openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Magnetic /MagneticContinuationPDE.lean
| import OAI.Probability.InvariantIsing.Magnetic.MagneticContinuationGenerator | |
| import OAI.Probability.InvariantIsing.Fields.FieldRealVariance | |
| namespace OAI | |
| /-! The actual bounded continuation solves the scalar heat equation with | |
| the nonlinear backward-field drift. The first coordinate is the amount | |
| of Gaussian variance already applied; reversing it gives the backward | |
| time convention in the manuscript. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron | |
| open scoped NNReal | |
| namespace InvariantIsing | |
| lemma magneticGaussianSpaceTwo_eq (P : MagneticContinuationJet) (A : MagneticContinuationTwoJet) | |
| (F : ℝ → ℝ) (hF : Measurable F) (ζ v z : ℝ) : | |
| magneticGaussianSpace ζ v F P.value A.value A.first z = | |
| fieldTiltSpatial ζ (Real.toNNReal v) F P.value A.value A.first z := by | |
| unfold magneticGaussianSpace fieldTiltSpatial | |
| rw [field_gaussianTiltAverage_eq_transition v ζ hF A.mFirst, | |
| field_gaussianTiltAverage_eq_transition v ζ (a := fun y => A.value y * P.value y) | |
| hF (A.mValue.mul P.mValue), | |
| field_gaussianTiltAverage_eq_transition v ζ hF A.mValue, | |
| field_gaussianTiltAverage_eq_transition v ζ hF P.mValue] | |
| lemma magneticGaussianGeneratorTwo_eq (P : MagneticContinuationJet) (A : MagneticContinuationTwoJet) | |
| (F : ℝ → ℝ) (hF : Measurable F) (hG : HasLinearGrowth F) (ζ v z : ℝ) : | |
| magneticGaussianGenerator ζ v F P.value P.first A.value A.first A.second z = | |
| magneticContinuationSecond ζ (Real.toNNReal v) F P.value P.first | |
| A.value A.first A.second z / 2 + | |
| ζ * fieldSpinTransition ζ (Real.toNNReal v) F P.value z * | |
| fieldTiltSpatial ζ (Real.toNNReal v) F P.value A.value A.first z := by | |
| rw [magneticContinuation_secondTwo_generator P A F hF hG] | |
| dsimp only [magneticGaussianGenerator] | |
| rw [field_gaussianTiltAverage_eq_transition v ζ hF A.mSecond, | |
| field_gaussianTiltAverage_eq_transition v ζ (a := fun y => A.first y * P.value y) | |
| hF (A.mFirst.mul P.mValue), | |
| field_gaussianTiltAverage_eq_transition v ζ (a := fun y => A.value y * P.first y) | |
| hF (A.mValue.mul P.mFirst), | |
| field_gaussianTiltAverage_eq_transition v ζ hF A.mValue, | |
| field_gaussianTiltAverage_eq_transition v ζ hF P.mFirst, | |
| field_gaussianTiltAverage_eq_transition v ζ (a := fun y => A.value y * (P.value y) ^ 2) | |
| hF (A.mValue.mul (P.mValue.pow_const 2)), | |
| field_gaussianTiltAverage_eq_transition v ζ (a := fun y => (P.value y) ^ 2) | |
| hF (P.mValue.pow_const 2)] | |
| theorem magneticGaussianAverageTwo_hasFDerivAt_generator (P : MagneticContinuationJet) (A : MagneticContinuationTwoJet) | |
| (F : ℝ → ℝ) (hF : Measurable F) (hG : HasLinearGrowth 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) | |
| (pairLinear | |
| (magneticContinuationSecond ζ (Real.toNNReal v) F P.value P.first | |
| A.value A.first A.second z / 2 + | |
| ζ * fieldSpinTransition ζ (Real.toNNReal v) F P.value z * | |
| fieldTiltSpatial ζ (Real.toNNReal v) F P.value A.value A.first z) | |
| (fieldTiltSpatial ζ (Real.toNNReal v) F P.value A.value A.first z)) (v, z) := by | |
| have hd := magneticGaussianAverageTwo_hasFDerivAt P A F hF dF ζ hv z | |
| rw [magneticGaussianAverageDifferentialTwo_eq_pair P A F hF hG, | |
| magneticGaussianTimeTwo_eq_generator P A F hF hG dF ζ hv, | |
| magneticGaussianGeneratorTwo_eq P A F hF hG, | |
| magneticGaussianSpaceTwo_eq P A F hF] at hd | |
| exact hd | |
| lemma magneticGaussianSpace_eq (P A : MagneticContinuationJet) | |
| (F : ℝ → ℝ) (hF : Measurable F) (ζ v z : ℝ) : | |
| magneticGaussianSpace ζ v F P.value A.value A.first z = | |
| fieldTiltSpatial ζ (Real.toNNReal v) F P.value A.value A.first z := by | |
| exact magneticGaussianSpaceTwo_eq P A.toTwoJet F hF ζ v z | |
| lemma magneticGaussianGenerator_eq (P A : MagneticContinuationJet) | |
| (F : ℝ → ℝ) (hF : Measurable F) (hG : HasLinearGrowth F) (ζ v z : ℝ) : | |
| magneticGaussianGenerator ζ v F P.value P.first A.value A.first A.second z = | |
| magneticContinuationSecond ζ (Real.toNNReal v) F P.value P.first | |
| A.value A.first A.second z / 2 + | |
| ζ * fieldSpinTransition ζ (Real.toNNReal v) F P.value z * | |
| fieldTiltSpatial ζ (Real.toNNReal v) F P.value A.value A.first z := by | |
| exact magneticGaussianGeneratorTwo_eq P A.toTwoJet F hF hG ζ v z | |
| theorem magneticGaussianAverage_hasFDerivAt_generator (P A : MagneticContinuationJet) | |
| (F : ℝ → ℝ) (hF : Measurable F) (hG : HasLinearGrowth 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) | |
| (pairLinear | |
| (magneticContinuationSecond ζ (Real.toNNReal v) F P.value P.first | |
| A.value A.first A.second z / 2 + | |
| ζ * fieldSpinTransition ζ (Real.toNNReal v) F P.value z * | |
| fieldTiltSpatial ζ (Real.toNNReal v) F P.value A.value A.first z) | |
| (fieldTiltSpatial ζ (Real.toNNReal v) F P.value A.value A.first z)) (v, z) := by | |
| exact magneticGaussianAverageTwo_hasFDerivAt_generator P A.toTwoJet F hF hG dF ζ hv z | |
| end InvariantIsing | |
| end | |
| end OAI | |