YES

Problem 1: 

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

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 G(g(x:S)) -> G(h(g(x:S)))
 G(g(x:S)) -> H(g(x:S))
 H(h(x:S)) -> H(f(h(x:S),x:S))
-> Rules:
 g(g(x:S)) -> g(h(g(x:S)))
 g(h(g(x:S))) -> g(x:S)
 h(h(x:S)) -> h(f(h(x:S),x:S))

Problem 1: 

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

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 G(g(x:S)) -> G(h(g(x:S)))
-> Rules:
 g(g(x:S)) -> g(h(g(x:S)))
 g(h(g(x:S))) -> g(x:S)
 h(h(x:S)) -> h(f(h(x:S),x:S))
-> Usable rules:
 g(g(x:S)) -> g(h(g(x:S)))
 g(h(g(x:S))) -> g(x:S)
 h(h(x:S)) -> h(f(h(x:S),x:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[g](X) = 1
[h](X) = 0
[f](X1,X2) = 2.X1 + 2.X2
[G](X) = 2.X

Problem 1: 

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

The problem is finite.