YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 2 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.