0.00/0.57 YES 0.00/0.57 0.00/0.57 Problem 1: 0.00/0.57 0.00/0.57 (VAR x) 0.00/0.57 (STRATEGY CONTEXTSENSITIVE 0.00/0.57 (*top*_0 1) 0.00/0.57 (f_0 1) 0.00/0.57 (f_1) 0.00/0.57 (g_0 1) 0.00/0.57 (b_0) 0.00/0.57 (g_1) 0.00/0.57 ) 0.00/0.57 (RULES 0.00/0.57 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.57 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.57 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.57 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.57 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.57 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.57 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.57 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.57 f_1(f_0(g_0(x))) -> x 0.00/0.57 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.57 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.57 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.57 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.57 ) 0.00/0.57 0.00/0.57 Problem 1: 0.00/0.57 0.00/0.57 Dependency Pairs Processor: 0.00/0.57 -> Pairs: 0.00/0.57 *TOP*_0(f_0(f_1(f_0(g_0(x))))) -> *TOP*_0(f_0(x)) 0.00/0.57 *TOP*_0(f_0(f_1(f_0(g_0(x))))) -> F_0(x) 0.00/0.57 *TOP*_0(f_0(f_1(f_0(g_0(x))))) -> x 0.00/0.57 *TOP*_0(f_1(f_0(g_0(x)))) -> *TOP*_0(x) 0.00/0.57 *TOP*_0(f_1(f_0(g_0(x)))) -> x 0.00/0.57 *TOP*_0(f_1(f_0(g_1(x)))) -> *TOP*_0(x) 0.00/0.57 *TOP*_0(f_1(f_0(g_1(x)))) -> x 0.00/0.57 *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) 0.00/0.57 *TOP*_0(g_1(b_0)) -> F_0(g_1(b_0)) 0.00/0.57 F_0(f_0(f_1(f_0(g_0(x))))) -> F_1(f_0(x)) 0.00/0.57 F_0(f_1(f_0(g_0(x)))) -> F_1(x) 0.00/0.57 F_0(f_1(f_0(g_1(x)))) -> F_0(x) 0.00/0.57 F_0(f_1(f_0(g_1(x)))) -> x 0.00/0.57 F_1(f_0(g_0(x))) -> x 0.00/0.57 G_0(f_0(f_1(f_0(g_0(x))))) -> F_0(x) 0.00/0.57 G_0(f_0(f_1(f_0(g_0(x))))) -> G_0(f_0(x)) 0.00/0.57 G_0(f_0(f_1(f_0(g_0(x))))) -> x 0.00/0.57 G_0(f_1(f_0(g_0(x)))) -> G_0(x) 0.00/0.57 G_0(f_1(f_0(g_0(x)))) -> x 0.00/0.57 G_0(g_1(b_0)) -> F_0(g_1(b_0)) 0.00/0.57 G_0(g_1(b_0)) -> G_0(f_0(g_1(b_0))) 0.00/0.57 -> Rules: 0.00/0.57 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.57 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.57 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.57 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.57 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.57 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.57 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.57 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.57 f_1(f_0(g_0(x))) -> x 0.00/0.57 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.57 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.57 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.57 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.57 -> Unhiding Rules: 0.00/0.57 f_0(g_1(b_0)) -> F_0(g_1(b_0)) 0.00/0.57 f_0(x) -> F_0(x) 0.00/0.57 f_0(x1) -> x1 0.00/0.57 0.00/0.57 Problem 1: 0.00/0.57 0.00/0.57 SCC Processor: 0.00/0.57 -> Pairs: 0.00/0.57 *TOP*_0(f_0(f_1(f_0(g_0(x))))) -> *TOP*_0(f_0(x)) 0.00/0.57 *TOP*_0(f_0(f_1(f_0(g_0(x))))) -> F_0(x) 0.00/0.57 *TOP*_0(f_0(f_1(f_0(g_0(x))))) -> x 0.00/0.57 *TOP*_0(f_1(f_0(g_0(x)))) -> *TOP*_0(x) 0.00/0.57 *TOP*_0(f_1(f_0(g_0(x)))) -> x 0.00/0.57 *TOP*_0(f_1(f_0(g_1(x)))) -> *TOP*_0(x) 0.00/0.57 *TOP*_0(f_1(f_0(g_1(x)))) -> x 0.00/0.57 *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) 0.00/0.57 *TOP*_0(g_1(b_0)) -> F_0(g_1(b_0)) 0.00/0.57 F_0(f_0(f_1(f_0(g_0(x))))) -> F_1(f_0(x)) 0.00/0.57 F_0(f_1(f_0(g_0(x)))) -> F_1(x) 0.00/0.57 F_0(f_1(f_0(g_1(x)))) -> F_0(x) 0.00/0.57 F_0(f_1(f_0(g_1(x)))) -> x 0.00/0.57 F_1(f_0(g_0(x))) -> x 0.00/0.57 G_0(f_0(f_1(f_0(g_0(x))))) -> F_0(x) 0.00/0.57 G_0(f_0(f_1(f_0(g_0(x))))) -> G_0(f_0(x)) 0.00/0.57 G_0(f_0(f_1(f_0(g_0(x))))) -> x 0.00/0.57 G_0(f_1(f_0(g_0(x)))) -> G_0(x) 0.00/0.57 G_0(f_1(f_0(g_0(x)))) -> x 0.00/0.57 G_0(g_1(b_0)) -> F_0(g_1(b_0)) 0.00/0.57 G_0(g_1(b_0)) -> G_0(f_0(g_1(b_0))) 0.00/0.57 -> Rules: 0.00/0.57 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.57 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.57 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.57 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.57 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.57 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.57 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.57 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.57 f_1(f_0(g_0(x))) -> x 0.00/0.57 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.57 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.57 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.57 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.57 -> Unhiding rules: 0.00/0.57 f_0(g_1(b_0)) -> F_0(g_1(b_0)) 0.00/0.57 f_0(x) -> F_0(x) 0.00/0.57 f_0(x1) -> x1 0.00/0.57 ->Strongly Connected Components: 0.00/0.57 ->->Cycle: 0.00/0.57 ->->-> Pairs: 0.00/0.57 F_0(f_0(f_1(f_0(g_0(x))))) -> F_1(f_0(x)) 0.00/0.57 F_0(f_1(f_0(g_0(x)))) -> F_1(x) 0.00/0.57 F_0(f_1(f_0(g_1(x)))) -> F_0(x) 0.00/0.57 F_0(f_1(f_0(g_1(x)))) -> x 0.00/0.57 F_1(f_0(g_0(x))) -> x 0.00/0.57 ->->-> Rules: 0.00/0.57 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.57 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.57 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.57 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.57 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.57 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.57 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.57 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.57 