YES Problem: a(a(x1)) -> a(b(b(b(x1)))) b(a(x1)) -> b(b(c(x1))) a(b(b(c(x1)))) -> a(a(a(b(x1)))) Proof: DP Processor: DPs: a#(a(x1)) -> b#(x1) a#(a(x1)) -> b#(b(x1)) a#(a(x1)) -> b#(b(b(x1))) a#(a(x1)) -> a#(b(b(b(x1)))) b#(a(x1)) -> b#(c(x1)) b#(a(x1)) -> b#(b(c(x1))) a#(b(b(c(x1)))) -> b#(x1) a#(b(b(c(x1)))) -> a#(b(x1)) a#(b(b(c(x1)))) -> a#(a(b(x1))) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) TRS: a(a(x1)) -> a(b(b(b(x1)))) b(a(x1)) -> b(b(c(x1))) a(b(b(c(x1)))) -> a(a(a(b(x1)))) TDG Processor: DPs: a#(a(x1)) -> b#(x1) a#(a(x1)) -> b#(b(x1)) a#(a(x1)) -> b#(b(b(x1))) a#(a(x1)) -> a#(b(b(b(x1)))) b#(a(x1)) -> b#(c(x1)) b#(a(x1)) -> b#(b(c(x1))) a#(b(b(c(x1)))) -> b#(x1) a#(b(b(c(x1)))) -> a#(b(x1)) a#(b(b(c(x1)))) -> a#(a(b(x1))) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) TRS: a(a(x1)) -> a(b(b(b(x1)))) b(a(x1)) -> b(b(c(x1))) a(b(b(c(x1)))) -> a(a(a(b(x1)))) graph: b#(a(x1)) -> b#(c(x1)) -> b#(a(x1)) -> b#(b(c(x1))) b#(a(x1)) -> b#(c(x1)) -> b#(a(x1)) -> b#(c(x1)) b#(a(x1)) -> b#(b(c(x1))) -> b#(a(x1)) -> b#(b(c(x1))) b#(a(x1)) -> b#(b(c(x1))) -> b#(a(x1)) -> b#(c(x1)) a#(b(b(c(x1)))) -> b#(x1) -> b#(a(x1)) -> b#(b(c(x1))) a#(b(b(c(x1)))) -> b#(x1) -> b#(a(x1)) -> b#(c(x1)) a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(b(b(c(x1)))) -> a#(a(b(x1))) a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(b(b(c(x1)))) -> a#(b(x1)) a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(b(b(c(x1)))) -> b#(x1) a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(b(b(x1)))) a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> b#(b(b(x1))) a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> b#(b(x1)) a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> b#(x1) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(b(b(c(x1)))) -> a#(a(b(x1))) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(b(b(c(x1)))) -> a#(b(x1)) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(b(b(c(x1)))) -> b#(x1) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(a(x1)) -> a#(b(b(b(x1)))) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(b(b(x1))) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(b(x1)) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(x1) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(b(b(c(x1)))) -> a#(a(b(x1))) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(b(b(c(x1)))) -> a#(b(x1)) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(b(b(c(x1)))) -> b#(x1) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(a(x1)) -> a#(b(b(b(x1)))) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(a(x1)) -> b#(b(b(x1))) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(a(x1)) -> b#(b(x1)) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(a(x1)) -> b#(x1) a#(a(x1)) -> b#(b(b(x1))) -> b#(a(x1)) -> b#(b(c(x1))) a#(a(x1)) -> b#(b(b(x1))) -> b#(a(x1)) -> b#(c(x1)) a#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(b(c(x1))) a#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(c(x1)) a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(b(c(x1))) a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(c(x1)) a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(b(b(c(x1)))) -> a#(a(b(x1))) a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(b(b(c(x1)))) -> a#(b(x1)) a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(b(b(c(x1)))) -> b#(x1) a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(a(x1)) -> a#(b(b(b(x1)))) a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(a(x1)) -> b#(b(b(x1))) a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(a(x1)) -> b#(b(x1)) a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(a(x1)) -> b#(x1) EDG