YES(?,O(n^3)) 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: RT Transformation Processor: strict: 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))))) weak: Matrix Interpretation Processor: dimension: 1 interpretation: [b](x0) = x0, [a](x0) = x0 + 8 orientation: a(x1) = x1 + 8 >= x1 = b(x1) a(a(x1)) = x1 + 16 >= x1 + 16 = a(b(a(x1))) a(b(x1)) = x1 + 8 >= x1 = b(b(b(x1))) a(a(a(x1))) = x1 + 24 >= x1 + 32 = a(a(b(a(a(x1))))) a(a(b(x1))) = x1 + 16 >= x1 + 16 = a(b(b(a(b(x1))))) a(b(a(x1))) = x1 + 16 >= x1 + 16 = b(a(b(b(a(x1))))) a(b(b(x1))) = x1 + 8 >= x1 = b(b(b(b(b(x1))))) b(a(x1)) = x1 + 8 >= x1 = b(b(b(x1))) a(b(a(x1))) = x1 + 16 >= x1 + 16 = a(b(b(a(b(x1))))) b(a(a(x1))) = x1 + 16 >= x1 + 16 = b(a(b(b(a(x1))))) b(b(a(x1))) = x1 + 8 >= x1 = b(b(b(b(b(x1))))) problem: strict: a(a(x1)) -> a(b(a(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(a(x1))) -> a(b(b(a(b(x1))))) b(a(a(x1))) -> b(a(b(b(a(x1))))) weak: a(x1) -> b(x1) a(b(x1)) -> b(b(b(x1))) a(b(b(x1))) -> b(b(b(b(b(x1))))) b(a(x1)) -> b(b(b(x1))) b(b(a(x1))) -> b(b(b(b(b(x1))))) Matrix Interpretation Processor: dimension: 2 interpretation: [1 0] [b](x0) = [0 0]x0, [1 1] [0] [a](x0) = [0 0]x0 + [3] orientation: [1 1] [3] [1 1] [0] a(a(x1)) = [0 0]x1 + [3] >= [0 0]x1 + [3] = a(b(a(x1))) [1 1] [6] [1 1] [6] a(a(a(x1))) = [0 0]x1 + [3] >= [0 0]x1 + [3] = a(a(b(a(a(x1))))) [1 0] [3] [1 0] [0] a(a(b(x1))) = [0 0]x1 + [3] >= [0 0]x1 + [3] = a(b(b(a(b(x1))))) [1 1] [0] [1 1] a(b(a(x1))) = [0 0]x1 + [3] >= [0 0]x1 = b(a(b(b(a(x1))))) [1 1] [0] [1 0] [0] a(b(a(x1))) = [0 0]x1 + [3] >= [0 0]x1 + [3] = a(b(b(a(b(x1))))) [1 1] [3] [1 1] b(a(a(x1))) = [0 0]x1 + [0] >= [0 0]x1 = b(a(b(b(a(x1))))) [1 1] [0] [1 0] a(x1) = [0 0]x1 + [3] >= [0 0]x1 = b(x1) [1 0] [0] [1 0] a(b(x1)) = [0 0]x1 + [3] >= [0 0]x1 = b(b(b(x1))) [1 0] [0] [1 0] a(b(b(x1))) = [0 0]x1 + [3] >= [0 0]x1 = b(b(b(b(b(x1))))) [1 1] [1 0] b(a(x1)) = [0 0]x1 >= [0 0]x1 = b(b(b(x1))) [1 1] [1 0] b(b(a(x1))) = [0 0]x1 >= [0 0]x1 = b(b(b(b(b(x1))))) problem: strict: a(a(a(x1))) -> a(a(b(a(a(x1))))) a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(a(x1))) -> a(b(b(a(b(x1))))) weak: a(a(x1)) -> a(b(a(x1))) a(a(b(x1))) -> a(b(b(a(b(x1))))) b(a(a(x1))) -> b(a(b(b(a(x1))))) a(x1) -> b(x1) a(b(x1)) -> b(b(b(x1))) a(b(b(x1))) -> b(b(b(b(b(x1))))) b(a(x1)) -> b(b(b(x1))) b(b(a(x1))) -> b(b(b(b(b(x1))))) Matrix Interpretation Processor: dimension: 2 interpretation: [1 0] [b](x0) = [0 0]x0, [1 2] [0] [a](x0) = [0 1]x0 + [1] orientation: [1 6] [6] [1 4] [4] a(a(a(x1))) = [0 1]x1 + [3] >= [0 0]x1 + [2] = a(a(b(a(a(x1))))) [1 2] [0] [1 