0.00/0.05	YES
0.00/0.05	
0.00/0.05	Problem 1: 
0.00/0.05	
0.00/0.05	(VAR v_NonEmpty:S x1:S)
0.00/0.05	(RULES
0.00/0.05	P(p(x1:S)) -> x1:S
0.00/0.05	R(r(x1:S)) -> x1:S
0.00/0.05	R(x1:S) -> r(x1:S)
0.00/0.05	p(P(x1:S)) -> x1:S
0.00/0.05	r(P(P(x1:S))) -> P(P(r(x1:S)))
0.00/0.05	r(R(x1:S)) -> x1:S
0.00/0.05	r(p(x1:S)) -> p(p(r(P(x1:S))))
0.00/0.05	r(r(x1:S)) -> x1:S
0.00/0.05	)
0.00/0.05	
0.00/0.05	Problem 1: 
0.00/0.05	
0.00/0.05	Dependency Pairs Processor:
0.00/0.05	-> Pairs:
0.00/0.05	 R#(x1:S) -> r#(x1:S)
0.00/0.05	 r#(P(P(x1:S))) -> P#(P(r(x1:S)))
0.00/0.05	 r#(P(P(x1:S))) -> P#(r(x1:S))
0.00/0.05	 r#(P(P(x1:S))) -> r#(x1:S)
0.00/0.05	 r#(p(x1:S)) -> P#(x1:S)
0.00/0.05	 r#(p(x1:S)) -> p#(p(r(P(x1:S))))
0.00/0.05	 r#(p(x1:S)) -> p#(r(P(x1:S)))
0.00/0.05	 r#(p(x1:S)) -> r#(P(x1:S))
0.00/0.05	-> Rules:
0.00/0.05	 P(p(x1:S)) -> x1:S
0.00/0.05	 R(r(x1:S)) -> x1:S
0.00/0.05	 R(x1:S) -> r(x1:S)
0.00/0.05	 p(P(x1:S)) -> x1:S
0.00/0.05	 r(P(P(x1:S))) -> P(P(r(x1:S)))
0.00/0.05	 r(R(x1:S)) -> x1:S
0.00/0.05	 r(p(x1:S)) -> p(p(r(P(x1:S))))
0.00/0.05	 r(r(x1:S)) -> x1:S
0.00/0.05	
0.00/0.05	Problem 1: 
0.00/0.05	
0.00/0.05	SCC Processor:
0.00/0.05	-> Pairs:
0.00/0.05	 R#(x1:S) -> r#(x1:S)
0.00/0.05	 r#(P(P(x1:S))) -> P#(P(r(x1:S)))
0.00/0.05	 r#(P(P(x1:S))) -> P#(r(x1:S))
0.00/0.05	 r#(P(P(x1:S))) -> r#(x1:S)
0.00/0.05	 r#(p(x1:S)) -> P#(x1:S)
0.00/0.05	 r#(p(x1:S)) -> p#(p(r(P(x1:S))))
0.00/0.05	 r#(p(x1:S)) -> p#(r(P(x1:S)))
0.00/0.05	 r#(p(x1:S)) -> r#(P(x1:S))
0.00/0.05	-> Rules:
0.00/0.05	 P(p(x1:S)) -> x1:S
0.00/0.05	 R(r(x1:S)) -> x1:S
0.00/0.05	 R(x1:S) -> r(x1:S)
0.00/0.05	 p(P(x1:S)) -> x1:S
0.00/0.05	 r(P(P(x1:S))) -> P(P(r(x1:S)))
0.00/0.05	 r(R(x1:S)) -> x1:S
0.00/0.05	 r(p(x1:S)) -> p(p(r(P(x1:S))))
0.00/0.05	 r(r(x1:S)) -> x1:S
0.00/0.05	->Strongly Connected Components:
0.00/0.05	->->Cycle:
0.00/0.05	->->-> Pairs:
0.00/0.05	 r#(P(P(x1:S))) -> r#(x1:S)
0.00/0.05	 r#(p(x1:S)) -> r#(P(x1:S))
0.00/0.05	->->-> Rules:
0.00/0.05	 P(p(x1:S)) -> x1:S
0.00/0.05	 R(r(x1:S)) -> x1:S
0.00/0.05	 R(x1:S) -> r(x1:S)
0.00/0.05	 p(P(x1:S)) -> x1:S
0.00/0.05	 r(P(P(x1:S))) -> P(P(r(x1:S)))
0.00/0.05	 r(R(x1:S)) -> x1:S
0.00/0.05	 r(p(x1:S)) -> p(p(r(P(x1:S))))
0.00/0.05	 r(r(x1:S)) -> x1:S
0.00/0.05	
0.00/0.05	Problem 1: 
0.00/0.05	
0.00/0.05	Reduction Pair Processor:
0.00/0.05	-> Pairs:
0.00/0.05	 r#(P(P(x1:S))) -> r#(x1:S)
0.00/0.05	 r#(p(x1:S)) -> r#(P(x1:S))
0.00/0.05	-> Rules:
0.00/0.05	 P(p(x1:S)) -> x1:S
0.00/0.05	 R(r(x1:S)) -> x1:S
0.00/0.05	 R(x1:S) -> r(x1:S)
0.00/0.05	 p(P(x1:S)) -> x1:S
0.00/0.05	 r(P(P(x1:S))) -> P(P(r(x1:S)))
