## Towards a formal mathematical vernacular (1992)

Venue: | Utrecht University |

Citations: | 2 - 0 self |

### BibTeX

@TECHREPORT{Groote92towardsa,

author = {Jan Friso Groote},

title = {Towards a formal mathematical vernacular},

institution = {Utrecht University},

year = {1992}

}

### OpenURL

### Abstract

Contemporary proof veri cators often use a command language to construct proofs. These commands are often called tactics. This new generation of theorem provers is a substantial improvement over earlier ones such asAUTOMATH. Based on experience with these new provers we feel the need to study these languages further, especially, because we think that these may be improved in their adequateness to express proofs closer to the established mathematical vernacular. We also feel that a systematic treatment of these vernaculars may lead to an improvement towards the automatic inference of trivial proof steps. In any case a systematic treatment will lead to a better understanding of the command languages. This exercise is carried out in the setting of Pure Type Systems (PTSs) in which a whole range of logics can be embedded. We rstidentify a subclass of PTSs, called the PTSs for logic. For this class we de ne a formal mathematical vernacular and we prove elementary sound- and completeness. Via an elaborate example we try to assess how easy proofs in mathematics can be written down in our vernacular along the lines of the original proofs. 1

### Citations

162 | A.: Rippling: A Heuristic for Guiding Inductive Proofs
- Bundy, Stevens, et al.
- 1993
(Show Context)
Citation Context ...ally and alternatively into so called meta-proofs. This last option is added to be able to express that several sequences of tactics may be used in an attempt to prove a proposition. See for instance =-=[7]-=- where a super tactic is defined that can apply several induction schemes at once. Roughly speaking one can say that these super tactics alternately try several induction schemes until one is successf... |

90 |
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(Show Context)
Citation Context ...se systems is that they interactively seek a proof of a certain theorem, while guaranteeing that the constructed proof is correct. These systems are a considerable improvement as compared to AUTOMATH =-=[5]-=- especially in their usability. The so called tactics that steer the construction of a proof are a relief compared to the explicit construction of a proof (actually a lambda term inhabiting a type) th... |

82 | An overview of the MIZAR project
- Rudnicki
- 1992
(Show Context)
Citation Context ...mathematical vernacular may require quite some artificial intelligence". 2 2 PURE TYPE SYSTEMS The Mizar project is an example of a project that attempts to move the first approach towards the se=-=cond [16]-=-. Based on classical logic and set theory a language has been developed for writing proofs in the `mathematicians style'. There is a whole family of Mizar tools to check these Mizar proof texts. It is... |

56 |
The Mathematical Vernacular, a language for mathematics with typed sets
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- 1987
(Show Context)
Citation Context ...s last point is slightly worrying, as the current `mathematical' form of proofs is the result of centuries of experience. The other approach to automatic theorem proving has been pursued by de Bruijn =-=[6, 15]-=-. He takes the `mathematician's vernacular' and orders and classifies it to make it more suitable for automatic handling. He calls a piece of mathematics a book. A book is an ordered sequence of lines... |

47 |
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(Show Context)
Citation Context ... called tactics that steer the construction of a proof are a relief compared to the explicit construction of a proof (actually a lambda term inhabiting a type) that was required by AUTOMATH (see e.g. =-=[4]-=-). Experiments with these systems led to a few questions. In the rst place the actual tactics, often quite substantial in number, appear to di er from system to system, except for a small core. Moreov... |

18 |
Benthem Jutting. Checking Landau's "Grundlagen
- van
- 1977
(Show Context)
Citation Context ... called tactics that steer the construction of a proof are a relief compared to the explicit construction of a proof (actually a lambda term inhabiting a type) that was required by AUTOMATH (see e.g. =-=[4]-=-). Experiments with these systems led to a few questions. In the first place the actual tactics, often quite substantial in number, appear to differ from system to system, except for a small core. Mor... |

16 |
Demonstration Automatique dans le Calcul des Constructions
- Dowek
- 1991
(Show Context)
Citation Context ...anisms, are incorporated into the systems, the exact effects and properties of the different tactics are not too clear. As far as we know, not too much attention to this matter has been paid (but see =-=[9, 10, 11]-=-). A striking point is that proofs formulated using tactics are hard to grasp. The reason for this is, besides the abundance of details that proofs consisting of tactics still contain, the imposition ... |

13 |
The Coq proof assistant version 5.6 user's guide
- Dowek, Felty, et al.
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(Show Context)
Citation Context ...are two main approaches towards the formalisation of proofs to enable automatic verification. The most well known one is based on LCF [13]. In this stream there are several proof checkers such as Coq =-=[12]-=-, LEGO [14], ALF [1] and NuPrl [8]. The main feature of these systems is that they interactively seek a proof of a certain theorem, while guaranteeing that the constructed proof is correct. These syst... |

10 |
How to use LEGO
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- 1989
(Show Context)
Citation Context ...n approaches towards the formalisation of proofs to enable automatic verification. The most well known one is based on LCF [13]. In this stream there are several proof checkers such as Coq [12], LEGO =-=[14]-=-, ALF [1] and NuPrl [8]. The main feature of these systems is that they interactively seek a proof of a certain theorem, while guaranteeing that the constructed proof is correct. These systems are a c... |

6 |
Naming and scoping in a mathematical vernacular
- Dowek
- 1990
(Show Context)
Citation Context ...anisms, are incorporated into the systems, the exact effects and properties of the different tactics are not too clear. As far as we know, not too much attention to this matter has been paid (but see =-=[9, 10, 11]-=-). A striking point is that proofs formulated using tactics are hard to grasp. The reason for this is, besides the abundance of details that proofs consisting of tactics still contain, the imposition ... |

2 |
De taal van de wiskunde | Een verkenning van wiskundig taalgebruik en logische redeneerpatronen. Versluys
- Nederpelt
- 1990
(Show Context)
Citation Context ...s last point is slightly worrying, as the current `mathematical' form of proofs is the result of centuries of experience. The other approach to automatic theorem proving has been pursued by de Bruijn =-=[6, 15]-=-. He takes the `mathematician's vernacular' and orders and classifies it to make it more suitable for automatic handling. He calls a piece of mathematics a book. A book is an ordered sequence of lines... |

1 |
Automatic proof checking and proof construction by tactics
- Dowek
- 1991
(Show Context)
Citation Context ...anisms, are incorporated into the systems, the exact effects and properties of the different tactics are not too clear. As far as we know, not too much attention to this matter has been paid (but see =-=[9, 10, 11]-=-). A striking point is that proofs formulated using tactics are hard to grasp. The reason for this is, besides the abundance of details that proofs consisting of tactics still contain, the imposition ... |