1.23/1.40	YES
1.23/1.40	
1.23/1.40	Problem 1: 
1.23/1.40	
1.23/1.40	(VAR v_NonEmpty:S X:S X1:S X2:S X3:S Y:S)
1.23/1.40	(RULES
1.23/1.40	a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.40	a__f(X:S) -> f(X:S)
1.23/1.40	a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.40	a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.40	a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.40	mark(c) -> c
1.23/1.40	mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.40	mark(ffalse) -> ffalse
1.23/1.40	mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.40	mark(ttrue) -> ttrue
1.23/1.40	) 
1.23/1.40	(STRATEGY INNERMOST)
1.23/1.40	
1.23/1.40	Problem 1: 
1.23/1.40	
1.23/1.40	Dependency Pairs Processor:
1.23/1.40	-> Pairs:
1.23/1.40	 A__F(X:S) -> A__IF(mark(X:S),c,f(ttrue))
1.23/1.40	 A__F(X:S) -> MARK(X:S)
1.23/1.40	 A__IF(ffalse,X:S,Y:S) -> MARK(Y:S)
1.23/1.40	 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
1.23/1.40	 MARK(f(X:S)) -> A__F(mark(X:S))
1.23/1.40	 MARK(f(X:S)) -> MARK(X:S)
1.23/1.40	 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S)
1.23/1.40	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.23/1.40	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S)
1.23/1.40	-> Rules:
1.23/1.40	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.40	 a__f(X:S) -> f(X:S)
1.23/1.40	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.40	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.40	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.40	 mark(c) -> c
1.23/1.40	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.40	 mark(ffalse) -> ffalse
1.23/1.40	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.40	 mark(ttrue) -> ttrue
1.23/1.40	
1.23/1.40	Problem 1: 
1.23/1.40	
1.23/1.40	SCC Processor:
1.23/1.40	-> Pairs:
1.23/1.40	 A__F(X:S) -> A__IF(mark(X:S),c,f(ttrue))
1.23/1.40	 A__F(X:S) -> MARK(X:S)
1.23/1.40	 A__IF(ffalse,X:S,Y:S) -> MARK(Y:S)
1.23/1.40	 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
1.23/1.40	 MARK(f(X:S)) -> A__F(mark(X:S))
1.23/1.40	 MARK(f(X:S)) -> MARK(X:S)
1.23/1.40	 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S)
1.23/1.40	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.23/1.40	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S)
1.23/1.40	-> Rules:
1.23/1.40	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.40	 a__f(X:S) -> f(X:S)
1.23/1.40	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.40	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.40	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.40	 mark(c) -> c
1.23/1.40	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.40	 mark(ffalse) -> ffalse
1.23/1.40	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.40	 mark(ttrue) -> ttrue
1.23/1.40	->Strongly Connected Components:
1.23/1.40	->->Cycle:
1.23/1.40	->->-> Pairs:
1.23/1.40	 A__F(X:S) -> A__IF(mark(X:S),c,f(ttrue))
1.23/1.40	 A__F(X:S) -> MARK(X:S)
1.23/1.40	 A__IF(ffalse,X:S,Y:S) -> MARK(Y:S)
1.23/1.40	 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
1.23/1.40	 MARK(f(X:S)) -> A__F(mark(X:S))
1.23/1.40	 MARK(f(X:S)) -> MARK(X:S)
1.23/1.40	 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S)
1.23/1.40	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.23/1.40	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S)
1.23/1.40	->->-> Rules:
1.23/1.40	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.40	 a__f(X:S) -> f(X:S)
1.23/1.40	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.40	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.40	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.40	 mark(c) -> c
1.23/1.40	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.40	 mark(ffalse) -> ffalse
1.23/1.40	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.40	 mark(ttrue) -> ttrue
1.23/1.40	
