34.96/9.16 YES 34.96/9.17 34.96/9.17 Problem: 34.96/9.17 strict: 34.96/9.17 a(a(b(x1))) -> c(b(c(x1))) 34.96/9.17 b(c(b(x1))) -> a(b(c(x1))) 34.96/9.17 a(a(b(x1))) -> c(a(b(x1))) 34.96/9.17 weak: 34.96/9.17 a(c(b(x1))) -> c(c(c(x1))) 34.96/9.17 c(a(c(x1))) -> c(b(b(x1))) 34.96/9.17 b(a(b(x1))) -> c(c(b(x1))) 34.96/9.17 c(b(c(x1))) -> c(c(b(x1))) 34.96/9.17 34.96/9.17 Proof: 34.96/9.17 Matrix Interpretation Processor: dim=2 34.96/9.17 34.96/9.17 interpretation: 34.96/9.17 [1 1] 34.96/9.17 [c](x0) = [0 0]x0, 34.96/9.17 34.96/9.17 [1 0] [1] 34.96/9.17 [a](x0) = [0 0]x0 + [1], 34.96/9.17 34.96/9.17 [1 1] [0] 34.96/9.17 [b](x0) = [0 0]x0 + [1] 34.96/9.17 orientation: 34.96/9.17 [1 1] [2] [1 1] [1] 34.96/9.17 a(a(b(x1))) = [0 0]x1 + [1] >= [0 0]x1 + [0] = c(b(c(x1))) 34.96/9.17 34.96/9.17 [1 1] [1] [1 1] [1] 34.96/9.17 b(c(b(x1))) = [0 0]x1 + [1] >= [0 0]x1 + [1] = a(b(c(x1))) 34.96/9.17 34.96/9.17 [1 1] [2] [1 1] [2] 34.96/9.17 a(a(b(x1))) = [0 0]x1 + [1] >= [0 0]x1 + [0] = c(a(b(x1))) 34.96/9.17 34.96/9.17 [1 1] [2] [1 1] 34.96/9.17 a(c(b(x1))) = [0 0]x1 + [1] >= [0 0]x1 = c(c(c(x1))) 34.96/9.17 34.96/9.17 [1 1] [2] [1 1] [2] 34.96/9.17 c(a(c(x1))) = [0 0]x1 + [0] >= [0 0]x1 + [0] = c(b(b(x1))) 34.96/9.17 34.96/9.17 [1 1] [2] [1 1] [1] 34.96/9.17 b(a(b(x1))) = [0 0]x1 + [1] >= [0 0]x1 + [0] = c(c(b(x1))) 34.96/9.17 34.96/9.17 [1 1] [1] [1 1] [1] 34.96/9.17 c(b(c(x1))) = [0 0]x1 + [0] >= [0 0]x1 + [0] = c(c(b(x1))) 34.96/9.17 problem: 34.96/9.17 strict: 34.96/9.17 b(c(b(x1))) -> a(b(c(x1))) 34.96/9.17 a(a(b(x1))) -> c(a(b(x1))) 34.96/9.17 weak: 34.96/9.17 c(a(c(x1))) -> c(b(b(x1))) 34.96/9.17 c(b(c(x1))) -> c(c(b(x1))) 34.96/9.17 RT Transformation Processor: 34.96/9.17 b(c(b(x1))) -> a(b(c(x1))) 34.96/9.17 a(a(b(x1))) -> c(a(b(x1))) 34.96/9.17 c(a(c(x1))) -> c(b(b(x1))) 34.96/9.17 c(b(c(x1))) -> c(c(b(x1))) 34.96/9.17 Matrix Interpretation Processor: dim=2 34.96/9.17 34.96/9.17 interpretation: 34.96/9.17 [1 1] [0] 34.96/9.17 [c](x0) = [0 0]x0 + [1], 34.96/9.17 34.96/9.17 [2 0] [1] 34.96/9.17 [a](x0) = [2 0]x0 + [2], 34.96/9.17 34.96/9.17 [1 0] [0] 34.96/9.17 [b](x0) = [1 2]x0 + [1] 34.96/9.17 orientation: 34.96/9.17 [2 2] [1] [2 2] [1] 34.96/9.17 b(c(b(x1))) = [2 2]x1 + [4] >= [2 2]x1 + [2] = a(b(c(x1))) 34.96/9.17 34.96/9.17 [4 0] [3] [4 0] [3] 34.96/9.17 a(a(b(x1))) = [4 0]x1 + [4] >= [0 0]x1 + [1] = c(a(b(x1))) 34.96/9.17 34.96/9.17 [4 4] [3] [4 4] [3] 34.96/9.17 c(a(c(x1))) = [0 0]x1 + [1] >= [0 0]x1 + [1] = c(b(b(x1))) 34.96/9.17 34.96/9.17 [2 2] [3] [2 2] [2] 34.96/9.17 c(b(c(x1))) = [0 0]x1 + [1] >= [0 0]x1 + [1] = c(c(b(x1))) 34.96/9.17 problem: 34.96/9.17 b(c(b(x1))) -> a(b(c(x1))) 34.96/9.17 a(a(b(x1))) -> c(a(b(x1))) 34.96/9.17 c(a(c(x1))) -> c(b(b(x1))) 34.96/9.17 Matrix Interpretation Processor: dim=1 34.96/9.17 34.96/9.17 interpretation: 34.96/9.17 [c](x0) = 2x0, 34.96/9.17 34.96/9.17 [a](x0) = 2x0 + 3, 34.96/9.17 34.96/9.17 [b](x0) = 2x0 + 1 34.96/9.17 orientation: 34.96/9.17 b(c(b(x1))) = 8x1 + 5 >= 8x1 + 5 = a(b(c(x1))) 34.96/9.17 34.96/9.17 a(a(b(x1))) = 8x1 + 13 >= 8x1 + 10 = c(a(b(x1))) 34.96/9.17 34.96/9.17 c(a(c(x1))) = 8x1 + 6 >= 8x1 + 6 = c(b(b(x1))) 34.96/9.17 problem: 34.96/9.17 b(c(b(x1))) -> a(b(c(x1))) 34.96/9.17 c(a(c(x1))) -> c(b(b(x1))) 34.96/9.17 Matrix Interpretation Processor: dim=3 34.96/9.17 34.96/9.17 interpretation: 34.96/9.17 [1 0 0] [0] 34.96/9.17 [c](x0) = [0 1 0]x0 + [1] 34.96/9.17 [0 0 0] [0], 34.96/9.17 34.96/9.17 [1 1 0] 34.96/9.17 [a](x0) = [0 0 0]x0 34.96/9.17 [0 1 0] , 34.96/9.17 34.96/9.17 [1 1 0] 34.96/9.17 [b](x0) = [0 0 0]x0 34.96/9.17 [0 0 0] 34.96/9.17 orientation: 34.96/9.17 [1 1 0] [1] [1 1 0] [1] 34.96/9.17 b(c(b(x1))) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = a(b(c(x1))) 34.96/9.17 [0 0 0] [0] [0 0 0] [0] 34.96/9.17 34.96/9.17 [1 1 0] [1] [1 1 0] [0] 34.96/9.17 c(a(c(x1))) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = c(b(b(x1))) 34.96/9.17 [0 0 0] [0] [0 0 0] [0] 34.96/9.17 problem: 34.96/9.17 b(c(b(x1))) -> a(b(c(x1))) 34.96/9.17 Matrix Interpretation Processor: dim=3 34.96/9.17 34.96/9.17 interpretation: 34.96/9.17 [1 0 0] 34.96/9.17 [c](x0) = [0 0 1]x0 34.96/9.17 [0 0 0] , 34.96/9.17 34.96/9.17 [1 0 0] 34.96/9.17 [a](x0) = [0 0 0]x0 34.96/9.17 [0 0 0] , 34.96/9.17 34.96/9.17 [1 1 0] [0] 34.96/9.17 [b](x0) = [0 0 0]x0 + [0] 34.96/9.17 [0 0 1] [1] 34.96/9.17 orientation: 34.96/9.17 [1 1 1] [1] [1 0 1] 34.96/9.17 b(c(b(x1))) = [0 0 0]x1 + [0] >= [0 0 0]x1 = a(b(c(x1))) 34.96/9.17 [0 0 0] [1] [0 0 0] 34.96/9.17 problem: 34.96/9.17 34.96/9.17 Qed 34.96/9.18 EOF