| import OAI.Combinatorics.BinaryEdit.LCSEditUpper |
| import OAI.Combinatorics.BinaryEdit.EditLCSLower |
|
|
| namespace OAI |
|
|
| /-! |
| Both metric comparisons in the preliminary section, for the actual unrestricted |
| unit-cost edit scripts and longest-common-subsequence definitions. |
| -/ |
| namespace TreeEdit.BinaryLower |
|
|
| / |
| theorem delta_le_twice_editDistance {α : Type*} (x y : List α) : |
| x.length + y.length - 2 * lcs x y ≤ 2 * ed x y := by |
| have hs : EditScript x y (ed x y) := |
| TreeEdit.ed_script x y |
| exact EditScript.lcs_loss_le hs |
|
|
| / |
| factor two, without imposing a cap on intermediate script words. -/ |
| theorem editDistance_delta_bounds {α : Type*} (x y : List α) : |
| ed x y ≤ x.length + y.length - 2 * lcs x y ∧ |
| x.length + y.length - 2 * lcs x y ≤ 2 * ed x y := |
| ⟨editDistance_le_delta x y, delta_le_twice_editDistance x y⟩ |
|
|
| / |
| theorem deficit_le_editDistance {α : Type*} {n : ℕ} {x y : List α} |
| (hx : x.length = n) (hy : y.length = n) : |
| deficit n x y ≤ ed x y := by |
| have h := delta_le_twice_editDistance x y |
| rw [hx, hy] at h |
| unfold deficit |
| omega |
|
|
| / |
| theorem editDistance_deficit_bounds {α : Type*} {n : ℕ} {x y : List α} |
| (hx : x.length = n) (hy : y.length = n) : |
| deficit n x y ≤ ed x y ∧ |
| ed x y ≤ 2 * deficit n x y := |
| ⟨deficit_le_editDistance hx hy, editDistance_le_twice_deficit hx hy⟩ |
|
|
| / |
| theorem editDistance_deficit_bounds_real {α : Type*} {n : ℕ} {x y : List α} |
| (hx : x.length = n) (hy : y.length = n) : |
| (deficit n x y : ℝ) ≤ (ed x y : ℝ) ∧ |
| (ed x y : ℝ) ≤ 2 * (deficit n x y : ℝ) := by |
| constructor |
| · exact_mod_cast deficit_le_editDistance hx hy |
| · exact_mod_cast editDistance_le_twice_deficit hx hy |
|
|
| end TreeEdit.BinaryLower |
|
|
| end OAI |
|
|
