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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 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 | import OAI.Analysis.FiniteFactor.Sampling.Selection
import OAI.Analysis.FiniteFactor.CrossedProduct.PhaseDecay
namespace OAI
namespace FiniteFactor
open Finset Filter
open scoped Topology
noncomputable section
def samplingQ (m : ℕ) : ℝ := (m : ℝ)^((7/10 : ℝ)*(m : ℝ))
def samplingH (m : ℕ) : ℕ := m^300
def samplingR (m K : ℕ) : ℕ := m*K*m^1000
theorem sampling_integer_candidates (Q : ℝ) (hQ : 4 ≤ Q) :
Q/4 ≤ ((Finset.Icc ⌈Q/2⌉ ⌊Q⌋).card : ℝ) := by
have hf := Int.sub_one_lt_floor Q
have hc := Int.ceil_lt_add_one (Q/2)
have horder : ⌈Q/2⌉ ≤ ⌊Q⌋ := by
have : (⌈Q/2⌉ : ℝ) < (⌊Q⌋ : ℝ)+1 := by linarith
have ht : ⌈Q/2⌉ < ⌊Q⌋+1 := by exact_mod_cast this
omega
rw [Int.card_Icc]
have hcast : ((⌊Q⌋+1-⌈Q/2⌉).toNat : ℝ) = (⌊Q⌋ : ℝ)+1-(⌈Q/2⌉ : ℝ) := by
exact_mod_cast Int.toNat_of_nonneg (by omega : 0 ≤ ⌊Q⌋+1-⌈Q/2⌉)
rw [hcast]
linarith
theorem sampling_forbidden_budget (m K : ℕ) (hm : 4 ≤ m) (hK : K ≤ m) :
(((2*samplingH m+1)^2*(2*samplingR m K+1) : ℕ) : ℝ) ≤ (m : ℝ)^1605 := by
have hm1 : 1 ≤ m := by omega
have hh : 1 ≤ samplingH m := one_le_pow₀ hm1
have hr : samplingR m K ≤ m^1002 := by
unfold samplingR
calc
m*K*m^1000 ≤ m*m*m^1000 := by gcongr
_ = m^1002 := by ring
have hp : 1 ≤ m^1002 := one_le_pow₀ hm1
have hD : (2*samplingH m+1)^2*(2*samplingR m K+1) ≤ 27*m^1602 := by
calc
_ ≤ (3*samplingH m)^2*(3*m^1002) := by gcongr <;> omega
_ = _ := by unfold samplingH; ring
have h27 : 27 ≤ m^3 := (by norm_num : 27 ≤ (4 : ℕ)^3).trans (Nat.pow_le_pow_left hm 3)
have hhD : (2*samplingH m+1)^2*(2*samplingR m K+1) ≤ m^1605 := by
calc
_ ≤ 27*m^1602 := hD
_ ≤ m^3*m^1602 := Nat.mul_le_mul_right _ h27
_ = _ := by ring
exact_mod_cast hhD
/-- An explicit sufficiently-large threshold discharges the source's greedy
asymptotics, retaining Q, H, R and the exponent 0.69m exactly. -/
theorem exists_source_sampling_powers (m K : ℕ) (hm : 200000 ≤ m) (hK : K ≤ m) :
∃ N : Finset ℤ,
(∀ n ∈ N, samplingQ m/2 ≤ (n : ℝ) ∧ (n : ℝ) ≤ samplingQ m ∧ 0 < n) ∧
(m : ℝ)^((69/100 : ℝ)*(m : ℝ)) ≤ (N.card : ℝ) ∧
(N : Set ℤ).Pairwise (fun n n' => ¬SamplingConflict (samplingH m) (samplingR m K) n n') := by
have hmR : (200000 : ℝ) ≤ m := by exact_mod_cast hm
have hm0 : (0 : ℝ) < m := by linarith
have hm1 : (1 : ℝ) ≤ m := by linarith
have hQ : (m : ℝ)^1606 ≤ samplingQ m := by
unfold samplingQ
rw [← Real.rpow_natCast]
apply Real.rpow_le_rpow_of_exponent_le hm1
norm_num
linarith
have hmQ : (m : ℝ) ≤ samplingQ m := by
calc
_ ≤ (m : ℝ)^1606 := le_self_pow₀ hm1 (by norm_num)
_ ≤ _ := hQ
have hQ4 : 4 ≤ samplingQ m := by linarith
have hR : (samplingR m K : ℝ) < samplingQ m/2 := by
have hr : samplingR m K ≤ m^1002 := by
unfold samplingR
calc
m*K*m^1000 ≤ m*m*m^1000 := by gcongr
_ = _ := by ring
have hsmall : (m : ℝ)^1002 * (m : ℝ) ≤ (m : ℝ)^1606 := by
rw [← pow_succ]
exact pow_le_pow_right₀ hm1 (by norm_num)
have hrR : (samplingR m K : ℝ) ≤ (m : ℝ)^1002 := by exact_mod_cast hr
have hpos : 0 < (m : ℝ)^1002 := by positivity
nlinarith
