YES

Problem 1: 

(VAR v_NonEmpty:S X:S X1:S X2:S Y:S)
(RULES
a__f(g(X:S),Y:S) -> a__f(mark(X:S),f(g(X:S),Y:S))
a__f(X1:S,X2:S) -> f(X1:S,X2:S)
mark(f(X1:S,X2:S)) -> a__f(mark(X1:S),X2:S)
mark(g(X:S)) -> g(mark(X:S))
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A__F(g(X:S),Y:S) -> A__F(mark(X:S),f(g(X:S),Y:S))
 A__F(g(X:S),Y:S) -> MARK(X:S)
 MARK(f(X1:S,X2:S)) -> A__F(mark(X1:S),X2:S)
 MARK(f(X1:S,X2:S)) -> MARK(X1:S)
 MARK(g(X:S)) -> MARK(X:S)
-> Rules:
 a__f(g(X:S),Y:S) -> a__f(mark(X:S),f(g(X:S),Y:S))
 a__f(X1:S,X2:S) -> f(X1:S,X2:S)
 mark(f(X1:S,X2:S)) -> a__f(mark(X1:S),X2:S)
 mark(g(X:S)) -> g(mark(X:S))

Problem 1: 

SCC Processor:
-> Pairs:
 A__F(g(X:S),Y:S) -> A__F(mark(X:S),f(g(X:S),Y:S))
 A__F(g(X:S),Y:S) -> MARK(X:S)
 MARK(f(X1:S,X2:S)) -> A__F(mark(X1:S),X2:S)
 MARK(f(X1:S,X2:S)) -> MARK(X1:S)
 MARK(g(X:S)) -> MARK(X:S)
-> Rules:
 a__f(g(X:S),Y:S) -> a__f(mark(X:S),f(g(X:S),Y:S))
 a__f(X1:S,X2:S) -> f(X1:S,X2:S)
 mark(f(X1:S,X2:S)) -> a__f(mark(X1:S),X2:S)
 mark(g(X:S)) -> g(mark(X:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A__F(g(X:S),Y:S) -> A__F(mark(X:S),f(g(X:S),Y:S))
 A__F(g(X:S),Y:S) -> MARK(X:S)
 MARK(f(X1:S,X2:S)) -> A__F(mark(X1:S),X2:S)
 MARK(f(X1:S,X2:S)) -> MARK(X1:S)
 MARK(g(X:S)) -> MARK(X:S)
->->-> Rules:
 a__f(g(X:S),Y:S) -> a__f(mark(X:S),f(g(X:S),Y:S))
 a__f(X1:S,X2:S) -> f(X1:S,X2:S)
 mark(f(X1:S,X2:S)) -> a__f(mark(X1:S),X2:S)
 mark(g(X:S)) -> g(mark(X:S))

Problem 1: 

Reduction Pairs Processor:
-> Pairs:
 A__F(g(X:S),Y:S) -> A__F(mark(X:S),f(g(X:S),Y:S))
 A__F(g(X:S),Y:S) -> MARK(X:S)
 MARK(f(X1:S,X2:S)) -> A__F(mark(X1:S),X2:S)
 MARK(f(X1:S,X2:S)) -> MARK(X1:S)
 MARK(g(X:S)) -> MARK(X:S)
-> Rules:
 a__f(g(X:S),Y:S) -> a__f(mark(X:S),f(g(X:S),Y:S))
 a__f(X1:S,X2:S) -> f(X1:S,X2:S)
 mark(f(X1:S,X2:S)) -> a__f(mark(X1:S),X2:S)
 mark(g(X:S)) -> g(mark(X:S))
-> Usable rules:
 a__f(g(X:S),Y:S) -> a__f(mark(X:S),f(g(X:S),Y:S))
 a__f(X1:S,X2:S) -> f(X1:S,X2:S)
 mark(f(X1:S,X2:S)) -> a__f(mark(X1:S),X2:S)
 mark(g(X:S)) -> g(mark(X:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a__f](X1,X2) = 2.X1 + 2
[mark](X) = X
[f](X1,X2) = 2.X1 + 2
[fSNonEmpty] = 0
[g](X) = X + 1
[A__F](X1,X2) = 2.X1 + 2
[MARK](X) = 2.X + 1

Problem 1: 

