34.91/9.30 YES 34.91/9.31 34.91/9.31 Problem: 34.91/9.31 strict: 34.91/9.31 a(a(c(x1))) -> c(c(b(x1))) 34.91/9.31 b(b(b(x1))) -> a(b(b(x1))) 34.91/9.31 weak: 34.91/9.31 b(b(a(x1))) -> a(a(c(x1))) 34.91/9.31 c(c(b(x1))) -> b(b(c(x1))) 34.91/9.31 b(b(a(x1))) -> a(b(b(x1))) 34.91/9.31 a(b(a(x1))) -> c(b(a(x1))) 34.91/9.31 34.91/9.31 Proof: 34.91/9.31 Matrix Interpretation Processor: dim=2 34.91/9.31 34.91/9.31 interpretation: 34.91/9.31 [1 1] [0] 34.91/9.31 [b](x0) = [1 0]x0 + [1], 34.91/9.31 34.91/9.31 [1 1] [0] 34.91/9.31 [a](x0) = [1 0]x0 + [1], 34.91/9.31 34.91/9.31 [1 1] 34.91/9.31 [c](x0) = [1 0]x0 34.91/9.31 orientation: 34.91/9.31 [3 2] [1] [3 2] [1] 34.91/9.31 a(a(c(x1))) = [2 1]x1 + [1] >= [2 1]x1 + [1] = c(c(b(x1))) 34.91/9.31 34.91/9.31 [3 2] [2] [3 2] [2] 34.91/9.31 b(b(b(x1))) = [2 1]x1 + [2] >= [2 1]x1 + [2] = a(b(b(x1))) 34.91/9.31 34.91/9.31 [3 2] [2] [3 2] [1] 34.91/9.31 b(b(a(x1))) = [2 1]x1 + [2] >= [2 1]x1 + [1] = a(a(c(x1))) 34.91/9.31 34.91/9.31 [3 2] [1] [3 2] [1] 34.91/9.31 c(c(b(x1))) = [2 1]x1 + [1] >= [2 1]x1 + [1] = b(b(c(x1))) 34.91/9.31 34.91/9.31 [3 2] [2] [3 2] [2] 34.91/9.31 b(b(a(x1))) = [2 1]x1 + [2] >= [2 1]x1 + [2] = a(b(b(x1))) 34.91/9.31 34.91/9.31 [3 2] [2] [3 2] [2] 34.91/9.31 a(b(a(x1))) = [2 1]x1 + [2] >= [2 1]x1 + [1] = c(b(a(x1))) 34.91/9.31 problem: 34.91/9.31 strict: 34.91/9.31 a(a(c(x1))) -> c(c(b(x1))) 34.91/9.31 b(b(b(x1))) -> a(b(b(x1))) 34.91/9.31 weak: 34.91/9.31 c(c(b(x1))) -> b(b(c(x1))) 34.91/9.31 b(b(a(x1))) -> a(b(b(x1))) 34.91/9.31 a(b(a(x1))) -> c(b(a(x1))) 34.91/9.31 RT Transformation Processor: 34.91/9.31 a(a(c(x1))) -> c(c(b(x1))) 34.91/9.31 b(b(b(x1))) -> a(b(b(x1))) 34.91/9.31 c(c(b(x1))) -> b(b(c(x1))) 34.91/9.31 b(b(a(x1))) -> a(b(b(x1))) 34.91/9.31 a(b(a(x1))) -> c(b(a(x1))) 34.91/9.31 Matrix Interpretation Processor: dim=2 34.91/9.31 34.91/9.31 interpretation: 34.91/9.31 [1 1] [1] 34.91/9.31 [b](x0) = [1 0]x0 + [0], 34.91/9.31 34.91/9.31 [1 1] [1] 34.91/9.31 [a](x0) = [1 0]x0 + [0], 34.91/9.31 34.91/9.31 [1 1] 34.91/9.31 [c](x0) = [1 0]x0 34.91/9.31 orientation: 34.91/9.31 [3 2] [2] [3 2] [2] 34.91/9.31 a(a(c(x1))) = [2 1]x1 + [1] >= [2 1]x1 + [1] = c(c(b(x1))) 34.91/9.31 34.91/9.31 [3 2] [4] [3 2] [4] 34.91/9.31 b(b(b(x1))) = [2 1]x1 + [2] >= [2 1]x1 + [2] = a(b(b(x1))) 34.91/9.31 34.91/9.31 [3 2] [2] [3 2] [2] 34.91/9.31 c(c(b(x1))) = [2 1]x1 + [1] >= [2 1]x1 + [1] = b(b(c(x1))) 34.91/9.31 34.91/9.31 [3 2] [4] [3 2] [4] 34.91/9.31 b(b(a(x1))) = [2 1]x1 + [2] >= [2 1]x1 + [2] = a(b(b(x1))) 34.91/9.31 34.91/9.31 [3 2] [4] [3 2] [3] 34.91/9.31 a(b(a(x1))) = [2 1]x1 + [2] >= [2 1]x1 + [2] = c(b(a(x1))) 34.91/9.31 problem: 34.91/9.31 a(a(c(x1))) -> c(c(b(x1))) 34.91/9.31 b(b(b(x1))) -> a(b(b(x1))) 34.91/9.31 c(c(b(x1))) -> b(b(c(x1))) 34.91/9.31 b(b(a(x1))) -> a(b(b(x1))) 34.91/9.31 Matrix Interpretation Processor: dim=3 34.91/9.31 34.91/9.31 interpretation: 34.91/9.31 [1 1 0] 34.91/9.31 [b](x0) = [0 0 0]x0 34.91/9.31 [0 0 0] , 34.91/9.31 34.91/9.31 [1 