[fireball-separation.git] / ocaml / lambda4.ml

open Util
open Util.Vars
open Pure
open Num

let bomb = ref(`Var ~-1);;

type problem =
 { freshno: int
 ; div: i_var option (* None = bomb *)
 ; conv: i_n_var list (* the inerts that must converge *)
 ; ps: i_n_var list (* the n-th inert must become n *)
 ; sigma: (int * nf) list (* the computed substitution *)
 ; deltas: (int * nf) list ref list (* collection of all branches *)
 ; steps: int (* how many steps the algorithm made until now *)
 }


(* let heads = Util.sort_uniq (List.map hd_of_i_var p.ps) in
for all head
  List.map (fun (_, bs) -> List.map (fun (x,_) -> List.nth p.ps x) !bs) p.deltas *)

(* let ($) f x = f x;; *)

let subterms tms freshno =
 let apply_var =
  let no = ref freshno in
  function t -> incr no; mk_app t (`Var !no) in
 let applicative hd args = snd (
    List.fold_left (fun (hd, acc) t -> let hd = mk_app hd t in hd, hd::acc) (hd, []) args) in
 let rec aux t = match t with
 | `Var _ -> []
 | `I(v,ts) ->
    (* applicative (`Var v) (Listx.to_list ts) @  *)
     Util.concat_map aux (Listx.to_list ts) @ List.map apply_var (Listx.to_list ts)
 | `Lam(_,t) -> aux (lift ~-1 t)
 | `Match(u,bs_lift,bs,args) ->
    aux (u :> nf) @
     (* applicative (`Match(u,bs_lift,bs,[])) args @ *)
      Util.concat_map aux args @ List.map apply_var args
      (* @ Util.concat_map (aux ++ (lift bs_lift) ++ snd) !bs *)
 | `N _ -> []
 in let tms' = (* Util.sort_uniq ~compare:eta_compare*) (Util.concat_map aux tms) in
 tms @ tms'
 (* List.map (fun (t, v) -> match t with `N _ -> t | _ -> mk_app t v) (List.combine tms (List.mapi (fun i _ -> `Var(freshno+i)) tms)) *)
;;

let all_terms p =
(match p.div with None -> [] | Some t -> [(t :> i_n_var)])
@ p.conv
@ p.ps
;;

let problem_measure p =
 let l = Array.to_list (Array.init (p.freshno + 1) string_of_var) in
 let open Listx in
 (* aux |t1;t2| e' numero di step per portare la diff in testa
    INVARIANTE: t1 <eta> t2
 *)
 let rec aux t1 t2 =
  match t1, t2 with
  | `I(v1,nfs1), `I(v2,nfs2) ->
    if v1 <> v2
     then 0 else 1 + find_first_diff (to_list nfs1, to_list nfs2)
  | `Match (t1,bs_lift,bs,args), `Match (t2,bs_lift',bs',args') ->
     if bs != bs' then 0 (* TODO *)
      else if eta_eq (t1 :> nf) (t2 :> nf) then 1 + find_first_diff (args, args') else aux (t1 :> nf) (t2 :> nf) (* TODO *)
  | `Match _, _
  | _, `Match _ -> 0 (* FIXME!!! *)
  | `Lam(_,t1), `Lam(_,t2) -> aux t1 t2
  | _ -> 0
 and find_first_diff = function
  | [], [] -> assert false
  | [], t::_
  | t::_, [] -> 1
  | t1::ts1, t2::ts2 ->
    if eta_eq (t1 :> nf) (t2 :> nf) then 1 + find_first_diff (ts1, ts2) else aux t1 t2
    (* no. di step da fare per separare t1 e t2 *)
 in let diff t1 t2 = (
  let res = if eta_eq t1 t2 then 0 else aux t1 t2 in
  if res <> 0 then prerr_endline ("diff (" ^ print ~l t1 ^ ") (" ^ print ~l t2 ^ ") = " ^ string_of_int res);
  res
  )
 (* aux calcola la somma delle differenze tra i termini in una lista (quadratico) *)
 in let rec sum = function
  | [] -> 0
  | x::xs -> List.fold_right ((+) ++ (diff (x :> nf))) (xs :> nf list) (sum xs)
 in let subterms = subterms ((all_terms p) :> nf list) p.freshno
 (* let subterms = sort_uniq ~compare:eta_compare subterms in *)
 in let a = sum subterms
 in let b = List.fold_right (fun bs -> (+) (sum (List.map ((List.nth p.ps) ++ fst) !bs))) p.deltas 0
 in let _ = prerr_endline ("Computed measure: " ^ string_of_int a ^ "," ^ string_of_int b)
 in a + b
;;

