YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 2 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 F(x:S,c(x:S),c(y:S)) -> F(y:S,x:S,y:S)
-> Rules:
 f(c(x:S),x:S,y:S) -> c(y:S)
 f(s(x:S),y:S,z:S) -> f(x:S,s(c(y:S)),c(z:S))
 f(x:S,c(x:S),c(y:S)) -> f(y:S,y:S,f(y:S,x:S,y:S))
 g(x:S,y:S) -> x:S
 g(x:S,y:S) -> y:S
->Projection:
 pi(F) = 2

Problem 1.2: 

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

The problem is finite.