4 exception AssertFailure of string
5 exception MetaSubstFailure of string
7 let debug_print = prerr_endline
9 type substitution = (int * Cic.term) list
14 (fun (idx, term) -> Printf.sprintf "?%d := %s" idx (CicPp.ppterm term))
18 (* the delift function takes in input a metavariable index, an ordered list of
19 * optional terms [t1,...,tn] and a term t, and substitutes every tk = Some
20 * (rel(nk)) with rel(k). Typically, the list of optional terms is the explicit
21 * substitution that is applied to a metavariable occurrence and the result of
22 * the delift function is a term the implicit variable can be substituted with
23 * to make the term [t] unifiable with the metavariable occurrence. In general,
24 * the problem is undecidable if we consider equivalence in place of alpha
25 * convertibility. Our implementation, though, is even weaker than alpha
26 * convertibility, since it replace the term [tk] if and only if [tk] is a Rel
27 * (missing all the other cases). Does this matter in practice?
28 * The metavariable index is the index of the metavariable that must not occur
29 * in the term (for occur check).
32 exception NotInTheList;;
37 [] -> raise NotInTheList
38 | (Some (Cic.Rel m))::_ when m=n -> k
39 | _::tl -> aux (k+1) tl in
45 let rec force_does_not_occur subst to_be_restricted t =
47 let more_to_be_restricted = ref [] in
48 let rec aux k = function
49 C.Rel r when List.mem (r+k) to_be_restricted -> raise Occur
52 | C.Implicit -> assert false
55 aux k (CicSubstitution.lift_meta l (List.assoc n subst))
67 more_to_be_restricted := (n,!i) :: !more_to_be_restricted;
72 | C.Cast (te,ty) -> C.Cast (aux k te, aux k ty)
73 | C.Prod (name,so,dest) -> C.Prod (name, aux k so, aux (k+1) dest)
74 | C.Lambda (name,so,dest) -> C.Lambda (name, aux k so, aux (k+1) dest)
75 | C.LetIn (name,so,dest) -> C.LetIn (name, aux k so, aux (k+1) dest)
76 | C.Appl l -> C.Appl (List.map (aux k) l)
77 | C.Var (uri,exp_named_subst) ->
78 let exp_named_subst' =
79 List.map (fun (uri,t) -> (uri, aux k t)) exp_named_subst
81 C.Var (uri, exp_named_subst')
82 | C.Const (uri, exp_named_subst) ->
83 let exp_named_subst' =
84 List.map (fun (uri,t) -> (uri, aux k t)) exp_named_subst
86 C.Const (uri, exp_named_subst')
87 | C.MutInd (uri,tyno,exp_named_subst) ->
88 let exp_named_subst' =
89 List.map (fun (uri,t) -> (uri, aux k t)) exp_named_subst
91 C.MutInd (uri, tyno, exp_named_subst')
92 | C.MutConstruct (uri,tyno,consno,exp_named_subst) ->
93 let exp_named_subst' =
94 List.map (fun (uri,t) -> (uri, aux k t)) exp_named_subst
96 C.MutConstruct (uri, tyno, consno, exp_named_subst')
97 | C.MutCase (uri,tyno,out,te,pl) ->
98 C.MutCase (uri, tyno, aux k out, aux k te, List.map (aux k) pl)
100 let len = List.length fl in
101 let k_plus_len = k + len in
104 (fun (name,j,ty,bo) -> (name, j, aux k ty, aux k_plus_len bo)) fl
108 let len = List.length fl in
109 let k_plus_len = k + len in
112 (fun (name,ty,bo) -> (name, aux k ty, aux k_plus_len bo)) fl
117 (!more_to_be_restricted, res)
119 let rec restrict subst to_be_restricted metasenv =
120 let names_of_context_indexes context indexes =
124 match List.nth context i with
125 | None -> assert false
