summaryrefslogtreecommitdiff
path: root/library/universes.ml
blob: 79070763ec1db9929aa4cc76e4f277eabd07046b (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
(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

open Util
open Pp
open Names
open Term
open Environ
open Univ

type universe_names = 
    Univ.universe_level Idmap.t * Id.t Univ.LMap.t

let global_universes = Summary.ref ~name:"Global universe names" 
  ((Idmap.empty, Univ.LMap.empty) : universe_names)

let global_universe_names () = !global_universes
let set_global_universe_names s = global_universes := s

let pr_with_global_universes l = 
  try Nameops.pr_id (LMap.find l (snd !global_universes))
  with Not_found -> Level.pr l

type universe_constraint_type = ULe | UEq | ULub

type universe_constraint = universe * universe_constraint_type * universe

module Constraints = struct
  module S = Set.Make(
  struct 
    type t = universe_constraint

    let compare_type c c' =
      match c, c' with
      | ULe, ULe -> 0
      | ULe, _ -> -1
      | _, ULe -> 1
      | UEq, UEq -> 0
      | UEq, _ -> -1
      | ULub, ULub -> 0
      | ULub, _ -> 1
      
    let compare (u,c,v) (u',c',v') =
      let i = compare_type c c' in
	if Int.equal i 0 then
	  let i' = Universe.compare u u' in
	    if Int.equal i' 0 then Universe.compare v v'
	    else 
	      if c != ULe && Universe.compare u v' = 0 && Universe.compare v u' = 0 then 0
	      else i'
	else i
  end)
  
  include S
  
  let add (l,d,r as cst) s = 
    if Universe.equal l r then s
    else add cst s

  let tr_dir = function
    | ULe -> Le
    | UEq -> Eq
    | ULub -> Eq

  let op_str = function ULe -> " <= " | UEq -> " = " | ULub -> " /\\ "

  let pr c =
    fold (fun (u1,op,u2) pp_std ->
	pp_std ++ Universe.pr u1 ++ str (op_str op) ++
	Universe.pr u2 ++ fnl ()) c (str "")

  let equal x y = 
    x == y || equal x y

end

type universe_constraints = Constraints.t
type 'a universe_constrained = 'a * universe_constraints

type 'a universe_constraint_function = 'a -> 'a -> universe_constraints -> universe_constraints

let enforce_eq_instances_univs strict x y c = 
  let d = if strict then ULub else UEq in
  let ax = Instance.to_array x and ay = Instance.to_array y in
    if Array.length ax != Array.length ay then
      Errors.anomaly (Pp.str "Invalid argument: enforce_eq_instances_univs called with" ++
	       Pp.str " instances of different lengths");
    CArray.fold_right2
      (fun x y -> Constraints.add (Universe.make x, d, Universe.make y))
      ax ay c

let subst_univs_universe_constraint fn (u,d,v) =
  let u' = subst_univs_universe fn u and v' = subst_univs_universe fn v in
    if Universe.equal u' v' then None
    else Some (u',d,v')

let subst_univs_universe_constraints subst csts =
  Constraints.fold 
    (fun c -> Option.fold_right Constraints.add (subst_univs_universe_constraint subst c))
    csts Constraints.empty 


let to_constraints g s = 
  let tr (x,d,y) acc =
    let add l d l' acc = Constraint.add (l,Constraints.tr_dir d,l') acc in
      match Universe.level x, d, Universe.level y with
      | Some l, (ULe | UEq | ULub), Some l' -> add l d l' acc
      | _, ULe, Some l' -> enforce_leq x y acc
      | _, ULub, _ -> acc
      | _, d, _ -> 
	let f = if d == ULe then check_leq else check_eq in
	  if f g x y then acc else 
	    raise (Invalid_argument 
		   "to_constraints: non-trivial algebraic constraint between universes")
  in Constraints.fold tr s Constraint.empty

let eq_constr_univs_infer univs m n =
  if m == n then true, Constraints.empty
  else 
    let cstrs = ref Constraints.empty in
    let eq_universes strict = Univ.Instance.check_eq univs in
    let eq_sorts s1 s2 = 
      if Sorts.equal s1 s2 then true
      else
	let u1 = Sorts.univ_of_sort s1 and u2 = Sorts.univ_of_sort s2 in
	  if Univ.check_eq univs u1 u2 then true
	  else
	    (cstrs := Constraints.add (u1, UEq, u2) !cstrs;
	     true)
    in
    let rec eq_constr' m n = 
      m == n ||	Constr.compare_head_gen eq_universes eq_sorts eq_constr' m n
    in
    let res = Constr.compare_head_gen eq_universes eq_sorts eq_constr' m n in
      res, !cstrs

