YES Problem: a(a(x1)) -> x1 b(b(x1)) -> c(c(c(c(x1)))) c(c(x1)) -> a(c(b(x1))) Proof: DP Processor: DPs: b#(b(x1)) -> c#(x1) b#(b(x1)) -> c#(c(x1)) b#(b(x1)) -> c#(c(c(x1))) b#(b(x1)) -> c#(c(c(c(x1)))) c#(c(x1)) -> b#(x1) c#(c(x1)) -> c#(b(x1)) c#(c(x1)) -> a#(c(b(x1))) TRS: a(a(x1)) -> x1 b(b(x1)) -> c(c(c(c(x1)))) c(c(x1)) -> a(c(b(x1))) TDG Processor: DPs: b#(b(x1)) -> c#(x1) b#(b(x1)) -> c#(c(x1)) b#(b(x1)) -> c#(c(c(x1))) b#(b(x1)) -> c#(c(c(c(x1)))) c#(c(x1)) -> b#(x1) c#(c(x1)) -> c#(b(x1)) c#(c(x1)) -> a#(c(b(x1))) TRS: a(a(x1)) -> x1 b(b(x1)) -> c(c(c(c(x1)))) c(c(x1)) -> a(c(b(x1))) graph: c#(c(x1)) -> c#(b(x1)) -> c#(c(x1)) -> a#(c(b(x1))) c#(c(x1)) -> c#(b(x1)) -> c#(c(x1)) -> c#(b(x1)) c#(c(x1)) -> c#(b(x1)) -> c#(c(x1)) -> b#(x1) c#(c(x1)) -> b#(x1) -> b#(b(x1)) -> c#(c(c(c(x1)))) c#(c(x1)) -> b#(x1) -> b#(b(x1)) -> c#(c(c(x1))) c#(c(x1)) -> b#(x1) -> b#(b(x1)) -> c#(c(x1)) c#(c(x1)) -> b#(x1) -> b#(b(x1)) -> c#(x1) b#(b(x1)) -> c#(c(c(c(x1)))) -> c#(c(x1)) -> a#(c(b(x1))) b#(b(x1)) -> c#(c(c(c(x1)))) -> c#(c(x1)) -> c#(b(x1)) b#(b(x1)) -> c#(c(c(c(x1)))) -> c#(c(x1)) -> b#(x1) b#(b(x1)) -> c#(c(c(x1))) -> c#(c(x1)) -> a#(c(b(x1))) b#(b(x1)) -> c#(c(c(x1))) -> c#(c(x1)) -> c#(b(x1)) b#(b(x1)) -> c#(c(c(x1))) -> c#(c(x1)) -> b#(x1) b#(b(x1)) -> c#(c(x1)) -> c#(c(x1)) -> a#(c(b(x1))) b#(b(x1)) -> c#(c(x1)) -> c#(c(x1)) -> c#(b(x1)) b#(b(x1)) -> c#(c(x1)) -> c#(c(x1)) -> b#(x1) b#(b(x1)) -> c#(x1) -> c#(c(x1)) -> a#(c(b(x1))) b#(b(x1)) -> c#(x1) -> c#(c(x1)) -> c#(b(x1)) b#(b(x1)) -> c#(x1) -> c#(c(x1)) -> b#(x1) SCC Processor: #sccs: 1 #rules: 6 #arcs: 19/49 DPs: c#(c(x1)) -> c#(b(x1)) c#(c(x1)) -> b#(x1) b#(b(x1)) -> c#(x1) b#(b(x1)) -> c#(c(x1)) b#(b(x1)) -> c#(c(c(x1))) b#(b(x1)) -> c#(c(c(c(x1)))) TRS: a(a(x1)) -> x1 b(b(x1)) -> c(c(c(c(x1)))) c(c(x1)) -> a(c(b(x1))) Matrix Interpretation Processor: dim=5 interpretation: [c#](x0) = [1 0 0 0 0]x0 + [1], [b#](x0) = [1 1 0 0 1]x0, [1 1 0 0 1] [0] [0 0 0 0 0] [0] [c](x0) = [1 0 0 1 0]x0 + [0] [0 0 0 0 0] [1] [0 0 0 0 0] [0], [1 1 0 0 0] [0] [0 0 0 0 1] [0] [b](x0) = [1 0 0 1 1]x0 + [0] [0 0 0 0 1] [0] [0 0 0 0 0] [1], [0 0 1 0 0] [0 1 0 0 0] [a](x0) = [1 0 0 0 0]x0 [0 0 0 1 0] [0 0 0 0 1] orientation: c#(c(x1)) = [1 1 0 0 1]x1 + [1] >= [1 1 0 0 0]x1 + [1] = c#(b(x1)) c#(c(x1)) = [1 1 0 0 1]x1 + [1] >= [1 1 0 0 1]x1 = b#(x1) b#(b(x1)) = [1 1 0 0 1]x1 + [1] >= [1 0 0 0 0]x1 + [1] = c#(x1) b#(b(x1)) = [1 1 0 0 1]x1 + [1] >= [1 1 0 0 