--- /dev/null
+(* *)
+
+module Private = struct
+ type var_t = int*string
+ let var v = (0, v)
+ let string_of_var (i, v) = v ^ String.make i '\''
+ let equal_var (i1, v1) (i2, v2) = i1 == i2 && (String.compare v1 v2 == 0)
+
+ type lambda_t = [ `Var of var_t | `Lam of var_t * lambda_t | `App of lambda_t * lambda_t ]
+
+(* DeBruijn terms
+ * substitution and translation algorithms from Chris Hankin, An Introduction to Lambda Calculi for Comptuer Scientists
+ *)
+
+ type debruijn_t = [ `Db_free of var_t | `Db_index of int | `Db_lam of debruijn_t | `Db_app of debruijn_t*debruijn_t ]
+
+ let debruijn_subst (expr : debruijn_t) (m : int) (new_term : debruijn_t) =
+ let rec renumber m i = function
+ | `Db_free _ as term -> term
+ | `Db_index j as term when j < i -> term
+ | `Db_index j -> `Db_index (j + m - 1)
+ | `Db_app(left, right) -> `Db_app(renumber m i left, renumber m i right)
+ | `Db_lam body -> `Db_lam(renumber m (i+1) body)
+ in let rec loop m = function
+ | `Db_free _ as term -> term
+ | `Db_index j as term when j < m -> term
+ | `Db_index j when j > m -> `Db_index (j-1)
+ | `Db_index j -> renumber j 1 new_term
+ | `Db_app(left, right) -> `Db_app(loop m left, loop m right)
+ | `Db_lam body -> `Db_lam(loop (m+1) body)
+ in loop m expr
+
+ let debruijn (expr : lambda_t) : debruijn_t =
+ let pos seq (target : var_t) =
+ let rec loop (i : int) = function
+ | [] -> `Db_free target
+ | x::xs when equal_var x target -> `Db_index i
+ | _::xs -> loop (i+1) xs
+ in loop 1 seq
+ in let rec loop seq = function
+ | `Var v -> pos seq v
+ | `Lam (v, body) -> `Db_lam(loop (v::seq) body)
+ | `App (left, right) -> `Db_app(loop seq left, loop seq right)
+ in loop [] expr
+
+ let rec dbruijn_equal (t1 : debruijn_t) (t2 : debruijn_t) = match (t1, t2) with
+ | (`Db_free v1, `Db_free v2) -> equal_var v1 v2
+ | (`Db_index j1, `Db_index j2) -> j1 == j2
+ | (`Db_app(left1, right1), `Db_app(left2, right2)) -> dbruijn_equal left1 left2 && dbruijn_equal right1 right2
+ | (`Db_lam(body1), `Db_lam(body2)) -> dbruijn_equal body1 body2
+ | _ -> false
+
+ let rec debruijn_contains (t1 : debruijn_t) (t2 : debruijn_t) = match (t1, t2) with
+ | (`Db_free v1, `Db_free v2) -> equal_var v1 v2
+ | (`Db_index j1, `Db_index j2) -> j1 == j2
+ | (`Db_app(left1, right1), `Db_app(left2, right2)) when dbruijn_equal left1 left2 && dbruijn_equal right1 right2 -> true
+ | (`Db_app(left, right), term2) -> debruijn_contains left term2 || debruijn_contains right term2
+ | (`Db_lam(body1), `Db_lam(body2)) when dbruijn_equal body1 body2 -> true
+ | (`Db_lam(body1), term2) -> debruijn_contains body1 term2
+ | _ -> false
+
+
+ (* non-normalizing string_of_lambda *)
+ let string_of_lambda (expr : lambda_t) =
+ let rec top = function
+ | `Var v -> string_of_var v
+ | `Lam _ as term -> "fun " ^ dotted term
+ | `App ((`App _ as left), right) -> top left ^ " " ^ atom right
+ | `App (left, right) -> atom left ^ " " ^ atom right
+ and atom = function
+ | `Var v -> string_of_var v
+ | `Lam _ as term -> "(fun " ^ dotted term ^ ")"
+ | `App _ as term -> "(" ^ top term ^ ")"
