21.44/21.66	YES
21.44/21.66	
21.44/21.66	Problem 1: 
21.44/21.66	
21.44/21.66	(VAR v_NonEmpty:S x1:S)
21.44/21.66	(RULES
21.44/21.66	a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	)
21.44/21.66	
21.44/21.66	Problem 1: 
21.44/21.66	
21.44/21.66	Dependency Pairs Processor:
21.44/21.66	-> Pairs:
21.44/21.66	 A(b(a(a(x1:S)))) -> A(a(a(a(x1:S))))
21.44/21.66	 A(b(a(a(x1:S)))) -> A(a(a(x1:S)))
21.44/21.66	 B(a(a(b(x1:S)))) -> A(b(a(x1:S)))
21.44/21.66	 B(a(a(b(x1:S)))) -> A(x1:S)
21.44/21.66	 B(a(a(b(x1:S)))) -> B(a(b(a(x1:S))))
21.44/21.66	 B(a(a(b(x1:S)))) -> B(a(x1:S))
21.44/21.66	 B(a(b(a(x1:S)))) -> A(a(b(b(x1:S))))
21.44/21.66	 B(a(b(a(x1:S)))) -> A(b(b(x1:S)))
21.44/21.66	 B(a(b(a(x1:S)))) -> B(b(x1:S))
21.44/21.66	 B(a(b(a(x1:S)))) -> B(x1:S)
21.44/21.66	-> Rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	
21.44/21.66	Problem 1: 
21.44/21.66	
21.44/21.66	SCC Processor:
21.44/21.66	-> Pairs:
21.44/21.66	 A(b(a(a(x1:S)))) -> A(a(a(a(x1:S))))
21.44/21.66	 A(b(a(a(x1:S)))) -> A(a(a(x1:S)))
21.44/21.66	 B(a(a(b(x1:S)))) -> A(b(a(x1:S)))
21.44/21.66	 B(a(a(b(x1:S)))) -> A(x1:S)
21.44/21.66	 B(a(a(b(x1:S)))) -> B(a(b(a(x1:S))))
21.44/21.66	 B(a(a(b(x1:S)))) -> B(a(x1:S))
21.44/21.66	 B(a(b(a(x1:S)))) -> A(a(b(b(x1:S))))
21.44/21.66	 B(a(b(a(x1:S)))) -> A(b(b(x1:S)))
21.44/21.66	 B(a(b(a(x1:S)))) -> B(b(x1:S))
21.44/21.66	 B(a(b(a(x1:S)))) -> B(x1:S)
21.44/21.66	-> Rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	->Strongly Connected Components:
21.44/21.66	->->Cycle:
21.44/21.66	->->-> Pairs:
21.44/21.66	 A(b(a(a(x1:S)))) -> A(a(a(a(x1:S))))
21.44/21.66	 A(b(a(a(x1:S)))) -> A(a(a(x1:S)))
21.44/21.66	->->-> Rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	->->Cycle:
21.44/21.66	->->-> Pairs:
21.44/21.66	 B(a(a(b(x1:S)))) -> B(a(b(a(x1:S))))
21.44/21.66	 B(a(a(b(x1:S)))) -> B(a(x1:S))
21.44/21.66	 B(a(b(a(x1:S)))) -> B(b(x1:S))
21.44/21.66	 B(a(b(a(x1:S)))) -> B(x1:S)
21.44/21.66	->->-> Rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	
21.44/21.66	
21.44/21.66	The problem is decomposed in 2 subproblems.
