YES Problem: a(x1) -> x1 a(a(x1)) -> a(b(x1)) b(x1) -> x1 c(b(x1)) -> a(b(c(c(x1)))) Proof: DP Processor: DPs: a#(a(x1)) -> b#(x1) a#(a(x1)) -> a#(b(x1)) c#(b(x1)) -> c#(x1) c#(b(x1)) -> c#(c(x1)) c#(b(x1)) -> b#(c(c(x1))) c#(b(x1)) -> a#(b(c(c(x1)))) TRS: a(x1) -> x1 a(a(x1)) -> a(b(x1)) b(x1) -> x1 c(b(x1)) -> a(b(c(c(x1)))) TDG Processor: DPs: a#(a(x1)) -> b#(x1) a#(a(x1)) -> a#(b(x1)) c#(b(x1)) -> c#(x1) c#(b(x1)) -> c#(c(x1)) c#(b(x1)) -> b#(c(c(x1))) c#(b(x1)) -> a#(b(c(c(x1)))) TRS: a(x1) -> x1 a(a(x1)) -> a(b(x1)) b(x1) -> x1 c(b(x1)) -> a(b(c(c(x1)))) graph: c#(b(x1)) -> c#(c(x1)) -> c#(b(x1)) -> a#(b(c(c(x1)))) c#(b(x1)) -> c#(c(x1)) -> c#(b(x1)) -> b#(c(c(x1))) c#(b(x1)) -> c#(c(x1)) -> c#(b(x1)) -> c#(c(x1)) c#(b(x1)) -> c#(c(x1)) -> c#(b(x1)) -> c#(x1) c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> a#(b(c(c(x1)))) c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> b#(c(c(x1))) c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(c(x1)) c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1) c#(b(x1)) -> a#(b(c(c(x1)))) -> a#(a(x1)) -> a#(b(x1)) c#(b(x1)) -> a#(b(c(c(x1)))) -> a#(a(x1)) -> b#(x1) a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> b#(x1) SCC Processor: #sccs: 2 #rules: 3 #arcs: 12/36 DPs: c#(b(x1)) -> c#(c(x1)) c#(b(x1)) -> c#(x1) TRS: a(x1) -> x1 a(a(x1)) -> a(b(x1)) b(x1) -> x1 c(b(x1)) -> a(b(c(c(x1)))) Root-Labeling Processor: DPs: c{#,(f6)}(f6(b)(b(f6)(x1))) -> c{#,(f6)}(f6(c)(c(f6)(x1))) c{#,(f6)}(f6(b)(b(a)(x1))) -> c{#,(f6)}(f6(c)(c(a)(x1))) c{#,(f6)}(f6(b)(b(b)(x1))) -> c{#,(f6)}(f6(c)(c(b)(x1))) c{#,(f6)}(f6(b)(b(c)(x1))) -> c{#,(f6)}(f6(c)(c(c)(x1))) c{#,(f6)}(f6(b)(b(f6)(x1))) -> c{#,(f6)}(f6(f6)(x1)) c{#,(f6)}(f6(b)(b(a)(x1))) -> c{#,(f6)}(f6(a)(x1)) c{#,(f6)}(f6(b)(b(b)(x1))) -> c{#,(f6)}(f6(b)(x1)) c{#,(f6)}(f6(b)(b(c)(x1))) -> c{#,(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) a(a)(a(f6)(x1)) -> a(b)(b(f6)(x1)) a(a)(a(a)(x1)) -> a(b)(b(a)(x1)) a(a)(a(b)(x1)) -> a(b)(b(b)(x1)) a(a)(a(c)(x1)) -> a(b)(b(c)(x1)) f6(b)(b(f6)(x1)) -> f6(f6)(x1) f6(b)(b(a)(x1)) -> f6(a)(x1) f6(b)(b(b)(x1)) -> f6(b)(x1) f6(b)(b(c)(x1)) -> f6(c)(x1) a(b)(b(f6)(x1)) -> a(f6)(x1) a(b)(b(a)(x1)) -> a(a)(x1) a(b)(b(b)(x1)) -> a(b)(x1) a(b)(b(c)(x1)) -> a(c)(x1) b(b)(b(f6)(x1)) -> b(f6)(x1) b(b)(b(a)(x1)) -> b(a)(x1) b(b)(b(b)(x1)) -> b(b)(x1) b(b)(b(c)(x1)) -> b(c)(x1) c(b)(b(f6)(x1)) -> c(f6)(x1) c(b)(b(a)(x1)) -> c(a)(x1) c(b)(b(b)(x1)) -> c(b)(x1) c(b)(b(c)(x1)) -> c(c)(x1) f6(c)(c(b)(b(f6)(x1))) -> f6(a)(a(b)(b(c)(c(c)(c(f6)(x1))))) f6(c)(c(b)(b(a)(x1))) -> f6(a)(a(b)(b(c)(c(c)(c(a)(x1))))) f6(c)(c(b)(b(b)(x1))) -> f6(a)(a(b)(b(c)(c(c)(c(b)(x1))))) f6(c)(c(b)(b(c)(x1))) -> f6(a)(a(b)(b(c)(c(c)(c(c)(x1))))) a(c)(c(b)(b(f6)(x1))) -> a(a)(a(b)(b(c)(c(c)(c(f6)(x1))