| 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 | |