44.49/45.67	YES
44.49/45.67	
44.49/45.67	Problem 1: 
44.49/45.67	
44.49/45.67	(VAR v_NonEmpty:S x1:S)
44.49/45.67	(RULES
44.49/45.67	2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	8(4(x1:S)) -> 6(x1:S)
44.49/45.67	9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	)
44.49/45.67	
44.49/45.67	Problem 1: 
44.49/45.67	
44.49/45.67	Dependency Pairs Processor:
44.49/45.67	-> Pairs:
44.49/45.67	 2#(6(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(5(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(5(x1:S)) -> 8#(9(7(x1:S)))
44.49/45.67	 3#(5(x1:S)) -> 9#(7(x1:S))
44.49/45.67	 3#(8(x1:S)) -> 2#(7(x1:S))
44.49/45.67	 3#(8(x1:S)) -> 3#(2(7(x1:S)))
44.49/45.67	 3#(8(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 5#(9(x1:S)) -> 2#(6(5(x1:S)))
44.49/45.67	 5#(9(x1:S)) -> 5#(x1:S)
44.49/45.67	 7#(1(x1:S)) -> 9#(x1:S)
44.49/45.67	 8#(8(4(x1:S))) -> 9#(x1:S)
44.49/45.67	 9#(x1:S) -> 2#(3(x1:S))
44.49/45.67	 9#(x1:S) -> 2#(x1:S)
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	 9#(x1:S) -> 5#(0(2(x1:S)))
44.49/45.67	-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	
44.49/45.67	Problem 1: 
44.49/45.67	
44.49/45.67	SCC Processor:
44.49/45.67	-> Pairs:
44.49/45.67	 2#(6(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(5(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(5(x1:S)) -> 8#(9(7(x1:S)))
44.49/45.67	 3#(5(x1:S)) -> 9#(7(x1:S))
44.49/45.67	 3#(8(x1:S)) -> 2#(7(x1:S))
44.49/45.67	 3#(8(x1:S)) -> 3#(2(7(x1:S)))
44.49/45.67	 3#(8(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 5#(9(x1:S)) -> 2#(6(5(x1:S)))
44.49/45.67	 5#(9(x1:S)) -> 5#(x1:S)
44.49/45.67	 7#(1(x1:S)) -> 9#(x1:S)
44.49/45.67	 8#(8(4(x1:S))) -> 9#(x1:S)
44.49/45.67	 9#(x1:S) -> 2#(3(x1:S))
44.49/45.67	 9#(x1:S) -> 2#(x1:S)
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	 9#(x1:S) -> 5#(0(2(x1:S)))
44.49/45.67	-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	->Strongly Connected Components:
44.49/45.67	->->Cycle:
44.49/45.67	->->-> Pairs:
44.49/45.67	 2#(6(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(5(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(5(x1:S)) -> 8#(9(7(x1:S)))
44.49/45.67	 3#(5(x1:S)) -> 9#(7(x1:S))
44.49/45.67	 3#(8(x1:S)) -> 2#(7(x1:S))
44.49/45.67	 3#(8(x1:S)) -> 3#(2(7(x1:S)))
44.49/45.67	 3#(8(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 7#(1(x1:S)) -> 9#(x1:S)
44.49/45.67	 8#(8(4(x1:S))) -> 9#(x1:S)
44.49/45.67	 9#(x1:S) -> 2#(3(x1:S))
44.49/45.67	 9#(x1:S) -> 2#(x1:S)
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	->->-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	->->Cycle:
44.49/45.67	->->-> Pairs:
44.49/45.67	 5#(9(x1:S)) -> 5#(x1:S)
44.49/45.67	->->-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	
44.49/45.67	
44.49/45.67	The problem is decomposed in 2 subproblems.
