YES Problem: a(b(a(x1))) -> b(b(b(a(x1)))) b(b(b(a(x1)))) -> a(a(a(b(x1)))) Proof: DP Processor: DPs: a#(b(a(x1))) -> b#(b(a(x1))) a#(b(a(x1))) -> b#(b(b(a(x1)))) b#(b(b(a(x1)))) -> b#(x1) b#(b(b(a(x1)))) -> a#(b(x1)) b#(b(b(a(x1)))) -> a#(a(b(x1))) b#(b(b(a(x1)))) -> a#(a(a(b(x1)))) TRS: a(b(a(x1))) -> b(b(b(a(x1)))) b(b(b(a(x1)))) -> a(a(a(b(x1)))) TDG Processor: DPs: a#(b(a(x1))) -> b#(b(a(x1))) a#(b(a(x1))) -> b#(b(b(a(x1)))) b#(b(b(a(x1)))) -> b#(x1) b#(b(b(a(x1)))) -> a#(b(x1)) b#(b(b(a(x1)))) -> a#(a(b(x1))) b#(b(b(a(x1)))) -> a#(a(a(b(x1)))) TRS: a(b(a(x1))) -> b(b(b(a(x1)))) b(b(b(a(x1)))) -> a(a(a(b(x1)))) graph: b#(b(b(a(x1)))) -> b#(x1) -> b#(b(b(a(x1)))) -> a#(a(a(b(x1)))) b#(b(b(a(x1)))) -> b#(x1) -> b#(b(b(a(x1)))) -> a#(a(b(x1))) b#(b(b(a(x1)))) -> b#(x1) -> b#(b(b(a(x1)))) -> a#(b(x1)) b#(b(b(a(x1)))) -> b#(x1) -> b#(b(b(a(x1)))) -> b#(x1) b#(b(b(a(x1)))) -> a#(b(x1)) -> a#(b(a(x1))) -> b#(b(b(a(x1)))) b#(b(b(a(x1)))) -> a#(b(x1)) -> a#(b(a(x1))) -> b#(b(a(x1))) b#(b(b(a(x1)))) -> a#(a(b(x1))) -> a#(b(a(x1))) -> b#(b(b(a(x1)))) b#(b(b(a(x1)))) -> a#(a(b(x1))) -> a#(b(a(x1))) -> b#(b(a(x1))) b#(b(b(a(x1)))) -> a#(a(a(b(x1)))) -> a#(b(a(x1))) -> b#(b(b(a(x1)))) b#(b(b(a(x1)))) -> a#(a(a(b(x1)))) -> a#(b(a(x1))) -> b#(b(a(x1))) a#(b(a(x1))) -> b#(b(b(a(x1)))) -> b#(b(b(a(x1)))) -> a#(a(a(b(x1)))) a#(b(a(x1))) -> b#(b(b(a(x1)))) -> b#(b(b(a(x1)))) -> a#(a(b(x1))) a#(b(a(x1))) -> b#(b(b(a(x1)))) -> b#(b(b(a(x1)))) -> a#(b(x1)) a#(b(a(x1))) -> b#(b(b(a(x1)))) -> b#(b(b(a(x1)))) -> b#(x1) a#(b(a(x1))) -> b#(b(a(x1))) -> b#(b(b(a(x1)))) -> a#(a(a(b(x1)))) a#(b(a(x1))) -> b#(b(a(x1))) -> b#(b(b(a(x1)))) -> a#(a(b(x1))) a#(b(a(x1))) -> b#(b(a(x1))) -> b#(b(b(a(x1)))) -> a#(b(x1)) a#(b(a(x1))) -> b#(b(a(x1))) -> b#(b(b(a(x1)))) -> b#(x1) Arctic Interpretation Processor: dimension: 4 interpretation: [b#](x0) = [0 0 0 0]x0 + [0], [a#](x0) = [0 0 0 0]x0 + [0], [0 0 -& 1 ] [0 ] [0 -& 0 1 ] [-&] [b](x0) = [1 0 -& 1 ]x0 + [1 ] [-& -& -& 0 ] [-&], [0 0 0 0 ] [-&] [0 0 0 0 ] [0 ] [a](x0) = [0 0 0 0 ]x0 + [-&] [-& -& -& -&] [-&] orientation: a#(b(a(x1))) = [1 1 1 1]x1 + [1] >= [1 1 1 1]x1 + [1] = b#(b(a(x1))) a#(b(a(x1))) = [1 1 1 1]x1 + [1] >= [1 1 1 1]x1 + [1] = b#(b(b(a(x1)))) b#(b(b(a(x1)))) = [1 1 1 1]x1 + [1] >= [0 0 0 0]x1 + [0] = b#(x1) b#(b(b(a(x1)))) = [1 1 1 1]x1 + [1] >= [1 0 0 1]x1 + [1] = a#(b(x1)) b#(b(b(a(x1)))) = [1 1 1 1]x1 + [1] >= [1 0 0 1]x1 + [1] = a#(a(b(x1))) b#(b(b(a(x1)))) = [1 1 1 1]x1 + [1] >= [1 0 0 1]x1 + [1] = a#(a(a(b(x1)))) [1 1 1 1 ] [1 ] [1 1 1 1 ] [1 ] [1 1 1 1 ] [1 ] [1 1 1 1 ] [1 ] a(b(a(x1))) = [1 1 1 1 ]x1 + [1 ] >= [1 1 1 1 ]x1 + [1 ] = b(b(b(a(x1)))) [-& -& -& -&] [-&] [-& -& -& -&] [-&] [1 1 1 1 ] [1 ] [1 0 0 1 ] [1 ] [1 1 1 1 ] [1 ] [1 0 0 1 ] [1 ] b(b(b(a(x1)))) = [1 1 1 1 ]x1 + [1 ] >= [1 0 0 1 ]x1 + [1 ] = a(a(a(b(x1)))) [-& -& -& -&] [-&] [-& -& -& -&] [-&] problem: DPs: a#(b(a(x1))) -> b#(b(a(x1))) a#(b(a(x1))) -> b#(b(b(a(x1)))) b#(b(b(a(x1)))) -> a#(b(x1)) b#(b(b(a(x1)))) -> a#(a(b(x1))) b#(b(b(a(x1)))) -> a#(a(a(b(x1)))) TRS: a(b(a(x1))) -> b(b(b(a(x1)))) b(b(b(a(x1)))) -> a(a(a(b(x1)))) EDG Processor: DPs: a#(b(a(x1))) -> b#(b(a(x1))) a#(b(a(x1))) -> b#(b(b(a(x1)))) b#(b(b(a(x1)))) -> a#(b(x1)) b#(b(b(a(x1)))) -> a#(a(b(x1))) b#(b(b(a(x1)))) -> a#(a(a(b(x1)))) TRS: a(b(a(x1))) -> b(b(b(a(x1)))) b(b(b(a(x1)))) -> a(a(a(b(x1)))) graph: b#(b(b(a(x1)))) -> a#(b(x1)) -> a#(b(a(x1))) -> b#(b(a(x1))) b#(b(b(a(x1)))) -> a#(b(x1)) -> a#(b(a(x1))) -> b#(b(b(a(x1)))) b#(b(b(a(x1)))) -> a#(a(b(x1))) -> a#(b(a(x1))) -> b#(b(a(x1))) b#(b(b(a(x1)))) -> a#(a(b(x1))) -> a#(b(a(x1))) -> b#(b(b(a(x1)))) b#(b(b(a(x1)))) -> a#(a(a(b(x1)))) -> a#(b(a(x1))) -> b#(b(a(x1))) b#(b(b(a(x1)))) -> a#(a(a(b(x1)))) -> a#(b(a(x1))) -> b#(b(b(a(x1)))) a#(b(a(x1))) -> b#(b(b(a(x1)))) -> b#(b(b(a(x1)))) -> a#(b(x1)) a#(b(a(x1))) -> b#(b(b(a(x1)))) -> b#(b(b(a(x1)))) -> a#(a(b(x1))) a#(b(a(x1))) -> b#(b(b(a(x1)))) -> b#(b(b(a(x1)))) -> a#(a(a(b(x1)))) a#(b(a(x1))) -> b#(b(a(x1))) -> b#(b(b(a(x1)))) -> a#(b(x1)) a#(b(a(x1))) -> b#(b(a(x1))) -> b#(b(b(a(x1)))) -> a#(a(b(x1))) a#(b(a(x1))) -> b#(b(a(x1))) -> b#(b(b(a(x1)))) -> a#(a(a(b(x1)))) Matrix Interpretation Processor: dim=4 interpretation: [b#](x0) = [0 0 1 0]x0, [a#](x0) = [1], [0 0 0 0] [0 0 1 0] [b](x0) = [0 0 0 1]x0 [0 1 0 0] , [0] [1] [a](x0) = [0] [0] orientation: a#(b(a(x1))) = [1] >= [0] = b#(b(a(x1))) a#(b(a(x1))) = [1] >= [1] = b#(b(b(a(x1)))) b#(b(b(a(x1)))) = [1] >= [1] = a#(b(x1)) b#(b(b(a(x1)))) = [1] >= [1] = a#(a(b(x1))) b#(b(b(a(x1)))) = [1] >= [1] = a#(a(a(b(x1)))) [0] [0] [1] [1] a(b(a(x1))) = [0] >= [0] = b(b(b(a(x1)))) [0] [0] [0] [0] [1] [1] b(b(b(a(x1)))) = [0] >= [0] = a(a(a(b(x1)))) [0] [0] problem: DPs: a#(b(a(x1))) -> b#(b(b(a(x1)))) b#(b(b(a(x1)))) -> a#(b(x1)) b#(b(b(a(x1)))) -> a#(a(b(x1))) b#(b(b(a(x1)))) -> a#(a(a(b(x1)))) TRS: a(b(a(x1))) -> b(b(b(a(x1)))) b(b(b(a(x1)))) -> a(a(a(b(x1)))) EDG Processor: DPs: a#(b(a(x1))) -> b#(b(b(a(x1)))) b#(b(b(a(x1)))) -> a#(b(x1)) b#(b(b(a(x1)))) -> a#(a(b(x1))) b#(b(b(a(x1)))) -> a#(a(a(b(x1)))) TRS: a(b(a(x1))) -> b(b(b(a(x1)))) b(b(b(a(x1)))) -> a(a(a(b(x1)))) graph: b#(b(b(a(x1)))) -> a#(b(x1)) -> a#(b(a(x1))) -> b#(b(b(a(x1)))) b#(b(b(a(x1)))) -> a#(a(b(x1))) -> a#(b(a(x1))) -> b#(b(b(a(x1)))) b#(b(b(a(x1)))) -> a#(a(a(b(x1)))) -> a#(b(a(x1))) -> b#(b(b(a(x1)))) a#(b(a(x1))) -> b#(b(b(a(x1)))) -> b#(b(b(a(x1)))) -> a#(a(a(b(x1)))) a#(b(a(x1))) -> b#(b(b(a(x1)))) -> b#(b(b(a(x1)))) -> a#(a(b(x1))) a#(b(a(x1))) -> b#(b(b(a(x1)))) -> b#(b(b(a(x1)))) -> a#(b(x1)) Arctic Interpretation Processor: dimension: 3 interpretation: [b#](x0) = [0 1 1]x0 + [0], [a#](x0) = [0 0 1]x0 + [0], [-& -& 0 ] [0] [b](x0) = [0 -& -&]x0 + [0] [-& 0 -&] [0], [0 0 0 ] [1] [a](x0) = [0 0 0 ]x0 + [1] [-& -& -&] [0] orientation: a#(b(a(x1))) = [1 1 1]x1 + [2] >= [1 1 1]x1 + [2] = b#(b(b(a(x1)))) b#(b(b(a(x1)))) = [1 1 1]x1 + [2] >= [0 1 0]x1 + [1] = a#(b(x1)) b#(b(b(a(x1)))) = [1 1 1]x1 + [2] >= [0 0 0]x1 + [1] = a#(a(b(x1))) b#(b(b(a(x1)))) = [1 1 1]x1 + [2] >= [0 0 0]x1 + [1] = a#(a(a(b(x1)))) [0 0 0 ] [1] [0 0 0 ] [1] a(b(a(x1))) = [0 0 0 ]x1 + [1] >= [0 0 0 ]x1 + [1] = b(b(b(a(x1)))) [-& -& -&] [0] [-& -& -&] [0] [0 0 0 ] [1] [0 0 0 ] [1] b(b(b(a(x1)))) = [0 0 0 ]x1 + [1] >= [0 0 0 ]x1 + [1] = a(a(a(b(x1)))) [-& -& -&] [0] [-& -& -&] [0] problem: DPs: a#(b(a(x1))) -> b#(b(b(a(x1)))) b#(b(b(a(x1)))) -> a#(b(x1)) TRS: a(b(a(x1))) -> b(b(b(a(x1)))) b(b(b(a(x1)))) -> a(a(a(b(x1)))) EDG Processor: DPs: a#(b(a(x1))) -> b#(b(b(a(x1)))) b#(b(b(a(x1)))) -> a#(b(x1)) TRS: a(b(a(x1))) -> b(b(b(a(x1)))) b(b(b(a(x1)))) -> a(a(a(b(x1)))) graph: b#(b(b(a(x1)))) -> a#(b(x1)) -> a#(b(a(x1))) -> b#(b(b(a(x1)))) a#(b(a(x1))) -> b#(b(b(a(x1)))) -> b#(b(b(a(x1)))) -> a#(b(x1)) Matrix Interpretation Processor: dim=3 interpretation: [b#](x0) = [0 2 0]x0, [a#](x0) = [2 0 0]x0, [0 0 3] [b](x0) = [1 0 0]x0 [0 1 0] , [0 0 0] [0] [a](x0) = [0 0 0]x0 + [0] [1 1 1] [1] orientation: a#(b(a(x1))) = [6 6 6]x1 + [6] >= [6 6 6]x1 + [6] = b#(b(b(a(x1)))) b#(b(b(a(x1)))) = [6 6 6]x1 + [6] >= [0 0 6]x1 = a#(b(x1)) [0 0 0] [0] [0 0 0] [0] a(b(a(x1))) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = b(b(b(a(x1)))) [3 3 3] [4] [3 3 3] [3] [0 0 0] [0] [0 0 0] [0] b(b(b(a(x1)))) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = a(a(a(b(x1)))) [3 3 3] [3] [1 1 3] [3] problem: DPs: a#(b(a(x1))) -> b#(b(b(a(x1)))) TRS: a(b(a(x1))) -> b(b(b(a(x1)))) b(b(b(a(x1)))) -> a(a(a(b(x1)))) EDG Processor: DPs: a#(b(a(x1))) -> b#(b(b(a(x1)))) TRS: a(b(a(x1))) -> b(b(b(a(x1)))) b(b(b(a(x1)))) -> a(a(a(b(x1)))) graph: Qed