YES Problem: a(a(x1)) -> c(b(a(b(a(x1))))) b(a(b(x1))) -> b(x1) a(a(a(x1))) -> c(c(a(x1))) c(c(x1)) -> a(b(c(b(a(x1))))) a(c(a(x1))) -> c(c(a(x1))) c(a(c(x1))) -> a(a(c(x1))) Proof: Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [b](x0) = [0 0 0]x0 [0 1 0] , [1 1 0] [0] [a](x0) = [0 1 1]x0 + [1] [0 0 0] [0], [1 1 0] [0] [c](x0) = [0 1 0]x0 + [1] [0 0 0] [0] orientation: [1 2 1] [1] [1 1 0] [0] a(a(x1)) = [0 1 1]x1 + [2] >= [0 0 0]x1 + [1] = c(b(a(b(a(x1))))) [0 0 0] [0] [0 0 0] [0] [1 0 0] [0] [1 0 0] b(a(b(x1))) = [0 0 0]x1 + [0] >= [0 0 0]x1 = b(x1) [0 1 0] [1] [0 1 0] [1 3 2] [3] [1 3 2] [3] a(a(a(x1))) = [0 1 1]x1 + [3] >= [0 1 1]x1 + [3] = c(c(a(x1))) [0 0 0] [0] [0 0 0] [0] [1 2 0] [1] [1 1 0] [0] c(c(x1)) = [0 1 0]x1 + [2] >= [0 0 0]x1 + [2] = a(b(c(b(a(x1))))) [0 0 0] [0] [0 0 0] [0] [1 3 2] [3] [1 3 2] [3] a(c(a(x1))) = [0 1 1]x1 + [3] >= [0 1 1]x1 + [3] = c(c(a(x1))) [0 0 0] [0] [0 0 0] [0] [1 3 0] [3] [1 3 0] [3] c(a(c(x1))) = [0 1 0]x1 + [3] >= [0 1 0]x1 + [3] = a(a(c(x1))) [0 0 0] [0] [0 0 0] [0] problem: b(a(b(x1))) -> b(x1) a(a(a(x1))) -> c(c(a(x1))) a(c(a(x1))) -> c(c(a(x1))) c(a(c(x1))) -> a(a(c(x1))) String Reversal Processor: b(a(b(x1))) -> b(x1) a(a(a(x1))) -> a(c(c(x1))) a(c(a(x1))) -> a(c(c(x1))) c(a(c(x1))) -> c(a(a(x1))) Matrix Interpretation Processor: dim=1 interpretation: [b](x0) = 4x0 + 2, [a](x0) = x0, [c](x0) = x0 orientation: b(a(b(x1))) = 16x1 + 10 >= 4x1 + 2 = b(x1) a(a(a(x1))) = x1 >= x1 = a(c(c(x1))) a(c(a(x1))) = x1 >= x1 = a(c(c(x1))) c(a(c(x1))) = x1 >= x1 = c(a(a(x1))) problem: a(a(a(x1))) -> a(c(c(x1))) a(c(a(x1))) -> a(c(c(x1))) c(a(c(x1))) -> c(a(a(x1))) DP Processor: DPs: a#(a(a(x1))) -> c#(x1) a#(a(a(x1))) -> c#(c(x1)) a#(a(a(x1))) -> a#(c(c(x1))) a#(c(a(x1))) -> c#(x1) a#(c(a(x1))) -> c#(c(x1)) a#(c(a(x1))) -> a#(c(c(x1))) c#(a(c(x1))) -> a#(x1) c#(a(c(x1))) -> a#(a(x1)) c#(a(c(x1))) -> c#(a(a(x1))) TRS: a(a(a(x1))) -> a(c(c(x1))) a(c(a(x1))) -> a(c(c(x1))) c(a(c(x1))) -> c(a(a(x1))) TDG Processor: DPs: a#(a(a(x1))) -> c#(x1) a#(a(a(x1))) -> c#(c(x1)) a#(a(a(x1))) -> a#(c(c(x1))) a#(c(a(x1))) -> c#(x1) a#(c(a(x1))) -> c#(c(x1)) a#(c(a(x1))) -> a#(c(c(x1))) c#(a(c(x1))) -> a#(x1) c#(a(c(x1))) -> a#(a(x1)) c#(a(c(x1))) -> c#(a(a(x1))) TRS: a(a(a(x1))) -> a(c(c(x1))) a(c(a(x1))) -> a(c(c(x1))) c(a(c(x1))) -> c(a(a(x1))) graph: c#(a(c(x1))) -> c#(a(a(x1))) -> c#(a(c(x1))) -> c#(a(a(x1))) c#(a(c(x1))) -> c#(a(a(x1))) -> c#(a(c(x1))) -> a#(a(x1)) c#(a(c(x1))) -> c#(a(a(x1))) -> c#(a(c(x1))) -> a#(x1) c#(a(c(x1))) -> a#(a(x1)) -> a#(c(a(x1))) -> a#(c(c(x1))) c#(a(c(x1))) -> a#(a(x1)) -> a#(c(a(x1))) -> c#(c(x1)) c#(a(c(x1))) -> a#(a(x1)) -> a#(c(a(x1))) -> c#(x1) c#(a(c(x1))) -> a#(a(x1)) -> a#(a(a(x1))) -> a#(c(c(x1))) c#(a(c(x1))) -> a#(a(x1)) -> a#(a(a(x1))) -> c#(c(x1)) c#(a(c(x1))) -> a#(a(x1)) -> a#(a(a(x1))) -> c#(x1) c#(a(c(x1))) -> a#(x1) -> a#(c(a(x1))) -> a#(c(c(x1))) c#(a(c(x1))) -> a#(x1) -> a#(c(a(x1))) -> c#(c(x1)) c#(a(c(x1))) -> a#(x1) -> a#(c(a(x1))) -> c#(x1) c#(a(c(x1))) -> a#(x1) -> a#(a(a(x1))) -> a#(c(c(x1))) c#(a(c(x1))) -> a#(x1) -> a#(a(a(x1))) -> c#(c(x1)) c#(a(c(x1))) -> a#(x1) -> a#(a(a(x1))) -> c#(x1) a#(c(a(x1))) -> c#(c(x1)) -> c#(a(c(x1))) -> c#(a(a(x1))) a#(c(a(x1))) -> c#(c(x1)) -> c#(a(c(x1))) -> a#(a(x1)) a#(c(a(x1))) -> c#(c(x1)) -> c#(a(c(x1))) -> a#(x1) a#(c(a(x1))) -> c#(x1) -> c#(a(c(x1))) -> c#(a(a(x1))) a#(c(a(x1))) -> c#(x1) -> c#(a(c(x1))) -> a#(a(x1)) a#(c(a(x1))) -> c#(x1) -> c#(a(c(x1))) -> a#(x1) a#(c(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> a#(c(c(x1))) a#(c(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> c#(c(x1)) a#(c(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> c#(x1) a#(c(a(x1))) -> a#(c(c(x1))) -> a#(a(a(x1))) -> a#(c(c(x1))) a#(c(a(x1))) -> a#(c(c(x1))) -> a#(a(a(x1))) -> c#(c(x1)) a#(c(a(x1))) -> a#(c(c(x1))) -> a#(a(a(x1))) -> c#(x1) a#(a(a(x1))) -> c#(c(x1)) -> c#(a(c(x1))) -> c#(a(a(x1))) a#(a(a(x1))) -> c#(c(x1)) -> c#(a(c(x1))) -> a#(a(x1)) a#(a(a(x1))) -> c#(c(x1)) -> c#(a(c(x1))) -> a#(x1) a#(a(a(x1))) -> c#(x1) -> c#(a(c(x1))) -> c#(a(a(x1))) a#(a(a(x1))) -> c#(x1) -> c#(a(c(x1))) -> a#(a(x1)) a#(a(a(x1))) -> c#(x1) -> c#(a(c(x1))) -> a#(x1) a#(a(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> a#(c(c(x1))) a#(a(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> c#(c(x1)) a#(a(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> c#(x1) a#(a(a(x1))) -> a#(c(c(x1))) -> a#(a(a(x1))) -> a#(c(c(x1))) a#(a(a(x1))) -> a#(c(c(x1))) -> a#(a(a(x1))) -> c#(c(x1)) a#(a(a(x1))) -> a#(c(c(x1))) -> a#(a(a(x1))) -> c#(x1) EDG Processor: DPs: a#(a(a(x1))) -> c#(x1) a#(a(a(x1))) -> c#(c(x1)) a#(a(a(x1))) -> a#(c(c(x1))) a#(c(a(x1))) -> c#(x1) a#(c(a(x1))) -> c#(c(x1)) a#(c(a(x1))) -> a#(c(c(x1))) c#(a(c(x1))) -> a#(x1) c#(a(c(x1))) -> a#(a(x1)) c#(a(c(x1))) -> c#(a(a(x1))) TRS: a(a(a(x1))) -> a(c(c(x1))) a(c(a(x1))) -> a(c(c(x1))) c(a(c(x1))) -> c(a(a(x1))) graph: c#(a(c(x1))) -> c#(a(a(x1))) -> c#(a(c(x1))) -> a#(x1) c#(a(c(x1))) -> c#(a(a(x1))) -> c#(a(c(x1))) -> a#(a(x1)) c#(a(c(x1))) -> c#(a(a(x1))) -> c#(a(c(x1))) -> c#(a(a(x1))) c#(a(c(x1))) -> a#(a(x1)) -> a#(a(a(x1))) -> c#(x1) c#(a(c(x1))) -> a#(a(x1)) -> a#(a(a(x1))) -> c#(c(x1)) c#(a(c(x1))) -> a#(a(x1)) -> a#(a(a(x1))) -> a#(c(c(x1))) c#(a(c(x1))) -> a#(x1) -> a#(a(a(x1))) -> c#(x1) c#(a(c(x1))) -> a#(x1) -> a#(a(a(x1))) -> c#(c(x1)) c#(a(c(x1))) -> a#(x1) -> a#(a(a(x1))) -> a#(c(c(x1))) c#(a(c(x1))) -> a#(x1) -> a#(c(a(x1))) -> c#(x1) c#(a(c(x1))) -> a#(x1) -> a#(c(a(x1))) -> c#(c(x1)) c#(a(c(x1))) -> a#(x1) -> a#(c(a(x1))) -> a#(c(c(x1))) a#(c(a(x1))) -> c#(x1) -> c#(a(c(x1))) -> a#(x1) a#(c(a(x1))) -> c#(x1) -> c#(a(c(x1))) -> a#(a(x1)) a#(c(a(x1))) -> c#(x1) -> c#(a(c(x1))) -> c#(a(a(x1))) a#(c(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> c#(x1) a#(c(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> c#(c(x1)) a#(c(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> a#(c(c(x1))) a#(a(a(x1))) -> c#(x1) -> c#(a(c(x1))) -> a#(x1) a#(a(a(x1))) -> c#(x1) -> c#(a(c(x1))) -> a#(a(x1)) a#(a(a(x1))) -> c#(x1) -> c#(a(c(x1))) -> c#(a(a(x1))) a#(a(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> c#(x1) a#(a(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> c#(c(x1)) a#(a(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> a#(c(c(x1))) SCC Processor: #sccs: 1 #rules: 7 #arcs: 24/81 DPs: c#(a(c(x1))) -> c#(a(a(x1))) c#(a(c(x1))) -> a#(a(x1)) a#(a(a(x1))) -> a#(c(c(x1))) a#(c(a(x1))) -> a#(c(c(x1))) a#(c(a(x1))) -> c#(x1) c#(a(c(x1))) -> a#(x1) a#(a(a(x1))) -> c#(x1) TRS: a(a(a(x1))) -> a(c(c(x1))) a(c(a(x1))) -> a(c(c(x1))) c(a(c(x1))) -> c(a(a(x1))) Arctic Interpretation Processor: dimension: 1 usable rules: a(a(a(x1))) -> a(c(c(x1))) a(c(a(x1))) -> a(c(c(x1))) c(a(c(x1))) -> c(a(a(x1))) interpretation: [c#](x0) = 1x0 + 0, [a#](x0) = x0 + 1, [a](x0) = 8x0 + 5, [c](x0) = 8x0 + 5 orientation: c#(a(c(x1))) = 17x1 + 14 >= 17x1 + 14 = c#(a(a(x1))) c#(a(c(x1))) = 17x1 + 14 >= 8x1 + 5 = a#(a(x1)) a#(a(a(x1))) = 16x1 + 13 >= 16x1 + 13 = a#(c(c(x1))) a#(c(a(x1))) = 