0.00/0.07	YES
0.00/0.07	
0.00/0.07	Problem 1: 
0.00/0.07	
0.00/0.07	(VAR v_NonEmpty:S x1:S)
0.00/0.07	(RULES
0.00/0.07	b(b(b(x1:S))) -> c(a(c(x1:S)))
0.00/0.07	b(c(a(x1:S))) -> a(b(x1:S))
0.00/0.07	c(b(x1:S)) -> d(a(x1:S))
0.00/0.07	c(d(b(x1:S))) -> d(c(c(x1:S)))
0.00/0.07	c(d(x1:S)) -> d(c(x1:S))
0.00/0.07	d(b(c(x1:S))) -> a(a(x1:S))
0.00/0.07	d(c(x1:S)) -> b(b(b(x1:S)))
0.00/0.07	d(a(x1:S)) -> b(x1:S)
0.00/0.07	)
0.00/0.07	
0.00/0.07	Problem 1: 
0.00/0.07	
0.00/0.07	Dependency Pairs Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 B(b(b(x1:S))) -> C(a(c(x1:S)))
0.00/0.07	 B(b(b(x1:S))) -> C(x1:S)
0.00/0.07	 B(c(a(x1:S))) -> B(x1:S)
0.00/0.07	 C(b(x1:S)) -> D(a(x1:S))
0.00/0.07	 C(d(b(x1:S))) -> C(c(x1:S))
0.00/0.07	 C(d(b(x1:S))) -> C(x1:S)
0.00/0.07	 C(d(b(x1:S))) -> D(c(c(x1:S)))
0.00/0.07	 C(d(x1:S)) -> C(x1:S)
0.00/0.07	 C(d(x1:S)) -> D(c(x1:S))
0.00/0.07	 D(c(x1:S)) -> B(b(b(x1:S)))
0.00/0.07	 D(c(x1:S)) -> B(b(x1:S))
0.00/0.07	 D(c(x1:S)) -> B(x1:S)
0.00/0.07	 D(a(x1:S)) -> B(x1:S)
0.00/0.07	-> Rules:
0.00/0.07	 b(b(b(x1:S))) -> c(a(c(x1:S)))
0.00/0.07	 b(c(a(x1:S))) -> a(b(x1:S))
0.00/0.07	 c(b(x1:S)) -> d(a(x1:S))
0.00/0.07	 c(d(b(x1:S))) -> d(c(c(x1:S)))
0.00/0.07	 c(d(x1:S)) -> d(c(x1:S))
0.00/0.07	 d(b(c(x1:S))) -> a(a(x1:S))
0.00/0.07	 d(c(x1:S)) -> b(b(b(x1:S)))
0.00/0.07	 d(a(x1:S)) -> b(x1:S)
0.00/0.07	
0.00/0.07	Problem 1: 
0.00/0.07	
0.00/0.07	SCC Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 B(b(b(x1:S))) -> C(a(c(x1:S)))
0.00/0.07	 B(b(b(x1:S))) -> C(x1:S)
0.00/0.07	 B(c(a(x1:S))) -> B(x1:S)
0.00/0.07	 C(b(x1:S)) -> D(a(x1:S))
0.00/0.07	 C(d(b(x1:S))) -> C(c(x1:S))
0.00/0.07	 C(d(b(x1:S))) -> C(x1:S)
0.00/0.07	 C(d(b(x1:S))) -> D(c(c(x1:S)))
0.00/0.07	 C(d(x1:S)) -> C(x1:S)
0.00/0.07	 C(d(x1:S)) -> D(c(x1:S))
0.00/0.07	 D(c(x1:S)) -> B(b(b(x1:S)))
0.00/0.07	 D(c(x1:S)) -> B(b(x1:S))
0.00/0.07	 D(c(x1:S)) -> B(x1:S)
0.00/0.07	 D(a(x1:S)) -> B(x1:S)
0.00/0.07	-> Rules:
0.00/0.07	 b(b(b(x1:S))) -> c(a(c(x1:S)))
0.00/0.07	 b(c(a(x1:S))) -> a(b(x1:S))
0.00/0.07	 c(b(x1:S)) -> d(a(x1:S))
0.00/0.07	 c(d(b(x1:S))) -> d(c(c(x1:S)))
0.00/0.07	 c(d(x1:S)) -> d(c(x1:S))
0.00/0.07	 d(b(c(x1:S))) -> a(a(x1:S))
