1 (* cOpyright (C) 2005, HELM Team.
3 * This file is part of HELM, an Hypertextual, Electronic
4 * Library of Mathematics, developed at the Computer Science
5 * Department, University of Bologna, Italy.
7 * HELM is free software; you can redistribute it and/or
8 * modify it under the terms of the GNU General Public License
9 * as published by the Free Software Foundation; either version 2
10 * of the License, or (at your option) any later version.
12 * HELM is distributed in the hope that it will be useful,
13 * but WITHOUT ANY WARRANTY; without even the implied warranty of
14 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
15 * GNU General Public License for more details.
17 * You should have received a copy of the GNU General Public License
18 * along with HELM; if not, write to the Free Software
19 * Foundation, Inc., 59 Temple Place - Suite 330, Boston,
22 * For details, see the HELM World-Wide-Web page,
23 * http://cs.unibo.it/helm/.
26 (* let _profiler = <:profiler<_profiler>>;; *)
28 (* $Id: inference.ml 6245 2006-04-05 12:07:51Z tassi $ *)
30 type rule = SuperpositionRight | SuperpositionLeft | Demodulation
31 type uncomparable = int -> int
34 uncomparable * (* trick to break structural equality *)
37 (Cic.term * (* type *)
38 Cic.term * (* left side *)
39 Cic.term * (* right side *)
40 Utils.comparison) * (* ordering *)
41 Cic.metasenv * (* environment for metas *)
45 | Step of Subst.substitution * (rule * int*(Utils.pos*int)* Cic.term)
46 (* subst, (rule,eq1, eq2,predicate) *)
47 and goal_proof = (rule * Utils.pos * int * Subst.substitution * Cic.term) list
49 (* the hashtbl eq_id -> proof, max_eq_id *)
50 type equality_bag = (int,equality) Hashtbl.t * int ref
52 type goal = goal_proof * Cic.metasenv * Cic.term
55 let mk_equality_bag () =
56 Hashtbl.create 1024, ref 0
63 let add_to_bag (id_to_eq,_) id eq =
64 Hashtbl.add id_to_eq id eq
67 let uncomparable = fun _ -> 0
69 let mk_equality bag (weight,p,(ty,l,r,o),m) =
70 let id = freshid bag in
71 let eq = (uncomparable,weight,p,(ty,l,r,o),m,id) in
76 let mk_tmp_equality (weight,(ty,l,r,o),m) =
78 uncomparable,weight,Exact (Cic.Implicit None),(ty,l,r,o),m,id
82 let open_equality (_,weight,proof,(ty,l,r,o),m,id) =
83 (weight,proof,(ty,l,r,o),m,id)
85 let string_of_rule = function
86 | SuperpositionRight -> "SupR"
87 | SuperpositionLeft -> "SupL"
88 | Demodulation -> "Demod"
91 let string_of_equality ?env eq =
94 let w, _, (ty, left, right, o), m , id = open_equality eq in
95 Printf.sprintf "Id: %d, Weight: %d, {%s}: %s =(%s) %s [%s]"
96 id w (CicPp.ppterm ty)
98 (Utils.string_of_comparison o) (CicPp.ppterm right)
99 (String.concat ", " (List.map (fun (i,_,_) -> string_of_int i) m))
101 | Some (_, context, _) ->
102 let names = Utils.names_of_context context in
103 let w, _, (ty, left, right, o), m , id = open_equality eq in
104 Printf.sprintf "Id: %d, Weight: %d, {%s}: %s =(%s) %s [%s]"
105 id w (CicPp.pp ty names)
106 (CicPp.pp left names) (Utils.string_of_comparison o)
107 (CicPp.pp right names)
108 (String.concat ", " (List.map (fun (i,_,_) -> string_of_int i) m))
112 let compare (_,_,_,s1,_,_) (_,_,_,s2,_,_) =
113 Pervasives.compare s1 s2
116 let rec max_weight_in_proof ((id_to_eq,_) as bag) current =
119 | Step (_, (_,id1,(_,id2),_)) ->
120 let eq1 = Hashtbl.find id_to_eq id1 in
121 let eq2 = Hashtbl.find id_to_eq id2 in
122 let (w1,p1,(_,_,_,_),_,_) = open_equality eq1 in
123 let (w2,p2,(_,_,_,_),_,_) = open_equality eq2 in
124 let current = max current w1 in
125 let current = max_weight_in_proof bag current p1 in
126 let current = max current w2 in
127 max_weight_in_proof bag current p2
129 let max_weight_in_goal_proof ((id_to_eq,_) as bag) =
131 (fun current (_,_,id,_,_) ->
132 let eq = Hashtbl.find id_to_eq id in
133 let (w,p,(_,_,_,_),_,_) = open_equality eq in
134 let current = max current w in
135 max_weight_in_proof bag current p)
137 let max_weight bag goal_proof proof =
138 let current = max_weight_in_proof bag 0 proof in
139 max_weight_in_goal_proof bag current goal_proof
141 let proof_of_id (id_to_eq,_) id =
143 let (_,p,(_,l,r,_),_,_) = open_equality (Hashtbl.find id_to_eq id) in
146 Not_found -> assert false
149 let string_of_proof ?(names=[]) bag p gp =
150 let str_of_pos = function
151 | Utils.Left -> "left"
152 | Utils.Right -> "right"
154 let fst3 (x,_,_) = x in
155 let rec aux margin name =
156 let prefix = String.make margin ' ' ^ name ^ ": " in function
158 Printf.sprintf "%sExact (%s)\n"
159 prefix (CicPp.pp t names)
160 | Step (subst,(rule,eq1,(pos,eq2),pred)) ->
161 Printf.sprintf "%s%s(%s|%d with %d dir %s pred %s))\n"
162 prefix (string_of_rule rule) (Subst.ppsubst ~names subst) eq1 eq2 (str_of_pos pos)
163 (CicPp.pp pred names)^
164 aux (margin+1) (Printf.sprintf "%d" eq1) (fst3 (proof_of_id bag eq1)) ^
