YES Problem: a12(a12(x1)) -> x1 a13(a13(x1)) -> x1 a14(a14(x1)) -> x1 a15(a15(x1)) -> x1 a16(a16(x1)) -> x1 a23(a23(x1)) -> x1 a24(a24(x1)) -> x1 a25(a25(x1)) -> x1 a26(a26(x1)) -> x1 a34(a34(x1)) -> x1 a35(a35(x1)) -> x1 a36(a36(x1)) -> x1 a45(a45(x1)) -> x1 a46(a46(x1)) -> x1 a56(a56(x1)) -> x1 a13(x1) -> a12(a23(a12(x1))) a14(x1) -> a12(a23(a34(a23(a12(x1))))) a15(x1) -> a12(a23(a34(a45(a34(a23(a12(x1))))))) a16(x1) -> a12(a23(a34(a45(a56(a45(a34(a23(a12(x1))))))))) a24(x1) -> a23(a34(a23(x1))) a25(x1) -> a23(a34(a45(a34(a23(x1))))) a26(x1) -> a23(a34(a45(a56(a45(a34(a23(x1))))))) a35(x1) -> a34(a45(a34(x1))) a36(x1) -> a34(a45(a56(a45(a34(x1))))) a46(x1) -> a45(a56(a45(x1))) a12(a23(a12(a23(a12(a23(x1)))))) -> x1 a23(a34(a23(a34(a23(a34(x1)))))) -> x1 a34(a45(a34(a45(a34(a45(x1)))))) -> x1 a45(a56(a45(a56(a45(a56(x1)))))) -> x1 a12(a34(x1)) -> a34(a12(x1)) a12(a45(x1)) -> a45(a12(x1)) a12(a56(x1)) -> a56(a12(x1)) a23(a45(x1)) -> a45(a23(x1)) a23(a56(x1)) -> a56(a23(x1)) a34(a56(x1)) -> a56(a34(x1)) Proof: DP Processor: DPs: a13#(x1) -> a12#(x1) a13#(x1) -> a23#(a12(x1)) a13#(x1) -> a12#(a23(a12(x1))) a14#(x1) -> a12#(x1) a14#(x1) -> a23#(a12(x1)) a14#(x1) -> a34#(a23(a12(x1))) a14#(x1) -> a23#(a34(a23(a12(x1)))) a14#(x1) -> a12#(a23(a34(a23(a12(x1))))) a15#(x1) -> a12#(x1) a15#(x1) -> a23#(a12(x1)) a15#(x1) -> a34#(a23(a12(x1))) a15#(x1) -> a45#(a34(a23(a12(x1)))) a15#(x1) -> a34#(a45(a34(a23(a12(x1))))) a15#(x1) -> a23#(a34(a45(a34(a23(a12(x1)))))) a15#(x1) -> a12#(a23(a34(a45(a34(a23(a12(x1))))))) a16#(x1) -> a12#(x1) a16#(x1) -> a23#(a12(x1)) a16#(x1) -> a34#(a23(a12(x1))) a16#(x1) -> a45#(a34(a23(a12(x1)))) a16#(x1) -> a56#(a45(a34(a23(a12(x1))))) a16#(x1) -> a45#(a56(a45(a34(a23(a12(x1)))))) a16#(x1) -> a34#(a45(a56(a45(a34(a23(a12(x1))))))) a16#(x1) -> a23#(a34(a45(a56(a45(a34(a23(a12(x1)))))))) a16#(x1) -> a12#(a23(a34(a45(a56(a45(a34(a23(a12(x1))))))))) a24#(x1) -> a23#(x1) a24#(x1) -> a34#(a23(x1)) a24#(x1) -> a23#(a34(a23(x1))) a25#(x1) -> a23#(x1) a25#(x1) -> a34#(a23(x1)) a25#(x1) -> a45#(a34(a23(x1))) a25#(x1) -> a34#(a45(a34(a23(x1)))) a25#(x1) -> a23#(a34(a45(a34(a23(x1))))) a26#(x1) -> a23#(x1) a26#(x1) -> a34#(a23(x1)) a26#(x1) -> a45#(a34(a23(x1))) a26#(x1) -> a56#(a45(a34(a23