YES Problem: a(b(a(x1))) -> a(a(b(b(a(a(x1)))))) b(a(a(b(x1)))) -> b(a(b(x1))) Proof: DP Processor: DPs: a#(b(a(x1))) -> a#(a(x1)) a#(b(a(x1))) -> b#(a(a(x1))) a#(b(a(x1))) -> b#(b(a(a(x1)))) a#(b(a(x1))) -> a#(b(b(a(a(x1))))) a#(b(a(x1))) -> a#(a(b(b(a(a(x1)))))) b#(a(a(b(x1)))) -> b#(a(b(x1))) TRS: a(b(a(x1))) -> a(a(b(b(a(a(x1)))))) b(a(a(b(x1)))) -> b(a(b(x1))) Matrix Interpretation Processor: dim=3 interpretation: [b#](x0) = [0 0 1]x0, [a#](x0) = [0 1 1]x0 + [1], [0 0 0] [b](x0) = [1 0 0]x0 [1 0 0] , [0 1 1] [1] [a](x0) = [0 0 0]x0 + [1] [0 1 1] [0] orientation: a#(b(a(x1))) = [0 2 2]x1 + [3] >= [0 1 1]x1 + [2] = a#(a(x1)) a#(b(a(x1))) = [0 2 2]x1 + [3] >= [0 1 1]x1 + [1] = b#(a(a(x1))) a#(b(a(x1))) = [0 2 2]x1 + [3] >= [0 1 1]x1 + [2] = b#(b(a(a(x1)))) a#(b(a(x1))) = [0 2 2]x1 + [3] >= [1] = a#(b(b(a(a(x1))))) a#(b(a(x1))) = [0 2 2]x1 + [3] >= [2] = a#(a(b(b(a(a(x1)))))) b#(a(a(b(x1)))) = [2 0 0]x1 + [1] >= [2 0 0]x1 = b#(a(b(x1))) [0 2 2] [3] [2] a(b(a(x1))) = [0 0 0]x1 + [1] >= [1] = a(a(b(b(a(a(x1)))))) [0 2 2] [2] [1] [0 0 0] [0] [0 0 0] [0] b(a(a(b(x1)))) = [2 0 0]x1 + [2] >= [2 0 0]x1 + [1] = b(a(b(x1))) [2 0 0] [2] [2 0 0] [1] problem: DPs: TRS: a(b(a(x1))) -> a(a(b(b(a(a(x1)))))) b(a(a(b(x1)))) -> b(a(b(x1))) Qed