| import OAI.Combinatorics.MatrixRemoval.OrderedAnchorRigidity |
| import OAI.Combinatorics.MatrixRemoval.OrderedHostCopy |
|
|
| namespace OAI |
|
|
| /-! |
| # Unconditional copy bound for the canonical host |
|
|
| Concrete anchor rigidity and six-mode body localization imply the integer |
| copy bound and its normalized form. |
| -/ |
|
|
| namespace Problem348.OrderedHost |
|
|
| open Construction |
|
|
| theorem anchor64Pinned (h : ℕ) : |
| Anchor64Pinned (rowIndex h) (columnIndex h) := |
| HostAnchorRigidity.representatives_of_indices |
|
|
| theorem matching_copy_leaf (h : ℕ) (r c : Fin 66 → Position h) |
| (hr : StrictMono (fun i => rowIndex h (r i))) |
| (hc : StrictMono (fun j => columnIndex h (c j))) |
| (hmatch : ∀ i j, host (r i) (c j) = fixedH i j) : |
| ∃ z, r 64 = countedLeaf z ∧ c 64 = countedLeaf z := |
| matching_copy_leaf_of_pinning h (anchor64Pinned h) r c hr hc hmatch |
|
|
| theorem canonical_copyCount_le (h : ℕ) : |
| copyCount fixedH (hostMatrix (rowIndex h) (columnIndex h)) ≤ |
| 2 ^ h * (size h) ^ 130 := |
| copyCount_le_of_pinning h (anchor64Pinned h) |
|
|
| theorem canonical_copyDensity_le (h : ℕ) : |
| (copyCount fixedH (hostMatrix (rowIndex h) (columnIndex h)) : ℝ) / |
| ((size h : ℝ) ^ 132) ≤ |
| (1 / (((386 * h + 2 : ℕ) : ℝ) ^ 2)) * (1 / ((2 : ℝ) ^ h)) := |
| copyDensity_le_of_pinning h (anchor64Pinned h) |
|
|
| end Problem348.OrderedHost |
|
|
| end OAI |
|
|
