------------------------------------------------------------------------
-- The Agda standard library
--
-- Properties related to Any
------------------------------------------------------------------------

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

module Data.List.Relation.Unary.Any.Properties where

open import Category.Monad
open import Data.Bool.Base using (Bool; false; true; T)
open import Data.Bool.Properties
open import Data.Empty using ()
open import Data.Fin.Base using (Fin) renaming (zero to fzero; suc to fsuc)
open import Data.List.Base as List
open import Data.List.Categorical using (monad)
open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
open import Data.List.Membership.Propositional
open import Data.List.Membership.Propositional.Properties.Core
  using (Any↔; find∘map; map∘find; lose∘find)
open import Data.List.Relation.Binary.Pointwise
  using (Pointwise; []; _∷_)
open import Data.Nat.Base using (zero; suc; _<_; z≤n; s≤s)
open import Data.Maybe.Base using (Maybe; just; nothing)
open import Data.Maybe.Relation.Unary.Any as MAny using (just)
open import Data.Product as Prod
  using (_×_; _,_; ; ∃₂; proj₁; proj₂; uncurry′)
open import Data.Product.Properties
open import Data.Product.Function.NonDependent.Propositional
  using (_×-cong_)
import Data.Product.Function.Dependent.Propositional as Σ
open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
open import Data.Sum.Function.Propositional using (_⊎-cong_)
open import Function.Base
open import Function.Equality using (_⟨$⟩_)
open import Function.Equivalence using (_⇔_; equivalence; Equivalence)
open import Function.Inverse as Inv using (_↔_; inverse; Inverse)
open import Function.Related as Related using (Kind; Related; SK-sym)
open import Level using (Level)
open import Relation.Binary as B
open import Relation.Binary.PropositionalEquality as P
  using (_≡_; refl; inspect)
open import Relation.Unary as U
  using (Pred; _⟨×⟩_; _⟨→⟩_) renaming (_⊆_ to _⋐_)
open import Relation.Nullary using (¬_; _because_; does; ofʸ; ofⁿ)
open import Relation.Nullary.Negation using (contradiction; ¬?; decidable-stable)

private
  open module ListMonad {} = RawMonad (monad { = })

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

------------------------------------------------------------------------
-- Equality properties

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

  lift-resp : P Respects _≈_  (Any P) Respects (Pointwise _≈_)
  lift-resp resp (x≈y  xs≈ys) (here px)   = here (resp x≈y px)
  lift-resp resp (x≈y  xs≈ys) (there pxs) =
    there (lift-resp resp xs≈ys pxs)

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

  here-injective :  {p q : P x} 
                   here {P = P} {xs = xs} p  here q  p  q
  here-injective refl = refl

  there-injective :  {p q : Any P xs} 
                    there {x = x} p  there q  p  q
  there-injective refl = refl

------------------------------------------------------------------------
-- Misc

module _ {P : A  Set p} where

  ¬Any[] : ¬ Any P []
  ¬Any[] ()

------------------------------------------------------------------------
-- Any is a congruence

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

  Any-cong :  {xs ys : List A} 
             (∀ x  Related k (P x) (Q x)) 
             (∀ {z}  Related k (z  xs) (z  ys)) 
             Related k (Any P xs) (Any Q ys)
  Any-cong {xs} {ys} P↔Q xs≈ys =
    Any P xs                ↔⟨ SK-sym Any↔ 
    ( λ x  x  xs × P x)  ∼⟨ Σ.cong Inv.id (xs≈ys ×-cong P↔Q _) 
    ( λ x  x  ys × Q x)  ↔⟨ Any↔ 
    Any Q ys                
    where open Related.EquationalReasoning

------------------------------------------------------------------------
-- map

map-id :  {P : A  Set p} (f : P  P) {xs} 
         (∀ {x} (p : P x)  f p  p) 
         (p : Any P xs)  Any.map f p  p
map-id f hyp (here  p) = P.cong here (hyp p)
map-id f hyp (there p) = P.cong there $ map-id f hyp p

map-∘ :  {P : A  Set p} {Q : A  Set q} {R : A  Set r}
        (f : Q  R) (g : P  Q)
        {xs} (p : Any P xs) 
        Any.map (f  g) p  Any.map f (Any.map g p)
map-∘ f g (here  p) = refl
map-∘ f g (there p) = P.cong there $ map-∘ f g p

