aboutsummaryrefslogtreecommitdiffhomepage
path: root/checker/inductive.ml
blob: 05ab5a846bbe7b1f1fffacc312205372e5ddf7ed (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

(* $Id: inductive.ml 10172 2007-10-04 13:02:03Z herbelin $ *)

open Util
open Names
open Univ
open Term
open Reduction
open Type_errors
open Declarations
open Environ

let inductive_of_constructor = fst
let index_of_constructor = snd
let ith_constructor_of_inductive ind i = (ind,i)

type mind_specif = mutual_inductive_body * one_inductive_body

(* raise Not_found if not an inductive type *)
let lookup_mind_specif env (kn,tyi) =
  let mib = lookup_mind kn env in
  if tyi >= Array.length mib.mind_packets then
    error "Inductive.lookup_mind_specif: invalid inductive index";
  (mib, mib.mind_packets.(tyi))

let find_rectype env c =
  let (t, l) = decompose_app (whd_betadeltaiota env c) in
  match t with
  | Ind ind -> (ind, l)
  | _ -> raise Not_found

let find_inductive env c =
  let (t, l) = decompose_app (whd_betadeltaiota env c) in
  match t with
    | Ind ind
        when (fst (lookup_mind_specif env ind)).mind_finite -> (ind, l)
    | _ -> raise Not_found

let find_coinductive env c =
  let (t, l) = decompose_app (whd_betadeltaiota env c) in
  match t with
    | Ind ind
        when not (fst (lookup_mind_specif env ind)).mind_finite -> (ind, l)
    | _ -> raise Not_found

let inductive_params (mib,_) = mib.mind_nparams

(************************************************************************)

(* Build the substitution that replaces Rels by the appropriate *)
(* inductives *)
let ind_subst mind mib =
  let ntypes = mib.mind_ntypes in
  let make_Ik k = Ind (mind,ntypes-k-1) in 
  list_tabulate make_Ik ntypes

(* Instantiate inductives in constructor type *)
let constructor_instantiate mind mib c =
  let s = ind_subst mind mib in
  substl s c

let instantiate_params full t args sign =
  let fail () = 
    anomaly "instantiate_params: type, ctxt and args mismatch" in
  let (rem_args, subs, ty) =
    fold_rel_context
      (fun (_,copt,_) (largs,subs,ty) ->
        match (copt, largs, ty) with
          | (None, a::args, Prod(_,_,t)) -> (args, a::subs, t)
          | (Some b,_,LetIn(_,_,_,t))    -> (largs, (substl subs b)::subs, t)
	  | (_,[],_)                -> if full then fail() else ([], subs, ty)
	  | _                       -> fail ())
      sign
      ~init:(args,[],t) 
  in
  if rem_args <> [] then fail();
  substl subs ty

let instantiate_partial_params = instantiate_params false

let full_inductive_instantiate mib params sign =
  let dummy = Prop Null in
  let t = mkArity (sign,dummy) in
  fst (destArity (instantiate_params true t params mib.mind_params_ctxt))

let full_constructor_instantiate ((mind,_),(mib,_),params) =
  let inst_ind = constructor_instantiate mind mib in
  (fun t ->
    instantiate_params true (inst_ind t) params mib.mind_params_ctxt)

(************************************************************************)
(************************************************************************)

(* Functions to build standard types related to inductive *)


let number_of_inductives mib = Array.length mib.mind_packets
let number_of_constructors mip = Array.length mip.mind_consnames

(* 
Computing the actual sort of an applied or partially applied inductive type:

I_i: forall uniformparams:utyps, forall otherparams:otyps, Type(a)
uniformargs : utyps 
otherargs : otyps
I_1:forall ...,s_1;...I_n:forall ...,s_n |- sort(C_kj(uniformargs)) = s_kj
s'_k = max(..s_kj..)
merge(..s'_k..) = ..s''_k..
--------------------------------------------------------------------
Gamma |- I_i uniformargs otherargs : phi(s''_i)

where

- if p=0, phi() = Prop
- if p=1, phi(s) = s
- if p<>1, phi(s) = sup(Set,s)

