| import OAI.Analysis.FiniteFactor.OrbitSampling.Cube |
| import Mathlib.Analysis.SpecialFunctions.Integrals.Basic |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter Finset |
| open scoped Classical ENNReal Topology |
| noncomputable section |
|
|
| theorem sourceSampling_triangle_mass : |
| ((unitIntervalMeasure.prod unitIntervalMeasure) |
| {p : ℝ × ℝ | p.1 ∈ Set.Icc (0 : ℝ) 1 ∧ p.2 ∈ Set.Icc (0 : ℝ) 1 ∧ p.1 < p.2}).toReal = 1/2 := by |
| rw [← samplingTestedFirstDensity_triangular_mass (1/2) 1 _ _ measurableSet_Icc measurableSet_Icc] |
| have he : samplingTestedFirstDensity true (1/2) 1 (Set.Icc 0 1) (Set.Icc 0 1) =ᵐ[unitIntervalMeasure] |
| (fun s => 1-s) := by |
| filter_upwards [ae_restrict_mem measurableSet_Ico] with s hs |
| simpa only [ite_true] using samplingTestedFirstDensity_whole true (1/2) 1 s |
| (by norm_num) (by norm_num) (by norm_num) hs |
| rw [integral_congr_ae he,unitIntervalMeasure,restrict_Ico_eq_restrict_Ioc, |
| ← intervalIntegral.integral_of_le (show (0 : ℝ) ≤ 1 by norm_num)] |
| rw [intervalIntegral.integral_sub (f := fun _ : ℝ => (1 : ℝ)) (g := fun x : ℝ => x) |
| intervalIntegrable_const (continuous_id.intervalIntegrable _ _)] |
| norm_num [intervalIntegral.integral_const] |
|
|
| theorem sourceSamplingLambda_mass (I : Finset SourceCoordinate) : |
| ((sourceSamplingLambda I) Set.univ).toReal = (1/2 : ℝ)^I.card := by |
| have he : sourceSamplingLambda I sourceSamplingCube = sourceSamplingLambda I Set.univ := |
| measure_congr (by filter_upwards [sourceSamplingLambda_cube I] with p hp; exact propext (iff_true_intro hp)) |
| rw [← he,← sourceSamplingEmptyTest_rectangle,sourceSamplingLambda_rectangle, |
| sourceSamplingLimitRectangleMass_formula] |
| have hE (i : SourceCoordinate) : sourceSamplingRationalX sourceSamplingEmptyTest i = Set.Icc (0 : ℝ) 1 ∧ |
| sourceSamplingRationalY sourceSamplingEmptyTest i = Set.Icc (0 : ℝ) 1 := by |
| simp [sourceSamplingRationalX,sourceSamplingRationalY,sourceSamplingRationalEndpoints,sourceSamplingEmptyTest] |
| change (∏ i ∈ I ∪ ∅, _) = _ |
| rw [Finset.union_empty] |
| calc |
| _ = ∏ _i ∈ I, (1/2 : ℝ) := by |
| apply Finset.prod_congr rfl |
| intro i hi |
| rw [ite_eq_left hi,(hE i).1,(hE i).2] |
| exact sourceSampling_triangle_mass |
| _ = _ := by simp |
|
|
| theorem sourceSamplingLambda_mass_pos (I : Finset SourceCoordinate) : |
| 0 < ((sourceSamplingLambda I) Set.univ).toReal := by |
| rw [sourceSamplingLambda_mass] |
| positivity |
|
|
| end |
| end FiniteFactor |
|
|
| end OAI |
|
|
