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module Generic where
import Category.Functor
import Category.Monad
open import Data.List using (List ; length ; replicate) renaming ([] to []L ; _∷_ to _∷L_)
open import Data.Maybe using (Maybe ; just ; nothing) renaming (setoid to MaybeEq)
open import Data.Nat using (ℕ ; zero ; suc)
open import Data.Product using (_×_ ; _,_)
open import Data.Vec using (Vec ; toList ; fromList ; map) renaming ([] to []V ; _∷_ to _∷V_)
open import Data.Vec.Equality using () renaming (module Equality to VecEq)
open import Function using (_∘_ ; id ; flip)
open import Function.Equality using (_⟶_)
open import Level using () renaming (zero to ℓ₀)
open import Relation.Binary using (Setoid ; module Setoid)
open import Relation.Binary.Core using (_≡_ ; refl)
open import Relation.Binary.Indexed using (_at_) renaming (Setoid to ISetoid)
open import Relation.Binary.PropositionalEquality using (_≗_ ; cong ; subst ; trans ; cong₂) renaming (setoid to EqSetoid)
open Setoid using () renaming (_≈_ to _∋_≈_)
open Category.Functor.RawFunctor {Level.zero} Data.Maybe.functor using (_<$>_)
open Category.Monad.RawMonad {Level.zero} Data.Maybe.monad using (_>>=_)
≡-to-Π : {A B : Set} → (A → B) → EqSetoid A ⟶ EqSetoid B
≡-to-Π f = record { _⟨$⟩_ = f; cong = cong f }
just-injective : {A : Set} → {x y : A} → Maybe.just x ≡ Maybe.just y → x ≡ y
just-injective refl = refl
length-replicate : {A : Set} {a : A} → (n : ℕ) → length (replicate n a) ≡ n
length-replicate zero = refl
length-replicate (suc n) = cong suc (length-replicate n)
mapMV : {A B : Set} {n : ℕ} → (A → Maybe B) → Vec A n → Maybe (Vec B n)
mapMV f []V = just []V
mapMV f (x ∷V xs) = (f x) >>= (λ y → (_∷V_ y) <$> (mapMV f xs))
mapMV-cong : {A B : Set} {f g : A → Maybe B} → f ≗ g → {n : ℕ} → mapMV {n = n} f ≗ mapMV g
mapMV-cong f≗g []V = refl
mapMV-cong f≗g (x ∷V xs) = cong₂ _>>=_ (f≗g x) (cong (flip (_<$>_ ∘ _∷V_)) (mapMV-cong f≗g xs))
mapMV-purity : {A B : Set} {n : ℕ} → (f : A → B) → (v : Vec A n) → mapMV (Maybe.just ∘ f) v ≡ just (map f v)
mapMV-purity f []V = refl
mapMV-purity f (x ∷V xs) = cong (_<$>_ (_∷V_ (f x))) (mapMV-purity f xs)
maybeEq-from-≡ : {A : Set} {a b : Maybe A} → a ≡ b → MaybeEq (EqSetoid A) ∋ a ≈ b
maybeEq-from-≡ {a = just x} {b = .(just x)} refl = just refl
maybeEq-from-≡ {a = nothing} {b = .nothing} refl = nothing
maybeEq-to-≡ : {A : Set} {a b : Maybe A} → MaybeEq (EqSetoid A) ∋ a ≈ b → a ≡ b
maybeEq-to-≡ (just refl) = refl
maybeEq-to-≡ nothing = refl
sequenceV : {A : Set} {n : ℕ} → Vec (Maybe A) n → Maybe (Vec A n)
sequenceV = mapMV id
sequence-map : {A B : Set} {n : ℕ} → (f : A → Maybe B) → sequenceV {n = n} ∘ map f ≗ mapMV f
sequence-map f []V = refl
sequence-map f (x ∷V xs) with f x
sequence-map f (x ∷V xs) | just y = cong (_<$>_ (_∷V_ y)) (sequence-map f xs)
sequence-map f (x ∷V xs) | nothing = refl
subst-cong : {A : Set} → (T : A → Set) → {g : A → A} → {a b : A} → (f : {c : A} → T c → T (g c)) → (p : a ≡ b) →
f ∘ subst T p ≗ subst T (cong g p) ∘ f
subst-cong T f refl _ = refl
subst-fromList : {A : Set} {x y : List A} → (p : y ≡ x) →
subst (Vec A) (cong length p) (fromList y) ≡ fromList x
subst-fromList refl = refl
subst-subst : {A : Set} (T : A → Set) {a b c : A} → (p : a ≡ b) → (p′ : b ≡ c) → (x : T a) →
subst T p′ (subst T p x) ≡ subst T (trans p p′) x
subst-subst T refl p′ x = refl
toList-fromList : {A : Set} → (l : List A) → toList (fromList l) ≡ l
toList-fromList []L = refl
toList-fromList (x ∷L xs) = cong (_∷L_ x) (toList-fromList xs)
toList-subst : {A : Set} → {n m : ℕ} (v : Vec A n) → (p : n ≡ m) →
toList (subst (Vec A) p v) ≡ toList v
toList-subst v refl = refl
VecISetoid : Setoid ℓ₀ ℓ₀ → ISetoid ℕ ℓ₀ ℓ₀
VecISetoid S = record
{ Carrier = Vec (Setoid.Carrier S)
; _≈_ = λ x → VecEq._≈_ S x
; isEquivalence = record
{ refl = VecEq.refl S _
; sym = VecEq.sym S
; trans = VecEq.trans S }
}
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