YES Problem: a(c(b(x1))) -> b(a(b(a(x1)))) b(x1) -> c(a(c(x1))) a(a(x1)) -> a(b(c(a(x1)))) Proof: DP Processor: DPs: a#(c(b(x1))) -> a#(x1) a#(c(b(x1))) -> b#(a(x1)) a#(c(b(x1))) -> a#(b(a(x1))) a#(c(b(x1))) -> b#(a(b(a(x1)))) b#(x1) -> a#(c(x1)) a#(a(x1)) -> b#(c(a(x1))) a#(a(x1)) -> a#(b(c(a(x1)))) TRS: a(c(b(x1))) -> b(a(b(a(x1)))) b(x1) -> c(a(c(x1))) a(a(x1)) -> a(b(c(a(x1)))) TDG Processor: DPs: a#(c(b(x1))) -> a#(x1) a#(c(b(x1))) -> b#(a(x1)) a#(c(b(x1))) -> a#(b(a(x1))) a#(c(b(x1))) -> b#(a(b(a(x1)))) b#(x1) -> a#(c(x1)) a#(a(x1)) -> b#(c(a(x1))) a#(a(x1)) -> a#(b(c(a(x1)))) TRS: a(c(b(x1))) -> b(a(b(a(x1)))) b(x1) -> c(a(c(x1))) a(a(x1)) -> a(b(c(a(x1)))) graph: b#(x1) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(c(a(x1)))) b#(x1) -> a#(c(x1)) -> a#(a(x1)) -> b#(c(a(x1))) b#(x1) -> a#(c(x1)) -> a#(c(b(x1))) -> b#(a(b(a(x1)))) b#(x1) -> a#(c(x1)) -> a#(c(b(x1))) -> a#(b(a(x1))) b#(x1) -> a#(c(x1)) -> a#(c(b(x1))) -> b#(a(x1)) b#(x1) -> a#(c(x1)) -> a#(c(b(x1))) -> a#(x1) a#(a(x1)) -> b#(c(a(x1))) -> b#(x1) -> a#(c(x1)) a#(a(x1)) -> a#(b(c(a(x1)))) -> a#(a(x1)) -> a#(b(c(a(x1)))) a#(a(x1)) -> a#(b(c(a(x1)))) -> a#(a(x1)) -> b#(c(a(x1))) a#(a(x1)) -> a#(b(c(a(x1)))) -> a#(c(b(x1))) -> b#(a(b(a(x1)))) a#(a(x1)) -> a#(b(c(a(x1)))) -> a#(c(b(x1))) -> a#(b(a(x1))) a#(a(x1)) -> a#(b(c(a(x1)))) -> a#(c(b(x1))) -> b#(a(x1)) a#(a(x1)) -> a#(b(c(a(x1)))) -> a#(c(b(x1))) -> a#(x1) a#(c(b(x1))) -> b#(a(b(a(x1)))) -> b#(x1) -> a#(c(x1)) a#(c(b(x1))) -> b#(a(x1)) -> b#(x1) -> a#(c(x1)) a#(c(b(x1))) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(c(a(x1)))) a#(c(b(x1))) -> a#(b(a(x1))) -> a#(a(x1)) -> b#(c(a(x1))) a#(c(b(x1))) -> a#(b(a(x1))) -> a#(c(b(x1))) -> b#(a(b(a(x1)))) a#(c(b(x1))) -> a#(b(a(x1))) -> a#(c(b(x1))) -> a#(b(a(x1))) a#(c(b(x1))) -> a#(b(a(x1))) -> a#(c(b(x1))) -> b#(a(x1)) a#(c(b(x1))) -> a#(b(a(x1))) -> a#(c(b(x1))) -> a#(x1) a#(c(b(x1))) -> a#(x1) -> a#(a(x1)) -> a#(b(c(a(x1)))) a#(c(b(x1))) -> a#(x1) -> a#(a(x1)) -> b#(c(a(x1))) a#(c(b(x1))) -> a#(x1) -> a#(c(b(x1))) -> b#(a(b(a(x1)))) a#(c(b(x1))) -> a#(x1) -> a#(c(b(x1))) -> a#(b(a(x1))) a#(c(b(x1))) -> a#(x1) -> a#(c(b(x1))) -> b#(a(x1)) a#(c(b(x1))) -> a#(x1) -> a#(c(b(x1))) -> a#(x1) EDG