# Extensional Crisis and Proving Identity

• Ashutosh Gupta
• Laura Kovács
• Bernhard Kragl
• Andrei Voronkov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8837)

## Abstract

Extensionality axioms are common when reasoning about data collections, such as arrays and functions in program analysis, or sets in mathematics. An extensionality axiom asserts that two collections are equal if they consist of the same elements at the same indices. Using extensionality is often required to show that two collections are equal. A typical example is the set theory theorem ( ∀ x)( ∀ y)x ∪ y = y ∪ x.Interestingly, while humans have no problem with proving such set identities using extensionality, they are very hard for superposition theorem provers because of the calculi they use.In this paper we show how addition of a new inference rule, called extensionality resolution, allows first-order theorem provers to easily solve problems no modern first-order theorem prover can solve. We illustrate this by running the Vampire theorem prover with extensionality resolution on a number of set theory and array problems. Extensionality resolution helps Vampire to solve problems from the TPTP library of first-order problems that were never solved before by any prover.

## Keywords

Inference Rule Theorem Prover Collection Type Extensionality Axiom Saturation Algorithm
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer International Publishing Switzerland 2014

## Authors and Affiliations

• Ashutosh Gupta
• 1
• Laura Kovács
• 2
• Bernhard Kragl
• 1
• 3
• Andrei Voronkov
• 4
1. 1.IST AustriaKlosterneuburgAustria
2. 2.Chalmers University of TechnologyGothenburgSweden
3. 3.Vienna University of TechnologyViennaAustria
4. 4.The University of ManchesterManchesterUK