YES Problem: a(a(a(x1))) -> a(c(a(a(x1)))) c(c(c(x1))) -> a(x1) a(x1) -> x1 Proof: DP Processor: DPs: a#(a(a(x1))) -> c#(a(a(x1))) a#(a(a(x1))) -> a#(c(a(a(x1)))) c#(c(c(x1))) -> a#(x1) TRS: a(a(a(x1))) -> a(c(a(a(x1)))) c(c(c(x1))) -> a(x1) a(x1) -> x1 TDG Processor: DPs: a#(a(a(x1))) -> c#(a(a(x1))) a#(a(a(x1))) -> a#(c(a(a(x1)))) c#(c(c(x1))) -> a#(x1) TRS: a(a(a(x1))) -> a(c(a(a(x1)))) c(c(c(x1))) -> a(x1) a(x1) -> x1 graph: c#(c(c(x1))) -> a#(x1) -> a#(a(a(x1))) -> a#(c(a(a(x1)))) c#(c(c(x1))) -> a#(x1) -> a#(a(a(x1))) -> c#(a(a(x1))) a#(a(a(x1))) -> c#(a(a(x1))) -> c#(c(c(x1))) -> a#(x1) a#(a(a(x1))) -> a#(c(a(a(x1)))) -> a#(a(a(x1))) -> a#(c(a(a(x1)))) a#(a(a(x1))) -> a#(c(a(a(x1)))) -> a#(a(a(x1))) -> c#(a(a(x1))) Root-Labeling Processor: DPs: a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(f4)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(a)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(c)(x1))))) c{#,(f4)}(f4(c)(c(c)(c(f4)(x1)))) -> a{#,(f4)}(f4(f4)(x1)) c{#,(f4)}(f4(c)(c(c)(c(a)(x1)))) -> a{#,(f4)}(f4(a)(x1)) c{#,(f4)}(f4(c)(c(c)(c(c)(x1)))) -> a{#,(f4)}(f4(c)(x1)) TRS: a(a)(a(a)(a(f4)(x1))) -> a(c)(c(a)(a(a)(a(f4)(x1)))) a(a)(a(a)(a(a)(x1))) -> a(c)(c(a)(a(a)(a(a)(x1)))) a(a)(a(a)(a(c)(x1))) -> a(c)(c(a)(a(a)(a(c)(x1)))) f4(c)(c(c)(c(c)(c(f4)(x1)))) -> f4(a)(a(f4)(x1)) f4(c)(c(c)(c(c)(c(a)(x1)))) -> f4(a)(a(a)(x1)) f4(c)(c(c)(c(c)(c(c)(x1)))) -> f4(a)(a(c)(x1)) a(c)(c(c)(c(c)(c(f4)(x1)))) -> a(a)(a(f4)(x1)) a(c)(c(c)(c(c)(c(a)(x1)))) -> a(a)(a(a)(x1)) a(c)(c(c)(c(c)(c(c)(x1)))) -> a(a)(a(c)(x1)) c(c)(c(c)(c(c)(c(f4)(x1)))) -> c(a)(a(f4)(x1)) c(c)(c(c)(c(c)(c(a)(x1)))) -> c(a)(a(a)(x1)) c(c)(c(c)(c(c)(c(c)(x1)))) -> c(a)(a(c)(x1)) f4(a)(a(f4)(x1)) -> f4(f4)(x1) f4(a)(a(a)(x1)) -> f4(a)(x1) f4(a)(a(c)(x1)) -> f4(c)(x1) a(a)(a(f4)(x1)) -> a(f4)(x1) a(a)(a(a)(x1)) -> a(a)(x1) a(a)(a(c)(x1)) -> a(c)(x1) c(a)(a(f4)(x1)) -> c(f4)(x1) c(a)(a(a)(x1)) -> c(a)(x1) c(a)(a(c)(x1)) -> c(c)(x1) TDG Processor: DPs: a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(f4)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(a)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(c)(x1))))) c{#,(f4)}(f4(c)(c(c)(c(f4)(x1)))) -> a{#,(f4)}(f4(f4)(x1)) c{#,(f4)}(f4(c)(c(c)(c(a)(x1)))) -> a{#,(f4)}(f4(a)(x1)) c{#,(f4)}(f4(c)(c(c)(c(c)(x1)))) -> a{#,(f4)}(f4(c)(x1)) TRS: a(a)(a(a)(a(f4)(x1))) -> a(c)(c(a)(a(a)(a(f4)(x1)))) a(a)(a(a)(a(a)(x1))) -> a(c)(c(a)(a(a)(a(a)(x1)))) a(a)(a(a)(a(c)(x1))) -> a(c)(c(a)(a(a)(a(c)(x1)))) f4(c)(c(c)(c(c)(c(f4)(x1)))) -> f4(a)(a(f4)(x1)) f4(c)(c(c)(c(c)(c(a)(x1)))) -> f4(a)(a(a)(x1)) f4(c)(c(c)(c(c)(c(c)(x1)))) -> f4(a)(a(c)(x1)) a(c)(c(c)(c(c)(c(f4)(x1)))) -> a(a)(a(f4)(x1)) a(c)(c(c)(c(c)(c(a)(x1)))) -> a(a)(a(a)(x1)) a(c)(c(c)(c(c)(c(c)(x1)))) -> a(a)(a(c)(x1)) c(c)(c(c)(c(c)(c(f4)(x1)))) -> c(a)(a(f4)(x1)) c(c)(c(c)(c(c)(c(a)(x1)))) -> c(a)(a(a)(x1)) c(c)(c(c)(c(c)(c(c)(x1)))) -> c(a)(a(c)(x1)) f4(a)(a(f4)(x1)) -> f4(f4)(x1) f4(a)(a(a)(x1)) -> f4(a)(x1) f4(a)(a(c)(x1)) -> f4(c)(x1) a(a)(a(f4)(x1)) -> a(f4)(x1) a(a)(a(a)(x1)) -> a(a)(x1) a(a)(a(c)(x1)) -> a(c)(x1) c(a)(a(f4)(x1)) -> c(f4)(x1) c(a)(a(a)(x1)) -> c(a)(x1) c(a)(a(c)(x1)) -> c(c)(x1) graph: c{#,(f4)}(f4(c)(c(c)(c(f4)(x1)))) -> a{#,(f4)}(f4(f4)(x1)) -> a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(c)(x1))))) c{#,(f4)}(f4(c)(c(c)(c(f4)(x1)))) -> a{#,(f4)}(f4(f4)(x1)) -> a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(a)(x1))))) c{#,(f4)}(f4(c)(c(c)(c(f4)(x1)))) -> a{#,(f4)}(f4(f4)(x1)) -> a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(f4)(x1))))) c{#,(f4)}(f4(c)(c(c)(c(f4)(x1)))) -> a{#,(f4)}(f4(f4)(x1)) -> a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) c{#,(f4)}(f4(c)(c(c)(c(f4)(x1)))) -> a{#,(f4)}(f4(f4)(x1)) -> a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) c{#,(f4)}(f4(c)(c(c)(c(f4)(x1)))) -> a{#,(f4)}(f4(f4)(x1)) -> a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) c{#,(f4)}(f4(c)(c(c)(c(c)(x1)))) -> a{#,(f4)}(f4(c)(x1)) -> a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(c)(x1))))) c{#,(f4)}(f4(c)(c(c)(c(c)(x1)))) -> a{#,(f4)}(f4(c)(x1)) -> a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(a)(x1))))) c{#,(f4)}(f4(c)(c(c)(c(c)(x1)))) -> a{#,(f4)}(f4(c)(x1)) -> a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(f4)(x1))))) c{#,(f4)}(f4(c)(c(c)(c(c)(x1)))) -> a{#,(f4)}(f4(c)(x1)) -> a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) c{#,(f4)}(f4(c)(c(c)(c(c)(x1)))) -> a{#,(f4)}(f4(c)(x1)) -> a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) c{#,(f4)}(f4(c)(c(c)(c(c)(x1)))) -> a{#,(f4)}(f4(c)(x1)) -> a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) c{#,(f4)}(f4(c)(c(c)(c(a)(x1)))) -> a{#,(f4)}(f4(a)(x1)) -> a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(c)(x1))))) c{#,(f4)}(f4(c)(c(c)(c(a)(x1)))) -> a{#,(f4)}(f4(a)(x1)) -> a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(a)(x1))))) c{#,(f4)}(f4(c)(c(c)(c(a)(x1)))