YES Problem: a(c(a(x1))) -> c(a(c(x1))) a(a(b(x1))) -> a(d(b(x1))) a(b(x1)) -> b(a(a(x1))) d(d(x1)) -> a(d(b(x1))) b(b(x1)) -> b(c(x1)) a(d(c(x1))) -> c(a(x1)) b(c(x1)) -> a(a(a(x1))) Proof: DP Processor: DPs: a#(c(a(x1))) -> a#(c(x1)) a#(a(b(x1))) -> d#(b(x1)) a#(a(b(x1))) -> a#(d(b(x1))) a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> b#(a(a(x1))) d#(d(x1)) -> b#(x1) d#(d(x1)) -> d#(b(x1)) d#(d(x1)) -> a#(d(b(x1))) b#(b(x1)) -> b#(c(x1)) a#(d(c(x1))) -> a#(x1) b#(c(x1)) -> a#(x1) b#(c(x1)) -> a#(a(x1)) b#(c(x1)) -> a#(a(a(x1))) TRS: a(c(a(x1))) -> c(a(c(x1))) a(a(b(x1))) -> a(d(b(x1))) a(b(x1)) -> b(a(a(x1))) d(d(x1)) -> a(d(b(x1))) b(b(x1)) -> b(c(x1)) a(d(c(x1))) -> c(a(x1)) b(c(x1)) -> a(a(a(x1))) TDG Processor: DPs: a#(c(a(x1))) -> a#(c(x1)) a#(a(b(x1))) -> d#(b(x1)) a#(a(b(x1))) -> a#(d(b(x1))) a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> b#(a(a(x1))) d#(d(x1)) -> b#(x1) d#(d(x1)) -> d#(b(x1)) d#(d(x1)) -> a#(d(b(x1))) b#(b(x1)) -> b#(c(x1)) a#(d(c(x1))) -> a#(x1) b#(c(x1)) -> a#(x1) b#(c(x1)) -> a#(a(x1)) b#(c(x1)) -> a#(a(a(x1))) TRS: a(c(a(x1))) -> c(a(c(x1))) a(a(b(x1))) -> a(d(b(x1))) a(b(x1)) -> b(a(a(x1))) d(d(x1)) -> a(d(b(x1))) b(b(x1)) -> b(c(x1)) a(d(c(x1))) -> c(a(x1)) b(c(x1)) -> a(a(a(x1))) graph: b#(b(x1)) -> b#(c(x1)) -> b#(c(x1)) -> a#(a(a(x1))) b#(b(x1)) -> b#(c(x1)) -> b#(c(x1)) -> a#(a(x1)) b#(b(x1)) -> b#(c(x1)) -> b#(c(x1)) -> a#(x1) b#(b(x1)) -> b#(c(x1)) -> b#(b(x1)) -> b#(c(x1)) b#(c(x1)) -> a#(a(a(x1))) -> a#(d(c(x1))) -> a#(x1) b#(c(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> b#(a(a(x1))) b#(c(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> a#(a(x1)) b#(c(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> a#(x1) b#(c(x1)) -> a#(a(a(x1))) -> a#(a(b(x1))) -> a#(d(b(x1))) b#(c(x1)) -> a#(a(a(x1))) -> a#(a(b(x1))) -> d#(b(x1)) b#(c(x1)) -> a#(a(a(x1))) -> a#(c(a(x1))) -> a#(c(x1)) b#(c(x1)) -> a#(a(x1)) -> a#(d(c(x1))) -> a#(x1) b#(c(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(a(a(x1))) b#(c(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) b#(c(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(x1) b#(c(x1)) -> a#(a(x1)) -> a#(a(b(x1))) -> a#(d(b(x1))) b#(c(x1)) -> a#(a(x1)) -> a#(a(b(x1))) -> d#(b(x1)) b#(c(x1)) -> a#(a(x1)) -> a#(c(a(x1))) -> a#(c(x1)) b#(c(x1)) -> a#(x1) -> a#(d(c(x1))) -> a#(x1) b#(c(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(a(x1))) b#(c(x1)) -> a#(x1) -> a#(b(x1)) -> a#(a(x1)) b#(c(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) b#(c(x1)) -> a#(x1) -> a#(a(b(x1))) -> a#(d(b(x1))) b#(c(x1)) -> a#(x1) -> a#(a(b(x1))) -> d#(b(x1)) b#(c(x1)) -> a#(x1) -> a#(c(a(x1))) -> a#(c(x1)) d#(d(x1)) -> b#(x1) -> b#(c(x1)) -> a#(a(a(x1))) d#(d(x1)) -> b#(x1) -> b#(c(x1)) -> a#(a(x1)) d#(d(x1)) -> b#(x1) -> b#(c(x1)) -> a#(x1) d#(d(x1)) -> b#(x1) -> b#(b(x1)) -> b#(c(x1)) d#(d(x1)) -> d#(b(x1)) -> d#(d(x1)) -> a#(d(b(x1))) d#(d(x1)) -> d#(b(x1)) -> d#(d(x1)) -> d#(b(x1)) d#(d(x1)) -> d#(b(x1)) -> d#(d(x1)) -> b#(x1) d#(d(x1)) -> a#(d(b(x1))) -> a#(d(c(x1))) -> a#(x1) d#(d(x1)) -> a#(d(b(x1))) -> a#(b(x1)) -> b#(a(a(x1))) d#(d(x1)) -> a#(d(b(x1))) -> a#(b(x1)) -> a#(a(x1)) d#(d(x1)) -> a#(d(b(x1))) -> a#(b(x1)) -> a#(x1) d#(d(x1)) -> a#(d(b(x1))) -> a#(a(b(x1))) -> a#(d(b(x1))) d#(d(x1)) -> a#(d(b(x1))) -> a#(a(b(x1))) -> d#(b(x1)) d#(d(x1)) -> a#(d(b(x1))) -> a#(c(a(x1))) -> a#(c(x1)) a#(d(c(x1))) -> a#(x1) -> a#(d(c(x1))) -> a#(x1) a#(d(c(x1))) -> a#(x1) -> a#(b(x1)) -> b#(a(a(x1))) a#(d(c(x1))) -> a#(x1) -> a#(b(x1)) -> a#(a(x1)) a#(d(c(x1))) -> a#(x1) -> a#(b(x1)) -> a#(x1) a#(d(c(x1))) -> a#(x1) -> a#(a(b(x1))) -> a#(d(b(x1))) a#(d(c(x1))) -> a#(x1) -> a#(a(b(x1))) -> d#(b(x1)) a#(d(c(x1))) -> a#(x1) -> a#(c(a(x1))) -> a#(c(x1)) a#(b(x1)) -> b#(a(a(x1))) -> b#(c(x1)) -> a#(a(a(x1))) a#(b(x1)) -> b#(a(a(x1))) -> b#(c(x1)) -> a#(a(x1)) a#(b(x1)) -> b#(a(a(x1))) -> b#(c(x1)) -> a#(x1) a#(b(x1)) -> b#(a(a(x1))) -> b#(b(x1)) -> b#(c(x1)) a#(b(x1)) -> a#(a(x1)) -> a#(d(c(x1))) -> a#(x1) a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(a(a(x1))) a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(a(x1)) -> a#(a(b(x1))) -> a#(d(b(x1))) a#(b(x1)) -> a#(a(x1)) -> a#(a(b(x1))) -> d#(b(x1)) a#(b(x1)) -> a#(a(x1)) -> a#(c(a(x1))) -> a#(c(x1)) a#(b(x1)) -> a#(x1) -> a#(d(c(x1))) -> a#(x1) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(a(x1))) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(x1) -> a#(a(b(x1))) -> a#(d(b(x1))) a#(b(x1)) -> a#(x1) -> a#(a(b(x1))) -> d#(b(x1)) a#(b(x1)) -> a#(x1) -> a#(c(a(x1))) -> a#(c(x1)) a#(c(a(x1))) -> a#(c(x1)) -> a#(d(c(x1))) -> a#(x1) a#(c(a(x1))) -> a#(c(x1)) -> a#(b(x1)) -> b#(a(a(x1))) a#(c(a(x1))) -> a#(c(x1)) -> a#(b(x1)) -> a#(a(x1)) a#(c(a(x1))) -> a#(c(x1)) -> a#(b(x1)) -> a#(x1) a#(c(a(x1))) -> a#(c(x1)) -> a#(a(b(x1))) -> a#(d(b(x1))) a#(c(a(x1))) -> a#(c(x1)) -> a#(a(b(x1))) -> d#(b(x1)) a#(c(a(x1))) -> a#(c(x1)) -> a#(c(a(x1))) -> a#(c(x1)) a#(a(b(x1))) -> d#(b(x1)) -> d#(d(x1)) -> a#(d(b(x1))) a#(a(b(x1))) -> d#(b(x1)) -> d#(d(x1)) -> d#(b(x1)) a#(a(b(x1))) -> d#(b(x1)) -> d#(d(x1)) -> b#(x1) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(d(c(x1))) -> a#(x1) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(b(x1)) -> b#(a(a(x1))) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(b(x1)) -> a#(a(x1)) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(b(x1)) -> a#(x1) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(a(b(x1))) -> a#(d(b(x1))) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(a(b(x1))) -> d#(b(x1)) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(c(a(x1))) -> a#(c(x1)) EDG Processor: DPs: a#(c(a(x1))) -> a#(c(x1)) a#(a(b(x1))) -> d#(b(x1)) a#(a(b(x1))) -> a#(d(b(x1))) a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> b#(a(a(x1))) d#(d(x1)) -> b#(x1) d#(d(x1)) -> d#(b(x1)) d#(d(x1)) -> a#(d(b(x1))) b#(b(x1)) -> b#(c(x1)) a#(d(c(x1))) -> a#(x1) b#(c(x1)) -> a#(x1) b#(c(x1)) -> a#(a(x1)) b#(c(x1)) -> a#(a(a(x1))) TRS: a(c(a(x1))) -> c(a(c(x1))) a(a(b(x1))) -> a(d(b(x1))) a(b(x1)) -> b(a(a(x1))) d(d(x1)) -> a(d(b(x1))) b(b(x1)) -> b(c(x1)) a(d(c(x1))) -> c(a(x1)) b(c(x1)) -> a(a(a(x1))) graph: b#(b(x1)) -> b#(c(x1)) -> b#(c(x1)) -> a#(x1) b#(b(x1)) -> b#(c(x1)) -> b#(c(x1)) -> a#(a(x1)) b#(b(x1)) -> b#(c(x1)) -> b#(c(x1)) -> a#(a(a(x1))) b#(c(x1)) -> a#(a(a(x1))) -> a#(c(a(x1))) -> a#(c(x1)) b#(c(x1)) -> a#(a(a(x1))) -> a#(a(b(x1))) -> d#(b(x1)) b#(c(x1)) -> a#(a(a(x1))) -> a#(a(b(x1))) -> a#(d(b(x1))) b#(c(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> a#(x1) b#(c(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> a#(a(x1)) b#(c(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> b#(a(a(x1))) b#(c(x1)) -> a#(a(x1)) -> a#(c(a(x1))) -> a#(c(x1)) b#(c(x1)) -> a#(a(x1)) -> a#(a(b(x1))) -> d#(b(x1)) b#(c(x1)) -> a#(a(x1)) -> a#(a(b(x1))) -> a#(d(b(x1))) b#(c(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(x1) b#(c(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) b#(c(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(a(a(x1))) b#(c(x1)) -> a#(x1) -> a#(c(a(x1))) -> a#(c(x1)) b#(c(x1)) -> a#(x1) -> a#(a(b(x1))) -> d#(b(x1)) b#(c(x1)) -> a#(x1) -> a#(a(b(x1))) -> a#(d(b(x1))) b#(c(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) b#(c(x1)) -> a#(x1) -> a#(b(x1)) -> a#(a(x1)) b#(c(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(a(x1))) b#(c(x1)) -> a#(x1) -> a#(d(c(x1))) -> a#(x1) d#(d(x1)) -> b#(x1) -> b#(b(x1)) -> b#(c(x1)) d#(d(x1)) -> b#(x1) -> b#(c(x1)) -> a#(x1) d#(d(x1)) -> b#(x1) -> b#(c(x1)) -> a#(a(x1)) d#(d(x1)) -> b#(x1) -> b#(c(x1)) -> a#(a(a(x1))) d#(d(x1)) -> a#(d(b(x1))) -> a#(c(a(x1))) -> a#(c(x1)) d#(d(x1)) -> a#(d(b(x1))) -> a#(a(b(x1))) -> d#(b(x1)) d#(d(x1)) -> a#(d(b(x1))) -> a#(a(b(x1))) -> a#(d(b(x1))) d#(d(x1)) -> a#(d(b(x1))) -> a#(b(x1)) -> a#(x1) d#(d(x1)) -> a#(d(b(x1))) -> a#(b(x1)) -> a#(a(x1)) d#(d(x1)) -> a#(d(b(x1))) -> a#(b(x1)) -> b#(a(a(x1))) d#(d(x1)) -> a#(d(b(x1))) -> a#(d(c(x1))) -> a#(x1) a#(d(c(x1))) -> a#(x1) -> a#(c(a(x1))) -> a#(c(x1)) a#(d(c(x1))) -> a#(x1) -> a#(a(b(x1))) -> d#(b(x1)) a#(d(c(x1))) -> a#(x1) -> a#(a(b(x1))) -> a#(d(b(x1))) a#(d(c(x1))) -> a#(x1) -> a#(b(x1)) -> a#(x1) a#(d(c(x1))) -> a#(x1) -> a#(b(x1)) -> a#(a(x1)) a#(d(c(x1))) -> a#(x1) -> a#(b(x1)) -> b#(a(a(x1))) a#(d(c(x1))) -> a#(x1) -> a#(d(c(x1))) -> a#(x1) a#(b(x1)) -> b#(a(a(x1))) -> b#(b(x1)) -> b#(c(x1)) a#(b(x1)) -> b#(a(a(x1))) -> b#(c(x1)) -> a#(x1) a#(b(x1)) -> b#(a(a(x1))) -> b#(c(x1)) -> a#(a(x1)) a#(b(x1)) -> b#(a(a(x1))) -> b#(c(x1)) -> a#(a(a(x1))) a#(b(x1)) -> a#(a(x1)) -> a#(c(a(x1))) -> a#(c(x1)) a#(b(x1)) -> a#(a(x1)) -> a#(a(b(x1))) -> d#(b(x1)) a#(b(x1)) -> a#(a(x1)) -> a#(a(b(x1))) -> a#(d(b(x1))) a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(a(a(x1))) a#(b(x1)) -> a#(x1) -> a#(c(a(x1))) -> a#(c(x1)) a#(b(x1)) -> a#(x1) -> a#(a(b(x1))) -> d#(b(x1)) a#(b(x1)) -> a#(x1) -> a#(a(b(x1))) -> a#(d(b(x1))) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(a(x1))) a#(b(x1)) -> a#(x1) -> a#(d(c(x1))) -> a#(x1) a#(c(a(x1))) -> a#(c(x1)) -> a#(c(a(x1))) -> a#(c(x1)) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(c(a(x1))) -> a#(c(x1)) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(a(b(x1))) -> d#(b(x1)) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(a(b(x1))) -> a#(d(b(x1))) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(b(x1)) -> a#(x1) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(b(x1)) -> a#(a(x1)) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(b(x1)) -> b#(a(a(x1))) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(d(c(x1))) -> a#(x1) CDG Processor: DPs: a#(c(a(x1))) -> a#(c(x1)) a#(a(b(x1))) -> d#(b(x1)) a#(a(b(x1))) -> a#(d(b(x1))) a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> b#(a(a(x1))) d#(d(x1)) -> b#(x1) d#(d(x1)) -> d#(b(x1)) d#(d(x1)) -> a#(d(b(x1))) b#(b(x1)) -> b#(c(x1)) a#(d(c(x1))) -> a#(x1) b#(c(x1)) -> a#(x1) b#(c(x1)) -> a#(a(x1)) b#(c(x1)) -> a#(a(a(x1))) TRS: a(c(a(x1))) -> c(a(c(x1))) a(a(b(x1))) -> a(d(b(x1))) a(b(x1)) -> b(a(a(x1))) d(d(x1)) -> a(d(b(x1))) b(b(x1)) -> b(c(x1)) a(d(c(x1))) -> c(a(x1)) b(c(x1)) -> a(a(a(x1))) graph: b#(b(x1)) -> b#(c(x1)) -> b#(c(x1)) -> a#(a(a(x1))) b#(b(x1)) -> b#(c(x1)) -> b#(c(x1)) -> a#(a(x1)) b#(b(x1)) -> b#(c(x1)) -> b#(c(x1)) -> a#(x1) b#(c(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> b#(a(a(x1))) b#(c(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> a#(a(x1)) b#(c(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> a#(x1) b#(c(x1)) -> a#(a(a(x1))) -> a#(a(b(x1))) -> a#(d(b(x1))) b#(c(x1)) -> a#(a(a(x1))) -> a#(a(b(x1))) -> d#(b(x1)) b#(c(x1)) -> a#(a(a(x1))) -> a#(c(a(x1))) -> a#(c(x1)) b#(c(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(a(a(x1))) b#(c(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) b#(c(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(x1) b#(c(x1)) -> a#(a(x1)) -> a#(a(b(x1))) -> a#(d(b(x1))) b#(c(x1)) -> a#(a(x1)) -> a#(a(b(x1))) -> d#(b(x1)) b#(c(x1)) -> a#(a(x1)) -> a#(c(a(x1))) -> a#(c(x1)) b#(c(x1)) -> a#(x1) -> a#(d(c(x1))) -> a#(x1) b#(c(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(a(x1))) b#(c(x1)) -> a#(x1) -> a#(b(x1)) -> a#(a(x1)) b#(c(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) b#(c(x1)) -> a#(x1) -> a#(a(b(x1))