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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 | import OAI.Probability.InvariantIsing.Magnetic.MagneticContinuationRegular
namespace OAI
/-! A bounded third-order spatial jet for the finite Ising continuation.
Only the square and actual Gaussian transition operations are required. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped NNReal
namespace InvariantIsing
structure MagneticContinuationJet where
value : ℝ → ℝ
first : ℝ → ℝ
second : ℝ → ℝ
third : ℝ → ℝ
mValue : Measurable value
mFirst : Measurable first
mSecond : Measurable second
mThird : Measurable third
bValue : MagneticContinuationBound value
bFirst : MagneticContinuationBound first
bSecond : MagneticContinuationBound second
bThird : MagneticContinuationBound third
dValue : ∀ z, HasDerivAt value (first z) z
dFirst : ∀ z, HasDerivAt first (second z) z
dSecond : ∀ z, HasDerivAt second (third z) z
namespace MagneticContinuationJet
def square (A : MagneticContinuationJet) : MagneticContinuationJet where
value := fun z => (A.value z) ^ 2
first := fun z => 2 * A.value z * A.first z
second := fun z => 2 * (A.first z) ^ 2 + 2 * A.value z * A.second z
third := fun z => 6 * A.first z * A.second z + 2 * A.value z * A.third z
mValue := A.mValue.pow_const 2
mFirst := (measurable_const.mul A.mValue).mul A.mFirst
mSecond := (measurable_const.mul (A.mFirst.pow_const 2)).add
((measurable_const.mul A.mValue).mul A.mSecond)
mThird := ((measurable_const.mul A.mFirst).mul A.mSecond).add
((measurable_const.mul A.mValue).mul A.mThird)
bValue := by simpa only [pow_two] using A.bValue.mul A.bValue
bFirst := ((MagneticContinuationBound.const 2).mul A.bValue).mul A.bFirst
bSecond := by
have hs : MagneticContinuationBound (fun z => (A.first z) ^ 2) := by
simpa only [pow_two] using A.bFirst.mul A.bFirst
exact ((MagneticContinuationBound.const 2).mul hs).add
(((MagneticContinuationBound.const 2).mul A.bValue).mul A.bSecond)
bThird := (((MagneticContinuationBound.const 6).mul A.bFirst).mul A.bSecond).add
(((MagneticContinuationBound.const 2).mul A.bValue).mul A.bThird)
dValue := fun z => by
convert (A.dValue z).pow 2 using 1
ring
dFirst := fun z => by
convert ((A.dValue z).const_mul 2).mul (A.dFirst z) using 1
ring
dSecond := fun z => by
convert (((A.dFirst z).pow 2).const_mul 2).add
(((A.dValue z).const_mul 2).mul (A.dSecond z)) using 1
ring
def transition (A P : MagneticContinuationJet) (ζ : ℝ) (v : ℝ≥0)
(F : ℝ → ℝ) (hF : Measurable F) (hG : HasLinearGrowth F)
(dF : ∀ z, HasDerivAt F (P.value z) z) : MagneticContinuationJet where
value := fieldSpinTransition ζ v F A.value
first := fieldTiltSpatial ζ v F P.value A.value A.first
second := magneticContinuationSecond ζ v F P.value P.first A.value A.first A.second
third := magneticContinuationThird ζ v F P.value P.first P.second
A.value A.first A.second A.third
mValue := measurable_fieldSpinTransition ζ v hF A.mValue
mFirst := measurable_fieldTiltSpatial ζ v hF P.mValue A.mValue A.mFirst
mSecond := measurable_magneticContinuationSecond ζ v hF P.mValue P.mFirst
A.mValue A.mFirst A.mSecond
mThird := measurable_magneticContinuationThird ζ v hF P.mValue P.mFirst P.mSecond
A.mValue A.mFirst A.mSecond A.mThird
bValue := A.bValue.transition ζ v hF hG
bFirst := MagneticContinuationBound.spatial ζ v hF hG P.bValue A.bValue A.bFirst
bSecond := magneticContinuationSecond_bounded ζ v hF hG P.bValue P.bFirst
A.bValue A.bFirst A.bSecond
bThird := magneticContinuationThird_bounded ζ v hF hG P.bValue P.bFirst P.bSecond
A.bValue A.bFirst A.bSecond A.bThird
dValue := fun z => by
obtain ⟨K, hK, bK⟩ := P.bValue
obtain ⟨L, hL, bL⟩ := A.bValue
obtain ⟨C, _, bC⟩ := A.bFirst
exact hasDerivAt_fieldSpinTransition ζ v hF hG P.mValue A.mValue A.mFirst
hK hL bK bL bC dF A.dValue z
dFirst := fun z => by
obtain ⟨K, hK, bK⟩ := P.bValue
obtain ⟨L, hL, bL⟩ := P.bFirst
obtain ⟨a, ha, ba⟩ := A.bValue
obtain ⟨b, hb, bb⟩ := A.bFirst
obtain ⟨c, _, bc⟩ := A.bSecond
exact hasDerivAt_magneticContinuationSecond ζ v hF hG P.mValue P.mFirst
A.mValue A.mFirst A.mSecond hK hL ha hb bK bL ba bb bc dF P.dValue
A.dValue A.dFirst z
dSecond := fun z => by
obtain ⟨K, hK, bK⟩ := P.bValue
obtain ⟨L, hL, bL⟩ := P.bFirst
obtain ⟨J, _, bJ⟩ := P.bSecond
obtain ⟨a, ha, ba⟩ := A.bValue
obtain ⟨b, hb, bb⟩ := A.bFirst
obtain ⟨c, hc, bc⟩ := A.bSecond
obtain ⟨d, _, bd⟩ := A.bThird
exact hasDerivAt_magneticContinuationThird ζ v hF hG P.mValue P.mFirst P.mSecond
A.mValue A.mFirst A.mSecond A.mThird hK hL ha hb hc bK bL bJ ba bb bc bd
dF P.dValue P.dFirst A.dValue A.dFirst A.dSecond z
end MagneticContinuationJet
end InvariantIsing
end
end OAI
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