YES Problem: a(a(b(a(x1)))) -> a(b(b(x1))) b(b(x1)) -> b(a(a(a(x1)))) Proof: DP Processor: DPs: a#(a(b(a(x1)))) -> b#(x1) a#(a(b(a(x1)))) -> b#(b(x1)) a#(a(b(a(x1)))) -> a#(b(b(x1))) b#(b(x1)) -> a#(x1) b#(b(x1)) -> a#(a(x1)) b#(b(x1)) -> a#(a(a(x1))) b#(b(x1)) -> b#(a(a(a(x1)))) TRS: a(a(b(a(x1)))) -> a(b(b(x1))) b(b(x1)) -> b(a(a(a(x1)))) TDG Processor: DPs: a#(a(b(a(x1)))) -> b#(x1) a#(a(b(a(x1)))) -> b#(b(x1)) a#(a(b(a(x1)))) -> a#(b(b(x1))) b#(b(x1)) -> a#(x1) b#(b(x1)) -> a#(a(x1)) b#(b(x1)) -> a#(a(a(x1))) b#(b(x1)) -> b#(a(a(a(x1)))) TRS: a(a(b(a(x1)))) -> a(b(b(x1))) b(b(x1)) -> b(a(a(a(x1)))) graph: b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> b#(a(a(a(x1)))) b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> a#(a(a(x1))) b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> a#(a(x1)) b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> a#(x1) b#(b(x1)) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> a#(b(b(x1))) b#(b(x1)) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(x1)) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> b#(x1) b#(b(x1)) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> a#(b(b(x1))) b#(b(x1)) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(x1)) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> b#(x1) b#(b(x1)) -> a#(x1) -> a#(a(b(a(x1)))) -> a#(b(b(x1))) b#(b(x1)) -> a#(x1) -> a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(x1)) -> a#(x1) -> a#(a(b(a(x1)))) -> b#(x1) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(x1)) -> b#(a(a(a(x1)))) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(x1)) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(x1)) -> a#(a(x1)) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(x1)) -> a#(x1) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(x1)) -> b#(a(a(a(x1)))) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(x1)) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(x1)) -> a#(a(x1)) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(x1)) -> a#(x1) a#(a(b(a(x1)))) -> a#(b(b(x1))) -> a#(a(b(a(x1)))) -> a#(b(b(x1))) a#(a(b(a(x1)))) -> a#(b(b(x1))) -> a#(a(b(a(x1)))) -> b#(b(x1)) a#(a(b(a(x1)))) -> a#(b(b(x1))) -> a#(a(b(a(x1)))) -> b#(x1) CDG Processor: DPs: a#(a(b(a(x1)))) -> b#(x1) a#(a(b(a(x1)))) -> b#(b(x1)) a#(a(b(a(x1)))) -> a#(b(b(x1))) b#(b(x1)) -> a#(x1) b#(b(x1)) -> a#(a(x1)) b#(b(x1)) -> a#(a(a(x1))) b#(b(x1)) -> b#(a(a(a(x1)))) TRS: a(a(b(a(x1)))) -> a(b(b(x1))) b(b(x1)) -> b(a(a(a(x1)))) graph: b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> a#(x1) b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> a#(a(x1)) b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> a#(a(a(x1))) b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> b#(a(a(a(x1)))) b#(b(x1)) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> b#(x1) b#(b(x1)) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(x1)) -> a#(a(a(x1))) -> a#(a(b(a(x1)))) -> a#(b(b(x1))) b#(b(x1)) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> b#(x1) b#(b(x1)) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(x1)) -> a#(a(x1)) -> a#(a(b(a(x1)))) -> a#(b(b(x1))) b#(b(x1)) -> a#(x1) -> a#(a(b(a(x1)))) -> b#(x1) b#(b(x1)) -> a#(x1) -> a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(x1)) -> a#(x1) -> a#(a(b(a(x1)))) -> a#(b(b(x1))) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(x1)) -> a#(x1) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(x1)) -> a#(a(x1)) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(x1)) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(b(x1)) -> b#(b(x1)) -> b#(a(a(a(x1)))) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(x1)) -> a#(x1) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(x1)) -> a#(a(x1)) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(x1)) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(x1) -> b#(b(x1)) -> b#(a(a(a(x1)))) SCC Processor: #sccs: 1 #rules: 6 #arcs: 21/49 DPs: b#(b(x1)) -> b#(a(a(a(x1)))) b#(b(x1)) -> a#(a(a(x1))) a#(a(b(a(x1)))) -> b#(b(x1)) b#(b(x1)) -> a#(a(x1)) a#(a(b(a(x1)))) -> b#(x1) b#(b(x1)) -> a#(x1) TRS: a(a(b(a(x1)))) -> a(b(b(x1))) b(b(x1)) -> b(a(a(a(x1)))) Arctic Interpretation Processor: dimension: 2 interpretation: [b#](x0) = [0 2]x0, [a#](x0) = [-& 2 ]x0 + [0], [0 2] [0] [b](x0) = [0 0]x0 + [2], [-& 0 ] [0] [a](x0) = [0 0 ]x0 + [2] orientation: b#(b(x1)) = [2 2]x1 + [4] >= [2 2]x1 + [4] = b#(a(a(a(x1)))) b#(b(x1)) = [2 2]x1 + [4] >= [2 2]x1 + [4] = a#(a(a(x1))) a#(a(b(a(x1)))) = [4 4]x1 + [6] >= [2 2]x1 + [4] = b#(b(x1)) b#(b(x1)) = [2 2]x1 + [4] >= [2 2]x1 + [4] = a#(a(x1)) a#(a(b(a(x1)))) = [4 4]x1 + [6] >= [0 2]x1 = b#(x1) b#(b(x1)) = [2 2]x1 + [4] >= [-& 2 ]x1 + [0] = a#(x1) [2 2] [4] [0 2] [2] a(a(b(a(x1)))) = [2 2]x1 + [4] >= [2 2]x1 + [4] = a(b(b(x1))) [2 2] [4] [2 2] [4] b(b(x1)) = [0 2]x1 + [2] >= [0 0]x1 + [2] = b(a(a(a(x1)))) problem: DPs: b#(b(x1)) -> b#(a(a(a(x1)))) b#(b(x1)) -> a#(a(a(x1))) b#(b(x1)) -> a#(a(x1)) b#(b(x1)) -> a#(x1) TRS: a(a(b(a(x1)))) -> a(b(b(x1))) b(b(x1)) -> b(a(a(a(x1)))) EDG Processor: DPs: b#(b(x1)) -> b#(a(a(a(x1)))) b#(b(x1)) -> a#(a(a(x1))) b#(b(x1)) -> a#(a(x1)) b#(b(x1)) -> a#(x1) TRS: a(a(b(a(x1)))) -> a(b(b(x1))) b(b(x1)) -> b(a(a(a(x1)))) graph: b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> b#(a(a(a(x1)))) b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> a#(a(a(x1))) b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> a#(a(x1)) b#(b(x1)) -> b#(a(a(a(x1)))) -> b#(b(x1)) -> a#(x1) SCC Processor: #sccs: 1 #rules: 1 #arcs: 4/16 DPs: b#(b(x1)) -> b#(a(a(a(x1)))) TRS: a(a(b(a(x1)))) -> a(b(b(x1))) b(b(x1)) -> b(a(a(a(x1)))) Arctic Interpretation Processor: dimension: 1 interpretation: [b#](x0) = x0 + 0, [b](x0) = x0 + 1, [a](x0) = 0 orientation: b#(b(x1)) = x1 + 1 >= 0 = b#(a(a(a(x1)))) a(a(b(a(x1)))) = 0 >= 0 = a(b(b(x1))) b(b(x1)) = x1 + 1 >= 1 = b(a(a(a(x1)))) problem: DPs: TRS: a(a(b(a(x1)))) -> a(b(b(x1))) b(b(x1)) -> b(a(a(a(x1)))) Qed