## First-order proof tactics in higher-order logic theorem provers (2003)

Venue: | Design and Application of Strategies/Tactics in Higher Order Logics, number NASA/CP-2003-212448 in NASA Technical Reports |

Citations: | 49 - 4 self |

### BibTeX

@INPROCEEDINGS{Hurd03first-orderproof,

author = {Joe Hurd},

title = {First-order proof tactics in higher-order logic theorem provers},

booktitle = {Design and Application of Strategies/Tactics in Higher Order Logics, number NASA/CP-2003-212448 in NASA Technical Reports},

year = {2003},

pages = {56--68}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. In this paper we evaluate the effectiveness of first-order proof procedures when used as tactics for proving subgoals in a higher-order logic interactive theorem prover. We first motivate why such first-order proof tactics are useful, and then describe the core integrating technology: an ‘LCFstyle’ logical kernel for clausal first-order logic. This allows the choice of different logical mappings between higher-order logic and first-order logic to be used depending on the subgoal, and also enables several different first-order proof procedures to cooperate on constructing the proof. This work was carried out using the HOL4 theorem prover; we comment on the ease of transferring the technology to other higher-order logic theorem provers. 1

### Citations

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(Show Context)
Citation Context ...m-u which proves 371 more TPTP problems. Similar results occur on the HOL problem sets. 4.2 Resolution Procedure The second proof procedure that we implemented is the resolution procedure of Robinson =-=[21]-=-. Our version uses the given clause algorithm, and we implement term nets to improve the speed of unification and subsumption checking. Additionally, unit clauses are used whenever possible to simplif... |

534 |
Theory of Statistics
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(Show Context)
Citation Context ...ounting the number of problems in each section that any two provers solve within the time limit, we can use the t-test to compute the statistical significance that one prover is better than the other =-=[9]-=-. Here is an example results table where we compare two hypothetical provers, foo and bar: foo bar +95 foo ∗ 99.5% bar +7 Since we do not compare provers with themselves, the diagonal entries are mark... |

227 | Purely Functional Data Structures
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- 1998
(Show Context)
Citation Context ...y either acting alone. This is compelling evidence that the combined procedure 14 We efficiently implement this alternation in ML by storing unused clauses as both queues and (leftist) heaps. Okasaki =-=[18]-=- implements functional versions of these and many more data structures. 15 In our experiments we set each time slice to be 1/3 second long. 9sTable 3. Comparing Combinations of Provers on the TPTP Pro... |

198 |
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(Show Context)
Citation Context ...using the HOL4 theorem prover; we comment on the ease of transferring the technology to other higher-order logic theorem provers. 1 Introduction Performing interactive proof in the HOL theorem prover =-=[12]-=- involves reducing goals to simpler subgoals. It turns out that many of these subgoals can be efficiently ‘finished off’ by an automatic first-order prover. To fill this niche, Harrison implemented a ... |

54 |
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- Loveland
- 1968
(Show Context)
Citation Context ...-order prover. To fill this niche, Harrison implemented a version of the MESON procedure [13] with the ability to translate proofs to higher-order logic. The original MESON procedure, due to Loveland =-=[17]-=-, is a sound and complete calculus for finding proofs in full first-order logic. This was integrated as a HOL tactic in 1996, and has since become a standard workhorse of interactive proof. Today, bui... |

50 |
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(Show Context)
Citation Context ...n is one of 1, 2, 3, 4 or 5, and is part of the prover name. This is called the ratio strategy, originally used in the Otter theorem prover [26]. The final parameter is Robinson’s positive refinement =-=[20]-=-, which we indicate with a final ‘ + ’ in the prover name. We found that the best prover for all three problem sets is r3 + : the default level of subsumption; picking 3 smallest clauses for every cla... |

49 | Caching and lemmaizing in model elimination theorem provers
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(Show Context)
Citation Context ... the model elimination procedure of Loveland [17]; our prover is essentially a ground-up reimplementation of Harrison’s MESON [13], incorporating some optimizations of Astrachan, Loveland and Stickel =-=[2, 3]-=-. 12 Moscow ML is available at http://www.dina.dk/ ∼ sestoft/mosml.html. 7 ∗sOur strategy is to first produce a naive implemention, and then incrementally optimize it. The starting point is a version ... |

42 | Integrating Gandalf and HOL
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- 1999
(Show Context)
Citation Context ...Since HOL4 is written in Standard ML, this is a convenient implementation language for our experiment, though in the past similar experiments have been performed by making calls to external C provers =-=[14]-=-. Therefore, this paper also provides a view of implementing first-order proof procedures in a functional programming language, and some interesting aspects of this are brought out in discussion. The ... |

38 | A Generic Tableau Prover and its Integration with Isabelle
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- 1999
(Show Context)
Citation Context ...matic first-order provers being used to prove problems in an interactive theorem prover: (in chronological order) FAUST in HOL [16]; SEDUCT in LAMBDA [5]; 3TAP in KIV [1]; Paulson’s blast in Isabelle =-=[19]-=-; Gandalf in HOL [14]; and Bliksem in Coq [4]. Various mappings are used from the theorem prover subgoals into problems of first-order logic, defining the scope of what can be automatically proved. Us... |

32 | An LCF-style interface between HOL and first-order logic
- Hurd
- 2002
(Show Context)
Citation Context ...as and proofs between higher-order logic and first-order logic, which plays a role in steps 2 and 4 of the above process. Much of this information appears in a previously published system description =-=[15]-=-; it is reproduced here because it is an essential part of our framework for creating first-order proof tactics. Before getting into the details, we first give an extended example of the whole process... |

