| import OAI.Probability.InvariantIsing.Fields.CascadeSeedRegular |
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| namespace OAI |
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| /-! The weight-preserving leaf map for ancestor-dependent vector marks. -/ |
| noncomputable section |
| open MeasureTheory ProbabilityTheory IsingPerceptron |
| namespace InvariantIsing |
| variable {ι : Type} |
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| def cascadeSeedLeaf : (n : ℕ) → |
| (ℕ → (ι → ℝ) → unitInterval → (ι → ℝ)) → (ι → ℝ) → |
| NoiseLeaf unitInterval n → NoiseLeaf (ι → ℝ) n |
| | 0, _, _, _ => PUnit.unit |
| | n+1, ψ, z, v => |
| (v.1,(ψ 0 z v.2.1, |
| cascadeSeedLeaf n (fun j => ψ (j+1)) (z+ψ 0 z v.2.1) v.2.2)) |
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| lemma measurable_cascadeSeedLeaf (n : ℕ) |
| (ψ : ℕ → (ι → ℝ) → unitInterval → (ι → ℝ)) |
| (hψ : ∀ i, Measurable (Function.uncurry (ψ i))) : |
| Measurable (fun p : (ι → ℝ) × NoiseLeaf unitInterval n => cascadeSeedLeaf n ψ p.1 p.2) := by |
| induction n generalizing ψ with |
| | zero => exact measurable_const |
| | succ n ih => |
| have hm : Measurable (fun p : (ι → ℝ) × NoiseLeaf unitInterval (n+1) => ψ 0 p.1 p.2.2.1) := |
| (hψ 0).comp (measurable_fst.prodMk |
| (measurable_fst.comp (measurable_snd.comp measurable_snd))) |
| exact (measurable_fst.comp measurable_snd).prodMk (hm.prodMk |
| ((ih _ (fun i => hψ (i+1))).comp |
| ((measurable_fst.add hm).prodMk (measurable_snd.comp (measurable_snd.comp measurable_snd))))) |
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| lemma cascadeSeedLeaf_forget (n : ℕ) |
| (ψ : ℕ → (ι → ℝ) → unitInterval → (ι → ℝ)) (z : ι → ℝ) (v : NoiseLeaf unitInterval n) : |
| noiseLeafForget (ι → ℝ) n (cascadeSeedLeaf n ψ z v) = noiseLeafForget unitInterval n v := by |
| induction n generalizing ψ z with |
| | zero => rfl |
| | succ n ih => |
| change (v.1,noiseLeafForget (ι → ℝ) n |
| (cascadeSeedLeaf n (fun j => ψ (j+1)) (z+ψ 0 z v.2.1) v.2.2)) = _ |
| rw [ih] |
| rfl |
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| end InvariantIsing |
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| end |
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| end OAI |
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