1124: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.
4105:
2553:
3932:
4565:
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
5583:
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
5545:
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.
5654:
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
5655:
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
104:. 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
5384:
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.
2382:
856:
5388:
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.
5729:
5908:
4100:{\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}}}
1234:
607:
528: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
1928:
5552:
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".
5270:
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
699:
2264:
5546:
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.
4325:
3861:
1648:
1114:
4180:
3285:
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 "
5405:
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
4225:
5541:
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
4393:
4132:
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:
3937:
648:
4270:
2548:{\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)}
1291:
2750:
759:
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:
5957:
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
2220:
5567:" 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
5584:
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
3820:
765:
1388:
193:
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
3104:
3066:
2656:
4445:
2969:
111:
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 (
4511:
4489:
3539:
3394:
2562:
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.
2908:
2877:
2846:
2815:
2784:
2718:
2687:
2603:
338:
2153:
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
616:
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
5448:. 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.
3430:
2061:
4557:
4469:
4425:
1512:
1439:
980:
907:
746:
2374:
1592:
2035:
1856:
1746:
5259:
1319:
137:
2096:
4736:
4590:
523:
417:
5205:
1798:
1708:
243:
5313:
5148:
5054:
5012:
2122:
5669:
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.
1345:
163:
5349:
5102:
4970:
4838:
3335:
3237:
3139:
2003:
299:
2336:
1541:
1468:
1257:
1038:
1009:
936:
3494:
3462:
3283:
3165:
1983:
1959:
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
1957:
1772:
1682:
1173:
217:
3906:
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
2307:
546:
6169:
4710:
4690:
4670:
4650:
4630:
4610:
4533:
4127:
3924:
3904:
3884:
3781:
3761:
3741:
3721:
3678:
3642:
3622:
3602:
3582:
3303:
3257:
3205:
3185:
3028:
3008:
2932:
2284:
2179:
1818:
1561:
1488:
1415:
1165:
1145:
1058:
956:
883:
722:
497:
477:
457:
437:
6462:
3109:
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
1868:
5577:, 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".
6699:
5490:. While many are based on Per Martin-Löf's ideas, many have added features, more axioms, or a different philosophical background. For instance, the
5436:
or similar constructions. There are many common mathematical objects that are hard to work with or cannot be represented without this, for example,
5563:
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 "
6735:
6740:
6111:
5712:
2574:
The universe types allow proofs to be written about all the types created with the other type constructors. Every term in the universe type
5409:
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
656:
6730:
5058:
of terms. The axioms for a functor require that these play harmoniously with substitution. Substitution is usually written in the form
1293:
is the type of functions from natural numbers to real numbers. Such Î -types correspond to logical implication. The logical proposition
6750:
6028:
2225:
6540:
6162:
6090:
6069:
5913:
5883:
5799:
4758:
5432:, but the representation of standard mathematical concepts is somewhat more cumbersome, since intensional reasoning requires using
4288:
3787:
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:
6455:
3826:
6578:
1600:
5986:. Logic, methodology and philosophy of science, VI (Hannover, 1979). Vol. 104. Amsterdam: North-Holland. pp. 153â175.
1066:
3648:
An object that depends on an object from another type can be done two ways. If the object is "abstracted", then it is written
6709:
5519:
5515:
4140:
3141:. It is a statement that two terms reduce to the same canonical term. There is also a judgement of type equality, say that
181:
Intuitionistic type theory has three finite types, which are then composed using five different type constructors. Unlike
4194:
2974:
Universe types are a tricky feature of type theories. Martin-Löf's original type theory had to be changed to account for
108:. An interesting consequence is that proofs become mathematical objects that can be examined, compared, and manipulated.
4347:
619:
166:
6155:
5527:
2158:
4243:
6704:
6530:
6448:
6316:
4776:
of contexts (in which the objects are contexts, and the context morphisms are substitutions), together with a functor
1347:, containing functions that take proofs-of-A and return proofs-of-B. This type could be written more consistently as:
1262:
369:
Propositions are instead represented by particular types. For instance, a true proposition can be represented by the
101:
2990:
The formal definition of intuitionistic type theory is written using judgements. For example, in the statement "if
2723:
851:{\displaystyle \sum _{m{\mathbin {:}}{\mathbb {Z} }}{\sum _{n{\mathbin {:}}{\mathbb {Z} }}((m<n)={\text{True}})}}
6472:
5511:
2191:
1748:. The terms of that new type represent proofs that the pair reduce to the same canonical term. Thus, since both
50:
6603:
5503:
3793:
756:
1353:
5495:
5459:
5417:. However, this does not prevent extensional type theory from being a basis for a practical tool; for example,
1394:
3071:
3033:
169:. Previous type theories had also followed this isomorphism, but Martin-Löf's was the first to extend it to
68:, who first published it in 1972. There are multiple versions of the type theory: Martin-Löf proposed both
6583:
6214:
2608:
6684:
6669:
6626:
6588:
6493:
5861:
5402:
5398:
4803:
4430:
2937:
1239:
When the output type does not depend on the input value, the function type is often simply written with a
862:
389:ÎŁ-types contain ordered pairs. As with typical ordered pair (or 2-tuple) types, a ÎŁ-type can describe the
378:
4496:
4474:
3505:
3360:
6755:
6646:
6621:
6419:
6199:
5683:
5573:
5523:
5507:
5455:
2882:
2851:
2820:
2789:
2758:
2692:
2661:
2577:
2563:
308:
6015:
3405:
2040:
358:
type contains 2 canonical terms. It represents a definite choice between two values. It is used for
6745:
6641:
6598:
6349:
6290:
6252:
6247:
6194:
5705:
Interactive theorem proving and program development: Coq'Art: the calculus of inductive constructions
5678:
5619:
5451:
2342:
2132:
5866:
4754:
4538:
4450:
4406:
3886:
is a constant object-depending-on-object. It is not associated with an abstraction. Constants like
1493:
1420:
961:
888:
727:
6651:
6429:
6262:
6178:
6127:
6052:
5410:
5406:
3340:
The description of judgements below is based on the discussion in
Nordström, Petersson, and Smith.
2975:
2348:
2154:
1566:
81:
5628:, but it is unclear how to declare them to be equal since there are no identity types connecting V
5486:
Different forms of type theory have been implemented as the formal systems underlying a number of
2008:
1823:
1713:
6573:
6520:
6480:
6424:
6339:
6204:
5940:
5889:
5823:
Allen, S.F.; Bickford, M.; Constable, R.L.; Eaton, R.; Kreitz, C.; Lorigo, L.; Moran, E. (2006).
5767:
5741:
5214:
2182:
1859:
1296:
114:
105:
5971:. Logic Colloquium '73 (Bristol, 1973). Vol. 80. Amsterdam: North-Holland. pp. 73â118.
2066:
4715:
4569:
502:
396:
6558:
6513:
6414:
6379:
6367:
6344:
6331:
6321:
6308:
6107:
6086:
6065:
6034:
6024:
5932:
5879:
5759:
5708:
5538:
5169:
3068:
is a type" there are judgements of "is a type", "and", and "if ... then ...". The expression
1777:
1687:
390:
222:
186:
69:
54:
6440:
6101:
5280:
5115:
5021:
4979:
2161:, etc.. In fact, the natural numbers type may be defined as an inductive type, either being
2101:
189:. So, each feature of the type theory does double duty as a feature of both math and logic.
6679:
6563:
6503:
6372:
5922:
5909:"Idris, a general-purpose dependently typed programming language: Design and implementation"
5871:
5858:
Proceedings of the 4th international workshop on Types in language design and implementation
5836:
5751:
5568:
5554:
5542:
5463:
5429:
1858:. In intuitionistic type theory, there is a single way to introduce =-types and that is by
1324:
1229:{\displaystyle \prod _{n{\mathbin {:}}{\mathbb {N} }}\operatorname {Vec} ({\mathbb {R} },n)}
142:
5322:
5075:
4943:
4811:
3308:
3210:
3112:
2558:
Inductive types in intuitionistic type theory are defined in terms of W-types, the type of
1988:
602:{\displaystyle \sum _{n{\mathbin {:}}{\mathbb {N} }}\operatorname {Vec} ({\mathbb {R} },n)}
284:
6661:
6359:
6275:
6270:
6232:
5487:
5467:
5441:
4750:
2911:
2312:
1517:
1444:
1242:
1014:
985:
912:
359:
301:
and represents anything unprovable. (That is, a proof of it cannot exist.) As a result,
170:
89:
6689:
5806:
4761:(LCCC) as the basic model of type theory. This has been refined by Hofmann and Dybjer to
3473:
3441:
3262:
3144:
1962:
1936:
1751:
1661:
196:
4741:
This can be done for other types (booleans, natural numbers, etc.) and their operators.
4632:. The objects in that dependent type are defined to exist for every pair of objects in
2289:
6631:
6508:
5730:"The biequivalence of locally cartesian closed categories and Martin-Löf type theories"
5499:
4695:
4675:
4655:
4635:
4615:
4595:
4518:
4112:
3909:
3889:
3869:
3766:
3746:
3726:
3689:
3654:
3627:
3607:
3587:
3550:
3288:
3242:
3190:
3170:
3013:
2993:
2917:
2269:
2164:
2149:
Inductive types allow the creation of complex, self-referential types. For example, a
2144:
1803:
1546:
1473:
1400:
1150:
1130:
1043:
941:
868:
707:
613:
529:
482:
462:
442:
422:
73:
17:
4129:
is manipulated as an opaque constant - it has no internal structure for substitution.
2658:
and the inductive type constructor. However, to avoid paradoxes, there is no term in
6724:
6498:
5564:
5425:
85:
77:
61:
6059:
5944:
2345:. Each new inductive type comes with its own inductive rule. To prove a predicate
6525:
6488:
6280:
6186:
5893:
2559:
2128:
6694:
6080:
5771:
5557:
has shown that this system was inconsistent and the preprint was never published.
2286:
does not have a definition and cannot be evaluated using substitution, terms like
1923:{\displaystyle \operatorname {refl} {\mathbin {:}}\prod _{a{\mathbin {:}}A}(a=a).}
245:
compute ("reduce") down to canonical terms like 4. For more, see the article on
88:
versions. However, all versions keep the core design of constructive logic using
6613:
6593:
6535:
6298:
6224:
6141:
5445:
2150:
537:
347:
type contains 1 canonical term and represents existence. It also is called the
246:
65:
46:
5967:
Martin-Löf, Per (1975). "An intuitionistic theory of types: predicative part".
938:
is proven" becomes the type of ordered pairs where the first item is the value
6568:
6550:
6237:
6133:
5927:
5841:
5824:
5755:
4403:
By convention, there is a type that represents all other types. It is called
2979:
278:
182:
5982:
Martin-Löf, Per (1982). "Constructive mathematics and computer programming".
