YES Problem: a(x1) -> x1 a(a(b(x1))) -> c(a(b(b(a(a(x1)))))) b(c(x1)) -> x1 Proof: DP Processor: DPs: a#(a(b(x1))) -> a#(x1) a#(a(b(x1))) -> a#(a(x1)) a#(a(b(x1))) -> b#(a(a(x1))) a#(a(b(x1))) -> b#(b(a(a(x1)))) a#(a(b(x1))) -> a#(b(b(a(a(x1))))) TRS: a(x1) -> x1 a(a(b(x1))) -> c(a(b(b(a(a(x1)))))) b(c(x1)) -> x1 TDG Processor: DPs: a#(a(b(x1))) -> a#(x1) a#(a(b(x1))) -> a#(a(x1)) a#(a(b(x1))) -> b#(a(a(x1))) a#(a(b(x1))) -> b#(b(a(a(x1)))) a#(a(b(x1))) -> a#(b(b(a(a(x1))))) TRS: a(x1) -> x1 a(a(b(x1))) -> c(a(b(b(a(a(x1)))))) b(c(x1)) -> x1 graph: a#(a(b(x1))) -> a#(b(b(a(a(x1))))) -> a#(a(b(x1))) -> a#(b(b(a(a(x1))))) a#(a(b(x1))) -> a#(b(b(a(a(x1))))) -> a#(a(b(x1))) -> b#(b(a(a(x1)))) a#(a(b(x1))) -> a#(b(b(a(a(x1))))) -> a#(a(b(x1))) -> b#(a(a(x1))) a#(a(b(x1))) -> a#(b(b(a(a(x1))))) -> a#(a(b(x1))) -> a#(a(x1)) a#(a(b(x1))) -> a#(b(b(a(a(x1))))) -> a#(a(b(x1))) -> a#(x1) a#(a(b(x1))) -> a#(a(x1)) -> a#(a(b(x1))) -> a#(b(b(a(a(x1))))) a#(a(b(x1))) -> a#(a(x1)) -> a#(a(b(x1))) -> b#(b(a(a(x1)))) a#(a(b(x1))) -> a#(a(x1)) -> a#(a(b(x1))) -> b#(a(a(x1))) a#(a(b(x1))) -> a#(a(x1)) -> a#(a(b(x1))) -> a#(a(x1)) a#(a(b(x1))) -> a#(a(x1)) -> a#(a(b(x1))) -> a#(x1) a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> a#(b(b(a(a(x1))))) a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> b#(b(a(a(x1)))) a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> b#(a(a(x1))) a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> a#(a(x1)) a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> a#(x1) SCC Processor: #sccs: 1 #rules: 3 #arcs: 15/25 DPs: a#(a(b(x1))) -> a#(b(b(a(a(x1))))) a#(a(b(x1))) -> a#(x1) a#(a(b(x1))) -> a#(a(x1)) TRS: a(x1) -> x1 a(a(b(x1))) -> c(a(b(b(a(a(x1)))))) b(c(x1)) -> x1 Arctic Interpretation Processor: dimension: 3 interpretation: [a#](x0) = [0 -& 0 ]x0, [-& -& -&] [0 ] [c](x0) = [-& -& -&]x0 + [-&] [0 0 0 ] [0 ], [0 0 0] [-&] [b](x0) = [1 1 0]x0 + [1 ] [0 0 0] [1 ], [0 0 -&] [a](x0) = [0 0 -&]x0 [1 0 0 ] orientation: a#(a(b(x1))) = [1 1 1]x1 + [1] >= [1 1 0]x1 + [1] = a#(b(b(a(a(x1))))) a#(a(b(x1))) = [1 1 1]x1 + [1] >= [0 -& 0 ]x1 = a#(x1) a#(a(b(x1))) = [1 1 1]x1 + [1] >= [1 0 0]x1 = a#(a(x1)) [0 0 -&] a(x1) = [0 0 -&]x1 >= x1 = x1 [1 0 0 ] [1 1 0] [1] [-& -& -&] [0 ] a(a(b(x1))) = [1 1 0]x1 + [1] >= [-& -& -&]x1 + [-&] = c(a(b(b(a(a(x1)))))) [2 