/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 XS:S Y:S YS:S ZS:S)
(RULES
U11(tt,N:S,X:S,XS:S) -> U12(splitAt(activate(N:S),activate(XS:S)),activate(X:S))
U12(pair(YS:S,ZS:S),X:S) -> pair(cons(activate(X:S),YS:S),ZS:S)
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))
and(tt,X:S) -> activate(X:S)
fst(pair(X:S,Y:S)) -> X: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(X:S,Y:S)) -> Y:S
splitAt(s(N:S),cons(X:S,XS:S)) -> U11(tt,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))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 U11#(tt,N:S,X:S,XS:S) -> U12#(splitAt(activate(N:S),activate(XS:S)),activate(X:S))
 U11#(tt,N:S,X:S,XS:S) -> ACTIVATE(N:S)
 U11#(tt,N:S,X:S,XS:S) -> ACTIVATE(X:S)
 U11#(tt,N:S,X:S,XS:S) -> ACTIVATE(XS:S)
 U11#(tt,N:S,X:S,XS:S) -> SPLITAT(activate(N:S),activate(XS:S))
 U12#(pair(YS:S,ZS:S),X:S) -> ACTIVATE(X:S)
 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)
 AND(tt,X:S) -> ACTIVATE(X: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)) -> U11#(tt,N:S,X:S,activate(XS:S))
 SPLITAT(s(N:S),cons(X:S,XS: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)
-> Rules:
 U11(tt,N:S,X:S,XS:S) -> U12(splitAt(activate(N:S),activate(XS:S)),activate(X:S))
 U12(pair(YS:S,ZS:S),X:S) -> pair(cons(activate(X:S),YS:S),ZS:S)
 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))
 and(tt,X:S) -> activate(X:S)
 fst(pair(X:S,Y:S)) -> X: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(X:S,Y:S)) -> Y:S
 splitAt(s(N:S),cons(X:S,XS:S)) -> U11(tt,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))

Problem 1: 

SCC Processor:
-> Pairs:
 U11#(tt,N:S,X:S,XS:S) -> U12#(splitAt(activate(N:S),activate(XS:S)),activate(X:S))
 U11#(tt,N:S,X:S,XS:S) -> ACTIVATE(N:S)
 U11#(tt,N:S,X:S,XS:S) -> ACTIVATE(X:S)
 U11#(tt,N:S,X:S,XS:S) -> ACTIVATE(XS:S)
 U11#(tt,N:S,X:S,XS:S) -> SPLITAT(activate(N:S),activate(XS:S))
 U12#(pair(YS:S,ZS:S),X:S) -> ACTIVATE(X:S)
 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)
 AND(tt,X:S) -> ACTIVATE(X: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)) -> U11#(tt,N:S,X:S,activate(XS:S))
 SPLITAT(s(N:S),cons(X:S,XS: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)
-> Rules:
 U11(tt,N:S,X:S,XS:S) -> U12(splitAt(activate(N:S),activate(XS:S)),activate(X:S))
 U12(pair(YS:S,ZS:S),X:S) -> pair(cons(activate(X:S),YS:S),ZS:S)
 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))
 and(tt,X:S) -> activate(X:S)
 fst(pair(X:S,Y:S)) -> X: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(X:S,Y:S)) -> Y:S
 splitAt(s(N:S),cons(X:S,XS:S)) -> U11(tt,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))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__natsFrom(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
->->-> Rules:
 U11(tt,N:S,X:S,XS:S) -> U12(splitAt(activate(N:S),activate(XS:S)),activate(X:S))
 U12(pair(YS:S,ZS:S),X:S) -> pair(cons(activate(X:S),YS:S),ZS:S)
 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))
 and(tt,X:S) -> activate(X:S)
 fst(pair(X:S,Y:S)) -> X: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(X:S,Y:S)) -> Y:S
 splitAt(s(N:S),cons(X:S,XS:S)) -> U11(tt,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))
->->Cycle:
->->-> Pairs:
 U11#(tt,N:S,X:S,XS:S) -> SPLITAT(activate(N:S),activate(XS:S))
 SPLITAT(s(N:S),cons(X:S,XS:S)) -> U11#(tt,N:S,X:S,activate(XS:S))
->->-> Rules:
 U11(tt,N:S,X:S,XS:S) -> U12(splitAt(activate(N:S),activate(XS:S)),activate(X:S))
 U12(pair(YS:S,ZS:S),X:S) -> pair(cons(activate(X:S),YS:S),ZS:S)
 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))
 and(tt,X:S) -> activate(X:S)
 fst(pair(X:S,Y:S)) -> X: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(X:S,Y:S)) -> Y:S
 splitAt(s(N:S),cons(X:S,XS:S)) -> U11(tt,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))


