0.00/0.19	YES
0.00/0.19	
0.00/0.19	Problem 1: 
0.00/0.19	
0.00/0.19	(VAR v_NonEmpty:S x1:S)
0.00/0.19	(RULES
0.00/0.19	0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	)
0.00/0.19	
0.00/0.19	Problem 1: 
0.00/0.19	
0.00/0.19	Dependency Pairs Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 0#(2(0(x1:S))) -> 0#(1(x1:S))
0.00/0.19	 0#(2(0(x1:S))) -> 1#(0(1(x1:S)))
0.00/0.19	 0#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 0#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
0.00/0.19	 1#(2(0(x1:S))) -> 0#(1(x1:S))
0.00/0.19	 1#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 1#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	 L#(2(0(x1:S))) -> 0#(1(x1:S))
0.00/0.19	 L#(2(0(x1:S))) -> 1#(0(1(x1:S)))
0.00/0.19	 L#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 L#(2(0(x1:S))) -> L#(1(0(1(x1:S))))
0.00/0.19	 L#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	 L#(2(1(x1:S))) -> 1#(0(2(x1:S)))
0.00/0.19	 L#(2(1(x1:S))) -> L#(1(0(2(x1:S))))
0.00/0.19	-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	
0.00/0.19	Problem 1: 
0.00/0.19	
0.00/0.19	SCC Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 0#(2(0(x1:S))) -> 0#(1(x1:S))
0.00/0.19	 0#(2(0(x1:S))) -> 1#(0(1(x1:S)))
0.00/0.19	 0#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 0#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
0.00/0.19	 1#(2(0(x1:S))) -> 0#(1(x1:S))
0.00/0.19	 1#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 1#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	 L#(2(0(x1:S))) -> 0#(1(x1:S))
0.00/0.19	 L#(2(0(x1:S))) -> 1#(0(1(x1:S)))
0.00/0.19	 L#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 L#(2(0(x1:S))) -> L#(1(0(1(x1:S))))
0.00/0.19	 L#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	 L#(2(1(x1:S))) -> 1#(0(2(x1:S)))
0.00/0.19	 L#(2(1(x1:S))) -> L#(1(0(2(x1:S))))
0.00/0.19	-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	->Strongly Connected Components:
0.00/0.19	->->Cycle:
0.00/0.19	->->-> Pairs:
0.00/0.19	 0#(2(0(x1:S))) -> 0#(1(x1:S))
0.00/0.19	 0#(2(0(x1:S))) -> 1#(0(1(x1:S)))
0.00/0.19	 0#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 0#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
0.00/0.19	 1#(2(0(x1:S))) -> 0#(1(x1:S))
0.00/0.19	 1#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 1#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	->->-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	->->Cycle:
0.00/0.19	->->-> Pairs:
0.00/0.19	 L#(2(0(x1:S))) -> L#(1(0(1(x1:S))))
0.00/0.19	 L#(2(1(x1:S))) -> L#(1(0(2(x1:S))))
0.00/0.19	->->-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	
0.00/0.19	
0.00/0.19	The problem is decomposed in 2 subproblems.
0.00/0.19	
0.00/0.19	Problem 1.1: 
0.00/0.19	
0.00/0.19	Reduction Pair Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 0#(2(0(x1:S))) -> 0#(1(x1:S))
0.00/0.19	 0#(2(0(x1:S))) -> 1#(0(1(x1:S)))
0.00/0.19	 0#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 0#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
0.00/0.19	 1#(2(0(x1:S))) -> 0#(1(x1:S))
0.00/0.19	 1#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 1#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	-> Usable rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	->Interpretation type:
0.00/0.19	 Linear
0.00/0.19	->Coefficients:
0.00/0.19	 Natural Numbers
0.00/0.19	->Dimension:
0.00/0.19	 1
0.00/0.19	->Bound:
0.00/0.19	 2
0.00/0.19	->Interpretation:
0.00/0.19	 
0.00/0.19	[0](X) = X
0.00/0.19	[1](X) = X + 1
0.00/0.19	[2](X) = X + 2
0.00/0.19	[R](X) = 2.X + 1
0.00/0.19	[0#](X) = 2.X + 2
0.00/0.19	[1#](X) = 2.X + 2
0.00/0.19	
0.00/0.19	Problem 1.1: 
0.00/0.19	
0.00/0.19	SCC Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 0#(2(0(x1:S))) -> 1#(0(1(x1:S)))
0.00/0.19	 0#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 0#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
0.00/0.19	 1#(2(0(x1:S))) -> 0#(1(x1:S))
0.00/0.19	 1#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 1#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	->Strongly Connected Components:
0.00/0.19	->->Cycle:
0.00/0.19	->->-> Pairs:
0.00/0.19	 0#(2(0(x1:S))) -> 1#(0(1(x1:S)))
0.00/0.19	 0#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 0#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
0.00/0.19	 1#(2(0(x1:S))) -> 0#(1(x1:S))
