MAYBE Problem: 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(0()) -> f(0()) s(s(0())) -> f(s(0())) f(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> f(s(s(0()))) f(s(s(0()))) -> s(s(s(s(s(0()))))) f(s(s(x))) -> h(f(x),g(h(x,x))) Proof: DP Processor: DPs: 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#(0()) -> f#(0()) s#(s(0())) -> f#(s(0())) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) f#(s(s(0()))) -> s#(s(s(0()))) f#(s(s(0()))) -> s#(s(s(s(0())))) f#(s(s(0()))) -> s#(s(s(s(s(0()))))) f#(s(s(x))) -> h#(x,x) f#(s(s(x))) -> g#(h(x,x)) f#(s(s(x))) -> f#(x) f#(s(s(x))) -> h#(f(x),g(h(x,x))) TRS: 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(0()) -> f(0()) s(s(0())) -> f(s(0())) f(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> f(s(s(0()))) f(s(s(0()))) -> s(s(s(s(s(0()))))) f(s(s(x))) -> h(f(x),g(h(x,x))) TDG Processor: DPs: 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#(0()) -> f#(0()) s#(s(0())) -> f#(s(0())) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) f#(s(s(0()))) -> s#(s(s(0()))) f#(s(s(0()))) -> s#(s(s(s(0())))) f#(s(s(0()))) -> s#(s(s(s(s(0()))))) f#(s(s(x))) -> h#(x,x) f#(s(s(x))) -> g#(h(x,x)) f#(s(s(x))) -> f#(x) f#(s(s(x))) -> h#(f(x),g(h(x,x))) TRS: 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(0()) -> f(0()) s(s(0())) -> f(s(0())) f(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> f(s(s(0()))) f(s(s(0()))) -> s(s(s(s(s(0()))))) f(s(s(x))) -> h(f(x),g(h(x,x))) graph: f#(s(s(0()))) -> s#(s(s(0()))) -> s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) f#(s(s(0()))) -> s#(s(s(0()))) -> s#(s(0())) -> f#(s(0())) f#(s(s(0()))) -> s#(s(s(0()))) -> s#(0()) -> f#(0()) f#(s(s(0()))) -> s#(s(s(0()))) -> s#(x) -> h#(0(),x) f#(s(s(0()))) -> s#(s(s(0()))) -> s#(x) -> h#(x,0()) f#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) f#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(s(0())) -> f#(s(0())) f#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(0()) -> f#(0()) f#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(x) -> h#(0(),x) f#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(x) -> h#(x,0()) f#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) f#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(s(0())) -> f#(s(0())) f#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(0()) -> f#(0()) f#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(x) -> h#(0(),x) f#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(x) -> h#(x,0()) f#(s(s(x))) -> f#(x) -> f#(s(s(x))) -> h#(f(x),g(h(x,x))) f#(s(s(x))) -> f#(x) -> f#(s(s(x))) -> f#(x) f#(s(s(x))) -> f#(x) -> f#(s(s(x))) -> g#(h(x,x)) f#(s(s(x))) -> f#(x) -> f#(s(s(x))) -> h#(x,x) f#(s(s(x))) -> f#(x) -> f#(s(s(0()))) -> s#(s(s(s(s(0()))))) f#(s(s(x))) -> f#(x) -> f#(s(s(0()))) -> s#(s(s(s(0())))) f#(s(s(x))) -> f#(x) -> f#(s(s(0()))) -> s#(s(s(0()))) f#(s(s(x))) -> f#