))) a(c)(c(b)(b(a)(x1))) -> a(a)(a(b)(b(c)(c(c)(c(a)(x1))))) a(c)(c(b)(b(b)(x1))) -> a(a)(a(b)(b(c)(c(c)(c(b)(x1))))) a(c)(c(b)(b(c)(x1))) -> a(a)(a(b)(b(c)(c(c)(c(c)(x1))))) b(c)(c(b)(b(f6)(x1))) -> b(a)(a(b)(b(c)(c(c)(c(f6)(x1))))) b(c)(c(b)(b(a)(x1))) -> b(a)(a(b)(b(c)(c(c)(c(a)(x1))))) b(c)(c(b)(b(b)(x1))) -> b(a)(a(b)(b(c)(c(c)(c(b)(x1))))) b(c)(c(b)(b(c)(x1))) -> b(a)(a(b)(b(c)(c(c)(c(c)(x1))))) c(c)(c(b)(b(f6)(x1))) -> c(a)(a(b)(b(c)(c(c)(c(f6)(x1))))) c(c)(c(b)(b(a)(x1))) -> c(a)(a(b)(b(c)(c(c)(c(a)(x1))))) c(c)(c(b)(b(b)(x1))) -> c(a)(a(b)(b(c)(c(c)(c(b)(x1))))) c(c)(c(b)(b(c)(x1))) -> c(a)(a(b)(b(c)(c(c)(c(c)(x1))))) Polynomial Interpretation Processor: dimension: 1 interpretation: [f6(f6)](x0) = x0, [c(b)](x0) = x0, [f6(b)](x0) = x0, [b(c)](x0) = x0, [a(a)](x0) = x0, [f6(a)](x0) = x0, [c(c)](x0) = x0, [a(b)](x0) = x0, [f6(c)](x0) = x0, [c{#,(f6)}](x0) = x0, [a(f6)](x0) = x0 + 1, [b(b)](x0) = x0, [b(a)](x0) = x0, [a(c)](x0) = x0, [c(f6)](x0) = x0, [b(f6)](x0) = x0 + 1, [c(a)](x0) = x0 orientation: c{#,(f6)}(f6(b)(b(f6)(x1))) = x1 + 1 >= x1 = c{#,(f6)}(f6(c)(c(f6)(x1))) c{#,(f6)}(f6(b)(b(a)(x1))) = x1 >= x1 = c{#,(f6)}(f6(c)(c(a)(x1))) c{#,(f6)}(f6(b)(b(b)(x1))) = x1 >= x1 = c{#,(f6)}(f6(c)(c(b)(x1))) c{#,(f6)}(f6(b)(b(c)(x1))) = x1 >= x1 = c{#,(f6)}(f6(c)(c(c)(x1))) c{#,(f6)}(f6(b)(b(f6)(x1))) = x1 + 1 >= x1 = c{#,(f6)}(f6(f6)(x1)) c{#,(f6)}(f6(b)(b(a)(x1))) = x1 >= x1 = c{#,(f6)}(f6(a)(x1)) c{#,(f6)}(f6(b)(b(b)(x1))) = x1 >= x1 = c{#,(f6)}(f6(b)(x1)) c{#,(f6)}(f6(b)(b(c)(x1))) = x1 >= x1 = c{#,(f6)}(f6(c)(x1)) f6(a)(a(f6)(x1)) = x1 + 1 >= x1 = f6(f6)(x1) f6(a)(a(a)(x1)) = x1 >= x1 = f6(a)(x1) f6(a)(a(b)(x1)) = x1 >= x1 = f6(b)(x1) f6(a)(a(c)(x1)) = x1 >= x1 = f6(c)(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 >= x1 = 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 >= x1 = 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 >= x1 = c(a)(x1) c(a)(a(b)(x1)) = x1 >= x1 = c(b)(x1) c(a)(a(c)(x1)) = x1 >= x1 = c(c)(x1) a(a)(a(f6)(x1)) = x1 + 1 >= x1 + 1 = a(b)(b(f6)(x1)) a(a)(a(a)(x1)) = x1 >= x1 = a(b)(b(a)(x1)) a(a)(a(b)(x1)) = x1 >= x1 = a(b)(b(b)(x1)) a(a)(a(c)(x1)) = x1 >= x1 = a(b)(b(c)(x1)) f6(b)(b(f6)(x1)) = x1 + 1 >= x1 = f6(f6)(x1) f6(b)(b(a)(x1)) = x1 >= x1 = f6(a)(x1) f6(b)(b(b)(x1)) = x1 >= x1 = f6(b)(x1) f6(b)(b(c)(x1)) = x1 >= x1 = f6(c)(x1) a(b)(b(f6)(x1)) = x1 + 1 >= x1 + 1 = a(f6)(x1) a(b)(b(a)(x1)) = x1 >= x1 = a(a)(x1) a(b)(b(b)(x1)) = x1 >= x1 = a(b)(x1) a(b)(b(c)(x1)) = x1 >= x1 = a(c)(x1) b(b)(b(f6)(x1)) = x1 + 1 >= x1 + 1 = b(f6)(x1) b(b)(b(a)(x1)) = x1 >= x1 = b(a)(x1) b(b)(b(b)(x1)) = x1 >= x1 = b(b)(x1) b(b)(b(c)(x1)) = x1 >= x1 = b(c)(x1) c(b)(b(f6)(x1)) = x1 + 1 >= x1 = c(f6)(x1) c(b)(b(a)(x1)) = x1 >= x1 = c(a)(x1) c(b)(b(b)(x1)) = x1 >= x1 = c(b)(x1) c(b)(b(c)(x1)) = x1 >= x1 = c(c)(x1) f6(c)(c(b)(b(f6)(x1))) = x1 + 1 >= x1 = f6(a)(a(b)(b(c)(c(c)(c(f6)(x1))))) f6(c)(c(b)(b(a)(x1))) = x1 >= x1 = f6(a)(a(b)(b(c)(c(c)(c(a)(x1))))) f6(c)(c(b)(b(b)(x1))) = x1 >= x1 = f6(a)(a(b)(b(c)(c(c)(c(b)(x1))))) f6(c)(c(b)(b(c)(x1))) = x1 >= x1 = f6(a)(a(b)(b(c)(c(c)(c(c)(x1))))) a(c)(c(b)(b(f6)(x1))) = x1 + 1 >= x1 = a(a)(a(b)(b(c)(c(c)(c(f6)(x1))))) a(c)(c(b)(b(a)(x1))) = x1 >= x1 = a(a)(a(b)(b(c)(c(c)(c(a)(x1))))) a(c)(c(b)(b(b)(x1))) = x1 >= x1 = a(a)(a(b)(b(c)(c(c)(c(b)(x1))))) a(c)(c(b)(b(c)(x1))) = x1 >= x1 = a(a)(a(b)(b(c)(c(c)(c(c)(x1))))) b(c)(c(b)(b(f6)(x1))) = x1 + 1 >= x1 = b(a)(a(b)(b(c)(c(c)(c(f6)(x1))))) b(c)(c(b)(b(a)(x1))) = x1 >= x1 = b(a)(a(b)(b(c)(c(c)(c(a)(x1))))) b(c)(c(b)(b(b)(x1))) = x1 >= x1 = b(a)(a(b)(b(c)(c(c)(c(b)(x1))))) b(c)(c(b)(b(c)(x1))) = x1 >= x1 = b(a)(a(b)(b(c)(c(c)(c(c)(x1))))) c(c)(c(b)(b(f6)(x1))) = x1 + 1 >= x1 = c(a)(a(b)(b(c)(c(c)(c(f6)(x1))))) c(c)(c(b)(b(a)(x1))) = x1 >= x1 = c(a)(a(b)(b(c)(c(c)(c(a)(x1))))) c(c)(c(b)(b(b)(x1))) = x1 >= x1 = c(a)(a(b)(b(c)(c(c)(c(b)(x1))))) c(c)(c(b)(b(c)(x1))) = x1 >= x1 = c(a)(a(b)(b(c)(c(c)(c(c)(x1))))) problem: DPs: c{#,(f6)}(f6(b)(b(a)(x1))) -> c{#,(f6)}(f6(c)(c(a)(x1))) c{#,(f6)}(f6(b)(b(b)(x1))) -> c{#,(f6)}(f6(c)(c(b)(x1))) c{#,(f6)}(f6(b)(b(c)(x1))) -> c{#,(f6)}(f6(c)(c(c)(x1))) c{#,(f6)}(f6(b)(b(a)(x1))) -> c{#,(f6)}(f6(a)(x1)) c{#,(f6)}(f6(b)(b(b)(x1))) -> c{#,(f6)}(f6(b)(x1)) c{#,(f6)}(f6(b)(b(c)(x1))) -> c{#,(f6)}(f6(c)(x1)) TRS: 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(a)(x1)) -> c(a)(x1) c(a)(a(b)(x1)) -> c(b)(x1) c(a)(a(c)(x1)) -> c(c)(x1) a(a)(a(f6)(x1)) -> a(b)(b(f6)(x1)) a(a)(a(a)(x1)) -> a(b)(b(a)(x1)) a(a)(a(b)(x1)) -> a(b)(b(b)(x1)) a(a)(a(c)(x1)) -> a(b)(b(c)(x1)) f6(b)(b(a)(x1)) -> f6(a)(x1) f6(b)(b(b)(x1)) -> f6(b)(x1) f6(b)(b(c)(x1)) -> f6(c)(x1) a(b)(b(f6)(x1)) -> a(f6)(x1) a(b)(b(a)(x1)) -> a(a)(x1) a(b)(b(b)(x1)) -> a(b)(x1) a(b)(b(c)(x1)) -> a(c)(x1) b(b)(b(f6)(x1)) -> b(f6)(x1) b(b)(b(a)(x1)) -> b(a)(x1) b(b)(b(b)(x1)) -> b(b)(x1) b(b)(b(c)(x1)) -> b(c)(x1) c(b)(b(a)(x1)) -> c(a)(x1) c(b)(b(b)(x1)) -> c(b)(x1) c(b)(b(c)(x1)) -> c(c)(x1) f6(c)(c(b)(b(a)(x1))) -> f6(a)(a(b)(b(c)(c(c)(c(a)(x1))))) f6(c)(c(b)(b(b)(x1))) -> f6(a)(a(b)(b(c)(c(c)(c(b)(x1))))) f6(c)(c(b)(b(c)(x1))) -> f6(a)(a(b)(b(c)(c(c)(c(c)(x1))))) a(c)(c(b)(b(a)(x1))) -> a(a)(a(b)(b(c)(c(c)(c(a)(x1))))) a(c)(c(b)(b(b)(x1))) -> a(a)(a(b)(b(c)(c(c)(c(b)(x1))))) a(c)(c(b)(b(c)(x1))) -> a(a)(a(b)(b(c)(c(c)(c(c)(x1))))) b(c)(c(b)(b(a)(x1))) -> b(a)(a(b)(b(c)(c(c)(c(a)(x1))))) b(c)(c(b)(b(b)(x1))) -> b(a)(a(b)(b(c)(c(c)(c(b)(x1))))) b(c)(c(b)(b(c)(x1))) -> b(a)(a(b)(b(c)(c(c)(c(c)(x1))))) c(c)(c(b)(b(a)(x1))) -> c(a)(a(b)(b(c)(c(c)(c(a)(x1))))) c(c)(c(b)(b(b)(x1))) -> c(a)(a(b)(b(c)(c(c)(c(b)(x1))))) c(c)(c(b)(b(c)(x1))) -> c(a)(a(b)(b(c)(c(c)(c(c)(x1))))) Polynomial