YES Problem: a(x1) -> b(x1) a(a(x1)) -> a(b(a(x1))) a(b(x1)) -> b(b(b(x1))) a(a(a(x1))) -> a(a(b(a(a(x1))))) a(a(b(x1))) -> a(b(b(a(b(x1))))) a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(b(x1))) -> b(b(b(b(b(x1))))) a(a(a(a(x1)))) -> a(a(a(b(a(a(a(x1))))))) a(a(a(b(x1)))) -> a(a(b(b(a(a(b(x1))))))) a(a(b(a(x1)))) -> a(b(a(b(a(b(a(x1))))))) a(a(b(b(x1)))) -> a(b(b(b(a(b(b(x1))))))) a(b(a(a(x1)))) -> b(a(a(b(b(a(a(x1))))))) a(b(a(b(x1)))) -> b(a(b(b(b(a(b(x1))))))) a(b(b(a(x1)))) -> b(b(a(b(b(b(a(x1))))))) a(b(b(b(x1)))) -> b(b(b(b(b(b(b(x1))))))) Proof: String Reversal Processor: a(x1) -> b(x1) a(a(x1)) -> a(b(a(x1))) b(a(x1)) -> b(b(b(x1))) a(a(a(x1))) -> a(a(b(a(a(x1))))) b(a(a(x1))) -> b(a(b(b(a(x1))))) a(b(a(x1))) -> a(b(b(a(b(x1))))) b(b(a(x1))) -> b(b(b(b(b(x1))))) a(a(a(a(x1)))) -> a(a(a(b(a(a(a(x1))))))) b(a(a(a(x1)))) -> b(a(a(b(b(a(a(x1))))))) a(b(a(a(x1)))) -> a(b(a(b(a(b(a(x1))))))) b(b(a(a(x1)))) -> b(b(a(b(b(b(a(x1))))))) a(a(b(a(x1)))) -> a(a(b(b(a(a(b(x1))))))) b(a(b(a(x1)))) -> b(a(b(b(b(a(b(x1))))))) a(b(b(a(x1)))) -> a(b(b(b(a(b(b(x1))))))) b(b(b(a(x1)))) -> b(b(b(b(b(b(b(x1))))))) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [b](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [0] [a](x0) = [0 0 0]x0 + [1] [0 1 0] [0] orientation: [1 0 1] [0] [1 0 0] a(x1) = [0 0 0]x1 + [1] >= [0 0 0]x1 = b(x1) [0 1 0] [0] [0 0 0] [1 1 1] [0] [1 0 1] [0] a(a(x1)) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = a(b(a(x1))) [0 0 0] [1] [0 0 0] [0] [1 0 1] [1 0 0] b(a(x1)) = [0 0 0]x1 >= [0 0 0]x1 = b(b(b(x1))) [0 0 0] [0 0 0] [1 1 1] [1] [1 1 1] [0] a(a(a(x1))) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = a(a(b(a(a(x1))))) [0 0 0] [1] [0 0 0] [1] [1 1 1] [1 0 1] b(a(a(x1))) = [0 0 0]x1 >= [0 0 0]x1 = b(a(b(b(a(x1))))) [0 0 0] [0 0 0] [1 0 1] [0] [1 0 0] [0] a(b(a(x1))) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = a(b(b(a(b(x1))))) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1 0 0] b(b(a(x1))) = [0 0 0]x1 >= [0 0 0]x1 = b(b(b(b(b(x1))))) [0 0 0] [0 0 0] [1 1 1] [2] [1 1 1] [2] a(a(a(a(x1)))) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = a(a(a(b(a(a(a(x1))))))) [0 0 0] [1] [0 0 0] [1] [1 1 1] [1] [1 1 1] b(a(a(a(x1)))) = [0 0 0]x1 + [0] >= [0 0 0]x1 = b(a(a(b(b(a(a(x1))))))) [0 0 0] [0] [0 0 0] [1 1 1] [0] [1 0 1] [0] a(b(a(a(x1)))) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = a(b(a(b(a(b(a(x1))))))) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1 0 1] b(b(a(a(x1)))) = [0 0 0]x1 >= [0 0 0]x1 = b(b(a(b(b(b(a(x1))))))) [0 0 0] [0 0 0] [1 0 1] [0] [1 0 0] [0] a(a(b(a(x1)))) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = a(a(b(b(a(a(b(x1))))))) [0 0 0] [1] [0 0 0] [1] [1 0 1] [1 0 0] b(a(b(a(x1)))) = [0 0 0]x1 >= [0 0 0]x1 = b(a(b(b(b(a(b(x1))))))) [0 0 0] [0 0 0] [1 0 1] [0] [1 0 