YES Problem: a(x1) -> x1 a(b(x1)) -> c(b(b(a(a(x1))))) b(b(x1)) -> x1 c(c(x1)) -> x1 Proof: DP Processor: DPs: a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> b#(a(a(x1))) a#(b(x1)) -> b#(b(a(a(x1)))) a#(b(x1)) -> c#(b(b(a(a(x1))))) TRS: a(x1) -> x1 a(b(x1)) -> c(b(b(a(a(x1))))) b(b(x1)) -> x1 c(c(x1)) -> x1 TDG Processor: DPs: a#(b(x1)) -> a#(x1) a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> b#(a(a(x1))) a#(b(x1)) -> b#(b(a(a(x1)))) a#(b(x1)) -> c#(b(b(a(a(x1))))) TRS: a(x1) -> x1 a(b(x1)) -> c(b(b(a(a(x1))))) b(b(x1)) -> x1 c(c(x1)) -> x1 graph: a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> c#(b(b(a(a(x1))))) a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> b#(b(a(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#(x1) -> a#(b(x1)) -> c#(b(b(a(a(x1))))) a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(b(a(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) SCC Processor: #sccs: 1 #rules: 2 #arcs: 10/25 DPs: a#(b(x1)) -> a#(a(x1)) a#(b(x1)) -> a#(x1) TRS: a(x1) -> x1 a(b(x1)) -> c(b(b(a(a(x1))))) b(b(x1)) -> x1 c(c(x1)) -> x1 Root-Labeling Processor: DPs: a{#,(f6)}(f6(b)(b(f6)(x1))) -> a{#,(f6)}(f6(a)(a(f6)(x1))) a{#,(f6)}(f6(b)(b(a)(x1))) -> a{#,(f6)}(f6(a)(a(a)(x1))) a{#,(f6)}(f6(b)(b(b)(x1))) -> a{#,(f6)}(f6(a)(a(b)(x1))) a{#,(f6)}(f6(b)(b(c)(x1))) -> a{#,(f6)}(f6(a)(a(c)(x1))) a{#,(f6)}(f6(b)(b(f6)(x1))) -> a{#,(f6)}(f6(f6)(x1)) a{#,(f6)}(f6(b)(b(a)(x1))) -> a{#,(f6)}(f6(a)(x1)) a{#,(f6)}(f6(b)(b(b)(x1))) -> a{#,(f6)}(f6(b)(x1)) a{#,(f6)}(f6(b)(b(c)(x1))) -> a{#,(f6)}(f6(c)(x1)) TRS: f6(a)(a(f6)(x1)) -> f6(f6)(x1) f6(a)(a(a)(x1)) -> f6(a)(x1) f6(a)(a(b)(x1)) -> f6(b)(x1) f6(a)(a(c)(x1)) -> f6(c)(x1) a(a)(a(f6)(x1)) -> a(f6)(x1) a(a)(a(a)(x1)) -> a(a)(x1) a(a)(a(b)(x1)) -> a(b)(x1) a(a)(a(c)(x1)) -> a(c)(x1) b(a)(a(f6)(x1)) -> b(f6)(x1) b(a)(a(a)(x1)) -> b(a)(x1) b(a)(a(b)(x1)) -> b(b)(x1) b(a)(a(c)(x1)) -> b(c)(x1) c(a)(a(f6)(x1)) -> c(f6)(x1) c(a)(a(a)(x1)) -> c(a)(x1) c(a)(a(b)(x1)) -> c(b)(x1) c(a)(a(c)(x1)) -> c(c)(x1) f6(a)(a(b)(b(f6)(x1))) -> f6(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) f6(a)(a(b)(b(a)(x1))) -> f6(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) f6(a)(a(b)(b(b)(x1))) -> f6(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) f6(a)(a(b)(b(c)(x1))) -> f6(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) a(a)(a(b)(b(f6)(x1))) -> a(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) a(a)(a(b)(b(a)(x1))) -> a(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) a(a)(a(b)(b(b)(x1))) -> a(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) a(a)(a(b)(b(c)(x1))) -> a(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) b(a)(a(b)(b(f6)(x1))) -> b(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) b(a)(a(b)(b(a)(x1))) -> b(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) b(a)(a(b)(b(b)(x1))) -> b(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) b(a)(a(b)(b(c)(x1))) -> b(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) c(a)(a(b)(b(f6)(x1))) -> c(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