YES Problem: s(s(0())) -> f(0()) f(0()) -> 0() s(s(s(0()))) -> f(s(0())) f(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> f(s(s(0()))) f(s(s(0()))) -> s(s(s(s(s(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)) Proof: DP Processor: DPs: s#(s(0())) -> f#(0()) s#(s(s(0()))) -> f#(s(0())) s#(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(0()))) -> s#(s(s(s(s(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)) TRS: s(s(0())) -> f(0()) f(0()) -> 0() s(s(s(0()))) -> f(s(0())) f(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> f(s(s(0()))) f(s(s(0()))) -> s(s(s(s(s(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)) TDG Processor: DPs: s#(s(0())) -> f#(0()) s#(s(s(0()))) -> f#(s(0())) s#(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(0()))) -> s#(s(s(s(s(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)) TRS: s(s(0())) -> f(0()) f(0()) -> 0() s(s(s(0()))) -> f(s(0())) f(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> f(s(s(0()))) f(s(s(0()))) -> s(s(s(s(s(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)) graph: 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)) -> f#(s(x)) -> f#(s(s(0()))) -> s#(s(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(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)) -> 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(s(s(s(s(s(s(0())))))))) -> f#(s(s(0()))) h#(f(x),g(x)) -> s#(x) -> s#(s(s(0()))) -> f#(s(0())) h#(f(x),g(x)) -> s#(x) -> s#(s(0())) -> f#(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#(x) -> h#(0(),x) g#(s(x)) -> s#(g(x)) -> s#(x) -> h#(x,0()) g#(s(x)) -> s#(g(x)) -> s#(s(s(s(s(s(s(s(0())))))))) -> f#(s(s(0()))) g#(s(x)) -> s#(g(x)) -> s#(s(s(0()))) -> f#(s(0())) g#(s(x)) -> s#(g(x)) -> s#(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#(s(g(x))) -> s#(s(s(s(s(s(s(s(0())))))))) -> f#(s(s(0()))) g#(s(x)) -> s#(s(g(x))) -> s#(s(s(0()))) -> f#(s(0())) g#(s(x)) -> s#(s(g(x))) -> s#(s(0())) -> f#(0()) 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) f#(g(x)) -> f#(x) -> f#(s(s(0()))) -> s#(s(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(0()))) -> s#(s(s(s(0())))) f#(g(x)) -> f#(x) -> f#(s(s(0()))) -> s#(s(s(0()))) f#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(x) -> h#(0(),x) f#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(x) -> h#(x,0()) f#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(s(s(s(s(s(s(s(0())))))))) -> f#(s(s(0()))) f#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(s(s(0()))) -> f#(s(0())) f#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(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(0()))) -> s#(s(s(s(s(0()))))) -> s#(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(0()))) -> f#(s(0())) f#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(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(0())))) -> s#(s(s(s(s(s(s(s(0())))))))) -> f#(s(s(0()))) f#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(s(s(0()))) -> f#(s(0())) f#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(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(0()))) -> s#(s(s(s(s(s(s(s(0())))))))) -> f#(s(s(0()))) f#(s(s(0()))) -> s#(s(s(0()))) -> s#(s(s(0()))) -> f#(s(0())) f#(s(s(0()))) -> s#(s(s(0()))) -> s#(s(0())) -> f#(0()) s#(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(s(0())))))))) -> f#(s(s(0()))) -> f#(g(x)) -> g#(f(x)) s#(s(s(s(s(s(s(s(0())))))))) -> f#(s(s(0()))) -> f#(g(x)) -> f#(x) s#(s(s(s(s(s(s(s(0())))))))) -> f#(s(s(0()))) -> f#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) s#(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(s(0())))))))) -> f#(s(s(0()))) -> f#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(s(s(s(s(s(s(0())))))))) -> f#(s(s(0()))) -> f#(s(s(0()))) -> s#(s(s(0()))) s#(s(s(0()))) -> f#(s(0())) -> f#(g(x)) -> g#(g(f(x))) s#(s(s(0()))) -> f#(s(0())) -> f#(g(x)) -> g#(f(x)) s#(s(s(0()))) -> f#(s(0())) -> f#(g(x)) -> f#(x) s#(s(s(0()))) -> f#(s(0())) -> f#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) s#(s(s(0()))) -> f#(s(0())) -> f#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(s(0()))) -> f#(s(0())) -> f#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(s(0()))) -> f#(s(0())) -> f#(s(s(0()))) -> s#(s(s(0()))) s#(s(0())) -> f#(0()) -> f#(g(x)) -> g#(g(f(x))) s#(s(0())) -> f#(0()) -> f#(g(x)) -> g#(f(x)) s#(s(0())) -> f#(0()) -> f#(g(x)) -> f#(x) s#(s(0())) -> f#(0()) -> f#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) s#(s(0())) -> f#(0()) -> f#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(0())) -> f#(0()) -> f#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(0())) -> f#(0()) -> f#(s(s(0()))) -> s#(s(s(0()))) 