YES Problem: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) c(b(x1)) -> b(a(x1)) L(a(a(x1))) -> L(a(b(c(x1)))) c(R(x1)) -> b(a(R(x1))) Proof: DP Processor: DPs: b#(a(a(x1))) -> c#(x1) b#(a(a(x1))) -> b#(c(x1)) c#(a(x1)) -> c#(x1) c#(b(x1)) -> b#(a(x1)) L#(a(a(x1))) -> c#(x1) L#(a(a(x1))) -> b#(c(x1)) L#(a(a(x1))) -> L#(a(b(c(x1)))) c#(R(x1)) -> b#(a(R(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) c(b(x1)) -> b(a(x1)) L(a(a(x1))) -> L(a(b(c(x1)))) c(R(x1)) -> b(a(R(x1))) TDG Processor: DPs: b#(a(a(x1))) -> c#(x1) b#(a(a(x1))) -> b#(c(x1)) c#(a(x1)) -> c#(x1) c#(b(x1)) -> b#(a(x1)) L#(a(a(x1))) -> c#(x1) L#(a(a(x1))) -> b#(c(x1)) L#(a(a(x1))) -> L#(a(b(c(x1)))) c#(R(x1)) -> b#(a(R(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) c(b(x1)) -> b(a(x1)) L(a(a(x1))) -> L(a(b(c(x1)))) c(R(x1)) -> b(a(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))) -> b#(c(x1)) L#(a(a(x1))) -> L#(a(b(c(x1)))) -> L#(a(a(x1))) -> c#(x1) L#(a(a(x1))) -> c#(x1) -> c#(R(x1)) -> b#(a(R(x1))) L#(a(a(x1))) -> c#(x1) -> c#(b(x1)) -> b#(a(x1)) L#(a(a(x1))) -> c#(x1) -> c#(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))) -> c#(x1) c#(R(x1)) -> b#(a(R(x1))) -> b#(a(a(x1))) -> b#(c(x1)) c#(R(x1)) -> b#(a(R(x1))) -> b#(a(a(x1))) -> c#(x1) c#(b(x1)) -> b#(a(x1)) -> b#(a(a(x1))) -> b#(c(x1)) c#(b(x1)) -> b#(a(x1)) -> b#(a(a(x1))) -> c#(x1) c#(a(x1)) -> c#(x1) -> c#(R(x1)) -> b#(a(R(x1))) c#(a(x1)) -> c#(x1) -> c#(b(x1)) -> b#(a(x1)) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1) b#(a(a(x1))) -> c#(x1) -> c#(R(x1)) -> b#(a(R(x1))) b#(a(a(x1))) -> c#(x1) -> c#(b(x1)) -> b#(a(x1)) b#(a(a(x1))) -> c#(x1) -> c#(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))) -> c#(x1) EDG Processor: DPs: b#(a(a(x1))) -> c#(x1) b#(a(a(x1))) -> b#(c(x1)) c#(a(x1)) -> c#(x1) c#(b(x1)) -> b#(a(x1)) L#(a(a(x1))) -> c#(x1) L#(a(a(x1))) -> b#(c(x1)) L#(a(a(x1))) -> L#(a(b(c(x1)))) c#(R(x1)) -> b#(a(R(x1))) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) c(b(x1)) -> b(a(x1)) L(a(a(x1))) -> L(a(b(c(x1)))) c(R(x1)) -> b(a(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))) -> L#(a(b(c(x1)))) L#(a(a(x1))) -> c#(x1) -> c#(a(x1)) -> c#(x1) L#(a(a(x1))) -> c#(x1) -> c#(b(x1)) -> b#(a(x1)) L#(a(a(x1))) -> c#(x1) -> c#(R(x1)) -> b#(a(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)) c#(b(x1)) -> b#(a(x1)) -> b#(a(a(x1))) -> c#(x1) c#(b(x1)) -> b#(a(x1)) -> b#(a(a(x1))) -> b#(c(x1)) c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1) c#(a(x1)) -> c#(x1) -> c#(b(x1)) -> b#(a(x1)) c#(a(x1)) -> c#(x1) -> c#(R(x1)) -> b#(a(R(x1))) b#(a(a(x1))) -> c#(x1) -> c#(a(x1)) -> c#(x1) b#(a(a(x1))) -> c#(x1) -> c#(b(x1)) -> b#(a(x1)) b#(a(a(x1))) -> c#(x1) -> c#(R(x1)) -> b#(a(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)) SCC