YES

Problem 1: 

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

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 Empty
-> Rules:
 f(f(x:S)) -> f(x:S)
 g(0) -> g(f(0))

Problem 1: 

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

The problem is finite.