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