YES Problem 1: (VAR v_NonEmpty:S x1:S) (RULES p(p(s(x1:S))) -> p(x1:S) p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) p(s(x1:S)) -> x1:S tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) twice(0(x1:S)) -> 0(x1:S) twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) twoto(0(x1:S)) -> s(0(x1:S)) twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) ) Problem 1: Dependency Pairs Processor: -> Pairs: P(p(s(x1:S))) -> P(x1:S) TOWER(0(x1:S)) -> P(s(p(s(x1:S)))) TOWER(0(x1:S)) -> P(s(x1:S)) TOWER(s(x1:S)) -> P(s(p(s(tower(p(s(p(s(x1:S))))))))) TOWER(s(x1:S)) -> P(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) TOWER(s(x1:S)) -> P(s(p(s(x1:S)))) TOWER(s(x1:S)) -> P(s(tower(p(s(p(s(x1:S))))))) TOWER(s(x1:S)) -> P(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))) TOWER(s(x1:S)) -> P(s(x1:S)) TOWER(s(x1:S)) -> TOWER(p(s(p(s(x1:S))))) TOWER(s(x1:S)) -> TWOTO(p(s(p(s(tower(p(s(p(s(x1:S)))))))))) TWICE(s(x1:S)) -> P(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) TWICE(s(x1:S)) -> P(p(p(s(s(s(x1:S)))))) TWICE(s(x1:S)) -> P(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S)))))))))))))) TWICE(s(x1:S)) -> P(p(s(s(s(x1:S))))) TWICE(s(x1:S)) -> P(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))) TWICE(s(x1:S)) -> P(s(s(s(x1:S)))) TWICE(s(x1:S)) -> TWICE(p(p(p(s(s(s(x1:S))))))) TWOTO(s(x1:S)) -> P(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))))))))))))))))))) TWOTO(s(x1:S)) -> P(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))) TWOTO(s(x1:S)) -> P(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) TWOTO(s(x1:S)) -> P(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))) TWOTO(s(x1:S)) -> P(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))) TWOTO(s(x1:S)) -> P(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))))))))))))))))))))) TWOTO(s(x1:S)) -> P(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))) TWOTO(s(x1:S)) -> P(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))))) TWOTO(s(x1:S)) -> P(s(p(s(x1:S)))) TWOTO(s(x1:S)) -> P(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))) TWOTO(s(x1:S)) -> P(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))))))))))) TWOTO(s(x1:S)) -> P(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))) TWOTO(s(x1:S)) -> P(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))))))))))))))))) TWOTO(s(x1:S)) -> P(s(s(s(twoto(p(s(p(s(x1:S))))))))) TWOTO(s(x1:S)) -> P(s(x1:S)) TWOTO(s(x1:S)) -> TWICE(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))) TWOTO(s(x1:S)) -> TWOTO(p(s(p(s(x1:S))))) -> Rules: p(p(s(x1:S))) -> p(x1:S) p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) p(s(x1:S)) -> x1:S tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) twice(0(x1:S)) -> 0(x1:S) twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) twoto(0(x1:S)) -> s(0(x1:S)) twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) Problem 1: SCC Processor: -> Pairs: P(p(s(x1:S))) -> P(x1:S) TOWER(0(x1:S)) -> P(s(p(s(x1:S)))) TOWER(0(x1:S)) -> P(s(x1:S)) TOWER(s(x1:S)) -> P(s(p(s(tower(p(s(p(s(x1:S))))))))) TOWER(s(x1:S)) -> P(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) TOWER(s(x1:S)) -> P(s(p(s(x1:S)))) TOWER(s(x1:S)) -> P(s(tower(p(s(p(s(x1:S))))))) TOWER(s(x1:S)) -> P(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))) TOWER(s(x1:S)) -> P(s(x1:S)) TOWER(s(x1:S)) -> TOWER(p(s(p(s(x1:S))))) TOWER(s(x1:S)) -> TWOTO(p(s(p(s(tower(p(s(p(s(x1:S)))))))))) TWICE(s(x1:S)) -> P(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) TWICE(s(x1:S)) -> P(p(p(s(s(s(x1:S)))))) TWICE(s(x1:S)) -> P(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S)))))))))))))) TWICE(s(x1:S)) -> P(p(s(s(s(x1:S))))) TWICE(s(x1:S)) -> P(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))) TWICE(s(x1:S)) -> P(s(s(s(x1:S)))) TWICE(s(x1:S)) -> TWICE(p(p(p(s(s(s(x1:S))))))) TWOTO(s(x1:S)) -> P(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))))))))))))))))))) TWOTO(s(x1:S)) -> P(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))) TWOTO(s(x1:S)) -> P(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) TWOTO(s(x1:S)) -> P(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))) TWOTO(s(x1:S)) -> P(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))) TWOTO(s(x1:S)) -> P(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))))))))))))))))))))) TWOTO(s(x1:S)) -> P(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))) TWOTO(s(x1:S)) -> P(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))))) TWOTO(s(x1:S)) -> P(s(p(s(x1:S)))) TWOTO(s(x1:S)) -> P(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))) TWOTO(s(x1:S)) -> P(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))))))))))) TWOTO(s(x1:S)) -> P(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))) TWOTO(s(x1:S)) -> P(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S))))))))))))))))))))))))))) TWOTO(s(x1:S)) -> P(s(s(s(twoto(p(s(p(s(x1:S))))))))) TWOTO(s(x1:S)) -> P(s(x1:S)) TWOTO(s(x1:S)) -> TWICE(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))) TWOTO(s(x1:S)) -> TWOTO(p(s(p(s(x1:S))))) -> Rules: p(p(s(x1:S))) -> p(x1:S) p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) p(s(x1:S)) -> x1:S tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) twice(0(x1:S)) -> 0(x1:S) twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) twoto(0(x1:S)) -> s(0(x1:S)) twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: P(p(s(x1:S))) -> P(x1:S) ->->-> Rules: p(p(s(x1:S))) -> p(x1:S) p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) p(s(x1:S)) -> x1:S tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) twice(0(x1:S)) -> 