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)))::context) t
55 (context',ty,C.LetIn(n,s,bo))
57 let irl = identity_relocation_list_for_metavariable context in
58 context, t, (C.Meta (newmeta,irl))
60 collect_context context ty
62 let eta_expand metasenv context t arg =
63 let module T = CicTypeChecker in
64 let module S = CicSubstitution in
68 t' when t' = S.lift n arg -> C.Rel (1 + n)
69 | C.Rel m -> if m <= n then C.Rel m else C.Rel (m+1)
70 | C.Var (uri,exp_named_subst) ->
71 let exp_named_subst' = aux_exp_named_subst n exp_named_subst in
72 C.Var (uri,exp_named_subst')
75 | C.Implicit as t -> t
76 | C.Cast (te,ty) -> C.Cast (aux n te, aux n ty)
77 | C.Prod (nn,s,t) -> C.Prod (nn, aux n s, aux (n+1) t)
78 | C.Lambda (nn,s,t) -> C.Lambda (nn, aux n s, aux (n+1) t)
79 | C.LetIn (nn,s,t) -> C.LetIn (nn, aux n s, aux (n+1) t)
80 | C.Appl l -> C.Appl (List.map (aux n) l)
81 | C.Const (uri,exp_named_subst) ->
82 let exp_named_subst' = aux_exp_named_subst n exp_named_subst in
83 C.Const (uri,exp_named_subst')
84 | C.MutInd (uri,i,exp_named_subst) ->
85 let exp_named_subst' = aux_exp_named_subst n exp_named_subst in
86 C.MutInd (uri,i,exp_named_subst')
87 | C.MutConstruct (uri,i,j,exp_named_subst) ->
88 let exp_named_subst' = aux_exp_named_subst n exp_named_subst in
89 C.MutConstruct (uri,i,j,exp_named_subst')
90 | C.MutCase (sp,i,outt,t,pl) ->
91 C.MutCase (sp,i,aux n outt, aux n t,
94 let tylen = List.length fl in
97 (fun (name,i,ty,bo) -> (name, i, aux n ty, aux (n+tylen) bo))
100 C.Fix (i, substitutedfl)
102 let tylen = List.length fl in
105 (fun (name,ty,bo) -> (name, aux n ty, aux (n+tylen) bo))
108 C.CoFix (i, substitutedfl)
109 and aux_exp_named_subst n =
110 List.map (function uri,t -> uri,aux n t)
113 T.type_of_aux' metasenv context arg
116 ProofEngineHelpers.mk_fresh_name context (Cic.Name "Heta") ~typ:argty
118 (C.Appl [C.Lambda (fresh_name,argty,aux 0 t) ; arg])
120 (*CSC: ma serve solamente la prima delle new_uninst e l'unione delle due!!! *)
121 let classify_metas newmeta in_subst_domain subst_in metasenv =
123 (fun (i,canonical_context,ty) (old_uninst,new_uninst) ->
124 if in_subst_domain i then
125 old_uninst,new_uninst
127 let ty' = subst_in canonical_context ty in
128 let canonical_context' =
130 (fun entry canonical_context' ->
133 Some (n,Cic.Decl s) ->
134 Some (n,Cic.Decl (subst_in canonical_context' s))
135 | Some (n,Cic.Def s) ->
136 Some (n,Cic.Def (subst_in canonical_context' s))
139 entry'::canonical_context'
140 ) canonical_context []
143 ((i,canonical_context',ty')::old_uninst),new_uninst
145 old_uninst,((i,canonical_context',ty')::new_uninst)
148 (* Auxiliary function for apply: given a type (a backbone), it returns its *)
149 (* head, a META environment in which there is new a META for each hypothesis,*)
150 (* a list of arguments for the new applications and the indexes of the first *)
151 (* and last new METAs introduced. The nth argument in the list of arguments *)
152 (* is just the nth new META. *)
153 let new_metasenv_for_apply newmeta proof context ty =
154 let module C = Cic in
155 let module S = CicSubstitution in
156 let rec aux newmeta =
158 C.Cast (he,_) -> aux newmeta he
159 | C.Prod (name,s,t) ->
160 let irl = identity_relocation_list_for_metavariable context in
