YES Problem: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) a(d(x1)) -> d(a(x1)) d(x1) -> b(a(x1)) L(a(a(x1))) -> L(a(b(c(x1)))) c(R(x1)) -> c(b(R(x1))) Proof: DP Processor: DPs: b#(a(a(x1))) -> c#(x1) b#(a(a(x1))) -> b#(c(x1)) b#(a(a(x1))) -> a#(b(c(x1))) c#(a(x1)) -> c#(x1) c#(a(x1)) -> a#(c(x1)) b#(c(a(x1))) -> c#(x1) b#(c(a(x1))) -> b#(c(x1)) b#(c(a(x1))) -> a#(b(c(x1))) c#(b(x1)) -> d#(x1) a#(d(x1)) -> a#(x1) a#(d(x1)) -> d#(a(x1)) d#(x1) -> a#(x1) d#(x1) -> b#(a(x1)) L#(a(a(x1))) -> c#(x1) L#(a(a(x1))) -> b#(c(x1)) L#(a(a(x1))) -> a#(b(c(x1))) L#(a(a(x1))) -> L#(a(b(c(x1)))) c#(R(x1)) -> b#(R(x1)) c#(R(x1)) -> c#(b(R(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) a(d(x1)) -> d(a(x1)) d(x1) -> b(a(x1)) L(a(a(x1))) -> L(a(b(c(x1)))) c(R(x1)) -> c(b(R(x1))) TDG Processor: DPs: b#(a(a(x1))) -> c#(x1) b#(a(a(x1))) -> b#(c(x1)) b#(a(a(x1))) -> a#(b(c(x1))) c#(a(x1)) -> c#(x1) c#(a(x1)) -> a#(c(x1)) b#(c(a(x1))) -> c#(x1) b#(c(a(x1))) -> b#(c(x1)) b#(c(a(x1))) -> a#(b(c(x1))) c#(b(x1)) -> d#(x1) a#(d(x1)) -> a#(x1) a#(d(x1)) -> d#(a(x1)) d#(x1) -> a#(x1) d#(x1) -> b#(a(x1)) L#(a(a(x1))) -> c#(x1) L#(a(a(x1))) -> b#(c(x1)) L#(a(a(x1))) -> a#(b(c(x1))) L#(a(a(x1))) -> L#(a(b(c(x1)))) c#(R(x1)) -> b#(R(x1)) c#(R(x1)) -> c#(b(R(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) a(d(x1)) -> d(a(x1)) d(x1) -> b(a(x1)) L(a(a(x1))) -> L(a(b(c(x1)))) c(R(x1)) -> c(b(R(x1))) graph: L#(a(a(x1))) -> L#(a(b(c(x1)))) -> L#(a(a(x1))) -> L#(a(b(c(x1)))) L#(a(a(x1))) -> L#(a(b(c(x1)))) -> L#(a(a(x1))) -> a#(b(c(x1))) L#(a(a(x1))) -> L#(a(b(c(x1)))) -> L#(a(a(x1))) -> b#(c(x1)) L#(a(a(x1))) -> L#(a(b(c(x1)))) -> L#(a(a(x1))) -> c#(x1) L#(a(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> d#(a(x1)) L#(a(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> a#(x1) L#(a(a(x1))) -> c#(x1) -> c#(R(x1)) -> c#(b(R(x1))) L#(a(a(x1))) -> c#(x1) -> c#(R(x1)) -> b#(R(x1)) L#(a(a(x1))) -> c#(x1) -> c#(b(x1)) -> d#(x1) L#(a(a(x1))) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) L#(a(a(x1))) -> c#(x1) -> c#(a(x1)) -> c#(x1) L#(a(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> a#(b(c(x1))) L#(a(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> b#(c(x1)) L#(a(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> c#(x1) L#(a(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> a#(b(c(x1))) L#(a(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> b#(c(x1)) L#(a(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> c#(x1) d#(x1) -> a#(x1) -> a#(d(x1)) -> d#(a(x1)) d#(x1) -> a#(x1) -> a#(d(x1)) -> a#(x1) d#(x1) -> b#(a(x1)) -> b#(c(a(x1))) -> a#(b(c(x1))) d#(x1) -> b#(a(x1)) -> b#(c(a(x1))) -> b#(c(x1)) d