172.13/44.03 YES 172.13/44.04 172.13/44.04 Problem: 172.13/44.04 b(a(a(x1))) -> a(b(c(x1))) 172.13/44.04 c(a(x1)) -> a(c(x1)) 172.13/44.04 c(b(x1)) -> b(a(x1)) 172.13/44.04 a(a(x1)) -> a(b(a(x1))) 172.13/44.04 172.13/44.04 Proof: 172.13/44.04 String Reversal Processor: 172.13/44.04 a(a(b(x1))) -> c(b(a(x1))) 172.13/44.04 a(c(x1)) -> c(a(x1)) 172.13/44.04 b(c(x1)) -> a(b(x1)) 172.13/44.04 a(a(x1)) -> a(b(a(x1))) 172.13/44.04 DP Processor: 172.13/44.04 DPs: 172.13/44.04 a#(a(b(x1))) -> a#(x1) 172.13/44.04 a#(a(b(x1))) -> b#(a(x1)) 172.13/44.04 a#(c(x1)) -> a#(x1) 172.13/44.04 b#(c(x1)) -> b#(x1) 172.13/44.04 b#(c(x1)) -> a#(b(x1)) 172.13/44.04 a#(a(x1)) -> b#(a(x1)) 172.13/44.04 a#(a(x1)) -> a#(b(a(x1))) 172.13/44.04 TRS: 172.13/44.04 a(a(b(x1))) -> c(b(a(x1))) 172.13/44.04 a(c(x1)) -> c(a(x1)) 172.13/44.04 b(c(x1)) -> a(b(x1)) 172.13/44.04 a(a(x1)) -> a(b(a(x1))) 172.13/44.04 TDG Processor: 172.13/44.04 DPs: 172.13/44.04 a#(a(b(x1))) -> a#(x1) 172.13/44.04 a#(a(b(x1))) -> b#(a(x1)) 172.13/44.04 a#(c(x1)) -> a#(x1) 172.13/44.04 b#(c(x1)) -> b#(x1) 172.13/44.04 b#(c(x1)) -> a#(b(x1)) 172.13/44.04 a#(a(x1)) -> b#(a(x1)) 172.13/44.04 a#(a(x1)) -> a#(b(a(x1))) 172.13/44.04 TRS: 172.13/44.04 a(a(b(x1))) -> c(b(a(x1))) 172.13/44.04 a(c(x1)) -> c(a(x1)) 172.13/44.04 b(c(x1)) -> a(b(x1)) 172.13/44.04 a(a(x1)) -> a(b(a(x1))) 172.13/44.04 graph: 172.13/44.04 b#(c(x1)) -> b#(x1) -> b#(c(x1)) -> a#(b(x1)) 172.13/44.04 b#(c(x1)) -> b#(x1) -> b#(c(x1)) -> b#(x1) 172.13/44.04 b#(c(x1)) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(a(x1))) 172.13/44.04 b#(c(x1)) -> a#(b(x1)) -> a#(a(x1)) -> b#(a(x1)) 172.13/44.04 b#(c(x1)) -> a#(b(x1)) -> a#(c(x1)) -> a#(x1) 172.13/44.04 b#(c(x1)) -> a#(b(x1)) -> a#(a(b(x1))) -> b#(a(x1)) 172.13/44.04 b#(c(x1)) -> a#(b(x1)) -> a#(a(b(x1))) -> a#(x1) 172.13/44.04 a#(c(x1)) -> a#(x1) -> a#(a(x1)) -> a#(b(a(x1))) 172.13/44.04 a#(c(x1)) -> a#(x1) -> a#(a(x1)) -> b#(a(x1)) 172.13/44.04 a#(c(x1)) -> a#(x1) -> a#(c(x1)) -> a#(x1) 172.13/44.04 a#(c(x1)) -> a#(x1) -> a#(a(b(x1))) -> b#(a(x1)) 172.13/44.04 a#(c(x1)) -> a#(x1) -> a#(a(b(x1))) -> a#(x1) 172.13/44.04 a#(a(b(x1))) -> b#(a(x1)) -> b#(c(x1)) -> a#(b(x1)) 172.13/44.04 a#(a(b(x1))) -> b#(a(x1)) -> b#(c(x1)) -> b#(x1) 172.13/44.04 a#(a(b(x1))) -> a#(x1) -> a#(a(x1)) -> a#(b(a(x1))) 