1 let (++) f g x = f (g x);;
\r
3 let rec fold_nat f x n = if n = 0 then x else f (fold_nat f x (n-1)) n ;;
\r
5 let print_hline = Console.print_hline;;
\r
13 | L of (t * t list (*garbage*))
\r
17 let delta = L(A(V 0, V 0),[]);;
\r
19 let rec is_stuck = function
\r
21 | A(t,_) -> is_stuck t
\r
26 let rec aux l1 l2 t1 t2 = match t1, t2 with
\r
27 | _, _ when is_stuck t1 || is_stuck t2 -> true
\r
28 | L t1, L t2 -> aux l1 l2 (fst t1) (fst t2)
\r
29 | L t1, t2 -> aux l1 (l2+1) (fst t1) t2
\r
30 | t1, L t2 -> aux (l1+1) l2 t1 (fst t2)
\r
31 | V a, V b -> a + l1 = b + l2
\r
32 | A(t1,t2), A(u1,u2) -> aux l1 l2 t1 u1 && aux l1 l2 t2 u2
\r
35 let eta_eq = eta_eq' 0 0;;
\r
37 (* is arg1 eta-subterm of arg2 ? *)
\r
39 let rec aux lev t = if t = C then false else (eta_eq' lev 0 u t || match t with
\r
40 | L(t,g) -> List.exists (aux (lev+1)) (t::g)
\r
41 | A(t1, t2) -> aux lev t1 || aux lev t2
\r
46 (* does NOT lift the argument *)
\r
47 let mk_lams = fold_nat (fun x _ -> L(x,[])) ;;
\r
50 let string_of_bvar =
\r
51 let bound_vars = ["x"; "y"; "z"; "w"; "q"] in
\r
52 let bvarsno = List.length bound_vars in
\r
53 fun nn -> if nn < bvarsno then List.nth bound_vars nn else "x" ^ (string_of_int (nn - bvarsno + 1)) in
\r
54 let rec string_of_term_w_pars level = function
\r
55 | V v -> if v >= level then "`" ^ string_of_int (v-level) else
\r
56 string_of_bvar (level - v-1)
\r
59 | L _ as t -> "(" ^ string_of_term_no_pars level t ^ ")"
\r
60 and string_of_term_no_pars_app level = function
\r
61 | A(t1,t2) -> string_of_term_no_pars_app level t1 ^ " " ^ string_of_term_w_pars level t2
\r
62 | _ as t -> string_of_term_w_pars level t
\r
63 and string_of_term_no_pars level = function
\r
64 | L(t,g) -> "λ" ^ string_of_bvar level ^ ". " ^ string_of_term_no_pars (level+1) t
\r
65 ^ (if g = [] then "" else String.concat ", " ("" :: List.map (string_of_term_w_pars (level+1)) g))
\r
66 | _ as t -> string_of_term_no_pars_app level t
\r
67 in string_of_term_no_pars 0
\r
75 ; sigma : (var * t) list (* substitutions *)
\r
78 let string_of_problem p =
\r
80 "[DV] " ^ string_of_t p.div;
\r
81 "[CV] " ^ string_of_t p.conv;
\r
83 String.concat "\n" lines
\r
87 exception Done of (var * t) list (* substitution *);;
\r
88 exception Fail of int * string;;
\r
90 let problem_fail p reason =
\r
91 print_endline "!!!!!!!!!!!!!!! FAIL !!!!!!!!!!!!!!!";
\r
92 print_endline (string_of_problem p);
\r
93 raise (Fail (-1, reason))
\r
96 let freshvar ({freshno} as p) =
\r
97 {p with freshno=freshno+1}, freshno+1
\r
100 (* CSC: rename? is an applied C an inert?
