| import Mathlib |
| import OAI.Probability.SKBarriers.Scalar.TiltedQuadraticIBP |
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| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open scoped BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| namespace SK.Analytic |
| attribute [local instance 2000] parameterNormedGroup parameterNormedSpace |
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| theorem spinGaussian_tent_error {S : Type} [Fintype S] [Nonempty S] |
| [MeasurableSpace S] [MeasurableSingletonClass S] |
| (n : ℕ) (V g : S → ParameterSpace n → ℝ) |
| (hV : ∀ s, BoundedDerivs (V s)) (hc : ∀ s, ContDiff ℝ 1 (g s)) |
| (hg : ∀ s, HasExpGrowth (g s)) (hdg : ∀ s, HasExpGrowth (fderiv ℝ (g s))) |
| (a : Fin n → ℝ) (hinv : ∀ s, TranslationInvariant (g s) (coordinateVector n a)) |
| (ε : ℝ) (hb : ∀ s z, |g s z| ≤ 1) |
| (h₁ : ∀ s z, |fderiv ℝ (V s) z (coordinateVector n a)| ≤ ε) |
| (h₂ : ∀ s z, |fderiv ℝ (fderiv ℝ (V s)) z (coordinateVector n a) (coordinateVector n a)| ≤ ε) : |
| |(∫ sz, (coordinateLinear n a sz.2)^2*g sz.1 sz.2 ∂spinGaussianLaw n V 0)- |
| (∑ i, (a i)^2)*(∫ sz, g sz.1 sz.2 ∂spinGaussianLaw n V 0)| ≤ ε^2+ε := by |
| let := spinGaussianLaw_probability n V hV |
| rw [spinGaussian_quadratic_invariant n V g hV hc hg hdg a hinv,add_sub_cancel_left] |
| have H := norm_integral_le_of_norm_le_const (μ := spinGaussianLaw n V 0) |
| (f := fun sz : S × ParameterSpace n => g sz.1 sz.2* |
| ((fderiv ℝ (V sz.1) sz.2 (coordinateVector n a))^2+ |
| fderiv ℝ (fderiv ℝ (V sz.1)) sz.2 (coordinateVector n a) (coordinateVector n a))) |
| (C := ε^2+ε) (ae_of_all _ (fun sz => by |
| rw [Real.norm_eq_abs,abs_mul] |
| have he : |(fderiv ℝ (V sz.1) sz.2 (coordinateVector n a))^2+ |
| fderiv ℝ (fderiv ℝ (V sz.1)) sz.2 (coordinateVector n a) (coordinateVector n a)| ≤ ε^2+ε := by |
| apply (abs_add_le _ _).trans |
| apply add_le_add _ (h₂ sz.1 sz.2) |
| rw [abs_pow] |
| exact pow_le_pow_left₀ (abs_nonneg _) (h₁ sz.1 sz.2) 2 |
| exact (mul_le_mul (hb sz.1 sz.2) he (abs_nonneg _) zero_le_one).trans_eq (one_mul _))) |
| simpa only [Real.norm_eq_abs,probReal_univ,mul_one] using H |
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| end SK.Analytic |
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| end |
| end |
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| end |
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| end OAI |
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