## Automation for interactive proof: First prototype

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Venue: | Information and Computation |

Citations: | 28 - 9 self |

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@ARTICLE{Meng_automationfor,

author = {Jia Meng and Claire Quigley and Lawrence C. Paulson},

title = {Automation for interactive proof: First prototype},

journal = {Information and Computation},

year = {},

pages = {2006}

}

### Years of Citing Articles

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### Abstract

Interactive theorem provers require too much effort from their users. We have been developing a system in which Isabelle users obtain automatic support from automatic theorem provers (ATPs) such as Vampire and SPASS. An ATP is invoked at suitable points in the interactive session, and any proof found is given to the user in a window displaying an Isar proof script. There are numerous differences between Isabelle (polymorphic higher-order logic with type classes, natural deduction rule format) and classical ATPs (first-order, untyped, clause form). Many of these differences have been bridged, and a working prototype that uses background processes already provides much of the desired functionality. 1

### Citations

716 |
Isabelle/HOL: A Proof Assistant for Higher-Order Logic
- Nipkow, Paulson, et al.
(Show Context)
Citation Context ...h of the desired functionality. 1 Introduction Automatic theorem provers (ATPs) such as Vampire [19], which work by resolution, are impressive in their power. Interactive proof tools such as Isabelle =-=[14]-=- and PVS [5] provide much less automation; proofs require substantial user effort. However, interactive tools are better suited for verification projects. They admit complicated definitions and specif... |

420 | Isabelle: A generic theorem prover
- Paulson
- 1994
(Show Context)
Citation Context ...log goal clause. When a user performs single-step proof checking, applying some rule to reduce a goal to subgoals, she is actually performing Isabelle resolution between that rule and the proof state =-=[16]-=-. Isabelle provides a variety of automatic tools—known as tactics—that can be used to construct proofs. They include • simp, which simplifies subgoals using rewriting and decision procedures, • blast,... |

127 |
The TPTP Problem Library: CNF Release v1.2.1
- G, Suttner
- 1998
(Show Context)
Citation Context ...o most resolution provers. Standardization of inputs and outputs would make this objective attainable. For input, the tptp2X utility can translate from TPTP (Thousands of Problems for Theorem Provers =-=[23]-=-) format into the input languages of all major resolution provers. Output of proofs is more problematical because systems differ in their inference rules, simplification steps, and problem transformat... |

114 |
A unification algorithm for typed λ-calculus
- Huet
- 1975
(Show Context)
Citation Context ...itted from this knowledge base are transitivity laws and similar lemmas that would blow up the search space. While integrating Isabelle with automatic theorem 1 Isabelle uses higher-order unification =-=[8]-=-. 4provers, we have sought to preserve this advantage: the user should only have to identify a few crucial lemmas, while the resolution search automatically finds other needed facts from the knowledg... |

90 | Computing small clause normal forms
- Nonnengart, Weidenbach
(Show Context)
Citation Context ...re were combined with axiom clauses obtained from the default classical and simplifier rules: the rules that Isabelle tactics such as auto would use. In some of the examples, we used formula renaming =-=[15]-=- before the CNF transformation in order to minimize the number of clauses. Overall, our experiments showed that our methods of translating ZF and HOL into first-order clauses were effective. Most of t... |

88 | Combining superposition, sorts and splitting
- Weidenbach
- 2001
(Show Context)
Citation Context ... (the CADE ATP System Competition). 2 We have even used a version of Vampire modified (by its developers) to support forward and backward inference, as Isabelle does. However, we have also used SPASS =-=[24]-=-, largely because its proof output is easier to interpret. We intend that our work should be applicable to most resolution provers. Standardization of inputs and outputs would make this objective atta... |

77 |
Efficiency and completeness of the set of support strategy in theorem proving
- wos, Robinson, et al.
- 1965
(Show Context)
Citation Context ...e’s full lemma library corresponds to about 1400 first-order clauses. Automatic provers often fail to prove even trivial results in the presence of so many clauses. The set of support heuristic (SOS) =-=[27]-=- is a classic means of improving performance by ignoring irrelevant axioms: it requires all inferences to involve the negated conjecture, preventing aimless forward inferences involving the axioms alo... |

