YES Problem: b(b(x1)) -> b(a(b(a(a(b(x1)))))) b(b(a(b(x1)))) -> b(a(b(b(x1)))) b(a(a(a(b(a(a(b(x1)))))))) -> b(a(a(a(b(b(x1)))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(x1))))) Proof: DP Processor: DPs: b#(b(x1)) -> b#(a(a(b(x1)))) b#(b(x1)) -> b#(a(b(a(a(b(x1)))))) b#(b(a(b(x1)))) -> b#(b(x1)) b#(b(a(b(x1)))) -> b#(a(b(b(x1)))) b#(a(a(a(b(a(a(b(x1)))))))) -> b#(b(x1)) b#(a(a(a(b(a(a(b(x1)))))))) -> b#(a(a(a(b(b(x1)))))) b#(b(a(a(b(x1))))) -> b#(b(x1)) b#(b(a(a(b(x1))))) -> b#(a(a(b(b(x1))))) TRS: b(b(x1)) -> b(a(b(a(a(b(x1)))))) b(b(a(b(x1)))) -> b(a(b(b(x1)))) b(a(a(a(b(a(a(b(x1)))))))) -> b(a(a(a(b(b(x1)))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(x1))))) EDG Processor: DPs: b#(b(x1)) -> b#(a(a(b(x1)))) b#(b(x1)) -> b#(a(b(a(a(b(x1)))))) b#(b(a(b(x1)))) -> b#(b(x1)) b#(b(a(b(x1)))) -> b#(a(b(b(x1)))) b#(a(a(a(b(a(a(b(x1)))))))) -> b#(b(x1)) b#(a(a(a(b(a(a(b(x1)))))))) -> b#(a(a(a(b(b(x1)))))) b#(b(a(a(b(x1))))) -> b#(b(x1)) b#(b(a(a(b(x1))))) -> b#(a(a(b(b(x1))))) TRS: b(b(x1)) -> b(a(b(a(a(b(x1)))))) b(b(a(b(x1)))) -> b(a(b(b(x1)))) b(a(a(a(b(a(a(b(x1)))))))) -> b(a(a(a(b(b(x1)))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(x1))))) graph: b#(a(a(a(b(a(a(b(x1)))))))) -> b#(a(a(a(b(b(x1)))))) -> b#(a(a(a(b(a(a(b(x1)))))))) -> b#(b(x1)) b#(a(a(a(b(a(a(b(x1)))))))) -> b#(a(a(a(b(b(x1)))))) -> b#(a(a(a(b(a(a(b(x1)))))))) -> b#(a(a(a(b(b(x1)))))) b#(a(a(a(b(a(a(b(x1)))))))) -> b#(b(x1)) -> b#(b(x1)) -> b#(a(a(b(x1)))) b#(a(a(a(b(a(a(b(x1)))))))) -> b#(b(x1)) -> b#(b(x1)) -> b#(a(b(a(a(b(x1)))))) b#(a(a(a(b(a(a(b(x1)))))))) -> b#(b(x1)) -> b#(b(a(b(x1)))) -> b#(b(x1)) b#(a(a(a(b(a(a(b(x1)))))))) -> b#(b(x1)) -> b#(b(a(b(x1)))) -> b#(a(b(b(x1)))) b#(a(a(a(b(a(a(b(x1)))))))) -> b#(b(x1)) -> b#(b(a(a(b(x1))))) -> b#(b(x1)) b#(a(a(a(b(a(a(b(x1)))))))) -> b#(b(x1)) -> b#(b(a(a(b(x1))))) -> b#(a(a(b(b(x1))))) b#(b(a(a(b(x1))))) -> b#(b(x1)) -> b#(b(x1)) -> b#(a(a(b(x1)))) b#(b(a(a(b(x1))))) -> b#(b(x1)) -> b#(b(x1)) -> b#(a(b(a(a(b(x1)))))) b#(b(a(a(b(x1))))) -> b#(b(x1)) -> b#(b(a(b(x1)))) -> b#(b(x1)) b#(b(a(a(b(x1))))) -> b#(b(x1)) -> b#(b(a(b(x1)))) -> b#(a(b(b(x1)))) b#(b(a(a(b(x1))))) -> b#(b(x1)) -> b#(b(a(a(b(x1))))) -> b#(b(x1)) b#(b(a(a(b(x1))))) -> b#(b(x1)) -> b#(b(a(a(b(x1))))) -> b#(a(a(b(b(x1))))) b#(b(a(b(x1)))) -> b#(b(x1)) -> b#(b(x1)) -> b#(a(a(b(x1)))) b#(b(a(b(x1)))) -> b#(b(x1)) -> b#(b(x1)) -> b#(a(b(a(a(b(x1)))))) b#(b(a(b(x1)))) -> b#(b(x1)) -> b#(b(a(b(x1)))) -> b#(b(x1)) b#(b(a(b(x1)))) -> b#(b(x1)) -> b#(b(a(b(x1)))) -> b#(a(b(b(x1)))) b#(b(a(b(x1)))) -> b#(b(x1)) -> b#(b(a(a(b(x1))))) -> b#(b(x1)) b#(b(a(b(x1)))) -> b#(b(x1)) -> b#(b(a(a(b(x1))))) -> b#(a(a(b(b(x1))))) SCC Processor: #sccs: 2 #rules: 3 #arcs: 20/64 DPs: b#(a(a(a(b(a(a(b(x1)))))))) -> b#(a(a(a(b(b(x1)))))) TRS: b(b(x1)) -> b(a(b(a(a(b(x1)))))) b(b(a(b(x1)))) -> b(a(b(b(x1)))) b(a(a(a(b(a(a(b(x1)))))))) -> b(a(a(a(b(b(x1)))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(x1))))) Matrix Interpretation Processor: dim=4 interpretation: [b#](x0) = [0 0 1 0]x0, [0 0 1 0] [0] [0 0 0 1] [0] [a](x0) = [1 0 0 0]x0 + [0] [0 0 0 0] [1], [1 1 0 0] [0 0 0 0] [b](x0) = [0 0 0 0]x0 [0 0 0 0] orientation: b#(a(a(a(b(a(a(b(x1)))))))) = [1 1 0 0]x1 + [1] >= [1 1 0 0]x1 = b#(a(a(a(b(b(x1)))))) [1 1 0 0] [0] [0 0 0 0] [0] b(b(x1)) = [0 0 0 0]x1 >= [0] = b(a(b(a(a(b(x1)))))) [0 0 0 0] [0] [0] [0] [0] [0] b(b(a(b(x1)))) = [0] >= [0] = b(a(b(b(x1)))) [0] [0] [1] [1] [0] [0] b(a(a(a(b(a(a(b(x1)))))))) = [0] >= [0] = b(a(a(a(b(b(x1)))))) [0] [0] [1 1 0 0] [1] [1 1 0 0] [1] [0 0 0 0] [0] [0 0 0 0] [0] b(b(a(a(b(x1))))) = [0 0 0 0]x1 + [0] >= [0 0 0 0]x1 + [0] = b(a(a(b(b(x1))))) [0 0 0 0] [0] [0 0 0 0] [0] problem: DPs: TRS: b(b(x1)) -> b(a(b(a(a(b(x1)))))) b(b(a(b(x1)))) -> b(a(b(b(x1)))) b(a(a(a(b(a(a(b(x1)))))))) -> b(a(a(a(b(b(x1)))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(x1))))) Qed DPs: b#(b(a(a(b(x1))))) -> b#(b(x1)) b#(b(a(b(x1)))) -> b#(b(x1)) TRS: b(b(x1)) -> b(a(b(a(a(b(x1)))))) b(b(a(b(x1)))) -> b(a(b(b(x1)))) b(a(a(a(b(a(a(b(x1)))))))) -> b(a(a(a(b(b(x1)))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(x1))))) Matrix Interpretation Processor: dim=4 interpretation: [b#](x0) = [0 1 0 1]x0, [0 0 0 0] [0 0 0 1] [a](x0) = [0 1 0 0]x0 [1 1 0 0] , [0 0 0 0] [1] [0 0 1 0] [0] [b](x0) = [1 1 0 0]x0 + [0] [1 1 0 0] [0] orientation: b#(b(a(a(b(x1))))) = [1 1 1 0]x1 + [1] >= [1 1 1 0]x1 = b#(b(x1)) b#(b(a(b(x1)))) = [1 1 1 0]x1 >= [1 1 1 0]x1 = b#(b(x1)) [0 0 0 0] [1] [0 0 0 0] [1] [1 1 0 0] [0] [1 1 0 