1.59/1.81	YES
1.59/1.81	
1.59/1.81	Problem 1: 
1.59/1.81	
1.59/1.81	(VAR v_NonEmpty:S x1:S)
1.59/1.81	(RULES
1.59/1.81	a(a(x1:S)) -> b(c(x1:S))
1.59/1.81	b(b(x1:S)) -> c(d(x1:S))
1.59/1.81	c(c(x1:S)) -> d(d(d(x1:S)))
1.59/1.81	d(d(d(x1:S))) -> a(c(x1:S))
1.59/1.81	)
1.59/1.81	
1.59/1.81	Problem 1: 
1.59/1.81	
1.59/1.81	Dependency Pairs Processor:
1.59/1.81	-> Pairs:
1.59/1.81	 A(a(x1:S)) -> B(c(x1:S))
1.59/1.81	 A(a(x1:S)) -> C(x1:S)
1.59/1.81	 B(b(x1:S)) -> C(d(x1:S))
1.59/1.81	 B(b(x1:S)) -> D(x1:S)
1.59/1.81	 C(c(x1:S)) -> D(d(d(x1:S)))
1.59/1.81	 C(c(x1:S)) -> D(d(x1:S))
1.59/1.81	 C(c(x1:S)) -> D(x1:S)
1.59/1.81	 D(d(d(x1:S))) -> A(c(x1:S))
1.59/1.81	 D(d(d(x1:S))) -> C(x1:S)
1.59/1.81	-> Rules:
1.59/1.81	 a(a(x1:S)) -> b(c(x1:S))
1.59/1.81	 b(b(x1:S)) -> c(d(x1:S))
1.59/1.81	 c(c(x1:S)) -> d(d(d(x1:S)))
1.59/1.81	 d(d(d(x1:S))) -> a(c(x1:S))
1.59/1.81	
1.59/1.81	Problem 1: 
1.59/1.81	
1.59/1.81	SCC Processor:
1.59/1.81	-> Pairs:
1.59/1.81	 A(a(x1:S)) -> B(c(x1:S))
1.59/1.81	 A(a(x1:S)) -> C(x1:S)
1.59/1.81	 B(b(x1:S)) -> C(d(x1:S))
1.59/1.81	 B(b(x1:S)) -> D(x1:S)
1.59/1.81	 C(c(x1:S)) -> D(d(d(x1:S)))
1.59/1.81	 C(c(x1:S)) -> D(d(x1:S))
1.59/1.81	 C(c(x1:S)) -> D(x1:S)
1.59/1.81	 D(d(d(x1:S))) -> A(c(x1:S))
1.59/1.81	 D(d(d(x1:S))) -> C(x1:S)
1.59/1.81	-> Rules:
1.59/1.81	 a(a(x1:S)) -> b(c(x1:S))
1.59/1.81	 b(b(x1:S)) -> c(d(x1:S))
1.59/1.81	 c(c(x1:S)) -> d(d(d(x1:S)))
1.59/1.81	 d(d(d(x1:S))) -> a(c(x1:S))
1.59/1.81	->Strongly Connected Components:
1.59/1.81	->->Cycle:
1.59/1.81	->->-> Pairs:
1.59/1.81	 A(a(x1:S)) -> B(c(x1:S))
1.59/1.81	 A(a(x1:S)) -> C(x1:S)
1.59/1.81	 B(b(x1:S)) -> C(d(x1:S))
1.59/1.81	 B(b(x1:S)) -> D(x1:S)
1.59/1.81	 C(c(x1:S)) -> D(d(d(x1:S)))
1.59/1.81	 C(c(x1:S)) -> D(d(x1:S))
1.59/1.81	 C(c(x1:S)) -> D(x1:S)
1.59/1.81	 D(d(d(x1:S))) -> A(c(x1:S))
1.59/1.81	 D(d(d(x1:S))) -> C(x1:S)
1.59/1.81	->->-> Rules:
1.59/1.81	 a(a(x1:S)) -> b(c(x1:S))
1.59/1.81	 b(b(x1:S)) -> c(d(x1:S))
1.59/1.81	 c(c(x1:S)) -> d(d(d(x1:S)))
1.59/1.81	 d(d(d(x1:S))) -> a(c(x1:S))
1.59/1.81	
1.59/1.81	Problem 1: 
