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

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

module Data.List.Relation.Unary.Any where

open import Data.Empty
open import Data.Fin.Base
open import Data.List.Base as List using (List; []; [_]; _∷_)
open import Data.Product as Prod using (; _,_)
open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
open import Level using (Level; _⊔_)
open import Relation.Nullary using (¬_; yes; no)
import Relation.Nullary.Decidable as Dec
open import Relation.Nullary.Negation using (contradiction)
open import Relation.Nullary.Sum using (_⊎-dec_)
open import Relation.Unary hiding (_∈_)

    a p q : Level
    A : Set a
    P Q : Pred A p
    x : A
    xs : List A

-- Definition

-- Given a predicate P, then Any P xs means that at least one element
-- in xs satisfies P. See `Relation.Unary` for an explanation of
-- predicates.

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

-- Operations on Any

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

tail : ¬ P x  Any P (x  xs)  Any P xs
tail ¬px (here  px)  = ⊥-elim (¬px px)
tail ¬px (there pxs) = pxs

map : P  Q  Any P  Any Q
map g (here px)   = here (g px)
map g (there pxs) = there (map g pxs)

-- `index x∈xs` is the list position (zero-based) which `x∈xs` points to.

index : Any P xs  Fin (List.length xs)
index (here  px)  = zero
index (there pxs) = suc (index pxs)

lookup : {P : Pred A p}  Any P xs  A
lookup {xs = xs} p = List.lookup xs (index p)

_∷=_ : {P : Pred A p}  Any P xs  A  List A
_∷=_ {xs = xs} x∈xs v = xs List.[ index x∈xs ]∷= v

infixl 4 _─_
_─_ : {P : Pred A p}   xs  Any P xs  List A
xs  x∈xs = xs List.─ index x∈xs

-- If any element satisfies P, then P is satisfied.

satisfied : Any P xs   P
satisfied (here px)   = _ , px
satisfied (there pxs) = satisfied pxs

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 (inj₁ px)  = here px
fromSum (inj₂ pxs) = there pxs

-- Properties of predicates preserved by Any

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

satisfiable : Satisfiable P  Satisfiable (Any P)
satisfiable (x , Px) = [ x ] , here Px

-- Please use the new names as continuing support for the old names is
-- not guaranteed.

-- Version 1.4

any = any?
"Warning: any was deprecated in v1.4.
Please use any? instead."