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9eeb9c7 | 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 | import OAI.Analysis.IrrationalRotation.Model.CircleCoupling
import Mathlib.Probability.Independence.Conditional
namespace OAI
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
end OAI
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