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) EDG Processor: DPs: s#(s(0())) -> f#(0()) s#(s(s(0()))) -> f#(s(0())) s#(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(0()))) -> s#(s(s(s(s(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)) TRS: s(s(0())) -> f(0()) f(0()) -> 0() s(s(s(0()))) -> f(s(0())) f(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> f(s(s(0()))) f(s(s(0()))) -> s(s(s(s(s(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)) graph: 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(0()))) -> s#(s(s(s(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#(0()) h#(f(x),g(x)) -> s#(x) -> s#(s(s(0()))) -> f#(s(0())) h#(f(x),g(x)) -> s#(x) -> s#(s(s(s(s(s(s(s(0())))))))) -> f#(s(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) 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#(0()) g#(s(x)) -> s#(g(x)) -> s#(s(s(0()))) -> f#(s(0())) g#(s(x)) -> s#(g(x)) -> s#(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#(s(g(x))) -> s#(s(0())) -> f#(0()) g#(s(x)) -> s#(s(g(x))) -> s#(s(s(0()))) -> f#(s(0())) g#(s(x)) -> s#(s(g(x))) -> s#(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#(x) -> h#(x,x) -> h#(f(x),g(x)) -> s#(x) g#(x) -> h#(x,x) -> h#(f(x),g(x)) -> f#(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#(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(0()))) -> s#(s(s(s(s(s(0())))))) 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#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(s(0())) -> f#(0()) f#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(s(s(0()))) -> f#(s(0())) f#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(s(s(s(s(s(s(s(0())))))))) -> f#(s(s(0()))) f#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(x) -> h#(x,0()) f#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(x) -> h#(0(),x) f#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(s(0())) -> f#(0()) f#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(s(s(0()))) -> f#(s(0())) f#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(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(0())))) -> s#(s(0())) -> f#(0()) f#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(s(s(0()))) -> f#(s(0())) f#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(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(0()))) -> s#(s(0())) -> f#(0()) f#(s(s(0()))) -> s#(s(s(0()))) -> s#(s(s(0()))) -> f#(s(0())) f#(s(s(0()))) -> s#(s(s(0()))) -> s#(s(s(s(s(s(s(s(0())))))))) -> f#(s(s(0()))) 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) s#(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(s(0())))))))) -> f#(s(s(0()))) -> f#(s(s(0()))) -> s#(s(s(s(0())))) s#(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(s(0())))))))) -> f#(s(s(0()))) -> f#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) s#(s(s(s(s(s(s(s(0())))))))) -> f#(s(s(0()))) -> f#(g(x)) -> f#(x) s#(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(s(0())))))))) -> f#(s(s(0()))) -> f#(g(x)) -> g#(g(f(x))) s#(s(s(0()))) -> f#(s(0())) -> f#(s(s(0()))) -> s#(s(s(0()))) s#(s(s(0()))) -> f#(s(0())) -> f#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(s(0()))) -> f#(s(0())) -> f#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(s(0()))) -> f#(s(0())) -> f#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) s#(s(s(0()))) -> f#(s(0())) -> f#(g(x)) -> f#(x) s#(s(s(0()))) -> f#(s(0())) -> f#(g(x)) -> g#(f(x)) s#(s(s(0()))) -> f#(s(0())) -> f#(g(x)) -> g#(g(f(x))) CDG Processor: DPs: s#(s(0())) -> f#(0()) s#(s(s(0()))) -> f#(s(0())) s#(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(0()))) -> s#(s(s(s(s(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)) TRS: s(s(0())) -> f(0()) f(0()) -> 0() s(s(s(0()))) -> f(s(0())) f(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> f(s(s(0()))) f(s(s(0()))) -> s(s(s(s(s(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)) graph: 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)) -> f#(s(x)) -> f#(s(s(0()))) -> s#(s(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(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)) -> 