YES Problem: a(a(a(b(x1)))) -> b(a(b(a(a(a(x1)))))) b(a(x1)) -> x1 Proof: DP Processor: DPs: a#(a(a(b(x1)))) -> a#(x1) a#(a(a(b(x1)))) -> a#(a(x1)) a#(a(a(b(x1)))) -> a#(a(a(x1))) a#(a(a(b(x1)))) -> b#(a(a(a(x1)))) a#(a(a(b(x1)))) -> a#(b(a(a(a(x1))))) a#(a(a(b(x1)))) -> b#(a(b(a(a(a(x1)))))) TRS: a(a(a(b(x1)))) -> b(a(b(a(a(a(x1)))))) b(a(x1)) -> x1 TDG Processor: DPs: a#(a(a(b(x1)))) -> a#(x1) a#(a(a(b(x1)))) -> a#(a(x1)) a#(a(a(b(x1)))) -> a#(a(a(x1))) a#(a(a(b(x1)))) -> b#(a(a(a(x1)))) a#(a(a(b(x1)))) -> a#(b(a(a(a(x1))))) a#(a(a(b(x1)))) -> b#(a(b(a(a(a(x1)))))) TRS: a(a(a(b(x1)))) -> b(a(b(a(a(a(x1)))))) b(a(x1)) -> x1 graph: a#(a(a(b(x1)))) -> a#(a(a(x1))) -> a#(a(a(b(x1)))) -> b#(a(b(a(a(a(x1)))))) a#(a(a(b(x1)))) -> a#(a(a(x1))) -> a#(a(a(b(x1)))) -> a#(b(a(a(a(x1))))) a#(a(a(b(x1)))) -> a#(a(a(x1))) -> a#(a(a(b(x1)))) -> b#(a(a(a(x1)))) a#(a(a(b(x1)))) -> a#(a(a(x1))) -> a#(a(a(b(x1)))) -> a#(a(a(x1))) a#(a(a(b(x1)))) -> a#(a(a(x1))) -> a#(a(a(b(x1)))) -> a#(a(x1)) a#(a(a(b(x1)))) -> a#(a(a(x1))) -> a#(a(a(b(x1)))) -> a#(x1) a#(a(a(b(x1)))) -> a#(a(x1)) -> a#(a(a(b(x1)))) -> b#(a(b(a(a(a(x1)))))) a#(a(a(b(x1)))) -> a#(a(x1)) -> a#(a(a(b(x1)))) -> a#(b(a(a(a(x1))))) a#(a(a(b(x1)))) -> a#(a(x1)) -> a#(a(a(b(x1)))) -> b#(a(a(a(x1)))) a#(a(a(b(x1)))) -> a#(a(x1)) -> a#(a(a(b(x1)))) -> a#(a(a(x1))) a#(a(a(b(x1)))) -> a#(a(x1)) -> a#(a(a(b(x1)))) -> a#(a(x1)) a#(a(a(b(x1)))) -> a#(a(x1)) -> a#(a(a(b(x1)))) -> a#(x1) a#(a(a(b(x1)))) -> a#(b(a(a(a(x1))))) -> a#(a(a(b(x1)))) -> b#(a(b(a(a(a(x1)))))) a#(a(a(b(x1)))) -> a#(b(a(a(a(x1))))) -> a#(a(a(b(x1)))) -> a#(b(a(a(a(x1))))) a#(a(a(b(x1)))) -> a#(b(a(a(a(x1))))) -> a#(a(a(b(x1)))) -> b#(a(a(a(x1)))) a#(a(a(b(x1)))) -> a#(b(a(a(a(x1))))) -> a#(a(a(b(x1)))) -> a#(a(a(x1))) a#(a(a(b(x1)))) -> a#(b(a(a(a(x1))))) -> a#(a(a(b(x1)))) -> a#(a(x1)) a#(a(a(b(x1)))) -> a#(b(a(a(a(x1))))) -> a#(a(a(b(x1)))) -> a#(x1) a#(a(a(b(x1)))) -> a#(x1) -> a#(a(a(b(x1)))) -> b#(a(b(a(a(a(x1)))))) a#(a(a(b(x1)))) -> a#(x1) -> a#(a(a(b(x1)))) -> a#(b(a(a(a(x1))))) a#(a(a(b(x1)))) -> a#(x1) -> a#(a(a(b(x1)))) -> b#(a(a(a(x1)))) a#(a(a(b(x1)))) -> a#(x1) -> a#(a(a(b(x1)))) -> a#(a(a(x1))) a#(a(a(b(x1)))) -> a#(x1) -> a#(a(a(b(x1)))) -> a#(a(x1)) a#(a(a(b(x1)))) -> a#(x1) -> a#(a(a(b(x1)))) -> a#(x1) SCC Processor: #sccs: 1 #rules: 4 #arcs: 24/36 DPs: a#(a(a(b(x1)))) -> a#(a(a(x1))) a#(a(a(b(x1)))) -> a#(x1) a#(a(a(b(x1)))) -> a#(a(x1)) a#(a(a(b(x1)))) -> a#(b(a(a(a(x1))))) TRS: a(a(a(b(x1)))) -> b(a(b(a(a(a(x1)))))) b(a(x1)) -> x1 Arctic Interpretation Processor: dimension: 3 interpretation: [a#](x0) = [-& 0 -&]x0 + [0], [0 1 0 ] [0] [a](x0) = [-& 0 0 ]x0 + [0] [0 0 0 ] [1], [0 1 0 ] [0 ] [b](x0) = [-& 0 -&]x0 + [-&] [-& 0 -&] [0 ] orientation: a#(a(a(b(x1)))) = [0 1 0]x1 + [1] >= [0 0 0]x1 + [1] = a#(a(a(x1))) a#(a(a(b(x1)))) = [0 1 0]x1 + [1] >= [-& 0 -&]x1 + [0] = a#(x1) a#(a(a(b(x1)))) = [0 1 0]x1 + [1] >= [-& 0 0 ]x1 + [0] = a#(a(x1)) a#(a(a(b(x1)))) = [0 1 0]x1 + [1] >= [0 1 0]x1 + [1] = a#(b(a(a(a(x1))))) [1 2 1] [2] [1 2 1] [2] a(a(a(b(x1)))) = [0 1 0]x1 + [1] >= [0 1 0]x1 + [1] = b(a(b(a(a(a(x1)))))) [0 1 0] [1] [0 1 0] [1] [0 1 1 ] [1] b(a(x1)) = [-& 0 0 ]x1 + [0] >= x1 = x1 [-& 0 0 ] [0] problem: DPs: a#(a(a(b(x1)))) -> a#(a(a(x1))) a#(a(a(b(x1)))) -> a#(a(x1)) a#(a(a(b(x1)))) -> a#(b(a(a(a(x1))))) TRS: a(a(a(b(x1)))) -> b(a(b(a(a(a(x1)))))) b(a(x1)) -> x1 EDG Processor: DPs: a#(a(a(b(x1)))) -> a#(a(a(x1))) a#(a(a(b(x1)))) -> a#(a(x1)) a#(a(a(b(x1)))) -> a#(b(a(a(a(x1))))) TRS: a(a(a(b(x1)))) -> b(a(b(a(a(a(x1)))))) b(a(x1)) -> x1 graph: a#(a(a(b(x1)))) -> a#(a(a(x1))) -> a#(a(a(b(x1)))) -> a#(a(x1)) a#(a(a(b(x1)))) -> a#(a(a(x1))) -> a#(a(a(b(x1)))) -> a#(a(a(x1))) a#(a(a(b(x1)))) -> a#(a(a(x1))) -> a#(a(a(b(x1)))) -> a#(b(a(a(a(x1))))) a#(a(a(b(x1)))) -> a#(a(x1)) -> a#(a(a(b(x1)))) -> a#(a(x1)) a#(a(a(b(x1)))) -> a#(a(x1)) -> a#(a(a(b(x1)))) -> a#(a(a(x1))) a#(a(a(b(x1)))) -> a#(a(x1)) -> a#(a(a(b(x1)))) -> a#(b(a(a(a(x1))))) a#(a(a(b(x1)))) -> a#(b(a(a(a(x1))))) -> a#(a(a(b(x1)))) -> a#(a(x1)) a#(a(a(b(x1)))) -> a#(b(a(a(a(x1))))) -> a#(a(a(b(x1)))) -> a#(a(a(x1))) a#(a(a(b(x1)))) -> a#(b(a(a(a(x1))))) -> a#(a(a(b(x1)))) -> a#(b(a(a(a(x1))))) Arctic Interpretation Processor: dimension: 3 interpretation: [a#](x0) = [1 -& 0 ]x0 + [0], [-& 0 1 ] [0 ] [a](x0) = [0 -& 0 ]x0 + [-&] [-& 0 -&] [0 ], [-& 0 -&] [0] [b](x0) = [0 1 0 ]x0 + [1] [-& 0 -&] [0] orientation: a#(a(a(b(x1)))) = [2 3 2]x1 + [3] >= [1 2 1]x1 + [2] = a#(a(a(x1))) a#(a(a(b(x1)))) = [2 3 2]x1 + [3] >= [-& 1 2 ]x1 + [1] = a#(a(x1)) a#(a(a(b(x1)))) = [2 3 2]x1 + [3] >= [1 2 1]x1 + [2] = a#(b(a(a(a(x1))))) [0 1 0] [1] [0 1 0] [1] a(a(a(b(x1)))) = [1 2 1]x1 + [2] >= [1 2 1]x1 + [2] = b(a(b(a(a(a(x1)))))) [0 1 0] [1] [0 1 0] [1] [0 -& 0 ] [0] b(a(x1)) = [1 0 1 ]x1 + [1] >= x1 = x1 [0 -& 0 ] [0] problem: DPs: a#(a(a(b(x1)))) -> a#(a(x1)) TRS: a(a(a(b(x1)))) -> b(a(b(a(a(a(x1)))))) b(a(x1)) -> x1 EDG Processor: DPs: a#(a(a(b(x1)))) -> a#(a(x1)) TRS: a(a(a(b(x1)))) -> b(a(b(a(a(a(x1)))))) b(a(x1)) -> x1 graph: a#(a(a(b(x1)))) -> a#(a(x1)) -> a#(a(a(b(x1)))) -> a#(a(x1)) Matrix Interpretation Processor: dim=1 interpretation: [a#](x0) = 2x0 + 1, [a](x0) = 2x0, [b](x0) = 1/2x0 + 4 orientation: a#(a(a(b(x1)))) = 4x1 + 33 >= 4x1 + 1 = a#(a(x1)) a(a(a(b(x1)))) = 4x1 + 32 >= 4x1 + 8 = b(a(b(a(a(a(x1)))))) b(a(x1)) = x1 + 4 >= x1 = x1 problem: DPs: TRS: a(a(a(b(x1)))) -> b(a(b(a(a(a(x1)))))) b(a(x1)) -> x1 Qed