YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
g(0,f(x:S,x:S)) -> x:S
g(f(x:S,y:S),0) -> f(g(x:S,0),g(y:S,0))
g(s(x:S),y:S) -> g(f(x:S,y:S),0)
g(x:S,s(y:S)) -> g(f(x:S,y:S),0)
)

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

The problem is finite.