openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Gaussian /GaussianColumnProductLaw.lean
| import OAI.Probability.InvariantIsing.Gaussian.GaussianGramColumns | |
| import Mathlib.Probability.ProductMeasure | |
| namespace OAI | |
| /-! The physical Gaussian matrix as independent Gaussian columns. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| namespace InvariantIsing | |
| def gaussianColumnArray {N m : ℕ} (g : Fin m → Fin N → ℝ) : | |
| EuclideanSpace ℝ (Fin N × Fin m) := WithLp.toLp 2 (fun p => g p.2 p.1) | |
| lemma gaussianColumnArray_hasLaw (N m : ℕ) : | |
| HasLaw (gaussianColumnArray (N := N) (m := m)) (stdGaussian _) | |
| (Measure.pi (fun _ : Fin m => Measure.pi (fun _ : Fin N => gaussianReal 0 1))) := by | |
| let μ := gaussianReal 0 1 | |
| have hc : HasLaw (MeasurableEquiv.curry (Fin m) (Fin N) ℝ).symm | |
| (Measure.pi (fun _ : Fin m × Fin N => μ)) | |
| (Measure.pi (fun _ : Fin m => Measure.pi (fun _ : Fin N => μ))) := by | |
| refine ⟨(MeasurableEquiv.curry _ _ _).symm.measurable.aemeasurable,?_⟩ | |
| simpa only [Measure.infinitePi_eq_pi] using Measure.infinitePi_map_curry_symm (fun (_ : Fin m) (_ : Fin N) => μ) | |
| have hs := (measurePreserving_piCongrLeft (fun _ : Fin N × Fin m => μ) | |
| (Equiv.prodComm (Fin m) (Fin N))).hasLaw | |
| have hz : HasLaw (WithLp.toLp 2 : (Fin N × Fin m → ℝ) → EuclideanSpace ℝ (Fin N × Fin m)) | |
| (stdGaussian _) (Measure.pi (fun _ : Fin N × Fin m => μ)) := | |
| ⟨(PiLp.continuous_toLp 2 (fun _ : Fin N × Fin m => ℝ)).measurable.aemeasurable, | |
| map_pi_eq_stdGaussian⟩ | |
| convert hz.comp (hs.comp hc) using 1 | |
| rfl | |
| lemma gaussianColumnArray_column {N m : ℕ} (g : Fin m → Fin N → ℝ) (j : Fin m) : | |
| gaussianGramRawColumn (gaussianColumnArray g) j = g j := rfl | |
| end InvariantIsing | |
| end | |
| end OAI | |