Interpretation Processor: dimension: 1 interpretation: [c(b)](x0) = x0 + 1, [f6(b)](x0) = x0, [b(c)](x0) = x0, [a(a)](x0) = x0 + 1, [f6(a)](x0) = x0, [c(c)](x0) = x0, [a(b)](x0) = x0 + 1, [f6(c)](x0) = x0, [c{#,(f6)}](x0) = x0, [a(f6)](x0) = x0, [b(b)](x0) = x0 + 1, [b(a)](x0) = x0, [a(c)](x0) = x0 + 1, [b(f6)](x0) = x0, [c(a)](x0) = x0 orientation: c{#,(f6)}(f6(b)(b(a)(x1))) = x1 >= x1 = c{#,(f6)}(f6(c)(c(a)(x1))) c{#,(f6)}(f6(b)(b(b)(x1))) = x1 + 1 >= x1 + 1 = c{#,(f6)}(f6(c)(c(b)(x1))) c{#,(f6)}(f6(b)(b(c)(x1))) = x1 >= x1 = c{#,(f6)}(f6(c)(c(c)(x1))) c{#,(f6)}(f6(b)(b(a)(x1))) = x1 >= x1 = c{#,(f6)}(f6(a)(x1)) c{#,(f6)}(f6(b)(b(b)(x1))) = x1 + 1 >= x1 = c{#,(f6)}(f6(b)(x1)) c{#,(f6)}(f6(b)(b(c)(x1))) = x1 >= x1 = c{#,(f6)}(f6(c)(x1)) f6(a)(a(a)(x1)) = x1 + 1 >= x1 = f6(a)(x1) f6(a)(a(b)(x1)) = x1 + 1 >= x1 = f6(b)(x1) f6(a)(a(c)(x1)) = x1 + 1 >= x1 = f6(c)(x1) a(a)(a(f6)(x1)) = x1 + 1 >= x1 = a(f6)(x1) a(a)(a(a)(x1)) = x1 + 2 >= x1 + 1 = a(a)(x1) a(a)(a(b)(x1)) = x1 + 2 >= x1 + 1 = a(b)(x1) a(a)(a(c)(x1)) = x1 + 2 >= x1 + 1 = a(c)(x1) b(a)(a(f6)(x1)) = x1 >= x1 = b(f6)(x1) b(a)(a(a)(x1)) = x1 + 1 >= x1 = b(a)(x1) b(a)(a(b)(x1)) = x1 + 1 >= x1 + 1 = b(b)(x1) b(a)(a(c)(x1)) = x1 + 1 >= x1 = b(c)(x1) c(a)(a(a)(x1)) = x1 + 1 >= x1 = c(a)(x1) c(a)(a(b)(x1)) = x1 + 1 >= x1 + 1 = c(b)(x1) c(a)(a(c)(x1)) = x1 + 1 >= x1 = c(c)(x1) a(a)(a(f6)(x1)) = x1 + 1 >= x1 + 1 = a(b)(b(f6)(x1)) a(a)(a(a)(x1)) = x1 + 2 >= x1 + 1 = a(b)(b(a)(x1)) a(a)(a(b)(x1)) = x1 + 2 >= x1 + 2 = a(b)(b(b)(x1)) a(a)(a(c)(x1)) = x1 + 2 >= x1 + 1 = a(b)(b(c)(x1)) f6(b)(b(a)(x1)) = x1 >= x1 = f6(a)(x1) f6(b)(b(b)(x1)) = x1 + 1 >= x1 = f6(b)(x1) f6(b)(b(c)(x1)) = x1 >= x1 = f6(c)(x1) a(b)(b(f6)(x1)) = x1 + 1 >= x1 = a(f6)(x1) a(b)(b(a)(x1)) = x1 + 1 >= x1 + 1 = a(a)(x1) a(b)(b(b)(x1)) = x1 + 2 >= x1 + 1 = a(b)(x1) a(b)(b(c)(x1)) = x1 + 1 >= x1 + 1 = a(c)(x1) b(b)(b(f6)(x1)) = x1 + 1 >= x1 = b(f6)(x1) b(b)(b(a)(x1)) = x1 + 1 >= x1 = b(a)(x1) b(b)(b(b)(x1)) = x1 + 2 >= x1 + 1 = b(b)(x1) b(b)(b(c)(x1)) = x1 + 1 >= x1 = b(c)(x1) c(b)(b(a)(x1)) = x1 + 1 >= x1 = c(a)(x1) c(b)(b(b)(x1)) = x1 + 2 >= x1 + 1 = c(b)(x1) c(b)(b(c)(x1)) = x1 + 1 >= x1 = c(c)(x1) f6(c)(c(b)(b(a)(x1))) = x1 + 1 >= x1 + 1 = f6(a)(a(b)(b(c)(c(c)(c(a)(x1))))) f6(c)(c(b)(b(b)(x1))) = x1 + 2 >= x1 + 2 = f6(a)(a(b)(b(c)(c(c)(c(b)(x1))))) f6(c)(c(b)(b(c)(x1))) = x1 + 1 >= x1 + 1 = f6(a)(a(b)(b(c)(c(c)(c(c)(x1))))) a(c)(c(b)(b(a)(x1))) = x1 + 2 >= x1 + 2 = a(a)(a(b)(b(c)(c(c)(c(a)(x1))))) a(c)(c(b)(b(b)(x1))) = x1 + 3 >= x1 + 3 = a(a)(a(b)(b(c)(c(c)(c(b)(x1))))) a(c)(c(b)(b(c)(x1))) = x1 + 2 >= x1 + 2 = a(a)(a(b)(b(c)(c(c)(c(c)(x1))))) b(c)(c(b)(b(a)(x1))) = x1 + 1 >= x1 + 1 = b(a)(a(b)(b(c)(c(c)(c(a)(x1))))) b(c)(c(b)(b(b)(x1))) = x1 + 2 >= x1 + 2 = b(a)(a(b)(b(c)(c(c)(c(b)(x1))))) b(c)(c(b)(b(c)(x1))) = x1 + 1 >= x1 + 1 = b(a)(a(b)(b(c)(c(c)(c(c)(x1))))) c(c)(c(b)(b(a)(x1))) = x1 + 1 >= x1 + 1 = c(a)(a(b)(b(c)(c(c)(c(a)(x1))))) c(c)(c(b)(b(b)(x1))) = x1 + 2 >= x1 + 2 = c(a)(a(b)(b(c)(c(c)(c(b)(x1))))) c(c)(c(b)(b(c)(x1))) = x1 + 1 >= x1 + 1 = c(a)(a(b)(b(c)(c(c)(c(c)(x1))))) problem: DPs: c{#,(f6)}(f6(b)(b(a)(x1))) -> c{#,(f6)}(f6(c)(c(a)(x1))) c{#,(f6)}(f6(b)(b(b)(x1))) -> c{#,(f6)}(f6(c)(c(b)(x1))) c{#,(f6)}(f6(b)(b(c)(x1))) -> c{#,(f6)}(f6(c)(c(c)(x1))) c{#,(f6)}(f6(b)(b(a)(x1))) -> c{#,(f6)}(f6(a)(x1)) c{#,(f6)}(f6(b)(b(c)(x1))) -> c{#,(f6)}(f6(c)(x1)) TRS: b(a)(a(f6)(x1)) -> b(f6)(x1) b(a)(a(b)(x1)) -> b(b)(x1) c(a)(a(b)(x1)) -> c(b)(x1) a(a)(a(f6)(x1)) -> a(b)(b(f6)(x1)) a(a)(a(b)(x1)) -> a(b)(b(b)(x1)) f6(b)(b(a)(x1)) -> f6(a)(x1) f6(b)(b(c)(x1)) -> f6(c)(x1) a(b)(b(a)(x1)) -> a(a)(x1) a(b)(b(c)(x1)) -> a(c)(x1) f6(c)(c(b)(b(a)(x1))) -> f6(a)(a(b)(b(c)(c(c)(c(a)(x1))))) f6(c)(c(b)(b(b)(x1))) -> f6(a)(a(b)(b(c)(c(c)(c(b)(x1))))) f6(c)(c(b)(b(c)(x1))) -> f6(a)(a(b)(b(c)(c(c)(c(c)(x1))))) a(c)(c(b)(b(a)(x1))) -> a(a)(a(b)(b(c)(c(c)(c(a)(x1))))) a(c)(c(b)(b(b)(x1))) -> a(a)(a(b)(b(c)(c(c)(c(b)(x1))))) a(c)(c(b)(b(c)(x1))) -> a(a)(a(b)(b(c)(c(c)(c(c)(x1))))) b(c)(c(b)(b(a)(x1))) -> b(a)(a(b)(b(c)(c(c)(c(a)(x1))))) b(c)(c(b)(b(b)(x1))) -> b(a)(a(b)(b(c)(c(c)(c(b)(x1))))) b(c)(c(b)(b(c)(x1))) -> b(a)(a(b)(b(c)(c(c)(c(c)(x1))))) c(c)(c(b)(b(a)(x1))) -> c(a)(a(b)(b(c)(c(c)(c(a)(x1))))) c(c)(c(b)(b(b)(x1))) -> c(a)(a(b)(b(c)(c(c)(c(b)(x1))))) c(c)(c(b)(b(c)(x1))) -> c(a)(a(b)(b(c)(c(c)(c(c)(x1))))) Usable Rule Processor: DPs: c{#,(f6)}(f6(b)(b(a)(x1))) -> c{#,(f6)}(f6(c)(c(a)(x1))) c{#,(f6)}(f6(b)(b(b)(x1))) -> c{#,(f6)}(f6(c)(c(b)(x1))) c{#,(f6)}(f6(b)(b(c)(x1))) -> c{#,(f6)}(f6(c)(c(c)(x1))) c{#,(f6)}(f6(b)(b(a)(x1))) -> c{#,(f6)}(f6(a)(x1)) c{#,(f6)}(f6(b)(b(c)(x1))) -> c{#,(f6)}(f6(c)(x1)) TRS: c(a)(a(b)(x1)) -> c(b)(x1) f6(c)(c(b)(b(a)(x1))) -> f6(a)(a(b)(b(c)(c(c)(c(a)(x1))))) f6(c)(c(b)(b(b)(x1))) -> f6(a)(a(b)(b(c)(c(c)(c(b)(x1))))) f6(c)(c(b)(b(c)(x1))) -> f6(a)(a(b)(b(c)(c(c)(c(c)(x1))))) c(c)(c(b)(b(a)(x1))) -> c(a)(a(b)(b(c)(c(c)(c(a)(x1))))) c(c)(c(b)(b(b)(x1))) -> c(a)(a(b)(b(c)(c(c)(c(b)(x1))))) c(c)(c(b)(b(c)(x1))) -> c(a)(a(b)(b(c)(c(c)(c(c)(x1))))) b(c)(c(b)(b(a)(x1))) -> b(a)(a(b)(b(c)(c(c)(c(a)(x1))))) b(c)(c(b)(b(b)(x1))) -> b(a)(a(b)(b(c)(c(c)(c(b)(x1))))) b(c)(c(b)(b(c)(x1))) -> b(a)(a(b)(b(c)(c(c)(c(c)(x1))))) b(a)(a(f6)(x1)) -> b(f6)(x1) b(a)(a(b)(x1)) -> b(b)(x1) a(b)(b(a)(x1)) -> a(a)(x1) a(b)(b(c)(x1)) -> a(c)(x1) a(a)(a(f6)(x1)) -> a(b)(b(f6)(x1)) a(a)(a(b)(x1)) -> a(b)(b(b)(x1)) a(c)(c(b)(b(a)(x1))) -> a(a)(a(b)(b(c)(c(c)(c(a)(x1))))) a(c)(c(b)(b(b)(x1))) -> a(a)(a(b)(b(c)(c(c)(c(b)(x1))))) a(c)(c(b)(b(c)(x1))) -> a(a)(a(b)(b(c)(c(c)(c(c)(x1))))) Polynomial