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f0f05de | 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 | import OAI.Combinatorics.BinaryEdit.CodeNumerics
import OAI.Combinatorics.BinaryEdit.SegmentCodeLCS
namespace OAI
/-! Closing the probabilistic segment-code construction at the exact source
parameters: block width M² and cutoff ceil(120 log₂ M). -/
namespace TreeEdit.BinaryLower.Code
universe u
/-- A single binary codebook separates all long segments of unequal labels.
The alphabet can be arbitrary finite; its size is only bounded by M. -/
theorem exists_fixed_width_separated_code {M : ℕ} (hM : 2 ≤ M)
(Γ : Type u) [Fintype Γ] (hΓ : Fintype.card Γ ≤ M) :
∃ c : Γ → Word Bool (M ^ 2),
∀ (k l : Γ), k ≠ l → ∀ (a i j : ℕ), segmentThreshold M ≤ a →
i + a ≤ M ^ 2 → j + a ≤ M ^ 2 →
100 * lcs (((c k).val.drop i).take a)
(((c l).val.drop j).take a) ≤ 99 * a := by
apply SegmentCode.exists_lcs_separated_code Γ (M ^ 2) (segmentThreshold M)
simpa only [Nat.cast_add, Nat.cast_pow, Nat.cast_one] using
code_union_bound_for_alphabet hΓ hM
/-- There is an absolute threshold at which the actual codebook and the
short-pair error budget needed by the deterministic proof coexist. -/
theorem exists_binary_code_with_error_budget :
∃ M₀ : ℕ, ∀ M : ℕ, M₀ ≤ M →
2 ≤ M ∧
2 * (M : ℝ) * segmentThreshold M ≤ (M : ℝ) ^ 2 / 2 ∧
∀ (Γ : Type u) [Fintype Γ], Fintype.card Γ ≤ M →
∃ c : Γ → Word Bool (M ^ 2),
∀ (k l : Γ), k ≠ l → ∀ (a i j : ℕ), segmentThreshold M ≤ a →
i + a ≤ M ^ 2 → j + a ≤ M ^ 2 →
100 * lcs (((c k).val.drop i).take a)
(((c l).val.drop j).take a) ≤ 99 * a := by
obtain ⟨M₀, hM₀⟩ := exists_code_numerical_threshold
refine ⟨M₀, fun M hM => ?_⟩
obtain ⟨hM2, herr, _⟩ := hM₀ M hM
exact ⟨hM2, herr, fun Γ _ hΓ => exists_fixed_width_separated_code hM2 Γ hΓ⟩
end TreeEdit.BinaryLower.Code
end OAI
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