44.49/45.67	
44.49/45.67	Problem 1.1: 
44.49/45.67	
44.49/45.67	Reduction Pair Processor:
44.49/45.67	-> Pairs:
44.49/45.67	 2#(6(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(5(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(5(x1:S)) -> 8#(9(7(x1:S)))
44.49/45.67	 3#(5(x1:S)) -> 9#(7(x1:S))
44.49/45.67	 3#(8(x1:S)) -> 2#(7(x1:S))
44.49/45.67	 3#(8(x1:S)) -> 3#(2(7(x1:S)))
44.49/45.67	 3#(8(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 7#(1(x1:S)) -> 9#(x1:S)
44.49/45.67	 8#(8(4(x1:S))) -> 9#(x1:S)
44.49/45.67	 9#(x1:S) -> 2#(3(x1:S))
44.49/45.67	 9#(x1:S) -> 2#(x1:S)
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	-> Usable rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	->Interpretation type:
44.49/45.67	 Linear
44.49/45.67	->Coefficients:
44.49/45.67	 Natural Numbers
44.49/45.67	->Dimension:
44.49/45.67	 1
44.49/45.67	->Bound:
44.49/45.67	 2
44.49/45.67	->Interpretation:
44.49/45.67	 
44.49/45.67	[2](X) = 0
44.49/45.67	[3](X) = 2.X + 1
44.49/45.67	[5](X) = 2.X + 1
44.49/45.67	[7](X) = 0
44.49/45.67	[8](X) = 2
44.49/45.67	[9](X) = 2.X + 1
44.49/45.67	[0](X) = 2.X
44.49/45.67	[1](X) = 0
44.49/45.67	[4](X) = 0
44.49/45.67	[6](X) = 0
44.49/45.67	[2#](X) = 2
44.49/45.67	[3#](X) = 2
44.49/45.67	[7#](X) = 2
44.49/45.67	[8#](X) = X
44.49/45.67	[9#](X) = 2
44.49/45.67	
44.49/45.67	Problem 1.1: 
44.49/45.67	
44.49/45.67	SCC Processor:
44.49/45.67	-> Pairs:
44.49/45.67	 2#(6(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(5(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(5(x1:S)) -> 9#(7(x1:S))
44.49/45.67	 3#(8(x1:S)) -> 2#(7(x1:S))
44.49/45.67	 3#(8(x1:S)) -> 3#(2(7(x1:S)))
44.49/45.67	 3#(8(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 7#(1(x1:S)) -> 9#(x1:S)
44.49/45.67	 8#(8(4(x1:S))) -> 9#(x1:S)
44.49/45.67	 9#(x1:S) -> 2#(3(x1:S))
44.49/45.67	 9#(x1:S) -> 2#(x1:S)
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	->Strongly Connected Components:
44.49/45.67	->->Cycle:
44.49/45.67	->->-> Pairs:
44.49/45.67	 2#(6(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(5(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(5(x1:S)) -> 9#(7(x1:S))
44.49/45.67	 3#(8(x1:S)) -> 2#(7(x1:S))
44.49/45.67	 3#(8(x1:S)) -> 3#(2(7(x1:S)))
44.49/45.67	 3#(8(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 7#(1(x1:S)) -> 9#(x1:S)
44.49/45.67	 9#(x1:S) -> 2#(3(x1:S))
44.49/45.67	 9#(x1:S) -> 2#(x1:S)
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	->->-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	
44.49/45.67	Problem 1.1: 
44.49/45.67	
44.49/45.67	Reduction Pair Processor:
44.49/45.67	-> Pairs:
44.49/45.67	 2#(6(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(5(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(5(x1:S)) -> 9#(7(x1:S))
44.49/45.67	 3#(8(x1:S)) -> 2#(7(x1:S))
44.49/45.67	 3#(8(x1:S)) -> 3#(2(7(x1:S)))
