終止檢查失敗

Question

我試圖在這個 kata 上訓練一個列表的最長公共子序列，我稍微修改了一下，以便它可以與我的 agda 版本和標准庫（Agda 2.6.2，stdlib 1.7）一起使用，這導致了這段代碼

{-# OPTIONS --safe #-}
module pg where

open import Data.List
open import Data.Nat
open import Data.Product
open import Relation.Nullary
open import Relation.Binary.PropositionalEquality
open import Relation.Binary

data Subseq {n} { A : Set n } : List A → List A → Set where
subseq-nil : Subseq [] []
subseq-take : ∀ a xs ys → Subseq xs ys → Subseq (a ∷ xs) (a ∷ ys)
subseq-drop : ∀ a xs ys → Subseq xs ys → Subseq xs (a ∷ ys)

is-lcs : ∀ {n} {A : Set n} → List A → List A → List A → Set n
is-lcs zs xs ys =
(Subseq zs xs × Subseq zs ys) ×
(∀ ts → Subseq ts xs → Subseq ts ys → length ts ≤ length zs)

longest : ∀ {n} {A : Set n} → List A → List A → List A
longest s1 s2 with length s1 ≤? length s2
... | yes _ = s2
... | no _ = s1

lcs : ∀ {n} {A : Set n} → Decidable {A = A} _≡_ → List A → List A → List A
lcs _ [] _ = []
lcs _ _ [] = []
lcs dec (x ∷ xs) (y ∷ ys) with dec x y
... | yes _ = x ∷ lcs dec xs ys
... | no _ = longest (lcs dec (x ∷ xs) ys) (lcs dec xs (y ∷ ys))

Unfort.netaly，Agda 無法識別 lcs 是一個終止 function，老實說我不明白：遞歸調用是在結構較小的 arguments 上進行的，如果我沒看錯的話？ 如果有人能向我解釋這里的問題是什么，那將大有幫助。 提前致謝 ！

Answer 1

在處理終止時盡量避免使用with 。 當您的代碼重構如下時，終止檢查器成功（請注意，我刪除了您的前兩個定義，因為它們與您的問題無關，也許您應該相應地對其進行編輯）：

{-# OPTIONS --safe #-}

open import Data.List
open import Data.Nat
open import Relation.Nullary
open import Relation.Binary.PropositionalEquality
open import Relation.Binary
open import Data.Bool using (if_then_else_)

module Term where

longest : ∀ {a} {A : Set a} → List A → List A → List A
longest s1 s2 with length s1 ≤? length s2
... | yes _ = s2
... | no _ = s1

lcs : ∀ {a} {A : Set a} → Decidable {A = A} _≡_ → List A → List A → List A
lcs _ [] _ = []
lcs _ _ [] = []
lcs dec (x ∷ xs) (y ∷ ys) = if does (dec x y) then
  x ∷ lcs dec xs ys else
  longest (lcs dec (x ∷ xs) ys) (lcs dec xs (y ∷ ys))

顯然，這是with抽象結合終止檢查器的已知限制，如 wiki 中所述：

https://agda.readthedocs.io/en/v2.6.2.1/language/with-abstraction.html#termination-checking

這是一個類似的問題：

使用 with-abstraction 的終止檢查失敗