aboutsummaryrefslogtreecommitdiffhomepage
path: root/kernel/univ.ml
blob: 9782312cae94289f940ead1bf776db2e5f7ce8de (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
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)

(* Created in Caml by Gérard Huet for CoC 4.8 [Dec 1988] *)
(* Functional code by Jean-Christophe Filliâtre for Coq V7.0 [1999] *)
(* Extension with algebraic universes by HH for Coq V7.0 [Sep 2001] *)
(* Additional support for sort-polymorphic inductive types by HH [Mar 2006] *)
(* Support for universe polymorphism by MS [2014] *)

(* Revisions by Bruno Barras, Hugo Herbelin, Pierre Letouzey, Matthieu
   Sozeau, Pierre-Marie Pédrot *)

open Pp
open CErrors
open Util

(* 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 *)

module RawLevel =
struct
  open Names
  type t =
    | Prop
    | Set
    | Level of int * DirPath.t
    | Var of int

  (* Hash-consing *)

  let equal x y =
    x == y ||
      match x, y with
      | Prop, Prop -> true
      | Set, Set -> true
      | Level (n,d), Level (n',d') ->
        Int.equal n n' && DirPath.equal d d'
      | Var n, Var n' -> Int.equal n n'
      | _ -> false

  let compare u v =
    match u, v with
    | Prop,Prop -> 0
    | Prop, _ -> -1
    | _, Prop -> 1
    | Set, Set -> 0
    | Set, _ -> -1
    | _, Set -> 1
    | Level (i1, dp1), Level (i2, dp2) ->
      if i1 < i2 then -1
      else if i1 > i2 then 1
      else DirPath.compare dp1 dp2
    | Level _, _ -> -1
    | _, Level _ -> 1
    | Var n, Var m -> Int.compare n m

  let hequal x y =
    x == y ||
      match x, y with
      | Prop, Prop -> true
      | Set, Set -> true
      | Level (n,d), Level (n',d') ->
        n == n' && d == d'
      | Var n, Var n' -> n == n'
      | _ -> false

  let hcons = function
    | Prop as x -> x
    | Set as x -> x
    | Level (n,d) as x -> 
      let d' = Names.DirPath.hcons d in
        if d' == d then x else Level (n,d')
    | Var n as x -> x

  open Hashset.Combine

  let hash = function
    | Prop -> combinesmall 1 0
    | Set -> combinesmall 1 1
    | Var n -> combinesmall 2 n
    | Level (n, d) -> combinesmall 3 (combine n (Names.DirPath.hash d))

end

module Level = struct

  open Names

  type raw_level = RawLevel.t =
  | Prop
  | Set
  | Level of int * DirPath.t
  | Var of int

  (** Embed levels with their hash value *)
  type t = { 
    hash : int;
    data : RawLevel.t }

  let equal x y = 
    x == y || Int.equal x.hash y.hash && RawLevel.equal x.data y.data

  let hash x = x.hash

  let data x = x.data

  (** Hashcons on levels + their hash *)

  module Self = struct
    type nonrec t = t
    type u = unit
    let eq x y = x.hash == y.hash && RawLevel.hequal x.data y.data
    let hash x = x.hash
    let hashcons () x =
      let data' = RawLevel.hcons x.data in
      if x.data == data' then x else { x with data = data' }
  end

  let hcons =
    let module H = Hashcons.Make(Self) in
    Hashcons.simple_hcons H.generate H.hcons ()

  let make l = hcons { hash = RawLevel.hash l; data = l }

  let set = make Set
  let prop = make Prop

  let is_small x = 
    match data x with
    | Level _ -> false
    | Var _ -> false
    | Prop -> true
    | Set -> true
 
  let is_prop x =
    match data x with
    | Prop -> true
    | _ -> false

  let is_set x =
    match data x with
    | Set -> true
    | _ -> false

  let compare u v =
    if u == v then 0
    else
      let c = Int.compare (hash u) (hash v) in
	if c == 0 then RawLevel.compare (data u) (data v)
	else c

  let natural_compare u v =
    if u == v then 0
    else RawLevel.compare (data u) (data v)
	    
  let to_string x = 
    match data x with
    | Prop -> "Prop"
    | Set -> "Set"
    | Level (n,d) -> Names.DirPath.to_string d^"."^string_of_int n
    | Var n -> "Var(" ^ string_of_int n ^ ")"

  let pr u = str (to_string u)

  let apart u v =
    match data u, data v with
    | Prop, Set | Set, Prop -> true
    | _ -> false

  let vars = Array.init 20 (fun i -> make (Var i))

  let var n = 
    if n < 20 then vars.(n) else make (Var n)

  let var_index u =
    match data u with
    | Var n -> Some n | _ -> None

  let make m n = make (Level (n, Names.DirPath.hcons m))

  let name u =
    match data u with
    | Level (n, d) -> Some (d, n)
    | _ -> None
end

(** Level maps *)
module LMap = struct 
  module M = HMap.Make (Level)
  include M

  let union l r = 
    merge (fun k l r -> 
      match l, r with
      | Some _, _ -> l
      | _, _ -> r) l r

  let subst_union l r = 
    merge (fun k l r -> 
      match l, r with
      | Some (Some _), _ -> l
      | Some None, None -> l
      | _, _ -> r) l r

  let diff ext orig =
    fold (fun u v acc -> 
      if mem u orig then acc 
      else add u v acc)
      ext empty

  let pr f m =
    h 0 (prlist_with_sep fnl (fun (u, v) ->
      Level.pr u ++ f v) (bindings m))
end

module LSet = struct
  include LMap.Set

  let pr prl s =
    str"{" ++ prlist_with_sep spc prl (elements s) ++ str"}"

  let of_array l =
    Array.fold_left (fun acc x -> add x acc) empty l

end


type 'a universe_map = 'a LMap.t

type universe_level = Level.t

type universe_level_subst_fn = universe_level -> universe_level

type universe_set = LSet.t

(* An algebraic universe [universe] is either a universe variable
   [Level.t] 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
*)

module Universe =
struct
  (* Invariants: non empty, sorted and without duplicates *)

  module Expr = 
  struct
    type t = Level.t * int

    (* Hashing of expressions *)
    module ExprHash = 
    struct
      type t = Level.t * int
      type u = Level.t -> Level.t
      let hashcons hdir (b,n as x) = 
	let b' = hdir b in 
	  if b' == b then x else (b',n)
      let eq l1 l2 =
        l1 == l2 || 
        match l1,l2 with
	| (b,n), (b',n') -> b == b' && n == n'

      let hash (x, n) = n + Level.hash x

    end

    module H = Hashcons.Make(ExprHash)

    let hcons =
      Hashcons.simple_hcons H.generate H.hcons Level.hcons

    let make l = (l, 0)

    let compare u v =
      if u == v then 0
      else 
	let (x, n) = u and (x', n') = v in
	  if Int.equal n n' then Level.compare x x'
	  else n - n'

    let prop = hcons (Level.prop, 0)
    let set = hcons (Level.set, 0)
    let type1 = hcons (Level.set, 1)

    let is_small = function
      | (l,0) -> Level.is_small l
      | _ -> false

    let equal x y = x == y ||
      (let (u,n) = x and (v,n') = y in
	 Int.equal n n' && Level.equal u v)

    let hash = ExprHash.hash

    let leq (u,n) (v,n') =
      let cmp = Level.compare u v in
	if Int.equal cmp 0 then n <= n'
	else if n <= n' then 
	  (Level.is_prop u && Level.is_small v)
	else false

    let successor (u,n) =
      if Level.is_prop u then type1
      else (u, n + 1)

    let addn k (u,n as x) = 
      if k = 0 then x 
      else if Level.is_prop u then
	(Level.set,n+k)
      else (u,n+k)

    type super_result =
	SuperSame of bool
        (* The level expressions are in cumulativity relation. boolean
           indicates if left is smaller than right?  *)
      | SuperDiff of int
        (* The level expressions are unrelated, the comparison result
           is canonical *)

    (** [super u v] compares two level expressions,
       returning [SuperSame] if they refer to the same level at potentially different
       increments or [SuperDiff] if they are different. The booleans indicate if the
       left expression is "smaller" than the right one in both cases. *)
    let super (u,n) (v,n') =
      let cmp = Level.compare u v in
	if Int.equal cmp 0 then SuperSame (n < n')
	else
          let open RawLevel in
          match Level.data u, n, Level.data v, n' with
          | Prop, _, Prop, _ -> SuperSame (n < n')
          | Prop, 0, _, _ -> SuperSame true
          | _, _, Prop, 0 -> SuperSame false
          | _, _, _, _ -> SuperDiff cmp

    let to_string (v, n) =
      if Int.equal n 0 then Level.to_string v
      else Level.to_string v ^ "+" ^ string_of_int n

    let pr x = str(to_string x)

    let pr_with f (v, n) = 
      if Int.equal n 0 then f v
      else f v ++ str"+" ++ int n

