| import Mathlib |
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| namespace OAI |
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| /-! |
| Analytic conclusion of the ordered-matrix construction. |
| The polynomial/geometric comparison uses |
| `tendsto_pow_const_div_const_pow_of_one_lt`. |
| -/ |
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| noncomputable section |
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| namespace Problem348 |
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| open Filter |
| open scoped Topology |
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| / |
| 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 |
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| end Problem348 |
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| end |
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| end OAI |
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