YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

The problem is finite.