1685:
2995:
1307:
2687:
1680:{\displaystyle {\begin{aligned}\mathrm {AND} &=\lambda x^{\mathsf {Boolean}}\lambda y^{\mathsf {Boolean}}{.}x\,{\mathsf {Boolean}}\,y\,\mathbf {F} \\\mathrm {OR} &=\lambda x^{\mathsf {Boolean}}\lambda y^{\mathsf {Boolean}}{.}x\,{\mathsf {Boolean}}\,\mathbf {T} \,y\\\mathrm {NOT} &=\lambda x^{\mathsf {Boolean}}{.}x\,{\mathsf {Boolean}}\,\mathbf {F} \,\mathbf {T} \end{aligned}}}
2990:{\displaystyle {\begin{aligned}0&:=\Lambda \alpha .\lambda x^{\alpha }.\lambda f^{\alpha \to \alpha }.x\\1&:=\Lambda \alpha .\lambda x^{\alpha }.\lambda f^{\alpha \to \alpha }.fx\\2&:=\Lambda \alpha .\lambda x^{\alpha }.\lambda f^{\alpha \to \alpha }.f(fx)\\3&:=\Lambda \alpha .\lambda x^{\alpha }.\lambda f^{\alpha \to \alpha }.f(f(fx))\end{aligned}}}
2257:
2516:
519:
2086:
2619:
1996:
422:
3054:
1299:
168:
2383:
995:
924:
429:
2675:
2252:{\displaystyle \mathrm {ISZERO} =\lambda n^{\forall \alpha .(\alpha \rightarrow \alpha )\rightarrow \alpha \rightarrow \alpha }{.}n\,{\mathsf {Boolean}}\,(\lambda x^{\mathsf {Boolean}}{.}\mathbf {F} )\,\mathbf {T} }
2332:
3158:: that every term in F2 satisfies a logical relation, which can be embedded into the logical relations P2. Reynolds proved that a Girard projection followed by a Reynolds embedding form the identity, i.e., the
2531:
3450:
1126:
1043:
710:
1866:
3154:: that in second-order intuitionistic predicate logic (P2), functions from the natural numbers to the natural numbers that can be proved total, form a projection from P2 into F2. Reynolds proved the
2692:
2536:
1312:
335:
4018:
1004:— not two — arguments. The latter two should be lambda expressions, but the first one should be a type. This fact is reflected in the fact that the type of these expressions is
1176:); otherwise, if such functions could be named (within System F), then there would be no need for the lambda-expressive apparatus capable of defining functions anonymously and for the
599:
2045:
1858:
1812:
1730:
1085:
776:
669:
3284:
778:
is the type of all functions which take as input a type α and two expressions of type α, and produce as output an expression of type α (note that we consider
551:
3323:
2071:
1170:
1148:
847:
825:
3389:
3356:
3006:
3540:
3520:
3500:
1206:
1201:
235:
215:
3126:
family. Over time, as the restrictions of HM-style type systems have become apparent, languages have steadily moved to more expressive logics for their type systems.
730:
619:
571:
279:
255:
191:
3256:
2511:{\displaystyle \forall \alpha .(K_{1}^{1}\rightarrow \dots \rightarrow \alpha )\dots \rightarrow (K_{1}^{m}\rightarrow \dots \rightarrow \alpha )\rightarrow \alpha }
104:
3234:
2367:
796:
1770:
1750:
930:
859:
3922:(1971). "Une Extension de l'Interpretation de Gödel à l'Analyse, et son Application à l'Élimination des Coupures dans l'Analyse et la Théorie des Types".
514:{\displaystyle {\frac {\Gamma ,\alpha ~{\text{type}}\vdash M{\mathbin {:}}\sigma }{\Gamma \vdash \Lambda \alpha .M{\mathbin {:}}\forall \alpha .\sigma }}}
3956:
3088:
The version of System F used in this article is as an explicitly typed, or Church-style, calculus. The typing information contained in λ-terms makes
4110:
3743:
289:
2627:
3150:, the second-order polymorphic lambda calculus (F2) was discovered by Girard (1972) and independently by Reynolds (1974). Girard proved the
3980:
3894:
2614:{\displaystyle {\begin{aligned}{\mathit {zero}}&:\mathrm {N} \\{\mathit {succ}}&:\mathrm {N} \rightarrow \mathrm {N} \end{aligned}}}
2285:
1045:; the universal quantifier binding the α corresponds to the Λ binding the alpha in the lambda expression itself. Also, note that
3664:
3397:
4067:
4045:
4011:
1991:{\displaystyle \mathrm {IFTHENELSE} =\Lambda \alpha .\lambda x^{\mathsf {Boolean}}\lambda y^{\alpha }\lambda z^{\alpha }.x\alpha yz}
1090:
1007:
674:
3130:, a Haskell compiler, goes beyond HM (as of 2008) and uses System F extended with non-syntactic type equality; non-HM features in
3572:
3135:
3115:
3848:
3119:
58:
2268:
417:{\displaystyle {\frac {\Gamma \vdash M{\mathbin {:}}\forall \alpha .\sigma }{\Gamma \vdash M\tau {\mathbin {:}}\sigma }}}
297:
3990:
Wells, J. B. (1994). "Typability and type checking in the second-order lambda-calculus are equivalent and undecidable".
