## A Survey on the Theorema Project (1997)

Venue: | In International Symposium on Symbolic and Algebraic Computation |

Citations: | 47 - 10 self |

### BibTeX

@INPROCEEDINGS{Buchberger97asurvey,

author = {Bruno Buchberger and Tudor Jebelean and Franz Kriftner and Mircea Marin and Elena Tomuta and Daniela Vasaru and A Schlo Hagenberg},

title = {A Survey on the Theorema Project},

booktitle = {In International Symposium on Symbolic and Algebraic Computation},

year = {1997},

pages = {384--391},

publisher = {ACM Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

The Theorema project aims at extending current computer algebra systems by facilities for supporting mathematical proving. The present early-prototype version of the Theorema software system is implemented in Mathematica 3.0. The system consists of a general higher-order predicate logic prover and a collection of special provers that call each other depending on the particular proof situations. The individual provers imitate the proof style of human mathematicians and aim at producing human-readable proofs in natural language presented in nested cells that facilitate studying the computer-generated proofs at various levels of detail. The special provers are intimately connected with the functors that build up the various mathematical domains. 1 The Objectives of the Theorema Project The Theorema project aims at providing a uniform (logic and software) frame for computing, solving, and proving. In a simplified view, given a "knowledge base" K of formulae (and a logical / computat...

### Citations

518 | Lambda calculi with types
- Barendregt
- 1992
(Show Context)
Citation Context ...notion which is sufficient for the practical purpose of our system. In a later stage of our project, we will study how these proof objects can be translated into the formal proof objects described in =-=[1]-=- (p. 202), in the frame of the Curry-Howard formalism. After having produced the proof object, our prover enters into the stage of producing the natural language version of the proof in nested cell re... |

395 |
A Computational Logic Handbook
- Boyer, Moore
- 1988
(Show Context)
Citation Context ...ography in [13]. The most important current projects combining computer algebra systems with theorem provers (in either of the above approaches) are the following: Isabelle [18], Analytica [7], NQTHM =-=[2]-=-, Nuprl [9], Coq [15], HOL [12], Oyster-Clam [14], [20], Redlog [11]. In the Theorema project, we decided to start from an existing computer algebra system, namely Mathematica (version 3.0) and to add... |

373 |
Quantifier elimination for real closed fields by cylindrical algebraic decomposition, Quantifier elimination and cylindrical algebraic decomposition
- Collins
- 1993
(Show Context)
Citation Context ... the proof problem to certain algebraic problems solved by purely algebraic algorithms (Groebner bases method, see [16]; characteristic sets method, see [22]; cylindrical algebraic decomposition, see =-=[8]-=-; Cayley factorization, see [19]). ffl Various special solvers including the ones already available in the current symbolic computation systems (notably the solvers for multivariate polynomial equatio... |

366 |
ML for the Working Programmer
- Paulson
- 1991
(Show Context)
Citation Context ... of Functor in Theorema Functors are a well known concept for building up mathematics in a structured way. In the frame of the Theorema project, we adopt the notion of functor in the sense of ML (see =-=[17]-=-), which is technically slightly different from the notion of a functor in the sense of category theory. It is important to observe that, although not explicitly mentioned in the Mathematica manual, f... |

135 |
Algorithms in Invariant Theory
- Sturmfels
- 1993
(Show Context)
Citation Context ...lgebraic problems solved by purely algebraic algorithms (Groebner bases method, see [16]; characteristic sets method, see [22]; cylindrical algebraic decomposition, see [8]; Cayley factorization, see =-=[19]-=-). ffl Various special solvers including the ones already available in the current symbolic computation systems (notably the solvers for multivariate polynomial equation systems). 2 ffl A general faci... |

44 |
The Mathematica book (Wolfram media & Cambridge press
- Wolfram
- 1999
(Show Context)
Citation Context ...e screen.) As soon as the notebook with the nested cells representing the proof is generated the user can click the desired parts of the proof "open" or "close" as described in the=-= Mathematica manual [21]: By the r-=-ecursive structure of the proof objects and the corresponding recursive call of the function "Nested-Presentation", entire subproofs may be compressed (by clicking the cell containing the su... |

41 | Introduction to Isabelle
- Paulson
- 1993
(Show Context)
Citation Context ... the one side and proving (theorem proving systems) on the other side one could start from either of the two sides. Efforts to close the gap starting from theorem provers are for instance, [9], [15], =-=[18]-=-. Also, there are some recent projects that are based either on adding proving facilities or on linking theorem provers to computer algebra systems, see the bibliography in [13]. The most important cu... |

36 | Analytica: A theorem prover for Mathematica
- Clarke, Zhao
- 1993
(Show Context)
Citation Context ...e the bibliography in [13]. The most important current projects combining computer algebra systems with theorem provers (in either of the above approaches) are the following: Isabelle [18], Analytica =-=[7]-=-, NQTHM [2], Nuprl [9], Coq [15], HOL [12], Oyster-Clam [14], [20], Redlog [11]. In the Theorema project, we decided to start from an existing computer algebra system, namely Mathematica (version 3.0)... |

20 |
Mathematica as a Rewrite Language
- Buchberger
- 1996
(Show Context)
Citation Context ... work on Mathematica 3.0 (at least in the prototype phase): ffl Essentially, the innermost part of the Mathematica language is identical to higher-order equational logic (see the detailed analysis in =-=[5]). As a co-=-nsequence, Mathematica can be viewed as a "logic-internal" programming language and, thus, the language gap between computing and proving is closed in a natural way. ffl The rule-based progr... |

