YES

Problem 1: 

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

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 g(f(x:S),y:S) -> f(h(x:S,y:S))
 h(x:S,y:S) -> g(x:S,f(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:
 G(f(x:S),y:S) -> H(x:S,y:S)
 H(x:S,y:S) -> G(x:S,f(y:S))
-> Rules:
 g(f(x:S),y:S) -> f(h(x:S,y:S))
 h(x:S,y:S) -> g(x:S,f(y:S))

Problem 1: 

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

Problem 1: 

Subterm Processor:
-> Pairs:
 G(f(x:S),y:S) -> H(x:S,y:S)
 H(x:S,y:S) -> G(x:S,f(y:S))
-> Rules:
 g(f(x:S),y:S) -> f(h(x:S,y:S))
 h(x:S,y:S) -> g(x:S,f(y:S))
->Projection:
 pi(G) = 1
 pi(H) = 1

Problem 1: 

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

The problem is finite.