0(x1:S) twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) twoto(0(x1:S)) -> s(0(x1:S)) twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) ->->Cycle: ->->-> Pairs: TWICE(s(x1:S)) -> TWICE(p(p(p(s(s(s(x1:S))))))) ->->-> Rules: p(p(s(x1:S))) -> p(x1:S) p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) p(s(x1:S)) -> x1:S tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) twice(0(x1:S)) -> 0(x1:S) twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) twoto(0(x1:S)) -> s(0(x1:S)) twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) ->->Cycle: ->->-> Pairs: TWOTO(s(x1:S)) -> TWOTO(p(s(p(s(x1:S))))) ->->-> Rules: p(p(s(x1:S))) -> p(x1:S) p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) p(s(x1:S)) -> x1:S tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) twice(0(x1:S)) -> 0(x1:S) twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) twoto(0(x1:S)) -> s(0(x1:S)) twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) ->->Cycle: ->->-> Pairs: TOWER(s(x1:S)) -> TOWER(p(s(p(s(x1:S))))) ->->-> Rules: p(p(s(x1:S))) -> p(x1:S) p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) p(s(x1:S)) -> x1:S tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) twice(0(x1:S)) -> 0(x1:S) twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) twoto(0(x1:S)) -> s(0(x1:S)) twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) The problem is decomposed in 4 subproblems. Problem 1.1: Subterm Processor: -> Pairs: P(p(s(x1:S))) -> P(x1:S) -> Rules: p(p(s(x1:S))) -> p(x1:S) p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) p(s(x1:S)) -> x1:S tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) twice(0(x1:S)) -> 0(x1:S) twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) twoto(0(x1:S)) -> s(0(x1:S)) twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) ->Projection: pi(P) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: p(p(s(x1:S))) -> p(x1:S) p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) p(s(x1:S)) -> x1:S tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) twice(0(x1:S)) -> 0(x1:S) twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) twoto(0(x1:S)) -> s(0(x1:S)) twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Reduction Pair Processor: -> Pairs: TWICE(s(x1:S)) -> TWICE(p(p(p(s(s(s(x1:S))))))) -> Rules: p(p(s(x1:S))) -> p(x1:S) p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) p(s(x1:S)) -> x1:S tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) twice(0(x1:S)) -> 0(x1:S) twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) twoto(0(x1:S)) -> s(0(x1:S)) twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) -> Usable rules: p(p(s(x1:S))) -> p(x1:S) p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) p(s(x1:S)) -> x1:S ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [p](X) = 1/2.X [0](X) = 0 [s](X) = 2.X + 1/2 [TWICE](X) = 1/2.X Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: p(p(s(x1:S))) -> p(x1:S) p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) p(s(x1:S)) -> x1:S tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) twice(0(x1:S)) -> 0(x1:S) twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) twoto(0(x1:S)) -> s(0(x1:S)) twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Reduction Pair Processor: -> Pairs: TWOTO(s(x1:S)) -> TWOTO(p(s(p(s(x1:S))))) -> Rules: p(p(s(x1:S))) -> p(x1:S) p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) p(s(x1:S)) -> x1:S tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) twice(0(x1:S)) -> 0(x1:S) twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) twoto(0(x1:S)) -> s(0(x1:S)) twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) -> Usable rules: p(p(s(x1:S))) -> p(x1:S) p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) p(s(x1:S)) -> x1:S ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 4 ->Interpretation: [p](X) = 1/4.X [0](X) = 0 [s](X) = 4.X + 1 [TWOTO](X) = 4.X Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: p(p(s(x1:S))) -> p(x1:S) p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) p(s(x1:S)) -> x1:S tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) twice(0(x1:S)) -> 0(x1:S) twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) twoto(0(x1:S)) -> s(0(x1:S)) twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Reduction Pair Processor: -> Pairs: TOWER(s(x1:S)) -> TOWER(p(s(p(s(x1:S))))) -> Rules: p(p(s(x1:S))) -> p(x1:S) p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) p(s(x1:S)) -> x1:S tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) twice(0(x1:S)) -> 0(x1:S) twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) twoto(0(x1:S)) -> s(0(x1:S)) twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) -> Usable rules: p(p(s(x1:S))) -> p(x1:S) p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) p(s(x1:S)) -> x1:S ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 4 ->Interpretation: [p](X) = 1/4.X [0](X) = 0 [s](X) = 4.X + 1 [TOWER](X) = 4.X Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: p(p(s(x1:S))) -> p(x1:S) p(0(x1:S)) -> 0(s(s(s(s(s(s(s(s(x1:S))))))))) p(s(x1:S)) -> x1:S tower(0(x1:S)) -> s(0(p(s(p(s(x1:S)))))) tower(s(x1:S)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1:S)))))))))))))) twice(0(x1:S)) -> 0(x1:S) twice(s(x1:S)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1:S))))))))))))))) twoto(0(x1:S)) -> s(0(x1:S)) twoto(s(x1:S)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1:S)))))))))))))))))))))))))))))))) ->Strongly Connected Components: There is no strongly connected component The problem is finite.