YES

Problem 1: 

(VAR v_NonEmpty:S x:S)
(RULES
f(s(0),g(x:S)) -> f(x:S,g(x:S))
g(s(x:S)) -> g(x:S)
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(s(0),g(x:S)) -> F(x:S,g(x:S))
 G(s(x:S)) -> G(x:S)
-> Rules:
 f(s(0),g(x:S)) -> f(x:S,g(x:S))
 g(s(x:S)) -> g(x:S)

Problem 1: 

SCC Processor:
-> Pairs:
 F(s(0),g(x:S)) -> F(x:S,g(x:S))
 G(s(x:S)) -> G(x:S)
-> Rules:
 f(s(0),g(x:S)) -> f(x:S,g(x:S))
 g(s(x:S)) -> g(x:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 G(s(x:S)) -> G(x:S)
->->-> Rules:
 f(s(0),g(x:S)) -> f(x:S,g(x:S))
 g(s(x:S)) -> g(x:S)

Problem 1: 

Subterm Processor:
-> Pairs:
 G(s(x:S)) -> G(x:S)
-> Rules:
 f(s(0),g(x:S)) -> f(x:S,g(x:S))
 g(s(x:S)) -> g(x:S)
->Projection:
 pi(G) = 1

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(s(0),g(x:S)) -> f(x:S,g(x:S))
 g(s(x:S)) -> g(x:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.