Remark: Set (predicative) is encoded as Type(0)
*)

let sort_as_univ = function
| Type u -> u
| Prop Null -> type0m_univ
| Prop Pos -> type0_univ

let cons_subst u su subst =
  try (u, sup su (List.assoc u subst)) :: List.remove_assoc u subst
  with Not_found -> (u, su) :: subst

let actualize_decl_level env lev t =
  let sign,s = dest_arity env t in
  mkArity (sign,lev)

let polymorphism_on_non_applied_parameters = false

(* Bind expected levels of parameters to actual levels *)
(* Propagate the new levels in the signature *)
let rec make_subst env = function
  | (_,Some _,_ as t)::sign, exp, args ->
      let ctx,subst = make_subst env (sign, exp, args) in
      t::ctx, subst
  | d::sign, None::exp, args ->
      let args = match args with _::args -> args | [] -> [] in
      let ctx,subst = make_subst env (sign, exp, args) in
      d::ctx, subst
  | d::sign, Some u::exp, a::args ->
      (* We recover the level of the argument, but we don't change the *)
      (* level in the corresponding type in the arity; this level in the *)
      (* arity is a global level which, at typing time, will be enforce *)
      (* to be greater than the level of the argument; this is probably *)
      (* a useless extra constraint *)
      let s = sort_as_univ (snd (dest_arity env a)) in
      let ctx,subst = make_subst env (sign, exp, args) in
      d::ctx, cons_subst u s subst
  | (na,None,t as d)::sign, Some u::exp, [] ->
      (* No more argument here: we instantiate the type with a fresh level *)
      (* which is first propagated to the corresponding premise in the arity *)
      (* (actualize_decl_level), then to the conclusion of the arity (via *)
      (* the substitution) *)
      let ctx,subst = make_subst env (sign, exp, []) in
      if polymorphism_on_non_applied_parameters then
	let s = fresh_local_univ () in
	let t = actualize_decl_level env (Type s) t in
	(na,None,t)::ctx, cons_subst u s subst
      else
	d::ctx, subst
  | sign, [], _ ->
      (* Uniform parameters are exhausted *)
      sign,[]
  | [], _, _ ->
      assert false

let instantiate_universes env ctx ar argsorts =
  let args = Array.to_list argsorts in
  let ctx,subst = make_subst env (ctx,ar.poly_param_levels,args) in
  let level = subst_large_constraints subst ar.poly_level in
  ctx,
  if is_type0m_univ level then Prop Null
  else if is_type0_univ level then Prop Pos
  else Type level

let type_of_inductive_knowing_parameters env mip paramtyps =
  match mip.mind_arity with
  | Monomorphic s ->
      s.mind_user_arity
  | Polymorphic ar ->
      let ctx = List.rev mip.mind_arity_ctxt in
      let ctx,s = instantiate_universes env ctx ar paramtyps in
      mkArity (List.rev ctx,s)

(* Type of a (non applied) inductive type *)

let type_of_inductive env (_,mip) =
  type_of_inductive_knowing_parameters env mip [||]

(* The max of an array of universes *)

let cumulate_constructor_univ u = function
  | Prop Null -> u
  | Prop Pos -> sup type0_univ u
  | Type u' -> sup u u'

let max_inductive_sort =
  Array.fold_left cumulate_constructor_univ type0m_univ

(************************************************************************)
(* Type of a constructor *)

let type_of_constructor cstr (mib,mip) =
  let ind = inductive_of_constructor cstr in
  let specif = mip.mind_user_lc in
  let i = index_of_constructor cstr in
  let nconstr = Array.length mip.mind_consnames in
  if i > nconstr then error "Not enough constructors in the type";
  constructor_instantiate (fst ind) mib specif.(i-1)

let arities_of_specif kn (mib,mip) = 
  let specif = mip.mind_nf_lc in
  Array.map (constructor_instantiate kn mib) specif



(************************************************************************)

let error_elim_expln kp ki =
  match kp,ki with
  | (InType | InSet), InProp -> NonInformativeToInformative
  | InType, InSet -> StrongEliminationOnNonSmallType (* if Set impredicative *)
  | _ -> WrongArity

(* Type of case predicates *)

