/export/starexec/sandbox/solver/bin/starexec_run_ttt2 /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES Problem: s(s(0())) -> f(s(0())) g(x) -> h(x,x) s(x) -> h(x,0()) s(x) -> h(0(),x) f(g(x)) -> g(g(f(x))) g(s(x)) -> s(s(g(x))) h(f(x),g(x)) -> f(s(x)) s(s(0())) -> k(0()) k(0()) -> 0() s(s(s(0()))) -> k(s(0())) k(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(0())))))) h(k(x),g(x)) -> k(s(x)) Proof: DP Processor: DPs: s#(s(0())) -> f#(s(0())) g#(x) -> h#(x,x) s#(x) -> h#(x,0()) s#(x) -> h#(0(),x) f#(g(x)) -> f#(x) f#(g(x)) -> g#(f(x)) f#(g(x)) -> g#(g(f(x))) g#(s(x)) -> g#(x) g#(s(x)) -> s#(g(x)) g#(s(x)) -> s#(s(g(x))) h#(f(x),g(x)) -> s#(x) h#(f(x),g(x)) -> f#(s(x)) s#(s(0())) -> k#(0()) s#(s(s(0()))) -> k#(s(0())) s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(0())))) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) h#(k(x),g(x)) -> s#(x) h#(k(x),g(x)) -> k#(s(x)) TRS: s(s(0())) -> f(s(0())) g(x) -> h(x,x) s(x) -> h(x,0()) s(x) -> h(0(),x) f(g(x)) -> g(g(f(x))) g(s(x)) -> s(s(g(x))) h(f(x),g(x)) -> f(s(x)) s(s(0())) -> k(0()) k(0()) -> 0() s(s(s(0()))) -> k(s(0())) k(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(0())))))) h(k(x),g(x)) -> k(s(x)) TDG Processor: DPs: s#(s(0())) -> f#(s(0())) g#(x) -> h#(x,x) s#(x) -> h#(x,0()) s#(x) -> h#(0(),x) f#(g(x)) -> f#(x) f#(g(x)) -> g#(f(x)) f#(g(x)) -> g#(g(f(x))) g#(s(x)) -> g#(x) g#(s(x)) -> s#(g(x)) g#(s(x)) -> s#(s(g(x))) h#(f(x),g(x)) -> s#(x) h#(f(x),g(x)) -> f#(s(x)) s#(s(0())) -> k#(0()) s#(s(s(0()))) -> k#(s(0())) s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(0())))) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) h#(k(x),g(x)) -> s#(x) h#(k(x),g(x)) -> k#(s(x)) TRS: s(s(0())) -> f(s(0())) g(x) -> h(x,x) s(x) -> h(x,0()) s(x) -> h(0(),x) f(g(x)) -> g(g(f(x))) g(s(x)) -> s(s(g(x))) h(f(x),g(x)) -> f(s(x)) s(s(0())) -> k(0()) k(0()) -> 0() s(s(s(0()))) -> k(s(0())) k(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(0())))))) h(k(x),g(x)) -> k(s(x)) graph: k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(s(s(0()))) -> k#(s(0())) k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(s(0())) -> k#(0()) k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(x) -> h#(0(),x) k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(x) -> h#(x,0()) k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(s(0())) -> f#(s(0())) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(s(s(0()))) -> k#(s(0())) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(s(0())) -> k#(0()) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(x) -> h#(0(),x) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(x) -> h#(x,0()) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(s(0())) -> f#(s(0())) k#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(s(s(0()))) -> k#(s(0())) k#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(s(0())) -> k#(0()) k#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(x) -> h#(0(),x) k#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(x) -> h#(x,0()) k#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(s(0())) -> f#(s(0())) k#(s(s(0()))) -> s#(s(s(0()))) -> s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(0()))) -> s#(s(s(0()))) -> k#(s(0())) k#(s(s(0()))) -> s#(s(s(0()))) -> s#(s(0())) -> k#(0()) k#(s(s(0()))) -> s#(s(s(0()))) -> s#(x) -> h#(0(),x) k#(s(s(0()))) -> s#(s(s(0()))) -> s#(x) -> h#(x,0()) k#(s(s(0()))) -> s#(s(s(0()))) -> s#(s(0())) -> f#(s(0())) h#(k(x),g(x)) -> k#(s(x)) -> k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) h#(k(x),g(x)) -> k#(s(x)) -> k#(s(s(0()))) -> s#(s(s(s(s(0()))))) h#(k(x),g(x)) -> k#(s(x)) -> k#(s(s(0()))) -> s#(s(s(s(0())))) h#(k(x),g(x)) -> k#(s(x)) -> k#(s(s(0()))) -> s#(s(s(0()))) h#(k(x),g(x)) -> s#(x) -> s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) h#(k(x),g(x)) -> s#(x) -> s#(s(s(0()))) -> k#(s(0())) h#(k(x),g(x)) -> s#(x) -> s#(s(0())) -> k#(0()) h#(k(x),g(x)) -> s#(x) -> s#(x) -> h#(0(),x) h#(k(x),g(x)) -> s#(x) -> s#(x) -> h#(x,0()) h#(k(x),g(x)) -> s#(x) -> s#(s(0())) -> f#(s(0())) h#(f(x),g(x)) -> f#(s(x)) -> f#(g(x)) -> g#(g(f(x))) h#(f(x),g(x)) -> f#(s(x)) -> f#(g(x)) -> g#(f(x)) h#(f(x),g(x)) -> f#(s(x)) -> f#(g(x)) -> f#(x) h#(f(x),g(x)) -> s#(x) -> s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) h#(f(x),g(x)) -> s#(x) -> s#(s(s(0()))) -> k#(s(0())) h#(f(x),g(x)) -> s#(x) -> s#(s(0())) -> k#(0()) h#(f(x),g(x)) -> s#(x) -> s#(x) -> h#(0(),x) h#(f(x),g(x)) -> s#(x) -> s#(x) -> h#(x,0()) h#(f(x),g(x)) -> s#(x) -> s#(s(0())) -> f#(s(0())) g#(s(x)) -> g#(x) -> g#(s(x)) -> s#(s(g(x))) g#(s(x)) -> g#(x) -> g#(s(x)) -> s#(g(x)) g#(s(x)) -> g#(x) -> g#(s(x)) -> g#(x) g#(s(x)) -> g#(x) -> g#(x) -> h#(x,x) g#(s(x)) -> s#(g(x)) -> s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) g#(s(x)) -> s#(g(x)) -> s#(s(s(0()))) -> k#(s(0())) g#(s(x)) -> s#(g(x)) -> s#(s(0())) -> k#(0()) g#(s(x)) -> s#(g(x)) -> s#(x) -> h#(0(),x) g#(s(x)) -> s#(g(x)) -> s#(x) -> h#(x,0()) g#(s(x)) -> s#(g(x)) -> s#(s(0())) -> f#(s(0())) g#(s(x)) -> s#(s(g(x))) -> s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) g#(s(x)) -> s#(s(g(x))) -> s#(s(s(0()))) -> k#(s(0())) g#(s(x)) -> s#(s(g(x))) -> s#(s(0())) -> k#(0()) g#(s(x)) -> s#(s(g(x))) -> s#(x) -> h#(0(),x) g#(s(x)) -> s#(s(g(x))) -> s#(x) -> h#(x,0()) g#(s(x)) -> s#(s(g(x))) -> s#(s(0())) -> f#(s(0())) g#(x) -> h#(x,x) -> h#(k(x),g(x)) -> k#(s(x)) g#(x) -> h#(x,x) -> h#(k(x),g(x)) -> s#(x) g#(x) -> h#(x,x) -> h#(f(x),g(x)) -> f#(s(x)) g#(x) -> h#(x,x) -> h#(f(x),g(x)) -> s#(x) f#(g(x)) -> g#(g(f(x))) -> g#(s(x)) -> s#(s(g(x))) f#(g(x)) -> g#(g(f(x))) -> g#(s(x)) -> s#(g(x)) f#(g(x)) -> g#(g(f(x))) -> g#(s(x)) -> g#(x) f#(g(x)) -> g#(g(f(x))) -> g#(x) -> h#(x,x) f#(g(x)) -> g#(f(x)) -> g#(s(x)) -> s#(s(g(x))) f#(g(x)) -> g#(f(x)) -> g#(s(x)) -> s#(g(x)) f#(g(x)) -> g#(f(x)) -> g#(s(x)) -> g#(x) f#(g(x)) -> g#(f(x)) -> g#(x) -> h#(x,x) f#(g(x)) -> f#(x) -> f#(g(x)) -> g#(g(f(x))) f#(g(x)) -> f#(x) -> f#(g(x)) -> g#(f(x)) f#(g(x)) -> f#(x) -> f#(g(x)) -> f#(x) s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