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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 | import OAI.Probability.IsingPerceptron.RetainedDisplacement
import Mathlib.Probability.Kernel.Representation
namespace OAI
/-! A conditional product-law bridge for ancestor-dependent cascade marks. -/
noncomputable section
open MeasureTheory
open scoped ENNReal
namespace InvariantIsing
/-- Replacing a mark by a seed realization and replacing each descendant law
by the same conditional law preserves the full marked subtree distribution. -/
lemma ancestor_fiber_map_product {S A B C D : Type*}
[MeasurableSpace S] [MeasurableSpace A] [MeasurableSpace B]
[MeasurableSpace C] [MeasurableSpace D]
(P : Measure S) (μ : Measure A) (Q : Measure B) (R : Measure C)
[IsProbabilityMeasure P] [IsProbabilityMeasure μ]
[IsProbabilityMeasure Q] [IsProbabilityMeasure R]
{f : S → A} (hf : Measurable f) (hlaw : P.map f = μ)
{g : A × B → D} {h : A × C → D} (hg : Measurable g) (hh : Measurable h)
(hsub : ∀ a, Q.map (fun b => g (a,b)) = R.map (fun c => h (a,c))) :
(P.prod Q).map (fun p => (f p.1,g (f p.1,p.2))) =
(μ.prod R).map (fun p => (p.1,h p)) := by
have hleft : Measurable (fun p : S × B => (f p.1,g (f p.1,p.2))) :=
(hf.comp measurable_fst).prodMk (hg.comp ((hf.comp measurable_fst).prodMk measurable_snd))
have hright : Measurable (fun p : A × C => (p.1,h p)) := measurable_fst.prodMk hh
apply Measure.ext_of_lintegral
intro F hF
rw [lintegral_map hF hleft, lintegral_map hF hright]
rw [lintegral_prod (fun p : S × B => F (f p.1,g (f p.1,p.2))) (hF.comp hleft).aemeasurable,
lintegral_prod (fun p : A × C => F (p.1,h p)) (hF.comp hright).aemeasurable]
have hi a : MeasureTheory.lintegral Q (fun b => F (a,g (a,b))) =
MeasureTheory.lintegral R (fun c => F (a,h (a,c))) := by
have hFa : Measurable (fun d : D => F (a,d)) := hF.comp (measurable_const.prodMk measurable_id)
rw [← lintegral_map (g := fun b : B => g (a,b)) hFa
(hg.comp (measurable_const.prodMk measurable_id)), hsub a,
lintegral_map (g := fun c : C => h (a,c)) hFa
(hh.comp (measurable_const.prodMk measurable_id))]
simp_rw [hi]
have hJ : Measurable (fun a => MeasureTheory.lintegral R (fun c => F (a,h (a,c)))) :=
(hF.comp hright).lintegral_prod_right'
rw [← lintegral_map hJ hf, hlaw]
end InvariantIsing
end
end OAI
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