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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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
49 | C.Anonymous -> C.Name (mknames ())
51 let (context',ty,bo) =
52 collect_context ((Some (n',(C.Decl s)))::context) t
54 (context',ty,C.Lambda(n',s,bo))
56 let (context',ty,bo) =
57 collect_context ((Some (n,(C.Def s)))::context) t
59 (context',ty,C.LetIn(n,s,bo))
61 let irl = identity_relocation_list_for_metavariable context in
62 context, t, (C.Meta (newmeta,irl))
64 collect_context context ty
66 let eta_expand metasenv context t arg =
67 let module T = CicTypeChecker in
68 let module S = CicSubstitution in
72 t' when t' = S.lift n arg -> C.Rel (1 + n)
73 | C.Rel m -> if m <= n then C.Rel m else C.Rel (m+1)
74 | C.Var (uri,exp_named_subst) ->
75 let exp_named_subst' = aux_exp_named_subst n exp_named_subst in
76 C.Var (uri,exp_named_subst')
79 | C.Implicit as t -> t
80 | C.Cast (te,ty) -> C.Cast (aux n te, aux n ty)
81 | C.Prod (nn,s,t) -> C.Prod (nn, aux n s, aux (n+1) t)
82 | C.Lambda (nn,s,t) -> C.Lambda (nn, aux n s, aux (n+1) t)
83 | C.LetIn (nn,s,t) -> C.LetIn (nn, aux n s, aux (n+1) t)
84 | C.Appl l -> C.Appl (List.map (aux n) l)
85 | C.Const (uri,exp_named_subst) ->
86 let exp_named_subst' = aux_exp_named_subst n exp_named_subst in
87 C.Const (uri,exp_named_subst')
88 | C.MutInd (uri,i,exp_named_subst) ->
89 let exp_named_subst' = aux_exp_named_subst n exp_named_subst in
90 C.MutInd (uri,i,exp_named_subst')
91 | C.MutConstruct (uri,i,j,exp_named_subst) ->
92 let exp_named_subst' = aux_exp_named_subst n exp_named_subst in
93 C.MutConstruct (uri,i,j,exp_named_subst')
94 | C.MutCase (sp,i,outt,t,pl) ->
95 C.MutCase (sp,i,aux n outt, aux n t,
98 let tylen = List.length fl in
101 (fun (name,i,ty,bo) -> (name, i, aux n ty, aux (n+tylen) bo))
104 C.Fix (i, substitutedfl)
106 let tylen = List.length fl in
109 (fun (name,ty,bo) -> (name, aux n ty, aux (n+tylen) bo))
112 C.CoFix (i, substitutedfl)
113 and aux_exp_named_subst n =
114 List.map (function uri,t -> uri,aux n t)
117 T.type_of_aux' metasenv context arg
119 (C.Appl [C.Lambda ((C.Name "dummy"),argty,aux 0 t) ; arg])
121 (*CSC: ma serve solamente la prima delle new_uninst e l'unione delle due!!! *)
122 let classify_metas newmeta in_subst_domain subst_in metasenv =
124 (fun (i,canonical_context,ty) (old_uninst,new_uninst) ->
125 if in_subst_domain i then
126 old_uninst,new_uninst
128 let ty' = subst_in canonical_context ty in
129 let canonical_context' =
131 (fun entry canonical_context' ->
134 Some (n,Cic.Decl s) ->
135 Some (n,Cic.Decl (subst_in canonical_context' s))
136 | Some (n,Cic.Def s) ->
137 Some (n,Cic.Def (subst_in canonical_context' s))
140 entry'::canonical_context'
141 ) canonical_context []
144 ((i,canonical_context',ty')::old_uninst),new_uninst
146 old_uninst,((i,canonical_context',ty')::new_uninst)
149 (* Auxiliary function for apply: given a type (a backbone), it returns its *)
150 (* head, a META environment in which there is new a META for each hypothesis,*)
151 (* a list of arguments for the new applications and the indexes of the first *)
152 (* and last new METAs introduced. The nth argument in the list of arguments *)
153 (* is just the nth new META. *)
154 let new_metasenv_for_apply newmeta proof context ty =
155 let module C = Cic in
156 let module S = CicSubstitution in
157 let rec aux newmeta =
