YES Problem: b(a(x1)) -> b(b(x1)) b(a(b(x1))) -> b(a(a(x1))) a(a(a(x1))) -> a(b(b(x1))) Proof: DP Processor: DPs: b#(a(x1)) -> b#(x1) b#(a(x1)) -> b#(b(x1)) b#(a(b(x1))) -> a#(x1) b#(a(b(x1))) -> a#(a(x1)) b#(a(b(x1))) -> b#(a(a(x1))) a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) TRS: b(a(x1)) -> b(b(x1)) b(a(b(x1))) -> b(a(a(x1))) a(a(a(x1))) -> a(b(b(x1))) TDG Processor: DPs: b#(a(x1)) -> b#(x1) b#(a(x1)) -> b#(b(x1)) b#(a(b(x1))) -> a#(x1) b#(a(b(x1))) -> a#(a(x1)) b#(a(b(x1))) -> b#(a(a(x1))) a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) TRS: b(a(x1)) -> b(b(x1)) b(a(b(x1))) -> b(a(a(x1))) a(a(a(x1))) -> a(b(b(x1))) graph: 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))) -> b#(b(x1)) -> b#(a(b(x1))) -> b#(a(a(x1))) a#(a(a(x1))) -> b#(b(x1)) -> b#(a(b(x1))) -> a#(a(x1)) a#(a(a(x1))) -> b#(b(x1)) -> b#(a(b(x1))) -> a#(x1) a#(a(a(x1))) -> b#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) a#(a(a(x1))) -> b#(b(x1)) -> b#(a(x1)) -> b#(x1) a#(a(a(x1))) -> b#(x1) -> b#(a(b(x1))) -> b#(a(a(x1))) a#(a(a(x1))) -> b#(x1) -> b#(a(b(x1))) -> a#(a(x1)) a#(a(a(x1))) -> b#(x1) -> b#(a(b(x1))) -> a#(x1) a#(a(a(x1))) -> b#(x1) -> b#(a(x1)) -> b#(b(x1)) a#(a(a(x1))) -> b#(x1) -> b#(a(x1)) -> b#(x1) b#(a(b(x1))) -> a#(a(x1)) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(a(b(x1))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(b(x1)) b#(a(b(x1))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(x1) b#(a(b(x1))) -> a#(x1) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(a(b(x1))) -> a#(x1) -> a#(a(a(x1))) -> b#(b(x1)) b#(a(b(x1))) -> a#(x1) -> a#(a(a(x1))) -> b#(x1) b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(b(x1))) -> b#(a(a(x1))) b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(b(x1))) -> a#(a(x1)) b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(b(x1))) -> a#(x1) b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(x1)) -> b#(b(x1)) b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(x1)) -> b#(x1) b#(a(x1)) -> b#(b(x1)) -> b#(a(b(x1))) -> b#(a(a(x1))) b#(a(x1)) -> b#(b(x1)) -> b#(a(b(x1))) -> a#(a(x1)) b#(a(x1)) -> b#(b(x1)) -> b#(a(b(x1))) -> a#(x1) b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(x1) b#(a(x1)) -> b#(x1) -> b#(a(b(x1))) -> b#(a(a(x1))) b#(a(x1)) -> b#(x1) -> b#(a(b(x1))) -> a#(a(x1)) b#(a(x1)) -> b#(x1) -> b#(a(b(x1))) -> a#(x1) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1) CDG Processor: DPs: b#(a(x1)) -> b#(x1) b#(a(x1)) -> b#(b(x1)) b#(a(b(x1))) -> a#(x1) b#(a(b(x1))) -> a#(a(x1)) b#(a(b(x1))) -> b#(a(a(x1))) a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) TRS: b(a(x1)) -> b(b(x1)) b(a(b(x1))) -> b(a(a(x1))) a(a(a(x1))) -> a(b(b(x1))) graph: a#(a(a(x1))) -> b#(b(x1)) -> b#(a(x1)) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) a#(a(a(x1))) -> b#(b(x1)) -> b#(a(b(x1))) -> a#(x1) a#(a(a(x1))) -> b#(b(x1)) -> b#(a(b(x1))) -> a#(a(x1)) a#(a(a(x1))) -> b#(b(x1)) -> b#(a(b(x1))) -> b#(a(a(x1))) a#(a(a(x1))) -> b#(x1) -> b#(a(x1)) -> b#(x1) a#(a(a(x1))) -> b#(x1) -> b#(a(x1)) -> b#(b(x1)) a#(a(a(x1))) -> b#(x1) -> b#(a(b(x1))) -> a#(x1) a#(a(a(x1))) -> b#(x1) -> b#(a(b(x1))) -> a#(a(x1)) a#(a(a(x1))) -> b#(x1) -> b#(a(b(x1))) -> b#(a(a(x1))) b#(a(b(x1))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(x1) b#(a(b(x1))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(b(x1)) b#(a(b(x1))) -> a#(a(x1)) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(a(b(x1))) -> a#(x1) -> a#(a(a(x1))) -> b#(x1) b#(a(b(x1))) -> a#(x1) -> a#(a(a(x1))) -> b#(b(x1)) b#(a(b(x1))) -> a#(x1) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(x1)) -> b#(x1) b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(x1)) -> b#(b(x1)) b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(b(x1))) -> a#(x1) b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(b(x1))) -> a#(a(x1)) b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(b(x1))) -> b#(a(a(x1))) b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(x1) b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> b#(b(x1)) -> b#(a(b(x1))) -> a#(x1) b#(a(x1)) -> b#(b(x1)) -> b#(a(b(x1))) -> a#(a(x1)) b#(a(x1)) -> b#(b(x1)) -> b#(a(b(x1))) -> b#(a(a(x1))) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> b#(x1) -> b#(a(b(x1))) -> a#(x1) b#(a(x1)) -> b#(x1) -> b#(a(b(x1))) -> a#(a(x1)) b#(a(x1)) -> b#(x1) -> b#(a(b(x1))) -> b#(a(a(x1))) SCC Processor: #sccs: 1 #rules: 7 #arcs: 31/64 DPs: a#(a(a(x1))) -> b#(b(x1)) b#(a(b(x1))) -> b#(a(a(x1))) b#(a(b(x1))) -> a#(a(x1)) a#(a(a(x1))) -> b#(x1) b#(a(b(x1))) -> a#(x1) b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> b#(x1) TRS: b(a(x1)) -> b(b(x1)) b(a(b(x1))) -> b(a(a(x1))) a(a(a(x1))) -> a(b(b(x1))) Arctic Interpretation Processor: dimension: 1 interpretation: [a#](x0) = x0, [b#](x0) = 2x0 + 6, [b](x0) = 8x0 + 4, [a](x0) = 8x0 + 4 orientation: a#(a(a(x1))) = 16x1 + 12 >= 10x1 + 6 = b#(b(x1)) b#(a(b(x1))) = 18x1 + 14 >= 18x1 + 14 = b#(a(a(x1))) b#(a(b(x1))) = 18x1 + 14 >= 8x1 + 4 = a#(a(x1)) a#(a(a(x1))) = 16x1 + 12 >= 2x1 + 6 = b#(x1) b#(a(b(x1))) = 18x1 + 14 >= x1 = a#(x1) b#(a(x1)) = 10x1 + 6 >= 10x1 + 6 = b#(b(x1)) b#(a(x1)) = 10x1 + 6 >= 2x1 + 6 = b#(x1) b(a(x1)) = 16x1 + 12 >= 16x1 + 12 = b(b(x1)) b(a(b(x1))) = 24x1 + 20 >= 24x1 + 20 = b(a(a(x1))) a(a(a(x1))) = 24x1 + 20 >= 24x1 + 20 = a(b(b(x1))) problem: DPs: b#(a(b(x1))) -> b#(a(a(x1))) b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> b#(x1) TRS: b(a(x1)) -> b(b(x1)) b(a(b(x1))) -> b(a(a(x1))) a(a(a(x1))) -> a(b(b(x1))) EDG