| import OAI.Combinatorics.MatrixRemoval.Model |
| import OAI.Combinatorics.MatrixRemoval.Asymptotics |
|
|
| namespace OAI |
|
|
| /-! |
| The sequence of matrix counterexamples rules out every polynomial |
| removal bound. |
| -/ |
|
|
| noncomputable section |
|
|
| namespace Problem348 |
|
|
| / |
| theorem no_polynomial_removal_bound_of_sequence |
| (hseq : ∀ h : ℕ, 1 ≤ h → |
| let d : ℕ := 386 * h + 2 |
| let n : ℕ := d * 2 ^ h |
| ∃ A : BinaryMatrix n, |
| fixedDistance A ≥ 1 / ((d : ℝ) ^ 2) ∧ |
| (copyCount fixedH A : ℝ) / ((n : ℝ) ^ 132) ≤ |
| (1 / ((d : ℝ) ^ 2)) * (1 / ((2 : ℝ) ^ h))) : |
| ∀ c C : ℝ, 0 < c → 0 < C → |
| ∃ n : ℕ, 1 ≤ n ∧ |
| ∃ ε : ℝ, 0 < ε ∧ ε < 1 ∧ |
| ∃ A : BinaryMatrix n, |
| fixedDistance A ≥ ε ∧ |
| (copyCount fixedH A : ℝ) < |
| c * Real.rpow ε C * ((n : ℝ) ^ 132) := by |
| intro c C hc _hC |
| obtain ⟨h, hh, hdecay⟩ := exists_depth_polynomial_decay c C hc |
| let d : ℕ := 386 * h + 2 |
| let n : ℕ := d * 2 ^ h |
| have hd : 1 < d := by dsimp [d]; omega |
| have hdR : (1 : ℝ) < d := by exact_mod_cast hd |
| have hdpos : (0 : ℝ) < d := by linarith |
| have hn : 0 < n := Nat.mul_pos (by omega) (pow_pos (by decide) _) |
| have hnR : (0 : ℝ) < n := by exact_mod_cast hn |
| obtain ⟨A, hdist, hcopies⟩ := hseq h hh |
| refine ⟨n, hn, 1 / ((d : ℝ) ^ 2), by positivity, ?_, A, hdist, ?_⟩ |
| · apply (div_lt_one (by positivity)).2 |
| nlinarith |
| · have hdecay' : (1 / ((d : ℝ) ^ 2)) * (1 / ((2 : ℝ) ^ h)) < |
| c * Real.rpow (1 / ((d : ℝ) ^ 2)) C := by |
| simpa [d] using hdecay |
| exact (div_lt_iff₀ (pow_pos hnR 132)).mp (hcopies.trans_lt hdecay') |
|
|
| end Problem348 |
|
|
| end |
|
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| end OAI |
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|