19 |
Mechanical Theorem Proving in Geometries
- Wu
- 1994
(Show Context)
Citation Context ...r certain geometrical theorems) by reduction of the proof problem to certain algebraic problems solved by purely algebraic algorithms (Groebner bases method, see [16]; characteristic sets method, see =-=[22]-=-; cylindrical algebraic decomposition, see [8]; Cayley factorization, see [19]). ffl Various special solvers including the ones already available in the current symbolic computation systems (notably t... |

16 |
Mathematics for Computer Science I - The Method of Mathematics (in German
- Buchberger, Lichtenberger
- 1981
(Show Context)
Citation Context ...sm, and the module concept, functors can be implemented elegantly in Mathematica. ffl A general predicate logic prover of a "natural style" introduced in earlier papers by Buchberger, see fo=-=r example [6]-=-, implemented in Mathematica. ffl Various special theorem provers (and / or interactive proof developers) corresponding, in a natural way, to the various functors that build up mathematical domains. T... |

12 |
On the application of Buchbergerâ€™s algorithm to automated geometry theorem proving
- Kutzler, Stifter
- 1986
(Show Context)
Citation Context ...ike the theory of real closed fields or certain geometrical theorems) by reduction of the proof problem to certain algebraic problems solved by purely algebraic algorithms (Groebner bases method, see =-=[16]-=-; characteristic sets method, see [22]; cylindrical algebraic decomposition, see [8]; Cayley factorization, see [19]). ffl Various special solvers including the ones already available in the current s... |

10 |
Reflections on Automath
- Bruijn
- 1990
(Show Context)
Citation Context ...quantifiers like: "8", "9", set braces "fg", summation "\Sigma", product "\Pi", etc. which all can be formulated in terms of "" which is ava=-=ilable as a standard construct in Mathematica. (In fact, in [10] it was shown that t-=-he availability of "" is crucial. ) Furthermore, we introduce a special category of symbols, called "sequence symbols". (This is a useful concept borrowed from Mathematica where th... |

9 |
and T.F.Melham, â€śIntroduction to HOL: a theorem proving environment for higher order logic
- Gordon
- 1993
(Show Context)
Citation Context ...rtant current projects combining computer algebra systems with theorem provers (in either of the above approaches) are the following: Isabelle [18], Analytica [7], NQTHM [2], Nuprl [9], Coq [15], HOL =-=[12]-=-, Oyster-Clam [14], [20], Redlog [11]. In the Theorema project, we decided to start from an existing computer algebra system, namely Mathematica (version 3.0) and to add proving facilities. Our projec... |

6 |
Mathematica: A System for Doing Mathematics by Computer
- Buchberger
- 1993
(Show Context)
Citation Context ...irection of the first author. It is an attempt at realizing the goal of integrating computer algebra and theorem proving, which has tentatively been formulated already in [3] and, more explicitly, in =-=[4]-=-. The co-authors of this paper are currently working jointly with the first author on the following parts of the system: syntax of the mathematical language (F. Kriftner), predicate logic prover (T. J... |

5 |
The NurPRL Proof Development System
- Horn
- 1988
(Show Context)
Citation Context ...ects combining computer algebra systems with theorem provers (in either of the above approaches) are the following: Isabelle [18], Analytica [7], NQTHM [2], Nuprl [9], Coq [15], HOL [12], Oyster-Clam =-=[14]-=-, [20], Redlog [11]. In the Theorema project, we decided to start from an existing computer algebra system, namely Mathematica (version 3.0) and to add proving facilities. Our project is different fro... |

4 |
The Coq proof assistant, A tutorial, version 5.10
- Huet, Kahn, et al.
- 1995
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Citation Context ...ms) on the one side and proving (theorem proving systems) on the other side one could start from either of the two sides. Efforts to close the gap starting from theorem provers are for instance, [9], =-=[15]-=-, [18]. Also, there are some recent projects that are based either on adding proving facilities or on linking theorem provers to computer algebra systems, see the bibliography in [13]. The most import... |

4 |
The Clam proof planner
- Harmelen
- 1989
(Show Context)
Citation Context ...ombining computer algebra systems with theorem provers (in either of the above approaches) are the following: Isabelle [18], Analytica [7], NQTHM [2], Nuprl [9], Coq [15], HOL [12], Oyster-Clam [14], =-=[20]-=-, Redlog [11]. In the Theorema project, we decided to start from an existing computer algebra system, namely Mathematica (version 3.0) and to add proving facilities. Our project is different from the ... |

3 |
REDLOG User Manual. Edition 1.0 for Version 1.0.Universitaet Passau
- Dolzmann, Sturm
- 1996
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Citation Context ...uter algebra systems with theorem provers (in either of the above approaches) are the following: Isabelle [18], Analytica [7], NQTHM [2], Nuprl [9], Coq [15], HOL [12], Oyster-Clam [14], [20], Redlog =-=[11]-=-. In the Theorema project, we decided to start from an existing computer algebra system, namely Mathematica (version 3.0) and to add proving facilities. Our project is different from the other project... |

3 |
Symbolisches Loesen mathematischer Probleme durch Kooperation algorithmischer und logischer Systeme
- Homann
- 1996
(Show Context)
Citation Context ...r instance, [9], [15], [18]. Also, there are some recent projects that are based either on adding proving facilities or on linking theorem provers to computer algebra systems, see the bibliography in =-=[13]-=-. The most important current projects combining computer algebra systems with theorem provers (in either of the above approaches) are the following: Isabelle [18], Analytica [7], NQTHM [2], Nuprl [9],... |