| import OAI.NumberTheory.Jacobsthal.Estimates.CoupledBoundaryBounds |
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| namespace OAI |
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| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| namespace NumberTheoryLean.AllLabelDiscrepancy |
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| open _root_.Set _root_.Finset _root_.MeasureTheory ProbabilityTheory |
| open FinitePathGeometry FinitePathMeasures PrimeHistories PrimeKilledChain |
| open MeshRatioLabels MeshBoundaryGeometry ActualSupportIntervals ActualProcessCoupling |
| open ContinuousKilledBins PrimeBinMembership RegeneratingInverseBands KernelBinConditioning |
| open CommonCellSum CouplingLabelEvents CoupledBoundaryEvents CoupledBoundaryBounds |
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| theorem all_label_discrepancy : ∃ 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, |
| normalizationThreshold ≤ w → 1 ≤ ell → 3 ≤ S → S ≤ (Real.log w)^3 → |
| 0 < start.gap → Valid start.side start.ratio → 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 → |
| (∑ 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*((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 |
| classical |
| obtain ⟨c₁,C₁,D,w₁,hc₁,hC₁,hD,hw₁,hcommon⟩ := common_cell_sum_error |
| obtain ⟨c₂,C₂,w₂,hc₂,hC₂,hw₂,hprime⟩ := prime_bad_cells_upper |
| obtain ⟨C₃,hC₃,hcontinuous⟩ := continuous_bad_cells_upper |
| refine ⟨c₁,c₂,C₁,C₂,C₃,D,max w₁ w₂,hc₁,hc₂,hC₁,hC₂,hC₃,hD, |
| hw₁.trans_le (le_max_left _ _),?_⟩ |
| 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 |
| 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) |
| let I := interiorLabels S mesh Ld Lc Ud Uc |
| let δ := |Ld-Lc|+|Ud-Uc|+2*mesh |
| let Bp := C₂*(1+S)^2*(δ+2*Real.exp (-c₂*Real.sqrt ((1/2:ℝ)*Real.log w))) |
| let Bc := C₃*(1+S)^2*δ |
| have hδ : 0 ≤ δ := by dsimp [δ]; positivity |
| have hBp : 0 ≤ Bp := by dsimp [Bp]; positivity |
| have hBc : 0 ≤ Bc := by dsimp [Bc]; positivity |
| have hi := hcommon w ((le_max_left _ _).trans hw) ell S start h v z mesh I |
| hell hS3 hr hs hsS hcons hzS hside hm (fun j hj => (Finset.mem_filter.mp hj).2) |
| have hp := hprime w ((le_max_right _ _).trans hw) ell S start hell hr hs hsS h mesh Lc Uc hm hL hU |
| have hc := hcontinuous v ell S z (by linarith) hzS mesh Ld Ud hm |
| have hpR : (chain w ell S start (some h) (primeBadCells mesh Ld Lc Ud Uc)).toReal ≤ Bp := by |
| have h := ENNReal.toReal_mono ENNReal.ofReal_ne_top hp |
| rwa [ENNReal.toReal_ofReal hBp] at h |
| have hcR : (continuousChain v ell S (.inl z) (continuousBadCells S mesh Ld Lc Ud Uc)).toReal ≤ Bc := by |
| have h := ENNReal.toReal_mono ENNReal.ofReal_ne_top hc |
| rwa [ENNReal.toReal_ofReal hBc] at h |
| have hb := actual_label_budget hnorm hell (by linarith) hS hr hs hsS h v z hm Ld Lc Ud Uc |
| change (∑ j,_) ≤ _ at hb |
| dsimp only [I,Bp,Bc,δ,Ld,Lc,Ud,Uc] at hi hpR hcR hb |
| linarith |
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| end NumberTheoryLean.AllLabelDiscrepancy |
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| end Erdos970 |
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| end OAI |
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