  Probabilistic Hoare logic (PHL) is an extension of Hoare logic and is
specifically useful in verifying randomized programs. It allows researchers to
formally reason about the behavior of programs with stochastic elements,
ensuring the desired probabilistic properties are upheld. The relative
completeness of satisfaction-based PHL has been an open problem ever since the
birth of the first PHL in 1979. More specifically, no satisfaction-based PHL
with While-loop has been proven to be relatively complete yet. This paper
solves this problem by establishing a new PHL with While-loop and prove its
relative completeness. The programming language concerned in our PHL is
expressively equivalent to the existing PHL systems but brings a lot of
convenience in showing completeness. The weakest preterm for While-loop command
reveals how it changes the probabilistic properties of computer states,
considering both execution branches that halt and infinite runs. We prove the
relative completeness of our PHL in two steps. We first establish a semantics
and proof system of Hoare triples with probabilistic programs and deterministic
assertions. Then, by utilizing the weakest precondition of deterministic
assertions, we construct the weakest preterm calculus of probabilistic
expressions. The relative completeness of our PHL is then obtained as a
consequence of the weakest preterm calculus.
