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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 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | 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
/-- All three errors in the original-prior norm bound fit the final common
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]
/-- The internal-prime comparison contributes one further error of the same
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]
/-- The literal giant comparison adds one further error to the two bulk and
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
end OAI
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