## Progress report on LEO-II, an automatic theorem prover for higher-order logic (2007)

Citations: | 10 - 6 self |

### BibTeX

@MISC{Benzmüller07progressreport,

author = {Christoph Benzmüller and Larry Paulson and Frank Theiss and Arnaud Fietzke},

title = {Progress report on LEO-II, an automatic theorem prover for higher-order logic},

year = {2007}

}

### OpenURL

### Abstract

Abstract. Leo-II, a resolution based theorem prover for classical higherorder logic, is currently being developed in a one year research project at the University of Cambridge, UK, with support from Saarland University, Germany. We report on the current stage of development of Leo-II. In particular, we sketch some main aspects of Leo-II’s automated proof search procedure, discuss its cooperation with first-order specialist provers, show that Leo-II is also an interactive proof assistant, and explain its shared term data structure and its term indexing mechanism. 1

### Citations

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Citation Context ...a structures and term indexing. Sec. 7 mentions related work and Sec. 8 concludes the paper. 2 Preliminaries Leo-II’s logic Leo-II’s logic is classical higher-order logic (Church’s simple type theory =-=[11]-=-), which is a logic built on top of the simply typed λ-calculus. The set of simple types T is usually freely generated from basic types o and ι using the function type constructor →. In Leo-II we allo... |

719 |
Isabelle/HOL: A Proof Assistant for Higher-Order Logic
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Citation Context ...nal logic. The idea is to combine the strengths of the different systems. On the other hand, Leo-II itself, as an external reasoner, wants to support interactive proof assistants such as Isabelle/HOL =-=[24]-=-, HOL [16], and OMEGA [28] by efficiently automating subproblems and thereby reducing user effort. Leo-II predominantly addresses higher-order aspects in its reasoning process with the aim to quickly ... |

308 | An open graph visualization system and its applications to software engineering
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Citation Context ...red terms), and term index. For example, the command termgraph-to-dot generates a graphical representation of Leo-II’s shared term data structure, which is a directed acyclic graph, in the DOT syntax =-=[15]-=- which can be processed by the program dot [15] in order obtain, for example, the ps-representation as given in Fig. 6. With the command analyze-index we may request useful statistical information abo... |

303 | N.: Lambda Calculus Notation with Nameless Dummies, a Tool for Automatic Formula Manipulation.; Indag
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Citation Context ... first-order theorem proving [26,27,31]: syntactically equal terms are represented by a single instance. For Leo-II, we have adapted this technique to the higher-order case. We use de Bruijn-notation =-=[12]-=- to avoid blurring of syntactical equality by α-conversion. A shared representation of terms has multiple benefits. The most obvious is the instant identification of all occurrences of a term or subte... |

198 |
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Citation Context ... The idea is to combine the strengths of the different systems. On the other hand, Leo-II itself, as an external reasoner, wants to support interactive proof assistants such as Isabelle/HOL [24], HOL =-=[16]-=-, and OMEGA [28] by efficiently automating subproblems and thereby reducing user effort. Leo-II predominantly addresses higher-order aspects in its reasoning process with the aim to quickly remove hig... |

181 |
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Citation Context ...ble (consistent with given brackets). We use infix notation A ∨ B for ((∨A)B) and binder notation ∀Xα A for (Π α (λXα Ao)). The target semantics for Leo-II in the first place is Henkin semantics; see =-=[8, 17]-=- for further details. Thus, in theory Leo-II aims at a Henkin complete calculus which includes Boolean and functional extensionality as required, for instance, to prove = ι = ι→ι→o (λXι.λYι.Y = ι X) I... |

127 |
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Citation Context ...ax transformations. Currently, the transformations supported in LEO are based on Kerber’s work [19] and on the work of Hurd [18]. For communication with first-order provers we use the TPTP FOF syntax =-=[29]-=-. – The specialist reasoners will run in parallel to Leo-II (cf. [5, 3,2,4]) and they will incrementally receive new clauses from Leo-II belonging to their fragment. Once they find a refutation, they ... |

125 | E – A Brainiac Theorem Prover
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(Show Context)
Citation Context ...proof assistant, and explain its shared term data structure and its term indexing mechanism. 1 Introduction Automatic theorem provers (ATPs) based on the resolution principle, such as Vampire [26], E =-=[27]-=-, and SPASS [31], have reached a high degree of sophistication. They can often find long proofs even for problems having thousands of axioms. However, they are limited to first-order logic. Higher-ord... |

54 | SPASS & FLOTTER, version 0.42
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(Show Context)
Citation Context ...ately, clause normalization as currently realized in Leo-II is quite naive. Future work therefore includes the development of a more efficient approach, for example, one based on the ideas of Flotter =-=[32]-=-. Fig. 2 shows the result of our running example after problem initialization, definition unfolding, and clause normalization. The clauses present in Leo-II’s search state then are 13, 25, and 31. In ... |

35 |
Tramp: Transformation of machine-found proofs into natural deduction proofs at the assertion level (system description
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Citation Context ...he cooperation of different reasoning strategies, and the Techs system [13], which realizes a cooperation between a set of heterogeneous first-order theorem provers. Related is also the work of Meier =-=[20]-=-, Hurd [18], and Meng/Paulson [21,23]. They realize interfaces between proof assistants (OMEGA, HOL, and Isabelle/HOL) and first-order theorem provers. All these approaches pass essentially first-orde... |

34 | OANTS – an open approach at combining interactive and automated theorem proving
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(Show Context)
Citation Context ...re based on Kerber’s work [19] and on the work of Hurd [18]. For communication with first-order provers we use the TPTP FOF syntax [29]. – The specialist reasoners will run in parallel to Leo-II (cf. =-=[5, 3,2,4]-=-) and they will incrementally receive new clauses from Leo-II belonging to their fragment. Once they find a refutation, they report this back to Leo-II, which then reports that a proof has been found.... |

32 | An LCF-style interface between HOL and first-order logic
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(Show Context)
Citation Context ... SPASS. Nevertheless our idea is generic and Leo-II may later support also other fragments, with propositional logic and monadic second-order logic as possible candidates. – As has been shown by Hurd =-=[18]-=- and Meng/Paulson [22] the particular syntax translations used to convert essentially higher-order clauses to essentially first-order ones may have a strong impact on the efficiency of the specialist ... |

28 | Experiments on supporting interactive proof using resolution
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(Show Context)
Citation Context ...ing strategies, and the Techs system [13], which realizes a cooperation between a set of heterogeneous first-order theorem provers. Related is also the work of Meier [20], Hurd [18], and Meng/Paulson =-=[21,23]-=-. They realize interfaces between proof assistants (OMEGA, HOL, and Isabelle/HOL) and first-order theorem provers. All these approaches pass essentially first-order clauses to first-order theorem prov... |

28 | Automation for interactive proof: First prototype
- Meng, Quigley, et al.
(Show Context)
Citation Context ...ing strategies, and the Techs system [13], which realizes a cooperation between a set of heterogeneous first-order theorem provers. Related is also the work of Meier [20], Hurd [18], and Meng/Paulson =-=[21,23]-=-. They realize interfaces between proof assistants (OMEGA, HOL, and Isabelle/HOL) and first-order theorem provers. All these approaches pass essentially first-order clauses to first-order theorem prov... |

25 |
Cooperation of Heterogeneous Provers
- Denzinger, Fuchs
- 1999
(Show Context)
Citation Context ... 7 Related Work The integration of reasoners and reasoning strategies was pioneered in the Teamwork system [14], which realizes the cooperation of different reasoning strategies, and the Techs system =-=[13]-=-, which realizes a cooperation between a set of heterogeneous first-order theorem provers. Related is also the work of Meier [20], Hurd [18], and Meng/Paulson [21,23]. They realize interfaces between ... |

25 | Goal oriented equational theorem proving using team work
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(Show Context)
Citation Context ...et for many other important components, including simplification, rewriting, and subsumption. 7 Related Work The integration of reasoners and reasoning strategies was pioneered in the Teamwork system =-=[14]-=-, which realizes the cooperation of different reasoning strategies, and the Techs system [13], which realizes a cooperation between a set of heterogeneous first-order theorem provers. Related is also ... |

18 |
Higher Order Semantics and Extensionality
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(Show Context)
Citation Context ...ble (consistent with given brackets). We use infix notation A ∨ B for ((∨A)B) and binder notation ∀Xα A for (Π α (λXα Ao)). The target semantics for Leo-II in the first place is Henkin semantics; see =-=[8, 17]-=- for further details. Thus, in theory Leo-II aims at a Henkin complete calculus which includes Boolean and functional extensionality as required, for instance, to prove = ι = ι→ι→o (λXι.λYι.Y = ι X) I... |

16 | Selectively instantiating definitions
- Bishop, Andrews
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(Show Context)
Citation Context ...nefits from the shared term data structures and indexing techniques as will be sketched in Sec. 6). Delayed and stepwise definition unfolding, which is required to successfully prove certain theorems =-=[10]-=-, is future work. When applied to our example clause C1, definition unfolding generates clause 3 as depicted in Fig. 2. Clause Normalization For clause normalization, Leo-II employs rules addressing t... |

16 | Higher-order substitution tree indexing
- Pientka
- 2003
(Show Context)
Citation Context ...nt on term analysis is compensated by the reusability of the results. The indexing approach of Leo-II has a strong focus on structural aspects. It differs in this respect from the approach by Pientka =-=[25]-=-, which is based on a discrimination tree and which allows for perfect filtering on the basis of higher order pattern unification. In contrast, we are particularly interested also in more relaxed sear... |

