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Cylindrical Algebraic Decomposition (CAD) was the first practical means fordoing real quantifier elimination (QE), and is still a major method, with manyimprovements since Collins' original method. Nevertheless, its complexity isinherently doubly exponential in the number of variables. Where applicable,virtual term substitution (VTS) is more effective, turning a QE problem in $n$variables to one in $n-1$ variables in one application, and so on. Hence thereis scope for hybrid methods: doing VTS where possible then using CAD.  This paper describes such a poly-algorithmic implementation, based on thesecond author's Ph.D. thesis. The version of CAD used is based on a newimplementation of Lazard's recently-justified method, with some improvements tohandle equational constraints.

A Poly-algorithmic Approach to Quantifier Elimination