f_1(f_0(g_0(x))) -> x 0.00/0.57 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.57 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.57 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.57 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.57 ->->-> Unhiding rules: 0.00/0.57 f_0(x) -> F_0(x) 0.00/0.57 f_0(x1) -> x1 0.00/0.57 ->->Cycle: 0.00/0.57 ->->-> Pairs: 0.00/0.57 G_0(f_0(f_1(f_0(g_0(x))))) -> G_0(f_0(x)) 0.00/0.57 G_0(f_1(f_0(g_0(x)))) -> G_0(x) 0.00/0.57 G_0(g_1(b_0)) -> G_0(f_0(g_1(b_0))) 0.00/0.57 ->->-> Rules: 0.00/0.57 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.57 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.57 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.57 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.57 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.57 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.57 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.57 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.57 f_1(f_0(g_0(x))) -> x 0.00/0.57 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.57 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.57 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.57 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.57 ->->-> Unhiding rules: 0.00/0.57 Empty 0.00/0.57 ->->Cycle: 0.00/0.57 ->->-> Pairs: 0.00/0.57 *TOP*_0(f_0(f_1(f_0(g_0(x))))) -> *TOP*_0(f_0(x)) 0.00/0.57 *TOP*_0(f_1(f_0(g_0(x)))) -> *TOP*_0(x) 0.00/0.57 *TOP*_0(f_1(f_0(g_1(x)))) -> *TOP*_0(x) 0.00/0.57 *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) 0.00/0.57 ->->-> Rules: 0.00/0.57 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.57 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 ->->-> Unhiding rules: 0.00/0.58 Empty 0.00/0.58 0.00/0.58 0.00/0.58 The problem is decomposed in 3 subproblems. 0.00/0.58 0.00/0.58 Problem 1.1: 0.00/0.58 0.00/0.58 Reduction Pairs Processor: 0.00/0.58 -> Pairs: 0.00/0.58 F_0(f_0(f_1(f_0(g_0(x))))) -> F_1(f_0(x)) 0.00/0.58 F_0(f_1(f_0(g_0(x)))) -> F_1(x) 0.00/0.58 F_0(f_1(f_0(g_1(x)))) -> F_0(x) 0.00/0.58 F_0(f_1(f_0(g_1(x)))) -> x 0.00/0.58 F_1(f_0(g_0(x))) -> x 0.00/0.58 -> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 -> Unhiding rules: 0.00/0.58 f_0(x) -> F_0(x) 0.00/0.58 f_0(x1) -> x1 0.00/0.58 -> Usable rules: 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 ->Interpretation type: 0.00/0.58 Linear 0.00/0.58 ->Coefficients: 0.00/0.58 Natural Numbers 0.00/0.58 ->Dimension: 0.00/0.58 1 0.00/0.58 ->Bound: 0.00/0.58 2 0.00/0.58 ->Interpretation: 0.00/0.58 0.00/0.58 [f_0](X) = 2.X 0.00/0.58 [f_1](X) = 2.X 0.00/0.58 [g_0](X) = 2.X + 2 0.00/0.58 [b_0] = 0 0.00/0.58 [g_1](X) = 2.X 0.00/0.58 [F_0](X) = X 0.00/0.58 [F_1](X) = 2.X + 2 0.00/0.58 0.00/0.58 Problem 1.1: 0.00/0.58 0.00/0.58 SCC Processor: 0.00/0.58 -> Pairs: 0.00/0.58 F_0(f_1(f_0(g_0(x)))) -> F_1(x) 0.00/0.58 F_0(f_1(f_0(g_1(x)))) -> F_0(x) 0.00/0.58 F_0(f_1(f_0(g_1(x)))) -> x 0.00/0.58 F_1(f_0(g_0(x))) -> x 0.00/0.58 -> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 -> Unhiding rules: 0.00/0.58 f_0(x) -> F_0(x) 0.00/0.58 f_0(x1) -> x1 0.00/0.58 ->Strongly Connected Components: 0.00/0.58 ->->Cycle: 0.00/0.58 ->->-> Pairs: 0.00/0.58 F_0(f_1(f_0(g_0(x)))) -> F_1(x) 0.00/0.58 F_0(f_1(f_0(g_1(x)))) -> F_0(x) 0.00/0.58 F_0(f_1(f_0(g_1(x)))) -> x 0.00/0.58 F_1(f_0(g_0(x))) -> x 0.00/0.58 ->->-> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 ->->-> Unhiding rules: 0.00/0.58 f_0(x) -> F_0(x) 0.00/0.58 f_0(x1) -> x1 0.00/0.58 0.00/0.58 Problem 1.1: 0.00/0.58 0.00/0.58 Reduction Pairs Processor: 0.00/0.58 -> Pairs: 0.00/0.58 F_0(f_1(f_0(g_0(x)))) -> F_1(x) 0.00/0.58 F_0(f_1(f_0(g_1(x)))) -> F_0(x) 0.00/0.58 F_0(f_1(f_0(g_1(x)))) -> x 0.00/0.58 F_1(f_0(g_0(x))) -> x 0.00/0.58 -> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 -> Unhiding rules: 0.00/0.58 f_0(x) -> F_0(x) 0.00/0.58 f_0(x1) -> x1 0.00/0.58 -> Usable rules: 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 ->Interpretation type: 0.00/0.58 Linear 0.00/0.58 ->Coefficients: 0.00/0.58 Natural Numbers 0.00/0.58 ->Dimension: 0.00/0.58 1 0.00/0.58 ->Bound: 0.00/0.58 2 0.00/0.58 ->Interpretation: 0.00/0.58 0.00/0.58 [f_0](X) = 2.X 0.00/0.58 [f_1](X) = 2.X 0.00/0.58 [g_0](X) = 2.X + 2 0.00/0.58 [b_0] = 0 0.00/0.58 [g_1](X) = 2.X 0.00/0.58 [F_0](X) = 2.X 0.00/0.58 [F_1](X) = X + 1 0.00/0.58 0.00/0.58 Problem 1.1: 0.00/0.58 0.00/0.58 SCC Processor: 0.00/0.58 -> Pairs: 0.00/0.58 F_0(f_1(f_0(g_1(x)))) -> F_0(x) 0.00/0.58 F_0(f_1(f_0(g_1(x)))) -> x 0.00/0.58 F_1(f_0(g_0(x))) -> x 0.00/0.58 -> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 -> Unhiding rules: 0.00/0.58 f_0(x) -> F_0(x) 0.00/0.58 f_0(x1) -> x1 0.00/0.58 ->Strongly Connected Components: 0.00/0.58 ->->Cycle: 0.00/0.58 ->->-> Pairs: 0.00/0.58 F_0(f_1(f_0(g_1(x)))) -> F_0(x) 0.00/0.58 F_0(f_1(f_0(g_1(x)))) -> x 0.00/0.58 ->->-> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 ->->-> Unhiding rules: 0.00/0.58 f_0(x) -> F_0(x) 0.00/0.58 f_0(x1) -> x1 0.00/0.58 0.00/0.58 Problem 1.1: 0.00/0.58 0.00/0.58 Reduction Pairs Processor: 0.00/0.58 -> Pairs: 0.00/0.58 F_0(f_1(f_0(g_1(x)))) -> F_0(x) 0.00/0.58 F_0(f_1(f_0(g_1(x)))) -> x 0.00/0.58 -> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 -> Unhiding rules: 0.00/0.58 f_0(x) -> F_0(x) 0.00/0.58 f_0(x1) -> x1 0.00/0.58 -> Usable rules: 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 ->Interpretation type: 0.00/0.58 Linear 0.00/0.58 ->Coefficients: 0.00/0.58 Natural Numbers 0.00/0.58 ->Dimension: 0.00/0.58 1 0.00/0.58 ->Bound: 0.00/0.58 2 0.00/0.58 ->Interpretation: 0.00/0.58 0.00/0.58 [f_0](X) = X 0.00/0.58 [f_1](X) = X 0.00/0.58 [g_0](X) = 2.X + 2 0.00/0.58 [b_0] = 2 0.00/0.58 [g_1](X) = 2.X + 2 0.00/0.58 [F_0](X) = X 0.00/0.58 0.00/0.58 Problem 1.1: 0.00/0.58 0.00/0.58 SCC Processor: 0.00/0.58 -> Pairs: 0.00/0.58 F_0(f_1(f_0(g_1(x)))) -> x 0.00/0.58 -> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 -> Unhiding rules: 0.00/0.58 f_0(x) -> F_0(x) 0.00/0.58 f_0(x1) -> x1 0.00/0.58 ->Strongly Connected Components: 0.00/0.58 ->->Cycle: 0.00/0.58 ->->-> Pairs: 0.00/0.58 F_0(f_1(f_0(g_1(x)))) -> x 0.00/0.58 ->->-> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 ->->-> Unhiding rules: 0.00/0.58 f_0(x) -> F_0(x) 0.00/0.58 f_0(x1) -> x1 0.00/0.58 0.00/0.58 Problem 1.1: 0.00/0.58 0.00/0.58 Reduction Pairs Processor: 0.00/0.58 -> Pairs: 0.00/0.58 F_0(f_1(f_0(g_1(x)))) -> x 0.00/0.58 -> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 -> Unhiding rules: 0.00/0.58 f_0(x) -> F_0(x) 0.00/0.58 f_0(x1) -> x1 0.00/0.58 -> Usable rules: 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 ->Interpretation type: 0.00/0.58 Linear 0.00/0.58 ->Coefficients: 0.00/0.58 Natural Numbers 0.00/0.58 ->Dimension: 0.00/0.58 1 0.00/0.58 ->Bound: 0.00/0.58 2 0.00/0.58 ->Interpretation: 0.00/0.58 0.00/0.58 [f_0](X) = X 0.00/0.58 [f_1](X) = X 0.00/0.58 [g_0](X) = 2.X + 2 0.00/0.58 [b_0] = 2 0.00/0.58 [g_1](X) = 2.X + 2 0.00/0.58 [F_0](X) = X 0.00/0.58 0.00/0.58 Problem 1.1: 0.00/0.58 0.00/0.58 Basic Processor: 0.00/0.58 -> Pairs: 0.00/0.58 Empty 0.00/0.58 -> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 -> Unhiding rules: 0.00/0.58 f_0(x) -> F_0(x) 0.00/0.58 f_0(x1) -> x1 0.00/0.58 -> Result: 0.00/0.58 Set P is empty 0.00/0.58 0.00/0.58 The problem is finite. 0.00/0.58 0.00/0.58 Problem 1.2: 0.00/0.58 0.00/0.58 Reduction Pairs Processor: 0.00/0.58 -> Pairs: 0.00/0.58 G_0(f_0(f_1(f_0(g_0(x))))) -> G_0(f_0(x)) 0.00/0.58 G_0(f_1(f_0(g_0(x)))) -> G_0(x) 0.00/0.58 G_0(g_1(b_0)) -> G_0(f_0(g_1(b_0))) 0.00/0.58 -> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 -> Unhiding rules: 0.00/0.58 Empty 0.00/0.58 -> Usable rules: 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 ->Interpretation type: 0.00/0.58 Linear 0.00/0.58 ->Coefficients: 0.00/0.58 Natural Numbers 0.00/0.58 ->Dimension: 0.00/0.58 1 0.00/0.58 ->Bound: 0.00/0.58 2 0.00/0.58 ->Interpretation: 0.00/0.58 0.00/0.58 [f_0](X) = 2.X 0.00/0.58 [f_1](X) = X 0.00/0.58 [g_0](X) = X + 2 0.00/0.58 [b_0] = 0 0.00/0.58 [g_1](X) = X 0.00/0.58 [G_0](X) = 2.X 0.00/0.58 0.00/0.58 Problem 1.2: 0.00/0.58 0.00/0.58 SCC Processor: 0.00/0.58 -> Pairs: 0.00/0.58 G_0(f_1(f_0(g_0(x)))) -> G_0(x) 0.00/0.58 G_0(g_1(b_0)) -> G_0(f_0(g_1(b_0))) 0.00/0.58 -> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 -> Unhiding rules: 0.00/0.58 Empty 0.00/0.58 ->Strongly