| import OAI.NumberTheory.Jacobsthal.Partitions.CommonCellSum |
| import OAI.NumberTheory.Jacobsthal.Partitions.FiniteLabelBudget |
| import OAI.NumberTheory.Jacobsthal.Primes.ConditionalPrimeBins |
|
|
| namespace OAI |
|
|
| namespace Erdos970 |
| open scoped _root_.Erdos970 |
|
|
| section |
|
|
| namespace NumberTheoryLean.CoupledBoundaryEvents |
|
|
| open _root_.Set _root_.MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
| open FinitePathGeometry FinitePathMeasures PrimeHistories PrimeKilledChain |
| open ActualSupportIntervals MeshBoundaryGeometry MeshRatioLabels |
| open PrimeCompletedIntervalUpper ContinuousIntervalUpper ActualCouplingSupport |
| open RegeneratingInverseBands ContinuousKilledBins |
| |
| noncomputable def lowerA (Ld Lc : ℝ) : ℝ := max (1/2) (min Ld Lc) |
| noncomputable def lowerB (mesh Ld Lc : ℝ) : ℝ := max (1/2) (max Ld Lc+mesh) |
| noncomputable def upperA (mesh Ud Uc : ℝ) : ℝ := max (1/2) (min Ud Uc-mesh) |
| noncomputable def upperB (Ud Uc : ℝ) : ℝ := max (1/2) (max Ud Uc) |
|
|
| variable {w ell S : ℝ} {start : Node} |
|
|
| def primeBadCells (mesh Ld Lc Ud Uc : ℝ) : Set (ChainState w ell S start) := |
| fun p => match p with |
| | none => False |
| | some h => ¬fullCell Ld Lc Ud Uc (ratioLabel S mesh h.node.ratio) |
|
|
| def continuousBadCells (S mesh Ld Lc Ud Uc : ℝ) : Set (CemeteryKernel.Space CostState) := |
| fun p => match p with |
| | .inr _ => False |
| | .inl z => ¬fullCell Ld Lc Ud Uc (ratioLabel S mesh (stateRatio z.1)) |
|
|
| theorem packet_widths {mesh : ℝ} (hm : 0 ≤ mesh) (Ld Lc Ud Uc : ℝ) : |
| (0 < lowerA Ld Lc ∧ lowerA Ld Lc ≤ lowerB mesh Ld Lc ∧ |
| lowerB mesh Ld Lc-lowerA Ld Lc ≤ |Ld-Lc|+mesh) ∧ |
| (0 < upperA mesh Ud Uc ∧ upperA mesh Ud Uc ≤ upperB Ud Uc ∧ |
| upperB Ud Uc-upperA mesh Ud Uc ≤ |Ud-Uc|+mesh) := by |
| have hl : min Ld Lc ≤ max Ld Lc+mesh := by linarith [min_le_max (a:=Ld) (b:=Lc)] |
| have hu : min Ud Uc-mesh ≤ max Ud Uc := by linarith [min_le_max (a:=Ud) (b:=Uc)] |
| exact ⟨clipped_packet_width hl (le_of_eq (lowerPacket_width mesh Ld Lc)), |
| clipped_packet_width hu (le_of_eq (upperPacket_width mesh Ud Uc))⟩ |
|
|
| theorem point_in_clipped_packets {mesh t Ld Lc Ud Uc : ℝ} (hm : 0 ≤ mesh) (ht : (1/2:ℝ) ≤ t) |
| (h : t ∈ lowerPacket mesh Ld Lc ∪ upperPacket mesh Ud Uc) : |
| t ∈ Icc (lowerA Ld Lc) (lowerB mesh Ld Lc) ∪ Icc (upperA mesh Ud Uc) (upperB Ud Uc) := by |
| rcases h with h | h |
| · exact Or.inl (clipPacket (show min Ld Lc ≤ max Ld Lc+mesh by linarith [min_le_max (a:=Ld) (b:=Lc)]) ht h).1 |
| · exact Or.inr (clipPacket (show min Ud Uc-mesh ≤ max Ud Uc by linarith [min_le_max (a:=Ud) (b:=Uc)]) ht h).1 |
|
|
| theorem prime_bad_cell_packets (h : History w ell S start) (hell : 0 < ell) |
