```------------------------------------------------------------------------
-- The Agda standard library
--
-- Lists where all elements satisfy a given property
------------------------------------------------------------------------

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

module Data.List.Relation.Unary.All where

open import Category.Applicative
open import Data.Empty using (⊥)
open import Data.List.Base as List using (List; []; _∷_)
open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
open import Data.List.Membership.Propositional using (_∈_)
open import Data.Product as Prod
using (∃; -,_; _×_; _,_; proj₁; proj₂; uncurry)
open import Function
open import Level
open import Relation.Nullary hiding (Irrelevant)
import Relation.Nullary.Decidable as Dec
open import Relation.Nullary.Product using (_×-dec_)
open import Relation.Unary hiding (_∈_)
open import Relation.Binary.PropositionalEquality as P

private
variable
a b p q r : Level
A : Set a
B : Set b

------------------------------------------------------------------------
-- Definitions

-- Given a predicate P, then All P xs means that every element in xs
-- satisfies P. See `Relation.Unary` for an explanation of predicates.

infixr 5 _∷_

data All {A : Set a} (P : Pred A p) : Pred (List A) (a ⊔ p) where
[]  : All P []
_∷_ : ∀ {x xs} (px : P x) (pxs : All P xs) → All P (x ∷ xs)

-- All P xs is a finite map from indices x ∈ xs to content P x.
-- Relation pxs [ i ]= px states that, in map pxs, key i : x ∈ xs points to value px.

infix 4 _[_]=_

data _[_]=_ {A : Set a} {P : Pred A p} :
∀ {x xs} → All P xs → x ∈ xs → P x → Set (a ⊔ p) where

here  : ∀ {x xs} {px : P x} {pxs : All P xs} →
px ∷ pxs [ here refl ]= px

there : ∀ {x xs y} {px : P x} {pxs : All P xs} {py : P y} {i : x ∈ xs} →
pxs [ i ]= px →
py ∷ pxs [ there i ]= px

-- A list is empty if having an element is impossible.

Null : Pred (List A) _
Null = All (λ _ → ⊥)

------------------------------------------------------------------------
-- Operations on All

module _ {P : Pred A p} where

uncons : ∀ {x xs} → All P (x ∷ xs) → P x × All P xs
uncons (px ∷ pxs) = px , pxs

head : ∀ {x xs} → All P (x ∷ xs) → P x

tail : ∀ {x xs} → All P (x ∷ xs) → All P xs
tail = proj₂ ∘ uncons

tabulate : ∀ {xs} → (∀ {x} → x ∈ xs → P x) → All P xs
tabulate {xs = []}     hyp = []
tabulate {xs = x ∷ xs} hyp = hyp (here refl) ∷ tabulate (hyp ∘ there)

reduce : (f : ∀ {x} → P x → B) → ∀ {xs} → All P xs → List B
reduce f []         = []
reduce f (px ∷ pxs) = f px ∷ reduce f pxs

construct : (f : B → ∃ P) (xs : List B) → ∃ (All P)
construct f []       = [] , []
construct f (x ∷ xs) = Prod.zip _∷_ _∷_ (f x) (construct f xs)

fromList : (xs : List (∃ P)) → All P (List.map proj₁ xs)
fromList []              = []
fromList ((x , p) ∷ xps) = p ∷ fromList xps

toList : ∀ {xs} → All P xs → List (∃ P)
toList pxs = reduce (λ {x} px → x , px) pxs

module _ {P : Pred A p} {Q : Pred A q} where

map : P ⊆ Q → All P ⊆ All Q
map g []         = []
map g (px ∷ pxs) = g px ∷ map g pxs

module _ {P : Pred A p} {Q : Pred A q} {R : Pred A r} where

zipWith : P ∩ Q ⊆ R → All P ∩ All Q ⊆ All R
zipWith f ([] , [])             = []
