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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 | 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
/-- Exponential decay of the copy density rules out polynomial removal bounds. -/
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
end OAI
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