+ and dotted = function
+ | `Lam (v, (`Lam _ as body)) -> (string_of_var v) ^ " " ^ dotted body
+ | `Lam (v, body) -> (string_of_var v) ^ " -> " ^ top body
+ in top expr
+
+(*
+ * substitution and normal-order evaluator based on Haskell version by Oleg Kisleyov
+ * http://okmij.org/ftp/Computation/lambda-calc.html#lambda-calculator-haskell
+ *)
+
+(* if v occurs free_in term, returns Some v' where v' is the highest-tagged
+ * variable with the same name as v occurring (free or bound) in term
+ *)
+ let free_in ((tag, name) as v) term =
+ let rec loop = function
+ | `Var((tag', name') as v') ->
+ if name <> name' then false, v
+ else if tag = tag' then true, v
+ else false, v'
+ | `App(left, right) ->
+ let left_bool, ((left_tag, _) as left_v) = loop left in
+ let right_bool, ((right_tag, _) as right_v) = loop right in
+ left_bool || right_bool, if left_tag > right_tag then left_v else right_v
+ | `Lam(v', _) when equal_var v v' -> (false, v)
+ | `Lam(_, body) -> loop body
+ in match loop term with
+ | false, _ -> None
+ | true, v -> Some v
+
+ let rec subst v new_term term = match new_term with
+ | `Var v' when equal_var v v' -> term
+ | _ -> (match term with
+ | `Var v' when equal_var v v' -> new_term
+ | `Var _ -> term
+ | `App(left, right) -> `App(subst v new_term left, subst v new_term right)
+ | `Lam(v', _) when equal_var v v' -> term
+ (* if x is free in the inserted term new_term, a capture is possible *)
+ | `Lam(v', body) ->
+ (match free_in v' new_term with
+ (* v' not free in new_term, can substitute new_term for v without any captures *)
+ | None -> `Lam(v', subst v new_term body)
+ (* v' free in new_term, need to alpha-convert *)
+ | Some max_x ->
+ let bump_tag (tag, name) (tag', _) =
+ (max tag tag') + 1, name in
+ let bump_tag' ((_, name) as v1) ((_, name') as v2) =
+ if (String.compare name name' == 0) then bump_tag v1 v2 else v1 in
+ (* bump v' > max_x from new_term, then check whether
+ * it also needs to be bumped > v
+ *)
+ let uniq_x = bump_tag' (bump_tag v' max_x) v in
+ let uniq_x' = (match free_in uniq_x body with
+ | None -> uniq_x
+ (* bump uniq_x > max_x' from body *)
+ | Some max_x' -> bump_tag uniq_x max_x'
+ ) in
+ (* alpha-convert body *)
+ let body' = subst v' (`Var uniq_x') body in
+ (* now substitute new_term for v *)
+ `Lam(uniq_x', subst v new_term body')
+ )
+ )
+
+ let check_eta = function
+ | `Lam(v, `App(body, `Var u)) when equal_var v u && free_in v body = None -> body
+ | (_ : lambda_t) as term -> term
+
+
+
+
+ exception Lambda_looping;;
+
+ let eval ?(eta=false) (expr : lambda_t) : lambda_t =
+ let rec looping (body : debruijn_t) = function
+ | [] -> false
+ | x::xs when dbruijn_equal body x -> true
+ | _::xs -> looping body xs
+ in let rec loop (stack : lambda_t list) (body : lambda_t) =
+ match body with
+ | `Var v as term -> unwind term stack
+ | `App(left, right) -> loop (right::stack) left
+ | `Lam(v, body) -> (match stack with
+ | [] ->
+ let term = (`Lam(v, loop [] body)) in
+ if eta then check_eta term else term
+ | x::xs -> loop xs (subst v x body)
+ )
