## Some Normalization Properties of Martin-Lof's Type Theory, and Applications (1991)

Venue: | in Proc. 1st Internat. Conf. on Theoretical Aspects of Computer Software, Lecture Notes in Computer Science |

Citations: | 3 - 1 self |

### BibTeX

@INPROCEEDINGS{Basin91somenormalization,

author = {David Basin and Douglas J. Howe},

title = {Some Normalization Properties of Martin-Lof's Type Theory, and Applications},

booktitle = {in Proc. 1st Internat. Conf. on Theoretical Aspects of Computer Software, Lecture Notes in Computer Science},

year = {1991},

pages = {475--494},

publisher = {Springer}

}

### OpenURL

### Abstract

For certain kinds of applications of type theories, the faithfulness of formalization in the theory depends on intensional, or structural, properties of objects constructed in the theory. For type theories such as LF, such properties can be established via an analysis of normal forms and types. In type theories such as Nuprl or Martin-Lof's polymorphic type theory, which are much more expressive than LF, the underlying programming language is essentially untyped, and terms proved to be in types do not necessarily have normal forms. Nevertheless, it is possible to show that for Martin-Lof's type theory, and a large class of extensions of it, a sufficient kind of normalization property does in fact hold in certain well-behaved subtheories. Applications of our results include the use of the type theory as a logical framework in the manner of LF, and an extension of the proofs-as-programs paradigm to the synthesis of verified computer hardware. For the latter application we point out some ...

### Citations

716 | A framework for defining logics
- Harper, Honsell, et al.
- 1993
(Show Context)
Citation Context ...ithfulness of formalization in the theory depends on intensional, or structural, properties of objects constructed in the theory. One example of this is the use of a typed lambda calculus, such as LF =-=[10], as a &qu-=-ot;general logic". One can embed a natural deduction logic in the calculus by introducing typed constants for each rule and for each term and formula constructor of the logic. The constants are t... |

280 | Constructive mathematics and computer programming - Martin-Löf - 1982 |

109 |
et al. Implementing Mathematics with the Nuprl Proof Development System
- Constable
- 1986
(Show Context)
Citation Context ...ics. It contains a functional programming language and a highly expressive type system. Some of its descendents have been used in a number of interactive proof development systems (for example, Nuprl =-=[6]-=-, Veritas [8], Isabelle [16] and a system based on [15] currently under development). Nuprl, in particular, has been extensively used to formalize mathematics and programming. Our results should apply... |

101 | Equality in lazy computation systems
- Howe
(Show Context)
Citation Context ...oof given is for encodings of first-order logic. The remainder of the paper is organized as follows. In the next section we supply some of the necessary background, summarize the relevant sections of =-=[12, 13]-=- and briefly describe the semantic approach of [1]. We then state and sketch the proofs of the results referred to above. In the last section we describe some applications. 2 Background 2.1 Structured... |

56 |
A Non-Type-Theoretic Semantics for Type-Theoretic Language
- Allen
- 1987
(Show Context)
Citation Context ...hen T 0 is a type which is equal to T . If t 2 S T and t S t 0 then t = t 0 2 S T . TT is also open-ended with respect to universe inhabitation. This property is rather difficult to characterize (see =-=[2]-=-), but a weak form is sufficient here. In particular, we can incorporate an arbitrary collection of base types meeting the following restrictions. Let S be a model of TT, and let OE S be the type syst... |

37 |
Computational Metatheory in Nuprl
- Howe
- 1988
(Show Context)
Citation Context ...heory for the practical applications mentioned above. For example, within stronger type theories, the type theory itself can serve as a meta-language for developing derived inference rules and tactics=-=[11]-=-; this is not possible within the weaker LF type theory. Another benefit to using a rich type theory is that one can reason about parameterized terms where the parameters come from arbitrary (not nece... |

30 |
Hardware verification using Higher-Order Logic
- Camilleri, Gordon, et al.
- 1986
(Show Context)
Citation Context ...phic to a proof. See [10] for a discussion of the advantages of the LF approach to encoding logics. Another example is a widely used method for formalizing the combinational logic of digital circuits =-=[4]-=-. Here one represents a circuit as a relation over the booleans, so that a circuit with n ports (inputs and outputs) is represented as an n-ary relation which is true exactly when the ports have value... |

