| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory | |
| namespace Problem367 | |
| abbrev TreeVertex (d : ℕ) := | |
| {word : List (Fin d) // List.IsChain (fun first second : Fin d => first ≠ second) word} | |
| def treeAdj {d : ℕ} (u v : TreeVertex d) : Prop := | |
| (∃ c : Fin d, v.1 = u.1 ++ [c]) ∨ | |
| (∃ c : Fin d, u.1 = v.1 ++ [c]) | |
| structure TreeAut (d : ℕ) where | |
| toEquiv : TreeVertex d ≃ TreeVertex d | |
| map_adj : ∀ u v, treeAdj (toEquiv u) (toEquiv v) ↔ treeAdj u v | |
| def relabel {d : ℕ} {α : Type*} (g : TreeAut d) | |
| (x : TreeVertex d → α) : TreeVertex d → α := | |
| fun v => x (g.toEquiv.symm v) | |
| inductive IsingSpin where | |
| | neg | |
| | pos | |
| deriving DecidableEq | |
| protected abbrev IsingSpin.enumList : List IsingSpin := [.neg, .pos] | |
| protected theorem IsingSpin.enumList_getElem?_ctorIdx_eq (x : IsingSpin) : | |
| IsingSpin.enumList[x.ctorIdx]? = some x := by | |
| cases x <;> rfl | |
| protected theorem IsingSpin.enumList_nodup : IsingSpin.enumList.Nodup := by decide | |
| instance : Fintype IsingSpin where | |
| elems := ⟨IsingSpin.enumList, IsingSpin.enumList_nodup⟩ | |
| complete x := by cases x <;> decide | |
| instance isingSpinMeasurable : MeasurableSpace IsingSpin := ⊤ | |
| def uniform01 : Measure ℝ := | |
| volume.restrict (Set.Icc (0 : ℝ) 1) | |
| instance uniform01Probability : IsProbabilityMeasure uniform01 where | |
| measure_univ := by simp [uniform01] | |
| def labelMeasure (d : ℕ) : Measure (TreeVertex d → ℝ) := by | |
| classical | |
| exact Measure.infinitePi (fun _ : TreeVertex d => uniform01) | |
| def ConnectedFinset {d : ℕ} (D : Finset (TreeVertex d)) : Prop := | |
| D.Nonempty ∧ | |
| ∀ u ∈ D, ∀ v ∈ D, | |
| Relation.ReflTransGen | |
| (fun x y : TreeVertex d => x ∈ D ∧ y ∈ D ∧ treeAdj x y) u v | |
| def inducedEdges {d : ℕ} (D : Finset (TreeVertex d)) : | |
| Finset (TreeVertex d × TreeVertex d) := by | |
| classical | |
| exact (D.product D).filter (fun e => | |
| treeAdj e.1 e.2 ∧ e.1.1.length < e.2.1.length) | |
| def freeCylinderWeight {d : ℕ} (theta : ℝ) | |
| (D : Finset (TreeVertex d)) (eta : TreeVertex d → IsingSpin) : ENNReal := | |
| ENNReal.ofReal | |
| ((1 / 2 : ℝ) * | |
| Finset.prod (inducedEdges D) (fun e => | |
| (1 + (if eta e.1 = eta e.2 then theta else -theta)) / 2)) | |
| def HasFreeIsingLaw {d : ℕ} (theta : ℝ) | |
| (mu : Measure (TreeVertex d → IsingSpin)) : Prop := | |
| ∀ (D : Finset (TreeVertex d)), ConnectedFinset D → | |
| ∀ eta : TreeVertex d → IsingSpin, | |
| mu {x | ∀ v ∈ D, x v = eta v} = freeCylinderWeight theta D eta | |
| def IsFactorOfIID (d : ℕ) (theta : ℝ) : Prop := | |
| ∃ Phi : (TreeVertex d → ℝ) → (TreeVertex d → IsingSpin), | |
| Measurable Phi ∧ | |
| HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d)) ∧ | |
| ∀ g : TreeAut d, | |
| ∀ᵐ U ∂labelMeasure d, Phi (relabel g U) = relabel g (Phi U) | |
| end Problem367 | |
| end | |
| end OAI | |