YES Problem: a(a(x1)) -> b(x1) a(a(a(x1))) -> a(b(a(x1))) a(b(a(x1))) -> b(b(b(x1))) a(a(a(a(x1)))) -> a(a(b(a(a(x1))))) a(a(b(a(x1)))) -> a(b(b(a(b(x1))))) a(b(a(a(x1)))) -> b(a(b(b(a(x1))))) a(b(b(a(x1)))) -> b(b(b(b(b(x1))))) a(a(a(a(a(x1))))) -> a(a(a(b(a(a(a(x1))))))) a(a(a(b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) a(a(b(a(a(x1))))) -> a(b(a(b(a(b(a(x1))))))) a(a(b(b(a(x1))))) -> a(b(b(b(a(b(b(x1))))))) a(b(a(a(a(x1))))) -> b(a(a(b(b(a(a(x1))))))) a(b(a(b(a(x1))))) -> b(a(b(b(b(a(b(x1))))))) a(b(b(a(a(x1))))) -> b(b(a(b(b(b(a(x1))))))) a(b(b(b(a(x1))))) -> b(b(b(b(b(b(b(x1))))))) Proof: String Reversal Processor: a(a(x1)) -> b(x1) a(a(a(x1))) -> a(b(a(x1))) a(b(a(x1))) -> b(b(b(x1))) a(a(a(a(x1)))) -> a(a(b(a(a(x1))))) a(b(a(a(x1)))) -> b(a(b(b(a(x1))))) a(a(b(a(x1)))) -> a(b(b(a(b(x1))))) a(b(b(a(x1)))) -> b(b(b(b(b(x1))))) a(a(a(a(a(x1))))) -> a(a(a(b(a(a(a(x1))))))) a(b(a(a(a(x1))))) -> b(a(a(b(b(a(a(x1))))))) a(a(b(a(a(x1))))) -> a(b(a(b(a(b(a(x1))))))) a(b(b(a(a(x1))))) -> b(b(a(b(b(b(a(x1))))))) a(a(a(b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) a(b(a(b(a(x1))))) -> b(a(b(b(b(a(b(x1))))))) a(a(b(b(a(x1))))) -> a(b(b(b(a(b(b(x1))))))) a(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 1 1] [0] [1 0 0] a(a(x1)) = [0 0 0]x1 + [1] >= [0 0 0]x1 = b(x1) [0 0 0] [1] [0 0 0] [1 1 1] [1] [1 0 1] [0] a(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] [0] [1 0 0] a(b(a(x1))) = [0 0 0]x1 + [1] >= [0 0 0]x1 = b(b(b(x1))) [0 0 0] [0] [0 0 0] [1 1 1] [2] [1 1 1] [0] a(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] [0] [1 0 1] a(b(a(a(x1)))) = [0 0 0]x1 + [1] >= [0 0 0]x1 = b(a(b(b(a(x1))))) [0 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(b(b(a(b(x1))))) [0 0 0] [1] [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(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(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(b(a(a(x1))))))) [0 0 0] [0] [0 0 0] [1 1 1] [0] [1 0 1] [0] a(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] [1] [0 0 0] [0] [1 1 1] [0] [1 0 1] a(b(b(a(a(x1))))) = [0 0 0]x1 + [1] >= [0 0 0]x1 = b(b(a(b(b(b(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(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(b(a(b(x1))))))) [0 0 0] [0] [0 0 0] [1 0 1] [0] [1 0 0] [0] a(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] [1] [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(b(b(b(x1))))))) [0 0 0] [0] [0 0 0] problem: a(a(x1)) -> b(x1) a(b(a(x1))) -> b(b(b(x1))) a(b(a(a(x1)))) -> b(a(b(b(a(x1))))) a(a(b(a(x1)))) -> a(b(b(a(b(x1))))) a(b(b(a(x1)))) -> b(b(b(b(b(x1))))) a(a(b(a(a(x1))))) -> a(b(a(b(a(b(a(x1))))))) a(b(b(a(a(x1))))) -> b(b(a(b(b(b(a(x1))))))) a(b(a(b(a(x1))))) -> b(a(b(b(b(a(b(x1))))))) a(a(b(b(a(x1))))) -> a(b(b(b(a(b(b(x1))))))) a(b(b(b(a(x1))))) -> b(b(b(b(b(b(b(x1))))))) Arctic Interpretation Processor: dimension: 1 interpretation: [b](x0) = x0, [a](x0) = 2x0 orientation: a(a(x1)) = 4x1 >= x1 = b(x1) a(b(a(x1))) = 4x1 >= x1 = b(b(b(x1))) a(b(a(a(x1)))) = 6x1 >= 4x1 = b(a(b(b(a(x1))))) a(a(b(a(x1)))) = 6x1 >= 4x1 = a(b(b(a(b(x1))))) a(b(b(a(x1)))) = 4x1 >= x1 = b(b(b(b(b(x1))))) a(a(b(a(a(x1))))) = 8x1 >= 8x1 = a(b(a(b(a(b(a(x1))))))) a(b(b(a(a(x1))))) = 6x1 >= 4x1 = b(b(a(b(b(b(a(x1))))))) a(b(a(b(a(x1))))) = 6x1 >= 4x1 = b(a(b(b(b(a(b(x1))))))) a(a(b(b(a(x1))))) = 6x1 >= 4x1 = a(b(b(b(a(b(b(x1))))))) a(b(b(b(a(x1))))) = 4x1 >= x1 = b(b(b(b(b(b(b(x1))))))) problem: a(a(b(a(a(x1))))) -> a(b(a(b(a(b(a(x1))))))) Bounds Processor: bound: 1 enrichment: match automaton: final states: {3} transitions: a1(12) -> 13* a1(14) -> 15* a1(16) -> 17* a1(18) -> 19* b1(20) -> 21* b1(15) -> 16* b1(17) -> 18* b1(13) -> 14* a0(3) -> 3* b0(3) -> 3* 3 -> 12* 19 -> 13,20,3 21 -> 14* problem: Qed