0] [0] a(b(b(a(x1)))) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = a(b(b(b(a(b(b(x1))))))) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1 0 0] b(b(b(a(x1)))) = [0 0 0]x1 >= [0 0 0]x1 = b(b(b(b(b(b(b(x1))))))) [0 0 0] [0 0 0] problem: a(x1) -> b(x1) a(a(x1)) -> a(b(a(x1))) b(a(x1)) -> b(b(b(x1))) b(a(a(x1))) -> b(a(b(b(a(x1))))) a(b(a(x1))) -> a(b(b(a(b(x1))))) b(b(a(x1))) -> b(b(b(b(b(x1))))) a(a(a(a(x1)))) -> a(a(a(b(a(a(a(x1))))))) a(b(a(a(x1)))) -> a(b(a(b(a(b(a(x1))))))) b(b(a(a(x1)))) -> b(b(a(b(b(b(a(x1))))))) a(a(b(a(x1)))) -> a(a(b(b(a(a(b(x1))))))) b(a(b(a(x1)))) -> b(a(b(b(b(a(b(x1))))))) a(b(b(a(x1)))) -> a(b(b(b(a(b(b(x1))))))) b(b(b(a(x1)))) -> b(b(b(b(b(b(b(x1))))))) String Reversal Processor: a(x1) -> b(x1) a(a(x1)) -> a(b(a(x1))) a(b(x1)) -> b(b(b(x1))) a(a(b(x1))) -> a(b(b(a(b(x1))))) a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(b(x1))) -> b(b(b(b(b(x1))))) a(a(a(a(x1)))) -> a(a(a(b(a(a(a(x1))))))) a(a(b(a(x1)))) -> a(b(a(b(a(b(a(x1))))))) a(a(b(b(x1)))) -> a(b(b(b(a(b(b(x1))))))) a(b(a(a(x1)))) -> b(a(a(b(b(a(a(x1))))))) a(b(a(b(x1)))) -> b(a(b(b(b(a(b(x1))))))) a(b(b(a(x1)))) -> b(b(a(b(b(b(a(x1))))))) a(b(b(b(x1)))) -> b(b(b(b(b(b(b(x1))))))) DP Processor: DPs: a#(a(x1)) -> a#(b(a(x1))) a#(a(b(x1))) -> a#(b(b(a(b(x1))))) a#(b(a(x1))) -> a#(b(b(a(x1)))) a#(a(a(a(x1)))) -> a#(b(a(a(a(x1))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) a#(a(b(a(x1)))) -> a#(b(a(b(a(x1))))) a#(a(b(a(x1)))) -> a#(b(a(b(a(b(a(x1))))))) a#(a(b(b(x1)))) -> a#(b(b(b(a(b(b(x1))))))) a#(b(a(a(x1)))) -> a#(b(b(a(a(x1))))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) a#(b(a(b(x1)))) -> a#(b(b(b(a(b(x1)))))) a#(b(b(a(x1)))) -> a#(b(b(b(a(x1))))) TRS: a(x1) -> b(x1) a(a(x1)) -> a(b(a(x1))) a(b(x1)) -> b(b(b(x1))) a(a(b(x1))) -> a(b(b(a(b(x1))))) a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(b(x1))) -> b(b(b(b(b(x1))))) a(a(a(a(x1)))) -> a(a(a(b(a(a(a(x1))))))) a(a(b(a(x1)))) -> a(b(a(b(a(b(a(x1))))))) a(a(b(b(x1)))) -> a(b(b(b(a(b(b(x1))))))) a(b(a(a(x1)))) -> b(a(a(b(b(a(a(x1))))))) a(b(a(b(x1)))) -> b(a(b(b(b(a(b(x1))))))) a(b(b(a(x1)))) -> b(b(a(b(b(b(a(x1))))))) a(b(b(b(x1)))) -> b(b(b(b(b(b(b(x1))))))) EDG Processor: DPs: a#(a(x1)) -> a#(b(a(x1))) a#(a(b(x1))) -> a#(b(b(a(b(x1))))) a#(b(a(x1))) -> a#(b(b(a(x1)))) a#(a(a(a(x1)))) -> a#(b(a(a(a(x1))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) a#(a(b(a(x1)))) -> a#(b(a(b(a(x1))))) a#(a(b(a(x1)))) -> a#(b(a(b(a(b(a(x1))))))) a#(a(b(b(x1)))) -> a#(b(b(b(a(b(b(x1))))))) a#(b(a(a(x1)))) -> a#(b(b(a(a(x1))))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) a#(b(a(b(x1)))) -> a#(b(b(b(a(b(x1)))))) a#(b(b(a(x1)))) -> a#(b(b(b(a(x1))))) TRS: a(x1) -> b(x1) a(a(x1)) -> a(b(a(x1))) a(b(x1)) -> b(b(b(x1))) a(a(b(x1))) -> a(b(b(a(b(x1))))) a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(b(x1))) -> b(b(b(b(b(x1))))) a(a(a(a(x1)))) -> a(a(a(b(a(a(a(x1))))))) a(a(b(a(x1)))) -> a(b(a(b(a(b(a(x1))))))) a(a(b(b(x1)))) -> a(b(b(b(a(b(b(x1))))))) a(b(a(a(x1)))) -> b(a(a(b(b(a(a(x1))))))) a(b(a(b(x1)))) -> b(a(b(b(b(a(b(x1))))))) a(b(b(a(x1)))) -> b(b(a(b(b(b(a(x1))))))) a(b(b(b(x1)))) -> b(b(b(b(b(b(b(x1))))))) graph: a#(b(a(a(x1)))) -> a#(b(b(a(a(x1))))) -> a#(b(b(a(x1)))) -> a#(b(b(b(a(x1))))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) -> a#(a(x1)) -> a#(b(a(x1))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) -> a#(a(b(x1))) -> a#(b(b(a(b(x1))))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) -> a#(b(a(x1))) -> a#(b(b(a(x1)))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) -> a#(a(a(a(x1)))) -> a#(b(a(a(a(x1))))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) -> a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) -> a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) -> a#(a(b(a(x1)))) -> a#(b(a(b(a(x1))))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) -> a#(a(b(a(x1)))) -> a#(b(a(b(a(b(a(x1))))))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) -> a#(a(b(b(x1)))) -> a#(b(b(b(a(b(b(x1))))))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) -> a#(b(a(a(x1)))) -> a#(b(b(a(a(x1))))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) -> a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) -> a#(b(a(b(x1)))) -> a#(b(b(b(a(b(x1)))))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) -> a#(b(b(a(x1)))) -> a#(b(b(b(a(x1))))) a#(b(a(x1))) -> a#(b(b(a(x1)))) -> a#(b(b(a(x1)))) -> a#(b(b(b(a(x1))))) a#(a(b(a(x1)))) -> a#(b(a(b(a(b(a(x1))))))) -> a#(b(a(x1))) -> a#(b(b(a(x1)))) a#(a(b(a(x1)))) -> a#(b(a(b(a(b(a(x1))))))) -> a#(b(a(a(x1)))) -> a#(b(b(a(a(x1))))) a#(a(b(a(x1)))) -> a#(b(a(b(a(b(a(x1))))))) -> a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) a#(a(b(a(x1)))) -> a#(b(a(b(a(b(a(x1))))))) -> a#(b(a(b(x1)))) -> a#(b(b(b(a(b(x1)))))) a#(a(b(a(x1)))) -> a#(b(a(b(a(b(a(x1))))))) -> a#(b(b(a(x1)))) -> a#(b(b(b(a(x1))))) a#(a(b(a(x1)))) -> a#(b(a(b(a(x1))))) -> a#(b(a(x1))) -> a#(b(b(a(x1)))) a#(a(b(a(x1)))) -> a#(b(a(b(a(x1))))) -> a#(b(a(a(x1)))) -> a#(b(b(a(a(x1))))) a#(a(b(a(x1)))) -> a#(b(a(b(a(x1))))) -> a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) a#(a(b(a(x1)))) -> a#(b(a(b(a(x1))))) -> a#(b(a(b(x1)))) -> a#(b(b(b(a(b(x1)))))) a#(a(b(a(x1)))) -> a#(b(a(b(a(x1))))) -> a#(b(b(a(x1)))) -> a#(b(b(b(a(x1))))) a#(a(b(x1))) -> a#(b(b(a(b(x1))))) -> a#(b(b(a(x1)))) -> a#(b(b(b(a(x1))))) a#(a(a(a(x1)))) -> a#(b(a(a(a(x1))))) -> a#(b(a(x1))) -> a#(b(b(a(x1)))) a#(a(a(a(x1)))) -> a#(b(a(a(a(x1))))) -> a#(b(a(a(x1)))) -> a#(b(b(a(a(x1))))) a#(a(a(a(x1)))) -> a#(b(a(a(a(x1))))) -> a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) a#(a(a(a(x1)))) -> a#(b(a(a(a(x1))))) -> a#(b(a(b(x1)))) -> a#(b(b(b(a(b(x1)))))) a#(a(a(a(x1)))) -> a#(b(a(a(a(x1))))) -> a#(b(b(a(x1)))) -> a#(b(b(b(a(x1))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