5936:
5763:
6391:
6242:
6137:
6038:
5875:
4562:
This is the complete foundation of the theory. Everything else is derived.
348:
4515:
From the context of the statement, a reader can almost always tell whether
377:
type. But we cannot assert that these are the only propositions, i.e. the
281:. It is used to represent anything that cannot exist. It is also written
5471:
533:
302:
3337:. Synonyms like these are called "definitionally equal" by Martin-Löf.
6384:
5437:
2914:
hierarchy of universes, so to quantify a proof over any fixed constant
6147:
5707:. Texts in theoretical computer science. Berlin Heidelberg: Springer.
5433:
4471:
is a type, the members of it are objects. There is a dependent type
58:
5466:), but higher-order constructors, i.e. equalities between elements (
5413:; a detailed example of this can be found in Nordstöm and Petersson
5618:. This means, for example, that it would be difficult to formulate
2063:
is how intuitionistic type theory defines negation, you would have
1127:
As an example, the type of a function that, given a natural number
694:{\displaystyle \sum _{n{\mathbin {:}}{\mathbb {N} }}{\mathbb {R} }}
6674:
5746:
5491:
5418:
4712:
has no proof and is an empty type, then the new type representing
2005:. Putting that into a function would generate a function of type
752:
540:
of length equal to the first term. Such a type would be written:
6396:
6444:
6151:
3499:
A type that depends on an object from another type is declared
2259:{\displaystyle S{\mathbin {:}}{\mathbb {N} }\to {\mathbb {N} }}
704:
It is important to note here that the value of the first term,
5594:, ... . The universes are predicative, Ă la Russell and
1040:) depends on the value in the first part of the ordered pair (
381:
does not hold for propositions in intuitionistic type theory.
6053:
Per Martin-Löf's Notes, as recorded by
Giovanni Sambin (1980)
5856:
Norell, Ulf (2009). "Dependently typed programming in Agda".
2978:. Later research covered topics such as "super universes", "
6130:â lecture notes and slides from the Types Summer School 2005
5622:
in this theoryâthere are contractible types in each of the V
4544:
4456:
4412:
4320:{\displaystyle \Gamma \vdash t\equiv u{\mathbin {:}}\sigma }
2944:
2889:
2858:
2827:
2796:
2765:
2729:
2699:
2668:
2584:
612:
Using set-theory terminology, this is similar to an indexed
6000:(lecture notes by Giovanni Sambin), vol. 1, pp. iv+91, 1984
3856:{\displaystyle S{\mathbin {:}}\mathbb {N} \to \mathbb {N} }
1658:=-types are created from two terms. Given two terms like
1643:{\displaystyle \prod _{n{\mathbin {:}}{\mathbb {N} }}P(n)}
373:
type, while a false proposition can be represented by the
6058:
Nordström, Bengt; Petersson, Kent; Smith, Jan M. (1990).
5598:. In fact, Corollary 3.10 on p. 115 says that if AâV
5454:
works on resolving this problem. It allows one to define
1109:{\displaystyle \sum _{n{\mathbin {:}}{\mathbb {N} }}P(n)}
751:ÎŁ-types can be used to build up longer dependently-typed
100:
Martin-Löf designed the type theory on the principles of
4535:
refers to a type, or whether it refers to the object in
2605:
can be mapped to a type created with any combination of
459:. Logically, such an ordered pair would hold a proof of
4175:{\displaystyle \Gamma \vdash \sigma \ {\mathsf {Type}}}
5825:"Innovations in computational type theory using Nuprl"
5784:
Bengt
Nordström; Kent Petersson; Jan M. Smith (1990).
3075:
3037:
2376:
for every natural number, you use the following rule:
1011:. Notice that the type of the second item (proofs of
5325:
5283:
5217:
5172:
5118:
5078:
5024:
4982:
4946:
4814:
4718:
4698:
4678:
4658:
4638:
4618:
4598:
4572:
4541:
4521:
4499:
4477:
4453:
4433:
4409:
4350:
4291:
4246:
4197:
4143:
4115:
3935:
3912:
3892:
3872:
3829:
3796:
3769:
3749:
3729:
3692:
3657:
3630:
3610:
3590:
3553:
3508:
3476:
3444:
3408:
3363:
3311:
3291:
3265:
3245:
3213:
3193:
3173:
3147:
3115:
3074:
3036:
3016:
2996:
2940:
2920:
2885:
2854:
2823:
2792:
2761:
2726:
2695:
2664:
2611:
2580:
2385:
2351:
2315:
2292:
2272:
2228:
2194:
2167:
2104:
2069:
2043:
2011:
1991:
1965:
1939:
1871:
1826:
1806:
1780:
1754:
1716:
1690:
1664:
1603:
1569:
1549:
1520:
1496:
1476:
1447:
1423:
1403:
1356:
1327:
1299:
1265:
1245:
1176:
1153:
1133:
1069:
1046:
1017:
988:
964:
944:
915:
891:
871:
861:
Dependent typing allows ÎŁ-types to serve the role of
768:
730:
724:, is not depended on by the type of the second term,
710:
659:
622:
549:
505:
485:
465:
445:
425:
399:
311:
287:
225:
199:
185:, type theories are not built on top of a logic like
145:
117:
4220:{\displaystyle \Gamma \vdash t{\mathbin {:}}\sigma }
3207:
and vice versa. At the type level, there is a type
2127:
Equality of proofs is an area of active research in
6660:
6612:
6549:
6479:
6407:
6358:
6330:
6307:
6289:
6261:
6223:
6185:
5984:
5969:
4388:{\displaystyle \vdash \Gamma \ {\mathsf {Context}}}
2338:become the canonical terms of the natural numbers.
2188:Inductive types define new constants, such as zero
643:{\displaystyle {\mathbb {N} }\times {\mathbb {R} }}
5860:. TLDI '09. New York, NY, USA: ACM. pp. 1â2.
5458:, which not only define first-order constructors (
5343:
5307:
5253:
5199:
5142:
5096:
5048:
5006:
4964:
4832:
4730:
4704:
4684:
4664:
4644:
4624:
4604:
4584:
4551:
4527:
4505:
4491:that maps each object to its corresponding type.
4483:
4463:
4439:
4419:
4397:Î is a well-formed context of typing assumptions.
4387:
4319:
4264:
4219:
4174:
4121:
4099:
3918:
3898:
3878:
3855:
3814:
3775:
3755:
3735:
3715:
3672:
3636:
3616:
3596:
3576:
3533:
3488:
3456:
3424:
3388:
3329:
3297:
3277:
3251:
3231:
3199:
3179:
3159:
3133:
3106:is not a judgement; it is the type being defined.
3098:
3060:
3022:
3002:
2963:
2926:
2902:
2871:
2840:
2809:
2778:
2744:
2712:
2681:
2650:
2597:
2547:
2368:
2330:
2301:
2278:
2258:
2214:
2173:
2116:
2090:
2055:
2029:
1997:
1977:
1951:
1922:
1850:
1812:
1792:
1766:
1740:
1702:
1676:
1642:
1586:
1555:
1535:
1506:
1482:
1462:
1433:
1409:
1382:
1339:
1313:
1285:
1251:
1228:
1159:
1139:
1108:
1052:
1032:
1003:
974:
950:
930:
901:
877:
850:
740:
716:
693:
642:
601:
517:
491:
471:
451:
431:
411:
332:
293:
237:
211:
157:
131:
5798:Altenkirch, Thorsten; Anberrée, Thomas; Li, Nuo.
4265:{\displaystyle \Gamma \vdash \sigma \equiv \tau }
5665:: there is a discussion of a type theory in the
5506:. Dependent types also feature in the design of
2755:To write proofs about all "the small types" and
1286:{\displaystyle {\mathbb {N} }\to {\mathbb {R} }}
3239:and it contains terms if there is a proof that
2341:Proofs on inductive types are made possible by
5996:Per Martin-Löf, "Intuitionistic type theory",
2745:{\displaystyle {\mathcal {n}}\in \mathbb {N} }
6456:
6163:
2215:{\displaystyle 0{\mathbin {:}}{\mathbb {N} }}
277:type contains 0 terms, it is also called the
8:
5728:Clairambault, Pierre; Dybjer, Peter (2014).
2566:allow equality to be defined between terms.
53:. Intuitionistic type theory was created by
5734:Mathematical Structures in Computer Science
5610:are such that A and B are convertible then
3815:{\displaystyle 0{\mathbin {:}}\mathbb {N} }
6463:
6449:
6441:
6170:
6156:
6148:
1383:{\displaystyle \prod _{a{\mathbin {:}}A}B}
1307:
1303:
125:
121:
6023:. Sambin, Giovanni. Napoli: Bibliopolis.
5926:
5865:
5840:
5745:
5324:
5282:
5216:
5171:
5117:
5077:
5023:
4981:
4945:
4813:
4717:
4697:
4677:
4657:
4637:
4617:
4597:
4571:
4543:
4542:
4540:
4520:
4498:
4476:
4455:
4454:
4452:
4432:
4411:
4410:
4408:
4361:
4360:
4349:
4308:
4307:
4290:
4245:
4208:
4207:
4196:
4157:
4156:
4142:
4114:
3980:
3979:
3969:
3968:
3961:
3960:
3948:
3947:
3936:
3934:
3911:
3891:
3871:
3849:
3848:
3841:
3840:
3834:
3833:
3828:
3808:
3807:
3801:
3800:
3795:
3768:
3748:
3728:
3702:
3691:
3656:
3629:
3609:
3589:
3563:
3552:
3516:
3515:
3507:
3475:
3443:
3413:
3412:
3407:
3371:
3370:
3362:
3310:
3290:
3264:
3244:
3212:
3192:
3172:
3146:
3114:
3080:
3073:
3042:
3035:
3015:
2995:
2982:universes", and impredicative universes.
2949:
2943:
2942:
2939:
2919:
2894:
2888:
2887:
2884:
2863:
2857:
2856:
2853:
2832:
2826:
2825:
2822:
2801:
2795:
2794:
2791:
2770:
2764:
2763:
2760:
2738:
2737:
2728:
2727:
2725:
2704:
2698:
2697:
2694:
2673:
2667:
2666:
2663:
2610:
2589:
2583:
2582:
2579:
2526:
2525:
2524:
2518:
2517:
2513:
2458:
2457:
2456:
2450:
2449:
2445:
2432:
2414:
2413:
2412:
2390:
2389:
2388:
2387:
2386:
2384:
2362:
2358:
2350:
2314:
2291:
2271:
2251:
2250:
2249:
2241:
2240:
2239:
2233:
2232:
2227:
2207:
2206:
2205:
2199:
2198:
2193:
2166:
2103:
2068:
2042:
2010:
1990:
1964:
1938:
1933:It is possible to create =-types such as
1891:
1890:
1886:
1876:
1875:
1870:
1825:
1805:
1779:
1753:
1715:
1689:
1663:
1621:
1620:
1619:
1613:
1612:
1608:
1602:
1580:
1576:
1568:
1548:
1519:
1499:
1498:
1497:
1495:
1475:
1446:
1426:
1425:
1424:
1422:
1402:
1366:
1365:
1361:
1355:
1326:
1298:
1278:
1277:
1276:
1268:
1267:
1266:
1264:
1244:
1212:
1211:
1210:
1194:
1193:
1192:
1186:
1185:
1181:
1175:
1152:
1132:
1087:
1086:
1085:
1079:
1078:
1074:
1068:
1045:
1016:
987:
967:
966:
965:
963:
943:
914:
894:
893:
892:
890:
870:
839:
811:
810:
809:
803:
802:
798:
793:
786:
785:
784:
778:
777:
773:
767:
733:
732:
731:
729:
709:
686:
685:
684:
677:
676:
675:
669:
668:
664:
658:
635:
634:
633:
625:
624:
623:
621:
585:
584:
583:
567:
566:
565:
559:
558:
554:
548:
504:
484:
464:
444:
424:
398:
310:
286:
266:type contains 1 canonical term. And the
224:
198:
144:
139:) corresponds to the type of a function (
116:
4187:is a well-formed type in the context Î.
4134:
1594:holds for that value. The type would be
499:, so one may see such a type written as
6061:Programming in Martin-Löf's Type Theory
5786:Programming in Martin-Löf's Type Theory
5703:Bertot, Yves; Castéran, Pierre (2004).
5695:
5504:calculus of (co)inductive constructions
5424:In contrast in intensional type theory
5415:Programming in Martin-Löf's Type Theory
4858:, and morphisms are pairs of functions
4772:A category with families is a category
3099:{\displaystyle \textstyle \sum _{a:A}B}
3061:{\displaystyle \textstyle \sum _{a:A}B}
6103:Treatise on Intuitionistic Type Theory
6082:Type Theory and Functional Programming
4380:
4377:
4374:
4371:
4368:
4365:
4362:
4167:
4164:
4161:
4158:
3399:An object exists and is in a type if:
3381:
3378:
3375:
3372:
532:and the second term's type might be a
80:versions, shown to be inconsistent by
6134:n-Categories - Sketch of a Definition
5421:is based on extensional type theory.
4336:are judgmentally equal terms of type
2651:{\displaystyle 0,1,2,\Sigma ,\Pi ,=,}
165:). This correspondence is called the
27:Alternative foundation of mathematics
7:
5800:"Definable Quotients in Type Theory"
4806:of Sets, in which objects are pairs
4440:{\displaystyle \operatorname {Set} }
2964:{\displaystyle {\mathcal {U}}_{k+1}}
1563:the function generates a proof that
4769:based on earlier work by Cartmell.
4592:is a type that depends on the type
4506:{\displaystyle \operatorname {El} }
4484:{\displaystyle \operatorname {El} }
3534:{\displaystyle (x{\mathbin {:}}A)B}
3389:{\displaystyle A\ {\mathsf {Type}}}
1820:, there will be a term of the type
1470:is proven" becomes a function from
1393:Î -types are also used in logic for
258:There are three finite types: The
5470:), equalities between equalities (
4354:
4292:
4247:
4198:
4144:
2903:{\displaystyle {\mathcal {U}}_{2}}
2872:{\displaystyle {\mathcal {U}}_{1}}
2841:{\displaystyle {\mathcal {U}}_{0}}
2810:{\displaystyle {\mathcal {U}}_{1}}
2779:{\displaystyle {\mathcal {U}}_{0}}
2713:{\displaystyle {\mathcal {U}}_{n}}
2682:{\displaystyle {\mathcal {U}}_{n}}
2636:
2630:
2598:{\displaystyle {\mathcal {U}}_{0}}
2407:
2404:
2401:
2398:
2131:and has led to the development of
2070:
2050:
2024:
1992:
982:and the second item is a proof of
865:. The statement "there exists an
333:{\displaystyle \neg A:=A\to \bot }
327:
312:
288:
25:
6541:List of mathematical logic topics
5914:Journal of Functional Programming
5788:. Oxford University Press, p. 90.
5317:, then there should be an object
4759:locally cartesian closed category
4745:Categorical models of type theory
3305:" to refer to the canonical term
270:type contains 2 canonical terms.
76:variants of the theory and early
3425:{\displaystyle a{\mathbin {:}}A}
2817:, which does contain a term for
2056:{\displaystyle \ldots \to \bot }
305:is defined as a function to it:
173:by introducing dependent types.
3167:, which means every element of
6710:List of category theory topics
5482:Implementations of type theory
5393:Extensional versus intensional
5338:
5326:
5302:
5296:
5248:
5233:
5194:
5188:
5137:
5125:
5091:
5085:
5043:
5031:
5001:
4995:
4959:
4953:
4827:
4815:
4559:that corresponds to the type.
4552:{\displaystyle {\mathcal {U}}}
4464:{\displaystyle {\mathcal {U}}}
4420:{\displaystyle {\mathcal {U}}}
4281:are equal types in context Î.
4232:is a well-formed term of type
4090:
4087:
4084:
4072:
4063:
4050:
4041:
4035:
4029:
4006:
3994:
3976:
3973:
3957:
3845:
3710:
3696:
3664:
3658:
3571:
3557:
3525:
3509:
2542:
2536:
2506:
2498:
2495:
2489:
2483:
2477:
2474:
2468:
2433:
2429:
2423:
2363:
2355:
2246:
2085:
2073:
2047:
2021:
1914:
1902:
1800:compute to the canonical term
1637:
1631:
1581:
1573:
1530:
1524:
1507:{\displaystyle {\mathbb {N} }}
1457:
1451:
1434:{\displaystyle {\mathbb {N} }}
1331:
1304:
1273:
1246:
1223:
1207:
1147:, returns a vector containing
1103:
1097:
1027:
1021:
998:
992:
975:{\displaystyle {\mathbb {N} }}
925:
919:
902:{\displaystyle {\mathbb {N} }}
844:
833:
821:
818:
741:{\displaystyle {\mathbb {R} }}
596:
580:
324:
149:
122:
1:
6736:Dependently typed programming
5397:A fundamental distinction is
3343:The formal theory works with
2369:{\displaystyle P(\,\cdot \,)}
1985:, you could create a term of
1587:{\displaystyle P(\,\cdot \,)}
1543:. Thus, given the value for
6741:Constructivism (mathematics)
6100:Granström, Johan G. (2011).
3683:and removed by substitution
3544:and removed by substitution
2030:{\displaystyle 1=2\to \bot }
1851:{\displaystyle 2+2=2\cdot 2}
1741:{\displaystyle 2+2=2\cdot 2}
1710:, you can create a new type
1397:. The statement "for every
755:used in mathematics and the
262:type contains 0 terms. The
41:, the latter abbreviated as
6705:Glossary of category theory
6579:ZermeloâFraenkel set theory
6531:Mathematical constructivism
6317:Ontology (computer science)
6128:EU Types Project: Tutorials
6064:. Oxford University Press.
5254:{\displaystyle af:Tm(D,Af)}
4757:introduced the notion of a
4566:types and new objects. So
2222:and the successor function
2185:of another natural number.
1314:{\displaystyle A\implies B}
132:{\displaystyle A\implies B}
102:mathematical constructivism
6772:
6731:Foundations of mathematics
6700:Mathematical structuralism
6637:Intuitionistic type theory
6473:Foundations of Mathematics
6210:Intuitionistic type theory
6017:Intuitionistic type theory
4767:Categories with Attributes
3187:is an element of the type
2142:
2091:{\displaystyle \neg (1=2)}
31:Intuitionistic type theory
6751:Logic in computer science
6604:List of set theory topics
5928:10.1017/S095679681300018X
5842:10.1016/j.jal.2005.10.005
5756:10.1017/S0960129513000881
5496:computational type theory
4731:{\displaystyle A\times B}
4585:{\displaystyle A\times B}
3723:, replacing the variable
3584:, replacing the variable
2135:and other type theories.
1167:real numbers is written:
518:{\displaystyle A\wedge B}
412:{\displaystyle A\times B}
254:0 type, 1 type and 2 type
51:foundation of mathematics
6079:Thompson, Simon (1991).
6014:Martin-Löf, Per (1984).
5829:Journal of Applied Logic
5534:Martin-Löf type theories
5200:{\displaystyle Af:Ty(D)}
4763:Categories with Families
1793:{\displaystyle 2\cdot 2}
1703:{\displaystyle 2\cdot 2}
1395:universal quantification
1321:corresponds to the type
238:{\displaystyle 2\cdot 2}
167:CurryâHoward isomorphism
35:constructive type theory
6584:Constructive set theory
6215:Constructive set theory
5998:Studies in Proof Theory
5876:10.1145/1481861.1481862
5308:{\displaystyle A:Ty(G)}
5156:is a substitution from
5143:{\displaystyle Tm(G,A)}
5049:{\displaystyle Tm(G,A)}
5007:{\displaystyle A:Ty(G)}
4974:of types, and for each
3467:and types can be equal
3354:A type is declared by:
2934:universes, you can use
2117:{\displaystyle 1\neq 2}
18:Extensional type theory
6685:Higher category theory
6589:Descriptive set theory
6494:Mathematical induction
5456:higher inductive types
5345:
5309:
5255:
5201:
5144:
5098:
5050:
5008:
4966:
4834:
4749:Using the language of
4732:
4706:
4686:
4666:
4646:
4626:
4606:
4586:
4553:
4529:
4507:
4485:
4465:
4441:
4421:
4389:
4321:
4266:
4221:
4176:
4123:
4101:
3920:
3900:
3880:
3857:
3816:
3777:
3757:
3737:
3717:
3674:
3638:
3618:
3598:
3578:
3535:
3490:
3458:
3426:
3390:
3331:
3299:
3279:
3253:
3233:
3201:
3181:
3161:
3135:
3100:
3062:
3024:
3004:
2965:
2928:
2904:
2873:
2842:
2811:
2780:
2746:
2714:
2683:
2652:
2599:
2564:Higher inductive types
2549:
2370:
2332:
2303:
2280:
2260:
2216:
2175:
2118:
2092:
2057:
2031:
1999:
1979:
1953:
1924:
1852:
1814:
1794:
1768:
1742:
1704:
1678:
1644:
1588:
1557:
1537:
1508:
1484:
1464:
1435:
1411:
1384:
1341:
1340:{\displaystyle A\to B}
1315:
1287:
1253:
1230:
1161:
1141:
1110:
1060:). Its type would be:
1054:
1034:
1005:
976:
952:
932:
903:
879:
863:existential quantifier
852:
742:
718:
695:
644:
603:
519:
493:
473:
453:
433:
419:, of two other types,
413:
379:law of excluded middle
334:
295:
239:
213:
159:
158:{\displaystyle A\to B}
133:
39:Martin-Löf type theory
6647:Univalent foundations
6632:Dependent type theory
6622:Axiom of reducibility
6200:Constructive analysis
5907:Brady, Edwin (2013).
