| import OAI.Combinatorics.MatrixRemoval.Model |
| import OAI.Combinatorics.MatrixRemoval.CopyBound |
|
|
| namespace OAI |
|
|
| namespace Problem348 |
|
|
| / |
| 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 |
|
|
| / |
| 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 |
|
|
| / |
| 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) |
|
|
| / |
| 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 |
|
|
| / |
| 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 |
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| end OAI |
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