16x1 + 13 >= 16x1 + 13 = a#(c(c(x1))) a#(c(a(x1))) = 16x1 + 13 >= 1x1 + 0 = c#(x1) c#(a(c(x1))) = 17x1 + 14 >= x1 + 1 = a#(x1) a#(a(a(x1))) = 16x1 + 13 >= 1x1 + 0 = c#(x1) a(a(a(x1))) = 24x1 + 21 >= 24x1 + 21 = a(c(c(x1))) a(c(a(x1))) = 24x1 + 21 >= 24x1 + 21 = a(c(c(x1))) c(a(c(x1))) = 24x1 + 21 >= 24x1 + 21 = c(a(a(x1))) problem: DPs: c#(a(c(x1))) -> c#(a(a(x1))) a#(a(a(x1))) -> a#(c(c(x1))) a#(c(a(x1))) -> a#(c(c(x1))) TRS: a(a(a(x1))) -> a(c(c(x1))) a(c(a(x1))) -> a(c(c(x1))) c(a(c(x1))) -> c(a(a(x1))) Restore Modifier: DPs: c#(a(c(x1))) -> c#(a(a(x1))) a#(a(a(x1))) -> a#(c(c(x1))) a#(c(a(x1))) -> a#(c(c(x1))) TRS: a(a(a(x1))) -> a(c(c(x1))) a(c(a(x1))) -> a(c(c(x1))) c(a(c(x1))) -> c(a(a(x1))) EDG Processor: DPs: c#(a(c(x1))) -> c#(a(a(x1))) a#(a(a(x1))) -> a#(c(c(x1))) a#(c(a(x1))) -> a#(c(c(x1))) TRS: a(a(a(x1))) -> a(c(c(x1))) a(c(a(x1))) -> a(c(c(x1))) c(a(c(x1))) -> c(a(a(x1))) graph: c#(a(c(x1))) -> c#(a(a(x1))) -> c#(a(c(x1))) -> c#(a(a(x1))) a#(c(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> a#(c(c(x1))) a#(a(a(x1))) -> a#(c(c(x1))) -> a#(c(a(x1))) -> a#(c(c(x1))) SCC Processor: #sccs: 2 #rules: 2 #arcs: 3/9 DPs: a#(c(a(x1))) -> a#(c(c(x1))) TRS: a(a(a(x1))) -> a(c(c(x1))) a(c(a(x1))) -> a(c(c(x1))) c(a(c(x1))) -> c(a(a(x1))) Bounds Processor: bound: 0 enrichment: match-dp automaton: final states: {4} transitions: c0(3) -> 5* c0(2) -> 2* c0(5) -> 6* c0(1) -> 2* a{#,0}(6) -> 4* a0(1) -> 1* a0(2) -> 1* 2 -> 3* 1 -> 3* problem: DPs: TRS: a(a(a(x1))) -> a(c(c(x1))) a(c(a(x1))) -> a(c(c(x1))) c(a(c(x1))) -> c(a(a(x1))) Qed DPs: c#(a(c(x1))) -> c#(a(a(x1))) TRS: a(a(a(x1))) -> a(c(c(x1))) a(c(a(x1))) -> a(c(c(x1))) c(a(c(x1))) -> c(a(a(x1))) Matrix Interpretation Processor: dim=3 interpretation: [c#](x0) = [0 0 1]x0 + [1], [0 1 0] [1] [a](x0) = [1 0 0]x0 + [0] [0 1 0] [0], [0 0 0] [0] [c](x0) = [1 0 0]x0 + [1] [0 0 1] [0] orientation: c#(a(c(x1))) = [1 0 0]x1 + [2] >= [1 0 0]x1 + [1] = c#(a(a(x1))) [0 1 0] [2] [2] a(a(a(x1))) = [1 0 0]x1 + [1] >= [0] = a(c(c(x1))) [0 1 0] [1] [1] [0 1 0] [3] [2] a(c(a(x1))) = [0 0 0]x1 + [0] >= [0] = a(c(c(x1))) [0 1 0] [2] [1] [0 0 0] [0] [0 0 0] [0] c(a(c(x1))) = [1 0 0]x1 + [3] >= [1 0 0]x1 + [2] = c(a(a(x1))) [1 0 0] [1] [1 0 0] [0] problem: DPs: TRS: a(a(a(x1))) -> a(c(c(x1))) a(c(a(x1))) -> a(c(c(x1))) c(a(c(x1))) -> c(a(a(x1))) Qed