YES Problem 1: (VAR v_NonEmpty:S x1:S) (RULES a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) ) Problem 1: Dependency Pairs Processor: -> Pairs: A(l(x1:S)) -> A(t(x1:S)) A(l(x1:S)) -> T(x1:S) N(s(x1:S)) -> A(t(x1:S)) N(s(x1:S)) -> A(l(a(t(x1:S)))) N(s(x1:S)) -> T(x1:S) O(m(a(x1:S))) -> N(x1:S) O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> A(t(o(m(a(t(e(x1:S))))))) S(a(x1:S)) -> A(t(e(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) S(a(x1:S)) -> T(o(m(a(t(e(x1:S)))))) S(a(x1:S)) -> T(e(x1:S)) T(o(x1:S)) -> A(x1:S) T(e(x1:S)) -> N(s(x1:S)) T(e(x1:S)) -> S(x1:S) -> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) Problem 1: SCC Processor: -> Pairs: A(l(x1:S)) -> A(t(x1:S)) A(l(x1:S)) -> T(x1:S) N(s(x1:S)) -> A(t(x1:S)) N(s(x1:S)) -> A(l(a(t(x1:S)))) N(s(x1:S)) -> T(x1:S) O(m(a(x1:S))) -> N(x1:S) O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> A(t(o(m(a(t(e(x1:S))))))) S(a(x1:S)) -> A(t(e(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) S(a(x1:S)) -> T(o(m(a(t(e(x1:S)))))) S(a(x1:S)) -> T(e(x1:S)) T(o(x1:S)) -> A(x1:S) T(e(x1:S)) -> N(s(x1:S)) T(e(x1:S)) -> S(x1:S) -> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A(l(x1:S)) -> A(t(x1:S)) A(l(x1:S)) -> T(x1:S) N(s(x1:S)) -> A(t(x1:S)) N(s(x1:S)) -> A(l(a(t(x1:S)))) N(s(x1:S)) -> T(x1:S) O(m(a(x1:S))) -> N(x1:S) O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> A(t(o(m(a(t(e(x1:S))))))) S(a(x1:S)) -> A(t(e(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) S(a(x1:S)) -> T(o(m(a(t(e(x1:S)))))) S(a(x1:S)) -> T(e(x1:S)) T(o(x1:S)) -> A(x1:S) T(e(x1:S)) -> N(s(x1:S)) T(e(x1:S)) -> S(x1:S) ->->-> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) Problem 1: Reduction Pair Processor: -> Pairs: A(l(x1:S)) -> A(t(x1:S)) A(l(x1:S)) -> T(x1:S) N(s(x1:S)) -> A(t(x1:S)) N(s(x1:S)) -> A(l(a(t(x1:S)))) N(s(x1:S)) -> T(x1:S) O(m(a(x1:S))) -> N(x1:S) O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> A(t(o(m(a(t(e(x1:S))))))) S(a(x1:S)) -> A(t(e(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) S(a(x1:S)) -> T(o(m(a(t(e(x1:S)))))) S(a(x1:S)) -> T(e(x1:S)) T(o(x1:S)) -> A(x1:S) T(e(x1:S)) -> N(s(x1:S)) T(e(x1:S)) -> S(x1:S) -> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) -> Usable rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [a](X) = X + 1/2 [n](X) = 1/2.X [o](X) = X + 1/2 [s](X) = 2.X + 2 [t](X) = X [e](X) = 2.X + 1 [l](X) = X [m](X) = X [A](X) = X [N](X) = 1/2.X [O](X) = X + 1/2 [S](X) = 2.X + 1 [T](X) = X Problem 1: SCC Processor: -> Pairs: A(l(x1:S)) -> A(t(x1:S)) A(l(x1:S)) -> T(x1:S) N(s(x1:S)) -> A(l(a(t(x1:S)))) N(s(x1:S)) -> T(x1:S) O(m(a(x1:S))) -> N(x1:S) O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> A(t(o(m(a(t(e(x1:S))))))) S(a(x1:S)) -> A(t(e(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) S(a(x1:S)) -> T(o(m(a(t(e(x1:S)))))) S(a(x1:S)) -> T(e(x1:S)) T(o(x1:S)) -> A(x1:S) T(e(x1:S)) -> N(s(x1:S)) T(e(x1:S)) -> S(x1:S) -> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A(l(x1:S)) -> A(t(x1:S)) A(l(x1:S)) -> T(x1:S) N(s(x1:S)) -> A(l(a(t(x1:S)))) N(s(x1:S)) -> T(x1:S) O(m(a(x1:S))) -> N(x1:S) O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> A(t(o(m(a(t(e(x1:S))))))) S(a(x1:S)) -> A(t(e(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) S(a(x1:S)) -> T(o(m(a(t(e(x1:S)))))) S(a(x1:S)) -> T(e(x1:S)) T(o(x1:S)) -> A(x1:S) T(e(x1:S)) -> N(s(x1:S)) T(e(x1:S)) -> S(x1:S) ->->-> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) Problem 1: Reduction Pair Processor: -> Pairs: A(l(x1:S)) -> A(t(x1:S)) A(l(x1:S)) -> T(x1:S) N(s(x1:S)) -> A(l(a(t(x1:S)))) N(s(x1:S)) -> T(x1:S) O(m(a(x1:S))) -> N(x1:S) O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> A(t(o(m(a(t(e(x1:S))))))) S(a(x1:S)) -> A(t(e(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) S(a(x1:S)) -> T(o(m(a(t(e(x1:S)))))) S(a(x1:S)) -> T(e(x1:S)) T(o(x1:S)) -> A(x1:S) T(e(x1:S)) -> N(s(x1:S)) T(e(x1:S)) -> S(x1:S) -> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) -> Usable rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [a](X) = X + 1/2 [n](X) = 1/2.X [o](X) = 2.X + 1/2 [s](X) = 2.X + 2 [t](X) = X [e](X) = 2.X + 1 [l](X) = X [m](X) = 1/2.X [A](X) = X [N](X) = 1/2.X [O](X) = 2.X + 1/2 [S](X) = 2.X + 1 [T](X) = X Problem 1: SCC Processor: -> Pairs: A(l(x1:S)) -> A(t(x1:S)) A(l(x1:S)) -> T(x1:S) N(s(x1:S)) -> T(x1:S) O(m(a(x1:S))) -> N(x1:S) O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> A(t(o(m(a(t(e(x1:S))))))) S(a(x1:S)) -> A(t(e(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) S(a(x1:S)) -> T(o(m(a(t(e(x1:S)))))) S(a(x1:S)) -> T(e(x1:S)) T(o(x1:S)) -> A(x1:S) T(e(x1:S)) -> N(s(x1:S)) T(e(x1:S)) -> S(x1:S) -> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A(l(x1:S)) -> A(t(x1:S)) A(l(x1:S)) -> T(x1:S) N(s(x1:S)) -> T(x1:S) O(m(a(x1:S))) -> N(x1:S) O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> A(t(o(m(a(t(e(x1:S))))))) S(a(x1:S)) -> A(t(e(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) S(a(x1:S)) -> T(o(m(a(t(e(x1:S)))))) S(a(x1:S)) -> T(e(x1:S)) T(o(x1:S)) -> A(x1:S) T(e(x1:S)) -> N(s(x1:S)) T(e(x1:S)) -> S(x1:S) ->->-> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) Problem 1: Reduction Pair Processor: -> Pairs: A(l(x1:S)) -> A(t(x1:S)) A(l(x1:S)) -> T(x1:S) N(s(x1:S)) -> T(x1:S) O(m(a(x1:S))) -> N(x1:S) O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> A(t(o(m(a(t(e(x1:S))))))) S(a(x1:S)) -> A(t(e(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) S(a(x1:S)) -> T(o(m(a(t(e(x1:S)))))) S(a(x1:S)) -> T(e(x1:S)) T(o(x1:S)) -> A(x1:S) T(e(x1:S)) -> N(s(x1:S)) T(e(x1:S)) -> S(x1:S) -> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) -> Usable rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [a](X) = X + 1/2 [n](X) = 