YES

Problem 1: 

(VAR v_NonEmpty:S N:S X:S X1:S X2:S Y:S Z:S)
(RULES
active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
active(add(0,X:S)) -> mark(X:S)
active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
active(dbl(0)) -> mark(0)
active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
active(first(0,X:S)) -> mark(nil)
active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
active(sqr(0)) -> mark(0)
active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
add(active(X1:S),X2:S) -> add(X1:S,X2:S)
add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
dbl(active(X:S)) -> dbl(X:S)
dbl(mark(X:S)) -> dbl(X:S)
first(active(X1:S),X2:S) -> first(X1:S,X2:S)
first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
mark(dbl(X:S)) -> active(dbl(mark(X:S)))
mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
mark(recip(X:S)) -> active(recip(mark(X:S)))
mark(s(X:S)) -> active(s(X:S))
mark(sqr(X:S)) -> active(sqr(mark(X:S)))
mark(terms(X:S)) -> active(terms(mark(X:S)))
mark(0) -> active(0)
mark(nil) -> active(nil)
recip(active(X:S)) -> recip(X:S)
recip(mark(X:S)) -> recip(X:S)
s(active(X:S)) -> s(X:S)
s(mark(X:S)) -> s(X:S)
sqr(active(X:S)) -> sqr(X:S)
sqr(mark(X:S)) -> sqr(X:S)
terms(active(X:S)) -> terms(X:S)
terms(mark(X:S)) -> terms(X:S)
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVE(add(s(X:S),Y:S)) -> ADD(X:S,Y:S)
 ACTIVE(add(s(X:S),Y:S)) -> MARK(s(add(X:S,Y:S)))
 ACTIVE(add(s(X:S),Y:S)) -> S(add(X:S,Y:S))
 ACTIVE(add(0,X:S)) -> MARK(X:S)
 ACTIVE(dbl(s(X:S))) -> DBL(X:S)
 ACTIVE(dbl(s(X:S))) -> MARK(s(s(dbl(X:S))))
 ACTIVE(dbl(s(X:S))) -> S(dbl(X:S))
 ACTIVE(dbl(s(X:S))) -> S(s(dbl(X:S)))
 ACTIVE(dbl(0)) -> MARK(0)
 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> CONS(Y:S,first(X:S,Z:S))
 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> FIRST(X:S,Z:S)
 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> MARK(cons(Y:S,first(X:S,Z:S)))
 ACTIVE(first(0,X:S)) -> MARK(nil)
 ACTIVE(sqr(s(X:S))) -> ADD(sqr(X:S),dbl(X:S))
 ACTIVE(sqr(s(X:S))) -> DBL(X:S)
 ACTIVE(sqr(s(X:S))) -> MARK(s(add(sqr(X:S),dbl(X:S))))
 ACTIVE(sqr(s(X:S))) -> S(add(sqr(X:S),dbl(X:S)))
 ACTIVE(sqr(s(X:S))) -> SQR(X:S)
 ACTIVE(sqr(0)) -> MARK(0)
 ACTIVE(terms(N:S)) -> CONS(recip(sqr(N:S)),terms(s(N:S)))
 ACTIVE(terms(N:S)) -> MARK(cons(recip(sqr(N:S)),terms(s(N:S))))
 ACTIVE(terms(N:S)) -> RECIP(sqr(N:S))
 ACTIVE(terms(N:S)) -> S(N:S)
 ACTIVE(terms(N:S)) -> SQR(N:S)
 ACTIVE(terms(N:S)) -> TERMS(s(N:S))
 ADD(active(X1:S),X2:S) -> ADD(X1:S,X2:S)
 ADD(mark(X1:S),X2:S) -> ADD(X1:S,X2:S)
 ADD(X1:S,active(X2:S)) -> ADD(X1:S,X2:S)
 ADD(X1:S,mark(X2:S)) -> ADD(X1:S,X2:S)
 CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S)
 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S)
 DBL(active(X:S)) -> DBL(X:S)
 DBL(mark(X:S)) -> DBL(X:S)
 FIRST(active(X1:S),X2:S) -> FIRST(X1:S,X2:S)
 FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S)
 FIRST(X1:S,active(X2:S)) -> FIRST(X1:S,X2:S)
 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S)
 MARK(add(X1:S,X2:S)) -> ACTIVE(add(mark(X1:S),mark(X2:S)))
 MARK(add(X1:S,X2:S)) -> ADD(mark(X1:S),mark(X2:S))
 MARK(add(X1:S,X2:S)) -> MARK(X1:S)
 MARK(add(X1:S,X2:S)) -> MARK(X2:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> CONS(mark(X1:S),X2:S)
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(dbl(X:S)) -> ACTIVE(dbl(mark(X:S)))
 MARK(dbl(X:S)) -> DBL(mark(X:S))
 MARK(dbl(X:S)) -> MARK(X:S)
 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S)))
 MARK(first(X1:S,X2:S)) -> FIRST(mark(X1:S),mark(X2:S))
 MARK(first(X1:S,X2:S)) -> MARK(X1:S)
 MARK(first(X1:S,X2:S)) -> MARK(X2:S)
 MARK(recip(X:S)) -> ACTIVE(recip(mark(X:S)))
 MARK(recip(X:S)) -> MARK(X:S)
 MARK(recip(X:S)) -> RECIP(mark(X:S))
 MARK(s(X:S)) -> ACTIVE(s(X:S))
 MARK(sqr(X:S)) -> ACTIVE(sqr(mark(X:S)))
 MARK(sqr(X:S)) -> MARK(X:S)
 MARK(sqr(X:S)) -> SQR(mark(X:S))
 MARK(terms(X:S)) -> ACTIVE(terms(mark(X:S)))
 MARK(terms(X:S)) -> MARK(X:S)
 MARK(terms(X:S)) -> TERMS(mark(X:S))
 RECIP(active(X:S)) -> RECIP(X:S)
 RECIP(mark(X:S)) -> RECIP(X:S)
 S(active(X:S)) -> S(X:S)
 S(mark(X:S)) -> S(X:S)
 SQR(active(X:S)) -> SQR(X:S)
 SQR(mark(X:S)) -> SQR(X:S)
 TERMS(active(X:S)) -> TERMS(X:S)
 TERMS(mark(X:S)) -> TERMS(X:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVE(add(s(X:S),Y:S)) -> ADD(X:S,Y:S)
 ACTIVE(add(s(X:S),Y:S)) -> MARK(s(add(X:S,Y:S)))
 ACTIVE(add(s(X:S),Y:S)) -> S(add(X:S,Y:S))
 ACTIVE(add(0,X:S)) -> MARK(X:S)
 ACTIVE(dbl(s(X:S))) -> DBL(X:S)
 ACTIVE(dbl(s(X:S))) -> MARK(s(s(dbl(X:S))))
 ACTIVE(dbl(s(X:S))) -> S(dbl(X:S))
 ACTIVE(dbl(s(X:S))) -> S(s(dbl(X:S)))
 ACTIVE(dbl(0)) -> MARK(0)
 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> CONS(Y:S,first(X:S,Z:S))
 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> FIRST(X:S,Z:S)
 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> MARK(cons(Y:S,first(X:S,Z:S)))
