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2c7d975 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 | import OAI.Algebra.DepthFive.AnalyticParameters
import OAI.Algebra.DepthFive.FractionRounding
namespace OAI
noncomputable section
namespace Problem335.LowerParameters
theorem sqrt_ge_four {n : ℕ} (hn : 16 ≤ n) : 4 ≤ Real.sqrt (n : ℝ) := by
apply (Real.le_sqrt (by norm_num) (by positivity)).mpr
exact_mod_cast hn
theorem a_ideal_le {n : ℕ} (_hn : 16 ≤ n) :
(v n : ℝ) / (Real.sqrt (n : ℝ) - 1) ≤ (a n : ℝ) := Nat.le_ceil _
theorem a_lt_ideal_add_one {n : ℕ} (hn : 16 ≤ n) :
(a n : ℝ) < (v n : ℝ) / (Real.sqrt (n : ℝ) - 1) + 1 := by
apply Nat.ceil_lt_add_one
have hs := sqrt_ge_four hn
exact div_nonneg (Nat.cast_nonneg _) (by linarith)
theorem a_scale_lower {n : ℕ} (hn : 16 ≤ n) :
(n : ℝ) ^ 2 * Real.sqrt (n : ℝ) / 4 ≤ (a n : ℝ) := by
have hs := sqrt_ge_four hn
have hden : 0 < Real.sqrt (n : ℝ) - 1 := by linarith
have hv : 3 * (n : ℝ) ^ 3 ≤ 8 * (v n : ℝ) := by
exact_mod_cast three_mul_cube_le_eight_mul_v hn
have heq : (n : ℝ) ^ 2 * Real.sqrt (n : ℝ) * Real.sqrt (n : ℝ) = (n : ℝ) ^ 3 := by
calc
_ = (n : ℝ) ^ 2 * (Real.sqrt (n : ℝ)) ^ 2 := by ring
_ = (n : ℝ) ^ 3 := by rw [Real.sq_sqrt (by positivity)]; ring
apply le_trans _ (a_ideal_le hn)
apply (le_div_iff₀ hden).mpr
have hnonneg : 0 ≤ (n : ℝ) ^ 2 * Real.sqrt (n : ℝ) := by positivity
nlinarith
theorem a_scale_upper {n : ℕ} (hn : 16 ≤ n) :
(a n : ℝ) ≤ 3 * (n : ℝ) ^ 2 * Real.sqrt (n : ℝ) := by
have hs := sqrt_ge_four hn
have hnR : (16 : ℝ) ≤ n := by exact_mod_cast hn
have hden : 0 < Real.sqrt (n : ℝ) - 1 := by linarith
have hv : (v n : ℝ) ≤ (n : ℝ) ^ 3 := by exact_mod_cast v_le_cube n
have heq : (n : ℝ) ^ 2 * Real.sqrt (n : ℝ) * Real.sqrt (n : ℝ) = (n : ℝ) ^ 3 := by
calc
_ = (n : ℝ) ^ 2 * (Real.sqrt (n : ℝ)) ^ 2 := by ring
_ = (n : ℝ) ^ 3 := by rw [Real.sq_sqrt (by positivity)]; ring
have hnonneg : 0 ≤ (n : ℝ) ^ 2 * Real.sqrt (n : ℝ) := by positivity
have hmul := mul_le_mul_of_nonneg_right (show (2 : ℝ) ≤ Real.sqrt (n : ℝ) by linarith) hnonneg
have hideal : (v n : ℝ) / (Real.sqrt (n : ℝ) - 1) ≤
2 * (n : ℝ) ^ 2 * Real.sqrt (n : ℝ) := by
apply (div_le_iff₀ hden).mpr
nlinarith
have hlarge : 1 ≤ (n : ℝ) ^ 2 * Real.sqrt (n : ℝ) := by
have hn2 : (1 : ℝ) ≤ (n : ℝ) ^ 2 := by nlinarith
nlinarith [mul_le_mul_of_nonneg_left (show (1 : ℝ) ≤ Real.sqrt (n : ℝ) by linarith)
(sq_nonneg (n : ℝ))]
have hround := a_lt_ideal_add_one hn
nlinarith
theorem alpha_bounds {n : ℕ} (hn : 16 ≤ n) :
1 / Real.sqrt (n : ℝ) ≤ alpha n ∧
alpha n ≤ 1 / Real.sqrt (n : ℝ) *
Real.exp ((Real.sqrt (n : ℝ) - 1) / (v n : ℝ)) := by
have hs := sqrt_ge_four hn
have hv : (0 : ℝ) < v n := by exact_mod_cast v_pos (by omega : 4 ≤ n)
exact fraction_natCeil_inverse_bounds (by linarith) hv
/-- The relative logarithmic rounding error is controlled by the rounded order. -/
theorem alpha_log_error_bounds {n : ℕ} (hn : 16 ≤ n) :
