0.00/0.01	YES
0.00/0.01	
0.00/0.01	Problem 1: 
0.00/0.01	
0.00/0.01	(VAR N X Y Z)
0.00/0.01	(STRATEGY CONTEXTSENSITIVE
0.00/0.01	(add 1 2)
0.00/0.01	(dbl 1)
0.00/0.01	(first 1 2)
0.00/0.01	(sqr 1)
0.00/0.01	(terms 1)
0.00/0.01	(0)
0.00/0.01	(cons 1)
0.00/0.01	(nil)
0.00/0.01	(recip 1)
0.00/0.01	(s 1)
0.00/0.01	)
0.00/0.01	(RULES
0.00/0.01	add(0,X) -> X
0.00/0.01	add(s(X),Y) -> s(add(X,Y))
0.00/0.01	dbl(0) -> 0
0.00/0.01	dbl(s(X)) -> s(s(dbl(X)))
0.00/0.01	first(0,X) -> nil
0.00/0.01	first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
0.00/0.01	sqr(0) -> 0
0.00/0.01	sqr(s(X)) -> s(add(sqr(X),dbl(X)))
0.00/0.01	terms(N) -> cons(recip(sqr(N)),terms(s(N)))
0.00/0.01	)
0.00/0.01	
0.00/0.01	Problem 1: 
0.00/0.01	
0.00/0.01	Innermost Equivalent Processor:
0.00/0.01	-> Rules:
0.00/0.01	 add(0,X) -> X
0.00/0.01	 add(s(X),Y) -> s(add(X,Y))
0.00/0.01	 dbl(0) -> 0
0.00/0.01	 dbl(s(X)) -> s(s(dbl(X)))
0.00/0.01	 first(0,X) -> nil
0.00/0.01	 first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
0.00/0.01	 sqr(0) -> 0
0.00/0.01	 sqr(s(X)) -> s(add(sqr(X),dbl(X)))
0.00/0.01	 terms(N) -> cons(recip(sqr(N)),terms(s(N)))
0.00/0.01	-> The context-sensitive term rewriting system is an orthogonal system. Therefore, innermost cs-termination implies cs-termination.
0.00/0.01	 
0.00/0.01	
0.00/0.01	Problem 1: 
0.00/0.01	
0.00/0.01	Dependency Pairs Processor:
0.00/0.01	-> Pairs:
0.00/0.01	 ADD(s(X),Y) -> ADD(X,Y)
0.00/0.01	 DBL(s(X)) -> DBL(X)
0.00/0.01	 SQR(s(X)) -> ADD(sqr(X),dbl(X))
0.00/0.01	 SQR(s(X)) -> DBL(X)
0.00/0.01	 SQR(s(X)) -> SQR(X)
0.00/0.01	 TERMS(N) -> SQR(N)
0.00/0.01	-> Rules:
0.00/0.01	 add(0,X) -> X
0.00/0.01	 add(s(X),Y) -> s(add(X,Y))
0.00/0.01	 dbl(0) -> 0
0.00/0.01	 dbl(s(X)) -> s(s(dbl(X)))
0.00/0.01	 first(0,X) -> nil
0.00/0.01	 first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
0.00/0.01	 sqr(0) -> 0
0.00/0.01	 sqr(s(X)) -> s(add(sqr(X),dbl(X)))
0.00/0.01	 terms(N) -> cons(recip(sqr(N)),terms(s(N)))
0.00/0.01	-> Unhiding Rules:
0.00/0.01	 Empty
0.00/0.01	
0.00/0.01	Problem 1: 
0.00/0.01	
0.00/0.01	SCC Processor:
0.00/0.01	-> Pairs:
0.00/0.01	 ADD(s(X),Y) -> ADD(X,Y)
0.00/0.01	 DBL(s(X)) -> DBL(X)
0.00/0.01	 SQR(s(X)) -> ADD(sqr(X),dbl(X))
0.00/0.01	 SQR(s(X)) -> DBL(X)
0.00/0.01	 SQR(s(X)) -> SQR(X)
0.00/0.01	 TERMS(N) -> SQR(N)
0.00/0.01	-> Rules:
0.00/0.01	 add(0,X) -> X
0.00/0.01	 add(s(X),Y) -> s(add(X,Y))
0.00/0.01	 dbl(0) -> 0
0.00/0.01	 dbl(s(X)) -> s(s(dbl(X)))
0.00/0.01	 first(0,X) -> nil
0.00/0.01	 first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
0.00/0.01	 sqr(0) -> 0
0.00/0.01	 sqr(s(X)) -> s(add(sqr(X),dbl(X)))
0.00/0.01	 terms(N) -> cons(recip(sqr(N)),terms(s(N)))
0.00/0.01	-> Unhiding rules:
0.00/0.01	 Empty
0.00/0.01	->Strongly Connected Components:
0.00/0.01	->->Cycle:
0.00/0.01	->->-> Pairs:
0.00/0.01	 DBL(s(X)) -> DBL(X)
0.00/0.01	->->-> Rules:
0.00/0.01	 add(0,X) -> X
0.00/0.01	 add(s(X),Y) -> s(add(X,Y))
0.00/0.01	 dbl(0) -> 0
0.00/0.01	 dbl(s(X)) -> s(s(dbl(X)))
0.00/0.01	 first(0,X) -> nil
0.00/0.01	 first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
0.00/0.01	 sqr(0) -> 0
0.00/0.01	 sqr(s(X)) -> s(add(sqr(X),dbl(X)))
0.00/0.01	 terms(N) -> cons(recip(sqr(N)),terms(s(N)))
0.00/0.01	->->-> Unhiding rules:
0.00/0.01	 Empty
0.00/0.01	->->Cycle:
0.00/0.01	->->-> Pairs:
0.00/0.01	 ADD(s(X),Y) -> ADD(X,Y)
0.00/0.01	->->-> Rules:
0.00/0.01	 add(0,X) -> X
0.00/0.01	 add(s(X),Y) -> s(add(X,Y))
0.00/0.01	 dbl(0) -> 0
0.00/0.01	 dbl(s(X)) -> s(s(dbl(X)))
0.00/0.01	 first(0,X) -> nil
0.00/0.01	 first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
0.00/0.01	 sqr(0) -> 0
0.00/0.01	 sqr(s(X)) -> s(add(sqr(X),dbl(X)))
0.00/0.01	 terms(N) -> cons(recip(sqr(N)),terms(s(N)))
0.00/0.01	->->-> Unhiding rules:
0.00/0.01	 Empty
0.00/0.01	->->Cycle:
0.00/0.01	->->-> Pairs:
0.00/0.01	 SQR(s(X)) -> SQR(X)
0.00/0.01	->->-> Rules:
0.00/0.01	 add(0,X) -> X
0.00/0.01	 add(s(X),Y) -> s(add(X,Y))
0.00/0.01	 dbl(0) -> 0
0.00/0.01	 dbl(s(X)) -> s(s(dbl(X)))
0.00/0.01	 first(0,X) -> nil
0.00/0.01	 first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
0.00/0.01	 sqr(0) -> 0
0.00/0.01	 sqr(s(X)) -> s(add(sqr(X),dbl(X)))
0.00/0.01	 terms(N) -> cons(recip(sqr(N)),terms(s(N)))
0.00/0.01	->->-> Unhiding rules:
0.00/0.01	 Empty
0.00/0.01	
0.00/0.01	
0.00/0.01	The problem is decomposed in 3 subproblems.
0.00/0.01	
0.00/0.01	Problem 1.1: 
0.00/0.01	
0.00/0.01	SubNColl Processor:
0.00/0.01	-> Pairs:
0.00/0.01	 DBL(s(X)) -> DBL(X)
0.00/0.01	-> Rules:
0.00/0.01	 add(0,X) -> X
0.00/0.01	 add(s(X),Y) -> s(add(X,Y))
0.00/0.01	 dbl(0) -> 0
0.00/0.01	 dbl(s(X)) -> s(s(dbl(X)))
0.00/0.01	 first(0,X) -> nil
0.00/0.01	 first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
0.00/0.01	 sqr(0) -> 0
0.00/0.01	 sqr(s(X)) -> s(add(sqr(X),dbl(X)))
0.00/0.01	 terms(N) -> cons(recip(sqr(N)),terms(s(N)))
0.00/0.01	-> Unhiding rules:
0.00/0.01	 Empty
0.00/0.01	->Projection:
0.00/0.01	 pi(DBL) = 1
0.00/0.01	
0.00/0.01	Problem 1.1: 
0.00/0.01	
0.00/0.01	Basic Processor:
0.00/0.01	-> Pairs:
0.00/0.01	 Empty
0.00/0.01	-> Rules:
0.00/0.01	 add(0,X) -> X
0.00/0.01	 add(s(X),Y) -> s(add(X,Y))
0.00/0.01	 dbl(0) -> 0
0.00/0.01	 dbl(s(X)) -> s(s(dbl(X)))
0.00/0.01	 first(0,X) -> nil
0.00/0.01	 first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
0.00/0.01	 sqr(0) -> 0
0.00/0.01	 sqr(s(X)) -> s(add(sqr(X),dbl(X)))
0.00/0.01	 terms(N) -> cons(recip(sqr(N)),terms(s(N)))
0.00/0.01	-> Unhiding rules:
0.00/0.01	 Empty
0.00/0.01	-> Result:
0.00/0.01	 Set P is empty
0.00/0.01	
0.00/0.01	The problem is finite.
