/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES

Problem 1: 

(VAR N X Y Z)
(RULES
2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
2ndsneg(0,Z) -> rnil
2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
2ndspos(0,Z) -> rnil
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(X) -> X
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
pi(X) -> 2ndspos(X,from(0))
plus(s(X),Y) -> s(plus(X,Y))
plus(0,Y) -> Y
s(X) -> n__s(X)
square(X) -> times(X,X)
times(s(X),Y) -> plus(Y,times(X,Y))
times(0,Y) -> 0
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 2NDSNEG(s(N),cons(X,Z)) -> 2NDSNEG(s(N),cons2(X,activate(Z)))
 2NDSNEG(s(N),cons(X,Z)) -> ACTIVATE(Z)
 2NDSNEG(s(N),cons2(X,cons(Y,Z))) -> 2NDSPOS(N,activate(Z))
 2NDSNEG(s(N),cons2(X,cons(Y,Z))) -> ACTIVATE(Z)
 2NDSPOS(s(N),cons(X,Z)) -> 2NDSPOS(s(N),cons2(X,activate(Z)))
 2NDSPOS(s(N),cons(X,Z)) -> ACTIVATE(Z)
 2NDSPOS(s(N),cons2(X,cons(Y,Z))) -> 2NDSNEG(N,activate(Z))
 2NDSPOS(s(N),cons2(X,cons(Y,Z))) -> ACTIVATE(Z)
 ACTIVATE(n__from(X)) -> ACTIVATE(X)
 ACTIVATE(n__from(X)) -> FROM(activate(X))
 ACTIVATE(n__s(X)) -> ACTIVATE(X)
 ACTIVATE(n__s(X)) -> S(activate(X))
 PI(X) -> 2NDSPOS(X,from(0))
 PI(X) -> FROM(0)
 PLUS(s(X),Y) -> PLUS(X,Y)
 PLUS(s(X),Y) -> S(plus(X,Y))
 SQUARE(X) -> TIMES(X,X)
 TIMES(s(X),Y) -> PLUS(Y,times(X,Y))
 TIMES(s(X),Y) -> TIMES(X,Y)
-> Rules:
 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
 2ndsneg(0,Z) -> rnil
 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
 2ndspos(0,Z) -> rnil
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 pi(X) -> 2ndspos(X,from(0))
 plus(s(X),Y) -> s(plus(X,Y))
 plus(0,Y) -> Y
 s(X) -> n__s(X)
 square(X) -> times(X,X)
 times(s(X),Y) -> plus(Y,times(X,Y))
 times(0,Y) -> 0

Problem 1: 

