File: indtypes.ml

package info (click to toggle)
coq-doc 8.2pl1-1
  • links: PTS, VCS
  • area: non-free
  • in suites: squeeze
  • size: 19,240 kB
  • ctags: 22,737
  • sloc: ml: 132,933; ansic: 1,960; sh: 1,366; lisp: 456; makefile: 327
file content (542 lines) | stat: -rw-r--r-- 20,232 bytes parent folder | download | duplicates (2)
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
(************************************************************************)
(*  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: indtypes.ml 10296 2007-11-07 11:02:42Z barras $ *)

open Util
open Names
open Univ
open Term
open Inductive
open Reduction
open Typeops
open Pp
open Declarations
open Environ

let rec debug_string_of_mp = function
  | MPfile sl -> string_of_dirpath sl
  | MPbound uid -> "bound("^string_of_mbid uid^")"
  | MPself uid -> "self("^string_of_msid uid^")"
  | MPdot (mp,l) -> string_of_mp mp ^ "." ^ string_of_label l

let rec string_of_mp = function
  | MPfile sl -> string_of_dirpath sl
  | MPbound uid -> string_of_mbid uid
  | MPself uid -> string_of_msid uid
  | MPdot (mp,l) -> string_of_mp mp ^ "." ^ string_of_label l

let string_of_mp mp =
  if !Flags.debug then debug_string_of_mp mp else string_of_mp mp

let prkn kn =
  let (mp,_,l) = repr_kn kn in
  str(string_of_mp mp ^ "." ^ string_of_label l)
let prcon c =
  let (mp,_,l) = repr_con c in
  str(string_of_mp mp ^ "." ^ string_of_label l)

(* Same as noccur_between but may perform reductions.
   Could be refined more...  *)
let weaker_noccur_between env x nvars t =
  if noccur_between x nvars t then Some t
  else
   let t' = whd_betadeltaiota env t in
   if noccur_between x nvars t' then Some t'
   else None

let is_constructor_head t =
  match fst(decompose_app t) with
  | Rel _ -> true
  | _ -> false

let conv_ctxt_prefix env (ctx1:rel_context) ctx2 =
  let rec chk env rctx1 rctx2 =
    match rctx1, rctx2 with
        (_,None,ty1 as d1)::rctx1', (_,None,ty2)::rctx2' ->
          conv env ty1 ty2;
          chk (push_rel d1 env) rctx1' rctx2'
      | (_,Some bd1,ty1 as d1)::rctx1', (_,Some bd2,ty2)::rctx2' ->
          conv env ty1 ty2;
          conv env bd1 bd2;
          chk (push_rel d1 env) rctx1' rctx2'
      | [],_ -> ()
      | _ -> failwith "non convertible contexts" in
  chk env (List.rev ctx1) (List.rev ctx2)

(************************************************************************)
(* Various well-formedness check for inductive declarations            *)

(* Errors related to inductive constructions *)
type inductive_error =
  | NonPos of env * constr * constr
  | NotEnoughArgs of env * constr * constr
  | NotConstructor of env * constr * constr
  | NonPar of env * constr * int * constr * constr
  | SameNamesTypes of identifier
  | SameNamesConstructors of identifier
  | SameNamesOverlap of identifier list
  | NotAnArity of identifier
  | BadEntry

exception InductiveError of inductive_error

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

(* Typing the arities and constructor types *)

let rec sorts_of_constr_args env t =
  let t = whd_betadeltaiota_nolet env t in
  match t with
    | Prod (name,c1,c2) ->
        let varj = infer_type env c1 in
	let env1 = push_rel (name,None,c1) env in
	varj :: sorts_of_constr_args env1 c2
    | LetIn (name,def,ty,c) ->
        let env1 = push_rel (name,Some def,ty) env in
	sorts_of_constr_args env1 c
    | _ when is_constructor_head t -> []
    | _ -> anomaly "infos_and_sort: not a positive constructor"


(* Prop and Set are small *)
let is_small_sort = function
  | Prop _ -> true
  | _ -> false

let is_logic_sort s = (s = Prop Null)

(* [infos] is a sequence of pair [islogic,issmall] for each type in
   the product of a constructor or arity *)

let is_small_constr infos = List.for_all (fun s -> is_small_sort s) infos
let is_logic_constr infos = List.for_all (fun s -> is_logic_sort s) infos

