YES Problem: a(b(x1)) -> b(a(a(a(x1)))) b(a(x1)) -> a(a(x1)) a(a(x1)) -> a(c(b(x1))) Proof: DP Processor: DPs: a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> a#(a(a(x1))) a#(b(x1)) -> b#(a(a(a(x1)))) b#(a(x1)) -> a#(a(x1)) a#(a(x1)) -> b#(x1) a#(a(x1)) -> a#(c(b(x1))) TRS: a(b(x1)) -> b(a(a(a(x1)))) b(a(x1)) -> a(a(x1)) a(a(x1)) -> a(c(b(x1))) TDG Processor: DPs: a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> a#(a(a(x1))) a#(b(x1)) -> b#(a(a(a(x1)))) b#(a(x1)) -> a#(a(x1)) a#(a(x1)) -> b#(x1) a#(a(x1)) -> a#(c(b(x1))) TRS: a(b(x1)) -> b(a(a(a(x1)))) b(a(x1)) -> a(a(x1)) a(a(x1)) -> a(c(b(x1))) graph: b#(a(x1)) -> a#(a(x1)) -> a#(a(x1)) -> a#(c(b(x1))) b#(a(x1)) -> a#(a(x1)) -> a#(a(x1)) -> b#(x1) b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(a(a(a(x1)))) b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(a(x1))) b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(x1) a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(a(x1)) a#(a(x1)) -> a#(c(b(x1))) -> a#(a(x1)) -> a#(c(b(x1))) a#(a(x1)) -> a#(c(b(x1))) -> a#(a(x1)) -> b#(x1) a#(a(x1)) -> a#(c(b(x1))) -> a#(b(x1)) -> b#(a(a(a(x1)))) a#(a(x1)) -> a#(c(b(x1))) -> a#(b(x1)) -> a#(a(a(x1))) a#(a(x1)) -> a#(c(b(x1))) -> a#(b(x1)) -> a#(a(x1)) a#(a(x1)) -> a#(c(b(x1))) -> a#(b(x1)) -> a#(x1) a#(b(x1)) -> b#(a(a(a(x1)))) -> b#(a(x1)) -> a#(a(x1)) a#(b(x1)) -> a#(a(a(x1))) -> a#(a(x1)) -> a#(c(b(x1))) a#(b(x1)) -> a#(a(a(x1))) -> a#(a(x1)) -> b#(x1) a#(b(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> b#(a(a(a(x1)))) a#(b(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> a#(a(a(x1))) a#(b(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> a#(c(b(x1))) a#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> b#(x1) a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(a(a(a(x1)))) a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(a(x1))) a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> a#(c(b(x1))) a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(a(a(x1)))) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(a(a(x1))) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) EDG Processor: DPs: a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> a#(a(a(x1))) a#(b(x1)) -> b#(a(a(a(x1)))) b#(a(x1)) -> a#(a(x1)) a#(a(x1)) -> b#(x1) a#(a(x1)) -> a#(c(b(x1))) TRS: a(b(x1)) -> b(a(a(a(x1)))) b(a(x1)) -> a(a(x1)) a(a(x1)) -> a(c(b(x1))) graph: b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(x1) b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(a(x1))) b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(a(a(a(x1)))) b#(a(x1)) -> a#(a(x1)) -> a#(a(x1)) -> b#(x1) b#(a(x1)) -> a#(a(x1)) -> a#(a(x1)) -> a#(c(b(x1))) a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(a(x1)) a#(b(x1)) -> b#(a(a(a(x1)))) -> b#(a(x1)) -> a#(a(x1)) a#(b(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> a#(a(a(x1))) a#(b(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> b#(a(a(a(x1)))) a#(b(x1)) -> a#(a(a(x1))) -> a#(a(x1)) -> b#(x1) a#(b(x1)) -> a#(a(a(x1))) -> a#(a(x1)) -> a#(c(b(x1))) a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(a(x1))) a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(a(a(a(x1)))) a#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> b#(x1) a#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> a#(c(b(x1))) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(a(a(x1))) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(a(a(x1)))) a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1) a#(b(x1)) -> a#(x1) -> a#(a(x1)) -> a#(c(b(x1))) SCC Processor: #sccs: 1 #rules: 6 #arcs: 26/49 DPs: b#(a(x1)) -> a#(a(x1)) a#(a(x1)) -> b#(x1) a#(b(x1)) -> b#(a(a(a(x1)))) a#(b(x1)) -> a#(a(a(x1))) a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> a#(x1) TRS: a(b(x1)) -> b(a(a(a(x1)))) b(a(x1)) -> a(a(x1)) a(a(x1)) -> a(c(b(x1))) Arctic Interpretation Processor: dimension: 1 interpretation: [b#](x0) = x0 + 0, [a#](x0) = x0 + 0, [c](x0) = 0, [a](x0) = x0 + 0, [b](x0) = 1x0 + 1 orientation: b#(a(x1)) = x1 + 0 >= x1 + 0 = a#(a(x1)) a#(a(x1)) = x1 + 0 >= x1 + 0 = b#(x1) a#(b(x1)) = 1x1 + 1 >= x1 + 0 = b#(a(a(a(x1)))) a#(b(x1)) = 1x1 + 1 >= x1 + 0 = a#(a(a(x1))) a#(b(x1)) = 1x1 + 1 >= x1 + 0 = a#(a(x1)) a#(b(x1)) = 1x1 + 1 >= x1 + 0 = a#(x1) a(b(x1)) = 1x1 + 1 >= 1x1 + 1 = b(a(a(a(x1)))) b(a(x1)) = 1x1 + 1 >= x1 + 0 = a(a(x1)) a(a(x1)) = x1 + 0 >= 0 = a(c(b(x1))) problem: DPs: b#(a(x1)) -> a#(a(x1)) a#(a(x1)) -> b#(x1) TRS: a(b(x1)) -> b(a(a(a(x1)))) b(a(x1)) -> a(a(x1)) a(a(x1)) -> a(c(b(x1))) EDG Processor: DPs: b#(a(x1)) -> a#(a(x1)) a#(a(x1)) -> b#(x1) TRS: a(b(x1)) -> b(a(a(a(x1)))) b(a(x1)) -> a(a(x1)) a(a(x1)) -> a(c(b(x1))) graph: b#(a(x1)) -> a#(a(x1)) -> a#(a(x1)) -> b#(x1) a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(a(x1)) KBO Processor: argument filtering: pi(b) = [0] pi(a) = [0] pi(c) = [] pi(a#) = 0 pi(b#) = 0 weight function: w0 = 1 w(a#) = w(c) = w(b) = 1 w(b#) = w(a) = 0 precedence: b# ~ a# ~ a > c ~ b problem: DPs: b#(a(x1)) -> a#(a(x1)) TRS: a(b(x1)) -> b(a(a(a(x1)))) b(a(x1)) -> a(a(x1)) a(a(x1)) -> a(c(b(x1))) EDG Processor: DPs: b#(a(x1)) -> a#(a(x1)) TRS: a(b(x1)) -> b(a(a(a(x1)))) b(a(x1)) -> a(a(x1)) a(a(x1)) -> a(c(b(x1))) graph: Qed