YES Problem: a(a(x1)) -> b(a(b(x1))) a(a(a(x1))) -> a(b(a(b(a(x1))))) a(b(a(x1))) -> b(b(a(b(b(x1))))) a(a(a(a(x1)))) -> a(a(b(a(b(a(a(x1))))))) a(a(b(a(x1)))) -> a(b(b(a(b(a(b(x1))))))) a(b(a(a(x1)))) -> b(a(b(a(b(b(a(x1))))))) a(b(b(a(x1)))) -> b(b(b(a(b(b(b(x1))))))) a(a(a(a(a(x1))))) -> a(a(a(b(a(b(a(a(a(x1))))))))) a(a(a(b(a(x1))))) -> a(a(b(b(a(b(a(a(b(x1))))))))) a(a(b(a(a(x1))))) -> a(b(a(b(a(b(a(b(a(x1))))))))) a(a(b(b(a(x1))))) -> a(b(b(b(a(b(a(b(b(x1))))))))) a(b(a(a(a(x1))))) -> b(a(a(b(a(b(b(a(a(x1))))))))) a(b(a(b(a(x1))))) -> b(a(b(b(a(b(b(a(b(x1))))))))) a(b(b(a(a(x1))))) -> b(b(a(b(a(b(b(b(a(x1))))))))) a(b(b(b(a(x1))))) -> b(b(b(b(a(b(b(b(b(x1))))))))) Proof: String Reversal Processor: a(a(x1)) -> b(a(b(x1))) a(a(a(x1))) -> a(b(a(b(a(x1))))) a(b(a(x1))) -> b(b(a(b(b(x1))))) a(a(a(a(x1)))) -> a(a(b(a(b(a(a(x1))))))) a(b(a(a(x1)))) -> b(a(b(a(b(b(a(x1))))))) a(a(b(a(x1)))) -> a(b(b(a(b(a(b(x1))))))) a(b(b(a(x1)))) -> b(b(b(a(b(b(b(x1))))))) a(a(a(a(a(x1))))) -> a(a(a(b(a(b(a(a(a(x1))))))))) a(b(a(a(a(x1))))) -> b(a(a(b(a(b(b(a(a(x1))))))))) a(a(b(a(a(x1))))) -> a(b(a(b(a(b(a(b(a(x1))))))))) a(b(b(a(a(x1))))) -> b(b(a(b(a(b(b(b(a(x1))))))))) a(a(a(b(a(x1))))) -> a(a(b(b(a(b(a(a(b(x1))))))))) a(b(a(b(a(x1))))) -> b(a(b(b(a(b(b(a(b(x1))))))))) a(a(b(b(a(x1))))) -> a(b(b(b(a(b(a(b(b(x1))))))))) a(b(b(b(a(x1))))) -> b(b(b(b(a(b(b(b(b(x1))))))))) Matrix Interpretation Processor: dim=2 interpretation: [1 0] [b](x0) = [0 0]x0, [1 1] [0] [a](x0) = [0 0]x0 + [1] orientation: [1 1] [1] [1 0] a(a(x1)) = [0 0]x1 + [1] >= [0 0]x1 = b(a(b(x1))) [1 1] [2] [1 1] [0] a(a(a(x1))) = [0 0]x1 + [1] >= [0 0]x1 + [1] = a(b(a(b(a(x1))))) [1 1] [0] [1 0] a(b(a(x1))) = [0 0]x1 + [1] >= [0 0]x1 = b(b(a(b(b(x1))))) [1 1] [3] [1 1] [2] a(a(a(a(x1)))) = [0 0]x1 + [1] >= [0 0]x1 + [1] = a(a(b(a(b(a(a(x1))))))) [1 1] [1] [1 1] a(b(a(a(x1)))) = [0 0]x1 + [1] >= [0 0]x1 = b(a(b(a(b(b(a(x1))))))) [1 1] [1] [1 0] [0] a(a(b(a(x1)))) = [0 0]x1 + [1] >= [0 0]x1 + [1] = a(b(b(a(b(a(b(x1))))))) [1 1] [0] [1 0] a(b(b(a(x1)))) = [0 0]x1 + [1] >= [0 0]x1 = b(b(b(a(b(b(b(x1))))))) [1 1] [4] [1 1] [4] a(a(a(a(a(x1))))) = [0 0]x1 + [1] >= [0 0]x1 + [1] = a(a(a(b(a(b(a(a(a(x1))))))))) [1 1] [2] [1 1] [2] a(b(a(a(a(x1))))) = [0 0]x1 + [1] >= [0 0]x1 + [0] = b(a(a(b(a(b(b(a(a(x1))))))))) [1 1] [2] [1 1] [0] a(a(b(a(a(x1))))) = [0 0]x1 + [1] >= [0 0]x1 + [1] = a(b(a(b(a(b(a(b(a(x1))))))))) [1 1] [1] [1 1] a(b(b(a(a(x1))))) = [0 0]x1 + [1] >= [0 0]x1 = b(b(a(b(a(b(b(b(a(x1))))))))) [1 