let print_problem label ({freshno; div; conv; ps; deltas; steps} as p) =
 Console.print_hline ();
 prerr_endline ("\n||||| Displaying problem: " ^ label ^ " |||||");
 let nl = "\n| " in
 let deltas = String.concat nl (List.map (fun r -> String.concat " <> " (List.map (fun (i,_) -> string_of_int i) !r)) deltas) in
 let l = Array.to_list (Array.init (freshno + 1) string_of_var) in
    nl ^ string_of_int steps ^ " steps left; measure="^string_of_int(problem_measure p)^" freshno = " ^ string_of_int freshno
 ^ nl ^ "\b> DISCRIMINATING SETS (deltas)"
 ^ nl ^ deltas ^ (if deltas = "" then "" else nl)
 ^ "\b> DIVERGENT" ^ nl
 ^ "*: " ^ (match div with None -> "*" | Some div -> print ~l (div :> nf)) ^ "\n| "
 ^ "\b> CONVERGENT" ^ nl
 ^ String.concat "\n| " (List.map (fun t -> "_: " ^ (if t = `N (-1) then "_" else print ~l (t :> nf))) conv) ^
 (if conv = [] then "" else "\n| ")
 ^ "\b> NUMERIC" ^ nl
 ^ String.concat "\n| " (List.mapi (fun i t -> string_of_int i ^ ": " ^ print ~l (t :> nf)) ps)
 ^ nl
;;


let failwithProblem p reason =
 print_endline (print_problem "FAIL" p);
 failwith reason
;;

let make_fresh_var p =
 let freshno = p.freshno + 1 in
 {p with freshno}, `Var freshno
;;

let make_fresh_vars p m =
 Array.fold_left
  (* fold_left vs. fold_right hides/shows the bug in problem q7 *)
  (fun (p, vars) _ -> let p, var = make_fresh_var p in p, var::vars)
  (p, [])
  (Array.make m ())
;;

let simple_expand_match ps =
  let rec aux level = function
  | #i_num_var as t -> aux_i_num_var level t
  | `Lam(b,t) -> `Lam(b, aux (level+1) t)
  and aux_i_num_var level = function
  | `Match(u,bs_lift,bs,args) as torig ->
    let u = aux_i_num_var level u in
    bs := List.map (fun (n, x) -> n, aux 0 x) !bs;
    (try
       (match u with
         | #i_n_var as u ->
            let i = index_of (lift (-level) u) (ps :> nf list) (* can raise Not_found *)
            in let t = mk_match (`N i) bs_lift bs args in
            if t <> torig then
            aux level (t :> nf)
           else raise Not_found
         | _ -> raise Not_found)
      with Not_found ->
       `Match(cast_to_i_num_var u,bs_lift,bs,List.map (aux level) args))
  | `I(k,args) -> `I(k,Listx.map (aux level) args)
  | `N _ | `Var _ as t -> t
in aux_i_num_var 0;;

let fixpoint f =
 let rec aux x = let x' = f x in if x <> x' then aux x' else x in aux
;;

let rec super_simplify_ps ps =
 fixpoint (List.map (cast_to_i_num_var ++ (simple_expand_match ps)))
;;

let super_simplify ({div; ps; conv} as p) =
  let ps = super_simplify_ps p.ps (p.ps :> i_num_var list) in
  let conv = super_simplify_ps ps (p.conv :> i_num_var list) in
  let div = option_map (fun div ->
   let divs = super_simplify_ps p.ps ([div] :> i_num_var list) in
    List.hd divs) div in
  {p with div=option_map cast_to_i_var div; ps=List.map cast_to_i_n_var ps; conv=List.map cast_to_i_n_var conv}

exception ExpandedToLambda;;

let subst_in_problem x inst ({freshno; div; conv; ps; sigma} as p) =
 let len_ps = List.length ps in
(*(let l = Array.to_list (Array.init (freshno + 1) string_of_var) in
prerr_endline ("# INST0: " ^ string_of_var x ^ " := " ^ print ~l inst));*)
 let rec aux ((freshno,acc_ps,acc_new_ps) as acc) =
  function
  | [] -> acc
  | t::todo_ps ->
(*prerr_endline ("EXPAND t:" ^ print (t :> nf));*)
     let t = subst false x inst (t :> nf) in
(*prerr_endline ("SUBSTITUTED t:" ^ print (t :> nf));*)
     let freshno,new_t,acc_new_ps =
      expand_match (freshno,acc_ps@`Var(max_int/3)::todo_ps,acc_new_ps) t
     in
      aux (freshno,acc_ps@[new_t],acc_new_ps) todo_ps

  (* cut&paste from aux above *)
  and aux' ps ((freshno,acc_conv,acc_new_ps) as acc) =
   function
   | [] -> acc
   | t::todo_conv ->
(*prerr_endline ("EXPAND t:" ^ print (t :> nf));*)
      (* try *)
       let t = subst false x inst (t :> nf) in
(*prerr_endline ("SUBSTITUTED t:" ^ print (t :> nf));*)
       let freshno,new_t,acc_new_ps =
        expand_match (freshno,ps,acc_new_ps) t
       in
        aux' ps (freshno,acc_conv@[new_t],acc_new_ps) todo_conv
      (* with ExpandedToLambda -> aux' ps (freshno,acc_conv@[`N(-1)],acc_new_ps) todo_conv *)

