open List;;
type lambda = Var of string
| App of (lambda * lambda)
| Abs of (string * lambda)
| Nil;;
let a_term = App (Abs ("x", App (Var "x", Var "x")),
App (Abs ("x", App (Var "x", Var "x")), Var "x"));;
let y_term = Abs ("f", App (Abs ("x", App (Var ("f"), (App (Var ("x"), Var ("x"))))),
Abs ("x", App (Var ("f"), (App (Var ("x"), Var ("x")))))));;
let b_term = Abs ("w", App (App (Var "x", Abs ("y", App (App (Var "x", Abs ("x", Var "x")), Var "y"))), Abs ("t", Var "y")));;
let redex_term = App (Abs ("x", Var "x"), Var "y");;
let xred = App (Abs ("x", Abs ("y", Var "x")), Var "y");;
let print_list = iter (print_string);;
let fst (x, y) = x;;
let scd (x, y) = y;;
let rec print_term t =
match t with
Var a -> a
|App (a, b) -> "(" ^ (print_term a) ^ " " ^ (print_term b) ^ ")"
|Abs (a, b) -> "(\\" ^ a ^ "." ^ (print_term b) ^ ")"
|Nil -> "NIL";;
let rec isin l a =
match l with
[] -> false
|l1::ls -> if l1 = a then true else isin ls a;;
let unique z =
let rec unique' l acc =
match l with
[] -> acc
|l1::ls -> if isin acc l1 then (unique' ls acc) else (unique' ls (l1::acc))
in unique' z [];;
let intersect c d =
fold_left (fun x y -> if isin d y then y::x else x) [] c;;
let fv z =
let rec fv' exp =
match exp with
Var a -> [a]
|App (a, b) -> (fv' a) @ (fv' b)
|Abs (a, b) -> filter (fun z -> (z <> a)) (fv' b)
|Nil -> []
in unique (fv' z);;
let cx y z =
let rec cx' a x =
if (not (isin (fv a) x)) then [] else (
match a with
Var b -> []
|App (b, c) -> (cx' b x) @ (cx' c x)
|Abs (b, c) -> (cx' c x) @ [b]
|Nil -> [] )
in unique (cx' y z);;
let replace c d f =
let rec rep' a x b =
if (not (isin (fv a) x)) then a else (
match a with
Var z -> b
|App (p, q) -> App (rep' p x b, rep' q x b)
|Abs (y, r) -> Abs (y, rep' r x b)
|Nil -> Nil )
in rep' c d f;;
let rec alpha_eq a b =
match (a, b) with
(Var x, Var y) -> x = y
|(App (p1, q1), App (p2, q2)) -> (alpha_eq p1 p2) && (alpha_eq q1 q2)
|(Abs (x, p1), Abs (y, p2)) -> alpha_eq (replace p1 x (Var y)) p2
|_ -> false;;
let rec bound_replace a x y =
match a with
Abs (z, q) -> if z = x then (Abs (y, replace q z (Var y))) else Abs (z, bound_replace q x y)
|App (p, q) -> App (bound_replace p x y, bound_replace q x y)
|Var z -> Var z
|Nil -> Nil;;
let rec conv_red a x b =
let cross = intersect (cx a x) (fv b) in
match cross with
[] -> a
|l1::ls -> conv_red (bound_replace a x (x ^ "1")) x b;;
let redex_reduction c =
match c with
App (Abs (x, a), b) -> if ((intersect (cx a x) (fv b)) = []) then (replace a x b) else (replace (conv_red a x b) x b)
|_ -> Nil;;
let reduct_once z =
let rec red' t =
match t with
Var a -> (false, Var a)
|Nil -> (false, Nil)
|(App (Abs (x, a), b)) as c -> let (p, q) = (red' a, red' b) in
if (not (fst p)) && (not (fst q)) then (true, redex_reduction c) else
(if (fst p) then (true, App (Abs (x, scd p), b)) else (true, App (Abs (x, a), scd q)))
|(App (a, b)) as c -> let (p, q) = (red' a, red' b) in
if (not (fst p)) && (not (fst q)) then
(false, c) else (if (fst p) then (true, App (scd p, b)) else (true, App (a, scd q)))
|(Abs (x, b)) as c -> let q = red' b in if not (fst q) then (false, c) else (true, Abs (x, scd q))
in scd (red' z);;
print_string ("Original: " ^ (print_term (xred)) ^ "\nReducted: ");;
print_string (print_term (conv_red (Abs ("x", Abs ("y", Var "x"))) "x" (Var "y")));;