) -> a{#,(f4)}(f4(a)(x1)) -> a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(f4)(x1))))) c{#,(f4)}(f4(c)(c(c)(c(a)(x1)))) -> a{#,(f4)}(f4(a)(x1)) -> a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) c{#,(f4)}(f4(c)(c(c)(c(a)(x1)))) -> a{#,(f4)}(f4(a)(x1)) -> a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) c{#,(f4)}(f4(c)(c(c)(c(a)(x1)))) -> a{#,(f4)}(f4(a)(x1)) -> a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> c{#,(f4)}(f4(c)(c(c)(c(c)(x1)))) -> a{#,(f4)}(f4(c)(x1)) a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> c{#,(f4)}(f4(c)(c(c)(c(a)(x1)))) -> a{#,(f4)}(f4(a)(x1)) a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> c{#,(f4)}(f4(c)(c(c)(c(f4)(x1)))) -> a{#,(f4)}(f4(f4)(x1)) a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(c)(x1))))) -> a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(c)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(c)(x1))))) -> a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(a)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(c)(x1))))) -> a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(f4)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(c)(x1))))) -> a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(c)(x1))))) -> a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(c)(x1))))) -> a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> c{#,(f4)}(f4(c)(c(c)(c(c)(x1)))) -> a{#,(f4)}(f4(c)(x1)) a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> c{#,(f4)}(f4(c)(c(c)(c(a)(x1)))) -> a{#,(f4)}(f4(a)(x1)) a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> c{#,(f4)}(f4(c)(c(c)(c(f4)(x1)))) -> a{#,(f4)}(f4(f4)(x1)) a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(f4)(x1))))) -> a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(c)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(f4)(x1))))) -> a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(a)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(f4)(x1))))) -> a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(f4)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(f4)(x1))))) -> a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(f4)(x1))))) -> a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(f4)(x1))))) -> a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> c{#,(f4)}(f4(c)(c(c)(c(c)(x1)))) -> a{#,(f4)}(f4(c)(x1)) a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> c{#,(f4)}(f4(c)(c(c)(c(a)(x1)))) -> a{#,(