1 (* Copyright (C) 2002, 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 open ProofEngineHelpers
29 exception NotAnInductiveTypeToEliminate
30 exception NotTheRightEliminatorShape
31 exception NoHypothesesFound
32 exception WrongUriToVariable of string
34 (* lambda_abstract newmeta ty *)
35 (* returns a triple [bo],[context],[ty'] where *)
36 (* [ty] = Pi/LetIn [context].[ty'] ([context] is a vector!) *)
37 (* and [bo] = Lambda/LetIn [context].(Meta [newmeta]) *)
38 (* So, lambda_abstract is the core of the implementation of *)
39 (* the Intros tactic. *)
40 let lambda_abstract context newmeta ty mk_fresh_name =
42 let rec collect_context context =
44 C.Cast (te,_) -> collect_context context te
46 let n' = mk_fresh_name context n ~typ:s in
47 let (context',ty,bo) =
48 collect_context ((Some (n',(C.Decl s)))::context) t
50 (context',ty,C.Lambda(n',s,bo))
52 let (context',ty,bo) =
53 collect_context ((Some (n,(C.Def (s,None))))::context) t
55 (context',ty,C.LetIn(n,s,bo))
58 CicMkImplicit.identity_relocation_list_for_metavariable context
60 context, t, (C.Meta (newmeta,irl))
62 collect_context context ty
64 let eta_expand metasenv context t arg =
65 let module T = CicTypeChecker in
66 let module S = CicSubstitution in
70 t' when t' = S.lift n arg -> C.Rel (1 + n)
71 | C.Rel m -> if m <= n then C.Rel m else C.Rel (m+1)
72 | C.Var (uri,exp_named_subst) ->
73 let exp_named_subst' = aux_exp_named_subst n exp_named_subst in
74 C.Var (uri,exp_named_subst')
77 | C.Implicit as t -> t
78 | C.Cast (te,ty) -> C.Cast (aux n te, aux n ty)
79 | C.Prod (nn,s,t) -> C.Prod (nn, aux n s, aux (n+1) t)
80 | C.Lambda (nn,s,t) -> C.Lambda (nn, aux n s, aux (n+1) t)
81 | C.LetIn (nn,s,t) -> C.LetIn (nn, aux n s, aux (n+1) t)
82 | C.Appl l -> C.Appl (List.map (aux n) l)
83 | C.Const (uri,exp_named_subst) ->
84 let exp_named_subst' = aux_exp_named_subst n exp_named_subst in
85 C.Const (uri,exp_named_subst')
86 | C.MutInd (uri,i,exp_named_subst) ->
87 let exp_named_subst' = aux_exp_named_subst n exp_named_subst in
88 C.MutInd (uri,i,exp_named_subst')
89 | C.MutConstruct (uri,i,j,exp_named_subst) ->
90 let exp_named_subst' = aux_exp_named_subst n exp_named_subst in
91 C.MutConstruct (uri,i,j,exp_named_subst')
92 | C.MutCase (sp,i,outt,t,pl) ->
93 C.MutCase (sp,i,aux n outt, aux n t,
96 let tylen = List.length fl in
99 (fun (name,i,ty,bo) -> (name, i, aux n ty, aux (n+tylen) bo))
102 C.Fix (i, substitutedfl)
104 let tylen = List.length fl in
107 (fun (name,ty,bo) -> (name, aux n ty, aux (n+tylen) bo))
110 C.CoFix (i, substitutedfl)
111 and aux_exp_named_subst n =
112 List.map (function uri,t -> uri,aux n t)
115 T.type_of_aux' metasenv context arg
118 ProofEngineHelpers.mk_fresh_name context (Cic.Name "Heta") ~typ:argty
120 (C.Appl [C.Lambda (fresh_name,argty,aux 0 t) ; arg])
122 (*CSC: ma serve solamente la prima delle new_uninst e l'unione delle due!!! *)
123 let classify_metas newmeta in_subst_domain subst_in metasenv =
125 (fun (i,canonical_context,ty) (old_uninst,new_uninst) ->
126 if in_subst_domain i then
127 old_uninst,new_uninst
129 let ty' = subst_in canonical_context ty in
130 let canonical_context' =
132 (fun entry canonical_context' ->
135 Some (n,Cic.Decl s) ->
136 Some (n,Cic.Decl (subst_in canonical_context' s))
137 | Some (n,Cic.Def (s,None)) ->
138 Some (n,Cic.Def ((subst_in canonical_context' s),None))
140 | Some (_,Cic.Def (_,Some _)) -> assert false
142 entry'::canonical_context'
143 ) canonical_context []
146 ((i,canonical_context',ty')::old_uninst),new_uninst
148 old_uninst,((i,canonical_context',ty')::new_uninst)
