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	(2ndsneg 1 2)
0.00/0.01	(2ndspos 1 2)
0.00/0.01	(from 1)
0.00/0.01	(pi 1)
0.00/0.01	(plus 1 2)
0.00/0.01	(square 1)
0.00/0.01	(times 1 2)
0.00/0.01	(0)
0.00/0.01	(cons 1)
0.00/0.01	(negrecip 1)
0.00/0.01	(posrecip 1)
0.00/0.01	(rcons 1 2)
0.00/0.01	(rnil)
0.00/0.01	(s 1)
0.00/0.01	)
0.00/0.01	(RULES
0.00/0.01	2ndsneg(0,Z) -> rnil
0.00/0.01	2ndsneg(s(N),cons(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z))
0.00/0.01	2ndspos(0,Z) -> rnil
0.00/0.01	2ndspos(s(N),cons(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z))
0.00/0.01	from(X) -> cons(X,from(s(X)))
0.00/0.01	pi(X) -> 2ndspos(X,from(0))
0.00/0.01	plus(0,Y) -> Y
0.00/0.01	plus(s(X),Y) -> s(plus(X,Y))
0.00/0.01	square(X) -> times(X,X)
0.00/0.01	times(0,Y) -> 0
0.00/0.01	times(s(X),Y) -> plus(Y,times(X,Y))
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	 2ndsneg(0,Z) -> rnil
0.00/0.01	 2ndsneg(s(N),cons(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z))
0.00/0.01	 2ndspos(0,Z) -> rnil
0.00/0.01	 2ndspos(s(N),cons(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z))
0.00/0.01	 from(X) -> cons(X,from(s(X)))
0.00/0.01	 pi(X) -> 2ndspos(X,from(0))
0.00/0.01	 plus(0,Y) -> Y
0.00/0.01	 plus(s(X),Y) -> s(plus(X,Y))
0.00/0.01	 square(X) -> times(X,X)
0.00/0.01	 times(0,Y) -> 0
0.00/0.01	 times(s(X),Y) -> plus(Y,times(X,Y))
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	 2NDSNEG(s(N),cons(X,cons(Y,Z))) -> 2NDSPOS(N,Z)
0.00/0.01	 2NDSNEG(s(N),cons(X,cons(Y,Z))) -> Y
0.00/0.01	 2NDSNEG(s(N),cons(X,cons(Y,Z))) -> Z
0.00/0.01	 2NDSPOS(s(N),cons(X,cons(Y,Z))) -> 2NDSNEG(N,Z)
0.00/0.01	 2NDSPOS(s(N),cons(X,cons(Y,Z))) -> Y
0.00/0.01	 2NDSPOS(s(N),cons(X,cons(Y,Z))) -> Z
0.00/0.01	 PI(X) -> 2NDSPOS(X,from(0))
0.00/0.01	 PI(X) -> FROM(0)
0.00/0.01	 PLUS(s(X),Y) -> PLUS(X,Y)
0.00/0.01	 SQUARE(X) -> TIMES(X,X)
0.00/0.01	 TIMES(s(X),Y) -> PLUS(Y,times(X,Y))
0.00/0.01	 TIMES(s(X),Y) -> TIMES(X,Y)
0.00/0.01	-> Rules:
0.00/0.01	 2ndsneg(0,Z) -> rnil
0.00/0.01	 2ndsneg(s(N),cons(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z))
0.00/0.01	 2ndspos(0,Z) -> rnil
0.00/0.01	 2ndspos(s(N),cons(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z))
0.00/0.01	 from(X) -> cons(X,from(s(X)))
0.00/0.01	 pi(X) -> 2ndspos(X,from(0))
0.00/0.01	 plus(0,Y) -> Y
0.00/0.01	 plus(s(X),Y) -> s(plus(X,Y))
0.00/0.01	 square(X) -> times(X,X)
0.00/0.01	 times(0,Y) -> 0
0.00/0.01	 times(s(X),Y) -> plus(Y,times(X,Y))
0.00/0.01	-> Unhiding Rules:
0.00/0.01	 from(s(X)) -> FROM(s(X))
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	 2NDSNEG(s(N),cons(X,cons(Y,Z))) -> 2NDSPOS(N,Z)
0.00/0.01	 2NDSNEG(s(N),cons(X,cons(Y,Z))) -> Y
0.00/0.01	 2NDSNEG(s(N),cons(X,cons(Y,Z))) -> Z
0.00/0.01	 2NDSPOS(s(N),cons(X,cons(Y,Z))) -> 2NDSNEG(N,Z)
0.00/0.01	 2NDSPOS(s(N),cons(X,cons(Y,Z))) -> Y
0.00/0.01	 2NDSPOS(s(N),cons(X,cons(Y,Z))) -> Z
0.00/0.01	 PI(X) -> 2NDSPOS(X,from(0))
0.00/0.01	 PI(X) -> FROM(0)
0.00/0.01	 PLUS(s(X),Y) -> PLUS(X,Y)
0.00/0.01	 SQUARE(X) -> TIMES(X,X)
0.00/0.01	 TIMES(s(X),Y) -> PLUS(Y,times(X,Y))
0.00/0.01	 TIMES(s(X),Y) -> TIMES(X,Y)
0.00/0.01	-> Rules:
0.00/0.01	 2ndsneg(0,Z) -> rnil