Processor: DPs: b#(a(b(x1))) -> b#(a(a(x1))) b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> b#(x1) TRS: b(a(x1)) -> b(b(x1)) b(a(b(x1))) -> b(a(a(x1))) a(a(a(x1))) -> a(b(b(x1))) graph: b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(b(x1))) -> b#(a(a(x1))) b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(x1)) -> b#(b(x1)) b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(x1)) -> b#(x1) b#(a(x1)) -> b#(b(x1)) -> b#(a(b(x1))) -> b#(a(a(x1))) b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(x1) b#(a(x1)) -> b#(x1) -> b#(a(b(x1))) -> b#(a(a(x1))) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1) Arctic Interpretation Processor: dimension: 1 interpretation: [b#](x0) = x0, [b](x0) = 2x0, [a](x0) = 2x0 orientation: b#(a(b(x1))) = 4x1 >= 4x1 = b#(a(a(x1))) b#(a(x1)) = 2x1 >= 2x1 = b#(b(x1)) b#(a(x1)) = 2x1 >= x1 = b#(x1) b(a(x1)) = 4x1 >= 4x1 = b(b(x1)) b(a(b(x1))) = 6x1 >= 6x1 = b(a(a(x1))) a(a(a(x1))) = 6x1 >= 6x1 = a(b(b(x1))) problem: DPs: b#(a(b(x1))) -> b#(a(a(x1))) b#(a(x1)) -> b#(b(x1)) TRS: b(a(x1)) -> b(b(x1)) b(a(b(x1))) -> b(a(a(x1))) a(a(a(x1))) -> a(b(b(x1))) EDG Processor: DPs: b#(a(b(x1))) -> b#(a(a(x1))) b#(a(x1)) -> b#(b(x1)) TRS: b(a(x1)) -> b(b(x1)) b(a(b(x1))) -> b(a(a(x1))) a(a(a(x1))) -> a(b(b(x1))) graph: b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(x1)) -> b#(b(x1)) b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(b(x1))) -> b#(a(a(x1))) b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) b#(a(x1)) -> b#(b(x1)) -> b#(a(b(x1))) -> b#(a(a(x1))) Arctic Interpretation Processor: dimension: 1 interpretation: [b#](x0) = x0 + -16, [b](x0) = 0, [a](x0) = 1 orientation: b#(a(b(x1))) = 1 >= 1 = b#(a(a(x1))) b#(a(x1)) = 1 >= 0 = b#(b(x1)) b(a(x1)) = 0 >= 0 = b(b(x1)) b(a(b(x1))) = 0 >= 0 = b(a(a(x1))) a(a(a(x1))) = 1 >= 1 = a(b(b(x1))) problem: DPs: b#(a(b(x1))) -> b#(a(a(x1))) TRS: b(a(x1)) -> b(b(x1)) b(a(b(x1))) -> b(a(a(x1))) a(a(a(x1))) -> a(b(b(x1))) EDG Processor: DPs: b#(a(b(x1))) -> b#(a(a(x1))) TRS: b(a(x1)) -> b(b(x1)) b(a(b(x1))) -> b(a(a(x1))) a(a(a(x1))) -> a(b(b(x1))) graph: b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(b(x1))) -> b#(a(a(x1))) Arctic Interpretation Processor: dimension: 2 interpretation: [b#](x0) = [-& 2 ]x0 + [0], [-& 3 ] [3 ] [b](x0) = [-& -&]x0 + [-&], [-& 2 ] [0] [a](x0) = [0 -&]x0 + [2] orientation: b#(a(b(x1))) = [-& 5 ]x1 + [5] >= [-& 4 ]x1 + [4] = b#(a(a(x1))) [3 -&] [5 ] [3 ] b(a(x1)) = [-& -&]x1 + [-&] >= [-&] = b(b(x1)) [-& 6 ] [6 ] [-& 5 ] [5 ] b(a(b(x1))) = [-& -&]x1 + [-&] >= [-& -&]x1 + [-&] = b(a(a(x1))) [-& 4 ] [4] [0] a(a(a(x1))) = [2 -&]x1 + [4] >= [3] = a(b(b(x1))) problem: DPs: TRS: b(a(x1)) -> b(b(x1)) b(a(b(x1))) -> b(a(a(x1))) a(a(a(x1))) -> a(b(b(x1))) Qed