An analytic calculus for intuitionistic belief

Intuitionistic belief has been axiomatized by Artemov and Protopopescu as an extension of intuitionistic propositional logic by means of the distributivity scheme K, and of co-reflection $A\rightarrow\Box A$. This way, belief is interpreted as a result of verification, and it fits an extended Brouwer-Heyting-Kolmogorov interpretation for intuitionistic propositional logic with an epistemic modality. In the present paper, structural properties of a natural deduction system $\mathsf{IEL}^{-}$ for intuitionistic belief are investigated. The focus is on the analyticity of the calculus, so that the normalization theorem and the subformula property are proven firstly. From these, decidability and consistency of the logic follow as corollaries. Finally, disjunction properties, $\Box$-primality, and admissibility of reflection rule are established by using purely proof-theoretic methods.
Comment: This is a very rough draft that is intended as the second part of work-in-progress started with [13]. For sure, many expository refinements are required to the present paper: it is basically a collection of rough results and reflections