| import OAI.Combinatorics.MatrixRemoval.CanonicalCopyBound | |
| import OAI.Combinatorics.MatrixRemoval.OrderedRepairDistance | |
| import OAI.Combinatorics.MatrixRemoval.SequenceAssembly | |
| import OAI.Combinatorics.MatrixRemoval.Consequence | |
| namespace OAI | |
| noncomputable section | |
| namespace Problem348 | |
| theorem counterexample_sequence : | |
| ∀ 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)) := by | |
| apply SequenceAssembly.counterexample_sequence_of_integer_bounds | |
| intro h hh | |
| refine ⟨Construction.hostMatrix (OrderedHost.rowIndex h) (OrderedHost.columnIndex h), ?_, | |
| OrderedHost.canonical_copyCount_le h⟩ | |
| intro B hfree | |
| exact OrderedHost.hammingDistance_ge hh B hfree | |
| theorem no_polynomial_removal_bound : | |
| ∀ 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 | |
| exact no_polynomial_removal_bound_of_sequence counterexample_sequence | |
| end Problem348 | |
| end | |
| end OAI | |