YES

Problem 1: 

(VAR v_NonEmpty:S X:S X1:S X2:S Y:S Z:S)
(RULES
activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
activate(n__from(X:S)) -> from(activate(X:S))
activate(n__s(X:S)) -> s(activate(X:S))
activate(X:S) -> X:S
first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
first(0,Z:S) -> nil
first(X1:S,X2:S) -> n__first(X1:S,X2:S)
from(X:S) -> cons(X:S,n__from(n__s(X:S)))
from(X:S) -> n__from(X:S)
s(X:S) -> n__s(X:S)
sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,activate(Z:S))
sel(0,cons(X:S,Z:S)) -> X:S
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S))
 ACTIVATE(n__from(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__from(X:S)) -> FROM(activate(X:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__s(X:S)) -> S(activate(X:S))
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S)
 SEL(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S)
 SEL(s(X:S),cons(Y:S,Z:S)) -> SEL(X:S,activate(Z:S))
-> Rules:
 activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
 activate(n__from(X:S)) -> from(activate(X:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(0,Z:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(n__s(X:S)))
 from(X:S) -> n__from(X:S)
 s(X:S) -> n__s(X:S)
 sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,activate(Z:S))
 sel(0,cons(X:S,Z:S)) -> X:S

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S))
 ACTIVATE(n__from(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__from(X:S)) -> FROM(activate(X:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__s(X:S)) -> S(activate(X:S))
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S)
 SEL(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S)
 SEL(s(X:S),cons(Y:S,Z:S)) -> SEL(X:S,activate(Z:S))
-> Rules:
 activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
 activate(n__from(X:S)) -> from(activate(X:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(0,Z:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(n__s(X:S)))
 from(X:S) -> n__from(X:S)
 s(X:S) -> n__s(X:S)
 sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,activate(Z:S))
 sel(0,cons(X:S,Z:S)) -> X:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S))
 ACTIVATE(n__from(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S)
->->-> Rules:
 activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
 activate(n__from(X:S)) -> from(activate(X:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(0,Z:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(n__s(X:S)))
 from(X:S) -> n__from(X:S)
 s(X:S) -> n__s(X:S)
 sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,activate(Z:S))
 sel(0,cons(X:S,Z:S)) -> X:S
->->Cycle:
->->-> Pairs:
 SEL(s(X:S),cons(Y:S,Z:S)) -> SEL(X:S,activate(Z:S))
->->-> Rules:
 activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
 activate(n__from(X:S)) -> from(activate(X:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(0,Z:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(n__s(X:S)))
 from(X:S) -> n__from(X:S)
 s(X:S) -> n__s(X:S)
 sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,activate(Z:S))
 sel(0,cons(X:S,Z:S)) -> X:S


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S))
 ACTIVATE(n__from(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S)
-> Rules:
 activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
 activate(n__from(X:S)) -> from(activate(X:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(0,Z:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(n__s(X:S)))
 from(X:S) -> n__from(X:S)
 s(X:S) -> n__s(X:S)
 sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,activate(Z:S))
 sel(0,cons(X:S,Z:S)) -> X:S
-> Usable rules:
 activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
 activate(n__from(X:S)) -> from(activate(X:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(0,Z:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(n__s(X:S)))
 from(X:S) -> n__from(X:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = X
[first](X1,X2) = 2.X1 + 2.X2 + 1
[from](X) = 2.X + 2
[s](X) = X
[0] = 2
[cons](X1,X2) = X2
[n__first](X1,X2) = 2.X1 + 2.X2 + 1
[n__from](X) = 2.X + 2
[n__s](X) = X
[nil] = 2
[ACTIVATE](X) = 2.X + 2
[FIRST](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S))
 ACTIVATE(n__from(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S)
-> Rules:
 activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
 activate(n__from(X:S)) -> from(activate(X:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(0,Z:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(n__s(X:S)))
 from(X:S) -> n__from(X:S)
 s(X:S) -> n__s(X:S)
 sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,activate(Z:S))
 sel(0,cons(X:S,Z:S)) -> X:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S))
 ACTIVATE(n__from(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S)
->->-> Rules:
 activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
 activate(n__from(X:S)) -> from(activate(X:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(0,Z:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(n__s(X:S)))
 from(X:S) -> n__from(X:S)
 s(X:S) -> n__s(X:S)
 sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,activate(Z:S))
 sel(0,cons(X:S,Z:S)) -> X:S

