## Abstract

We present a principle for introducing new types in type theory which generalises strictly positive indexed inductive data types. In this new principle a set A is defined inductively simultaneously with an A-indexed set B, which is also defined inductively. Compared to indexed inductive definitions, the novelty is that the index set A is generated inductively simultaneously with B. In other words, we mutually define two inductive sets, of which one depends on the other.

Instances of this principle have previously been used in order to formalise type theory inside type theory. However the consistency of the framework used (the theorem prover Agda) is not so clear, as it allows the definition of a universe containing a code for itself. We give an axiomatisation of the new principle in such a way that the resulting type theory is consistent, which we prove by constructing a set-theoretic model.

Instances of this principle have previously been used in order to formalise type theory inside type theory. However the consistency of the framework used (the theorem prover Agda) is not so clear, as it allows the definition of a universe containing a code for itself. We give an axiomatisation of the new principle in such a way that the resulting type theory is consistent, which we prove by constructing a set-theoretic model.

Original language | English |
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Title of host publication | Computer Science Logic |

Subtitle of host publication | 24th International Workshop, CSL 2010, 19th Annual Conference of the EACSL, Brno, Czech Republic, August 23-27, 2010. Proceedings |

Editors | Anuj Dawar, Helmut Veith |

Place of Publication | Berlin |

Pages | 454-468 |

Number of pages | 15 |

DOIs | |

Publication status | Published - 11 Aug 2010 |

Event | 24th International Workshop on Computer Science Logic, CSL 2010, and 19th Annual Conference of the EACSL - Brno, Czech Republic Duration: 23 Aug 2010 → 27 Aug 2010 |

### Publication series

Name | Lecture Notes in Computer Science |
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Publisher | Springer Berlin Heidelberg |

Volume | 6247 |

ISSN (Print) | 0302-9743 |

### Conference

Conference | 24th International Workshop on Computer Science Logic, CSL 2010, and 19th Annual Conference of the EACSL |
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Country | Czech Republic |

City | Brno |

Period | 23/08/10 → 27/08/10 |

## Keywords

- axiomatisation
- data type
- inductive definitions
- theorem provers
- theoretical model
- type theory