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(0()) -> k(0()) s(k(0())) -> 0() s(s(0())) -> k(s(0())) k(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> k(s(s(0()))) k(s(s(0()))) -> 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#(0()) -> k#(0()) s#(s(0())) -> k#(s(0())) 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()))))) 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(0()) -> k(0()) s(k(0())) -> 0() s(s(0())) -> k(s(0())) k(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> k(s(s(0()))) k(s(s(0()))) -> 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#(0()) -> k#(0()) s#(s(0())) -> k#(s(0())) 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()))))) 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(0()) -> k(0()) s(k(0())) -> 0() s(s(0())) -> k(s(0())) k(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(0()))))) h(k(x),g(x)) -> k(s(x)) graph: k#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> 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())) -> k#(s(0())) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> 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(0()))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(s(0())) -> k#(s(0())) k#(s(s(0()))) -> s#(s(s(s(0())))) -> 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(0()))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(0()))) -> s#(s(0())) -> k#(s(0())) k#(s(s(0()))) -> s#(s(s(0()))) -> 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(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(0()))))))) -> k#(s(s(0()))) h#(k(x),g(x)) -> s#(x) -> s#(s(0())) -> k#(s(0())) h#(k(x),g(x)) -> s#(x) -> 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(0()))))))) -> k#(s(s(0()))) h#(f(x),g(x)) -> s#(x) -> s#(s(0())) -> k#(s(0())) h#(f(x),g(x)) -> s#(x) -> 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(0()))))))) -> k#(s(s(0()))) g#(s(x)) -> s#(g(x)) -> s#(s(0())) -> k#(s(0())) g#(s(x)) -> s#(g(x)) -> 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(0()))))))) -> k#(s(s(0()))) g#(s(x)) -> s#(s(g(x))) -> s#(s(0())) -> k#(s(0())) g#(s(x)) -> s#(s(g(x))) -> 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(0()))))))) -> k#(s(s(0()))) -> k#(s(s(0()))) -> s#(s(s(s(s(0()))))) 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(0()))))))) -> k#(s(s(0()))) -> k#(s(s(0()))) -> s#(s(s(0()))) s#(s(0())) -> k#(s(0())) -> k#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(0())) -> k#(s(0())) -> k#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(0())) -> k#(s(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#(0()) -> k#(0()) -> k#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(0()) -> k#(0()) -> k#(s(s(0()))) -> s#(s(s(s(0())))) s#(0()) -> k#(0()) -> k#(s(s(0()))) -> s#(s(s(0()))) 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#(0()) -> k#(0()) s#(s(0())) -> k#(s(0())) 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()))))) 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(0()) -> k(0()) s(k(0())) -> 0() s(s(0())) -> k(s(0())) k(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(0()))))) h(k(x),g(x)) -> k(s(x)) graph: 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#(0()) -> k#(0()) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> s#(s(0())) -> k#(s(0())) k#(s(s(0()))) -> s#(s(s(s(s(0()))))) -> 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#(0()) -> k#(0()) k#(s(s(0()))) -> s#(s(s(s(0())))) -> s#(s(0())) -> k#(s(0())) k#(s(s(0()))) -> s#(s(s(s(0())))) -> 