0.64/0.88	YES
0.64/0.88	
0.64/0.88	Problem 1: 
0.64/0.88	
0.64/0.88	(VAR v_NonEmpty:S x1:S)
0.64/0.88	(RULES
0.64/0.88	a(a(a(x1:S))) -> b(b(x1:S))
0.64/0.88	b(b(b(x1:S))) -> c(x1:S)
0.64/0.88	c(x1:S) -> d(d(x1:S))
0.64/0.88	d(x1:S) -> a(a(x1:S))
0.64/0.88	)
0.64/0.88	
0.64/0.88	Problem 1: 
0.64/0.88	
0.64/0.88	Dependency Pairs Processor:
0.64/0.88	-> Pairs:
0.64/0.88	 A(a(a(x1:S))) -> B(b(x1:S))
0.64/0.88	 A(a(a(x1:S))) -> B(x1:S)
0.64/0.88	 B(b(b(x1:S))) -> C(x1:S)
0.64/0.88	 C(x1:S) -> D(d(x1:S))
0.64/0.88	 C(x1:S) -> D(x1:S)
0.64/0.88	 D(x1:S) -> A(a(x1:S))
0.64/0.88	 D(x1:S) -> A(x1:S)
0.64/0.88	-> Rules:
0.64/0.88	 a(a(a(x1:S))) -> b(b(x1:S))
0.64/0.88	 b(b(b(x1:S))) -> c(x1:S)
0.64/0.88	 c(x1:S) -> d(d(x1:S))
0.64/0.88	 d(x1:S) -> a(a(x1:S))
0.64/0.88	
0.64/0.88	Problem 1: 
0.64/0.88	
0.64/0.88	SCC Processor:
0.64/0.88	-> Pairs:
0.64/0.88	 A(a(a(x1:S))) -> B(b(x1:S))
0.64/0.88	 A(a(a(x1:S))) -> B(x1:S)
0.64/0.88	 B(b(b(x1:S))) -> C(x1:S)
0.64/0.88	 C(x1:S) -> D(d(x1:S))
0.64/0.88	 C(x1:S) -> D(x1:S)
0.64/0.88	 D(x1:S) -> A(a(x1:S))
0.64/0.88	 D(x1:S) -> A(x1:S)
0.64/0.88	-> Rules:
0.64/0.88	 a(a(a(x1:S))) -> b(b(x1:S))
0.64/0.88	 b(b(b(x1:S))) -> c(x1:S)
0.64/0.88	 c(x1:S) -> d(d(x1:S))
0.64/0.88	 d(x1:S) -> a(a(x1:S))
0.64/0.88	->Strongly Connected Components:
0.64/0.88	->->Cycle:
0.64/0.88	->->-> Pairs:
0.64/0.88	 A(a(a(x1:S))) -> B(b(x1:S))
0.64/0.88	 A(a(a(x1:S))) -> B(x1:S)
0.64/0.88	 B(b(b(x1:S))) -> C(x1:S)
0.64/0.88	 C(x1:S) -> D(d(x1:S))
0.64/0.88	 C(x1:S) -> D(x1:S)
0.64/0.88	 D(x1:S) -> A(a(x1:S))
0.64/0.88	 D(x1:S) -> A(x1:S)
0.64/0.88	->->-> Rules:
0.64/0.88	 a(a(a(x1:S))) -> b(b(x1:S))
0.64/0.88	 b(b(b(x1:S))) -> c(x1:S)
0.64/0.88	 c(x1:S) -> d(d(x1:S))
0.64/0.88	 d(x1:S) -> a(a(x1:S))
0.64/0.88	
0.64/0.88	Problem 1: 
0.64/0.88	
0.64/0.88	Reduction Pair Processor:
0.64/0.88	-> Pairs:
0.64/0.88	 A(a(a(x1:S))) -> B(b(x1:S))
0.64/0.88	 A(a(a(x1:S))) -> B(x1:S)
0.64/0.88	 B(b(b(x1:S))) -> C(x1:S)
0.64/0.88	 C(x1:S) -> D(d(x1:S))
0.64/0.88	 C(x1:S) -> D(x1:S)
0.64/0.88	 D(x1:S) -> A(a(x1:S))
0.64/0.88	 D(x1:S) -> A(x1:S)
0.64/0.88	-> Rules:
0.64/0.88	 a(a(a(x1:S))) -> b(b(x1:S))
0.64/0.88	 b(b(b(x1:S))) -> c(x1:S)
0.64/0.88	 c(x1:S) -> d(d(x1:S))
0.64/0.88	 d(x1:S) -> a(a(x1:S))
0.64/0.88	-> Usable rules:
0.64/0.88	 a(a(a(x1:S))) -> b(b(x1:S))
0.64/0.88	 b(b(b(x1:S))) -> c(x1:S)
0.64/0.88	 c(x1:S) -> d(d(x1:S))
0.64/0.88	 d(x1:S) -> a(a(x1:S))
0.64/0.88	->Interpretation type:
0.64/0.88	 Linear
0.64/0.88	->Coefficients:
0.64/0.88	 All rationals
0.64/0.88	->Dimension:
0.64/0.88	 1
0.64/0.88	->Bound:
0.64/0.88	 4
0.64/0.88	->Interpretation:
0.64/0.88	 
0.64/0.88	[a](X) = X + 1
0.64/0.88	[b](X) = X + 3/2
