0.00/0.06	YES
0.00/0.06	
0.00/0.06	Problem 1: 
0.00/0.06	
0.00/0.06	(VAR x)
0.00/0.06	(STRATEGY CONTEXTSENSITIVE
0.00/0.06	(*top*_0 1)
0.00/0.06	(f_0 1)
0.00/0.06	(f_1)
0.00/0.06	(g_1)
0.00/0.06	(s_0 1)
0.00/0.06	(a_0)
0.00/0.06	(b_0)
0.00/0.06	)
0.00/0.06	(RULES
0.00/0.06	*top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
0.00/0.06	f_0(g_1(x)) -> f_1(f_0(g_1(x)))
0.00/0.06	f_1(f_0(x)) -> b_0
0.00/0.06	f_1(f_1(x)) -> b_0
0.00/0.06	f_1(g_1(a_0)) -> f_0(s_0(g_1(b_0)))
0.00/0.06	g_1(x) -> f_1(g_1(x))
0.00/0.06	s_0(g_1(x)) -> s_0(f_0(g_1(x)))
0.00/0.06	)
0.00/0.06	
0.00/0.06	Problem 1: 
0.00/0.06	
0.00/0.06	Dependency Pairs Processor:
0.00/0.06	-> Pairs:
0.00/0.06	 *TOP*_0(g_1(x)) -> *TOP*_0(f_0(g_1(x)))
0.00/0.06	 *TOP*_0(g_1(x)) -> F_0(g_1(x))
0.00/0.06	 F_0(g_1(x)) -> F_1(f_0(g_1(x)))
0.00/0.06	 F_1(g_1(a_0)) -> F_0(s_0(g_1(b_0)))
0.00/0.06	 F_1(g_1(a_0)) -> G_1(b_0)
0.00/0.06	 F_1(g_1(a_0)) -> S_0(g_1(b_0))
0.00/0.06	 G_1(x) -> F_1(g_1(x))
0.00/0.06	 S_0(g_1(x)) -> F_0(g_1(x))
0.00/0.06	 S_0(g_1(x)) -> S_0(f_0(g_1(x)))
0.00/0.06	-> Rules:
0.00/0.06	 *top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
0.00/0.06	 f_0(g_1(x)) -> f_1(f_0(g_1(x)))
0.00/0.06	 f_1(f_0(x)) -> b_0
0.00/0.06	 f_1(f_1(x)) -> b_0
0.00/0.06	 f_1(g_1(a_0)) -> f_0(s_0(g_1(b_0)))
0.00/0.06	 g_1(x) -> f_1(g_1(x))
0.00/0.06	 s_0(g_1(x)) -> s_0(f_0(g_1(x)))
0.00/0.06	-> Unhiding Rules:
0.00/0.06	 Empty
0.00/0.06	
0.00/0.06	Problem 1: 
0.00/0.06	
0.00/0.06	SCC Processor:
0.00/0.06	-> Pairs:
0.00/0.06	 *TOP*_0(g_1(x)) -> *TOP*_0(f_0(g_1(x)))
0.00/0.06	 *TOP*_0(g_1(x)) -> F_0(g_1(x))
0.00/0.06	 F_0(g_1(x)) -> F_1(f_0(g_1(x)))
0.00/0.06	 F_1(g_1(a_0)) -> F_0(s_0(g_1(b_0)))
0.00/0.06	 F_1(g_1(a_0)) -> G_1(b_0)
0.00/0.06	 F_1(g_1(a_0)) -> S_0(g_1(b_0))
0.00/0.06	 G_1(x) -> F_1(g_1(x))
0.00/0.06	 S_0(g_1(x)) -> F_0(g_1(x))
0.00/0.06	 S_0(g_1(x)) -> S_0(f_0(g_1(x)))