| (hr : 0 < start.gap) (hs : Valid start.side start.ratio) (hsS : start.ratio ≤ S) |
| {mesh : ℝ} (hm : 0 < mesh) (Lc Uc : ℝ) : |
| let Ld := minRatio h.node.side h.node.ratio |
| let Ud := upperSupport S ell h.node.gap |
| chain w ell S start (some h) (primeBadCells mesh Ld Lc Ud Uc) ≤ |
| chain w ell S start (some h) (closedLiveBin (lowerA Ld Lc) (lowerB mesh Ld Lc))+ |
| chain w ell S start (some h) (closedLiveBin (upperA mesh Ud Uc) (upperB Ud Uc)) := by |
| dsimp only |
| have hvalid := terminal_valid hs h.admissible |
| have hgap := terminal_gap_positive hell.le hr hs h.admissible |
| apply le_trans (measure_mono_ae ?_) (measure_union_le _ _) |
| filter_upwards [prime_row_interval h hell hgap hvalid] with k hk |
| intro hbad |
| cases k with |
| | none => exact False.elim hbad |
| | some k => |
| have hhalf : (1/2:ℝ) ≤ k.node.ratio := PrimeTiltBounds.valid_ge_half (terminal_valid hs k.admissible) |
| have hS := terminal_ratio_le hsS k.admissible |
| have hp := noninterior_in_packets hm (by linarith : 0 ≤ k.node.ratio) hS (Or.inl hk) hbad |
| exact point_in_clipped_packets hm.le hhalf hp |
|
|
| theorem continuous_bad_cell_packets (v : ℝ) (z : CostState) (hell : 0 < ell) |
| {mesh : ℝ} (hm : 0 < mesh) (Ld Ud : ℝ) : |
| let Lc := minRatio (stateSide z.1) (stateRatio z.1) |
| let Uc := upperSupport S ell (gapValue v z) |
| continuousChain v ell S (.inl z) (continuousBadCells S mesh Ld Lc Ud Uc) ≤ |
| continuousChain v ell S (.inl z) (Sum.inl '' closedCostBin (lowerA Ld Lc) (lowerB mesh Ld Lc))+ |
| continuousChain v ell S (.inl z) (Sum.inl '' closedCostBin (upperA mesh Ud Uc) (upperB Ud Uc)) := by |
| dsimp only |
| apply le_trans (measure_mono_ae ?_) (measure_union_le _ _) |
| filter_upwards [continuous_row_interval v S hell z,continuous_supported v ell S (.inl z)] with y hy hS |
| intro hbad |
| cases y with |
| | inr u => exact False.elim hbad |
| | inl y => |
| have hhalf := CostPrefixTransport.stateRatio_ge_half y.1 |
| have hp := noninterior_in_packets hm (by linarith : 0 ≤ stateRatio y.1) hS (Or.inr hy) hbad |
| rcases point_in_clipped_packets hm.le hhalf hp with hp | hp |
| · exact Or.inl (Set.mem_image_of_mem Sum.inl hp) |
| · exact Or.inr (Set.mem_image_of_mem Sum.inl hp) |
| |
| end NumberTheoryLean.CoupledBoundaryEvents |
| |
| end |
| |
| section |
| |
| |
| namespace NumberTheoryLean.CouplingLabelEvents |
| |
| open _root_.Set _root_.Finset _root_.MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
| open FinitePathGeometry FinitePathMeasures PrimeHistories PrimeKilledChain |
| open MeshRatioLabels MeshBoundaryGeometry ActualProcessCoupling ActualCouplingSupport |
| open CoupledBoundaryEvents KernelBinConditioning FiniteLabelBudget ContinuousKilledBins |
