YES

Problem 1: 

(VAR v_NonEmpty:S X:S)
(RULES
f(X:S,g(X:S)) -> f(1,g(X:S))
g(1) -> g(0)
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(X:S,g(X:S)) -> F(1,g(X:S))
-> Rules:
 f(X:S,g(X:S)) -> f(1,g(X:S))
 g(1) -> g(0)

Problem 1: 

SCC Processor:
-> Pairs:
 F(X:S,g(X:S)) -> F(1,g(X:S))
-> Rules:
 f(X:S,g(X:S)) -> f(1,g(X:S))
 g(1) -> g(0)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.