| import OAI.Combinatorics.MatrixRemoval.OrderedSampledSelection |
| import OAI.Combinatorics.MatrixRemoval.HostRepairDistance |
|
|
| namespace OAI |
|
|
| /-! |
| # Unconditional repair lower bound for the ordered canonical host |
|
|
| The sampler order certificate follows from the independent increasing |
| row and column enumerations, giving the arbitrary-repair lower bound. |
| -/ |
|
|
| noncomputable section |
| namespace Problem348.OrderedHost |
|
|
| open Construction |
|
|
| / |
| cells. In particular no pinning premise or preservation condition is assumed. -/ |
| theorem hammingDistance_ge {h : ℕ} (hh : 1 ≤ h) |
| (B : BinaryMatrix (size h)) (hfree : HFree fixedH B) : |
| (2 ^ h) ^ 2 ≤ |
| hammingDistance (SelectedAnchor.matrix (rowIndex h) (columnIndex h)) B := |
| SampledSelection.hammingDistance_ge hh (rowIndex h) (columnIndex h) |
| (sampledAxisConditions h) B hfree |
|
|
| / |
| theorem sampledSelection_guard_frequency {h : ℕ} (hh : 1 ≤ h) |
| (rc : Fin (size h) × Fin (size h)) : |
| (Finset.univ.filter fun s : TreeSampling.Seeds (2 ^ h) => |
| rc ∈ (sampledSelection hh s.1 s.2.1 s.2.2).guards h).card * (2 ^ h) ^ 2 ≤ |
| Fintype.card (TreeSampling.Seeds (2 ^ h)) := |
| SampledSelection.selection_guard_frequency hh (rowIndex h) (columnIndex h) |
| (sampledAxisConditions h) rc |
|
|
| end Problem348.OrderedHost |
|
|
| end |
|
|
| end OAI |
| |
