| import OAI.Probability.InvariantIsing.Fields.PriorContactContinuity |
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| namespace OAI |
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| /-! A prior-independent joint modulus for centered contact pressures. -/ |
| noncomputable section |
| open MeasureTheory ProbabilityTheory IsingPerceptron |
| open scoped BigOperators Topology NNReal |
| namespace InvariantIsing |
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| 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 |
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| lemma contactModulus_self {N m n : ℕ} (K : ℝ) (p : TensorContactParameter N m n) : |
| contactModulus K p p=0 := by |
| simp [contactModulus,tensorAmplitudeModulus] |
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| 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 |
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| 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) |
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| end InvariantIsing |
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| end |
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| end OAI |
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