) -> k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) -> k#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) -> k#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) -> k#(s(s(0()))) -> s#(s(s(0()))) s#(s(s(0()))) -> k#(s(0())) -> k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) s#(s(s(0()))) -> k#(s(0())) -> k#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(s(0()))) -> k#(s(0())) -> k#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(s(0()))) -> k#(s(0())) -> k#(s(s(0()))) -> s#(s(s(0()))) s#(s(0())) -> k#(0()) -> k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) s#(s(0())) -> k#(0()) -> k#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(0())) -> k#(0()) -> k#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(0())) -> k#(0()) -> k#(s(s(0()))) -> s#(s(s(0()))) s#(s(0())) -> f#(s(0())) -> f#(g(x)) -> g#(g(f(x))) s#(s(0())) -> f#(s(0())) -> f#(g(x)) -> g#(f(x)) s#(s(0())) -> f#(s(0())) -> f#(g(x)) -> f#(x) s#(x) -> h#(0(),x) -> h#(k(x),g(x)) -> k#(s(x)) s#(x) -> h#(0(),x) -> h#(k(x),g(x)) -> s#(x) s#(x) -> h#(0(),x) -> h#(f(x),g(x)) -> f#(s(x)) s#(x) -> h#(0(),x) -> h#(f(x),g(x)) -> s#(x) s#(x) -> h#(x,0()) -> h#(k(x),g(x)) -> k#(s(x)) s#(x) -> h#(x,0()) -> h#(k(x),g(x)) -> s#(x) s#(x) -> h#(x,0()) -> h#(f(x),g(x)) -> f#(s(x)) s#(x) -> h#(x,0()) -> h#(f(x),g(x)) -> s#(x) EDG Processor: DPs: s#(s(0())) -> f#(s(0())) g#(x) -> h#(x,x) s#(x) -> h#(x,0()) s#(x) -> h#(0(),x) f#(g(x)) -> f#(x) f#(g(x)) -> g#(f(x)) f#(g(x)) -> g#(g(f(x))) g#(s(x)) -> g#(x) g#(s(x)) -> s#(g(x)) g#(s(x)) -> s#(s(g(x))) h#(f(x),g(x)) -> s#(x) h#(f(x),g(x)) -> f#(s(x)) s#(s(0())) -> k#(0()) s#(s(s(0()))) -> k#(s(0())) s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(0())))) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) h#(k(x),g(x)) -> s#(x) h#(k(x),g(x)) -> k#(s(x)) TRS: s(s(0())) -> f(s(0())) g(x) -> h(x,x) s(x) -> h(x,0()) s(x) -> h(0(),x) f(g(x)) -> g(g(f(x))) g(s(x)) -> s(s(g(x))) h(f(x),g(x)) -> f(s(x)) s(s(0())) -> k(0()) k(0()) -> 0() s(s(s(0()))) -> k(s(0())) k(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(0())))))) h(k(x),g(x)) -> k(s(x)) graph: k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(s(0())) -> f#(s(0())) k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(x) -> h#(x,0()) k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(x) -> h#(0(),x) k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(s(0())) -> k#(0()) k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(s(s(0()))) -> k#(s(0())) k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(s(0())) -> f#(s(0())) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(x) -> h#(x,0()) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(x) -> h#(0(),x) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(s(0())) -> k#(0()) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(s(s(0()))) -> k#(s(0())) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(s(0())) -> f#(s(0())) k#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(x) -> h#(x,0()) k#