YES

Problem 1: 

(VAR v_NonEmpty:S X:S)
(RULES
activate(n__d(X:S)) -> d(X:S)
activate(n__f(X:S)) -> f(X:S)
activate(X:S) -> X:S
c(X:S) -> d(activate(X:S))
d(X:S) -> n__d(X:S)
f(f(X:S)) -> c(n__f(g(n__f(X:S))))
f(X:S) -> n__f(X:S)
h(X:S) -> c(n__d(X:S))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVATE(n__d(X:S)) -> D(X:S)
 ACTIVATE(n__f(X:S)) -> F(X:S)
 C(X:S) -> ACTIVATE(X:S)
 C(X:S) -> D(activate(X:S))
 F(f(X:S)) -> C(n__f(g(n__f(X:S))))
 H(X:S) -> C(n__d(X:S))
-> Rules:
 activate(n__d(X:S)) -> d(X:S)
 activate(n__f(X:S)) -> f(X:S)
 activate(X:S) -> X:S
 c(X:S) -> d(activate(X:S))
 d(X:S) -> n__d(X:S)
 f(f(X:S)) -> c(n__f(g(n__f(X:S))))
 f(X:S) -> n__f(X:S)
 h(X:S) -> c(n__d(X:S))

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__d(X:S)) -> D(X:S)
 ACTIVATE(n__f(X:S)) -> F(X:S)
 C(X:S) -> ACTIVATE(X:S)
 C(X:S) -> D(activate(X:S))
 F(f(X:S)) -> C(n__f(g(n__f(X:S))))
 H(X:S) -> C(n__d(X:S))
-> Rules:
 activate(n__d(X:S)) -> d(X:S)
 activate(n__f(X:S)) -> f(X:S)
 activate(X:S) -> X:S
 c(X:S) -> d(activate(X:S))
 d(X:S) -> n__d(X:S)
 f(f(X:S)) -> c(n__f(g(n__f(X:S))))
 f(X:S) -> n__f(X:S)
 h(X:S) -> c(n__d(X:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__f(X:S)) -> F(X:S)
 C(X:S) -> ACTIVATE(X:S)
 F(f(X:S)) -> C(n__f(g(n__f(X:S))))
->->-> Rules:
 activate(n__d(X:S)) -> d(X:S)
 activate(n__f(X:S)) -> f(X:S)
 activate(X:S) -> X:S
 c(X:S) -> d(activate(X:S))
 d(X:S) -> n__d(X:S)
 f(f(X:S)) -> c(n__f(g(n__f(X:S))))
 f(X:S) -> n__f(X:S)
 h(X:S) -> c(n__d(X:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__f(X:S)) -> F(X:S)
 C(X:S) -> ACTIVATE(X:S)
 F(f(X:S)) -> C(n__f(g(n__f(X:S))))
-> Rules:
 activate(n__d(X:S)) -> d(X:S)
 activate(n__f(X:S)) -> f(X:S)
 activate(X:S) -> X:S
 c(X:S) -> d(activate(X:S))
 d(X:S) -> n__d(X:S)
 f(f(X:S)) -> c(n__f(g(n__f(X:S))))
 f(X:S) -> n__f(X:S)
 h(X:S) -> c(n__d(X:S))
-> Usable rules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X) = X + 1
[g](X) = 0
[n__f](X) = 2.X + 2
[ACTIVATE](X) = X + 2
[C](X) = X + 2
[F](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 C(X:S) -> ACTIVATE(X:S)
 F(f(X:S)) -> C(n__f(g(n__f(X:S))))
-> Rules:
 activate(n__d(X:S)) -> d(X:S)
 activate(n__f(X:S)) -> f(X:S)
 activate(X:S) -> X:S
 c(X:S) -> d(activate(X:S))
 d(X:S) -> n__d(X:S)
 f(f(X:S)) -> c(n__f(g(n__f(X:S))))
 f(X:S) -> n__f(X:S)
 h(X:S) -> c(n__d(X:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.