1073:
274:
944:
161:
561:
717:
114:
169:
1118:
905:
449:
310:
501:
869:
667:
403:
833:
797:
769:
423:
377:
743:
353:
615:
1138:
641:
589:
333:
1068:{\displaystyle \lambda k^{\square }\lambda \alpha ^{k\to k}\lambda \beta ^{k}\!.\alpha (\alpha \beta )\;:\;\Pi k:\square .((k\to k)\to k\to k)}
1221:
1145:
40:
122:
515:
1245:
679:
81:
60:
1309:
723:
is a function from types to types”, that is, a polymorphic type). The rules restrict how we can form new kinds.
269:{\displaystyle \{(\ast ,\ast ),(\square ,\ast ),(\square ,\square ),(\triangle ,\ast ),(\triangle ,\square )\}}
1253:
920:
872:
1082:
878:
428:
1314:
1269:
283:
36:
469:
1148:, this is equivalent to all logical propositions being provable, which makes the system inconsistent.
842:
1319:
1159:
836:
646:
382:
806:
923:
356:
63:
was inconsistent as it allowed the same "Type in Type" behaviour that Girard's paradox exploits.
20:
782:
748:
408:
362:
1217:
1185:"Interprétation fonctionnelle et Élimination des coupures de l'arithmétique d'ordre supérieur"
1141:
728:
338:
76:
56:
594:
1288:
1209:
1163:
908:
48:
32:
1123:
626:
574:
318:
1261:
1241:
930:
in analogy to polymorphic types of terms in classical polymorphic lambda calculi, such as
1277:
Hurkens, Antonius J. C. (1995). Dezani-Ciancaglini, Mariangiola; Plotkin, Gordon (eds.).
1204:
Sørensen, Morten Heine; Urzyczyn, Paweł (2006). "Pure type systems and the lambda cube".
379:
doesn't have a specific name. The two axioms describe the containment of types in kinds (
1278:
1213:
1184:
1303:
777:
is a kind”). Similarly we can build related terms, according to what the rules allow.
1284:
1285:
Second
International Conference on Typed Lambda Calculi and Applications (TLCA '95)
1155:
931:
1292:
1079:
This mechanism is sufficient to construct a term with the type
934:. An example of such a generic constructor might be (where
919:
The definitions of System U and U allow the assignment of
55:
was formulated). This result led to the realization that
43:, axioms and rules (or dependencies between the sorts).
1199:
1197:
280:
System U is defined the same with the exception of the
451:). Intuitively, the sorts describe a hierarchy in the
156:{\displaystyle \{\ast :\square ,\square :\triangle \}}
1126:
1085:
947:
881:
845:
809:
803:
The rules govern the dependencies between the sorts:
785:
751:
731:
682:
649:
629:
597:
577:
518:
472:
431:
411:
385:
365:
341:
321:
286:
172:
125:
84:
556:{\displaystyle f:\mathrm {Nat} \to \mathrm {Bool} }
1132:
1112:
1067:
899:
863:
827:
791:
763:
737:
711:
661:
635:
609:
583:
555:
495:
443:
417:
397:
371:
347:
327:
304:
268:
155:
108:
993:
569:is a function from natural numbers to booleans”).
1248:. In S. Abramsky; D. Gabbay; T. Maibaum (eds.).
509:is a boolean”) or a (dependent) function type (
71:System U is defined as a pure type system with
712:{\displaystyle \mathrm {List} :\ast \to \ast }
109:{\displaystyle \{\ast ,\square ,\triangle \}}
8:
263:
173:
150:
126:
103:
85:
51:in 1972 (and the question of consistency of
1264:(1986). "An analysis of Girard's paradox".
