YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S z:S)
(RULES
f(.(.(x:S,y:S),z:S)) -> f(.(x:S,.(y:S,z:S)))
f(.(nil,y:S)) -> .(nil,f(y:S))
f(nil) -> nil
g(.(x:S,.(y:S,z:S))) -> g(.(.(x:S,y:S),z:S))
g(.(x:S,nil)) -> .(g(x:S),nil)
g(nil) -> nil
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 f(.(.(x:S,y:S),z:S)) -> f(.(x:S,.(y:S,z:S)))
 f(.(nil,y:S)) -> .(nil,f(y:S))
 f(nil) -> nil
 g(.(x:S,.(y:S,z:S))) -> g(.(.(x:S,y:S),z:S))
 g(.(x:S,nil)) -> .(g(x:S),nil)
 g(nil) -> nil
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(.(.(x:S,y:S),z:S)) -> F(.(x:S,.(y:S,z:S)))
 F(.(nil,y:S)) -> F(y:S)
 G(.(x:S,.(y:S,z:S))) -> G(.(.(x:S,y:S),z:S))
 G(.(x:S,nil)) -> G(x:S)
-> Rules:
 f(.(.(x:S,y:S),z:S)) -> f(.(x:S,.(y:S,z:S)))
 f(.(nil,y:S)) -> .(nil,f(y:S))
 f(nil) -> nil
 g(.(x:S,.(y:S,z:S))) -> g(.(.(x:S,y:S),z:S))
 g(.(x:S,nil)) -> .(g(x:S),nil)
 g(nil) -> nil

Problem 1: 

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


The problem is decomposed in 2 subproblems.

Problem 1.1: 

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

Problem 1.1: 

SCC Processor:
-> Pairs:
 G(.(x:S,nil)) -> G(x:S)
-> Rules:
 f(.(.(x:S,y:S),z:S)) -> f(.(x:S,.(y:S,z:S)))
 f(.(nil,y:S)) -> .(nil,f(y:S))
 f(nil) -> nil
 g(.(x:S,.(y:S,z:S))) -> g(.(.(x:S,y:S),z:S))
 g(.(x:S,nil)) -> .(g(x:S),nil)
 g(nil) -> nil
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 G(.(x:S,nil)) -> G(x:S)
->->-> Rules:
 f(.(.(x:S,y:S),z:S)) -> f(.(x:S,.(y:S,z:S)))
 f(.(nil,y:S)) -> .(nil,f(y:S))
 f(nil) -> nil
 g(.(x:S,.(y:S,z:S))) -> g(.(.(x:S,y:S),z:S))
 g(.(x:S,nil)) -> .(g(x:S),nil)
 g(nil) -> nil

Problem 1.1: 

Subterm Processor:
-> Pairs:
 G(.(x:S,nil)) -> G(x:S)
-> Rules:
 f(.(.(x:S,y:S),z:S)) -> f(.(x:S,.(y:S,z:S)))
 f(.(nil,y:S)) -> .(nil,f(y:S))
 f(nil) -> nil
 g(.(x:S,.(y:S,z:S))) -> g(.(.(x:S,y:S),z:S))
 g(.(x:S,nil)) -> .(g(x:S),nil)
 g(nil) -> nil
->Projection:
 pi(G) = 1

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.