YES Problem: a(a(x1)) -> b(a(c(b(x1)))) b(b(x1)) -> a(a(x1)) c(a(x1)) -> x1 Proof: DP Processor: DPs: a#(a(x1)) -> b#(x1) a#(a(x1)) -> c#(b(x1)) a#(a(x1)) -> a#(c(b(x1))) a#(a(x1)) -> b#(a(c(b(x1)))) b#(b(x1)) -> a#(x1) b#(b(x1)) -> a#(a(x1)) TRS: a(a(x1)) -> b(a(c(b(x1)))) b(b(x1)) -> a(a(x1)) c(a(x1)) -> x1 TDG Processor: DPs: a#(a(x1)) -> b#(x1) a#(a(x1)) -> c#(b(x1)) a#(a(x1)) -> a#(c(b(x1))) a#(a(x1)) -> b#(a(c(b(x1)))) b#(b(x1)) -> a#(x1) b#(b(x1)) -> a#(a(x1)) TRS: a(a(x1)) -> b(a(c(b(x1)))) b(b(x1)) -> a(a(x1)) c(a(x1)) -> x1 graph: b#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> b#(a(c(b(x1)))) b#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> a#(c(b(x1))) b#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> c#(b(x1)) b#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> b#(x1) b#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(a(c(b(x1)))) b#(b(x1)) -> a#(x1) -> a#(a(x1)) -> a#(c(b(x1))) b#(b(x1)) -> a#(x1) -> a#(a(x1)) -> c#(b(x1)) b#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(x1) a#(a(x1)) -> b#(a(c(b(x1)))) -> b#(b(x1)) -> a#(a(x1)) a#(a(x1)) -> b#(a(c(b(x1)))) -> b#(b(x1)) -> a#(x1) a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> a#(a(x1)) a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> a#(x1) a#(a(x1)) -> a#(c(b(x1))) -> a#(a(x1)) -> b#(a(c(b(x1)))) a#(a(x1)) -> a#(c(b(x1))) -> a#(a(x1)) -> a#(c(b(x1))) a#(a(x1)) -> a#(c(b(x1))) -> a#(a(x1)) -> c#(b(x1)) a#(a(x1)) -> a#(c(b(x1))) -> a#(a(x1)) -> b#(x1) SCC Processor: #sccs: 1 #rules: 5 #arcs: 16/36 DPs: b#(b(x1)) -> a#(a(x1)) a#(a(x1)) -> b#(x1) b#(b(x1)) -> a#(x1) a#(a(x1)) -> a#(c(b(x1))) a#(a(x1)) -> b#(a(c(b(x1)))) TRS: a(a(x1)) -> b(a(c(b(x1)))) b(b(x1)) -> a(a(x1)) c(a(x1)) -> x1 Arctic Interpretation Processor: dimension: 3 usable rules: a(a(x1)) -> b(a(c(b(x1)))) b(b(x1)) -> a(a(x1)) c(a(x1)) -> x1 interpretation: [b#](x0) = [0 0 0]x0 + [0], [a#](x0) = [0 0 0]x0 + [0], [-& 0 -&] [0] [c](x0) = [-& 0 0 ]x0 + [0] [-& 0 0 ] [0], [1 1 1 ] [1] [b](x0) = [0 -& -&]x0 + [0] [-& 0 0 ] [1], [1 0 0 ] [1] [a](x0) = [0 -& 0 ]x0 + [1] [-& 0 0 ] [1] orientation: b#(b(x1)) = [1 1 1]x1 + [1] >= [1 0 0]x1 + [1] = a#(a(x1)) a#(a(x1)) = [1 0 0]x1 + [1] >= [0 0 0]x1 + [0] = b#(x1) b#(b(x1)) = [1 1 1]x1 + [1] >= [0 0 0]x1 + [0] = a#(x1) a#(a(x1)) = [1 0 0]x1 + [1] >= [0 0 0]x1 + [1] = a#(c(b(x1))) a#(a(x1)) = [1 0 0]x1 + [1] >= [1 0 0]x1 + [1] = b#(a(c(b(x1)))) [2 1 1] [2] [2 1 1] [2] a(a(x1)) = [1 0 0]x1 + [1] >= [1 0 0]x1 + [1] = b(a(c(b(x1)))) [0 0 0] [1] [0 0 0] [1] [2 2 2] [2] [2 1 1] [2] b(b(x1)) = [1 1 1]x1 + [1] >= [1 0 0]x1 + [1] = a(a(x1)) [0 0 0] [1] [0 0 0] [1] [0 -& 0 ] [1] c(a(x1)) = [0 0 0 ]x1 + [1] >= x1 = x1 [0 0 0 ] [1] problem: DPs: b#(b(x1)) -> a#(a(x1)) a#(a(x1)) -> b#(x1) a#(a(x1)) -> a#(c(b(x1))) a#(a(x1)) -> b#(a(c(b(x1)))) TRS: a(a(x1)) -> b(a(c(b(x1)))) b(b(x1)) -> a(a(x1)) c(a(x1)) -> x1 Restore Modifier: DPs: b#(b(x1)) -> a#(a(x1)) a#(a(x1)) -> b#(x1) a#(a(x1)) -> a#(c(b(x1))) a#(a(x1)) -> b#(a(c(b(x1)))) TRS: a(a(x1)) -> b(a(c(b(x1)))) b(b(x1)) -> a(a(x1)) c(a(x1)) -> x1 EDG Processor: DPs: b#(b(x1)) -> a#(a(x1)) a#(a(x1)) -> b#(x1) a#(a(x1)) -> a#(c(b(x1))) a#(a(x1)) -> b#(a(c(b(x1)))) TRS: a(a(x1)) -> b(a(c(b(x1)))) b(b(x1)) -> a(a(x1)) c(a(x1)) -> x1 graph: b#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> b#(x1) b#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> a#(c(b(x1))) b#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> b#(a(c(b(x1)))) a#(a(x1)) -> b#(a(c(b(x1)))) -> b#(b(x1)) -> a#(a(x1)) a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> a#(a(x1)) a#(a(x1)) -> a#(c(b(x1))) -> a#(a(x1)) -> b#(x1) a#(a(x1)) -> a#(c(b(x1))) -> a#(a(x1)) -> a#(c(b(x1))) a#(a(x1)) -> a#(c(b(x1))) -> a#(a(x1)) -> b#(a(c(b(x1)))) Arctic Interpretation Processor: dimension: 3 usable rules: a(a(x1)) -> b(a(c(b(x1)))) b(b(x1)) -> a(a(x1)) c(a(x1)) -> x1 interpretation: [b#](x0) = [0 0 0]x0 + [0], [a#](x0) = [0 -& -&]x0 + [0], [-& 0 -&] [0] [c](x0) = [0 0 0 ]x0 + [0] [-& -& 0 ] [0], [1 0 0 ] [1] [b](x0) = [0 -& -&]x0 + [0] [0 0 0 ] [1], [1 0 0] [1] [a](x0) = [0 0 0]x0 + [0] [0 0 0] [0] orientation: b#(b(x1)) = [1 0 0]x1 + [1] >= [1 0 0]x1 + [1] = a#(a(x1)) a#(a(x1)) = [1 0 0]x1 + [1] >= [0 0 0]x1 + [0] = b#(x1) a#(a(x1)) = [1 0 0]x1 + [1] >= [0 -& -&]x1 + [0] = a#(c(b(x1))) a#(a(x1)) = [1 0 0]x1 + [1] >= [1 0 0]x1 + [1] = b#(a(c(b(x1)))) [2 1 1] [2] [2 1 1] [2] a(a(x1)) = [1 0 0]x1 + [1] >= [1 0 0]x1 + [1] = b(a(c(b(x1)))) [1 0 0] [1] [1 0 0] [1] [2 1 1] [2] [2 1 1] [2] b(b(x1)) = [1 0 0]x1 + [1] >= [1 0 0]x1 + [1] = a(a(x1)) [1 0 0] [1] [1 0 0] [1] [0 0 0] [0] c(a(x1)) = [1 0 0]x1 + [1] >= x1 = x1 [0 0 0] [0] problem: DPs: b#(b(x1)) -> a#(a(x1)) a#(a(x1)) -> b#(x1) a#(a(x1)) -> b#(a(c(b(x1)))) TRS: a(a(x1)) -> b(a(c(b(x1)))) b(b(x1)) -> a(a(x1)) c(a(x1)) -> x1 Restore Modifier: DPs: b#(b(x1)) -> a#(a(x1)) a#(a(x1)) -> b#(x1) a#(a(x1)) -> b#(a(c(b(x1)))) TRS: a(a(x1)) -> b(a(c(b(x1)))) b(b(x1)) -> a(a(x1)) c(a(x1)) -> x1 EDG Processor: DPs: b#(b(x1)) -> a#(a(x1)) a#(a(x1)) -> b#(x1) a#(a(x1)) -> b#(a(c(b(x1)))) TRS: a(a(x1)) -> b(a(c(b(x1)))) b(b(x1)) -> a(a(x1)) c(a(x1)) -> x1 graph: b#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> b#(a(c(b(x1)))) b#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> b#(x1) a#(a(x1)) -> b#(a(c(b(x1)))) -> b#(b(x1)) -> a#(a(x1)) a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> a#(a(x1)) Arctic