openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Couplings /CircleConditionalProbabilities.lean
| 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 | |