1/2.X [o](X) = X + 1/2 [s](X) = 2.X + 2 [t](X) = X [e](X) = 2.X + 1 [l](X) = X [m](X) = X [A](X) = X [N](X) = 1/2.X [O](X) = X + 1/2 [S](X) = 2.X + 1 [T](X) = X Problem 1: SCC Processor: -> Pairs: A(l(x1:S)) -> A(t(x1:S)) A(l(x1:S)) -> T(x1:S) O(m(a(x1:S))) -> N(x1:S) O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> A(t(o(m(a(t(e(x1:S))))))) S(a(x1:S)) -> A(t(e(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) S(a(x1:S)) -> T(o(m(a(t(e(x1:S)))))) S(a(x1:S)) -> T(e(x1:S)) T(o(x1:S)) -> A(x1:S) T(e(x1:S)) -> N(s(x1:S)) T(e(x1:S)) -> S(x1:S) -> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A(l(x1:S)) -> A(t(x1:S)) A(l(x1:S)) -> T(x1:S) O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> A(t(o(m(a(t(e(x1:S))))))) S(a(x1:S)) -> A(t(e(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) S(a(x1:S)) -> T(o(m(a(t(e(x1:S)))))) S(a(x1:S)) -> T(e(x1:S)) T(o(x1:S)) -> A(x1:S) T(e(x1:S)) -> S(x1:S) ->->-> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) Problem 1: Reduction Pair Processor: -> Pairs: A(l(x1:S)) -> A(t(x1:S)) A(l(x1:S)) -> T(x1:S) O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> A(t(o(m(a(t(e(x1:S))))))) S(a(x1:S)) -> A(t(e(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) S(a(x1:S)) -> T(o(m(a(t(e(x1:S)))))) S(a(x1:S)) -> T(e(x1:S)) T(o(x1:S)) -> A(x1:S) T(e(x1:S)) -> S(x1:S) -> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) -> Usable rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a](X) = 0 [n](X) = 0 [o](X) = 2.X [s](X) = 2.X + 2 [t](X) = 2.X [e](X) = 0 [l](X) = 2.X + 2 [m](X) = 2.X [A](X) = 2.X + 1 [O](X) = 2.X + 2 [S](X) = 2 [T](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A(l(x1:S)) -> T(x1:S) O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> A(t(o(m(a(t(e(x1:S))))))) S(a(x1:S)) -> A(t(e(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) S(a(x1:S)) -> T(o(m(a(t(e(x1:S)))))) S(a(x1:S)) -> T(e(x1:S)) T(o(x1:S)) -> A(x1:S) T(e(x1:S)) -> S(x1:S) -> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A(l(x1:S)) -> T(x1:S) O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> A(t(o(m(a(t(e(x1:S))))))) S(a(x1:S)) -> A(t(e(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) S(a(x1:S)) -> T(o(m(a(t(e(x1:S)))))) S(a(x1:S)) -> T(e(x1:S)) T(o(x1:S)) -> A(x1:S) T(e(x1:S)) -> S(x1:S) ->->-> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) Problem 1: Reduction Pair Processor: -> Pairs: A(l(x1:S)) -> T(x1:S) O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> A(t(o(m(a(t(e(x1:S))))))) S(a(x1:S)) -> A(t(e(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) S(a(x1:S)) -> T(o(m(a(t(e(x1:S)))))) S(a(x1:S)) -> T(e(x1:S)) T(o(x1:S)) -> A(x1:S) T(e(x1:S)) -> S(x1:S) -> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) -> Usable rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a](X) = 2 [n](X) = 2 [o](X) = X [s](X) = 2.X + 2 [t](X) = 2 [e](X) = 2 [l](X) = 2.X + 2 [m](X) = 2 [A](X) = X [O](X) = 2 [S](X) = 2 [T](X) = X Problem 1: SCC Processor: -> Pairs: O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> A(t(o(m(a(t(e(x1:S))))))) S(a(x1:S)) -> A(t(e(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) S(a(x1:S)) -> T(o(m(a(t(e(x1:S)))))) S(a(x1:S)) -> T(e(x1:S)) T(o(x1:S)) -> A(x1:S) T(e(x1:S)) -> S(x1:S) -> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) S(a(x1:S)) -> T(o(m(a(t(e(x1:S)))))) S(a(x1:S)) -> T(e(x1:S)) T(e(x1:S)) -> S(x1:S) ->->-> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) Problem 1: Reduction Pair Processor: -> Pairs: O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) S(a(x1:S)) -> T(o(m(a(t(e(x1:S)))))) S(a(x1:S)) -> T(e(x1:S)) T(e(x1:S)) -> S(x1:S) -> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) -> Usable rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a](X) = 2.X [n](X) = 0 [o](X) = 0 [s](X) = 2.X + 2 [t](X) = 0 [e](X) = X + 1 [l](X) = 0 [m](X) = 0 [O](X) = 2 [S](X) = X + 2 [T](X) = X + 1 Problem 1: SCC Processor: -> Pairs: O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) S(a(x1:S)) -> T(e(x1:S)) T(e(x1:S)) -> S(x1:S) -> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) S(a(x1:S)) -> T(e(x1:S)) T(e(x1:S)) -> S(x1:S) ->->-> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) Problem 1: Reduction Pair Processor: -> Pairs: O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) S(a(x1:S)) -> T(e(x1:S)) T(e(x1:S)) -> S(x1:S) -> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) -> Usable rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [a](X) = X + 1/2 [n](X) = 1/2.X [o](X) = 2.X + 2 [s](X) = 2.X + 2 [t](X) = 1/2.X [e](X) = 2.X + 2 [l](X) = 1/2.X [m](X) = X + 1/2 [O](X) = 1/2.X + 1/2 [S](X) = X + 1 [T](X) = 1/2.X Problem 1: SCC Processor: -> Pairs: O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) T(e(x1:S)) -> S(x1:S) -> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) T(e(x1:S)) -> S(x1:S) ->->-> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) Problem 1: Reduction Pair Processor: -> Pairs: O(m(a(x1:S))) -> T(e(n(x1:S))) S(a(x1:S)) -> O(m(a(t(e(x1:S))))) T(e(x1:S)) -> S(x1:S) -> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) -> Usable rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 4 ->Interpretation: [a](X) = 1/2.X + 2/3 [n](X) = 1/3.X [o](X) = 4.X + 4 [s](X) = 3.X + 4 [t](X) = 1/4.X + 1 [e](X) = 4.X + 2 [l](X) = 1/4.X + 1 [m](X) = 2.X + 2/3 [O](X) = 4/3.X + 3/2 [S](X) = 4.X + 4 [T](X) = X + 2 Problem 1: SCC Processor: -> Pairs: S(a(x1:S)) -> O(m(a(t(e(x1:S))))) T(e(x1:S)) -> S(x1:S) -> Rules: a(l(x1:S)) -> a(t(x1:S)) n(s(x1:S)) -> a(l(a(t(x1:S)))) o(m(a(x1:S))) -> t(e(n(x1:S))) s(a(x1:S)) -> l(a(t(o(m(a(t(e(x1:S)))))))) t(o(x1:S)) -> m(a(x1:S)) t(e(x1:S)) -> n(s(x1:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite.