| import OAI.Probability.FreeIsing.Model | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory | |
| namespace Problem367 | |
| theorem connectedFinset_singleton {d : ℕ} (v : TreeVertex d) : | |
| ConnectedFinset {v} := by | |
| classical | |
| refine ⟨Finset.singleton_nonempty v, ?_⟩ | |
| intro x hx y hy | |
| simp only [Finset.mem_singleton] at hx hy | |
| subst x | |
| subst y | |
| exact Relation.ReflTransGen.refl | |
| @[simp] theorem inducedEdges_singleton {d : ℕ} (v : TreeVertex d) : | |
| inducedEdges {v} = ∅ := by | |
| classical | |
| ext e | |
| simp [inducedEdges] | |
| rintro rfl | |
| simp | |
| @[simp] theorem freeCylinderWeight_singleton {d : ℕ} (θ : ℝ) | |
| (v : TreeVertex d) (η : TreeVertex d → IsingSpin) : | |
| freeCylinderWeight θ {v} η = ENNReal.ofReal (1 / 2 : ℝ) := by | |
| classical | |
| simp [freeCylinderWeight] | |
| theorem HasFreeIsingLaw.marginal {d : ℕ} {θ : ℝ} | |
| {μ : Measure (TreeVertex d → IsingSpin)} (hμ : HasFreeIsingLaw θ μ) | |
| (v : TreeVertex d) (s : IsingSpin) : | |
| μ {x | x v = s} = ENNReal.ofReal (1 / 2 : ℝ) := by | |
| classical | |
| simpa using hμ {v} (connectedFinset_singleton v) (fun _ => s) | |
| end Problem367 | |
| end | |
| end OAI | |