1113:Î -types contain functions. As with typical function types, they consist of an input type and an output type. They are more powerful than typical function types however, in that the return type can depend on the input value. Functions in type theory are different from set theory. In set theory, you look up the argument's value in a set of ordered pairs. In type theory, the argument is substituted into a term and then computation ("reduction") is applied to the term.
4094:
2542:
3921:
4554:
To implement logic, each proposition is given its own type. The objects in those types represent the different possible ways to prove the proposition. If there is no proof for the proposition, then the type has no objects in it. Operators like "and" and "or" that work on propositions introduce new
5572:
was the first definition of a type theory that Per Martin-Löf published (it was presented at the Logic
Colloquium '73 and published in 1975). There are identity types, which he describes as "propositions", but since no real distinction between propositions and the rest of the types is introduced the
5534:
and
Giovanni Sambin). The list below attempts to list all the theories that have been described in a printed form and to sketch the key features that distinguished them from each other. All of these theories had dependent products, dependent sums, disjoint unions, finite types and natural numbers.
5643:
was presented in 1979 and published in 1982. In this paper, Martin-Löf introduced the four basic types of judgement for the dependent type theory that has since become fundamental in the study of the meta-theory of such systems. He also introduced contexts as a separate concept in it (see
5644:
p. 161). There are identity types with the J-eliminator (which already appeared in MLTT73 but did not have this name there) but also with the rule that makes the theory "extensional" (p. 169). There are W-types. There is an infinite sequence of predicative universes that
93:. Constructivism requires any existence proof to contain a "witness". So, any proof of "there exists a prime greater than 1000" must identify a specific number that is both prime and greater than 1000. Intuitionistic type theory accomplished this design goal by internalizing the
5373:
A logical framework, such as Martin-Löf's takes the form of closure conditions on the context-dependent sets of types and terms: that there should be a type called Set, and for each set a type, that the types should be closed under forms of dependent sum and product, and so forth.
2371:
845:
5377:
A theory such as that of predicative set theory expresses closure conditions on the types of sets and their elements: that they should be closed under operations that reflect dependent sum and product, and under various forms of inductive definition.
5718:
5897:
4089:{\displaystyle {\begin{aligned}\operatorname {add} &{\mathbin {:}}\ (\mathbb {N} \times \mathbb {N} )\to \mathbb {N} \\\operatorname {add} (0,b)&=b\\\operatorname {add} (S(a),b)&=S(\operatorname {add} (a,b)))\end{aligned}}}
1223:
596:
517:ÎŁ-types are more powerful than typical ordered pair types because of dependent typing. In the ordered pair, the type of the second term can depend on the value of the first term. For example, the first term of the pair might be a
1917:
5541:
was the first type theory created by Per Martin-Löf. It appeared in a preprint in 1971. It had one universe, but this universe had a name in itself, i.e. it was a type theory with, as it is called today, "Type in Type".
5259:
must contain a terminal object (the empty context), and a final object for a form of product called comprehension, or context extension, in which the right element is a type in the context of the left element. If
688:
2253:
5535:
All the theories had the same reduction rules that did not include η-reduction either for dependent products or for dependent sums, except for MLTT79 where the η-reduction for dependent products is added.
4314:
3850:
1637:
1103:
4169:
3274:
reduce to the same value. (Terms of this type are generated using the term-equality judgement.) Lastly, there is an
English-language level of equality, because we use the word "four" and symbol "
5394:
type theory. In extensional type theory, definitional (i.e., computational) equality is not distinguished from propositional equality, which requires proof. As a consequence type checking becomes
4214:
5530:
constructed several type theories that were published at various times, some of them much later than when the preprints with their description became accessible to the specialists (among others
4382:
4121:
So, objects and types and these relations are used to express formulae in the theory. The following styles of judgements are used to create new objects, types and relations from existing ones:
3926:
637:
4259:
2537:{\displaystyle {\operatorname {{\mathbb {N} }-elim} }\,{\mathbin {:}}P(0)\,\to \left(\prod _{n{\mathbin {:}}{\mathbb {N} }}P(n)\to P(S(n))\right)\to \prod _{n{\mathbin {:}}{\mathbb {N} }}P(n)}
1280:
2739:
748:
used in most programming languages. An example of a dependently-typed 3-tuple is two integers and a proof that the first integer is smaller than the second integer, described by the type:
5946:
Per Martin-Löf, An intuitionistic theory of types, Twenty-five years of constructive type theory (Venice,1995), Oxford Logic Guides, v. 36, pp. 127--172, Oxford Univ. Press, New York, 1998
2209:
5556:" in the sense that the dependent product of a family of objects from V over an object that was not in V such as, for example, V itself, was not assumed to be in V. The universe was Ă la
5573:
meaning of this is unclear. There is what later acquires the name of J-eliminator but yet without a name (see pp. 94â95). There is in this theory an infinite sequence of universes V
3809:
754:
1377:
182:
If you are unfamiliar with type theory and know set theory, a quick summary is: Types contain terms just like sets contain elements. Terms belong to one and only one type. Terms like
3093:
3055:
2645:
4434:
2958:
100:
Intuitionistic type theory's type constructors were built to follow a one-to-one correspondence with logical connectives. For example, the logical connective called implication (
4500:
4478:
3528:
3383:
2551:
trees. Later work in type theory generated coinductive types, induction-recursion, and induction-induction for working on types with more obscure kinds of self-referentiality.
2897:
2866:
2835:
2804:
2773:
2707:
2676:
2592:
327:
2142:
of natural numbers is either an empty list or a pair of a natural number and another linked list. Inductive types can be used to define unbounded mathematical structures like
605:
of sets. In the case of the usual cartesian product, the type of the second term does not depend on the value of the first term. Thus the type describing the cartesian product
5437:. Integers and rational numbers can be represented without setoids, but this representation is difficult to work with. Cauchy real numbers cannot be represented without this.
3419:
2050:
4546:
4458:
4414:
1501:
1428:
969:
896:
735:
2363:
1581:
2024:
1845:
1735:
5248:
1308:
126:
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2085:
4725:
4579:
512:
406:
5194:
1787:
1697:
232:
5302:
5137:
5043:
5001:
2111:
5658:
book from 1984, but it is somewhat open-ended and does not seem to represent a particular set of choices and so there is no specific type theory associated with it.
1334:
152:
5338:
5091:
4959:
4827:
3324:
3226:
3128:
1992:
288:
2325:
1530:
1457:
1246:
1027:
998:
925:
3483:
3451:
3272:
3154:
1972:
1948:
where the terms do not reduce to the same canonical term, but you will be unable to create terms of that new type. In fact, if you were able to create a term of
1946:
1761:
1671:
1162:
206:
3895:
can be removed by defining equality. Here the relationship with addition is defined using equality and using pattern matching to handle the recursive aspect of
2296:
535:
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4679:
4659:
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4619:
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3913:
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3017:
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1807:
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1404:
1154:
1134:
1047:
945:
872:
711:
486:
466:
446:
426:
4792:
3098:
This second level of the type theory can be confusing, particularly where it comes to equality. There is a judgement of term equality, which might say
1857:
5566:, i.e., one would write directly "TâV" and "tâT" (Martin-Löf uses the sign "â" instead of modern ":") without the additional constructor such as "El".
5479:. While many are based on Per Martin-Löf's ideas, many have added features, more axioms, or a different philosophical background. For instance, the
5425:
or similar constructions. There are many common mathematical objects that are hard to work with or cannot be represented without this, for example,
4743:
5552:
was presented in a 1972 preprint that has now been published. That theory had one universe V and no identity types (=-types). The universe was "
6158:
2563:
The universe types allow proofs to be written about all the types created with the other type constructors. Every term in the universe type
6451:
5398:
in extensional type theory because programs in the theory might not terminate. For example, such a theory allows one to give a type to the
645:
5047:
of terms. The axioms for a functor require that these play harmoniously with substitution. Substitution is usually written in the form
1282:
is the type of functions from natural numbers to real numbers. Such Î -types correspond to logical implication. The logical proposition
6688:
2214:
5788:
5421:, but the representation of standard mathematical concepts is somewhat more cumbersome, since intensional reasoning requires using
4277:
3776:
The object-depending-on-object can also be declared as a constant as part of a recursive type. An example of a recursive type is:
3815:
1589:
6724:
5975:. Logic, methodology and philosophy of science, VI (Hannover, 1979). Vol. 104. Amsterdam: North-Holland. pp. 153â175.
1055:
3637:
An object that depends on an object from another type can be done two ways. If the object is "abstracted", then it is written
6729:
6100:
5701:
4129:
3130:. It is a statement that two terms reduce to the same canonical term. There is also a judgement of type equality, say that
170:
Intuitionistic type theory has three finite types, which are then composed using five different type constructors. Unlike
4183:
2963:
Universe types are a tricky feature of type theories. Martin-Löf's original type theory had to be changed to account for
97:. An interesting consequence is that proofs become mathematical objects that can be examined, compared, and manipulated.
6719:
4336:
608:
6739:
6017:
4232:
6615:
4765:
of contexts (in which the objects are contexts, and the context morphisms are substitutions), together with a functor
1336:, containing functions that take proofs-of-A and return proofs-of-B. This type could be written more consistently as:
1251:
358:
Propositions are instead represented by particular types. For instance, a true proposition can be represented by the
6529:
6151:
6079:
6058:
5902:
5872:
4747:
2979:
The formal definition of intuitionistic type theory is written using judgements. For example, in the statement "if
2712:
840:{\displaystyle \sum _{m{\mathbin {:}}{\mathbb {Z} }}{\sum _{n{\mathbin {:}}{\mathbb {Z} }}((m<n)={\text{True}})}}
6444:
2180:
1737:. The terms of that new type represent proofs that the pair reduce to the same canonical term. Thus, since both
6567:
3782:
5608:
1342:
6698:
5508:
5504:
5406:. However, this does not prevent extensional type theory from being a basis for a practical tool; for example,
3060:
3022:
158:. Previous type theories had also followed this isomorphism, but Martin-Löf's was the first to extend it to
57:, who first published it in 1972. There are multiple versions of the type theory: Martin-Löf proposed both
6144:
5516:
2597:
2147:
155:
4419:
2926:
1228:
When the output type does not depend on the input value, the function type is often simply written with a
378:ÎŁ-types contain ordered pairs. As with typical ordered pair (or 2-tuple) types, a ÎŁ-type can describe the
6693:
6519:
6437:
6305:
4485:
4463:
3494:
3349:
90:
2871:
2840:
2809:
2778:
2747:
2681:
2650:
2566:
297:
6461:
5500:
39:
6004:
3394:
2029:
347:
type contains 2 canonical terms. It represents a definite choice between two values. It is used for
6592:
5694:
Interactive theorem proving and program development: Coq'Art: the calculus of inductive constructions
5492:
745:
5855:
4527:
4439:
4395:
3875:
is a constant object-depending-on-object. It is not associated with an abstraction. Constants like
1482:
1409:
950:
877:
716:
6116:
6041:
5448:
3329:
The description of judgements below is based on the discussion in
Nordström, Petersson, and Smith.
