YES Problem: active(f(a(),X,X)) -> mark(f(X,b(),b())) active(b()) -> mark(a()) active(f(X1,X2,X3)) -> f(X1,active(X2),X3) f(X1,mark(X2),X3) -> mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(f(a(),X,X)) -> f#(X,b(),b()) active#(f(X1,X2,X3)) -> active#(X2) active#(f(X1,X2,X3)) -> f#(X1,active(X2),X3) f#(X1,mark(X2),X3) -> f#(X1,X2,X3) proper#(f(X1,X2,X3)) -> proper#(X3) proper#(f(X1,X2,X3)) -> proper#(X2) proper#(f(X1,X2,X3)) -> proper#(X1) proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) 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(a(),X,X)) -> mark(f(X,b(),b())) active(b()) -> mark(a()) active(f(X1,X2,X3)) -> f(X1,active(X2),X3) f(X1,mark(X2),X3) -> mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(f(a(),X,X)) -> f#(X,b(),b()) active#(f(X1,X2,X3)) -> active#(X2) active#(f(X1,X2,X3)) -> f#(X1,active(X2),X3) f#(X1,mark(X2),X3) -> f#(X1,X2,X3) proper#(f(X1,X2,X3)) -> proper#(X3) proper#(f(X1,X2,X3)) -> proper#(X2) proper#(f(X1,X2,X3)) -> proper#(X1) proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) 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(a(),X,X)) -> mark(f(X,b(),b())) active(b()) -> mark(a()) active(f(X1,X2,X3)) -> f(X1,active(X2),X3) f(X1,mark(X2),X3) -> mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) 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(X1,X2,X3)) -> f#(X1,active(X2),X3) top#(ok(X)) -> active#(X) -> active#(f(X1,X2,X3)) -> active#(X2) top#(ok(X)) -> active#(X) -> active#(f(a(),X,X)) -> f#(X,b(),b()) 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#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2,X3)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2,X3)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2,X3)) -> proper#(X3) proper#(f(X1,X2,X3)) -> proper#(X3) -> proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) proper#(f(X1,X2,X3)) -> proper#(X3) -> proper#(f(X1,X2,X3)) -> proper#(X1) proper#(f(X1,X2,X3)) -> proper#(X3) -> proper#(f(X1,X2,X3)) -> proper#(X2) proper#(f(X1,X2,X3)) -> proper#(X3) -> proper#(f(X1,X2,X3)) -> proper#(X3) proper#(f(X1,X2,X3)) -> proper#(X2) -> proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) proper#(f(X1,X2,X3)) -> proper#(X2) -> proper#(f(X1,X2,X3)) -> proper#(X1) proper#(f(X1,X2,X3)) -> proper#(X2) -> proper#(f(X1,X2,X3)) -> proper#(X2) proper#(f(X1,X2,X3)) -> proper#(X2) -> proper#(f(X1,X2,X3)) -> proper#(X3) proper#(f(X1,X2,X3)) -> proper#(X1) -> proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) proper#(f(X1,X2,X3)) -> proper#(X1) -> proper#(f(X1,X2,X3)) -> proper#(X1) proper#(f(X1,X2,X3)) -> proper#(X1) -> proper#(f(X1,X2,X3)) -> proper#(X2) proper#(f(X1,X2,X3)) -> proper#(X1) -> proper#(f(X1,X2,X3)) -> proper#(X3) proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) -> f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) -> f#(X1,mark(X2),X3) -> f#(X1,X2,X3) f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) -> f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) -> f#(X1,mark(X2),X3) -> f#(X1,X2,X3) f#(X1,mark(X2),X3) -> f#(X1,X2,X3) -> f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) f#(X1,mark(X2),