(x) -> f#(g(x)) -> g#(g(f(x))) f#(s(s(x))) -> f#(x) -> f#(g(x)) -> g#(f(x)) f#(s(s(x))) -> f#(x) -> f#(g(x)) -> f#(x) f#(s(s(x))) -> h#(f(x),g(h(x,x))) -> h#(f(x),g(x)) -> f#(s(x)) f#(s(s(x))) -> h#(f(x),g(h(x,x))) -> h#(f(x),g(x)) -> s#(x) f#(s(s(x))) -> h#(x,x) -> h#(f(x),g(x)) -> f#(s(x)) f#(s(s(x))) -> h#(x,x) -> h#(f(x),g(x)) -> s#(x) f#(s(s(x))) -> g#(h(x,x)) -> g#(s(x)) -> s#(s(g(x))) f#(s(s(x))) -> g#(h(x,x)) -> g#(s(x)) -> s#(g(x)) f#(s(s(x))) -> g#(h(x,x)) -> g#(s(x)) -> g#(x) f#(s(s(x))) -> g#(h(x,x)) -> g#(x) -> h#(x,x) f#(g(x)) -> f#(x) -> f#(s(s(x))) -> h#(f(x),g(h(x,x))) f#(g(x)) -> f#(x) -> f#(s(s(x))) -> f#(x) f#(g(x)) -> f#(x) -> f#(s(s(x))) -> g#(h(x,x)) f#(g(x)) -> f#(x) -> f#(s(s(x))) -> h#(x,x) f#(g(x)) -> f#(x) -> f#(s(s(0()))) -> s#(s(s(s(s(0()))))) f#(g(x)) -> f#(x) -> f#(s(s(0()))) -> s#(s(s(s(0())))) f#(g(x)) -> f#(x) -> f#(s(s(0()))) -> s#(s(s(0()))) 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) 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)) -> 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) s#(0()) -> f#(0()) -> f#(s(s(x))) -> h#(f(x),g(h(x,x))) s#(0()) -> f#(0()) -> f#(s(s(x))) -> f#(x) s#(0()) -> f#(0()) -> f#(s(s(x))) -> g#(h(x,x)) s#(0()) -> f#(0()) -> f#(s(s(x))) -> h#(x,x) s#(0()) -> f#(0()) -> f#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(0()) -> f#(0()) -> f#(s(s(0()))) -> s#(s(s(s(0())))) s#(0()) -> f#(0()) -> f#(s(s(0()))) -> s#(s(s(0()))) s#(0()) -> f#(0()) -> f#(g(x)) -> g#(g(f(x))) s#(0()) -> f#(0()) -> f#(g(x)) -> g#(f(x)) s#(0()) -> f#(0()) -> f#(g(x)) -> f#(x) s#(s(0())) -> f#(s(0())) -> f#(s(s(x))) -> h#(f(x),g(h(x,x))) s#(s(0())) -> f#(s(0())) -> f#(s(s(x))) -> f#(x) s#(s(0())) -> f#(s(0())) -> f#(s(s(x))) -> g#(h(x,x)) s#(s(0())) -> f#(s(0())) -> f#(s(s(x))) -> h#(x,x) s#(s(0())) -> f#(s(0())) -> f#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(0())) -> f#(s(0())) -> f#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(0())) -> f#(s(0())) -> f#(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#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(s(s(x))) -> h#(f(x),g(h(x,x))) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(s(s(x))) -> f#(x) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(s(s(x))) -> g#(h(x,x)) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(s(s(x))) -> h#(x,x) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(s(s(0()))) -> s#(s(s(0()))) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(g(x)) -> g#(g(f(x))) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(g(x)) -> g#(f(x)) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(g(x)) -> f#(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#(f(x),g(x)) -> f#(s(x)) s#(x) -> h#(x,0()) -> h#(f(x),g(x)) -> s#(x) h#(f(x),g(x)) -> f#(s(x)) -> f#(s(s(x))) -> h#(f(x),g(h(x,x))) h#(f(x),g(x)) -> f#(s(x)) -> f#(s(s(x))) -> f#(x) h#(f(x),g(x)) -> f#(s