165 aux (margin+1) (Printf.sprintf "%d" eq2) (fst3 (proof_of_id bag eq2))
170 (fun (r,pos,i,s,t) ->
172 "GOAL: %s %s %d %s %s\n" (string_of_rule r)
173 (str_of_pos pos) i (Subst.ppsubst ~names s) (CicPp.pp t names)) ^
174 aux 1 (Printf.sprintf "%d " i) (fst3 (proof_of_id bag i)))
178 let rec depend ((id_to_eq,_) as bag) eq id seen =
179 let (_,p,(_,_,_,_),_,ideq) = open_equality eq in
180 if List.mem ideq seen then
187 | Exact _ -> false,seen
188 | Step (_,(_,id1,(_,id2),_)) ->
189 let seen = ideq::seen in
190 let eq1 = Hashtbl.find id_to_eq id1 in
191 let eq2 = Hashtbl.find id_to_eq id2 in
192 let b1,seen = depend bag eq1 id seen in
193 if b1 then b1,seen else depend bag eq2 id seen
196 let depend bag eq id = fst (depend bag eq id []);;
198 let ppsubst = Subst.ppsubst ~names:[];;
200 (* returns an explicit named subst and a list of arguments for sym_eq_URI *)
201 let build_ens uri termlist =
202 let obj, _ = CicEnvironment.get_obj CicUniv.empty_ugraph uri in
204 | Cic.Constant (_, _, _, uris, _) ->
205 (* assert (List.length uris <= List.length termlist); *)
206 let rec aux = function
208 | (uri::uris), (term::tl) ->
209 let ens, args = aux (uris, tl) in
210 (uri, term)::ens, args
211 | _, _ -> assert false
217 let mk_sym uri ty t1 t2 p =
218 let ens, args = build_ens uri [ty;t1;t2;p] in
219 Cic.Appl (Cic.Const(uri, ens) :: args)
222 let mk_trans uri ty t1 t2 t3 p12 p23 =
223 let ens, args = build_ens uri [ty;t1;t2;t3;p12;p23] in
224 Cic.Appl (Cic.Const (uri, ens) :: args)
227 let mk_eq_ind uri ty what pred p1 other p2 =
228 let ens, args = build_ens uri [ty; what; pred; p1; other; p2] in
229 Cic.Appl (Cic.Const (uri, ens) :: args)
232 let p_of_sym ens tl =
233 let args = List.map snd ens @ tl in
239 let open_trans ens tl =
240 let args = List.map snd ens @ tl in
242 | [ty;l;m;r;p1;p2] -> ty,l,m,r,p1,p2
246 let open_sym ens tl =
247 let args = List.map snd ens @ tl in
249 | [ty;l;r;p] -> ty,l,r,p
253 let open_eq_ind args =
255 | [ty;l;pred;pl;r;pleqr] -> ty,l,pred,pl,r,pleqr
261 | Cic.Lambda (_,_,(Cic.Appl [Cic.MutInd (uri, 0,_);ty;l;r]))
262 when LibraryObjects.is_eq_URI uri -> ty,uri,l,r
263 | _ -> Utils.debug_print (lazy (CicPp.ppterm pred)); assert false
267 CicSubstitution.subst (Cic.Implicit None) t <>
268 CicSubstitution.subst (Cic.Rel 1) t
271 let canonical t context menv =
272 let remove_cycles t =
275 Cic.Appl (Cic.Const (uri_trans,_)::_)
276 when LibraryObjects.is_trans_eq_URI uri_trans ->
281 Cic.Appl (Cic.Const (uri_trans,ens)::tl)
282 when LibraryObjects.is_trans_eq_URI uri_trans ->
283 let ty,l,m,r,p1,p2 = open_trans ens tl in
284 (if is_transitive p1 then fst (collect p1) else [l,p1]) @
285 (if is_transitive p2 then fst (collect p2) else [m,p2]),
287 | t -> assert false in
288 let rec cut_to_last_duplicate l acc =
291 | (l',p)::tl when l=l' ->
293 Utils.debug_print (lazy ("!!! RISPARMIO " ^ string_of_int (List.length acc) ^ " PASSI"));
294 cut_to_last_duplicate l [l',p] tl
296 cut_to_last_duplicate l ((l',p)::acc) tl
300 (l,_)::_::_ as steps, ((r,uri_trans,ty) as last) ->
301 (match cut_to_last_duplicate l [] steps with
302 (l,p1)::((m,_)::_::_ as tl) ->
303 mk_trans uri_trans ty l m r p1 (rebuild (tl,last))
304 | [l,p1 ; m,p2] -> mk_trans uri_trans ty l m r p1 p2
306 | [] -> assert false)
309 if is_transitive t then
314 let rec remove_refl t =
316 | Cic.Appl (((Cic.Const(uri_trans,ens))::tl) as args)
317 when LibraryObjects.is_trans_eq_URI uri_trans ->
318 let ty,l,m,r,p1,p2 = open_trans ens tl in
320 | Cic.Appl [Cic.MutConstruct (uri, 0, 1,_);_;_],p2 ->
322 | p1,Cic.Appl [Cic.MutConstruct (uri, 0, 1,_);_;_] ->
324 | _ -> Cic.Appl (List.map remove_refl args))
325 | Cic.Appl l -> Cic.Appl (List.map remove_refl l)
326 | Cic.LetIn (name,bo,ty,rest) ->
327 Cic.LetIn (name,remove_refl bo,remove_refl ty,remove_refl rest)
330 let rec canonical_trough_lambda context = function
331 | Cic.Lambda(name,ty,bo) ->
332 let context' = (Some (name,Cic.Decl ty))::context in
333 Cic.Lambda(name,ty,canonical_trough_lambda context' bo)
334 | t -> canonical context t
336 and canonical context t =
338 | Cic.LetIn(name,bo,ty,rest) ->
339 let bo = canonical_trough_lambda context bo in
340 let ty = canonical_trough_lambda context ty in
341 let context' = (Some (name,Cic.Def (bo,ty)))::context in
342 Cic.LetIn(name,bo,ty,canonical context' rest)
343 | Cic.Appl (((Cic.Const(uri_sym,ens))::tl) as args)
344 when LibraryObjects.is_sym_eq_URI uri_sym ->
345 (match p_of_sym ens tl with
346 | Cic.Appl ((Cic.Const(uri,ens))::tl)
347 when LibraryObjects.is_sym_eq_URI uri ->
348 canonical context (p_of_sym ens tl)
349 | Cic.Appl ((Cic.Const(uri_trans,ens))::tl)
350 when LibraryObjects.is_trans_eq_URI uri_trans ->
351 let ty,l,m,r,p1,p2 = open_trans ens tl in
352 mk_trans uri_trans ty r m l
353 (canonical context (mk_sym uri_sym ty m r p2))