X3) -> f#(X1,X2,X3) -> f#(X1,mark(X2),X3) -> f#(X1,X2,X3) active#(f(a(),X,X)) -> f#(X,b(),b()) -> f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) active#(f(a(),X,X)) -> f#(X,b(),b()) -> f#(X1,mark(X2),X3) -> f#(X1,X2,X3) active#(f(X1,X2,X3)) -> f#(X1,active(X2),X3) -> f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) active#(f(X1,X2,X3)) -> f#(X1,active(X2),X3) -> f#(X1,mark(X2),X3) -> f#(X1,X2,X3) active#(f(X1,X2,X3)) -> active#(X2) -> active#(f(X1,X2,X3)) -> f#(X1,active(X2),X3) active#(f(X1,X2,X3)) -> active#(X2) -> active#(f(X1,X2,X3)) -> active#(X2) active#(f(X1,X2,X3)) -> active#(X2) -> active#(f(a(),X,X)) -> f#(X,b(),b()) EDG Processor: DPs: active#(f(a(),X,X)) -> f#(X,b(),b()) active#(f(X1,X2,X3)) -> active#(X2) active#(f(X1,X2,X3)) -> f#(X1,active(X2),X3) f#(X1,mark(X2),X3) -> f#(X1,X2,X3) proper#(f(X1,X2,X3)) -> proper#(X3) proper#(f(X1,X2,X3)) -> proper#(X2) proper#(f(X1,X2,X3)) -> proper#(X1) proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) 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(a(),X,X)) -> mark(f(X,b(),b())) active(b()) -> mark(a()) active(f(X1,X2,X3)) -> f(X1,active(X2),X3) f(X1,mark(X2),X3) -> mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) 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(a(),X,X)) -> f#(X,b(),b()) top#(ok(X)) -> active#(X) -> active#(f(X1,X2,X3)) -> active#(X2) top#(ok(X)) -> active#(X) -> active#(f(X1,X2,X3)) -> f#(X1,active(X2),X3) 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(X1,X2,X3)) -> proper#(X3) top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2,X3)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2,X3)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) proper#(f(X1,X2,X3)) -> proper#(X3) -> proper#(f(X1,X2,X3)) -> proper#(X3) proper#(f(X1,X2,X3)) -> proper#(X3) -> proper#(f(X1,X2,X3)) -> proper#(X2) proper#(f(X1,X2,X3)) -> proper#(X3) -> proper#(f(X1,X2,X3)) -> proper#(X1) proper#(f(X1,X2,X3)) -> proper#(X3) -> proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) proper#(f(X1,X2,X3)) -> proper#(X2) -> proper#(f(X1,X2,X3)) -> proper#(X3) proper#(f(X1,X2,X3)) -> proper#(X2) -> proper#(f(X1,X2,X3)) -> proper#(X2) proper#(f(X1,X2,X3)) -> proper#(X2) -> proper#(f(X1,X2,X3)) -> proper#(X1) proper#(f(X1,X2,X3)) -> proper#(X2) -> proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) proper#(f(X1,X2,X3)) -> proper#(X1) -> proper#(f(X1,X2,X3)) -> proper#(X3) proper#(f(X1,X2,X3)) -> proper#(X1) -> proper#(f(X1,X2,X3)) -> proper#(X2) proper#(f(X1,X2,X3)) -> proper#(X1) -> proper#(f(X1,X2,X3)) -> proper#(X1) proper#(f(X1,X2,X3)) -> proper#(X1) -> proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) -> f#(X1,mark(X2),X3) -> f#(X1,X2,X3) proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) -> f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) -> f#(X1,mark(X2),X3) -> f#(X1,X2,X3) f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) -> f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) f#(X1,mark(X2),X3) -> f#(X1,X2,X3) -> f#(X1,mark(X2),X3) -> f#(X1,X2,X3) f#(X1,mark(X2),X3) -> f#(X1,X2,X3) -> f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) active#(f(X1,X2,X3)) -> f#(X1,active(X2),X3) -> f#(X1,mark(X2),X3) -> f#(X1,X2,X3) active#(f(X1,X2,X3)) -> f#(X1,active(X2),X3) -> f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) active#(f(X1,X2,X3)) -> active#(X2) -> active#(f(a(),X,X)) -> f#(X,b(),b()) active#(f(X1,X2,X3)) -> active#(X2) -> active#(f(X1,X2,X3)) -> active#(X2) active#(f(X1,X2,X3)) -> active#(X2) -> active#(f(X1,X2,X3)) -> f#(X1,active(X2),X3) CDG