f4)}(f4(a)(x1)) a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> c{#,(f4)}(f4(c)(c(c)(c(f4)(x1)))) -> a{#,(f4)}(f4(f4)(x1)) a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(a)(x1))))) -> a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(c)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(a)(x1))))) -> a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(a)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(a)(x1))))) -> a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(f4)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(a)(x1))))) -> a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(a)(x1))))) -> a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(a)(x1))))) -> a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) Polynomial Interpretation Processor: dimension: 1 interpretation: [f4(f4)](x0) = x0, [a(f4)](x0) = x0, [a{#,(f4)}](x0) = x0, [a(c)](x0) = x0, [a(a)](x0) = x0, [c(f4)](x0) = x0, [c(a)](x0) = x0, [f4(c)](x0) = x0 + 1, [c{#,(f4)}](x0) = x0, [c(c)](x0) = x0, [f4(a)](x0) = x0 + 1 orientation: a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) = x1 + 1 >= x1 + 1 = c{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) = x1 + 1 >= x1 + 1 = c{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) = x1 + 1 >= x1 + 1 = c{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) = x1 + 1 >= x1 + 1 = a{#,(f4)}(f4(c)(c(a)(a(a)(a(f4)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) = x1 + 1 >= x1 + 1 = a{#,(f4)}(f4(c)(c(a)(a(a)(a(a)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) = x1 + 1 >= x1 + 1 = a{#,(f4)}(f4(c)(c(a)(a(a)(a(c)(x1))))) c{#,(f4)}(f4(c)(c(c)(c(f4)(x1)))) = x1 + 1 >= x1 = a{#,(f4)}(f4(f4)(x1)) c{#,(f4)}(f4(c)(c(c)(c(a)(x1)))) = x1 + 1 >= x1 + 1 = a{#,(f4)}(f4(a)(x1)) c{#,(f4)}(f4(c)(c(c)(c(c)(x1)))) = x1 + 1 >= x1 + 1 = a{#,(f4)}(f4(c)(x1)) a(a)(a(a)(a(f4)(x1))) = x1 >= x1 = a(c)(c(a)(a(a)(a(f4)(x1)))) a(a)(a(a)(a(a)(x1))) = x1 >= x1 = a(c)(c(a)(a(a)(a(a)(x1)))) a(a)(a(a)(a(c)(x1))) = x1 >= x1 = a(c)(c(a)(a(a)(a(c)(x1)))) f4(c)(c(c)(c(c)(c(f4)(x1)))) = x1 + 1 >= x1 + 1 = f4(a)(a(f4)(x1)) f4(c)(c(c)(c(c)(c(a)(x1)))) = x1 + 1 >= x1 + 1 = f4(a)(a(a)(x1)) f4(c)(c(c)(c(c)(c(c)(x1)))) = x1 + 1 >= x1 + 1 = f4(a)(a(c)(x1)) a(c)(c(c)(c(c)(c(f4)(x1)))) = x1 >= x1 = a(a)(a(f4)(x1)) a(c)(c(c)(c(c)(c(a)(x1)))) = x1 >= x1 = a(a)(a(a)(x1)) a(c)(c(c)(c(c)(c(c)(x1)))) = x1 >= x1 = a(a)(a(c)(x1)) c(c)(c(c)(c(c)(c(f4)(x1)))) = x1 >= x1 = c(a)(a(f4)(x1)) c(c)(c(c)(c(c)(c(a)(x1)))) = x1 >= x1 = c(a)(a(a)(x1)) c(c)(c(c)(c(c)(c(c)(x1)))) = x1 >= x1 = c(a)(a(c)(x1)) f4(a)(a(f4)(x1)) = x1 + 1 >= x1 = f4(f4)(x1) f4(a)(a(a)(x1)) = x1 + 1 >= x1 + 1 = f4(a)(x1) f4(a)(a(c)(x1)) = x1 + 1 >= x1 + 1 = f4(c)(x1) a(a)(a(f4)(x1)) = x1 >= x1 = a(f4)(x1) a(a)(a(a)(x1)) = x1 >= x1 = a(a)(x1) a(a)(a(c)(x1)) = x1 >= x1 = a(c)(x1) c(a)(a(f4)(x1)) = x1 >= x1 = c(f4)(x1) c(a)(a(a)(x1)) = x1 >= x1 = c(a)(x1) c(a)(a(c)(x1)) = x1 >= x1 = c(c)(x1) problem: DPs: a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(f4)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(a)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(c)(x1))))) c{#,(f4)}(f4(c)(c(c)(c(a)(x1)))) -> a{#,(f4)}(f4(a)(x1)) c{#,(f4)}(f4(c)(c(c)(c(c)(x1)))) -> a{#,(f4)}(f4(c)(x1)) TRS: a(a)(a(a)(a(f4)(x1))) -> a(c)(c(a)(a(a)(a(f4)(x1)))) a(a)(a(a)(a(a)(x1))) -> a(c)(c(a)(a(a)(a(a)(x1)))) a(a)(a(a)(a(c)(x1))) -> a(c)(c(a)(a(a)(a(c)(x1)))) f4(c)(c(c)(c(c)(c(f4)(x1)))) -> f4(a)(a(f4)(x1)) f4(c)(c(c)(c(c)(c(a)(x1)))) -> f4(a)(a(a)(x1)) f4(c)(c(c)(c(c)(c(c)(x1)))) -> f4(a)(a(c)(x1)) a(c)(c(c)(c(c)(c(f4)(x1)))) -> a(a)(a(f4)(x1)) a(c)(c(c)(c(c)(c(a)(x1)))) -> a(a)(a(a)(x1)) a(c)(c(c)(c(c)(c(c)(x1)))) -> a(a)(a(c)(x1)) c(c)(c(c)(c(c)(c(f4)(x1)))) -> c(a)(a(f4)(x1)) c(c)(c(c)(c(c)(c(a)(x1)))) -> c(a)(a(a)(x1)) c(c)(c(c)(c(c)(c(c)(x1)))) -> c(a)(a(c)(x1)) f4(a)(a(a)(x1)) -> f4(a)(x1) f4(a)(a(c)(x1)) -> f4(c)(x1) a(a)(a(f4)(x1)) -> a(f4)(x1) a(a)(a(a)(x1)) -> a(a)(x1) a(a)(a(c)(x1)) -> a(c)(x1) c(a)(a(f4)(x1)) -> c(f4)(x1) c(a)(a(a)(x1)) -> c(a)(x1) c(a)(a(c)(x1)) -> c(c)(x1) Polynomial Interpretation Processor: dimension: 1 usable rules: a(a)(a(a)(a(f4)(x1))) -> a(c)(c(a)(a(a)(a(f4)(x1)))) a(a)(a(a)(a(a)(x1))) -> a(c)(c(a)(a(a)(a(a)(x1)))) a(a)(a(a)(a(c)(x1))) -> a(c)(c(a)(a(a)(a(c)(x1)))) f4(c)(c(c)(c(c)(c(f4)(x1)))) -> f4(a)(a(f4)(x1)) f4(c)(c(c)(c(c)(c(a)(x1)))) -> f4(a)(a(a)(x1)) f4(c)(c(c)(c(c)(c(c)(x1)))) -> f4(a)(a(c)(x1)) a(c)(c(c)(c(c)(c(f4)(x1)))) -> a(a)(a(f4)(x1)) a(c)(c(c)(c(c)(c(a)(x1)))) -> a(a)(a(a)(x1)) a(c)(c(c)(c(c)(c(c)(x1)))) -> a(a)(a(c)(x1)) c(c)(c(c)(c(c)(c(f4)(x1)))) -> c(a)(a(f4)(x1)) c(c)(c(c)(c(c)(c(a)(x1)))) -> c(a)(a(a)(x1)) c(c)(c(c)(c(c)(c(c)(x1)))) -> c(a)(a(c)(x1)) f4(a)(a(a)(x1)) -> f4(a)(x1) f4(a)(a(c)(x1)) -> f4(c)(x1) a(a)(a(f4)(x1)) -> a(f4)(x1) a(a)(a(a)(x1)) -> a(a)(x1) a(a)(a(c)(x1)) -> a(c)(x1) c(a)(a(f4)(x1)) -> c(f4)(x1) c(a)(a(a)(x1)) -> c(a)(x1) c(a)(a(c)(x1)) -> c(c)(x1) interpretation: [a(f4)](x0) = 1, [a{#,(f4)}](x0) = x0, [a(c)](x0) = x0 + 1, [a(a)](x0) = x0 + 1, [c(f4)](x0) = 1, [c(a)](x0) = x0, [f4(c)](x0) = x0, [c{#,(f4)}](x0) = x0, [c(c)](x0) = x0 + 1, [f4(a)](x0) = x0 orientation: a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) = 2 >= 2 = c{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) = x1 + 2 >= x1 + 2 = c{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) = x1 + 2 >= x1 + 2 = c{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) = 2 >= 2 = a{#,(f4)}(f4(c)(c(a)(a(a)(a(f4)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) = x1 + 2 >= x1 + 2 = a{#,(f4)}(f4(c)(c(a)(a(a)(a(a)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) = x1 + 2 >= x1 + 2 = a{#,(f4)}(f4(c)(c(a)(a(a)(a(c)(x1))))) c{#,(f4)}(f4(c)(c(c)(c(a)(x1)))) = x1 + 1 >= x1 = a{#,(f4)}(f4(a)(x1)) c{#,(f4)}(f4(c)(c(c)(c(c)(x1)))) = x1 + 2 >= x1 = a{#,(f4)}(f4(c)(x1)) a(a)(a(a)(a(f4)(x1))) = 3 >= 3 = a(c)(c(a)(a(a)(a(f4)(x1)))) a(a)(a(a)(a(a)(x1))) = x1 + 3 >= x1 + 3 = a(c)(c(a)(a(a)(a(a)(x1)))) a(a)(a(a)(a(c)(x1))) = x1 + 3 >= x1 + 3 = a(c)(c(a)(a(a)(a(c)(x1)))) f4(c)(c(c)(c(c)(c(f4)(x1)))) = 3 >= 1 = f4(a)(a(f4)(x1)) f4(c)(c(c)(c(c)(c(a)(x1)))) = x1 + 2 >= x1 + 1 = f4(a)(a(a)(x1)) f4(c)(c(c)(c(c)(c(c)(x1)))) = x1 + 3 >= x1 + 1 = f4(a)(a(c)(x1)) a(c)(c(c)(c(c)(c(f4)(x1)))) = 4 >= 2 = a(a)(a(f4)(x1)) a(c)(c(c)(c(c)(c(a)(x1)))) = x1 + 3 >= x1 + 2 = a(a)(a(a)(x1)) a(c)(c(c)(c(c)(c(c)(x1)))) = x1 + 4 >= x1 + 2 = a(a)(a(c)(x1)) c(c)(c(c)(c(c)(c(f4)(x1)))) = 4 >= 1 = c(a)(a(f4)(x1)) c(c)(c(c)(c(c)(c(a)(x1)))) = x1 + 3 >= x1 + 1 = c(a)(a(a)(x1)) c(c)(c(c)(c(c)(c(c)(x1)))) = x1 + 4 >= x1 + 1 = c(a)(a(c)(x1)) f4(a)(a(a)(x1)) = x1 + 1 >= x1 = f4(a)(x1) f4(a)(a(c)(x1)) = x1 + 1 >= x1 = f4(c)(x1) a(a)(a(f4)(x1)) = 2 >= 1 = a(f4)(x1) a(a)(a(a)(x1)) = x1 + 2 >= x1 + 1 = a(a)(x1) a(a)(a(c)(x1)) = x1 + 2 >= x1 + 1 = a(c)(x1) c(a)(a(f4)(x1)) = 1 >= 1 = c(f4)(x1) c(a)(a(a)(x1)) = x1 + 1 >= x1 = c(a)(x1) c(a)(a(c)(x1)) = x1 + 1 >= x1 + 1 = c(c)(x1) problem: DPs: a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> c{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(f4)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(a)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(c)(x1))))) TRS: a(a)(a(a)(a(f4)(x1))) -> a(c)(c(a)(a(a)(a(f4)(x1)))) a(a)(a(a)(a(a)(x1))) -> a(c)(c(a)(a(a)(a(a)(x1)))) a(a)(a(a)(a(c)(x1))) -> a(c)(c(a)(a(a)(a(c)(x1)))) f4(c)(c(c)(c(c)(c(f4)(x1)))) -> f4(a)(a(f4)(x1)) f4(c)(c(c)(c(c)(c(a)(x1)))) -> f4(a)(a(a)(x1)) f4(c)(c(c)(c(c)(c(c)(x1)))) -> f4(a)(a(c)(x1)) a(c)(c(c)(c(c)(c(f4)(x1)))) -> a(a)(a(f4)(x1)) a(c)(c(c)(c(c)(c(a)(x1)))) -> a(a)(a(a)(x1)) a(c)(c(c)(c(c)(c(c)(x1)))) -> a(a)(a(c)(x1)) c(c)(c(c)(c(c)(c(f4)(x1)))) -> c(a)(a(f4)(x1)) c(c)(c(c)(c(c)(c(a)(x1)))) -> c(a)(a(a)(x1)) c(c)(c(c)(c(c)(c(c)(x1)))) -> c(a)(a(c)(x1)) f4(a)(a(a)(x1)) -> f4(a)(x1) f4(a)(a(c)(x1)) -> f4(c)(x1) a(a)(a(f4)(x1)) -> a(f4)(x1) a(a)(a(a)(x1)) -> a(a)(x1) a(a)(a(c)(x1)) -> a(c)(x1) c(a)(a(f4)(x1)) -> c(f4)(x1) c(a)(a(a)(x1)) -> c(a)(x1) c(a)(a(c)(x1)) -> c(c)(x1) SCC Processor: #sccs: 1 #rules: 3 #arcs: 45/36 DPs: a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(c)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(f4)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) -> a{#,(f4)}(f4(c)(c(a)(a(a)(a(a)(x1))))) TRS: a(a)(a(a)(a(f4)(x1))) -> a(c)(c(a)(a(a)(a(f4)(x1)))) a(a)(a(a)(a(a)(x1))) -> a(c)(c(a)(a(a)(a(a)(x1)))) a(a)(a(a)(a(c)(x1))) -> a(c)(c(a)(a(a)(a(c)(x1)))) f4(c)(c(c)(c(c)(c(f4)(x1)))) -> f4(a)(a(f4)(x1)) f4(c)(c(c)(c(c)(c(a)(x1)))) -> f4(a)(a(a)(x1)) f4(c)(c(c)(c(c)(c(c)(x1)))) -> f4(a)(a(c)(x1)) a(c)(c(c)(c(c)(c(f4)(x1)))) -> a(a)(a(f4)(x1)) a(c)(c(c)(c(c)(c(a)(x1)))) -> a(a)(a(a)(x1)) a(c)(c(c)(c(c)(c(c)(x1)))) -> a(a)(a(c)(x1)) c(c)(c(c)(c(c)(c(f4)(x1)))) -> c(a)(a(f4)(x1)) c(c)(c(c)(c(c)(c(a)(x1)))) -> c(a)(a(a)(x1)) c(c)(c(c)(c(c)(c(c)(x1)))) -> c(a)(a(c)(x1)) f4(a)(a(a)(x1)) -> f4(a)(x1) f4(a)(a(c)(x1)) -> f4(c)(x1) a(a)(a(f4)(x1)) -> a(f4)(x1) a(a)(a(a)(x1)) -> a(a)(x1) a(a)(a(c)(x1)) -> a(c)(x1) c(a)(a(f4)(x1)) -> c(f4)(x1) c(a)(a(a)(x1)) -> c(a)(x1) c(a)(a(c)(x1)) -> c(c)(x1) Polynomial Interpretation Processor: dimension: 1 usable rules: a(a)(a(a)(a(f4)(x1))) -> a(c)(c(a)(a(a)(a(f4)(x1)))) a(a)(a(a)(a(a)(x1))) -> a(c)(c(a)(a(a)(a(a)(x1)))) a(a)(a(a)(a(c)(x1))) -> a(c)(c(a)(a(a)(a(c)(x1)))) f4(c)(c(c)(c(c)(c(f4)(x1)))) -> f4(a)(a(f4)(x1)) f4(c)(c(c)(c(c)(c(a)(x1)))) -> f4(a)(a(a)(x1)) f4(c)(c(c)(c(c)(c(c)(x1)))) -> f4(a)(a(c)(x1)) a(c)(c(c)(c(c)(c(f4)(x1)))) -> a(a)(a(f4)(x1)) a(c)(c(c)(c(c)(c(a)(x1)))) -> a(a)(a(a)(x1)) a(c)(c(c)(c(c)(c(c)(x1)))) -> a(a)(a(c)(x1)) c(c)(c(c)(c(c)(c(f4)(x1)))) -> c(a)(a(f4)(x1)) c(c)(c(c)(c(c)(c(a)(x1)))) -> c(a)(a(a)(x1)) c(c)(c(c)(c(c)(c(c)(x1)))) -> c(a)(a(c)(x1)) f4(a)(a(a)(x1)) -> f4(a)(x1) f4(a)(a(c)(x1)) -> f4(c)(x1) a(a)(a(f4)(x1)) -> a(f4)(x1) a(a)(a(a)(x1)) -> a(a)(x1) a(a)(a(c)(x1)) -> a(c)(x1) c(a)(a(f4)(x1)) -> c(f4)(x1) c(a)(a(a)(x1)) -> c(a)(x1) c(a)(a(c)(x1)) -> c(c)(x1) interpretation: [a(f4)](x0) = 1, [a{#,(f4)}](x0) = x0, [a(c)](x0) = x0 + 1, [a(a)](x0) = x0 + 1, [c(f4)](x0) = 1, [c(a)](x0) = x0, [f4(c)](x0) = x0, [c(c)](x0) = x0 + 1, [f4(a)](x0) = x0 + 1 orientation: a{#,(f4)}(f4(a)(a(a)(a(c)(x1)))) = x1 + 3 >= x1 + 2 = a{#,(f4)}(f4(c)(c(a)(a(a)(a(c)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(f4)(x1)))) = 3 >= 2 = a{#,(f4)}(f4(c)(c(a)(a(a)(a(f4)(x1))))) a{#,(f4)}(f4(a)(a(a)(a(a)(x1)))) = x1 + 3 >= x1 + 2 = a{#,(f4)}(f4(c)(c(a)(a(a)(a(a)(x1))))) a(a)(a(a)(a(f4)(x1))) = 3 >= 3 = a(c)(c(a)(a(a)(a(f4)(x1)))) a(a)(a(a)(a(a)(x1))) = x1 + 3 >= x1 + 3 = a(c)(c(a)(a(a)(a(a)(x1)))) a(a)(a(a)(a(c)(x1))) = x1 + 3 >= x1 + 3 = a(c)(c(a)(a(a)(a(c)(x1)))) f4(c)(c(c)(c(c)(c(f4)(x1)))) = 3 >= 2 = f4(a)(a(f4)(x1)) f4(c)(c(c)(c(c)(c(a)(x1)))) = x1 + 2 >= x1 + 2 = f4(a)(a(a)(x1)) f4(c)(c(c)(c(c)(c(c)(x1)))) = x1 + 3 >= x1 + 2 = f4(a)(a(c)(x1)) a(c)(c(c)(c(c)(c(f4)(x1)))) = 4 >= 2 = a(a)(a(f4)(x1)) a(c)(c(c)(c(c)(c(a)(x1)))) = x1 + 3 >= x1 + 2 = a(a)(a(a)(x1)) a(c)(c(c)(c(c)(c(c)(x1)))) = x1 + 4 >= x1 + 2 = a(a)(a(c)(x1)) c(c)(c(c)(c(c)(c(f4)(x1)))) = 4 >= 1 = c(a)(a(f4)(x1)) c(c)(c(c)(c(c)(c(a)(x1)))) = x1 + 3 >= x1 + 1 = c(a)(a(a)(x1)) c(c)(c(c)(c(c)(c(c)(x1)))) = x1 + 4 >= x1 + 1 = c(a)(a(c)(x1)) f4(a)(a(a)(x1)) = x1 + 2 >= x1 + 1 = f4(a)(x1) f4(a)(a(c)(x1)) = x1 + 2 >= x1 = f4(c)(x1) a(a)(a(f4)(x1)) = 2 >= 1 = a(f4)(x1) a(a)(a(a)(x1)) = x1 + 2 >= x1 + 1 = a(a)(x1) a(a)(a(c)(x1)) = x1 + 2 >= x1 + 1 = a(c)(x1) c(a)(a(f4)(x1)) = 1 >= 1 = c(f4)(x1) c(a)(a(a)(x1)) = x1 + 1 >= x1 = c(a)(x1) c(a)(a(c)(x1)) = x1 + 1 >= x1 + 1 = c(c)(x1) problem: DPs: TRS: a(a)(a(a)(a(f4)(x1))) -> a(c)(c(a)(a(a)(a(f4)(x1)))) a(a)(a(a)(a(a)(x1))) -> a(c)(c(a)(a(a)(a(a)(x1)))) a(a)(a(a)(a(c)(x1))) -> a(c)(c(a)(a(a)(a(c)(x1)))) f4(c)(c(c)(c(c)(c(f4)(x1)))) -> f4(a)(a(f4)(x1)) f4(c)(c(c)(c(c)(c(a)(x1)))) -> f4(a)(a(a)(x1)) f4(c)(c(c)(c(c)(c(c)(x1)))) -> f4(a)(a(c)(x1)) a(c)(c(c)(c(c)(c(f4)(x1)))) -> a(a)(a(f4)(x1)) a(c)(c(c)(c(c)(c(a)(x1)))) -> a(a)(a(a)(x1)) a(c)(c(c)(c(c)(c(c)(x1)))) -> a(a)(a(c)(x1)) c(c)(c(c)(c(c)(c(f4)(x1)))) -> c(a)(a(f4)(x1)) c(c)(c(c)(c(c)(c(a)(x1)))) -> c(a)(a(a)(x1)) c(c)(c(c)(c(c)(c(c)(x1)))) -> c(a)(a(c)(x1)) f4(a)(a(a)(x1)) -> f4(a)(x1) f4(a)(a(c)(x1)) -> f4(c)(x1) a(a)(a(f4)(x1)) -> a(f4)(x1) a(a)(a(a)(x1)) -> a(a)(x1) a(a)(a(c)(x1)) -> a(c)(x1) c(a)(a(f4)(x1)) -> c(f4)(x1) c(a)(a(a)(x1)) -> c(a)(x1) c(a)(a(c)(x1)) -> c(c)(x1) Qed