YES

Problem 1: 

(VAR v_NonEmpty:S X:S X1:S X2:S Y:S Z:S)
(RULES
activate(n__add(X1:S,X2:S)) -> add(X1:S,X2:S)
activate(n__first(X1:S,X2:S)) -> first(X1:S,X2:S)
activate(n__from(X:S)) -> from(X:S)
activate(n__s(X:S)) -> s(X:S)
activate(X:S) -> X:S
add(s(X:S),Y:S) -> s(n__add(activate(X:S),activate(Y:S)))
add(0,X:S) -> activate(X:S)
add(X1:S,X2:S) -> n__add(X1:S,X2:S)
and(ffalse,Y:S) -> ffalse
and(ttrue,X:S) -> activate(X:S)
first(s(X:S),cons(Y:S,Z:S)) -> cons(activate(Y:S),n__first(activate(X:S),activate(Z:S)))
first(0,X:S) -> nil
first(X1:S,X2:S) -> n__first(X1:S,X2:S)
from(X:S) -> cons(activate(X:S),n__from(n__s(activate(X:S))))
from(X:S) -> n__from(X:S)
if(ffalse,X:S,Y:S) -> activate(Y:S)
if(ttrue,X:S,Y:S) -> activate(X:S)
s(X:S) -> n__s(X:S)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVATE(n__add(X1:S,X2:S)) -> ADD(X1:S,X2:S)
 ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(X1:S,X2:S)
 ACTIVATE(n__from(X:S)) -> FROM(X:S)
 ACTIVATE(n__s(X:S)) -> S(X:S)
 ADD(s(X:S),Y:S) -> ACTIVATE(X:S)
 ADD(s(X:S),Y:S) -> ACTIVATE(Y:S)
 ADD(s(X:S),Y:S) -> S(n__add(activate(X:S),activate(Y:S)))
 ADD(0,X:S) -> ACTIVATE(X:S)
 AND(ttrue,X:S) -> ACTIVATE(X:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Y:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S)
 FROM(X:S) -> ACTIVATE(X:S)
 IF(ffalse,X:S,Y:S) -> ACTIVATE(Y:S)
 IF(ttrue,X:S,Y:S) -> ACTIVATE(X:S)
-> Rules:
 activate(n__add(X1:S,X2:S)) -> add(X1:S,X2:S)
 activate(n__first(X1:S,X2:S)) -> first(X1:S,X2:S)
 activate(n__from(X:S)) -> from(X:S)
 activate(n__s(X:S)) -> s(X:S)
 activate(X:S) -> X:S
 add(s(X:S),Y:S) -> s(n__add(activate(X:S),activate(Y:S)))
 add(0,X:S) -> activate(X:S)
 add(X1:S,X2:S) -> n__add(X1:S,X2:S)
 and(ffalse,Y:S) -> ffalse
 and(ttrue,X:S) -> activate(X:S)
 first(s(X:S),cons(Y:S,Z:S)) -> cons(activate(Y:S),n__first(activate(X:S),activate(Z:S)))
 first(0,X:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(activate(X:S),n__from(n__s(activate(X:S))))
 from(X:S) -> n__from(X:S)
 if(ffalse,X:S,Y:S) -> activate(Y:S)
 if(ttrue,X:S,Y:S) -> activate(X:S)
 s(X:S) -> n__s(X:S)

