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))) d#(x1) -> b#(a(x1)) -> b#(c(a(x1))) -> c#(x1) d#(x1) -> b#(a(x1)) -> b#(c(a(x1))) -> b#(c(x1)) d#(x1) -> b#(a(x1)) -> b#(c(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: 65/361 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))) Arctic Interpretation Processor: dimension: 1 interpretation: [L#](x0) = 1x0 + 0, [R](x0) = -15x0 + 9, [L](x0) = x0 + 1, [d](x0) = x0 + 2, [c](x0) = 1x0 + 2, [b](x0) = -1x0 + 0, [a](x0) = 1x0 + 3 orientation: L#(a(a(x1))) = 3x1 + 5 >= 2x1 + 4 = L#(a(b(c(x1)))) b(a(a(x1))) = 1x1 + 3 >= 1x1 + 3 = a(b(c(x1))) c(a(x1)) = 2x1 + 4 >= 2x1 + 3 = a(c(x1)) b(c(a(x1))) = 1x1 + 3 >= 1x1 + 3 = a(b(c(x1))) c(b(x1)) = x1 + 2 >= x1 + 2 = d(x1) a(d(x1)) = 1x1 + 3 >= 1x1 + 3 = d(a(x1)) d(x1) = x1 + 2 >= x1 + 2 = b(a(x1)) L(a(a(x1))) = 2x1 + 4 >= 1x1 + 3 = L(a(b(c(x1)))) c(R(x1)) = -14x1 + 10 >= -15x1 + 9 = c(b(R(x1))) problem: DPs: 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))) Qed DPs: a#(d(x1)) -> d#(a(x1)) d#(x1) -> b#(a(x1)) b#(c(a(x1))) -> a#(b(c(x1))) a#(d(x1)) -> a#(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))) -> a#(b(c(x1))) b#(a(a(x1))) -> b#(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))) Arctic Interpretation Processor: dimension: 1 interpretation: [d#](x0) = 2x0, [a#](x0) = 2x0, [c#](x0) = 3x0, [b#](x0) = x0, [R](x0) = 0, [L](x0) = -4x0 + 10, [d](x0) = 2x0, [c](x0) = 2x0, [b](x0) = x0, [a](x0) = 2x0 orientation: a#(d(x1)) = 4x1 >= 4x1 = d#(a(x1)) d#(x1) = 2x1 >= 2x1 = b#(a(x1)) b#(c(a(x1))) = 4x1 >= 4x1 = a#(b(c(x1))) a#(d(x1)) = 4x1 >= 2x1 = a#(x1) b#(c(a(x1))) = 4x1 >= 2x1 = b#(c(x1)) b#(c(a(x1))) = 4x1 >= 3x1 = c#(x1) c#(R(x1)) = 3 >= 3 = c#(b(R(x1))) c#(b(x1)) = 3x1 >= 2x1 = d#(x1) d#(x1) = 2x1 >= 2x1 = a#(x1) c#(a(x1)) = 5x1 >= 4x1 = a#(c(x1)) c#(a(x1)) = 5x1 >= 3x1 = c#(x1) b#(a(a(x1))) = 4x1 >= 4x1 = a#(b(c(x1))) b#(a(a(x1))) = 4x1 >= 2x1 = b#(c(x1)) b#(a(a(x1))) = 4x1 >= 3x1 = c#(x1) b(a(a(x1))) = 4x1 >= 4x1 = a(b(c(x1))) c(a(x1)) = 4x1 >= 4x1 = a(c(x1)) b(c(a(x1))) = 4x1 >= 4x1 = a(b(c(x1))) c(b(x1)) = 2x1 >= 2x1 = d(x1) a(d(x1)) = 4x1 >= 4x1 = d(a(x1)) d(x1) = 2x1 >= 2x1 = b(a(x1)) L(a(a(x1))) = x1 + 10 >= x1 + 10 = L(a(b(c(x1)))) c(R(x1)) = 2 >= 2 = c(b(R(x1))) problem: DPs: a#(d(x1)) -> d#(a(x1)) d#(x1) -> b#(a(x1)) b#(c(a(x1))) -> a#(b(c(x1))) c#(R(x1)) -> c#(b(R(x1))) d#(x1) -> 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))) EDG