(* *)
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
let alpha_eq x f = db_equal (db x) (db y)