let S := Finset.Icc ⌈samplingQ m/2⌉ ⌊samplingQ m⌋
have hS (n : ℤ) (hn : n ∈ S) :
samplingQ m/2 ≤ (n : ℝ) ∧ (n : ℝ) ≤ samplingQ m := by
have hh := mem_Icc.mp hn
constructor
· exact (Int.le_ceil _).trans (by exact_mod_cast hh.1)
· exact (show (n : ℝ) ≤ (⌊samplingQ m⌋ : ℝ) by exact_mod_cast hh.2).trans (Int.floor_le _)
obtain ⟨N,hNS,hpair,hcard⟩ := exists_sampling_separated (samplingH m) (samplingR m K)
(one_le_pow₀ (by omega)) S (fun n hn => by exact_mod_cast hR.trans_le (hS n hn).1)
refine ⟨N, fun n hn => ⟨(hS n (hNS hn)).1, (hS n (hNS hn)).2, ?_⟩, ?_, hpair⟩
· have hh := (hS n (hNS hn)).1
have : (0 : ℝ) < n := by linarith
exact_mod_cast this
· have hcount : samplingQ m/4 ≤
(((2*samplingH m+1)^2*(2*samplingR m K+1) : ℕ) : ℝ)*(N.card : ℝ) :=
(sampling_integer_candidates _ hQ4).trans (by exact_mod_cast hcard)
have hDb := sampling_forbidden_budget m K (by omega) hK
have hcount' : samplingQ m/4 ≤ (m : ℝ)^1605*(N.card : ℝ) :=
hcount.trans (mul_le_mul_of_nonneg_right hDb (Nat.cast_nonneg _))
have hexp : (m : ℝ)^1605*(m : ℝ)^((69/100 : ℝ)*(m : ℝ))*(m : ℝ) ≤ samplingQ m := by
unfold samplingQ
calc
_ = (m : ℝ)^(1605+(69/100 : ℝ)*(m : ℝ)+1) := by
rw [Real.rpow_add hm0, Real.rpow_one, Real.rpow_add hm0]
norm_num [Real.rpow_ofNat]
_ ≤ _ := by
apply Real.rpow_le_rpow_of_exponent_le hm1
norm_num
linarith
have hprod : 0 < (m : ℝ)^1605 := by positivity
have htarget : 0 ≤ (m : ℝ)^((69/100 : ℝ)*(m : ℝ)) := by positivity
apply (mul_le_mul_iff_right₀ hprod).mp
nlinarith
/-- A deterministic finite set fixed from the numerical stages alone. -/
def sourceSamplingPowers (s : ℕ) : Finset ℤ :=
if h : 200000 ≤ sourceBlockSize s then
Classical.choose (exists_source_sampling_powers (sourceBlockSize s)
(sourceStageState s).K h (sourceStage s).m_ge_K)
else ∅
theorem sourceSamplingPowers_spec (s : ℕ) (hs : 200000 ≤ sourceBlockSize s) :
(∀ n ∈ sourceSamplingPowers s,
samplingQ (sourceBlockSize s)/2 ≤ (n : ℝ) ∧
(n : ℝ) ≤ samplingQ (sourceBlockSize s) ∧ 0 < n) ∧
(sourceBlockSize s : ℝ)^((69/100 : ℝ)*(sourceBlockSize s : ℝ)) ≤
((sourceSamplingPowers s).card : ℝ) ∧
((sourceSamplingPowers s : Finset ℤ) : Set ℤ).Pairwise (fun n n' =>
¬SamplingConflict (samplingH (sourceBlockSize s))
(samplingR (sourceBlockSize s) (sourceStageState s).K) n n') := by
unfold sourceSamplingPowers
rw [dite_eq_left hs]
exact Classical.choose_spec (exists_source_sampling_powers _ _ hs (sourceStage s).m_ge_K)
theorem sourceSamplingPowers_eventually : ∀ᶠ s in atTop,
(∀ n ∈ sourceSamplingPowers s,
samplingQ (sourceBlockSize s)/2 ≤ (n : ℝ) ∧
(n : ℝ) ≤ samplingQ (sourceBlockSize s) ∧ 0 < n) ∧
(sourceBlockSize s : ℝ)^((69/100 : ℝ)*(sourceBlockSize s : ℝ)) ≤
((sourceSamplingPowers s).card : ℝ) ∧
((sourceSamplingPowers s : Finset ℤ) : Set ℤ).Pairwise (fun n n' =>
¬SamplingConflict (samplingH (sourceBlockSize s))
(samplingR (sourceBlockSize s) (sourceStageState s).K) n n') := by
filter_upwards [sourceBlockSize_tendsto.eventually (eventually_ge_atTop 200000)] with s hs
exact sourceSamplingPowers_spec s hs
end
end FiniteFactor
end OAI
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