summaryrefslogtreecommitdiff
path: root/kernel/univ.ml
blob: 3d254ce6d7e875bf893e9bc7afd15a9c51082956 (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
(************************************************************************)
(*  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: univ.ml 11596 2008-11-16 15:34:06Z letouzey $ *)

(* Initial Caml version originates from CoC 4.8 [Dec 1988] *)
(* Extension with algebraic universes by HH [Sep 2001] *)
(* Additional support for sort-polymorphic inductive types by HH [Mar 2006] *)

(* Universes are stratified by a partial ordering $\le$.
   Let $\~{}$ be the associated equivalence. We also have a strict ordering
   $<$ between equivalence classes, and we maintain that $<$ is acyclic,
   and contained in $\le$ in the sense that $[U]<[V]$ implies $U\le V$.

   At every moment, we have a finite number of universes, and we
   maintain the ordering in the presence of assertions $U<V$ and $U\le V$.

   The equivalence $\~{}$ is represented by a tree structure, as in the
   union-find algorithm. The assertions $<$ and $\le$ are represented by
   adjacency lists *)

open Pp
open Util

(* An algebraic universe [universe] is either a universe variable
   [universe_level] or a formal universe known to be greater than some
   universe variables and strictly greater than some (other) universe
   variables

   Universes variables denote universes initially present in the term
   to type-check and non variable algebraic universes denote the
   universes inferred while type-checking: it is either the successor
   of a universe present in the initial term to type-check or the
   maximum of two algebraic universes
 *)

type universe_level =
  | Set
  | Level of Names.dir_path * int

(* A specialized comparison function: we compare the [int] part first.
   This way, most of the time, the [dir_path] part is not considered. *)

let cmp_univ_level u v = match u,v with
  | Set, Set -> 0
  | Set, _ -> -1
  | _, Set -> 1
  | Level (dp1,i1), Level (dp2,i2) ->
      if i1 < i2 then -1
      else if i1 > i2 then 1
      else compare dp1 dp2

type universe =
  | Atom of universe_level
  | Max of universe_level list * universe_level list
  
module UniverseOrdered = struct
  type t = universe_level
  let compare = cmp_univ_level
end

let string_of_univ_level = function
  | Set -> "0"
  | Level (d,n) -> Names.string_of_dirpath d^"."^string_of_int n

let make_univ (m,n) = Atom (Level (m,n))

let pr_uni_level u = str (string_of_univ_level u)

let pr_uni = function
  | Atom u -> 
      pr_uni_level u
  | Max ([],[u]) ->
      str "(" ++ pr_uni_level u ++ str ")+1"
  | Max (gel,gtl) ->
      str "max(" ++ hov 0
       (prlist_with_sep pr_coma pr_uni_level gel ++
	  (if gel <> [] & gtl <> [] then pr_coma () else mt ()) ++
	prlist_with_sep pr_coma
	  (fun x -> str "(" ++ pr_uni_level x ++ str ")+1") gtl) ++
      str ")"

(* Returns the formal universe that lies juste above the universe variable u.
   Used to type the sort u. *)
let super = function
  | Atom u -> 
      Max ([],[u])
  | Max _ ->
      anomaly ("Cannot take the successor of a non variable universe:\n"^
               "(maybe a bugged tactic)")

(* Returns the formal universe that is greater than the universes u and v.
   Used to type the products. *)
let sup u v =
  match u,v with
    | Atom u, Atom v ->
	if cmp_univ_level u v = 0 then Atom u else Max ([u;v],[])
    | u, Max ([],[]) -> u
    | Max ([],[]), v -> v
    | Atom u, Max (gel,gtl) -> Max (list_add_set u gel,gtl)
    | Max (gel,gtl), Atom v -> Max (list_add_set v gel,gtl)
    | Max (gel,gtl), Max (gel',gtl') ->
	let gel'' = list_union gel gel' in
	let gtl'' = list_union gtl gtl' in
	Max (list_subtract gel'' gtl'',gtl'')

(* Comparison on this type is pointer equality *)
type canonical_arc =
    { univ: universe_level; lt: universe_level list; le: universe_level list }

let terminal u = {univ=u; lt=[]; le=[]}

(* A universe_level is either an alias for another one, or a canonical one,
   for which we know the universes that are above *)
type univ_entry =
    Canonical of canonical_arc
  | Equiv of universe_level * universe_level

module UniverseMap = Map.Make(UniverseOrdered)

type universes = univ_entry UniverseMap.t
		   
let enter_equiv_arc u v g =
  UniverseMap.add u (Equiv(u,v)) g

let enter_arc ca g =
  UniverseMap.add ca.univ (Canonical ca) g

let declare_univ u g =
  if not (UniverseMap.mem u g) then
    enter_arc (terminal u) g
  else
    g

