openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Fields /FieldAffineAverageDerivative.lean
| import OAI.Probability.InvariantIsing.Fields.FieldSecondBounds | |
| import OAI.Probability.InvariantIsing.Fields.FieldGaussianPairDerivative | |
| namespace OAI | |
| /-! Applying the quadratic-envelope differentiation theorem to the | |
| actual affine Gaussian shift of a finite scalar-field family. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron Filter Set | |
| open scoped Topology | |
| namespace InvariantIsing | |
| namespace FieldSmoothFamily | |
| variable {I : Set ℝ} (F : FieldSmoothFamily I) | |
| def affineShift (a v : ℝ) (p : ℝ × ℝ) (u : ℝ) : ℝ := | |
| F.U (p.1, p.2 + Real.sqrt (a + v * p.1) * u) | |
| def affineDifferential (a v : ℝ) (p : ℝ × ℝ) (u : ℝ) : (ℝ × ℝ) →L[ℝ] ℝ := | |
| pairLinear | |
| (F.T (p.1, p.2 + Real.sqrt (a + v * p.1) * u) + | |
| F.X (p.1, p.2 + Real.sqrt (a + v * p.1) * u) * | |
| (fieldAmplitudeSlope a v p.1 * u)) | |
| (F.X (p.1, p.2 + Real.sqrt (a + v * p.1) * u)) | |
| lemma affine_average_hasFDerivAt (hI : IsOpen I) | |
| (A : (ℝ × ℝ) → ℝ → ℝ) | |
| (DA : (ℝ × ℝ) → ℝ → (ℝ × ℝ) →L[ℝ] ℝ) | |
| (a v ζ : ℝ) {m V : ℝ} (hm : 0 < m) | |
| (hlo : ∀ t ∈ I, m ≤ a + v * t) (hhi : ∀ t ∈ I, a + v * t ≤ V) | |
| {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 : ℝ × ℝ} (hp : p.1 ∈ I) : | |
| HasFDerivAt (fun q => ∫ u, A q u | |
| ∂(gaussianReal 0 1).tilted (fun u => ζ * F.affineShift a v q u)) | |
| ((∫ u, DA p u + (ζ * A p u) • F.affineDifferential a v p u | |
| ∂(gaussianReal 0 1).tilted (fun u => ζ * F.affineShift a v p u)) - | |
| (∫ u, A p u | |
| ∂(gaussianReal 0 1).tilted (fun u => ζ * F.affineShift a v p u)) • | |
| (∫ u, ζ • F.affineDifferential a v p u | |
| ∂(gaussianReal 0 1).tilted (fun u => ζ * F.affineShift a v p u))) p := by | |
| let R := |v| / (2 * Real.sqrt m) | |
| let CD := F.KT + F.KX * R + F.KX | |
| let K := |F.U (p.1, 0)| + 1 | |
| let C := |ζ| * (K + F.KX * (|p.2| + 1)) | |
| let L := |ζ| * F.KX * Real.sqrt V | |
| have hR : 0 ≤ R := by dsimp only [R]; positivity | |
| have hX := F.kx_nonneg | |
| have hT : 0 ≤ F.KT := (abs_nonneg _).trans (F.bT (p.1, 0) hp) | |
| have hCD : 0 ≤ CD := by dsimp only [CD]; positivity | |
| have hpair : ContinuousAt (fun t : ℝ => (t, (0 : ℝ))) p.1 := | |
| continuousAt_id.prodMk continuousAt_const | |
| have hc : ContinuousAt (fun t => F.U (t, 0)) p.1 := | |
| ContinuousAt.comp (f := fun t : ℝ => (t, (0 : ℝ))) (x := p.1) | |
| (F.derivative (p.1, 0) hp).continuousAt hpair | |
| have hg := field_parameter_growth_eventually hX F.bX F.spatial hc | |
| have hset : {t : ℝ | t ∈ I ∧ ∀ y, | |
| |F.U (t, y)| ≤ K + F.KX * |y|} ∈ 𝓝 p.1 := by | |
| filter_upwards [hI.mem_nhds hp, hg] with t ht hgt | |
| exact ⟨ht, hgt⟩ | |
| obtain ⟨J, hJsub, hJopen, hpJ⟩ := mem_nhds_iff.mp hset | |
