File size: 2,022 Bytes
f61e547 | 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 56 57 58 | import OAI.Probability.FreeIsing.Cavity.MomentTransport
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367
variable {Ω ι : Type*} {b : ℕ}
def ownChildFinFamily (e : Fin b ≃ ι) (Z : Option ι → Ω → ℝ) :
Fin (b + 1) → Ω → ℝ :=
fun i => Z (((finSuccEquiv b).trans e.optionCongr) i)
@[simp] theorem ownChildFinFamily_zero (e : Fin b ≃ ι)
(Z : Option ι → Ω → ℝ) : ownChildFinFamily e Z 0 = Z none := rfl
@[simp] theorem ownChildFinFamily_succ (e : Fin b ≃ ι)
(Z : Option ι → Ω → ℝ) (i : Fin b) :
ownChildFinFamily e Z i.succ = Z (some (e i)) := by
simp [ownChildFinFamily]
theorem ownChildFinFamily_eq_cases (e : Fin b ≃ ι)
(Z : Option ι → Ω → ℝ) :
ownChildFinFamily e Z = Fin.cases (Z none) (fun j => Z (some (e j))) := by
funext i
induction i using Fin.cases <;> simp
theorem iIndepFun_ownChildFinFamily [MeasurableSpace Ω] (e : Fin b ≃ ι)
{Z : Option ι → Ω → ℝ} {μ : Measure Ω} (hZ : iIndepFun Z μ) :
iIndepFun (ownChildFinFamily e Z) μ := by
exact iIndepFun.precomp (f := Z) (g := (finSuccEquiv b).trans e.optionCongr)
((finSuccEquiv b).trans e.optionCongr).injective hZ
theorem measurable_ownChildFinFamily [MeasurableSpace Ω] (e : Fin b ≃ ι)
{Z : Option ι → Ω → ℝ} (hm : ∀ i, Measurable (Z i)) (i : Fin (b + 1)) :
Measurable (ownChildFinFamily e Z i) :=
hm _
theorem sum_response_ownChildFinFamily [Fintype ι] (e : Fin b ≃ ι)
(Z : Option ι → Ω → ℝ) (psi : ℝ → ℝ) (ω : Ω) :
(∑ i : Fin (b + 1), Fin.cases (ownChildFinFamily e Z 0 ω)
(fun j => psi (ownChildFinFamily e Z j.succ ω)) i) =
Z none ω + ∑ a : ι, psi (Z (some a) ω) := by
rw [Fin.sum_univ_succ]
simp only [Fin.cases_zero, Fin.cases_succ, ownChildFinFamily_zero,
ownChildFinFamily_succ]
exact congrArg (fun x : ℝ => Z none ω + x)
(e.sum_comp (fun a => psi (Z (some a) ω)))
end Problem367
end
end OAI
|