174.12/44.51 YES 174.12/44.52 174.12/44.52 Problem: 174.12/44.52 a(x1) -> x1 174.12/44.52 a(x1) -> b(c(x1)) 174.12/44.52 a(b(b(x1))) -> b(b(a(a(x1)))) 174.12/44.52 b(x1) -> x1 174.12/44.52 174.12/44.52 Proof: 174.12/44.52 DP Processor: 174.12/44.52 DPs: 174.12/44.52 a#(x1) -> b#(c(x1)) 174.12/44.52 a#(b(b(x1))) -> a#(x1) 174.12/44.52 a#(b(b(x1))) -> a#(a(x1)) 174.12/44.52 a#(b(b(x1))) -> b#(a(a(x1))) 174.12/44.52 a#(b(b(x1))) -> b#(b(a(a(x1)))) 174.12/44.52 TRS: 174.12/44.52 a(x1) -> x1 174.12/44.52 a(x1) -> b(c(x1)) 174.12/44.52 a(b(b(x1))) -> b(b(a(a(x1)))) 174.12/44.52 b(x1) -> x1 174.12/44.52 TDG Processor: 174.12/44.52 DPs: 174.12/44.52 a#(x1) -> b#(c(x1)) 174.12/44.52 a#(b(b(x1))) -> a#(x1) 174.12/44.52 a#(b(b(x1))) -> a#(a(x1)) 174.12/44.52 a#(b(b(x1))) -> b#(a(a(x1))) 174.12/44.52 a#(b(b(x1))) -> b#(b(a(a(x1)))) 174.12/44.52 TRS: 174.12/44.52 a(x1) -> x1 174.12/44.52 a(x1) -> b(c(x1)) 174.12/44.52 a(b(b(x1))) -> b(b(a(a(x1)))) 174.12/44.52 b(x1) -> x1 174.12/44.52 graph: 174.12/44.52 a#(b(b(x1))) -> a#(a(x1)) -> a#(b(b(x1))) -> b#(b(a(a(x1)))) 174.12/44.52 a#(b(b(x1))) -> a#(a(x1)) -> a#(b(b(x1))) -> b#(a(a(x1))) 174.12/44.52 a#(b(b(x1))) -> a#(a(x1)) -> a#(b(b(x1))) -> a#(a(x1)) 174.12/44.52 a#(b(b(x1))) -> a#(a(x1)) -> a#(b(b(x1))) -> a#(x1) 174.12/44.52 a#(b(b(x1))) -> a#(a(x1)) -> a#(x1) -> b#(c(x1)) 174.12/44.52 a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> b#(b(a(a(x1)))) 174.12/44.52 a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> b#(a(a(x1))) 174.12/44.52 a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> a#(a(x1)) 174.12/44.52 a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> a#(x1) 174.12/44.52 a#(b(b(x1))) -> a#(x1) -> a#(x1) -> b#(c(x1)) 174.12/44.52 SCC Processor: 174.12/44.52 #sccs: 1 174.12/44.52 #rules: 2 174.12/44.52 #arcs: 10/25 174.12/44.52 DPs: 174.12/44.52 a#(b(b(x1))) -> a#(a(x1)) 174.12/44.52 a#(b(b(x1))) -> a#(x1) 174.12/44.52 TRS: 174.12/44.52 a(x1) -> x1 174.12/44.52 a(x1) -> b(c(x1)) 174.12/44.52 a(b(b(x1))) -> b(b(a(a(x1)))) 174.12/44.52 b(x1) -> x1 174.12/44.52 Arctic Interpretation Processor: 174.12/44.52 dimension: 2 174.12/44.52 usable rules: 174.12/44.52 a(x1) -> x1 174.12/44.52 a(x1) -> b(c(x1)) 174.12/44.52 a(b(b(x1))) -> b(b(a(a(x1)))) 174.12/44.52 b(x1) -> x1 174.12/44.52 interpretation: 174.12/44.52 [a#](x0) = [0 0]x0, 174.12/44.52 174.12/44.52 [0 0] [-&] 174.12/44.52 [b](x0) = [2 0]x0 + [0 ], 174.12/44.52 174.12/44.52 [0 -&] 174.12/44.52 [c](x0) = [0 -&]x0, 174.12/44.52 174.12/44.52 [0 -&] [-&] 174.12/44.52 [a](x0) = [2 0 ]x0 + [0 ] 174.12/44.52 