1.59/1.81	
1.59/1.81	Reduction Pair Processor:
1.59/1.81	-> Pairs:
1.59/1.81	 A(a(x1:S)) -> B(c(x1:S))
1.59/1.81	 A(a(x1:S)) -> C(x1:S)
1.59/1.81	 B(b(x1:S)) -> C(d(x1:S))
1.59/1.81	 B(b(x1:S)) -> D(x1:S)
1.59/1.81	 C(c(x1:S)) -> D(d(d(x1:S)))
1.59/1.81	 C(c(x1:S)) -> D(d(x1:S))
1.59/1.81	 C(c(x1:S)) -> D(x1:S)
1.59/1.81	 D(d(d(x1:S))) -> A(c(x1:S))
1.59/1.81	 D(d(d(x1:S))) -> C(x1:S)
1.59/1.81	-> Rules:
1.59/1.81	 a(a(x1:S)) -> b(c(x1:S))
1.59/1.81	 b(b(x1:S)) -> c(d(x1:S))
1.59/1.81	 c(c(x1:S)) -> d(d(d(x1:S)))
1.59/1.81	 d(d(d(x1:S))) -> a(c(x1:S))
1.59/1.81	-> Usable rules:
1.59/1.81	 a(a(x1:S)) -> b(c(x1:S))
1.59/1.81	 b(b(x1:S)) -> c(d(x1:S))
1.59/1.81	 c(c(x1:S)) -> d(d(d(x1:S)))
1.59/1.81	 d(d(d(x1:S))) -> a(c(x1:S))
1.59/1.81	->Interpretation type:
1.59/1.81	 Linear
1.59/1.81	->Coefficients:
1.59/1.81	 All rationals
1.59/1.81	->Dimension:
1.59/1.81	 1
1.59/1.81	->Bound:
1.59/1.81	 4
1.59/1.81	->Interpretation:
1.59/1.81	 
1.59/1.81	[a](X) = X + 1/2
1.59/1.81	[b](X) = X + 1/2
1.59/1.81	[c](X) = X + 1/2
1.59/1.81	[d](X) = X + 1/3
1.59/1.81	[A](X) = 4.X + 3/2
1.59/1.81	[B](X) = 4.X + 4/3
1.59/1.81	[C](X) = 4.X + 2
1.59/1.81	[D](X) = 4.X + 1
1.59/1.81	
1.59/1.81	Problem 1: 
1.59/1.81	
1.59/1.81	SCC Processor:
1.59/1.81	-> Pairs:
1.59/1.81	 A(a(x1:S)) -> C(x1:S)
1.59/1.81	 B(b(x1:S)) -> C(d(x1:S))
1.59/1.81	 B(b(x1:S)) -> D(x1:S)
1.59/1.81	 C(c(x1:S)) -> D(d(d(x1:S)))
1.59/1.81	 C(c(x1:S)) -> D(d(x1:S))
1.59/1.81	 C(c(x1:S)) -> D(x1:S)
1.59/1.81	 D(d(d(x1:S))) -> A(c(x1:S))
1.59/1.81	 D(d(d(x1:S))) -> C(x1:S)
1.59/1.81	-> Rules:
1.59/1.81	 a(a(x1:S)) -> b(c(x1:S))
1.59/1.81	 b(b(x1:S)) -> c(d(x1:S))
1.59/1.81	 c(c(x1:S)) -> d(d(d(x1:S)))
1.59/1.81	 d(d(d(x1:S))) -> a(c(x1:S))
1.59/1.81	->Strongly Connected Components:
1.59/1.81	->->Cycle:
1.59/1.81	->->-> Pairs:
1.59/1.81	 A(a(x1:S)) -> C(x1:S)
1.59/1.81	 C(c(x1:S)) -> D(d(d(x1:S)))
1.59/1.81	 C(c(x1:S)) -> D(d(x1:S))
1.59/1.81	 C(c(x1:S)) -> D(x1:S)
1.59/1.81	 D(d(d(x1:S))) -> A(c(x1:S))
1.59/1.81	 D(d(d(x1:S))) -> C(x1:S)
1.59/1.81	->->-> Rules:
1.59/1.81	 a(a(x1:S)) -> b(c(x1:S))
1.59/1.81	 b(b(x1:S)) -> c(d(x1:S))
1.59/1.81	 c(c(x1:S)) -> d(d(d(x1:S)))
1.59/1.81	 d(d(d(x1:S))) -> a(c(x1:S))
1.59/1.81	
1.59/1.81	Problem 1: 
1.59/1.81	
1.59/1.81	Reduction Pair Processor:
1.59/1.81	-> Pairs:
1.59/1.81	 A(a(x1:S)) -> C(x1:S)
1.59/1.81	 C(c(x1:S)) -> D(d(d(x1:S)))
1.59/1.81	 C(c(x1:S)) -> D(d(x1:S))
1.59/1.81	 C(c(x1:S)) -> D(x1:S)
1.59/1.81	 D(d(d(x1:S))) -> A(c(x1:S))
1.59/1.81	 D(d(d(x1:S))) -> C(x1:S)
1.59/1.81	-> Rules:
1.59/1.81	 a(a(x1:S)) -> b(c(x1:S))
1.59/1.81	 b(b(x1:S)) -> c(d(x1:S))
1.59/1.81	 c(c(x1:S)) -> d(d(d(x1:S)))
1.59/1.81	 d(d(d(x1:S))) -> a(c(x1:S))
1.59/1.81	-> Usable rules:
1.59/1.81	 a(a(x1:S)) -> b(c(x1:S))
1.59/1.81	 b(b(x1:S)) -> c(d(x1:S))
1.59/1.81	 c(c(x1:S)) -> d(d(d(x1:S)))
1.59/1.81	 d(d(d(x1:S))) -> a(c(x1:S))
1.59/1.81	->Interpretation type:
1.59/1.81	 Linear
1.59/1.81	->Coefficients:
1.59/1.81	 All rationals
1.59/1.81	->Dimension:
1.59/1.81	 1
1.59/1.81	->Bound:
1.59/1.81	 4
1.59/1.81	->Interpretation:
1.59/1.81	 
1.59/1.81	[a](X) = X + 1
1.59/1.81	[b](X) = X + 1
1.59/1.81	[c](X) = X + 1
1.59/1.81	[d](X) = X + 2/3
1.59/1.81	[A](X) = 3.X + 1/2
1.59/1.81	[C](X) = 3.X + 4/3
1.59/1.81	[D](X) = 3.X + 1/3
1.59/1.81	
1.59/1.81	Problem 1: 
1.59/1.81	
1.59/1.81	SCC Processor:
1.59/1.81	-> Pairs:
1.59/1.81	 C(c(x1:S)) -> D(d(d(x1:S)))
1.59/1.81	 C(c(x1:S)) -> D(d(x1:S))
1.59/1.81	 C(c(x1:S)) -> D(x1:S)
1.59/1.81	 D(d(d(x1:S))) -> A(c(x1:S))
1.59/1.81	 D(d(d(x1:S))) -> C(x1:S)
1.59/1.81	-> Rules:
1.59/1.81	 a(a(x1:S)) -> b(c(x1:S))
1.59/1.81	 b(b(x1:S)) -> c(d(x1:S))
1.59/1.81	 c(c(x1:S)) -> d(d(d(x1:S)))
1.59/1.81	 d(d(d(x1:S))) -> a(c(x1:S))
1.59/1.81	->Strongly Connected Components:
1.59/1.81	->->Cycle:
1.59/1.81	->->-> Pairs:
1.59/1.81	 C(c(x1:S)) -> D(d(d(x1:S)))
1.59/1.81	 C(c(x1:S)) -> D(d(x1:S))
1.59/1.81	 C(c(x1:S)) -> D(x1:S)
1.59/1.81	 D(d(d(x1:S))) -> C(x1:S)
1.59/1.81	->->-> Rules:
1.59/1.81	 a(a(x1:S)) -> b(c(x1:S))
1.59/1.81	 b(b(x1:S)) -> c(d(x1:S))
1.59/1.81	 c(c(x1:S)) -> d(d(d(x1:S)))
1.59/1.81	 d(d(d(x1:S))) -> a(c(x1:S))
