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a89d282 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 | 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
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