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.
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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.
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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 mknames =
42 let rec collect_context context =
44 C.Cast (te,_) -> collect_context context te
46 let n' = C.Name (mknames n) 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
115 (C.Appl [C.Lambda ((C.Name "dummy"),argty,aux 0 t) ; arg])
117 (*CSC: ma serve solamente la prima delle new_uninst e l'unione delle due!!! *)
118 let classify_metas newmeta in_subst_domain subst_in metasenv =
120 (fun (i,canonical_context,ty) (old_uninst,new_uninst) ->
121 if in_subst_domain i then
122 old_uninst,new_uninst
124 let ty' = subst_in canonical_context ty in
125 let canonical_context' =
127 (fun entry canonical_context' ->
130 Some (n,Cic.Decl s) ->
131 Some (n,Cic.Decl (subst_in canonical_context' s))
132 | Some (n,Cic.Def s) ->
133 Some (n,Cic.Def (subst_in canonical_context' s))
136 entry'::canonical_context'
137 ) canonical_context []
140 ((i,canonical_context',ty')::old_uninst),new_uninst
142 old_uninst,((i,canonical_context',ty')::new_uninst)
145 (* Auxiliary function for apply: given a type (a backbone), it returns its *)
146 (* head, a META environment in which there is new a META for each hypothesis,*)
147 (* a list of arguments for the new applications and the indexes of the first *)
148 (* and last new METAs introduced. The nth argument in the list of arguments *)
149 (* is just the nth new META. *)
150 let new_metasenv_for_apply newmeta proof context ty =
151 let module C = Cic in
152 let module S = CicSubstitution in
153 let rec aux newmeta =
155 C.Cast (he,_) -> aux newmeta he
156 | C.Prod (name,s,t) ->
157 let irl = identity_relocation_list_for_metavariable context in
158 let newargument = C.Meta (newmeta,irl) in
159 let (res,newmetasenv,arguments,lastmeta) =
160 aux (newmeta + 1) (S.subst newargument t)
162 res,(newmeta,context,s)::newmetasenv,newargument::arguments,lastmeta
163 | t -> t,[],[],newmeta
165 (* WARNING: here we are using the invariant that above the most *)
166 (* recente new_meta() there are no used metas. *)
167 let (res,newmetasenv,arguments,lastmeta) = aux newmeta ty in
168 res,newmetasenv,arguments,lastmeta
170 (* Useful only inside apply_tac *)
172 generalize_exp_named_subst_with_fresh_metas context newmeta uri exp_named_subst
174 let module C = Cic in
176 match CicEnvironment.get_obj uri with
177 C.Constant (_,_,_,params)
178 | C.CurrentProof (_,_,_,_,params)
179 | C.Variable (_,_,_,params)
180 | C.InductiveDefinition (_,params,_) -> params
182 let exp_named_subst_diff,new_fresh_meta,newmetasenvfragment,exp_named_subst'=
183 let next_fresh_meta = ref newmeta in
184 let newmetasenvfragment = ref [] in
185 let exp_named_subst_diff = ref [] in
191 match CicEnvironment.get_obj uri with
192 C.Variable (_,_,ty,_) ->
193 CicSubstitution.subst_vars !exp_named_subst_diff ty
194 | _ -> raise (WrongUriToVariable (UriManager.string_of_uri uri))
196 let irl = identity_relocation_list_for_metavariable context in
197 let subst_item = uri,C.Meta (!next_fresh_meta,irl) in
198 newmetasenvfragment :=
199 (!next_fresh_meta,context,ty)::!newmetasenvfragment ;
200 exp_named_subst_diff := !exp_named_subst_diff @ [subst_item] ;
201 incr next_fresh_meta ;
202 subst_item::(aux (tl,[]))
203 | uri::tl1,((uri',_) as s)::tl2 ->
204 assert (UriManager.eq uri uri') ;
206 | [],_ -> assert false
208 let exp_named_subst' = aux (params,exp_named_subst) in
209 !exp_named_subst_diff,!next_fresh_meta,
210 List.rev !newmetasenvfragment, exp_named_subst'
212 prerr_endline ("@@@ " ^ CicPp.ppterm (Cic.Var (uri,exp_named_subst)) ^ " |--> " ^ CicPp.ppterm (Cic.Var (uri,exp_named_subst'))) ;