    let is_level = function
      | (v, 0) -> true
      | _ -> false

    let level = function
      | (v,0) -> Some v
      | _ -> None
	
    let get_level (v,n) = v

    let map f (v, n as x) = 
      let v' = f v in 
	if v' == v then x
	else if Level.is_prop v' && n != 0 then
	  (Level.set, n)
	else (v', n)

  end

  type t = Expr.t list

  let tip l = [l]
  let cons x l = x :: l

  let rec hash = function
  | [] -> 0
  | e :: l -> Hashset.Combine.combinesmall (Expr.ExprHash.hash e) (hash l)

  let equal x y = x == y || List.equal Expr.equal x y

  let compare x y = if x == y then 0 else List.compare Expr.compare x y

  module Huniv = Hashcons.Hlist(Expr)

  let hcons = Hashcons.recursive_hcons Huniv.generate Huniv.hcons Expr.hcons

  let make l = tip (Expr.make l)
  let tip x = tip x

  let pr l = match l with
    | [u] -> Expr.pr u
    | _ -> 
      str "max(" ++ hov 0
	(prlist_with_sep pr_comma Expr.pr l) ++
        str ")"

  let pr_with f l = match l with
    | [u] -> Expr.pr_with f u
    | _ -> 
      str "max(" ++ hov 0
	(prlist_with_sep pr_comma (Expr.pr_with f) l) ++
        str ")"

  let is_level l = match l with
    | [l] -> Expr.is_level l
    | _ -> false

  let rec is_levels l = match l with
    | l :: r -> Expr.is_level l && is_levels r
    | [] -> true

  let level l = match l with
    | [l] -> Expr.level l
    | _ -> None

  let levels l = 
    List.fold_left (fun acc x -> LSet.add (Expr.get_level x) acc) LSet.empty l

  let is_small u = 
    match u with
    | [l] -> Expr.is_small l
    | _ -> false

  (* The lower predicative level of the hierarchy that contains (impredicative)
     Prop and singleton inductive types *)
  let type0m = tip Expr.prop

  (* The level of sets *)
  let type0 = tip Expr.set

  (* When typing [Prop] and [Set], there is no constraint on the level,
     hence the definition of [type1_univ], the type of [Prop] *)    
  let type1 = tip (Expr.successor Expr.set)

  let is_type0m x = equal type0m x
  let is_type0 x = equal type0 x

  (* Returns the formal universe that lies just above the universe variable u.
     Used to type the sort u. *)
  let super l = 
    if is_small l then type1
    else
      List.Smart.map (fun x -> Expr.successor x) l

  let addn n l =
    List.Smart.map (fun x -> Expr.addn n x) l

  let rec merge_univs l1 l2 =
    match l1, l2 with
    | [], _ -> l2
    | _, [] -> l1
    | h1 :: t1, h2 :: t2 ->
       let open Expr in
       (match super h1 h2 with
	| SuperSame true (* h1 < h2 *) -> merge_univs t1 l2
	| SuperSame false -> merge_univs l1 t2
	| SuperDiff c ->
           if c <= 0 (* h1 < h2 is name order *)
	   then cons h1 (merge_univs t1 l2)
	   else cons h2 (merge_univs l1 t2))

  let sort u =
    let rec aux a l = 
      match l with
      | b :: l' ->
	let open Expr in
        (match super a b with
	 | SuperSame false -> aux a l'
	 | SuperSame true -> l
	 | SuperDiff c ->
	    if c <= 0 then cons a l
	    else cons b (aux a l'))
      | [] -> cons a l
    in 
      List.fold_right (fun a acc -> aux a acc) u []

  (* Returns the formal universe that is greater than the universes u and v.
     Used to type the products. *)
  let sup x y = merge_univs x y

  let empty = []

  let exists = List.exists

  let for_all = List.for_all

  let smart_map = List.Smart.map

  let map = List.map
end

type universe = Universe.t

(* The level of predicative Set *)
let type0m_univ = Universe.type0m
let type0_univ = Universe.type0
let type1_univ = Universe.type1
let is_type0m_univ = Universe.is_type0m
let is_type0_univ = Universe.is_type0
let is_univ_variable l = Universe.level l != None
let is_small_univ = Universe.is_small
let pr_uni = Universe.pr

let sup = Universe.sup
let super = Universe.super

open Universe

let universe_level = Universe.level


type constraint_type = Lt | Le | Eq

type explanation = (constraint_type * universe) list

let constraint_type_ord c1 c2 = match c1, c2 with
| Lt, Lt -> 0
| Lt, _ -> -1
| Le, Lt -> 1
| Le, Le -> 0
| Le, Eq -> -1
| Eq, Eq -> 0
| Eq, _ -> 1