let leq_constr_univs_infer univs m n =
  if m == n then true, Constraints.empty
  else 
    let cstrs = ref Constraints.empty in
    let eq_universes strict l l' = Univ.Instance.check_eq univs l l' in
    let eq_sorts s1 s2 = 
      if Sorts.equal s1 s2 then true
      else
	let u1 = Sorts.univ_of_sort s1 and u2 = Sorts.univ_of_sort s2 in
	  if Univ.check_eq univs u1 u2 then true
	  else (cstrs := Constraints.add (u1, UEq, u2) !cstrs;
		true)
    in
    let leq_sorts s1 s2 = 
      if Sorts.equal s1 s2 then true
      else 
	let u1 = Sorts.univ_of_sort s1 and u2 = Sorts.univ_of_sort s2 in
	  if Univ.check_leq univs u1 u2 then true
	  else
	    (cstrs := Constraints.add (u1, ULe, u2) !cstrs; 
	     true)
    in
    let rec eq_constr' m n = 
      m == n ||	Constr.compare_head_gen eq_universes eq_sorts eq_constr' m n
    in
    let rec compare_leq m n =
      Constr.compare_head_gen_leq eq_universes eq_sorts leq_sorts 
	eq_constr' leq_constr' m n
    and leq_constr' m n = m == n || compare_leq m n in
    let res = compare_leq m n in
      res, !cstrs

let eq_constr_universes m n =
  if m == n then true, Constraints.empty
  else 
    let cstrs = ref Constraints.empty in
    let eq_universes strict l l' = 
      cstrs := enforce_eq_instances_univs strict l l' !cstrs; true in
    let eq_sorts s1 s2 = 
      if Sorts.equal s1 s2 then true
      else
	(cstrs := Constraints.add 
	   (Sorts.univ_of_sort s1, UEq, Sorts.univ_of_sort s2) !cstrs;
	 true)
    in
    let rec eq_constr' m n = 
      m == n ||	Constr.compare_head_gen eq_universes eq_sorts eq_constr' m n
    in
    let res = Constr.compare_head_gen eq_universes eq_sorts eq_constr' m n in
      res, !cstrs

let leq_constr_universes m n =
  if m == n then true, Constraints.empty
  else 
    let cstrs = ref Constraints.empty in
    let eq_universes strict l l' = 
      cstrs := enforce_eq_instances_univs strict l l' !cstrs; true in
    let eq_sorts s1 s2 = 
      if Sorts.equal s1 s2 then true
      else (cstrs := Constraints.add 
	      (Sorts.univ_of_sort s1,UEq,Sorts.univ_of_sort s2) !cstrs; 
	    true)
    in
    let leq_sorts s1 s2 = 
      if Sorts.equal s1 s2 then true
      else 
	(cstrs := Constraints.add 
	   (Sorts.univ_of_sort s1,ULe,Sorts.univ_of_sort s2) !cstrs; 
	 true)
    in
    let rec eq_constr' m n = 
      m == n ||	Constr.compare_head_gen eq_universes eq_sorts eq_constr' m n
    in
    let rec compare_leq m n =
      Constr.compare_head_gen_leq eq_universes eq_sorts leq_sorts eq_constr' leq_constr' m n
    and leq_constr' m n = m == n || compare_leq m n in
    let res = compare_leq m n in
      res, !cstrs

let compare_head_gen_proj env equ eqs eqc' m n =
  match kind_of_term m, kind_of_term n with
  | Proj (p, c), App (f, args)
  | App (f, args), Proj (p, c) -> 
    (match kind_of_term f with
    | Const (p', u) when eq_constant (Projection.constant p) p' -> 
      let pb = Environ.lookup_projection p env in
      let npars = pb.Declarations.proj_npars in
	if Array.length args == npars + 1 then
	  eqc' c args.(npars)
	else false
    | _ -> false)
  | _ -> Constr.compare_head_gen equ eqs eqc' m n
    
let eq_constr_universes_proj env m n =
  if m == n then true, Constraints.empty
  else 
    let cstrs = ref Constraints.empty in
    let eq_universes strict l l' = 
      cstrs := enforce_eq_instances_univs strict l l' !cstrs; true in
    let eq_sorts s1 s2 = 
      if Sorts.equal s1 s2 then true
      else
	(cstrs := Constraints.add 
	   (Sorts.univ_of_sort s1, UEq, Sorts.univ_of_sort s2) !cstrs;
	 true)
    in
    let rec eq_constr' m n = 
      m == n ||	compare_head_gen_proj env eq_universes eq_sorts eq_constr' m n
    in
    let res = eq_constr' m n in
      res, !cstrs

(* Generator of levels *)
let new_univ_level, set_remote_new_univ_level =
  RemoteCounter.new_counter ~name:"Universes" 0 ~incr:((+) 1)
    ~build:(fun n -> Univ.Level.make (Global.current_dirpath ()) n)

let new_univ_level _ = new_univ_level ()
  (* Univ.Level.make db (new_univ_level ()) *)

let fresh_level () = new_univ_level (Global.current_dirpath ())