2337:
1555:
1383:
5617:, but it is unclear how to declare them to be equal since there are no identity types connecting V
5475:
Different forms of type theory have been implemented as the formal systems underlying a number of
1997:
1812:
1702:
6572:
6203:
5812:
Allen, S.F.; Bickford, M.; Constable, R.L.; Eaton, R.; Kreitz, C.; Lorigo, L.; Moran, E. (2006).
5203:
1285:
103:
5960:. Logic Colloquium '73 (Bristol, 1973). Vol. 80. Amsterdam: North-Holland. pp. 73â118.
2055:
6673:
6658:
6577:
6482:
5850:
5391:
5387:
4704:
4558:
851:
491:
385:
367:
5158:
3057:
is a type" there are judgements of "is a type", "and", and "if ... then ...". The expression
1766:
1676:
211:
6744:
6635:
6610:
6429:
6408:
6188:
6090:
5672:
5562:
5512:
5496:
5444:
5269:
5104:
5010:
4968:
2552:
2150:, etc.. In fact, the natural numbers type may be defined as an inductive type, either being
2090:
178:. So, each feature of the type theory does double duty as a feature of both math and logic.
6557:
5898:"Idris, a general-purpose dependently typed programming language: Design and implementation"
5847:
Proceedings of the 4th international workshop on Types in language design and implementation
1847:. In intuitionistic type theory, there is a single way to introduce =-types and that is by
1313:
1218:{\displaystyle \prod _{n{\mathbin {:}}{\mathbb {N} }}\operatorname {Vec} ({\mathbb {R} },n)}
131:
6734:
6630:
6587:
6279:
6241:
6236:
6183:
5667:
5440:
5311:
5064:
4932:
4800:
3297:
3199:
3101:
2547:
Inductive types in intuitionistic type theory are defined in terms of W-types, the type of
2331:
2121:
1977:
591:{\displaystyle \sum _{n{\mathbin {:}}{\mathbb {N} }}\operatorname {Vec} ({\mathbb {R} },n)}
273:
6338:
2301:
1506:
1433:
1231:
1003:
974:
901:
290:
and represents anything unprovable. (That is, a proof of it cannot exist.) As a result,
8:
6640:
6418:
6251:
6167:
5795:
5399:
5395:
4750:(LCCC) as the basic model of type theory. This has been refined by Hofmann and Dybjer to
3462:
3430:
3251:
3133:
2964:
2143:
1951:
1925:
1740:
1650:
185:
70:
4730:
This can be done for other types (booleans, natural numbers, etc.) and their operators.
4621:. The objects in that dependent type are defined to exist for every pair of objects in
2278:
6562:
6509:
6469:
6413:
6328:
6193:
5929:
5878:
5756:
5730:
5719:"The biequivalence of locally cartesian closed categories and Martin-Löf type theories"
4684:
4664:
4644:
4624:
4604:
4584:
4507:
4101:
3898:
3878:
3858:
3755:
3735:
3715:
3678:
3643:
3616:
3596:
3576:
3539:
3277:
3231:
3179:
3159:
3002:
2982:
2906:
2258:
2171:
2153:
2138:
Inductive types allow the creation of complex, self-referential types. For example, a
1848:
1792:
1535:
1462:
1389:
1139:
1119:
1032:
930:
857:
696:
471:
451:
431:
411:
94:
4118:
is manipulated as an opaque constant - it has no internal structure for substitution.
2647:
and the inductive type constructor. However, to avoid paradoxes, there is no term in
6547:
6502:
6403:
6368:
6356:
6333:
6320:
6310:
6297:
6096:
6075:
6054:
6023:
6013:
5921:
5868:
5748:
5697:
379:
175:
58:
6048:
5933:
5527:
2334:. Each new inductive type comes with its own inductive rule. To prove a predicate
43:
6668:
6552:
6492:
6361:
5911:
5882:
5860:
5825:
5740:
5557:
5543:
5531:
5452:
5418:
6069:
5760:
5546:
has shown that this system was inconsistent and the preprint was never published.
2275:
does not have a definition and cannot be evaluated using substitution, terms like
1912:{\displaystyle \operatorname {refl} {\mathbin {:}}\prod _{a{\mathbin {:}}A}(a=a).}
6650:
6348:
6264:
6259:
6221:
5476:
5456:
5430:
4739:
2900:
348:
234:
compute ("reduce") down to canonical terms like 4. For more, see the article on
159:
78:
77:
versions. However, all versions keep the core design of constructive logic using
336:
type contains 1 canonical term and represents existence. It also is called the
6678:
6620:
6497:
5956:
Martin-Löf, Per (1975). "An intuitionistic theory of types: predicative part".
5488:
2133:
927:
is proven" becomes the type of ordered pairs where the first item is the value
602:
518:
62:
6122:
5916:
5830:
5813:
5744:
4392:
By convention, there is a type that represents all other types. It is called
6713:
6487:
5971:
Martin-Löf, Per (1982). "Constructive mathematics and computer programming".
5925:
5752:
5553:
5414:
74:
66:
50:
6027:
5864:
4551:
This is the complete foundation of the theory. Everything else is derived.
6514:
6477:
6269:
6175:
4504:
From the context of the statement, a reader can almost always tell whether
2548:
2117:
366:
type. But we cannot assert that these are the only propositions, i.e. the
270:. It is used to represent anything that cannot exist. It is also written
6683:
6602:
6582:
6524:
6287:
6213:
6130:
5434:
2139:
526:
235:
54:
35:
3326:. Synonyms like these are called "definitionally equal" by Martin-Löf.
6539:
6226:
2968:
2903:
hierarchy of universes, so to quantify a proof over any fixed constant
267:
171:
6136:
5696:. Texts in theoretical computer science. Berlin Heidelberg: Springer.
6380:
6231:
6126:
4460:
is a type, the members of it are objects. There is a dependent type
337:
5455:), but higher-order constructors, i.e. equalities between elements (
5402:; a detailed example of this can be found in Nordstöm and Petersson
5607:. This means, for example, that it would be difficult to formulate
5460:
2052:
is how intuitionistic type theory defines negation, you would have
1116:
As an example, the type of a function that, given a natural number
683:{\displaystyle \sum _{n{\mathbin {:}}{\mathbb {N} }}{\mathbb {R} }}
522:
291:
5735:
4701:
has no proof and is an empty type, then the new type representing
1994:. Putting that into a function would generate a function of type
6373:
5426:
529:
of length equal to the first term. Such a type would be written:
5422:
3488:
A type that depends on an object from another type is declared
2248:{\displaystyle S{\mathbin {:}}{\mathbb {N} }\to {\mathbb {N} }}
693:
It is important to note here that the value of the first term,
47:
6459:
5583:, ... . The universes are predicative, Ă la Russell and
1029:) depends on the value in the first part of the ordered pair (
370:
does not hold for propositions in intuitionistic type theory.
6663:
6042:
Per Martin-Löf's Notes, as recorded by
Giovanni Sambin (1980)
5845:
Norell, Ulf (2009). "Dependently typed programming in Agda".
5480:
5407:
2967:. Later research covered topics such as "super universes", "
741:
6119:â lecture notes and slides from the Types Summer School 2005
5611:
in this theoryâthere are contractible types in each of the V
4309:{\displaystyle \Gamma \vdash t\equiv u{\mathbin {:}}\sigma }
601:
Using set-theory terminology, this is similar to an indexed
6385:
5989:(lecture notes by Giovanni Sambin), vol. 1, pp. iv+91, 1984
5811:
3845:{\displaystyle S{\mathbin {:}}\mathbb {N} \to \mathbb {N} }
1647:=-types are created from two terms. Given two terms like
1632:{\displaystyle \prod _{n{\mathbin {:}}{\mathbb {N} }}P(n)}
362:
type, while a false proposition can be represented by the
6047:
Nordström, Bengt; Petersson, Kent; Smith, Jan M. (1990).
5587:. In fact, Corollary 3.10 on p. 115 says that if AâV
5443:
works on resolving this problem. It allows one to define
1098:{\displaystyle \sum _{n{\mathbin {:}}{\mathbb {N} }}P(n)}
740:ÎŁ-types can be used to build up longer dependently-typed
89:
Martin-Löf designed the type theory on the principles of
4524:
refers to a type, or whether it refers to the object in
2594:
can be mapped to a type created with any combination of
448:. Logically, such an ordered pair would hold a proof of
6046:
4164:{\displaystyle \Gamma \vdash \sigma \ {\mathsf {Type}}}
5814:"Innovations in computational type theory using Nuprl"
5773:
Bengt
Nordström; Kent Petersson; Jan M. Smith (1990).
3064:
3026:
2365:
for every natural number, you use the following rule:
1000:. Notice that the type of the second item (proofs of
5314:
5272:
5206:
5161:
5107:
5067:
5013:
4971:
4935:
4803:
4707:
4687:
4667:
4647:
4627:
4607:
4587:
4561:
4530:
4510:
4488:
4466:
4442:
4422:
4398:
4339:
4280:
4235:
4186:
4132:
4104:
3924:
3901:
3881:
3861:
3818:
3785:
3758:
3738:
3718:
3681:
3646:
3619:
3599:
3579:
3542:
3497:
3465:
3433:
3397:
3352:
3300:
3280:
3254:
3234:
3202:
3182:
3162:
3136:
3104:
3063:
3025:
3005:
2985:
2929:
2909:
2874:
2843:
2812:
2781:
2750:
2715:
2684:
2653:
2600:
2569:
2374:
2340:
2304:
2281:
2261:
2217:
2183:
2156:
2093:
2058:
2032:
2000:
1980:
1954:
1928:
1860:
1815:
1795:
1769:
1743:
1705:
1679:
1653:
1592:
1558:
1538:
1509:
1485:
1465:
1436:
1412:
1392:
1345:
1316:
1288:
1254:
1234:
1165:
1142:
1122:
1058:
1035:
1006:
977:
953:
933:
904:
880:
860:
850:
Dependent typing allows ÎŁ-types to serve the role of
757:
719:
713:, is not depended on by the type of the second term,
699:
648:
611:
538:
494:
474:
454:
434:
414:
388:
300:
276:
214:
188:
174:, type theories are not built on top of a logic like
134:
106:
4733:
4209:{\displaystyle \Gamma \vdash t{\mathbin {:}}\sigma }
3196:
and vice versa. At the type level, there is a type
2116:
Equality of proofs is an area of active research in
5973:
5958:
4377:{\displaystyle \vdash \Gamma \ {\mathsf {Context}}}
2327:become the canonical terms of the natural numbers.
2177:Inductive types define new constants, such as zero
632:{\displaystyle {\mathbb {N} }\times {\mathbb {R} }}
5849:. TLDI '09. New York, NY, USA: ACM. pp. 1â2.
5716:
5447:, which not only define first-order constructors (
5332:
5296:
5242:
5188:
5131:
5085:
5037:
4995:
4953:
4821:
4719:
4693:
4673:
4653:
4633:
4613:
4593:
4573:
4540:
4516:
4494:
4480:that maps each object to its corresponding type.
4472:
4452:
4428:
4408:
4386:Î is a well-formed context of typing assumptions.
4376:
4308:
4253:
4208:
4163:
4110:
4088:
3907:
3887:
3867:
3844:
3803:
3764:
3744:
3724:
3704:
3661:
3625:
3605:
3585:
3565:
3522:
3477:
3445:
3413:
3377:
3318:
3286:
3266:
3240:
3220:
3188:
3168:
3148:
3122:
3095:is not a judgement; it is the type being defined.
