20.10/20.16	YES
20.10/20.16	
20.10/20.16	Problem 1: 
20.10/20.16	
20.10/20.16	(VAR v_NonEmpty:S X:S Y:S)
20.10/20.16	(RULES
20.10/20.16	f(0,1,X:S) -> f(g(X:S,X:S),X:S,X:S)
20.10/20.16	g(X:S,Y:S) -> X:S
20.10/20.16	g(X:S,Y:S) -> Y:S
20.10/20.16	) 
20.10/20.16	(STRATEGY INNERMOST)
20.10/20.16	
20.10/20.16	Problem 1: 
20.10/20.16	
20.10/20.16	Dependency Pairs Processor:
20.10/20.16	-> Pairs:
20.10/20.16	 F(0,1,X:S) -> F(g(X:S,X:S),X:S,X:S)
20.10/20.16	 F(0,1,X:S) -> G(X:S,X:S)
20.10/20.16	-> Rules:
20.10/20.16	 f(0,1,X:S) -> f(g(X:S,X:S),X:S,X:S)
20.10/20.16	 g(X:S,Y:S) -> X:S
20.10/20.16	 g(X:S,Y:S) -> Y:S
20.10/20.16	
20.10/20.16	Problem 1: 
20.10/20.16	
20.10/20.16	SCC Processor:
20.10/20.16	-> Pairs:
20.10/20.16	 F(0,1,X:S) -> F(g(X:S,X:S),X:S,X:S)
20.10/20.16	 F(0,1,X:S) -> G(X:S,X:S)
20.10/20.16	-> Rules:
20.10/20.16	 f(0,1,X:S) -> f(g(X:S,X:S),X:S,X:S)
20.10/20.16	 g(X:S,Y:S) -> X:S
20.10/20.16	 g(X:S,Y:S) -> Y:S
20.10/20.16	->Strongly Connected Components:
20.10/20.16	->->Cycle:
20.10/20.16	->->-> Pairs:
20.10/20.16	 F(0,1,X:S) -> F(g(X:S,X:S),X:S,X:S)
20.10/20.16	->->-> Rules:
20.10/20.16	 f(0,1,X:S) -> f(g(X:S,X:S),X:S,X:S)
20.10/20.16	 g(X:S,Y:S) -> X:S
20.10/20.16	 g(X:S,Y:S) -> Y:S
20.10/20.16	
20.10/20.16	Problem 1: 
20.10/20.16	
20.10/20.16	Instantiation Processor:
20.10/20.16	-> Pairs:
20.10/20.16	 F(0,1,X:S) -> F(g(X:S,X:S),X:S,X:S)
20.10/20.16	-> Rules:
20.10/20.16	 f(0,1,X:S) -> f(g(X:S,X:S),X:S,X:S)
20.10/20.16	 g(X:S,Y:S) -> X:S
20.10/20.16	 g(X:S,Y:S) -> Y:S
20.10/20.16	->Instantiated Pairs:
20.10/20.16	->->Original Pair:
20.10/20.16	 F(0,1,X:S) -> F(g(X:S,X:S),X:S,X:S)
20.10/20.16	->-> Instantiated pairs:
20.10/20.16	 F(0,1,1) -> F(g(1,1),1,1)
20.10/20.16	
20.10/20.16	Problem 1: 
20.10/20.16	
20.10/20.16	SCC Processor:
20.10/20.16	-> Pairs:
20.10/20.16	 F(0,1,1) -> F(g(1,1),1,1)
20.10/20.16	-> Rules:
20.10/20.16	 f(0,1,X:S) -> f(g(X:S,X:S),X:S,X:S)
20.10/20.16	 g(X:S,Y:S) -> X:S
20.10/20.16	 g(X:S,Y:S) -> Y:S
20.10/20.16	->Strongly Connected Components:
20.10/20.16	->->Cycle:
20.10/20.16	->->-> Pairs:
20.10/20.16	 F(0,1,1) -> F(g(1,1),1,1)
20.10/20.16	->->-> Rules:
20.10/20.16	 f(0,1,X:S) -> f(g(X:S,X:S),X:S,X:S)
20.10/20.16	 g(X:S,Y:S) -> X:S
20.10/20.16	 g(X:S,Y:S) -> Y:S
20.10/20.16	
20.10/20.16	Problem 1: 
20.10/20.16	
20.10/20.16	Reduction Pairs Processor:
20.10/20.16	-> Pairs:
20.10/20.16	 F(0,1,1) -> F(g(1,1),1,1)
20.10/20.16	-> Rules:
20.10/20.16	 f(0,1,X:S) -> f(g(X:S,X:S),X:S,X:S)
20.10/20.16	 g(X:S,Y:S) -> X:S
20.10/20.16	 g(X:S,Y:S) -> Y:S
20.10/20.16	-> Usable rules:
20.10/20.16	 g(X:S,Y:S) -> X:S
20.10/20.16	 g(X:S,Y:S) -> Y:S
20.10/20.16	->Interpretation type:
20.10/20.16	 Linear
20.10/20.16	->Coefficients:
20.10/20.16	 Natural Numbers
20.10/20.16	->Dimension:
20.10/20.16	 1
20.10/20.16	->Bound:
20.10/20.16	 2
20.10/20.16	->Interpretation:
20.10/20.16	 
20.10/20.16	[f](X1,X2,X3) = 0
20.10/20.16	[g](X1,X2) = X1 + 2.X2 + 1
20.10/20.16	[0] = 2
20.10/20.16	[1] = 0
20.10/20.16	[fSNonEmpty] = 0
20.10/20.16	[F](X1,X2,X3) = 2.X1
20.10/20.16	[G](X1,X2) = 0
20.10/20.16	
20.10/20.16	Problem 1: 
20.10/20.16	
20.10/20.16	SCC Processor:
20.10/20.16	-> Pairs:
20.10/20.16	 Empty
20.10/20.16	-> Rules:
20.10/20.16	 f(0,1,X:S) -> f(g(X:S,X:S),X:S,X:S)
20.10/20.16	 g(X:S,Y:S) -> X:S
20.10/20.16	 g(X:S,Y:S) -> Y:S
20.10/20.16	->Strongly Connected Components:
20.10/20.16	 There is no strongly connected component
20.10/20.16	
20.10/20.16	The problem is finite.
20.10/20.16	EOF