| import OAI.Combinatorics.MatrixRemoval.OrderedBodyLocalization |
| import OAI.Combinatorics.MatrixRemoval.PinnedHostCopy |
|
|
| namespace OAI |
|
|
| /-! |
| # Ordered-copy localization and counting |
|
|
| The six-mode body theorem places the distinguished cell at a shared leaf. |
| The count needs only anchor pinning, not a further order certificate. |
| -/ |
|
|
| namespace Problem348.OrderedHost |
|
|
| open Construction |
|
|
| / |
| theorem orderedBodyLocalizes (h : ℕ) : |
| OrderedBodyLocalizes (rowIndex h) (columnIndex h) := by |
| intro t r₀ r₁ c₀ c₁ hr₀ hr₁ hc₀ hc₁ hrow hcol hb |
| exact body_leaf_diagonal hr₀ hr₁ hc₀ hc₁ hrow hcol hb |
|
|
| / |
| theorem matching_copy_leaf_of_pinning (h : ℕ) |
| (hpin : Anchor64Pinned (rowIndex h) (columnIndex h)) |
| (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 := |
| host_raw_localization_of_anchor_pinning (rowIndex h) (columnIndex h) |
| hpin (orderedBodyLocalizes h) r c hr hc hmatch |
|
|
| / |
| theorem copyCount_le_of_pinning (h : ℕ) |
| (hpin : Anchor64Pinned (rowIndex h) (columnIndex h)) : |
| copyCount fixedH (hostMatrix (rowIndex h) (columnIndex h)) ≤ |
| 2 ^ h * (size h) ^ 130 := |
| host_copyCount_le_of_pinning (rowIndex h) (columnIndex h) |
| hpin (orderedBodyLocalizes h) |
|
|
| / |
| theorem copyDensity_le_of_pinning (h : ℕ) |
| (hpin : Anchor64Pinned (rowIndex h) (columnIndex h)) : |
| (copyCount fixedH (hostMatrix (rowIndex h) (columnIndex h)) : ℝ) / |
| ((size h : ℝ) ^ 132) ≤ |
| (1 / (((386 * h + 2 : ℕ) : ℝ) ^ 2)) * (1 / ((2 : ℝ) ^ h)) := |
| host_copyDensity_le_of_pinning h (rowIndex h) (columnIndex h) |
| hpin (orderedBodyLocalizes h) |
|
|
| end Problem348.OrderedHost |
|
|
| end OAI |
|
|
