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	a(b(x1:S)) -> b(r(x1:S))
0.00/0.05	d(a(x1:S)) -> a(a(d(x1:S)))
0.00/0.05	d(x1:S) -> a(x1:S)
0.00/0.05	r(a(x1:S)) -> d(r(x1:S))
0.00/0.05	r(x1:S) -> d(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	 A(b(x1:S)) -> R(x1:S)
0.00/0.05	 D(a(x1:S)) -> A(a(d(x1:S)))
0.00/0.05	 D(a(x1:S)) -> A(d(x1:S))
0.00/0.05	 D(a(x1:S)) -> D(x1:S)
0.00/0.05	 D(x1:S) -> A(x1:S)
0.00/0.05	 R(a(x1:S)) -> D(r(x1:S))
0.00/0.05	 R(a(x1:S)) -> R(x1:S)
0.00/0.05	 R(x1:S) -> D(x1:S)
0.00/0.05	-> Rules:
0.00/0.05	 a(b(x1:S)) -> b(r(x1:S))
0.00/0.05	 d(a(x1:S)) -> a(a(d(x1:S)))
0.00/0.05	 d(x1:S) -> a(x1:S)
0.00/0.05	 r(a(x1:S)) -> d(r(x1:S))
0.00/0.05	 r(x1:S) -> d(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	 A(b(x1:S)) -> R(x1:S)
0.00/0.05	 D(a(x1:S)) -> A(a(d(x1:S)))
0.00/0.05	 D(a(x1:S)) -> A(d(x1:S))
0.00/0.05	 D(a(x1:S)) -> D(x1:S)
0.00/0.05	 D(x1:S) -> A(x1:S)
0.00/0.05	 R(a(x1:S)) -> D(r(x1:S))
0.00/0.05	 R(a(x1:S)) -> R(x1:S)
0.00/0.05	 R(x1:S) -> D(x1:S)
0.00/0.05	-> Rules:
0.00/0.05	 a(b(x1:S)) -> b(r(x1:S))
0.00/0.05	 d(a(x1:S)) -> a(a(d(x1:S)))
0.00/0.05	 d(x1:S) -> a(x1:S)
0.00/0.05	 r(a(x1:S)) -> d(r(x1:S))
0.00/0.05	 r(x1:S) -> d(x1:S)
0.00/0.05	->Strongly Connected Components:
0.00/0.05	->->Cycle:
0.00/0.05	->->-> Pairs:
0.00/0.05	 A(b(x1:S)) -> R(x1:S)
0.00/0.05	 D(a(x1:S)) -> A(a(d(x1:S)))
0.00/0.05	 D(a(x1:S)) -> A(d(x1:S))
0.00/0.05	 D(a(x1:S)) -> D(x1:S)
0.00/0.05	 D(x1:S) -> A(x1:S)
0.00/0.05	 R(a(x1:S)) -> D(r(x1:S))
0.00/0.05	 R(a(x1:S)) -> R(x1:S)
0.00/0.05	 R(x1:S) -> D(x1:S)
0.00/0.05	->->-> Rules:
0.00/0.05	 a(b(x1:S)) -> b(r(x1:S))
0.00/0.05	 d(a(x1:S)) -> a(a(d(x1:S)))
0.00/0.05	 d(x1:S) -> a(x1:S)
0.00/0.05	 r(a(x1:S)) -> d(r(x1:S))
0.00/0.05	 r(x1:S) -> d(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	 A(b(x1:S)) -> R(x1:S)
0.00/0.05	 D(a(x1:S)) -> A(a(d(x1:S)))
0.00/0.05	 D(a(x1:S)) -> A(d(x1:S))
0.00/0.05	 D(a(x1:S)) -> D(x1:S)
0.00/0.05	 D(x1:S) -> A(x1:S)
0.00/0.05	 R(a(x1:S)) -> D(r(x1:S))
0.00/0.05	 R(a(x1:S)) -> R(x1:S)
0.00/0.05	 R(x1:S) -> D(x1:S)
0.00/0.05	-> Rules:
0.00/0.05	 a(b(x1:S)) -> b(r(x1:S))
0.00/0.05	 d(a(x1:S)) -> a(a(d(x1:S)))
0.00/0.05	 d(x1:S) -> a(x1:S)
0.00/0.05	 r(a(x1:S)) -> d(r(x1:S))
0.00/0.05	 r(x1:S) -> d(x1:S)
0.00/0.05	-> Usable rules:
0.00/0.05	 a(b(x1:S)) -> b(r(x1:S))
0.00/0.05	 d(a(x1:S)) -> a(a(d(x1:S)))
0.00/0.05	 d(x1:S) -> a(x1:S)
0.00/0.05	 r(a(x1:S)) -> d(r(x1:S))
0.00/0.05	 r(x1:S) -> d(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	[a](X) = X
0.00/0.05	[d](X) = X
0.00/0.05	[r](X) = X
0.00/0.05	[b](X) = X + 2