161 let newargument = C.Meta (newmeta,irl) in
162 let (res,newmetasenv,arguments,lastmeta) =
163 aux (newmeta + 1) (S.subst newargument t)
165 res,(newmeta,context,s)::newmetasenv,newargument::arguments,lastmeta
166 | t -> t,[],[],newmeta
168 (* WARNING: here we are using the invariant that above the most *)
169 (* recente new_meta() there are no used metas. *)
170 let (res,newmetasenv,arguments,lastmeta) = aux newmeta ty in
171 res,newmetasenv,arguments,lastmeta
173 (* Useful only inside apply_tac *)
175 generalize_exp_named_subst_with_fresh_metas context newmeta uri exp_named_subst
177 let module C = Cic in
179 match CicEnvironment.get_obj uri with
180 C.Constant (_,_,_,params)
181 | C.CurrentProof (_,_,_,_,params)
182 | C.Variable (_,_,_,params)
183 | C.InductiveDefinition (_,params,_) -> params
185 let exp_named_subst_diff,new_fresh_meta,newmetasenvfragment,exp_named_subst'=
186 let next_fresh_meta = ref newmeta in
187 let newmetasenvfragment = ref [] in
188 let exp_named_subst_diff = ref [] in
194 match CicEnvironment.get_obj uri with
195 C.Variable (_,_,ty,_) ->
196 CicSubstitution.subst_vars !exp_named_subst_diff ty
197 | _ -> raise (WrongUriToVariable (UriManager.string_of_uri uri))
199 let irl = identity_relocation_list_for_metavariable context in
200 let subst_item = uri,C.Meta (!next_fresh_meta,irl) in
201 newmetasenvfragment :=
202 (!next_fresh_meta,context,ty)::!newmetasenvfragment ;
203 exp_named_subst_diff := !exp_named_subst_diff @ [subst_item] ;
204 incr next_fresh_meta ;
205 subst_item::(aux (tl,[]))
206 | uri::tl1,((uri',_) as s)::tl2 ->
207 assert (UriManager.eq uri uri') ;
209 | [],_ -> assert false
211 let exp_named_subst' = aux (params,exp_named_subst) in
212 !exp_named_subst_diff,!next_fresh_meta,
213 List.rev !newmetasenvfragment, exp_named_subst'
215 prerr_endline ("@@@ " ^ CicPp.ppterm (Cic.Var (uri,exp_named_subst)) ^ " |--> " ^ CicPp.ppterm (Cic.Var (uri,exp_named_subst'))) ;
216 new_fresh_meta,newmetasenvfragment,exp_named_subst',exp_named_subst_diff
219 let apply_tac ~term ~status:(proof, goal) =
220 (* Assumption: The term "term" must be closed in the current context *)
221 let module T = CicTypeChecker in
222 let module R = CicReduction in
223 let module C = Cic in
224 let (_,metasenv,_,_) = proof in
225 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
226 let newmeta = new_meta ~proof in
227 let exp_named_subst_diff,newmeta',newmetasenvfragment,term' =
229 C.Var (uri,exp_named_subst) ->
230 let newmeta',newmetasenvfragment,exp_named_subst',exp_named_subst_diff =
231 generalize_exp_named_subst_with_fresh_metas context newmeta uri
234 exp_named_subst_diff,newmeta',newmetasenvfragment,
235 C.Var (uri,exp_named_subst')
236 | C.Const (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.Const (uri,exp_named_subst')
243 | C.MutInd (uri,tyno,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.MutInd (uri,tyno,exp_named_subst')
250 | C.MutConstruct (uri,tyno,consno,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.MutConstruct (uri,tyno,consno,exp_named_subst')
257 | _ -> [],newmeta,[],term
259 let metasenv' = metasenv@newmetasenvfragment in
260 prerr_endline ("^^^^^TERM': " ^ CicPp.ppterm term') ;
262 CicSubstitution.subst_vars exp_named_subst_diff
263 (CicTypeChecker.type_of_aux' metasenv' context term)