#(x1) -> b#(a(x1)) -> b#(c(a(x1))) -> c#(x1) d#(x1) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(b(c(x1))) d#(x1) -> b#(a(x1)) -> b#(a(a(x1))) -> b#(c(x1)) d#(x1) -> b#(a(x1)) -> b#(a(a(x1))) -> c#(x1) a#(d(x1)) -> d#(a(x1)) -> d#(x1) -> b#(a(x1)) a#(d(x1)) -> d#(a(x1)) -> d#(x1) -> a#(x1) a#(d(x1)) -> a#(x1) -> a#(d(x1)) -> d#(a(x1)) a#(d(x1)) -> a#(x1) -> a#(d(x1)) -> a#(x1) c#(R(x1)) -> c#(b(R(x1))) -> c#(R(x1)) -> c#(b(R(x1))) c#(R(x1)) -> c#(b(R(x1))) -> c#(R(x1)) -> b#(R(x1)) c#(R(x1)) -> c#(b(R(x1))) -> c#(b(x1)) -> d#(x1) c#(R(x1)) -> c#(b(R(x1))) -> c#(a(x1)) -> a#(c(x1)) c#(R(x1)) -> c#(b(R(x1))) -> c#(a(x1)) -> c#(x1) c#(R(x1)) -> b#(R(x1)) -> b#(c(a(x1))) -> a#(b(c(x1))) c#(R(x1)) -> b#(R(x1)) -> b#(c(a(x1))) -> b#(c(x1)) c#(R(x1)) -> b#(R(x1)) -> b#(c(a(x1))) -> c#(x1) c#(R(x1)) -> b#(R(x1)) -> b#(a(a(x1))) -> a#(b(c(x1))) c#(R(x1)) -> b#(R(x1)) -> b#(a(a(x1))) -> b#(c(x1)) c#(R(x1)) -> b#(R(x1)) -> b#(a(a(x1))) -> c#(x1) c#(b(x1)) -> d#(x1) -> d#(x1) -> b#(a(x1)) c#(b(x1)) -> d#(x1) -> d#(x1) -> a#(x1) c#(a(x1)) -> a#(c(x1)) -> a#(d(x1)) -> d#(a(x1)) c#(a(x1)) -> a#(c(x1)) -> a#(d(x1)) -> a#(x1) c#(a(x1)) -> c#(x1) -> c#(R(x1)) -> c#(b(R(x1))) c#(a(x1)) -> c#(x1) -> c#(R(x1)) -> b#(R(x1)) c#(a(x1)) -> c#(x1) -> c#(b(x1)) -> d#(x1) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1) b#(c(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> d#(a(x1)) b#(c(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> a#(x1) b#(c(a(x1))) -> c#(x1) -> c#(R(x1)) -> c#(b(R(x1))) b#(c(a(x1))) -> c#(x1) -> c#(R(x1)) -> b#(R(x1)) b#(c(a(x1))) -> c#(x1) -> c#(b(x1)) -> d#(x1) b#(c(a(x1))) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) b#(c(a(x1))) -> c#(x1) -> c#(a(x1)) -> c#(x1) b#(c(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> a#(b(c(x1))) b#(c(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> b#(c(x1)) b#(c(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> c#(x1) b#(c(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> a#(b(c(x1))) b#(c(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> b#(c(x1)) b#(c(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> c#(x1) b#(a(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> d#(a(x1)) b#(a(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> a#(x1) b#(a(a(x1))) -> c#(x1) -> c#(R(x1)) -> c#(b(R(x1))) b#(a(a(x1))) -> c#(x1) -> c#(R(x1)) -> b#(R(x1)) b#(a(a(x1))) -> c#(x1) -> c#(b(x1)) -> d#(x1) b#(a(a(x1))) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) b#(a(a(x1))) -> c#(x1) -> c#(a(x1)) -> c#(x1) b#(a(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> a#(b(c(x1))) b#(a(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> b#(c(x1)) b#(a(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> c#(x1) b#(a(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> a#(b(c(x1))) b#(a(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> b#(c(x1)) b#(a(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> c#(x1) EDG