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c4cdbbd | 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 48 49 50 51 52 53 54 55 | import OAI.Probability.InvariantIsing.Fields.CascadeSeedLeaf
import OAI.Probability.InvariantIsing.Fields.MarkForestRootLaw
namespace OAI
/-! The ancestor-state increment path generated by an independent seed forest. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
namespace InvariantIsing
variable {ι : Type}
def cascadeSeedPath : (n : ℕ) → (ℕ → (ι → ℝ) → unitInterval → (ι → ℝ)) →
(ι → ℝ) → MarkForest unitInterval n → LabeledLeaf n → Fin n → (ι → ℝ)
| 0, _, _, _, _ => Fin.elim0
| n+1, ψ, z, g, α =>
Fin.cons (ψ 0 z (g α.1.1 α.1.2).1)
(cascadeSeedPath n (fun j => ψ (j+1)) (z+ψ 0 z (g α.1.1 α.1.2).1)
(g α.1.1 α.1.2).2 α.2)
lemma measurable_cascadeSeedPath (n : ℕ)
(ψ : ℕ → (ι → ℝ) → unitInterval → (ι → ℝ))
(hψ : ∀ i, Measurable (Function.uncurry (ψ i))) (α : LabeledLeaf n) :
Measurable (fun p : (ι → ℝ) × MarkForest unitInterval n => cascadeSeedPath n ψ p.1 p.2 α) := by
induction n generalizing ψ with
| zero => exact Measurable.of_eval fun i => i.elim0
| succ n ih =>
have hm : Measurable (fun p : (ι → ℝ) × MarkForest unitInterval (n+1) =>
ψ 0 p.1 (p.2 α.1.1 α.1.2).1) :=
(hψ 0).comp (measurable_fst.prodMk (by fun_prop))
apply Measurable.of_eval
intro i
refine Fin.cases hm ?_ i
intro j
exact ((measurable_pi_apply j).comp (ih _ (fun i => hψ (i+1)) α.2)).comp
((measurable_fst.add hm).prodMk (by fun_prop))
lemma cascadeSeedPath_labeled (n : ℕ)
(ψ : ℕ → (ι → ℝ) → unitInterval → (ι → ℝ)) (z : ι → ℝ)
(T : LabeledTree n) (g : MarkForest unitInterval n) (α : LabeledLeaf n) :
noiseLeafMark n (cascadeSeedLeaf n ψ z (labeledNoiseLeaf unitInterval n (T,g) α)) =
cascadeSeedPath n ψ z g α := by
induction n generalizing ψ z with
| zero => exact funext fun i => i.elim0
| succ n ih =>
funext i
refine Fin.cases rfl (fun j => ?_) i
exact congrFun (ih (fun k => ψ (k+1)) (z+ψ 0 z (g α.1.1 α.1.2).1)
(T.2 α.1.1 α.1.2) (g α.1.1 α.1.2).2 α.2) j
end InvariantIsing
end
end OAI
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