  (* cut&paste from aux' above *)
  and aux'' ps (freshno,acc_new_ps) =
   function
   | None -> freshno, None, acc_new_ps
   | Some t ->
      let t = subst false x inst (t :> nf) in
      let freshno,new_t,acc_new_ps =
       expand_match (freshno,ps,acc_new_ps) t
      in
       freshno,Some new_t,acc_new_ps

  and expand_match ((freshno,acc_ps,acc_new_ps) as acc) t =
   match t with
   | `Match(u',bs_lift,bs,args) ->
        let freshno,u,acc_new_ps = expand_match acc (u' :> nf) in
        let acc_new_ps,i =
         match u with
         | `N i -> acc_new_ps,i
         | _ ->
            let ps = List.map (fun t -> cast_to_i_num_var (subst false x inst (t:> nf))) (acc_ps@acc_new_ps) in
            let super_simplified_ps = super_simplify_ps ps ps in
(*prerr_endline ("CERCO u:" ^ print (fst u :> nf));
List.iter (fun x -> prerr_endline ("IN: " ^ print (fst x :> nf))) ps;
List.iter (fun x -> prerr_endline ("IN2: " ^ print (fst x :> nf))) super_simplified_ps;*)
            match index_of_opt ~eq:eta_eq super_simplified_ps u with
               Some i -> acc_new_ps, i
             | None -> acc_new_ps@[u], len_ps + List.length acc_new_ps
        in
         let freshno=
          if List.exists (fun (j,_) -> i=j) !bs then
           freshno
          else
           let freshno,v = freshno+1, `Var (freshno+1) in (* make_fresh_var freshno in *)
           bs := !bs @ [i, v] ;
           freshno in
(*prerr_endlie ("t DA RIDURRE:" ^ print (`Match(`N i,arity,bs_lift,bs,args) :> nf) ^ " more_args=" ^ string_of_int more_args);*)
         let t = mk_match (`N i) bs_lift bs args in
(*prerr_endline ("NUOVO t:" ^ print (fst t :> nf) ^ " more_args=" ^ string_of_int (snd t));*)
          expand_match (freshno,acc_ps,acc_new_ps) t
   | `Lam _ -> raise ExpandedToLambda
   | #i_n_var as x ->
      let x = simple_expand_match (acc_ps@acc_new_ps) x in
      freshno,cast_to_i_num_var x,acc_new_ps in

 let freshno,old_ps,new_ps = aux (freshno,[],[]) (ps :> i_num_var list) in
 let freshno,conv,new_ps = aux' old_ps (freshno,[],new_ps) (conv :> i_num_var list) in
 let freshno,div,new_ps = aux'' old_ps (freshno,new_ps) (div :> i_num_var option) in
 let div = option_map cast_to_i_var div in
 let ps = List.map cast_to_i_n_var (old_ps @ new_ps) in
 let conv = List.map cast_to_i_n_var conv in
(let l = Array.to_list (Array.init (freshno + 1) string_of_var) in
prerr_endline ("# INST: " ^ string_of_var x ^ " := " ^ print ~l inst));
 let p = {p with freshno; div; conv; ps} in
 ( (* check if double substituting a variable *)
  if List.exists (fun (x',_) -> x = x') sigma
   then failwithProblem p "Variable replaced twice"
 );
 let p = {p with sigma = sigma@[x,inst]} in
 let p = super_simplify p in
 prerr_endline (print_problem "instantiate" p);
 p
;;

exception Dangerous

let arity_of arities k =
 let _,pos,y = List.find (fun (v,_,_) -> v=k) arities in
 let arity = match y with `Var _ -> 0 | `I(_,args) -> Listx.length args | _ -> assert false in
 arity + if pos = -1 then - 1 else 0
;;

let rec dangerous arities showstoppers =
 function
    `N _
  | `Var _
  | `Lam _ -> ()
  | `Match(t,liftno,bs,args) ->
      (* CSC: XXX partial dependency on the encoding *)
      (match t with
          `N _ -> List.iter (dangerous arities showstoppers) args
        | `Match _ as t -> dangerous arities showstoppers t ; List.iter (dangerous arities showstoppers) args
        | `Var x -> dangerous_inert arities showstoppers x args 2 (* 2 coming from Scott's encoding *)
        | `I(x,args') -> dangerous_inert arities showstoppers x (Listx.to_list args' @ args) 2 (* 2 coming from Scott's encoding *)
      )
  | `I(k,args) -> dangerous_inert arities showstoppers k (Listx.to_list args) 0

and dangerous_inert arities showstoppers k args more_args =
 List.iter (dangerous arities showstoppers) args ;
 if List.mem k showstoppers then raise Dangerous else
 try
  let arity = arity_of arities k in
  if List.length args + more_args > arity then raise Dangerous else ()
 with
  Not_found -> ()

