NO

Problem 1: 

(VAR v_NonEmpty:S x:S)
(RULES
f(x:S) -> g(f(x:S))
g(f(x:S)) -> x:S
g(x:S) -> a
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(x:S) -> F(x:S)
 F(x:S) -> G(f(x:S))
-> Rules:
 f(x:S) -> g(f(x:S))
 g(f(x:S)) -> x:S
 g(x:S) -> a

Problem 1: 

Infinite Processor:
-> Pairs:
 F(x:S) -> F(x:S)
 F(x:S) -> G(f(x:S))
-> Rules:
 f(x:S) -> g(f(x:S))
 g(f(x:S)) -> x:S
 g(x:S) -> a
-> Pairs in cycle:
 F(x:S) -> F(x:S)

The problem is infinite.