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(x) -> h#(0(),x) k#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(s(0())) -> k#(0()) k#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(s(s(0()))) -> k#(s(0())) k#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(0()))) -> s#(s(0())) -> f#(s(0())) k#(s(s(0()))) -> s#(s(s(0()))) -> s#(x) -> h#(x,0()) k#(s(s(0()))) -> s#(s(s(0()))) -> s#(x) -> h#(0(),x) k#(s(s(0()))) -> s#(s(s(0()))) -> s#(s(0())) -> k#(0()) k#(s(s(0()))) -> s#(s(s(0()))) -> s#(s(s(0()))) -> k#(s(0())) k#(s(s(0()))) -> s#(s(s(0()))) -> s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) h#(k(x),g(x)) -> k#(s(x)) -> k#(s(s(0()))) -> s#(s(s(0()))) h#(k(x),g(x)) -> k#(s(x)) -> k#(s(s(0()))) -> s#(s(s(s(0())))) h#(k(x),g(x)) -> k#(s(x)) -> k#(s(s(0()))) -> s#(s(s(s(s(0()))))) h#(k(x),g(x)) -> k#(s(x)) -> k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) h#(k(x),g(x)) -> s#(x) -> s#(s(0())) -> f#(s(0())) h#(k(x),g(x)) -> s#(x) -> s#(x) -> h#(x,0()) h#(k(x),g(x)) -> s#(x) -> s#(x) -> h#(0(),x) h#(k(x),g(x)) -> s#(x) -> s#(s(0())) -> k#(0()) h#(k(x),g(x)) -> s#(x) -> s#(s(s(0()))) -> k#(s(0())) h#(k(x),g(x)) -> s#(x) -> s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) h#(f(x),g(x)) -> f#(s(x)) -> f#(g(x)) -> f#(x) h#(f(x),g(x)) -> f#(s(x)) -> f#(g(x)) -> g#(f(x)) h#(f(x),g(x)) -> f#(s(x)) -> f#(g(x)) -> g#(g(f(x))) h#(f(x),g(x)) -> s#(x) -> s#(s(0())) -> f#(s(0())) h#(f(x),g(x)) -> s#(x) -> s#(x) -> h#(x,0()) h#(f(x),g(x)) -> s#(x) -> s#(x) -> h#(0(),x) h#(f(x),g(x)) -> s#(x) -> s#(s(0())) -> k#(0()) h#(f(x),g(x)) -> s#(x) -> s#(s(s(0()))) -> k#(s(0())) h#(f(x),g(x)) -> s#(x) -> s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) g#(s(x)) -> g#(x) -> g#(x) -> h#(x,x) g#(s(x)) -> g#(x) -> g#(s(x)) -> g#(x) g#(s(x)) -> g#(x) -> g#(s(x)) -> s#(g(x)) g#(s(x)) -> g#(x) -> g#(s(x)) -> s#(s(g(x))) g#(s(x)) -> s#(g(x)) -> s#(s(0())) -> f#(s(0())) g#(s(x)) -> s#(g(x)) -> s#(x) -> h#(x,0()) g#(s(x)) -> s#(g(x)) -> s#(x) -> h#(0(),x) g#(s(x)) -> s#(g(x)) -> s#(s(0())) -> k#(0()) g#(s(x)) -> s#(g(x)) -> s#(s(s(0()))) -> k#(s(0())) g#(s(x)) -> s#(g(x)) -> s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) g#(s(x)) -> s#(s(g(x))) -> s#(s(0())) -> f#(s(0())) g#(s(x)) -> s#(s(g(x))) -> s#(x) -> h#(x,0()) g#(s(x)) -> s#(s(g(x))) -> s#(x) -> h#(0(),x) g#(s(x)) -> s#(s(g(x))) -> s#(s(0())) -> k#(0()) g#(s(x)) -> s#(s(g(x))) -> s#(s(s(0()))) -> k#(s(0())) g#(s(x)) -> s#(s(g(x))) -> s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) g#(x) -> h#(x,x) -> h#(f(x),g(x)) -> s#(x) g#(x) -> h#(x,x) -> h#(f(x),g(x)) -> f#(s(x)) g#(x) -> h#(x,x) -> h#(k(x),g(x)) -> s#(x) g#(x) -> h#(x,x) -> h#(k(x),g(x)) -> k#(s(x)) f#(g(x)) -> g#(g(f(x))) -> g#(x) -> h#(x,x) f#(g(x)) -> g#(g(f(x))) -> g#(s(x)) -> g#(x) f#(g(x)) -> g#(g(f(x))) -> g#(s(x)) -> s#(g(x)) f#(g(x)) -> g#(g(f(x))) -> g#(s(x)) -> s#(s(g(x))) f#(g(x)) -> g#(f(x)) -> g#(x) -> h#(x,x) f#(g(x)) -> g#(f(x)) -> g#(s(x)) -> g#(x) f#(g(x)) -> g#(f(x)) -> g#(s(x)) -> s#(g(x)) f#(g(x)) -> g#(f(x)) -> g#(s(x)) -> s#