 ACTIVE(first(0,X:S)) -> MARK(nil)
 ACTIVE(sqr(s(X:S))) -> ADD(sqr(X:S),dbl(X:S))
 ACTIVE(sqr(s(X:S))) -> DBL(X:S)
 ACTIVE(sqr(s(X:S))) -> MARK(s(add(sqr(X:S),dbl(X:S))))
 ACTIVE(sqr(s(X:S))) -> S(add(sqr(X:S),dbl(X:S)))
 ACTIVE(sqr(s(X:S))) -> SQR(X:S)
 ACTIVE(sqr(0)) -> MARK(0)
 ACTIVE(terms(N:S)) -> CONS(recip(sqr(N:S)),terms(s(N:S)))
 ACTIVE(terms(N:S)) -> MARK(cons(recip(sqr(N:S)),terms(s(N:S))))
 ACTIVE(terms(N:S)) -> RECIP(sqr(N:S))
 ACTIVE(terms(N:S)) -> S(N:S)
 ACTIVE(terms(N:S)) -> SQR(N:S)
 ACTIVE(terms(N:S)) -> TERMS(s(N:S))
 ADD(active(X1:S),X2:S) -> ADD(X1:S,X2:S)
 ADD(mark(X1:S),X2:S) -> ADD(X1:S,X2:S)
 ADD(X1:S,active(X2:S)) -> ADD(X1:S,X2:S)
 ADD(X1:S,mark(X2:S)) -> ADD(X1:S,X2:S)
 CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S)
 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S)
 DBL(active(X:S)) -> DBL(X:S)
 DBL(mark(X:S)) -> DBL(X:S)
 FIRST(active(X1:S),X2:S) -> FIRST(X1:S,X2:S)
 FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S)
 FIRST(X1:S,active(X2:S)) -> FIRST(X1:S,X2:S)
 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S)
 MARK(add(X1:S,X2:S)) -> ACTIVE(add(mark(X1:S),mark(X2:S)))
 MARK(add(X1:S,X2:S)) -> ADD(mark(X1:S),mark(X2:S))
 MARK(add(X1:S,X2:S)) -> MARK(X1:S)
 MARK(add(X1:S,X2:S)) -> MARK(X2:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> CONS(mark(X1:S),X2:S)
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(dbl(X:S)) -> ACTIVE(dbl(mark(X:S)))
 MARK(dbl(X:S)) -> DBL(mark(X:S))
 MARK(dbl(X:S)) -> MARK(X:S)
 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S)))
 MARK(first(X1:S,X2:S)) -> FIRST(mark(X1:S),mark(X2:S))
 MARK(first(X1:S,X2:S)) -> MARK(X1:S)
 MARK(first(X1:S,X2:S)) -> MARK(X2:S)
 MARK(recip(X:S)) -> ACTIVE(recip(mark(X:S)))
 MARK(recip(X:S)) -> MARK(X:S)
 MARK(recip(X:S)) -> RECIP(mark(X:S))
 MARK(s(X:S)) -> ACTIVE(s(X:S))
 MARK(sqr(X:S)) -> ACTIVE(sqr(mark(X:S)))
 MARK(sqr(X:S)) -> MARK(X:S)
 MARK(sqr(X:S)) -> SQR(mark(X:S))
 MARK(terms(X:S)) -> ACTIVE(terms(mark(X:S)))
 MARK(terms(X:S)) -> MARK(X:S)
 MARK(terms(X:S)) -> TERMS(mark(X:S))
 RECIP(active(X:S)) -> RECIP(X:S)
 RECIP(mark(X:S)) -> RECIP(X:S)
 S(active(X:S)) -> S(X:S)
 S(mark(X:S)) -> S(X:S)
 SQR(active(X:S)) -> SQR(X:S)
 SQR(mark(X:S)) -> SQR(X:S)
 TERMS(active(X:S)) -> TERMS(X:S)
 TERMS(mark(X:S)) -> TERMS(X:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TERMS(active(X:S)) -> TERMS(X:S)
 TERMS(mark(X:S)) -> TERMS(X:S)
->->-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->->Cycle:
->->-> Pairs:
 SQR(active(X:S)) -> SQR(X:S)
 SQR(mark(X:S)) -> SQR(X:S)
->->-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->->Cycle:
->->-> Pairs:
 S(active(X:S)) -> S(X:S)
 S(mark(X:S)) -> S(X:S)
->->-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->->Cycle:
->->-> Pairs:
 RECIP(active(X:S)) -> RECIP(X:S)
 RECIP(mark(X:S)) -> RECIP(X:S)
->->-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->->Cycle:
->->-> Pairs:
 FIRST(active(X1:S),X2:S) -> FIRST(X1:S,X2:S)
 FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S)
 FIRST(X1:S,active(X2:S)) -> FIRST(X1:S,X2:S)
 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S)
->->-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->->Cycle:
->->-> Pairs:
 DBL(active(X:S)) -> DBL(X:S)
 DBL(mark(X:S)) -> DBL(X:S)
->->-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->->Cycle:
->->-> Pairs:
 CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S)
 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S)
->->-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->->Cycle:
->->-> Pairs:
 ADD(active(X1:S),X2:S) -> ADD(X1:S,X2:S)
 ADD(mark(X1:S),X2:S) -> ADD(X1:S,X2:S)
 ADD(X1:S,active(X2:S)) -> ADD(X1:S,X2:S)
 ADD(X1:S,mark(X2:S)) -> ADD(X1:S,X2:S)
->->-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->->Cycle:
->->-> Pairs:
 ACTIVE(add(s(X:S),Y:S)) -> MARK(s(add(X:S,Y:S)))
 ACTIVE(add(0,X:S)) -> MARK(X:S)
 ACTIVE(dbl(s(X:S))) -> MARK(s(s(dbl(X:S))))
 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> MARK(cons(Y:S,first(X:S,Z:S)))
 ACTIVE(sqr(s(X:S))) -> MARK(s(add(sqr(X:S),dbl(X:S))))
 ACTIVE(terms(N:S)) -> MARK(cons(recip(sqr(N:S)),terms(s(N:S))))
 MARK(add(X1:S,X2:S)) -> ACTIVE(add(mark(X1:S),mark(X2:S)))
 MARK(add(X1:S,X2:S)) -> MARK(X1:S)
 MARK(add(X1:S,X2:S)) -> MARK(X2:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(dbl(X:S)) -> ACTIVE(dbl(mark(X:S)))
 MARK(dbl(X:S)) -> MARK(X:S)
 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S)))
 MARK(first(X1:S,X2:S)) -> MARK(X1:S)
 MARK(first(X1:S,X2:S)) -> MARK(X2:S)
 MARK(recip(X:S)) -> ACTIVE(recip(mark(X:S)))
 MARK(recip(X:S)) -> MARK(X:S)
 MARK(sqr(X:S)) -> ACTIVE(sqr(mark(X:S)))
 MARK(sqr(X:S)) -> MARK(X:S)
 MARK(terms(X:S)) -> ACTIVE(terms(mark(X:S)))
 MARK(terms(X:S)) -> MARK(X:S)
->->-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)