Processor: DPs: a#(d(x1)) -> d#(a(x1)) d#(x1) -> b#(a(x1)) b#(c(a(x1))) -> a#(b(c(x1))) c#(R(x1)) -> c#(b(R(x1))) d#(x1) -> 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))) graph: d#(x1) -> a#(x1) -> a#(d(x1)) -> d#(a(x1)) d#(x1) -> b#(a(x1)) -> b#(c(a(x1))) -> a#(b(c(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) 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: 5 #arcs: 7/36 DPs: d#(x1) -> a#(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))) 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))) Arctic Interpretation Processor: dimension: 1 interpretation: [d#](x0) = x0 + 0, [a#](x0) = x0, [b#](x0) = x0 + 0, [R](x0) = 2x0, [L](x0) = 6x0 + 0, [d](x0) = 0, [c](x0) = -13x0 + 1, [b](x0) = 0, [a](x0) = 0 orientation: d#(x1) = x1 + 0 >= x1 = a#(x1) a#(d(x1)) = 0 >= 0 = d#(a(x1)) d#(x1) = x1 + 0 >= 0 = b#(a(x1)) b#(a(a(x1))) = 0 >= 0 = a#(b(c(x1))) b#(c(a(x1))) = 1 >= 0 = a#(b(c(x1))) b(a(a(x1))) = 0 >= 0 = a(b(c(x1))) c(a(x1)) = 1 >= 0 = a(c(x1)) b(c(a(x1))) = 0 >= 0 = a(b(c(x1))) c(b(x1)) = 1 >= 0 = d(x1) a(d(x1)) = 0 >= 0 = d(a(x1)) d(x1) = 0 >= 0 = b(a(x1)) L(a(a(x1))) = 6 >= 6 = L(a(b(c(x1)))) c(R(x1)) = -11x1 + 1 >= 1 = c(b(R(x1))) problem: 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))) EDG 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: 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) -> 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)) Arctic Interpretation Processor: dimension: 2 interpretation: [d#](x0) = [-& 2 ]x0 + [2], [a#](x0) = [-& 2 ]x0 + [0], [b#](x0) = [-& 1 ]x0 + [2], [2 1] [3] [R](x0) = [0 1]x0 + [2], [0 2 ] [0 ] [L](x0) = [-& -&]x0 + [-&], [0 0] [0] [d](x0) = [0 0]x0 + [0], [0 -&] [c](x0) = [0 0 ]x0, [0 0 ] [0 ] [b](x0) = [-& -&]x0 + [-&], [0 0 ] [0 ] [a](x0) = [-& 0 ]x0 + [-&] orientation: d#(x1) = [-& 2 ]x1 + [2] >= [-& 2 ]x1 + [0] = a#(x1) a#(d(x1)) = [2 2]x1 + [2] >= [-& 2 ]x1 + [2] = d#(a(x1)) d#(x1) = [-& 2 ]x1 + [2] >= [-& 1 ]x1 + [2] = b#(a(x1)) b#(a(a(x1))) = [-& 1 ]x1 + [2] >= [0] = a#(b(c(x1))) [0 0 ] [0 ] [0 0 ] [0 ] b(a(a(x1))) = [-& -&]x1 + [-&] >= [-& -&]x1 + [-&] = a(b(c(x1))) [0 0] [0] [0 0] [0 ] c(a(x1)) = [0 0]x1 + [0] >= [0 0]x1 + [-&] = a(c(x1)) [0 0 ] [0 ] [0 0 ] [0 ] b(c(a(x1))) = [-& -&]x1 + [-&] >= [-& -&]x1 + [-&] = a(b(c(x1))) [0 0] [0] [0 0] [0] c(b(x1)) = [0 0]x1 + [0] >= [0 0]x1 + [0] = d(x1) [0 0] [0] [0 0] [0] a(d(x1)) = [0 0]x1 + [0] >= [0 0]x1 + [0] = d(a(x1)) [0 0] [0] [0 