/export/starexec/sandbox2/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem: log(s(x1)) -> s(log(half(s(x1)))) half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) p(s(s(s(x1)))) -> s(p(s(s(x1)))) s(s(p(s(x1)))) -> s(s(x1)) 0(x1) -> x1 Proof: Matrix Interpretation Processor: dim=1 interpretation: [p](x0) = x0, [0](x0) = 2x0 + 1, [half](x0) = x0, [log](x0) = x0, [s](x0) = x0 orientation: log(s(x1)) = x1 >= x1 = s(log(half(s(x1)))) half(0(x1)) = 2x1 + 1 >= 2x1 + 1 = 0(s(s(half(x1)))) half(s(0(x1))) = 2x1 + 1 >= 2x1 + 1 = 0(x1) half(s(s(x1))) = x1 >= x1 = s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) = x1 >= x1 = s(s(half(half(x1)))) p(s(s(s(x1)))) = x1 >= x1 = s(p(s(s(x1)))) s(s(p(s(x1)))) = x1 >= x1 = s(s(x1)) 0(x1) = 2x1 + 1 >= x1 = x1 problem: log(s(x1)) -> s(log(half(s(x1)))) half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) p(s(s(s(x1)))) -> s(p(s(s(x1)))) s(s(p(s(x1)))) -> s(s(x1)) DP Processor: DPs: log#(s(x1)) -> half#(s(x1)) log#(s(x1)) -> log#(half(s(x1))) log#(s(x1)) -> s#(log(half(s(x1)))) half#(0(x1)) -> half#(x1) half#(0(x1)) -> s#(half(x1)) half#(0(x1)) -> s#(s(half(x1))) half#(s(s(x1))) -> p#(s(s(x1))) half#(s(s(x1))) -> half#(p(s(s(x1)))) half#(s(s(x1))) -> s#(half(p(s(s(x1))))) half#(half(s(s(s(s(x1)))))) -> half#(x1) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) half#(half(s(s(s(s(x1)))))) -> s#(half(half(x1))) half#(half(s(s(s(s(x1)))))) -> s#(s(half(half(x1)))) p#(s(s(s(x1)))) -> p#(s(s(x1))) p#(s(s(s(x1)))) -> s#(p(s(s(x1)))) s#(s(p(s(x1)))) -> s#(s(x1)) TRS: log(s(x1)) -> s(log(half(s(x1)))) half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) p(s(s(s(x1)))) -> s(p(s(s(x1)))) s(s(p(s(x1)))) -> s(s(x1)) TDG Processor: DPs: log#(s(x1)) -> half#(s(x1)) log#(s(x1)) -> log#(half(s(x1))) log#(s(x1)) -> s#(log(half(s(x1)))) half#(0(x1)) -> half#(x1) half#(0(x1)) -> s#(half(x1)) half#(0(x1)) -> s#(s(half(x1))) half#(s(s(x1))) -> p#(s(s(x1))) half#(s(s(x1))) -> half#(p(s(s(x1)))) half#(s(s(x1))) -> s#(half(p(s(s(x1))))) half#(half(s(s(s(s(x1)))))) -> half#(x1) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) half#(half(s(s(s(s(x1)))))) -> s#(half(half(x1))) half#(half(s(s(s(s(x1)))))) -> s#(s(half(half(x1)))) p#(s(s(s(x1)))) -> p#(s(s(x1))) p#(s(s(s(x1)))) -> s#(p(s(s(x1)))) s#(s(p(s(x1)))) -> s#(s(x1)) TRS: log(s(x1)) -> s(log(half(s(x1)))) half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) p(s(s(s(x1)))) -> s(p(s(s(x1)))) s(s(p(s(x1)))) -> s(s(x1)) graph: p#(s(s(s(x1)))) -> p#(s(s(x1))) -> p#(s(s(s(x1)))) -> s#(p(s(s(x1)))) p#(s(s(s(x1)))) -> p#(s(s(x1))) -> p#(s(s(s(x1)))) -> p#(s(s(x1))) p#(s(s(s(x1)))) -> s#(p(s(s(x1)))) -> s#(s(p(s(x1)))) -> s#(s(x1)) s#(s(p(s(x1)))) -> s#(s(x1)) -> s#(s(p(s(x1)))) -> s#(s(x1)) half#(0(x1)) -> s#(half(x1)) -> s#(s(p(s(x1)))) -> s#(s(x1)) half#(0(x1)) -> s#(s(half(x1))) -> s#(s(p(s(x1)))) -> s#(s(x1)) half#(0(x1)) -> half#(x1) -> half#(half(s(s(s(s