/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/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(X1:S,X2:S)
activate(n__terms(X:S)) -> terms(X:S)
activate(X:S) -> X:S
add(0,X:S) -> X:S
add(s(X:S),Y:S) -> s(add(X:S,Y:S))
dbl(0) -> 0
dbl(s(X:S)) -> s(s(dbl(X:S)))
first(0,X:S) -> nil
first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
first(X1:S,X2:S) -> n__first(X1:S,X2:S)
sqr(0) -> 0
sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S)))
terms(N:S) -> cons(recip(sqr(N:S)),n__terms(s(N:S)))
terms(X:S) -> n__terms(X:S)
)

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 4 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.2: 

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

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 activate(n__first(X1:S,X2:S)) -> first(X1:S,X2:S)
 activate(n__terms(X:S)) -> terms(X:S)
 activate(X:S) -> X:S
 add(0,X:S) -> X:S
 add(s(X:S),Y:S) -> s(add(X:S,Y:S))
 dbl(0) -> 0
 dbl(s(X:S)) -> s(s(dbl(X:S)))
 first(0,X:S) -> nil
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 sqr(0) -> 0
 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S)))
 terms(N:S) -> cons(recip(sqr(N:S)),n__terms(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:
 ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(X1:S,X2:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S)
-> Rules:
 activate(n__first(X1:S,X2:S)) -> first(X1:S,X2:S)
 activate(n__terms(X:S)) -> terms(X:S)
 activate(X:S) -> X:S
 add(0,X:S) -> X:S
 add(s(X:S),Y:S) -> s(add(X:S,Y:S))
 dbl(0) -> 0
 dbl(s(X:S)) -> s(s(dbl(X:S)))
 first(0,X:S) -> nil
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 sqr(0) -> 0
 sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S)))
 terms(N:S) -> cons(recip(sqr(N:S)),n__terms(s(N:S)))
 terms(X:S) -> n__terms(X:S)
->Projection:
 pi(ACTIVATE) = 1
 pi(FIRST) = 2

Problem 1.4: 

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

The problem is finite.