159 C.Cast (he,_) -> aux newmeta he
160 | C.Prod (name,s,t) ->
161 let irl = identity_relocation_list_for_metavariable context in
162 let newargument = C.Meta (newmeta,irl) in
163 let (res,newmetasenv,arguments,lastmeta) =
164 aux (newmeta + 1) (S.subst newargument t)
166 res,(newmeta,context,s)::newmetasenv,newargument::arguments,lastmeta
167 | t -> t,[],[],newmeta
169 (* WARNING: here we are using the invariant that above the most *)
170 (* recente new_meta() there are no used metas. *)
171 let (res,newmetasenv,arguments,lastmeta) = aux newmeta ty in
172 res,newmetasenv,arguments,lastmeta
174 (* Useful only inside apply_tac *)
176 generalize_exp_named_subst_with_fresh_metas context newmeta uri exp_named_subst
178 let module C = Cic in
180 match CicEnvironment.get_obj uri with
181 C.Constant (_,_,_,params)
182 | C.CurrentProof (_,_,_,_,params)
183 | C.Variable (_,_,_,params)
184 | C.InductiveDefinition (_,params,_) -> params
186 let exp_named_subst_diff,new_fresh_meta,newmetasenvfragment,exp_named_subst'=
187 let next_fresh_meta = ref newmeta in
188 let newmetasenvfragment = ref [] in
189 let exp_named_subst_diff = ref [] in
195 match CicEnvironment.get_obj uri with
196 C.Variable (_,_,ty,_) ->
197 CicSubstitution.subst_vars !exp_named_subst_diff ty
198 | _ -> raise (WrongUriToVariable (UriManager.string_of_uri uri))
200 let irl = identity_relocation_list_for_metavariable context in
201 let subst_item = uri,C.Meta (!next_fresh_meta,irl) in
202 newmetasenvfragment :=
203 (!next_fresh_meta,context,ty)::!newmetasenvfragment ;
204 exp_named_subst_diff := !exp_named_subst_diff @ [subst_item] ;
205 incr next_fresh_meta ;
206 subst_item::(aux (tl,[]))
207 | uri::tl1,((uri',_) as s)::tl2 ->
208 assert (UriManager.eq uri uri') ;
210 | [],_ -> assert false
212 let exp_named_subst' = aux (params,exp_named_subst) in
213 !exp_named_subst_diff,!next_fresh_meta,
214 List.rev !newmetasenvfragment, exp_named_subst'
216 prerr_endline ("@@@ " ^ CicPp.ppterm (Cic.Var (uri,exp_named_subst)) ^ " |--> " ^ CicPp.ppterm (Cic.Var (uri,exp_named_subst'))) ;
217 new_fresh_meta,newmetasenvfragment,exp_named_subst',exp_named_subst_diff
220 let apply_tac ~term ~status:(proof, goal) =
221 (* Assumption: The term "term" must be closed in the current context *)
222 let module T = CicTypeChecker in
223 let module R = CicReduction in
224 let module C = Cic in
225 let (_,metasenv,_,_) = proof in
226 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
227 let newmeta = new_meta ~proof in
228 let exp_named_subst_diff,newmeta',newmetasenvfragment,term' =
230 C.Var (uri,exp_named_subst) ->
231 let newmeta',newmetasenvfragment,exp_named_subst',exp_named_subst_diff =
232 generalize_exp_named_subst_with_fresh_metas context newmeta uri
235 exp_named_subst_diff,newmeta',newmetasenvfragment,
236 C.Var (uri,exp_named_subst')
237 | C.Const (uri,exp_named_subst) ->
238 let newmeta',newmetasenvfragment,exp_named_subst',exp_named_subst_diff =
239 generalize_exp_named_subst_with_fresh_metas context newmeta uri
242 exp_named_subst_diff,newmeta',newmetasenvfragment,
243 C.Const (uri,exp_named_subst')
244 | C.MutInd (uri,tyno,exp_named_subst) ->
245 let newmeta',newmetasenvfragment,exp_named_subst',exp_named_subst_diff =
246 generalize_exp_named_subst_with_fresh_metas context newmeta uri
249 exp_named_subst_diff,newmeta',newmetasenvfragment,
250 C.MutInd (uri,tyno,exp_named_subst')