0.64/0.88	[c](X) = X + 4
0.64/0.88	[d](X) = X + 2
0.64/0.88	[A](X) = 1/3.X
0.64/0.88	[B](X) = 1/3.X
0.64/0.88	[C](X) = 1/3.X + 1
0.64/0.88	[D](X) = 1/3.X + 1/3
0.64/0.88	
0.64/0.88	Problem 1: 
0.64/0.88	
0.64/0.88	SCC Processor:
0.64/0.88	-> Pairs:
0.64/0.88	 A(a(a(x1:S))) -> B(x1:S)
0.64/0.88	 B(b(b(x1:S))) -> C(x1:S)
0.64/0.88	 C(x1:S) -> D(d(x1:S))
0.64/0.88	 C(x1:S) -> D(x1:S)
0.64/0.88	 D(x1:S) -> A(a(x1:S))
0.64/0.88	 D(x1:S) -> A(x1:S)
0.64/0.88	-> Rules:
0.64/0.88	 a(a(a(x1:S))) -> b(b(x1:S))
0.64/0.88	 b(b(b(x1:S))) -> c(x1:S)
0.64/0.88	 c(x1:S) -> d(d(x1:S))
0.64/0.88	 d(x1:S) -> a(a(x1:S))
0.64/0.88	->Strongly Connected Components:
0.64/0.88	->->Cycle:
0.64/0.88	->->-> Pairs:
0.64/0.88	 A(a(a(x1:S))) -> B(x1:S)
0.64/0.88	 B(b(b(x1:S))) -> C(x1:S)
0.64/0.88	 C(x1:S) -> D(d(x1:S))
0.64/0.88	 C(x1:S) -> D(x1:S)
0.64/0.88	 D(x1:S) -> A(a(x1:S))
0.64/0.88	 D(x1:S) -> A(x1:S)
0.64/0.88	->->-> Rules:
0.64/0.88	 a(a(a(x1:S))) -> b(b(x1:S))
0.64/0.88	 b(b(b(x1:S))) -> c(x1:S)
0.64/0.88	 c(x1:S) -> d(d(x1:S))
0.64/0.88	 d(x1:S) -> a(a(x1:S))
0.64/0.88	
0.64/0.88	Problem 1: 
0.64/0.88	
0.64/0.88	Reduction Pair Processor:
0.64/0.88	-> Pairs:
0.64/0.88	 A(a(a(x1:S))) -> B(x1:S)
0.64/0.88	 B(b(b(x1:S))) -> C(x1:S)
0.64/0.88	 C(x1:S) -> D(d(x1:S))
0.64/0.88	 C(x1:S) -> D(x1:S)
0.64/0.88	 D(x1:S) -> A(a(x1:S))
0.64/0.88	 D(x1:S) -> A(x1:S)
0.64/0.88	-> Rules:
0.64/0.88	 a(a(a(x1:S))) -> b(b(x1:S))
0.64/0.88	 b(b(b(x1:S))) -> c(x1:S)
0.64/0.88	 c(x1:S) -> d(d(x1:S))
0.64/0.88	 d(x1:S) -> a(a(x1:S))
0.64/0.88	-> Usable rules:
0.64/0.88	 a(a(a(x1:S))) -> b(b(x1:S))
0.64/0.88	 b(b(b(x1:S))) -> c(x1:S)
0.64/0.88	 c(x1:S) -> d(d(x1:S))
0.64/0.88	 d(x1:S) -> a(a(x1:S))
0.64/0.88	->Interpretation type:
0.64/0.88	 Linear
0.64/0.88	->Coefficients:
0.64/0.88	 All rationals
0.64/0.88	->Dimension:
0.64/0.88	 1
0.64/0.88	->Bound:
0.64/0.88	 4
0.64/0.88	->Interpretation:
0.64/0.88	 
0.64/0.88	[a](X) = X + 1/4
0.64/0.88	[b](X) = X + 1/3
0.64/0.88	[c](X) = X + 1
0.64/0.88	[d](X) = X + 1/2
0.64/0.88	[A](X) = 4.X + 1/2
0.64/0.88	[B](X) = 4.X + 3/2
0.64/0.88	[C](X) = 4.X + 4
0.64/0.88	[D](X) = 4.X + 2
0.64/0.88	
0.64/0.88	Problem 1: 
0.64/0.88	
0.64/0.88	SCC Processor:
0.64/0.88	-> Pairs:
0.64/0.88	 B(b(b(x1:S))) -> C(x1:S)
0.64/0.88	 C(x1:S) -> D(d(x1:S))
0.64/0.88	 C(x1:S) -> D(x1:S)
0.64/0.88	 D(x1:S) -> A(a(x1:S))
0.64/0.88	 D(x1:S) -> A(x1:S)
0.64/0.88	-> Rules:
0.64/0.88	 a(a(a(x1:S))) -> b(b(x1:S))
0.64/0.88	 b(b(b(x1:S))) -> c(x1:S)
0.64/0.88	 c(x1:S) -> d(d(x1:S))
0.64/0.88	 d(x1:S) -> a(a(x1:S))
0.64/0.88	->Strongly Connected Components:
0.64/0.88	 There is no strongly connected component
0.64/0.88	
0.64/0.88	The problem is finite.
0.64/0.88	EOF