YES Problem: active(f(f(a()))) -> mark(f(g(f(a())))) mark(f(X)) -> active(f(X)) mark(a()) -> active(a()) mark(g(X)) -> active(g(mark(X))) f(mark(X)) -> f(X) f(active(X)) -> f(X) g(mark(X)) -> g(X) g(active(X)) -> g(X) Proof: DP Processor: DPs: active#(f(f(a()))) -> g#(f(a())) active#(f(f(a()))) -> f#(g(f(a()))) active#(f(f(a()))) -> mark#(f(g(f(a())))) mark#(f(X)) -> active#(f(X)) mark#(a()) -> active#(a()) mark#(g(X)) -> mark#(X) mark#(g(X)) -> g#(mark(X)) mark#(g(X)) -> active#(g(mark(X))) f#(mark(X)) -> f#(X) f#(active(X)) -> f#(X) g#(mark(X)) -> g#(X) g#(active(X)) -> g#(X) TRS: active(f(f(a()))) -> mark(f(g(f(a())))) mark(f(X)) -> active(f(X)) mark(a()) -> active(a()) mark(g(X)) -> active(g(mark(X))) f(mark(X)) -> f(X) f(active(X)) -> f(X) g(mark(X)) -> g(X) g(active(X)) -> g(X) TDG Processor: DPs: active#(f(f(a()))) -> g#(f(a())) active#(f(f(a()))) -> f#(g(f(a()))) active#(f(f(a()))) -> mark#(f(g(f(a())))) mark#(f(X)) -> active#(f(X)) mark#(a()) -> active#(a()) mark#(g(X)) -> mark#(X) mark#(g(X)) -> g#(mark(X)) mark#(g(X)) -> active#(g(mark(X))) f#(mark(X)) -> f#(X) f#(active(X)) -> f#(X) g#(mark(X)) -> g#(X) g#(active(X)) -> g#(X) TRS: active(f(f(a()))) -> mark(f(g(f(a())))) mark(f(X)) -> active(f(X)) mark(a()) -> active(a()) mark(g(X)) -> active(g(mark(X))) f(mark(X)) -> f(X) f(active(X)) -> f(X) g(mark(X)) -> g(X) g(active(X)) -> g(X) graph: mark#(g(X)) -> mark#(X) -> mark#(g(X)) -> active#(g(mark(X))) mark#(g(X)) -> mark#(X) -> mark#(g(X)) -> g#(mark(X)) mark#(g(X)) -> mark#(X) -> mark#(g(X)) -> mark#(X) mark#(g(X)) -> mark#(X) -> mark#(a()) -> active#(a()) mark#(g(X)) -> mark#(X) -> mark#(f(X)) -> active#(f(X)) mark#(g(X)) -> g#(mark(X)) -> g#(active(X)) -> g#(X) mark#(g(X)) -> g#(mark(X)) -> g#(mark(X)) -> g#(X) mark#(g(X)) -> active#(g(mark(X))) -> active#(f(f(a()))) -> mark#(f(g(f(a())))) mark#(g(X)) -> active#(g(mark(X))) -> active#(f(f(a()))) -> f#(g(f(a()))) mark#(g(X)) -> active#(g(mark(X))) -> active#(f(f(a()))) -> g#(f(a())) mark#(f(X)) -> active#(f(X)) -> active#(f(f(a()))) -> mark#(f(g(f(a())))) mark#(f(X)) -> active#(f(X)) -> active#(f(f(a()))) -> f#(g(f(a()))) mark#(f(X)) -> active#(f(X)) -> active#(f(f(a()))) -> g#(f(a())) mark#(a()) -> active#(a()) -> active#(f(f(a()))) -> mark#(f(g(f(a())))) mark#(a()) -> active#(a()) -> active#(f(f(a()))) -> f#(g(f(a()))) mark#(a()) -> active#(a()) -> active#(f(f(a()))) -> g#(f(a())) f#(mark(X)) -> f#(X) -> f#(active(X)) -> f#(X) f#(mark(X)) -> f#(X) -> f#(mark(X)) -> f#(X) f#(active(X)) -> f#(X) -> f#(active(X)) -> f#(X) f#(active(X)) -> f#(X) -> f#(mark(X)) -> f#(X) g#(mark(X)) -> g#(X) -> g#(active(X)) -> g#(X) g#(mark(X)) -> g#(X) -> g#(mark(X)) -> g#(X) g#(active(X)) -> g#(X) -> g#(active(X)) -> g#(X) g#(active(X)) -> g#(X) -> g#(mark(X)) -> g#(X) active#(f(f(a()))) -> mark#(f(g(f(a())))) -> mark#(g(X)) -> active#(g(mark(X))) active#(f(f(a()))) -> mark#(f(g(f(a())))) -> mark#(g(X)) -> g#(mark(X)) active#(f(f(a()))) -> mark#(f(g(f(a())))) -> mark#(g(X)) -> mark#(X) active#(f(f(a()))) -> mark#(f(g(f(a())))) -> mark#(a()) -> active#(a()) active#(f(f(a()))) -> mark#(f(g(f(a())))) -> mark#(f(X)) -> active#(f(X)) active#(f(f(a()))) -> f#(g(f(a()))) -> f#(active(X)) -> f#(X) active#(f(f(a()))) -> f#(g(f(a()))) -> f#(mark(X)) -> f#(X) active#(f(f(a()))) -> g#(f(a())) -> g#(active(X)) -> g#(X) active#(f(f(a()))) -> g#(f(a())) -> g#(mark(X)) -> g#(X) SCC