/export/starexec/sandbox/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- MAYBE We consider the system theBenchmark. We are asked to determine termination of the following first-order TRS. 0 : [] --> o cond1 : [o * o * o] --> o cond2 : [o * o * o] --> o false : [] --> o gr : [o * o] --> o gr0 : [o] --> o p : [o] --> o s : [o] --> o true : [] --> o cond1(true, X, Y) => cond2(gr(X, Y), X, Y) cond2(true, X, Y) => cond1(gr0(X), Y, Y) cond2(false, X, Y) => cond1(gr0(X), p(X), Y) gr(0, X) => false gr(s(X), 0) => true gr(s(X), s(Y)) => gr(X, Y) gr0(0) => false gr0(s(X)) => true p(0) => 0 p(s(X)) => X As the system is orthogonal, it is terminating if it is innermost terminating by [Gra95]. Then, by [FuhGieParSchSwi11], it suffices to prove (innermost) termination of the typed system, with sort annotations chosen to respect the rules, as follows: 0 : [] --> kc cond1 : [bc * kc * kc] --> xa cond2 : [bc * kc * kc] --> xa false : [] --> bc gr : [kc * kc] --> bc gr0 : [kc] --> bc p : [kc] --> kc s : [kc] --> kc true : [] --> bc +++ Citations +++ [FuhGieParSchSwi11] C. Fuhs, J. Giesl, M. Parting, P. Schneider-Kamp, and S. Swiderski. Proving Termination by Dependency Pairs and Inductive Theorem Proving. In volume 47(2) of Journal of Automated Reasoning. 133--160, 2011. [Gra95] B. Gramlich. Abstract Relations Between Restricted Termination and Confluence Properties of Rewrite Systems. In volume 24(1-2) of Fundamentae Informaticae. 3--23, 1995.