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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 | 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
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