The problem is decomposed in 9 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 TERMS(active(X:S)) -> TERMS(X:S)
 TERMS(mark(X:S)) -> TERMS(X:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Projection:
 pi(TERMS) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 SQR(active(X:S)) -> SQR(X:S)
 SQR(mark(X:S)) -> SQR(X:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Projection:
 pi(SQR) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 S(active(X:S)) -> S(X:S)
 S(mark(X:S)) -> S(X:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Projection:
 pi(S) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Subterm Processor:
-> Pairs:
 RECIP(active(X:S)) -> RECIP(X:S)
 RECIP(mark(X:S)) -> RECIP(X:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Projection:
 pi(RECIP) = 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.5: 

Subterm Processor:
-> Pairs:
 FIRST(active(X1:S),X2:S) -> FIRST(X1:S,X2:S)
 FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S)
 FIRST(X1:S,active(X2:S)) -> FIRST(X1:S,X2:S)
 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Projection:
 pi(FIRST) = 1

Problem 1.5: 

SCC Processor:
-> Pairs:
 FIRST(X1:S,active(X2:S)) -> FIRST(X1:S,X2:S)
 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 FIRST(X1:S,active(X2:S)) -> FIRST(X1:S,X2:S)
 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S)
->->-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)

Problem 1.5: 

Subterm Processor:
-> Pairs:
 FIRST(X1:S,active(X2:S)) -> FIRST(X1:S,X2:S)
 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Projection:
 pi(FIRST) = 2

Problem 1.5: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.6: 

Subterm Processor:
-> Pairs:
 DBL(active(X:S)) -> DBL(X:S)
 DBL(mark(X:S)) -> DBL(X:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Projection:
 pi(DBL) = 1

Problem 1.6: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.7: 

Subterm Processor:
-> Pairs:
 CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S)
 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Projection:
 pi(CONS) = 1

Problem 1.7: 

SCC Processor:
-> Pairs:
 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S)
 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S)
 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S)
->->-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)

Problem 1.7: 

Subterm Processor:
-> Pairs:
 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S)
 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Projection:
 pi(CONS) = 2

Problem 1.7: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.8: 

Subterm Processor:
-> Pairs:
 ADD(active(X1:S),X2:S) -> ADD(X1:S,X2:S)
 ADD(mark(X1:S),X2:S) -> ADD(X1:S,X2:S)
 ADD(X1:S,active(X2:S)) -> ADD(X1:S,X2:S)
 ADD(X1:S,mark(X2:S)) -> ADD(X1:S,X2:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Projection:
 pi(ADD) = 1

Problem 1.8: 

SCC Processor:
-> Pairs:
 ADD(X1:S,active(X2:S)) -> ADD(X1:S,X2:S)
 ADD(X1:S,mark(X2:S)) -> ADD(X1:S,X2:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ADD(X1:S,active(X2:S)) -> ADD(X1:S,X2:S)
 ADD(X1:S,mark(X2:S)) -> ADD(X1:S,X2:S)
->->-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)

Problem 1.8: 

Subterm Processor:
-> Pairs:
 ADD(X1:S,active(X2:S)) -> ADD(X1:S,X2:S)
 ADD(X1:S,mark(X2:S)) -> ADD(X1:S,X2:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Projection:
 pi(ADD) = 2