(* An inductive definition is a "unit" if it has only one constructor
   and that all arguments expected by this constructor are 
   logical, this is the case for equality, conjunction of logical properties 
*)
let is_unit constrsinfos =
  match constrsinfos with  (* One info = One constructor *)
   | [|constrinfos|] -> is_logic_constr constrinfos 
   | [||] -> (* type without constructors *) true
   | _ -> false

let small_unit constrsinfos =
  let issmall = array_for_all is_small_constr constrsinfos 
  and isunit = is_unit constrsinfos in
  issmall, isunit

(* check information related to inductive arity *)
let typecheck_arity env params inds =
  let nparamargs = rel_context_nhyps params in
  let check_arity arctxt = function
      Monomorphic mar ->
        let ar = mar.mind_user_arity in
        let _ = infer_type env ar in
        conv env (it_mkProd_or_LetIn (Sort mar.mind_sort) arctxt) ar;
        ar
    | Polymorphic par ->
        check_polymorphic_arity env params par;
        it_mkProd_or_LetIn (Sort(Type par.poly_level)) arctxt in
  let env_arities =
    Array.fold_left
      (fun env_ar ind ->
        let ar_ctxt = ind.mind_arity_ctxt in
        let _ = check_ctxt env ar_ctxt in
        conv_ctxt_prefix env params ar_ctxt;
        (* Arities (with params) are typed-checked here *)
        let arity = check_arity ar_ctxt ind.mind_arity in
        (* mind_nrealargs *)
        if ind.mind_nrealargs <> rel_context_nhyps ar_ctxt - nparamargs then
             failwith "bad number of real inductive arguments";
	(* We do not need to generate the universe of full_arity; if
	   later, after the validation of the inductive definition,
	   full_arity is used as argument or subject to cast, an
	   upper universe will be generated *)
	let id = ind.mind_typename in
	let env_ar' = push_rel (Name id, None, arity) env_ar in
        env_ar')
      env
      inds in
  env_arities

(* Allowed eliminations *)

let check_predicativity env s small level =
  match s, engagement env with
      Type u, _ ->
        let u' = fresh_local_univ () in
        let cst =
          merge_constraints (enforce_geq u' u Constraint.empty)
            (universes env) in
        if not (check_geq cst u' level) then
          failwith "impredicative Type inductive type"
    | Prop Pos, Some ImpredicativeSet -> ()
    | Prop Pos, _ ->
        if not small then failwith "impredicative Set inductive type"
    | Prop Null,_ -> ()


let sort_of_ind = function
    Monomorphic mar -> mar.mind_sort
  | Polymorphic par -> Type par.poly_level

let all_sorts = [InProp;InSet;InType]
let small_sorts = [InProp;InSet]
let logical_sorts = [InProp]

let allowed_sorts issmall isunit s =
  match family_of_sort s with
  (* Type: all elimination allowed *)
  | InType -> all_sorts

  (* Small Set is predicative: all elimination allowed *)
  | InSet when issmall -> all_sorts

  (* Large Set is necessarily impredicative: forbids large elimination *)
  | InSet -> small_sorts

  (* Unitary/empty Prop: elimination to all sorts are realizable *)
  (* unless the type is large. If it is large, forbids large elimination *)
  (* which otherwise allows to simulate the inconsistent system Type:Type *)
  | InProp when isunit -> if issmall then all_sorts else small_sorts

  (* Other propositions: elimination only to Prop *)
  | InProp -> logical_sorts



let compute_elim_sorts env_ar params mib arity lc =
  let inst = extended_rel_list 0 params in
  let env_params = push_rel_context params env_ar in
  let lc = Array.map
    (fun c ->
      hnf_prod_applist env_params (lift (rel_context_length params) c) inst)
    lc in
  let s = sort_of_ind arity in
  let infos = Array.map (sorts_of_constr_args env_params) lc in
  let (small,unit) = small_unit infos in
  (* We accept recursive unit types... *)
  let unit = unit && mib.mind_ntypes = 1 in
  (* compute the max of the sorts of the products of the constructor type *)
  let level = max_inductive_sort
    (Array.concat (Array.to_list (Array.map Array.of_list infos))) in
  check_predicativity env_ar s small level;
  allowed_sorts small unit s