------------------------------------------------------------------------
-- lookup

lookup-result :  {P : Pred A p} {xs}  (p : Any P xs)  P (Any.lookup p)
lookup-result (here px) = px
lookup-result (there p) = lookup-result p

------------------------------------------------------------------------
-- Swapping

-- Nested occurrences of Any can sometimes be swapped. See also ×↔.

swap :  {P : A  B  Set } {xs ys} 
       Any  x  Any (P x) ys) xs  Any  y  Any (flip P y) xs) ys
swap (here  pys)  = Any.map here pys
swap (there pxys) = Any.map there (swap pxys)

swap-there :  {P : A  B  Set } {x xs ys} 
             (any : Any  x  Any (P x) ys) xs) 
             swap (Any.map (there {x = x}) any)  there (swap any)
swap-there (here  pys)  = refl
swap-there (there pxys) = P.cong (Any.map there) (swap-there pxys)

swap-invol :  {P : A  B  Set } {xs ys} 
             (any : Any  x  Any (P x) ys) xs) 
             swap (swap any)  any
swap-invol (here (here px))   = refl
swap-invol (here (there pys)) =
  P.cong (Any.map there) (swap-invol (here pys))
swap-invol (there pxys)       =
  P.trans (swap-there (swap pxys)) (P.cong there (swap-invol pxys))

swap↔ :  {P : A  B  Set } {xs ys} 
       Any  x  Any (P x) ys) xs  Any  y  Any (flip P y) xs) ys
swap↔ = inverse swap swap swap-invol swap-invol

------------------------------------------------------------------------
-- Lemmas relating Any to ⊥

⊥↔Any⊥ :  {xs : List A}    Any (const ) xs
⊥↔Any⊥ {A = A} = inverse (λ())  p  from p) (λ())  p  from p)
  where
  from : {xs : List A}  Any (const ) xs   {b} {B : Set b}  B
  from (there p) = from p

⊥↔Any[] :  {P : A  Set p}    Any P []
⊥↔Any[] = inverse (λ()) (λ()) (λ()) (λ())

------------------------------------------------------------------------
-- Lemmas relating Any to ⊤

-- These introduction and elimination rules are not inverses, though.

any⁺ :  (p : A  Bool) {xs}  Any (T  p) xs  T (any p xs)
any⁺ p (here  px)          = Equivalence.from T-∨ ⟨$⟩ inj₁ px
any⁺ p (there {x = x} pxs) with p x
... | true  = _
... | false = any⁺ p pxs

any⁻ :  (p : A  Bool) xs  T (any p xs)  Any (T  p) xs
any⁻ p (x  xs) px∷xs with p x | inspect p x
... | true  | P.[ eq ] = here (Equivalence.from T-≡ ⟨$⟩ eq)
... | false | _        = there (any⁻ p xs px∷xs)

any⇔ :  {p : A  Bool} {xs}  Any (T  p) xs  T (any p xs)
any⇔ = equivalence (any⁺ _) (any⁻ _ _)

------------------------------------------------------------------------
-- Sums commute with Any

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

  Any-⊎⁺ :  {xs}  Any P xs  Any Q xs  Any  x  P x  Q x) xs
  Any-⊎⁺ = [ Any.map inj₁ , Any.map inj₂ ]′

  Any-⊎⁻ :  {xs}  Any  x  P x  Q x) xs  Any P xs  Any Q xs
  Any-⊎⁻ (here (inj₁ p)) = inj₁ (here p)
  Any-⊎⁻ (here (inj₂ q)) = inj₂ (here q)
  Any-⊎⁻ (there p)       = Sum.map there there (Any-⊎⁻ p)

  ⊎↔ :  {xs}  (Any P xs  Any Q xs)  Any  x  P x  Q x) xs
  ⊎↔ = inverse Any-⊎⁺ Any-⊎⁻ from∘to to∘from
    where
    from∘to :  {xs} (p : Any P xs  Any Q xs)  Any-⊎⁻ (Any-⊎⁺ p)  p
    from∘to (inj₁ (here  p)) = refl
    from∘to (inj₁ (there p)) rewrite from∘to (inj₁ p) = refl
    from∘to (inj₂ (here  q)) = refl
    from∘to (inj₂ (there q)) rewrite from∘to (inj₂ q) = refl

    to∘from :  {xs} (p : Any  x  P x  Q x) xs) 
              Any-⊎⁺ (Any-⊎⁻ p)  p
    to∘from (here (inj₁ p)) = refl
    to∘from (here (inj₂ q)) = refl
    to∘from (there p) with Any-⊎⁻ p | to∘from p
    to∘from (there .(Any.map inj₁ p)) | inj₁ p | refl = refl
    to∘from (there .(Any.map inj₂ q)) | inj₂ q | refl = refl

------------------------------------------------------------------------
-- Products "commute" with Any.

