YES

Problem 1: 

(VAR v_NonEmpty:S X:S)
(RULES
active(f(g(X:S))) -> mark(g(X:S))
active(c) -> mark(f(g(c)))
f(active(X:S)) -> f(X:S)
f(mark(X:S)) -> f(X:S)
g(active(X:S)) -> g(X:S)
g(mark(X:S)) -> g(X:S)
mark(f(X:S)) -> active(f(X:S))
mark(g(X:S)) -> active(g(X:S))
mark(c) -> active(c)
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVE(f(g(X:S))) -> MARK(g(X:S))
 ACTIVE(c) -> MARK(f(g(c)))
 F(active(X:S)) -> F(X:S)
 F(mark(X:S)) -> F(X:S)
 G(active(X:S)) -> G(X:S)
 G(mark(X:S)) -> G(X:S)
 MARK(f(X:S)) -> ACTIVE(f(X:S))
 MARK(g(X:S)) -> ACTIVE(g(X:S))
 MARK(c) -> ACTIVE(c)
-> Rules:
 active(f(g(X:S))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(active(X:S)) -> f(X:S)
 f(mark(X:S)) -> f(X:S)
 g(active(X:S)) -> g(X:S)
 g(mark(X:S)) -> g(X:S)
 mark(f(X:S)) -> active(f(X:S))
 mark(g(X:S)) -> active(g(X:S))
 mark(c) -> active(c)

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVE(f(g(X:S))) -> MARK(g(X:S))
 ACTIVE(c) -> MARK(f(g(c)))
 F(active(X:S)) -> F(X:S)
 F(mark(X:S)) -> F(X:S)
 G(active(X:S)) -> G(X:S)
 G(mark(X:S)) -> G(X:S)
 MARK(f(X:S)) -> ACTIVE(f(X:S))
 MARK(g(X:S)) -> ACTIVE(g(X:S))
 MARK(c) -> ACTIVE(c)
-> Rules:
 active(f(g(X:S))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(active(X:S)) -> f(X:S)
 f(mark(X:S)) -> f(X:S)
 g(active(X:S)) -> g(X:S)
 g(mark(X:S)) -> g(X:S)
 mark(f(X:S)) -> active(f(X:S))
 mark(g(X:S)) -> active(g(X:S))
 mark(c) -> active(c)
->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))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(active(X:S)) -> f(X:S)
 f(mark(X:S)) -> f(X:S)
 g(active(X:S)) -> g(X:S)
 g(mark(X:S)) -> g(X:S)
 mark(f(X:S)) -> active(f(X:S))
 mark(g(X:S)) -> active(g(X:S))
 mark(c) -> active(c)
->->Cycle:
->->-> Pairs:
 F(active(X:S)) -> F(X:S)
 F(mark(X:S)) -> F(X:S)
->->-> Rules:
 active(f(g(X:S))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(active(X:S)) -> f(X:S)
 f(mark(X:S)) -> f(X:S)
 g(active(X:S)) -> g(X:S)
 g(mark(X:S)) -> g(X:S)
 mark(f(X:S)) -> active(f(X:S))
 mark(g(X:S)) -> active(g(X:S))
 mark(c) -> active(c)


The problem is decomposed in 2 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))) -> mark(g(X:S))
 active(c) -> mark(f(g(c)))
 f(active(X:S)) -> f(X:S)
 f(mark(X:S)) -> f(X:S)
 g(active(X:S)) -> g(X:S)
 g(mark(X:S)) -> g(X:S)
 mark(f(X:S)) -> active(f(X:S))
 mark(g(X:S)) -> active(g(X:S))
 mark(c) -> active(c)
->Projection:
 pi(G) = 1

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.