0] [0] b(b(x1)) = [0 0 1 0]x1 + [1] >= [0 0 1 0]x1 + [1] = b(a(b(a(a(b(x1)))))) [0 0 1 0] [1] [0 0 1 0] [1] [0 0 0 0] [1] [0 0 0 0] [1] [1 1 0 0] [0] [1 1 0 0] [0] b(b(a(b(x1)))) = [0 0 1 0]x1 + [1] >= [0 0 1 0]x1 + [1] = b(a(b(b(x1)))) [0 0 1 0] [1] [0 0 1 0] [1] [0 0 0 0] [1] [0 0 0 0] [1] [1 1 0 0] [1] [1 1 0 0] [1] b(a(a(a(b(a(a(b(x1)))))))) = [0 0 1 0]x1 + [1] >= [0 0 1 0]x1 + [1] = b(a(a(a(b(b(x1)))))) [0 0 1 0] [1] [0 0 1 0] [1] [0 0 0 0] [1] [0 0 0 0] [1] [0 0 1 0] [1] [0 0 1 0] [1] b(b(a(a(b(x1))))) = [1 1 0 0]x1 + [1] >= [1 1 0 0]x1 + [1] = b(a(a(b(b(x1))))) [1 1 0 0] [1] [1 1 0 0] [1] problem: DPs: b#(b(a(b(x1)))) -> b#(b(x1)) TRS: b(b(x1)) -> b(a(b(a(a(b(x1)))))) b(b(a(b(x1)))) -> b(a(b(b(x1)))) b(a(a(a(b(a(a(b(x1)))))))) -> b(a(a(a(b(b(x1)))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(x1))))) Restore Modifier: DPs: b#(b(a(b(x1)))) -> b#(b(x1)) TRS: b(b(x1)) -> b(a(b(a(a(b(x1)))))) b(b(a(b(x1)))) -> b(a(b(b(x1)))) b(a(a(a(b(a(a(b(x1)))))))) -> b(a(a(a(b(b(x1)))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(x1))))) EDG Processor: DPs: b#(b(a(b(x1)))) -> b#(b(x1)) TRS: b(b(x1)) -> b(a(b(a(a(b(x1)))))) b(b(a(b(x1)))) -> b(a(b(b(x1)))) b(a(a(a(b(a(a(b(x1)))))))) -> b(a(a(a(b(b(x1)))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(x1))))) graph: b#(b(a(b(x1)))) -> b#(b(x1)) -> b#(b(a(b(x1)))) -> b#(b(x1)) Arctic Interpretation Processor: dimension: 2 usable rules: b(b(x1)) -> b(a(b(a(a(b(x1)))))) b(b(a(b(x1)))) -> b(a(b(b(x1)))) b(a(a(a(b(a(a(b(x1)))))))) -> b(a(a(a(b(b(x1)))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(x1))))) interpretation: [b#](x0) = [-& 0 ]x0 + [0], [-& -&] [0] [a](x0) = [1 -&]x0 + [0], [0 0] [1] [b](x0) = [0 1]x0 + [2] orientation: b#(b(a(b(x1)))) = [2 2]x1 + [3] >= [0 1]x1 + [2] = b#(b(x1)) [0 1] [2] [2] b(b(x1)) = [1 2]x1 + [3] >= [3] = b(a(b(a(a(b(x1)))))) [2 2] [3] [1 2] [3] b(b(a(b(x1)))) = [3 3]x1 + [4] >= [2 3]x1 + [4] = b(a(b(b(x1)))) [1] [1] b(a(a(a(b(a(a(b(x1)))))))) = [2] >= [2] = b(a(a(a(b(b(x1)))))) [2] [1] b(b(a(a(b(x1))))) = [3] >= [2] = b(a(a(b(b(x1))))) problem: DPs: TRS: b(b(x1)) -> b(a(b(a(a(b(x1)))))) b(b(a(b(x1)))) -> b(a(b(b(x1)))) b(a(a(a(b(a(a(b(x1)))))))) -> b(a(a(a(b(b(x1)))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(x1))))) Qed