151 (* Auxiliary function for apply: given a type (a backbone), it returns its *)
152 (* head, a META environment in which there is new a META for each hypothesis,*)
153 (* a list of arguments for the new applications and the indexes of the first *)
154 (* and last new METAs introduced. The nth argument in the list of arguments *)
155 (* is just the nth new META. *)
156 let new_metasenv_for_apply newmeta proof context ty =
157 let module C = Cic in
158 let module S = CicSubstitution in
159 let rec aux newmeta =
161 C.Cast (he,_) -> aux newmeta he
162 | C.Prod (name,s,t) ->
164 CicMkImplicit.identity_relocation_list_for_metavariable context
166 let newargument = C.Meta (newmeta,irl) in
167 let (res,newmetasenv,arguments,lastmeta) =
168 aux (newmeta + 1) (S.subst newargument t)
170 res,(newmeta,context,s)::newmetasenv,newargument::arguments,lastmeta
171 | t -> t,[],[],newmeta
173 (* WARNING: here we are using the invariant that above the most *)
174 (* recente new_meta() there are no used metas. *)
175 let (res,newmetasenv,arguments,lastmeta) = aux newmeta ty in
176 res,newmetasenv,arguments,lastmeta
178 (* Useful only inside apply_tac *)
180 generalize_exp_named_subst_with_fresh_metas context newmeta uri exp_named_subst
182 let module C = Cic in
184 match CicEnvironment.get_obj uri with
185 C.Constant (_,_,_,params)
186 | C.CurrentProof (_,_,_,_,params)
187 | C.Variable (_,_,_,params)
188 | C.InductiveDefinition (_,params,_) -> params
190 let exp_named_subst_diff,new_fresh_meta,newmetasenvfragment,exp_named_subst'=
191 let next_fresh_meta = ref newmeta in
192 let newmetasenvfragment = ref [] in
193 let exp_named_subst_diff = ref [] in
199 match CicEnvironment.get_obj uri with
200 C.Variable (_,_,ty,_) ->
201 CicSubstitution.subst_vars !exp_named_subst_diff ty
202 | _ -> raise (WrongUriToVariable (UriManager.string_of_uri uri))
205 CicMkImplicit.identity_relocation_list_for_metavariable context
207 let subst_item = uri,C.Meta (!next_fresh_meta,irl) in
208 newmetasenvfragment :=
209 (!next_fresh_meta,context,ty)::!newmetasenvfragment ;
210 exp_named_subst_diff := !exp_named_subst_diff @ [subst_item] ;
211 incr next_fresh_meta ;
212 subst_item::(aux (tl,[]))
213 | uri::tl1,((uri',_) as s)::tl2 ->
214 assert (UriManager.eq uri uri') ;
216 | [],_ -> assert false
218 let exp_named_subst' = aux (params,exp_named_subst) in
219 !exp_named_subst_diff,!next_fresh_meta,
220 List.rev !newmetasenvfragment, exp_named_subst'
222 prerr_endline ("@@@ " ^ CicPp.ppterm (Cic.Var (uri,exp_named_subst)) ^ " |--> " ^ CicPp.ppterm (Cic.Var (uri,exp_named_subst'))) ;
223 new_fresh_meta,newmetasenvfragment,exp_named_subst',exp_named_subst_diff
226 let apply_tac ~term ~status:(proof, goal) =
227 (* Assumption: The term "term" must be closed in the current context *)
228 let module T = CicTypeChecker in
229 let module R = CicReduction in
230 let module C = Cic in
231 let (_,metasenv,_,_) = proof in
232 let metano,context,ty = CicUtil.lookup_meta goal metasenv in
233 let newmeta = new_meta_of_proof ~proof in
234 let exp_named_subst_diff,newmeta',newmetasenvfragment,term' =
236 C.Var (uri,exp_named_subst) ->
237 let newmeta',newmetasenvfragment,exp_named_subst',exp_named_subst_diff =
238 generalize_exp_named_subst_with_fresh_metas context newmeta uri
241 exp_named_subst_diff,newmeta',newmetasenvfragment,
242 C.Var (uri,exp_named_subst')
243 | C.Const (uri,exp_named_subst) ->
244 let newmeta',newmetasenvfragment,exp_named_subst',exp_named_subst_diff =
245 generalize_exp_named_subst_with_fresh_metas context newmeta uri
248 exp_named_subst_diff,newmeta',newmetasenvfragment,
249 C.Const (uri,exp_named_subst')
