YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 2 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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


The problem is decomposed in 2 subproblems.

Problem 1.2.1: 

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

Problem 1.2.1: 

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

The problem is finite.

Problem 1.2.2: 

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

Problem 1.2.2: 

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

Problem 1.2.2: 

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

Problem 1.2.2: 

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

The problem is finite.