265 prerr_endline ("^^^^^TERMTY: " ^ CicPp.ppterm termty) ;
266 (* newmeta is the lowest index of the new metas introduced *)
267 let (consthead,newmetas,arguments,_) =
268 new_metasenv_for_apply newmeta' proof context termty
270 let newmetasenv = metasenv'@newmetas in
271 let subst,newmetasenv' =
272 CicUnification.fo_unif newmetasenv context consthead ty
274 let in_subst_domain i = List.exists (function (j,_) -> i=j) subst in
275 let apply_subst = CicUnification.apply_subst subst in
276 let old_uninstantiatedmetas,new_uninstantiatedmetas =
277 (* subst_in doesn't need the context. Hence the underscore. *)
278 let subst_in _ = CicUnification.apply_subst subst in
279 classify_metas newmeta in_subst_domain subst_in newmetasenv'
283 (if List.length newmetas = 0 then
286 Cic.Appl (term'::arguments)
289 prerr_endline ("XXXX " ^ CicPp.ppterm (if List.length newmetas = 0 then term' else Cic.Appl (term'::arguments)) ^ " |>>> " ^ CicPp.ppterm bo') ;
290 let newmetasenv'' = new_uninstantiatedmetas@old_uninstantiatedmetas in
291 let (newproof, newmetasenv''') =
292 let subst_in = CicUnification.apply_subst ((metano,bo')::subst) in
293 subst_meta_and_metasenv_in_proof
294 proof metano subst_in newmetasenv''
296 (newproof, List.map (function (i,_,_) -> i) new_uninstantiatedmetas)
298 (* TODO per implementare i tatticali e' necessario che tutte le tattiche
299 sollevino _solamente_ Fail *)
300 let apply_tac ~term ~status =
302 apply_tac ~term ~status
303 (* TODO cacciare anche altre eccezioni? *)
304 with CicUnification.UnificationFailed as e ->
305 raise (Fail (Printexc.to_string e))
308 ?(mk_fresh_name_callback = ProofEngineHelpers.mk_fresh_name) ()
309 ~status:(proof, goal)
311 let module C = Cic in
312 let module R = CicReduction in
313 let (_,metasenv,_,_) = proof in
314 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
315 let newmeta = new_meta ~proof in
316 let (context',ty',bo') =
317 lambda_abstract context newmeta ty mk_fresh_name_callback
320 subst_meta_in_proof proof metano bo' [newmeta,context',ty']
322 (newproof, [newmeta])
325 ?(mk_fresh_name_callback = ProofEngineHelpers.mk_fresh_name)
326 term ~status:(proof, goal)
328 let module C = Cic in
329 let curi,metasenv,pbo,pty = proof in
330 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
331 let newmeta1 = new_meta ~proof in
332 let newmeta2 = newmeta1 + 1 in
334 mk_fresh_name_callback context (Cic.Name "Hcut") ~typ:term in
335 let context_for_newmeta1 =
336 (Some (fresh_name,C.Decl term))::context in
338 identity_relocation_list_for_metavariable context_for_newmeta1 in
339 let irl2 = identity_relocation_list_for_metavariable context in
340 let newmeta1ty = CicSubstitution.lift 1 ty in
343 [C.Lambda (fresh_name,term,C.Meta (newmeta1,irl1)) ;
344 C.Meta (newmeta2,irl2)]
347 subst_meta_in_proof proof metano bo'
348 [newmeta2,context,term; newmeta1,context_for_newmeta1,newmeta1ty];
350 (newproof, [newmeta1 ; newmeta2])
353 ?(mk_fresh_name_callback = ProofEngineHelpers.mk_fresh_name)
354 term ~status:(proof, goal)
356 let module C = Cic in
357 let curi,metasenv,pbo,pty = proof in
358 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
359 let _ = CicTypeChecker.type_of_aux' metasenv context term in
360 let newmeta = new_meta ~proof in
362 mk_fresh_name_callback context (Cic.Name "Hletin") ~typ:term in