(x1)))) a26#(x1) -> a45#(a56(a45(a34(a23(x1))))) a26#(x1) -> a34#(a45(a56(a45(a34(a23(x1)))))) a26#(x1) -> a23#(a34(a45(a56(a45(a34(a23(x1))))))) a35#(x1) -> a34#(x1) a35#(x1) -> a45#(a34(x1)) a35#(x1) -> a34#(a45(a34(x1))) a36#(x1) -> a34#(x1) a36#(x1) -> a45#(a34(x1)) a36#(x1) -> a56#(a45(a34(x1))) a36#(x1) -> a45#(a56(a45(a34(x1)))) a36#(x1) -> a34#(a45(a56(a45(a34(x1))))) a46#(x1) -> a45#(x1) a46#(x1) -> a56#(a45(x1)) a46#(x1) -> a45#(a56(a45(x1))) a12#(a34(x1)) -> a12#(x1) a12#(a34(x1)) -> a34#(a12(x1)) a12#(a45(x1)) -> a12#(x1) a12#(a45(x1)) -> a45#(a12(x1)) a12#(a56(x1)) -> a12#(x1) a12#(a56(x1)) -> a56#(a12(x1)) a23#(a45(x1)) -> a23#(x1) a23#(a45(x1)) -> a45#(a23(x1)) a23#(a56(x1)) -> a23#(x1) a23#(a56(x1)) -> a56#(a23(x1)) a34#(a56(x1)) -> a34#(x1) a34#(a56(x1)) -> a56#(a34(x1)) TRS: a12(a12(x1)) -> x1 a13(a13(x1)) -> x1 a14(a14(x1)) -> x1 a15(a15(x1)) -> x1 a16(a16(x1)) -> x1 a23(a23(x1)) -> x1 a24(a24(x1)) -> x1 a25(a25(x1)) -> x1 a26(a26(x1)) -> x1 a34(a34(x1)) -> x1 a35(a35(x1)) -> x1 a36(a36(x1)) -> x1 a45(a45(x1)) -> x1 a46(a46(x1)) -> x1 a56(a56(x1)) -> x1 a13(x1) -> a12(a23(a12(x1))) a14(x1) -> a12(a23(a34(a23(a12(x1))))) a15(x1) -> a12(a23(a34(a45(a34(a23(a12(x1))))))) a16(x1) -> a12(a23(a34(a45(a56(a45(a34(a23(a12(x1))))))))) a24(x1) -> a23(a34(a23(x1))) a25(x1) -> a23(a34(a45(a34(a23(x1))))) a26(x1) -> a23(a34(a45(a56(a45(a34(a23(x1))))))) a35(x1) -> a34(a45(a34(x1))) a36(x1) -> a34(a45(a56(a45(a34(x1))))) a46(x1) -> a45(a56(a45(x1))) a12(a23(a12(a23(a12(a23(x1)))))) -> x1 a23(a34(a23(a34(a23(a34(x1)))))) -> x1 a34(a45(a34(a45(a34(a45(x1)))))) -> x1 a45(a56(a45(a56(a45(a56(x1)))))) -> x1 a12(a34(x1)) -> a34(a12(x1)) a12(a45(x1)) -> a45(a12(x1)) a12(a56(x1)) -> a56(a12(x1)) a23(a45(x1)) -> a45(a23(x1)) a23(a56(x1)) -> a56(a23(x1)) a34(a56(x1)) -> a56(a34(x1)) Usable Rule Processor: DPs: a13#(x1) -> a12#(x1) a13#(x1) -> a23#(a12(x1)) a13#(x1) -> a12#(a23(a12(x1))) a14#(x1) -> a12#(x1) a14#(x1) -> a23#(a12(x1)) a14#(x1) -> a34#(a23(a12(x1))) a14#(x1) -> a23#(a34(a23(a12(x1)))) a14#(x1) -> a12#(a23(a34(a23(a12(x1))))) a15#(x1) -> a12#(x1) a15#(x1) -> a23#(a12(x1)) a15#(x1) -> a34#(a23(a12(x1))) a15#(x1) -> a45#(a34(a23(a12(x1)))) a15#(x1) -> a34#(a45(a34(a23(a12(x1))))) a15#(x1) -