YES Problem: g(h(x1)) -> g(f(s(x1))) f(s(s(s(x1)))) -> h(f(s(h(x1)))) f(h(x1)) -> h(f(s(h(x1)))) h(x1) -> x1 f(f(s(s(x1)))) -> s(s(s(f(f(x1))))) b(a(x1)) -> a(b(x1)) a(a(a(x1))) -> b(a(a(b(x1)))) b(b(b(b(x1)))) -> a(x1) Proof: DP Processor: DPs: g#(h(x1)) -> f#(s(x1)) g#(h(x1)) -> g#(f(s(x1))) f#(s(s(s(x1)))) -> h#(x1) f#(s(s(s(x1)))) -> f#(s(h(x1))) f#(s(s(s(x1)))) -> h#(f(s(h(x1)))) f#(h(x1)) -> f#(s(h(x1))) f#(h(x1)) -> h#(f(s(h(x1)))) f#(f(s(s(x1)))) -> f#(x1) f#(f(s(s(x1)))) -> f#(f(x1)) b#(a(x1)) -> b#(x1) b#(a(x1)) -> a#(b(x1)) a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(b(x1)) a#(a(a(x1))) -> a#(a(b(x1))) a#(a(a(x1))) -> b#(a(a(b(x1)))) b#(b(b(b(x1)))) -> a#(x1) TRS: g(h(x1)) -> g(f(s(x1))) f(s(s(s(x1)))) -> h(f(s(h(x1)))) f(h(x1)) -> h(f(s(h(x1)))) h(x1) -> x1 f(f(s(s(x1)))) -> s(s(s(f(f(x1))))) b(a(x1)) -> a(b(x1)) a(a(a(x1))) -> b(a(a(b(x1)))) b(b(b(b(x1)))) -> a(x1) TDG Processor: DPs: g#(h(x1)) -> f#(s(x1)) g#(h(x1)) -> g#(f(s(x1))) f#(s(s(s(x1)))) -> h#(x1) f#(s(s(s(x1)))) -> f#(s(h(x1))) f#(s(s(s(x1)))) -> h#(f(s(h(x1)))) f#(h(x1)) -> f#(s(h(x1))) f#(h(x1)) -> h#(f(s(h(x1)))) f#(f(s(s(x1)))) -> f#(x1) f#(f(s(s(x1)))) -> f#(f(x1)) b#(a(x1)) -> b#(x1) b#(a(x1)) -> a#(b(x1)) a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(b(x1)) a#(a(a(x1))) -> a#(a(b(x1))) a#(a(a(x1))) -> b#(a(a(b(x1)))) b#(b(b(b(x1)))) -> a#(x1) TRS: g(h(x1)) -> g(f(s(x1))) f(s(s(s(x1)))) -> h(f(s(h(x1)))) f(h(x1)) -> h(f(s(h(x1)))) h(x1) -> x1 f(f(s(s(x1)))) -> s(s(s(f(f(x1))))) b(a(x1)) -> a(b(x1)) a(a(a(x1))) -> b(a(a(b(x1)))) b(b(b(b(x1)))) -> a(x1) graph: a#(a(a(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(a(a(b(x1)))) a#(a(a(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> a#(a(b(x1))) a#(a(a(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> a#(b(x1)) a#(a(a(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(a(x1))) -> b#(a(a(b(x1)))) a#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(a(x1))) -> a#(a(b(x1))) a#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(a(x1))) -> a#(b(x1)) a#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(a(a(b(x1)))) -> b#(b(b(b(x1)))) -> a#(x1) a#(a(a(x1))) -> b#(a(a(b(x1)))) -> b#(a(x1)) -> a#(b(x1)) a#(a(a(x1))) -> b#(a(a(b(x1)))) -> b#(a(x1)) -> b#(x1) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