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


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YES

Problem 1: 

(VAR v_NonEmpty:S N:S X:S X1:S X2:S Y:S Z:S)
(RULES
activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
activate(n__s(X:S)) -> s(activate(X:S))
activate(n__terms(X:S)) -> terms(activate(X:S))
activate(X:S) -> X:S
add(s(X:S),Y:S) -> s(add(X:S,Y:S))
add(0,X:S) -> X:S
dbl(s(X:S)) -> s(s(dbl(X:S)))
dbl(0) -> 0
first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
first(0,X:S) -> nil
first(X1:S,X2:S) -> n__first(X1:S,X2:S)
half(dbl(X:S)) -> X:S
half(s(s(X:S))) -> s(half(X:S))
half(s(0)) -> 0
half(0) -> 0
s(X:S) -> n__s(X:S)
sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S)))
sqr(0) -> 0
terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S)))
terms(X:S) -> n__terms(X:S)
)

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 5 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 HALF(s(s(X:S))) -> HALF(X:S)
-> Rules:
 activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(n__terms(X:S)) -> terms(activate(X:S))
 activate(X:S) -> X:S
 add(s(X:S),Y:S) -> s(add(X:S,Y:S))
 add(0,X:S) -> X:S
 dbl(s(X:S)) -> s(s(dbl(X:S)))
 dbl(0) -> 0
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(0,X:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 half(dbl(X:S)) -> X:S
 half(s(s(X:S))) -> s(half(X:S))
 half(s(0)) -> 0
 half(0) -> 0
 s(X:S) -> n__s(X:S)
 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S)))
 sqr(0) -> 0
 terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S)))
 terms(X:S) -> n__terms(X:S)
->Projection:
 pi(HALF) = 1

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.

Problem 1.3: 

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

Problem 1.3: 

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

The problem is finite.

Problem 1.4: 

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

Problem 1.4: 

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

The problem is finite.

Problem 1.5: 

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

Problem 1.5: 

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

Problem 1.5: 

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

Problem 1.5: 

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

Problem 1.5: 

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

Problem 1.5: 

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

Problem 1.5: 

Subterm Processor:
-> Pairs:
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S)
-> Rules:
 activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(n__terms(X:S)) -> terms(activate(X:S))
 activate(X:S) -> X:S
 add(s(X:S),Y:S) -> s(add(X:S,Y:S))
 add(0,X:S) -> X:S
 dbl(s(X:S)) -> s(s(dbl(X:S)))
 dbl(0) -> 0
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(0,X:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 half(dbl(X:S)) -> X:S
 half(s(s(X:S))) -> s(half(X:S))
 half(s(0)) -> 0
 half(0) -> 0
 s(X:S) -> n__s(X:S)
 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S)))
 sqr(0) -> 0
 terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S)))
 terms(X:S) -> n__terms(X:S)
->Projection:
 pi(ACTIVATE) = 1

Problem 1.5: 

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

The problem is finite.