## Representing Nested Inductive Types Using W-types

Citations: | 6 - 3 self |

### BibTeX

@MISC{Abbott_representingnested,

author = {Michael Abbott and Thorsten Altenkirch and Neil Ghani},

title = {Representing Nested Inductive Types Using W-types},

year = {}

}

### OpenURL

### Abstract

We show that strictly positive inductive types, constructed from polynomial functors, constant exponentiation and arbitrarily nested inductive

### Citations

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(Show Context)
Citation Context ...bra of F[−] :GI → GI. We will show in this paper that the functor μ :GI+1 → GI exists, and that the initial algebra of a container functor is a container functor. W-Types In Martin-Löf’s Type Theory (=-=Martin-Löf, 1984-=-; Nordström et al., 1990) the building block for inductive constructions is the W-type. Given a family of constructors A ⊢ B the type Wa : A.B(a) (or WAB) should be regarded as the type of “well found... |

185 |
Topos Theory
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Citation Context ...ly there cannot be any infinite paths. The structure of the functor X ↦→ P + ∑Q ε ∗ X respects the structure of the initial algebra ψ, which forces B to be unique. Compare this with Wraith’s theorem (=-=Johnstone, 1977-=-, theorem 6.19), for the special case A = N. Of course, it remains to prove the hypothesis of the theorem above, that a family A ⊢ B with the given isomorphism ϕ exists; we do this below in propositio... |

43 | Indexed induction-recursion - Dybjer, Setzer - 2006 |

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40 | Categories of containers
- Abbott, Altenkirch, et al.
- 2003
(Show Context)
Citation Context ...pair (u, f) for u : A → C in C and f :(u ∗ ) I D → B in (C/A) I .sNote that the alternative of defining an n+1-ary container as an indexed family of n-ary containers is equivalent to this definition (=-=Abbott, 2003-=-, proposition 4.1.1). A container (A ⊲ B) ∈ GI can be written using type theoretic notation as ⊢ A i : I, a : A ⊢ Bi(a) . A morphism (u, f):(A ⊲ B) → (C ⊲ D) can be written in type theoretic notation ... |

40 | Syntax and semantics of dependent types - Hofmann - 1997 |

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35 | Categorical Logic and Type Theory. Number 141 - Jacobs - 1999 |

31 | Handbook of categorical algebra. 2, volume 51 of Encyclopedia of Mathematics and its Applications - Borceux - 1994 |

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15 | Representing inductively defined sets by wellorderings in Martin-Löf’s type theory - Dybjer - 1997 |

13 | Least fixed point of a functor - Adámek, Koubek - 1979 |

8 | A predicative strong normalisation proof for a λcalculus with interleaving inductive types
- Abel, Altenkirch
- 2000
(Show Context)
Citation Context ...ctive Types We now have enough machinery in place to observe that all strictly positive types can be described as containers. Definition 6.1. A strictly positive inductive type (SPIT) in n variables (=-=Abel and Altenkirch, 2000-=-) is a type expression (with type variables X1,...,Xn) built up inductively according to the following rules: – if K is a constant type (with no type variables) then K is a SPIT; – each type variable ... |

6 | Wellfounded trees in categories. Annals of pure and applied logic - Moerdijk, Palmgren - 2000 |

1 |
Representing strictly positive types. Presented at APPSEM annual meeting, invited for submission to Theoretical Computer Science
- Abbott, Altenkirch, et al.
- 2004
(Show Context)
Citation Context ...bras, to cover non-well founded data structures such as streams (Stream A = νX.A × X), which are used extensively in lazy functional programming. We have also established (see Abbott, 2003, p. 78 and =-=Abbott et al., 2004-=-), that Martin-Löf categories are closed under ν-types—this can be reduced to constructing the dual of W-types which we dub M-types. Another interesting extension would be to consider inductive and co... |