\r
101 is_inert and get_inert work inconsistently *)
\r
104 | A(t,_) -> is_inert t
\r
110 let rec is_constant =
\r
115 | L(t,_) -> is_constant t
\r
118 let rec get_inert = function
\r
119 | V _ | C as t -> (t,0)
\r
120 | A(t, _) -> let hd,args = get_inert t in hd,args+1
\r
121 | _ -> assert false
\r
124 let args_of_inert =
\r
128 | A(t, a) -> aux (a::acc) t
\r
129 | _ -> assert false
\r
134 (* precomputes the number of leading lambdas in a term,
\r
135 after replacing _v_ w/ a term starting with n lambdas *)
\r
136 let rec no_leading_lambdas v n = function
\r
137 | L(t,_) -> 1 + no_leading_lambdas (v+1) n t
\r
138 | A _ as t -> let v', m = get_inert t in if V v = v' then max 0 (n - m) else 0
\r
139 | V v' -> if v = v' then n else 0
\r
143 let rec subst level delift sub =
\r
145 | V v -> (if v = level + fst sub then lift level (snd sub) else V (if delift && v > level then v-1 else v)), []
\r
146 | L x -> let t, g = subst_in_lam (level+1) delift sub x in L(t, g), []
\r
148 let t1, g1 = subst level delift sub t1 in
\r
149 let t2, g2 = subst level delift sub t2 in
\r
150 let t3, g3 = mk_app t1 t2 in
\r
153 and subst_in_lam level delift sub (t, g) =
\r
154 let t', g' = subst level delift sub t in
\r
155 let g'' = List.fold_left
\r
157 let x,y = subst level delift sub t in
\r
158 (x :: y @ xs)) g' g in t', g''
\r
159 and mk_app t1 t2 = if t1 = delta && t2 = delta then raise B
\r
161 | L x -> subst_in_lam 0 true (0, t2) x
\r
162 | _ -> A (t1, t2), []
\r
166 | V m -> V (if m >= lev then m + n else m)
\r
167 | L(t,g) -> L (aux (lev+1) t, List.map (aux (lev+1)) g)
\r
168 | A (t1, t2) -> A (aux lev t1, aux lev t2)
\r
172 let subst = subst 0 false;;
\r
174 let subst_in_problem ((v, t) as sub) p =
\r
175 print_endline ("-- SUBST " ^ string_of_t (V v) ^ " |-> " ^ string_of_t t);
\r
176 let sigma = sub :: p.sigma in
\r
177 let div, g = try subst sub p.div with B -> raise (Done sigma) in
\r
179 let conv, f = try subst sub p.conv with B -> raise (Fail(-1, "p.conv diverged")) in
\r
181 {p with div; conv; sigma}
\r
184 let get_subterms_with_head hd_var =
\r
185 let rec aux lev inert_done g = function
\r
186 | L(t,g') -> List.fold_left (aux (lev+1) false) g (t::g')
\r
189 let hd_var', n_args' = get_inert t1 in
\r
190 if not inert_done && hd_var' = V (hd_var + lev)
\r
191 then lift ~-lev t :: aux lev false (aux lev true g t1) t2
\r
192 else aux lev false (aux lev true g t1) t2
\r
197 let rec aux = function
\r
199 let t = aux (lift (List.length g) t) in
\r
200 let t = List.fold_left (fun t g -> Pure.A(Pure.L t, aux g)) t g in
\r
202 | A (t1,t2) -> Pure.A (aux t1, aux t2)
\r
203 | V n -> Pure.V (n)
\r
204 | C -> Pure.V (min_int/2)
\r
208 let check p sigma =
\r
209 print_endline "Checking...";
\r
210 let div = purify p.div in
\r
211 let conv = purify p.conv in
\r
212 let sigma = List.map (fun (v,t) -> v, purify t) sigma in
\r
213 let freshno = List.fold_right (max ++ fst) sigma 0 in
\r
214 let env = Pure.env_of_sigma freshno sigma in
\r
215 assert (Pure.diverged (Pure.mwhd (env,div,[])));
\r
216 print_endline " D diverged.";
\r
217 assert (not (Pure.diverged (Pure.mwhd (env,conv,[]))));
\r
218 print_endline " C converged.";
\r