Processor: #sccs: 3 #rules: 9 #arcs: 33/144 DPs: mark#(g(X)) -> mark#(X) mark#(f(X)) -> active#(f(X)) active#(f(f(a()))) -> mark#(f(g(f(a())))) mark#(a()) -> active#(a()) mark#(g(X)) -> active#(g(mark(X))) TRS: active(f(f(a()))) -> mark(f(g(f(a())))) mark(f(X)) -> active(f(X)) mark(a()) -> active(a()) mark(g(X)) -> active(g(mark(X))) f(mark(X)) -> f(X) f(active(X)) -> f(X) g(mark(X)) -> g(X) g(active(X)) -> g(X) EDG Processor: DPs: mark#(g(X)) -> mark#(X) mark#(f(X)) -> active#(f(X)) active#(f(f(a()))) -> mark#(f(g(f(a())))) mark#(a()) -> active#(a()) mark#(g(X)) -> active#(g(mark(X))) TRS: active(f(f(a()))) -> mark(f(g(f(a())))) mark(f(X)) -> active(f(X)) mark(a()) -> active(a()) mark(g(X)) -> active(g(mark(X))) f(mark(X)) -> f(X) f(active(X)) -> f(X) g(mark(X)) -> g(X) g(active(X)) -> g(X) graph: mark#(g(X)) -> mark#(X) -> mark#(f(X)) -> active#(f(X)) mark#(g(X)) -> mark#(X) -> mark#(a()) -> active#(a()) mark#(g(X)) -> mark#(X) -> mark#(g(X)) -> mark#(X) mark#(g(X)) -> mark#(X) -> mark#(g(X)) -> active#(g(mark(X))) mark#(g(X)) -> active#(g(mark(X))) -> active#(f(f(a()))) -> mark#(f(g(f(a())))) mark#(f(X)) -> active#(f(X)) -> active#(f(f(a()))) -> mark#(f(g(f(a())))) active#(f(f(a()))) -> mark#(f(g(f(a())))) -> mark#(f(X)) -> active#(f(X)) CDG Processor: DPs: mark#(g(X)) -> mark#(X) mark#(f(X)) -> active#(f(X)) active#(f(f(a()))) -> mark#(f(g(f(a())))) mark#(a()) -> active#(a()) mark#(g(X)) -> active#(g(mark(X))) TRS: active(f(f(a()))) -> mark(f(g(f(a())))) mark(f(X)) -> active(f(X)) mark(a()) -> active(a()) mark(g(X)) -> active(g(mark(X))) f(mark(X)) -> f(X) f(active(X)) -> f(X) g(mark(X)) -> g(X) g(active(X)) -> g(X) graph: mark#(g(X)) -> mark#(X) -> mark#(f(X)) -> active#(f(X)) active#(f(f(a()))) -> mark#(f(g(f(a())))) -> mark#(f(X)) -> active#(f(X)) SCC Processor: #sccs: 0 #rules: 0 #arcs: 2/25 DPs: f#(mark(X)) -> f#(X) f#(active(X)) -> f#(X) TRS: active(f(f(a()))) -> mark(f(g(f(a())))) mark(f(X)) -> active(f(X)) mark(a()) -> active(a()) mark(g(X)) -> active(g(mark(X))) f(mark(X)) -> f(X) f(active(X)) -> f(X) g(mark(X)) -> g(X) g(active(X)) -> g(X) Subterm Criterion Processor: simple projection: pi(f#) = 0 problem: DPs: TRS: active(f(f(a()))) -> mark(f(g(f(a())))) mark(f(X)) -> active(f(X)) mark(a()) -> active(a()) mark(g(X)) -> active(g(mark(X))) f(mark(X)) -> f(X) f(active(X)) -> f(X) g(mark(X)) -> g(X) g(active(X)) -> g(X) Qed DPs: g#(mark(X)) -> g#(X) g#(active(X)) -> g#(X) TRS: active(f(f(a()))) -> mark(f(g(f(a())))) mark(f(X)) -> active(f(X)) mark(a()) -> active(a()) mark(g(X)) -> active(g(mark(X))) f(mark(X)) -> f(X) f(active(X)) -> f(X) g(mark(X)) -> g(X) g(active(X)) -> g(X) Subterm Criterion Processor: simple projection: pi(g#) = 0 problem: DPs: TRS: active(f(f(a()))) -> mark(f(g(f(a())))) mark(f(X)) -> active(f(X)) mark(a()) -> active(a()) mark(g(X)) -> active(g(mark(X))) f(mark(X)) -> f(X) f(active(X)) -> f(X) g(mark(X)) -> g(X) g(active(X)) -> g(X) Qed