(x)) -> f#(s(s(x))) -> g#(h(x,x)) h#(f(x),g(x)) -> f#(s(x)) -> f#(s(s(x))) -> h#(x,x) h#(f(x),g(x)) -> f#(s(x)) -> f#(s(s(0()))) -> s#(s(s(s(s(0()))))) h#(f(x),g(x)) -> f#(s(x)) -> f#(s(s(0()))) -> s#(s(s(s(0())))) h#(f(x),g(x)) -> f#(s(x)) -> f#(s(s(0()))) -> s#(s(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(0()))))))) -> f#(s(s(0()))) h#(f(x),g(x)) -> s#(x) -> s#(s(0())) -> f#(s(0())) h#(f(x),g(x)) -> s#(x) -> s#(0()) -> f#(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()) g#(s(x)) -> s#(s(g(x))) -> s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) g#(s(x)) -> s#(s(g(x))) -> s#(s(0())) -> f#(s(0())) g#(s(x)) -> s#(s(g(x))) -> s#(0()) -> f#(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#(g(x)) -> s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) g#(s(x)) -> s#(g(x)) -> s#(s(0())) -> f#(s(0())) g#(s(x)) -> s#(g(x)) -> s#(0()) -> f#(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)) -> 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#(x) -> h#(x,x) -> h#(f(x),g(x)) -> f#(s(x)) g#(x) -> h#(x,x) -> h#(f(x),g(x)) -> s#(x) EDG Processor: DPs: 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#(0()) -> f#(0()) s#(s(0())) -> f#(s(0())) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) f#(s(s(0()))) -> s#(s(s(0()))) f#(s(s(0()))) -> s#(s(s(s(0())))) f#(s(s(0()))) -> s#(s(s(s(s(0()))))) f#(s(s(x))) -> h#(x,x) f#(s(s(x))) -> g#(h(x,x)) f#(s(s(x))) -> f#(x) f#(s(s(x))) -> h#(f(x),g(h(x,x))) TRS: 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(0()) -> f(0()) s(s(0())) -> f(s(0())) f(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> f(s(s(0()))) f(s(s(0()))) -> s(s(s(s(s(0()))))) f(s(s(x))) -> h(f(x),g(h(x,x))) graph: f#(s(s(0()))) -> s#(s(s(0()))) -> s#(x) -> h#(x,0()) f#(s(s(0()))) -> s#(s(s(0()))) -> s#(x) -> h#(0(),x) f#(s(s(0()))) -> s#(s(s(0()))) -> s#(0()) -> f#(0()) f#(s(s(0()))) -> s#(s(s(0()))) -> s#(s(0())) -> f#(s(0())) f#(s(s(0()))) -> s#(s(s(0()))) -> s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) f#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(x) -> h#(x,0()) f#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(x) -> h#(0(),x) f#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(0()) -> f#(0()) f#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(s(0())) -> f#(s(0())) f#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) f#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(x) -> h#(x,0()) f#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(x) -> h#(0(),x) f#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(0()) -> f#(0()) f#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(s(0())) -> f#(s(0())) f#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) f#(s(s(x))) -> f#(x) -> f#(g(x)) -> f#(x) f#(s(s(x))) -> f#(x) -> f#(g(x)) -> g#(f(x)) f#(s(s(x))) -> f#(x) -> f#(g(x)) -> g#(g(f(x))) f#(s(s(x))) -> f#(x) -> f#(s(s(0()))) -> s#(s(s(0()))) f#(s(s(x))) -> f#(x) -> f#(s(s(0()))) -> s#(s(s(s(0())))) f#(s(s(x))) -> f#(x) -> f#(s(s(0()))) -> s#(s(s(s(s(0()))))) f#(s(s(x))) -> f#(x) -> f#(s(s(x))) -> h#(x,x) f#(s(s(x))) -> f#(x) -> f#(s(s(x))) -> g#(h(x,x)) f#(s(s(x))) -> f#(x) -> f#(s(s(x))) -> f#(x) f#(s(s(x))) -> f#(x) -> f#(s(s(x))) -> h#(f(x),g(h(x,x))) f#(s(s(x))) -> h#(f(x),g(h(x,x))) -> h#(f(x),g(x)) -> s#(x) f#(s(s(x))) -> h#(f(x),g(h(x,x))) -> h#(f(x),g(x)) -> f#(s(x)) f#(s(s(x))) -> h#(x,x) -> h#(f(x),g(x)) -> s#(x) f#(s(s(x))) -> h#(x,x) -> h#(f(x),g(x)) -> f#(s(x)) f#(s(s(x))) -> g#(h(x,x)) -> g#(x) -> h#(x,x) f#(s(s(x))) -> g#(h(x,x)) -> g#(s(x)) -> g#(x) f#(s(s(x))) -> g#(h(x,x)) -> g#(s(x)) -> s#(g(x)) f#(s(s(x))) -> g#(h(x,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))) f#(g(x)) -> f#(x) -> f#(s(s(0()))) -> s#(s(s(0()))) f#(g(x)) -> f#(x) -> f#(s(s(0()))) -> s#(s(s(s(0())))) f#(g(x)) -> f#(x) -> f#(s(s(0()))) -> s#(s(s(s(s(0()))))) f#(g(x)) -> f#(x) -> f#(s(s(x))) -> h#(x,x) f#(g(x)) -> f#(x) -> f#(s(s(x))) -> g#(h(x,x)) f#(g(x)) -> f#(x) -> f#(s(s(x))) -> f#(x) f#(g(x)) -> f#(x) -> f#(s(s(x))) -> h#(f(x),g(h(x,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)) -> 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))) 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))) s#(s(0())) -> f#(s(0())) -> f#(s(s(0()))) -> s#(s(s(0()))) s#(s(0())) -> f#(s(0())) -> f#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(0())) -> f#(s(0())) -> f#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(0())) -> f#(s(0())) -> f#(s(s(x))) -> h#(x,x) s#(s(0())) -> f#(s(0())) -> f#(s(s(x))) -> g#(h(x,x)) s#(s(0())) -> f#(s(0())) -> f#(s(s(x))) -> f#(x) s#(s(0())) -> f#(s(0())) -> f#(s(s(x))) -> h#(f(x),g(h(x,x))) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(g(x)) -> f#(x) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(g(x)) -> g#(f(x)) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(g(x)) -> g#(g(f(x))) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(s(s(0()))) -> s#(s(s(0()))) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(s(s(x))) -> h#(x,x) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(s(s(x))) -> g#(h(x,x)) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(s(s(x))) -> f#(x) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(s(s(x))) -> h#(f(x),g(h(x,x))) 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)) -> f#(s(x)) -> f#(s(s(0()))) -> s#(s(s(0()))) h#(f(x),g(x)) -> f#(s(x)) -> f#(s(s(0()))) -> s#(s(s(s(0())))) h#(f(x),g(x)) -> f#(s(x)) -> f#(s(s(0()))) -> s#(s(s(s(s(0()))))) h#(f(x),g(x)) -> f#(s(x)) -> f#(s(s(x))) -> h#(x,x) h#(f(x),g(x)) -> f#(s(x)) -> f#(s(s(x))) -> g#(h(x,x)) h#(f(x),g(x)) -> f#(s(x)) -> f#(s(s(x))) -> f#(x) h#(f(x),g(x)) -> f#(s(x)) -> f#(s(s(x))) -> h#(f(x),g(h(x,x))) 