(x1) a#(a(a(x1))) -> b#(x1) -> b#(a(x1)) -> a#(b(x1)) a#(a(a(x1))) -> b#(x1) -> b#(a(x1)) -> b#(x1) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> b#(a(a(b(x1)))) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> a#(a(b(x1))) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> a#(b(x1)) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> b#(x1) b#(a(x1)) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(a(a(b(x1)))) b#(a(x1)) -> a#(b(x1)) -> a#(a(a(x1))) -> a#(a(b(x1))) b#(a(x1)) -> a#(b(x1)) -> a#(a(a(x1))) -> a#(b(x1)) b#(a(x1)) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(x1) b#(a(x1)) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(x1) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(b(x1)) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1) f#(f(s(s(x1)))) -> f#(f(x1)) -> f#(f(s(s(x1)))) -> f#(f(x1)) f#(f(s(s(x1)))) -> f#(f(x1)) -> f#(f(s(s(x1)))) -> f#(x1) f#(f(s(s(x1)))) -> f#(f(x1)) -> f#(h(x1)) -> h#(f(s(h(x1)))) f#(f(s(s(x1)))) -> f#(f(x1)) -> f#(h(x1)) -> f#(s(h(x1))) f#(f(s(s(x1)))) -> f#(f(x1)) -> f#(s(s(s(x1)))) -> h#(f(s(h(x1)))) f#(f(s(s(x1)))) -> f#(f(x1)) -> f#(s(s(s(x1)))) -> f#(s(h(x1))) f#(f(s(s(x1)))) -> f#(f(x1)) -> f#(s(s(s(x1)))) -> h#(x1) f#(f(s(s(x1)))) -> f#(x1) -> f#(f(s(s(x1)))) -> f#(f(x1)) f#(f(s(s(x1)))) -> f#(x1) -> f#(f(s(s(x1)))) -> f#(x1) f#(f(s(s(x1)))) -> f#(x1) -> f#(h(x1)) -> h#(f(s(h(x1)))) f#(f(s(s(x1)))) -> f#(x1) -> f#(h(x1)) -> f#(s(h(x1))) f#(f(s(s(x1)))) -> f#(x1) -> f#(s(s(s(x1)))) -> h#(f(s(h(x1)))) f#(f(s(s(x1)))) -> f#(x1) -> f#(s(s(s(x1)))) -> f#(s(h(x1))) f#(f(s(s(x1)))) -> f#(x1) -> f#(s(s(s(x1)))) -> h#(x1) f#(s(s(s(x1)))) -> f#(s(h(x1))) -> f#(f(s(s(x1)))) -> f#(f(x1)) f#(s(s(s(x1)))) -> f#(s(h(x1))) -> f#(f(s(s(x1)))) -> f#(x1) f#(s(s(s(x1)))) -> f#(s(h(x1))) -> f#(h(x1)) -> h#(f(s(h(x1)))) f#(s(s(s(x1)))) -> f#(s(h(x1))) -> f#(h(x1)) -> f#(s(h(x1))) f#(s(s(s(x1)))) -> f#(s(h(x1))) -> f#(s(s(s(x1)))) -> h#(f(s(h(x1)))) f#(s(s(s(x1)))) -> f#(s(h(x1))) -> f#(s(s(s(x1)))) -> f#(s(h(x1))) f#(s(s(s(x1)))) -> f#(s(h(x1))) -> f#(s(s(s(x1)))) -> h#(x1) f#(h(x1)) -> f#(s(h(x1))) -> f#(f(s(s(x1)))) -> f#(f(x1)) f#(h(x1)) -> f#(s(h(x1))) -> f#(f(s(s(x1)))) -> f#(x1) f#(h(x1)) -> f#(s(h(x1))) -> f#(h(x1)) -> h#(f(s(h(x1)))) f#(h(x1)) -> f#(s(h(x1))) -> f#(