------------------------------------------------------------------------
-- 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 ℓ} (S : Setoid c ℓ) 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  _⊆_ _⊇_ _⊈_ _⊉_
 
_⊆_ : Rel (List A) (c ⊔ ℓ)
xs ⊆ ys = ∀ {x} → x ∈ xs → x ∈ ys
 
_⊇_ : Rel (List A) (c ⊔ ℓ)
_⊇_ = flip _⊆_
 
_⊈_ : Rel (List A) (c ⊔ ℓ)
xs ⊈ ys = ¬ xs ⊆ ys
 
_⊉_ : Rel (List A) (c ⊔ ℓ)
xs ⊉ ys = ¬ xs ⊇ ys