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First Order Stålmarck

Universal Lemmas Through Branch Merges


We present a proof method with a novel way of introducing universal lemmas. The method is a first order extension of Stålmarck’s method, containing a branch-and-merge rule known as the dilemma rule. The dilemma rule creates two branches in a tableau-like way, but later recombines the two branches, keeping the common consequences. While the propositional version uses normal set intersection in the merges, the first order version searches for pairwise unifiable formulae in the two branches. Within branches, the system uses a special kind of variables that may not be substituted. At branch merges, these variables are replaced by universal variables, and in this way universal lemmas can be introduced. Relevant splitting formulae are found through failed unifications of variables in branches. This article presents the calculus and proof procedure, and shows soundness and completeness. Benchmarks of an implementation are also presented.

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Correspondence to Magnus Björk.

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The research was chiefly carried out during the author’s PhD studies at Chalmers University of Technology, partially funded by Prover Technology. Large parts of the article was written during the author’s employment at Oxford University Computing Laboratory.

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Björk, M. First Order Stålmarck. J Autom Reasoning 42, 99–122 (2009).

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  • Automated theorem proving
  • First order logic
  • Stålmarck’s method
  • Universal lemmas
  • Intersections