0.00/0.19	 1#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 1#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	->->-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	
0.00/0.19	Problem 1.1: 
0.00/0.19	
0.00/0.19	Reduction Pair Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 0#(2(0(x1:S))) -> 1#(0(1(x1:S)))
0.00/0.19	 0#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 0#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
0.00/0.19	 1#(2(0(x1:S))) -> 0#(1(x1:S))
0.00/0.19	 1#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 1#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	-> Usable rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	->Interpretation type:
0.00/0.19	 Linear
0.00/0.19	->Coefficients:
0.00/0.19	 Natural Numbers
0.00/0.19	->Dimension:
0.00/0.19	 1
0.00/0.19	->Bound:
0.00/0.19	 2
0.00/0.19	->Interpretation:
0.00/0.19	 
0.00/0.19	[0](X) = X
0.00/0.19	[1](X) = X + 1
0.00/0.19	[2](X) = X + 2
0.00/0.19	[R](X) = 2.X + 1
0.00/0.19	[0#](X) = 2.X + 2
0.00/0.19	[1#](X) = 2.X + 2
0.00/0.19	
0.00/0.19	Problem 1.1: 
0.00/0.19	
0.00/0.19	SCC Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 0#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 0#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
0.00/0.19	 1#(2(0(x1:S))) -> 0#(1(x1:S))
0.00/0.19	 1#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 1#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	->Strongly Connected Components:
0.00/0.19	->->Cycle:
0.00/0.19	->->-> Pairs:
0.00/0.19	 0#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 0#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
0.00/0.19	 1#(2(0(x1:S))) -> 0#(1(x1:S))
0.00/0.19	 1#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 1#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	->->-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	
0.00/0.19	Problem 1.1: 
0.00/0.19	
0.00/0.19	Reduction Pair Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 0#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 0#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
0.00/0.19	 1#(2(0(x1:S))) -> 0#(1(x1:S))
0.00/0.19	 1#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 1#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	-> Usable rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	->Interpretation type:
0.00/0.19	 Linear
0.00/0.19	->Coefficients:
0.00/0.19	 Natural Numbers
0.00/0.19	->Dimension:
0.00/0.19	 1
0.00/0.19	->Bound:
0.00/0.19	 2
0.00/0.19	->Interpretation:
0.00/0.19	 
0.00/0.19	[0](X) = X
0.00/0.19	[1](X) = X + 1
0.00/0.19	[2](X) = X + 2
0.00/0.19	[R](X) = 2.X + 1
0.00/0.19	[0#](X) = X
0.00/0.19	[1#](X) = X
0.00/0.19	
0.00/0.19	Problem 1.1: 
0.00/0.19	
0.00/0.19	SCC Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 0#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
0.00/0.19	 1#(2(0(x1:S))) -> 0#(1(x1:S))
0.00/0.19	 1#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 1#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	->Strongly Connected Components:
0.00/0.19	->->Cycle:
0.00/0.19	->->-> Pairs:
0.00/0.19	 0#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
0.00/0.19	 1#(2(0(x1:S))) -> 0#(1(x1:S))
0.00/0.19	 1#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 1#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	->->-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	
0.00/0.19	Problem 1.1: 
0.00/0.19	
0.00/0.19	Reduction Pair Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 0#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
0.00/0.19	 1#(2(0(x1:S))) -> 0#(1(x1:S))
0.00/0.19	 1#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 1#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	-> Usable rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	->Interpretation type:
0.00/0.19	 Linear
0.00/0.19	->Coefficients:
0.00/0.19	 Natural Numbers
0.00/0.19	->Dimension:
0.00/0.19	 1
0.00/0.19	->Bound:
0.00/0.19	 2
0.00/0.19	->Interpretation:
0.00/0.19	 
0.00/0.19	[0](X) = X
0.00/0.19	[1](X) = X + 1
0.00/0.19	[2](X) = X + 2
0.00/0.19	[R](X) = 2.X + 2
0.00/0.19	[0#](X) = 2.X + 2
0.00/0.19	[1#](X) = 2.X + 1
0.00/0.19	
0.00/0.19	Problem 1.1: 
0.00/0.19	
0.00/0.19	SCC Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
0.00/0.19	 1#(2(0(x1:S))) -> 0#(1(x1:S))
0.00/0.19	 1#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 1#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	->Strongly Connected Components:
0.00/0.19	->->Cycle:
0.00/0.19	->->-> Pairs:
0.00/0.19	 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
0.00/0.19	 1#(2(0(x1:S))) -> 0#(1(x1:S))
0.00/0.19	 1#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 1#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	->->-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	
0.00/0.19	Problem 1.1: 
0.00/0.19	
0.00/0.19	Reduction Pair Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 0#(2(1(x1:S))) -> 1#(0(2(x1:S)))
0.00/0.19	 1#(2(0(x1:S))) -> 0#(1(x1:S))
0.00/0.19	 1#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 1#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	-> Usable rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	->Interpretation type:
0.00/0.19	 Linear
0.00/0.19	->Coefficients:
0.00/0.19	 Natural Numbers
0.00/0.19	->Dimension:
0.00/0.19	 1
0.00/0.19	->Bound:
0.00/0.19	 2
0.00/0.19	->Interpretation:
0.00/0.19	 
0.00/0.19	[0](X) = X
0.00/0.19	[1](X) = X + 1
0.00/0.19	[2](X) = X + 2
0.00/0.19	[R](X) = 2.X + 1
0.00/0.19	[0#](X) = 2.X + 2
0.00/0.19	[1#](X) = 2.X + 1
0.00/0.19	
0.00/0.19	Problem 1.1: 
0.00/0.19	
0.00/0.19	SCC Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 1#(2(0(x1:S))) -> 0#(1(x1:S))
0.00/0.19	 1#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	 1#(2(1(x1:S))) -> 0#(2(x1:S))
0.00/0.19	-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	->Strongly Connected Components:
0.00/0.19	->->Cycle:
0.00/0.19	->->-> Pairs:
0.00/0.19	 1#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	->->-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	
0.00/0.19	Problem 1.1: 
0.00/0.19	
0.00/0.19	Subterm Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 1#(2(0(x1:S))) -> 1#(x1:S)
0.00/0.19	-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	->Projection:
0.00/0.19	 pi(1#) = 1
0.00/0.19	
0.00/0.19	Problem 1.1: 
0.00/0.19	
0.00/0.19	SCC Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 Empty
0.00/0.19	-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	->Strongly Connected Components:
0.00/0.19	 There is no strongly connected component
0.00/0.19	
0.00/0.19	The problem is finite.
0.00/0.19	
0.00/0.19	Problem 1.2: 
0.00/0.19	
0.00/0.19	Reduction Pair Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 L#(2(0(x1:S))) -> L#(1(0(1(x1:S))))
0.00/0.19	 L#(2(1(x1:S))) -> L#(1(0(2(x1:S))))
0.00/0.19	-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	-> Usable rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	->Interpretation type:
0.00/0.19	 Linear
0.00/0.19	->Coefficients:
0.00/0.19	 Natural Numbers
0.00/0.19	->Dimension:
0.00/0.19	 1
0.00/0.19	->Bound:
0.00/0.19	 2
0.00/0.19	->Interpretation:
0.00/0.19	 
0.00/0.19	[0](X) = 0
0.00/0.19	[1](X) = 2.X
0.00/0.19	[2](X) = 2.X + 2
0.00/0.19	[R](X) = 2
0.00/0.19	[L#](X) = 2.X
0.00/0.19	
0.00/0.19	Problem 1.2: 
0.00/0.19	
0.00/0.19	SCC Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 L#(2(1(x1:S))) -> L#(1(0(2(x1:S))))
0.00/0.19	-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	->Strongly Connected Components:
0.00/0.19	->->Cycle:
0.00/0.19	->->-> Pairs:
0.00/0.19	 L#(2(1(x1:S))) -> L#(1(0(2(x1:S))))
0.00/0.19	->->-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	
0.00/0.19	Problem 1.2: 
0.00/0.19	
0.00/0.19	Reduction Pair Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 L#(2(1(x1:S))) -> L#(1(0(2(x1:S))))
0.00/0.19	-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	-> Usable rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	->Interpretation type:
0.00/0.19	 Linear
0.00/0.19	->Coefficients:
0.00/0.19	 Natural Numbers
0.00/0.19	->Dimension:
0.00/0.19	 1
0.00/0.19	->Bound:
0.00/0.19	 2
0.00/0.19	->Interpretation:
0.00/0.19	 
0.00/0.19	[0](X) = 1
0.00/0.19	[1](X) = X
0.00/0.19	[2](X) = 2
0.00/0.19	[R](X) = 0
0.00/0.19	[L#](X) = 2.X
0.00/0.19	
0.00/0.19	Problem 1.2: 
0.00/0.19	
0.00/0.19	SCC Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 Empty
0.00/0.19	-> Rules:
0.00/0.19	 0(2(0(x1:S))) -> 1(0(1(x1:S)))
0.00/0.19	 0(2(1(x1:S))) -> 1(0(2(x1:S)))
0.00/0.19	 0(2(R(x1:S))) -> 1(0(1(R(x1:S))))
0.00/0.19	 1(2(0(x1:S))) -> 2(0(1(x1:S)))
0.00/0.19	 1(2(1(x1:S))) -> 2(0(2(x1:S)))
0.00/0.19	 1(2(R(x1:S))) -> 2(0(1(R(x1:S))))
0.00/0.19	 L(2(0(x1:S))) -> L(1(0(1(x1:S))))
0.00/0.19	 L(2(1(x1:S))) -> L(1(0(2(x1:S))))
0.00/0.19	->Strongly Connected Components:
0.00/0.19	 There is no strongly connected component
0.00/0.19	
0.00/0.19	The problem is finite.
0.00/0.19	EOF