Irrelevant clauses in resolution problems increase the search space, making it hard to find proofs in a reasonable time. Simple relevance filtering methods, based on counting function symbols in clauses, improve the success rate for a variety of automatic theorem provers and with various initial settings. We have designed these techniques as part of a project to link automatic theorem provers to the interactive theorem prover Isabelle. They should be applicable to other situations where the resolution problems are produced mechanically and where completeness is less important than achieving a high success rate with limited processor time. 1

The prime number theorem, established by Hadamard and de la Vallée Poussin independently in 1896, asserts that the density of primes in the positive integers is asymptotic to 1 / ln x. Whereas their proofs made serious use of the methods of complex analysis, elementary proofs were provided by Selberg and Erdös in 1948. We describe a formally verified version of Selberg’s proof, obtained using the Isabelle proof assistant. 1

Let F be the set of functions from an infinite set, S, to an ordered ring, R. For f, g, and h in F, the assertion f = g + O(h) means that for some constant C, |f(x) − g(x) | ≤ C|h(x) | for every x in S. Let L be the first-order language with variables ranging over such functions, symbols for 0,+, −,min,max, and absolute value, and a ternary relation f = g + O(h). We show that the set of quantifier-free formulas in this language that are valid in the intended class of interpretations is decidable, and does not depend on the underlying set, S, or the ordered ring, R. If R is a subfield of the real numbers, we can add a constant 1 function, as well as multiplication by constants from any computable subfield. We obtain further decidability results for certain situations in which one adds symbols denoting the elements of a fixed sequence of functions of strictly increasing rates of growth. 1

1.1 Specification of GCD on nats.................. 9 1.2 GCD on nat by Euclid’s algorithm............... 9

1.1 Specification of GCD on nats.................. 9 1.2 GCD on nat by Euclid’s algorithm............... 9

1 Abstract-Rat: Abstract rational numbers 11

1 Abstract-Rat: Abstract rational numbers 11

1 Abstract-Rat: Abstract rational numbers 13 2 Multiset: (Finite) multisets 18 2.1 The type of multisets....................... 18

1 Abstract-Rat: Abstract rational numbers 13 2 Multiset: (Finite) multisets 24 2.1 The type of multisets....................... 24