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S))
 ACTIVATE(n__from(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S)
-> Rules:
 activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
 activate(n__from(X:S)) -> from(activate(X:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(0,Z:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(n__s(X:S)))
 from(X:S) -> n__from(X:S)
 s(X:S) -> n__s(X:S)
 sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,activate(Z:S))
 sel(0,cons(X:S,Z:S)) -> X:S
-> Usable rules:
 activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
 activate(n__from(X:S)) -> from(activate(X:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(0,Z:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(n__s(X:S)))
 from(X:S) -> n__from(X:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = X
[first](X1,X2) = 2.X1 + 2.X2 + 2
[from](X) = 2.X + 1
[s](X) = X
[0] = 2
[cons](X1,X2) = X2
[n__first](X1,X2) = 2.X1 + 2.X2 + 2
[n__from](X) = 2.X + 1
[n__s](X) = X
[nil] = 1
[ACTIVATE](X) = 2.X + 2
[FIRST](X1,X2) = X1 + 2.X2 + 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S))
 ACTIVATE(n__from(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S)
-> Rules:
 activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
 activate(n__from(X:S)) -> from(activate(X:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(0,Z:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(n__s(X:S)))
 from(X:S) -> n__from(X:S)
 s(X:S) -> n__s(X:S)
 sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,activate(Z:S))
 sel(0,cons(X:S,Z:S)) -> X:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S))
 ACTIVATE(n__from(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S)
->->-> Rules:
 activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
 activate(n__from(X:S)) -> from(activate(X:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(0,Z:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(n__s(X:S)))
 from(X:S) -> n__from(X:S)
 s(X:S) -> n__s(X:S)
 sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,activate(Z:S))
 sel(0,cons(X:S,Z:S)) -> X:S

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S))
 ACTIVATE(n__from(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S)
-> Rules:
 activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
 activate(n__from(X:S)) -> from(activate(X:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(0,Z:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(n__s(X:S)))
 from(X:S) -> n__from(X:S)
 s(X:S) -> n__s(X:S)
 sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,activate(Z:S))
 sel(0,cons(X:S,Z:S)) -> X:S
-> Usable rules:
 activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
 activate(n__from(X:S)) -> from(activate(X:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(0,Z:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(n__s(X:S)))
 from(X:S) -> n__from(X:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = X
[first](X1,X2) = 2.X1 + 2.X2 + 2
[from](X) = 2.X + 2
[s](X) = X
[0] = 0
[cons](X1,X2) = X2
[n__first](X1,X2) = 2.X1 + 2.X2 + 2
[n__from](X) = 2.X + 2
[n__s](X) = X
[nil] = 1
[ACTIVATE](X) = 2.X + 2
[FIRST](X1,X2) = X1 + 2.X2 + 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__from(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S)
-> Rules:
 activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
 activate(n__from(X:S)) -> from(activate(X:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(0,Z:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(n__s(X:S)))
 from(X:S) -> n__from(X:S)
 s(X:S) -> n__s(X:S)
 sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,activate(Z:S))
 sel(0,cons(X:S,Z:S)) -> X:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__from(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
->->-> Rules:
 activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
 activate(n__from(X:S)) -> from(activate(X:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(0,Z:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(n__s(X:S)))
 from(X:S) -> n__from(X:S)
 s(X:S) -> n__s(X:S)
 sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,activate(Z:S))
 sel(0,cons(X:S,Z:S)) -> X:S

Problem 1.1: 

Subterm Processor:
-> Pairs:
 ACTIVATE(n__from(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
-> Rules:
 activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
 activate(n__from(X:S)) -> from(activate(X:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(0,Z:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(n__s(X:S)))
 from(X:S) -> n__from(X:S)
 s(X:S) -> n__s(X:S)
 sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,activate(Z:S))
 sel(0,cons(X:S,Z:S)) -> X:S
->Projection:
 pi(ACTIVATE) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
 activate(n__from(X:S)) -> from(activate(X:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(0,Z:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(n__s(X:S)))
 from(X:S) -> n__from(X:S)
 s(X:S) -> n__s(X:S)
 sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,activate(Z:S))
 sel(0,cons(X:S,Z:S)) -> X:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 SEL(s(X:S),cons(Y:S,Z:S)) -> SEL(X:S,activate(Z:S))
-> Rules:
 activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
 activate(n__from(X:S)) -> from(activate(X:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(0,Z:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(n__s(X:S)))
 from(X:S) -> n__from(X:S)
 s(X:S) -> n__s(X:S)
 sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,activate(Z:S))
 sel(0,cons(X:S,Z:S)) -> X:S
->Projection:
 pi(SEL) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S))
 activate(n__from(X:S)) -> from(activate(X:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(X:S,activate(Z:S)))
 first(0,Z:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(X:S,n__from(n__s(X:S)))
 from(X:S) -> n__from(X:S)
 s(X:S) -> n__s(X:S)
 sel(s(X:S),cons(Y:S,Z:S)) -> sel(X:S,activate(Z:S))
 sel(0,cons(X:S,Z:S)) -> X:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.