1016:
1012:
1125:
1084:
987:
968:
955:
946:
880:
844:
808:
784:
750:
730:
683:
681:
648:
628:
596:
576:
539:
525:
517:
479:
471:
430:
410:
384:
364:
340:
320:
285:
171:
124:
83:
1206:Lectures on the Curry–Howard isomorphism
16:Special forms of a typed lambda calculus
1175:
835:says that values may depend on values (
355:are conventionally called “Type” and “
1250:Handbook of Logic in Computer Science
7:
1280:A simplification of Girard's paradox
1140:), which implies that every type is
1113:{\displaystyle (\forall p:\ast ,p)}
900:{\displaystyle (\square ,\square )}
444:{\displaystyle \square :\triangle }
1127:
1089:
1017:
871:allows values to depend on types (
786:
693:
690:
687:
684:
549:
546:
543:
540:
532:
529:
526:
489:
486:
483:
480:
438:
412:
366:
305:{\displaystyle (\triangle ,\ast )}
290:
251:
233:
147:
100:
14:
907:allows types to depend on types (
643:we can build more terms, such as
496:{\displaystyle b:\mathrm {Bool} }
864:{\displaystyle (\square ,\ast )}
1287:. Edinburgh. pp. 266–278.
745:is the sort of all such kinds (
673:of unary type-level operators (
591:is the sort of all such types (
1107:
1086:
1062:
1056:
1050:
1047:
1041:
1035:
1032:
1009:
1000:
972:
894:
882:
858:
846:
822:
810:
799:is the sort of all such terms.
703:
662:{\displaystyle \ast \to \ast }
653:
536:
398:{\displaystyle \ast :\square }
299:
287:
260:
248:
242:
230:
224:
212:
206:
194:
188:
176:
1:
1214:10.1016/S0049-237X(06)80015-7
828:{\displaystyle (\ast ,\ast )}
39:with an arbitrary number of
1254:Oxford Science Publications
1246:"Lambda calculi with types"
1146:Curry–Howard correspondence
47:was proved inconsistent by
1336:
1183:Girard, Jean-Yves (1972).
792:{\displaystyle \triangle }
764:{\displaystyle k:\square }
418:{\displaystyle \triangle }
372:{\displaystyle \triangle }
359:”, respectively; the sort
35:, i.e. special forms of a
1266:Logic in Computer Science
938:denotes a kind variable)
1272:Press. pp. 227–236.
1120:(equivalent to the type
738:{\displaystyle \square }
348:{\displaystyle \square }
914:
610:{\displaystyle t:\ast }
463:, such as a base type (
1134:
1114:
1069:
901:
865:
829:
793:
765:
739:
713:
663:
637:
611:
585:
557:
497:
445:
419:
399:
373:
349:
329:
306:
270:
157:
110:
1270:IEEE Computer Society
1135:
1133:{\displaystyle \bot }
1115:
1070:
902:
866:
830:
794:
766:
740:
714:
664:
638:
636:{\displaystyle \ast }
612:
586:
584:{\displaystyle \ast }
558:
498:
446:
420:
400:
374:
350:
330:
328:{\displaystyle \ast }
307:
271:
158:
111:
37:typed lambda calculus
1124:
1083:
945:
928:generic constructors
879:
843:
807:
783:
749:
729:
680:
647:
627:
595:
575:
516:
470:
429:
409:
383:
363:
339:
319:
284:
170:
123:
82:
1256:. pp. 117–309.
