174.98/44.69 YES 174.98/44.70 174.98/44.70 Problem: 174.98/44.70 a(x1) -> x1 174.98/44.70 a(b(b(x1))) -> b(b(b(c(x1)))) 174.98/44.70 b(c(x1)) -> a(a(x1)) 174.98/44.70 174.98/44.70 Proof: 174.98/44.70 DP Processor: 174.98/44.70 DPs: 174.98/44.70 a#(b(b(x1))) -> b#(c(x1)) 174.98/44.70 a#(b(b(x1))) -> b#(b(c(x1))) 174.98/44.70 a#(b(b(x1))) -> b#(b(b(c(x1)))) 174.98/44.70 b#(c(x1)) -> a#(x1) 174.98/44.70 b#(c(x1)) -> a#(a(x1)) 174.98/44.70 TRS: 174.98/44.70 a(x1) -> x1 174.98/44.70 a(b(b(x1))) -> b(b(b(c(x1)))) 174.98/44.70 b(c(x1)) -> a(a(x1)) 174.98/44.70 TDG Processor: 174.98/44.70 DPs: 174.98/44.70 a#(b(b(x1))) -> b#(c(x1)) 174.98/44.70 a#(b(b(x1))) -> b#(b(c(x1))) 174.98/44.70 a#(b(b(x1))) -> b#(b(b(c(x1)))) 174.98/44.70 b#(c(x1)) -> a#(x1) 174.98/44.70 b#(c(x1)) -> a#(a(x1)) 174.98/44.70 TRS: 174.98/44.70 a(x1) -> x1 174.98/44.70 a(b(b(x1))) -> b(b(b(c(x1)))) 174.98/44.70 b(c(x1)) -> a(a(x1)) 174.98/44.70 graph: 174.98/44.70 b#(c(x1)) -> a#(a(x1)) -> a#(b(b(x1))) -> b#(b(b(c(x1)))) 174.98/44.70 b#(c(x1)) -> a#(a(x1)) -> a#(b(b(x1))) -> b#(b(c(x1))) 174.98/44.70 b#(c(x1)) -> a#(a(x1)) -> a#(b(b(x1))) -> b#(c(x1)) 174.98/44.70 b#(c(x1)) -> a#(x1) -> a#(b(b(x1))) -> b#(b(b(c(x1)))) 174.98/44.70 b#(c(x1)) -> a#(x1) -> a#(b(b(x1))) -> b#(b(c(x1))) 174.98/44.70 b#(c(x1)) -> a#(x1) -> a#(b(b(x1))) -> b#(c(x1)) 174.98/44.70 a#(b(b(x1))) -> b#(c(x1)) -> b#(c(x1)) -> a#(a(x1)) 174.98/44.70 a#(b(b(x1))) -> b#(c(x1)) -> b#(c(x1)) -> a#(x1) 174.98/44.70 a#(b(b(x1))) -> b#(b(c(x1))) -> b#(c(x1)) -> a#(a(x1)) 174.98/44.70 a#(b(b(x1))) -> b#(b(c(x1))) -> b#(c(x1)) -> a#(x1) 174.98/44.70 a#(b(b(x1))) -> b#(b(b(c(x1)))) -> b#(c(x1)) -> a#(a(x1)) 174.98/44.70 a#(b(b(x1))) -> b#(b(b(c(x1)))) -> b#(c(x1)) -> a#(x1) 174.98/44.70 Arctic Interpretation Processor: 174.98/44.70 dimension: 2 174.98/44.70 usable rules: 174.98/44.70 a(x1) -> x1 174.98/44.70 a(b(b(x1))) -> b(b(b(c(x1)))) 174.98/44.70 b(c(x1)) -> a(a(x1)) 174.98/44.70 interpretation: 174.98/44.70 [b#](x0) = [-& 1 ]x0 + [0], 174.98/44.70 174.98/44.70 [a#](x0) = [-& 1 ]x0 + [0], 174.98/44.70 174.98/44.70 [-& -&] [1] 174.98/44.70 [c](x0) = [0 0 ]x0 + [3], 174.98/44.70 174.98/44.70 [-& 0 ] [3] 174.98/44.70 [b](x0) = [1 0 ]x0 + [3], 174.98/44.70 174.98/44.70 [0 0] [3] 174.98/44.70 [a](x0) = [0 0]x0 + [0] 174.98/44.70 orientation: 174.98/44.70 a#(b(b(x1))) = [2 2]x1 + [5] >= [1 1]x1 + [4] = b#(c(x1)) 174.98/44.70 174.98/44.70 a#(b(b(x1))) = [2 2]x1 + [5] >= [1 1]x1 + [4] = b#(b(c(x1))) 174.98/44.70 174.98/44.70 a#(b(b(x1))) = [2 2]x1 + [5] >= [2 2]x1 + [5] = b#(b(b(c(x1)))) 174.98/44.70 174.98/44.70 b#(c(x1)) = [1 1]x1 + [4] >= [-& 1 ]x1 + [0] = a#(x1) 174.98/44.70 174.98/44.70 b#(c(x1)) = [1 1]x1 + [4] >= [1 1]x1 + [1] = a#(a(x1)) 174.98/44.70 174.98/44.70 [0 0] [3] 174.98/44.70 a(x1) = [0 0]x1 + [0] >= x1 = x1 174.98/44.70 174.98/44.70 [1 1] [4] [1 1] [4] 174.98/44.70 a(b(b(x1))) = [1 1]x1 + [4] >= [1 1]x1 + [4] = b(b(b(c(x1)))) 174.98/44.70 174.98/44.70 [0 0] [3] [0 0] [3] 174.98/44.70 b(c(x1)) = [0 0]x1 + [3] >= [0 0]x1 + [3] = a(a(x1)) 174.98/44.70 problem: 174.98/44.70 DPs: 174.98/44.70 a#(b(b(x1))) -> b#(b(b(c(x1)))) 174.98/44.70 b#(c(x1)) -> a#(x1) 174.98/44.70 b#(c(x1)) -> a#(a(x1)) 174.98/44.70 TRS: 174.98/44.70 a(x1) -> x1 174.98/44.70 a(b(b(x1))) -> b(b(b(c(x1)))) 174.98/44.70 b(c(x1)) -> a(a(x1)) 174.98/44.70 Restore Modifier: 174.98/44.70 DPs: 174.98/44.70 a#(b(b(x1))) -> b#(b(b(c(x1)))) 174.98/44.70 b#(c(x1)) -> a#(x1) 174.98/44.70 b#(c(x1)) -> a#(a(x1)) 174.98/44.70 TRS: 174.98/44.70 a(x1) -> x1 174.98/44.70 a(b(b(x1))) -> b(b(b(c(x1)))) 174.98/44.70 b(c(x1)) -> a(a(x1)) 174.98/44.70 EDG Processor: 174.98/44.70 DPs: 174.98/44.70 a#(b(b(x1))) -> b#(b(b(c(x1)))) 174.98/44.70 b#(c(x1)) -> a#(x1) 174.98/44.70 b#(c(x1)) -> a#(a(x1)) 174.98/44.70 TRS: 174.98/44.70 a(x1) -> x1 174.98/44.70 a(b(b(x1))) -> b(b(b(c(x1)))) 174.98/44.70 b(c(x1)) -> a(a(x1)) 174.98/44.70 graph: 174.98/44.70 b#(c(x1)) -> a#(a(x1)) -> a#(b(b(x1))) -> b#(b(b(c(x1)))) 174.98/44.70 b#(c(x1)) -> a#(x1) -> a#(b(b(x1))) -> b#(b(b(c(x1)))) 174.98/44.70 a#(b(b(x1))) -> b#(b(b(c(x1)))) -> b#(c(x1)) -> a#(x1) 174.98/44.70 a#(b(b(x1))) -> b#(b(b(c(x1)))) -> b#(c(x1)) -> a#(a(x1)) 174.98/44.70 Matrix Interpretation Processor: dim=1 174.98/44.70 174.98/44.70 usable rules: 174.98/44.70 a(x1) -> x1 174.98/44.70 a(b(b(x1))) -> b(b(b(c(x1)))) 174.98/44.70 b(c(x1)) -> a(a(x1)) 174.98/44.70 interpretation: 174.98/44.70 [b#](x0) = 2x0 + 15/2, 174.98/44.70 174.98/44.70 [a#](x0) = 4x0, 174.98/44.70 174.98/44.70 [c](x0) = 2x0, 174.98/44.70 174.98/44.70 [b](x0) = 1/2x0 + 3, 174.98/44.70 174.98/44.70 [a](x0) = x0 + 1 174.98/44.70 orientation: 174.98/44.70 a#(b(b(x1))) = x1 + 18 >= x1 + 33/2 = b#(b(b(c(x1)))) 174.98/44.70 174.98/44.70 b#(c(x1)) = 4x1 + 15/2 >= 4x1 = a#(x1) 174.98/44.70 174.98/44.70 b#(c(x1)) = 4x1 + 15/2 >= 4x1 + 4 = a#(a(x1)) 174.98/44.70 174.98/44.70 a(x1) = x1 + 1 >= x1 = x1 174.98/44.70 174.98/44.70 a(b(b(x1))) = 1/4x1 + 11/2 >= 1/4x1 + 21/4 = b(b(b(c(x1)))) 174.98/44.70 174.98/44.70 b(c(x1)) = x1 + 3 >= x1 + 2 = a(a(x1)) 174.98/44.70 problem: 174.98/44.70 DPs: 174.98/44.70 174.98/44.70 TRS: 174.98/44.70 a(x1) -> x1 174.98/44.70 a(b(b(x1))) -> b(b(b(c(x1)))) 174.98/44.70 b(c(x1)) -> a(a(x1)) 174.98/44.70 Qed 174.98/44.70 EOF