YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

SCC Processor:
-> Pairs:
 F(g(x:S)) -> F(x:S)
-> Rules:
 f(g(x:S)) -> g(g(f(x:S)))
 f(g(x:S)) -> g(g(g(x:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(g(x:S)) -> F(x:S)
->->-> Rules:
 f(g(x:S)) -> g(g(f(x:S)))
 f(g(x:S)) -> g(g(g(x:S)))

Problem 1: 

Subterm Processor:
-> Pairs:
 F(g(x:S)) -> F(x:S)
-> Rules:
 f(g(x:S)) -> g(g(f(x:S)))
 f(g(x:S)) -> g(g(g(x:S)))
->Projection:
 pi(F) = 1

Problem 1: 

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

The problem is finite.