1293:10.1007/BFb0014058
1130:
1110:
1065:
897:
861:
825:
789:
761:
735:
709:
659:
633:
623:is a type”). From
607:
581:
553:
493:
459:All values have a
441:
415:
395:
369:
345:
325:
302:
266:
153:
106:
21:mathematical logic
1160:Russell's paradox
67:Formal definition
33:pure type systems
1327:
1296:
1273:
1262:Coquand, Thierry
1257:
1242:Barendregt, Henk
1228:
1227:
1201:
1192:
1191:
1189:
1180:
1152:Girard's paradox
1139:
1137:
1136:
1131:
1119:
1117:
1116:
1111:
1074:
1072:
1071:
1066:
992:
991:
979:
978:
960:
959:
915:Girard's paradox
906:
904:
903:
898:
870:
868:
867:
862:
834:
832:
831:
826:
798:
796:
795:
790:
776:
770:
768:
767:
762:
744:
742:
741:
736:
722:
718:
716:
715:
710:
696:
668:
666:
665:
660:
642:
640:
639:
634:
622:
616:
614:
613:
608:
590:
588:
587:
582:
568:
562:
560:
559:
554:
552:
535:
508:
502:
500:
499:
494:
492:
450:
448:
447:
442:
424:
422:
421:
416:
404:
402:
401:
396:
378:
376:
375:
370:
354:
352:
351:
346:
334:
332:
331:
326:
311:
309:
308:
303:
275:
273:
272:
267:
162:
160:
159:
154:
115:
113:
112:
107:
61:1971 type theory
49:Jean-Yves Girard
1335:
1334:
1330:
1329:
1328:
1326:
1325:
1324:
1310:Lambda calculus
1300:
1299:
1276:
1260:
1240:
1237:
1235:Further reading
1232:
1231:
1224:
1203:
1202:
1195:
1187:
1182:
1181:
1177:
1172:
1122:
1121:
1081:
1080:
983:
964:
951:
943:
942:
917:
877:
876:
841:
840:
805:
804:
781:
780:
772:
747:
746:
727:
726:
720:
678:
677:
645:
644:
625:
624:
618:
593:
592:
573:
572:
564:
514:
513:
504:
468:
467:
427:
426:
407:
406:
405:) and kinds in
381:
380:
361:
360:
337:
336:
317:
316:
282:
281:
168:
167:
121:
120:
80:
79:
69:
17:
12:
11:
5:
1333:
1331:
1323:
1322:
1317:
1312:
1302:
1301:
1298:
1297:
1274:
1258:
1236:
1233:
1230:
1229:
1222:
1193:
1174:
1173:
1171:
1168:
1156:type-theoretic
1129:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1077:
1076:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1015:
1011:
1008:
1005:
1002:
999:
996:
990:
986:
982:
977:
974:
971:
967:
963:
958:
954:
950:
916:
913:
911:), and so on.
909:type operators
896:
893:
890:
887:
884:
860:
857:
854:
851:
848:
824:
821:
818:
815:
812:
801:
800:
788:
778:
760:
757:
754:
734:
724:
708:
705:
702:
699:
695:
692:
689:
686:
658:
655:
652:
632:
606:
603:
600:
580:
570:
551:
548:
545:
542:
538:
534:
531:
528:
524:
521:
491:
488:
485:
482:
478:
475:
455:of the terms.
440:
437:
434:
414:
394:
391:
388:
368:
344:
324:
301:
298:
295:
292:
289:
278:
277:
265:
262:
259:
256:
253:
250:
247:
244:
241:
238:
235:
232:
229:
226:
223:
220:
217:
214:
211:
208:
205:
202:
199:
196:
193:
190:
187:
184:
181:
178:
175:
164:
152:
149:
146:
143:
140:
137:
134:
131:
128:
117:
105:
102:
99:
96:
93:
90:
87:
68:
65:
15:
13:
10:
9:
6:
4:
3:
2:
1332:
1321:
1318:
1316:
1313:
1311:
1308:
1307:
1305:
1294:
1290:
1286:
1282:
1281:
1275:
1271:
1267:
1263:
1259:
1255:
1251:
1247:
1243:
1239:
1238:
1234:
1225:
1223:0-444-52077-5
1219:
1215:
1211:
1207:
1200:
1198:
1194:
1186:
1179:
1176:
1169:
1167:
1165:
1161:
1157:
1153:
1149:
1147:
1143:
1104:
1101:
1098:
1095:
1092:
1059:
1053:
1044:
1038:
1029:
1026:
1023:
1020:
1013:
1006:
1003:
997:
994:
988:
984:
980:
975:
969:
965:
961:
956:
952:
948:
941:
940:
939:
937:
933:
929:
925:
922:
912:
910:
891:
888:
885:
874:
855:
852:
849:
838:
819:
816:
813:
779:
775:
758:
755:
752:
732:
725:
706:
700:
697:
676:
672:
669:which is the
656:
650:
630:
621:
604:
601:
598:
578:
571:
567:
522:
519:
512:
507:
476:
473:
466:
462:
458:
457:
456:
454:
435:
432:
392:
389:
386:
358:
342:
322:
313:
296:
293:
257:
254:
245:
239:
236:
227:
221:
218:
215:
209:
203:
200:
197:
191:
185:
182:
179:
165:
144:
141:
138:
135:
132:
129:
118:
97:
94:
91:
88:
78:
74:
73:
72:
66:
64:
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
1315:Proof theory
1279:
1265:
1249:
1208:. Elsevier.