3087:
3049:
3011:
2991:
2952:
2915:
2891:
2860:
2829:
2798:
2767:
2733:
2701:
2670:
2639:
2586:
2536:
2357:
2319:
2290:
2267:
2247:
2203:
2162:
2105:
2079:
2044:
2018:
1986:
1966:
1940:
1911:
1839:
1801:
1781:
1755:
1729:
1691:
1665:
1631:
1575:
1544:
1524:
1495:
1471:
1451:
1422:
1398:
1371:
1328:
1302:
1274:
1240:
1217:
1148:
1128:
1097:
1041:
1021:
992:
963:
939:
919:
890:
866:
839:
729:
705:
682:
631:
590:
506:
480:
460:
440:
420:
400:
321:
282:
226:
200:
146:
120:
5787:Altenkirch, Thorsten; Anberrée, Thomas; Li, Nuo.
5470:
5381:
4254:{\displaystyle \Gamma \vdash \sigma \equiv \tau }
6711:
5786:
5654:: there is a discussion of a type theory in the
5495:. Dependent types also feature in the design of
2744:To write proofs about all "the small types" and
1275:{\displaystyle {\mathbb {N} }\to {\mathbb {R} }}
3228:and it contains terms if there is a proof that
2330:Proofs on inductive types are made possible by
5985:Per Martin-Löf, "Intuitionistic type theory",
5691:
2734:{\displaystyle {\mathcal {n}}\in \mathbb {N} }
6445:
6152:
5979:
2204:{\displaystyle 0{\mathbin {:}}{\mathbb {N} }}
266:type contains 0 terms, it is also called the
5717:Clairambault, Pierre; Dybjer, Peter (2014).
2555:allow equality to be defined between terms.
242:
42:. Intuitionistic type theory was created by
5723:Mathematical Structures in Computer Science
5599:are such that A and B are convertible then
5522:
3804:{\displaystyle 0{\mathbin {:}}\mathbb {N} }
6452:
6438:
6159:
6145:
6002:
5970:
5955:
1372:{\displaystyle \prod _{a{\mathbin {:}}A}B}
1296:
1292:
114:
110:
6088:
6012:. Sambin, Giovanni. Napoli: Bibliopolis.
5915:
5854:
5829:
5734:
3969:
3958:
3950:
3838:
3830:
3797:
2971:universes", and impredicative universes.
2727:
2515:
2447:
2421:
2401:
2379:
2351:
2347:
2240:
2230:
2196:
1922:It is possible to create =-types such as
1610:
1569:
1565:
1488:
1415:
1267:
1257:
1201:
1183:
1076:
956:
883:
800:
775:
722:
675:
666:
624:
614:
574:
556:
255:type contains 1 canonical term. And the
128:) corresponds to the type of a function (
6067:
4176:is a well-formed type in the context Î.
1583:holds for that value. The type would be
488:, so one may see such a type written as
6166:
6050:Programming in Martin-Löf's Type Theory
5775:Programming in Martin-Löf's Type Theory
5692:Bertot, Yves; Castéran, Pierre (2004).
5493:calculus of (co)inductive constructions
5413:In contrast in intensional type theory
5404:Programming in Martin-Löf's Type Theory
4847:, and morphisms are pairs of functions
4761:A category with families is a category
3088:{\displaystyle \textstyle \sum _{a:A}B}
3050:{\displaystyle \textstyle \sum _{a:A}B}
6712:
6092:Treatise on Intuitionistic Type Theory
6071:Type Theory and Functional Programming
5844:
4369:
4366:
4363:
4360:
4357:
4354:
4351:
4156:
4153:
4150:
4147:
3388:An object exists and is in a type if:
3370:
3367:
3364:
3361:
521:and the second term's type might be a
69:versions, shown to be inconsistent by
6433:
6140:
6123:n-Categories - Sketch of a Definition
5895:
5410:is based on extensional type theory.
4325:are judgmentally equal terms of type
2640:{\displaystyle 0,1,2,\Sigma ,\Pi ,=,}
1642:
1108:
373:
154:). This correspondence is called the
16:Alternative foundation of mathematics
5789:"Definable Quotients in Type Theory"
4795:of Sets, in which objects are pairs
4429:{\displaystyle \operatorname {Set} }
2953:{\displaystyle {\mathcal {U}}_{k+1}}
1552:the function generates a proof that
4758:based on earlier work by Cartmell.
4581:is a type that depends on the type
4495:{\displaystyle \operatorname {El} }
4473:{\displaystyle \operatorname {El} }
3523:{\displaystyle (x{\mathbin {:}}A)B}
3378:{\displaystyle A\ {\mathsf {Type}}}
1809:, there will be a term of the type
1459:is proven" becomes a function from
1382:Î -types are also used in logic for
247:There are three finite types: The
13:
6035:
5459:), equalities between equalities (
4533:
4445:
4401:
4343:
4281:
4236:
4187:
4133:
2933:
2892:{\displaystyle {\mathcal {U}}_{2}}
2878:
2861:{\displaystyle {\mathcal {U}}_{1}}
2847:
2830:{\displaystyle {\mathcal {U}}_{0}}
2816:
2799:{\displaystyle {\mathcal {U}}_{1}}
2785:
2768:{\displaystyle {\mathcal {U}}_{0}}
2754:
2718:
2702:{\displaystyle {\mathcal {U}}_{n}}
2688:
2671:{\displaystyle {\mathcal {U}}_{n}}
2657:
2625:
2619:
2587:{\displaystyle {\mathcal {U}}_{0}}
2573:
2396:
2393:
2390:
2387:
2127:
2120:and has led to the development of
2059:
2039:
2013:
1981:
971:and the second item is a proof of
854:. The statement "there exists an
322:{\displaystyle \neg A:=A\to \bot }
316:
301:
277:
14:
6756:
6530:List of mathematical logic topics
6110:
5903:Journal of Functional Programming
5777:. Oxford University Press, p. 90.
5306:, then there should be an object
4748:locally cartesian closed category
4734:Categorical models of type theory
3294:" to refer to the canonical term
2558:
259:type contains 2 canonical terms.
65:variants of the theory and early
3414:{\displaystyle a{\mathbin {:}}A}
2806:, which does contain a term for
2045:{\displaystyle \ldots \to \bot }
294:is defined as a function to it:
162:by introducing dependent types.
5964:
3156:, which means every element of
6699:List of category theory topics
5949:
5940:
5889:
5838:
5805:
5780:
5767:
5710:
5685:
5471:Implementations of type theory
5382:Extensional versus intensional
5327:
5315:
5291:
5285:
5237:
5222:
5183:
5177:
5126:
5114:
5080:
5074:
5032:
5020:
4990:
4984:
4948:
4942:
4816:
4804:
4548:that corresponds to the type.
4541:{\displaystyle {\mathcal {U}}}
4453:{\displaystyle {\mathcal {U}}}
4409:{\displaystyle {\mathcal {U}}}
4270:are equal types in context Î.
4221:is a well-formed term of type
4079:
4076:
4073:
4061:
4052:
4039:
4030:
4024:
4018:
3995:
3983:
3965:
3962:
3946:
3834:
3699:
3685:
3653:
3647:
3560:
3546:
3514:
3498:
2531:
2525:
2495:
2487:
2484:
2478:
2472:
2466:
2463:
2457:
2422:
2418:
2412:
2352:
2344:
2235:
2074:
2062:
2036:
2010:
1903:
1891:
1789:compute to the canonical term
1626:
1620:
1570:
1562:
1519:
1513:
1496:{\displaystyle {\mathbb {N} }}
1446:
1440:
1423:{\displaystyle {\mathbb {N} }}
1320:
1293:
1262:
1235:
1212:
1196:
1136:, returns a vector containing
1092:
1086:
1016:
1010:
987:
981:
964:{\displaystyle {\mathbb {N} }}
914:
908:
891:{\displaystyle {\mathbb {N} }}
833:
822:
810:
807:
730:{\displaystyle {\mathbb {R} }}
585:
569:
313:
165:
138:
111:
1:
6725:Dependently typed programming
5996:
5386:A fundamental distinction is
3332:The formal theory works with
2974:
2358:{\displaystyle P(\,\cdot \,)}
1974:, you could create a term of
1576:{\displaystyle P(\,\cdot \,)}
1532:. Thus, given the value for
6730:Constructivism (mathematics)
6089:Granström, Johan G. (2011).
3672:and removed by substitution
3533:and removed by substitution
2019:{\displaystyle 1=2\to \bot }
1840:{\displaystyle 2+2=2\cdot 2}
1730:{\displaystyle 2+2=2\cdot 2}
1699:, you can create a new type
1386:. The statement "for every
744:used in mathematics and the
251:type contains 0 terms. The
30:, the latter abbreviated as
7:
6694:Glossary of category theory
6568:ZermeloâFraenkel set theory
6520:Mathematical constructivism
6306:Ontology (computer science)
6117:EU Types Project: Tutorials
6053:. Oxford University Press.
5661:
5243:{\displaystyle af:Tm(D,Af)}
4746:introduced the notion of a
4555:types and new objects. So
2211:and the successor function
2174:of another natural number.
1303:{\displaystyle A\implies B}
121:{\displaystyle A\implies B}
91:mathematical constructivism
10:
6761:
6720:Foundations of mathematics
6689:Mathematical structuralism
6626:Intuitionistic type theory
6462:Foundations of Mathematics
6199:Intuitionistic type theory
6006:Intuitionistic type theory
4756:Categories with Attributes
3176:is an element of the type
2131:
2080:{\displaystyle \neg (1=2)}
20:Intuitionistic type theory
6740:Logic in computer science
6649:
6601:
6593:List of set theory topics
6538:
6468:
6396:
6347:
6319:
6296:
6278:
6250:
6212:
6174:
5917:10.1017/S095679681300018X
5831:10.1016/j.jal.2005.10.005
5745:10.1017/S0960129513000881
5485:computational type theory
4720:{\displaystyle A\times B}
4574:{\displaystyle A\times B}
3712:, replacing the variable
3573:, replacing the variable
2124:and other type theories.
1156:real numbers is written:
507:{\displaystyle A\wedge B}
401:{\displaystyle A\times B}
243:0 type, 1 type and 2 type
84:
40:foundation of mathematics
6068:Thompson, Simon (1991).
6003:Martin-Löf, Per (1984).
