YES Problem: active(f(f(a()))) -> mark(f(g(f(a())))) active(f(X)) -> f(active(X)) f(mark(X)) -> mark(f(X)) proper(f(X)) -> f(proper(X)) proper(a()) -> ok(a()) proper(g(X)) -> g(proper(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(f(f(a()))) -> g#(f(a())) active#(f(f(a()))) -> f#(g(f(a()))) active#(f(X)) -> active#(X) active#(f(X)) -> f#(active(X)) f#(mark(X)) -> f#(X) proper#(f(X)) -> proper#(X) proper#(f(X)) -> f#(proper(X)) proper#(g(X)) -> proper#(X) proper#(g(X)) -> g#(proper(X)) f#(ok(X)) -> f#(X) g#(ok(X)) -> g#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(f(f(a()))) -> mark(f(g(f(a())))) active(f(X)) -> f(active(X)) f(mark(X)) -> mark(f(X)) proper(f(X)) -> f(proper(X)) proper(a()) -> ok(a()) proper(g(X)) -> g(proper(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(f(f(a()))) -> g#(f(a())) active#(f(f(a()))) -> f#(g(f(a()))) active#(f(X)) -> active#(X) active#(f(X)) -> f#(active(X)) f#(mark(X)) -> f#(X) proper#(f(X)) -> proper#(X) proper#(f(X)) -> f#(proper(X)) proper#(g(X)) -> proper#(X) proper#(g(X)) -> g#(proper(X)) f#(ok(X)) -> f#(X) g#(ok(X)) -> g#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(f(f(a()))) -> mark(f(g(f(a())))) active(f(X)) -> f(active(X)) f(mark(X)) -> mark(f(X)) proper(f(X)) -> f(proper(X)) proper(a()) -> ok(a()) proper(g(X)) -> g(proper(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X)) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X) top#(ok(X)) -> active#(X) -> active#(f(X)) -> f#(active(X)) top#(ok(X)) -> active#(X) -> active#(f(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(f(f(a()))) -> f#(g(f(a()))) top#(ok(X)) -> active#(X) -> active#(f(f(a()))) -> g#(f(a())) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> top#(proper(X)) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(g(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X)) proper#(g(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) proper#(g(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) proper#(g(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(g(X)) -> g#(proper(X)) -> g#(ok(X)) -> g#(X) proper#(f(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X)) proper#(f(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) proper#(f(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) proper#(f(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(f(X)) -> f#(proper(X)) -> f#(ok(X)) -> f#(X) proper#(f(X)) -> f#(proper(X)) -> f#(mark(X)) -> f#(X) f#(ok(X)) -> f#(X) -> f#(ok(X)) -> f#(X) f#(ok(X)) -> f#(X) -> f#(mark(X)) -> f#(X) f#(mark(X)) -> f#(X) -> f#(ok(X)) -> f#(X) f#(mark(X)) -> f#(X) -> f#(mark(X)) -> f#(X) g#(ok(X)) -> g#(X) -> g#(ok(X)) -> g#(X) active#(f(f(a()))) -> f#(g(f(a()))) -> f#(ok(X)) -> f#(X) active#(f(f(a()))) -> f#(g(f(a()))) -> f#(mark(X)) -> f#(X) active#(f(f(a()))) -> g#(f(a())) -> g#(ok(X)) -> g#(X) active#(f(X)) -> f#(active(X)) -> f#(ok(X)) -> f#(X) active#(f(X)) -> f#(active(X)) -> f#(mark(X)) -> f#(X) active#(f(X)) -> active#(X) -> active#(f(X)) -> f#(active(X)) active#(f(X)) -> active#(X) -> active#(f(X)) -> active#(X) active#(f(X)) -> active#(X) -> active#(f(f(a()))) -> f#(g(f(a()))) active#(f(X)) -> active#(X) -> active#(f(f(a()))) -> g#(f(a())) EDG Processor: DPs: active#(f(f(a()))) -> g#(f(a())) active#(f(f(a()))) -> f#(g(f(a()))) active#(f(X)) -> active#(X) active#(f(X)) -> f#(active(X)) f#(mark(X)) -> f#(X) proper#(f(X)) -> proper#(X) proper#(f(X)) -> f#(proper(X)) proper#(g(X)) -> proper#(X) proper#(g(X)) -> g#(proper(X)) f#(ok(X)) -> f#(X) g#(ok(X)) -> g#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(f(f(a()))) -> mark(f(g(f(a())))) active(f(X)) -> f(active(X)) f(mark(X)) -> mark(f(X)) proper(f(X)) -> f(proper(X)) proper(a()) -> ok(a()) proper(g(X)) -> g(proper(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X)) top#(ok(X)) -> active#(X) -> active#(f(f(a()))) -> g#(f(a())) top#(ok(X)) -> active#(X) -> active#(f(f(a()))) -> f#(g(f(a()))) top#(ok(X)) -> active#(X) -> active#(f(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(f(X)) -> f#(active(X)) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> top#(proper(X)) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X)) proper#(g(