(* The lower predicative level of the hierarchy that contains (impredicative)
   Prop and singleton inductive types *)
let type0m_univ = Max ([],[])

let is_type0m_univ = function
  | Max ([],[]) -> true
  | _ -> false

(* The level of predicative Set *)
let type0_univ = Atom Set

let is_type0_univ = function
  | Atom Set -> true
  | Max ([Set],[]) -> warning "Non canonical Set"; true
  | u -> false

let is_univ_variable = function
  | Atom a when a<>Set -> true
  | _ -> false

(* When typing [Prop] and [Set], there is no constraint on the level,
   hence the definition of [type1_univ], the type of [Prop] *)

let type1_univ = Max ([],[Set])

let initial_universes = UniverseMap.empty

(* Every universe_level has a unique canonical arc representative *)

(* repr : universes -> universe_level -> canonical_arc *)
(* canonical representative : we follow the Equiv links *)
let repr g u = 
  let rec repr_rec u =
    let a =
      try UniverseMap.find u g
      with Not_found -> anomalylabstrm "Univ.repr"
	  (str"Universe " ++ pr_uni_level u ++ str" undefined") 
    in
    match a with 
      | Equiv(_,v) -> repr_rec v
      | Canonical arc -> arc
  in
  repr_rec u

let can g = List.map (repr g)

(* transitive closure : we follow the Less links *)

(* collect : canonical_arc -> canonical_arc list * canonical_arc list *)
(* collect u = (V,W) iff V={v canonical | u<v} W={w canonical | u<=w}-V *)
(* i.e. collect does the transitive upward closure of what is known about u *)
let collect g arcu =
  let rec coll_rec lt le = function
    | [],[] -> (lt, list_subtractq le lt)
    | arcv::lt', le' ->
	if List.memq arcv lt then 
	  coll_rec lt le (lt',le')
	else
          coll_rec (arcv::lt) le ((can g (arcv.lt@arcv.le))@lt',le')
    | [], arcw::le' -> 
	if (List.memq arcw lt) or (List.memq arcw le) then 
	  coll_rec lt le ([],le')
	else
          coll_rec lt (arcw::le) (can g arcw.lt, (can g arcw.le)@le')
  in 
  coll_rec [] [] ([],[arcu])

(* reprleq : canonical_arc -> canonical_arc list *)
(* All canonical arcv such that arcu<=arcv with arcv#arcu *)
let reprleq g arcu =
  let rec searchrec w = function
    | [] -> w
    | v :: vl ->
	let arcv = repr g v in
        if List.memq arcv w || arcu==arcv then 
	  searchrec w vl
        else 
	  searchrec (arcv :: w) vl
  in 
  searchrec [] arcu.le


(* between : universe_level -> canonical_arc -> canonical_arc list *)
(* between u v = {w|u<=w<=v, w canonical}          *)     
(* between is the most costly operation *)

let between g u arcv = 
  (* good are all w | u <= w <= v  *)
  (* bad are all w | u <= w ~<= v *)
    (* find good and bad nodes in {w | u <= w} *)
    (* explore b u = (b or "u is good") *)
  let rec explore ((good, bad, b) as input) arcu =
    if List.memq arcu good then
      (good, bad, true) (* b or true *)
    else if List.memq arcu bad then
      input    (* (good, bad, b or false) *)
    else 
      let leq = reprleq g arcu in 
	(* is some universe >= u good ? *)
      let good, bad, b_leq = 
	List.fold_left explore (good, bad, false) leq
      in
	if b_leq then
	  arcu::good, bad, true (* b or true *)
	else 
	  good, arcu::bad, b    (* b or false *)
  in
  let good,_,_ = explore ([arcv],[],false) (repr g u) in
    good
      
(* We assume  compare(u,v) = LE with v canonical (see compare below).
   In this case List.hd(between g u v) = repr u
   Otherwise, between g u v = [] 
 *)


type order = EQ | LT | LE | NLE

(* compare : universe_level -> universe_level -> order *)
let compare g u v = 
  let arcu = repr g u 
  and arcv = repr g v in
  if arcu==arcv then 
    EQ
  else 
    let (lt,leq) = collect g arcu in
    if List.memq arcv lt then 
      LT
    else if List.memq arcv leq then 
      LE
    else 
      NLE