223 print_endline (string_of_problem p); (* non cancellare *)
\r
224 if not (is_inert p.div) then problem_fail p "p.div converged";
\r
225 (* Trailing constant args can be removed because do not contribute to eta-diff *)
\r
226 let rec remove_trailing_constant_args = function
\r
227 | A(t1, t2) when is_constant t2 -> remove_trailing_constant_args t1
\r
229 let p = {p with div=remove_trailing_constant_args p.div} in
\r
233 (* drops the arguments of t after the n-th *)
\r
234 let inert_cut_at n t =
\r
239 let k', t' = aux t1 in
\r
240 if k' = n then n, t'
\r
242 | _ -> assert false
\r
246 (* return the index of the first argument with a difference
\r
247 (the first argument is 0) *)
\r
248 let find_eta_difference p t =
\r
249 let divargs = args_of_inert p.div in
\r
250 let conargs = args_of_inert t in
\r
251 let rec aux k divargs conargs =
\r
252 match divargs,conargs with
\r
255 | t1::divargs,t2::conargs ->
\r
256 (if not (eta_eq t1 t2) then [k] else []) @ aux (k+1) divargs conargs
\r
258 aux 0 divargs conargs
\r
261 let compute_max_lambdas_at hd_var j =
\r
262 let rec aux hd = function
\r
264 (if get_inert t1 = (V hd, j)
\r
265 then max ( (*FIXME*)
\r
266 if is_inert t2 && let hd', j' = get_inert t2 in hd' = V hd
\r
267 then let hd', j' = get_inert t2 in j - j'
\r
268 else no_leading_lambdas hd_var j t2)
\r
269 else id) (max (aux hd t1) (aux hd t2))
\r
270 | L(t,_) -> aux (hd+1) t
\r
275 let print_cmd s1 s2 = print_endline (">> " ^ s1 ^ " " ^ s2);;
\r
277 (* returns Some i if i is the smallest integer s.t. p holds for the i-th
\r
278 element of the list in input *)
\r
279 let smallest_such_that p =
\r
283 | hd::_ when (print_endline (string_of_t hd) ; p hd) -> Some i
\r
284 | _::tl -> aux (i+1) tl
\r
289 (* step on the head of div, on the k-th argument, with n fresh vars *)
\r
291 let hd, _ = get_inert p.div in
\r
293 | C | L _ | A _ -> assert false
\r
295 print_cmd "STEP" ("on " ^ string_of_t (V var) ^ " (on " ^ string_of_int (k+1) ^ "th)");
\r
296 let p, t = (* apply fresh vars *)
\r
297 fold_nat (fun (p, t) _ ->
\r
298 let p, v = freshvar p in
\r
299 p, A(t, V (v + k + 1))
\r
301 let t = (* apply unused bound variables V_{k-1}..V_1 *)
\r
302 fold_nat (fun t m -> A(t, V (k-m+1))) t k in
\r
303 let t = mk_lams t (k+1) in (* make leading lambdas *)
\r
304 let subst = var, t in
\r
305 let p = subst_in_problem subst p in
\r
310 (* one-step version of eat *)
\r
311 let compute_max_arity =
\r
312 let rec aux n = function
\r
313 | A(t1,t2) -> max (aux (n+1) t1) (aux 0 t2)
\r
314 | L(t,g) -> List.fold_right (max ++ (aux 0)) (t::g) 0
\r
317 print_cmd "FINISH" "";
\r
318 (* First, a step on the last argument of the divergent.
\r
319 Because of the sanity check, it will never be a constant term. *)
\r
320 let div_hd, div_nargs = get_inert p.div in
\r
321 let div_hd = match div_hd with V n -> n | _ -> assert false in
\r
322 let j = div_nargs - 1 in
\r
323 let arity = compute_max_arity p.conv in
\r
324 let n = 1 + arity + max
\r
325 (compute_max_lambdas_at div_hd j p.div)
\r
326 (compute_max_lambdas_at div_hd j p.conv) in
\r
327 let p = step j n p in
\r
328 (* Now, find first argument of div that is a variable never applied anywhere.
