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(s(s(0())))))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(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(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())))))) k#(s(s(0()))) -> s#(s(s(s(s(s(s(0()))))))) k#(s(s(0()))) -> s#(s(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(s(s(0())))))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(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(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())))))) k#(s(s(0()))) -> s#(s(s(s(s(s(s(0()))))))) k#(s(s(0()))) -> s#(s(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(s(s(0())))))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(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(s(s(0())))))))) -> s#(s(s(s(s(s(s(s(s(s(0())))))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(s(s(s(s(0())))))))) -> s#(s(s(0()))) -> k#(s(0())) k#(s(s(0()))) -> s#(s(s(s(s(s(s(s(0())))))))) -> s#(s(0())) -> k#(0()) k#(s(s(0()))) -> s#(s(s(s(s(s(s(s(0())))))))) -> s#(x) -> h#(0(),x) k#(s(s(0()))) -> s#(s(s(s(s(s(s(s(0())))))))) -> s#(x) -> h#(x,0()) k#(s(s(0()))) -> s#(s(s(s(s(s(s(s(0())))))))) -> s#(s(0())) -> f#(s(0())) k#(s(s(0()))) -> s#(s(s(s(s(s(s(0()))))))) -> s#(s(s(s(s(s(s(s(s(s(0())))))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(s(s(s(0()))))))) -> s#(s(s(0()))) -> k#(s(0())) k#(s(s(0()))) -> s#(s(s(s(s(s(s(0()))))))) -> s#(s(0())) -> k#(0()) k#(s(s(0()))) -> s#(s(s(s(s(s(s(0()))))))) -> s#(x) -> h#(0(),x) k#(s(s(0()))) -> s#(s(s(s(s(s(s(0()))))))) -> s#(x) -> h#(x,0()) k#(s(s(0()))) -> s#(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#(s(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(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(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(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(s(s(0())))))))) h#(k(x),g(x)) -> k#(s(x)) -> k#(s(s(0()))) -> s#(s(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)) -> 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(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(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(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(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(s(s(0())))))))))) -> k#(s(s(0()))) -> k#(s(s(0()))) -> s#(s(s(s(s(s(s(s(0())))))))) s#(s(s(s(s(s(s(s(s(s(0())))))))))) -> k#(s(s(0()))) -> k#(s(s(0()))) -> s#(s(s(s(s(s(s(0()))))))) s#(s(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(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(s(s(0())))))))))) -> k#(s(s(0()))) -> k#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(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(s(s(0())))))))) s#(s(s(0()))) -> k#(s(0())) -> k#(s(s(0()))) -> s#(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(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(s(s(0())))))))) s#(s(0())) -> k#(0()) -> k#(s(s(0()))) -> s#(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(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(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())))))) k#(s(s(0()))) -> s#(s(s(s(s(s(s(0()))))))) k#(s(s(0()))) -> s#(s(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(s(s(0())))))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(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(s(s(0())))))))) -> s#(s(0())) -> f#(s(0())) k#(s(s(0()))) -> s#(s(s(s(s(s(s(s(0())))))))) -> s#(x) -> h#(x,0()) k#(s(s(0()))) -> s#(s(s(s(s(s(s(s(0())))))))) -> s#(x) -> h#(0(),x) k#(s(s(0()))) -> s#(s(s(s(s(s(s(s(0())))))))) -> s#(s(0())) -> k#(0()) k#(s(s(0()))) -> s#(s(s(s(s(s(s(s(0())))))))) -> s#(s(s(0()))) -> k#(s(0())) k#(s(s(0()))) -> s#(s(s(s(s(s(s(s(0())))))))) -> s#(s(s(s(s(s(s(s(s(s(0())))))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(s(s(s(0()))))))) -> s#(s(0())) -> f#(s(0())) k#(s(s(0()))) -> s#(s(s(s(s(s(s(0()))))))) -> s#(x) -> h#(x,0()) k#(s(s(0()))) -> s#(s(s(s(s(s(s(0()))))))) -> s#(x) -> h#(0(),x) k#(s(s(0()))) -> s#(s(s(s(s(s(s(0()