354 (canonical context (mk_sym uri_sym ty l m p1))
355 | Cic.Appl (([Cic.Const(uri_feq,ens);ty1;ty2;f;x;y;p]))
356 when LibraryObjects.is_eq_f_URI uri_feq ->
357 let eq = LibraryObjects.eq_URI_of_eq_f_URI uri_feq in
359 Cic.Const (LibraryObjects.eq_f_sym_URI ~eq, [])
361 let rc = Cic.Appl [eq_f_sym;ty1;ty2;f;x;y;p] in
362 Utils.debug_print (lazy ("CANONICAL " ^ CicPp.ppterm rc));
364 | Cic.Appl [Cic.MutConstruct (uri, 0, 1,_);_;_] as t
365 when LibraryObjects.is_eq_URI uri -> t
366 | _ -> Cic.Appl (List.map (canonical context) args))
367 | Cic.Appl l -> Cic.Appl (List.map (canonical context) l)
370 remove_cycles (remove_refl (canonical context t))
373 let compose_contexts ctx1 ctx2 =
374 ProofEngineReduction.replace_lifting
375 ~equality:(fun _ ->(=)) ~context:[] ~what:[Cic.Implicit(Some `Hole)] ~with_what:[ctx2] ~where:ctx1
378 let put_in_ctx ctx t =
379 ProofEngineReduction.replace_lifting
380 ~equality:(fun _ -> (=)) ~context:[] ~what:[Cic.Implicit (Some `Hole)] ~with_what:[t] ~where:ctx
383 let mk_eq uri ty l r =
384 let ens, args = build_ens uri [ty; l; r] in
385 Cic.Appl (Cic.MutInd(uri,0,ens) :: args)
388 let mk_refl uri ty t =
389 let ens, args = build_ens uri [ty; t] in
390 Cic.Appl (Cic.MutConstruct(uri,0,1,ens) :: args)
393 let open_eq = function
394 | Cic.Appl [Cic.MutInd(uri,0,[]);ty;l;r] when LibraryObjects.is_eq_URI uri ->
399 let mk_feq uri_feq ty ty1 left pred right t =
400 let ens, args = build_ens uri_feq [ty;ty1;pred;left;right;t] in
401 Cic.Appl (Cic.Const(uri_feq,ens) :: args)
404 let rec look_ahead aux = function
405 | Cic.Appl ((Cic.Const(uri_ind,ens))::tl) as t
406 when LibraryObjects.is_eq_ind_URI uri_ind ||
407 LibraryObjects.is_eq_ind_r_URI uri_ind ->
408 let ty1,what,pred,p1,other,p2 = open_eq_ind tl in
409 let ty2,eq,lp,rp = open_pred pred in
410 let hole = Cic.Implicit (Some `Hole) in
411 let ty2 = CicSubstitution.subst hole ty2 in
412 aux ty1 (CicSubstitution.subst other lp) (CicSubstitution.subst other rp) hole ty2 t
413 | Cic.Lambda (n,s,t) -> Cic.Lambda (n,s,look_ahead aux t)
417 let contextualize uri ty left right t =
418 let hole = Cic.Implicit (Some `Hole) in
419 (* aux [uri] [ty] [left] [right] [ctx] [ctx_ty] [t]
421 * the parameters validate this invariant
422 * t: eq(uri) ty left right
423 * that is used only by the base case
425 * ctx is a term with an hole. Cic.Implicit(Some `Hole) is the empty context
426 * ctx_ty is the type of ctx
428 let rec aux uri ty left right ctx_d ctx_ty t =
430 | Cic.Appl ((Cic.Const(uri_sym,ens))::tl)
431 when LibraryObjects.is_sym_eq_URI uri_sym ->
432 let ty,l,r,p = open_sym ens tl in
433 mk_sym uri_sym ty l r (aux uri ty l r ctx_d ctx_ty p)
434 | Cic.LetIn (name,body,bodyty,rest) ->
436 (name,look_ahead (aux uri) body, bodyty,
437 aux uri ty left right ctx_d ctx_ty rest)
438 | Cic.Appl ((Cic.Const(uri_ind,ens))::tl)
439 when LibraryObjects.is_eq_ind_URI uri_ind ||
440 LibraryObjects.is_eq_ind_r_URI uri_ind ->
441 let ty1,what,pred,p1,other,p2 = open_eq_ind tl in
442 let ty2,eq,lp,rp = open_pred pred in
443 let uri_trans = LibraryObjects.trans_eq_URI ~eq:uri in
444 let uri_sym = LibraryObjects.sym_eq_URI ~eq:uri in
445 let is_not_fixed_lp = is_not_fixed lp in
446 let avoid_eq_ind = LibraryObjects.is_eq_ind_URI uri_ind in
447 (* extract the context and the fixed term from the predicate *)
449 let m, ctx_c = if is_not_fixed_lp then rp,lp else lp,rp in
450 (* they were under a lambda *)
451 let m = CicSubstitution.subst hole m in
452 let ctx_c = CicSubstitution.subst hole ctx_c in
453 let ty2 = CicSubstitution.subst hole ty2 in
456 (* create the compound context and put the terms under it *)
457 let ctx_dc = compose_contexts ctx_d ctx_c in
458 let dc_what = put_in_ctx ctx_dc what in
459 let dc_other = put_in_ctx ctx_dc other in
460 (* m is already in ctx_c so it is put in ctx_d only *)
461 let d_m = put_in_ctx ctx_d m in
462 (* we also need what in ctx_c *)
463 let c_what = put_in_ctx ctx_c what in
464 (* now put the proofs in the compound context *)
465 let p1 = (* p1: dc_what = d_m *)
466 if is_not_fixed_lp then
467 aux uri ty2 c_what m ctx_d ctx_ty p1
469 mk_sym uri_sym ctx_ty d_m dc_what
470 (aux uri ty2 m c_what ctx_d ctx_ty p1)
472 let p2 = (* p2: dc_other = dc_what *)
474 mk_sym uri_sym ctx_ty dc_what dc_other
475 (aux uri ty1 what other ctx_dc ctx_ty p2)
477 aux uri ty1 other what ctx_dc ctx_ty p2
479 (* if pred = \x.C[x]=m --> t : C[other]=m --> trans other what m
480 if pred = \x.m=C[x] --> t : m=C[other] --> trans m what other *)
481 let a,b,c,paeqb,pbeqc =
482 if is_not_fixed_lp then
483 dc_other,dc_what,d_m,p2,p1
485 d_m,dc_what,dc_other,
486 (mk_sym uri_sym ctx_ty dc_what d_m p1),
487 (mk_sym uri_sym ctx_ty dc_other dc_what p2)
489 mk_trans uri_trans ctx_ty a b c paeqb pbeqc
490 | t when ctx_d = hole -> t
