This paper presents the proofs of the strong normalization, subject reduction, and Church-Rosser theorems for a presentation of the intuitionistic modal lambda calculus S4. It is adapted from Healfdene Goguen's thesis, where these properties are shown for the simply-typed lambda calculus and for UTT. Following this method, we introduce the notion of typed operational semantics for our system. We define a notion of typed substitution for our system, which has context stacks instead of usual contexts. This latter peculiarity leads to the main difficulties and consequently to the main original features in our proofs. Since the original proof was extended to an inductive setting, we expect our proof could also be extended to a calculus with higher order abstract syntax and induction
