We propose an incremental approach for safety proofs that decomposes a proof with a complex inductive invariant into a sequence of simpler proof steps. Our proof system combines rules for (i) forward reasoning using inductive invariants, (ii) backward reasoning using inductive invariants of a time-reversed system, and (iii) prophecy steps that add witnesses for existentially quantified properties. We prove each rule sound and give a construction that recovers a single safe inductive invariant from an incremental proof. The construction of the invariant demonstrates the increased complexity of a single inductive invariant compared to the invariant formulas used in an incremental proof, which may have simpler Boolean structures and fewer quantifiers and quantifier alternations. Under natural restrictions on the available invariant formulas, each proof rule strictly increases proof power. That is, each rule allows to prove more safety problems with the same set of formulas. Thus, the incremental approach is able to reduce the search space of invariant formulas needed to prove safety of a given system. A case study on Paxos, several of its variants, and Raft demonstrates that forward-backward steps can remove complex Boolean structure while prophecy eliminates quantifiers and quantifier alternations.

Abstract This paper lays a practical foundation for using abstract interpretation with an abstract domain that consists of sets of quantified first-order logic formulas. This abstract domain seems infeasible at first sight due to the complexity of the formulas involved and the enormous size of sets of formulas (abstract elements). We introduce an efficient representation of abstract elements, which eliminates redundancies based on a novel syntactic subsumption relation that under-approximates semantic entailment. We develop algorithms and data structures to efficiently compute the join of an abstract element with the abstraction of a concrete state, operating on the representation of abstract elements. To demonstrate feasibility of the domain, we use our data structures and algorithms to implement a symbolic abstraction algorithm that computes the least fixpoint of the best abstract transformer of a transition system, which corresponds to the strongest inductive invariant. We succeed at finding, for example, the least fixpoint for Paxos (which in our representation has 1,438 formulas with $$\forall ^*\exists ^*\forall ^*$$ ∀ ∗ ∃ ∗ ∀ ∗ quantification) in time comparable to state-of-the-art property-directed approaches.