openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Fields /FieldFiniteAverageDerivative.lean
| import OAI.Probability.InvariantIsing.Fields.FieldFiniteLocal | |
| import OAI.Probability.InvariantIsing.Fields.FieldFiniteMeasurable | |
| namespace OAI | |
| /-! Differentiation of the actual finite-height Gaussian transform and | |
| its tilted observables. The local envelope is derived from the family. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron Filter Set | |
| open scoped Topology | |
| namespace InvariantIsing | |
| namespace FieldFiniteFamily | |
| variable {n : ℕ} {I : Set (Fin n → ℝ)} (F : FieldFiniteFamily n I) | |
| lemma affine_average_hasFDerivAt (hI : IsOpen I) | |
| (A : FieldCovariate n → ℝ → ℝ) | |
| (DA : FieldCovariate n → ℝ → FieldCovariate n →L[ℝ] ℝ) | |
| (a : ℝ) (v : Fin n → ℝ) (ζ : ℝ) {V R : ℝ} (hR : 0 ≤ R) | |
| (hlo : ∀ t ∈ I, 0 < fieldFiniteVariance a v t) | |
| (hhi : ∀ t ∈ I, fieldFiniteVariance a v t ≤ V) | |
| (hc : ∀ t ∈ I, ∀ i, |fieldFiniteSlope a v t i| ≤ R) | |
| {CA CE : ℝ} (hCA : 0 ≤ CA) (hCE : 0 ≤ CE) | |
| (hA : ∀ q, Measurable (A q)) | |
| (hDA : ∀ q, AEStronglyMeasurable (DA q) (gaussianReal 0 1)) | |
| (hAb : ∀ q, q.1 ∈ I → ∀ u, |A q u| ≤ CA * (1 + |u|)) | |
| (hEb : ∀ q, q.1 ∈ I → ∀ u, ‖DA q u‖ ≤ CE * (1 + |u|) ^ 2) | |
| (dA : ∀ q, q.1 ∈ I → ∀ u, HasFDerivAt (fun r => A r u) (DA q u) q) | |
| {p : FieldCovariate n} (hp : p.1 ∈ I) : | |
| HasFDerivAt (fun q => ∫ u, A q u ∂F.affineLaw a v ζ q) | |
| ((∫ u, DA p u + (ζ * A p u) • F.affineDifferential a v p u ∂F.affineLaw a v ζ p) - | |
| (∫ u, A p u ∂F.affineLaw a v ζ p) • | |
| (∫ u, ζ • F.affineDifferential a v p u ∂F.affineLaw a v ζ p)) p := by | |
| obtain ⟨S, C, L, CD, hS, hpS, hSI, hCD, hExp, hD⟩ := | |
| F.local_envelope hI a v ζ hR hhi hc hp | |
| apply field_finite_gaussianPair_hasFDerivAt (F.affineShift a v) A | |
| (F.affineDifferential a v) DA hS hpS ζ C L CA CD CE hCA hCD hCE | |
| (fun q => F.mU.comp (by dsimp only [shiftPoint]; fun_prop)) hA | |
| (F.measurable_affineDifferential a v p).aestronglyMeasurable (hDA p) hExp | |
| (fun q hq => hAb q (hSI q hq)) hD (fun q hq => hEb q (hSI q hq)) | |
| · intro q hq u | |
| exact field_finite_affine_shift_hasFDerivAt F.derivative a v u | |
| (hSI q hq) (hlo q.1 (hSI q hq)) | |
| · exact fun q hq => dA q (hSI q hq) | |
| def value (a : ℝ) (v : Fin n → ℝ) (ζ : ℝ) (p : FieldCovariate n) : ℝ := | |
| gaussianTransform (fieldFiniteVariance a v p.1) ζ (fun y => F.U (p.1, y)) p.2 | |
| lemma value_hasFDerivAt (hI : IsOpen I) (a : ℝ) (v : Fin n → ℝ) (ζ : ℝ) | |
| {V R : ℝ} (hR : 0 ≤ R) | |
| (hlo : ∀ t ∈ I, 0 < fieldFiniteVariance a v t) | |
| (hhi : ∀ t ∈ I, fieldFiniteVariance a v t ≤ V) | |
| (hc : ∀ t ∈ I, ∀ i, |fieldFiniteSlope a v t i| ≤ R) | |
| {p : FieldCovariate n} (hp : p.1 ∈ I) : | |
| HasFDerivAt (F.value a v ζ) | |
| (∫ u, F.affineDifferential a v p u ∂F.affineLaw a v ζ p) p := by | |
| obtain ⟨S, C, L, CD, hS, _hpS, hSI, hCD, hExp, hD⟩ := | |
| F.local_envelope hI a v ζ hR hhi hc hp | |
| let B : ℝ → ℝ := fun u => CD * (1 + |u|) | |
| let E : ℝ → ℝ := fun u => Real.exp (C + L * |u|) | |
| have hBg : HasLinearGrowth B := by | |
| refine ⟨CD, CD, hCD, hCD, fun u => ?_⟩ | |
| dsimp only [B] | |
| rw [abs_of_nonneg (by positivity), Real.norm_eq_abs] | |
| exact le_of_eq (by ring) | |
| have hBi : Integrable B (gaussianReal 0 1) := | |
| field_gaussian_linear_integrable 1 0 (by fun_prop) hBg | |
| have hEBi : Integrable (fun u => E u * B u) (gaussianReal 0 1) := by | |
| apply ((field_gaussian_quadratic_envelope_integrable C L).const_mul CD).mono' (by fun_prop) | |
| apply ae_of_all | |
| intro u | |
| change |Real.exp (C + L * |u|) * (CD * (1 + |u|))| ≤ | |
| CD * (Real.exp (C + L * |u|) * (1 + |u|) ^ 2) | |
| rw [abs_of_nonneg (by positivity)] | |
| calc | |
| _ = CD * Real.exp (C + L * |u|) * (1 + |u|) := by ring | |
| _ ≤ CD * Real.exp (C + L * |u|) * (1 + |u|) ^ 2 := | |
| mul_le_mul_of_nonneg_left (field_mark_one_add_le_sq u) (by positivity) | |
| _ = _ := by ring | |
| have hm : Measurable (F.affineShift a v p) := F.mU.comp (by dsimp only [shiftPoint]; fun_prop) | |
| have hg : HasLinearGrowth (F.affineShift a v p) := | |
| (F.growth p.1 hp).add_left p.2 |>.scale_argument (Real.sqrt (fieldFiniteVariance a v p.1)) | |
| exact field_finite_average_hasFDerivAt_of_envelope (gaussianReal 0 1) | |
| (F.affineShift a v) (F.affineDifferential a v) hS | |
| (fun q => F.mU.comp (by dsimp only [shiftPoint]; fun_prop)) | |
| (F.measurable_affineDifferential a v p).aestronglyMeasurable | |
| (field_gaussian_linear_integrable 1 0 hm hg) ζ | |
| (integrable_exp_of_linearGrowth _ (gaussianReal_exponentialNormMoments 0 1) hm hg ζ) | |
| B E hBi hEBi hD hExp (fun q hq u => | |
| field_finite_affine_shift_hasFDerivAt F.derivative a v u | |
| (hSI q hq) (hlo q.1 (hSI q hq))) | |
| end FieldFiniteFamily | |
| end InvariantIsing | |
| end | |
| end OAI | |