0.00/0.06	-> Rules:
0.00/0.06	 *top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
0.00/0.06	 f_0(g_1(x)) -> f_1(f_0(g_1(x)))
0.00/0.06	 f_1(f_0(x)) -> b_0
0.00/0.06	 f_1(f_1(x)) -> b_0
0.00/0.06	 f_1(g_1(a_0)) -> f_0(s_0(g_1(b_0)))
0.00/0.06	 g_1(x) -> f_1(g_1(x))
0.00/0.06	 s_0(g_1(x)) -> s_0(f_0(g_1(x)))
0.00/0.06	-> Unhiding rules:
0.00/0.06	 Empty
0.00/0.06	->Strongly Connected Components:
0.00/0.06	->->Cycle:
0.00/0.06	->->-> Pairs:
0.00/0.06	 S_0(g_1(x)) -> S_0(f_0(g_1(x)))
0.00/0.06	->->-> Rules:
0.00/0.06	 *top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
0.00/0.06	 f_0(g_1(x)) -> f_1(f_0(g_1(x)))
0.00/0.06	 f_1(f_0(x)) -> b_0
0.00/0.06	 f_1(f_1(x)) -> b_0
0.00/0.06	 f_1(g_1(a_0)) -> f_0(s_0(g_1(b_0)))
0.00/0.06	 g_1(x) -> f_1(g_1(x))
0.00/0.06	 s_0(g_1(x)) -> s_0(f_0(g_1(x)))
0.00/0.06	->->-> Unhiding rules:
0.00/0.06	 Empty
0.00/0.06	->->Cycle:
0.00/0.06	->->-> Pairs:
0.00/0.06	 F_1(g_1(a_0)) -> G_1(b_0)
0.00/0.06	 G_1(x) -> F_1(g_1(x))
0.00/0.06	->->-> Rules:
0.00/0.06	 *top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
0.00/0.06	 f_0(g_1(x)) -> f_1(f_0(g_1(x)))
0.00/0.06	 f_1(f_0(x)) -> b_0
0.00/0.06	 f_1(f_1(x)) -> b_0
0.00/0.06	 f_1(g_1(a_0)) -> f_0(s_0(g_1(b_0)))
0.00/0.06	 g_1(x) -> f_1(g_1(x))
0.00/0.06	 s_0(g_1(x)) -> s_0(f_0(g_1(x)))
0.00/0.06	->->-> Unhiding rules:
0.00/0.06	 Empty
0.00/0.06	->->Cycle:
0.00/0.06	->->-> Pairs:
0.00/0.06	 *TOP*_0(g_1(x)) -> *TOP*_0(f_0(g_1(x)))
0.00/0.06	->->-> Rules:
0.00/0.06	 *top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
0.00/0.06	 f_0(g_1(x)) -> f_1(f_0(g_1(x)))
0.00/0.06	 f_1(f_0(x)) -> b_0
0.00/0.06	 f_1(f_1(x)) -> b_0
0.00/0.06	 f_1(g_1(a_0)) -> f_0(s_0(g_1(b_0)))
0.00/0.06	 g_1(x) -> f_1(g_1(x))
0.00/0.06	 s_0(g_1(x)) -> s_0(f_0(g_1(x)))
0.00/0.06	->->-> Unhiding rules:
0.00/0.06	 Empty
0.00/0.06	
0.00/0.06	
0.00/0.06	The problem is decomposed in 3 subproblems.
0.00/0.06	
0.00/0.06	Problem 1.1: 
0.00/0.06	
0.00/0.06	Reduction Pairs Processor:
0.00/0.06	-> Pairs:
0.00/0.06	 S_0(g_1(x)) -> S_0(f_0(g_1(x)))
0.00/0.06	-> Rules:
0.00/0.06	 *top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
0.00/0.06	 f_0(g_1(x)) -> f_1(f_0(g_1(x)))
0.00/0.06	 f_1(f_0(x)) -> b_0
0.00/0.06	 f_1(f_1(x)) -> b_0
0.00/0.06	 f_1(g_1(a_0)) -> f_0(s_0(g_1(b_0)))
0.00/0.06	 g_1(x) -> f_1(g_1(x))
0.00/0.06	 s_0(g_1(x)) -> s_0(f_0(g_1(x)))
0.00/0.06	-> Unhiding rules:
0.00/0.06	 Empty
0.00/0.06	-> Usable rules:
0.00/0.06	 f_0(g_1(x)) -> f_1(f_0(g_1(x)))
0.00/0.06	 f_1(f_0(x)) -> b_0
0.00/0.06	 f_1(f_1(x)) -> b_0
0.00/0.06	 f_1(g_1(a_0)) -> f_0(s_0(g_1(b_0)))
0.00/0.06	 g_1(x) -> f_1(g_1(x))
0.00/0.06	 s_0(g_1(x)) -> s_0(f_0(g_1(x)))
0.00/0.06	->Interpretation type:
0.00/0.06	 Linear
0.00/0.06	->Coefficients:
0.00/0.06	 Natural Numbers
0.00/0.06	->Dimension:
0.00/0.06	 1
0.00/0.06	->Bound:
0.00/0.06	 2
0.00/0.06	->Interpretation:
0.00/0.06	 
0.00/0.06	[f_0](X) = 1
0.00/0.06	[f_1](X) = 1
0.00/0.06	[g_1](X) = 2.X + 2
0.00/0.06	[s_0](X) = 2.X
0.00/0.06	[a_0] = 1
0.00/0.06	[b_0] = 1
0.00/0.06	[S_0](X) = 2.X
0.00/0.06	
0.00/0.06	Problem 1.1: 
0.00/0.06	
0.00/0.06	Basic Processor:
0.00/0.06	-> Pairs:
0.00/0.06	 Empty
0.00/0.06	-> Rules:
0.00/0.06	 *top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
0.00/0.06	 f_0(g_1(x)) -> f_1(f_0(g_1(x)))
0.00/0.06	 f_1(f_0(x)) -> b_0
0.00/0.06	 f_1(f_1(x)) -> b_0
0.00/0.06	 f_1(g_1(a_0)) -> f_0(s_0(g_1(b_0)))
0.00/0.06	 g_1(x) -> f_1(g_1(x))
0.00/0.06	 s_0(g_1(x)) -> s_0(f_0(g_1(x)))
0.00/0.06	-> Unhiding rules:
0.00/0.06	 Empty
0.00/0.06	-> Result:
0.00/0.06	 Set P is empty
0.00/0.06	
0.00/0.06	The problem is finite.
0.00/0.06	
0.00/0.06	Problem 1.2: 
0.00/0.06	
0.00/0.06	Reduction Pairs Processor:
0.00/0.06	-> Pairs:
0.00/0.06	 F_1(g_1(a_0)) -> G_1(b_0)
0.00/0.06	 G_1(x) -> F_1(g_1(x))
0.00/0.06	-> Rules:
0.00/0.06	 *top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
0.00/0.06	 f_0(g_1(x)) -> f_1(f_0(g_1(x)))
0.00/0.06	 f_1(f_0(x)) -> b_0
0.00/0.06	 f_1(f_1(x)) -> b_0
0.00/0.06	 f_1(g_1(a_0)) -> f_0(s_0(g_1(b_0)))
0.00/0.06	 g_1(x) -> f_1(g_1(x))
0.00/0.06	 s_0(g_1(x)) -> s_0(f_0(g_1(x)))
0.00/0.06	-> Unhiding rules:
0.00/0.06	 Empty
0.00/0.06	-> Usable rules:
0.00/0.06	 Empty
0.00/0.06	->Interpretation type:
0.00/0.06	 Linear
0.00/0.06	->Coefficients:
0.00/0.06	 Natural Numbers
0.00/0.06	->Dimension:
0.00/0.06	 1
0.00/0.06	->Bound:
0.00/0.06	 2
0.00/0.06	->Interpretation:
0.00/0.06	 
0.00/0.06	[g_1](X) = 2.X + 2
0.00/0.06	[a_0] = 2
0.00/0.06	[b_0] = 1
0.00/0.06	[F_1](X) = X
0.00/0.06	[G_1](X) = 2.X + 2
0.00/0.06	
0.00/0.06	Problem 1.2: 
0.00/0.06	
0.00/0.06	SCC Processor:
0.00/0.06	-> Pairs:
0.00/0.06	 G_1(x) -> F_1(g_1(x))
0.00/0.06	-> Rules:
0.00/0.06	 *top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
0.00/0.06	 f_0(g_1(x)) -> f_1(f_0(g_1(x)))
0.00/0.06	 f_1(f_0(x)) -> b_0
0.00/0.06	 f_1(f_1(x)) -> b_0
0.00/0.06	 f_1(g_1(a_0)) -> f_0(s_0(g_1(b_0)))
0.00/0.06	 g_1(x) -> f_1(g_1(x))
0.00/0.06	 s_0(g_1(x)) -> s_0(f_0(g_1(x)))
0.00/0.06	-> Unhiding rules:
0.00/0.06	 Empty
0.00/0.06	->Strongly Connected Components:
0.00/0.06	 There is no strongly connected component
0.00/0.06	
0.00/0.06	The problem is finite.
