YES Problem: r1(a(x1)) -> a(a(a(r1(x1)))) r2(a(x1)) -> a(a(a(r2(x1)))) a(l1(x1)) -> l1(a(a(a(x1)))) a(a(l2(x1))) -> l2(a(a(x1))) r1(b(x1)) -> l1(b(x1)) r2(b(x1)) -> l2(a(b(x1))) b(l1(x1)) -> b(r2(x1)) b(l2(x1)) -> b(r1(x1)) a(a(x1)) -> x1 Proof: DP Processor: DPs: r1#(a(x1)) -> r1#(x1) r1#(a(x1)) -> a#(r1(x1)) r1#(a(x1)) -> a#(a(r1(x1))) r1#(a(x1)) -> a#(a(a(r1(x1)))) r2#(a(x1)) -> r2#(x1) r2#(a(x1)) -> a#(r2(x1)) r2#(a(x1)) -> a#(a(r2(x1))) r2#(a(x1)) -> a#(a(a(r2(x1)))) a#(l1(x1)) -> a#(x1) a#(l1(x1)) -> a#(a(x1)) a#(l1(x1)) -> a#(a(a(x1))) a#(a(l2(x1))) -> a#(x1) a#(a(l2(x1))) -> a#(a(x1)) r2#(b(x1)) -> a#(b(x1)) b#(l1(x1)) -> r2#(x1) b#(l1(x1)) -> b#(r2(x1)) b#(l2(x1)) -> r1#(x1) b#(l2(x1)) -> b#(r1(x1)) TRS: r1(a(x1)) -> a(a(a(r1(x1)))) r2(a(x1)) -> a(a(a(r2(x1)))) a(l1(x1)) -> l1(a(a(a(x1)))) a(a(l2(x1))) -> l2(a(a(x1))) r1(b(x1)) -> l1(b(x1)) r2(b(x1)) -> l2(a(b(x1))) b(l1(x1)) -> b(r2(x1)) b(l2(x1)) -> b(r1(x1)) a(a(x1)) -> x1 TDG Processor: DPs: r1#(a(x1)) -> r1#(x1) r1#(a(x1)) -> a#(r1(x1)) r1#(a(x1)) -> a#(a(r1(x1))) r1#(a(x1)) -> a#(a(a(r1(x1)))) r2#(a(x1)) -> r2#(x1) r2#(a(x1)) -> a#(r2(x1)) r2#(a(x1)) -> a#(a(r2(x1))) r2#(a(x1)) -> a#(a(a(r2(x1)))) a#(l1(x1)) -> a#(x1) a#(l1(x1)) -> a#(a(x1)) a#(l1(x1)) -> a#(a(a(x1))) a#(a(l2(x1))) -> a#(x1) a#(a(l2(x1))) -> a#(a(x1)) r2#(b(x1)) -> a#(b(x1)) b#(l1(x1)) -> r2#(x1) b#(l1(x1)) -> b#(r2(x1)) b#(l2(x1)) -> r1#(x1) b#(l2(x1)) -> b#(r1(x1)) TRS: r1(a(x1)) -> a(a(a(r1(x1)))) r2(a(x1)) -> a(a(a(r2(x1)))) a(l1(x1)) -> l1(a(a(a(x1)))) a(a(l2(x1))) -> l2(a(a(x1))) r1(b(x1)) -> l1(b(x1)) r2(b(x1)) -> l2(a(b(x1))) b(l1(x1)) -> b(r2(x1)) b(l2(x1)) -> b(r1(x1)) a(a(x1)) -> x1 graph: b#(l2(x1)) -> b#(r1(x1)) -> b#(l2(x1)) -> b#(r1(x1)) b#(l2(x1)) -> b#(r1(x1)) -> b#(l2(x1)) -> r1#(x1) b#(l2(x1)) -> b#(r1(x1)) -> b#(l1(x1)) -> b#(r2(x1)) b#(l2(x1)) -> b#(r1(x1)) -> b#(l1(x1)) -> r2#(x1) b#(l2(x1)) -> r1#(x1) -> r1#(a(x1)) -> a#(a(a(r1(x1)))) b#(l2(x1)) -> r1#(x1) -> r1#(a(x1)) -> a#(a(r1(x1))) b#(l2(x1)) -> r1#(x