2 1] [2] [2 2 1 ] [2 ] [0 0 0] [0] b(c(x1)) = [0 0 0]x1 + [1] >= x1 = x1 [0 0 0] [1] problem: DPs: a#(a(b(x1))) -> a#(b(b(a(a(x1))))) a#(a(b(x1))) -> a#(a(x1)) TRS: a(x1) -> x1 a(a(b(x1))) -> c(a(b(b(a(a(x1)))))) b(c(x1)) -> x1 EDG Processor: DPs: a#(a(b(x1))) -> a#(b(b(a(a(x1))))) a#(a(b(x1))) -> a#(a(x1)) TRS: a(x1) -> x1 a(a(b(x1))) -> c(a(b(b(a(a(x1)))))) b(c(x1)) -> x1 graph: a#(a(b(x1))) -> a#(b(b(a(a(x1))))) -> a#(a(b(x1))) -> a#(a(x1)) a#(a(b(x1))) -> a#(b(b(a(a(x1))))) -> a#(a(b(x1))) -> a#(b(b(a(a(x1))))) a#(a(b(x1))) -> a#(a(x1)) -> a#(a(b(x1))) -> a#(a(x1)) a#(a(b(x1))) -> a#(a(x1)) -> a#(a(b(x1))) -> a#(b(b(a(a(x1))))) Arctic Interpretation Processor: dimension: 3 interpretation: [a#](x0) = [0 -& -&]x0 + [0], [-& -& -&] [0] [c](x0) = [-& -& -&]x0 + [1] [0 0 0 ] [1], [0 0 0] [0] [b](x0) = [1 1 0]x0 + [1] [1 0 0] [0], [0 0 -&] [0 ] [a](x0) = [-& 0 -&]x0 + [-&] [1 0 0 ] [1 ] orientation: a#(a(b(x1))) = [1 1 0]x1 + [1] >= [1 1 0]x1 + [1] = a#(b(b(a(a(x1))))) a#(a(b(x1))) = [1 1 0]x1 + [1] >= [0 0 -&]x1 + [0] = a#(a(x1)) [0 0 -&] [0 ] a(x1) = [-& 0 -&]x1 + [-&] >= x1 = x1 [1 0 0 ] [1 ] [1 1 0] [1] [-& -& -&] [0] a(a(b(x1))) = [1 1 0]x1 + [1] >= [-& -& -&]x1 + [1] = c(a(b(b(a(a(x1)))))) [2 2 1] [2] [2 2 1 ] [2] [0 0 0] [1] b(c(x1)) = [0 0 0]x1 + [2] >= x1 = x1 [0 0 0] [1] problem: DPs: a#(a(b(x1))) -> a#(b(b(a(a(x1))))) TRS: a(x1) -> x1 a(a(b(x1))) -> c(a(b(b(a(a(x1)))))) b(c(x1)) -> x1 EDG Processor: DPs: a#(a(b(x1))) -> a#(b(b(a(a(x1))))) TRS: a(x1) -> x1 a(a(b(x1))) -> c(a(b(b(a(a(x1)))))) b(c(x1)) -> x1 graph: a#(a(b(x1))) -> a#(b(b(a(a(x1))))) -> a#(a(b(x1))) -> a#(b(b(a(a(x1))))) Arctic Interpretation Processor: dimension: 3 interpretation: [a#](x0) = [0 0 0]x0, [0 -& 0 ] [-&] [c](x0) = [0 0 0 ]x0 + [0 ] [0 0 1 ] [0 ], [0 -& -&] [b](x0) = [0 -& 0 ]x0 [0 -& -&] , [0 -& -&] [-&] [a](x0) = [1 0 0 ]x0 + [0 ] [0 0 0 ] [-&] orientation: a#(a(b(x1))) = [1 -& 0 ]x1 + [0] >= [0 -& -&]x1 = a#(b(b(a(a(x1))))) [0 -& -&] [-&] a(x1) = [1 0 0 ]x1 + [0 ] >= x1 = x1 [0 0 0 ] [-&] [0 -& -&] [-&] [0 -& -&] [-&] a(a(b(x1))) = [1 -& 0 ]x1 + [0 ] >= [1 -& -&]x1 + [0 ] = c(a(b(b(a(a(x1)))))) [1 -& 0 ] [0 ] [1 -& -&] [0 ] [0 -& 0 ] [-&] b(c(x1)) = [0 0 1 ]x1 + [0 ] >= x1 = x1 [0 -& 0 ] [-&] problem: DPs: TRS: a(x1) -> x1 a(a(b(x1))) -> c(a(b(b(a(a(x1)))))) b(c(x1)) -> x1 Qed