The formal verification of security protocols can be carried out in two categories of models. Symbolic models, pioneered by Dolev and Yao, represent messages by first-order terms and attacker capabilities by inference rules or equational theories. This approach allows automatic verification, notably using techniques from rewriting and logic. The computational model, however, represents messages as bitstrings and attackers as probabilistic polynomial-time (PTIME) Turing machines. It is the standard model for provable security in cryptography, but formal verification in that model remains challenging. Bana and Comon have recently proposed a compelling approach to derive guarantees in the computational model, which they call the computationally complete symbolic attacker (CCSA). It is based on first-order logic, with complex axioms derived from cryptographic assumptions. Verification in the original CCSA approach involves a translation of protocol traces into first-order terms that necessitates to bound the length of traces. The generated goals are difficult to process by humans and, so far, cannot be handled automatically either. We have proposed a refinement of the approach based on a meta-logic which conveniently replaces the translation step. We have implemented this refined approach in an interactive theorem prover, SQUIRREL, and validated it on a first set of case studies. In this paper, we present three improvements of the foundations of the SQUIRREL meta-logic and of its proof system, which are required as we are considering more complex case studies. First, we extend our model to support memory cells in order to allow the analysis of security protocols that manipulate states. Second, we adapt the notion of trace model, which provides the semantics of our meta-logic formulas, to enable more natural and expressive modelling. Finally, we present a generalized proof system which allows to derive more protocol properties.