83:
42:
3860:
Cardelli, Luca; Martini, Simone; Mitchell, John C.; Scedrov, Andre (1994). "An extension of system F with subtyping".
3568:
3686:
3652:
However, in it was shown that the obvious rules of conversion for this system, called F by chance, were converging.
4095:
4090:
3584:
3576:
62:
4100:
3127:
46:
3459:
is the system which allows functions from types to types where the argument (and result) may be of any order.
3844:
3580:
3123:
50:
3810:
576:
327:
The typing rules of System F are those of simply typed lambda calculus with the addition of the following:
3902:
1177:
3747:
2008:
1821:
1775:
1693:
1048:
739:
632:
90:, and binders for them. As an example, the fact that the identity function can have any type of the form
86:
has variables ranging over terms, and binders for them, System F additionally has variables ranging over
3073:
1173:
285:
38:
4061:
4105:
3147:
301:
54:
3263:
530:
3761:
3289:
3108:
3049:{\displaystyle \forall \alpha .(\alpha \rightarrow \alpha )\rightarrow \alpha \rightarrow \alpha }
2054:
1153:
1131:
830:
808:
3744:"The Church Project: Typability and type checking in {S}ystem {F} are equivalent and undecidable"
3367:
3203:
1294:{\displaystyle {\mathsf {Boolean}}\rightarrow {\mathsf {Boolean}}\rightarrow {\mathsf {Boolean}}}
73:
3328:
3093:
2267:
System F allows recursive constructions to be embedded in a natural manner, related to that in
163:{\displaystyle \vdash \Lambda \alpha .\lambda x^{\alpha }.x:\forall \alpha .\alpha \to \alpha }
4041:
4033:
4007:
3976:
3890:
3603:
3525:
3505:
3485:
1186:
799:
220:
200:
715:
604:
556:
264:
240:
176:
3999:
3952:
3944:
Interprétation fonctionnelle et élimination des coupures de l'arithmétique d'ordre supérieur
3939:
3927:
3919:
3865:
3731:
3643:
3241:
3112:
3100:
for a Curry-style variant of System F, that is, one that lacks explicit typing annotations.
3097:
76:
69:
3212:
2345:
3841:
3479:
3174:
2074:
850:
781:
309:
990:{\displaystyle \mathbf {F} =\Lambda \alpha {.}\lambda x^{\alpha }\lambda y^{\alpha }{.}y}
919:{\displaystyle \mathbf {T} =\Lambda \alpha {.}\lambda x^{\alpha }\lambda y^{\alpha }{.}x}
3358:(the kind of functions from types to types, where the argument type is of a lower order)
621:
is bound. The first rule is that of application, and the second is that of abstraction.
4115:
4075:
3104:
2678:
1755:
1735:
293:
3970:
3931:
3736:
3719:
3704:
4084:
3837:
3647:
3474:
of the arguments for these mappings: they must be types rather than values. System F
3192:
3089:
733:
281:; the expression after the colon is the type of the lambda expression preceding it.)
194:
20:
3202:
can be defined inductively on a family of systems, where induction is based on the
3111:", or simply "HM", does have an easy type inference algorithm and is used for many
1128:, but it is not a symbol of System F itself, but rather a "meta-symbol". Likewise,
217:
is traditionally used to denote type-level functions, as opposed to the lower-case
4034:"V Polymorphism Ch. 23. Universal Types, Ch. 25. An ML Implementation of System F"
3634:
Girard, Jean-Yves (1986). "The system F of variable types, fifteen years later".
1172:
are also "meta-symbols", convenient shorthands, of System F "assemblies" (in the
3614:
3177:
308:, together with even more expressive typed lambda calculi, including those with
305:
3096:(1994) settled an "embarrassing open problem" by proving that type checking is
4003:
3668:
304:
that uses only universal quantification. System F can be seen as part of the
3560:
3870:
2670:{\displaystyle \forall \alpha .\alpha \to (\alpha \to \alpha )\to \alpha }
2521:
For instance, the natural numbers can be defined as an inductive datatype
296:
in System F (without explicit type annotations) is undecidable. Under the
3785:
3720:"Typability and type checking in System F are equivalent and undecidable"
3609:
66:
2327:{\displaystyle K_{1}\rightarrow K_{2}\rightarrow \dots \rightarrow S}
3131:
1860:-typed terms as decision functions. However, if one is requested:
3669:"Practical Foundations for Programming Languages, Second Edition"
3993:
3445:{\displaystyle F_{\omega }={\underset {1\leq i}{\bigcup }}F_{i}}
3107:
for System F is impossible. A restriction of System F known as "
3606:— the existentially quantified counterparts of universal types
1203:-terms, we can define some logic operators (which are of type
3191:
combines the first axis (polymorphism) with the second axis (
1752:, specifying that the other two parameters that are given to
1121:{\displaystyle \forall \alpha .\alpha \to \alpha \to \alpha }
1038:{\displaystyle \forall \alpha .\alpha \to \alpha \to \alpha }
705:{\displaystyle \forall \alpha .\alpha \to \alpha \to \alpha }
2582:
2579:
2576:
2573:
2550:
2547:
2544:
2541:
237:
which is used for value-level functions. (The superscripted
2677:. The terms of this type comprise a typed version of the
3482:), though it does permit mappings from values to values (
3470:
of the arguments in these mappings, it does restrict the
57:, thus forming a theoretical basis for languages such as
3559:, pronounced "F-sub", is an extension of system F with
300:, System F corresponds to the fragment of second-order
3969:
Girard, Jean-Yves; Lafont, Yves; Taylor, Paul (1989).
