YES

Problem 1: 

(VAR v_NonEmpty:S N:S X:S X1:S X2:S Y:S Z:S)
(RULES
2ndsneg(0,Z:S) -> rnil
2ndsneg(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(negrecip(activate(Y:S)),2ndspos(N:S,activate(Z:S)))
2ndspos(0,Z:S) -> rnil
2ndspos(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(posrecip(activate(Y:S)),2ndsneg(N:S,activate(Z:S)))
activate(n__cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
activate(n__from(X:S)) -> from(X:S)
activate(X:S) -> X:S
cons(X1:S,X2:S) -> n__cons(X1:S,X2:S)
from(X:S) -> cons(X:S,n__from(s(X:S)))
from(X:S) -> n__from(X:S)
pi(X:S) -> 2ndspos(X:S,from(0))
plus(0,Y:S) -> Y:S
plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
square(X:S) -> times(X:S,X:S)
times(0,Y:S) -> 0
times(s(X:S),Y:S) -> plus(Y:S,times(X:S,Y:S))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 2NDSNEG(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> 2NDSPOS(N:S,activate(Z:S))
 2NDSNEG(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> ACTIVATE(Y:S)
 2NDSNEG(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> ACTIVATE(Z:S)
 2NDSPOS(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> 2NDSNEG(N:S,activate(Z:S))
 2NDSPOS(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> ACTIVATE(Y:S)
 2NDSPOS(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> ACTIVATE(Z:S)
 ACTIVATE(n__cons(X1:S,X2:S)) -> CONS(X1:S,X2:S)
 ACTIVATE(n__from(X:S)) -> FROM(X:S)
 FROM(X:S) -> CONS(X:S,n__from(s(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)
 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(0,Z:S) -> rnil
 2ndsneg(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(negrecip(activate(Y:S)),2ndspos(N:S,activate(Z:S)))
 2ndspos(0,Z:S) -> rnil
 2ndspos(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(posrecip(activate(Y:S)),2ndsneg(N:S,activate(Z:S)))
 activate(n__cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
 activate(n__from(X:S)) -> from(X:S)
 activate(X:S) -> X:S
 cons(X1:S,X2:S) -> n__cons(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(s(X:S)))
 from(X:S) -> n__from(X:S)
 pi(X:S) -> 2ndspos(X:S,from(0))
 plus(0,Y:S) -> Y:S
 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
 square(X:S) -> times(X:S,X:S)
 times(0,Y:S) -> 0
 times(s(X:S),Y:S) -> plus(Y:S,times(X:S,Y:S))

Problem 1: 

