3.14/3.22	YES
3.14/3.22	
3.14/3.22	Problem 1: 
3.14/3.22	
3.14/3.22	(VAR v_NonEmpty:S x1:S)
3.14/3.22	(RULES
3.14/3.22	a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	)
3.14/3.22	
3.14/3.22	Problem 1: 
3.14/3.22	
3.14/3.22	Dependency Pairs Processor:
3.14/3.22	-> Pairs:
3.14/3.22	 A(a(a(x1:S))) -> A(b(b(x1:S)))
3.14/3.22	 A(a(a(x1:S))) -> B(b(x1:S))
3.14/3.22	 A(a(a(x1:S))) -> B(x1:S)
3.14/3.22	 B(a(b(x1:S))) -> A(a(x1:S))
3.14/3.22	 B(a(b(x1:S))) -> A(x1:S)
3.14/3.22	 B(a(b(x1:S))) -> B(a(a(x1:S)))
3.14/3.22	 B(a(x1:S)) -> B(b(x1:S))
3.14/3.22	 B(a(x1:S)) -> B(x1:S)
3.14/3.22	-> Rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	
3.14/3.22	Problem 1: 
3.14/3.22	
3.14/3.22	SCC Processor:
3.14/3.22	-> Pairs:
3.14/3.22	 A(a(a(x1:S))) -> A(b(b(x1:S)))
3.14/3.22	 A(a(a(x1:S))) -> B(b(x1:S))
3.14/3.22	 A(a(a(x1:S))) -> B(x1:S)
3.14/3.22	 B(a(b(x1:S))) -> A(a(x1:S))
3.14/3.22	 B(a(b(x1:S))) -> A(x1:S)
3.14/3.22	 B(a(b(x1:S))) -> B(a(a(x1:S)))
3.14/3.22	 B(a(x1:S)) -> B(b(x1:S))
3.14/3.22	 B(a(x1:S)) -> B(x1:S)
3.14/3.22	-> Rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	->Strongly Connected Components:
3.14/3.22	->->Cycle:
3.14/3.22	->->-> Pairs:
3.14/3.22	 A(a(a(x1:S))) -> A(b(b(x1:S)))
3.14/3.22	 A(a(a(x1:S))) -> B(b(x1:S))
3.14/3.22	 A(a(a(x1:S))) -> B(x1:S)
3.14/3.22	 B(a(b(x1:S))) -> A(a(x1:S))
3.14/3.22	 B(a(b(x1:S))) -> A(x1:S)
3.14/3.22	 B(a(b(x1:S))) -> B(a(a(x1:S)))
3.14/3.22	 B(a(x1:S)) -> B(b(x1:S))
3.14/3.22	 B(a(x1:S)) -> B(x1:S)
3.14/3.22	->->-> Rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	
3.14/3.22	Problem 1: 
3.14/3.22	
3.14/3.22	Reduction Pair Processor:
3.14/3.22	-> Pairs:
3.14/3.22	 A(a(a(x1:S))) -> A(b(b(x1:S)))
3.14/3.22	 A(a(a(x1:S))) -> B(b(x1:S))
3.14/3.22	 A(a(a(x1:S))) -> B(x1:S)
3.14/3.22	 B(a(b(x1:S))) -> A(a(x1:S))
3.14/3.22	 B(a(b(x1:S))) -> A(x1:S)
3.14/3.22	 B(a(b(x1:S))) -> B(a(a(x1:S)))
3.14/3.22	 B(a(x1:S)) -> B(b(x1:S))
3.14/3.22	 B(a(x1:S)) -> B(x1:S)
3.14/3.22	-> Rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	-> Usable rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	->Interpretation type:
3.14/3.22	 Linear
3.14/3.22	->Coefficients:
3.14/3.22	 Natural Numbers
3.14/3.22	->Dimension:
3.14/3.22	 1
3.14/3.22	->Bound:
3.14/3.22	 2
3.14/3.22	->Interpretation:
3.14/3.22	 
3.14/3.22	[a](X) = X + 2
3.14/3.22	[b](X) = X + 2
3.14/3.22	[A](X) = 2.X + 2
3.14/3.22	[B](X) = 2.X
3.14/3.22	
3.14/3.22	Problem 1: 
3.14/3.22	
3.14/3.22	SCC Processor:
3.14/3.22	-> Pairs:
3.14/3.22	 A(a(a(x1:S))) -> A(b(b(x1:S)))
3.14/3.22	 A(a(a(x1:S))) -> B(x1:S)
3.14/3.22	 B(a(b(x1:S))) -> A(a(x1:S))
3.14/3.22	 B(a(b(x1:S))) -> A(x1:S)
3.14/3.22	 B(a(b(x1:S))) -> B(a(a(x1:S)))
3.14/3.22	 B(a(x1:S)) -> B(b(x1:S))
3.14/3.22	 B(a(x1:S)) -> B(x1:S)
3.14/3.22	-> Rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	->Strongly Connected Components:
3.14/3.22	->->Cycle:
3.14/3.22	->->-> Pairs:
3.14/3.22	 A(a(a(x1:S))) -> A(b(b(x1:S)))
3.14/3.22	 A(a(a(x1:S))) -> B(x1:S)
3.14/3.22	 B(a(b(x1:S))) -> A(a(x1:S))
3.14/3.22	 B(a(b(x1:S))) -> A(x1:S)
3.14/3.22	 B(a(b(x1:S))) -> B(a(a(x1:S)))
3.14/3.22	 B(a(x1:S)) -> B(b(x1:S))
3.14/3.22	 B(a(x1:S)) -> B(x1:S)
3.14/3.22	->->-> Rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	
3.14/3.22	Problem 1: 
3.14/3.22	
3.14/3.22	Reduction Pair Processor:
3.14/3.22	-> Pairs:
3.14/3.22	 A(a(a(x1:S))) -> A(b(b(x1:S)))
3.14/3.22	 A(a(a(x1:S))) -> B(x1:S)
3.14/3.22	 B(a(b(x1:S))) -> A(a(x1:S))
3.14/3.22	 B(a(b(x1:S))) -> A(x1:S)
3.14/3.22	 B(a(b(x1:S))) -> B(a(a(x1:S)))
3.14/3.22	 B(a(x1:S)) -> B(b(x1:S))
3.14/3.22	 B(a(x1:S)) -> B(x1:S)
3.14/3.22	-> Rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	-> Usable rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	->Interpretation type:
3.14/3.22	 Linear
3.14/3.22	->Coefficients:
3.14/3.22	 Natural Numbers
3.14/3.22	->Dimension:
3.14/3.22	 1
3.14/3.22	->Bound:
3.14/3.22	 2
3.14/3.22	->Interpretation:
3.14/3.22	 
3.14/3.22	[a](X) = 2.X + 2
3.14/3.22	[b](X) = 2.X + 2
3.14/3.22	[A](X) = X + 2
3.14/3.22	[B](X) = 2.X + 2
3.14/3.22	
3.14/3.22	Problem 1: 
3.14/3.22	
3.14/3.22	SCC Processor:
3.14/3.22	-> Pairs:
3.14/3.22	 A(a(a(x1:S))) -> A(b(b(x1:S)))
3.14/3.22	 B(a(b(x1:S))) -> A(a(x1:S))
3.14/3.22	 B(a(b(x1:S))) -> A(x1:S)
3.14/3.22	 B(a(b(x1:S))) -> B(a(a(x1:S)))
3.14/3.22	 B(a(x1:S)) -> B(b(x1:S))
3.14/3.22	 B(a(x1:S)) -> B(x1:S)
3.14/3.22	-> Rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	->Strongly Connected Components:
3.14/3.22	->->Cycle:
3.14/3.22	->->-> Pairs:
3.14/3.22	 A(a(a(x1:S))) -> A(b(b(x1:S)))
3.14/3.22	->->-> Rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	->->Cycle:
3.14/3.22	->->-> Pairs:
3.14/3.22	 B(a(b(x1:S))) -> B(a(a(x1:S)))
3.14/3.22	 B(a(x1:S)) -> B(b(x1:S))
3.14/3.22	 B(a(x1:S)) -> B(x1:S)
3.14/3.22	->->-> Rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	
3.14/3.22	
3.14/3.22	The problem is decomposed in 2 subproblems.
3.14/3.22	
3.14/3.22	Problem 1.1: 
3.14/3.22	
3.14/3.22	Reduction Pair Processor:
3.14/3.22	-> Pairs:
3.14/3.22	 A(a(a(x1:S))) -> A(b(b(x1:S)))
3.14/3.22	-> Rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	-> Usable rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	->Interpretation type:
3.14/3.22	 Linear
3.14/3.22	->Coefficients:
3.14/3.22	 Natural Numbers
3.14/3.22	->Dimension:
3.14/3.22	 1
3.14/3.22	->Bound:
3.14/3.22	 2
3.14/3.22	->Interpretation:
3.14/3.22	 
3.14/3.22	[a](X) = 2.X + 2
3.14/3.22	[b](X) = 2
3.14/3.22	[A](X) = 2.X
3.14/3.22	
3.14/3.22	Problem 1.1: 
3.14/3.22	
3.14/3.22	SCC Processor:
3.14/3.22	-> Pairs:
3.14/3.22	 Empty
3.14/3.22	-> Rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	->Strongly Connected Components:
3.14/3.22	 There is no strongly connected component
3.14/3.22	
3.14/3.22	The problem is finite.