3924:
3687:"Proofs of Programs and Formalisation of Mathematics"
3528:
3508:
3488:
3400:
3370:
3331:
3292:
3266:
3244:
3215:
3009:
2690:
2630:
2624:
The System F type corresponding to this structure is
2534:
2386:
2348:
2288:
2089:
2057:
2011:
1869:
1824:
1778:
1758:
1738:
1696:
1310:
1209:
1189:
1156:
1134:
1093:
1051:
1010:
933:
862:
833:
811:
805:
The following two definitions for the boolean values
784:
742:
718:
677:
635:
607:
579:
559:
533:
432:
338:
267:
243:
223:
203:
179:
107:
3849:Programming Languages and Foundations at Edinburgh
3534:
3514:
3494:
3444:
3383:
3350:
3317:
3278:
3250:
3228:
3048:
3000:If we reverse the order of the curried arguments (
2989:
2669:
2613:
2510:
2373:of these constructors, you can define the type of
2361:
2326:
2251:
2065:
2039:
1990:
1852:
1814:. As in Church encodings, there is no need for an
1806:
1764:
1744:
1724:
1679:
1293:
1195:
1164:
1142:
1120:
1079:
1037:
989:
918:
841:
819:
790:
770:
724:
704:
663:
613:
593:
565:
545:
513:
416:
273:
249:
229:
209:
185:
162:
98:would be formalized in System F as the judgement
3842:The Girard-Reynolds Isomorphism (second edition)
3522:abstraction), and mappings from types to types (
3173:While System F corresponds to the first axis of
3080:, and returns another single-argument function.
2047:-typed value. The most fundamental predicate is
3762:"System FC: equality constraints and coercions"
3478:does not permit mappings from values to types (
329:
3946:(Ph.D. thesis) (in French), Université Paris 7
3996:Symposium on Logic in Computer Science (LICS)
3502:abstraction), mappings from types to values (
8:
19:For the electronic trance music artist, see
3864:. North Holland, Amsterdam. pp. 4–56.
3195:); it is a different, more complex system.
1000:(Note that the above two functions require
3869:
3735:
3527:
3507:
3487:
3436:
3414:
3405:
3399:
3375:
3369:
3342:
3330:
3303:
3291:
3265:
3243:
3220:
3214:
3008:
2944:
2928:
2868:
2852:
2801:
2785:
2737:
2721:
2691:
2689:
2629:
2602:
2594:
2572:
2571:
2562:
2540:
2539:
2535:
2533:
2476:
2464:
2459:
2420:
2408:
2403:
2385:
2353:
2347:
2306:
2293:
2287:
2244:
2243:
2235:
2230:
2205:
2204:
2193:
2169:
2168:
2167:
2159:
2120:
2090:
2088:
2058:
2056:
2013:
2012:
2010:
1967:
1954:
1922:
1921:
1870:
1868:
1826:
1825:
1823:
1780:
1779:
1777:
1757:
1737:
1698:
1697:
1695:
1668:
1667:
1662:
1661:
1637:
1636:
1635:
1627:
1602:
1601:
1576:
1568:
1563:
1562:
1538:
1537:
1536:
1528:
1503:
1502:
1470:
1469:
1447:
1438:
1437:
1433:
1409:
1408:
1407:
1399:
1374:
1373:
1341:
1340:
1315:
1311:
1309:
1267:
1266:
1239:
1238:
1211:
1210:
1208:
1188:
1157:
1155:
1135:
1133:
1092:
1053:
1052:
1050:
1009:
979:
973:
960:
948:
934:
932:
908:
902:
889:
877:
863:
861:
834:
832:
812:
810:
783:
744:
743:
741:
717:
676:
637:
636:
634:
606:
586:
578:
558:
532:
490:
489:
460:
459:
448:
433:
431:
400:
385:
384:
352:
351:
339:
337:
266:
242:
222:
202:
178:
127:
106:
4040:. MIT Press. pp. 339–362, 381–388.
3189:higher-order polymorphic lambda calculus
3072:. That is to say, a Church numeral is a
1180:, which works around that restriction.)
3626:
2342:itself appears within one of the types
3076:– it takes a single-argument function
2224:
2221:
2218:
2215:
2212:
2209:
2206:
2188:
2185:
2182:
2179:
2176:
2173:
2170:
2032:
2029:
2026:
2023:
2020:
2017:
2014:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1799:
1796:
1793:
1790:
1787:
1784:
1781:
1717:
1714:
1711:
1708:
1705:
1702:
1699:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1557:
1554:
1551:
1548:
1545:
1542:
1539:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1393:
1390:
1387:
1384:
1381:
1378:
1375:
1360:
1357:
1354:
1351:
1348:
1345:
1342:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
849:are used, extending the definition of
763:
760:
757:
754:
751:
748:
745:
656:
653:
650:
647:
644:
641:
638:
3833:
3831:
3829:
3827:
3587:subtyping, which can be expressed in
594:{\displaystyle \alpha ~{\text{type}}}
65:. It was discovered independently by
7:
3571:since the 1980s because the core of
3542:abstraction at the level of types).