(a(x1)) -> a#(b(a(x1))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(a(b(x1))) -> a#(b(b(a(b(x1))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(b(a(x1))) -> a#(b(b(a(x1)))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(a(a(a(x1)))) -> a#(b(a(a(a(x1))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(a(b(a(x1)))) -> a#(b(a(b(a(x1))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(a(b(a(x1)))) -> a#(b(a(b(a(b(a(x1))))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(a(b(b(x1)))) -> a#(b(b(b(a(b(b(x1))))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(b(a(a(x1)))) -> a#(b(b(a(a(x1))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(b(a(b(x1)))) -> a#(b(b(b(a(b(x1)))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(b(b(a(x1)))) -> a#(b(b(b(a(x1))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(a(x1)) -> a#(b(a(x1))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(a(b(x1))) -> a#(b(b(a(b(x1))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(b(a(x1))) -> a#(b(b(a(x1)))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(a(a(a(x1)))) -> a#(b(a(a(a(x1))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(a(b(a(x1)))) -> a#(b(a(b(a(x1))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(a(b(a(x1)))) -> a#(b(a(b(a(b(a(x1))))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(a(b(b(x1)))) -> a#(b(b(b(a(b(b(x1))))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(b(a(a(x1)))) -> a#(b(b(a(a(x1))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(b(a(b(x1)))) -> a#(b(b(b(a(b(x1)))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(b(b(a(x1)))) -> a#(b(b(b(a(x1))))) a#(a(x1)) -> a#(b(a(x1))) -> a#(b(a(x1))) -> a#(b(b(a(x1)))) a#(a(x1)) -> a#(b(a(x1))) -> a#(b(a(a(x1)))) -> a#(b(b(a(a(x1))))) a#(a(x1)) -> a#(b(a(x1))) -> a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) a#(a(x1)) -> a#(b(a(x1))) -> a#(b(a(b(x1)))) -> a#(b(b(b(a(b(x1)))))) a#(a(x1)) -> a#(b(a(x1))) -> a#(b(b(a(x1)))) -> a#(b(b(b(a(x1))))) CDG Processor: DPs: a#(a(x1)) -> a#(b(a(x1))) a#(a(b(x1))) -> a#(b(b(a(b(x1))))) a#(b(a(x1))) -> a#(b(b(a(x1)))) a#(a(a(a(x1)))) -> a#(b(a(a(a(x1))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) a#(a(b(a(x1)))) -> a#(b(a(b(a(x1))))) a#(a(b(a(x1)))) -> a#(b(a(b(a(b(a(x1))))))) a#(a(b(b(x1)))) -> a#(b(b(b(a(b(b(x1))))))) a#(b(a(a(x1)))) -> a#(b(b(a(a(x1))))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) a#(b(a(b(x1)))) -> a#(b(b(b(a(b(x1)))))) a#(b(b(a(x1)))) -> a#(b(b(b(a(x1))))) TRS: a(x1) -> b(x1) a(a(x1)) -> a(b(a(x1))) a(b(x1)) -> b(b(b(x1))) a(a(b(x1))) -> a(b(b(a(b(x1))))) a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(b(x1))) -> b(b(b(b(b(x1))))) a(a(a(a(x1)))) -> a(a(a(b(a(a(a(x1))))))) a(a(b(a(x1)))) -> a(b(a(b(a(b(a(x1))))))) a(a(b(b(x1)))) -> a(b(b(b(a(b(b(x1))))))) a(b(a(a(x1)))) -> b(a(a(b(b(a(a(x1))))))) a(b(a(b(x1)))) -> b(a(b(b(b(a(b(x1))))))) a(b(b(a(x1)))) -> b(b(a(b(b(b(a(x1))))))) a(b(b(b(x1)))) -> b(b(b(b(b(b(b(x1))))))) graph: a#(b(a(a(x1)))) -> a#(b(b(a(a(x1))))) -> a#(b(b(a(x1)))) -> a#(b(b(b(a(x1))))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) -> a#(b(b(a(x1)))) -> a#(b(b(b(a(x1))))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) -> a#(b(a(b(x1)))) -> a#(b(b(b(a(b(x1)))))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) -> a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) -> a#(b(a(a(x1)))) -> a#(b(b(a(a(x1))))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) -> a#(a(b(b(x1)))) -> a#(b(b(b(a(b(b(x1))))))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) -> a#(a(b(a(x1)))) -> a#(b(a(b(a(b(a(x1))))))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) -> a#(a(b(a(x1)))) -> a#(b(a(b(a(x1))))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) -> a#(b(a(x1))) -> a#(b(b(a(x1)))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) -> a#(a(b(x1))) -> a#(b(b(a(b(x1))))) a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) -> a#(a(x1)) -> a#(b(a(x1))) a#(b(a(x1))) -> a#(b(b(a(x1)))) -> a#(b(b(a(x1)))) -> a#(b(b(b(a(x1))))) a#(a(b(a(x1)))) -> a#(b(a(b(a(b(a(x1))))))) -> a#(b(b(a(x1)))) -> a#(b(b(b(a(x1))))) a#(a(b(a(x1)))) -> a#(b(a(b(a(b(a(x1))))))) -> a#(b(a(b(x1)))) -> a#(b(b(b(a(b(x1)))))) a#(a(b(a(x1)))) -> a#(b(a(b(a(b(a(x1))))))) -> a#(b(a(x1))) -> a#(b(b(a(x1)))) a#(a(b(a(x1)))) -> a#(b(a(b(a(x1))))) -> a#(b(b(a(x1)))) -> a#(b(b(b(a(x1))))) a#(a(b(a(x1)))) -> a#(b(a(b(a(x1))))) -> a#(b(a(b(x1)))) -> a#(b(b(b(a(b(x1)))))) a#(a(b(a(x1)))) -> a#(b(a(b(a(x1))))) -> a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) a#(a(b(a(x1)))) -> a#(b(a(b(a(x1))))) -> a#(b(a(a(x1)))) -> a#(b(b(a(a(x1))))) a#(a(b(a(x1)))) -> a#(b(a(b(a(x1))))) -> a#(b(a(x1))) -> a#(b(b(a(x1)))) a#(a(b(x1))) -> a#(b(b(a(b(x1))))) -> a#(b(b(a(x1)))) -> a#(b(b(b(a(x1))))) a#(a(a(a(x1)))) -> a#(b(a(a(a(x1))))) -> a#(b(b(a(x1)))) -> a#(b(b(b(a(x1))))) a#(a(a(a(x1)))) -> a#(b(a(a(a(x1))))) -> a#(b(a(b(x1)))) -> a#(b(b(b(a(b(x1)))))) a#(a(a(a(x1)))) -> a#(b(a(a(a(x1))))) -> a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) a#(a(a(a(x1)))) -> a#(b(a(a(a(x1))))) -> a#(b(a(a(x1)))) -> a#(b(b(a(a(x1))))) a#(a(a(a(x1)))) -> a#(b(a(a(a(x1))))) -> a#(b(a(x1))) -> a#(b(b(a(x1)))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(b(b(a(x1)))) -> a#(b(b(b(a(x1))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(b(a(b(x1)))) -> a#(b(b(b(a(b(x1)))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(b(a(a(x1)))) -> a#(b(b(a(a(x1))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(a(b(b(x1)))) -> a#(b(b(b(a(b(b(x1))))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(a(b(a(x1)))) -> a#(b(a(b(a(b(a(x1))))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(a(b(a(x1)))) -> a#(b(a(b(a(x1))