YES Problem: a(x1) -> x1 a(a(b(a(x1)))) -> b(b(a(a(a(x1))))) b(x1) -> a(x1) Proof: DP Processor: DPs: a#(a(b(a(x1)))) -> a#(a(x1)) a#(a(b(a(x1)))) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(a(a(a(x1)))) a#(a(b(a(x1)))) -> b#(b(a(a(a(x1))))) b#(x1) -> a#(x1) TRS: a(x1) -> x1 a(a(b(a(x1)))) -> b(b(a(a(a(x1))))) b(x1) -> a(x1) TDG Processor: DPs: a#(a(b(a(x1)))) -> a#(a(x1)) a#(a(b(a(x1)))) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(a(a(a(x1)))) a#(a(b(a(x1)))) -> b#(b(a(a(a(x1))))) b#(x1) -> a#(x1) TRS: a(x1) -> x1 a(a(b(a(x1)))) -> b(b(a(a(a(x1))))) b(x1) -> a(x1) graph: b#(x1) -> a#(x1) -> a#(a(b(a(x1)))) -> b#(b(a(a(a(x1))))) b#(x1) -> a#(x1) -> a#(a(b(a(x1)))) -> b#(a(a(a(x1)))) b#(x1) -> a#(x1) -> a#(a(b(a(x1)))) -> a#(a(a(x1))) b#(x1) -> a#(x1) -> a#(a(b(a(x1)))) -> a#(a(x1)) a#(a(b(a(x1)))) -> b#(b(a(a(a(x1))))) -> b#(x1) -> a#(x1) a#(a(b(a(x1)))) -> b#(a(a(a(x1)))) -> b#(x1) -> a#(x1) a#(a(b(a(x1)))) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> b#(b(a(a(a(x1))))) a#(a(b(a(x1)))) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> b#(a(a(a(x1)))) a#(a(b(a(x1)))) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> a#(a(x1)) a#(a(b(a(x1)))) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> b#(b(a(a(a(x1))))) a#(a(b(a(x1)))) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> b#(a(a(a(x1)))) a#(a(b(a(x1)))) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> a#(a(x1)) Arctic Interpretation Processor: dimension: 2 interpretation: [b#](x0) = [2 -&]x0 + [3], [a#](x0) = [2 -&]x0 + [2], [0 2] [0] [b](x0) = [0 0]x0 + [1], [0 -&] [-&] [a](x0) = [0 0 ]x0 + [1 ] orientation: a#(a(b(a(x1)))) = [4 4]x1 + [5] >= [2 -&]x1 + [2] = a#(a(x1)) a#(a(b(a(x1)))) = [4 4]x1 + [5] >= [2 -&]x1 + [2] = a#(a(a(x1))) a#(a(b(a(x1)))) = [4 4]x1 + [5] >= [2 -&]x1 + [3] = b#(a(a(a(x1)))) a#(a(b(a(x1)))) = [4 4]x1 + [5] >= [4 4]x1 + [5] = b#(b(a(a(a(x1))))) b#(x1) = [2 -&]x1 + [3] >= [2 -&]x1 + [2] = a#(x1) [0 -&] [-&] a(x1) = [0 0 ]x1 + [1 ] >= x1 = x1 [2 2] [3] [2 2] [3] a(a(b(a(x1)))) = [2 2]x1 + [3] >= [2 2]x1 + [3] = b(b(a(a(a(x1))))) [0 2] [0] [0 -&] [-&] b(x1) = [0 0]x1 + [1] >= [0 0 ]x1 + [1 ] = a(x1) problem: DPs: a#(a(b(a(x1)))) -> b#(b(a(a(a(x1))))) b#(x1) -> a#(x1) TRS: a(x1) -> x1 a(a(b(a(x1)))) -> b(b(a(a(a(x1))))) b(x1) -> a(x1) EDG Processor: DPs: a#(a(b(a(x1)))) -> b#(b(a(a(a(x1))))) b#(x1) -> a#(x1) TRS: a(x1) -> x1 a(a(b(a(x1)))) -> b(b(a(a(a(x1))))) b(x1) -> a(x1) graph: b#(x1) -> a#(x1) -> a#(a(b(a(x1)))) -> b#(b(a(a(a(x1))))) a#(a(b(a(x1)))) -> b#(b(a(a(a(x1))))) -> b#(x1) -> a#(x1) Arctic Interpretation Processor: dimension: 2 interpretation: [b#](x0) = [1 2]x0, [a#](x0) = [0 1]x0, [0 2] [b](x0) = [0 0]x0, [0 0] [a](x0) = [0 0]x0 orientation: a#(a(b(a(x1)))) = [3 3]x1 >= [3 3]x1 = b#(b(a(a(a(x1))))) b#(x1) = [1 2]x1 >= [0 1]x1 = a#(x1) [0 0] a(x1) = [0 0]x1 >= x1 = x1 [2 2] [2 2] a(a(b(a(x1)))) = [2 2]x1 >= [2 2]x1 = b(b(a(a(a(x1))))) [0 2] [0 0] b(x1) = [0 0]x1 >= [0 0]x1 = a(x1) problem: DPs: a#(a(b(a(x1)))) -> b#(b(a(a(a(x1))))) TRS: a(x1) -> x1 a(a(b(a(x1)))) -> b(b(a(a(a(x1))))) b(x1) -> a(x1) EDG Processor: DPs: a#(a(b(a(x1)))) -> b#(b(a(a(a(x1))))) TRS: a(x1) -> x1 a(a(b(a(x1)))) -> b(b(a(a(a(x1))))) b(x1) -> a(x1) graph: Qed