492 (* let uri_sym = LibraryObjects.sym_eq_URI ~eq:uri in *)
493 (* let uri_ind = LibraryObjects.eq_ind_URI ~eq:uri in *)
495 let uri_feq = LibraryObjects.eq_f_URI ~eq:uri in
497 (* let r = CicSubstitution.lift 1 (put_in_ctx ctx_d left) in *)
499 let ctx_d = CicSubstitution.lift 1 ctx_d in
500 put_in_ctx ctx_d (Cic.Rel 1)
502 (* let lty = CicSubstitution.lift 1 ctx_ty in *)
503 (* Cic.Lambda (Cic.Name "foo",ty,(mk_eq uri lty l r)) *)
504 Cic.Lambda (Cic.Name "foo",ty,l)
506 (* let d_left = put_in_ctx ctx_d left in *)
507 (* let d_right = put_in_ctx ctx_d right in *)
508 (* let refl_eq = mk_refl uri ctx_ty d_left in *)
509 (* mk_sym uri_sym ctx_ty d_right d_left *)
510 (* (mk_eq_ind uri_ind ty left pred refl_eq right t) *)
511 (mk_feq uri_feq ty ctx_ty left pred right t)
513 aux uri ty left right hole ty t
516 let contextualize_rewrites t ty =
517 let eq,ty,l,r = open_eq ty in
518 contextualize eq ty l r t
521 let add_subst subst =
523 | Exact t -> Exact (Subst.apply_subst subst t)
524 | Step (s,(rule, id1, (pos,id2), pred)) ->
525 Step (Subst.concat subst s,(rule, id1, (pos,id2), pred))
528 let build_proof_step eq lift subst p1 p2 pos l r pred =
529 let p1 = Subst.apply_subst_lift lift subst p1 in
530 let p2 = Subst.apply_subst_lift lift subst p2 in
531 let l = CicSubstitution.lift lift l in
532 let l = Subst.apply_subst_lift lift subst l in
533 let r = CicSubstitution.lift lift r in
534 let r = Subst.apply_subst_lift lift subst r in
535 let pred = CicSubstitution.lift lift pred in
536 let pred = Subst.apply_subst_lift lift subst pred in
539 | Cic.Lambda (_,ty,body) -> ty,body
543 if pos = Utils.Left then l,r else r,l
548 mk_eq_ind (LibraryObjects.eq_ind_URI ~eq) ty what pred p1 other p2
550 mk_eq_ind (LibraryObjects.eq_ind_r_URI ~eq) ty what pred p1 other p2
555 let parametrize_proof p l r =
556 let uniq l = HExtlib.list_uniq (List.sort (fun (i,_) (j,_) -> Pervasives.compare i j) l) in
557 let mot = CicUtil.metas_of_term_set in
558 let parameters = uniq (mot p @ mot l @ mot r) in
559 (* ?if they are under a lambda? *)
562 HExtlib.list_uniq (List.sort Pervasives.compare parameters)
565 (* resorts l such that *hopefully* dependencies can be inferred *)
566 let guess_dependency p l =
568 | Cic.Appl ((Cic.Const(uri_ind,ens))::tl)
569 when LibraryObjects.is_eq_ind_URI uri_ind ||
570 LibraryObjects.is_eq_ind_r_URI uri_ind ->
571 let ty,_,_,_,_,_ = open_eq_ind tl in
572 let metas = CicUtil.metas_of_term ty in
574 List.partition (fun (i,_) -> List.exists (fun (j,_) -> j=i) metas) l
579 let parameters = guess_dependency p parameters in
580 let what = List.map (fun (i,l) -> Cic.Meta (i,l)) parameters in
581 let with_what, lift_no =
582 List.fold_right (fun _ (acc,n) -> ((Cic.Rel n)::acc),n+1) what ([],1)
584 let p = CicSubstitution.lift (lift_no-1) p in
586 ProofEngineReduction.replace_lifting
587 ~equality:(fun _ t1 t2 ->
588 match t1,t2 with Cic.Meta (i,_),Cic.Meta(j,_) -> i=j | _ -> false)
590 ~what ~with_what ~where:p
592 let ty_of_m _ = Cic.Implicit (Some `Type) in
595 (fun (instance,p,n) m ->
598 (Cic.Name ("X"^string_of_int n),
599 CicSubstitution.lift (lift_no - n - 1) (ty_of_m m),
605 let instance = match args with | [x] -> x | _ -> Cic.Appl args in
609 let wfo bag goalproof proof id =
611 let p,_,_ = proof_of_id bag id in
613 | Exact _ -> if (List.mem id acc) then acc else id :: acc
614 | Step (_,(_,id1, (_,id2), _)) ->
615 let acc = if not (List.mem id1 acc) then aux acc id1 else acc in
616 let acc = if not (List.mem id2 acc) then aux acc id2 else acc in
622 | Step (_,(_,id1, (_,id2), _)) -> aux (aux [id] id1) id2
624 List.fold_left (fun acc (_,_,id,_,_) -> aux acc id) acc goalproof
627 let string_of_id (id_to_eq,_) names id =
628 if id = 0 then "" else
630 let (_,p,(t,l,r,_),m,_) = open_equality (Hashtbl.find id_to_eq id) in
633 Printf.sprintf "%d = %s: %s = %s [%s]" id
634 (CicPp.pp t names) (CicPp.pp l names) (CicPp.pp r names)
636 (String.concat ", " (List.map (fun (i,_,_) -> string_of_int i) m))
637 | Step (_,(step,id1, (dir,id2), p) ) ->
638 Printf.sprintf "%6d: %s %6d %6d %s =(%s) %s [%s]" id
639 (string_of_rule step)
640 id1 id2 (CicPp.pp l names) (CicPp.pp t names) (CicPp.pp r names)
641 (String.concat ", " (List.map (fun (i,_,_) -> string_of_int i) m))
644 Not_found -> assert false
646 let pp_proof bag names goalproof proof subst id initial_goal =
647 String.concat "\n" (List.map (string_of_id bag names) (wfo bag goalproof proof id)) ^
650 (fst (List.fold_right
651 (fun (r,pos,i,s,pred) (acc,g) ->
652 let _,_,left,right = open_eq g in
655 | Utils.Left -> CicReduction.head_beta_reduce (Cic.Appl[pred;right])
656 | Utils.Right -> CicReduction.head_beta_reduce (Cic.Appl[pred;left])
658 let ty = Subst.apply_subst s ty in
659 ("("^ string_of_rule r ^ " " ^ string_of_int i^") -> "