(* Get type of inductive, with parameters instantiated *)

let inductive_sort_family mip =
  match mip.mind_arity with
   | Monomorphic s -> family_of_sort s.mind_sort 
   | Polymorphic _ -> InType

let mind_arity mip =
  mip.mind_arity_ctxt, inductive_sort_family mip

let get_instantiated_arity (mib,mip) params =
  let sign, s = mind_arity mip in
  full_inductive_instantiate mib params sign, s

let elim_sorts (_,mip) = mip.mind_kelim

let rel_list n m = 
  let rec reln l p = 
    if p>m then l else reln (Rel(n+p)::l) (p+1)
  in 
  reln [] 1

let build_dependent_inductive ind (_,mip) params =
  let nrealargs = mip.mind_nrealargs in
  applist 
    (Ind ind, (List.map (lift nrealargs) params)@(rel_list 0 nrealargs))

(* This exception is local *)
exception LocalArity of (sorts_family * sorts_family * arity_error) option

let check_allowed_sort ksort specif =
  if not (List.exists ((=) ksort) (elim_sorts specif)) then 
    let s = inductive_sort_family (snd specif) in
    raise (LocalArity (Some(ksort,s,error_elim_expln ksort s)))

let is_correct_arity env c (p,pj) ind specif params = 
  let arsign,_ = get_instantiated_arity specif params in
  let rec srec env pt ar =
    let pt' = whd_betadeltaiota env pt in
    match pt', ar with
      | Prod (na1,a1,t), (_,None,a1')::ar' ->
          (try conv env a1 a1'
          with NotConvertible -> raise (LocalArity None));
          srec (push_rel (na1,None,a1) env) t ar'
      | Prod (_,a1,a2), [] -> (* whnf of t was not needed here! *)
          let ksort = match (whd_betadeltaiota env a2) with
            | Sort s -> family_of_sort s 
	    | _ -> raise (LocalArity None) in
	  let dep_ind = build_dependent_inductive ind specif params in 
          (try conv env a1 dep_ind
          with NotConvertible -> raise (LocalArity None));
	  check_allowed_sort ksort specif;
	  true
      | Sort s', [] ->
	  check_allowed_sort (family_of_sort s') specif;
	  false
      | _ ->
	  raise (LocalArity None)
  in 
  try srec env pj (List.rev arsign)
  with LocalArity kinds ->
    error_elim_arity env ind (elim_sorts specif) c (p,pj) kinds


(************************************************************************)
(* Type of case branches *)

(* [p] is the predicate, [i] is the constructor number (starting from 0),
   and [cty] is the type of the constructor (params not instantiated) *)
let build_branches_type ind (_,mip as specif) params dep p =
  let build_one_branch i cty =
    let typi = full_constructor_instantiate (ind,specif,params) cty in
    let (args,ccl) = decompose_prod_assum typi in
    let nargs = rel_context_length args in
    let (_,allargs) = decompose_app ccl in
    let (lparams,vargs) = list_chop (inductive_params specif) allargs in
    let cargs =
      if dep then
        let cstr = ith_constructor_of_inductive ind (i+1) in
        let dep_cstr =
          applist (Construct cstr,lparams@extended_rel_list 0 args) in
        vargs @ [dep_cstr]
      else
        vargs in
    let base = beta_appvect (lift nargs p) (Array.of_list cargs) in
    it_mkProd_or_LetIn base args in
  Array.mapi build_one_branch mip.mind_nf_lc

(* [p] is the predicate, [c] is the match object, [realargs] is the
   list of real args of the inductive type *)
let build_case_type dep p c realargs =
  let args = if dep then realargs@[c] else realargs in
  beta_appvect p (Array.of_list args)

let type_case_branches env (ind,largs) (p,pj) c =
  let specif = lookup_mind_specif env ind in 
  let nparams = inductive_params specif in
  let (params,realargs) = list_chop nparams largs in
  let dep = is_correct_arity env c (p,pj) ind specif params in
  let lc = build_branches_type ind specif params dep p in
  let ty = build_case_type dep p c realargs in
  (lc, ty)


(************************************************************************)
(* Checking the case annotation is relevent *)

let check_case_info env indsp ci =
  let (mib,mip) = lookup_mind_specif env indsp in
  if
    not (mind_equiv env indsp ci.ci_ind) or
    (mib.mind_nparams <> ci.ci_npar) or
    (mip.mind_consnrealdecls <> ci.ci_cstr_nargs)
  then raise (TypeError(env,WrongCaseInfo(indsp,ci)))

(************************************************************************)
(************************************************************************)

(* Guard conditions for fix and cofix-points *)

