YES

Problem 1: 

(VAR v_NonEmpty:S X:S)
(RULES
a__c -> c
a__c -> d
a__g(X:S) -> a__h(X:S)
a__g(X:S) -> g(X:S)
a__h(d) -> a__g(c)
a__h(X:S) -> h(X:S)
mark(c) -> a__c
mark(d) -> d
mark(g(X:S)) -> a__g(X:S)
mark(h(X:S)) -> a__h(X:S)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A__G(X:S) -> A__H(X:S)
 A__H(d) -> A__G(c)
 MARK(c) -> A__C
 MARK(g(X:S)) -> A__G(X:S)
 MARK(h(X:S)) -> A__H(X:S)
-> Rules:
 a__c -> c
 a__c -> d
 a__g(X:S) -> a__h(X:S)
 a__g(X:S) -> g(X:S)
 a__h(d) -> a__g(c)
 a__h(X:S) -> h(X:S)
 mark(c) -> a__c
 mark(d) -> d
 mark(g(X:S)) -> a__g(X:S)
 mark(h(X:S)) -> a__h(X:S)

Problem 1: 

SCC Processor:
-> Pairs:
 A__G(X:S) -> A__H(X:S)
 A__H(d) -> A__G(c)
 MARK(c) -> A__C
 MARK(g(X:S)) -> A__G(X:S)
 MARK(h(X:S)) -> A__H(X:S)
-> Rules:
 a__c -> c
 a__c -> d
 a__g(X:S) -> a__h(X:S)
 a__g(X:S) -> g(X:S)
 a__h(d) -> a__g(c)
 a__h(X:S) -> h(X:S)
 mark(c) -> a__c
 mark(d) -> d
 mark(g(X:S)) -> a__g(X:S)
 mark(h(X:S)) -> a__h(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A__G(X:S) -> A__H(X:S)
 A__H(d) -> A__G(c)
->->-> Rules:
 a__c -> c
 a__c -> d
 a__g(X:S) -> a__h(X:S)
 a__g(X:S) -> g(X:S)
 a__h(d) -> a__g(c)
 a__h(X:S) -> h(X:S)
 mark(c) -> a__c
 mark(d) -> d
 mark(g(X:S)) -> a__g(X:S)
 mark(h(X:S)) -> a__h(X:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A__G(X:S) -> A__H(X:S)
 A__H(d) -> A__G(c)
-> Rules:
 a__c -> c
 a__c -> d
 a__g(X:S) -> a__h(X:S)
 a__g(X:S) -> g(X:S)
 a__h(d) -> a__g(c)
 a__h(X:S) -> h(X:S)
 mark(c) -> a__c
 mark(d) -> d
 mark(g(X:S)) -> a__g(X:S)
 mark(h(X:S)) -> a__h(X:S)
-> Usable rules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[c] = 1
[d] = 2
[A__G](X) = 2.X + 2
[A__H](X) = 2.X + 1

Problem 1: 

SCC Processor:
-> Pairs:
 A__H(d) -> A__G(c)
-> Rules:
 a__c -> c
 a__c -> d
 a__g(X:S) -> a__h(X:S)
 a__g(X:S) -> g(X:S)
 a__h(d) -> a__g(c)
 a__h(X:S) -> h(X:S)
 mark(c) -> a__c
 mark(d) -> d
 mark(g(X:S)) -> a__g(X:S)
 mark(h(X:S)) -> a__h(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.