126 | Some (n, _) -> CicPp.ppname n)
129 let force_does_not_occur_in_context to_be_restricted = function
131 | Some (name, Cic.Decl t) ->
132 let (more_to_be_restricted, t') =
133 force_does_not_occur subst to_be_restricted t
135 more_to_be_restricted, Some (name, Cic.Decl t)
136 | Some (name, Cic.Def (bo, ty)) ->
137 let (more_to_be_restricted, bo') =
138 force_does_not_occur subst to_be_restricted bo
140 let more_to_be_restricted, ty' =
142 | None -> more_to_be_restricted, None
144 let more_to_be_restricted', ty' =
145 force_does_not_occur subst to_be_restricted ty
147 more_to_be_restricted @ more_to_be_restricted',
150 more_to_be_restricted, Some (name, Cic.Def (bo', ty'))
152 let rec erase i to_be_restricted n = function
153 | [] -> [], to_be_restricted, []
155 let restrict_me = List.mem i to_be_restricted in
157 let more_to_be_restricted, restricted, new_tl =
158 erase (i+1) (i :: to_be_restricted) n tl
160 more_to_be_restricted, restricted, None :: new_tl
163 let more_to_be_restricted, hd' =
164 force_does_not_occur_in_context to_be_restricted hd
166 let more_to_be_restricted', restricted, tl' =
167 erase (i+1) to_be_restricted n tl
169 more_to_be_restricted @ more_to_be_restricted',
170 restricted, hd' :: tl'
172 let more_to_be_restricted, restricted, tl' =
173 erase (i+1) (i :: to_be_restricted) n tl
175 more_to_be_restricted, restricted, None :: tl')
177 let (more_to_be_restricted, metasenv, subst) =
179 (fun (n, context, t) (more, metasenv, subst) ->
180 let to_be_restricted =
181 List.map snd (List.filter (fun (m, _) -> m = n) to_be_restricted)
183 let (more_to_be_restricted, restricted, context') =
184 erase 1 to_be_restricted n context
187 let more_to_be_restricted', t' =
188 force_does_not_occur subst restricted t
190 let metasenv' = (n, context', t') :: metasenv in
192 let s = List.assoc n subst in
194 let more_to_be_restricted'', s' =
195 force_does_not_occur subst restricted s
197 let subst' = (n, s') :: (List.remove_assoc n subst) in
199 more @ more_to_be_restricted @ more_to_be_restricted' @
200 more_to_be_restricted''
202 (more, metasenv', subst')
204 raise (MetaSubstFailure (sprintf
205 "Cannot restrict the context of the metavariable ?%d over the hypotheses %s since ?%d is already instantiated with %s and at least one of the hypotheses occurs in the substituted term"
206 n (names_of_context_indexes context to_be_restricted) n
208 with Not_found -> (more @ more_to_be_restricted, metasenv', subst))
210 raise (MetaSubstFailure (sprintf
211 "Cannot restrict the context of the metavariable ?%d over the hypotheses %s since metavariable's type depends on at least one of them"
212 n (names_of_context_indexes context to_be_restricted))))
213 metasenv ([], [], subst)
215 match more_to_be_restricted with
216 | [] -> (metasenv, subst)
217 | _ -> restrict subst more_to_be_restricted metasenv
220 (*CSC: maybe we should rename delift in abstract, as I did in my dissertation *)
221 let delift n subst context metasenv l t =
223 let (_, canonical_context, _) = CicUtil.lookup_meta n metasenv in
224 List.map2 (fun ct lt ->
230 let module S = CicSubstitution in
231 let to_be_restricted = ref [] in
232 let rec deliftaux k =
