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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 | import OAI.Probability.InvariantIsing.Magnetic.MagneticScalarFourth
namespace OAI
/-! Three spatial derivatives of a bounded continuation through a single
actual Gaussian transition. These are the derivatives needed to close the
inverse-coordinate continuation at the two mean-spin endpoints. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped NNReal
namespace InvariantIsing
def magneticContinuationSecond (ζ : ℝ) (v : ℝ≥0)
(F M Q A B C : ℝ → ℝ) (z : ℝ) : ℝ :=
let T := fieldSpinTransition ζ v F
let D := fieldTiltSpatial ζ v F M
D B C z + ζ * (D (fun y => A y * M y) (fun y => B y * M y + A y * Q y) z -
(D A B z * T M z + T A z * D M Q z))
def magneticContinuationThird (ζ : ℝ) (v : ℝ≥0)
(F M Q R A B C D : ℝ → ℝ) (z : ℝ) : ℝ :=
let T := fieldSpinTransition ζ v F
let S := fieldTiltSpatial ζ v F M
magneticContinuationSecond ζ v F M Q B C D z + ζ *
(magneticContinuationSecond ζ v F M Q
(fun y => A y * M y) (fun y => B y * M y + A y * Q y)
(fun y => C y * M y + 2 * B y * Q y + A y * R y) z -
(magneticContinuationSecond ζ v F M Q A B C z * T M z +
2 * S A B z * S M Q z +
T A z * magneticContinuationSecond ζ v F M Q M Q R z))
lemma hasDerivAt_magneticContinuationSecond (ζ : ℝ) (v : ℝ≥0)
{F M Q A B C : ℝ → ℝ} (hF : Measurable F) (hG : HasLinearGrowth F)
(hM : Measurable M) (hQ : Measurable Q)
(hA : Measurable A) (hB : Measurable B) (hC : Measurable C)
{K L a b c : ℝ} (hK : 0 ≤ K) (_hL : 0 ≤ L) (ha : 0 ≤ a) (hb : 0 ≤ b)
(bM : ∀ y, |M y| ≤ K) (bQ : ∀ y, |Q y| ≤ L)
(bA : ∀ y, |A y| ≤ a) (bB : ∀ y, |B y| ≤ b) (bC : ∀ y, |C y| ≤ c)
(dF : ∀ y, HasDerivAt F (M y) y) (dM : ∀ y, HasDerivAt M (Q y) y)
(dA : ∀ y, HasDerivAt A (B y) y) (dB : ∀ y, HasDerivAt B (C y) y)
(z : ℝ) :
HasDerivAt (fieldTiltSpatial ζ v F M A B)
(magneticContinuationSecond ζ v F M Q A B C z) z := by
have bAM : ∀ y, |A y * M y| ≤ a * K := by
intro y
rw [abs_mul]
exact mul_le_mul (bA y) (bM y) (abs_nonneg _) ha
have bAM' : ∀ y, |B y * M y + A y * Q y| ≤ b * K + a * L := by
intro y
refine (abs_add_le _ _).trans (add_le_add ?_ ?_)
· rw [abs_mul]
exact mul_le_mul (bB y) (bM y) (abs_nonneg _) hb
· rw [abs_mul]
exact mul_le_mul (bA y) (bQ y) (abs_nonneg _) ha
have htA := hasDerivAt_fieldSpinTransition ζ v hF hG hM hA hB hK ha bM bA bB dF dA z
have htB := hasDerivAt_fieldSpinTransition ζ v hF hG hM hB hC hK hb bM bB bC dF dB z
have htM := hasDerivAt_fieldSpinTransition ζ v hF hG hM hM hQ hK hK bM bM bQ dF dM z
have htAM := hasDerivAt_fieldSpinTransition ζ v hF hG hM (hA.mul hM)
((hB.mul hM).add (hA.mul hQ)) hK (mul_nonneg ha hK) bM bAM bAM'
dF (fun y => (dA y).mul (dM y)) z
exact htB.add ((htAM.sub (htA.mul htM)).const_mul ζ)
lemma hasDerivAt_magneticContinuationThird (ζ : ℝ) (v : ℝ≥0)
{F M Q R A B C D : ℝ → ℝ} (hF : Measurable F) (hG : HasLinearGrowth F)
(hM : Measurable M) (hQ : Measurable Q) (hR : Measurable R)
(hA : Measurable A) (hB : Measurable B) (hC : Measurable C) (hD : Measurable D)
{K L J a b c d : ℝ} (hK : 0 ≤ K) (hL : 0 ≤ L)
(ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c)
(bM : ∀ y, |M y| ≤ K) (bQ : ∀ y, |Q y| ≤ L) (bR : ∀ y, |R y| ≤ J)
(bA : ∀ y, |A y| ≤ a) (bB : ∀ y, |B y| ≤ b)
(bC : ∀ y, |C y| ≤ c) (bD : ∀ y, |D y| ≤ d)
(dF : ∀ y, HasDerivAt F (M y) y) (dM : ∀ y, HasDerivAt M (Q y) y)
(dQ : ∀ y, HasDerivAt Q (R y) y)
(dA : ∀ y, HasDerivAt A (B y) y) (dB : ∀ y, HasDerivAt B (C y) y)
(dC : ∀ y, HasDerivAt C (D y) y) (z : ℝ) :
HasDerivAt (magneticContinuationSecond ζ v F M Q A B C)
(magneticContinuationThird ζ v F M Q R A B C D z) z := by
have bprod {U V : ℝ → ℝ} {u w : ℝ} (hu : 0 ≤ u)
(bu : ∀ y, |U y| ≤ u) (bv : ∀ y, |V y| ≤ w) (y : ℝ) :
|U y * V y| ≤ u * w := by
rw [abs_mul]
exact mul_le_mul (bu y) (bv y) (abs_nonneg _) hu
have bAM := bprod ha bA bM
have bAM' (y : ℝ) : |B y * M y + A y * Q y| ≤ b * K + a * L :=
(abs_add_le _ _).trans (add_le_add (bprod hb bB bM y) (bprod ha bA bQ y))
have bAM'' (y : ℝ) : |C y * M y + 2 * B y * Q y + A y * R y| ≤
c * K + 2 * b * L + a * J := by
have h2 : |2 * B y * Q y| ≤ 2 * b * L := by
rw [show 2 * B y * Q y = 2 * (B y * Q y) by ring, abs_mul]
rw [abs_of_pos (by norm_num : (0 : ℝ) < 2)]
simpa only [mul_assoc] using
mul_le_mul_of_nonneg_left (bprod hb bB bQ y) (by norm_num : (0 : ℝ) ≤ 2)
exact (abs_add_le _ _).trans (add_le_add
((abs_add_le _ _).trans (add_le_add (bprod hc bC bM y) h2)) (bprod ha bA bR y))
have dAM (y : ℝ) : HasDerivAt (fun x => A x * M x)
(B y * M y + A y * Q y) y := (dA y).mul (dM y)
have dAM' (y : ℝ) : HasDerivAt (fun x => B x * M x + A x * Q x)
(C y * M y + 2 * B y * Q y + A y * R y) y := by
convert ((dB y).mul (dM y)).add ((dA y).mul (dQ y)) using 1
ring
have hSA := hasDerivAt_magneticContinuationSecond ζ v hF hG hM hQ hA hB hC
hK hL ha hb bM bQ bA bB bC dF dM dA dB z
have hSB := hasDerivAt_magneticContinuationSecond ζ v hF hG hM hQ hB hC hD
hK hL hb hc bM bQ bB bC bD dF dM dB dC z
have hSM := hasDerivAt_magneticContinuationSecond ζ v hF hG hM hQ hM hQ hR
hK hL hK hL bM bQ bM bQ bR dF dM dM dQ z
have hSAM := hasDerivAt_magneticContinuationSecond ζ v hF hG hM hQ
(hA.mul hM) ((hB.mul hM).add (hA.mul hQ))
(((hC.mul hM).add ((measurable_const.mul hB).mul hQ)).add (hA.mul hR))
hK hL (mul_nonneg ha hK) (by positivity) bM bQ bAM bAM' bAM''
dF dM dAM dAM' z
change HasDerivAt (fieldTiltSpatial ζ v F M (fun y => A y * M y)
(fun y => B y * M y + A y * Q y))
(magneticContinuationSecond ζ v F M Q (fun y => A y * M y)
(fun y => B y * M y + A y * Q y)
(fun y => C y * M y + 2 * B y * Q y + A y * R y) z) z at hSAM
have hTA : HasDerivAt (fieldSpinTransition ζ v F A)
(fieldTiltSpatial ζ v F M A B z) z :=
hasDerivAt_fieldSpinTransition ζ v hF hG hM hA hB hK ha bM bA bB dF dA z
have hTM : HasDerivAt (fieldSpinTransition ζ v F M)
(fieldTiltSpatial ζ v F M M Q z) z :=
hasDerivAt_fieldSpinTransition ζ v hF hG hM hM hQ hK hK bM bM bQ dF dM z
have hh := hSB.add ((hSAM.sub ((hSA.mul hTM).add (hTA.mul hSM))).const_mul ζ)
convert hh using 1
· rfl
· dsimp only [magneticContinuationThird]
ring
end InvariantIsing
end
end OAI
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