175.92/44.95 YES 175.92/44.96 175.92/44.96 Problem: 175.92/44.96 a(x1) -> x1 175.92/44.96 a(b(b(x1))) -> b(b(b(c(a(x1))))) 175.92/44.96 b(c(x1)) -> a(x1) 175.92/44.96 175.92/44.96 Proof: 175.92/44.96 String Reversal Processor: 175.92/44.96 a(x1) -> x1 175.92/44.96 b(b(a(x1))) -> a(c(b(b(b(x1))))) 175.92/44.96 c(b(x1)) -> a(x1) 175.92/44.96 DP Processor: 175.92/44.96 DPs: 175.92/44.96 b#(b(a(x1))) -> b#(x1) 175.92/44.96 b#(b(a(x1))) -> b#(b(x1)) 175.92/44.96 b#(b(a(x1))) -> b#(b(b(x1))) 175.92/44.96 b#(b(a(x1))) -> c#(b(b(b(x1)))) 175.92/44.96 b#(b(a(x1))) -> a#(c(b(b(b(x1))))) 175.92/44.96 c#(b(x1)) -> a#(x1) 175.92/44.96 TRS: 175.92/44.96 a(x1) -> x1 175.92/44.96 b(b(a(x1))) -> a(c(b(b(b(x1))))) 175.92/44.96 c(b(x1)) -> a(x1) 175.92/44.96 TDG Processor: 175.92/44.96 DPs: 175.92/44.96 b#(b(a(x1))) -> b#(x1) 175.92/44.96 b#(b(a(x1))) -> b#(b(x1)) 175.92/44.96 b#(b(a(x1))) -> b#(b(b(x1))) 175.92/44.96 b#(b(a(x1))) -> c#(b(b(b(x1)))) 175.92/44.96 b#(b(a(x1))) -> a#(c(b(b(b(x1))))) 175.92/44.96 c#(b(x1)) -> a#(x1) 175.92/44.96 TRS: 175.92/44.96 a(x1) -> x1 175.92/44.96 b(b(a(x1))) -> a(c(b(b(b(x1))))) 175.92/44.96 c(b(x1)) -> a(x1) 175.92/44.96 graph: 175.92/44.96 b#(b(a(x1))) -> c#(b(b(b(x1)))) -> c#(b(x1)) -> a#(x1) 175.92/44.96 b#(b(a(x1))) -> b#(b(b(x1))) -> b#(b(a(x1))) -> a#(c(b(b(b(x1))))) 175.92/44.96 b#(b(a(x1))) -> b#(b(b(x1))) -> b#(b(a(x1))) -> c#(b(b(b(x1)))) 175.92/44.96 b#(b(a(x1))) -> b#(b(b(x1))) -> b#(b(a(x1))) -> b#(b(b(x1))) 175.92/44.96 b#(b(a(x1))) -> b#(b(b(x1))) -> b#(b(a(x1))) -> b#(b(x1)) 175.92/44.96 b#(b(a(x1))) -> b#(b(b(x1))) -> b#(b(a(x1))) -> b#(x1) 175.92/44.96 b#(b(a(x1))) -> b#(b(x1)) -> b#(b(a(x1))) -> a#(c(b(b(b(x1))))) 175.92/44.96 b#(b(a(x1))) -> b#(b(x1)) -> b#(b(a(x1))) -> c#(b(b(b(x1)))) 175.92/44.96 b#(b(a(x1))) -> b#(b(x1)) -> b#(b(a(x1))) -> b#(b(b(x1))) 175.92/44.96 b#(b(a(x1))) -> b#(b(x1)) -> b#(b(a(x1))) -> b#(b(x1)) 175.92/44.96 b#(b(a(x1))) -> b#(b(x1)) -> b#(b(a(x1))) -> b#(x1) 175.92/44.96 b#(b(a(x1))) -> b#(x1) -> b#(b(a(x1))) -> a#(c(b(b(b(x1))))) 175.92/44.96 b#(b(a(x1))) -> b#(x1) -> b#(b(a(x1))) -> c#(b(b(b(x1)))) 175.92/44.96 b#(b(a(x1))) -> b#(x1) -> b#(b(a(x1))) -> b#(b(b(x1))) 175.92/44.96 b#(b(a(x1))) -> b#(x1) -> b#(b(a(x1))) -> b#(b(x1)) 175.92/44.96 b#(b(a(x1))) -> b#(x1) -> b#(b(a(x1))) -> b#(x1) 175.92/44.96 SCC Processor: 175.92/44.96 #sccs: 1 175.92/44.96 #rules: 3 175.92/44.96 #arcs: 16/36 175.92/44.96 DPs: 175.92/44.96 b#(b(a(x1))) -> b#(b(b(x1))) 175.92/44.96 b#(b(a(x1))) -> b#(x1) 175.92/44.96 b#(b(a(x1))) -> b#(b(x1)) 175.92/44.96 TRS: 175.92/44.96 a(x1) -> x1 175.92/44.96 b(b(a(x1))) -> a(c(b(b(b(x1))))) 175.92/44.96 c(b(x1)) -> a(x1) 175.92/44.96 Arctic