/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 X1 X2 XS)
(RULES
2nd(cons(X,XS)) -> head(activate(XS))
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
activate(X) -> X
from(X) -> cons(X,n__from(n__s(X)))
from(X) -> n__from(X)
head(cons(X,XS)) -> X
s(X) -> n__s(X)
sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
sel(0,cons(X,XS)) -> X
take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
take(0,XS) -> nil
take(X1,X2) -> n__take(X1,X2)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 2ND(cons(X,XS)) -> ACTIVATE(XS)
 2ND(cons(X,XS)) -> HEAD(activate(XS))
 ACTIVATE(n__from(X)) -> ACTIVATE(X)
 ACTIVATE(n__from(X)) -> FROM(activate(X))
 ACTIVATE(n__s(X)) -> ACTIVATE(X)
 ACTIVATE(n__s(X)) -> S(activate(X))
 ACTIVATE(n__take(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__take(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__take(X1,X2)) -> TAKE(activate(X1),activate(X2))
 SEL(s(N),cons(X,XS)) -> ACTIVATE(XS)
 SEL(s(N),cons(X,XS)) -> SEL(N,activate(XS))
 TAKE(s(N),cons(X,XS)) -> ACTIVATE(XS)
-> Rules:
 2nd(cons(X,XS)) -> head(activate(XS))
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 head(cons(X,XS)) -> X
 s(X) -> n__s(X)
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 sel(0,cons(X,XS)) -> X
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)

Problem 1: 

SCC Processor:
-> Pairs:
 2ND(cons(X,XS)) -> ACTIVATE(XS)
 2ND(cons(X,XS)) -> HEAD(activate(XS))
 ACTIVATE(n__from(X)) -> ACTIVATE(X)
 ACTIVATE(n__from(X)) -> FROM(activate(X))
 ACTIVATE(n__s(X)) -> ACTIVATE(X)
 ACTIVATE(n__s(X)) -> S(activate(X))
 ACTIVATE(n__take(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__take(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__take(X1,X2)) -> TAKE(activate(X1),activate(X2))
 SEL(s(N),cons(X,XS)) -> ACTIVATE(XS)
 SEL(s(N),cons(X,XS)) -> SEL(N,activate(XS))
 TAKE(s(N),cons(X,XS)) -> ACTIVATE(XS)
-> Rules:
 2nd(cons(X,XS)) -> head(activate(XS))
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 head(cons(X,XS)) -> X
 s(X) -> n__s(X)
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 sel(0,cons(X,XS)) -> X
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__from(X)) -> ACTIVATE(X)
 ACTIVATE(n__s(X)) -> ACTIVATE(X)
 ACTIVATE(n__take(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__take(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__take(X1,X2)) -> TAKE(activate(X1),activate(X2))
 TAKE(s(N),cons(X,XS)) -> ACTIVATE(XS)
->->-> Rules:
 2nd(cons(X,XS)) -> head(activate(XS))
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 head(cons(X,XS)) -> X
 s(X) -> n__s(X)
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 sel(0,cons(X,XS)) -> X
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)
->->Cycle:
->->-> Pairs:
 SEL(s(N),cons(X,XS)) -> SEL(N,activate(XS))
->->-> Rules:
 2nd(cons(X,XS)) -> head(activate(XS))
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 head(cons(X,XS)) -> X
 s(X) -> n__s(X)
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 sel(0,cons(X,XS)) -> X
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__from(X)) -> ACTIVATE(X)
 ACTIVATE(n__s(X)) -> ACTIVATE(X)
 ACTIVATE(n__take(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__take(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__take(X1,X2)) -> TAKE(activate(X1),activate(X2))
 TAKE(s(N),cons(X,XS)) -> ACTIVATE(XS)
-> Rules:
 2nd(cons(X,XS)) -> head(activate(XS))
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 head(cons(X,XS)) -> X
 s(X) -> n__s(X)
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 sel(0,cons(X,XS)) -> X
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)
-> Usable rules:
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 s(X) -> n__s(X)
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = X
[from](X) = 2.X + 2
[s](X) = X
[take](X1,X2) = X1 + 2.X2 + 1
[0] = 2
[cons](X1,X2) = X2
[n__from](X) = 2.X + 2
[n__s](X) = X
[n__take](X1,X2) = X1 + 2.X2 + 1
[nil] = 2
[ACTIVATE](X) = X
[TAKE](X1,X2) = 2.X2

Problem 1.1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__s(X)) -> ACTIVATE(X)
