YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

The problem is finite.