YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S z:S)
(RULES
f(g(x:S,y:S),x:S,z:S) -> f(z:S,z:S,z:S)
g(x:S,y:S) -> x:S
g(x:S,y:S) -> y:S
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 Empty
-> Rules:
 f(g(x:S,y:S),x:S,z:S) -> f(z:S,z:S,z:S)
 g(x:S,y:S) -> x:S
 g(x:S,y:S) -> y:S

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(g(x:S,y:S),x:S,z:S) -> f(z:S,z:S,z: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.