YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

SCC Processor:
-> Pairs:
 F(a) -> F(b)
 F(x:S) -> G(x:S)
-> Rules:
 f(a) -> f(b)
 f(x:S) -> g(x:S)
 g(b) -> g(a)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.