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There are several ways to define the notion of submodel for Kripke models of intuitionistic first-order logic. In our approach, a Kripke model A is a submodel of a Kripke model B if the frame of A is a subframe of the frame of B and for each two corresponding worlds AÎ± and BÎ± of them, AÎ± is a classical submodel of BÎ±. In this case, B is called an extension of A. We characterize formulas that are preserved under taking extensions of Kripke models.
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