A simple blame calculus for explicit nulls

Abstract Gradual typing provides a model for when a legacy language with less precise types interacts with a newer language with more precise types. Casts mediate between types of different precision, allocating blame when a value fails to conform to a type. The blame theorem asserts that blame always falls on the less-precisely typed side of a cast. One instance of interest is when a legacy language (such as Java) permits null values at every type, while a newer language (such as Scala or Kotlin) explicitly indicates which types permit null values. Nieto et al. in 2020 introduced a gradually typed calculus for just this purpose. The calculus requires three distinct constructors for function types and a non-standard proof of the blame theorem; it can embed terms from the legacy language into the newer language (or vice versa) only when they are closed. Here, we define a simpler calculus that is more orthogonal, with one constructor for function types and one for possibly nullable types, and with an entirely standard proof of the blame theorem; it can embed terms from the legacy language into the newer language (and vice versa) even if they are open. All results in the paper have been mechanized in Coq.