File size: 4,120 Bytes
67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 | import OAI.Analysis.FiniteFactor.Sampling.OrientedMoments
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter
noncomputable section
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)
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
/-- All tested source weights have the logarithmic self-pair bound in both
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)
end
end FiniteFactor
end OAI
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