{-# 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 using (Fin) renaming (zero to fzero; suc to fsuc)
open import Data.List 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 using (zero; suc; _<_; z≤n; s≤s)
open import Data.Maybe 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.Relation.Binary.Pointwise.NonDependent
using (_×-cong_)
import Data.Product.Relation.Binary.Pointwise.Dependent as Σ
open import Data.Sum as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
open import Data.Sum.Relation.Binary.Pointwise using (_⊎-cong_)
open import Function
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 (Related; SK-sym)
open import Relation.Binary
open import Relation.Binary.PropositionalEquality as P
using (_≡_; refl; inspect)
open import Relation.Unary
using (Pred; _⟨×⟩_; _⟨→⟩_) renaming (_⊆_ to _⋐_)
open import Relation.Nullary using (¬_)
open Related.EquationalReasoning
private
open module ListMonad {ℓ} = RawMonad (monad {ℓ = ℓ})
module _ {a p ℓ} {A : Set a} {P : A → Set p} {_≈_ : Rel A ℓ} where
lift-resp : P Respects _≈_ → (Any P) Respects (Pointwise _≈_)
lift-resp resp [] ()
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 _ {a p} {A : Set a} {P : A → Set p} where
here-injective : ∀ {x xs} {p q : P x} →
here {P = P} {xs = xs} p ≡ here q → p ≡ q
here-injective refl = refl
there-injective : ∀ {x xs} {p q : Any P xs} →
there {x = x} p ≡ there q → p ≡ q
there-injective refl = refl
module _ {a p} {A : Set a} {P : A → Set p} where
¬Any[] : ¬ Any P []
¬Any[] ()
module _ {a k p q} {A : Set a} {P : Pred A p} {Q : Pred A q} where
Any-cong : ∀ {xs ys : List A} →
(∀ x → Related k (P x) (Q x)) →
Preorder._∼_ (Related.InducedPreorder₂ k {A = A} _∈_) xs 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 ∎
map-id : ∀ {a p} {A : Set a} {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-∘ : ∀ {a p q r}
{A : Set a} {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
swap : ∀ {a b ℓ} {A : Set a} {B : Set b} {P : A → B → Set ℓ} {xs ys} →
Any (λ x → Any (P x) ys) xs → Any (λ y → Any (flip P y) xs) ys
swap (here pys) = Any.map here pys
swap (there pxys) = Any.map there (swap pxys)
swap-there : ∀ {a b ℓ} {A : Set a} {B : Set b} {P : A → B → Set ℓ}
{x xs ys} → (any : Any (λ 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 : ∀ {a b ℓ} {A : Set a} {B : Set b} {P : A → B → Set ℓ}
{xs ys} → (any : Any (λ 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↔ : ∀ {ℓ} {A B : Set ℓ} {P : A → B → Set ℓ} {xs ys} →
Any (λ x → Any (P x) ys) xs ↔ Any (λ y → Any (flip P y) xs) ys
swap↔ {P = P} = inverse swap swap swap-invol swap-invol
⊥↔Any⊥ : ∀ {a} {A : Set a} {xs : List A} → ⊥ ↔ Any (const ⊥) xs
⊥↔Any⊥ {A = A} = inverse (λ()) (λ p → from p) (λ()) (λ p → from p)
where
from : {xs : List A} → Any (const ⊥) xs → ∀ {b} {B : Set b} → B
from (here ())
from (there p) = from p
⊥↔Any[] : ∀ {a p} {A : Set a} {P : A → Set p} → ⊥ ↔ Any P []
⊥↔Any[] = inverse (λ()) (λ()) (λ()) (λ())
module _ {a} {A : Set a} where
