## Dependent Record Types, Subtyping and Proof Reutilization

Citations: | 4 - 1 self |

### BibTeX

@MISC{Betarte_dependentrecord,

author = {Gustavo Betarte and Gustavo Betarte (draft},

title = {Dependent Record Types, Subtyping and Proof Reutilization},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

. We present an example of formalization of systems of algebras using an extension of Martin-Lof's theory of types with record types and subtyping. This extension has been presented in [5]. In this paper we intend to illustrate all the features of the extended theory that we consider relevant for the task of formalizing algebraic constructions. We also provide code of the formalization as accepted by a type checker that has been implemented. 1. Introduction We shall use an extension of Martin-Lof's theory of logical types [14] with dependent record types and subtyping as the formal language in which constructions concerning systems of algebras are going to be represented. The original formulation of Martin-Lof's theory of types, from now on referred to as the logical framework, has been presented in [15, 7]. The system of types that this calculus embodies are the type Set (the type of inductively defined sets), dependent function types and for each set A, the type of the elements of A...

### Citations

274 | A course in universal algebra - Burris, Sankappanavar - 1981 |

108 |
Computation and Reasoning, A Type Theory for Computer Science
- Luo
- 1994
(Show Context)
Citation Context ...at allows to leave the coercions implicit. This is formulated for pure type systems and extends ideas given originally in [1]. Then applications are shown using the extended calculus of constructions =-=[11]-=-, which contains \Sigma types. In [2] and [16], mechanisms for defining coercions between types or classes of types are developed for the proof-assistants LEGO [15] and Coq [3], respectively. In [12],... |

68 | The theory of LEGO: A proof checker for the extended calculus of constructions
- Pollack
- 1994
(Show Context)
Citation Context ... extended calculus of constructions [11], which contains \Sigma types. In [2] and [16], mechanisms for defining coercions between types or classes of types are developed for the proof-assistants LEGO =-=[15]-=- and Coq [3], respectively. In [12], a calculus in the spirit of Martin-Lof's theory of types is presented, where forms of judgement are introduced, among others, that express the concepts of a kind K... |

45 | An algorithm for type-checking dependent types
- Coquand
- 1996
(Show Context)
Citation Context ...pe, not only alleviates notation, but may also render the process of proof checking more efficient. The way let expressions are checked in our system is heavily influenced by a proposal by Coquand in =-=[7]-=-. Namely, in order to check that an expression let x : ff 1 = e in e 1 has a certain type ff in a environment E proceed as follows: check first that x : ff 1 = e 1 is a valid declaration in E . If thi... |

44 | Enhancing the Nuprl proof-development system and applying it to computational abstract algebra
- Jackson
- 1995
(Show Context)
Citation Context ...rty. As expected, the proof DeMorgan(ii) of the second law is obtained by applying DeMorgan(i) to the object dualBoolAlg Ba. 4. Related work and conclusions In the context of variants of type theory, =-=[10] and [4] have co-=-nsidered using (dependent) pair types (\Sigma types) for expressing tuple types as required by the formalization of algebraic % \Gamma 8B:8x; y 2 jBj:��(xsy) �� (��xs��y) \Gamma % DeMo... |

41 |
Lattice Theory.' First Concepts and Distributive Lattices
- Gratzer
- 1971
(Show Context)
Citation Context ... Boolean algebras in type theory extended with record types and subtyping. The definitions introduced in the next section as well as the enunciation of some of the propositions are taken from [6] and =-=[9]-=-. 3.1. Informal presentation. There are two standard ways of defining lattices: one is to grasp them as an algebraic system and the other is based on the notion of order. Here, we shall follow the fir... |

41 | Typing algorithm in type theory with inheritance
- Saïbi
- 1997
(Show Context)
Citation Context ...s is formulated for pure type systems and extends ideas given originally in [1]. Then applications are shown using the extended calculus of constructions [11], which contains \Sigma types. In [2] and =-=[16]-=-, mechanisms for defining coercions between types or classes of types are developed for the proof-assistants LEGO [15] and Coq [3], respectively. In [12], a calculus in the spirit of Martin-Lof's theo... |

25 | Coercive subtyping in type theory
- Luo
- 1996
(Show Context)
Citation Context ... [11], which contains \Sigma types. In [2] and [16], mechanisms for defining coercions between types or classes of types are developed for the proof-assistants LEGO [15] and Coq [3], respectively. In =-=[12]-=-, a calculus in the spirit of Martin-Lof's theory of types is presented, where forms of judgement are introduced, among others, that express the concepts of a kind K being a principal kind of an objec... |

24 | Extension of Martin Löf’s type theory with record types and subtyping
- Betarte, Tasistro
- 1999
(Show Context)
Citation Context ...TEBORG Abstract. We present an example of formalization of systems of algebras using an extension of Martin-Lof's theory of types with record types and subtyping. This extension has been presented in =-=[18, 5]-=-. In this paper we intend to illustrate all the features of the extended theory that we consider relevant for the task of formalizing algebraic constructions. We also provide code of the formalization... |

