E (theorem prover)

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E is a high performance theorem prover for full first-order logic with equality.[1] It is based on the equational superposition calculus and uses a purely equational paradigm. It has been integrated into other theorem provers and it has been among the best-placed systems in several theorem proving competitions. E is developed by Stephan Schulz, originally in the Automated Reasoning Group at TU Munich, now at Baden-Württemberg Cooperative State University Stuttgart.


The system is based on the equational superposition calculus. In contrast to most other current provers, the implementation actually uses a purely equational paradigm, and simulates non-equational inferences via appropriate equality inferences. Significant innovations include shared term rewriting (where many possible equational simplifications are carried out in a single operation),[2] several efficient term indexing data structures for speeding up inferences, advanced inference literal selection strategies, and various uses of machine learning techniques to improve the search behaviour.[2][3][4] Since version 2.0, E supports many-sorted logic. [5]

E is implemented in C and portable to most UNIX variants and the Cygwin environment. It is available under the GNU GPL.[6]


The prover has consistently performed well in the CADE ATP System Competition, winning the CNF/MIX category in 2000 and finishing among the top systems ever since.[7] In 2008 it came in second place.[8] In 2009 it won second place in the FOF (full first order logic) and UEQ (unit equational logic) categories and third place (after two versions of Vampire) in CNF (clausal logic).[9] It repeated the performance in FOF and CNF in 2010, and won a special award as "overall best" system.[10] In the 2011 CASC-23 E won the CNF division and achieved second places in UEQ and LTB.[11]


E has been integrated into several other theorem provers. It is, with Vampire, SPASS, CVC4, and Z3, at the core of Isabelle's Sledgehammer strategy.[12][13] E also is the reasoning engine in SInE[14] and LEO-II[15] and used as the clausification system for iProver.[16]

Applications of E include reasoning on large ontologies,[17] software verification,[18] and software certification.[19]


  1. ^ Schulz, Stephan (2002). "E - A Brainiac Theorem Prover". Journal of AI Communications. 15 (2/3): 111–126.
  2. ^ a b Schulz, Stephan (2008). "Entrants System Descriptions: E 1.0pre and EP 1.0pre". Retrieved 24 March 2009.
  3. ^ Schulz, Stephan (2004). "System Description: E 0.81". Proceedings of the 2nd International Joint Conference on Automated Reasoning. Lecture Notes in Computer Science. Springer. LNAI 3097: 223–228. doi:10.1007/978-3-540-25984-8_15. ISBN 978-3-540-22345-0.
  4. ^ Schulz, Stephan (2001). "Learning Search Control Knowledge for Equational Theorem Proving". Proceedings of the Joint German/Austrian Conference on Artificial Intelligence (KI-2001). Lecture Notes in Computer Science. Springer. LNAI 2174: 320–334. doi:10.1007/3-540-45422-5_23. ISBN 978-3-540-42612-7.
  5. ^ "news on E's website". Retrieved 10 July 2017.
  6. ^ Schulz, Stephan (2008). "The E Equational Theorem Prover". Retrieved 24 March 2009.
  7. ^ Sutcliffe, Geoff. "The CADE ATP System Competition". Archived from the original on 2 March 2009. Retrieved 24 March 2009.
  8. ^ FOF division of CASC in 2008
  9. ^ Sutcliffe, Geoff (2009). "The 4th IJCAR Automated Theorem Proving System Competition--CASC-J4". AI Communications. 22 (1): 59–72. doi:10.3233/AIC-2009-0441. Retrieved 16 December 2009.
  10. ^ Sutcliffe, Geoff (2010). "The CADE ATP System Competition". University of Miami. Retrieved 20 July 2010.
  11. ^ Sutcliffe, Geoff (2011). "The CADE ATP System Competition". University of Miami. Retrieved 14 August 2011.
  12. ^ Paulson, Lawrence C. (2008). "Automation for Interactive Proof: Techniques, Lessons and Prospects" (PDF). Tools and Techniques for Verification of System Infrastructure - A Festschrift in Honour of Professor Michael J. C. Gordon FRS: 29–30. Retrieved 19 December 2009.
  13. ^ Meng, Jia; Lawrence C. Paulson (2004). Experiments on Supporting Interactive Proof Using Resolution. Lecture Notes in Computer Science. Vol. 3097. Springer. pp. 372–384. CiteSeerX doi:10.1007/978-3-540-25984-8_28. ISBN 978-3-540-22345-0.
  14. ^ Sutcliffe, Geoff; et al. (2009). The 4th IJCAR ATP System Competition (PDF). Retrieved 18 December 2009.
  15. ^ Benzmüller, Christoph; Lawrence C. Paulson; Frank Theiss; Arnaud Fietzke (2008). LEO-II - A Cooperative Automatic Theorem Prover for Classical Higher-Order Logic (System Description) (PDF). Lecture Notes in Computer Science. Vol. 5195. 4th International Joint Conference on Automated Reasoning, IJCAR 2008 Sydney, Australia: Springer. pp. 162–170. doi:10.1007/978-3-540-71070-7_14. ISBN 978-3-540-71069-1. Archived from the original (PDF) on 15 June 2011. Retrieved 20 December 2009.{{cite book}}: CS1 maint: location (link)
  16. ^ Korovin, Konstantin (2008). "iProver—an instantiation-based theorem prover for first-order logic (system description)". Automated Reasoning (PDF). Lecture Notes in Computer Science. Vol. 5195. Springer. pp. 292–298. doi:10.1007/978-3-540-71070-7_24. ISBN 978-3-540-71069-1. Retrieved 18 December 2009.[permanent dead link]
  17. ^ Ramachandran, Deepak; Pace Reagan; Keith Goolsbery (2005). "First-Orderized ResearchCyc : Expressivity and Efficiency in a Common-Sense Ontology" (PDF). AAAI Workshop on Contexts and Ontologies: Theory, Practice and Applications. AAAI.
  18. ^ Ranise, Silvio; David Déharbe (2003). "Applying Light-Weight Theorem Proving to Debugging and Verifying Pointer Programs". Electronic Notes in Theoretical Computer Science. 4th International Workshop on First-Order Theorem Proving: Elsevier. 86 (1): 109–119. doi:10.1016/S1571-0661(04)80656-X.{{cite journal}}: CS1 maint: location (link)
  19. ^ Denney, Ewen; Bernd Fischer; Johan Schumann (2006). "An Empirical Evaluation of Automated Theorem Provers in Software Certification". International Journal on Artificial Intelligence Tools. 15 (1): 81–107. CiteSeerX doi:10.1142/s0218213006002576.

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