) -> a#(d(b(x1))) b#(c(x1)) -> a#(x1) -> a#(a(b(x1))) -> d#(b(x1)) b#(c(x1)) -> a#(x1) -> a#(c(a(x1))) -> a#(c(x1)) d#(d(x1)) -> b#(x1) -> b#(c(x1)) -> a#(a(a(x1))) d#(d(x1)) -> b#(x1) -> b#(c(x1)) -> a#(a(x1)) d#(d(x1)) -> b#(x1) -> b#(c(x1)) -> a#(x1) d#(d(x1)) -> b#(x1) -> b#(b(x1)) -> b#(c(x1)) d#(d(x1)) -> a#(d(b(x1))) -> a#(d(c(x1))) -> a#(x1) d#(d(x1)) -> a#(d(b(x1))) -> a#(b(x1)) -> b#(a(a(x1))) d#(d(x1)) -> a#(d(b(x1))) -> a#(b(x1)) -> a#(a(x1)) d#(d(x1)) -> a#(d(b(x1))) -> a#(b(x1)) -> a#(x1) d#(d(x1)) -> a#(d(b(x1))) -> a#(a(b(x1))) -> a#(d(b(x1))) d#(d(x1)) -> a#(d(b(x1))) -> a#(a(b(x1))) -> d#(b(x1)) d#(d(x1)) -> a#(d(b(x1))) -> a#(c(a(x1))) -> a#(c(x1)) a#(d(c(x1))) -> a#(x1) -> a#(d(c(x1))) -> a#(x1) a#(d(c(x1))) -> a#(x1) -> a#(b(x1)) -> b#(a(a(x1))) a#(d(c(x1))) -> a#(x1) -> a#(b(x1)) -> a#(a(x1)) a#(d(c(x1))) -> a#(x1) -> a#(b(x1)) -> a#(x1) a#(d(c(x1))) -> a#(x1) -> a#(a(b(x1))) -> a#(d(b(x1))) a#(d(c(x1))) -> a#(x1) -> a#(a(b(x1))) -> d#(b(x1)) a#(d(c(x1))) -> a#(x1) -> a#(c(a(x1))) -> a#(c(x1)) a#(b(x1)) -> b#(a(a(x1))) -> b#(c(x1)) -> a#(a(a(x1))) a#(b(x1)) -> b#(a(a(x1))) -> b#(c(x1)) -> a#(a(x1)) a#(b(x1)) -> b#(a(a(x1))) -> b#(c(x1)) -> a#(x1) a#(b(x1)) -> b#(a(a(x1))) -> b#(b(x1)) -> b#(c(x1)) a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(a(a(x1))) a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(a(x1)) -> a#(a(b(x1))) -> a#(d(b(x1))) a#(b(x1)) -> a#(a(x1)) -> a#(a(b(x1))) -> d#(b(x1)) a#(b(x1)) -> a#(a(x1)) -> a#(c(a(x1))) -> a#(c(x1)) a#(b(x1)) -> a#(x1) -> a#(d(c(x1))) -> a#(x1) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(a(x1))) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(x1) -> a#(a(b(x1))) -> a#(d(b(x1))) a#(b(x1)) -> a#(x1) -> a#(a(b(x1))) -> d#(b(x1)) a#(b(x1)) -> a#(x1) -> a#(c(a(x1))) -> a#(c(x1)) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(d(c(x1))) -> a#(x1) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(b(x1)) -> b#(a(a(x1))) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(b(x1)) -> a#(a(x1)) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(b(x1)) -> a#(x1) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(a(b(x1))) -> a#(d(b(x1))) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(a(b(x1))) -> d#(b(x1)) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(c(a(x1))) -> a#(c(x1)) SCC Processor: #sccs: 1 #rules: 9 #arcs: 64/196 DPs: b#(b(x1)) -> b#(c(x1)) b#(c(x1)) -> a#(x1) a#(a(b(x1))) -> a#(d(b(x1))) a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> b#(a(a(x1))) b#(c(x1)) -> a#(a(x1)) b#(c(x1)) -> a#(a(a(x1))) a#(d(c(x1))) -> a#(x1) TRS: a(c(a(x1))) -> c(a(c(x1))) a(a(b(x1))) -> a(d(b(x1))) a(b(x1)) -> b(a(a(x1))) d(d(x1)) -> a(d(b(x1))) b(b(x1)) -> b(c(x1)) a(d(c(x1))) -> c(a(x1)) b(c(x1)) -> a(a(a(x1))) Matrix Interpretation Processor: dim=3 