YES

Problem 1: 

(VAR v_NonEmpty:S N:S X:S X1:S X2:S XS:S Y:S)
(RULES
activate(n__fib1(X1:S,X2:S)) -> fib1(X1:S,X2:S)
activate(X:S) -> X:S
add(0,X:S) -> X:S
add(s(X:S),Y:S) -> s(add(X:S,Y:S))
fib(N:S) -> sel(N:S,fib1(s(0),s(0)))
fib1(X:S,Y:S) -> cons(X:S,n__fib1(Y:S,add(X:S,Y:S)))
fib1(X1:S,X2:S) -> n__fib1(X1:S,X2:S)
sel(0,cons(X:S,XS:S)) -> X:S
sel(s(N:S),cons(X:S,XS:S)) -> sel(N:S,activate(XS:S))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVATE(n__fib1(X1:S,X2:S)) -> FIB1(X1:S,X2:S)
 ADD(s(X:S),Y:S) -> ADD(X:S,Y:S)
 FIB(N:S) -> FIB1(s(0),s(0))
 FIB(N:S) -> SEL(N:S,fib1(s(0),s(0)))
 FIB1(X:S,Y:S) -> ADD(X:S,Y:S)
 SEL(s(N:S),cons(X:S,XS:S)) -> ACTIVATE(XS:S)
 SEL(s(N:S),cons(X:S,XS:S)) -> SEL(N:S,activate(XS:S))
-> Rules:
 activate(n__fib1(X1:S,X2:S)) -> fib1(X1:S,X2:S)
 activate(X:S) -> X:S
 add(0,X:S) -> X:S
 add(s(X:S),Y:S) -> s(add(X:S,Y:S))
 fib(N:S) -> sel(N:S,fib1(s(0),s(0)))
 fib1(X:S,Y:S) -> cons(X:S,n__fib1(Y:S,add(X:S,Y:S)))
 fib1(X1:S,X2:S) -> n__fib1(X1:S,X2:S)
 sel(0,cons(X:S,XS:S)) -> X:S
 sel(s(N:S),cons(X:S,XS:S)) -> sel(N:S,activate(XS:S))

