YES Problem: a(a(b(b(x1)))) -> b(b(c(c(a(a(x1)))))) b(b(c(c(x1)))) -> c(c(b(b(b(b(x1)))))) b(b(a(a(x1)))) -> a(a(c(c(b(b(x1)))))) Proof: DP Processor: DPs: a#(a(b(b(x1)))) -> a#(x1) a#(a(b(b(x1)))) -> a#(a(x1)) a#(a(b(b(x1)))) -> b#(c(c(a(a(x1))))) a#(a(b(b(x1)))) -> b#(b(c(c(a(a(x1)))))) b#(b(c(c(x1)))) -> b#(x1) b#(b(c(c(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(b(b(x1))) b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) b#(b(a(a(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> a#(c(c(b(b(x1))))) b#(b(a(a(x1)))) -> a#(a(c(c(b(b(x1)))))) TRS: a(a(b(b(x1)))) -> b(b(c(c(a(a(x1)))))) b(b(c(c(x1)))) -> c(c(b(b(b(b(x1)))))) b(b(a(a(x1)))) -> a(a(c(c(b(b(x1)))))) TDG Processor: DPs: a#(a(b(b(x1)))) -> a#(x1) a#(a(b(b(x1)))) -> a#(a(x1)) a#(a(b(b(x1)))) -> b#(c(c(a(a(x1))))) a#(a(b(b(x1)))) -> b#(b(c(c(a(a(x1)))))) b#(b(c(c(x1)))) -> b#(x1) b#(b(c(c(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(b(b(x1))) b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) b#(b(a(a(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> a#(c(c(b(b(x1))))) b#(b(a(a(x1)))) -> a#(a(c(c(b(b(x1)))))) TRS: a(a(b(b(x1)))) -> b(b(c(c(a(a(x1)))))) b(b(c(c(x1)))) -> c(c(b(b(b(b(x1)))))) b(b(a(a(x1)))) -> a(a(c(c(b(b(x1)))))) graph: b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) -> b#(b(a(a(x1)))) -> a#(a(c(c(b(b(x1)))))) b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) -> b#(b(a(a(x1)))) -> a#(c(c(b(b(x1))))) b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) -> b#(b(a(a(x1)))) -> b#(x1) b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) -> b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) -> b#(b(c(c(x1)))) -> b#(b(b(x1))) b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) -> b#(b(c(c(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) -> b#(b(c(c(x1)))) -> b#(x1) b#(b(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(x1)))) -> a#(a(c(c(b(b(x1)))))) b#(b(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(x1)))) -> a#(c(c(b(b(x1))))) b#(b(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(x1)))) -> b#(x1) b#(b(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) b#(b(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(c(c(x1)))) -> b#(b(b(x1))) b#(b(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(c(c(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(c(c(x1)))) -> b#(x1) b#(b(c(c(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> a#(a(c(c(b(b(x1)))))) b#(b(c(c(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> a#(c(c(b(b(x1))))) b#(b(c(c(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> b#(x1) b#(b(c(c(x1)))) -> b#(b(x1)) -> b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) b#(b(c(c(x1)))) -> b#(b(x1)) -> b#(b(c(c(x1)))) -> b#(b(b(x1))) b#(b(c(c(x1)))) -> b#(b(x1)) -> b#(b(c(c(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(b(x1)) -> b#(b(c(c(x1)))) -> b#(x1) b#(b(c(c(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> a#(a(c(c(b(b(x1)))))) b#(b(c(c(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> a#(c(c(b(b(x1))))) b#(b(c(c(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(x1) b#(b(c(c(x1)))) -> b#(x1) -> b