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