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f61e547 | 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 | 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
open MeasureTheory
namespace Problem367
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
end
end OAI
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