2] a(b(a(x1))) = [0 0]x1 + [1] >= [0 0]x1 = b(a(b(b(a(x1))))) [1 2] [0] [1 0] [0] a(b(a(x1))) = [0 0]x1 + [1] >= [0 0]x1 + [1] = a(b(b(a(b(x1))))) [1 4] [2] [1 2] [0] a(a(x1)) = [0 1]x1 + [2] >= [0 0]x1 + [1] = a(b(a(x1))) [1 0] [2] [1 0] [0] a(a(b(x1))) = [0 0]x1 + [2] >= [0 0]x1 + [1] = a(b(b(a(b(x1))))) [1 4] [2] [1 2] b(a(a(x1))) = [0 0]x1 + [0] >= [0 0]x1 = b(a(b(b(a(x1))))) [1 2] [0] [1 0] a(x1) = [0 1]x1 + [1] >= [0 0]x1 = b(x1) [1 0] [0] [1 0] a(b(x1)) = [0 0]x1 + [1] >= [0 0]x1 = b(b(b(x1))) [1 0] [0] [1 0] a(b(b(x1))) = [0 0]x1 + [1] >= [0 0]x1 = b(b(b(b(b(x1))))) [1 2] [1 0] b(a(x1)) = [0 0]x1 >= [0 0]x1 = b(b(b(x1))) [1 2] [1 0] b(b(a(x1))) = [0 0]x1 >= [0 0]x1 = b(b(b(b(b(x1))))) problem: strict: a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(a(x1))) -> a(b(b(a(b(x1))))) weak: a(a(a(x1))) -> a(a(b(a(a(x1))))) a(a(x1)) -> a(b(a(x1))) a(a(b(x1))) -> a(b(b(a(b(x1))))) b(a(a(x1))) -> b(a(b(b(a(x1))))) a(x1) -> b(x1) a(b(x1)) -> b(b(b(x1))) a(b(b(x1))) -> b(b(b(b(b(x1))))) b(a(x1)) -> b(b(b(x1))) b(b(a(x1))) -> b(b(b(b(b(x1))))) Matrix Interpretation Processor: dimension: 3 interpretation: [1 0 0] [b](x0) = [0 0 2]x0 [0 0 0] , [1 1 4] [0] [a](x0) = [0 1 4]x0 + [0] [0 0 0] [1] orientation: [1 1 4] [2] [1 1 4] [0] a(b(a(x1))) = [0 0 0]x1 + [2] >= [0 0 0]x1 + [2] = b(a(b(b(a(x1))))) [0 0 0] [1] [0 0 0] [0] [1 1 4] [2] [1 0 2] [0] a(b(a(x1))) = [0 0 0]x1 + [2] >= [0 0 0]x1 + [0] = a(b(b(a(b(x1))))) [0 0 0] [1] [0 0 0] [1] [1 3 12] [12] [1 2 8] [12] a(a(a(x1))) = [0 1 4 ]x1 + [8 ] >= [0 0 0]x1 + [6 ] = a(a(b(a(a(x1))))) [0 0 0 ] [1 ] [0 0 0] [1 ] [1 2 8] [4] [1 1 4] [2] a(a(x1)) = [0 1 4]x1 + [4] >= [0 0 0]x1 + [2] = a(b(a(x1))) [0 0 0] [1] [0 0 0] [1] [1 0 4] [4] [1 0 2] [0] a(a(b(x1))) = [0 0 2]x1 + [4] >= [0 0 0]x1 + [0] = a(b(b(a(b(x1))))) [0 0 0] [1] [0 0 0] [1] [1 2 8] [4] [1 1 4] [0] b(a(a(x1))) = [0 0 0]x1 + [2] >= [0 0 0]x1 + [2] = b(a(b(b(a(x1))))) [0 0 0] [0] [0 0 0] [0] [1 1 4] [0] [1 0 0] a(x1) = [0 1 4]x1 + [0] >= [0 0 2]x1 = b(x1) [0 0 0] [1] [0 0 0] [1 0 2] [0] [1 0 0] a(b(x1)) = [0 0 2]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 1 4] [0] [1 0 0] b(a(x1)) = [0 0 0]x1 + [2] >= [0 0 0]x1 = b(b(b(x1))) [0 0 0] [0] [0 0 0] [1 1 4] [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: strict: weak: a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(a(x1))) -> a(b(b(a(b(x1))))) a(a(a(x1))) -> a(a(b(a(a(x1))))) a(a(x1)) -> a(b(a(x1))) a(a(b(x1))) -> a(b(b(a(b(x1))))) b(a(a(x1))) -> b(a(b(b(a(x1))))) a(x1) -> b(x1) a(b(x1)) -> b(b(b(x1))) a(b(b(x1))) -> b(b(b(b(b(x1))))) b(a(x1)) -> b(b(b(x1))) b(b(a(x1))) -> b(b(b(b(b(x1))))) Qed