0.00/0.11	YES
0.00/0.11	
0.00/0.11	Problem 1: 
0.00/0.11	
0.00/0.11	(VAR v_NonEmpty:S x1:S)
0.00/0.11	(RULES
0.00/0.11	d(0(x1:S)) -> 0(x1:S)
0.00/0.11	d(s(x1:S)) -> s(s(d(p(s(x1:S)))))
0.00/0.11	f(0(x1:S)) -> s(0(x1:S))
0.00/0.11	f(s(x1:S)) -> d(f(p(s(x1:S))))
0.00/0.11	p(s(x1:S)) -> x1:S
0.00/0.11	)
0.00/0.11	
0.00/0.11	Problem 1: 
0.00/0.11	
0.00/0.11	Innermost Equivalent Processor:
0.00/0.11	-> Rules:
0.00/0.11	 d(0(x1:S)) -> 0(x1:S)
0.00/0.11	 d(s(x1:S)) -> s(s(d(p(s(x1:S)))))
0.00/0.11	 f(0(x1:S)) -> s(0(x1:S))
0.00/0.11	 f(s(x1:S)) -> d(f(p(s(x1:S))))
0.00/0.11	 p(s(x1:S)) -> x1:S
0.00/0.11	-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
0.00/0.11	 
0.00/0.11	
0.00/0.11	Problem 1: 
0.00/0.11	
0.00/0.11	Dependency Pairs Processor:
0.00/0.11	-> Pairs:
0.00/0.11	 D(s(x1:S)) -> D(p(s(x1:S)))
0.00/0.11	 D(s(x1:S)) -> P(s(x1:S))
0.00/0.11	 F(s(x1:S)) -> D(f(p(s(x1:S))))
0.00/0.11	 F(s(x1:S)) -> F(p(s(x1:S)))
0.00/0.11	 F(s(x1:S)) -> P(s(x1:S))
0.00/0.11	-> Rules:
0.00/0.11	 d(0(x1:S)) -> 0(x1:S)
0.00/0.11	 d(s(x1:S)) -> s(s(d(p(s(x1:S)))))
0.00/0.11	 f(0(x1:S)) -> s(0(x1:S))
0.00/0.11	 f(s(x1:S)) -> d(f(p(s(x1:S))))
0.00/0.11	 p(s(x1:S)) -> x1:S
0.00/0.11	
0.00/0.11	Problem 1: 
0.00/0.11	
0.00/0.11	SCC Processor:
0.00/0.11	-> Pairs:
0.00/0.11	 D(s(x1:S)) -> D(p(s(x1:S)))
0.00/0.11	 D(s(x1:S)) -> P(s(x1:S))
0.00/0.11	 F(s(x1:S)) -> D(f(p(s(x1:S))))
0.00/0.11	 F(s(x1:S)) -> F(p(s(x1:S)))
0.00/0.11	 F(s(x1:S)) -> P(s(x1:S))
0.00/0.11	-> Rules:
0.00/0.11	 d(0(x1:S)) -> 0(x1:S)
0.00/0.11	 d(s(x1:S)) -> s(s(d(p(s(x1:S)))))
0.00/0.11	 f(0(x1:S)) -> s(0(x1:S))
0.00/0.11	 f(s(x1:S)) -> d(f(p(s(x1:S))))
0.00/0.11	 p(s(x1:S)) -> x1:S
0.00/0.11	->Strongly Connected Components:
0.00/0.11	->->Cycle:
0.00/0.11	->->-> Pairs:
0.00/0.11	 D(s(x1:S)) -> D(p(s(x1:S)))
0.00/0.11	->->-> Rules:
0.00/0.11	 d(0(x1:S)) -> 0(x1:S)
0.00/0.11	 d(s(x1:S)) -> s(s(d(p(s(x1:S)))))
0.00/0.11	 f(0(x1:S)) -> s(0(x1:S))
0.00/0.11	 f(s(x1:S)) -> d(f(p(s(x1:S))))
0.00/0.11	 p(s(x1:S)) -> x1:S
0.00/0.11	->->Cycle:
0.00/0.11	->->-> Pairs:
0.00/0.11	 F(s(x1:S)) -> F(p(s(x1:S)))
0.00/0.11	->->-> Rules:
0.00/0.11	 d(0(x1:S)) -> 0(x1:S)
0.00/0.11	 d(s(x1:S)) -> s(s(d(p(s(x1:S)))))
0.00/0.11	 f(0(x1:S)) -> s(0(x1:S))
0.00/0.11	 f(s(x1:S)) -> d(f(p(s(x1:S))))
0.00/0.11	 p(s(x1:S)) -> x1:S
0.00/0.11	
0.00/0.11	
0.00/0.11	The problem is decomposed in 2 subproblems.
