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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 | 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
/-- Every fixedH-free repair of the actual ordered host changes at least m²
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
/-- The concrete three-seed sampler has the exact uniform guard bound. -/
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
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