5684:Typed lambda calculus
5574:Principia Mathematica
5508:programming languages
5353:final among contexts
5346:
5344:{\displaystyle (G,A)}
5310:
5256:
5202:
5145:
5099:
5097:{\displaystyle Ty(G)}
5051:
5009:
4967:
4965:{\displaystyle Ty(G)}
4934:assigns to a context
4835:
4833:{\displaystyle (A,B)}
4733:
4707:
4687:
4667:
4647:
4627:
4607:
4587:
4554:
4530:
4508:
4486:
4466:
4442:
4422:
4390:
4322:
4267:
4222:
4177:
4124:
4102:
3921:
3901:
3881:
3858:
3817:
3778:
3758:
3738:
3718:
3675:
3639:
3619:
3599:
3579:
3536:
3491:
3459:
3435:Objects can be equal
3427:
3391:
3332:
3330:{\displaystyle SSSS0}
3300:
3280:
3254:
3234:
3232:{\displaystyle 4=2+2}
3202:
3182:
3162:
3136:
3134:{\displaystyle 4=2+2}
3101:
3063:
3025:
3005:
2966:
2929:
2905:
2874:
2848:, but not for itself
2843:
2812:
2781:
2747:
2715:
2684:
2653:
2600:
2550:
2371:
2333:
2304:
2281:
2261:
2217:
2176:
2119:
2093:
2058:
2032:
2000:
1998:{\displaystyle \bot }
1980:
1954:
1925:
1853:
1815:
1795:
1769:
1743:
1705:
1679:
1645:
1589:
1558:
1538:
1509:
1485:
1465:
1436:
1412:
1385:
1342:
1316:
1288:
1254:
1231:
1162:
1142:
1111:
1055:
1035:
1006:
977:
953:
933:
904:
880:
853:
743:
719:
696:
645:
604:
520:
494:
474:
454:
434:
414:
335:
296:
294:{\displaystyle \bot }
240:
214:
160:
134:
6642:Homotopy type theory
6569:Axiomatic set theory
6253:Fuzzy set operations
6248:Fuzzy finite element
6195:Intuitionistic logic
5679:Intuitionistic logic
5452:Homotopy type theory
5323:
5281:
5215:
5170:
5116:
5076:
5022:
4980:
4944:
4812:
4804:category of families
4716:
4696:
4676:
4656:
4636:
4616:
4596:
4570:
4539:
4519:
4497:
4475:
4451:
4431:
4407:
4348:
4289:
4244:
4195:
4141:
4113:
3933:
3910:
3890:
3870:
3827:
3794:
3767:
3747:
3727:
3690:
3655:
3628:
3608:
3588:
3551:
3506:
3474:
3442:
3406:
3361:
3309:
3289:
3263:
3243:
3211:
3191:
3171:
3145:
3113:
3072:
3034:
3014:
2994:
2938:
2918:
2883:
2852:
2821:
2790:
2759:
2724:
2693:
2662:
2609:
2578:
2383:
2349:
2331:{\displaystyle SSS0}
2313:
2290:
2270:
2226:
2192:
2165:
2133:homotopy type theory
2102:
2067:
2041:
2009:
1989:
1963:
1937:
1869:
1824:
1804:
1778:
1752:
1714:
1688:
1662:
1601:
1567:
1547:
1536:{\displaystyle P(n)}
1518:
1494:
1474:
1463:{\displaystyle P(n)}
1445:
1421:
1401:
1354:
1325:
1297:
1263:
1252:{\displaystyle \to }
1243:
1174:
1151:
1131:
1067:
1044:
1033:{\displaystyle P(n)}
1015:
1004:{\displaystyle P(n)}
986:
962:
942:
931:{\displaystyle P(n)}
913:
889:
869:
766:
728:
708:
657:
620:
547:
503:
483:
463:
443:
423:
397:
309:
285:
223:
197:
143:
115:
6430:Non-monotonic logic
6179:Non-classical logic
6144:, November 29, 1995
6140:and James Dolan to
5494:system is based on
3489:{\displaystyle A=B}
3457:{\displaystyle a=b}
3278:{\displaystyle 2+2}
3160:{\displaystyle A=B}
1978:{\displaystyle 1=2}
1952:{\displaystyle 1=2}
1767:{\displaystyle 2+2}
1677:{\displaystyle 2+2}
212:{\displaystyle 2+2}
49:and an alternative
6627:Simple type theory
6574:Zermelo set theory
6521:Mathematical proof
6481:Mathematical logic
6425:Intermediate logic
6205:Heyting arithmetic
6085:. Addison-Wesley.
5341:
5305:
5275:is a context, and
5251:
5197:
5140:
5094:
5046:
5004:
4962:
4902:â in other words,
4842:of an "index set"
4830:
4728:
4702:
4682:
4662:
4642:
4622:
4602:
4582:
4549:
4525:
4503:
4481:
4461:
4437:
4417:
4385:
4317:
4262:
4217:
4172:
4119:
4097:
4095:
3916:
3896:
3876:
3853:
3812:
3773:
3753:
3733:
3713:
3670:
3634:
3614:
3594:
3574:
3531:
3486:
3454:
3422:
3386:
3327:
3295:
3275:
3249:
3229:
3197:
3177:
3157:
3131:
3096:
3095:
3091:
3058:
3057:
3053:
3020:
3000:
2961:
2924:
2900:
2879:. Similarly, for
2869:
2838:
2807:
2776:
2742:
2710:
2679:
2648:
2595:
2545:
2532:
2464:
2366:
2328:
2302:{\displaystyle S0}
2299:
2276:
2256:
2212:
2171:
2114:
2088:
2053:
2027:
1995:
1975:
1949:
1920:
1901:
1848:
1810:
1790:
1764:
1738:
1700:
1674:
1654:= type constructor
1640:
1627:
1584:
1553:
1533:
1504:
1480:
1460:
1431:
1407:
1380:
1376:
1337:
1311:
1283:
1249:
1226:
1200:
1157:
1137:
1120:Î type constructor
1106:
1093:
1050:
1030:
1001:
972:
948:
928:
899:
875:
848:
817:
792:
757:records or structs
738:
714:
691:
683:
640:
599:
573:
515:
489:
469:
449:
429:
409:
385:ÎŁ type constructor
330:
291:
235:
209:
155:
129:
106:BHK interpretation
6718:
6717:
6599:Russell's paradox
6514:Natural deduction
6438:
6437:
6420:Inquisitive logic
6415:Dynamic semantics
6368:Three-state logic
6322:Ontology language
6113:978-94-007-1735-0
5714:978-3-540-20854-9
4705:{\displaystyle B}
4685:{\displaystyle A}
4665:{\displaystyle B}
4645:{\displaystyle A}
4625:{\displaystyle B}
4605:{\displaystyle A}
4528:{\displaystyle A}
4513:is never written.
4401:
4400:
4359:
4155:
4122:{\displaystyle S}
3956:
3919:{\displaystyle S}
3899:{\displaystyle S}
3879:{\displaystyle S}
3776:{\displaystyle b}
3756:{\displaystyle a}
3736:{\displaystyle x}
3716:{\displaystyle b}
3673:{\displaystyle b}
3637:{\displaystyle B}
3617:{\displaystyle a}
3597:{\displaystyle x}
3577:{\displaystyle B}
3369:
3298:{\displaystyle 4}
3252:{\displaystyle 4}
3200:{\displaystyle B}
3180:{\displaystyle A}
3076:
3038:
3023:{\displaystyle B}
3003:{\displaystyle A}
2927:{\displaystyle k}
2509:
2441:
2397:
2279:{\displaystyle S}
2174:{\displaystyle 0}
1882:
1813:{\displaystyle 4}
1604:
1556:{\displaystyle n}
1483:{\displaystyle n}
1410:{\displaystyle n}
1357:
1177:
1160:{\displaystyle n}
1140:{\displaystyle n}
1070:
1053:{\displaystyle n}
951:{\displaystyle n}
878:{\displaystyle n}
842:
794:
769:
717:{\displaystyle n}
660:
550:
492:{\displaystyle B}
472:{\displaystyle A}
452:{\displaystyle B}
432:{\displaystyle A}
391:Cartesian product
16:(Redirected from
6763:
6680:Category of sets
6652:Girard's paradox
6564:Naive set theory
6504:Axiomatic system
6471:Major topics in
6465:
6458:
6451:
6442:
6373:Tri-state buffer
6172:
6165:
6158:
6149:
6117:
6096:
6075:
6042:
6022:
6001:
5994:
5988:
5987:
5979:
5973:
5972:
5964:
5958:
5955:
5949:
5948:
5930:
5904:
5898:
5897:
5869:
5853:
5847:
5846:
5844:
5820:
5814:
5813:
5811:
5805:. Archived from
5804:
5795:
5789:
5782:
5776:
5775:
5749:
5725:
5719:
5718:
5700:
5620:univalence axiom
5555:Jean-Yves Girard
5543:Jean-Yves Girard
5502:is based on the
5488:proof assistants
5442:rational numbers
5352:
5350:
5348:
5347:
5342:
5316:
5314:
5312:
5311:
5306:
5262:
5260:
5258:
5257:
5252:
5208:
5206:
5204:
5203:
5198:
5151:
5149:
5147:
5146:
5141:
5105:
5103:
5101:
5100:
5095:
5057:
5055:
5053:
5052:
5047:
5015:
5013:
5011:
5010:
5005:
4973:
4971:
4969:
4968:
4963:
4841:
4839:
4837:
4836:
4831:
4737:
4735:
4734:
4729:
4711:
4709:
4708:
4703:
4691:
4689:
4688:
4683:
4671:
4669:
4668:
4663:
4651:
4649:
4648:
4643:
4631:
4629:
4628:
4623:
4611:
4609:
4608:
4603:
4591:
4589:
4588:
4583:
4558:
4556:
4555:
4550:
4548:
4547:
4534:
4532:
4531:
4526:
4512:
4510:
4509:
4504:
4490:
4488:
4487:
4482:
4470:
4468:
4467:
4462:
4460:
4459:
4446:
4444:
4443:
4438:
4426:
4424:
4423:
4418:
4416:
4415:
4394:
4392:
4391:
4386:
4384:
4383:
4357:
4326:
4324:
4323:
4318:
4313:
4312:
4271:
4269:
4268:
4263:
4226:
4224:
4223:
4218:
4213:
4212:
4181:
4179:
4178:
4173:
4171:
4170:
4153:
4135:
4128:
4126:
4125:
4120:
4106:
4104:
4103:
4098:
4096:
3983:
3972:
3964:
3954:
3953:
3952:
3925:
3923:
3922:
3917:
3905:
3903:
3902:
3897:
3885:
3883:
3882:
3877:
3862:
3860:
3859:
3854:
3852:
3844:
3839:
3838:
3821:
3819:
3818:
3813:
3811:
3806:
3805:
3782:
3780:
3779:
3774:
3762:
3760:
3759:
3754:
3743:with the object
3742:
3740:
3739:
3734:
3722:
3720:
3719:
3714:
3706:
3679:
3677:
3676:
3671:
3643:
3641:
3640:
3635:
3623:
3621:
3620:
3615:
3604:with the object
3603:
3601:
3600:
3595:
3583:
3581:
3580:
3575:
3567:
3540:
3538:
3537:
3532:
3521:
3520:
3495:
3493:
3492:
3487:
3463:
3461:
3460:
3455:
3431:
3429:
3428:
3423:
3418:
3417:
3395:
3393:
3392:
3387:
3385:
3384:
3367:
3336:
3334:
3333:
3328:
3304:
3302:
3301:
3296:
3284:
3282:
3281:
3276:
3258:
3256:
3255:
3250:
3238:
3236:
3235:
3230:
3206:
3204:
3203:
3198:
3186:
3184:
3183:
3178:
3166:
3164:
3163:
3158:
3140:
3138:
3137:
3132:
3105:
3103:
3102:
3097:
3090:
3067:
3065:
3064:
3059:
3052:
3029:
3027:
3026:
3021:
3009:
3007:
3006:
3001:
2976:Girard's paradox
2970:
2968:
2967:
2962:
2960:
2959:
2948:
2947:
2933:
2931:
2930:
2925:
2909:
2907:
2906:
2901:
2899:
2898:
2893:
2892:
2878:
2876:
2875:
2870:
2868:
2867:
2862:
2861:
2847:
2845:
2844:
2839:
2837:
2836:
2831:
2830:
2816:
2814:
2813:
2808:
2806:
2805:
2800:
2799:
2785:
2783:
2782:
2777:
2775:
2774:
2769:
2768:
2751:
2749:
2748:
2743:
2741:
2733:
2732:
2719:
2717:
2716:
2711:
2709:
2708:
2703:
2702:
2688:
2686:
2685:
2680:
2678:
2677:
2672:
2671:
2657:
2655:
2654:
2649:
2604:
2602:
2601:
2596:
2594:
2593:
2588:
2587:
2554:
2552:
2551:
2546:
2531:
2530:
2529:
2523:
2522:
2505:
2501:
2463:
2462:
2461:
2455:
2454:
2419:
2418:
2411:
2410:
2395:
2394:
2393:
2375:
2373:
2372:
2367:
2337:
2335:
2334:
2329:
2308:
2306:
2305:
2300:
2285:
2283:
2282:
2277:
2265:
2263:
2262:
2257:
2255:
2254:
2245:
2244:
2238:
2237:
2221:
2219:
2218:
2213:
2211:
2210:
2204:
2203:
2180:
2178:
2177:
2172:
2123:
2121:
2120:
2115:
2097:
2095:
2094:
2089:
2062:
2060:
2059:
2054:
2036:
2034:
2033:
2028:
2004:
2002:
2001:
1996:
1984:
1982:
1981:
1976:
1958:
1956:
1955:
1950:
1929:
1927:
1926:
1921:
1900:
1896:
1895:
1881:
1880:
1857:
1855:
1854:
1849:
1819:
1817:
1816:
1811:
1799:
1797:
1796:
1791:
1773:
1771:
1770:
1765:
1747:
1745:
1744:
1739:
1709:
1707:
1706:
1701:
1683:
1681:
1680:
1675:
1649:
1647:
1646:
1641:
1626:
1625:
1624:
1618:
1617:
1593:
1591:
1590:
1585:
1562:
1560:
1559:
1554:
1542:
1540:
1539:
1534:
1513:
1511:
1510:
1505:
1503:
1502:
1489:
1487:
1486:
1481:
1469:
1467:
1466:
1461:
1440:
1438:
1437:
1432:
1430:
1429:
1416:
1414:
1413:
1408:
1389:
1387:
1386:
1381:
1375:
1371:
1370:
1346:
1344:
1343:
1338:
1320:
1318:
1317:
1312:
1292:
1290:
1289:
1284:
1282:
1281:
1272:
1271:
1258:
1256:
1255:
1250:
1235:
1233:
1232:
1227:
1216:
1215:
1199:
1198:
1197:
1191:
1190:
1166:
1164:
1163:
1158:
1146:
1144:
1143:
1138:
1115:
1113:
1112:
1107:
1092:
1091:
1090:
1084:
1083:
1059:
1057:
1056:
1051:
1039:
1037:
1036:
1031:
1010:
1008:
1007:
1002:
981:
979:
978:
973:
971:
970:
957:
955:
954:
949:
937:
935:
934:
929:
908:
906:
905:
900:
898:
897:
884:
882:
881:
876:
857:
855:
854:
849:
847:
843:
840:
816:
815:
814:
808:
807:
791:
790:
789:
783:
782:
747:
745:
744:
739:
737:
736:
723:
721:
720:
715:
700:
698:
697:
692:
690:
689:
682:
681:
680:
674:
673:
649:
647:
646:
641:
639:
638:
629:
628:
608:
606:
605:
600:
589:
588:
572:
571:
570:
564:
563:
524:
522:
521:
516:
498:
496:
495:
490:
478:
476:
475:
470:
458:
456:
455:
450:
438:
436:
435:
430:
418:
416:
415:
410:
339:
337:
336:
331:
300:
298:
297:
292:
244:
242:
241:
236:
218:
216:
215:
210:
164:
162:
161:
156:
138:
136:
135:
130:
82:Girard's paradox
21:
6771:
6770:
6766:
6765:
6764:
6762:
6761:
6760:
6721:
6720:
6719:
6714:
6662:Category theory
6656:
6608:
6545:
6475:
6469:
6439:
6434:
6403:
6354:
6326:
6303:
6285:
6276:Relevance logic
6271:Structural rule
6257:
6233:Degree of truth
6219:
6181:
6176:
6124:
6114:
6099:
6093:
6078:
6072:
6057:
6049:
6047:Further reading
6031:
6020:
6013:
6010:
6005:
6004:
5995:
5991:
5981:
5980:
5976:
5966:
5965:
5961:
5956:
5952:
5906:
5905:
5901:
5886:
5867:10.1.1.163.7149
5855:
5854:
5850:
5822:
5821:
5817:
5809:
5802:
5797:
5796:
5792:
5783:
5779:
5727:
5726:
5722:
5715:
5702:
5701:
5697:
5692:
5675:
5639:
5633:
5627:
5614: =
5609:
5603:
5593:
5587:
5536:
5484:
5438:integer numbers
5395:
5321:
5320:
5318:
5279:
5278:
5276:
5213:
5212:
5210:
5168:
5167:
5165:
5114:
5113:
5111:
5074:
5073:
5071:
5020:
5019:
5017:
4978:
4977:
4975:
4942:
4941:
4939:
4926:
4911:
4898:
4888:
4846:and a function
4810:
4809:
4807:
4751:category theory
4747:
4738:is also empty.
4714:
4713:
4694:
4693:
4674:
4673:
4654:
4653:
4634:
4633:
4614:
4613:
4594:
4593:
4568:
4567:
4537:
4536:
4517:
4516:
4495:
4494:
4473:
4472:
4449:
4448:
4429:
4428:
4405:
4404:
4346:
4345:
4287:
4286:
4242:
4241:
4193:
4192:
4139:
4138:
4111:
4110:
4094:
4093:
4053:
4020:
4019:
4009:
3985:
3984:
3943:
3931:
3930:
3908:
3907:
3888:
3887:
3868:
3867:
3825:
3824:
3792:
3791:
3765:
3764:
3745:
3744:
3725:
3724:
3688:
3687:
3653:
3652:
3626:
3625:
3606:
3605:
3586:
3585:
3549:
3548:
3504:
3503:
3472:
3471:
3440:
3439:
3404:
3403:
3359:
3358:
3307:
3306:
3287:
3286:
3261:
3260:
3241:
3240:
3209:
3208:
3189:
3188:
3169:
3168:
3143:
3142:
3111:
3110:
3070:
3069:
3032:
3031:
3030:is a type then
3012:
3011:
2992:
2991:
2988:
2941:
2936:
2935:
2916:
2915:
2886:
2881:
2880:
2855:
2850:
2849:
2824:
2819:
2818:
2793:
2788:
2787:
2786:, you must use
2762:
2757:
2756:
2722:
2721:
2696:
2691:
2690:
2665:
2660:
2659:
2607:
2606:
2581:
2576:
2575:
2572:
2440:
2436:
2381:
2380:
2347:
2346:
2311:
2310:
2288:
2287:
2268:
2267:
2224:
2223:
2190:
2189:
2163:
2162:
2147:
2141:
2139:Inductive types
2100:
2099:
2065:
2064:
2039:
2038:
2007:
2006:
1987:
1986:
1961:
1960:
1935:
1934:
1867:
1866:
1822:
1821:
1802:
1801:
1776:
1775:
1750:
1749:
1712:
1711:
1686:
1685:
1660:
1659:
1656:
1599:
1598:
1565:
1564:
1545:
1544:
1516:
1515:
1492:
1491:
1472:
1471:
1443:
1442:
1419:
1418:
1399:
1398:
1352:
1351:
1323:
1322:
1295:
1294:
1261:
1260:
1241:
1240:
1172:
1171:
1149:
1148:
1129:
1128:
1122:
1065:
1064:
1042:
1041:
1013:
1012:
984:
983:
960:
959:
940:
939:
911:
910:
887:
886:
867:
866:
764:
763:
726:
725:
706:
705:
655:
654:
618:
617:
545:
544:
501:
500:
481:
480:
479:and a proof of
461:
460:
441:
440:
421:
420:
395:
394:
387:
307:
306:
283:
282:
256:
221:
220:
195:
194:
179:
171:predicate logic
141:
140:
113:
112:
98:
90:dependent types
33:(also known as
28:
23:
22:
15:
12:
11:
5:
6769:
6767:
6759:
6758:
6753:
6748:
6743:
6738:
6733:
6723:
6722:
6716:
6715:
6713:
6712:
6707:
6702:
6697:
6695:â-topos theory
6692:
6687:
6682:
6677:
6672:
6666:
6664:
6658:
6657:
6655:
6654:
6649:
6644:
6639:
6634:
6629:
6624:
6618:
6616:
6610:
6609:
6607:
6606:
6601:
6596:
6591:
6586:
6581:
6576:
6571:
6566:
6561:
6555:
6553:
6547:
6546:
6544:
6543:
6538:
6533:
6528:
6523:
6518:
6517:
6516:
6511:
6509:Hilbert system
6506:
6496:
6491:
6485:
6483:
6477:
6476:
6470:
6468:
6467:
6460:
6453:
6445:
6436:
6435:
6433:
6432:
6427:
6422:
6417:
6411:
6409:
6405:
6404:
6402:
6401:
6400:
6399:
6389:
6388:
6387:
6377:
6376:
6375:
6364:
6362:
6356:
6355:
6353:
6352:
6347:
6342:
6336:
6334:
6328:
6327:
6325:
6324:
6319:
6313:
6311:
6305:
6304:
6302:
6301:
6295:
6293:
6291:Paraconsistent
6287:
6286:
6284:
6283:
6278:
6273:
6267:
6265:
6259:
6258:
6256:
6255:
6250:
6245:
6240:
6235:
6229:
6227:
6221:
6220:
6218:
6217:
6212:
6207:
6202:
6197:
6191:
6189:
6187:Intuitionistic
6183:
6182:
6177:
6175:
6174:
6167:
6160:
6152:
6146:
6145:
6136:â letter from
6131:
6123:
6122:External links
6120:
6119:
6118:
6112:
6097:
6091:
6076:
6070:
6055:
6048:
6045:
6044:
6043:
6030:978-8870881059
6029:
6009:
6006:
6003:
6002:
5989:
5974:
5959:
5950:
5921:(5): 552â593.
5899:
5884:
5848:
5835:(4): 428â469.
5815:
5812:on 2024-04-19.
5790:
5777:
5720:
5713:
5694:
5693:
5691:
5688:
5687:
5686:
5681:
5674:
5671:
5657:are cumulative
5635:
5629:
5623:
5605:
5599:
5596:non-cumulative
5589:
5585:
5539:Per Martin-Löf
5535:
5532:
5483:
5480:
5394:
5391:
5357:with mappings
5340:
5337:
5334:
5331:
5328:
5304:
5301:
5298:
5295:
5292:
5289:
5286:
5250:
5247:
5244:
5241:
5238:
5235:
5232:
5229:
5226:
5223:
5220:
5196:
5193:
5190:
5187:
5184:
5181:
5178:
5175:
5139:
5136:
5133:
5130:
5127:
5124:
5121:
5093:
5090:
5087:
5084:
5081:
5045:
5042:
5039:
5036:
5033:
5030:
5027:
5003:
5000:
4997:
4994:
4991:
4988:
4985:
4961:
4958:
4955:
4952:
4949:
4917:
4909:
4896:
4886:
4829:
4826:
4823:
4820:
4817:
4755:R. A. G. Seely
4746:
4743:
4727:
4724:
4721:
4701:
4681:
4661:
4641:
4621:
4601:
4581:
4578:
4575:
4546:
4524:
4502:
4493:In most texts
4480:
4458:
4436:
4414:
4399:
4398:
4395:
4382:
4379:
4376:
4373:
4370:
4367:
4364:
4356:
4353:
4342:
4341:
4340:in context Î.
4327:
4316:
4311:
4306:
4303:
4300:
4297:
4294:
4283:
4282:
4272:
4261:
4258:
4255:
4252:
4249:
4238:
4237:
4236:in context Î.
4227:
4216:
4211:
4206:
4203:
4200:
4189:
4188:
4182:
4169:
4166:
4163:
4160:
4152:
4149:
4146:
4118:
4108:
4107:
4092:
4089:
4086:
4083:
4080:
4077:
4074:
4071:
4068:
4065:
4062:
4059:
4056:
4054:
4052:
4049:
4046:
4043:
4040:
4037:
4034:
4031:
4028:
4025:
4022:
4021:
4018:
4015:
4012:
4010:
4008:
4005:
4002:
3999:
3996:
3993:
3990:
3987:
3986:
3982:
3978:
3975:
3971:
3967:
3963:
3959:
3951:
3946:
3944:
3942:
3939:
3938:
3915:
3895:
3875:
3864:
3863:
3851:
3847:
3843:
3837:
3832:
3822:
3810:
3804:
3799:
3785:
3784:
3772:
3752:
3732:
3712:
3709:
3705:
3701:
3698:
3695:
3681:
3680:
3669:
3666:
3663:
3660:
3646:
3645:
3633:
3613:
3593:
3573:
3570:
3566:
3562:
3559:
3556:
3542:
3541:
3530:
3527:
3524:
3519:
3514:
3511:
3497:
3496:
3485:
3482:
3479:
3465:
3464:
3453:
3450:
3447:
3433:
3432:
3421:
3416:
3411:
3397:
3396:
3383:
3380:
3377:
3374:
3366:
3326:
3323:
3320:
3317:
3314:
3294:
3274:
3271:
3268:
3248:
3228:
3225:
3222:
3219:
3216:
3196:
3176:
3156:
3153:
3150:
3130:
3127:
3124:
3121:
3118:
3094:
3089:
3086:
3083:
3079:
3056:
3051:
3048:
3045:
3041:
3019:
3010:is a type and
2999:
2987:
2984:
2958:
2955:
2952:
2946:
2923:
2910:. There is a
2897:
2891:
2866:
2860:
2835:
2829:
2804:
2798:
2773:
2767:
2740:
2736:
2731:
2707:
2701:
2676:
2670:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2614:
2592:
2586:
2571:
2570:Universe types
2568:
2556:
2555:
2544:
2541:
2538:
2535:
2528:
2521:
2516:
2512:
2508:
2504:
2500:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2460:
2453:
2448:
2444:
2439:
2435:
2431:
2428:
2425:
2422:
2417:
2409:
2406:
2403:
2400:
2392:
2365:
2361:
2357:
2354:
2327:
2324:
2321:
2318:
2298:
2295:
2275:
2253:
2248:
2243:
2236:
2231:
2209:
2202:
2197:
2170:
2145:Inductive type
2143:Main article:
2140:
2137:
2113:
2110:
2107:
2087:
2084:
2081:
2078:
2075:
2072:
2052:
2049:
2046:
2026:
2023:
2020:
2017:
2014:
1994:
1974:
1971:
1968:
1948:
1945:
1942:
1931:
1930:
1919:
1916:
1913:
1910:
1907:
1904:
1899:
1894:
1889:
1885:
1879:
1874:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1809:
1789:
1786:
1783:
1763:
1760:
1757:
1737:
1734:
1731:
1728:
1725:
1722:
1719:
1699:
1696:
1693:
1673:
1670:
1667:
1655:
1652:
1651:
1650:
1639:
1636:
1633:
1630:
1623:
1616:
1611:
1607:
1583:
1579:
1575:
1572:
1552:
1532:
1529:
1526:
1523:
1501:
1479:
1459:
1456:
1453:
1450:
1428:
1406:
1391:
1390:
1379:
1374:
1369:
1364:
1360:
1336:
1333:
1330:
1310:
1306:
1302:
1280:
1275:
1270:
1248:
1237:
1236:
1225:
1222:
1219:
1214:
1209:
1206:
1203:
1196:
1189:
1184:
1180:
1156:
1136:
1121:
1118:
1117:
1116:
1105:
1102:
1099:
1096:
1089:
1082:
1077:
1073:
1049:
1029:
1026:
1023:
1020:
1000:
997:
994:
991:
969:
947:
927:
924:
921:
918:
896:
874:
859:
858:
846:
838:
835:
832:
829:
826:
823:
820:
813:
806:
801:
797:
788:
781:
776:
772:
735:
713:
702:
701:
688:
679:
672:
667:
663:
637:
632:
627:
614:disjoint union
610:
609:
598:
595:
592:
587:
582:
579:
576:
569:
562:
557:
553:
530:natural number
514:
511:
508:
488:
468:
448:
428:
408:
405:
402:
386:
383:
366:propositions.
