YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
f(c(s(x:S),y:S)) -> f(c(x:S,s(y:S)))
g(c(x:S,s(y:S))) -> g(c(s(x:S),y:S))
g(s(f(x:S))) -> g(f(x:S))
) 
(STRATEGY INNERMOST)

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 2 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.