aboutsummaryrefslogtreecommitdiffhomepage
path: root/checker/univ.ml
blob: 46b3ce6808190c6e8525b674c717503d05bcdc09 (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
(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

(* 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] *)

(* Revisions by Bruno Barras, Hugo Herbelin, Pierre Letouzey *)

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

  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

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

  let set = make Set
  let prop = make Prop
  let var i = make (Var i)
		  
  let is_small x = 
    match data x with
    | Level _ -> false
    | _ -> 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 to_string x = 
    match data x with
    | Prop -> "Prop"
    | Set -> "Set"
    | Level (n,d) -> Names.DirPath.to_string d^"."^string_of_int n
    | Var i -> "Var("^string_of_int i^")"

  let pr u = str (to_string u)

  let make m n = make (Level (n, m))

end

(** Level sets and maps *)
module LMap = HMap.Make (Level)
module LSet = LMap.Set

type 'a universe_map = 'a LMap.t

type universe_level = Level.t

type universe_level_subst_fn = universe_level -> universe_level

(* 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
	
    let make l = (l, 0)

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

    let is_prop = function
      | (l,0) -> Level.is_prop 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 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)
	
    let super (u,n as x) (v,n' as y) =
      let cmp = Level.compare u v in
	if Int.equal cmp 0 then 
	  if n < n' then Inl true
	  else Inl false
	else if is_prop x then Inl true
	else if is_prop y then Inl false
	else Inr 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 level = function
      | (v,0) -> Some v
      | _ -> None

    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 u = [u]
  let cons u v = u :: v

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

  let make l = tip (Expr.make l)

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

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

  (* 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 juste above the universe variable u.
     Used to type the sort u. *)
  let super l = 
    List.map (fun x -> Expr.successor x) l

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

  let rec merge_univs l1 l2 =
    match l1, l2 with
    | [], _ -> l2
    | _, [] -> l1
    | h1 :: t1, h2 :: t2 ->
      (match Expr.super h1 h2 with
      | Inl true (* h1 < h2 *) -> merge_univs t1 l2
      | Inl false -> merge_univs l1 t2
      | Inr 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' ->
        (match Expr.super a b with
	| Inl false -> aux a l'
	| Inl true -> l
	| Inr 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 smartmap = List.smartmap

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 pr_uni = Universe.pr

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

open Universe

(* Comparison on this type is pointer equality *)
type canonical_arc =
    { univ: Level.t;
      lt: Level.t list;
      le: Level.t list;
      rank : int;
      predicative : bool}

let terminal u = {univ=u; lt=[]; le=[]; rank=0; predicative=false}

module UMap :
sig
  type key = Level.t
  type +'a t
  val empty : 'a t
  val add : key -> 'a -> 'a t -> 'a t
  val find : key -> 'a t -> 'a
  val fold : (key -> 'a -> 'b -> 'b) -> 'a t -> 'b -> 'b
end = HMap.Make(Level)

(* A Level.t 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 Level.t

type universes = univ_entry UMap.t

let enter_equiv_arc u v g =
  UMap.add u (Equiv v) g

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

(* Every Level.t has a unique canonical arc representative *)

(* repr : universes -> Level.t -> canonical_arc *)
(* canonical representative : we follow the Equiv links *)

let repr g u =
  let rec repr_rec u =
    let a =
      try UMap.find u g
      with Not_found -> anomaly ~label:"Univ.repr"
	  (str"Universe " ++ Level.pr u ++ str" undefined.")
    in
    match a with
      | Equiv v -> repr_rec v
      | Canonical arc -> arc
  in
  repr_rec u

let get_set_arc g = repr g Level.set

exception AlreadyDeclared
	    
let add_universe vlev strict g =
  try
    let _arcv = UMap.find vlev g in
      raise AlreadyDeclared
  with Not_found -> 
    let v = terminal vlev in
    let arc =
      let arc = get_set_arc g in
	if strict then
	  { arc with lt=vlev::arc.lt}
	else 
	  { arc with le=vlev::arc.le}
    in
    let g = enter_arc arc g in
      enter_arc v g
		       
(* 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 : Level.t -> canonical_arc -> canonical_arc list *)
(* between u v = { w | u<=w<=v, w canonical }          *)
(* between is the most costly operation *)

let between g arcu 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) arcu 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 constraint_type = Lt | Le | Eq

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

(** [compare_neq] : is [arcv] in the transitive upward closure of [arcu] ?

