YES

Problem 1: 

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

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVATE(n__f(X:S)) -> F(X:S)
-> Rules:
 activate(n__f(X:S)) -> f(X:S)
 activate(X:S) -> X:S
 f(f(a)) -> c(n__f(g(f(a))))
 f(X:S) -> n__f(X:S)

Problem 1: 

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

The problem is finite.