Problem 1.8: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.9: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(add(s(X:S),Y:S)) -> MARK(s(add(X:S,Y:S)))
 ACTIVE(add(0,X:S)) -> MARK(X:S)
 ACTIVE(dbl(s(X:S))) -> MARK(s(s(dbl(X:S))))
 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> MARK(cons(Y:S,first(X:S,Z:S)))
 ACTIVE(sqr(s(X:S))) -> MARK(s(add(sqr(X:S),dbl(X:S))))
 ACTIVE(terms(N:S)) -> MARK(cons(recip(sqr(N:S)),terms(s(N:S))))
 MARK(add(X1:S,X2:S)) -> ACTIVE(add(mark(X1:S),mark(X2:S)))
 MARK(add(X1:S,X2:S)) -> MARK(X1:S)
 MARK(add(X1:S,X2:S)) -> MARK(X2:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(dbl(X:S)) -> ACTIVE(dbl(mark(X:S)))
 MARK(dbl(X:S)) -> MARK(X:S)
 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S)))
 MARK(first(X1:S,X2:S)) -> MARK(X1:S)
 MARK(first(X1:S,X2:S)) -> MARK(X2:S)
 MARK(recip(X:S)) -> ACTIVE(recip(mark(X:S)))
 MARK(recip(X:S)) -> MARK(X:S)
 MARK(sqr(X:S)) -> ACTIVE(sqr(mark(X:S)))
 MARK(sqr(X:S)) -> MARK(X:S)
 MARK(terms(X:S)) -> ACTIVE(terms(mark(X:S)))
 MARK(terms(X:S)) -> MARK(X:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
-> Usable rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = X
[add](X1,X2) = 2.X1 + 2.X2 + 1
[cons](X1,X2) = X1
[dbl](X) = 2.X + 2
[first](X1,X2) = 2.X1 + 2.X2
[mark](X) = 2.X
[recip](X) = X
[s](X) = 1
[sqr](X) = X + 1
[terms](X) = 2.X + 2
[0] = 0
[fSNonEmpty] = 0
[nil] = 0
[ACTIVE](X) = X + 2
[ADD](X1,X2) = 0
[CONS](X1,X2) = 0
[DBL](X) = 0
[FIRST](X1,X2) = 0
[MARK](X) = 2.X + 2
[RECIP](X) = 0
[S](X) = 0
[SQR](X) = 0
[TERMS](X) = 0

Problem 1.9: 

SCC Processor:
-> Pairs:
 ACTIVE(add(0,X:S)) -> MARK(X:S)
 ACTIVE(dbl(s(X:S))) -> MARK(s(s(dbl(X:S))))
 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> MARK(cons(Y:S,first(X:S,Z:S)))
 ACTIVE(sqr(s(X:S))) -> MARK(s(add(sqr(X:S),dbl(X:S))))
 ACTIVE(terms(N:S)) -> MARK(cons(recip(sqr(N:S)),terms(s(N:S))))
 MARK(add(X1:S,X2:S)) -> ACTIVE(add(mark(X1:S),mark(X2:S)))
 MARK(add(X1:S,X2:S)) -> MARK(X1:S)
 MARK(add(X1:S,X2:S)) -> MARK(X2:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(dbl(X:S)) -> ACTIVE(dbl(mark(X:S)))
 MARK(dbl(X:S)) -> MARK(X:S)
 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S)))
 MARK(first(X1:S,X2:S)) -> MARK(X1:S)
 MARK(first(X1:S,X2:S)) -> MARK(X2:S)
 MARK(recip(X:S)) -> ACTIVE(recip(mark(X:S)))
 MARK(recip(X:S)) -> MARK(X:S)
 MARK(sqr(X:S)) -> ACTIVE(sqr(mark(X:S)))
 MARK(sqr(X:S)) -> MARK(X:S)
 MARK(terms(X:S)) -> ACTIVE(terms(mark(X:S)))
 MARK(terms(X:S)) -> MARK(X:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(add(0,X:S)) -> MARK(X:S)
 ACTIVE(dbl(s(X:S))) -> MARK(s(s(dbl(X:S))))
 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> MARK(cons(Y:S,first(X:S,Z:S)))
 ACTIVE(sqr(s(X:S))) -> MARK(s(add(sqr(X:S),dbl(X:S))))
 ACTIVE(terms(N:S)) -> MARK(cons(recip(sqr(N:S)),terms(s(N:S))))
 MARK(add(X1:S,X2:S)) -> ACTIVE(add(mark(X1:S),mark(X2:S)))
 MARK(add(X1:S,X2:S)) -> MARK(X1:S)
 MARK(add(X1:S,X2:S)) -> MARK(X2:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(dbl(X:S)) -> ACTIVE(dbl(mark(X:S)))
 MARK(dbl(X:S)) -> MARK(X:S)
 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S)))
 MARK(first(X1:S,X2:S)) -> MARK(X1:S)
 MARK(first(X1:S,X2:S)) -> MARK(X2:S)
 MARK(recip(X:S)) -> ACTIVE(recip(mark(X:S)))
 MARK(recip(X:S)) -> MARK(X:S)
 MARK(sqr(X:S)) -> ACTIVE(sqr(mark(X:S)))
 MARK(sqr(X:S)) -> MARK(X:S)
 MARK(terms(X:S)) -> ACTIVE(terms(mark(X:S)))
 MARK(terms(X:S)) -> MARK(X:S)
->->-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)