(* Check if t is a subterm of Rel n, and gives its specification, 
   assuming lst already gives index of
   subterms with corresponding specifications of recursive arguments *)

(* A powerful notion of subterm *)

(* To each inductive definition corresponds an array describing the
   structure of recursive arguments for each constructor, we call it
   the recursive spec of the type (it has type recargs vect).  For
   checking the guard, we start from the decreasing argument (Rel n)
   with its recursive spec.  During checking the guardness condition,
   we collect patterns variables corresponding to subterms of n, each
   of them with its recursive spec.  They are organised in a list lst
   of type (int * recargs) list which is sorted with respect to the
   first argument.
*)

(*************************************************************)
(* Environment annotated with marks on recursive arguments *)

(* tells whether it is a strict or loose subterm *)
type size = Large | Strict

(* merging information *)
let size_glb s1 s2 =
  match s1,s2 with
      Strict, Strict -> Strict
    | _ -> Large

(* possible specifications for a term:
   - Not_subterm: when the size of a term is not related to the
     recursive argument of the fixpoint
   - Subterm: when the term is a subterm of the recursive argument
       the wf_paths argument specifies which subterms are recursive
   - Dead_code: when the term has been built by elimination over an
       empty type
 *)

type subterm_spec =
    Subterm of (size * wf_paths)
  | Dead_code
  | Not_subterm

let spec_of_tree t =
  if Rtree.eq_rtree (=) t mk_norec then Not_subterm else Subterm(Strict,t)

let subterm_spec_glb =
  let glb2 s1 s2 =
    match s1,s2 with
        _, Dead_code -> s1
      | Dead_code, _ -> s2
      | Not_subterm, _ -> Not_subterm
      | _, Not_subterm -> Not_subterm
      | Subterm (a1,t1), Subterm (a2,t2) ->
          if Rtree.eq_rtree (=) t1 t2 then Subterm (size_glb a1 a2, t1)
          (* branches do not return objects with same spec *)
          else Not_subterm in
  Array.fold_left glb2 Dead_code
          
type guard_env =
  { env     : env;
    (* dB of last fixpoint *)
    rel_min : int;
    (* inductive of recarg of each fixpoint *)
    inds    : inductive array;
    (* the recarg information of inductive family *)
    recvec  : wf_paths array;
    (* dB of variables denoting subterms *)
    genv    : subterm_spec list;
  }

let make_renv env minds recarg (kn,tyi) =
  let mib = lookup_mind kn env in
  let mind_recvec =
    Array.map (fun mip -> mip.mind_recargs) mib.mind_packets in
  { env = env;
    rel_min = recarg+2;
    inds = minds;
    recvec = mind_recvec;
    genv = [Subterm(Large,mind_recvec.(tyi))] }

let push_var renv (x,ty,spec) =
  { renv with 
    env = push_rel (x,None,ty) renv.env;
    rel_min = renv.rel_min+1;
    genv = spec:: renv.genv }

let assign_var_spec renv (i,spec) =
  { renv with genv = list_assign renv.genv (i-1) spec }

let push_var_renv renv (x,ty) =
  push_var renv (x,ty,Not_subterm)

(* Fetch recursive information about a variable p *)
let subterm_var p renv = 
  try List.nth renv.genv (p-1)
  with Failure _ | Invalid_argument _ -> Not_subterm

(* Add a variable and mark it as strictly smaller with information [spec]. *)
let add_subterm renv (x,a,spec) =
  push_var renv (x,a,spec_of_tree spec)

let push_ctxt_renv renv ctxt =
  let n = rel_context_length ctxt in
  { renv with 
    env = push_rel_context ctxt renv.env;
    rel_min = renv.rel_min+n;
    genv = iterate (fun ge -> Not_subterm::ge) n renv.genv }

let push_fix_renv renv (_,v,_ as recdef) =
  let n = Array.length v in
  { renv with
    env = push_rec_types recdef renv.env;
    rel_min = renv.rel_min+n;
    genv = iterate (fun ge -> Not_subterm::ge) n renv.genv }


(******************************)
(* Computing the recursive subterms of a term (propagation of size
   information through Cases). *)

(*
   c is a branch of an inductive definition corresponding to the spec
   lrec.  mind_recvec is the recursive spec of the inductive
   definition of the decreasing argument n.