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(a(a(a(x1)))) -> a#(b(a(a(a(x1))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(b(a(x1))) -> a#(b(b(a(x1)))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(a(b(x1))) -> a#(b(b(a(b(x1))))) a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) -> a#(a(x1)) -> a#(b(a(x1))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(b(b(a(x1)))) -> a#(b(b(b(a(x1))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(b(a(b(x1)))) -> a#(b(b(b(a(b(x1)))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(b(a(a(x1)))) -> a#(b(b(a(a(x1))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(a(b(b(x1)))) -> a#(b(b(b(a(b(b(x1))))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(a(b(a(x1)))) -> a#(b(a(b(a(b(a(x1))))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(a(b(a(x1)))) -> a#(b(a(b(a(x1))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(a(a(a(x1)))) -> a#(b(a(a(a(x1))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(b(a(x1))) -> a#(b(b(a(x1)))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(a(b(x1))) -> a#(b(b(a(b(x1))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(a(x1)) -> a#(b(a(x1))) a#(a(x1)) -> a#(b(a(x1))) -> a#(b(b(a(x1)))) -> a#(b(b(b(a(x1))))) a#(a(x1)) -> a#(b(a(x1))) -> a#(b(a(b(x1)))) -> a#(b(b(b(a(b(x1)))))) a#(a(x1)) -> a#(b(a(x1))) -> a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) a#(a(x1)) -> a#(b(a(x1))) -> a#(b(a(a(x1)))) -> a#(b(b(a(a(x1))))) a#(a(x1)) -> a#(b(a(x1))) -> a#(b(a(x1))) -> a#(b(b(a(x1)))) SCC Processor: #sccs: 2 #rules: 5 #arcs: 57/169 DPs: a#(a(a(a(x1)))) -> a#(a(b(a(a(a(x1)))))) a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) TRS: a(x1) -> b(x1) a(a(x1)) -> a(b(a(x1))) a(b(x1)) -> b(b(b(x1))) a(a(b(x1))) -> a(b(b(a(b(x1))))) a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(b(x1))) -> b(b(b(b(b(x1))))) a(a(a(a(x1)))) -> a(a(a(b(a(a(a(x1))))))) a(a(b(a(x1)))) -> a(b(a(b(a(b(a(x1))))))) a(a(b(b(x1)))) -> a(b(b(b(a(b(b(x1))))))) a(b(a(a(x1)))) -> b(a(a(b(b(a(a(x1))))))) a(b(a(b(x1)))) -> b(a(b(b(b(a(b(x1))))))) a(b(b(a(x1)))) -> b(b(a(b(b(b(a(x1))))))) a(b(b(b(x1)))) -> b(b(b(b(b(b(b(x1))))))) Arctic Interpretation Processor: dimension: 1 interpretation: [a#](x0) = x0 + 0, [b](x0) = 8, [a](x0) = 4x0 + 8 orientation: a#(a(a(a(x1)))) = 12x1 + 16 >= 12 = a#(a(b(a(a(a(x1)))))) a#(a(a(a(x1)))) = 12x1 + 16 >= 16 = a#(a(a(b(a(a(a(x1))))))) a(x1) = 4x1 + 8 >= 8 = b(x1) a(a(x1)) = 8x1 + 12 >= 12 = a(b(a(x1))) a(b(x1)) = 12 >= 8 = b(b(b(x1))) a(a(b(x1))) = 16 >= 12 = a(b(b(a(b(x1))))) a(b(a(x1))) = 12 >= 8 = b(a(b(b(a(x1))))) a(b(b(x1))) = 12 >= 8 = b(b(b(b(b(x1))))) a(a(a(a(x1)))) = 16x1 + 20 >= 20 = a(a(a(b(a(a(a(x1))))))) a(a(b(a(x1)))) = 16 >= 12 = a(b(a(b(a(b(a(x1))))))) a(a(b(b(x1)))) = 16 >= 12 = a(b(b(b(a(b(b(x1))))))) a(b(a(a(x1)))) = 12 >= 8 = b(a(a(b(b(a(a(x1))))))) a(b(a(b(x1)))) = 12 >= 8 = b(a(b(b(b(a(b(x1))))))) a(b(b(a(x1)))) = 12 >= 8 = b(b(a(b(b(b(a(x1))))))) a(b(b(b(x1)))) = 12 >= 8 = b(b(b(b(b(b(b(x1))))))) problem: DPs: a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) TRS: a(x1) -> b(x1) a(a(x1)) -> a(b(a(x1))) a(b(x1)) -> b(b(b(x1))) a(a(b(x1))) -> a(b(b(a(b(x1))))) a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(b(x1))) -> b(b(b(b(b(x1))))) a(a(a(a(x1)))) -> a(a(a(b(a(a(a(x1))))))) a(a(b(a(x1)))) -> a(b(a(b(a(b(a(x1))))))) a(a(b(b(x1)))) -> a(b(b(b(a(b(b(x1))))))) a(b(a(a(x1)))) -> b(a(a(b(b(a(a(x1))))))) a(b(a(b(x1)))) -> b(a(b(b(b(a(b(x1))))))) a(b(b(a(x1)))) -> b(b(a(b(b(b(a(x1))))))) a(b(b(b(x1)))) -> b(b(b(b(b(b(b(x1))))))) EDG Processor: DPs: a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) TRS: a(x1) -> b(x1) a(a(x1)) -> a(b(a(x1))) a(b(x1)) -> b(b(b(x1))) a(a(b(x1))) -> a(b(b(a(b(x1))))) a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(b(x1))) -> b(b(b(b(b(x1))))) a(a(a(a(x1)))) -> a(a(a(b(a(a(a(x1))))))) a(a(b(a(x1)))) -> a(b(a(b(a(b(a(x1))))))) a(a(b(b(x1)))) -> a(b(b(b(a(b(b(x1))))))) a(b(a(a(x1)))) -> b(a(a(b(b(a(a(x1))))))) a(b(a(b(x1)))) -> b(a(b(b(b(a(b(x1))))))) a(b(b(a(x1)))) -> b(b(a(b(b(b(a(x1))))))) a(b(b(b(x1)))) -> b(b(b(b(b(b(b(x1))))))) graph: a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) -> a#(a(a(a(x1)))) -> a#(a(a(b(a(a(a(x1))))))) Arctic Interpretation Processor: dimension: 1 interpretation: [a#](x0) = -16x0 + 4, [b](x0) = 0, [a](x0) = 5x0 + 11 orientation: a#(a(a(a(x1)))) = -1x1 + 5 >= 4 = a#(a(a(b(a(a(a(x1))))))) a(x1) = 5x1 + 11 >= 0 = b(x1) a(a(x1)) = 10x1 + 16 >= 11 = a(b(a(x1))) a(b(x1)) = 11 >= 0 = b(b(b(x1))) a(a(b(x1))) = 16 >= 11 = a(b(b(a(b(x1))))) a(b(a(x1))) = 11 >= 0 = b(a(b(b(a(x1))))) a(b(b(x1))) = 11 >= 0 = b(b(b(b(b(x1))))) a(a(a(a(x1)))) = 20x1 + 26 >= 21 = a(a(a(b(a(a(a(x1))))))) a(a(b(a(x1)))) = 16 >= 11 = a(b(a(b(a(b(a(x1))))))) a(a(b(b(x1)))) = 16 >= 11 = a(b(b(b(a(b(b(x1))))))) a(b(a(a(x1)))) = 11 >= 0 = b(a(a(b(b(a(a(x1))))))) a(b(a(b(x1)))) = 11 >= 0 = b(a(b(b(b(a(b(x1))))))) a(b(b(a(x1)))) = 11 >= 0 = b(b(a(b(b(b(a(x1))))))) a(b(b(b(x1)))) = 11 >= 0 = b(b(b(b(b(b(b(x1))))))) problem: DPs: TRS: a(x1) -> b(x1) a(a(x1)) -> a(b(a(x1))) a(b(x1)) -> b(b(b(x1))) a(a(b(x1))) -> a(b(b(a(b(x1))))) a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(b(x1))) -> b(b(b(b(b(x1))))) a(a(a(a(x1)))) -> a(a(a(b(a(a(a(x1))))))) a(a(b(a(x1)))) -> a(b(a(b(a(b(a(x1))))))) a(a(b(b(x1)))) -> a(b(b(b(a(b(b(x1))))))) a(b(a(a(x1)))) -> b(a(a(b(b(a(a(x1))))))) a(b(a(b(x1)))) -> b(a(b(b(b(a(b(x1))))))) a(b(b(a(x1)))) -> b(b(a(b(b(b(a(x1))))))) a(b(b(b(x1)))) -> b(b(b(b(b(b(b(x1))))))) Qed DPs: a#(b(a(a(x1)))) -> a#(a(b(b(a(a(x1)))))) a#(a(x1)) -> a#(b(a(x1))) a#(a(b(a(x1)))) -> a#(b(a(b(a(x1))))) TRS: a(x1) -> b(x1) a(a(x1)) -> a(b(a(x1))) a(b(x1)) -> b(b(b(x1))) a(a(b(x1))) -> a(b(b(a(b(x1))))) a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(b(x1))) -> b(b(b(b(b(x1))))) a(a(a(a(x1)))) -> a(a(a(b(a(a(a(x1))))))) a(a(b(a(x1)))) -> a(b(a(b(a(b(a(x1))))))) a(a(b(b(x1)))) -> a(b(b(b(a(b(b(x1))))))) a(b(a(a(x1)))) -> b(a(a(b(b(a(a(x1))))))) a(b(a(b(x1)))) -> b(a(b(b(b(a(b(x1))))))) a(b(b(a(x1)))) -> b(b(a(b(b(b(a(x1))))))) a(b(b(b(x1)))) -> b(b(b(b(b(b(b(x1))))))) Matrix Interpretation Processor: dim=3 interpretation: [a#](x0) = [0 1 1]x0 + [1], [0 0 0] [b](x0) = [0 0 1]x0 [0 0 0] , [0 0 0] [1] [a](x0) = [0 1 1]x0 + [0] [1 0 0] [0] orientation: a#(b(a(a(x1)))) = [2] >= [1] = a#(a(b(b(a(a(x1)))))) a#(a(x1)) = [1 1 1]x1 + [1] >= [1 0 0]x1 + [1] = a#(b(a(x1))) a#(a(b(a(x1)))) = [1 0 0]x1 + [1] >= [1] = a#(b(a(b(a(x1))))) [0 0 0] [1] [0 0 0] a(x1) = [0 1 1]x1 + [0] >= [0 0 1]x1 = b(x1) [1 0 0] [0] [0 0 0] [0 0 0] [1] [0 0 0] [1] a(a(x1)) = [1 1 1]x1 + [0] >= [1 0 0]x1 + [0] = a(b(a(x1))) [0 0 0] [1] [0 0 0] [0] [0 0 0] [1] [0] a(b(x1)) = [0 0 1]x1 + [0] >= [0] = b(b(b(x1))) [0 0 0] [0] [0] [0 0 0] [1] [1] a(a(b(x1))) = [0 0 1]x1 + [0] >= [0] = a(b(b(a(b(x1))))) [0 0 0] [1] [0] [0 0 0] [1] [0] a(b(a(x1))) = [1 0 0]x1 + [0] >= [0] = b(a(b(b(a(x1))))) [0 0 0] [0] [0] [1] [0] a(b(b(x1))) = [0] >= [0] = b(b(b(b(b(x1))))) [0] [0] [0 0 0] [1] [1] a(a(a(a(x1)))) = [1 1 1]x1 + [2] >= [2] = a(a(a(b(a(a(a(x1))))))) [0 0 0] [1] [1] [0 0 0] [1] [1] a(a(b(a(x1)))) = [1 0 0]x1 + [0] >= [0] = a(b(a(b(a(b(a(x1))))))) [0 0 0] [1] [0] [1] [1] a(a(b(b(x1)))) = [0] >= [0] = a(b(b(b(a(b(b(x1))))))) [1] [0] [1] [0] a(b(a(a(x1)))) = [1] >= [1] = b(a(a(b(b(a(a(x1))))))) [0] [0] [1] [0] a(b(a(b(x1)))) = [0] >= [0] = b(a(b(b(b(a(b(x1))))))) [0] [0] [1] [0] a(b(b(a(x1)))) = [0] >= [0] = b(b(a(b(b(b(a(x1))))))) [0] [0] [1] [0] a(b(b(b(x1)))) = [0] >= [0] = b(b(b(b(b(b(b(x1))))))) [0] [0] problem: DPs: a#(a(x1)) -> a#(b(a(x1))) a#(a(b(a(x1)))) -> a#(b(a(b(a(x1))))) TRS: a(x1) -> b(x1) a(a(x1)) -> a(b(a(x1))) a(b(x1)) -> b(b(b(x1))) a(a(b(x1))) -> a(b(b(a(b(x1))))) a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(b(x1))) -> b(b(b(b(b(x1))))) a(a(a(a(x1)))) -> a(a(a(b(a(a(a(x1))))))) a(a(b(a(x1)))) -> a(b(a(b(a(b(a(x1))))))) a(a(b(b(x1)))) -> a(b(b(b(a(b(b(x1))))))) a(b(a(a(x1)))) -> b(a(a(b(b(a(a(x1))))))) a(b(a(b(x1)))) -> b(a(b(b(b(a(b(x1))))))) a(b(b(a(x1)))) -> b(b(a(b(b(b(a(x1))))))) a(b(b(b(x1)))) -> b(b(b(b(b(b(b(x1))))))) EDG Processor: DPs: a#(a(x1)) -> a#(b(a(x1))) a#(a(b(a(x1)))) -> a#(b(a(b(a(x1))))) TRS: a(x1) -> b(x1) a(a(x1)) -> a(b(a(x1))) a(b(x1)) -> b(b(b(x1))) a(a(b(x1))) -> a(b(b(a(b(x1))))) a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(b(x1))) -> b(b(b(b(b(x1))))) a(a(a(a(x1)))) -> a(a(a(b(a(a(a(x1))))))) a(a(b(a(x1)))) -> a(b(a(b(a(b(a(x1))))))) a(a(b(b(x1)))) -> a(b(b(b(a(b(b(x1))))))) a(b(a(a(x1)))) -> b(a(a(b(b(a(a(x1))))))) a(b(a(b(x1)))) -> b(a(b(b(b(a(b(x1))))))) a(b(b(a(x1)))) -> b(b(a(b(b(b(a(x1))))))) a(b(b(b(x1)))) -> b(b(b(b(b(b(b(x1))))))) graph: Qed