0.00/0.07	 d(c(x1:S)) -> b(b(b(x1:S)))
0.00/0.07	 d(a(x1:S)) -> b(x1:S)
0.00/0.07	->Strongly Connected Components:
0.00/0.07	->->Cycle:
0.00/0.07	->->-> Pairs:
0.00/0.07	 B(b(b(x1:S))) -> C(x1:S)
0.00/0.07	 B(c(a(x1:S))) -> B(x1:S)
0.00/0.07	 C(b(x1:S)) -> D(a(x1:S))
0.00/0.07	 C(d(b(x1:S))) -> C(c(x1:S))
0.00/0.07	 C(d(b(x1:S))) -> C(x1:S)
0.00/0.07	 C(d(b(x1:S))) -> D(c(c(x1:S)))
0.00/0.07	 C(d(x1:S)) -> C(x1:S)
0.00/0.07	 C(d(x1:S)) -> D(c(x1:S))
0.00/0.07	 D(c(x1:S)) -> B(b(b(x1:S)))
0.00/0.07	 D(c(x1:S)) -> B(b(x1:S))
0.00/0.07	 D(c(x1:S)) -> B(x1:S)
0.00/0.07	 D(a(x1:S)) -> B(x1:S)
0.00/0.07	->->-> Rules:
0.00/0.07	 b(b(b(x1:S))) -> c(a(c(x1:S)))
0.00/0.07	 b(c(a(x1:S))) -> a(b(x1:S))
0.00/0.07	 c(b(x1:S)) -> d(a(x1:S))
0.00/0.07	 c(d(b(x1:S))) -> d(c(c(x1:S)))
0.00/0.07	 c(d(x1:S)) -> d(c(x1:S))
0.00/0.07	 d(b(c(x1:S))) -> a(a(x1:S))
0.00/0.07	 d(c(x1:S)) -> b(b(b(x1:S)))
0.00/0.07	 d(a(x1:S)) -> b(x1:S)
0.00/0.07	
0.00/0.07	Problem 1: 
0.00/0.07	
0.00/0.07	Reduction Pair Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 B(b(b(x1:S))) -> C(x1:S)
0.00/0.07	 B(c(a(x1:S))) -> B(x1:S)
0.00/0.07	 C(b(x1:S)) -> D(a(x1:S))
0.00/0.07	 C(d(b(x1:S))) -> C(c(x1:S))
0.00/0.07	 C(d(b(x1:S))) -> C(x1:S)
0.00/0.07	 C(d(b(x1:S))) -> D(c(c(x1:S)))
0.00/0.07	 C(d(x1:S)) -> C(x1:S)
0.00/0.07	 C(d(x1:S)) -> D(c(x1:S))
0.00/0.07	 D(c(x1:S)) -> B(b(b(x1:S)))
0.00/0.07	 D(c(x1:S)) -> B(b(x1:S))
0.00/0.07	 D(c(x1:S)) -> B(x1:S)
0.00/0.07	 D(a(x1:S)) -> B(x1:S)
0.00/0.07	-> Rules:
0.00/0.07	 b(b(b(x1:S))) -> c(a(c(x1:S)))
0.00/0.07	 b(c(a(x1:S))) -> a(b(x1:S))
0.00/0.07	 c(b(x1:S)) -> d(a(x1:S))
0.00/0.07	 c(d(b(x1:S))) -> d(c(c(x1:S)))
0.00/0.07	 c(d(x1:S)) -> d(c(x1:S))
0.00/0.07	 d(b(c(x1:S))) -> a(a(x1:S))
0.00/0.07	 d(c(x1:S)) -> b(b(b(x1:S)))
0.00/0.07	 d(a(x1:S)) -> b(x1:S)
0.00/0.07	-> Usable rules:
0.00/0.07	 b(b(b(x1:S))) -> c(a(c(x1:S)))
0.00/0.07	 b(c(a(x1:S))) -> a(b(x1:S))
0.00/0.07	 c(b(x1:S)) -> d(a(x1:S))
0.00/0.07	 c(d(b(x1:S))) -> d(c(c(x1:S)))
0.00/0.07	 c(d(x1:S)) -> d(c(x1:S))
0.00/0.07	 d(b(c(x1:S))) -> a(a(x1:S))
0.00/0.07	 d(c(x1:S)) -> b(b(b(x1:S)))
0.00/0.07	 d(a(x1:S)) -> b(x1:S)
0.00/0.07	->Interpretation type:
0.00/0.07	 Linear