1.23/1.40	Problem 1: 
1.23/1.40	
1.23/1.40	Reduction Pairs Processor:
1.23/1.40	-> Pairs:
1.23/1.40	 A__F(X:S) -> A__IF(mark(X:S),c,f(ttrue))
1.23/1.40	 A__F(X:S) -> MARK(X:S)
1.23/1.40	 A__IF(ffalse,X:S,Y:S) -> MARK(Y:S)
1.23/1.40	 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
1.23/1.40	 MARK(f(X:S)) -> A__F(mark(X:S))
1.23/1.40	 MARK(f(X:S)) -> MARK(X:S)
1.23/1.40	 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S)
1.23/1.40	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.23/1.40	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S)
1.23/1.40	-> Rules:
1.23/1.40	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.40	 a__f(X:S) -> f(X:S)
1.23/1.40	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.40	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.40	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.40	 mark(c) -> c
1.23/1.40	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.40	 mark(ffalse) -> ffalse
1.23/1.40	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.40	 mark(ttrue) -> ttrue
1.23/1.40	-> Usable rules:
1.23/1.40	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.40	 a__f(X:S) -> f(X:S)
1.23/1.40	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.40	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.40	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.40	 mark(c) -> c
1.23/1.40	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.40	 mark(ffalse) -> ffalse
1.23/1.40	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.40	 mark(ttrue) -> ttrue
1.23/1.40	->Interpretation type:
1.23/1.40	 Linear
1.23/1.40	->Coefficients:
1.23/1.40	 Natural Numbers
1.23/1.40	->Dimension:
1.23/1.40	 1
1.23/1.40	->Bound:
1.23/1.40	 2
1.23/1.40	->Interpretation:
1.23/1.40	 
1.23/1.40	[a__f](X) = 2.X + 2
1.23/1.40	[a__if](X1,X2,X3) = X1 + 2.X2 + 2.X3
1.23/1.40	[mark](X) = 2.X
1.23/1.40	[c] = 0
1.23/1.40	[f](X) = 2.X + 1
1.23/1.40	[fSNonEmpty] = 0
1.23/1.40	[false] = 1
1.23/1.40	[if](X1,X2,X3) = X1 + 2.X2 + 2.X3
1.23/1.40	[true] = 0
1.23/1.40	[A__F](X) = 2.X + 2
1.23/1.40	[A__IF](X1,X2,X3) = 2.X2 + 2.X3
1.23/1.40	[MARK](X) = 2.X
1.23/1.40	
1.23/1.40	Problem 1: 
1.23/1.40	
1.23/1.40	SCC Processor:
1.23/1.40	-> Pairs:
1.23/1.40	 A__F(X:S) -> A__IF(mark(X:S),c,f(ttrue))
1.23/1.40	 A__IF(ffalse,X:S,Y:S) -> MARK(Y:S)
1.23/1.40	 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
1.23/1.40	 MARK(f(X:S)) -> A__F(mark(X:S))
1.23/1.40	 MARK(f(X:S)) -> MARK(X:S)
1.23/1.40	 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S)
1.23/1.40	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.23/1.40	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S)
1.23/1.40	-> Rules:
1.23/1.40	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.40	 a__f(X:S) -> f(X:S)
1.23/1.40	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.40	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.40	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.40	 mark(c) -> c
1.23/1.40	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.40	 mark(ffalse) -> ffalse
1.23/1.40	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.40	 mark(ttrue) -> ttrue
1.23/1.40	->Strongly Connected Components:
1.23/1.40	->->Cycle:
1.23/1.40	->->-> Pairs:
1.23/1.40	 A__F(X:S) -> A__IF(mark(X:S),c,f(ttrue))
1.23/1.40	 A__IF(ffalse,X:S,Y:S) -> MARK(Y:S)
1.23/1.40	 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
1.23/1.40	 MARK(f(X:S)) -> A__F(mark(X:S))
1.23/1.40	 MARK(f(X:S)) -> MARK(X:S)
1.23/1.40	 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S)
1.23/1.40	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.23/1.40	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S)
1.23/1.40	->->-> Rules:
1.23/1.40	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.40	 a__f(X:S) -> f(X:S)
