Internal proof calculi for modal logics with separating conjunction

Abstract Modal separation logics are formalisms that combine modal operators to reason locally, with separating connectives allow perform global updates on the models. In this work, we design Hilbert-style proof systems for $\text {MSL}(\ast ,\langle \neq \rangle )$ and ,\Diamond )$, where $\ast $ is conjunction, $\Diamond standard operator $\langle difference modality. The calculi only use logical languages at hand (no external features such as labels) can be divided in two main parts. First, normal forms formulae designed transform every formula into a form. Second, another part of dedicated axiomatization form, which may still require non-trivial developments but more manageable.