44.49/45.67	 3#(8(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 7#(1(x1:S)) -> 9#(x1:S)
44.49/45.67	 9#(x1:S) -> 2#(3(x1:S))
44.49/45.67	 9#(x1:S) -> 2#(x1:S)
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	-> Usable rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	->Interpretation type:
44.49/45.67	 Linear
44.49/45.67	->Coefficients:
44.49/45.67	 Natural Numbers
44.49/45.67	->Dimension:
44.49/45.67	 2
44.49/45.67	->Bound:
44.49/45.67	 1
44.49/45.67	->Interpretation:
44.49/45.67	 
44.49/45.67	[2](X) = [0 0;0 1].X
44.49/45.67	[3](X) = [1 0;1 0].X
44.49/45.67	[5](X) = [1 1;1 1].X + [1;0]
44.49/45.67	[7](X) = [0 1;1 1].X + [1;0]
44.49/45.67	[8](X) = [0 1;0 1].X + [1;1]
44.49/45.67	[9](X) = [1 0;0 1].X + [1;0]
44.49/45.67	[0](X) = [1 0;0 0].X
44.49/45.67	[1](X) = [0 0;1 0].X + [1;1]
44.49/45.67	[4](X) = [0 0;1 0].X
44.49/45.67	[6](X) = [1 0;1 0].X + [1;0]
44.49/45.67	[2#](X) = [1 0;1 0].X
44.49/45.67	[3#](X) = [1 0;1 0].X + [0;1]
44.49/45.67	[7#](X) = [0 1;0 1].X + [1;1]
44.49/45.67	[9#](X) = [1 0;1 0].X + [0;1]
44.49/45.67	
44.49/45.67	Problem 1.1: 
44.49/45.67	
44.49/45.67	SCC Processor:
44.49/45.67	-> Pairs:
44.49/45.67	 3#(5(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(5(x1:S)) -> 9#(7(x1:S))
44.49/45.67	 3#(8(x1:S)) -> 2#(7(x1:S))
44.49/45.67	 3#(8(x1:S)) -> 3#(2(7(x1:S)))
44.49/45.67	 3#(8(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 7#(1(x1:S)) -> 9#(x1:S)
44.49/45.67	 9#(x1:S) -> 2#(3(x1:S))
44.49/45.67	 9#(x1:S) -> 2#(x1:S)
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	->Strongly Connected Components:
44.49/45.67	->->Cycle:
44.49/45.67	->->-> Pairs:
44.49/45.67	 3#(5(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(5(x1:S)) -> 9#(7(x1:S))
44.49/45.67	 3#(8(x1:S)) -> 3#(2(7(x1:S)))
44.49/45.67	 3#(8(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 7#(1(x1:S)) -> 9#(x1:S)
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	->->-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	
44.49/45.67	Problem 1.1: 
44.49/45.67	
44.49/45.67	Reduction Pair Processor:
44.49/45.67	-> Pairs:
44.49/45.67	 3#(5(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(5(x1:S)) -> 9#(7(x1:S))
44.49/45.67	 3#(8(x1:S)) -> 3#(2(7(x1:S)))
44.49/45.67	 3#(8(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 7#(1(x1:S)) -> 9#(x1:S)
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	-> Usable rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	->Interpretation type:
44.49/45.67	 Linear
44.49/45.67	->Coefficients:
44.49/45.67	 All rationals
44.49/45.67	->Dimension:
44.49/45.67	 1
44.49/45.67	->Bound:
44.49/45.67	 4
44.49/45.67	->Interpretation:
44.49/45.67	 
44.49/45.67	[2](X) = 3/4.X
44.49/45.67	[3](X) = X + 2/3
44.49/45.67	[5](X) = 3.X + 4/3
44.49/45.67	[7](X) = 2.X
44.49/45.67	[8](X) = 3/2.X
44.49/45.67	[9](X) = X + 4/3
44.49/45.67	[0](X) = 0