Interpretation Processor: dimension: 1 interpretation: [c(b)](x0) = x0, [f6(b)](x0) = x0, [b(c)](x0) = x0, [a(a)](x0) = x0, [f6(a)](x0) = x0, [c(c)](x0) = x0, [a(b)](x0) = x0, [f6(c)](x0) = x0, [c{#,(f6)}](x0) = x0, [a(f6)](x0) = x0 + 1, [b(b)](x0) = x0, [b(a)](x0) = x0, [a(c)](x0) = x0, [b(f6)](x0) = x0, [c(a)](x0) = x0 orientation: c{#,(f6)}(f6(b)(b(a)(x1))) = x1 >= x1 = c{#,(f6)}(f6(c)(c(a)(x1))) c{#,(f6)}(f6(b)(b(b)(x1))) = x1 >= x1 = c{#,(f6)}(f6(c)(c(b)(x1))) c{#,(f6)}(f6(b)(b(c)(x1))) = x1 >= x1 = c{#,(f6)}(f6(c)(c(c)(x1))) c{#,(f6)}(f6(b)(b(a)(x1))) = x1 >= x1 = c{#,(f6)}(f6(a)(x1)) c{#,(f6)}(f6(b)(b(c)(x1))) = x1 >= x1 = c{#,(f6)}(f6(c)(x1)) c(a)(a(b)(x1)) = x1 >= x1 = c(b)(x1) f6(c)(c(b)(b(a)(x1))) = x1 >= x1 = f6(a)(a(b)(b(c)(c(c)(c(a)(x1))))) f6(c)(c(b)(b(b)(x1))) = x1 >= x1 = f6(a)(a(b)(b(c)(c(c)(c(b)(x1))))) f6(c)(c(b)(b(c)(x1))) = x1 >= x1 = f6(a)(a(b)(b(c)(c(c)(c(c)(x1))))) c(c)(c(b)(b(a)(x1))) = x1 >= x1 = c(a)(a(b)(b(c)(c(c)(c(a)(x1))))) c(c)(c(b)(b(b)(x1))) = x1 >= x1 = c(a)(a(b)(b(c)(c(c)(c(b)(x1))))) c(c)(c(b)(b(c)(x1))) = x1 >= x1 = c(a)(a(b)(b(c)(c(c)(c(c)(x1))))) b(c)(c(b)(b(a)(x1))) = x1 >= x1 = b(a)(a(b)(b(c)(c(c)(c(a)(x1))))) b(c)(c(b)(b(b)(x1))) = x1 >= x1 = b(a)(a(b)(b(c)(c(c)(c(b)(x1))))) b(c)(c(b)(b(c)(x1))) = x1 >= x1 = b(a)(a(b)(b(c)(c(c)(c(c)(x1))))) b(a)(a(f6)(x1)) = x1 + 1 >= x1 = b(f6)(x1) b(a)(a(b)(x1)) = x1 >= x1 = b(b)(x1) a(b)(b(a)(x1)) = x1 >= x1 = a(a)(x1) a(b)(b(c)(x1)) = x1 >= x1 = a(c)(x1) a(a)(a(f6)(x1)) = x1 + 1 >= x1 = a(b)(b(f6)(x1)) a(a)(a(b)(x1)) = x1 >= x1 = a(b)(b(b)(x1)) a(c)(c(b)(b(a)(x1))) = x1 >= x1 = a(a)(a(b)(b(c)(c(c)(c(a)(x1))))) a(c)(c(b)(b(b)(x1))) = x1 >= x1 = a(a)(a(b)(b(c)(c(c)(c(b)(x1))))) a(c)(c(b)(b(c)(x1))) = x1 >= x1 = a(a)(a(b)(b(c)(c(c)(c(c)(x1))))) problem: DPs: c{#,(f6)}(f6(b)(b(a)(x1))) -> c{#,(f6)}(f6(c)(c(a)(x1))) c{#,(f6)}(f6(b)(b(b)(x1))) -> c{#,(f6)}(f6(c)(c(b)(x1))) c{#,(f6)}(f6(b)(b(c)(x1))) -> c{#,(f6)}(f6(c)(c(c)(x1))) c{#,(f6)}(f6(b)(b(a)(x1))) -> c{#,(f6)}(f6(a)(x1)) c{#,(f6)}(f6(b)(b(c)(x1))) -> c{#,(f6)}(f6(c)(x1)) TRS: c(a)(a(b)(x1)) -> c(b)(x1) f6(c)(c(b)(b(a)(x1))) -> f6(a)(a(b)(b(c)(c(c)(c(a)(x1))))) f6(c)(c(b)(b(b)(x1))) -> f6(a)(a(b)(b(c)(c(c)(c(b)(x1))))) f6(c)(c(b)(b(c)(x1))) -> f6(a)(a(b)(b(c)(c(c)(c(c)(x1))))) c(c)(c(b)(b(a)(x1))) -> c(a)(a(b)(b(c)(c(c)(c(a)(x1))))) c(c)(c(b)(b(b)(x1))) -> c(a)(a(b)(b(c)(c(c)(c(b)(x1))))) c(c)(c(b)(b(c)(x1))) -> c(a)(a(b)(b(c)(c(c)(c(c)(x1))))) b(c)(c(b)(b(a)(x1))) -> b(a)(a(b)(b(c)(c(c)(c(a)(x1))))) b(c)(c(b)(b(b)(x1))) -> b(a)(a(b)(b(c)(c(c)(c(b)(x1))))) b(c)(c(b)(b(c)(x1))) -> b(a)(a(b)(b(c)(c(c)(c(c)(x1))))) b(a)(a(b)(x1)) -> b(b)(x1) a(b)(b(a)(x1)) -> a(a)(x1) a(b)(b(c)(x1)) -> a(c)(x1) a(a)(a(b)(x1)) -> a(b)(b(b)(x1)) a(c)(c(b)(b(a)(x1))) -> a(a)(a(b)(b(c)(c(c)(c(a)(x1))))) a(c)(c(b)(b(b)(x1))) -> a(a)(a(b)(b(c)(c(c)(c(b)(x1))))) a(c)(c(b)(b(c)(x1))) -> a(a)(a(b)(b(c)(c(c)(c(c)(x1))))) Polynomial