| import OAI.Probability.InvariantIsing.Fields.NoiseReplicaPairData |
| import OAI.Probability.IsingPerceptron.SamplingIndependent |
|
|
| namespace OAI |
|
|
| /-! The measurable labeled two-path observable in the original forest coordinates. -/ |
| noncomputable section |
| open MeasureTheory ProbabilityTheory IsingPerceptron |
| namespace InvariantIsing |
|
|
| def coordinatePairData {A : Type} (n : ℕ) |
| (p : (ForestVertex n → A) × (ℕ → LabeledLeaf n)) : ℕ × (Fin n → A × A) := |
| (labeledCommonDepth n (p.2 0) (p.2 1), |
| fun i => (p.1 (edgeAt n (p.2 0) i),p.1 (edgeAt n (p.2 1) i))) |
|
|
| lemma measurable_coordinatePairData {A : Type} [MeasurableSpace A] (n : ℕ) : |
| Measurable (coordinatePairData (A := A) n) := by |
| have hE : Measurable (fun p : (ForestVertex n → A) × LabeledLeaf n => |
| fun i => p.1 (edgeAt n p.2 i)) := by |
| apply measurable_from_prod_countable_left |
| intro α |
| exact Measurable.of_eval fun i => measurable_pi_apply (edgeAt n α i) |
| have hp (j : ℕ) : Measurable (fun p : (ForestVertex n → A) × (ℕ → LabeledLeaf n) => |
| (p.1,p.2 j)) := measurable_fst.prodMk ((measurable_pi_apply j).comp measurable_snd) |
| have hd : Measurable (fun p : (ForestVertex n → A) × (ℕ → LabeledLeaf n) => |
| (p.2 0,p.2 1)) := ((measurable_pi_apply 0).comp measurable_snd).prodMk |
| ((measurable_pi_apply 1).comp measurable_snd) |
| exact ((measurable_of_countable (fun q : LabeledLeaf n × LabeledLeaf n => |
| labeledCommonDepth n q.1 q.2)).comp hd).prodMk (Measurable.of_eval fun i => |
| ((measurable_pi_apply i).comp (hE.comp (hp 0))).prodMk |
| ((measurable_pi_apply i).comp (hE.comp (hp 1)))) |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
|