213 new_fresh_meta,newmetasenvfragment,exp_named_subst',exp_named_subst_diff
216 let apply_tac ~term ~status:(proof, goal) =
217 (* Assumption: The term "term" must be closed in the current context *)
218 let module T = CicTypeChecker in
219 let module R = CicReduction in
220 let module C = Cic in
221 let (_,metasenv,_,_) = proof in
222 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
223 let newmeta = new_meta ~proof in
224 let exp_named_subst_diff,newmeta',newmetasenvfragment,term' =
226 C.Var (uri,exp_named_subst) ->
227 let newmeta',newmetasenvfragment,exp_named_subst',exp_named_subst_diff =
228 generalize_exp_named_subst_with_fresh_metas context newmeta uri
231 exp_named_subst_diff,newmeta',newmetasenvfragment,
232 C.Var (uri,exp_named_subst')
233 | C.Const (uri,exp_named_subst) ->
234 let newmeta',newmetasenvfragment,exp_named_subst',exp_named_subst_diff =
235 generalize_exp_named_subst_with_fresh_metas context newmeta uri
238 exp_named_subst_diff,newmeta',newmetasenvfragment,
239 C.Const (uri,exp_named_subst')
240 | C.MutInd (uri,tyno,exp_named_subst) ->
241 let newmeta',newmetasenvfragment,exp_named_subst',exp_named_subst_diff =
242 generalize_exp_named_subst_with_fresh_metas context newmeta uri
245 exp_named_subst_diff,newmeta',newmetasenvfragment,
246 C.MutInd (uri,tyno,exp_named_subst')
247 | C.MutConstruct (uri,tyno,consno,exp_named_subst) ->
248 let newmeta',newmetasenvfragment,exp_named_subst',exp_named_subst_diff =
249 generalize_exp_named_subst_with_fresh_metas context newmeta uri
252 exp_named_subst_diff,newmeta',newmetasenvfragment,
253 C.MutConstruct (uri,tyno,consno,exp_named_subst')
254 | _ -> [],newmeta,[],term
256 let metasenv' = metasenv@newmetasenvfragment in
257 prerr_endline ("^^^^^TERM': " ^ CicPp.ppterm term') ;
259 CicSubstitution.subst_vars exp_named_subst_diff
260 (CicTypeChecker.type_of_aux' metasenv' context term)
262 prerr_endline ("^^^^^TERMTY: " ^ CicPp.ppterm termty) ;
263 (* newmeta is the lowest index of the new metas introduced *)
264 let (consthead,newmetas,arguments,_) =
265 new_metasenv_for_apply newmeta' proof context termty
267 let newmetasenv = metasenv'@newmetas in
268 let subst,newmetasenv' =
269 CicUnification.fo_unif newmetasenv context consthead ty
271 let in_subst_domain i = List.exists (function (j,_) -> i=j) subst in
272 let apply_subst = CicUnification.apply_subst subst in
273 let old_uninstantiatedmetas,new_uninstantiatedmetas =
274 (* subst_in doesn't need the context. Hence the underscore. *)
275 let subst_in _ = CicUnification.apply_subst subst in
276 classify_metas newmeta in_subst_domain subst_in newmetasenv'
280 (if List.length newmetas = 0 then
283 Cic.Appl (term'::arguments)
286 prerr_endline ("XXXX " ^ CicPp.ppterm (if List.length newmetas = 0 then term' else Cic.Appl (term'::arguments)) ^ " |>>> " ^ CicPp.ppterm bo') ;
287 let newmetasenv'' = new_uninstantiatedmetas@old_uninstantiatedmetas in
288 let (newproof, newmetasenv''') =
289 let subst_in = CicUnification.apply_subst ((metano,bo')::subst) in
290 subst_meta_and_metasenv_in_proof
291 proof metano subst_in newmetasenv''
293 (newproof, List.map (function (i,_,_) -> i) new_uninstantiatedmetas)
295 (* TODO per implementare i tatticali e' necessario che tutte le tattiche
296 sollevino _solamente_ Fail *)
297 let apply_tac ~term ~status =
299 apply_tac ~term ~status
300 (* TODO cacciare anche altre eccezioni? *)
301 with CicUnification.UnificationFailed as e ->
302 raise (Fail (Printexc.to_string e))
304 let intros_tac ~status:(proof, goal) =
305 let module C = Cic in
306 let module R = CicReduction in
307 let (_,metasenv,_,_) = proof in
308 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
309 let newmeta = new_meta ~proof in
310 let (context',ty',bo') =
311 lambda_abstract context newmeta ty (ProofEngineHelpers.fresh_name)
