18.54/5.01 YES 18.54/5.01 18.54/5.01 Problem: 18.54/5.01 strict: 18.54/5.01 a(a(b(x1))) -> a(c(b(x1))) 18.54/5.01 c(c(c(x1))) -> a(b(a(x1))) 18.54/5.01 b(c(b(x1))) -> b(a(c(x1))) 18.54/5.01 weak: 18.54/5.01 b(b(c(x1))) -> c(a(b(x1))) 18.54/5.01 b(b(b(x1))) -> a(b(b(x1))) 18.54/5.01 b(b(c(x1))) -> b(c(c(x1))) 18.54/5.01 b(a(a(x1))) -> a(c(c(x1))) 18.54/5.01 18.54/5.01 Proof: 18.54/5.01 Matrix Interpretation Processor: dim=2 18.54/5.01 18.54/5.01 interpretation: 18.54/5.01 18.54/5.01 [c](x0) = x0, 18.54/5.01 18.54/5.01 [1 0] 18.54/5.01 [a](x0) = [0 0]x0, 18.54/5.01 18.54/5.01 [1 1] [0] 18.54/5.01 [b](x0) = [0 1]x0 + [1] 18.54/5.01 orientation: 18.54/5.01 [1 1] [1 1] 18.54/5.01 a(a(b(x1))) = [0 0]x1 >= [0 0]x1 = a(c(b(x1))) 18.54/5.01 18.54/5.01 [1 0] 18.54/5.01 c(c(c(x1))) = x1 >= [0 0]x1 = a(b(a(x1))) 18.54/5.01 18.54/5.01 [1 2] [1] [1 0] [0] 18.54/5.01 b(c(b(x1))) = [0 1]x1 + [2] >= [0 0]x1 + [1] = b(a(c(x1))) 18.54/5.01 18.54/5.01 [1 2] [1] [1 1] 18.54/5.01 b(b(c(x1))) = [0 1]x1 + [2] >= [0 0]x1 = c(a(b(x1))) 18.54/5.01 18.54/5.01 [1 3] [3] [1 2] [1] 18.54/5.01 b(b(b(x1))) = [0 1]x1 + [3] >= [0 0]x1 + [0] = a(b(b(x1))) 18.54/5.01 18.54/5.01 [1 2] [1] [1 1] [0] 18.54/5.01 b(b(c(x1))) = [0 1]x1 + [2] >= [0 1]x1 + [1] = b(c(c(x1))) 18.54/5.01 18.54/5.01 [1 0] [0] [1 0] 18.54/5.01 b(a(a(x1))) = [0 0]x1 + [1] >= [0 0]x1 = a(c(c(x1))) 18.54/5.01 problem: 18.54/5.01 strict: 18.54/5.01 a(a(b(x1))) -> a(c(b(x1))) 18.54/5.01 c(c(c(x1))) -> a(b(a(x1))) 18.54/5.01 weak: 18.54/5.01 b(a(a(x1))) -> a(c(c(x1))) 18.54/5.01 Matrix Interpretation Processor: dim=1 18.54/5.01 18.54/5.01 interpretation: 18.54/5.01 [c](x0) = 2x0 + 7, 18.54/5.01 18.54/5.01 [a](x0) = 2x0 + 8, 18.54/5.01 18.54/5.01 [b](x0) = 2x0 + 4 18.54/5.01 orientation: 18.54/5.01 a(a(b(x1))) = 8x1 + 40 >= 8x1 + 38 = a(c(b(x1))) 18.54/5.01 18.54/5.01 c(c(c(x1))) = 8x1 + 49 >= 8x1 + 48 = a(b(a(x1))) 18.54/5.01 18.54/5.01 b(a(a(x1))) = 8x1 + 52 >= 8x1 + 50 = a(c(c(x1))) 18.54/5.01 problem: 18.54/5.01 strict: 18.54/5.01 18.54/5.01 weak: 18.54/5.01 18.54/5.01 Qed 18.54/5.02 EOF