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f4b3ad1 | 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 | import OAI.NumberTheory.Ostmann.Arithmetic.ArithmeticPatternErrorRate
namespace OAI
/-! # Full equality-pattern and frequency sums in the good-matching estimate -/
namespace Ostmann
open Filter
open scoped Classical BigOperators
/-- The signed gain can be chosen before the local quartet tolerance. This
amount absorbs every original pattern and paired-frequency count. -/
def goodPatternGain (n : ℕ) (Cprior Cfreq gain : ℝ) : ℝ :=
2 * (((4 * n * 2 ^ n) ^ 2 + 4 * n * 2 ^ n : ℕ) : ℝ) * Cprior +
(2 * (2 ^ (n + 1) - 1 : ℕ) : ℝ) * Cfreq + gain
theorem arithmetic_good_pattern_rate (n k : ℕ) (hk : 0 < k)
(Cprior Cfreq gain : ℝ) (hCprior : 1 ≤ Cprior) (hCfreq : 0 ≤ Cfreq) :
∀ᶠ L : ℝ in atTop, let m := spectatorBulkCount k L
∀ S : Finset ℤ, (S.card : ℝ) ≤ Real.exp (Cfreq * m) →
let _ := sampleSetoidFintype (Bool × MovingSampleIndex n)
∀ R : Setoid (Bool × MovingSampleIndex n) → FrequencyTree (S × S) n → ℂ,
(∀ s t, ‖R s t‖ ≤
((2 : ℝ) ^ Fintype.card (Quotient s) *
(Real.exp (Cprior * L)) ^ (4 * n * 2 ^ n - Fintype.card (Quotient s))) *
(Real.exp (-goodPatternGain n Cprior Cfreq gain * m) +
Real.exp (-Real.exp ((125 / 100000 : ℝ) * L)) +
4 * Real.exp (-Real.exp ((2 / 1000 : ℝ) * L)))) →
‖∑ s, ∑ t, R s t‖ ≤ Real.exp (-gain * m) +
5 * Real.exp (-Real.exp ((12 / 10000 : ℝ) * L)) := by
filter_upwards [movingPattern_five_errors_rate n k hk Cprior Cfreq hCprior hCfreq,
eventually_ge_atTop (4 : ℝ)] with L herrors hL
dsimp only
intro S hS R hR
let _ := sampleSetoidFintype (Bool × MovingSampleIndex n)
let m := spectatorBulkCount k L
let A : ℝ := (((4 * n * 2 ^ n) ^ 2 + 4 * n * 2 ^ n : ℕ) : ℝ) * Cprior
let B : ℝ := (2 * (2 ^ (n + 1) - 1 : ℕ) : ℝ) * Cfreq
let cost := (2 : ℝ) ^ ((4 * n * 2 ^ n) ^ 2) *
(max 2 (Real.exp (Cprior * L))) ^ (4 * n * 2 ^ n)
let err := Real.exp (-Real.exp ((125 / 100000 : ℝ) * L)) +
4 * Real.exp (-Real.exp ((2 / 1000 : ℝ) * L))
have hA : 0 ≤ A := by dsimp only [A]; positivity
have hB : 0 ≤ B := by dsimp only [B]; positivity
have hL0 : 0 ≤ L := by linarith
have hkpow : (1 : ℝ) ≤ (k : ℝ) ^ 4 := one_le_pow₀ (by exact_mod_cast hk)
have hLL : L ≤ (k : ℝ) ^ 4 * L := by nlinarith only [hkpow, hL0]
have hhalf := spectatorBulkCount_half k L (show 4 ≤ (k : ℝ) ^ 4 * L by linarith)
have hLm : L ≤ 2 * (m : ℝ) := by dsimp only [m]; linarith
have hcost : cost ≤ Real.exp (2 * A * m) := by
apply (movingPattern_total_cost_le_exp n Cprior L hCprior (by linarith)).trans
apply Real.exp_le_exp.mpr
change A * L ≤ 2 * A * m
nlinarith only [hLm, hA]
have hcard : (Fintype.card (FrequencyTree (S × S) n) : ℝ) ≤ Real.exp (B * m) := by
rw [pairedFrequencyTree_card, Nat.cast_pow]
exact frequency_history_pair_card_bound S n Cfreq m hS
have hgain : cost * (Fintype.card (FrequencyTree (S × S) n) : ℝ) *
Real.exp (-goodPatternGain n Cprior Cfreq gain * m) ≤ Real.exp (-gain * m) := by
apply (mul_le_mul_of_nonneg_right
(mul_le_mul hcost hcard (Nat.cast_nonneg _) (Real.exp_nonneg _)) (Real.exp_nonneg _)).trans_eq
rw [← Real.exp_add, ← Real.exp_add]
congr 1
dsimp only [goodPatternGain, A, B]
ring
have hsum : ‖∑ s, ∑ t, R s t‖ ≤
cost * (Fintype.card (FrequencyTree (S × S) n) : ℝ) *
(Real.exp (-goodPatternGain n Cprior Cfreq gain * m) + err) := by
have hs (s : Setoid (Bool × MovingSampleIndex n)) :
‖∑ t, R s t‖ ≤
((Fintype.card (FrequencyTree (S × S) n) : ℝ) *
(Real.exp (-goodPatternGain n Cprior Cfreq gain * m) + err)) *
((2 : ℝ) ^ Fintype.card (Quotient s) *
(Real.exp (Cprior * L)) ^ (4 * n * 2 ^ n - Fintype.card (Quotient s))) := by
apply (norm_sum_le_of_le _ (fun t _ => hR s t)).trans_eq
rw [Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
dsimp only [err]
ring
have hh := movingPattern_error_sum_le n (Real.exp (Cprior * L))
((Fintype.card (FrequencyTree (S × S) n) : ℝ) *
(Real.exp (-goodPatternGain n Cprior Cfreq gain * m) + err))
(Real.exp_nonneg _) (by dsimp only [err]; positivity) (fun s => ∑ t, R s t) hs
exact hh.trans_eq (by dsimp only [cost]; ring)
have he : cost * (Fintype.card (FrequencyTree (S × S) n) : ℝ) * err ≤
5 * Real.exp (-Real.exp ((12 / 10000 : ℝ) * L)) := herrors S hS
calc
_ ≤ _ := hsum
_ = cost * (Fintype.card (FrequencyTree (S × S) n) : ℝ) *
Real.exp (-goodPatternGain n Cprior Cfreq gain * m) +
cost * (Fintype.card (FrequencyTree (S × S) n) : ℝ) * err := by ring
_ ≤ _ := add_le_add hgain he
end Ostmann
end OAI
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