------------------------------------------------------------------------
-- The Agda standard library
--
-- The extensional sublist relation over setoid equality.
------------------------------------------------------------------------

{-# OPTIONS --cubical-compatible --safe #-}

open import Relation.Binary.Core using (Rel)
open import Relation.Binary.Bundles using (Setoid)

module Data.List.Relation.Binary.Subset.Setoid
  {c } (S : Setoid c ) where

open import Data.List.Base using (List)
open import Data.List.Membership.Setoid S using (_∈_)
open import Function.Base using (flip)
open import Level using (_⊔_)
open import Relation.Nullary.Negation using (¬_)

open Setoid S renaming (Carrier to A)

------------------------------------------------------------------------
-- Definitions

infix 4 _⊆_ _⊇_ _⊈_ _⊉_

_⊆_ : Rel (List A) (c  )
xs  ys =  {x}  x  xs  x  ys

_⊇_ : Rel (List A) (c  )
_⊇_ = flip _⊆_

_⊈_ : Rel (List A) (c  )
xs  ys = ¬ xs  ys

_⊉_ : Rel (List A) (c  )
xs  ys = ¬ xs  ys