22 |
Formal synthesis of digital systems
- Hanna, Longley, et al.
- 1989
(Show Context)
Citation Context ...eous synthesis and verification of hardware can simplify the design process. This methodology can put to particularly good advantage when circuit terms can be constructed and reasoned about in parts. =-=[9]-=- and [7] give examples of this kind of development methodology applied to the top down synthesis of proven correct combinational logic and sequential circuits. [3] provides examples of using type theo... |

21 |
Programming in Martin-Lof's Type Theory, volume 7
- Nordstrom, Petersson, et al.
- 1990
(Show Context)
Citation Context ... a highly expressive type system. Some of its descendents have been used in a number of interactive proof development systems (for example, Nuprl [6], Veritas [8], Isabelle [16] and a system based on =-=[15]-=- currently under development). Nuprl, in particular, has been extensively used to formalize mathematics and programming. Our results should apply, with minor modifications, to the type theory of Nuprl... |

17 |
On computational open-endedness in Martin-Löf’s type theory
- Howe
- 1991
(Show Context)
Citation Context ... with a certain kind of structural operational semantics. The proofs of the technical lemmas involve a detailed analysis of these presentations. The technical work of this paper builds on the work in =-=[13]-=-, which introduces the operational semantics formalism and constructs the classical extension of TT. There are some advantages to being able to use a highly expressive type theory for the practical ap... |

14 |
Veritas+: A Specification Language Based on Type Theory
- Hanna, Daeche, et al.
- 1989
(Show Context)
Citation Context ...ins a functional programming language and a highly expressive type system. Some of its descendents have been used in a number of interactive proof development systems (for example, Nuprl [6], Veritas =-=[8]-=-, Isabelle [16] and a system based on [15] currently under development). Nuprl, in particular, has been extensively used to formalize mathematics and programming. Our results should apply, with minor ... |

11 |
NuPRL as a General Logic
- Constable, Howe
- 1990
(Show Context)
Citation Context ...r instantiating the parameters to produce a term representing a specific circuit. We provide an example of this kind of development at the end of this paper. Some closely related work is presented in =-=[5]-=-. The main result there is that LF-like encodings in Nuprl are faithful in the case where the encoded logic has a complete semantics that can be introduced into the Nuprl semantics. The notion of fait... |

11 |
Natural deduction proof as higher-order resolution
- Paulson
- 1986
(Show Context)
Citation Context ...al programming language and a highly expressive type system. Some of its descendents have been used in a number of interactive proof development systems (for example, Nuprl [6], Veritas [8], Isabelle =-=[16]-=- and a system based on [15] currently under development). Nuprl, in particular, has been extensively used to formalize mathematics and programming. Our results should apply, with minor modifications, ... |

9 |
A non-type theoretic definition of Martin-Lof's types
- Allen
- 1987
(Show Context)
Citation Context ...property by a metatheoretic analysis of TT's inference system. The rules of this type theory are too numerous and complex. Instead, we take a semantic approach, using the semantics developed by Allen =-=[1]-=-. In this semantics, the meaning of a type is a set of closed terms (together with an equivalence relation over the set). A term belongs in a type if and only if it evaluates to a value of the right f... |

6 |
Extracting circuits from constructive proofs
- Basin
- 1991
(Show Context)
Citation Context ...to a circuit. Providing a formal proof of the above specification would take us outside the scope of this paper; the reader may find details of proofs conducted in Nuprl for similar specifications in =-=[3]-=-. Such specifications are proved by induction and different proofs will construct terms representing different kinds of parameterized adders (e.g., ripple-carry, carry look ahead, etc.). As with softw... |

2 |
Formal methods for modelling design
- Fourman
- 1990
(Show Context)
Citation Context ...thesis and verification of hardware can simplify the design process. This methodology can put to particularly good advantage when circuit terms can be constructed and reasoned about in parts. [9] and =-=[7]-=- give examples of this kind of development methodology applied to the top down synthesis of proven correct combinational logic and sequential circuits. [3] provides examples of using type theory to sy... |