YES Problem: a(x1) -> x1 a(a(x1)) -> b(c(x1)) b(x1) -> x1 c(x1) -> x1 c(b(x1)) -> a(b(c(x1))) Proof: DP Processor: DPs: a#(a(x1)) -> c#(x1) a#(a(x1)) -> b#(c(x1)) c#(b(x1)) -> c#(x1) c#(b(x1)) -> b#(c(x1)) c#(b(x1)) -> a#(b(c(x1))) TRS: a(x1) -> x1 a(a(x1)) -> b(c(x1)) b(x1) -> x1 c(x1) -> x1 c(b(x1)) -> a(b(c(x1))) TDG Processor: DPs: a#(a(x1)) -> c#(x1) a#(a(x1)) -> b#(c(x1)) c#(b(x1)) -> c#(x1) c#(b(x1)) -> b#(c(x1)) c#(b(x1)) -> a#(b(c(x1))) TRS: a(x1) -> x1 a(a(x1)) -> b(c(x1)) b(x1) -> x1 c(x1) -> x1 c(b(x1)) -> a(b(c(x1))) graph: c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> a#(b(c(x1))) c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> b#(c(x1)) c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1) c#(b(x1)) -> a#(b(c(x1))) -> a#(a(x1)) -> b#(c(x1)) c#(b(x1)) -> a#(b(c(x1))) -> a#(a(x1)) -> c#(x1) a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> a#(b(c(x1))) a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> b#(c(x1)) a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1) SCC Processor: #sccs: 1 #rules: 3 #arcs: 8/25 DPs: c#(b(x1)) -> c#(x1) c#(b(x1)) -> a#(b(c(x1))) a#(a(x1)) -> c#(x1) TRS: a(x1) -> x1 a(a(x1)) -> b(c(x1)) b(x1) -> x1 c(x1) -> x1 c(b(x1)) -> a(b(c(x1))) Arctic Interpretation Processor: dimension: 2 interpretation: [c#](x0) = [1 -&]x0 + [0], [a#](x0) = [0 -&]x0 + [0], [0 0] [0] [b](x0) = [1 1]x0 + [3], [0 0] [0] [c](x0) = [0 0]x0 + [0], [1 0] [-&] [a](x0) = [1 0]x0 + [3 ] orientation: c#(b(x1)) = [1 1]x1 + [1] >= [1 -&]x1 + [0] = c#(x1) c#(b(x1)) = [1 1]x1 + [1] >= [0 0]x1 + [0] = a#(b(c(x1))) a#(a(x1)) = [1 0]x1 + [0] >= [1 -&]x1 + [0] = c#(x1) [1 0] [-&] a(x1) = [1 0]x1 + [3 ] >= x1 = x1 [2 1] [3] [0 0] [0] a(a(x1)) = [2 1]x1 + [3] >= [1 1]x1 + [3] = b(c(x1)) [0 0] [0] b(x1) = [1 1]x1 + [3] >= x1 = x1 [0 0] [0] c(x1) = [0 0]x1 + [0] >= x1 = x1 [1 1] [3] [1 1] [3] c(b(x1)) = [1 1]x1 + [3] >= [1 1]x1 + [3] = a(b(c(x1))) problem: DPs: c#(b(x1)) -> c#(x1) a#(a(x1)) -> c#(x1) TRS: a(x1) -> x1 a(a(x1)) -> b(c(x1)) b(x1) -> x1 c(x1) -> x1 c(b(x1)) -> a(b(c(x1))) EDG Processor: DPs: c#(b(x1)) -> c#(x1) a#(a(x1)) -> c#(x1) TRS: a(x1) -> x1 a(a(x1)) -> b(c(x1)) b(x1) -> x1 c(x1) -> x1 c(b(x1)) -> a(b(c(x1))) graph: c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1) a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1) CDG Processor: DPs: c#(b(x1)) -> c#(x1) a#(a(x1)) -> c#(x1) TRS: a(x1) -> x1 a(a(x1)) -> b(c(x1)) b(x1) -> x1 c(x1) -> x1 c(b(x1)) -> a(b(c(x1))) graph: Qed