0.00/0.07	->Coefficients:
0.00/0.07	 Natural Numbers
0.00/0.07	->Dimension:
0.00/0.07	 1
0.00/0.07	->Bound:
0.00/0.07	 2
0.00/0.07	->Interpretation:
0.00/0.07	 
0.00/0.07	[b](X) = X + 1
0.00/0.07	[c](X) = X + 1
0.00/0.07	[d](X) = X + 2
0.00/0.07	[a](X) = X
0.00/0.07	[B](X) = 2.X
0.00/0.07	[C](X) = 2.X
0.00/0.07	[D](X) = 2.X + 2
0.00/0.07	
0.00/0.07	Problem 1: 
0.00/0.07	
0.00/0.07	SCC Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 B(c(a(x1:S))) -> B(x1:S)
0.00/0.07	 C(b(x1:S)) -> D(a(x1:S))
0.00/0.07	 C(d(b(x1:S))) -> C(c(x1:S))
0.00/0.07	 C(d(b(x1:S))) -> C(x1:S)
0.00/0.07	 C(d(b(x1:S))) -> D(c(c(x1:S)))
0.00/0.07	 C(d(x1:S)) -> C(x1:S)
0.00/0.07	 C(d(x1:S)) -> D(c(x1:S))
0.00/0.07	 D(c(x1:S)) -> B(b(b(x1:S)))
0.00/0.07	 D(c(x1:S)) -> B(b(x1:S))
0.00/0.07	 D(c(x1:S)) -> B(x1:S)
0.00/0.07	 D(a(x1:S)) -> B(x1:S)
0.00/0.07	-> Rules:
0.00/0.07	 b(b(b(x1:S))) -> c(a(c(x1:S)))
0.00/0.07	 b(c(a(x1:S))) -> a(b(x1:S))
0.00/0.07	 c(b(x1:S)) -> d(a(x1:S))
0.00/0.07	 c(d(b(x1:S))) -> d(c(c(x1:S)))
0.00/0.07	 c(d(x1:S)) -> d(c(x1:S))
0.00/0.07	 d(b(c(x1:S))) -> a(a(x1:S))
0.00/0.07	 d(c(x1:S)) -> b(b(b(x1:S)))
0.00/0.07	 d(a(x1:S)) -> b(x1:S)
0.00/0.07	->Strongly Connected Components:
0.00/0.07	->->Cycle:
0.00/0.07	->->-> Pairs:
0.00/0.07	 B(c(a(x1:S))) -> B(x1:S)
0.00/0.07	->->-> Rules:
0.00/0.07	 b(b(b(x1:S))) -> c(a(c(x1:S)))
0.00/0.07	 b(c(a(x1:S))) -> a(b(x1:S))
0.00/0.07	 c(b(x1:S)) -> d(a(x1:S))
0.00/0.07	 c(d(b(x1:S))) -> d(c(c(x1:S)))
0.00/0.07	 c(d(x1:S)) -> d(c(x1:S))
0.00/0.07	 d(b(c(x1:S))) -> a(a(x1:S))
0.00/0.07	 d(c(x1:S)) -> b(b(b(x1:S)))
0.00/0.07	 d(a(x1:S)) -> b(x1:S)
0.00/0.07	->->Cycle:
0.00/0.07	->->-> Pairs:
0.00/0.07	 C(d(b(x1:S))) -> C(c(x1:S))
0.00/0.07	 C(d(b(x1:S))) -> C(x1:S)
0.00/0.07	 C(d(x1:S)) -> C(x1:S)
0.00/0.07	->->-> Rules:
0.00/0.07	 b(b(b(x1:S))) -> c(a(c(x1:S)))
0.00/0.07	 b(c(a(x1:S))) -> a(b(x1:S))
0.00/0.07	 c(b(x1:S)) -> d(a(x1:S))
0.00/0.07	 c(d(b(x1:S))) -> d(c(c(x1:S)))
0.00/0.07	 c(d(x1:S)) -> d(c(x1:S))
0.00/0.07	 d(b(c(x1:S))) -> a(a(x1:S))
0.00/0.07	 d(c(x1:S)) -> b(b(b(x1:S)))
0.00/0.07	 d(a(x1:S)) -> b(x1:S)
0.00/0.07	
0.00/0.07	
0.00/0.07	The problem is decomposed in 2 subproblems.
