------------------------------------------------------------------------
-- 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)
open import Relation.Nullary.Negation using (contradiction)
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
  head ¬pxs (there pxs) = contradiction pxs ¬pxs

-- 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))