| |
| noncomputable def interiorLabels (S mesh Ld Lc Ud Uc : ℝ) : Finset (Label S mesh) := by |
| classical |
| exact Finset.univ.filter (fullCell Ld Lc Ud Uc) |
| |
| noncomputable def boundaryLabels (S mesh Ld Lc Ud Uc : ℝ) : Finset (Label S mesh) := by |
| classical |
| exact (Finset.univ.erase (cemeteryLabel S mesh)).filter (fun j => ¬fullCell Ld Lc Ud Uc j) |
| |
| variable {w ell S : ℝ} {start : Node} |
| |
| theorem prime_boundary_event {mesh : ℝ} (hm : 0 < mesh) |
| (hsS : start.ratio ≤ S) (Ld Lc Ud Uc : ℝ) : |
| (primeLabel w ell S mesh start) ⁻¹' (boundaryLabels S mesh Ld Lc Ud Uc : Set (Label S mesh)) = |
| primeBadCells mesh Ld Lc Ud Uc := by |
| classical |
| ext p |
| cases p with |
| | none => |
| change cemeteryLabel S mesh ∈ boundaryLabels S mesh Ld Lc Ud Uc ↔ False |
| simp [boundaryLabels] |
| | some h => |
| have hne : ratioLabel S mesh h.node.ratio ≠ cemeteryLabel S mesh := ratioLabel_ne_cemetery hm (terminal_ratio_le hsS h.admissible) |
| change ratioLabel S mesh h.node.ratio ∈ boundaryLabels S mesh Ld Lc Ud Uc ↔ ¬fullCell Ld Lc Ud Uc (ratioLabel S mesh h.node.ratio) |
| simp [boundaryLabels,hne] |
|
|
| theorem continuous_boundary_event_mass (v ell S : ℝ) {mesh : ℝ} (hm : 0 < mesh) |
| (z : CostState) (Ld Lc Ud Uc : ℝ) : |
| continuousChain v ell S (.inl z) |
| ((continuousLabel S mesh) ⁻¹' (boundaryLabels S mesh Ld Lc Ud Uc : Set (Label S mesh))) = |
| continuousChain v ell S (.inl z) (continuousBadCells S mesh Ld Lc Ud Uc) := by |
| classical |
| apply measure_congr |
| filter_upwards [continuous_supported v ell S (.inl z)] with y hy |
| apply propext |
| cases y with |
| | inr u => |
| change cemeteryLabel S mesh ∈ boundaryLabels S mesh Ld Lc Ud Uc ↔ False |
| simp [boundaryLabels] |
| | inl y => |
| have hne := ratioLabel_ne_cemetery hm hy |
| change ratioLabel S mesh (stateRatio y.1) ∈ boundaryLabels S mesh Ld Lc Ud Uc ↔ ¬fullCell Ld Lc Ud Uc (ratioLabel S mesh (stateRatio y.1)) |
| simp [boundaryLabels,hne] |
| |
| theorem prime_boundary_mass [IsMarkovKernel (chain w ell S start)] {mesh : ℝ} (hm : 0 < mesh) |
| (hsS : start.ratio ≤ S) |
| (h : History w ell S start) (Ld Lc Ud Uc : ℝ) : |
| (∑ j ∈ boundaryLabels S mesh Ld Lc Ud Uc,labelMass (chain w ell S start) (primeLabel w ell S mesh start) (some h) j) = |
| (chain w ell S start (some h) (primeBadCells mesh Ld Lc Ud Uc)).toReal := by |
| rw [selected_label_mass _ (primeLabel_measurable w ell S mesh start),prime_boundary_event hm hsS] |
| |
| theorem continuous_boundary_mass (v ell S : ℝ) {mesh : ℝ} (hm : 0 < mesh) |
| (z : CostState) (Ld Lc Ud Uc : ℝ) : |
| (∑ j ∈ boundaryLabels S mesh Ld Lc Ud Uc,labelMass (continuousChain v ell S) (continuousLabel S mesh) (.inl z) j) = |
