| import OAI.Combinatorics.BinaryEdit.Model |
| import OAI.Combinatorics.BinaryEdit.SubsequenceIntersection |
| import Mathlib.Tactic |
|
|
| namespace OAI |
|
|
| namespace TreeEdit.BinaryLower |
|
|
| /-- Every elementary edit preserves a common subsequence after at most two |
| unmatched symbols in total. This includes substitutions by the same letter. -/ |
| theorem EditStep.common_subsequence {α : Type*} {x y : List α} |
| (h : EditStep x y) : |
| ∃ z : List α, z.Sublist x ∧ z.Sublist y ∧ |
| x.length + y.length ≤ 2 + 2 * z.length := by |
| cases h with |
| | insert p s a => |
| refine ⟨p ++ s, List.Sublist.refl _, ?_, ?_⟩ |
| · exact (List.Sublist.refl p).append ((List.Sublist.refl s).cons a) |
| · simp only [List.length_append, List.length_cons] |
| omega |
| | delete p s a => |
| refine ⟨p ++ s, ?_, List.Sublist.refl _, ?_⟩ |
| · exact (List.Sublist.refl p).append ((List.Sublist.refl s).cons a) |
| · simp only [List.length_append, List.length_cons] |
| omega |
| | substitute p s a b => |
| refine ⟨p ++ s, ?_, ?_, ?_⟩ |
| · exact (List.Sublist.refl p).append ((List.Sublist.refl s).cons a) |
| · exact (List.Sublist.refl p).append ((List.Sublist.refl s).cons b) |
| · simp only [List.length_append, List.length_cons] |
| omega |
|
|
| /-- A script of cost `n` has a common subsequence with at most `2*n` unmatched |
| symbols in its two endpoints. Intermediate words are unrestricted. -/ |
| theorem EditScript.common_subsequence {α : Type*} {x y : List α} {n : ℕ} |
| (h : EditScript x y n) : |
| ∃ z : List α, z.Sublist x ∧ z.Sublist y ∧ |
| x.length + y.length ≤ 2 * n + 2 * z.length := by |
| induction h with |
| | nil x => |
| exact ⟨x, List.Sublist.refl _, List.Sublist.refl _, by omega⟩ |
| | cons hstep hscript ih => |
| obtain ⟨z, hzx, hzy, hlen⟩ := |
| common_subsequence_compose (EditStep.common_subsequence hstep) ih |
| exact ⟨z, hzx, hzy, by omega⟩ |
|
|
| /-- The insertion/deletion deficit of a script's endpoints is at most twice |
| its ordinary unit-edit cost. -/ |
| theorem EditScript.lcs_loss_le {α : Type*} {x y : List α} {n : ℕ} |
| (h : EditScript x y n) : |
| x.length + y.length - 2 * lcs x y ≤ 2 * n := by |
| obtain ⟨z, hzx, hzy, hlen⟩ := EditScript.common_subsequence h |
| have hz : z.length ≤ lcs x y := by |
| apply le_csSup |
| · refine ⟨x.length, ?_⟩ |
| rintro k ⟨w, hwx, hwy, rfl⟩ |
| exact hwx.length_le |
| · exact ⟨z, hzx, hzy, rfl⟩ |
| omega |
|
|
| end TreeEdit.BinaryLower |
|
|
| end OAI |
|
|