0.00/0.05	 r(R(x1:S)) -> x1:S
0.00/0.05	 r(p(x1:S)) -> p(p(r(P(x1:S))))
0.00/0.05	 r(r(x1:S)) -> x1:S
0.00/0.05	-> Usable rules:
0.00/0.05	 P(p(x1:S)) -> x1:S
0.00/0.05	->Interpretation type:
0.00/0.05	 Linear
0.00/0.05	->Coefficients:
0.00/0.05	 Natural Numbers
0.00/0.05	->Dimension:
0.00/0.05	 1
0.00/0.05	->Bound:
0.00/0.05	 2
0.00/0.05	->Interpretation:
0.00/0.05	 
0.00/0.05	[P](X) = 2.X + 2
0.00/0.05	[p](X) = 2.X + 2
0.00/0.05	[r#](X) = 2.X
0.00/0.05	
0.00/0.05	Problem 1: 
0.00/0.05	
0.00/0.05	SCC Processor:
0.00/0.05	-> Pairs:
0.00/0.05	 r#(p(x1:S)) -> r#(P(x1:S))
0.00/0.05	-> Rules:
0.00/0.05	 P(p(x1:S)) -> x1:S
0.00/0.05	 R(r(x1:S)) -> x1:S
0.00/0.05	 R(x1:S) -> r(x1:S)
0.00/0.05	 p(P(x1:S)) -> x1:S
0.00/0.05	 r(P(P(x1:S))) -> P(P(r(x1:S)))
0.00/0.05	 r(R(x1:S)) -> x1:S
0.00/0.05	 r(p(x1:S)) -> p(p(r(P(x1:S))))
0.00/0.05	 r(r(x1:S)) -> x1:S
0.00/0.05	->Strongly Connected Components:
0.00/0.05	->->Cycle:
0.00/0.05	->->-> Pairs:
0.00/0.05	 r#(p(x1:S)) -> r#(P(x1:S))
0.00/0.05	->->-> Rules:
0.00/0.05	 P(p(x1:S)) -> x1:S
0.00/0.05	 R(r(x1:S)) -> x1:S
0.00/0.05	 R(x1:S) -> r(x1:S)
0.00/0.05	 p(P(x1:S)) -> x1:S
0.00/0.05	 r(P(P(x1:S))) -> P(P(r(x1:S)))
0.00/0.05	 r(R(x1:S)) -> x1:S
0.00/0.05	 r(p(x1:S)) -> p(p(r(P(x1:S))))
0.00/0.05	 r(r(x1:S)) -> x1:S
0.00/0.05	
0.00/0.05	Problem 1: 
0.00/0.05	
0.00/0.05	Reduction Pair Processor:
0.00/0.05	-> Pairs:
0.00/0.05	 r#(p(x1:S)) -> r#(P(x1:S))
0.00/0.05	-> Rules:
0.00/0.05	 P(p(x1:S)) -> x1:S
0.00/0.05	 R(r(x1:S)) -> x1:S
0.00/0.05	 R(x1:S) -> r(x1:S)
0.00/0.05	 p(P(x1:S)) -> x1:S
0.00/0.05	 r(P(P(x1:S))) -> P(P(r(x1:S)))
0.00/0.05	 r(R(x1:S)) -> x1:S
0.00/0.05	 r(p(x1:S)) -> p(p(r(P(x1:S))))
0.00/0.05	 r(r(x1:S)) -> x1:S
0.00/0.05	-> Usable rules:
0.00/0.05	 P(p(x1:S)) -> x1:S
0.00/0.05	->Interpretation type:
0.00/0.05	 Linear
0.00/0.05	->Coefficients:
0.00/0.05	 Natural Numbers
0.00/0.05	->Dimension:
0.00/0.05	 1
0.00/0.05	->Bound:
0.00/0.05	 2
0.00/0.05	->Interpretation:
0.00/0.05	 
0.00/0.05	[P](X) = X + 1
0.00/0.05	[p](X) = 2.X + 2
0.00/0.05	[r#](X) = 2.X
0.00/0.05	
0.00/0.05	Problem 1: 
0.00/0.05	
0.00/0.05	SCC Processor:
0.00/0.05	-> Pairs:
0.00/0.05	 Empty
0.00/0.05	-> Rules:
0.00/0.05	 P(p(x1:S)) -> x1:S
0.00/0.05	 R(r(x1:S)) -> x1:S
0.00/0.05	 R(x1:S) -> r(x1:S)
0.00/0.05	 p(P(x1:S)) -> x1:S
0.00/0.05	 r(P(P(x1:S))) -> P(P(r(x1:S)))
0.00/0.05	 r(R(x1:S)) -> x1:S
0.00/0.05	 r(p(x1:S)) -> p(p(r(P(x1:S))))
0.00/0.05	 r(r(x1:S)) -> x1:S
0.00/0.05	->Strongly Connected Components:
0.00/0.05	 There is no strongly connected component
0.00/0.05	
0.00/0.05	The problem is finite.
0.00/0.05	EOF