(* cut & paste from above *)
let rec dangerous_conv arities showstoppers =
 function
    `N _
  | `Var _
  | `Lam _ -> []
  | `Match(t,liftno,bs,args) ->
      (* CSC: XXX partial dependency on the encoding *)
      (match t with
          `N _ -> concat_map (dangerous_conv arities showstoppers) args
        | `Match _ as t -> dangerous_conv arities showstoppers t @ concat_map (dangerous_conv arities showstoppers) args
        | `Var x -> dangerous_inert_conv arities showstoppers x args 2 (* 2 coming from Scott's encoding *)
        | `I(x,args') -> dangerous_inert_conv arities showstoppers x (Listx.to_list args' @ args) 2 (* 2 coming from Scott's encoding *)
      )
  | `I(k,args) -> dangerous_inert_conv arities showstoppers k (Listx.to_list args) 0

and dangerous_inert_conv arities showstoppers k args more_args =
 concat_map (dangerous_conv arities showstoppers) args @
 if List.mem k showstoppers then k :: concat_map free_vars args else
 try
  let arity = arity_of arities k in
  prerr_endline ("dangerous_inert_conv: ar=" ^ string_of_int arity ^ " k="^string_of_var k ^ " listlenargs=" ^ (string_of_int (List.length args)) );
  if List.length args + more_args > arity then k :: concat_map free_vars args else []
 with
  Not_found -> []

(* inefficient algorithm *)
let rec edible arities div ps conv showstoppers =
 let rec aux showstoppers =
  function
     [] -> showstoppers
   | x::xs when List.exists (fun y -> hd_of x = Some y) showstoppers ->
      (* se la testa di x e' uno show-stopper *)
      let new_showstoppers = sort_uniq (showstoppers @ free_vars (x :> nf)) in
      (* aggiungi tutte le variabili libere di x *)
      if List.length showstoppers <> List.length new_showstoppers then
       aux new_showstoppers ps
      else
       aux showstoppers xs
   | x::xs ->
      match hd_of x with
         None -> aux showstoppers xs
       | Some h ->
          try
           dangerous arities showstoppers (x : i_n_var :> nf) ;
           aux showstoppers xs
          with
           Dangerous ->
            aux (sort_uniq (h::showstoppers)) ps
  in
    let showstoppers = sort_uniq (aux showstoppers ps) in
    let dangerous_conv =
     List.map (dangerous_conv arities showstoppers) (conv :> nf list) in

    prerr_endline ("dangerous_conv lenght:" ^ string_of_int (List.length dangerous_conv));
    List.iter (fun l -> prerr_endline (String.concat " " (List.map string_of_var l))) dangerous_conv;
    let showstoppers' = showstoppers @ List.concat dangerous_conv in
    let showstoppers' = sort_uniq (match div with
     | None -> showstoppers'
     | Some div ->
       if List.exists ((=) (hd_of_i_var div)) showstoppers'
       then showstoppers' @ free_vars (div :> nf) else showstoppers') in
    if showstoppers <> showstoppers' then edible arities div ps conv showstoppers' else showstoppers', dangerous_conv
;;

let precompute_edible_data {ps; div} xs =
 (match div with None -> [] | Some div -> [hd_of_i_var div, -1, (div :> i_n_var)]) @
  List.map (fun hd ->
   let i, tm = Util.findi (fun y -> hd_of y = Some hd) ps in
    hd, i, tm
   ) xs
;;

let critical_showstoppers p =
  let p = super_simplify p in
  let hd_of_div = match p.div with None -> [] | Some t -> [hd_of_i_var t] in
  let showstoppers_step =
  concat_map (fun bs ->
    let heads = List.map (fun (i,_) -> List.nth p.ps i) !bs in
    let heads = List.sort compare (hd_of_div @ filter_map hd_of heads) in
    snd (split_duplicates heads)
    ) p.deltas @
     if List.exists (fun t -> [hd_of t] = List.map (fun x -> Some x) hd_of_div) p.conv
     then hd_of_div else [] in
  let showstoppers_step = sort_uniq showstoppers_step in
  let showstoppers_eat =
   let heads_and_arities =
    List.sort (fun (k,_) (h,_) -> compare k h)
     (filter_map (function `Var k -> Some (k,0) | `I(k,args) -> Some (k,Listx.length args) | _ -> None ) p.ps) in
   let rec multiple_arities =
    function
       []
     | [_] -> []
     | (x,i)::(y,j)::tl when x = y && i <> j ->
         x::multiple_arities tl
     | _::tl -> multiple_arities tl in
   multiple_arities heads_and_arities in

  let showstoppers_eat = sort_uniq showstoppers_eat in
  let showstoppers_eat = List.filter
    (fun x -> not (List.mem x showstoppers_step))
    showstoppers_eat in
  List.iter (fun v -> prerr_endline ("DANGEROUS STEP: " ^ string_of_var v)) showstoppers_step;
  List.iter (fun v -> prerr_endline ("DANGEROUS EAT: " ^ string_of_var v)) showstoppers_eat;
  p, showstoppers_step, showstoppers_eat
  ;;