251 | C.MutConstruct (uri,tyno,consno,exp_named_subst) ->
252 let newmeta',newmetasenvfragment,exp_named_subst',exp_named_subst_diff =
253 generalize_exp_named_subst_with_fresh_metas context newmeta uri
256 exp_named_subst_diff,newmeta',newmetasenvfragment,
257 C.MutConstruct (uri,tyno,consno,exp_named_subst')
258 | _ -> [],newmeta,[],term
260 let metasenv' = metasenv@newmetasenvfragment in
261 prerr_endline ("^^^^^TERM': " ^ CicPp.ppterm term') ;
263 CicSubstitution.subst_vars exp_named_subst_diff
264 (CicTypeChecker.type_of_aux' metasenv' context term)
266 prerr_endline ("^^^^^TERMTY: " ^ CicPp.ppterm termty) ;
267 (* newmeta is the lowest index of the new metas introduced *)
268 let (consthead,newmetas,arguments,_) =
269 new_metasenv_for_apply newmeta' proof context termty
271 let newmetasenv = metasenv'@newmetas in
272 let subst,newmetasenv' =
273 CicUnification.fo_unif newmetasenv context consthead ty
275 let in_subst_domain i = List.exists (function (j,_) -> i=j) subst in
276 let apply_subst = CicUnification.apply_subst subst in
277 let old_uninstantiatedmetas,new_uninstantiatedmetas =
278 (* subst_in doesn't need the context. Hence the underscore. *)
279 let subst_in _ = CicUnification.apply_subst subst in
280 classify_metas newmeta in_subst_domain subst_in newmetasenv'
284 (if List.length newmetas = 0 then
287 Cic.Appl (term'::arguments)
290 prerr_endline ("XXXX " ^ CicPp.ppterm (if List.length newmetas = 0 then term' else Cic.Appl (term'::arguments)) ^ " |>>> " ^ CicPp.ppterm bo') ;
291 let newmetasenv'' = new_uninstantiatedmetas@old_uninstantiatedmetas in
292 let (newproof, newmetasenv''') =
293 let subst_in = CicUnification.apply_subst ((metano,bo')::subst) in
294 subst_meta_and_metasenv_in_proof
295 proof metano subst_in newmetasenv''
297 (newproof, List.map (function (i,_,_) -> i) new_uninstantiatedmetas)
299 (* TODO per implementare i tatticali e' necessario che tutte le tattiche
300 sollevino _solamente_ Fail *)
301 let apply_tac ~term ~status =
303 apply_tac ~term ~status
304 (* TODO cacciare anche altre eccezioni? *)
305 with CicUnification.UnificationFailed as e ->
306 raise (Fail (Printexc.to_string e))
308 let intros_tac ~mknames ~status:(proof, goal) =
309 let module C = Cic in
310 let module R = CicReduction in
311 let (_,metasenv,_,_) = proof in
312 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
313 let newmeta = new_meta ~proof in
314 let (context',ty',bo') = lambda_abstract context newmeta ty mknames in
316 subst_meta_in_proof proof metano bo' [newmeta,context',ty']
318 (newproof, [newmeta])
320 let cut_tac ~term ~status:(proof, goal) =
321 let module C = Cic in
322 let curi,metasenv,pbo,pty = proof in
323 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
324 let newmeta1 = new_meta ~proof in
325 let newmeta2 = newmeta1 + 1 in
326 let context_for_newmeta1 =
327 (Some (C.Name "dummy_for_cut",C.Decl term))::context in
329 identity_relocation_list_for_metavariable context_for_newmeta1 in
330 let irl2 = identity_relocation_list_for_metavariable context in
331 let newmeta1ty = CicSubstitution.lift 1 ty in
334 [C.Lambda (C.Name "dummy_for_cut",term,C.Meta (newmeta1,irl1)) ;
335 C.Meta (newmeta2,irl2)]
338 subst_meta_in_proof proof metano bo'
339 [newmeta2,context,term; newmeta1,context_for_newmeta1,newmeta1ty];
341 (newproof, [newmeta1 ; newmeta2])
343 let letin_tac ~term ~status:(proof, goal) =