orientation: 174.12/44.52 a#(b(b(x1))) = [2 2]x1 + [0] >= [2 0]x1 + [0] = a#(a(x1)) 174.12/44.52 174.12/44.52 a#(b(b(x1))) = [2 2]x1 + [0] >= [0 0]x1 = a#(x1) 174.12/44.52 174.12/44.52 [0 -&] [-&] 174.12/44.52 a(x1) = [2 0 ]x1 + [0 ] >= x1 = x1 174.12/44.52 174.12/44.52 [0 -&] [-&] [0 -&] [-&] 174.12/44.52 a(x1) = [2 0 ]x1 + [0 ] >= [2 -&]x1 + [0 ] = b(c(x1)) 174.12/44.52 174.12/44.52 [2 0] [0] [2 0] [0] 174.12/44.52 a(b(b(x1))) = [4 2]x1 + [2] >= [4 2]x1 + [2] = b(b(a(a(x1)))) 174.12/44.52 174.12/44.52 [0 0] [-&] 174.12/44.52 b(x1) = [2 0]x1 + [0 ] >= x1 = x1 174.12/44.52 problem: 174.12/44.52 DPs: 174.12/44.52 a#(b(b(x1))) -> a#(a(x1)) 174.12/44.52 TRS: 174.12/44.52 a(x1) -> x1 174.12/44.52 a(x1) -> b(c(x1)) 174.12/44.52 a(b(b(x1))) -> b(b(a(a(x1)))) 174.12/44.52 b(x1) -> x1 174.12/44.52 Restore Modifier: 174.12/44.52 DPs: 174.12/44.52 a#(b(b(x1))) -> a#(a(x1)) 174.12/44.52 TRS: 174.12/44.52 a(x1) -> x1 174.12/44.52 a(x1) -> b(c(x1)) 174.12/44.52 a(b(b(x1))) -> b(b(a(a(x1)))) 174.12/44.52 b(x1) -> x1 174.12/44.52 EDG Processor: 174.12/44.52 DPs: 174.12/44.52 a#(b(b(x1))) -> a#(a(x1)) 174.12/44.52 TRS: 174.12/44.52 a(x1) -> x1 174.12/44.52 a(x1) -> b(c(x1)) 174.12/44.52 a(b(b(x1))) -> b(b(a(a(x1)))) 174.12/44.52 b(x1) -> x1 174.12/44.52 graph: 174.12/44.52 a#(b(b(x1))) -> a#(a(x1)) -> a#(b(b(x1))) -> a#(a(x1)) 174.12/44.52 Arctic Interpretation Processor: 174.12/44.52 dimension: 2 174.12/44.52 usable rules: 174.12/44.52 a(x1) -> x1 174.12/44.52 a(x1) -> b(c(x1)) 174.12/44.52 a(b(b(x1))) -> b(b(a(a(x1)))) 174.12/44.52 b(x1) -> x1 174.12/44.52 interpretation: 174.12/44.52 [a#](x0) = [0 -&]x0, 174.12/44.52 174.12/44.52 [0 0] [-&] 174.12/44.52 [b](x0) = [1 0]x0 + [0 ], 174.12/44.52 174.12/44.52 [0 -&] 174.12/44.52 [c](x0) = [0 -&]x0, 174.12/44.52 174.12/44.52 [0 -&] [-&] 174.12/44.52 [a](x0) = [1 0 ]x0 + [0 ] 174.12/44.52 orientation: 174.12/44.52 a#(b(b(x1))) = [1 0]x1 + [0] >= [0 -&]x1 = a#(a(x1)) 174.12/44.52 174.12/44.52 [0 -&] [-&] 174.12/44.52 a(x1) = [1 0 ]x1 + [0 ] >= x1 = x1 174.12/44.52 174.12/44.52 [0 -&] [-&] [0 -&] [-&] 174.12/44.52 a(x1) = [1 0 ]x1 + [0 ] >= [1 -&]x1 + [0 ] = b(c(x1)) 174.12/44.52 174.12/44.52 [1 0] [0] [1 0] [0] 174.12/44.52 a(b(b(x1))) = [2 1]x1 + [1] >= [2 1]x1 + [1] = b(b(a(a(x1)))) 174.12/44.52 174.12/44.52 [0 0] [-&] 174.12/44.52 b(x1) = [1 0]x1 + [0 ] >= x1 = x1 174.12/44.52 problem: 174.12/44.52 DPs: 174.12/44.52 174.12/44.52 TRS: 174.12/44.52 a(x1) -> x1 174.12/44.52 a(x1) -> b(c(x1)) 174.12/44.52 a(b(b(x1))) -> b(b(a(a(x1)))) 174.12/44.52 b(x1) -> x1 174.12/44.52 Qed 174.12/44.52 EOF