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135052a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 | import Mathlib
import OAI.Probability.SKBarriers.Scalar.TiltedQuadraticIBP
namespace OAI
section
section
noncomputable section
open scoped BigOperators Topology
open MeasureTheory ProbabilityTheory Filter
namespace SK.Analytic
attribute [local instance 2000] parameterNormedGroup parameterNormedSpace
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
end SK.Analytic
end
end
end
end OAI
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