21.44/21.66	
21.44/21.66	Problem 1.1: 
21.44/21.66	
21.44/21.66	Reduction Pair Processor:
21.44/21.66	-> Pairs:
21.44/21.66	 A(b(a(a(x1:S)))) -> A(a(a(a(x1:S))))
21.44/21.66	 A(b(a(a(x1:S)))) -> A(a(a(x1:S)))
21.44/21.66	-> Rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	-> Usable rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	->Interpretation type:
21.44/21.66	 Linear
21.44/21.66	->Coefficients:
21.44/21.66	 Natural Numbers
21.44/21.66	->Dimension:
21.44/21.66	 1
21.44/21.66	->Bound:
21.44/21.66	 2
21.44/21.66	->Interpretation:
21.44/21.66	 
21.44/21.66	[a](X) = 2.X
21.44/21.66	[b](X) = 2.X + 2
21.44/21.66	[A](X) = 2.X
21.44/21.66	
21.44/21.66	Problem 1.1: 
21.44/21.66	
21.44/21.66	SCC Processor:
21.44/21.66	-> Pairs:
21.44/21.66	 A(b(a(a(x1:S)))) -> A(a(a(x1:S)))
21.44/21.66	-> Rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	->Strongly Connected Components:
21.44/21.66	->->Cycle:
21.44/21.66	->->-> Pairs:
21.44/21.66	 A(b(a(a(x1:S)))) -> A(a(a(x1:S)))
21.44/21.66	->->-> Rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	
21.44/21.66	Problem 1.1: 
21.44/21.66	
21.44/21.66	Subterm Processor:
21.44/21.66	-> Pairs:
21.44/21.66	 A(b(a(a(x1:S)))) -> A(a(a(x1:S)))
21.44/21.66	-> Rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	->Projection:
21.44/21.66	 pi(A) = 1
21.44/21.66	
21.44/21.66	Problem 1.1: 
21.44/21.66	
21.44/21.66	SCC Processor:
21.44/21.66	-> Pairs:
21.44/21.66	 Empty
21.44/21.66	-> Rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	->Strongly Connected Components:
21.44/21.66	 There is no strongly connected component
21.44/21.66	
21.44/21.66	The problem is finite.
21.44/21.66	
21.44/21.66	Problem 1.2: 
21.44/21.66	
21.44/21.66	Reduction Pair Processor:
21.44/21.66	-> Pairs:
21.44/21.66	 B(a(a(b(x1:S)))) -> B(a(b(a(x1:S))))
21.44/21.66	 B(a(a(b(x1:S)))) -> B(a(x1:S))
21.44/21.66	 B(a(b(a(x1:S)))) -> B(b(x1:S))
21.44/21.66	 B(a(b(a(x1:S)))) -> B(x1:S)
21.44/21.66	-> Rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	-> Usable rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	->Interpretation type:
21.44/21.66	 Linear
21.44/21.66	->Coefficients:
21.44/21.66	 Natural Numbers
21.44/21.66	->Dimension:
21.44/21.66	 1
21.44/21.66	->Bound:
21.44/21.66	 2
21.44/21.66	->Interpretation:
21.44/21.66	 
21.44/21.66	[a](X) = 2.X + 2
21.44/21.66	[b](X) = 2.X + 2
21.44/21.66	[B](X) = 2.X
21.44/21.66	
21.44/21.66	Problem 1.2: 
21.44/21.66	
21.44/21.66	SCC Processor:
21.44/21.66	-> Pairs:
21.44/21.66	 B(a(a(b(x1:S)))) -> B(a(b(a(x1:S))))
21.44/21.66	 B(a(b(a(x1:S)))) -> B(b(x1:S))
21.44/21.66	 B(a(b(a(x1:S)))) -> B(x1:S)
21.44/21.66	-> Rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	->Strongly Connected Components:
21.44/21.66	->->Cycle:
21.44/21.66	->->-> Pairs:
21.44/21.66	 B(a(a(b(x1:S)))) -> B(a(b(a(x1:S))))
21.44/21.66	 B(a(b(a(x1:S)))) -> B(b(x1:S))
21.44/21.66	 B(a(b(a(x1:S)))) -> B(x1:S)
21.44/21.66	->->-> Rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	
21.44/21.66	Problem 1.2: 
21.44/21.66	
21.44/21.66	Reduction Pair Processor:
21.44/21.66	-> Pairs:
21.44/21.66	 B(a(a(b(x1:S)))) -> B(a(b(a(x1:S))))
