------------------------------------------------------------------------
-- The Agda standard library
--
-- All predicate transformer for fresh lists
------------------------------------------------------------------------
{-# OPTIONS --cubical-compatible --safe #-}
module Data.List.Fresh.Relation.Unary.All where
open import Data.List.Fresh using (List#; []; cons; _∷#_; _#_)
open import Data.List.Fresh.Relation.Unary.Any as Any using (Any; here; there)
open import Data.Product.Base using (_×_; _,_; proj₁; uncurry)
open import Data.Sum.Base as Sum using (inj₁; inj₂; [_,_]′)
open import Function.Base using (_∘_; _$_)
open import Level using (Level; _⊔_; Lift)
open import Relation.Nullary.Decidable as Dec using (Dec; yes; no; _×-dec_)
open import Relation.Unary as U
using (Pred; IUniversal; Universal; Decidable; _⇒_; _∪_; _∩_)
open import Relation.Binary.Core using (Rel)
private
variable
a p q r : Level
A : Set a
module _ {A : Set a} {R : Rel A r} (P : Pred A p) where
infixr 5 _∷_
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)
------------------------------------------------------------------------
-- Generalised decidability procedure
module _ {R : Rel A r} {P : Pred A p} {Q : Pred A q} where
decide : Π[ P ∪ Q ] → Π[ All {R = R} P ∪ Any Q ]
decide p∪q [] = inj₁ []
decide p∪q (x ∷# xs) =
[ (λ px → Sum.map (px ∷_) there (decide p∪q xs))
, inj₂ ∘ here
]′ $ p∪q x
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