5.00/5.14	YES
5.00/5.14	
5.00/5.14	Problem 1: 
5.00/5.14	
5.00/5.14	(VAR X Y)
5.00/5.14	(STRATEGY CONTEXTSENSITIVE
5.00/5.14	(add 1 2)
5.00/5.14	(fact 1)
5.00/5.14	(if 1)
5.00/5.14	(p 1)
5.00/5.14	(prod 1 2)
5.00/5.14	(zero 1)
5.00/5.14	(0)
5.00/5.14	(false)
5.00/5.14	(s 1)
5.00/5.14	(true)
5.00/5.14	)
5.00/5.14	(RULES
5.00/5.14	add(0,X) -> X
5.00/5.14	add(s(X),Y) -> s(add(X,Y))
5.00/5.14	fact(X) -> if(zero(X),s(0),prod(X,fact(p(X))))
5.00/5.14	if(false,X,Y) -> Y
5.00/5.14	if(true,X,Y) -> X
5.00/5.14	p(s(X)) -> X
5.00/5.14	prod(0,X) -> 0
5.00/5.14	prod(s(X),Y) -> add(Y,prod(X,Y))
5.00/5.14	zero(0) -> true
5.00/5.14	zero(s(X)) -> false
5.00/5.14	)
5.00/5.14	
5.00/5.14	Problem 1: 
5.00/5.14	
5.00/5.14	Innermost Equivalent Processor:
5.00/5.14	-> Rules:
5.00/5.14	 add(0,X) -> X
5.00/5.14	 add(s(X),Y) -> s(add(X,Y))
5.00/5.14	 fact(X) -> if(zero(X),s(0),prod(X,fact(p(X))))
5.00/5.14	 if(false,X,Y) -> Y
5.00/5.14	 if(true,X,Y) -> X
5.00/5.14	 p(s(X)) -> X
5.00/5.14	 prod(0,X) -> 0
5.00/5.14	 prod(s(X),Y) -> add(Y,prod(X,Y))
5.00/5.14	 zero(0) -> true
5.00/5.14	 zero(s(X)) -> false
5.00/5.14	-> The context-sensitive term rewriting system is an orthogonal system. Therefore, innermost cs-termination implies cs-termination.
5.00/5.14	 
5.00/5.14	
5.00/5.14	Problem 1: 
5.00/5.14	
5.00/5.14	Dependency Pairs Processor:
5.00/5.14	-> Pairs:
5.00/5.14	 ADD(s(X),Y) -> ADD(X,Y)
5.00/5.14	 FACT(X) -> IF(zero(X),s(0),prod(X,fact(p(X))))
5.00/5.14	 FACT(X) -> ZERO(X)
5.00/5.14	 IF(false,X,Y) -> Y
5.00/5.14	 IF(true,X,Y) -> X
5.00/5.14	 PROD(s(X),Y) -> ADD(Y,prod(X,Y))
5.00/5.14	 PROD(s(X),Y) -> PROD(X,Y)
5.00/5.14	-> Rules:
5.00/5.14	 add(0,X) -> X
5.00/5.14	 add(s(X),Y) -> s(add(X,Y))
5.00/5.14	 fact(X) -> if(zero(X),s(0),prod(X,fact(p(X))))
5.00/5.14	 if(false,X,Y) -> Y
5.00/5.14	 if(true,X,Y) -> X
5.00/5.14	 p(s(X)) -> X
5.00/5.14	 prod(0,X) -> 0
5.00/5.14	 prod(s(X),Y) -> add(Y,prod(X,Y))
5.00/5.14	 zero(0) -> true
5.00/5.14	 zero(s(X)) -> false
5.00/5.14	-> Unhiding Rules:
5.00/5.14	 prod(X,fact(p(X))) -> FACT(p(X))
5.00/5.14	 prod(X,fact(p(X))) -> P(X)
5.00/5.14	 prod(X,fact(p(X))) -> PROD(X,fact(p(X)))
5.00/5.14	
5.00/5.14	Problem 1: 
5.00/5.14	
5.00/5.14	SCC Processor:
5.00/5.14	-> Pairs:
5.00/5.14	 ADD(s(X),Y) -> ADD(X,Y)
5.00/5.14	 FACT(X) -> IF(zero(X),s(0),prod(X,fact(p(X))))
5.00/5.14	 FACT(X) -> ZERO(X)
