File size: 3,945 Bytes
f89a930 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 | import OAI.NumberTheory.Jacobsthal.Estimates.CoupledBoundaryBounds
namespace OAI
namespace Erdos970
open scoped _root_.Erdos970
namespace NumberTheoryLean.AllLabelDiscrepancy
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
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
end NumberTheoryLean.AllLabelDiscrepancy
end Erdos970
end OAI
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