> a23#(a34(a45(a34(a23(a12(x1)))))) a15#(x1) -> a12#(a23(a34(a45(a34(a23(a12(x1))))))) a16#(x1) -> a12#(x1) a16#(x1) -> a23#(a12(x1)) a16#(x1) -> a34#(a23(a12(x1))) a16#(x1) -> a45#(a34(a23(a12(x1)))) a16#(x1) -> a56#(a45(a34(a23(a12(x1))))) a16#(x1) -> a45#(a56(a45(a34(a23(a12(x1)))))) a16#(x1) -> a34#(a45(a56(a45(a34(a23(a12(x1))))))) a16#(x1) -> a23#(a34(a45(a56(a45(a34(a23(a12(x1)))))))) a16#(x1) -> a12#(a23(a34(a45(a56(a45(a34(a23(a12(x1))))))))) a24#(x1) -> a23#(x1) a24#(x1) -> a34#(a23(x1)) a24#(x1) -> a23#(a34(a23(x1))) a25#(x1) -> a23#(x1) a25#(x1) -> a34#(a23(x1)) a25#(x1) -> a45#(a34(a23(x1))) a25#(x1) -> a34#(a45(a34(a23(x1)))) a25#(x1) -> a23#(a34(a45(a34(a23(x1))))) a26#(x1) -> a23#(x1) a26#(x1) -> a34#(a23(x1)) a26#(x1) -> a45#(a34(a23(x1))) a26#(x1) -> a56#(a45(a34(a23(x1)))) a26#(x1) -> a45#(a56(a45(a34(a23(x1))))) a26#(x1) -> a34#(a45(a56(a45(a34(a23(x1)))))) a26#(x1) -> a23#(a34(a45(a56(a45(a34(a23(x1))))))) a35#(x1) -> a34#(x1) a35#(x1) -> a45#(a34(x1)) a35#(x1) -> a34#(a45(a34(x1))) a36#(x1) -> a34#(x1) a36#(x1) -> a45#(a34(x1)) a36#(x1) -> a56#(a45(a34(x1))) a36#(x1) -> a45#(a56(a45(a34(x1)))) a36#(x1) -> a34#(a45(a56(a45(a34(x1))))) a46#(x1) -> a45#(x1) a46#(x1) -> a56#(a45(x1)) a46#(x1) -> a45#(a56(a45(x1))) a12#(a34(x1)) -> a12#(x1) a12#(a34(x1)) -> a34#(a12(x1)) a12#(a45(x1)) -> a12#(x1) a12#(a45(x1)) -> a45#(a12(x1)) a12#(a56(x1)) -> a12#(x1) a12#(a56(x1)) -> a56#(a12(x1)) a23#(a45(x1)) -> a23#(x1) a23#(a45(x1)) -> a45#(a23(x1)) a23#(a56(x1)) -> a23#(x1) a23#(a56(x1)) -> a56#(a23(x1)) a34#(a56(x1)) -> a34#(x1) a34#(a56(x1)) -> a56#(a34(x1)) TRS: a12(a12(x1)) -> x1 a12(a23(a12(a23(a12(a23(x1)))))) -> x1 a12(a34(x1)) -> a34(a12(x1)) a12(a45(x1)) -> a45(a12(x1)) a12(a56(x1)) -> a56(a12(x1)) a34(a34(x1)) -> x1 a34(a45(a34(a45(a34(a45(x1)))))) -> x1 a34(a56(x1)) -> a56(a34(x1)) a56(a56(x1)) -> x1 a45(a45(x1)) -> x1 a45(a56(a45(a56(a45(a56(x1)))))) -> x1 a23(a23(x1)) -> x1 a23(a34(a23(a34(a23(a34(x1)))))) -> x1 a23(a45(x1)) -> a45(a23(x1)) a23(a56(x1)) -> a56(a23(x1)) Matrix Interpretation Processor: dim=1 interpretation: [a56#](x0) = 0, [a46#](x0) = 8x0 + 25, [a45#](x0) = 0, [a36#](x0) = 4x0 + 23, [a35#](x0) = 20x0 + 20, [a34#](x0) = x0, [a26#](x0) = 16x0 + 31, [a25#](x0) = 