  In [strict] mode, we fully distinguish between LE and LT, while in
  non-strict mode, we simply answer LE for both situations.

  If [arcv] is encountered in a LT part, we could directly answer
  without visiting unneeded parts of this transitive closure.
  In [strict] mode, if [arcv] is encountered in a LE part, we could only
  change the default answer (1st arg [c]) from NLE to LE, since a strict
  constraint may appear later. During the recursive traversal,
  [lt_done] and [le_done] are universes we have already visited,
  they do not contain [arcv]. The 4rd arg is [(lt_todo,le_todo)],
  two lists of universes not yet considered, known to be above [arcu],
  strictly or not.

  We use depth-first search, but the presence of [arcv] in [new_lt]
  is checked as soon as possible : this seems to be slightly faster
  on a test.
*)

type fast_order = FastEQ | FastLT | FastLE | FastNLE

let fast_compare_neq strict g arcu arcv =
  (* [c] characterizes whether arcv has already been related
     to arcu among the lt_done,le_done universe *)
  let rec cmp c lt_done le_done lt_todo le_todo = match lt_todo, le_todo with
  | [],[] -> c
  | arc::lt_todo, le_todo ->
    if List.memq arc lt_done then
      cmp c lt_done le_done lt_todo le_todo
    else
      let rec find lt_todo lt le = match le with
      | [] ->
        begin match lt with
        | [] -> cmp c (arc :: lt_done) le_done lt_todo le_todo
        | u :: lt ->
          let arc = repr g u in
          if arc == arcv then
            if strict then FastLT else FastLE
          else find (arc :: lt_todo) lt le
        end
      | u :: le ->
        let arc = repr g u in
        if arc == arcv then
          if strict then FastLT else FastLE
        else find (arc :: lt_todo) lt le
      in
      find lt_todo arc.lt arc.le
  | [], arc::le_todo ->
    if arc == arcv then
      (* No need to continue inspecting universes above arc:
	 if arcv is strictly above arc, then we would have a cycle.
         But we cannot answer LE yet, a stronger constraint may
	 come later from [le_todo]. *)
      if strict then cmp FastLE lt_done le_done [] le_todo else FastLE
    else
      if (List.memq arc lt_done) || (List.memq arc le_done) then
        cmp c lt_done le_done [] le_todo
      else
        let rec find lt_todo lt = match lt with
        | [] ->
          let fold accu u =
            let node = repr g u in
            node :: accu
          in
          let le_new = List.fold_left fold le_todo arc.le in
          cmp c lt_done (arc :: le_done) lt_todo le_new
        | u :: lt ->
          let arc = repr g u in
          if arc == arcv then
            if strict then FastLT else FastLE
          else find (arc :: lt_todo) lt
        in
        find [] arc.lt
  in
  cmp FastNLE [] [] [] [arcu]

let fast_compare g arcu arcv =
  if arcu == arcv then FastEQ else fast_compare_neq true g arcu arcv

let is_leq g arcu arcv =
  arcu == arcv ||
    (match fast_compare_neq false g arcu arcv with
    | FastNLE -> false
    | (FastEQ|FastLE|FastLT) -> true)
    
let is_lt g arcu arcv =
  if arcu == arcv then false
  else
    match fast_compare_neq true g arcu arcv with
    | FastLT -> true
    | (FastEQ|FastLE|FastNLE) -> false

(* 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 *)

(** * Universe checks [check_eq] and [check_leq], used in coqchk *)

(** First, checks on universe levels *)

let check_equal g u v =
  let arcu = repr g u in
  let arcv = repr g v in
  arcu == arcv

let check_eq_level g u v = u == v || check_equal g u v

let is_set_arc u = Level.is_set u.univ
let is_prop_arc u = Level.is_prop u.univ

let check_smaller g strict u v =
  let arcu = repr g u in
  let arcv = repr g v in
  if strict then
    is_lt g arcu arcv
  else
    is_prop_arc arcu 
    || (is_set_arc arcu && arcv.predicative) 
    || is_leq g arcu arcv

(** Then, checks on universes *)

type 'a check_function = universes -> 'a -> 'a -> bool

let check_equal_expr g x y =
  x == y || (let (u, n) = x and (v, m) = y in 
	       Int.equal n m && check_equal g u v)

let check_eq_univs g l1 l2 =
  let f x1 x2 = check_equal_expr g x1 x2 in
  let exists x1 l = List.exists (fun x2 -> f x1 x2) l in
    List.for_all (fun x1 -> exists x1 l2) l1
    && List.for_all (fun x2 -> exists x2 l1) l2

let check_eq g u v =
  Universe.equal u v || check_eq_univs g u v

let check_smaller_expr g (u,n) (v,m) =
  let diff = n - m in
    match diff with
    | 0 -> check_smaller g false u v
    | 1 -> check_smaller g true u v
    | x when x < 0 -> check_smaller g false u v
    | _ -> false

let exists_bigger g ul l =
  Universe.exists (fun ul' -> 
    check_smaller_expr g ul ul') l

let real_check_leq g u v =
  Universe.for_all (fun ul -> exists_bigger g ul v) u
    
let check_leq g u v =
  Universe.equal u v ||
    Universe.is_type0m u ||
    check_eq_univs g u v || real_check_leq g u v

(** Enforcing new constraints : [setlt], [setleq], [merge], [merge_disc] *)

(** To speed up tests of Set </<= i *)
let set_predicative g arcv = 
  enter_arc {arcv with predicative = true} g

(* setlt : Level.t -> Level.t -> reason -> unit *)
(* forces u > v *)
(* this is normally an update of u in g rather than a creation. *)
let setlt g arcu arcv =
  let arcu' = {arcu with lt=arcv.univ::arcu.lt} in
  let g = 
    if is_set_arc arcu then set_predicative g arcv
    else g
  in
    enter_arc arcu' g, arcu'

(* checks that non-redundant *)
let setlt_if (g,arcu) v =
  let arcv = repr g v in
  if is_lt g arcu arcv then g, arcu
  else setlt g arcu arcv

(* setleq : Level.t -> Level.t -> unit *)
(* forces u >= v *)
(* this is normally an update of u in g rather than a creation. *)
let setleq g arcu arcv =
  let arcu' = {arcu with le=arcv.univ::arcu.le} in
  let g = 
    if is_set_arc arcu' then
      set_predicative g arcv
    else g
  in
    enter_arc arcu' g, arcu'

(* checks that non-redundant *)
let setleq_if (g,arcu) v =
  let arcv = repr g v in
  if is_leq g arcu arcv then g, arcu
  else setleq g arcu arcv

(* merge : Level.t -> Level.t -> unit *)
(* we assume  compare(u,v) = LE *)
(* merge u v  forces u ~ v with repr u as canonical repr *)
let merge g arcu arcv =
  (* we find the arc with the biggest rank, and we redirect all others to it *)
  let arcu, g, v =
    let best_ranked (max_rank, old_max_rank, best_arc, rest) arc =
      if Level.is_small arc.univ || arc.rank >= max_rank
      then (arc.rank, max_rank, arc, best_arc::rest)
      else (max_rank, old_max_rank, best_arc, arc::rest)
    in
      match between g arcu arcv with
      | [] -> anomaly (str "Univ.between.")
      | arc::rest ->
        let (max_rank, old_max_rank, best_arc, rest) =
          List.fold_left best_ranked (arc.rank, min_int, arc, []) rest in
          if max_rank > old_max_rank then best_arc, g, rest
          else begin
              (* one redirected node also has max_rank *)
            let arcu = {best_arc with rank = max_rank + 1} in
	      arcu, enter_arc arcu g, rest
          end 
  in
  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_arcu = (g',arcu) in
  let g_arcu = List.fold_left setlt_if g_arcu w in
  let g_arcu = List.fold_left setleq_if g_arcu w' in
  fst g_arcu

