The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

f(X) → g(n__h(n__f(X)))
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → X

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
activate(n__h(X)) →⁺ h(activate(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / n__h(X)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

f(X) → g(n__h(n__f(X)))
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → X

S is empty.
Rewrite Strategy: FULL

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
Rules:
f(X) → g(n__h(n__f(X)))
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → X

Types:
f :: n__f:n__h:g → n__f:n__h:g
g :: n__f:n__h:g → n__f:n__h:g
n__h :: n__f:n__h:g → n__f:n__h:g
n__f :: n__f:n__h:g → n__f:n__h:g
h :: n__f:n__h:g → n__f:n__h:g
activate :: n__f:n__h:g → n__f:n__h:g
hole_n__f:n__h:g1_0 :: n__f:n__h:g
gen_n__f:n__h:g2_0 :: Nat → n__f:n__h:g

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
activate

(8) Obligation:

TRS:
Rules:
f(X) → g(n__h(n__f(X)))
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → X

Types:
f :: n__f:n__h:g → n__f:n__h:g
g :: n__f:n__h:g → n__f:n__h:g
n__h :: n__f:n__h:g → n__f:n__h:g
n__f :: n__f:n__h:g → n__f:n__h:g
h :: n__f:n__h:g → n__f:n__h:g
activate :: n__f:n__h:g → n__f:n__h:g
hole_n__f:n__h:g1_0 :: n__f:n__h:g
gen_n__f:n__h:g2_0 :: Nat → n__f:n__h:g

Generator Equations:
gen_n__f:n__h:g2_0(0) ⇔ hole_n__f:n__h:g1_0
gen_n__f:n__h:g2_0(+(x, 1)) ⇔ n__h(gen_n__f:n__h:g2_0(x))

The following defined symbols remain to be analysed:
activate

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol activate.

(10) Obligation:

TRS:
Rules:
f(X) → g(n__h(n__f(X)))
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → X

Types:
f :: n__f:n__h:g → n__f:n__h:g
g :: n__f:n__h:g → n__f:n__h:g
n__h :: n__f:n__h:g → n__f:n__h:g
n__f :: n__f:n__h:g → n__f:n__h:g
h :: n__f:n__h:g → n__f:n__h:g
activate :: n__f:n__h:g → n__f:n__h:g
hole_n__f:n__h:g1_0 :: n__f:n__h:g
gen_n__f:n__h:g2_0 :: Nat → n__f:n__h:g

Generator Equations:
gen_n__f:n__h:g2_0(0) ⇔ hole_n__f:n__h:g1_0
gen_n__f:n__h:g2_0(+(x, 1)) ⇔ n__h(gen_n__f:n__h:g2_0(x))

No more defined symbols left to analyse.