1) -> r1#(a(x1)) -> a#(r1(x1)) b#(l2(x1)) -> r1#(x1) -> r1#(a(x1)) -> r1#(x1) b#(l1(x1)) -> b#(r2(x1)) -> b#(l2(x1)) -> b#(r1(x1)) b#(l1(x1)) -> b#(r2(x1)) -> b#(l2(x1)) -> r1#(x1) b#(l1(x1)) -> b#(r2(x1)) -> b#(l1(x1)) -> b#(r2(x1)) b#(l1(x1)) -> b#(r2(x1)) -> b#(l1(x1)) -> r2#(x1) b#(l1(x1)) -> r2#(x1) -> r2#(b(x1)) -> a#(b(x1)) b#(l1(x1)) -> r2#(x1) -> r2#(a(x1)) -> a#(a(a(r2(x1)))) b#(l1(x1)) -> r2#(x1) -> r2#(a(x1)) -> a#(a(r2(x1))) b#(l1(x1)) -> r2#(x1) -> r2#(a(x1)) -> a#(r2(x1)) b#(l1(x1)) -> r2#(x1) -> r2#(a(x1)) -> r2#(x1) r2#(b(x1)) -> a#(b(x1)) -> a#(a(l2(x1))) -> a#(a(x1)) r2#(b(x1)) -> a#(b(x1)) -> a#(a(l2(x1))) -> a#(x1) r2#(b(x1)) -> a#(b(x1)) -> a#(l1(x1)) -> a#(a(a(x1))) r2#(b(x1)) -> a#(b(x1)) -> a#(l1(x1)) -> a#(a(x1)) r2#(b(x1)) -> a#(b(x1)) -> a#(l1(x1)) -> a#(x1) r2#(a(x1)) -> r2#(x1) -> r2#(b(x1)) -> a#(b(x1)) r2#(a(x1)) -> r2#(x1) -> r2#(a(x1)) -> a#(a(a(r2(x1)))) r2#(a(x1)) -> r2#(x1) -> r2#(a(x1)) -> a#(a(r2(x1))) r2#(a(x1)) -> r2#(x1) -> r2#(a(x1)) -> a#(r2(x1)) r2#(a(x1)) -> r2#(x1) -> r2#(a(x1)) -> r2#(x1) r2#(a(x1)) -> a#(r2(x1)) -> a#(a(l2(x1))) -> a#(a(x1)) r2#(a(x1)) -> a#(r2(x1)) -> a#(a(l2(x1))) -> a#(x1) r2#(a(x1)) -> a#(r2(x1)) -> a#(l1(x1)) -> a#(a(a(x1))) r2#(a(x1)) -> a#(r2(x1)) -> a#(l1(x1)) -> a#(a(x1)) r2#(a(x1)) -> a#(r2(x1)) -> a#(l1(x1)) -> a#(x1) r2#(a(x1)) -> a#(a(r2(x1))) -> a#(a(l2(x1))) -> a#(a(x1)) r2#(a(x1)) -> a#(a(r2(x1))) -> a#(a(l2(x1))) -> a#(x1) r2#(a(x1)) -> a#(a(r2(x1))) -> a#(l1(x1)) -> a#(a(a(x1))) r2#(a(x1)) -> a#(a(r2(x1))) -> a#(l1(x1)) -> a#(a(x1)) r2#(a(x1)) -> a#(a(r2(x1))) -> a#(l1(x1)) -> a#(x1) r2#(a(x1)) -> a#(a(a(r2(x1)))) -> a#(a(l2(x1))) -> a#(a(x1)) r2#(a(x1)) -> a#(a(a(r2(x1)))) -> a#(a(l2(x1))) -> a#(x1) r2#(a(x1)) -> a#(a(a(r2(x1)))) -> a#(l1(x1)) -> a#(a(a(x1))) r2#(a(x1)) -> a#(a(a(r2(x1)))) -> a#(l1(x1)) -> a#(a(x1)) r2#(a(x1)) -> a#(a(a(r2(x1)))) -> a#(l1(x1)) -> a#(x1) a#(l1(x1)) -> a#(a(a(x1))) -> a#(a(l2(x1))) -> a#(a(x1)) a#(l1(x1)) -> a#(a(a(x1))) -> a#(a(l2(x1))) -> a#(