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


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YES

Problem 1: 

(VAR N X X1 X2 Y Z)
(RULES
activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
activate(n__dbl(X)) -> dbl(activate(X))
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__s(X)) -> s(X)
activate(n__sqr(X)) -> sqr(activate(X))
activate(n__terms(X)) -> terms(activate(X))
activate(X) -> X
add(s(X),Y) -> s(n__add(activate(X),Y))
add(0,X) -> X
add(X1,X2) -> n__add(X1,X2)
dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
dbl(0) -> 0
dbl(X) -> n__dbl(X)
first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
first(0,X) -> nil
first(X1,X2) -> n__first(X1,X2)
s(X) -> n__s(X)
sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
sqr(0) -> 0
sqr(X) -> n__sqr(X)
terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
terms(X) -> n__terms(X)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVATE(n__add(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__add(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__add(X1,X2)) -> ADD(activate(X1),activate(X2))
 ACTIVATE(n__dbl(X)) -> ACTIVATE(X)
 ACTIVATE(n__dbl(X)) -> DBL(activate(X))
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__s(X)) -> S(X)
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 ADD(s(X),Y) -> ACTIVATE(X)
 ADD(s(X),Y) -> S(n__add(activate(X),Y))
 DBL(s(X)) -> ACTIVATE(X)
 DBL(s(X)) -> S(n__s(n__dbl(activate(X))))
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 SQR(s(X)) -> S(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__add(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__add(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__add(X1,X2)) -> ADD(activate(X1),activate(X2))
 ACTIVATE(n__dbl(X)) -> ACTIVATE(X)
 ACTIVATE(n__dbl(X)) -> DBL(activate(X))
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__s(X)) -> S(X)
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 ADD(s(X),Y) -> ACTIVATE(X)
 ADD(s(X),Y) -> S(n__add(activate(X),Y))
 DBL(s(X)) -> ACTIVATE(X)
 DBL(s(X)) -> S(n__s(n__dbl(activate(X))))
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 SQR(s(X)) -> S(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__add(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__add(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__add(X1,X2)) -> ADD(activate(X1),activate(X2))
 ACTIVATE(n__dbl(X)) -> ACTIVATE(X)
 ACTIVATE(n__dbl(X)) -> DBL(activate(X))
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 ADD(s(X),Y) -> ACTIVATE(X)
 DBL(s(X)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
->->-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__add(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__add(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__add(X1,X2)) -> ADD(activate(X1),activate(X2))
 ACTIVATE(n__dbl(X)) -> ACTIVATE(X)
 ACTIVATE(n__dbl(X)) -> DBL(activate(X))
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 ADD(s(X),Y) -> ACTIVATE(X)
 DBL(s(X)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
-> Usable rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = X
[add](X1,X2) = X1 + X2 + 1/2
[dbl](X) = 2.X + 1
[first](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1/2
[s](X) = X + 2
[sqr](X) = 2.X.X + X + 2
[terms](X) = X + 2
[0] = 0
[cons](X1,X2) = 1/2.X2
[n__add](X1,X2) = X1 + X2 + 1/2
[n__dbl](X) = 2.X + 1
[n__first](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1/2
[n__s](X) = X + 2
[n__sqr](X) = 2.X.X + X + 2
[n__terms](X) = X + 2
[nil] = 0
[recip](X) = 0
[ACTIVATE](X) = 1/2.X + 2
[ADD](X1,X2) = 1/2.X1 + 1
[DBL](X) = X + 1/2
[FIRST](X1,X2) = 1/2.X1.X2 + 1/2.X1 + 2
[SQR](X) = 1/2.X + 2
[TERMS](X) = 1/2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__add(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__add(X1,X2)) -> ADD(activate(X1),activate(X2))
 ACTIVATE(n__dbl(X)) -> ACTIVATE(X)
 ACTIVATE(n__dbl(X)) -> DBL(activate(X))
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 ADD(s(X),Y) -> ACTIVATE(X)
 DBL(s(X)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__add(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__add(X1,X2)) -> ADD(activate(X1),activate(X2))
 ACTIVATE(n__dbl(X)) -> ACTIVATE(X)
 ACTIVATE(n__dbl(X)) -> DBL(activate(X))
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 ADD(s(X),Y) -> ACTIVATE(X)
 DBL(s(X)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
->->-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__add(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__add(X1,X2)) -> ADD(activate(X1),activate(X2))
 ACTIVATE(n__dbl(X)) -> ACTIVATE(X)
 ACTIVATE(n__dbl(X)) -> DBL(activate(X))
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 ADD(s(X),Y) -> ACTIVATE(X)
 DBL(s(X)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
-> Usable rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = X
[add](X1,X2) = X1 + X2 + 1/2
[dbl](X) = 2.X + 1
[first](X1,X2) = 1/2.X1.X2 + 2.X1 + 2.X2
[s](X) = X + 2
[sqr](X) = 1/2.X.X + X
[terms](X) = X + 2
[0] = 0
[cons](X1,X2) = 1/2.X2
[n__add](X1,X2) = X1 + X2 + 1/2
[n__dbl](X) = 2.X + 1
[n__first](X1,X2) = 1/2.X1.X2 + 2.X1 + 2.X2
[n__s](X) = X + 2
[n__sqr](X) = 1/2.X.X + X
[n__terms](X) = X + 2
[nil] = 0
[recip](X) = 1
[ACTIVATE](X) = 2.X + 1
[ADD](X1,X2) = 2.X1 + 1
[DBL](X) = 2.X
[FIRST](X1,X2) = X1.X2 + 2.X1 + 2.X2
[SQR](X) = 2.X
[TERMS](X) = 2.X

