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We show that simultaneous rigid E-uniÿcation, 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 classiÿcation of decidability of the prenex fragment of intuitionistic logic with equality, in terms of the quantiÿer preÿx. It is also proved that SREU with onedoi:10.1016/s0304-3975(98)00185-6 fatcat:boefe4gcbzhmvk7y7rqrioehuq