314 subst_meta_in_proof proof metano bo' [newmeta,context',ty']
316 (newproof, [newmeta])
318 let cut_tac ~term ~status:(proof, goal) =
319 let module C = Cic in
320 let curi,metasenv,pbo,pty = proof in
321 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
322 let newmeta1 = new_meta ~proof in
323 let newmeta2 = newmeta1 + 1 in
324 let context_for_newmeta1 =
325 (Some (C.Name "dummy_for_cut",C.Decl term))::context in
327 identity_relocation_list_for_metavariable context_for_newmeta1 in
328 let irl2 = identity_relocation_list_for_metavariable context in
329 let newmeta1ty = CicSubstitution.lift 1 ty in
332 [C.Lambda (C.Name "dummy_for_cut",term,C.Meta (newmeta1,irl1)) ;
333 C.Meta (newmeta2,irl2)]
336 subst_meta_in_proof proof metano bo'
337 [newmeta2,context,term; newmeta1,context_for_newmeta1,newmeta1ty];
339 (newproof, [newmeta1 ; newmeta2])
341 let letin_tac ~term ~status:(proof, goal) =
342 let module C = Cic in
343 let curi,metasenv,pbo,pty = proof in
344 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
345 let _ = CicTypeChecker.type_of_aux' metasenv context term in
346 let newmeta = new_meta ~proof in
347 let context_for_newmeta =
348 (Some (C.Name "dummy_for_letin",C.Def term))::context in
350 identity_relocation_list_for_metavariable context_for_newmeta in
351 let newmetaty = CicSubstitution.lift 1 ty in
352 let bo' = C.LetIn (C.Name "dummy_for_letin",term,C.Meta (newmeta,irl)) in
355 proof metano bo'[newmeta,context_for_newmeta,newmetaty]
357 (newproof, [newmeta])
359 (** functional part of the "exact" tactic *)
360 let exact_tac ~term ~status:(proof, goal) =
361 (* Assumption: the term bo must be closed in the current context *)
362 let (_,metasenv,_,_) = proof in
363 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
364 let module T = CicTypeChecker in
365 let module R = CicReduction in
366 if R.are_convertible context (T.type_of_aux' metasenv context term) ty then
368 let (newproof, metasenv') =
369 subst_meta_in_proof proof metano term [] in
373 raise (Fail "The type of the provided term is not the one expected.")
376 (* not really "primitive" tactics .... *)
378 let elim_tac ~term ~status:(proof, goal) =
379 let module T = CicTypeChecker in
380 let module U = UriManager in
381 let module R = CicReduction in
382 let module C = Cic in
383 let (curi,metasenv,_,_) = proof in
384 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
385 let termty = T.type_of_aux' metasenv context term in
386 let uri,exp_named_subst,typeno,args =
388 C.MutInd (uri,typeno,exp_named_subst) -> (uri,exp_named_subst,typeno,[])
389 | C.Appl ((C.MutInd (uri,typeno,exp_named_subst))::args) ->
390 (uri,exp_named_subst,typeno,args)
391 | _ -> raise NotAnInductiveTypeToEliminate
394 let buri = U.buri_of_uri uri in
396 match CicEnvironment.get_obj uri with
397 C.InductiveDefinition (tys,_,_) ->
398 let (name,_,_,_) = List.nth tys typeno in
403 match T.type_of_aux' metasenv context ty with
404 C.Sort C.Prop -> "_ind"
405 | C.Sort C.Set -> "_rec"
406 | C.Sort C.Type -> "_rect"
409 U.uri_of_string (buri ^ "/" ^ name ^ ext ^ ".con")
411 let eliminator_ref = C.Const (eliminator_uri,exp_named_subst) in
412 let ety = T.type_of_aux' metasenv context eliminator_ref in
413 let newmeta = new_meta ~proof in
414 let (econclusion,newmetas,arguments,lastmeta) =
415 new_metasenv_for_apply newmeta proof context ety
417 (* Here we assume that we have only one inductive hypothesis to *)
418 (* eliminate and that it is the last hypothesis of the theorem. *)
419 (* A better approach would be fingering the hypotheses in some *)
422 let (_,canonical_context,_) =
423 List.find (function (m,_,_) -> m=(lastmeta - 1)) newmetas
426 identity_relocation_list_for_metavariable canonical_context
428 Cic.Meta (lastmeta - 1, irl)
430 let newmetasenv = newmetas @ metasenv in
431 let subst1,newmetasenv' =
