YES Problem: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) c(b(x1)) -> b(a(x1)) a(a(x1)) -> a(b(a(x1))) Proof: DP Processor: DPs: b#(a(a(x1))) -> c#(x1) b#(a(a(x1))) -> b#(c(x1)) b#(a(a(x1))) -> a#(b(c(x1))) c#(a(x1)) -> c#(x1) c#(a(x1)) -> a#(c(x1)) c#(b(x1)) -> a#(x1) c#(b(x1)) -> b#(a(x1)) a#(a(x1)) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) c(b(x1)) -> b(a(x1)) a(a(x1)) -> a(b(a(x1))) TDG Processor: DPs: b#(a(a(x1))) -> c#(x1) b#(a(a(x1))) -> b#(c(x1)) b#(a(a(x1))) -> a#(b(c(x1))) c#(a(x1)) -> c#(x1) c#(a(x1)) -> a#(c(x1)) c#(b(x1)) -> a#(x1) c#(b(x1)) -> b#(a(x1)) a#(a(x1)) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) c(b(x1)) -> b(a(x1)) a(a(x1)) -> a(b(a(x1))) graph: a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(a(x1))) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> b#(a(x1)) a#(a(x1)) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(b(c(x1))) a#(a(x1)) -> b#(a(x1)) -> b#(a(a(x1))) -> b#(c(x1)) a#(a(x1)) -> b#(a(x1)) -> b#(a(a(x1))) -> c#(x1) c#(b(x1)) -> a#(x1) -> a#(a(x1)) -> a#(b(a(x1))) c#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(a(x1)) c#(b(x1)) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(b(c(x1))) c#(b(x1)) -> b#(a(x1)) -> b#(a(a(x1))) -> b#(c(x1)) c#(b(x1)) -> b#(a(x1)) -> b#(a(a(x1))) -> c#(x1) c#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(a(x1))) c#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(a(x1)) c#(a(x1)) -> c#(x1) -> c#(b(x1)) -> b#(a(x1)) c#(a(x1)) -> c#(x1) -> c#(b(x1)) -> a#(x1) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1) b#(a(a(x1))) -> a#(b(c(x1))) -> a#(a(x1)) -> a#(b(a(x1))) b#(a(a(x1))) -> a#(b(c(x1))) -> a#(a(x1)) -> b#(a(x1)) b#(a(a(x1))) -> c#(x1) -> c#(b(x1)) -> b#(a(x1)) b#(a(a(x1))) -> c#(x1) -> c#(b(x1)) -> a#(x1) b#(a(a(x1))) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) b#(a(a(x1))) -> c#(x1) -> c#(a(x1)) -> c#(x1) b#(a(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> a#(b(c(x1))) b#(a(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> b#(c(x1)) b#(a(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> c#(x1) Arctic Interpretation Processor: dimension: 1 interpretation: [a#](x0) = x0, [c#](x0) = 4x0, [b#](x0) = x0, [c](x0) = 2x0, [b](x0) = x0, [a](x0) = 2x0 orientation: b#(a(a(x1))) = 4x1 >= 4x1 = c#(x1) b#(a(a(x1))) = 4x1 >= 2x1 = b#(c(x1)) b#(a(a(x1))) = 4x1 >= 2x1 = a#(b(c(x1))) c#(a(x1)) = 6x1 >= 4x1 = c#(x1) c#(a(x1)) = 6x1 >= 2x1 = a#(c(x1)) c#(b(x1)) = 4x1 >= x1 = a#(x1) c#(b(x1)) = 4x1 >= 2x1 = b#(a(x1)) a#(a(x1)) = 2x1 >= 2x1 = b#(a(x1)) a#(a(x1)) = 2x1 >= 2x1 = a#(b(a(x1))) b(a(a(x1))) = 4x1 >= 4x1 = a(b(c(x1))) c(a(x1)) = 4x1 >= 4x1 = a(c(x1)) c(b(x1)) = 2x1 >= 2x1 = b(a(x1)) a(a(x1)) = 4x1 >= 4x1 = a(b(a(x1))) problem: DPs: b#(a(a(x1))) -> c#(x1) a#(a(x1)) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) c(b(x1)) -> b(a(x1)) a(a(x1)) -> a(b(a(x1))) EDG Processor: DPs: b#(a(a(x1))) -> c#(x1) a#(a(x1)) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) c(b(x1)) -> b(a(x1)) a(a(x1)) -> a(b(a(x1))) graph: a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> b#(a(x1)) a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(a(x1))) a#(a(x1)) -> b#(a(x1)) -> b#(a(a(x1))) -> c#(x1) SCC Processor: #sccs: 1 #rules: 1 #arcs: 3/9 DPs: a#(a(x1)) -> a#(b(a(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) c(b(x1)) -> b(a(x1)) a(a(x1)) -> a(b(a(x1))) Arctic Interpretation Processor: dimension: 1 interpretation: [a#](x0) = x0 + 0, [c](x0) = 2x0 + 4, [b](x0) = -2x0 + 0, [a](x0) = 2x0 + 4 orientation: a#(a(x1)) = 2x1 + 4 >= x1 + 2 = a#(b(a(x1))) b(a(a(x1))) = 2x1 + 4 >= 2x1 + 4 = a(b(c(x1))) c(a(x1)) = 4x1 + 6 >= 4x1 + 6 = a(c(x1)) c(b(x1)) = x1 + 4 >= x1 + 2 = b(a(x1)) a(a(x1)) = 4x1 + 6 >= 2x1 + 4 = a(b(a(x1))) problem: DPs: TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) c(b(x1)) -> b(a(x1)) a(a(x1)) -> a(b(a(x1))) Qed