openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Magnetic /MagneticContinuationTime.lean
| import OAI.Probability.InvariantIsing.Magnetic.MagneticContinuationTwoJet | |
| import OAI.Probability.InvariantIsing.Magnetic.MagneticStationaryFamily | |
| import OAI.Probability.InvariantIsing.Fields.FieldAffineAverageDerivative | |
| namespace OAI | |
| /-! Joint variance/spatial differentiation of the actual scalar | |
| continuation. The covariance differential is left as integrals here; | |
| Gaussian integration by parts then identifies its time generator. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron Filter Set | |
| open scoped NNReal Topology | |
| namespace InvariantIsing | |
| def magneticGaussianAverageDifferential (ζ v : ℝ) (F M A B : ℝ → ℝ) (z : ℝ) : | |
| (ℝ × ℝ) →L[ℝ] ℝ := | |
| let ν := (gaussianReal 0 1).tilted (fun u => ζ * F (z + Real.sqrt v * u)) | |
| let D := fun u => pairLinear | |
| (M (z + Real.sqrt v * u) * ((1 / (2 * Real.sqrt v)) * u)) | |
| (M (z + Real.sqrt v * u)) | |
| (∫ u, pairLinear | |
| (B (z + Real.sqrt v * u) * ((1 / (2 * Real.sqrt v)) * u)) | |
| (B (z + Real.sqrt v * u)) + (ζ * A (z + Real.sqrt v * u)) • D u ∂ν) - | |
| (∫ u, A (z + Real.sqrt v * u) ∂ν) • (∫ u, ζ • D u ∂ν) | |
| theorem magneticGaussianAverageTwo_hasFDerivAt (P : MagneticContinuationJet) (A : MagneticContinuationTwoJet) | |
| (F : ℝ → ℝ) (hF : Measurable F) (dF : ∀ z, HasDerivAt F (P.value z) z) | |
| (ζ : ℝ) {v : ℝ} (hv : 0 < v) (z : ℝ) : | |
| HasFDerivAt (fun q : ℝ × ℝ => gaussianTiltAverage q.1 ζ F A.value q.2) | |
| (magneticGaussianAverageDifferential ζ v F P.value A.value A.first z) (v, z) := by | |
| obtain ⟨K, hK, bK⟩ := P.bValue | |
| obtain ⟨L, _, bL⟩ := P.bFirst | |
| obtain ⟨CA, hCA, bCA⟩ := A.bValue | |
| obtain ⟨CB, hCB, bCB⟩ := A.bFirst | |
| let I := Ioo (v / 2) (v + 1) | |
| let S := (magneticStationaryFamily I F P hF K L bK bL dF).toFieldSmoothFamily | |
| let α : (ℝ × ℝ) → ℝ → ℝ := fun q u => A.value (q.2 + Real.sqrt q.1 * u) | |
| let Dα : (ℝ × ℝ) → ℝ → (ℝ × ℝ) →L[ℝ] ℝ := fun q u => pairLinear | |
| (A.first (q.2 + Real.sqrt q.1 * u) * ((1 / (2 * Real.sqrt q.1)) * u)) | |
| (A.first (q.2 + Real.sqrt q.1 * u)) | |
| let R := 1 / (2 * Real.sqrt (v / 2)) | |
| have hR : 0 ≤ R := by dsimp only [R]; positivity | |
| have hm : 0 < v / 2 := by linarith | |
| have hbound (q : ℝ × ℝ) (hq : q.1 ∈ I) : | |
| |1 / (2 * Real.sqrt q.1)| ≤ R := by | |
| simpa only [abs_one] using abs_div_two_sqrt_le hm hq.1.le 1 | |
| have hdα (q : ℝ × ℝ) (hq : q.1 ∈ I) (u : ℝ) : | |
| HasFDerivAt (fun r => α r u) (Dα q u) q := by | |
| have hb (p : ℝ × ℝ) (_ : p.1 ∈ I) : | |