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

  Any-×⁺ :  {xs ys}  Any P xs × Any Q ys 
           Any  x  Any  y  P x × Q y) ys) xs
  Any-×⁺ (p , q) = Any.map  p  Any.map  q  (p , q)) q) p

  Any-×⁻ :  {xs ys}  Any  x  Any  y  P x × Q y) ys) xs 
           Any P xs × Any Q ys
  Any-×⁻ pq with Prod.map id (Prod.map id find) (find pq)
  ... | (x , x∈xs , y , y∈ys , p , q) = (lose x∈xs p , lose y∈ys q)

  ×↔ :  {xs ys} 
       (Any P xs × Any Q ys)  Any  x  Any  y  P x × Q y) ys) xs
  ×↔ {xs} {ys} = inverse Any-×⁺ Any-×⁻ from∘to to∘from
    where
    open P.≡-Reasoning

    from∘to :  pq  Any-×⁻ (Any-×⁺ pq)  pq
    from∘to (p , q) =

      Any-×⁻ (Any-×⁺ (p , q))

        ≡⟨⟩

      (let (x , x∈xs , pq)    = find (Any-×⁺ (p , q))
           (y , y∈ys , p , q) = find pq
       in  lose x∈xs p , lose y∈ys q)

       ≡⟨ P.cong    let (x , x∈xs , pq)    = 
                            (y , y∈ys , p , q) = find pq
                        in  lose x∈xs p , lose y∈ys q)
                 (find∘map p  p  Any.map (p ,_) q)) 

      (let (x , x∈xs , p)     = find p
           (y , y∈ys , p , q) = find (Any.map (p ,_) q)
       in  lose x∈xs p , lose y∈ys q)

       ≡⟨ P.cong    let (x , x∈xs , p)     = find p
                            (y , y∈ys , p , q) = 
                        in  lose x∈xs p , lose y∈ys q)
                 (find∘map q (proj₂ (proj₂ (find p)) ,_)) 

      (let (x , x∈xs , p) = find p
           (y , y∈ys , q) = find q
       in  lose x∈xs p , lose y∈ys q)

       ≡⟨ P.cong₂ _,_ (lose∘find p) (lose∘find q) 

      (p , q) 

    to∘from :  pq  Any-×⁺ {xs} (Any-×⁻ pq)  pq
    to∘from pq
      with find pq
        |  (f : (proj₁ (find pq) ≡_)  _)  map∘find pq {f})
    ... | (x , x∈xs , pq′) | lem₁
      with find pq′
        |  (f : (proj₁ (find pq′) ≡_)  _)  map∘find pq′ {f})
    ... | (y , y∈ys , p , q) | lem₂
      rewrite P.sym $ map-∘ {R = λ x  Any  y  P x × Q y) ys}
                             p  Any.map  q  p , q) (lose y∈ys q))
                             y  P.subst P y p)
                            x∈xs
              = lem₁ _ helper
      where
      helper : Any.map  q  p , q) (lose y∈ys q)  pq′
      helper rewrite P.sym $ map-∘  q  p , q)
                                    y  P.subst Q y q)
                                   y∈ys
             = lem₂ _ refl

------------------------------------------------------------------------
-- Half-applied product commutes with Any.

module _ {_~_ : REL A B r} where

  Any-Σ⁺ʳ :  {xs}  ( λ x  Any (_~ x) xs)  Any (  _~_) xs
  Any-Σ⁺ʳ (b , here px) = here (b , px)
  Any-Σ⁺ʳ (b , there pxs) = there (Any-Σ⁺ʳ (b , pxs))

  Any-Σ⁻ʳ :  {xs}  Any (  _~_) xs   λ x  Any (_~ x) xs
  Any-Σ⁻ʳ (here (b , x)) = b , here x
  Any-Σ⁻ʳ (there xs) = Prod.map₂ there $ Any-Σ⁻ʳ xs

