YES

Problem 1: 

(VAR v_NonEmpty:S x1:S)
(RULES
f(f(g(x1:S))) -> g(f(x1:S))
g(f(c(x1:S))) -> g(f(f(c(x1:S))))
g(g(x1:S)) -> g(f(g(x1:S)))
g(c(x1:S)) -> g(f(c(x1:S)))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(f(g(x1:S))) -> F(x1:S)
 F(f(g(x1:S))) -> G(f(x1:S))
 G(g(x1:S)) -> F(g(x1:S))
 G(g(x1:S)) -> G(f(g(x1:S)))
 G(c(x1:S)) -> G(f(c(x1:S)))
-> Rules:
 f(f(g(x1:S))) -> g(f(x1:S))
 g(f(c(x1:S))) -> g(f(f(c(x1:S))))
 g(g(x1:S)) -> g(f(g(x1:S)))
 g(c(x1:S)) -> g(f(c(x1:S)))

Problem 1: 

SCC Processor:
-> Pairs:
 F(f(g(x1:S))) -> F(x1:S)
 F(f(g(x1:S))) -> G(f(x1:S))
 G(g(x1:S)) -> F(g(x1:S))
 G(g(x1:S)) -> G(f(g(x1:S)))
 G(c(x1:S)) -> G(f(c(x1:S)))
-> Rules:
 f(f(g(x1:S))) -> g(f(x1:S))
 g(f(c(x1:S))) -> g(f(f(c(x1:S))))
 g(g(x1:S)) -> g(f(g(x1:S)))
 g(c(x1:S)) -> g(f(c(x1:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(f(g(x1:S))) -> F(x1:S)
 F(f(g(x1:S))) -> G(f(x1:S))
 G(g(x1:S)) -> F(g(x1:S))
 G(g(x1:S)) -> G(f(g(x1:S)))
->->-> Rules:
 f(f(g(x1:S))) -> g(f(x1:S))
 g(f(c(x1:S))) -> g(f(f(c(x1:S))))
 g(g(x1:S)) -> g(f(g(x1:S)))
 g(c(x1:S)) -> g(f(c(x1:S)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 F(f(g(x1:S))) -> F(x1:S)
 F(f(g(x1:S))) -> G(f(x1:S))
 G(g(x1:S)) -> F(g(x1:S))
 G(g(x1:S)) -> G(f(g(x1:S)))
-> Rules:
 f(f(g(x1:S))) -> g(f(x1:S))
 g(f(c(x1:S))) -> g(f(f(c(x1:S))))
 g(g(x1:S)) -> g(f(g(x1:S)))
 g(c(x1:S)) -> g(f(c(x1:S)))
-> Usable rules:
 f(f(g(x1:S))) -> g(f(x1:S))
 g(f(c(x1:S))) -> g(f(f(c(x1:S))))
 g(g(x1:S)) -> g(f(g(x1:S)))
 g(c(x1:S)) -> g(f(c(x1:S)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X) = X
[g](X) = 2.X + 2
[c](X) = 2.X + 2
[F](X) = X
[G](X) = X

Problem 1: 

SCC Processor:
-> Pairs:
 F(f(g(x1:S))) -> G(f(x1:S))
 G(g(x1:S)) -> F(g(x1:S))
 G(g(x1:S)) -> G(f(g(x1:S)))
-> Rules:
 f(f(g(x1:S))) -> g(f(x1:S))
 g(f(c(x1:S))) -> g(f(f(c(x1:S))))
 g(g(x1:S)) -> g(f(g(x1:S)))
 g(c(x1:S)) -> g(f(c(x1:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(f(g(x1:S))) -> G(f(x1:S))
 G(g(x1:S)) -> F(g(x1:S))
 G(g(x1:S)) -> G(f(g(x1:S)))
->->-> Rules:
 f(f(g(x1:S))) -> g(f(x1:S))
 g(f(c(x1:S))) -> g(f(f(c(x1:S))))
 g(g(x1:S)) -> g(f(g(x1:S)))
 g(c(x1:S)) -> g(f(c(x1:S)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 F(f(g(x1:S))) -> G(f(x1:S))
 G(g(x1:S)) -> F(g(x1:S))
 G(g(x1:S)) -> G(f(g(x1:S)))
-> Rules:
 f(f(g(x1:S))) -> g(f(x1:S))
 g(f(c(x1:S))) -> g(f(f(c(x1:S))))
 g(g(x1:S)) -> g(f(g(x1:S)))
 g(c(x1:S)) -> g(f(c(x1:S)))
-> Usable rules:
 f(f(g(x1:S))) -> g(f(x1:S))
 g(f(c(x1:S))) -> g(f(f(c(x1:S))))
 g(g(x1:S)) -> g(f(g(x1:S)))
 g(c(x1:S)) -> g(f(c(x1:S)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X) = X
[g](X) = 2.X + 1
[c](X) = 0
[F](X) = X + 1
[G](X) = 2.X + 1

Problem 1: 

SCC Processor:
-> Pairs:
 G(g(x1:S)) -> F(g(x1:S))
 G(g(x1:S)) -> G(f(g(x1:S)))
-> Rules:
 f(f(g(x1:S))) -> g(f(x1:S))
 g(f(c(x1:S))) -> g(f(f(c(x1:S))))
 g(g(x1:S)) -> g(f(g(x1:S)))
 g(c(x1:S)) -> g(f(c(x1:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 G(g(x1:S)) -> G(f(g(x1:S)))
->->-> Rules:
 f(f(g(x1:S))) -> g(f(x1:S))
 g(f(c(x1:S))) -> g(f(f(c(x1:S))))
 g(g(x1:S)) -> g(f(g(x1:S)))
 g(c(x1:S)) -> g(f(c(x1:S)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 G(g(x1:S)) -> G(f(g(x1:S)))
-> Rules:
 f(f(g(x1:S))) -> g(f(x1:S))
 g(f(c(x1:S))) -> g(f(f(c(x1:S))))
 g(g(x1:S)) -> g(f(g(x1:S)))
 g(c(x1:S)) -> g(f(c(x1:S)))
-> Usable rules:
 f(f(g(x1:S))) -> g(f(x1:S))
 g(f(c(x1:S))) -> g(f(f(c(x1:S))))
 g(g(x1:S)) -> g(f(g(x1:S)))
 g(c(x1:S)) -> g(f(c(x1:S)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 2
->Bound:
 1
->Interpretation:
 
[f](X) = [0 1;1 0].X
[g](X) = [1 1;0 0].X + [1;0]
[c](X) = [1 1;1 1].X + [1;1]
[G](X) = [1 0;1 1].X

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(f(g(x1:S))) -> g(f(x1:S))
 g(f(c(x1:S))) -> g(f(f(c(x1:S))))
 g(g(x1:S)) -> g(f(g(x1:S)))
 g(c(x1:S)) -> g(f(c(x1:S)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.