   case_branches_specif renv lrec lc will pass the lambdas
   of c corresponding to pattern variables and collect possibly new
   subterms variables and returns the bodies of the branches with the
   correct envs and decreasing args.
*)

let lookup_subterms env ind =
  let (_,mip) = lookup_mind_specif env ind in
  mip.mind_recargs

(*********************************)

(* Propagation of size information through Cases: if the matched
   object is a recursive subterm then compute the information
   associated to its own subterms.
   Rq: if branch is not eta-long, then the recursive information
   is not propagated to the missing abstractions *)
let case_branches_specif renv c_spec ind lbr = 
  let rec push_branch_args renv lrec c = 
    match lrec with
        ra::lr ->
          let c' = whd_betadeltaiota renv.env c in
          (match c' with
              Lambda(x,a,b) ->
                let renv' = push_var renv (x,a,ra) in
                push_branch_args renv' lr b
            | _ -> (* branch not in eta-long form: cannot perform rec. calls *)
                (renv,c'))
      | [] -> (renv, c) in
  match c_spec with
      Subterm (_,t) ->
        let sub_spec = Array.map (List.map spec_of_tree) (dest_subterms t) in
        assert (Array.length sub_spec = Array.length lbr);
        array_map2 (push_branch_args renv) sub_spec lbr
    | Dead_code -> 
        let t = dest_subterms (lookup_subterms renv.env ind) in
        let sub_spec = Array.map (List.map (fun _ -> Dead_code)) t in
        assert (Array.length sub_spec = Array.length lbr);
        array_map2 (push_branch_args renv) sub_spec lbr
    | Not_subterm -> Array.map (fun c -> (renv,c)) lbr

(* [subterm_specif renv t] computes the recursive structure of [t] and
   compare its size with the size of the initial recursive argument of
   the fixpoint we are checking. [renv] collects such information
   about variables.
*)

let rec subterm_specif renv t = 
  (* maybe reduction is not always necessary! *)
  let f,l = decompose_app (whd_betadeltaiota renv.env t) in
  match f with 
    | Rel k -> subterm_var k renv

    | Case (ci,_,c,lbr) ->
        if Array.length lbr = 0 then Dead_code
        else
          let c_spec = subterm_specif renv c in
          let lbr_spec = case_branches_specif renv c_spec ci.ci_ind lbr in
          let stl  =
            Array.map (fun (renv',br') -> subterm_specif renv' br')
              lbr_spec in
          subterm_spec_glb stl
	       
    | Fix ((recindxs,i),(_,typarray,bodies as recdef)) ->
(* when proving that the fixpoint f(x)=e is less than n, it is enough
   to prove that e is less than n assuming f is less than n
   furthermore when f is applied to a term which is strictly less than
   n, one may assume that x itself is strictly less than n
*)
        let (ctxt,clfix) = dest_prod renv.env typarray.(i) in
        let oind =
          let env' = push_rel_context ctxt renv.env in
          try Some(fst(find_inductive env' clfix))
          with Not_found -> None in
        (match oind with
            None -> Not_subterm (* happens if fix is polymorphic *)
          | Some ind ->
              let nbfix = Array.length typarray in
              let recargs = lookup_subterms renv.env ind in
              (* pushing the fixpoints *)
              let renv' = push_fix_renv renv recdef in
              let renv' =
                (* Why Strict here ? To be general, it could also be
                   Large... *)
                assign_var_spec renv' (nbfix-i, Subterm(Strict,recargs)) in
              let decrArg = recindxs.(i) in 
              let theBody = bodies.(i)   in
              let nbOfAbst = decrArg+1 in
              let sign,strippedBody = decompose_lam_n_assum nbOfAbst theBody in
              (* pushing the fix parameters *)
              let renv'' = push_ctxt_renv renv' sign in
              let renv'' =
                if List.length l < nbOfAbst then renv''
                else
                  let theDecrArg  = List.nth l decrArg in
	          let arg_spec    = subterm_specif renv theDecrArg in
                  assign_var_spec renv'' (1, arg_spec) in
	      subterm_specif renv'' strippedBody)

    | Lambda (x,a,b) -> 
        assert (l=[]);
        subterm_specif (push_var_renv renv (x,a)) b