3.14/3.22	
3.14/3.22	Problem 1.2: 
3.14/3.22	
3.14/3.22	Reduction Pair Processor:
3.14/3.22	-> Pairs:
3.14/3.22	 B(a(b(x1:S))) -> B(a(a(x1:S)))
3.14/3.22	 B(a(x1:S)) -> B(b(x1:S))
3.14/3.22	 B(a(x1:S)) -> B(x1:S)
3.14/3.22	-> Rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	-> Usable rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	->Interpretation type:
3.14/3.22	 Linear
3.14/3.22	->Coefficients:
3.14/3.22	 Natural Numbers
3.14/3.22	->Dimension:
3.14/3.22	 1
3.14/3.22	->Bound:
3.14/3.22	 2
3.14/3.22	->Interpretation:
3.14/3.22	 
3.14/3.22	[a](X) = 2.X + 2
3.14/3.22	[b](X) = 2.X + 2
3.14/3.22	[B](X) = 2.X
3.14/3.22	
3.14/3.22	Problem 1.2: 
3.14/3.22	
3.14/3.22	SCC Processor:
3.14/3.22	-> Pairs:
3.14/3.22	 B(a(b(x1:S))) -> B(a(a(x1:S)))
3.14/3.22	 B(a(x1:S)) -> B(b(x1:S))
3.14/3.22	-> Rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	->Strongly Connected Components:
3.14/3.22	->->Cycle:
3.14/3.22	->->-> Pairs:
3.14/3.22	 B(a(b(x1:S))) -> B(a(a(x1:S)))
3.14/3.22	 B(a(x1:S)) -> B(b(x1:S))
3.14/3.22	->->-> Rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	
3.14/3.22	Problem 1.2: 
3.14/3.22	
3.14/3.22	Reduction Pair Processor:
3.14/3.22	-> Pairs:
3.14/3.22	 B(a(b(x1:S))) -> B(a(a(x1:S)))
3.14/3.22	 B(a(x1:S)) -> B(b(x1:S))
3.14/3.22	-> Rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	-> Usable rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	->Interpretation type:
3.14/3.22	 Linear
3.14/3.22	->Coefficients:
3.14/3.22	 Natural Numbers
3.14/3.22	->Dimension:
3.14/3.22	 1
3.14/3.22	->Bound:
3.14/3.22	 2
3.14/3.22	->Interpretation:
3.14/3.22	 
3.14/3.22	[a](X) = 2
3.14/3.22	[b](X) = 0
3.14/3.22	[B](X) = 2.X
3.14/3.22	
3.14/3.22	Problem 1.2: 
3.14/3.22	
3.14/3.22	SCC Processor:
3.14/3.22	-> Pairs:
3.14/3.22	 B(a(b(x1:S))) -> B(a(a(x1:S)))
3.14/3.22	-> Rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	->Strongly Connected Components:
3.14/3.22	->->Cycle:
3.14/3.22	->->-> Pairs:
3.14/3.22	 B(a(b(x1:S))) -> B(a(a(x1:S)))
3.14/3.22	->->-> Rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	
3.14/3.22	Problem 1.2: 
3.14/3.22	
3.14/3.22	Reduction Pair Processor:
3.14/3.22	-> Pairs:
3.14/3.22	 B(a(b(x1:S))) -> B(a(a(x1:S)))
3.14/3.22	-> Rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	-> Usable rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	->Interpretation type:
3.14/3.22	 Linear
3.14/3.22	->Coefficients:
3.14/3.22	 Natural Numbers
3.14/3.22	->Dimension:
3.14/3.22	 2
3.14/3.22	->Bound:
3.14/3.22	 1
3.14/3.22	->Interpretation:
3.14/3.22	 
3.14/3.22	[a](X) = [0 1;1 0].X + [1;0]
3.14/3.22	[b](X) = [0 0;1 0].X + [0;1]
3.14/3.22	[B](X) = [1 0;1 0].X
3.14/3.22	
3.14/3.22	Problem 1.2: 
3.14/3.22	
3.14/3.22	SCC Processor:
3.14/3.22	-> Pairs:
3.14/3.22	 Empty
3.14/3.22	-> Rules:
3.14/3.22	 a(a(a(x1:S))) -> a(b(b(x1:S)))
3.14/3.22	 b(a(b(x1:S))) -> b(a(a(x1:S)))
3.14/3.22	 b(a(x1:S)) -> b(b(x1:S))
3.14/3.22	->Strongly Connected Components:
3.14/3.22	 There is no strongly connected component
3.14/3.22	
3.14/3.22	The problem is finite.
3.14/3.22	EOF