660 ^ CicPp.pp ty names) :: acc,ty) goalproof ([],initial_goal)))) ^
661 "\nand then subsumed by " ^ string_of_int id ^ " when " ^ Subst.ppsubst subst
667 let compare = Pervasives.compare
670 module M = Map.Make(OT)
672 let rec find_deps bag m i =
675 let p,_,_ = proof_of_id bag i in
677 | Exact _ -> M.add i [] m
678 | Step (_,(_,id1,(_,id2),_)) ->
679 let m = find_deps bag m id1 in
680 let m = find_deps bag m id2 in
681 (* without the uniq there is a stack overflow doing concatenation *)
682 let xxx = [id1;id2] @ M.find id1 m @ M.find id2 m in
683 let xxx = HExtlib.list_uniq (List.sort Pervasives.compare xxx) in
687 let topological_sort bag l =
688 (* build the partial order relation *)
689 let m = List.fold_left (fun m i -> find_deps bag m i) M.empty l in
690 let m = (* keep only deps inside l *)
693 M.add i (List.filter (fun x -> List.mem x l) (M.find i m)) m')
696 let m = M.map (fun x -> Some x) m in
698 let keys m = M.fold (fun i _ acc -> i::acc) m [] in
699 let split l m = List.filter (fun i -> M.find i m = Some []) l in
702 (fun k v -> if List.mem k l then None else
705 | Some ll -> Some (List.filter (fun i -> not (List.mem i l)) ll))
710 let ok = split keys m in
711 let m = purge ok m in
712 let res = ok @ res in
713 if ok = [] then res else aux m res
715 let rc = List.rev (aux m []) in
720 (* returns the list of ids that should be factorized *)
721 let get_duplicate_step_in_wfo bag l p =
722 let ol = List.rev l in
723 let h = Hashtbl.create 13 in
724 (* NOTE: here the n parameter is an approximation of the dependency
725 between equations. To do things seriously we should maintain a
726 dependency graph. This approximation is not perfect. *)
728 let p,_,_ = proof_of_id bag i in
733 let no = Hashtbl.find h i in
734 Hashtbl.replace h i (no+1);
736 with Not_found -> Hashtbl.add h i 1;true
738 let rec aux = function
740 | Step (_,(_,i1,(_,i2),_)) ->
741 let go_on_1 = add i1 in
742 let go_on_2 = add i2 in
743 if go_on_1 then aux (let p,_,_ = proof_of_id bag i1 in p);
744 if go_on_2 then aux (let p,_,_ = proof_of_id bag i2 in p)
748 (fun (_,_,id,_,_) -> aux (let p,_,_ = proof_of_id bag id in p))
750 (* now h is complete *)
751 let proofs = Hashtbl.fold (fun k count acc-> (k,count)::acc) h [] in
752 let proofs = List.filter (fun (_,c) -> c > 1) proofs in
753 let res = topological_sort bag (List.map (fun (i,_) -> i) proofs) in
757 let build_proof_term bag eq h lift proof =
758 let proof_of_id aux id =
759 let p,l,r = proof_of_id bag id in
760 try List.assoc id h,l,r with Not_found -> aux p, l, r
762 let rec aux = function
764 CicSubstitution.lift lift term
765 | Step (subst,(rule, id1, (pos,id2), pred)) ->
766 let p1,_,_ = proof_of_id aux id1 in
767 let p2,l,r = proof_of_id aux id2 in
770 | SuperpositionRight -> Cic.Name ("SupR" ^ Utils.string_of_pos pos)
771 | Demodulation -> Cic.Name ("DemEq"^ Utils.string_of_pos pos)
776 | Cic.Lambda (_,a,b) -> Cic.Lambda (varname,a,b)
779 let p = build_proof_step eq lift subst p1 p2 pos l r pred in
780 (* let cond = (not (List.mem 302 (Utils.metas_of_term p)) || id1 = 8 || id1 = 132) in
782 prerr_endline ("ERROR " ^ string_of_int id1 ^ " " ^ string_of_int id2);
789 let build_goal_proof ?(contextualize=true) ?(forward=false) bag eq l initial ty se context menv =
790 let se = List.map (fun i -> Cic.Meta (i,[])) se in
791 let lets = get_duplicate_step_in_wfo bag l initial in
792 let letsno = List.length lets in
793 let l = if forward then List.rev l else l in
794 let lift_list l = List.map (fun (i,t) -> i,CicSubstitution.lift 1 t) l in
798 let p,l,r = proof_of_id bag id in
799 let cic = build_proof_term bag eq h n p in
800 let real_cic,instance =
801 parametrize_proof cic l r
803 let h = (id, instance)::lift_list h in
804 acc@[id,real_cic],n+1,h)
808 List.map (fun (id,cic) -> id,cic,Cic.Implicit (Some `Type)) lets
811 let rec aux se current_proof = function
812 | [] -> current_proof,se
813 | (rule,pos,id,subst,pred)::tl ->
814 let p,l,r = proof_of_id bag id in
815 let p = build_proof_term bag eq h letsno p in
816 let pos = if forward then pos else
817 if pos = Utils.Left then Utils.Right else Utils.Left in
820 | SuperpositionLeft -> Cic.Name ("SupL" ^ Utils.string_of_pos pos)
821 | Demodulation -> Cic.Name ("DemG"^ Utils.string_of_pos pos)
826 | Cic.Lambda (_,a,b) -> Cic.Lambda (varname,a,b)
830 build_proof_step eq letsno subst current_proof p pos l r pred
832 let proof,se = aux se proof tl in
833 Subst.apply_subst_lift letsno subst proof,
834 List.map (fun x -> Subst.apply_subst(*_lift letsno*) subst x) se
836 aux se (build_proof_term bag eq h letsno initial) l
839 let initial = proof in
841 (fun (id,cic,ty) (n,p) ->
844 Cic.Name ("H"^string_of_int id),
848 lets (letsno-1,initial)
852 then contextualize_rewrites proof (CicSubstitution.lift letsno ty)
854 canonical proof context menv, se
