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e63eb7f | 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 | import OAI.Combinatorics.EditApproximation.Geometry.CommonArrayFinal
namespace OAI
namespace EditApproximation
open MeasureTheory ProbabilityTheory
section Law
variable {Ω : Type} [Fintype Ω] [MeasurableSpace Ω] [MeasurableSingletonClass Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
theorem finite_init_last_law (n : ℕ) :
(Measure.pi fun _ : Fin (n + 1) => μ).map
(fun draw => ((fun i : Fin n => draw i.castSucc), draw (Fin.last n))) =
(Measure.pi fun _ : Fin n => μ).prod μ := by
classical
apply Measure.ext_of_singleton
rintro ⟨front, last⟩
rw [Measure.map_apply (measurable_of_finite _) (measurableSet_singleton _)]
have hpre : (fun draw : Fin (n + 1) → Ω =>
((fun i : Fin n => draw i.castSucc), draw (Fin.last n))) ⁻¹' {(front, last)} =
{Fin.snoc front last} := by
ext draw
simp only [Set.mem_preimage, Set.mem_singleton_iff, Prod.mk.injEq]
constructor
· rintro ⟨hp, hl⟩
funext i
refine Fin.lastCases ?_ (fun j => ?_) i
· simpa only [Fin.snoc_last] using hl
· simpa only [Fin.snoc_castSucc] using congrFun hp j
· rintro rfl
constructor
· funext i; simp only [Fin.snoc_castSucc]
· simp only [Fin.snoc_last]
rw [hpre, Measure.pi_singleton, Fin.prod_univ_castSucc]
simp only [Fin.snoc_castSucc, Fin.snoc_last]
rw [← Set.singleton_prod_singleton, Measure.prod_prod, Measure.pi_singleton]
theorem finite_init_law (n : ℕ) :
(Measure.pi fun _ : Fin (n + 1) => μ).map
(fun draw (i : Fin n) => draw i.castSucc) = Measure.pi fun _ : Fin n => μ := by
have h := congrArg (fun law : Measure ((Fin n → Ω) × Ω) => law.map Prod.fst)
(finite_init_last_law μ n)
rw [Measure.map_map (measurable_of_finite _) (measurable_of_finite _), Measure.map_fst_prod] at h
simpa only [Function.comp_def, measure_univ, one_smul] using h
end Law
end EditApproximation
end OAI
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