YES Problem: a(a(b(a(x1)))) -> a(b(b(x1))) b(b(x1)) -> b(a(a(a(x1)))) Proof: DP Processor: DPs: a#(a(b(a(x1)))) -> b#(x1) a#(a(b(a(x1)))) -> b#(b(x1)) a#(a(b(a(x1)))) -> a#(b(b(x1))) b#(b(x1)) -> a#(x1) b#(b(x1)) -> a#(a(x1)) b#(b(x1)) -> a#(a(a(x1))) b#(b(x1)) -> b#(a(a(a(x1)))) TRS: a(a(b(a(x1)))) -> a(b(b(x1))) b(b(x1)) -> b(a(a(a(x1)))) TDG Processor: DPs: a#(a(b(a(x1)))) -> b#(x1) a#(a(b(a(x1)))) -> b#(b(x1)) a#(a(b(a(x1)))) -> a#(b(b(x1))) b#(b(x1)) -> a#(x1) b#(b(x1)) -> a#(a(x1)) b#(b(x1)) -> a#(a(a(x1))) b#(b(x1)) -> b#(a(a(a(x1)))) TRS: a(a(b(a(x1)))) -> a(b(b(x1))) b(b(x1)) -> b(a(a(a(x1)))) graph: b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> b#(a(a(a(x1)))) b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> a#(a(a(x1))) b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> a#(a(x1)) b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> a#(x1) b#(b(x1)) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> a#(b(b(x1))) b#(b(x1)) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(x1)) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> b#(x1) b#(b(x1)) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> a#(b(b(x1))) b#(b(x1)) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(x1)) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> b#(x1) b#(b(x1)) -> a#(x1) -> a#(a(b(a(x1)))) -> a#(b(b(x1))) b#(b(x1)) -> a#(x1) -> a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(x1)) -> a#(x1) -> a#(a(b(a(x1)))) -> b#(x1) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(x1)) -> b#(a(a(a(x1)))) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(x1)) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(x1)) -> a#(a(x1)) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(x1)) -> a#(x1) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(x1)) -> b#(a(a(a(x1)))) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(x1)) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(x1)) -> a#(a(x1)) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(x1)) -> a#(x1) a#(a(b(a(x1)))) -> a#(b(b(x1))) -> a#(a(b(a(x1)))) -> a#(b(b(x1))) a#(a(b(a(x1)))) -> a#(b(b(x1))) -> a#(a(b(a(x1)))) -> b#(b(x1)) a#(a(b(a(x1)))) -> a#(b(b(x1))) -> a#(a(b(a(x1)))) -> b#(x1) CDG Processor: DPs: a#(a(b(a(x1)))) -> b#(x1) a#(a(b(a(x1)))) -> b#(b(x1)) a#(a(b(a(x1)))) -> a#(b(b(x1))) b#(b(x1)) -> a#(x1) b#(b(x1)) -> a#(a(x1)) b#(b(x1)) -> a#(a(a(x1))) b#(b(x1)) -> b#(a(a(a(x1)))) TRS: a(a(b(a(x1)))) -> a(b(b(x1))) b(b(x1)) -> b(a(a(a(x1)))) graph: b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> a#(x1) b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> a#(a(x1)) b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> a#(a(a(x1))) b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> b#(a(a(a(x1)))) b#(b(x1)) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> b#(x1) b#(b(x1)) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(x1)) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> a#(b(b(x1))) b#(b(x1)) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> b#(x1) b#(b(x1)) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(x1)) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> a#(b(b(x1))) b#(b(x1)) -> a#(x1) -> a#(a(b(a(x1