1.23/1.40	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.40	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.40	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.40	 mark(c) -> c
1.23/1.40	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.40	 mark(ffalse) -> ffalse
1.23/1.40	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.40	 mark(ttrue) -> ttrue
1.23/1.40	
1.23/1.40	Problem 1: 
1.23/1.40	
1.23/1.40	Reduction Pairs Processor:
1.23/1.40	-> Pairs:
1.23/1.40	 A__F(X:S) -> A__IF(mark(X:S),c,f(ttrue))
1.23/1.40	 A__IF(ffalse,X:S,Y:S) -> MARK(Y:S)
1.23/1.40	 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
1.23/1.40	 MARK(f(X:S)) -> A__F(mark(X:S))
1.23/1.40	 MARK(f(X:S)) -> MARK(X:S)
1.23/1.40	 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S)
1.23/1.40	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.23/1.40	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S)
1.23/1.40	-> Rules:
1.23/1.40	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.40	 a__f(X:S) -> f(X:S)
1.23/1.40	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.40	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.40	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.40	 mark(c) -> c
1.23/1.40	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.40	 mark(ffalse) -> ffalse
1.23/1.40	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.40	 mark(ttrue) -> ttrue
1.23/1.40	-> Usable rules:
1.23/1.40	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.40	 a__f(X:S) -> f(X:S)
1.23/1.40	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.40	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.40	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.40	 mark(c) -> c
1.23/1.40	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.40	 mark(ffalse) -> ffalse
1.23/1.40	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.40	 mark(ttrue) -> ttrue
1.23/1.40	->Interpretation type:
1.23/1.40	 Linear
1.23/1.40	->Coefficients:
1.23/1.40	 Natural Numbers
1.23/1.40	->Dimension:
1.23/1.40	 1
1.23/1.40	->Bound:
1.23/1.40	 2
1.23/1.40	->Interpretation:
1.23/1.40	 
1.23/1.40	[a__f](X) = 2.X + 2
1.23/1.40	[a__if](X1,X2,X3) = X1 + 2.X2 + 2.X3
1.23/1.40	[mark](X) = 2.X
1.23/1.40	[c] = 0
1.23/1.40	[f](X) = 2.X + 1
1.23/1.40	[fSNonEmpty] = 0
1.23/1.40	[false] = 2
1.23/1.40	[if](X1,X2,X3) = X1 + 2.X2 + X3
1.23/1.40	[true] = 0
1.23/1.40	[A__F](X) = 2.X + 2
1.23/1.40	[A__IF](X1,X2,X3) = X1 + 2.X2 + 2.X3
1.23/1.40	[MARK](X) = 2.X
1.23/1.40	
1.23/1.40	Problem 1: 
1.23/1.40	
1.23/1.40	SCC Processor:
1.23/1.40	-> Pairs:
1.23/1.40	 A__F(X:S) -> A__IF(mark(X:S),c,f(ttrue))
1.23/1.40	 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
1.23/1.40	 MARK(f(X:S)) -> A__F(mark(X:S))
1.23/1.40	 MARK(f(X:S)) -> MARK(X:S)
1.23/1.40	 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S)
1.23/1.40	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S)
1.23/1.41	-> Rules:
1.23/1.41	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.41	 a__f(X:S) -> f(X:S)
1.23/1.41	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.41	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.41	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.41	 mark(c) -> c
1.23/1.41	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.41	 mark(ffalse) -> ffalse
1.23/1.41	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 mark(ttrue) -> ttrue
1.23/1.41	->Strongly Connected Components:
1.23/1.41	->->Cycle:
1.23/1.41	->->-> Pairs:
1.23/1.41	 A__F(X:S) -> A__IF(mark(X:S),c,f(ttrue))
1.23/1.41	 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
1.23/1.41	 MARK(f(X:S)) -> A__F(mark(X:S))
1.23/1.41	 MARK(f(X:S)) -> MARK(X:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S)
1.23/1.41	->->-> Rules:
1.23/1.41	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.41	 a__f(X:S) -> f(X:S)
