YES Problem: a(b(x1)) -> b(b(b(x1))) a(c(x1)) -> c(b(b(x1))) b(a(x1)) -> a(a(a(x1))) b(c(x1)) -> c(c(c(x1))) a(x1) -> x1 b(x1) -> x1 c(x1) -> x1 Proof: DP Processor: DPs: a#(b(x1)) -> b#(b(x1)) a#(b(x1)) -> b#(b(b(x1))) a#(c(x1)) -> b#(x1) a#(c(x1)) -> b#(b(x1)) a#(c(x1)) -> c#(b(b(x1))) b#(a(x1)) -> a#(a(x1)) b#(a(x1)) -> a#(a(a(x1))) b#(c(x1)) -> c#(c(x1)) b#(c(x1)) -> c#(c(c(x1))) TRS: a(b(x1)) -> b(b(b(x1))) a(c(x1)) -> c(b(b(x1))) b(a(x1)) -> a(a(a(x1))) b(c(x1)) -> c(c(c(x1))) a(x1) -> x1 b(x1) -> x1 c(x1) -> x1 TDG Processor: DPs: a#(b(x1)) -> b#(b(x1)) a#(b(x1)) -> b#(b(b(x1))) a#(c(x1)) -> b#(x1) a#(c(x1)) -> b#(b(x1)) a#(c(x1)) -> c#(b(b(x1))) b#(a(x1)) -> a#(a(x1)) b#(a(x1)) -> a#(a(a(x1))) b#(c(x1)) -> c#(c(x1)) b#(c(x1)) -> c#(c(c(x1))) TRS: a(b(x1)) -> b(b(b(x1))) a(c(x1)) -> c(b(b(x1))) b(a(x1)) -> a(a(a(x1))) b(c(x1)) -> c(c(c(x1))) a(x1) -> x1 b(x1) -> x1 c(x1) -> x1 graph: b#(a(x1)) -> a#(a(a(x1))) -> a#(c(x1)) -> c#(b(b(x1))) b#(a(x1)) -> a#(a(a(x1))) -> a#(c(x1)) -> b#(b(x1)) b#(a(x1)) -> a#(a(a(x1))) -> a#(c(x1)) -> b#(x1) b#(a(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> b#(b(b(x1))) b#(a(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> b#(b(x1)) b#(a(x1)) -> a#(a(x1)) -> a#(c(x1)) -> c#(b(b(x1))) b#(a(x1)) -> a#(a(x1)) -> a#(c(x1)) -> b#(b(x1)) b#(a(x1)) -> a#(a(x1)) -> a#(c(x1)) -> b#(x1) b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(b(b(x1))) b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(b(x1)) a#(c(x1)) -> b#(b(x1)) -> b#(c(x1)) -> c#(c(c(x1))) a#(c(x1)) -> b#(b(x1)) -> b#(c(x1)) -> c#(c(x1)) a#(c(x1)) -> b#(b(x1)) -> b#(a(x1)) -> a#(a(a(x1))) a#(c(x1)) -> b#(b(x1)) -> b#(a(x1)) -> a#(a(x1)) a#(c(x1)) -> b#(x1) -> b#(c(x1)) -> c#(c(c(x1))) a#(c(x1)) -> b#(x1) -> b#(c(x1)) -> c#(c(x1)) a#(c(x1)) -> b#(x1) -> b#(a(x1)) -> a#(a(a(x1))) a#(c(x1)) -> b#(x1) -> b#(a(x1)) -> a#(a(x1)) a#(b(x1)) -> b#(b(b(x1))) -> b#(c(x1)) -> c#(c(c(x1))) a#(b(x1)) -> b#(b(b(x1))) -> b#(c(x1)) -> c#(c(x1)) a#(b(x1)) -> b#(b(b(x1))) -> b#(a(x1)) -> a#(a(a(x1))) a#(b(x1)) -> b#(b(b(x1))) -> b#(a(x1)) -> a#(a(x1)) a#(b(x1)) -> b#(b(x1)) -> b#(c(x1)) -> c#(c(c(x1))) a#(b(x1)) -> b#(b(x1)) -> b#(c(x1)) -> c#(c(x1)) a#(b(x1)) -> b#(b(x1)) -> b#(a(x1)) -> a#(a(a(x1))) a#(b(x1)) -> b#(b(x1)) -> b#(a(x1)) -> a#(a(x1)) SCC Processor: #sccs: 1 #rules: 6 #arcs: 26/81 DPs: b#(a(x1)) -> a#(a(a(x1))) a#(b(x1)) -> b#(b(x1)) b#(a(x1)) -> a#(a(x1)) a#(b(x1)) -> b#(b(b(x1))) a#(c(x1)) -> b#(x1) a#(c(x1)) -> b#(b(x1)) TRS: a(b(x1)) -> b(b(b(x1))) a(c(x1)) -> c(b(b(x1))) b(a(x1)) -> a(a(a(x1))) b(c(x1)) -> c(c(c(x1))) a(x1) -> x1 b(x1) -> x1 c(x1) -> x1 Arctic Interpretation Processor: dimension: 2 interpretation: [b#](x0) = [-& 0 ]x0 + [0], [a#](x0) = [0 0]x0 + [0], [0 1 ] [1] [c](x0) = [-& 0 ]x0 + [0], [0 -&] [0] [a](x0) = [1 0 ]x0 + [1], [0 0 ] [1 ] [b](x0) = [-& 0 ]x0 + [-&] orientation: b#(a(x1)) = [1 0]x1 + [1] >= [1 0]x1 + [1] = a#(a(a(x1))) a#(b(x1)) = [0 0]x1 + [1] >= [-& 0 ]x1 + [0] = b#(b(x1)) b#(a(x1)) = [1 