openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Couplings /CircleConditionalLabels.lean
| import OAI.Analysis.IrrationalRotation.Model.CircleConditionalLabels | |
| import OAI.Analysis.IrrationalRotation.Couplings.CircleConditionalProbabilities | |
| namespace OAI | |
| noncomputable section | |
| namespace Invariant233.CircleCoupling | |
| theorem conditionalLabelLaws {θ : ℝ} {p : CircleParameterData} | |
| (hp : CircleParameterLaws θ p) (j : ℕ) | |
| (hr : ∀ i, 0 < MixedRadix.radices p.q i) : ConditionalLabelLaws p j hr where | |
| probability := law_isProbabilityMeasure hp j | |
| endpoint_measurable := measurable_point hp hr | |
| independent := iCondIndepFun_labels_point hp j hr | |
| finite_posterior := condExp_labels_point hp j hr | |
| top_probability := fun _ hs _ hi which => ae_top_conditional_probability hp j hr hs hi which | |
| bottom_probability := fun _ hs _ hi which => ae_bottom_conditional_probability hp j hr hs hi which | |
| top_forcing := fun _ hs _ hi which => ae_top_forces hp j hr hs hi which | |
| bottom_forcing := fun _ hs _ hi which => ae_bottom_forces hp j hr hs hi which | |
| theorem exists_conditionalLabelLaws {θ : ℝ} {p : CircleParameterData} | |
| (hp : CircleParameterLaws θ p) (j : ℕ) : | |
| ∃ hr : ∀ i, 0 < MixedRadix.radices p.q i, ConditionalLabelLaws p j hr := | |
| ⟨MixedRadix.radices_pos hp.q_pos hp.frequency_increasing hp.frequency_divisible, | |
| conditionalLabelLaws hp j _⟩ | |
| end Invariant233.CircleCoupling | |
| end | |
| end OAI | |