5818:Journal of Applied Logic
5678:
5523:Martin-Löf type theories
5189:{\displaystyle Af:Ty(D)}
4752:Categories with Families
1782:{\displaystyle 2\cdot 2}
1692:{\displaystyle 2\cdot 2}
1384:universal quantification
1310:corresponds to the type
227:{\displaystyle 2\cdot 2}
156:CurryâHoward isomorphism
24:constructive type theory
6573:Constructive set theory
6204:Constructive set theory
5987:Studies in Proof Theory
5865:10.1145/1481861.1481862
5297:{\displaystyle A:Ty(G)}
5145:is a substitution from
5132:{\displaystyle Tm(G,A)}
5038:{\displaystyle Tm(G,A)}
4996:{\displaystyle A:Ty(G)}
4963:of types, and for each
3456:and types can be equal
3343:A type is declared by:
2923:universes, you can use
2106:{\displaystyle 1\neq 2}
6674:Higher category theory
6578:Descriptive set theory
6483:Mathematical induction
5445:higher inductive types
5334:
5298:
5244:
5190:
5133:
5087:
5039:
4997:
4955:
4823:
4738:Using the language of
4721:
4695:
4675:
4655:
4635:
4615:
4595:
4575:
4542:
4518:
4496:
4474:
4454:
4430:
4410:
4378:
4310:
4255:
4210:
4165:
4112:
4090:
3909:
3889:
3869:
3846:
3805:
3766:
3746:
3726:
3706:
3663:
3627:
3607:
3587:
3567:
3524:
3479:
3447:
3415:
3379:
3320:
3288:
3268:
3242:
3222:
3190:
3170:
3150:
3124:
3089:
3051:
3013:
2993:
2954:
2917:
2893:
2862:
2831:
2800:
2769:
2735:
2703:
2672:
2641:
2588:
2553:Higher inductive types
2538:
2359:
2321:
2292:
2269:
2249:
2205:
2164:
2107:
2081:
2046:
2020:
1988:
1968:
1942:
1913:
1841:
1803:
1783:
1757:
1731:
1693:
1667:
1633:
1577:
1546:
1526:
1497:
1473:
1453:
1424:
1400:
1373:
1330:
1329:{\displaystyle A\to B}
1304:
1276:
1242:
1219:
1150:
1130:
1099:
1049:). Its type would be:
1043:
1023:
994:
965:
941:
921:
892:
868:
852:existential quantifier
841:
731:
707:
684:
633:
592:
508:
482:
462:
442:
422:
408:, of two other types,
402:
368:law of excluded middle
323:
284:
228:
202:
148:
147:{\displaystyle A\to B}
122:
28:Martin-Löf type theory
6636:Univalent foundations
6621:Dependent type theory
6611:Axiom of reducibility
6189:Constructive analysis
5896:Brady, Edwin (2013).
5673:Typed lambda calculus
5563:Principia Mathematica
5497:programming languages
5342:final among contexts
5335:
5333:{\displaystyle (G,A)}
5299:
5245:
5191:
5134:
5088:
5086:{\displaystyle Ty(G)}
5040:
4998:
4956:
4954:{\displaystyle Ty(G)}
4923:assigns to a context
4824:
4822:{\displaystyle (A,B)}
4722:
4696:
4676:
4656:
4636:
4616:
4596:
4576:
4543:
4519:
4497:
4475:
4455:
4431:
4411:
4379:
4311:
4256:
4211:
4166:
4113:
4091:
3910:
3890:
3870:
3847:
3806:
3767:
3747:
3727:
3707:
3664:
3628:
3608:
3588:
3568:
3525:
3480:
3448:
3424:Objects can be equal
3416:
3380:
3321:
3319:{\displaystyle SSSS0}
3289:
3269:
3243:
3223:
3221:{\displaystyle 4=2+2}
3191:
3171:
3151:
3125:
3123:{\displaystyle 4=2+2}
3090:
3052:
3014:
2994:
2955:
2918:
2894:
2863:
2837:, but not for itself
2832:
2801:
2770:
2736:
2704:
2673:
2642:
2589:
2539:
2360:
2322:
2293:
2270:
2250:
2206:
2165:
2108:
2082:
2047:
2021:
1989:
1987:{\displaystyle \bot }
1969:
1943:
1914:
1842:
1804:
1784:
1758:
1732:
1694:
1668:
1634:
1578:
1547:
1527:
1498:
1474:
1454:
1425:
1401:
1374:
1331:
1305:
1277:
1243:
1220:
1151:
1131:
1100:
1044:
1024:
995:
966:
942:
922:
893:
869:
842:
732:
708:
685:
634:
593:
509:
483:
463:
443:
423:
403:
324:
285:
283:{\displaystyle \bot }
229:
203:
149:
123:
6631:Homotopy type theory
6558:Axiomatic set theory
6242:Fuzzy set operations
6237:Fuzzy finite element
6184:Intuitionistic logic
5668:Intuitionistic logic
5441:Homotopy type theory
5312:
5270:
5204:
5159:
5105:
5065:
5011:
4969:
4933:
4801:
4793:category of families
4705:
4685:
4665:
4645:
4625:
4605:
4585:
4559:
4528:
4508:
4486:
4464:
4440:
4420:
4396:
4337:
4278:
4233:
4184:
4130:
4102:
3922:
3899:
3879:
3859:
3816:
3783:
3756:
3736:
3716:
3679:
3644:
3617:
3597:
3577:
3540:
3495:
3463:
3431:
3395:
3350:
3298:
3278:
3252:
3232:
3200:
3180:
3160:
3134:
3102:
3061:
3023:
3003:
2983:
2927:
2907:
2872:
2841:
2810:
2779:
2748:
2713:
2682:
2651:
2598:
2567:
2372:
2338:
2320:{\displaystyle SSS0}
2302:
2279:
2259:
2215:
2181:
2154:
2122:homotopy type theory
2091:
2056:
2030:
1998:
1978:
1952:
1926:
1858:
1813:
1793:
1767:
1741:
1703:
1677:
1651:
1590:
1556:
1536:
1525:{\displaystyle P(n)}
1507:
1483:
1463:
1452:{\displaystyle P(n)}
1434:
1410:
1390:
1343:
1314:
1286:
1252:
1241:{\displaystyle \to }
1232:
1163:
1140:
1120:
1056:
1033:
1022:{\displaystyle P(n)}
1004:
993:{\displaystyle P(n)}
975:
951:
931:
920:{\displaystyle P(n)}
902:
878:
858:
755:
717:
697:
646:
609:
536:
492:
472:
452:
432:
412:
386:
298:
274:
212:
186:
132:
104:
6419:Non-monotonic logic
6168:Non-classical logic
6133:, November 29, 1995
6129:and James Dolan to
5483:system is based on
3478:{\displaystyle A=B}
3446:{\displaystyle a=b}
3267:{\displaystyle 2+2}
3149:{\displaystyle A=B}
1967:{\displaystyle 1=2}
1941:{\displaystyle 1=2}
1756:{\displaystyle 2+2}
1666:{\displaystyle 2+2}
201:{\displaystyle 2+2}
38:and an alternative
6616:Simple type theory
6563:Zermelo set theory
6510:Mathematical proof
6470:Mathematical logic
6414:Intermediate logic
6194:Heyting arithmetic
6074:. Addison-Wesley.
5330:
5294:
5264:is a context, and
5240:
5186:
5129:
5083:
5035:
4993:
4951:
4891:â in other words,
4831:of an "index set"
4819:
4717:
4691:
4671:
4651:
4631:
4611:
4591:
4571:
4538:
4514:
4492:
4470:
4450:
4426:
4406:
4374:
4306:
4251:
4206:
4161:
4108:
4086:
4084:
3905:
3885:
3865:
3842:
3801:
3762:
3742:
3722:
3702:
3659:
3623:
3603:
3583:
3563:
3520:
3475:
3443:
3411:
3375:
3316:
3284:
3264:
3238:
3218:
3186:
3166:
3146:
3120:
3085:
3084:
3080:
3047:
3046:
3042:
3009:
2989:
2950:
2913:
2889:
2868:. Similarly, for
2858:
2827:
2796:
2765:
2731:
2699:
2668:
2637:
2584:
2534:
2521:
2453:
2355:
2317:
2291:{\displaystyle S0}
2288:
2265:
2245:
2201:
2160:
2103:
2077:
2042:
2016:
1984:
1964:
1938:
1909:
1890:
1837:
1799:
1779:
1753:
1727:
1689:
1663:
1643:= type constructor
1629:
1616:
1573:
1542:
1522:
1493:
1469:
1449:
1420:
1396:
1369:
1365:
1326:
1300:
1272:
1238:
1215:
1189:
1146:
1126:
1109:Î type constructor
1095:
1082:
1039:
1019:
990:
961:
937:
917:
888:
864:
837:
806:
781:
746:records or structs
727:
703:
680:
672:
629:
588:
562:
504:
478:
458:
438:
418:
398:
374:ÎŁ type constructor
319:
280:
224:
198:
144:
118:
95:BHK interpretation
6707:
6706:
6588:Russell's paradox
6503:Natural deduction
6427:
6426:
6409:Inquisitive logic
6404:Dynamic semantics
6357:Three-state logic
6311:Ontology language
6102:978-94-007-1735-0
5703:978-3-540-20854-9
4694:{\displaystyle B}
4674:{\displaystyle A}
4654:{\displaystyle B}
4634:{\displaystyle A}
4614:{\displaystyle B}
4594:{\displaystyle A}
4517:{\displaystyle A}
4502:is never written.