 ACTIVATE(n__take(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__take(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__take(X1,X2)) -> TAKE(activate(X1),activate(X2))
 TAKE(s(N),cons(X,XS)) -> ACTIVATE(XS)
-> Rules:
 2nd(cons(X,XS)) -> head(activate(XS))
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 head(cons(X,XS)) -> X
 s(X) -> n__s(X)
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 sel(0,cons(X,XS)) -> X
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__s(X)) -> ACTIVATE(X)
 ACTIVATE(n__take(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__take(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__take(X1,X2)) -> TAKE(activate(X1),activate(X2))
 TAKE(s(N),cons(X,XS)) -> ACTIVATE(XS)
->->-> Rules:
 2nd(cons(X,XS)) -> head(activate(XS))
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 head(cons(X,XS)) -> X
 s(X) -> n__s(X)
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 sel(0,cons(X,XS)) -> X
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__s(X)) -> ACTIVATE(X)
 ACTIVATE(n__take(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__take(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__take(X1,X2)) -> TAKE(activate(X1),activate(X2))
 TAKE(s(N),cons(X,XS)) -> ACTIVATE(XS)
-> Rules:
 2nd(cons(X,XS)) -> head(activate(XS))
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 head(cons(X,XS)) -> X
 s(X) -> n__s(X)
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 sel(0,cons(X,XS)) -> X
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)
-> Usable rules:
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 s(X) -> n__s(X)
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = X
[from](X) = 0
[s](X) = 2.X + 2
[take](X1,X2) = 2.X1 + 2.X2 + 2
[0] = 0
[cons](X1,X2) = 2.X2
[n__from](X) = 0
[n__s](X) = 2.X + 2
[n__take](X1,X2) = 2.X1 + 2.X2 + 2
[nil] = 2
[ACTIVATE](X) = 2.X
[TAKE](X1,X2) = X1 + X2

Problem 1.1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__take(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__take(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__take(X1,X2)) -> TAKE(activate(X1),activate(X2))
 TAKE(s(N),cons(X,XS)) -> ACTIVATE(XS)
-> Rules:
 2nd(cons(X,XS)) -> head(activate(XS))
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 head(cons(X,XS)) -> X
 s(X) -> n__s(X)
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 sel(0,cons(X,XS)) -> X
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__take(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__take(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__take(X1,X2)) -> TAKE(activate(X1),activate(X2))
 TAKE(s(N),cons(X,XS)) -> ACTIVATE(XS)
->->-> Rules:
 2nd(cons(X,XS)) -> head(activate(XS))
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 head(cons(X,XS)) -> X
 s(X) -> n__s(X)
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 sel(0,cons(X,XS)) -> X
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__take(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__take(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__take(X1,X2)) -> TAKE(activate(X1),activate(X2))
 TAKE(s(N),cons(X,XS)) -> ACTIVATE(XS)
-> Rules:
 2nd(cons(X,XS)) -> head(activate(XS))
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 head(cons(X,XS)) -> X
 s(X) -> n__s(X)
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 sel(0,cons(X,XS)) -> X
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)
-> Usable rules:
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 s(X) -> n__s(X)
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = X
[from](X) = 2.X + 2
[s](X) = X
[take](X1,X2) = 2.X1 + 2.X2 + 2
[0] = 2
[cons](X1,X2) = X2
[n__from](X) = 2.X + 2
[n__s](X) = X
[n__take](X1,X2) = 2.X1 + 2.X2 + 2
[nil] = 0
[ACTIVATE](X) = X + 2
[TAKE](X1,X2) = 2.X1 + X2 + 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__take(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__take(X1,X2)) -> TAKE(activate(X1),activate(X2))
 TAKE(s(N),cons(X,XS)) -> ACTIVATE(XS)
-> Rules:
 2nd(cons(X,XS)) -> head(activate(XS))
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 head(cons(X,XS)) -> X
