openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Magnetic /MagneticContinuationGenerator.lean
| import OAI.Probability.InvariantIsing.Magnetic.MagneticGaussianCovariance | |
| import OAI.Probability.InvariantIsing.Fields.FieldSpinLinear | |
| namespace OAI | |
| /-! The exact heat generator of the scalar continuation. Integration by | |
| parts removes the apparent inverse square-root singularity from the | |
| positive-variance differential. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron | |
| open scoped NNReal | |
| namespace InvariantIsing | |
| def magneticGaussianGenerator (ζ v : ℝ) (F M Q A B C : ℝ → ℝ) (z : ℝ) : ℝ := | |
| let T := fun f => gaussianTiltAverage v ζ F f z | |
| T C / 2 + ζ * T (fun y => B y * M y) + | |
| ζ / 2 * (T (fun y => A y * Q y) - T A * T Q) + | |
| ζ ^ 2 / 2 * (T (fun y => A y * (M y) ^ 2) - T A * T (fun y => (M y) ^ 2)) | |
| lemma magneticGaussianTimeTwo_eq_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) : | |
| magneticGaussianTime ζ v F P.value A.value A.first z = | |
| magneticGaussianGenerator ζ v F P.value P.first A.value A.first A.second z := by | |
| let r := Real.sqrt v | |
| let ν := (gaussianReal 0 1).tilted (fun u => ζ * F (z + r * u)) | |
| obtain ⟨KM, _, bM⟩ := P.bValue | |
| obtain ⟨KQ, _, bQ⟩ := P.bFirst | |
| obtain ⟨KA, hKA, bA⟩ := A.bValue | |
| obtain ⟨KB, hKB, bB⟩ := A.bFirst | |
| obtain ⟨KC, _, bC⟩ := A.bSecond | |
| have hr : r ≠ 0 := (Real.sqrt_pos.mpr hv).ne' | |
| have bAM (y : ℝ) : |A.value y * P.value y| ≤ KA * KM := by | |
| rw [abs_mul] | |
| exact mul_le_mul (bA y) (bM y) (abs_nonneg _) hKA | |
| have bBM (y : ℝ) : |A.first y * P.value y| ≤ KB * KM := by | |
| rw [abs_mul] | |
| exact mul_le_mul (bB y) (bM y) (abs_nonneg _) hKB | |
| have bAQ (y : ℝ) : |A.value y * P.first y| ≤ KA * KQ := by | |
| rw [abs_mul] | |
| exact mul_le_mul (bA y) (bQ y) (abs_nonneg _) hKA | |
| have bdAM (y : ℝ) : |A.first y * P.value y + A.value y * P.first y| ≤ | |
| KB * KM + KA * KQ := (abs_add_le _ _).trans (add_le_add (bBM y) (bAQ y)) | |
| have iBM : Integrable (fun u => A.first (z + r * u) * P.value (z + r * u)) ν := | |
| magnetic_tilted_shift_integrable hF hG (A.mFirst.mul P.mValue) bBM r ζ z | |
| have iAQ : Integrable (fun u => A.value (z + r * u) * P.first (z + r * u)) ν := | |
| magnetic_tilted_shift_integrable hF hG (A.mValue.mul P.mFirst) bAQ r ζ z | |
| have hB := magnetic_tilted_scaled_test_ibp hF hG P.mValue A.mFirst A.mSecond | |
| bM bB bC dF A.dFirst hr ζ z | |
| have hM := magnetic_tilted_scaled_test_ibp hF hG P.mValue P.mValue P.mFirst | |
| bM bM bQ dF P.dValue hr ζ z | |
| have hAM := magnetic_tilted_scaled_test_ibp hF hG P.mValue | |
| (A.mValue.mul P.mValue) ((A.mFirst.mul P.mValue).add (A.mValue.mul P.mFirst)) | |
| bM bAM bdAM dF (fun y => (A.dValue y).mul (P.dValue y)) hr ζ z | |
| dsimp only [Pi.mul_apply, Pi.add_apply] at hAM | |
| have hAMleft : (∫ u, A.value (z + r * u) * | |
| (P.value (z + r * u) * ((1 / (2 * r)) * u)) ∂ν) = | |
| (∫ u, (A.value (z + r * u) * P.value (z + r * u)) * | |
| ((1 / (2 * r)) * u) ∂ν) := by | |
| congr 1 | |
| funext u | |
| ring | |
| have hAMright : (∫ u, (A.value (z + r * u) * P.value (z + r * u)) * | |
| P.value (z + r * u) ∂ν) = | |
| ∫ u, A.value (z + r * u) * (P.value (z + r * u)) ^ 2 ∂ν := by | |
| congr 1 | |
| funext u | |
| ring | |
| have hMright : (∫ u, P.value (z + r * u) * P.value (z + r * u) ∂ν) = | |
| ∫ u, (P.value (z + r * u)) ^ 2 ∂ν := by | |
| congr 1 | |
| funext u | |
| ring | |
| change magneticGaussianTime ζ v F P.value A.value A.first z = _ | |
| dsimp only [magneticGaussianTime] | |
| rw [hB, hAMleft, hAM, hM, hAMright, hMright, integral_add iBM iAQ] | |
| dsimp only [magneticGaussianGenerator, gaussianTiltAverage] | |
| ring | |
| lemma magneticContinuation_secondTwo_generator (P : MagneticContinuationJet) (A : MagneticContinuationTwoJet) | |
| (F : ℝ → ℝ) (hF : Measurable F) (hG : HasLinearGrowth F) | |
| (ζ : ℝ) (v : ℝ≥0) (z : ℝ) : | |
| magneticContinuationSecond ζ v F P.value P.first A.value A.first A.second z / 2 + | |
| ζ * fieldSpinTransition ζ v F P.value z * | |
| fieldTiltSpatial ζ v F P.value A.value A.first z = | |
| (let T := fun f => fieldSpinTransition ζ v F f z | |
| T A.second / 2 + ζ * T (fun y => A.first y * P.value y) + | |
| ζ / 2 * (T (fun y => A.value y * P.first y) - T A.value * T P.first) + | |
| ζ ^ 2 / 2 * (T (fun y => A.value y * (P.value y) ^ 2) - | |
| T A.value * T (fun y => (P.value y) ^ 2))) := by | |
| obtain ⟨KM, _, bM⟩ := P.bValue | |
| obtain ⟨KQ, _, bQ⟩ := P.bFirst | |
| obtain ⟨KA, hKA, bA⟩ := A.bValue | |
| obtain ⟨KB, hKB, bB⟩ := A.bFirst | |
| have bBM (y : ℝ) : |A.first y * P.value y| ≤ KB * KM := by | |
| rw [abs_mul] | |
| exact mul_le_mul (bB y) (bM y) (abs_nonneg _) hKB | |
| have bAQ (y : ℝ) : |A.value y * P.first y| ≤ KA * KQ := by | |
| rw [abs_mul] | |
| exact mul_le_mul (bA y) (bQ y) (abs_nonneg _) hKA | |
| have hadd := fieldSpinTransition_add ζ v hF hG (A.mFirst.mul P.mValue) | |
| (A.mValue.mul P.mFirst) bBM bAQ z | |
| change fieldSpinTransition ζ v F (fun y => A.first y * P.value y + A.value y * P.first y) z = | |
| fieldSpinTransition ζ v F (fun y => A.first y * P.value y) z + | |
| fieldSpinTransition ζ v F (fun y => A.value y * P.first y) z at hadd | |
| have hassoc : (fun y => (A.value y * P.value y) * P.value y) = | |
| (fun y => A.value y * (P.value y) ^ 2) := by | |
| funext y | |
| ring | |
| have hsquare : (fun y => P.value y * P.value y) = (fun y => (P.value y) ^ 2) := by | |
| funext y | |
| ring | |
| dsimp only [magneticContinuationSecond, fieldTiltSpatial] | |
| rw [hadd, hassoc, hsquare] | |
| ring | |
| lemma magneticGaussianTime_eq_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) : | |
| magneticGaussianTime ζ v F P.value A.value A.first z = | |
| magneticGaussianGenerator ζ v F P.value P.first A.value A.first A.second z := by | |
| exact magneticGaussianTimeTwo_eq_generator P A.toTwoJet F hF hG dF ζ hv z | |
| lemma magneticContinuation_second_generator (P A : MagneticContinuationJet) | |
| (F : ℝ → ℝ) (hF : Measurable F) (hG : HasLinearGrowth F) | |
| (ζ : ℝ) (v : ℝ≥0) (z : ℝ) : | |
| magneticContinuationSecond ζ v F P.value P.first A.value A.first A.second z / 2 + | |
| ζ * fieldSpinTransition ζ v F P.value z * | |
| fieldTiltSpatial ζ v F P.value A.value A.first z = | |
| (let T := fun f => fieldSpinTransition ζ v F f z | |
| T A.second / 2 + ζ * T (fun y => A.first y * P.value y) + | |
| ζ / 2 * (T (fun y => A.value y * P.first y) - T A.value * T P.first) + | |
| ζ ^ 2 / 2 * (T (fun y => A.value y * (P.value y) ^ 2) - | |
| T A.value * T (fun y => (P.value y) ^ 2))) := by | |
| exact magneticContinuation_secondTwo_generator P A.toTwoJet F hF hG ζ v z | |
| end InvariantIsing | |
| end | |
| end OAI | |