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__fib1(X1:S,X2:S)) -> FIB1(X1:S,X2:S)
 ADD(s(X:S),Y:S) -> ADD(X:S,Y:S)
 FIB(N:S) -> FIB1(s(0),s(0))
 FIB(N:S) -> SEL(N:S,fib1(s(0),s(0)))
 FIB1(X:S,Y:S) -> ADD(X:S,Y:S)
 SEL(s(N:S),cons(X:S,XS:S)) -> ACTIVATE(XS:S)
 SEL(s(N:S),cons(X:S,XS:S)) -> SEL(N:S,activate(XS:S))
-> Rules:
 activate(n__fib1(X1:S,X2:S)) -> fib1(X1:S,X2:S)
 activate(X:S) -> X:S
 add(0,X:S) -> X:S
 add(s(X:S),Y:S) -> s(add(X:S,Y:S))
 fib(N:S) -> sel(N:S,fib1(s(0),s(0)))
 fib1(X:S,Y:S) -> cons(X:S,n__fib1(Y:S,add(X:S,Y:S)))
 fib1(X1:S,X2:S) -> n__fib1(X1:S,X2:S)
 sel(0,cons(X:S,XS:S)) -> X:S
 sel(s(N:S),cons(X:S,XS:S)) -> sel(N:S,activate(XS:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ADD(s(X:S),Y:S) -> ADD(X:S,Y:S)
->->-> Rules:
 activate(n__fib1(X1:S,X2:S)) -> fib1(X1:S,X2:S)
 activate(X:S) -> X:S
 add(0,X:S) -> X:S
 add(s(X:S),Y:S) -> s(add(X:S,Y:S))
 fib(N:S) -> sel(N:S,fib1(s(0),s(0)))
 fib1(X:S,Y:S) -> cons(X:S,n__fib1(Y:S,add(X:S,Y:S)))
 fib1(X1:S,X2:S) -> n__fib1(X1:S,X2:S)
 sel(0,cons(X:S,XS:S)) -> X:S
 sel(s(N:S),cons(X:S,XS:S)) -> sel(N:S,activate(XS:S))
->->Cycle:
->->-> Pairs:
 SEL(s(N:S),cons(X:S,XS:S)) -> SEL(N:S,activate(XS:S))
->->-> Rules:
 activate(n__fib1(X1:S,X2:S)) -> fib1(X1:S,X2:S)
 activate(X:S) -> X:S
 add(0,X:S) -> X:S
 add(s(X:S),Y:S) -> s(add(X:S,Y:S))
 fib(N:S) -> sel(N:S,fib1(s(0),s(0)))
 fib1(X:S,Y:S) -> cons(X:S,n__fib1(Y:S,add(X:S,Y:S)))
 fib1(X1:S,X2:S) -> n__fib1(X1:S,X2:S)
 sel(0,cons(X:S,XS:S)) -> X:S
 sel(s(N:S),cons(X:S,XS:S)) -> sel(N:S,activate(XS:S))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 ADD(s(X:S),Y:S) -> ADD(X:S,Y:S)
-> Rules:
 activate(n__fib1(X1:S,X2:S)) -> fib1(X1:S,X2:S)
 activate(X:S) -> X:S
 add(0,X:S) -> X:S
 add(s(X:S),Y:S) -> s(add(X:S,Y:S))
 fib(N:S) -> sel(N:S,fib1(s(0),s(0)))
 fib1(X:S,Y:S) -> cons(X:S,n__fib1(Y:S,add(X:S,Y:S)))
 fib1(X1:S,X2:S) -> n__fib1(X1:S,X2:S)
 sel(0,cons(X:S,XS:S)) -> X:S
 sel(s(N:S),cons(X:S,XS:S)) -> sel(N:S,activate(XS:S))
->Projection:
 pi(ADD) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 activate(n__fib1(X1:S,X2:S)) -> fib1(X1:S,X2:S)
 activate(X:S) -> X:S
 add(0,X:S) -> X:S
 add(s(X:S),Y:S) -> s(add(X:S,Y:S))
 fib(N:S) -> sel(N:S,fib1(s(0),s(0)))
 fib1(X:S,Y:S) -> cons(X:S,n__fib1(Y:S,add(X:S,Y:S)))
 fib1(X1:S,X2:S) -> n__fib1(X1:S,X2:S)
 sel(0,cons(X:S,XS:S)) -> X:S
 sel(s(N:S),cons(X:S,XS:S)) -> sel(N:S,activate(XS:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 SEL(s(N:S),cons(X:S,XS:S)) -> SEL(N:S,activate(XS:S))
-> Rules:
 activate(n__fib1(X1:S,X2:S)) -> fib1(X1:S,X2:S)
 activate(X:S) -> X:S
 add(0,X:S) -> X:S
 add(s(X:S),Y:S) -> s(add(X:S,Y:S))
 fib(N:S) -> sel(N:S,fib1(s(0),s(0)))
 fib1(X:S,Y:S) -> cons(X:S,n__fib1(Y:S,add(X:S,Y:S)))
 fib1(X1:S,X2:S) -> n__fib1(X1:S,X2:S)
 sel(0,cons(X:S,XS:S)) -> X:S
 sel(s(N:S),cons(X:S,XS:S)) -> sel(N:S,activate(XS:S))
->Projection:
 pi(SEL) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 activate(n__fib1(X1:S,X2:S)) -> fib1(X1:S,X2:S)
 activate(X:S) -> X:S
 add(0,X:S) -> X:S
 add(s(X:S),Y:S) -> s(add(X:S,Y:S))
 fib(N:S) -> sel(N:S,fib1(s(0),s(0)))
 fib1(X:S,Y:S) -> cons(X:S,n__fib1(Y:S,add(X:S,Y:S)))
 fib1(X1:S,X2:S) -> n__fib1(X1:S,X2:S)
 sel(0,cons(X:S,XS:S)) -> X:S
 sel(s(N:S),cons(X:S,XS:S)) -> sel(N:S,activate(XS:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.