233 let module C = Cic in
237 C.Rel m (*CSC: che succede se c'e' un Def? Dovrebbe averlo gia' *)
238 (*CSC: deliftato la regola per il LetIn *)
239 (*CSC: FALSO! La regola per il LetIn non lo fa *)
241 (match List.nth context (m-k-1) with
242 Some (_,C.Def (t,_)) ->
243 (*CSC: Hmmm. This bit of reduction is not in the spirit of *)
244 (*CSC: first order unification. Does it help or does it harm? *)
245 deliftaux k (S.lift m t)
246 | Some (_,C.Decl t) ->
247 (*CSC: The following check seems to be wrong! *)
248 (*CSC: B:Set |- ?2 : Set *)
249 (*CSC: A:Set ; x:?2[A/B] |- ?1[x/A] =?= x *)
250 (*CSC: Why should I restrict ?2 over B? The instantiation *)
251 (*CSC: ?1 := A is perfectly reasonable and well-typed. *)
252 (*CSC: Thus I comment out the following two lines that *)
253 (*CSC: are the incriminated ones. *)
254 (*(* It may augment to_be_restricted *)
255 ignore (deliftaux k (S.lift m t)) ;*)
256 (*CSC: end of bug commented out *)
257 C.Rel ((position (m-k) l) + k)
258 | None -> raise (MetaSubstFailure "RelToHiddenHypothesis"))
259 | C.Var (uri,exp_named_subst) ->
260 let exp_named_subst' =
261 List.map (function (uri,t) -> uri,deliftaux k t) exp_named_subst
263 C.Var (uri,exp_named_subst')
264 | C.Meta (i, l1) as t ->
266 raise (MetaSubstFailure (sprintf
267 "Cannot unify the metavariable ?%d with a term that has as subterm %s in which the same metavariable occurs (occur check)"
271 deliftaux k (S.lift_meta l (List.assoc i subst))
276 | None::tl -> None::(deliftl (j+1) tl)
278 let l1' = (deliftl (j+1) tl) in
280 Some (deliftaux k t)::l1'
283 | MetaSubstFailure _ ->
284 to_be_restricted := (i,j)::!to_be_restricted ; None::l1'
286 let l' = deliftl 1 l1 in
289 | C.Implicit as t -> t
290 | C.Cast (te,ty) -> C.Cast (deliftaux k te, deliftaux k ty)
291 | C.Prod (n,s,t) -> C.Prod (n, deliftaux k s, deliftaux (k+1) t)
292 | C.Lambda (n,s,t) -> C.Lambda (n, deliftaux k s, deliftaux (k+1) t)
293 | C.LetIn (n,s,t) -> C.LetIn (n, deliftaux k s, deliftaux (k+1) t)
294 | C.Appl l -> C.Appl (List.map (deliftaux k) l)
295 | C.Const (uri,exp_named_subst) ->
296 let exp_named_subst' =
297 List.map (function (uri,t) -> uri,deliftaux k t) exp_named_subst
299 C.Const (uri,exp_named_subst')
300 | C.MutInd (uri,typeno,exp_named_subst) ->
301 let exp_named_subst' =
302 List.map (function (uri,t) -> uri,deliftaux k t) exp_named_subst
304 C.MutInd (uri,typeno,exp_named_subst')
305 | C.MutConstruct (uri,typeno,consno,exp_named_subst) ->
306 let exp_named_subst' =
307 List.map (function (uri,t) -> uri,deliftaux k t) exp_named_subst
309 C.MutConstruct (uri,typeno,consno,exp_named_subst')
310 | C.MutCase (sp,i,outty,t,pl) ->
311 C.MutCase (sp, i, deliftaux k outty, deliftaux k t,
312 List.map (deliftaux k) pl)
314 let len = List.length fl in
317 (fun (name, i, ty, bo) ->
318 (name, i, deliftaux k ty, deliftaux (k+len) bo))
323 let len = List.length fl in
326 (fun (name, ty, bo) -> (name, deliftaux k ty, deliftaux (k+len) bo))
329 C.CoFix (i, liftedfl)
336 (* This is the case where we fail even first order unification. *)
337 (* The reason is that our delift function is weaker than first *)
338 (* order (in the sense of alpha-conversion). See comment above *)
339 (* related to the delift function. *)
340 debug_print "!!!!!!!!!!! First Order UnificationFailure, but maybe it could have been successful even in a first order setting (no conversion, only alpha convertibility)! Please, implement a better delift function !!!!!!!!!!!!!!!!" ;
341 raise (MetaSubstFailure (sprintf
342 "Error trying to abstract %s over [%s]: the algorithm only tried to abstract over bound variables"
346 (function Some t -> CicPp.ppterm t | None -> "_")
349 let (metasenv, subst) = restrict subst !to_be_restricted metasenv in
353 (**** END OF DELIFT ****)
355 let apply_subst_gen ~appl_fun subst term =
357 let module C = Cic in
358 let module S = CicSubstitution in
364 let t = List.assoc i subst in
365 um_aux (S.lift_meta l t)
366 with Not_found -> (* not constrained variable, i.e. free in subst*)
368 List.map (function None -> None | Some t -> Some (um_aux t)) l
372 | C.Implicit -> assert false
373 | C.Cast (te,ty) -> C.Cast (um_aux te, um_aux ty)
374 | C.Prod (n,s,t) -> C.Prod (n, um_aux s, um_aux t)
375 | C.Lambda (n,s,t) -> C.Lambda (n, um_aux s, um_aux t)
376 | C.LetIn (n,s,t) -> C.LetIn (n, um_aux s, um_aux t)
377 | C.Appl (hd :: tl) -> appl_fun um_aux hd tl
378 | C.Appl _ -> assert false
379 | C.Const (uri,exp_named_subst) ->
380 let exp_named_subst' =
381 List.map (fun (uri, t) -> (uri, um_aux t)) exp_named_subst
383 C.Const (uri, exp_named_subst')
384 | C.MutInd (uri,typeno,exp_named_subst) ->
385 let exp_named_subst' =
386 List.map (fun (uri, t) -> (uri, um_aux t)) exp_named_subst
388 C.MutInd (uri,typeno,exp_named_subst')
389 | C.MutConstruct (uri,typeno,consno,exp_named_subst) ->
390 let exp_named_subst' =
391 List.map (fun (uri, t) -> (uri, um_aux t)) exp_named_subst
393 C.MutConstruct (uri,typeno,consno,exp_named_subst')
394 | C.MutCase (sp,i,outty,t,pl) ->
395 let pl' = List.map um_aux pl in
396 C.MutCase (sp, i, um_aux outty, um_aux t, pl')
399 List.map (fun (name, i, ty, bo) -> (name, i, um_aux ty, um_aux bo)) fl
404 List.map (fun (name, ty, bo) -> (name, um_aux ty, um_aux bo)) fl
411 let appl_fun um_aux he tl =
412 let tl' = List.map um_aux tl in
415 Cic.Appl l -> Cic.Appl (l@tl')
416 | he' -> Cic.Appl (he'::tl')
419 apply_subst_gen ~appl_fun
421 let ppterm subst term = CicPp.ppterm (apply_subst subst term)
423 (* apply_subst_reducing subst (Some (mtr,reductions_no)) t *)
424 (* performs as (apply_subst subst t) until it finds an application of *)
425 (* (META [meta_to_reduce]) that, once unwinding is performed, creates *)
426 (* a new beta-redex; in this case up to [reductions_no] consecutive *)
427 (* beta-reductions are performed. *)
428 (* Hint: this function is usually called when [reductions_no] *)
429 (* eta-expansions have been performed and the head of the new *)
430 (* application has been unified with (META [meta_to_reduce]): *)
431 (* during the unwinding the eta-expansions are undone. *)
433 let apply_subst_reducing meta_to_reduce =
434 let appl_fun um_aux he tl =
435 let tl' = List.map um_aux tl in
438 Cic.Appl l -> Cic.Appl (l@tl')
439 | he' -> Cic.Appl (he'::tl')
442 match meta_to_reduce, he with
443 Some (mtr,reductions_no), Cic.Meta (m,_) when m = mtr ->
444 let rec beta_reduce =
446 (n,(Cic.Appl (Cic.Lambda (_,_,t)::he'::tl'))) when n > 0 ->
447 let he'' = CicSubstitution.subst he' t in
451 beta_reduce (n-1,Cic.Appl(he''::tl'))
454 beta_reduce (reductions_no,t')
458 apply_subst_gen ~appl_fun
460 let rec apply_subst_context subst context =
464 | Some (n, Cic.Decl t) ->
465 let t' = apply_subst subst t in
466 Some (n, Cic.Decl t') :: context
467 | Some (n, Cic.Def (t, ty)) ->
471 | Some ty -> Some (apply_subst subst ty)
473 let t' = apply_subst subst t in
474 Some (n, Cic.Def (t', ty')) :: context
475 | None -> None :: context)
478 let apply_subst_metasenv subst metasenv =
480 (fun (n, context, ty) ->
481 (n, apply_subst_context subst context, apply_subst subst ty))
483 (fun (i, _, _) -> not (List.exists (fun (j, _) -> (j = i)) subst))
486 let ppterm subst term = CicPp.ppterm (apply_subst subst term)
488 let ppcontext ?(sep = "\n") subst context =
490 (List.rev_map (function
491 | Some (n, Cic.Decl t) ->
492 sprintf "%s : %s" (CicPp.ppname n) (ppterm subst t)
493 | Some (n, Cic.Def (t, ty)) ->
494 sprintf "%s : %s := %s"
496 (match ty with None -> "_" | Some ty -> ppterm subst ty)
501 let ppmetasenv ?(sep = "\n") metasenv subst =
505 sprintf "%s |- ?%d: %s" (ppcontext ~sep:"; " subst c) i
508 (fun (i, _, _) -> not (List.exists (fun (j, _) -> (j = i)) subst))
511 (* UNWIND THE MGU INSIDE THE MGU *)
513 let unwind_subst metasenv subst =
515 (fun (unwinded,metasenv) (i,_) ->
516 let (_,canonical_context,_) = CicUtil.lookup_meta i metasenv in
517 let identity_relocation_list =
518 CicMkImplicit.identity_relocation_list_for_metavariable canonical_context
520 let (_,metasenv',subst') =
521 unwind metasenv subst unwinded (Cic.Meta (i,identity_relocation_list))
524 ) ([],metasenv) subst
527 (* From now on we recreate a kernel abstraction where substitutions are part of
530 let lift subst n term =
531 let term = apply_subst subst term in
533 CicSubstitution.lift n term
535 raise (MetaSubstFailure ("Lift failure: " ^ Printexc.to_string e))
537 let subst subst t1 t2 =
538 let t1 = apply_subst subst t1 in
539 let t2 = apply_subst subst t2 in
541 CicSubstitution.subst t1 t2
543 raise (MetaSubstFailure ("Subst failure: " ^ Printexc.to_string e))
545 let whd subst context term =
546 let term = apply_subst subst term in
547 let context = apply_subst_context subst context in
549 CicReduction.whd context term
551 raise (MetaSubstFailure ("Weak head reduction failure: " ^
552 Printexc.to_string e))
554 let are_convertible subst context t1 t2 =
555 let context = apply_subst_context subst context in
556 let t1 = apply_subst subst t1 in
557 let t2 = apply_subst subst t2 in
558 CicReduction.are_convertible context t1 t2
560 let type_of_aux' metasenv subst context term =
561 let term = apply_subst subst term in
562 let context = apply_subst_context subst context in
565 (fun (i, c, t) -> (i, apply_subst_context subst c, apply_subst subst t))
567 (fun (i, _, _) -> not (List.exists (fun (j, _) -> (j = i)) subst))
571 CicTypeChecker.type_of_aux' metasenv context term
572 with CicTypeChecker.TypeCheckerFailure msg ->
573 raise (MetaSubstFailure ("Type checker failure: " ^ msg))