any⁺ : ∀ (p : A → Bool) {xs} → Any (T ∘ p) xs → T (any p xs)
any⁺ p (here px) = Equivalence.from T-∨ ⟨$⟩ inj₁ px
any⁺ p (there {x = x} pxs) with p x
... | true = _
... | false = any⁺ p pxs
any⁻ : ∀ (p : A → Bool) xs → T (any p xs) → Any (T ∘ p) xs
any⁻ p [] ()
any⁻ p (x ∷ xs) px∷xs with p x | inspect p x
any⁻ p (x ∷ xs) px∷xs | true | P.[ eq ] = here (Equivalence.from T-≡ ⟨$⟩ eq)
any⁻ p (x ∷ xs) px∷xs | false | _ = there (any⁻ p xs px∷xs)
any⇔ : ∀ {p : A → Bool} {xs} → Any (T ∘ p) xs ⇔ T (any p xs)
any⇔ = equivalence (any⁺ _) (any⁻ _ _)
module _ {a p q} {A : Set a} {P : A → Set p} {Q : A → Set q} where
Any-⊎⁺ : ∀ {xs} → Any P xs ⊎ Any Q xs → Any (λ x → P x ⊎ Q x) xs
Any-⊎⁺ = [ Any.map inj₁ , Any.map inj₂ ]′
Any-⊎⁻ : ∀ {xs} → Any (λ 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 (λ 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 (λ 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
module _ {a b p q} {A : Set a} {B : Set b}
{P : Pred A p} {Q : Pred B q} where
Any-×⁺ : ∀ {xs ys} → Any P xs × Any Q ys →
Any (λ x → Any (λ y → P x × Q y) ys) xs
Any-×⁺ (p , q) = Any.map (λ p → Any.map (λ q → (p , q)) q) p
Any-×⁻ : ∀ {xs ys} → Any (λ x → Any (λ 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 (λ x → Any (λ y → P x × Q y) ys) xs
×↔ {xs} {ys} = inverse Any-×⁺ Any-×⁻ from∘to to∘from
where
from∘to : ∀ pq → Any-×⁻ (Any-×⁺ pq) ≡ pq
from∘to (p , q) rewrite
find∘map p (λ p → Any.map (λ q → (p , q)) q)
| find∘map q (λ q → proj₂ (proj₂ (find p)) , q)
| lose∘find p
| lose∘find q
= refl
to∘from : ∀ pq → Any-×⁺ (Any-×⁻ pq) ≡ pq
to∘from pq
with find pq
| (λ (f : (proj₁ (find pq) ≡_) ⋐ _) → map∘find pq {f})
... | (x , x∈xs , pq′) | lem₁
with find pq′
| (λ (f : (proj₁ (find pq′) ≡_) ⋐ _) → map∘find pq′ {f})
... | (y , y∈ys , p , q) | lem₂
rewrite P.sym $ map-∘ {R = λ x → Any (λ y → P x × Q y) ys}
(λ p → Any.map (λ q → p , q) (lose y∈ys q))
(λ y → P.subst P y p)
x∈xs
= lem₁ _ helper
where
helper : Any.map (λ q → p , q) (lose y∈ys q) ≡ pq′
helper rewrite P.sym $ map-∘ (λ q → p , q)
(λ y → P.subst Q y q)
y∈ys
= lem₂ _ refl
module _ {a p} {A : Set a} {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
singleton⁻ (there ())
module _ {a b} {A : Set a} {B : Set b} {f : A → B} where
map⁺ : ∀ {p} {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} {P : B → Set p} {xs} →
Any P (List.map f xs) → Any (P ∘ f) xs
map⁻ {xs = []} ()
map⁻ {xs = x ∷ xs} (here p) = here p
map⁻ {xs = x ∷ xs} (there p) = there $ map⁻ p
map⁺∘map⁻ : ∀ {p} {P : B → Set p} {xs} →
(p : Any P (List.map f xs)) → map⁺ (map⁻ p) ≡ p
map⁺∘map⁻ {xs = []} ()
map⁺∘map⁻ {xs = x ∷ xs} (here p) = refl
map⁺∘map⁻ {xs = x ∷ xs} (there p) = P.cong there (map⁺∘map⁻ p)
map⁻∘map⁺ : ∀ {p} (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} {P : B → Set p} {xs} →
Any (P ∘ f) xs ↔ Any P (List.map f xs)
map↔ = inverse map⁺ map⁻ (map⁻∘map⁺ _) map⁺∘map⁻
module _ {a b p} {A : Set a} {B : Set b} {P : B → Set p}
(f : A → Maybe B) where
mapMaybe⁺ : ∀ xs → Any (MAny.Any P) (map f xs) →
Any P (mapMaybe f xs)
mapMaybe⁺ [] ()
