YES Problem: a(a(b(a(x1)))) -> a(b(b(b(x1)))) b(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)))) -> b#(b(b(x1))) a#(a(b(a(x1)))) -> a#(b(b(b(x1)))) b#(b(b(x1))) -> a#(x1) b#(b(b(x1))) -> a#(a(x1)) b#(b(b(x1))) -> a#(a(a(x1))) b#(b(b(x1))) -> b#(a(a(a(x1)))) TRS: a(a(b(a(x1)))) -> a(b(b(b(x1)))) b(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)))) -> b#(b(b(x1))) a#(a(b(a(x1)))) -> a#(b(b(b(x1)))) b#(b(b(x1))) -> a#(x1) b#(b(b(x1))) -> a#(a(x1)) b#(b(b(x1))) -> a#(a(a(x1))) b#(b(b(x1))) -> b#(a(a(a(x1)))) TRS: a(a(b(a(x1)))) -> a(b(b(b(x1)))) b(b(b(x1))) -> b(a(a(a(x1)))) graph: b#(b(b(x1))) -> b#(a(a(a(x1)))) -> b#(b(b(x1))) -> b#(a(a(a(x1)))) b#(b(b(x1))) -> b#(a(a(a(x1)))) -> b#(b(b(x1))) -> a#(a(a(x1))) b#(b(b(x1))) -> b#(a(a(a(x1)))) -> b#(b(b(x1))) -> a#(a(x1)) b#(b(b(x1))) -> b#(a(a(a(x1)))) -> b#(b(b(x1))) -> a#(x1) b#(b(b(x1))) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> a#(b(b(b(x1)))) b#(b(b(x1))) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> b#(b(b(x1))) b#(b(b(x1))) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(b(x1))) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> b#(x1) b#(b(b(x1))) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> a#(b(b(b(x1)))) b#(b(b(x1))) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> b#(b(b(x1))) b#(b(b(x1))) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(b(x1))) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> b#(x1) b#(b(b(x1))) -> a#(x1) -> a#(a(b(a(x1)))) -> a#(b(b(b(x1)))) b#(b(b(x1))) -> a#(x1) -> a#(a(b(a(x1)))) -> b#(b(b(x1))) b#(b(b(x1))) -> a#(x1) -> a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(b(x1))) -> a#(x1) -> a#(a(b(a(x1)))) -> b#(x1) a#(a(b(a(x1)))) -> b#(b(b(x1))) -> b#(b(b(x1))) -> b#(a(a(a(x1)))) a#(a(b(a(x1)))) -> b#(b(b(x1))) -> b#(b(b(x1))) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(b(b(x1))) -> b#(b(b(x1))) -> a#(a(x1)) a#(a(b(a(x1)))) -> b#(b(b(x1))) -> b#(b(b(x1))) -> a#(x1) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(b(x1))) -> b#(a(a(a(x1)))) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(b(x1))) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(b(x1))) -> a#(a(x1)) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(b(x1))) -> a#(x1) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(b(x1))) -> b#(a(a(a(x1)))) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(b(x1))) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(b(x1))) -> a#(a(x1)) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(b(x1))) -> a#(x1) a#(a(b(a(x1)))) -> a#(b(b(b(x1)))) -> a#(a(b(a(x1)))) -> a#(b(b(b(x1)))) a#(a(b(a(x1)))) -> a#(b(b(b(x1)))) -> a#(a(b(a(x1)))) -> b#(b(b(x1))) a#(a(b(a(x1)))) -> a#(b(b(b(x1)))) -> a#(a(b(a(x1)))) -> b#(b(x1)) a#(a(b(a(x1)))) -> a#(b(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)))) -> b#(b(b(x1))) a#(a(b(a(x1)))) -> a#(b(b(b(x1)))) b#(b(b(x1))) -> a#(x1) b#(b(b(x1))) -> a#(a(x1)) b#(b(b(x1))) -> a#(a(a(x1))) b#(b(b(x1))) -> b#(a(a(a(x1)))) TRS: a(a(b(a(x1)))) -> a(b(b(b(x1)))) b(b(b(x1))) -> b(a(a(a(x1)))) graph: b#(b(b(x1))) -> b#(a(a(a(x1)))) -> b#(b(b(x1))) -> a#(x1) b#(b(b(x1))) -> b#(a(a(a(x1)))) -> b#(b(b(x1))) -> a#(a(x1)) b#(b(b(x1))) -> b#(a(a(a(x1)))) -> b#(b(b(x1))) -> a#(a(a(x1))) b#(b(b(x1))) -> b#(a(a(a(x1)))) -> b#(b(b(x1))) -> b#(a(a(a(x1)))) b#(b(b(x1))) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> b#(x1) b#(b(b(x1))) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(b(x1))) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> b#(b(b(x1))) b#(b(b(x1))) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> a#(b(b(b(x1)))) b#(b(b(x1))) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> b#(x1) b#(b(b(x1))) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(b(x1))) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> b#(b(b(x1))) b#(b(b(x1))) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> a#(b(b(b(x1)))) b#(b(b(x1))) -> a#(x1) -> a#(a(b(a(x1)))) -> b#(x1) b#(b(b(x1))) -> a#(x1) -> a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(b(x1))) -> a#(x1) -> a#(a(b(a(x1)))) -> b#(b(b(x1))) b#(b(b(x1))) -> a#(x1) -> a#(a(b(a(x1)))) -> a#(b(b(b(x1)))) a#(a(b(a(x1)))) -> b#(b(b(x1))) -> b#(b(b(x1))) -> a#(x1) a#(a(b(a(x1)))) -> b#(b(b(x1))) -> b#(b(b(x1))) -> a#(a(x1)) a#(a(b(a(x1)))) -> b#(b(b(x1))) -> b#(b(b(x1))) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(b(b(x1))) -> b#(b(b(x1))) -> b#(a(a(a(x1)))) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(b(x1))) -> a#(x1) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(b(x1))) -> a#(a(x1)) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(b(x1))) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(b(x1))) -> b#(a(a(a(x1)))) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(b(x1))) -> a#(x1) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(b(x1))) -> a#(a(x1)) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(b(x1))) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(b(x1))) -> b#(a(a(a(x1)))) SCC Processor: #sccs: 1 #rules: 7 #arcs: 28/64 DPs: b#(b(b(x1))) -> b#(a(a(a(x1)))) b#(b(b(x1))) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(b(b(x1))) b#(b(b(x1))) -> a#(a(x1)) a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(b(x1))) -> a#(x1) a#(a(b(a(x1)))) -> b#(x1) TRS: a(a(b(a(x1)))) -> a(b(b(b(x1)))) b(b(b(x1))) -> b(a(a(a(x1)))) Arctic Interpretation Processor: dimension: 2 interpretation: [b#](x0) = [0 1]x0 + [0], [a#](x0) = [1 0]x0 + [0], [-4 0 ] [0] [b](x0) = [0 0 ]x0 + [0], [0 0 ] [-4] [a](x0) = [-& -&]x0 + [-1] orientation: b#(b(b(x1))) = [1 1]x1 + [1] >= [0 0]x1 + [0] = b#(a(a(a(x1)))) b#(b(b(x1))) = [1 1]x1 + [1] >= [1 1]x1 + [0] = a#(a(a(x1))) a#(a(b(a(x1)))) = [1 1]x1 + [1] >= [1 1]x1 + [1] = b#(b(b(x1))) b#(b(b(x1))) = [1 1]x1 + [1] >= [1 1]x1 + [0] = a#(a(x1)) a#(a(b(a(x1)))) = [1 1]x1 + [1] >= [1 1]x1 + [1] = b#(b(x1)) b#(b(b(x1))) = [1 1]x1 + [1] >= [1 0]x1 + [0] = a#(x1) a#(a(b(a(x1)))) = [1 1]x1 + [1] >= [0 1]x1 + [0] = b#(x1) [0 0 ] [0 ] [0 0 ] [0 ] a(a(b(a(x1)))) = [-& -&]x1 + [-1] >= [-& -&]x1 + [-1] = a(b(b(b(x1)))) [0 0] [0] [-4 -4] [0] b(b(b(x1))) = [0 0]x1 + [0] >= [0 0 ]x1 + [0] = b(a(a(a(x1)))) problem: DPs: b#(b(b(x1))) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(b(b(x1))) b#(b(b(x1))) -> a#(a(x1)) a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(b(x1))) -> a#(x1) a#(a(b(a(x1)))) -> b#(x1) TRS: a(a(b(a(x1)))) -> a(b(b(b(x1)))) b(b(b(x1))) -> b(a(a(a(x1)))) EDG Processor: DPs: b#(b(b(x1))) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(b(b(x1))) b#(b(b(x1))) -> a#(a(x1)) a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(b(x1))) -> a#(x1) a#(a(b(a(x1)))) -> b#(x1) TRS: a(a(b(a(x1)))) -> a(b(b(b(x1)))) b(b(b(x1))) -> b(a(a(a(x1)))) graph: b#(b(b(x1))) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> b#(b(b(x1))) b#(b(b(x1))) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(b(x1))) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> b#(x1) b#(b(b(x1))) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> b#(b(b(x1))) b#(b(b(x1))) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(b(x1))) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> b#(x1) b#(b(b(x1))) -> a#(x1) -> a#(a(b(a(x1)))) -> b#(b(b(x1))) b#(b(b(x1))) -> a#(x1) -> a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(b(x1))) -> a#(x1) -> a#(a(b(a(x1)))) -> b#(x1) a#(a(b(a(x1)))) -> b#(b(b(x1))) -> b#(b(b(x1))) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(b(b(x1))) -> b#(b(b(x1))) -> a#(a(x1)) a#(a(b(a(x1)))) -> b#(b(b(x1))) -> b#(b(b(x1))) -> a#(x1) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(b(x1))) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(b(x1))) -> a#(a(x1)) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(b(x1))) -> a#(x1) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(b(x1))) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(b(x1))) -> a#(a(x1)) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(b(x1))) -> a#(x1) Arctic Interpretation Processor: dimension: 4 interpretation: [b#](x0) = [0 -& -& -&]x0, [a#](x0) = [-& 0 -& 0 ]x0 + [0], [0 0 -& -&] [0 ] [0 0 0 0 ] [-&] [b](x0) = [1 0 -& 0 ]x0 + [0 ] [1 0 0 0 ] [-&], [0 0 0 0 ] [0] [0 0 -& 0 ] [0] [a](x0) = [0 0 0 0 ]x0 + [0] [0 0 0 0 ] [0] orientation: b#(b(b(x1))) = [0 0 0 0]x1 + [0] >= [0 0 0 0]x1 + [0] = a#(a(a(x1))) a#(a(b(a(x1)))) = [1 1 1 1]x1 + [1] >= [0 0 0 0]x1 + [0] = b#(b(b(x1))) b#(b(b(x1))) = [0 0 0 0]x1 + [0] >= [0 0 0 0]x1 + [0] = a#(a(x1)) a#(a(b(a(x1)))) = [1 1 1 1]x1 + [1] >= [0 0 -& -&]x1 + [0] = b#(b(x1)) b#(b(b(x1))) = [0 0 0 0]x1 + [0] >= [-& 0 -& 0 ]x1 + [0] = a#(x1) a#(a(b(a(x1)))) = [1 1 1 1]x1 + [1] >= [0 -& -& -&]x1 = b#(x1) [1 1 1 1] [1] [1 1 1 1] [1] [1 1 1 1] [1] [1 1 1 1] [1] a(a(b(a(x1)))) = [1 1 1 1]x1 + [1] >= [1 1 1 1]x1 + [1] = a(b(b(b(x1)))) [1 1 1 1] [1] [1 1 1 1] [1] [1 0 0 0] [0] [0 0 0 0] [0] [1 1 0 0] [1] [0 0 0 0] [0] b(b(b(x1))) = [1 1 1 1]x1 + [1] >= [1 1 1 1]x1 + [1] = b(a(a(a(x1)))) [1 1 1 1] [1] [1 1 1 1] [1] problem: DPs: b#(b(b(x1))) -> a#(a(a(x1))) b#(b(b(x1))) -> a#(a(x1)) b#(b(b(x1))) -> a#(x1) TRS: a(a(b(a(x1)))) -> a(b(b(b(x1)))) b(b(b(x1))) -> b(a(a(a(x1)))) EDG Processor: DPs: b#(b(b(x1))) -> a#(a(a(x1))) b#(b(b(x1))) -> a#(a(x1)) b#(b(b(x1))) -> a#(x1) TRS: a(a(b(a(x1)))) -> a(b(b(b(x1)))) b(b(b(x1))) -> b(a(a(a(x1)))) graph: Qed