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

type univ_inconsistency = constraint_type * universe * universe * explanation Lazy.t option

exception UniverseInconsistency of univ_inconsistency

let error_inconsistency o u v p =
  raise (UniverseInconsistency (o,make u,make v,p))

(* Constraints and sets of constraints. *)    

type univ_constraint = Level.t * constraint_type * Level.t

let pr_constraint_type op = 
  let op_str = match op with
    | Lt -> " < "
    | Le -> " <= "
    | Eq -> " = "
  in str op_str

module UConstraintOrd =
struct
  type t = univ_constraint
  let compare (u,c,v) (u',c',v') =
    let i = constraint_type_ord c c' in
    if not (Int.equal i 0) then i
    else
      let i' = Level.compare u u' in
      if not (Int.equal i' 0) then i'
      else Level.compare v v'
end

module Constraint = 
struct 
  module S = Set.Make(UConstraintOrd)
  include S

  let pr prl c =
    fold (fun (u1,op,u2) pp_std ->
      pp_std ++ prl u1 ++ pr_constraint_type op ++
	prl u2 ++ fnl () )  c (str "")

  let universes_of c =
    fold (fun (u1, op, u2) unvs -> LSet.add u2 (LSet.add u1 unvs)) c LSet.empty
end

let universes_of_constraints = Constraint.universes_of
let empty_constraint = Constraint.empty
let union_constraint = Constraint.union
let eq_constraint = Constraint.equal

type constraints = Constraint.t

module Hconstraint =
  Hashcons.Make(
    struct
      type t = univ_constraint
      type u = universe_level -> universe_level
      let hashcons hul (l1,k,l2) = (hul l1, k, hul l2)
      let eq (l1,k,l2) (l1',k',l2') =
	l1 == l1' && k == k' && l2 == l2'
      let hash = Hashtbl.hash
    end)

module Hconstraints =
  Hashcons.Make(
    struct
      type t = constraints
      type u = univ_constraint -> univ_constraint
      let hashcons huc s =
	Constraint.fold (fun x -> Constraint.add (huc x)) s Constraint.empty
      let eq s s' =
	List.for_all2eq (==)
	  (Constraint.elements s)
	  (Constraint.elements s')
      let hash = Hashtbl.hash
    end)

let hcons_constraint = Hashcons.simple_hcons Hconstraint.generate Hconstraint.hcons Level.hcons
let hcons_constraints = Hashcons.simple_hcons Hconstraints.generate Hconstraints.hcons hcons_constraint


(** A value with universe constraints. *)
type 'a constrained = 'a * constraints

let constraints_of (_, cst) = cst

(** Constraint functions. *)

type 'a constraint_function = 'a -> 'a -> constraints -> constraints

let enforce_eq_level u v c =
  (* We discard trivial constraints like u=u *)
  if Level.equal u v then c 
  else if Level.apart u v then
    error_inconsistency Eq u v None
  else Constraint.add (u,Eq,v) c

let enforce_eq u v c =
  match Universe.level u, Universe.level v with
    | Some u, Some v -> enforce_eq_level u v c
    | _ -> anomaly (Pp.str "A universe comparison can only happen between variables.")

let check_univ_eq u v = Universe.equal u v

let enforce_eq u v c =
  if check_univ_eq u v then c
  else enforce_eq u v c

let constraint_add_leq v u c =
  (* We just discard trivial constraints like u<=u *)
  if Expr.equal v u then c
  else 
    match v, u with
    | (x,n), (y,m) -> 
    let j = m - n in
      if j = -1 (* n = m+1, v+1 <= u <-> v < u *) then
	Constraint.add (x,Lt,y) c
      else if j <= -1 (* n = m+k, v+k <= u <-> v+(k-1) < u *) then
	if Level.equal x y then (* u+(k+1) <= u *)
	  raise (UniverseInconsistency (Le, Universe.tip v, Universe.tip u, None))
	else anomaly (Pp.str"Unable to handle arbitrary u+k <= v constraints.")
      else if j = 0 then
	Constraint.add (x,Le,y) c
      else (* j >= 1 *) (* m = n + k, u <= v+k *)
	if Level.equal x y then c (* u <= u+k, trivial *)
	else if Level.is_small x then c (* Prop,Set <= u+S k, trivial *)
	else anomaly (Pp.str"Unable to handle arbitrary u <= v+k constraints.")
	  
let check_univ_leq_one u v = Universe.exists (Expr.leq u) v

let check_univ_leq u v = 
  Universe.for_all (fun u -> check_univ_leq_one u v) u

let enforce_leq u v c =
  let rec aux acc v =
  match v with
  | v :: l ->
    aux (List.fold_right (fun u -> constraint_add_leq u v) u c) l
  | [] -> acc
  in aux c v

let enforce_leq u v c =
  if check_univ_leq u v then c
  else enforce_leq u v c

let enforce_leq_level u v c =
  if Level.equal u v then c else Constraint.add (u,Le,v) c

(* Miscellaneous functions to remove or test local univ assumed to
   occur in a universe *)

let univ_level_mem u v =
  List.exists (fun (l, n) -> Int.equal n 0 && Level.equal u l) v

let univ_level_rem u v min = 
  match Universe.level v with
  | Some u' -> if Level.equal u u' then min else v
  | None -> List.filter (fun (l, n) -> not (Int.equal n 0 && Level.equal u l)) v

(* Is u mentionned in v (or equals to v) ? *)


(**********************************************************************)
(** Universe polymorphism                                             *)
(**********************************************************************)

(** A universe level substitution, note that no algebraic universes are
    involved *)

type universe_level_subst = universe_level universe_map

(** A full substitution might involve algebraic universes *)
type universe_subst = universe universe_map

module Variance =
struct
  (** A universe position in the instance given to a cumulative
     inductive can be the following. Note there is no Contravariant
     case because [forall x : A, B <= forall x : A', B'] requires [A =
     A'] as opposed to [A' <= A]. *)
  type t = Irrelevant | Covariant | Invariant

  let sup x y =
    match x, y with
    | Irrelevant, s | s, Irrelevant -> s
    | Invariant, _ | _, Invariant -> Invariant
    | Covariant, Covariant -> Covariant

  let check_subtype x y = match x, y with
  | (Irrelevant | Covariant | Invariant), Irrelevant -> true
  | Irrelevant, Covariant -> false
  | (Covariant | Invariant), Covariant -> true
  | (Irrelevant | Covariant), Invariant -> false
  | Invariant, Invariant -> true

  let pr = function
    | Irrelevant -> str "*"
    | Covariant -> str "+"
    | Invariant -> str "="

  let leq_constraint csts variance u u' =
    match variance with
    | Irrelevant -> csts
    | Covariant -> enforce_leq_level u u' csts
    | Invariant -> enforce_eq_level u u' csts

  let eq_constraint csts variance u u' =
    match variance with
    | Irrelevant -> csts
    | Covariant | Invariant -> enforce_eq_level u u' csts

  let leq_constraints variance u u' csts =
    let len = Array.length u in
    assert (len = Array.length u' && len = Array.length variance);
    Array.fold_left3 leq_constraint csts variance u u'

  let eq_constraints variance u u' csts =
    let len = Array.length u in
    assert (len = Array.length u' && len = Array.length variance);
    Array.fold_left3 eq_constraint csts variance u u'
end

module Instance : sig
    type t = Level.t array

    val empty : t
    val is_empty : t -> bool
      
    val of_array : Level.t array -> t
    val to_array : t -> Level.t array

    val append : t -> t -> t
    val equal : t -> t -> bool
    val length : t -> int

    val hcons : t -> t
    val hash : t -> int

    val share : t -> t * int

    val subst_fn : universe_level_subst_fn -> t -> t
    
    val pr : (Level.t -> Pp.t) -> ?variance:Variance.t array -> t -> Pp.t
    val levels : t -> LSet.t
end = 
struct
  type t = Level.t array

  let empty : t = [||]

  module HInstancestruct =
  struct
    type nonrec t = t
    type u = Level.t -> Level.t

    let hashcons huniv a = 
      let len = Array.length a in
	if Int.equal len 0 then empty
	else begin
	  for i = 0 to len - 1 do
	    let x = Array.unsafe_get a i in
	    let x' = huniv x in
	      if x == x' then ()
	      else Array.unsafe_set a i x'
	  done;
	  a
	end

    let eq t1 t2 =
      t1 == t2 ||
	(Int.equal (Array.length t1) (Array.length t2) &&
	   let rec aux i =
	     (Int.equal i (Array.length t1)) || (t1.(i) == t2.(i) && aux (i + 1))
	   in aux 0)
	
    let hash a = 
      let accu = ref 0 in
	for i = 0 to Array.length a - 1 do
	  let l = Array.unsafe_get a i in
	  let h = Level.hash l in
	    accu := Hashset.Combine.combine !accu h;
	done;
	(* [h] must be positive. *)
	let h = !accu land 0x3FFFFFFF in
	  h
  end