let eat p =
  let ({ps} as p), showstoppers_step, showstoppers_eat = critical_showstoppers p in
  let showstoppers = showstoppers_step @ showstoppers_eat in
  let heads = List.sort compare (filter_map hd_of ps) in
  let arities = precompute_edible_data p (uniq heads) in
  let showstoppers, showstoppers_conv =
   edible arities p.div ps p.conv showstoppers in
  let l = List.filter (fun (x,_,_) -> not (List.mem x showstoppers)) arities in
  let p =
  List.fold_left (fun p (x,pos,(xx : i_n_var)) -> if pos = -1 then p else
   let n = match xx with `I(_,args) -> Listx.length args | _ -> 0 in
   let v = `N(pos) in
   let inst = make_lams v n in
(let l = Array.to_list (Array.init (p.freshno + 1) string_of_var) in
prerr_endline ("# INST_IN_EAT: " ^ string_of_var x ^ " := " ^ print ~l inst));
   { p with sigma = p.sigma @ [x,inst] }
   ) p l in
  (* to avoid applied numbers in safe positions that
     trigger assert failures subst_in_problem x inst p*)
 let ps =
  List.map (fun t ->
   try
    let _,j,_ = List.find (fun (h,_,_) -> hd_of t = Some h) l in
    `N j
   with Not_found -> t
  ) ps in
 let p = match p.div with
  | None -> p
  | Some div ->
   if List.mem (hd_of_i_var div) showstoppers
   then p
   else
    let n = match div with `I(_,args) -> Listx.length args | `Var _ -> 0 in
    let p, bomb' = make_fresh_var p in
    (if !bomb <> `Var (-1) then
     failwithProblem p
      ("Bomb was duplicated! It was " ^ string_of_nf !bomb ^
       ", tried to change it to " ^ string_of_nf bomb'));
    bomb := bomb';
    prerr_endline ("Just created bomb var: " ^ string_of_nf !bomb);
    let x = hd_of_i_var div in
    let inst = make_lams !bomb n in
    prerr_endline ("# INST (div): " ^ string_of_var x ^ " := " ^ string_of_nf inst);
    let p = {p with div=None} in
    (* subst_in_problem (hd_of_i_var div) inst p in *)
     {p with sigma=p.sigma@[x,inst]} in
     let dangerous_conv = showstoppers_conv in
let _ = prerr_endline ("dangerous_conv lenght:" ^ string_of_int (List.length dangerous_conv));
List.iter (fun l -> prerr_endline (String.concat " " (List.map string_of_var l))) dangerous_conv; in
 let conv =
   List.map (function s,t ->
    try
     if s <> [] then t else (
     (match t with | `Var _ -> raise Not_found | _ -> ());
     let _ = List.find (fun h -> hd_of t = Some h) showstoppers in
      t)
    with Not_found -> match hd_of t with
     | None -> assert (t = `N ~-1); t
     | Some h ->
      prerr_endline ("FREEZING " ^ string_of_var h);
      `N ~-1 (* convergent dummy*)
   ) (List.combine showstoppers_conv p.conv) in
 List.iter
  (fun bs ->
    bs :=
     List.map
      (fun (n,t as res) ->
        match List.nth ps n with
           `N m -> m,t
         | _ -> res
      ) !bs
  ) p.deltas ;
 let old_conv = p.conv in
 let p = { p with ps; conv } in
 if l <> [] || old_conv <> conv
  then prerr_endline (print_problem "eat" p);
 if List.for_all (function `N _ -> true | _ -> false) ps && p.div = None then
  `Finished p
 else
  `Continue p

let instantiate p x n =
 (if `Var x = !bomb
   then failwithProblem p ("BOMB (" ^ string_of_nf !bomb ^ ") cannot be instantiated!"));
 let p,vars = make_fresh_vars p n in
 let p,zero = make_fresh_var p in
 let zero = Listx.Nil zero in
 let args = if n = 0 then zero else Listx.append zero (Listx.from_list vars) in
 let bs = ref [] in
 let inst = `Lam(false,`Match(`I(0,Listx.map (lift 1) args),1,bs,[])) in
 let p = {p with deltas=bs::p.deltas} in
 subst_in_problem x inst p
;;

let compute_special_k tms =
  let rec aux k (t: nf) = Pervasives.max k (match t with
    | `Lam(b,t) -> aux (k + if b then 1 else 0) t
    | `I(n, tms) -> Listx.max (Listx.map (aux 0) tms)
    | `Match(t, liftno, bs, args) ->
        List.fold_left max 0 (List.map (aux 0) ((t :> nf)::args@List.map snd !bs))
    | `N _ -> 0
    | `Var _ -> 0
  ) in Listx.max (Listx.map (aux 0) tms)
;;

