```------------------------------------------------------------------------
-- The Agda standard library
--
-- List membership and some related definitions
------------------------------------------------------------------------

{-# OPTIONS --without-K --safe #-}

open import Relation.Binary

module Data.List.Membership.Setoid {c ℓ} (S : Setoid c ℓ) where

open import Function.Base using (_∘_; id; flip)
open import Data.Fin.Base using (Fin; zero; suc)
open import Data.List.Base as List using (List; []; _∷_; length; lookup)
open import Data.List.Relation.Unary.Any
using (Any; index; map; here; there)
open import Data.Product as Prod using (∃; _×_; _,_)
open import Relation.Unary using (Pred)
open import Relation.Nullary using (¬_)

open Setoid S renaming (Carrier to A)

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

infix 4 _∈_ _∉_

_∈_ : A → List A → Set _
x ∈ xs = Any (x ≈_) xs

_∉_ : A → List A → Set _
x ∉ xs = ¬ x ∈ xs

------------------------------------------------------------------------
-- Operations

open Data.List.Relation.Unary.Any using (_∷=_; _─_) public

mapWith∈ : ∀ {b} {B : Set b}
(xs : List A) → (∀ {x} → x ∈ xs → B) → List B
mapWith∈ []       f = []
mapWith∈ (x ∷ xs) f = f (here refl) ∷ mapWith∈ xs (f ∘ there)

------------------------------------------------------------------------
-- Finding and losing witnesses

module _ {p} {P : Pred A p} where

find : ∀ {xs} → Any P xs → ∃ λ x → x ∈ xs × P x
find (here px)   = (_ , here refl , px)
find (there pxs) = Prod.map id (Prod.map there id) (find pxs)

lose : P Respects _≈_ →  ∀ {x xs} → x ∈ xs → P x → Any P xs
lose resp x∈xs px = map (flip resp px) x∈xs
```