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(** Copyright 2026, Kirill K. Smirnov *)
(** SPDX-License-Identifier: LGPL-3.0-or-later *)
type lterm =
| Var of string
| Int of int
| Unit
| App of lterm * lterm
| Abs of string * lterm
| Exc of lterm
| Try of lterm * (string * lterm) list
let ltrue = Abs ("x", Abs ("y", Var "x"))
let lfalse = Abs ("x", Abs ("y", Var "y"))
let fix_comb_no =
let helper = Abs ("x", App (Var "f", App (Var "x", Var "x"))) in
Abs ("f", App (helper, helper))
;;
let fix_comb_cbv =
let helper =
Abs ("x", App (Var "f", Abs ("v", App (App (Var "x", Var "x"), Var "v"))))
in
Abs ("f", App (helper, helper))
;;
let fix_comb_cbn = fix_comb_no
let try_stdlib = function
| App (Var "println_int", Int i) ->
let () = Printf.printf "%d\n%!" i in
Some Unit
| App (App (Var "+", Int x), Int y) -> Some (Int (x + y))
| App (App (Var "-", Int x), Int y) -> Some (Int (x - y))
| App (App (Var "*", Int x), Int y) -> Some (Int (x * y))
| App (App (Var "/", Int _), Int 0) -> Some (Exc (Var "DivisionByZero"))
| App (App (Var "/", Int x), Int y) -> Some (Int (x / y))
| App (App (Var "=", Int x), Int y) -> Some (if x = y then ltrue else lfalse)
| App (App (Var ">", Int x), Int y) -> Some (if x > y then ltrue else lfalse)
| App (App (Var "<", Int x), Int y) -> Some (if x < y then ltrue else lfalse)
| _ -> None
;;
let fresh_var vars =
let cands = List.init (List.length vars + 1) (fun i -> "fv" ^ string_of_int i) in
let res = List.filter (fun x -> not (List.mem x vars)) cands in
match res with
| h :: _ -> h
| [] -> raise (Failure "fresh: should not happen!")
;;
let rec lterm_to_string = function
| Var x -> x
| Int i -> Printf.sprintf "%d" i
| Unit -> "()"
| App (t1, t2) -> "(" ^ lterm_to_string t1 ^ " " ^ lterm_to_string t2 ^ ")"
| Abs (x, t2) -> "(\\" ^ x ^ "." ^ lterm_to_string t2 ^ ")"
| Exc t -> "Exception " ^ lterm_to_string t
| Try (t, l) ->
"try "
^ lterm_to_string t
^ " with "
^ String.concat " " (List.map (fun (s, t) -> "| " ^ s ^ " -> " ^ lterm_to_string t) l)
;;
let rec lterm_to_string_typed term tp =
assert (snd tp = []);
match term, fst tp with
| Var x, _ -> x
| Int i, Ast.Basetype "int" -> Printf.sprintf "%d" i
| Unit, Ast.Basetype "unit" -> "()"
| Abs (x, App (App (Var y, t1), t2)), Ast.Prod (tp1, tp2) when x = y ->
"("
^ lterm_to_string_typed t1 (tp1, [])
^ ", "
^ lterm_to_string_typed t2 (tp2, [])
^ ")"
| Abs (x, Abs (_, Var z)), Ast.Basetype "bool" when z = x -> "true"
| Abs (_, Abs (y, Var z)), Ast.Basetype "bool" when z = y -> "false"
| Abs ("f", Abs ("g", App (Var "f", t))), Ast.Sum (tp1, _) ->
"(inl " ^ lterm_to_string_typed t (tp1, []) ^ ")"
| Abs ("f", Abs ("g", App (Var "g", t))), Ast.Sum (_, tp2) ->
"(inr " ^ lterm_to_string_typed t (tp2, []) ^ ")"
| Abs (_, _), _ -> "Fun"
| Exc t, _ -> "Exception: " ^ lterm_to_string t
| t, _ -> lterm_to_string t
;;
let vars t =
let rec helper = function
| Var x -> [ x ]
| App (t1, t2) -> helper t1 @ helper t2