8x0 + 27, [a24#](x0) = 8x0 + 24, [a23#](x0) = x0 + 4, [a16#](x0) = 16x0 + 23, [a15#](x0) = 8x0 + 27, [a14#](x0) = 5x0 + 22, [a13#](x0) = 7x0 + 9, [a12#](x0) = x0, [a56](x0) = 2x0 + 1, [a45](x0) = x0 + 1, [a34](x0) = 2x0 + 1, [a23](x0) = x0, [a12](x0) = 2x0 orientation: a13#(x1) = 7x1 + 9 >= x1 = a12#(x1) a13#(x1) = 7x1 + 9 >= 2x1 + 4 = a23#(a12(x1)) a13#(x1) = 7x1 + 9 >= 2x1 = a12#(a23(a12(x1))) a14#(x1) = 5x1 + 22 >= x1 = a12#(x1) a14#(x1) = 5x1 + 22 >= 2x1 + 4 = a23#(a12(x1)) a14#(x1) = 5x1 + 22 >= 2x1 = a34#(a23(a12(x1))) a14#(x1) = 5x1 + 22 >= 4x1 + 5 = a23#(a34(a23(a12(x1)))) a14#(x1) = 5x1 + 22 >= 4x1 + 1 = a12#(a23(a34(a23(a12(x1))))) a15#(x1) = 8x1 + 27 >= x1 = a12#(x1) a15#(x1) = 8x1 + 27 >= 2x1 + 4 = a23#(a12(x1)) a15#(x1) = 8x1 + 27 >= 2x1 = a34#(a23(a12(x1))) a15#(x1) = 8x1 + 27 >= 0 = a45#(a34(a23(a12(x1)))) a15#(x1) = 8x1 + 27 >= 4x1 + 2 = a34#(a45(a34(a23(a12(x1))))) a15#(x1) = 8x1 + 27 >= 8x1 + 9 = a23#(a34(a45(a34(a23(a12(x1)))))) a15#(x1) = 8x1 + 27 >= 8x1 + 5 = a12#(a23(a34(a45(a34(a23(a12(x1))))))) a16#(x1) = 16x1 + 23 >= x1 = a12#(x1) a16#(x1) = 16x1 + 23 >= 2x1 + 4 = a23#(a12(x1)) a16#(x1) = 16x1 + 23 >= 2x1 = a34#(a23(a12(x1))) a16#(x1) = 16x1 + 23 >= 0 = a45#(a34(a23(a12(x1)))) a16#(x1) = 16x1 + 23 >= 0 = a56#(a45(a34(a23(a12(x1))))) a16#(x1) = 16x1 + 23 >= 0 = a45#(a56(a45(a34(a23(a12(x1)))))) a16#(x1) = 16x1 + 23 >= 8x1 + 6 = a34#(a45(a56(a45(a34(a23(a12(x1))))))) a16#(x1) = 16x1 + 23 >= 16x1 + 17 = a23#(a34(a45(a56(a45(a34(a23(a12(x1)))))))) a16#(x1) = 16x1 + 23 >= 16x1 + 13 = a12#(a23(a34(a45(a56(a45(a34(a23(a12(x1))))))))) a24#(x1) = 8x1 + 24 >= x1 + 4 = a23#(x1) a24#(x1) = 8x1 + 24 >= x1 = a34#(a23(x1)) a24#(x1) = 8x1 + 24 >= 2x1 + 5 = a23#(a34(a23(x1))) a25#(x1) = 8x1 + 27 >= x1 + 4 = a23#(x1) a25#(x1) = 8x1 + 27 >= x1 = a34#(a23(x1)) a25#(x1) = 8x1 + 27 >= 0 = a45#(a34(a23(x1))) a25#(x1) = 8x1 + 27 >= 2x1 + 2 = a34#(a45(a34(a23(x1)))) a25#(x1) = 8x1 + 27 >= 4x1 + 9 = a23#(a34(a45(a34(a23(x1))))) a26#(x1) = 16x1 + 31 >= x1 + 4 = a23#(x1) a26#(x1) = 16x1 + 31 >= x1 = a34#(a23(x1)) a26#(x1) = 16x1 + 31 >= 0 = a45#(a34(a23(x1))) a26#(x1) = 16x1 + 31 >= 0 = a56#(a45(a34(a23(x1)))) a26#(x1) = 16x1 + 31 >= 0 = a45#(a56(a45(a34(a23(x1))))) a26#(x1) = 16x1 + 31 >= 4x1 + 6 = a34#(a45(a56(a45(a34(a23(x1)))))) a26#(x1) = 16x1 + 31 >= 8x1 + 17 = a23#(a34(a45(a56(a45(a34(a23(x1))))))) a35#(x1) = 20x1 + 20 >= x1 = a34#(x1) a35#(x1) = 20x1 + 20 >= 0 = a45#(a34(x1)) a35#(x1) = 20x1 + 20 >= 2x1 + 2 = a34#(a45(a34(x1))) a36#(x1) = 4x1 + 23 >= x1 = a34#(x1) a36#(x1) = 4x1 + 23 >= 0 = a45#(a34(x1)) a36#(x1) = 4x1 + 23 >= 0 = a56#(a45(a34(x1))) a36#(x1) = 4x1 + 23 >= 0 = a45#(a56(a45(a34(x1)))) a36#(x1) = 4x1 + 23 >= 4x1 + 6 = a34#(a45(a56(a45(a34(x1))))) a46#(x1) = 8x1 + 25 >= 0 = a45#(x1) a46#(x1) = 8x1 + 25 >= 0 = a56#(a45(x1)) a46#(x1) = 8x1 + 25 >= 0 = a45#(a56(a45(x1))) a12#(a34(x1)) = 2x1 + 1 >= x1 = a12#(x1) a12#(a34(x1)) = 2x1 + 1 >= 2x1 = a34#(a12(x1)) a12#(a45(x1)) = x1 + 1 >= x1 = a12#(x1) a12#(a45(x1)) = x1 + 1 >= 0 = a45#(a12(x1)) a12#(a56(x1)) = 2x1 + 1 >= x1 = a12#(x1) a12#(a56(x1)) = 2x1 + 1 >= 0 = a56#(a12(x1)) a23#(a45(x1)) = x1 + 5 >= x1 + 4 = a23#(x1) a23#(a45(x1)) = x1 + 5 >= 0 = a45#(a23(x1)) a23#(a56(x1)) = 2x1 + 5 >= x1 + 4 = a23#(x1) a23#(a56(x1)) = 2x1 + 5 >= 0 = a56#(a23(x1)) a34#(a56(x1)) = 2x1 + 1 >= x1 = a34#(x1) a34#(a56(x1)) = 2x1 + 1 >= 0 = a56#(a34(x1)) a12(a12(x1)) = 4x1 >= x1 = x1 a12(a23(a12(a23(a12(a23(x1)))))) = 8x1 >= x1 = x1 a12(a34(x1)) = 4x1 + 2 >= 4x1 + 1 = a34(a12(x1)) a12(a45(x1)) = 2x1 + 2 >= 2x1 + 1 = a45(a12(x1)) a12(a56(x1)) = 4x1 + 2 >= 4x1 + 1 = a56(a12(x1)) a34(a34(x1)) = 4x1 + 3 >= x1 = x1 a34(a45(a34(a45(a34(a45(x1)))))) = 8x1 + 21 >= x1 = x1 a34(a56(x1)) = 4x1 + 3 >= 4x1 + 3 = a56(a34(x1)) a56(a56(x1)) = 4x1 + 3 >= x1 = x1 a45(a45(x1)) = x1 + 2 >= x1 = x1 a45(a56(a45(a56(a45(a56(x1)))))) = 8x1 + 14 >= x1 = x1 a23(a23(x1)) = x1 >= x1 = x1 a23(a34(a23(a34(a23(a34(x1)))))) = 8x1 + 7 >= x1 = x1 a23(a45(x1)) = x1 + 1 >= x1 + 1 = a45(a23(x1)) a23(a56(x1)) = 2x1 + 1 >= 2x1 + 1 = a56(a23(x1)) problem: DPs: TRS: a12(a12(x1)) -> x1 a12(a23(a12(a23(a12(a23(x1)))))) -> x1 a12(a34(x1)) -> a34(a12(x1)) a12(a45(x1)) -> a45(a12(x1)) a12(a56(x1)) -> a56(a12(x1)) a34(a34(x1)) -> x1 a34(a45(a34(a45(a34(a45(x1)))))) -> x1 a34(a56(x1)) -> a56(a34(x1)) a56(a56(x1)) -> x1 a45(a45(x1)) -> x1 a45(a56(a45(a56(a45(a56(x1)))))) -> x1 a23(a23(x1)) -> x1 a23(a34(a23(a34(a23(a34(x1)))))) -> x1 a23(a45(x1)) -> a45(a23(x1)) a23(a56(x1)) -> a56(a23(x1)) Qed