0]x1 + [1] >= [1 0]x1 + [1] = a#(a(x1)) a#(b(x1)) = [0 0]x1 + [1] >= [-& 0 ]x1 + [0] = b#(b(b(x1))) a#(c(x1)) = [0 1]x1 + [1] >= [-& 0 ]x1 + [0] = b#(x1) a#(c(x1)) = [0 1]x1 + [1] >= [-& 0 ]x1 + [0] = b#(b(x1)) [0 0] [1] [0 0 ] [1 ] a(b(x1)) = [1 1]x1 + [2] >= [-& 0 ]x1 + [-&] = b(b(b(x1))) [0 1] [1] [0 1 ] [1] a(c(x1)) = [1 2]x1 + [2] >= [-& 0 ]x1 + [0] = c(b(b(x1))) [1 0] [1] [0 -&] [0] b(a(x1)) = [1 0]x1 + [1] >= [1 0 ]x1 + [1] = a(a(a(x1))) [0 1 ] [1] [0 1 ] [1] b(c(x1)) = [-& 0 ]x1 + [0] >= [-& 0 ]x1 + [0] = c(c(c(x1))) [0 -&] [0] a(x1) = [1 0 ]x1 + [1] >= x1 = x1 [0 0 ] [1 ] b(x1) = [-& 0 ]x1 + [-&] >= x1 = x1 [0 1 ] [1] c(x1) = [-& 0 ]x1 + [0] >= x1 = x1 problem: DPs: b#(a(x1)) -> a#(a(a(x1))) a#(b(x1)) -> b#(b(x1)) b#(a(x1)) -> a#(a(x1)) a#(b(x1)) -> b#(b(b(x1))) TRS: a(b(x1)) -> b(b(b(x1))) a(c(x1)) -> c(b(b(x1))) b(a(x1)) -> a(a(a(x1))) b(c(x1)) -> c(c(c(x1))) a(x1) -> x1 b(x1) -> x1 c(x1) -> x1 EDG Processor: DPs: b#(a(x1)) -> a#(a(a(x1))) a#(b(x1)) -> b#(b(x1)) b#(a(x1)) -> a#(a(x1)) a#(b(x1)) -> b#(b(b(x1))) TRS: a(b(x1)) -> b(b(b(x1))) a(c(x1)) -> c(b(b(x1))) b(a(x1)) -> a(a(a(x1))) b(c(x1)) -> c(c(c(x1))) a(x1) -> x1 b(x1) -> x1 c(x1) -> x1 graph: b#(a(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> b#(b(x1)) b#(a(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> b#(b(b(x1))) b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(b(x1)) b#(a(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(b(b(x1))) a#(b(x1)) -> b#(b(b(x1))) -> b#(a(x1)) -> a#(a(x1)) a#(b(x1)) -> b#(b(b(x1))) -> b#(a(x1)) -> a#(a(a(x1))) a#(b(x1)) -> b#(b(x1)) -> b#(a(x1)) -> a#(a(x1)) a#(b(x1)) -> b#(b(x1)) -> b#(a(x1)) -> a#(a(a(x1))) Arctic Interpretation Processor: dimension: 2 interpretation: [b#](x0) = [0 0]x0 + [0], [a#](x0) = [-& 0 ]x0 + [0], [0 0] [1] [c](x0) = [0 0]x0 + [1], [0 2 ] [2] [a](x0) = [-& 0 ]x0 + [0], [0 -&] [0] [b](x0) = [0 0 ]x0 + [0] orientation: b#(a(x1)) = [0 2]x1 + [2] >= [-& 0 ]x1 + [0] = a#(a(a(x1))) a#(b(x1)) = [0 0]x1 + [0] >= [0 0]x1 + [0] = b#(b(x1)) b#(a(x1)) = [0 2]x1 + [2] >= [-& 0 ]x1 + [0] = a#(a(x1)) a#(b(x1)) = [0 0]x1 + [0] >= [0 0]x1 + [0] = b#(b(b(x1))) [2 2] [2] [0 -&] [0] a(b(x1)) = [0 0]x1 + [0] >= [0 0 ]x1 + [0] = b(b(b(x1))) [2 2] [3] [0 0] [1] a(c(x1)) = [0 0]x1 + [1] >= [0 0]x1 + [1] = c(b(b(x1))) [0 2] [2] [0 2 ] [2] b(a(x1)) = [0 2]x1 + [2] >= [-& 0 ]x1 + [0] = a(a(a(x1))) [0 0] [1] [0 0] [1] b(c(x1)) = [0 0]x1 + [1] >= [0 0]x1 + [1] = c(c(c(x1))) [0 2 ] [2] a(x1) = [-& 0 ]x1 + [0] >= x1 = x1 [0 -&] [0] b(x1) = [0 0 ]x1 + [0] >= x1 = x1 [0 0] [1] c(x1) = [0 0]x1 + [1] >= x1 = x1 problem: DPs: a#(b(x1)) -> b#(b(x1)) a#(b(x1)) -> b#(b(b(x1))) TRS: a(b(x1)) -> b(b(b(x1))) a(c(x1)) -> c(b(b(x1))) b(a(x1)) -> a(a(a(x1))) b(c(x1)) -> c(c(c(x1))) a(x1) -> x1 b(x1) -> x1 c(x1) -> x1 EDG Processor: DPs: a#(b(x1)) -> b#(b(x1)) a#(b(x1)) -> b#(b(b(x1))) TRS: a(b(x1)) -> b(b(b(x1))) a(c(x1)) -> c(b(b(x1))) b(a(x1)) -> a(a(a(x1))) b(c(x1)) -> c(c(c(x1))) a(x1) -> x1 b(x1) -> x1 c(x1) -> x1 graph: Qed