| Abs (x, t2) -> [ x ] @ helper t2
| Try (t, l) -> helper t @ List.concat_map (fun (_, t) -> helper t) l
| _ -> []
in
List.sort_uniq compare (helper t)
;;
let fvars t =
let rec helper = function
| Var x -> [ x ]
| App (t1, t2) -> helper t1 @ helper t2
| Abs (x, t2) -> List.filter (fun y -> x <> y) (helper t2)
| Try (t, l) -> helper t @ List.concat_map (fun (_, t) -> helper t) l
| _ -> []
in
List.sort_uniq compare (helper t)
;;
let rec rename_free_var term x y =
match term with
| Var u when u = x -> Var y
| App (t1, t2) -> App (rename_free_var t1 x y, rename_free_var t2 x y)
| Abs (u, _) when u = x -> term
| Abs (u, t) -> Abs (u, rename_free_var t x y)
| Try (t, l) ->
Try (rename_free_var t x y, List.map (fun (s, t) -> s, rename_free_var t x y) l)
| _ -> term
;;
(* term[x |-> t] *)
let subst term x t =
let rec helper term x t vars =
match term with
| Var y -> if x = y then t, vars else term, vars
| Int _ -> term, vars
| Exc t1 ->
let t2, vars = helper t1 x t vars in
Exc t2, vars
| Unit -> term, vars
| App (t1, t2) ->
let r1 = helper t1 x t vars in
let r2 = helper t2 x t (snd r1) in
App (fst r1, fst r2), snd r2
| Try (t1, l) ->
let t1, vars = helper t1 x t vars in
let l1, vars =
List.fold_left
(fun acc (e1, t1) ->
let t2, vars = helper t1 x t (snd acc) in
fst acc @ [ e1, t2 ], vars)
([], vars)
l
in
Try (t1, l1), vars
| Abs (y, _) when y = x -> term, vars
| Abs (y, t2) ->
(match List.find_opt (( = ) y) (fvars t) with
| Some _ ->
let u = fresh_var vars in
let r = helper (rename_free_var t2 y u) x t ([ u ] @ vars) in
Abs (u, fst r), snd r
| None ->
let r = helper t2 x t vars in
Abs (y, fst r), snd r)
in
fst (helper term x t ([ x ] @ vars term @ vars t))
;;
let rec beta_step_in_cbv term =
match try_stdlib term with
| Some t -> Some t
| None ->
(match term with
| Try (t, l) ->
(match beta_step_in_cbv t with
| Some t1 -> Some (Try (t1, l))
| None ->
(match t with
| Exc (App (Var c, t1)) ->
(match List.assoc_opt c l with
| Some t -> Some (App (t, t1))
| None -> Some t)
| Exc tt -> raise (Failure ("Try: should not happen " ^ lterm_to_string tt))
| _ -> Some t))
| Abs (_, _) -> None
| App (Var "raise", e) ->
(match beta_step_in_cbv e with
| Some e -> Some (App (Var "raise", e))
| None ->
(match e with
| Exc t -> Some (Exc t)
| _ -> Some (Exc e)))
| App (Abs (_, _), Exc e) -> Some (Exc e)
| App (Abs (x, y), z) ->
(match beta_step_in_cbv z with
| Some t -> Some (App (Abs (x, y), t))
| None -> Some (subst y x z))
| App (t1, t2) ->
(match beta_step_in_cbv t1 with
| Some t -> Some (App (t, t2))
| None ->
(match t2 with
| Exc e -> Some (Exc e)
| _ ->
(match beta_step_in_cbv t2 with
| Some t -> Some (App (t1, t))
| None ->
(match t1 with
| Exc e -> Some (Exc e)
| _ -> None))))
| _ -> None)
;;
let rec beta_reduce strategy x =
(* let () = Printf.printf "%s\n" (lterm_to_string x) in *)
match strategy x with
| Some y -> beta_reduce strategy y
| None -> x
;;