(* TODO: remove *)
let new_univ dp = Univ.Universe.make (new_univ_level dp)
let new_Type dp = mkType (new_univ dp)
let new_Type_sort dp = Type (new_univ dp)

let fresh_universe_instance ctx =
  Instance.subst_fn (fun _ -> new_univ_level (Global.current_dirpath ())) 
    (UContext.instance ctx)

let fresh_instance_from_context ctx =
  let inst = fresh_universe_instance ctx in
  let constraints = instantiate_univ_constraints inst ctx in
    inst, constraints

let fresh_instance ctx =
  let ctx' = ref LSet.empty in
  let inst = 
    Instance.subst_fn (fun v -> 
      let u = new_univ_level (Global.current_dirpath ()) in
	ctx' := LSet.add u !ctx'; u) 
      (UContext.instance ctx)
  in !ctx', inst

let existing_instance ctx inst = 
  let () = 
    let a1 = Instance.to_array inst 
    and a2 = Instance.to_array (UContext.instance ctx) in
    let len1 = Array.length a1 and len2 = Array.length a2 in 
      if not (len1 == len2) then
	Errors.errorlabstrm "Universes"
	  (str "Polymorphic constant expected " ++ int len2 ++ 
	     str" levels but was given " ++ int len1)
      else ()
  in LSet.empty, inst

let fresh_instance_from ctx inst =
  let ctx', inst = 
    match inst with 
    | Some inst -> existing_instance ctx inst
    | None -> fresh_instance ctx 
  in
  let constraints = instantiate_univ_constraints inst ctx in
    inst, (ctx', constraints)

let unsafe_instance_from ctx =
  (Univ.UContext.instance ctx, ctx)
    
(** Fresh universe polymorphic construction *)

let fresh_constant_instance env c inst =
  let cb = lookup_constant c env in
    if cb.Declarations.const_polymorphic then
      let inst, ctx =
        fresh_instance_from
          (Declareops.universes_of_constant (Environ.opaque_tables env) cb) inst
      in
	((c, inst), ctx)
    else ((c,Instance.empty), ContextSet.empty)

let fresh_inductive_instance env ind inst = 
  let mib, mip = Inductive.lookup_mind_specif env ind in
    if mib.Declarations.mind_polymorphic then
      let inst, ctx = fresh_instance_from mib.Declarations.mind_universes inst in
	((ind,inst), ctx)
    else ((ind,Instance.empty), ContextSet.empty)

let fresh_constructor_instance env (ind,i) inst = 
  let mib, mip = Inductive.lookup_mind_specif env ind in
    if mib.Declarations.mind_polymorphic then
      let inst, ctx = fresh_instance_from mib.Declarations.mind_universes inst in
	(((ind,i),inst), ctx)
    else (((ind,i),Instance.empty), ContextSet.empty)

let unsafe_constant_instance env c =
  let cb = lookup_constant c env in
    if cb.Declarations.const_polymorphic then
      let inst, ctx = unsafe_instance_from
        (Declareops.universes_of_constant (Environ.opaque_tables env) cb) in
	((c, inst), ctx)
    else ((c,Instance.empty), UContext.empty)

let unsafe_inductive_instance env ind = 
  let mib, mip = Inductive.lookup_mind_specif env ind in
    if mib.Declarations.mind_polymorphic then
      let inst, ctx = unsafe_instance_from mib.Declarations.mind_universes in
	((ind,inst), ctx)
    else ((ind,Instance.empty), UContext.empty)

let unsafe_constructor_instance env (ind,i) = 
  let mib, mip = Inductive.lookup_mind_specif env ind in
    if mib.Declarations.mind_polymorphic then
      let inst, ctx = unsafe_instance_from mib.Declarations.mind_universes in
	(((ind,i),inst), ctx)
    else (((ind,i),Instance.empty), UContext.empty)

open Globnames

let fresh_global_instance ?names env gr =
  match gr with
  | VarRef id -> mkVar id, ContextSet.empty
  | ConstRef sp -> 
     let c, ctx = fresh_constant_instance env sp names in
       mkConstU c, ctx
  | ConstructRef sp ->
     let c, ctx = fresh_constructor_instance env sp names in
       mkConstructU c, ctx
  | IndRef sp -> 
     let c, ctx = fresh_inductive_instance env sp names in
       mkIndU c, ctx

let fresh_constant_instance env sp = 
  fresh_constant_instance env sp None

let fresh_inductive_instance env sp = 
  fresh_inductive_instance env sp None

let fresh_constructor_instance env sp = 
  fresh_constructor_instance env sp None

let unsafe_global_instance env gr =
  match gr with
  | VarRef id -> mkVar id, UContext.empty
  | ConstRef sp -> 
     let c, ctx = unsafe_constant_instance env sp in
       mkConstU c, ctx
  | ConstructRef sp ->
     let c, ctx = unsafe_constructor_instance env sp in
       mkConstructU c, ctx
  | IndRef sp -> 
     let c, ctx = unsafe_inductive_instance env sp in
       mkIndU c, ctx

