Partial synchrony is a model of computation in many distributed algorithms and modern blockchains. Correctness of these algorithms requires the existence of bounds on message delays and on the relative speed of processes after reaching Global Stabilization Time (GST). This makes partially synchronous algorithms parametric in time bounds, which renders automated verification of partially synchronous algorithms challenging. In this paper, we present a case study on formal verification of both safety and liveness of a Chandra and Toueg failure detector that is based on partial synchrony. To this end, we specify the algorithm and the partial synchrony assumptions in three frameworks: $$\textsc {TLA}^+$$TLA+, Ivy, and counter automata. Importantly, we tune our modeling to use the strength of each method: (1) We are using counters to encode message buffers with counter automata, (2) we are using first-order relations to encode message buffers in Ivy, and (3) we are using both approaches in $$\textsc {TLA}^+$$TLA+. By running the tools for $$\textsc {TLA}^+$$TLA+ (TLC and APALACHE) and counter automata (FAST), we demonstrate safety for fixed time bounds. This helped us to find the inductive invariants for fixed parameters, which we used as a starting point for the proofs with Ivy. By running Ivy, we prove safety for arbitrary time bounds. Moreover, we show how to verify liveness of the failure detector by reducing the verification problem to safety verification. Thus, both properties are verified by developing inductive invariants with Ivy. We conjecture that correctness of other partially synchronous algorithms may be proven by following the presented methodology.