## A new implementation of Automath

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Venue: | Journal of Automated Reasoning |

Citations: | 3 - 0 self |

### BibTeX

@ARTICLE{Wiedijk_anew,

author = {Freek Wiedijk},

title = {A new implementation of Automath},

journal = {Journal of Automated Reasoning},

year = {},

volume = {29},

pages = {2002}

}

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### Abstract

Abstract. This paper presents aut, a modern Automath checker. It is a straightforward re-implementation of the Zandleven Automath checker from the seventies. It was implemented about five years ago, in the programming language C. It accepts both the AUT-68 and AUT-QE dialects of Automath. This program was written to restore a damaged version of Jutting’s translation of Landau’s Grundlagen. Some notable features: − It is fast. On a 1GHz machine it will check the full Jutting formalization (736K of non-whitespace Automath source) in 0.6 seconds. − Its implementation of λ-terms does not use named variables or de Bruijn indices (the two common approaches) but instead uses a graph representation. In this representation variables are represented by pointers to a binder. − The program can compile an Automath text into one big ‘Automath single line’ style λ-term. It outputs such a term using de Bruijn indices. (These λ-terms cannot be checked by modern systems like Coq or Agda, because the λ-typed λ-calculi of de Bruijn are different from the Π-typed λ-calculi of modern type theory.) The source of aut is freely available on the Web at the address

### Citations

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711 | A framework for defining logics
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(Show Context)
Citation Context ...ed to be ‘just syntax’ and to have ‘no meaning’. 23 aut.tex; 18/09/2002; 15:32; p.23s24 • The theory of LF as a logical framework is well developed, but the corresponding theory of AUT-∆Λ is not. In (=-=Harper et al., 1991-=-) it has been proved that the proofs that one can do with first order logic encoded in LF correspond exactly to the proofs of first order logic itself. A similar result for AUT-∆Λ has never been prove... |

544 | Lambda Calculi with Types
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- 1992
(Show Context)
Citation Context ...hen present an example of such an Automath proof object. 6.1. λ-typed versus Π-typed type theory Automath is fundamentally different from the modern type theories, which are called pure type systems (=-=Barendregt, 1992-=-). In Automath the type of a λ-expression is itself again a λ-expression. In a pure type system the type of a λ-expression is a product type or Π-expression. The Automath type theories traditionally h... |

433 | Isabelle : a generic theorem prover
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(Show Context)
Citation Context ...ne different logics. In modern terminology this means that he proposes to use Automath as a logical framework (Pfenning, 1996). Of the modern systems for formalizing mathematics, the Isabelle system (=-=Paulson, 1994-=-; Nipkow et al., 2002) is the primary one that is based on a logical framework. It consists of the logical framework called Isabelle/Pure, and on top of this a higher order logic is defined called Isa... |

315 | Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem - Bruijn - 1972 |

222 | The practice of logical frameworks
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(Show Context)
Citation Context ...st as in a restaurant one can order different kinds of food, one can use Automath to define different logics. In modern terminology this means that he proposes to use Automath as a logical framework (=-=Pfenning, 1996-=-). Of the modern systems for formalizing mathematics, the Isabelle system (Paulson, 1994; Nipkow et al., 2002) is the primary one that is based on a logical framework. It consists of the logical frame... |

89 |
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Citation Context ...f the difference between the type system of Automath and that of the modern systems. Several of these systems – HOL, PVS, Coq, Isabelle, PhoX, Lego and Nuprl – are also descendants of the LCF system (=-=Gordon et al., 1979-=-). They are interactive systems that operate on a ‘proof state’ by applying ‘tactics’. This is very different from Automath, where one writes λ-terms that are checked by a non-interactive checker. The... |

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32 |
Grundlagen der Analysis
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(Show Context)
Citation Context ... texts, but for compiling texts into. 1.2. Grundlagen The largest existing Automath formalization is the translation of a small mathematics book. The book is Grundlagen der Analysis by Edmund Landau (=-=Landau, 1965-=-). It treats the definition of the field of complex numbers. It consists of 161 pages of German text, which is divided in 301 ‘Sätze’ (theorems). The translation was done by Bert van Benthem Jutting. ... |

13 | Telescopic mapping in typed lambda calculus - Bruijn - 1991 |

11 |
On π-conversion in the λ-cube and the combination with abbreviations. APAL
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Citation Context ...x:A.B (application) (This rule suggests to add Π-application to pure type systems, with a reduction rule: (Πx:A.B)a → B[x := a] Such a system is investigated in (Kamareddine and Nederpelt, 1996) and (=-=Kamareddine et al., 1999-=-). In a sense this system is in between Automath and the pure type systems.) We will briefly discuss the lack of popularity in the type theoretic community of the λ-typed type theories in Section 7.2 ... |

8 |
Canonical Typing and π-conversion in the Barendregt Cube
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(Show Context)
Citation Context ...Γ ⊢ F : G Γ ⊢ a : A Γ ⊢ F a : Ga G ≡ λx:A.B (application) (This rule suggests to add Π-application to pure type systems, with a reduction rule: (Πx:A.B)a → B[x := a] Such a system is investigated in (=-=Kamareddine and Nederpelt, 1996-=-) and (Kamareddine et al., 1999). In a sense this system is in between Automath and the pure type systems.) We will briefly discuss the lack of popularity in the type theoretic community of the λ-type... |

7 | Metamath: A Computer Language for Pure Mathematics
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- 1997
(Show Context)
Citation Context ...d by the difference in reduction strategies of those checkers. Clearly the computers have become much faster since the seventies! The speed of aut is compatible with the speed of the Metamath system (=-=Megill, 1997-=-), which is able to check its whole library in seconds. This system also is a one pass batch checker for a simple proof language and it is also written in C. 13 aut.tex; 18/09/2002; 15:32; p.13s14 4.2... |

6 |
Lambda calculus extended with segments
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- 1994
(Show Context)
Citation Context ... be extended with telescopes, which are variables for lists of λ-binders. This extension of the Automath language is described in (de Bruijn, 1991). A generalization of telescopes is called segments (=-=Balsters, 1986-=-). The aut checker currently does not implement telescopes or segments. It is not immediately clear how to implement them in the term graph representation that is used by aut. − AUT-SYNT, AUT-Π. There... |

6 | Typing in Pure Type Systems - Jutting, S - 1993 |

6 | A Description of Automath and Some Aspects of its Language Theory - Daalen |

5 | Checking Landau’s ‘Grundlagen - Jutting - 1979 |

3 | On -conversion in the -cube and the combination with abbreviations. Annals of Pure and Applied Logic - Kamareddine, Bloo, et al. - 1999 |

3 | Canonical typing and -conversion in the Barendregt Cube - Kamareddine, Nederpelt - 1996 |

2 | Defining λ-Typed λ-Calculi by Axiomatizing the Typing Relation - Groote - 1993 |

1 | A translation of Landau’s “Grundlagen” in AUTOMATH - Jutting, L - 1976 |

1 | 2000, `The De Bruijn Factor'. factor.ps.gz - Wiedijk |

1 | 2002, `The Fifteen Provers of the World'. ~freek/notes/comparison.ps.gz - Wiedijk |

1 | De -Typed -Calculi by Axiomatizing the Typing Relation - Groote - 1993 |

1 | A translation of Landau's \Grundlagen" in AUTOMATH - Jutting, L - 1976 |