250 | C.MutInd (uri,tyno,exp_named_subst) ->
251 let newmeta',newmetasenvfragment,exp_named_subst',exp_named_subst_diff =
252 generalize_exp_named_subst_with_fresh_metas context newmeta uri
255 exp_named_subst_diff,newmeta',newmetasenvfragment,
256 C.MutInd (uri,tyno,exp_named_subst')
257 | C.MutConstruct (uri,tyno,consno,exp_named_subst) ->
258 let newmeta',newmetasenvfragment,exp_named_subst',exp_named_subst_diff =
259 generalize_exp_named_subst_with_fresh_metas context newmeta uri
262 exp_named_subst_diff,newmeta',newmetasenvfragment,
263 C.MutConstruct (uri,tyno,consno,exp_named_subst')
264 | _ -> [],newmeta,[],term
266 let metasenv' = metasenv@newmetasenvfragment in
267 prerr_endline ("^^^^^TERM': " ^ CicPp.ppterm term') ;
269 CicSubstitution.subst_vars exp_named_subst_diff
270 (CicTypeChecker.type_of_aux' metasenv' context term)
272 prerr_endline ("^^^^^TERMTY: " ^ CicPp.ppterm termty) ;
273 (* newmeta is the lowest index of the new metas introduced *)
274 let (consthead,newmetas,arguments,_) =
275 new_metasenv_for_apply newmeta' proof context termty
277 let newmetasenv = metasenv'@newmetas in
278 let subst,newmetasenv' =
279 CicUnification.fo_unif newmetasenv context consthead ty
281 let in_subst_domain i = List.exists (function (j,_) -> i=j) subst in
282 let apply_subst = CicMetaSubst.apply_subst subst in
283 let old_uninstantiatedmetas,new_uninstantiatedmetas =
284 (* subst_in doesn't need the context. Hence the underscore. *)
285 let subst_in _ = CicMetaSubst.apply_subst subst in
286 classify_metas newmeta in_subst_domain subst_in newmetasenv'
290 (if List.length newmetas = 0 then
293 Cic.Appl (term'::arguments)
296 prerr_endline ("XXXX " ^ CicPp.ppterm (if List.length newmetas = 0 then term' else Cic.Appl (term'::arguments)) ^ " |>>> " ^ CicPp.ppterm bo') ;
297 let newmetasenv'' = new_uninstantiatedmetas@old_uninstantiatedmetas in
298 let (newproof, newmetasenv''') =
299 let subst_in = CicMetaSubst.apply_subst ((metano,bo')::subst) in
300 subst_meta_and_metasenv_in_proof
301 proof metano subst_in newmetasenv''
303 (newproof, List.map (function (i,_,_) -> i) new_uninstantiatedmetas)
305 (* TODO per implementare i tatticali e' necessario che tutte le tattiche
306 sollevino _solamente_ Fail *)
307 let apply_tac ~term ~status =
309 apply_tac ~term ~status
310 (* TODO cacciare anche altre eccezioni? *)
311 with CicUnification.UnificationFailure _ as e ->
312 raise (Fail (Printexc.to_string e))
315 ?(mk_fresh_name_callback = ProofEngineHelpers.mk_fresh_name) ()
316 ~status:(proof, goal)
318 let module C = Cic in
319 let module R = CicReduction in
320 let (_,metasenv,_,_) = proof in
321 let metano,context,ty = CicUtil.lookup_meta goal metasenv in
322 let newmeta = new_meta_of_proof ~proof in
323 let (context',ty',bo') =
324 lambda_abstract context newmeta ty mk_fresh_name_callback
327 subst_meta_in_proof proof metano bo' [newmeta,context',ty']
329 (newproof, [newmeta])
332 ?(mk_fresh_name_callback = ProofEngineHelpers.mk_fresh_name)
333 term ~status:(proof, goal)
335 let module C = Cic in
336 let curi,metasenv,pbo,pty = proof in
337 let metano,context,ty = CicUtil.lookup_meta goal metasenv in
338 let newmeta1 = new_meta_of_proof ~proof in
339 let newmeta2 = newmeta1 + 1 in
341 mk_fresh_name_callback context (Cic.Name "Hcut") ~typ:term in
342 let context_for_newmeta1 =
343 (Some (fresh_name,C.Decl term))::context in
345 CicMkImplicit.identity_relocation_list_for_metavariable
349 CicMkImplicit.identity_relocation_list_for_metavariable context
351 let newmeta1ty = CicSubstitution.lift 1 ty in
354 [C.Lambda (fresh_name,term,C.Meta (newmeta1,irl1)) ;
