## Constructing strictly positive families (2007)

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Venue: | In The Australasian Theory Symposium (CATS2007 |

Citations: | 11 - 4 self |

### BibTeX

@INPROCEEDINGS{Morris07constructingstrictly,

author = {Peter Morris and Thorsten Altenkirch},

title = {Constructing strictly positive families},

booktitle = {In The Australasian Theory Symposium (CATS2007},

year = {2007}

}

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### Abstract

We present an inductive definition of a universe containing codes for strictly positive families (SPFs) such as vectors or simply typed lambda terms. This construction extends the usual definition of inductive strictly positive types as given in previous joint work with McBride. We relate this to Indexed Containers, which were recently proposed in joint work with Ghani, Hancock and McBride. We demonstrate by example how dependent types can be encoded in this universe and give examples for generic programs.

### Citations

354 |
Intuitionistic Type Theory
- Martin-Löf
- 1984
(Show Context)
Citation Context ...universe and give examples for generic programs. Keywords: datatypes, containers, universes, generic programming, dependent types, Epigram 1. INTRODUCTION In a dependently typed language like Epigram =-=[9, 8, 5]-=- generic programming is normal programing. This is achieved by defining a universe [?, ?] consisting of a type of names U : ⋆ and a family of elements El : U → ⋆ indexed by type names. We have exploit... |

274 | Programming in Martin-Löf’s Type Theory: an introduction - Nordström, Petersson, et al. - 1990 |

164 |
The view from the left
- McBride, McKinna
(Show Context)
Citation Context ... normal programing. This is achieved by defining a universe [?, ?] consisting of a type of names U : ⋆ and a family of elements El : U → ⋆ indexed by type names. We have exploited this opportunity in =-=[10]-=- by defining the universe of regular tree types and developing generic programs and proofs for this universe. However, there is an obvious asymmetry in our previous definitions where we exploit the po... |

149 | Intuitionistic type theory," Bibliopolis - Martin-Lof - 1984 |

65 |
Containers - constructing strictly positive types
- Abbott, Altenkirch, et al.
- 2005
(Show Context)
Citation Context ...family of types by an inductive definition, e.g. by giving the constructions under which the universe is closed we can alternatively give a semantic characterisation of a class of types by containers =-=[1]-=-. A simple 1-ary container is given by a type S : ⋆ of shapes and a family of positions P : S → ⋆. To every 1-ary container we can assign an endofunctor on the category of types. In loc.cit. we show t... |

63 | A semantics for shape
- Jay
- 1995
(Show Context)
Citation Context ...inductive recursive definitions. This has been the base for generic programming within the AGDA system (Benke, Dybjer & Jansson 2003). Containers are a natural generalisation of the shapely types of (=-=Jay 1995-=-) while indexed containers have been studied under the name of dependent polynomials in (Gambino & Hyland 2004). Structure of the Paper: The rest of the paper is structured as follows: Since our use o... |

45 | Universes for generic programs and proofs in dependent type theory
- Benke, Dybjer, et al.
(Show Context)
Citation Context ...container. In related work, [7] presents an alternative approach to defining a universe of indexed strictly positive definitions. This has been the base for generic programming within the AGDA system =-=[6]-=-. Mathematically Structured Functional Programming 1sConstructing Strictly Positive Families 2. STRICTLY POSITIVE FAMILIES, REGULAR FAMILIES We start by defining the type of strictly positive families... |

45 | Indexed induction-recursion
- Dybjer, Setzer
- 2001
(Show Context)
Citation Context ...hose objects are families O → ⋆). In the present paper we relate the two approaches showing that any strictly positive type gives rise to a semantically equivalent indexed container. In related work, =-=[7]-=- presents an alternative approach to defining a universe of indexed strictly positive definitions. This has been the base for generic programming within the AGDA system [6]. Mathematically Structured ... |

28 | Wellfounded trees and dependent polynomial functors. available from °±fl±nuQ´fl´µ£µflµu¶·n¸fl¸d¶Qº'» ¸T¶»rºd¶?r´¿flÀrÁnÂflÂ
- Gambino, Hyland
- 2004
(Show Context)
Citation Context ...ystem (Benke, Dybjer & Jansson 2003). Containers are a natural generalisation of the shapely types of (Jay 1995) while indexed containers have been studied under the name of dependent polynomials in (=-=Gambino & Hyland 2004-=-). Structure of the Paper: The rest of the paper is structured as follows: Since our use of dependent types, universes and containers are a novel approach to the theory of data types, we begin in sect... |

18 | Exploring the regular tree types - Morris, Altenkirch, et al. - 2006 |

15 |
Why dependent types matter. Manuscript, available online. http://www.cs.nott.ac.uk/∼txa/publ/ydtm.pdf
- Altenkirch, McBride, et al.
- 2005
(Show Context)
Citation Context ...universe and give examples for generic programs. Keywords: datatypes, containers, universes, generic programming, dependent types, Epigram 1. INTRODUCTION In a dependently typed language like Epigram =-=[9, 8, 5]-=- generic programming is normal programing. This is achieved by defining a universe [?, ?] consisting of a type of names U : ⋆ and a family of elements El : U → ⋆ indexed by type names. We have exploit... |

9 |
Towards observational type theory. Manuscript, available online
- Altenkirch, McBride
- 2006
(Show Context)
Citation Context ...tem. This approach also has the benefit that it allows a more flexible and extensible positivity test as we have demonstrated in the example of n-branching trees. Exploiting Observational Type Theory =-=[4]-=- we are also planning to include coinductive definitions in the universe. It turns out that the Epigram ‘gadgets’ that build the structural recursion, and case analysis principals (⇐ rec and ⇐ case) f... |

6 |
Indexed containers. Manuscript, available online
- Altenkirch, Ghani, et al.
- 2006
(Show Context)
Citation Context ...lternative access to generic programming: a generic program is simply one which works on all containers. We have exploited this approach to study derivatives of datatypes [2]. In yet unpublished work =-=[3]-=- we have generalized containers to indexed containers which capture dependent types. Given types I , O : ⋆ an indexed container IC I O represents an endofunctor form the slice category over I (whose o... |