YES Problem: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) Proof: DP Processor: DPs: a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(a(b(b(x1)))) b#(b(b(b(x1)))) -> a#(x1) b#(b(b(b(x1)))) -> a#(a(x1)) b#(b(b(b(x1)))) -> b#(a(a(x1))) b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) TRS: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) TDG Processor: DPs: a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(a(b(b(x1)))) b#(b(b(b(x1)))) -> a#(x1) b#(b(b(b(x1)))) -> a#(a(x1)) b#(b(b(b(x1)))) -> b#(a(a(x1))) b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) TRS: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) graph: b#(b(b(b(x1)))) -> b#(a(a(x1))) -> b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) b#(b(b(b(x1)))) -> b#(a(a(x1))) -> b#(b(b(b(x1)))) -> b#(a(a(x1))) b#(b(b(b(x1)))) -> b#(a(a(x1))) -> b#(b(b(b(x1)))) -> a#(a(x1)) b#(b(b(b(x1)))) -> b#(a(a(x1))) -> b#(b(b(b(x1)))) -> a#(x1) b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) -> a#(a(a(x1))) -> b#(b(x1)) b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) -> a#(a(a(x1))) -> b#(x1) b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(b(x1)) b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(x1) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> b#(b(x1)) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> b#(a(a(x1))) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> a#(a(x1)) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> a#(x1) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> b#(a(a(x1))) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(a(x1)) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(x1) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> b#(x1) EDG Processor: DPs: a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(a(b(b(x1)))) b#(b(b(b(x1)))) -> a#(x1) b#(b(b(b(x1)))) -> a#(a(x1)) b#(b(b(b(x1)))) -> b#(a(a(x1))) b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) TRS: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) graph: b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) -> a#(a(a(x1))) -> b#(x1) b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) -> a#(a(a(x1))) -> b#(b(x1)) b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(x1) b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(b(x1)) b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> b#(x1) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> b#(b(x1)) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> a#(x1) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> a#(a(x1)) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> b#(a(a(x1))) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(x1) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(a(x1)) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> b#(a(a(x1))) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) CDG Processor: DPs: a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(a(b(b(x1)))) b#(b(b(b(x1)))) -> a#(x1) b#(b(b(b(x1)))) -> a#(a(x1)) b#(b(b(b(x1)))) -> b#(a(a(x1))) b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) TRS: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) graph: b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(b(x1)) b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(x1) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> b#(b(x1)) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> b#(a(a(x1))) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> a#(a(x1)) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> a#(x1) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> b#(a(a(x1))) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(a(x1)) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(x1) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> b#(x1) SCC Processor: #sccs: 1 #rules: 6 #arcs: 24/64 DPs: b#(b(b(b(x1)))) -> a#(a(x1)) a#(a(a(x1))) -> b#(x1) b#(b(b(b(x1)))) -> a#(x1) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(a(b(b(x1)))) TRS: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) Arctic Interpretation Processor: dimension: 3 usable rules: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) interpretation: [b#](x0) = [0 1 0]x0 + [0], [a#](x0) = [-& 1 0 ]x0 + [0], [-& 1 0 ] [1] [b](x0) = [0 -& -&]x0 + [0] [1 0 -&] [0], [-& -& -&] [0] [a](x0) = [-& 1 0 ]x0 + [1] [0 -& -&] [0] orientation: b#(b(b(b(x1)))) = [2 2 1]x1 + [2] >= [0 2 1]x1 + [2] = a#(a(x1)) a#(a(a(x1))) = [1 3 2]x1 + [3] >= [0 1 0]x1 + [0] = b#(x1) b#(b(b(b(x1)))) = [2 2 1]x1 + [2] >= [-& 1 0 ]x1 + [0] = a#(x1) a#(a(a(x1))) = [1 3 2]x1 + [3] >= [1 1 0]x1 + [1] = b#(b(x1)) a#(a(a(x1))) = [1 3 2]x1 + [3] >= [0 2 1]x1 + [2] = a#(b(b(x1))) a#(a(a(x1))) = [1 3 2]x1 + [3] >= [1 3 2]x1 + [3] = a#(a(b(b(x1)))) [-& -& -&] [0] [-& -& -&] [0] a(a(a(x1))) = [1 3 2 ]x1 + [3] >= [1 3 2 ]x1 + [3] = a(a(b(b(x1)))) [-& -& -&] [0] [-& -& -&] [0] [2 1 0] [2] [-& -& -&] [0] b(b(b(b(x1)))) = [0 2 1]x1 + [2] >= [0 2 1 ]x1 + [2] = a(b(a(a(x1)))) [1 3 2] [3] [1 3 2 ] [3] problem: DPs: b#(b(b(b(x1)))) -> a#(a(x1)) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(a(b(b(x1)))) TRS: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) Restore Modifier: DPs: b#(b(b(b(x1)))) -> a#(a(x1)) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(a(b(b(x1)))) TRS: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) EDG Processor: DPs: b#(b(b(b(x1)))) -> a#(a(x1)) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(a(b(b(x1)))) TRS: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) graph: b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(b(x1)) b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> a#(a(x1)) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) Matrix Interpretation Processor: dim=3 interpretation: [b#](x0) = [0 0 1]x0 + [1], [a#](x0) = [1 0 0]x0, [0 0 1] [0] [b](x0) = [0 0 0]x0 + [0] [1 0 0] [1], [1 0 0] [1] [a](x0) = [0 0 0]x0 + [0] [0 0 0] [0] orientation: b#(b(b(b(x1)))) = [1 0 0]x1 + [3] >= [1 0 0]x1 + [1] = a#(a(x1)) a#(a(a(x1))) = [1 0 0]x1 + [2] >= [1 0 0]x1 + [2] = b#(b(x1)) a#(a(a(x1))) = [1 0 0]x1 + [2] >= [1 0 0]x1 + [2] = a#(a(b(b(x1)))) [1 0 0] [3] [1 0 0] [3] a(a(a(x1))) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = a(a(b(b(x1)))) [0 0 0] [0] [0 0 0] [0] [1 0 0] [2] [1] b(b(b(b(x1)))) = [0 0 0]x1 + [0] >= [0] = a(b(a(a(x1)))) [0 0 1] [2] [0] problem: DPs: a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(a(b(b(x1)))) TRS: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) Restore Modifier: DPs: a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(a(b(b(x1)))) TRS: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) EDG Processor: DPs: a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(a(b(b(x1)))) TRS: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) graph: a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> b#(b(x1)) SCC Processor: #sccs: 1 #rules: 1 #arcs: 2/4 DPs: a#(a(a(x1))) -> a#(a(b(b(x1)))) TRS: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) Arctic Interpretation Processor: dimension: 2 usable rules: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) interpretation: [a#](x0) = [-& 0 ]x0 + [0], [-& 1 ] [2] [b](x0) = [0 -&]x0 + [1], [-& -&] [0] [a](x0) = [-& 2 ]x0 + [3] orientation: a#(a(a(x1))) = [-& 4 ]x1 + [5] >= [-& 3 ]x1 + [4] = a#(a(b(b(x1)))) [-& -&] [0] [-& -&] [0] a(a(a(x1))) = [-& 6 ]x1 + [7] >= [-& 5 ]x1 + [6] = a(a(b(b(x1)))) [2 -&] [3] [0] b(b(b(b(x1)))) = [-& 2 ]x1 + [3] >= [3] = a(b(a(a(x1)))) problem: DPs: TRS: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) Qed