0 ] [0 ] d(x1) = [0 0]x1 + [0] >= [-& -&]x1 + [-&] = b(a(x1)) [0 2 ] [0 ] [0 0 ] [0 ] L(a(a(x1))) = [-& -&]x1 + [-&] >= [-& -&]x1 + [-&] = L(a(b(c(x1)))) [2 1] [3] [2 1] [3] c(R(x1)) = [2 1]x1 + [3] >= [2 1]x1 + [3] = c(b(R(x1))) problem: DPs: d#(x1) -> a#(x1) a#(d(x1)) -> d#(a(x1)) d#(x1) -> b#(a(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: d#(x1) -> a#(x1) a#(d(x1)) -> d#(a(x1)) d#(x1) -> b#(a(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)) a#(d(x1)) -> d#(a(x1)) -> d#(x1) -> b#(a(x1)) a#(d(x1)) -> d#(a(x1)) -> d#(x1) -> a#(x1) SCC Processor: #sccs: 1 #rules: 2 #arcs: 3/9 DPs: d#(x1) -> a#(x1) a#(d(x1)) -> d#(a(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))) Arctic Interpretation Processor: dimension: 3 interpretation: [d#](x0) = [-& 0 0 ]x0 + [0], [a#](x0) = [-& 0 0 ]x0 + [0], [-& 0 1 ] [0 ] [R](x0) = [-& 0 0 ]x0 + [-&] [-& 0 0 ] [0 ], [0 1 1 ] [1 ] [L](x0) = [-& -& -&]x0 + [-&] [-& -& 1 ] [0 ], [1 0 0 ] [0] [d](x0) = [-& 0 0 ]x0 + [0] [-& 0 1 ] [1], [1 0 0] [0] [c](x0) = [1 0 0]x0 + [0] [1 1 0] [0], [0 0 -&] [0 ] [b](x0) = [-& 0 0 ]x0 + [-&] [-& -& -&] [-&], [1 0 0 ] [0 ] [a](x0) = [-& -& 0 ]x0 + [-&] [-& -& 0 ] [-&] orientation: d#(x1) = [-& 0 0 ]x1 + [0] >= [-& 0 0 ]x1 + [0] = a#(x1) a#(d(x1)) = [-& 0 1 ]x1 + [1] >= [-& -& 0 ]x1 + [0] = d#(a(x1)) [2 1 1 ] [1 ] [2 1 1 ] [1 ] b(a(a(x1))) = [-& -& 0 ]x1 + [-&] >= [-& -& -&]x1 + [-&] = a(b(c(x1))) [-& -& -&] [-&] [-& -& -&] [-&] [2 1 1] [1] [2 1 1] [1] c(a(x1)) = [2 1 1]x1 + [1] >= [1 1 0]x1 + [0] = a(c(x1)) [2 1 1] [1] [1 1 0] [0] [2 1 1 ] [1 ] [2 1 1 ] [1 ] b(c(a(x1))) = [2 1 1 ]x1 + [1 ] >= [-& -& -&]x1 + [-&] = a(b(c(x1))) [-& -& -&] [-&] [-& -& -&] [-&] [1 1 0] [1] [1 0 0 ] [0] c(b(x1)) = [1 1 0]x1 + [1] >= [-& 0 0 ]x1 + [0] = d(x1) [1 1 1] [1] [-& 0 1 ] [1] [2 1 1 ] [1] [2 1 1 ] [1] a(d(x1)) = [-& 0 1 ]x1 + [1] >= [-& -& 0 ]x1 + [0] = d(a(x1)) [-& 0 1 ] [1] [-& -& 1 ] [1] [1 0 0 ] [0] [1 0 0 ] [0 ] d(x1) = [-& 0 0 ]x1 + [0] >= [-& -& 0 ]x1 + [-&] = b(a(x1)) [-& 0 1 ] [1] [-& -& -&] [-&] [2 1 1 ] [1 ] [2 1 1 ] [1 ] L(a(a(x1))) = [-& -& -&]x1 + [-&] >= [-& -& -&]x1 + [-&] = L(a(b(c(x1)))) [-& -& 1 ] [0 ] [-& -& -&] [0 ] [-& 1 2 ] [1] [-& 1 2 ] [1] c(R(x1)) = [-& 1 2 ]x1 + [1] >= [-& 1 2 ]x1 + [1] = c(b(R(x1))) [-& 1 2 ] [1] [-& 1 2 ] [1] problem: DPs: d#(x1) -> a#(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: d#(x1) -> a#(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: Qed