## The structure of nuprl’s type theory (1997)

Citations: | 9 - 3 self |

### BibTeX

@MISC{Constable97thestructure,

author = {Robert L. Constable},

title = {The structure of nuprl’s type theory},

year = {1997}

}

### Years of Citing Articles

### OpenURL

### Abstract

on the World Wide Web (\the Web") (www.cs.cornell.edu/Info/NuPrl/nuprl.html)

### Citations

454 | The formulas-as-types notion of construction - Howard - 1980 |

421 | Constructive analysis
- Bishop, Bridges
- 1985
(Show Context)
Citation Context ...make them understandable. 5 Conclusion At the summer school I presented the details of Max Forester's constructive proof of the Intermediate Value Theorem [20] which was taken from Bishop and Bridges =-=[9]-=-. I also discussed the stamps problem from the Nuprl 4.2 library. I related this to Sam Buss' account of feasible arithmetic by using the efficient induction tactic (complete nat ind with y at the end... |

347 | Intuitionistic Type Theory - Martin-Löf - 1984 |

272 |
Constructive mathematics and computer programming
- Martin-Lof
- 1982
(Show Context)
Citation Context ...verse of sets. In type theory we allow a different class of predicates --- those involving predicative higher-order logic in a sense. This topic is discussed in many articles and books on type theory =-=[33, 13, 36, 43, 12, 11]-=- and is beyond the scope of this article, so here we will just assume that the reader is familiar with one account of propositionsas -types or representing logic in type theory. The Nuprl style is to ... |

268 | Foundations of Constructive Mathematics - Beeson - 1985 |

176 | An Introduction to Inductive Definitions - Aczel - 1977 |

151 |
Proving and applying program transformations expressed with secondorder patterns
- Huet, Lang
(Show Context)
Citation Context ... j an index and the f i an index family name. alternatives --- calculus with operator names Another representation of terms that is natural is based on using the lambda calculus to define all binding =-=[30]-=-. For example, 8x : A: B(x) would be represented as all(x : A: B(x)). And spread(p; u; v: t) would be spread(p; (u; v: t)). This approach is natural in Lisp where terms are coded as S-expressions, but... |

119 | The type theoretic interpretation of constructive set theory, Logic colloquium ’77 - Aczel - 1978 |

106 | Proofs and Types. Cambridge Tracts - Girard, Taylor, et al. - 1989 |

97 | Equality in lazy computation systems - Howe - 1989 |

94 | A survey of the project Automath - Bruijn - 1980 |

90 | A language and axioms for explicit mathematics - Feferman - 1975 |

77 | Inductive sets and families in Martin-Löf’s Type Theory and their set-theoretic semantics: An inversion principle for Martin-Löf’s type theory - Dybjer - 1991 |

75 |
Implementing mathematics with the NuPrl development system
- Constable, Allen, et al.
- 1986
(Show Context)
Citation Context ...verse of sets. In type theory we allow a different class of predicates --- those involving predicative higher-order logic in a sense. This topic is discussed in many articles and books on type theory =-=[33, 13, 36, 43, 12, 11]-=- and is beyond the scope of this article, so here we will just assume that the reader is familiar with one account of propositionsas -types or representing logic in type theory. The Nuprl style is to ... |

70 |
Notes on data structuring
- Hoare
- 1972
(Show Context)
Citation Context ...tivated both by mathematical and computational considerations. So we will see a tight relationship to the notion of type in programming languages. The notes by C.A.R. Hoare, Notes on Data Structuring =-=[23]-=-, make the point well. 2.2 Cartesian Products If A and B are types, then so is their product, written A \Theta B. There will be many formation rules of this form, so we adopt a simple convention for s... |

63 | Metamathematical investigations of a calculus of constructions - Coquand - 1990 |

57 | Program extraction from classical proofs - Berger, Schwicthenberg - 1995 |

56 |
A Non- Type-Theoretic Semantics for Type-Theoretic Language
- Allen
- 1987
(Show Context)
Citation Context ...Type Theory In this section we look at some features of the Nuprl type theory. The first section is a discussion of the uniform syntax of Nuprl 4 terms. The second section considers Allen's semantics =-=[4, 3]-=- for Nuprl without recursive types. Mendler [35] provides a semantics for recursive types as well, but it is more involved than what we present here. 4.1 Nuprl Term Syntax Following Frege, Church, and... |

46 | Enhancing the Nuprl Proof Development System and Applying it to Computational Abstract Algebra
- Jackson
- 1995
(Show Context)
Citation Context ...rial both at the Nuprl home page on the World Wide Web ("the Web") (www.cs.cornell.edu/Info/NuPrl/nuprl.html) and in publications [22], some soon to appear [11]. At the Web site the thesis o=-=f Jackson [31]-=- and the article by Forester [20] are especially relevant to my lecture. Also, the 1986 Nuprl book is now available on line at the Web site as is the Nuprl 4 reference manual and a host of Nuprl libra... |

33 |
A non-type-theoretic definition of Martin-Lof's types
- Allen
- 1987
(Show Context)
Citation Context ...Type Theory In this section we look at some features of the Nuprl type theory. The first section is a discussion of the uniform syntax of Nuprl 4 terms. The second section considers Allen's semantics =-=[4, 3]-=- for Nuprl without recursive types. Mendler [35] provides a semantics for recursive types as well, but it is more involved than what we present here. 4.1 Nuprl Term Syntax Following Frege, Church, and... |

