  Dynamic reliability block diagrams (DRBDs) are introduced to overcome the
modeling limitations of traditional reliability block diagrams, such as the
inability to capture redundant components. However, so far there is no
algebraic framework that allows conducting the analysis of a given DRBD based
on its structure function and enables verifying its soundness using
higher-order logic (HOL) theorem proving. In this work, we propose a new
algebra to formally express the structure function and the reliability of a
DRBD with spare constructs based on basic system blocks and newly introduced
DRBD operators. We present several simplification properties that allow
reducing the structure of a given DRBD. We provide the HOL formalization of the
proposed algebra, and formally verify its corresponding properties using the
HOL4 theorem prover. This includes formally verifying generic reliability
expressions of the spare construct, series, parallel and deeper structures in
an extensible manner that allows verifying the reliability of complex systems.
Finally, we demonstrate the applicability of this algebra by formally analyzing
the terminal reliability analysis of a shuffle-exchange network in HOL4.
