YES Problem: a(a(a(x1))) -> b(b(a(x1))) a(b(a(x1))) -> b(b(a(x1))) b(a(b(x1))) -> a(a(b(x1))) Proof: String Reversal Processor: a(a(a(x1))) -> a(b(b(x1))) a(b(a(x1))) -> a(b(b(x1))) b(a(b(x1))) -> b(a(a(x1))) DP Processor: DPs: a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) a#(b(a(x1))) -> b#(x1) a#(b(a(x1))) -> b#(b(x1)) a#(b(a(x1))) -> a#(b(b(x1))) b#(a(b(x1))) -> a#(x1) b#(a(b(x1))) -> a#(a(x1)) b#(a(b(x1))) -> b#(a(a(x1))) TRS: a(a(a(x1))) -> a(b(b(x1))) a(b(a(x1))) -> a(b(b(x1))) b(a(b(x1))) -> b(a(a(x1))) TDG Processor: DPs: a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) a#(b(a(x1))) -> b#(x1) a#(b(a(x1))) -> b#(b(x1)) a#(b(a(x1))) -> a#(b(b(x1))) b#(a(b(x1))) -> a#(x1) b#(a(b(x1))) -> a#(a(x1)) b#(a(b(x1))) -> b#(a(a(x1))) TRS: a(a(a(x1))) -> a(b(b(x1))) a(b(a(x1))) -> a(b(b(x1))) b(a(b(x1))) -> b(a(a(x1))) graph: b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(b(x1))) -> b#(a(a(x1))) b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(b(x1))) -> a#(a(x1)) b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(b(x1))) -> a#(x1) b#(a(b(x1))) -> a#(a(x1)) -> a#(b(a(x1))) -> a#(b(b(x1))) b#(a(b(x1))) -> a#(a(x1)) -> a#(b(a(x1))) -> b#(b(x1)) b#(a(b(x1))) -> a#(a(x1)) -> a#(b(a(x1))) -> b#(x1) b#(a(b(x1))) -> a#(a(x1)) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(a(b(x1))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(b(x1)) b#(a(b(x1))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(x1) b#(a(b(x1))) -> a#(x1) -> a#(b(a(x1))) -> a#(b(b(x1))) b#(a(b(x1))) -> a#(x1) -> a#(b(a(x1))) -> b#(b(x1)) b#(a(b(x1))) -> a#(x1) -> a#(b(a(x1))) -> b#(x1) b#(a(b(x1))) -> a#(x1) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(a(b(x1))) -> a#(x1) -> a#(a(a(x1))) -> b#(b(x1)) b#(a(b(x1))) -> a#(x1) -> a#(a(a(x1))) -> b#(x1) a#(b(a(x1))) -> b#(b(x1)) -> b#(a(b(x1))) -> b#(a(a(x1))) a#(b(a(x1))) -> b#(b(x1)) -> b#(a(b(x1))) -> a#(a(x1)) a#(b(a(x1))) -> b#(b(x1)) -> b#(a(b(x1))) -> a#(x1) a#(b(a(x1))) -> b#(x1) -> b#(a(b(x1))) -> b#(a(a(x1))) a#(b(a(x1))) -> b#(x1) -> b#(a(b(x1))) -> a#(a(x1)) a#(b(a(x1))) -> b#(x1) -> b#(a(b(x1))) -> a#(x1) a#(b(a(x1))) -> a#(b(b(x1))) -> a#(b(a(x1))) -> a#(b(b(x1))) a#(b(a(x1))) -> a#(b(b(x1))) -> a#(b(a(x1))) -> b#(b(x1)) a#(b(a(x1))) -> a#(b(b(x1))) -> a#(b(a(x1))) -> b#(x1) a#(b(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(b(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(b(x1)) a#(b(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) -> b#(a(b(x1))) -> b#(a(a(x1))) a#(a(a(x1))) -> b#(b(x1)) -> b#(a(b(x1))) -> a#(a(x1)) a#(a(a(x1))) -> b#(b(x1)) -> b#(a(b(x1))) -> a#(x1) a#(a(a(x1))) -> b#(x1) -> b#(a(b(x1))) -> b#(a(a(x1))) a#(a(a(x1))) -> b#(x1) -> b#(a(b(x1))) -> a#(a(x1)) a#(a(a(x1))) -> b#(x1) -> b#(a(b(x1))) -> a#(x1) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(b(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(b(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(b(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(x1) EDG