)))) -> b#(x1) b#(b(x1)) -> a#(x1) -> a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(x1)) -> a#(x1) -> a#(a(b(a(x1)))) -> a#(b(b(x1))) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(x1)) -> a#(x1) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(x1)) -> a#(a(x1)) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(x1)) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(x1)) -> b#(a(a(a(x1)))) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(x1)) -> a#(x1) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(x1)) -> a#(a(x1)) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(x1)) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(x1)) -> b#(a(a(a(x1)))) SCC Processor: #sccs: 1 #rules: 6 #arcs: 21/49 DPs: b#(b(x1)) -> b#(a(a(a(x1)))) b#(b(x1)) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(x1)) -> a#(a(x1)) a#(a(b(a(x1)))) -> b#(x1) b#(b(x1)) -> a#(x1) TRS: a(a(b(a(x1)))) -> a(b(b(x1))) b(b(x1)) -> b(a(a(a(x1)))) Arctic Interpretation Processor: dimension: 2 usable rules: a(a(b(a(x1)))) -> a(b(b(x1))) b(b(x1)) -> b(a(a(a(x1)))) interpretation: [b#](x0) = [0 -&]x0 + [0], [a#](x0) = [0 0]x0 + [0], [0 0] [0] [b](x0) = [1 0]x0 + [2], [0 0] [0] [a](x0) = [0 0]x0 + [0] orientation: b#(b(x1)) = [0 0]x1 + [0] >= [0 0]x1 + [0] = b#(a(a(a(x1)))) b#(b(x1)) = [0 0]x1 + [0] >= [0 0]x1 + [0] = a#(a(a(x1))) a#(a(b(a(x1)))) = [1 1]x1 + [2] >= [0 0]x1 + [0] = b#(b(x1)) b#(b(x1)) = [0 0]x1 + [0] >= [0 0]x1 + [0] = a#(a(x1)) a#(a(b(a(x1)))) = [1 1]x1 + [2] >= [0 -&]x1 + [0] = b#(x1) b#(b(x1)) = [0 0]x1 + [0] >= [0 0]x1 + [0] = a#(x1) [1 1] [2] [1 1] [2] a(a(b(a(x1)))) = [1 1]x1 + [2] >= [1 1]x1 + [2] = a(b(b(x1))) [1 0] [2] [0 0] [0] b(b(x1)) = [1 1]x1 + [2] >= [1 1]x1 + [2] = b(a(a(a(x1)))) problem: DPs: b#(b(x1)) -> b#(a(a(a(x1)))) b#(b(x1)) -> a#(a(a(x1))) b#(b(x1)) -> a#(a(x1)) b#(b(x1)) -> a#(x1) TRS: a(a(b(a(x1)))) -> a(b(b(x1))) b(b(x1)) -> b(a(a(a(x1)))) Restore Modifier: DPs: b#(b(x1)) -> b#(a(a(a(x1)))) b#(b(x1)) -> a#(a(a(x1))) b#(b(x1)) -> a#(a(x1)) b#(b(x1)) -> a#(x1) TRS: a(a(b(a(x1)))) -> a(b(b(x1))) b(b(x1)) -> b(a(a(a(x1)))) EDG Processor: DPs: b#(b(x1)) -> b#(a(a(a(x1)))) b#(b(x1)) -> a#(a(a(x1))) b#(b(x1)) -> a#(a(x1)) b#(b(x1)) -> a#(x1) TRS: a(a(b(a(x1)))) -> a(b(b(x1))) b(b(x1)) -> b(a(a(a(x1)))) graph: b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> b#(a(a(a(x1)))) b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> a#(a(a(x1))) b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> a#(a(x1)) b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> a#(x1) SCC Processor: #sccs: 1 #rules: 1 #arcs: 4/16 DPs: b#(b(x1)) -> b#(a(a(a(x1)))) TRS: a(a(b(a(x1)))) -> a(b(b(x1))) b(b(x1)) -> b(a(a(a(x1)))) Arctic Interpretation Processor: dimension: 2 usable rules: a(a(b(a(x1)))) -> a(b(b(x1))) b(b(x1)) -> b(a(a(a(x1)))) interpretation: [b#](x0) = [2 -&]x0 + [0], [3 0 ] [1] [b](x0) = [-& -&]x0 + [1], [-& -&] [0] [a](x0) = [-& 1 ]x0 + [0] orientation: b#(b(x1)) = [5 2]x1 + [3] >= [2] = b#(a(a(a(x1)))) [0] [0] a(a(b(a(x1)))) = [3] >= [2] = a(b(b(x1))) [6 3 ] [4] [-& 3 ] [3] b(b(x1)) = [-& -&]x1 + [1] >= [-& -&]x1 + [1] = b(a(a(a(x1)))) problem: DPs: TRS: a(a(b(a(x1)))) -> a(b(b(x1))) b(b(x1)) -> b(a(a(a(x1)))) Qed