| let S : Set (ℝ × ℝ) := (Prod.fst ⁻¹' J) ∩ | |
| (Prod.snd ⁻¹' Ioo (p.2 - 1) (p.2 + 1)) | |
| have hS : S ∈ 𝓝 p := | |
| Filter.inter_mem ((hJopen.preimage continuous_fst).mem_nhds hpJ) | |
| ((isOpen_Ioo.preimage continuous_snd).mem_nhds ⟨by linarith, by linarith⟩) | |
| have hpS : p ∈ S := ⟨hpJ, by constructor <;> linarith⟩ | |
| apply field_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 fun_prop)) hA | |
| ((measurable_pairLinear | |
| ((F.mT.comp (by fun_prop)).add ((F.mX.comp (by fun_prop)).mul (by | |
| dsimp only [fieldAmplitudeSlope] | |
| fun_prop))) (F.mX.comp (by fun_prop))).aestronglyMeasurable) | |
| (hDA p) | |
| · intro q hq u | |
| have hqI := (hJsub hq.1).1 | |
| have hq2 : |q.2| ≤ |p.2| + 1 := by | |
| have hh : |q.2 - p.2| ≤ 1 := abs_le.mpr ⟨by linarith [hq.2.1], by linarith [hq.2.2]⟩ | |
| calc | |
| |q.2| = |q.2 - p.2 + p.2| := by congr 1; ring | |
| _ ≤ |q.2 - p.2| + |p.2| := abs_add_le _ _ | |
| _ ≤ _ := by linarith | |
| have hshift : |q.2 + Real.sqrt (a + v * q.1) * u| ≤ | |
| |p.2| + 1 + Real.sqrt V * |u| := by | |
| calc | |
| _ ≤ |q.2| + |Real.sqrt (a + v * q.1) * u| := abs_add_le _ _ | |
| _ = |q.2| + Real.sqrt (a + v * q.1) * |u| := by | |
| rw [abs_mul, abs_of_nonneg (Real.sqrt_nonneg _)] | |
| _ ≤ _ := add_le_add hq2 | |
| (mul_le_mul_of_nonneg_right (Real.sqrt_le_sqrt (hhi q.1 hqI)) (abs_nonneg _)) | |
| apply Real.exp_le_exp.mpr | |
| calc | |
| ζ * F.affineShift a v q u ≤ |ζ| * |F.affineShift a v q u| := by | |
| simpa only [abs_mul] using le_abs_self (ζ * F.affineShift a v q u) | |
| _ ≤ |ζ| * (K + F.KX * |q.2 + Real.sqrt (a + v * q.1) * u|) := | |
| mul_le_mul_of_nonneg_left ((hJsub hq.1).2 _) (abs_nonneg _) | |
| _ ≤ |ζ| * (K + F.KX * (|p.2| + 1 + Real.sqrt V * |u|)) := by gcongr | |
| _ = C + L * |u| := by dsimp only [C, L]; ring | |
| · intro q hq u | |
| exact hAb q (hJsub hq.1).1 u | |
| · intro q hq u | |
| have hqI := (hJsub hq.1).1 | |
| have hc : |fieldAmplitudeSlope a v q.1| ≤ R := abs_div_two_sqrt_le hm (hlo q.1 hqI) v | |
| unfold affineDifferential | |
| refine (norm_pairLinear_le _ _).trans ?_ | |
| calc | |
| _ ≤ F.KT + F.KX * (R * |u|) + F.KX := by | |
| refine add_le_add ((abs_add_le _ _).trans (add_le_add (F.bT _ hqI) ?_)) (F.bX _) | |
| rw [abs_mul, abs_mul] | |
| gcongr | |
| exact F.bX _ | |
| _ ≤ CD * (1 + |u|) := by | |
| dsimp only [CD] | |
| nlinarith [mul_nonneg hT (abs_nonneg u), mul_nonneg hX (abs_nonneg u)] | |
| · intro q hq u | |
| exact hEb q (hJsub hq.1).1 u | |
| · intro q hq u | |
| exact affine_gaussian_shift_hasFDerivAt F.derivative a v u (hJsub hq.1).1 | |
| (hm.trans_le (hlo q.1 (hJsub hq.1).1)) | |
| · intro q hq u | |
| exact dA q (hJsub hq.1).1 u | |
| end FieldSmoothFamily | |
| end InvariantIsing | |
| end | |
| end OAI | |