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 ... Once an ssh key has been set up on the calling machine, ssh connections can be authenticated automatically. 156.2 Interaction with the User The standard user interface for Isabelle is Proof General =-=[2]-=-. Based on the Emacs text editor, Proof General connects the evaluation of a proof to the editing of a proof script containing a series of commands. Once a command has been executed by Isabelle, the c... |

69 | Type classes and overloading in higher-order logic
- Wenzel
- 1922
(Show Context)
Citation Context ... ATP’s search space: theorems will take part in proof attempts only if the types are appropriate. 4.1 Type Classes In Isabelle, a type class is a set of types for which certain operations are defined =-=[25]-=-. An axiomatic type class has a set of axioms that must be satisfied by its instances, namely the types belonging to that class. If a type τ belongs to a class C then it is written as τ :: C. A type c... |

66 | Isabelle/Isar— a versatile environment for human-readable formal proof documents
- Wenzel
- 2002
(Show Context)
Citation Context ...des two proof styles, linear and structured. Linear proofs resemble the tactic scripts of HOL and PVS, and consist of commands that manipulate the proof state. Structured proofs (of the Isar language =-=[26]-=-) are an attempt at formalizing mathematical style. We have concentrated on supporting structured proofs, chiefly because they work on different subgoals independently. In the linear style, an Isabell... |

49 | First-order proof tactics in higher-order logic theorem provers
- Hurd
- 2003
(Show Context)
Citation Context ...thers have attempted to integrate interactive and automatic provers. • Coq has been integrated with Bliksem [4]. • HOL has integrated with various first-order provers, including Gandalf [9] and Metis =-=[10]-=-, the latter designed specifically for that integration. • Isabelle has been integrated with a purpose-built prover, blast [17]. • KIV has been integrated with a tableau prover, 3T A P [1]. Closest to... |

42 | Integrating Gandalf and HOL
- Hurd
- 1999
(Show Context)
Citation Context ...d work. Many others have attempted to integrate interactive and automatic provers. • Coq has been integrated with Bliksem [4]. • HOL has integrated with various first-order provers, including Gandalf =-=[9]-=- and Metis [10], the latter designed specifically for that integration. • Isabelle has been integrated with a purpose-built prover, blast [17]. • KIV has been integrated with a tableau prover, 3T A P ... |

38 | A Generic Tableau Prover and its Integration with Isabelle
- Paulson
- 1999
(Show Context)
Citation Context ...grated with various first-order provers, including Gandalf [9] and Metis [10], the latter designed specifically for that integration. • Isabelle has been integrated with a purpose-built prover, blast =-=[17]-=-. • KIV has been integrated with a tableau prover, 3T A P [1]. Closest to our conception is Ωmega [21]. It shares with our work the idea that automatic provers can run in the background without being ... |

37 |
Vampire 1.1 (system description
- Riazanov, Voronkov
- 2001
(Show Context)
Citation Context ...fferences have been bridged, and a working prototype that uses background processes already provides much of the desired functionality. 1 Introduction Automatic theorem provers (ATPs) such as Vampire =-=[19]-=-, which work by resolution, are impressive in their power. Interactive proof tools such as Isabelle [14] and PVS [5] provide much less automation; proofs require substantial user effort. However, inte... |

35 |
Tramp: Transformation of machine-found proofs into natural deduction proofs at the assertion level (system description
- Meier
- 2000
(Show Context)
Citation Context ...ivity of ≤ to the set of lemmas permitted in proofs. The proofs given to the user are chains of resolution-style inference rules. Much could be done to make them more concise and more readable. TRAMP =-=[11]-=- generates natural proofs, but integrating it with our system would be a major project. An alternative approach to proof reconstruction is to use the ATP’s output merely to extract the names of the le... |

31 | Integrating automated and interactive theorem proving
- Ahrendt, Beckert, et al.
- 1998
(Show Context)
Citation Context ... and Metis [10], the latter designed specifically for that integration. • Isabelle has been integrated with a purpose-built prover, blast [17]. • KIV has been integrated with a tableau prover, 3T A P =-=[1]-=-. Closest to our conception is Ωmega [21]. It shares with our work the idea that automatic provers can run in the background without being invoked by the user. However, there are also some important d... |