4390:
4389:
4348:
4144:
4111:{\displaystyle S}
3945:
3908:{\displaystyle S}
3888:{\displaystyle S}
3868:{\displaystyle S}
3765:{\displaystyle b}
3745:{\displaystyle a}
3725:{\displaystyle x}
3705:{\displaystyle b}
3662:{\displaystyle b}
3626:{\displaystyle B}
3606:{\displaystyle a}
3586:{\displaystyle x}
3566:{\displaystyle B}
3358:
3287:{\displaystyle 4}
3241:{\displaystyle 4}
3189:{\displaystyle B}
3169:{\displaystyle A}
3065:
3027:
3012:{\displaystyle B}
2992:{\displaystyle A}
2916:{\displaystyle k}
2498:
2430:
2386:
2268:{\displaystyle S}
2163:{\displaystyle 0}
1871:
1802:{\displaystyle 4}
1593:
1545:{\displaystyle n}
1472:{\displaystyle n}
1399:{\displaystyle n}
1346:
1166:
1149:{\displaystyle n}
1129:{\displaystyle n}
1059:
1042:{\displaystyle n}
940:{\displaystyle n}
867:{\displaystyle n}
831:
783:
758:
706:{\displaystyle n}
649:
539:
481:{\displaystyle B}
461:{\displaystyle A}
441:{\displaystyle B}
421:{\displaystyle A}
380:Cartesian product
6752:
6669:Category of sets
6641:Girard's paradox
6553:Naive set theory
6493:Axiomatic system
6460:Major topics in
6454:
6447:
6440:
6431:
6430:
6362:Tri-state buffer
6161:
6154:
6147:
6138:
6137:
6106:
6085:
6064:
6031:
6011:
5990:
5983:
5977:
5976:
5968:
5962:
5961:
5953:
5947:
5944:
5938:
5937:
5919:
5893:
5887:
5886:
5858:
5842:
5836:
5835:
5833:
5809:
5803:
5802:
5800:
5794:. Archived from
5793:
5784:
5778:
5771:
5765:
5764:
5738:
5714:
5708:
5707:
5689:
5609:univalence axiom
5544:Jean-Yves Girard
5532:Jean-Yves Girard
5491:is based on the
5477:proof assistants
5431:rational numbers
5341:
5339:
5337:
5336:
5331:
5305:
5303:
5301:
5300:
5295:
5251:
5249:
5247:
5246:
5241:
5197:
5195:
5193:
5192:
5187:
5140:
5138:
5136:
5135:
5130:
5094:
5092:
5090:
5089:
5084:
5046:
5044:
5042:
5041:
5036:
5004:
5002:
5000:
4999:
4994:
4962:
4960:
4958:
4957:
4952:
4830:
4828:
4826:
4825:
4820:
4726:
4724:
4723:
4718:
4700:
4698:
4697:
4692:
4680:
4678:
4677:
4672:
4660:
4658:
4657:
4652:
4640:
4638:
4637:
4632:
4620:
4618:
4617:
4612:
4600:
4598:
4597:
4592:
4580:
4578:
4577:
4572:
4547:
4545:
4544:
4539:
4537:
4536:
4523:
4521:
4520:
4515:
4501:
4499:
4498:
4493:
4479:
4477:
4476:
4471:
4459:
4457:
4456:
4451:
4449:
4448:
4435:
4433:
4432:
4427:
4415:
4413:
4412:
4407:
4405:
4404:
4383:
4381:
4380:
4375:
4373:
4372:
4346:
4315:
4313:
4312:
4307:
4302:
4301:
4260:
4258:
4257:
4252:
4215:
4213:
4212:
4207:
4202:
4201:
4170:
4168:
4167:
4162:
4160:
4159:
4142:
4124:
4123:
4117:
4115:
4114:
4109:
4095:
4093:
4092:
4087:
4085:
3972:
3961:
3953:
3943:
3942:
3941:
3914:
3912:
3911:
3906:
3894:
3892:
3891:
3886:
3874:
3872:
3871:
3866:
3851:
3849:
3848:
3843:
3841:
3833:
3828:
3827:
3810:
3808:
3807:
3802:
3800:
3795:
3794:
3771:
3769:
3768:
3763:
3751:
3749:
3748:
3743:
3732:with the object
3731:
3729:
3728:
3723:
3711:
3709:
3708:
3703:
3695:
3668:
3666:
3665:
3660:
3632:
3630:
3629:
3624:
3612:
3610:
3609:
3604:
3593:with the object
3592:
3590:
3589:
3584:
3572:
3570:
3569:
3564:
3556:
3529:
3527:
3526:
3521:
3510:
3509:
3484:
3482:
3481:
3476:
3452:
3450:
3449:
3444:
3420:
3418:
3417:
3412:
3407:
3406:
3384:
3382:
3381:
3376:
3374:
3373:
3356:
3325:
3323:
3322:
3317:
3293:
3291:
3290:
3285:
3273:
3271:
3270:
3265:
3247:
3245:
3244:
3239:
3227:
3225:
3224:
3219:
3195:
3193:
3192:
3187:
3175:
3173:
3172:
3167:
3155:
3153:
3152:
3147:
3129:
3127:
3126:
3121:
3094:
3092:
3091:
3086:
3079:
3056:
3054:
3053:
3048:
3041:
3018:
3016:
3015:
3010:
2998:
2996:
2995:
2990:
2965:Girard's paradox
2959:
2957:
2956:
2951:
2949:
2948:
2937:
2936:
2922:
2920:
2919:
2914:
2898:
2896:
2895:
2890:
2888:
2887:
2882:
2881:
2867:
2865:
2864:
2859:
2857:
2856:
2851:
2850:
2836:
2834:
2833:
2828:
2826:
2825:
2820:
2819:
2805:
2803:
2802:
2797:
2795:
2794:
2789:
2788:
2774:
2772:
2771:
2766:
2764:
2763:
2758:
2757:
2740:
2738:
2737:
2732:
2730:
2722:
2721:
2708:
2706:
2705:
2700:
2698:
2697:
2692:
2691:
2677:
2675:
2674:
2669:
2667:
2666:
2661:
2660:
2646:
2644:
2643:
2638:
2593:
2591:
2590:
2585:
2583:
2582:
2577:
2576:
2543:
2541:
2540:
2535:
2520:
2519:
2518:
2512:
2511:
2494:
2490:
2452:
2451:
2450:
2444:
2443:
2408:
2407:
2400:
2399:
2384:
2383:
2382:
2364:
2362:
2361:
2356:
2326:
2324:
2323:
2318:
2297:
2295:
2294:
2289:
2274:
2272:
2271:
2266:
2254:
2252:
2251:
2246:
2244:
2243:
2234:
2233:
2227:
2226:
2210:
2208:
2207:
2202:
2200:
2199:
2193:
2192:
2169:
2167:
2166:
2161:
2112:
2110:
2109:
2104:
2086:
2084:
2083:
2078:
2051:
2049:
2048:
2043:
2025:
2023:
2022:
2017:
1993:
1991:
1990:
1985:
1973:
1971:
1970:
1965:
1947:
1945:
1944:
1939:
1918:
1916:
1915:
1910:
1889:
1885:
1884:
1870:
1869:
1846:
1844:
1843:
1838:
1808:
1806:
1805:
1800:
1788:
1786:
1785:
1780:
1762:
1760:
1759:
1754:
1736:
1734:
1733:
1728:
1698:
1696:
1695:
1690:
1672:
1670:
1669:
1664:
1638:
1636:
1635:
1630:
1615:
1614:
1613:
1607:
1606:
1582:
1580:
1579:
1574:
1551:
1549:
1548:
1543:
1531:
1529:
1528:
1523:
1502:
1500:
1499:
1494:
1492:
1491:
1478:
1476:
1475:
1470:
1458:
1456:
1455:
1450:
1429:
1427:
1426:
1421:
1419:
1418:
1405:
1403:
1402:
1397:
1378:
1376:
1375:
1370:
1364:
1360:
1359:
1335:
1333:
1332:
1327:
1309:
1307:
1306:
1301:
1281:
1279:
1278:
1273:
1271:
1270:
1261:
1260:
1247:
1245:
1244:
1239:
1224:
1222:
1221:
1216:
1205:
1204:
1188:
1187:
1186:
1180:
1179:
1155:
1153:
1152:
1147:
1135:
1133:
1132:
1127:
1104:
1102:
1101:
1096:
1081:
1080:
1079:
1073:
1072:
1048:
1046:
1045:
1040:
1028:
1026:
1025:
1020:
999:
997:
996:
991:
970:
968:
967:
962:
960:
959:
946:
944:
943:
938:
926:
924:
923:
918:
897:
895:
894:
889:
887:
886:
873:
871:
870:
865:
846:
844:
843:
838:
836:
832:
829:
805:
804:
803:
797:
796:
780:
779:
778:
772:
771:
736:
734:
733:
728:
726:
725:
712:
710:
709:
704:
689:
687:
686:
681:
679:
678:
671:
670:
669:
663:
662:
638:
636:
635:
630:
628:
627:
618:
617:
597:
595:
594:
589:
578:
577:
561:
560:
559:
553:
552:
513:
511:
510:
505:
487:
485:
484:
479:
467:
465:
464:
459:
447:
445:
444:
439:
427:
425:
424:
419:
407:
405:
404:
399:
328:
326:
325:
320:
289:
287:
286:
281:
233:
231:
230:
225:
207:
205:
204:
199:
153:
151:
150:
145:
127:
125:
124:
119:
71:Girard's paradox
6760:
6759:
6755:
6754:
6753:
6751:
6750:
6749:
6710:
6709:
6708:
6703:
6651:Category theory
6645:
6597:
6534:
6464:
6458:
6428:
6423:
6392:
6343:
6315:
6292:
6274:
6265:Relevance logic
6260:Structural rule
6246:
6222:Degree of truth
6208:
6170:
6165:
6113:
6103:
6082:
6061:
6038:
6036:Further reading
6020:
6009:
5999:
5994:
5993:
5984:
5980:
5969:
5965:
5954:
5950:
5945:
5941:
5894:
5890:
5875:
5856:10.1.1.163.7149
5843:
5839:
5810:
5806:
5798:
5791:
5785:
5781:
5772:
5768:
5715:
5711:
5704:
5690:
5686:
5681:
5664:
5628:
5622:
5616:
5603: =
5598:
5592:
5582:
5576:
5525:
5473:
5427:integer numbers
5384:
5313:
5310:
5309:
5307:
5271:
5268:
5267:
5265:
5205:
5202:
5201:
5199:
5160:
5157:
5156:
5154:
5106:
5103:
5102:
5100:
5066:
5063:
5062:
5060:
5012:
5009:
5008:
5006:
4970:
4967:
4966:
4964:
4934:
4931:
4930:
4928:
4915:
4900:
4887:
4877:
4835:and a function
4802:
4799:
4798:
4796:
4740:category theory
4736:
4727:is also empty.
4706:
4703:
4702:
4686:
4683:
4682:
4666:
4663:
4662:
4646:
4643:
4642:
4626:
4623:
4622:
4606:
4603:
4602:
4586:
4583:
4582:
4560:
4557:
4556:
4532:
4531:
4529:
4526:
4525:
4509:
4506:
4505:
4487:
4484:
4483:
4465:
4462:
4461:
4444:
4443:
4441:
4438:
4437:
4421:
4418:
4417:
4400:
4399:
4397:
4394:
4393:
4350:
4349:
4338:
4335:
4334:
4297:
4296:
4279:
4276:
4275:
4234:
4231:
4230:
4197:
4196:
4185:
4182:
4181:
4146:
4145:
4131:
4128:
4127:
4103:
4100:
4099:
4083:
4082:
4042:
4009:
4008:
3998:
3974:
3973:
3968:
3957:
3949:
3937:
3936:
3932:
3925:
3923:
3920:
3919:
3900:
3897:
3896:
3880:
3877:
3876:
3860:
3857:
3856:
3837:
3829:
3823:
3822:
3817:
3814:
3813:
3796:
3790:
3789:
3784:
3781:
3780:
3757:
3754:
3753:
3737:
3734:
3733:
3717:
3714:
3713:
3691:
3680:
3677:
3676:
3645:
3642:
3641:
3618:
3615:
3614:
3598:
3595:
3594:
3578:
3575:
3574:
3552:
3541:
3538:
3537:
3505:
3504:
3496:
3493:
3492:
3464:
3461:
3460:
3432:
3429:
3428:
3402:
3401:
3396:
3393:
3392:
3360:
3359:
3351:
3348:
3347:
3299:
3296:
3295:
3279:
3276:
3275:
3253:
3250:
3249:
3233:
3230:
3229:
3201:
3198:
3197:
3181:
3178:
3177:
3161:
3158:
3157:
3135:
3132:
3131:
3103:
3100:
3099:
3069:
3062:
3059:
3058:
3031:
3024:
3021:
3020:
3019:is a type then
3004:
3001:
3000:
2984:
2981:
2980:
2977:
2938:
2932:
2931:
2930:
2928:
2925:
2924:
2908:
2905:
2904:
2883:
2877:
2876:
2875:
2873:
2870:
2869:
2852:
2846:
2845:
2844:
2842:
2839:
2838:
2821:
2815:
2814:
2813:
2811:
2808:
2807:
2790:
2784:
2783:
2782:
2780:
2777:
2776:
2775:, you must use
2759:
2753:
2752:
2751:
2749:
2746:
2745:
2726:
2717:
2716:
2714:
2711:
2710:
2693:
2687:
2686:
2685:
2683:
2680:
2679:
2662:
2656:
2655:
2654:
2652:
2649:
2648:
2599:
2596:
2595:
2578:
2572:
2571:
2570:
2568:
2565:
2564:
2561:
2514:
2513:
2507:
2506:
2502:
2446:
2445:
2439:
2438:
2434:
2429:
2425:
2403:
2402:
2378:
2377:
2376:
2375:
2373:
2370:
2369:
2339:
2336:
2335:
2303:
2300:
2299:
2280:
2277:
2276:
2260:
2257:
2256:
2239:
2238:
2229:
2228:
2222:
2221:
2216:
2213:
2212:
2195:
2194:
2188:
2187:
2182:
2179:
2178:
2155:
2152:
2151:
2136:
2130:
2128:Inductive types
2092:
2089:
2088:
2057:
2054:
2053:
2031:
2028:
2027:
1999:
1996:
1995:
1979:
1976:
1975:
1953:
1950:
1949:
1927:
1924:
1923:
1880:
1879:
1875:
1865:
1864:
1859:
1856:
1855:
1814:
1811:
1810:
1794:
1791:
1790:
1768:
1765:
1764:
1742:
1739:
1738:
1704:
1701:
1700:
1678:
1675:
1674:
1652:
1649:
1648:
1645:
1609:
1608:
1602:
1601:
1597:
1591:
1588:
1587:
1557:
1554:
1553:
1537:
1534:
1533:
1508:
1505:
1504:
1487:
1486:
1484:
1481:
1480:
1464:
1461:
1460:
1435:
1432:
1431:
1414:
1413:
1411:
1408:
1407:
1391:
1388:
1387:
1355:
1354:
1350:
1344:
1341:
1340:
1315:
1312:
1311:
1287:
1284:
1283:
1266:
1265:
1256:
1255:
1253:
1250:
1249:
1233:
1230:
1229:
1200:
1199:
1182:
1181:
1175:
1174:
1170:
1164:
1161:
1160:
1141:
1138:
1137:
1121:
1118:
1117:
1111:
1075:
1074:
1068:
1067:
1063:
1057:
1054:
1053:
1034:
1031:
1030:
1005:
1002:
1001:
976:
973:
972:
955:
954:
952:
949:
948:
932:
929:
928:
903:
900:
899:
882:
881:
879:
876:
875:
859:
856:
855:
828:
799:
798:
792:
791:
787:
782:
774:
773:
767:
766:
762:
756:
753:
752:
721:
720:
718:
715:
714:
698:
695:
694:
674:
673:
665:
664:
658:
657:
653:
647:
644:
643:
623:
622:
613:
612:
610:
607:
606:
573:
572:
555:
554:
548:
547:
543:
537:
534:
533:
493:
490:
489:
473:
470:
469:
468:and a proof of
453:
450:
449:
433:
430:
429:
413:
410:
409:
387:
384:
383:
376:
299:
296:
295:
275:
272:
271:
245:
213:
210:
209:
187:
184:
183:
168:
160:predicate logic
133:
130:
129:
105:
102:
101:
87:
79:dependent types
22:(also known as
17:
12:
11:
5:
6758:
6748:
6747:
6742:
6737:
6732:
6727:
6722:
6705:
6704:
6702:
6701:
6696:
6691:
6686:
6684:â-topos theory
6681:
6676:
6671:
6666:
6661:
6655:
6653:
6647:
6646:
6644:
6643:
6638:
6633:
6628:
6623:
6618:
6613:
6607:
6605:
6599:
6598:
6596:
6595:
6590:
6585:
6580:
6575:
6570:
6565:
6560:
6555:
6550:
6544:
6542:
6536:
6535:
6533:
6532:
6527:
6522:
6517:
6512:
6507:
6506:
6505:
6500:
6498:Hilbert system
6495:
6485:
6480:
6474:
6472:
6466:
6465:
6457:
6456:
6449:
6442:
6434:
6425:
6424:
6422:
6421:
6416:
6411:
6406:
6400:
6398:
6394:
6393:
6391:
6390:
6389:
6388:
6378:
6377:
6376:
6366:
6365:
6364:
6353:
6351:
6345:
6344:
6342:
6341:
6336:
6331:
6325:
6323:
6317:
6316:
6314:
6313:
6308:
6302:
6300:
6294:
6293:
6291:
6290:
6284:
6282:
6280:Paraconsistent
6276:
6275:
6273:
6272:
6267:
6262:
6256:
6254:
6248:
6247:
6245:
6244:
6239:
6234:
6229:
6224:
6218:
6216:
6210:
6209:
6207:
6206:
6201:
6196:
6191:
6186:
6180:
6178:
6176:Intuitionistic
6172:
6171:
6164:
6163:
6156:
6149:
6141:
6135:
6134:
6125:â letter from
6120:
6112:
6111:External links
6109:
6108:
6107:
6101:
6086:
6080:
6065:
6059:
6044:
6037:
6034:
6033:
6032:
6019:978-8870881059
6018:
5998:
5995:
5992:
5991:
5978:
5963:
5948:
5939:
5910:(5): 552â593.
5888:
5873:
5837:
5824:(4): 428â469.
5804:
5801:on 2024-04-19.
5779:
5766:
5709:
5702:
5683:
5682:
5680:
5677:
5676:
5675:
5670:
5663:
5660:
5646:are cumulative
5624:
5618:
5612:
5594:
5588:
5585:non-cumulative
5578:
5574:
5528:Per Martin-Löf
5524:
5521:
5472:
5469:
5383:
5380:
5346:with mappings
5329:
5326:
5323:
5320:
5317:
5293:
5290:
5287:
5284:
5281:
5278:
5275:
5239:
5236:
5233:
5230:
5227:
5224:
5221:
5218:
5215:
5212:
5209:
5185:
5182:
5179:
5176:
5173:
5170:
5167:
5164:
5128:
5125:
5122:
5119:
5116:
5113:
5110:
5082:
5079:
5076:
5073:
5070:
5034:
5031:
5028:
5025:
5022:
5019:
5016:
4992:
4989:
4986:
4983:
4980:
4977:
4974:
4950:
4947:
4944:
4941:
4938:
4906:
4898:
4885:
4875:
4818:
4815:
4812:
4809:
4806:
4744:R. A. G. Seely
4735:
4732:
4716:
4713:
4710:
4690:
4670:
4650:
4630:
4610:
4590:
4570:
4567:
4564:
4535:
4513:
4491:
4482:In most texts
4469:
4447:
4425:
4403:
4388:
4387:
4384:
4371:
4368:
4365:
4362:
4359:
4356:
4353:
4345:
4342:
4331:
4330:
4329:in context Î.
4316:
4305:
4300:
4295:
4292:
4289:
4286:
4283:
4272:
4271:
4261:
4250:
4247:
4244:
4241:
4238:
4227:
4226:
4225:in context Î.
4216:
4205:
4200:
4195:
4192:
4189:
4178:
4177:
4171:
4158:
4155:
4152:
4149:
4141:
4138:
4135:
4107:
4097:
4096:
4081:
4078:
4075:
4072:
4069:
4066:
4063:
4060:
4057:
4054:
4051:
4048:
4045:
4043:
4041:
4038:
4035:
4032:
4029:
4026:
4023:
4020:
4017:
4014:
4011:
4010:
4007:
4004:
4001:
3999:
3997:
3994:
3991:
3988:
3985:
3982:
3979:
3976:
3975:
3971:
3967:
3964:
3960:
3956:
3952:
3948:
3940:
3935:
3933:
3931:
3928:
3927:
3904:
3884:
3864:
3853:
3852:
3840:
3836:
3832:
3826:
3821:
3811:
3799:
3793:
3788:
3774:
3773:
3761:
3741:
3721:
3701:
3698:
3694:
3690:
3687:
3684:
3670:
3669:
3658:
3655:
3652:
3649:
3635:
3634:
3622:
3602:
3582:
3562:
3559:
3555:
3551:
3548:
3545:
3531:
3530:
3519:
3516:
3513:
3508:
3503:
3500:
3486:
3485:
3474:
3471:
3468:
3454:
3453:
3442:
3439:
3436:
3422:
3421:
3410:
3405:
3400:
3386:
3385:
3372:
3369:
3366:
3363:
3355:
3315:
3312:
3309:
3306:
3303:
3283:
3263:
3260:
3257:
3237:
3217:
3214:
3211:
3208:
3205:
3185:
3165:
3145:
3142:
3139:
3119:
3116:
3113:
3110:
3107:
3083:
3078:
3075:
3072:
3068:
3045:
3040:
3037:
3034:
3030:
3008:
2999:is a type and
2988:
2976:
2973:
2947:
2944:
2941:
2935:
2912:
2899:. There is a
2886:
2880:
2855:
2849:
2824:
2818:
2793:
2787:
2762:
2756:
2729:
2725:
2720:
2696:
2690:
2665:
2659:
2636:
2633:
2630:
2627:
2624:
2621:
2618:
2615:
2612:
2609:
2606:
2603:
2581:
2575:
2560:
2559:Universe types
2557:
2545:
2544:
2533:
2530:
2527:
2524:
2517:
2510:
2505:
2501:
2497:
2493:
2489:
2486:
2483:
2480:
2477:
2474:
2471:
2468:
2465:
2462:
2459:
2456:
2449:
2442:
2437:
2433:
2428:
2424:
2420:
2417:
2414:
2411:
2406:
2398:
2395:
2392:
2389:
2381:
2354:
2350:
2346:
2343:
2316:
2313:
2310:
2307:
2287:
2284:
2264:
2242:
2237:
2232:
2225:
2220:
2198:
2191:
2186:
2159:
2134:Inductive type
2132:Main article:
2129:
2126:
2102:
2099:
2096:
2076:
2073:
2070:
2067:
2064:
2061:
2041:
2038:
2035:
2015:
2012:
2009:
2006:
2003:
1983:
1963:
1960:
1957:
1937:
1934:
1931:
1920:
1919:
1908:
1905:
1902:
1899:
1896:
1893:
1888:
1883:
1878:
1874:
1868:
1863:
1836:
1833:
1830:
1827:
1824:
1821:
1818:
1798:
1778:
1775:
1772:
1752:
1749:
1746:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1688:
1685:
1682:
1662:
1659:
1656:
1644:
1641:
1640:
1639:
1628:
1625:
1622:
1619:
1612:
1605:
1600:
1596:
1572:
1568:
1564:
1561:
1541:
1521:
1518:
1515:
1512:
1490:
1468:
1448:
1445:
1442:
1439:
1417:
1395:
1380:
1379:
1368:
1363:
1358:
1353:
1349:
1325:
1322:
1319:
1299:
1295:
1291:
1269:
1264:
1259:
1237:
1226:
1225:
1214:
1211:
1208:
1203:
1198:
1195:
1192:
1185:
1178:
1173:
1169:
1145:
1125:
1110:
1107:
1106:
1105:
1094:
1091:
1088:
1085:
1078:
1071:
1066:
1062:
1038:
1018:
1015:
1012:
1009:
989:
986:
983:
980:
958:
936:
916:
913:
910:
907:
885:
863:
848:
847:
835:
827:
824:
821:
818:
815:
812:
809:
802:
795:
790:
786:
777:
770:
765:
761:
724:
702:
691:
690:
677:
668:
661:
656:
652:
626:
621:
616:
603:disjoint union
599:
598:
587:
584:
581:
576:
571:
568:
565:
558:
551:
546:
542:
519:natural number
503:
500:
497:
477:
457:
437:
417:
397:
394:
391:
375:
372:
355:propositions.
