2.64/3.22	YES
2.64/3.22	
2.64/3.22	Problem 1: 
2.64/3.22	
2.64/3.22	(VAR v_NonEmpty:S x1:S)
2.64/3.22	(RULES
2.64/3.22	a(a(x1:S)) -> b(b(b(x1:S)))
2.64/3.22	b(c(d(x1:S))) -> a(x1:S)
2.64/3.22	b(c(x1:S)) -> c(b(x1:S))
2.64/3.22	b(x1:S) -> c(c(d(x1:S)))
2.64/3.22	c(x1:S) -> d(d(d(x1:S)))
2.64/3.22	)
2.64/3.22	
2.64/3.22	Problem 1: 
2.64/3.22	
2.64/3.22	Dependency Pairs Processor:
2.64/3.22	-> Pairs:
2.64/3.22	 A(a(x1:S)) -> B(b(b(x1:S)))
2.64/3.22	 A(a(x1:S)) -> B(b(x1:S))
2.64/3.22	 A(a(x1:S)) -> B(x1:S)
2.64/3.22	 B(c(d(x1:S))) -> A(x1:S)
2.64/3.22	 B(c(x1:S)) -> B(x1:S)
2.64/3.22	 B(c(x1:S)) -> C(b(x1:S))
2.64/3.22	 B(x1:S) -> C(c(d(x1:S)))
2.64/3.22	 B(x1:S) -> C(d(x1:S))
2.64/3.22	-> Rules:
2.64/3.22	 a(a(x1:S)) -> b(b(b(x1:S)))
2.64/3.22	 b(c(d(x1:S))) -> a(x1:S)
2.64/3.22	 b(c(x1:S)) -> c(b(x1:S))
2.64/3.22	 b(x1:S) -> c(c(d(x1:S)))
2.64/3.22	 c(x1:S) -> d(d(d(x1:S)))
2.64/3.22	
2.64/3.22	Problem 1: 
2.64/3.22	
2.64/3.22	SCC Processor:
2.64/3.22	-> Pairs:
2.64/3.22	 A(a(x1:S)) -> B(b(b(x1:S)))
2.64/3.22	 A(a(x1:S)) -> B(b(x1:S))
2.64/3.22	 A(a(x1:S)) -> B(x1:S)
2.64/3.22	 B(c(d(x1:S))) -> A(x1:S)
2.64/3.22	 B(c(x1:S)) -> B(x1:S)
2.64/3.22	 B(c(x1:S)) -> C(b(x1:S))
2.64/3.22	 B(x1:S) -> C(c(d(x1:S)))
2.64/3.22	 B(x1:S) -> C(d(x1:S))
2.64/3.22	-> Rules:
2.64/3.22	 a(a(x1:S)) -> b(b(b(x1:S)))
2.64/3.22	 b(c(d(x1:S))) -> a(x1:S)
2.64/3.22	 b(c(x1:S)) -> c(b(x1:S))
2.64/3.22	 b(x1:S) -> c(c(d(x1:S)))
2.64/3.22	 c(x1:S) -> d(d(d(x1:S)))
2.64/3.22	->Strongly Connected Components:
2.64/3.22	->->Cycle:
2.64/3.22	->->-> Pairs:
2.64/3.22	 A(a(x1:S)) -> B(b(b(x1:S)))
2.64/3.22	 A(a(x1:S)) -> B(b(x1:S))
2.64/3.22	 A(a(x1:S)) -> B(x1:S)
2.64/3.22	 B(c(d(x1:S))) -> A(x1:S)
2.64/3.22	 B(c(x1:S)) -> B(x1:S)
2.64/3.22	->->-> Rules:
2.64/3.22	 a(a(x1:S)) -> b(b(b(x1:S)))
2.64/3.22	 b(c(d(x1:S))) -> a(x1:S)
2.64/3.22	 b(c(x1:S)) -> c(b(x1:S))
2.64/3.22	 b(x1:S) -> c(c(d(x1:S)))
2.64/3.22	 c(x1:S) -> d(d(d(x1:S)))
2.64/3.22	
2.64/3.22	Problem 1: 
2.64/3.22	
2.64/3.22	Reduction Pair Processor:
2.64/3.22	-> Pairs:
2.64/3.22	 A(a(x1:S)) -> B(b(b(x1:S)))
2.64/3.22	 A(a(x1:S)) -> B(b(x1:S))
2.64/3.22	 A(a(x1:S)) -> B(x1:S)
2.64/3.22	 B(c(d(x1:S))) -> A(x1:S)
2.64/3.22	 B(c(x1:S)) -> B(x1:S)
