YES Problem: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) Proof: DP Processor: DPs: a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(a(b(b(x1)))) b#(b(b(b(x1)))) -> a#(x1) b#(b(b(b(x1)))) -> a#(a(x1)) b#(b(b(b(x1)))) -> b#(a(a(x1))) b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) TRS: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) TDG Processor: DPs: a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(a(b(b(x1)))) b#(b(b(b(x1)))) -> a#(x1) b#(b(b(b(x1)))) -> a#(a(x1)) b#(b(b(b(x1)))) -> b#(a(a(x1))) b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) TRS: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) graph: b#(b(b(b(x1)))) -> b#(a(a(x1))) -> b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) b#(b(b(b(x1)))) -> b#(a(a(x1))) -> b#(b(b(b(x1)))) -> b#(a(a(x1))) b#(b(b(b(x1)))) -> b#(a(a(x1))) -> b#(b(b(b(x1)))) -> a#(a(x1)) b#(b(b(b(x1)))) -> b#(a(a(x1))) -> b#(b(b(b(x1)))) -> a#(x1) b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) -> a#(a(a(x1))) -> b#(b(x1)) b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) -> a#(a(a(x1))) -> b#(x1) b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(b(x1)) b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(x1) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> b#(b(x1)) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> b#(a(a(x1))) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> a#(a(x1)) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> a#(x1) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> b#(a(a(x1))) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(a(x1)) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(x1) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> b#(x1) EDG Processor: DPs: a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(a(b(b(x1)))) b#(b(b(b(x1)))) -> a#(x1) b#(b(b(b(x1)))) -> a#(a(x1)) b#(b(b(b(x1)))) -> b#(a(a(x1))) b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) TRS: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) graph: b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) -> a#(a(a(x1))) -> b#(x1) b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) -> a#(a(a(x1))) -> b#(b(x1)) b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(x1) b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(b(x1)) b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> b#(x1) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> b#(b(x1)) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> a#(x1) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> a#(a(x1)) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> b#(a(a(x1))) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(x1) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(a(x1)) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> b#(a(a(x1))) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) CDG Processor: DPs: a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(a(b(b(x1)))) b#(b(b(b(x1)))) -> a#(x1) b#(b(b(b(x1)))) -> a#(a(x1)) b#(b(b(b(x1)))) -> b#(a(a(x1))) b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) TRS: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) graph: b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(b(x1)) b#(b(b(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(x1) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> b#(b(x1)) b#(b(b(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> b#(a(a(x1))) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> a#(a(x1)) a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(b(x1)))) -> a#(x1) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(b(a(a(x1)))) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> b#(a(a(x1))) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(a(x1)) a#(a(a(x1))) -> b#(x1) -> b#(b(b(b(x1)))) -> a#(x1) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> b#(x1) SCC Processor: #sccs: 1 #rules: 6 #arcs: 24/64 DPs: b#(b(b(b(x1)))) -> a#(a(x1)) a#(a(a(x1))) -> b#(x1) b#(b(b(b(x1)))) -> a#(x1) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(a(b(b(x1)))) TRS: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) Arctic Interpretation Processor: dimension: 2 interpretation: [b#](x0) = [-4 -2]x0 + [0], [a#](x0) = [-2 -4]x0 + [0], [-& 0 ] [3 ] [b](x0) = [1 -1]x0 + [-4], [1 -1] [3] [a](x0) = [-& -&]x0 + [0] orientation: b#(b(b(b(x1)))) = [0 -2]x1 + [2] >= [-1 -3]x1 + [1] = a#(a(x1)) a#(a(a(x1))) = [0 -2]x1 + [2] >= [-4 -2]x1 + [0] = b#(x1) b#(b(b(b(x1)))) = [0 -2]x1 + [2] >= [-2 -4]x1 + [0] = a#(x1) a#(a(a(x1))) = [0 -2]x1 + [2] >= [-1 -3]x1 + [0] = b#(b(x1)) a#(a(a(x1))) = [0 -2]x1 + [2] >= [-1 -3]x1 + [1] = a#(b(b(x1))) a#(a(a(x1))) = [0 -2]x1 + [2] >= [0 -2]x1 + [2] = a#(a(b(b(x1)))) [3 1 ] [5] [3 1 ] [5] a(a(a(x1))) = [-& -&]x1 + [0] >= [-& -&]x1 + [0] = a(a(b(b(x1)))) [2 0] [4] [2 0 ] [4] b(b(b(b(x1)))) = [1 2]x1 + [5] >= [-& -&]x1 + [0] = a(b(a(a(x1)))) problem: DPs: a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(a(b(b(x1)))) TRS: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) EDG Processor: DPs: a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(a(b(b(x1)))) TRS: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) graph: a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(a(b(b(x1)))) -> a#(a(a(x1))) -> a#(a(b(b(x1)))) SCC Processor: #sccs: 1 #rules: 1 #arcs: 2/4 DPs: a#(a(a(x1))) -> a#(a(b(b(x1)))) TRS: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) Matrix Interpretation Processor: dim=2 interpretation: [a#](x0) = [1 0]x0, [0 1] [0] [b](x0) = [1 0]x0 + [1], [1 0] [2] [a](x0) = [0 0]x0 + [0] orientation: a#(a(a(x1))) = [1 0]x1 + [4] >= [1 0]x1 + [3] = a#(a(b(b(x1)))) [1 0] [6] [1 0] [5] a(a(a(x1))) = [0 0]x1 + [0] >= [0 0]x1 + [0] = a(a(b(b(x1)))) [2] [2] b(b(b(b(x1)))) = x1 + [2] >= [0] = a(b(a(a(x1)))) problem: DPs: TRS: a(a(a(x1))) -> a(a(b(b(x1)))) b(b(b(b(x1)))) -> a(b(a(a(x1)))) Qed