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b317718 | 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 | import OAI.Probability.FreeIsing.Posterior.ComponentPosteriorStationarity
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367
theorem componentObservationField_eq_own_sup_children
{d : ℕ} {A B : Type*} [MeasurableSpace A] [MeasurableSpace B]
(F : IsingSpin → A → B) (c : Fin d) :
selectedObservationField F (componentIndexVertex (c := c)) =
selectedObservationField F (fun _ : Unit =>
(⟨[c], List.IsChain.singleton c⟩ : TreeVertex d)) ⊔
⨆ a : {a : Fin d // c ≠ a}, selectedObservationField F (childComponentIndexVertex c a.1) := by
simp only [selectedObservationField, MeasurableSpace.comap_process_pi, iSup_const]
apply le_antisymm
· apply iSup_le
rintro ⟨n, w, hlen, hchain⟩
cases w with
| nil => exact le_sup_left
| cons a w =>
have hca : c ≠ a := (List.isChain_cons_cons.mp hchain).1
let u : ComponentIndex a := ⟨w.length, ⟨w, rfl, hchain.tail⟩⟩
have hv : componentIndexVertex (c := c) ⟨n, ⟨a :: w, hlen, hchain⟩⟩ =
childComponentIndexVertex c a u := by
apply Subtype.ext
simp [childComponentIndexVertex_val hca, componentIndexVertex, componentVertex, u]
rw [hv]
exact le_sup_of_le_right (le_iSup_of_le (⟨a, hca⟩ : {a : Fin d // c ≠ a})
(le_iSup_of_le u le_rfl))
· apply sup_le
· exact le_iSup_of_le (⟨0, ⟨[], rfl, List.IsChain.singleton c⟩⟩ : ComponentIndex c) le_rfl
· apply iSup_le
intro a
apply iSup_le
intro u
let w : ComponentIndex c := ⟨u.1 + 1, ⟨a.1 :: u.2.1,
by simp [u.2.2.1], List.isChain_cons_cons.mpr ⟨a.2, u.2.2.2⟩⟩⟩
have hv : childComponentIndexVertex c a.1 u = componentIndexVertex w := by
apply Subtype.ext
simp [childComponentIndexVertex_val a.2, componentIndexVertex, componentVertex, w]
rw [hv]
exact le_iSup_of_le w le_rfl
end Problem367
end
end OAI
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