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e320e82 | 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 47 48 49 50 51 52 53 54 55 56 57 58 59 60 | 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
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