YES Problem: a(a(x1)) -> b(b(b(x1))) b(x1) -> c(c(d(x1))) c(x1) -> d(d(d(x1))) b(c(x1)) -> c(b(x1)) b(c(d(x1))) -> a(x1) Proof: DP Processor: DPs: a#(a(x1)) -> b#(x1) a#(a(x1)) -> b#(b(x1)) a#(a(x1)) -> b#(b(b(x1))) b#(x1) -> c#(d(x1)) b#(x1) -> c#(c(d(x1))) b#(c(x1)) -> b#(x1) b#(c(x1)) -> c#(b(x1)) b#(c(d(x1))) -> a#(x1) TRS: a(a(x1)) -> b(b(b(x1))) b(x1) -> c(c(d(x1))) c(x1) -> d(d(d(x1))) b(c(x1)) -> c(b(x1)) b(c(d(x1))) -> a(x1) Matrix Interpretation Processor: dim=1 interpretation: [c#](x0) = 0, [b#](x0) = 2x0 + 5, [a#](x0) = 2x0 + 16, [c](x0) = x0 + 5, [d](x0) = x0 + 1, [b](x0) = x0 + 11, [a](x0) = x0 + 17 orientation: a#(a(x1)) = 2x1 + 50 >= 2x1 + 5 = b#(x1) a#(a(x1)) = 2x1 + 50 >= 2x1 + 27 = b#(b(x1)) a#(a(x1)) = 2x1 + 50 >= 2x1 + 49 = b#(b(b(x1))) b#(x1) = 2x1 + 5 >= 0 = c#(d(x1)) b#(x1) = 2x1 + 5 >= 0 = c#(c(d(x1))) b#(c(x1)) = 2x1 + 15 >= 2x1 + 5 = b#(x1) b#(c(x1)) = 2x1 + 15 >= 0 = c#(b(x1)) b#(c(d(x1))) = 2x1 + 17 >= 2x1 + 16 = a#(x1) a(a(x1)) = x1 + 34 >= x1 + 33 = b(b(b(x1))) b(x1) = x1 + 11 >= x1 + 11 = c(c(d(x1))) c(x1) = x1 + 5 >= x1 + 3 = d(d(d(x1))) b(c(x1)) = x1 + 16 >= x1 + 16 = c(b(x1)) b(c(d(x1))) = x1 + 17 >= x1 + 17 = a(x1) problem: DPs: TRS: a(a(x1)) -> b(b(b(x1))) b(x1) -> c(c(d(x1))) c(x1) -> d(d(d(x1))) b(c(x1)) -> c(b(x1)) b(c(d(x1))) -> a(x1) Qed