344 let module C = Cic in
345 let curi,metasenv,pbo,pty = proof in
346 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
347 let _ = CicTypeChecker.type_of_aux' metasenv context term in
348 let newmeta = new_meta ~proof in
349 let context_for_newmeta =
350 (Some (C.Name "dummy_for_letin",C.Def term))::context in
352 identity_relocation_list_for_metavariable context_for_newmeta in
353 let newmetaty = CicSubstitution.lift 1 ty in
354 let bo' = C.LetIn (C.Name "dummy_for_letin",term,C.Meta (newmeta,irl)) in
357 proof metano bo'[newmeta,context_for_newmeta,newmetaty]
359 (newproof, [newmeta])
361 (** functional part of the "exact" tactic *)
362 let exact_tac ~term ~status:(proof, goal) =
363 (* Assumption: the term bo must be closed in the current context *)
364 let (_,metasenv,_,_) = proof in
365 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
366 let module T = CicTypeChecker in
367 let module R = CicReduction in
368 if R.are_convertible context (T.type_of_aux' metasenv context term) ty then
370 let (newproof, metasenv') =
371 subst_meta_in_proof proof metano term [] in
375 raise (Fail "The type of the provided term is not the one expected.")
378 (* not really "primitive" tactics .... *)
380 let elim_tac ~term ~status:(proof, goal) =
381 let module T = CicTypeChecker in
382 let module U = UriManager in
383 let module R = CicReduction in
384 let module C = Cic in
385 let (curi,metasenv,_,_) = proof in
386 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
387 let termty = T.type_of_aux' metasenv context term in
388 let uri,exp_named_subst,typeno,args =
390 C.MutInd (uri,typeno,exp_named_subst) -> (uri,exp_named_subst,typeno,[])
391 | C.Appl ((C.MutInd (uri,typeno,exp_named_subst))::args) ->
392 (uri,exp_named_subst,typeno,args)
393 | _ -> raise NotAnInductiveTypeToEliminate
396 let buri = U.buri_of_uri uri in
398 match CicEnvironment.get_obj uri with
399 C.InductiveDefinition (tys,_,_) ->
400 let (name,_,_,_) = List.nth tys typeno in
405 match T.type_of_aux' metasenv context ty with
406 C.Sort C.Prop -> "_ind"
407 | C.Sort C.Set -> "_rec"
408 | C.Sort C.Type -> "_rect"
411 U.uri_of_string (buri ^ "/" ^ name ^ ext ^ ".con")
413 let eliminator_ref = C.Const (eliminator_uri,exp_named_subst) in
414 let ety = T.type_of_aux' metasenv context eliminator_ref in
415 let newmeta = new_meta ~proof in
416 let (econclusion,newmetas,arguments,lastmeta) =
417 new_metasenv_for_apply newmeta proof context ety
419 (* Here we assume that we have only one inductive hypothesis to *)
420 (* eliminate and that it is the last hypothesis of the theorem. *)
421 (* A better approach would be fingering the hypotheses in some *)
424 let (_,canonical_context,_) =
425 List.find (function (m,_,_) -> m=(lastmeta - 1)) newmetas
428 identity_relocation_list_for_metavariable canonical_context
430 Cic.Meta (lastmeta - 1, irl)
432 let newmetasenv = newmetas @ metasenv in
433 let subst1,newmetasenv' =
434 CicUnification.fo_unif newmetasenv context term meta_of_corpse
436 let ueconclusion = CicUnification.apply_subst subst1 econclusion in
437 (* The conclusion of our elimination principle is *)
438 (* (?i farg1 ... fargn) *)
439 (* The conclusion of our goal is ty. So, we can *)
440 (* eta-expand ty w.r.t. farg1 .... fargn to get *)
441 (* a new ty equal to (P farg1 ... fargn). Now *)
442 (* ?i can be instantiated with P and we are ready *)