2.64/3.22	-> Rules:
2.64/3.22	 a(a(x1:S)) -> b(b(b(x1:S)))
2.64/3.22	 b(c(d(x1:S))) -> a(x1:S)
2.64/3.22	 b(c(x1:S)) -> c(b(x1:S))
2.64/3.22	 b(x1:S) -> c(c(d(x1:S)))
2.64/3.22	 c(x1:S) -> d(d(d(x1:S)))
2.64/3.22	-> Usable rules:
2.64/3.22	 a(a(x1:S)) -> b(b(b(x1:S)))
2.64/3.22	 b(c(d(x1:S))) -> a(x1:S)
2.64/3.22	 b(c(x1:S)) -> c(b(x1:S))
2.64/3.22	 b(x1:S) -> c(c(d(x1:S)))
2.64/3.22	 c(x1:S) -> d(d(d(x1:S)))
2.64/3.22	->Interpretation type:
2.64/3.22	 Linear
2.64/3.22	->Coefficients:
2.64/3.22	 All rationals
2.64/3.22	->Dimension:
2.64/3.22	 1
2.64/3.22	->Bound:
2.64/3.22	 4
2.64/3.22	->Interpretation:
2.64/3.22	 
2.64/3.22	[a](X) = X + 3/2
2.64/3.22	[b](X) = X + 1
2.64/3.22	[c](X) = X + 1/2
2.64/3.22	[d](X) = X
2.64/3.22	[A](X) = 3.X + 2
2.64/3.22	[B](X) = 3.X + 1/2
2.64/3.22	
2.64/3.22	Problem 1: 
2.64/3.22	
2.64/3.22	SCC Processor:
2.64/3.22	-> Pairs:
2.64/3.22	 A(a(x1:S)) -> B(b(b(x1:S)))
2.64/3.22	 A(a(x1:S)) -> B(x1:S)
2.64/3.22	 B(c(d(x1:S))) -> A(x1:S)
2.64/3.22	 B(c(x1:S)) -> B(x1:S)
2.64/3.22	-> Rules:
2.64/3.22	 a(a(x1:S)) -> b(b(b(x1:S)))
2.64/3.22	 b(c(d(x1:S))) -> a(x1:S)
2.64/3.22	 b(c(x1:S)) -> c(b(x1:S))
2.64/3.22	 b(x1:S) -> c(c(d(x1:S)))
2.64/3.22	 c(x1:S) -> d(d(d(x1:S)))
2.64/3.22	->Strongly Connected Components:
2.64/3.22	->->Cycle:
2.64/3.22	->->-> Pairs:
2.64/3.22	 A(a(x1:S)) -> B(b(b(x1:S)))
2.64/3.22	 A(a(x1:S)) -> B(x1:S)
2.64/3.22	 B(c(d(x1:S))) -> A(x1:S)
2.64/3.22	 B(c(x1:S)) -> B(x1:S)
2.64/3.22	->->-> Rules:
2.64/3.22	 a(a(x1:S)) -> b(b(b(x1:S)))
2.64/3.22	 b(c(d(x1:S))) -> a(x1:S)
2.64/3.22	 b(c(x1:S)) -> c(b(x1:S))
2.64/3.22	 b(x1:S) -> c(c(d(x1:S)))
2.64/3.22	 c(x1:S) -> d(d(d(x1:S)))
2.64/3.22	
2.64/3.22	Problem 1: 
2.64/3.22	
2.64/3.22	Reduction Pair Processor:
2.64/3.22	-> Pairs:
2.64/3.22	 A(a(x1:S)) -> B(b(b(x1:S)))
2.64/3.22	 A(a(x1:S)) -> B(x1:S)
2.64/3.22	 B(c(d(x1:S))) -> A(x1:S)
2.64/3.22	 B(c(x1:S)) -> B(x1:S)
2.64/3.22	-> Rules:
2.64/3.22	 a(a(x1:S)) -> b(b(b(x1:S)))
2.64/3.22	 b(c(d(x1:S))) -> a(x1:S)
2.64/3.22	 b(c(x1:S)) -> c(b(x1:S))
2.64/3.22	 b(x1:S) -> c(c(d(x1:S)))
2.64/3.22	 c(x1:S) -> d(d(d(x1:S)))
2.64/3.22	-> Usable rules:
2.64/3.22	 a(a(x1:S)) -> b(b(b(x1:S)))
2.64/3.22	 b(c(d(x1:S))) -> a(x1:S)
2.64/3.22	 b(c(x1:S)) -> c(b(x1:S))