------------------------------------------------------------------------
-- Invertible introduction (⁺) and elimination (⁻) rules for various
-- list functions
------------------------------------------------------------------------

------------------------------------------------------------------------
-- singleton

module _ {P : Pred A p} where

  singleton⁺ :  {x}  P x  Any P [ x ]
  singleton⁺ Px = here Px

  singleton⁻ :  {x}  Any P [ x ]  P x
  singleton⁻ (here Px) = Px

------------------------------------------------------------------------
-- map

module _ {f : A  B} where

  map⁺ :  {P : B  Set p} {xs}  Any (P  f) xs  Any P (List.map f xs)
  map⁺ (here p)  = here p
  map⁺ (there p) = there $ map⁺ p

  map⁻ :  {P : B  Set p} {xs}  Any P (List.map f xs)  Any (P  f) xs
  map⁻ {xs = x  xs} (here p)  = here p
  map⁻ {xs = x  xs} (there p) = there $ map⁻ p

  map⁺∘map⁻ :  {P : B  Set p} {xs} 
              (p : Any P (List.map f xs))  map⁺ (map⁻ p)  p
  map⁺∘map⁻ {xs = x  xs} (here  p) = refl
  map⁺∘map⁻ {xs = x  xs} (there p) = P.cong there (map⁺∘map⁻ p)

  map⁻∘map⁺ :  (P : B  Set p) {xs} 
              (p : Any (P  f) xs)  map⁻ {P = P} (map⁺ p)  p
  map⁻∘map⁺ P (here  p) = refl
  map⁻∘map⁺ P (there p) = P.cong there (map⁻∘map⁺ P p)

  map↔ :  {P : B  Set p} {xs} 
         Any (P  f) xs  Any P (List.map f xs)
  map↔ = inverse map⁺ map⁻ (map⁻∘map⁺ _) map⁺∘map⁻

module _ {f : A  B} {P : A  Set p} {Q : B  Set q} where
  gmap : P  Q  f  Any P  Any Q  map f
  gmap g = map⁺  Any.map g

------------------------------------------------------------------------
-- mapMaybe

module _ {P : B  Set p} (f : A  Maybe B) where

  mapMaybe⁺ :  xs  Any (MAny.Any P) (map f xs) 
              Any P (mapMaybe f xs)
  mapMaybe⁺ (x  xs) ps with f x | ps
  ... | nothing | here  ()
  ... | nothing | there pxs      = mapMaybe⁺ xs pxs
  ... | just _  | here (just py) = here py
  ... | just _  | there pxs      = there (mapMaybe⁺ xs pxs)

------------------------------------------------------------------------
-- _++_

module _ {P : A  Set p} where

  ++⁺ˡ :  {xs ys}  Any P xs  Any P (xs ++ ys)
  ++⁺ˡ (here p)  = here p
  ++⁺ˡ (there p) = there (++⁺ˡ p)

  ++⁺ʳ :  xs {ys}  Any P ys  Any P (xs ++ ys)
  ++⁺ʳ []       p = p
  ++⁺ʳ (x  xs) p = there (++⁺ʳ xs p)

  ++⁻ :  xs {ys}  Any P (xs ++ ys)  Any P xs  Any P ys
  ++⁻ []       p         = inj₂ p
  ++⁻ (x  xs) (here p)  = inj₁ (here p)
  ++⁻ (x  xs) (there p) = Sum.map there id (++⁻ xs p)

  ++⁺∘++⁻ :  xs {ys} (p : Any P (xs ++ ys)) 
            [ ++⁺ˡ , ++⁺ʳ xs ]′ (++⁻ xs p)  p
  ++⁺∘++⁻ []       p         = refl
  ++⁺∘++⁻ (x  xs) (here  p) = refl
  ++⁺∘++⁻ (x  xs) (there p) with ++⁻ xs p | ++⁺∘++⁻ xs p
  ++⁺∘++⁻ (x  xs) (there p) | inj₁ p′ | ih = P.cong there ih
  ++⁺∘++⁻ (x  xs) (there p) | inj₂ p′ | ih = P.cong there ih

