0.44/2.80	YES
0.44/2.80	
0.44/2.80	Problem 1: 
0.44/2.80	
0.44/2.80	(VAR v_NonEmpty:S x1:S)
0.44/2.80	(RULES
0.44/2.80	a(p(x1:S)) -> p(a(A(x1:S)))
0.44/2.80	a(A(x1:S)) -> A(a(x1:S))
0.44/2.80	p(A(A(x1:S))) -> a(p(x1:S))
0.44/2.80	)
0.44/2.80	
0.44/2.80	Problem 1: 
0.44/2.80	
0.44/2.80	Dependency Pairs Processor:
0.44/2.80	-> Pairs:
0.44/2.80	 a#(p(x1:S)) -> a#(A(x1:S))
0.44/2.80	 a#(p(x1:S)) -> P(a(A(x1:S)))
0.44/2.80	 a#(A(x1:S)) -> a#(x1:S)
0.44/2.80	 P(A(A(x1:S))) -> a#(p(x1:S))
0.44/2.80	 P(A(A(x1:S))) -> P(x1:S)
0.44/2.80	-> Rules:
0.44/2.80	 a(p(x1:S)) -> p(a(A(x1:S)))
0.44/2.80	 a(A(x1:S)) -> A(a(x1:S))
0.44/2.80	 p(A(A(x1:S))) -> a(p(x1:S))
0.44/2.80	
0.44/2.80	Problem 1: 
0.44/2.80	
0.44/2.80	SCC Processor:
0.44/2.80	-> Pairs:
0.44/2.80	 a#(p(x1:S)) -> a#(A(x1:S))
0.44/2.80	 a#(p(x1:S)) -> P(a(A(x1:S)))
0.44/2.80	 a#(A(x1:S)) -> a#(x1:S)
0.44/2.80	 P(A(A(x1:S))) -> a#(p(x1:S))
0.44/2.80	 P(A(A(x1:S))) -> P(x1:S)
0.44/2.80	-> Rules:
0.44/2.80	 a(p(x1:S)) -> p(a(A(x1:S)))
0.44/2.80	 a(A(x1:S)) -> A(a(x1:S))
0.44/2.80	 p(A(A(x1:S))) -> a(p(x1:S))
0.44/2.80	->Strongly Connected Components:
0.44/2.80	->->Cycle:
0.44/2.80	->->-> Pairs:
0.44/2.80	 a#(p(x1:S)) -> a#(A(x1:S))
0.44/2.80	 a#(p(x1:S)) -> P(a(A(x1:S)))
0.44/2.80	 a#(A(x1:S)) -> a#(x1:S)
0.44/2.80	 P(A(A(x1:S))) -> a#(p(x1:S))
0.44/2.80	 P(A(A(x1:S))) -> P(x1:S)
0.44/2.80	->->-> Rules:
0.44/2.80	 a(p(x1:S)) -> p(a(A(x1:S)))
0.44/2.80	 a(A(x1:S)) -> A(a(x1:S))
0.44/2.80	 p(A(A(x1:S))) -> a(p(x1:S))
0.44/2.80	
0.44/2.80	Problem 1: 
0.44/2.80	
0.44/2.80	Reduction Pair Processor:
0.44/2.80	-> Pairs:
0.44/2.80	 a#(p(x1:S)) -> a#(A(x1:S))
0.44/2.80	 a#(p(x1:S)) -> P(a(A(x1:S)))
0.44/2.80	 a#(A(x1:S)) -> a#(x1:S)
0.44/2.80	 P(A(A(x1:S))) -> a#(p(x1:S))
0.44/2.80	 P(A(A(x1:S))) -> P(x1:S)
0.44/2.80	-> Rules:
0.44/2.80	 a(p(x1:S)) -> p(a(A(x1:S)))
0.44/2.80	 a(A(x1:S)) -> A(a(x1:S))
0.44/2.80	 p(A(A(x1:S))) -> a(p(x1:S))
0.44/2.80	-> Usable rules:
0.44/2.80	 a(p(x1:S)) -> p(a(A(x1:S)))
0.44/2.80	 a(A(x1:S)) -> A(a(x1:S))
0.44/2.80	 p(A(A(x1:S))) -> a(p(x1:S))
0.44/2.80	->Interpretation type:
0.44/2.80	 Linear
0.44/2.80	->Coefficients:
0.44/2.80	 Natural Numbers
0.44/2.80	->Dimension:
0.44/2.80	 1
0.44/2.80	->Bound:
0.44/2.80	 2
0.44/2.80	->Interpretation:
0.44/2.80	 
0.44/2.80	[a](X) = X
0.44/2.80	[p](X) = X + 1
0.44/2.80	[A](X) = X
0.44/2.80	[a#](X) = X + 1
0.44/2.80	[P](X) = X + 2
0.44/2.80	
0.44/2.80	Problem 1: 
0.44/2.80	
0.44/2.80	SCC Processor:
