Converted to use reverse de bruijn representation.
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@ -1,19 +1,26 @@
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{-# LANGUAGE TemplateHaskell, TypeFamilies, DeriveFunctor, DeriveFoldable, DeriveTraversable, MultiWayIf #-}
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{-# LANGUAGE TemplateHaskell, TypeFamilies, DeriveFunctor, DeriveFoldable, DeriveTraversable, MultiWayIf, LambdaCase, BlockArguments #-}
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module UntypedLambdaCalculus (Expr (Free, Var, Lam, App), ReaderAlg, eval, cataReader) where
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import Control.Monad.Reader (Reader, runReader, local, reader)
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import Control.Monad.Reader (Reader, runReader, local, reader, ask, asks)
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import Control.Monad.Writer (Writer, runWriter, listen, tell)
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import Data.Foldable (fold)
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import Data.Functor ((<&>))
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import Data.Functor.Foldable (Base, Recursive, cata, embed, project)
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import Data.Functor.Foldable.TH (makeBaseFunctor)
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import Data.Monoid (Any (Any, getAny))
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-- | A lambda calculus expression where variables are identified
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-- | by their distance from their binding site (De Bruijn indices).
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-- | An expression using the Reverse De Bruijn representation.
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-- | Like De Bruijn representation, variables are named according to their binding site.
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-- | However, instead of being named by the number of binders between variable and its binder,
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-- | here variables are represented by the distance between their binder and the top level.
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data Expr = Free String
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-- Var index is bound `index` in from the outermost binder.
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-- The outermost binder's name is the last element of the list.
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| Var Int
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| Lam String Expr
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-- This lambda is `index` bindings away from the outermost binding.
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-- If the index is `0`, then this is the outermost binder.
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| Lam String Int Expr
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| App Expr Expr
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| Subst Int Expr Expr
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makeBaseFunctor ''Expr
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@ -23,76 +30,68 @@ type ReaderAlg f s a = Algebra f (Reader s a)
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cataReader :: Recursive r => ReaderAlg (Base r) s a -> s -> r -> a
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cataReader f initialState x = runReader (cata f x) initialState
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indexOr :: a -> Int -> [a] -> a
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indexOr def index xs
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| index < length xs && index >= 0 = xs !! index
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| otherwise = def
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instance Show Expr where
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show = cataReader alg []
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where alg :: ReaderAlg ExprF [String] String
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alg (FreeF v) = return v
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alg (VarF i) = reader (\vars -> vars !! i ++ ':' : show i)
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alg (LamF v e) = do
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body <- local (v :) e
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return $ "(\\" ++ v ++ ". " ++ body ++ ")"
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alg (AppF f' x') = do
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f <- f'
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x <- x'
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return $ "(" ++ f ++ " " ++ x ++ ")"
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alg (SubstF index val' body') = do
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body <- local ("SUBSTVAR" :) body'
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val <- val'
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return $ body ++ "[ " ++ show index ++ " := " ++ val ++ " ]"
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-- | Is the innermost bound variable of this subexpression (`Var 0`) used in its body?
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-- | For example: in `\x. a:1 x:0 b:2`, `x:0` is bound in `a:1 x:0 b:2`.
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-- | On the other hand, in `\x. a:3 b:2 c:1`, it is not.
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bound :: Expr -> Bool
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bound = getAny . cataReader alg 0
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where alg :: ReaderAlg ExprF Int Any
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alg (VarF index) = reader (Any . (== index))
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alg (LamF _ e) = local (+ 1) e
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alg x = fold <$> sequenceA x
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-- | Opposite of `bound`.
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unbound :: Expr -> Bool
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unbound = not . bound
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-- | When we bind a new variable, we enter a new scope.
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-- | Since variables are identified by their distance from their binder,
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-- | we must increment them to account for the incremented distance,
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-- | thus embedding them into the new expression.
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liftExpr :: Int -> Expr -> Expr
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liftExpr n (Var i) = Var $ i + n
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liftExpr _ o@(Lam _ _) = o
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liftExpr n x = embed $ fmap (liftExpr n) $ project x
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substitute :: Int -> Expr -> Expr -> Expr
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substitute index val v@(Var index')
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| index == index' = val
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| index < index' = Var $ index' - 1
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| otherwise = v
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substitute index val (Lam name body) = Lam name $ Subst (index + 1) (liftExpr 1 val) body
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substitute index val (Subst index2 val2 body) =
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substitute index val $ substitute index2 val2 body
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substitute index val x = embed $ fmap (Subst index val) $ project x
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etaReduce :: String -> Expr -> Expr
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-- `\x. f x -> f` if `x` is not bound in `f`.
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etaReduce name o@(App f (Var 0))
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| unbound f = eval $ Subst 0 undefined f
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| otherwise = Lam name $ o
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-- `\x y. f y -> \x. f` if `y` is not bound in `f`;
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-- the resultant term may itself be eta-reducible.