14 |
On the Representation of Mathematical Concepts and their Translation into First-Order Logic
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- 1992
(Show Context)
Citation Context ...act on the efficiency of the specialist provers. Thus, we later want to support a wide range of such syntax transformations. Currently, the transformations supported in LEO are based on Kerber’s work =-=[19]-=- and on the work of Hurd [18]. For communication with first-order provers we use the TPTP FOF syntax [29]. – The specialist reasoners will run in parallel to Leo-II (cf. [5, 3,2,4]) and they will incr... |

14 |
Computer supported mathematics with Omega
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Citation Context ...combine the strengths of the different systems. On the other hand, Leo-II itself, as an external reasoner, wants to support interactive proof assistants such as Isabelle/HOL [24], HOL [16], and OMEGA =-=[28]-=- by efficiently automating subproblems and thereby reducing user effort. Leo-II predominantly addresses higher-order aspects in its reasoning process with the aim to quickly remove higher-order clause... |

13 | Equality and Extensionality in Automated Higher-Order Theorem Proving
- Benzmüller
- 1999
(Show Context)
Citation Context ...tion, unfolding of definitions, and exhaustive clause normalization. The Vi are free variables. Extensional Pre-Unification Pre-unification in Leo-II is based on the rules as presented in former work =-=[6]-=-. It is well known, that pre-unification is not decidable (remember our discussion of flex-rigid unification constraints in the beginning) which is why Leo-II introduces a depth limit for the generati... |

12 | Experiments with an AgentOriented Reasoning System
- Benzmüller, Jamnik, et al.
- 2001
(Show Context)
Citation Context ...re based on Kerber’s work [19] and on the work of Hurd [18]. For communication with first-order provers we use the TPTP FOF syntax [29]. – The specialist reasoners will run in parallel to Leo-II (cf. =-=[5, 3,2,4]-=-) and they will incrementally receive new clauses from Leo-II belonging to their fragment. Once they find a refutation, they report this back to Leo-II, which then reports that a proof has been found.... |

11 | Can a higher-order and a first-order theorem prover cooperate
- Benzmüller, Sorge, et al.
- 2005
(Show Context)
Citation Context ...re based on Kerber’s work [19] and on the work of Hurd [18]. For communication with first-order provers we use the TPTP FOF syntax [29]. – The specialist reasoners will run in parallel to Leo-II (cf. =-=[5, 3,2,4]-=-) and they will incrementally receive new clauses from Leo-II belonging to their fragment. Once they find a refutation, they report this back to Leo-II, which then reports that a proof has been found.... |

11 | Combined reasoning by automated cooperation
- Benzmüller, Sorge, et al.
(Show Context)
Citation Context |

6 |
The design and implementation of Vampire. AICOM
- Riazanov, Voronkov
- 2002
(Show Context)
Citation Context ...ractive proof assistant, and explain its shared term data structure and its term indexing mechanism. 1 Introduction Automatic theorem provers (ATPs) based on the resolution principle, such as Vampire =-=[26]-=-, E [27], and SPASS [31], have reached a high degree of sophistication. They can often find long proofs even for problems having thousands of axioms. However, they are limited to first-order logic. Hi... |

6 |
Term indexing for the LEO-II prover
- Theiss, Benzmüller
- 2006
(Show Context)
Citation Context ...ize / term size: 0.504699009398 ------------- End Termset Analysis -------------s6 Shared Term Data Structures and Term Indexing Operations on terms in Leo-II are supported by term indexing (see also =-=[30]-=-). Key features of Leo-II’s term indexing are the representation of terms in a perfectly shared graph structure and the indexing of various structural properties, such as the occurrence of subterms an... |

6 |
et al. Spass version 2.0
- Weidenbach
- 2002
(Show Context)
Citation Context ... and explain its shared term data structure and its term indexing mechanism. 1 Introduction Automatic theorem provers (ATPs) based on the resolution principle, such as Vampire [26], E [27], and SPASS =-=[31]-=-, have reached a high degree of sophistication. They can often find long proofs even for problems having thousands of axioms. However, they are limited to first-order logic. Higher-order logic extends... |

4 | System description: Leo – a resolution based higher-order theorem prover
- Benzmüller
- 2005
(Show Context)
Citation Context ...ssful in automated first-order theorem proving, such as shared data structures and term indexing, can be lifted to the higher-order setting. Leo-II is implemented in OCAML; it is the successor of LEO =-=[9,7]-=-, which was implemented in LISP and hardwired to the OMEGA proof assistant. This paper is structured as follows: Sec. 2 presents some preliminaries. Sec. 3 illustrates Leo-II’s main proof search proce... |

3 | System description: LEO — a higher-order theorem prover
- Benzmüller, Kohlhase
- 1998
(Show Context)
Citation Context ...ssful in automated first-order theorem proving, such as shared data structures and term indexing, can be lifted to the higher-order setting. Leo-II is implemented in OCAML; it is the successor of LEO =-=[9,7]-=-, which was implemented in LISP and hardwired to the OMEGA proof assistant. This paper is structured as follows: Sec. 2 presents some preliminaries. Sec. 3 illustrates Leo-II’s main proof search proce... |