Connected Components: 0.00/0.58 ->->Cycle: 0.00/0.58 ->->-> Pairs: 0.00/0.58 G_0(f_1(f_0(g_0(x)))) -> G_0(x) 0.00/0.58 G_0(g_1(b_0)) -> G_0(f_0(g_1(b_0))) 0.00/0.58 ->->-> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 ->->-> Unhiding rules: 0.00/0.58 Empty 0.00/0.58 0.00/0.58 Problem 1.2: 0.00/0.58 0.00/0.58 Reduction Pairs Processor: 0.00/0.58 -> Pairs: 0.00/0.58 G_0(f_1(f_0(g_0(x)))) -> G_0(x) 0.00/0.58 G_0(g_1(b_0)) -> G_0(f_0(g_1(b_0))) 0.00/0.58 -> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 -> Unhiding rules: 0.00/0.58 Empty 0.00/0.58 -> Usable rules: 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 ->Interpretation type: 0.00/0.58 Linear 0.00/0.58 ->Coefficients: 0.00/0.58 Natural Numbers 0.00/0.58 ->Dimension: 0.00/0.58 1 0.00/0.58 ->Bound: 0.00/0.58 2 0.00/0.58 ->Interpretation: 0.00/0.58 0.00/0.58 [f_0](X) = 2.X 0.00/0.58 [f_1](X) = 2.X 0.00/0.58 [g_0](X) = 2.X + 2 0.00/0.58 [b_0] = 0 0.00/0.58 [g_1](X) = 2.X 0.00/0.58 [G_0](X) = 2.X 0.00/0.58 0.00/0.58 Problem 1.2: 0.00/0.58 0.00/0.58 SCC Processor: 0.00/0.58 -> Pairs: 0.00/0.58 G_0(g_1(b_0)) -> G_0(f_0(g_1(b_0))) 0.00/0.58 -> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 -> Unhiding rules: 0.00/0.58 Empty 0.00/0.58 ->Strongly Connected Components: 0.00/0.58 ->->Cycle: 0.00/0.58 ->->-> Pairs: 0.00/0.58 G_0(g_1(b_0)) -> G_0(f_0(g_1(b_0))) 0.00/0.58 ->->-> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 ->->-> Unhiding rules: 0.00/0.58 Empty 0.00/0.58 0.00/0.58 Problem 1.2: 0.00/0.58 0.00/0.58 Reduction Pairs Processor: 0.00/0.58 -> Pairs: 0.00/0.58 G_0(g_1(b_0)) -> G_0(f_0(g_1(b_0))) 0.00/0.58 -> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 -> Unhiding rules: 0.00/0.58 Empty 0.00/0.58 -> Usable rules: 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 ->Interpretation type: 0.00/0.58 Linear 0.00/0.58 ->Coefficients: 0.00/0.58 All rationals 0.00/0.58 ->Dimension: 0.00/0.58 1 0.00/0.58 ->Bound: 0.00/0.58 4 0.00/0.58 ->Interpretation: 0.00/0.58 0.00/0.58 [f_0](X) = 3/4.X + 3/2 0.00/0.58 [f_1](X) = 2/3.X + 1/2 0.00/0.58 [g_0](X) = 3.X + 3/2 0.00/0.58 [b_0] = 3 0.00/0.58 [g_1](X) = 2.X + 1 0.00/0.58 [G_0](X) = 2.X 0.00/0.58 0.00/0.58 Problem 1.2: 0.00/0.58 0.00/0.58 Basic Processor: 0.00/0.58 -> Pairs: 0.00/0.58 Empty 0.00/0.58 -> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 -> Unhiding rules: 0.00/0.58 Empty 0.00/0.58 -> Result: 0.00/0.58 Set P is empty 0.00/0.58 0.00/0.58 The problem is finite. 