))))))) -> s#(s(0())) -> k#(0()) k#(s(s(0()))) -> s#(s(s(s(s(s(s(0()))))))) -> s#(s(s(0()))) -> k#(s(0())) k#(s(s(0()))) -> s#(s(s(s(s(s(s(0()))))))) -> s#(s(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(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(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(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(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(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)) -> k#(s(x)) -> k#(s(s(0()))) -> s#(s(s(s(s(s(s(0()))))))) h#(k(x),g(x)) -> k#(s(x)) -> k#(s(s(0()))) -> s#(s(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(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(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(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(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(s(s(0())))))))))) -> k#(s(s(0()))) -> k#(s(s(0()))) -> s#(s(s(0()))) s#(s(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(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(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(s(s(0())))))))))) -> k#(s(s(0()))) -> k#(s(s(0()))) -> s#(s(s(s(s(s(s(0()))))))) s#(s(s(s(s(s(s(s(s(s(0())))))))))) -> k#(s(s(0()))) -> k#(s(s(0()))) -> s#(s(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(s(0()))) -> k#(s(0())) -> k#(s(s(0()))) -> s#(s(s(s(s(s(s(0()))))))) s#(s(s(0()))) -> k#(s(0())) -> k#(s(s(0()))) -> s#(s(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: 20 #arcs: 103/529 DPs: k#(s(s(0()))) -> s#(s(s(s(s(s(s(s(0())))))))) s#(s(s(s(s(s(s(s(s(s(0())))))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(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())) -> 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(s(s(0()))))) k#(s(s(0()))) -> s#(s(s(s(0())))) 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(s(s(0())))))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(s(s(0())))))))) h(k(x),g(x)) -> k(s(x)) Arctic Interpretation Processor: dimension: 1 interpretation: [k#](x0) = x0 + 0, [h#](x0, x1) = 0, [g#](x0) = 0, [f#](x0) = x0 + 0, [s#](x0) = 0, [k](x0) = 0, [h](x0, x1) = 0, [g](x0) = x0 + 4, [f](x0) = x0 + 0, [s](x0) = 0, [0] = 0 orientation: k#(s(s(0()))) = 0 >= 0 = s#(s(s(s(s(s(s(s(0())))))))) s#(s(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(s(0()))))))) s#(s(s(0()))) = 0 >= 0 = k#(s(0())) k#(s(s(0()))) = 0 >= 0 = s#(s(s(s(s(s(0())))))) s#(s(0())) = 0 >= 0 = f#(s(0())) f#(g(x)) = x + 4 >= 0 = g#(g(f(x))) g#(s(x)) = 0 >= 0 = s#(s(g(x))) g#(s(x)) = 0 >= 0 = s#(g(x)) g#(s(x)) = 0 >= 0 = g#(x) g#(x) = 0 >= 0 = h#(x,x) h#(k(x),g(x)) = 0 >= 0 = k#(s(x)) k#(s(s(0()))) = 0 >= 0 = s#(s(s(s(s(0()))))) k#(s(s(0()))) = 0 >= 0 = s#(s(s(s(0())))) k#(s(s(0()))) = 0 >= 0 = s#(s(s(0()))) h#(k(x),g(x)) = 0 >= 0 = s#(x) h#(f(x),g(x)) = 0 >= 0 = f#(s(x)) f#(g(x)) = x + 4 >= 0 = g#(f(x)) f#(g(x)) = x + 4 >= x + 0 = f#(x) h#(f(x),g(x)) = 0 >= 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(s(s(0())))))))))) = 0 >= 0 = k(s(s(0()))) k(s(s(0()))) = 0 >= 0 = s(s(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(s(s(0())))))))) s#(s(s(s(s(s(s(s(s(s(0())))))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(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())) -> f#(s(0())) 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(s(s(0()))))) k#(s(s(0()))) -> s#(s(s(s(0())))) 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)) -> 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(s(s(0())))))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(s(s(0())))))))) h(k(x),g(x)) -> k(s(x)) SCC Processor: #sccs: 3 #rules: 10 #arcs: 73/324 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(s(s(0())))))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(s(s(0())))))))) h(k(x),g(x)) -> k(s(x)) Subterm Criterion Processor: simple projection: pi(g#) = 0 problem: 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(s(s(0())))))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(s(s(0())))))))) h(k(x),g(x)) -> k(s(x)) Qed DPs: k#(s(s(0()))) -> s#(s(s(s(s(s(s(s(0())))))))) s#(s(s(s(s(s(s(s(s(s(0())))))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(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())))))) 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(s(s(0())))))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(s(s(0())))))))) h(k(x),g(x)) -> k(s(x)) Matrix Interpretation Processor: dim=2 interpretation: [k#](x0) = [1 0]x0, [s#](x0) = [0 2]x0, [0 2] [k](x0) = [0 1]x0, [0 2] [0] [h](x0, x1) = [0 0]x1 + [2], [0 2] [0] [g](x0) = [0 0]x0 + [2], [0 2] [0] [f](x0) = [0 0]x0 + [2], [0 2] [0] [s](x0) = [0 0]x0 + [2], [0] [0] = [0] orientation: k#(s(s(0()))) = 4 >= 4 = s#(s(s(s(s(s(s(s(0())))))))) s#(s(s(s(s(s(s(s(s(s(0())))))))))) = 4 >= 4 = k#(s(s(0()))) k#(s(s(0()))) = 4 >= 4 = s#(s(s(s(s(s(s(0()))))))) s#(s(s(0()))) = 4 >= 0 = k#(s(0())) k#(s(s(0()))) = 4 >= 4 = s#(s(s(s(s(s(0())))))) k#(s(s(0()))) = 4 >= 4 = s#(s(s(s(s(0()))))) k#(s(s(0()))) = 4 >= 4 = s#(s(s(s(0())))) k#(s(s(0()))) = 4 >= 4 = s#(s(s(0()))) [4] [4] s(s(0())) = [2] >= [2] = f(s(0())) [0 2] [0] [0 2] [0] g(x) = [0 0]x + [2] >= [0 0]x + [2] = h(x,x) [0 2] [0] [0] s(x) = [0 0]x + [2] >= [2] = h(x,0()) [0 2] [0] [0 2] [0] s(x) = [0 0]x + [2] >= [0 0]x + [2] = h(0(),x) [4] [4] f(g(x)) = [2] >= [2] = g(g(f(x))) [4] [4] g(s(x)) = [2] >= [2] = s(s(g(x))) [4] [4] h(f(x),g(x)) = [2] >= [2] = f(s(x)) [4] [0] s(s(0())) = [2] >= [0] = k(0()) [0] [0] k(0()) = [0] >= [0] = 0() [4] [4] s(s(s(0()))) = [2] >= [2] = k(s(0())) [4] [0] k(s(0())) = [2] >= [2] = s(0()) [4] [4] s(s(s(s(s(s(s(s(s(s(0())))))))))) = [2] >= [2] = k(s(s(0()))) [4] [4] k(s(s(0()))) = [2] >= [2] = s(s(s(s(s(s(s(s(0())))))))) [4] [4] h(k(x),g(x)) = [2] >= [2] = k(s(x)) problem: DPs: k#(s(s(0()))) -> s#(s(s(s(s(s(s(s(0())))))))) s#(s(s(s(s(s(s(s(s(s(0())))))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(s(s(s(0()))))))) k#(s(s(0()))) -> s#(s(s(s(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(s(s(0())))))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(s(s(0())))))))) h(k(x),g(x)) -> k(s(x)) Matrix Interpretation Processor: dim=2 interpretation: [k#](x0) = [3 1]x0 + [1], [s#](x0) = [0 1]x0, [3 0] [1] [k](x0) = [3 0]x0 + [1], [0 1] [0 1] [h](x0, x1) = [1 0]x0 + [1 0]x1, [3 3] [g](x0) = [3 3]x0, [0] [f](x0) = [0], [0 1] [0] [s](x0) = [1 0]x0 + [1], [0] [0] = [0] orientation: k#(s(s(0()))) = 5 >= 4 = s#(s(s(s(s(s(s(s(0())))))))) s#(s(s(s(s(s(s(s(s(s(0())))))))))) = 5 >= 5 = k#(s(s(0()))) k#(s(s(0()))) = 5 >= 3 = s#(s(s(s(s(s(s(0()))))))) k#(s(s(0()))) = 5 >= 3 = s#(s(s(s(s(s(0())))))) k#(s(s(0()))) = 5 >= 2 = s#(s(s(s(s(0()))))) k#(s(s(0()))) = 5 >= 2 = s#(s(s(s(0())))) k#(s(s(0()))) = 5 >= 1 = s#(s(s(0()))) [1] [0] s(s(0())) = [1] >= [0] = f(s(0())) [3 3] [0 2] g(x) = [3 3]x >= [2 0]x = h(x,x) [0 1] [0] [0 1] s(x) = [1 0]x + [1] >= [1 0]x = h(x,0()) [0 1] [0] [0 1] s(x) = [1 0]x + [1] >= [1 0]x = h(0(),x) [0] [0] f(g(x)) = [0] >= [0] = g(g(f(x))) [3 3] [3] [3 3] [1] g(s(x)) = [3 3]x + [3] >= [3 3]x + [1] = s(s(g(x))) [3 3] [0] h(f(x),g(x)) = [3 3]x >= [0] = f(s(x)) [1] [1] s(s(0())) = [1] >= [1] = k(0()) [1] [0] k(0()) = [1] >= [0] = 0() [1] [1] s(s(s(0()))) = [2] >= [1] = k(s(0())) [1] [0] k(s(0())) = [1] >= [1] = s(0()) [5] [4] s(s(s(s(s(s(s(s(s(s(0())))))))))) = [5] >= [4] = k(s(s(0()))) [4] [4] k(s(s(0()))) = [4] >= [4] = s(s(s(s(s(s(s(s(0())))))))) [6 3] [1] [0 3] [1] h(k(x),g(x)) = [6 3]x + [1] >= [0 3]x + [1] = k(s(x)) problem: DPs: s#(s(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(s(s(0())))))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(s(s(0())))))))) h(k(x),g(x)) -> k(s(x)) SCC Processor: #sccs: 0 #rules: 0 #arcs: 24/1 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(s(s(0())))))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(s(s(0())))))))) h(k(x),g(x)) -> k(s(x)) Subterm Criterion Processor: simple projection: pi(f#) = 0 problem: 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(s(s(0())))))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(s(s(s(0())))))))) h(k(x),g(x)) -> k(s(x)) Qed