YES Problem: a(a(b(d(b(d(a(x1))))))) -> a(a(c(a(a(b(d(x1))))))) a(a(c(x1))) -> c(c(a(a(x1)))) c(c(c(x1))) -> b(d(c(b(d(x1))))) Proof: DP Processor: DPs: a#(a(b(d(b(d(a(x1))))))) -> a#(b(d(x1))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) a#(a(b(d(b(d(a(x1))))))) -> c#(a(a(b(d(x1))))) a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) a#(a(c(x1))) -> a#(x1) a#(a(c(x1))) -> a#(a(x1)) a#(a(c(x1))) -> c#(a(a(x1))) a#(a(c(x1))) -> c#(c(a(a(x1)))) c#(c(c(x1))) -> c#(b(d(x1))) TRS: a(a(b(d(b(d(a(x1))))))) -> a(a(c(a(a(b(d(x1))))))) a(a(c(x1))) -> c(c(a(a(x1)))) c(c(c(x1))) -> b(d(c(b(d(x1))))) TDG Processor: DPs: a#(a(b(d(b(d(a(x1))))))) -> a#(b(d(x1))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) a#(a(b(d(b(d(a(x1))))))) -> c#(a(a(b(d(x1))))) a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) a#(a(c(x1))) -> a#(x1) a#(a(c(x1))) -> a#(a(x1)) a#(a(c(x1))) -> c#(a(a(x1))) a#(a(c(x1))) -> c#(c(a(a(x1)))) c#(c(c(x1))) -> c#(b(d(x1))) TRS: a(a(b(d(b(d(a(x1))))))) -> a(a(c(a(a(b(d(x1))))))) a(a(c(x1))) -> c(c(a(a(x1)))) c(c(c(x1))) -> b(d(c(b(d(x1))))) graph: c#(c(c(x1))) -> c#(b(d(x1))) -> c#(c(c(x1))) -> c#(b(d(x1))) a#(a(c(x1))) -> c#(c(a(a(x1)))) -> c#(c(c(x1))) -> c#(b(d(x1))) a#(a(c(x1))) -> c#(a(a(x1))) -> c#(c(c(x1))) -> c#(b(d(x1))) a#(a(c(x1))) -> a#(a(x1)) -> a#(a(c(x1))) -> c#(c(a(a(x1)))) a#(a(c(x1))) -> a#(a(x1)) -> a#(a(c(x1))) -> c#(a(a(x1))) a#(a(c(x1))) -> a#(a(x1)) -> a#(a(c(x1))) -> a#(a(x1)) a#(a(c(x1))) -> a#(a(x1)) -> a#(a(c(x1))) -> a#(x1) a#(a(c(x1))) -> a#(a(x1)) -> a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) a#(a(c(x1))) -> a#(a(x1)) -> a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) a#(a(c(x1))) -> a#(a(x1)) -> a#(a(b(d(b(d(a(x1))))))) -> c#(a(a(b(d(x1))))) a#(a(c(x1))) -> a#(a(x1)) -> a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) a#(a(c(x1))) -> a#(a(x1)) -> a#(a(b(d(b(d(a(x1))))))) -> a#(b(d(x1))) a#(a(c(x1))) -> a#(x1) -> a#(a(c(x1))) -> c#(c(a(a(x1)))) a#(a(c(x1))) -> a#(x1) -> a#(a(c(x1))) -> c#(a(a(x1))) a#(a(c(x1))) -> a#(x1) -> a#(a(c(x1))) -> a#(a(x1)) a#(a(c(x1))) -> a#(x1) -> a#(a(c(x1))) -> a#(x1) a#(a(c(x1))) -> a#(x1) -> a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) a#(a(c(x1))) -> a#(x1) -> a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) a#(a(c(x1))) -> a#(x1) -> a#(a(b(d(b(d(a(x1))))))) -> c#(a(a(b(d(x1))))) a#(a(c(x1))) -> a#(x1) -> a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) a#(a(c(x1))) -> a#(x1) -> a#(a(b(d(b(d(a(x1))))))) -> a#(b(d(x1))) a#(a(b(d(b(d(a(x1))))))) -> c#(a(a(b(d(x1))))) -> c#(c(c(x1))) -> c#(b(d(x1))) a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) -> a#(a(c(x1))) -> c#(c(a(a(x1)))) a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) -> a#(a(c(x1))) -> c#(a(a(x1))) a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) -> a#(a(c(x1))) -> a#(a(x1)) a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) -> a#(a(c(x1))) -> a#(x1) a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) -> a#(a(b(d(b(d(a(x1))))))) -> c#(a(a(b(d(x1))))) a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(b(d(x1))) a#(a(b(d(b(d(a(x1))))))) -> a#(b(d(x1))) -> a#(a(c(x1))) -> c#(c(a(a(x1)))) a#(a(b(d(b(d(a(x1))))))) -> a#(b(d(x1))) -> a#(a(c(x1))) -> c#(a(a(x1))) a#(a(b(d(b(d(a(x1))))))) -> a#(b(d(x1))) -> a#(a(c(x1))) -> a#(a(x1)) a#(a(b(d(