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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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 | import OAI.Combinatorics.EditDistance.UpperRecursion
import OAI.Combinatorics.EditDistance.UpperParameters
import OAI.Combinatorics.EditDistance.CutRealization
import OAI.Combinatorics.EditDistance.DistortionMonotonicity
namespace OAI
/-!
# Alphabet-uniform upper bounds for least distortion
-/
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
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