| import Mathlib |
| import OAI.Probability.SKBarriers.Hierarchy.AverageOverlap |
| import OAI.Probability.SKBarriers.Hierarchy.BlockAdaptiveOverlap |
| import OAI.Probability.SKBarriers.Interpolation.AdaptivePositivePath |
|
|
| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| namespace SK.Analytic |
| attribute [local instance 2000] parameterNormedGroup parameterNormedSpace |
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| theorem exists_block_adaptive_path {D N k : ℕ} (hN : 0 < N) (hk : 0 < k) |
| (I : Fin D → Finset (Fin N)) (a : Fin D → ℝ) |
| {β η : ℝ} (hβ : 0 < β) (hη : 0 < η) |
| (hsmall : (k:ℝ)/Real.sqrt ((N:ℝ)*(β^2*η)) ≤ 1) : |
| ∃ q : ℝ → (Fin (k+1) → ℝ), |
| q 0 = (fun j => ((j.val:ℝ)+1)*η) ∧ |
| (∀ u ∈ Set.Icc (0:ℝ) 1, HasDerivWithinAt q |
| (blockAdaptiveOverlap I a β (Real.sqrt (1-u),fun j => Real.sqrt (cumulativeGapMap k (q u) j))) |
| (Set.Icc (0:ℝ) 1) u) ∧ |
| (∀ u ∈ Set.Icc (0:ℝ) 1, ∀ j, η ≤ cumulativeGapMap k (q u) j) ∧ |
| (∀ u ∈ Set.Icc (0:ℝ) 1, ∀ j, q u j ≤ ((j.val:ℝ)+1)*η+u) ∧ |
| (∀ u ∈ Set.Icc (0:ℝ) 1, |
| let U := blockExponent I (fun i => Real.sqrt (1-u)*a i) |
| (fun b => β*Real.sqrt (cumulativeGapMap k (q u) b)) |
| (∑ j : Fin (k+1), ((k+1:ℕ):ℝ)⁻¹ * |
| hierarchyOverlapError (blockDimension D N k) (blockMass D N k) U |
| (fun s i => spin (s i)) (blockLevel D N k j)) ≤ |
| 20*((k:ℝ)/Real.sqrt ((N:ℝ)*(β^2*η))+1/Real.sqrt (k:ℝ))) := by |
| obtain ⟨q,h0,hD,hgap,hupper⟩ := exists_positive_adaptive_path (k := k) |
| (blockAdaptiveOverlap I a β) ((blockAdaptiveOverlap_contDiff I a β).of_le (by norm_num)) |
| (blockAdaptiveOverlap_bounds I a β) (blockAdaptiveOverlap_monotone I a β) η hη |
| refine ⟨q,h0,hD,hgap,hupper,?_⟩ |
| intro u hu |
| have hs (b : Fin (k+1)) : |
| Real.sqrt (β^2*cumulativeGapMap k (q u) b) = β*Real.sqrt (cumulativeGapMap k (q u) b) := by |
| rw [Real.sqrt_mul (sq_nonneg β),Real.sqrt_sq_eq_abs,abs_of_nonneg hβ.le] |
| have H := block_average_overlap_concentration hN hk I (fun i => Real.sqrt (1-u)*a i) |
| (fun b => β^2*cumulativeGapMap k (q u) b) (by positivity : 0 < β^2*η) |
| (fun b => mul_le_mul_of_nonneg_left (hgap u hu b) (sq_nonneg β)) hsmall |
| simpa only [hs] using H |
| end SK.Analytic |
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| end |
| end |
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| end |
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| end OAI |
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