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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 | import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Ring
namespace OAI
/-! # Absorbing the fixed-depth arithmetic comparison costs
The polynomial and spectator costs in Section 8 are dominated by the
double-exponential savings. The strict numerical gaps in the manuscript
therefore survive the finite history and equality-pattern sums.
-/
namespace Ostmann
open Filter Asymptotics
theorem arithmetic_exponent_absorption (a b d C c : ℝ) (n : ℕ)
(hb : 0 < b) (hab : a < b) (hdb : d < b) (hc : 0 < c) :
∀ᶠ L : ℝ in atTop,
C * L ^ n + C * L * Real.exp (a * L) - c * Real.exp (b * L) ≤
-Real.exp (d * L) := by
have hp := (isLittleO_pow_exp_pos_mul_atTop n hb).const_mul_left C
have hm : (fun L : ℝ => C * (Real.exp (a * L) * L)) =o[atTop]
(fun L => Real.exp (b * L)) := by
simpa only [Real.rpow_one] using (isLittleO_exp_mul_rpow_of_lt 1 hab).const_mul_left C
have he : (fun L : ℝ => Real.exp (d * L)) =o[atTop] (fun L => Real.exp (b * L)) := by
simpa only [Real.rpow_zero, mul_one] using isLittleO_exp_mul_rpow_of_lt 0 hdb
have h := ((hp.add hm).add he).bound hc
filter_upwards [h] with L hL
simp only [Real.norm_eq_abs, abs_of_pos (Real.exp_pos _)] at hL
have hle := (le_abs_self _).trans hL
nlinarith
theorem arithmetic_error_absorption (a b d C c : ℝ) (n : ℕ)
(hb : 0 < b) (hab : a < b) (hdb : d < b) (hc : 0 < c) :
∀ᶠ L : ℝ in atTop,
Real.exp (C * L ^ n + C * L * Real.exp (a * L)) *
Real.exp (-c * Real.exp (b * L)) ≤ Real.exp (-Real.exp (d * L)) := by
filter_upwards [arithmetic_exponent_absorption a b d C c n hb hab hdb hc] with L hL
rw [← Real.exp_add]
apply Real.exp_le_exp.mpr
linarith
/-- The mesh variation error in the bulk idealization, including the
spectator and polynomial history costs. -/
theorem bulk_variation_error (C : ℝ) :
∀ᶠ L : ℝ in atTop,
Real.exp (C * L ^ 2 + C * L * Real.exp ((1 / 1000 : ℝ) * L)) *
Real.exp (-Real.exp ((14 / 10000 : ℝ) * L)) ≤
Real.exp (-Real.exp ((125 / 100000 : ℝ) * L)) := by
simpa only [one_mul, neg_mul] using
arithmetic_error_absorption (1 / 1000) (14 / 10000) (125 / 100000) C 1 2
(by norm_num) (by norm_num) (by norm_num) (by norm_num)
/-- The polynomial-root loss remains negligible after the same costs. -/
theorem symbolic_root_error (C c : ℝ) (hc : 0 < c) :
∀ᶠ L : ℝ in atTop,
Real.exp (C * L ^ 2 + C * L * Real.exp ((1 / 1000 : ℝ) * L)) *
Real.exp (-c * Real.exp ((4 / 1000 : ℝ) * L)) ≤
Real.exp (-Real.exp ((2 / 1000 : ℝ) * L)) :=
arithmetic_error_absorption (1 / 1000) (4 / 1000) (2 / 1000) C c 2
(by norm_num) (by norm_num) (by norm_num) hc
/-- The error from the bulk progression formula after counting joint boxes. -/
theorem bulk_progression_error (C c : ℝ) (hc : 0 < c) :
∀ᶠ L : ℝ in atTop,
Real.exp (C * L ^ 2 + C * L * Real.exp ((14 / 10000 : ℝ) * L)) *
Real.exp (-c * Real.exp ((16 / 10000 : ℝ) * L)) ≤
Real.exp (-Real.exp ((125 / 100000 : ℝ) * L)) :=
arithmetic_error_absorption (14 / 10000) (16 / 10000) (125 / 100000) C c 2
(by norm_num) (by norm_num) (by norm_num) hc
/-- Both the prime-square exclusions and the nonzero-polynomial prime-divisor
loss have the stronger internal-prime saving. -/
theorem internal_prime_error (C c : ℝ) (hc : 0 < c) :
∀ᶠ L : ℝ in atTop,
Real.exp (C * L ^ 2 + C * L * Real.exp ((1 / 1000 : ℝ) * L)) *
Real.exp (-c * Real.exp ((1 / 100 : ℝ) * L)) ≤
Real.exp (-Real.exp ((2 / 1000 : ℝ) * L)) :=
arithmetic_error_absorption (1 / 1000) (1 / 100) (2 / 1000) C c 2
(by norm_num) (by norm_num) (by norm_num) hc
/-- Freezing the two giant log coordinates has ample reserve. -/
theorem giant_variation_error (C : ℝ) :
∀ᶠ L : ℝ in atTop,
Real.exp (C * L ^ 2 + C * L * Real.exp ((12 / 1000 : ℝ) * L)) *
Real.exp (-Real.exp ((15 / 1000 : ℝ) * L)) ≤
Real.exp (-Real.exp ((13 / 1000 : ℝ) * L)) := by
simpa only [one_mul, neg_mul] using
arithmetic_error_absorption (12 / 1000) (15 / 1000) (13 / 1000) C 1 2
(by norm_num) (by norm_num) (by norm_num) (by norm_num)
/-- Summing the giant progression errors over all boxes and residues. -/
theorem giant_progression_error (C c : ℝ) (hc : 0 < c) :
∀ᶠ L : ℝ in atTop,
Real.exp (C * L ^ 2 + C * L * Real.exp ((15 / 1000 : ℝ) * L)) *
Real.exp (-c * Real.exp ((18 / 1000 : ℝ) * L)) ≤
Real.exp (-Real.exp ((13 / 1000 : ℝ) * L)) :=
arithmetic_error_absorption (15 / 1000) (18 / 1000) (13 / 1000) C c 2
(by norm_num) (by norm_num) (by norm_num) hc
/-- The elementary external-integer error has a stronger saving. -/
theorem giant_integer_error (C c : ℝ) (hc : 0 < c) :
∀ᶠ L : ℝ in atTop,
Real.exp (C * L ^ 2 + C * L * Real.exp ((15 / 1000 : ℝ) * L)) *
Real.exp (-c * Real.exp ((49 / 1000 : ℝ) * L)) ≤
Real.exp (-Real.exp ((13 / 1000 : ℝ) * L)) :=
arithmetic_error_absorption (15 / 1000) (49 / 1000) (13 / 1000) C c 2
(by norm_num) (by norm_num) (by norm_num) hc
end Ostmann
end OAI
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