3060:is a function that takes a function
1690:Note that in the definitions above,
4072:: the workhorse of modern compilers
3862:Information and Computation, vol. 9
3746:. 29 September 2007. Archived from
3364:In the limit, we can define system
2073:if and only if its argument is the
2040:{\displaystyle {\mathsf {Boolean}}}
1853:{\displaystyle {\mathsf {Boolean}}}
1807:{\displaystyle {\mathsf {Boolean}}}
1725:{\displaystyle {\mathsf {Boolean}}}
1080:{\displaystyle {\mathsf {Boolean}}}
771:{\displaystyle {\mathsf {Boolean}}}
664:{\displaystyle {\mathsf {Boolean}}}
3958:Towards a Theory of Type Structure
3567:has been of central importance to
3509:
3010:
2912:
2836:
2769:
2705:
2631:
2603:
2595:
2563:
2387:
2121:
2106:
2103:
2100:
2097:
2094:
2091:
1905:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1874:
1871:
1583:
1580:
1577:
1451:
1448:
1322:
1319:
1316:
1094:
1011:
942:
871:
678:
496:
477:
471:
436:
372:
358:
342:
204:
142:
111:
14:
3685:Geuvers H, Nordström B, Dowek G.
1818:function as one can just use raw
3705:"Joe Wells's Research Interests"
3573:functional programming languages
3116:functional programming languages
2279:. These are functions typed as:
2245:
2236:
2059:
1669:
1663:
1564:
1439:
1158:
1136:
935:
864:
835:
813:
315:According to Girard, the "F" in
49:over types. System F formalizes
4111:Polymorphism (computer science)
4038:Types and Programming Languages
3887:Types and Programming Languages
3142:The Girard-Reynolds Isomorphism
3056:), then the Church numeral for
2338:Recursivity is manifested when
3992:Proceedings of the 9th Annual
3975:. Cambridge University Press.
3466:places no restrictions on the
3279:{\displaystyle J\Rightarrow K}
3270:
3040:
3034:
3031:
3025:
3019:
2980:
2977:
2968:
2962:
2948:
2895:
2886:
2872:
2805:
2741:
2681:, the first few of which are:
2661:
2658:
2652:
2646:
2643:
2599:
2502:
2499:
2493:
2487:
2484:
2470:
2452:
2449:
2443:
2437:
2431:
2428:
2414:
2396:
2318:
2312:
2299:
2240:
2194:
2151:
2145:
2142:
2136:
2130:
2005:is a function which returns a
1263:
1235:
1112:
1106:
1087:is a convenient shorthand for
1029:
1023:
785:
696:
690:
601:in the context indicates that
408:
394:
154:
1:
3932:10.1016/S0049-237X(08)70843-7
3926:. Amsterdam. pp. 63–92.
3811:"OCaml 4.09 reference manual"
3737:10.1016/S0168-0072(98)00047-5
546:{\displaystyle \sigma ,\tau }
3786:"OCaml 4.00.1 release notes"
3648:10.1016/0304-3975(86)90044-7
3636:Theoretical Computer Science
3318:{\displaystyle J\in F_{n-1}}
3103:Wells's result implies that
3084:Use in programming languages
3064:as argument and returns the
2066:{\displaystyle \mathbf {T} }
1165:{\displaystyle \mathbf {F} }
1143:{\displaystyle \mathbf {T} }
842:{\displaystyle \mathbf {F} }
820:{\displaystyle \mathbf {T} }
426:
332:
84:simply typed lambda calculus
43:simply typed lambda calculus
35:second-order lambda calculus
3569:programming language theory
3384:{\displaystyle F_{\omega }}
3160:Girard-Reynolds Isomorphism
31:polymorphic lambda calculus
4132:
3703:Wells, J.B. (2005-01-20).
3351:{\displaystyle K\in F_{n}}
3206:permitted in each system:
18:
4032:Pierce, Benjamin (2002).
3885:Pierce, Benjamin (2002).
3707:. Heriot-Watt University.
4004:10.1109/LICS.1994.316068
3535:{\displaystyle \lambda }
3515:{\displaystyle \Lambda }
3495:{\displaystyle \lambda }
2269:Martin-Löf's type theory
1196:{\displaystyle \lambda }
573:is a type variable, and
298:Curry–Howard isomorphism
230:{\displaystyle \lambda }
210:{\displaystyle \Lambda }
47:universal quantification
3845:University of Edinburgh
3581:parametric polymorphism
3258:(the kind of types) and
3134:'s type system include
2271:. Abstract structures (
725:{\displaystyle \alpha }
614:{\displaystyle \alpha }
566:{\displaystyle \alpha }
274:{\displaystyle \alpha }
250:{\displaystyle \alpha }
186:{\displaystyle \alpha }
51:parametric polymorphism
3903:Bounded quantification
3871:10.1006/inco.1994.1013
3536:
3516:
3496:
3446:
3385:
3352:
3319:
3280:
3252:
3251:{\displaystyle \star }
3230:
3152:Representation Theorem
3050:
2991:
2671:
2615:
2512:
2363:
2328:
2253:
2067:
2041:
1992:
1854:
1808:
1766:
1746:
1732:is a type argument to
1726:
1681:
1295:
1197:
1178:fixed-point combinator
1166:
1144:
1122:
1081:
1039:
991:
920:
843:
821:
792:
772:
726:
706:
665:
615:
595:
567:
547:
515:
418:
319:was picked by chance.
