189.92/47.82 YES 189.92/47.83 189.92/47.83 Problem: 189.92/47.83 a(x1) -> x1 189.92/47.83 a(a(x1)) -> b(a(b(c(x1)))) 189.92/47.83 b(x1) -> x1 189.92/47.83 c(b(x1)) -> a(c(x1)) 189.92/47.83 189.92/47.83 Proof: 189.92/47.83 DP Processor: 189.92/47.83 DPs: 189.92/47.83 a#(a(x1)) -> c#(x1) 189.92/47.83 a#(a(x1)) -> b#(c(x1)) 189.92/47.83 a#(a(x1)) -> a#(b(c(x1))) 189.92/47.83 a#(a(x1)) -> b#(a(b(c(x1)))) 189.92/47.83 c#(b(x1)) -> c#(x1) 189.92/47.83 c#(b(x1)) -> a#(c(x1)) 189.92/47.83 TRS: 189.92/47.83 a(x1) -> x1 189.92/47.83 a(a(x1)) -> b(a(b(c(x1)))) 189.92/47.83 b(x1) -> x1 189.92/47.83 c(b(x1)) -> a(c(x1)) 189.92/47.83 TDG Processor: 189.92/47.83 DPs: 189.92/47.83 a#(a(x1)) -> c#(x1) 189.92/47.83 a#(a(x1)) -> b#(c(x1)) 189.92/47.83 a#(a(x1)) -> a#(b(c(x1))) 189.92/47.83 a#(a(x1)) -> b#(a(b(c(x1)))) 189.92/47.83 c#(b(x1)) -> c#(x1) 189.92/47.83 c#(b(x1)) -> a#(c(x1)) 189.92/47.83 TRS: 189.92/47.83 a(x1) -> x1 189.92/47.83 a(a(x1)) -> b(a(b(c(x1)))) 189.92/47.83 b(x1) -> x1 189.92/47.83 c(b(x1)) -> a(c(x1)) 189.92/47.83 graph: 189.92/47.83 c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> a#(c(x1)) 189.92/47.83 c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1) 189.92/47.83 c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(a(b(c(x1)))) 189.92/47.83 c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(c(x1))) 189.92/47.83 c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> b#(c(x1)) 189.92/47.83 c#(b(x1)) -> a#(c(x1)) -> a#(a(x1)) -> c#(x1) 189.92/47.83 a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> a#(c(x1)) 189.92/47.83 a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1) 189.92/47.83 a#(a(x1)) -> a#(b(c(x1))) -> a#(a(x1)) -> b#(a(b(c(x1)))) 189.92/47.83 a#(a(x1)) -> a#(b(c(x1))) -> a#(a(x1)) -> a#(b(c(x1))) 189.92/47.83 a#(a(x1)) -> a#(b(c(x1))) -> a#(a(x1)) -> b#(c(x1)) 189.92/47.83 a#(a(x1)) -> a#(b(c(x1))) -> a#(a(x1)) -> c#(x1) 189.92/47.83 SCC Processor: 189.92/47.83 #sccs: 1 189.92/47.83 #rules: 4 189.92/47.83 #arcs: 12/36 189.92/47.83 DPs: 189.92/47.83 c#(b(x1)) -> c#(x1) 189.92/47.83 c#(b(x1)) -> a#(c(x1)) 189.92/47.83 a#(a(x1)) -> c#(x1) 189.92/47.83 a#(a(x1)) -> a#(b(c(x1))) 189.92/47.83 TRS: 189.92/47.83 a(x1) -> x1 189.92/47.83 a(a(x1)) -> b(a(b(c(x1)))) 189.92/47.83 b(x1) -> x1 189.92/47.83 c(b(x1)) -> a(c(x1)) 189.92/47.83 Arctic Interpretation Processor: 189.92/47.83 dimension: 3 189.92/47.83 usable rules: 189.92/47.83 a(x1) -> x1 189.92/47.83 a(a(x1)) -> b(a(b(c(x1)))) 189.92/47.83 b(x1) -> x1 189.92/47.83 c(b(x1)) -> a(c(x1)) 189.92/47.83 interpretation: 189.92/47.83 [c#](x0) = [-& -& 0 ]x0 + [1], 189.92/47.83 189.92/47.83 [a#](x0) = [1 0 0]x0 + [0], 189.92/47.83 189.92/47.83 [0 -& 0 ] [0] 189.92/47.83 [b](x0) = [-& 0 0 ]x0 + [1] 189.92/47.83 [0 0 1 ] [1], 189.92/47.83 189.92/47.83 [-& -& 0 ] [0] 189.92/47.83 [c](x0) = [0 -& 0 ]x0 + [0] 189.92/47.83 [-& -& 0 ] [0], 189.92/47.83 189.92/47.83 [1 -& 1 ] [1] 189.92/47.83 [a](x0) = [0 0 0 ]x0 + [0] 189.92/47.83 [1 -& 0 ] [0] 189.92/47.83 orientation: 189.92/47.83 c#(b(x1)) = [0 0 1]x1 + [1] >= [-& -& 0 ]x1 + [1] = c#(x1) 189.92/47.83 189.92/47.83 c#(b(x1)) = [0 0 1]x1 + [1] >= [0 -& 1 ]x1 + [1] = a#(c(x1)) 189.92/47.83 189.92/47.83 a#(a(x1)) = [2 0 2]x1 + [2] >= [-& -& 0 ]x1 + [1] = c#(x1) 189.92/47.83 189.92/47.83 a#(a(x1)) = [2 0 2]x1 + [2] >= [0 -& 1 ]x1 + [1] = a#(b(c(x1))) 189.92/47.83 189.92/47.83 [1 -& 1 ] [1] 189.92/47.83 a(x1) = [0 0 0 ]x1 + [0] >= x1 = x1 189.92/47.83 [1 -& 0 ] [0] 189.92/47.83 189.92/47.83 [2 -& 2 ] [2] [1 -& 2 ] [2] 189.92/47.83 a(a(x1)) = [1 0 1 ]x1 + [1] >= [0 -& 1 ]x1 + [1] = b(a(b(c(x1)))) 189.92/47.83 [2 -& 2 ] [2] [1 -& 2 ] [2] 189.92/47.83 189.92/47.83 [0 -& 0 ] [0] 189.92/47.83 b(x1) = [-& 0 0 ]x1 + [1] >= x1 = x1 189.92/47.83 [0 0 1 ] [1] 189.92/47.83 189.92/47.83 [0 0 1] [1] [-& -& 1 ] [1] 189.92/47.83 c(b(x1)) = [0 0 1]x1 + [1] >= [0 -& 0 ]x1 + [0] = a(c(x1)) 189.92/47.83 [0 0 1] [1] [-& -& 1 ] [1] 189.92/47.83 problem: 189.92/47.83 DPs: 189.92/47.83 c#(b(x1)) -> c#(x1) 189.92/47.83 c#(b(x1)) -> a#(c(x1)) 189.92/47.83 TRS: 189.92/47.83 a(x1) -> x1 189.92/47.83 a(a(x1)) -> b(a(b(c(x1)))) 189.92/47.83 b(x1) -> x1 189.92/47.83 c(b(x1)) -> a(c(x1)) 189.92/47.83 Restore Modifier: 189.92/47.83 DPs: 189.92/47.83 c#(b(x1)) -> c#(x1) 189.92/47.83 c#(b(x1)) -> a#(c(x1)) 189.92/47.83 TRS: 189.92/47.83 a(x1) -> x1 189.92/47.83 a(a(x1)) -> b(a(b(c(x1)))) 189.92/47.83 b(x1) -> x1 189.92/47.83 c(b(x1)) -> a(c(x1)) 189.92/47.83 EDG Processor: 189.92/47.83 DPs: 189.92/47.83 c#(b(x1)) -> c#(x1) 189.92/47.83 c#(b(x1)) -> a#(c(x1)) 189.92/47.83 TRS: 189.92/47.83 a(x1) -> x1 189.92/47.83 a(a(x1)) -> b(a(b(c(x1)))) 189.92/47.83 b(x1) -> x1 189.92/47.83 c(b(x1)) -> a(c(x1)) 189.92/47.83 graph: 189.92/47.83 c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1) 189.92/47.83 c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> a#(c(x1)) 189.92/47.83 SCC Processor: 189.92/47.83 #sccs: 1 189.92/47.83 #rules: 1 189.92/47.83 #arcs: 2/4 189.92/47.83 DPs: 189.92/47.83 c#(b(x1)) -> c#(x1) 189.92/47.83 TRS: 189.92/47.83 a(x1) -> x1 189.92/47.83 a(a(x1)) -> b(a(b(c(x1)))) 189.92/47.83 b(x1) -> x1 189.92/47.83 c(b(x1)) -> a(c(x1)) 189.92/47.83 Usable Rule Processor: 189.92/47.83 DPs: 189.92/47.83 c#(b(x1)) -> c#(x1) 189.92/47.83 TRS: 189.92/47.83 189.92/47.83 Arctic Interpretation Processor: 189.92/47.83 dimension: 4 189.92/47.83 usable rules: 189.92/47.83 189.92/47.83 interpretation: 189.92/47.83 [c#](x0) = [0 -& -& -&]x0, 189.92/47.83 189.92/47.83 [1 0 0 0] [0] 189.92/47.83 [0 0 0 0] [0] 189.92/47.83 [b](x0) = [1 0 0 0]x0 + [0] 189.92/47.83 [0 0 0 0] [0] 189.92/47.83 orientation: 189.92/47.83 c#(b(x1)) = [1 0 0 0]x1 + [0] >= [0 -& -& -&]x1 = c#(x1) 189.92/47.83 problem: 189.92/47.83 DPs: 189.92/47.83 189.92/47.83 TRS: 189.92/47.83 189.92/47.83 Qed 189.92/47.83 EOF