let auto_instantiate (n,p) =
  let p, showstoppers_step, showstoppers_eat = critical_showstoppers p in
 let x =
  match showstoppers_step, showstoppers_eat with
    | [], y::_ ->
      prerr_endline ("INSTANTIATING CRITICAL TO EAT " ^ string_of_var y); y
    | [], [] ->
     let heads = List.sort compare (filter_map (fun t -> match t with `Var _ -> None | x -> hd_of x) ((match p.div with Some t -> [(t :> i_n_var)] | _ -> []) @ p.ps)) in
     (match heads with
         [] -> assert false
       | x::_ ->
          prerr_endline ("INSTANTIATING TO EAT " ^ string_of_var x);
          x)
  | x::_, _ ->
      prerr_endline ("INSTANTIATING " ^ string_of_var x);
      x in
(* Strategy that decreases the special_k to 0 first (round robin)
1:11m42 2:14m5 3:11m16s 4:14m46s 5:12m7s 6:6m31s *)
let x =
 try
  match hd_of (List.find (fun t -> compute_special_k (Listx.Nil (t :> nf)) > 0) (all_terms p)) with
      None -> assert false
    | Some x ->
       prerr_endline ("INSTANTIATING AND HOPING " ^ string_of_var x);
       x
 with
  Not_found -> x
in
(* Instantiate in decreasing order of compute_special_k
1:15m14s 2:13m14s 3:4m55s 4:4m43s 5:4m34s 6:6m28s 7:3m31s
let x =
 try
  (match hd_of (snd (List.hd (List.sort (fun c1 c2 -> - compare (fst c1) (fst c2)) (filter_map (function `I _ as t -> Some (compute_special_k (Listx.Nil (t :> nf)),t) | _ -> None) (all_terms p))))) with
      None -> assert false
    | Some x ->
       prerr_endline ("INSTANTIATING AND HOPING " ^ string_of_var x);
       x)
 with
  Not_found -> x
in*)
 let special_k =
     compute_special_k (Listx.from_list (all_terms p :> nf list) )in
 if special_k < n then
  prerr_endline ("@@@@ NEW INSTANTIATE PHASE (" ^ string_of_int special_k ^ ") @@@@");
 let p = instantiate p x special_k in
 special_k,p


let rec auto_eat (n,({ps} as p)) =
 prerr_endline "{{{{{{{{ Computing measure before auto_instantiate }}}}}}";
 let m = problem_measure p in
 let (n,p') = auto_instantiate (n,p) in
 match eat p' with
   `Finished p -> p
 | `Continue p ->
     prerr_endline "{{{{{{{{ Computing measure inafter auto_instantiate }}}}}}";
     let delta = m - problem_measure p' in
     if delta <= 0 then (
      (* failwithProblem p' *)
      prerr_endline
      ("Measure did not decrease (delta=" ^ string_of_int delta ^ ")"))
     else prerr_endline ("$ Measure decreased by " ^ string_of_int delta);
     let p' = {p' with steps=(p'.steps - 1)} in
     (if p'.steps < 0 then prerr_endline ">>>>>>>>>> STEPS ARE OVER <<<<<<<<<"
     (*failwithProblem p' "steps are over. sorry."*) );
     auto_eat (n,p)
;;

let auto p n =
 prerr_endline ("@@@@ FIRST INSTANTIATE PHASE (" ^ string_of_int n ^ ") @@@@");
 match eat p with
   `Finished p -> p
 | `Continue p -> auto_eat (n,p)
;;

(*
0 = snd

      x y = y 0    a y = k  k z = z 0  c y = k   y u = u h1 h2 0          h2 a = h3
1 x a c    1 a 0 c  1 k c    1 c 0      1 k        1 k                     1 k
2 x a y    2 a 0 y  2 k y    2 y 0      2 y 0      2 h2 0                  2 h3
3 x b y    3 b 0 y  3 b 0 y  3 b 0 y    3 b 0 y    3 b 0 (\u. u h1 h2 0)   3 b 0 (\u. u h1 (\w.h3) 0)
4 x b c    4 b 0 c  4 b 0 c  4 b 0 c    4 b 0 c    4 b 0 c                 4 b 0 c
5 x (b e)  5 b e 0  5 b e 0  5 b e 0    5 b e 0    5 b e 0                 5 b e 0
6 y y      6 y y    6 y y    6 y y      6 y y      6 h1 h1 h2 0 h2 0       6 h1 h1 (\w. h3) 0 (\w. h3) 0

                                l2 _ = l3
b u = u l1 l2 0                 e _ _ _ _ = f                         l3 n = n j 0
1 k                             1 k                                  1 k
2 h3                            2 h3                                 2 h3
3 l2 0 (\u. u h1 (\w. h3) 0)    3 l3 (\u. u h1 (\w. h3) 0)           3 j h1 (\w. h3) 0 0
4 l2 0 c                        4 l3 c                               4 c j 0
5 e l1 l2 0 0                   5 f                                  5 f
6 h1 h1 (\w. h3) 0 (\w. h3) 0   6 h1 h1 (\w. h3) 0 (\w. h3) 0        6 h1 h1 (\w. h3) 0 (\w. h3) 0
*)

