YES Problem 1: (VAR v_NonEmpty:S x1:S) (RULES a(s(x1:S)) -> s(s(s(p(s(b(p(p(s(s(x1:S)))))))))) b(s(x1:S)) -> s(s(s(p(p(s(s(c(p(s(p(s(x1:S)))))))))))) c(s(x1:S)) -> p(s(p(s(a(p(s(p(s(x1:S))))))))) p(p(s(x1:S))) -> p(x1:S) p(s(x1:S)) -> x1:S ) Problem 1: Dependency Pairs Processor: -> Pairs: A(s(x1:S)) -> B(p(p(s(s(x1:S))))) A(s(x1:S)) -> P(p(s(s(x1:S)))) A(s(x1:S)) -> P(s(b(p(p(s(s(x1:S))))))) A(s(x1:S)) -> P(s(s(x1:S))) B(s(x1:S)) -> C(p(s(p(s(x1:S))))) B(s(x1:S)) -> P(p(s(s(c(p(s(p(s(x1:S))))))))) B(s(x1:S)) -> P(s(p(s(x1:S)))) B(s(x1:S)) -> P(s(s(c(p(s(p(s(x1:S)))))))) B(s(x1:S)) -> P(s(x1:S)) C(s(x1:S)) -> A(p(s(p(s(x1:S))))) C(s(x1:S)) -> P(s(a(p(s(p(s(x1:S))))))) C(s(x1:S)) -> P(s(p(s(a(p(s(p(s(x1:S))))))))) C(s(x1:S)) -> P(s(p(s(x1:S)))) C(s(x1:S)) -> P(s(x1:S)) P(p(s(x1:S))) -> P(x1:S) -> Rules: a(s(x1:S)) -> s(s(s(p(s(b(p(p(s(s(x1:S)))))))))) b(s(x1:S)) -> s(s(s(p(p(s(s(c(p(s(p(s(x1:S)))))))))))) c(s(x1:S)) -> p(s(p(s(a(p(s(p(s(x1:S))))))))) p(p(s(x1:S))) -> p(x1:S) p(s(x1:S)) -> x1:S Problem 1: SCC Processor: -> Pairs: A(s(x1:S)) -> B(p(p(s(s(x1:S))))) A(s(x1:S)) -> P(p(s(s(x1:S)))) A(s(x1:S)) -> P(s(b(p(p(s(s(x1:S))))))) A(s(x1:S)) -> P(s(s(x1:S))) B(s(x1:S)) -> C(p(s(p(s(x1:S))))) B(s(x1:S)) -> P(p(s(s(c(p(s(p(s(x1:S))))))))) B(s(x1:S)) -> P(s(p(s(x1:S)))) B(s(x1:S)) -> P(s(s(c(p(s(p(s(x1:S)))))))) B(s(x1:S)) -> P(s(x1:S)) C(s(x1:S)) -> A(p(s(p(s(x1:S))))) C(s(x1:S)) -> P(s(a(p(s(p(s(x1:S))))))) C(s(x1:S)) -> P(s(p(s(a(p(s(p(s(x1:S))))))))) C(s(x1:S)) -> P(s(p(s(x1:S)))) C(s(x1:S)) -> P(s(x1:S)) P(p(s(x1:S))) -> P(x1:S) -> Rules: a(s(x1:S)) -> s(s(s(p(s(b(p(p(s(s(x1:S)))))))))) b(s(x1:S)) -> s(s(s(p(p(s(s(c(p(s(p(s(x1:S)))))))))))) c(s(x1:S)) -> p(s(p(s(a(p(s(p(s(x1:S))))))))) p(p(s(x1:S))) -> p(x1:S) p(s(x1:S)) -> x1:S ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: P(p(s(x1:S))) -> P(x1:S) ->->-> Rules: a(s(x1:S)) -> s(s(s(p(s(b(p(p(s(s(x1:S)))))))))) b(s(x1:S)) -> s(s(s(p(p(s(s(c(p(s(p(s(x1:S)))))))))))) c(s(x1:S)) -> p(s(p(s(a(p(s(p(s(x1:S))))))))) p(p(s(x1:S))) -> p(x1:S) p(s(x1:S)) -> x1:S ->->Cycle: ->->-> Pairs: A(s(x1:S)) -> B(p(p(s(s(x1:S))))) B(s(x1:S)) -> C(p(s(p(s(x1:S))))) C(s(x1:S)) -> A(p(s(p(s(x1:S))))) ->->-> Rules: a(s(x1:S)) -> s(s(s(p(s(b(p(p(s(s(x1:S)))))))))) b(s(x1:S)) -> s(s(s(p(p(s(s(c(p(s(p(s(x1:S)))))))))))) c(s(x1:S)) -> p(s(p(s(a(p(s(p(s(x1:S))))))))) p(p(s(x1:S))) -> p(x1:S) p(s(x1:S)) -> x1:S The problem is decomposed in 2 subproblems. Problem 1.1: Subterm Processor: -> Pairs: P(p(s(x1:S))) -> P(x1:S) -> Rules: a(s(x1:S)) -> s(s(s(p(s(b(p(p(s(s(x1:S)))))))))) b(s(x1:S)) -> s(s(s(p(p(s(s(c(p(s(p(s(x1:S)))))))))))) c(s(x1:S)) -> p(s(p(s(a(p(s(p(s(x1:S))))))))) p(p(s(x1:S))) -> p(x1:S) p(s(x1:S)) -> x1:S ->Projection: pi(P) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: a(s(x1:S)) -> s(s(s(p(s(b(p(p(s(s(x1:S)))))))))) b(s(x1:S)) -> s(s(s(p(p(s(s(c(p(s(p(s(x1:S)))))))))))) c(s(x1:S)) -> p(s(p(s(a(p(s(p(s(x1:S))))))))) p(p(s(x1:S))) -> p(x1:S) p(s(x1:S)) -> x1:S ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Reduction Pair Processor: -> Pairs: A(s(x1:S)) -> B(p(p(s(s(x1:S))))) B(s(x1:S)) -> C(p(s(p(s(x1:S))))) C(s(x1:S)) -> A(p(s(p(s(x1:S))))) -> Rules: a(s(x1:S)) -> s(s(s(p(s(b(p(p(s(s(x1:S)))))))))) b(s(x1:S)) -> s(s(s(p(p(s(s(c(p(s(p(s(x1:S)))))))))))) c(s(x1:S)) -> p(s(p(s(a(p(s(p(s(x1:S))))))))) p(p(s(x1:S))) -> p(x1:S) p(s(x1:S)) -> x1:S -> Usable rules: p(p(s(x1:S))) -> p(x1:S) p(s(x1:S)) -> x1:S ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [p](X) = 1/2.X [s](X) = 2.X + 2 [A](X) = 2.X [B](X) = 2.X + 1/2 [C](X) = X + 2 Problem 1.2: SCC Processor: -> Pairs: B(s(x1:S)) -> C(p(s(p(s(x1:S))))) C(s(x1:S)) -> A(p(s(p(s(x1:S))))) -> Rules: a(s(x1:S)) -> s(s(s(p(s(b(p(p(s(s(x1:S)))))))))) b(s(x1:S)) -> s(s(s(p(p(s(s(c(p(s(p(s(x1:S)))))))))))) c(s(x1:S)) -> p(s(p(s(a(p(s(p(s(x1:S))))))))) p(p(s(x1:S))) -> p(x1:S) p(s(x1:S)) -> x1:S ->Strongly Connected Components: There is no strongly connected component The problem is finite.