  We develop a theory of decidable inductive invariants for an infinite-state
variant of the Applied pi-calculus, with applications to automatic verification
of stateful cryptographic protocols with unbounded sessions/nonces. Since the
problem is undecidable in general, we introduce depth-bounded protocols, a
strict generalisation of a class from the literature, for which our decidable
analysis is sound and complete. Our core contribution is a procedure to check
that an invariant is inductive, which implies that every reachable
configuration satisfies it. Our invariants can capture security properties like
secrecy, can be inferred automatically, and represent an independently
checkable certificate of correctness.
  We provide a prototype implementation and we report on its performance on
some textbook examples.
