module Soat.Data.Product

import Control.Relation
import Data.List.Elem

%default total

infix 10 ^
infix 5 ++

-- Definitions

public export
data (^) : (0 _ : a -> Type) -> List a -> Type where
  Nil  : x ^ []
  (::) : {0 x : a -> Type} -> x i -> x ^ is -> x ^ (i :: is)

public export
ary : List Type -> Type -> Type
ary []        y = y
ary (x :: xs) y = x -> ary xs y

-- Conversions

public export
uncurry : map x is `ary` a -> x ^ is -> a
uncurry f []        = f
uncurry f (x :: xs) = uncurry (f x) xs

-- Destructors

public export
head : x ^ (i :: is) -> x i
head (x :: xs) = x

public export
tail : x ^ (i :: is) -> x ^ is
tail (x :: xs) = xs

public export
consHeadTail : (xs : x ^ (i :: is)) -> head xs :: tail xs = xs
consHeadTail (x :: xs) = Refl

public export
index : x ^ is -> Elem i is -> x i
index xs Here         = head xs
index xs (There elem) = index (tail xs) elem

-- Constructors

public export
tabulate : {is : List a} -> ({i : a} -> Elem i is -> x i) -> x ^ is
tabulate {is = []}        f = []
tabulate {is = (i :: is)} f = f Here :: tabulate (f . There)

public export
indexTabulate : forall x . (0 f : {i : a} -> Elem i is -> x i) -> (elem : Elem i is)
  -> index (tabulate f) elem = f elem
indexTabulate f Here         = Refl
indexTabulate f (There elem) = indexTabulate (f . There) elem

-- Operations

public export
map : (f : (i : a) -> x i -> y i) -> {is : List a} -> x ^ is -> y ^ is
map f []        = []
map f (x :: xs) = f _ x :: map f xs

public export
(++) : x ^ is -> x ^ js -> x ^ (is ++ js)
(++) []        ys = ys
(++) (x :: xs) ys = x :: (xs ++ ys)

-- Relations

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

  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

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

  public export
  equalImpliesPwEq : {is : List a} -> {0 xs, ys : b ^ is}
    -> xs = ys -> Pointwise (\_ => Equal) {is} xs ys
  equalImpliesPwEq {is = []}        Refl = []
  equalImpliesPwEq {is = (x :: xs)} eq   = cong head eq :: equalImpliesPwEq (cong tail eq)

  -- FIXME: work out how to expose interfaces

  public export
  pwRefl : {0 x : a -> Type} -> {0 rel : (i : a) -> Rel (x i)}
    -> ((i : a) -> Reflexive (x i) (rel i))
    -> {is : List a} -> {xs : x ^ is} -> Pointwise rel xs xs
  pwRefl f {xs = []}        = []
  pwRefl f {xs = (x :: xs)} = reflexive :: pwRefl f

  public export
  pwSym : {0 x : a -> Type} -> {0 rel : (i : a) -> Rel (x i)}
    -> ((i : a) -> Symmetric (x i) (rel i))
    -> {is : List a} -> {xs, ys : x ^ is} -> Pointwise rel xs ys -> Pointwise rel ys xs
  pwSym f []        = []
  pwSym f (r :: rs) = symmetric r :: pwSym f rs

  public export
  pwTrans : {0 x : a -> Type} -> {0 rel : (i : a) -> Rel (x i)}
    -> ((i : a) -> Transitive (x i) (rel i))
    -> {is : List a} -> {xs, ys, zs : x ^ is}
    -> Pointwise rel xs ys -> Pointwise rel ys zs -> Pointwise rel xs zs
  pwTrans f []        []        = []
  pwTrans f (r :: rs) (s :: ss) = transitive r s :: pwTrans f rs ss