YES Problem 1: (VAR v_NonEmpty:S x:S) (RULES f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) ) Problem 1: Dependency Pairs Processor: -> Pairs: F(g(x:S)) -> F(x:S) F(g(x:S)) -> G(f(x:S)) F(g(x:S)) -> G(g(f(x:S))) F(s(s(0))) -> S(s(s(s(s(s(0)))))) F(s(s(0))) -> S(s(s(s(s(0))))) F(s(s(0))) -> S(s(s(s(0)))) F(s(s(0))) -> S(s(s(0))) G(s(x:S)) -> G(x:S) G(s(x:S)) -> S(g(x:S)) G(s(x:S)) -> S(s(g(x:S))) G(x:S) -> H(x:S,x:S) H(f(x:S),g(x:S)) -> F(s(x:S)) H(f(x:S),g(x:S)) -> S(x:S) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) S(s(0)) -> F(0) -> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) Problem 1: SCC Processor: -> Pairs: F(g(x:S)) -> F(x:S) F(g(x:S)) -> G(f(x:S)) F(g(x:S)) -> G(g(f(x:S))) F(s(s(0))) -> S(s(s(s(s(s(0)))))) F(s(s(0))) -> S(s(s(s(s(0))))) F(s(s(0))) -> S(s(s(s(0)))) F(s(s(0))) -> S(s(s(0))) G(s(x:S)) -> G(x:S) G(s(x:S)) -> S(g(x:S)) G(s(x:S)) -> S(s(g(x:S))) G(x:S) -> H(x:S,x:S) H(f(x:S),g(x:S)) -> F(s(x:S)) H(f(x:S),g(x:S)) -> S(x:S) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) S(s(0)) -> F(0) -> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: F(g(x:S)) -> F(x:S) F(g(x:S)) -> G(f(x:S)) F(g(x:S)) -> G(g(f(x:S))) F(s(s(0))) -> S(s(s(s(s(s(0)))))) F(s(s(0))) -> S(s(s(s(s(0))))) F(s(s(0))) -> S(s(s(s(0)))) F(s(s(0))) -> S(s(s(0))) G(s(x:S)) -> G(x:S) G(s(x:S)) -> S(g(x:S)) G(s(x:S)) -> S(s(g(x:S))) G(x:S) -> H(x:S,x:S) H(f(x:S),g(x:S)) -> F(s(x:S)) H(f(x:S),g(x:S)) -> S(x:S) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) ->->-> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) Problem 1: Reduction Pair Processor: -> Pairs: F(g(x:S)) -> F(x:S) F(g(x:S)) -> G(f(x:S)) F(g(x:S)) -> G(g(f(x:S))) F(s(s(0))) -> S(s(s(s(s(s(0)))))) F(s(s(0))) -> S(s(s(s(s(0))))) F(s(s(0))) -> S(s(s(s(0)))) F(s(s(0))) -> S(s(s(0))) G(s(x:S)) -> G(x:S) G(s(x:S)) -> S(g(x:S)) G(s(x:S)) -> S(s(g(x:S))) G(x:S) -> H(x:S,x:S) H(f(x:S),g(x:S)) -> F(s(x:S)) H(f(x:S),g(x:S)) -> S(x:S) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) -> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) -> Usable rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [f](X) = 2.X [g](X) = X + 2 [h](X1,X2) = X1 [s](X) = X [0] = 0 [F](X) = 2.X + 2 [G](X) = X + 2 [H](X1,X2) = X1 + 2 [S](X) = 2 Problem 1: SCC Processor: -> Pairs: F(g(x:S)) -> G(f(x:S)) F(g(x:S)) -> G(g(f(x:S))) F(s(s(0))) -> S(s(s(s(s(s(0)))))) F(s(s(0))) -> S(s(s(s(s(0))))) F(s(s(0))) -> S(s(s(s(0)))) F(s(s(0))) -> S(s(s(0))) G(s(x:S)) -> G(x:S) G(s(x:S)) -> S(g(x:S)) G(s(x:S)) -> S(s(g(x:S))) G(x:S) -> H(x:S,x:S) H(f(x:S),g(x:S)) -> F(s(x:S)) H(f(x:S),g(x:S)) -> S(x:S) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) -> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: F(g(x:S)) -> G(f(x:S)) F(g(x:S)) -> G(g(f(x:S))) F(s(s(0))) -> S(s(s(s(s(s(0)))))) F(s(s(0))) -> S(s(s(s(s(0))))) F(s(s(0))) -> S(s(s(s(0)))) F(s(s(0))) -> S(s(s(0))) G(s(x:S)) -> G(x:S) G(s(x:S)) -> S(g(x:S)) G(s(x:S)) -> S(s(g(x:S))) G(x:S) -> H(x:S,x:S) H(f(x:S),g(x:S)) -> F(s(x:S)) H(f(x:S),g(x:S)) -> S(x:S) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) ->->-> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) Problem 