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e5a97c6 | 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 | import Mathlib
namespace OAI
/-!
Analytic conclusion of the ordered-matrix construction.
The polynomial/geometric comparison uses
`tendsto_pow_const_div_const_pow_of_one_lt`.
-/
noncomputable section
namespace Problem348
open Filter
open scoped Topology
/-- Exponential decay beats the inverse-polynomial distance parameter. -/
theorem exists_depth_polynomial_decay (c C : ℝ) (hc : 0 < c) :
∃ h : ℕ, 1 ≤ h ∧
(1 / ((386 * (h : ℝ) + 2) ^ 2)) * (1 / ((2 : ℝ) ^ h)) <
c * Real.rpow (1 / ((386 * (h : ℝ) + 2) ^ 2)) C := by
obtain ⟨k, hk⟩ := exists_nat_ge C
have ht := (tendsto_pow_const_div_const_pow_of_one_lt (2 * k)
(by norm_num : (1 : ℝ) < 2)).const_mul ((388 : ℝ) ^ (2 * k))
have hev : ∀ᶠ h : ℕ in atTop,
(388 : ℝ) ^ (2 * k) * ((h : ℝ) ^ (2 * k) / (2 : ℝ) ^ h) < c := by
exact ht.eventually (gt_mem_nhds (by simpa using hc))
obtain ⟨h, hh, hsmall⟩ := (eventually_ge_atTop 1 |>.and hev).exists
refine ⟨h, hh, ?_⟩
have hhR : (1 : ℝ) ≤ h := by exact_mod_cast hh
have hd : 0 < 386 * (h : ℝ) + 2 := by positivity
have hd1 : (1 : ℝ) < 386 * (h : ℝ) + 2 := by linarith
have htwo : 0 < (2 : ℝ) ^ h := by positivity
have hpoly : (386 * (h : ℝ) + 2) ^ (2 * k) ≤
(388 : ℝ) ^ (2 * k) * (h : ℝ) ^ (2 * k) := by
rw [← mul_pow]
exact pow_le_pow_left₀ hd.le (by linarith) _
have hratio : (386 * (h : ℝ) + 2) ^ (2 * k) / (2 : ℝ) ^ h < c := by
calc
_ ≤ ((388 : ℝ) ^ (2 * k) * (h : ℝ) ^ (2 * k)) / (2 : ℝ) ^ h :=
div_le_div_of_nonneg_right hpoly htwo.le
_ = (388 : ℝ) ^ (2 * k) * ((h : ℝ) ^ (2 * k) / (2 : ℝ) ^ h) := by ring
_ < c := hsmall
have hepspos : 0 < 1 / ((386 * (h : ℝ) + 2) ^ 2) := by positivity
have hepsle : 1 / ((386 * (h : ℝ) + 2) ^ 2) ≤ 1 := by
apply (div_le_one (by positivity)).2
nlinarith
have hexponent : (1 / ((386 * (h : ℝ) + 2) ^ 2)) ^ k ≤
Real.rpow (1 / ((386 * (h : ℝ) + 2) ^ 2)) C := by
simpa using Real.rpow_le_rpow_of_exponent_ge hepspos hepsle hk
have hsmall' : 1 / ((2 : ℝ) ^ h) <
c * (1 / ((386 * (h : ℝ) + 2) ^ 2)) ^ k := by
rw [div_pow, one_pow, ← pow_mul, mul_one_div]
apply (lt_div_iff₀ (pow_pos hd (2 * k))).2
simpa [div_eq_mul_inv, mul_comm] using hratio
calc
_ ≤ 1 / ((2 : ℝ) ^ h) := by
exact mul_le_of_le_one_left (by positivity) hepsle
_ < c * (1 / ((386 * (h : ℝ) + 2) ^ 2)) ^ k := hsmall'
_ ≤ c * Real.rpow (1 / ((386 * (h : ℝ) + 2) ^ 2)) C :=
mul_le_mul_of_nonneg_left hexponent hc.le
end Problem348
end
end OAI
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