Problem 1.9: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(add(0,X:S)) -> MARK(X:S)
 ACTIVE(dbl(s(X:S))) -> MARK(s(s(dbl(X:S))))
 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> MARK(cons(Y:S,first(X:S,Z:S)))
 ACTIVE(sqr(s(X:S))) -> MARK(s(add(sqr(X:S),dbl(X:S))))
 ACTIVE(terms(N:S)) -> MARK(cons(recip(sqr(N:S)),terms(s(N:S))))
 MARK(add(X1:S,X2:S)) -> ACTIVE(add(mark(X1:S),mark(X2:S)))
 MARK(add(X1:S,X2:S)) -> MARK(X1:S)
 MARK(add(X1:S,X2:S)) -> MARK(X2:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(dbl(X:S)) -> ACTIVE(dbl(mark(X:S)))
 MARK(dbl(X:S)) -> MARK(X:S)
 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S)))
 MARK(first(X1:S,X2:S)) -> MARK(X1:S)
 MARK(first(X1:S,X2:S)) -> MARK(X2:S)
 MARK(recip(X:S)) -> ACTIVE(recip(mark(X:S)))
 MARK(recip(X:S)) -> MARK(X:S)
 MARK(sqr(X:S)) -> ACTIVE(sqr(mark(X:S)))
 MARK(sqr(X:S)) -> MARK(X:S)
 MARK(terms(X:S)) -> ACTIVE(terms(mark(X:S)))
 MARK(terms(X:S)) -> MARK(X:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
-> Usable rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = X
[add](X1,X2) = 2.X1 + 2.X2 + 2
[cons](X1,X2) = X1
[dbl](X) = X + 1
[first](X1,X2) = 2.X1 + 2.X2 + 2
[mark](X) = 2.X
[recip](X) = X
[s](X) = 1
[sqr](X) = X + 1
[terms](X) = 2.X + 2
[0] = 0
[fSNonEmpty] = 0
[nil] = 1
[ACTIVE](X) = X + 2
[ADD](X1,X2) = 0
[CONS](X1,X2) = 0
[DBL](X) = 0
[FIRST](X1,X2) = 0
[MARK](X) = 2.X + 2
[RECIP](X) = 0
[S](X) = 0
[SQR](X) = 0
[TERMS](X) = 0

Problem 1.9: 

SCC Processor:
-> Pairs:
 ACTIVE(dbl(s(X:S))) -> MARK(s(s(dbl(X:S))))
 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> MARK(cons(Y:S,first(X:S,Z:S)))
 ACTIVE(sqr(s(X:S))) -> MARK(s(add(sqr(X:S),dbl(X:S))))
 ACTIVE(terms(N:S)) -> MARK(cons(recip(sqr(N:S)),terms(s(N:S))))
 MARK(add(X1:S,X2:S)) -> ACTIVE(add(mark(X1:S),mark(X2:S)))
 MARK(add(X1:S,X2:S)) -> MARK(X1:S)
 MARK(add(X1:S,X2:S)) -> MARK(X2:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(dbl(X:S)) -> ACTIVE(dbl(mark(X:S)))
 MARK(dbl(X:S)) -> MARK(X:S)
 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S)))
 MARK(first(X1:S,X2:S)) -> MARK(X1:S)
 MARK(first(X1:S,X2:S)) -> MARK(X2:S)
 MARK(recip(X:S)) -> ACTIVE(recip(mark(X:S)))
 MARK(recip(X:S)) -> MARK(X:S)
 MARK(sqr(X:S)) -> ACTIVE(sqr(mark(X:S)))
 MARK(sqr(X:S)) -> MARK(X:S)
 MARK(terms(X:S)) -> ACTIVE(terms(mark(X:S)))
 MARK(terms(X:S)) -> MARK(X:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(dbl(s(X:S))) -> MARK(s(s(dbl(X:S))))
 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> MARK(cons(Y:S,first(X:S,Z:S)))
 ACTIVE(sqr(s(X:S))) -> MARK(s(add(sqr(X:S),dbl(X:S))))
 ACTIVE(terms(N:S)) -> MARK(cons(recip(sqr(N:S)),terms(s(N:S))))
 MARK(add(X1:S,X2:S)) -> ACTIVE(add(mark(X1:S),mark(X2:S)))
 MARK(add(X1:S,X2:S)) -> MARK(X1:S)
 MARK(add(X1:S,X2:S)) -> MARK(X2:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(dbl(X:S)) -> ACTIVE(dbl(mark(X:S)))
 MARK(dbl(X:S)) -> MARK(X:S)
 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S)))
 MARK(first(X1:S,X2:S)) -> MARK(X1:S)
 MARK(first(X1:S,X2:S)) -> MARK(X2:S)
 MARK(recip(X:S)) -> ACTIVE(recip(mark(X:S)))
 MARK(recip(X:S)) -> MARK(X:S)
 MARK(sqr(X:S)) -> ACTIVE(sqr(mark(X:S)))
 MARK(sqr(X:S)) -> MARK(X:S)
 MARK(terms(X:S)) -> ACTIVE(terms(mark(X:S)))
 MARK(terms(X:S)) -> MARK(X:S)
->->-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)

Problem 1.9: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(dbl(s(X:S))) -> MARK(s(s(dbl(X:S))))
 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> MARK(cons(Y:S,first(X:S,Z:S)))
 ACTIVE(sqr(s(X:S))) -> MARK(s(add(sqr(X:S),dbl(X:S))))
 ACTIVE(terms(N:S)) -> MARK(cons(recip(sqr(N:S)),terms(s(N:S))))
 MARK(add(X1:S,X2:S)) -> ACTIVE(add(mark(X1:S),mark(X2:S)))
 MARK(add(X1:S,X2:S)) -> MARK(X1:S)
 MARK(add(X1:S,X2:S)) -> MARK(X2:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(dbl(X:S)) -> ACTIVE(dbl(mark(X:S)))
 MARK(dbl(X:S)) -> MARK(X:S)
 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S)))
 MARK(first(X1:S,X2:S)) -> MARK(X1:S)
 MARK(first(X1:S,X2:S)) -> MARK(X2:S)
 MARK(recip(X:S)) -> ACTIVE(recip(mark(X:S)))
 MARK(recip(X:S)) -> MARK(X:S)
 MARK(sqr(X:S)) -> ACTIVE(sqr(mark(X:S)))
 MARK(sqr(X:S)) -> MARK(X:S)
 MARK(terms(X:S)) -> ACTIVE(terms(mark(X:S)))
 MARK(terms(X:S)) -> MARK(X:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
-> Usable rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = X
[add](X1,X2) = X1 + 2.X2
[cons](X1,X2) = X1
[dbl](X) = 2.X + 2
[first](X1,X2) = 2.X1 + 2.X2 + 2
[mark](X) = 2.X
[recip](X) = X
[s](X) = 0
[sqr](X) = X + 1
[terms](X) = 2.X + 2
[0] = 1
[fSNonEmpty] = 0
[nil] = 2
[ACTIVE](X) = X + 2
[ADD](X1,X2) = 0
[CONS](X1,X2) = 0
[DBL](X) = 0
[FIRST](X1,X2) = 0
[MARK](X) = 2.X + 2
[RECIP](X) = 0
[S](X) = 0
[SQR](X) = 0
[TERMS](X) = 0

