openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Sampling /CircleDigitSecondMoment.lean
| import OAI.Analysis.IrrationalRotation.Sampling.CircleDigitEnergy | |
| import OAI.Analysis.IrrationalRotation.Couplings.ProductDensity | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped ENNReal | |
| namespace Invariant233.CircleDigitCoupling | |
| open OneCoordinateCoupling | |
| theorem marginalDensity_second_moment_le {κ m a b t : ℝ} (hm : 100 ≤ m) | |
| (ha : 1 / m ≤ a) (ha' : a ≤ 1 / 4) | |
| (hb : 1 / m ≤ b) (hb' : b ≤ 1 / 4) | |
| (ht0 : 0 ≤ t) (ht : t ≤ 1) (side : Bool) : | |
| (∫ x, marginalDensity side κ m a b t x ^ 2 ∂baseLaw) ≤ | |
| 1 + 50 * (κ ^ 2 / m + t / m) := by | |
| rw [ProductDensity.integral_sq_eq_one_add_centered_sq | |
| (marginalDensity_memLp side κ m a b t) | |
| (marginalDensity_integral hm ha ha' hb hb' side)] | |
| exact add_le_add_right (marginalDensity_energy hm ha ha' hb hb' ht0 ht side) 1 | |
| theorem binAverage_marginalDensity_second_moment_le | |
| {κ m a b t : ℝ} (hm : 100 ≤ m) | |
| (ha : 1 / m ≤ a) (ha' : a ≤ 1 / 4) | |
| (hb : 1 / m ≤ b) (hb' : b ≤ 1 / 4) | |
| (ht0 : 0 ≤ t) (ht : t ≤ 1) (side : Bool) (r : ℕ) [NeZero r] : | |
| (∫ d, (UniformQuantization.binAverage r (marginalDensity side κ m a b t) d) ^ 2 | |
| ∂(PMF.uniformOfFintype (Fin r)).toMeasure) ≤ | |
| 1 + 50 * (κ ^ 2 / m + t / m) := by | |
| have hh : MemLp (marginalDensity side κ m a b t) 2 | |
| (volume.restrict (Ico (0 : ℝ) 1)) := by | |
| simpa only [baseLaw_eq_restrict] using marginalDensity_memLp side κ m a b t | |
| have hmean : (∫ d, | |
| UniformQuantization.binAverage r (marginalDensity side κ m a b t) d | |
| ∂(PMF.uniformOfFintype (Fin r)).toMeasure) = 1 := by | |
| rw [UniformQuantization.integral_binAverage r (hh.integrable (by norm_num))] | |
| simpa only [baseLaw_eq_restrict] using | |
| (marginalDensity_integral (κ := κ) (t := t) hm ha ha' hb hb' side) | |
| rw [ProductDensity.integral_sq_eq_one_add_centered_sq MemLp.of_discrete hmean] | |
| apply add_le_add_right | |
| apply (UniformQuantization.integral_binAverage_energy_le r hh).trans | |
| simpa only [baseLaw_eq_restrict] using | |
| (marginalDensity_energy (κ := κ) hm ha ha' hb hb' ht0 ht side) | |
| end Invariant233.CircleDigitCoupling | |
| end | |
| end OAI | |