0.00/0.05	[A](X) = 2.X
0.00/0.05	[D](X) = 2.X
0.00/0.05	[R](X) = 2.X + 1
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	 D(a(x1:S)) -> A(a(d(x1:S)))
0.00/0.06	 D(a(x1:S)) -> A(d(x1:S))
0.00/0.06	 D(a(x1:S)) -> D(x1:S)
0.00/0.06	 D(x1:S) -> A(x1:S)
0.00/0.06	 R(a(x1:S)) -> D(r(x1:S))
0.00/0.06	 R(a(x1:S)) -> R(x1:S)
0.00/0.06	 R(x1:S) -> D(x1:S)
0.00/0.06	-> Rules:
0.00/0.06	 a(b(x1:S)) -> b(r(x1:S))
0.00/0.06	 d(a(x1:S)) -> a(a(d(x1:S)))
0.00/0.06	 d(x1:S) -> a(x1:S)
0.00/0.06	 r(a(x1:S)) -> d(r(x1:S))
0.00/0.06	 r(x1:S) -> d(x1:S)
0.00/0.06	->Strongly Connected Components:
0.00/0.06	->->Cycle:
0.00/0.06	->->-> Pairs:
0.00/0.06	 D(a(x1:S)) -> D(x1:S)
0.00/0.06	->->-> Rules:
0.00/0.06	 a(b(x1:S)) -> b(r(x1:S))
0.00/0.06	 d(a(x1:S)) -> a(a(d(x1:S)))
0.00/0.06	 d(x1:S) -> a(x1:S)
0.00/0.06	 r(a(x1:S)) -> d(r(x1:S))
0.00/0.06	 r(x1:S) -> d(x1:S)
0.00/0.06	->->Cycle:
0.00/0.06	->->-> Pairs:
0.00/0.06	 R(a(x1:S)) -> R(x1:S)
0.00/0.06	->->-> Rules:
0.00/0.06	 a(b(x1:S)) -> b(r(x1:S))
0.00/0.06	 d(a(x1:S)) -> a(a(d(x1:S)))
0.00/0.06	 d(x1:S) -> a(x1:S)
0.00/0.06	 r(a(x1:S)) -> d(r(x1:S))
0.00/0.06	 r(x1:S) -> d(x1:S)
0.00/0.06	
0.00/0.06	
0.00/0.06	The problem is decomposed in 2 subproblems.
0.00/0.06	
0.00/0.06	Problem 1.1: 
0.00/0.06	
0.00/0.06	Subterm Processor:
0.00/0.06	-> Pairs:
0.00/0.06	 D(a(x1:S)) -> D(x1:S)
0.00/0.06	-> Rules:
0.00/0.06	 a(b(x1:S)) -> b(r(x1:S))
0.00/0.06	 d(a(x1:S)) -> a(a(d(x1:S)))
0.00/0.06	 d(x1:S) -> a(x1:S)
0.00/0.06	 r(a(x1:S)) -> d(r(x1:S))
0.00/0.06	 r(x1:S) -> d(x1:S)
0.00/0.06	->Projection:
0.00/0.06	 pi(D) = 1
0.00/0.06	
0.00/0.06	Problem 1.1: 
0.00/0.06	
0.00/0.06	SCC Processor:
0.00/0.06	-> Pairs:
0.00/0.06	 Empty
0.00/0.06	-> Rules:
0.00/0.06	 a(b(x1:S)) -> b(r(x1:S))
0.00/0.06	 d(a(x1:S)) -> a(a(d(x1:S)))
0.00/0.06	 d(x1:S) -> a(x1:S)
0.00/0.06	 r(a(x1:S)) -> d(r(x1:S))
0.00/0.06	 r(x1:S) -> d(x1:S)
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 1.2: 
0.00/0.06	
0.00/0.06	Subterm Processor:
0.00/0.06	-> Pairs:
0.00/0.06	 R(a(x1:S)) -> R(x1:S)
0.00/0.06	-> Rules:
0.00/0.06	 a(b(x1:S)) -> b(r(x1:S))
0.00/0.06	 d(a(x1:S)) -> a(a(d(x1:S)))
0.00/0.06	 d(x1:S) -> a(x1:S)
0.00/0.06	 r(a(x1:S)) -> d(r(x1:S))
0.00/0.06	 r(x1:S) -> d(x1:S)
0.00/0.06	->Projection:
0.00/0.06	 pi(R) = 1
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	 Empty
0.00/0.06	-> Rules:
0.00/0.06	 a(b(x1:S)) -> b(r(x1:S))
0.00/0.06	 d(a(x1:S)) -> a(a(d(x1:S)))
0.00/0.06	 d(x1:S) -> a(x1:S)
0.00/0.06	 r(a(x1:S)) -> d(r(x1:S))
0.00/0.06	 r(x1:S) -> d(x1:S)
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	EOF