2.64/3.22	 b(x1:S) -> c(c(d(x1:S)))
2.64/3.22	 c(x1:S) -> d(d(d(x1:S)))
2.64/3.22	->Interpretation type:
2.64/3.22	 Linear
2.64/3.22	->Coefficients:
2.64/3.22	 All rationals
2.64/3.22	->Dimension:
2.64/3.22	 1
2.64/3.22	->Bound:
2.64/3.22	 4
2.64/3.22	->Interpretation:
2.64/3.22	 
2.64/3.22	[a](X) = X + 1
2.64/3.22	[b](X) = X + 2/3
2.64/3.22	[c](X) = X + 1/3
2.64/3.22	[d](X) = X
2.64/3.22	[A](X) = 3/2.X + 2
2.64/3.22	[B](X) = 3/2.X + 3/2
2.64/3.22	
2.64/3.22	Problem 1: 
2.64/3.22	
2.64/3.22	SCC Processor:
2.64/3.22	-> Pairs:
2.64/3.22	 A(a(x1:S)) -> B(b(b(x1:S)))
2.64/3.22	 B(c(d(x1:S))) -> A(x1:S)
2.64/3.22	 B(c(x1:S)) -> B(x1:S)
2.64/3.22	-> Rules:
2.64/3.22	 a(a(x1:S)) -> b(b(b(x1:S)))
2.64/3.22	 b(c(d(x1:S))) -> a(x1:S)
2.64/3.22	 b(c(x1:S)) -> c(b(x1:S))
2.64/3.22	 b(x1:S) -> c(c(d(x1:S)))
2.64/3.22	 c(x1:S) -> d(d(d(x1:S)))
2.64/3.22	->Strongly Connected Components:
2.64/3.22	->->Cycle:
2.64/3.22	->->-> Pairs:
2.64/3.22	 A(a(x1:S)) -> B(b(b(x1:S)))
2.64/3.22	 B(c(d(x1:S))) -> A(x1:S)
2.64/3.22	 B(c(x1:S)) -> B(x1:S)
2.64/3.22	->->-> Rules:
2.64/3.22	 a(a(x1:S)) -> b(b(b(x1:S)))
2.64/3.22	 b(c(d(x1:S))) -> a(x1:S)
2.64/3.22	 b(c(x1:S)) -> c(b(x1:S))
2.64/3.22	 b(x1:S) -> c(c(d(x1:S)))
2.64/3.22	 c(x1:S) -> d(d(d(x1:S)))
2.64/3.22	
2.64/3.22	Problem 1: 
2.64/3.22	
2.64/3.22	Reduction Pair Processor:
2.64/3.22	-> Pairs:
2.64/3.22	 A(a(x1:S)) -> B(b(b(x1:S)))
2.64/3.22	 B(c(d(x1:S))) -> A(x1:S)
2.64/3.22	 B(c(x1:S)) -> B(x1:S)
2.64/3.22	-> Rules:
2.64/3.22	 a(a(x1:S)) -> b(b(b(x1:S)))
2.64/3.22	 b(c(d(x1:S))) -> a(x1:S)
2.64/3.22	 b(c(x1:S)) -> c(b(x1:S))
2.64/3.22	 b(x1:S) -> c(c(d(x1:S)))
2.64/3.22	 c(x1:S) -> d(d(d(x1:S)))
2.64/3.22	-> Usable rules:
2.64/3.22	 a(a(x1:S)) -> b(b(b(x1:S)))
2.64/3.22	 b(c(d(x1:S))) -> a(x1:S)
2.64/3.22	 b(c(x1:S)) -> c(b(x1:S))
2.64/3.22	 b(x1:S) -> c(c(d(x1:S)))
2.64/3.22	 c(x1:S) -> d(d(d(x1:S)))
2.64/3.22	->Interpretation type:
2.64/3.22	 Linear
2.64/3.22	->Coefficients:
2.64/3.22	 All rationals
2.64/3.22	->Dimension:
2.64/3.22	 1
2.64/3.22	->Bound:
2.64/3.22	 4
2.64/3.22	->Interpretation:
2.64/3.22	 
2.64/3.22	[a](X) = X + 3
2.64/3.22	[b](X) = X + 2
2.64/3.22	[c](X) = X + 1
2.64/3.22	[d](X) = X
2.64/3.22	[A](X) = 4.X + 4
2.64/3.22	[B](X) = 4.X