349:Boolean values
332:Likewise, the
318:
315:
312:
309:
306:
303:
279:
244:
241:
223:
220:
217:
197:
194:
191:
167:
164:
143:
140:
137:
117:
113:
109:
86:
83:
73:, gave way to
44:Per Martin-Löf
15:
9:
6:
4:
3:
2:
6757:
6746:
6743:
6741:
6738:
6736:
6733:
6731:
6728:
6726:
6723:
6721:
6718:
6717:
6715:
6700:
6697:
6695:
6692:
6690:
6687:
6685:
6682:
6680:
6677:
6675:
6672:
6670:
6667:
6665:
6662:
6660:
6657:
6656:
6654:
6652:
6648:
6642:
6639:
6637:
6634:
6632:
6629:
6627:
6624:
6622:
6619:
6617:
6614:
6612:
6609:
6608:
6606:
6604:
6600:
6594:
6591:
6589:
6586:
6584:
6581:
6579:
6576:
6574:
6571:
6569:
6566:
6564:
6561:
6559:
6556:
6554:
6551:
6549:
6546:
6545:
6543:
6541:
6537:
6531:
6528:
6526:
6523:
6521:
6518:
6516:
6513:
6511:
6508:
6504:
6501:
6499:
6496:
6494:
6491:
6490:
6489:
6488:Formal system
6486:
6484:
6481:
6479:
6476:
6475:
6473:
6471:
6467:
6463:
6455:
6450:
6448:
6443:
6441:
6436:
6435:
6432:
6420:
6417:
6415:
6412:
6410:
6407:
6405:
6402:
6401:
6399:
6395:
6387:
6384:
6383:
6382:
6379:
6375:
6372:
6371:
6370:
6367:
6363:
6360:
6359:
6358:
6355:
6354:
6352:
6350:
6349:Digital logic
6346:
6340:
6337:
6335:
6332:
6330:
6327:
6326:
6324:
6322:
6318:
6312:
6309:
6307:
6304:
6303:
6301:
6299:
6295:
6289:
6286:
6285:
6283:
6281:
6277:
6271:
6268:
6266:
6263:
6261:
6258:
6257:
6255:
6253:
6252:Substructural
6249:
6243:
6240:
6238:
6235:
6233:
6230:
6228:
6225:
6223:
6220:
6219:
6217:
6215:
6211:
6205:
6202:
6200:
6197:
6195:
6192:
6190:
6187:
6185:
6182:
6181:
6179:
6177:
6173:
6169:
6162:
6157:
6155:
6150:
6148:
6143:
6142:
6139:
6132:
6128:
6124:
6121:
6118:
6115:
6114:
6104:
6098:
6094:
6093:
6087:
6083:
6081:0-201-41667-0
6077:
6073:
6072:
6066:
6062:
6060:9780198538141
6056:
6052:
6051:
6045:
6043:
6040:
6039:
6029:
6025:
6021:
6015:
6008:
6007:
6001:
6000:
5988:
5982:
5974:
5967:
5959:
5952:
5943:
5935:
5931:
5927:
5923:
5918:
5913:
5909:
5905:
5904:
5899:
5892:
5884:
5880:
5876:
5874:9781605584201
5870:
5866:
5862:
5857:
5852:
5848:
5841:
5832:
5827:
5823:
5819:
5815:
5808:
5797:
5790:
5783:
5776:
5770:
5762:
5758:
5754:
5750:
5746:
5742:
5737:
5732:
5728:
5724:
5720:
5713:
5705:
5699:
5695:
5688:
5684:
5674:
5671:
5669:
5666:
5665:
5659:
5657:
5653:
5649:
5647:
5642:
5638:
5636:
5632:
5627:
5621:
5615:
5610:
5606:
5602:
5597:
5591:
5586:
5581:
5571:
5567:
5565:
5564:
5559:
5555:
5551:
5547:
5545:
5540:
5536:
5533:
5529:
5520:
5518:
5514:
5510:
5506:
5502:
5498:
5494:
5490:
5486:
5482:
5478:
5468:
5466:
5462:
5458:
5454:
5450:
5446:
5442:
5438:
5436:
5432:
5428:
5424:
5420:
5416:
5415:type checking
5411:
5409:
5405:
5401:
5397:
5393:
5389:
5379:
5375:
5371:
5369:
5365:
5361:
5357:
5353:
5349:
5345:
5324:
5321:
5318:
5288:
5282:
5279:
5276:
5273:
5263:
5258:
5255:The category
5253:
5234:
5231:
5228:
5225:
5219:
5216:
5213:
5210:
5207:
5180:
5174:
5171:
5168:
5165:
5162:
5152:
5148:
5144:
5123:
5120:
5117:
5111:
5108:
5099:is a term in
5098:
5077:
5071:
5068:
5059:is a type in
5058:
5054:
5050:
5029:
5026:
5023:
5017:
5014:
4987:
4981:
4978:
4975:
4972:
4945:
4939:
4936:
4926:
4922:
4917:
4913:
4909:
4905:
4901:
4894:
4890:
4884:
4880:
4874:
4870:
4866:
4862:
4858:
4854:
4850:
4846:
4842:
4838:
4834:
4813:
4810:
4807:
4794:
4790:
4786:
4782:
4780:
4776:
4772:
4768:
4764:
4759:
4757:
4753:
4749:
4745:
4741:
4731:
4728:
4714:
4711:
4708:
4688:
4668:
4648:
4628:
4608:
4601:and the type
4588:
4568:
4565:
4562:
4552:
4549:
4511:
4503:
4489:
4467:
4423:
4385:
4340:
4333:
4332:
4328:
4324:
4320:
4317:
4303:
4298:
4293:
4290:
4287:
4284:
4274:
4273:
4269:
4265:
4262:
4248:
4245:
4242:
4239:
4229:
4228:
4224:
4220:
4217:
4203:
4198:
4193:
4190:
4180:
4179:
4175:
4172:
4139:
4136:
4126:
4125:
4122:
4119:
4105:
4070:
4067:
4064:
4058:
4055:
4049:
4046:
4044:
4036:
4033:
4027:
4021:
4015:
4012:
4005:
4002:
4000:
3992:
3989:
3986:
3980:
3977:
3954:
3938:
3934:
3929:
3918:
3917:
3916:
3902:
3882:
3862:
3824:
3819:
3812:
3791:
3786:
3779:
3778:
3777:
3759:
3739:
3719:
3696:
3692:
3688:
3682:
3675:
3674:
3673:
3656:
3650:
3640:
3639:
3638:
3620:
3600:
3580:
3557:
3553:
3549:
3543:
3536:
3535:
3534:
3517:
3511:
3506:
3501:
3491:
3490:
3489:
3472:
3469:
3466:
3459:
3458:
3457:
3440:
3437:
3434:
3427:
3426:
3425:
3408:
3403:
3398:
3391:
3390:
3389:
3353:
3346:
3345:
3344:
3341:
3339:
3335:
3330:
3327:
3313:
3310:
3307:
3304:
3301:
3281:
3261:
3258:
3255:
3235:
3215:
3212:
3209:
3206:
3203:
3183:
3163:
3143:
3140:
3137:
3117:
3114:
3111:
3108:
3105:
3096:
3081:
3076:
3073:
3070:
3066:
3043:
3038:
3035:
3032:
3028:
3006:
2986:
2972:
2970:
2966:
2961:
2945:
2942:
2939:
2910:
2902:
2884:
2853:
2822:
2791:
2760:
2742:
2723:
2694:
2678:that maps to
2663:
2634:
2631:
2628:
2622:
2616:
2613:
2610:
2607:
2604:
2601:
2579:
2556:
2554:
2550:
2528:
2522:
2508:
2503:
2499:
2491:
2481:
2475:
2469:
2460:
2454:
2440:
2435:
2431:
2426:
2415:
2409:
2404:
2368:
2367:
2366:
2348:
2341:
2333:
2328:
2314:
2311:
2308:
2305:
2285:
2282:
2262:
2223:
2218:
2189:
2184:
2175:
2173:
2157:
2149:
2145:
2141:
2135:
2125:
2123:
2119:
2114:
2100:
2097:
2094:
2087:or, finally,
2071:
2068:
2065:
2033:
2007:
2004:
2001:
1961:
1958:
1955:
1935:
1932:
1929:
1906:
1900:
1897:
1894:
1886:
1881:
1876:
1872:
1866:
1861:
1854:
1853:
1852:
1850:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1796:
1776:
1773:
1770:
1750:
1747:
1744:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1686:
1683:
1680:
1660:
1657:
1654:
1623:
1617:
1603:
1598:
1594:
1586:
1585:
1584:
1566:
1559:
1539:
1516:
1510:
1503:to proofs of
1466:
1443:
1437:
1393:
1385:
1366:
1361:
1356:
1351:
1347:
1339:
1338:
1337:
1323:
1317:
1297:
1289:
1209:
1206:
1193:
1190:
1176:
1171:
1167:
1159:
1158:
1157:
1143:
1123:
1114:
1089:
1083:
1069:
1064:
1060:
1052:
1051:
1050:
1036:
1013:
1007:
984:
978:
934:
911:
905:
861:
853:
825:
819:
816:
813:
793:
788:
784:
768:
763:
759:
751:
750:
749:
747:
743:
738:
700:
659:
654:
650:
642:
641:
640:
619:
604:
582:
579:
566:
563:
549:
544:
540:
532:
531:
530:
528:
524:
520:
515:
501:
498:
495:
475:
455:
435:
415:
395:
392:
389:
381:
371:
369:
365:
361:
356:
354:
350:
346:
343:Finally, the
341:
339:
335:
330:
310:
307:
304:
293:
269:
265:
260:
258:
254:
250:
240:
239:
237:
221:
218:
215:
195:
192:
189:
179:
177:
173:
163:
161:
157:
141:
135:
115:
107:
98:
96:
92:
82:
80:
76:
72:
68:
67:impredicative
64:
60:
56:
52:
51:mathematician
49:
45:
41:
37:
33:
29:
25:
21:
6745:Intuitionism
6664:Topos theory
6625:
6515:Model theory
6478:Peano axioms
6329:Three-valued
6270:Linear logic
6198:
6095:. Springer.
