YES Problem: a(x1) -> x1 a(b(x1)) -> c(x1) a(c(c(x1))) -> c(b(a(c(a(x1))))) Proof: String Reversal Processor: a(x1) -> x1 b(a(x1)) -> c(x1) c(c(a(x1))) -> a(c(a(b(c(x1))))) DP Processor: DPs: b#(a(x1)) -> c#(x1) c#(c(a(x1))) -> c#(x1) c#(c(a(x1))) -> b#(c(x1)) c#(c(a(x1))) -> a#(b(c(x1))) c#(c(a(x1))) -> c#(a(b(c(x1)))) c#(c(a(x1))) -> a#(c(a(b(c(x1))))) TRS: a(x1) -> x1 b(a(x1)) -> c(x1) c(c(a(x1))) -> a(c(a(b(c(x1))))) TDG Processor: DPs: b#(a(x1)) -> c#(x1) c#(c(a(x1))) -> c#(x1) c#(c(a(x1))) -> b#(c(x1)) c#(c(a(x1))) -> a#(b(c(x1))) c#(c(a(x1))) -> c#(a(b(c(x1)))) c#(c(a(x1))) -> a#(c(a(b(c(x1))))) TRS: a(x1) -> x1 b(a(x1)) -> c(x1) c(c(a(x1))) -> a(c(a(b(c(x1))))) graph: c#(c(a(x1))) -> c#(a(b(c(x1)))) -> c#(c(a(x1))) -> a#(c(a(b(c(x1))))) c#(c(a(x1))) -> c#(a(b(c(x1)))) -> c#(c(a(x1))) -> c#(a(b(c(x1)))) c#(c(a(x1))) -> c#(a(b(c(x1)))) -> c#(c(a(x1))) -> a#(b(c(x1))) c#(c(a(x1))) -> c#(a(b(c(x1)))) -> c#(c(a(x1))) -> b#(c(x1)) c#(c(a(x1))) -> c#(a(b(c(x1)))) -> c#(c(a(x1))) -> c#(x1) c#(c(a(x1))) -> c#(x1) -> c#(c(a(x1))) -> a#(c(a(b(c(x1))))) c#(c(a(x1))) -> c#(x1) -> c#(c(a(x1))) -> c#(a(b(c(x1)))) c#(c(a(x1))) -> c#(x1) -> c#(c(a(x1))) -> a#(b(c(x1))) c#(c(a(x1))) -> c#(x1) -> c#(c(a(x1))) -> b#(c(x1)) c#(c(a(x1))) -> c#(x1) -> c#(c(a(x1))) -> c#(x1) c#(c(a(x1))) -> b#(c(x1)) -> b#(a(x1)) -> c#(x1) b#(a(x1)) -> c#(x1) -> c#(c(a(x1))) -> a#(c(a(b(c(x1))))) b#(a(x1)) -> c#(x1) -> c#(c(a(x1))) -> c#(a(b(c(x1)))) b#(a(x1)) -> c#(x1) -> c#(c(a(x1))) -> a#(b(c(x1))) b#(a(x1)) -> c#(x1) -> c#(c(a(x1))) -> b#(c(x1)) b#(a(x1)) -> c#(x1) -> c#(c(a(x1))) -> c#(x1) SCC Processor: #sccs: 1 #rules: 4 #arcs: 16/36 DPs: c#(c(a(x1))) -> c#(a(b(c(x1)))) c#(c(a(x1))) -> c#(x1) c#(c(a(x1))) -> b#(c(x1)) b#(a(x1)) -> c#(x1) TRS: a(x1) -> x1 b(a(x1)) -> c(x1) c(c(a(x1))) -> a(c(a(b(c(x1))))) Arctic Interpretation Processor: dimension: 3 interpretation: [c#](x0) = [1 -& 0 ]x0 + [1], [b#](x0) = [0 -& 0 ]x0 + [0], [-& 0 -&] [0] [c](x0) = [1 0 0 ]x0 + [0] [0 0 0 ] [1], [0 -& -&] [0] [b](x0) = [0 -& 0 ]x0 + [0] [0 -& 0 ] [0], [0 0 0] [1] [a](x0) = [0 0 0]x0 + [1] [1 0 0] [0] orientation: c#(c(a(x1))) = [1 1 1]x1 + [2] >= [1 1 1]x1 + [2] = c#(a(b(c(x1)))) c#(c(a(x1))) = [1 1 1]x1 + [2] >= [1 -& 0 ]x1 + [1] = c#(x1) c#(c(a(x1))) = [1 1 1]x1 + [2] >= [0 0 0]x1 + [1] = b#(c(x1)) b#(a(x1)) = [1 0 0]x1 + [1] >= [1 -& 0 ]x1 + [1] = c#(x1) [0 0 0] [1] a(x1) = [0 0 0]x1 + [1] >= x1 = x1 [1 0 0] [0] [0 0 0] [1] [-& 0 -&] [0] b(a(x1)) = [1 0 0]x1 + [1] >= [1 0 0 ]x1 + [0] = c(x1) [1 0 0] [1] [0 0 0 ] [1] [1 1 1] [2] [1 1 1] [2] c(c(a(x1))) = [1 1 1]x1 + [2] >= [1 1 1]x1 + [2] = a(c(a(b(c(x1))))) [1 1 1] [2] [1 1 1] [2] problem: DPs: c#(c(a(x1))) -> c#(a(b(c(x1)))) c#(c(a(x1))) -> c#(x1) b#(a(x1)) -> c#(x1) TRS: a(x1) -> x1 b(a(x1)) -> c(x1) c(c(a(x1))) -> a(c(a(b(c(x1))))) EDG