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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 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 | 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)
/-- A concrete logarithmic estimate sufficient for the source exponent gap. -/
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
/-- The actual block growth makes the inverse-square stage sequence summable. -/
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
/-- Summability of the actual full-product self-pair budget. -/
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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