857 let refl_proof eq_uri ty term =
858 Cic.Appl [Cic.MutConstruct (eq_uri, 0, 1, []); ty; term]
861 let metas_of_proof bag p =
863 match LibraryObjects.eq_URI () with
867 (ProofEngineTypes.Fail
868 (lazy "No default equality defined when calling metas_of_proof"))
870 let p = build_proof_term bag eq [] 0 p in
871 Utils.metas_of_term p
874 let remove_local_context eq =
875 let w, p, (ty, left, right, o), menv,id = open_equality eq in
876 let p = Utils.remove_local_context p in
877 let ty = Utils.remove_local_context ty in
878 let left = Utils.remove_local_context left in
879 let right = Utils.remove_local_context right in
880 w, p, (ty, left, right, o), menv, id
883 let relocate newmeta menv to_be_relocated =
884 let subst, newmetasenv, newmeta =
886 (fun i (subst, metasenv, maxmeta) ->
887 let _,context,ty = CicUtil.lookup_meta i menv in
889 let newmeta = Cic.Meta(maxmeta,irl) in
890 let newsubst = Subst.buildsubst i context newmeta ty subst in
891 (* newsubst, (maxmeta,context,ty)::metasenv, maxmeta+1) *)
892 newsubst, (maxmeta,[],ty)::metasenv, maxmeta+1)
893 to_be_relocated (Subst.empty_subst, [], newmeta+1)
895 (* let subst = Subst.flatten_subst subst in *)
896 let menv = Subst.apply_subst_metasenv subst (menv @ newmetasenv) in
899 let fix_metas_goal newmeta goal =
900 let (proof, menv, ty) = goal in
901 let to_be_relocated =
902 HExtlib.list_uniq (List.sort Pervasives.compare (Utils.metas_of_term ty))
904 let subst, menv, newmeta = relocate newmeta menv to_be_relocated in
905 let ty = Subst.apply_subst subst ty in
908 | [] -> assert false (* is a nonsense to relocate the initial goal *)
909 | (r,pos,id,s,p) :: tl -> (r,pos,id,Subst.concat subst s,p) :: tl
911 newmeta+1,(proof, menv, ty)
914 let fix_metas bag newmeta eq =
915 let w, p, (ty, left, right, o), menv,_ = open_equality eq in
916 let to_be_relocated =
917 List.map (fun i ,_,_ -> i) menv
920 (List.sort Pervasives.compare
921 (Utils.metas_of_term left @ Utils.metas_of_term right @
922 Utils.metas_of_term ty))
925 let subst, metasenv, newmeta = relocate newmeta menv to_be_relocated in
926 let ty = Subst.apply_subst subst ty in
927 let left = Subst.apply_subst subst left in
928 let right = Subst.apply_subst subst right in
929 let fix_proof = function
930 | Exact p -> Exact (Subst.apply_subst subst p)
931 | Step (s,(r,id1,(pos,id2),pred)) ->
932 Step (Subst.concat s subst,(r,id1,(pos,id2), pred))
934 let p = fix_proof p in
935 let eq' = mk_equality bag (w, p, (ty, left, right, o), metasenv) in
938 exception NotMetaConvertible;;
940 let meta_convertibility_aux table t1 t2 =
941 let module C = Cic in
942 let rec aux ((table_l,table_r) as table) t1 t2 =
944 | C.Meta (m1, tl1), C.Meta (m2, tl2) when m1 = m2 -> table
945 | C.Meta (m1, tl1), C.Meta (m2, tl2) when m1 < m2 -> aux table t2 t1
946 | C.Meta (m1, tl1), C.Meta (m2, tl2) ->
947 let m1_binding, table_l =
948 try List.assoc m1 table_l, table_l
949 with Not_found -> m2, (m1, m2)::table_l
950 and m2_binding, table_r =
951 try List.assoc m2 table_r, table_r
952 with Not_found -> m1, (m2, m1)::table_r
954 if (m1_binding <> m2) || (m2_binding <> m1) then
955 raise NotMetaConvertible
957 | C.Var (u1, ens1), C.Var (u2, ens2)
958 | C.Const (u1, ens1), C.Const (u2, ens2) when (UriManager.eq u1 u2) ->
959 aux_ens table ens1 ens2
960 | C.Cast (s1, t1), C.Cast (s2, t2)
961 | C.Prod (_, s1, t1), C.Prod (_, s2, t2)
962 | C.Lambda (_, s1, t1), C.Lambda (_, s2, t2) ->
963 let table = aux table s1 s2 in
965 | C.LetIn (_, s1, ty1, t1), C.LetIn (_, s2, ty2, t2) ->
966 let table = aux table s1 s2 in
967 let table = aux table ty1 ty2 in
969 | C.Appl l1, C.Appl l2 -> (
970 try List.fold_left2 (fun res t1 t2 -> (aux res t1 t2)) table l1 l2
971 with Invalid_argument _ -> raise NotMetaConvertible
973 | C.MutInd (u1, i1, ens1), C.MutInd (u2, i2, ens2)
974 when (UriManager.eq u1 u2) && i1 = i2 -> aux_ens table ens1 ens2
975 | C.MutConstruct (u1, i1, j1, ens1), C.MutConstruct (u2, i2, j2, ens2)
976 when (UriManager.eq u1 u2) && i1 = i2 && j1 = j2 ->
977 aux_ens table ens1 ens2
978 | C.MutCase (u1, i1, s1, t1, l1), C.MutCase (u2, i2, s2, t2, l2)
979 when (UriManager.eq u1 u2) && i1 = i2 ->
980 let table = aux table s1 s2 in
981 let table = aux table t1 t2 in (
982 try List.fold_left2 (fun res t1 t2 -> (aux res t1 t2)) table l1 l2
983 with Invalid_argument _ -> raise NotMetaConvertible
985 | C.Fix (i1, il1), C.Fix (i2, il2) when i1 = i2 -> (
988 (fun res (n1, i1, s1, t1) (n2, i2, s2, t2) ->
989 if i1 <> i2 then raise NotMetaConvertible
991 let res = (aux res s1 s2) in aux res t1 t2)
993 with Invalid_argument _ -> raise NotMetaConvertible
995 | C.CoFix (i1, il1), C.CoFix (i2, il2) when i1 = i2 -> (
998 (fun res (n1, s1, t1) (n2, s2, t2) ->
999 let res = aux res s1 s2 in aux res t1 t2)
1001 with Invalid_argument _ -> raise NotMetaConvertible