(* Invariants : compare(u,v) = EQ <=> compare(v,u) = EQ
                compare(u,v) = LT or LE => compare(v,u) = NLE
                compare(u,v) = NLE => compare(v,u) = NLE or LE or LT

   Adding u>=v is consistent iff compare(v,u) # LT 
    and then it is redundant iff compare(u,v) # NLE
   Adding u>v is consistent iff compare(v,u) = NLE 
    and then it is redundant iff compare(u,v) = LT *)

let compare_eq g u v =
  let g = declare_univ u g in
  let g = declare_univ v g in
  repr g u == repr g v


type check_function = universes -> universe -> universe -> bool

let incl_list cmp l1 l2 =
  List.for_all (fun x1 -> List.exists (fun x2 -> cmp x1 x2) l2) l1 

let compare_list cmp l1 l2 =
  incl_list cmp l1 l2 && incl_list cmp l2 l1

let rec check_eq g u v =
  match (u,v) with
    | Atom ul, Atom vl -> compare_eq g ul vl
    | Max(ule,ult), Max(vle,vlt) ->
        (* TODO: remove elements of lt in le! *)
        compare_list (compare_eq g) ule vle &&
        compare_list (compare_eq g) ult vlt
    | _ -> anomaly "check_eq" (* not complete! (Atom(u) = Max([u],[]) *)

let check_eq g u v =
  check_eq g u v

let compare_greater g strict u v =
  let g = declare_univ u g in
  let g = declare_univ v g in
  if not strict && compare_eq g v Set then true else
  match compare g v u with
    | (EQ|LE) -> not strict
    | LT -> true
    | NLE -> false
(*
let compare_greater g strict u v =
  let b = compare_greater g strict u v in
  ppnl(str (if b then if strict then ">" else ">=" else "NOT >="));
  b
*)
let rec check_greater g strict u v =
  match u, v with
    | Atom ul, Atom vl -> compare_greater g strict ul vl
    | Atom ul, Max(le,lt) ->
        List.for_all (fun vl -> compare_greater g strict ul vl) le &&
        List.for_all (fun vl -> compare_greater g true ul vl) lt
    | _ -> anomaly "check_greater"

let check_geq g = check_greater g false

(* setlt : universe_level -> universe_level -> unit *)
(* forces u > v *)
let setlt g u v =
  let arcu = repr g u in
  enter_arc {arcu with lt=v::arcu.lt} g

(* checks that non-redundant *)
let setlt_if g u v = match compare g u v with
  | LT -> g
  | _ -> setlt g u v

(* setleq : universe_level -> universe_level -> unit *)
(* forces u >= v *)
let setleq g u v =
  let arcu = repr g u in
  enter_arc {arcu with le=v::arcu.le} g


(* checks that non-redundant *)
let setleq_if g u v = match compare g u v with
  | NLE -> setleq g u v
  | _ -> g

(* merge : universe_level -> universe_level -> unit *)
(* we assume  compare(u,v) = LE *)
(* merge u v  forces u ~ v with repr u as canonical repr *)
let merge g u v =
  match between g u (repr g v) with
    | arcu::v -> (* arcu is chosen as canonical and all others (v) are *)
                 (* redirected to it *)
	let redirect (g,w,w') arcv =
 	  let g' = enter_equiv_arc arcv.univ arcu.univ g in
 	  (g',list_unionq arcv.lt w,arcv.le@w') 
	in
	let (g',w,w') = List.fold_left redirect (g,[],[]) v in
	let g'' = List.fold_left (fun g -> setlt_if g arcu.univ) g' w in
	let g''' = List.fold_left (fun g -> setleq_if g arcu.univ) g'' w' in
	g'''
    | [] -> anomaly "Univ.between"

(* merge_disc : universe_level -> universe_level -> unit *)
(* we assume  compare(u,v) = compare(v,u) = NLE *)
(* merge_disc u v  forces u ~ v with repr u as canonical repr *)
let merge_disc g u v =
  let arcu = repr g u in
  let arcv = repr g v in
  let g' = enter_equiv_arc arcv.univ arcu.univ g in
  let g'' = List.fold_left (fun g -> setlt_if g arcu.univ) g' arcv.lt in
  let g''' = List.fold_left (fun g -> setleq_if g arcu.univ) g'' arcv.le in
  g'''

(* Universe inconsistency: error raised when trying to enforce a relation
   that would create a cycle in the graph of universes. *)

type order_request = Lt | Le | Eq

exception UniverseInconsistency of order_request * universe * universe

let error_inconsistency o u v = raise (UniverseInconsistency (o,Atom u,Atom v))