h(x1)) -> f#(s(h(x1))) f#(h(x1)) -> f#(s(h(x1))) -> f#(s(s(s(x1)))) -> h#(f(s(h(x1)))) f#(h(x1)) -> f#(s(h(x1))) -> f#(s(s(s(x1)))) -> f#(s(h(x1))) f#(h(x1)) -> f#(s(h(x1))) -> f#(s(s(s(x1)))) -> h#(x1) g#(h(x1)) -> f#(s(x1)) -> f#(f(s(s(x1)))) -> f#(f(x1)) g#(h(x1)) -> f#(s(x1)) -> f#(f(s(s(x1)))) -> f#(x1) g#(h(x1)) -> f#(s(x1)) -> f#(h(x1)) -> h#(f(s(h(x1)))) g#(h(x1)) -> f#(s(x1)) -> f#(h(x1)) -> f#(s(h(x1))) g#(h(x1)) -> f#(s(x1)) -> f#(s(s(s(x1)))) -> h#(f(s(h(x1)))) g#(h(x1)) -> f#(s(x1)) -> f#(s(s(s(x1)))) -> f#(s(h(x1))) g#(h(x1)) -> f#(s(x1)) -> f#(s(s(s(x1)))) -> h#(x1) g#(h(x1)) -> g#(f(s(x1))) -> g#(h(x1)) -> g#(f(s(x1))) g#(h(x1)) -> g#(f(s(x1))) -> g#(h(x1)) -> f#(s(x1)) EDG Processor: DPs: g#(h(x1)) -> f#(s(x1)) g#(h(x1)) -> g#(f(s(x1))) f#(s(s(s(x1)))) -> h#(x1) f#(s(s(s(x1)))) -> f#(s(h(x1))) f#(s(s(s(x1)))) -> h#(f(s(h(x1)))) f#(h(x1)) -> f#(s(h(x1))) f#(h(x1)) -> h#(f(s(h(x1)))) f#(f(s(s(x1)))) -> f#(x1) f#(f(s(s(x1)))) -> f#(f(x1)) b#(a(x1)) -> b#(x1) b#(a(x1)) -> a#(b(x1)) a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(b(x1)) a#(a(a(x1))) -> a#(a(b(x1))) a#(a(a(x1))) -> b#(a(a(b(x1)))) b#(b(b(b(x1)))) -> a#(x1) TRS: g(h(x1)) -> g(f(s(x1))) f(s(s(s(x1)))) -> h(f(s(h(x1)))) f(h(x1)) -> h(f(s(h(x1)))) h(x1) -> x1 f(f(s(s(x1)))) -> s(s(s(f(f(x1))))) b(a(x1)) -> a(b(x1)) a(a(a(x1))) -> b(a(a(b(x1)))) b(b(b(b(x1)))) -> a(x1) graph: a#(a(a(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> a#(b(x1)) a#(a(a(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> a#(a(b(x1))) a#(a(a(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(a(a(b(x1)))) a#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(a(x1))) -> a#(b(x1)) a#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(a(x1))) -> a#(a(b(x1))) a#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(a(x1))) -> b#(a(a(b(x1)))) a#(a(a(x1))) -> b#(a(a(b(x1)))) -> b#(a(x1)) -> b#(x1) a#(a(a(x1))) -> b#(a(a(b(x1)))) -> b#(a(x1)) -> a#(b(x1)) a#(a(a(x1))) -> b#(a(a(b(x1)))) -> b#(b(b(b(x1)))) -> a#(x1) a#(a(a(x1))) -> b#(x1) -> b#(a(x1)) -> b#(x1) a#(a(a(x1))) -> b#(x1) -> b#(a(x1)) -> a#(b(x1)) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(x1) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> b#(x1) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> a#(b(x1)) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> a#(a(b(x1))) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> b#(a(a(b(x1)))) b#(a(x1)) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(x1) b#(a(x1)) -> a#(b(x1)) -> a#(a(a(x1))) -> a#(b(x1)) b#(a(x1)) -> a#(b(x1)) -> a#(a(a(x1))) -> a#(a(b(x1))) b#(a(x1)) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(a(a(b(x1)))) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(b(x1)) b#(a(x1)) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(x1) f#(f(s(s(x1)))) -> f#(f(x1)) -> f#(s(s(s(x1)))) -> h#(x1) f#(f(s(s(x1)))) -> f#(f(x1)) -> f#(s(s(s(x1)))) -> f#(s(h(x1))) f#(f(s(s(x1)))) -> f#(f(x1)) -> f#(s(s(s(x1)))) -> h#(f(s(h(x1)))) f#(f(s(s(x1)))) -> f#(f(x1)) -> f#(h(x1)) -> f#(s(h(x1))) f#(f(s(s(x1)))) -> f#(f(x1)) -> f#(h(x1)) -> h#(f(s(h(x1)))) f#(f(s(s(x1)))) -> f#(f(x1)) -> f#(f(s(s(x1)))) -> f#(x1) f#(f(s(s(x1)))) -> f#(f(x1)) -> f#(f(s(s(x1)))) -> f#(f(x1)) f#(f(s(s(x1)))) -> f#(x1) -> f#(s(s(s(x1)))) -> h#(x1) f#(f(s(s(x1)))) -> f#(x1) -> f#(s(s(s(x1)))) -> f#(s(h(x1))) f#(f(s(s(x1)))) -> f#(x1) -> f#(s(s(s(x1)))) -> h#(f(s(h(x1)))) f#(f(s(s(x1)))) -> f#(x1) -> f#(h(x1)) -> f#(s(h(x1))) f#(f(s(s(x1)))) -> f#(x1) -> f#(h(x1)) -> h#(f(s(h(x1)))) f#(f(s(s(x1)))) -> f#(x1) -> f#(f(s(s(x1)))) -> f#(x1) f#(f(s(s(x1)))) -> f#(x1) -> f#(f(s(s(x1)))) -> f#(f(x1)) f#(s(s(s(x1)))) -> f#(s(h(x1))) -> f#(s(s(s(x1)))) -> h#(x1) f#(s(s(s(x1)))) -> f#(s(h(x1))) -> f#(s(s(s(x1)))) -> f#(s(h(x1))) f#(s(s(s(x1)))) -> f#(s(h(x1))) -> f#(s(s(s(x1)))) -> h#(f(s(h(x1)))) f#(h(x1)) -> f#(s(h(x1))) -> f#(s(s(s(x1)))) -> h#(x1) f#(h(x1)) -> f#(s(h(x1))) -> f#(s(s(s(x1)))) -> f#(s(h(x1))) f#(h(x1)) -> f#(s(h(x1))) -> f#(s(s(s(x1)))) -> h#(f(s(h(x1)))) g#(h(x1)) -> f#(s(x1)) -> f#(s(s(s(x1)))) -> h#(x1) g#(h(x1)) -> f#(s(x1)) -> f#(s(s(s(x1)))) -> f#(s(h(x1))) g#(h(x1)) -> f#(s(x1)) -> f#(s(s(s(x1)))) -> h#(f(s(h(x1)))) g#(h(x1)) -> g#(f(s(x1))) -> g#(h(x1)) -> f#(s(x1)) g#(h(x1)) -> g#(f(s(x1))) -> g#(h(x1)) -> g#(f(s(x1))) CDG Processor: DPs: g#(h(x1)) -> f#(s(x1)) g#(h(x1)) -> g#(f(s(x1))) f#(s(s(s(x1)))) -> h#(x1) f#(s(s(s(x1)))) -> f#(s(h(x1))) f#(s(s(s(x1)))) -> h#(f(s(h(x1)))) f#(h(x1)) -> f#(s(h(x1))) f#(h(x1)) -> h#(f(s(h(x1)))) f#(f(s(s(x1)))) -> f#(x1) f#(f(s(s(x1)))) -> f#(f(x1)) b#(a(x1)) -> b#(x1) b#(a(x1)) -> a#(b(x1)) a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(b(x1)) a#(a(a(x1))) -> a#(a(b(x1))) a#(a(a(x1))) -> b#(a(a(b(x1)))) b#(b(b(b(x1)))) -> a#(x1) TRS: g(h(x1)) -> g(f(s(x1))) f(s(s(s(x1)))) -> h(f(s(h(x1)))) f(h(x1)) -> h(f(s(h(x1)))) h(x1) -> x1 f(f(s(s(x1)))) -> s(s(s(f(f(x1))))) b(a(x1)) -> a(b(x1)) a(a(a(x1))) -> b(a(a(b(x1)))) b(b(b(b(x1)))) -> a(x1) graph: a#(a(a(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(a(a(b(x1)))) a#(a(a(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> a#(a(b(x1))) a#(a(a(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> a#(b(x1)) a#(a(a(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(a(x1))) -> b#(a(a(b(x1)))) a#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(a(x1))) -> a#(a(b(x1))) a#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(a(x1))) -> a#(b(x1)) a#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(a(a(b(x1)))) -> b#(b(b(b(x1)))) -> a#(x1) a#(a(a(x1))) -> b#(a(a(b(x1)))) -> b#(a(x1)) -> a#(b(x1)) a#(a(a(x1))) -> b#(a(a(b(x1)))) -> b#(a(x1)) -> b#(x1) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(x1) a#(a(a(x1))) -> b#(x1) -> b#(a(x1)) -> a#(b(x1)) a#(a(a(x1))) -> b#(x1) -> b#(a(x1)) -> b#(x1) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> b#(a(a(b(x1)))) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> a#(a(b(x1))) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> a#(b(x1)) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> b#(x1) b#(a(x1)) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(a(a(b(x1)))) b#(a(x1)) -> a#(b(x1)) -> a#(a(a(x1))) -> a#(a(b(x1))) b#(a(x1)) -> a#(b(x1)) -> a#(a(a(x1))) -> a#(b(x1)) b#(a(x1)) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(x1) b#(a(x1)) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(x1) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(b(x1)) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1) f#(f(s(s(x1)))) -> f#(f(x1)) -> f#(f(s(s(x1)))) -> f#(f(x1)) f#(f(s(s(x1)))) -> f#(f(x1)) -> f#(f(s(s(x1)))) -> f#(x1) f#(f(s(s(x1)))) -> f#(f(x1)) -> f#(h(x1)) -> h#(f(s(h(x1)))) f#(f(s(s(x1)))) -> f#(f(x1)) -> f#(h(x1)) -> f#(s(h(x1))) f#(f(s(s(x1)))) -> f#(f(x1)) -> f#(s(s(s(x1)))) -> h#(f(s(h(x1)))) f#(f(s(s(x1)))) -> f#(f(x1)) -> f#(s(s(s(x1)))) -> f#(s(h(x1))) f#(f(s(s(x1)))) -> f#