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#(0()) -> k#(0()) k#(s(s(0()))) -> s#(s(s(0()))) -> s#(s(0())) -> k#(s(0())) k#(s(s(0()))) -> s#(s(s(0()))) -> 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)) -> 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#(0()) -> k#(0()) h#(k(x),g(x)) -> s#(x) -> s#(s(0())) -> k#(s(0())) h#(k(x),g(x)) -> s#(x) -> 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#(0()) -> k#(0()) h#(f(x),g(x)) -> s#(x) -> s#(s(0())) -> k#(s(0())) h#(f(x),g(x)) -> s#(x) -> 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#(0()) -> k#(0()) g#(s(x)) -> s#(g(x)) -> s#(s(0())) -> k#(s(0())) g#(s(x)) -> s#(g(x)) -> 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#(0()) -> k#(0()) g#(s(x)) -> s#(s(g(x))) -> s#(s(0())) -> k#(s(0())) g#(s(x)) -> s#(s(g(x))) -> 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(0()))))))) -> k#(s(s(0()))) -> k#(s(s(0()))) -> s#(s(s(0()))) 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(0()))))))) -> k#(s(s(0()))) -> k#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(0())) -> k#(s(0())) -> k#(s(s(0()))) -> s#(s(s(0()))) s#(s(0())) -> k#(s(0())) -> k#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(0())) -> k#(s(0())) -> k#(s(s(0()))) -> 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: 17 #arcs: 76/400 DPs: k#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(s(s(s(s(s(0()))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(0())) -> k#(s(0())) k#(s(s(0()))) -> 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)) 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(0()) -> k(0()) s(k(0())) -> 0() s(s(0())) -> k(s(0())) k(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> k(s(s(0()))) k(s(s(0()))) -> 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(0()) -> k(0()) s(k(0())) -> 0() s(s(0())) -> k(s(0())) k(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(0()))))) h(k(x),g(x)) -> k(s(x)) interpretation: [h](x0, x1) = x0 + x1, [g#](x0) = 2x0 + 3, [s](x0) = x0 + 0, [g](x0) = x0 + 2, [k](x0) = x0 + 0, [k#](x0) = 2x0 + 3, [0] = 0, [f#](x0) = 2x0 + 0, [h#](x0, x1) = x0 + 2x1, [s#](x0) = x0 + 3, [f](x0) = x0 orientation: k#(s(s(0()))) = 3 >= 3 = s#(s(s(s(s(0()))))) s#(s(s(s(s(s(s(0()))))))) = 3 >= 3 = k#(s(s(0()))) k#(s(s(0()))) = 3 >= 3 = s#(s(s(s(0())))) s#(s(0())) = 3 >= 3 = k#(s(0())) k#(s(s(0()))) = 3 >= 3 = s#(s(s(0()))) s#(s(0())) = 3 >= 2 = f#(s(0())) f#(g(x)) = 2x + 4 >= 2x + 4 = g#(g(f(x))) g#(s(x)) = 2x + 3 >= x + 3 = s#(s(g(x))) g#(s(x)) = 2x + 3 >= x + 3 = s#(g(x)) g#(s(x)) = 2x + 3 >= 2x + 3 = g#(x) g#(x) = 2x + 3 >= 2x = h#(x,x) h#(k(x),g(x)) = 2x + 4 >= 2x + 3 = k#(s(x)) h#(k(x),g(x)) = 2x + 4 >= x + 3 = s#(x) h#(f(x),g(x)) = 2x + 4 >= 2x + 2 = f#(s(x)) f#(g(x)) = 2x + 4 >= 2x + 3 = g#(f(x)) f#(g(x)) = 2x + 4 >= 2x + 0 = f#(x) h#(f(x),g(x)) = 2x + 4 >= x + 3 = s#(x) s(s(0())) = 0 >= 0 = f(s(0())) g(x) = x + 2 >= x = h(x,x) s(x) = x + 0 >= x + 0 = h(x,0()) s(x) = x + 0 >= x + 0 = h(0(),x) f(g(x)) = x + 2 >= x + 2 = g(g(f(x))) g(s(x)) = x + 2 >= x + 2 = s(s(g(x))) h(f(x),g(x)) = x + 2 >= x + 0 = f(s(x)) s(0()) = 0 >= 0 = k(0()) s(k(0())) = 0 >= 0 = 0() s(s(0())) = 0 >= 0 = k(s(0())) k(s(0())) = 0 >= 0 = 0() 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()))))) h(k(x),g(x)) = x + 2 >= x + 0 = k(s(x)) problem: DPs: k#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(s(s(s(s(s(0()))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(0())) -> k#(s(0())) k#(s(s(0()))) -> s#(s(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)) h#(f(x),g(x)) -> f#(s(x)) 