0.00/0.01	 2ndsneg(s(N),cons(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z))
0.00/0.01	 2ndspos(0,Z) -> rnil
0.00/0.01	 2ndspos(s(N),cons(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z))
0.00/0.01	 from(X) -> cons(X,from(s(X)))
0.00/0.01	 pi(X) -> 2ndspos(X,from(0))
0.00/0.01	 plus(0,Y) -> Y
0.00/0.01	 plus(s(X),Y) -> s(plus(X,Y))
0.00/0.01	 square(X) -> times(X,X)
0.00/0.01	 times(0,Y) -> 0
0.00/0.01	 times(s(X),Y) -> plus(Y,times(X,Y))
0.00/0.01	-> Unhiding rules:
0.00/0.01	 from(s(X)) -> FROM(s(X))
0.00/0.01	->Strongly Connected Components:
0.00/0.01	->->Cycle:
0.00/0.01	->->-> Pairs:
0.00/0.01	 PLUS(s(X),Y) -> PLUS(X,Y)
0.00/0.01	->->-> Rules:
0.00/0.01	 2ndsneg(0,Z) -> rnil
0.00/0.01	 2ndsneg(s(N),cons(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z))
0.00/0.01	 2ndspos(0,Z) -> rnil
0.00/0.01	 2ndspos(s(N),cons(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z))
0.00/0.01	 from(X) -> cons(X,from(s(X)))
0.00/0.01	 pi(X) -> 2ndspos(X,from(0))
0.00/0.01	 plus(0,Y) -> Y
0.00/0.01	 plus(s(X),Y) -> s(plus(X,Y))
0.00/0.01	 square(X) -> times(X,X)
0.00/0.01	 times(0,Y) -> 0
0.00/0.01	 times(s(X),Y) -> plus(Y,times(X,Y))
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	 TIMES(s(X),Y) -> TIMES(X,Y)
0.00/0.01	->->-> Rules:
0.00/0.01	 2ndsneg(0,Z) -> rnil
0.00/0.01	 2ndsneg(s(N),cons(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z))
0.00/0.01	 2ndspos(0,Z) -> rnil
0.00/0.01	 2ndspos(s(N),cons(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z))
0.00/0.01	 from(X) -> cons(X,from(s(X)))
0.00/0.01	 pi(X) -> 2ndspos(X,from(0))
0.00/0.01	 plus(0,Y) -> Y
0.00/0.01	 plus(s(X),Y) -> s(plus(X,Y))
0.00/0.01	 square(X) -> times(X,X)
0.00/0.01	 times(0,Y) -> 0
0.00/0.01	 times(s(X),Y) -> plus(Y,times(X,Y))
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	 2NDSNEG(s(N),cons(X,cons(Y,Z))) -> 2NDSPOS(N,Z)
0.00/0.01	 2NDSPOS(s(N),cons(X,cons(Y,Z))) -> 2NDSNEG(N,Z)
0.00/0.01	->->-> Rules:
0.00/0.01	 2ndsneg(0,Z) -> rnil
0.00/0.01	 2ndsneg(s(N),cons(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z))
0.00/0.01	 2ndspos(0,Z) -> rnil
0.00/0.01	 2ndspos(s(N),cons(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z))
0.00/0.01	 from(X) -> cons(X,from(s(X)))
0.00/0.01	 pi(X) -> 2ndspos(X,from(0))
0.00/0.01	 plus(0,Y) -> Y
0.00/0.01	 plus(s(X),Y) -> s(plus(X,Y))
0.00/0.01	 square(X) -> times(X,X)
0.00/0.01	 times(0,Y) -> 0
0.00/0.01	 times(s(X),Y) -> plus(Y,times(X,Y))
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	 PLUS(s(X),Y) -> PLUS(X,Y)
0.00/0.01	-> Rules:
0.00/0.01	 2ndsneg(0,Z) -> rnil
0.00/0.01	 2ndsneg(s(N),cons(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z))
0.00/0.01	 2ndspos(0,Z) -> rnil
0.00/0.01	 2ndspos(s(N),cons(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z))
0.00/0.01	 from(X) -> cons(X,from(s(X)))
0.00/0.01	 pi(X) -> 2ndspos(X,from(0))
0.00/0.01	 plus(0,Y) -> Y
0.00/0.01	 plus(s(X),Y) -> s(plus(X,Y))
0.00/0.01	 square(X) -> times(X,X)
0.00/0.01	 times(0,Y) -> 0
0.00/0.01	 times(s(X),Y) -> plus(Y,times(X,Y))
0.00/0.01	-> Unhiding rules:
0.00/0.01	 Empty
0.00/0.01	->Projection:
0.00/0.01	 pi(PLUS) = 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	 2ndsneg(0,Z) -> rnil
0.00/0.01	 2ndsneg(s(N),cons(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z))
0.00/0.01	 2ndspos(0,Z) -> rnil
0.00/0.01	 2ndspos(s(N),cons(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z))
0.00/0.01	 from(X) -> cons(X,from(s(X)))
0.00/0.01	 pi(X) -> 2ndspos(X,from(0))
0.00/0.01	 plus(0,Y) -> Y
0.00/0.01	 plus(s(X),Y) -> s(plus(X,Y))
0.00/0.01	 square(X) -> times(X,X)
0.00/0.01	 times(0,Y) -> 0
0.00/0.01	 times(s(X),Y) -> plus(Y,times(X,Y))
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	 TIMES(s(X),Y) -> TIMES(X,Y)
0.00/0.01	-> Rules:
0.00/0.01	 2ndsneg(0,Z) -> rnil
0.00/0.01	 2ndsneg(s(N),cons(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z))
0.00/0.01	 2ndspos(0,Z) -> rnil
0.00/0.01	 2ndspos(s(N),cons(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z))
0.00/0.01	 from(X) -> cons(X,from(s(X)))
0.00/0.01	 pi(X) -> 2ndspos(X,from(0))
0.00/0.01	 plus(0,Y) -> Y
0.00/0.01	 plus(s(X),Y) -> s(plus(X,Y))
0.00/0.01	 square(X) -> times(X,X)
0.00/0.01	 times(0,Y) -> 0
0.00/0.01	 times(s(X),Y) -> plus(Y,times(X,Y))
0.00/0.01	-> Unhiding rules:
0.00/0.01	 Empty
0.00/0.01	->Projection:
0.00/0.01	 pi(TIMES) = 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	 2ndsneg(0,Z) -> rnil
0.00/0.01	 2ndsneg(s(N),cons(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z))
0.00/0.01	 2ndspos(0,Z) -> rnil
0.00/0.01	 2ndspos(s(N),cons(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z))
0.00/0.01	 from(X) -> cons(X,from(s(X)))
0.00/0.01	 pi(X) -> 2ndspos(X,from(0))
0.00/0.01	 plus(0,Y) -> Y
0.00/0.01	 plus(s(X),Y) -> s(plus(X,Y))
0.00/0.01	 square(X) -> times(X,X)
0.00/0.01	 times(0,Y) -> 0
0.00/0.01	 times(s(X),Y) -> plus(Y,times(X,Y))
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	 2NDSNEG(s(N),cons(X,cons(Y,Z))) -> 2NDSPOS(N,Z)
0.00/0.01	 2NDSPOS(s(N),cons(X,cons(Y,Z))) -> 2NDSNEG(N,Z)
0.00/0.01	-> Rules:
0.00/0.01	 2ndsneg(0,Z) -> rnil
0.00/0.01	 2ndsneg(s(N),cons(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z))
0.00/0.01	 2ndspos(0,Z) -> rnil
0.00/0.01	 2ndspos(s(N),cons(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z))
0.00/0.01	 from(X) -> cons(X,from(s(X)))
0.00/0.01	 pi(X) -> 2ndspos(X,from(0))
0.00/0.01	 plus(0,Y) -> Y
0.00/0.01	 plus(s(X),Y) -> s(plus(X,Y))
0.00/0.01	 square(X) -> times(X,X)
0.00/0.01	 times(0,Y) -> 0
0.00/0.01	 times(s(X),Y) -> plus(Y,times(X,Y))
0.00/0.01	-> Unhiding rules:
0.00/0.01	 Empty
0.00/0.01	->Projection:
0.00/0.01	 pi(2NDSNEG) = 1
0.00/0.01	 pi(2NDSPOS) = 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	 2ndsneg(0,Z) -> rnil
0.00/0.01	 2ndsneg(s(N),cons(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z))
0.00/0.01	 2ndspos(0,Z) -> rnil
0.00/0.01	 2ndspos(s(N),cons(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z))
0.00/0.01	 from(X) -> cons(X,from(s(X)))
0.00/0.01	 pi(X) -> 2ndspos(X,from(0))
0.00/0.01	 plus(0,Y) -> Y
0.00/0.01	 plus(s(X),Y) -> s(plus(X,Y))
0.00/0.01	 square(X) -> times(X,X)
0.00/0.01	 times(0,Y) -> 0
0.00/0.01	 times(s(X),Y) -> plus(Y,times(X,Y))
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.
0.00/0.02	EOF