    (* Metas and evars are considered OK *)
    | (Meta _|Evar _) -> Dead_code

    (* Other terms are not subterms *)
    | _ -> Not_subterm


(* Check term c can be applied to one of the mutual fixpoints. *)
let check_is_subterm renv c = 
  match subterm_specif renv c with
    Subterm (Strict,_) | Dead_code -> true
  |  _ -> false

(************************************************************************)

exception FixGuardError of env * guard_error

let error_illegal_rec_call renv fx arg =
  let (_,le_vars,lt_vars) =
    List.fold_left
      (fun (i,le,lt) sbt ->
        match sbt with
            (Subterm(Strict,_) | Dead_code) -> (i+1, le, i::lt)
          | (Subterm(Large,_)) -> (i+1, i::le, lt)
          | _ -> (i+1, le ,lt))
      (1,[],[]) renv.genv in
  raise (FixGuardError (renv.env,
                        RecursionOnIllegalTerm(fx,arg,le_vars,lt_vars)))

let error_partial_apply renv fx =
  raise (FixGuardError (renv.env,NotEnoughArgumentsForFixCall fx))

(* Check if [def] is a guarded fixpoint body with decreasing arg.
   given [recpos], the decreasing arguments of each mutually defined
   fixpoint. *)
let check_one_fix renv recpos def =
  let nfi = Array.length recpos in 

  (* Checks if [t] only make valid recursive calls *)
  let rec check_rec_call renv t = 
    (* if [t] does not make recursive calls, it is guarded: *)
    if noccur_with_meta renv.rel_min nfi t then ()
    else
      let (f,l) = decompose_app (whd_betaiotazeta renv.env t) in
      match f with
        | Rel p -> 
            (* Test if [p] is a fixpoint (recursive call) *) 
	    if renv.rel_min <= p & p < renv.rel_min+nfi then
              begin
                List.iter (check_rec_call renv) l;
                (* the position of the invoked fixpoint: *) 
	        let glob = renv.rel_min+nfi-1-p in
                (* the decreasing arg of the rec call: *)
	        let np = recpos.(glob) in
                if List.length l <= np then error_partial_apply renv glob
                else
                  (* Check the decreasing arg is smaller *)
                  let z = List.nth l np in
	          if not (check_is_subterm renv z) then
                    error_illegal_rec_call renv glob z
              end
            else
              begin
                match pi2 (lookup_rel p renv.env) with
                | None ->
                    List.iter (check_rec_call renv) l
                | Some c ->
                    try List.iter (check_rec_call renv) l
                    with FixGuardError _ -> check_rec_call renv (applist(c,l))
              end

        | Case (ci,p,c_0,lrest) ->
            List.iter (check_rec_call renv) (c_0::p::l);
            (* compute the recarg information for the arguments of
               each branch *)
            let c_spec = subterm_specif renv c_0 in
            let lbr = case_branches_specif renv c_spec ci.ci_ind lrest in
            Array.iter (fun (renv',br') -> check_rec_call renv' br') lbr

        (* Enables to traverse Fixpoint definitions in a more intelligent
           way, ie, the rule :
           if - g = Fix g/p := [y1:T1]...[yp:Tp]e &
              - f is guarded with respect to the set of pattern variables S 
                in a1 ... am        &
              - f is guarded with respect to the set of pattern variables S 
                in T1 ... Tp        &
              - ap is a sub-term of the formal argument of f &
              - f is guarded with respect to the set of pattern variables
                S+{yp} in e
           then f is guarded with respect to S in (g a1 ... am).
           Eduardo 7/9/98 *)

        | Fix ((recindxs,i),(_,typarray,bodies as recdef)) ->
            List.iter (check_rec_call renv) l;
            Array.iter (check_rec_call renv) typarray;
            let decrArg = recindxs.(i) in
            let renv' = push_fix_renv renv recdef in 
            if (List.length l < (decrArg+1)) then
	      Array.iter (check_rec_call renv') bodies
            else 
              Array.iteri
                (fun j body ->
                  if i=j then
                    let theDecrArg  = List.nth l decrArg in
	            let arg_spec = subterm_specif renv theDecrArg in
	            check_nested_fix_body renv' (decrArg+1) arg_spec body
                  else check_rec_call renv' body)
                bodies

        | Const kn -> 
            if evaluable_constant kn renv.env then 
              try List.iter (check_rec_call renv) l
              with (FixGuardError _ ) ->
	        check_rec_call renv(applist(constant_value renv.env kn, l))
	    else List.iter (check_rec_call renv) l

