openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Sampling /CircleDigitMarginalLaws.lean
| import OAI.Analysis.IrrationalRotation.Rotation.CircleMarginalBudget | |
| import OAI.Analysis.IrrationalRotation.Rotation.CircleMarginalDensity | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory | |
| open scoped ENNReal BigOperators | |
| namespace Invariant233.CircleCoupling | |
| open CircleParameterData | |
| theorem digitMarginalLaws {θ : ℝ} {p : CircleParameterData} | |
| (hp : CircleParameterLaws θ p) | |
| (hr : ∀ i, 0 < MixedRadix.radices p.q i) : DigitMarginalLaws p hr := by | |
| refine ⟨Real.exp_pos _, fun j ε => ?_⟩ | |
| obtain ⟨hs, hC⟩ := hp.circle_marginal_budget j | |
| have hs' : Summable (fun i : ℕ => if i = 0 then 0 else 50 * | |
| (kappa (p.blockIndex i) ^ 2 / (p.m (p.blockIndex i) : ℝ) + | |
| tag j (p.blockIndex i) / (p.m (p.blockIndex i) : ℝ))) := by | |
| simpa only [tag] using hs | |
| have hC' : (∑' i : ℕ, if i = 0 then 0 else 50 * | |
| (kappa (p.blockIndex i) ^ 2 / (p.m (p.blockIndex i) : ℝ) + | |
| tag j (p.blockIndex i) / (p.m (p.blockIndex i) : ℝ))) ≤ | |
| 50 * ((∑' s : ℕ, kappa s ^ 2) + 1) := by | |
| simpa only [tag] using hC | |
| have h := hybrid_density_of_coordinate_budget hp j hr ε hs' hC' | |
| have he : 50 * ((∑' s : ℕ, kappa s ^ 2) + 1) / 2 = | |
| 25 * ((∑' s : ℕ, kappa s ^ 2) + 1) := by ring | |
| simpa only [marginalBound, he] using h | |
| theorem exists_digitMarginalLaws {θ : ℝ} {p : CircleParameterData} | |
| (hp : CircleParameterLaws θ p) : | |
| ∃ hr : ∀ i, 0 < MixedRadix.radices p.q i, DigitMarginalLaws p hr := | |
| ⟨MixedRadix.radices_pos hp.q_pos hp.frequency_increasing hp.frequency_divisible, | |
| digitMarginalLaws hp _⟩ | |
| theorem endpoint_density {θ : ℝ} {p : CircleParameterData} | |
| (hp : CircleParameterLaws θ p) (j : ℕ) | |
| (hr : ∀ i, 0 < MixedRadix.radices p.q i) (side : Bool) : | |
| ∃ H : Circle → ℝ, MemLp H 2 volume ∧ (0 ≤ᵐ[volume] H) ∧ | |
| (∫ x, H x) = 1 ∧ eLpNorm H 2 volume ≤ ENNReal.ofReal marginalBound ∧ | |
| (law p j).map (point p hr side) = | |
| volume.withDensity (fun x => ENNReal.ofReal (H x)) := by | |
| simpa only [hybridPoint_const] using (digitMarginalLaws hp hr).2 j (fun _ => side) | |
| end Invariant233.CircleCoupling | |
| end | |
| end OAI | |