(*
                x n = n 0 ?
x a (b (a c))   a 0 = 1 ? (b (a c))   8
x a (b d')      a 0 = 1 ? (b d')      7
x b (a c)       b 0 = 1 ? (a c)       4
x b (a c')      b 0 = 1 ? (a c')      5

c = 2
c' = 3
a 2 = 4  (* a c *)
a 3 = 5  (* a c' *)
d' = 6
b 6 = 7  (* b d' *)
b 4 = 8  (* b (a c) *)
b 0 = 1
a 0 = 1
*)

(************** Tests ************************)

let optimize_numerals p =
  let replace_in_sigma perm =
    let rec aux = function
    | `N n -> `N (List.nth perm n)
    | `I _ -> assert false
    | `Var _ as t -> t
    | `Lam(v,t) -> `Lam(v, aux t)
    | `Match(_,_,bs,_) as t -> (bs := List.map (fun (n,t) -> (List.nth perm n, t)) !bs); t
    in List.map (fun (n,t) -> (n,aux t))
  in
  let deltas' = List.mapi (fun n d -> (n, List.map fst !d)) p.deltas in
  let maxs = Array.to_list (Array.init (List.length deltas') (fun _ -> 0)) in
  let max = List.fold_left max 0 (concat_map snd deltas') in
  let perm,_ = List.fold_left (fun (perm, maxs) (curr_n:int) ->
      let containing = filter_map (fun (i, bs) -> if List.mem curr_n bs then Some i else None) deltas' in
      (* (prerr_endline (string_of_int curr_n ^ " occurs in: " ^ (String.concat " " (List.map string_of_int containing)))); *)
      let neww = List.fold_left Pervasives.max 0 (List.mapi (fun n max -> if List.mem n containing then max else 0) maxs) in
      let maxs = List.mapi (fun i m -> if List.mem i containing then neww+1 else m) maxs in
      (neww::perm, maxs)
    ) ([],maxs) (Array.to_list (Array.init (max+1) (fun x -> x))) in
  replace_in_sigma (List.rev perm) p.sigma
;;

let env_of_sigma freshno sigma should_explode =
 let rec aux n =
  if n > freshno then
   []
  else
   let e = aux (n+1) in
   (try
    e,Pure.lift (-n-1) (snd (List.find (fun (i,_) -> i = n) sigma)),[]
   with
    Not_found ->
     if should_explode && `Var n = !bomb
      then ([], (let f t = Pure.A(t,t) in f (Pure.L (f (Pure.V 0)))), [])
      else ([],Pure.V n,[]))::e
 in aux 0
;;

prerr_endline "########## main ##########";;

(* Commands:
    v ==> v := \a. a k1 .. kn \^m.0
    + ==> v := \^k. numero  for every v such that ...
    * ==> tries v as long as possible and then +v as long as possible
*)
let main problems =
 let rec aux ({ps} as p) n l =
  if List.for_all (function `N _ -> true | _ -> false) ps && p.div = None then begin
   p
  end else
   let _ = prerr_endline (print_problem "main" p) in
   let x,l =
    match l with
     | cmd::l -> cmd,l
     | [] -> read_line (),[] in
   let cmd =
    if x = "+" then
     `DoneWith
    else if x = "*" then
     `Auto
    else
     `Step x in
   match cmd with
    | `DoneWith -> assert false (*aux (eat p) n l*) (* CSC: TODO *)
    | `Step x ->
        let x = var_of_string x in
        aux (instantiate p x n) n l
    | `Auto -> aux (auto p n) n l
 in
  List.iter
   (fun (p,n,cmds) ->
    Console.print_hline();
    bomb := `Var (-1);
    let p_finale = aux p n cmds in
    let freshno,sigma = p_finale.freshno, p_finale.sigma in
    prerr_endline ("------- <DONE> ------\n " ^ (string_of_int (p.steps - p_finale.steps)) ^ " steps of "^ (string_of_int p.steps) ^".");
    (* prerr_endline (print_problem "Original problem" p); *)
    prerr_endline "---------------------";
    let l = Array.to_list (Array.init (freshno + 1) string_of_var) in
    prerr_endline (" BOMB == " ^ print ~l !bomb);
    prerr_endline "---------------------";
    List.iter (fun (x,inst) -> prerr_endline (string_of_var x ^ " := " ^ print ~l inst)) sigma;
(*
     prerr_endline "----------------------";
     let ps =
      List.fold_left (fun ps (x,inst) ->
       (* CSC: XXXX Is the subst always sorted correctly? Otherwise, implement a recursive subst *)
       (* In this non-recursive version, the intermediate states may containt Matchs *)
       List.map (fun t -> let t = subst false x inst (t :> nf) in cast_to_i_num_var t) ps)
       (p.ps :> i_num_var list) sigma in
     prerr_endline (print_problem {p with ps= List.map (function t -> cast_to_i_n_var t) ps; freshno});
     List.iteri (fun i (n,more_args) -> assert (more_args = 0 && n = `N i)) ps ;
*)
    prerr_endline "---------<OPT>----------";
    let sigma = optimize_numerals p_finale in (* optimize numerals *)
    let l = Array.to_list (Array.init (freshno + 1) string_of_var) in
    List.iter (fun (x,inst) -> prerr_endline (string_of_var x ^ " := " ^ print ~l inst)) sigma;
    prerr_endline "---------<PURE>---------";
    let div = option_map (fun div -> ToScott.t_of_nf (div :> nf)) p.div in
    let conv = List.map (fun t -> ToScott.t_of_nf (t :> nf)) p.conv in
    let ps = List.map (fun t -> ToScott.t_of_nf (t :> nf)) p.ps in
    let sigma = List.map (fun (x,inst) -> x, ToScott.t_of_nf inst) sigma in
    (*let ps_ok = List.fold_left (fun ps (x,inst) ->
      List.map (Pure.subst false x inst) ps) ps sigma in*)
    let e = env_of_sigma freshno sigma true in
    let e' = env_of_sigma freshno sigma false in