let constr_of_global gr =
  let c, ctx = fresh_global_instance (Global.env ()) gr in
    if not (Univ.ContextSet.is_empty ctx) then
      if Univ.LSet.is_empty (Univ.ContextSet.levels ctx) then 
	(* Should be an error as we might forget constraints, allow for now
	   to make firstorder work with "using" clauses *)
	c
      else raise (Invalid_argument 
		    ("constr_of_global: globalization of polymorphic reference " ^ 
			Pp.string_of_ppcmds (Nametab.pr_global_env Id.Set.empty gr) ^
			" would forget universes."))
    else c

let constr_of_reference = constr_of_global

let unsafe_constr_of_global gr =
  unsafe_global_instance (Global.env ()) gr

let constr_of_global_univ (gr,u) =
  match gr with
  | VarRef id -> mkVar id
  | ConstRef sp -> mkConstU (sp,u)
  | ConstructRef sp -> mkConstructU (sp,u)
  | IndRef sp -> mkIndU (sp,u)

let fresh_global_or_constr_instance env = function
  | IsConstr c -> c, ContextSet.empty
  | IsGlobal gr -> fresh_global_instance env gr

let global_of_constr c =
  match kind_of_term c with
  | Const (c, u) -> ConstRef c, u
  | Ind (i, u) -> IndRef i, u
  | Construct (c, u) -> ConstructRef c, u
  | Var id -> VarRef id, Instance.empty
  | _ -> raise Not_found

let global_app_of_constr c =
  match kind_of_term c with
  | Const (c, u) -> (ConstRef c, u), None
  | Ind (i, u) -> (IndRef i, u), None
  | Construct (c, u) -> (ConstructRef c, u), None
  | Var id -> (VarRef id, Instance.empty), None
  | Proj (p, c) -> (ConstRef (Projection.constant p), Instance.empty), Some c
  | _ -> raise Not_found

open Declarations

let type_of_reference env r =
  match r with
  | VarRef id -> Environ.named_type id env, ContextSet.empty
  | ConstRef c ->
     let cb = Environ.lookup_constant c env in
     let ty = Typeops.type_of_constant_type env cb.const_type in
       if cb.const_polymorphic then
	 let inst, ctx = fresh_instance_from (Declareops.universes_of_constant (Environ.opaque_tables env) cb) None in
	   Vars.subst_instance_constr inst ty, ctx
       else ty, ContextSet.empty

  | IndRef ind ->
     let (mib, oib as specif) = Inductive.lookup_mind_specif env ind in
       if mib.mind_polymorphic then
	 let inst, ctx = fresh_instance_from mib.mind_universes None in
	 let ty = Inductive.type_of_inductive env (specif, inst) in
	   ty, ctx
       else
	 let ty = Inductive.type_of_inductive env (specif, Univ.Instance.empty) in
	   ty, ContextSet.empty
  | ConstructRef cstr ->
     let (mib,oib as specif) = Inductive.lookup_mind_specif env (inductive_of_constructor cstr) in
       if mib.mind_polymorphic then
	 let inst, ctx = fresh_instance_from mib.mind_universes None in
	   Inductive.type_of_constructor (cstr,inst) specif, ctx
       else Inductive.type_of_constructor (cstr,Instance.empty) specif, ContextSet.empty

let type_of_global t = type_of_reference (Global.env ()) t

let unsafe_type_of_reference env r =
  match r with
  | VarRef id -> Environ.named_type id env
  | ConstRef c ->
     let cb = Environ.lookup_constant c env in
       Typeops.type_of_constant_type env cb.const_type

  | IndRef ind ->
     let (mib, oib as specif) = Inductive.lookup_mind_specif env ind in
     let (_, inst), _ = unsafe_inductive_instance env ind in
       Inductive.type_of_inductive env (specif, inst)

  | ConstructRef (ind, _ as cstr) ->
     let (mib,oib as specif) = Inductive.lookup_mind_specif env (inductive_of_constructor cstr) in
     let (_, inst), _ = unsafe_inductive_instance env ind in
       Inductive.type_of_constructor (cstr,inst) specif

let unsafe_type_of_global t = unsafe_type_of_reference (Global.env ()) t

let fresh_sort_in_family env = function
  | InProp -> prop_sort, ContextSet.empty
  | InSet -> set_sort, ContextSet.empty
  | InType -> 
    let u = fresh_level () in
      Type (Univ.Universe.make u), ContextSet.singleton u

let new_sort_in_family sf =
  fst (fresh_sort_in_family (Global.env ()) sf)

let extend_context (a, ctx) (ctx') =
  (a, ContextSet.union ctx ctx')

let new_global_univ () =
  let u = fresh_level () in
    (Univ.Universe.make u, ContextSet.singleton u)