  module HInstance = Hashcons.Make(HInstancestruct)

  let hcons = Hashcons.simple_hcons HInstance.generate HInstance.hcons Level.hcons
    
  let hash = HInstancestruct.hash
    
  let share a = (hcons a, hash a)
	      
  let empty = hcons [||]

  let is_empty x = Int.equal (Array.length x) 0

  let append x y =
    if Array.length x = 0 then y
    else if Array.length y = 0 then x 
    else Array.append x y

  let of_array a =
    assert(Array.for_all (fun x -> not (Level.is_prop x)) a);
    a

  let to_array a = a

  let length a = Array.length a

  let subst_fn fn t = 
    let t' = CArray.Smart.map fn t in
      if t' == t then t else of_array t'

  let levels x = LSet.of_array x

  let pr prl ?variance =
    let ppu i u =
      let v = Option.map (fun v -> v.(i)) variance in
      pr_opt_no_spc Variance.pr v ++ prl u
    in
    prvecti_with_sep spc ppu

  let equal t u = 
    t == u ||
      (Array.is_empty t && Array.is_empty u) ||
      (CArray.for_all2 Level.equal t u 
	 (* Necessary as universe instances might come from different modules and 
	    unmarshalling doesn't preserve sharing *))

end

let enforce_eq_instances x y = 
  let ax = Instance.to_array x and ay = Instance.to_array y in
    if Array.length ax != Array.length ay then
      anomaly (Pp.(++) (Pp.str "Invalid argument: enforce_eq_instances called with")
		 (Pp.str " instances of different lengths."));
    CArray.fold_right2 enforce_eq_level ax ay

let enforce_eq_variance_instances = Variance.eq_constraints
let enforce_leq_variance_instances = Variance.leq_constraints

let subst_instance_level s l =
  match l.Level.data with
  | Level.Var n -> s.(n) 
  | _ -> l

let subst_instance_instance s i = 
  Array.Smart.map (fun l -> subst_instance_level s l) i

let subst_instance_universe s u =
  let f x = Universe.Expr.map (fun u -> subst_instance_level s u) x in
  let u' = Universe.smart_map f u in
    if u == u' then u
    else Universe.sort u'

let subst_instance_constraint s (u,d,v as c) =
  let u' = subst_instance_level s u in
  let v' = subst_instance_level s v in
    if u' == u && v' == v then c
    else (u',d,v')

let subst_instance_constraints s csts =
  Constraint.fold 
    (fun c csts -> Constraint.add (subst_instance_constraint s c) csts)
    csts Constraint.empty 

type universe_instance = Instance.t

type 'a puniverses = 'a * Instance.t
let out_punivs (x, y) = x
let in_punivs x = (x, Instance.empty)
let eq_puniverses f (x, u) (y, u') =
  f x y && Instance.equal u u'

(** A context of universe levels with universe constraints,
    representing local universe variables and constraints *)

module UContext =
struct
  type t = Instance.t constrained

  let make x = x

  (** Universe contexts (variables as a list) *)
  let empty = (Instance.empty, Constraint.empty)
  let is_empty (univs, cst) = Instance.is_empty univs && Constraint.is_empty cst

  let pr prl ?variance (univs, cst as ctx) =
    if is_empty ctx then mt() else
      h 0 (Instance.pr prl ?variance univs ++ str " |= ") ++ h 0 (v 0 (Constraint.pr prl cst))

  let hcons (univs, cst) =
    (Instance.hcons univs, hcons_constraints cst)

  let instance (univs, cst) = univs
  let constraints (univs, cst) = cst

  let union (univs, cst) (univs', cst') =
    Instance.append univs univs', Constraint.union cst cst'

  let dest x = x

  let size (x,_) = Instance.length x

end

type universe_context = UContext.t
let hcons_universe_context = UContext.hcons

module AUContext =
struct
  include UContext

  let repr (inst, cst) =
    (Array.mapi (fun i l -> Level.var i) inst, cst)

  let instantiate inst (u, cst) =
    assert (Array.length u = Array.length inst);
    subst_instance_constraints inst cst

end

type abstract_universe_context = AUContext.t
let hcons_abstract_universe_context = AUContext.hcons

(** Universe info for cumulative inductive types: A context of
   universe levels with universe constraints, representing local
   universe variables and constraints, together with an array of
   Variance.t.

