YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
a(b) -> c
f(a(x:S),y:S) -> g(x:S,y:S)
g(b,y:S) -> f(y:S,y:S)
) 
(STRATEGY INNERMOST)

Problem 1: 

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

Problem 1: 

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

The problem is finite.