0.00/0.58 0.00/0.58 Problem 1.3: 0.00/0.58 0.00/0.58 Reduction Pairs Processor: 0.00/0.58 -> Pairs: 0.00/0.58 *TOP*_0(f_0(f_1(f_0(g_0(x))))) -> *TOP*_0(f_0(x)) 0.00/0.58 *TOP*_0(f_1(f_0(g_0(x)))) -> *TOP*_0(x) 0.00/0.58 *TOP*_0(f_1(f_0(g_1(x)))) -> *TOP*_0(x) 0.00/0.58 *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) 0.00/0.58 -> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 -> Unhiding rules: 0.00/0.58 Empty 0.00/0.58 -> Usable rules: 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 ->Interpretation type: 0.00/0.58 Linear 0.00/0.58 ->Coefficients: 0.00/0.58 Natural Numbers 0.00/0.58 ->Dimension: 0.00/0.58 1 0.00/0.58 ->Bound: 0.00/0.58 2 0.00/0.58 ->Interpretation: 0.00/0.58 0.00/0.58 [f_0](X) = 2.X 0.00/0.58 [f_1](X) = 2.X 0.00/0.58 [g_0](X) = 2.X + 2 0.00/0.58 [b_0] = 0 0.00/0.58 [g_1](X) = 2.X 0.00/0.58 [*TOP*_0](X) = X 0.00/0.58 0.00/0.58 Problem 1.3: 0.00/0.58 0.00/0.58 SCC Processor: 0.00/0.58 -> Pairs: 0.00/0.58 *TOP*_0(f_1(f_0(g_0(x)))) -> *TOP*_0(x) 0.00/0.58 *TOP*_0(f_1(f_0(g_1(x)))) -> *TOP*_0(x) 0.00/0.58 *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) 0.00/0.58 -> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 -> Unhiding rules: 0.00/0.58 Empty 0.00/0.58 ->Strongly Connected Components: 0.00/0.58 ->->Cycle: 0.00/0.58 ->->-> Pairs: 0.00/0.58 *TOP*_0(f_1(f_0(g_0(x)))) -> *TOP*_0(x) 0.00/0.58 *TOP*_0(f_1(f_0(g_1(x)))) -> *TOP*_0(x) 0.00/0.58 *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) 0.00/0.58 ->->-> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 ->->-> Unhiding rules: 0.00/0.58 Empty 0.00/0.58 0.00/0.58 Problem 1.3: 0.00/0.58 0.00/0.58 Reduction Pairs Processor: 0.00/0.58 -> Pairs: 0.00/0.58 *TOP*_0(f_1(f_0(g_0(x)))) -> *TOP*_0(x) 0.00/0.58 *TOP*_0(f_1(f_0(g_1(x)))) -> *TOP*_0(x) 0.00/0.58 *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) 0.00/0.58 -> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 -> Unhiding rules: 0.00/0.58 Empty 0.00/0.58 -> Usable rules: 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 ->Interpretation type: 0.00/0.58 Linear 0.00/0.58 ->Coefficients: 0.00/0.58 Natural Numbers 0.00/0.58 ->Dimension: 0.00/0.58 1 0.00/0.58 ->Bound: 0.00/0.58 2 0.00/0.58 ->Interpretation: 0.00/0.58 0.00/0.58 [f_0](X) = 2.X 0.00/0.58 [f_1](X) = 2.X 0.00/0.58 [g_0](X) = 2.X + 2 0.00/0.58 [b_0] = 0 0.00/0.58 [g_1](X) = X 0.00/0.58 [*TOP*_0](X) = 2.X 0.00/0.58 0.00/0.58 Problem 1.3: 0.00/0.58 0.00/0.58 SCC Processor: 0.00/0.58 -> Pairs: 0.00/0.58 *TOP*_0(f_1(f_0(g_1(x)))) -> *TOP*_0(x) 0.00/0.58 *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) 0.00/0.58 -> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 -> Unhiding rules: 0.00/0.58 Empty 0.00/0.58 ->Strongly Connected Components: 0.00/0.58 ->->Cycle: 0.00/0.58 ->->-> Pairs: 0.00/0.58 *TOP*_0(f_1(f_0(g_1(x)))) -> *TOP*_0(x) 0.00/0.58 *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) 0.00/0.58 ->->-> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 ->->-> Unhiding rules: 0.00/0.58 Empty 0.00/0.58 0.00/0.58 Problem 1.3: 0.00/0.58 0.00/0.58 Reduction Pairs Processor: 0.00/0.58 -> Pairs: 0.00/0.58 *TOP*_0(f_1(f_0(g_1(x)))) -> *TOP*_0(x) 