190.46/48.50 YES 190.46/48.50 190.46/48.50 Problem: 190.46/48.50 a(a(x1)) -> a(b(b(c(x1)))) 190.46/48.50 b(c(x1)) -> x1 190.46/48.50 c(b(x1)) -> a(c(x1)) 190.46/48.50 190.46/48.50 Proof: 190.46/48.50 DP Processor: 190.46/48.50 DPs: 190.46/48.50 a#(a(x1)) -> c#(x1) 190.46/48.50 a#(a(x1)) -> b#(c(x1)) 190.46/48.50 a#(a(x1)) -> b#(b(c(x1))) 190.46/48.50 a#(a(x1)) -> a#(b(b(c(x1)))) 190.46/48.50 c#(b(x1)) -> c#(x1) 190.46/48.50 c#(b(x1)) -> a#(c(x1)) 190.46/48.50 TRS: 190.46/48.50 a(a(x1)) -> a(b(b(c(x1)))) 190.46/48.50 b(c(x1)) -> x1 190.46/48.50 c(b(x1)) -> a(c(x1)) 190.46/48.50 TDG Processor: 190.46/48.50 DPs: 190.46/48.50 a#(a(x1)) -> c#(x1) 190.46/48.50 a#(a(x1)) -> b#(c(x1)) 190.46/48.50 a#(a(x1)) -> b#(b(c(x1))) 190.46/48.50 a#(a(x1)) -> a#(b(b(c(x1)))) 190.46/48.50 c#(b(x1)) -> c#(x1) 190.46/48.50 c#(b(x1)) -> a#(c(x1)) 190.46/48.50 TRS: 190.46/48.50 a(a(x1)) -> a(b(b(c(x1)))) 190.46/48.50 b(c(x1)) -> x1 190.46/48.50 c(b(x1)) -> a(c(x1)) 190.46/48.50 graph: 190.46/48.50 c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> a#(c(x1)) 190.46/48.50 c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1) 190.46/48.50 c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(b(c(x1)))) 190.46/48.50 c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(b(c(x1))) 190.46/48.50 c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(c(x1)) 190.46/48.50 c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> c#(x1) 190.46/48.50 a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> a#(c(x1)) 190.46/48.50 a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1) 190.46/48.50 a#(a(x1)) -> a#(b(b(c(x1)))) -> a#(a(x1)) -> a#(b(b(c(x1)))) 190.46/48.50 a#(a(x1)) -> a#(b(b(c(x1)))) -> a#(a(x1)) -> b#(b(c(x1))) 190.46/48.50 a#(a(x1)) -> a#(b(b(c(x1)))) -> a#(a(x1)) -> b#(c(x1)) 190.46/48.50 a#(a(x1)) -> a#(b(b(c(x1)))) -> a#(a(x1)) -> c#(x1) 190.46/48.50 SCC Processor: 190.46/48.50 #sccs: 1 190.46/48.50 #rules: 4 190.46/48.50 #arcs: 12/36 190.46/48.50 DPs: 190.46/48.50 c#(b(x1)) -> c#(x1) 190.46/48.50 c#(b(x1)) -> a#(c(x1)) 190.46/48.50 a#(a(x1)) -> c#(x1) 190.46/48.50 a#(a(x1)) -> a#(b(b(c(x1)))) 190.46/48.50 TRS: 190.46/48.50 a(a(x1)) -> a(b(b(c(x1)))) 190.46/48.50 b(c(x1)) -> x1 190.46/48.50 c(b(x1)) -> a(c(x1)) 190.46/48.50 Arctic Interpretation Processor: 190.46/48.50 dimension: 2 190.46/48.50 usable rules: 190.46/48.50 a(a(x1)) -> a(b(b(c(x1)))) 190.46/48.50 b(c(x1)) -> x1 190.46/48.50 c(b(x1)) -> a(c(x1)) 190.46/48.50 interpretation: 190.46/48.50 [c#](x0) = [2 -&]x0 + [0], 190.46/48.50 190.46/48.50 [a#](x0) = [2 0]x0 + [0], 190.46/48.50 190.46/48.50 [0 0] [2] 190.46/48.50 [b](x0) = [0 1]x0 + [2], 190.46/48.50 190.46/48.50 [-& 0 ] [0] 190.46/48.50 [c](x0) = [0 -&]x0 + [1], 190.46/48.50 190.46/48.50 [1 0 ] [2] 190.46/48.50 [a](x0) = [0 -&]x0 + [2] 190.46/48.50 orientation: 190.46/48.50 c#(b(x1)) = [2 2]x1 + [4] >= [2 -&]x1 + [0] = c#(x1) 190.46/48.50 190.46/48.50 c#(b(x1)) = [2 2]x1 + [4] >= [0 2]x1 + [2] = a#(c(x1)) 190.46/48.50 190.46/48.50 a#(a(x1)) = [3 2]x1 + [4] >= [2 -&]x1 + [0] = c#(x1) 190.46/48.50 190.46/48.50 a#(a(x1)) = [3 2]x1 + [4] >= [3 2]x1 + [4] = a#(b(b(c(x1)))) 190.46/48.50 190.46/48.50 [2 1] [3] [2 1] [3] 190.46/48.50 a(a(x1)) = [1 0]x1 + [2] >= [1 0]x1 + [2] = a(b(b(c(x1)))) 190.46/48.50 190.46/48.50 [0 0] [2] 