{-# LANGUAGE GADTs #-}

module Expr where

import Data.Function (on)

data Expr where
    Var :: Integer -> String -> Expr
    Star :: Expr
    Square :: Expr
    App :: Expr -> Expr -> Expr
    Abs :: String -> Expr -> Expr -> Expr
    Pi :: String -> Expr -> Expr -> Expr
    deriving (Show, Eq)

occursFree :: Integer -> Expr -> Bool
occursFree n (Var k _) = n == k
occursFree _ Star = False
occursFree _ Square = False
occursFree n (App a b) = on (||) (occursFree n) a b
occursFree n (Abs _ a b) = occursFree n a || occursFree (n + 1) b
occursFree n (Pi _ a b) = occursFree n a || occursFree (n + 1) b

{- --------------------- PRETTY PRINTING ----------------------------- -}

parenthesize :: String -> String
parenthesize s = "(" ++ s ++ ")"

helper :: Integer -> Expr -> String
helper _ (Var _ s) = s
helper _ Star = "*"
helper _ Square = "□"
helper k (App e1 e2) = if k >= 3 then parenthesize res else res
  where
    res = helper 3 e1 ++ " " ++ helper 4 e2
helper k (Pi x ty b) = if k >= 2 then parenthesize res else res
  where
    res =
        if occursFree 0 b
            then "∏" ++ x ++ " : " ++ helper 0 ty ++ " . " ++ helper 0 b
            else helper 3 ty ++ " -> " ++ helper 2 b
helper k (Abs x ty b) = if k >= 1 then parenthesize res else res
  where
    res = "λ" ++ x ++ " : " ++ helper 0 ty ++ " . " ++ helper 0 b

pretty :: Expr -> String
pretty = helper 0

{- --------------- ACTUAL MATH STUFF ---------------- -}

isSort :: Expr -> Bool
isSort Star = True
isSort Square = True
isSort _ = False

incIndices :: Expr -> Expr
incIndices (Var n x) = Var (n + 1) x
incIndices Star = Star
incIndices Square = Square
incIndices (App m n) = App (incIndices m) (incIndices n)
incIndices (Abs x m n) = Abs x (incIndices m) (incIndices n)
incIndices (Pi x m n) = Pi x (incIndices m) (incIndices n)

-- substitute s for 0 *AND* decrement indices; only use after reducing a redex.
subst :: Expr -> Expr -> Expr
subst s (Var 0 _) = s
subst _ (Var n s) = Var (n - 1) s
subst _ Star = Star
subst _ Square = Square
subst s (App m n) = App (subst s m) (subst s n)
subst s (Abs x m n) = Abs x (subst s m) (subst s n)
subst s (Pi x m n) = Pi x (subst s m) (subst s n)

substnd :: Expr -> Expr -> Expr
substnd s (Var 0 _) = s
substnd _ (Var n s) = Var (n - 1) s
substnd _ Star = Star
substnd _ Square = Square
substnd s (App m n) = App (substnd s m) (substnd s n)
substnd s (Abs x m n) = Abs x (substnd s m) (substnd s n)
substnd s (Pi x m n) = Pi x (substnd s m) (substnd s n)

betaReduce :: Expr -> Expr
betaReduce (Var k s) = Var k s
betaReduce Star = Star
betaReduce Square = Square
betaReduce (App (Abs _ _ v) n) = subst n v
betaReduce (App m n) = App (betaReduce m) (betaReduce n)
betaReduce (Abs x t v) = Abs x (betaReduce t) (betaReduce v)
betaReduce (Pi x t v) = Pi x (betaReduce t) (betaReduce v)

betaNF :: Expr -> Expr
betaNF e = if e == e' then e else betaNF e'
  where
    e' = betaReduce e

betaEquiv :: Expr -> Expr -> Bool
betaEquiv = on (==) betaNF