275:
251:
231:
211:
187:
164:
3750:on 29 September 2007.
3724:Ann. Pure Appl. Logic
3579:family, support both
3537:
3517:
3497:
3447:
3386:
3353:
3320:
3281:
3253:
3231:
3229:{\displaystyle F_{n}}
3074:higher-order function
3051:
2992:
2672:
2616:
2513:
2364:
2362:{\displaystyle K_{i}}
2329:
2254:
2068:
2042:
1993:
1855:
1809:
1767:
1747:
1727:
1682:
1296:
1198:
1183:Then, with these two
1167:
1145:
1123:
1082:
1040:
992:
921:
844:
822:
793:
773:
727:
707:
666:
616:
596:
568:
548:
516:
419:
286:term rewriting system
276:
257:means that the bound
252:
232:
212:
188:
165:
55:programming languages
39:typed lambda calculus
16:Typed lambda calculus
3998:. pp. 176–185.
3718:Wells, J.B. (1999).
3575:, like those in the
3526:
3506:
3486:
3462:Note that although F
3398:
3368:
3329:
3290:
3264:
3242:
3213:
3148:intuitionistic logic
3007:
2688:
2628:
2532:
2384:
2346:
2286:
2275:) are created using
2087:
2055:
2009:
1867:
1822:
1776:
1756:
1736:
1694:
1308:
1207:
1187:
1154:
1132:
1091:
1049:
1008:
931:
860:
831:
809:
791:{\displaystyle \to }
782:
740:
716:
675:
671:type is defined as:
633:
625:Logic and predicates
605:
577:
557:
531:
430:
336:
302:intuitionistic logic
290:strongly normalizing
265:
241:
221:
201:
177:
105:
41:that introduces, to
4062:Summary of System F
3156:Abstraction Theorem
2469:
2413:
2263:System F structures
4019:Postscript version
3766:gitlab.haskell.org
3532:
3512:
3492:
3442:
3430:
3381:
3348:
3315:
3276:
3248:
3226:
3092:straightforward.
3046:
2987:
2985:
2667:
2611:
2609:
2525:with constructors
2508:
2455:
2399:
2359:
2324:
2249:
2063:
2037:
1988:
1850:
1804:
1762:
1742:
1722:
1677:
1675:
1291:
1193:
1162:
1140:
1118:
1077:
1035:
987:
916:
839:
817:
788:
768:
722:
702:
661:
611:
591:
563:
543:
511:
414:
271:
247:
227:
207:
197:. The upper-case
183:
160:
74:computer scientist
4096:1974 in computing
4091:1971 in computing
4064:by Franck Binard.
3982:978-0-521-37181-0
3940:Girard, Jean-Yves
3920:Girard, Jean-Yves
3896:978-0-262-16209-8
3674:. pp. 142–3.
3604:Existential types
3415:
1765:{\displaystyle x}
1745:{\displaystyle x}
800:right-associative
589:
585:
525:
524:
509:
451:
447:
412:
45:, a mechanism of
4123:
4051:
4017:
3986:
3972:Proofs and Types
3965:
3963:
3947:
3935:
3905:
3900:
3882:
3876:
3875:
3873:
3857:
3851:
3835:
3822:
3821:
3819:
3818:
3807:
3801:
3800:
3798:
3797:
3782:
3776:
3775:
3773:
3772:
3758:
3752:
3751:
3741:
3739:
3730:(1–3): 111–156.
3715:
3709:
3708:
3700:
3694:
3693:
3691:
3682:
3676:
3675:
3673:
3661:
3655:
3654:
3631:
3541:
3539:
3538:
3533:
3521:
3519:
3518:
3513:
3501:
3499:
3498:
3493:
3451:
3449:
3448:
3443:
3441:
3440:
3431:
3429:
3410:
3409:
3390:
3388:
3387:
3382:
3380:
3379:
3357:
3355:
3354:
3349:
3347:
3346:
3324:
3322:
3321:
3316:
3314:
3313:
3285:
3283:
3282:
3277:
3257:
3255:
3254:
3249:
3235:
3233:
3232:
3227:
3225:
3224:
3146:In second-order
3113:statically typed
3079:
3071:
3067:
3063:
3059:
3055:
3053:
3052:
3047:
2996:
2994:
2993:
2988:
2986:
2955:
2954:
2933:
2932:
2879:
2878:
2857:
2856:
2812:
2811:
2790:
2789:
2748:
2747:
2726:
2725:
2676:
2674:
2673:
2668:
2620:
2618:
2617:
2612:
2610:
2606:
2598:
2586:
2585:
2566:
2554:
2553:
2524:
2517:
2515:
2514:
2509:
2480:
2468:
2463:
2424:
2412:
2407:
2376:
2372:
2368:
2366:
2365:
2360:
2358:
2357:
2341:
2333:
2331:
2330:
2325:
2311:
2310:
2298:
2297:
2274:
2258:
2256:
2255:
2250:
2248:
2239:
2234:
2229:
2228:
2227:
2192:
2191:
2163:
2158:
2157:
2109:
2079:
2072:
2070:
2069:
2064:
2062:
2050:
2046:
2044:
2043:
2038:
2036:
2035:
1997:
1995:
1994:
1989:
1972:
1971:
1959:
1958:
1946:
1945:
1944:
1901:
1859:
1857:
1856:
1851:
1849:
1848:
1817:
1813:
1811:
1810:
1805:
1803:
1802:
1771:
1769:
1768:
1763:
1751:
1749:
1748:
1743:
1731:
1729:
1728:
1723:
1721:
1720:
1686:
1684:
1683:
1678:
1676:
1672:
1666:
1660:
1659:
1631:
1626:
1625:
1624:
1586:
1567:
1561:
1560:
1532:
1527:
1526:
1525:
1494:
1493:
1492:
1454:
1442:
1432:
1431:
1403:
1398:
1397:
1396:
1365:
1364:
1363:
1325:
1300:
1298:
1297:
1292:
1290:
1289:
1262:
1261:
1234:
1233:
1202:
1200:
1199:
1194:
1171:
1169:
1168:
1163:
1161:
1149:
1147:
1146:
1141:
1139:
1127:
1125:
1124:
1119:
1086:
1084:
1083:
1078:
1076:
1075:
1044:
1042:
1041:
1036:
996:
994:
993:
988:
983:
978:
977:
965:
964:
952:
938:
925:
923:
922:
917:
912:
907:
906:
894:
893:
881:
867:
848:
846:
845:
840:
838:
826:
824:
823:
818:
816:
797:
795:
794:
789:
777:
775:
774:
769:
767:
766:
731:
729:
728:
723:
711:
709:
708:
703:
670:
668:
667:
662:
660:
659:
620:
618:
617:
612:
600:
598:
597:
592:
590:
587:
583:
572:
570:
569:
564:
552:
550:
549:
544:
520:
518:
517:
512:
510:
508:
495:
494:
469:
465:
464:
452:
449:
445:
434:
423:
421:
420:
415:
413:
411:
404:
390:
389:
370:
357:
356:
340:
330:
280:
278:
277:
272:
256:
254:
253:
248:
236:
234:
233:
228:
216:
214:
213:
208:
192:
190:
189:
184:
169:
167:
166:
161:
132:
131:
77:John C. Reynolds
70:Jean-Yves Girard
4131:
4130:
4126:
4125:
4124:
4122:
4121:
4120:
4101:Lambda calculus
4081:
4080:
4071:
4058:
4048:
4031:
4028:
4026:Further reading
4023:
4014:
3989:
3983:
3968:
3961:
3951:
3938:
3918:
3914:
3909:
3908:
3897:
3884:
3883:
3879:
3859:
3858:
3854:
3836:
3825:
3816:
3814:
3809:
3808:
3804:
3795:
3793:
3784:
3783:
3779:
3770:
3768:
3760:
3759:
3755:
3742:
3717:
3716:
3712:
3702:
3701:
3697:
3689:
3684:
3683:
3679:
3671:
3663:
3662:
3658:
3633:
3632:
3628:
3623:
3600:
3592:
3566:
3557:
3551:
3549:
3524:
3523:
3504:
3503:
3484:
3483:
3480:dependent types
3477:
3465:
3458:
3432:
3419:
3401:
3396:
3395:
3371:
3366:
3365:
3338:
3327:
3326:
3299:
3288:
3287:
3262:
3261:
3240:
3239:
3236:permits kinds:
3216:
3211:
3210:
3201:
3185:
3171:
3169:
3144:
3086:
3077:
3069:
3065:
3061:
3057:
3005:
3004:
2984:
2983:
2940:
2924:
2905:
2899:
2898:
2864:
2848:
2829:
2823:
2822:
2797:
2781:
2762:
2756:
2755:
2733:
2717:
2698:
2686:
2685:
2679:Church numerals
2626:
2625:
2608:
2607:
2587:
2568:
2567:
2555:
2530:
2529:
2522:
2382:
2381:
2374:
2370:
2349:
2344:
2343:
2339:
2302:
2289:
2284:
2283:
2272:
2265:
2200:
2116:
2085:
2084:
2077:
2053:
2052:
2048:
2007:
2006:
1963:
1950:
1917:
1865:
1864:
1820:
1819:
1815:
1774:
1773:
1754:
1753:
1734:
1733:
1692:
1691:
1674:
1673:
1597:
1587:
1573:
1572:
1498:
1465:
1455:
1444:
1443:
1369:
1336:
1326:
1306:
1305:
1205:
1204:
1185:
1184:
1152:
1151:
1130:
1129:
1089:
1088:
1047:
1046:
1006:
1005:
969:
956:
929:
928:
898:
885:
858:
857:
851:Church booleans
829:
828:
807:
806:
780:
779:
738:
737:
714:
713:
673:
672:
631:
630:
627:
603:
602:
575:
574:
555:
554:
529:
528:
470:
435:
428:
427:
371:
341:
334:
333:
325:
310:dependent types
263:
262:
239:
238:
219:
218:
199:
198:
175:
174:
123:
103:
102:
24:
17:
12:
11:
5:
4129:
4127:
4119:
4118:
4113:
4108:
4103:
4098:
4093:
4083:
4082:
4079:
4078:
4076:Greg Morrisett
4069:
4065:
4057:
4056:External links
4054:
4053:
4052:
4046:
4027:
4024:
4022:
4021:
4012:
3987:
3981:
3966:
3953:Reynolds, John
3949:
3936:
3915:
3913:
3910:
3907:
3906:
3901:, Chapter 26:
3895:
3877:
3852:
3823:
3802:
3777:
3753:
3710:
3695:
3677:
3656:
3625:
3624:
3622:
3619:
3618:
3617:
3612:
3607:
3599:
3596:
3590:
3564:
3555:
3550:
3547:
3544:
3531:
3511:
3491:
3475:
3463:
3456:
3453:
3452:
3439:
3435:
3428:
3425:
3422:
3418:
3413:
3408:
3404:
3378:
3374:
3362:
3361:
3360:
3359:
3345:
3341:
3337:
3334:
3312:
3309:
3306:
3302:
3298:
3295:
3275:
3272:
3269:
3259:
3247:
3223:
3219:
3199:
3193:type operators
3183:
3170:
3167:
3164:
3143:
3140:
3109:Hindley–Milner
3105:type inference
3085:
3082:
3045:
3042:
3039:
3036:
3033:
3030:
3027:
3024:
3021:
3018:
3015:
3012:
2998:
2997:
2982:
2979:
2976:
2973:
2970:
2967:
2964:
2961:
2958:
2953:
2950:
2947:
2943:
2939:
2936:
2931:
2927:
2923:
2920:
2917:
2914:
2911:
2908:
2906:
2904:
2901:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2877:
2874:
2871:
2867:
2863:
2860:
2855:
2851:
2847:
2844:
2841:
2838:
2835:
2832:
2830:
2828:
2825:
2824:
2821:
2818:
2815:
2810:
2807:
2804:
2800:
2796:
2793:
2788:
2784:
2780:
2777:
2774:
2771:
2768:
2765:
2763:
2761:
2758:
2757:
2754:
2751:
2746:
2743:
2740:
2736:
2732:
2729:
2724:
2720:
2716:
2713:
2710:
2707:
2704:
2701:
2699:
2697:
2694:
2693:
2666:
2663:
2660:
2657:
2654:
2651:
2648:
2645:
2642:
2639:
2636:
2633:
2622:
2621:
2605:
2601:
2597:
2593:
2590:
2588:
2584:
2581:
2578:
2575:
2570:
2569:
2565:
2561:
2558:
2556:
2552:
2549:
2546:
2543:
2538:
2537:
2519:
2518:
2507:
2504:
2501:
2498:
2495:
2492:
2489:
2486:
2483:
2479:
2475:
2472:
2467:
2462:
2458:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2423:
2419:
2416:
2411:
2406:
2402:
2398:
2395:
2392:
2389:
2369:. If you have
2356:
2352:
2336:
2335:
2323:
2320:
2317:
2314:
2309:
2305:
2301:
2296:
2292:
2264:
2261:
2260:
2259:
2247:
2242:
2238:
2233:
2226:
2223:
2220:
2217:
2214:
2211:
2208:
2203:
2199:
2196:
2190:
2187:
2184:
2181:
2178:
2175:
2172:
2166:
2162:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2119:
2115:
2112:
2108:
2105:
2102:
2099:
2096:
2093:
2075:Church numeral
2061:
2051:which returns
2034:
2031:
2028:
2025:
2022:
2019:
2016:
1999:
1998:
1987:
1984:
1981:
1978:
1975:
1970:
1966:
1962:
1957:
1953:
1949:
1943:
1940:
1937:
1934:
1931:
1928:
1925:
1920:
1916:
1913:
1910:
1907:
1904:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1761:
1741:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1688:
1687:
1671:
1665:
1658:
1655:
1652:
1649:
1646:
1643:
1640:
1634:
1630:
1623:
1620:
1617:
1614:
1611:
1608:
1605:
1600:
1596:
1593:
1590:
1588:
1585:
1582:
1579:
1575:
1574:
1571:
1566:
1559:
1556:
1553:
1550:
1547:
1544:
1541:
1535:
1531:
1524:
1521:
1518:
1515:
1512:
1509:
1506:
1501:
1497:
1491:
1488:
1485:
1482:
1479:
1476:
1473:
1468:
1464:
1461:
1458:
1456:
1453:
1450:
1446:
1445:
1441:
1436:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1406:
1402:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1372:
1368:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1339:
1335:
1332:
1329:
1327:
1324:
1321:
1318:
1314:
1313:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1265:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1237:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1192:
1174:Bourbaki sense
1160:
1138:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
998:
997:
986:
982:
976:
972:
968:
963:
959:
955:
951:
947:
944:
941:
937:
926:
915:
911:
905:
901:
897:
892:
888:
884:
880:
876:
873:
870:
866:
837:
815:
787:
765:
762:
759:
756:
753:
750:
747:
736:. This means:
721:
701:
698:
695:
692:
689:
686:
683:
680:
658:
655:
652:
649:
646:
643:
640:
626:
623:
610:
582:
562:
542:
539:
536:
523:
522:
507:
504:
501:
498:
493:
488:
485:
482:
479:
476:
473:
468:
463:
458:
455:
444:
441:
438:
425:
410:
407:
403:
399:
396:
393:
388:
383:
380:
377:
374:
369:
366:
363:
360:
355:
350:
347:
344:
324:
321:
294:type inference
288:, System F is
270:
246:
226:
206:
182:
171:
170:
159:
156:
153:
150:
147:
144:
141:
138:
135:
130:
126:
122:
119:
116:
113:
110:
15:
13:
10:
9:
6:
4:
3:
2:
4128:
4117:
4114:
4112:
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4047:0-262-16209-1
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3889:. MIT Press.