(s(g(x))) f#(g(x)) -> f#(x) -> f#(g(x)) -> f#(x) f#(g(x)) -> f#(x) -> f#(g(x)) -> g#(f(x)) f#(g(x)) -> f#(x) -> f#(g(x)) -> g#(g(f(x))) s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) -> k#(s(s(0()))) -> s#(s(s(0()))) s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) -> k#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) -> k#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) -> k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) s#(s(s(0()))) -> k#(s(0())) -> k#(s(s(0()))) -> s#(s(s(0()))) s#(s(s(0()))) -> k#(s(0())) -> k#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(s(0()))) -> k#(s(0())) -> k#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(s(0()))) -> k#(s(0())) -> k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) s#(s(0())) -> f#(s(0())) -> f#(g(x)) -> f#(x) s#(s(0())) -> f#(s(0())) -> f#(g(x)) -> g#(f(x)) s#(s(0())) -> f#(s(0())) -> f#(g(x)) -> g#(g(f(x))) SCC Processor: #sccs: 1 #rules: 18 #arcs: 85/441 DPs: k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(s(0()))) -> k#(s(0())) k#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(0())) -> f#(s(0())) f#(g(x)) -> g#(g(f(x))) g#(s(x)) -> s#(s(g(x))) g#(s(x)) -> s#(g(x)) g#(s(x)) -> g#(x) g#(x) -> h#(x,x) h#(k(x),g(x)) -> k#(s(x)) k#(s(s(0()))) -> s#(s(s(0()))) h#(k(x),g(x)) -> s#(x) h#(f(x),g(x)) -> f#(s(x)) f#(g(x)) -> g#(f(x)) f#(g(x)) -> f#(x) h#(f(x),g(x)) -> s#(x) TRS: s(s(0())) -> f(s(0())) g(x) -> h(x,x) s(x) -> h(x,0()) s(x) -> h(0(),x) f(g(x)) -> g(g(f(x))) g(s(x)) -> s(s(g(x))) h(f(x),g(x)) -> f(s(x)) s(s(0())) -> k(0()) k(0()) -> 0() s(s(s(0()))) -> k(s(0())) k(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(0())))))) h(k(x),g(x)) -> k(s(x)) Arctic Interpretation Processor: dimension: 1 usable rules: s(s(0())) -> f(s(0())) g(x) -> h(x,x) s(x) -> h(x,0()) s(x) -> h(0(),x) f(g(x)) -> g(g(f(x))) g(s(x)) -> s(s(g(x))) h(f(x),g(x)) -> f(s(x)) s(s(0())) -> k(0()) k(0()) -> 0() s(s(s(0()))) -> k(s(0())) k(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(0())))))) h(k(x),g(x)) -> k(s(x)) interpretation: [h](x0, x1) = 0, [g#](x0) = 4, [s](x0) = 0, [g](x0) = x0 + 4, [k](x0) = x0 + 0, [k#](x0) = x0 + 0, [0] = 0, [f#](x0) = x0 + 0, [h#](x0, x1) = 4, [s#](x0) = 0, [f](x0) = x0 + 0 orientation: k#(s(s(0()))) = 0 >= 0 = s#(s(s(s(s(s(0())))))) s#(s(s(s(s(s(s(s(0())))))))) = 0 >= 0 = k#(s(s(0()))) k#(s(s(0()))) = 0 >= 0 = s#(s(s(s(s(0()))))) s#(s(s(0()))) = 0 >= 0 = k#(s(0())) k#(s(s(0()))) = 0 >= 0 = s#(s(s(s(0())))) s#(s(0())) = 0 >= 0 = f#(s(0())) f#(g(x)) = x + 4 >= 4 = g#(g(f(x))) g#(s(x)) = 4 >= 0 = s#(s(g(x))) g#(s(x)) = 4 >= 0 = s#(g(x)) g#(s(x)) = 4 >= 4 = g#(x) g#(x) = 4 >= 4 = h#(x,x) h#(k(x),g(x)) = 4 >= 0 = k#(s(x)) k#(s(s(0()))) = 0 >= 0 = s#(s(s(0()))) h#(k(x),g(x)) = 4 >= 0 = s#(x) h#(f(x),g(x)) = 4 >= 0 = f#(s(x)) f#(g(x)) = x + 4 >= 4 = g#(f(x)) f#(g(x)) = x + 4 >= x + 0 = f#(x) h#(f(x),g(x)) = 4 >= 0 = s#(x) s(s(0())) = 0 >= 0 = f(s(0())) g(x) = x + 4 >= 0 = h(x,x) s(x) = 0 >= 0 = h(x,0()) s(x) = 0 >= 0 = h(0(),x) f(g(x)) = x + 4 >= x + 4 = g(g(f(x))) g(s(x)) = 4 >= 0 = s(s(g(x))) h(f(x),g(x)) = 0 >= 0 = f(s(x)) s(s(0())) = 0 >= 0 = k(0()) k(0()) = 0 >= 0 = 0() s(s(s(0()))) = 0 >= 0 = k(s(0())) k(s(0())) = 0 >= 0 = s(0()) s(s(s(s(s(s(s(s(0())))))))) = 0 >= 0 = k(s(s(0()))) k(s(s(0()))) = 0 >= 0 = s(s(s(s(s(s(0())))))) h(k(x),g(x)) = 0 >= 0 = k(s(x)) problem: DPs: k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(s(0()))) -> k#(s(0())) k#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(0())) -> f#(s(0())) f#(g(x)) -> g#(g(f(x))) g#(s(x)) -> g#(x) g#(x) -> h#(x,x) k#(s(s(0()))) -> s#(s(s(0()))) f#(g(x)) -> g#(f(x)) f#(g(x)) -> f#(x) TRS: s(s(0())) -> f(s(0())) g(x) -> h(x,x) s(x) -> h(x,0()) s(x) -> h(0(),x) f(g(x)) -> g(g(f(x))) g(s(x)) -> s(s(g(x))) h(f(x),g(x)) -> f(s(x)) s(s(0())) -> k(0()) k(0()) -> 0() s(s(s(0()))) -> k(s(0())) k(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(0())))))) h(k(x),g(x)) -> k(s(x)) Restore Modifier: DPs: k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(s(0()))) -> k#(s(0())) k#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(0())) -> f#(s(0())) f#(g(x)) -> g#(g(f(x))) g#(s(x)) -> g#(x) g#(x) -> h#(x,x) k#(s(s(0()))) -> s#(s(s(0()))) f#(g(x)) -> g#(f(x)) f#(g(x)) -> f#(x) TRS: s(s(0())) -> f(s(0())) g(x) -> h(x,x) s(x) -> h(x,0()) s(x) -> h(0(),x) f(g(x)) -> g(g(f(x))) g(s(x)) -> s(s(g(x))) h(f(x),g(x)) -> f(s(x)) s(s(0())) -> k(0()) k(0()) -> 0() s(s(s(0()))) -> k(s(0())) k(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(0())))))) h(k(x),g(x)) -> k(s(x)) SCC Processor: #sccs: 3 #rules: 8 #arcs: 61/144 DPs: k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(s(0()))) -> k#(s(0())) k#(s(s(0()))) -> s#(s(s(s(0())))) k#(s(s(0()))) -> s#(s(s(0()))) TRS: s(s(0())) -> f(s(0())) g(x) -> h(x,x) s(x) -> h(x,0()) s(x) -> h(0(),x) f(g(x)) -> g(g(f(x))) g(s(x)) -> s(s(g(x))) h(f(x),g(x)) -> f(s(x)) s(s(0())) -> k(0()) k(0()) -> 0() s(s(s(0()))) -> k(s(0())) k(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(0())))))) h(k(x),g(x)) -> k(s(x)) Bounds Processor: bound: 0 enrichment: top-dp automaton: final states: {22} transitions: 00() -> 23* s0(23) -> 24* k{#,0}(24) -> 22* h0(23,23) -> 24* problem: DPs: k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) k#(s(s(0()))) -> s#(s(s(s(0())))) k#(s(s(0()))) -> s#(s(s(0()))) TRS: s(s(0())) -> f(s(0())) g(x) -> h(x,x) s(x) -> h(x,0()) s(x) -> h(0(),x) f(g(x)) -> g(g(f(x))) g(s(x)) -> s(s(g(x))) h(f(x),g(x)) -> f(s(x)) s(s(0())) -> k(0()) k(0()) -> 0() s(s(s(0()))) -> k(s(0())) k(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(0())))))) h(k(x),g(x)) -> k(s(x)) Bounds Processor: bound: 0 enrichment: top-dp automaton: final states: {28} transitions: s{#,0}(35) -> 28* s{#,0}(31) -> 28* 00() -> 35*,31,29 s0(30) -> 31* s0(35) -> 30* s0(29) -> 30* h0(35,35) -> 30* h0(29,29) -> 30* h0(29,30) -> 31* h0(30,29) -> 31* h0(29,35) -> 30* h0(35,30) -> 31* h0(35,29) -> 30* h0(30,35) -> 31* k0(35) -> 31* k0(29) -> 31* f0(30) -> 31* problem: DPs: k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) k#(s(s(0()))) -> s#(s(s(s(0())))) TRS: s(s(0())) -> f(s(0())) g(x) -> h(x,x) s(x) -> h(x,0()) s(x) -> h(0(),x) f(g(x)) -> g(g(f(x))) g(s(x)) -> s(s(g(x))) h(f(x),g(x)) -> f(s(x)) s(s(0())) -> k(0()) k(0()) -> 0() s(s(s(0()))) -> k(s(0())) k(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(0())))))) h(k(x),g(x)) -> k(s(x)) Bounds Processor: bound: 0 enrichment: top-dp automaton: final states: {22} transitions: s{#,0}(31) -> 22* s{#,0}(26) -> 22* 00() -> 30*,25,23 s0(23) -> 24* s0(24) -> 25* s0(30) -> 31*,24,26 s0(31) -> 25* s0(25) -> 26* h0(23,23) -> 24* h0(30,30) -> 31*,24,26 h0(25,30) -> 26* h0(31,23) -> 25* h0(31,30) -> 25* h0(23,31) -> 25* h0(25,23) -> 26* h0(23,25) -> 26* h0(24,23) -> 25* h0(24,30) -> 25* h0(23,24) -> 25* h0(30,23) -> 31*,24,26 h0(30,31) -> 25* h0(30,24) -> 25* h0(30,25) -> 26* h0(23,30) -> 31*,24,26 k0(31) -> 26* k0(30) -> 25* k0(24) -> 26* k0(23) -> 25* f0(31) -> 25* f0(24) -> 25* problem: DPs: k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) TRS: s(s(0())) -> f(s(0())) g(x) -> h(x,x) s(x) -> h(x,0()) s(x) -> h(0(),x) f(g(x)) -> g(g(f(x))) g(s(x)) -> s(s(g(x))) h(f(x),g(x)) -> f(s(x)) s(s(0())) -> k(0()) k(0()) -> 0() s(s(s(0()))) -> k(s(0())) k(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(0())))))) h(k(x),g(x)) -> k(s(x)) Bounds Processor: bound: 0 enrichment: top-dp automaton: final states: {15} transitions: s{#,0}(20) -> 15* s{#,0}(35) -> 15* s{#,0}(26) -> 15* 00() -> 24,26,35*,18,16 s0(24) -> 25*,19,17 s0(18) -> 19* s0(35) -> 25*,17,19 s0(16) -> 17* s0(26) -> 19* s0(17) -> 18* s0(25) -> 26*,18,20 s0(19) -> 20* h0(26,24) -> 19* h0(16,19) -> 20* h0(17,35) -> 18* h0(19,35) -> 20* h0(16,18) -> 19* h0(26,35) -> 19* h0(25,16) -> 26*,18,20 h0(16,17) -> 18* h0(24,24) -> 25*,19,17 h0(24,26) -> 19* h0(17,24) -> 18* h0(35,17) -> 18* h0(35,35) -> 25*,17,19 h0(35,25) -> 20,18,26* h0(16,24) -> 25*,19,17 h0(18,24) -> 19* h0(35,18) -> 19* h0(18,16) -> 19* h0(24,25) -> 26*,20,18 h0(35,16) -> 25*,17,19 h0(25,24) -> 26*,18,20 h0(17,16) -> 18* h0(24,18) -> 19* h0(35,24) -> 25*,17,19 h0(16,35) -> 25*,17,19 h0(24,19) -> 20* h0(24,17) -> 18* h0(24,16) -> 25*,19,17 h0(16,16) -> 17* h0(26,16) -> 19* h0(35,19) -> 20* h0(25,35) -> 26*,18,20 h0(16,26) -> 19* h0(16,25) -> 26*,20,18 h0(19,16) -> 20* h0(19,24) -> 20* h0(18,35) -> 19* h0(35,26) -> 19* h0(24,35) -> 25*,17,19 k0(25) -> 19* k0(17) -> 19* k0(35) -> 26* k0(24) -> 26*,18 k0(16) -> 26*,18 f0(25) -> 26*,18 f0(17) -> 26*,18 problem: DPs: k#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) TRS: s(s(0())) -> f(s(0())) g(x) -> h(x,x) s(x) -> h(x,0()) s(x) -> h(0(),x) f(g(x)) -> g(g(f(x))) g(s(x)) -> s(s(g(x))) h(f(x),g(x)) -> f(s(x)) s(s(0())) -> k(0()) k(0()) -> 0() s(s(s(0()))) -> k(s(0())) k(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(0())))))) h(k(x),g(x)) -> k(s(x)) Bounds Processor: bound: 0 enrichment: top-dp automaton: final states: {2} transitions: s{#,0}(8) -> 2* s{#,0}(24) -> 2* s{#,0}(26) -> 2* 00() -> 21,23,25*,5,3 s0(23) -> 24*,8,6 s0(24) -> 7* s0(3) -> 4* s0(22) -> 23*,7,5 s0(6) -> 7* s0(7) -> 