| import OAI.Analysis.IrrationalRotation.Model.CircleCoupling |
| import Mathlib.Probability.Independence.Conditional |
|
|
| namespace OAI |
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|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal BigOperators ProbabilityTheory |
| namespace Invariant233.CircleCoupling |
| open CircleDigitCoupling CircleParameterData CouplingMarginalBounds |
|
|
| structure ConditionalLabelLaws (p : CircleParameterData) (j : ℕ) |
| (hr : ∀ i, 0 < MixedRadix.radices p.q i) : Prop where |
| probability : IsProbabilityMeasure (law p j) |
| endpoint_measurable : ∀ side, Measurable (point p hr side) |
| independent : letI := probability |
| ∀ side, iCondIndepFun (MeasurableSpace.comap (point p hr side) inferInstance) |
| (endpoint_measurable side).comap_le label (law p j) |
| finite_posterior : ∀ side (S : Finset ℕ) (C : ℕ → Set Label), |
| (∀ i, MeasurableSet (C i)) → |
| (law p j)⟦(⋂ i ∈ S, label i ⁻¹' C i) | |
| MeasurableSpace.comap (point p hr side) inferInstance⟧ =ᵐ[law p j] |
| (fun ω => ∏ i ∈ S, (posterior p j side i |
| (MixedRadix.circleCanonical (MixedRadix.radices p.q) hr (point p hr side ω) i)).real (C i)) |
| top_probability : ∀ s, 1 ≤ s → ∀ i ∈ p.block s, ∀ which, |
| ∀ᵐ ω ∂law p j, ∀ u : ℝ, |
| p.coordinate i (point p hr false ω) < u → u ≤ 1 - width (p.a s) (p.b s) which → |
| rho (kappa s) (p.m s) (width (p.a s) (p.b s) which) / 10 ≤ |
| ((law p j)⟦label i ⁻¹' {topLabel which} | |
| MeasurableSpace.comap (point p hr false) inferInstance⟧) ω |
| bottom_probability : ∀ s, 1 ≤ s → ∀ i ∈ p.block s, ∀ which, |
| ∀ᵐ ω ∂law p j, ∀ u : ℝ, |
| u ≤ p.coordinate i (point p hr true ω) → width (p.a s) (p.b s) which ≤ u → |
| rho (kappa s) (p.m s) (width (p.a s) (p.b s) which) / 10 ≤ |
| ((law p j)⟦label i ⁻¹' {bottomLabel which} | |
| MeasurableSpace.comap (point p hr true) inferInstance⟧) ω |
| top_forcing : ∀ s, 1 ≤ s → ∀ i ∈ p.block s, ∀ which, |
| ∀ᵐ ω ∂law p j, ∀ u : ℝ, label i ω = topLabel which → |
| u ≤ 1 - width (p.a s) (p.b s) which → u ≤ p.coordinate i (point p hr true ω) |
| bottom_forcing : ∀ s, 1 ≤ s → ∀ i ∈ p.block s, ∀ which, |
| ∀ᵐ ω ∂law p j, ∀ u : ℝ, label i ω = bottomLabel which → |
| width (p.a s) (p.b s) which ≤ u → p.coordinate i (point p hr false ω) < u |
|
|
| end Invariant233.CircleCoupling |
|
|
| end |
|
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| end OAI |
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|
