openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Magnetic /MagneticContinuationIBP.lean
| import OAI.Probability.InvariantIsing.Magnetic.MagneticContinuationEndpoint | |
| import OAI.Probability.InvariantIsing.Fields.FieldGaussianIntegrability | |
| namespace OAI | |
| /-! Gaussian integration by parts for the bounded continuation test, | |
| with the actual linearly growing scalar field as tilted potential. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron | |
| open scoped NNReal | |
| namespace InvariantIsing | |
| lemma magnetic_gaussian_tilted_test_ibp {F M A B : ℝ → ℝ} | |
| (hF : Measurable F) (hG : HasLinearGrowth F) | |
| (hM : Measurable M) (hA : Measurable A) (hB : Measurable B) | |
| {K C D : ℝ} (bM : ∀ x, |M x| ≤ K) (bA : ∀ x, |A x| ≤ C) | |
| (bB : ∀ x, |B x| ≤ D) (dF : ∀ x, HasDerivAt F (M x) x) | |
| (dA : ∀ x, HasDerivAt A (B x) x) (r ζ z : ℝ) : | |
| (∫ u, u * A (z + r * u) | |
| ∂(gaussianReal 0 1).tilted (fun u => ζ * F (z + r * u))) = | |
| r * (∫ u, B (z + r * u) + ζ * A (z + r * u) * M (z + r * u) | |
| ∂(gaussianReal 0 1).tilted (fun u => ζ * F (z + r * u))) := by | |
| have hC : 0 ≤ C := (abs_nonneg _).trans (bA 0) | |
| let U := fun u => F (z + r * u) | |
| let V := fun u => Real.exp (ζ * U u) * A (z + r * u) | |
| let DV := fun u => r * (Real.exp (ζ * U u) * | |
| (B (z + r * u) + ζ * A (z + r * u) * M (z + r * u))) | |
| have hUm : Measurable U := hF.comp (by fun_prop) | |
| have hUg : HasLinearGrowth U := (hG.add_left z).scale_argument r | |
| have hw := integrable_exp_of_linearGrowth (gaussianReal 0 1) | |
| (gaussianReal_exponentialNormMoments 0 1) hUm hUg ζ | |
| have bgen (x : ℝ) : |B x + ζ * A x * M x| ≤ D + |ζ| * C * K := by | |
| refine (abs_add_le _ _).trans (add_le_add (bB x) ?_) | |
| rw [abs_mul, abs_mul] | |
| exact mul_le_mul (mul_le_mul_of_nonneg_left (bA x) (abs_nonneg ζ)) | |
| (bM x) (abs_nonneg _) (mul_nonneg (abs_nonneg ζ) hC) | |
| have hV : Integrable V (gaussianReal 0 1) := | |
| hw.mul_bdd (hA.comp (by fun_prop)).aestronglyMeasurable | |
| (Filter.Eventually.of_forall fun u => by | |
| simpa only [Real.norm_eq_abs] using bA (z + r * u)) | |
| have hDV : Integrable DV (gaussianReal 0 1) := by | |
| apply Integrable.const_mul | |
| exact hw.mul_bdd | |
| ((hB.comp (by fun_prop)).add | |
| (((hA.comp (by fun_prop)).const_mul ζ).mul (hM.comp (by fun_prop)))).aestronglyMeasurable | |
| (Filter.Eventually.of_forall fun u => by | |
| simpa only [Real.norm_eq_abs] using bgen (z + r * u)) | |
| have hderiv (u : ℝ) : HasDerivAt V (DV u) u := by | |
| have hs : HasDerivAt (fun x : ℝ => z + r * x) r u := by | |
| convert ((hasDerivAt_id u).const_mul r).const_add z using 1 <;> first | rfl | simp | |
| have hf : HasDerivAt (fun x => F (z + r * x)) (M (z + r * u) * r) u := | |
| (dF (z + r * u)).comp u hs | |
| have ha : HasDerivAt (fun x => A (z + r * x)) (B (z + r * u) * r) u := | |
| (dA (z + r * u)).comp u hs | |
| convert ((hf.const_mul ζ).exp).mul ha using 1 | |
| dsimp only [V, DV, U] | |
| ring | |
| have hlinear : HasLinearGrowth (fun u => u * A (z + r * u)) := by | |
| refine ⟨0, C, le_rfl, hC, fun u => ?_⟩ | |
| rw [abs_mul, Real.norm_eq_abs, zero_add] | |
| simpa only [mul_comm] using mul_le_mul_of_nonneg_left (bA (z + r * u)) (abs_nonneg u) | |
| have hUV : Integrable (fun u => u * V u) (gaussianReal 0 1) := by | |
| convert field_exp_mul_linear_integrable 1 0 ζ hUm hUg | |
| (measurable_id.mul (hA.comp (by fun_prop))) hlinear using 1 | |
| funext u | |
| dsimp only [V, Pi.mul_apply, Function.comp_apply, id_eq] | |
| ring | |
| have hibp := gaussian_integration_by_parts hderiv hV hDV hUV | |
| simp only [integral_tilted_eq_div] | |
| have hn : (∫ u, Real.exp (ζ * F (z + r * u)) * (u * A (z + r * u)) | |
| ∂gaussianReal 0 1) = | |
| r * ∫ u, Real.exp (ζ * F (z + r * u)) * | |
| (B (z + r * u) + ζ * A (z + r * u) * M (z + r * u)) ∂gaussianReal 0 1 := by | |
| rw [← integral_const_mul] | |
| convert hibp using 1 | |
| congr 1 | |
| funext u | |
| dsimp only [V, DV, U] | |
| ring | |
| rw [hn] | |
| ring | |
| end InvariantIsing | |
| end | |
| end OAI | |