5.00/5.14	 IF(false,X,Y) -> Y
5.00/5.14	 IF(true,X,Y) -> X
5.00/5.14	 PROD(s(X),Y) -> ADD(Y,prod(X,Y))
5.00/5.14	 PROD(s(X),Y) -> PROD(X,Y)
5.00/5.14	-> Rules:
5.00/5.14	 add(0,X) -> X
5.00/5.14	 add(s(X),Y) -> s(add(X,Y))
5.00/5.14	 fact(X) -> if(zero(X),s(0),prod(X,fact(p(X))))
5.00/5.14	 if(false,X,Y) -> Y
5.00/5.14	 if(true,X,Y) -> X
5.00/5.14	 p(s(X)) -> X
5.00/5.14	 prod(0,X) -> 0
5.00/5.14	 prod(s(X),Y) -> add(Y,prod(X,Y))
5.00/5.14	 zero(0) -> true
5.00/5.14	 zero(s(X)) -> false
5.00/5.14	-> Unhiding rules:
5.00/5.14	 prod(X,fact(p(X))) -> FACT(p(X))
5.00/5.14	 prod(X,fact(p(X))) -> P(X)
5.00/5.14	 prod(X,fact(p(X))) -> PROD(X,fact(p(X)))
5.00/5.14	->Strongly Connected Components:
5.00/5.14	->->Cycle:
5.00/5.14	->->-> Pairs:
5.00/5.14	 ADD(s(X),Y) -> ADD(X,Y)
5.00/5.14	->->-> Rules:
5.00/5.14	 add(0,X) -> X
5.00/5.14	 add(s(X),Y) -> s(add(X,Y))
5.00/5.14	 fact(X) -> if(zero(X),s(0),prod(X,fact(p(X))))
5.00/5.14	 if(false,X,Y) -> Y
5.00/5.14	 if(true,X,Y) -> X
5.00/5.14	 p(s(X)) -> X
5.00/5.14	 prod(0,X) -> 0
5.00/5.14	 prod(s(X),Y) -> add(Y,prod(X,Y))
5.00/5.14	 zero(0) -> true
5.00/5.14	 zero(s(X)) -> false
5.00/5.14	->->-> Unhiding rules:
5.00/5.14	 Empty
5.00/5.14	->->Cycle:
5.00/5.14	->->-> Pairs:
5.00/5.14	 PROD(s(X),Y) -> PROD(X,Y)
5.00/5.14	->->-> Rules:
5.00/5.14	 add(0,X) -> X
5.00/5.14	 add(s(X),Y) -> s(add(X,Y))
5.00/5.14	 fact(X) -> if(zero(X),s(0),prod(X,fact(p(X))))
5.00/5.14	 if(false,X,Y) -> Y
5.00/5.14	 if(true,X,Y) -> X
5.00/5.14	 p(s(X)) -> X
5.00/5.14	 prod(0,X) -> 0
5.00/5.14	 prod(s(X),Y) -> add(Y,prod(X,Y))
5.00/5.14	 zero(0) -> true
5.00/5.14	 zero(s(X)) -> false
5.00/5.14	->->-> Unhiding rules:
5.00/5.14	 Empty
5.00/5.14	->->Cycle:
5.00/5.14	->->-> Pairs:
5.00/5.14	 FACT(X) -> IF(zero(X),s(0),prod(X,fact(p(X))))
5.00/5.14	 IF(false,X,Y) -> Y
5.00/5.14	 IF(true,X,Y) -> X
5.00/5.14	->->-> Rules:
5.00/5.14	 add(0,X) -> X
5.00/5.14	 add(s(X),Y) -> s(add(X,Y))
5.00/5.14	 fact(X) -> if(zero(X),s(0),prod(X,fact(p(X))))
5.00/5.14	 if(false,X,Y) -> Y
5.00/5.14	 if(true,X,Y) -> X
5.00/5.14	 p(s(X)) -> X
5.00/5.14	 prod(0,X) -> 0
5.00/5.14	 prod(s(X),Y) -> add(Y,prod(X,Y))
5.00/5.14	 zero(0) -> true
5.00/5.14	 zero(s(X)) -> false
5.00/5.14	->->-> Unhiding rules:
5.00/5.14	 prod(X,fact(p(X))) -> FACT(p(X))
5.00/5.14	
5.00/5.14	
5.00/5.14	The problem is decomposed in 3 subproblems.