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) b#(b(c(c(x1)))) -> b#(x1) -> b#(b(c(c(x1)))) -> b#(b(b(x1))) b#(b(c(c(x1)))) -> b#(x1) -> b#(b(c(c(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(x1) -> b#(b(c(c(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> a#(a(c(c(b(b(x1)))))) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> a#(c(c(b(b(x1))))) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(c(c(x1)))) -> b#(b(b(x1))) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(c(c(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(c(c(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> a#(a(c(c(b(b(x1)))))) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> a#(c(c(b(b(x1))))) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(c(c(x1)))) -> b#(b(b(x1))) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(c(c(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(c(c(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> a#(c(c(b(b(x1))))) -> a#(a(b(b(x1)))) -> b#(b(c(c(a(a(x1)))))) b#(b(a(a(x1)))) -> a#(c(c(b(b(x1))))) -> a#(a(b(b(x1)))) -> b#(c(c(a(a(x1))))) b#(b(a(a(x1)))) -> a#(c(c(b(b(x1))))) -> a#(a(b(b(x1)))) -> a#(a(x1)) b#(b(a(a(x1)))) -> a#(c(c(b(b(x1))))) -> a#(a(b(b(x1)))) -> a#(x1) b#(b(a(a(x1)))) -> a#(a(c(c(b(b(x1)))))) -> a#(a(b(b(x1)))) -> b#(b(c(c(a(a(x1)))))) b#(b(a(a(x1)))) -> a#(a(c(c(b(b(x1)))))) -> a#(a(b(b(x1)))) -> b#(c(c(a(a(x1))))) b#(b(a(a(x1)))) -> a#(a(c(c(b(b(x1)))))) -> a#(a(b(b(x1)))) -> a#(a(x1)) b#(b(a(a(x1)))) -> a#(a(c(c(b(b(x1)))))) -> a#(a(b(b(x1)))) -> a#(x1) a#(a(b(b(x1)))) -> b#(c(c(a(a(x1))))) -> b#(b(a(a(x1)))) -> a#(a(c(c(b(b(x1)))))) a#(a(b(b(x1)))) -> b#(c(c(a(a(x1))))) -> b#(b(a(a(x1)))) -> a#(c(c(b(b(x1))))) a#(a(b(b(x1)))) -> b#(c(c(a(a(x1))))) -> b#(b(a(a(x1)))) -> b#(b(x1)) a#(a(b(b(x1)))) -> b#(c(c(a(a(x1))))) -> b#(b(a(a(x1)))) -> b#(x1) a#(a(b(b(x1)))) -> b#(c(c(a(a(x1))))) -> b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) a#(a(b(b(x1)))) -> b#(c(c(a(a(x1))))) -> b#(b(c(c(x1)))) -> b#(b(b(x1))) a#(a(b(b(x1)))) -> b#(c(c(a(a(x1))))) -> b#(b(c(c(x1)))) -> b#(b(x1)) a#(a(b(b(x1)))) -> b#(c(c(a(a(x1))))) -> b#(b(c(c(x1)))) -> b#(x1) a#(a(b(b(x1)))) -> b#(b(c(c(a(a(x1)))))) -> b#(b(a(a(x1)))) -> a#(a(c(c(b(b(x1)))))) a#(a(b(b(x1)))) -> b#(b(c(c(a(a(x1)))))) -> b#(b(a(a(x1)))) -> a#(c(c(b(b(x1))))) a#(a(b(b(x1)))) -> b#(b(c(c(a(a(x1)))))) -> b#(b(a(a(x1)))) -> b#(b(x1)) a#(a(b(b(x1)))) -> b#(b(c(c(a(a(x1)))))) -> b#(b(a(a(x1)))) -> b#(x1) a#(a(b(b(x1)))) -> b#(b(c(c(a(a(x1)))))) -> b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) a#(a(b(b(x1)))) -> b#(b(c(c(a(a(x1)))))) -> b#(b(c(c(x1)))) -> b#(b(b(x1))) a#(a(b(b(x1)))) -> b#(b(c(c(a(a(x1)))))) -> b#(b(c(c(x1)))) -> b#(b(x1)) a#(a(b(b(x1)))) -> b#(b(c(c(a(a(x1)))))) -> b#(b(c(c(x1)))) -> b#(x1) a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> b#(b(c(c(a(a(x1)))))) a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> b#(c(c(a(a(x1))))) a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> a#(a(x1)) a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> a#(x1) a#(a(b(b(x1)))) -> a#(x1) -> a#(a(b(b(x1)))) -> b#(b(c(c(a(a(x1)))))) a#(a(b(b(x1)))) -> a#(x1) -> a#(a(b(b(x1)))) -> b#(c(c(a(a(x1))))) a#(a(b(b(x1)))) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(a(x1)) a#(a(b(b(x1)))) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(x1) EDG