File size: 2,102 Bytes
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 43 44 45 46 47 48 49 50 51 52 53 54 | 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
/-- The six-mode theorem supplies the exact body-localization counting input. -/
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
/-- Every matching ordered copy has its top-left body cell at a shared leaf. -/
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
/-- Integer copy bound for the actual two-axis enumeration. -/
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)
/-- The normalized copy bound, conditional only on anchor pinning. -/
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
|