/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) (STRATEGY CONTEXTSENSITIVE (*top*_0 1) (f_0 1) (f_1) (g_0 1) (b_0) (g_1) ) (RULES *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) ) Problem 1: Dependency Pairs Processor: -> Pairs: *TOP*_0(f_0(f_1(f_0(g_0(x))))) -> *TOP*_0(f_0(x)) *TOP*_0(f_0(f_1(f_0(g_0(x))))) -> F_0(x) *TOP*_0(f_0(f_1(f_0(g_0(x))))) -> x *TOP*_0(f_1(f_0(g_0(x)))) -> *TOP*_0(x) *TOP*_0(f_1(f_0(g_0(x)))) -> x *TOP*_0(f_1(f_0(g_1(x)))) -> *TOP*_0(x) *TOP*_0(f_1(f_0(g_1(x)))) -> x *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) *TOP*_0(g_1(b_0)) -> F_0(g_1(b_0)) F_0(f_0(f_1(f_0(g_0(x))))) -> F_1(f_0(x)) F_0(f_1(f_0(g_0(x)))) -> F_1(x) F_0(f_1(f_0(g_1(x)))) -> F_0(x) F_0(f_1(f_0(g_1(x)))) -> x F_1(f_0(g_0(x))) -> x G_0(f_0(f_1(f_0(g_0(x))))) -> F_0(x) G_0(f_0(f_1(f_0(g_0(x))))) -> G_0(f_0(x)) G_0(f_0(f_1(f_0(g_0(x))))) -> x G_0(f_1(f_0(g_0(x)))) -> G_0(x) G_0(f_1(f_0(g_0(x)))) -> x G_0(g_1(b_0)) -> F_0(g_1(b_0)) G_0(g_1(b_0)) -> G_0(f_0(g_1(b_0))) -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding Rules: f_0(g_1(b_0)) -> F_0(g_1(b_0)) f_0(x) -> F_0(x) f_0(x1) -> x1 Problem 1: SCC Processor: -> Pairs: *TOP*_0(f_0(f_1(f_0(g_0(x))))) -> *TOP*_0(f_0(x)) *TOP*_0(f_0(f_1(f_0(g_0(x))))) -> F_0(x) *TOP*_0(f_0(f_1(f_0(g_0(x))))) -> x *TOP*_0(f_1(f_0(g_0(x)))) -> *TOP*_0(x) *TOP*_0(f_1(f_0(g_0(x)))) -> x *TOP*_0(f_1(f_0(g_1(x)))) -> *TOP*_0(x) *TOP*_0(f_1(f_0(g_1(x)))) -> x *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) *TOP*_0(g_1(b_0)) -> F_0(g_1(b_0)) F_0(f_0(f_1(f_0(g_0(x))))) -> F_1(f_0(x)) F_0(f_1(f_0(g_0(x)))) -> F_1(x) F_0(f_1(f_0(g_1(x)))) -> F_0(x) F_0(f_1(f_0(g_1(x)))) -> x F_1(f_0(g_0(x))) -> x G_0(f_0(f_1(f_0(g_0(x))))) -> F_0(x) G_0(f_0(f_1(f_0(g_0(x))))) -> G_0(f_0(x)) G_0(f_0(f_1(f_0(g_0(x))))) -> x G_0(f_1(f_0(g_0(x)))) -> G_0(x) G_0(f_1(f_0(g_0(x)))) -> x G_0(g_1(b_0)) -> F_0(g_1(b_0)) G_0(g_1(b_0)) -> G_0(f_0(g_1(b_0))) -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding rules: f_0(g_1(b_0)) -> F_0(g_1(b_0)) f_0(x) -> F_0(x) f_0(x1) -> x1 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: F_0(f_0(f_1(f_0(g_0(x))))) -> F_1(f_0(x)) F_0(f_1(f_0(g_0(x)))) -> F_1(x) F_0(f_1(f_0(g_1(x)))) -> F_0(x) F_0(f_1(f_0(g_1(x)))) -> x F_1(f_0(g_0(x))) -> x ->->-> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) ->->-> Unhiding rules: f_0(x) -> F_0(x) f_0(x1) -> x1 ->->Cycle: ->->-> Pairs: G_0(f_0(f_1(f_0(g_0(x))))) -> G_0(f_0(x)) G_0(f_1(f_0(g_0(x)))) -> G_0(x) G_0(g_1(b_0)) -> G_0(f_0(g_1(b_0))) ->->-> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) ->->-> Unhiding rules: Empty ->->Cycle: ->->-> Pairs: *TOP*_0(f_0(f_1(f_0(g_0(x))))) -> *TOP*_0(f_0(x)) *TOP*_0(f_1(f_0(g_0(x)))) -> *TOP*_0(x) *TOP*_0(f_1(f_0(g_1(x)))) -> *TOP*_0(x) *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) ->->-> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) ->->-> Unhiding rules: Empty The problem is decomposed in 3 subproblems. Problem 1.1: Reduction Pairs Processor: -> Pairs: F_0(f_0(f_1(f_0(g_0(x))))) -> F_1(f_0(x)) F_0(f_1(f_0(g_0(x)))) -> F_1(x) F_0(f_1(f_0(g_1(x)))) -> F_0(x) F_0(f_1(f_0(g_1(x)))) -> x F_1(f_0(g_0(x))) -> x -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding rules: f_0(x) -> F_0(x) f_0(x1) -> x1 -> Usable rules: f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [f_0](X) = 2.X [f_1](X) = 2.X [g_0](X) = 2.X + 2 [b_0] = 0 [g_1](X) = 2.X [F_0](X) = X [F_1](X) = 2.X + 2 Problem 1.1: SCC Processor: -> Pairs: F_0(f_1(f_0(g_0(x)))) -> F_1(x) F_0(f_1(f_0(g_1(x)))) -> F_0(x) F_0(f_1(f_0(g_1(x)))) -> x F_1(f_0(g_0(x))) -> x -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding rules: f_0(x) -> F_0(x) f_0(x1) -> x1 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: F_0(f_1(f_0(g_0(x)))) -> F_1(x) F_0(f_1(f_0(g_1(x)))) -> F_0(x) F_0(f_1(f_0(g_1(x)))) -> x F_1(f_0(g_0(x))) -> x ->->-> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) ->->-> Unhiding rules: f_0(x) -> F_0(x) f_0(x1) -> x1 Problem 1.1: Reduction Pairs Processor: -> Pairs: F_0(f_1(f_0(g_0(x)))) -> F_1(x) F_0(f_1(f_0(g_1(x)))) -> F_0(x) F_0(f_1(f_0(g_1(x)))) -> x F_1(f_0(g_0(x))) -> x -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding rules: f_0(x) -> F_0(x) f_0(x1) -> x1 -> Usable rules: f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [f_0](X) = 2.X [f_1](X) = 2.X [g_0](X) = 2.X + 2 [b_0] = 0 [g_1](X) = 2.X [F_0](X) = 2.X [F_1](X) = X + 1 Problem 1.1: SCC Processor: -> Pairs: F_0(f_1(f_0(g_1(x)))) -> F_0(x) F_0(f_1(f_0(g_1(x)))) -> x F_1(f_0(g_0(x))) -> x -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding rules: f_0(x) -> F_0(x) f_0(x1) -> x1 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: F_0(f_1(f_0(g_1(x)))) -> F_0(x) F_0(f_1(f_0(g_1(x)))) -> x ->->-> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) ->->-> Unhiding rules: f_0(x) -> F_0(x) f_0(x1) -> x1 Problem 1.1: Reduction Pairs Processor: -> Pairs: F_0(f_1(f_0(g_1(x)))) -> F_0(x) F_0(f_1(f_0(g_1(x)))) -> x -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding rules: f_0(x) -> F_0(x) f_0(x1) -> x1 -> Usable rules: f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [f_0](X) = X [f_1](X) = X [g_0](X) = 2.X + 2 [b_0] = 2 [g_1](X) = 2.X + 2 [F_0](X) = X Problem 1.1: SCC Processor: -> Pairs: F_0(f_1(f_0(g_1(x)))) -> x -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding rules: f_0(x) -> F_0(x) f_0(x1) -> x1 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: F_0(f_1(f_0(g_1(x)))) -> x ->->-> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) ->->-> Unhiding rules: f_0(x) -> F_0(x) f_0(x1) -> x1 Problem 1.1: Reduction Pairs Processor: -> Pairs: F_0(f_1(f_0(g_1(x)))) -> x -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding rules: f_0(x) -> F_0(x) f_0(x1) -> x1 -> Usable rules: f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [f_0](X) = X [f_1](X) = X [g_0](X) = 2.X + 2 [b_0] = 2 [g_1](X) = 2.X + 2 [F_0](X) = X Problem 1.1: Basic Processor: -> Pairs: Empty -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding rules: f_0(x) -> F_0(x) f_0(x1) -> x1 -> Result: Set P is empty The problem is finite. Problem 1.2: Reduction Pairs Processor: -> Pairs: G_0(f_0(f_1(f_0(g_0(x))))) -> G_0(f_0(x)) G_0(f_1(f_0(g_0(x)))) -> G_0(x) G_0(g_1(b_0)) -> G_0(f_0(g_1(b_0))) -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding rules: Empty -> Usable rules: f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [f_0](X) = 2.X [f_1](X) = X [g_0](X) = X + 2 [b_0] = 0 [g_1](X) = X [G_0](X) = 2.X Problem 1.2: SCC Processor: -> Pairs: G_0(f_1(f_0(g_0(x)))) -> G_0(x) G_0(g_1(b_0)) -> G_0(f_0(g_1(b_0))) -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding rules: Empty ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: G_0(f_1(f_0(g_0(x)))) -> G_0(x) G_0(g_1(b_0)) -> G_0(f_0(g_1(b_0))) ->->-> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) ->->-> Unhiding rules: Empty Problem 1.2: Reduction Pairs Processor: -> Pairs: G_0(f_1(f_0(g_0(x)))) -> G_0(x) G_0(g_1(b_0)) -> G_0(f_0(g_1(b_0))) -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding rules: Empty -> Usable rules: f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [f_0](X) = 2.X [f_1](X) = 2.X [g_0](X) = 2.X + 2 [b_0] = 0 [g_1](X) = 2.X [G_0](X) = 2.X Problem 1.2: SCC Processor: -> Pairs: G_0(g_1(b_0)) -> G_0(f_0(g_1(b_0))) -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding rules: Empty ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: G_0(g_1(b_0)) -> G_0(f_0(g_1(b_0))) ->->-> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) ->->-> Unhiding rules: Empty Problem 1.2: Reduction Pairs Processor: -> Pairs: G_0(g_1(b_0)) -> G_0(f_0(g_1(b_0))) -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding rules: Empty -> Usable rules: f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 4 ->Interpretation: [f_0](X) = 3/4.X + 3/2 [f_1](X) = 2/3.X + 1/2 [g_0](X) = 3.X + 3/2 [b_0] = 3 [g_1](X) = 2.X + 1 [G_0](X) = 2.X Problem 1.2: Basic Processor: -> Pairs: Empty -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding rules: Empty -> Result: Set P is empty The problem is finite. Problem 1.3: Reduction Pairs Processor: -> Pairs: *TOP*_0(f_0(f_1(f_0(g_0(x))))) -> *TOP*_0(f_0(x)) *TOP*_0(f_1(f_0(g_0(x)))) -> *TOP*_0(x) *TOP*_0(f_1(f_0(g_1(x)))) -> *TOP*_0(x) *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding rules: Empty -> Usable rules: f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [f_0](X) = 2.X [f_1](X) = 2.X [g_0](X) = 2.X + 2 [b_0] = 0 [g_1](X) = 2.X [*TOP*_0](X) = X Problem 1.3: SCC Processor: -> Pairs: *TOP*_0(f_1(f_0(g_0(x)))) -> *TOP*_0(x) *TOP*_0(f_1(f_0(g_1(x)))) -> *TOP*_0(x) *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding rules: Empty ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: *TOP*_0(f_1(f_0(g_0(x)))) -> *TOP*_0(x) *TOP*_0(f_1(f_0(g_1(x)))) -> *TOP*_0(x) *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) ->->-> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) ->->-> Unhiding rules: Empty Problem 1.3: Reduction Pairs Processor: -> Pairs: *TOP*_0(f_1(f_0(g_0(x)))) -> *TOP*_0(x) *TOP*_0(f_1(f_0(g_1(x)))) -> *TOP*_0(x) *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding rules: Empty -> Usable rules: f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [f_0](X) = 2.X [f_1](X) = 2.X [g_0](X) = 2.X + 2 [b_0] = 0 [g_1](X) = X [*TOP*_0](X) = 2.X Problem 1.3: SCC Processor: -> Pairs: *TOP*_0(f_1(f_0(g_1(x)))) -> *TOP*_0(x) *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding rules: Empty ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: *TOP*_0(f_1(f_0(g_1(x)))) -> *TOP*_0(x) *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) ->->-> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) ->->-> Unhiding rules: Empty Problem 1.3: Reduction Pairs Processor: -> Pairs: *TOP*_0(f_1(f_0(g_1(x)))) -> *TOP*_0(x) *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding rules: Empty -> Usable rules: f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [f_0](X) = X [f_1](X) = X [g_0](X) = X + 2 [b_0] = 0 [g_1](X) = 2.X + 1 [*TOP*_0](X) = X Problem 1.3: SCC Processor: -> Pairs: *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding rules: Empty ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) ->->-> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) ->->-> Unhiding rules: Empty Problem 1.3: Reduction Pairs Processor: -> Pairs: *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding rules: Empty -> Usable rules: f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 4 ->Interpretation: [f_0](X) = 3/4.X + 3/2 [f_1](X) = 2/3.X + 1/2 [g_0](X) = 3.X + 3/2 [b_0] = 3 [g_1](X) = 2.X + 1 [*TOP*_0](X) = 2.X Problem 1.3: Basic Processor: -> Pairs: Empty -> Rules: *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) f_0(f_1(f_0(g_0(x)))) -> f_1(x) f_0(f_1(f_0(g_1(x)))) -> f_0(x) f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) f_1(f_0(g_0(x))) -> x g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) g_0(f_1(f_0(g_0(x)))) -> g_0(x) g_0(f_1(f_0(g_1(x)))) -> g_1(x) g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) -> Unhiding rules: Empty -> Result: Set P is empty The problem is finite.