88 lines
2.4 KiB
OCaml
88 lines
2.4 KiB
OCaml
type id = string
|
|
|
|
type ty_annot = id * Types.ty
|
|
|
|
type lam =
|
|
Fun of ty_annot * lam
|
|
| App of lam * lam
|
|
| Var of id
|
|
| Exf of lam * Types.ty
|
|
|
|
(** alpha renaming in a deterministic way
|
|
if readable is set to true, original variable's names are used to rename *)
|
|
let alpha_convert ?(readable=false) (e : lam) : lam =
|
|
let cpt_var = ref 0 in (* id cpt for variables *)
|
|
let generator (v : id) (cpt : int ref) : id = (* id generator *)
|
|
let start = if readable then v else "" in
|
|
let v = start ^ "__"^(string_of_int !cpt) in
|
|
incr cpt; v
|
|
|
|
in let get_ren_var (var : id) (g : (id * id) list): id = (* returns var if not found *)
|
|
match List.assoc_opt var g with
|
|
| Some var' -> var'
|
|
| None -> var
|
|
|
|
in let rec alpha_aux (e : lam) (g : (id * id) list): lam = match e with (* actual renaming *)
|
|
Fun ((v, t), e) ->
|
|
let v' = generator v cpt_var in
|
|
Fun ((v', t), alpha_aux e ((v, v')::g))
|
|
| App (e1, e2) -> App (alpha_aux e1 g, alpha_aux e2 g)
|
|
| Var v -> Var (get_ren_var v g)
|
|
| Exf (e, t) -> Exf (alpha_aux e g, t)
|
|
in alpha_aux e []
|
|
|
|
|
|
(** alpha equivalence *)
|
|
let (=~) (e1 : lam) (e2 : lam) : bool =
|
|
alpha_convert e1 = alpha_convert e2
|
|
|
|
(** beta-reduction *)
|
|
(* subst m n x substitutes x for n in m *)
|
|
let rec subst (m : lam) (n : lam) (x : id) : lam =
|
|
match m with
|
|
Fun ((y, t), e) ->
|
|
if x = y
|
|
then m
|
|
else Fun ((y, t), subst e n x)
|
|
| App (e1, e2) ->
|
|
let e1 = subst e1 n x in
|
|
let e2 = subst e2 n x in
|
|
App (e1, e2)
|
|
| Var y ->
|
|
if x = y then n else m
|
|
| Exf (e, t) ->
|
|
let e = subst e n x in
|
|
Exf (e, t)
|
|
|
|
(* INVARIANT : e has already been alpha-converted *)
|
|
let rec betastep (e : lam) : lam option =
|
|
match e with
|
|
Fun ((x, t), e) ->
|
|
begin match betastep e with
|
|
None -> None
|
|
| Some e -> Some (Fun ((x, t), e))
|
|
end
|
|
| App (e1, e2) ->
|
|
(* reduce leftmost redex.
|
|
e1 e2 -> e1' e2 then
|
|
e1_nf e2 -> e1_nf e2 then
|
|
(fun x -> e_nf) e2_nf -> e_nf[e2/x] *)
|
|
begin match betastep e1 with
|
|
None ->
|
|
begin match betastep e2 with
|
|
None ->
|
|
begin match e1 with
|
|
Fun ((x, _), e) -> Some (subst e e2 x)
|
|
| _ -> None
|
|
end
|
|
| Some e2 -> Some (App (e1, e2))
|
|
end
|
|
| Some e1 -> Some (App (e1, e2))
|
|
end
|
|
| Var _ -> None
|
|
| Exf (e, t) ->
|
|
begin match betastep e with
|
|
None -> None
|
|
| Some e -> Some (Exf (e, t))
|
|
end
|