| import OAI.Combinatorics.EditDistance.DepthInversion |
| import OAI.Combinatorics.EditDistance.FiniteLower |
| import OAI.Combinatorics.EditDistance.DistortionMonotonicity |
|
|
| namespace OAI |
|
|
| /-! |
| # Binary witnesses for every sufficiently large length bound |
| -/ |
|
|
| namespace EditLower |
|
|
| universe u |
|
|
| open EditDistortion EditParameters Filter |
|
|
| theorem eventually_length_witness : ∀ᶠ d : ℕ in atTop, |
| ∃ n : ℕ, n ≤ d ∧ ∃ S : Set (List Bool), S.Finite ∧ S.Nonempty ∧ S.Nontrivial ∧ |
| (∀ w ∈ S, w.length = n) ∧ |
| exponentScale (1/1200) d ≤ leastDistortion (fun x y : S => (edit x.val y.val : ℝ)) := by |
| obtain ⟨K,hK⟩ := eventually_atTop.mp eventually_depth_witness |
| let J := max (max K (24^4)) 2 |
| have hlim : Tendsto (fun d : ℕ => Real.log (d : ℝ)) atTop atTop := |
| Real.tendsto_log_atTop.comp tendsto_natCast_atTop_atTop |
| have hchoose := hlim.eventually (EditDepth.eventually_choose_depth J) |
| filter_upwards [hchoose,eventually_ge_atTop 1] with d hd hd1 |
| let k := EditDepth.depth (Real.log (d : ℝ)) |
| have hk : J ≤ k := hd.1 |
| have hkK : K ≤ k := (le_max_left K (24^4)).trans ((le_max_left _ _).trans hk) |
| have hk24 : 24^4 ≤ k := (le_max_right K (24^4)).trans ((le_max_left _ _).trans hk) |
| have hk2 : 2 ≤ k := (le_max_right _ _).trans hk |
| have hk0 : 0 < k := by omega |
| obtain ⟨S,hfin,hne,hnt,hlen,hD⟩ := hK k hkK |
| have hn : encodedLength k ≤ d := by |
| have hlog : Real.log (encodedLength k : ℝ) ≤ Real.log (d : ℝ) := |
| (log_length_bound hk2).trans hd.2.1 |
| have hnp : (0 : ℝ) < encodedLength k := by exact_mod_cast encodedLength_pos hk0 |
| have hdp : (0 : ℝ) < d := by exact_mod_cast hd1 |
| exact_mod_cast (Real.log_le_log_iff hnp hdp).mp hlog |
| refine ⟨encodedLength k,hn,S,hfin,hne,hnt,hlen,?_⟩ |
| apply le_trans ?_ ((ratio_exponential hk24).trans hD) |
| apply Real.exp_le_exp.mpr |
| simpa only [div_eq_mul_inv,mul_comm,one_mul,k] using hd.2.2 |
|
|
| theorem lower_main : ∃ c : ℝ, 0 < c ∧ ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d → |
| (∀ (Alpha : Type u) [Fintype Alpha], 2 ≤ Fintype.card Alpha → |
| exponentScale c d ≤ E Alpha d) ∧ BinaryWitness c d := by |
| obtain ⟨d₀,hd₀⟩ := eventually_atTop.mp eventually_length_witness |
| refine ⟨1/1200,by norm_num,d₀,?_⟩ |
| intro d hd |
| obtain ⟨n,hn,S,hfin,hne,hnt,hlen,hD⟩ := hd₀ d hd |
| constructor |
| · intro Alpha inst hAlpha |
| exact hD.trans (binary_subset_lower Alpha hAlpha hn S hfin hnt hlen) |
| · exact ⟨n,hn,S,hfin,hne,hlen,hD⟩ |
|
|
| end EditLower |
|
|
| end OAI |
|
|