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(g(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) proper#(g(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) proper#(g(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X)) proper#(g(X)) -> g#(proper(X)) -> g#(ok(X)) -> g#(X) proper#(f(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X) proper#(f(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X)) proper#(f(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X) proper#(f(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X)) proper#(f(X)) -> f#(proper(X)) -> f#(mark(X)) -> f#(X) proper#(f(X)) -> f#(proper(X)) -> f#(ok(X)) -> f#(X) f#(ok(X)) -> f#(X) -> f#(mark(X)) -> f#(X) f#(ok(X)) -> f#(X) -> f#(ok(X)) -> f#(X) f#(mark(X)) -> f#(X) -> f#(mark(X)) -> f#(X) f#(mark(X)) -> f#(X) -> f#(ok(X)) -> f#(X) g#(ok(X)) -> g#(X) -> g#(ok(X)) -> g#(X) active#(f(X)) -> f#(active(X)) -> f#(mark(X)) -> f#(X) active#(f(X)) -> f#(active(X)) -> f#(ok(X)) -> f#(X) active#(f(X)) -> active#(X) -> active#(f(f(a()))) -> g#(f(a())) active#(f(X)) -> active#(X) -> active#(f(f(a()))) -> f#(g(f(a()))) active#(f(X)) -> active#(X) -> active#(f(X)) -> active#(X) active#(f(X)) -> active#(X) -> active#(f(X)) -> f#(active(X)) CDG Processor: DPs: active#(f(f(a()))) -> g#(f(a())) active#(f(f(a()))) -> f#(g(f(a()))) active#(f(X)) -> active#(X) active#(f(X)) -> f#(active(X)) f#(mark(X)) -> f#(X) proper#(f(X)) -> proper#(X) proper#(f(X)) -> f#(proper(X)) proper#(g(X)) -> proper#(X) proper#(g(X)) -> g#(proper(X)) f#(ok(X)) -> f#(X) g#(ok(X)) -> g#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(f(f(a()))) -> mark(f(g(f(a())))) active(f(X)) -> f(active(X)) f(mark(X)) -> mark(f(X)) proper(f(X)) -> f(proper(X)) proper(a()) -> ok(a()) proper(g(X)) -> g(proper(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X)) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> top#(proper(X)) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X) proper#(g(X)) -> g#(proper(X)) -> g#(ok(X)) -> g#(X) proper#(f(X)) -> f#(proper(X)) -> f#(ok(X)) -> f#(X) active#(f(X)) -> f#(active(X)) -> f#(mark(X)) -> f#(X) SCC Processor: #sccs: 1 #rules: 2 #arcs: 11/225 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(f(f(a()))) -> mark(f(g(f(a())))) active(f(X)) -> f(active(X)) f(mark(X)) -> mark(f(X)) proper(f(X)) -> f(proper(X)) proper(a()) -> ok(a()) proper(g(X)) -> g(proper(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [top#](x0) = [1 1 0]x0, [1] [top](x0) = [0] [0], [1 0 0] [1] [ok](x0) = [0 1 0]x0 + [0] [0 0 0] [0], [0 0 1] [1] [proper](x0) = [0 1 0]x0 + [0] [1 0 0] [0], [0 0 1] [1] [mark](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 0] [g](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [active](x0) = [0 1 0]x0 [1 0 0] , [1 1 0] [0] [f](x0) = [0 1 0]x0 + [1] [0 1 1] [0], [0] [a] = [0] [0] orientation: top#(ok(X)) = [1 1 0]X + [1] >= [1 1 0]X = top#(active(X)) top#(mark(X)) = [0 1 1]X + [2] >= [0 1 1]X + [1] = top#(proper(X)) [1] [1] active(f(f(a()))) = [2] >= [2] = mark(f(g(f(a())))) [1] [0] [1 1 0] [0] [1 1 0] [0] active(f(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = f(active(X)) [1 1 0] [0] [1 1 0] [0] [0 1 1] [2] [0 1 1] [1] f(mark(X)) = [0 1 0]X + [2] >= [0 1 0]X + [2] = mark(f(X)) [0 1 0] [1] [0 0 0] [0] [0 1 1] [1] [0 1 1] [1] proper(f(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = f(proper(X)) [1 1 0] [0] [1 1 0] [0] [1] [1] proper(a()) = [0] >= [0] = ok(a()) [0] [0] [0 0 1] [1] [0 0 1] [1] proper(g(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = g(proper(X)) [1 0 0] [0] [1 0 0] [0] [1 1 0] [1] [1 1 0] [1] f(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(f(X)) [0 1 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] g(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(g(X)) [0 0 0] [0] [0 0 0] [0] [1] [1] top(mark(X)) = [0] >= [0] = top(proper(X)) [0] [0] [1] [1] top(ok(X)) = [0] >= [0] = top(active(X)) [0] [0] problem: DPs: TRS: active(f(f(a()))) -> mark(f(g(f(a())))) active(f(X)) -> f(active(X)) f(mark(X)) -> mark(f(X)) proper(f(X)) -> f(proper(X)) proper(a()) -> ok(a()) proper(g(X)) -> g(proper(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed