  Kozen and Tiuryn have introduced the substructural logic $\mathsf{S}$ for
reasoning about correctness of while programs (ACM TOCL, 2003). The logic
$\mathsf{S}$ distinguishes between tests and partial correctness assertions,
representing the latter by special implicational formulas. Kozen and Tiuryn's
logic extends Kleene altebra with tests, where partial correctness assertions
are represented by equations, not terms. Kleene algebra with codomain,
$\mathsf{KAC}$, is a one-sorted alternative to Kleene algebra with tests that
expands Kleene algebra with an operator that allows to construct a Boolean
subalgebra of tests. In this paper we show that Kozen and Tiuryn's logic embeds
into the equational theory of the expansion of $\mathsf{KAC}$ with residuals of
Kleene algebra multiplication and the upper adjoint of the codomain operator.
