YES

Problem 1: 

(VAR v_NonEmpty:S x:S)
(RULES
f(f(x:S)) -> f(x:S)
f(s(x:S)) -> f(x:S)
g(s(0)) -> g(f(s(0)))
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(s(x:S)) -> F(x:S)
 G(s(0)) -> F(s(0))
 G(s(0)) -> G(f(s(0)))
-> Rules:
 f(f(x:S)) -> f(x:S)
 f(s(x:S)) -> f(x:S)
 g(s(0)) -> g(f(s(0)))

Problem 1: 

SCC Processor:
-> Pairs:
 F(s(x:S)) -> F(x:S)
 G(s(0)) -> F(s(0))
 G(s(0)) -> G(f(s(0)))
-> Rules:
 f(f(x:S)) -> f(x:S)
 f(s(x:S)) -> f(x:S)
 g(s(0)) -> g(f(s(0)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(s(x:S)) -> F(x:S)
->->-> Rules:
 f(f(x:S)) -> f(x:S)
 f(s(x:S)) -> f(x:S)
 g(s(0)) -> g(f(s(0)))
->->Cycle:
->->-> Pairs:
 G(s(0)) -> G(f(s(0)))
->->-> Rules:
 f(f(x:S)) -> f(x:S)
 f(s(x:S)) -> f(x:S)
 g(s(0)) -> g(f(s(0)))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 F(s(x:S)) -> F(x:S)
-> Rules:
 f(f(x:S)) -> f(x:S)
 f(s(x:S)) -> f(x:S)
 g(s(0)) -> g(f(s(0)))
->Projection:
 pi(F) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(f(x:S)) -> f(x:S)
 f(s(x:S)) -> f(x:S)
 g(s(0)) -> g(f(s(0)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 G(s(0)) -> G(f(s(0)))
-> Rules:
 f(f(x:S)) -> f(x:S)
 f(s(x:S)) -> f(x:S)
 g(s(0)) -> g(f(s(0)))
-> Usable rules:
 f(f(x:S)) -> f(x:S)
 f(s(x:S)) -> f(x:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X) = 1
[g](X) = 0
[0] = 2
[fSNonEmpty] = 0
[s](X) = 2
[F](X) = 0
[G](X) = 2.X

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(f(x:S)) -> f(x:S)
 f(s(x:S)) -> f(x:S)
 g(s(0)) -> g(f(s(0)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.