(* merge_disc : Level.t -> Level.t -> 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 arc1 arc2 =
  let arcu, arcv = if arc1.rank < arc2.rank then arc2, arc1 else arc1, arc2 in
  let arcu, g = 
    if not (Int.equal arc1.rank arc2.rank) then arcu, g
    else
      let arcu = {arcu with rank = succ arcu.rank} in 
      arcu, enter_arc arcu g
  in
  let g' = enter_equiv_arc arcv.univ arcu.univ g in
  let g_arcu = (g',arcu) in
  let g_arcu = List.fold_left setlt_if g_arcu arcv.lt in
  let g_arcu = List.fold_left setleq_if g_arcu arcv.le in
  fst g_arcu

(* 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

exception UniverseInconsistency of univ_inconsistency

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

(* enforc_univ_eq : Level.t -> Level.t -> unit *)
(* enforc_univ_eq u v will force u=v if possible, will fail otherwise *)

let enforce_univ_eq u v g =
  let arcu = repr g u in
  let arcv = repr g v in
    match fast_compare g arcu arcv with
    | FastEQ -> g
    | FastLT -> error_inconsistency Eq v u
    | FastLE -> merge g arcu arcv
    | FastNLE ->
      (match fast_compare g arcv arcu with
      | FastLT -> error_inconsistency Eq u v
      | FastLE -> merge g arcv arcu
      | FastNLE -> merge_disc g arcu arcv
      | FastEQ -> anomaly (Pp.str "Univ.compare."))

(* enforce_univ_leq : Level.t -> Level.t -> unit *)
(* enforce_univ_leq u v will force u<=v if possible, will fail otherwise *)
let enforce_univ_leq u v g =
  let arcu = repr g u in
  let arcv = repr g v in
  if is_leq g arcu arcv then g
  else
    match fast_compare g arcv arcu with
    | FastLT -> error_inconsistency Le u v
    | FastLE  -> merge g arcv arcu
    | FastNLE -> fst (setleq g arcu arcv)
    | FastEQ -> anomaly (Pp.str "Univ.compare.")

(* enforce_univ_lt u v will force u<v if possible, will fail otherwise *)
let enforce_univ_lt u v g =
  let arcu = repr g u in
  let arcv = repr g v in
    match fast_compare g arcu arcv with
    | FastLT -> g
    | FastLE -> fst (setlt g arcu arcv)
    | FastEQ -> error_inconsistency Lt u v
    | FastNLE ->
      match fast_compare_neq false g arcv arcu with
	FastNLE -> fst (setlt g arcu arcv)
      | FastEQ -> anomaly (Pp.str "Univ.compare.")
      | FastLE | FastLT -> error_inconsistency Lt u v

(* Prop = Set is forbidden here. *)
let initial_universes =
  let g = enter_arc (terminal Level.set) UMap.empty in
  let g = enter_arc (terminal Level.prop) g in
    enforce_univ_lt Level.prop Level.set g

(* Constraints and sets of constraints. *)    

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

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

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

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

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 "")
end

let empty_constraint = Constraint.empty
let merge_constraints c g =
  Constraint.fold enforce_constraint c g
		  
type constraints = Constraint.t

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

(** Constraint functions. *)

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

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))
	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 =
  match v with
  | [v] ->
    List.fold_right (fun u -> constraint_add_leq u v) u c
  | _ -> anomaly (Pp.str"A universe bound can only be a variable.")

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

let check_constraint g (l,d,r) =
  match d with
  | Eq -> check_equal g l r
  | Le -> check_smaller g false l r
  | Lt -> check_smaller g true l r

let check_constraints c g =
  Constraint.for_all (check_constraint g) c

(**********************************************************************)
(** 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 Instance : sig 
    type t = Level.t array

    val empty : t
    val is_empty : t -> bool
    val equal : t -> t -> bool
    val subst_fn : universe_level_subst_fn -> t -> t
    val subst : universe_level_subst -> t -> t
    val pr : t -> Pp.t
    val check_eq : t check_function
    val length : t -> int
    val append : t -> t -> t
    val of_array : Level.t array -> t
end = 
struct
  type t = Level.t array

  let empty = [||]

