| import OAI.NumberTheory.TwoPoint.Walks.ForestCode |
|
|
| namespace OAI |
|
|
| /-! |
| # Counting the finite column-code universe |
|
|
| The shape is a typed plane forest on at most `N` vertices. Binary marks |
| record boundaries and run types. Omitted runs and imperfect positions |
| store representative indices; regular segments store their two endpoints. |
| -/ |
|
|
| namespace TwoPointCorrelations |
| |
| instance boundedForestFinite (N : ℕ) : Finite {t : BinaryTree Bool // t.numNodes ≤ N} := |
| Finite.of_injective (boundedForestCode N) (boundedForestCode_injective N) |
| |
| noncomputable instance boundedForestFintype (N : ℕ) : |
| Fintype {t : BinaryTree Bool // t.numNodes ≤ N} := Fintype.ofFinite _ |
| |
| / |
| abbrev ColumnCode (N segments omitted imperfect : ℕ) := |
| {t : BinaryTree Bool // t.numNodes ≤ N} × |
| (Fin (2 * N) → Bool) × (Fin segments → Fin N × Fin N) × |
| (Fin omitted → Fin N) × (Fin imperfect → Fin N) |
| |
| / |
| theorem card_columnCode_le (N segments omitted imperfect : ℕ) : |
| Fintype.card (ColumnCode N segments omitted imperfect) ≤ |
| 2 ^ (5 * N + 1) * N ^ (2 * segments + omitted + imperfect) := by |
| have hshape : Fintype.card {t : BinaryTree Bool // t.numNodes ≤ N} ≤ 2 ^ (3 * N + 1) := by |
| simpa only [Nat.card_eq_fintype_card] using card_bounded_forests N |
| simp only [ColumnCode, Fintype.card_prod, Fintype.card_fun, Fintype.card_fin, |
| Fintype.card_bool] |
| calc |
| _ ≤ 2 ^ (3 * N + 1) * |
| (2 ^ (2 * N) * ((N * N) ^ segments * (N ^ omitted * N ^ imperfect))) := by |
| exact Nat.mul_le_mul_right _ hshape |
| _ = _ := by |
| rw [mul_pow] |
| simp only [← pow_add] |
| ring_nf |
| |
| / |
| no numerical coefficients present in the finite counting universe. -/ |
| theorem patterns_le_columnCode {Pattern : Type*} [Fintype Pattern] |
| (N segments omitted imperfect : ℕ) |
| (encode : Pattern → ColumnCode N segments omitted imperfect) |
| (decode : ColumnCode N segments omitted imperfect → Pattern) |
| (hdecode : ∀ p, decode (encode p) = p) : |
| Fintype.card Pattern ≤ 2 ^ (5 * N + 1) * N ^ (2 * segments + omitted + imperfect) := by |
| apply (Fintype.card_le_of_injective encode ?_).trans (card_columnCode_le _ _ _ _) |
| intro p q heq |
| have h := congrArg decode heq |
| simpa only [hdecode] using h |
| |
| end TwoPointCorrelations |
| |
| end OAI |
| |