Interpretation Processor: dimension: 1 interpretation: [c(b)](x0) = x0 + 1, [f6(b)](x0) = x0, [b(c)](x0) = x0, [a(a)](x0) = x0 + 1, [f6(a)](x0) = x0, [c(c)](x0) = x0, [a(b)](x0) = x0, [f6(c)](x0) = x0, [c{#,(f6)}](x0) = x0, [b(b)](x0) = x0 + 1, [b(a)](x0) = x0 + 1, [a(c)](x0) = x0, [c(a)](x0) = x0 + 1 orientation: c{#,(f6)}(f6(b)(b(a)(x1))) = x1 + 1 >= x1 + 1 = c{#,(f6)}(f6(c)(c(a)(x1))) c{#,(f6)}(f6(b)(b(b)(x1))) = x1 + 1 >= x1 + 1 = c{#,(f6)}(f6(c)(c(b)(x1))) c{#,(f6)}(f6(b)(b(c)(x1))) = x1 >= x1 = c{#,(f6)}(f6(c)(c(c)(x1))) c{#,(f6)}(f6(b)(b(a)(x1))) = x1 + 1 >= x1 = c{#,(f6)}(f6(a)(x1)) c{#,(f6)}(f6(b)(b(c)(x1))) = x1 >= x1 = c{#,(f6)}(f6(c)(x1)) c(a)(a(b)(x1)) = x1 + 1 >= x1 + 1 = c(b)(x1) f6(c)(c(b)(b(a)(x1))) = x1 + 2 >= x1 + 1 = f6(a)(a(b)(b(c)(c(c)(c(a)(x1))))) f6(c)(c(b)(b(b)(x1))) = x1 + 2 >= x1 + 1 = f6(a)(a(b)(b(c)(c(c)(c(b)(x1))))) f6(c)(c(b)(b(c)(x1))) = x1 + 1 >= x1 = f6(a)(a(b)(b(c)(c(c)(c(c)(x1))))) c(c)(c(b)(b(a)(x1))) = x1 + 2 >= x1 + 2 = c(a)(a(b)(b(c)(c(c)(c(a)(x1))))) c(c)(c(b)(b(b)(x1))) = x1 + 2 >= x1 + 2 = c(a)(a(b)(b(c)(c(c)(c(b)(x1))))) c(c)(c(b)(b(c)(x1))) = x1 + 1 >= x1 + 1 = c(a)(a(b)(b(c)(c(c)(c(c)(x1))))) b(c)(c(b)(b(a)(x1))) = x1 + 2 >= x1 + 2 = b(a)(a(b)(b(c)(c(c)(c(a)(x1))))) b(c)(c(b)(b(b)(x1))) = x1 + 2 >= x1 + 2 = b(a)(a(b)(b(c)(c(c)(c(b)(x1))))) b(c)(c(b)(b(c)(x1))) = x1 + 1 >= x1 + 1 = b(a)(a(b)(b(c)(c(c)(c(c)(x1))))) b(a)(a(b)(x1)) = x1 + 1 >= x1 + 1 = b(b)(x1) a(b)(b(a)(x1)) = x1 + 1 >= x1 + 1 = a(a)(x1) a(b)(b(c)(x1)) = x1 >= x1 = a(c)(x1) a(a)(a(b)(x1)) = x1 + 1 >= x1 + 1 = a(b)(b(b)(x1)) a(c)(c(b)(b(a)(x1))) = x1 + 2 >= x1 + 2 = a(a)(a(b)(b(c)(c(c)(c(a)(x1))))) a(c)(c(b)(b(b)(x1))) = x1 + 2 >= x1 + 2 = a(a)(a(b)(b(c)(c(c)(c(b)(x1))))) a(c)(c(b)(b(c)(x1))) = x1 + 1 >= x1 + 1 = a(a)(a(b)(b(c)(c(c)(c(c)(x1))))) problem: DPs: c{#,(f6)}(f6(b)(b(a)(x1))) -> c{#,(f6)}(f6(c)(c(a)(x1))) c{#,(f6)}(f6(b)(b(b)(x1))) -> c{#,(f6)}(f6(c)(c(b)(x1))) c{#,(f6)}(f6(b)(b(c)(x1))) -> c{#,(f6)}(f6(c)(c(c)(x1))) c{#,(f6)}(f6(b)(b(c)(x1))) -> c{#,(f6)}(f6(c)(x1)) TRS: c(a)(a(b)(x1)) -> c(b)(x1) c(c)(c(b)(b(a)(x1))) -> c(a)(a(b)(b(c)(c(c)(c(a)(x1))))) c(c)(c(b)(b(b)(x1))) -> c(a)(a(b)(b(c)(c(c)(c(b)(x1))))) c(c)(c(b)(b(c)(x1))) -> c(a)(a(b)(b(c)(c(c)(c(c)(x1))))) b(c)(c(b)(b(a)(x1))) -> b(a)(a(b)(b(c)(c(c)(c(a)(x1))))) b(c)(c(b)(b(b)(x1))) -> b(a)(a(b)(b(c)(c(c)(c(b)(x1))))) b(c)(c(b)(b(c)(x1))) -> b(a)(a(b)(b(c)(c(c)(c(c)(x1))))) b(a)(a(b)(x1)) -> b(b)(x1) a(b)(b(a)(x1)) -> a(a)(x1) a(b)(b(c)(x1)) -> a(c)(x1) a(a)(a(b)(x1)) -> a(b)(b(b)(x1)) a(c)(c(b)(b(a)(x1))) -> a(a)(a(b)(b(c)(c(c)(c(a)(x1))))) a(c)(c(b)(b(b)(x1))) -> a(a)(a(b)(b(c)(c(c)(c(b)(x1))))) a(c)(c(b)(b(c)(x1))) -> a(a)(a(b)(b(c)(c(c)(c(c)(x1))))) Polynomial