1.59/1.81	
1.59/1.81	Problem 1: 
1.59/1.81	
1.59/1.81	Reduction Pair Processor:
1.59/1.81	-> Pairs:
1.59/1.81	 C(c(x1:S)) -> D(d(d(x1:S)))
1.59/1.81	 C(c(x1:S)) -> D(d(x1:S))
1.59/1.81	 C(c(x1:S)) -> D(x1:S)
1.59/1.81	 D(d(d(x1:S))) -> C(x1:S)
1.59/1.81	-> Rules:
1.59/1.81	 a(a(x1:S)) -> b(c(x1:S))
1.59/1.81	 b(b(x1:S)) -> c(d(x1:S))
1.59/1.81	 c(c(x1:S)) -> d(d(d(x1:S)))
1.59/1.81	 d(d(d(x1:S))) -> a(c(x1:S))
1.59/1.81	-> Usable rules:
1.59/1.81	 a(a(x1:S)) -> b(c(x1:S))
1.59/1.81	 b(b(x1:S)) -> c(d(x1:S))
1.59/1.81	 c(c(x1:S)) -> d(d(d(x1:S)))
1.59/1.81	 d(d(d(x1:S))) -> a(c(x1:S))
1.59/1.81	->Interpretation type:
1.59/1.81	 Linear
1.59/1.81	->Coefficients:
1.59/1.81	 All rationals
1.59/1.81	->Dimension:
1.59/1.81	 1
1.59/1.81	->Bound:
1.59/1.81	 4
1.59/1.81	->Interpretation:
1.59/1.81	 
1.59/1.81	[a](X) = X + 2
1.59/1.81	[b](X) = X + 2
1.59/1.81	[c](X) = X + 2
1.59/1.81	[d](X) = X + 4/3
1.59/1.81	[C](X) = 4.X + 4
1.59/1.81	[D](X) = 4.X + 1/4
1.59/1.81	
1.59/1.81	Problem 1: 
1.59/1.81	
1.59/1.81	SCC Processor:
1.59/1.81	-> Pairs:
1.59/1.81	 C(c(x1:S)) -> D(d(x1:S))
1.59/1.81	 C(c(x1:S)) -> D(x1:S)
1.59/1.81	 D(d(d(x1:S))) -> C(x1:S)
1.59/1.81	-> Rules:
1.59/1.81	 a(a(x1:S)) -> b(c(x1:S))
1.59/1.81	 b(b(x1:S)) -> c(d(x1:S))
1.59/1.81	 c(c(x1:S)) -> d(d(d(x1:S)))
1.59/1.81	 d(d(d(x1:S))) -> a(c(x1:S))
1.59/1.81	->Strongly Connected Components:
1.59/1.81	->->Cycle:
1.59/1.81	->->-> Pairs:
1.59/1.81	 C(c(x1:S)) -> D(d(x1:S))
1.59/1.81	 C(c(x1:S)) -> D(x1:S)
1.59/1.81	 D(d(d(x1:S))) -> C(x1:S)
1.59/1.81	->->-> Rules:
1.59/1.81	 a(a(x1:S)) -> b(c(x1:S))
1.59/1.81	 b(b(x1:S)) -> c(d(x1:S))
1.59/1.81	 c(c(x1:S)) -> d(d(d(x1:S)))
1.59/1.81	 d(d(d(x1:S))) -> a(c(x1:S))
1.59/1.81	
1.59/1.81	Problem 1: 
1.59/1.81	
1.59/1.81	Reduction Pair Processor:
1.59/1.81	-> Pairs:
1.59/1.81	 C(c(x1:S)) -> D(d(x1:S))
1.59/1.81	 C(c(x1:S)) -> D(x1:S)
1.59/1.81	 D(d(d(x1:S))) -> C(x1:S)
1.59/1.81	-> Rules:
1.59/1.81	 a(a(x1:S)) -> b(c(x1:S))
1.59/1.81	 b(b(x1:S)) -> c(d(x1:S))
1.59/1.81	 c(c(x1:S)) -> d(d(d(x1:S)))
