module Quox.OPE.Length

import Quox.OPE.Basics

%default total


public export
data Length : Scope a -> Type where
  Z : Length [<]
  S : (s : Length xs) -> Length (xs :< x)
%name Length s
%builtin Natural Length

public export
(.nat) : Length xs -> Nat
(Z).nat   = Z
(S s).nat = S s.nat
%transform "Length.nat" Length.(.nat) xs = believe_me xs

export
0 ok : (s : Length xs) -> s.nat = length xs
ok Z     = Refl
ok (S s) = cong S $ ok s

public export %hint
toLength : (xs : Scope a) -> Length xs
toLength [<]       = Z
toLength (sx :< x) = S (toLength sx)

public export %hint
lengthApp : Length xs -> Length ys -> Length (xs ++ ys)
lengthApp sx Z      = sx
lengthApp sx (S sy) = S (lengthApp sx sy)

public export
0 lengthIrrel : (sx1, sx2 : Length xs) -> sx1 = sx2
lengthIrrel Z       Z       = Refl
lengthIrrel (S sx1) (S sx2) = cong S $ lengthIrrel sx1 sx2