module Data.Vec.All where
open import Data.Nat using (zero; suc)
open import Data.Fin using (Fin; zero; suc)
open import Data.Product as Prod using (_,_)
open import Data.Vec as Vec using (Vec; []; _∷_)
open import Function using (_∘_)
open import Level using (_⊔_)
open import Relation.Nullary
import Relation.Nullary.Decidable as Dec
open import Relation.Unary
open import Relation.Binary.PropositionalEquality as P using (subst)
infixr 5 _∷_
data All {a p} {A : Set a}
(P : A → Set p) : ∀ {k} → Vec A k → Set (p ⊔ a) where
[] : All P []
_∷_ : ∀ {k x} {xs : Vec A k} (px : P x) (pxs : All P xs) → All P (x ∷ xs)
head : ∀ {a p} {A : Set a} {P : A → Set p} {k x} {xs : Vec A k} →
All P (x ∷ xs) → P x
head (px ∷ pxs) = px
tail : ∀ {a p} {A : Set a} {P : A → Set p} {k x} {xs : Vec A k} →
All P (x ∷ xs) → All P xs
tail (px ∷ pxs) = pxs
lookup : ∀ {a p} {A : Set a} {P : A → Set p} {k} {xs : Vec A k} →
(i : Fin k) → All P xs → P (Vec.lookup i xs)
lookup () []
lookup zero (px ∷ pxs) = px
lookup (suc i) (px ∷ pxs) = lookup i pxs
tabulate : ∀ {a p} {A : Set a} {P : A → Set p} {k xs} →
(∀ i → P (Vec.lookup i xs)) → All P {k} xs
tabulate {xs = []} pxs = []
tabulate {xs = _ ∷ _} pxs = pxs zero ∷ tabulate (pxs ∘ suc)
map : ∀ {a p q} {A : Set a} {P : A → Set p} {Q : A → Set q} {k} →
P ⊆ Q → All P {k} ⊆ All Q {k}
map g [] = []
map g (px ∷ pxs) = g px ∷ map g pxs
zipWith : ∀ {a b c p q r} {A : Set a} {B : Set b} {C : Set c} {_⊕_ : A → B → C}
{P : A → Set p} {Q : B → Set q} {R : C → Set r} →
(∀ {x y} → P x → Q y → R (x ⊕ y)) →
∀ {k xs ys} → All P {k} xs → All Q {k} ys →
All R {k} (Vec.zipWith _⊕_ xs ys)
zipWith _⊕_ {xs = []} {[]} [] [] = []
zipWith _⊕_ {xs = x ∷ xs} {y ∷ ys} (px ∷ pxs) (qy ∷ qys) =
px ⊕ qy ∷ zipWith _⊕_ pxs qys
zip : ∀ {a p q k} {A : Set a} {P : A → Set p} {Q : A → Set q} →
All P ∩ All Q ⊆ All (P ∩ Q) {k}
zip ([] , []) = []
zip (px ∷ pxs , qx ∷ qxs) = (px , qx) ∷ zip (pxs , qxs)
unzip : ∀ {a p q k} {A : Set a} {P : A → Set p} {Q : A → Set q} →
All (P ∩ Q) {k} ⊆ All P ∩ All Q
unzip [] = [] , []
unzip (pqx ∷ pqxs) = Prod.zip _∷_ _∷_ pqx (unzip pqxs)
module _ {a p} {A : Set a} {P : A → Set p} where
all : ∀ {k} → Decidable P → Decidable (All P {k})
all P? [] = yes []
all P? (x ∷ xs) with P? x
... | yes px = Dec.map′ (px ∷_) tail (all P? xs)
... | no ¬px = no (¬px ∘ head)
universal : Universal P → ∀ {k} → Universal (All P {k})
universal u [] = []
universal u (x ∷ xs) = u x ∷ universal u xs
irrelevant : Irrelevant P → ∀ {k} → Irrelevant (All P {k})
irrelevant irr [] [] = P.refl
irrelevant irr (px₁ ∷ pxs₁) (px₂ ∷ pxs₂) =
P.cong₂ _∷_ (irr px₁ px₂) (irrelevant irr pxs₁ pxs₂)
satisfiable : Satisfiable P → ∀ {k} → Satisfiable (All P {k})
satisfiable (x , p) {zero} = [] , []
satisfiable (x , p) {suc k} = Prod.map (x ∷_) (p ∷_) (satisfiable (x , p))