Processor: #sccs: 2 #rules: 5 #arcs: 18/64 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)) c(b(x1)) -> b(a(x1)) L(a(a(x1))) -> L(a(b(c(x1)))) c(R(x1)) -> b(a(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)) -> b(a(x1)) c(R(x1)) -> b(a(R(x1))) b(a(a(x1))) -> a(b(c(x1))) Arctic Interpretation Processor: dimension: 1 usable rules: c(a(x1)) -> a(c(x1)) c(b(x1)) -> b(a(x1)) c(R(x1)) -> b(a(R(x1))) b(a(a(x1))) -> a(b(c(x1))) interpretation: [L#](x0) = x0 + 0, [R](x0) = -14x0 + 12, [c](x0) = 5x0 + 13, [b](x0) = -2x0 + 0, [a](x0) = 5x0 + 13 orientation: L#(a(a(x1))) = 10x1 + 18 >= 8x1 + 16 = L#(a(b(c(x1)))) c(a(x1)) = 10x1 + 18 >= 10x1 + 18 = a(c(x1)) c(b(x1)) = 3x1 + 13 >= 3x1 + 11 = b(a(x1)) c(R(x1)) = -9x1 + 17 >= -11x1 + 15 = b(a(R(x1))) b(a(a(x1))) = 8x1 + 16 >= 8x1 + 16 = a(b(c(x1))) problem: DPs: TRS: c(a(x1)) -> a(c(x1)) c(b(x1)) -> b(a(x1)) c(R(x1)) -> b(a(R(x1))) b(a(a(x1))) -> a(b(c(x1))) Qed DPs: b#(a(a(x1))) -> b#(c(x1)) b#(a(a(x1))) -> c#(x1) c#(b(x1)) -> b#(a(x1)) c#(a(x1)) -> c#(x1) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) c(b(x1)) -> b(a(x1)) L(a(a(x1))) -> L(a(b(c(x1)))) c(R(x1)) -> b(a(R(x1))) Usable Rule Processor: DPs: b#(a(a(x1))) -> b#(c(x1)) b#(a(a(x1))) -> c#(x1) c#(b(x1)) -> b#(a(x1)) c#(a(x1)) -> c#(x1) TRS: c(a(x1)) -> a(c(x1)) c(b(x1)) -> b(a(x1)) c(R(x1)) -> b(a(R(x1))) b(a(a(x1))) -> a(b(c(x1))) Arctic Interpretation Processor: dimension: 1 usable rules: c(a(x1)) -> a(c(x1)) c(b(x1)) -> b(a(x1)) c(R(x1)) -> b(a(R(x1))) b(a(a(x1))) -> a(b(c(x1))) interpretation: [c#](x0) = 9x0, [b#](x0) = 7x0, [R](x0) = 4x0, [c](x0) = 1x0, [b](x0) = x0, [a](x0) = 1x0 orientation: b#(a(a(x1))) = 9x1 >= 8x1 = b#(c(x1)) b#(a(a(x1))) = 9x1 >= 9x1 = c#(x1) c#(b(x1)) = 9x1 >= 8x1 = b#(a(x1)) c#(a(x1)) = 10x1 >= 9x1 = c#(x1) c(a(x1)) = 2x1 >= 2x1 = a(c(x1)) c(b(x1)) = 1x1 >= 1x1 = b(a(x1)) c(R(x1)) = 5x1 >= 5x1 = b(a(R(x1))) b(a(a(x1))) = 2x1 >= 2x1 = a(b(c(x1))) problem: DPs: b#(a(a(x1))) -> c#(x1) TRS: c(a(x1)) -> a(c(x1)) c(b(x1)) -> b(a(x1)) c(R(x1)) -> b(a(R(x1))) b(a(a(x1))) -> a(b(c(x1))) Restore Modifier: DPs: b#(a(a(x1))) -> c#(x1) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) c(b(x1)) -> b(a(x1)) L(a(a(x1))) -> L(a(b(c(x1)))) c(R(x1)) -> b(a(R(x1))) EDG Processor: DPs: b#(a(a(x1))) -> c#(x1) TRS: b(a(a(x1))) -> a(b(c(x1))) c(a(x1)) -> a(c(x1)) c(b(x1)) -> b(a(x1)) L(a(a(x1))) -> L(a(b(c(x1)))) c(R(x1)) -> b(a(R(x1))) graph: Qed