openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Magnetic /MagneticContinuationJet.lean
| 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 | |