0.00/0.06	
0.00/0.06	Problem 1.3: 
0.00/0.06	
0.00/0.06	Reduction Pairs Processor:
0.00/0.06	-> Pairs:
0.00/0.06	 *TOP*_0(g_1(x)) -> *TOP*_0(f_0(g_1(x)))
0.00/0.06	-> Rules:
0.00/0.06	 *top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
0.00/0.06	 f_0(g_1(x)) -> f_1(f_0(g_1(x)))
0.00/0.06	 f_1(f_0(x)) -> b_0
0.00/0.06	 f_1(f_1(x)) -> b_0
0.00/0.06	 f_1(g_1(a_0)) -> f_0(s_0(g_1(b_0)))
0.00/0.06	 g_1(x) -> f_1(g_1(x))
0.00/0.06	 s_0(g_1(x)) -> s_0(f_0(g_1(x)))
0.00/0.06	-> Unhiding rules:
0.00/0.06	 Empty
0.00/0.06	-> Usable rules:
0.00/0.06	 f_0(g_1(x)) -> f_1(f_0(g_1(x)))
0.00/0.06	 f_1(f_0(x)) -> b_0
0.00/0.06	 f_1(f_1(x)) -> b_0
0.00/0.06	 f_1(g_1(a_0)) -> f_0(s_0(g_1(b_0)))
0.00/0.06	 g_1(x) -> f_1(g_1(x))
0.00/0.06	 s_0(g_1(x)) -> s_0(f_0(g_1(x)))
0.00/0.06	->Interpretation type:
0.00/0.06	 Linear
0.00/0.06	->Coefficients:
0.00/0.06	 Natural Numbers
0.00/0.06	->Dimension:
0.00/0.06	 1
0.00/0.06	->Bound:
0.00/0.06	 2
0.00/0.06	->Interpretation:
0.00/0.06	 
0.00/0.06	[f_0](X) = 1
0.00/0.06	[f_1](X) = 1
0.00/0.06	[g_1](X) = 2.X + 2
0.00/0.06	[s_0](X) = X
0.00/0.06	[a_0] = 1
0.00/0.06	[b_0] = 1
0.00/0.06	[*TOP*_0](X) = 2.X
0.00/0.06	
0.00/0.06	Problem 1.3: 
0.00/0.06	
0.00/0.06	Basic Processor:
0.00/0.06	-> Pairs:
0.00/0.06	 Empty
0.00/0.06	-> Rules:
0.00/0.06	 *top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
0.00/0.06	 f_0(g_1(x)) -> f_1(f_0(g_1(x)))
0.00/0.06	 f_1(f_0(x)) -> b_0
0.00/0.06	 f_1(f_1(x)) -> b_0
0.00/0.06	 f_1(g_1(a_0)) -> f_0(s_0(g_1(b_0)))
0.00/0.06	 g_1(x) -> f_1(g_1(x))
0.00/0.06	 s_0(g_1(x)) -> s_0(f_0(g_1(x)))
0.00/0.06	-> Unhiding rules:
0.00/0.06	 Empty
0.00/0.06	-> Result:
0.00/0.06	 Set P is empty
0.00/0.06	
0.00/0.06	The problem is finite.
0.00/0.06	EOF