(x1)))))) -> s#(s(half(half(x1)))) half#(0(x1)) -> half#(x1) -> half#(half(s(s(s(s(x1)))))) -> s#(half(half(x1))) half#(0(x1)) -> half#(x1) -> half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) half#(0(x1)) -> half#(x1) -> half#(half(s(s(s(s(x1)))))) -> half#(x1) half#(0(x1)) -> half#(x1) -> half#(s(s(x1))) -> s#(half(p(s(s(x1))))) half#(0(x1)) -> half#(x1) -> half#(s(s(x1))) -> half#(p(s(s(x1)))) half#(0(x1)) -> half#(x1) -> half#(s(s(x1))) -> p#(s(s(x1))) half#(0(x1)) -> half#(x1) -> half#(0(x1)) -> s#(s(half(x1))) half#(0(x1)) -> half#(x1) -> half#(0(x1)) -> s#(half(x1)) half#(0(x1)) -> half#(x1) -> half#(0(x1)) -> half#(x1) half#(half(s(s(s(s(x1)))))) -> s#(half(half(x1))) -> s#(s(p(s(x1)))) -> s#(s(x1)) half#(half(s(s(s(s(x1)))))) -> s#(s(half(half(x1)))) -> s#(s(p(s(x1)))) -> s#(s(x1)) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) -> half#(half(s(s(s(s(x1)))))) -> s#(s(half(half(x1)))) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) -> half#(half(s(s(s(s(x1)))))) -> s#(half(half(x1))) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) -> half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) -> half#(half(s(s(s(s(x1)))))) -> half#(x1) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) -> half#(s(s(x1))) -> s#(half(p(s(s(x1))))) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) -> half#(s(s(x1))) -> half#(p(s(s(x1)))) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) -> half#(s(s(x1))) -> p#(s(s(x1))) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) -> half#(0(x1)) -> s#(s(half(x1))) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) -> half#(0(x1)) -> s#(half(x1)) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) -> half#(0(x1)) -> half#(x1) half#(half(s(s(s(s(x1)))))) -> half#(x1) -> half#(half(s(s(s(s(x1)))))) -> s#(s(half(half(x1)))) half#(half(s(s(s(s(x1)))))) -> half#(x1) -> half#(half(s(s(s(s(x1)))))) -> s#(half(half(x1))) half#(half(s(s(s(s(x1)))))) -> half#(x1) -> half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) half#(half(s(s(s(s(x1)))))) -> half#(x1) -> half#(half(s(s(s(s(x1)))))) -> half#(x1) half#(half(s(s(s(s(x1)))))) -> half#(x1) -> half#(s(s(x1))) -> s#(half(p(s(s(x1))))) half#(half(s(s(s(s(x1)))))) -> half#(x1) -> half#(s(s(x1))) -> half#(p(s(s(x1)))) half#(half(s(s(s(s(x1)))))) -> half#(x1) -> half#(s(s(x1))) -> p#(s(s(x1))) half#(half(s(s(s(s(x1)))))) -> half#(x1) -> half#(0(x1)) -> s#(s(half(x1))) half#(half(s(s(s(s(x1)))))) -> half#(x1) -> half#(0(x1)) -> s#(half(x1)) half#(half(s(s(s(s(x1)))))) -> half#(x1) -> half#(0(x1)) -> half#(x1) half#(s(s(x1))) -> p#(s(s(x1))) -> p#(s(s(s(x1)))) -> s#(p(s(s(x1)))) half#(s(s(x1))) -> p#(s(s(x1))) -> p#(s(s(s(x1)))) -> p#(s(s(x1))) half#(s(s(x1))) -> s#(half(p(s(s(x1))))) -> s#(s(p(s(x1)))) -> s#(s(x1)) half#(s(s(x1))) -> half#(p(s(s(x1)))) -> half#(half(s(s(s(s(x1)))))) -> s#(s(half(half(x1)))) half#(s(s(x1))) -> half#(p(s(s(x1)