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