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started as a study into the limits of ITT. Once found, the limits were turned into rules that allowed defining new inductive types. These types could depend upon a function and the function on the type, as long as both were defined simultaneously.
1977:(HoTT). HoTT differs from ITT by its identity type (equality). Higher inductive types not only define a new type with constants and functions that create elements of the type, but also new instances of the identity type that relate them.
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In words: for any predicate "P" over natural numbers, given a proof of "P 0" and a proof of "P n -> P (n+1)", we get back a proof of "forall n, P n". This is the familiar induction principle for natural numbers.
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1664:{\displaystyle {\frac {w:{\mathsf {W}}_{a:A}B(a)\qquad a:A,\;f:B(a)\to {\mathsf {W}}_{x:A}B(x),\;c:\prod _{b:B(a)}C(f(b))\;\vdash \;h(a,f,c):C({\mathsf {sup}}(a,f))}{{\mathsf {elim}}(w,h):C(w)}}}
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1685:. In this case, the property of initiality (res. finality) corresponds directly to the appropriate induction principle. In intensional type theories with the
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827:{\displaystyle {\begin{aligned}f(\operatorname {inl} (1_{\mathbf {1} }))&=\mathbf {0} \\f(\operatorname {inr} (a))&=\mathbf {1} \end{aligned}}}
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of each constructor. W-types (resp. M-types) may also be understood as well-founded (resp. non-well-founded) trees with nodes labeled by elements
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if it has facilities for creating a new type from constants and functions that create terms of that type. The feature serves a role similar to
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Here, a natural number is created either from the constant "0" or by applying the function "S" to another natural number. "S" is the
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which represents adding 1 to a number. Thus, "0" is zero, "S 0" is one, "S (S 0)" is two, "S (S (S 0))" is three, and so on.
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Inductive types usually come with a function to prove properties about them. Thus, "nat" may come with (in Coq syntax):
1132:{\displaystyle {\mathsf {sup}}:\prod _{a:A}{\Big (}B(a)\to {\mathsf {W}}_{x:A}B(x){\Big )}\to {\mathsf {W}}_{x:A}B(x).}
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1290:{\displaystyle {\frac {a:A\qquad f:B(a)\to {\mathsf {W}}_{x:A}B(x)}{{\mathsf {sup}}(a,f):{\mathsf {W}}_{x:A}B(x)}}.}
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Since their introduction, inductive types have been extended to encode more and more structures, while still being
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Awodey, Steve; Gambino, Nicola; Sojakova, Kristina (2015-04-21). "Homotopy-initial algebras in type theory".
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Awodey, Steve; Gambino, Nicola; Sojakova, Kristina (2012-01-18). "Inductive types in homotopy type theory".
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Dybjer, Peter (1997). "Representing inductively defined sets by wellorderings in Martin-Löf's type theory".
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149:. As the name suggests, inductive types can be self-referential, but usually only in a way that permits
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376:(ITT). They generalize natural numbers, lists, binary trees, and other "tree-shaped" data types. Let
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holds for all subtrees of a given tree it also holds for that tree, then it holds for all trees.
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2198:. Leibniz International Proceedings in Informatics (LIPIcs). Vol. 38. pp. 17–30.
2097:. Leibniz International Proceedings in Informatics (LIPIcs). Vol. 38. pp. 17–30.
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van den Berg, Benno; Marchi, Federico De (2007). "Non-well-founded trees in categories".
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definitions of multiple types that depend on each other. For example, defining two
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488:)-many subtrees. Each W-type is isomorphic to the initial algebra of a so-called
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allows definition of a type and a family of types at the same time. So, a type
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717:{\displaystyle \operatorname {List} (A):={\mathsf {W}}_{(x:\mathbf {1} +A)}f(x)}
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In extensional type theories, W-types (resp. M-types) can be defined up to
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corresponds to the constructor for the empty list, whereas the value of
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in a programming language and allows a type theory to add concepts like
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1689:, this correspondence holds up to homotopy (propositional equality).
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Ahrens, Benedikt; Capriotti, Paolo; Spadotti, Régis (2015-04-12).
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Ahrens, Benedikt; Capriotti, Paolo; Spadotti, Régis (2015-04-12).
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632:(representing the successor function, which takes one argument).
585:{\displaystyle \mathbb {N} :={\mathsf {W}}_{x:\mathbf {2} }f(x)}
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The existence of a new constructor for the identity type makes
22:
2313:
Abbott, Michael; Altenkirch, Thorsten; Ghani, Neil (2005).
1984:
type, which is defined with two constructors, a basepoint;
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Homotopy Type Theory: Univalent
Foundations of Mathematics
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permits (effectively) infinite structures in type theory.