Processor: DPs: b#(a(a(x1))) -> c#(x1) b#(a(a(x1))) -> b#(c(x1)) b#(a(a(x1))) -> a#(b(c(x1))) c#(a(x1)) -> c#(x1) c#(a(x1)) -> a#(c(x1)) b#(c(a(x1))) -> c#(x1) b#(c(a(x1))) -> b#(c(x1)) b#(c(a(x1))) -> a#(b(c(x1))) c#(b(x1)) -> d#(x1) a#(d(x1)) -> a#(x1) a#(d(x1)) -> d#(a(x1)) d#(x1) -> a#(x1) d#(x1) -> b#(a(x1)) L#(a(a(x1))) -> c#(x1) L#(a(a(x1))) -> b#(c(x1)) L#(a(a(x1))) -> a#(b(c(x1))) L#(a(a(x1))) -> L#(a(b(c(x1)))) c#(R(x1)) -> b#(R(x1)) c#(R(x1)) -> c#(b(R(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) a(d(x1)) -> d(a(x1)) d(x1) -> b(a(x1)) L(a(a(x1))) -> L(a(b(c(x1)))) c(R(x1)) -> c(b(R(x1))) graph: L#(a(a(x1))) -> L#(a(b(c(x1)))) -> L#(a(a(x1))) -> c#(x1) L#(a(a(x1))) -> L#(a(b(c(x1)))) -> L#(a(a(x1))) -> b#(c(x1)) L#(a(a(x1))) -> L#(a(b(c(x1)))) -> L#(a(a(x1))) -> a#(b(c(x1))) L#(a(a(x1))) -> L#(a(b(c(x1)))) -> L#(a(a(x1))) -> L#(a(b(c(x1)))) L#(a(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> a#(x1) L#(a(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> d#(a(x1)) L#(a(a(x1))) -> c#(x1) -> c#(a(x1)) -> c#(x1) L#(a(a(x1))) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) L#(a(a(x1))) -> c#(x1) -> c#(b(x1)) -> d#(x1) L#(a(a(x1))) -> c#(x1) -> c#(R(x1)) -> b#(R(x1)) L#(a(a(x1))) -> c#(x1) -> c#(R(x1)) -> c#(b(R(x1))) L#(a(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> c#(x1) L#(a(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> b#(c(x1)) L#(a(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> a#(b(c(x1))) L#(a(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> c#(x1) L#(a(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> b#(c(x1)) L#(a(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> a#(b(c(x1))) d#(x1) -> a#(x1) -> a#(d(x1)) -> a#(x1) d#(x1) -> a#(x1) -> a#(d(x1)) -> d#(a(x1)) d#(x1) -> b#(a(x1)) -> b#(a(a(x1))) -> c#(x1) d#(x1) -> b#(a(x1)) -> b#(a(a(x1))) -> b#(c(x1)) d#(x1) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(b(c(x1))) a#(d(x1)) -> d#(a(x1)) -> d#(x1) -> a#(x1) a#(d(x1)) -> d#(a(x1)) -> d#(x1) -> b#(a(x1)) a#(d(x1)) -> a#(x1) -> a#(d(x1)) -> a#(x1) a#(d(x1)) -> a#(x1) -> a#(d(x1)) -> d#(a(x1)) c#(R(x1)) -> c#(b(R(x1))) -> c#(b(x1)) -> d#(x1) c#(b(x1)) -> d#(x1) -> d#(x1) -> a#(x1) c#(b(x1)) -> d#(x1) -> d#(x1) -> b#(a(x1)) c#(a(x1)) -> a#(c(x1)) -> a#(d(x1)) -> a#(x1) c#(a(x1)) -> a#(c(x1)) -> a#(d(x1)) -> d#(a(x1)) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) c#(a(x1)) -> c#(x1) -> c#(b(x1)) -> d#(x1) c#(a(x1)) -> c#(x1) -> c#(R(x1)) -> b#(R(x1)) c#(a(x1)) -> c#(x1) -> c#(R(x1)) -> c#(b(R(x1))) b#(c(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> a#(x1) b#(c(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> d#(a(x1)) b#(c(a(x1))) -> c#(x1) -> c#(a(x1)) -> c#(x1) b#(c(a(x1))) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) b#(c(a(x1))) -> c#(x1) -> c#(b(x1)) -> d#(x1) b#(c(a(x1))) -> c#(x1) -> c#(R(x1)) -> b#(R(x1)) b#(c(a(x1))) -> c#(x1) -> c#(R(x1)) -> c#(b(R(x1))) b#(c(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> c#(x1) b#(c(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> b#(c(x1)) b#(c(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> a#(b(c(x1))) b#(c(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> c#(x1) b#(c(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> b#(c(x1)) b#(c(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> a#(b(c(x1))) b#(a(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> a#(x1) b#(a(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> d#(a(x1)) b#(a(a(x1))) -> c#(x1) -> c#(a(x1)) -> c#(x1) b#(a(a(x1))) -> c#(x1) -> c#(a(x1)) -> a#(c(x1)) b#(a(a(x1))) -> c#(x1) -> c#(b(x1)) -> d#(x1) b#(a(a(x1))) -> c#(x1) -> c#(R(x1)) -> b#(R(x1)) b#(a(a(x1))) -> c#(x1) -> c#(R(x1)) -> c#(b(R(x1))) b#(a(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> c#(x1) b#(a(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> b#(c(x1)) b#(a(a(x1))) -> b#(c(x1)) -> b#(a(a(x1))) -> a#(b(c(x1))) b#(a(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> c#(x1) b#(a(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> b#(c(x1)) b#(a(a(x1))) -> b#(c(x1)) -> b#(c(a(x1))) -> a#(b(c(x1))) SCC Processor: #sccs: 2 #rules: 15 #arcs: 62/361 DPs: L#(a(a(x1))) -> L#(a(b(c(x1)))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) a(d(x1)) -> d(a(x1)) d(x1) -> b(a(x1)) L(a(a(x1))) -> L(a(b(c(x1)))) c(R(x1)) -> c(b(R(x1))) Usable Rule Processor: DPs: L#(a(a(x1))) -> L#(a(b(c(x1)))) TRS: c(a(x1)) -> a(c(x1)) c(b(x1)) -> d(x1) c(R(x1)) -> c(b(R(x1))) a(d(x1)) -> d(a(x1)) d(x1) -> b(a(x1)) b(a(a(x1))) -> a(b(c(x1))) b(c(a(x1))) -> a(b(c(x1))) Arctic Interpretation Processor: dimension: 1 usable rules: c(a(x1)) -> a(c(x1)) c(b(x1)) -> d(x1) c(R(x1)) -> c(b(R(x1))) a(d(x1)) -> d(a(x1)) d(x1) -> b(a(x1)) b(a(a(x1))) -> a(b(c(x1))) b(c(a(x1))) -> a(b(c(x1))) interpretation: [b](x0) = -1x0 + 12, [d](x0) = 3x0 + 12, [R](x0) = 4x0 + 12, [L#](x0) = x0 + 0, [a](x0) = 4x0 + 13, [c](x0) = 4x0 + 0 orientation: L#(a(a(x1))) = 8x1 + 17 >= 7x1 + 16 = L#(a(b(c(x1)))) c(a(x1)) = 8x1 + 17 >= 8x1 + 13 = a(c(x1)) c(b(x1)) = 3x1 + 16 >= 3x1 + 12 = d(x1) c(R(x1)) = 8x1 + 16 >= 7x1 + 16 = c(b(R(x1))) a(d(x1)) = 7x1 + 16 >= 7x1 + 16 = d(a(x1)) d(x1) = 3x1 + 12 >= 3x1 + 12 = b(a(x1)) b(a(a(x1))) = 7x1 + 16 >= 7x1 + 16 = a(b(c(x1))) b(c(a(x1))) = 7x1 + 16 >= 7x1 + 16 = a(b(c(x1))) problem: DPs: TRS: c(a(x1)) -> a(c(x1)) c(b(x1)) -> d(x1) c(R(x1)) -> c(b(R(x1))) a(d(x1)) -> d(a(x1)) d(x1) -> b(a(x1)) b(a(a(x1))) -> a(b(c(x1))) b(c(a(x1))) -> a(b(c(x1))) Qed DPs: a#(d(x1)) -> d#(a(x1)) d#(x1) -> b#(a(x1)) b#(a(a(x1))) -> a#(b(c(x1))) a#(d(x1)) -> a#(x1) b#(a(a(x1))) -> b#(c(x1)) b#(c(a(x1))) -> a#(b(c(x1))) b#(c(a(x1))) -> b#(c(x1)) b#(c(a(x1))) -> c#(x1) c#(R(x1)) -> c#(b(R(x1))) c#(b(x1)) -> d#(x1) d#(x1) -> a#(x1) c#(a(x1)) -> a#(c(x1)) c#(a(x1)) -> c#(x1) b#(a(a(x1))) -> c#(x1) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) a(d(x1)) -> d(a(x1)) d(x1) -> b(a(x1)) L(a(a(x1))) -> L(a(b(c(x1)))) c(R(x1)) -> c(b(R(x1))) Usable Rule Processor: DPs: a#(d(x1)) -> d#(a(x1)) d#(x1) -> b#(a(x1)) b#(a(a(x1))) -> a#(b(c(x1))) a#(d(x1)) -> a#(x1) b#(a(a(x1))) -> b#(c(x1)) b#(c(a(x1))) -> a#(b(c(x1))) b#(c(a(x1))) -> b#(c(x1)) b#(c(a(x1))) -> c#(x1) c#(R(x1)) -> c#(b(R(x1))) c#(b(x1)) -> d#(x1) d#(x1) -> a#(x1) c#(a(x1)) -> a#(c(x1)) c#(a(x1)) -> c#(x1) b#(a(a(x1))) -> c#(x1) TRS: a(d(x1)) -> d(a(x1)) d(x1) -> b(a(x1)) b(a(a(x1))) -> a(b(c(x1))) b(c(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) c(b(x1)) -> d(x1) c(R(x1)) -> c(b(R(x1))) Arctic Interpretation Processor: dimension: 1 usable rules: a(d(x1)) -> d(a(x1)) d(x1) -> b(a(x1)) b(a(a(x1))) -> a(b(c(x1))) b(c(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) c(b(x1)) -> d(x1) c(R(x1)) -> c(b(R(x1))) interpretation: [a#](x0) = 2x0, [b](x0) = x0, [d](x0) = 2x0, [R](x0) = x0, [a](x0) = 2x0, [c#](x0) = 2x0, [d#](x0) = 2x0, [b#](x0) = x0, [c](x0) = 2x0 orientation: a#(d(x1)) = 4x1 >= 4x1 = d#(a(x1)) d#(x1) = 2x1 >= 2x1 = b#(a(x1)) b#(a(a(x1))) = 4x1 >= 4x1 = a#(b(c(x1))) a#(d(x1)) = 4x1 >= 2x1 = a#(x1) b#(a(a(x1))) = 4x1 >= 2x1 = b#(c(x1)) b#(c(a(x1))) = 4x1 >= 4x1 = a#(b(c(x1))) b#(c(a(x1))) = 4x1 >= 2x1 = b#(c(x1)) b#(c(a(x1))) = 4x1 >= 2x1 = c#(x1) c#(R(x1)) = 2x1 >= 2x1 = c#(b(R(x1))) c#(b(x1)) = 2x1 >= 2x1 = d#(x1) d#(x1) = 2x1 >= 2x1 = a#(x1) c#(a(x1)) = 4x1 >= 4x1 = a#(c(x1)) c#(a(x1)) = 4x1 >= 2x1 = c#(x1) b#(a(a(x1))) = 4x1 >= 2x1 = c#(x1) a(d(x1)) = 4x1 >= 4x1 = d(a(x1)) d(x1) = 2x1 >= 2x1 = b(a(x1)) b(a(a(x1))) = 4x1 >= 4x1 = a(b(c(x1))) b(c(a(x1))) = 4x1 >= 4x1 = a(b(c(x1))) c(a(x1)) = 4x1 >= 4x1 = a(c(x1)) c(b(x1)) = 2x1 >= 2x1 = d(x1) c(R(x1)) = 2x1 >= 2x1 = c(b(R(x1))) problem: DPs: a#(d(x1)) -> d#(a(x1)) d#(x1) -> b#(a(x1)) b#(a(a(x1))) -> a#(b(c(x1))) b#(c(a(x1))) -> a#(b(c(x1))) c#(R(x1)) -> c#(b(R(x1))) c#(b(x1)) -> d#(x1) d#(x1) -> a#(x1) c#(a(x1)) -> a#(c(x1)) TRS: a(d(x1)) -> d(a(x1)) d(x1) -> b(a(x1)) b(a(a(x1))) -> a(b(c(x1))) b(c(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) c(b(x1)) -> d(x1) c(R(x1)) -> c(b(R(x1))) Restore Modifier: DPs: a#(d(x1)) -> d#(a(x1)) d#(x1) -> b#(a(x1)) b#(a(a(x1))) -> a#(b(c(x1))) b#(c(a(x1))) -> a#(b(c(x1))) c#(R(x1)) -> c#(b(R(x1))) c#(b(x1)) -> d#(x1) d#(x1) -> a#(x1) c#(a(x1)) -> a#(c(x1)) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) a(d(x1)) -> d(a(x1)) d(x1) -> b(a(x1)) L(a(a(x1))) -> L(a(b(c(x1)))) c(R(x1)) -> c(b(R(x1))) EDG Processor: DPs: a#(d(x1)) -> d#(a(x1)) d#(x1) -> b#(a(x1)) b#(a(a(x1))) -> a#(b(c(x1))) b#(c(a(x1))) -> a#(b(c(x1))) c#(R(x1)) -> c#(b(R(x1))) c#(b(x1)) -> d#(x1) d#(x1) -> a#(x1) c#(a(x1)) -> a#(c(x1)) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) a(d(x1)) -> d(a(x1)) d(x1) -> b(a(x1)) L(a(a(x1))) -> L(a(b(c(x1)))) c(R(x1)) -> c(b(R(x1))) graph: d#(x1) -> a#(x1) -> a#(d(x1)) -> d#(a(x1)) d#(x1) -> b#(a(x1)) -> b#(a(a(x1))) -> a#(b(c(x1))) a#(d(x1)) -> d#(a(x1)) -> d#(x1) -> b#(a(x1)) a#(d(x1)) -> d#(a(x1)) -> d#(x1) -> a#(x1) c#(R(x1)) -> c#(b(R(x1))) -> c#(b(x1)) -> d#(x1) c#(b(x1)) -> d#(x1) -> d#(x1) -> b#(a(x1)) c#(b(x1)) -> d#(x1) -> d#(x1) -> a#(x1) c#(a(x1)) -> a#(c(x1)) -> a#(d(x1)) -> d#(a(x1)) b#(c(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> d#(a(x1)) b#(a(a(x1))) -> a#(b(c(x1))) -> a#(d(x1)) -> d#(a(x1)) SCC Processor: #sccs: 1 #rules: 4 #arcs: 10/64 DPs: d#(x1) -> a#(x1) a#(d(x1)) -> d#(a(x1)) d#(x1) -> b#(a(x1)) b#(a(a(x1))) -> a#(b(c(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) b(c(a(x1))) -> a(b(c(x1))) c(b(x1)) -> d(x1) a(d(x1)) -> d(a(x1)) d(x1) -> b(a(x1)) L(a(a(x1))) -> L(a(b(c(x1)))) c(R(x1)) -> c(b(R(x1))) Usable Rule Processor: DPs: d#(x1) -> a#(x1) a#(d(x1)) -> d#(a(x1)) d#(x1) -> b#(a(x1)) b#(a(a(x1))) -> a#(b(c(x1))) TRS: a(d(x1)) -> d(a(x1)) d(x1) -> b(a(x1)) b(a(a(x1))) -> a(b(c(x1))) b(c(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) c(b(x1)) -> d(x1) c(R(x1)) -> c(b(R(x1))) KBO Processor: argument filtering: pi(a) = 0 pi(b) = [] pi(c) = 0 pi(d) = [0] pi(R) = 0 pi(b#) = [0] pi(a#) = [0] pi(d#) = [0] usable rules: a(d(x1)) -> d(a(x1)) d(x1) -> b(a(x1)) b(a(a(x1))) -> a(b(c(x1))) b(c(a(x1))) -> a(b(c(x1))) weight function: w0 = 1 w(d#) = w(a#) = w(b#) = w(d) = w(b) = 1 w(R) = w(c) = w(a) = 0 precedence: d# > R > b > b# > d > a > c > a# problem: DPs: TRS: a(d(x1)) -> d(a(x1)) d(x1) -> b(a(x1)) b(a(a(x1))) -> a(b(c(x1))) b(c(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) c(b(x1)) -> d(x1) c(R(x1)) -> c(b(R(x1))) Qed