Problem 1.9: 

SCC Processor:
-> Pairs:
 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> MARK(cons(Y:S,first(X:S,Z:S)))
 ACTIVE(sqr(s(X:S))) -> MARK(s(add(sqr(X:S),dbl(X:S))))
 ACTIVE(terms(N:S)) -> MARK(cons(recip(sqr(N:S)),terms(s(N:S))))
 MARK(add(X1:S,X2:S)) -> ACTIVE(add(mark(X1:S),mark(X2:S)))
 MARK(add(X1:S,X2:S)) -> MARK(X1:S)
 MARK(add(X1:S,X2:S)) -> MARK(X2:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(dbl(X:S)) -> ACTIVE(dbl(mark(X:S)))
 MARK(dbl(X:S)) -> MARK(X:S)
 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S)))
 MARK(first(X1:S,X2:S)) -> MARK(X1:S)
 MARK(first(X1:S,X2:S)) -> MARK(X2:S)
 MARK(recip(X:S)) -> ACTIVE(recip(mark(X:S)))
 MARK(recip(X:S)) -> MARK(X:S)
 MARK(sqr(X:S)) -> ACTIVE(sqr(mark(X:S)))
 MARK(sqr(X:S)) -> MARK(X:S)
 MARK(terms(X:S)) -> ACTIVE(terms(mark(X:S)))
 MARK(terms(X:S)) -> MARK(X:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> MARK(cons(Y:S,first(X:S,Z:S)))
 ACTIVE(sqr(s(X:S))) -> MARK(s(add(sqr(X:S),dbl(X:S))))
 ACTIVE(terms(N:S)) -> MARK(cons(recip(sqr(N:S)),terms(s(N:S))))
 MARK(add(X1:S,X2:S)) -> ACTIVE(add(mark(X1:S),mark(X2:S)))
 MARK(add(X1:S,X2:S)) -> MARK(X1:S)
 MARK(add(X1:S,X2:S)) -> MARK(X2:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(dbl(X:S)) -> ACTIVE(dbl(mark(X:S)))
 MARK(dbl(X:S)) -> MARK(X:S)
 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S)))
 MARK(first(X1:S,X2:S)) -> MARK(X1:S)
 MARK(first(X1:S,X2:S)) -> MARK(X2:S)
 MARK(recip(X:S)) -> ACTIVE(recip(mark(X:S)))
 MARK(recip(X:S)) -> MARK(X:S)
 MARK(sqr(X:S)) -> ACTIVE(sqr(mark(X:S)))
 MARK(sqr(X:S)) -> MARK(X:S)
 MARK(terms(X:S)) -> ACTIVE(terms(mark(X:S)))
 MARK(terms(X:S)) -> MARK(X:S)
->->-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)

Problem 1.9: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> MARK(cons(Y:S,first(X:S,Z:S)))
 ACTIVE(sqr(s(X:S))) -> MARK(s(add(sqr(X:S),dbl(X:S))))
 ACTIVE(terms(N:S)) -> MARK(cons(recip(sqr(N:S)),terms(s(N:S))))
 MARK(add(X1:S,X2:S)) -> ACTIVE(add(mark(X1:S),mark(X2:S)))
 MARK(add(X1:S,X2:S)) -> MARK(X1:S)
 MARK(add(X1:S,X2:S)) -> MARK(X2:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(dbl(X:S)) -> ACTIVE(dbl(mark(X:S)))
 MARK(dbl(X:S)) -> MARK(X:S)
 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S)))
 MARK(first(X1:S,X2:S)) -> MARK(X1:S)
 MARK(first(X1:S,X2:S)) -> MARK(X2:S)
 MARK(recip(X:S)) -> ACTIVE(recip(mark(X:S)))
 MARK(recip(X:S)) -> MARK(X:S)
 MARK(sqr(X:S)) -> ACTIVE(sqr(mark(X:S)))
 MARK(sqr(X:S)) -> MARK(X:S)
 MARK(terms(X:S)) -> ACTIVE(terms(mark(X:S)))
 MARK(terms(X:S)) -> MARK(X:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
-> Usable rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = X
[add](X1,X2) = X1 + 2.X2 + 2
[cons](X1,X2) = X1
[dbl](X) = 2.X + 1
[first](X1,X2) = 2.X1 + 2.X2 + 2
[mark](X) = 2.X
[recip](X) = X + 1
[s](X) = 0
[sqr](X) = X
[terms](X) = 2.X + 2
[0] = 0
[fSNonEmpty] = 0
[nil] = 1
[ACTIVE](X) = X + 1
[ADD](X1,X2) = 0
[CONS](X1,X2) = 0
[DBL](X) = 0
[FIRST](X1,X2) = 0
[MARK](X) = 2.X + 1
[RECIP](X) = 0
[S](X) = 0
[SQR](X) = 0
[TERMS](X) = 0

Problem 1.9: 

