YES Problem: a(a(b(c(x1)))) -> b(b(a(a(x1)))) b(x1) -> c(c(a(a(x1)))) b(c(x1)) -> a(x1) a(a(c(x1))) -> x1 Proof: DP Processor: DPs: a#(a(b(c(x1)))) -> a#(x1) a#(a(b(c(x1)))) -> a#(a(x1)) a#(a(b(c(x1)))) -> b#(a(a(x1))) a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) b#(x1) -> a#(x1) b#(x1) -> a#(a(x1)) b#(c(x1)) -> a#(x1) TRS: a(a(b(c(x1)))) -> b(b(a(a(x1)))) b(x1) -> c(c(a(a(x1)))) b(c(x1)) -> a(x1) a(a(c(x1))) -> x1 TDG Processor: DPs: a#(a(b(c(x1)))) -> a#(x1) a#(a(b(c(x1)))) -> a#(a(x1)) a#(a(b(c(x1)))) -> b#(a(a(x1))) a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) b#(x1) -> a#(x1) b#(x1) -> a#(a(x1)) b#(c(x1)) -> a#(x1) TRS: a(a(b(c(x1)))) -> b(b(a(a(x1)))) b(x1) -> c(c(a(a(x1)))) b(c(x1)) -> a(x1) a(a(c(x1))) -> x1 graph: b#(c(x1)) -> a#(x1) -> a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) b#(c(x1)) -> a#(x1) -> a#(a(b(c(x1)))) -> b#(a(a(x1))) b#(c(x1)) -> a#(x1) -> a#(a(b(c(x1)))) -> a#(a(x1)) b#(c(x1)) -> a#(x1) -> a#(a(b(c(x1)))) -> a#(x1) b#(x1) -> a#(a(x1)) -> a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) b#(x1) -> a#(a(x1)) -> a#(a(b(c(x1)))) -> b#(a(a(x1))) b#(x1) -> a#(a(x1)) -> a#(a(b(c(x1)))) -> a#(a(x1)) b#(x1) -> a#(a(x1)) -> a#(a(b(c(x1)))) -> a#(x1) b#(x1) -> a#(x1) -> a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) b#(x1) -> a#(x1) -> a#(a(b(c(x1)))) -> b#(a(a(x1))) b#(x1) -> a#(x1) -> a#(a(b(c(x1)))) -> a#(a(x1)) b#(x1) -> a#(x1) -> a#(a(b(c(x1)))) -> a#(x1) a#(a(b(c(x1)))) -> b#(a(a(x1))) -> b#(c(x1)) -> a#(x1) a#(a(b(c(x1)))) -> b#(a(a(x1))) -> b#(x1) -> a#(a(x1)) a#(a(b(c(x1)))) -> b#(a(a(x1))) -> b#(x1) -> a#(x1) a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) -> b#(c(x1)) -> a#(x1) a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) -> b#(x1) -> a#(a(x1)) a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) -> b#(x1) -> a#(x1) a#(a(b(c(x1)))) -> a#(a(x1)) -> a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) a#(a(b(c(x1)))) -> a#(a(x1)) -> a#(a(b(c(x1)))) -> b#(a(a(x1))) a#(a(b(c(x1)))) -> a#(a(x1)) -> a#(a(b(c(x1)))) -> a#(a(x1)) a#(a(b(c(x1)))) -> a#(a(x1)) -> a#(a(b(c(x1)))) -> a#(x1) a#(a(b(c(x1)))) -> a#(x1) -> a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) a#(a(b(c(x1)))) -> a#(x1) -> a#(a(b(c(x1)))) -> b#(a(a(x1))) a#(a(b(c(x1)))) -> a#(x1) -> a#(a(b(c(x1)))) -> a#(a(x1)) a#(a(b(c(x1)))) -> a#(x1) -> a#(a(b(c(x1)))) -> a#(x1) Arctic Interpretation Processor: dimension: 2 interpretation: [b#](x0) = [0 1]x0, [a#](x0) = [0 0]x0, [0 0] [a](x0) = [0 0]x0, [0 1] [b](x0) = [0 0]x0, [0 0] [c](x0) = [0 0]x0 orientation: a#(a(b(c(x1)))) = [1 1]x1 >= [0 0]x1 = a#(x1) a#(a(b(c(x1)))) = [1 1]x1 >= [0 0]x1 = a#(a(x1)) a#(a(b(c(x1)))) = [1 1]x1 >= [1 1]x1 = b#(a(a(x1))) a#(a(b(c(x1)))) = [1 1]x1 >= [1 1]x1 = b#(b(a(a(x1)))) b#(x1) = [0 1]x1 >= [0 0]x1 = a#(x1) b#(x1) = [0 1]x1 >= [0 0]x1 = a#(a(x1)) b#(c(x1)) = [1 1]x1 >= [0 0]x1 = a#(x1) [1 1] [1 1] a(a(b(c(x1)))) = [1 1]x1 >= [1 1]x1 = b(b(a(a(x1)))) [0 1] [0 0] b(x1) = [0 0]x1 >= [0 0]x1 = c(c(a(a(x1)))) [1 1] [0 0] b(c(x1)) = [0 0]x1 >= [0 0]x1 = a(x1) [0 0] a(a(c(x1))) = [0 0]x1 >= x1 = x1 problem: DPs: a#(a(b(c(x1)))) -> b#(a(a(x1))) a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) b#(x1) -> a#(x1) b#(x1) -> a#(a(x1)) TRS: a(a(b(c(x1)))) -> b(b(a(a(x1)))) b(x1) -> c(c(a(a(x1)))) b(c(x1)) -> a(x1) a(a(c(x1))) -> x1 