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etaReduce name (Lam name' body') = case etaReduce name' body' of
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body@(Lam _ _) -> Lam name body
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body -> etaReduce name body
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etaReduce name body = Lam name body
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betaReduce :: Expr -> Expr -> Expr
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betaReduce f' x = case eval f' of
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Lam _ e -> eval $ Subst 0 x e
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f -> App f $ eval x
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-- | Evaluate an expression to normal form.
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show = flip cataReader [] \case
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FreeF name -> return name
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VarF index -> asks (\boundNames -> indexOr "" (length boundNames - index - 1) boundNames)
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<&> \name -> name ++ ":" ++ show index
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LamF name index body' -> local (name :) body' <&> \body ->
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"(\\" ++ name ++ ":" ++ show index ++ ". " ++ body ++ ")"
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AppF f' x' -> f' >>= \f -> x' <&> \x ->
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"(" ++ f ++ " " ++ x ++ ")"
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eval :: Expr -> Expr
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eval (Subst index val body) = eval $ substitute index val body
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eval (App f x) = betaReduce f x
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eval (Lam name body) = etaReduce name body
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eval o = o
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eval x = case reduce innerReduced of
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Just expr -> eval expr
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Nothing -> x
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where innerReduced = embed $ fmap eval $ project x
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reduce :: Expr -> Maybe Expr
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reduce (Lam name index body) = etaReduce name index body
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reduce (App f x) = betaReduce f x
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reduce _ = Nothing
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betaReduce :: Expr -> Expr -> Maybe Expr
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betaReduce (Lam name index body) x = Just $ subst index x body
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betaReduce _ _ = Nothing
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etaReduce :: String -> Int -> Expr -> Maybe Expr
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etaReduce name index body@(App f (Var index'))
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-- If the variable bound by this lambda is only used in the right hand of the outermost app,
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-- then we may delete this function. The absolute position of all binding terms inside
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-- this one has been decreased by the removal of this lambda, and must be renumbered.
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| index == index' && unbound index f = Just $ subst index undefined f
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| otherwise = Nothing
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etaReduce _ _ _ = Nothing
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unbound :: Int -> Expr -> Bool
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unbound index = not . bound index
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bound :: Int -> Expr -> Bool
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bound index = getAny . cata \case
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VarF index' -> Any $ index == index'
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expr -> fold expr
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embedExpr :: Int -> Expr -> Expr
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embedExpr index (Free name) = Free name
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embedExpr index (Var index')
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| index' >= index = Var $ index' + 1
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| otherwise = Var index'
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embedExpr index (App f x) = App (embedExpr index f) (embedExpr index x)
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embedExpr index (Lam name index' body) = Lam name (index' + 1) $ embedExpr index body
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subst :: Int -> Expr -> Expr -> Expr
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subst index val (Free name) = Free name
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subst index val (Var index')
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| index == index' = val
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-- There is now one fewer binding site between the innermost binding site and `index`,
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-- thus if the binding site is further in than ours, it must be decremented.
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| index < index' = Var $ index' - 1
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| otherwise = Var index'
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subst index val (App f x) = App (subst index val f) (subst index val x)
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subst index val (Lam name index' body) = Lam name (index' - 1) $ subst index (embedExpr index val) body
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@ -2,7 +2,7 @@
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module UntypedLambdaCalculus.Parser (parseExpr) where
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import Control.Applicative (liftA2)
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import Control.Monad.Reader (local, asks)
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import Control.Monad.Reader (local, ask)
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import Data.List (foldl', elemIndex)
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import Data.Functor.Foldable.TH (makeBaseFunctor)
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import Text.Parsec (SourceName, ParseError, (<|>), many, sepBy, letter, alphaNum, char, between, spaces, parse, string)
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@ -61,19 +61,25 @@ consumesInput :: Parser Ast
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consumesInput = let_ <|> var <|> lam <|> parens app
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toExpr :: Ast -> Expr
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toExpr = cataReader alg []
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toExpr = cataReader alg (0, [])
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where
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alg :: ReaderAlg AstF [String] Expr
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alg :: ReaderAlg AstF (Int, [String]) Expr
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alg (AstVarF varName) = do
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bindingSite <- asks (elemIndex varName)
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return $ case bindingSite of
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Just index -> Var index
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(_, bound) <- ask
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return $ case varName `elemIndex` bound of
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Just index -> Var $ length bound - index - 1
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Nothing -> Free varName
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alg (AstLamF vars body) = foldr (\v e -> Lam v <$> local (v :) e) body vars
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alg (AstAppF es) = foldl' App (Lam "x" (Var 0)) <$> sequenceA es
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alg (AstLamF vars body) =
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foldr (\v e -> do
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(index, bound) <- ask
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Lam v index <$> local (\_ -> (index + 1, v : bound)) e) body vars
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alg (AstAppF es) = do
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(index, _) <- ask
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foldl' App (Lam "x" index (Var index)) <$> sequenceA es
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alg (AstLetF var val body) = do
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body' <- local (var :) body
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App (Lam var body') <$> val
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(index, bound) <- ask
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body' <- local (\_ -> (index + 1, var : bound)) body
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App (Lam var index body') <$> val
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-- | Since applications do not require parentheses and can contain only a single item,
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-- | the `app` parser is sufficient to parse any expression at all.
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