1 0] 34.91/9.31 [a](x0) = [0 0 1]x0 34.91/9.31 [0 1 0] , 34.91/9.31 34.91/9.31 [1 0 0] [0] 34.91/9.31 [c](x0) = [0 0 0]x0 + [0] 34.91/9.31 [0 1 0] [1] 34.91/9.31 orientation: 34.91/9.31 [1 1 0] [1] [1 1 0] [0] 34.91/9.31 a(a(c(x1))) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = c(c(b(x1))) 34.91/9.31 [0 1 0] [1] [0 0 0] [1] 34.91/9.31 34.91/9.31 [1 1 0] [1 1 0] 34.91/9.31 b(b(b(x1))) = [0 0 0]x1 >= [0 0 0]x1 = a(b(b(x1))) 34.91/9.31 [0 0 0] [0 0 0] 34.91/9.31 34.91/9.31 [1 1 0] [0] [1 0 0] 34.91/9.31 c(c(b(x1))) = [0 0 0]x1 + [0] >= [0 0 0]x1 = b(b(c(x1))) 34.91/9.31 [0 0 0] [1] [0 0 0] 34.91/9.31 34.91/9.31 [1 1 1] [1 1 0] 34.91/9.31 b(b(a(x1))) = [0 0 0]x1 >= [0 0 0]x1 = a(b(b(x1))) 34.91/9.31 [0 0 0] [0 0 0] 34.91/9.31 problem: 34.91/9.31 b(b(b(x1))) -> a(b(b(x1))) 34.91/9.31 c(c(b(x1))) -> b(b(c(x1))) 34.91/9.31 b(b(a(x1))) -> a(b(b(x1))) 34.91/9.31 Matrix Interpretation Processor: dim=3 34.91/9.31 34.91/9.31 interpretation: 34.91/9.31 [1 1 0] [0] 34.91/9.31 [b](x0) = [0 0 1]x0 + [0] 34.91/9.31 [0 0 0] [1], 34.91/9.31 34.91/9.31 34.91/9.31 [a](x0) = x0 34.91/9.31 , 34.91/9.31 34.91/9.31 [1 1 0] [0] 34.91/9.31 [c](x0) = [0 0 1]x0 + [0] 34.91/9.31 [0 0 0] [1] 34.91/9.31 orientation: 34.91/9.31 [1 1 1] [1] [1 1 1] [0] 34.91/9.31 b(b(b(x1))) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = a(b(b(x1))) 34.91/9.31 [0 0 0] [1] [0 0 0] [1] 34.91/9.31 34.91/9.31 [1 1 1] [1] [1 1 1] [1] 34.91/9.31 c(c(b(x1))) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = b(b(c(x1))) 34.91/9.31 [0 0 0] [1] [0 0 0] [1] 34.91/9.31 34.91/9.31 [1 1 1] [0] [1 1 1] [0] 34.91/9.31 b(b(a(x1))) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = a(b(b(x1))) 34.91/9.31 [0 0 0] [1] [0 0 0] [1] 34.91/9.31 problem: 34.91/9.31 c(c(b(x1))) -> b(b(c(x1))) 34.91/9.31 b(b(a(x1))) -> a(b(b(x1))) 34.91/9.31 Matrix Interpretation Processor: dim=3 34.91/9.31 34.91/9.31 interpretation: 34.91/9.31 34.91/9.31 [b](x0) = x0 34.91/9.31 , 34.91/9.31 34.91/9.31 [1 1 0] [1] 34.91/9.31 [a](x0) = [1 0 0]x0 + [0] 34.91/9.31 [0 0 1] [0], 34.91/9.31 34.91/9.31 [1 0 1] [0] 34.91/9.31 [c](x0) = [0 1 0]x0 + [0] 34.91/9.31 [0 1 0] [1] 34.91/9.31 orientation: 34.91/9.31 [1 1 1] [1] [1 0 1] [0] 34.91/9.31 c(c(b(x1))) = [0 1 0]x1 + [0] >= [0 1 0]x1 + [0] = b(b(c(x1))) 34.91/9.31 [0 1 0] [1] [0 1 0] [1] 34.91/9.31 34.91/9.31 [1 1 0] [1] [1 1 0] [1] 34.91/9.31 b(b(a(x1))) = [1 0 0]x1 + [0] >= [1 0 0]x1 + [0] = a(b(b(x1))) 34.91/9.31 [0 0 1] [0] [0 0 1] [0] 34.91/9.31 problem: 34.91/9.31 b(b(a(x1))) -> a(b(b(x1))) 34.91/9.31 Matrix Interpretation Processor: dim=3 34.91/9.31 34.91/9.31 interpretation: 34.91/9.31 [1 1 0] 34.91/9.31 [b](x0) = [0 0 1]x0 34.91/9.31 [1 1 0] , 34.91/9.31 34.91/9.31 [0] 34.91/9.31 [a](x0) = x0 + [1] 34.91/9.31 [0] 34.91/9.31 orientation: 34.91/9.31 [1 1 1] [1] [1 1 1] [0] 34.91/9.31 b(b(a(x1))) = [1 1 0]x1 + [1] >= [1 1 0]x1 + [1] = a(b(b(x1))) 34.91/9.31 [1 1 1] [1] [1 1 1] [0] 34.91/9.31 problem: 34.91/9.31 34.91/9.31 Qed 34.91/9.32 EOF