(** Simplification *)

module LevelUnionFind = Unionfind.Make (Univ.LSet) (Univ.LMap)

let add_list_map u t map =
  try
    let l = LMap.find u map in
    LMap.update u (t :: l) map
  with Not_found ->
    LMap.add u [t] map

module UF = LevelUnionFind

(** Precondition: flexible <= ctx *)
let choose_canonical ctx flexible algs s =
  let global = LSet.diff s ctx in
  let flexible, rigid = LSet.partition flexible (LSet.inter s ctx) in
    (** If there is a global universe in the set, choose it *)
    if not (LSet.is_empty global) then
      let canon = LSet.choose global in
	canon, (LSet.remove canon global, rigid, flexible)
    else (** No global in the equivalence class, choose a rigid one *)
	if not (LSet.is_empty rigid) then
	  let canon = LSet.choose rigid in
	    canon, (global, LSet.remove canon rigid, flexible)
	else (** There are only flexible universes in the equivalence
		 class, choose a non-algebraic. *)
	  let algs, nonalgs = LSet.partition (fun x -> LSet.mem x algs) flexible in
	    if not (LSet.is_empty nonalgs) then
	      let canon = LSet.choose nonalgs in
		canon, (global, rigid, LSet.remove canon flexible)
	    else
	      let canon = LSet.choose algs in
		canon, (global, rigid, LSet.remove canon flexible)

let subst_univs_fn_puniverses lsubst (c, u as cu) =
  let u' = Instance.subst_fn lsubst u in
    if u' == u then cu else (c, u')

let nf_evars_and_universes_opt_subst f subst =
  let subst = fun l -> match LMap.find l subst with None -> raise Not_found | Some l' -> l' in
  let lsubst = Univ.level_subst_of subst in
  let rec aux c =
    match kind_of_term c with
    | Evar (evk, args) ->
      let args = Array.map aux args in
      (match try f (evk, args) with Not_found -> None with
      | None -> c
      | Some c -> aux c)
    | Const pu -> 
      let pu' = subst_univs_fn_puniverses lsubst pu in
	if pu' == pu then c else mkConstU pu'
    | Ind pu ->
      let pu' = subst_univs_fn_puniverses lsubst pu in
	if pu' == pu then c else mkIndU pu'
    | Construct pu ->
      let pu' = subst_univs_fn_puniverses lsubst pu in
	if pu' == pu then c else mkConstructU pu'
    | Sort (Type u) ->
      let u' = Univ.subst_univs_universe subst u in
	if u' == u then c else mkSort (sort_of_univ u')
    | _ -> map_constr aux c
  in aux

let fresh_universe_context_set_instance ctx =
  if ContextSet.is_empty ctx then LMap.empty, ctx
  else
    let (univs, cst) = ContextSet.levels ctx, ContextSet.constraints ctx in
    let univs',subst = LSet.fold
      (fun u (univs',subst) ->
	let u' = fresh_level () in
	  (LSet.add u' univs', LMap.add u u' subst))
      univs (LSet.empty, LMap.empty)
    in
    let cst' = subst_univs_level_constraints subst cst in
      subst, (univs', cst')

let normalize_univ_variable ~find ~update =
  let rec aux cur =
    let b = find cur in
    let b' = subst_univs_universe aux b in
      if Universe.equal b' b then b
      else update cur b'
  in aux

let normalize_univ_variable_opt_subst ectx =
  let find l = 
    match Univ.LMap.find l !ectx with
    | Some b -> b
    | None -> raise Not_found
  in
  let update l b =
    assert (match Universe.level b with Some l' -> not (Level.equal l l') | None -> true);
    ectx := Univ.LMap.add l (Some b) !ectx; b
  in normalize_univ_variable ~find ~update

let normalize_univ_variable_subst subst =
  let find l = Univ.LMap.find l !subst in
  let update l b =
    assert (match Universe.level b with Some l' -> not (Level.equal l l') | None -> true);
    subst := Univ.LMap.add l b !subst; b in
    normalize_univ_variable ~find ~update

let normalize_universe_opt_subst subst =
  let normlevel = normalize_univ_variable_opt_subst subst in
    subst_univs_universe normlevel

let normalize_universe_subst subst =
  let normlevel = normalize_univ_variable_subst subst in
    subst_univs_universe normlevel

let normalize_opt_subst ctx = 
  let ectx = ref ctx in
  let normalize = normalize_univ_variable_opt_subst ectx in
  let () =
    Univ.LMap.iter (fun u v ->
      if Option.is_empty v then ()
      else try ignore(normalize u) with Not_found -> assert(false)) ctx 
  in !ectx

type universe_opt_subst = universe option universe_map
	  
let make_opt_subst s = 
  fun x -> 
    (match Univ.LMap.find x s with
    | Some u -> u
    | None -> raise Not_found)

let subst_opt_univs_constr s = 
  let f = make_opt_subst s in
    Vars.subst_univs_fn_constr f