))) -> half#(half(s(s(s(s(x1)))))) -> s#(half(half(x1))) half#(s(s(x1))) -> half#(p(s(s(x1)))) -> half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) half#(s(s(x1))) -> half#(p(s(s(x1)))) -> half#(half(s(s(s(s(x1)))))) -> half#(x1) half#(s(s(x1))) -> half#(p(s(s(x1)))) -> half#(s(s(x1))) -> s#(half(p(s(s(x1))))) half#(s(s(x1))) -> half#(p(s(s(x1)))) -> half#(s(s(x1))) -> half#(p(s(s(x1)))) half#(s(s(x1))) -> half#(p(s(s(x1)))) -> half#(s(s(x1))) -> p#(s(s(x1))) half#(s(s(x1))) -> half#(p(s(s(x1)))) -> half#(0(x1)) -> s#(s(half(x1))) half#(s(s(x1))) -> half#(p(s(s(x1)))) -> half#(0(x1)) -> s#(half(x1)) half#(s(s(x1))) -> half#(p(s(s(x1)))) -> half#(0(x1)) -> half#(x1) log#(s(x1)) -> s#(log(half(s(x1)))) -> s#(s(p(s(x1)))) -> s#(s(x1)) log#(s(x1)) -> half#(s(x1)) -> half#(half(s(s(s(s(x1)))))) -> s#(s(half(half(x1)))) log#(s(x1)) -> half#(s(x1)) -> half#(half(s(s(s(s(x1)))))) -> s#(half(half(x1))) log#(s(x1)) -> half#(s(x1)) -> half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) log#(s(x1)) -> half#(s(x1)) -> half#(half(s(s(s(s(x1)))))) -> half#(x1) log#(s(x1)) -> half#(s(x1)) -> half#(s(s(x1))) -> s#(half(p(s(s(x1))))) log#(s(x1)) -> half#(s(x1)) -> half#(s(s(x1))) -> half#(p(s(s(x1)))) log#(s(x1)) -> half#(s(x1)) -> half#(s(s(x1))) -> p#(s(s(x1))) log#(s(x1)) -> half#(s(x1)) -> half#(0(x1)) -> s#(s(half(x1))) log#(s(x1)) -> half#(s(x1)) -> half#(0(x1)) -> s#(half(x1)) log#(s(x1)) -> half#(s(x1)) -> half#(0(x1)) -> half#(x1) log#(s(x1)) -> log#(half(s(x1))) -> log#(s(x1)) -> s#(log(half(s(x1)))) log#(s(x1)) -> log#(half(s(x1))) -> log#(s(x1)) -> log#(half(s(x1))) log#(s(x1)) -> log#(half(s(x1))) -> log#(s(x1)) -> half#(s(x1)) EDG Processor: DPs: log#(s(x1)) -> half#(s(x1)) log#(s(x1)) -> log#(half(s(x1))) log#(s(x1)) -> s#(log(half(s(x1)))) half#(0(x1)) -> half#(x1) half#(0(x1)) -> s#(half(x1)) half#(0(x1)) -> s#(s(half(x1))) half#(s(s(x1))) -> p#(s(s(x1))) half#(s(s(x1))) -> half#(p(s(s(x1)))) half#(s(s(x1))) -> s#(half(p(s(s(x1))))) half#(half(s(s(s(s(x1)))))) -> half#(x1) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) half#(half(s(s(s(s(x1)))))) -> s#(half(half(x1))) half#(half(s(s(s(s(x1)))))) -> s#(s(half(half(x1)))) p#(s(s(s(x1)))) -> p#(s(s(x1))) p#(s(s(s(x1)))) -> s#(p(s(s(x1)))) s#(s(p(s(x1)))) -> s#(s(x1)) TRS: log(s(x1)) -> s(log(half(s(x1)))) half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) p(s(s(s(x1)))) -> s(p(s(s(x1)))) s(s(p(s(x1)))) -> s(s(x1)) graph: p#(s(s(s(x1)))) -> p#(s(s(x1))) -> p#(s(s(s(x1)))) -> p#(s(s(x1))) p#(s(s(s(x1)))) -> p#(s(s(x1))) -> p#(s(s(s(x1)))) -> s#(p(s(s(x1)))) p#(s(s(s(x1)))) -> s#(p(s(s(x1)))) -> s#(s(p(s(x1)))) -> s#(s(x1)) s#(s(p(s(x1)))) -> s#(s(x1)) -> s#(s(p(s(x1)))) -> s#(s(x1)) half#(0(x1)) -> s#(half(x1)) -> s#(s(p(s(x1)))) -> s#(s(x1)) half#(0(x1)) -> s#(s(half(x1))) -> s#(s(p(s(x1)))) -> s#(s(x1)) half#(0(x1)) -> half#(x1) -> half#(0(x1)) -> half#(x1) half#(0(x1)) -> half#