1.23/1.41	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.41	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.41	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.41	 mark(c) -> c
1.23/1.41	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.41	 mark(ffalse) -> ffalse
1.23/1.41	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 mark(ttrue) -> ttrue
1.23/1.41	
1.23/1.41	Problem 1: 
1.23/1.41	
1.23/1.41	Reduction Pairs Processor:
1.23/1.41	-> Pairs:
1.23/1.41	 A__F(X:S) -> A__IF(mark(X:S),c,f(ttrue))
1.23/1.41	 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
1.23/1.41	 MARK(f(X:S)) -> A__F(mark(X:S))
1.23/1.41	 MARK(f(X:S)) -> MARK(X:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S)
1.23/1.41	-> Rules:
1.23/1.41	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.41	 a__f(X:S) -> f(X:S)
1.23/1.41	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.41	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.41	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.41	 mark(c) -> c
1.23/1.41	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.41	 mark(ffalse) -> ffalse
1.23/1.41	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 mark(ttrue) -> ttrue
1.23/1.41	-> Usable rules:
1.23/1.41	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.41	 a__f(X:S) -> f(X:S)
1.23/1.41	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.41	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.41	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.41	 mark(c) -> c
1.23/1.41	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.41	 mark(ffalse) -> ffalse
1.23/1.41	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 mark(ttrue) -> ttrue
1.23/1.41	->Interpretation type:
1.23/1.41	 Linear
1.23/1.41	->Coefficients:
1.23/1.41	 Natural Numbers
1.23/1.41	->Dimension:
1.23/1.41	 1
1.23/1.41	->Bound:
1.23/1.41	 2
1.23/1.41	->Interpretation:
1.23/1.41	 
1.23/1.41	[a__f](X) = 2.X + 2
1.23/1.41	[a__if](X1,X2,X3) = X1 + 2.X2 + 2.X3
1.23/1.41	[mark](X) = 2.X
1.23/1.41	[c] = 0
1.23/1.41	[f](X) = 2.X + 1
1.23/1.41	[fSNonEmpty] = 0
1.23/1.41	[false] = 1
1.23/1.41	[if](X1,X2,X3) = X1 + 2.X2 + X3
1.23/1.41	[true] = 0
1.23/1.41	[A__F](X) = 2.X + 2
1.23/1.41	[A__IF](X1,X2,X3) = 2.X2 + X3
1.23/1.41	[MARK](X) = 2.X
1.23/1.41	
1.23/1.41	Problem 1: 
1.23/1.41	
1.23/1.41	SCC Processor:
1.23/1.41	-> Pairs:
1.23/1.41	 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
1.23/1.41	 MARK(f(X:S)) -> A__F(mark(X:S))
1.23/1.41	 MARK(f(X:S)) -> MARK(X:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S)
1.23/1.41	-> Rules:
1.23/1.41	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.41	 a__f(X:S) -> f(X:S)
1.23/1.41	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.41	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.41	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.41	 mark(c) -> c
1.23/1.41	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.41	 mark(ffalse) -> ffalse
1.23/1.41	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 mark(ttrue) -> ttrue
1.23/1.41	->Strongly Connected Components:
1.23/1.41	->->Cycle:
1.23/1.41	->->-> Pairs:
1.23/1.41	 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
1.23/1.41	 MARK(f(X:S)) -> MARK(X:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S)
1.23/1.41	->->-> Rules:
1.23/1.41	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.41	 a__f(X:S) -> f(X:S)
1.23/1.41	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.41	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.41	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.41	 mark(c) -> c
1.23/1.41	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.41	 mark(ffalse) -> ffalse
1.23/1.41	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 mark(ttrue) -> ttrue
1.23/1.41	
1.23/1.41	Problem 1: 
1.23/1.41	
1.23/1.41	Reduction Pairs Processor:
1.23/1.41	-> Pairs:
1.23/1.41	 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
1.23/1.41	 MARK(f(X:S)) -> MARK(X:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S)
1.23/1.41	-> Rules:
1.23/1.41	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.41	 a__f(X:S) -> f(X:S)
1.23/1.41	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.41	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.41	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.41	 mark(c) -> c
1.23/1.41	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.41	 mark(ffalse) -> ffalse
1.23/1.41	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 mark(ttrue) -> ttrue
1.23/1.41	-> Usable rules:
1.23/1.41	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.41	 a__f(X:S) -> f(X:S)
1.23/1.41	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.41	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.41	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.41	 mark(c) -> c
1.23/1.41	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.41	 mark(ffalse) -> ffalse
1.23/1.41	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 mark(ttrue) -> ttrue
1.23/1.41	->Interpretation type:
1.23/1.41	 Linear
1.23/1.41	->Coefficients:
1.23/1.41	 Natural Numbers
1.23/1.41	->Dimension:
1.23/1.41	 1
1.23/1.41	->Bound:
1.23/1.41	 2
1.23/1.41	->Interpretation:
1.23/1.41	 
1.23/1.41	[a__f](X) = 2.X + 2
1.23/1.41	[a__if](X1,X2,X3) = X1 + 2.X2 + 2.X3
1.23/1.41	[mark](X) = 2.X
1.23/1.41	[c] = 0
1.23/1.41	[f](X) = 2.X + 1
1.23/1.41	[fSNonEmpty] = 0
1.23/1.41	[false] = 2
1.23/1.41	[if](X1,X2,X3) = X1 + 2.X2 + 2.X3
1.23/1.41	[true] = 0
1.23/1.41	[A__F](X) = 0
1.23/1.41	[A__IF](X1,X2,X3) = X1 + 2.X2 + X3 + 2
1.23/1.41	[MARK](X) = 2.X + 2
1.23/1.41	
1.23/1.41	Problem 1: 
1.23/1.41	
1.23/1.41	SCC Processor:
1.23/1.41	-> Pairs:
1.23/1.41	 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S)
1.23/1.41	-> Rules:
1.23/1.41	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.41	 a__f(X:S) -> f(X:S)
1.23/1.41	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.41	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.41	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.41	 mark(c) -> c
1.23/1.41	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.41	 mark(ffalse) -> ffalse
1.23/1.41	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 mark(ttrue) -> ttrue
1.23/1.41	->Strongly Connected Components:
1.23/1.41	->->Cycle:
1.23/1.41	->->-> Pairs:
1.23/1.41	 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S)
1.23/1.41	->->-> Rules:
1.23/1.41	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.41	 a__f(X:S) -> f(X:S)
1.23/1.41	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.41	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.41	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.41	 mark(c) -> c
1.23/1.41	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.41	 mark(ffalse) -> ffalse
1.23/1.41	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 mark(ttrue) -> ttrue
1.23/1.41	
1.23/1.41	Problem 1: 
1.23/1.41	
1.23/1.41	Reduction Pairs Processor:
1.23/1.41	-> Pairs:
1.23/1.41	 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S)
1.23/1.41	-> Rules:
1.23/1.41	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.41	 a__f(X:S) -> f(X:S)
1.23/1.41	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.41	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.41	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.41	 mark(c) -> c
1.23/1.41	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.41	 mark(ffalse) -> ffalse
1.23/1.41	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 mark(ttrue) -> ttrue
1.23/1.41	-> Usable rules:
1.23/1.41	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.41	 a__f(X:S) -> f(X:S)