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

  let subst_fn fn t = 
    let t' = CArray.smartmap fn t in
      if t' == t then t else t'

  let subst s t =
    let t' = 
      CArray.smartmap (fun x -> try LMap.find x s with Not_found -> x) t
    in if t' == t then t else t'

  let pr =
    prvect_with_sep spc Level.pr

  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 *))

  let check_eq g t1 t2 =
    t1 == t2 ||
      (Int.equal (Array.length t1) (Array.length t2) &&
	 let rec aux i =
	   (Int.equal i (Array.length t1)) || (check_eq_level g t1.(i) t2.(i) && aux (i + 1))
	 in aux 0)

  let length = Array.length

  let append = Array.append

  let of_array i = i

end

(** Substitute instance inst for ctx in csts *)

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

let subst_instance_instance s i = 
  Array.smartmap (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.smartmap 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
(** A context of universe levels with universe constraints,
    representiong local universe variables and constraints *)

module UContext =
struct
  type t = Instance.t constrained

  (** Universe contexts (variables as a list) *)
  let empty = (Instance.empty, Constraint.empty)
  let make x = x
  let instance (univs, cst) = univs
  let constraints (univs, cst) = cst
  let size (univs, _) = Instance.length univs

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

type universe_context = UContext.t

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

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 leq_constraint csts variance u u' =
    match variance with
    | Irrelevant -> csts
    | Covariant -> Constraint.add (u, Le, u') csts
    | Invariant -> Constraint.add (u, Eq, u') csts

  let eq_constraint csts variance u u' =
    match variance with
    | Irrelevant -> csts
    | Covariant | Invariant -> Constraint.add (u, Eq, 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 CumulativityInfo =
struct
  type t = universe_context * Variance.t array

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

end

module ACumulativityInfo = CumulativityInfo
type abstract_cumulativity_info = ACumulativityInfo.t

module ContextSet =
struct
  type t = LSet.t constrained
  let empty = LSet.empty, Constraint.empty
  let constraints (_, cst) = cst
  let levels (ctx, _) = ctx
  let make ctx cst = (ctx, cst)
end
type universe_context_set = ContextSet.t

(** Instance subtyping *)

let check_subtype univs ctxT ctx =
  if AUContext.size ctx == AUContext.size ctx then
    let (inst, cst) = AUContext.repr ctx in
    let cstT = UContext.constraints (AUContext.repr ctxT) in
    let push accu v = add_universe v false accu in
    let univs = Array.fold_left push univs inst in
    let univs = merge_constraints cstT univs in
    check_constraints cst univs
  else false

(** Substitutions. *)

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.smartmap f u in
    if u == u' then u
    else Universe.sort u'

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

(** 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 merge_context strict ctx g =
  let g = Array.fold_left
   (* Be lenient, module typing reintroduces universes and 
      constraints due to includes *)
	    (fun g v -> try add_universe v strict g with AlreadyDeclared -> g)
	    g (UContext.instance ctx)
  in merge_constraints (UContext.constraints ctx) g

let merge_context_set strict ctx g =
  let g = LSet.fold
	    (fun v g -> try add_universe v strict g with AlreadyDeclared -> g)
	    (ContextSet.levels ctx) g
  in merge_constraints (ContextSet.constraints ctx) g

(** Pretty-printing *)

let pr_constraints prl = Constraint.pr prl
    
let pr_universe_context = UContext.pr

let pr_arc = function
  | _, Canonical {univ=u; lt=[]; le=[]} ->
      mt ()
  | _, Canonical {univ=u; lt=lt; le=le} ->
      let opt_sep = match lt, le with
      | [], _ | _, [] -> mt ()
      | _ -> spc ()
      in
      Level.pr u ++ str " " ++
      v 0
        (pr_sequence (fun v -> str "< " ++ Level.pr v) lt ++
	 opt_sep ++
         pr_sequence (fun v -> str "<= " ++ Level.pr v) le) ++
      fnl ()
  | u, Equiv v ->
      Level.pr u  ++ str " = " ++ Level.pr v ++ fnl ()

let pr_universes g =
  let graph = UMap.fold (fun u a l -> (u,a)::l) g [] in
  prlist pr_arc graph