| import Mathlib |
| import OAI.Probability.Perceptron.Variational.CovarianceMatrix |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped Classical ENNReal NNReal BigOperators Topology |
| namespace SphericalPerceptronFreeEnergy |
|
|
| lemma array_law_eq_of_blocks {K : Type*} [MeasurableSpace K] |
| (μ ν : Measure (CompactArray K)) [IsFiniteMeasure μ] [IsFiniteMeasure ν] |
| (h : ∀ n, μ.map (compactBlock (n+1)) = ν.map (compactBlock (n+1))) : μ = ν := by |
| let e : CompactArray K ≃ᵐ (ℕ × ℕ → K) := (MeasurableEquiv.curry ℕ ℕ K).symm |
| apply e.map_measurableEquiv_injective |
| let P (I : Finset (ℕ × ℕ)) := (ν.map e).map I.restrict |
| have hp : IsProjectiveLimit (ν.map e) P := fun _ => rfl |
| apply IsProjectiveLimit.unique (P := P) ?_ hp |
| intro I |
| let n := I.sup (fun a => max a.1 a.2) |
| have hb (a : I) : a.val.1 < n+1 ∧ a.val.2 < n+1 := by |
| have he : max a.val.1 a.val.2 ≤ n := Finset.le_sup (f := fun a : ℕ × ℕ => max a.1 a.2) a.property |
| exact ⟨by omega,by omega⟩ |
| let G : CompactBlock K (n+1) → (I → K) := |
| fun B a => B ⟨a.val.1,(hb a).1⟩ ⟨a.val.2,(hb a).2⟩ |
| have hG : Measurable G := by |
| apply Measurable.of_eval |
| intro a |
| exact (measurable_pi_apply _).comp (measurable_pi_apply _) |
| have hm : Measurable (compactBlock (K:=K) (n+1)) := by |
| unfold compactBlock |
| exact Measurable.of_eval fun _ => Measurable.of_eval fun _ => by fun_prop |
| change (μ.map e).map I.restrict = (ν.map e).map I.restrict |
| have he : I.restrict ∘ e = G ∘ compactBlock (n+1) := by |
| funext Q a |
| rfl |
| rw [Measure.map_map I.measurable_restrict e.measurable, |
| Measure.map_map I.measurable_restrict e.measurable,he, |
| ← Measure.map_map hG hm,← Measure.map_map hG hm,h n] |
|
|
| end SphericalPerceptronFreeEnergy |
|
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| end |
|
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| end OAI |
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