0.00/0.07	
0.00/0.07	Problem 1.1: 
0.00/0.07	
0.00/0.07	Subterm Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 B(c(a(x1:S))) -> B(x1:S)
0.00/0.07	-> Rules:
0.00/0.07	 b(b(b(x1:S))) -> c(a(c(x1:S)))
0.00/0.07	 b(c(a(x1:S))) -> a(b(x1:S))
0.00/0.07	 c(b(x1:S)) -> d(a(x1:S))
0.00/0.07	 c(d(b(x1:S))) -> d(c(c(x1:S)))
0.00/0.07	 c(d(x1:S)) -> d(c(x1:S))
0.00/0.07	 d(b(c(x1:S))) -> a(a(x1:S))
0.00/0.07	 d(c(x1:S)) -> b(b(b(x1:S)))
0.00/0.07	 d(a(x1:S)) -> b(x1:S)
0.00/0.07	->Projection:
0.00/0.07	 pi(B) = 1
0.00/0.07	
0.00/0.07	Problem 1.1: 
0.00/0.07	
0.00/0.07	SCC Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 Empty
0.00/0.07	-> Rules:
0.00/0.07	 b(b(b(x1:S))) -> c(a(c(x1:S)))
0.00/0.07	 b(c(a(x1:S))) -> a(b(x1:S))
0.00/0.07	 c(b(x1:S)) -> d(a(x1:S))
0.00/0.07	 c(d(b(x1:S))) -> d(c(c(x1:S)))
0.00/0.07	 c(d(x1:S)) -> d(c(x1:S))
0.00/0.07	 d(b(c(x1:S))) -> a(a(x1:S))
0.00/0.07	 d(c(x1:S)) -> b(b(b(x1:S)))
0.00/0.07	 d(a(x1:S)) -> b(x1:S)
0.00/0.07	->Strongly Connected Components:
0.00/0.07	 There is no strongly connected component
0.00/0.07	
0.00/0.07	The problem is finite.
0.00/0.07	
0.00/0.07	Problem 1.2: 
0.00/0.07	
0.00/0.07	Reduction Pair Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 C(d(b(x1:S))) -> C(c(x1:S))
0.00/0.07	 C(d(b(x1:S))) -> C(x1:S)
0.00/0.07	 C(d(x1:S)) -> C(x1:S)
0.00/0.07	-> Rules:
0.00/0.07	 b(b(b(x1:S))) -> c(a(c(x1:S)))
0.00/0.07	 b(c(a(x1:S))) -> a(b(x1:S))
0.00/0.07	 c(b(x1:S)) -> d(a(x1:S))
0.00/0.07	 c(d(b(x1:S))) -> d(c(c(x1:S)))
0.00/0.07	 c(d(x1:S)) -> d(c(x1:S))
0.00/0.07	 d(b(c(x1:S))) -> a(a(x1:S))
0.00/0.07	 d(c(x1:S)) -> b(b(b(x1:S)))
0.00/0.07	 d(a(x1:S)) -> b(x1:S)
0.00/0.07	-> Usable rules:
0.00/0.07	 b(b(b(x1:S))) -> c(a(c(x1:S)))
0.00/0.07	 b(c(a(x1:S))) -> a(b(x1:S))
0.00/0.07	 c(b(x1:S)) -> d(a(x1:S))
0.00/0.07	 c(d(b(x1:S))) -> d(c(c(x1:S)))
0.00/0.07	 c(d(x1:S)) -> d(c(x1:S))
0.00/0.07	 d(b(c(x1:S))) -> a(a(x1:S))
0.00/0.07	 d(c(x1:S)) -> b(b(b(x1:S)))
0.00/0.07	 d(a(x1:S)) -> b(x1:S)
0.00/0.07	->Interpretation type:
0.00/0.07	 Linear
0.00/0.07	->Coefficients:
0.00/0.07	 Natural Numbers
0.00/0.07	->Dimension:
0.00/0.07	 1
0.00/0.07	->Bound:
0.00/0.07	 2
0.00/0.07	->Interpretation:
0.00/0.07	 
0.00/0.07	[b](X) = X + 1
