## Some considerations on the usability of Interactive Provers

Citations: | 3 - 1 self |

### BibTeX

@MISC{Asperti_someconsiderations,

author = {Andrea Asperti and Claudio Sacerdoti Coen},

title = {Some considerations on the usability of Interactive Provers},

year = {}

}

### OpenURL

### Abstract

Abstract. In spite of the remarkable achievements recently obtained in the field of mechanization of formal reasoning, the overall usability of interactive provers does not seem to be sensibly improved since the advent of the “second generation ” of systems, in the mid of the eighties. We try to analyze the reasons of such a slow progress, pointing out the main problems and suggesting some possible research directions. 1

### Citations

186 | Fast decision procedures based on congruence closure
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- 1980
(Show Context)
Citation Context ...oint of view, the most challenging aspect of small scale automation is by far the management of equational reasoning, and many interesting techniques addressing this issue (comprising e.g. congruence =-=[17]-=-, narrowing [6] or superposition [18]) have been developed over the years. Although the problem of e-unification is, in general, undecidable, in practice we have at present sufficient knowhow to deal ... |

74 |
Implementing Mathematics with the Nuprl Development System
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(Show Context)
Citation Context ...ty, derived by training and experience, of recognizing specific patterns and to follow different lines of reasoning in different contexts. This line of research was already traced by Constable et. al =-=[5]-=- more than 20 years ago, but went almost neglected The natural growth path for a system like Nuprl tends toward increased “intelligence”. [...] For example, it is helpful if the system is aware of wha... |

47 |
Benthem Jutting. Checking Landau’s “Grundlagen” in the Automath system
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- 1977
(Show Context)
Citation Context ...presented a modern re-implementation of DeBruijn’s Automath checker from the seventies (see [16]). The program was written to restore a damaged version of Jutting’s translation of Landau’s Grundlagen =-=[20]-=-, and the interest of this development is that it is one of the first examples of a large piece of mathematics ever formalized and checked by a machine. In particular, it looks like a good touchstone ... |

24 | Paul(2007) ‘A formally verified proof of the prime number theorem
- Avigad, Donnelly, et al.
(Show Context)
Citation Context ...ns that are investigated in [21], sensibly more recent than the Automath effort, the de Bruijn factor lies around 4. On even more recent works, some authors point out even higher factors (8 and more) =-=[4,2,15]-=-.2 Andrea Asperti, Claudio Sacerdoti Coen A more explicit indicator for measuring the progress of the field is the average amount of time required to formalize a given quantity of text (a page, say).... |

24 | From LCF to HOL: a short history
- Gordon
- 2000
(Show Context)
Citation Context ...vements, the overall usability of interactive provers has not sensibly improved over the last 25 years, since the advent of the current “second generation” of systems 1 : Coq, Hol, Isabelle, PVS (see =-=[10,14,11,7]-=- for some interesting historical surveys). This is certainly also due, in part, to backward compatibility issues: 70 80 90 00 10 HOL HOL88 ProofPower HOL90 HOL light LCF Stanford Edinburgh Cambridge I... |

21 |
Formal proof – the four color theorem
- Gonthier
(Show Context)
Citation Context ...ization and automatic verification of complex mathematical results such as the asymptotic distribution of prime numbers (both in its elementary [4] and analytic [15] versions), the four color theorem =-=[8,9]-=- or the Jordan curve theorem [13]; similar achievements have been also obtained in the field of automatic verification of software (see e.g. [1] for a discussion). However, it is also true that these ... |

13 | Formal proof
- Hales
- 2008
(Show Context)
Citation Context ...some of these figures, computed by different people on different mathematical sources and using different systems. source formalization cost (weeks per page) Van Benthem [20] 1 Wiedijk [22] 1.5 Hales =-=[12]-=- 1 Asperti [2] 1.5 Fig. 1. Formalization cost In the case of Van Benthem Jutting’s work, the cost factor is easily estimated: the Grundlagen are 161 pages long, and he worked at their formalization fo... |

11 |
The Four Colour Theorem: Engineering of a Formal Proof
- Gonthier
- 2007
(Show Context)
Citation Context ...ization and automatic verification of complex mathematical results such as the asymptotic distribution of prime numbers (both in its elementary [4] and analytic [15] versions), the four color theorem =-=[8,9]-=- or the Jordan curve theorem [13]; similar achievements have been also obtained in the field of automatic verification of software (see e.g. [1] for a discussion). However, it is also true that these ... |

9 |
Formalizing an Analytic Proof of the Prime Number Theorem
- Harrison
- 2008
(Show Context)
Citation Context ...ns that are investigated in [21], sensibly more recent than the Automath effort, the de Bruijn factor lies around 4. On even more recent works, some authors point out even higher factors (8 and more) =-=[4,2,15]-=-.2 Andrea Asperti, Claudio Sacerdoti Coen A more explicit indicator for measuring the progress of the field is the average amount of time required to formalize a given quantity of text (a page, say).... |

8 | About the formalization of some results by Chebyshev
- Asperti, Ricciotti
- 2008
(Show Context)
Citation Context ...ns that are investigated in [21], sensibly more recent than the Automath effort, the de Bruijn factor lies around 4. On even more recent works, some authors point out even higher factors (8 and more) =-=[4,2,15]-=-.2 Andrea Asperti, Claudio Sacerdoti Coen A more explicit indicator for measuring the progress of the field is the average amount of time required to formalize a given quantity of text (a page, say).... |

