| import Mathlib |
| import OAI.Probability.SKBarriers.SpinGlass.SpinParameterMoments |
| import OAI.Probability.SKBarriers.Hierarchy.AverageMeanVariance |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| namespace SK.Analytic |
| attribute [local instance 2000] parameterNormedGroup parameterNormedSpace |
|
|
| def blockAdaptiveOverlap {D N k : ℕ} (I : Fin D → Finset (Fin N)) (a : Fin D → ℝ) |
| (β : ℝ) (p : ℝ × (Fin (k+1) → ℝ)) (j : Fin (k+1)) : ℝ := |
| hierarchyMeanOverlap (blockDimension D N k) (blockMass D N k) |
| (blockExponent I (fun i => p.1*a i) (fun b => β*p.2 b)) |
| (fun i s => spin (s i)) (blockLevel D N k j) |
|
|
| theorem blockCoefficients_contDiff {D N k : ℕ} (a : Fin D → ℝ) (β : ℝ) : |
| ContDiff ℝ 2 (fun p : ℝ × (Fin (k+1) → ℝ) => |
| blockCoefficients (N := N) (fun i => p.1*a i) (fun b => β*p.2 b)) := by |
| apply contDiff_pi.mpr |
| intro t |
| refine Fin.addCases (fun i => ?_) (fun i => ?_) t |
| · simp only [blockCoefficients,Fin.addCases_left] |
| exact contDiff_fst.mul contDiff_const |
| · simp only [blockCoefficients,Fin.addCases_right] |
| exact contDiff_const.mul ((contDiff_apply ℝ ℝ _).comp contDiff_snd) |
|
|
| theorem blockAdaptiveOverlap_contDiff {D N k : ℕ} (I : Fin D → Finset (Fin N)) |
| (a : Fin D → ℝ) (β : ℝ) : ContDiff ℝ 2 (blockAdaptiveOverlap (k := k) I a β) := by |
| apply contDiff_pi.mpr |
| intro j |
| exact (spinHierarchyOverlap_contDiff (blockDimension D N k) (blockMass D N k) |
| (blockInteraction I) (fun i s => spin (s i)) (fun i s => by cases s i <;> norm_num [spin]) |
| (blockLevel D N k j)).comp (blockCoefficients_contDiff a β) |
|
|
| theorem blockAdaptiveOverlap_bounds {D N k : ℕ} (I : Fin D → Finset (Fin N)) |
| (a : Fin D → ℝ) (β : ℝ) (p : ℝ × (Fin (k+1) → ℝ)) (j : Fin (k+1)) : |
| blockAdaptiveOverlap I a β p j ∈ Set.Icc (0:ℝ) 1 := |
| hierarchyMeanOverlap_bounds _ _ _ _ (fun i s => by cases s i <;> norm_num [spin]) _ |
|
|
| theorem blockAdaptiveOverlap_monotone {D N k : ℕ} (I : Fin D → Finset (Fin N)) |
| (a : Fin D → ℝ) (β : ℝ) (p : ℝ × (Fin (k+1) → ℝ)) : |
| Monotone (blockAdaptiveOverlap I a β p) := |
| (hierarchyMeanOverlap_mono _ _ _ _ (fun i s => by cases s i <;> norm_num [spin])).comp |
| (blockLevel_mono D N k) |
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end |
|
|
| end OAI |
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|