Interpretation Processor: dimension: 3 usable rules: a(a(x1)) -> b(a(c(b(x1)))) b(b(x1)) -> a(a(x1)) c(a(x1)) -> x1 interpretation: [b#](x0) = [0 1 0]x0 + [0], [a#](x0) = [0 1 -&]x0 + [0], [-& -& 0 ] [0] [c](x0) = [0 -& 0 ]x0 + [0] [0 -& 0 ] [0], [0 0 0] [0] [b](x0) = [1 1 1]x0 + [1] [0 0 0] [0], [0 0 0] [-&] [a](x0) = [1 1 1]x0 + [1 ] [0 0 0] [0 ] orientation: b#(b(x1)) = [2 2 2]x1 + [2] >= [2 2 2]x1 + [2] = a#(a(x1)) a#(a(x1)) = [2 2 2]x1 + [2] >= [0 1 0]x1 + [0] = b#(x1) a#(a(x1)) = [2 2 2]x1 + [2] >= [2 2 2]x1 + [2] = b#(a(c(b(x1)))) [1 1 1] [1] [1 1 1] [1] a(a(x1)) = [2 2 2]x1 + [2] >= [2 2 2]x1 + [2] = b(a(c(b(x1)))) [1 1 1] [1] [1 1 1] [1] [1 1 1] [1] [1 1 1] [1] b(b(x1)) = [2 2 2]x1 + [2] >= [2 2 2]x1 + [2] = a(a(x1)) [1 1 1] [1] [1 1 1] [1] [0 0 0] [0] c(a(x1)) = [0 0 0]x1 + [0] >= x1 = x1 [0 0 0] [0] problem: DPs: b#(b(x1)) -> a#(a(x1)) a#(a(x1)) -> b#(a(c(b(x1)))) TRS: a(a(x1)) -> b(a(c(b(x1)))) b(b(x1)) -> a(a(x1)) c(a(x1)) -> x1 Restore Modifier: DPs: b#(b(x1)) -> a#(a(x1)) a#(a(x1)) -> b#(a(c(b(x1)))) TRS: a(a(x1)) -> b(a(c(b(x1)))) b(b(x1)) -> a(a(x1)) c(a(x1)) -> x1 EDG Processor: DPs: b#(b(x1)) -> a#(a(x1)) a#(a(x1)) -> b#(a(c(b(x1)))) TRS: a(a(x1)) -> b(a(c(b(x1)))) b(b(x1)) -> a(a(x1)) c(a(x1)) -> x1 graph: b#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> b#(a(c(b(x1)))) a#(a(x1)) -> b#(a(c(b(x1)))) -> b#(b(x1)) -> a#(a(x1)) Arctic Interpretation Processor: dimension: 3 usable rules: a(a(x1)) -> b(a(c(b(x1)))) b(b(x1)) -> a(a(x1)) c(a(x1)) -> x1 interpretation: [b#](x0) = [0 0 1]x0 + [0], [a#](x0) = [0 0 0]x0 + [0], [0 -& -&] [0] [c](x0) = [0 -& -&]x0 + [0] [0 -& -&] [0], [0 -& 0 ] [0] [b](x0) = [0 0 1 ]x0 + [0] [1 0 1 ] [1], [0 0 0] [0] [a](x0) = [1 0 1]x0 + [1] [0 0 0] [0] orientation: b#(b(x1)) = [2 1 2]x1 + [2] >= [1 0 1]x1 + [1] = a#(a(x1)) a#(a(x1)) = [1 0 1]x1 + [1] >= [1 -& 1 ]x1 + [1] = b#(a(c(b(x1)))) [1 0 1] [1] [0 -& 0 ] [0] a(a(x1)) = [1 1 1]x1 + [1] >= [1 -& 1 ]x1 + [1] = b(a(c(b(x1)))) [1 0 1] [1] [1 -& 1 ] [1] [1 0 1] [1] [1 0 1] [1] b(b(x1)) = [2 1 2]x1 + [2] >= [1 1 1]x1 + [1] = a(a(x1)) [2 1 2] [2] [1 0 1] [1] [0 0 0] [0] c(a(x1)) = [0 0 0]x1 + [0] >= x1 = x1 [0 0 0] [0] problem: DPs: a#(a(x1)) -> b#(a(c(b(x1)))) TRS: a(a(x1)) -> b(a(c(b(x1)))) b(b(x1)) -> a(a(x1)) c(a(x1)) -> x1 Restore Modifier: DPs: a#(a(x1)) -> b#(a(c(b(x1)))) TRS: a(a(x1)) -> b(a(c(b(x1)))) b(b(x1)) -> a(a(x1)) c(a(x1)) -> x1 EDG Processor: DPs: a#(a(x1)) -> b#(a(c(b(x1)))) TRS: a(a(x1)) -> b(a(c(b(x1)))) b(b(x1)) -> a(a(x1)) c(a(x1)) -> x1 graph: Qed