  ++⁻∘++⁺ :  xs {ys} (p : Any P xs  Any P ys) 
            ++⁻ xs ([ ++⁺ˡ , ++⁺ʳ xs ]′ p)  p
  ++⁻∘++⁺ []            (inj₂ p)         = refl
  ++⁻∘++⁺ (x  xs)      (inj₁ (here  p)) = refl
  ++⁻∘++⁺ (x  xs) {ys} (inj₁ (there p)) rewrite ++⁻∘++⁺ xs {ys} (inj₁ p) = refl
  ++⁻∘++⁺ (x  xs)      (inj₂ p)         rewrite ++⁻∘++⁺ xs      (inj₂ p) = refl

  ++↔ :  {xs ys}  (Any P xs  Any P ys)  Any P (xs ++ ys)
  ++↔ {xs = xs} = inverse [ ++⁺ˡ , ++⁺ʳ xs ]′ (++⁻ xs) (++⁻∘++⁺ xs) (++⁺∘++⁻ xs)

  ++-comm :  xs ys  Any P (xs ++ ys)  Any P (ys ++ xs)
  ++-comm xs ys = [ ++⁺ʳ ys , ++⁺ˡ ]′  ++⁻ xs

  ++-comm∘++-comm :  xs {ys} (p : Any P (xs ++ ys)) 
                    ++-comm ys xs (++-comm xs ys p)  p
  ++-comm∘++-comm [] {ys} p
    rewrite ++⁻∘++⁺ ys {ys = []} (inj₁ p) = refl
  ++-comm∘++-comm (x  xs) {ys} (here p)
    rewrite ++⁻∘++⁺ ys {ys = x  xs} (inj₂ (here p)) = refl
  ++-comm∘++-comm (x  xs)      (there p) with ++⁻ xs p | ++-comm∘++-comm xs p
  ++-comm∘++-comm (x  xs) {ys} (there .([ ++⁺ʳ xs , ++⁺ˡ ]′ (++⁻ ys (++⁺ʳ ys p))))
    | inj₁ p | refl
    rewrite ++⁻∘++⁺ ys (inj₂                 p)
            | ++⁻∘++⁺ ys (inj₂ $ there {x = x} p) = refl
  ++-comm∘++-comm (x  xs) {ys} (there .([ ++⁺ʳ xs , ++⁺ˡ ]′ (++⁻ ys (++⁺ˡ p))))
    | inj₂ p | refl
    rewrite ++⁻∘++⁺ ys {ys =     xs} (inj₁ p)
            | ++⁻∘++⁺ ys {ys = x  xs} (inj₁ p) = refl

  ++↔++ :  xs ys  Any P (xs ++ ys)  Any P (ys ++ xs)
  ++↔++ xs ys = inverse (++-comm xs ys) (++-comm ys xs)
                        (++-comm∘++-comm xs) (++-comm∘++-comm ys)

  ++-insert :  xs {ys x}  P x  Any P (xs ++ [ x ] ++ ys)
  ++-insert xs Px = ++⁺ʳ xs (++⁺ˡ (singleton⁺ Px))

------------------------------------------------------------------------
-- concat

module _ {P : A  Set p} where

  concat⁺ :  {xss}  Any (Any P) xss  Any P (concat xss)
  concat⁺ (here p)           = ++⁺ˡ p
  concat⁺ (there {x = xs} p) = ++⁺ʳ xs (concat⁺ p)

  concat⁻ :  xss  Any P (concat xss)  Any (Any P) xss
  concat⁻ ([]        xss) p         = there $ concat⁻ xss p
  concat⁻ ((x  xs)  xss) (here  p) = here (here p)
  concat⁻ ((x  xs)  xss) (there p) with concat⁻ (xs  xss) p
  ... | here  p′ = here (there p′)
  ... | there p′ = there p′

  concat⁻∘++⁺ˡ :  {xs} xss (p : Any P xs) 
                 concat⁻ (xs  xss) (++⁺ˡ p)  here p
  concat⁻∘++⁺ˡ xss (here  p) = refl
  concat⁻∘++⁺ˡ xss (there p) rewrite concat⁻∘++⁺ˡ xss p = refl

  concat⁻∘++⁺ʳ :  xs xss (p : Any P (concat xss)) 
                   concat⁻ (xs  xss) (++⁺ʳ xs p)  there (concat⁻ xss p)
  concat⁻∘++⁺ʳ []       xss p = refl
  concat⁻∘++⁺ʳ (x  xs) xss p rewrite concat⁻∘++⁺ʳ xs xss p = refl

