let (++) f g x = f (g x);;
let id x = x;;
let rec fold_nat f x n = if n = 0 then x else f (fold_nat f x (n-1)) n ;;

let print_hline = Console.print_hline;;

open Pure

type var = int;;
type t =
 | V of var
 | A of t * t
 | L of t
;;

let delta = L(A(V 0, V 0));;

let eta_eq' =
 let rec aux l1 l2 t1 t2 = match t1, t2 with
  | L t1, L t2 -> aux l1 l2 t1 t2
  | L t1, t2 -> aux l1 (l2+1) t1 t2
  | t1, L t2 -> aux (l1+1) l2 t1 t2
  | V a, V b -> a + l1 = b + l2
  | A(t1,t2), A(u1,u2) -> aux l1 l2 t1 u1 && aux l1 l2 t2 u2
  | _, _ -> false
 in aux ;;
let eta_eq = eta_eq' 0 0;;

(* is arg1 eta-subterm of arg2 ? *)
let eta_subterm u =
 let rec aux lev t = eta_eq' lev 0 u t || match t with
 | L t -> aux (lev+1) t
 | A(t1, t2) -> aux lev t1 || aux lev t2
 | _ -> false
 in aux 0
;;

(* does NOT lift the argument *)
let mk_lams = fold_nat (fun x _ -> L x) ;;

let string_of_t =
  let string_of_bvar =
   let bound_vars = ["x"; "y"; "z"; "w"; "q"] in
   let bvarsno = List.length bound_vars in
   fun nn -> if nn < bvarsno then List.nth bound_vars nn else "x" ^ (string_of_int (nn - bvarsno + 1)) in
  let rec string_of_term_w_pars level = function
    | V v -> if v >= level then "`" ^ string_of_int (v-level) else
       string_of_bvar (level - v-1)
    | A _
    | L _ as t -> "(" ^ string_of_term_no_pars level t ^ ")"
  and string_of_term_no_pars_app level = function
    | A(t1,t2) -> string_of_term_no_pars_app level t1 ^ " " ^ string_of_term_w_pars level t2
    | _ as t -> string_of_term_w_pars level t
  and string_of_term_no_pars level = function
    | L t -> "λ" ^ string_of_bvar level ^ ". " ^ string_of_term_no_pars (level+1) t
    | _ as t -> string_of_term_no_pars_app level t
  in string_of_term_no_pars 0
;;

type problem = {
   orig_freshno: int
 ; freshno : int
 ; div : t
 ; conv : t
 ; sigma : (var * t) list (* substitutions *)
 ; phase : [`One | `Two] (* :'( *)
}

let string_of_problem p =
 let lines = [
  "[DV] " ^ string_of_t p.div;
  "[CV] " ^ string_of_t p.conv;
 ] in
 String.concat "\n" lines
;;

exception B;;
exception Done of (var * t) list (* substitution *);;
exception Fail of int * string;;

let problem_fail p reason =
 print_endline "!!!!!!!!!!!!!!! FAIL !!!!!!!!!!!!!!!";
 print_endline (string_of_problem p);
 raise (Fail (-1, reason))
;;

let freshvar ({freshno} as p) =
 {p with freshno=freshno+1}, freshno+1
;;

let rec is_inert =
 function
 | A(t,_) -> is_inert t
 | V _ -> true
 | L _ -> false
;;

let is_var = function V _ -> true | _ -> false;;
let is_lambda = function L _ -> true | _ -> false;;

let rec get_inert = function
 | V n -> (n,0)
 | A(t, _) -> let hd,args = get_inert t in hd,args+1
 | _ -> assert false
;;

(* precomputes the number of leading lambdas in a term,
   after replacing _v_ w/ a term starting with n lambdas *)
let rec no_leading_lambdas v n = function
 | L t -> 1 + no_leading_lambdas (v+1) n t
 | A _ as t -> let v', m = get_inert t in if v = v' then max 0 (n - m) else 0
 | V v' -> if v = v' then n else 0
;;

