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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 50 51 52 53 54 | import OAI.Analysis.IrrationalRotation.Couplings.CircleCouplingForcing
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal ProbabilityTheory
namespace Invariant233.CircleCoupling
open CircleDigitCoupling CircleParameterData OneCoordinateCoupling CouplingMarginalBounds
variable {θ : ℝ} {p : CircleParameterData} (hp : CircleParameterLaws θ p)
include hp
lemma ae_top_conditional_probability (j : ℕ) (hr : ∀ i, 0 < MixedRadix.radices p.q i)
{s i : ℕ} (hs : 1 ≤ s) (hi : i ∈ p.block s) (which : Bool) :
∀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 := by
let := posterior_isMarkov hp j false i
have hρ0 : 0 ≤ rho (kappa s) (p.m s) (width (p.a s) (p.b s) which) :=
rho_nonneg (CircleParameterLaws.coupling_kappa_bounds s hs).1
filter_upwards [ae_top_posterior_lower hp j hr hs hi which,
condExp_single_label_point hp j hr false i (topLabel which)] with ω hlo heq
intro u hxu hu
rw [heq]
have h := ENNReal.toReal_mono (measure_ne_top _ _) (hlo u hxu hu)
simpa only [measureReal_def, ENNReal.toReal_div, ENNReal.toReal_ofReal hρ0,
ENNReal.toReal_ofNat] using h
lemma ae_bottom_conditional_probability (j : ℕ) (hr : ∀ i, 0 < MixedRadix.radices p.q i)
{s i : ℕ} (hs : 1 ≤ s) (hi : i ∈ p.block s) (which : Bool) :
∀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 := by
let := posterior_isMarkov hp j true i
have hρ0 : 0 ≤ rho (kappa s) (p.m s) (width (p.a s) (p.b s) which) :=
rho_nonneg (CircleParameterLaws.coupling_kappa_bounds s hs).1
filter_upwards [ae_bottom_posterior_lower hp j hr hs hi which,
condExp_single_label_point hp j hr true i (bottomLabel which)] with ω hlo heq
intro u huy hu
rw [heq]
have h := ENNReal.toReal_mono (measure_ne_top _ _) (hlo u huy hu)
simpa only [measureReal_def, ENNReal.toReal_div, ENNReal.toReal_ofReal hρ0,
ENNReal.toReal_ofNat] using h
end Invariant233.CircleCoupling
end
end OAI
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