| import OAI.Analysis.FiniteFactor.Sampling.OrientedMoments |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory Filter |
| noncomputable section |
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| theorem sourceSamplingCutoff_le_half (s : ℕ) : sourceSamplingCutoff s ≤ 1/2 := by |
| have hm : (100 : ℝ) ≤ sourceBlockSize s := by exact_mod_cast (sourceStage s).m_ge_100 |
| have hp := sourceBlockSize_pos s |
| calc |
| sourceSamplingCutoff s ≤ (sourceBlockSize s : ℝ)^(-1 : ℝ) := |
| Real.rpow_le_rpow_of_exponent_le (by linarith) (by norm_num) |
| _ = 1/(sourceBlockSize s : ℝ) := by rw [Real.rpow_neg_one,one_div] |
| _ ≤ 1/2 := one_div_le_one_div_of_le (by norm_num) (by linarith) |
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| theorem sourceSampling_log_self_bound (s : ℕ) : |
| (2/sourceSamplingWidth s)*Real.log (1/(2*sourceSamplingCutoff s)) ≤ |
| 4*(sourceBlockSize s : ℝ)^(3/5 : ℝ)*Real.log (sourceBlockSize s : ℝ) := by |
| have hp := sourceSampling_parameters s |
| have hm := sourceBlockSize_pos s |
| have hl : Real.log (1/(2*sourceSamplingCutoff s)) ≤ Real.log (1/sourceSamplingCutoff s) := |
| Real.log_le_log (one_div_pos.mpr (mul_pos (by norm_num) hp.2.2)) (one_div_le_one_div_of_le hp.2.2 (by linarith [hp.2.2])) |
| have he : Real.log (1/sourceSamplingCutoff s) = 2*Real.log (sourceBlockSize s : ℝ) := by |
| unfold sourceSamplingCutoff |
| rw [one_div,Real.log_inv,Real.log_rpow hm] |
| ring |
| have hw : (sourceSamplingWidth s)⁻¹ = (sourceBlockSize s : ℝ)^(3/5 : ℝ) := by |
| unfold sourceSamplingWidth |
| rw [show (-3/5 : ℝ) = -(3/5) by ring,Real.rpow_neg hm.le,inv_inv] |
| calc |
| _ ≤ (2/sourceSamplingWidth s)*(2*Real.log (sourceBlockSize s : ℝ)) := |
| mul_le_mul_of_nonneg_left (hl.trans_eq he) (div_nonneg (by norm_num) hp.1.le) |
| _ = _ := by rw [div_eq_mul_inv,hw]; ring |
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| / |
| orientations. The harmless extra 1 includes triangular coordinates. -/ |
| theorem sourceSampling_oriented_self_moment (stage : ℕ) (r b : Bool) (E G : Set ℝ) |
| (hE : MeasurableSet E) (hG : MeasurableSet G) : |
| (∫ s, ∫ t, samplingOrientedWeight r b (sourceSamplingWidth stage) (sourceSamplingCutoff stage) E G s t ^ 2 |
| ∂unitIntervalMeasure ∂unitIntervalMeasure) ≤ |
| 1+4*(sourceBlockSize stage : ℝ)^(3/5 : ℝ)*Real.log (sourceBlockSize stage : ℝ) := by |
| have hp := sourceSampling_parameters stage |
| have hc := sourceSamplingCutoff_le_half stage |
| have hlog : 0 ≤ Real.log (sourceBlockSize stage : ℝ) := Real.log_nonneg (by |
| have hm : (100 : ℝ) ≤ sourceBlockSize stage := by exact_mod_cast (sourceStage stage).m_ge_100 |
| linarith) |
| have herr : (0 : ℝ) ≤ 4*(sourceBlockSize stage : ℝ)^(3/5 : ℝ)*Real.log (sourceBlockSize stage : ℝ) := by positivity |
| cases b |
| · have hb : (∫ s, ∫ t, samplingOrientedWeight r false (sourceSamplingWidth stage) (sourceSamplingCutoff stage) E G s t ^ 2 |
| ∂unitIntervalMeasure ∂unitIntervalMeasure) ≤ |
| (2/sourceSamplingWidth stage)*Real.log (1/(2*sourceSamplingCutoff stage)) := by |
| cases r |
| · exact samplingTestedFirstWeight_square_integral _ _ E G hp.1 hp.2.1 hp.2.2 hc hE hG |
| · exact samplingTestedSecondWeight_square_integral _ _ E G hp.1 hp.2.1 hp.2.2 hc hE hG |
| exact (hb.trans (sourceSampling_log_self_bound stage)).trans (by linarith) |
| · have hm := samplingOrientedWeight_measurable r true (sourceSamplingWidth stage) |
| (sourceSamplingCutoff stage) E G hE hG |
| have hb := samplingOrientedWeight_triangular_bounds r (sourceSamplingWidth stage) |
| (sourceSamplingCutoff stage) E G |
| have hi := sampling_square_integrable (unitIntervalMeasure.prod unitIntervalMeasure) _ 1 hm |
| (fun p => hb p.1 p.2) |
| rw [← integral_prod _ hi] |
| have hle : (∫ p : ℝ × ℝ, samplingOrientedWeight r true (sourceSamplingWidth stage) |
| (sourceSamplingCutoff stage) E G p.1 p.2 ^ 2 ∂unitIntervalMeasure.prod unitIntervalMeasure) ≤ 1 := by |
| simpa using integral_mono_ae hi (integrable_const (1 : ℝ)) |
| (Eventually.of_forall (fun p => by nlinarith [(hb p.1 p.2).1,(hb p.1 p.2).2])) |
| exact hle.trans (by linarith) |
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| end |
| end FiniteFactor |
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| end OAI |
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