let typecheck_one_inductive env params mib mip =
  (* mind_typename and mind_consnames not checked *)
  (* mind_reloc_tbl, mind_nb_constant, mind_nb_args not checked (VM) *)
  (* mind_arity_ctxt, mind_arity, mind_nrealargs DONE (typecheck_arity) *)
  (* mind_user_lc *)
  let _ = Array.map (infer_type env) mip.mind_user_lc in
  (* mind_nf_lc *)
  let _ = Array.map (infer_type env) mip.mind_nf_lc in
  array_iter2 (conv env) mip.mind_nf_lc mip.mind_user_lc;
  (* mind_consnrealdecls *)
  let check_cons_args c n =
    let ctx,_ = decompose_prod_assum c in
    if n <> rel_context_length ctx - rel_context_length params then
      failwith "bad number of real constructor arguments" in
  array_iter2 check_cons_args mip.mind_nf_lc mip.mind_consnrealdecls;
  (* mind_kelim: checked by positivity criterion ? *)
  let sorts =
    compute_elim_sorts env params mib mip.mind_arity mip.mind_nf_lc in
  if List.exists (fun s -> not (List.mem s sorts)) mip.mind_kelim then
    failwith "elimination not allowed";
  (* mind_recargs: checked by positivity criterion *)
  ()

(************************************************************************)
(************************************************************************)
(* Positivity *)

type ill_formed_ind =
  | LocalNonPos of int
  | LocalNotEnoughArgs of int
  | LocalNotConstructor
  | LocalNonPar of int * int

exception IllFormedInd of ill_formed_ind

(* [mind_extract_params mie] extracts the params from an inductive types
   declaration, and checks that they are all present (and all the same)
   for all the given types. *)

let mind_extract_params = decompose_prod_n_assum

let explain_ind_err ntyp env0 nbpar c err = 
  let (lpar,c') = mind_extract_params nbpar c in
  let env = push_rel_context lpar env0 in
  match err with
    | LocalNonPos kt -> 
	raise (InductiveError (NonPos (env,c',Rel (kt+nbpar))))
    | LocalNotEnoughArgs kt -> 
	raise (InductiveError 
		 (NotEnoughArgs (env,c',Rel (kt+nbpar))))
    | LocalNotConstructor ->
	raise (InductiveError 
		 (NotConstructor (env,c',Rel (ntyp+nbpar))))
    | LocalNonPar (n,l) ->
	raise (InductiveError 
		 (NonPar (env,c',n,Rel (nbpar-n+1), Rel (l+nbpar))))

let failwith_non_pos n ntypes c =
  for k = n to n + ntypes - 1 do
    if not (noccurn k c) then raise (IllFormedInd (LocalNonPos (k-n+1)))
  done

let failwith_non_pos_vect n ntypes v =
  Array.iter (failwith_non_pos n ntypes) v;
  anomaly "failwith_non_pos_vect: some k in [n;n+ntypes-1] should occur"

let failwith_non_pos_list n ntypes l =
  List.iter (failwith_non_pos n ntypes) l;
  anomaly "failwith_non_pos_list: some k in [n;n+ntypes-1] should occur"

(* Conclusion of constructors: check the inductive type is called with
   the expected parameters *)
let check_correct_par (env,n,ntypes,_) hyps l largs =
  let nparams = rel_context_nhyps hyps in
  let largs = Array.of_list largs in
  if Array.length largs < nparams then 
    raise (IllFormedInd (LocalNotEnoughArgs l));
  let (lpar,largs') = array_chop nparams largs in
  let nhyps = List.length hyps in
  let rec check k index = function
    | [] -> ()
    | (_,Some _,_)::hyps -> check k (index+1) hyps
    | _::hyps ->
        match whd_betadeltaiota env lpar.(k) with
	  | Rel w when w = index -> check (k-1) (index+1) hyps
	  | _ -> raise (IllFormedInd (LocalNonPar (k+1,l)))
  in check (nparams-1) (n-nhyps) hyps;
  if not (array_for_all (noccur_between n ntypes) largs') then 
    failwith_non_pos_vect n ntypes largs'

(* Arguments of constructor: check the number of recursive parameters nrecp.
    the first parameters which are constant in recursive arguments 
    n is the current depth, nmr is the maximum number of possible 
    recursive parameters *)

let check_rec_par (env,n,_,_) hyps nrecp largs = 
  let (lpar,_) = list_chop nrecp largs in
  let rec find index = 
    function 
      | ([],_) -> ()
      | (_,[]) ->
          failwith "number of recursive parameters cannot be greater than the number of parameters."
      | (lp,(_,Some _,_)::hyps) -> find (index-1) (lp,hyps)
      | (p::lp,_::hyps) ->
          (match whd_betadeltaiota env p with
	    | Rel w when w = index -> find (index-1) (lp,hyps)
            | _ -> failwith "bad number of recursive parameters")
  in find (n-1) (lpar,List.rev hyps)

let lambda_implicit_lift n a =
  let lambda_implicit a = Lambda(Anonymous,Evar(0,[||]),a) in
  iterate lambda_implicit n (lift n a)

