--- /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 ]
+
+ type debruijn_t = [ `Var of var_t | `DVar of int | `DLam of debruijn_t | `DApp of debruijn_t*debruijn_t ]
+
+ let db_subst (expr : debruijn_t) (m : int) (repl : debruijn_t) =
+ let rec rename m i = function
+ | `Var _ as term -> term
+ | `DVar j as term when j < i -> term
+ | `DVar j -> `DVar (j + m - 1)
+ | `DApp(n1,n2) -> `DApp(rename m i n1, rename m i n2)
+ | `DLam n -> `DLam(rename m (i+1) n)
+ in let rec loop m = function
+ | `Var _ as term -> term
+ | `DVar n as term when n < m -> term
+ | `DVar n when n > m -> `DVar (n-1)
+ | `DVar n -> rename n 1 repl
+ | `DApp(m1,m2) -> `DApp(loop m m1, loop m m2)
+ | `DLam mterm -> `DLam(loop (m+1) mterm)
+ in loop m expr
+
+ let db (expr : lambda_t) : debruijn_t =
+ let pos seq (target : var_t) handler default =
+ let rec loop (i : int) = function
+ | [] -> default
+ | x::xs when equal_var x target -> handler i
+ | _::xs -> loop (i+1) xs
+ in loop 1 seq
+ in let rec loop seq = function
+ | `Var v as term -> pos seq v (fun i -> `DVar i) term
+ | `Lam (v,t) -> `DLam(loop (v::seq) t)
+ | `App (t1,t2) -> `DApp(loop seq t1, loop seq t2)
+ in loop [] expr
+
+ let rec db_equal (t1 : debruijn_t) (t2 : debruijn_t) = match (t1,t2) with
+ | (`Var v1,`Var v2) -> equal_var v1 v2
+ | (`DVar i1, `DVar i2) -> i1 == i2
+ | (`DApp(m1,m2),`DApp(n1,n2)) -> db_equal m1 n1 && db_equal m2 n2
+ | (`DLam(t1),`DLam(t2)) -> db_equal t1 t2
+ | _ -> false
+
+ let rec db_contains (t1 : debruijn_t) (t2 : debruijn_t) = match (t1,t2) with
+ | (`Var v1,`Var v2) -> equal_var v1 v2
+ | (`DVar i1, `DVar i2) -> i1 == i2
+ | (`DApp(m1,m2),`DApp(n1,n2)) when db_equal m1 n1 && db_equal m2 n2 -> true
+ | (`DApp(m1,m2), term) -> db_contains m1 term || db_contains m2 term
+ | (`DLam(t1),`DLam(t2)) when db_equal t1 t2 -> true
+ | (`DLam(t1), term) -> db_contains t1 term
+ | _ -> 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 t -> "fun " ^ funct t
+ | `App ((`App _ as t1),t2) -> top t1 ^ " " ^ atom t2
+ | `App (t1,t2) -> atom t1 ^ " " ^ atom t2
+ and atom = function
+ | `Var v -> string_of_var v
+ | `Lam _ as t -> "(fun " ^ funct t ^ ")"
+ | `App _ as t -> "(" ^ top t ^ ")"
+ and funct = function
+ | `Lam (v,(`Lam _ as t)) -> (string_of_var v) ^ " " ^ funct t
+ | `Lam (v,t) -> (string_of_var v) ^ " -> " ^ top t
+ in top expr
+
+
+ (* evaluator based on http://okmij.org/ftp/Haskell/Lambda_calc.lhs *)
+
+ (* 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(t1, t2) ->
+ let b1, ((tag1, _) as v1) = loop t1 in
+ let b2, ((tag2, _) as v2) = loop t2 in
+ b1 || b2, if tag1 > tag2 then v1 else v2
+ | `Lam(x, _) when x = v -> (false, v)
+ | `Lam(_, body) -> loop body
+ in match loop term with
+ | false, _ -> None
+ | true, v -> Some v
+
+ let rec subst v st = function
+ | term when st = `Var v -> term
+ | `Var x when x = v -> st
+ | `Var _ as term -> term
+ | `App(t1,t2) -> `App(subst v st t1, subst v st t2)
+ | `Lam(x, _) as term when x = v -> term
+ (* if x is free in the inserted term st, a capture is possible
+ * we handle by ...
+ *)
+ | `Lam(x, body) ->
+ (match free_in x st with
+ (* x not free in st, can substitute st for v without any captures *)
+ | None -> `Lam(x, subst v st body)
+ (* x free in st, need to alpha-convert `Lam(x, body) *)
+ | 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 name = name' then bump_tag v1 v2 else v1 in
+ (* bump x > max_x from st, then check whether
+ * it also needs to be bumped > v
+ *)
+ let uniq_x = bump_tag' (bump_tag x 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 x (`Var uniq_x') body in
+ (* now substitute st for v *)
+ `Lam(uniq_x', subst v st body')
+ )
+
+ let check_eta = function
+ | `Lam(v, `App(t, `Var u)) when v = u && free_in v t = None -> t
+ | (_ : 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 db_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(t1, t2) as term -> loop (t2::stack) t1
+ | `Lam(v, body) -> (match stack with
+ | [] ->
+ let term = (`Lam(v, loop [] body)) in
+ if eta then check_eta term else term
+ | t::rest -> loop rest (subst v t body)
+ )
+ and unwind t1 = function
+ | [] -> t1
+ | t2::ts -> unwind (`App(t1, loop [] t2)) ts
+ 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 t1, t2) ->
+ (match loop t1 with
+ | `Lam _ as redux -> loop (`App(redux, t2))
+ | nonred_head -> `App(nonred_head, loop t2)
+ )
+ | `App(t1, t2) -> `App(t1, loop t2)
+ in loop expr
+
+ let 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
+
+
+
+ (*
+ 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 st) -> `Lam(y, subst v st body)
+ | `Lam(y, body) ->
+ let z = new_var () in
+ subst v st (`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 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 db, db_equal, db_contains = db, db_equal, db_contains
+