432 CicUnification.fo_unif newmetasenv context term meta_of_corpse
434 let ueconclusion = CicUnification.apply_subst subst1 econclusion in
435 (* The conclusion of our elimination principle is *)
436 (* (?i farg1 ... fargn) *)
437 (* The conclusion of our goal is ty. So, we can *)
438 (* eta-expand ty w.r.t. farg1 .... fargn to get *)
439 (* a new ty equal to (P farg1 ... fargn). Now *)
440 (* ?i can be instantiated with P and we are ready *)
441 (* to refine the term. *)
443 match ueconclusion with
444 C.Appl ((C.Meta (emeta,_))::fargs) -> emeta,fargs
445 | C.Meta (emeta,_) -> emeta,[]
446 | _ -> raise NotTheRightEliminatorShape
448 let ty' = CicUnification.apply_subst subst1 ty in
449 let eta_expanded_ty =
450 (*CSC: newmetasenv' era metasenv ??????????? *)
451 List.fold_left (eta_expand newmetasenv' context) ty' fargs
453 let subst2,newmetasenv'' =
454 (*CSC: passo newmetasenv', ma alcune variabili sono gia' state sostituite
455 da subst1!!!! Dovrei rimuoverle o sono innocue?*)
456 CicUnification.fo_unif
457 newmetasenv' context ueconclusion eta_expanded_ty
459 let in_subst_domain i =
460 let eq_to_i = function (j,_) -> i=j in
461 List.exists eq_to_i subst1 ||
462 List.exists eq_to_i subst2
464 (* When unwinding the META that corresponds to the elimination *)
465 (* predicate (which is emeta), we must also perform one-step *)
466 (* beta-reduction. apply_subst doesn't need the context. Hence *)
467 (* the underscore. *)
468 let apply_subst _ t =
469 let t' = CicUnification.apply_subst subst1 t in
470 CicUnification.apply_subst_reducing
471 subst2 (Some (emeta,List.length fargs)) t'
473 let old_uninstantiatedmetas,new_uninstantiatedmetas =
474 classify_metas newmeta in_subst_domain apply_subst
477 let arguments' = List.map (apply_subst context) arguments in
478 let bo' = Cic.Appl (eliminator_ref::arguments') in
480 new_uninstantiatedmetas@old_uninstantiatedmetas
482 let (newproof, newmetasenv'''') =
483 (* When unwinding the META that corresponds to the *)
484 (* elimination predicate (which is emeta), we must *)
485 (* also perform one-step beta-reduction. *)
486 (* The only difference w.r.t. apply_subst is that *)
487 (* we also substitute metano with bo'. *)
488 (*CSC: Nota: sostituire nuovamente subst1 e' superfluo, *)
491 let t' = CicUnification.apply_subst subst1 t in
492 CicUnification.apply_subst_reducing
493 ((metano,bo')::subst2)
494 (Some (emeta,List.length fargs)) t'
496 subst_meta_and_metasenv_in_proof
497 proof metano apply_subst' newmetasenv'''
500 List.map (function (i,_,_) -> i) new_uninstantiatedmetas)
503 (* The simplification is performed only on the conclusion *)
504 let elim_intros_simpl_tac ~term =
505 Tacticals.then_ ~start:(elim_tac ~term)
510 [ReductionTactics.simpl_tac ~also_in_hypotheses:false ~term:None])
513 exception NotConvertible
515 (*CSC: Bug (or feature?). [with_what] is parsed in the context of the goal, *)
516 (*CSC: while [what] can have a richer context (because of binders) *)
517 (*CSC: So it is _NOT_ possible to use those binders in the [with_what] term. *)
518 (*CSC: Is that evident? Is that right? Or should it be changed? *)
519 let change_tac ~what ~with_what ~status:(proof, goal) =
520 let curi,metasenv,pbo,pty = proof in
521 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
522 (* are_convertible works only on well-typed terms *)
523 ignore (CicTypeChecker.type_of_aux' metasenv context with_what) ;
524 if CicReduction.are_convertible context what with_what then
527 ProofEngineReduction.replace ~equality:(==) ~what ~with_what
529 let ty' = replace ty in
533 Some (name,Cic.Def t) -> Some (name,Cic.Def (replace t))
534 | Some (name,Cic.Decl t) -> Some (name,Cic.Decl (replace t))
541 (n,_,_) when n = metano -> (metano,context',ty')
545 (curi,metasenv',pbo,pty), [metano]
548 raise (ProofEngineTypes.Fail "Not convertible")