| HasFDerivAt (fun r : ℝ × ℝ => A.value r.2) (pairLinear 0 (A.first p.2)) p := by | |
| convert (A.dValue p.2).hasFDerivAt.comp p hasFDerivAt_snd using 1 | |
| · rfl | |
| · apply ContinuousLinearMap.ext | |
| intro w | |
| simp [pairLinear, mul_comm] | |
| simpa only [zero_add, one_mul] using | |
| affine_gaussian_shift_hasFDerivAt hb 0 1 u hq (by | |
| dsimp only [I] at hq | |
| linarith [hq.1]) | |
| have hbα (q : ℝ × ℝ) (_ : q.1 ∈ I) (u : ℝ) : |α q u| ≤ CA * (1 + |u|) := by | |
| exact (bCA _).trans (le_mul_of_one_le_right hCA (by linarith [abs_nonneg u])) | |
| have hbDα (q : ℝ × ℝ) (hq : q.1 ∈ I) (u : ℝ) : | |
| ‖Dα q u‖ ≤ (CB * (R + 1)) * (1 + |u|) ^ 2 := by | |
| refine (norm_pairLinear_le _ _).trans ?_ | |
| have hf : |A.first (q.2 + Real.sqrt q.1 * u) * ((1 / (2 * Real.sqrt q.1)) * u)| ≤ | |
| CB * R * |u| := by | |
| rw [abs_mul, abs_mul] | |
| exact (mul_le_mul (bCB _) (mul_le_mul_of_nonneg_right (hbound q hq) (abs_nonneg u)) | |
| (mul_nonneg (abs_nonneg _) (abs_nonneg _)) hCB).trans_eq (by ring) | |
| have hg := bCB (q.2 + Real.sqrt q.1 * u) | |
| have h1 : 1 ≤ (1 + |u|) ^ 2 := by nlinarith [abs_nonneg u] | |
| have h2 : |u| ≤ (1 + |u|) ^ 2 := by nlinarith [sq_nonneg |u|, abs_nonneg u] | |
| nlinarith [mul_le_mul_of_nonneg_left h1 hCB, | |
| mul_le_mul_of_nonneg_left h2 (mul_nonneg hCB hR)] | |
| have hp : (v, z) ∈ I ×ˢ univ := ⟨by dsimp only [I]; constructor <;> linarith, mem_univ z⟩ | |
| have hh := S.affine_average_hasFDerivAt isOpen_Ioo α Dα 0 1 ζ hm | |
| (fun t ht => by simpa only [zero_add, one_mul] using ht.1.le) | |
| (fun t ht => by simpa only [zero_add, one_mul] using ht.2.le) | |
| hCA (mul_nonneg hCB (by linarith : 0 ≤ R + 1)) | |
| (fun q => A.mValue.comp (by fun_prop)) | |
| (fun q => (measurable_pairLinear | |
| (A.mFirst.comp (by fun_prop) |>.mul (by fun_prop)) | |
| (A.mFirst.comp (by fun_prop))).aestronglyMeasurable) | |
| hbα hbDα hdα hp.1 | |
| simpa only [α, Dα, S, magneticStationaryFamily, FieldSmoothFamily.affineShift, | |
| FieldSmoothFamily.affineDifferential, fieldAmplitudeSlope, zero_add, one_mul, | |
| gaussianTiltAverage, magneticGaussianAverageDifferential] using hh | |
| theorem magneticGaussianAverage_hasFDerivAt (P A : MagneticContinuationJet) | |
| (F : ℝ → ℝ) (hF : Measurable F) (dF : ∀ z, HasDerivAt F (P.value z) z) | |
| (ζ : ℝ) {v : ℝ} (hv : 0 < v) (z : ℝ) : | |
| HasFDerivAt (fun q : ℝ × ℝ => gaussianTiltAverage q.1 ζ F A.value q.2) | |
| (magneticGaussianAverageDifferential ζ v F P.value A.value A.first z) (v, z) := by | |
| exact magneticGaussianAverageTwo_hasFDerivAt P A.toTwoJet F hF dF ζ hv z | |
| end InvariantIsing | |
| end | |
| end OAI | |