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Abstract. In order to support the verification of programs, verification tools such as ACL2 or Isabelle try to extract suitable induction axioms from the definitions of terminating, recursively defined procedures. However, these extraction techniques have difficulties with procedures that are defined by second-order recursion: There a first-order procedure f passes itself as an argument to a second-order procedure like map, every, foldl , etc., which leads to indirect recursive calls. For instance, secondorder recursion is commonly used in algorithms on data structures such as terms (variadic trees). We present a method to automatically extract induction axioms from such procedures. Furthermore, we describe how the induction axioms can be optimized (i. e., generalized and simplified). An implementation of our methods demonstrates that the approach facilitates straightforward inductive proofs in a verification tool.

Abstract. Many algorithms on data structures such as terms (finitely branching trees) are naturally implemented by second-order recursion: A first-order procedure f passes itself as an argument to a second-order procedure like map, every, foldl , foldr , etc. to recursively apply f to the direct subterms of a term. We present a method for automated termination analysis of such procedures. It extends the approach of argumentbounded functions (i) by inspecting type components and (ii) by adding a facility to take care of second-order recursion. Our method has been implemented and automatically solves the examples considered in the literature. This improves the state of the art of inductive theorem provers, which (without our approach) require user interaction even for termination proofs of simple second-order recursive procedures.

Abstract. In the implementation of procedures, developers often assume that the input satisfies certain properties; for example, binary search assumes the array to be sorted. Such requirements on the input can be formally expressed as preconditions of procedures. If a secondorder procedure p (e.g., map or foldl ) is called with a first-order procedure f that has a precondition, the question arises whether p will call f only with arguments that satisfy the precondition of f . In this paper, we propose a method to statically analyze if all procedure calls in a given second-order program satisfy the respective preconditions. In particular, we consider indirect calls of procedures that are passed as an argument to a second-order procedure.