44.49/45.67	[1](X) = 3/4.X + 2
44.49/45.67	[4](X) = 1/3.X + 2
44.49/45.67	[6](X) = 1/2.X + 3
44.49/45.67	[3#](X) = X
44.49/45.67	[7#](X) = 4/3.X
44.49/45.67	[9#](X) = X + 2/3
44.49/45.67	
44.49/45.67	Problem 1.1: 
44.49/45.67	
44.49/45.67	SCC Processor:
44.49/45.67	-> Pairs:
44.49/45.67	 3#(5(x1:S)) -> 9#(7(x1:S))
44.49/45.67	 3#(8(x1:S)) -> 3#(2(7(x1:S)))
44.49/45.67	 3#(8(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 7#(1(x1:S)) -> 9#(x1:S)
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	->Strongly Connected Components:
44.49/45.67	->->Cycle:
44.49/45.67	->->-> Pairs:
44.49/45.67	 3#(5(x1:S)) -> 9#(7(x1:S))
44.49/45.67	 3#(8(x1:S)) -> 3#(2(7(x1:S)))
44.49/45.67	 3#(8(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 7#(1(x1:S)) -> 9#(x1:S)
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	->->-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	
44.49/45.67	Problem 1.1: 
44.49/45.67	
44.49/45.67	Reduction Pair Processor:
44.49/45.67	-> Pairs:
44.49/45.67	 3#(5(x1:S)) -> 9#(7(x1:S))
44.49/45.67	 3#(8(x1:S)) -> 3#(2(7(x1:S)))
44.49/45.67	 3#(8(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 7#(1(x1:S)) -> 9#(x1:S)
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	-> Usable rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	->Interpretation type:
44.49/45.67	 Linear
44.49/45.67	->Coefficients:
44.49/45.67	 All rationals
44.49/45.67	->Dimension:
44.49/45.67	 1
44.49/45.67	->Bound:
44.49/45.67	 4
44.49/45.67	->Interpretation:
44.49/45.67	 
44.49/45.67	[2](X) = X
44.49/45.67	[3](X) = X + 2
44.49/45.67	[5](X) = 2.X + 4
44.49/45.67	[7](X) = 4/3.X
44.49/45.67	[8](X) = 3/2.X
44.49/45.67	[9](X) = X + 4
44.49/45.67	[0](X) = 1/4.X
44.49/45.67	[1](X) = 2/3.X + 4
44.49/45.67	[4](X) = 1/3.X + 3
44.49/45.67	[6](X) = 1/3.X + 4
44.49/45.67	[3#](X) = 1/2.X
44.49/45.67	[7#](X) = 3/4.X
44.49/45.67	[9#](X) = 1/2.X + 1
44.49/45.67	
44.49/45.67	Problem 1.1: 
44.49/45.67	
44.49/45.67	SCC Processor:
44.49/45.67	-> Pairs:
44.49/45.67	 3#(8(x1:S)) -> 3#(2(7(x1:S)))
44.49/45.67	 3#(8(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 7#(1(x1:S)) -> 9#(x1:S)
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	->Strongly Connected Components:
44.49/45.67	->->Cycle:
44.49/45.67	->->-> Pairs:
44.49/45.67	 3#(8(x1:S)) -> 3#(2(7(x1:S)))
44.49/45.67	 3#(8(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 7#(1(x1:S)) -> 9#(x1:S)
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	->->-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	
44.49/45.67	Problem 1.1: 
44.49/45.67	
44.49/45.67	Reduction Pair Processor:
44.49/45.67	-> Pairs:
44.49/45.67	 3#(8(x1:S)) -> 3#(2(7(x1:S)))
44.49/45.67	 3#(8(x1:S)) -> 7#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 7#(1(x1:S)) -> 9#(x1:S)
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	-> Usable rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	->Interpretation type:
44.49/45.67	 Linear
44.49/45.67	->Coefficients:
44.49/45.67	 All rationals
44.49/45.67	->Dimension:
44.49/45.67	 1
44.49/45.67	->Bound:
44.49/45.67	 4
44.49/45.67	->Interpretation:
44.49/45.67	 
44.49/45.67	[2](X) = 3/4.X
44.49/45.67	[3](X) = X + 1/2
44.49/45.67	[5](X) = 3.X + 1
44.49/45.67	[7](X) = 2.X
44.49/45.67	[8](X) = 3/2.X
44.49/45.67	[9](X) = X + 1
44.49/45.67	[0](X) = 1/3.X
44.49/45.67	[1](X) = 3/4.X + 2
44.49/45.67	[4](X) = 1/3.X + 4/3
44.49/45.67	[6](X) = 1/2.X + 2
44.49/45.67	[3#](X) = 3/4.X + 1/3
44.49/45.67	[7#](X) = X + 1/4
44.49/45.67	[9#](X) = 3/4.X + 2/3
44.49/45.67	
44.49/45.67	Problem 1.1: 
44.49/45.67	
44.49/45.67	SCC Processor:
44.49/45.67	-> Pairs:
44.49/45.67	 3#(8(x1:S)) -> 3#(2(7(x1:S)))
44.49/45.67	 3#(9(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 7#(1(x1:S)) -> 9#(x1:S)
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	->Strongly Connected Components:
44.49/45.67	->->Cycle:
44.49/45.67	->->-> Pairs:
44.49/45.67	 3#(8(x1:S)) -> 3#(2(7(x1:S)))
44.49/45.67	 3#(9(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	->->-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	
44.49/45.67	Problem 1.1: 
44.49/45.67	
44.49/45.67	Reduction Pair Processor:
44.49/45.67	-> Pairs:
44.49/45.67	 3#(8(x1:S)) -> 3#(2(7(x1:S)))
44.49/45.67	 3#(9(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	-> Usable rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	->Interpretation type:
44.49/45.67	 Linear
44.49/45.67	->Coefficients:
44.49/45.67	 Natural Numbers
44.49/45.67	->Dimension:
44.49/45.67	 1
44.49/45.67	->Bound:
44.49/45.67	 2
44.49/45.67	->Interpretation:
44.49/45.67	 
44.49/45.67	[2](X) = 0
44.49/45.67	[3](X) = 2.X + 1
44.49/45.67	[5](X) = X + 1
44.49/45.67	[7](X) = 2.X
44.49/45.67	[8](X) = 2
44.49/45.67	[9](X) = 2.X + 2
44.49/45.67	[0](X) = 2.X + 1
44.49/45.67	[1](X) = 2
44.49/45.67	[4](X) = 0
44.49/45.67	[6](X) = 2
44.49/45.67	[3#](X) = 2.X
44.49/45.67	[9#](X) = 2.X + 2
44.49/45.67	
44.49/45.67	Problem 1.1: 
44.49/45.67	
44.49/45.67	SCC Processor:
44.49/45.67	-> Pairs:
44.49/45.67	 3#(9(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	->Strongly Connected Components:
44.49/45.67	->->Cycle:
44.49/45.67	->->-> Pairs:
44.49/45.67	 3#(9(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	->->-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	
44.49/45.67	Problem 1.1: 
44.49/45.67	
44.49/45.67	Reduction Pair Processor:
44.49/45.67	-> Pairs:
44.49/45.67	 3#(9(x1:S)) -> 3#(x1:S)
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	-> Usable rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	->Interpretation type:
44.49/45.67	 Linear
44.49/45.67	->Coefficients:
44.49/45.67	 Natural Numbers
44.49/45.67	->Dimension:
44.49/45.67	 1
44.49/45.67	->Bound:
44.49/45.67	 2
44.49/45.67	->Interpretation:
44.49/45.67	 