Processor: DPs: a#(a(a(x1))) -> b#(x1) a#(a(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) a#(b(a(x1))) -> b#(x1) a#(b(a(x1))) -> b#(b(x1)) a#(b(a(x1))) -> a#(b(b(x1))) b#(a(b(x1))) -> a#(x1) b#(a(b(x1))) -> a#(a(x1)) b#(a(b(x1))) -> b#(a(a(x1))) TRS: a(a(a(x1))) -> a(b(b(x1))) a(b(a(x1))) -> a(b(b(x1))) b(a(b(x1))) -> b(a(a(x1))) graph: b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(b(x1))) -> a#(x1) b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(b(x1))) -> a#(a(x1)) b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(b(x1))) -> b#(a(a(x1))) b#(a(b(x1))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(x1) b#(a(b(x1))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(b(x1)) b#(a(b(x1))) -> a#(a(x1)) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(a(b(x1))) -> a#(x1) -> a#(a(a(x1))) -> b#(x1) b#(a(b(x1))) -> a#(x1) -> a#(a(a(x1))) -> b#(b(x1)) b#(a(b(x1))) -> a#(x1) -> a#(a(a(x1))) -> a#(b(b(x1))) b#(a(b(x1))) -> a#(x1) -> a#(b(a(x1))) -> b#(x1) b#(a(b(x1))) -> a#(x1) -> a#(b(a(x1))) -> b#(b(x1)) b#(a(b(x1))) -> a#(x1) -> a#(b(a(x1))) -> a#(b(b(x1))) a#(b(a(x1))) -> b#(x1) -> b#(a(b(x1))) -> a#(x1) a#(b(a(x1))) -> b#(x1) -> b#(a(b(x1))) -> a#(a(x1)) a#(b(a(x1))) -> b#(x1) -> b#(a(b(x1))) -> b#(a(a(x1))) a#(b(a(x1))) -> a#(b(b(x1))) -> a#(b(a(x1))) -> b#(x1) a#(b(a(x1))) -> a#(b(b(x1))) -> a#(b(a(x1))) -> b#(b(x1)) a#(b(a(x1))) -> a#(b(b(x1))) -> a#(b(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> b#(x1) -> b#(a(b(x1))) -> a#(x1) a#(a(a(x1))) -> b#(x1) -> b#(a(b(x1))) -> a#(a(x1)) a#(a(a(x1))) -> b#(x1) -> b#(a(b(x1))) -> b#(a(a(x1))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(b(a(x1))) -> b#(x1) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(b(a(x1))) -> b#(b(x1)) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(b(a(x1))) -> a#(b(b(x1))) SCC Processor: #sccs: 1 #rules: 7 #arcs: 24/81 DPs: b#(a(b(x1))) -> b#(a(a(x1))) b#(a(b(x1))) -> a#(a(x1)) a#(a(a(x1))) -> a#(b(b(x1))) a#(b(a(x1))) -> a#(b(b(x1))) a#(b(a(x1))) -> b#(x1) b#(a(b(x1))) -> a#(x1) a#(a(a(x1))) -> b#(x1) TRS: a(a(a(x1))) -> a(b(b(x1))) a(b(a(x1))) -> a(b(b(x1))) b(a(b(x1))) -> b(a(a(x1))) Arctic Interpretation Processor: dimension: 1 usable rules: a(a(a(x1))) -> a(b(b(x1))) a(b(a(x1))) -> a(b(b(x1))) b(a(b(x1))) -> b(a(a(x1))) interpretation: [b](x0) = 8x0 + 5, [b#](x0) = 1x0 + 0, [a](x0) = 8x0 + 5, [a#](x0) = x0 + 1 orientation: b#(a(b(x1))) = 17x1 + 14 >= 17x1 + 14 = b#(a(a(x1))) b#(a(b(x1))) = 17x1 + 14 >= 8x1 + 5 = a#(a(x1)) a#(a(a(x1))) = 16x1 + 13 >= 16x1 + 13 = a#(b(b(x1))) a#(b(a(x1))) = 16x1 + 13 >= 16x1 + 13 = a#(b(b(x1))) a#(b(a(x1))) = 16x1 + 13 >= 1x1 + 0 = b#(x1) b#(a(b(x1))) = 17x1 + 14 >= x1 + 1 = a#(x1) a#(a(a(x1))) = 16x1 + 13 >= 1x1 + 0 = b#(x1) a(a(a(x1))) = 24x1 + 21 >= 24x1 + 21 = a(b(b(x1))) a(b(a(x1))) = 24x1 + 21 >= 24x1 + 21 = a(b(b(x1))) b(a(b(x1))) = 24x1 + 21 >= 24x1 + 21 = b(a(a(x1))) problem: DPs: b#(a(b(x1))) -> b#(a(a(x1))) a#(a(a(x1))) -> a#(b(b(x1))) a#(b(a(x1))) -> a#(b(b(x1))) TRS: a(a(a(x1))) -> a(b(b(x1))) a(b(a(x1))) -> a(b(b(x1))) b(a(b(x1))) -> b(a(a(x1))) Restore Modifier: DPs: b#(a(b(x1))) -> b#(a(a(x1))) a#(a(a(x1))) -> a#(b(b(x1))) a#(b(a(x1))) -> a#(b(b(x1))) TRS: a(a(a(x1))) -> a(b(b(x1))) a(b(a(x1))) -> a(b(b(x1))) b(a(b(x1))) -> b(a(a(x1))) EDG Processor: DPs: b#(a(b(x1))) -> b#(a(a(x1))) a#(a(a(x1))) -> a#(b(b(x1))) a#(b(a(x1))) -> a#(b(b(x1))) TRS: a(a(a(x1))) -> a(b(b(x1))) a(b(a(x1))) -> a(b(b(x1))) b(a(b(x1))) -> b(a(a(x1))) graph: b#(a(b(x1))) -> b#(a(a(x1))) -> b#(a(b(x1))) -> b#(a(a(x1))) a#(b(a(x1))) -> a#(b(b(x1))) -> a#(b(a(x1))) -> a#(b(b(x1))) a#(a(a(x1))) -> a#(b(b(x1))) -> a#(b(a(x1))) -> a#(b(b(x1))) SCC Processor: #sccs: 2 #rules: 2 #arcs: 3/9 DPs: a#(b(a(x1))) -> a#(b(b(x1))) TRS: a(a(a(x1))) -> a(b(b(x1))) a(b(a(x1))) -> a(b(b(x1))) b(a(b(x1))) -> b(a(a(x1))) Bounds Processor: bound: 0 enrichment: match-dp automaton: final states: {4} transitions: a0(2) -> 1* a0(1) -> 1* a{#,0}(6) -> 4* b0(1) -> 2* b0(3) -> 5* b0(2) -> 2* b0(5) -> 6* 2 -> 3* 1 -> 3* problem: DPs: TRS: a(a(a(x1))) -> a(b(b(x1))) a(b(a(x1))) -> a(b(b(x1))) b(a(b(x1))) -> b(a(a(x1))) Qed DPs: b#(a(b(x1))) -> b#(a(a(x1))) TRS: a(a(a(x1))) -> a(b(b(x1))) a(b(a(x1))) -> a(b(b(x1))) b(a(b(x1))) -> b(a(a(x1))) Matrix Interpretation Processor: dim=3 interpretation: [0 0 0] [0] [b](x0) = [1 0 0]x0 + [1] [0 0 1] [0], [b#](x0) = [0 0 1]x0 + [1], [0 1 0] [1] [a](x0) = [1 0 0]x0 + [0] [0 1 0] [0] orientation: b#(a(b(x1))) = [1 0 0]x1 + [2] >= [1 0 0]x1 + [1] = b#(a(a(x1))) [0 1 0] [2] [2] a(a(a(x1))) = [1 0 0]x1 + [1] >= [0] = a(b(b(x1))) [0 1 0] [1] [1] [0 1 0] [3] [2] a(b(a(x1))) = [0 0 0]x1 + [0] >= [0] = a(b(b(x1))) [0 1 0] [2] [1] [0 0 0] [0] [0 0 0] [0] b(a(b(x1))) = [1 0 0]x1 + [3] >= [1 0 0]x1 + [2] = b(a(a(x1))) [1 0 0] [1] [1 0 0] [0] problem: DPs: TRS: a(a(a(x1))) -> a(b(b(x1))) a(b(a(x1))) -> a(b(b(x1))) b(a(b(x1))) -> b(a(a(x1))) Qed