x1) a#(l1(x1)) -> a#(a(a(x1))) -> a#(l1(x1)) -> a#(a(a(x1))) a#(l1(x1)) -> a#(a(a(x1))) -> a#(l1(x1)) -> a#(a(x1)) a#(l1(x1)) -> a#(a(a(x1))) -> a#(l1(x1)) -> a#(x1) a#(l1(x1)) -> a#(a(x1)) -> a#(a(l2(x1))) -> a#(a(x1)) a#(l1(x1)) -> a#(a(x1)) -> a#(a(l2(x1))) -> a#(x1) a#(l1(x1)) -> a#(a(x1)) -> a#(l1(x1)) -> a#(a(a(x1))) a#(l1(x1)) -> a#(a(x1)) -> a#(l1(x1)) -> a#(a(x1)) a#(l1(x1)) -> a#(a(x1)) -> a#(l1(x1)) -> a#(x1) a#(l1(x1)) -> a#(x1) -> a#(a(l2(x1))) -> a#(a(x1)) a#(l1(x1)) -> a#(x1) -> a#(a(l2(x1))) -> a#(x1) a#(l1(x1)) -> a#(x1) -> a#(l1(x1)) -> a#(a(a(x1))) a#(l1(x1)) -> a#(x1) -> a#(l1(x1)) -> a#(a(x1)) a#(l1(x1)) -> a#(x1) -> a#(l1(x1)) -> a#(x1) a#(a(l2(x1))) -> a#(a(x1)) -> a#(a(l2(x1))) -> a#(a(x1)) a#(a(l2(x1))) -> a#(a(x1)) -> a#(a(l2(x1))) -> a#(x1) a#(a(l2(x1))) -> a#(a(x1)) -> a#(l1(x1)) -> a#(a(a(x1))) a#(a(l2(x1))) -> a#(a(x1)) -> a#(l1(x1)) -> a#(a(x1)) a#(a(l2(x1))) -> a#(a(x1)) -> a#(l1(x1)) -> a#(x1) a#(a(l2(x1))) -> a#(x1) -> a#(a(l2(x1))) -> a#(a(x1)) a#(a(l2(x1))) -> a#(x1) -> a#(a(l2(x1))) -> a#(x1) a#(a(l2(x1))) -> a#(x1) -> a#(l1(x1)) -> a#(a(a(x1))) a#(a(l2(x1))) -> a#(x1) -> a#(l1(x1)) -> a#(a(x1)) a#(a(l2(x1))) -> a#(x1) -> a#(l1(x1)) -> a#(x1) r1#(a(x1)) -> a#(r1(x1)) -> a#(a(l2(x1))) -> a#(a(x1)) r1#(a(x1)) -> a#(r1(x1)) -> a#(a(l2(x1))) -> a#(x1) r1#(a(x1)) -> a#(r1(x1)) -> a#(l1(x1)) -> a#(a(a(x1))) r1#(a(x1)) -> a#(r1(x1)) -> a#(l1(x1)) -> a#(a(x1)) r1#(a(x1)) -> a#(r1(x1)) -> a#(l1(x1)) -> a#(x1) r1#(a(x1)) -> a#(a(r1(x1))) -> a#(a(l2(x1))) -> a#(a(x1)) r1#(a(x1)) -> a#(a(r1(x1))) -> a#(a(l2(x1))) -> a#(x1) r1#(a(x1)) -> a#(a(r1(x1))) -> a#(l1(x1)) -> a#(a(a(x1))) r1#(a(x1)) -> a#(a(r1(x1))) -> a#(l1(x1)) -> a#(a(x1)) r1#(a(x1)) -> a#(a(r1(x1))) -> a#(l1(x1)) -> a#(x1) r1#(a(x1)) -> a#(a(a(r1(x1)))) -> a#(a(l2(x1))) -> a#(a(x1)) r1#(a(x1)) -> a#(a(a(r1(x1)))) -> a#(a(l2(x1))) -> a#(x1) r1#(a(x1)) -> a#(a(a(r1(x1)))) -> a#(l1(x1)) -> a#(a(a(x1))) r1#(a(x1)) -> a#(a(a(r1(x1)))) -> a#(l1(x1)) -> a#(a(x1)) r1#(a(x1)) -> a#(a(a(r1(x1)))) -> a#(l1(x1)) -> a#(x1) r1#(a(x1)) -> r1#(x1) -> r1#(a(x1)) -> a#(a(a(r1(x1)))) r1#(a(x1)) -> r1#(x1) -> r1#(a(x1)) -> a#(a(r1(x1))) r1#(a(x1)) -> r1#(x1) -> r1#(a(x1)) -> a#(r1(x1)) r1#(a(x1)) -> r1#(x1) -> r1#(a(x1)) -> r1#(x1) SCC