172.13/44.04 a#(a(b(x1))) -> a#(x1) -> a#(a(x1)) -> b#(a(x1)) 172.13/44.04 a#(a(b(x1))) -> a#(x1) -> a#(c(x1)) -> a#(x1) 172.13/44.04 a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> b#(a(x1)) 172.13/44.04 a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> a#(x1) 172.13/44.04 a#(a(x1)) -> b#(a(x1)) -> b#(c(x1)) -> a#(b(x1)) 172.13/44.04 a#(a(x1)) -> b#(a(x1)) -> b#(c(x1)) -> b#(x1) 172.13/44.04 a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(a(x1))) 172.13/44.04 a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> b#(a(x1)) 172.13/44.04 a#(a(x1)) -> a#(b(a(x1))) -> a#(c(x1)) -> a#(x1) 172.13/44.04 a#(a(x1)) -> a#(b(a(x1))) -> a#(a(b(x1))) -> b#(a(x1)) 172.13/44.04 a#(a(x1)) -> a#(b(a(x1))) -> a#(a(b(x1))) -> a#(x1) 172.13/44.04 Arctic Interpretation Processor: 172.13/44.04 dimension: 1 172.13/44.04 usable rules: 172.13/44.04 a(a(b(x1))) -> c(b(a(x1))) 172.13/44.04 a(c(x1)) -> c(a(x1)) 172.13/44.04 b(c(x1)) -> a(b(x1)) 172.13/44.04 a(a(x1)) -> a(b(a(x1))) 172.13/44.04 interpretation: 172.13/44.04 [b#](x0) = x0 + 0, 172.13/44.04 172.13/44.04 [a#](x0) = x0 + 0, 172.13/44.04 172.13/44.04 [c](x0) = 4x0 + 4, 172.13/44.04 172.13/44.04 [b](x0) = x0 + 0, 172.13/44.04 172.13/44.04 [a](x0) = 4x0 + 2 172.13/44.04 orientation: 172.13/44.04 a#(a(b(x1))) = 4x1 + 4 >= x1 + 0 = a#(x1) 172.13/44.04 172.13/44.04 a#(a(b(x1))) = 4x1 + 4 >= 4x1 + 2 = b#(a(x1)) 172.13/44.04 172.13/44.04 a#(c(x1)) = 4x1 + 4 >= x1 + 0 = a#(x1) 172.13/44.04 172.13/44.04 b#(c(x1)) = 4x1 + 4 >= x1 + 0 = b#(x1) 172.13/44.04 172.13/44.04 b#(c(x1)) = 4x1 + 4 >= x1 + 0 = a#(b(x1)) 172.13/44.04 172.13/44.04 a#(a(x1)) = 4x1 + 2 >= 4x1 + 2 = b#(a(x1)) 172.13/44.04 172.13/44.04 a#(a(x1)) = 4x1 + 2 >= 4x1 + 2 = a#(b(a(x1))) 172.13/44.04 172.13/44.04 a(a(b(x1))) = 8x1 + 8 >= 8x1 + 6 = c(b(a(x1))) 172.13/44.04 172.13/44.04 a(c(x1)) = 8x1 + 8 >= 8x1 + 6 = c(a(x1)) 172.13/44.04 172.13/44.04 b(c(x1)) = 4x1 + 4 >= 4x1 + 4 = a(b(x1)) 172.13/44.04 172.13/44.04 a(a(x1)) = 8x1 + 6 >= 8x1 + 6 = a(b(a(x1))) 172.13/44.04 problem: 172.13/44.04 DPs: 172.13/44.04 a#(a(b(x1))) -> b#(a(x1)) 172.13/44.04 a#(a(x1)) -> b#(a(x1)) 172.13/44.04 a#(a(x1)) -> a#(b(a(x1))) 172.13/44.04 TRS: 172.13/44.04 a(a(b(x1))) -> c(b(a(x1))) 172.13/44.04 a(c(x1)) -> c(a(x1)) 172.13/44.04 b(c(x1)) -> a(b(x1)) 172.13/44.04 