\r
329 It must exist because of some invariant, since we just did a step,
\r
330 and because of the arity of the divergent *)
\r
331 let div_hd, div_nargs = get_inert p.div in
\r
332 let div_hd = match div_hd with V n -> n | _ -> assert false in
\r
333 let rec aux m = function
\r
334 | A(t, V delta_var) ->
\r
335 if delta_var <> div_hd && get_subterms_with_head delta_var p.conv = []
\r
338 | A(t,_) -> aux (m-1) t
\r
339 | _ -> assert false in
\r
340 let m, delta_var = aux div_nargs p.div in
\r
341 let p = subst_in_problem (delta_var, delta) p in
\r
342 let p = subst_in_problem (div_hd, mk_lams delta (m-1)) p in
\r
348 let hd, n_args = get_inert p.div in
\r
350 | C | L _ | A _ -> assert false
\r
352 let tms = get_subterms_with_head hd_var p.conv in
\r
353 if List.exists (fun t -> snd (get_inert t) >= n_args) tms
\r
355 (* let tms = List.sort (fun t1 t2 -> - compare (snd (get_inert t1)) (snd (get_inert t2))) tms in *)
\r
356 List.iter (fun t -> try
\r
357 let js = find_eta_difference p t in
\r
358 (* print_endline (String.concat ", " (List.map string_of_int js)); *)
\r
359 if js = [] then problem_fail p "no eta difference found (div subterm of conv?)";
\r
360 let js = List.rev js in
\r
365 (compute_max_lambdas_at hd_var j p.div)
\r
366 (compute_max_lambdas_at hd_var j p.conv) in
\r
367 ignore (aux (step j k p))
\r
369 print_endline ("Backtracking (eta_diff) because: " ^ s)) js;
\r
370 raise (Fail(-1, "no eta difference"))
\r
372 print_endline ("Backtracking (get_subterms) because: " ^ s)) tms;
\r
373 raise (Fail(-1, "no similar terms"))
\r
376 problem_fail (finish p) "Finish did not complete the problem"
\r
380 with Done sigma -> sigma
\r
383 let problem_of (label, div, convs, ps, var_names) =
\r
385 let rec aux lev = function
\r
386 | `Lam(_, t) -> L (aux (lev+1) t, [])
\r
387 | `I (v, args) -> Listx.fold_left (fun x y -> fst (mk_app x (aux lev y))) (aux lev (`Var v)) args
\r
388 | `Var(v,_) -> if v >= lev && List.nth var_names (v-lev) = "C" then C else V v
\r
389 | `N _ | `Match _ -> assert false in
\r
390 assert (List.length ps = 0);
\r
391 let convs = List.rev convs in
\r
392 let conv = List.fold_left (fun x y -> fst (mk_app x (aux 0 (y :> Num.nf)))) (V (List.length var_names)) convs in
\r
393 let var_names = "@" :: var_names in
\r
394 let div = match div with
\r
395 | Some div -> aux 0 (div :> Num.nf)
\r
396 | None -> assert false in
\r
397 let varno = List.length var_names in
\r
398 {orig_freshno=varno; freshno=1+varno; div; conv; sigma=[]}
\r
402 if is_constant p.div
\r
403 then print_endline "!!! div is stuck. Problem was not run !!!"
\r
404 else if eta_subterm p.div p.conv
\r
405 then print_endline "!!! div is subterm of conv. Problem was not run !!!"
\r
406 else let p = sanity p (* initial sanity check *) in check p (auto p)
\r
409 Problems.main (solve ++ problem_of);
\r
411 (* Example usage of interactive: *)
\r
413 (* let interactive div conv cmds =
\r
414 let p = problem_of div conv in
\r
416 let p = List.fold_left (|>) p cmds in
\r
418 let nth spl n = int_of_string (List.nth spl n) in
\r
420 let s = read_line () in
\r
421 let spl = Str.split (Str.regexp " +") s in
\r
422 s, let uno = List.hd spl in
\r
423 try if uno = "eat" then eat
\r
424 else if uno = "step" then step (nth spl 1) (nth spl 2)
\r
425 else failwith "Wrong input."
\r
426 with Failure s -> print_endline s; (fun x -> x) in
\r
427 let str, cmd = read_cmd () in
\r
428 let cmds = (" " ^ str ^ ";")::cmds in
\r
430 let p = cmd p in f p cmds
\r
432 | Done _ -> print_endline "Done! Commands history: "; List.iter print_endline (List.rev cmds)
\r
434 ) with Done _ -> ()
\r
437 (* interactive "x y"
\r
438 "@ (x x) (y x) (y z)" [step 0 1; step 0 2; eat]
\r