        (* The cases below simply check recursively the condition on the
           subterms *)
        | Cast (a,_, b) -> 
            List.iter (check_rec_call renv) (a::b::l)

        | Lambda (x,a,b) ->
            List.iter (check_rec_call renv) (a::l);
            check_rec_call (push_var_renv renv (x,a)) b

        | Prod (x,a,b) -> 
            List.iter (check_rec_call renv) (a::l);
            check_rec_call (push_var_renv renv (x,a)) b

        | CoFix (i,(_,typarray,bodies as recdef)) ->
            List.iter (check_rec_call renv) l;
	    Array.iter (check_rec_call renv) typarray;
	    let renv' = push_fix_renv renv recdef in
	    Array.iter (check_rec_call renv') bodies

        | (Ind _ | Construct _ | Sort _) ->
            List.iter (check_rec_call renv) l

        | Var id ->
            begin
              match pi2 (lookup_named id renv.env) with
              | None ->
                  List.iter (check_rec_call renv) l
              | Some c ->
                  try List.iter (check_rec_call renv) l
                  with (FixGuardError _) -> check_rec_call renv (applist(c,l))
            end

        (* l is not checked because it is considered as the meta's context *)
        | (Evar _ | Meta _) -> ()

        | (App _|LetIn _) -> assert false (* beta zeta reduction *)

  and check_nested_fix_body renv decr recArgsDecrArg body =
    if decr = 0 then
      check_rec_call (assign_var_spec renv (1,recArgsDecrArg)) body
    else
      match body with
	| Lambda (x,a,b) ->
	    check_rec_call renv a;
            let renv' = push_var_renv renv (x,a) in
	    check_nested_fix_body renv' (decr-1) recArgsDecrArg b
	| _ -> anomaly "Not enough abstractions in fix body"

  in
  check_rec_call renv def


let inductive_of_mutfix env ((nvect,bodynum),(names,types,bodies as recdef)) =
  let nbfix = Array.length bodies in 
  if nbfix = 0
    or Array.length nvect <> nbfix 
    or Array.length types <> nbfix
    or Array.length names <> nbfix
    or bodynum < 0
    or bodynum >= nbfix
  then anomaly "Ill-formed fix term";
  let fixenv = push_rec_types recdef env in
  let raise_err env i err =
    error_ill_formed_rec_body env err names i in
  (* Check the i-th definition with recarg k *)
  let find_ind i k def = 
    (* check fi does not appear in the k+1 first abstractions, 
       gives the type of the k+1-eme abstraction (must be an inductive)  *)
    let rec check_occur env n def = 
      match (whd_betadeltaiota env def) with
        | Lambda (x,a,b) -> 
	    if noccur_with_meta n nbfix a then
	      let env' = push_rel (x, None, a) env in
              if n = k+1 then
                (* get the inductive type of the fixpoint *)
                let (mind, _) = 
                  try find_inductive env a 
                  with Not_found ->
		    raise_err env i (RecursionNotOnInductiveType a) in
                (mind, (env', b))
	      else check_occur env' (n+1) b
            else anomaly "check_one_fix: Bad occurrence of recursive call"
        | _ -> raise_err env i NotEnoughAbstractionInFixBody in
    check_occur fixenv 1 def in
  (* Do it on every fixpoint *)
  let rv = array_map2_i find_ind nvect bodies in
  (Array.map fst rv, Array.map snd rv)


let check_fix env ((nvect,_),(names,_,bodies as _recdef) as fix) =
  let (minds, rdef) = inductive_of_mutfix env fix in
  for i = 0 to Array.length bodies - 1 do
    let (fenv,body) = rdef.(i) in
    let renv = make_renv fenv minds nvect.(i) minds.(i) in
    try check_one_fix renv nvect body
    with FixGuardError (fixenv,err) ->
      error_ill_formed_rec_body fixenv err names i
  done

(*
let cfkey = Profile.declare_profile "check_fix";;
let check_fix env fix = Profile.profile3 cfkey check_fix env fix;;
*)