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#(0()) -> f#(0()) h#(f(x),g(x)) -> s#(x) -> s#(s(0())) -> f#(s(0())) h#(f(x),g(x)) -> s#(x) -> s#(s(s(s(s(s(s(0()))))))) -> f#(s(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#(0()) -> f#(0()) g#(s(x)) -> s#(s(g(x))) -> s#(s(0())) -> f#(s(0())) g#(s(x)) -> s#(s(g(x))) -> s#(s(s(s(s(s(s(0()))))))) -> f#(s(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#(0()) -> f#(0()) g#(s(x)) -> s#(g(x)) -> s#(s(0())) -> f#(s(0())) g#(s(x)) -> s#(g(x)) -> s#(s(s(s(s(s(s(0()))))))) -> f#(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#(x) -> h#(x,x) -> h#(f(x),g(x)) -> s#(x) g#(x) -> h#(x,x) -> h#(f(x),g(x)) -> f#(s(x)) CDG Processor: DPs: 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#(0()) -> f#(0()) s#(s(0())) -> f#(s(0())) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) f#(s(s(0()))) -> s#(s(s(0()))) f#(s(s(0()))) -> s#(s(s(s(0())))) f#(s(s(0()))) -> s#(s(s(s(s(0()))))) f#(s(s(x))) -> h#(x,x) f#(s(s(x))) -> g#(h(x,x)) f#(s(s(x))) -> f#(x) f#(s(s(x))) -> h#(f(x),g(h(x,x))) TRS: 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(0()) -> f(0()) s(s(0())) -> f(s(0())) f(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> f(s(s(0()))) f(s(s(0()))) -> s(s(s(s(s(0()))))) f(s(s(x))) -> h(f(x),g(h(x,x))) graph: f#(s(s(0()))) -> s#(s(s(0()))) -> s#(0()) -> f#(0()) f#(s(s(0()))) -> s#(s(s(0()))) -> s#(x) -> h#(0(),x) f#(s(s(0()))) -> s#(s(s(0()))) -> s#(x) -> h#(x,0()) f#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(s(0())) -> f#(s(0())) f#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(x) -> h#(0(),x) f#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(x) -> h#(x,0()) f#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(0()) -> f#(0()) f#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(x) -> h#(0(),x) f#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(x) -> h#(x,0()) f#(s(s(x))) -> f#(x) -> f#(s(s(x))) -> h#(f(x),g(h(x,x))) f#(s(s(x))) -> f#(x) -> f#(s(s(x))) -> f#(x) f#(s(s(x))) -> f#(x) -> f#(s(s(x))) -> g#(h(x,x)) f#(s(s(x))) -> f#(x) -> f#(s(s(x))) -> h#(x,x) f#(s(s(x))) -> f#(x) -> f#(s(s(0()))) -> s#(s(s(s(s(0()))))) f#(s(s(x))) -> f#(x) -> f#(s(s(0()))) -> s#(s(s(s(0())))) f#(s(s(x))) -> f#(x) -> f#(s(s(0()))) -> s#(s(s(0()))) f#(s(s(x))) -> f#(x) -> f#(g(x)) -> g#(g(f(x))) f#(s(s(x))) -> f#(x) -> f#(g(x)) -> g#(f(x)) f#(s(s(x))) -> f#(x) -> f#(g(x)) -> f#(x) f#(s(s(x))) -> h#(f(x),g(h(x,x))) -> h#(f(x),g(x)) -> f#(s(x)) f#(s(s(x))) -> h#(f(x),g(h(x,x))) -> h#(f(x),g(x)) -> s#(x) f#(s(s(x))) -> h#(x,x) -> h#(f(x),g(x)) -> f#(s(x)) f#(s(s(x))) -> h#(x,x) -> h#(f(x),g(x)) -> s#(x) f#(s(s(x))) -> g#(h(x,x)) -> g#(s(x)) -> s#(s(g(x))) f#(s(s(x))) -> g#(h(x,x)) -> g#(s(x)) -> s#(g(x)) f#(s(s(x))) -> g#(h(x,x)) -> g#(s(x)) -> g#(x) f#(s(s(x))) -> g#(h(x,x)) -> g#(x) -> h#(x,x) f#(g(x)) -> f#(x) -> f#(s(s(x))) -> h#(f(x),g(h(x,x))) f#(g(x)) -> f#(x) -> f#(s(s(x))) -> f#(x) f#(g(x)) -> f#(x) -> f#(s(s(x))) -> g#(h(x,x)) f#(g(x)) -> f#(x) -> f#(s(s(x))) -> h#(x,x) f#(g(x)) -> f#(x) -> f#(s(s(0()))) -> s#(s(s(s(s(0()))))) f#(g(x)) -> f#(x) -> f#(s(s(0()))) -> s#(s(s(s(0())))) f#(g(x)) -> f#(x) -> f#(s(s(0()))) -> s#(s(s(0()))) 