175.70/44.88 YES 175.70/44.89 175.70/44.89 Problem: 175.70/44.89 a(x1) -> x1 175.70/44.89 a(b(x1)) -> b(b(a(c(x1)))) 175.70/44.89 b(x1) -> a(c(x1)) 175.70/44.89 c(c(x1)) -> x1 175.70/44.89 175.70/44.89 Proof: 175.70/44.89 String Reversal Processor: 175.70/44.89 a(x1) -> x1 175.70/44.89 b(a(x1)) -> c(a(b(b(x1)))) 175.70/44.89 b(x1) -> c(a(x1)) 175.70/44.89 c(c(x1)) -> x1 175.70/44.89 DP Processor: 175.70/44.89 DPs: 175.70/44.89 b#(a(x1)) -> b#(x1) 175.70/44.89 b#(a(x1)) -> b#(b(x1)) 175.70/44.89 b#(a(x1)) -> a#(b(b(x1))) 175.70/44.89 b#(a(x1)) -> c#(a(b(b(x1)))) 175.70/44.89 b#(x1) -> a#(x1) 175.70/44.89 b#(x1) -> c#(a(x1)) 175.70/44.89 TRS: 175.70/44.89 a(x1) -> x1 175.70/44.89 b(a(x1)) -> c(a(b(b(x1)))) 175.70/44.89 b(x1) -> c(a(x1)) 175.70/44.89 c(c(x1)) -> x1 175.70/44.89 TDG Processor: 175.70/44.89 DPs: 175.70/44.89 b#(a(x1)) -> b#(x1) 175.70/44.89 b#(a(x1)) -> b#(b(x1)) 175.70/44.89 b#(a(x1)) -> a#(b(b(x1))) 175.70/44.89 b#(a(x1)) -> c#(a(b(b(x1)))) 175.70/44.89 b#(x1) -> a#(x1) 175.70/44.89 b#(x1) -> c#(a(x1)) 175.70/44.89 TRS: 175.70/44.89 a(x1) -> x1 175.70/44.89 b(a(x1)) -> c(a(b(b(x1)))) 175.70/44.89 b(x1) -> c(a(x1)) 175.70/44.89 c(c(x1)) -> x1 175.70/44.89 graph: 175.70/44.89 b#(a(x1)) -> b#(b(x1)) -> b#(x1) -> c#(a(x1)) 175.70/44.89 b#(a(x1)) -> b#(b(x1)) -> b#(x1) -> a#(x1) 175.70/44.89 b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> c#(a(b(b(x1)))) 175.70/44.89 b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> a#(b(b(x1))) 175.70/44.89 b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) 175.70/44.89 b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(x1) 175.70/44.89 b#(a(x1)) -> b#(x1) -> b#(x1) -> c#(a(x1)) 175.70/44.89 b#(a(x1)) -> b#(x1) -> b#(x1) -> a#(x1) 175.70/44.89 b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> c#(a(b(b(x1)))) 175.70/44.89 b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(b(b(x1))) 175.70/44.89 b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(b(x1)) 175.70/44.89 b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1) 175.70/44.89 SCC Processor: 175.70/44.89 #sccs: 1 175.70/44.89 #rules: 2 175.70/44.89 #arcs: 12/36 175.70/44.89 DPs: 175.70/44.89 b#(a(x1)) -> b#(b(x1)) 175.70/44.89 b#(a(x1)) -> b#(x1) 175.70/44.89 TRS: 175.70/44.89 a(x1) -> x1 175.70/44.89 b(a(x1)) -> c(a(b(b(x1)))) 175.70/44.89 b(x1) -> c(a(x1)) 175.70/44.89 c(c(x1)) -> x1 175.70/44.89 Arctic Interpretation Processor: 175.70/44.89 dimension: 2 175.70/44.89 usable rules: 175.70/44.89 a(x1) -> x1 175.70/44.89 b(a(x1)) -> c(a(b(b(x1)))) 175.70/44.89 b(x1) -> c(a(x1)) 175.70/44.89 c(c(x1)) -> x1 175.70/44.89 interpretation: 175.70/44.89 [b#](x0) = [0 0]x0 + [0], 175.70/44.89 175.70/44.89 [-& 0 ] [0] 175.70/44.89 [c](x0) = [0 1 ]x0 + [0], 175.70/44.89 175.70/44.89 [1 0] [0] 175.70/44.89 [b](x0) = [2 1]x0 + [1], 175.70/44.89 175.70/44.89 [2 1 ] [1] 175.70/44.89 [a](x0) = [-& 0 ]x0 + [0] 175.70/44.89 orientation: 175.70/44.89 b#(a(x1)) = [2 1]x1 + [1] >= [2 1]x1 + [1] = b#(b(x1)) 175.70/44.89 175.70/44.89 b#(a(x1)) = [2 1]x1 + [1] >= [0 0]x1 + [0] = b#(x1) 175.70/44.89 175.70/44.89 [2 1 ] [1] 175.70/44.89 a(x1) = [-& 0 ]x1 + [0] >= x1 = x1 175.70/44.89 175.70/44.89 [3 2] [2] [3 2] [2] 175.70/44.89 b(a(x1)) = [4 3]x1 + [3] >= [4 3]x1 + [3] = c(a(b(b(x1)))) 175.70/44.89 175.70/44.89 [1 0] [0] [-& 0 ] [0] 175.70/44.89 b(x1) = [2 1]x1 + [1] >= [2 1 ]x1 + [1] = c(a(x1)) 175.70/44.89 175.70/44.89 [0 1] [0] 175.70/44.89 c(c(x1)) = [1 2]x1 + [1] >= x1 = x1 175.70/44.89 problem: 175.70/44.89 DPs: 175.70/44.89 b#(a(x1)) -> b#(b(x1)) 175.70/44.89 TRS: 175.70/44.89 a(x1) -> x1 175.70/44.89 b(a(x1)) -> c(a(b(b(x1)))) 175.70/44.89 b(x1) -> c(a(x1)) 175.70/44.89 c(c(x1)) -> x1 175.70/44.89 Restore Modifier: 175.70/44.89 DPs: 175.70/44.89 b#(a(x1)) -> b#(b(x1)) 175.70/44.89 TRS: 175.70/44.89 a(x1) -> x1 175.70/44.89 b(a(x1)) -> c(a(b(b(x1)))) 175.70/44.89 b(x1) -> c(a(x1)) 175.70/44.89 c(c(x1)) -> x1 175.70/44.89 EDG Processor: 175.70/44.89 DPs: 175.70/44.89 b#(a(x1)) -> b#(b(x1)) 175.70/44.89 TRS: 175.70/44.89 a(x1) -> x1 175.70/44.89 b(a(x1)) -> c(a(b(b(x1)))) 175.70/44.89 b(x1) -> c(a(x1)) 175.70/44.89 c(c(x1)) -> x1 175.70/44.89 graph: 175.70/44.89 b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) 175.70/44.89 Arctic Interpretation Processor: 175.70/44.89 dimension: 2 175.70/44.89 usable rules: 175.70/44.89 a(x1) -> x1 175.70/44.89 b(a(x1)) -> c(a(b(b(x1)))) 175.70/44.89 b(x1) -> c(a(x1)) 175.70/44.89 c(c(x1)) -> x1 175.70/44.89 interpretation: 175.70/44.89 [b#](x0) = [-& 0 ]x0 + [0], 175.70/44.89 175.70/44.89 [0 0 ] [1] 175.70/44.89 [c](x0) = [0 -&]x0 + [2], 175.70/44.89 175.70/44.89 [1 2] [3] 175.70/44.89 [b](x0) = [0 1]x0 + [2], 175.70/44.89 175.70/44.89 [0 0] [0] 175.70/44.89 [a](x0) = [1 2]x0 + [3] 175.70/44.89 orientation: 175.70/44.89 b#(a(x1)) = [1 2]x1 + [3] >= [0 1]x1 + [2] = b#(b(x1)) 175.70/44.89 175.70/44.89 [0 0] [0] 175.70/44.89 a(x1) = [1 2]x1 + [3] >= x1 = x1 175.70/44.89 175.70/44.89 [3 4] [5] [3 4] [5] 175.70/44.89 b(a(x1)) = [2 3]x1 + [4] >= [2 3]x1 + [4] = c(a(b(b(x1)))) 175.70/44.89 175.70/44.89 [1 2] [3] [1 2] [3] 175.70/44.89 b(x1) = [0 1]x1 + [2] >= [0 0]x1 + [2] = c(a(x1)) 175.70/44.89 175.70/44.89 [0 0] [2] 175.70/44.89 c(c(x1)) = [0 0]x1 + [2] >= x1 = x1 175.70/44.89 problem: 175.70/44.89 DPs: 175.70/44.89 175.70/44.89 TRS: 175.70/44.89 a(x1) -> x1 175.70/44.89 b(a(x1)) -> c(a(b(b(x1)))) 175.70/44.89 b(x1) -> c(a(x1)) 175.70/44.89 c(c(x1)) -> x1 175.70/44.89 Qed 175.70/44.89 EOF