0.00/0.01	
0.00/0.01	Problem 1.2: 
0.00/0.01	
0.00/0.01	SubNColl Processor:
0.00/0.01	-> Pairs:
0.00/0.01	 ADD(s(X),Y) -> ADD(X,Y)
0.00/0.01	-> Rules:
0.00/0.01	 add(0,X) -> X
0.00/0.01	 add(s(X),Y) -> s(add(X,Y))
0.00/0.01	 dbl(0) -> 0
0.00/0.01	 dbl(s(X)) -> s(s(dbl(X)))
0.00/0.01	 first(0,X) -> nil
0.00/0.01	 first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
0.00/0.01	 sqr(0) -> 0
0.00/0.01	 sqr(s(X)) -> s(add(sqr(X),dbl(X)))
0.00/0.01	 terms(N) -> cons(recip(sqr(N)),terms(s(N)))
0.00/0.01	-> Unhiding rules:
0.00/0.01	 Empty
0.00/0.01	->Projection:
0.00/0.01	 pi(ADD) = 1
0.00/0.01	
0.00/0.01	Problem 1.2: 
0.00/0.01	
0.00/0.01	Basic Processor:
0.00/0.01	-> Pairs:
0.00/0.01	 Empty
0.00/0.01	-> Rules:
0.00/0.01	 add(0,X) -> X
0.00/0.01	 add(s(X),Y) -> s(add(X,Y))
0.00/0.01	 dbl(0) -> 0
0.00/0.01	 dbl(s(X)) -> s(s(dbl(X)))
0.00/0.01	 first(0,X) -> nil
0.00/0.01	 first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
0.00/0.01	 sqr(0) -> 0
0.00/0.01	 sqr(s(X)) -> s(add(sqr(X),dbl(X)))
0.00/0.01	 terms(N) -> cons(recip(sqr(N)),terms(s(N)))
0.00/0.01	-> Unhiding rules:
0.00/0.01	 Empty
0.00/0.01	-> Result:
0.00/0.01	 Set P is empty
0.00/0.01	
0.00/0.01	The problem is finite.
0.00/0.01	
0.00/0.01	Problem 1.3: 
0.00/0.01	
0.00/0.01	SubNColl Processor:
0.00/0.01	-> Pairs:
0.00/0.01	 SQR(s(X)) -> SQR(X)
0.00/0.01	-> Rules:
0.00/0.01	 add(0,X) -> X
0.00/0.01	 add(s(X),Y) -> s(add(X,Y))
0.00/0.01	 dbl(0) -> 0
0.00/0.01	 dbl(s(X)) -> s(s(dbl(X)))
0.00/0.01	 first(0,X) -> nil
0.00/0.01	 first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
0.00/0.01	 sqr(0) -> 0
0.00/0.01	 sqr(s(X)) -> s(add(sqr(X),dbl(X)))
0.00/0.01	 terms(N) -> cons(recip(sqr(N)),terms(s(N)))
0.00/0.01	-> Unhiding rules:
0.00/0.01	 Empty
0.00/0.01	->Projection:
0.00/0.01	 pi(SQR) = 1
0.00/0.01	
0.00/0.01	Problem 1.3: 
0.00/0.01	
0.00/0.01	Basic Processor:
0.00/0.01	-> Pairs:
0.00/0.01	 Empty
0.00/0.01	-> Rules:
0.00/0.01	 add(0,X) -> X
0.00/0.01	 add(s(X),Y) -> s(add(X,Y))
0.00/0.01	 dbl(0) -> 0
0.00/0.01	 dbl(s(X)) -> s(s(dbl(X)))
0.00/0.01	 first(0,X) -> nil
0.00/0.01	 first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
0.00/0.01	 sqr(0) -> 0
0.00/0.01	 sqr(s(X)) -> s(add(sqr(X),dbl(X)))
0.00/0.01	 terms(N) -> cons(recip(sqr(N)),terms(s(N)))
0.00/0.01	-> Unhiding rules:
0.00/0.01	 Empty
0.00/0.01	-> Result:
0.00/0.01	 Set P is empty
0.00/0.01	
0.00/0.01	The problem is finite.
0.00/0.01	EOF