1205:
1178:
1158:analogue of
1151:
1150:
1078:
935:
927:
918:
873:polymorphism
802:
773:
771:is read as “
719:is read as “
674:
670:
619:
617:is read as “
565:
563:is read as “
510:
505:
503:is read as “
464:
460:
452:
314:
279:
70:
59:'s original
52:
44:
28:
24:
18:
1320:Type theory
921:polymorphic
166:five rules
119:two axioms
1304:Categories
1170:References
1164:set theory
315:The sorts
57:Martin-Löf
1144:. By the
1142:inhabited
1128:⊥
1099:∗
1090:∀
1057:→
1051:→
1042:→
1027:◻
1018:Π
1007:β
1004:α
998:α
985:β
981:λ
973:→
966:α
962:λ
957:◻
949:λ
892:◻
886:◻
856:∗
850:◻
837:functions
820:∗
814:∗
787:△
759:◻
733:◻
707:∗
704:→
701:∗
657:∗
654:→
651:∗
631:∗
605:∗
579:∗
537:→
439:△
433:◻
413:△
393:◻
387:∗
367:△
343:◻
323:∗
297:∗
291:△
258:◻
252:△
240:∗
234:△
222:◻
216:◻
204:∗
198:◻
186:∗
180:∗
148:△
142:◻
136:◻
130:∗
101:△
95:◻
89:∗
1244:(1992).
932:System F
53:System U
45:System U
29:System U
25:System U
1154:is the
1220:
453:nature
312:rule.
75:three
1188:(PDF)
924:kinds
163:; and
77:sorts
41:sorts
1218:ISBN
721:List
675:e.g.
671:kind
511:e.g.
465:e.g.
461:type
357:Kind
335:and
31:are
27:and
1289:doi
1210:doi
1162:in
926:to
875:),
839:),
19:In
1306::
1283:.
1268:.
1252:.
1216:.
1196:^
1166:.
23:,
1295:.
1291::
1226:.
1212::
1190:.
1108:)
1105:p
1102:,
1096::
1093:p
1087:(
1075:.
1063:)
1060:k
1054:k
1048:)
1045:k
1039:k
1036:(
1033:(
1030:.
1024::
1021:k
1014::
1010:)
1001:(
995:.
989:k
976:k
970:k
953:k
936:k
895:)
889:,
883:(
859:)
853:,
847:(
823:)
817:,
811:(
774:k
756::
753:k
698::
694:t
691:s
688:i
685:L
620:t
602::
599:t
566:f
550:l
547:o
544:o
541:B
533:t
530:a
527:N
523::
520:f
506:b
490:l
487:o
484:o
481:B
477::
474:b
436::
425:(
390::
300:)
294:,
288:(
276:.
264:}
261:)
255:,
249:(
246:,
243:)
237:,
231:(
228:,
225:)
219:,
213:(
210:,
207:)
201:,
195:(
192:,
189:)
183:,
177:(
174:{
151:}
145::
139:,
133::
127:{
116:;
104:}
98:,
92:,
86:{
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.