| import Mathlib |
| import OAI.Probability.SKBarriers.Scalar.ScalarMomentSquare |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators NNReal Topology |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
| attribute [local instance 2000] parameterNormedGroup parameterNormedSpace |
|
|
| theorem scalarMomentSquare_eq_parameterAverage (n : ℕ) (m v : Fin n → ℝ) |
| (F H : ℝ → ℝ → ℝ) (x a : ℝ) |
| (hf : BoundedDerivs (scalarParameterTerminal n v F x)) (j : Fin (n+1)) : |
| scalarMomentSquare n m v (F a) (H a) j x = |
| hierarchyAverage n m (scalarParameterTerminal n v F x) |
| (fun z => (hierarchyMomentLevel n m (scalarParameterTerminal n v F x) |
| (scalarParameterTerminal n v H x) j z)^2) a := by |
| induction n generalizing F H with |
| | zero => simp [scalarMomentSquare,hierarchyAverage,hierarchyMomentLevel, |
| scalarParameterTerminal,coordinateLinear,parameter] |
| | succ n ih => |
| refine Fin.lastCases ?_ (fun j => ?_) j |
| · simp only [scalarMomentSquare,Fin.lastCases_last,hierarchyMomentLevel] |
| exact (hierarchyAverage_scalarParameterTerminal (n+1) m v F |
| (fun a x => (H a x)^2) x a).symm |
| · simp only [scalarMomentSquare,hierarchyMomentLevel,Fin.lastCases_castSucc,hierarchyAverage] |
| rw [gaussianAverage_const_prefix hf (m (Fin.last n)) (fun z => |
| (hierarchyMomentLevel n (fun i => m i.castSucc) |
| (gaussianStep (m (Fin.last n)) (scalarParameterTerminal (n+1) v F x)) |
| (gaussianAverage (m (Fin.last n)) (scalarParameterTerminal (n+1) v F x) |
| (scalarParameterTerminal (n+1) v H x)) j z)^2)] |
| simp only [gaussianStep_scalarParameterTerminal,gaussianAverage_scalarParameterTerminal] |
| have hf' := hf.gaussianStep (m (Fin.last n)) |
| rw [gaussianStep_scalarParameterTerminal] at hf' |
| exact ih (fun i => m i.castSucc) (fun i => v i.castSucc) |
| (fun a => scalarStep (m (Fin.last n)) (v (Fin.last n)) (F a)) |
| (fun a => scalarStepAverage (m (Fin.last n)) (v (Fin.last n)) (F a) (H a)) hf' j |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
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|
