阿格达：返回空列表的头尾

Question

I am learning agda and practicing on lists to get a better understanding. Right now I am trying to write functions for list. I am confused on how to return the head and tail of an empty list. Here is my code:

data list (A : Set) : Set where
[]  : list A
_∷_ : A → list A → list A

Null : {A : Set} → (list A) → Bool
Null [] = true
Null (x ∷ a) = false

tail :  {A : Set} → (list A) → A
tail [] = {!!}
tail (x ∷ []) = x  
tail (x ∷ a) = tail a

head : {A : Set} → (list A) →  A
head [] = {!!}
head (x ∷ a) = x

A work around that I found was that instead of returning the first and last members I return a list containing the first and last members which is as follows:

tail :  {A : Set} → (list A) → (list A)
tail [] = []
tail (x ∷ []) = x ∷ []  
tail (x ∷ a) = tail a

head : {A : Set} → (list A) → (list A)
head [] = []
head (x ∷ a) = (x ∷ [])

But I am still confused about how to return the head and tail values. How can I do this?

PS Not an assignment. Miles ahead of this stuff in class

Answer 1

In Agda, functions are total: if you have head : {A : Set} -> list A -> A , then it will need to be defined over all lists. However, for head [] you can't conjure up an element for some arbitrary type A (imagine head ([] : list Void) ...)

The problem is that your type of head promises too much. It is not, in fact, true that you can return the first element of any list; you can only do it for non-empty lists. So you need to change head to either take a separate proof of non- emptiness , or to take a non-empty list as argument:

module SeparateProof where
  open import Data.List
  open import Data.Bool
  open import Data.Unit

  head : {A : Set} → (xs : List A) → {{nonEmpty : T (not (null xs))}} → A
  head [] {{nonEmpty = ()}} -- There's no way to pass a proof of non-emptiness for an empty list!
  head (x ∷ xs) = x

module NonEmptyType where
  open import Data.List.NonEmpty hiding (head)

  head : {A : Set} → List⁺ A → A
  head (x ∷ xs) = x -- This is the only pattern matching a List⁺ A!

Answer 2

This is a misnomer, it is last that you wanted not tail, and for the correct tail function, you could just look at drop in Data\\List\\Base.agda of the standard library:

drop : ∀ {a} {A : Set a} → ℕ → List A → List A
drop zero    xs       = xs
drop (suc n) []       = []
drop (suc n) (x ∷ xs) = drop n xs
--e.g. drop 1
tail : ∀ {a} {A : Set a} → List A → List A
tail []       = []
tail (x ∷ xs) = xs