interpretation: [b#](x0) = [0 0 1]x0, [a#](x0) = [1 0 0]x0, [0 1 0] [d](x0) = [0 1 1]x0 [0 1 0] , [1 0 1] [0] [b](x0) = [0 0 1]x0 + [0] [1 0 1] [1], [0 0 0] [c](x0) = [1 0 1]x0 [1 0 1] , [1 0 0] [a](x0) = [0 0 1]x0 [0 0 1] orientation: b#(b(x1)) = [1 0 1]x1 + [1] >= [1 0 1]x1 = b#(c(x1)) b#(c(x1)) = [1 0 1]x1 >= [1 0 0]x1 = a#(x1) a#(a(b(x1))) = [1 0 1]x1 >= [0 0 1]x1 = a#(d(b(x1))) a#(b(x1)) = [1 0 1]x1 >= [1 0 0]x1 = a#(x1) a#(b(x1)) = [1 0 1]x1 >= [1 0 0]x1 = a#(a(x1)) a#(b(x1)) = [1 0 1]x1 >= [0 0 1]x1 = b#(a(a(x1))) b#(c(x1)) = [1 0 1]x1 >= [1 0 0]x1 = a#(a(x1)) b#(c(x1)) = [1 0 1]x1 >= [1 0 0]x1 = a#(a(a(x1))) a#(d(c(x1))) = [1 0 1]x1 >= [1 0 0]x1 = a#(x1) [0 0 0] [0 0 0] a(c(a(x1))) = [1 0 1]x1 >= [1 0 1]x1 = c(a(c(x1))) [1 0 1] [1 0 1] [1 0 1] [0] [0 0 1] a(a(b(x1))) = [1 0 1]x1 + [1] >= [0 0 1]x1 = a(d(b(x1))) [1 0 1] [1] [0 0 1] [1 0 1] [0] [1 0 1] [0] a(b(x1)) = [1 0 1]x1 + [1] >= [0 0 1]x1 + [0] = b(a(a(x1))) [1 0 1] [1] [1 0 1] [1] [0 1 1] [0 0 1] d(d(x1)) = [0 2 1]x1 >= [0 0 1]x1 = a(d(b(x1))) [0 1 1] [0 0 1] [2 0 2] [1] [1 0 1] [0] b(b(x1)) = [1 0 1]x1 + [1] >= [1 0 1]x1 + [0] = b(c(x1)) [2 0 2] [2] [1 0 1] [1] [1 0 1] [0 0 0] a(d(c(x1))) = [1 0 1]x1 >= [1 0 1]x1 = c(a(x1)) [1 0 1] [1 0 1] [1 0 1] [0] [1 0 0] b(c(x1)) = [1 0 1]x1 + [0] >= [0 0 1]x1 = a(a(a(x1))) [1 0 1] [1] [0 0 1] problem: DPs: b#(c(x1)) -> a#(x1) a#(a(b(x1))) -> a#(d(b(x1))) a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> b#(a(a(x1))) b#(c(x1)) -> a#(a(x1)) b#(c(x1)) -> a#(a(a(x1))) a#(d(c(x1))) -> a#(x1) TRS: a(c(a(x1))) -> c(a(c(x1))) a(a(b(x1))) -> a(d(b(x1))) a(b(x1)) -> b(a(a(x1))) d(d(x1)) -> a(d(b(x1))) b(b(x1)) -> b(c(x1)) a(d(c(x1))) -> c(a(x1)) b(c(x1)) -> a(a(a(x1))) EDG Processor: DPs: b#(c(x1)) -> a#(x1) a#(a(b(x1))) -> a#(d(b(x1))) a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> b#(a(a(x1))) b#(c(x1)) -> a#(a(x1)) b#(c(x1)) -> a#(a(a(x1))) a#(d(c(x1))) -> a#(x1) TRS: a(c(a(x1))) -> c(a(c(x1))) a(a(b(x1))) -> a(d(b(x1))) a(b(x1)) -> b(a(a(x1))) d(d(x1)) -> a(d(b(x1))) b(b(x1)) -> b(c(x1)) a(d(c(x1))) -> c(a(x1)) b(c(x1)) -> a(a(a(x1))) graph: b#(c(x1)) -> a#(a(a(x1))) -> a#(a(b(x1))) -> a#(d(b(x1))) b#(c(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> a#(x1) b#(c(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> a#(a(x1)) b#(c(x1)) -> a#(a(a(x1))) -> a#(b(x1)) -> b#(a(a(x1))) b#(c(x1)) -> a#(a(x1)) -> a#(a(b(x1))) -> a#(d(b(x1))) b#(c(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(x1) b#(c(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) b#(c(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(a(a(x1))) b#(c(x1)) -> a#(x1) -> a#(a(b(x1))) -> a#(d(b(x1))) b#(c(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) b#(c(x1)) -> a#(x1) -> a#(b(x1)) -> a#(a(x1)) b#(c(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(a(x1))) b#(c(x1)) -> a#(x1) -> a#(d(c(x1))) -> a#(x1) a#(d(c(x1))) -> a#(x1) -> a#(a(b(x1))) -> a#(d(b(x1))) a#(d(c(x1))) -> a#(x1) -> a#(b(x1)) -> a#(x1) a#(d(c(x1))) -> a#(x1) -> a#(b(x1)) -> a#(a(x1)) a#(d(c(x1))) -> a#(x1) -> a#(b(x1)) -> b#(a(a(x1))) a#(d(c(x1))) -> a#(x1) -> a#(d(c(x1))) -> a#(x1) a#(b(x1)) -> b#(a(a(x1))) -> b#(c(x1)) -> a#(x1) a#(b(x1)) -> b#(a(a(x1))) -> b#(c(x1)) -> a#(a(x1)) a#(b(x1)) -> b#(a(a(x1))) -> b#(c(x1)) -> a#(a(a(x1))) a#(b(x1)) -> a#(a(x1)) -> a#(a(b(x1))) -> a#(d(b(x1))) a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(a(a(x1))) a#(b(x1)) -> a#(x1) -> a#(a(b(x1))) -> a#(d(b(x1))) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(a(x1))) a#(b(x1)) -> a#(x1) -> a#(d(c(x1))) -> a#(x1) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(a(b(x1))) -> a#(d(b(x1))) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(b(x1)) -> a#(x1) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(b(x1)) -> a#(a(x1)) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(b(x1)) -> b#(a(a(x1))) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(d(c(x1))) -> a#(x1) Matrix Interpretation Processor: dim=3 interpretation: [b#](x0) = [0 0 1]x0, [a#](x0) = [0 0 1]x0, [0 0 0] [0] [d](x0) = [0 0 1]x0 + [1] [0 1 1] [0], [0 0 0] [0] [b](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [0 0 0] [c](x0) = [0 0 0]x0 [0 0 1] , [0 0 0] [a](x0) = [0 0 0]x0 [0 0 1] orientation: b#(c(x1)) = [0 0 1]x1 >= [0 0 1]x1 = a#(x1) a#(a(b(x1))) = [0 0 1]x1 + [1] >= [0 0 1]x1 + [1] = a#(d(b(x1))) a#(b(x1)) = [0 0 1]x1 + [1] >= [0 0 1]x1 = a#(x1) a#(b(x1)) = [0 0 1]x1 + [1] >= [0 0 1]x1 = a#(a(x1)) a#(b(x1)) = [0 0 1]x1 + [1] >= [0 0 1]x1 = b#(a(a(x1))) b#(c(x1)) = [0 0 1]x1 >= [0 0 1]x1 = a#(a(x1)) b#(c(x1)) = [0 0 1]x1 >= [0 0 1]x1 = a#(a(a(x1))) a#(d(c(x1))) = [0 0 1]x1 >= [0 0 1]x1 = a#(x1) [0 0 0] [0 0 0] a(c(a(x1))) = [0 0 0]x1 >= [0 0 0]x1 = c(a(c(x1))) [0 0 1] [0 0 1] [0 0 0] [0] [0 0 0] [0] a(a(b(x1))) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = a(d(b(x1))) [0 0 1] [1] [0 0 1] [1] [0 0 0] [0] [0 0 0] [0] a(b(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = b(a(a(x1))) [0 0 1] [1] [0 0 1] [1] [0 0 0] [0] [0 0 0] [0] d(d(x1)) = [0 1 1]x1 + [1] >= [0 0 0]x1 + [0] = a(d(b(x1))) [0 1 2] [1] [0 0 1] [1] [0 0 0] [0] [0 0 0] [0] b(b(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = b(c(x1)) [0 0 1] [2] [0 0 1] [1] [0 0 0] [0 0 0] a(d(c(x1))) = [0 0 0]x1 >= [0 0 0]x1 = c(a(x1)) [0 0 1] [0 0 1] [0 0 0] [0] [0 0 0] b(c(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 = a(a(a(x1))) [0 0 1] [1] [0 0 1] problem: DPs: b#(c(x1)) -> a#(x1) a#(a(b(x1))) -> a#(d(b(x1))) b#(c(x1)) -> a#(a(x1)) b#(c(x1)) -> a#(a(a(x1))) a#(d(c(x1))) -> a#(x1) TRS: a(c(a(x1))) -> c(a(c(x1))) a(a(b(x1))) -> a(d(b(x1))) a(b(x1)) -> b(a(a(x1))) d(d(x1)) -> a(d(b(x1))) b(b(x1)) -> b(c(x1)) a(d(c(x1))) -> c(a(x1)) b(c(x1)) -> a(a(a(x1))) EDG