YES

Problem 1: 

(VAR v_NonEmpty:S X:S X1:S X2:S X3:S Y:S)
(RULES
a__b -> b
a__b -> c
a__f(X:S,g(X:S),Y:S) -> a__f(Y:S,Y:S,Y:S)
a__f(X1:S,X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
a__g(b) -> c
a__g(X:S) -> g(X:S)
mark(b) -> a__b
mark(c) -> c
mark(f(X1:S,X2:S,X3:S)) -> a__f(X1:S,X2:S,X3:S)
mark(g(X:S)) -> a__g(mark(X:S))
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A__F(X:S,g(X:S),Y:S) -> A__F(Y:S,Y:S,Y:S)
 MARK(b) -> A__B
 MARK(f(X1:S,X2:S,X3:S)) -> A__F(X1:S,X2:S,X3:S)
 MARK(g(X:S)) -> A__G(mark(X:S))
 MARK(g(X:S)) -> MARK(X:S)
-> Rules:
 a__b -> b
 a__b -> c
 a__f(X:S,g(X:S),Y:S) -> a__f(Y:S,Y:S,Y:S)
 a__f(X1:S,X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 a__g(b) -> c
 a__g(X:S) -> g(X:S)
 mark(b) -> a__b
 mark(c) -> c
 mark(f(X1:S,X2:S,X3:S)) -> a__f(X1:S,X2:S,X3:S)
 mark(g(X:S)) -> a__g(mark(X:S))

Problem 1: 

SCC Processor:
-> Pairs:
 A__F(X:S,g(X:S),Y:S) -> A__F(Y:S,Y:S,Y:S)
 MARK(b) -> A__B
 MARK(f(X1:S,X2:S,X3:S)) -> A__F(X1:S,X2:S,X3:S)
 MARK(g(X:S)) -> A__G(mark(X:S))
 MARK(g(X:S)) -> MARK(X:S)
-> Rules:
 a__b -> b
 a__b -> c
 a__f(X:S,g(X:S),Y:S) -> a__f(Y:S,Y:S,Y:S)
 a__f(X1:S,X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 a__g(b) -> c
 a__g(X:S) -> g(X:S)
 mark(b) -> a__b
 mark(c) -> c
 mark(f(X1:S,X2:S,X3:S)) -> a__f(X1:S,X2:S,X3:S)
 mark(g(X:S)) -> a__g(mark(X:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MARK(g(X:S)) -> MARK(X:S)
->->-> Rules:
 a__b -> b
 a__b -> c
 a__f(X:S,g(X:S),Y:S) -> a__f(Y:S,Y:S,Y:S)
 a__f(X1:S,X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 a__g(b) -> c
 a__g(X:S) -> g(X:S)
 mark(b) -> a__b
 mark(c) -> c
 mark(f(X1:S,X2:S,X3:S)) -> a__f(X1:S,X2:S,X3:S)
 mark(g(X:S)) -> a__g(mark(X:S))

Problem 1: 

Subterm Processor:
-> Pairs:
 MARK(g(X:S)) -> MARK(X:S)
-> Rules:
 a__b -> b
 a__b -> c
 a__f(X:S,g(X:S),Y:S) -> a__f(Y:S,Y:S,Y:S)
 a__f(X1:S,X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 a__g(b) -> c
 a__g(X:S) -> g(X:S)
 mark(b) -> a__b
 mark(c) -> c
 mark(f(X1:S,X2:S,X3:S)) -> a__f(X1:S,X2:S,X3:S)
 mark(g(X:S)) -> a__g(mark(X:S))
->Projection:
 pi(MARK) = 1

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a__b -> b
 a__b -> c
 a__f(X:S,g(X:S),Y:S) -> a__f(Y:S,Y:S,Y:S)
 a__f(X1:S,X2:S,X3:S) -> f(X1:S,X2:S,X3:S)
 a__g(b) -> c
 a__g(X:S) -> g(X:S)
 mark(b) -> a__b
 mark(c) -> c
 mark(f(X1:S,X2:S,X3:S)) -> a__f(X1:S,X2:S,X3:S)
 mark(g(X:S)) -> a__g(mark(X:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.