| import OAI.Combinatorics.BinaryEdit.FiniteSourceFamily | |
| import OAI.Combinatorics.BinaryEdit.FixedLengthBinaryConversion | |
| namespace OAI | |
| namespace TreeEdit | |
| open BinaryLower | |
| /-- Introduction, `thm:binary`: actual equal-length binary witnesses for every | |
| sufficiently large cap, from the finite-tree family and quadratic block code. -/ | |
| theorem binary_lower_bound : | |
| ∃ c : ℝ, 0 < c ∧ ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d → | |
| ∃ n : ℕ, 1 ≤ n ∧ n ≤ d ∧ | |
| ∃ W : Finset (Word Bool n), 2 ≤ W.card ∧ | |
| growth c d ≤ binarySetDistortion W := by | |
| apply binary_lower_bound_of_finite_source_family ?_ exists_finite_source_family | |
| obtain ⟨M₀, hcode⟩ := fixed_length_binary_conversion.{0} | |
| refine ⟨M₀, ?_⟩ | |
| intro M hM Γ inst hΓ1 hΓM | |
| obtain ⟨c, _, hc⟩ := hcode M hM Γ hΓ1 hΓM | |
| exact ⟨c, fun x y => (hc x y).1⟩ | |
| end TreeEdit | |
| end OAI | |