(* enforce_univ_leq : universe_level -> universe_level -> unit *)
(* enforce_univ_leq u v will force u<=v if possible, will fail otherwise *)
let enforce_univ_leq u v g =
  let g = declare_univ u g in
  let g = declare_univ v g in
  match compare g u v with
    | NLE -> 
	(match compare g v u with
           | LT -> error_inconsistency Le u v
           | LE -> merge g v u
           | NLE -> setleq g u v
           | EQ -> anomaly "Univ.compare")
    | _ -> g

(* enforc_univ_eq : universe_level -> universe_level -> unit *)
(* enforc_univ_eq u v will force u=v if possible, will fail otherwise *)
let enforce_univ_eq u v g =
  let g = declare_univ u g in
  let g = declare_univ v g in
  match compare g u v with
    | EQ -> g
    | LT -> error_inconsistency Eq u v
    | LE -> merge g u v
    | NLE -> 
	(match compare g v u with
           | LT -> error_inconsistency Eq u v
           | LE -> merge g v u
           | NLE -> merge_disc g u v
           | EQ -> anomaly "Univ.compare")

(* enforce_univ_lt u v will force u<v if possible, will fail otherwise *)
let enforce_univ_lt u v g =
  let g = declare_univ u g in
  let g = declare_univ v g in
  match compare g u v with
    | LT -> g
    | LE -> setlt g u v
    | EQ -> error_inconsistency Lt u v
    | NLE -> 
	(match compare g v u with
           | NLE -> setlt g u v
           | _ -> error_inconsistency Lt u v)

(*
let enforce_univ_relation g = function 
  | Equiv (u,v) -> enforce_univ_eq u v g
  | Canonical {univ=u; lt=lt; le=le} ->
      let g' = List.fold_right (enforce_univ_lt u) lt g in
      List.fold_right (enforce_univ_leq u) le g'
*)

(* Merging 2 universe graphs *)
(*
let merge_universes sp u1 u2 =
  UniverseMap.fold (fun _ a g -> enforce_univ_relation g a) u1 u2
*)


(* Constraints and sets of consrtaints. *)

type constraint_type = Lt | Leq | Eq

type univ_constraint = universe_level * constraint_type * universe_level

let enforce_constraint cst g =
  match cst with
    | (u,Lt,v) -> enforce_univ_lt u v g
    | (u,Leq,v) -> enforce_univ_leq u v g
    | (u,Eq,v) -> enforce_univ_eq u v g


module Constraint = Set.Make(
  struct 
    type t = univ_constraint 
    let compare = Pervasives.compare 
  end)
		      
type constraints = Constraint.t

type constraint_function = 
    universe -> universe -> constraints -> constraints

let constraint_add_leq v u c =
  if v = Set then c else Constraint.add (v,Leq,u) c

let enforce_geq u v c =
  match u, v with
  | Atom u, Atom v -> constraint_add_leq v u c
  | Atom u, Max (gel,gtl) ->
      let d = List.fold_right (fun v -> constraint_add_leq v u) gel c in
      List.fold_right (fun v -> Constraint.add (v,Lt,u)) gtl d
  | _ -> anomaly "A universe bound can only be a variable"

let enforce_eq u v c =
  match (u,v) with
    | Atom u, Atom v -> Constraint.add (u,Eq,v) c
    | _ -> anomaly "A universe comparison can only happen between variables"

let merge_constraints c g =
  Constraint.fold enforce_constraint c g

(**********************************************************************)
(* Tools for sort-polymorphic inductive types                         *)

(* Temporary inductive type levels *)

let fresh_level =
  let n = ref 0 in fun () -> incr n; Level (Names.make_dirpath [],!n)

let fresh_local_univ () = Atom (fresh_level ())

(* Miscellaneous functions to remove or test local univ assumed to
   occur only in the le constraints *)

let make_max = function
  | ([u],[]) -> Atom u
  | (le,lt) -> Max (le,lt)

let remove_large_constraint u = function
  | Atom u' as x -> if u = u' then Max ([],[]) else x
  | Max (le,lt) -> make_max (list_remove u le,lt)

let is_direct_constraint u = function
  | Atom u' -> u = u'
  | Max (le,lt) -> List.mem u le