(f(x1)) -> f#(s(s(s(x1)))) -> h#(x1) f#(f(s(s(x1)))) -> f#(x1) -> f#(f(s(s(x1)))) -> f#(f(x1)) f#(f(s(s(x1)))) -> f#(x1) -> f#(f(s(s(x1)))) -> f#(x1) f#(f(s(s(x1)))) -> f#(x1) -> f#(h(x1)) -> h#(f(s(h(x1)))) f#(f(s(s(x1)))) -> f#(x1) -> f#(h(x1)) -> f#(s(h(x1))) f#(f(s(s(x1)))) -> f#(x1) -> f#(s(s(s(x1)))) -> h#(f(s(h(x1)))) f#(f(s(s(x1)))) -> f#(x1) -> f#(s(s(s(x1)))) -> f#(s(h(x1))) f#(f(s(s(x1)))) -> f#(x1) -> f#(s(s(s(x1)))) -> h#(x1) g#(h(x1)) -> g#(f(s(x1))) -> g#(h(x1)) -> g#(f(s(x1))) g#(h(x1)) -> g#(f(s(x1))) -> g#(h(x1)) -> f#(s(x1)) SCC Processor: #sccs: 3 #rules: 10 #arcs: 41/256 DPs: g#(h(x1)) -> g#(f(s(x1))) TRS: g(h(x1)) -> g(f(s(x1))) f(s(s(s(x1)))) -> h(f(s(h(x1)))) f(h(x1)) -> h(f(s(h(x1)))) h(x1) -> x1 f(f(s(s(x1)))) -> s(s(s(f(f(x1))))) b(a(x1)) -> a(b(x1)) a(a(a(x1))) -> b(a(a(b(x1)))) b(b(b(b(x1)))) -> a(x1) Arctic Interpretation Processor: dimension: 3 interpretation: [g#](x0) = [0 0 0]x0 + [0], [0 0 -&] [0] [b](x0) = [0 -& 0 ]x0 + [0] [0 -& 0 ] [0], [0 0 0 ] [0 ] [a](x0) = [-& 0 0 ]x0 + [-&] [0 0 0 ] [0 ], [-& 0 -&] [0] [f](x0) = [-& 0 0 ]x0 + [0] [-& 0 -&] [0], [0 0 0 ] [1] [s](x0) = [-& -& -&]x0 + [0] [0 -& -&] [0], [-& 1 1 ] [1] [g](x0) = [1 -& -&]x0 + [0] [1 1 -&] [0], [0 -& 0 ] [0] [h](x0) = [1 0 0 ]x0 + [1] [0 -& 0 ] [0] orientation: g#(h(x1)) = [1 0 0]x1 + [1] >= [0 -& -&]x1 + [0] = g#(f(s(x1))) [2 1 1 ] [2] [1 -& -&] [1] g(h(x1)) = [1 -& 1 ]x1 + [1] >= [-& -& -&]x1 + [1] = g(f(s(x1))) [2 1 1 ] [2] [1 -& -&] [1] [-& -& -&] [0] [-& -& -&] [0] f(s(s(s(x1)))) = [0 0 0 ]x1 + [1] >= [0 -& 0 ]x1 + [1] = h(f(s(h(x1)))) [-& -& -&] [0] [-& -& -&] [0] [1 0 0] [1] [-& -& -&] [0] f(h(x1)) = [1 0 0]x1 + [1] >= [0 -& 0 ]x1 + [1] = h(f(s(h(x1)))) [1 0 0] [1] [-& -& -&] [0] [0 -& 0 ] [0] h(x1) = [1 0 0 ]x1 + [1] >= x1 = x1 [0 -& 0 ] [0] [0 0 0] [1] [-& 0 0 ] [1] f(f(s(s(x1)))) = [0 0 0]x1 + [1] >= [-& -& -&]x1 + [0] = s(s(s(f(f(x1))))) [0 0 0] [1] [-& 0 0 ] [1] [0 0 0] [0] [0 0 0 ] [0] b(a(x1)) = [0 0 0]x1 + [0] >= [0 -& 0 ]x1 + [0] = a(b(x1)) [0 0 0] [0] [0 0 0 ] [0] [0 0 0] [0] [0 0 0] [0] a(a(a(x1))) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = b(a(a(b(x1)))) [0 0 0] [0] [0 0 0] [0] [0 0 0] [0] [0 0 0 ] [0 ] b(b(b(b(x1)))) = [0 0 0]x1 + [0] >= [-& 0 0 ]x1 + [-&] = a(x1) [0 0 0] [0] [0 0 0 ] [0 ] problem: DPs: TRS: g(h(x1)) -> g(f(s(x1))) f(s(s(s(x1)))) -> h(f(s(h(x1)))) f(h(x1)) -> h(f(s(h(x1)))) h(x1) -> x1 f(f(s(s(x1)))) -> s(s(s(f(f(x1))))) b(a(x1)) -> a(b(x1)) a(a(a(x1))) -> b(a(a(b(x1)))) b(b(b(b(x1)))) -> a(x1) Qed DPs: f#(f(s(s(x1)))) -> f#(f(x1)) f#(f(s(s(x1)))) -> f#(x1) TRS: g(h(x1)) -> g(f(s(x1))) f(s(s(s(x1)))) -> h(f(s(h(x1)))) f(h(x1)) -> h(f(s(h(x1)))) h(x1) -> x1 f(f(s(s(x1)))) -> s(s(s(f(f(x1))))) b(a(x1)) -> a(b(x1)) a(a(a(x1))) -> b(a(a(b(x1)))) b(b(b(b(x1)))) -> a(x1) Arctic Interpretation Processor: dimension: 1 interpretation: [f#](x0) = x0, [b](x0) = x0 + 0, [a](x0) = x0 + 0, [f](x0) = 1x0, [s](x0) = x0, [g](x0) = 2, [h](x0) = x0 orientation: f#(f(s(s(x1)))) = 1x1 >= 1x1 = f#(f(x1)) f#(f(s(s(x1)))) = 1x1 >= x1 = f#(x1) g(h(x1)) = 2 >= 2 = g(f(s(x1))) f(s(s(s(x1)))) = 1x1 >= 1x1 = h(f(s(h(x1)))) f(h(x1)) = 1x1 >= 1x1 = h(f(s(h(x1)))) h(x1) = x1 >= x1 = x1 f(f(s(s(x1)))) = 2x1 >= 2x1 = s(s(s(f(f(x1))))) b(a(x1)) = x1 + 0 >= x1 + 0 = a(b(x1)) a(a(a(x1))) = x1 + 0 >= x1 + 0 = b(a(a(b(x1)))) b(b(b(b(x1)))) = x1 + 0 >= x1 + 0 = a(x1) problem: DPs: f#(f(s(s(x1)))) -> f#(f(x1)) TRS: g(h(x1)) -> g(f(s(x1))) f(s(s(s(x1)))) -> h(f(s(h(x1)))) f(h(x1)) -> h(f(s(h(x1)))) h(x1) -> x1 f(f(s(s(x1)))) -> s(s(s(f(f(x1))))) b(a(x1)) -> a(b(x1)) a(a(a(x1))) -> b(a(a(b(x1)))) b(b(b(b(x1)))) -> a(x1) EDG Processor: DPs: f#(f(s(s(x1)))) -> f#(f(x1)) TRS: g(h(x1)) -> g(f(s(x1))) f(s(s(s(x1)))) -> h(f(s(h(x1)))) f(h(x1)) -> h(f(s(h(x1)))) h(x1) -> x1 f(f(s(s(x1)))) -> s(s(s(f(f(x1))))) b(a(x1)) -> a(b(x1)) a(a(a(x1))) -> b(a(a(b(x1)))) b(b(b(b(x1)))) -> a(x1) graph: f#(f(s(s(x1)))) -> f#(f(x1)) -> f#(f(s(s(x1)))) -> f#(f(x1)) CDG Processor: DPs: f#(f(s(s(x1)))) -> f#(f(x1)) TRS: g(h(x1)) -> g(f(s(x1))) f(s(s(s(x1)))) -> h(f(s(h(x1)))) f(h(x1)) -> h(f(s(h(x1)))) h(x1) -> x1 f(f(s(s(x1)))) -> s(s(s(f(f(x1))))) b(a(x1)) -> a(b(x1)) a(a(a(x1))) -> b(a(a(b(x1)))) b(b(b(b(x1)))) -> a(x1) graph: Qed DPs: a#(a(a(x1))) -> a#(b(x1)) a#(a(a(x1))) -> b#(x1) b#(a(x1)) -> b#(x1) b#(a(x1)) -> a#(b(x1)) a#(a(a(x1))) -> a#(a(b(x1))) a#(a(a(x1))) -> b#(a(a(b(x1)))) b#(b(b(b(x1)))) -> a#(x1) TRS: g(h(x1)) -> g(f(s(x1))) f(s(s(s(x1)))) -> h(f(s(h(x1)))) f(h(x1)) -> h(f(s(h(x1)))) h(x1) -> x1 f(f(s(s(x1)))) -> s(s(s(f(f(x1))))) b(a(x1)) -> a(b(x1)) a(a(a(x1))) -> b(a(a(b(x1)))) b(b(b(b(x1)))) -> a(x1) Arctic