(*
     prerr_endline "---------<PPP>---------";
let rec print_e e =
 "[" ^ String.concat ";" (List.map (fun (e,t,[]) -> print_e e ^ ":" ^ Pure.print t) e) ^ "]"
in
     prerr_endline (print_e e);
     List.iter (fun (t,t_ok) ->
      prerr_endline ("T0= " ^ Pure.print t ^ "\nTM= " ^ Pure.print (Pure.unwind (e,t,[])) ^ "\nOM= " ^ Pure.print t_ok);
      (*assert (Pure.unwind (e,t,[]) = t_ok)*)
     ) (List.combine ps ps_ok);
*)
     prerr_endline "--------<REDUCE>---------";
     (function Some div ->
      print_endline (Pure.print div);
      let t = Pure.mwhd (e',div,[]) in
      prerr_endline ("*:: " ^ (Pure.print t));
      prerr_endline (print !bomb);
      assert (t = ToScott.t_of_nf (!bomb:>nf))
     | None -> ()) div;
     List.iter (fun n ->
       prerr_endline ("_::: " ^ (Pure.print n));
       let t = Pure.mwhd (e,n,[]) in
       prerr_endline ("_:: " ^ (Pure.print t))
     ) conv ;
     List.iteri (fun i n ->
       prerr_endline ((string_of_int i) ^ "::: " ^ (Pure.print n));
       let t = Pure.mwhd (e,n,[]) in
       prerr_endline ((string_of_int i) ^ ":: " ^ (Pure.print t));
       assert (t = Scott.mk_n i)
     ) ps ;
     prerr_endline "-------- </DONE> --------"
   ) problems

(********************** problems *******************)

let zero = `Var 0;;

let append_zero =
 function
  | `I _
  | `Var _ as i ->  cast_to_i_n_var (mk_app i zero)
  | _ -> assert false
;;

let bounds_on_steps all_tms =
 let rec aux = function
 | `I(k,args) -> Listx.fold_left (fun acc t -> 1 + acc + (aux t)) 0 args
 | `Var _ -> 1
 | `Lam (_, t) -> 1 + aux t
 | _ -> assert false
 in List.fold_right ((+) ++ aux) all_tms 0
;;

type t = problem * int * string list;;

let magic_conv ~div ~conv ~nums cmds =
 let all_tms = (match div with None -> [] | Some div -> [div]) @ nums @ conv in
  let all_tms, var_names = parse' all_tms in
  let steps = bounds_on_steps all_tms in
  let div, (tms, conv) = match div with
    | None -> None, list_cut (List.length nums, all_tms)
    | Some _ -> Some (List.hd all_tms), list_cut (List.length nums, List.tl all_tms) in

 if match div with None -> false | Some div -> List.exists (eta_subterm div) (tms@conv)
 then (
  prerr_endline "--- TEST SKIPPED ---";
  {freshno=0; div=None; conv=[]; ps=[]; sigma=[]; deltas=[]; steps=(-1)}, 0, []
 ) else
  let tms = sort_uniq ~compare:eta_compare tms in
  let special_k = compute_special_k (Listx.from_list all_tms) in (* compute initial special K *)
  (* casts *)
  let div = option_map cast_to_i_var div in
  let conv = List.map cast_to_i_n_var conv in
  let tms = List.map cast_to_i_n_var tms in

  let ps = List.map append_zero tms in (* crea lista applicando zeri o dummies *)
  let freshno = List.length var_names in
  let deltas =
   let dummy = `Var (max_int / 2) in
    [ ref (Array.to_list (Array.init (List.length ps) (fun i -> i, dummy))) ] in

  {freshno; div; conv; ps; sigma=[] ; deltas; steps}, special_k, cmds
;;

let magic strings cmds = magic_conv None [] strings cmds;;