5.00/5.14	
5.00/5.14	Problem 1.1: 
5.00/5.14	
5.00/5.14	SubNColl Processor:
5.00/5.14	-> Pairs:
5.00/5.14	 ADD(s(X),Y) -> ADD(X,Y)
5.00/5.14	-> Rules:
5.00/5.14	 add(0,X) -> X
5.00/5.14	 add(s(X),Y) -> s(add(X,Y))
5.00/5.14	 fact(X) -> if(zero(X),s(0),prod(X,fact(p(X))))
5.00/5.14	 if(false,X,Y) -> Y
5.00/5.14	 if(true,X,Y) -> X
5.00/5.14	 p(s(X)) -> X
5.00/5.14	 prod(0,X) -> 0
5.00/5.14	 prod(s(X),Y) -> add(Y,prod(X,Y))
5.00/5.14	 zero(0) -> true
5.00/5.14	 zero(s(X)) -> false
5.00/5.14	-> Unhiding rules:
5.00/5.14	 Empty
5.00/5.14	->Projection:
5.00/5.14	 pi(ADD) = 1
5.00/5.14	
5.00/5.14	Problem 1.1: 
5.00/5.14	
5.00/5.14	Basic Processor:
5.00/5.14	-> Pairs:
5.00/5.14	 Empty
5.00/5.14	-> Rules:
5.00/5.14	 add(0,X) -> X
5.00/5.14	 add(s(X),Y) -> s(add(X,Y))
5.00/5.14	 fact(X) -> if(zero(X),s(0),prod(X,fact(p(X))))
5.00/5.14	 if(false,X,Y) -> Y
5.00/5.14	 if(true,X,Y) -> X
5.00/5.14	 p(s(X)) -> X
5.00/5.14	 prod(0,X) -> 0
5.00/5.14	 prod(s(X),Y) -> add(Y,prod(X,Y))
5.00/5.14	 zero(0) -> true
5.00/5.14	 zero(s(X)) -> false
5.00/5.14	-> Unhiding rules:
5.00/5.14	 Empty
5.00/5.14	-> Result:
5.00/5.14	 Set P is empty
5.00/5.14	
5.00/5.14	The problem is finite.
5.00/5.14	
5.00/5.14	Problem 1.2: 
5.00/5.14	
5.00/5.14	SubNColl Processor:
5.00/5.14	-> Pairs:
5.00/5.14	 PROD(s(X),Y) -> PROD(X,Y)
5.00/5.14	-> Rules:
5.00/5.14	 add(0,X) -> X
5.00/5.14	 add(s(X),Y) -> s(add(X,Y))
5.00/5.14	 fact(X) -> if(zero(X),s(0),prod(X,fact(p(X))))
5.00/5.14	 if(false,X,Y) -> Y
5.00/5.14	 if(true,X,Y) -> X
5.00/5.14	 p(s(X)) -> X
5.00/5.14	 prod(0,X) -> 0
5.00/5.14	 prod(s(X),Y) -> add(Y,prod(X,Y))
5.00/5.14	 zero(0) -> true
5.00/5.14	 zero(s(X)) -> false
5.00/5.14	-> Unhiding rules:
5.00/5.14	 Empty
5.00/5.14	->Projection:
5.00/5.14	 pi(PROD) = 1
5.00/5.14	
5.00/5.14	Problem 1.2: 
5.00/5.14	
5.00/5.14	Basic Processor:
5.00/5.14	-> Pairs:
5.00/5.14	 Empty
5.00/5.14	-> Rules:
5.00/5.14	 add(0,X) -> X
5.00/5.14	 add(s(X),Y) -> s(add(X,Y))
5.00/5.14	 fact(X) -> if(zero(X),s(0),prod(X,fact(p(X))))
5.00/5.14	 if(false,X,Y) -> Y
5.00/5.14	 if(true,X,Y) -> X
5.00/5.14	 p(s(X)) -> X
5.00/5.14	 prod(0,X) -> 0
5.00/5.14	 prod(s(X),Y) -> add(Y,prod(X,Y))
5.00/5.14	 zero(0) -> true
5.00/5.14	 zero(s(X)) -> false
5.00/5.14	-> Unhiding rules:
5.00/5.14	 Empty
5.00/5.14	-> Result:
5.00/5.14	 Set P is empty
5.00/5.14	
5.00/5.14	The problem is finite.
