YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
+(x:S,0) -> x:S
+(x:S,s(y:S)) -> s(+(x:S,y:S))
f(0) -> 1
f(s(x:S)) -> g(x:S,s(x:S))
g(0,y:S) -> y:S
g(s(x:S),y:S) -> g(x:S,+(y:S,s(x:S)))
g(s(x:S),y:S) -> g(x:S,s(+(y:S,x:S)))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
 F(s(x:S)) -> G(x:S,s(x:S))
 G(s(x:S),y:S) -> +#(y:S,s(x:S))
 G(s(x:S),y:S) -> +#(y:S,x:S)
 G(s(x:S),y:S) -> G(x:S,+(y:S,s(x:S)))
 G(s(x:S),y:S) -> G(x:S,s(+(y:S,x:S)))
-> Rules:
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 f(0) -> 1
 f(s(x:S)) -> g(x:S,s(x:S))
 g(0,y:S) -> y:S
 g(s(x:S),y:S) -> g(x:S,+(y:S,s(x:S)))
 g(s(x:S),y:S) -> g(x:S,s(+(y:S,x:S)))

Problem 1: 

SCC Processor:
-> Pairs:
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
 F(s(x:S)) -> G(x:S,s(x:S))
 G(s(x:S),y:S) -> +#(y:S,s(x:S))
 G(s(x:S),y:S) -> +#(y:S,x:S)
 G(s(x:S),y:S) -> G(x:S,+(y:S,s(x:S)))
 G(s(x:S),y:S) -> G(x:S,s(+(y:S,x:S)))
-> Rules:
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 f(0) -> 1
 f(s(x:S)) -> g(x:S,s(x:S))
 g(0,y:S) -> y:S
 g(s(x:S),y:S) -> g(x:S,+(y:S,s(x:S)))
 g(s(x:S),y:S) -> g(x:S,s(+(y:S,x:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
->->-> Rules:
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 f(0) -> 1
 f(s(x:S)) -> g(x:S,s(x:S))
 g(0,y:S) -> y:S
 g(s(x:S),y:S) -> g(x:S,+(y:S,s(x:S)))
 g(s(x:S),y:S) -> g(x:S,s(+(y:S,x:S)))
->->Cycle:
->->-> Pairs:
 G(s(x:S),y:S) -> G(x:S,+(y:S,s(x:S)))
 G(s(x:S),y:S) -> G(x:S,s(+(y:S,x:S)))
->->-> Rules:
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 f(0) -> 1
 f(s(x:S)) -> g(x:S,s(x:S))
 g(0,y:S) -> y:S
 g(s(x:S),y:S) -> g(x:S,+(y:S,s(x:S)))
 g(s(x:S),y:S) -> g(x:S,s(+(y:S,x:S)))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
-> Rules:
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 f(0) -> 1
 f(s(x:S)) -> g(x:S,s(x:S))
 g(0,y:S) -> y:S
 g(s(x:S),y:S) -> g(x:S,+(y:S,s(x:S)))
 g(s(x:S),y:S) -> g(x:S,s(+(y:S,x:S)))
->Projection:
 pi(+#) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 f(0) -> 1
 f(s(x:S)) -> g(x:S,s(x:S))
 g(0,y:S) -> y:S
 g(s(x:S),y:S) -> g(x:S,+(y:S,s(x:S)))
 g(s(x:S),y:S) -> g(x:S,s(+(y:S,x:S)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 G(s(x:S),y:S) -> G(x:S,+(y:S,s(x:S)))
 G(s(x:S),y:S) -> G(x:S,s(+(y:S,x:S)))
-> Rules:
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 f(0) -> 1
 f(s(x:S)) -> g(x:S,s(x:S))
 g(0,y:S) -> y:S
 g(s(x:S),y:S) -> g(x:S,+(y:S,s(x:S)))
 g(s(x:S),y:S) -> g(x:S,s(+(y:S,x:S)))
->Projection:
 pi(G) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 f(0) -> 1
 f(s(x:S)) -> g(x:S,s(x:S))
 g(0,y:S) -> y:S
 g(s(x:S),y:S) -> g(x:S,+(y:S,s(x:S)))
 g(s(x:S),y:S) -> g(x:S,s(+(y:S,x:S)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.