We present a framework for formal refutational completeness proofs of abstract provers that implement saturation calculi, such as ordered resolution or superposition. The framework relies on modular extensions of lifted redundancy criteria. It allows us to extend redundancy criteria so that they cover subsumption, and also to model entire prover architectures in such a way that the static refutational completeness of a calculus immediately implies the dynamic refutational completeness of a prover implementing the calculus, for instance within an Otter or DISCOUNT loop. Our framework is mechanized in Isabelle/HOL.

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We describe symbol elimination and consequence nding in the rst-order theorem prover Vampire for automatic generation of quantied invariants, possibly with quantier alternations, of loops with arrays. Unlike the previous implementation of symbol elimination in Vampire, our work is not limited to a specic programming language but provides a generic framework by relying on a simple guarded command representation of the input loop. We also improve the loop analysis part in Vampire by generating loop properties more easily handled by the saturation engine of Vampire. Our experiments show that, with our changes, the number of generated invariants is decreased, in some cases, by a factor of 20. We also provide a framework to use our approach to invariant generation in conjunction with pre-and post-conditions of program loops. We use the program specication to nd relevant invariants as well as to verify the partial correctness of the loop. As a case study, we demonstrate how symbol elimination in Vampire can be used as an interface for realistic imperative languages, by integrating our tool in the KeY verication system, thus allowing reasoning about loops in Java programs in a fully automated way, without any user guidance.

We present a framework to analyze and verify programs containing loops by using a first-order language of so-called extended expressions. This language can express both functional and temporal properties of loops. We prove soundness and completeness of our framework and use our approach to automate the tasks of partial correctness verification, termination analysis and invariant generation. For doing so, we express the loop semantics as a set of first-order properties over extended expressions and use theorem provers and/or SMT solvers to reason about these properties. Our approach supports full first-order reasoning, including proving program properties with alternation of quantifiers. Our work is implemented in the tool QuIt and successfully evaluated on benchmarks coming from software verification.

The theory of finite term algebras provides a natural framework to describe the semantics of functional languages. The ability to efficiently reason about term algebras is essential to automate program analysis and verification for functional or imperative programs over algebraic data types such as lists and trees. However, as the theory of finite term algebras is not finitely axiomatizable, reasoning about quantified properties over term algebras is challenging.In this paper we address full first-order reasoning about properties of programs manipulating term algebras, and describe two approaches for doing so by using first-order theorem proving. Our first method is a conservative extension of the theory of term algebras using a finite number of statements, while our second method relies on extending the superposition calculus of first-order theorem provers with additional inference rules.We implemented our work in the first-order theorem prover Vampire and evaluated it on a large number of algebraic data type benchmarks, as well as game theory constraints. Our experimental results show that our methods are able to find proofs for many hard problems previously unsolved by state-of-the-art methods. We also show that Vampire implementing our methods outperforms existing SMT solvers able to deal with algebraic data types.