  concat⁺∘concat⁻ :  xss (p : Any P (concat xss)) 
                      concat⁺ (concat⁻ xss p)  p
  concat⁺∘concat⁻ ([]        xss) p         = concat⁺∘concat⁻ xss p
  concat⁺∘concat⁻ ((x  xs)  xss) (here p)  = refl
  concat⁺∘concat⁻ ((x  xs)  xss) (there p)
                with concat⁻ (xs  xss) p | concat⁺∘concat⁻ (xs  xss) p
  concat⁺∘concat⁻ ((x  xs)  xss) (there .(++⁺ˡ p′))              | here  p′ | refl = refl
  concat⁺∘concat⁻ ((x  xs)  xss) (there .(++⁺ʳ xs (concat⁺ p′))) | there p′ | refl = refl

  concat⁻∘concat⁺ :  {xss} (p : Any (Any P) xss)  concat⁻ xss (concat⁺ p)  p
  concat⁻∘concat⁺ (here                      p) = concat⁻∘++⁺ˡ _ p
  concat⁻∘concat⁺ (there {x = xs} {xs = xss} p)
    rewrite concat⁻∘++⁺ʳ xs xss (concat⁺ p) =
      P.cong there $ concat⁻∘concat⁺ p

  concat↔ :  {xss}  Any (Any P) xss  Any P (concat xss)
  concat↔ {xss} = inverse concat⁺ (concat⁻ xss) concat⁻∘concat⁺ (concat⁺∘concat⁻ xss)

------------------------------------------------------------------------
-- applyUpTo

module _ {P : A  Set p} where

  applyUpTo⁺ :  f {i n}  P (f i)  i < n  Any P (applyUpTo f n)
  applyUpTo⁺ _ p (s≤s z≤n)       = here p
  applyUpTo⁺ f p (s≤s (s≤s i<n)) =
    there (applyUpTo⁺ (f  suc) p (s≤s i<n))

  applyUpTo⁻ :  f {n}  Any P (applyUpTo f n) 
                λ i  i < n × P (f i)
  applyUpTo⁻ f {suc n} (here p)  = zero , s≤s z≤n , p
  applyUpTo⁻ f {suc n} (there p) with applyUpTo⁻ (f  suc) p
  ... | i , i<n , q = suc i , s≤s i<n , q

------------------------------------------------------------------------
-- tabulate

module _ {P : A  Set p} where

  tabulate⁺ :  {n} {f : Fin n  A} i  P (f i)  Any P (tabulate f)
  tabulate⁺ fzero    p = here p
  tabulate⁺ (fsuc i) p = there (tabulate⁺ i p)

  tabulate⁻ :  {n} {f : Fin n  A} 
              Any P (tabulate f)   λ i  P (f i)
  tabulate⁻ {suc n} (here p)   = fzero , p
  tabulate⁻ {suc n} (there p) = Prod.map fsuc id (tabulate⁻ p)

------------------------------------------------------------------------
-- filter

module _ {P : A  Set p} {Q : A  Set q} (Q? : U.Decidable Q) where

  filter⁺ :  {xs}  (p : Any P xs)  Any P (filter Q? xs)  ¬ Q (Any.lookup p)
  filter⁺ {x  xs} (here px) with Q? x
  ... | true  because _       = inj₁ (here px)
  ... | false because ofⁿ ¬Qx = inj₂ ¬Qx
  filter⁺ {x  xs} (there p) with Q? x
  ... | true  because _       = Sum.map₁ there (filter⁺ p)
  ... | false because _       = filter⁺ p

  filter⁻ :  {xs}  Any P (filter Q? xs)  Any P xs
  filter⁻ {x  xs} p with does (Q? x)
  filter⁻ {x  xs} (here px) | true  = here px
  filter⁻ {x  xs} (there p) | true  = there (filter⁻ p)
  filter⁻ {x  xs} p         | false = there (filter⁻ p)

