YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

The problem is finite.