363 let context_for_newmeta =
364 (Some (fresh_name,C.Def term))::context in
366 identity_relocation_list_for_metavariable context_for_newmeta in
367 let newmetaty = CicSubstitution.lift 1 ty in
368 let bo' = C.LetIn (fresh_name,term,C.Meta (newmeta,irl)) in
371 proof metano bo'[newmeta,context_for_newmeta,newmetaty]
373 (newproof, [newmeta])
375 (** functional part of the "exact" tactic *)
376 let exact_tac ~term ~status:(proof, goal) =
377 (* Assumption: the term bo must be closed in the current context *)
378 let (_,metasenv,_,_) = proof in
379 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
380 let module T = CicTypeChecker in
381 let module R = CicReduction in
382 if R.are_convertible context (T.type_of_aux' metasenv context term) ty then
384 let (newproof, metasenv') =
385 subst_meta_in_proof proof metano term [] in
389 raise (Fail "The type of the provided term is not the one expected.")
392 (* not really "primitive" tactics .... *)
394 let elim_tac ~term ~status:(proof, goal) =
395 let module T = CicTypeChecker in
396 let module U = UriManager in
397 let module R = CicReduction in
398 let module C = Cic in
399 let (curi,metasenv,_,_) = proof in
400 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
401 let termty = T.type_of_aux' metasenv context term in
402 let uri,exp_named_subst,typeno,args =
404 C.MutInd (uri,typeno,exp_named_subst) -> (uri,exp_named_subst,typeno,[])
405 | C.Appl ((C.MutInd (uri,typeno,exp_named_subst))::args) ->
406 (uri,exp_named_subst,typeno,args)
407 | _ -> raise NotAnInductiveTypeToEliminate
410 let buri = U.buri_of_uri uri in
412 match CicEnvironment.get_obj uri with
413 C.InductiveDefinition (tys,_,_) ->
414 let (name,_,_,_) = List.nth tys typeno in
419 match T.type_of_aux' metasenv context ty with
420 C.Sort C.Prop -> "_ind"
421 | C.Sort C.Set -> "_rec"
422 | C.Sort C.Type -> "_rect"
425 U.uri_of_string (buri ^ "/" ^ name ^ ext ^ ".con")
427 let eliminator_ref = C.Const (eliminator_uri,exp_named_subst) in
428 let ety = T.type_of_aux' metasenv context eliminator_ref in
429 let newmeta = new_meta ~proof in
430 let (econclusion,newmetas,arguments,lastmeta) =
431 new_metasenv_for_apply newmeta proof context ety
433 (* Here we assume that we have only one inductive hypothesis to *)
434 (* eliminate and that it is the last hypothesis of the theorem. *)
435 (* A better approach would be fingering the hypotheses in some *)
438 let (_,canonical_context,_) =
439 List.find (function (m,_,_) -> m=(lastmeta - 1)) newmetas
442 identity_relocation_list_for_metavariable canonical_context
444 Cic.Meta (lastmeta - 1, irl)
446 let newmetasenv = newmetas @ metasenv in
447 let subst1,newmetasenv' =
448 CicUnification.fo_unif newmetasenv context term meta_of_corpse
450 let ueconclusion = CicUnification.apply_subst subst1 econclusion in
451 (* The conclusion of our elimination principle is *)
452 (* (?i farg1 ... fargn) *)
453 (* The conclusion of our goal is ty. So, we can *)
454 (* eta-expand ty w.r.t. farg1 .... fargn to get *)
455 (* a new ty equal to (P farg1 ... fargn). Now *)
456 (* ?i can be instantiated with P and we are ready *)
457 (* to refine the term. *)
459 match ueconclusion with
460 C.Appl ((C.Meta (emeta,_))::fargs) -> emeta,fargs
461 | C.Meta (emeta,_) -> emeta,[]