SCC Processor:
-> Pairs:
 2NDSNEG(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> 2NDSPOS(N:S,activate(Z:S))
 2NDSNEG(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> ACTIVATE(Y:S)
 2NDSNEG(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> ACTIVATE(Z:S)
 2NDSPOS(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> 2NDSNEG(N:S,activate(Z:S))
 2NDSPOS(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> ACTIVATE(Y:S)
 2NDSPOS(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> ACTIVATE(Z:S)
 ACTIVATE(n__cons(X1:S,X2:S)) -> CONS(X1:S,X2:S)
 ACTIVATE(n__from(X:S)) -> FROM(X:S)
 FROM(X:S) -> CONS(X:S,n__from(s(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)
 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(0,Z:S) -> rnil
 2ndsneg(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(negrecip(activate(Y:S)),2ndspos(N:S,activate(Z:S)))
 2ndspos(0,Z:S) -> rnil
 2ndspos(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(posrecip(activate(Y:S)),2ndsneg(N:S,activate(Z:S)))
 activate(n__cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
 activate(n__from(X:S)) -> from(X:S)
 activate(X:S) -> X:S
 cons(X1:S,X2:S) -> n__cons(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(s(X:S)))
 from(X:S) -> n__from(X:S)
 pi(X:S) -> 2ndspos(X:S,from(0))
 plus(0,Y:S) -> Y:S
 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
 square(X:S) -> times(X:S,X:S)
 times(0,Y:S) -> 0
 times(s(X:S),Y:S) -> plus(Y:S,times(X:S,Y:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(s(X:S),Y:S) -> PLUS(X:S,Y:S)
->->-> Rules:
 2ndsneg(0,Z:S) -> rnil
 2ndsneg(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(negrecip(activate(Y:S)),2ndspos(N:S,activate(Z:S)))
 2ndspos(0,Z:S) -> rnil
 2ndspos(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(posrecip(activate(Y:S)),2ndsneg(N:S,activate(Z:S)))
 activate(n__cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
 activate(n__from(X:S)) -> from(X:S)
 activate(X:S) -> X:S
 cons(X1:S,X2:S) -> n__cons(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(s(X:S)))
 from(X:S) -> n__from(X:S)
 pi(X:S) -> 2ndspos(X:S,from(0))
 plus(0,Y:S) -> Y:S
 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
 square(X:S) -> times(X:S,X:S)
 times(0,Y:S) -> 0
 times(s(X:S),Y:S) -> plus(Y:S,times(X:S,Y:S))
->->Cycle:
->->-> Pairs:
 TIMES(s(X:S),Y:S) -> TIMES(X:S,Y:S)
->->-> Rules:
 2ndsneg(0,Z:S) -> rnil
 2ndsneg(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(negrecip(activate(Y:S)),2ndspos(N:S,activate(Z:S)))
 2ndspos(0,Z:S) -> rnil
 2ndspos(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(posrecip(activate(Y:S)),2ndsneg(N:S,activate(Z:S)))
 activate(n__cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
 activate(n__from(X:S)) -> from(X:S)
 activate(X:S) -> X:S
 cons(X1:S,X2:S) -> n__cons(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(s(X:S)))
 from(X:S) -> n__from(X:S)
 pi(X:S) -> 2ndspos(X:S,from(0))
 plus(0,Y:S) -> Y:S
 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
 square(X:S) -> times(X:S,X:S)
 times(0,Y:S) -> 0
 times(s(X:S),Y:S) -> plus(Y:S,times(X:S,Y:S))
->->Cycle:
->->-> Pairs:
 2NDSNEG(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> 2NDSPOS(N:S,activate(Z:S))
 2NDSPOS(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> 2NDSNEG(N:S,activate(Z:S))
->->-> Rules:
 2ndsneg(0,Z:S) -> rnil
 2ndsneg(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(negrecip(activate(Y:S)),2ndspos(N:S,activate(Z:S)))
 2ndspos(0,Z:S) -> rnil
 2ndspos(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(posrecip(activate(Y:S)),2ndsneg(N:S,activate(Z:S)))
 activate(n__cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
 activate(n__from(X:S)) -> from(X:S)
 activate(X:S) -> X:S
 cons(X1:S,X2:S) -> n__cons(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(s(X:S)))
 from(X:S) -> n__from(X:S)
 pi(X:S) -> 2ndspos(X:S,from(0))
 plus(0,Y:S) -> Y:S
 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
 square(X:S) -> times(X:S,X:S)
 times(0,Y:S) -> 0
 times(s(X:S),Y:S) -> plus(Y:S,times(X:S,Y:S))