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(s(s(s(s(s(s(0())))))))) -> f#(s(s(0()))) h#(f(x),g(x)) -> s#(x) -> s#(s(s(0()))) -> f#(s(0())) h#(f(x),g(x)) -> s#(x) -> s#(s(0())) -> f#(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#(x) -> h#(0(),x) g#(s(x)) -> s#(g(x)) -> s#(x) -> h#(x,0()) g#(s(x)) -> s#(g(x)) -> s#(s(s(s(s(s(s(s(0())))))))) -> f#(s(s(0()))) g#(s(x)) -> s#(g(x)) -> s#(s(s(0()))) -> f#(s(0())) g#(s(x)) -> s#(g(x)) -> s#(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#(s(g(x))) -> s#(s(s(s(s(s(s(s(0())))))))) -> f#(s(s(0()))) g#(s(x)) -> s#(s(g(x))) -> s#(s(s(0()))) -> f#(s(0())) g#(s(x)) -> s#(s(g(x))) -> s#(s(0())) -> f#(0()) 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) f#(g(x)) -> f#(x) -> f#(s(s(0()))) -> s#(s(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(0()))) -> s#(s(s(s(0())))) f#(g(x)) -> f#(x) -> f#(s(s(0()))) -> s#(s(s(0()))) f#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(x) -> h#(0(),x) f#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(x) -> h#(x,0()) f#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) -> s#(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(0()))) -> s#(s(s(s(s(0()))))) -> s#(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(0())))) -> s#(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(0()))) -> s#(s(s(0()))) -> f#(s(0())) s#(s(s(s(s(s(s(s(0())))))))) -> f#(s(s(0()))) -> f#(s(s(0()))) -> s#(s(s(s(s(s(0())))))) s#(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(s(0())))))))) -> f#(s(s(0()))) -> f#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(s(s(s(s(s(s(0())))))))) -> f#(s(s(0()))) -> f#(s(s(0()))) -> s#(s(s(0()))) SCC Processor: #sccs: 1 #rules: 6 #arcs: 59/324 DPs: h#(f(x),g(x)) -> f#(s(x)) f#(g(x)) -> f#(x) f#(g(x)) -> g#(f(x)) g#(x) -> h#(x,x) g#(s(x)) -> g#(x) f#(g(x)) -> g#(g(f(x))) TRS: s(s(0())) -> f(0()) f(0()) -> 0() s(s(s(0()))) -> f(s(0())) f(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> f(s(s(0()))) f(s(s(0()))) -> s(s(s(s(s(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)) Arctic Interpretation Processor: dimension: 1 usable rules: s(s(0())) -> f(0()) f(0()) -> 0() s(s(s(0()))) -> f(s(0())) f(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> f(s(s(0()))) f(s(s(0()))) -> s(s(s(s(s(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)) interpretation: [h#](x0, x1) = 1, [g#](x0) = 2, [f#](x0) = x0 + 0, [h](x0, x1) = 0, [g](x0) = x0 + 2, [f](x0) = x0, [s](x0) = 0, [0] = 0 orientation: h#(f(x),g(x)) = 1 >= 0 = f#(s(x)) f#(g(x)) = x + 2 >= x + 0 = f#(x) f#(g(x)) = x + 2 >= 2 = g#(f(x)) g#(x) = 2 >= 1 = h#(x,x) g#(s(x)) = 2 >= 2 = g#(x) f#(g(x)) = x + 2 >= 2 = g#(g(f(x))) s(s(0())) = 0 >= 0 = f(0()) f(0()) = 0 >= 0 = 0() s(s(s(0()))) = 0 >= 0 = f(s(0())) f(s(0())) = 0 >= 0 = s(0()) s(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(s(0())))))) g(x) = x + 2 >= 0 = h(x,x) s(x) = 0 >= 0 = h(x,0()) s(x) = 0 >= 0 = h(0(),x) f(g(x)) = x + 2 >= x + 2 = g(g(f(x))) g(s(x)) = 2 >= 0 = s(s(g(x))) h(f(x),g(x)) = 0 >= 0 = f(s(x)) problem: DPs: f#(g(x)) -> f#(x) f#(g(x)) -> g#(f(x)) g#(s(x)) -> g#(x) f#(g(x)) -> g#(g(f(x))) TRS: s(s(0())) -> f(0()) f(0()) -> 0() s(s(s(0()))) -> f(s(0())) f(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> f(s(s(0()))) f(s(s(0()))) -> s(s(s(s(s(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)) Restore Modifier: DPs: f#(g(x)) -> f#(x) f#(g(x)) -> g#(f(x)) g#(s(x)) -> g#(x) f#(g(x)) -> g#(g(f(x))) TRS: s(s(0())) -> f(0()) f(0()) -> 0() s(s(s(0()))) -> f(s(0())) f(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> f(s(s(0()))) f(s(s(0()))) -> s(s(s(s(s(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)) SCC Processor: #sccs: 2 #rules: 2 #arcs: 13/16 DPs: f#(g(x)) -> f#(x) TRS: s(s(0())) -> f(0()) f(0()) -> 0() s(s(s(0()))) -> f(s(0())) f(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> f(s(s(0()))) f(s(s(0()))) -> s(s(s(s(s(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)) Subterm Criterion Processor: simple projection: pi(f#) = 0 problem: DPs: TRS: s(s(0())) -> f(0()) f(0()) -> 0() s(s(s(0()))) -> f(s(0())) f(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> f(s(s(0()))) f(s(s(0()))) -> s(s(s(s(s(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)) Qed DPs: g#(s(x)) -> g#(x) TRS: s(s(0())) -> f(0()) f(0()) -> 0() s(s(s(0()))) -> f(s(0())) f(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> f(s(s(0()))) f(s(s(0()))) -> s(s(s(s(s(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)) Subterm Criterion Processor: simple projection: pi(g#) = 0 problem: DPs: TRS: s(s(0())) -> f(0()) f(0()) -> 0() s(s(s(0()))) -> f(s(0())) f(s(0())) -> s(0()) s(s(s(s(s(s(s(s(0())))))))) -> f(s(s(0()))) f(s(s(0()))) -> s(s(s(s(s(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)) Qed