190.46/48.50 b(c(x1)) = [1 0]x1 + [2] >= x1 = x1 190.46/48.50 190.46/48.50 [0 1] [2] [0 1 ] [2] 190.46/48.50 c(b(x1)) = [0 0]x1 + [2] >= [-& 0 ]x1 + [2] = a(c(x1)) 190.46/48.50 problem: 190.46/48.50 DPs: 190.46/48.50 c#(b(x1)) -> c#(x1) 190.46/48.50 c#(b(x1)) -> a#(c(x1)) 190.46/48.50 a#(a(x1)) -> a#(b(b(c(x1)))) 190.46/48.50 TRS: 190.46/48.50 a(a(x1)) -> a(b(b(c(x1)))) 190.46/48.50 b(c(x1)) -> x1 190.46/48.50 c(b(x1)) -> a(c(x1)) 190.46/48.50 Restore Modifier: 190.46/48.50 DPs: 190.46/48.50 c#(b(x1)) -> c#(x1) 190.46/48.50 c#(b(x1)) -> a#(c(x1)) 190.46/48.50 a#(a(x1)) -> a#(b(b(c(x1)))) 190.46/48.50 TRS: 190.46/48.50 a(a(x1)) -> a(b(b(c(x1)))) 190.46/48.50 b(c(x1)) -> x1 190.46/48.50 c(b(x1)) -> a(c(x1)) 190.46/48.50 EDG Processor: 190.46/48.50 DPs: 190.46/48.50 c#(b(x1)) -> c#(x1) 190.46/48.50 c#(b(x1)) -> a#(c(x1)) 190.46/48.50 a#(a(x1)) -> a#(b(b(c(x1)))) 190.46/48.50 TRS: 190.46/48.50 a(a(x1)) -> a(b(b(c(x1)))) 190.46/48.50 b(c(x1)) -> x1 190.46/48.50 c(b(x1)) -> a(c(x1)) 190.46/48.50 graph: 190.46/48.50 c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1) 190.46/48.50 c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> a#(c(x1)) 190.46/48.50 c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(b(c(x1)))) 190.46/48.50 a#(a(x1)) -> a#(b(b(c(x1)))) -> a#(a(x1)) -> a#(b(b(c(x1)))) 190.46/48.50 SCC Processor: 190.46/48.50 #sccs: 2 190.46/48.50 #rules: 2 190.46/48.50 #arcs: 4/9 190.46/48.50 DPs: 190.46/48.50 c#(b(x1)) -> c#(x1) 190.46/48.50 TRS: 190.46/48.50 a(a(x1)) -> a(b(b(c(x1)))) 190.46/48.50 b(c(x1)) -> x1 190.46/48.50 c(b(x1)) -> a(c(x1)) 190.46/48.50 Usable Rule Processor: 190.46/48.50 DPs: 190.46/48.50 c#(b(x1)) -> c#(x1) 190.46/48.50 TRS: 190.46/48.50 190.46/48.50 Arctic Interpretation Processor: 190.46/48.50 dimension: 4 190.46/48.50 usable rules: 190.46/48.50 190.46/48.50 interpretation: 190.46/48.50 [c#](x0) = [0 -& -& -&]x0, 190.46/48.50 190.46/48.50 [1 0 0 0] [0] 190.46/48.51 [0 0 0 0] [0] 190.46/48.51 [b](x0) = [1 0 0 0]x0 + [0] 190.46/48.51 [0 0 0 0] [0] 190.46/48.51 orientation: 190.46/48.51 c#(b(x1)) = [1 0 0 0]x1 + [0] >= [0 -& -& -&]x1 = c#(x1) 190.46/48.51 problem: 190.46/48.51 DPs: 190.46/48.51 190.46/48.51 TRS: 190.46/48.51 190.46/48.51 Qed 190.46/48.51 190.46/48.51 DPs: 190.46/48.51 a#(a(x1)) -> a#(b(b(c(x1)))) 190.46/48.51 TRS: 190.46/48.51 a(a(x1)) -> a(b(b(c(x1)))) 190.46/48.51 b(c(x1)) -> x1 190.46/48.51 c(b(x1)) -> a(c(x1)) 190.46/48.51 Matrix Interpretation Processor: dim=1 190.46/48.51 190.46/48.51 usable rules: 190.46/48.51 a(a(x1)) -> a(b(b(c(x1)))) 190.46/48.51 b(c(x1)) -> x1 190.46/48.51 c(b(x1)) -> a(c(x1)) 190.46/48.51 interpretation: 190.46/48.51 [a#](x0) = 2x0 + 3/2, 190.46/48.51 190.46/48.51 [b](x0) = 1/2x0, 190.46/48.51 190.46/48.51 [c](x0) = 2x0 + 2, 190.46/48.51 190.46/48.51 [a](x0) = 1/2x0 + 1 190.46/48.51 orientation: 190.46/48.51 a#(a(x1)) = x1 + 7/2 >= x1 + 5/2 = a#(b(b(c(x1)))) 190.46/48.51 190.46/48.51 a(a(x1)) = 1/4x1 + 3/2 >= 1/4x1 + 5/4 = a(b(b(c(x1)))) 190.46/48.51 190.46/48.51 b(c(x1)) = x1 + 1 >= x1 = x1 190.46/48.51 190.46/48.51 c(b(x1)) = x1 + 2 >= x1 + 2 = a(c(x1)) 190.46/48.51 problem: 190.46/48.51 DPs: 190.46/48.51 190.46/48.51 TRS: 190.46/48.51 a(a(x1)) -> a(b(b(c(x1)))) 190.46/48.51 b(c(x1)) -> x1 190.46/48.51 c(b(x1)) -> a(c(x1)) 190.46/48.51 Qed 190.46/48.51 EOF