Basic Ideas
To show constructively that a theory has quantifier elimination, it suffices to show that we can eliminate an existential quantifier applied to a conjunction of literals, that is, show that each formula of the form:
where each is a literal, is equivalent to a quantifier-free formula. Indeed, suppose we know how to eliminate quantifiers from conjunctions of formulae, then if is a quantifier-free formula, we can write it in disjunctive normal form
and use the fact that
is equivalent to
Finally, to eliminate a universal quantifier
where is quantifier-free, we transform into disjunctive normal form, and use the fact that is equivalent to
Read more about this topic: Quantifier Elimination
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