File size: 3,335 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 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 | import OAI.Combinatorics.MatrixRemoval.Model
import OAI.Combinatorics.MatrixRemoval.CopyBound
namespace OAI
namespace Problem348
/-- Ordered-copy specialization of the generic constrained-cell bound. -/
theorem copyCount_le_of_cell_mem {k n : ℕ}
(H : BinaryMatrix (k + 1)) (A : BinaryMatrix n)
(i j : Fin (k + 1)) (L : Finset (Fin n × Fin n))
(hL : ∀ rc ∈ orderedCopies H A, (rc.1.val i, rc.2.val j) ∈ L) :
copyCount H A ≤ L.card * n ^ (2 * k) :=
CopyBound.card_le_card_mul_pow (orderedCopies H A) i j L hL
/-- Locate the top-left body entry (zero-based index 64) to count fixed-H copies. -/
theorem fixedCopyCount_le_of_body_cell_mem {n : ℕ}
(A : BinaryMatrix n) (L : Finset (Fin n × Fin n))
(hL : ∀ rc ∈ orderedCopies fixedH A, (rc.1.val 64, rc.2.val 64) ∈ L) :
copyCount fixedH A ≤ L.card * n ^ 130 :=
copyCount_le_of_cell_mem fixedH A 64 64 L hL
/-- The copy bound reduced to concrete leaf localization. -/
theorem fixedCopyDensity_le_of_body_cell_mem {d m : ℕ}
(hd : 0 < d) (hm : 0 < m) (A : BinaryMatrix (d * m))
(L : Finset (Fin (d * m) × Fin (d * m))) (hcard : L.card ≤ m)
(hL : ∀ rc ∈ orderedCopies fixedH A, (rc.1.val 64, rc.2.val 64) ∈ L) :
(copyCount fixedH A : ℝ) / (((d * m : ℕ) : ℝ) ^ 132) ≤
(1 / ((d : ℝ) ^ 2)) * (1 / (m : ℝ)) := by
apply CopyBound.normalized_count_le hd hm
exact (fixedCopyCount_le_of_body_cell_mem A L hL).trans
(Nat.mul_le_mul_right _ hcard)
/-- The exact numerical upper bound appearing in counterexample_sequence. -/
theorem sequenceCopyDensity_le_of_body_cell_mem (h : ℕ)
(A : BinaryMatrix ((386 * h + 2) * 2 ^ h))
(L : Finset
(Fin ((386 * h + 2) * 2 ^ h) × Fin ((386 * h + 2) * 2 ^ h)))
(hcard : L.card ≤ 2 ^ h)
(hL : ∀ rc ∈ orderedCopies fixedH A, (rc.1.val 64, rc.2.val 64) ∈ L) :
(copyCount fixedH A : ℝ) / ((((386 * h + 2) * 2 ^ h : ℕ) : ℝ) ^ 132) ≤
(1 / (((386 * h + 2 : ℕ) : ℝ) ^ 2)) * (1 / ((2 : ℝ) ^ h)) := by
simpa only [Nat.cast_pow, Nat.cast_ofNat] using
fixedCopyDensity_le_of_body_cell_mem (by omega : 0 < 386 * h + 2)
(by positivity : 0 < 2 ^ h) A L hcard hL
/-- A localization witness is enough; no separate leaf-cell set or injectivity
proof is required for the upper bound. Images can only decrease cardinality. -/
theorem sequenceCopyDensity_le_of_leaf_witness (h : ℕ)
(A : BinaryMatrix ((386 * h + 2) * 2 ^ h))
(rowLeaf colLeaf : Fin (2 ^ h) → Fin ((386 * h + 2) * 2 ^ h))
(hloc : ∀ r c : IncreasingMap 66 ((386 * h + 2) * 2 ^ h),
(∀ i j, A (r.val i) (c.val j) = fixedH i j) →
∃ z, r.val 64 = rowLeaf z ∧ c.val 64 = colLeaf z) :
(copyCount fixedH A : ℝ) / ((((386 * h + 2) * 2 ^ h : ℕ) : ℝ) ^ 132) ≤
(1 / (((386 * h + 2 : ℕ) : ℝ) ^ 2)) * (1 / ((2 : ℝ) ^ h)) := by
classical
let L := Finset.univ.image (fun z => (rowLeaf z, colLeaf z))
apply sequenceCopyDensity_le_of_body_cell_mem h A L
· exact Finset.card_image_le.trans (by simp)
· intro rc hrc
have hentries : ∀ i j, A (rc.1.val i) (rc.2.val j) = fixedH i j :=
(Finset.mem_filter.mp hrc).2
obtain ⟨z, hrow, hcol⟩ := hloc rc.1 rc.2 hentries
exact Finset.mem_image.mpr ⟨z, Finset.mem_univ _, by simp [hrow, hcol]⟩
end Problem348
end OAI
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