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__add(X1:S,X2:S)) -> ADD(X1:S,X2:S)
 ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(X1:S,X2:S)
 ACTIVATE(n__from(X:S)) -> FROM(X:S)
 ACTIVATE(n__s(X:S)) -> S(X:S)
 ADD(s(X:S),Y:S) -> ACTIVATE(X:S)
 ADD(s(X:S),Y:S) -> ACTIVATE(Y:S)
 ADD(s(X:S),Y:S) -> S(n__add(activate(X:S),activate(Y:S)))
 ADD(0,X:S) -> ACTIVATE(X:S)
 AND(ttrue,X:S) -> ACTIVATE(X:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Y:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S)
 FROM(X:S) -> ACTIVATE(X:S)
 IF(ffalse,X:S,Y:S) -> ACTIVATE(Y:S)
 IF(ttrue,X:S,Y:S) -> ACTIVATE(X:S)
-> Rules:
 activate(n__add(X1:S,X2:S)) -> add(X1:S,X2:S)
 activate(n__first(X1:S,X2:S)) -> first(X1:S,X2:S)
 activate(n__from(X:S)) -> from(X:S)
 activate(n__s(X:S)) -> s(X:S)
 activate(X:S) -> X:S
 add(s(X:S),Y:S) -> s(n__add(activate(X:S),activate(Y:S)))
 add(0,X:S) -> activate(X:S)
 add(X1:S,X2:S) -> n__add(X1:S,X2:S)
 and(ffalse,Y:S) -> ffalse
 and(ttrue,X:S) -> activate(X:S)
 first(s(X:S),cons(Y:S,Z:S)) -> cons(activate(Y:S),n__first(activate(X:S),activate(Z:S)))
 first(0,X:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(activate(X:S),n__from(n__s(activate(X:S))))
 from(X:S) -> n__from(X:S)
 if(ffalse,X:S,Y:S) -> activate(Y:S)
 if(ttrue,X:S,Y:S) -> activate(X:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__add(X1:S,X2:S)) -> ADD(X1:S,X2:S)
 ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(X1:S,X2:S)
 ACTIVATE(n__from(X:S)) -> FROM(X:S)
 ADD(s(X:S),Y:S) -> ACTIVATE(X:S)
 ADD(s(X:S),Y:S) -> ACTIVATE(Y:S)
 ADD(0,X:S) -> ACTIVATE(X:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Y:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S)
 FROM(X:S) -> ACTIVATE(X:S)
->->-> Rules:
 activate(n__add(X1:S,X2:S)) -> add(X1:S,X2:S)
 activate(n__first(X1:S,X2:S)) -> first(X1:S,X2:S)
 activate(n__from(X:S)) -> from(X:S)
 activate(n__s(X:S)) -> s(X:S)
 activate(X:S) -> X:S
 add(s(X:S),Y:S) -> s(n__add(activate(X:S),activate(Y:S)))
 add(0,X:S) -> activate(X:S)
 add(X1:S,X2:S) -> n__add(X1:S,X2:S)
 and(ffalse,Y:S) -> ffalse
 and(ttrue,X:S) -> activate(X:S)
 first(s(X:S),cons(Y:S,Z:S)) -> cons(activate(Y:S),n__first(activate(X:S),activate(Z:S)))
 first(0,X:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(activate(X:S),n__from(n__s(activate(X:S))))
 from(X:S) -> n__from(X:S)
 if(ffalse,X:S,Y:S) -> activate(Y:S)
 if(ttrue,X:S,Y:S) -> activate(X:S)
 s(X:S) -> n__s(X:S)

Problem 1: 