5.00/5.14	
5.00/5.14	Problem 1.3: 
5.00/5.14	
5.00/5.14	Reduction Pairs Processor:
5.00/5.14	-> Pairs:
5.00/5.14	 FACT(X) -> IF(zero(X),s(0),prod(X,fact(p(X))))
5.00/5.14	 IF(false,X,Y) -> Y
5.00/5.14	 IF(true,X,Y) -> X
5.00/5.14	-> Rules:
5.00/5.14	 add(0,X) -> X
5.00/5.14	 add(s(X),Y) -> s(add(X,Y))
5.00/5.14	 fact(X) -> if(zero(X),s(0),prod(X,fact(p(X))))
5.00/5.14	 if(false,X,Y) -> Y
5.00/5.14	 if(true,X,Y) -> X
5.00/5.14	 p(s(X)) -> X
5.00/5.14	 prod(0,X) -> 0
5.00/5.14	 prod(s(X),Y) -> add(Y,prod(X,Y))
5.00/5.14	 zero(0) -> true
5.00/5.14	 zero(s(X)) -> false
5.00/5.14	-> Unhiding rules:
5.00/5.14	 prod(X,fact(p(X))) -> FACT(p(X))
5.00/5.14	-> Usable rules:
5.00/5.14	 p(s(X)) -> X
5.00/5.14	 zero(0) -> true
5.00/5.14	 zero(s(X)) -> false
5.00/5.14	->Interpretation type:
5.00/5.14	 Simple mixed
5.00/5.14	->Coefficients:
5.00/5.14	 All rationals
5.00/5.14	->Dimension:
5.00/5.14	 1
5.00/5.14	->Bound:
5.00/5.14	 2
5.00/5.14	->Interpretation:
5.00/5.14	 
5.00/5.14	[fact](X) = 2.X
5.00/5.14	[p](X) = 1/2.X
5.00/5.14	[prod](X1,X2) = 1/2.X1.X2 + 2
5.00/5.14	[zero](X) = 1/2.X.X
5.00/5.14	[0] = 2
5.00/5.14	[false] = 1/2
5.00/5.14	[s](X) = 2.X + 1
5.00/5.14	[true] = 2
5.00/5.14	[FACT](X) = 2.X.X + 2
5.00/5.14	[IF](X1,X2,X3) = 1/2.X1.X2 + 1/2.X1 + X3
5.00/5.14	
5.00/5.14	Problem 1.3: 
5.00/5.14	
5.00/5.14	SCC Processor:
5.00/5.14	-> Pairs:
5.00/5.14	 FACT(X) -> IF(zero(X),s(0),prod(X,fact(p(X))))
5.00/5.14	 IF(true,X,Y) -> X
5.00/5.14	-> Rules:
5.00/5.14	 add(0,X) -> X
5.00/5.14	 add(s(X),Y) -> s(add(X,Y))
5.00/5.14	 fact(X) -> if(zero(X),s(0),prod(X,fact(p(X))))
5.00/5.14	 if(false,X,Y) -> Y
5.00/5.14	 if(true,X,Y) -> X
5.00/5.14	 p(s(X)) -> X
5.00/5.14	 prod(0,X) -> 0
5.00/5.14	 prod(s(X),Y) -> add(Y,prod(X,Y))
5.00/5.14	 zero(0) -> true
5.00/5.14	 zero(s(X)) -> false
5.00/5.14	-> Unhiding rules:
5.00/5.14	 prod(X,fact(p(X))) -> FACT(p(X))
5.00/5.14	->Strongly Connected Components:
5.00/5.14	->->Cycle:
5.00/5.14	->->-> Pairs:
5.00/5.14	 FACT(X) -> IF(zero(X),s(0),prod(X,fact(p(X))))
5.00/5.14	 IF(true,X,Y) -> X
5.00/5.14	->->-> Rules:
5.00/5.14	 add(0,X) -> X
5.00/5.14	 add(s(X),Y) -> s(add(X,Y))
5.00/5.14	 fact(X) -> if(zero(X),s(0),prod(X,fact(p(X))))