0 ≤ Real.log (alpha n * Real.sqrt (n : ℝ)) ∧
Real.log (alpha n * Real.sqrt (n : ℝ)) ≤ 2 / (a n : ℝ) := by
have hs := sqrt_ge_four hn
have hsm : 0 < Real.sqrt (n : ℝ) - 1 := by linarith
have hspos : 0 < Real.sqrt (n : ℝ) := by linarith
have hv : (0 : ℝ) < v n := by exact_mod_cast v_pos (by omega : 4 ≤ n)
have hx : 0 < (v n : ℝ) / (Real.sqrt (n : ℝ) - 1) := div_pos hv hsm
have hxa := a_ideal_le hn
have hax := a_lt_ideal_add_one hn
have ha : 4 * (n : ℝ) ≤ (a n : ℝ) := by exact_mod_cast four_mul_n_le_a hn
have hnR : (16 : ℝ) ≤ n := by exact_mod_cast hn
have hxone : 1 ≤ (v n : ℝ) / (Real.sqrt (n : ℝ) - 1) := by linarith
have hbase : ((v n : ℝ) / (Real.sqrt (n : ℝ) - 1)) /
((v n : ℝ) / (Real.sqrt (n : ℝ) - 1) + (v n : ℝ)) =
1 / Real.sqrt (n : ℝ) := by
field_simp
ring
have heq : ((a n : ℝ) / ((a n : ℝ) + (v n : ℝ))) /
(((v n : ℝ) / (Real.sqrt (n : ℝ) - 1)) /
((v n : ℝ) / (Real.sqrt (n : ℝ) - 1) + (v n : ℝ))) =
alpha n * Real.sqrt (n : ℝ) := by
rw [hbase, alpha]
field_simp
constructor
· rw [← heq]
exact (fraction_rounding_log_bounds hx hv hxa hax.le).1
· rw [← heq]
exact fraction_rounding_log_le_two_div hxone hv hxa hax.le
/-- The occupation-moment error from differentiation is of order `n^(-1/2)`. -/
theorem sq_div_a_le {n : ℕ} (hn : 16 ≤ n) :
(n : ℝ) ^ 2 / (a n : ℝ) ≤ 4 / Real.sqrt (n : ℝ) := by
have ha : (0 : ℝ) < a n := by exact_mod_cast a_pos (by omega : 4 ≤ n)
have hs : 0 < Real.sqrt (n : ℝ) := by have := sqrt_ge_four hn; linarith
apply (div_le_div_iff₀ ha hs).mpr
have h := a_scale_lower hn
linarith
/-- The ambient-variable occupation error has the better order `1/n`. -/
theorem sq_div_v_le {n : ℕ} (hn : 16 ≤ n) :
(n : ℝ) ^ 2 / (v n : ℝ) ≤ 8 / (3 * (n : ℝ)) := by
have hv : (0 : ℝ) < v n := by exact_mod_cast v_pos (by omega : 4 ≤ n)
have hnR : (0 : ℝ) < n := by exact_mod_cast (by omega : 0 < n)
apply (div_le_div_iff₀ hv (by positivity)).mpr
have h : 3 * (n : ℝ) ^ 3 ≤ 8 * (v n : ℝ) := by
exact_mod_cast three_mul_cube_le_eight_mul_v hn
nlinarith
/-- Uniform two-sided growth of the chosen differentiation order. -/
theorem a_isTheta :
Asymptotics.IsTheta Filter.atTop (fun n : ℕ => (a n : ℝ))
(fun n : ℕ => (n : ℝ) ^ 2 * Real.sqrt (n : ℝ)) := by
constructor
· apply Asymptotics.isBigO_iff.mpr
refine ⟨3, ?_⟩
filter_upwards [Filter.eventually_ge_atTop 16] with n hn
have hnonneg : 0 ≤ (n : ℝ) ^ 2 * Real.sqrt (n : ℝ) := by positivity
simpa only [Real.norm_eq_abs, abs_of_nonneg (show (0 : ℝ) ≤ (a n : ℝ) by positivity),
abs_of_nonneg hnonneg, mul_assoc] using a_scale_upper hn
· apply Asymptotics.isBigO_iff.mpr
refine ⟨4, ?_⟩
filter_upwards [Filter.eventually_ge_atTop 16] with n hn
have hnonneg : 0 ≤ (n : ℝ) ^ 2 * Real.sqrt (n : ℝ) := by positivity
simp only [Real.norm_eq_abs, abs_of_nonneg (show (0 : ℝ) ≤ (a n : ℝ) by positivity), abs_of_nonneg hnonneg]
have h := a_scale_lower hn
linarith
end Problem335.LowerParameters
end
end OAI
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