let normalize_univ_variables ctx = 
  let ctx = normalize_opt_subst ctx in
  let undef, def, subst =
    Univ.LMap.fold (fun u v (undef, def, subst) -> 
      match v with
      | None -> (Univ.LSet.add u undef, def, subst)
      | Some b -> (undef, Univ.LSet.add u def, Univ.LMap.add u b subst))
    ctx (Univ.LSet.empty, Univ.LSet.empty, Univ.LMap.empty)
  in ctx, undef, def, subst

let pr_universe_body = function
  | None -> mt ()
  | Some v -> str" := " ++ Univ.Universe.pr v

let pr_universe_opt_subst = Univ.LMap.pr pr_universe_body

exception Found of Level.t
let find_inst insts v =
  try LMap.iter (fun k (enf,alg,v') ->
    if not alg && enf && Universe.equal v' v then raise (Found k))
	insts; raise Not_found
  with Found l -> l

let compute_lbound left =
 (** The universe variable was not fixed yet.
     Compute its level using its lower bound. *)
  let sup l lbound = 
    match lbound with
    | None -> Some l
    | Some l' -> Some (Universe.sup l l')
  in
    List.fold_left (fun lbound (d, l) -> 
      if d == Le (* l <= ?u *) then sup l lbound
      else (* l < ?u *) 
	(assert (d == Lt); 
	 if not (Universe.level l == None) then
	   sup (Universe.super l) lbound
	 else None))
      None left
  
let instantiate_with_lbound u lbound alg enforce (ctx, us, algs, insts, cstrs) =
  if enforce then
    let inst = Universe.make u in
    let cstrs' = enforce_leq lbound inst cstrs in
      (ctx, us, LSet.remove u algs, 
       LMap.add u (enforce,alg,lbound) insts, cstrs'), (enforce, alg, inst)
  else (* Actually instantiate *)
    (Univ.LSet.remove u ctx, Univ.LMap.add u (Some lbound) us, algs,
     LMap.add u (enforce,alg,lbound) insts, cstrs), (enforce, alg, lbound)

type constraints_map = (Univ.constraint_type * Univ.LMap.key) list Univ.LMap.t

let pr_constraints_map cmap =
  LMap.fold (fun l cstrs acc -> 
    Level.pr l ++ str " => " ++ 
      prlist_with_sep spc (fun (d,r) -> pr_constraint_type d ++ Level.pr r) cstrs ++ fnl ()
    ++ acc)
    cmap (mt ())
	
let minimize_univ_variables ctx us algs left right cstrs =
  let left, lbounds = 
    Univ.LMap.fold (fun r lower (left, lbounds as acc)  ->
      if Univ.LMap.mem r us || not (Univ.LSet.mem r ctx) then acc
      else (* Fixed universe, just compute its glb for sharing *)
	let lbounds' = 
	  match compute_lbound (List.map (fun (d,l) -> d, Universe.make l) lower) with
	  | None -> lbounds
	  | Some lbound -> LMap.add r (true, false, lbound) lbounds
	in (Univ.LMap.remove r left, lbounds'))
      left (left, Univ.LMap.empty)
  in
  let rec instance (ctx', us, algs, insts, cstrs as acc) u =
    let acc, left = 
      try let l = LMap.find u left in
	    List.fold_left (fun (acc, left') (d, l) -> 
	      let acc', (enf,alg,l') = aux acc l in
		(* if alg then assert(not alg); *)
		let l' = 
		  if enf then Universe.make l 
		  else l' 
		(* match Universe.level l' with Some _ -> l' | None -> Universe.make l  *)
		in
		  acc', (d, l') :: left') (acc, []) l
      with Not_found -> acc, []
    and right =
      try Some (LMap.find u right)
      with Not_found -> None
    in
    let instantiate_lbound lbound =
      let alg = LSet.mem u algs in
      if alg then
	 (* u is algebraic and has no upper bound constraints: we
	    instantiate it with it's lower bound, if any *)
	 instantiate_with_lbound u lbound true false acc
      else (* u is non algebraic *)
	match Universe.level lbound with
	| Some l -> (* The lowerbound is directly a level *) 
	  (* u is not algebraic but has no upper bounds,
	     we instantiate it with its lower bound if it is a 
	     different level, otherwise we keep it. *)
	  if not (Level.equal l u) && not (LSet.mem l algs) then
	    (* if right = None then. Should check that u does not 
	       have upper constraints that are not already in right *)
	      instantiate_with_lbound u lbound false false acc
	    (* else instantiate_with_lbound u lbound false true acc *)
	  else
	    (* assert false: l can't be alg *)
	    acc, (true, false, lbound)
	| None -> 
	  try 
	    (* if right <> None then raise Not_found; *)
	    (* Another universe represents the same lower bound, 
	       we can share them with no harm. *)
	    let can = find_inst insts lbound in
	      instantiate_with_lbound u (Universe.make can) false false acc
	  with Not_found -> 
	    (* We set u as the canonical universe representing lbound *)
	    instantiate_with_lbound u lbound false true acc
    in
    let acc' acc = 
      match right with
      | None -> acc
      | Some cstrs -> 
	let dangling = List.filter (fun (d, r) -> not (LMap.mem r us)) cstrs in
	  if List.is_empty dangling then acc
	  else
	    let ((ctx', us, algs, insts, cstrs), (enf,_,inst as b)) = acc in
	    let cstrs' = List.fold_left (fun cstrs (d, r) -> 
	      if d == Univ.Le then
		enforce_leq inst (Universe.make r) cstrs
	      else 
		try let lev = Option.get (Universe.level inst) in
		      Constraint.add (lev, d, r) cstrs
		with Option.IsNone -> assert false)
	      cstrs dangling
	    in
	      (ctx', us, algs, insts, cstrs'), b
    in
      if not (LSet.mem u ctx) then acc' (acc, (true, false, Universe.make u))
      else
	let lbound = compute_lbound left in
	  match lbound with
	  | None -> (* Nothing to do *)
	    acc' (acc, (true, false, Universe.make u))
	  | Some lbound ->
	    acc' (instantiate_lbound lbound)
  and aux (ctx', us, algs, seen, cstrs as acc) u =
    try acc, LMap.find u seen 
    with Not_found -> instance acc u
  in
    LMap.fold (fun u v (ctx', us, algs, seen, cstrs as acc) -> 
      if v == None then fst (aux acc u)
      else LSet.remove u ctx', us, LSet.remove u algs, seen, cstrs)
      us (ctx, us, algs, lbounds, cstrs)
      