21.44/21.66	 B(a(b(a(x1:S)))) -> B(b(x1:S))
21.44/21.66	 B(a(b(a(x1:S)))) -> B(x1:S)
21.44/21.66	-> Rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	-> Usable rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	->Interpretation type:
21.44/21.66	 Linear
21.44/21.66	->Coefficients:
21.44/21.66	 Natural Numbers
21.44/21.66	->Dimension:
21.44/21.66	 1
21.44/21.66	->Bound:
21.44/21.66	 2
21.44/21.66	->Interpretation:
21.44/21.66	 
21.44/21.66	[a](X) = 2.X + 1
21.44/21.66	[b](X) = 2.X + 1
21.44/21.66	[B](X) = 2.X
21.44/21.66	
21.44/21.66	Problem 1.2: 
21.44/21.66	
21.44/21.66	SCC Processor:
21.44/21.66	-> Pairs:
21.44/21.66	 B(a(a(b(x1:S)))) -> B(a(b(a(x1:S))))
21.44/21.66	 B(a(b(a(x1:S)))) -> B(x1:S)
21.44/21.66	-> Rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	->Strongly Connected Components:
21.44/21.66	->->Cycle:
21.44/21.66	->->-> Pairs:
21.44/21.66	 B(a(a(b(x1:S)))) -> B(a(b(a(x1:S))))
21.44/21.66	 B(a(b(a(x1:S)))) -> B(x1:S)
21.44/21.66	->->-> Rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	
21.44/21.66	Problem 1.2: 
21.44/21.66	
21.44/21.66	Reduction Pair Processor:
21.44/21.66	-> Pairs:
21.44/21.66	 B(a(a(b(x1:S)))) -> B(a(b(a(x1:S))))
21.44/21.66	 B(a(b(a(x1:S)))) -> B(x1:S)
21.44/21.66	-> Rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	-> Usable rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	->Interpretation type:
21.44/21.66	 Linear
21.44/21.66	->Coefficients:
21.44/21.66	 Natural Numbers
21.44/21.66	->Dimension:
21.44/21.66	 1
21.44/21.66	->Bound:
21.44/21.66	 2
21.44/21.66	->Interpretation:
21.44/21.66	 
21.44/21.66	[a](X) = 2.X + 2
21.44/21.66	[b](X) = 2.X + 2
21.44/21.66	[B](X) = 2.X
21.44/21.66	
21.44/21.66	Problem 1.2: 
21.44/21.66	
21.44/21.66	SCC Processor:
21.44/21.66	-> Pairs:
21.44/21.66	 B(a(a(b(x1:S)))) -> B(a(b(a(x1:S))))
21.44/21.66	-> Rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	->Strongly Connected Components:
21.44/21.66	->->Cycle:
21.44/21.66	->->-> Pairs:
21.44/21.66	 B(a(a(b(x1:S)))) -> B(a(b(a(x1:S))))
21.44/21.66	->->-> Rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	
21.44/21.66	Problem 1.2: 
21.44/21.66	
21.44/21.66	Reduction Pair Processor:
21.44/21.66	-> Pairs:
21.44/21.66	 B(a(a(b(x1:S)))) -> B(a(b(a(x1:S))))
21.44/21.66	-> Rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	-> Usable rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	->Interpretation type:
21.44/21.66	 Linear
21.44/21.66	->Coefficients:
21.44/21.66	 Natural Numbers
21.44/21.66	->Dimension:
21.44/21.66	 2
21.44/21.66	->Bound:
21.44/21.66	 1
21.44/21.66	->Interpretation:
21.44/21.66	 
21.44/21.66	[a](X) = [0 1;1 0].X
21.44/21.66	[b](X) = [1 1;1 1].X + [0;1]
21.44/21.66	[B](X) = [0 1;1 1].X
21.44/21.66	
21.44/21.66	Problem 1.2: 
21.44/21.66	
21.44/21.66	SCC Processor:
21.44/21.66	-> Pairs:
21.44/21.66	 Empty
21.44/21.66	-> Rules:
21.44/21.66	 a(b(a(a(x1:S)))) -> a(a(a(a(x1:S))))
21.44/21.66	 b(a(a(b(x1:S)))) -> b(a(b(a(x1:S))))
21.44/21.66	 b(a(b(a(x1:S)))) -> a(a(b(b(x1:S))))
21.44/21.66	->Strongly Connected Components:
21.44/21.66	 There is no strongly connected component
21.44/21.66	
21.44/21.66	The problem is finite.
21.44/67.15	EOF