### BibTeX

@MISC{Asperti_smartmatching,

author = {Andrea Asperti and Enrico Tassi},

title = {Smart matching},

year = {}

}

### OpenURL

### Abstract

One of the most annoying aspects in the formalization of mathematics is the need of transforming notions to match a given, existing result. This kind of transformations, often based on a conspicuous background knowledge in the given scientific domain (mostly expressed in the form of equalities or isomorphisms), are usually implicit in the mathematical discourse, and it would be highly desirable to obtain a similar behaviour in interactive provers. The paper describes the superposition-based implementation of this feature inside the Matita interactive theorem prover, focusing in particular on the so called smart application tactic, supporting smart matching between a goal and a given result.

### Citations

489 |
Simple word problems in universal algebras
- Knuth, Bendix
- 1970
(Show Context)
Citation Context ...ensibly easier. Techniques for equational reasoning have been broadly investigated in the realm of automated theorem proving (see eg [7,21,10]). The main deductive mechanism is a completion technique =-=[17]-=- attempting to transform a given set of equations into a confluent rewriting system so that two terms are equal if and only if they have identical normal forms. Not every equational theory can be pres... |

453 | Termination of rewriting
- Dershowitz
- 1985
(Show Context)
Citation Context ...ll probably require implementing more elaborate data structures, such as substitution [14] or context trees [13], or even adopt an indexing technique that works modulo associativity and commutativity =-=[12]-=-, that looks very promising when working on algebraic structures. Demodulation Another important issue for performance is demodulation: the given clause algorithm spends most of its time (up to 80%) i... |

265 | Orderings for term rewriting systems
- DERSHOWITZ
- 1982
(Show Context)
Citation Context ...An ordering is stable under substitution if s ≺ t implies sσ ≺ tσ for all terms t, s and substitutions σ. A well founded monotonic ordering stable under substitution is called reduction ordering (see =-=[11]-=-). The intuition behind the use of reduction orderings for limiting the combinatorial explosion of new equations during inference, is to only rewrite big terms to smaller ones. superposition left supe... |

197 |
Rewrite-based equational theorem proving with selection and simplification
- Bachmair, Ganzinger
- 1994
(Show Context)
Citation Context ...igence, making the communication between the user and the machine sensibly easier. Techniques for equational reasoning have been broadly investigated in the realm of automated theorem proving (see eg =-=[7,21,10]-=-). The main deductive mechanism is a completion technique [17] attempting to transform a given set of equations into a confluent rewriting system so that two terms are equal if and only if they have i... |

186 | Fast decision procedures based on congruence closure
- Nelson, Oppen
- 1980
(Show Context)
Citation Context ... until no further reduction is possible. The tactic does not perform narrowing, nor any form of completion. The congruence tactic implements the standard Nelson and Oppen congruence closure algorithm =-=[21]-=-, which is a decision procedure for ground equalities with uninterpreted symbols; the COQ tactic only deals with equalities in the local context. Both COQ tactics are sensibly weaker than superpositio... |

172 |
A.: The design and implementation of VAMPIRE
- Riazanov, Voronkov
- 2002
(Show Context)
Citation Context ...f new equations generated at each step grows extremely fast, in practice preventing the iteratation of the main loop more than a few times. To avoid this problem, all modern theorem provers (see e.g. =-=[23]-=-) adopt the opposite solution. According to some heuristics, like size and goal similarity for example, they select only one passive clause at each step. Not to loose completeness, some fairness condi... |

49 | First-order proof tactics in higher-order logic theorem provers
- Hurd
- 2003
(Show Context)
Citation Context ...its abstraction and performance. The tool took part to the 22nd CADE ATP System Competition, in the unit equality division, scoring in fourth position, beating glorious systems such as Otter or Metis =-=[16]-=-, and being awarded as the best new entrant tool of the competion [27]. In the rest of this section we shall give an outline, as concise as possible, of the theory and the architecture of the tool. Th... |

46 | User interaction with the Matita proof assistant
- Asperti, Coen, et al.
- 2007
(Show Context)
Citation Context ...h some available tool. This paper discusses our experience of implementation of a (first order) superposition calculus (Section 2), its integration within the (higher-order) Matita interactive prover =-=[5]-=- (Section 3), and in particular its use for the implementation of a smart application tactic, supporting smart matching between a goal and a given results (Section 4). We shall conclude with a large n... |

42 | Integrating Gandalf and HOL
- Hurd
- 1999
(Show Context)
Citation Context ...y efforts have been already done in this direction: for instance, KIV has been integrated with the tableau prover 3T A P [1]; HOL has been integrated with various first order provers, such as Gandalf =-=[15]-=- and Metis; Coq has been integrated with Bliksem [8]; Isabelle was first integrated with a purpose-built prover [22] and more recently with Vampire [20]. The problems of these integrations are usually... |

