YES Problem: active(f(a(),X,X)) -> mark(f(X,b(),b())) active(b()) -> mark(a()) mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3)) mark(a()) -> active(a()) mark(b()) -> active(b()) f(mark(X1),X2,X3) -> f(X1,X2,X3) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(X1,X2,mark(X3)) -> f(X1,X2,X3) f(active(X1),X2,X3) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) f(X1,X2,active(X3)) -> f(X1,X2,X3) Proof: DP Processor: DPs: active#(f(a(),X,X)) -> f#(X,b(),b()) active#(f(a(),X,X)) -> mark#(f(X,b(),b())) active#(b()) -> mark#(a()) mark#(f(X1,X2,X3)) -> mark#(X2) mark#(f(X1,X2,X3)) -> f#(X1,mark(X2),X3) mark#(f(X1,X2,X3)) -> active#(f(X1,mark(X2),X3)) mark#(a()) -> active#(a()) mark#(b()) -> active#(b()) f#(mark(X1),X2,X3) -> f#(X1,X2,X3) f#(X1,mark(X2),X3) -> f#(X1,X2,X3) f#(X1,X2,mark(X3)) -> f#(X1,X2,X3) f#(active(X1),X2,X3) -> f#(X1,X2,X3) f#(X1,active(X2),X3) -> f#(X1,X2,X3) f#(X1,X2,active(X3)) -> f#(X1,X2,X3) TRS: active(f(a(),X,X)) -> mark(f(X,b(),b())) active(b()) -> mark(a()) mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3)) mark(a()) -> active(a()) mark(b()) -> active(b()) f(mark(X1),X2,X3) -> f(X1,X2,X3) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(X1,X2,mark(X3)) -> f(X1,X2,X3) f(active(X1),X2,X3) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) f(X1,X2,active(X3)) -> f(X1,X2,X3) Matrix Interpretation Processor: dim=3 usable rules: active(f(a(),X,X)) -> mark(f(X,b(),b())) active(b()) -> mark(a()) mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3)) mark(a()) -> active(a()) mark(b()) -> active(b()) f(mark(X1),X2,X3) -> f(X1,X2,X3) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(X1,X2,mark(X3)) -> f(X1,X2,X3) f(active(X1),X2,X3) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) f(X1,X2,active(X3)) -> f(X1,X2,X3) interpretation: [mark#](x0) = [0 2 0]x0, [f#](x0, x1, x2) = [2 0 0]x0 + [0 1 0]x1 + [1 0 0]x2, [active#](x0) = [0 0 2]x0 + [1], [1 0 1] [1] [mark](x0) = [0 2 0]x0 + [1] [0 0 0] [0], [0] [b] = [2] [1], [1 0 1] [1] [active](x0) = [1 1 0]x0 + [1] [0 0 0] [0], [2 0 0] [0 0 0] [3 0 0] [0] [f](x0, x1, x2) = [3 0 0]x0 + [0 1 0]x1 + [3 0 0]x2 + [1] [3 0 0] [0 0 0] [3 0 0] [0], [1] [a] = [1] [0] orientation: active#(f(a(),X,X)) = [6 0 0]X + [7] >= [2 0 0]X + [2] = f#(X,b(),b()) active#(f(a(),X,X)) = [6 0 0]X + [7] >= [6 0 0]X + [6] = mark#(f(X,b(),b())) active#(b()) = 3 >= 2 = mark#(a()) mark#(f(X1,X2,X3)) = [6 0 0]X1 + [0 2 0]X2 + [6 0 0]X3 + [2] >= [0 2 0]X2 = mark#(X2) mark#(f(X1,X2,X3)) = [6 0 0]X1 + [0 2 0]X2 + [6 0 0]X3 + [2] >= [2 0 0]X1 + [0 2 0]X2 + [1 0 0]X3 + [1] = f#(X1,mark(X2),X3) mark#(f(X1,X2,X3)) = [6 0 0]X1 + [0 2 0]X2 + [6 0 0]X3 + [2] >= [6 0 0]X1 + [6 0 0]X3 + [1] = active#(f(X1,mark(X2),X3)) mark#(a()) = 2 >= 1 = active#(a()) mark#(b()) = 4 >= 3 = active#(b()) f#(mark(X1),X2,X3) = [2 0 2]X1 + [0 1 0]X2 + [1 0 0]X3 + [2] >= [2 0 0]X1 + [0 1 0]X2 + [1 0 0]X3 = f#(X1,X2,X3) f#(X1,mark(X2),X3) = [2 0 0]X1 + [0 2 0]X2 + [1 0 0]X3 + [1] >= [2 0 0]X1 + [0 1 0]X2 + [1 0 0]X3 = f#(X1,X2,X3) f#(X1,X2,mark(X3)) = [2 0 0]X1 + [0 1 0]X2 + [1 0 1]X3 + [1] >= [2 0 0]X1 + [0 1 0]X2 + [1 0 0]X3 = f#(X1,X2,X3) f#(active(X1),X2,X3) = [2 0 2]X1 + [0 1 0]X2 + [1 0 0]X3 + [2] >= [2 0 0]X1 + [0 1 0]X2 + [1 0 0]X3 = f#(X1,X2,X3) f#(X1,active(X2),X3) = [2 0 0]X1 + [1 1 0]X2 + [1 0 0]X3 + [1] >= [2 0 0]X1 + [0 1 0]X2 + [1 0 0]X3 = f#(X1,X2,X3) f#(X1,X2,active(X3)) = [2 0 0]X1 + [0 1 0]X2 + [1 0 1]X3 + [1] >= [2 0 0]X1 + [0 1 0]X2 + [1 0 0]X3 = f#(X1,X2,X3) [6 0 0] [6] [5 0 0] [1] active(f(a(),X,X)) = [6 1 0]X + [7] >= [6 0 0]X + [7] = mark(f(X,b(),b())) [0 0 0] [0] [0 0 0] [0] [2] [2] active(b()) = [3] >= [3] = mark(a()) [0] [0] [5 0 0] [0 0 0] [6 0 0] [1] [5 0 0] [0 0 0] [6 0 0] [1] mark(f(X1,X2,X3)) = [6 0 0]X1 + [0 2 0]X2 + [6 0 0]X3 + [3] >= [5 0 0]X1 + [0 2 0]X2 + [6 0 0]X3 + [3] = active(f(X1,mark(X2),X3)) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [2] [2] mark(a()) = [3] >= [3] = active(a()) [0] [0] [2] [2] mark(b()) = [5] >= [3] = active(b()) [0] [0] [2 0 2] [0 0 0] [3 0 0] [2] [2 0 0] [0 0 0] [3 0 0] [0] f(mark(X1),X2,X3) = [3 0 3]X1 + [0 1 0]X2 + [3 0 0]X3 + [4] >= [3 0 0]X1 + [0 1 0]X2 + [3 0 0]X3 + [1] = f(X1,X2,X3) [3 0 3] [0 0 0] [3 0 0] [3] [3 0 0] [0 0 0] [3 0 0] [0] [2 0 0] [0 0 0] [3 0 0] [0] [2 0 0] [0 0 0] [3 0 0] [0] f(X1,mark(X2),X3) = [3 0 0]X1 + [0 2 0]X2 + [3 0 0]X3 + [2] >= [3 0 0]X1 + [0 1 0]X2 + [3 0 0]X3 + [1] = f(X1,X2,X3) [3 0 0] [0 0 0] [3 0 0] [0] [3 0 0] [0 0 0] [3 0 0] [0] [2 0 0] [0 0 0] [3 0 3] [3] [2 0 0] [0 0 0] [3 0 0] [0] f(X1,X2,mark(X3)) = [3 0 0]X1 + [0 1 0]X2 + [3 0 3]X3 + [4] >= [3 0 0]X1 + [0 1 0]X2 + [3 0 0]X3 + [1] = f(X1,X2,X3) [3 0 0] [0 0 0] [3 0 3] [3] [3 0 0] [0 0 0] [3 0 0] [0] [2 0 2] [0 0 0] [3 0 0] [2] [2 0 0] [0 0 0] [3 0 0] [0] f(active(X1),X2,X3) = [3 0 3]X1 + [0 1 0]X2 + [3 0 0]X3 + [4] >= [3 0 0]X1 + [0 1 0]X2 + [3 0 0]X3 + [1] = f(X1,X2,X3) [3 0 3] [0 0 0] [3 0 0] [3] [3 0 0] [0 0 0] [3 0 0] [0] [2 0 0] [0 0 0] [3 0 0] [0] [2 0 0] [0 0 0] [3 0 0] [0] f(X1,active(X2),X3) = [3 0 0]X1 + [1 1 0]X2 + [3 0 0]X3 + [2] >= [3 0 0]X1 + [0 1 0]X2 + [3 0 0]X3 + [1] = f(X1,X2,X3) [3 0 0] [0 0 0] [3 0 0] [0] [3 0 0] [0 0 0] [3 0 0] [0] [2 0 0] [0 0 0] [3 0 3] [3] [2 0 0] [0 0 0] [3 0 0] [0] f(X1,X2,active(X3)) = [3 0 0]X1 + [0 1 0]X2 + [3 0 3]X3 + [4] >= [3 0 0]X1 + [0 1 0]X2 + [3 0 0]X3 + [1] = f(X1,X2,X3) [3 0 0] [0 0 0] [3 0 3] [3] [3 0 0] [0 0 0] [3 0 0] [0] problem: DPs: TRS: active(f(a(),X,X)) -> mark(f(X,b(),b())) active(b()) -> mark(a()) mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3)) mark(a()) -> active(a()) mark(b()) -> active(b()) f(mark(X1),X2,X3) -> f(X1,X2,X3) f(X1,mark(X2),X3) -> f(X1,X2,X3) f(X1,X2,mark(X3)) -> f(X1,X2,X3) f(active(X1),X2,X3) -> f(X1,X2,X3) f(X1,active(X2),X3) -> f(X1,X2,X3) f(X1,X2,active(X3)) -> f(X1,X2,X3) Qed