355 C.Meta (newmeta2,irl2)]
358 subst_meta_in_proof proof metano bo'
359 [newmeta2,context,term; newmeta1,context_for_newmeta1,newmeta1ty];
361 (newproof, [newmeta1 ; newmeta2])
364 ?(mk_fresh_name_callback = ProofEngineHelpers.mk_fresh_name)
365 term ~status:(proof, goal)
367 let module C = Cic in
368 let curi,metasenv,pbo,pty = proof in
369 let metano,context,ty = CicUtil.lookup_meta goal metasenv in
370 let _ = CicTypeChecker.type_of_aux' metasenv context term in
371 let newmeta = new_meta_of_proof ~proof in
373 mk_fresh_name_callback context (Cic.Name "Hletin") ~typ:term in
374 let context_for_newmeta =
375 (Some (fresh_name,C.Def (term,None)))::context in
377 CicMkImplicit.identity_relocation_list_for_metavariable
380 let newmetaty = CicSubstitution.lift 1 ty in
381 let bo' = C.LetIn (fresh_name,term,C.Meta (newmeta,irl)) in
384 proof metano bo'[newmeta,context_for_newmeta,newmetaty]
386 (newproof, [newmeta])
388 (** functional part of the "exact" tactic *)
389 let exact_tac ~term ~status:(proof, goal) =
390 (* Assumption: the term bo must be closed in the current context *)
391 let (_,metasenv,_,_) = proof in
392 let metano,context,ty = CicUtil.lookup_meta goal metasenv in
393 let module T = CicTypeChecker in
394 let module R = CicReduction in
395 if R.are_convertible context (T.type_of_aux' metasenv context term) ty then
397 let (newproof, metasenv') =
398 subst_meta_in_proof proof metano term [] in
402 raise (Fail "The type of the provided term is not the one expected.")
405 (* not really "primitive" tactics .... *)
407 let elim_tac ~term ~status:(proof, goal) =
408 let module T = CicTypeChecker in
409 let module U = UriManager in
410 let module R = CicReduction in
411 let module C = Cic in
412 let (curi,metasenv,_,_) = proof in
413 let metano,context,ty = CicUtil.lookup_meta goal metasenv in
414 let termty = T.type_of_aux' metasenv context term in
415 let uri,exp_named_subst,typeno,args =
417 C.MutInd (uri,typeno,exp_named_subst) -> (uri,exp_named_subst,typeno,[])
418 | C.Appl ((C.MutInd (uri,typeno,exp_named_subst))::args) ->
419 (uri,exp_named_subst,typeno,args)
420 | _ -> raise NotAnInductiveTypeToEliminate
423 let buri = U.buri_of_uri uri in
425 match CicEnvironment.get_obj uri with
426 C.InductiveDefinition (tys,_,_) ->
427 let (name,_,_,_) = List.nth tys typeno in
432 match T.type_of_aux' metasenv context ty with
433 C.Sort C.Prop -> "_ind"
434 | C.Sort C.Set -> "_rec"
435 | C.Sort C.CProp -> "_rec"
436 | C.Sort C.Type -> "_rect"
439 U.uri_of_string (buri ^ "/" ^ name ^ ext ^ ".con")
441 let eliminator_ref = C.Const (eliminator_uri,exp_named_subst) in
442 let ety = T.type_of_aux' metasenv context eliminator_ref in
443 let newmeta = new_meta_of_proof ~proof in
444 let (econclusion,newmetas,arguments,lastmeta) =
445 new_metasenv_for_apply newmeta proof context ety
447 (* Here we assume that we have only one inductive hypothesis to *)
448 (* eliminate and that it is the last hypothesis of the theorem. *)
449 (* A better approach would be fingering the hypotheses in some *)
452 let (_,canonical_context,_) =
453 CicUtil.lookup_meta (lastmeta - 1) newmetas
456 CicMkImplicit.identity_relocation_list_for_metavariable
459 Cic.Meta (lastmeta - 1, irl)
461 let newmetasenv = newmetas @ metasenv in
462 let subst1,newmetasenv' =
463 CicUnification.fo_unif newmetasenv context term meta_of_corpse
465 let ueconclusion = CicMetaSubst.apply_subst subst1 econclusion in
466 (* The conclusion of our elimination principle is *)
467 (* (?i farg1 ... fargn) *)
468 (* The conclusion of our goal is ty. So, we can *)
469 (* eta-expand ty w.r.t. farg1 .... fargn to get *)