3888:
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3850:
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3838:Philip Wadler
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3692:. p. 51.
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3090:type-checking
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735:
734:type variable
719:
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624:
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180:
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48:
44:
40:
36:
32:
28:
22:
21:Ferry Corsten
4037:
3991:
3971:
3957:
3943:
3923:
3886:
3880:
3861:
3855:
3815:. Retrieved
3813:. 2012-09-11
3805:
3794:. Retrieved
3792:. 2012-10-05
3789:
3780:
3769:. Retrieved
3765:
3756:
3748:the original
3727:
3723:
3713:
3698:
3680:
3659:
3651:
3639:
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3629:
3588:
3553:
3552:
3471:
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3461:
3454:
3363:
3197:
3188:
3181:
3175:Barendregt's
3172:
3159:
3155:
3151:
3145:
3102:
3087:
3001:
2999:
2623:
2520:
2337:
2277:constructors
2276:
2266:
2002:
2000:
1772:are of type
1689:
1182:
1001:
999:
804:
628:
526:
326:
323:Typing rules
316:
314:
283:
258:
172:
95:
91:
87:
81:
34:
30:
26:
25:
4106:Type theory
3615:Lambda cube
3178:lambda cube
3098:undecidable
2001:will do. A
553:are types,
306:lambda cube
292:. However,
261:is of type
72:(1972) and
4085:Categories
3912:References
3817:2019-09-23
3796:2019-09-23
3771:2019-07-08
3563:. System F
3455:That is, F
3120:Haskell 98
1816:IFTHENELSE
3790:ocaml.org
3561:subtyping
3530:λ
3510:Λ
3490:λ
3424:≤
3417:⋃
3407:ω
3377:ω
3336:∈
3308:−
3297:∈
3271:⇒
3246:⋆
3094:Joe Wells
3068:power of
3044:α
3041:→
3038:α
3035:→
3029:α
3026:→
3023:α
3014:α
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2952:α
2949:→
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2938:λ
2930:α
2922:λ
2916:α
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2876:α
2873:→
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2862:λ
2854:α
2846:λ
2840:α
2837:Λ
2809:α
2806:→
2803:α
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2787:α
2779:λ
2773:α
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2745:α
2742:→
2739:α
2731:λ
2723:α
2715:λ
2709:α
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2665:α
2662:→
2656:α
2653:→
2650:α
2644:→
2641:α
2635:α
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2600:→
2506:α
2503:→
2497:α
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2491:⋯
2488:→
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2441:α
2438:→
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2432:→
2418:α
2391:α
2388:∀
2319:→
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2313:→
2300:→
2198:λ
2155:α
2152:→
2149:α
2146:→
2140:α
2137:→
2134:α
2125:α
2122:∀
2114:λ
2003:predicate
1980:α
1969:α
1961:λ
1956:α
1948:λ
1915:λ
1909:α
1906:Λ
1595:λ
1496:λ
1463:λ
1367:λ
1334:λ
1264:→
1236:→
1191:λ
1116:α
1113:→
1110:α
1107:→
1104:α
1098:α
1095:∀
1033:α
1030:→
1027:α
1024:→
1021:α
1015:α
1012:∀
975:α
967:λ
962:α
954:λ
946:α
943:Λ
904:α
896:λ
891:α
883:λ
875:α
872:Λ
786:→
720:α
700:α
697:→
694:α
691:→
688:α
682:α
679:∀
609:α
581:α
561:α
541:τ
535:σ
506:σ
500:α
497:∀
481:α
478:Λ
475:⊢
472:Γ
467:σ
454:⊢
443:α
437:Γ
406:α
398:τ
392:σ
382:τ
376:⊢
373:Γ
368:σ
362:α
359:∀
346:⊢
343:Γ
269:α
245:α
225:λ
205:Λ
181:α
158:α
155:→
152:α
146:α
143:∀
129:α
121:λ
115:α
112:Λ
109:⊢
4068:System F
3955:(1974).
3942:(1972),
3665:Harper R
3610:System U
3598:See also
3589:System F
3554:System F
3546:System F
3472:universe
3198:System F
3182:System F
3166:System F
3122:and the
3118:such as
712:, where
317:System F
82:Whereas
67:logician
27:System F
3840:(2005)
3642:: 160.
3187:or the
59:Haskell
37:) is a
4070:ω
4044:
4010:
3979:
3893:
3585:record
3391:to be
3286:where
2049:ISZERO
798:to be
584:
527:where
446:
173:where
29:(also
4116:Logic
3962:(PDF)
3690:(PDF)
3672:(PDF)
3621:Notes
3591:<:
3565:<:
3556:<:
3548:<:
3468:order
3204:kinds
3132:OCaml
3002:i.e.,
1002:three
732:is a
284:As a
193:is a
88:types
4042:ISBN
4008:ISBN
3994:IEEE
3977:ISBN
3891:ISBN
3583:and
3325:and
3136:GADT
2377:as:
1150:and
827:and
629:The
588:type
521:(2)
450:type
424:(1)
61:and
4074:by
4000:doi
3928:doi
3866:doi
3732:doi
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33:or
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4006:.
3847:,
3826:^
3788:.
3764:.
3728:98
3726:.
3722:.
3667:.
3650:.
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3594:.
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3138:.
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63:ML
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