/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR x) (RULES f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) ) Problem 1: Dependency Pairs Processor: -> Pairs: F(g(x)) -> F(x) F(g(x)) -> G(f(x)) F(g(x)) -> G(g(f(x))) 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)) -> G(x) G(s(x)) -> S(g(x)) G(s(x)) -> S(s(g(x))) G(x) -> H(x,x) H(f(x),g(x)) -> F(s(x)) H(f(x),g(x)) -> S(x) 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)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) Problem 1: SCC Processor: -> Pairs: F(g(x)) -> F(x) F(g(x)) -> G(f(x)) F(g(x)) -> G(g(f(x))) 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)) -> G(x) G(s(x)) -> S(g(x)) G(s(x)) -> S(s(g(x))) G(x) -> H(x,x) H(f(x),g(x)) -> F(s(x)) H(f(x),g(x)) -> S(x) 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)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: F(g(x)) -> F(x) F(g(x)) -> G(f(x)) F(g(x)) -> G(g(f(x))) 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)) -> G(x) G(s(x)) -> S(g(x)) G(s(x)) -> S(s(g(x))) G(x) -> H(x,x) H(f(x),g(x)) -> F(s(x)) H(f(x),g(x)) -> S(x) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) ->->-> Rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) Problem 1: Reduction Pair Processor: -> Pairs: F(g(x)) -> F(x) F(g(x)) -> G(f(x)) F(g(x)) -> G(g(f(x))) 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)) -> G(x) G(s(x)) -> S(g(x)) G(s(x)) -> S(s(g(x))) G(x) -> H(x,x) H(f(x),g(x)) -> F(s(x)) H(f(x),g(x)) -> S(x) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) -> Rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) -> Usable rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [f](X) = 2.X [g](X) = X + 1 [h](X1,X2) = 0 [s](X) = 0 [0] = 0 [F](X) = X + 1 [G](X) = 2 [H](X1,X2) = 1 [S](X) = 1 Problem 1: SCC Processor: -> Pairs: F(g(x)) -> G(f(x)) F(g(x)) -> G(g(f(x))) 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)) -> G(x) G(s(x)) -> S(g(x)) G(s(x)) -> S(s(g(x))) G(x) -> H(x,x) H(f(x),g(x)) -> F(s(x)) H(f(x),g(x)) -> S(x) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) -> Rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: F(g(x)) -> G(f(x)) F(g(x)) -> G(g(f(x))) 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)) -> G(x) G(s(x)) -> S(g(x)) G(s(x)) -> S(s(g(x))) G(x) -> H(x,x) H(f(x),g(x)) -> F(s(x)) H(f(x),g(x)) -> S(x) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) ->->-> Rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) Problem 1: Reduction Pair Processor: -> Pairs: F(g(x)) -> G(f(x)) F(g(x)) -> G(g(f(x))) 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)) -> G(x) G(s(x)) -> S(g(x)) G(s(x)) -> S(s(g(x))) G(x) -> H(x,x) H(f(x),g(x)) -> F(s(x)) H(f(x),g(x)) -> S(x) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) -> Rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) -> Usable rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,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) = 2.X + 1 [G](X) = 2 [H](X1,X2) = 1 [S](X) = 1 Problem 1: SCC Processor: -> Pairs: F(g(x)) -> G(g(f(x))) 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)) -> G(x) G(s(x)) -> S(g(x)) G(s(x)) -> S(s(g(x))) G(x) -> H(x,x) H(f(x),g(x)) -> F(s(x)) H(f(x),g(x)) -> S(x) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) -> Rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: F(g(x)) -> G(g(f(x))) 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)) -> G(x) G(s(x)) -> S(g(x)) G(s(x)) -> S(s(g(x))) G(x) -> H(x,x) H(f(x),g(x)) -> F(s(x)) H(f(x),g(x)) -> S(x) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) ->->-> Rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) Problem 1: Reduction Pair Processor: -> Pairs: F(g(x)) -> G(g(f(x))) 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)) -> G(x) G(s(x)) -> S(g(x)) G(s(x)) -> S(s(g(x))) G(x) -> H(x,x) H(f(x),g(x)) -> F(s(x)) H(f(x),g(x)) -> S(x) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) -> Rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) -> Usable rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,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(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)) -> G(x) G(s(x)) -> S(g(x)) G(s(x)) -> S(s(g(x))) G(x) -> H(x,x) H(f(x),g(x)) -> F(s(x)) H(f(x),g(x)) -> S(x) S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0))) S(s(s(0))) -> F(s(0)) -> Rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,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)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) ->->Cycle: ->->-> Pairs: G(s(x)) -> G(x) ->->-> Rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,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)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) -> Usable rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 4 ->Interpretation: [f](X) = 2.X [g](X) = X + 3 [h](X1,X2) = 1/2.X1 + 1/2.X2 + 4/3 [s](X) = 3/4.X + 4/3 [0] = 0 [F](X) = 1/2.X + 2/3 [S](X) = 1/4.X + 3/4 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)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,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)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,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)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) -> Usable rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,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 [s](X) = 3/4.X + 1 [0] = 0 [F](X) = 2.X [S](X) = 1/2.X + 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)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,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)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,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)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) -> Usable rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,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 + 1/3 [s](X) = 3/4.X + 2/3 [0] = 0 [F](X) = 3.X + 4/3 [S](X) = 2.X + 1 Problem 1.1: SCC Processor: -> Pairs: F(s(s(0))) -> S(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)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: F(s(s(0))) -> S(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)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) Problem 1.1: Reduction Pair Processor: -> Pairs: F(s(s(0))) -> S(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)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) -> Usable rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,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 + 1 [s](X) = 3/4.X + 1 [0] = 0 [F](X) = X + 3 [S](X) = 1/4.X + 4 Problem 1.1: SCC Processor: -> Pairs: F(s(s(0))) -> S(s(s(s(0)))) S(s(s(0))) -> F(s(0)) -> Rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: F(s(s(0))) -> S(s(s(s(0)))) S(s(s(0))) -> F(s(0)) ->->-> Rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) Problem 1.1: Reduction Pair Processor: -> Pairs: F(s(s(0))) -> S(s(s(s(0)))) S(s(s(0))) -> F(s(0)) -> Rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) -> Usable rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 4 ->Interpretation: [f](X) = 2.X [g](X) = X + 3 [h](X1,X2) = 3/4.X1 + 1/4.X2 [s](X) = 3/4.X + 1/3 [0] = 0 [F](X) = 4.X + 4/3 [S](X) = 3/4.X + 3 Problem 1.1: SCC Processor: -> Pairs: S(s(s(0))) -> F(s(0)) -> Rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: G(s(x)) -> G(x) -> Rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) ->Projection: pi(G) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: f(g(x)) -> g(g(f(x))) f(s(s(0))) -> s(s(s(s(s(s(0)))))) f(s(0)) -> s(0) f(0) -> 0 g(s(x)) -> s(s(g(x))) g(x) -> h(x,x) h(f(x),g(x)) -> f(s(x)) 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) -> h(0,x) s(x) -> h(x,0) ->Strongly Connected Components: There is no strongly connected component The problem is finite.