## The theorema environment for interactive proof development. Contributed talk at (2005)

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Venue: | 12th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning, LPAR’05 |

Citations: | 2 - 1 self |

### BibTeX

@INPROCEEDINGS{Piroi05thetheorema,

author = {Florina Piroi and Temur Kutsia},

title = {The theorema environment for interactive proof development. Contributed talk at},

booktitle = {12th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning, LPAR’05},

year = {2005}

}

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### Abstract

Abstract. We describe an environment that allows the users of the Theorema system to flexibly control aspects of computer-supported proof development. The environment supports the display and manipulation of proof trees and proof situations, logs the user activities (commands communicated with the system during the proving session), and presents (also unfinished) proofs in a human-oriented style. In particular, the user can navigate through the proof object, expand/remove proof branches, provide witness terms, develop several proofs concurrently, proceed step by step or automatically and so on. The environment enhances the effectiveness and flexibility of the reasoners of the Theorema system. 1

### Citations

500 |
T.: Introduction to HOL: A Theorem Proving Environment for Higher Order Logic: Cambridge
- Melham
- 1993
(Show Context)
Citation Context ...literature (e.g. [29], or the forthcoming issue on Mathematics Assistance Systems of the Journal of Applied Logic [4]). One of the interactive systems with the largest pool of users is the Hol system =-=[14]-=-, now at version 4. It is an environment for interactive theorem proving in higher-order logic and has a wide variety of uses from formalizing mathematics (see for example [15]) to verification of ind... |

470 |
P.: Interactive Theorem Proving and Program Development. Coq’Art: The Calculus of Inductive Constructions
- Bertot, Castéran
- 2004
(Show Context)
Citation Context ... the proof is available (the one shown in the Proof General window). In its latest version, Isabelle provides a tool for searching theorems in the system’s library of theories by simple patterns. Coq =-=[5]-=-, another logical framework system, is a proof assistant for the Calculus of Inductive Constructions. It allows the interactive construction of formal proofs, and also the manipulation of functional p... |

373 |
Quantifier elimination for real closed fields by cylindrical algebraic decomposition, Quantifier elimination and cylindrical algebraic decomposition
- Collins
- 1993
(Show Context)
Citation Context ...ave been developed, e.g. the Pcs prover [7] (standing for ‘Prove Compute Solve’) which implements a heuristics for elementary analysis and uses Collins’s Cylindrical Algebraic Decomposition algorithm =-=[10]-=- as a solver. Another example of a special reasoner is the solver and simplifier for two-point linear boundary value problems [22]. Theorema currently contains 19 reasoners and is linked to 11 externa... |

232 |
The Mathematica Book
- Wolfram
- 1999
(Show Context)
Citation Context ...ire users to) make selections in notebooks and, on button clicks, manipulate the notebooks in the Mathematica kernel. Within any open notebook, the front end always maintains a current selection (see =-=[30]-=-, Section 2.11.3). Selections can be done by user clicks or by issuing commands from the kernel. Mathematica also provides commands for extracting the content of a selections in a notebook. So we are ... |

182 |
Isabelle: The next 700 theorem provers
- Paulson
- 1990
(Show Context)
Citation Context ...terfaces provide theory browsing and searching, graphical views of the proof state (similar to the schematic proof tree representation in Theorema’s interactive environment), etc. The Isabelle system =-=[20]-=- is a generic proof assistant which allows defining different logical calculi and using them for proving. It is closely integrated with the Proof General [2] for editing and developing proofs and, sim... |

75 | Proof General: A generic tool for proof development. Tools and Algorithms for the Construction and Analysis of Systems
- Aspinall
- 2000
(Show Context)
Citation Context ...ve environment), etc. The Isabelle system [20] is a generic proof assistant which allows defining different logical calculi and using them for proving. It is closely integrated with the Proof General =-=[2]-=- for editing and developing proofs and, similar to the Hol system, allows proof storage and replay, undo and revert operations, proof states display, etc. (Proof General [2] is a generic tool for proo... |

60 |
Theorema: Towards Computer-Aided Mathematical Theory Exploration
- Buchberger, Craciun, et al.
(Show Context)
Citation Context ...interactive provers, which require human assistance in developing the proof [18]. An extensive list of both automatic and interactive provers can be inspected at [3]. The goal of the Theorema project =-=[8]-=- is to provide support to the entire process of mathematical theory exploration. By default, Theorema tries to solve given reasoning problems automatically. However, since many mathematical theorems a... |

23 | A new interface for HOL - ideas, issues and implementation
- Syme
- 1995
(Show Context)
Citation Context ...e is also a restart possibility by backtracking to the root. Switching between goals is possible both in Hol and in Theorema. Several graphical interfaces were implemented for the system, like Tk-Hol =-=[26]-=-, xhol [23], and Emacs modes are widely used. The interfaces provide theory browsing and searching, graphical views of the proof state (similar to the schematic proof tree representation in Theorema’s... |