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 PLUS(s(X:S),Y:S) -> PLUS(X:S,Y:S)
-> Rules:
 2ndsneg(0,Z:S) -> rnil
 2ndsneg(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(negrecip(activate(Y:S)),2ndspos(N:S,activate(Z:S)))
 2ndspos(0,Z:S) -> rnil
 2ndspos(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(posrecip(activate(Y:S)),2ndsneg(N:S,activate(Z:S)))
 activate(n__cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
 activate(n__from(X:S)) -> from(X:S)
 activate(X:S) -> X:S
 cons(X1:S,X2:S) -> n__cons(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(s(X:S)))
 from(X:S) -> n__from(X:S)
 pi(X:S) -> 2ndspos(X:S,from(0))
 plus(0,Y:S) -> Y:S
 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
 square(X:S) -> times(X:S,X:S)
 times(0,Y:S) -> 0
 times(s(X:S),Y:S) -> plus(Y:S,times(X:S,Y:S))
->Projection:
 pi(PLUS) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 2ndsneg(0,Z:S) -> rnil
 2ndsneg(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(negrecip(activate(Y:S)),2ndspos(N:S,activate(Z:S)))
 2ndspos(0,Z:S) -> rnil
 2ndspos(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(posrecip(activate(Y:S)),2ndsneg(N:S,activate(Z:S)))
 activate(n__cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
 activate(n__from(X:S)) -> from(X:S)
 activate(X:S) -> X:S
 cons(X1:S,X2:S) -> n__cons(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(s(X:S)))
 from(X:S) -> n__from(X:S)
 pi(X:S) -> 2ndspos(X:S,from(0))
 plus(0,Y:S) -> Y:S
 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
 square(X:S) -> times(X:S,X:S)
 times(0,Y:S) -> 0
 times(s(X:S),Y:S) -> plus(Y:S,times(X:S,Y:S))
->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(0,Z:S) -> rnil
 2ndsneg(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(negrecip(activate(Y:S)),2ndspos(N:S,activate(Z:S)))
 2ndspos(0,Z:S) -> rnil
 2ndspos(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(posrecip(activate(Y:S)),2ndsneg(N:S,activate(Z:S)))
 activate(n__cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
 activate(n__from(X:S)) -> from(X:S)
 activate(X:S) -> X:S
 cons(X1:S,X2:S) -> n__cons(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(s(X:S)))
 from(X:S) -> n__from(X:S)
 pi(X:S) -> 2ndspos(X:S,from(0))
 plus(0,Y:S) -> Y:S
 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
 square(X:S) -> times(X:S,X:S)
 times(0,Y:S) -> 0
 times(s(X:S),Y:S) -> plus(Y:S,times(X:S,Y:S))
->Projection:
 pi(TIMES) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 2ndsneg(0,Z:S) -> rnil
 2ndsneg(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(negrecip(activate(Y:S)),2ndspos(N:S,activate(Z:S)))
 2ndspos(0,Z:S) -> rnil
 2ndspos(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(posrecip(activate(Y:S)),2ndsneg(N:S,activate(Z:S)))
 activate(n__cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
 activate(n__from(X:S)) -> from(X:S)
 activate(X:S) -> X:S
 cons(X1:S,X2:S) -> n__cons(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(s(X:S)))
 from(X:S) -> n__from(X:S)
 pi(X:S) -> 2ndspos(X:S,from(0))
 plus(0,Y:S) -> Y:S
 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
 square(X:S) -> times(X:S,X:S)
 times(0,Y:S) -> 0
 times(s(X:S),Y:S) -> plus(Y:S,times(X:S,Y:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 2NDSNEG(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> 2NDSPOS(N:S,activate(Z:S))
 2NDSPOS(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> 2NDSNEG(N:S,activate(Z:S))
-> Rules:
 2ndsneg(0,Z:S) -> rnil
 2ndsneg(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(negrecip(activate(Y:S)),2ndspos(N:S,activate(Z:S)))
 2ndspos(0,Z:S) -> rnil
 2ndspos(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(posrecip(activate(Y:S)),2ndsneg(N:S,activate(Z:S)))
 activate(n__cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
 activate(n__from(X:S)) -> from(X:S)
 activate(X:S) -> X:S
 cons(X1:S,X2:S) -> n__cons(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(s(X:S)))
 from(X:S) -> n__from(X:S)
 pi(X:S) -> 2ndspos(X:S,from(0))
 plus(0,Y:S) -> Y:S
 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
 square(X:S) -> times(X:S,X:S)
 times(0,Y:S) -> 0
 times(s(X:S),Y:S) -> plus(Y:S,times(X:S,Y:S))
->Projection:
 pi(2NDSNEG) = 1
 pi(2NDSPOS) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 2ndsneg(0,Z:S) -> rnil
 2ndsneg(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(negrecip(activate(Y:S)),2ndspos(N:S,activate(Z:S)))
 2ndspos(0,Z:S) -> rnil
 2ndspos(s(N:S),cons(X:S,n__cons(Y:S,Z:S))) -> rcons(posrecip(activate(Y:S)),2ndsneg(N:S,activate(Z:S)))
 activate(n__cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
 activate(n__from(X:S)) -> from(X:S)
 activate(X:S) -> X:S
 cons(X1:S,X2:S) -> n__cons(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(s(X:S)))
 from(X:S) -> n__from(X:S)
 pi(X:S) -> 2ndspos(X:S,from(0))
 plus(0,Y:S) -> Y:S
 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
 square(X:S) -> times(X:S,X:S)
 times(0,Y:S) -> 0
 times(s(X:S),Y:S) -> plus(Y:S,times(X:S,Y:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.