1]x1 + [1] = c#(c(x1)) b#(b(x1)) = [1 1 0 0 1]x1 + [1] >= [1 1 0 0 1]x1 + [1] = c#(c(c(x1))) b#(b(x1)) = [1 1 0 0 1]x1 + [1] >= [1 1 0 0 1]x1 + [1] = c#(c(c(c(x1)))) a(a(x1)) = x1 >= x1 = x1 [1 1 0 0 1] [0] [1 1 0 0 1] [0] [0 0 0 0 0] [1] [0 0 0 0 0] [0] b(b(x1)) = [1 1 0 0 1]x1 + [1] >= [1 1 0 0 1]x1 + [1] = c(c(c(c(x1)))) [0 0 0 0 0] [1] [0 0 0 0 0] [1] [0 0 0 0 0] [1] [0 0 0 0 0] [0] [1 1 0 0 1] [0] [1 1 0 0 1] [0] [0 0 0 0 0] [0] [0 0 0 0 0] [0] c(c(x1)) = [1 1 0 0 1]x1 + [1] >= [1 1 0 0 1]x1 + [1] = a(c(b(x1))) [0 0 0 0 0] [1] [0 0 0 0 0] [1] [0 0 0 0 0] [0] [0 0 0 0 0] [0] problem: DPs: c#(c(x1)) -> c#(b(x1)) b#(b(x1)) -> c#(x1) b#(b(x1)) -> c#(c(x1)) b#(b(x1)) -> c#(c(c(x1))) b#(b(x1)) -> c#(c(c(c(x1)))) TRS: a(a(x1)) -> x1 b(b(x1)) -> c(c(c(c(x1)))) c(c(x1)) -> a(c(b(x1))) Restore Modifier: DPs: c#(c(x1)) -> c#(b(x1)) b#(b(x1)) -> c#(x1) b#(b(x1)) -> c#(c(x1)) b#(b(x1)) -> c#(c(c(x1))) b#(b(x1)) -> c#(c(c(c(x1)))) TRS: a(a(x1)) -> x1 b(b(x1)) -> c(c(c(c(x1)))) c(c(x1)) -> a(c(b(x1))) EDG Processor: DPs: c#(c(x1)) -> c#(b(x1)) b#(b(x1)) -> c#(x1) b#(b(x1)) -> c#(c(x1)) b#(b(x1)) -> c#(c(c(x1))) b#(b(x1)) -> c#(c(c(c(x1)))) TRS: a(a(x1)) -> x1 b(b(x1)) -> c(c(c(c(x1)))) c(c(x1)) -> a(c(b(x1))) graph: c#(c(x1)) -> c#(b(x1)) -> c#(c(x1)) -> c#(b(x1)) b#(b(x1)) -> c#(c(c(c(x1)))) -> c#(c(x1)) -> c#(b(x1)) b#(b(x1)) -> c#(c(c(x1))) -> c#(c(x1)) -> c#(b(x1)) b#(b(x1)) -> c#(c(x1)) -> c#(c(x1)) -> c#(b(x1)) b#(b(x1)) -> c#(x1) -> c#(c(x1)) -> c#(b(x1)) SCC Processor: #sccs: 1 #rules: 1 #arcs: 5/25 DPs: c#(c(x1)) -> c#(b(x1)) TRS: a(a(x1)) -> x1 b(b(x1)) -> c(c(c(c(x1)))) c(c(x1)) -> a(c(b(x1))) Arctic Interpretation Processor: dimension: 3 usable rules: a(a(x1)) -> x1 b(b(x1)) -> c(c(c(c(x1)))) c(c(x1)) -> a(c(b(x1))) interpretation: [c#](x0) = [0 0 1]x0 + [0], [-& -& -&] [0] [c](x0) = [0 -& 1 ]x0 + [0] [1 -& 0 ] [1], [1 -& 0 ] [1] [b](x0) = [1 -& 0 ]x0 + [0] [0 -& -&] [0], [0 -& -&] [0] [a](x0) = [-& -& 0 ]x0 + [0] [0 0 -&] [0] orientation: c#(c(x1)) = [2 -& 1 ]x1 + [2] >= [1 -& 0 ]x1 + [1] = c#(b(x1)) [0 -& -&] [0] a(a(x1)) = [0 0 -&]x1 + [0] >= x1 = x1 [0 -& 0 ] [0] [2 -& 1 ] [2] [-& -& -&] [0] b(b(x1)) = [2 -& 1 ]x1 + [2] >= [2 -& 1 ]x1 + [2] = c(c(c(c(x1)))) [1 -& 0 ] [1] [1 -& 0 ] [1] [-& -& -&] [0] [-& -& -&] [0] c(c(x1)) = [2 -& 1 ]x1 + [2] >= [2 -& 1 ]x1 + [2] = a(c(b(x1))) [1 -& 0 ] [1] [1 -& 0 ] [1] problem: DPs: TRS: a(a(x1)) -> x1 b(b(x1)) -> c(c(c(c(x1)))) c(c(x1)) -> a(c(b(x1))) Qed