Processor: DPs: a#(a(b(b(x1)))) -> a#(x1) a#(a(b(b(x1)))) -> a#(a(x1)) a#(a(b(b(x1)))) -> b#(c(c(a(a(x1))))) a#(a(b(b(x1)))) -> b#(b(c(c(a(a(x1)))))) b#(b(c(c(x1)))) -> b#(x1) b#(b(c(c(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(b(b(x1))) b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) b#(b(a(a(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> a#(c(c(b(b(x1))))) b#(b(a(a(x1)))) -> a#(a(c(c(b(b(x1)))))) TRS: a(a(b(b(x1)))) -> b(b(c(c(a(a(x1)))))) b(b(c(c(x1)))) -> c(c(b(b(b(b(x1)))))) b(b(a(a(x1)))) -> a(a(c(c(b(b(x1)))))) graph: b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) -> b#(b(c(c(x1)))) -> b#(x1) b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) -> b#(b(c(c(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) -> b#(b(c(c(x1)))) -> b#(b(b(x1))) b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) -> b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) -> b#(b(a(a(x1)))) -> b#(x1) b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) -> b#(b(a(a(x1)))) -> a#(c(c(b(b(x1))))) b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) -> b#(b(a(a(x1)))) -> a#(a(c(c(b(b(x1)))))) b#(b(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(c(c(x1)))) -> b#(x1) b#(b(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(c(c(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(c(c(x1)))) -> b#(b(b(x1))) b#(b(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) b#(b(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(x1)))) -> b#(x1) b#(b(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(x1)))) -> a#(c(c(b(b(x1))))) b#(b(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(a(a(x1)))) -> a#(a(c(c(b(b(x1)))))) b#(b(c(c(x1)))) -> b#(b(x1)) -> b#(b(c(c(x1)))) -> b#(x1) b#(b(c(c(x1)))) -> b#(b(x1)) -> b#(b(c(c(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(b(x1)) -> b#(b(c(c(x1)))) -> b#(b(b(x1))) b#(b(c(c(x1)))) -> b#(b(x1)) -> b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) b#(b(c(c(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> b#(x1) b#(b(c(c(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> a#(c(c(b(b(x1))))) b#(b(c(c(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> a#(a(c(c(b(b(x1)))))) b#(b(c(c(x1)))) -> b#(x1) -> b#(b(c(c(x1)))) -> b#(x1) b#(b(c(c(x1)))) -> b#(x1) -> b#(b(c(c(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(x1) -> b#(b(c(c(x1)))) -> b#(b(b(x1))) b#(b(c(c(x1)))) -> b#(x1) -> b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) b#(b(c(c(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(x1) b#(b(c(c(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> a#(c(c(b(b(x1))))) b#(b(c(c(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> a#(a(c(c(b(b(x1)))))) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(c(c(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(c(c(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(c(c(x1)))) -> b#(b(b(x1))) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> a#(c(c(b(b(x1))))) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> a#(a(c(c(b(b(x1)))))) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(c(c(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(c(c(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(c(c(x1)))) -> b#(b(b(x1))) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> a#(c(c(b(b(x1))))) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> a#(a(c(c(b(b(x1)))))) a#(a(b(b(x1)))) -> b#(b(c(c(a(a(x1)))))) -> b#(b(c(c(x1)))) -> b#(x1) a#(a(b(b(x1)))) -> b#(b(c(c(a(a(x1)))))) -> b#(b(c(c(x1)))) -> b#(b(x1)) a#(a(b(b(x1)))) -> b#(b(c(c(a(a(x1)))))) -> b#(b(c(c(x1)))) -> b#(b(b(x1))) a#(a(b(b(x1)))) -> b#(b(c(c(a(a(x1)))))) -> b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> a#(x1) a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> a#(a(x1)) a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> b#(c(c(a(a(x1))))) a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> b#(b(c(c(a(a(x1)))))) a#(a(b(b(x1)))) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(x1) a#(a(b(b(x1)))) -> a#(x1) -> a#(a(b(b(x1)))) -> a#(a(x1)) a#(a(b(b(x1)))) -> a#(x1) -> a#(a(b(b(x1)))) -> b#(c(c(a(a(x1))))) a#(a(b(b(x1)))) -> a#(x1) -> a#(a(b(b(x1)))) -> b#(b(c(c(a(a(x1)))))) SCC Processor: #sccs: 2 #rules: 8 #arcs: 60/144 DPs: a#(a(b(b(x1)))) -> a#(a(x1)) a#(a(b(b(x1)))) -> a#(x1) TRS: a(a(b(b(x1)))) -> b(b(c(c(a(a(x1)))))) b(b(c(c(x1)))) -> c(c(b(b(b(b(x1)))))) b(b(a(a(x1)))) -> a(a(c(c(b(b(x1)))))) Arctic Interpretation Processor: dimension: 1 interpretation: [a#](x0) = x0 + 0, [c](x0) = 7, [a](x0) = 5x0 + 12, [b](x0) = 2x0 + 0 orientation: a#(a(b(b(x1)))) = 9x1 + 12 >= 5x1 + 12 = a#(a(x1)) a#(a(b(b(x1)))) = 9x1 + 12 >= x1 + 0 = a#(x1) a(a(b(b(x1)))) = 14x1 + 17 >= 11 = b(b(c(c(a(a(x1)))))) b(b(c(c(x1)))) = 11 >= 7 = c(c(b(b(b(b(x1)))))) b(b(a(a(x1)))) = 14x1 + 21 >= 17 = a(a(c(c(b(b(x1)))))) problem: DPs: a#(a(b(b(x1)))) -> a#(a(x1)) TRS: a(a(b(b(x1)))) -> b(b(c(c(a(a(x1)))))) b(b(c(c(x1)))) -> c(c(b(b(b(b(x1)))))) b(b(a(a(x1)))) -> a(a(c(c(b(b(x1)))))) EDG Processor: DPs: a#(a(b(b(x1)))) -> a#(a(x1)) TRS: a(a(b(b(x1)))) -> b(b(c(c(a(a(x1)))))) b(b(c(c(x1)))) -> c(c(b(b(b(b(x1)))))) b(b(a(a(x1)))) -> a(a(c(c(b(b(x1)))))) graph: a#(a(b(b(x1)))) -> a#(a(x1)) -> a#(a(b(b(x1)))) -> a#(a(x1)) Arctic Interpretation Processor: dimension: 1 interpretation: [a#](x0) = x0, [c](x0) = 5, [a](x0) = x0, [b](x0) = 1x0 + 15 orientation: a#(a(b(b(x1)))) = 2x1 + 16 >= x1 = a#(a(x1)) a(a(b(b(x1)))) = 2x1 + 16 >= 16 = b(b(c(c(a(a(x1)))))) b(b(c(c(x1)))) = 16 >= 5 = c(c(b(b(b(b(x1)))))) b(b(a(a(x1)))) = 2x1 + 16 >= 5 = a(a(c(c(b(b(x1)))))) problem: DPs: TRS: a(a(b(b(x1)))) -> b(b(c(c(a(a(x1)))))) b(b(c(c(x1)))) -> c(c(b(b(b(b(x1)))))) b(b(a(a(x1)))) -> a(a(c(c(b(b(x1)))))) Qed DPs: b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(x1) b#(b(c(c(x1)))) -> b#(b(b(x1))) b#(b(c(c(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(x1) TRS: a(a(b(b(x1)))) -> b(b(c(c(a(a(x1)))))) b(b(c(c(x1)))) -> c(c(b(b(b(b(x1)))))) b(b(a(a(x1)))) -> a(a(c(c(b(b(x1)))))) Arctic Interpretation Processor: dimension: 1 interpretation: [b#](x0) = x0 + 1, [c](x0) = x0, [a](x0) = 6x0 + 6, [b](x0) = x0 orientation: b#(b(c(c(x1)))) = x1 + 