    This data structure maintains the invariant that the variance
   array has the same length as the universe instance. *)
module CumulativityInfo =
struct
  type t = universe_context * Variance.t array

  let make x =
    if (Instance.length (UContext.instance (fst x))) =
       (Array.length (snd x)) then x
    else anomaly (Pp.str "Invalid subtyping information encountered!")

  let empty = (UContext.empty, [||])
  let is_empty (univs, variance) = UContext.is_empty univs && Array.is_empty variance

  let pr prl (univs, variance) =
    UContext.pr prl ~variance univs

  let hcons (univs, variance) = (* should variance be hconsed? *)
    (UContext.hcons univs, variance)

  let univ_context (univs, subtypcst) = univs
  let variance (univs, variance) = variance

  (** This function takes a universe context representing constraints
     of an inductive and produces a CumulativityInfo.t with the
     trivial subtyping relation. *)
  let from_universe_context univs =
    (univs, Array.init (UContext.size univs) (fun _ -> Variance.Invariant))

  let leq_constraints (_,variance) u u' csts = Variance.leq_constraints variance u u' csts
  let eq_constraints (_,variance) u u' csts = Variance.eq_constraints variance u u' csts

end

type cumulativity_info = CumulativityInfo.t
let hcons_cumulativity_info = CumulativityInfo.hcons

module ACumulativityInfo = CumulativityInfo

type abstract_cumulativity_info = ACumulativityInfo.t
let hcons_abstract_cumulativity_info = ACumulativityInfo.hcons

(** A set of universes with universe constraints.
    We linearize the set to a list after typechecking. 
    Beware, representation could change.
*)

module ContextSet =
struct
  type t = universe_set constrained

  let empty = (LSet.empty, Constraint.empty)
  let is_empty (univs, cst) = LSet.is_empty univs && Constraint.is_empty cst

  let equal (univs, cst as x) (univs', cst' as y) =
    x == y || (LSet.equal univs univs' && Constraint.equal cst cst')
									
  let of_set s = (s, Constraint.empty)
  let singleton l = of_set (LSet.singleton l)
  let of_instance i = of_set (Instance.levels i)

  let union (univs, cst as x) (univs', cst' as y) =
    if x == y then x
    else LSet.union univs univs', Constraint.union cst cst'

  let append (univs, cst) (univs', cst') =
    let univs = LSet.fold LSet.add univs univs' in
    let cst = Constraint.fold Constraint.add cst cst' in
    (univs, cst)

  let diff (univs, cst) (univs', cst') =
    LSet.diff univs univs', Constraint.diff cst cst'

  let add_universe u (univs, cst) =
    LSet.add u univs, cst

  let add_constraints cst' (univs, cst) =
    univs, Constraint.union cst cst'

  let add_instance inst (univs, cst) =
    let v = Instance.to_array inst in
    let fold accu u = LSet.add u accu in
    let univs = Array.fold_left fold univs v in
    (univs, cst)

  let sort_levels a = 
    Array.sort Level.natural_compare a; a

  let to_context (ctx, cst) =
    (Instance.of_array (sort_levels (Array.of_list (LSet.elements ctx))), cst)

  let of_context (ctx, cst) =
    (Instance.levels ctx, cst)

  let pr prl (univs, cst as ctx) =
    if is_empty ctx then mt() else
      h 0 (LSet.pr prl univs ++ str " |= ") ++ h 0 (v 0 (Constraint.pr prl cst))

  let constraints (univs, cst) = cst
  let levels (univs, cst) = univs

  let size (univs,_) = LSet.cardinal univs
end

type universe_context_set = ContextSet.t

(** A value in a universe context (resp. context set). *)
type 'a in_universe_context = 'a * universe_context
type 'a in_universe_context_set = 'a * universe_context_set

(** Substitutions. *)

let empty_subst = LMap.empty
let is_empty_subst = LMap.is_empty

let empty_level_subst = LMap.empty
let is_empty_level_subst = LMap.is_empty

(** Substitution functions *)

(** With level to level substitutions. *)
let subst_univs_level_level subst l =
  try LMap.find l subst
  with Not_found -> l

let subst_univs_level_universe subst u =
  let f x = Universe.Expr.map (fun u -> subst_univs_level_level subst u) x in
  let u' = Universe.smart_map f u in
    if u == u' then u
    else Universe.sort u'

let subst_univs_level_instance subst i =
  let i' = Instance.subst_fn (subst_univs_level_level subst) i in
    if i == i' then i
    else i'
	
let subst_univs_level_constraint subst (u,d,v) =
  let u' = subst_univs_level_level subst u 
  and v' = subst_univs_level_level subst v in
    if d != Lt && Level.equal u' v' then None
    else Some (u',d,v')

let subst_univs_level_constraints subst csts =
  Constraint.fold 
    (fun c -> Option.fold_right Constraint.add (subst_univs_level_constraint subst c))
    csts Constraint.empty 

let subst_univs_level_abstract_universe_context subst (inst, csts) =
  inst, subst_univs_level_constraints subst csts

