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: 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 0] [0] [1 0 0] a(b(x1)) = [0 0 0]x1 + [1] >= [0 0 0]x1 = b(b(b(x1))) [0 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 0 0] [0] [1 0 0] [0] a(a(b(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 1] a(b(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 0] [0] [1 0 0] a(b(b(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] [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 0 0] [1] [1 0 0] [0] a(a(a(b(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 1] [0] a(a(b(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 0 0] [0] [1 0 0] [0] a(a(b(b(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 1 1] [0] [1 1 1] a(b(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 0 0] [0] [1 0 0] a(b(a(b(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 1] a(b(b(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 0] [0] [1 0 0] a(b(b(b(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(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=4 interpretation: [1 0 0 0] [0 0 0 0] [b](x0) = [0 0 0 0]x0 [0 0 0 0] , [1 0 0 1] [0] [0 0 0 0] [1] [a](x0) = [0 1 0 0]x0 + [0] [0 0 1 0] [0] orientation: [1 0 0 1] [0] [1 0 0 0] [0 0 0 0] [1] [0 0 0 0] a(x1) = [0 1 0 0]x1 + [0] >= [0 0 0 0]x1 = b(x1) [0 0 1 0] [0] [0 0 0 0] [1 0 1 1] [0] [1 0 0 1] [0] [0 0 0 0] [1] [0 0 0 0] [1] a(a(x1)) = [0 0 0 0]x1 + [1] >= [0 0 0 0]x1 + [0] = a(b(a(x1))) [0 1 0 0] [0] [0 0 0 0] [0] [1 0 0 0] [0] [1 0 0 0] [0 0 0 0] [1] [0 0 0 0] a(b(x1)) = [0 0 0 0]x1 + [0] >= [0 0 0 0]x1 = b(b(b(x1))) [0 0 0 0] [0] [0 0 0 0] [1 0 0 0] [0] [1 0 0 0] [0] [0 0 0 0] [1] [0 0 0 0] [1] a(a(b(x1))) = [0 0 0 0]x1 + [1] >= [0 0 0 0]x1 + [0] = a(b(b(a(b(x1))))) [0 0 0 0] [0] [0 0 0 0] [0] [1 0 0 1] [0] [1 0 0 1] [0 0 0 0] [1] [0 0 0 0] a(b(a(x1))) = [0 0 0 0]x1 + [0] >= [0 0 0 0]x1 = b(a(b(b(a(x1))))) [0 0 0 0] [0] [0 0 0 0] [1 0 0 0] [0] [1 0 0 0] [0 0 0 0] [1] [0 0 0 0] a(b(b(x1))) = [0 0 0 0]x1 + [0] >= [0 0 0 0]x1 = b(b(b(b(b(x1))))) [0 0 0 0] [0] [0 0 0 0] [1 1 1 1] [1] [1 1 1 1] [0] [0 0 0 0] [1] [0 0 0 0] [1] a(a(a(a(x1)))) = [0 0 0 0]x1 + [1] >= [0 0 0 0]x1 + [1] = a(a(a(b(a(a(a(x1))))))) [0 0 0 0] [1] [0 0 0 0] [1] [1 0 0 1] [0] [1 0 0 1] [0] [0 0 0 0] [1] [0 0 0 0] [1] a(a(b(a(x1)))) = [0 0 0 0]x1 + [1] >= [0 0 0 0]x1 + [0] = a(b(a(b(a(b(a(x1))))))) [0 0 0 0] [0] [0 0 0 0] [0] [1 0 0 0] [0] [1 0 0 0] [0] [0 0 0 0] [1] [0 0 0 0] [1] a(a(b(b(x1)))) = [0 0 0 0]x1 + [1] >= [0 0 0 0]x1 + [0] = a(b(b(b(a(b(b(x1))))))) [0 0 0 0] [0] [0 0 0 0] [0] [1 0 1 1] [0] [1 0 1 1] [0 0 0 0] [1] [0 0 0 0] a(b(a(a(x1)))) = [0 0 0 0]x1 + [0] >= [0 0 0 0]x1 = b(a(a(b(b(a(a(x1))))))) [0 0 0 0] [0] [0 0 0 0] [1 0 0 0] [0] [1 0 0 0] [0 0 0 0] [1] [0 0 0 0] a(b(a(b(x1)))) = [0 0 0 0]x1 + [0] >= [0 0 0 0]x1 = b(a(b(b(b(a(b(x1))))))) [0 0 0 0] [0] [0 0 0 0] [1 0 0 1] [0] [1 0 0 1] [0 0 0 0] [1] [0 0 0 0] a(b(b(a(x1)))) = [0 0 0 0]x1 + [0] >= [0 0 0 0]x1 = b(b(a(b(b(b(a(x1))))))) [0 0 0 0] [0] [0 0 0 0] [1 0 0 0] [0] [1 0 0 0] [0 0 0 0] [1] [0 0 0 0] a(b(b(b(x1)))) = [0 0 0 0]x1 + [0] >= [0 0 0 0]x1 = b(b(b(b(b(b(b(x1))))))) [0 0 0 0] [0] [0 0 0 0] problem: 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(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))))))) String Reversal Processor: 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(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 1]x0 [0 0 0] , [1 1 0] [0] [a](x0) = [0 0 1]x0 + [1] [0 0 1] [1] orientation: [1 1 0] [0] [1 0 0] a(x1) = [0 0 1]x1 + [1] >= [0 0 1]x1 = b(x1) [0 0 1] [1] [0 0 0] [1 1 1] [1] [1 1 1] [1] a(a(x1)) = [0 0 1]x1 + [2] >= [0 0 0]x1 + [1] = a(b(a(x1))) [0 0 1] [2] [0 0 0] [1] [1 1 0] [0] [1 0 0] b(a(x1)) = [0 0 1]x1 + [1] >= [0 0 0]x1 = b(b(b(x1))) [0 0 0] [0] [0 0 0] [1 1 1] [1] [1 1 0] [0] b(a(a(x1))) = [0 0 1]x1 + [2] >= [0 0 0]x1 + [1] = b(a(b(b(a(x1))))) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1] [1 0 1] [0] 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] [1] [1 1 0] [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 2] [3] [1 1 1] [3] 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] [1] [1 1 1] [1] [1 1 0] b(b(a(a(x1)))) = [0 0 0]x1 + [0] >= [0 0 0]x1 = b(b(a(b(b(b(a(x1))))))) [0 0 0] [0] [0 0 0] [1 1 1] [2] [1 0 1] [2] a(a(b(a(x1)))) = [0 0 0]x1 + [2] >= [0 0 0]x1 + [2] = a(a(b(b(a(a(b(x1))))))) [0 0 0] [2] [0 0 0] [2] [1 1 1] [1] [1 0 1] [0] b(a(b(a(x1)))) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = b(a(b(b(b(a(b(x1))))))) [0 0 0] [0] [0 0 0] [0] [1 1 0] [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] [1] [0 0 0] [1] [1 1 0] [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(b(a(x1))) -> b(b(b(b(b(x1))))) a(b(a(a(x1)))) -> a(b(a(b(a(b(a(x1))))))) a(a(b(a(x1)))) -> a(a(b(b(a(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=4 interpretation: [1 0 0 0] [0 0 0 1] [b](x0) = [0 0 0 1]x0 [0 0 0 0] , [1 0 1 0] [0] [0 1 0 1] [1] [a](x0) = [0 1 0 1]x0 + [0] [0 1 0 1] [0] orientation: [1 0 1 0] [0] [1 0 0 0] [0 1 0 1] [1] [0 0 0 1] a(x1) = [0 1 0 1]x1 + [0] >= [0 0 0 1]x1 = b(x1) [0 1 0 1] [0] [0 0 0 0] [1 1 1 1] [0] [1 1 1 1] [0] [0 2 0 2] [2] [0 1 0 1] [1] a(a(x1)) = [0 2 0 2]x1 + [1] >= [0 1 0 1]x1 + [0] = a(b(a(x1))) [0 2 0 2] [1] [0 1 0 1] [0] [1 0 1 0] [1 0 0 0] [0 1 0 1] [0 0 0 0] b(a(x1)) = [0 1 0 1]x1 >= [0 0 0 0]x1 = b(b(b(x1))) [0 0 0 0] [0 0 0 0] [1 0 1 0] [1 0 0 0] [0 0 0 0] [0 0 0 0] b(b(a(x1))) = [0 0 0 0]x1 >= [0 0 0 0]x1 = b(b(b(b(b(x1))))) [0 0 0 0] [0 0 0 0] [1 3 1 3] [1] [1 3 1 3] [0] [0 2 0 2] [2] [0 1 0 1] [1] a(b(a(a(x1)))) = [0 2 0 2]x1 + [1] >= [0 1 0 1]x1 + [0] = a(b(a(b(a(b(a(x1))))))) [0 2 0 2] [1] [0 1 0 1] [0] [1 2 1 2] [0] [1 0 0 2] [0] [0 2 0 2] [2] [0 0 0 0] [2] a(a(b(a(x1)))) = [0 2 0 2]x1 + [1] >= [0 0 0 0]x1 + [1] = a(a(b(b(a(a(b(x1))))))) [0 2 0 2] [1] [0 0 0 0] [1] [1 0 1 0] [0] [1 0 0 0] [0] [0 0 0 0] [1] [0 0 0 0] [1] a(b(b(a(x1)))) = [0 0 0 0]x1 + [0] >= [0 0 0 0]x1 + [0] = a(b(b(b(a(b(b(x1))))))) [0 0 0 0] [0] [0 0 0 0] [0] [1 0 1 0] [1 0 0 0] [0 0 0 0] [0 0 0 0] b(b(b(a(x1)))) = [0 0 0 0]x1 >= [0 0 0 0]x1 = b(b(b(b(b(b(b(x1))))))) [0 0 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(b(a(x1))) -> b(b(b(b(b(x1))))) a(a(b(a(x1)))) -> a(a(b(b(a(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 1]x0 [0 0 0] , [1 0 1] [0] [a](x0) = [0 0 1]x0 + [0] [0 1 1] [1] orientation: [1 0 1] [0] [1 0 0] a(x1) = [0 0 1]x1 + [0] >= [0 0 1]x1 = b(x1) [0 1 1] [1] [0 0 0] [1 1 2] [1] [1 0 1] [0] a(a(x1)) = [0 1 1]x1 + [1] >= [0 0 0]x1 + [0] = a(b(a(x1))) [0 1 2] [2] [0 1 1] [2] [1 0 1] [0] [1 0 0] b(a(x1)) = [0 1 1]x1 + [1] >= [0 0 0]x1 = b(b(b(x1))) [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 2] [2] [1 0 1] [2] a(a(b(a(x1)))) = [0 1 1]x1 + [2] >= [0 0 0]x1 + [1] = a(a(b(b(a(a(b(x1))))))) [0 1 1] [3] [0 0 0] [2] [1 0 1] [0] [1 0 0] [0] a(b(b(a(x1)))) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = a(b(b(b(a(b(b(x1))))))) [0 0 0] [1] [0 0 0] [1] [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) b(a(x1)) -> b(b(b(x1))) b(b(a(x1))) -> b(b(b(b(b(x1))))) a(a(b(a(x1)))) -> a(a(b(b(a(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(b(x1)) -> b(b(b(x1))) a(b(b(x1))) -> b(b(b(b(b(x1))))) a(b(a(a(x1)))) -> b(a(a(b(b(a(a(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: [1 0 0] [b](x0) = [0 0 1]x0 [0 0 0] , [1 1 0] [0] [a](x0) = [0 1 1]x0 + [0] [0 0 0] [1] orientation: [1 1 0] [0] [1 0 0] a(x1) = [0 1 1]x1 + [0] >= [0 0 1]x1 = b(x1) [0 0 0] [1] [0 0 0] [1 0 1] [0] [1 0 0] a(b(x1)) = [0 0 1]x1 + [0] >= [0 0 0]x1 = b(b(b(x1))) [0 0 0] [1] [0 0 0] [1 0 0] [0] [1 0 0] a(b(b(x1))) = [0 0 0]x1 + [0] >= [0 0 0]x1 = b(b(b(b(b(x1))))) [0 0 0] [1] [0 0 0] [1 2 1] [1] [1 2 1] [0] a(b(a(a(x1)))) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = b(a(a(b(b(a(a(x1))))))) [0 0 0] [1] [0 0 0] [0] [1 1 0] [0] [1 1 0] a(b(b(a(x1)))) = [0 0 0]x1 + [0] >= [0 0 0]x1 = b(b(a(b(b(b(a(x1))))))) [0 0 0] [1] [0 0 0] [1 0 0] [0] [1 0 0] a(b(b(b(x1)))) = [0 0 0]x1 + [0] >= [0 0 0]x1 = b(b(b(b(b(b(b(x1))))))) [0 0 0] [1] [0 0 0] problem: a(x1) -> b(x1) a(b(x1)) -> b(b(b(x1))) a(b(b(x1))) -> b(b(b(b(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))))))) String Reversal Processor: a(x1) -> b(x1) b(a(x1)) -> b(b(b(x1))) b(b(a(x1))) -> b(b(b(b(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))))))) Bounds Processor: bound: 1 enrichment: match automaton: final states: {12,7,5,3,1} transitions: b1(16) -> 17* b1(40) -> 41* b1(25) -> 26* b1(37) -> 38* b1(28) -> 29* b1(18) -> 19* b1(29) -> 30* b1(26) -> 27* b1(38) -> 39* f20() -> 2* b0(10) -> 11* b0(5) -> 13* b0(6) -> 5* b0(2) -> 1* b0(1) -> 4* b0(4) -> 3* b0(9) -> 10* b0(8) -> 9* b0(3) -> 6* b0(13) -> 12* a0(11) -> 7* a0(4) -> 8* 27 -> 28,9 19 -> 40,7 17 -> 25,8 30 -> 37,10 7 -> 8* 4 -> 16* 11 -> 18* 3 -> 4,16,1 39 -> 11* 41 -> 26* problem: Qed