This paper presents a formally verified quantifier elimination (QE) algorithm for first-order real arithmetic by linear and quadratic virtual substitution (VS) in Isabelle/HOL. The Tarski-Seidenberg theorem established that the logic of is decidable QE. However, practice, QE algorithms are highly complicated often combine multiple methods performance. VS practically successful method targets fo...

We generalize and improve the schemes of [4]. We introduce analogues of exponentiation and cliscrele logarithms in the principle cycle of real quadratic orders. This enables us t o implement many cryptographic protocols based on discrete logarithms, e.g. a variant of the signature scheme of ElGamal [S].

It is shown that counting certain differences of overpartition functions is equivalent to counting elements of a given norm in appropriate real quadratic fields.