{-# OPTIONS --without-K --safe #-}
module Data.List.Fresh.Relation.Unary.All where
open import Level using (Level; _⊔_; Lift)
open import Data.Product using (_×_; _,_; proj₁; uncurry)
open import Relation.Nullary
import Relation.Nullary.Decidable as Dec
open import Relation.Nullary.Product using (_×-dec_)
open import Relation.Unary as U
open import Relation.Binary as B using (Rel)
open import Data.List.Fresh using (List#; []; cons; _∷#_; _#_)
open import Data.List.Fresh.Relation.Unary.Any as Any using (Any; here; there)
private
variable
a p q r : Level
A : Set a
module _ {A : Set a} {R : Rel A r} (P : Pred A p) where
data All : List# A R → Set (p ⊔ a ⊔ r) where
[] : All []
_∷_ : ∀ {x xs pr} → P x → All xs → All (cons x xs pr)
module _ {R : Rel A r} {P : Pred A p} where
uncons : ∀ {x} {xs : List# A R} {pr} →
All P (cons x xs pr) → P x × All P xs
uncons (p ∷ ps) = p , ps
module _ {R : Rel A r} where
append : (xs ys : List# A R) → All (_# ys) xs → List# A R
append-# : ∀ {x} xs ys {ps} → x # xs → x # ys → x # append xs ys ps
append [] ys _ = ys
append (cons x xs pr) ys ps =
let (p , ps) = uncons ps in
cons x (append xs ys ps) (append-# xs ys pr p)
append-# [] ys x#xs x#ys = x#ys
append-# (cons x xs pr) ys (r , x#xs) x#ys = r , append-# xs ys x#xs x#ys
module _ {R : Rel A r} {P : Pred A p} {Q : Pred A q} where
map : ∀ {xs : List# A R} → ∀[ P ⇒ Q ] → All P xs → All Q xs
map p⇒q [] = []
map p⇒q (p ∷ ps) = p⇒q p ∷ map p⇒q ps
lookup : ∀ {xs : List# A R} → All Q xs → (ps : Any P xs) →
Q (proj₁ (Any.witness ps))
lookup (q ∷ _) (here _) = q
lookup (_ ∷ qs) (there k) = lookup qs k
module _ {R : Rel A r} {P : Pred A p} (P? : Decidable P) where
all? : (xs : List# A R) → Dec (All P xs)
all? [] = yes []
all? (x ∷# xs) = Dec.map′ (uncurry _∷_) uncons (P? x ×-dec all? xs)