(* This removes global parameters of the inductive types in lc (for
   nested inductive types only ) *)
let abstract_mind_lc env ntyps npars lc = 
  if npars = 0 then 
    lc
  else 
    let make_abs = 
      list_tabulate
	(function i -> lambda_implicit_lift npars (Rel (i+1))) ntyps 
    in 
    Array.map (substl make_abs) lc

(* [env] is the typing environment
   [n] is the dB of the last inductive type
   [ntypes] is the number of inductive types in the definition
     (i.e. range of inductives is [n; n+ntypes-1])
   [lra] is the list of recursive tree of each variable
 *) 
let ienv_push_var (env, n, ntypes, lra) (x,a,ra) =
 (push_rel (x,None,a) env, n+1, ntypes, (Norec,ra)::lra)

let ienv_push_inductive (env, n, ntypes, ra_env) (mi,lpar) =
  let auxntyp = 1 in
  let specif = lookup_mind_specif env mi in
  let env' =
    push_rel (Anonymous,None,
              hnf_prod_applist env (type_of_inductive env specif) lpar) env in
  let ra_env' = 
    (Imbr mi,(Rtree.mk_rec_calls 1).(0)) ::
    List.map (fun (r,t) -> (r,Rtree.lift 1 t)) ra_env in
  (* New index of the inductive types *)
  let newidx = n + auxntyp in
  (env', newidx, ntypes, ra_env')

(* The recursive function that checks positivity and builds the list
   of recursive arguments *)
let check_positivity_one (env, _,ntypes,_ as ienv) hyps nrecp i indlc =
  let lparams = rel_context_length hyps in
  (* check the inductive types occur positively in [c] *)
  let rec check_pos (env, n, ntypes, ra_env as ienv) c = 
    let x,largs = decompose_app (whd_betadeltaiota env c) in
      match x with
	| Prod (na,b,d) ->
	    assert (largs = []);
            (match weaker_noccur_between env n ntypes b with
		None -> failwith_non_pos_list n ntypes [b]
              | Some b ->
	          check_pos (ienv_push_var ienv (na, b, mk_norec)) d)
	| Rel k ->
            (try
              let (ra,rarg) = List.nth ra_env (k-1) in 
	      (match ra with
                  Mrec _ -> check_rec_par ienv hyps nrecp largs
		|  _ -> ());
	      if not (List.for_all (noccur_between n ntypes) largs)
	      then failwith_non_pos_list n ntypes largs
	      else rarg
            with Failure _ | Invalid_argument _ -> mk_norec)
	| Ind ind_kn ->
            (* If the inductive type being defined appears in a
               parameter, then we have an imbricated type *)
            if List.for_all (noccur_between n ntypes) largs then mk_norec
            else check_positive_imbr ienv (ind_kn, largs)
	| err -> 
	    if noccur_between n ntypes x &&
              List.for_all (noccur_between n ntypes) largs 
	    then mk_norec
	    else failwith_non_pos_list n ntypes (x::largs)

  (* accesses to the environment are not factorised, but is it worth it? *)
  and check_positive_imbr (env,n,ntypes,ra_env as ienv) (mi, largs) =
    let (mib,mip) = lookup_mind_specif env mi in
    let auxnpar = mib.mind_nparams_rec in
    let (lpar,auxlargs) =
      try list_chop auxnpar largs 
      with Failure _ -> raise (IllFormedInd (LocalNonPos n)) in 
      (* If the inductive appears in the args (non params) then the
	 definition is not positive. *)
      if not (List.for_all (noccur_between n ntypes) auxlargs) then
	raise (IllFormedInd (LocalNonPos n));
      (* We do not deal with imbricated mutual inductive types *)
      let auxntyp = mib.mind_ntypes in 
	if auxntyp <> 1 then raise (IllFormedInd (LocalNonPos n));
	(* The nested inductive type with parameters removed *)
	let auxlcvect = abstract_mind_lc env auxntyp auxnpar mip.mind_nf_lc in
	  (* Extends the environment with a variable corresponding to
	     the inductive def *)
	let (env',_,_,_ as ienv') = ienv_push_inductive ienv (mi,lpar) in
	  (* Parameters expressed in env' *)
	let lpar' = List.map (lift auxntyp) lpar in
	let irecargs = 
	  (* fails if the inductive type occurs non positively *)
	  (* when substituted *) 
	  Array.map 
	    (function c -> 
	      let c' = hnf_prod_applist env' c lpar' in 
		check_constructors ienv' false c') 
	    auxlcvect in 
	(Rtree.mk_rec [|mk_paths (Imbr mi) irecargs|]).(0)
	    
