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