(x1) -> half#(0(x1)) -> s#(half(x1)) half#(0(x1)) -> half#(x1) -> half#(0(x1)) -> s#(s(half(x1))) half#(0(x1)) -> half#(x1) -> half#(s(s(x1))) -> p#(s(s(x1))) half#(0(x1)) -> half#(x1) -> half#(s(s(x1))) -> half#(p(s(s(x1)))) half#(0(x1)) -> half#(x1) -> half#(s(s(x1))) -> s#(half(p(s(s(x1))))) half#(0(x1)) -> half#(x1) -> half#(half(s(s(s(s(x1)))))) -> half#(x1) half#(0(x1)) -> half#(x1) -> half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) half#(0(x1)) -> half#(x1) -> half#(half(s(s(s(s(x1)))))) -> s#(half(half(x1))) half#(0(x1)) -> half#(x1) -> half#(half(s(s(s(s(x1)))))) -> s#(s(half(half(x1)))) half#(half(s(s(s(s(x1)))))) -> s#(half(half(x1))) -> s#(s(p(s(x1)))) -> s#(s(x1)) half#(half(s(s(s(s(x1)))))) -> s#(s(half(half(x1)))) -> s#(s(p(s(x1)))) -> s#(s(x1)) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) -> half#(0(x1)) -> half#(x1) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) -> half#(0(x1)) -> s#(half(x1)) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) -> half#(0(x1)) -> s#(s(half(x1))) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) -> half#(s(s(x1))) -> p#(s(s(x1))) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) -> half#(s(s(x1))) -> half#(p(s(s(x1)))) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) -> half#(s(s(x1))) -> s#(half(p(s(s(x1))))) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) -> half#(half(s(s(s(s(x1)))))) -> half#(x1) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) -> half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) -> half#(half(s(s(s(s(x1)))))) -> s#(half(half(x1))) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) -> half#(half(s(s(s(s(x1)))))) -> s#(s(half(half(x1)))) half#(half(s(s(s(s(x1)))))) -> half#(x1) -> half#(0(x1)) -> half#(x1) half#(half(s(s(s(s(x1)))))) -> half#(x1) -> half#(0(x1)) -> s#(half(x1)) half#(half(s(s(s(s(x1)))))) -> half#(x1) -> half#(0(x1)) -> s#(s(half(x1))) half#(half(s(s(s(s(x1)))))) -> half#(x1) -> half#(s(s(x1))) -> p#(s(s(x1))) half#(half(s(s(s(s(x1)))))) -> half#(x1) -> half#(s(s(x1))) -> half#(p(s(s(x1)))) half#(half(s(s(s(s(x1)))))) -> half#(x1) -> half#(s(s(x1))) -> s#(half(p(s(s(x1))))) half#(half(s(s(s(s(x1)))))) -> half#(x1) -> half#(half(s(s(s(s(x1)))))) -> half#(x1) half#(half(s(s(s(s(x1)))))) -> half#(x1) -> half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) half#(half(s(s(s(s(x1)))))) -> half#(x1) -> half#(half(s(s(s(s(x1)))))) -> s#(half(half(x1))) half#(half(s(s(s(s(x1)))))) -> half#(x1) -> half#(half(s(s(s(s(x1)))))) -> s#(s(half(half(x1)))) half#(s(s(x1))) -> p#(s(s(x1))) -> p#(s(s(s(x1)))) -> p#(s(s(x1))) half#(s(s(x1))) -> p#(s(s(x1))) -> p#(s(s(s(x1)))) -> s#(p(s(s(x1)))) half#(s(s(x1))) -> s#(half(p(s(s(x1))))) -> s#(s(p(s(x1)))) -> s#(s(x1)) half#(s(s(x1))) -> half#(p(s(s(x1)))) -> half#(s(s(x1))) -> p#(s(s(x1))) half#(s(s(x1))) -> half#(p(s(s(x1)))) -> half#(s(s(x1))) -> half#(p(s(s(x1)))) half#