887:{\displaystyle f(\operatorname {inl} (1_{\mathbf {1} }))}
533:, etc. One may define the natural numbers as the W-type
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The elimination rule for W-types works similarly to
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2383:Higher Inductive Types: a tour of the menagerie
1361:{\displaystyle C:{\mathsf {W}}_{x:A}B(x)\to U}
1139:We can also write this rule in the style of a
2196:Non-wellfounded trees in Homotopy Type Theory
2095:Non-wellfounded trees in Homotopy Type Theory
1303:on trees. If, whenever a property (under the
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2064:. Sambin, Giovanni. Napoli: Bibliopolis.
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113:Learn how and when to remove this message
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1704:. M-types can be derived from W-types.
984:{\displaystyle {\mathsf {W}}_{x:A}B(x)}
449:{\displaystyle {\mathsf {W}}_{a:A}B(a)}
2355:Univalent Foundations Program (2013).
1724:using two mutually inductive types in
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1973:This is a current research area in
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635:One may define lists over a type
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1681:(resp. final coalgebras) for
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2148:Theoretical Computer Science
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1958:{\displaystyle B:A\to Type}
1696:to W-types, they represent
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2378:Induction-Induction Slides
2373:Induction-Recursion Slides
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2058:Intuitionistic type theory
840:is the sole inhabitant of
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2055:Martin-Löf, Per (1984).
1980:A simple example is the
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664:
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623:
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604:is defined by
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364:W- and M-types
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33:
26:
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10:
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64: –
63:
59:
58:Find sources:
52:
48:
42:
41:
36:This article
34:
30:
25:
24:
19:
2357:
2322:
2318:
2308:
2283:math/0409158
2276:(1): 40–59.
2273:
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2195:
2189:
2168:
2151:
2147:
2141:
2094:
2088:
2057:
2050:
2020:
2013:
2009:
2005:
2000:and a loop;
1999:
1993:
1989:
1979:
1972:
1916:
1905:
1898:
1757:zero_is_even
1713:
1711:
1692:M-types are
1691:
1672:
1370:
1298:
941:
841:
836:
634:
629:
624:
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597:
530:
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519:
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370:well-founded
368:W-types are
367:
353:
241:
229:
222:
168:as follows:
155:
130:
124:
109:
103:January 2018
100:
90:
83:
76:
69:
57:
45:Please help
40:verification
37:
2398:Type theory
2325:(1): 3–27.
2035:Coinduction
1698:coinductive
1675:isomorphism
456:. The type
238:Elimination
232:predicative
127:type theory
2254:1504.05531
2205:1504.02949
2104:1504.02949
2071:8870881059
2042:References
73:newspapers
2327:CiteSeerX
2180:1201.3898
1941:→
1733:Inductive
1543:⊢
1497:∏
1451:→
1353:→
1182:→
1092:→
1050:→
1016:∏
991:has type
911:
861:
794:
745:
654:
372:types in
173:Inductive
143:relations
2392:Category
2234:15020752
2133:15020752
2080:12731401
2029:See also
2008: :
1992: :
639: :
596: :
513: :
472: :
399: :
388: :
247:nat_elim
1702:streams
1143:proof,
139:numbers
87:scholar
2329:
2300:360990
2298:
2232:
2222:
2131:
2121:
2078:
2068:
2023:circle
1994:circle
1982:circle
1853:forall
1781:forall
1718:parity
724:where
392:and a
334:forall
295:forall
256:forall
160:using
145:, and
89:
82:
75:
68:
60:
2296:S2CID
2278:arXiv
2249:arXiv
2230:S2CID
2200:arXiv
2175:arXiv
2129:S2CID
2099:arXiv
2062:(PDF)
1874:->
1832:->
1802:->
1745:->
834:and 1
592:with
466:arity
380:be a
328:->
310:->
289:->
268:->
212:->
147:trees
94:JSTOR
80:books
2220:ISBN
2119:ISBN
2076:OCLC
2066:ISBN
2014:base
2010:base
2006:loop
1990:base
1868:even
1835:Prop
1820:with
1805:even
1763:even
1748:Prop
1736:even
1714:some
1694:dual
651:List
628:) =
614:) =
495:Let
480:has
271:Prop
182:Type
66:news
2337:doi
2323:342
2288:doi
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2210:doi
2156:doi
2152:176
2109:doi
1889:)).
1877:odd
1862:nat
1829:nat
1823:odd
1796:odd
1790:nat
1742:nat
1726:Coq
1677:as
908:inr
858:inl
791:inr
742:inl
643:as
523:, 2
517:, 1
349:)).
265:nat
215:nat
209:nat
197:nat
176:nat
166:Coq
125:In
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