(** With level to universe substitutions. *)
type universe_subst_fn = universe_level -> universe

let make_subst subst = fun l -> LMap.find l subst

let subst_univs_expr_opt fn (l,n) =
  Universe.addn n (fn l)

let subst_univs_universe fn ul =
  let subst, nosubst = 
    List.fold_right (fun u (subst,nosubst) -> 
      try let a' = subst_univs_expr_opt fn u in
	    (a' :: subst, nosubst)
      with Not_found -> (subst, u :: nosubst))
      ul ([], [])
  in 
    if CList.is_empty subst then ul
    else 
      let substs = 
	List.fold_left Universe.merge_univs Universe.empty subst
      in
	List.fold_left (fun acc u -> Universe.merge_univs acc (Universe.tip u))
	  substs nosubst

let make_instance_subst i = 
  let arr = Instance.to_array i in
    Array.fold_left_i (fun i acc l ->
      LMap.add l (Level.var i) acc)
      LMap.empty arr

let make_inverse_instance_subst i = 
  let arr = Instance.to_array i in
    Array.fold_left_i (fun i acc l ->
      LMap.add (Level.var i) l acc)
      LMap.empty arr

let make_abstract_instance (ctx, _) = 
  Array.mapi (fun i l -> Level.var i) ctx

let abstract_universes ctx =
  let instance = UContext.instance ctx in
  let subst = make_instance_subst instance in
  let cstrs = subst_univs_level_constraints subst 
      (UContext.constraints ctx)
  in
  let ctx = UContext.make (instance, cstrs) in
  instance, ctx

let abstract_cumulativity_info (univs, variance) =
  let subst, univs = abstract_universes univs in
  subst, (univs, variance)

let rec compact_univ s vars i u =
  match u with
  | [] -> (s, List.rev vars)
  | (lvl, _) :: u ->
    match Level.var_index lvl with
    | Some k when not (LMap.mem lvl s) ->
      let lvl' = Level.var i in
      compact_univ (LMap.add lvl lvl' s) (k :: vars) (i+1) u
    | _ -> compact_univ s vars i u

let compact_univ u =
  let (s, s') = compact_univ LMap.empty [] 0 u in
  (subst_univs_level_universe s u, s')

(** Pretty-printing *)

let pr_constraints prl = Constraint.pr prl

let pr_universe_context = UContext.pr

let pr_cumulativity_info = CumulativityInfo.pr

let pr_abstract_universe_context = AUContext.pr

let pr_abstract_cumulativity_info = ACumulativityInfo.pr

let pr_universe_context_set = ContextSet.pr

let pr_universe_subst = 
  LMap.pr (fun u -> str" := " ++ Universe.pr u ++ spc ())

let pr_universe_level_subst = 
  LMap.pr (fun u -> str" := " ++ Level.pr u ++ spc ())

module Huniverse_set = 
  Hashcons.Make(
    struct
      type t = universe_set
      type u = universe_level -> universe_level
      let hashcons huc s =
	LSet.fold (fun x -> LSet.add (huc x)) s LSet.empty
      let eq s s' =
	LSet.equal s s'
      let hash = Hashtbl.hash
    end)

let hcons_universe_set = 
  Hashcons.simple_hcons Huniverse_set.generate Huniverse_set.hcons Level.hcons

let hcons_universe_context_set (v, c) = 
  (hcons_universe_set v, hcons_constraints c)

let hcons_univ x = Universe.hcons x

let explain_universe_inconsistency prl (o,u,v,p) =
  let pr_uni = Universe.pr_with prl in
  let pr_rel = function
    | Eq -> str"=" | Lt -> str"<" | Le -> str"<=" 
  in
  let reason = match p with
    | None -> mt()
    | Some p ->
      let p = Lazy.force p in
      if p = [] then mt ()
      else
        str " because" ++ spc() ++ pr_uni v ++
	prlist (fun (r,v) -> spc() ++ pr_rel r ++ str" " ++ pr_uni v)
          p ++
	(if Universe.equal (snd (List.last p)) u then mt() else
           (spc() ++ str "= " ++ pr_uni u))
  in
    str "Cannot enforce" ++ spc() ++ pr_uni u ++ spc() ++
      pr_rel o ++ spc() ++ pr_uni v ++ reason

let compare_levels = Level.compare
let eq_levels = Level.equal
let equal_universes = Universe.equal