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

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(X1:S,X2:S)
 ACTIVATE(n__from(X:S)) -> FROM(X:S)
 ADD(s(X:S),Y:S) -> ACTIVATE(X:S)
 ADD(s(X:S),Y:S) -> ACTIVATE(Y:S)
 ADD(0,X:S) -> ACTIVATE(X:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Y:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S)
 FROM(X:S) -> ACTIVATE(X:S)
-> Rules:
 activate(n__add(X1:S,X2:S)) -> add(X1:S,X2:S)
 activate(n__first(X1:S,X2:S)) -> first(X1:S,X2:S)
 activate(n__from(X:S)) -> from(X:S)
 activate(n__s(X:S)) -> s(X:S)
 activate(X:S) -> X:S
 add(s(X:S),Y:S) -> s(n__add(activate(X:S),activate(Y:S)))
 add(0,X:S) -> activate(X:S)
 add(X1:S,X2:S) -> n__add(X1:S,X2:S)
 and(ffalse,Y:S) -> ffalse
 and(ttrue,X:S) -> activate(X:S)
 first(s(X:S),cons(Y:S,Z:S)) -> cons(activate(Y:S),n__first(activate(X:S),activate(Z:S)))
 first(0,X:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(activate(X:S),n__from(n__s(activate(X:S))))
 from(X:S) -> n__from(X:S)
 if(ffalse,X:S,Y:S) -> activate(Y:S)
 if(ttrue,X:S,Y:S) -> activate(X:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(X1:S,X2:S)
 ACTIVATE(n__from(X:S)) -> FROM(X:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Y:S)
 FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S)
 FROM(X:S) -> ACTIVATE(X:S)
->->-> Rules:
 activate(n__add(X1:S,X2:S)) -> add(X1:S,X2:S)
 activate(n__first(X1:S,X2:S)) -> first(X1:S,X2:S)
 activate(n__from(X:S)) -> from(X:S)
 activate(n__s(X:S)) -> s(X:S)
 activate(X:S) -> X:S
 add(s(X:S),Y:S) -> s(n__add(activate(X:S),activate(Y:S)))
 add(0,X:S) -> activate(X:S)
 add(X1:S,X2:S) -> n__add(X1:S,X2:S)
 and(ffalse,Y:S) -> ffalse
 and(ttrue,X:S) -> activate(X:S)
 first(s(X:S),cons(Y:S,Z:S)) -> cons(activate(Y:S),n__first(activate(X:S),activate(Z:S)))
 first(0,X:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(activate(X:S),n__from(n__s(activate(X:S))))
 from(X:S) -> n__from(X:S)
 if(ffalse,X:S,Y:S) -> activate(Y:S)
 if(ttrue,X:S,Y:S) -> activate(X:S)
 s(X:S) -> n__s(X:S)

Problem 1: 

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

Problem 1: 

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

Problem 1: 

Subterm Processor:
-> Pairs:
 ACTIVATE(n__from(X:S)) -> FROM(X:S)
 FROM(X:S) -> ACTIVATE(X:S)
-> Rules:
 activate(n__add(X1:S,X2:S)) -> add(X1:S,X2:S)
 activate(n__first(X1:S,X2:S)) -> first(X1:S,X2:S)
 activate(n__from(X:S)) -> from(X:S)
 activate(n__s(X:S)) -> s(X:S)
 activate(X:S) -> X:S
 add(s(X:S),Y:S) -> s(n__add(activate(X:S),activate(Y:S)))
 add(0,X:S) -> activate(X:S)
 add(X1:S,X2:S) -> n__add(X1:S,X2:S)
 and(ffalse,Y:S) -> ffalse
 and(ttrue,X:S) -> activate(X:S)
 first(s(X:S),cons(Y:S,Z:S)) -> cons(activate(Y:S),n__first(activate(X:S),activate(Z:S)))
 first(0,X:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(activate(X:S),n__from(n__s(activate(X:S))))
 from(X:S) -> n__from(X:S)
 if(ffalse,X:S,Y:S) -> activate(Y:S)
 if(ttrue,X:S,Y:S) -> activate(X:S)
 s(X:S) -> n__s(X:S)
->Projection:
 pi(ACTIVATE) = 1
 pi(FROM) = 1

Problem 1: 

SCC Processor:
-> Pairs:
 FROM(X:S) -> ACTIVATE(X:S)
-> Rules:
 activate(n__add(X1:S,X2:S)) -> add(X1:S,X2:S)
 activate(n__first(X1:S,X2:S)) -> first(X1:S,X2:S)
 activate(n__from(X:S)) -> from(X:S)
 activate(n__s(X:S)) -> s(X:S)
 activate(X:S) -> X:S
 add(s(X:S),Y:S) -> s(n__add(activate(X:S),activate(Y:S)))
 add(0,X:S) -> activate(X:S)
 add(X1:S,X2:S) -> n__add(X1:S,X2:S)
 and(ffalse,Y:S) -> ffalse
 and(ttrue,X:S) -> activate(X:S)
 first(s(X:S),cons(Y:S,Z:S)) -> cons(activate(Y:S),n__first(activate(X:S),activate(Z:S)))
 first(0,X:S) -> nil
 first(X1:S,X2:S) -> n__first(X1:S,X2:S)
 from(X:S) -> cons(activate(X:S),n__from(n__s(activate(X:S))))
 from(X:S) -> n__from(X:S)
 if(ffalse,X:S,Y:S) -> activate(Y:S)
 if(ttrue,X:S,Y:S) -> activate(X:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.