470 (* a new ty equal to (P farg1 ... fargn). Now *)
471 (* ?i can be instantiated with P and we are ready *)
472 (* to refine the term. *)
474 match ueconclusion with
475 C.Appl ((C.Meta (emeta,_))::fargs) -> emeta,fargs
476 | C.Meta (emeta,_) -> emeta,[]
477 | _ -> raise NotTheRightEliminatorShape
479 let ty' = CicMetaSubst.apply_subst subst1 ty in
480 let eta_expanded_ty =
481 (*CSC: newmetasenv' era metasenv ??????????? *)
482 List.fold_left (eta_expand newmetasenv' context) ty' fargs
484 let subst2,newmetasenv'' =
485 (*CSC: passo newmetasenv', ma alcune variabili sono gia' state sostituite
486 da subst1!!!! Dovrei rimuoverle o sono innocue?*)
487 CicUnification.fo_unif
488 newmetasenv' context ueconclusion eta_expanded_ty
490 let in_subst_domain i =
491 let eq_to_i = function (j,_) -> i=j in
492 List.exists eq_to_i subst1 ||
493 List.exists eq_to_i subst2
495 (* When unwinding the META that corresponds to the elimination *)
496 (* predicate (which is emeta), we must also perform one-step *)
497 (* beta-reduction. apply_subst doesn't need the context. Hence *)
498 (* the underscore. *)
499 let apply_subst _ t =
500 let t' = CicMetaSubst.apply_subst subst1 t in
501 CicMetaSubst.apply_subst_reducing
502 (Some (emeta,List.length fargs)) subst2 t'
504 let old_uninstantiatedmetas,new_uninstantiatedmetas =
505 classify_metas newmeta in_subst_domain apply_subst
508 let arguments' = List.map (apply_subst context) arguments in
509 let bo' = Cic.Appl (eliminator_ref::arguments') in
511 new_uninstantiatedmetas@old_uninstantiatedmetas
513 let (newproof, newmetasenv'''') =
514 (* When unwinding the META that corresponds to the *)
515 (* elimination predicate (which is emeta), we must *)
516 (* also perform one-step beta-reduction. *)
517 (* The only difference w.r.t. apply_subst is that *)
518 (* we also substitute metano with bo'. *)
519 (*CSC: Nota: sostituire nuovamente subst1 e' superfluo, *)
522 let t' = CicMetaSubst.apply_subst subst1 t in
523 CicMetaSubst.apply_subst_reducing
524 (Some (emeta,List.length fargs))
525 ((metano,bo')::subst2) t'
527 subst_meta_and_metasenv_in_proof
528 proof metano apply_subst' newmetasenv'''
531 List.map (function (i,_,_) -> i) new_uninstantiatedmetas)
534 (* The simplification is performed only on the conclusion *)
535 let elim_intros_simpl_tac ~term =
536 Tacticals.then_ ~start:(elim_tac ~term)
539 ~start:(intros_tac ())
541 [ReductionTactics.simpl_tac ~also_in_hypotheses:false ~terms:None])
544 exception NotConvertible
546 (*CSC: Bug (or feature?). [with_what] is parsed in the context of the goal, *)
547 (*CSC: while [what] can have a richer context (because of binders) *)
548 (*CSC: So it is _NOT_ possible to use those binders in the [with_what] term. *)
549 (*CSC: Is that evident? Is that right? Or should it be changed? *)
550 let change_tac ~what ~with_what ~status:(proof, goal) =
551 let curi,metasenv,pbo,pty = proof in
552 let metano,context,ty = CicUtil.lookup_meta goal metasenv in
553 (* are_convertible works only on well-typed terms *)
554 ignore (CicTypeChecker.type_of_aux' metasenv context with_what) ;
555 if CicReduction.are_convertible context what with_what then
558 ProofEngineReduction.replace
559 ~equality:(==) ~what:[what] ~with_what:[with_what]
561 let ty' = replace ty in
565 Some (name,Cic.Def (t,None)) -> Some (name,Cic.Def ((replace t),None))
566 | Some (name,Cic.Decl t) -> Some (name,Cic.Decl (replace t))
568 | Some (_,Cic.Def (_,Some _)) -> assert false
574 (n,_,_) when n = metano -> (metano,context',ty')
578 (curi,metasenv',pbo,pty), [metano]
581 raise (ProofEngineTypes.Fail "Not convertible")