Interpretation Processor: dimension: 1 interpretation: [c(b)](x0) = x0, [f6(b)](x0) = x0 + 1, [b(c)](x0) = x0, [a(a)](x0) = x0, [c(c)](x0) = x0, [a(b)](x0) = x0, [f6(c)](x0) = x0, [c{#,(f6)}](x0) = x0, [b(b)](x0) = x0, [b(a)](x0) = x0, [a(c)](x0) = x0, [c(a)](x0) = x0 orientation: c{#,(f6)}(f6(b)(b(a)(x1))) = x1 + 1 >= x1 = c{#,(f6)}(f6(c)(c(a)(x1))) c{#,(f6)}(f6(b)(b(b)(x1))) = x1 + 1 >= x1 = c{#,(f6)}(f6(c)(c(b)(x1))) c{#,(f6)}(f6(b)(b(c)(x1))) = x1 + 1 >= x1 = c{#,(f6)}(f6(c)(c(c)(x1))) c{#,(f6)}(f6(b)(b(c)(x1))) = x1 + 1 >= x1 = c{#,(f6)}(f6(c)(x1)) c(a)(a(b)(x1)) = x1 >= x1 = c(b)(x1) c(c)(c(b)(b(a)(x1))) = x1 >= x1 = c(a)(a(b)(b(c)(c(c)(c(a)(x1))))) c(c)(c(b)(b(b)(x1))) = x1 >= x1 = c(a)(a(b)(b(c)(c(c)(c(b)(x1))))) c(c)(c(b)(b(c)(x1))) = x1 >= x1 = c(a)(a(b)(b(c)(c(c)(c(c)(x1))))) b(c)(c(b)(b(a)(x1))) = x1 >= x1 = b(a)(a(b)(b(c)(c(c)(c(a)(x1))))) b(c)(c(b)(b(b)(x1))) = x1 >= x1 = b(a)(a(b)(b(c)(c(c)(c(b)(x1))))) b(c)(c(b)(b(c)(x1))) = x1 >= x1 = b(a)(a(b)(b(c)(c(c)(c(c)(x1))))) b(a)(a(b)(x1)) = x1 >= x1 = b(b)(x1) a(b)(b(a)(x1)) = x1 >= x1 = a(a)(x1) a(b)(b(c)(x1)) = x1 >= x1 = a(c)(x1) a(a)(a(b)(x1)) = x1 >= x1 = a(b)(b(b)(x1)) a(c)(c(b)(b(a)(x1))) = x1 >= x1 = a(a)(a(b)(b(c)(c(c)(c(a)(x1))))) a(c)(c(b)(b(b)(x1))) = x1 >= x1 = a(a)(a(b)(b(c)(c(c)(c(b)(x1))))) a(c)(c(b)(b(c)(x1))) = x1 >= x1 = a(a)(a(b)(b(c)(c(c)(c(c)(x1))))) problem: DPs: TRS: c(a)(a(b)(x1)) -> c(b)(x1) c(c)(c(b)(b(a)(x1))) -> c(a)(a(b)(b(c)(c(c)(c(a)(x1))))) c(c)(c(b)(b(b)(x1))) -> c(a)(a(b)(b(c)(c(c)(c(b)(x1))))) c(c)(c(b)(b(c)(x1))) -> c(a)(a(b)(b(c)(c(c)(c(c)(x1))))) b(c)(c(b)(b(a)(x1))) -> b(a)(a(b)(b(c)(c(c)(c(a)(x1))))) b(c)(c(b)(b(b)(x1))) -> b(a)(a(b)(b(c)(c(c)(c(b)(x1))))) b(c)(c(b)(b(c)(x1))) -> b(a)(a(b)(b(c)(c(c)(c(c)(x1))))) b(a)(a(b)(x1)) -> b(b)(x1) a(b)(b(a)(x1)) -> a(a)(x1) a(b)(b(c)(x1)) -> a(c)(x1) a(a)(a(b)(x1)) -> a(b)(b(b)(x1)) a(c)(c(b)(b(a)(x1))) -> a(a)(a(b)(b(c)(c(c)(c(a)(x1))))) a(c)(c(b)(b(b)(x1))) -> a(a)(a(b)(b(c)(c(c)(c(b)(x1))))) a(c)(c(b)(b(c)(x1))) -> a(a)(a(b)(b(c)(c(c)(c(c)(x1))))) Qed DPs: a#(a(x1)) -> a#(b(x1)) TRS: a(x1) -> x1 a(a(x1)) -> a(b(x1)) b(x1) -> x1 c(b(x1)) -> a(b(c(c(x1)))) Usable Rule Processor: DPs: a#(a(x1)) -> a#(b(x1)) TRS: b(x1) -> x1 Polynomial Interpretation Processor: dimension: 1 interpretation: [b](x0) = x0 + 1, [a#](x0) = x0, [a](x0) = x0 + 1 orientation: a#(a(x1)) = x1 + 1 >= x1 + 1 = a#(b(x1)) b(x1) = x1 + 1 >= x1 = x1 problem: DPs: a#(a(x1)) -> a#(b(x1)) TRS: Polynomial Interpretation Processor: dimension: 1 interpretation: [b](x0) = x0, [a#](x0) = x0, [a](x0) = x0 + 1 orientation: a#(a(x1)) = x1 + 1 >= x1 = a#(b(x1)) problem: DPs: TRS: Qed