1 >= x1 + 1 = b#(b(b(b(x1)))) b#(b(a(a(x1)))) = 12x1 + 12 >= x1 + 1 = b#(b(x1)) b#(b(a(a(x1)))) = 12x1 + 12 >= x1 + 1 = b#(x1) b#(b(c(c(x1)))) = x1 + 1 >= x1 + 1 = b#(b(b(x1))) b#(b(c(c(x1)))) = x1 + 1 >= x1 + 1 = b#(b(x1)) b#(b(c(c(x1)))) = x1 + 1 >= x1 + 1 = b#(x1) a(a(b(b(x1)))) = 12x1 + 12 >= 12x1 + 12 = b(b(c(c(a(a(x1)))))) b(b(c(c(x1)))) = x1 >= x1 = c(c(b(b(b(b(x1)))))) b(b(a(a(x1)))) = 12x1 + 12 >= 12x1 + 12 = a(a(c(c(b(b(x1)))))) problem: DPs: b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) b#(b(c(c(x1)))) -> b#(b(b(x1))) b#(b(c(c(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(x1) TRS: a(a(b(b(x1)))) -> b(b(c(c(a(a(x1)))))) b(b(c(c(x1)))) -> c(c(b(b(b(b(x1)))))) b(b(a(a(x1)))) -> a(a(c(c(b(b(x1)))))) EDG Processor: DPs: b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) b#(b(c(c(x1)))) -> b#(b(b(x1))) b#(b(c(c(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(x1) TRS: a(a(b(b(x1)))) -> b(b(c(c(a(a(x1)))))) b(b(c(c(x1)))) -> c(c(b(b(b(b(x1)))))) b(b(a(a(x1)))) -> a(a(c(c(b(b(x1)))))) graph: b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) -> b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) -> b#(b(c(c(x1)))) -> b#(b(b(x1))) b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) -> b#(b(c(c(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) -> b#(b(c(c(x1)))) -> b#(x1) b#(b(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) b#(b(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(c(c(x1)))) -> b#(b(b(x1))) b#(b(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(c(c(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(b(b(x1))) -> b#(b(c(c(x1)))) -> b#(x1) b#(b(c(c(x1)))) -> b#(b(x1)) -> b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) b#(b(c(c(x1)))) -> b#(b(x1)) -> b#(b(c(c(x1)))) -> b#(b(b(x1))) b#(b(c(c(x1)))) -> b#(b(x1)) -> b#(b(c(c(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(b(x1)) -> b#(b(c(c(x1)))) -> b#(x1) b#(b(c(c(x1)))) -> b#(x1) -> b#(b(c(c(x1)))) -> b#(b(b(b(x1)))) b#(b(c(c(x1)))) -> b#(x1) -> b#(b(c(c(x1)))) -> b#(b(b(x1))) b#(b(c(c(x1)))) -> b#(x1) -> b#(b(c(c(x1)))) -> b#(b(x1)) b#(b(c(c(x1)))) -> b#(x1) -> b#(b(c(c(x1)))) -> b#(x1) Matrix Interpretation Processor: dim=3 interpretation: [b#](x0) = [0 0 1]x0, [0 0 0] [0] [c](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 1 0] [0] [a](x0) = [0 0 0]x0 + [1] [1 1 0] [0], [0 1 0] [0] [b](x0) = [1 0 0]x0 + [1] [0 0 1] [0] orientation: b#(b(c(c(x1)))) = [0 0 1]x1 + [2] >= [0 0 1]x1 = b#(b(b(b(x1)))) b#(b(c(c(x1)))) = [0 0 1]x1 + [2] >= [0 0 1]x1 = b#(b(b(x1))) b#(b(c(c(x1)))) = [0 0 1]x1 + [2] >= [0 0 1]x1 = b#(b(x1)) b#(b(c(c(x1)))) = [0 0 1]x1 + [2] >= [0 0 1]x1 = b#(x1) [1 1 0] [3] [0 0 0] [1] a(a(b(b(x1)))) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = b(b(c(c(a(a(x1)))))) [1 1 0] [3] [1 1 0] [3] [0 0 0] [1] [0 0 0] [0] b(b(c(c(x1)))) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [0] = c(c(b(b(b(b(x1)))))) [0 0 1] [2] [0 0 1] [2] [1 1 0] [2] [1] b(b(a(a(x1)))) = [0 0 0]x1 + [2] >= [1] = a(a(c(c(b(b(x1)))))) [1 1 0] [1] [1] problem: DPs: TRS: a(a(b(b(x1)))) -> b(b(c(c(a(a(x1)))))) b(b(c(c(x1)))) -> c(c(b(b(b(b(x1)))))) b(b(a(a(x1)))) -> a(a(c(c(b(b(x1)))))) Qed