let rec subst level delift sub =
 function
 | V v -> if v = level + fst sub then lift level (snd sub) else V (if delift && v > level then v-1 else v)
 | L t -> let t = subst (level + 1) delift sub t in if t = B then B else L t
 | A (t1,t2) ->
  let t1 = subst level delift sub t1 in
  let t2 = subst level delift sub t2 in
  mk_app t1 t2
and mk_app t1 t2 = if t1 = delta && t2 = delta then raise B
 else match t1 with
 | L t1 -> subst 0 true (0, t2) t1
 | _ -> A (t1, t2)
and lift n =
 let rec aux lev =
  function
  | V m -> V (if m >= lev then m + n else m)
  | L t -> L (aux (lev+1) t)
  | A (t1, t2) -> A (aux lev t1, aux lev t2)
 in aux 0
;;
let subst = subst 0 false;;

let subst_in_problem ((v, t) as sub) p =
print_endline ("-- SUBST " ^ string_of_t (V v) ^ " |-> " ^ string_of_t t);
 let sigma = sub::p.sigma in
 let div = try subst sub p.div with B -> raise (Done sigma) in
 let conv = try subst sub p.conv with B -> raise (Fail(-1,"p.conv diverged")) in
 {p with div; conv; sigma}
;;

let get_subterm_with_head_and_args hd_var n_args =
 let rec aux lev = function
 | V _ -> None
 | L t -> aux (lev+1) t
 | A(t1,t2) as t ->
   let hd_var', n_args' = get_inert t1 in
   if hd_var' = hd_var + lev && n_args <= 1 + n_args'
    (* the `+1` above is because of t2 *)
    then Some (lift ~-lev t)
    else match aux lev t2 with
    | None -> aux lev t1
    | Some _ as res -> res
 in aux 0
;;

let rec purify = function
 | L t -> Pure.L (purify t)
 | A (t1,t2) -> Pure.A (purify t1, purify t2)
 | V n -> Pure.V n
;;

let check p sigma =
 print_endline "Checking...";
 let div = purify p.div in
 let conv = purify p.conv in
 let sigma = List.map (fun (v,t) -> v, purify t) sigma in
 let freshno = List.fold_right (max ++ fst) sigma 0 in
 let env = Pure.env_of_sigma freshno sigma in
 assert (Pure.diverged (Pure.mwhd (env,div,[])));
 print_endline " D diverged.";
 assert (not (Pure.diverged (Pure.mwhd (env,conv,[]))));
 print_endline " C converged.";
 ()
;;

let sanity p =
 print_endline (string_of_problem p); (* non cancellare *)
 if p.phase = `Two && p.div = delta then raise (Done p.sigma);
 if not (is_inert p.div) then problem_fail p "p.div converged";
 p
;;

(* drops the arguments of t after the n-th *)
(* FIXME! E' usato in modo improprio contando sul fatto
   errato che ritorna un inerte lungo esattamente n *)
let inert_cut_at n t =
 let rec aux t =
  match t with
  | V _ as t -> 0, t
  | A(t1,_) as t ->
    let k', t' = aux t1 in
     if k' = n then n, t'
      else k'+1, t
  | _ -> assert false
 in snd (aux t)
;;

(* return the index of the first argument with a difference
   (the first argument is 0)
   precondition: p.div and t have n+1 arguments
   *)
let find_eta_difference p t argsno =
 let t = inert_cut_at argsno t in
 let rec aux t u k = match t, u with
 | V _, V _ -> problem_fail p "no eta difference found (div subterm of conv?)"
 | A(t1,t2), A(u1,u2) ->
    if not (eta_eq t2 u2) then (k-1)
    else aux t1 u1 (k-1)
 | _, _ -> assert false
 in aux p.div t argsno
;;

let compute_max_lambdas_at hd_var j =
 let rec aux hd = function
 | A(t1,t2) ->
    (if get_inert t1 = (hd, j)
      then max ( (*FIXME*)
       if is_inert t2 && let hd', j' = get_inert t2 in hd' = hd
        then let hd', j' = get_inert t2 in j - j'
        else no_leading_lambdas hd_var j t2)
      else id) (max (aux hd t1) (aux hd t2))
 | L t -> aux (hd+1) t
 | V _ -> 0
 in aux hd_var
;;

let print_cmd s1 s2 = print_endline (">> " ^ s1 ^ " " ^ s2);;

