| import Mathlib |
| import OAI.Probability.SKBarriers.Scalar.PrefixAlgebra |
|
|
| 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 |
|
|
| section SpinProduct |
| variable {S : Type} [Fintype S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] |
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| theorem hierarchySpinMean_terminal_product (n : ℕ) (m : Fin n → ℝ) |
| (U : S → ParameterSpace n →L[ℝ] ℝ) (c : S → ℝ) {C : ℝ} |
| (hC : 0 ≤ C) (hc : ∀ s, ‖c s‖ ≤ C) (j : Fin (n+1)) : |
| (∫ sz, c sz.1*hierarchySpinMean n m U c j sz.2 ∂hierarchyGaussianLaw n m U) = |
| ∫ z, (hierarchySpinMean n m U c j z)^2 |
| ∂hierarchyPathLaw n m (affineLogPartition (fun _ => 0) U) 0 := by |
| let f := affineLogPartition (fun _ => 0) U |
| have hf := affineLogPartition_boundedDerivs (fun _ => 0) U |
| have hr := hierarchySpinMean_regular n m U c hC hc j |
| rw [hierarchyGaussian_terminal_marginal n m U c _ hr.1 (HasExpGrowth.of_bounded hC hr.2)] |
| have he : hierarchyMomentLevel n m f (hierarchySpinMean n m U c j) j = |
| hierarchySpinMean n m U c j := hierarchyMomentLevel_retained n m f _ hf j j le_rfl |
| have H := hierarchyPathLaw_fixed_mul n m f (affineMoment (fun _ => 0) U c) |
| (hierarchySpinMean n m U c j) hf (affineMoment_continuous _ _ _) hr.1 hC hC |
| (affineMoment_norm_le _ _ _ hc) hr.2 j he 0 |
| calc |
| _ = ∫ z, hierarchySpinMean n m U c j z*affineMoment (fun _ => 0) U c z |
| ∂hierarchyPathLaw n m f 0 := by congr 1; funext z; ring |
| _ = _ := H.trans (by congr 1; funext z; change hierarchySpinMean n m U c j z * hierarchySpinMean n m U c j z = _; ring) |
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| end SpinProduct |
| end SK.Analytic |
|
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| end |
| end |
|
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| end |
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| end OAI |
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