1: Reduction Pair Processor: -> Pairs: F(g(x:S)) -> G(f(x:S)) F(g(x:S)) -> G(g(f(x:S))) F(s(s(0))) -> S(s(s(s(s(s(0)))))) F(s(s(0))) -> S(s(s(s(s(0))))) F(s(s(0))) -> S(s(s(s(0)))) F(s(s(0))) -> S(s(s(0))) G(s(x:S)) -> G(x:S) G(s(x:S)) -> S(g(x:S)) G(s(x:S)) -> S(s(g(x:S))) G(x:S) -> H(x:S,x:S) H(f(x:S),g(x:S)) -> F(s(x:S)) H(f(x:S),g(x:S)) -> S(x:S) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) -> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) -> Usable rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [f](X) = 2.X [g](X) = 2 [h](X1,X2) = 0 [s](X) = 0 [0] = 0 [F](X) = X + 2 [G](X) = 2 [H](X1,X2) = 2 [S](X) = 2 Problem 1: SCC Processor: -> Pairs: F(g(x:S)) -> G(g(f(x:S))) F(s(s(0))) -> S(s(s(s(s(s(0)))))) F(s(s(0))) -> S(s(s(s(s(0))))) F(s(s(0))) -> S(s(s(s(0)))) F(s(s(0))) -> S(s(s(0))) G(s(x:S)) -> G(x:S) G(s(x:S)) -> S(g(x:S)) G(s(x:S)) -> S(s(g(x:S))) G(x:S) -> H(x:S,x:S) H(f(x:S),g(x:S)) -> F(s(x:S)) H(f(x:S),g(x:S)) -> S(x:S) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) -> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: F(g(x:S)) -> G(g(f(x:S))) F(s(s(0))) -> S(s(s(s(s(s(0)))))) F(s(s(0))) -> S(s(s(s(s(0))))) F(s(s(0))) -> S(s(s(s(0)))) F(s(s(0))) -> S(s(s(0))) G(s(x:S)) -> G(x:S) G(s(x:S)) -> S(g(x:S)) G(s(x:S)) -> S(s(g(x:S))) G(x:S) -> H(x:S,x:S) H(f(x:S),g(x:S)) -> F(s(x:S)) H(f(x:S),g(x:S)) -> S(x:S) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) ->->-> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) Problem 1: Reduction Pair Processor: -> Pairs: F(g(x:S)) -> G(g(f(x:S))) F(s(s(0))) -> S(s(s(s(s(s(0)))))) F(s(s(0))) -> S(s(s(s(s(0))))) F(s(s(0))) -> S(s(s(s(0)))) F(s(s(0))) -> S(s(s(0))) G(s(x:S)) -> G(x:S) G(s(x:S)) -> S(g(x:S)) G(s(x:S)) -> S(s(g(x:S))) G(x:S) -> H(x:S,x:S) H(f(x:S),g(x:S)) -> F(s(x:S)) H(f(x:S),g(x:S)) -> S(x:S) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) -> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) -> Usable rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [f](X) = X [g](X) = 2 [h](X1,X2) = 1 [s](X) = 1 [0] = 0 [F](X) = 2.X [G](X) = 2 [H](X1,X2) = 2 [S](X) = 2 Problem 1: SCC Processor: -> Pairs: F(s(s(0))) -> S(s(s(s(s(s(0)))))) F(s(s(0))) -> S(s(s(s(s(0))))) F(s(s(0))) -> S(s(s(s(0)))) F(s(s(0))) -> S(s(s(0))) G(s(x:S)) -> G(x:S) G(s(x:S)) -> S(g(x:S)) G(s(x:S)) -> S(s(g(x:S))) G(x:S) -> H(x:S,x:S) H(f(x:S),g(x:S)) -> F(s(x:S)) H(f(x:S),g(x:S)) -> S(x:S) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) -> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: F(s(s(0))) -> S(s(s(s(s(s(0)))))) F(s(s(0))) -> S(s(s(s(s(0))))) F(s(s(0))) -> S(s(s(s(0)))) F(s(s(0))) -> S(s(s(0))) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) ->->-> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) ->->Cycle: ->->-> Pairs: G(s(x:S)) -> G(x:S) ->->-> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) The problem is decomposed in 2 subproblems. Problem 1.1: Reduction Pair Processor: -> Pairs: F(s(s(0))) -> S(s(s(s(s(s(0)))))) F(s(s(0))) -> S(s(s(s(s(0))))) F(s(s(0))) -> S(s(s(s(0)))) F(s(s(0))) -> S(s(s(0))) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) -> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) -> Usable rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 4 ->Interpretation: [f](X) = 2.X [g](X) = X + 4/3 [h](X1,X2) = 3/4.X1 + 1/4.X2 + 1/4 [s](X) = 3/4.X + 1/4 [0] = 0 [F](X) = 3/2.X + 1/4 [S](X) = 3/4.X + 1/3 Problem 1.1: SCC Processor: -> Pairs: F(s(s(0))) -> S(s(s(s(s(0))))) F(s(s(0))) -> S(s(s(s(0)))) F(s(s(0))) -> S(s(s(0))) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) -> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: F(s(s(0))) -> S(s(s(s(s(0))))) F(s(s(0))) -> S(s(s(s(0)))) F(s(s(0))) -> S(s(s(0))) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) ->->-> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) Problem 1.1: Reduction Pair Processor: -> Pairs: F(s(s(0))) -> S(s(s(s(s(0))))) F(s(s(0))) -> S(s(s(s(0)))) F(s(s(0))) -> S(s(s(0))) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) -> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) -> Usable rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 4 ->Interpretation: [f](X) = 2.X [g](X) = X + 4 [h](X1,X2) = 3/4.X1 + 1/4.X2 + 3/4 [s](X) = 3/4.X + 3/4 [0] = 0 [F](X) = 4/3.X + 1/3 [S](X) = 1/4.X + 3/2 Problem 1.1: SCC Processor: -> Pairs: F(s(s(0))) -> S(s(s(s(0)))) F(s(s(0))) -> S(s(s(0))) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) -> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: F(s(s(0))) -> S(s(s(s(0)))) F(s(s(0))) -> S(s(s(0))) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) ->->-> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) Problem 1.1: Reduction Pair Processor: -> Pairs: F(s(s(0))) -> S(s(s(s(0)))) F(s(s(0))) -> S(s(s(0))) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) -> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) -> Usable rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 4 ->Interpretation: [f](X) = 2.X [g](X) = X + 4 [h](X1,X2) = 1/2.X1 + 1/2.X2 + 2 [s](X) = 3/4.X + 2 [0] = 0 [F](X) = X + 2 [S](X) = X + 1/2 Problem 1.1: SCC Processor: -> Pairs: F(s(s(0))) -> S(s(s(0))) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) -> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: F(s(s(0))) -> S(s(s(0))) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) ->->-> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) Problem 1.1: Subterm Processor: -> Pairs: F(s(s(0))) -> S(s(s(0))) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) -> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) ->Projection: pi(F) = 1 pi(S) = 1 Problem 1.1: SCC Processor: -> Pairs: F(s(s(0))) -> S(s(s(0))) -> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: G(s(x:S)) -> G(x:S) -> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) ->Projection: pi(G) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: f(g(x:S)) -> g(g(f(x:S))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x:S)) -> s(s(g(x:S))) g(x:S) -> h(x:S,x:S) h(f(x:S),g(x:S)) -> f(s(x:S)) s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0))) s(s(s(0))) -> f(s(0)) s(s(0)) -> f(0) s(x:S) -> h(0,x:S) s(x:S) -> h(x:S,0) ->Strongly Connected Components: There is no strongly connected component The problem is finite.