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))))) b(a(x1)) -> b(b(b(x1))) a(b(a(x1))) -> a(b(b(a(b(x1))))) b(a(a(x1))) -> b(a(b(b(a(x1))))) b(b(a(x1))) -> 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(b(x1)) -> b(b(b(x1))) a(b(a(x1))) -> b(a(b(b(a(x1))))) a(a(b(x1))) -> a(b(b(a(b(x1))))) a(b(b(x1))) -> b(b(b(b(b(x1))))) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [b](x0) = [0 1 0]x0 [0 0 0] , [1 0 1] [0] [a](x0) = [0 1 0]x0 + [0] [0 0 0] [1] orientation: [1 0 1] [0] [1 0 0] a(x1) = [0 1 0]x1 + [0] >= [0 1 0]x1 = b(x1) [0 0 0] [1] [0 0 0] [1 0 1] [1] [1 0 1] [0] a(a(x1)) = [0 1 0]x1 + [0] >= [0 1 0]x1 + [0] = a(b(a(x1))) [0 0 0] [1] [0 0 0] [1] [1 0 1] [1 0 0] b(a(x1)) = [0 1 0]x1 >= [0 1 0]x1 = b(b(b(x1))) [0 0 0] [0 0 0] [1 0 1] [2] [1 0 1] [2] a(a(a(x1))) = [0 1 0]x1 + [0] >= [0 1 0]x1 + [0] = a(a(b(a(a(x1))))) [0 0 0] [1] [0 0 0] [1] [1 0 1] [1] [1 0 1] b(a(a(x1))) = [0 1 0]x1 + [0] >= [0 1 0]x1 = b(a(b(b(a(x1))))) [0 0 0] [0] [0 0 0] [1 0 1] [0] [1 0 0] [0] a(b(a(x1))) = [0 1 0]x1 + [0] >= [0 1 0]x1 + [0] = a(b(b(a(b(x1))))) [0 0 0] [1] [0 0 0] [1] [1 0 1] [1 0 0] b(b(a(x1))) = [0 1 0]x1 >= [0 1 0]x1 = b(b(b(b(b(x1))))) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] a(b(x1)) = [0 1 0]x1 + [0] >= [0 1 0]x1 = b(b(b(x1))) [0 0 0] [1] [0 0 0] [1 0 1] [0] [1 0 1] a(b(a(x1))) = [0 1 0]x1 + [0] >= [0 1 0]x1 = b(a(b(b(a(x1))))) [0 0 0] [1] [0 0 0] [1 0 0] [1] [1 0 0] [0] a(a(b(x1))) = [0 1 0]x1 + [0] >= [0 1 0]x1 + [0] = a(b(b(a(b(x1))))) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] a(b(b(x1))) = [0 1 0]x1 + [0] >= [0 1 0]x1 = b(b(b(b(b(x1))))) [0 0 0] [1] [0 0 0] problem: a(x1) -> b(x1) b(a(x1)) -> b(b(b(x1))) a(a(a(x1))) -> a(a(b(a(a(x1))))) a(b(a(x1))) -> a(b(b(a(b(x1))))) b(b(a(x1))) -> b(b(b(b(b(x1))))) a(b(x1)) -> b(b(b(x1))) a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(b(x1))) -> b(b(b(b(b(x1))))) String Reversal Processor: a(x1) -> b(x1) a(b(x1)) -> b(b(b(x1))) a(a(a(x1))) -> a(a(b(a(a(x1))))) a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(b(x1))) -> b(b(b(b(b(x1))))) b(a(x1)) -> b(b(b(x1))) a(b(a(x1))) -> a(b(b(a(b(x1))))) b(b(a(x1))) -> 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 1 0]x0 + [1] [0 1 0] [0] orientation: [1 0 1] [0] [1 0 0] a(x1) = [0 1 0]x1 + [1] >= [0 0 0]x1 = b(x1) [0 1 0] [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 2 1] [1] [1 1 1] [0] a(a(a(x1))) = [0 1 0]x1 + [3] >= [0 0 0]x1 + [2] = a(a(b(a(a(x1))))) [0 1 0] [2] [0 0 0] [1] [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 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 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] problem: a(x1) -> b(x1) a(b(x1)) -> b(b(b(x1))) a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(b(x1))) -> b(b(b(b(b(x1))))) b(a(x1)) -> b(b(b(x1))) a(b(a(x1))) -> a(b(b(a(b(x1))))) b(b(a(x1))) -> b(b(b(b(b(x1))))) Matrix Interpretation Processor: dim=1 interpretation: [b](x0) = x0, [a](x0) = 4x0 + 2 orientation: a(x1) = 4x1 + 2 >= x1 = b(x1) a(b(x1)) = 4x1 + 2 >= x1 = b(b(b(x1))) a(b(a(x1))) = 16x1 + 10 >= 16x1 + 10 = b(a(b(b(a(x1))))) a(b(b(x1))) = 4x1 + 2 >= x1 = b(b(b(b(b(x1))))) b(a(x1)) = 4x1 + 2 >= x1 = b(b(b(x1))) a(b(a(x1))) = 16x1 + 10 >= 16x1 + 10 = a(b(b(a(b(x1))))) b(b(a(x1))) = 4x1 + 2 >= x1 = b(b(b(b(b(x1))))) problem: a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(a(x1))) -> a(b(b(a(b(x1))))) Bounds Processor: bound: 0 enrichment: match automaton: final states: {7,1} transitions: f20() -> 2* b0(10) -> 11* b0(6) -> 1* b0(2) -> 8* b0(4) -> 5* b0(9) -> 10* b0(3) -> 4* a0(11) -> 7* a0(2) -> 3* a0(8) -> 9* a0(5) -> 6* 7 -> 3,9 1 -> 3,9 problem: Qed