  (* check the inductive types occur positively in the products of C, if
     check_head=true, also check the head corresponds to a constructor of
     the ith type *) 
	    
  and check_constructors ienv check_head c = 
    let rec check_constr_rec (env,n,ntypes,ra_env as ienv) lrec c = 
      let x,largs = decompose_app (whd_betadeltaiota env c) in
	match x with
          | Prod (na,b,d) -> 
	      assert (largs = []);
              let recarg = check_pos ienv b in 
              let ienv' = ienv_push_var ienv (na,b,mk_norec) in
		check_constr_rec ienv' (recarg::lrec) d
		  
	  | hd ->
	      if check_head then
		if hd = Rel (n+ntypes-i-1) then
		  check_correct_par ienv hyps (ntypes-i) largs
		else
		  raise (IllFormedInd LocalNotConstructor)
	      else
		if not (List.for_all (noccur_between n ntypes) largs)
              then raise (IllFormedInd (LocalNonPos n));
	      List.rev lrec
    in check_constr_rec ienv [] c
  in
  let irecargs =
    Array.map
      (fun c ->
        let _,rawc = mind_extract_params lparams c in
          try
	    check_constructors ienv true rawc
          with IllFormedInd err -> 
            explain_ind_err (ntypes-i) env lparams c err)
      indlc
  in mk_paths (Mrec i) irecargs

let check_subtree (t1:'a) (t2:'a) =
  if not (Rtree.compare_rtree (fun t1 t2 ->
    let l1 = fst(Rtree.dest_node t1) in
    let l2 = fst(Rtree.dest_node t2) in
    if l1 = Norec || l1 = l2 then 0 else -1)
    t1 t2) then
    failwith "bad recursive trees"
(* if t1=t2 then () else msg_warning (str"TODO: check recursive positions")*)

let check_positivity env_ar params nrecp inds =
  let ntypes = Array.length inds in
  let rc = Array.mapi (fun j t -> (Mrec j,t)) (Rtree.mk_rec_calls ntypes) in
  let lra_ind = List.rev (Array.to_list rc) in
  let lparams = rel_context_length params in
  let check_one i mip =
    let ra_env =
      list_tabulate (fun _ -> (Norec,mk_norec)) lparams @ lra_ind in
    let ienv = (env_ar, 1+lparams, ntypes, ra_env) in
      check_positivity_one ienv params nrecp i mip.mind_nf_lc 
  in
  let irecargs = Array.mapi check_one inds in 
  let wfp = Rtree.mk_rec irecargs in
  array_iter2 (fun ind wfpi -> check_subtree ind.mind_recargs wfpi) inds wfp

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

let check_inductive env kn mib =
  Flags.if_verbose msgnl (str "  checking ind: " ++ prkn kn);
  (* check mind_constraints: should be consistent with env *)
  let env = add_constraints mib.mind_constraints env in
  (* check mind_record : TODO ? check #constructor = 1 ? *)
  (* check mind_finite : always OK *)
  (* check mind_ntypes *)
  if Array.length mib.mind_packets <> mib.mind_ntypes then
    error "not the right number of packets";
  (* check mind_hyps: should be empty *)
  if mib.mind_hyps <> empty_named_context then
    error "section context not empty";
  (* check mind_params_ctxt *)
  let params = mib.mind_params_ctxt in
  let _ = check_ctxt env params in
  (* check mind_nparams *)
  if rel_context_nhyps params <> mib.mind_nparams then
    error "number the right number of parameters";
  (* mind_packets *)
  (*  - check arities *)
  let env_ar = typecheck_arity env params mib.mind_packets in
  (*  - check constructor types *)
  Array.iter (typecheck_one_inductive env_ar params mib) mib.mind_packets;
  (* check mind_nparams_rec: positivity condition *)
  check_positivity env_ar params mib.mind_nparams_rec mib.mind_packets;
  (* check mind_equiv... *)
  if mib.mind_equiv <> None then
    msg_warning (str"TODO: mind_equiv not checked");
  (* Now we can add the inductive *)
  add_mind kn mib env