------------------------------------------------------------------------ -- The Agda standard library -- -- List membership and some related definitions ------------------------------------------------------------------------ {-# OPTIONS --cubical-compatible --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.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.Negation 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