8* s0(4) -> 5* s0(26) -> 23*,7,5 s0(5) -> 6* s0(21) -> 22*,4,6 s0(25) -> 26*,4,8,6 h0(7,25) -> 8* h0(26,3) -> 23*,7,5 h0(23,21) -> 24*,8,6 h0(21,7) -> 8* h0(6,25) -> 7* h0(3,3) -> 4* h0(3,22) -> 23*,5,7 h0(3,5) -> 6* h0(3,7) -> 8* h0(22,3) -> 23*,7,5 h0(6,3) -> 7* h0(3,4) -> 5* h0(3,21) -> 22*,4,6 h0(7,3) -> 8* h0(21,23) -> 24*,8,6 h0(21,22) -> 23*,7,5 h0(21,24) -> 7* h0(24,25) -> 7* h0(22,25) -> 23*,7,5 h0(5,25) -> 6* h0(3,26) -> 23*,5,7 h0(26,21) -> 23*,5,7 h0(25,24) -> 7* h0(21,3) -> 22*,4,6 h0(7,21) -> 8* h0(21,26) -> 23*,7,5 h0(3,23) -> 24*,6,8 h0(3,24) -> 7* h0(3,25) -> 26*,4,6,8 h0(25,23) -> 8,24*,6 h0(24,3) -> 7* h0(5,21) -> 6* h0(23,25) -> 8,6,24* h0(4,25) -> 5* h0(5,3) -> 6* h0(22,21) -> 23*,7,5 h0(4,3) -> 5* h0(25,25) -> 26*,4,8,6 h0(25,5) -> 6* h0(25,21) -> 26*,4,8,6 h0(25,3) -> 26*,4,8,6 h0(25,26) -> 7,23*,5 h0(21,5) -> 6* h0(21,25) -> 26*,4,8,6 h0(25,6) -> 7* h0(21,6) -> 7* h0(3,6) -> 7* h0(6,21) -> 7* h0(4,21) -> 5* h0(26,25) -> 7,23*,5 h0(25,22) -> 23*,7,5 h0(21,4) -> 5* h0(25,7) -> 8* h0(23,3) -> 24*,8,6 h0(25,4) -> 5* h0(24,21) -> 7* h0(21,21) -> 22*,4,6 k0(21) -> 23*,5 k0(25) -> 23* k0(26) -> 24* k0(22) -> 24*,6 k0(3) -> 23*,5 k0(4) -> 24*,6 f0(26) -> 23* f0(4) -> 23*,5 f0(22) -> 23*,5 problem: DPs: s#(s(s(s(s(s(s(s(0())))))))) -> k#(s(s(0()))) TRS: s(s(0())) -> f(s(0())) g(x) -> h(x,x) s(x) -> h(x,0()) s(x) -> h(0(),x) f(g(x)) -> g(g(f(x))) g(s(x)) -> s(s(g(x))) h(f(x),g(x)) -> f(s(x)) s(s(0())) -> k(0()) k(0()) -> 0() s(s(s(0()))) -> k(s(0())) k(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(0())))))) h(k(x),g(x)) -> k(s(x)) SCC Processor: #sccs: 0 #rules: 0 #arcs: 16/1 DPs: f#(g(x)) -> f#(x) TRS: s(s(0())) -> f(s(0())) g(x) -> h(x,x) s(x) -> h(x,0()) s(x) -> h(0(),x) f(g(x)) -> g(g(f(x))) g(s(x)) -> s(s(g(x))) h(f(x),g(x)) -> f(s(x)) s(s(0())) -> k(0()) k(0()) -> 0() s(s(s(0()))) -> k(s(0())) k(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(0())))))) h(k(x),g(x)) -> k(s(x)) Size-Change Termination Processor: DPs: TRS: s(s(0())) -> f(s(0())) g(x) -> h(x,x) s(x) -> h(x,0()) s(x) -> h(0(),x) f(g(x)) -> g(g(f(x))) g(s(x)) -> s(s(g(x))) h(f(x),g(x)) -> f(s(x)) s(s(0())) -> k(0()) k(0()) -> 0() s(s(s(0()))) -> k(s(0())) k(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(0())))))) h(k(x),g(x)) -> k(s(x)) The DP: f#(g(x)) -> f#(x) has the edges: 0 > 0 Qed DPs: g#(s(x)) -> g#(x) TRS: s(s(0())) -> f(s(0())) g(x) -> h(x,x) s(x) -> h(x,0()) s(x) -> h(0(),x) f(g(x)) -> g(g(f(x))) g(s(x)) -> s(s(g(x))) h(f(x),g(x)) -> f(s(x)) s(s(0())) -> k(0()) k(0()) -> 0() s(s(s(0()))) -> k(s(0())) k(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(0())))))) h(k(x),g(x)) -> k(s(x)) Size-Change Termination Processor: DPs: TRS: s(s(0())) -> f(s(0())) g(x) -> h(x,x) s(x) -> h(x,0()) s(x) -> h(0(),x) f(g(x)) -> g(g(f(x))) g(s(x)) -> s(s(g(x))) h(f(x),g(x)) -> f(s(x)) s(s(0())) -> k(0()) k(0()) -> 0() s(s(s(0()))) -> k(s(0())) k(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(0())))))) h(k(x),g(x)) -> k(s(x)) The DP: g#(s(x)) -> g#(x) has the edges: 0 > 0 Qed