Processor: #sccs: 4 #rules: 9 #arcs: 86/324 DPs: b#(l2(x1)) -> b#(r1(x1)) b#(l1(x1)) -> b#(r2(x1)) TRS: r1(a(x1)) -> a(a(a(r1(x1)))) r2(a(x1)) -> a(a(a(r2(x1)))) a(l1(x1)) -> l1(a(a(a(x1)))) a(a(l2(x1))) -> l2(a(a(x1))) r1(b(x1)) -> l1(b(x1)) r2(b(x1)) -> l2(a(b(x1))) b(l1(x1)) -> b(r2(x1)) b(l2(x1)) -> b(r1(x1)) a(a(x1)) -> x1 Matrix Interpretation Processor: dim=3 interpretation: [b#](x0) = [0 1 0]x0, [0 0 0] [0] [b](x0) = [0 0 0]x0 + [1] [0 1 0] [0], [0 0 0] [0] [l2](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 0] [l1](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [r2](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [r1](x0) = [0 1 0]x0 [0 0 0] , [0 1 0] [a](x0) = [1 0 0]x0 [0 0 1] orientation: b#(l2(x1)) = [0 1 0]x1 + [1] >= [0 1 0]x1 = b#(r1(x1)) b#(l1(x1)) = [0 1 0]x1 >= [0 1 0]x1 = b#(r2(x1)) [0 1 0] [0 1 0] r1(a(x1)) = [1 0 0]x1 >= [1 0 0]x1 = a(a(a(r1(x1)))) [0 0 0] [0 0 0] [0 1 0] [0 1 0] r2(a(x1)) = [1 0 0]x1 >= [1 0 0]x1 = a(a(a(r2(x1)))) [0 0 0] [0 0 0] [0 1 0] [0 1 0] a(l1(x1)) = [1 0 0]x1 >= [1 0 0]x1 = l1(a(a(a(x1)))) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0] a(a(l2(x1))) = [0 1 0]x1 + [1] >= [0 1 0]x1 + [1] = l2(a(a(x1))) [0 0 0] [0] [0 0 0] [0] [0] [0] r1(b(x1)) = [1] >= [1] = l1(b(x1)) [0] [0] [0] [0] r2(b(x1)) = [1] >= [1] = l2(a(b(x1))) [0] [0] [0 0 0] [0] [0 0 0] [0] b(l1(x1)) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = b(r2(x1)) [0 1 0] [0] [0 1 0] [0] [0 0 0] [0] [0 0 0] [0] b(l2(x1)) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = b(r1(x1)) [0 1 0] [1] [0 1 0] [0] a(a(x1)) = x1 >= x1 = x1 problem: DPs: b#(l1(x1)) -> b#(r2(x1)) TRS: r1(a(x1)) -> a(a(a(r1(x1)))) r2(a(x1)) -> a(a(a(r2(x1)))) a(l1(x1)) -> l1(a(a(a(x1)))) a(a(l2(x1))) -> l2(a(a(x1))) r1(b(x1)) -> l1(b(x1)) r2(b(x1)) -> l2(a(b(x1))) b(l1(x1)) -> b(r2(x1)) b(l2(x1)) -> b(r1(x1)) a(a(x1)) -> x1 EDG Processor: DPs: b#(l1(x1)) -> b#(r2(x1)) TRS: r1(a(x1)) -> a(a(a(r1(x1)))) r2(a(x1)) -> a(a(a(r2(x1)))) a(l1(x1)) -> l1(a(a(a(x1)))) a(a(l2(x1))) -> l2(a(a(x1))) r1(b(x1)) -> l1(b(x1)) r2(b(x1)) -> l2(a(b(x1))) b(l1(x1)) -> b(r2(x1)) b(l2(x1)) -> b(r1(x1)) a(a(x1)) -> x1 graph: b#(l1(x1)) -> b#(r2(x1)) -> b#(l1(x1)) -> b#(r2(x1)) CDG Processor: DPs: b#(l1(x1)) -> b#(r2(x1)) TRS: r1(a(x1)) -> a(a(a(r1(x1)))) r2(a(x1)) -> a(a(a(r2(x1)))) a(l1(x1)) -> l1(a(a(a(x1)))) a(a(l2(x1))) -> l2(a(a(x1))) r1(b(x1)) -> l1(b(x1)) r2(b(x1)) -> l2(a(b(x1))) b(l1(x1)) -> b(r2(x1)) b(l2(x1)) -> b(r1(x1)) a(a(x1)) -> x1 graph: Qed DPs: r1#(a(x1)) -> r1#(x1) TRS: r1(a(x1)) -> a(a(a(r1(x1)))) r2(a(x1)) -> a(a(a(r2(x1)))) a(l1(x1)) -> l1(a(a(a(x1)))) a(a(l2(x1))) -> l2(a(a(x1))) r1(b(x1)) -> l1(b(x1)) r2(b(x1)) -> l2(a(b(x1))) b(l1(x1)) -> b(r2(x1)) b(l2(x1)) -> b(r1(x1)) a(a(x1)) -> x1 LPO Processor: argument filtering: pi(a) = [0] pi(r1) = [0] pi(r2) = [0] pi(l1) = [] pi(l2) = [] pi(b) = [] pi(r1#) = 0 precedence: r1 > b > r2 > r1# ~ l2 ~ l1 ~ a problem: DPs: TRS: r1(a(x1)) -> a(a(a(r1(x1)))) r2(a(x1)) -> a(a(a(r2(x1)))) a(l1(x1)) -> l1(a(a(a(x1)))) a(a(l2(x1))) -> l2(a(a(x1))) r1(b(x1)) -> l1(b(x1)) r2(b(x1)) -> l2(a(b(x1))) b(l1(x1)) -> b(r2(x1)) b(l2(x1)) -> b(r1(x1)) a(a(x1)) -> x1 Qed DPs: r2#(a(x1)) -> r2#(x1) TRS: r1(a(x1)) -> a(a(a(r1(x1)))) r2(a(x1)) -> a(a(a(r2(x1)))) a(l1(x1)) -> l1(a(a(a(x1)))) a(a(l2(x1))) -> l2(a(a(x1))) r1(b(x1)) -> l1(b(x1)) r2(b(x1)) -> l2(a(b(x1))) b(l1(x1)) -> b(r2(x1)) b(l2(x1)) -> b(r1(x1)) a(a(x1)) -> x1 LPO Processor: argument filtering: pi(a) = [0] pi(r1) = [0] pi(r2) = [0] pi(l1) = [] pi(l2) = [] pi(b) = [] pi(r2#) = 0 precedence: r1 > b > r2 > r2# ~ l2 ~ l1 ~ a problem: DPs: TRS: r1(a(x1)) -> a(a(a(r1(x1)))) r2(a(x1)) -> a(a(a(r2(x1)))) a(l1(x1)) -> l1(a(a(a(x1)))) a(a(l2(x1))) -> l2(a(a(x1))) r1(b(x1)) -> l1(b(x1)) r2(b(x1)) -> l2(a(b(x1))) b(l1(x1)) -> b(r2(x1)) b(l2(x1)) -> b(r1(x1)) a(a(x1)) -> x1 