| import OAI.NumberTheory.Jacobsthal.Partitions.AllLabelDiscrepancy |
|
|
| namespace OAI |
|
|
| namespace Erdos970 |
| open scoped _root_.Erdos970 |
|
|
|
|
| namespace NumberTheoryLean.ActualCouplingError |
|
|
| open _root_.Set _root_.MeasureTheory ProbabilityTheory |
| open FinitePathGeometry FinitePathMeasures PrimeHistories PrimeKilledChain |
| open MeshRatioLabels ActualSupportIntervals ActualProcessCoupling |
| open PrimeBinMembership RegeneratingInverseBands AllLabelDiscrepancy |
|
|
| theorem actual_one_step_failure : ∃ c₁ c₂ C₁ C₂ C₃ D w₀ : ℝ, |
| 0 < c₁ ∧ 0 < c₂ ∧ 0 < C₁ ∧ 0 < C₂ ∧ 0 < C₃ ∧ 0 < D ∧ 1 < w₀ ∧ |
| ∀ w : ℝ, w₀ ≤ w → ∀ ell S : ℝ, ∀ start : Node, ∀ h : History w ell S start, |
| ∀ v : ℝ, ∀ z : CostState, ∀ mesh : ℝ, |
| ∀ (hnorm : normalizationThreshold ≤ w) (hell : 1 ≤ ell) (hS3 : 3 ≤ S) (hS : S ≤ (Real.log w)^3) |
| (hr : 0 < start.gap) (hs : Valid start.side start.ratio) (hsS : start.ratio ≤ S), Consistent start → |
| stateRatio z.1 ≤ S → h.node.side = stateSide z.1 → 0 < mesh → |
| let Ld := minRatio h.node.side h.node.ratio |
| let Lc := minRatio (stateSide z.1) (stateRatio z.1) |
| let Ud := upperSupport S ell h.node.gap |
| let Uc := upperSupport S ell (gapValue v z) |
| |Ld-Lc|+mesh ≤ 1 → |Ud-Uc|+mesh ≤ 1 → |
| jointKernel hnorm hell (show 0 ≤ S from (by linarith)) hS hr hs hsS v mesh (some h,.inl z) |
| {q | primeLabel w ell S mesh start q.1 ≠ continuousLabel S mesh q.2} ≤ |
| ENNReal.ofReal (2*((S/mesh+2)*(C₁*(1+S)^5*(mesh^2+Real.exp (-c₁*Real.sqrt ((1/2:ℝ)*Real.log w))))+ |
| epsilon w+(Real.exp (D*(1+S)^3*|h.node.ratio-stateRatio z.1|)-1)+ |
| C₂*(1+S)^2*(|Ld-Lc|+|Ud-Uc|+2*mesh+2*Real.exp (-c₂*Real.sqrt ((1/2:ℝ)*Real.log w)))+ |
| C₃*(1+S)^2*(|Ld-Lc|+|Ud-Uc|+2*mesh))) := by |
| obtain ⟨c₁,c₂,C₁,C₂,C₃,D,w₀,hc₁,hc₂,hC₁,hC₂,hC₃,hD,hw₀,hbound⟩ := all_label_discrepancy |
| refine ⟨c₁,c₂,C₁,C₂,C₃,D,w₀,hc₁,hc₂,hC₁,hC₂,hC₃,hD,hw₀,?_⟩ |
| intro w hw ell S start h v z mesh hnorm hell hS3 hS hr hs hsS hcons hzS hside hm |
| dsimp only |
| intro hL hU |
| exact (jointKernel_mismatch_bound hnorm hell (by linarith) hS hr hs hsS v mesh (some h) (.inl z)).trans |
| (ENNReal.ofReal_le_ofReal (hbound w hw ell S start h v z mesh hnorm hell hS3 hS hr hs hsS hcons hzS hside hm hL hU)) |
|
|
| end NumberTheoryLean.ActualCouplingError |
|
|
|
|
| end Erdos970 |
|
|
| end OAI |
|
|