5.00/5.14	 if(false,X,Y) -> Y
5.00/5.14	 if(true,X,Y) -> X
5.00/5.14	 p(s(X)) -> X
5.00/5.14	 prod(0,X) -> 0
5.00/5.14	 prod(s(X),Y) -> add(Y,prod(X,Y))
5.00/5.14	 zero(0) -> true
5.00/5.14	 zero(s(X)) -> false
5.00/5.14	->->-> Unhiding rules:
5.00/5.14	 prod(X,fact(p(X))) -> FACT(p(X))
5.00/5.14	
5.00/5.14	Problem 1.3: 
5.00/5.14	
5.00/5.14	Reduction Pairs Processor:
5.00/5.14	-> Pairs:
5.00/5.14	 FACT(X) -> IF(zero(X),s(0),prod(X,fact(p(X))))
5.00/5.14	 IF(true,X,Y) -> X
5.00/5.14	-> Rules:
5.00/5.14	 add(0,X) -> X
5.00/5.14	 add(s(X),Y) -> s(add(X,Y))
5.00/5.14	 fact(X) -> if(zero(X),s(0),prod(X,fact(p(X))))
5.00/5.14	 if(false,X,Y) -> Y
5.00/5.14	 if(true,X,Y) -> X
5.00/5.14	 p(s(X)) -> X
5.00/5.14	 prod(0,X) -> 0
5.00/5.14	 prod(s(X),Y) -> add(Y,prod(X,Y))
5.00/5.14	 zero(0) -> true
5.00/5.14	 zero(s(X)) -> false
5.00/5.14	-> Unhiding rules:
5.00/5.14	 prod(X,fact(p(X))) -> FACT(p(X))
5.00/5.14	-> Usable rules:
5.00/5.14	 p(s(X)) -> X
5.00/5.14	 zero(0) -> true
5.00/5.14	 zero(s(X)) -> false
5.00/5.14	->Interpretation type:
5.00/5.14	 Linear
5.00/5.14	->Coefficients:
5.00/5.14	 Natural Numbers
5.00/5.14	->Dimension:
5.00/5.14	 1
5.00/5.14	->Bound:
5.00/5.14	 2
5.00/5.14	->Interpretation:
5.00/5.14	 
5.00/5.14	[fact](X) = 1
5.00/5.14	[p](X) = 2.X + 1
5.00/5.14	[prod](X1,X2) = 2.X1 + 2
5.00/5.14	[zero](X) = 2.X + 1
5.00/5.14	[0] = 0
5.00/5.14	[false] = 1
5.00/5.14	[s](X) = 2.X
5.00/5.14	[true] = 1
5.00/5.14	[FACT](X) = 2
5.00/5.14	[IF](X1,X2,X3) = X2 + 1
5.00/5.14	
5.00/5.14	Problem 1.3: 
5.00/5.14	
5.00/5.14	Basic Processor:
5.00/5.14	-> Pairs:
5.00/5.14	 IF(true,X,Y) -> X
5.00/5.14	-> Rules:
5.00/5.14	 add(0,X) -> X
5.00/5.14	 add(s(X),Y) -> s(add(X,Y))
5.00/5.14	 fact(X) -> if(zero(X),s(0),prod(X,fact(p(X))))
5.00/5.14	 if(false,X,Y) -> Y
5.00/5.14	 if(true,X,Y) -> X
5.00/5.14	 p(s(X)) -> X
5.00/5.14	 prod(0,X) -> 0
5.00/5.14	 prod(s(X),Y) -> add(Y,prod(X,Y))
5.00/5.14	 zero(0) -> true
5.00/5.14	 zero(s(X)) -> false
5.00/5.14	-> Unhiding rules:
5.00/5.14	 prod(X,fact(p(X))) -> FACT(p(X))
5.00/5.14	-> Result:
5.00/5.14	 All pairs P are from Px1
5.00/5.14	
5.00/5.14	The problem is finite.
5.00/5.15	EOF