(s(s(x1))) -> half#(p(s(s(x1)))) -> half#(s(s(x1))) -> s#(half(p(s(s(x1))))) log#(s(x1)) -> s#(log(half(s(x1)))) -> s#(s(p(s(x1)))) -> s#(s(x1)) log#(s(x1)) -> half#(s(x1)) -> half#(s(s(x1))) -> p#(s(s(x1))) log#(s(x1)) -> half#(s(x1)) -> half#(s(s(x1))) -> half#(p(s(s(x1)))) log#(s(x1)) -> half#(s(x1)) -> half#(s(s(x1))) -> s#(half(p(s(s(x1))))) log#(s(x1)) -> log#(half(s(x1))) -> log#(s(x1)) -> half#(s(x1)) log#(s(x1)) -> log#(half(s(x1))) -> log#(s(x1)) -> log#(half(s(x1))) log#(s(x1)) -> log#(half(s(x1))) -> log#(s(x1)) -> s#(log(half(s(x1)))) SCC Processor: #sccs: 5 #rules: 7 #arcs: 51/256 DPs: log#(s(x1)) -> log#(half(s(x1))) TRS: log(s(x1)) -> s(log(half(s(x1)))) half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) p(s(s(s(x1)))) -> s(p(s(s(x1)))) s(s(p(s(x1)))) -> s(s(x1)) Usable Rule Processor: DPs: log#(s(x1)) -> log#(half(s(x1))) TRS: s(s(p(s(x1)))) -> s(s(x1)) half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) p(s(s(s(x1)))) -> s(p(s(s(x1)))) Arctic Interpretation Processor: dimension: 1 usable rules: s(s(p(s(x1)))) -> s(s(x1)) half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) p(s(s(s(x1)))) -> s(p(s(s(x1)))) interpretation: [log#](x0) = x0 + 1, [p](x0) = -4x0 + 0, [0](x0) = 0, [half](x0) = -2x0 + 0, [s](x0) = 4x0 + 2 orientation: log#(s(x1)) = 4x1 + 2 >= 2x1 + 1 = log#(half(s(x1))) s(s(p(s(x1)))) = 8x1 + 8 >= 8x1 + 6 = s(s(x1)) half(0(x1)) = 0 >= 0 = 0(s(s(half(x1)))) half(s(0(x1))) = 2 >= 0 = 0(x1) half(s(s(x1))) = 6x1 + 4 >= 6x1 + 4 = s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) = 12x1 + 10 >= 4x1 + 8 = s(s(half(half(x1)))) p(s(s(s(x1)))) = 8x1 + 6 >= 8x1 + 6 = s(p(s(s(x1)))) problem: DPs: TRS: s(s(p(s(x1)))) -> s(s(x1)) half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) p(s(s(s(x1)))) -> s(p(s(s(x1)))) Qed DPs: half#(0(x1)) -> half#(x1) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) half#(half(s(s(s(s(x1)))))) -> half#(x1) TRS: log(s(x1)) -> s(log(half(s(x1)))) half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) p(s(s(s(x1)))) -> s(p(s(s(x1)))) s(s(p(s(x1)))) -> s(s(x1)) Usable Rule Processor: DPs: half#(0(x1)) -> half#(x1) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) half#(half(s(s(s(s(x1)))))) -> half#(x1) TRS: half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) s(s(p(s(x1)))) -> s(s(x1)) p(s(s(s(x1)))) -> s(p(s(s(x1)))) KBO Processor: argument filtering: pi(s) = 0 pi(half) = 0 pi(0) = [0] pi(p) = 0 pi(half#) = 0 usable rules: half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) s(s(p(s(x1)))) -> s(s(x1)) p(s(s(s(x1)))) -> s(p(s(s(x1)))) weight function: w0 = 1 w(0) = 1 w(half#) = w(p) = w(half) = w(s) = 0 precedence: half# ~ p ~ 0 ~ half ~ s problem: DPs: half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) half#(half(s(s(s(s(x1)))))) -> half#(x1) TRS: half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) s(s(p(s(x1)))) -> s(s(x1)) p(s(s(s(x1)))) -> s(p(s(s(x1)))) Restore