Qed DPs: a#(l1(x1)) -> a#(x1) a#(l1(x1)) -> a#(a(x1)) a#(l1(x1)) -> a#(a(a(x1))) a#(a(l2(x1))) -> a#(x1) a#(a(l2(x1))) -> a#(a(x1)) TRS: r1(a(x1)) -> a(a(a(r1(x1)))) r2(a(x1)) -> a(a(a(r2(x1)))) a(l1(x1)) -> l1(a(a(a(x1)))) a(a(l2(x1))) -> l2(a(a(x1))) r1(b(x1)) -> l1(b(x1)) r2(b(x1)) -> l2(a(b(x1))) b(l1(x1)) -> b(r2(x1)) b(l2(x1)) -> b(r1(x1)) a(a(x1)) -> x1 KBO Processor: argument filtering: pi(a) = 0 pi(r1) = [0] pi(r2) = 0 pi(l1) = [0] pi(l2) = 0 pi(b) = [] pi(a#) = 0 weight function: w0 = 1 w(a#) = w(b) = w(l2) = w(r2) = w(r1) = w(a) = 1 w(l1) = 0 precedence: a# ~ b ~ l2 ~ l1 ~ r2 ~ r1 ~ a problem: DPs: a#(a(l2(x1))) -> a#(x1) a#(a(l2(x1))) -> a#(a(x1)) TRS: r1(a(x1)) -> a(a(a(r1(x1)))) r2(a(x1)) -> a(a(a(r2(x1)))) a(l1(x1)) -> l1(a(a(a(x1)))) a(a(l2(x1))) -> l2(a(a(x1))) r1(b(x1)) -> l1(b(x1)) r2(b(x1)) -> l2(a(b(x1))) b(l1(x1)) -> b(r2(x1)) b(l2(x1)) -> b(r1(x1)) a(a(x1)) -> x1 EDG Processor: DPs: a#(a(l2(x1))) -> a#(x1) a#(a(l2(x1))) -> a#(a(x1)) TRS: r1(a(x1)) -> a(a(a(r1(x1)))) r2(a(x1)) -> a(a(a(r2(x1)))) a(l1(x1)) -> l1(a(a(a(x1)))) a(a(l2(x1))) -> l2(a(a(x1))) r1(b(x1)) -> l1(b(x1)) r2(b(x1)) -> l2(a(b(x1))) b(l1(x1)) -> b(r2(x1)) b(l2(x1)) -> b(r1(x1)) a(a(x1)) -> x1 graph: a#(a(l2(x1))) -> a#(a(x1)) -> a#(a(l2(x1))) -> a#(x1) a#(a(l2(x1))) -> a#(a(x1)) -> a#(a(l2(x1))) -> a#(a(x1)) a#(a(l2(x1))) -> a#(x1) -> a#(a(l2(x1))) -> a#(x1) a#(a(l2(x1))) -> a#(x1) -> a#(a(l2(x1))) -> a#(a(x1)) KBO Processor: argument filtering: pi(a) = 0 pi(r1) = 0 pi(r2) = [0] pi(l1) = 0 pi(l2) = [0] pi(b) = [] pi(a#) = 0 weight function: w0 = 1 w(a#) = w(b) = w(l2) = w(r2) = 1 w(l1) = w(r1) = w(a) = 0 precedence: a# ~ r2 > b ~ l2 ~ l1 ~ r1 ~ a problem: DPs: TRS: r1(a(x1)) -> a(a(a(r1(x1)))) r2(a(x1)) -> a(a(a(r2(x1)))) a(l1(x1)) -> l1(a(a(a(x1)))) a(a(l2(x1))) -> l2(a(a(x1))) r1(b(x1)) -> l1(b(x1)) r2(b(x1)) -> l2(a(b(x1))) b(l1(x1)) -> b(r2(x1)) b(l2(x1)) -> b(r1(x1)) a(a(x1)) -> x1 Qed