0.44/2.80	-> Pairs:
0.44/2.80	 a#(p(x1:S)) -> P(a(A(x1:S)))
0.44/2.80	 a#(A(x1:S)) -> a#(x1:S)
0.44/2.80	 P(A(A(x1:S))) -> a#(p(x1:S))
0.44/2.80	 P(A(A(x1:S))) -> P(x1:S)
0.44/2.80	-> Rules:
0.44/2.80	 a(p(x1:S)) -> p(a(A(x1:S)))
0.44/2.80	 a(A(x1:S)) -> A(a(x1:S))
0.44/2.80	 p(A(A(x1:S))) -> a(p(x1:S))
0.44/2.80	->Strongly Connected Components:
0.44/2.80	->->Cycle:
0.44/2.80	->->-> Pairs:
0.44/2.80	 a#(p(x1:S)) -> P(a(A(x1:S)))
0.44/2.80	 a#(A(x1:S)) -> a#(x1:S)
0.44/2.80	 P(A(A(x1:S))) -> a#(p(x1:S))
0.44/2.80	 P(A(A(x1:S))) -> P(x1:S)
0.44/2.80	->->-> Rules:
0.44/2.80	 a(p(x1:S)) -> p(a(A(x1:S)))
0.44/2.80	 a(A(x1:S)) -> A(a(x1:S))
0.44/2.80	 p(A(A(x1:S))) -> a(p(x1:S))
0.44/2.80	
0.44/2.80	Problem 1: 
0.44/2.80	
0.44/2.80	Reduction Pair Processor:
0.44/2.80	-> Pairs:
0.44/2.80	 a#(p(x1:S)) -> P(a(A(x1:S)))
0.44/2.80	 a#(A(x1:S)) -> a#(x1:S)
0.44/2.80	 P(A(A(x1:S))) -> a#(p(x1:S))
0.44/2.80	 P(A(A(x1:S))) -> P(x1:S)
0.44/2.80	-> Rules:
0.44/2.80	 a(p(x1:S)) -> p(a(A(x1:S)))
0.44/2.80	 a(A(x1:S)) -> A(a(x1:S))
0.44/2.80	 p(A(A(x1:S))) -> a(p(x1:S))
0.44/2.80	-> Usable rules:
0.44/2.80	 a(p(x1:S)) -> p(a(A(x1:S)))
0.44/2.80	 a(A(x1:S)) -> A(a(x1:S))
0.44/2.80	 p(A(A(x1:S))) -> a(p(x1:S))
0.44/2.80	->Interpretation type:
0.44/2.80	 Linear
0.44/2.80	->Coefficients:
0.44/2.80	 Natural Numbers
0.44/2.80	->Dimension:
0.44/2.80	 1
0.44/2.80	->Bound:
0.44/2.80	 2
0.44/2.80	->Interpretation:
0.44/2.80	 
0.44/2.80	[a](X) = X
0.44/2.80	[p](X) = 1
0.44/2.80	[A](X) = 2.X + 2
0.44/2.80	[a#](X) = 2.X
0.44/2.80	[P](X) = 2
0.44/2.80	
0.44/2.80	Problem 1: 
0.44/2.80	
0.44/2.80	SCC Processor:
0.44/2.80	-> Pairs:
0.44/2.80	 a#(p(x1:S)) -> P(a(A(x1:S)))
0.44/2.80	 P(A(A(x1:S))) -> a#(p(x1:S))
0.44/2.80	 P(A(A(x1:S))) -> P(x1:S)
0.44/2.80	-> Rules:
0.44/2.80	 a(p(x1:S)) -> p(a(A(x1:S)))
0.44/2.80	 a(A(x1:S)) -> A(a(x1:S))
0.44/2.80	 p(A(A(x1:S))) -> a(p(x1:S))
0.44/2.80	->Strongly Connected Components:
0.44/2.80	->->Cycle:
0.44/2.80	->->-> Pairs:
0.44/2.80	 a#(p(x1:S)) -> P(a(A(x1:S)))
0.44/2.80	 P(A(A(x1:S))) -> a#(p(x1:S))
0.44/2.80	 P(A(A(x1:S))) -> P(x1:S)
0.44/2.80	->->-> Rules:
0.44/2.80	 a(p(x1:S)) -> p(a(A(x1:S)))
0.44/2.80	 a(A(x1:S)) -> A(a(x1:S))
0.44/2.80	 p(A(A(x1:S))) -> a(p(x1:S))
0.44/2.80	
0.44/2.80	Problem 1: 
0.44/2.80	
0.44/2.80	Reduction Pair Processor:
0.44/2.80	-> Pairs:
0.44/2.80	 a#(p(x1:S)) -> P(a(A(x1:S)))
0.44/2.80	 P(A(A(x1:S))) -> a#(p(x1:S))
0.44/2.80	 P(A(A(x1:S))) -> P(x1:S)
