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67aee1d | 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 | import OAI.Analysis.FreeEntropy.GaussianEntropyIntegrability
namespace OAI
universe u1
noncomputable section
open MeasureTheory
namespace FiniteEntropySeparation
variable {ι : Type u1} [Fintype ι] [DecidableEq ι]
theorem integral_coordinate_ibp (i : ι) (f f' g g' : EuclideanSpace ℝ ι → ℝ)
(hf : ∀x r,HasDerivAt (fun t : ℝ => f (x+t • EuclideanSpace.single i 1)) (f' (x+r • EuclideanSpace.single i 1)) r)
(hg : ∀x r,HasDerivAt (fun t : ℝ => g (x+t • EuclideanSpace.single i 1)) (g' (x+r • EuclideanSpace.single i 1)) r)
(hfg' : Integrable (fun x => f x*g' x)) (hf'g : Integrable (fun x => f' x*g x))
(hfg : Integrable (fun x => f x*g x)) :
(∫x,f x*g' x) = -(∫x,f' x*g x) := by
apply integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable
(B := ContinuousLinearMap.mul ℝ ℝ) hf'g hfg' hfg
· intro x hx
simpa only [HasLineDerivAt,zero_smul,add_zero] using hf x 0
· intro x hx
simpa only [HasLineDerivAt,zero_smul,add_zero] using hg x 0
end FiniteEntropySeparation
end
end OAI
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