let normalize_context_set ctx us algs = 
  let (ctx, csts) = ContextSet.levels ctx, ContextSet.constraints ctx in
  let uf = UF.create () in
  let csts = 
    (* We first put constraints in a normal-form: all self-loops are collapsed
       to equalities. *)
    let g = Univ.merge_constraints csts Univ.empty_universes in
      Univ.constraints_of_universes g
  in
  let noneqs =
    Constraint.fold (fun (l,d,r) noneqs ->
      if d == Eq then (UF.union l r uf; noneqs) 
      else Constraint.add (l,d,r) noneqs)
    csts Constraint.empty
  in
  let partition = UF.partition uf in
  let flex x = LMap.mem x us in
  let ctx, subst, eqs = List.fold_left (fun (ctx, subst, cstrs) s -> 
    let canon, (global, rigid, flexible) = choose_canonical ctx flex algs s in
    (* Add equalities for globals which can't be merged anymore. *)
    let cstrs = LSet.fold (fun g cst -> 
      Constraint.add (canon, Univ.Eq, g) cst) global
      cstrs 
    in
    let subst = LSet.fold (fun f -> LMap.add f canon) rigid subst in 
    let subst = LSet.fold (fun f -> LMap.add f canon) flexible subst in 
      (LSet.diff (LSet.diff ctx rigid) flexible, subst, cstrs))
    (ctx, LMap.empty, Constraint.empty) partition
  in
  (* Noneqs is now in canonical form w.r.t. equality constraints, 
     and contains only inequality constraints. *)
  let noneqs = subst_univs_level_constraints subst noneqs in
  let us = LMap.fold (fun u v acc -> LMap.add u (Some (Universe.make v)) acc) subst us in
  (* Compute the left and right set of flexible variables, constraints
     mentionning other variables remain in noneqs. *)
  let noneqs, ucstrsl, ucstrsr = 
    Constraint.fold (fun (l,d,r as cstr) (noneq, ucstrsl, ucstrsr) -> 
      let lus = LMap.mem l us 
      and rus = LMap.mem r us
      in
      let ucstrsl' = 
	if lus then add_list_map l (d, r) ucstrsl
	else ucstrsl
      and ucstrsr' = 
	add_list_map r (d, l) ucstrsr
      in 
      let noneqs = 
	if lus || rus then noneq 
	else Constraint.add cstr noneq
      in (noneqs, ucstrsl', ucstrsr'))
    noneqs (Constraint.empty, LMap.empty, LMap.empty)
  in
  (* Now we construct the instantiation of each variable. *)
  let ctx', us, algs, inst, noneqs = 
    minimize_univ_variables ctx us algs ucstrsr ucstrsl noneqs
  in
  let us = normalize_opt_subst us in
    (us, algs), (ctx', Constraint.union noneqs eqs)

