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

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 4 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.

Problem 1.3: 

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

Problem 1.3: 

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

The problem is finite.

Problem 1.4: 

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

Problem 1.4: 

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

The problem is finite.