2.64/3.22	
2.64/3.22	Problem 1: 
2.64/3.22	
2.64/3.22	SCC Processor:
2.64/3.22	-> Pairs:
2.64/3.22	 A(a(x1:S)) -> B(b(b(x1:S)))
2.64/3.22	 B(c(d(x1:S))) -> A(x1:S)
2.64/3.22	-> Rules:
2.64/3.22	 a(a(x1:S)) -> b(b(b(x1:S)))
2.64/3.22	 b(c(d(x1:S))) -> a(x1:S)
2.64/3.22	 b(c(x1:S)) -> c(b(x1:S))
2.64/3.22	 b(x1:S) -> c(c(d(x1:S)))
2.64/3.22	 c(x1:S) -> d(d(d(x1:S)))
2.64/3.22	->Strongly Connected Components:
2.64/3.22	->->Cycle:
2.64/3.22	->->-> Pairs:
2.64/3.22	 A(a(x1:S)) -> B(b(b(x1:S)))
2.64/3.22	 B(c(d(x1:S))) -> A(x1:S)
2.64/3.22	->->-> Rules:
2.64/3.22	 a(a(x1:S)) -> b(b(b(x1:S)))
2.64/3.22	 b(c(d(x1:S))) -> a(x1:S)
2.64/3.22	 b(c(x1:S)) -> c(b(x1:S))
2.64/3.22	 b(x1:S) -> c(c(d(x1:S)))
2.64/3.22	 c(x1:S) -> d(d(d(x1:S)))
2.64/3.22	
2.64/3.22	Problem 1: 
2.64/3.22	
2.64/3.22	Reduction Pair Processor:
2.64/3.22	-> Pairs:
2.64/3.22	 A(a(x1:S)) -> B(b(b(x1:S)))
2.64/3.22	 B(c(d(x1:S))) -> A(x1:S)
2.64/3.22	-> Rules:
2.64/3.22	 a(a(x1:S)) -> b(b(b(x1:S)))
2.64/3.22	 b(c(d(x1:S))) -> a(x1:S)
2.64/3.22	 b(c(x1:S)) -> c(b(x1:S))
2.64/3.22	 b(x1:S) -> c(c(d(x1:S)))
2.64/3.22	 c(x1:S) -> d(d(d(x1:S)))
2.64/3.22	-> Usable rules:
2.64/3.22	 a(a(x1:S)) -> b(b(b(x1:S)))
2.64/3.22	 b(c(d(x1:S))) -> a(x1:S)
2.64/3.22	 b(c(x1:S)) -> c(b(x1:S))
2.64/3.22	 b(x1:S) -> c(c(d(x1:S)))
2.64/3.22	 c(x1:S) -> d(d(d(x1:S)))
2.64/3.22	->Interpretation type:
2.64/3.22	 Linear
2.64/3.22	->Coefficients:
2.64/3.22	 Natural Numbers
2.64/3.22	->Dimension:
2.64/3.22	 2
2.64/3.22	->Bound:
2.64/3.22	 1
2.64/3.22	->Interpretation:
2.64/3.22	 
2.64/3.22	[a](X) = [1;1]
2.64/3.22	[b](X) = [1;1]
2.64/3.22	[c](X) = [0 1;0 0].X + [0;1]
2.64/3.22	[d](X) = [0 0;1 0].X + [0;1]
2.64/3.22	[A](X) = [1 0;1 0].X + [1;1]
2.64/3.22	[B](X) = [1 0;1 1].X
2.64/3.22	
2.64/3.22	Problem 1: 
2.64/3.22	
2.64/3.22	SCC Processor:
2.64/3.22	-> Pairs:
2.64/3.22	 B(c(d(x1:S))) -> A(x1:S)
2.64/3.22	-> Rules:
2.64/3.22	 a(a(x1:S)) -> b(b(b(x1:S)))
2.64/3.22	 b(c(d(x1:S))) -> a(x1:S)
2.64/3.22	 b(c(x1:S)) -> c(b(x1:S))
2.64/3.22	 b(x1:S) -> c(c(d(x1:S)))
2.64/3.22	 c(x1:S) -> d(d(d(x1:S)))
2.64/3.22	->Strongly Connected Components:
2.64/3.22	 There is no strongly connected component
2.64/3.22	
2.64/3.22	The problem is finite.
2.64/3.22	EOF