31 |
Rushby, Natarajan Shankar, and Mandayam Srivas. A tutorial introduction to PVS
- Crow, Owre, et al.
- 1995
(Show Context)
Citation Context ...red functionality. 1 Introduction Automatic theorem provers (ATPs) such as Vampire [19], which work by resolution, are impressive in their power. Interactive proof tools such as Isabelle [14] and PVS =-=[5]-=- provide much less automation; proofs require substantial user effort. However, interactive tools are better suited for verification projects. They admit complicated definitions and specifications, in... |

30 | TSTP Data-Exchange Formats for Automated Theorem Proving Tools
- Sutcliffe, Zimmer, et al.
- 2004
(Show Context)
Citation Context ...ence rules, simplification steps, and problem transformations. We hope that many theorem provers will eventually produce complete, explicit proofs in TSTP (Thousands of Solutions from Theorem Provers =-=[22]-=-) format. 4 Coding Isabelle Types in First-Order Logic Isabelle/HOL implements classical higher-order logic, whose complex type system is not supported by standard ATPs. Therefore, we need to model Is... |

28 | Experiments on supporting interactive proof using resolution
- Meng, Paulson
(Show Context)
Citation Context ...as, while the resolution search automatically finds other needed facts from the knowledge base. Separating relevant facts from irrelevant ones is a task that resolution theorem provers find difficult =-=[13]-=-. Isabelle provides two proof styles, linear and structured. Linear proofs resemble the tactic scripts of HOL and PVS, and consist of commands that manipulate the proof state. Structured proofs (of th... |

23 |
On skolemization and proof complexity
- Baaz
- 1994
(Show Context)
Citation Context ... use of the axiom of choice by instead pulling out and discarding the quantifier ∀x whenever we encounter a formula of the form ∀x ∃y P (x, y), at the cost of substantially increased proof complexity =-=[3]-=-. 9equivalence yields a function of one variable. Repeated application—to move an existential variable past several universal variables—results in a function of all of those variables. Rewriting with... |

10 |
de Nivelle. Automatic proof construction in type theory using resolution
- Bezem, Hendriks, et al.
- 2002
(Show Context)
Citation Context ... in need of many refinements to make it generally usable. There is much related work. Many others have attempted to integrate interactive and automatic provers. • Coq has been integrated with Bliksem =-=[4]-=-. • HOL has integrated with various first-order provers, including Gandalf [9] and Metis [10], the latter designed specifically for that integration. • Isabelle has been integrated with a purpose-buil... |

5 | Integration of interactive and automatic provers
- Meng
(Show Context)
Citation Context ...setting boolean flags. 5.4 Summary of Preliminary Experiments Before undertaking any implementation, we carried out a series of experiments in order to examine whether our approach would be practical =-=[12,13]-=-. The experiments consisted of taking basic tactic invocations (the tactics were blast, fast, clarify, auto and simp) from existing proofs. We attempted to reproduce them using Vampire and SPASS, with... |

4 | Isabelle’s logics: FOL and ZF
- Paulson
- 2003
(Show Context)
Citation Context ... Isabelle is generic: it supports a wide range of formalisms. The most important object-logic is higher-order logic (HOL), but several others are available, including Zermelo-Fraenkel set theory (ZF) =-=[18]-=-. The version of higher-order logic in Isabelle has polymorphic types (here we mean logical types, not ML types). Unlike other implementations of HOL, Isabelle also provides the concept of axiomatic t... |

2 |
editors. The Standard ML Basis Library
- Gansner, Reppy
- 2004
(Show Context)
Citation Context ...OSIX is an international standard that defines how an application obtains the basic services of an operating system. 4 The watcher calls the POSIX functions specified in the Standard ML Basis Library =-=[6]-=-. Both Poly/ML and Standard ML of New Jersey implement these functions. 4 See http://www.pasc.org/. 146.1 Watcher Process From the main Isabelle process, a watcher process can be created with the com... |