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) 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)) -> 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) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(s(s(x))) -> h#(f(x),g(h(x,x))) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(s(s(x))) -> f#(x) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(s(s(x))) -> g#(h(x,x)) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(s(s(x))) -> h#(x,x) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) -> f#(s(s(0()))) -> s#(s(s(0()))) h#(f(x),g(x)) -> f#(s(x)) -> f#(s(s(x))) -> h#(f(x),g(h(x,x))) h#(f(x),g(x)) -> f#(s(x)) -> f#(s(s(x))) -> f#(x) h#(f(x),g(x)) -> f#(s(x)) -> f#(s(s(x))) -> g#(h(x,x)) h#(f(x),g(x)) -> f#(s(x)) -> f#(s(s(x))) -> h#(x,x) h#(f(x),g(x)) -> f#(s(x)) -> f#(s(s(0()))) -> s#(s(s(s(s(0()))))) h#(f(x),g(x)) -> f#(s(x)) -> f#(s(s(0()))) -> s#(s(s(s(0())))) h#(f(x),g(x)) -> f#(s(x)) -> f#(s(s(0()))) -> s#(s(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(0()))))))) -> f#(s(s(0()))) h#(f(x),g(x)) -> s#(x) -> s#(s(0())) -> f#(s(0())) h#(f(x),g(x)) -> s#(x) -> s#(0()) -> f#(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()) g#(s(x)) -> s#(s(g(x))) -> s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) g#(s(x)) -> s#(s(g(x))) -> s#(s(0())) -> f#(s(0())) g#(s(x)) -> s#(s(g(x))) -> s#(0()) -> f#(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#(g(x)) -> s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) g#(s(x)) -> s#(g(x)) -> s#(s(0())) -> f#(s(0())) g#(s(x)) -> s#(g(x)) -> s#(0()) -> f#(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)) -> 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#(x) -> h#(x,x) -> h#(f(x),g(x)) -> f#(s(x)) g#(x) -> h#(x,x) -> h#(f(x),g(x)) -> s#(x) SCC Processor: #sccs: 1 #rules: 14 #arcs: 83/441 DPs: f#(s(s(x))) -> f#(x) f#(g(x)) -> f#(x) f#(g(x)) -> g#(f(x)) g#(x) -> h#(x,x) h#(f(x),g(x)) -> s#(x) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) f#(s(s(x))) -> h#(x,x) h#(f(x),g(x)) -> f#(s(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))) f#(s(s(x))) -> g#(h(x,x)) f#(s(s(x))) -> h#(f(x),g(h(x,x))) TRS: 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(0()) -> f(0()) s(s(0())) -> f(s(0())) f(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> f(s(s(0()))) f(s(s(0()))) -> s(s(s(s(s(0()))))) f(s(s(x))) -> h(f(x),g(h(x,x))) Matrix Interpretation Processor: dim=1 interpretation: [f#](x0) = x0, [s#](x0) = 0, [h#](x0, x1) = 1/2x0, [g#](x0) = 1/2x0, [f](x0) = 2x0, [0] = 0, [s](x0) = x0, [h](x0, x1) = x0, [g](x0) = x0 + 2 orientation: f#(s(s(x))) = x >= x = f#(x) f#(g(x)) = x + 2 >= x = f#(x) f#(g(x)) = x + 2 >= x = g#(f(x)) g#(x) = 1/2x >= 1/2x = h#(x,x) h#(f(x),g(x)) = x >= 0 = s#(x) s#(s(s(s(s(s(s(0()))))))) = 0 >= 0 = f#(s(s(0()))) f#(s(s(x))) = x >= 1/2x = h#(x,x) h#(f(x),g(x)) = x >= x = f#(s(x)) f#(g(x)) = x + 2 >= x + 1 = g#(g(f(x))) g#(s(x)) = 1/2x >= 1/2x = g#(x) g#(s(x)) = 1/2x >= 0 = s#(g(x)) g#(s(x)) = 1/2x >= 0 = s#(s(g(x))) f#(s(s(x))) = x >= 1/2x = g#(h(x,x)) f#(s(s(x))) = x >= x = h#(f(x),g(h(x,x))) g(x) = x + 2 >= x = h(x,x) s(x) = x >= x = h(x,0()) s(x) = x >= 0 = h(0(),x) f(g(x)) = 2x + 4 >= 2x + 4 = g(g(f(x))) g(s(x)) = x + 2 >= x + 2 = s(s(g(x))) h(f(x),g(x)) = 2x >= 2x = f(s(x)) s(0()) = 0 >= 0 = f(0()) s(s(0())) = 0 >= 0 = f(s(0())) f(s(0())) = 0 >= 0 = 0() s(s(s(s(s(s(s(0()))))))) = 0 >= 0 = f(s(s(0()))) f(s(s(0()))) = 0 >= 0 = s(s(s(s(s(0()))))) f(s(s(x))) = 2x >= 2x = h(f(x),g(h(x,x))) problem: DPs: f#(s(s(x))) -> f#(x) g#(x) -> h#(x,x) h#(f(x),g(x)) -> s#(x) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) f#(s(s(x))) -> h#(x,x) h#(f(x),g(x)) -> f#(s(x)) g#(s(x)) -> g#(x) g#(s(x)) -> s#(g(x)) g#(s(x)) -> s#(s(g(x))) f#(s(s(x))) -> g#(h(x,x)) f#(s(s(x))) -> h#(f(x),g(h(x,x))) TRS: 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(0()) -> f(0()) s(s(0())) -> f(s(0())) f(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> f(s(s(0()))) f(s(s(0()))) -> s(s(s(s(s(0()))))) f(s(s(x))) -> h(f(x),g(h(x,x))) Matrix Interpretation Processor: dim=1 interpretation: [f#](x0) = x0 + 1, [s#](x0) = 1, [h#](x0, x1) = x1, [g#](x0) = x0 + 1, [f](x0) = 2x0, [0] = 0, [s](x0) = x0, [h](x0, x1) = x0, [g](x0) = x0 + 1 orientation: f#(s(s(x))) = x + 1 >= x + 1 = f#(x) g#(x) = x + 1 >= x = h#(x,x) h#(f(x),g(x)) = x + 1 >= 1 = s#(x) s#(s(s(s(s(s(s(0()))))))) = 1 >= 1 = f#(s(s(0()))) f#(s(s(x))) = x + 1 >= x = h#(x,x) h#(f(x),g(x)) = x + 1 >= x + 1 = f#(s(x)) g#(s(x)) = x + 1 >= x + 1 = g#(x) g#(s(x)) = x + 1 >= 1 = s#(g(x)) g#(s(x)) = x + 1 >= 1 = s#(s(g(x))) f#(s(s(x))) = x + 1 >= x + 1 = g#(h(x,x)) f#(s(s(x))) = x + 1 >= x + 1 = h#(f(x),g(h(x,x))) g(x) = x + 1 >= x = h(x,x) s(x) = x >= x = h(x,0()) s(x) = x >= 0 = h(0(),x) f(g(x)) = 2x + 2 >= 2x + 2 = g(g(f(x))) g(s(x)) = x + 1 >= x + 1 = s(s(g(x))) h(f(x),g(x)) = 2x >= 2x = f(s(x)) s(0()) = 0 >= 0 = f(0()) s(s(0())) = 0 >= 0 = f(s(0())) f(s(0())) = 0 >= 0 = 0() s(s(s(s(s(s(s(0()))))))) = 0 >= 0 = f(s(s(0()))) f(s(s(0()))) = 0 >= 0 = s(s(s(s(s(0()))))) f(s(s(x))) = 2x >= 2x = h(f(x),g(h(x,x))) problem: DPs: f#(s(s(x))) -> f#(x) h#(f(x),g(x)) -> s#(x) s#(s(s(s(s(s(s(0()))))))) -> f#(s(s(0()))) h#(f(x),g(x)) -> f#(s(x)) g#(s(x)) -> g#(x) g#(s(x)) -> s#(g(x)) g#(s(x)) -> s#(s(g(x))) f#(s(s(x))) -> g#(h(x,x)) f#(s(s(x))) -> h#(f(x),g(h(x,x))) TRS: 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(0()) -> f(0()) s(s(0())) -> f(s(0())) f(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> f(s(s(0()))) f(s(s(0()))) -> s(s(s(s(s(0()))))) f(s(s(x))) -> h(f(x),g(h(x,x))) Open