NO

Problem 1: 

(VAR v_NonEmpty:S X:S)
(RULES
activate(n__c) -> c
activate(n__g(X:S)) -> g(X:S)
activate(X:S) -> X:S
c -> f(n__g(n__c))
c -> n__c
f(n__g(X:S)) -> g(activate(X:S))
g(X:S) -> n__g(X:S)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVATE(n__c) -> C
 ACTIVATE(n__g(X:S)) -> G(X:S)
 C -> F(n__g(n__c))
 F(n__g(X:S)) -> ACTIVATE(X:S)
 F(n__g(X:S)) -> G(activate(X:S))
-> Rules:
 activate(n__c) -> c
 activate(n__g(X:S)) -> g(X:S)
 activate(X:S) -> X:S
 c -> f(n__g(n__c))
 c -> n__c
 f(n__g(X:S)) -> g(activate(X:S))
 g(X:S) -> n__g(X:S)

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__c) -> C
 ACTIVATE(n__g(X:S)) -> G(X:S)
 C -> F(n__g(n__c))
 F(n__g(X:S)) -> ACTIVATE(X:S)
 F(n__g(X:S)) -> G(activate(X:S))
-> Rules:
 activate(n__c) -> c
 activate(n__g(X:S)) -> g(X:S)
 activate(X:S) -> X:S
 c -> f(n__g(n__c))
 c -> n__c
 f(n__g(X:S)) -> g(activate(X:S))
 g(X:S) -> n__g(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__c) -> C
 C -> F(n__g(n__c))
 F(n__g(X:S)) -> ACTIVATE(X:S)
->->-> Rules:
 activate(n__c) -> c
 activate(n__g(X:S)) -> g(X:S)
 activate(X:S) -> X:S
 c -> f(n__g(n__c))
 c -> n__c
 f(n__g(X:S)) -> g(activate(X:S))
 g(X:S) -> n__g(X:S)

Problem 1: 

Infinite Processor:
-> Pairs:
 ACTIVATE(n__c) -> C
 C -> F(n__g(n__c))
 F(n__g(X:S)) -> ACTIVATE(X:S)
-> Rules:
 activate(n__c) -> c
 activate(n__g(X:S)) -> g(X:S)
 activate(X:S) -> X:S
 c -> f(n__g(n__c))
 c -> n__c
 f(n__g(X:S)) -> g(activate(X:S))
 g(X:S) -> n__g(X:S)
-> Pairs in cycle:
 ACTIVATE(n__c) -> C
 C -> F(n__g(n__c))
 F(n__g(n__c)) -> ACTIVATE(n__c)

The problem is infinite.