YES

Problem 1: 

(VAR v_NonEmpty:S u:S v:S x:S y:S z:S)
(RULES
f(f(x:S,y:S,z:S),u:S,f(x:S,y:S,v:S)) -> f(x:S,y:S,f(z:S,u:S,v:S))
f(g(x:S),x:S,y:S) -> y:S
f(x:S,x:S,y:S) -> x:S
f(x:S,y:S,g(y:S)) -> x:S
f(x:S,y:S,y:S) -> y:S
)

Problem 1: 

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

Problem 1: 

SCC Processor:
-> Pairs:
 F(f(x:S,y:S,z:S),u:S,f(x:S,y:S,v:S)) -> F(x:S,y:S,f(z:S,u:S,v:S))
 F(f(x:S,y:S,z:S),u:S,f(x:S,y:S,v:S)) -> F(z:S,u:S,v:S)
-> Rules:
 f(f(x:S,y:S,z:S),u:S,f(x:S,y:S,v:S)) -> f(x:S,y:S,f(z:S,u:S,v:S))
 f(g(x:S),x:S,y:S) -> y:S
 f(x:S,x:S,y:S) -> x:S
 f(x:S,y:S,g(y:S)) -> x:S
 f(x:S,y:S,y:S) -> y:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(f(x:S,y:S,z:S),u:S,f(x:S,y:S,v:S)) -> F(x:S,y:S,f(z:S,u:S,v:S))
 F(f(x:S,y:S,z:S),u:S,f(x:S,y:S,v:S)) -> F(z:S,u:S,v:S)
->->-> Rules:
 f(f(x:S,y:S,z:S),u:S,f(x:S,y:S,v:S)) -> f(x:S,y:S,f(z:S,u:S,v:S))
 f(g(x:S),x:S,y:S) -> y:S
 f(x:S,x:S,y:S) -> x:S
 f(x:S,y:S,g(y:S)) -> x:S
 f(x:S,y:S,y:S) -> y:S

Problem 1: 

Subterm Processor:
-> Pairs:
 F(f(x:S,y:S,z:S),u:S,f(x:S,y:S,v:S)) -> F(x:S,y:S,f(z:S,u:S,v:S))
 F(f(x:S,y:S,z:S),u:S,f(x:S,y:S,v:S)) -> F(z:S,u:S,v:S)
-> Rules:
 f(f(x:S,y:S,z:S),u:S,f(x:S,y:S,v:S)) -> f(x:S,y:S,f(z:S,u:S,v:S))
 f(g(x:S),x:S,y:S) -> y:S
 f(x:S,x:S,y:S) -> x:S
 f(x:S,y:S,g(y:S)) -> x:S
 f(x:S,y:S,y:S) -> y:S
->Projection:
 pi(F) = 1

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(f(x:S,y:S,z:S),u:S,f(x:S,y:S,v:S)) -> f(x:S,y:S,f(z:S,u:S,v:S))
 f(g(x:S),x:S,y:S) -> y:S
 f(x:S,x:S,y:S) -> x:S
 f(x:S,y:S,g(y:S)) -> x:S
 f(x:S,y:S,y:S) -> y:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.