YES Problem: a(x1) -> b(c(x1)) b(b(x1)) -> x1 c(c(b(x1))) -> a(a(a(c(x1)))) Proof: DP Processor: DPs: a#(x1) -> c#(x1) a#(x1) -> b#(c(x1)) c#(c(b(x1))) -> c#(x1) c#(c(b(x1))) -> a#(c(x1)) c#(c(b(x1))) -> a#(a(c(x1))) c#(c(b(x1))) -> a#(a(a(c(x1)))) TRS: a(x1) -> b(c(x1)) b(b(x1)) -> x1 c(c(b(x1))) -> a(a(a(c(x1)))) TDG Processor: DPs: a#(x1) -> c#(x1) a#(x1) -> b#(c(x1)) c#(c(b(x1))) -> c#(x1) c#(c(b(x1))) -> a#(c(x1)) c#(c(b(x1))) -> a#(a(c(x1))) c#(c(b(x1))) -> a#(a(a(c(x1)))) TRS: a(x1) -> b(c(x1)) b(b(x1)) -> x1 c(c(b(x1))) -> a(a(a(c(x1)))) graph: c#(c(b(x1))) -> c#(x1) -> c#(c(b(x1))) -> a#(a(a(c(x1)))) c#(c(b(x1))) -> c#(x1) -> c#(c(b(x1))) -> a#(a(c(x1))) c#(c(b(x1))) -> c#(x1) -> c#(c(b(x1))) -> a#(c(x1)) c#(c(b(x1))) -> c#(x1) -> c#(c(b(x1))) -> c#(x1) c#(c(b(x1))) -> a#(c(x1)) -> a#(x1) -> b#(c(x1)) c#(c(b(x1))) -> a#(c(x1)) -> a#(x1) -> c#(x1) c#(c(b(x1))) -> a#(a(c(x1))) -> a#(x1) -> b#(c(x1)) c#(c(b(x1))) -> a#(a(c(x1))) -> a#(x1) -> c#(x1) c#(c(b(x1))) -> a#(a(a(c(x1)))) -> a#(x1) -> b#(c(x1)) c#(c(b(x1))) -> a#(a(a(c(x1)))) -> a#(x1) -> c#(x1) a#(x1) -> c#(x1) -> c#(c(b(x1))) -> a#(a(a(c(x1)))) a#(x1) -> c#(x1) -> c#(c(b(x1))) -> a#(a(c(x1))) a#(x1) -> c#(x1) -> c#(c(b(x1))) -> a#(c(x1)) a#(x1) -> c#(x1) -> c#(c(b(x1))) -> c#(x1) SCC Processor: #sccs: 1 #rules: 5 #arcs: 14/36 DPs: c#(c(b(x1))) -> c#(x1) c#(c(b(x1))) -> a#(c(x1)) a#(x1) -> c#(x1) c#(c(b(x1))) -> a#(a(c(x1))) c#(c(b(x1))) -> a#(a(a(c(x1)))) TRS: a(x1) -> b(c(x1)) b(b(x1)) -> x1 c(c(b(x1))) -> a(a(a(c(x1)))) Arctic Interpretation Processor: dimension: 2 interpretation: [c#](x0) = [0 1]x0, [a#](x0) = [0 1]x0 + [0], [1 0 ] [2] [b](x0) = [0 -&]x0 + [0], [-& 0 ] [0] [c](x0) = [0 -&]x0 + [1], [0 1 ] [2] [a](x0) = [-& 0 ]x0 + [0] orientation: c#(c(b(x1))) = [2 1]x1 + [3] >= [0 1]x1 = c#(x1) c#(c(b(x1))) = [2 1]x1 + [3] >= [1 0]x1 + [2] = a#(c(x1)) a#(x1) = [0 1]x1 + [0] >= [0 1]x1 = c#(x1) c#(c(b(x1))) = [2 1]x1 + [3] >= [1 0]x1 + [2] = a#(a(c(x1))) c#(c(b(x1))) = [2 1]x1 + [3] >= [1 0]x1 + [2] = a#(a(a(c(x1)))) [0 1 ] [2] [0 1 ] [2] a(x1) = [-& 0 ]x1 + [0] >= [-& 0 ]x1 + [0] = b(c(x1)) [2 1] [3] b(b(x1)) = [1 0]x1 + [2] >= x1 = x1 [1 0 ] [2] [1 0 ] [2] c(c(b(x1))) = [0 -&]x1 + [1] >= [0 -&]x1 + [1] = a(a(a(c(x1)))) problem: DPs: c#(c(b(x1))) -> c#(x1) a#(x1) -> c#(x1) TRS: a(x1) -> b(c(x1)) b(b(x1)) -> x1 c(c(b(x1))) -> a(a(a(c(x1)))) EDG Processor: DPs: c#(c(b(x1))) -> c#(x1) a#(x1) -> c#(x1) TRS: a(x1) -> b(c(x1)) b(b(x1)) -> x1 c(c(b(x1))) -> a(a(a(c(x1)))) graph: c#(c(b(x1))) -> c#(x1) -> c#(c(b(x1))) -> c#(x1) a#(x1) -> c#(x1) -> c#(c(b(x1))) -> c#(x1) CDG Processor: DPs: c#(c(b(x1))) -> c#(x1) a#(x1) -> c#(x1) TRS: a(x1) -> b(c(x1)) b(b(x1)) -> x1 c(c(b(x1))) -> a(a(a(c(x1)))) graph: Qed