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__add(X1,X2)) -> ADD(activate(X1),activate(X2))
 ACTIVATE(n__dbl(X)) -> ACTIVATE(X)
 ACTIVATE(n__dbl(X)) -> DBL(activate(X))
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 ADD(s(X),Y) -> ACTIVATE(X)
 DBL(s(X)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__add(X1,X2)) -> ADD(activate(X1),activate(X2))
 ACTIVATE(n__dbl(X)) -> ACTIVATE(X)
 ACTIVATE(n__dbl(X)) -> DBL(activate(X))
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 ADD(s(X),Y) -> ACTIVATE(X)
 DBL(s(X)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
->->-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__add(X1,X2)) -> ADD(activate(X1),activate(X2))
 ACTIVATE(n__dbl(X)) -> ACTIVATE(X)
 ACTIVATE(n__dbl(X)) -> DBL(activate(X))
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 ADD(s(X),Y) -> ACTIVATE(X)
 DBL(s(X)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
-> Usable rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = X
[add](X1,X2) = X1 + X2 + 1/2
[dbl](X) = 2.X
[first](X1,X2) = 2.X1 + 2.X2 + 2
[s](X) = X + 2
[sqr](X) = 2.X.X + 2.X + 2
[terms](X) = X + 2
[0] = 0
[cons](X1,X2) = 1/2.X2
[n__add](X1,X2) = X1 + X2 + 1/2
[n__dbl](X) = 2.X
[n__first](X1,X2) = 2.X1 + 2.X2 + 2
[n__s](X) = X + 2
[n__sqr](X) = 2.X.X + 2.X + 2
[n__terms](X) = X + 2
[nil] = 1
[recip](X) = 0
[ACTIVATE](X) = 1/2.X + 1
[ADD](X1,X2) = 1/2.X1 + 1/2.X2 + 1
[DBL](X) = 1/2.X + 1
[FIRST](X1,X2) = 1/2.X1 + X2 + 1
[SQR](X) = 1/2.X
[TERMS](X) = 1/2.X