1] [2] [1 0] [2] a(a(a(b(a(x1))))) = [0 0]x1 + [1] >= [0 0]x1 + [1] = a(a(b(b(a(b(a(a(b(x1))))))))) [1 1] [0] [1 0] a(b(a(b(a(x1))))) = [0 0]x1 + [1] >= [0 0]x1 = b(a(b(b(a(b(b(a(b(x1))))))))) [1 1] [1] [1 0] [0] a(a(b(b(a(x1))))) = [0 0]x1 + [1] >= [0 0]x1 + [1] = a(b(b(b(a(b(a(b(b(x1))))))))) [1 1] [0] [1 0] a(b(b(b(a(x1))))) = [0 0]x1 + [1] >= [0 0]x1 = b(b(b(b(a(b(b(b(b(x1))))))))) problem: a(b(a(x1))) -> b(b(a(b(b(x1))))) a(b(b(a(x1)))) -> b(b(b(a(b(b(b(x1))))))) a(a(a(a(a(x1))))) -> a(a(a(b(a(b(a(a(a(x1))))))))) a(b(a(a(a(x1))))) -> b(a(a(b(a(b(b(a(a(x1))))))))) a(a(a(b(a(x1))))) -> a(a(b(b(a(b(a(a(b(x1))))))))) a(b(a(b(a(x1))))) -> b(a(b(b(a(b(b(a(b(x1))))))))) a(b(b(b(a(x1))))) -> b(b(b(b(a(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(b(a(x1))) = [0 0 0]x1 + [1] >= [0 0 0]x1 = b(b(a(b(b(x1))))) [0 0 0] [0] [0 0 0] [1 0 1] [0] [1 0 0] a(b(b(a(x1)))) = [0 0 0]x1 + [1] >= [0 0 0]x1 = b(b(b(a(b(b(b(x1))))))) [0 0 0] [0] [0 0 0] [1 1 1] [3] [1 1 1] [2] a(a(a(a(a(x1))))) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = a(a(a(b(a(b(a(a(a(x1))))))))) [0 0 0] [1] [0 0 0] [1] [1 1 1] [1] [1 1 1] a(b(a(a(a(x1))))) = [0 0 0]x1 + [1] >= [0 0 0]x1 = b(a(a(b(a(b(b(a(a(x1))))))))) [0 0 0] [0] [0 0 0] [1 0 1] [1] [1 0 0] [0] a(a(a(b(a(x1))))) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = a(a(b(b(a(b(a(a(b(x1))))))))) [0 0 0] [1] [0 0 0] [1] [1 0 1] [0] [1 0 0] a(b(a(b(a(x1))))) = [0 0 0]x1 + [1] >= [0 0 0]x1 = b(a(b(b(a(b(b(a(b(x1))))))))) [0 0 0] [0] [0 0 0] [1 0 1] [0] [1 0 0] a(b(b(b(a(x1))))) = [0 0 0]x1 + [1] >= [0 0 0]x1 = b(b(b(b(a(b(b(b(b(x1))))))))) [0 0 0] [0] [0 0 0] problem: a(b(a(x1))) -> b(b(a(b(b(x1))))) a(b(b(a(x1)))) -> b(b(b(a(b(b(b(x1))))))) a(b(a(b(a(x1))))) -> b(a(b(b(a(b(b(a(b(x1))))))))) a(b(b(b(a(x1))))) -> b(b(b(b(a(b(b(b(b(x1))))))))) Arctic Interpretation Processor: dimension: 1 interpretation: [b](x0) = x0, [a](x0) = 1x0 orientation: a(b(a(x1))) = 2x1 >= 1x1 = b(b(a(b(b(x1))))) a(b(b(a(x1)))) = 2x1 >= 1x1 = b(b(b(a(b(b(b(x1))))))) a(b(a(b(a(x1))))) = 3x1 >= 3x1 = b(a(b(b(a(b(b(a(b(x1))))))))) a(b(b(b(a(x1))))) = 2x1 >= 1x1 = b(b(b(b(a(b(b(b(b(x1))))))))) problem: a(b(a(b(a(x1))))) -> b(a(b(b(a(b(b(a(b(x1))))))))) Bounds Processor: bound: 1 enrichment: match automaton: final states: {3} transitions: b1(20) -> 21* b1(22) -> 23* b1(17) -> 18* b1(24) -> 25* b1(19) -> 20* b1(14) -> 15* b1(16) -> 17* a1(15) -> 16* a1(21) -> 22* a1(18) -> 19* a0(3) -> 3* b0(3) -> 3* 3 -> 14* 21 -> 24* 23 -> 16,3 25 -> 15* problem: Qed