English
The asymptotic bound combines p(a,b) power and sumTransform contributions to bound growth.
Русский
Асимптотическое ограничение сочетает степень p(a,b) и вклад sumTransform для границы роста.
LaTeX
$$$\text{asympBound}(g,a,b;n) = n^{p(a,b)} + \text{sumTransform}(p(a,b),g;0,n)$$$
Lean4
theorem eventually_atTop_sumTransform_le :
∃ c > 0, ∀ᶠ (n : ℕ) in atTop, ∀ i, sumTransform (p a b) g (r i n) n ≤ c * g n :=
by
obtain ⟨c₁, hc₁_mem, hc₁⟩ := R.exists_eventually_const_mul_le_r
obtain ⟨c₂, hc₂_mem, hc₂⟩ := R.g_grows_poly.eventually_atTop_le_nat hc₁_mem
have hc₁_pos : 0 < c₁ := hc₁_mem.1
refine ⟨max c₂ (c₂ / c₁ ^ (p a b + 1)), by positivity, ?_⟩
filter_upwards [hc₁, hc₂, R.eventually_r_pos, R.eventually_r_lt_n, eventually_gt_atTop 0] with n hn₁ hn₂ hrpos hr_lt_n
hn_pos i
have hrpos_i := hrpos i
have g_nonneg : 0 ≤ g n := R.g_nonneg n (by positivity)
cases le_or_gt 0 (p a b + 1) with
| inl hp => -- 0 ≤ p a b + 1
calc
sumTransform (p a b) g (r i n) n = n ^ (p a b) * (∑ u ∈ Finset.Ico (r i n) n, g u / u ^ ((p a b) + 1)) := by rfl
_ ≤ n ^ (p a b) * (∑ u ∈ Finset.Ico (r i n) n, c₂ * g n / u ^ ((p a b) + 1)) :=
by
gcongr with u hu
rw [Finset.mem_Ico] at hu
have hu' : u ∈ Set.Icc (r i n) n := ⟨hu.1, by cutsat⟩
refine hn₂ u ?_
rw [Set.mem_Icc]
refine ⟨?_, by norm_cast; cutsat⟩
calc
c₁ * n ≤ r i n := by exact hn₁ i
_ ≤ u := by exact_mod_cast hu'.1
_ ≤ n ^ (p a b) * (∑ _u ∈ Finset.Ico (r i n) n, c₂ * g n / (r i n) ^ ((p a b) + 1)) := by gcongr with u hu;
rw [Finset.mem_Ico] at hu; exact hu.1
_ ≤ n ^ p a b * #(Ico (r i n) n) • (c₂ * g n / r i n ^ (p a b + 1)) := by gcongr;
exact Finset.sum_le_card_nsmul _ _ _ (fun x _ => by rfl)
_ = n ^ p a b * #(Ico (r i n) n) * (c₂ * g n / r i n ^ (p a b + 1)) := by rw [nsmul_eq_mul, mul_assoc]
_ = n ^ (p a b) * (n - r i n) * (c₂ * g n / (r i n) ^ ((p a b) + 1)) := by congr;
rw [Nat.card_Ico, Nat.cast_sub (le_of_lt <| hr_lt_n i)]
_ ≤ n ^ (p a b) * n * (c₂ * g n / (r i n) ^ ((p a b) + 1)) := by gcongr;
simp only [tsub_le_iff_right, le_add_iff_nonneg_right, Nat.cast_nonneg]
_ ≤ n ^ (p a b) * n * (c₂ * g n / (c₁ * n) ^ ((p a b) + 1)) := by gcongr; exact hn₁ i
_ = c₂ * g n * n ^ ((p a b) + 1) / (c₁ * n) ^ ((p a b) + 1) := by
rw [← Real.rpow_add_one (by positivity) (p a b)]; ring
_ = c₂ * g n * n ^ ((p a b) + 1) / (n ^ ((p a b) + 1) * c₁ ^ ((p a b) + 1)) := by
rw [mul_comm c₁, Real.mul_rpow (by positivity) (by positivity)]
_ = c₂ * g n * (n ^ ((p a b) + 1) / (n ^ ((p a b) + 1))) / c₁ ^ ((p a b) + 1) := by ring
_ = c₂ * g n / c₁ ^ ((p a b) + 1) := by rw [div_self (by positivity), mul_one]
_ = (c₂ / c₁ ^ ((p a b) + 1)) * g n := by ring
_ ≤ max c₂ (c₂ / c₁ ^ ((p a b) + 1)) * g n := by gcongr; exact le_max_right _ _
| inr hp => -- p a b + 1 < 0
calc
sumTransform (p a b) g (r i n) n
_ = n ^ (p a b) * (∑ u ∈ Finset.Ico (r i n) n, g u / u ^ ((p a b) + 1)) := by rfl
_ ≤ n ^ (p a b) * (∑ u ∈ Finset.Ico (r i n) n, c₂ * g n / u ^ ((p a b) + 1)) :=
by
gcongr with u hu
rw [Finset.mem_Ico] at hu
have hu' : u ∈ Set.Icc (r i n) n := ⟨hu.1, by cutsat⟩
refine hn₂ u ?_
rw [Set.mem_Icc]
refine ⟨?_, by norm_cast; cutsat⟩
calc
c₁ * n ≤ r i n := by exact hn₁ i
_ ≤ u := by exact_mod_cast hu'.1
_ ≤ n ^ (p a b) * (∑ _u ∈ Finset.Ico (r i n) n, c₂ * g n / n ^ ((p a b) + 1)) :=
by
gcongr n ^ (p a b) * (Finset.Ico (r i n) n).sum (fun _ => c₂ * g n / ?_) with u hu
rw [Finset.mem_Ico] at hu
have : 0 < u :=
calc
0 < r i n := by exact hrpos_i
_ ≤ u := by exact hu.1
exact rpow_le_rpow_of_exponent_nonpos (by positivity) (by exact_mod_cast (le_of_lt hu.2)) (le_of_lt hp)
_ ≤ n ^ p a b * #(Ico (r i n) n) • (c₂ * g n / n ^ (p a b + 1)) := by gcongr;
exact Finset.sum_le_card_nsmul _ _ _ (fun x _ => by rfl)
_ = n ^ p a b * #(Ico (r i n) n) * (c₂ * g n / n ^ (p a b + 1)) := by rw [nsmul_eq_mul, mul_assoc]
_ = n ^ (p a b) * (n - r i n) * (c₂ * g n / n ^ ((p a b) + 1)) := by congr;
rw [Nat.card_Ico, Nat.cast_sub (le_of_lt <| hr_lt_n i)]
_ ≤ n ^ (p a b) * n * (c₂ * g n / n ^ ((p a b) + 1)) := by gcongr;
simp only [tsub_le_iff_right, le_add_iff_nonneg_right, Nat.cast_nonneg]
_ = c₂ * (n ^ ((p a b) + 1) / n ^ ((p a b) + 1)) * g n := by rw [← Real.rpow_add_one (by positivity) (p a b)];
ring
_ = c₂ * g n := by rw [div_self (by positivity), mul_one]
_ ≤ max c₂ (c₂ / c₁ ^ ((p a b) + 1)) * g n := by gcongr; exact le_max_left _ _
