### Abstract

the introduction rules for A may also refer to B. So we have formation rules A ∶ Set, B ∶ A → Set and typical introduction rules might take the form a ∶ A b ∶ B(a)... introA(a, b,...) ∶ A a0 ∶ A b ∶ B(a0) a1 ∶ A...

### Citations

150 | Intuitionistic Type Theory, Bibliopolis - Martin-Löf - 1984 |

67 | A general formulation of simultaneous inductive-recursive definitions in type theory - Dybjer |

45 | Universes for generic programs and proofs in dependent type theory - Benke, Dybjer, et al. - 2003 |

44 | A finite axiomatization of inductiverecursive definitions
- Dybjer, Setzer
- 1999
(Show Context)
Citation Context ... (Our approach could also be straightforwardly extended to allow several constructors for A, where later constructors make use of earlier ones.) 4 An Axiomatisation We proceed as in Dybjer and Setzer =-=[9]-=- and introduce a datatype of codes for constructors. In other words, we define a type SP (for strictly positive) whose elements represent the inductively defined sets, together with a way to construct... |

40 | Internal type theory
- Dybjer
- 1996
(Show Context)
Citation Context ... b⟩), i.e. hangingUnder(⟨p, b⟩) ∶ Building(extension(⟨p, b⟩)). In other words, it is not possible to have a building hanging under the ground. Inductive-inductive definitions have been used by Dybjer =-=[7]-=-, Danielsson [5] and Chapman [4] to internalise the syntax and semantics of type theory. Slightly simplified, they define a set Ctxt of contexts, a family Ty ∶ Ctxt → Set of types in a given context, ... |

33 | On universes in type theory - Palmgren - 1998 |

29 | Induction-recursion and initial algebras - Dybjer, Setzer - 2003 |

23 | On relating type theories and set theories
- Aczel
- 2000
(Show Context)
Citation Context ...dinals i0 < i1 in order to interpret Set and Type. Our model will be a simpler version of the models developed in [9, 11]. Here we present the main ideas; more details can be found in [14]. See Aczel =-=[1]-=- for a more detailed treatment of interpreting type theory in set theory. For every expression A of our type theory, we will give an interpretation �A�ρ, which might be undefined. Open terms will be i... |

23 |
Inductive Families. Formal Aspects of Computing 6(4
- Dybjer
(Show Context)
Citation Context ...and in the case of B, we also need the index of the codomain of the constructor. From this, we can write down the introduction rules, and the elimination rules should be determined by these (see e.g. =-=[6]-=-). Thus, the codes in SPA and SPB will be codes for the domain of the constructors, and we will have functions Arg A, Arg B that map the code to the domain it represents. For B, there will also be a f... |

17 | Constructing Universes for Generic Programming - Morris - 2007 |

13 | A Formalisation of a Dependently Typed Language as an Inductive-Recursive Family
- Danielsson
- 2006
(Show Context)
Citation Context ...ngUnder(⟨p, b⟩) ∶ Building(extension(⟨p, b⟩)). In other words, it is not possible to have a building hanging under the ground. Inductive-inductive definitions have been used by Dybjer [7], Danielsson =-=[5]-=- and Chapman [4] to internalise the syntax and semantics of type theory. Slightly simplified, they define a set Ctxt of contexts, a family Ty ∶ Ctxt → Set of types in a given context, and a family Ter... |

11 |
Type theory should eat itself
- Chapman
(Show Context)
Citation Context ...∶ Building(extension(⟨p, b⟩)). In other words, it is not possible to have a building hanging under the ground. Inductive-inductive definitions have been used by Dybjer [7], Danielsson [5] and Chapman =-=[4]-=- to internalise the syntax and semantics of type theory. Slightly simplified, they define a set Ctxt of contexts, a family Ty ∶ Ctxt → Set of types in a given context, and a family Term ∶ (Γ ∶ Ctxt) →... |

11 |
Investigations into intensional type theory. Habilitiation Thesis
- Streicher
- 1993
(Show Context)
Citation Context ...vers what can be defined in Agda. However, just as for ordinary induction, we do not expect dependent pattern matching to follow from our elimination rules without the addition of Streicher’s Axiom K =-=[16]-=-. On the theoretical side, work is underway to show that inductive-inductive definitions can be reduced to indexed inductive definitions. This would show that the proof theoretical strength does not i... |

9 | Do-it-yourself type theory, Formal Aspects of Computing 1 - Backhouse, Chisholm, et al. - 1989 |

4 |
A.: Indexed induction–recursion. Journal of logic and algebraic programming 66(1
- Dybjer, Setzer
- 2006
(Show Context)
Citation Context ...theory can be constructed in ZFC set theory, extended by two strongly inaccessible cardinals i0 < i1 in order to interpret Set and Type. Our model will be a simpler version of the models developed in =-=[9, 11]-=-. Here we present the main ideas; more details can be found in [14]. See Aczel [1] for a more detailed treatment of interpreting type theory in set theory. For every expression A of our type theory, w... |

1 |
A.: Induction-induction: Agda development and extended version. http://cs.swan.ac.uk/~csfnf/induction-induction
- Forsberg, Setzer
- 2010
(Show Context)
Citation Context ...atform from it by building an extension. We can always build a building on top of any platform, and if we have an extension, we can also construct a building hanging from it. See the extended version =-=[14]-=- of this article for an illustration.) This gives rise to the following inductive-inductive definition of Platform ∶ Set, Building ∶ Platform → Set (where p ∶ Platform means that p is a platform and b... |