YES Problem: a(x1) -> x1 a(a(b(b(x1)))) -> b(b(a(a(b(a(x1)))))) Proof: String Reversal Processor: a(x1) -> x1 b(b(a(a(x1)))) -> a(b(a(a(b(b(x1)))))) DP Processor: DPs: b#(b(a(a(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> a#(b(b(x1))) b#(b(a(a(x1)))) -> a#(a(b(b(x1)))) b#(b(a(a(x1)))) -> b#(a(a(b(b(x1))))) b#(b(a(a(x1)))) -> a#(b(a(a(b(b(x1)))))) TRS: a(x1) -> x1 b(b(a(a(x1)))) -> a(b(a(a(b(b(x1)))))) TDG Processor: DPs: b#(b(a(a(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> a#(b(b(x1))) b#(b(a(a(x1)))) -> a#(a(b(b(x1)))) b#(b(a(a(x1)))) -> b#(a(a(b(b(x1))))) b#(b(a(a(x1)))) -> a#(b(a(a(b(b(x1)))))) TRS: a(x1) -> x1 b(b(a(a(x1)))) -> a(b(a(a(b(b(x1)))))) graph: b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> a#(b(a(a(b(b(x1)))))) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> b#(a(a(b(b(x1))))) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> a#(a(b(b(x1)))) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> a#(b(b(x1))) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(a(a(b(b(x1))))) -> b#(b(a(a(x1)))) -> a#(b(a(a(b(b(x1)))))) b#(b(a(a(x1)))) -> b#(a(a(b(b(x1))))) -> b#(b(a(a(x1)))) -> b#(a(a(b(b(x1))))) b#(b(a(a(x1)))) -> b#(a(a(b(b(x1))))) -> b#(b(a(a(x1)))) -> a#(a(b(b(x1)))) b#(b(a(a(x1)))) -> b#(a(a(b(b(x1))))) -> b#(b(a(a(x1)))) -> a#(b(b(x1))) b#(b(a(a(x1)))) -> b#(a(a(b(b(x1))))) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(a(a(b(b(x1))))) -> b#(b(a(a(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> a#(b(a(a(b(b(x1)))))) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(a(a(b(b(x1))))) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> a#(a(b(b(x1)))) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> a#(b(b(x1))) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(x1) SCC Processor: #sccs: 1 #rules: 3 #arcs: 18/36 DPs: b#(b(a(a(x1)))) -> b#(b(x1)) b#(b(a(a(x1)))) -> b#(x1) b#(b(a(a(x1)))) -> b#(a(a(b(b(x1))))) TRS: a(x1) -> x1 b(b(a(a(x1)))) -> a(b(a(a(b(b(x1)))))) Arctic Interpretation Processor: dimension: 3 interpretation: [b#](x0) = [-& -& 0 ]x0 + [0], [0 0 1 ] [0] [b](x0) = [0 -& -&]x0 + [0] [-& -& 0 ] [0], [0 0 -&] [0] [a](x0) = [0 0 1 ]x0 + [1] [-& 1 0 ] [0] orientation: b#(b(a(a(x1)))) = [1 1 2]x1 + [2] >= [-& -& 0 ]x1 + [0] = b#(b(x1)) b#(b(a(a(x1)))) = [1 1 2]x1 + [2] >= [-& -& 0 ]x1 + [0] = b#(x1) b#(b(a(a(x1)))) = [1 1 2]x1 + [2] >= [1 1 2]x1 + [2] = b#(a(a(b(b(x1))))) [0 0 -&] [0] a(x1) = [0 0 1 ]x1 + [1] >= x1 = x1 [-& 1 0 ] [0] [2 2 3] [3] [2 2 3] [3] b(b(a(a(x1)))) = [2 2 3]x1 + [3] >= [2 2 3]x1 + [3] = a(b(a(a(b(b(x1)))))) [1 1 2] [2] [1 1 2] [2] problem: DPs: b#(b(a(a(x1)))) -> b#(a(a(b(b(x1))))) TRS: a(x1) -> x1 b(b(a(a(x1)))) -> a(b(a(a(b(b(x1)))))) EDG Processor: DPs: b#(b(a(a(x1)))) -> b#(a(a(b(b(x1))))) TRS: a(x1) -> x1 b(b(a(a(x1)))) -> a(b(a(a(b(b(x1)))))) graph: b#(b(a(a(x1)))) -> b#(a(a(b(b(x1))))) -> b#(b(a(a(x1)))) -> b#(a(a(b(b(x1))))) Arctic Interpretation Processor: dimension: 3 interpretation: [b#](x0) = [0 0 1]x0 + [0], [-& 0 0 ] [0] [b](x0) = [-& -& 0 ]x0 + [0] [0 0 0 ] [0], [0 1 0 ] [1] [a](x0) = [0 0 0 ]x0 + [0] [-& 0 0 ] [0] orientation: b#(b(a(a(x1)))) = [2 2 2]x1 + [2] >= [1 1 1]x1 + [1] = b#(a(a(b(b(x1))))) [0 1 0 ] [1] a(x1) = [0 0 0 ]x1 + [0] >= x1 = x1 [-& 0 0 ] [0] [1 1 1] [1] [1 1 1] [1] b(b(a(a(x1)))) = [1 1 1]x1 + [1] >= [1 1 1]x1 + [1] = a(b(a(a(b(b(x1)))))) [1 1 1] [1] [1 1 1] [1] problem: DPs: TRS: a(x1) -> x1 b(b(a(a(x1)))) -> a(b(a(a(b(b(x1)))))) Qed