SCC Processor:
-> Pairs:
 A__F(g(X:S),Y:S) -> MARK(X:S)
 MARK(f(X1:S,X2:S)) -> A__F(mark(X1:S),X2:S)
 MARK(f(X1:S,X2:S)) -> MARK(X1:S)
 MARK(g(X:S)) -> MARK(X:S)
-> Rules:
 a__f(g(X:S),Y:S) -> a__f(mark(X:S),f(g(X:S),Y:S))
 a__f(X1:S,X2:S) -> f(X1:S,X2:S)
 mark(f(X1:S,X2:S)) -> a__f(mark(X1:S),X2:S)
 mark(g(X:S)) -> g(mark(X:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A__F(g(X:S),Y:S) -> MARK(X:S)
 MARK(f(X1:S,X2:S)) -> A__F(mark(X1:S),X2:S)
 MARK(f(X1:S,X2:S)) -> MARK(X1:S)
 MARK(g(X:S)) -> MARK(X:S)
->->-> Rules:
 a__f(g(X:S),Y:S) -> a__f(mark(X:S),f(g(X:S),Y:S))
 a__f(X1:S,X2:S) -> f(X1:S,X2:S)
 mark(f(X1:S,X2:S)) -> a__f(mark(X1:S),X2:S)
 mark(g(X:S)) -> g(mark(X:S))

Problem 1: 

Reduction Pairs Processor:
-> Pairs:
 A__F(g(X:S),Y:S) -> MARK(X:S)
 MARK(f(X1:S,X2:S)) -> A__F(mark(X1:S),X2:S)
 MARK(f(X1:S,X2:S)) -> MARK(X1:S)
 MARK(g(X:S)) -> MARK(X:S)
-> Rules:
 a__f(g(X:S),Y:S) -> a__f(mark(X:S),f(g(X:S),Y:S))
 a__f(X1:S,X2:S) -> f(X1:S,X2:S)
 mark(f(X1:S,X2:S)) -> a__f(mark(X1:S),X2:S)
 mark(g(X:S)) -> g(mark(X:S))
-> Usable rules:
 a__f(g(X:S),Y:S) -> a__f(mark(X:S),f(g(X:S),Y:S))
 a__f(X1:S,X2:S) -> f(X1:S,X2:S)
 mark(f(X1:S,X2:S)) -> a__f(mark(X1:S),X2:S)
 mark(g(X:S)) -> g(mark(X:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a__f](X1,X2) = X1 + 2
[mark](X) = 2.X + 2
[f](X1,X2) = X1 + 2
[fSNonEmpty] = 0
[g](X) = 2.X + 2
[A__F](X1,X2) = X1 + 2
[MARK](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 MARK(f(X1:S,X2:S)) -> A__F(mark(X1:S),X2:S)
 MARK(f(X1:S,X2:S)) -> MARK(X1:S)
 MARK(g(X:S)) -> MARK(X:S)
-> Rules:
 a__f(g(X:S),Y:S) -> a__f(mark(X:S),f(g(X:S),Y:S))
 a__f(X1:S,X2:S) -> f(X1:S,X2:S)
 mark(f(X1:S,X2:S)) -> a__f(mark(X1:S),X2:S)
 mark(g(X:S)) -> g(mark(X:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MARK(f(X1:S,X2:S)) -> MARK(X1:S)
 MARK(g(X:S)) -> MARK(X:S)
->->-> Rules:
 a__f(g(X:S),Y:S) -> a__f(mark(X:S),f(g(X:S),Y:S))
 a__f(X1:S,X2:S) -> f(X1:S,X2:S)
 mark(f(X1:S,X2:S)) -> a__f(mark(X1:S),X2:S)
 mark(g(X:S)) -> g(mark(X:S))

Problem 1: 

Subterm Processor:
-> Pairs:
 MARK(f(X1:S,X2:S)) -> MARK(X1:S)
 MARK(g(X:S)) -> MARK(X:S)
-> Rules:
 a__f(g(X:S),Y:S) -> a__f(mark(X:S),f(g(X:S),Y:S))
 a__f(X1:S,X2:S) -> f(X1:S,X2:S)
 mark(f(X1:S,X2:S)) -> a__f(mark(X1:S),X2:S)
 mark(g(X:S)) -> g(mark(X:S))
->Projection:
 pi(MARK) = 1

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a__f(g(X:S),Y:S) -> a__f(mark(X:S),f(g(X:S),Y:S))
 a__f(X1:S,X2:S) -> f(X1:S,X2:S)
 mark(f(X1:S,X2:S)) -> a__f(mark(X1:S),X2:S)
 mark(g(X:S)) -> g(mark(X:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.