b(d(a(x1))))))) -> a#(b(d(x1))) -> a#(a(c(x1))) -> a#(x1) a#(a(b(d(b(d(a(x1))))))) -> a#(b(d(x1))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) a#(a(b(d(b(d(a(x1))))))) -> a#(b(d(x1))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) a#(a(b(d(b(d(a(x1))))))) -> a#(b(d(x1))) -> a#(a(b(d(b(d(a(x1))))))) -> c#(a(a(b(d(x1))))) a#(a(b(d(b(d(a(x1))))))) -> a#(b(d(x1))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) a#(a(b(d(b(d(a(x1))))))) -> a#(b(d(x1))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(b(d(x1))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) -> a#(a(c(x1))) -> c#(c(a(a(x1)))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) -> a#(a(c(x1))) -> c#(a(a(x1))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) -> a#(a(c(x1))) -> a#(a(x1)) a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) -> a#(a(c(x1))) -> a#(x1) a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) -> a#(a(b(d(b(d(a(x1))))))) -> c#(a(a(b(d(x1))))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(b(d(x1))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) -> a#(a(c(x1))) -> c#(c(a(a(x1)))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) -> a#(a(c(x1))) -> c#(a(a(x1))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) -> a#(a(c(x1))) -> a#(a(x1)) a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) -> a#(a(c(x1))) -> a#(x1) a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) -> a#(a(b(d(b(d(a(x1))))))) -> c#(a(a(b(d(x1))))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(b(d(x1))) EDG Processor: DPs: a#(a(b(d(b(d(a(x1))))))) -> a#(b(d(x1))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) a#(a(b(d(b(d(a(x1))))))) -> c#(a(a(b(d(x1))))) a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) a#(a(c(x1))) -> a#(x1) a#(a(c(x1))) -> a#(a(x1)) a#(a(c(x1))) -> c#(a(a(x1))) a#(a(c(x1))) -> c#(c(a(a(x1)))) c#(c(c(x1))) -> c#(b(d(x1))) TRS: a(a(b(d(b(d(a(x1))))))) -> a(a(c(a(a(b(d(x1))))))) a(a(c(x1))) -> c(c(a(a(x1)))) c(c(c(x1))) -> b(d(c(b(d(x1))))) graph: a#(a(c(x1))) -> c#(c(a(a(x1)))) -> c#(c(c(x1))) -> c#(b(d(x1))) a#(a(c(x1))) -> c#(a(a(x1))) -> c#(c(c(x1))) -> c#(b(d(x1))) a#(a(c(x1))) -> a#(a(x1)) -> a#(a(b(d(b(d(a(x1))))))) -> a#(b(d(x1))) a#(a(c(x1))) -> a#(a(x1)) -> a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) a#(a(c(x1))) -> a#(a(x1)) -> a#(a(b(d(b(d(a(x1))))))) -> c#(a(a(b(d(x1))))) a#(a(c(x1))) -> a#(a(x1)) -> a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) a#(a(c(x1))) -> a#(a(x1)) -> a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) a#(a(c(x1))) -> a#(a(x1)) -> a#(a(c(x1))) -> a#(x1) a#(a(c(x1))) -> a#(a(x1)) -> a#(a(c(x1))) -> a#(a(x1)) a#(a(c(x1))) -> a#(a(x1)) -> a#(a(c(x1))) -> c#(a(a(x1))) a#(a(c(x1))) -> a#(a(x1)) -> a#(a(c(x1))) -> c#(c(a(a(x1)))) a#(a(c(x1))) -> a#(x1) -> a#(a(b(d(b(d(a(x1))))))) -> a#(b(d(x1))) a#(a(c(x1))) -> a#(x1) -> a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) a#(a(c(x1))) -> a#(x1) -> a#(a(b(d(b(d(a(x1))))))) -> c#(a(a(b(d(x1))))) a#(a(c(x1))) -> a#(x1) -> a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) a#(a(c(x1))) -> a#(x1) -> a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) a#(a(c(x1))) -> a#(x1) -> a#(a(c(x1))) -> a#(x1) a#(a(c(x1))) -> a#(x1) -> a#(a(c(x1))) -> a#(a(x1)) a#(a(c(x1))) -> a#(x1) -> a#(a(c(x1))) -> c#(a(a(x1))) a#(a(c(x