| 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 |
|
|
