YES

Problem 1: 

(VAR v_NonEmpty:S u:S v:S x:S y:S)
(RULES
f(g(x:S,y:S),g(u:S,v:S)) -> g(f(x:S,u:S),f(y:S,v:S))
f(a,y:S) -> y:S
f(x:S,a) -> x:S
g(a,a) -> a
s(f(x:S,y:S)) -> f(s(y:S),s(x:S))
s(g(x:S,y:S)) -> g(s(x:S),s(y:S))
s(s(x:S)) -> x:S
s(a) -> a
)

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 F(g(x:S,y:S),g(u:S,v:S)) -> F(x:S,u:S)
 F(g(x:S,y:S),g(u:S,v:S)) -> F(y:S,v:S)
-> Rules:
 f(g(x:S,y:S),g(u:S,v:S)) -> g(f(x:S,u:S),f(y:S,v:S))
 f(a,y:S) -> y:S
 f(x:S,a) -> x:S
 g(a,a) -> a
 s(f(x:S,y:S)) -> f(s(y:S),s(x:S))
 s(g(x:S,y:S)) -> g(s(x:S),s(y:S))
 s(s(x:S)) -> x:S
 s(a) -> a
->Projection:
 pi(F) = 1

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 S(f(x:S,y:S)) -> S(x:S)
 S(f(x:S,y:S)) -> S(y:S)
 S(g(x:S,y:S)) -> S(x:S)
 S(g(x:S,y:S)) -> S(y:S)
-> Rules:
 f(g(x:S,y:S),g(u:S,v:S)) -> g(f(x:S,u:S),f(y:S,v:S))
 f(a,y:S) -> y:S
 f(x:S,a) -> x:S
 g(a,a) -> a
 s(f(x:S,y:S)) -> f(s(y:S),s(x:S))
 s(g(x:S,y:S)) -> g(s(x:S),s(y:S))
 s(s(x:S)) -> x:S
 s(a) -> a
->Projection:
 pi(S) = 1

Problem 1.2: 

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

The problem is finite.