0.00/0.07	[c](X) = X + 1
0.00/0.07	[d](X) = X + 2
0.00/0.07	[a](X) = X
0.00/0.07	[C](X) = 2.X
0.00/0.07	
0.00/0.07	Problem 1.2: 
0.00/0.07	
0.00/0.07	SCC Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 C(d(b(x1:S))) -> C(x1:S)
0.00/0.07	 C(d(x1:S)) -> C(x1:S)
0.00/0.07	-> Rules:
0.00/0.07	 b(b(b(x1:S))) -> c(a(c(x1:S)))
0.00/0.07	 b(c(a(x1:S))) -> a(b(x1:S))
0.00/0.07	 c(b(x1:S)) -> d(a(x1:S))
0.00/0.07	 c(d(b(x1:S))) -> d(c(c(x1:S)))
0.00/0.07	 c(d(x1:S)) -> d(c(x1:S))
0.00/0.07	 d(b(c(x1:S))) -> a(a(x1:S))
0.00/0.07	 d(c(x1:S)) -> b(b(b(x1:S)))
0.00/0.07	 d(a(x1:S)) -> b(x1:S)
0.00/0.07	->Strongly Connected Components:
0.00/0.07	->->Cycle:
0.00/0.07	->->-> Pairs:
0.00/0.07	 C(d(b(x1:S))) -> C(x1:S)
0.00/0.07	 C(d(x1:S)) -> C(x1:S)
0.00/0.07	->->-> Rules:
0.00/0.07	 b(b(b(x1:S))) -> c(a(c(x1:S)))
0.00/0.07	 b(c(a(x1:S))) -> a(b(x1:S))
0.00/0.07	 c(b(x1:S)) -> d(a(x1:S))
0.00/0.07	 c(d(b(x1:S))) -> d(c(c(x1:S)))
0.00/0.07	 c(d(x1:S)) -> d(c(x1:S))
0.00/0.07	 d(b(c(x1:S))) -> a(a(x1:S))
0.00/0.07	 d(c(x1:S)) -> b(b(b(x1:S)))
0.00/0.07	 d(a(x1:S)) -> b(x1:S)
0.00/0.07	
0.00/0.07	Problem 1.2: 
0.00/0.07	
0.00/0.07	Subterm Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 C(d(b(x1:S))) -> C(x1:S)
0.00/0.07	 C(d(x1:S)) -> C(x1:S)
0.00/0.07	-> Rules:
0.00/0.07	 b(b(b(x1:S))) -> c(a(c(x1:S)))
0.00/0.07	 b(c(a(x1:S))) -> a(b(x1:S))
0.00/0.07	 c(b(x1:S)) -> d(a(x1:S))
0.00/0.07	 c(d(b(x1:S))) -> d(c(c(x1:S)))
0.00/0.07	 c(d(x1:S)) -> d(c(x1:S))
0.00/0.07	 d(b(c(x1:S))) -> a(a(x1:S))
0.00/0.07	 d(c(x1:S)) -> b(b(b(x1:S)))
0.00/0.07	 d(a(x1:S)) -> b(x1:S)
0.00/0.07	->Projection:
0.00/0.07	 pi(C) = 1
0.00/0.07	
0.00/0.07	Problem 1.2: 
0.00/0.07	
0.00/0.07	SCC Processor:
0.00/0.07	-> Pairs:
0.00/0.07	 Empty
0.00/0.07	-> Rules:
0.00/0.07	 b(b(b(x1:S))) -> c(a(c(x1:S)))
0.00/0.07	 b(c(a(x1:S))) -> a(b(x1:S))
0.00/0.07	 c(b(x1:S)) -> d(a(x1:S))
0.00/0.07	 c(d(b(x1:S))) -> d(c(c(x1:S)))
0.00/0.07	 c(d(x1:S)) -> d(c(x1:S))
0.00/0.07	 d(b(c(x1:S))) -> a(a(x1:S))
0.00/0.07	 d(c(x1:S)) -> b(b(b(x1:S)))
0.00/0.07	 d(a(x1:S)) -> b(x1:S)
0.00/0.07	->Strongly Connected Components:
0.00/0.07	 There is no strongly connected component
0.00/0.07	
0.00/0.07	The problem is finite.
0.00/0.07	EOF