YES

Problem 1: 

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

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 f(g(x:S,y:S),f(y:S,y:S)) -> f(g(y:S,x:S),y:S)
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

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

Problem 1: 

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

Problem 1: 

Subterm Processor:
-> Pairs:
 F(g(x:S,y:S),f(y:S,y:S)) -> F(g(y:S,x:S),y:S)
-> Rules:
 f(g(x:S,y:S),f(y:S,y:S)) -> f(g(y:S,x:S),y:S)
->Projection:
 pi(F) = 2

Problem 1: 

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

The problem is finite.