0.44/2.80	-> Rules:
0.44/2.80	 a(p(x1:S)) -> p(a(A(x1:S)))
0.44/2.80	 a(A(x1:S)) -> A(a(x1:S))
0.44/2.80	 p(A(A(x1:S))) -> a(p(x1:S))
0.44/2.80	-> Usable rules:
0.44/2.80	 a(p(x1:S)) -> p(a(A(x1:S)))
0.44/2.80	 a(A(x1:S)) -> A(a(x1:S))
0.44/2.80	 p(A(A(x1:S))) -> a(p(x1:S))
0.44/2.80	->Interpretation type:
0.44/2.80	 Linear
0.44/2.80	->Coefficients:
0.44/2.80	 All rationals
0.44/2.80	->Dimension:
0.44/2.80	 1
0.44/2.80	->Bound:
0.44/2.80	 2
0.44/2.80	->Interpretation:
0.44/2.80	 
0.44/2.80	[a](X) = X + 1
0.44/2.80	[p](X) = 1/2.X + 1
0.44/2.80	[A](X) = X + 1
0.44/2.80	[a#](X) = 2.X + 1/2
0.44/2.80	[P](X) = X + 1/2
0.44/2.80	
0.44/2.80	Problem 1: 
0.44/2.80	
0.44/2.80	SCC Processor:
0.44/2.80	-> Pairs:
0.44/2.80	 a#(p(x1:S)) -> P(a(A(x1:S)))
0.44/2.80	 P(A(A(x1:S))) -> a#(p(x1:S))
0.44/2.80	-> Rules:
0.44/2.80	 a(p(x1:S)) -> p(a(A(x1:S)))
0.44/2.80	 a(A(x1:S)) -> A(a(x1:S))
0.44/2.80	 p(A(A(x1:S))) -> a(p(x1:S))
0.44/2.80	->Strongly Connected Components:
0.44/2.80	->->Cycle:
0.44/2.80	->->-> Pairs:
0.44/2.80	 a#(p(x1:S)) -> P(a(A(x1:S)))
0.44/2.80	 P(A(A(x1:S))) -> a#(p(x1:S))
0.44/2.80	->->-> Rules:
0.44/2.80	 a(p(x1:S)) -> p(a(A(x1:S)))
0.44/2.80	 a(A(x1:S)) -> A(a(x1:S))
0.44/2.80	 p(A(A(x1:S))) -> a(p(x1:S))
0.44/2.80	
0.44/2.80	Problem 1: 
0.44/2.80	
0.44/2.80	Reduction Pair Processor:
0.44/2.80	-> Pairs:
0.44/2.80	 a#(p(x1:S)) -> P(a(A(x1:S)))
0.44/2.80	 P(A(A(x1:S))) -> a#(p(x1:S))
0.44/2.80	-> Rules:
0.44/2.80	 a(p(x1:S)) -> p(a(A(x1:S)))
0.44/2.80	 a(A(x1:S)) -> A(a(x1:S))
0.44/2.80	 p(A(A(x1:S))) -> a(p(x1:S))
0.44/2.80	-> Usable rules:
0.44/2.80	 a(p(x1:S)) -> p(a(A(x1:S)))
0.44/2.80	 a(A(x1:S)) -> A(a(x1:S))
0.44/2.80	 p(A(A(x1:S))) -> a(p(x1:S))
0.44/2.80	->Interpretation type:
0.44/2.80	 Linear
0.44/2.80	->Coefficients:
0.44/2.80	 All rationals
0.44/2.80	->Dimension:
0.44/2.80	 1
0.44/2.80	->Bound:
0.44/2.80	 4
0.44/2.80	->Interpretation:
0.44/2.80	 
0.44/2.80	[a](X) = X + 2/3
0.44/2.80	[p](X) = 1/4.X + 1
0.44/2.80	[A](X) = X + 2
0.44/2.80	[a#](X) = 4/3.X + 3
0.44/2.80	[P](X) = 1/3.X + 3
0.44/2.80	
0.44/2.80	Problem 1: 
0.44/2.80	
0.44/2.80	SCC Processor:
0.44/2.80	-> Pairs:
0.44/2.80	 P(A(A(x1:S))) -> a#(p(x1:S))
0.44/2.80	-> Rules:
0.44/2.80	 a(p(x1:S)) -> p(a(A(x1:S)))
0.44/2.80	 a(A(x1:S)) -> A(a(x1:S))
0.44/2.80	 p(A(A(x1:S))) -> a(p(x1:S))
0.44/2.80	->Strongly Connected Components:
0.44/2.80	 There is no strongly connected component
0.44/2.80	
0.44/2.80	The problem is finite.
0.44/2.81	EOF