| import OAI.Probability.FreeIsing.Estimates.IndependentLabelSampler |
| import OAI.Probability.FreeIsing.Factors.ZeroFactor |
| import OAI.Probability.FreeIsing.Ising.ParameterRange |
| import OAI.Probability.FreeIsing.Spins.ProductLaw |
|
|
| namespace OAI |
|
|
| noncomputable section |
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|
| open MeasureTheory |
|
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| namespace Problem367 |
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|
| theorem IsFactorOfIID.mul_of_bounds {d : ℕ} {a b : ℝ} |
| (hfa : IsFactorOfIID d a) (hfb : IsFactorOfIID d b) |
| (ha0 : -1 ≤ a) (ha1 : a ≤ 1) (hb0 : -1 ≤ b) (hb1 : b ≤ 1) : |
| IsFactorOfIID d (a * b) := by |
| obtain ⟨F, hF, hFlaw, hFeq⟩ := hfa |
| obtain ⟨G, hG, hGlaw, hGeq⟩ := hfb |
| let : IsProbabilityMeasure (labelMeasure d) := by |
| unfold labelMeasure |
| infer_instance |
| let H := fun p : (TreeVertex d → ℝ) × (TreeVertex d → ℝ) => |
| spinProduct (F p.1, G p.2) |
| have hH : Measurable H := measurable_spinProduct.comp (hF.prodMap hG) |
| apply isFactorOfIID_of_independent_product_inputs uniform01 uniform01 H hH |
| · change HasFreeIsingLaw (a * b) (((labelMeasure d).prod (labelMeasure d)).map H) |
| have hmap : ((labelMeasure d).prod (labelMeasure d)).map H = |
| (((labelMeasure d).map F).prod ((labelMeasure d).map G)).map spinProduct := by |
| rw [Measure.map_prod_map _ _ hF hG, |
| Measure.map_map measurable_spinProduct (hF.prodMap hG)] |
| rfl |
| rw [hmap] |
| exact hFlaw.mul_product hGlaw ha0 ha1 hb0 hb1 |
| · intro g |
| change ∀ᵐ p ∂(labelMeasure d).prod (labelMeasure d), |
| H (relabel g p.1, relabel g p.2) = relabel g (H p) |
| have hfirst : ∀ᵐ p ∂(labelMeasure d).prod (labelMeasure d), |
| F (relabel g p.1) = relabel g (F p.1) := by |
| apply ae_of_ae_map (p := fun x => F (relabel g x) = relabel g (F x)) |
| measurable_fst.aemeasurable |
| simpa using hFeq g |
| have hsecond : ∀ᵐ p ∂(labelMeasure d).prod (labelMeasure d), |
| G (relabel g p.2) = relabel g (G p.2) := by |
| apply ae_of_ae_map (p := fun x => G (relabel g x) = relabel g (G x)) |
| measurable_snd.aemeasurable |
| simpa using hGeq g |
| filter_upwards [hfirst, hsecond] with p hp hq |
| change spinProduct (F (relabel g p.1), G (relabel g p.2)) = _ |
| rw [hp, hq, spinProduct_relabel] |
|
|
| theorem isFactorOfIID_degree_zero (theta : ℝ) : IsFactorOfIID 0 theta := by |
| obtain ⟨F, hF, hLaw, hEquiv⟩ := isFactorOfIID_zero 0 |
| refine ⟨F, hF, ?_, hEquiv⟩ |
| intro D hD eta |
| rw [hLaw D hD eta] |
| have hadj (u v : TreeVertex 0) : ¬treeAdj u v := by |
| rintro (⟨c, _⟩ | ⟨c, _⟩) <;> exact Fin.elim0 c |
| have hedge : inducedEdges D = ∅ := by |
| classical |
| simp [inducedEdges, hadj] |
| simp [freeCylinderWeight, hedge] |
|
|
| theorem IsFactorOfIID.mul {d : ℕ} {a b : ℝ} |
| (hfa : IsFactorOfIID d a) (hfb : IsFactorOfIID d b) : |
| IsFactorOfIID d (a * b) := by |
| by_cases hd : d = 0 |
| · subst d |
| exact isFactorOfIID_degree_zero _ |
| · have hdpos : 0 < d := Nat.pos_of_ne_zero hd |
| have ha := hfa.parameter_mem_Icc hdpos |
| have hb := hfb.parameter_mem_Icc hdpos |
| exact hfa.mul_of_bounds hfb ha.1 ha.2 hb.1 hb.2 |
|
|
| end Problem367 |
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| end |
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| end OAI |
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|