Processor: DPs: a#(c(b(x1))) -> a#(x1) a#(c(b(x1))) -> b#(a(x1)) a#(c(b(x1))) -> a#(b(a(x1))) a#(c(b(x1))) -> b#(a(b(a(x1)))) b#(x1) -> a#(c(x1)) a#(a(x1)) -> b#(c(a(x1))) a#(a(x1)) -> a#(b(c(a(x1)))) TRS: a(c(b(x1))) -> b(a(b(a(x1)))) b(x1) -> c(a(c(x1))) a(a(x1)) -> a(b(c(a(x1)))) graph: b#(x1) -> a#(c(x1)) -> a#(c(b(x1))) -> a#(x1) b#(x1) -> a#(c(x1)) -> a#(c(b(x1))) -> b#(a(x1)) b#(x1) -> a#(c(x1)) -> a#(c(b(x1))) -> a#(b(a(x1))) b#(x1) -> a#(c(x1)) -> a#(c(b(x1))) -> b#(a(b(a(x1)))) a#(a(x1)) -> b#(c(a(x1))) -> b#(x1) -> a#(c(x1)) a#(a(x1)) -> a#(b(c(a(x1)))) -> a#(c(b(x1))) -> a#(x1) a#(a(x1)) -> a#(b(c(a(x1)))) -> a#(c(b(x1))) -> b#(a(x1)) a#(a(x1)) -> a#(b(c(a(x1)))) -> a#(c(b(x1))) -> a#(b(a(x1))) a#(a(x1)) -> a#(b(c(a(x1)))) -> a#(c(b(x1))) -> b#(a(b(a(x1)))) a#(a(x1)) -> a#(b(c(a(x1)))) -> a#(a(x1)) -> b#(c(a(x1))) a#(a(x1)) -> a#(b(c(a(x1)))) -> a#(a(x1)) -> a#(b(c(a(x1)))) a#(c(b(x1))) -> b#(a(b(a(x1)))) -> b#(x1) -> a#(c(x1)) a#(c(b(x1))) -> b#(a(x1)) -> b#(x1) -> a#(c(x1)) a#(c(b(x1))) -> a#(b(a(x1))) -> a#(c(b(x1))) -> a#(x1) a#(c(b(x1))) -> a#(b(a(x1))) -> a#(c(b(x1))) -> b#(a(x1)) a#(c(b(x1))) -> a#(b(a(x1))) -> a#(c(b(x1))) -> a#(b(a(x1))) a#(c(b(x1))) -> a#(b(a(x1))) -> a#(c(b(x1))) -> b#(a(b(a(x1)))) a#(c(b(x1))) -> a#(b(a(x1))) -> a#(a(x1)) -> b#(c(a(x1))) a#(c(b(x1))) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(c(a(x1)))) a#(c(b(x1))) -> a#(x1) -> a#(c(b(x1))) -> a#(x1) a#(c(b(x1))) -> a#(x1) -> a#(c(b(x1))) -> b#(a(x1)) a#(c(b(x1))) -> a#(x1) -> a#(c(b(x1))) -> a#(b(a(x1))) a#(c(b(x1))) -> a#(x1) -> a#(c(b(x1))) -> b#(a(b(a(x1)))) a#(c(b(x1))) -> a#(x1) -> a#(a(x1)) -> b#(c(a(x1))) a#(c(b(x1))) -> a#(x1) -> a#(a(x1)) -> a#(b(c(a(x1)))) Matrix Interpretation Processor: dim=4 interpretation: [b#](x0) = [0 1 0 1]x0, [a#](x0) = [1 0 1 0]x0, [1 0 0 0] [0 