| import OAI.Combinatorics.BinaryEdit.Model |
| import OAI.Combinatorics.BinaryEdit.FiniteParameters |
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|
| namespace OAI |
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| namespace TreeEdit.BinaryLower |
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| / |
| words. The hypothesis is exactly the output needed from the finite-tree |
| obstruction followed by the fixed-length binary code. -/ |
| theorem binary_lower_bound_of_finite_family |
| (hfamily : ∃ c₀ : ℝ, 0 < c₀ ∧ ∃ K : ℕ, ∀ k : ℕ, K ≤ k → |
| ∃ W : Finset (Word Bool (Parameters.finiteLength k)), |
| 2 ≤ W.card ∧ c₀ * ((k : ℝ) / 4) ^ k ≤ binarySetDistortion W) : |
| ∃ 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 |
| obtain ⟨c₀, hc₀, K, hfamily⟩ := hfamily |
| obtain ⟨d₀, hd₀⟩ := Parameters.exists_finite_cap_parameters hc₀ K |
| refine ⟨1 / 160, by norm_num, d₀, ?_⟩ |
| intro d hd |
| obtain ⟨k, hkK, hk20, hkcap, hkgrowth⟩ := hd₀ d hd |
| obtain ⟨W, hWcard, hWdist⟩ := hfamily k hkK |
| refine ⟨Parameters.finiteLength k, ?_, hkcap, W, hWcard, ?_⟩ |
| · exact Parameters.finiteLength_pos (by omega : 0 < k) |
| · exact hkgrowth.trans hWdist |
|
|
| end TreeEdit.BinaryLower |
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| end OAI |
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|