------------------------------------------------------------------------
-- derun and deduplicate

module _ {P : A  Set p} {R : A  A  Set r} (R? : B.Decidable R) where

  private
    derun⁺-aux :  x xs  P Respects R  P x  Any P (derun R? (x  xs))
    derun⁺-aux x [] P-resp-R Px = here Px
    derun⁺-aux x (y  xs) P-resp-R Px with R? x y
    ... | true  because ofʸ Rxy = derun⁺-aux y xs P-resp-R (P-resp-R Rxy Px)
    ... | false because _       = here Px

  derun⁺ :  {xs}  P Respects R  Any P xs  Any P (derun R? xs)
  derun⁺ {x  xs} P-resp-R (here px) = derun⁺-aux x xs P-resp-R px
  derun⁺ {x  y  xs} P-resp-R (there any[P,xs]) with R? x y
  ... | true  because _ = derun⁺ P-resp-R any[P,xs]
  ... | false because _ = there (derun⁺ P-resp-R any[P,xs])

  deduplicate⁺ :  {xs}  P Respects (flip R)  Any P xs  Any P (deduplicate R? xs)
  deduplicate⁺ {x  xs} P-resp-R (here px) = here px
  deduplicate⁺ {x  xs} P-resp-R (there any[P,xs]) with filter⁺ (¬?  R? x) (deduplicate⁺ {xs} P-resp-R any[P,xs])
  ... | inj₁ p = there p
  ... | inj₂ ¬¬q with decidable-stable (R? x (Any.lookup (deduplicate⁺ P-resp-R any[P,xs]))) ¬¬q
  ...  | q = here (P-resp-R q (lookup-result (deduplicate⁺ P-resp-R any[P,xs])))

  private
    derun⁻-aux :  {x xs}  Any P (derun R? (x  xs))  Any P (x  xs)
    derun⁻-aux {x} {[]} (here px) = here px
    derun⁻-aux {x} {y  xs} any[P,derun[x∷y∷xs]] with R? x y
    derun⁻-aux {x} {y  xs} any[P,derun[y∷xs]]         | true  because _ = there (derun⁻-aux any[P,derun[y∷xs]])
    derun⁻-aux {x} {y  xs} (here px)                  | false because _ = here px
    derun⁻-aux {x} {y  xs} (there any[P,derun[y∷xs]]) | false because _ = there (derun⁻-aux any[P,derun[y∷xs]])

  derun⁻ :  {xs}  Any P (derun R? xs)  Any P xs
  derun⁻ {x  xs} any[P,derun[x∷xs]] = derun⁻-aux any[P,derun[x∷xs]]

  deduplicate⁻ :  {xs}  Any P (deduplicate R? xs)  Any P xs
  deduplicate⁻ {x  xs} (here px) = here px
  deduplicate⁻ {x  xs} (there any[P,dedup[xs]]) = there (deduplicate⁻ (filter⁻ (¬?  R? x) any[P,dedup[xs]]))

------------------------------------------------------------------------
-- map-with-∈.

module _ {P : B  Set p} where

  map-with-∈⁺ :  {xs : List A} (f :  {x}  x  xs  B) 
                (∃₂ λ x (x∈xs : x  xs)  P (f x∈xs)) 
                Any P (map-with-∈ xs f)
  map-with-∈⁺ f (_ , here refl  , p) = here p
  map-with-∈⁺ f (_ , there x∈xs , p) =
    there $ map-with-∈⁺ (f  there) (_ , x∈xs , p)

  map-with-∈⁻ :  (xs : List A) (f :  {x}  x  xs  B) 
                Any P (map-with-∈ xs f) 
                ∃₂ λ x (x∈xs : x  xs)  P (f x∈xs)
  map-with-∈⁻ (y  xs) f (here  p) = (y , here refl , p)
  map-with-∈⁻ (y  xs) f (there p) =
    Prod.map id (Prod.map there id) $ map-with-∈⁻ xs (f  there) p

  map-with-∈↔ :   {xs : List A} {f :  {x}  x  xs  B} 
    (∃₂ λ x (x∈xs : x  xs)  P (f x∈xs))  Any P (map-with-∈ xs f)
  map-with-∈↔ = inverse (map-with-∈⁺ _) (map-with-∈⁻ _ _) (from∘to _) (to∘from _ _)
    where
    from∘to :  {xs : List A} (f :  {x}  x  xs  B)
              (p : ∃₂ λ x (x∈xs : x  xs)  P (f x∈xs)) 
              map-with-∈⁻ xs f (map-with-∈⁺ f p)  p
    from∘to f (_ , here refl  , p) = refl
    from∘to f (_ , there x∈xs , p)
      rewrite from∘to (f  there) (_ , x∈xs , p) = refl

    to∘from :  (xs : List A) (f :  {x}  x  xs  B)
              (p : Any P (map-with-∈ xs f)) 
              map-with-∈⁺ f (map-with-∈⁻ xs f p)  p
    to∘from (y  xs) f (here  p) = refl
    to∘from (y  xs) f (there p) =
      P.cong there $ to∘from xs (f  there) p

