| import OAI.Combinatorics.EditDistance.UpperRecursion |
| import OAI.Combinatorics.EditDistance.UpperParameters |
| import OAI.Combinatorics.EditDistance.CutRealization |
| import OAI.Combinatorics.EditDistance.DistortionMonotonicity |
|
|
| namespace OAI |
|
|
| /-! |
| |
| -/ |
|
|
| namespace EditUpper |
| open EditDistortion EditCutModel |
| universe u |
|
|
| lemma model_E_bound (A : Type u) [Fintype A] [Nonempty A] {d : ℕ} (hd : 1 ≤ d) |
| (M : Model (List A)) {D : ℝ} (hD : 0 ≤ D) |
| (hlo : ∀ x y : List A, x.length ≤ d → y.length ≤ d → (indel x y : ℝ) ≤ M.dist x y) |
| (hup : ∀ x y : List A, M.dist x y ≤ D*(indel x y : ℝ)) : E A d ≤ 2*D := by |
| classical |
| let : Finite (Words A d) := finite_words A d |
| let x : Words A d := ⟨[],by simp⟩ |
| let y : Words A d := ⟨[Classical.arbitrary A],by simpa using hd⟩ |
| let : Nontrivial (Words A d) := ⟨⟨x,y,by simp [x,y]⟩⟩ |
| let f : Words A d ↪ L1 := |
| { toFun := fun z => M.vector z.val |
| inj' := by |
| intro v w he |
| change M.vector v.val = M.vector w.val at he |
| have hh := hlo v.val w.val v.property w.property |
| rw [← M.vector_distance,he,sub_self,norm_zero] at hh |
| have hz : indel v.val w.val = 0 := by exact_mod_cast le_antisymm hh (Nat.cast_nonneg _) |
| exact Subtype.ext ((cost_eq_zero_iff (s := false)).mp hz) } |
| apply (leastDistortion_le _ (fun z => edit_real_diag z.val) |
| (fun v w h => edit_real_pos v.val w.val (fun hh => h (Subtype.ext hh))) f).trans |
| have hbound : distortion (fun v w : Words A d => (edit v.val w.val : ℝ)) f ≤ (2*D)*1 := by |
| apply distortion_le_of_bounds _ |
| (fun v w h => edit_real_pos v.val w.val (fun hh => h (Subtype.ext hh))) f (by positivity) |
| · intro v w |
| change ‖M.vector v.val-M.vector w.val‖ ≤ _ |
| rw [M.vector_distance] |
| have hi : (indel v.val w.val : ℝ) ≤ 2*(edit v.val w.val : ℝ) := by |
| exact_mod_cast indel_le_two_edit v.val w.val |
| calc |
| _ ≤ D*(indel v.val w.val : ℝ) := hup _ _ |
| _ ≤ D*(2*(edit v.val w.val : ℝ)) := mul_le_mul_of_nonneg_left hi hD |
| _ = _ := by ring |
| · intro v w |
| change (edit v.val w.val : ℝ) ≤ 1*‖M.vector v.val-M.vector w.val‖ |
| rw [one_mul,M.vector_distance] |
| exact (show (edit v.val w.val : ℝ) ≤ indel v.val w.val by |
| exact_mod_cast edit_le_indel v.val w.val).trans (hlo _ _ v.property w.property) |
| simpa using hbound |
|
|
| lemma level_E_bound (b N : ℕ) (hb : 1 ≤ b) (A : Type u) [Fintype A] [Nonempty A] |
| {d : ℕ} (hd : 1 ≤ d) (hcap : d ≤ b^N) : |
| E A d ≤ 2*(((b : ℝ)+2)*(growth b N)^N) := by |
| obtain ⟨M,hlo,hup⟩ := exists_level b N hb N le_rfl A |
| have hD : 0 ≤ ((b : ℝ)+2)*(growth b N)^N := by |
| have hh := growth_ge (N := N) hb |
| positivity |
| exact model_E_bound A hd M hD |
| (fun x y hx hy => hlo x y (hx.trans hcap) (hy.trans hcap)) hup |
|
|
| theorem eventual_upper : ∀ᶠ d : ℕ in Filter.atTop, |
| (∀ (A : Type u) [Fintype A], 2 ≤ Fintype.card A → E A d ≤ exponentScale 32 d) ∧ |
| alphabetSup d ≤ exponentScale 32 d := by |
| filter_upwards [EditUpperParameters.eventual_choice,Filter.eventually_ge_atTop (1 : ℕ)] with d hc hd |
| rcases hc with ⟨hN,hb,hcap,hbound⟩ |
| let N := EditUpperParameters.level (Real.log (d : ℝ)) |
| let b := EditUpperParameters.branching (Real.log (d : ℝ)) |
| have hnum : 2*(((b : ℝ)+2)*(growth b N)^N) ≤ exponentScale 32 d := by |
| simpa [growth,exponentScale,b,N,mul_assoc] using hbound |
| constructor |
| · intro A inst hA |
| let : Nonempty A := Fintype.card_pos_iff.mp (by omega) |
| exact (level_E_bound b N (by omega) A hd hcap).trans hnum |
| · apply csSup_le (s := {r : ℝ | ∃ n : ℕ, 2 ≤ n ∧ r = E (Fin n) d}) |
| ⟨E (Fin 2) d,2,by omega,rfl⟩ |
| rintro r ⟨n,hn,rfl⟩ |
| let : NeZero n := ⟨by omega⟩ |
| exact (level_E_bound b N (by omega) (Fin n) hd hcap).trans hnum |
|
|
| end EditUpper |
|
|
| end OAI |
|
|