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 _ {a p} {A : Set a} {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
++⁻∘++⁺ [] (inj₁ ())
++⁻∘++⁺ [] (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))
module _ {a p} {A : Set a} {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⁻ [] ()
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⁻ [] ()
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)
module _ {a p} {A : Set a} {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 {zero} ()
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
module _ {a p} {A : Set a} {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⁻ {zero} ()
tabulate⁻ {suc n} (here p) = fzero , p
tabulate⁻ {suc n} (there p) = Prod.map fsuc id (tabulate⁻ p)
module _ {a b p} {A : Set a} {B : Set b} {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-∈⁻ [] f ()
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 [] f ()
to∘from (y ∷ xs) f (here p) = refl
to∘from (y ∷ xs) f (there p) =
P.cong there $ to∘from xs (f ∘ there) p
module _ {a p} {A : Set a} {P : A → Set p} where
return⁺ : ∀ {x} → P x → Any P (return x)
return⁺ = here
return⁻ : ∀ {x} → Any P (return x) → P x
return⁻ (here p) = p
return⁻ (there ())
return⁺∘return⁻ : ∀ {x} (p : Any P (return x)) →
return⁺ (return⁻ p) ≡ p
return⁺∘return⁻ (here p) = refl
return⁺∘return⁻ (there ())
return⁻∘return⁺ : ∀ {x} (p : P x) → return⁻ (return⁺ p) ≡ p
return⁻∘return⁺ p = refl
return↔ : ∀ {x} → P x ↔ Any P (return x)
return↔ = inverse return⁺ return⁻ return⁻∘return⁺ return⁺∘return⁻
module _ {a p} {A : Set a} where
∷↔ : ∀ (P : Pred A p) {x xs} → (P x ⊎ Any P xs) ↔ Any P (x ∷ xs)
∷↔ P {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) ∎
module _ {ℓ p} {A B : Set ℓ} {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) ∎
module _ {ℓ} {A B : Set ℓ} where
⊛↔ : ∀ {P : B → Set ℓ} {fs : List (A → B)} {xs : List A} →
Any (λ f → Any (P ∘ f) xs) fs ↔ Any P (fs ⊛ xs)
⊛↔ {P = P} {fs} {xs} =
Any (λ f → Any (P ∘ f) xs) fs ↔⟨ Any-cong (λ _ → Any-cong (λ _ → return↔) (_ ∎)) (_ ∎) ⟩
Any (λ f → Any (Any P ∘ return ∘ f) xs) fs ↔⟨ Any-cong (λ _ → >>=↔ ) (_ ∎) ⟩
Any (λ f → Any P (xs >>= return ∘ f)) fs ↔⟨ >>=↔ ⟩
Any P (fs ⊛ xs) ∎
⊛⁺′ : ∀ {P : A → Set ℓ} {Q : B → Set ℓ} {fs : List (A → B)} {xs} →
Any (P ⟨→⟩ Q) fs → Any P xs → Any Q (fs ⊛ xs)
⊛⁺′ pq p =
Inverse.to ⊛↔ ⟨$⟩
Any.map (λ pq → Any.map (λ {x} → pq {x}) p) pq
module _ {ℓ} {A B : Set ℓ} where
⊗↔ : {P : A × B → Set ℓ} {xs : List A} {ys : List B} →
Any (λ x → Any (λ y → P (x , y)) ys) xs ↔ Any P (xs ⊗ ys)
⊗↔ {P} {xs} {ys} =
Any (λ x → Any (λ y → P (x , y)) ys) xs ↔⟨ return↔ ⟩
Any (λ _,_ → Any (λ x → Any (λ y → P (x , y)) ys) xs) (return _,_) ↔⟨ ⊛↔ ⟩
Any (λ x, → Any (P ∘ x,) ys) (_,_ <$> xs) ↔⟨ ⊛↔ ⟩
Any P (xs ⊗ ys) ∎
⊗↔′ : {P : A → Set ℓ} {Q : B → Set ℓ} {xs : List A} {ys : List B} →
(Any P xs × Any Q ys) ↔ Any (P ⟨×⟩ Q) (xs ⊗ ys)
⊗↔′ {P} {Q} {xs} {ys} =
(Any P xs × Any Q ys) ↔⟨ ×↔ ⟩
Any (λ x → Any (λ y → P x × Q y) ys) xs ↔⟨ ⊗↔ ⟩
Any (P ⟨×⟩ Q) (xs ⊗ ys) ∎