 s(X) -> n__s(X)
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 sel(0,cons(X,XS)) -> X
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__take(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__take(X1,X2)) -> TAKE(activate(X1),activate(X2))
 TAKE(s(N),cons(X,XS)) -> ACTIVATE(XS)
->->-> Rules:
 2nd(cons(X,XS)) -> head(activate(XS))
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 head(cons(X,XS)) -> X
 s(X) -> n__s(X)
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 sel(0,cons(X,XS)) -> X
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__take(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__take(X1,X2)) -> TAKE(activate(X1),activate(X2))
 TAKE(s(N),cons(X,XS)) -> ACTIVATE(XS)
-> Rules:
 2nd(cons(X,XS)) -> head(activate(XS))
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 head(cons(X,XS)) -> X
 s(X) -> n__s(X)
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 sel(0,cons(X,XS)) -> X
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)
-> Usable rules:
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 s(X) -> n__s(X)
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = X
[from](X) = 0
[s](X) = 2.X + 1
[take](X1,X2) = 2.X1 + 2.X2 + 1
[0] = 2
[cons](X1,X2) = 2.X2
[n__from](X) = 0
[n__s](X) = 2.X + 1
[n__take](X1,X2) = 2.X1 + 2.X2 + 1
[nil] = 2
[ACTIVATE](X) = 2.X
[TAKE](X1,X2) = 2.X1 + X2 + 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__take(X1,X2)) -> TAKE(activate(X1),activate(X2))
 TAKE(s(N),cons(X,XS)) -> ACTIVATE(XS)
-> Rules:
 2nd(cons(X,XS)) -> head(activate(XS))
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 head(cons(X,XS)) -> X
 s(X) -> n__s(X)
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 sel(0,cons(X,XS)) -> X
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__take(X1,X2)) -> TAKE(activate(X1),activate(X2))
 TAKE(s(N),cons(X,XS)) -> ACTIVATE(XS)
->->-> Rules:
 2nd(cons(X,XS)) -> head(activate(XS))
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 head(cons(X,XS)) -> X
 s(X) -> n__s(X)
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 sel(0,cons(X,XS)) -> X
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__take(X1,X2)) -> TAKE(activate(X1),activate(X2))
 TAKE(s(N),cons(X,XS)) -> ACTIVATE(XS)
-> Rules:
 2nd(cons(X,XS)) -> head(activate(XS))
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 head(cons(X,XS)) -> X
 s(X) -> n__s(X)
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 sel(0,cons(X,XS)) -> X
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)
-> Usable rules:
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 s(X) -> n__s(X)
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = X
[from](X) = 0
[s](X) = 2.X + 1
[take](X1,X2) = 2.X1 + 2.X2 + 2
[0] = 2
[cons](X1,X2) = 2.X2
[n__from](X) = 0
[n__s](X) = 2.X + 1
[n__take](X1,X2) = 2.X1 + 2.X2 + 2
[nil] = 2
[ACTIVATE](X) = 2.X + 2
[TAKE](X1,X2) = 2.X1 + 2.X2 + 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 TAKE(s(N),cons(X,XS)) -> ACTIVATE(XS)
-> Rules:
 2nd(cons(X,XS)) -> head(activate(XS))
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 head(cons(X,XS)) -> X
 s(X) -> n__s(X)
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 sel(0,cons(X,XS)) -> X
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 SEL(s(N),cons(X,XS)) -> SEL(N,activate(XS))
-> Rules:
 2nd(cons(X,XS)) -> head(activate(XS))
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 head(cons(X,XS)) -> X
 s(X) -> n__s(X)
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 sel(0,cons(X,XS)) -> X
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)
->Projection:
 pi(SEL) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 2nd(cons(X,XS)) -> head(activate(XS))
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
 activate(X) -> X
 from(X) -> cons(X,n__from(n__s(X)))
 from(X) -> n__from(X)
 head(cons(X,XS)) -> X
 s(X) -> n__s(X)
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 sel(0,cons(X,XS)) -> X
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 take(0,XS) -> nil
 take(X1,X2) -> n__take(X1,X2)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.