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(0()) -> k(0()) s(k(0())) -> 0() s(s(0())) -> k(s(0())) k(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> k(s(s(0()))) k(s(s(0()))) -> 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(0()))))) s#(s(s(s(s(s(s(0()))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(0())) -> k#(s(0())) k#(s(s(0()))) -> s#(s(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)) h#(f(x),g(x)) -> f#(s(x)) 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(0()) -> k(0()) s(k(0())) -> 0() s(s(0())) -> k(s(0())) k(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(0()))))) h(k(x),g(x)) -> k(s(x)) SCC Processor: #sccs: 2 #rules: 11 #arcs: 55/196 DPs: h#(f(x),g(x)) -> f#(s(x)) f#(g(x)) -> g#(g(f(x))) g#(s(x)) -> g#(x) g#(x) -> h#(x,x) 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(0()) -> k(0()) s(k(0())) -> 0() s(s(0())) -> k(s(0())) k(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> k(s(s(0()))) k(s(s(0()))) -> 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(0()) -> k(0()) s(k(0())) -> 0() s(s(0())) -> k(s(0())) k(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(0()))))) h(k(x),g(x)) -> k(s(x)) interpretation: [h](x0, x1) = 0, [g#](x0) = 0, [s](x0) = 0, [g](x0) = x0 + 5, [k](x0) = x0 + 0, [0] = 0, [f#](x0) = x0 + 0, [h#](x0, x1) = 0, [f](x0) = x0 + 0 orientation: h#(f(x),g(x)) = 0 >= 0 = f#(s(x)) f#(g(x)) = x + 5 >= 0 = g#(g(f(x))) g#(s(x)) = 0 >= 0 = g#(x) g#(x) = 0 >= 0 = h#(x,x) f#(g(x)) = x + 5 >= 0 = g#(f(x)) f#(g(x)) = x + 5 >= x + 0 = f#(x) s(s(0())) = 0 >= 0 = f(s(0())) g(x) = x + 5 >= 0 = h(x,x) s(x) = 0 >= 0 = h(x,0()) s(x) = 0 >= 0 = h(0(),x) f(g(x)) = x + 5 >= x + 5 = g(g(f(x))) g(s(x)) = 5 >= 0 = s(s(g(x))) h(f(x),g(x)) = 0 >= 0 = f(s(x)) s(0()) = 0 >= 0 = k(0()) s(k(0())) = 0 >= 0 = 0() s(s(0())) = 0 >= 0 = k(s(0())) k(s(0())) = 0 >= 0 = 0() 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()))))) h(k(x),g(x)) = 0 >= 0 = k(s(x)) problem: DPs: h#(f(x),g(x)) -> f#(s(x)) g#(s(x)) -> g#(x) g#(x) -> h#(x,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(0()) -> k(0()) s(k(0())) -> 0() s(s(0())) -> k(s(0())) k(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(0()))))) h(k(x),g(x)) -> k(s(x)) Restore Modifier: DPs: h#(f(x),g(x)) -> f#(s(x)) g#(s(x)) -> g#(x) g#(x) -> h#(x,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(0()) -> k(0()) s(k(0())) -> 0() s(s(0())) -> k(s(0())) k(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(0()))))) h(k(x),g(x)) -> k(s(x)) SCC Processor: #sccs: 2 #rules: 2 #arcs: 13/16 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(0()) -> k(0()) s(k(0())) -> 0() s(s(0())) -> k(s(0())) k(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(0()))))) h(k(x),g(x)) -> k(s(x)) Usable Rule Processor: DPs: g#(s(x)) -> g#(x) TRS: Semantic Labeling Processor: dimension: 2 usable rules: interpretation: [1 1] [1] [s](x0) = [1 1]x0 + [0] labeled: g# usable (for model): g# s argument filtering: pi(s) = [] pi(g#) = [] precedence: g# ~ s problem: DPs: TRS: Qed 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(0()) -> k(0()) s(k(0())) -> 0() s(s(0())) -> k(s(0())) k(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> k(s(s(0()))) k(s(s(0()))) -> 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(0()) -> k(0()) s(k(0())) -> 0() s(s(0())) -> k(s(0())) k(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> k(s(s(0()))) k(s(s(0()))) -> 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: k#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(s(s(s(s(s(0()))))))) -> k#(s(s(0()))) k#(s(s(0()))) -> s#(s(s(s(0())))) s#(s(0())) -> k#(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(0()) -> k(0()) s(k(0())) -> 0() s(s(0())) -> k(s(0())) k(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(0()))))) h(k(x),g(x)) -> k(s(x)) Bounds Processor: bound: 0 enrichment: top-dp automaton: final states: {20} transitions: h0(21,21) -> 22* k{#,0}(22) -> 20* k0(21) -> 22* 00() -> 21* s0(21) -> 22* problem: DPs: k#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(s(s(s(s(s(0()))))))) -> k#(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(0()) -> k(0()) s(k(0())) -> 0() s(s(0())) -> k(s(0())) k(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(0()))))) h(k(x),g(x)) -> k(s(x)) Bounds Processor: bound: 0 enrichment: top-dp automaton: final states: {20} transitions: h0(21,26) -> 22* h0(21,21) -> 22* h0(26,21) -> 22* h0(22,26) -> 23* h0(22,21) -> 23* h0(26,26) -> 22* h0(26,22) -> 23* h0(21,22) -> 23* k0(22) -> 23* k0(21) -> 22* k0(26) -> 22* 00() -> 26*,23,21 s{#,0}(26) -> 20* s{#,0}(23) -> 20* f0(22) -> 23* s0(26) -> 22* s0(21) -> 22* s0(22) -> 23* problem: DPs: k#(s(s(0()))) -> s#(s(s(s(s(0()))))) s#(s(s(s(s(s(s(0()))))))) -> k#(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(0()) -> k(0()) s(k(0())) -> 0() s(s(0())) -> k(s(0())) k(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(0()))))) h(k(x),g(x)) -> k(s(x)) Bounds Processor: bound: 0 enrichment: top-dp automaton: final states: {14} transitions: h0(21,21) -> 22*,16,18 h0(15,15) -> 16* h0(17,21) -> 18* h0(15,21) -> 22*,16,18 h0(22,21) -> 17* h0(16,21) -> 17* h0(15,17) -> 18* h0(17,15) -> 18* h0(15,16) -> 17* h0(21,15) -> 22*,16,18 h0(21,17) -> 18* h0(21,16) -> 17* h0(16,15) -> 17* h0(15,22) -> 17* h0(21,22) -> 17* h0(22,15) -> 17* k0(16) -> 17* k0(22) -> 17* k0(21) -> 22*,16 k0(15) -> 22*,16 00() -> 21*,17,15 s{#,0}(18) -> 14* s{#,0}(22) -> 14* f0(22) -> 17* f0(16) -> 17* s0(21) -> 22*,18,16 s0(16) -> 17* s0(22) -> 17* s0(17) -> 18* s0(15) -> 16* problem: DPs: k#(s(s(0()))) -> s#(s(s(s(s(0()))))) 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(0()) -> k(0()) s(k(0())) -> 0() s(s(0())) -> k(s(0())) k(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> k(s(s(0()))) k(s(s(0()))) -> 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: h0(15,5) -> 6* h0(17,3) -> 6* h0(21,21) -> 4,6,16* h0(6,15) -> 7* h0(5,3) -> 6* h0(15,3) -> 16*,6,4 h0(15,15) -> 16*,4,6 h0(21,6) -> 7* h0(17,21) -> 6* h0(3,3) -> 4* h0(3,21) -> 4,6,16* h0(15,4) -> 5* h0(3,17) -> 6* h0(5,21) -> 6* h0(15,21) -> 4,6,16* h0(5,15) -> 6* h0(16,21) -> 17*,5,7 h0(17,15) -> 6* h0(15,17) -> 6* h0(21,4) -> 5* h0(4,3) -> 5* h0(15,16) -> 17*,5,7 h0(3,6) -> 7* h0(21,15) -> 4,6,16* h0(21,3) -> 6,16*,4 h0(21,5) -> 6* h0(15,6) -> 7* h0(16,3) -> 17*,5,7 h0(21,17) -> 6* h0(21,16) -> 7,17*,5 h0(16,15) -> 17*,7,5 h0(4,15) -> 5* h0(3,5) -> 6* h0(6,3) -> 7* h0(6,21) -> 7* h0(4,21) -> 5* h0(3,15) -> 16*,4,6 h0(3,16) -> 17*,7,5 h0(3,4) -> 5* k0(3) -> 16*,4 k0(16) -> 17*,5 k0(21) -> 16* k0(4) -> 17*,5 k0(15) -> 16*,4 00() -> 15,17,21*,5,3 s{#,0}(17) -> 2* s{#,0}(7) -> 2* s{#,0}(21) -> 2* f0(4) -> 17*,5 f0(16) -> 17*,5 s0(5) -> 6* s0(21) -> 4,6,16* s0(6) -> 7* s0(3) -> 4* s0(16) -> 17*,7,5 s0(4) -> 5* s0(17) -> 6* s0(15) -> 16*,6,4 problem: DPs: 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(0()) -> k(0()) s(k(0())) -> 0() s(s(0())) -> k(s(0())) k(s(0())) -> 0() s(s(s(s(s(s(s(0()))))))) -> k(s(s(0()))) k(s(s(0()))) -> s(s(s(s(s(0()))))) h(k(x),g(x)) -> k(s(x)) SCC Processor: #sccs: 0 #rules: 0 #arcs: 12/1