38 | A Generic Tableau Prover and its Integration with Isabelle
- Paulson
- 1999
(Show Context)
Citation Context ... A P [1]; HOL has been integrated with various first order provers, such as Gandalf [15] and Metis; Coq has been integrated with Bliksem [8]; Isabelle was first integrated with a purpose-built prover =-=[22]-=- and more recently with Vampire [20]. The problems of these integrations are usually of two kinds: (a) there is a technical difficulty in the forward and backward translation of the information betwee... |

36 |
The concept of demodulation in theorem proving
- wos, Robinson, et al.
- 1967
(Show Context)
Citation Context ...σ t1 = t2 ⊢ □ if ∃σ = mgu(t1, t2). Fig. 1. Inference rules For efficiency reasons, the calculus must be integrated with a few additional optimization rules, the most important one being demodulation (=-=[28]-=-). 2.2 The main algorithm A naive implementation of the superposition calculus could just combine (superpose) all known clauses in all (admitted) ways, and repeat that process until the desired clause... |

31 | Integrating automated and interactive theorem proving
- Ahrendt, Beckert, et al.
- 1998
(Show Context)
Citation Context ...ring paramodulation) and interactive ones is a major research challenge and many efforts have been already done in this direction: for instance, KIV has been integrated with the tableau prover 3T A P =-=[1]-=-; HOL has been integrated with various first order provers, such as Gandalf [15] and Metis; Coq has been integrated with Bliksem [8]; Isabelle was first integrated with a purpose-built prover [22] and... |

31 | Substitution tree indexing
- Graf
- 1995
(Show Context)
Citation Context ...ance, we did not invest resources in adopting better data structures, but we believe that further optimizations will probably require implementing more elaborate data structures, such as substitution =-=[14]-=- or context trees [13], or even adopt an indexing technique that works modulo associativity and commutativity [12], that looks very promising when working on algebraic structures. Demodulation Another... |

28 | Automation for interactive proof: First prototype
- Meng, Quigley, et al.
(Show Context)
Citation Context ...th various first order provers, such as Gandalf [15] and Metis; Coq has been integrated with Bliksem [8]; Isabelle was first integrated with a purpose-built prover [22] and more recently with Vampire =-=[20]-=-. The problems of these integrations are usually of two kinds: (a) there is a technical difficulty in the forward and backward translation of the information between systems, due to the different unde... |

26 | L.C.: Translating higher-order clauses to first-order clauses
- Meng, Paulson
- 2008
(Show Context)
Citation Context ...communication between the two components is hence far from trivial. Instead of attempting a complex, faithful encoding of CIC in first order logic (that is essentially the approach adopted for HOL in =-=[19]-=-) we choose to follow a more naif approach, based on a forgetful translation that remove types and just keeps the first order applicative skeleton of CIC-terms. In the opposite direction, we try to re... |

20 | Equality reasoning in sequentbased calculi
- Degtyarev, Voronkov
- 2001
(Show Context)
Citation Context ...igence, making the communication between the user and the machine sensibly easier. Techniques for equational reasoning have been broadly investigated in the realm of automated theorem proving (see eg =-=[7,21,10]-=-). The main deductive mechanism is a completion technique [17] attempting to transform a given set of equations into a confluent rewriting system so that two terms are equal if and only if they have i... |

20 |
Experiments with discrimination tree indexing and path indexing for term retrieval
- McCune
- 1992
(Show Context)
Citation Context ... steps if the resulting clauses are unlikely to be processed, mainly because of a lack of time. Data structures and code optimization We adopted relatively simple data structures (like discrimination =-=[18]-=- trees for term indexing), and a purely functional (in the sense of functional programming) implementation of them. After some code optimisation, we reached a point where very fast functions are the m... |

17 | 2001, ‘Uniform Derivation of Decision Procedures by Superposition
- Armando, Ranise, et al.
(Show Context)
Citation Context ...q tactics are sensibly weaker than superposition that seems to provide a good surrogate for several decision procedures for various theories, as well as a simple framework for composing them (see e.g =-=[2]-=-). Comparing the integration of superposition in Matita with similar functionalities provided by Isabelle is twofold complex, due not only to the different approaches, but also to the different underl... |

17 | Limited resource strategy in resolution theorem proving
- Riazanov, Voronkov
- 2003
(Show Context)
Citation Context ...s that would slow down the main loop too much due to their excessive size. Another similar optimization we did not implement but we could consider as a future development is Limited Resource Strategy =-=[24]-=-, which basically allows the procedure to skip some inference steps if the resulting clauses are unlikely to be processed, mainly because of a lack of time. Data structures and code optimization We ad... |

16 | Nivelle. Automated Proof Construction in Type Theory using Resolution
- Bezem, de
- 2002
(Show Context)
Citation Context ...or instance, KIV has been integrated with the tableau prover 3T A P [1]; HOL has been integrated with various first order provers, such as Gandalf [15] and Metis; Coq has been integrated with Bliksem =-=[8]-=-; Isabelle was first integrated with a purpose-built prover [22] and more recently with Vampire [20]. The problems of these integrations are usually of two kinds: (a) there is a technical difficulty i... |

