Decidability and complexity of simultaneous rigid E-unification with one variable and related results

We show that simultaneous rigid E-unification, or SREU for short, is decidable and in fact EXPTIME-complete in the case of one variable. This result implies that the ∀*∃∀* fragment of intuitionistic logic with equality is decidable. Together with a previous result regarding the undecidability of the ∃∃-fragment, we obtain a complete classification of decidability of the prenex fragment of intuitionistic logic with equality, in terms of the quantifier prefix. It is also proved that SREU with one variable and a constant bound on the number of rigid equations is P-complete. Moreover, we consider a case of SREU where one allows several variables, but each rigid equation either contains one variable, or has a ground left-hand side and an equality between two variables as a right-hand side. We show that SREU is decidable also in this restricted case.