Interpretation Processor: 175.92/44.96 dimension: 2 175.92/44.96 usable rules: 175.92/44.96 a(x1) -> x1 175.92/44.96 b(b(a(x1))) -> a(c(b(b(b(x1))))) 175.92/44.96 c(b(x1)) -> a(x1) 175.92/44.96 interpretation: 175.92/44.96 [b#](x0) = [0 1]x0 + [2], 175.92/44.96 175.92/44.96 [1 0 ] [3] 175.92/44.96 [c](x0) = [0 -&]x0 + [0], 175.92/44.96 175.92/44.96 [-& 0 ] [1] 175.92/44.96 [b](x0) = [0 -&]x0 + [0], 175.92/44.96 175.92/44.96 [0 1 ] [3] 175.92/44.96 [a](x0) = [-& 0 ]x0 + [1] 175.92/44.96 orientation: 175.92/44.96 b#(b(a(x1))) = [1 2]x1 + [4] >= [0 1]x1 + [2] = b#(b(b(x1))) 175.92/44.96 175.92/44.96 b#(b(a(x1))) = [1 2]x1 + [4] >= [0 1]x1 + [2] = b#(x1) 175.92/44.96 175.92/44.96 b#(b(a(x1))) = [1 2]x1 + [4] >= [1 0]x1 + [2] = b#(b(x1)) 175.92/44.96 175.92/44.96 [0 1 ] [3] 175.92/44.96 a(x1) = [-& 0 ]x1 + [1] >= x1 = x1 175.92/44.96 175.92/44.96 [0 1 ] [3] [0 1 ] [3] 175.92/44.96 b(b(a(x1))) = [-& 0 ]x1 + [1] >= [-& 0 ]x1 + [1] = a(c(b(b(b(x1))))) 175.92/44.96 175.92/44.96 [0 1 ] [3] [0 1 ] [3] 175.92/44.96 c(b(x1)) = [-& 0 ]x1 + [1] >= [-& 0 ]x1 + [1] = a(x1) 175.92/44.96 problem: 175.92/44.96 DPs: 175.92/44.96 b#(b(a(x1))) -> b#(b(x1)) 175.92/44.96 TRS: 175.92/44.96 a(x1) -> x1 175.92/44.96 b(b(a(x1))) -> a(c(b(b(b(x1))))) 175.92/44.96 c(b(x1)) -> a(x1) 175.92/44.96 Restore Modifier: 175.92/44.96 DPs: 175.92/44.96 b#(b(a(x1))) -> b#(b(x1)) 175.92/44.96 TRS: 175.92/44.96 a(x1) -> x1 175.92/44.96 b(b(a(x1))) -> a(c(b(b(b(x1))))) 175.92/44.96 c(b(x1)) -> a(x1) 175.92/44.96 EDG Processor: 175.92/44.96 DPs: 175.92/44.96 b#(b(a(x1))) -> b#(b(x1)) 175.92/44.96 TRS: 175.92/44.96 a(x1) -> x1 175.92/44.96 b(b(a(x1))) -> a(c(b(b(b(x1))))) 175.92/44.96 c(b(x1)) -> a(x1) 175.92/44.96 graph: 175.92/44.96 b#(b(a(x1))) -> b#(b(x1)) -> b#(b(a(x1))) -> b#(b(x1)) 175.92/44.96 Matrix Interpretation Processor: dim=1 175.92/44.96 175.92/44.96 usable rules: 175.92/44.96 a(x1) -> x1 175.92/44.96 b(b(a(x1))) -> a(c(b(b(b(x1))))) 175.92/44.96 c(b(x1)) -> a(x1) 175.92/44.96 interpretation: 175.92/44.96 [b#](x0) = 11/2x0 + 4, 175.92/44.96 175.92/44.96 [c](x0) = 1/2x0 + 15/2, 175.92/44.96 175.92/44.96 [b](x0) = 2x0 + 3/2, 175.92/44.96 175.92/44.96 [a](x0) = x0 + 3 175.92/44.96 orientation: 175.92/44.96 b#(b(a(x1))) = 11x1 + 181/4 >= 11x1 + 49/4 = b#(b(x1)) 175.92/44.96 175.92/44.96 a(x1) = x1 + 3 >= x1 = x1 175.92/44.96 175.92/44.96 b(b(a(x1))) = 4x1 + 33/2 >= 4x1 + 63/4 = a(c(b(b(b(x1))))) 175.92/44.96 175.92/44.96 c(b(x1)) = x1 + 33/4 >= x1 + 3 = a(x1) 175.92/44.96 problem: 175.92/44.96 DPs: 175.92/44.96 175.92/44.96 TRS: 175.92/44.96 a(x1) -> x1 175.92/44.96 b(b(a(x1))) -> a(c(b(b(b(x1))))) 175.92/44.96 c(b(x1)) -> a(x1) 175.92/44.96 Qed 175.92/44.97 EOF