zipWith f (px ∷ pxs , qx ∷ qxs) = f (px , qx) ∷ zipWith f (pxs , qxs)

unzipWith : R ⊆ P ∩ Q → All R ⊆ All P ∩ All Q
unzipWith f []         = [] , []
unzipWith f (rx ∷ rxs) = Prod.zip _∷_ _∷_ (f rx) (unzipWith f rxs)

module _ {P : Pred A p} {Q : Pred A q} where

zip : All P ∩ All Q ⊆ All (P ∩ Q)
zip = zipWith id

unzip : All (P ∩ Q) ⊆ All P ∩ All Q
unzip = unzipWith id

self : ∀ {xs : List A} → All (const A) xs
self = tabulate (λ {x} _ → x)

------------------------------------------------------------------------
-- (weak) updateAt

module _ {P : Pred A p} where

infixl 6 _[_]%=_ _[_]≔_

updateAt : ∀ {x xs} → x ∈ xs → (P x → P x) → All P xs → All P xs
updateAt () f []
updateAt (here refl) f (px ∷ pxs) = f px ∷ pxs
updateAt (there i)   f (px ∷ pxs) =   px ∷ updateAt i f pxs

_[_]%=_ : ∀ {x xs} → All P xs → x ∈ xs → (P x → P x) → All P xs
pxs [ i ]%= f = updateAt i f pxs

_[_]≔_ : ∀ {x xs} → All P xs → x ∈ xs → P x → All P xs
pxs [ i ]≔ px = pxs [ i ]%= const px

------------------------------------------------------------------------
-- Traversable-like functions

module _ (p : Level) {A : Set a} {P : Pred A (a ⊔ p)}
{F : Set (a ⊔ p) → Set (a ⊔ p)}
(App : RawApplicative F)
where

open RawApplicative App

sequenceA : All (F ∘′ P) ⊆ F ∘′ All P
sequenceA []       = pure []
sequenceA (x ∷ xs) = _∷_ <\$> x ⊛ sequenceA xs

mapA : ∀ {Q : Pred A q} → (Q ⊆ F ∘′ P) → All Q ⊆ (F ∘′ All P)
mapA f = sequenceA ∘′ map f

forA : ∀ {Q : Pred A q} {xs} → All Q xs → (Q ⊆ F ∘′ P) → F (All P xs)
forA qxs f = mapA f qxs

module _ (p : Level) {A : Set a} {P : Pred A (a ⊔ p)}
{M : Set (a ⊔ p) → Set (a ⊔ p)}
where

sequenceM : All (M ∘′ P) ⊆ M ∘′ All P
sequenceM = sequenceA p App

mapM : ∀ {Q : Pred A q} → (Q ⊆ M ∘′ P) → All Q ⊆ (M ∘′ All P)
mapM = mapA p App

forM : ∀ {Q : Pred A q} {xs} → All Q xs → (Q ⊆ M ∘′ P) → M (All P xs)
forM = forA p App

------------------------------------------------------------------------
-- Generalised lookup based on a proof of Any

module _ {P : Pred A p} {Q : Pred A q} where

lookupAny : ∀ {xs} → All P xs → (i : Any Q xs) → (P ∩ Q) (Any.lookup i)
lookupAny (px ∷ pxs) (here qx) = px , qx
lookupAny (px ∷ pxs) (there i) = lookupAny pxs i

module _ {P : Pred A p} {Q : Pred A q} {R : Pred A r} where

lookupWith : ∀[ P ⇒ Q ⇒ R ] → ∀ {xs} → All P xs → (i : Any Q xs) →
R (Any.lookup i)
lookupWith f pxs i = Prod.uncurry f (lookupAny pxs i)

module _ {P : Pred A p} where

lookup : ∀ {xs} → All P xs → (∀ {x} → x ∈ xs → P x)
lookup pxs = lookupWith (λ { px refl → px }) pxs

------------------------------------------------------------------------
-- Properties of predicates preserved by All

module _ {P : Pred A p} where

all : Decidable P → Decidable (All P)
all p []       = yes []
all p (x ∷ xs) = Dec.map′ (uncurry _∷_) uncons (p x ×-dec all p xs)

universal : Universal P → Universal (All P)
universal u []       = []
universal u (x ∷ xs) = u x ∷ universal u xs

irrelevant : Irrelevant P → Irrelevant (All P)
irrelevant irr []           []           = P.refl
irrelevant irr (px₁ ∷ pxs₁) (px₂ ∷ pxs₂) =
P.cong₂ _∷_ (irr px₁ px₂) (irrelevant irr pxs₁ pxs₂)

satisfiable : Satisfiable (All P)
satisfiable = [] , []
```