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:
 U11(tt,N:S,X:S,XS:S) -> U12(splitAt(activate(N:S),activate(XS:S)),activate(X:S))
 U12(pair(YS:S,ZS:S),X:S) -> pair(cons(activate(X:S),YS:S),ZS:S)
 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))
 and(tt,X:S) -> activate(X:S)
 fst(pair(X:S,Y:S)) -> X: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(X:S,Y:S)) -> Y:S
 splitAt(s(N:S),cons(X:S,XS:S)) -> U11(tt,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))
->Projection:
 pi(ACTIVATE) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(tt,N:S,X:S,XS:S) -> U12(splitAt(activate(N:S),activate(XS:S)),activate(X:S))
 U12(pair(YS:S,ZS:S),X:S) -> pair(cons(activate(X:S),YS:S),ZS:S)
 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))
 and(tt,X:S) -> activate(X:S)
 fst(pair(X:S,Y:S)) -> X: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(X:S,Y:S)) -> Y:S
 splitAt(s(N:S),cons(X:S,XS:S)) -> U11(tt,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))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 U11#(tt,N:S,X:S,XS:S) -> SPLITAT(activate(N:S),activate(XS:S))
 SPLITAT(s(N:S),cons(X:S,XS:S)) -> U11#(tt,N:S,X:S,activate(XS:S))
-> Rules:
 U11(tt,N:S,X:S,XS:S) -> U12(splitAt(activate(N:S),activate(XS:S)),activate(X:S))
 U12(pair(YS:S,ZS:S),X:S) -> pair(cons(activate(X:S),YS:S),ZS:S)
 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))
 and(tt,X:S) -> activate(X:S)
 fst(pair(X:S,Y:S)) -> X: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(X:S,Y:S)) -> Y:S
 splitAt(s(N:S),cons(X:S,XS:S)) -> U11(tt,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))
-> Usable rules:
 activate(n__natsFrom(X:S)) -> natsFrom(activate(X:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X: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)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = 2.X + 1
[natsFrom](X) = X + 2
[s](X) = 2.X + 2
[cons](X1,X2) = X1 + 2
[n__natsFrom](X) = X + 2
[n__s](X) = 2.X + 2
[tt] = 2
[U11#](X1,X2,X3,X4) = X1 + 2.X2 + 2
[SPLITAT](X1,X2) = X1 + 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 SPLITAT(s(N:S),cons(X:S,XS:S)) -> U11#(tt,N:S,X:S,activate(XS:S))
-> Rules:
 U11(tt,N:S,X:S,XS:S) -> U12(splitAt(activate(N:S),activate(XS:S)),activate(X:S))
 U12(pair(YS:S,ZS:S),X:S) -> pair(cons(activate(X:S),YS:S),ZS:S)
 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))
 and(tt,X:S) -> activate(X:S)
 fst(pair(X:S,Y:S)) -> X: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(X:S,Y:S)) -> Y:S
 splitAt(s(N:S),cons(X:S,XS:S)) -> U11(tt,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))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.