SCC Processor:
-> Pairs:
 2NDSNEG(s(N),cons(X,Z)) -> 2NDSNEG(s(N),cons2(X,activate(Z)))
 2NDSNEG(s(N),cons(X,Z)) -> ACTIVATE(Z)
 2NDSNEG(s(N),cons2(X,cons(Y,Z))) -> 2NDSPOS(N,activate(Z))
 2NDSNEG(s(N),cons2(X,cons(Y,Z))) -> ACTIVATE(Z)
 2NDSPOS(s(N),cons(X,Z)) -> 2NDSPOS(s(N),cons2(X,activate(Z)))
 2NDSPOS(s(N),cons(X,Z)) -> ACTIVATE(Z)
 2NDSPOS(s(N),cons2(X,cons(Y,Z))) -> 2NDSNEG(N,activate(Z))
 2NDSPOS(s(N),cons2(X,cons(Y,Z))) -> ACTIVATE(Z)
 ACTIVATE(n__from(X)) -> ACTIVATE(X)
 ACTIVATE(n__from(X)) -> FROM(activate(X))
 ACTIVATE(n__s(X)) -> ACTIVATE(X)
 ACTIVATE(n__s(X)) -> S(activate(X))
 PI(X) -> 2NDSPOS(X,from(0))
 PI(X) -> FROM(0)
 PLUS(s(X),Y) -> PLUS(X,Y)
 PLUS(s(X),Y) -> S(plus(X,Y))
 SQUARE(X) -> TIMES(X,X)
 TIMES(s(X),Y) -> PLUS(Y,times(X,Y))
 TIMES(s(X),Y) -> TIMES(X,Y)
-> Rules:
 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
 2ndsneg(0,Z) -> rnil
 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
 2ndspos(0,Z) -> rnil
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 pi(X) -> 2ndspos(X,from(0))
 plus(s(X),Y) -> s(plus(X,Y))
 plus(0,Y) -> Y
 s(X) -> n__s(X)
 square(X) -> times(X,X)
 times(s(X),Y) -> plus(Y,times(X,Y))
 times(0,Y) -> 0
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(s(X),Y) -> PLUS(X,Y)
->->-> Rules:
 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
 2ndsneg(0,Z) -> rnil
 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
 2ndspos(0,Z) -> rnil
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 pi(X) -> 2ndspos(X,from(0))
 plus(s(X),Y) -> s(plus(X,Y))
 plus(0,Y) -> Y
 s(X) -> n__s(X)
 square(X) -> times(X,X)
 times(s(X),Y) -> plus(Y,times(X,Y))
 times(0,Y) -> 0
->->Cycle:
->->-> Pairs:
 TIMES(s(X),Y) -> TIMES(X,Y)
->->-> Rules:
 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
 2ndsneg(0,Z) -> rnil
 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
 2ndspos(0,Z) -> rnil
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 pi(X) -> 2ndspos(X,from(0))
 plus(s(X),Y) -> s(plus(X,Y))
 plus(0,Y) -> Y
 s(X) -> n__s(X)
 square(X) -> times(X,X)
 times(s(X),Y) -> plus(Y,times(X,Y))
 times(0,Y) -> 0
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__from(X)) -> ACTIVATE(X)
 ACTIVATE(n__s(X)) -> ACTIVATE(X)
->->-> Rules:
 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
 2ndsneg(0,Z) -> rnil
 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
 2ndspos(0,Z) -> rnil
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 pi(X) -> 2ndspos(X,from(0))
 plus(s(X),Y) -> s(plus(X,Y))
 plus(0,Y) -> Y
 s(X) -> n__s(X)
 square(X) -> times(X,X)
 times(s(X),Y) -> plus(Y,times(X,Y))
 times(0,Y) -> 0
->->Cycle:
->->-> Pairs:
 2NDSNEG(s(N),cons(X,Z)) -> 2NDSNEG(s(N),cons2(X,activate(Z)))
 2NDSNEG(s(N),cons2(X,cons(Y,Z))) -> 2NDSPOS(N,activate(Z))
 2NDSPOS(s(N),cons(X,Z)) -> 2NDSPOS(s(N),cons2(X,activate(Z)))
 2NDSPOS(s(N),cons2(X,cons(Y,Z))) -> 2NDSNEG(N,activate(Z))
->->-> Rules:
 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
 2ndsneg(0,Z) -> rnil
 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
 2ndspos(0,Z) -> rnil
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 pi(X) -> 2ndspos(X,from(0))
 plus(s(X),Y) -> s(plus(X,Y))
 plus(0,Y) -> Y
 s(X) -> n__s(X)
 square(X) -> times(X,X)
 times(s(X),Y) -> plus(Y,times(X,Y))
 times(0,Y) -> 0