21 |
et al. The Coq Proof Assistant Reference Manual (Version 6.3.1). INRIA-Rocquencourt
- Barras
- 2000
(Show Context)
Citation Context ...culus of constructions [11], which contains \Sigma types. In [2] and [16], mechanisms for defining coercions between types or classes of types are developed for the proof-assistants LEGO [15] and Coq =-=[3]-=-, respectively. In [12], a calculus in the spirit of Martin-Lof's theory of types is presented, where forms of judgement are introduced, among others, that express the concepts of a kind K being a pri... |

21 | Type Theory and Programming
- Coquand, Nordstrom, et al.
- 1994
(Show Context)
Citation Context ...tructions concerning systems of algebras are going to be represented. The original formulation of Martin-Lof's theory of types, from now on referred to as the logical framework, has been presented in =-=[14, 17, 8]-=-. The system of types that this calculus embodies are the type Set (the type of inductively defined sets), dependent function types and for each set A, the type of the elements of A. The extension of ... |

17 | Formulation of Martin-Löf’s theory of types with explicit substitutions
- Tasistro
- 1993
(Show Context)
Citation Context ...tructions concerning systems of algebras are going to be represented. The original formulation of Martin-Lof's theory of types, from now on referred to as the logical framework, has been presented in =-=[14, 17, 8]-=-. The system of types that this calculus embodies are the type Set (the type of inductively defined sets), dependent function types and for each set A, the type of the elements of A. The extension of ... |

15 | Implicit coercions in type systems
- Barthe
- 1995
(Show Context)
Citation Context ...xpected, the proof DeMorgan(ii) of the second law is obtained by applying DeMorgan(i) to the object dualBoolAlg Ba. 4. Related work and conclusions In the context of variants of type theory, [10] and =-=[4] have considered-=- using (dependent) pair types (\Sigma types) for expressing tuple types as required by the formalization of algebraic % \Gamma 8B:8x; y 2 jBj:��(xsy) �� (��xs��y) \Gamma % DeMorgan(i) ... |

10 |
Record Types and Subtyping in Type Theory, with Applications to the Theory of Programming
- Substitution
- 1997
(Show Context)
Citation Context ...TEBORG Abstract. We present an example of formalization of systems of algebras using an extension of Martin-Lof's theory of types with record types and subtyping. This extension has been presented in =-=[18, 5]-=-. In this paper we intend to illustrate all the features of the extended theory that we consider relevant for the task of formalizing algebraic constructions. We also provide code of the formalization... |

8 | Galois - a theory development project. Report for the 1993 Turin meeting on the Representation of Mathematics in Logical Frameworks. Available at URL: www.cs.man.ac.uk/˜petera/papers.html - Aczel |

6 |
A notion of class for theory development in algebra in a predicative type theory
- Aczel
- 1994
(Show Context)
Citation Context ...en the types that represent those structures. But a mechanism is introduced that allows to leave the coercions implicit. This is formulated for pure type systems and extends ideas given originally in =-=[1]-=-. Then applications are shown using the extended calculus of constructions [11], which contains \Sigma types. In [2] and [16], mechanisms for defining coercions between types or classes of types are d... |

6 |
Programming in Martin-Lof's Type Theory.An Introduction
- Nordstrom, Petersson, et al.
- 1989
(Show Context)
Citation Context ...tructions concerning systems of algebras are going to be represented. The original formulation of Martin-Lof's theory of types, from now on referred to as the logical framework, has been presented in =-=[14, 17, 8]-=-. The system of types that this calculus embodies are the type Set (the type of inductively defined sets), dependent function types and for each set A, the type of the elements of A. The extension of ... |

3 | Philosophical Implications of Type Theory. Lectures given at the Facolt'a de Lettere e Filosofia, Universit'a degli Studi di - Martin-Lof |

2 |
Philosophical Implications of Type Theory
- Martin-Lof
- 1987
(Show Context)
Citation Context ...aic constructions. We also provide code of the formalization as accepted by a type checker that has been implemented. 1. Introduction We shall use an extension of Martin-Lof's theory of logical types =-=[13]-=- with dependent record types and subtyping as the formal language in which constructions concerning systems of algebras are going to be represented. The original formulation of Martin-Lof's theory of ... |

2 | Lego with implicit coercions. Available at ftp.cs.man.ac.uk/pub/baileya/Coercions - Bailey - 1996 |

1 |
Lego with implicit coercions. Documentation report, available at ftp.cs.man.ac.uk/pub/baileya/Coercions
- Bailey
(Show Context)
Citation Context ...cit. This is formulated for pure type systems and extends ideas given originally in [1]. Then applications are shown using the extended calculus of constructions [11], which contains \Sigma types. In =-=[2]-=- and [16], mechanisms for defining coercions between types or classes of types are developed for the proof-assistants LEGO [15] and Coq [3], respectively. In [12], a calculus in the spirit of Martin-L... |