YES

Problem 1: 

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

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A__C -> A__F(g(c))
 MARK(c) -> A__C
 MARK(f(X:S)) -> A__F(X:S)
-> Rules:
 a__c -> a__f(g(c))
 a__c -> c
 a__f(g(X:S)) -> g(X:S)
 a__f(X:S) -> f(X:S)
 mark(c) -> a__c
 mark(f(X:S)) -> a__f(X:S)
 mark(g(X:S)) -> g(X:S)

Problem 1: 

SCC Processor:
-> Pairs:
 A__C -> A__F(g(c))
 MARK(c) -> A__C
 MARK(f(X:S)) -> A__F(X:S)
-> Rules:
 a__c -> a__f(g(c))
 a__c -> c
 a__f(g(X:S)) -> g(X:S)
 a__f(X:S) -> f(X:S)
 mark(c) -> a__c
 mark(f(X:S)) -> a__f(X:S)
 mark(g(X:S)) -> g(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.