrectionFactor NumberTheoryLean.PrimeFamilyOccupation |
| open NumberTheoryLean.SourceNodeCoordinates |
| open NumberTheoryLean.PrimeBinMembership |
| open ErdosPrimeInputs.PrimePrefixMass |
|
|
| noncomputable def transferConstant : ℝ := 2*weight .even (199/100) |
|
|
| theorem transferConstant_positive : 0 < transferConstant := by |
| unfold transferConstant |
| exact mul_pos (by norm_num) (weight_pos (by norm_num [Valid])) |
|
|
| theorem scaled_normalized_reward_bound {d w ell B delta : ℝ} {start : Node} |
| (hB : 0 < B) (hell : 1 ≤ ell) (hlogB : 2 ≤ Real.log B) |
| (hcomp : Real.log B ≤ d*Real.log w) (hbudget : correctionBudget d w ≤ 1) |
| (hi : start.side=.even) (h199 : 199/100 ≤ start.ratio) (h23 : start.ratio ≤ 23/10) |
| (hcons : Consistent start) (hcut : start.cutoff=B) (_hdelta : 0 ≤ delta) |
| (h : History w ell ((Real.log B)^2) start) (F : List ℕ → ℝ) |
| (hF : h.node.gap^2*|F h.primes|/weight h.node.side h.node.ratio ≤ delta) : |
| scaledWeight w B start (some h)*|F h.primes| ≤ transferConstant*delta := by |
| obtain ⟨hs,hr,hBr,hsize⟩ := source_node_bounds hB start hi h199 h23 hcons hcut |
| have hphi : 0 < weight h.node.side h.node.ratio := weight_pos (terminal_valid hs h.admissible) |
| have hp0 : 0 ≤ weight start.side start.ratio := (weight_pos hs).le |
| have hW : 0 ≤ weight .even (199/100) := (weight_pos (by norm_num [Valid])).le |
| have hinit : weight start.side start.ratio ≤ weight .even (199/100) := by |
| rw [hi] |
| exact NumberTheoryLean.TwoStepDensityBounds.weight_antitone (by norm_num [Valid]) (hi ▸ hs) h199 |
| have hc : (1+epsilon w)^h.primes.length ≤ 2 := by |
| have hh := (history_correction hlogB hcomp hell hr hs hB hsize h).2 |
| linarith |
| have hc0 : 0 ≤ (1+epsilon w)^h.primes.length := pow_nonneg (normalizer_pos w).le _ |
| have hratio : 0 ≤ B/start.gap := div_nonneg hB.le hr.le |
| have hratio1 : B/start.gap ≤ 1 := (div_le_one hr).mpr hBr |
| have hsq : (B/start.gap)^2 ≤ 1 := by nlinarith |
| have hF0 : 0 ≤ h.node.gap^2*|F h.primes|/weight h.node.side h.node.ratio := by positivity |
| calc |
| _ = ((B/start.gap)^2*(1+epsilon w)^h.primes.length*weight start.side start.ratio)* |
| (h.node.gap^2*|F h.primes|/weight h.node.side h.node.ratio) := by unfold scaledWeight; ring |
| _ ≤ (1*2*weight .even (199/100))*delta := |
| mul_le_mul (mul_le_mul (mul_le_mul hsq hc hc0 (by norm_num)) hinit hp0 (by norm_num)) hF hF0 (by positivity) |
| _ = _ := by unfold transferConstant; ring |
| |
| theorem retained_normalized_reward_sum {d w ell B delta : ℝ} {start : Node} |
| (hw : normalizationThreshold ≤ w) (hell : 1 ≤ ell) |
| (hS : (Real.log B)^2 ≤ (Real.log w)^3) (hsS : start.ratio ≤ (Real.log B)^2) |
| (hB : 0 < B) (hlogB : 2 ≤ Real.log B) (hcomp : Real.log B ≤ d*Real.log w) |
| (hbudget : correctionBudget d w ≤ 1) |
| (hi : start.side=.even) (h199 : 199/100 ≤ start.ratio) (h23 : start.ratio ≤ 23/10) |
| (hcons : Consistent start) (hcut : start.cutoff=B) (hdelta : 0 ≤ delta) (F : List ℕ → ℝ) |
| (hF : ∀ ps ∈ retainedPrefixes w ell ((Real.log B)^2) start, |
| (terminal w start ps).gap^2*|F ps|/weight (terminal w start ps).side (terminal w start ps).ratio ≤ delta) : |
| B^2*(∑ ps ∈ retainedPrefixes w ell ((Real.log B)^2) start,prefixWeight ps*|F ps|) ≤ |
| transferConstant*(NumberTheoryLean.LowStateHorizon.sourceHorizon ((Real.log B)^2) B : ℝ)*delta := by |
| classical |
| obtain ⟨hs,hr,_hBr,hsize⟩ := source_node_bounds hB start hi h199 h23 hcons hcut |
| have hw1 : 1 < w := normalizationThreshold_gt_one.trans_le hw |
| let := historyFintype (ell:=ell) (S:=(Real.log B)^2) hw1 start |
| have hpoint (z : ChainState w ell ((Real.log B)^2) start) : |
| ‖scaledWeight w B start z*listReward (fun ps => |F ps|) z‖ ≤ transferConstant*delta := by |
| cases z with |
| | none => |
| simp only [scaledWeight,listReward,mul_zero,norm_zero] |
| exact mul_nonneg transferConstant_positive.le hdelta |
| | some h => |
| have hnorm : 0 ≤ scaledWeight w B start (some h)*|F h.primes| := |
| mul_nonneg (scaledWeight_nonneg hs _) (abs_nonneg _) |
| change ‖scaledWeight w B start (some h)*|F h.primes|‖ ≤ transferConstant*delta |
| rw [Real.norm_of_nonneg hnorm] |
| apply scaled_normalized_reward_bound hB hell hlogB hcomp hbudget hi h199 h23 hcons hcut hdelta h F |
| exact hF h.primes ((mem_retainedPrefixes hw1 start h.primes).mpr h.admissible) |
| have hterm (n : ℕ) : |
| (∫ z,scaledWeight w B start z*listReward (fun ps => |F ps|) z ∂pathLaw w ell ((Real.log B)^2) start n) ≤ |
| transferConstant*delta := by |
| let := pathLaw_isProbability hw hell (sq_nonneg _) hS hr hs hsS n |
| have hn := norm_integral_le_of_norm_le_const (μ:=pathLaw w ell ((Real.log B)^2) start n) |
| (Filter.Eventually.of_forall hpoint) |
| rw [Real.norm_eq_abs] at hn |
| have hle := le_abs_self (∫ z,scaledWeight w B start z*listReward (fun ps => |F ps|) z ∂pathLaw w ell ((Real.log B)^2) start n) |
| exact hle.trans (by simpa using hn) |
| rw [← finite_signed_original_occupation hw hell (sq_nonneg _) hS hr hs hsS hB hsize (fun ps => |F ps|)] |
| calc |
| _ ≤ ∑ _n ∈ range (NumberTheoryLean.LowStateHorizon.sourceHorizon ((Real.log B)^2) B),transferConstant*delta := |
| Finset.sum_le_sum (fun n _ => hterm n) |
| _ = _ := by simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul]; ring |
| |
| end ErdosRetainedRewardTransfer |
| |
| end |
| |
| end Erdos970 |
| |
| end OAI |
| |
