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)) 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#(mark(X)) -> proper#(X) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> top#(proper(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#(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#(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)) -> 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)) SCC Processor: #sccs: 5 #rules: 8 #arcs: 32/225 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: dimension: 1 interpretation: [top#](x0) = x0, [top](x0) = x0 + 1, [ok](x0) = x0, [proper](x0) = x0, [mark](x0) = x0 + 1, [g](x0) = 0, [active](x0) = x0, [f](x0) = x0 + 1, [a] = 0 orientation: top#(ok(X)) = X >= X = top#(active(X)) top#(mark(X)) = X + 1 >= X = top#(proper(X)) active(f(f(a()))) = 2 >= 2 = mark(f(g(f(a())))) active(f(X)) = X + 1 >= X + 1 = f(active(X)) f(mark(X)) = X + 2 >= X + 2 = mark(f(X)) proper(f(X)) = X + 1 >= X + 1 = f(proper(X)) proper(a()) = 0 >= 0 = ok(a()) proper(g(X)) = 0 >= 0 = g(proper(X)) f(ok(X)) = X + 1 >= X + 1 = ok(f(X)) g(ok(X)) = 0 >= 0 = ok(g(X)) top(mark(X)) = X + 2 >= X + 1 = top(proper(X)) top(ok(X)) = X + 1 >= X + 1 = top(active(X)) problem: DPs: 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [top#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = 1, [proper](x0) = x0 + 1, [mark](x0) = 0, [g](x0) = x0, [active](x0) = 0, [f](x0) = x0, [a] = 1 orientation: top#(ok(X)) = 2 >= 1 = top#(active(X)) active(f(f(a()))) = 0 >= 0 = mark(f(g(f(a())))) active(f(X)) = 0 >= 0 = f(active(X)) f(mark(X)) = 0 >= 0 = mark(f(X)) proper(f(X)) = X + 1 >= X + 1 = f(proper(X)) proper(a()) = 2 >= 1 = ok(a()) proper(g(X)) = X + 1 >= X + 1 = g(proper(X)) f(ok(X)) = 1 >= 1 = ok(f(X)) g(ok(X)) = 1 >= 1 = ok(g(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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 DPs: proper#(g(X)) -> proper#(X) proper#(f(X)) -> 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: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [mark](x0) = 0, [g](x0) = x0 + 1, [active](x0) = 1, [f](x0) = x0, [a] = 0 orientation: proper#(g(X)) = X + 2 >= X + 1 = proper#(X) proper#(f(X)) = X + 1 >= X + 1 = proper#(X) active(f(f(a()))) = 1 >= 0 = mark(f(g(f(a())))) active(f(X)) = 1 >= 1 = f(active(X)) f(mark(X)) = 0 >= 0 = mark(f(X)) proper(f(X)) = X >= X = f(proper(X)) proper(a()) = 0 >= 0 = ok(a()) proper(g(X)) = X + 1 >= X + 1 = g(proper(X)) f(ok(X)) = 0 >= 0 = ok(f(X)) g(ok(X)) = 1 >= 0 = ok(g(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(f(X)) -> 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: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0 + 1, [mark](x0) = 0, [g](x0) = 1, [active](x0) = x0, [f](x0) = x0 + 1, [a] = 0 orientation: proper#(f(X)) = X + 2 >= X + 1 = proper#(X) active(f(f(a()))) = 2 >= 0 = mark(f(g(f(a())))) active(f(X)) = X + 1 >= X + 1 = f(active(X)) f(mark(X)) = 1 >= 0 = mark(f(X)) proper(f(X)) = X + 2 >= X + 2 = f(proper(X)) proper(a()) = 1 >= 0 = ok(a()) proper(g(X)) = 2 >= 1 = g(proper(X)) f(ok(X)) = X + 1 >= X + 1 = ok(f(X)) g(ok(X)) = 1 >= 1 = ok(g(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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 DPs: g#(ok(X)) -> g#(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: dimension: 1 interpretation: [g#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [mark](x0) = 0, [g](x0) = x0, [active](x0) = 0, [f](x0) = x0, [a] = 0 orientation: g#(ok(X)) = X + 2 >= X + 1 = g#(X) active(f(f(a()))) = 0 >= 0 = mark(f(g(f(a())))) active(f(X)) = 0 >= 0 = f(active(X)) f(mark(X)) = 0 >= 0 = mark(f(X)) proper(f(X)) = 1 >= 1 = f(proper(X)) proper(a()) = 1 >= 1 = ok(a()) proper(g(X)) = 1 >= 1 = g(proper(X)) f(ok(X)) = X + 1 >= X + 1 = ok(f(X)) g(ok(X)) = X + 1 >= X + 1 = ok(g(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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 DPs: active#(f(X)) -> 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0 + 1, [mark](x0) = 0, [g](x0) = 1, [active](x0) = x0, [f](x0) = x0 + 1, [a] = 0 orientation: active#(f(X)) = X + 2 >= X + 1 = active#(X) active(f(f(a()))) = 2 >= 0 = mark(f(g(f(a())))) active(f(X)) = X + 1 >= X + 1 = f(active(X)) f(mark(X)) = 1 >= 0 = mark(f(X)) proper(f(X)) = X + 2 >= X + 2 = f(proper(X)) proper(a()) = 1 >= 0 = ok(a()) proper(g(X)) = 2 >= 1 = g(proper(X)) f(ok(X)) = X + 1 >= X + 1 = ok(f(X)) g(ok(X)) = 1 >= 1 = ok(g(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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 DPs: f#(mark(X)) -> f#(X) f#(ok(X)) -> f#(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: dimension: 1 interpretation: [f#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [mark](x0) = x0 + 1, [g](x0) = x0, [active](x0) = 1, [f](x0) = x0, [a] = 0 orientation: f#(mark(X)) = X + 2 >= X + 1 = f#(X) f#(ok(X)) = X + 2 >= X + 1 = f#(X) active(f(f(a()))) = 1 >= 1 = mark(f(g(f(a())))) active(f(X)) = 1 >= 1 = f(active(X)) f(mark(X)) = X + 1 >= X + 1 = mark(f(X)) proper(f(X)) = 1 >= 1 = f(proper(X)) proper(a()) = 1 >= 1 = ok(a()) proper(g(X)) = 1 >= 1 = g(proper(X)) f(ok(X)) = X + 1 >= X + 1 = ok(f(X)) g(ok(X)) = X + 1 >= X + 1 = ok(g(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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