| import OAI.NumberTheory.Ostmann.Arithmetic.ArithmeticPatternPriorRate |
| import OAI.NumberTheory.Ostmann.Construction.FinalHistoryCount |
|
|
| namespace OAI |
|
|
| /-! # The original pattern and frequency counts preserve the arithmetic error -/ |
|
|
| namespace Ostmann |
| open Filter |
|
|
| theorem pairedFrequencyTree_card (S : Finset ℤ) (n : ℕ) : |
| Fintype.card (FrequencyTree (S × S) n) = (Fintype.card (FrequencyTree S n)) ^ 2 := by |
| simp only [card_frequencyTree, Fintype.card_prod, pow_two, mul_pow] |
|
|
| theorem movingPattern_frequency_error_rate (n k : ℕ) (hk : 0 < k) |
| (Cprior Cfreq α β : ℝ) (hCprior : 1 ≤ Cprior) (hCfreq : 0 ≤ Cfreq) |
| (hα : 0 < α) (hβα : β < α) : |
| ∀ᶠ L : ℝ in atTop, ∀ S : Finset ℤ, |
| (S.card : ℝ) ≤ Real.exp (Cfreq * spectatorBulkCount k L) → |
| ((2 : ℝ) ^ ((4 * n * 2 ^ n) ^ 2) * |
| (max 2 (Real.exp (Cprior * L))) ^ (4 * n * 2 ^ n)) * |
| (Fintype.card (FrequencyTree (S × S) n) : ℝ) * Real.exp (-Real.exp (α * L)) ≤ |
| Real.exp (-Real.exp (β * L)) := by |
| let J := 4 * n * 2 ^ n |
| let A : ℝ := ((J ^ 2 + J : ℕ) : ℝ) * Cprior |
| let B : ℝ := (2 * (2 ^ (n + 1) - 1 : ℕ) : ℝ) * Cfreq |
| let H := A + B * (k : ℝ) ^ 4 |
| have hA : 0 ≤ A := by dsimp only [A]; positivity |
| have hB : 0 ≤ B := by dsimp only [B]; positivity |
| have hH : 0 ≤ H := by dsimp only [H]; positivity |
| filter_upwards [arithmetic_exponent_absorption 0 α β H 1 0 hα hα hβα (by norm_num), |
| eventually_ge_atTop (1 : ℝ)] with L hrate hL S hS |
| have hL0 : 0 ≤ L := by linarith |
| have hp := movingPattern_total_cost_le_exp n Cprior L hCprior hL |
| have hc := frequency_history_pair_card_bound S n Cfreq (spectatorBulkCount k L) hS |
| have hc' : (Fintype.card (FrequencyTree (S × S) n) : ℝ) ≤ |
| Real.exp (B * (k : ℝ) ^ 4 * L) := by |
| rw [pairedFrequencyTree_card, Nat.cast_pow] |
| apply hc.trans |
| apply Real.exp_le_exp.mpr |
| have hm := mul_le_mul_of_nonneg_left (spectatorBulkCount_upper k L hL0) hB |
| exact hm.trans_eq (by ring) |
| have hprod := mul_le_mul hp hc' (Nat.cast_nonneg _) (Real.exp_nonneg _) |
| have hexp : ((2 : ℝ) ^ ((4 * n * 2 ^ n) ^ 2) * |
| (max 2 (Real.exp (Cprior * L))) ^ (4 * n * 2 ^ n)) * |
| (Fintype.card (FrequencyTree (S × S) n) : ℝ) ≤ Real.exp (H * L) := by |
| apply hprod.trans_eq |
| rw [← Real.exp_add] |
| congr 1 |
| dsimp only [H, A, J] |
| ring |
| apply (mul_le_mul_of_nonneg_right hexp (Real.exp_nonneg _)).trans |
| rw [← Real.exp_add] |
| apply Real.exp_le_exp.mpr |
| simp only [pow_zero, mul_one, zero_mul, Real.exp_zero, one_mul] at hrate |
| linarith |
|
|
| / |
| scale after the exact equality-pattern and two-history sums. -/ |
| theorem movingPattern_three_errors_rate (n k : ℕ) (hk : 0 < k) |
| (Cprior Cfreq : ℝ) (hCprior : 1 ≤ Cprior) (hCfreq : 0 ≤ Cfreq) : |
| ∀ᶠ L : ℝ in atTop, ∀ S : Finset ℤ, |
| (S.card : ℝ) ≤ Real.exp (Cfreq * spectatorBulkCount k L) → |
| ((2 : ℝ) ^ ((4 * n * 2 ^ n) ^ 2) * |
| (max 2 (Real.exp (Cprior * L))) ^ (4 * n * 2 ^ n)) * |
| (Fintype.card (FrequencyTree (S × S) n) : ℝ) * |
| (Real.exp (-Real.exp ((125 / 100000 : ℝ) * L)) + |
| 2 * Real.exp (-Real.exp ((2 / 1000 : ℝ) * L))) ≤ |
| 3 * Real.exp (-Real.exp ((12 / 10000 : ℝ) * L)) := by |
| filter_upwards [movingPattern_frequency_error_rate n k hk Cprior Cfreq |
| (125 / 100000) (12 / 10000) hCprior hCfreq (by norm_num) (by norm_num), |
| movingPattern_frequency_error_rate n k hk Cprior Cfreq |
| (2 / 1000) (12 / 10000) hCprior hCfreq (by norm_num) (by norm_num)] |
| with L hfirst hsecond S hS |
| have h := add_le_add (hfirst S hS) (mul_le_mul_of_nonneg_left (hsecond S hS) |
| (by norm_num : (0 : ℝ) ≤ 2)) |
| nlinarith only [h] |
|
|
| / |
| scale. All four errors survive both finite sums with a common saving. -/ |
| theorem movingPattern_four_errors_rate (n k : ℕ) (hk : 0 < k) |
| (Cprior Cfreq : ℝ) (hCprior : 1 ≤ Cprior) (hCfreq : 0 ≤ Cfreq) : |
| ∀ᶠ L : ℝ in atTop, ∀ S : Finset ℤ, |
| (S.card : ℝ) ≤ Real.exp (Cfreq * spectatorBulkCount k L) → |
| ((2 : ℝ) ^ ((4 * n * 2 ^ n) ^ 2) * |
| (max 2 (Real.exp (Cprior * L))) ^ (4 * n * 2 ^ n)) * |
| (Fintype.card (FrequencyTree (S × S) n) : ℝ) * |
| (Real.exp (-Real.exp ((125 / 100000 : ℝ) * L)) + |
| 3 * Real.exp (-Real.exp ((2 / 1000 : ℝ) * L))) ≤ |
| 4 * Real.exp (-Real.exp ((12 / 10000 : ℝ) * L)) := by |
| filter_upwards [movingPattern_three_errors_rate n k hk Cprior Cfreq hCprior hCfreq, |
| movingPattern_frequency_error_rate n k hk Cprior Cfreq |
| (2 / 1000) (12 / 10000) hCprior hCfreq (by norm_num) (by norm_num)] |
| with L hthree hone S hS |
| have h := add_le_add (hthree S hS) (hone S hS) |
| nlinarith only [h] |
|
|
| / |
| one internal comparison errors. All five survive the exact finite sums. -/ |
| theorem movingPattern_five_errors_rate (n k : ℕ) (hk : 0 < k) |
| (Cprior Cfreq : ℝ) (hCprior : 1 ≤ Cprior) (hCfreq : 0 ≤ Cfreq) : |
| ∀ᶠ L : ℝ in atTop, ∀ S : Finset ℤ, |
| (S.card : ℝ) ≤ Real.exp (Cfreq * spectatorBulkCount k L) → |
| ((2 : ℝ) ^ ((4 * n * 2 ^ n) ^ 2) * |
| (max 2 (Real.exp (Cprior * L))) ^ (4 * n * 2 ^ n)) * |
| (Fintype.card (FrequencyTree (S × S) n) : ℝ) * |
| (Real.exp (-Real.exp ((125 / 100000 : ℝ) * L)) + |
| 4 * Real.exp (-Real.exp ((2 / 1000 : ℝ) * L))) ≤ |
| 5 * Real.exp (-Real.exp ((12 / 10000 : ℝ) * L)) := by |
| filter_upwards [movingPattern_four_errors_rate n k hk Cprior Cfreq hCprior hCfreq, |
| movingPattern_frequency_error_rate n k hk Cprior Cfreq |
| (2 / 1000) (12 / 10000) hCprior hCfreq (by norm_num) (by norm_num)] |
| with L hfour hone S hS |
| have h := add_le_add (hfour S hS) (hone S hS) |
| nlinarith only [h] |
|
|
| end Ostmann |
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| end OAI |
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