Modifier: DPs: half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) half#(half(s(s(s(s(x1)))))) -> half#(x1) TRS: log(s(x1)) -> s(log(half(s(x1)))) half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) p(s(s(s(x1)))) -> s(p(s(s(x1)))) s(s(p(s(x1)))) -> s(s(x1)) EDG Processor: DPs: half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) half#(half(s(s(s(s(x1)))))) -> half#(x1) TRS: log(s(x1)) -> s(log(half(s(x1)))) half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) p(s(s(s(x1)))) -> s(p(s(s(x1)))) s(s(p(s(x1)))) -> s(s(x1)) graph: half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) -> half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) -> half#(half(s(s(s(s(x1)))))) -> half#(x1) half#(half(s(s(s(s(x1)))))) -> half#(x1) -> half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) half#(half(s(s(s(s(x1)))))) -> half#(x1) -> half#(half(s(s(s(s(x1)))))) -> half#(x1) Usable Rule Processor: DPs: half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) half#(half(s(s(s(s(x1)))))) -> half#(x1) TRS: half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) s(s(p(s(x1)))) -> s(s(x1)) p(s(s(s(x1)))) -> s(p(s(s(x1)))) Arctic Interpretation Processor: dimension: 1 usable rules: half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) s(s(p(s(x1)))) -> s(s(x1)) p(s(s(s(x1)))) -> s(p(s(s(x1)))) interpretation: [half#](x0) = x0 + 0, [p](x0) = x0 + 0, [0](x0) = 7, [half](x0) = 1x0 + 0, [s](x0) = x0 + 7 orientation: half#(half(s(s(s(s(x1)))))) = 1x1 + 8 >= 1x1 + 0 = half#(half(x1)) half#(half(s(s(s(s(x1)))))) = 1x1 + 8 >= x1 + 0 = half#(x1) half(0(x1)) = 8 >= 7 = 0(s(s(half(x1)))) half(s(0(x1))) = 8 >= 7 = 0(x1) half(s(s(x1))) = 1x1 + 8 >= 1x1 + 8 = s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) = 2x1 + 9 >= 2x1 + 7 = s(s(half(half(x1)))) s(s(p(s(x1)))) = x1 + 7 >= x1 + 7 = s(s(x1)) p(s(s(s(x1)))) = x1 + 7 >= x1 + 7 = s(p(s(s(x1)))) problem: DPs: half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) TRS: half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) s(s(p(s(x1)))) -> s(s(x1)) p(s(s(s(x1)))) -> s(p(s(s(x1)))) Restore Modifier: DPs: half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) TRS: log(s(x1)) -> s(log(half(s(x1)))) half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) p(s(s(s(x1)))) -> s(p(s(s(x1)))) s(s(p(s(x1)))) -> s(s(x1)) EDG Processor: DPs: half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) TRS: log(s(x1)) -> s(log(half(s(x1)))) half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) p(s(s(s(x1)))) -> s(p(s(s(x1)))) s(s(p(s(x1)))) -> s(s(x1)) graph: half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) -> half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) Usable Rule Processor: DPs: half#(half(s(s(s(s(x1)))))) -> half#(half(x1)) TRS: half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) s(s(p(s(x1)))) -> s(s(x1)) p(s(s(s(x1)))) -> s(p(s(s(x1)))) Arctic