/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 XS YS ZS)
(RULES
activate(n__natsFrom(X)) -> natsFrom(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(X) -> X
afterNth(N,XS) -> snd(splitAt(N,XS))
fst(pair(XS,YS)) -> XS
head(cons(N,XS)) -> N
natsFrom(N) -> cons(N,n__natsFrom(n__s(N)))
natsFrom(X) -> n__natsFrom(X)
s(X) -> n__s(X)
sel(N,XS) -> head(afterNth(N,XS))
snd(pair(XS,YS)) -> YS
splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS))
splitAt(0,XS) -> pair(nil,XS)
tail(cons(N,XS)) -> activate(XS)
take(N,XS) -> fst(splitAt(N,XS))
u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVATE(n__natsFrom(X)) -> ACTIVATE(X)
 ACTIVATE(n__natsFrom(X)) -> NATSFROM(activate(X))
 ACTIVATE(n__s(X)) -> ACTIVATE(X)
 ACTIVATE(n__s(X)) -> S(activate(X))
 AFTERNTH(N,XS) -> SND(splitAt(N,XS))
 AFTERNTH(N,XS) -> SPLITAT(N,XS)
 SEL(N,XS) -> AFTERNTH(N,XS)
 SEL(N,XS) -> HEAD(afterNth(N,XS))
 SPLITAT(s(N),cons(X,XS)) -> ACTIVATE(XS)
 SPLITAT(s(N),cons(X,XS)) -> SPLITAT(N,activate(XS))
 SPLITAT(s(N),cons(X,XS)) -> U(splitAt(N,activate(XS)),N,X,activate(XS))
 TAIL(cons(N,XS)) -> ACTIVATE(XS)
 TAKE(N,XS) -> FST(splitAt(N,XS))
 TAKE(N,XS) -> SPLITAT(N,XS)
 U(pair(YS,ZS),N,X,XS) -> ACTIVATE(X)
-> Rules:
 activate(n__natsFrom(X)) -> natsFrom(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(X) -> X
 afterNth(N,XS) -> snd(splitAt(N,XS))
 fst(pair(XS,YS)) -> XS
 head(cons(N,XS)) -> N
 natsFrom(N) -> cons(N,n__natsFrom(n__s(N)))
 natsFrom(X) -> n__natsFrom(X)
 s(X) -> n__s(X)
 sel(N,XS) -> head(afterNth(N,XS))
 snd(pair(XS,YS)) -> YS
 splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS))
 splitAt(0,XS) -> pair(nil,XS)
 tail(cons(N,XS)) -> activate(XS)
 take(N,XS) -> fst(splitAt(N,XS))
 u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__natsFrom(X)) -> ACTIVATE(X)
 ACTIVATE(n__natsFrom(X)) -> NATSFROM(activate(X))
 ACTIVATE(n__s(X)) -> ACTIVATE(X)
 ACTIVATE(n__s(X)) -> S(activate(X))
 AFTERNTH(N,XS) -> SND(splitAt(N,XS))
 AFTERNTH(N,XS) -> SPLITAT(N,XS)
 SEL(N,XS) -> AFTERNTH(N,XS)
 SEL(N,XS) -> HEAD(afterNth(N,XS))
 SPLITAT(s(N),cons(X,XS)) -> ACTIVATE(XS)
 SPLITAT(s(N),cons(X,XS)) -> SPLITAT(N,activate(XS))
 SPLITAT(s(N),cons(X,XS)) -> U(splitAt(N,activate(XS)),N,X,activate(XS))
 TAIL(cons(N,XS)) -> ACTIVATE(XS)
 TAKE(N,XS) -> FST(splitAt(N,XS))
 TAKE(N,XS) -> SPLITAT(N,XS)
 U(pair(YS,ZS),N,X,XS) -> ACTIVATE(X)
-> Rules:
 activate(n__natsFrom(X)) -> natsFrom(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(X) -> X
 afterNth(N,XS) -> snd(splitAt(N,XS))
 fst(pair(XS,YS)) -> XS
 head(cons(N,XS)) -> N
 natsFrom(N) -> cons(N,n__natsFrom(n__s(N)))
 natsFrom(X) -> n__natsFrom(X)
 s(X) -> n__s(X)
 sel(N,XS) -> head(afterNth(N,XS))
 snd(pair(XS,YS)) -> YS
 splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS))
 splitAt(0,XS) -> pair(nil,XS)
 tail(cons(N,XS)) -> activate(XS)
 take(N,XS) -> fst(splitAt(N,XS))
 u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__natsFrom(X)) -> ACTIVATE(X)
 ACTIVATE(n__s(X)) -> ACTIVATE(X)
->->-> Rules:
 activate(n__natsFrom(X)) -> natsFrom(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(X) -> X
 afterNth(N,XS) -> snd(splitAt(N,XS))
 fst(pair(XS,YS)) -> XS
 head(cons(N,XS)) -> N
 natsFrom(N) -> cons(N,n__natsFrom(n__s(N)))
 natsFrom(X) -> n__natsFrom(X)
 s(X) -> n__s(X)
 sel(N,XS) -> head(afterNth(N,XS))
 snd(pair(XS,YS)) -> YS
 splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS))
 splitAt(0,XS) -> pair(nil,XS)
 tail(cons(N,XS)) -> activate(XS)
 take(N,XS) -> fst(splitAt(N,XS))
 u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
->->Cycle:
->->-> Pairs:
 SPLITAT(s(N),cons(X,XS)) -> SPLITAT(N,activate(XS))
->->-> Rules:
 activate(n__natsFrom(X)) -> natsFrom(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(X) -> X
 afterNth(N,XS) -> snd(splitAt(N,XS))
 fst(pair(XS,YS)) -> XS
 head(cons(N,XS)) -> N
 natsFrom(N) -> cons(N,n__natsFrom(n__s(N)))
 natsFrom(X) -> n__natsFrom(X)
 s(X) -> n__s(X)
 sel(N,XS) -> head(afterNth(N,XS))
 snd(pair(XS,YS)) -> YS
 splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS))
 splitAt(0,XS) -> pair(nil,XS)
 tail(cons(N,XS)) -> activate(XS)
 take(N,XS) -> fst(splitAt(N,XS))
 u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 ACTIVATE(n__natsFrom(X)) -> ACTIVATE(X)
 ACTIVATE(n__s(X)) -> ACTIVATE(X)
-> Rules:
 activate(n__natsFrom(X)) -> natsFrom(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(X) -> X
 afterNth(N,XS) -> snd(splitAt(N,XS))
 fst(pair(XS,YS)) -> XS
 head(cons(N,XS)) -> N
 natsFrom(N) -> cons(N,n__natsFrom(n__s(N)))
 natsFrom(X) -> n__natsFrom(X)
 s(X) -> n__s(X)
 sel(N,XS) -> head(afterNth(N,XS))
 snd(pair(XS,YS)) -> YS
 splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS))
 splitAt(0,XS) -> pair(nil,XS)
 tail(cons(N,XS)) -> activate(XS)
 take(N,XS) -> fst(splitAt(N,XS))
 u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
->Projection:
 pi(ACTIVATE) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 activate(n__natsFrom(X)) -> natsFrom(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(X) -> X
 afterNth(N,XS) -> snd(splitAt(N,XS))
 fst(pair(XS,YS)) -> XS
 head(cons(N,XS)) -> N
 natsFrom(N) -> cons(N,n__natsFrom(n__s(N)))
 natsFrom(X) -> n__natsFrom(X)
 s(X) -> n__s(X)
 sel(N,XS) -> head(afterNth(N,XS))
 snd(pair(XS,YS)) -> YS
 splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS))
 splitAt(0,XS) -> pair(nil,XS)
 tail(cons(N,XS)) -> activate(XS)
 take(N,XS) -> fst(splitAt(N,XS))
 u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 SPLITAT(s(N),cons(X,XS)) -> SPLITAT(N,activate(XS))
-> Rules:
 activate(n__natsFrom(X)) -> natsFrom(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(X) -> X
 afterNth(N,XS) -> snd(splitAt(N,XS))
 fst(pair(XS,YS)) -> XS
 head(cons(N,XS)) -> N
 natsFrom(N) -> cons(N,n__natsFrom(n__s(N)))
 natsFrom(X) -> n__natsFrom(X)
 s(X) -> n__s(X)
 sel(N,XS) -> head(afterNth(N,XS))
 snd(pair(XS,YS)) -> YS
 splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS))
 splitAt(0,XS) -> pair(nil,XS)
 tail(cons(N,XS)) -> activate(XS)
 take(N,XS) -> fst(splitAt(N,XS))
 u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
->Projection:
 pi(SPLITAT) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 activate(n__natsFrom(X)) -> natsFrom(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(X) -> X
 afterNth(N,XS) -> snd(splitAt(N,XS))
 fst(pair(XS,YS)) -> XS
 head(cons(N,XS)) -> N
 natsFrom(N) -> cons(N,n__natsFrom(n__s(N)))
 natsFrom(X) -> n__natsFrom(X)
 s(X) -> n__s(X)
 sel(N,XS) -> head(afterNth(N,XS))
 snd(pair(XS,YS)) -> YS
 splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS))
 splitAt(0,XS) -> pair(nil,XS)
 tail(cons(N,XS)) -> activate(XS)
 take(N,XS) -> fst(splitAt(N,XS))
 u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.