0.00/0.11	
0.00/0.11	Problem 1.1: 
0.00/0.11	
0.00/0.11	Reduction Pairs Processor:
0.00/0.11	-> Pairs:
0.00/0.11	 D(s(x1:S)) -> D(p(s(x1:S)))
0.00/0.11	-> Rules:
0.00/0.11	 d(0(x1:S)) -> 0(x1:S)
0.00/0.11	 d(s(x1:S)) -> s(s(d(p(s(x1:S)))))
0.00/0.11	 f(0(x1:S)) -> s(0(x1:S))
0.00/0.11	 f(s(x1:S)) -> d(f(p(s(x1:S))))
0.00/0.11	 p(s(x1:S)) -> x1:S
0.00/0.11	-> Usable rules:
0.00/0.11	 p(s(x1:S)) -> x1:S
0.00/0.11	->Interpretation type:
0.00/0.11	 Linear
0.00/0.11	->Coefficients:
0.00/0.11	 All rationals
0.00/0.11	->Dimension:
0.00/0.11	 1
0.00/0.11	->Bound:
0.00/0.11	 2
0.00/0.11	->Interpretation:
0.00/0.11	 
0.00/0.11	[d](X) = 0
0.00/0.11	[f](X) = 0
0.00/0.11	[p](X) = 1/2.X
0.00/0.11	[0](X) = 0
0.00/0.11	[fSNonEmpty] = 0
0.00/0.11	[s](X) = 2.X + 1/2
0.00/0.11	[D](X) = 2.X
0.00/0.11	[F](X) = 0
0.00/0.11	[P](X) = 0
0.00/0.11	
0.00/0.11	Problem 1.1: 
0.00/0.11	
0.00/0.11	SCC Processor:
0.00/0.11	-> Pairs:
0.00/0.11	 Empty
0.00/0.11	-> Rules:
0.00/0.11	 d(0(x1:S)) -> 0(x1:S)
0.00/0.11	 d(s(x1:S)) -> s(s(d(p(s(x1:S)))))
0.00/0.11	 f(0(x1:S)) -> s(0(x1:S))
0.00/0.11	 f(s(x1:S)) -> d(f(p(s(x1:S))))
0.00/0.11	 p(s(x1:S)) -> x1:S
0.00/0.11	->Strongly Connected Components:
0.00/0.11	 There is no strongly connected component
0.00/0.11	
0.00/0.11	The problem is finite.
0.00/0.11	
0.00/0.11	Problem 1.2: 
0.00/0.11	
0.00/0.11	Reduction Pairs Processor:
0.00/0.11	-> Pairs:
0.00/0.11	 F(s(x1:S)) -> F(p(s(x1:S)))
0.00/0.11	-> Rules:
0.00/0.11	 d(0(x1:S)) -> 0(x1:S)
0.00/0.11	 d(s(x1:S)) -> s(s(d(p(s(x1:S)))))
0.00/0.11	 f(0(x1:S)) -> s(0(x1:S))
0.00/0.11	 f(s(x1:S)) -> d(f(p(s(x1:S))))
0.00/0.11	 p(s(x1:S)) -> x1:S
0.00/0.11	-> Usable rules:
0.00/0.11	 p(s(x1:S)) -> x1:S
0.00/0.11	->Interpretation type:
0.00/0.11	 Linear
0.00/0.11	->Coefficients:
0.00/0.11	 All rationals
0.00/0.11	->Dimension:
0.00/0.11	 1
0.00/0.11	->Bound:
0.00/0.11	 2
0.00/0.11	->Interpretation:
0.00/0.11	 
0.00/0.11	[d](X) = 0
0.00/0.11	[f](X) = 0
0.00/0.11	[p](X) = 1/2.X
0.00/0.11	[0](X) = 0
0.00/0.11	[fSNonEmpty] = 0
0.00/0.11	[s](X) = 2.X + 1/2
0.00/0.11	[D](X) = 0
0.00/0.11	[F](X) = 2.X
0.00/0.11	[P](X) = 0
0.00/0.11	
0.00/0.11	Problem 1.2: 
0.00/0.11	
0.00/0.11	SCC Processor:
0.00/0.11	-> Pairs:
0.00/0.11	 Empty
0.00/0.11	-> Rules:
0.00/0.11	 d(0(x1:S)) -> 0(x1:S)
0.00/0.11	 d(s(x1:S)) -> s(s(d(p(s(x1:S)))))
0.00/0.11	 f(0(x1:S)) -> s(0(x1:S))
0.00/0.11	 f(s(x1:S)) -> d(f(p(s(x1:S))))
0.00/0.11	 p(s(x1:S)) -> x1:S
0.00/0.11	->Strongly Connected Components:
0.00/0.11	 There is no strongly connected component
0.00/0.11	
0.00/0.11	The problem is finite.
0.00/0.11	EOF