) c(a)(a(b)(b(a)(x1))) -> c(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) c(a)(a(b)(b(b)(x1))) -> c(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) c(a)(a(b)(b(c)(x1))) -> c(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) f6(b)(b(b)(b(f6)(x1))) -> f6(f6)(x1) f6(b)(b(b)(b(a)(x1))) -> f6(a)(x1) f6(b)(b(b)(b(b)(x1))) -> f6(b)(x1) f6(b)(b(b)(b(c)(x1))) -> f6(c)(x1) a(b)(b(b)(b(f6)(x1))) -> a(f6)(x1) a(b)(b(b)(b(a)(x1))) -> a(a)(x1) a(b)(b(b)(b(b)(x1))) -> a(b)(x1) a(b)(b(b)(b(c)(x1))) -> a(c)(x1) b(b)(b(b)(b(f6)(x1))) -> b(f6)(x1) b(b)(b(b)(b(a)(x1))) -> b(a)(x1) b(b)(b(b)(b(b)(x1))) -> b(b)(x1) b(b)(b(b)(b(c)(x1))) -> b(c)(x1) c(b)(b(b)(b(f6)(x1))) -> c(f6)(x1) c(b)(b(b)(b(a)(x1))) -> c(a)(x1) c(b)(b(b)(b(b)(x1))) -> c(b)(x1) c(b)(b(b)(b(c)(x1))) -> c(c)(x1) f6(c)(c(c)(c(f6)(x1))) -> f6(f6)(x1) f6(c)(c(c)(c(a)(x1))) -> f6(a)(x1) f6(c)(c(c)(c(b)(x1))) -> f6(b)(x1) f6(c)(c(c)(c(c)(x1))) -> f6(c)(x1) a(c)(c(c)(c(f6)(x1))) -> a(f6)(x1) a(c)(c(c)(c(a)(x1))) -> a(a)(x1) a(c)(c(c)(c(b)(x1))) -> a(b)(x1) a(c)(c(c)(c(c)(x1))) -> a(c)(x1) b(c)(c(c)(c(f6)(x1))) -> b(f6)(x1) b(c)(c(c)(c(a)(x1))) -> b(a)(x1) b(c)(c(c)(c(b)(x1))) -> b(b)(x1) b(c)(c(c)(c(c)(x1))) -> b(c)(x1) c(c)(c(c)(c(f6)(x1))) -> c(f6)(x1) c(c)(c(c)(c(a)(x1))) -> c(a)(x1) c(c)(c(c)(c(b)(x1))) -> c(b)(x1) c(c)(c(c)(c(c)(x1))) -> c(c)(x1) Polynomial Interpretation Processor: dimension: 1 interpretation: [f6(f6)](x0) = x0, [a(b)](x0) = x0 + 1, [f6(b)](x0) = x0, [b(c)](x0) = x0, [c(f6)](x0) = x0, [f6(c)](x0) = x0, [a(c)](x0) = x0, [c(b)](x0) = x0, [f6(a)](x0) = x0, [a{#,(f6)}](x0) = x0, [c(a)](x0) = x0 + 1, [b(b)](x0) = x0 + 1, [b(a)](x0) = x0, [c(c)](x0) = x0 + 1, [a(f6)](x0) = x0, [b(f6)](x0) = x0, [a(a)](x0) = x0 orientation: a{#,(f6)}(f6(b)(b(f6)(x1))) = x1 >= x1 = a{#,(f6)}(f6(a)(a(f6)(x1))) a{#,(f6)}(f6(b)(b(a)(x1))) = x1 >= x1 = a{#,(f6)}(f6(a)(a(a)(x1))) a{#,(f6)}(f6(b)(b(b)(x1))) = x1 + 1 >= x1 + 1 = a{#,(f6)}(f6(a)(a(b)(x1))) a{#,(f6)}(f6(b)(b(c)(x1))) = x1 >= x1 = a{#,(f6)}(f6(a)(a(c)(x1))) a{#,(f6)}(f6(b)(b(f6)(x1))) = x1 >= x1 = a{#,(f6)}(f6(f6)(x1)) a{#,(f6)}(f6(b)(b(a)(x1))) = x1 >= x1 = a{#,(f6)}(f6(a)(x1)) a{#,(f6)}(f6(b)(b(b)(x1))) = x1 + 1 >= x1 = a{#,(f6)}(f6(b)(x1)) a{#,(f6)}(f6(b)(b(c)(x1))) = x1 >= x1 = a{#,(f6)}(f6(c)(x1)) f6(a)(a(f6)(x1)) = x1 >= x1 = f6(f6)(x1) f6(a)(a(a)(x1)) = x1 >= x1 = f6(a)(x1) f6(a)(a(b)(x1)) = x1 + 1 >= x1 = f6(b)(x1) f6(a)(a(c)(x1)) = x1 >= x1 = f6(c)(x1) a(a)(a(f6)(x1)) = x1 >= x1 = a(f6)(x1) a(a)(a(a)(x1)) = x1 >= x1 = a(a)(x1) a(a)(a(b)(x1)) = x1 + 1 >= x1 + 1 = a(b)(x1) a(a)(a(c)(x1)) = x1 >= x1 = a(c)(x1) b(a)(a(f6)(x1)) = x1 >= x1 = b(f6)(x1) b(a)(a(a)(x1)) = x1 >= x1 = b(a)(x1) b(a)(a(b)(x1)) = x1 + 1 >= x1 + 1 = b(b)(x1) b(a)(a(c)(x1)) = x1 >= x1 = b(c)(x1) c(a)(a(f6)(x1)) = x1 + 1 >= x1 = c(f6)(x1) c(a)(a(a)(x1)) = x1 + 1 >= x1 + 1 = c(a)(x1) c(a)(a(b)(x1)) = x1 + 2 >= x1 = c(b)(x1) c(a)(a(c)(x1)) = x1 + 1 >= x1 + 1 = c(c)(x1) f6(a)(a(b)(b(f6)(x1))) = x1 + 1 >= x1 + 1 = f6(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) f6(a)(a(b)(b(a)(x1))) = x1 + 1 >= x1 + 1 = f6(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) f6(a)(a(b)(b(b)(x1))) = x1 + 2 >= x1 + 2 = f6(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) f6(a)(a(b)(b(c)(x1))) = x1 + 1 >= x1 + 1 = f6(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) a(a)(a(b)(b(f6)(x1))) = x1 + 1 >= x1 + 1 = a(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) a(a)(a(b)(b(a)(x1))) = x1 + 1 >= x1 + 1 = a(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) a(a)(a(b)(b(b)(x1))) = x1 + 2 >= x1 + 2 = a(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) a(a)(a(b)(b(c)(x1))) = x1 + 1 >= x1 + 1 = a(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) b(a)(a(b)(b(f6)(x1))) = x1 + 1 >= x1 + 1 = b(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) b(a)(a(b)(b(a)(x1))) = x1 + 1 >= x1 + 1 = b(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) b(a)(a(b)(b(b)(x1))) = x1 + 2 >= x1 + 2 = b(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) b(a)(a(b)(b(c)(x1))) = x1 + 1 >= x1 + 1 = b(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) c(a)(a(b)(b(f6)(x1))) = x1 + 2 >= x1 + 2 = c(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) c(a)(a(b)(b(a)(x1))) = x1 + 2 >= x1 + 2 = c(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) c(a)(a(b)(b(b)(x1))) = x1 + 3 >= x1 + 3 = c(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) c(a)(a(b)(b(c)(x1))) = x1 + 2 >= x1 + 2 = c(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) f6(b)(b(b)(b(f6)(x1))) = x1 + 1 >= x1 = f6(f6)(x1) f6(b)(b(b)(b(a)(x1))) = x1 + 1 >= x1 = f6(a)(x1) f6(b)(b(b)(b(b)(x1))) = x1 + 2 >= x1 = f6(b)(x1) f6(b)(b(b)(b(c)(x1))) = x1 + 1 >= x1 = f6(c)(x1) a(b)(b(b)(b(f6)(x1))) = x1 + 2 >= x1 = a(f6)(x1) a(b)(b(b)(b(a)(x1))) = x1 + 2 >= x1 = a(a)(x1) a(b)(b(b)(b(b)(x1))) = x1 + 3 >= x1 + 1 = a(b)(x1) a(b)(b(b)(b(c)(x1))) = x1 + 2 >= x1 = a(c)(x1) b(b)(b(b)(b(f6)(x1))) = x1 + 2 >= x1 = b(f6)(x1) b(b)(b(b)(b(a)(x1))) = x1 + 2 >= x1 = b(a)(x1) b(b)(b(b)(b(b)(x1))) = x1 + 3 >= x1 + 1 = b(b)(x1) b(b)(b(b)(b(c)(x1))) = x1 + 2 >= x1 = b(c)(x1) c(b)(b(b)(b(f6)(x1))) = x1 + 1 >= x1 = c(f6)(x1) c(b)(b(b)(b(a)(x1))) = x1 + 1 >= x1 + 1 = c(a)(x1) c(b)(b(b)(b(b)(x1))) = x1 + 2 >= x1 = c(b)(x1) c(b)(b(b)(b(c)(x1))) = x1 + 1 >= x1 + 1 = c(c)(x1) f6(c)(c(c)(c(f6)(x1))) = x1 + 1 >= x1 = f6(f6)(x1) f6(c)(c(c)(c(a)(x1))) = x1 + 2 >= x1 = f6(a)(x1) f6(c)(c(c)(c(b)(x1))) = x1 + 1 >= x1 = f6(b)(x1) f6(c)(c(c)(c(c)(x1))) = x1 + 2 >= x1 = f6(c)(x1) a(c)(c(c)(c(f6)(x1))) = x1 + 1 >= x1 = a(f6)(x1) a(c)(c(c)(c(a)(x1))) = x1 + 2 >= x1 = a(a)(x1) a(c)(c(c)(c(b)(x1))) = x1 + 1 >= x1 + 1 = a(b)(x1) a(c)(c(c)(c(c)(x1))) = x1 + 2 >= x1 = a(c)(x1) b(c)(c(c)(c(f6)(x1))) = x1 + 1 >= x1 = b(f6)(x1) b(c)(c(c)(c(a)(x1))) = x1 + 2 >= x1 = b(a)(x1) b(c)(c(c)(c(b)(x1))) = x1 + 1 >= x1 + 1 = b(b)(x1) b(c)(c(c)(c(c)(x1))) = x1 + 2 >= x1 = b(c)(x1) c(c)(c(c)(c(f6)(x1))) = x1 + 2 >= x1 = c(f6)(x1) c(c)(c(c)(c(a)(x1))) = x1 + 3 >= x1 + 1 = c(a)(x1) c(c)(c(c)(c(b)(x1))) = x1 + 2 >= x1 = c(b)(x1) c(c)(c(c)(c(c)(x1))) = x1 + 3 >= x1 + 1 = c(c)(x1) problem: DPs: a{#,(f6)}(f6(b)(b(f6)(x1))) -> a{#,(f6)}(f6(a)(a(f6)(x1))) a{#,(f6)}(f6(b)(b(a)(x1))) -> a{#,(f6)}(f6(a)(a(a)(x1))) a{#,(f6)}(f6(b)(b(b)(x1))) -> a{#,(f6)}(f6(a)(a(b)(x1))) a{#,(f6)}(f6(b)(b(c)(x1))) -> a{#,(f6)}(f6(a)(a(c)(x1))) a{#,(f6)}(f6(b)(b(f6)(x1))) -> a{#,(f6)}(f6(f6)(x1)) a{#,(f6)}(f6(b)(b(a)(x1))) -> a{#,(f6)}(f6(a)(x1)) a{#,(f6)}(f6(b)(b(c)(x1))) -> a{#,(f6)}(f6(c)(x1)) TRS: f6(a)(a(f6)(x1)) -> f6(f6)(x1) f6(a)(a(a)(x1)) -> f6(a)(x1) f6(a)(a(c)(x1)) -> f6(c)(x1) a(a)(a(f6)(x1)) -> a(f6)(x1) a(a)(a(a)(x1)) -> a(a)(x1) a(a)(a(b)(x1)) -> a(b)(x1) a(a)(a(c)(x1)) -> a(c)(x1) b(a)(a(f6)(x1)) -> b(f6)(x1) b(a)(a(a)(x1)) -> b(a)(x1) b(a)(a(b)(x1)) -> b(b)(x1) b(a)(a(c)(x1)) -> b(c)(x1) c(a)(a(a)(x1)) -> c(a)(x1) c(a)(a(c)(x1)) -> c(c)(x1) f6(a)(a(b)(b(f6)(x1))) -> f6(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) f6(a)(a(b)(b(a)(x1))) -> f6(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) f6(a)(a(b)(b(b)(x1))) -> f6(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) f6(a)(a(b)(b(c)(x1))) -> f6(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) a(a)(a(b)(b(f6)(x1))) -> a(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) a(a)(a(b)(b(a)(x1))) -> a(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) a(a)(a(b)(b(b)(x1))) -> a(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) a(a)(a(b)(b(c)(x1))) -> a(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) b(a)(a(b)(b(f6)(x1))) -> b(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) b(a)(a(b)(b(a)(x1))) -> b(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) b(a)(a(b)(b(b)(x1))) -> b(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) b(a)(a(b)(b(c)(x1))) -> b(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) c(a)(a(b)(b(f6)(x1))) -> c(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) c(a)(a(b)(b(a)(x1))) -> c(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) c(a)(a(b)(b(b)(x1))) -> c(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) c(a)(a(b)(b(c)(x1))) -> c(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) c(b)(b(b)(b(a)(x1))) -> c(a)(x1) c(b)(b(b)(b(c)(x1))) -> c(c)(x1) a(c)(c(c)(c(b)(x1))) -> a(b)(x1) b(c)(c(c)(c(b)(x1))) -> b(b)(x1) Polynomial Interpretation Processor: dimension: 1 interpretation: [f6(f6)](x0) = x0 + 1, [a(b)](x0) = x0, [f6(b)](x0) = x0 + 1, [b(c)](x0) = x0, [f6(c)](x0) = x0, [a(c)](x0) = x0, [c(b)](x0) = x0, [f6(a)](x0) = x0 + 1, [a{#,(f6)}](x0) = x0 + 1, [c(a)](x0) = x0, [b(b)](x0) = x0, [b(a)](x0) = x0, [c(c)](x0) = x0, [a(f6)](x0) = x0, [b(f6)](x0) = x0, [a(a)](x0) = x0 orientation: a{#,(f6)}(f6(b)(b(f6)(x1))) = x1 + 2 >= x1 + 2 = a{#,(f6)}(f6(a)(a(f6)(x1))) a{#,(f6)}(f6(b)(b(a)(x1))) = x1 + 2 >= x1 + 2 = a{#,(f6)}(f6(a)(a(a)(x1))) a{#,(f6)}(f6(b)(b(b)(x1))) = x1 + 2 >= x1 + 2 = a{#,(f6)}(f6(a)(a(b)(x1))) a{#,(f6)}(f6(b)(b(c)(x1))) = x1 + 2 >= x1 + 2 = a{#,(f6)}(f6(a)(a(c)(x1))) a{#,(f6)}(f6(b)(b(f6)(x1))) = x1 + 2 >= x1 + 2 = a{#,(f6)}(f6(f6)(x1)) a{#,(f6)}(f6(b)(b(a)(x1))) = x1 + 2 >= x1 + 2 = a{#,(f6)}(f6(a)(x1)) a{#,(f6)}(f6(b)(b(c)(x1))) = x1 + 2 >= x1 + 1 = a{#,(f6)}(f6(c)(x1)) f6(a)(a(f6)(x1)) = x1 + 1 >= x1 + 1 = f6(f6)(x1) f6(a)(a(a)(x1)) = x1 + 1 >= x1 + 1 = f6(a)(x1) f6(a)(a(c)(x1)) = x1 + 1 >= x1 = f6(c)(x1) a(a)(a(f6)(x1)) = x1 >= x1 = a(f6)(x1) a(a)(a(a)(x1)) = x1 >= x1 = a(a)(x1) a(a)(a(b)(x1)) = x1 >= x1 = a(b)(x1) a(a)(a(c)(x1)) = x1 >= x1 = a(c)(x1) b(a)(a(f6)(x1)) = x1 >= x1 = b(f6)(x1) b(a)(a(a)(x1)) = x1 >= x1 = b(a)(x1) b(a)(a(b)(x1)) = x1 >= x1 = b(b)(x1) b(a)(a(c)(x1)) = x1 >= x1 = b(c)(x1) c(a)(a(a)(x1)) = x1 >= x1 = c(a)(x1) c(a)(a(c)(x1)) = x1 >= x1 = c(c)(x1) f6(a)(a(b)(b(f6)(x1))) = x1 + 1 >= x1 = f6(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) f6(a)(a(b)(b(a)(x1))) = x1 + 1 >= x1 = f6(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) f6(a)(a(b)(b(b)(x1))) = x1 + 1 >= x1 = f6(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) f6(a)(a(b)(b(c)(x1))) = x1 + 1 >= x1 = f6(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) a(a)(a(b)(b(f6)(x1))) = x1 >= x1 = a(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) a(a)(a(b)(b(a)(x1))) = x1 >= x1 = a(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) a(a)(a(b)(b(b)(x1))) = x1 >= x1 = a(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) a(a)(a(b)(b(c)(x1))) = x1 >= x1 = a(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) b(a)(a(b)(b(f6)(x1))) = x1 >= x1 = b(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) b(a)(a(b)(b(a)(x1))) = x1 >= x1 = b(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) b(a)(a(b)(b(b)(x1))) = x1 >= x1 = b(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) b(a)(a(b)(b(c)(x1))) = x1 >= x1 = b(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) c(a)(a(b)(b(f6)(x1))) = x1 >= x1 = c(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) c(a)(a(b)(b(a)(x1))) = x1 >= x1 = c(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) c(a)(a(b)(b(b)(x1))) = x1 >= x1 = c(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) c(a)(a(b)(b(c)(x1))) = x1 >= x1 = c(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) c(b)(b(b)(b(a)(x1))) = x1 >= x1 = c(a)(x1) c(b)(b(b)(b(c)(x1))) = x1 >= x1 = c(c)(x1) a(c)(c(c)(c(b)(x1))) = x1 >= x1 = a(b)(x1) b(c)(c(c)(c(b)(x1))) = x1 >= x1 = b(b)(x1) problem: DPs: a{#,(f6)}(f6(b)(b(f6)(x1))) -> a{#,(f6)}(f6(a)(a(f6)(x1))) a{#,(f6)}(f6(b)(b(a)(x1))) -> a{#,(f6)}(f6(a)(a(a)(x1))) a{#,(f6)}(f6(b)(b(b)(x1))) -> a{#,(f6)}(f6(a)(a(b)(x1))) a{#,(f6)}(f6(b)(b(c)(x1))) -> a{#,(f6)}(f6(a)(a(c)(x1))) a{#,(f6)}(f6(b)(b(f6)(x1))) -> a{#,(f6)}(f6(f6)(x1)) a{#,(f6)}(f6(b)(b(a)(x1))) -> a{#,(f6)}(f6(a)(x1)) TRS: f6(a)(a(f6)(x1)) -> f6(f6)(x1) f6(a)(a(a)(x1)) -> f6(a)(x1) a(a)(a(f6)(x1)) -> a(f6)(x1) a(a)(a(a)(x1)) -> a(a)(x1) a(a)(a(b)(x1)) -> a(b)(x1) a(a)(a(c)(x1)) -> a(c)(x1) b(a)(a(f6)(x1)) -> b(f6)(x1) b(a)(a(a)(x1)) -> b(a)(x1) b(a)(a(b)(x1)) -> b(b)(x1) b(a)(a(c)(x1)) -> b(c)(x1) c(a)(a(a)(x1)) -> c(a)(x1) c(a)(a(c)(x1)) -> c(c)(x1) a(a)(a(b)(b(f6)(x1))) -> a(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) a(a)(a(b)(b(a)(x1))) -> a(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) a(a)(a(b)(b(b)(x1))) -> a(