SCC Processor:
-> Pairs:
 ACTIVE(sqr(s(X:S))) -> MARK(s(add(sqr(X:S),dbl(X:S))))
 ACTIVE(terms(N:S)) -> MARK(cons(recip(sqr(N:S)),terms(s(N:S))))
 MARK(add(X1:S,X2:S)) -> ACTIVE(add(mark(X1:S),mark(X2:S)))
 MARK(add(X1:S,X2:S)) -> MARK(X1:S)
 MARK(add(X1:S,X2:S)) -> MARK(X2:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(dbl(X:S)) -> ACTIVE(dbl(mark(X:S)))
 MARK(dbl(X:S)) -> MARK(X:S)
 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S)))
 MARK(first(X1:S,X2:S)) -> MARK(X1:S)
 MARK(first(X1:S,X2:S)) -> MARK(X2:S)
 MARK(recip(X:S)) -> ACTIVE(recip(mark(X:S)))
 MARK(recip(X:S)) -> MARK(X:S)
 MARK(sqr(X:S)) -> ACTIVE(sqr(mark(X:S)))
 MARK(sqr(X:S)) -> MARK(X:S)
 MARK(terms(X:S)) -> ACTIVE(terms(mark(X:S)))
 MARK(terms(X:S)) -> MARK(X:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(sqr(s(X:S))) -> MARK(s(add(sqr(X:S),dbl(X:S))))
 ACTIVE(terms(N:S)) -> MARK(cons(recip(sqr(N:S)),terms(s(N:S))))
 MARK(add(X1:S,X2:S)) -> ACTIVE(add(mark(X1:S),mark(X2:S)))
 MARK(add(X1:S,X2:S)) -> MARK(X1:S)
 MARK(add(X1:S,X2:S)) -> MARK(X2:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(dbl(X:S)) -> ACTIVE(dbl(mark(X:S)))
 MARK(dbl(X:S)) -> MARK(X:S)
 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S)))
 MARK(first(X1:S,X2:S)) -> MARK(X1:S)
 MARK(first(X1:S,X2:S)) -> MARK(X2:S)
 MARK(recip(X:S)) -> ACTIVE(recip(mark(X:S)))
 MARK(recip(X:S)) -> MARK(X:S)
 MARK(sqr(X:S)) -> ACTIVE(sqr(mark(X:S)))
 MARK(sqr(X:S)) -> MARK(X:S)
 MARK(terms(X:S)) -> ACTIVE(terms(mark(X:S)))
 MARK(terms(X:S)) -> MARK(X:S)
->->-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)

Problem 1.9: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(sqr(s(X:S))) -> MARK(s(add(sqr(X:S),dbl(X:S))))
 ACTIVE(terms(N:S)) -> MARK(cons(recip(sqr(N:S)),terms(s(N:S))))
 MARK(add(X1:S,X2:S)) -> ACTIVE(add(mark(X1:S),mark(X2:S)))
 MARK(add(X1:S,X2:S)) -> MARK(X1:S)
 MARK(add(X1:S,X2:S)) -> MARK(X2:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(dbl(X:S)) -> ACTIVE(dbl(mark(X:S)))
 MARK(dbl(X:S)) -> MARK(X:S)
 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S)))
 MARK(first(X1:S,X2:S)) -> MARK(X1:S)
 MARK(first(X1:S,X2:S)) -> MARK(X2:S)
 MARK(recip(X:S)) -> ACTIVE(recip(mark(X:S)))
 MARK(recip(X:S)) -> MARK(X:S)
 MARK(sqr(X:S)) -> ACTIVE(sqr(mark(X:S)))
 MARK(sqr(X:S)) -> MARK(X:S)
 MARK(terms(X:S)) -> ACTIVE(terms(mark(X:S)))
 MARK(terms(X:S)) -> MARK(X:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
-> Usable rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = X
[add](X1,X2) = X1 + 2.X2 + 2
[cons](X1,X2) = X1
[dbl](X) = 2.X + 2
[first](X1,X2) = 2.X1 + 2.X2
[mark](X) = 2.X
[recip](X) = X
[s](X) = 0
[sqr](X) = X + 1
[terms](X) = 2.X + 2
[0] = 0
[fSNonEmpty] = 0
[nil] = 0
[ACTIVE](X) = X + 2
[ADD](X1,X2) = 0
[CONS](X1,X2) = 0
[DBL](X) = 0
[FIRST](X1,X2) = 0
[MARK](X) = 2.X + 2
[RECIP](X) = 0
[S](X) = 0
[SQR](X) = 0
[TERMS](X) = 0

Problem 1.9: 

SCC Processor:
-> Pairs:
 ACTIVE(terms(N:S)) -> MARK(cons(recip(sqr(N:S)),terms(s(N:S))))
 MARK(add(X1:S,X2:S)) -> ACTIVE(add(mark(X1:S),mark(X2:S)))
 MARK(add(X1:S,X2:S)) -> MARK(X1:S)
 MARK(add(X1:S,X2:S)) -> MARK(X2:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(dbl(X:S)) -> ACTIVE(dbl(mark(X:S)))
 MARK(dbl(X:S)) -> MARK(X:S)
 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S)))
 MARK(first(X1:S,X2:S)) -> MARK(X1:S)
 MARK(first(X1:S,X2:S)) -> MARK(X2:S)
 MARK(recip(X:S)) -> ACTIVE(recip(mark(X:S)))
 MARK(recip(X:S)) -> MARK(X:S)
 MARK(sqr(X:S)) -> ACTIVE(sqr(mark(X:S)))
 MARK(sqr(X:S)) -> MARK(X:S)
 MARK(terms(X:S)) -> ACTIVE(terms(mark(X:S)))
 MARK(terms(X:S)) -> MARK(X:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(terms(N:S)) -> MARK(cons(recip(sqr(N:S)),terms(s(N:S))))
 MARK(add(X1:S,X2:S)) -> ACTIVE(add(mark(X1:S),mark(X2:S)))
 MARK(add(X1:S,X2:S)) -> MARK(X1:S)
 MARK(add(X1:S,X2:S)) -> MARK(X2:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(dbl(X:S)) -> ACTIVE(dbl(mark(X:S)))
 MARK(dbl(X:S)) -> MARK(X:S)
 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S)))
 MARK(first(X1:S,X2:S)) -> MARK(X1:S)
 MARK(first(X1:S,X2:S)) -> MARK(X2:S)
 MARK(recip(X:S)) -> ACTIVE(recip(mark(X:S)))
 MARK(recip(X:S)) -> MARK(X:S)
 MARK(sqr(X:S)) -> ACTIVE(sqr(mark(X:S)))
 MARK(sqr(X:S)) -> MARK(X:S)
 MARK(terms(X:S)) -> ACTIVE(terms(mark(X:S)))
 MARK(terms(X:S)) -> MARK(X:S)
->->-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)

