YES

Problem 1: 

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

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVE(f(g(X:S),Y:S)) -> F(X:S,f(g(X:S),Y:S))
 ACTIVE(f(g(X:S),Y:S)) -> MARK(f(X:S,f(g(X:S),Y:S)))
 F(active(X1:S),X2:S) -> F(X1:S,X2:S)
 F(mark(X1:S),X2:S) -> F(X1:S,X2:S)
 F(X1:S,active(X2:S)) -> F(X1:S,X2:S)
 F(X1:S,mark(X2:S)) -> F(X1:S,X2:S)
 G(active(X:S)) -> G(X:S)
 G(mark(X:S)) -> G(X:S)
 MARK(f(X1:S,X2:S)) -> ACTIVE(f(mark(X1:S),X2:S))
 MARK(f(X1:S,X2:S)) -> F(mark(X1:S),X2:S)
 MARK(f(X1:S,X2:S)) -> MARK(X1:S)
 MARK(g(X:S)) -> ACTIVE(g(mark(X:S)))
 MARK(g(X:S)) -> G(mark(X:S))
 MARK(g(X:S)) -> MARK(X:S)
-> Rules:
 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S)))
 f(active(X1:S),X2:S) -> f(X1:S,X2:S)
 f(mark(X1:S),X2:S) -> f(X1:S,X2:S)
 f(X1:S,active(X2:S)) -> f(X1:S,X2:S)
 f(X1:S,mark(X2:S)) -> f(X1:S,X2:S)
 g(active(X:S)) -> g(X:S)
 g(mark(X:S)) -> g(X:S)
 mark(f(X1:S,X2:S)) -> active(f(mark(X1:S),X2:S))
 mark(g(X:S)) -> active(g(mark(X:S)))

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVE(f(g(X:S),Y:S)) -> F(X:S,f(g(X:S),Y:S))
 ACTIVE(f(g(X:S),Y:S)) -> MARK(f(X:S,f(g(X:S),Y:S)))
 F(active(X1:S),X2:S) -> F(X1:S,X2:S)
 F(mark(X1:S),X2:S) -> F(X1:S,X2:S)
 F(X1:S,active(X2:S)) -> F(X1:S,X2:S)
 F(X1:S,mark(X2:S)) -> F(X1:S,X2:S)
 G(active(X:S)) -> G(X:S)
 G(mark(X:S)) -> G(X:S)
 MARK(f(X1:S,X2:S)) -> ACTIVE(f(mark(X1:S),X2:S))
 MARK(f(X1:S,X2:S)) -> F(mark(X1:S),X2:S)
 MARK(f(X1:S,X2:S)) -> MARK(X1:S)
 MARK(g(X:S)) -> ACTIVE(g(mark(X:S)))
 MARK(g(X:S)) -> G(mark(X:S))
 MARK(g(X:S)) -> MARK(X:S)
-> Rules:
 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S)))
 f(active(X1:S),X2:S) -> f(X1:S,X2:S)
 f(mark(X1:S),X2:S) -> f(X1:S,X2:S)
 f(X1:S,active(X2:S)) -> f(X1:S,X2:S)
 f(X1:S,mark(X2:S)) -> f(X1:S,X2:S)
 g(active(X:S)) -> g(X:S)
 g(mark(X:S)) -> g(X:S)
 mark(f(X1:S,X2:S)) -> active(f(mark(X1:S),X2:S))
 mark(g(X:S)) -> active(g(mark(X:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 G(active(X:S)) -> G(X:S)
 G(mark(X:S)) -> G(X:S)
->->-> Rules:
 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S)))
 f(active(X1:S),X2:S) -> f(X1:S,X2:S)
 f(mark(X1:S),X2:S) -> f(X1:S,X2:S)
 f(X1:S,active(X2:S)) -> f(X1:S,X2:S)
 f(X1:S,mark(X2:S)) -> f(X1:S,X2:S)
 g(active(X:S)) -> g(X:S)
 g(mark(X:S)) -> g(X:S)
 mark(f(X1:S,X2:S)) -> active(f(mark(X1:S),X2:S))
 mark(g(X:S)) -> active(g(mark(X:S)))
->->Cycle:
->->-> Pairs:
 F(active(X1:S),X2:S) -> F(X1:S,X2:S)
 F(mark(X1:S),X2:S) -> F(X1:S,X2:S)
 F(X1:S,active(X2:S)) -> F(X1:S,X2:S)
 F(X1:S,mark(X2:S)) -> F(X1:S,X2:S)
->->-> Rules:
 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S)))
 f(active(X1:S),X2:S) -> f(X1:S,X2:S)
 f(mark(X1:S),X2:S) -> f(X1:S,X2:S)
 f(X1:S,active(X2:S)) -> f(X1:S,X2:S)
 f(X1:S,mark(X2:S)) -> f(X1:S,X2:S)
 g(active(X:S)) -> g(X:S)
 g(mark(X:S)) -> g(X:S)
 mark(f(X1:S,X2:S)) -> active(f(mark(X1:S),X2:S))
 mark(g(X:S)) -> active(g(mark(X:S)))
->->Cycle:
->->-> Pairs:
 ACTIVE(f(g(X:S),Y:S)) -> MARK(f(X:S,f(g(X:S),Y:S)))
 MARK(f(X1:S,X2:S)) -> ACTIVE(f(mark(X1:S),X2:S))
 MARK(f(X1:S,X2:S)) -> MARK(X1:S)
 MARK(g(X:S)) -> ACTIVE(g(mark(X:S)))
 MARK(g(X:S)) -> MARK(X:S)
->->-> Rules:
 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S)))
 f(active(X1:S),X2:S) -> f(X1:S,X2:S)
 f(mark(X1:S),X2:S) -> f(X1:S,X2:S)
 f(X1:S,active(X2:S)) -> f(X1:S,X2:S)
 f(X1:S,mark(X2:S)) -> f(X1:S,X2:S)
 g(active(X:S)) -> g(X:S)
 g(mark(X:S)) -> g(X:S)
 mark(f(X1:S,X2:S)) -> active(f(mark(X1:S),X2:S))
 mark(g(X:S)) -> active(g(mark(X:S)))


