openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Sampling /MixedRadixConditioning.lean
| import OAI.Analysis.IrrationalRotation.Couplings.ConditionalObservationTransport | |
| import OAI.Analysis.IrrationalRotation.Sampling.MixedRadixHaarLaw | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped MeasureTheory | |
| namespace Invariant233.MixedRadix | |
| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} | |
| {r : ℕ → ℕ} | |
| theorem ae_circleCanonical_circleValue_of_map_ac | |
| (hr : ∀ n, 2 ≤ r n) (hp : ∀ n, 0 < r n) | |
| {D : Ω → Digits r} (hD : Measurable D) | |
| (hAC : μ.map D ≪ digitLaw r hp) : | |
| ∀ᵐ ω ∂μ, circleCanonical r hp (circleValue (D ω)) = D ω := by | |
| exact ae_of_ae_map hD.aemeasurable | |
| (hAC.ae_le (ae_circleCanonical_circleValue hr hp)) | |
| variable [IsFiniteMeasure μ] | |
| theorem condExp_circleValue_eq_digits | |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] | |
| (hr : ∀ n, 2 ≤ r n) (hp : ∀ n, 0 < r n) | |
| {D : Ω → Digits r} (hD : Measurable D) | |
| (hAC : μ.map D ≪ digitLaw r hp) (f : Ω → E) : | |
| μ[f | MeasurableSpace.comap (circleValue ∘ D) inferInstance] =ᵐ[μ] | |
| μ[f | MeasurableSpace.comap D inferInstance] := | |
| ConditionalObservationTransport.condExp_encoding hD (measurable_circleValue hr) | |
| (measurable_circleCanonical r hp) | |
| (ae_circleCanonical_circleValue_of_map_ac hr hp hD hAC) f | |
| theorem condExp_circleValue_coordinate_formula | |
| (hr : ∀ n, 2 ≤ r n) (hp : ∀ n, 0 < r n) | |
| {D : Ω → Digits r} (hD : Measurable D) | |
| (hAC : μ.map D ≪ digitLaw r hp) (i : ℕ) | |
| (f : Ω → ℝ) (posterior : Fin (r i) → ℝ) | |
| (hposterior : μ[f | MeasurableSpace.comap D inferInstance] =ᵐ[μ] | |
| fun ω => posterior (D ω i)) : | |
| μ[f | MeasurableSpace.comap (circleValue ∘ D) inferInstance] =ᵐ[μ] | |
| fun ω => posterior (circleCanonical r hp (circleValue (D ω)) i) := | |
| ConditionalObservationTransport.condExp_encoding_formula hD (measurable_circleValue hr) | |
| (measurable_circleCanonical r hp) | |
| (ae_circleCanonical_circleValue_of_map_ac hr hp hD hAC) f | |
| (fun d => posterior (d i)) hposterior | |
| theorem iCondIndepFun_circleValue_iff_digits | |
| [StandardBorelSpace Ω] {ι : Type*} {β : ι → Type*} | |
| [∀ i, MeasurableSpace (β i)] | |
| (hr : ∀ n, 2 ≤ r n) (hp : ∀ n, 0 < r n) | |
| {D : Ω → Digits r} (hD : Measurable D) | |
| (hAC : μ.map D ≪ digitLaw r hp) | |
| (L : ∀ i, Ω → β i) (hL : ∀ i, Measurable (L i)) : | |
| iCondIndepFun (MeasurableSpace.comap (circleValue ∘ D) inferInstance) | |
| ((measurable_circleValue hr).comp hD).comap_le L μ ↔ | |
| iCondIndepFun (MeasurableSpace.comap D inferInstance) hD.comap_le L μ := | |
| ConditionalObservationTransport.iCondIndepFun_encoding_iff hD (measurable_circleValue hr) | |
| (measurable_circleCanonical r hp) | |
| (ae_circleCanonical_circleValue_of_map_ac hr hp hD hAC) L hL | |
| end Invariant233.MixedRadix | |
| end | |
| end OAI | |