------------------------------------------------------------------------
-- The Agda standard library
--
-- Vectors where at least one element satisfies a given property
------------------------------------------------------------------------

{-# OPTIONS --without-K --safe #-}

module Data.Vec.Relation.Unary.Any {a} {A : Set a} where

open import Data.Empty
open import Data.Fin
open import Data.Nat using (zero; suc)
open import Data.Sum using (_⊎_; inj₁; inj₂; [_,_]′)
open import Data.Vec as Vec using (Vec; []; [_]; _∷_)
open import Data.Product as Prod using (; _,_)
open import Level using (_⊔_)
open import Relation.Nullary using (¬_; yes; no)
import Relation.Nullary.Decidable as Dec
open import Relation.Nullary.Sum using (_⊎-dec_)
open import Relation.Unary

------------------------------------------------------------------------
-- Any P xs means that at least one element in xs satisfies P.

data Any {p} (P : A  Set p) :  {n}  Vec A n  Set (a  p) where
here  :  {n x} {xs : Vec A n} (px  : P x)       Any P (x  xs)
there :  {n x} {xs : Vec A n} (pxs : Any P xs)  Any P (x  xs)

------------------------------------------------------------------------
-- Operations on Any

module _ {p} {P : A  Set p} {n x} {xs : Vec A n} where

-- If the tail does not satisfy the predicate, then the head will.

head : ¬ Any P xs  Any P (x  xs)  P x
head ¬pxs (here px)   = px

-- If the head does not satisfy the predicate, then the tail will.
tail : ¬ P x  Any P (x  xs)  Any P xs
tail ¬px (here  px)  = ⊥-elim (¬px px)
tail ¬px (there pxs) = pxs

-- Convert back and forth with sum
toSum : Any P (x  xs)  P x  Any P xs
toSum (here px)   = inj₁ px
toSum (there pxs) = inj₂ pxs

fromSum : P x  Any P xs  Any P (x  xs)
fromSum = [ here , there ]′

map :  {p q} {P : A  Set p} {Q : A  Set q}
P  Q   {n}  Any P {n}  Any Q {n}
map g (here px)   = here (g px)
map g (there pxs) = there (map g pxs)

index :  {p} {P : A  Set p} {n} {xs : Vec A n}  Any P xs  Fin n
index (here  px)  = zero
index (there pxs) = suc (index pxs)

-- If any element satisfies P, then P is satisfied.
satisfied :  {p} {P : A  Set p} {n} {xs : Vec A n}  Any P xs   P
satisfied (here px)   = _ , px
satisfied (there pxs) = satisfied pxs

------------------------------------------------------------------------
-- Properties of predicates preserved by Any

module _ {p} {P : A  Set p} where

any : Decidable P   {n}  Decidable (Any P {n})
any P? []       = no λ()
any P? (x  xs) = Dec.map′ fromSum toSum (P? x ⊎-dec any P? xs)

satisfiable : Satisfiable P   {n}  Satisfiable (Any P {suc n})
satisfiable (x , p) {zero}  = x  [] , here p
satisfiable (x , p) {suc n} = Prod.map (x ∷_) there (satisfiable (x , p))