(* let normalize_conkey = Profile.declare_profile "normalize_context_set" *)
(* let normalize_context_set a b c = Profile.profile3 normalize_conkey normalize_context_set a b c *)

let universes_of_constr c =
  let rec aux s c = 
    match kind_of_term c with
    | Const (_, u) | Ind (_, u) | Construct (_, u) ->
      LSet.union (Instance.levels u) s
    | Sort u when not (Sorts.is_small u) -> 
      let u = univ_of_sort u in
	LSet.union (Universe.levels u) s
    | _ -> fold_constr aux s c
  in aux LSet.empty c

let restrict_universe_context (univs,csts) s =
  (* Universes that are not necessary to typecheck the term.
     E.g. univs introduced by tactics and not used in the proof term. *)
  let diff = LSet.diff univs s in
  let rec aux diff candid univs ness = 
    let (diff', candid', univs', ness') = 
      Constraint.fold
	(fun (l, d, r as c) (diff, candid, univs, csts) ->
	  if not (LSet.mem l diff) then
	    (LSet.remove r diff, candid, univs, Constraint.add c csts)
	  else if not (LSet.mem r diff) then
	    (LSet.remove l diff, candid, univs, Constraint.add c csts)
	  else (diff, Constraint.add c candid, univs, csts))
	candid (diff, Constraint.empty, univs, ness)
    in
      if ness' == ness then (LSet.diff univs diff', ness)
      else aux diff' candid' univs' ness'
  in aux diff csts univs Constraint.empty

let simplify_universe_context (univs,csts) =
  let uf = UF.create () in
  let noneqs =
    Constraint.fold (fun (l,d,r) noneqs ->
      if d == Eq && (LSet.mem l univs || LSet.mem r univs) then 
	(UF.union l r uf; noneqs)
      else Constraint.add (l,d,r) noneqs)
      csts Constraint.empty
  in
  let partition = UF.partition uf in
  let flex x = LSet.mem x univs in
  let subst, univs', csts' = List.fold_left (fun (subst, univs, cstrs) s -> 
    let canon, (global, rigid, flexible) = choose_canonical univs flex LSet.empty s in
    (* Add equalities for globals which can't be merged anymore. *)
    let cstrs = LSet.fold (fun g cst -> 
      Constraint.add (canon, Univ.Eq, g) cst) (LSet.union global rigid)
      cstrs 
    in
    let subst = LSet.fold (fun f -> LMap.add f canon)
      flexible subst
    in (subst, LSet.diff univs flexible, cstrs))
    (LMap.empty, univs, noneqs) partition
  in
  (* Noneqs is now in canonical form w.r.t. equality constraints, 
     and contains only inequality constraints. *)
  let csts' = subst_univs_level_constraints subst csts' in
    (univs', csts'), subst

let is_small_leq (l,d,r) =
  Level.is_small l && d == Univ.Le

(* Prop < i <-> Set+1 <= i <-> Set < i *)
let translate_cstr (l,d,r as cstr) =
  if Level.equal Level.prop l && d == Univ.Lt then
    (Level.set, d, r)
  else cstr

let refresh_constraints univs (ctx, cstrs) =
  let cstrs', univs' = 
    Univ.Constraint.fold (fun c (cstrs', univs as acc) -> 
      let c = translate_cstr c in
      if Univ.check_constraint univs c && not (is_small_leq c) then acc
      else (Univ.Constraint.add c cstrs', Univ.enforce_constraint c univs))
      cstrs (Univ.Constraint.empty, univs)
  in ((ctx, cstrs'), univs')


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

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

(*
   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 -> univ_level_mem u v
  | None -> false

let solve_constraints_system levels level_bounds level_min =
  let open Univ in
  let levels =
    Array.mapi (fun i o ->
      match o with
      | Some u ->
	(match Universe.level u with 
	| Some u -> Some u 
	| _ -> level_bounds.(i) <- Universe.sup level_bounds.(i) u; None)
      | None -> None)
      levels in
  let v = Array.copy level_bounds in
  let nind = Array.length v in
  let clos = Array.map (fun _ -> Int.Set.empty) levels in
  (* First compute the transitive closure of the levels dependencies *)
  for i=0 to nind-1 do
    for j=0 to nind-1 do
      if not (Int.equal i j) && is_direct_sort_constraint levels.(j) v.(i) then
	clos.(i) <- Int.Set.add j clos.(i);
    done;
  done;
  let rec closure () = 
    let continue = ref false in
      Array.iteri (fun i deps -> 
	let deps' = 
	  Int.Set.fold (fun j acc -> Int.Set.union acc clos.(j)) deps deps
	in 
	  if Int.Set.equal deps deps' then ()
	  else (clos.(i) <- deps'; continue := true))
	clos;
      if !continue then closure ()
      else ()
  in 
  closure ();
  for i=0 to nind-1 do
    for j=0 to nind-1 do
      if not (Int.equal i j) && Int.Set.mem j clos.(i) then
	(v.(i) <- Universe.sup v.(i) level_bounds.(j);
	 level_min.(i) <- Universe.sup level_min.(i) level_min.(j))
    done;
    for j=0 to nind-1 do
      match levels.(j) with
      | Some u -> v.(i) <- univ_level_rem u v.(i) level_min.(i)
      | None -> ()
    done
  done;
  v