Interpretation Processor: dimension: 1 usable rules: half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) s(s(p(s(x1)))) -> s(s(x1)) p(s(s(s(x1)))) -> s(p(s(s(x1)))) interpretation: [half#](x0) = x0 + 0, [p](x0) = -1x0 + 1, [0](x0) = 0, [half](x0) = x0, [s](x0) = 1x0 + 1 orientation: half#(half(s(s(s(s(x1)))))) = 4x1 + 4 >= x1 + 0 = half#(half(x1)) half(0(x1)) = 0 >= 0 = 0(s(s(half(x1)))) half(s(0(x1))) = 1 >= 0 = 0(x1) half(s(s(x1))) = 2x1 + 2 >= 2x1 + 2 = s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) = 4x1 + 4 >= 2x1 + 2 = s(s(half(half(x1)))) s(s(p(s(x1)))) = 2x1 + 3 >= 2x1 + 2 = s(s(x1)) p(s(s(s(x1)))) = 2x1 + 2 >= 2x1 + 2 = s(p(s(s(x1)))) problem: DPs: TRS: half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) s(s(p(s(x1)))) -> s(s(x1)) p(s(s(s(x1)))) -> s(p(s(s(x1)))) Qed DPs: half#(s(s(x1))) -> half#(p(s(s(x1)))) TRS: log(s(x1)) -> s(log(half(s(x1)))) half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) p(s(s(s(x1)))) -> s(p(s(s(x1)))) s(s(p(s(x1)))) -> s(s(x1)) Usable Rule Processor: DPs: half#(s(s(x1))) -> half#(p(s(s(x1)))) TRS: s(s(p(s(x1)))) -> s(s(x1)) p(s(s(s(x1)))) -> s(p(s(s(x1)))) Arctic Interpretation Processor: dimension: 1 usable rules: s(s(p(s(x1)))) -> s(s(x1)) p(s(s(s(x1)))) -> s(p(s(s(x1)))) interpretation: [half#](x0) = -15x0 + 0, [p](x0) = -7x0 + 0, [s](x0) = 7x0 + 9 orientation: half#(s(s(x1))) = -1x1 + 1 >= -8x1 + 0 = half#(p(s(s(x1)))) s(s(p(s(x1)))) = 14x1 + 16 >= 14x1 + 16 = s(s(x1)) p(s(s(s(x1)))) = 14x1 + 16 >= 14x1 + 16 = s(p(s(s(x1)))) problem: DPs: TRS: s(s(p(s(x1)))) -> s(s(x1)) p(s(s(s(x1)))) -> s(p(s(s(x1)))) Qed DPs: p#(s(s(s(x1)))) -> p#(s(s(x1))) TRS: log(s(x1)) -> s(log(half(s(x1)))) half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) p(s(s(s(x1)))) -> s(p(s(s(x1)))) s(s(p(s(x1)))) -> s(s(x1)) Usable Rule Processor: DPs: p#(s(s(s(x1)))) -> p#(s(s(x1))) TRS: s(s(p(s(x1)))) -> s(s(x1)) Arctic Interpretation Processor: dimension: 1 usable rules: s(s(p(s(x1)))) -> s(s(x1)) interpretation: [p#](x0) = x0, [p](x0) = 5x0, [s](x0) = 4x0 orientation: p#(s(s(s(x1)))) = 12x1 >= 8x1 = p#(s(s(x1))) s(s(p(s(x1)))) = 17x1 >= 8x1 = s(s(x1)) problem: DPs: TRS: s(s(p(s(x1)))) -> s(s(x1)) Qed DPs: s#(s(p(s(x1)))) -> s#(s(x1)) TRS: log(s(x1)) -> s(log(half(s(x1)))) half(0(x1)) -> 0(s(s(half(x1)))) half(s(0(x1))) -> 0(x1) half(s(s(x1))) -> s(half(p(s(s(x1))))) half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1)))) p(s(s(s(x1)))) -> s(p(s(s(x1)))) s(s(p(s(x1)))) -> s(s(x1)) Usable Rule Processor: DPs: s#(s(p(s(x1)))) -> s#(s(x1)) TRS: s(s(p(s(x1)))) -> s(s(x1)) Arctic Interpretation Processor: dimension: 1 usable rules: s(s(p(s(x1)))) -> s(s(x1)) interpretation: [s#](x0) = x0 + 3, [p](x0) = 4x0 + 4, [s](x0) = 1x0 + 2 orientation: s#(s(p(s(x1)))) = 6x1 + 7 >= 1x1 + 3 = s#(s(x1)) s(s(p(s(x1)))) = 7x1 + 8 >= 2x1 + 3 = s(s(x1)) problem: DPs: TRS: s(s(p(s(x1)))) -> s(s(x1)) Qed