1003 | t1, t2 when t1 = t2 -> table
1004 | _, _ -> raise NotMetaConvertible
1006 and aux_ens table ens1 ens2 =
1007 let cmp (u1, t1) (u2, t2) =
1008 Pervasives.compare (UriManager.string_of_uri u1) (UriManager.string_of_uri u2)
1010 let ens1 = List.sort cmp ens1
1011 and ens2 = List.sort cmp ens2 in
1014 (fun res (u1, t1) (u2, t2) ->
1015 if not (UriManager.eq u1 u2) then raise NotMetaConvertible
1018 with Invalid_argument _ -> raise NotMetaConvertible
1024 let meta_convertibility_eq eq1 eq2 =
1025 let _, _, (ty, left, right, _), _,_ = open_equality eq1 in
1026 let _, _, (ty', left', right', _), _,_ = open_equality eq2 in
1029 else if (left = left') && (right = right') then
1031 else if (left = right') && (right = left') then
1035 let table = meta_convertibility_aux ([],[]) left left' in
1036 let _ = meta_convertibility_aux table right right' in
1038 with NotMetaConvertible ->
1040 let table = meta_convertibility_aux ([],[]) left right' in
1041 let _ = meta_convertibility_aux table right left' in
1043 with NotMetaConvertible ->
1047 let meta_convertibility t1 t2 =
1052 ignore(meta_convertibility_aux ([],[]) t1 t2);
1054 with NotMetaConvertible ->
1058 let meta_convertibility_subst t1 t2 menv =
1063 let (l,_) = meta_convertibility_aux ([],[]) t1 t2 in
1068 let (_,c,t) = CicUtil.lookup_meta x menv in
1070 CicMkImplicit.identity_relocation_list_for_metavariable c in
1071 (y,(c,Cic.Meta(x,irl),t))
1072 with CicUtil.Meta_not_found _ ->
1074 let (_,c,t) = CicUtil.lookup_meta y menv in
1076 CicMkImplicit.identity_relocation_list_for_metavariable c in
1077 (x,(c,Cic.Meta(y,irl),t))
1078 with CicUtil.Meta_not_found _ -> assert false) l in
1080 with NotMetaConvertible ->
1084 exception TermIsNotAnEquality;;
1086 let term_is_equality term =
1088 | Cic.Appl [Cic.MutInd (uri, _, _); _; _; _]
1089 when LibraryObjects.is_eq_URI uri -> true
1093 let equality_of_term bag proof term =
1095 | Cic.Appl [Cic.MutInd (uri, _, _); ty; t1; t2]
1096 when LibraryObjects.is_eq_URI uri ->
1097 let o = !Utils.compare_terms t1 t2 in
1098 let stat = (ty,t1,t2,o) in
1099 let w = Utils.compute_equality_weight stat in
1100 let e = mk_equality bag (w, Exact proof, stat,[]) in
1103 raise TermIsNotAnEquality
1106 let is_weak_identity eq =
1107 let _,_,(_,left, right,_),_,_ = open_equality eq in
1109 (* doing metaconv here is meaningless *)
1112 let is_identity (_, context, ugraph) eq =
1113 let _,_,(ty,left,right,_),menv,_ = open_equality eq in
1114 (* doing metaconv here is meaningless *)
1116 (* fst (CicReduction.are_convertible ~metasenv:menv context left right ugraph)
1121 let term_of_equality eq_uri equality =
1122 let _, _, (ty, left, right, _), menv, _= open_equality equality in
1123 let eq i = function Cic.Meta (j, _) -> i = j | _ -> false in
1124 let argsno = List.length menv in
1126 CicSubstitution.lift argsno
1127 (Cic.Appl [Cic.MutInd (eq_uri, 0, []); ty; left; right])
1131 (fun (i,_,ty) (n, t) ->
1132 let name = Cic.Name ("X" ^ (string_of_int n)) in
1133 let ty = CicSubstitution.lift (n-1) ty in
1135 ProofEngineReduction.replace
1136 ~equality:eq ~what:[i]
1137 ~with_what:[Cic.Rel (argsno - (n - 1))] ~where:t
1139 (n-1, Cic.Prod (name, ty, t)))
1143 let symmetric bag eq_ty l id uri m =
1144 let eq = Cic.MutInd(uri,0,[]) in
1146 Cic.Lambda (Cic.Name "Sym",eq_ty,
1147 Cic.Appl [CicSubstitution.lift 1 eq ;
1148 CicSubstitution.lift 1 eq_ty;
1149 Cic.Rel 1;CicSubstitution.lift 1 l])
1153 [Cic.MutConstruct(uri,0,1,[]);eq_ty;l])
1156 let eq = mk_equality bag (0,prefl,(eq_ty,l,l,Utils.Eq),m) in
1157 let (_,_,_,_,id) = open_equality eq in
1160 Step(Subst.empty_subst,
1161 (Demodulation,id1,(Utils.Left,id),pred))
1164 module IntOT = struct
1166 let compare = Pervasives.compare
1169 module IntSet = Set.Make(IntOT);;
1171 let n_purged = ref 0;;
1173 let collect ((id_to_eq,_) as bag) alive1 alive2 alive3 =
1174 (* let _ = <:start<collect>> in *)
1176 let p,_,_ = proof_of_id bag id in
1178 | Exact _ -> IntSet.empty
1179 | Step (_,(_,id1,(_,id2),_)) ->
1180 IntSet.add id1 (IntSet.add id2 IntSet.empty)
1183 let news = IntSet.fold (fun id s -> IntSet.union (deps_of id) s) s s in
1184 if IntSet.equal news s then s else close news
1186 let l_to_s s l = List.fold_left (fun s x -> IntSet.add x s) s l in
1187 let alive_set = l_to_s (l_to_s (l_to_s IntSet.empty alive2) alive1) alive3 in
1188 let closed_alive_set = close alive_set in
1192 if not (IntSet.mem k closed_alive_set) then
1193 k::s else s) id_to_eq []
1195 n_purged := !n_purged + List.length to_purge;
1196 List.iter (Hashtbl.remove id_to_eq) to_purge;
1197 (* let _ = <:stop<collect>> in () *)
1201 let _,_,_,_,id = open_equality e in id
1204 let get_stats () = ""
1206 <:show<Equality.>> ^
1207 "# of purged eq by the collector: " ^ string_of_int !n_purged ^ "\n"
1211 let rec pp_proofterm name t context =