(************************************************************************)
(* Co-fixpoints. *)

exception CoFixGuardError of env * guard_error

let anomaly_ill_typed () =
  anomaly "check_one_cofix: too many arguments applied to constructor"

let rec codomain_is_coind env c =
  let b = whd_betadeltaiota env c in
  match b with
    | Prod (x,a,b) ->
	codomain_is_coind (push_rel (x, None, a) env) b 
    | _ -> 
	(try find_coinductive env b
        with Not_found ->
	  raise (CoFixGuardError (env, CodomainNotInductiveType b)))

let check_one_cofix env nbfix def deftype = 
  let rec check_rec_call env alreadygrd n vlra  t =
    if not (noccur_with_meta n nbfix t) then
      let c,args = decompose_app (whd_betadeltaiota env t) in
      match c with 
	| Rel p when  n <= p && p < n+nbfix ->
	    (* recursive call: must be guarded and no nested recursive
               call allowed *)
            if not alreadygrd then
	      raise (CoFixGuardError (env,UnguardedRecursiveCall t))
            else if not(List.for_all (noccur_with_meta n nbfix) args) then
	      raise (CoFixGuardError (env,NestedRecursiveOccurrences))
		 
	| Construct (_,i as cstr_kn)  ->
            let lra = vlra.(i-1) in 
            let mI = inductive_of_constructor cstr_kn in
	    let (mib,mip) = lookup_mind_specif env mI in
            let realargs = list_skipn mib.mind_nparams args in
            let rec process_args_of_constr = function
              | (t::lr), (rar::lrar) -> 
                  if rar = mk_norec then
                    if noccur_with_meta n nbfix t
                    then process_args_of_constr (lr, lrar)
                    else raise (CoFixGuardError
		                 (env,RecCallInNonRecArgOfConstructor t))
                  else
                    let spec = dest_subterms rar in
                    check_rec_call env true n spec t;
                    process_args_of_constr (lr, lrar)
              | [],_ -> ()
              | _ -> anomaly_ill_typed () 
            in process_args_of_constr (realargs, lra)
		 
	| Lambda (x,a,b) ->
	     assert (args = []);
            if noccur_with_meta n nbfix a then
              let env' = push_rel (x, None, a) env in
              check_rec_call env' alreadygrd (n+1)  vlra b
            else 
	      raise (CoFixGuardError (env,RecCallInTypeOfAbstraction a))
	      
	| CoFix (j,(_,varit,vdefs as recdef)) ->
            if (List.for_all (noccur_with_meta n nbfix) args)
            then 
              let nbfix = Array.length vdefs in
	      if (array_for_all (noccur_with_meta n nbfix) varit) then
		let env' = push_rec_types recdef env in
		(Array.iter (check_rec_call env' alreadygrd (n+1) vlra) vdefs;
		 List.iter (check_rec_call env alreadygrd n vlra) args)
              else 
		raise (CoFixGuardError (env,RecCallInTypeOfDef c))
	    else
	      raise (CoFixGuardError (env,UnguardedRecursiveCall c))

	| Case (_,p,tm,vrest) ->
            if (noccur_with_meta n nbfix p) then
              if (noccur_with_meta n nbfix tm) then
		if (List.for_all (noccur_with_meta n nbfix) args) then
		  Array.iter (check_rec_call env alreadygrd n vlra) vrest
		else 
		  raise (CoFixGuardError (env,RecCallInCaseFun c))
              else 
		raise (CoFixGuardError (env,RecCallInCaseArg c))
            else 
	      raise (CoFixGuardError (env,RecCallInCasePred c))
              
	| Meta _ -> ()
        | Evar _ ->
	    List.iter (check_rec_call env alreadygrd n vlra) args
	      
	| _    -> raise (CoFixGuardError (env,NotGuardedForm t)) in 

  let (mind, _) = codomain_is_coind env deftype in
  let vlra = lookup_subterms env mind in
  check_rec_call env false 1 (dest_subterms vlra) def

(* The  function which checks that the whole block of definitions 
   satisfies the guarded condition *)

let check_cofix env (bodynum,(names,types,bodies as recdef)) = 
  let nbfix = Array.length bodies in 
  for i = 0 to nbfix-1 do
    let fixenv = push_rec_types recdef env in
    try check_one_cofix fixenv nbfix bodies.(i) types.(i)
    with CoFixGuardError (errenv,err) -> 
      error_ill_formed_rec_body errenv err names i
  done