An intuitionistic axiomatization of `eventually'

Boudou and the authors have recently introduced the intuitionistic temporal logic $\mathsf{I}\mathsf{T}{\mathsf{L}}^{\mathsf{e}}$ and shown it to be decidable. In this article we show that the `henceforth’-free fragment of this logic is complete for the class of non-deterministic quasimodels introduced by Fern'andez-Duque. From this and results of Boudou, Romero and the authors, we conclude that this fragment is also complete for the standard semantics of $\mathsf{I}\mathsf{T}{\mathsf{L}}^{\mathsf{e}}$ over the class of expanding posets.