462 | _ -> raise NotTheRightEliminatorShape
464 let ty' = CicUnification.apply_subst subst1 ty in
465 let eta_expanded_ty =
466 (*CSC: newmetasenv' era metasenv ??????????? *)
467 List.fold_left (eta_expand newmetasenv' context) ty' fargs
469 let subst2,newmetasenv'' =
470 (*CSC: passo newmetasenv', ma alcune variabili sono gia' state sostituite
471 da subst1!!!! Dovrei rimuoverle o sono innocue?*)
472 CicUnification.fo_unif
473 newmetasenv' context ueconclusion eta_expanded_ty
475 let in_subst_domain i =
476 let eq_to_i = function (j,_) -> i=j in
477 List.exists eq_to_i subst1 ||
478 List.exists eq_to_i subst2
480 (* When unwinding the META that corresponds to the elimination *)
481 (* predicate (which is emeta), we must also perform one-step *)
482 (* beta-reduction. apply_subst doesn't need the context. Hence *)
483 (* the underscore. *)
484 let apply_subst _ t =
485 let t' = CicUnification.apply_subst subst1 t in
486 CicUnification.apply_subst_reducing
487 subst2 (Some (emeta,List.length fargs)) t'
489 let old_uninstantiatedmetas,new_uninstantiatedmetas =
490 classify_metas newmeta in_subst_domain apply_subst
493 let arguments' = List.map (apply_subst context) arguments in
494 let bo' = Cic.Appl (eliminator_ref::arguments') in
496 new_uninstantiatedmetas@old_uninstantiatedmetas
498 let (newproof, newmetasenv'''') =
499 (* When unwinding the META that corresponds to the *)
500 (* elimination predicate (which is emeta), we must *)
501 (* also perform one-step beta-reduction. *)
502 (* The only difference w.r.t. apply_subst is that *)
503 (* we also substitute metano with bo'. *)
504 (*CSC: Nota: sostituire nuovamente subst1 e' superfluo, *)
507 let t' = CicUnification.apply_subst subst1 t in
508 CicUnification.apply_subst_reducing
509 ((metano,bo')::subst2)
510 (Some (emeta,List.length fargs)) t'
512 subst_meta_and_metasenv_in_proof
513 proof metano apply_subst' newmetasenv'''
516 List.map (function (i,_,_) -> i) new_uninstantiatedmetas)
519 (* The simplification is performed only on the conclusion *)
520 let elim_intros_simpl_tac ~term =
521 Tacticals.then_ ~start:(elim_tac ~term)
524 ~start:(intros_tac ())
526 [ReductionTactics.simpl_tac ~also_in_hypotheses:false ~terms:None])
529 exception NotConvertible
531 (*CSC: Bug (or feature?). [with_what] is parsed in the context of the goal, *)
532 (*CSC: while [what] can have a richer context (because of binders) *)
533 (*CSC: So it is _NOT_ possible to use those binders in the [with_what] term. *)
534 (*CSC: Is that evident? Is that right? Or should it be changed? *)
535 let change_tac ~what ~with_what ~status:(proof, goal) =
536 let curi,metasenv,pbo,pty = proof in
537 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
538 (* are_convertible works only on well-typed terms *)
539 ignore (CicTypeChecker.type_of_aux' metasenv context with_what) ;
540 if CicReduction.are_convertible context what with_what then
543 ProofEngineReduction.replace
544 ~equality:(==) ~what:[what] ~with_what:[with_what]
546 let ty' = replace ty in
550 Some (name,Cic.Def t) -> Some (name,Cic.Def (replace t))
551 | Some (name,Cic.Decl t) -> Some (name,Cic.Decl (replace t))
558 (n,_,_) when n = metano -> (metano,context',ty')
562 (curi,metasenv',pbo,pty), [metano]
565 raise (ProofEngineTypes.Fail "Not convertible")