EDG Processor: DPs: a#(a(b(c(x1)))) -> b#(a(a(x1))) a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) b#(x1) -> a#(x1) b#(x1) -> a#(a(x1)) TRS: a(a(b(c(x1)))) -> b(b(a(a(x1)))) b(x1) -> c(c(a(a(x1)))) b(c(x1)) -> a(x1) a(a(c(x1))) -> x1 graph: b#(x1) -> a#(a(x1)) -> a#(a(b(c(x1)))) -> b#(a(a(x1))) b#(x1) -> a#(a(x1)) -> a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) b#(x1) -> a#(x1) -> a#(a(b(c(x1)))) -> b#(a(a(x1))) b#(x1) -> a#(x1) -> a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) a#(a(b(c(x1)))) -> b#(a(a(x1))) -> b#(x1) -> a#(x1) a#(a(b(c(x1)))) -> b#(a(a(x1))) -> b#(x1) -> a#(a(x1)) a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) -> b#(x1) -> a#(x1) a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) -> b#(x1) -> a#(a(x1)) Arctic Interpretation Processor: dimension: 2 interpretation: [b#](x0) = [0 2]x0 + [2], [a#](x0) = [0 2]x0, [0 2 ] [0] [a](x0) = [-& 0 ]x0 + [0], [0 2] [2] [b](x0) = [2 0]x0 + [0], [0 2 ] [2 ] [c](x0) = [-& 0 ]x0 + [-&] orientation: a#(a(b(c(x1)))) = [4 6]x1 + [6] >= [0 2]x1 + [2] = b#(a(a(x1))) a#(a(b(c(x1)))) = [4 6]x1 + [6] >= [4 6]x1 + [6] = b#(b(a(a(x1)))) b#(x1) = [0 2]x1 + [2] >= [0 2]x1 = a#(x1) b#(x1) = [0 2]x1 + [2] >= [0 2]x1 + [2] = a#(a(x1)) [4 6] [6] [4 6] [6] a(a(b(c(x1)))) = [2 4]x1 + [4] >= [2 4]x1 + [4] = b(b(a(a(x1)))) [0 2] [2] [0 2 ] [2] b(x1) = [2 0]x1 + [0] >= [-& 0 ]x1 + [0] = c(c(a(a(x1)))) [0 2] [2] [0 2 ] [0] b(c(x1)) = [2 4]x1 + [4] >= [-& 0 ]x1 + [0] = a(x1) [0 2 ] [2] a(a(c(x1))) = [-& 0 ]x1 + [0] >= x1 = x1 problem: DPs: a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) b#(x1) -> a#(x1) b#(x1) -> a#(a(x1)) TRS: a(a(b(c(x1)))) -> b(b(a(a(x1)))) b(x1) -> c(c(a(a(x1)))) b(c(x1)) -> a(x1) a(a(c(x1))) -> x1 EDG Processor: DPs: a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) b#(x1) -> a#(x1) b#(x1) -> a#(a(x1)) TRS: a(a(b(c(x1)))) -> b(b(a(a(x1)))) b(x1) -> c(c(a(a(x1)))) b(c(x1)) -> a(x1) a(a(c(x1))) -> x1 graph: b#(x1) -> a#(a(x1)) -> a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) b#(x1) -> a#(x1) -> a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) -> b#(x1) -> a#(a(x1)) a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) -> b#(x1) -> a#(x1) Arctic Interpretation Processor: dimension: 2 interpretation: [b#](x0) = [-& 1 ]x0 + [1], [a#](x0) = [-& 0 ]x0 + [0], [-& 0 ] [0] [a](x0) = [-& 0 ]x0 + [0], [0 1] [1] [b](x0) = [1 1]x0 + [1], [0 1] [1] [c](x0) = [0 0]x0 + [0] orientation: a#(a(b(c(x1)))) = [1 2]x1 + [2] >= [-& 2 ]x1 + [2] = b#(b(a(a(x1)))) b#(x1) = [-& 1 ]x1 + [1] >= [-& 0 ]x1 + [0] = a#(x1) b#(x1) = [-& 1 ]x1 + [1] >= [-& 0 ]x1 + [0] = a#(a(x1)) [1 2] [2] [-& 2 ] [2] a(a(b(c(x1)))) = [1 2]x1 + [2] >= [-& 2 ]x1 + [2] = b(b(a(a(x1)))) [0 1] [1] [-& 1 ] [1] b(x1) = [1 1]x1 + [1] >= [-& 1 ]x1 + [1] = c(c(a(a(x1)))) [1 1] [1] [-& 0 ] [0] b(c(x1)) = [1 2]x1 + [2] >= [-& 0 ]x1 + [0] = a(x1) [0 0] [0] a(a(c(x1))) = [0 0]x1 + [0] >= x1 = x1 problem: DPs: a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) TRS: a(a(b(c(x1)))) -> b(b(a(a(x1)))) b(x1) -> c(c(a(a(x1)))) b(c(x1)) -> a(x1) a(a(c(x1))) -> x1 EDG Processor: DPs: a#(a(b(c(x1)))) -> b#(b(a(a(x1)))) TRS: a(a(b(c(x1)))) -> b(b(a(a(x1)))) b(x1) -> c(c(a(a(x1)))) b(c(x1)) -> a(x1) a(a(c(x1))) -> x1 graph: Qed