+ and unwind left = function
+ | [] -> left
+ | x::xs -> unwind (`App(left, loop [] x)) xs
+ in loop [] expr
+
+
+ let cbv ?(aggressive=true) (expr : lambda_t) : lambda_t =
+ let rec loop = function
+ | `Var v as term -> term
+ | `App(left, right) ->
+ let right' = loop right in
+ (match loop left with
+ | `Lam(v, body) -> loop (subst v right' body)
+ | _ as left' -> `App(left', right')
+ )
+ | `Lam(v, body) as term ->
+ if aggressive then `Lam(v, loop body)
+ else term
+ in loop expr
+
+
+
+
+
+ (*
+
+ (* (Oleg's version of) Ken's evaluator; doesn't seem to work -- requires laziness? *)
+ let eval' ?(eta=false) (expr : lambda_t) : lambda_t =
+ let rec loop = function
+ | `Var v as term -> term
+ | `Lam(v, body) ->
+ let term = (`Lam(v, loop body)) in
+ if eta then check_eta term else term
+ | `App(`App _ as left, right) ->
+ (match loop left with
+ | `Lam _ as redux -> loop (`App(redux, right))
+ | nonred_head -> `App(nonred_head, loop right)
+ )
+ | `App(left, right) -> `App(left, loop right)
+ in loop expr
+
+
+ module Sorted = struct
+ let rec cons y = function
+ | x :: _ as xs when x = y -> xs
+ | x :: xs when x < y -> x :: cons y xs
+ | xs [* [] or x > y *] -> y :: xs
+
+ let rec mem y = function
+ | x :: _ when x = y -> true
+ | x :: xs when x < y -> mem y xs
+ | _ [* [] or x > y *] -> false
+
+ let rec remove y = function
+ | x :: xs when x = y -> xs
+ | x :: xs when x < y -> x :: remove y xs
+ | xs [* [] or x > y *] -> xs
+
+ let rec merge x' y' = match x', y' with
+ | [], ys -> ys
+ | xs, [] -> xs
+ | x::xs, y::ys ->
+ if x < y then x :: merge xs y'
+ else if x = y then x :: merge xs ys
+ else [* x > y *] y :: merge x' ys
+ end
+
+ let free_vars (expr : lambda_t) : string list =
+ let rec loop = function
+ | `Var x -> [x]
+ | `Lam(x, t) -> Sorted.remove x (loop t)
+ | `App(t1, t2) -> Sorted.merge (loop t1) (loop t2)
+ in loop expr
+
+ let free_in v (expr : lambda_t) =
+ Sorted.mem v (free_vars t)
+
+ let new_var =
+ let counter = ref 0 in
+ fun () -> (let z = !counter in incr counter; "_v"^(string_of_int z))
+
+ ...
+ | `Lam(x, body) as term when not (free_in v body) -> term
+ | `Lam(y, body) when not (free_in y new_term) -> `Lam(y, subst v new_term body)
+ | `Lam(y, body) ->
+ let z = new_var () in
+ subst v new_term (`Lam(z, subst y (`Var z) body))
+ *)
+
+
+
+ (*
+
+ let bound_vars (expr : lambda_t) : string list =
+ let rec loop = function
+ | `Var x -> []
+ | `Lam(x, t) -> Sorted.cons x (loop t)
+ | `App(t1, t2) -> Sorted.merge (loop t1) (loop t2)
+ in loop expr
+
+ let reduce_cbv ?(aggressive=true) (expr : lambda_t) : lambda_t =
+ let rec loop = function
+ | `Var x as term -> term
+ | `App(t1, t2) ->
+ let t2' = loop t2 in
+ (match loop t1 with
+ | `Lam(x, t) -> loop (subst x t2' t)
+ | _ as term -> `App(term, t2')
+ )
+ | `Lam(x, t) as term ->
+ if aggressive then `Lam(x, loop t)
+ else term
+ in loop expr
+
+ let reduce_cbn (expr : lambda_t) : lambda_t =
+ let rec loop = function
+ | `Var x as term -> term
+ | `Lam(v, body) ->
+ check_eta (`Lam(v, loop body))
+ | `App(t1, t2) ->
+ (match loop t1 with