23 | Comparing mathematical provers
- Wiedijk
- 2003
(Show Context)
Citation Context ...of these systems (mathematical assistants), insisting on those features similar to the ones present in Theorema. For more details on the described systems we direct the reader to the literature (e.g. =-=[29]-=-, or the forthcoming issue on Mathematics Assistance Systems of the Journal of Applied Logic [4]). One of the interactive systems with the largest pool of users is the Hol system [14], now at version ... |

19 | Social and semiotic analyses for theorem prover user interface design. Formal Aspects of Computing, 11:272–301. Special issue on user interfaces for theorem provers
- Goguen
- 1999
(Show Context)
Citation Context ... for Proof Development Design principles for interfaces to (interactive) provers, as well as the functionalities such interfaces should offer, have already been formulated by a number of authors; see =-=[6, 12, 13, 28]-=-. We do not intend to give yet another set of principles, but we will just gather user actions that correspond to the already formulated principles and classify them into logical and abstract interact... |

17 |
A new symbolic method for solving linear two-point boundary value problems on the level of operators
- Rosenkranz
- 2005
(Show Context)
Citation Context ...ysis and uses Collins’s Cylindrical Algebraic Decomposition algorithm [10] as a solver. Another example of a special reasoner is the solver and simplifier for two-point linear boundary value problems =-=[22]-=-. Theorema currently contains 19 reasoners and is linked to 11 external reasoning systems and to the Tptp library; see [8]. During a Theorema session reasoners are accessed by a call of the form Reaso... |

14 |
The PCS Prover in Theorema
- Buchberger
- 2001
(Show Context)
Citation Context ...in special theories—like geometry, analysis, combinatorics—using algebraic algorithms as black box inference rules. For this purpose several special reasoners have been developed, e.g. the Pcs prover =-=[7]-=- (standing for ‘Prove Compute Solve’) which implements a heuristics for elementary analysis and uses Collins’s Cylindrical Algebraic Decomposition algorithm [10] as a solver. Another example of a spec... |

12 | Interactive Theorem Proving: An Empirical Study of User Activity
- Aitken
- 1998
(Show Context)
Citation Context ...he effectiveness and flexibility of the reasoners of the Theorema system. 1 Introduction In general terms, it is agreed that mechanized theorem proving is about using computers to find a formal proof =-=[1]-=-. A rough classification of theorem provers divides them in automatic provers, where close to no human assistance is needed, and interactive provers, which require human assistance in developing the p... |

10 | LΩUI: Lovely ΩMEGA user interface
- Siekmann, Hess, et al.
- 1999
(Show Context)
Citation Context ...rtext facilities for term browsing, proof and proof plan presentation in natural language. It also has an editor for adding and maintaining the knowledge base, and a command suggestion mechanism; see =-=[25]-=-. NuPRL [11] supports the interactive creation of proofs, formulas, and terms. Based on Martin-Löf type theory, it is a system for implementing mathematics. NuPRL has a multi-window graphical environm... |

9 | Jape’s Quiet Interface
- Bornat, Sufrin
- 1996
(Show Context)
Citation Context ... for Proof Development Design principles for interfaces to (interactive) provers, as well as the functionalities such interfaces should offer, have already been formulated by a number of authors; see =-=[6, 12, 13, 28]-=-. We do not intend to give yet another set of principles, but we will just gather user actions that correspond to the already formulated principles and classify them into logical and abstract interact... |

8 |
Tools for Using Automated Provers in Mathematical Theory Exploration
- Piroi
- 2004
(Show Context)
Citation Context .... In this paper, we do not consider the actions at the concrete interaction level. We give, however, some considerations in this respect in Section 3.4. (For more usage and implementation details see =-=[21]-=-.) At the logical level the user actions are sketched only in terms of logical concepts [1], like for example the activity of reducing a mathematical expression to its canonical form. Other actions th... |

7 |
Theorema: A System for Formal Scientific Training in Natural Language Presentation
- Buchberger, Jebelean, et al.
- 1998
(Show Context)
Citation Context ...orema developers group, for which the environment is used as a tool for testing the provers that are still in development. The first attempts to integrate interactivity into Theorema are described in =-=[9]-=- and some of those ideas were a starting point for the current interactive environment. Prior to this work, in [27] it is shown how interactive proving was to be integrated in the architecture of Theo... |

6 | Support for Interactive Theorem Proving: Some Design Principles and Their Application
- Eastaughffe
- 1998
(Show Context)
Citation Context ... for Proof Development Design principles for interfaces to (interactive) provers, as well as the functionalities such interfaces should offer, have already been formulated by a number of authors; see =-=[6, 12, 13, 28]-=-. We do not intend to give yet another set of principles, but we will just gather user actions that correspond to the already formulated principles and classify them into logical and abstract interact... |

