YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

The problem is finite.