44.49/45.67	[2](X) = 0
44.49/45.67	[3](X) = 2.X + 1
44.49/45.67	[5](X) = X + 1
44.49/45.67	[7](X) = 0
44.49/45.67	[8](X) = 0
44.49/45.67	[9](X) = 2.X + 2
44.49/45.67	[0](X) = 2.X + 1
44.49/45.67	[1](X) = 0
44.49/45.67	[4](X) = 0
44.49/45.67	[6](X) = 0
44.49/45.67	[3#](X) = 2.X + 2
44.49/45.67	[9#](X) = 2.X + 2
44.49/45.67	
44.49/45.67	Problem 1.1: 
44.49/45.67	
44.49/45.67	SCC Processor:
44.49/45.67	-> Pairs:
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	->Strongly Connected Components:
44.49/45.67	->->Cycle:
44.49/45.67	->->-> Pairs:
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	->->-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	
44.49/45.67	Problem 1.1: 
44.49/45.67	
44.49/45.67	Reduction Pair Processor:
44.49/45.67	-> Pairs:
44.49/45.67	 3#(9(x1:S)) -> 9#(3(x1:S))
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	-> Usable rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	->Interpretation type:
44.49/45.67	 Linear
44.49/45.67	->Coefficients:
44.49/45.67	 Natural Numbers
44.49/45.67	->Dimension:
44.49/45.67	 1
44.49/45.67	->Bound:
44.49/45.67	 2
44.49/45.67	->Interpretation:
44.49/45.67	 
44.49/45.67	[2](X) = 0
44.49/45.67	[3](X) = 2.X + 1
44.49/45.67	[5](X) = 2.X + 2
44.49/45.67	[7](X) = X
44.49/45.67	[8](X) = 2.X + 1
44.49/45.67	[9](X) = 2.X + 2
44.49/45.67	[0](X) = 2.X
44.49/45.67	[1](X) = 1
44.49/45.67	[4](X) = 0
44.49/45.67	[6](X) = 1
44.49/45.67	[3#](X) = 2.X + 2
44.49/45.67	[9#](X) = 2.X + 2
44.49/45.67	
44.49/45.67	Problem 1.1: 
44.49/45.67	
44.49/45.67	SCC Processor:
44.49/45.67	-> Pairs:
44.49/45.67	 9#(x1:S) -> 3#(2(3(x1:S)))
44.49/45.67	 9#(x1:S) -> 3#(x1:S)
44.49/45.67	-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	->Strongly Connected Components:
44.49/45.67	 There is no strongly connected component
44.49/45.67	
44.49/45.67	The problem is finite.
44.49/45.67	
44.49/45.67	Problem 1.2: 
44.49/45.67	
44.49/45.67	Subterm Processor:
44.49/45.67	-> Pairs:
44.49/45.67	 5#(9(x1:S)) -> 5#(x1:S)
44.49/45.67	-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	->Projection:
44.49/45.67	 pi(5#) = 1
44.49/45.67	
44.49/45.67	Problem 1.2: 
44.49/45.67	
44.49/45.67	SCC Processor:
44.49/45.67	-> Pairs:
44.49/45.67	 Empty
44.49/45.67	-> Rules:
44.49/45.67	 2(6(x1:S)) -> 4(3(x1:S))
44.49/45.67	 3(5(x1:S)) -> 8(9(7(x1:S)))
44.49/45.67	 3(8(x1:S)) -> 3(2(7(x1:S)))
44.49/45.67	 3(9(x1:S)) -> 9(3(x1:S))
44.49/45.67	 3(1(x1:S)) -> 4(1(x1:S))
44.49/45.67	 5(9(x1:S)) -> 2(6(5(x1:S)))
44.49/45.67	 7(5(x1:S)) -> 1(0(x1:S))
44.49/45.67	 7(1(x1:S)) -> 6(9(x1:S))
44.49/45.67	 8(8(4(x1:S))) -> 1(9(x1:S))
44.49/45.67	 8(4(x1:S)) -> 6(x1:S)
44.49/45.67	 9(x1:S) -> 3(2(3(x1:S)))
44.49/45.67	 9(x1:S) -> 5(0(2(x1:S)))
44.49/45.67	->Strongly Connected Components:
44.49/45.67	 There is no strongly connected component
44.49/45.67	
44.49/45.67	The problem is finite.
44.49/45.67	EOF