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__dbl(X)) -> ACTIVATE(X)
 ACTIVATE(n__dbl(X)) -> DBL(activate(X))
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 ADD(s(X),Y) -> ACTIVATE(X)
 DBL(s(X)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__dbl(X)) -> ACTIVATE(X)
 ACTIVATE(n__dbl(X)) -> DBL(activate(X))
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 DBL(s(X)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
->->-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__dbl(X)) -> ACTIVATE(X)
 ACTIVATE(n__dbl(X)) -> DBL(activate(X))
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 DBL(s(X)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
-> Usable rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = X
[add](X1,X2) = X2
[dbl](X) = 2.X + 2
[first](X1,X2) = 2.X1 + 2.X2 + 2
[s](X) = X
[sqr](X) = 2.X + 2
[terms](X) = 2.X
[0] = 0
[cons](X1,X2) = X2
[n__add](X1,X2) = X2
[n__dbl](X) = 2.X + 2
[n__first](X1,X2) = 2.X1 + 2.X2 + 2
[n__s](X) = X
[n__sqr](X) = 2.X + 2
[n__terms](X) = 2.X
[nil] = 1
[recip](X) = 2
[ACTIVATE](X) = 2.X + 2
[DBL](X) = 2.X + 2
[FIRST](X1,X2) = 2.X1 + 2.X2 + 2
[SQR](X) = 2.X + 2
[TERMS](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__dbl(X)) -> DBL(activate(X))
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 DBL(s(X)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__dbl(X)) -> DBL(activate(X))
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 DBL(s(X)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
->->-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__dbl(X)) -> DBL(activate(X))
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 DBL(s(X)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
-> Usable rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = X
[add](X1,X2) = 2.X2
[dbl](X) = X + 1
[first](X1,X2) = 2.X1 + X2 + 2
[s](X) = X
[sqr](X) = 2.X + 2
[terms](X) = 2.X + 1
[0] = 0
[cons](X1,X2) = X2
[n__add](X1,X2) = 2.X2
[n__dbl](X) = X + 1
[n__first](X1,X2) = 2.X1 + X2 + 2
[n__s](X) = X
[n__sqr](X) = 2.X + 2
[n__terms](X) = 2.X + 1
[nil] = 2
[recip](X) = X + 2
[ACTIVATE](X) = 2.X + 1
[DBL](X) = 2.X + 1
[FIRST](X1,X2) = 2.X1 + 2.X2 + 2
[SQR](X) = 2.X + 2
[TERMS](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 DBL(s(X)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
->->-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
-> Usable rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = X
[add](X1,X2) = X2
[dbl](X) = 2
[first](X1,X2) = 2.X1 + 2.X2 + 2
[s](X) = X
[sqr](X) = 2.X + 2
[terms](X) = 2.X + 2
[0] = 0
[cons](X1,X2) = 2.X1 + X2
[n__add](X1,X2) = X2
[n__dbl](X) = 2
[n__first](X1,X2) = 2.X1 + 2.X2 + 2
[n__s](X) = X
[n__sqr](X) = 2.X + 2
[n__terms](X) = 2.X + 2
[nil] = 2
[recip](X) = 0
[ACTIVATE](X) = 2.X + 2
[FIRST](X1,X2) = 2.X1 + 2.X2 + 2
[SQR](X) = 2.X + 2
[TERMS](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
->->-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
-> Usable rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = X
[add](X1,X2) = X2 + 1
[dbl](X) = 2.X + 1
[first](X1,X2) = 2.X1 + 2.X2 + 1
[s](X) = X
[sqr](X) = 2.X + 2
[terms](X) = X + 1
[0] = 2
[cons](X1,X2) = X1 + X2
[n__add](X1,X2) = X2 + 1
[n__dbl](X) = 2.X + 1
[n__first](X1,X2) = 2.X1 + 2.X2 + 1
[n__s](X) = X
[n__sqr](X) = 2.X + 2
[n__terms](X) = X + 1
[nil] = 2
[recip](X) = 0
[ACTIVATE](X) = 2.X
[FIRST](X1,X2) = 2.X1 + 2.X2
[SQR](X) = 2.X + 1
[TERMS](X) = 2.X + 1

