File size: 2,766 Bytes
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 | import OAI.Combinatorics.EditDistance.WindowRecursion
namespace OAI
/-!
# Base finite-cut embedding for short words
-/
namespace EditUpper
attribute [local instance] Classical.propDecidable
open scoped BigOperators
open Finset EditDistortion EditWindow EditCutModel
variable {A : Type*} [Fintype A]
noncomputable def firstKey (x : List A) : Fin (Fintype.card (Option A)) :=
Fintype.equivFin (Option A) (x[0]?)
noncomputable def base : Model (List A) where
Index := Fin (Fintype.card (Option A))
finiteIndex := inferInstance
weight _ := 1
nonneg _ := by norm_num
cut i x := decide (firstKey x = i)
lemma base_distance (x y : List A) :
(base (A := A)).dist x y = 2*EditCutHash.sep (x[0]?) (y[0]?) := by
have hcuts (i : Fin (Fintype.card (Option A))) :
sep (decide (firstKey x = i)) (decide (firstKey y = i)) =
|EditCutHash.hit (firstKey x) i-EditCutHash.hit (firstKey y) i| := by
by_cases hx : firstKey x = i <;> by_cases hy : firstKey y = i <;>
simp [sep,EditCutHash.hit,hx,hy]
change (∑ i, 1*sep (decide (firstKey x = i)) (decide (firstKey y = i))) = _
simp_rw [one_mul,hcuts]
rw [EditCutHash.hit_distance]
congr 1
have he : firstKey x = firstKey y ↔ x[0]? = y[0]? := (Fintype.equivFin (Option A)).injective.eq_iff
by_cases hxy : x[0]? = y[0]? <;> simp only [EditCutHash.sep,he,hxy,ite_true,ite_false]
lemma base_le_two (x y : List A) : (base (A := A)).dist x y ≤ 2 := by
rw [base_distance]
have h := EditCutHash.sep_le_one (x[0]?) (y[0]?)
linarith
omit [Fintype A] in
lemma first_eq_of_small (x y : List A) (hx : x.length ≤ 1) (hy : y.length ≤ 1)
(he : x[0]? = y[0]?) : x = y := by
cases x with
| nil => cases y <;> simp_all
| cons a x =>
have hxn : x = [] := List.eq_nil_iff_length_eq_zero.mpr (by simp only [List.length_cons] at hx; omega)
subst x
cases y with
| nil => simp at he
| cons b y =>
have hyn : y = [] := List.eq_nil_iff_length_eq_zero.mpr (by simp only [List.length_cons] at hy; omega)
subst y
simpa using he
lemma base_lower (x y : List A) (hx : x.length ≤ 1) (hy : y.length ≤ 1) :
(indel x y : ℝ) ≤ (base (A := A)).dist x y := by
by_cases he : x[0]? = y[0]?
· have hxy := first_eq_of_small x y hx hy he
subst y
rw [Model.dist_self]
norm_cast
exact cost_le (.nil _)
· rw [base_distance,EditCutHash.sep_eq_one he,mul_one]
have hlen : indel x y ≤ x.length+y.length :=
cost_le ((Script.to_nil (s := false) x).trans (Script.to_nil y).symm)
exact_mod_cast (show indel x y ≤ 2 by omega)
lemma base_upper (x y : List A) : (base (A := A)).dist x y ≤ 2*(indel x y : ℝ) :=
model_indel_lipschitz base (fun p q a => base_le_two (p++q) (p++a::q)) x y
end EditUpper
end OAI
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