6 | A tree-based, graphical interface for large proof development
- Schubert, Biggs
- 1994
(Show Context)
Citation Context ... restart possibility by backtracking to the root. Switching between goals is possible both in Hol and in Theorema. Several graphical interfaces were implemented for the system, like Tk-Hol [26], xhol =-=[23]-=-, and Emacs modes are widely used. The interfaces provide theory browsing and searching, graphical views of the proof state (similar to the schematic proof tree representation in Theorema’s interactiv... |

5 | A computer verification of the Kepler conjecture
- Hales
(Show Context)
Citation Context ...users is the Hol system [14], now at version 4. It is an environment for interactive theorem proving in higher-order logic and has a wide variety of uses from formalizing mathematics (see for example =-=[15]-=-) to verification of industrial hardware. It has high degree of programmability through the meta language Ml which allows extending the system to provide more functionality. Thus, packages for proof t... |

5 |
An architecture for combining provers and its applications in the Theorema system
- Tomuta
- 1998
(Show Context)
Citation Context ...lopment. The first attempts to integrate interactivity into Theorema are described in [9] and some of those ideas were a starting point for the current interactive environment. Prior to this work, in =-=[27]-=- it is shown how interactive proving was to be integrated in the architecture of Theorema, but no implementation was done. Another attempt to provide user-system interaction is described in [17] and [... |

4 |
Computer supported mathematics with Ωmega
- Siekmann, Benzmüller, et al.
(Show Context)
Citation Context ...d commands, and has an autocomplete facility for command articulation. Pcoq, CtCoq, and Proof General are other interfaces for Coq. Lately, an integration of Coq into TeXmacs is also available. Ωmega =-=[24]-=- is an interactive proof development system. The system has two main components: a proof planner, and an integrated collection of tools for formulating problems, proving subproblems, and proof present... |

3 |
Implementing Mathematics Using the Nuprl Proof Development System
- Constable
- 1986
(Show Context)
Citation Context ...ties for term browsing, proof and proof plan presentation in natural language. It also has an editor for adding and maintaining the knowledge base, and a command suggestion mechanism; see [25]. NuPRL =-=[11]-=- supports the interactive creation of proofs, formulas, and terms. Based on Martin-Löf type theory, it is a system for implementing mathematics. NuPRL has a multi-window graphical environment and a ke... |

3 |
User System Interaction Within Theorema
- Kossak, Nakagawa
(Show Context)
Citation Context ...k, in [27] it is shown how interactive proving was to be integrated in the architecture of Theorema, but no implementation was done. Another attempt to provide user-system interaction is described in =-=[17]-=- and [16]. Shortly, our main contribution to the previous implementations are improved proof tree and proof situation management, a schematic representation of the proof tree, and multiple interconnec... |

2 | Thoughts on Requirements and Design Issues of User Interfaces for Proof Assistants
- Völker
- 2004
(Show Context)
Citation Context |

1 |
Mathematics Assistance Systems
- Benzmüller, editor
- 2005
(Show Context)
Citation Context ...nt in Theorema. For more details on the described systems we direct the reader to the literature (e.g. [29], or the forthcoming issue on Mathematics Assistance Systems of the Journal of Applied Logic =-=[4]-=-). One of the interactive systems with the largest pool of users is the Hol system [14], now at version 4. It is an environment for interactive theorem proving in higher-order logic and has a wide var... |

1 |
An interface for interactive proving with the mathematical software system Theorema
- Kossak
- 1999
(Show Context)
Citation Context ...] it is shown how interactive proving was to be integrated in the architecture of Theorema, but no implementation was done. Another attempt to provide user-system interaction is described in [17] and =-=[16]-=-. Shortly, our main contribution to the previous implementations are improved proof tree and proof situation management, a schematic representation of the proof tree, and multiple interconnected views... |

1 |
Just the facts, Jack: Truths and myths of automated theorem provers
- Moten
- 1999
(Show Context)
Citation Context ...ough classification of theorem provers divides them in automatic provers, where close to no human assistance is needed, and interactive provers, which require human assistance in developing the proof =-=[18]-=-. An extensive list of both automatic and interactive provers can be inspected at [3]. The goal of the Theorema project [8] is to provide support to the entire process of mathematical theory explorati... |

1 |
An introduction to interactive theorem proving
- Nipkow, Reif
- 1998
(Show Context)
Citation Context ...th the system: cutting a branch and backtracking, changing the prover, adding assumptions, and providing witness terms. 5 Related Work A concise historical overview of interactive systems is given in =-=[19]-=-. Though in most of the cases the design principles listed in [6, 12, 13] or [28] were not specifically followed, many interfaces have common functionalities. We briefly describe some of these systems... |