------------------------------------------------------------------------
-- return

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

  return⁺ : P x  Any P (return x)
  return⁺ = here

  return⁻ : Any P (return x)  P x
  return⁻ (here p)   = p

  return⁺∘return⁻ : (p : Any P (return x))  return⁺ (return⁻ p)  p
  return⁺∘return⁻ (here p)   = refl

  return⁻∘return⁺ : (p : P x)  return⁻ (return⁺ p)  p
  return⁻∘return⁺ p = refl

  return↔ : P x  Any P (return x)
  return↔ = inverse return⁺ return⁻ return⁻∘return⁺ return⁺∘return⁻

------------------------------------------------------------------------
-- _∷_

module _ (P : Pred A p) where

  ∷↔ :  {x xs}  (P x  Any P xs)  Any P (x  xs)
  ∷↔ {x} {xs} =
    (P x          Any P xs)  ↔⟨ return↔ {P = P} ⊎-cong (Any P xs ) 
    (Any P [ x ]  Any P xs)  ↔⟨ ++↔ {P = P} {xs = [ x ]} 
    Any P (x  xs)            
    where open Related.EquationalReasoning

------------------------------------------------------------------------
-- _>>=_

module _ {A B : Set } {P : B  Set p} {f : A  List B} where

  >>=↔ :  {xs}  Any (Any P  f) xs  Any P (xs >>= f)
  >>=↔ {xs} =
    Any (Any P  f) xs           ↔⟨ map↔ 
    Any (Any P) (List.map f xs)  ↔⟨ concat↔ 
    Any P (xs >>= f)             
    where open Related.EquationalReasoning

------------------------------------------------------------------------
-- _⊛_

⊛↔ :  {P : B  Set } {fs : List (A  B)} {xs : List A} 
     Any  f  Any (P  f) xs) fs  Any P (fs  xs)
⊛↔ {P = P} {fs} {xs} =
  Any  f  Any (P  f) xs) fs               ↔⟨ Any-cong  _  Any-cong  _  return↔) (_ )) (_ ) 
  Any  f  Any (Any P  return  f) xs) fs  ↔⟨ Any-cong  _  >>=↔ ) (_ ) 
  Any  f  Any P (xs >>= return  f)) fs    ↔⟨ >>=↔ 
  Any P (fs  xs)                             
  where open Related.EquationalReasoning

-- An alternative introduction rule for _⊛_

⊛⁺′ :  {P : A  Set } {Q : B  Set } {fs : List (A  B)} {xs} 
      Any (P ⟨→⟩ Q) fs  Any P xs  Any Q (fs  xs)
⊛⁺′ pq p =
  Inverse.to ⊛↔ ⟨$⟩
    Any.map  pq  Any.map  {x}  pq {x}) p) pq

------------------------------------------------------------------------
-- _⊗_

⊗↔ : {P : A × B  Set } {xs : List A} {ys : List B} 
     Any  x  Any  y  P (x , y)) ys) xs  Any P (xs  ys)
⊗↔ {P = P} {xs} {ys} =
  Any  x  Any  y  P (x , y)) ys) xs                             ↔⟨ return↔ 
  Any  _,_  Any  x  Any  y  P (x , y)) ys) xs) (return _,_)  ↔⟨ ⊛↔ 
  Any  x,  Any (P  x,) ys) (_,_ <$> xs)                           ↔⟨ ⊛↔ 
  Any P (xs  ys)                                                     
  where open Related.EquationalReasoning

⊗↔′ : {P : A  Set } {Q : B  Set } {xs : List A} {ys : List B} 
      (Any P xs × Any Q ys)  Any (P ⟨×⟩ Q) (xs  ys)
⊗↔′ {P = P} {Q} {xs} {ys} =
  (Any P xs × Any Q ys)                    ↔⟨ ×↔ 
  Any  x  Any  y  P x × Q y) ys) xs  ↔⟨ ⊗↔ 
  Any (P ⟨×⟩ Q) (xs  ys)                  
  where open Related.EquationalReasoning