31 | Integrating automated and interactive theorem proving
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- 1998
(Show Context)
Citation Context ...e are many other examples of automatic first-order provers being used to prove problems in an interactive theorem prover: (in chronological order) FAUST in HOL [16]; SEDUCT in LAMBDA [5]; 3TAP in KIV =-=[1]-=-; Paulson’s blast in Isabelle [19]; Gandalf in HOL [14]; and Bliksem in Coq [4]. Various mappings are used from the theorem prover subgoals into problems of first-order logic, defining the scope of wh... |

20 | Optimizing proof search in model elimination
- Harrison
- 1996
(Show Context)
Citation Context ...pler subgoals. It turns out that many of these subgoals can be efficiently ‘finished off’ by an automatic first-order prover. To fill this niche, Harrison implemented a version of the MESON procedure =-=[13]-=- with the ability to translate proofs to higher-order logic. The original MESON procedure, due to Loveland [17], is a sound and complete calculus for finding proofs in full first-order logic. This was... |

14 | The use of lemmas in the model elimination procedure
- Astrachan, Loveland
- 1997
(Show Context)
Citation Context ... the model elimination procedure of Loveland [17]; our prover is essentially a ground-up reimplementation of Harrison’s MESON [13], incorporating some optimizations of Astrachan, Loveland and Stickel =-=[2, 3]-=-. 12 Moscow ML is available at http://www.dina.dk/ ∼ sestoft/mosml.html. 7 ∗sOur strategy is to first produce a naive implemention, and then incrementally optimize it. The starting point is a version ... |

13 | Integrating a first-order automatic prover in the HOL environment
- Kumar, Kropf, et al.
- 1991
(Show Context)
Citation Context ...d Work In addition to MESON in HOL, there are many other examples of automatic first-order provers being used to prove problems in an interactive theorem prover: (in chronological order) FAUST in HOL =-=[16]-=-; SEDUCT in LAMBDA [5]; 3TAP in KIV [1]; Paulson’s blast in Isabelle [19]; Gandalf in HOL [14]; and Bliksem in Coq [4]. Various mappings are used from the theorem prover subgoals into problems of firs... |

12 | C.: First order proof problems extracted from an article
- Dahn, Wernhard
- 1997
(Show Context)
Citation Context ...’ logical kernel for clausal first-order logic, then this would further simplify their integration into interactive theorem provers. As part of the ILF Mathematical Library Project, Dahn and Wernhard =-=[8]-=- extracted 97 first-order problems from the article Boolean Properties of Sets in the Mizar Mathematical Library. Later, Dahn [7] added the ability to represent Mizar type information, and extracted 4... |

7 |
First-order automation for higher-order-logic theorem proving
- Busch
- 1994
(Show Context)
Citation Context ...ESON in HOL, there are many other examples of automatic first-order provers being used to prove problems in an interactive theorem prover: (in chronological order) FAUST in HOL [16]; SEDUCT in LAMBDA =-=[5]-=-; 3TAP in KIV [1]; Paulson’s blast in Isabelle [19]; Gandalf in HOL [14]; and Bliksem in Coq [4]. Various mappings are used from the theorem prover subgoals into problems of first-order logic, definin... |

7 |
A note on mechanizing higher order logic
- ROBINSON
- 1969
(Show Context)
Citation Context ...vers, as opposed to our system that simply shares unit clauses. Further investigation is needed to decide the best way of combining proof procedures in our application. Finally, we note that Robinson =-=[22]-=- proposed a version of higher-order logic in terms of combinators (though it is typeless and therefore unsound due to the ‘Russell combinator’ we defined in Section 2.3). 17 However, the motivation be... |

6 | Interpretation of a Mizar-like logic in first-order logic
- Dahn
- 1998
(Show Context)
Citation Context ...rs. As part of the ILF Mathematical Library Project, Dahn and Wernhard [8] extracted 97 first-order problems from the article Boolean Properties of Sets in the Mizar Mathematical Library. Later, Dahn =-=[7]-=- added the ability to represent Mizar type information, and extracted 47 problems from the article Relations Defined on Sets. However, there has been no published study of the comparitive effectivenes... |

6 | Knowledge-Based Cooperation between Theorem Provers by Techs
- Fuchs
- 1997
(Show Context)
Citation Context ...iveness of first-order provers on this problem set. Several projects have aimed to create combination first-order provers that are better than the individual components. For example, the TECHS system =-=[10]-=- uses automatic referees to decide which clauses to exchange between provers, as opposed to our system that simply shares unit clauses. Further investigation is needed to decide the best way of combin... |

3 |
Hendriks and Hans de Nivelle. Automated Proof Construction in Type Theory using Resolution
- Bezem, Dimitri
- 2000
(Show Context)
Citation Context ...problems in an interactive theorem prover: (in chronological order) FAUST in HOL [16]; SEDUCT in LAMBDA [5]; 3TAP in KIV [1]; Paulson’s blast in Isabelle [19]; Gandalf in HOL [14]; and Bliksem in Coq =-=[4]-=-. Various mappings are used from the theorem prover subgoals into problems of first-order logic, defining the scope of what can be automatically proved. Using the architecture presented in this paper ... |

1 |
DELTA — A bottom-up processor for top-down theorem provers (system abstract
- Schumann
(Show Context)
Citation Context ...rameter setting is better than any other with confidence at least 95%. 4.3 Delta-style Procedure The third and final proof procedure that we implemented is based on the Delta preprocessor of Schumann =-=[23]-=-. Put simply, for every n-ary relation R present in the problem, we generate the ‘Delta goals’ R(X1, . . . , Xn) and ¬R(Y1, . . . , Yn) (with fresh variables Xi and Yi). We then use the model eliminat... |