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

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 2 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.