openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Gaussian /GaussianExponentialShift.lean
| import OAI.Probability.InvariantIsing.Gaussian.GaussianTiltLinearGrowth | |
| namespace OAI | |
| /-! Completing the square under the finite scalar Gaussian law. This is | |
| the shift identity used to control canonical increments uniformly in the | |
| number and size of the cascade steps. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| open scoped NNReal | |
| namespace InvariantIsing | |
| lemma gaussianPDFReal_exp_linear {v : ℝ≥0} (hv : v ≠ 0) (t x : ℝ) : | |
| gaussianPDFReal 0 v x * Real.exp (t * x) = | |
| Real.exp (t ^ 2 * v / 2) * gaussianPDFReal (t * v) v x := by | |
| have hvR : (v : ℝ) ≠ 0 := by exact_mod_cast hv | |
| have he : -(x ^ 2) / (2 * v) + t * x = | |
| t ^ 2 * v / 2 + -(x - t * v) ^ 2 / (2 * v) := by | |
| field_simp | |
| ring | |
| simp only [gaussianPDFReal, sub_zero] | |
| rw [mul_assoc, ← Real.exp_add, he, Real.exp_add] | |
| ring | |
| theorem gaussian_exp_linear_shift (v : ℝ≥0) (t : ℝ) (F : ℝ → ℝ) (hF : Measurable F) : | |
| (∫ x, Real.exp (t * x) * F x ∂gaussianReal 0 v) = | |
| Real.exp (t ^ 2 * v / 2) * | |
| ∫ x, F (x + t * v) ∂gaussianReal 0 v := by | |
| by_cases hv : v = 0 | |
| · subst v | |
| simp | |
| have hshift : (∫ x, F x ∂gaussianReal (t * v) v) = | |
| ∫ x, F (x + t * v) ∂gaussianReal 0 v := by | |
| have he := integral_map (μ := gaussianReal 0 v) | |
| (f := F) (by fun_prop : AEMeasurable (fun x : ℝ => x + t * v) | |
| (gaussianReal 0 v)) hF.aestronglyMeasurable | |
| simpa only [gaussianReal_map_add_const, zero_add] using he | |
| rw [← hshift, integral_gaussianReal_eq_integral_smul hv, | |
| integral_gaussianReal_eq_integral_smul hv, ← integral_const_mul] | |
| apply integral_congr_ae | |
| apply ae_of_all | |
| intro x | |
| simp only [smul_eq_mul] | |
| rw [← mul_assoc, gaussianPDFReal_exp_linear hv] | |
| ring | |
| end InvariantIsing | |
| end | |
| end OAI | |