| import OAI.NumberTheory.Ostmann.Arithmetic.ArithmeticPatternErrorRate |
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| namespace OAI |
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| /-! # Full equality-pattern and frequency sums in the good-matching estimate -/ |
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| namespace Ostmann |
| open Filter |
| open scoped Classical BigOperators |
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| / |
| 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 |
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| 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 |
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| end Ostmann |
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| end OAI |
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