YES

Problem 1: 

(VAR v_NonEmpty:S X:S)
(RULES
activate(n__f(X:S)) -> f(activate(X:S))
activate(n__h(X:S)) -> h(activate(X:S))
activate(X:S) -> X:S
f(X:S) -> g(n__h(n__f(X:S)))
f(X:S) -> n__f(X:S)
h(X:S) -> n__h(X:S)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVATE(n__f(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__f(X:S)) -> F(activate(X:S))
 ACTIVATE(n__h(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__h(X:S)) -> H(activate(X:S))
-> Rules:
 activate(n__f(X:S)) -> f(activate(X:S))
 activate(n__h(X:S)) -> h(activate(X:S))
 activate(X:S) -> X:S
 f(X:S) -> g(n__h(n__f(X:S)))
 f(X:S) -> n__f(X:S)
 h(X:S) -> n__h(X:S)

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__f(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__f(X:S)) -> F(activate(X:S))
 ACTIVATE(n__h(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__h(X:S)) -> H(activate(X:S))
-> Rules:
 activate(n__f(X:S)) -> f(activate(X:S))
 activate(n__h(X:S)) -> h(activate(X:S))
 activate(X:S) -> X:S
 f(X:S) -> g(n__h(n__f(X:S)))
 f(X:S) -> n__f(X:S)
 h(X:S) -> n__h(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__f(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__h(X:S)) -> ACTIVATE(X:S)
->->-> Rules:
 activate(n__f(X:S)) -> f(activate(X:S))
 activate(n__h(X:S)) -> h(activate(X:S))
 activate(X:S) -> X:S
 f(X:S) -> g(n__h(n__f(X:S)))
 f(X:S) -> n__f(X:S)
 h(X:S) -> n__h(X:S)

Problem 1: 

Subterm Processor:
-> Pairs:
 ACTIVATE(n__f(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__h(X:S)) -> ACTIVATE(X:S)
-> Rules:
 activate(n__f(X:S)) -> f(activate(X:S))
 activate(n__h(X:S)) -> h(activate(X:S))
 activate(X:S) -> X:S
 f(X:S) -> g(n__h(n__f(X:S)))
 f(X:S) -> n__f(X:S)
 h(X:S) -> n__h(X:S)
->Projection:
 pi(ACTIVATE) = 1

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 activate(n__f(X:S)) -> f(activate(X:S))
 activate(n__h(X:S)) -> h(activate(X:S))
 activate(X:S) -> X:S
 f(X:S) -> g(n__h(n__f(X:S)))
 f(X:S) -> n__f(X:S)
 h(X:S) -> n__h(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.