(* eat the arguments of the divergent and explode.
 It does NOT perform any check, may fail if done unsafely *)
let eat p =
print_cmd "EAT" "";
 let var, k = get_inert p.div in
 let phase = p.phase in
 let p =
  match phase with
  | `One ->
      let n = 1 + max
       (compute_max_lambdas_at var (k-1) p.div)
       (compute_max_lambdas_at var (k-1) p.conv) in
      (* apply fresh vars *)
      let p, t = fold_nat (fun (p, t) _ ->
        let p, v = freshvar p in
        p, A(t, V (v + k))
      ) (p, V 0) n in
      let p = {p with phase=`Two} in
      let t = A(t, delta) in
      let t = fold_nat (fun t m -> A(t, V (k-m))) t (k-1) in
      let subst = var, mk_lams t k in
      let p = subst_in_problem subst p in
      let _, args = get_inert p.div in
      {p with div = inert_cut_at (args-k) p.div}
  | `Two ->
      let subst = var, mk_lams delta k in
      subst_in_problem subst p in
 sanity p
;;

(* step on the head of div, on the k-th argument, with n fresh vars *)
let step k n p =
 let var, _ = get_inert p.div in
print_cmd "STEP" ("on " ^ string_of_t (V var) ^ " (of:" ^ string_of_int n ^ ")");
 let p, t = (* apply fresh vars *)
  fold_nat (fun (p, t) _ ->
    let p, v = freshvar p in
    p, A(t, V (v + k + 1))
  ) (p, V 0) n in
 let t = (* apply unused bound variables V_{k-1}..V_1 *)
  fold_nat (fun t m -> A(t, V (k-m+1))) t k in
 let t = mk_lams t (k+1) in (* make leading lambdas *)
 let subst = var, t in
 let p = subst_in_problem subst p in
 sanity p
;;

let rec auto p =
 let hd_var, n_args = get_inert p.div in
 match get_subterm_with_head_and_args hd_var n_args p.conv with
 | None ->
   (try
    let phase = p.phase in
     let p = eat p in
     if phase = `Two
      then problem_fail p "Auto.2 did not complete the problem"
      else auto p
    with Done sigma -> sigma)
 | Some t ->
  let j = find_eta_difference p t n_args in
  let k = 1 + max
   (compute_max_lambdas_at hd_var j p.div)
    (compute_max_lambdas_at hd_var j p.conv) in
  let p = step j k p in
  auto p
;;

let problem_of (label, div, convs, ps, var_names) =
 print_hline ();
 let rec aux = function
 | `Lam(_, t) -> L (aux t)
 | `I ((v,_), args) -> Listx.fold_left (fun x y -> mk_app x (aux y)) (V v) args
 | `Var(v,_) -> V v
 | `N _ | `Match _ -> assert false in
 assert (List.length ps = 0);
 let convs = List.rev convs in
 let conv = List.fold_left (fun x y -> mk_app x (aux (y :> Num.nf))) (V (List.length var_names)) convs in
 let var_names = "@" :: var_names in
 let div = match div with
 | Some div -> aux (div :> Num.nf)
 | None -> assert false in
 let varno = List.length var_names in
 let p = {orig_freshno=varno; freshno=1+varno; div; conv; sigma=[]; phase=`One} in
 (* initial sanity check *)
 sanity p
;;

let solve p =
 if eta_subterm p.div p.conv
  then print_endline "!!! div is subterm of conv. Problem was not run !!!"
  else check p (auto p)
;;

Problems.main (solve ++ problem_of);

(* Example usage of interactive: *)

(* let interactive div conv cmds =
 let p = problem_of div conv in
 try (
 let p = List.fold_left (|>) p cmds in
 let rec f p cmds =
  let nth spl n = int_of_string (List.nth spl n) in
  let read_cmd () =
   let s = read_line () in
   let spl = Str.split (Str.regexp " +") s in
   s, let uno = List.hd spl in
    try if uno = "eat" then eat
    else if uno = "step" then step (nth spl 1) (nth spl 2)
    else failwith "Wrong input."
    with Failure s -> print_endline s; (fun x -> x) in
  let str, cmd = read_cmd () in
  let cmds = (" " ^ str ^ ";")::cmds in
  try
   let p = cmd p in f p cmds
  with
  | Done _ -> print_endline "Done! Commands history: "; List.iter print_endline (List.rev cmds)
 in f p []
 ) with Done _ -> ()
;; *)

(* interactive "x y"
 "@ (x x) (y x) (y z)" [step 0 1; step 0 2; eat]
;; *)