| import OAI.Analysis.FiniteFactor.OrbitSampling.ProductMoments |
| import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter Finset |
| noncomputable section |
|
|
| def sourceSamplingSelfBudget (stage : ℕ) : ℝ := |
| (2+8*(sourceBlockSize stage : ℝ)^(3/5 : ℝ)*Real.log (sourceBlockSize stage : ℝ))^(sourceSamplingPrefix stage).card / |
| ((sourceSamplingPowers stage).card : ℝ) |
|
|
| / |
| theorem sampling_self_base_bound (m : ℝ) (hm1 : 1 ≤ m) (hlarge : 802 ≤ m^(1/50 : ℝ)) : |
| 2+8*m^(3/5 : ℝ)*Real.log m ≤ m^(63/100 : ℝ) := by |
| have hm : 0 < m := lt_of_lt_of_le zero_lt_one hm1 |
| have hl := Real.log_le_rpow_div hm.le (by norm_num : (0 : ℝ) < 1/100) |
| have he : m^(3/5 : ℝ)*m^(1/100 : ℝ) = m^(61/100 : ℝ) := by |
| rw [← Real.rpow_add hm]; norm_num |
| have hpow : 1 ≤ m^(61/100 : ℝ) := Real.one_le_rpow hm1 (by norm_num) |
| have hmul : m^(3/5 : ℝ)*Real.log m ≤ 100*m^(61/100 : ℝ) := by |
| have hh := mul_le_mul_of_nonneg_left hl (Real.rpow_nonneg hm.le (3/5 : ℝ)) |
| rw [← mul_div_assoc,he] at hh |
| linarith |
| calc |
| _ ≤ 802*m^(61/100 : ℝ) := by linarith |
| _ ≤ m^(1/50 : ℝ)*m^(61/100 : ℝ) := mul_le_mul_of_nonneg_right hlarge (by positivity) |
| _ = _ := by rw [← Real.rpow_add hm]; norm_num |
|
|
| theorem sourceSamplingSelfBudget_eventually : ∀ᶠ stage in atTop, |
| sourceSamplingSelfBudget stage ≤ (sourceBlockSize stage : ℝ)^(-(1/25 : ℝ)*(sourceBlockSize stage : ℝ)) := by |
| have ht : Tendsto (fun stage => (sourceBlockSize stage : ℝ)^(1/50 : ℝ)) atTop atTop := |
| (tendsto_rpow_atTop (by norm_num : (0 : ℝ) < 1/50)).comp |
| (tendsto_natCast_atTop_atTop.comp sourceBlockSize_tendsto) |
| filter_upwards [ht.eventually (eventually_ge_atTop 802),sourceSamplingPowers_eventually] with stage hlarge hspec |
| let m : ℝ := sourceBlockSize stage |
| let B := 2+8*m^(3/5 : ℝ)*Real.log m |
| let P : ℝ := (sourceSamplingPowers stage).card |
| have hm : 0 < m := sourceBlockSize_pos stage |
| have hm1 : 1 ≤ m := by |
| have hh : (100 : ℝ) ≤ sourceBlockSize stage := by exact_mod_cast (sourceStage stage).m_ge_100 |
| dsimp [m]; linarith |
| have hB0 : 0 ≤ B := by dsimp [B]; positivity [Real.log_nonneg hm1] |
| have hB : B ≤ m^(63/100 : ℝ) := sampling_self_base_bound m hm1 hlarge |
| have hnum : B^(sourceSamplingPrefix stage).card ≤ m^((65/100 : ℝ)*m) := by |
| calc |
| _ ≤ (m^(63/100 : ℝ))^(sourceSamplingPrefix stage).card := pow_le_pow_left₀ hB0 hB _ |
| _ = m^((63/100 : ℝ)*((sourceSamplingPrefix stage).card : ℝ)) := by |
| rw [← Real.rpow_natCast,← Real.rpow_mul hm.le] |
| _ ≤ _ := Real.rpow_le_rpow_of_exponent_le hm1 (by |
| have hd := sourceSamplingPrefix_card_bound stage |
| change ((sourceSamplingPrefix stage).card : ℝ) ≤ (101/100 : ℝ)*m at hd |
| nlinarith) |
| have hP : m^((69/100 : ℝ)*m) ≤ P := hspec.2.1 |
| have hPpos : 0 < P := (Real.rpow_pos_of_pos hm _).trans_le hP |
| change B^(sourceSamplingPrefix stage).card/P ≤ m^(-(1/25 : ℝ)*m) |
| calc |
| _ ≤ m^((65/100 : ℝ)*m)/P := div_le_div_of_nonneg_right hnum hPpos.le |
| _ ≤ m^((65/100 : ℝ)*m)/m^((69/100 : ℝ)*m) := |
| div_le_div_of_nonneg_left (by positivity) (by positivity) hP |
| _ = _ := by rw [← Real.rpow_sub hm]; congr 1; ring |
|
|
| / |
| theorem sourceSampling_inverse_square_summable : Summable (fun stage => (sourceBlockSize stage : ℝ)^(-2 : ℝ)) := by |
| have hg : Summable (fun stage : ℕ => (1/4 : ℝ)^stage) := |
| summable_geometric_of_lt_one (by norm_num) (by norm_num) |
| apply hg.of_nonneg_of_le (fun _ => by positivity) |
| intro stage |
| have hp : (2 : ℝ)^stage ≤ sourceBlockSize stage := by |
| have hh := (sourceStage stage).m_ge_pow |
| have hle : 2^stage ≤ 2^(stage+1) := Nat.pow_le_pow_right (by norm_num) (by omega) |
| exact_mod_cast hle.trans hh |
| calc |
| _ ≤ ((2 : ℝ)^stage)^(-2 : ℝ) := Real.rpow_le_rpow_of_nonpos (by positivity) hp (by norm_num) |
| _ = _ := by |
| rw [Real.rpow_neg (by positivity),Real.rpow_two] |
| rw [← pow_mul, Nat.mul_comm stage 2, pow_mul, ← inv_pow] |
| norm_num |
|
|
| / |
| theorem sourceSamplingSelfBudget_summable : Summable sourceSamplingSelfBudget := by |
| apply sourceSampling_inverse_square_summable.of_norm_bounded_eventually_nat |
| filter_upwards [sourceSamplingSelfBudget_eventually] with stage hb |
| have hm : (100 : ℝ) ≤ sourceBlockSize stage := by exact_mod_cast (sourceStage stage).m_ge_100 |
| have hn : 0 ≤ sourceSamplingSelfBudget stage := by |
| unfold sourceSamplingSelfBudget |
| have hl := Real.log_nonneg (show (1 : ℝ) ≤ sourceBlockSize stage by linarith) |
| positivity |
| rw [Real.norm_eq_abs,abs_of_nonneg hn] |
| exact hb.trans (Real.rpow_le_rpow_of_exponent_le (by linarith) (by nlinarith)) |
|
|
| end |
| end FiniteFactor |
|
|
| end OAI |
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|