1.23/1.41	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.41	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.41	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.41	 mark(c) -> c
1.23/1.41	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.41	 mark(ffalse) -> ffalse
1.23/1.41	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 mark(ttrue) -> ttrue
1.23/1.41	->Interpretation type:
1.23/1.41	 Simple mixed
1.23/1.41	->Coefficients:
1.23/1.41	 Natural Numbers
1.23/1.41	->Dimension:
1.23/1.41	 1
1.23/1.41	->Bound:
1.23/1.41	 2
1.23/1.41	->Interpretation:
1.23/1.41	 
1.23/1.41	[a__f](X) = 2.X.X + 2.X + 1
1.23/1.41	[a__if](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + X1.X3 + 2.X2.X3 + X1 + 2.X2 + 1
1.23/1.41	[mark](X) = X
1.23/1.41	[c] = 0
1.23/1.41	[f](X) = 2.X.X + 2.X + 1
1.23/1.41	[fSNonEmpty] = 0
1.23/1.41	[false] = 1
1.23/1.41	[if](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + X1.X3 + 2.X2.X3 + X1 + 2.X2 + 1
1.23/1.41	[true] = 0
1.23/1.41	[A__F](X) = 0
1.23/1.41	[A__IF](X1,X2,X3) = 2.X1.X2 + 2.X2.X3 + 2.X2 + 2
1.23/1.41	[MARK](X) = 2.X
1.23/1.41	
1.23/1.41	Problem 1: 
1.23/1.41	
1.23/1.41	SCC Processor:
1.23/1.41	-> Pairs:
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S)
1.23/1.41	-> Rules:
1.23/1.41	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.41	 a__f(X:S) -> f(X:S)
1.23/1.41	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.41	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.41	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.41	 mark(c) -> c
1.23/1.41	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.41	 mark(ffalse) -> ffalse
1.23/1.41	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 mark(ttrue) -> ttrue
1.23/1.41	->Strongly Connected Components:
1.23/1.41	->->Cycle:
1.23/1.41	->->-> Pairs:
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S)
1.23/1.41	->->-> Rules:
1.23/1.41	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.41	 a__f(X:S) -> f(X:S)
1.23/1.41	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.41	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.41	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.41	 mark(c) -> c
1.23/1.41	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.41	 mark(ffalse) -> ffalse
1.23/1.41	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 mark(ttrue) -> ttrue
1.23/1.41	
1.23/1.41	Problem 1: 
1.23/1.41	
1.23/1.41	Subterm Processor:
1.23/1.41	-> Pairs:
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
1.23/1.41	 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S)
1.23/1.41	-> Rules:
1.23/1.41	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.41	 a__f(X:S) -> f(X:S)
1.23/1.41	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.41	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.41	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.41	 mark(c) -> c
1.23/1.41	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.41	 mark(ffalse) -> ffalse
1.23/1.41	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 mark(ttrue) -> ttrue
1.23/1.41	->Projection:
1.23/1.41	 pi(MARK) = 1
1.23/1.41	
1.23/1.41	Problem 1: 
1.23/1.41	
1.23/1.41	SCC Processor:
1.23/1.41	-> Pairs:
1.23/1.41	 Empty
1.23/1.41	-> Rules:
1.23/1.41	 a__f(X:S) -> a__if(mark(X:S),c,f(ttrue))
1.23/1.41	 a__f(X:S) -> f(X:S)
1.23/1.41	 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
1.23/1.41	 a__if(ttrue,X:S,Y:S) -> mark(X:S)
1.23/1.41	 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
1.23/1.41	 mark(c) -> c
1.23/1.41	 mark(f(X:S)) -> a__f(mark(X:S))
1.23/1.41	 mark(ffalse) -> ffalse
1.23/1.41	 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S)
1.23/1.41	 mark(ttrue) -> ttrue
1.23/1.41	->Strongly Connected Components:
1.23/1.41	 There is no strongly connected component
1.23/1.41	
1.23/1.41	The problem is finite.
1.23/1.41	EOF