443 (* to refine the term. *)
445 match ueconclusion with
446 C.Appl ((C.Meta (emeta,_))::fargs) -> emeta,fargs
447 | C.Meta (emeta,_) -> emeta,[]
448 | _ -> raise NotTheRightEliminatorShape
450 let ty' = CicUnification.apply_subst subst1 ty in
451 let eta_expanded_ty =
452 (*CSC: newmetasenv' era metasenv ??????????? *)
453 List.fold_left (eta_expand newmetasenv' context) ty' fargs
455 let subst2,newmetasenv'' =
456 (*CSC: passo newmetasenv', ma alcune variabili sono gia' state sostituite
457 da subst1!!!! Dovrei rimuoverle o sono innocue?*)
458 CicUnification.fo_unif
459 newmetasenv' context ueconclusion eta_expanded_ty
461 let in_subst_domain i =
462 let eq_to_i = function (j,_) -> i=j in
463 List.exists eq_to_i subst1 ||
464 List.exists eq_to_i subst2
466 (* When unwinding the META that corresponds to the elimination *)
467 (* predicate (which is emeta), we must also perform one-step *)
468 (* beta-reduction. apply_subst doesn't need the context. Hence *)
469 (* the underscore. *)
470 let apply_subst _ t =
471 let t' = CicUnification.apply_subst subst1 t in
472 CicUnification.apply_subst_reducing
473 subst2 (Some (emeta,List.length fargs)) t'
475 let old_uninstantiatedmetas,new_uninstantiatedmetas =
476 classify_metas newmeta in_subst_domain apply_subst
479 let arguments' = List.map (apply_subst context) arguments in
480 let bo' = Cic.Appl (eliminator_ref::arguments') in
482 new_uninstantiatedmetas@old_uninstantiatedmetas
484 let (newproof, newmetasenv'''') =
485 (* When unwinding the META that corresponds to the *)
486 (* elimination predicate (which is emeta), we must *)
487 (* also perform one-step beta-reduction. *)
488 (* The only difference w.r.t. apply_subst is that *)
489 (* we also substitute metano with bo'. *)
490 (*CSC: Nota: sostituire nuovamente subst1 e' superfluo, *)
493 let t' = CicUnification.apply_subst subst1 t in
494 CicUnification.apply_subst_reducing
495 ((metano,bo')::subst2)
496 (Some (emeta,List.length fargs)) t'
498 subst_meta_and_metasenv_in_proof
499 proof metano apply_subst' newmetasenv'''
502 List.map (function (i,_,_) -> i) new_uninstantiatedmetas)
505 let elim_simpl_intros_tac ~term =
506 Tacticals.then_ ~start:(elim_tac ~term)
509 ~start:(ReductionTactics.simpl_tac ~also_in_hypotheses:false ~term:None)
510 ~continuation:(intros_tac ~mknames:(function () -> "FOO")))
514 exception NotConvertible
516 (*CSC: Bug (or feature?). [with_what] is parsed in the context of the goal, *)
517 (*CSC: while [what] can have a richer context (because of binders) *)
518 (*CSC: So it is _NOT_ possible to use those binders in the [with_what] term. *)
519 (*CSC: Is that evident? Is that right? Or should it be changed? *)
520 let change_tac ~what ~with_what ~status:(proof, goal) =
521 let curi,metasenv,pbo,pty = proof in
522 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
523 (* are_convertible works only on well-typed terms *)
524 ignore (CicTypeChecker.type_of_aux' metasenv context with_what) ;
525 if CicReduction.are_convertible context what with_what then
528 ProofEngineReduction.replace ~equality:(==) ~what ~with_what
530 let ty' = replace ty in
534 Some (name,Cic.Def t) -> Some (name,Cic.Def (replace t))
535 | Some (name,Cic.Decl t) -> Some (name,Cic.Decl (replace t))
542 (n,_,_) when n = metano -> (metano,context',ty')
546 (curi,metasenv',pbo,pty), [metano]
549 raise (ProofEngineTypes.Fail "Not convertible")