The problem is decomposed in 3 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

Subterm Processor:
-> Pairs:
 F(X1:S,active(X2:S)) -> F(X1:S,X2:S)
 F(X1:S,mark(X2:S)) -> F(X1:S,X2:S)
-> Rules:
 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S)))
 f(active(X1:S),X2:S) -> f(X1:S,X2:S)
 f(mark(X1:S),X2:S) -> f(X1:S,X2:S)
 f(X1:S,active(X2:S)) -> f(X1:S,X2:S)
 f(X1:S,mark(X2:S)) -> f(X1:S,X2:S)
 g(active(X:S)) -> g(X:S)
 g(mark(X:S)) -> g(X:S)
 mark(f(X1:S,X2:S)) -> active(f(mark(X1:S),X2:S))
 mark(g(X:S)) -> active(g(mark(X:S)))
->Projection:
 pi(F) = 2

Problem 1.2: 

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

The problem is finite.

Problem 1.3: 

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

Problem 1.3: 

SCC Processor:
-> Pairs:
 MARK(f(X1:S,X2:S)) -> ACTIVE(f(mark(X1:S),X2:S))
 MARK(f(X1:S,X2:S)) -> MARK(X1:S)
 MARK(g(X:S)) -> ACTIVE(g(mark(X:S)))
 MARK(g(X:S)) -> MARK(X:S)
-> Rules:
 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S)))
 f(active(X1:S),X2:S) -> f(X1:S,X2:S)
 f(mark(X1:S),X2:S) -> f(X1:S,X2:S)
 f(X1:S,active(X2:S)) -> f(X1:S,X2:S)
 f(X1:S,mark(X2:S)) -> f(X1:S,X2:S)
 g(active(X:S)) -> g(X:S)
 g(mark(X:S)) -> g(X:S)
 mark(f(X1:S,X2:S)) -> active(f(mark(X1:S),X2:S))
 mark(g(X:S)) -> active(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:
 active(f(g(X:S),Y:S)) -> mark(f(X:S,f(g(X:S),Y:S)))
 f(active(X1:S),X2:S) -> f(X1:S,X2:S)
 f(mark(X1:S),X2:S) -> f(X1:S,X2:S)
 f(X1:S,active(X2:S)) -> f(X1:S,X2:S)
 f(X1:S,mark(X2:S)) -> f(X1:S,X2:S)
 g(active(X:S)) -> g(X:S)
 g(mark(X:S)) -> g(X:S)
 mark(f(X1:S,X2:S)) -> active(f(mark(X1:S),X2:S))
 mark(g(X:S)) -> active(g(mark(X:S)))

Problem 1.3: 

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

Problem 1.3: 

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

The problem is finite.