openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Fields /FieldAffineSecondDerivative.lean
| import OAI.Probability.InvariantIsing.Fields.FieldAffineNeighborhood | |
| import OAI.Probability.InvariantIsing.Fields.FieldAdjacentDerivative | |
| namespace OAI | |
| /-! Actual second covariance derivatives of every finite strict scalar | |
| recursion. All derivative data are constructed from the Gaussian integrals. -/ | |
| noncomputable section | |
| open Filter Set | |
| open scoped Topology | |
| namespace InvariantIsing | |
| namespace FieldSecondFamily | |
| variable {I : Set ℝ} (F : FieldSecondFamily I) | |
| lemma parameter_tangent_derivative (t z : ℝ) (ht : t ∈ I) : | |
| HasDerivAt (fun s => F.T (s, z)) (F.TT (t, z)) t := by | |
| have hmap : HasDerivAt (fun s : ℝ => (s, z)) ((1 : ℝ), (0 : ℝ)) t := | |
| (hasDerivAt_id t).prodMk (hasDerivAt_const t z) | |
| have hd := (F.derivativeT (t, z) ht).comp_hasDerivAt t hmap | |
| simpa [IsingPerceptron.pairLinear, Function.comp_def] using hd | |
| lemma parameter_mean_derivative (t z : ℝ) (ht : t ∈ I) : | |
| HasDerivAt (fun s => F.X (s, z)) (F.TX (t, z)) t := by | |
| have hmap : HasDerivAt (fun s : ℝ => (s, z)) ((1 : ℝ), (0 : ℝ)) t := | |
| (hasDerivAt_id t).prodMk (hasDerivAt_const t z) | |
| have hd := (F.derivativeX (t, z) ht).comp_hasDerivAt t hmap | |
| simpa [IsingPerceptron.pairLinear, Function.comp_def] using hd | |
| end FieldSecondFamily | |
| theorem fieldAffineValue_second_derivative (L : List FieldAffineStep) (t z : ℝ) | |
| (hpos : ∀ av ∈ L, 0 < av.base + av.slope * t) : | |
| ∃ D : ℝ, HasDerivAt (fun s => deriv (fun r => fieldAffineValue L (r, z)) s) D t := by | |
| obtain ⟨I, hI, ht, m, V, hm, hL⟩ := fieldAffine_variance_neighborhood L t hpos | |
| let F := fieldAffineSecondFamily I hI m V hm L hL | |
| have hval : F.U = fieldAffineValue L := fieldAffineSecondFamily_value I hI m V hm L hL | |
| have he : (fun s => deriv (fun r => fieldAffineValue L (r, z)) s) =ᶠ[𝓝 t] | |
| (fun s => F.T (s, z)) := by | |
| filter_upwards [hI.mem_nhds ht] with s hs | |
| have hd := F.toFieldSmoothFamily.parameter_derivative s z hs | |
| rw [hval] at hd | |
| exact hd.deriv | |
| exact ⟨F.TT (t, z), (F.parameter_tangent_derivative t z ht).congr_of_eventuallyEq he⟩ | |
| theorem fieldAffineValue_spatial_derivative_parameter_differentiable | |
| (L : List FieldAffineStep) (t z : ℝ) | |
| (hpos : ∀ av ∈ L, 0 < av.base + av.slope * t) : | |
| DifferentiableAt ℝ (fun s => deriv (fun y => fieldAffineValue L (s, y)) z) t := by | |
| obtain ⟨I, hI, ht, m, V, hm, hL⟩ := fieldAffine_variance_neighborhood L t hpos | |
| let F := fieldAffineSecondFamily I hI m V hm L hL | |
| have hval : F.U = fieldAffineValue L := fieldAffineSecondFamily_value I hI m V hm L hL | |
| have he : (fun s => deriv (fun y => fieldAffineValue L (s, y)) z) =ᶠ[𝓝 t] | |
| (fun s => F.X (s, z)) := by | |
| apply Eventually.of_forall | |
| intro s | |
| have hd := F.spatial s z | |
| rw [hval] at hd | |
| exact hd.deriv | |
| exact ((F.parameter_mean_derivative t z ht).congr_of_eventuallyEq he).differentiableAt | |
| end InvariantIsing | |
| end | |
| end OAI | |