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f0f05de | 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 | 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
/-- Second inequality of source equation `eq:metric-conversion`. -/
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
/-- Ordinary edit distance and insertion--deletion distance differ by at most
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⟩
/-- Lower inequality of source equation `eq:deficit`, on equal-length words. -/
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
/-- The complete equal-length deficit comparison, including empty words. -/
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⟩
/-- Real-valued form used in the fixed-length binary conversion inequalities. -/
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
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