Co-Induction Simply: Automatic Co-Inductive Proofs in a Program Verifier

Program verification relies heavily on induction, which has received decades of attention in mechanical verification tools. When program correctness is best described by infinite structures, program verification is usefully aided also by co-induction, which has not benefited from the same degree of tool support. Co-induction is complicated to work with in interactive proof assistants and has had no previous support in dedicated program verifiers. This paper shows that an SMT-based program verifier can support reasoning about co-induction—handling infinite data structures, lazy function calls, and user-defined properties defined as greatest fix-points, as well as letting users write co-inductive proofs. Moreover, the support can be packaged to provide a simple user experience. The paper describes the features for co-induction in the language and verifier Dafny, defines their translation into input for a first-order SMT solver, and reports on some encouraging initial experience.