13 |
The 5th IJCAR Automated Theorem Proving System Competition - CASC-J5
- Sutcliffe
(Show Context)
Citation Context ...TP System Competition, in the unit equality division, scoring in fourth position, beating glorious systems such as Otter or Metis [16], and being awarded as the best new entrant tool of the competion =-=[27]-=-. In the rest of this section we shall give an outline, as concise as possible, of the theory and the architecture of the tool. This is important in order to understand its integration with the intera... |

8 |
A new look at generalized rewriting in type theory
- Sozeau
- 2009
(Show Context)
Citation Context ...mentation of more sophisticated data structures for indexes (we currently use discrimination trees). Another interesting research direction is to extend the management of equality to setoid rewriting =-=[26]-=-. Indeed, the current version of the superposition tool just works with an intensional equality, and it would be interesting to try to figure out how to handle more general binary relations. The hard ... |

8 |
Sacerdoti Coen, and Enrico Tassi. Hints in unification
- Asperti, Ricciotti, et al.
- 2009
(Show Context)
Citation Context ...first order applicative skeleton of CIC-terms. In the opposite direction, we try to reconstruct the missing information by just exploiting the sophisticated inference capability of the Matita refiner =-=[3]-=-, that is the tool in charge of transforming the user input into a machine understandable low-level CIC term. Automation is thus a best effort service, in the sense that not only it may obviously fail... |

7 |
2009a) A compact kernel for the Calculus of Inductive Constructions
- Asperti, Ricciotti, et al.
(Show Context)
Citation Context ...s and systems Matita was essentially conceived as a light version of Coq [9], sharing the same foundational logic (the Calculus of Inductive Constructions) and being partially compatible with it (see =-=[4]-=- for a discussion of the main differences between the 3 The lifting operation ↑ n (M) is meant to relocate the term M under n additional levels of bindings: in other words, it increases by n all free ... |

7 | Fast term indexing with coded context trees
- Ganzinger, Nieuwenhuis, et al.
(Show Context)
Citation Context ...t resources in adopting better data structures, but we believe that further optimizations will probably require implementing more elaborate data structures, such as substitution [14] or context trees =-=[13]-=-, or even adopt an indexing technique that works modulo associativity and commutativity [12], that looks very promising when working on algebraic structures. Demodulation Another important issue for p... |

7 | A constructive and formal proof of Lebesgue’s dominated convergence theorem in the interactive theorem prover Matita
- Coen, C, et al.
- 2008
(Show Context)
Citation Context ... refiner [3] to automatically check the legality of the rewriting operation (i.e. the monotonicity of the context inside which rewriting has to be performed), exploiting some of the ideas outlined in =-=[25]-=-. One of the most promising uses of smart application is inside the backwardbased automation tactic of Matita. In fact, smart application allows a smooth integration of equational reasoning with the p... |

6 |
Higher order proof reconstruction from paramodulation-based refutations: The unit equality case
- Asperti, Tassi
- 2007
(Show Context)
Citation Context ...lines of OCaml code), well separated by the rest of the system. It was entirely rewritten during the summer 2009 starting from a previous prototype (some of whose functionalities had been outlined in =-=[6]-=-), with the aim to improve both its abstraction and performance. The tool took part to the 22nd CADE ATP System Competition, in the unit equality division, scoring in fourth position, beating glorious... |

3 |
Paramodulation-based thorem proving
- Nieuwenhuis, Rubio
- 2001
(Show Context)
Citation Context ...igence, making the communication between the user and the machine sensibly easier. Techniques for equational reasoning have been broadly investigated in the realm of automated theorem proving (see eg =-=[7,21,10]-=-). The main deductive mechanism is a completion technique [17] attempting to transform a given set of equations into a confluent rewriting system so that two terms are equal if and only if they have i... |

3 |
Michaël Rusinowitch. Uniform derivation of decision procedures by superposition
- Armando, Ranise
- 2001
(Show Context)
Citation Context ...Q tactics are sensibly weaker than superposition that seems to provide a good surrogate for several decision procedures for various theories, as well as a simple framework for composing them (see e.g =-=[2]-=-). Comparing the integration of superposition in Matita with similar functionalities provided by Isabelle is twofold complex. Isabelle’s support for equational reasoning is both delegated to external ... |

1 | Hints in unification
- Asperti, Ricciotti, et al.
(Show Context)
Citation Context ...first order applicative skeleton of CIC-terms. In the opposite direction, we try to reconstruct the missing information by just exploiting the sophisticated inference capability of the Matita refiner =-=[3]-=-, that is the tool in charge of transforming the user input into a machine understandable low-level CIC term. Automation is thus a best effort service, in the sense that not only it may obviously fail... |