6091:
6070:
6049:
6005:
5986:
5981:
5972:
5966:
5957:
5951:
5942:
5907:
5901:
5891:
5846:
5840:
5821:
5817:
5807:
5796:the original
5782:
5774:
5769:
5726:
5722:
5712:
5693:
5687:
5655:
5651:
5650:
5645:
5640:
5639:
5634:
5630:
5625:
5619:
5613:
5604:
5600:
5595:
5589:
5584:
5579:
5569:
5568:
5561:
5549:
5548:
5538:
5537:
5526:
5474:
5465:ad infinitum
5464:
5439:
5435:real numbers
5412:
5403:
5400:Y-combinator
5385:
5376:
5372:
5367:
5363:
5359:
5355:
5351:
5347:
5343:
5261:
5256:
5254:
5150:
5146:
5142:
5096:
5056:
5052:
5048:
4924:
4920:
4919:The functor
4918:
4911:
4907:
4903:
4896:
4892:
4888:
4882:
4878:
4872:
4871:, such that
4868:
4864:
4860:
4856:
4852:
4848:
4844:
4840:
4836:
4832:
4788:
4784:
4783:
4778:
4774:
4770:
4766:
4762:
4760:
4755:
4751:
4737:
4729:
4553:
4550:
4481:
4391:
4326:
4322:
4318:
4267:
4263:
4222:
4218:
4173:
4120:
4098:
3854:
3775:
3671:
3636:
3532:
3487:
3455:
3423:
3387:
3342:
3337:
3333:
3331:
3328:
3097:
2978:
2962:
2743:
2562:
2549:well-founded
2546:
2329:
2176:
2137:
2118:proof theory
2115:
1921:
1646:
1381:
1227:
1115:
1112:
898:, such that
849:
739:
692:
639:is written:
600:
516:
377:
363:
359:
357:
352:
344:
342:
333:
331:
263:
262:Because the
261:
256:
252:
248:
246:
181:
180:
172:set theories
169:
99:
88:
31:
27:
23:
19:
18:
6735:Type theory
6603:Type theory
6583:Determinacy
6525:Modal logic
6369:Four-valued
6339:Ćukasiewicz
6334:Four-valued
6321:Many-valued
6298:Description
6288:Dialetheism
6131:Ross Street
5656:Bibliopolis
5652:Bibliopolis
5554:predicative
5396:undecidable
5392:intensional
5388:extensional
2901:predicative
2140:linked list
1849:reflexivity
236:type theory
166:Type theory
75:predicative
63:extensional
59:intensional
55:philosopher
36:type theory
6714:Categories
6679:â-groupoid
6540:Set theory
6227:Fuzzy rule
5997:References
5461:homotopies
4436:). Since
2975:Judgements
268:empty type
6381:IEEE 1164
6232:Fuzzy set
6127:John Baez
5926:0956-7968
5851:CiteSeerX
5753:0960-1295
5736:1112.3456
5419:decidable
4791:) is the
4712:×
4566:×
4344:Γ
4341:⊢
4304:σ
4291:≡
4285:⊢
4282:Γ
4249:τ
4246:≡
4243:σ
4240:⊢
4237:Γ
4204:σ
4191:⊢
4188:Γ
4140:σ
4137:⊢
4134:Γ
4059:
4016:
3981:
3966:→
3955:×
3835:→
3067:∑
3029:∑
2724:∈
2626:Π
2620:Σ
2500:∏
2496:→
2467:→
2432:∏
2423:→
2349:⋅
2332:induction
2255:. Since
2236:→
2172:successor
2098:≠
2060:¬
2040:⊥
2037:→
2034:…
2014:⊥
2011:→
1982:⊥
1873:∏
1832:⋅
1774:⋅
1722:⋅
1684:⋅
1595:∏
1567:⋅
1348:∏
1321:→
1294:⟹
1263:→
1236:→
1194:
1168:∏
1061:∑
785:∑
760:∑
651:∑
620:×
567:
541:∑
499:∧
393:×
338:unit type
317:⊥
314:→
302:¬
278:⊥
219:⋅
139:→
112:⟹
6659:Category
6028:12731401
5934:19895964
5662:See also
5577:, ..., V
5499:such as
5362: :
5350: :
5153:. Here
5055:, where
5005:, a set
4863: :
4851: :
4769: :
2709:for any
2026:. Since
1479:of type
1406:of type
1248:. Thus,
947:of type
874:of type
523:sequence
292:negation
6374:Verilog
5883:1777213
5593:and BâV
5558:Russell
5509:Epigram
5505:Cayenne
5423:setoids
5340:
5308:
5304:
5266:
5250:
5200:
5196:
5155:
5139:
5101:
5093:
5061:
5045:
5007:
5003:
4965:
4961:
4929:
4829:
4797:
3338:objects
2170:or the
176:Frege's
48:Swedish
34:) is a
6397:Others
6099:
6078:
6057:
6026:
6016:
5932:
5924:
5881:
5871:
5853:
5761:416274
5759:
5751:
5700:
5641:MLTT79
5570:MLTT73
5550:MLTT72
5539:MLTT71
5515:, and
5453:points
5449:values
5433:, and
5141:, and
4927:a set
4661:. If
4347:
4143:
3944:
3855:Here,
3357:
2148:graphs
742:tuples
85:Design
6214:Fuzzy
6010:(PDF)
5930:S2CID
5879:S2CID
5799:(PDF)
5792:(PDF)
5757:S2CID
5731:arXiv
5729:(6).
5679:Notes
5623:and V
5517:Idris
5481:Nuprl
5457:paths
5408:Nuprl
4895:maps
3334:types
2969:Mahlo
2144:trees
527:reals
26:, or
6386:VHDL
6097:ISBN
6076:ISBN
6055:ISBN
6024:OCLC
6014:ISBN
5922:ISSN
5869:ISBN
5749:ISSN
5698:ISBN
5629:for
5513:Agda
5487:and
5368:D,Ap
5198:and
5095:and
4859:and
4641:and
4416:(or
4321:and
4266:and
3336:and
3248:and
2298:and
1862:refl
1763:and
1673:and
830:True
817:<
428:and
351:but
208:and
61:and
53:and
46:, a
32:MLTT
6548:Set
5912:doi
5861:doi
5826:doi
5741:doi
5560:'s
5501:ATS
5489:Coq
5463:),
5451:or
5417:is
5390:vs
5370:).
5149:to
5051:or
4902:to
4873:B'
4869:X'
4857:A'
4789:Set
4785:Fam
4781:).
4779:Set
4775:Fam
4754:or
4681:or
4424:Set
4056:add
4013:add
3978:add
3930:add
3752:in
3613:in
1191:Vec
564:Vec
525:of
353:not
6716::
6022:.
5928:.
5920:.
5908:23
5906:.
5900:.
5877:.
5867:.
5859:.
5820:.
5816:.
5755:.
5747:.
5739:.
5727:24
5725:.
5721:.
5648:.
5637:.
5633:â
5519:.
5511:,
5507:,
5503:,
5467:.
5429:,
5364:Tm
5358:,
5354:â
5252:.
5053:af
5049:Af
4916:.
4881:=
4867:â
4855:â
4843:â
4839::
4773:â
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4490:El
4468:El
3915::
3340:.
2960:.
2741:.
2146:,
2113:.
1851::
1430:,
737:.
514:.
382:,
340:.
329:.
308::=
81:.
6453:e
6446:t
6439:v
6160:e
6153:t
6146:v
6105:.
6084:.
6063:.
6030:.
5936:.
5914::
5885:.
5863::
5834:.
5828::
5822:4
5763:.
5743::
5733::
5706:.
5635:j
5631:i
5626:j
5620:i
5614:i
5605:n
5601:m
5596:n
5590:m
5580:n
5575:0
5366:(
5360:q
5356:G
5352:D
5348:p
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5325:A
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5319:G
5316:(
5292:)
5289:G
5286:(
5283:y
5280:T
5277::
5274:A
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5235:f
5232:A
5229:,
5226:D
5223:(
5220:m
5217:T
5214::
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5121:,
5118:G
5115:(
5112:m
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5078:G
5075:(
5072:y
5069:T
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5033:)
5030:A
5027:,
5024:G
5021:(
5018:m
5015:T
4991:)
4988:G
4985:(
4982:y
4979:T
4976::
4973:A
4949:)
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4943:(
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4912:a
4910:(
4908:g
4904:B
4899:a
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4893:f
4889:B
4886:°
4883:f
4879:g
4876:°
4865:X
4861:g
4853:A
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4833:A
4817:)
4814:B
4811:,
4808:A
4805:(
4787:(
4777:(
4771:C
4767:T
4763:C
4715:B
4709:A
4689:B
4669:A
4649:B
4629:A
4609:B
4589:A
4569:B
4563:A
4534:U
4512:A
4446:U
4402:U
4370:t
4367:x
4364:e
4361:t
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4319:t
4299::
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4077:)
4074:)
4071:b
4068:,
4065:a
4062:(
4053:(
4050:S
4047:=
4040:)
4037:b
4034:,
4031:)
4028:a
4025:(
4022:S
4019:(
4006:b
4003:=
3996:)
3993:b
3990:,
3987:0
3984:(
3970:N
3963:)
3959:N
3951:N
3947:(
3939::
3903:S
3883:S
3863:S
3839:N
3831:N
3825::
3820:S
3798:N
3792::
3787:0
3772:.
3760:b
3740:a
3720:x
3700:]
3697:a
3693:/
3689:x
3686:[
3683:b
3657:b
3654:]
3651:x
3648:[
3633:.
3621:B
3601:a
3581:x
3561:]
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3554:/
3550:x
3547:[
3544:B
3518:B
3515:)
3512:A
3507::
3502:x
3499:(
3473:B
3470:=
3467:A
3441:b
3438:=
3435:a
3409:A
3404::
3399:a
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3302:S
3282:4
3262:2
3259:+
3256:2
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3216:2
3213:+
3210:2
3207:=
3204:4
3184:B
3164:A
3144:B
3141:=
3138:A
3118:2
3115:+
3112:2
3109:=
3106:4
3082:B
3077:A
3074::
3071:a
3044:B
3039:A
3036::
3033:a
3007:B
2987:A
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2879:U
2854:1
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2823:0
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2761:0
2755:U
2728:N
2719:n
2695:n
2689:U
2664:n
2658:U
2635:,
2632:=
2629:,
2623:,
2617:,
2614:2
2611:,
2608:1
2605:,
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2532:)
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2523:P
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2509::
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2413:(
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2315:0
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2309:S
2306:S
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2263:S
2241:N
2231:N
2224::
2219:S
2197:N
2190::
2185:0
2158:0
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2095:1
2075:)
2072:2
2069:=
2066:1
2063:(
2008:2
2005:=
2002:1
1962:2
1959:=
1956:1
1936:2
1933:=
1930:1
1907:.
1904:)
1901:a
1898:=
1895:a
1892:(
1887:A
1882::
1877:a
1867::
1835:2
1829:2
1826:=
1823:2
1820:+
1817:2
1797:4
1777:2
1771:2
1751:2
1748:+
1745:2
1725:2
1719:2
1716:=
1713:2
1710:+
1707:2
1687:2
1681:2
1661:2
1658:+
1655:2
1627:)
1624:n
1621:(
1618:P
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1604::
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1357::
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1177::
1172:n
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1011:(
1008:P
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979:P
957:N
935:n
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912:n
909:(
906:P
884:N
862:n
834:)
826:=
823:)
820:n
814:m
811:(
808:(
801:Z
794::
789:n
776:Z
769::
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723:R
701:n
676:R
667:N
660::
655:n
625:R
615:N
586:)
583:n
580:,
575:R
570:(
557:N
550::
545:n
502:B
496:A
476:B
456:A
436:B
416:A
396:B
390:A
364:0
360:1
345:2
334:1
311:A
305:A
264:0
257:2
253:1
249:0
238:.
222:2
216:2
196:2
193:+
190:2
142:B
136:A
116:B
108:A
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