YES

Problem 1: 

(VAR v_NonEmpty:S X:S)
(RULES
a -> n__a
activate(n__a) -> a
activate(n__f(X:S)) -> f(X:S)
activate(n__g(X:S)) -> g(activate(X:S))
activate(X:S) -> X:S
f(n__f(n__a)) -> f(n__g(n__f(n__a)))
f(X:S) -> n__f(X:S)
g(X:S) -> n__g(X:S)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVATE(n__a) -> A
 ACTIVATE(n__f(X:S)) -> F(X:S)
 ACTIVATE(n__g(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__g(X:S)) -> G(activate(X:S))
 F(n__f(n__a)) -> F(n__g(n__f(n__a)))
-> Rules:
 a -> n__a
 activate(n__a) -> a
 activate(n__f(X:S)) -> f(X:S)
 activate(n__g(X:S)) -> g(activate(X:S))
 activate(X:S) -> X:S
 f(n__f(n__a)) -> f(n__g(n__f(n__a)))
 f(X:S) -> n__f(X:S)
 g(X:S) -> n__g(X:S)

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__a) -> A
 ACTIVATE(n__f(X:S)) -> F(X:S)
 ACTIVATE(n__g(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__g(X:S)) -> G(activate(X:S))
 F(n__f(n__a)) -> F(n__g(n__f(n__a)))
-> Rules:
 a -> n__a
 activate(n__a) -> a
 activate(n__f(X:S)) -> f(X:S)
 activate(n__g(X:S)) -> g(activate(X:S))
 activate(X:S) -> X:S
 f(n__f(n__a)) -> f(n__g(n__f(n__a)))
 f(X:S) -> n__f(X:S)
 g(X:S) -> n__g(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__g(X:S)) -> ACTIVATE(X:S)
->->-> Rules:
 a -> n__a
 activate(n__a) -> a
 activate(n__f(X:S)) -> f(X:S)
 activate(n__g(X:S)) -> g(activate(X:S))
 activate(X:S) -> X:S
 f(n__f(n__a)) -> f(n__g(n__f(n__a)))
 f(X:S) -> n__f(X:S)
 g(X:S) -> n__g(X:S)

Problem 1: 

Subterm Processor:
-> Pairs:
 ACTIVATE(n__g(X:S)) -> ACTIVATE(X:S)
-> Rules:
 a -> n__a
 activate(n__a) -> a
 activate(n__f(X:S)) -> f(X:S)
 activate(n__g(X:S)) -> g(activate(X:S))
 activate(X:S) -> X:S
 f(n__f(n__a)) -> f(n__g(n__f(n__a)))
 f(X:S) -> n__f(X:S)
 g(X:S) -> n__g(X:S)
->Projection:
 pi(ACTIVATE) = 1

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a -> n__a
 activate(n__a) -> a
 activate(n__f(X:S)) -> f(X:S)
 activate(n__g(X:S)) -> g(activate(X:S))
 activate(X:S) -> X:S
 f(n__f(n__a)) -> f(n__g(n__f(n__a)))
 f(X:S) -> n__f(X:S)
 g(X:S) -> n__g(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.