0.00/0.58 *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) 0.00/0.58 -> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 -> Unhiding rules: 0.00/0.58 Empty 0.00/0.58 -> Usable rules: 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 ->Interpretation type: 0.00/0.58 Linear 0.00/0.58 ->Coefficients: 0.00/0.58 Natural Numbers 0.00/0.58 ->Dimension: 0.00/0.58 1 0.00/0.58 ->Bound: 0.00/0.58 2 0.00/0.58 ->Interpretation: 0.00/0.58 0.00/0.58 [f_0](X) = X 0.00/0.58 [f_1](X) = X 0.00/0.58 [g_0](X) = X + 2 0.00/0.58 [b_0] = 0 0.00/0.58 [g_1](X) = 2.X + 1 0.00/0.58 [*TOP*_0](X) = X 0.00/0.58 0.00/0.58 Problem 1.3: 0.00/0.58 0.00/0.58 SCC Processor: 0.00/0.58 -> Pairs: 0.00/0.58 *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) 0.00/0.58 -> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 -> Unhiding rules: 0.00/0.58 Empty 0.00/0.58 ->Strongly Connected Components: 0.00/0.58 ->->Cycle: 0.00/0.58 ->->-> Pairs: 0.00/0.58 *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) 0.00/0.58 ->->-> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 ->->-> Unhiding rules: 0.00/0.58 Empty 0.00/0.58 0.00/0.58 Problem 1.3: 0.00/0.58 0.00/0.58 Reduction Pairs Processor: 0.00/0.58 -> Pairs: 0.00/0.58 *TOP*_0(g_1(b_0)) -> *TOP*_0(f_0(g_1(b_0))) 0.00/0.58 -> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 -> Unhiding rules: 0.00/0.58 Empty 0.00/0.58 -> Usable rules: 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 ->Interpretation type: 0.00/0.58 Linear 0.00/0.58 ->Coefficients: 0.00/0.58 All rationals 0.00/0.58 ->Dimension: 0.00/0.58 1 0.00/0.58 ->Bound: 0.00/0.58 4 0.00/0.58 ->Interpretation: 0.00/0.58 0.00/0.58 [f_0](X) = 3/4.X + 3/2 0.00/0.58 [f_1](X) = 2/3.X + 1/2 0.00/0.58 [g_0](X) = 3.X + 3/2 0.00/0.58 [b_0] = 3 0.00/0.58 [g_1](X) = 2.X + 1 0.00/0.58 [*TOP*_0](X) = 2.X 0.00/0.58 0.00/0.58 Problem 1.3: 0.00/0.58 0.00/0.58 Basic Processor: 0.00/0.58 -> Pairs: 0.00/0.58 Empty 0.00/0.58 -> Rules: 0.00/0.58 *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x)) 0.00/0.58 *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x) 0.00/0.58 *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0))) 0.00/0.58 f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x)) 0.00/0.58 f_0(f_1(f_0(g_0(x)))) -> f_1(x) 0.00/0.58 f_0(f_1(f_0(g_1(x)))) -> f_0(x) 0.00/0.58 f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0))) 0.00/0.58 f_1(f_0(g_0(x))) -> x 0.00/0.58 g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x)) 0.00/0.58 g_0(f_1(f_0(g_0(x)))) -> g_0(x) 0.00/0.58 g_0(f_1(f_0(g_1(x)))) -> g_1(x) 0.00/0.58 g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0))) 0.00/0.58 -> Unhiding rules: 0.00/0.58 Empty 0.00/0.58 -> Result: 0.00/0.58 Set P is empty 0.00/0.58 0.00/0.58 The problem is finite. 0.00/0.58 EOF