File size: 3,517 Bytes
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 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 | import Mathlib
namespace OAI
/-! Restricting independent finite key streams preserves their exact
product law. Unused keys need not be allocated by a lazy implementation. -/
namespace EditApproximation
open MeasureTheory ProbabilityTheory
theorem finite_pi_restrict_map {ι κ : Type*} [Fintype ι] [Fintype κ]
{X : ι → Type*} [∀ i, MeasurableSpace (X i)]
(mu : ∀ i, Measure (X i)) [∀ i, IsProbabilityMeasure (mu i)]
(key : κ → ι) (hinjective : Function.Injective key) :
(Measure.pi mu).map (fun draw k => draw (key k)) =
Measure.pi (fun k => mu (key k)) := by
have hindependent : iIndepFun (fun i (draw : ∀ j, X j) => draw i) (Measure.pi mu) :=
iIndepFun_pi (fun _ => measurable_id.aemeasurable)
have hrestricted := hindependent.precomp hinjective
have hmap := hrestricted.map_fun_eq_pi_map
(fun k => (measurable_pi_apply (key k)).aemeasurable)
simpa only [(measurePreserving_eval mu _).map_eq] using hmap
theorem finite_pi_restrict_image_map {ι κ : Type*} [Fintype ι] [Fintype κ]
{X : ι → Type*} [∀ i, MeasurableSpace (X i)]
{Y : κ → Type*} [∀ k, MeasurableSpace (Y k)]
(mu : ∀ i, Measure (X i)) [∀ i, IsProbabilityMeasure (mu i)]
(key : κ → ι) (hinjective : Function.Injective key)
(image : ∀ k, X (key k) → Y k) (himage : ∀ k, Measurable (image k)) :
(Measure.pi mu).map (fun draw k => image k (draw (key k))) =
Measure.pi (fun k => (mu (key k)).map (image k)) := by
have hrestrict := finite_pi_restrict_map mu key hinjective
have hproject : Measurable (fun draw : ∀ i, X i => fun k => draw (key k)) :=
Measurable.of_eval (fun k => measurable_pi_apply (key k))
have htransform : Measurable (fun draw : ∀ k, X (key k) => fun k => image k (draw k)) :=
Measurable.of_eval (fun k => (himage k).comp (measurable_pi_apply k))
refine (Measure.map_map htransform hproject).symm.trans ?_
rw [hrestrict]
exact Measure.pi_map_pi (fun k => (himage k).aemeasurable)
theorem finite_uniform_array_cast (R n m : ℕ) [NeZero n] [NeZero m] (h : n = m) :
(Measure.pi (fun _ : Fin R => (PMF.uniformOfFintype (Fin n)).toMeasure)).map
(fun draw t => Fin.cast h (draw t)) =
Measure.pi (fun _ : Fin R => (PMF.uniformOfFintype (Fin m)).toMeasure) := by
subst m
simp only [Fin.cast_refl, id_eq, Measure.map_id']
/-- The accepted finite sampler outputs can be keyed by a larger bounded
universe. Restriction and the range equalities give the original independent
time arrays exactly. -/
theorem finite_uniform_arrays_restrict_cast {ι κ : Type*} [Fintype ι] [Fintype κ]
(R : ℕ) (n : ι → ℕ) (m : κ → ℕ)
[∀ i, NeZero (n i)] [∀ k, NeZero (m k)]
(key : κ → ι) (hinjective : Function.Injective key)
(hrange : ∀ k, n (key k) = m k) :
(Measure.pi (fun i => Measure.pi (fun _ : Fin R =>
(PMF.uniformOfFintype (Fin (n i))).toMeasure))).map
(fun draw k t => Fin.cast (hrange k) (draw (key k) t)) =
Measure.pi (fun k => Measure.pi (fun _ : Fin R =>
(PMF.uniformOfFintype (Fin (m k))).toMeasure)) := by
have hmap := finite_pi_restrict_image_map
(fun i => Measure.pi (fun _ : Fin R => (PMF.uniformOfFintype (Fin (n i))).toMeasure))
key hinjective
(fun k (draw : Fin R → Fin (n (key k))) t => Fin.cast (hrange k) (draw t))
(fun _ => measurable_of_finite _)
refine hmap.trans ?_
congr 1
funext k
exact finite_uniform_array_cast R (n (key k)) (m k) (hrange k)
end EditApproximation
end OAI
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