Processor: DPs: c#(c(a(x1))) -> c#(a(b(c(x1)))) c#(c(a(x1))) -> c#(x1) b#(a(x1)) -> c#(x1) TRS: a(x1) -> x1 b(a(x1)) -> c(x1) c(c(a(x1))) -> a(c(a(b(c(x1))))) graph: c#(c(a(x1))) -> c#(a(b(c(x1)))) -> c#(c(a(x1))) -> c#(x1) c#(c(a(x1))) -> c#(a(b(c(x1)))) -> c#(c(a(x1))) -> c#(a(b(c(x1)))) c#(c(a(x1))) -> c#(x1) -> c#(c(a(x1))) -> c#(x1) c#(c(a(x1))) -> c#(x1) -> c#(c(a(x1))) -> c#(a(b(c(x1)))) b#(a(x1)) -> c#(x1) -> c#(c(a(x1))) -> c#(x1) b#(a(x1)) -> c#(x1) -> c#(c(a(x1))) -> c#(a(b(c(x1)))) SCC Processor: #sccs: 1 #rules: 2 #arcs: 6/9 DPs: c#(c(a(x1))) -> c#(a(b(c(x1)))) c#(c(a(x1))) -> c#(x1) TRS: a(x1) -> x1 b(a(x1)) -> c(x1) c(c(a(x1))) -> a(c(a(b(c(x1))))) Arctic Interpretation Processor: dimension: 4 interpretation: [c#](x0) = [0 0 0 0]x0 + [0], [0 -& -& 0 ] [-&] [0 -& -& 0 ] [-&] [c](x0) = [0 -& 0 0 ]x0 + [-&] [1 0 -& -&] [0 ], [0 0 0 -&] [-&] [0 -& -& -&] [-&] [b](x0) = [0 0 0 -&]x0 + [0 ] [0 0 0 -&] [0 ], [0 -& 0 0 ] [0] [1 0 0 0 ] [0] [a](x0) = [0 1 0 0 ]x0 + [0] [0 0 0 0 ] [0] orientation: c#(c(a(x1))) = [1 1 1 1]x1 + [1] >= [1 -& 1 1 ]x1 + [0] = c#(a(b(c(x1)))) c#(c(a(x1))) = [1 1 1 1]x1 + [1] >= [0 0 0 0]x1 + [0] = c#(x1) [0 -& 0 0 ] [0] [1 0 0 0 ] [0] a(x1) = [0 1 0 0 ]x1 + [0] >= x1 = x1 [0 0 0 0 ] [0] [1 1 0 0 ] [0] [0 -& -& 0 ] [-&] [0 -& 0 0 ] [0] [0 -& -& 0 ] [-&] b(a(x1)) = [1 1 0 0 ]x1 + [0] >= [0 -& 0 0 ]x1 + [-&] = c(x1) [1 1 0 0 ] [0] [1 0 -& -&] [0 ] [1 0 1 1] [1] [1 -& 1 1 ] [1] [1 0 1 1] [1] [1 -& 1 1 ] [1] c(c(a(x1))) = [1 1 1 1]x1 + [1] >= [1 -& 1 1 ]x1 + [1] = a(c(a(b(c(x1))))) [1 1 1 1] [1] [1 -& 1 1 ] [1] problem: DPs: c#(c(a(x1))) -> c#(a(b(c(x1)))) TRS: a(x1) -> x1 b(a(x1)) -> c(x1) c(c(a(x1))) -> a(c(a(b(c(x1))))) EDG Processor: DPs: c#(c(a(x1))) -> c#(a(b(c(x1)))) TRS: a(x1) -> x1 b(a(x1)) -> c(x1) c(c(a(x1))) -> a(c(a(b(c(x1))))) graph: c#(c(a(x1))) -> c#(a(b(c(x1)))) -> c#(c(a(x1))) -> c#(a(b(c(x1)))) Arctic Interpretation Processor: dimension: 4 interpretation: [c#](x0) = [-& 0 0 0 ]x0 + [0], [0 0 0 0 ] [0 ] [0 0 0 1 ] [-&] [c](x0) = [0 0 -& 0 ]x0 + [0 ] [-& 0 0 -&] [0 ], [-& -& 0 0 ] [0] [0 -& 0 0 ] [0] [b](x0) = [0 -& 0 0 ]x0 + [0] [-& -& -& 0 ] [0], [0 0 0 1] [0] [0 0 0 0] [0] [a](x0) = [0 0 0 0]x0 + [0] [0 0 0 0] [0] orientation: c#(c(a(x1))) = [1 1 1 1]x1 + [1] >= [0 0 0 0]x1 + [0] = c#(a(b(c(x1)))) [0 0 0 1] [0] [0 0 0 0] [0] a(x1) = [0 0 0 0]x1 + [0] >= x1 = x1 [0 0 0 0] [0] [0 0 0 0] [0] [0 0 0 0 ] [0 ] [0 0 0 1] [0] [0 0 0 1 ] [-&] b(a(x1)) = [0 0 0 1]x1 + [0] >= [0 0 -& 0 ]x1 + [0 ] = c(x1) [0 0 0 0] [0] [-& 0 0 -&] [0 ] [1 1 1 1] [1] [1 1 1 1] [1] [1 1 1 1] [1] [1 1 1 1] [1] c(c(a(x1))) = [1 1 1 1]x1 + [1] >= [1 1 1 1]x1 + [1] = a(c(a(b(c(x1))))) [1 1 1 1] [1] [1 1 1 1] [1] problem: DPs: TRS: a(x1) -> x1 b(a(x1)) -> c(x1) c(c(a(x1))) -> a(c(a(b(c(x1))))) Qed