module Data.Setoid.Product

import Data.List.Elem
import public Data.Product
import Data.Setoid.Indexed

public export
data Pointwise : (0 _ : (i : a) -> Rel (x i)) -> forall is . Rel (x ^ is) where
  Nil  : Pointwise rel [] []
  (::) : forall b, x, y . {0 xs, ys : b ^ is} -> {0 rel : (i : a) -> Rel (b i)}
    -> (r : rel i x y) -> (rs : Pointwise rel xs ys) -> Pointwise rel (x :: xs) (y :: ys)

-- Destructors

public export
index : Pointwise rel xs ys -> (elem : Elem i is) -> rel i (index xs elem) (index ys elem)
index (r :: rs) Here         = r
index (r :: rs) (There elem) = index rs elem

-- Operators

public export
map : forall b . {0 r, s : (i : a) -> Rel (b i)}
  -> ((i : a) -> {x, y : b i} -> r i x y -> s i x y)
  -> {is : List a} -> {xs, ys : b ^ is} -> Pointwise r {is} xs ys -> Pointwise s {is} xs ys
map f []        = []
map f (r :: rs) = f _ r :: map f rs

-- Relational Properties

pwEqImpliesEqual : Pointwise (\_ => Equal) xs ys -> xs = ys
pwEqImpliesEqual []        = Refl
pwEqImpliesEqual (r :: rs) = trans
  (sym $ consHeadTail _)
  (trans (cong2 (::) r (pwEqImpliesEqual rs)) (consHeadTail _))

equalImpliesPwEq : {xs, ys : b ^ is} -> xs = ys -> Pointwise (\_ => Equal) xs ys
equalImpliesPwEq {xs = []}        {ys = []}        eq = []
equalImpliesPwEq {xs = (x :: xs)} {ys = (y :: ys)} eq =
  cong head eq :: equalImpliesPwEq (cong tail eq)

pwRefl : {0 x : a -> Type} -> {0 rel : (i : a) -> Rel (x i)}
  -> ((i : a) -> (u : x i) -> rel i u u)
  -> {is : List a} -> (xs : x ^ is) -> Pointwise rel xs xs
pwRefl refl []        = []
pwRefl refl (x :: xs) = refl _ x :: pwRefl refl xs

pwSym : {0 x : a -> Type} -> {0 rel : (i : a) -> Rel (x i)}
  -> ((i : a) -> (u, v : x i) -> rel i u v -> rel i v u)
  -> {is : List a} -> {xs, ys : x ^ is} -> Pointwise rel xs ys -> Pointwise rel ys xs
pwSym sym []        = []
pwSym sym (r :: rs) = sym _ _ _ r :: pwSym sym rs

pwTrans : {0 x : a -> Type} -> {0 rel : (i : a) -> Rel (x i)}
  -> ((i : a) -> (u, v, w : x i) -> rel i u v -> rel i v w -> rel i u w)
  -> {is : List a} -> {xs, ys, zs : x ^ is}
  -> Pointwise rel xs ys -> Pointwise rel ys zs -> Pointwise rel xs zs
pwTrans trans []        []        = []
pwTrans trans (r :: rs) (s :: ss) = trans _ _ _ _ r s :: pwTrans trans rs ss

public export
pwEquivalence : IndexedEquivalence a x -> IndexedEquivalence (List a) ((^) x)
pwEquivalence eq = MkIndexedEquivalence
  { relation   = \_ => Pointwise eq.relation
  , reflexive  = \_ => pwRefl eq.reflexive
  , symmetric  = \_, _, _ => pwSym eq.symmetric
  , transitive = \_, _, _, _ => pwTrans eq.transitive
  }

public export
Product : IndexedSetoid a -> IndexedSetoid (List a)
Product x = MkIndexedSetoid ((^) x.U) (pwEquivalence x.equivalence)

-- Introductors and Eliminators

public export
mapIntro'' : forall x, y . {0 rel : (i : a) -> Rel (x i)} -> {0 rel' : (i : a) -> Rel (y i)}
  -> {0 f, g : (i : a) -> x i -> y i}
  -> (cong : (i : a) -> (u, v : x i) -> rel i u v -> rel' i (f i u) (g i v))
  -> {is : _} -> {xs, ys : x ^ is} -> Pointwise rel xs ys -> Pointwise rel' (map f xs) (map g ys)
mapIntro'' cong []        = []
mapIntro'' cong (r :: rs) = cong _ _ _ r :: mapIntro'' cong rs

public export
mapIntro' : forall x, y . {0 rel : (i : a) -> Rel (x i)} -> {0 rel' : (i : a) -> Rel (y i)}
  -> {0 f, g : (i : a) -> x i -> y i}
  -> (cong : (i : a) -> {u, v : x i} -> rel i u v -> rel' i (f i u) (g i v))
  -> {is : _} -> {xs, ys : x ^ is} -> Pointwise rel xs ys -> Pointwise rel' (map f xs) (map g ys)
mapIntro' cong = mapIntro'' (\t, _, _ => cong t)

public export
mapIntro : (f : x ~> y) -> {is : _} -> {xs, ys : x.U ^ is}
  -> Pointwise x.equivalence.relation xs ys
  -> Pointwise y.equivalence.relation (map f.H xs) (map f.H ys)
mapIntro f = mapIntro' f.homomorphic

public export
wrapIntro : forall f . {0 rel : (i : a) -> Rel (x i)} -> {0 xs, ys : (x . f) ^ is}
  -> Pointwise (\i => rel (f i)) xs ys -> Pointwise rel (wrap f xs) (wrap f ys)
wrapIntro []        = []
wrapIntro (r :: rs) = r :: wrapIntro rs

public export
wrapFunc : (0 x : IndexedSetoid b) -> (0 f : a -> b)
  -> Product (reindex f x) ~> reindex (map f) (Product x)
wrapFunc x f = MkIndexedSetoidHomomorphism (\_ => wrap f) (\_, _, _ => wrapIntro)

public export
unwrapIntro : forall f . {0 rel : (i : b) -> Rel (x i)} -> {is : _} -> {0 xs, ys : x ^ map f is}
  -> Pointwise rel xs ys -> Pointwise (\i => rel (f i)) {is = is} (unwrap f xs) (unwrap f ys)
unwrapIntro {is = []}        []        = []
unwrapIntro {is = (i :: is)} (r :: rs) = r :: unwrapIntro rs