The problem is decomposed in 4 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 PLUS(s(X),Y) -> PLUS(X,Y)
-> Rules:
 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
 2ndsneg(0,Z) -> rnil
 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
 2ndspos(0,Z) -> rnil
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 pi(X) -> 2ndspos(X,from(0))
 plus(s(X),Y) -> s(plus(X,Y))
 plus(0,Y) -> Y
 s(X) -> n__s(X)
 square(X) -> times(X,X)
 times(s(X),Y) -> plus(Y,times(X,Y))
 times(0,Y) -> 0
->Projection:
 pi(PLUS) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
 2ndsneg(0,Z) -> rnil
 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
 2ndspos(0,Z) -> rnil
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 pi(X) -> 2ndspos(X,from(0))
 plus(s(X),Y) -> s(plus(X,Y))
 plus(0,Y) -> Y
 s(X) -> n__s(X)
 square(X) -> times(X,X)
 times(s(X),Y) -> plus(Y,times(X,Y))
 times(0,Y) -> 0
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 TIMES(s(X),Y) -> TIMES(X,Y)
-> Rules:
 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
 2ndsneg(0,Z) -> rnil
 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
 2ndspos(0,Z) -> rnil
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 pi(X) -> 2ndspos(X,from(0))
 plus(s(X),Y) -> s(plus(X,Y))
 plus(0,Y) -> Y
 s(X) -> n__s(X)
 square(X) -> times(X,X)
 times(s(X),Y) -> plus(Y,times(X,Y))
 times(0,Y) -> 0
->Projection:
 pi(TIMES) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
 2ndsneg(0,Z) -> rnil
 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
 2ndspos(0,Z) -> rnil
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 pi(X) -> 2ndspos(X,from(0))
 plus(s(X),Y) -> s(plus(X,Y))
 plus(0,Y) -> Y
 s(X) -> n__s(X)
 square(X) -> times(X,X)
 times(s(X),Y) -> plus(Y,times(X,Y))
 times(0,Y) -> 0
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 ACTIVATE(n__from(X)) -> ACTIVATE(X)
 ACTIVATE(n__s(X)) -> ACTIVATE(X)
-> Rules:
 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
 2ndsneg(0,Z) -> rnil
 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
 2ndspos(0,Z) -> rnil
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 pi(X) -> 2ndspos(X,from(0))
 plus(s(X),Y) -> s(plus(X,Y))
 plus(0,Y) -> Y
 s(X) -> n__s(X)
 square(X) -> times(X,X)
 times(s(X),Y) -> plus(Y,times(X,Y))
 times(0,Y) -> 0
->Projection:
 pi(ACTIVATE) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
 2ndsneg(0,Z) -> rnil
 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
 2ndspos(0,Z) -> rnil
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 pi(X) -> 2ndspos(X,from(0))
 plus(s(X),Y) -> s(plus(X,Y))
 plus(0,Y) -> Y
 s(X) -> n__s(X)
 square(X) -> times(X,X)
 times(s(X),Y) -> plus(Y,times(X,Y))
 times(0,Y) -> 0
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Subterm Processor:
-> Pairs:
 2NDSNEG(s(N),cons(X,Z)) -> 2NDSNEG(s(N),cons2(X,activate(Z)))
 2NDSNEG(s(N),cons2(X,cons(Y,Z))) -> 2NDSPOS(N,activate(Z))
 2NDSPOS(s(N),cons(X,Z)) -> 2NDSPOS(s(N),cons2(X,activate(Z)))
 2NDSPOS(s(N),cons2(X,cons(Y,Z))) -> 2NDSNEG(N,activate(Z))
-> Rules:
 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
 2ndsneg(0,Z) -> rnil
 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
 2ndspos(0,Z) -> rnil
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 pi(X) -> 2ndspos(X,from(0))
 plus(s(X),Y) -> s(plus(X,Y))
 plus(0,Y) -> Y
 s(X) -> n__s(X)
 square(X) -> times(X,X)
 times(s(X),Y) -> plus(Y,times(X,Y))
 times(0,Y) -> 0
->Projection:
 pi(2NDSNEG) = 1
 pi(2NDSPOS) = 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 2NDSNEG(s(N),cons(X,Z)) -> 2NDSNEG(s(N),cons2(X,activate(Z)))
 2NDSPOS(s(N),cons(X,Z)) -> 2NDSPOS(s(N),cons2(X,activate(Z)))
-> Rules:
 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
 2ndsneg(0,Z) -> rnil
 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
 2ndspos(0,Z) -> rnil
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 pi(X) -> 2ndspos(X,from(0))
 plus(s(X),Y) -> s(plus(X,Y))
 plus(0,Y) -> Y
 s(X) -> n__s(X)
 square(X) -> times(X,X)
 times(s(X),Y) -> plus(Y,times(X,Y))
 times(0,Y) -> 0
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.