7 | The Jordan curve theorem, formally and informally
- Hales
- 2007
(Show Context)
Citation Context ... of complex mathematical results such as the asymptotic distribution of prime numbers (both in its elementary [4] and analytic [15] versions), the four color theorem [8,9] or the Jordan curve theorem =-=[13]-=-; similar achievements have been also obtained in the field of automatic verification of software (see e.g. [1] for a discussion). However, it is also true that these accomplishments can be justified ... |

7 | A constructive and formal proof of Lebesgue’s dominated convergence theorem in the interactive theorem prover Matita
- Coen, C, et al.
- 2008
(Show Context)
Citation Context ...ibution of prime numbers (subsuming, as a corollary, Bertrand’s postulate) [2]; the third development (Lebesgue) is a formalisation of a constructive proof of Lebesgue’s Dominated Convergence Theorem =-=[19]-=-; finally, the lastSome considerations on the usability of Interactive Provers 5 development is a solution to part-1 of the POPLmark challenge in different styles (with names, locally nameless and wi... |

6 | Estimating the cost of a standard library for a mathematical proof checker
- Wiedijk
- 2001
(Show Context)
Citation Context ...gure 1 reports some of these figures, computed by different people on different mathematical sources and using different systems. source formalization cost (weeks per page) Van Benthem [20] 1 Wiedijk =-=[22]-=- 1.5 Hales [12] 1 Asperti [2] 1.5 Fig. 1. Formalization cost In the case of Van Benthem Jutting’s work, the cost factor is easily estimated: the Grundlagen are 161 pages long, and he worked at their f... |

5 |
The ”de Bruijn factor”. http://www.cs.ru.nl/~freek/factor
- Wiedijk
(Show Context)
Citation Context ...now what a modern system could do for us supposing to grant him 35 minutes, as in the seventies. A different measure that is sometimes used to compare formalizations is the so called de Bruijn factor =-=[21]-=-. This is defined as the quotient between the dimension of the formalization and the dimension of the source mathematical text (sometimes computed on compressed files), and it is supposed to give evid... |

3 | Smart matching
- Asperti, Tassi
(Show Context)
Citation Context ...h the problem of e-unification is, in general, undecidable, in practice we have at present sufficient knowhow to deal with it reasonably well (but apart from a few experimental exceptions like Matita =-=[3]-=-, no major interactive prover provides, at present, a strong native support for narrowing or superposition). In principle, case analysis and the management of background knowledge is another part of t... |

3 | Proof Assistants: history, ideas and future
- Geuvers
(Show Context)
Citation Context ...vements, the overall usability of interactive provers has not sensibly improved over the last 25 years, since the advent of the current “second generation” of systems 1 : Coq, Hol, Isabelle, PVS (see =-=[10,14,11,7]-=- for some interesting historical surveys). This is certainly also due, in part, to backward compatibility issues: 70 80 90 00 10 HOL HOL88 ProofPower HOL90 HOL light LCF Stanford Edinburgh Cambridge I... |

3 |
Paramodulation-based thorem proving
- Nieuwenhuis, Rubio
- 2001
(Show Context)
Citation Context ...pect of small scale automation is by far the management of equational reasoning, and many interesting techniques addressing this issue (comprising e.g. congruence [17], narrowing [6] or superposition =-=[18]-=-) have been developed over the years. Although the problem of e-unification is, in general, undecidable, in practice we have at present sufficient knowhow to deal with it reasonably well (but apart fr... |

3 | A new implementation of Automath
- Wiedijk
(Show Context)
Citation Context ...tion” of systems, in the mid of the eighties. We try to analyze the reasons of such a slow progress, pointing out the main problems and suggesting some possible research directions. 1 Introduction In =-=[23]-=-, Wiedijk presented a modern re-implementation of DeBruijn’s Automath checker from the seventies (see [16]). The program was written to restore a damaged version of Jutting’s translation of Landau’s G... |

2 | Social processes, program verification and all that
- Asperti, Geuvers, et al.
(Show Context)
Citation Context ...] and analytic [15] versions), the four color theorem [8,9] or the Jordan curve theorem [13]; similar achievements have been also obtained in the field of automatic verification of software (see e.g. =-=[1]-=- for a discussion). However, it is also true that these accomplishments can be justified in many other different ways, quite independent from the improvements of systems: a) the already mentioned prog... |

2 |
José Meseguer, and Prasanna Thati. Narrowing and rewriting logic: from foundations to applications
- Escobar
(Show Context)
Citation Context ...e most challenging aspect of small scale automation is by far the management of equational reasoning, and many interesting techniques addressing this issue (comprising e.g. congruence [17], narrowing =-=[6]-=- or superposition [18]) have been developed over the years. Although the problem of e-unification is, in general, undecidable, in practice we have at present sufficient knowhow to deal with it reasona... |

2 |
Twenty years of theorem proving for HOLs: Past, present and future
- Gordon
(Show Context)
Citation Context ...vements, the overall usability of interactive provers has not sensibly improved over the last 25 years, since the advent of the current “second generation” of systems 1 : Coq, Hol, Isabelle, PVS (see =-=[10,14,11,7]-=- for some interesting historical surveys). This is certainly also due, in part, to backward compatibility issues: 70 80 90 00 10 HOL HOL88 ProofPower HOL90 HOL light LCF Stanford Edinburgh Cambridge I... |

2 | A Short Survey of Automated Reasoning
- Harrison
- 2007
(Show Context)
Citation Context |