a(a(x1)) -> a(b(a(x1))) 172.13/44.04 Restore Modifier: 172.13/44.04 DPs: 172.13/44.04 a#(a(b(x1))) -> b#(a(x1)) 172.13/44.04 a#(a(x1)) -> b#(a(x1)) 172.13/44.04 a#(a(x1)) -> a#(b(a(x1))) 172.13/44.04 TRS: 172.13/44.04 a(a(b(x1))) -> c(b(a(x1))) 172.13/44.04 a(c(x1)) -> c(a(x1)) 172.13/44.04 b(c(x1)) -> a(b(x1)) 172.13/44.04 a(a(x1)) -> a(b(a(x1))) 172.13/44.04 EDG Processor: 172.13/44.04 DPs: 172.13/44.04 a#(a(b(x1))) -> b#(a(x1)) 172.13/44.04 a#(a(x1)) -> b#(a(x1)) 172.13/44.04 a#(a(x1)) -> a#(b(a(x1))) 172.13/44.04 TRS: 172.13/44.04 a(a(b(x1))) -> c(b(a(x1))) 172.13/44.04 a(c(x1)) -> c(a(x1)) 172.13/44.04 b(c(x1)) -> a(b(x1)) 172.13/44.04 a(a(x1)) -> a(b(a(x1))) 172.13/44.04 graph: 172.13/44.04 a#(a(x1)) -> a#(b(a(x1))) -> a#(a(b(x1))) -> b#(a(x1)) 172.13/44.04 a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> b#(a(x1)) 172.13/44.04 a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(a(x1))) 172.13/44.04 SCC Processor: 172.13/44.04 #sccs: 1 172.13/44.04 #rules: 1 172.13/44.04 #arcs: 3/9 172.13/44.04 DPs: 172.13/44.04 a#(a(x1)) -> a#(b(a(x1))) 172.13/44.04 TRS: 172.13/44.04 a(a(b(x1))) -> c(b(a(x1))) 172.13/44.04 a(c(x1)) -> c(a(x1)) 172.13/44.04 b(c(x1)) -> a(b(x1)) 172.13/44.04 a(a(x1)) -> a(b(a(x1))) 172.13/44.04 Arctic Interpretation Processor: 172.13/44.04 dimension: 1 172.13/44.04 usable rules: 172.13/44.04 a(a(b(x1))) -> c(b(a(x1))) 172.13/44.04 a(c(x1)) -> c(a(x1)) 172.13/44.04 b(c(x1)) -> a(b(x1)) 172.13/44.04 a(a(x1)) -> a(b(a(x1))) 172.13/44.04 interpretation: 172.13/44.04 [a#](x0) = x0 + 1, 172.13/44.04 172.13/44.04 [c](x0) = 8x0 + 14, 172.13/44.04 172.13/44.04 [b](x0) = -6x0 + 0, 172.13/44.04 172.13/44.04 [a](x0) = 8x0 + 2 172.13/44.04 orientation: 172.13/44.04 a#(a(x1)) = 8x1 + 2 >= 2x1 + 1 = a#(b(a(x1))) 172.13/44.04 172.13/44.04 a(a(b(x1))) = 10x1 + 16 >= 10x1 + 14 = c(b(a(x1))) 172.13/44.04 172.13/44.04 a(c(x1)) = 16x1 + 22 >= 16x1 + 14 = c(a(x1)) 172.13/44.04 172.13/44.04 b(c(x1)) = 2x1 + 8 >= 2x1 + 8 = a(b(x1)) 172.13/44.04 172.13/44.04 a(a(x1)) = 16x1 + 10 >= 10x1 + 8 = a(b(a(x1))) 172.13/44.04 problem: 172.13/44.04 DPs: 172.13/44.04 172.13/44.04 TRS: 172.13/44.04 a(a(b(x1))) -> c(b(a(x1))) 172.13/44.04 a(c(x1)) -> c(a(x1)) 172.13/44.04 b(c(x1)) -> a(b(x1)) 172.13/44.04 a(a(x1)) -> a(b(a(x1))) 172.13/44.04 Qed 172.13/44.05 EOF