30 | Semantic foundations for embedding HOL in Nuprl - Howe - 1996 |

20 | Inductively defined types (preliminary version - Coquand, Paulin - 1990 |

17 | The polynomial hierarchy and intuitionistic bounded arithmetic, Structure in Complexity Theory - Buss - 1986 |

15 |
Types in logic, mathematics, and programming
- Constable
(Show Context)
Citation Context ... and I made available much more related material both at the Nuprl home page on the World Wide Web ("the Web") (www.cs.cornell.edu/Info/NuPrl/nuprl.html) and in publications [22], some soon =-=to appear [11]-=-. At the Web site the thesis of Jackson [31] and the article by Forester [20] are especially relevant to my lecture. Also, the 1986 Nuprl book is now available on line at the Web site as is the Nuprl ... |

14 | Intrinsic theories and computational complexity - Leivant - 1995 |

12 | Reasoning about functional programs in Nuprl - Howe |

11 | Using reflection to explain and enhance type theory
- Constable
- 1994
(Show Context)
Citation Context ...verse of sets. In type theory we allow a different class of predicates --- those involving predicative higher-order logic in a sense. This topic is discussed in many articles and books on type theory =-=[33, 13, 36, 43, 12, 11]-=- and is beyond the scope of this article, so here we will just assume that the reader is familiar with one account of propositionsas -types or representing logic in type theory. The Nuprl style is to ... |

10 | A uniform approach to type theory - Huet - 1990 |

10 | An introduction to inductive de nitions - Aczel - 1977 |

8 |
Formalizing constructive mathematics: why and how
- Beeson
- 1980
(Show Context)
Citation Context ....e. the integers with equality taken mod n so that x = y mod n iff n divides (x \Gamma y). The only difference between Zand Z=mod n is the equality relation on the integer constants. Following Beeson =-=[6]-=- we might speak of the constants without an equality relation as a pre-type. Then a type arises by pairing an equality with a pre-type, say hT; Ei where E is an equivalence relation on T . But when we... |

6 | Do-it-yourself type theory (part I - Backhouse, Chisholm, et al. - 1989 |

5 | Formalizing constructive real analysis
- Forester
- 1993
(Show Context)
Citation Context ...on the World Wide Web ("the Web") (www.cs.cornell.edu/Info/NuPrl/nuprl.html) and in publications [22], some soon to appear [11]. At the Web site the thesis of Jackson [31] and the article by=-= Forester [20]-=- are especially relevant to my lecture. Also, the 1986 Nuprl book is now available on line at the Web site as is the Nuprl 4 reference manual and a host of Nuprl libraries. As a result of the ready ac... |

5 |
A non-type-theoretic de nition of Martin-Lof's types
- Allen
- 1987
(Show Context)
Citation Context ...l Type Theory In this section we look at some features of the Nuprl type theory. The rst section is a discussion of the uniform syntax of Nuprl 4 terms. The second section considers Allen's semantics =-=[4, 3]-=- for Nuprl without recursive types. Mendler [35] provides a semantics for recursive types as well,butitismoreinvolved than what we present here. 154.1 Nuprl Term Syntax Following Frege, Church, and M... |

4 | An operational approach to combining classical set theory and functional programming languages - Howe, Stoller - 1994 |

3 |
Objects and theories as very dependent types
- Hickey
- 1996
(Show Context)
Citation Context ...rktoberdorf, my colleagues and I made available much more related material both at the Nuprl home page on the World Wide Web ("the Web") (www.cs.cornell.edu/Info/NuPrl/nuprl.html) and in pub=-=lications [22]-=-, some soon to appear [11]. At the Web site the thesis of Jackson [31] and the article by Forester [20] are especially relevant to my lecture. Also, the 1986 Nuprl book is now available on line at the... |

2 |
Using re ection to explain and enhance type theory
- Constable
- 1994
(Show Context)
Citation Context ...e universe of sets. In type theory we allowadi erent class of predicates | those involving predicative higher-order logic in a sense. This topic is discussed in many articles and books on type theory =-=[33, 13, 36, 43, 12, 11]-=- and is beyond the scope of this article, so here we will just assume that the reader is familiar with one account of propositionsas-types or representing logic in type theory. The Nuprl style is to u... |

2 | Inductively de ned types, preliminary version - Coquand, Paulin - 1990 |

1 |
Formalizing constructive mathematics: Why andhow
- Beeson
- 1981
(Show Context)
Citation Context ...egers, i.e. the integers with equality taken mod n so that x = y mod n i n divides (x ; y). The only di erence between Zand Z=mod n is the equality relation on the integer constants. Following Beeson =-=[6]-=- we might speak of the constants without an equality relation as a pre-type. Then a type arises by pairing an equality with a pre-type, say hT�Ei where E is an equivalence relation on T . But when we ... |

1 | The polynomial hierarchy andintuitionistic bounded arithmetic - Buss - 1986 |

1 |
30 Thierry Coquand. Metamathematical investigations of a calculus of constructions
- Constable, Allen, et al.
- 1986
(Show Context)
Citation Context ...e universe of sets. In type theory we allowadi erent class of predicates | those involving predicative higher-order logic in a sense. This topic is discussed in many articles and books on type theory =-=[33, 13, 36, 43, 12, 11]-=- and is beyond the scope of this article, so here we will just assume that the reader is familiar with one account of propositionsas-types or representing logic in type theory. The Nuprl style is to u... |

1 | An operational approach tocombining classical set theory and functional programming languages - Howe, Stoller - 1994 |

1 | A uniform approach totype theory - Huet - 1990 |