Formally verified complete quantifier instatiation strategy for the theory of bounded linear integer arithmetic

The Isabelle/HOL proof assistant provides quite advanced and reliable support for discharging proof goals with external SMT solvers by reconstructing the resulting proof tree within the Isabelle/Pure inference kernel. In particular, currently Isabelle fully and efficiently supports proof reconstruction for the quantifier-free fragments of the theories of equality and uninterpreted functions (QF_UF) and linear integer arithmetic (QF_LIA). Yet there are a number of practically relevant theories that can be decided using relatively efficient algorithms, but are not supported either by the Isabelle proof reconstruction procedure or by the SMT solvers themselves. Meanwhile, if the decision procedure for an extended logical theory can be expressed as a complete quantifier instantiation strategy that reduces the given decision problem into the supported logical fragment (QF_UFLIA), the theory can be fully supported within the existing Isabelle/HOL infrastructure with relatively little additional effort. The goal of our work was to develop a fully formal completeness proof for an efficient quantifier instantiation strategy that translates a satisfiability problem in the logic of bounded linear integer arithmetic (we call it QF_BLIA) into the equisatisfiable problem in the logic of linear integer arithmetic and uninterpreted functions (QF_UFLIA). In this paper we first give the initial informal presentation of the proof, then discuss the most relevant issues of our resulting corresponding formal development of the completeness proof within Isabelle/HOL. The resulting formal proofs tightly follow the provided informal presentation, but at the same time they also contain quite significant amount of auxiliary developments that can potentially be generalized for other similar proof formalizations and automated using theory generation facilities of Isabelle/ML.