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) a(a)(a(b)(b(c)(x1))) -> a(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) b(a)(a(b)(b(f6)(x1))) -> b(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) b(a)(a(b)(b(a)(x1))) -> b(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) b(a)(a(b)(b(b)(x1))) -> b(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) b(a)(a(b)(b(c)(x1))) -> b(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) c(a)(a(b)(b(f6)(x1))) -> c(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) c(a)(a(b)(b(a)(x1))) -> c(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) c(a)(a(b)(b(b)(x1))) -> c(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) c(a)(a(b)(b(c)(x1))) -> c(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) c(b)(b(b)(b(a)(x1))) -> c(a)(x1) c(b)(b(b)(b(c)(x1))) -> c(c)(x1) a(c)(c(c)(c(b)(x1))) -> a(b)(x1) b(c)(c(c)(c(b)(x1))) -> b(b)(x1) Polynomial Interpretation Processor: dimension: 1 interpretation: [f6(f6)](x0) = x0, [a(b)](x0) = x0 + 1, [f6(b)](x0) = x0, [b(c)](x0) = x0, [a(c)](x0) = x0, [c(b)](x0) = x0, [f6(a)](x0) = x0, [a{#,(f6)}](x0) = x0, [c(a)](x0) = x0 + 1, [b(b)](x0) = x0 + 1, [b(a)](x0) = x0, [c(c)](x0) = x0 + 1, [a(f6)](x0) = x0 + 1, [b(f6)](x0) = x0 + 1, [a(a)](x0) = x0 orientation: a{#,(f6)}(f6(b)(b(f6)(x1))) = x1 + 1 >= x1 + 1 = a{#,(f6)}(f6(a)(a(f6)(x1))) a{#,(f6)}(f6(b)(b(a)(x1))) = x1 >= x1 = a{#,(f6)}(f6(a)(a(a)(x1))) a{#,(f6)}(f6(b)(b(b)(x1))) = x1 + 1 >= x1 + 1 = a{#,(f6)}(f6(a)(a(b)(x1))) a{#,(f6)}(f6(b)(b(c)(x1))) = x1 >= x1 = a{#,(f6)}(f6(a)(a(c)(x1))) a{#,(f6)}(f6(b)(b(f6)(x1))) = x1 + 1 >= x1 = a{#,(f6)}(f6(f6)(x1)) a{#,(f6)}(f6(b)(b(a)(x1))) = x1 >= x1 = a{#,(f6)}(f6(a)(x1)) f6(a)(a(f6)(x1)) = x1 + 1 >= x1 = f6(f6)(x1) f6(a)(a(a)(x1)) = x1 >= x1 = f6(a)(x1) a(a)(a(f6)(x1)) = x1 + 1 >= x1 + 1 = a(f6)(x1) a(a)(a(a)(x1)) = x1 >= x1 = a(a)(x1) a(a)(a(b)(x1)) = x1 + 1 >= x1 + 1 = a(b)(x1) a(a)(a(c)(x1)) = x1 >= x1 = a(c)(x1) b(a)(a(f6)(x1)) = x1 + 1 >= x1 + 1 = b(f6)(x1) b(a)(a(a)(x1)) = x1 >= x1 = b(a)(x1) b(a)(a(b)(x1)) = x1 + 1 >= x1 + 1 = b(b)(x1) b(a)(a(c)(x1)) = x1 >= x1 = b(c)(x1) c(a)(a(a)(x1)) = x1 + 1 >= x1 + 1 = c(a)(x1) c(a)(a(c)(x1)) = x1 + 1 >= x1 + 1 = c(c)(x1) a(a)(a(b)(b(f6)(x1))) = x1 + 2 >= x1 + 2 = a(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) a(a)(a(b)(b(a)(x1))) = x1 + 1 >= x1 + 1 = a(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) a(a)(a(b)(b(b)(x1))) = x1 + 2 >= x1 + 2 = a(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) a(a)(a(b)(b(c)(x1))) = x1 + 1 >= x1 + 1 = a(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) b(a)(a(b)(b(f6)(x1))) = x1 + 2 >= x1 + 2 = b(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) b(a)(a(b)(b(a)(x1))) = x1 + 1 >= x1 + 1 = b(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) b(a)(a(b)(b(b)(x1))) = x1 + 2 >= x1 + 2 = b(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) b(a)(a(b)(b(c)(x1))) = x1 + 1 >= x1 + 1 = b(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) c(a)(a(b)(b(f6)(x1))) = x1 + 3 >= x1 + 3 = c(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) c(a)(a(b)(b(a)(x1))) = x1 + 2 >= x1 + 2 = c(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) c(a)(a(b)(b(b)(x1))) = x1 + 3 >= x1 + 3 = c(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) c(a)(a(b)(b(c)(x1))) = x1 + 2 >= x1 + 2 = c(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) c(b)(b(b)(b(a)(x1))) = x1 + 1 >= x1 + 1 = c(a)(x1) c(b)(b(b)(b(c)(x1))) = x1 + 1 >= x1 + 1 = c(c)(x1) a(c)(c(c)(c(b)(x1))) = x1 + 1 >= x1 + 1 = a(b)(x1) b(c)(c(c)(c(b)(x1))) = x1 + 1 >= x1 + 1 = b(b)(x1) problem: DPs: a{#,(f6)}(f6(b)(b(f6)(x1))) -> a{#,(f6)}(f6(a)(a(f6)(x1))) a{#,(f6)}(f6(b)(b(a)(x1))) -> a{#,(f6)}(f6(a)(a(a)(x1))) a{#,(f6)}(f6(b)(b(b)(x1))) -> a{#,(f6)}(f6(a)(a(b)(x1))) a{#,(f6)}(f6(b)(b(c)(x1))) -> a{#,(f6)}(f6(a)(a(c)(x1))) a{#,(f6)}(f6(b)(b(a)(x1))) -> a{#,(f6)}(f6(a)(x1)) TRS: f6(a)(a(a)(x1)) -> f6(a)(x1) a(a)(a(f6)(x1)) -> a(f6)(x1) a(a)(a(a)(x1)) -> a(a)(x1) a(a)(a(b)(x1)) -> a(b)(x1) a(a)(a(c)(x1)) -> a(c)(x1) b(a)(a(f6)(x1)) -> b(f6)(x1) b(a)(a(a)(x1)) -> b(a)(x1) b(a)(a(b)(x1)) -> b(b)(x1) b(a)(a(c)(x1)) -> b(c)(x1) c(a)(a(a)(x1)) -> c(a)(x1) c(a)(a(c)(x1)) -> c(c)(x1) a(a)(a(b)(b(f6)(x1))) -> a(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) a(a)(a(b)(b(a)(x1))) -> a(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) a(a)(a(b)(b(b)(x1))) -> a(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) a(a)(a(b)(b(c)(x1))) -> a(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) b(a)(a(b)(b(f6)(x1))) -> b(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) b(a)(a(b)(b(a)(x1))) -> b(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) b(a)(a(b)(b(b)(x1))) -> b(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) b(a)(a(b)(b(c)(x1))) -> b(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) c(a)(a(b)(b(f6)(x1))) -> c(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) c(a)(a(b)(b(a)(x1))) -> c(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) c(a)(a(b)(b(b)(x1))) -> c(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) c(a)(a(b)(b(c)(x1))) -> c(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) c(b)(b(b)(b(a)(x1))) -> c(a)(x1) c(b)(b(b)(b(c)(x1))) -> c(c)(x1) a(c)(c(c)(c(b)(x1))) -> a(b)(x1) b(c)(c(c)(c(b)(x1))) -> b(b)(x1) Polynomial Interpretation Processor: dimension: 1 interpretation: [a(b)](x0) = x0 + 1, [f6(b)](x0) = x0 + 1, [b(c)](x0) = x0, [a(c)](x0) = x0, [c(b)](x0) = x0, [f6(a)](x0) = x0, [a{#,(f6)}](x0) = x0, [c(a)](x0) = x0 + 1, [b(b)](x0) = x0 + 1, [b(a)](x0) = x0, [c(c)](x0) = x0 + 1, [a(f6)](x0) = x0, [b(f6)](x0) = x0, [a(a)](x0) = x0 orientation: a{#,(f6)}(f6(b)(b(f6)(x1))) = x1 + 1 >= x1 = a{#,(f6)}(f6(a)(a(f6)(x1))) a{#,(f6)}(f6(b)(b(a)(x1))) = x1 + 1 >= x1 = a{#,(f6)}(f6(a)(a(a)(x1))) a{#,(f6)}(f6(b)(b(b)(x1))) = x1 + 2 >= x1 + 1 = a{#,(f6)}(f6(a)(a(b)(x1))) a{#,(f6)}(f6(b)(b(c)(x1))) = x1 + 1 >= x1 = a{#,(f6)}(f6(a)(a(c)(x1))) a{#,(f6)}(f6(b)(b(a)(x1))) = x1 + 1 >= x1 = a{#,(f6)}(f6(a)(x1)) f6(a)(a(a)(x1)) = x1 >= x1 = f6(a)(x1) a(a)(a(f6)(x1)) = x1 >= x1 = a(f6)(x1) a(a)(a(a)(x1)) = x1 >= x1 = a(a)(x1) a(a)(a(b)(x1)) = x1 + 1 >= x1 + 1 = a(b)(x1) a(a)(a(c)(x1)) = x1 >= x1 = a(c)(x1) b(a)(a(f6)(x1)) = x1 >= x1 = b(f6)(x1) b(a)(a(a)(x1)) = x1 >= x1 = b(a)(x1) b(a)(a(b)(x1)) = x1 + 1 >= x1 + 1 = b(b)(x1) b(a)(a(c)(x1)) = x1 >= x1 = b(c)(x1) c(a)(a(a)(x1)) = x1 + 1 >= x1 + 1 = c(a)(x1) c(a)(a(c)(x1)) = x1 + 1 >= x1 + 1 = c(c)(x1) a(a)(a(b)(b(f6)(x1))) = x1 + 1 >= x1 + 1 = a(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) a(a)(a(b)(b(a)(x1))) = x1 + 1 >= x1 + 1 = a(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) a(a)(a(b)(b(b)(x1))) = x1 + 2 >= x1 + 2 = a(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) a(a)(a(b)(b(c)(x1))) = x1 + 1 >= x1 + 1 = a(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) b(a)(a(b)(b(f6)(x1))) = x1 + 1 >= x1 + 1 = b(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) b(a)(a(b)(b(a)(x1))) = x1 + 1 >= x1 + 1 = b(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) b(a)(a(b)(b(b)(x1))) = x1 + 2 >= x1 + 2 = b(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) b(a)(a(b)(b(c)(x1))) = x1 + 1 >= x1 + 1 = b(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) c(a)(a(b)(b(f6)(x1))) = x1 + 2 >= x1 + 2 = c(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) c(a)(a(b)(b(a)(x1))) = x1 + 2 >= x1 + 2 = c(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) c(a)(a(b)(b(b)(x1))) = x1 + 3 >= x1 + 3 = c(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) c(a)(a(b)(b(c)(x1))) = x1 + 2 >= x1 + 2 = c(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) c(b)(b(b)(b(a)(x1))) = x1 + 1 >= x1 + 1 = c(a)(x1) c(b)(b(b)(b(c)(x1))) = x1 + 1 >= x1 + 1 = c(c)(x1) a(c)(c(c)(c(b)(x1))) = x1 + 1 >= x1 + 1 = a(b)(x1) b(c)(c(c)(c(b)(x1))) = x1 + 1 >= x1 + 1 = b(b)(x1) problem: DPs: TRS: f6(a)(a(a)(x1)) -> f6(a)(x1) a(a)(a(f6)(x1)) -> a(f6)(x1) a(a)(a(a)(x1)) -> a(a)(x1) a(a)(a(b)(x1)) -> a(b)(x1) a(a)(a(c)(x1)) -> a(c)(x1) b(a)(a(f6)(x1)) -> b(f6)(x1) b(a)(a(a)(x1)) -> b(a)(x1) b(a)(a(b)(x1)) -> b(b)(x1) b(a)(a(c)(x1)) -> b(c)(x1) c(a)(a(a)(x1)) -> c(a)(x1) c(a)(a(c)(x1)) -> c(c)(x1) a(a)(a(b)(b(f6)(x1))) -> a(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) a(a)(a(b)(b(a)(x1))) -> a(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) a(a)(a(b)(b(b)(x1))) -> a(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) a(a)(a(b)(b(c)(x1))) -> a(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) b(a)(a(b)(b(f6)(x1))) -> b(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) b(a)(a(b)(b(a)(x1))) -> b(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) b(a)(a(b)(b(b)(x1))) -> b(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) b(a)(a(b)(b(c)(x1))) -> b(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) c(a)(a(b)(b(f6)(x1))) -> c(c)(c(b)(b(b)(b(a)(a(a)(a(f6)(x1)))))) c(a)(a(b)(b(a)(x1))) -> c(c)(c(b)(b(b)(b(a)(a(a)(a(a)(x1)))))) c(a)(a(b)(b(b)(x1))) -> c(c)(c(b)(b(b)(b(a)(a(a)(a(b)(x1)))))) c(a)(a(b)(b(c)(x1))) -> c(c)(c(b)(b(b)(b(a)(a(a)(a(c)(x1)))))) c(b)(b(b)(b(a)(x1))) -> c(a)(x1) c(b)(b(b)(b(c)(x1))) -> c(c)(x1) a(c)(c(c)(c(b)(x1))) -> a(b)(x1) b(c)(c(c)(c(b)(x1))) -> b(b)(x1) Qed