360:Boolean values
343:Likewise, the
329:
326:
323:
320:
317:
314:
290:
255:
252:
234:
231:
228:
208:
205:
202:
178:
175:
154:
151:
148:
128:
124:
120:
97:
94:
84:, gave way to
55:Per Martin-Löf
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6768:
6757:
6754:
6752:
6749:
6747:
6744:
6742:
6739:
6737:
6734:
6732:
6729:
6728:
6726:
6711:
6708:
6706:
6703:
6701:
6698:
6696:
6693:
6691:
6688:
6686:
6683:
6681:
6678:
6676:
6673:
6671:
6668:
6667:
6665:
6663:
6659:
6653:
6650:
6648:
6645:
6643:
6640:
6638:
6635:
6633:
6630:
6628:
6625:
6623:
6620:
6619:
6617:
6615:
6611:
6605:
6602:
6600:
6597:
6595:
6592:
6590:
6587:
6585:
6582:
6580:
6577:
6575:
6572:
6570:
6567:
6565:
6562:
6560:
6557:
6556:
6554:
6552:
6548:
6542:
6539:
6537:
6534:
6532:
6529:
6527:
6524:
6522:
6519:
6515:
6512:
6510:
6507:
6505:
6502:
6501:
6500:
6499:Formal system
6497:
6495:
6492:
6490:
6487:
6486:
6484:
6482:
6478:
6474:
6466:
6461:
6459:
6454:
6452:
6447:
6446:
6443:
6431:
6428:
6426:
6423:
6421:
6418:
6416:
6413:
6412:
6410:
6406:
6398:
6395:
6394:
6393:
6390:
6386:
6383:
6382:
6381:
6378:
6374:
6371:
6370:
6369:
6366:
6365:
6363:
6361:
6360:Digital logic
6357:
6351:
6348:
6346:
6343:
6341:
6338:
6337:
6335:
6333:
6329:
6323:
6320:
6318:
6315:
6314:
6312:
6310:
6306:
6300:
6297:
6296:
6294:
6292:
6288:
6282:
6279:
6277:
6274:
6272:
6269:
6268:
6266:
6264:
6263:Substructural
6260:
6254:
6251:
6249:
6246:
6244:
6241:
6239:
6236:
6234:
6231:
6230:
6228:
6226:
6222:
6216:
6213:
6211:
6208:
6206:
6203:
6201:
6198:
6196:
6193:
6192:
6190:
6188:
6184:
6180:
6173:
6168:
6166:
6161:
6159:
6154:
6153:
6150:
6143:
6139:
6135:
6132:
6129:
6126:
6125:
6121:
6115:
6109:
6105:
6104:
6098:
6094:
6092:0-201-41667-0
6088:
6084:
6083:
6077:
6073:
6071:9780198538141
6067:
6063:
6062:
6056:
6054:
6051:
6050:
6046:
6040:
6036:
6032:
6026:
6019:
6018:
6012:
6011:
6007:
5999:
5993:
5990:
5985:
5978:
5975:
5970:
5963:
5960:
5954:
5951:
5946:
5942:
5938:
5934:
5929:
5924:
5920:
5916:
5915:
5910:
5903:
5900:
5895:
5891:
5887:
5885:9781605584201
5881:
5877:
5873:
5868:
5863:
5859:
5852:
5849:
5843:
5838:
5834:
5830:
5826:
5819:
5816:
5808:
5801:
5794:
5791:
5787:
5781:
5778:
5773:
5769:
5765:
5761:
5757:
5753:
5748:
5743:
5739:
5735:
5731:
5724:
5721:
5716:
5710:
5706:
5699:
5696:
5689:
5685:
5682:
5680:
5677:
5676:
5672:
5670:
5668:
5664:
5660:
5658:
5653:
5649:
5647:
5643:
5638:
5632:
5626:
5621:
5617:
5613:
5608:
5602:
5597:
5592:
5582:
5578:
5576:
5575:
5570:
5566:
5562:
5558:
5556:
5551:
5547:
5544:
5540:
5533:
5531:
5529:
5525:
5521:
5517:
5513:
5509:
5505:
5501:
5497:
5493:
5489:
5481:
5479:
5477:
5473:
5469:
5465:
5461:
5457:
5453:
5449:
5447:
5443:
5439:
5435:
5431:
5427:
5426:type checking
5422:
5420:
5416:
5412:
5408:
5404:
5400:
5392:
5390:
5386:
5382:
5380:
5376:
5372:
5368:
5364:
5360:
5356:
5335:
5332:
5329:
5299:
5293:
5290:
5287:
5284:
5274:
5269:
5266:The category
5264:
5245:
5242:
5239:
5236:
5230:
5227:
5224:
5221:
5218:
5191:
5185:
5182:
5179:
5176:
5173:
5163:
5159:
5155:
5134:
5131:
5128:
5122:
5119:
5110:is a term in
5109:
5088:
5082:
5079:
5070:is a type in
5069:
5065:
5061:
5040:
5037:
5034:
5028:
5025:
4998:
4992:
4989:
4986:
4983:
4956:
4950:
4947:
4937:
4933:
4928:
4924:
4920:
4916:
4912:
4905:
4901:
4895:
4891:
4885:
4881:
4877:
4873:
4869:
4865:
4861:
4857:
4853:
4849:
4845:
4824:
4821:
4818:
4805:
4801:
4797:
4793:
4791:
4787:
4783:
4779:
4775:
4770:
4768:
4764:
4760:
4756:
4752:
4744:
4742:
4739:
4725:
4722:
4719:
4699:
4679:
4659:
4639:
4619:
4612:and the type
4599:
4579:
4576:
4573:
4563:
4560:
4522:
4514:
4500:
4478:
4434:
4396:
4351:
4344:
4343:
4339:
4335:
4331:
4328:
4314:
4309:
4304:
4301:
4298:
4295:
4285:
4284:
4280:
4276:
4273:
4259:
4256:
4253:
4250:
4240:
4239:
4235:
4231:
4228:
4214:
4209:
4204:
4201:
4191:
4190:
4186:
4183:
4150:
4147:
4137:
4136:
4133:
4130:
4116:
4081:
4078:
4075:
4069:
4066:
4060:
4057:
4055:
4047:
4044:
4038:
4032:
4026:
4023:
4016:
4013:
4011:
4003:
4000:
3997:
3991:
3988:
3965:
3949:
3945:
3940:
3929:
3928:
3927:
3913:
3893:
3873:
3835:
3830:
3823:
3802:
3797:
3790:
3789:
3788:
3770:
3750:
3730:
3707:
3703:
3699:
3693:
3686:
3685:
3684:
3667:
3661:
3651:
3650:
3649:
3631:
3611:
3591:
3568:
3564:
3560:
3554:
3547:
3546:
3545:
3528:
3522:
3517:
3512:
3502:
3501:
3500:
3483:
3480:
3477:
3470:
3469:
3468:
3451:
3448:
3445:
3438:
3437:
3436:
3419:
3414:
3409:
3402:
3401:
3400:
3364:
3357:
3356:
3355:
3352:
3350:
3346:
3341:
3338:
3324:
3321:
3318:
3315:
3312:
3292:
3272:
3269:
3266:
3246:
3226:
3223:
3220:
3217:
3214:
3194:
3174:
3154:
3151:
3148:
3128:
3125:
3122:
3119:
3116:
3107:
3092:
3087:
3084:
3081:
3077:
3054:
3049:
3046:
3043:
3039:
3017:
2997:
2985:
2983:
2981:
2977:
2972:
2956:
2953:
2950:
2921:
2913:
2895:
2864:
2833:
2802:
2771:
2753:
2734:
2705:
2689:that maps to
2674:
2645:
2642:
2639:
2633:
2627:
2624:
2621:
2618:
2615:
2612:
2590:
2569:
2567:
2565:
2561:
2539:
2533:
2519:
2514:
2510:
2502:
2492:
2486:
2480:
2471:
2465:
2451:
2446:
2442:
2437:
2426:
2420:
2415:
2379:
2378:
2377:
2359:
2352:
2344:
2339:
2325:
2322:
2319:
2316:
2296:
2293:
2273:
2234:
2229:
2200:
2195:
2186:
2184:
2168:
2160:
2156:
2152:
2146:
2138:
2136:
2134:
2130:
2125:
2111:
2108:
2105:
2098:or, finally,
2082:
2079:
2076:
2044:
2018:
2015:
2012:
1972:
1969:
1966:
1946:
1943:
1940:
1917:
1911:
1908:
1905:
1897:
1892:
1887:
1883:
1877:
1872:
1865:
1864:
1863:
1861:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1807:
1787:
1784:
1781:
1761:
1758:
1755:
1735:
1732:
1729:
1726:
1723:
1720:
1717:
1697:
1694:
1691:
1671:
1668:
1665:
1653:
1634:
1628:
1614:
1609:
1605:
1597:
1596:
1595:
1577:
1570:
1550:
1527:
1521:
1514:to proofs of
1477:
1454:
1448:
1404:
1396:
1377:
1372:
1367:
1362:
1358:
1350:
1349:
1348:
1334:
1328:
1308:
1300:
1220:
1217:
1204:
1201:
1187:
1182:
1178:
1170:
1169:
1168:
1154:
1134:
1125:
1119:
1100:
1094:
1080:
1075:
1071:
1063:
1062:
1061:
1047:
1024:
1018:
995:
989:
945:
922:
916:
872:
864:
836:
830:
827:
824:
804:
799:
795:
779:
774:
770:
762:
761:
760:
758:
754:
749:
711:
670:
665:
661:
653:
652:
651:
630:
615:
593:
590:
577:
574:
560:
555:
551:
543:
542:
541:
539:
535:
531:
526:
512:
509:
506:
486:
466:
446:
426:
406:
403:
400:
392:
384:
382:
380:
376:
372:
367:
365:
361:
357:
354:Finally, the
352:
350:
346:
341:
321:
318:
315:
304:
280:
276:
271:
269:
265:
261:
253:
251:
250:
248:
232:
229:
226:
206:
203:
200:
190:
188:
184:
176:
174:
172:
168:
152:
146:
126:
118:
109:
107:
103:
95:
93:
91:
87:
83:
79:
78:impredicative
75:
71:
67:
63:
62:mathematician
60:
56:
52:
48:
44:
40:
36:
32:
19:
6756:Intuitionism
6675:Topos theory
6636:
6526:Model theory
6489:Peano axioms
6340:Three-valued
6281:Linear logic
6209:
6106:. Springer.
6102:
6081:
6060:
6016:
5997:
5992:
5983:
5977:
5968:
5962:
5953:
5918:
5912:
5902:
5857:
5851:
5832:
5828:
5818:
5807:the original
5793:
5785:
5780:
5737:
5733:
5723:
5704:
5698:
5666:
5662:
5661:
5656:
5651:
5650:
5645:
5641:
5636:
5630:
5624:
5615:
5611:
5606:
5600:
5595:
5590:
5580:
5579:
5572:
5560:
5559:
5549:
5548:
5537:
5485:
5476:ad infinitum
5475:
5450:
5446:real numbers
5423:
5414:
5411:Y-combinator
5396:
5387:
5383:
5378:
5374:
5370:
5366:
5362:
5358:
5354:
5272:
5267:
5265:
5161:
5157:
5153:
5107:
5067:
5063:
5059:
4935:
4931:
4930:The functor
4929:
4922:
4918:
4914:
4907:
4903:
4899:
4893:
4889:
4883:
4882:, such that
4879:
4875:
4871:
4867:
4863:
4859:
4855:
4851:
4847:
4843:
4799:
4795:
4794:
4789:
4785:
4781:
4777:
4773:
4771:
4766:
4762:
4748:
4740:
4564:
4561:
4492:
4402:
4337:
4333:
4329:
4278:
4274:
4233:
4229:
4184:
4131:
4109:
3865:
3786:
3682:
3647:
3543:
3498:
3466:
3434:
3398:
3353:
3348:
3344:
3342:
3339:
3108:
2989:
2973:
2754:
2573:
2560:well-founded
2557:
2340:
2187:
2148:
2129:proof theory
2126:
1932:
1657:
1392:
1238:
1126:
1123:
909:, such that
860:
750:
703:
650:is written:
611:
527:
388:
374:
370:
368:
363:
355:
353:
344:
342:
274:
273:Because the
272:
267:
263:
259:
257:
192:
191:
183:set theories
180:
110:
99:
42:
38:
34:
30:
29:
6746:Type theory
6614:Type theory
6594:Determinacy
6536:Modal logic
6380:Four-valued
6350:Ćukasiewicz
6345:Four-valued
6332:Many-valued
6309:Description
6299:Dialetheism
6142:Ross Street
5667:Bibliopolis
5663:Bibliopolis
5565:predicative
5407:undecidable
5403:intensional
5399:extensional
2912:predicative
2151:linked list
1860:reflexivity
247:type theory
177:Type theory
86:predicative
74:extensional
70:intensional
66:philosopher
47:type theory
6725:Categories
6690:â-groupoid
6551:Set theory
6238:Fuzzy rule
6008:References
5472:homotopies
4447:). Since
2986:Judgements
279:empty type
6392:IEEE 1164
6243:Fuzzy set
6138:John Baez
5937:0956-7968
5862:CiteSeerX
5764:0960-1295
5747:1112.3456
5430:decidable
4802:) is the
4723:×
4577:×
4355:Γ
4352:⊢
4315:σ
4302:≡
4296:⊢
4293:Γ
4260:τ
4257:≡
4254:σ
4251:⊢
4248:Γ
4215:σ
4202:⊢
4199:Γ
4151:σ
4148:⊢
4145:Γ
4070:
4027:
3992:
3977:→
3966:×
3846:→
3078:∑
3040:∑
2735:∈
2637:Π
2631:Σ
2511:∏
2507:→
2478:→
2443:∏
2434:→
2360:⋅
2343:induction
2266:. Since
2247:→
2183:successor
2109:≠
2071:¬
2051:⊥
2048:→
2045:…
2025:⊥
2022:→
1993:⊥
1884:∏
1843:⋅
1785:⋅
1733:⋅
1695:⋅
1606:∏
1578:⋅
1359:∏
1332:→
1305:⟹
1274:→
1247:→
1205:
1179:∏
1072:∑
796:∑
771:∑
662:∑
631:×
578:
552:∑
510:∧
404:×
349:unit type
328:⊥
325:→
313:¬
289:⊥
230:⋅
150:→
123:⟹
6670:Category
6039:12731401
5945:19895964
5673:See also
5588:, ..., V
5510:such as
5373: :
5361: :
5164:. Here
5066:, where
5016:, a set
4874: :
4862: :
4780: :
2720:for any
2037:. Since
1490:of type
1417:of type
1259:. Thus,
958:of type
885:of type
534:sequence
303:negation
6385:Verilog
5894:1777213
5604:and BâV
5569:Russell
5520:Epigram
5516:Cayenne
5434:setoids
5351:
5319:
5315:
5277:
5261:
5211:
5207:
5166:
5150:
5112:
5104:
5072:
5056:
5018:
5014:
4976:
4972:
4940:
4840:
4808:
3349:objects
2181:or the
187:Frege's
59:Swedish
45:) is a
6408:Others
6110:
6089:
6068:
6037:
6027:
5943:
5935:
5892:
5882:
5864:
5772:416274
5770:
5762:
5711:
5652:MLTT79
5581:MLTT73
5561:MLTT72
5550:MLTT71
5526:, and
5464:points
5460:values
5444:, and
5152:, and
4938:a set
4672:. If
4358:
4154:
3955:
3866:Here,
3368:
2159:graphs
753:tuples
96:Design
6225:Fuzzy
6021:(PDF)
5941:S2CID
5890:S2CID
5810:(PDF)
5803:(PDF)
5768:S2CID
5742:arXiv
5740:(6).
5690:Notes
5634:and V
5528:Idris
5492:Nuprl
5468:paths
5419:Nuprl
4906:maps
3345:types
2980:Mahlo
2155:trees
538:reals
37:, or
6397:VHDL
6108:ISBN
6087:ISBN
6066:ISBN
6035:OCLC
6025:ISBN
5933:ISSN
5880:ISBN
5760:ISSN
5709:ISBN
5640:for
5524:Agda
5498:and
5379:D,Ap
5209:and
5106:and
4870:and
4652:and
4427:(or
4332:and
4277:and
3347:and
3259:and
2309:and
1873:refl
1774:and
1684:and
841:True
828:<
439:and
362:but
219:and
72:and
64:and
57:, a
43:MLTT
6559:Set
5923:doi
5872:doi
5837:doi
5752:doi
5571:'s
5512:ATS
5500:Coq
5474:),
5462:or
5428:is
5401:vs
5381:).
5160:to
5062:or
4913:to
4884:B'
4880:X'
4868:A'
4800:Set
4796:Fam
4792:).
4790:Set
4786:Fam
4765:or
4692:or
4435:Set
4067:add
4024:add
3989:add
3941:add
3763:in
3624:in
1202:Vec
575:Vec
536:of
364:not
6727::
6033:.
5939:.
5931:.
5919:23
5917:.
5911:.
5888:.
5878:.
5870:.
5831:.
5827:.
5766:.
5758:.
5750:.
5738:24
5736:.
5732:.
5659:.
5648:.
5644:â
5530:.
5522:,
5518:,
5514:,
5478:.
5440:,
5375:Tm
5369:,
5365:â
5263:.
5064:af
5060:Af
4927:.
4892:=
4878:â
4866:â
4854:â
4850::
4784:â
4753:,
4501:El
4479:El
3926::
3351:.
2971:.
2752:.
2157:,
2124:.
1862::
1441:,
748:.
525:.
393:,
351:.
340:.
319::=
92:.
6464:e
6457:t
6450:v
6171:e
6164:t
6157:v
6116:.
6095:.
6074:.
6041:.
5947:.
5925::
5896:.
5874::
5845:.
5839::
5833:4
5774:.
5754::
5744::
5717:.
5646:j
5642:i
5637:j
5631:i
5625:i
5616:n
5612:m
5607:n
5601:m
5591:n
5586:0
5377:(
5371:q
5367:G
5363:D
5359:p
5355:D
5339:)
5336:A
5333:,
5330:G
5327:(
5303:)
5300:G
5297:(
5294:y
5291:T
5288::
5285:A
5273:G
5268:C
5249:)
5246:f
5243:A
5240:,
5237:D
5234:(
5231:m
5228:T
5225::
5222:f
5219:a
5195:)
5192:D
5189:(
5186:y
5183:T
5180::
5177:f
5174:A
5162:G
5158:D
5154:f
5138:)
5135:A
5132:,
5129:G
5126:(
5123:m
5120:T
5108:a
5092:)
5089:G
5086:(
5083:y
5080:T
5068:A
5044:)
5041:A
5038:,
5035:G
5032:(
5029:m
5026:T
5002:)
4999:G
4996:(
4993:y
4990:T
4987::
4984:A
4960:)
4957:G
4954:(
4951:y
4948:T
4936:G
4932:T
4925:)
4923:a
4921:(
4919:g
4915:B
4910:a
4908:B
4904:f
4900:B
4897:°
4894:f
4890:g
4887:°
4876:X
4872:g
4864:A
4860:f
4856:A
4852:X
4848:B
4844:A
4828:)
4825:B
4822:,
4819:A
4816:(
4798:(
4788:(
4782:C
4778:T
4774:C
4726:B
4720:A
4700:B
4680:A
4660:B
4640:A
4620:B
4600:A
4580:B
4574:A
4545:U
4523:A
4457:U
4413:U
4381:t
4378:x
4375:e
4372:t
4369:n
4366:o
4363:C
4338:Ï
4334:u
4330:t
4310::
4305:u
4299:t
4279:Ï
4275:Ï
4234:Ï
4230:t
4210::
4205:t
4185:Ï
4168:e
4165:p
4162:y
4159:T
4117:S
4091:)
4088:)
4085:)
4082:b
4079:,
4076:a
4073:(
4064:(
4061:S
4058:=
4051:)
4048:b
4045:,
4042:)
4039:a
4036:(
4033:S
4030:(
4017:b
4014:=
4007:)
4004:b
4001:,
3998:0
3995:(
3981:N
3974:)
3970:N
3962:N
3958:(
3950::
3914:S
3894:S
3874:S
3850:N
3842:N
3836::
3831:S
3809:N
3803::
3798:0
3783:.
3771:b
3751:a
3731:x
3711:]
3708:a
3704:/
3700:x
3697:[
3694:b
3668:b
3665:]
3662:x
3659:[
3644:.
3632:B
3612:a
3592:x
3572:]
3569:a
3565:/
3561:x
3558:[
3555:B
3529:B
3526:)
3523:A
3518::
3513:x
3510:(
3484:B
3481:=
3478:A
3452:b
3449:=
3446:a
3420:A
3415::
3410:a
3382:e
3379:p
3376:y
3373:T
3365:A
3325:0
3322:S
3319:S
3316:S
3313:S
3293:4
3273:2
3270:+
3267:2
3247:4
3227:2
3224:+
3221:2
3218:=
3215:4
3195:B
3175:A
3155:B
3152:=
3149:A
3129:2
3126:+
3123:2
3120:=
3117:4
3093:B
3088:A
3085::
3082:a
3055:B
3050:A
3047::
3044:a
3018:B
2998:A
2957:1
2954:+
2951:k
2945:U
2922:k
2896:2
2890:U
2865:1
2859:U
2834:0
2828:U
2803:1
2797:U
2772:0
2766:U
2739:N
2730:n
2706:n
2700:U
2675:n
2669:U
2646:,
2643:=
2640:,
2634:,
2628:,
2625:2
2622:,
2619:1
2616:,
2613:0
2591:0
2585:U
2543:)
2540:n
2537:(
2534:P
2527:N
2520::
2515:n
2503:)
2499:)
2496:)
2493:n
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2487:S
2484:(
2481:P
2475:)
2472:n
2469:(
2466:P
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2452::
2447:n
2438:(
2430:)
2427:0
2424:(
2421:P
2416::
2408:m
2405:i
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2391:N
2364:)
2356:(
2353:P
2326:0
2323:S
2320:S
2317:S
2297:0
2294:S
2274:S
2252:N
2242:N
2235::
2230:S
2208:N
2201::
2196:0
2169:0
2112:2
2106:1
2086:)
2083:2
2080:=
2077:1
2074:(
2019:2
2016:=
2013:1
1973:2
1970:=
1967:1
1947:2
1944:=
1941:1
1918:.
1915:)
1912:a
1909:=
1906:a
1903:(
1898:A
1893::
1888:a
1878::
1846:2
1840:2
1837:=
1834:2
1831:+
1828:2
1808:4
1788:2
1782:2
1762:2
1759:+
1756:2
1736:2
1730:2
1727:=
1724:2
1721:+
1718:2
1698:2
1692:2
1672:2
1669:+
1666:2
1638:)
1635:n
1632:(
1629:P
1622:N
1615::
1610:n
1582:)
1574:(
1571:P
1551:n
1531:)
1528:n
1525:(
1522:P
1500:N
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1455:n
1452:(
1449:P
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1378:B
1373:A
1368::
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1301:A
1279:R
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1224:)
1221:n
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1195:N
1188::
1183:n
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1101:n
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1081::
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1025:n
1022:(
1019:P
999:)
996:n
993:(
990:P
968:N
946:n
926:)
923:n
920:(
917:P
895:N
873:n
845:)
837:=
834:)
831:n
825:m
822:(
819:(
812:Z
805::
800:n
787:Z
780::
775:m
734:R
712:n
687:R
678:N
671::
666:n
636:R
626:N
597:)
594:n
591:,
586:R
581:(
568:N
561::
556:n
513:B
507:A
487:B
467:A
447:B
427:A
407:B
401:A
375:0
371:1
356:2
345:1
322:A
316:A
275:0
268:2
264:1
260:0
249:.
233:2
227:2
207:2
204:+
201:2
153:B
147:A
127:B
119:A
20:)
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