| (continuousChain v ell S (.inl z) (continuousBadCells S mesh Ld Lc Ud Uc)).toReal := by |
| rw [selected_label_mass _ (continuousLabel_measurable S mesh),continuous_boundary_event_mass v ell S hm z] |
| |
| theorem actual_label_budget (hw : normalizationThreshold ≤ w) (hell : 1 ≤ ell) |
| (hS0 : 0 ≤ S) (hS : S ≤ (Real.log w)^3) (hr : 0 < start.gap) |
| (hs : Valid start.side start.ratio) (hsS : start.ratio ≤ S) |
| (h : History w ell S start) (v : ℝ) (z : CostState) {mesh : ℝ} (hm : 0 < mesh) |
| (Ld Lc Ud Uc : ℝ) : |
| (∑ j : Label S mesh, |labelMass (chain w ell S start) (primeLabel w ell S mesh start) (some h) j- |
| labelMass (continuousChain v ell S) (continuousLabel S mesh) (.inl z) j|) ≤ |
| 2*((∑ j ∈ interiorLabels S mesh Ld Lc Ud Uc, |
| |labelMass (chain w ell S start) (primeLabel w ell S mesh start) (some h) j- |
| labelMass (continuousChain v ell S) (continuousLabel S mesh) (.inl z) j|)+ |
| (chain w ell S start (some h) (primeBadCells mesh Ld Lc Ud Uc)).toReal+ |
| (continuousChain v ell S (.inl z) (continuousBadCells S mesh Ld Lc Ud Uc)).toReal) := by |
| classical |
| let := chain_isMarkov hw hell hS0 hS hr hs hsS |
| let a := labelMass (chain w ell S start) (primeLabel w ell S mesh start) (some h) |
| let b := labelMass (continuousChain v ell S) (continuousLabel S mesh) (.inl z) |
| let good := fullCell (S:=S) (mesh:=mesh) Ld Lc Ud Uc |
| have hbudget := interior_boundary_budget a b (cemeteryLabel S mesh) good |
| (labelMass_nonneg (chain w ell S start) (primeLabel w ell S mesh start) (some h)) |
| (labelMass_nonneg (continuousChain v ell S) (continuousLabel S mesh) (.inl z)) |
| (labelMass_sum (chain w ell S start) (primeLabel_measurable w ell S mesh start) (some h)) |
| (labelMass_sum (continuousChain v ell S) (continuousLabel_measurable S mesh) (.inl z)) |
| have hsub : (Finset.univ.erase (cemeteryLabel S mesh)).filter good ⊆ interiorLabels S mesh Ld Lc Ud Uc := by |
| intro j hj |
| exact Finset.mem_filter.mpr ⟨Finset.mem_univ _,(Finset.mem_filter.mp hj).2⟩ |
| have hsum := Finset.sum_le_sum_of_subset_of_nonneg hsub (fun j _ _ => abs_nonneg (a j-b j)) |
| change (∑ j,|a j-b j|) ≤ _ |
| have hbad : (∑ j ∈ (Finset.univ.erase (cemeteryLabel S mesh)).filter (fun j => ¬good j),(a j+b j)) = |
| (chain w ell S start (some h) (primeBadCells mesh Ld Lc Ud Uc)).toReal+ |
| (continuousChain v ell S (.inl z) (continuousBadCells S mesh Ld Lc Ud Uc)).toReal := by |
| rw [Finset.sum_add_distrib] |
| exact congrArg₂ (·+·) (prime_boundary_mass hm hsS h Ld Lc Ud Uc) |
| (continuous_boundary_mass v ell S hm z Ld Lc Ud Uc) |
| rw [hbad] at hbudget |
| linarith |
| |
| end NumberTheoryLean.CouplingLabelEvents |
| |
| end |
| |
| end Erdos970 |
| |
| end OAI |
| |