Processor: DPs: b#(c(x1)) -> a#(x1) a#(a(b(x1))) -> a#(d(b(x1))) b#(c(x1)) -> a#(a(x1)) b#(c(x1)) -> a#(a(a(x1))) a#(d(c(x1))) -> a#(x1) TRS: a(c(a(x1))) -> c(a(c(x1))) a(a(b(x1))) -> a(d(b(x1))) a(b(x1)) -> b(a(a(x1))) d(d(x1)) -> a(d(b(x1))) b(b(x1)) -> b(c(x1)) a(d(c(x1))) -> c(a(x1)) b(c(x1)) -> a(a(a(x1))) graph: b#(c(x1)) -> a#(a(a(x1))) -> a#(a(b(x1))) -> a#(d(b(x1))) b#(c(x1)) -> a#(a(x1)) -> a#(a(b(x1))) -> a#(d(b(x1))) b#(c(x1)) -> a#(x1) -> a#(d(c(x1))) -> a#(x1) b#(c(x1)) -> a#(x1) -> a#(a(b(x1))) -> a#(d(b(x1))) a#(d(c(x1))) -> a#(x1) -> a#(d(c(x1))) -> a#(x1) a#(d(c(x1))) -> a#(x1) -> a#(a(b(x1))) -> a#(d(b(x1))) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(d(c(x1))) -> a#(x1) a#(a(b(x1))) -> a#(d(b(x1))) -> a#(a(b(x1))) -> a#(d(b(x1))) SCC Processor: #sccs: 1 #rules: 2 #arcs: 8/25 DPs: a#(a(b(x1))) -> a#(d(b(x1))) a#(d(c(x1))) -> a#(x1) TRS: a(c(a(x1))) -> c(a(c(x1))) a(a(b(x1))) -> a(d(b(x1))) a(b(x1)) -> b(a(a(x1))) d(d(x1)) -> a(d(b(x1))) b(b(x1)) -> b(c(x1)) a(d(c(x1))) -> c(a(x1)) b(c(x1)) -> a(a(a(x1))) Matrix Interpretation Processor: dim=3 interpretation: [a#](x0) = [0 0 1]x0, [0 1 0] [d](x0) = [1 0 0]x0 [1 0 0] , [0 0 0] [0] [b](x0) = [0 1 0]x0 + [0] [1 0 1] [1], [0 0 1] [c](x0) = [0 1 0]x0 [0 0 0] , [0 1 0] [a](x0) = [0 0 0]x0 [0 0 1] orientation: a#(a(b(x1))) = [1 0 1]x1 + [1] >= [0] = a#(d(b(x1))) a#(d(c(x1))) = [0 0 1]x1 >= [0 0 1]x1 = a#(x1) [0] [0] a(c(a(x1))) = [0] >= [0] = c(a(c(x1))) [0] [0] [0 0 0] [0] [0] a(a(b(x1))) = [0 0 0]x1 + [0] >= [0] = a(d(b(x1))) [1 0 1] [1] [0] [0 1 0] [0] [0 0 0] [0] a(b(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = b(a(a(x1))) [1 0 1] [1] [0 0 1] [1] [1 0 0] [0] d(d(x1)) = [0 1 0]x1 >= [0] = a(d(b(x1))) [0 1 0] [0] [0 0 0] [0] [0 0 0] [0] b(b(x1)) = [0 1 0]x1 + [0] >= [0 1 0]x1 + [0] = b(c(x1)) [1 0 1] [2] [0 0 1] [1] [0 0 1] [0 0 1] a(d(c(x1))) = [0 0 0]x1 >= [0 0 0]x1 = c(a(x1)) [0 0 1] [0 0 0] [0 0 0] [0] [0 0 0] b(c(x1)) = [0 1 0]x1 + [0] >= [0 0 0]x1 = a(a(a(x1))) [0 0 1] [1] [0 0 1] problem: DPs: a#(d(c(x1))) -> a#(x1) TRS: a(c(a(x1))) -> c(a(c(x1))) a(a(b(x1))) -> a(d(b(x1))) a(b(x1)) -> b(a(a(x1))) d(d(x1)) -> a(d(b(x1))) b(b(x1)) -> b(c(x1)) a(d(c(x1))) -> c(a(x1)) b(c(x1)) -> a(a(a(x1))) EDG Processor: DPs: a#(d(c(x1))) -> a#(x1) TRS: a(c(a(x1))) -> c(a(c(x1))) a(a(b(x1))) -> a(d(b(x1))) a(b(x1)) -> b(a(a(x1))) d(d(x1)) -> a(d(b(x1))) b(b(x1)) -> b(c(x1)) a(d(c(x1))) -> c(a(x1)) b(c(x1)) -> a(a(a(x1))) graph: a#(d(c(x1))) -> a#(x1) -> a#(d(c(x1))) -> a#(x1) CDG Processor: DPs: a#(d(c(x1))) -> a#(x1) TRS: a(c(a(x1))) -> c(a(c(x1))) a(a(b(x1))) -> a(d(b(x1))) a(b(x1)) -> b(a(a(x1))) d(d(x1)) -> a(d(b(x1))) b(b(x1)) -> b(c(x1)) a(d(c(x1))) -> c(a(x1)) b(c(x1)) -> a(a(a(x1))) graph: Qed