Processor: DPs: a#(a(x1)) -> b#(x1) a#(a(x1)) -> b#(b(x1)) a#(a(x1)) -> b#(b(b(x1))) a#(a(x1)) -> a#(b(b(b(x1)))) b#(a(x1)) -> b#(c(x1)) b#(a(x1)) -> b#(b(c(x1))) a#(b(b(c(x1)))) -> b#(x1) a#(b(b(c(x1)))) -> a#(b(x1)) a#(b(b(c(x1)))) -> a#(a(b(x1))) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) TRS: a(a(x1)) -> a(b(b(b(x1)))) b(a(x1)) -> b(b(c(x1))) a(b(b(c(x1)))) -> a(a(a(b(x1)))) graph: a#(b(b(c(x1)))) -> b#(x1) -> b#(a(x1)) -> b#(c(x1)) a#(b(b(c(x1)))) -> b#(x1) -> b#(a(x1)) -> b#(b(c(x1))) a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> b#(x1) a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> b#(b(x1)) a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> b#(b(b(x1))) a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(b(b(x1)))) a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(b(b(c(x1)))) -> b#(x1) a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(b(b(c(x1)))) -> a#(b(x1)) a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(b(b(c(x1)))) -> a#(a(b(x1))) a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(x1) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(b(x1)) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(b(b(x1))) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(a(x1)) -> a#(b(b(b(x1)))) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(b(b(c(x1)))) -> b#(x1) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(b(b(c(x1)))) -> a#(b(x1)) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(b(b(c(x1)))) -> a#(a(b(x1))) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(a(x1)) -> b#(x1) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(a(x1)) -> b#(b(x1)) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(a(x1)) -> b#(b(b(x1))) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(a(x1)) -> a#(b(b(b(x1)))) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(b(b(c(x1)))) -> b#(x1) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(b(b(c(x1)))) -> a#(b(x1)) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(b(b(c(x1)))) -> a#(a(b(x1))) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) a#(a(x1)) -> b#(b(b(x1))) -> b#(a(x1)) -> b#(c(x1)) a#(a(x1)) -> b#(b(b(x1))) -> b#(a(x1)) -> b#(b(c(x1))) a#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(c(x1)) a#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(b(c(x1))) a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(c(x1)) a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(b(c(x1))) a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(a(x1)) -> b#(x1) a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(a(x1)) -> b#(b(x1)) a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(a(x1)) -> b#(b(b(x1))) a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(a(x1)) -> a#(b(b(b(x1)))) a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(b(b(c(x1)))) -> b#(x1) a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(b(b(c(x1)))) -> a#(b(x1)) a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(b(b(c(x1)))) -> a#(a(b(x1))) a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) CDG Processor: DPs: a#(a(x1)) -> b#(x1) a#(a(x1)) -> b#(b(x1)) a#(a(x1)) -> b#(b(b(x1))) a#(a(x1)) -> a#(b(b(b(x1)))) b#(a(x1)) -> b#(c(x1)) b#(a(x1)) -> b#(b(c(x1))) a#(b(b(c(x1)))) -> b#(x1) a#(b(b(c(x1)))) -> a#(b(x1)) a#(b(b(c(x1)))) -> a#(a(b(x1))) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) TRS: a(a(x1)) -> a(b(b(b(x1)))) b(a(x1)) -> b(b(c(x1))) a(b(b(c(x1)))) -> a(a(a(b(x1)))) graph: a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(b(b(c(x1)))) -> a#(a(b(x1))) a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(b(b(c(x1)))) -> a#(b(x1)) a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(b(b(c(x1)))) -> b#(x1) a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(b(b(x1)))) a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> b#(b(b(x1))) a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> b#(b(x1)) a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> b#(x1) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(b(b(c(x1)))) -> a#(a(b(x1))) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(b(b(c(x1)))) -> a#(b(x1)) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(b(b(c(x1)))) -> b#(x1) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(a(x1)) -> a#(b(b(b(x1)))) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(b(b(x1))) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(b(x1)) a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(x1) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(b(b(c(x1)))) -> a#(a(b(x1))) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(b(b(c(x1)))) -> a#(b(x1)) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(b(b(c(x1)))) -> b#(x1) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(a(x1)) -> a#(b(b(b(x1)))) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(a(x1)) -> b#(b(b(x1))) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(a(x1)) -> b#(b(x1)) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(a(x1)) -> b#(x1) SCC Processor: #sccs: 1 #rules: 3 #arcs: 24/100 DPs: a#(b(b(c(x1)))) -> a#(b(x1)) a#(b(b(c(x1)))) -> a#(a(b(x1))) a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) TRS: a(a(x1)) -> a(b(b(b(x1)))) b(a(x1)) -> b(b(c(x1))) a(b(b(c(x1)))) -> a(a(a(b(x1)))) Arctic Interpretation Processor: dimension: 1 interpretation: [a#](x0) = 8x0 + 0, [c](x0) = 8x0 + 9, [b](x0) = 8, [a](x0) = 0 orientation: a#(b(b(c(x1)))) = 16 >= 16 = a#(b(x1)) a#(b(b(c(x1)))) = 16 >= 8 = a#(a(b(x1))) a#(b(b(c(x1)))) = 16 >= 8 = a#(a(a(b(x1)))) a(a(x1)) = 0 >= 0 = a(b(b(b(x1)))) b(a(x1)) = 8 >= 8 = b(b(c(x1))) a(b(b(c(x1)))) = 0 >= 0 = a(a(a(b(x1)))) problem: DPs: a#(b(b(c(x1)))) -> a#(b(x1)) TRS: a(a(x1)) -> a(b(b(b(x1)))) b(a(x1)) -> b(b(c(x1))) a(b(b(c(x1)))) -> a(a(a(b(x1)))) EDG Processor: DPs: a#(b(b(c(x1)))) -> a#(b(x1)) TRS: a(a(x1)) -> a(b(b(b(x1)))) b(a(x1)) -> b(b(c(x1))) a(b(b(c(x1)))) -> a(a(a(b(x1)))) graph: a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(b(b(c(x1)))) -> a#(b(x1)) Matrix Interpretation Processor: dim=5 interpretation: [a#](x0) = [0 0 0 1 0]x0, [0 0 0 0 0] [0] [1 1 0 0 0] [1] [c](x0) = [0 1 0 0 0]x0 + [0] [0 1 0 1 0] [0] [0 0 0 0 0] [0], [0 1 0 0 0] [0 0 0 0 0] [b](x0) = [0 0 1 0 0]x0 [1 0 0 1 0] [1 0 0 0 0] , [1 1 0 1 1] [1] [0 0 0 0 0] [0] [a](x0) = [0 1 0 0 0]x0 + [0] [0 1 0 0 0] [0] [0 0 0 0 0] [0] orientation: a#(b(b(c(x1)))) = [1 2 0 1 0]x1 + [1] >= [1 0 0 1 0]x1 = a#(b(x1)) [1 2 0 1 1] [2] [1 1 0 1 0] [1] [0 0 0 0 0] [0] [0 0 0 0 0] [0] a(a(x1)) = [0 0 0 0 0]x1 + [0] >= [0 0 0 0 0]x1 + [0] = a(b(b(b(x1)))) [0 0 0 0 0] [0] [0 0 0 0 0] [0] [0 0 0 0 0] [0] [0 0 0 0 0] [0] [0 0 0 0 0] [0] [0 0 0 0 0] [0] [0 0 0 0 0] [0] [0 0 0 0 0] [0] b(a(x1)) = [0 1 0 0 0]x1 + [0] >= [0 1 0 0 0]x1 + [0] = b(b(c(x1))) [1 2 0 1 1] [1] [1 2 0 1 0] [1] [1 1 0 1 1] [1] [1 1 0 0 0] [1] [2 3 0 1 0] [3] [2 1 0 1 0] [3] [0 0 0 0 0] [0] [0 0 0 0 0] [0] a(b(b(c(x1)))) = [0 0 0 0 0]x1 + [0] >= [0 0 0 0 0]x1 + [0] = a(a(a(b(x1)))) [0 0 0 0 0] [0] [0 0 0 0 0] [0] [0 0 0 0 0] [0] [0 0 0 0 0] [0] problem: DPs: TRS: a(a(x1)) -> a(b(b(b(x1)))) b(a(x1)) -> b(b(c(x1))) a(b(b(c(x1)))) -> a(a(a(b(x1)))) Qed