1))) -> a#(x1) -> a#(a(c(x1))) -> c#(c(a(a(x1)))) a#(a(b(d(b(d(a(x1))))))) -> c#(a(a(b(d(x1))))) -> c#(c(c(x1))) -> c#(b(d(x1))) a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(b(d(x1))) a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) -> a#(a(b(d(b(d(a(x1))))))) -> c#(a(a(b(d(x1))))) a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) -> a#(a(c(x1))) -> a#(x1) a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) -> a#(a(c(x1))) -> a#(a(x1)) a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) -> a#(a(c(x1))) -> c#(a(a(x1))) a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) -> a#(a(c(x1))) -> c#(c(a(a(x1)))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(b(d(x1))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) -> a#(a(b(d(b(d(a(x1))))))) -> c#(a(a(b(d(x1))))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) -> a#(a(c(x1))) -> a#(x1) a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) -> a#(a(c(x1))) -> a#(a(x1)) a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) -> a#(a(c(x1))) -> c#(a(a(x1))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) -> a#(a(c(x1))) -> c#(c(a(a(x1)))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(b(d(x1))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) -> a#(a(b(d(b(d(a(x1))))))) -> c#(a(a(b(d(x1))))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) -> a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) SCC Processor: #sccs: 1 #rules: 5 #arcs: 44/100 DPs: a#(a(c(x1))) -> a#(a(x1)) a#(a(c(x1))) -> a#(x1) a#(a(b(d(b(d(a(x1))))))) -> a#(a(c(a(a(b(d(x1))))))) a#(a(b(d(b(d(a(x1))))))) -> a#(c(a(a(b(d(x1)))))) a#(a(b(d(b(d(a(x1))))))) -> a#(a(b(d(x1)))) TRS: a(a(b(d(b(d(a(x1))))))) -> a(a(c(a(a(b(d(x1))))))) a(a(c(x1))) -> c(c(a(a(x1)))) c(c(c(x1))) -> b(d(c(b(d(x1))))) Matrix Interpretation Processor: dim=5 interpretation: [a#](x0) = [0 1 0 0 0]x0, [0 0 0 0 0] [0] [0 1 0 0 0] [1] [c](x0) = [0 0 0 0 1]x0 + [0] [0 0 0 0 0] [0] [0 0 1 1 0] [0], [0 0 0 0 0] [0 0 0 0 0] [b](x0) = [0 0 0 0 0]x0 [0 1 1 0 1] [0 0 0 1 0] , [0 0 0 1 0] [1 0 0 0 0] [d](x0) = [1 0 0 0 0]x0 [0 0 0 1 0] [1 0 0 0 0] , [1 0 0 1 0] [1] [0 1 1 0 1] [0] [a](x0) = [0 1 0 0 0]x0 + [0] [0 0 0 0 0] [0] [0 0 0 0 0] [0] orientation: a#(a(c(x1))) = [0 1 1 1 1]x1 + [1] >= [0 1 1 0 1]x1 = a#(a(x1)) a#(a(c(x1))) = [0 1 1 1 1]x1 + [1] >= [0 1 0 0 0]x1 = a#(x1) a#(a(b(d(b(d(a(x1))))))) = [3 0 0 3 0]x1 + [3] >= [0 0 0 2 0]x1 + [1] = a#(a(c(a(a(b(d(x1))))))) a#(a(b(d(b(d(a(x1))))))) = [3 0 0 3 0]x1 + [3] >= [0 0 0 1 0]x1 + [1] = a#(c(a(a(b(d(x1)))))) a#(a(b(d(b(d(a(x1))))))) = [3 0 0 3 0]x1 + [3] >= [0 0 0 1 0]x1 = a#(a(b(d(x1)))) [0 0 0 0 0] [2] [0 0 0 0 0] [2] [3 0 0 3 0] [3] [0 0 0 3 0] [2] a(a(b(d(b(d(a(x1))))))) = [3 0 0 3 0]x1 + [3] >= [0 0 0 2 0]x1 + [1] = a(a(c(a(a(b(d(x1))))))) [0 0 0 0 0] [0] [0 0 0 0 0] [0] [0 0 0 0 0] [0] [0 0 0 0 0] [0] [0 0 0 0 0] [2] [0 0 0 0 0] [0] [0 2 1 1 1] [2] [0 2 1 0 1] [2] a(a(c(x1))) = [0 1 1 1 1]x1 + [1] >= [0 1 1 0 1]x1 + [0] = c(c(a(a(x1)))) [0 0 0 0 0] [0] [0 0 0 0 0] [0] [0 0 0 0 0] [0] [0 0 0 0 0] [0] [0 0 0 0 0] [0] [0] [0 1 0 0 0] [3] [0] c(c(c(x1))) = [0 0 0 0 1]x1 + [0] >= [0] = b(d(c(b(d(x1))))) [0 0 0 0 0] [0] [0] [0 0 1 1 0] [0] [0] problem: DPs: TRS: a(a(b(d(b(d(a(x1))))))) -> a(a(c(a(a(b(d(x1))))))) a(a(c(x1))) -> c(c(a(a(x1)))) c(c(c(x1))) -> b(d(c(b(d(x1))))) Qed