+ | `Lam(x, t) -> loop (subst x t2 t)
+ | _ as term -> `App(term, loop t2)
+ )
+ in loop expr
+
+ *)
+
+
+ (*
+
+ type env_t = (string * lambda_t) list
+
+ let subst body x value =
+ ((fun env ->
+ let new_env = (x, value) :: env in
+ body new_env) : env_t -> lambda_t)
+
+ type strategy_t = By_value | By_name
+
+ let eval (strategy : strategy_t) (expr : lambda_t) : lambda_t =
+ in let rec inner = function
+ | `Var x as t ->
+ (fun env ->
+ try List.assoc x env with
+ | Not_found -> t)
+ | `App(t1, value) ->
+ (fun env ->
+ let value' =
+ if strategy = By_value then inner value env else value in
+ (match inner t1 env with
+ | `Lam(x, body) ->
+ let body' = (subst (inner body) x value' env) in
+ if strategy = By_value then body' else inner body' env
+ | (t1' : lambda_t) -> `App(t1', inner value env)
+ )
+ )
+ | `Lam(x, body) ->
+ (fun env ->
+ let v = new_var () in
+ `Lam(v, inner body ((x, `Var v) :: env)))
+ in inner expr ([] : env_t)
+
+ let pp_env env =
+ let rec loop acc = function
+ | [] -> acc
+ | (x, term)::es -> loop ((x ^ "=" ^ string_of_lambda term) :: acc) es
+ in "[" ^ (String.concat ", " (loop [] (List.rev env))) ^ "]"
+
+ let eval (strategy : strategy_t) (expr : lambda_t) : lambda_t =
+ let new_var =
+ let counter = ref 0 in
+ fun () -> (let z = !counter in incr counter; "_v"^(string_of_int z))
+ in let rec inner term =
+ begin
+ Printf.printf "starting [ %s ]\n" (string_of_lambda term);
+ let res = match term with
+ | `Var x as t ->
+ (fun env ->
+ try List.assoc x env with
+ | Not_found -> t)
+ | `App(t1, value) ->
+ (fun env ->
+ let value' =
+ if strategy = By_value then inner value env else value in
+ (match inner t1 env with
+ | `Lam(x, body) ->
+ let body' = (subst (inner body) x value' env) in
+ if strategy = By_value then body' else inner body' env
+ | (t1' : lambda_t) -> `App(t1', inner value env)
+ )
+ )
+ | `Lam(x, body) ->
+ (fun env ->
+ let v = new_var () in
+ `Lam(v, inner body ((x, `Var v) :: env)))
+ in
+ (fun env ->
+ (Printf.printf "%s with %s => %s\n" (string_of_lambda term) (pp_env env) (string_of_lambda (res env)); res env))
+ end
+ in inner expr ([] : env_t)
+
+ *)
+
+ let normal ?(eta=false) expr = eval ~eta expr
+
+ let normal_string_of_lambda ?(eta=false) (expr : lambda_t) =
+ string_of_lambda (normal ~eta expr)
+
+ let rec to_int expr = match expr with
+ | `Lam(s, `Lam(z, `Var z')) when z' = z -> 0
+ | `Lam(s, `Var s') when equal_var s s' -> 1
+ | `Lam(s, `Lam(z, `App (`Var s', t))) when s' = s -> 1 + to_int (`Lam(s, `Lam(z, t)))
+ | _ -> failwith (normal_string_of_lambda expr ^ " is not a church numeral")
+
+ let int_of_lambda ?(eta=false) (expr : lambda_t) =
+ to_int (normal ~eta expr)
+
+end
+
+type lambda_t = Private.lambda_t
+open Private
+let var = var
+let pp, pn, pi = string_of_lambda, normal_string_of_lambda, int_of_lambda
+let pnv, piv= (fun expr -> string_of_lambda (cbv expr)), (fun expr -> to_int (cbv expr))
+let debruijn, dbruijn_equal, debruijn_contains = debruijn, dbruijn_equal, debruijn_contains
+
+let alpha_eq x y = dbruijn_equal (debruijn x) (debruijn y)
+