------------------------------------------------------------------------ -- The Agda standard library -- -- Pointwise lifting of relations to lists ------------------------------------------------------------------------ {-# OPTIONS --cubical-compatible --safe #-} module Data.List.Relation.Binary.Pointwise.Base where open import Data.Product using (_×_; <_,_>) open import Data.List.Base using (List; []; _∷_) open import Level open import Relation.Binary.Core using (REL; _⇒_) private variable a b c ℓ : Level A : Set a B : Set b x y : A xs ys : List A R S : REL A B ℓ ------------------------------------------------------------------------ -- Definition ------------------------------------------------------------------------ infixr 5 _∷_ data Pointwise {A : Set a} {B : Set b} (R : REL A B ℓ) : List A → List B → Set (a ⊔ b ⊔ ℓ) where [] : Pointwise R [] [] _∷_ : (x∼y : R x y) (xs∼ys : Pointwise R xs ys) → Pointwise R (x ∷ xs) (y ∷ ys) ------------------------------------------------------------------------ -- Operations ------------------------------------------------------------------------ head : Pointwise R (x ∷ xs) (y ∷ ys) → R x y head (x∼y ∷ xs∼ys) = x∼y tail : Pointwise R (x ∷ xs) (y ∷ ys) → Pointwise R xs ys tail (x∼y ∷ xs∼ys) = xs∼ys uncons : Pointwise R (x ∷ xs) (y ∷ ys) → R x y × Pointwise R xs ys uncons = < head , tail > rec : ∀ (P : ∀ {xs ys} → Pointwise R xs ys → Set c) → (∀ {x y xs ys} {Rxsys : Pointwise R xs ys} → (Rxy : R x y) → P Rxsys → P (Rxy ∷ Rxsys)) → P [] → ∀ {xs ys} (Rxsys : Pointwise R xs ys) → P Rxsys rec P c n [] = n rec P c n (Rxy ∷ Rxsys) = c Rxy (rec P c n Rxsys) map : R ⇒ S → Pointwise R ⇒ Pointwise S map R⇒S [] = [] map R⇒S (Rxy ∷ Rxsys) = R⇒S Rxy ∷ map R⇒S Rxsys