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c4cdbbd | 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 | import OAI.Probability.InvariantIsing.Fields.PriorContactContinuity
namespace OAI
/-! A prior-independent joint modulus for centered contact pressures. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators Topology NNReal
namespace InvariantIsing
def contactModulus {N m n : ℕ} (K : ℝ) (p q : TensorContactParameter N m n) : ℝ :=
(K/2)*|p.1-q.1|+
2*∑ i : Fin (n+1),
|(NNReal.sqrt (varianceIncrement (finiteFieldPath p.2.1) i) : ℝ)-
(NNReal.sqrt (varianceIncrement (finiteFieldPath q.2.1) i) : ℝ)| *
((NNReal.sqrt (varianceIncrement (finiteFieldPath p.2.1) i) : ℝ)+
(NNReal.sqrt (varianceIncrement (finiteFieldPath q.2.1) i) : ℝ))+
2*(N : ℝ)⁻¹*tensorAmplitudeModulus (tensorPerturbationAmplitude N p.2.2.1)
(tensorPerturbationAmplitude N q.2.2.1)+perturbationScale N*∑ a, |p.2.2.2 a-q.2.2.2 a|+
|finiteFieldPath p.2.1 n-finiteFieldPath q.2.1 n|/2
lemma contactModulus_self {N m n : ℕ} (K : ℝ) (p : TensorContactParameter N m n) :
contactModulus K p p=0 := by
simp [contactModulus,tensorAmplitudeModulus]
lemma continuous_contactModulus {N m n : ℕ} (K : ℝ) (q : TensorContactParameter N m n) :
Continuous (fun p : TensorContactParameter N m n => contactModulus K p q) := by
have hs (i : Fin (n+1)) : Continuous (fun p : TensorContactParameter N m n =>
varianceIncrement (finiteFieldPath p.2.1) i) :=
(continuous_finiteFieldVariance i).comp (by fun_prop)
have hf : Continuous (fun p : TensorContactParameter N m n => finiteFieldPath p.2.1 n) :=
(continuous_finiteFieldPath n).comp (by fun_prop)
unfold contactModulus tensorAmplitudeModulus tensorPerturbationAmplitude
fun_prop
lemma priorContactPressure_modulus
(hhaar : HaarConcentrationInput) (hgauss : GaussianLipschitzVarianceInput)
{N m n : ℕ} (hN : 3 ≤ N)
(μ : Measure (SpecialOrthogonal N)) [IsProbabilityMeasure μ] (hμ : μ.IsMulLeftInvariant)
(ν : Measure (Spin N × LabeledLeaf n)) [IsProbabilityMeasure ν]
(eig c : Fin N → ℝ) (I : Fin m → Finset (Fin N))
(K : ℝ) (hK : ∀ i, |eig i| ≤ K) (p q : TensorContactParameter N m n)
(hp : ∀ i, 0 ≤ p.2.1 i) (hq : ∀ i, 0 ≤ q.2.1 i) :
|priorContactPressure μ ν eig c I p-priorContactPressure μ ν eig c I q| ≤ contactModulus K p q := by
have hraw := priorContact_raw_modulus hhaar hgauss hN μ hμ ν eig c I K hK p q hp hq
let A := priorPerturbationPressureMean μ ν eig c I p.1 (finiteFieldPath p.2.1) p.2.2.1 p.2.2.2
let B := priorPerturbationPressureMean μ ν eig c I q.1 (finiteFieldPath q.2.1) q.2.2.1 q.2.2.2
have he : (A-finiteFieldPath p.2.1 n/2)-(B-finiteFieldPath q.2.1 n/2) =
(A-B)-(finiteFieldPath p.2.1 n-finiteFieldPath q.2.1 n)/2 := by ring
change |(A-finiteFieldPath p.2.1 n/2)-(B-finiteFieldPath q.2.1 n/2)| ≤ _
rw [he]
have ht := abs_sub (A-B) ((finiteFieldPath p.2.1 n-finiteFieldPath q.2.1 n)/2)
rw [abs_div,abs_of_pos (show 0 < (2 : ℝ) by norm_num)] at ht
exact ht.trans (add_le_add hraw le_rfl)
end InvariantIsing
end
end OAI
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