(* 
   Solve a system of universe constraint of the form

   u_s11, ..., u_s1p1, w1 <= u1
   ...
   u_sn1, ..., u_snpn, wn <= un

where 

  - the ui (1 <= i <= n) are universe variables,
  - the sjk select subsets of the ui for each equations, 
  - the wi are arbitrary complex universes that do not mention the ui.
*)

let is_direct_sort_constraint s v = match s with
  | Some u -> is_direct_constraint u v
  | None -> false

let solve_constraints_system levels level_bounds =
  let levels = 
    Array.map (Option.map (function Atom u -> u | _ -> anomaly "expects Atom"))
      levels in
  let v = Array.copy level_bounds in
  let nind = Array.length v in
  for i=0 to nind-1 do
    for j=0 to nind-1 do
      if i<>j & is_direct_sort_constraint levels.(j) v.(i) then
	v.(i) <- sup v.(i) level_bounds.(j)
    done;
    for j=0 to nind-1 do
      match levels.(j) with
      | Some u -> v.(i) <- remove_large_constraint u v.(i)
      | None -> ()
    done
  done;
  v

let subst_large_constraint u u' v =
  match u with 
  | Atom u ->
      if is_direct_constraint u v then sup u' (remove_large_constraint u v)
      else v
  | _ ->
      anomaly "expect a universe level"

let subst_large_constraints =
  List.fold_right (fun (u,u') -> subst_large_constraint u u')

let no_upper_constraints u cst =
  match u with
  | Atom u -> Constraint.for_all (fun (u1,_,_) -> u1 <> u) cst
  | Max _ -> anomaly "no_upper_constraints"

(* Pretty-printing *)

let num_universes g =
  UniverseMap.fold (fun _ _ -> succ) g 0

let num_edges g =
  let reln_len = function
    | Equiv _ -> 1
    | Canonical {lt=lt;le=le} -> List.length lt + List.length le
  in
  UniverseMap.fold (fun _ a n -> n + (reln_len a)) g 0
    
let pr_arc = function 
  | Canonical {univ=u; lt=[]; le=[]} ->
      mt ()
  | Canonical {univ=u; lt=lt; le=le} ->
      pr_uni_level u ++ str " " ++
      v 0
        (prlist_with_sep pr_spc (fun v -> str "< " ++ pr_uni_level v) lt ++
	 (if lt <> [] & le <> [] then spc () else mt()) ++
         prlist_with_sep pr_spc (fun v -> str "<= " ++ pr_uni_level v) le) ++
      fnl ()
  | Equiv (u,v) -> 
      pr_uni_level u  ++ str " = " ++ pr_uni_level v ++ fnl ()

let pr_universes g =
  let graph = UniverseMap.fold (fun k a l -> (k,a)::l) g [] in
  prlist (function (_,a) -> pr_arc a) graph
    
let pr_constraints c =
  Constraint.fold (fun (u1,op,u2) pp_std -> 
		     let op_str = match op with 
		       | Lt -> " < "
		       | Leq -> " <= "
		       | Eq -> " = "
		     in pp_std ++  pr_uni_level u1 ++ str op_str ++
			  pr_uni_level u2 ++ fnl () )  c (str "")
    
(* Dumping constrains to a file *)

let dump_universes output g = 
  let dump_arc _ = function
    | Canonical {univ=u; lt=lt; le=le} -> 
	let u_str = string_of_univ_level u in
	  List.iter 
	    (fun v -> 
	       Printf.fprintf output "%s < %s ;\n" u_str
		 (string_of_univ_level v)) 
	    lt;
	  List.iter 
	    (fun v -> 
	       Printf.fprintf output "%s <= %s ;\n" u_str
		 (string_of_univ_level v)) 
	    le
    | Equiv (u,v) ->
	Printf.fprintf output "%s = %s ;\n"
	  (string_of_univ_level u) (string_of_univ_level v)
  in
    UniverseMap.iter dump_arc g 

(* Hash-consing *)

module Huniv =
  Hashcons.Make(
    struct
      type t = universe
      type u = Names.dir_path -> Names.dir_path
      let hash_aux hdir = function
	| Set -> Set
	| Level (d,n) -> Level (hdir d,n)
      let hash_sub hdir = function
	| Atom u -> Atom (hash_aux hdir u)
	| Max (gel,gtl) ->
	    Max (List.map (hash_aux hdir) gel, List.map (hash_aux hdir) gtl)
      let equal u v =
	match u, v with
	  | Atom u, Atom v -> u == v
	  | Max (gel,gtl), Max (gel',gtl') ->
	      (list_for_all2eq (==) gel gel') &&
              (list_for_all2eq (==) gtl gtl')
	  | _ -> false
      let hash = Hashtbl.hash
    end)

let hcons1_univ u =
  let _,_,hdir,_,_,_ = Names.hcons_names() in
  Hashcons.simple_hcons Huniv.f hdir u