Interpretation Processor: dimension: 1 interpretation: [a#](x0) = 4x0, [b#](x0) = x0, [b](x0) = 4x0, [a](x0) = 8x0, [f](x0) = 0, [s](x0) = x0 + -8, [g](x0) = 3x0 + 3, [h](x0) = x0 orientation: a#(a(a(x1))) = 20x1 >= 8x1 = a#(b(x1)) a#(a(a(x1))) = 20x1 >= x1 = b#(x1) b#(a(x1)) = 8x1 >= x1 = b#(x1) b#(a(x1)) = 8x1 >= 8x1 = a#(b(x1)) a#(a(a(x1))) = 20x1 >= 16x1 = a#(a(b(x1))) a#(a(a(x1))) = 20x1 >= 20x1 = b#(a(a(b(x1)))) b#(b(b(b(x1)))) = 12x1 >= 4x1 = a#(x1) g(h(x1)) = 3x1 + 3 >= 3 = g(f(s(x1))) f(s(s(s(x1)))) = 0 >= 0 = h(f(s(h(x1)))) f(h(x1)) = 0 >= 0 = h(f(s(h(x1)))) h(x1) = x1 >= x1 = x1 f(f(s(s(x1)))) = 0 >= 0 = s(s(s(f(f(x1))))) b(a(x1)) = 12x1 >= 12x1 = a(b(x1)) a(a(a(x1))) = 24x1 >= 24x1 = b(a(a(b(x1)))) b(b(b(b(x1)))) = 16x1 >= 8x1 = a(x1) problem: DPs: b#(a(x1)) -> a#(b(x1)) a#(a(a(x1))) -> b#(a(a(b(x1)))) TRS: g(h(x1)) -> g(f(s(x1))) f(s(s(s(x1)))) -> h(f(s(h(x1)))) f(h(x1)) -> h(f(s(h(x1)))) h(x1) -> x1 f(f(s(s(x1)))) -> s(s(s(f(f(x1))))) b(a(x1)) -> a(b(x1)) a(a(a(x1))) -> b(a(a(b(x1)))) b(b(b(b(x1)))) -> a(x1) EDG Processor: DPs: b#(a(x1)) -> a#(b(x1)) a#(a(a(x1))) -> b#(a(a(b(x1)))) TRS: g(h(x1)) -> g(f(s(x1))) f(s(s(s(x1)))) -> h(f(s(h(x1)))) f(h(x1)) -> h(f(s(h(x1)))) h(x1) -> x1 f(f(s(s(x1)))) -> s(s(s(f(f(x1))))) b(a(x1)) -> a(b(x1)) a(a(a(x1))) -> b(a(a(b(x1)))) b(b(b(b(x1)))) -> a(x1) graph: a#(a(a(x1))) -> b#(a(a(b(x1)))) -> b#(a(x1)) -> a#(b(x1)) b#(a(x1)) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(a(a(b(x1)))) Arctic Interpretation Processor: dimension: 1 interpretation: [a#](x0) = 8x0, [b#](x0) = 4x0, [b](x0) = 2x0, [a](x0) = 8x0, [f](x0) = 2, [s](x0) = x0 + -12, [g](x0) = 10x0 + 12, [h](x0) = x0 + -11 orientation: b#(a(x1)) = 12x1 >= 10x1 = a#(b(x1)) a#(a(a(x1))) = 24x1 >= 22x1 = b#(a(a(b(x1)))) g(h(x1)) = 10x1 + 12 >= 12 = g(f(s(x1))) f(s(s(s(x1)))) = 2 >= 2 = h(f(s(h(x1)))) f(h(x1)) = 2 >= 2 = h(f(s(h(x1)))) h(x1) = x1 + -11 >= x1 = x1 f(f(s(s(x1)))) = 2 >= 2 = s(s(s(f(f(x1))))) b(a(x1)) = 10x1 >= 10x1 = a(b(x1)) a(a(a(x1))) = 24x1 >= 20x1 = b(a(a(b(x1)))) b(b(b(b(x1)))) = 8x1 >= 8x1 = a(x1) problem: DPs: TRS: g(h(x1)) -> g(f(s(x1))) f(s(s(s(x1)))) -> h(f(s(h(x1)))) f(h(x1)) -> h(f(s(h(x1)))) h(x1) -> x1 f(f(s(s(x1)))) -> s(s(s(f(f(x1))))) b(a(x1)) -> a(b(x1)) a(a(a(x1))) -> b(a(a(b(x1)))) b(b(b(b(x1)))) -> a(x1) Qed