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
->->-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
-> Usable rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = X
[add](X1,X2) = 2.X2 + 1
[dbl](X) = X
[first](X1,X2) = 2.X1 + X2 + 2
[s](X) = X
[sqr](X) = 2.X + 1
[terms](X) = X + 2
[0] = 2
[cons](X1,X2) = X2
[n__add](X1,X2) = 2.X2 + 1
[n__dbl](X) = X
[n__first](X1,X2) = 2.X1 + X2 + 2
[n__s](X) = X
[n__sqr](X) = 2.X + 1
[n__terms](X) = X + 2
[nil] = 1
[recip](X) = 2.X + 2
[ACTIVATE](X) = 2.X + 1
[FIRST](X1,X2) = 2.X1 + 2.X2 + 2
[SQR](X) = 2.X + 2
[TERMS](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
->->-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__sqr(X)) -> ACTIVATE(X)
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
-> Usable rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = 2.X + 2
[add](X1,X2) = X2 + 2
[dbl](X) = 2
[first](X1,X2) = 1
[s](X) = X + 1
[sqr](X) = 2.X + 2
[terms](X) = 2.X + 2
[0] = 2
[cons](X1,X2) = X2
[n__add](X1,X2) = X2 + 1
[n__dbl](X) = 0
[n__first](X1,X2) = 1
[n__s](X) = X
[n__sqr](X) = 2.X + 2
[n__terms](X) = 2.X + 2
[nil] = 1
[recip](X) = 2.X
[ACTIVATE](X) = 2.X + 2
[SQR](X) = 2.X + 2
[TERMS](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
->->-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__sqr(X)) -> SQR(activate(X))
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
-> Usable rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = 2.X + 1
[add](X1,X2) = X2 + 2
[dbl](X) = 2
[first](X1,X2) = X1 + 2.X2 + 2
[s](X) = X + 1
[sqr](X) = 2.X + 2
[terms](X) = 2.X + 2
[0] = 2
[cons](X1,X2) = 2.X1 + 1
[n__add](X1,X2) = X2 + 1
[n__dbl](X) = 1
[n__first](X1,X2) = X1 + 2.X2 + 2
[n__s](X) = X
[n__sqr](X) = 2.X + 2
[n__terms](X) = 2.X + 2
[nil] = 1
[recip](X) = 0
[ACTIVATE](X) = X + 2
[SQR](X) = X + 2
[TERMS](X) = X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
->->-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__terms(X)) -> ACTIVATE(X)
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
-> Usable rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = 2.X + 2
[add](X1,X2) = X2 + 2
[dbl](X) = 2
[first](X1,X2) = 2.X1 + 2
[s](X) = X + 1
[sqr](X) = 2.X + 2
[terms](X) = 2.X + 2
[0] = 2
[cons](X1,X2) = 0
[n__add](X1,X2) = X2 + 1
[n__dbl](X) = 0
[n__first](X1,X2) = 2.X1 + 2
[n__s](X) = X + 1
[n__sqr](X) = 2.X + 2
[n__terms](X) = 2.X + 2
[nil] = 2
[recip](X) = 2
[ACTIVATE](X) = 2.X + 2
[SQR](X) = 2.X
[TERMS](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
->->-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__terms(X)) -> TERMS(activate(X))
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
-> Usable rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = 2.X + 1
[add](X1,X2) = X2 + 2
[dbl](X) = 2
[first](X1,X2) = X1
[s](X) = X + 1
[sqr](X) = X + 2
[terms](X) = 2.X + 2
[0] = 2
[cons](X1,X2) = 0
[n__add](X1,X2) = X2 + 1
[n__dbl](X) = 1
[n__first](X1,X2) = X1
[n__s](X) = X
[n__sqr](X) = X + 2
[n__terms](X) = 2.X + 2
[nil] = 2
[recip](X) = 2.X + 2
[ACTIVATE](X) = 2.X + 2
[SQR](X) = 2.X + 1
[TERMS](X) = 2.X + 1

Problem 1: 

SCC Processor:
-> Pairs:
 SQR(s(X)) -> ACTIVATE(X)
 TERMS(N) -> SQR(N)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
 activate(n__dbl(X)) -> dbl(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__s(X)) -> s(X)
 activate(n__sqr(X)) -> sqr(activate(X))
 activate(n__terms(X)) -> terms(activate(X))
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 add(0,X) -> X
 add(X1,X2) -> n__add(X1,X2)
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 dbl(0) -> 0
 dbl(X) -> n__dbl(X)
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 s(X) -> n__s(X)
 sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X))))
 sqr(0) -> 0
 sqr(X) -> n__sqr(X)
 terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
 terms(X) -> n__terms(X)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.