Processor: DPs: active#(f(a(),X,X)) -> f#(X,b(),b()) active#(f(X1,X2,X3)) -> active#(X2) active#(f(X1,X2,X3)) -> f#(X1,active(X2),X3) f#(X1,mark(X2),X3) -> f#(X1,X2,X3) proper#(f(X1,X2,X3)) -> proper#(X3) proper#(f(X1,X2,X3)) -> proper#(X2) proper#(f(X1,X2,X3)) -> proper#(X1) proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) 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(a(),X,X)) -> mark(f(X,b(),b())) active(b()) -> mark(a()) active(f(X1,X2,X3)) -> f(X1,active(X2),X3) f(X1,mark(X2),X3) -> mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) 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#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) -> f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) active#(f(X1,X2,X3)) -> f#(X1,active(X2),X3) -> f#(X1,mark(X2),X3) -> f#(X1,X2,X3) SCC Processor: #sccs: 1 #rules: 2 #arcs: 10/169 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(f(a(),X,X)) -> mark(f(X,b(),b())) active(b()) -> mark(a()) active(f(X1,X2,X3)) -> f(X1,active(X2),X3) f(X1,mark(X2),X3) -> mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=2 interpretation: [top#](x0) = [2 0]x0 + [2], [1] [top](x0) = [3], [1] [ok](x0) = x0 + [0], [1] [proper](x0) = x0 + [0], [1 0] [2] [mark](x0) = [0 0]x0 + [0], [2] [b] = [0], [1 0] [active](x0) = [0 0]x0, [0 2] [1 0] [0 2] [f](x0, x1, x2) = [0 0]x0 + [0 0]x1 + [0 0]x2, [0] [a] = [2] orientation: top#(ok(X)) = [2 0]X + [4] >= [2 0]X + [2] = top#(active(X)) top#(mark(X)) = [2 0]X + [6] >= [2 0]X + [4] = top#(proper(X)) [1 2] [4] [0 2] [4] active(f(a(),X,X)) = [0 0]X + [0] >= [0 0]X + [0] = mark(f(X,b(),b())) [2] [2] active(b()) = [0] >= [0] = mark(a()) [0 2] [1 0] [0 2] [0 2] [1 0] [0 2] active(f(X1,X2,X3)) = [0 0]X1 + [0 0]X2 + [0 0]X3 >= [0 0]X1 + [0 0]X2 + [0 0]X3 = f(X1,active(X2),X3) [0 2] [1 0] [0 2] [2] [0 2] [1 0] [0 2] [2] f(X1,mark(X2),X3) = [0 0]X1 + [0 0]X2 + [0 0]X3 + [0] >= [0 0]X1 + [0 0]X2 + [0 0]X3 + [0] = mark(f(X1,X2,X3)) [0 2] [1 0] [0 2] [1] [0 2] [1 0] [0 2] [1] proper(f(X1,X2,X3)) = [0 0]X1 + [0 0]X2 + [0 0]X3 + [0] >= [0 0]X1 + [0 0]X2 + [0 0]X3 + [0] = f(proper(X1),proper(X2),proper(X3)) [1] [1] proper(a()) = [2] >= [2] = ok(a()) [3] [3] proper(b()) = [0] >= [0] = ok(b()) [0 2] [1 0] [0 2] [1] [0 2] [1 0] [0 2] [1] f(ok(X1),ok(X2),ok(X3)) = [0 0]X1 + [0 0]X2 + [0 0]X3 + [0] >= [0 0]X1 + [0 0]X2 + [0 0]X3 + [0] = ok(f(X1,X2,X3)) [1] [1] top(mark(X)) = [3] >= [3] = top(proper(X)) [1] [1] top(ok(X)) = [3] >= [3] = top(active(X)) problem: DPs: TRS: active(f(a(),X,X)) -> mark(f(X,b(),b())) active(b()) -> mark(a()) active(f(X1,X2,X3)) -> f(X1,active(X2),X3) f(X1,mark(X2),X3) -> mark(f(X1,X2,X3)) proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed