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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 42 | 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
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