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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 | import OAI.Combinatorics.BinaryEdit.Model
import OAI.Combinatorics.BinaryEdit.FiniteParameters
namespace OAI
namespace TreeEdit.BinaryLower
/-- Apply the numerical cap interpolation to actual finite sets of binary
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
end OAI
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