1212 let rec skip_lambda tys ctx = function
1213 | Cic.Lambda (n,s,t) -> skip_lambda (s::tys) ((Some n)::ctx) t
1218 | Cic.Name s1 -> Cic.Name (s ^ s1)
1221 let rec skip_letin ctx = function
1222 | Cic.LetIn (n,b,_,t) ->
1223 pp_proofterm (Some (rename "Lemma " n)) b ctx::
1224 skip_letin ((Some n)::ctx) t
1226 let ppterm t = CicPp.pp t ctx in
1227 let rec pp inner = function
1228 | Cic.Appl [Cic.Const (uri,[]);_;l;m;r;p1;p2]
1229 when Pcre.pmatch ~pat:"trans_eq" (UriManager.string_of_uri uri)->
1231 (" " ^ ppterm l) :: pp true p1 @
1232 [ " = " ^ ppterm m ] @ pp true p2 @
1233 [ " = " ^ ppterm r ]
1236 [ " = " ^ ppterm m ] @ pp true p2
1237 | Cic.Appl [Cic.Const (uri,[]);_;l;m;p]
1238 when Pcre.pmatch ~pat:"sym_eq" (UriManager.string_of_uri uri)->
1240 | Cic.Appl [Cic.Const (uri,[]);_;_;_;_;_;p]
1241 when Pcre.pmatch ~pat:"eq_f" (UriManager.string_of_uri uri)->
1243 | Cic.Appl [Cic.Const (uri,[]);_;_;_;_;_;p]
1244 when Pcre.pmatch ~pat:"eq_OF_eq" (UriManager.string_of_uri uri)->
1246 | Cic.Appl [Cic.MutConstruct (uri,_,_,[]);_;_;t;p]
1247 when Pcre.pmatch ~pat:"ex.ind" (UriManager.string_of_uri uri)->
1248 [ "witness " ^ ppterm t ] @ pp true p
1249 | Cic.Appl (t::_) ->[ " [by " ^ ppterm t ^ "]"]
1250 | t ->[ " [by " ^ ppterm t ^ "]"]
1252 let rec compat = function
1253 | a::b::tl -> (b ^ a) :: compat tl
1257 let compat l = List.hd l :: compat (List.tl l) in
1258 compat (pp false t) @ ["";""]
1260 let names, tys, body = skip_lambda [] context t in
1261 let ppname name = (match name with Some (Cic.Name s) -> s | _ -> "") in
1262 ppname name ^ ":\n" ^
1263 (if context = [] then
1264 let rec pp_l ctx = function
1266 " " ^ ppname name ^ ": " ^ CicPp.pp t ctx ^ "\n" ^
1270 pp_l [] (List.rev (List.combine tys names))
1273 String.concat "\n" (skip_letin names body)
1276 let pp_proofterm t =
1278 pp_proofterm (Some (Cic.Name "Hypothesis")) t []
1281 let initial_nameset_list = [
1282 "x"; "y"; "z"; "t"; "u"; "v"; "a"; "b"; "c"; "d";
1283 "e"; "l"; "m"; "n"; "o"; "p"; "q"; "r";
1286 module S = Set.Make(String)
1288 let initial_nameset = List.fold_right S.add initial_nameset_list S.empty, [];;
1290 let freshname (nameset, subst) term =
1291 let m = CicUtil.metas_of_term term in
1292 let nameset, subst =
1294 (fun (set,rc) (m,_) ->
1295 if List.mem_assoc m rc then set,rc else
1296 let name = S.choose set in
1297 let set = S.remove name set in
1299 (m,Cic.Const(UriManager.uri_of_string
1300 ("cic:/"^name^".con"),[]))::rc)
1304 ProofEngineReduction.replace
1305 ~equality:(fun i t -> match t with Cic.Meta (j,_) -> i=j| _ -> false)
1306 ~what:(List.map fst subst)
1307 ~with_what:(List.map snd subst) ~where:term
1309 (nameset, subst), term
1312 let remove_names_in_context (set,subst) names =
1315 match n with Some (Cic.Name n) -> S.remove n s | _ -> s)
1319 let string_of_id2 (id_to_eq,_) names nameset id =
1320 if id = 0 then "" else
1322 let (_,_,(_,l,r,_),_,_) = open_equality (Hashtbl.find id_to_eq id) in
1323 let nameset, l = freshname nameset l in
1324 let nameset, r = freshname nameset r in
1325 Printf.sprintf "%s = %s" (CicPp.pp l names) (CicPp.pp r names)
1327 Not_found -> assert false
1330 let draw_proof bag names goal_proof proof id =
1331 let b = Buffer.create 100 in
1332 let fmt = Format.formatter_of_buffer b in
1333 let sint = string_of_int in
1334 let fst3 (x,_,_) = x in
1335 let visited = ref [] in
1336 let nameset = remove_names_in_context initial_nameset names in
1337 let rec fact id = function
1339 if not (List.mem id !visited) then
1341 visited := id :: !visited;
1342 let nameset, t = freshname nameset t in
1343 let t = CicPp.pp t names in
1344 GraphvizPp.Dot.node (sint id)
1345 ~attrs:["label",t^":"^string_of_id2 bag names nameset id;
1346 "shape","rectangle"] fmt;
1348 | Step (_,(_,id1,(_,id2),_)) ->
1349 GraphvizPp.Dot.edge (sint id) (sint id1) fmt;
1350 GraphvizPp.Dot.edge (sint id) (sint id2) fmt;
1351 let p1,_,_ = proof_of_id bag id1 in
1352 let p2,_,_ = proof_of_id bag id2 in
1355 if not (List.mem id !visited); then
1357 visited := id :: !visited;
1358 GraphvizPp.Dot.node (sint id)
1359 ~attrs:["label",sint id^":"^string_of_id2 bag names nameset id;
1360 "shape","ellipse"] fmt
1363 let sleft acc (_,_,id,_,_) =
1364 if acc != 0 then GraphvizPp.Dot.edge (sint acc) (sint id) fmt;
1365 fact id (fst3 (proof_of_id bag id));
1368 GraphvizPp.Dot.header ~node_attrs:["fontsize","10"; ] fmt;
1369 ignore(List.fold_left sleft id goal_proof);
1370 GraphvizPp.Dot.trailer fmt;
1371 let oc = open_out "/tmp/matita_paramod.dot" in
1372 Buffer.output_buffer oc b;
1374 Utils.debug_print (lazy "dot!");
1376 "dot -Tps -o /tmp/matita_paramod.eps /tmp/matita_paramod.dot"
1377 (* "cat /tmp/matita_paramod.dot| tred | dot -Tps -o /tmp/matita_paramod.eps" *)
1379 ignore(Unix.system "gv /tmp/matita_paramod.eps");