1.59/1.81	 d(d(d(x1:S))) -> a(c(x1:S))
1.59/1.81	-> Usable rules:
1.59/1.81	 a(a(x1:S)) -> b(c(x1:S))
1.59/1.81	 b(b(x1:S)) -> c(d(x1:S))
1.59/1.81	 c(c(x1:S)) -> d(d(d(x1:S)))
1.59/1.81	 d(d(d(x1:S))) -> a(c(x1:S))
1.59/1.81	->Interpretation type:
1.59/1.81	 Linear
1.59/1.81	->Coefficients:
1.59/1.81	 All rationals
1.59/1.81	->Dimension:
1.59/1.81	 1
1.59/1.81	->Bound:
1.59/1.81	 4
1.59/1.81	->Interpretation:
1.59/1.81	 
1.59/1.81	[a](X) = X + 3/2
1.59/1.81	[b](X) = X + 4/3
1.59/1.81	[c](X) = X + 3/2
1.59/1.81	[d](X) = X + 1
1.59/1.81	[C](X) = X + 2
1.59/1.81	[D](X) = X + 3/2
1.59/1.81	
1.59/1.81	Problem 1: 
1.59/1.81	
1.59/1.81	SCC Processor:
1.59/1.81	-> Pairs:
1.59/1.81	 C(c(x1:S)) -> D(x1:S)
1.59/1.81	 D(d(d(x1:S))) -> C(x1:S)
1.59/1.81	-> Rules:
1.59/1.81	 a(a(x1:S)) -> b(c(x1:S))
1.59/1.81	 b(b(x1:S)) -> c(d(x1:S))
1.59/1.81	 c(c(x1:S)) -> d(d(d(x1:S)))
1.59/1.81	 d(d(d(x1:S))) -> a(c(x1:S))
1.59/1.81	->Strongly Connected Components:
1.59/1.81	->->Cycle:
1.59/1.81	->->-> Pairs:
1.59/1.81	 C(c(x1:S)) -> D(x1:S)
1.59/1.81	 D(d(d(x1:S))) -> C(x1:S)
1.59/1.81	->->-> Rules:
1.59/1.81	 a(a(x1:S)) -> b(c(x1:S))
1.59/1.81	 b(b(x1:S)) -> c(d(x1:S))
1.59/1.81	 c(c(x1:S)) -> d(d(d(x1:S)))
1.59/1.81	 d(d(d(x1:S))) -> a(c(x1:S))
1.59/1.81	
1.59/1.81	Problem 1: 
1.59/1.81	
1.59/1.81	Subterm Processor:
1.59/1.81	-> Pairs:
1.59/1.81	 C(c(x1:S)) -> D(x1:S)
1.59/1.81	 D(d(d(x1:S))) -> C(x1:S)
1.59/1.81	-> Rules:
1.59/1.81	 a(a(x1:S)) -> b(c(x1:S))
1.59/1.81	 b(b(x1:S)) -> c(d(x1:S))
1.59/1.81	 c(c(x1:S)) -> d(d(d(x1:S)))
1.59/1.81	 d(d(d(x1:S))) -> a(c(x1:S))
1.59/1.81	->Projection:
1.59/1.81	 pi(C) = 1
1.59/1.81	 pi(D) = 1
1.59/1.81	
1.59/1.81	Problem 1: 
1.59/1.81	
1.59/1.81	SCC Processor:
1.59/1.81	-> Pairs:
1.59/1.81	 Empty
1.59/1.81	-> Rules:
1.59/1.81	 a(a(x1:S)) -> b(c(x1:S))
1.59/1.81	 b(b(x1:S)) -> c(d(x1:S))
1.59/1.81	 c(c(x1:S)) -> d(d(d(x1:S)))
1.59/1.81	 d(d(d(x1:S))) -> a(c(x1:S))
1.59/1.81	->Strongly Connected Components:
1.59/1.81	 There is no strongly connected component
1.59/1.81	
1.59/1.81	The problem is finite.
1.59/1.81	EOF