Local variable scoping and Kleene algebra with tests

We explore the power of relational semantics and equational reasoning in the style of Kleene algebra for analyzing programs with mutable, statically scoped local variables. We provide (i) a fully compositional relational semantics for a first-order programming language with constructs for local variable declaration and destructive update; and (ii) an equational proof system based on Kleene algebra with tests for proving the equivalence of programs in this language. We show that the proof system is sound and complete relative to the underlying equational theory without local variables. We illustrate the use of the system with several examples.