0 0 0] [a](x0) = [0 0 0 0]x0 [1 0 0 0] , [0 0 0 1] [0 0 0 0] [c](x0) = [0 1 0 0]x0 [0 0 0 0] , [0 1 0 1] [0] [0 0 0 0] [0] [b](x0) = [0 0 0 0]x0 + [0] [1 0 1 0] [1] orientation: a#(c(b(x1))) = [1 0 1 0]x1 + [1] >= [1 0 1 0]x1 = a#(x1) a#(c(b(x1))) = [1 0 1 0]x1 + [1] >= [1 0 0 0]x1 = b#(a(x1)) a#(c(b(x1))) = [1 0 1 0]x1 + [1] >= [1 0 0 0]x1 = a#(b(a(x1))) a#(c(b(x1))) = [1 0 1 0]x1 + [1] >= [1 0 0 0]x1 = b#(a(b(a(x1)))) b#(x1) = [0 1 0 1]x1 >= [0 1 0 1]x1 = a#(c(x1)) a#(a(x1)) = [1 0 0 0]x1 >= [0] = b#(c(a(x1))) a#(a(x1)) = [1 0 0 0]x1 >= [0] = a#(b(c(a(x1)))) [1 0 1 0] [1] [1 0 0 0] [0] [0 0 0 0] [0] [0 0 0 0] [0] a(c(b(x1))) = [0 0 0 0]x1 + [0] >= [0 0 0 0]x1 + [0] = b(a(b(a(x1)))) [1 0 1 0] [1] [1 0 0 0] [1] [0 1 0 1] [0] [0 0 0 1] [0 0 0 0] [0] [0 0 0 0] b(x1) = [0 0 0 0]x1 + [0] >= [0 0 0 0]x1 = c(a(c(x1))) [1 0 1 0] [1] [0 0 0 0] [1 0 0 0] [0] [0 0 0 0] [0] a(a(x1)) = [0 0 0 0]x1 >= [0] = a(b(c(a(x1)))) [1 0 0 0] [0] problem: DPs: b#(x1) -> a#(c(x1)) a#(a(x1)) -> b#(c(a(x1))) a#(a(x1)) -> a#(b(c(a(x1)))) TRS: a(c(b(x1))) -> b(a(b(a(x1)))) b(x1) -> c(a(c(x1))) a(a(x1)) -> a(b(c(a(x1)))) EDG Processor: DPs: b#(x1) -> a#(c(x1)) a#(a(x1)) -> b#(c(a(x1))) a#(a(x1)) -> a#(b(c(a(x1)))) TRS: a(c(b(x1))) -> b(a(b(a(x1)))) b(x1) -> c(a(c(x1))) a(a(x1)) -> a(b(c(a(x1)))) graph: a#(a(x1)) -> b#(c(a(x1))) -> b#(x1) -> a#(c(x1)) a#(a(x1)) -> a#(b(c(a(x1)))) -> a#(a(x1)) -> a#(b(c(a(x1)))) a#(a(x1)) -> a#(b(c(a(x1)))) -> a#(a(x1)) -> b#(c(a(x1))) SCC Processor: #sccs: 1 #rules: 1 #arcs: 3/9 DPs: a#(a(x1)) -> a#(b(c(a(x1)))) TRS: a(c(b(x1))) -> b(a(b(a(x1)))) b(x1) -> c(a(c(x1))) a(a(x1)) -> a(b(c(a(x1)))) Arctic Interpretation Processor: dimension: 1 interpretation: [a#](x0) = x0 + 0, [a](x0) = x0 + 1, [c](x0) = 0, [b](x0) = 0 orientation: a#(a(x1)) = x1 + 1 >= 0 = a#(b(c(a(x1)))) a(c(b(x1))) = 1 >= 0 = b(a(b(a(x1)))) b(x1) = 0 >= 0 = c(a(c(x1))) a(a(x1)) = x1 + 1 >= 1 = a(b(c(a(x1)))) problem: DPs: TRS: a(c(b(x1))) -> b(a(b(a(x1)))) b(x1) -> c(a(c(x1))) a(a(x1)) -> a(b(c(a(x1)))) Qed