Problem 1.9: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(terms(N:S)) -> MARK(cons(recip(sqr(N:S)),terms(s(N:S))))
 MARK(add(X1:S,X2:S)) -> ACTIVE(add(mark(X1:S),mark(X2:S)))
 MARK(add(X1:S,X2:S)) -> MARK(X1:S)
 MARK(add(X1:S,X2:S)) -> MARK(X2:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(dbl(X:S)) -> ACTIVE(dbl(mark(X:S)))
 MARK(dbl(X:S)) -> MARK(X:S)
 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S)))
 MARK(first(X1:S,X2:S)) -> MARK(X1:S)
 MARK(first(X1:S,X2:S)) -> MARK(X2:S)
 MARK(recip(X:S)) -> ACTIVE(recip(mark(X:S)))
 MARK(recip(X:S)) -> MARK(X:S)
 MARK(sqr(X:S)) -> ACTIVE(sqr(mark(X:S)))
 MARK(sqr(X:S)) -> MARK(X:S)
 MARK(terms(X:S)) -> ACTIVE(terms(mark(X:S)))
 MARK(terms(X:S)) -> MARK(X:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
-> Usable rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = X
[add](X1,X2) = X1 + 2.X2 + 2
[cons](X1,X2) = X1
[dbl](X) = 2.X + 2
[first](X1,X2) = 2.X1 + 2.X2 + 1
[mark](X) = 2.X
[recip](X) = X
[s](X) = 0
[sqr](X) = X
[terms](X) = 2.X + 2
[0] = 0
[fSNonEmpty] = 0
[nil] = 0
[ACTIVE](X) = X + 2
[ADD](X1,X2) = 0
[CONS](X1,X2) = 0
[DBL](X) = 0
[FIRST](X1,X2) = 0
[MARK](X) = 2.X + 2
[RECIP](X) = 0
[S](X) = 0
[SQR](X) = 0
[TERMS](X) = 0

Problem 1.9: 

SCC Processor:
-> Pairs:
 MARK(add(X1:S,X2:S)) -> ACTIVE(add(mark(X1:S),mark(X2:S)))
 MARK(add(X1:S,X2:S)) -> MARK(X1:S)
 MARK(add(X1:S,X2:S)) -> MARK(X2:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(dbl(X:S)) -> ACTIVE(dbl(mark(X:S)))
 MARK(dbl(X:S)) -> MARK(X:S)
 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S)))
 MARK(first(X1:S,X2:S)) -> MARK(X1:S)
 MARK(first(X1:S,X2:S)) -> MARK(X2:S)
 MARK(recip(X:S)) -> ACTIVE(recip(mark(X:S)))
 MARK(recip(X:S)) -> MARK(X:S)
 MARK(sqr(X:S)) -> ACTIVE(sqr(mark(X:S)))
 MARK(sqr(X:S)) -> MARK(X:S)
 MARK(terms(X:S)) -> ACTIVE(terms(mark(X:S)))
 MARK(terms(X:S)) -> MARK(X:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MARK(add(X1:S,X2:S)) -> MARK(X1:S)
 MARK(add(X1:S,X2:S)) -> MARK(X2:S)
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(dbl(X:S)) -> MARK(X:S)
 MARK(first(X1:S,X2:S)) -> MARK(X1:S)
 MARK(first(X1:S,X2:S)) -> MARK(X2:S)
 MARK(recip(X:S)) -> MARK(X:S)
 MARK(sqr(X:S)) -> MARK(X:S)
 MARK(terms(X:S)) -> MARK(X:S)
->->-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)

Problem 1.9: 

Subterm Processor:
-> Pairs:
 MARK(add(X1:S,X2:S)) -> MARK(X1:S)
 MARK(add(X1:S,X2:S)) -> MARK(X2:S)
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(dbl(X:S)) -> MARK(X:S)
 MARK(first(X1:S,X2:S)) -> MARK(X1:S)
 MARK(first(X1:S,X2:S)) -> MARK(X2:S)
 MARK(recip(X:S)) -> MARK(X:S)
 MARK(sqr(X:S)) -> MARK(X:S)
 MARK(terms(X:S)) -> MARK(X:S)
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Projection:
 pi(MARK) = 1

Problem 1.9: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S)))
 active(add(0,X:S)) -> mark(X:S)
 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S))))
 active(dbl(0)) -> mark(0)
 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S)))
 active(first(0,X:S)) -> mark(nil)
 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S))))
 active(sqr(0)) -> mark(0)
 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S))))
 add(active(X1:S),X2:S) -> add(X1:S,X2:S)
 add(mark(X1:S),X2:S) -> add(X1:S,X2:S)
 add(X1:S,active(X2:S)) -> add(X1:S,X2:S)
 add(X1:S,mark(X2:S)) -> add(X1:S,X2:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 dbl(active(X:S)) -> dbl(X:S)
 dbl(mark(X:S)) -> dbl(X:S)
 first(active(X1:S),X2:S) -> first(X1:S,X2:S)
 first(mark(X1:S),X2:S) -> first(X1:S,X2:S)
 first(X1:S,active(X2:S)) -> first(X1:S,X2:S)
 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S)
 mark(add(X1:S,X2:S)) -> active(add(mark(X1:S),mark(X2:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(dbl(X:S)) -> active(dbl(mark(X:S)))
 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S)))
 mark(recip(X:S)) -> active(recip(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(sqr(X:S)) -> active(sqr(mark(X:S)))
 mark(terms(X:S)) -> active(terms(mark(X:S)))
 mark(0) -> active(0)
 mark(nil) -> active(nil)
 recip(active(X:S)) -> recip(X:S)
 recip(mark(X:S)) -> recip(X:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 sqr(active(X:S)) -> sqr(X:S)
 sqr(mark(X:S)) -> sqr(X:S)
 terms(active(X:S)) -> terms(X:S)
 terms(mark(X:S)) -> terms(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.